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AIRCRAFT NAVIGATION
.

by SI SI Fedcbipz

i

‘7.

.

I

‘Transport” Press
Moscow, 1966

N A T I O N A L A E R O N A U T I C S A N D SPACE A D M I N I S T R A T I O N

W A S H I N G T O N , D. C.

FEBRUARY 1 9 6 9

k

I

{

TECH LIBRARY KAFB,

NM

I111 111l0068952
llIWI 111Hlliltl1
lI
1


AIRCRAFT NAVIGATION
By S. S. Fedchin

Translation of: "Samoletovozhdeniye . I t
"Transport" Press, Moscow, 1966

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

. -- -_ - For sale by the Clearinghouse for Federal Scientific and Technical Information
Springfield, Virginia 22151
CFSTI price $3.00

-

TABLE OF CONTENTS

.....................................................
INTRODUCTION .................................................
CHAPTER O N E . C O O R D I N A T E S Y S T E M S A N D ELEMENTS OF

AIRCRAFT NAVIGATION ........................................
1. E l e m e n t s o f A i r c r a f t M o v e m e n t i n S p a c e ...............
2 . C o n c e p t s o f S t a b l e and U n s t a b l e F l i g h t

C o n d i t i o n s ...........................................
3 . Form a n d D i m e n s i o n s o f t h e E a r t h .....................
4 . E l e m e n t s Which C o n n e c t t h e E a r t h ' s S u r f a c e

w i t h T h r e e - D i m e n s i o n a l S p a c e .........................
5 . C h a r t s . Maps. a n d C a r t o g r a p h i c P r o j e c t i o n s ...........
D i s t o r t i o n s o f C a r t o g r a p h i c P r o j e c t i o n s ............
E Z Z i p s e o f D i s t o r t i o n s ...........................
D i s t o r t i o n o f L e n g t h s ............................
D i s t o r t i o n of D i r e c t i o n s .........................
D i s t o r t i o n of A r e a s ..............................
ABSTRACT

.........
...............................
c o n f o r m a l p r o j e c t i o n s ..............

Classification of

Cartographic Projections

D i v i s i o n o f P r o j e c t i o n s by t h e Nature

of the Distortions

.
2.
3.
4.
1

Isogonal

or

......
........
..........................

E q u a l l y spaced o r e q u i d i s t a n t p r o j e c t i o n s
Equally large or equivalent projections
Arbitrary projections
D i v i s i o n o f P r o j e c t i o n s According t o t h e

Method o f C o n s t r u c t i o n ( A c c o r d i n g t o t h e

A p p e a r a n c e o f t h e Normal G r i d )

...................
............................
N o r m a l ( e q u i v a l e n t ) c y l i n d r i c a l p r o j e c t i o n .........
S i m p l e e q u a l l y s p a c e d c y l i n d r i c a l p r o j e c t i o n .......
I s o g o n a l c y l i n d r i c a l p r o j e c t i o n ....................
I s o g o n a l o b l i q u e c y l i n d r i c a l p r o j e c t i o n s ...........
I s o g o n a l t r a n s v e r s e and c y l i n d r i c a l G a u s s i a n

p r o j e c t i o n .........................................
C o n i c P r o j e c t i o n s ..................................
S i m p l e n o r m a l c o n i c p r o j e c t i o n .....................
I s o g o n a l c o n i c p r o j e c t i o n ..........................
C o n v e r g e n c e a n g l e o f t h e m e r i d i a n s .................
P o l y c o n i c p r o j e c t i o n s ..............................
Cylindrical

Projections

...........................
................
C e n t r a l p o l a r ( g n o m o n i c p r o j e c t i o n ) ................
E q u a l l y s p a c e d a z i m u t h a l ( c e n t . r a 1 ) p r o j e c t i o n ......
S t e r e o g r a p h i c p o l a r p r o j e c t i o n .....................
N o m e n c l a t u r e o f M a p s ...............................
International projection

Azimuthal

(Perspective)

Projections

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..............................................
O r t h o d r o m e o n t h e E a r t h ' s S u r f a c e ..................
O r t h o d r o m e o n T o p o g r a p h i c a l Maps o f D i f f e r e n t

P r o j e c t i o n s ........................................
L o x o d r o m e o n t h e E a r t h ' s S u r f a c e ...................
General Recommendations f o r M e a s u r i n g D i r e c t i o n s

a n d D i s t a n c e s ......................................
7 . S p e c i a l C o o r d i n a t e S y s t e m s o n t h e E a r t h ' s S u r f a c e ....
O r t h o d r o m i c C o o r d i n a t e S y s t e m ......................
A r b i t r a r y (Ob1 i q u e a n d T r a n s v e r s e ) S p h e r i c a l a n d

P o l a r C o o r d i n a t e S y s t e m s ...........................
P o s i t i o n L i n e s o f an A i r c r a f t on t h e E a r t h ' s

S u r f a c e ............................................
B i p o l a r A z i m u t h a l C o o r d i n a t e S y s t e m ................
G o n i o m e t r i c Range-Finding C o o r d i n a t e System ........
B i p o l a r Range-F ind ing ( C i r c u 1 a r ) C o o r d i n a t e

S y s t e m .............................................
L i n e s o f E q u a l A z i m u t h s ............................
6

.

Maps U s e d f o r A i r c r a f t N a v i g a t i o n

M e a s u r i n g D i r e c t i o n s and D i s t a n c e s on t h e E a r t h ' s

Surface

Difference-Range-Finding
(Hyperbolic)

C o o r d i n a t e System
Overall-Range-Finding
( E l l i p t i c a l ) Coordinate

System

..................................
.............................................
8 . E l e m e n t s o f A i r c r a f t N a v i g a t i o n ......................
E l e m e n t s w h i c h d e t e r m i n e F l i g h t D i r e c t i o n ..........
1 . Assymetry o f the Engine T h r u s t o r A i r c r a f t

D r a g ( F i g . 1 . 5 9 ) ...............................

.
3.

42


45

45

55

60

65

66

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71

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81

85

88

88


94


2

A l l o w a b l e L a t e r a l B a n k i n g o f an A i r c r a f t i n

Horizontal Flight
C o r i o l i s Force
4
Two-dimensional F l u c t u a t i o n s i n the A i r c r a f t

Course
5
G l i d i n g D u r i n g Changes i n t h e L a t e r a l Wind
Speed Component a t F l i g h t A l t i t u d e
E l e m e n t s Which C h a r a c t e r i z e t h e F l i g h t Speed o f

an A i r c r a f t
N a v i g a t i o n a l Speed T r i a n g l e
Elements Which D e t e r m i n e F l i g h t A l t i t u d e

..............................
.................................
.
.........................................
.
.............
........................................
.........................
...........
Calculating F l i g h t A l t i t u d e i n Determining

D i s t a n c e s o n t h e E a r t h ' s S u r f a c e ...................
E l e m e n t s o f A i r c r a f t R o l l ..........................
1 . C o m b i n a t i o n o f R o l l w i t h ' a S t r a i g h t L i n e .......
2 . C o m b i n a t i o n o f t w o r o l l s .......................
3 . L i n e a r p r e d i c t i o n o f r o l l ( L P R ) ................
C H A P T E R TWO . A I R C R A F T N A V I G A T I O N U S I N G MISCELLANEOUS

D E V I C E S ....................................................
1. G e o t e c h n i c a 7 Means o f A i r c r a f t N a v i g a t i o n ............
2 . C o u r s e I n s t r u m e n t s a n d S y s t e m s .......................
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Methods o f U s i n g t h e M a g n e t i c F i e l d o f t h e

Earth t o Determine D i r e c t i o n
V a r i a t i o n s and O s c i l l a t i o n s i n t h e E a r t h ' s

Magnetic F i e l d
M a g n e t i c Compasses
D e v i a t i o n o f M a g n e t i c Compasses a n d i t s

Compensation

.......................
.....................................
.................................
.......................................
E q u a l i z i n g t h e M a g n e t i c F i e l d of t h e A i r c r a f t ....
D e v i a t i o n FormuZas ...............................
C a Z c u Z a t i o n of A p p r o x i m a t e D e v i a t i o n

C o e f f i c i e n t s .....................................

Change i n D e v i a t i o n of M a g n e t i c Compasses a s a

F u n c t i o n o f t h e M a g n e t i c L a t i t u d e of t h e L o c u s

of t h e A i r c r a f t
E Z i m i n a t i o n of D e v i a t i o n i n t h e M a g n e t i c

Compasses

..................................
........................................
G y r o s c o p i c C o u r s e D e v i c e s ..........................
P r 5 n c i p Ze of O p e r a t i o n of G y r o s c o p i c

I n s t r u m e n t s ......................................
Degree of Freedom of t h e G y r o s c o p e . . . . . . . . . . . . . . .
D i r e c t i o n of P r e c e s s i o n of t h e G y r o s c o p e A x i s . . . .
A p p a r e n t R o t a t i o n of G y r o s c o p e A x i s on t h e

E a r t h ' s S u r f a c e ..................................

.............................
......................
G y r o i n d u c t i o n Compass ..............................
D e t a i l s o f D e v i a t i o n O p e r a t i o n s on D i s t a n c e

G y r o m a g n e t i c a n d G y r o i n d u c t i o n Compasses . . . . . . . . . . .
Methods o f U s i n g Course Devices f o r Purposes

o f A i r c r a f t N a v i g a t i o n .............................
Methods of U s i n g C o u r s e D e v i c e s Under C o n d i t i o n s

I n c l u d e d i n t h e F i r s t Group ......................
Methods of U s i n g C o u r s e D e v i c e s Under C o n d i t i o n s

of t h e S e c o n d Group ..............................
Methods of U s i n g C o u r s e D e v i c e s Under t h e

C o n d i t i o n s of t h e T h i r d Group ....................
3 . B a r o m e t r i c A l t i m e t e r s ................................
Description o f a Barometric A l t i m e t e r ..............
Errors i n Measuring A l t i t u d e w i t h a Barometric

A l t i m e t e r ..........................................
4 . A i r s p e e d I n d i c a t o r s ..................................
E r r o r s i n M e a s u r i n g A i r s p e e d .......................
R e l a t i o n s h i p B e t w e e n E r r o r s i n Speed I n d i c a t o r s

a n d F l i g h t A l t i t u d e ................................
G y r o s c o p i c Semicompass
D i s t a n c e G y r o m a g n e t i c Compass

.
.
7.
8.
9.
5
6

....
......................................
S p e c i a l R e q u i r e m e n t s f o r A v i a t i o n C l o c k s ...........
N a v i g a t i o n a l S i g h t s ..................................

Measurement o f t h e T e m p e r a t u r e o f t h e O u t s i d e Air
Aviation Clocks

A u t o m a t i c N a v i g a t i o n I n s t r u m e n t s .....................
P r a c t i c a l Methods o f A i r c r a f t N a v i g a t i o n U s i n g

Geotechnical Devices

.................................
V


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T a k e o f f of the A i r c r a f t at the S t a r t i n g Point

of the Route.......................................
S e l e c t i n g t h e C o u r s e ' t o be F o l l o w e d for the

F l i g h t Route............
.Change i n Navigational Elements D u r i n g F l i g h t
M e a s u r i n g the W i n d a t F l i g h t A l t i t u d e a n d

C a l c u l a t i n g Navigational Elements a t S u c c e s s i v e

Stages
C a l c u l a t i o n o f the P a t h of the A i r c r a f t and

Monitoring Aircraft Navigation i n Terms of

D i s t a n c e s and Direction..
Use of A u t o m a t i c Navigational Devices for

C a l c u l a t i n g the A i r c r a f t P a t h a n d M e a s u r i n g

the W i n d P a r a m e t e r s
Details of A i r c r a f t N a v i g a t i o n U s i n g Geotechnical

M e t h o d s i n Various F l i g h t Conditions.....

...........................
......

224


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227


....
Purpose of Calculating a n d Measuring Pilotage

.........................
Instruments...............
Navigational S l i d e Rule N L - l O M . . . . .................

10. Calculating and Measuring Pilotage Instruments..

CHAPTER THREE. AIRCRAFT NAVIGATION USING RADIO-ENGINEERING

DEVICES ....................................................
Principles o f the Theory o f Radionavigational
Instruments .........................................

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250

250

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Goniometric and G o n i o m e t r i c - R a n g e f i n d i n g Systems ....
A i r c r a f t N a v i g a t i o n U s i n g G r o u n - B a s e d Radio

Direction-Finders....................
.............
S e Z e c t i o n of t h e C o u r s e t o b e FoZZowed and

C o n t r o Z of F Z i g h t D i r e c t i o n .....................
P a t h C o n t r o l . i n Terms of D i s t a n c e and D e t e r ­

m i n a t i o n of t h e A i r c r a f t ' s L o c a t i o n .............
D e t e r m i n a t i o n of t h e Ground S p e e d , D r i f t AngZe,

and Wind ........................................
A u t o m a t i c A i r c r a f t Radio D i s t a n c e - F i n d e r s

( R a d i o c o m p a s s e s ) ..................................
R a d i o c o m p a s s D e v i a t i o n . .........................
A i r c r a f t N a v i g a t i o n U s i n g R a d i o c o m p a s s e s on

Board t h e A i r c r a f t ..............................

283


S p e c i a l F e a t u r e s o f U s i n g Radiocompasses on

Board A i r c r a f t a t High A Z t i t u d e s and F Z i g h t

Speeds

292


W a v e Polarization....
P r o p a g a t i o n of E l e c t r o m a g n e t i c O s c i l l a t i o n s i n

H o m o g e n e o u s Medi'a
P r i n c i p l e s of S u p e r p o s i t i o n a n d I n t e r f e r e n c e

of Radio W a v e s .
P r i n c i p l e C h a r a c t e r i s t i c s of R a d i o n a v i g a t i o n a l

Instruments..
O p e r a t i n g P r i n c i p l e s of R a d i o n a v i g a t i o n a l

Instruments..

2.

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D e t a i Z s of U s i n g R a d i o c o m p a s s e s i n Mak<ng

Maneuvers i n t h e V i c i n i t y of t h e A i r p o r t

a t W h i c h a L a n d i n g is t o b e Made

295


U l t r a - S h o r t w a v e G o n i o m e t r i c and G o n i o m e t r i c Range F i n d i n g S y s t e m s

296


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D e t a i 2s of U s i n g G o n i o m e t r i c - R a n g e F i n d i n g

S y s t e m s a t D i f f e r e n t F Z i g h t A Z t i t u d e d ...........
F a n - S h a p e d G o n i o m e t r i c R a d i o B e a c o n s ..............

Difference-Rangefinding (Hyperbolic) Navigational
S y s t e m s ..............................................
O p e r a t i n g P r i n c i p l e s o f D i f f e r e n t i a l Range-

f i n d i n g Systems
Navigational Applications of Differential-

R a n g e f i n d i n g Systems
Methods o f Improving D i f f e r e n t i a l RAngefinding

N a v i g a t i o n a l Systems

...................................
..............................
..............................
4. Autonomous Radio-Navigational Instruments . . . . . . . . . . . .
A i r c r a f t N a v i g a t i o n a l R a d a r .......................
I n d i c a t o r s of A i r c r a f t N a v i g a t i o n a l R a d a r s . . . . . .
N a t u r e of t h e V i s i b i Z i t y of Landmarks o n t h e

S c r e e n of an A i r c r a f t Radar .....................
Use of A i r c r a f t Radar f o r P u r p o s e s of A i r ­

c r a f t N a v i g a t i o n and A v o i d a n c e of Dangerous

M e t e o r o Z o g i c a Z Phenomena ........................
Autonomous D o p p l e r M e t e r s f o r
Ground Speed

S c h e m a t i c Diagram of t h e O p e r a t i o n of a

M e t e r w i t h C o n t i n u o u s R a d i a t i o n Regime . . . . . . . . . .
Use of D o p p l e r M e t e r s f o r P u r p o s e s of

A i r c r a f t N a v i g a t i o n .............................
P r e p a r a t i o n f o r F l i g h t and C o r r e c t i o n of
E r r o r s i n A i r c r a f t Alavigation by Using
D o p p l e r M e t e r s ..................................

Principles o f Combining Navigational Instruments . . . . .

CHAPTER FOUR . DEVICES AND METHODS FOR MAKING AN

INSTRUMENT LANDING .........................................
S Y S T E M S F O R M A K I N G AN I N S T R U M E N T L A N D I N G . . . . . . . . . . . . . . . . .
S i m p l i f i e d System f o r Making an
Landing

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Instrument


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Marker D e v i c e s ..................................
Low-AZtitude Radio A Z t i m e t e r s . . . . . . . . . . . . . . . . . . .
G y r o h o r i z o n .....................................
V a r i o m e t e r ......................................

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..........
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C a Z c u Z a t i o n of C o r r e c t i o n s f o r t h e T i m e f o r

B e g i n n i n g t h e T h i r d T u r n ........................

Angle o f Slope f o r A i r c r a f t G l i d e
T y p i c a l Maneuvers i n L a n d i n g an A i r c r a f t
C a l c u l a t i o n o f Landing Approach Parameters

f o r a S i m p l i f i e d System

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D r i f t Angle and


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5.

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386

387


C a Z c u Z a t i o n of t h e C o r r e c t i o n for t h e Time

of S t a r t i n g t h e F o u r t h T u r n
C a Z c u Z a t i o n of t h e Moment f o r B e g i n n i n g

Descent Along t h e Landing Course
C a Z c u Z a t i o n o f t h e V e r t i c a l R a t e of D e s c e n t
A l o n g t h e GZide P a t h
D e t e r m i n a t i o n of t h e Lead AngZe for t h e

Landing Path

.....................
................
............................
....................................

L a n d i n g t h e A i r c r a f t o n t h e Runway a n d F l i g h t
along a Given T r a j e c t o r y w i t h a S i m p l i f i e d

L a n d i n g System
C o u r s e - G l i d e L a n d i n g Systems

....................................
......................
Ground ControZ of Course-GZide S y s t e m s . . . . . . . . . .
A i r c r a f t - M o u n t e d Equipment f o r t h e Course-

GZide L a n d i n g S y s t e m ............................

L o c a t i o n and P a r a m e t e r s f o r R e g u Z a t i n g t h e

Equipment for t h e Course-GZide Landing

System
L a n d i n g a n A i r c r a f t w i t h t h e Course-GZide

S y s t e m ..........................................
D i r e c t i o n a Z P r o p e r t i e s of t h e L a n d i n g

S y s t e m A p p a r a t u s ................................
DirectionaZ Devices f o r Landing A i r c r a f t . . . . . . . .

..........................................

.............................
..............................
CHAPTER F I V E . AVIATION ASTRONOMY ............................
1. T h e C e l e s t i a l S p h e r e .................................
S p e c i a l P o i n t s . P l a n e s . and C i r c l e s i n t h e

C e l e s t i a l S p h e r e ..................................
S y s t e m s o f C o o r d i n a t e s ............................
Apparent System o f Coordinates . . . . . . . . . . . . . . . . . .
E q u a t o r i a Z S y s t e m of C o o r d i n a t e s . . . . . . . . . . . . . . . .
G r a p h i c R e p r e s e n t a t i o n o f t h e C e l e s t i a l Sphere . . . .
2 . Diurnal Motion of the Stars ...........................
Motion o f the Stars a t D i f f e r e n t Latitudes ........
R i s i n g and S e t t i n g . N e v e r - R i s i n g and

N e v e r - S e t t i n g S t a r s ...............................
Motion o f Stars a t the Ter-restrial Poles ..........
Motion o f Stars a t Middle Latitudes ...............
M o t i o n o f S t a r s a t t h e E q u a t o r ....................
C u l m i n a t i o n o f S t a r s ..............................
ProbZems and E x e r c i s e s ..........................
3 . T h e M o t i o n o f t h e Sun ................................
T h e A n n u a l M o t i o n o f t h e Sun ......................
M o t i o n of t h e Sun AZong t h e E c Z i p t i c . . . . . . . . . . . .
D i u r n a l M o t i o n o f t h e Sun .........................
The M o t i o n of t h e Sun a t t h e N o r t h PoZe . . . . . . . . .
M o t i o n of t h e Sun b e t w e e n t h e N o r t h PoZe and

t h e A r c t i c C i r c Z e ...............................
Radar L a n d i n g Systems

B r i n g i n g an A i r c r a f t In for a L a n d i n g

w i t h L a n d i n g Radar

viii


.

388

389


390

391


391

394

396

400


401

403


406

408

410

415

418


418

418
421
421
422
424
426
427

















428

431

432

433

433

435

436

436

437

439

439

439


I

1 . 1 . 1 1 1 1 . 1 1

.

4

..........
1-1.1.1-1

.........
.
111

11.1

I,

.

I, I I. 111 I, I

.
.
I

.I

I,

111

I

..
.I

111. .I. 8.8

I I

. .... .
I.

-- .
II. -I

I.

M o t i o n of t h e Sun a b o v e t h e A r c t i c C i r c l e . . . . . . .
M o t i o n of t h e S u n a t M i d d l e L a t i t u d e s . . . . . . . . . . .
M o t i o n of t h e S u n a t t h e T e r r e s t r i a l

E q u a t o r .........................................
Motion of t h e Moon ...................................
I n t r i n s i c Motion of

t h e Moon

......................

D i r e c t i o n and R a t e of t h e Moon's M o t i o n . . . . . . . . .
P h a s e s of t h e Moon ..............................

N a t u r e o f t h e M o t i o n o f t h e Moon a r o u n d

the Earth
L o c a t i o n o f t h e Moon A b o v e t h e H o r i z o n

.........................................
............
5 . Measurement o f Time ..................................
E s s e n c e o f C a l c u l a t i n g T i m e .......................
S i d e r e a l T i m e .....................................
T r u e S o l a r T i m e ...................................
Mean S o l a r T i m e ...................................
L o c a l C i v i l T i m e ..................................
G r e e n w i c h T i m e ....................................
Z o n e T i m e . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
S t a n d a r d T i m e .....................................
R e l a t i o n B e t w e e n G r e e n w i c h , L o c a l and Zone

( S t a n d a r d ) Time ...................................
M e a s u r i n g A n g l e s i n T i m e U n i t s ....................
T i m e S i g n a l s ......................................
O r g a n i z a t i o n of T i m e S i g n a l s i n A v i a t i o n . . . . . . . .
A B r i e f H i s t o r y o f Time Reckoning . . . . . . . . . . . . . . . . .
6 . Use o f A s t r o n o m i c a l Devices ..........................
A s t r o n o m i c a l C o m p a s s e s ............................
A s t r o n o m i c a l S e x t a n t s .............................
C H A P T E R SIX

.
2.
3.
4.
1

5

.

6

.

7.

8.

.

A C C U R A C Y IN AIRCRAFT NAVIGATION . . . . . . . . . . . . . . . .

A c c u r a c y i n M e a s u r i n g N a v i g a t i o n a l E l e m e n t s and

i n A i r c r a f t N a v i g a t i o n a s a Whole ....................
Methods o f E v a l u a t i n g t h e A c c u r a c y o f A i r c r a f t

N a v i g a t i o n . . . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. .
L i n e a r and Two-Dimensional P r o b l e m s o f

P r o b a b i l i t y T h e o r y ...................................
C o m b i n a t i o n o f Methods o f M a t h e m a t i c a l A n a l y s i s

and M a t h e m a t i c a l S t a t i s t i c s i n E v a 7 u a t i n g t h e

Accuracy o f N a v i g a t i o n a l Measurements . . . . . . . . . . . . . . . .
I n f l u e n c e o f t h e Geometry o f a N a v i g a t i o n a l

S y s t e m on t h e A c c u r a c y o f D e t e r m i n i n g A i r c r a f t

C o o r d i n a t e s ..........................................
E v a l u a t i o n of t h e Accuracy of Measuring a

N a v i g a t i o n a l P a r a m e t e r ...............................
C a l c u l a t i o n o f t h e Wind w i t h an E v a l u a t i o n o f t h e

A c c u r a c y o f A i r c r a f t N a v i g a t i o n ......................
Consideration o f the Polar F l a t t e n i n g of the Earth

i n t h e D e t e r m i n a t i o n o f D i r e c t i o n s and D i s t a n c e s

on t h e E a r t h ' s s u r f a c e ...............................

ix


441

441

442
442
442
443
443












445
445
446
446
446
44'7
448
449
449
451
453
























454
455
457
458
459
461
467
469


















470

470


474

478

490

493


497

499

501


.

1
2.
3.

4

.

5.

6

.

2

.

FLIGHT PREPARATION

C a l c u l a t i n g t h e F u e l Scrpply f o r F l i g h t o n

A i r c r a f t w i t h L o w - A l t i t u d e P i s t o n Engines
C a l c u l a t i n g t h e Fuel Supply f o r F l i g h t i n A i r ­

c r a f t w i t h H i g h - A l t i t u d e P i s t o n Engines
C a l c u l a t i n g t h e F u e l Supply f o r F l i g h t on

A i r c r a f t w i t h Gas T u r b i n e E n g i n e s
C a l c u l a t i n g t h e Greatest Distance of t h e
A i r c r a f t ' s P o i n t o f Closest Approach t o a

Reserve A i r p o r t

.

507

508

514


517

518


521


...................................
........

530

532


....

536


G E N E R A L P R O C E D U R E F O R AIRCRAFT NAVIGATION

G e n e r a l Methods of A i r c r a f t N a v i g a t i o n a l o n g

A i r R o u t e s ...........................................
S t a g e s i n E x e c u t i n g t h e F l i g h t .......................

................................
..................
Descent and E n t r a n c e t o t h e Region o f t h e

L a n d i n g A i r p o r t b y a n A i r c r a f t ....................
Maneuvering i n t h e V i c i n i t y o f t h e A i r p o r t
a n d t h e L a n d i n g A p p r o a c h ..........................
S u p p l e m e n t 1 .
C o m p o s i t e C h a r t o f T o p o g r a p h i c a l Maps . . . . .
Supplement 2 .
S p h e r i c a l Trigonometry Formulas . . . . . . . . . . .
S u p p l e m e n t 3 .
Map of t h e Heavens ........................
S u p p l e m e n t 4 .
Map o f Time Zones .........................
S u p p l e m e n t 5 .
T a b l e o f Greenwich Hour A n g l e s o f t h e

S u n and C h a r t o f T h e i r C o r r e c t i o n s f o r

t h e F l i g h t Date ...........................
S u p p l e m e n t 6 .
T a b l e o f V a l u e s o f t h e F u n c t i o n @ (Z - a) .
Supplement 7 .
Units o f t e n Encountered i n A i r c r a f t

N a v i g a t i o n and T h e i r V a l u e s ...............
Take-Off

507

.........
...........
.................

Pre-flight Preparation and Flight Calculation

C H A P T E R EIGHT
1.

.

...........................
G o a l s and P r o b l e m s o f F l i g h t P r e p a r a t i o n .............
P r e p a r i n g F l i g h t C h a r t s a n d Marking t h e Route . . . . . . . .
S t u d y i n g t h e Route and C a l c u l a t i n g a S a f e

F l i g h t A l t i t u d e ......................................
S p e c i a l P r e p a r a t i o n of C h a r t s and A i d s f o r

Using V a r i o u s N a v i g a t i o n a l D e v i c e s i n F l i g h t .........
Calculating t h e Distance and Duration o f F l i g h t ......

CHAPTER SEVEN

and Climb

E x e c u t i n g a F l i g h t Along a R o u t e

X


518


521


536
538
539
540










542

543

545

547

549

550

551

552

554


T h e t h e o r y and p r a c t i c e of a i r c r a f t
n a v i g a t i o n a t t h e modern ZeveZ o f a v i a t i o n t e c h ­
noZogy a r e s u m m a r i z e d i n t h i s b o o k ; t h e m o s t i m ­
p o r t a n t p r a c t i c a Z probZems of t h e u t i Z i z a t i o n of
g e n e r a Z , r a d i o - e n g i n e e r i n g , and a s t r o n o m i c a Z means
of a i r c r a f t n a v i g a t i o n a r e s e t f o r t h ; t h e p r o c e ­
d u r e of t h e p i Z o t ' s p r e p a r a t i o n f o r f Z i g h t , t h e
means of c a Z c u Z a t i n g t h e d i s t a n c e and d u r a t i o n
of a f l i g h t , and t h e c a r r y i n g o u t of p r e - l a n d i n g
m a n e u v e r i n g and l a n d i n g of t h e a i r c r a f t u n d e r
compZex m e t e o r o Z o g i c a Z c o n d i t i o n s d u r i n g t h e day
o r a t night are elucidated.
The basic m a t e r i a l o f t h e b o o k , s u f f i c i e n t
f o r t h e p r a c t i c a Z m a s t e r y of t h e means and m e t h o d s
of a i r c r a f t n a v i g a t i o n , is p r e s e n t e d w i t h t h e ap­
p Z i c a t i o n of m a t h e m a t i c s w i t h i n t h e Z i m i t s of a
s e c o n d a r y schooZ c o u r s e .
The p r o b l e m s w h i c h a r e
n e c e s s a r y f o r a d e e p e r s t u d y of t h e m a t e r i a 2 a r e
d i s c u s s e d i n t e r m s of p r i n c i p l e s of h i g h e r m a t h ­
ematics.
The book is i n t e n d e d f o r p i Z o t s and n a v i g a ­
t o r s . I t can be used as a t e x t b o o k f o r s t u d e n t s
of c i v i Z a v i a t i o n e d u c a t i o n a Z i n s t i t u t i o n s .
ABSTRACT:

xi


I NT RO DU CT I ON
A i r c r a f t n a v i g a t i o n or a e r i a l n a v i g a t i o n i s a s c i e n c e w h i c h
s t u d i e s t h e t h e o r y and p r a c t i c a l methods of t h e safe n a v i g a t i o n o f
a i r p l a n e s as w e l l as o t h e r a i r c r a f t ( h e l i c o p t e r s , d i r i g i b l e s , e t c . )
i n t h e a i r s p a c e above t h e E a r t h ' s s u r f a c e .
By t h e p r o c e s s o f a i r c r a f t n a v i g a t i o n , w e
a c t i v i t i e s o f t h e a i r c r a f t crew a n d t h e g r o u n d
which are d i r e c t e d t o w a r d a c o n s t a n t knowledge
l o c a t i o n and which e n s u r e s a f e and a c c u r a t e f l i
as w e l l as a r r i v a l a t t h e p o i n t o f d e s t i n a t i o n
a t an e s t a b l i s h e d t i m e .

mean t h e c o m p l e x o f
traffic control,
of t h e aircraft's
g h t along a set course
a t a s e t a l t i t u d e and

During t h e i n i t i a l p e r i o d of t h e development of a v i a t i o n , a i r ­
c r a f t d i d n o t h a v e e q u i p m e n t f o r p i l o t i n g when t h e n a t u r a l h o r i z o n
w a s n o t v i s i b l e a n d f o r o r i e n t a t i o n when t h e g r o u n d w a s n o t v i s i b l e ,
s o t h a t v i s u a l o r i e n t a t i o n w a s t h e b a s i c method o f a i r c r a f t n a v i g a ­
tion.
The p o s i t i o n o f t h e a i r c r a f t w a s d e t e r m i n e d by c o m p a r i n g v i s ­
i b l e landmarks i n t h e area o v e r which t h e a i r c r a f t w a s f l y i n g , w i t h
t h e i r r e p r e s e n t a t i o n o n a map.
However, a t t h i s t i m e t h e n e c e s s i t y f o r i n s t r u m e n t a l methods
of aircraft navigation w a s already f e l t .
The most s i m p l e d e v i c e s
f o r measuring a i r s p e e d , f l i g h t a l t i t u d e , t h e a i r c r a f t ' s c o u r s e , and
This
s e v e r a l o t h e r f l i g h t p a r a m e t e r s w e r e i n s t a l l e d on a i r c r a f t .
period s a w t h e appearance of t h e first navigator's c a l c u l a t i n g in­
s t r u m e n t s (wind-speed i n d i c a t o r s and n a v i g a t i o n a l s l i d e r u l e s ) .
A t t h e b e g i n n i n g o f t h e 1 9 2 0 ' ~t h~ e f i r s t h y d r o s c o p i c d e v i c e s
a p p e a r e d on a i r c r a f t ; t h e y were t u r n and g l i d e i n d i c a t o r s which ( i n
combination w i t h i n d i c a t o r s o f a i r s p e e d and v e r t i c a l v e l o c i t y ) i n ­
d i c a t o r s ( v a r i o m e t e r s ) made i t p o s s i b l e t o j u d g e i n a r a t h e r p r i m i ­
t i v e way t h e p o s i t i o n o f t h e a i r c r a f t i n s p a c e w h e n t h e n a t u r a l
By m e a n s o f t h e s e d e v i c e s , t h e a i r c r a f t
horizon w a s not v i si b l e .
crews ( a f t e r s p e c i a l t r a i n i n g ) were a l r e a d y a b l e t o c a r r y o u t
f l i g h t s i n t h e clouds and above t h e clouds.

A t t h e end o f t h e 2 0 ' s and t h e b e g i n n i n g o f t h e ~ O ' S , more r e ­
f i n e d p i l o t a g e d e v i c e s were d e v e l o p e d :
gyrohorizons and gyrosemi­
compasses, which a t t h i s t i m e r e l i a b l y ensured p i l o t a g e of a i r c r a f t
w i t h a d e q u a t e s a f e t y when t h e g r o u n d w a s n o t v i s i b l e .
L a t e r , on
t h e basis of these devices, devices f o r automatic aircraft pilotage
(automatic p i l o t s ) w e r e created.

xiii

J

I

A c h i e v e m e n t s i n t h e a r e a o f p i l o t i n g a i r c r a f t when t h e E a r t h
w a s n o t v i s i b l e , a s w e l l as t h e g r o w t h by t h a t t i m e o f t h e s p e e d ,
a l t i t u d e , and d i s t a n c e o f a i r c r a f t f l i g h t s , r e q u i r e d t h e c r e a t i o n
of means t o e n s u r e a i r c r a f t n a v i g a t i o n i n d e p e n d e n t o f t h e v i s i b i l i t y
of t e r r e s t r i a l landmarks.
During t h e s e y e a r s , zone r a d i o beacons which allowed t h e a i r ­
c r a f t ' s f l i g h t d i r e c t i o n t o be maintained along a narrowly d i r e c t e d
r a d i a l l i n e which c o i n c i d e s w i t h t h e d i r e c t i o n of t h e s t r a i g h t p a r t
o f an a e r i a l r o u t e began t o appear.
Ground r a d i o g o n i o m e t e r s a l s o
a p p e a r e d , by means o f w h i c h d i r e c t i o n w a s d e t e r m i n e d i n a n a i r c r a f t ,
a s w e l l as t h e p o s i t i o n o f t h e a i r c r a f t a l o n g two i n t e r s e c t i n g d i ­
rections.
Another a s p e c t o f t h e development of a i r c r a f t n a v i g a t i o n a t
t h i s time w a s astronomical o r i e n t a t i o n .
To determine the location
o f a n a i r c r a f t , v a r i o u s s e x t a n t s w e r e c o n s t r u c t e d a n d s p e c i a l com­
p u t a t i o n t a b l e s a n d g r a p h s o f t h e movement o f h e a v e n l y b o d i e s w e r e
compiled f o r use w i t h t h e s e x t a n t s .
I n t h e m i d - 3 0 ' ~ d~e v i c e s a p ­
p e a r e d f o r d e t e r m i n i n g t h e c o u r s e o f an a i r c r a f t a c c o r d i n g t o t h e
heavenly bodies.
A t t h e same t i m e , o p t i c a l s i g h t i n g d e v i c e s w e r e u s e d , by means
of w h i c h ( d u r i n g v i s i b i l i t y o f t h e t e r r e s t r i a l l a n d m a r k s ) t h e g r o u n d ­
s p e e d , f l i g h t d i r e c t i o n and d r i f t a n g l e o f t h e a i r c r a f t were meas­
u r e d , a l l o f w h i c h w e r e l a t e r u s e d f o r some t i m e a s c o n s t a n t s f o r
c a l c u l a t i n g t h e p'ath o f an a i r c r a f t accordring t o f l i g h t t i m e and
direction.
A v e r y i m p o r t a n t s t a g e i n t h e d e v e l o p m e n t o f means o f a i r c r a f t
n a v i g a t i o n o f t h e m i d - 3 0 ' s was t h e a p p e a r a n c e o f a i r c r a f t r a d i o ­
goniometers ( r a d i o s e m i c o m p a s s e s ) , a f u r t h e r m o d i f i c a t i o n of which
were t h e a u t o m a t i c a i r c r a f t radiocompasses.
Radiosemicompasses and
r a d i o c o m p a s s e s w e r e , f o r a p e r i o d o f more t h a n 2 0 y e a r s , t h e b a s i c
means o f a i r c r a f t n a v i g a t i o n i n a i r c r a f t w i t h p i s t o n e n g i n e s .
D u r i n g W o r l d War I1 a n d e s p e c i a l l y i n t h e p o s t w a r y e a r s , r a d i o e n g i n e e r i n g s y s t e m s o f l o n g a n d s h o r t - d i s t a n c e n a v i g a t i o n of a d i f ­
f e r e n t k i n d a s w e l l a s r a d i o - n a v i g a t i o n l a n d i n g s y s t e m s became w i d e ­
spread.
E s s e n t i a l l y , t h e s e were n o t autonomous means o f a e r i a l
n a v i g a t i o n b u t systems which i n c l u d e d b o t h ground-based f a c i l i t i e s
f o r t h e - s e c u r i t y of a i r c r a f t n a v i g a t i o n and a i r c r a f t equipment.
R a d i c a l c h a n g e s i n t h e a r e a o f means a n d m e t h o d s o f a i r c r a f t
n a v i g a t i o n o c c u r r e d (and a r e o c c u r r i n g a t t h e p r e s e n t t i m e ) i n con­
n e c t i o n with t h e development of j e t a v i a t i o n technology.
The s h a r p l y g r o w i n g s p e e d , a l t i t u d e , a n d d i s t a n c e o f f l i g h t s
h a v e r e q u i r e d a u t o m a t i o n o f t h e most l a b o r i o u s p r o c e s s e s o f a i r c r a f t
navigation.
Magnetic course d e v i c e s and non-automatic r a d i o navig a t i o n a l systems were of l i t t l e use f o r e n s u r i n g t h e automation of
a i r c r a f t n a v i g a t i o n and t h e p i l o t i n g of high-speed aircraft.
There
xiv

­
/5

arose a n e c e s s i t y f o r developing highly s t a b l e gyroscopic compasses,
autonomous s p e e d and f l i g h t d i r e c t i o n meters, and s t r i c t e r c o n s i d e r ­
a t i o n o f t h e a i r c r a f t ' s f l i g h t ' d y n a m i c s t o e n s u r e t h e r a p i d a n d ac­
c u r a t e s o l u t i o n o f n a v i g a t i o n a l problems by c o m p u t e r s .
The s c i e n c e o f " a i r c r a f t n a v i g a t i o n " grew and d e v e l o p e d a l o n g
w i t h t h e development o f a v i a t i o n and n a v i g a t i o n t e c h n o l o g y .
The
works o f t h e o u t s t a n d i n g R u s s i a n s c i e n t i s t s and i n v e n t o r s , M . V.
L o m o n o s o v , N . Ye. Z h u k o v s k i y , K . E , T s i o l k o v s k i y , a n d A . S . P o p o v
were t h e b a s i s o f a i r c r a f t n a v i g a t i o n t h e o r y .

A l a r g e c o n t r i b u t i o n t o t h e s c i e n c e o f a i r c r a f t n a v i g a t i o n was
made b y t h e f o l l o w i n g S o v i e t n a v i g a t o r s a n d s c i e n t i s t s :
B. V .
S t e r l i g o v , S . A . D a n i l i n , I . T . S p i r i n , G. S. F r e n k e l ' , A . V. Bely­
a k o v , L . P . S e r g e y e v , R . V . K u n i t s k i y , G . 0 . F r i d l e n d e r , G . F.
M o l o k a n o v , B . G . R a t s , V . Yu. P o l y a k , e t a l .
The s u c c e s s e s a c h i e v e d i n t h e d e v e l o p m e n t o f a i r c r a f t n a v i g a ­
t i o n a s a s c i e n c e made i t p o s s i b l e , e v e n i n 1 9 2 5 - 1 9 2 9 , t o a c c o m p l i s h
l o n g f l i g h t s by S o v i e t a i r c r a f t a l o n g t h e r o u t e s :
Moscow-Peking
( M . M . G r o m o v ) , Moscow-Tokyo a n d Moscow-New Y o r k ( S . A . S h e s t a k o v ) .
F u r t h e r n o n s t o p f l i g h t s by S o v i e t a v i a t o r s , o r g a n i z e d from
1 9 3 6 - 1 9 3 9 (V. P . C h k a l o v , M . M . G r o m o v , a n d V . K . K o k k i n a k i ) b o t h
o v e r t h e t e r r i t o r y o f t h e S o v i e t Union and e s p e c i a l l y o v e r t h e N o r t h
P o l e t o t h e USA, w e r e l i k e a g r e a t s c h o o l , i n w h i c h t h e e x a m i n a t i o n s
were t h e s u c c e s s e s a c h i e v e d b y S o v i e t s c i e n t i s t s i n t h e m c . a 2 5
aircraft navigation.
W o r l d War I1 w a s a v e r i f i c a t i o n o f a l l t h e a c h i e v e m e n t s i n t h e
t h e o r y and p r a c t i c e o f a i r c r a f t n a v i g a t i o n , e s p e c i a l l y i n t h e f i e l d
of long-distance aviation, with t h e carrying out of long-distance
night flights.
During t h i s period, a r i c h s t o r e o f experience w a s
a c c u m u l a t e d and f u r t h e r improvements i n a i r c r a f t n a v i g a t i o n methods
were c a r r i e d o u t .
I n t h e postwar p e r i o d , t h e s c i e n c e of a i r c r a f t n a v i g a t i o n under­
,ent an e s p e c i a l l y v i g o r o u s development i n c o n n e c t i o n with t h e ap­
e a r a n c e o f h i g h - s p e e d j e t a i r c r a f t , and a l s o i n c o n n e c t i o n w i t h
he g r e a t a c h i e v e m e n t s o f t h e r a d i o and e l e c t r o n i c s i n d u s t r y .
L o n g - d i s t a n c e f l i g h t s of h i g h - s p e e d a i r c r a f t a l o n g a e r i a l
l u t e s which i n c l u d e i n t e r n a t i o n a l and i n t e r c o n t i n e n t a l f l i g h t s , as
11 a s f l i g h t s t o t h e A r c t i c a n d A n t a r c t i c , a r e b e c o m i n g r o u t i n e
r c i v i l a v i a t i o n crews.

A t t h e p r e s e n t t i m e , a i r c r a f t navigation s c i e n c e has been d i s ­
g u i s h e d as an i n d e p e n d e n t and o r d e r l y s c i e n c e i n which t h e
i e v e m e n t s o f a number o f t h e g e n e r a l and s p e c i a l b r a n c h e s o f
wledge a r e employed:
p h y s i c s , mathematics, geodesy, astronomy,
>hysics, aerodynamics, r a d i o engineering, r a d i o e l e c t r o n i c s , etc.

Navigation technology is developing at a rapid pace; a i r c r a f t
a n d g r o u n d f a c i l i t i e s for a i r c r a f t n a v i g a t i o n a r e ' c o n t i n u a l l y b e i n g
p e r f e c t e d and t h e p r o f e s s i o n a l t r a i n i n g and n a v i g a t i o n a l p r e p a r a ­
t i o n o f f l i g h t and ground p e r s o n n e l h a s improved.
A l l t h i s has
r a d i c a l l y r a i s e d t h e r e l i a b i l i t y o f aircraft n a v i g a t i o n , i t s accu­
r a c y , and i t s c h i e f c r i t e r i o n , s a f e t y .
Modern t e c h n i c a l means o f a i r c r a f t n a v i g a t i o n a r e d i v i d e d i n t o
f o u r b a s i c groups according t o t h e p r i n c i p l e o f o p e r a t i o n .
1.
G e o t e c h n i c a Z means of a i r c r a f t n a v i g a t i o n , w h i c h a r e b a s e d
on t h e p r i n c i p l e o f m e a s u r i n g d i f f e r e n t p a r a m e t e r s o f t h e E a r t h ' s
fields.
They i n c l u d e :
magnetic compasses, gyroscopic n a v i g a t i o n
and p i l o t i n g d e v i c e s , gyromagnetic and g y r o i n d u c t i o n t e l e c o m p a s s e s ,
course systems, airspeed indicators, barometric altimeters, exter­
n a l a i r thermometers, navigation i n d i c a t o r s , i n e r t i a l i n d i c a t o r s ,
mechanical clocks, e t c .
2.
R a d i o - e n g i n e e r i n g means of a i r c r a f t n a v i g a t i o n , w h i c h a r e
b a s e d on t h e o p e r a t i n g p r i n c i p l e o f r a d i o - e l e c t r o n i c t e c h n o l o g y .
These i n c l u d e goniometer r a d i o - e n g i n e e r i n g systems ( r a d i o compasses
w i t h ground t r a n s m i t t i n g r a d i o s t a t i o n s , ground radiogoniometers
w i t h a i r c r a f t r e c e i v i n g - t r a n s m i t t i n g r a d i o s t a t i o n s , and r a d i o bea­
cons w i t h a i r c r a f t r e c e i v i n g r a d i o equipment), r a n g e f i n d i n g systems,
g o n i o m e t e r - r a n g e f i n d i n g s y s t e m s , ground and a i r c r a f t r a d a r , Doppler
m e t e r s a n d s y s t e m s , r a d i o a l t i m e t e r s , c o u r s e - l a n d i n g beam s y s t e m s
w i t h t h e i r ground and a i r c r a f t e q u i p m e n t , e t c .
3.
A s t r o n o m i c a l ( r a d i o a s t r o n o m i c a l ) means of a i r c r a f t n a v i ­
g a t i o n , w h i c h a r e b a s e d on t h e p r i n c i p l e o f m e a s u r i n g t h e m o t i o n '

parameters of heavenly bodies.
These i n c l u d e a v i a t i o n s e x t a n t s ,
astrocompasses, astronomical o r i e n t a t o r s , e t c .
4.
L i g h t e n g i n e e r i n g means of a i r c r a f t n a v i g a t i o n , w h i c h a r e
b a s e d on t h e p r i n c i p l e o f u s i n g l i g h t e n e r g y r a d i a t i o n .
These i n ­
c l u d e ground l i g h t beams, l i g h t and p u l s e - l i g h t equipment f o r t a k e ­
o f f and l a n d i n g s t r i p s as w e l l as a i r c r a f t , e n c l o s u r e s f o r t h e l i g h t ­
i n g equipment of t h e r o u t e s and a i r p o r t s (housings f o r ground i n ­
s t a l l a t i o n s ) , various pyrotechnic devices, etc.

A t t h e h e a r t o f a s a f e and a c c u r a t e f l i g h t a c c o r d i n g t o a s e t
r o u t e , i n t h e v i c i n i t y o f t h e a i r p o r t , or d u r i n g t a k e - o f f a n d l a n d ­
ing, lies t h e p r i n c i p l e of t h e o v e r a l l usage of a l l t h e available
t e c h n i c a l means o f a i r c r a f t n a v i g a t i o n , b o t h g r o u n d f a c i l i t i e s and
those aboard t h e a i r c r a f t .

/6

NASA T T F-524


C H A P T E R O.NE
COORDINATE SYSTEMS AND ELEMENTS O F A I R C R A F T N A V I G A T I O N
1.

Elements o f Aircraft

Movement

in Space

The f u n d a m e n t a l p r o b l e m o f a i r c r a f t n a v i g a t i o n i n a l l s t a g e s
o f f l i g h t i s m a i n t a i n i n g a g i v e n t r a j e c t o r y o f a i r c r a f t movement i n
a l t i t u d e , d i r e c t i o n a n d t i m e by means o f a complex u t i l i z a t i o n o f
n a v i g a t i o n a l means a n d m e t h o d s .
A successful solution t o these
p r o b l e m s d e p e n d s on c o n s t a n t a n d a c c u r a t e i n f o r m a t i o n c o n c e r n i n g
t h e position of t h e craft r e l a t i v e t o a given f l i g h t t r a j e c t o r y ,
t h e n a t u r e of t h e a i r c r a f t movement, and t h e a c t i o n s of t h e c r e w .
A s a r e s u l t of t h e c u r v a t u r e of t h e E a r t h ' s s u r f a c e ,
f l i g h t t r a j e c t o r y of an aircraft i s c u r v i l i n e a r .
However,
i n t o account t h e l a r g e r a d i u s of curvature of t h e Earth's
a s m a l l a r e a c a n a l w a y s b e d e l i n e a t e d on i t whose s u r f a c e
a s s u m e d t o b e p l a n e ( F i g . 1.1).

any given
by t a k i n g
surface,
can be

L e t us e r e c t a perpendicular 01Y from
t h e c e n t e r o f t h e s m a l l area which w e have
chosen and c o n t i n u e it u n t i l it i n t e r s e c t s
the center of the Earth.
Obviously, t h i s
w i l l be a p e r p e n d i c u l a r l i n e , which w e can
c a l l t h e v e r t i c a z of t h e Z o c u s .

I n t h e p l a n e o f t h e s m a l l area which
w e have c h o s e n , l e t u s draw a s t r a i g h t
l i n e t h r o u g h t h e p o i n t 01 a n d t a k e i t a s
t h e X a x i s ; t h e n l e t u s draw a n o t h e r
s t r a i g h t l i n e t h r o u g h t h e p o i n t 01 i n t h e
plane of t h e area, perpendicular t o t h e
f i r s t , and c a l l it t h e Z a x i s .
T h u s , a t p o i n t 0 1 o n t h e E a r t h ' s surface, w e w i l l o b t a i n a r e c t a n g u l a r system
of space coordinates X, Y, Z.

F i g . 1.1. R e c t a n g u l a r
Coordinate System on
t h e Earth's Surface.

-.

;t.

~-

.

-

__

.

~~

..

Numbers i n t h e m a r g i n i n d i c a t e p a g i n a t i o n i n t h e f o r e i g n t e x t .

1

The t r a v e l of a n a i r c r a f t o v e r t h e E a r t h ' s s u r f a c e w i l l i n ­
v o l v e b o t h a s h i f t i n t h e p o i n t 01 ( o r i g i n o f t h e c o o r d i n a t e s ) a n d
t h e r o t a t i o n of t h e axes o f t h e c o o r d i n a t e system around t h e c e n t e r
of t h e E a r t h ( p o i n t 0 ) .
However, t h e s y s t e m o f c o o r d i n a t e s which w e have o b t a i n e d can
b e u s e d for d e t e r m i n i n g t h e d i r e c t i o n s o f t h e a i r c r a f t a x e s a n d t h e
component f l i g h t s p e e d v e c t o r s .
Since t h e o r i g i n of t h i s system is
b e i n g c o n t i n u o u s l y s h i f t e d , l e t u s d e s i g n a t e it as a g l i d i n g r e c ­

t a n g u l a r s y s t e m of c o o r d i n a t e s .
In t h i s coordinate system, t h e
f o l l o w i n g elements can be d i s t i n ­
guished:

;

44

(a) P o s i t i o n o f t h e l o n g i t u ­
d i n a l axis of t h e a i r c r a f t i n t h e
horizontal plane (aircraft course),

F i g . 1 . 2 . Dip Angle of t h e
T r a j e c t o r y of A l t i t u d e Gain.

(b) Position of the longitu­
d i n a l a x i s of t h e a i r c r a f t i n t h e
v e r t i c a l p l a n e ( a n g l e of p i t c h o f

the a i r c r a f t ) ,

( c > Position of t h e l a t e r a l a x i s of t h e a i r c r a f t i n t h e ver­
t i c a l plane (lateral banking),
( d ) D i s t a n c e a l o n g t h e v e r t i c a l from t h e E a r t h ' s s u r f a c e ( t h e
a r e a w h i c h we h a v e c h o s e n ) t o t h e a i r c r a f t ( f l i g h t a l t i t u d e ) ,
( e ) V e r t i c a l s p e e d ( a l t i t u d e g a i n and

loss),

( f ) C o m p o n e n t f l i g h t s p e e d a l o n g t h e X a n d Z a x e s or t h e v e c ­
t o r o f g r o u n d s p e e d a n d i t s d i r e c t i o n ( g r o u n d s p e e d and f l i g h t a n g l e ) ,
( g ) A n g u l a r v e l o c i t y o f a i r c r a f t roll,
( h ) Component wind

speed along t h e

X and

Z axes of

t h e system,

or t h e w i n d v e c t o r a n d i t s d i r e c t i o n ( w i n d s p e e d and d i r e c t i o n ) .
U s u a l l y t h e p o s i t i o n o f t h e c r a f t on t h e E a r t h ' s s u r f a c e i s
t r e a t e d i n s u r f a c e - c o o r d i n a t e s y s t e m s , t h e most widely used of which
a r e t h e g e o g r a p h i c s y s t e m a n d t h e r e f e r e n c e s y s t e m whose m a j o r a x i s
c o i n c i d e s w i t h a g i v e n f l i g h t t r a j e c t o r y on t h e E a r t h ' s s u r f a c e .
The p o s i t i o n o f t h e a i r c r a f t i n s u r f a c e - c o o r d i n a t e s y s t e m s i s
a s s u m e d t o b e t h e p o s i t i o n of t h e o r i g i n o f t h e g l i d i n g s y s t e m .
To
a n a l y z e t h e e l e m e n t s o f a i r c r a f t n a v i g a t i o n , l e t u s combine t h e X
a x i s of t h e gliding-coordinate system with a given f l i g h t t r a j e c t o r y
of the aircraft.
I n o r d e r t o keep t h e a i r c r a f t i n t h e r e c t i l i n e a r h o r i z o n t a l
2

­
/8


segment o f t h i s t r a j e c t o r y , t h e crew must m a i n t a i n a f l i g h t c o n d i ­
t i o n i n which t h e a i r c r a f t w i l l n o t be s h i f t e d a l o n g t h e v e r t i c a l
( a l t i t u d e g a i n and l o s s ) , t h e r e w i l l be no l a t e r a l d e v i a t i o n ( t o
t h e r i g h t or l e f t ) , i . e . , t h e v e r t i c a l v e l o c i t y Vy a n d t h e l a t e r a l
component o f t h e v e l o c i t y V z , w i l l b e equal t o z e r o , and t h e l o n g i ­
t u d i n a l f l i g h t v e l o c i t y Vx ( a l o n g t h e X a x i s ) w i l l b e a s g i v e n .
If t h e f l i g h t t r a j e c t o r y i s i n c l i n e d ( s e g m e n t s o f a l t i t u d e g a i n
a n d loss), t h e c r e w m u s t h o l d t h i s t r a j e c t o r y b y m a i n t a i n i n g t h e
/9
v e r t i c a l and l o n g i t u d i n a l f l i g h t v e l o c i t i e s ( V y and Vx>, i . e . ,
m a i n t a i n a given d i p a n g l e of t h e t r a j e c t o r y 0 ( F i g . 1 . 2 ) .

-

Obviously, a t a constant dip angle of t h e f l i g h t t r a j e c t o r y ,
t h e l a t t e r w i l l have a c u r v a t u r e i n t h e v e r t i c a l p l a n e j u s t as i n
horizontal flight.
T h e r e f o r e , i f we n e g l e c t t h e c u r v a t u r e o f t h e
h o r i z o n t a l f l i g h t t r a j e c t o r y , we may a s s u m e

(1.1)

where 0 i s t h e d i p a n g l e of t h e f l i g h t t r a j e c t o r y ; X I , X 2 a r e t h e
c o o r d i n a t e s of t h e i n i t i a l and f i n a l p o i n t s o f t h e s l o p i n g segment
o f t h e t r a j e c t o r y ; H I , H2 r e p r e s e n t a g i v e n a l t i t u d e a t t h e i n i t i a l
and f i n a l p o i n t s .
When t h e a i r c r a f t t r a v e l s f r o m t h e i n i t i a l p o i n t X I t o t h e mov­
i n g p o i n t X , t h e f l i g h t a l t i t u d e i s changed by t h e v a l u e
AH= (X.- XI) tg 8,

(1.2)

and t h e v a l u e o f t h e moving f l i g h t a l t i t u d e i s

or i f we t a k e F o r m u l a (1.1) i n t o a c c o u n t ,

Since the a l t i t u d e during a sloping trajectory i s a variable
value, a given f l i g h t t r a j e c t o r y is maintained a t a constant value
of t h e v e r t i c a l velocity

vy*v,tgeo'r

v -v
y-

H2-Hl
X,-X,

,

(1.5)

C h e c k i n g of t h e p o s i t i o n o f t h e a i r c r a f t a t g i v e n v a l u e s o f
the varying f l i g h t a l t i t u d e is carried out only at specific points
on t h e s l o p i n g t r a j e c t o r y .

Translator's note:

tg = tan.
3


Concepts o f S t a b l e and U n s t a b l e F l i g h t C o n d i t i o n s

2.

A n a v i g a t i o n a l f l i g h t c o n d i t i o n i s d e t e r m i n e d by t h e m o t i o n
p a r a m e t e r s o f a n a i r c r a f t a l o n g a t r a j e c t o r y or b y n a v i g a t i o n a l
elements of f l i g h t :
course, speed, and a l t i t u d e .
The m o t i o n p a r a m e t e r s o f a n a i r c r a f t a r e u s u a l l y m e a s u r e d r e l ­
ative t o airspace.
However, c o n s i d e r i n g t h a t t h e a i r s p a c e a l s o
s h i f t s , t h e y a r e s e l e c t e d i n s u c h a way a s t o e n s u r e r e t a i n i n g t h e
given f l i g h t t r a j e c t o r y r e l a t i v e t o t h e E a r t h l s surface.
B a s e d on t h e n a t u r e o f t h e t r a j e c t o r y a n d t h e c o n d i t i o n s o f
a i r c r a f t n a v i g a t i o n , f o u r main f l i g h t c o n d i t i o n s a r e d i s t i n g u i s h e d :
h o r i z o n t a l r e c t i l i n e a r f l i g h t , a l t i t u d e g a i n , a l t i t u d e loss, a n d

roll.
H o r i z o n t a l r e c t i l i n e a r f l i g h t i s c h a r a c t e r i z e d by t w o c o n s t a n t
parameters:
h e i g h t and f l i g h t d i r e c t i o n .
A l t i t u d e g a i n and l o s s c o n d i t i o n s e a c h have two c o n s t a n t param­
eters:
f l i g h t d i r e c t i o n , a n d v e r t i c a l v e l o c i t y or d i p a n g l e o f t h e
t r aj e c to r y

.

The c o n d i t i o n o f roll i s a l w a y s c o m b i n e d w i t h o n e o f t h e f i r s t
t h r e e f l i g h t c o n d i t i o n s , s o t h a t t h e f l i g h t d i r e c t i o n becomes v a r i ­
a b l e and can be r e p l a c e d by a p a r a m e t e r which c h a r a c t e r i z e s t h e
c u r v a t u r e o f t h e roll t r a j e c t o r y t h r o u g h t h e r a d i u s o f roll or t h e
angular velocity.

A f l i g h t condition is s t a b l e i f i t s p a r a m e t e r s a c q u i r e c o n s t a n t
v a l u e s , and unstable i f i t s parameters a r e v a r i a b l e .
F l i g h t p r a c t i c e shows t h a t f l i g h t c o n d i t i o n s , s t r i c t l y s p e a k ­
i n g , a r e n e v e r f i x e d f o r any p r o l o n g e d t i m e , s i n c e t h e r e a r e always
f a c t o r s changing t h e a i r c r a f t ' s motion parameters.
The m a i n s i g n o f a s t a b l e f l i g h t c o n d i t i o n i s t h e e q u a l i t y t o
zero of t h e first derivative of t h e given parameter with time
d2S
or o f t h e s e c o n d d e r i v a t i v e p a t h w i t h t i m e -

(g)

d.t

.

For e x a m p l e , f o r t h e v e l o c i t y p a r a m e t e r V = c o n s t , i f

-dv

-0

df

o r -dzS
- - 0.
dt2

Analogously, f o r t h e f l i g h t d i r e c t i o n parameter
a l t i t u d e parameter (HI:
+=consf,

4

if dd,l -­
0,
dt

H=const,

dH
i f --- 0.

dt

( 9 ) and t h e

/10

~~~

I f f o r c e s a r i s e d u r i n g f l i g h t w h i c h c h a n g e t h e a i r c r a f t ' s mo­
t i o n parameters, t h e extreme v a l u e s o f t h e motion parameters ( ? . e . ,
t h e p o i n t s o f t h e maxima a n d m i n i m a o n t h e c u r v e w h i c h c h a r a c t e r i z e s
t h e change o f t h e g i v e n p a r a m e t e r w i t h t i m e ) i n d i c a t e e q u i l i b r i u m
of these forces.
A stable f l i g h t condition based
on a given parameter e x i s t s o n l y a t
t h e extreme p o i n t s , s i n c e t h e first
d e r i v a t i v e p a r a m e t e r s b a s e d on t i m e
a t these points are equal t o zero
while t h e d i s t u r b i n g f o r c e s are absent.

The d i s t u r b i n g f o r c e s a c q u i r e a
maximum v a l u e a t p o i n t s o f i n f l e c t i o n ,
i . e . , when t h e s e c o n d d e r i v a t i v e p a ­
r a m e t e r s b a s e d on t i m e a r e e q u a l t o
On a c u r v e c o n ­
zero (Fig. 1.3).
structed f o r the velocity parameter,
t h e p o i n t s of a s t a b l e condition are
d e s i g n a t e d b y o n e l i n e , w h i l e p o i n t s of maximum d i s t u r b i n g f o r c e s
a r e d e s i g n a t e d by two l i n e s .

F i g . 1 . 3 . Graph o f t h e
Changes o f a N a v i g a t i o n a l
P a r a m e t e r and P o i n t s w i t h
a Stable Flight Condition.

/11
-

From a e r o d y n a m i c s , w e know t h a t i n h o r i z o n t a l f l i g h t a t a v e -
l o c i t y s i g n i f i c a n t l y less than t h e speed of sound, t h e drag of an
aircraft i n a counterflow is

w h e r e ex i s t h e c o e f f i c i e n t o f d r a g o f t h e a i r c r a f t , S i s t h e c r o s s s e c t i o n a l a r e a o f t h e m i d s h i p s e c t i o n , and p i s t h e a i r d e n s i t y a t
flight altitude.
I t i s obvious t h a t t h e a i r s p e e d w i l l be s t a b l e i f t h e t h r u s t
o f t h e e n g i n e s ( P ) i s e q u a l t o t h e d r a g o f t h e a i r c r a f t P = Qx.

With a d i s t u r b a n c e o f t h i s e q u i l i b r i u m , t h e r e a r i s e s a d i s t u r b ­
i n g f o r c e which changes t h e f l i g h t v e l o c i t y .
For example, w i t h an
i n c r e a s e i n t h e t h r u s t of t h e engines t h e d i s t u r b i n g f o r c e w i l l be
equal t o :
AP=P'-c~

p",
2

'

which c a u s e s an i n i t i a l a c c e l e r a t i o n o f t h e a i r c r a f t
dV
-=­
dt

AP
m '

where m i s t h e m a s s of t h e a i r c r a f t i n kg.

5

L a t e r , with an increase i n v e l o c i t y , t h e drag of t h e aircraft
w i l l also increase.
T h e v a l u e of t h i s d r a g w i l l a p p r o a c h t h e v a l u e
of t h e t h r u s t o f t h e e n g i n e s , i . e . , t h e v e l o c i t y v e r y s l o w l y ap­
proaches a s t a b l e value logarithmically.
Changes i n a i r s p e e d which a r e a n a l o g o u s i n n a t u r e a r i s e d u r i n g
c h a n g e s i n t h e v e l o c i t y o f t h e h e a d w i n d or t h e i n c i d e n t a i r f l o w a t
flight altitude.
For e x a m p l e , w i t h a n i n c r e a s e i n t h e v e l o c i t y o f
t h e incident airflow, t h e airspeed diminishes.
This provides a
surplus of engine t h r u s t .
Subsequently, an increase i n airspeed
occurs logarithmically.

If t h e l a t e r a l component o f t h e wind s p e e d c h a n g e s , a l a t e r a l
p r e s s u r e on t h e s u r f a c e o f t h e a i r c r a f t a r i s e s :

w h e r e c z i s t h e c o e f f i c i e n t o f l a t e r a l d r a g o f t h e a i r c r a f t ; S, i s
t h e c r o s s - s e c t i o n a l a r e a o f t h e a i r c r a f t i n t h e XY p l a n e ; Y , i s t h e
l a t e r a l v e l o c i t y c o m p o n e n t e q u a l t o uz.
The i n i t i a l l a t e r a l a c c e l e r a t i o n o f t h e a i r c r a f t i s :

Subsequently, the lateral velocity of t h e aircraft w i l l log­
a r i t h m i c a l l y a p p r o a c h t h e l a t e r a l component o f t h e wind v e l o c i t y ,
i . e . , t h e f l i g h t c o n d i t i o n w i l l approach a c o n d i t i o n which i s s t a b l e
in direction.
Usually, during navigational c a l c u l a t i o n s f o r each parameter,
i t s mean v a l u e f o r a d e f i n i t e l e n g t h o f t i m e i s c a l l e d a s t a b l e
flight condition:
mean v e l o c i t y , mean v e r t i c a l v e l o c i t y , mean d i ­
rection, etc.
From t h e p o i n t o f v i e w o f m a i n t a i n i n g f l i g h t d i r e c t i o n , a i r ­
craft r o l l is an u n s t a b l e condition.
If a g i v e n t r a j e c t o r y i s c u r v ­
i l i n e a r , t h e r o l l c o n d i t i o n i s a l s o e x a m i n e d a s s t a b l e or u n s t a b l e .
T h e e n t r a n c e or e x i t o f a n a i r c r a f t f r o m roll, a s w e l l a s roll w i t h
v a r i a b l e b a n k i n g , c a n s e r v e a s e x a m p l e s o f u n s t a b l e roll c o n d i t i o n s .
The r o l l i n g o f a n a i r c r a f t i s c o n s i d e r e d t o b e c o o r d i n a t e d i f
t h e l o n g i t u d i n a l a x i s o f t h e a i r c r a f t c o n s t a n t l y c o i n c i d e s w i t h t.he
t a n g e n t t o t h e t r a j e c t o r y o f i t s m o v e m e n t , ? . e . , e x t e r n a l or i n t e r ­
T h i s i s a c h i e v e d by t i l t i n g t h e r u d ­
n a l a i r c r a f t glide is absent.
d e r o f t h e a i r c r a f t f o r b a n k i n g i n a roll.
D u r i n g b a n k i n g o f a n a i r c r a f t , i t s l i f t (Y) i s d i r e c t e d n o t
a l o n g t h e v e r t i c a l p l a n e b u t a l o n g t-he a x i s o f t h e a i r c r a f t , which
i s d e f l e c t e d from it ( F i g . 1 . 4 ) .

6


­
/12

R o l l i n g o f an a i r c r a f t w i t h o u t d e s c e n t o r w i t h s t a b l e v e r t i c a l
v e l o c i t y i s p o s s i b l e o n l y when t h e v e r t i c a l c o m p o n e n t o f t h e l i f t
( Y 1 ) i s e q u a l t o t h e w e i g h t of t h e a i r c r a f t G.
I n t h i s case, t h e h o r i z o n t a l ( c e n t r i p ­
e t a l ) component o f t h e l i f t i s :
Yr=Gtg8.

w h e r e f3
craft.

is t h e banking angle of t h e air­

Since w e a r e examining a c o o r d i n a t e
r o l l (without gliding of the aircraft),
t h e c e n t r i f u g a l f o r c e i n t h e roll
Fig. 1.4. Resolution
of Forces During

R o l l i n g of an A i r ­

craft.

F,=R

m V2

w i l l be e q u a l t o t h e c e n t r i p e t a l f o r c e ,

i.e.,

where m i s t h e m a s s o f t h e a i r c r a f t ; and R i s t h e r a d i u s o f t h e co­
o r d i n a t e d roll.

G
Transforming t h i s equation, taking i n t o account t h a t m = -

9 ,

w e w i l l o b t a i n formulas f o r d e t e r m i n i n g b o t h t h e r a d i u s and p a t h o f
t h e a i r c r a f t w i t h c o o r d i n a t e d roll:

F o r m u l a s ( 1 . 6 ) r e l a t e t h e r a d i u s o f s t a b l e c o o r d i n a t e d roll o f
t h e a i r c r a f t w i t h t h e a i r s p e e d and a l s o w i t h banking i n r o l l i n g ,
and t h e y a r e used i n c a l c u l a t i o n s o f t h e r a d i u s and p a t h o f t h e a i r ­
craft along a curvilinear f l i g h t trajectory.

3.

/13

Form and D i m e n s i o n s o f t h e E a r t h

I n t h e p r a c t i c e of aircraft navigation, it i s necessary first
of a l l t o d e a l w i t h d i s t a n c e s a n d d i r e c t i o n s on t h e E a r t h ' s s u r f a c e
which are t h e r e s u l t o f t h e mutual d i s t r i b u t i o n o f o b j e c t s through
which t h e f l i g h t p a t h p a s s e s .
The E a r t h ' s s u r f a c e , i t s r e l i e f a n d m u t u a l d i s t r i b u t i o n o f o b ­
j e c t s c a n b e m o s t a c c u r a t e l y e x p r e s s e d on a m o d e l o f t h e E a r t h ( a
globe).
H o w e v e r , a g l o b e w i t h a r e p r e s e n t a t i o n o f t h e E a r t h ' s sur­
f a c e t h a t s a t i s f i e s t h e demands o f a i r c r a f t n a v i g a t i o n would b e s o
l a r g e t h a t i t s u s e i n f l i g h t would be i m p o s s i b l e .
Therefore, dif­
f e r e n t means o f r e p r e s e n t i n g t h e s u r f a c e o f t h e E a r t h , which i s
curved i n a l l d i r e c t i o n s , on a p l a n e ( s h e e t s of p a p e r ) are used.
7

The E a r t h h a s a c o m p l e x f o r m c a l l e d a g e o i d ( w i t h o u t c o n s i d e r ­
i n g t h e l o c a l r e l i e f , i f w e imagine t h a t i t s e n t i r e s u r f a c e i s cov­
The s u r f a c e o f a g e o i d a t a n y p o i n t
e r e d w i t h w a t e r a t sea l e v e l ) .
i s p e r p e n d i c u l a r t o t h e d i r e c t i o n o f t h e a c t i o n of g r a v i t y .
A de­
s c r i p t i o n of a g e o i d by m a t h e m a t i c a l e x p r e s s i o n s i s v e r y c o m p l e x ,
and i f w e c o n s i d e r t h e f o l d s i n t h e r e l i e f o f t h e E a r t h ' s s u r f a c e ,
t h e n it i s p r a c t i c a l l y impossible t o e x p r e s s i t s form mathematically,
T h e r e f o r e , i n c a l c u l a t i o n s t h e form o f t h e E a r t h i s t a k e n as an
e Z Z i p s o i d of revoZu+.ion, t h e f o r m c l o s e s t t o a g e o i d .

Fig.

1.5.
G r e a t a n d S m a l l C i r c l e s on t h e E a r t h ' s S u r f a c e .
a x i s o f t h e E a r t h a n d Great C i r c l e ; b ) S m a l l C i r c l e .

a ) Semi-

A c c o r d i n g t o m e a s u r e m e n t s made b y S o v i e t s c i e n t i s t s u n d e r t h e
s u p e r v i s i o n of F . N . K r a s o v s k i y , t h e m a j o r s e m i a x i s o f t h i s e l l i p s o i d ( a ) , which c o i n c i d e s w i t h t h e r a d i u s of t h e e q u a t o r , i s e q u a l
t o 6 , 3 7 8 , 2 4 5 km.
The m i n o r s e m i a x i s o f t h e e l l i p s o i d ( b ) , w h i c h
coincides with t h e a x i s of t h e E a r t h ' s r o t a t i o n , is equal t o
6 , 3 5 6 , 8 6 3 km ( F i g . 1 . 5 , a ) .
The f l a t t e n i n g o f t h e E a r t h a t t h e p o l e s i s
e = - -a-- b

a

­

1
298.3

'

T h e s e d i m e n s i o n s show t h a t t h e E a r t h ' s e l l i p s o i d o f r e v o l u t i o n
i s p r a c t i c a l l y c l o s e t o a s p h e r e ; t o s i m p l i f y t h e s o l u t i o n of t h e
m a j o r i t y o f p r o b l e m s i n a i r c r a f t n a v i g a t i o n , i t i s t a k e n as a t r u e
s p h e r e , e q u i v a l e n t i n volume t o t h e E a r t h ' s e l l i p s o i d .
The r a d i u s
o f s u c h a s p h e r e i s e q u a l t o 6 3 7 1 km.
T h e maximum d i s t o r t i o n o f d i s t a n c e s c a u s e d b y t h e r e p l a c e m e n t

8

­
/14

o f t h e E a r t h ' s e l l i p s o i d b y a s p h e r e d o e s n o t e x c e e d 0.5%, a n d t h e
d i s t o r t i o n o f d i r e c t i o n s i s n o t more t h a n 1 2 m i n u t e s o f a n g l e .

I n g e o d e s y a n d c a r t o g r a p h y , t h e p l o t t i n g o f maps, as w e l l as
i n o t h e r b r a n c h e s o f s c i e n c e w h e r e more a c c u r a t e c a l c u l a t i o n s o f
d i s t a n c e s and d i r e c t i o n s a r e n e c e s s a r y , t h e E a r t h ' s s u r f a c e i s t a k e n
a s a n e l l i p s o i d of r e v o l u t i o n .

4. E l e m e n t s W h i c h C o n n e c t t h e E a r t h ' s S u r f a c e

with Three-Dimensional Space

T a k i n g t h e E a r t h as a t r u e s p h e r e , w e w i l l l o c a t e a p e r p e n d i c u ­
l a r ( a r e s t i n g pendulum) a t any p o i n t above t h e E a r t h ' s s u r f a c e .
Then, d i s r e g a r d i n g t h e p o s s i b l e i n s i g n i f i c a n t d e v i a t i o n s c a u s e d by
t h e varying r e l i e f , t h e i r r e g u l a r i t y of d i s t r i b u t i o n of t h e densest
masses i n t h e E a r t h ' s c r u s t , and t h e t a n g e n t i a l a c c e l e r a t i o n s con­
nected with t h e E a r t h ' s r o t a t i o n , it i s p o s s i b l e t o consider t h a t
t h e l i n e of t h e perpendicular runs i n t h e d i r e c t i o n of t h e c e n t e r
of the Earth.
The p e r p e n d i c u l a r l i n e ( s e e F i g . 1 . 5 , a ) j o i n i n g t h e c e n t e r o f
t h e E a r t h w i t h t h e p o i n t o f t h e o b s e r v e r ' s p o s i t i o n , and c o n t i n u e d
i n t h e d i r e c t i o n o f t h e c e l e s t i a l s p h e r e (Y), i s c a l l e d t h e geo­

c e n t r i c v e r t i c a l of t h e l o c u s .
The p l a n e on t h e E a r t h ' s s u r f a c e , t a n g e n t t o t h e s p h e r e a t t h e
p o i n t of t h e o b s e r v e r and p e r p e n d i c u l a r t o t h e t r u e v e r t i c a l o f t h e
l o c u s , i s c a l l e d t h e p l a n e of t h e t r u e h o r i z o n .
The d i r e c t i o n a n d v e l o c i t y o f a i r c r a f t movement a t e v e r y p o i n t
on t h e E a r t h ' s s u r f a c e a r e examined i n t h e p l a n e o f t h e t r u e h o r i ­
zon, w h i l e t h e a l t i t u d e change i s examined i n t h e d i r e c t i o n o f t h e
true vertical.
If
another
Earth),
Earth's
which w

we c u t t h e p l a n e o f t h i s t r u e h o r i z o n i n a n y d i r e c t i o n b y
plane along the t r u e v e r t i c a l (through t h e center of t h e
t h e l i n e formed by t h e i n t e r s e c t i o n o f t h i s p l a n e w i t h t h e
s u r f a c e f o r m s a c l o s e d g r e a t c i r c l e , t h e mean r a d i u s o f
i l l be equal t o t h e r a d i u s o f t h e Earth.

The s h o r t e s t d i s t a n c e b e t w e e n two p o i n t s A B on t h e E a r t h ' s s u r ­
f a c e or p a r t o f t h e a r c o f a g r e a t c i r c l e i s c a l l e d t h e o r t h o d r o m e
( s e e Fig. 1.5, a ) .
T h e mean r a d i u s o f a g r e a t c i r c l e i s a s s u m e d t o b e e q u a l t o
6 3 7 1 km.
The l e n g t h o f t h e c i r c u m f e r e n c e o f s u c h a r a d i u s i s e q u a l
t o 4 0 , 0 0 0 km.
One d e g r e e o f a r c o f a g r e a t c i r c l e i s e q u a l t o
111.1 km, w h i l e o n e m i n u t e o f a r c i s e q u a l t o 1 , 8 5 2 km.
The l e n g t h
o f a segment of t h e arc o f a g r e a t c i r c l e a t one minute o f a n g l e i s
c a l l e d a nautical mile.
With an i n t e r s e c t i o n o f t h e E a r t h ' s

s p h e r e by a p l a n e which
9


/15
-

d o e s n o t p a s s t h r o u g h t h e c e n t e r of t h e E a r t h , t h e l i n e o f i n t e r ­
s e c t i o n of t h i s p l a n e w i t h t h e E a r t h ' s s u r f a c e forms a c l o s e d s m a l l
c i r c l e , t h e r a d i u s o f w h i c h w i l l a l w a y s b e l e s s t h a n t h e mean r a d i u s
of t h e E a r t h .
The s m a l l c i r c l e s p a r a l l e l t o t h e p l a n e o f t h e e q u a ­
t o r are c a l l e d parallels ( s e e Fig. 1 . 5 , b ) .
For t h e hurposes o f a i r c r a f t
navigation, a coordinate system
which unequivocally determines
t h e p o s i t i o n of an a i r c r a f t and
o b j e c t s on t h e E a r t h ' s s u r f a c e
is necessary.
Obviously, a
s p h e r i c a l coordinate system w i l l
be t h e most convenient ( F i g . 1 . 6 ) .
A spherical coordinate sys­
tem i s d i s t i n g u i s h e d from a r e c ­
t a n g u l a r s y s t e m ( C a r t e s i a n ) by
the fact t h a t instead of deterF i g . 1 . 6 . R e l a t i o n s h i p Between
a S p h e r i c a l System of Coordimining t h r e e distances t o a
n a t e s and a R e c t a n g u l a r System.
p o i n t i n t h e d i r e c t i o n s of t h e
X , Y , a n d Z a x e s , we d e t e r m i n e
the length of the radius-vector
R from t h e c e n t e r o f t h e c o o r d i n a t e s y s t e m t o a p o i n t , and two a n ­
gles:
a n g l e X b e t w e e n t h e XY p l a n e and t h e p r o j e c t i o n o f t h e 'ra­
d i u s - v e c t o r (R) t o t h e p l a n e X Z , a n d a n g l e 4 b e t w e e n t h e X Z p l a n e
and t h e d i r e c t i o n o f t h e r a d i u s - v e c t o r ( R ) .
T h e r e i s an o b v i o u s r e l a t i o n b e t w e e n s p h e r i c a l a n d r e c t a n g u l a r .
coordinate systems:

With a c o n s t a n t l e n g t h o f t h e r a d i u s - v e c t o r R , i f a n g l e s X and
4 a s s u m e all p o s s i b l e v a l u e s , t h e g e o m e t r i c l o c a t i o n o f t h e p o i n t s
of

t h e end of t h e v e c t o r r a d i u s w i l l be a s p h e r e .

To d e t e r m i n e c o o r d i n a t e s on t h e E a r t h ' s s u r f a c e , t h e r e i s n o
This coorneed t o i n d i c a t e t h e r a d i u s of t h e E a r t h ( R ) each time.
d i n a t e i s c o n s i d e r e d , o n c e and f o r a l l , c o n s t a n t .
Thus, t h e s p h e r i c a l coordinate system is transformed i n t o a
two-dimensional s u r f a c e system which i s c a l l e d a geographic system
of coordinates.

The p l a n e o f t h e e q u a t o r a n d t h e p l a n e o f t h e p r i m e ( G r e e n w i c h )
meridian a r e taken as t h e i n i t i a l r e f e r e n c e p l a n e s i n a geographic
coordinate system.
The p o i n t c o o r d i n a t e s o n t h e E a r t h ' s s u r f a c e
b e a r t h e name " l o n g i t u d e o f t h e l o c u s " a n d " l a t i t u d e of t h e l o c u s "
(Fig. 1.7).

10


/16

The d i h e d r a l a n g l e b e t w e e n t h e p l a n e o f t h e p r i m e m e r i d i a n a n d
t h e p l a n e of t h e m e r i d i a n o f a g i v e n p o i n t i s c a l l e d t h e l o n g i t u d e
of t h e p o i n t ( A ) .
Determination of t h e longitude can be given i n
arc values:
t h e length of the
a r c o f t h e e q u a t o r (or t h e p a r a l ­
l e l ) , expressed i n degrees, be­
PN
tween t h e prime m e r i d i a n and t h e
meridian of a given point is
called the longitude of the point.

ps

Fig. 1.7. Spherical Coordinate
S y s t e m on t h e E a r t h ' s S u r f a c e .

Reading o f t h e l o n g i t u d e i s
c a r r i e d o u t from 0 t o 180° e a s t
o f t h e rpime m e r i d i a n ( e a s t long­
i t u d e ) a n d f r o m 0 t o 180° w e s t
o f t h e rpime m e r i d i a n (west long­
i t u d e ) . I n navigational calcu­
l a t i o n s , east longitude is taken
as p o s i t i v e and i s d e s i g n a t e d by
a plus sign, while w e s t longitude
i s n e g a t i v e a n d i s d e s i g n a t e d by
a minus s i g n .
However, i n c a r r y ­
ing out navigational calculations,
i t i s more c o n v e n i e n t t o c a r r y
o u t a reading of longitude i n t h e
e a s t e r l y d i r e c t i o n from z e r o t o
360O.

The a n g l e b e t w e e n t h e p l a n e of t h e e q u a t o r a n d t h e t r u e v e r t i ­
c a l o f a g i v e n p o i n t (or t h e l e n g t h o f t h e m e r i d i a n a r c , e x p r e s s e d
i n d e g r e e s , from t h e p l a n e of t h e e q u a t o r t o t h e p a r a l l e l o f a g i v e n
p o i n t ) i s c a l l e d t h e Z a t i t u d e of t h e p o i n t ( 9 ) .
Since a set of t r u e
v e r t i c a l s a t a c o n s t a n t l a t i t u d e forms a cone w i t h t h e v e r t e x i n t h e
c e n t e r of t h e E a r t h and an a n g l e a t t h e v e r t e x e q u a l t o 90°-4,
then
i n c o n t r a s t t o t h e d i h e d r a l angle between t h e p l a n e s of t h e meridians,
we s h a l l call a similar angle i n other spherical systems, t h e conic

angle.
Reading of t h e l a t i t u d e i s c a r r i e d o u t from t h e plane of t h e
e q u a t o r t o t h e n o r t h a n d s o u t h f r o m 0 t o 9 0 ° ( n o r t h and s o u t h Z a t i ­
tude). In navigational calculations, north l a t i t u d e is considered
p o s i t i v e and s o u t h , n e g a t i v e .
A geographic coordinate system i s a surface c u r v i l i n e a r system,
;.e.,
t h e m e r i d i a n s of t h e c o o r d i n a t e g r i d on t h e E a r t h a r e n o t
parallel.
However, i f w e e x a m i n e t h e m e r i d i a n s a n d p a r a l l e l s on
any u n i t area of t h e E a r t h ' s s u r f a c e , t h e y t u r n o u t t o be o r t h o g o n a l
( p e r p e n d i c u l a r i n one p l a n e ) .
Two s p e c i a l p o i n t s o n t h e E a r t h ' s
surface ( t h e geographic poles) are an exception.

A geographic coordinate system is used n o t only t o determine
t h e l o c a t i o n o f a p o i n t ( o b j e c t ) on t h e E a r t h , b u t t o d e t e r m i n e
d i r e c t i o n from one p o i n t t o a n o t h e r .

11


/17

The a n g l e i n c l u d e d b e t w e e n t h e n o r t h e r n d i r e c t i o n o f t h e m e r i ­
d i a n which p a s s e s t h r o u g h a g i v e n p o i n t and t h e orthodrome d i r e c t i o n
Read­
t o a p o i n t s e t t i n g a c o u r s e i s c a l l e d t h e b e a r i n g or a z i m u t h .
i n g o f t h e a n g l e s o f b e a r i n g or a z i m u t h i s d o n e c l o c k w i s e f r o m 0 t o
360O.
S i n c e t h e m e r i d i a n s on t h e E a r t h ' s s u r f a c e a r e g e n e r a l l y n o t
p a r a l l e l , t h e v a l u e o f t h e a z i m u t h c h a n g e s w i t h a c h a n g e i n t h e mov­
i n g l o n g i t u d e a l o n g t h e l i n e which j o i n s t h e two p o i n t s ; t h e g r e a t e r
Therefore, f o r t h e orthodrome
t h e l a t i t u d e , t h e more i t c h a n g e s .
d i r e c t i o n t o g e t h e r w i t h an i n d i c a t i o n o f t h e a z i m u t h , it i s n e c e s ­
s a r y t o mention from which meridian t h i a d i r e c t i o n i s measured.
The c h a n g e i n a z i m u t h w i t h a c h a n g e i n t h e moving l o n g i t u d e
d o e s n o t make i t p o s s i b l e t o u s e m a g n e t i c c o m p a s s e s f o r moving a l o n g
t h e orthodrome without i n t r o d u c i n g corresponding c o r r e c t i o n s , espe­
c i a l l y when t h e t w o p o i n t s a r e f a r a p a r t .
If t h e m a g n e t i c d e c l i n a t i o n d o e s n o t c h a n g e , f o l l o w i n g a con­
s t a n t magnetic course w i l l cause t h e meridians t o i n t e r s e c t a t iden­
tical angles.
The l i n e w h i c h i n t e r s e c t s t h e m e r i d i a n s a t a c o n s t a n t
a n g l e i s c a l l e d t h e loxodrome.

I n o r d e r t o p r o c e e d t o a more d e t a i l e d e x a m i n a t i o n o f t h e e l e ­
ments o f a i r c r a f t n a v i g a t i o n and t h e i r measurement, it i s n e c e s s a r y
t o become a c q u a i n t e d w i t h t h e m a k i n g o f m a p s , t h e i r s c a l e s , a n d
some f e a t u r e s o f c a r t o g r a p h i c p r o j e c t i o n s .

5.

C h a r t s , Maps

,

and C a r t o g r a p h i c P r o j e c t i o n s

The r e p r e s e n t a t i o n o f a s m a l l p a r t o f t h e E a r t h ' s s u r f a c e on a
plane is called a chart.
D i s t o r t i o n as a r e s u l t o f t h e c u r v a t u r e
o f t h e E a r t h ' s s u r f a c e i s p r a c t i c a l l y a b s e n t on a c h a r t .
The c o n v e n t i o n a l r e p r e s e n t a t i o n o f t h e E a r t h ' s
p l a n e i s c a l l e d a map.

surface in a

A map i s a c o n t i n u o u s r e p r e s e n t a t i o n o f t h e s u r f a c e o f t h e
E a r t h or a p a r t o f i t w i t h o u t d i s c o n t i n u i t i e s a n d f o l d s , made w i t h
a variable scale according t o a d e f i n i t e r u l e .
The s p h e r i c i t y o f
t h e E a r t h ' s s u r f a c e d o e s n o t a l l o w i t t o b e r e p r e s e n t e d w i t h com­
p l e t e a c c u r a c y on a p l a n e s u r f a c e .
T h e r e f o r e , t h e r e a r e many w a y s
o f p r o j e c t i n g t h e E a r t h ' s s u r f a c e o n t o a p l a n e w h i c h make i t p o s s i ­
b l e t o r e p r e s e n t m o s t a c c u r a t e l y on t h e map o n l y t h o s e p a r a m e t e r s
( e l e m e n t s ) which a r e most n e c e s s a r y under t h e g i v e n c o n d i t i o n s of
application.
M e t h o d s or l a w s o f r e p r e s e n t i n g t h e E a r t h ' s
are called cartographic projections.

s u r f a c e on a p l a n e

A common g e o m e t r i c a l p r o j e c t i o n i s t h e p o i n t o f i n t e r s e c t i o n
o f t h e l i n e of s i g h t ( w h i c h p a s s e s t h r o u g h t h e e y e o f t h e o b s e r v e r

12

/18

and t h e p r o j e c t e d p o i n t ) w i t h t h e p l a n e o n t o which t h e g i v e n p o i n t
is projected.
I t i s a s p e c i a l case o f c a r t o g r a p h i c p r o j e c t i o n .
A c a r t o g r a p h i c p r o j e c t i o n i s set a n a l y t i c a l l y as a f u n c t i o n o f
g e o g r a p h i c a l c o o r d i n a t e s on t h e E a r t h ( s p h e r e ) b e t w e e n t h e c o o r d i ­
n a t e s o f a p o i n t on a p l a n e .

If w e c a l l o n e o f t h e m a i n d i r e c t i o n s o n a map t h e X a x i s a n d
t h e perpendicular t o it t h e Z a x i s , t h e n
X = F l ( r p ; A) a n d z=&('9; 1);

or

P=

1) an d ii = F, (~p; A),

w h e r e p a n d 6 a r e t h e m a i n d i r e c t i o n s o n maps o f c o n i c a n d a z i m u t h a l
p r o j e c t i o n s , a n d I$ a n d X a r e t h e g e o g r a p h i c a l c o o r d i n a t e s o f a p o i n t
on t h e E a r t h ( s p h e r e ) .
The p r o p e r t i e s o f t h e p r o j e c t i o n s w i l l d e p e n d on t h e p r o p e r t i e s
of t h e s e f u n c t i o n s (F1, F2, F 3 , and F 4 ) , which must be c o n t i n u o u s
a n d w e l l - d e f i n e d , s i n c e t h e map i s made w i t h o u t d i s c o n t i n u i t i e s s o
t h a t a s i n g l e p o i n t o n t h e map c o r r e s p o n d s t o e v e r y p o i n t i n t h e
location.
Map S c a l e s
The map-making
a)
globe.
b)

p r o c e s s i s d i v i d e d i n t o two s t a g e s .

The E a r t h i s d e c r e a s e d t o t h e d e f i n i t e d i m e n s i o n s o f a

The g l o b e i s u n r o l l e d t o f o r m a p l a n e .

The e x t e n t of t h e o v e r a l l d e c r e a s e i n t h e E a r t h ' s d i m e n s i o n s
t o t h e f i x e d dimensions of a globe i s c a l l e d a principaz scale.
A p r i n c i p a l s c a l e i s a l w a y s i n d i c a t e d o n t h e e d g e o f a map a n d
makes it p o s s i b l e t o j u d g e t h e d e c r e a s e o f t h e l e n g t h o f a s e g m e n t
i n t r a n s f e r r i n g i t from t h e E a r t h ' s s u r f a c e t o t h e g l o b e .
A principal scale is numerically equal t o t h e r a t i o of t h e
d i s t a n c e on t h e g l o b e t o t h e a c t u a l d i s t a n c e a t a l o c a t i o n :

A. s a
M= ASe.s.
4

w h e r e M i s t h e p r i n c i p a l s c a l e , A S g i s a s e g m e n t on t h e g l o b e , a n d
i s a s e g m e n t on t h e E a r t h ' s s u r f a c e w h i c h c o r r e s p o n d s t o t h e
segment on t h e g l o b e .
On m a p s , t h e p r i n c i p a l s c a l e i s u s u a l l y s h o w n a s a f r a c t i o n
( n u m e r i c a l s c a l e ) a n d by means o f a s p e c i a l s c a l e ( l i n e a r s c a l e ) .
13


The n u m e r i c a l s c a l e . i s a f r a c t i o n , t h e n u m e r a t o r o f w h i c h i s
o n e , w h i l e t h e d e n o m i n a t o r s h o w s how many s u c h u n i t s o f m e a s u r e m e n t
fit i n t o t h e location.

-

/19
F o r e x a m p l e , 1 : 1 , 0 0 0 , 0 0 0 means t h a t i f w e t a k e 1 c m on a maps
t h e n 1 , 0 0 0 , 0 0 0 c m a t a l o c a t i o n ( i . e . , 1 0 km) w i l $ c o r r e s p o n d t o i t .

A l i n e a r s c a l e i s a s c a l e on a map i n w h i c h a d e f i n i t e n u m b e r
o f k i l o m e t e r s a t a l o c a t i o n correspond t o s p e c i a l segments of t h e
scale.
However, a p r i n c i p a l s c a l e ( n u m e r i c a l and l i n e a r ) i s i n s u f f i ­
c i e n t f o r a c c u r a t e l y m e a s u r i n g d i s t a n c e s on t h e e n t i r e f i e l d o f a
map.
I t i s n e c e s s a r y t o know t h e l a w s o f d i s t o r t i o n o f d i s t a n c e s
and d i r e c t i o n s .
The l a w s o f c h a n g e i n t h e p r i n c i p a l s c a l e a l o n g
t h e map f i e l d a r e d e t e r m i n e d by a s p e c i a l s c a l e .

A special scale i s t h e r a t i o of an i n f i n i t e l y small segment i n
a g i v e n p l a c e on t h e map i n a g i v e n d i r e c t i o n , t o a n a n a l o g o u s s e g ­
A t e a c h p o i n t on t h e m a p , t h e s p e c i a l
ment i n a l o c a t i o n ( g l o b e ) .
scale is different.
I t i s e i t h e r s o m e w h a t l a r g e r or s o m e w h a t s m a l l ­
e r than the principal scale.
Distort ions o f

Cartographic Project ions

E l l i p s e of D i s t o P t i o n s
L e t u s draw on a s p h e r e ( g l o b e ) , a n i n f i n i t e l y s m a l l c i r c l e
with radius r ; l e t us a l s o designate a rectangular coordinate system'
Then
on t h e s p h e r e by x a n d z ( F i g . 1 . 8 , a ) .
f2

Fig.

14


1.8.

= x2

+ P.

D i s t o r t i o n o f S c a l e s on a P l a n e :
( b ) S c a l e on a P l a n e .

(1.8)

( a ) S c a l e on a G l o b e ;

I n t h e t r a n s f e r o f t h e coordinate system from t h e sphere
(globe) t o the plane, the direction of the coordinate,axes is dis­
torted (Fig. 1.8, b).
Having d e s i g n a t e d t h e s p e c i a l s c a l e s on a p l a n e (map) b y m i n
t h e d i r e c t i o n X and n i n t h e d i r e c t i o n z , w e o b t a i n :
XI = mx;
21 = nz

S u b s t i t u t i n g t h e l a t t e r i n (1.81,

a n d t h e n d i v i d i n g b o t h s i d e s o f t h e e q u a t i o n b y r2, w e o b t a i n

mr

+ (")

nr

=

,

(1.9)

From m a t h e m a t i c s , i t i s known t h a t t h i s i s t h e f o r m u l a o f a n
e l l i p s e with conjugate diameters; therefore:
a)
Any i n f i n i t e l y s m a l l c i r c l e o n t h e s u r f a c e o f t h e E a r t h ' s
s p h e r e i n any p r o j e c t i o n i s r e p r e s e n t e d by a n i n f i n i t e l y s m a l l e l ­
lipse.
b)
On t h e s u r f a c e o f t h e E a r t h ' s s p h e r e ( g l o b e ) , i t i s p o s ­
s i b l e t o choose two m u t u a l l y p e r p e n d i c u l a r d i r e c t i o n s which w i l l be
t r a n s f e r r e d t o a map w i t h o u t a n y d i s t o r t i o n s .
These d i r e c t i o n s are c a l l e d p r i n c i p a l d i r e c t i o n s .
Knowing t h e s p e c i a l s c a l e s ( m a n d n ) i n t h e p r i n c i p a l d i r e c ­
t i o n s , it i s always p o s s i b l e t o c o n s t r u c t an e l l i p s e o f d i s t o r t i o n s
which w i l l make it p o s s i b l e t o judge t h e n a t u r e o f t h e d i s t o r t i o n s
o f t h e p r o j e c t i o n as a whole.
In the majority of projections, the
d i r e c t i o n s along t h e m e r i d i a n s and p a r a l l e l s are t a k e n as t h e p r i n ­
cipal directions.

Distortion of L e n g t h s
I f a n i n f i n i t e l y s m a l l c i r c l e on t h e E a r t h i s r e p r e s e n t e d by
an e l l i p s e (Fig. 1 . 9 , b) with i t s t r a n s f e r t o a p l a n e , t h e d i s t o r ­
t i o n o f t h e s p e c i a l scale i n any d i r e c t i o n ( A S , )
can be expressed
as f o l l o w s :
ASa=-- 0 1 4 - V ( W +( n i p
D
(1.10)
OM
r

15


­
/21


b u t from t h e c i r c l e i n F i g u r e 1 . 9 ,

a:

~ = s ! n a r , ~wh i 1 e t t = c o s a r ,
then
ASa=

Fig.

1.9.

Distortion

mz sinza + n2 cos2 a ,

(1.11)

i n a Plane:
( a ) L e n g t h on a G l o b e ;
on a P l a n e .

(b) L e n g t h

Xl

Fig.

1.10.

D i s t o r t i o n of D i r e c t i o n s o n a Map.
( a ) D i r e c t i o n on a
G l o b e ; ( b ) D i r e c t i o n o n a Map

i . e . , k n o w i n g t h e s p e c i a l s c a l e s for t h e p r i n c i p a l d i r e c t i o n s , we
can always judge t h e value o f t h e d i s t o r t i o n o f t h e s p e c i a l s c a l e
i n any d i r e c t i o n ( a n d t h e r e f o r e , t h e d i s t o r t i o n of t h e l e n g t h o f t h e
segment as a w h o l e ) .

D i s t o r t i o n of D i r e c t i o n s
Let us take t h e radius r = 1 (Fig.
c i r c l e on t h e E a r t h ; t h e n
I

tga=%-,
X

16

1.10)

o f an i n f i n i t e l y small

nz

w h i le t g e = - . mr

(1.12)

Dividing Equations

(1.12)

i n t o one a n o t h e r , w e o b t a i n :
tgB

=mtg a.
II

(1.13)

Obviously, knowing t h e s p e c i a l s c a l e s f o r t h e p r i n c i p a l d i r e c ­
t i o n s , i t i s a l w a y s p o s s i b l e t o f i n d a n a n g l e f3 o n a map f o r a n a n ­
g i e a i n a l o c a t i o n , and v i c e v e r s a .

­
/22

D i s t o r t i o n of A r e a s
T h e d i s t o r t i o n o f a r e a s AP c a n b e d e t e r m i n e d b y a c o m p a r i s o n
o r d i v i s i o n o f t h e a r e a of t h e e l l i p s e ( S e l l by t h e a r e a o f a c i r c l e
(sei); s e e F i g u r e 1.11:
(1.14)
b u t i f we t a k e t h e r a d i u s o f t h e c i r c l e o n t h e E a r t h a s e q u a l t o 1 ,
then
AP = ab

or, i f w e e x p r e s s a a n d b b y s p e c i a l s c a l e s f o r t h e p r i n c i p a l d i r e c ­
t i o n s , we o b t a i n :
LIP = mn,,
(1.15)

x
Fig.

1.11.

D i s t o r t i o n o f A r e a s o n a Map.
( a ) Area on a G l o b e ;
( b ) A r e a o n a Map.

The d i s t o r t i o n o f a r e a s i s e q u a l t o t h e p r o d u c t o f t h e s p e c i a l
scales f o r t h e p r i n c i p a l d i r e c t i o n s .
H e n c e , w e s e e t h a t i f w e know t h e s p e c i a l s c a l e s for t h e p r i n ­
c i p a l d i r e c t i o n s , w e can give t h e complete c h a r a c t e r i s t i c s of any
map p r o j e c t i o n .
, .

17

C 1 a s s i f i c a t i o n o f C a r t o g r a p h i c P r o j e c t i,on s

T h e r e a r e many c a r t o g r a p h i c p r o j e c t i o n s .
a c c o r d i n g t o two b a s i c c h a r a c t e r i s t i c s :

They c a n b e d i v i d e d

(a)
a c c o r d i n g t o t h e n a t u r e o f t h e d i s t o r t i o n s , and
(b)
a c c o r d i n g t o t h e means o f c o n s t r u c t i o n ' o r t h e a p p e a r a n c e
o f t h e normal g r i d .
By norma2 grid w e mean t h e c o o r d i n a t e s y s t e m o n a g l o b e w h i c h
i s m o s t s i m p l y r e p r e s e n t e d o n a map.
Obviously, t h i s i s a system
o f m e r i d i a n s and p a r a l l e l s .

Division of Projections b y the Nature of the Distortions

The c h o i c e o f c a r t o g r a p h i c p r o j e c t i o n s d e p e n d s on t h e p r o b l e m s
f o r whose s o l u t i o n t h e y a r e i n t e n d e d .
According t o t h e n a t u r e of
t h e e l e m e n t s w h i c h h a v e t h e l e a s t d i s t o r t i o n on a map, c a r t o g r a p h i c
projections are divided i n t o t h e following groups:

1.

Isogonal o r conformal p r o j e c t i o n s

These p r o j e c t i o n s must s a t i s f y t h e r e q u i r e m e n t o f e q u a l i t y o f
a n g l e s and s i m i l a r i t y o f f i g u r e s ( c o n f o r m a b i l i t y ) w i t h i n t h e l i m i t s
o f u n i t areas of t h e E a r t h ' s s u r f a c e , i . e . , s o t h a t i n p r o j e c t i n g a
s u r f a c e o f a g l o b e o n t o a p l a n e (map)., t h e a n g l e s a n d s i m i l a r f i g ­
u r e s do n o t change.
b)

I xf

IX

I
41

CI
c

z




Fig. 1.12.
C o n f o r m a b i l i t y o f F i g u r e s o n Maps.
( a ) Preserving the
C o n f o r m a b i l i t y of a U n i t A r e a ; ( b ) D e s t r o y i n g t h e C o n f o r m a b i l i t y o �
a Long S t r i p .
A c c o r d i n g t o t h e s t i p u l a t i o n , t h e a n g l e o n a map m u s t b e e q u a l
t o t h e a n g l e a t t h e l o c a t i o n : L 6 = L a , b u t from ( 1 . 1 3 ) it i s ob­
v i o u s t h a t i n t h i s c a s e m = n.
Therefore, t h e equation of s p e c i a l scales f o r p r i n c i p a l d i r e c ­
tions is a condition for isogonality.

18


On l a r g e p a r t s o f t h e s u r f a c e , w i t h i n t h e l i m i t s o f w h i c h i t i s
impossible t o d i s r e g a r d t h e change i n scale, t h e c o n f o r m a b i l i t y
(and therefore t h e isogonality) are not preserved.
Figure 1.12
g i v e s a n example of p r e s e r v i n g t h e c o n f o r m a b i l i t y o f a u n i t area
and d e s t r o y i n g t h e c o n f o r m a b i l i t y o f a l o n g s t r i p .
T h e u n i t a r e a ( F i g . 1 . 1 2 , a ) i s t r a n s f e r r e d t o t h e map o n a
d e f i n i t e scale without d i s t o r t i o n s .
The l o n g s t r i p ( F i g . 1 . 1 2 , b )
c a n b e d i v i d e d i n t o a number o f u n i t a r e a s , e a c h o f w h i c h w i l l b e
t r a n s f e r r e d t o t h e map o n a s o m e w h a t c h a n g e d s c a l e .
Since t h e scales
mx,and nz a r e i n c r e a s e d p r o p o r t i o n a l l y i n t h e d i r e c t i o n o f t h e s t r i p ,
e a c h o f t h e s m a l l a r e a s i s r e p r e s e n t e d o n t h e map w i t h t h e c o n f o r m a ­
By e q u a t i n g t h e
b i l i t y b e i n g p r e s e r v e d , o n l y on a d i f f e r e n t s c a l e .
/24
l a t e r a l l i m i t s o f t h e s m a l l a r e a s , w e do n o t o b t a i n a c o n f o r m a l
figure, i.e., the similarity of small figures i n isogonal projec­
tions is preserved, while the similarity of large figures (large
l a k e s , seas, e t c . ) is destroyed.

2.

Equally spaced o r equidistant projections


The e q u i v a l e n c e t o u n i t y o f t h e s p e c i a l s c a l e s f o r a p r i n c i p a l
d i r e c t i o n ( m = 1 o r n = 1) i s a n e c e s s a r y c o n d i t i o n o f t h i s g r o u p
of projections.

4

z

T h i s m e a n s t h a t t h e map s c a l e w i l l
be p r e s e r v e d i n one o f t h e p r i n c i p a l
directions.
T h e r e f o r e , when u s i n g s u c h
a map w e c a n m e a s u r e t h e d i s t a n c e i n
o n e o f t h e d i r e c t i o n s by means o f a
scale.
The n a t u r e o f t h e d i s t o r t i o n
of c o n f o r m a b i l i t y i n t h e s e
projections
i s shown i n F i g . 1 . 1 3 .
Here m = c o n s t ,
while n is a function of Z.

$?,

X

J,

3.

Fig. 1.13. Distortion of
Conformability i n Equally
Spaced P r o j e c t i o n s :
( a ) Appearance o f a Figu r e i n a L o c a t i o n ; (b)
Appearance of t h e F i g u r e
o n a Map.

Equally large or equivalent
projections

T h i s group o f p r o j e c t i o n s must
s a t i s f y the condition of equivalence
of a r e a s
i.e. , the product of t h e
s p e c i a l scales f o r t h e p r i n c i p a l d i r e c ­
t i o n s m u s t e q u a l u n i t y ( m n = 1); t h e r e ­
f o r e , t h e r e l a t i o n between t h e s p e c i a l
scales f o r t h e p r i n c i p a l d i r e c t i o n s w i l l be i n v e r s e l y p r o p o r t i o n a l :


m=-;

1

n

1
m

n=-.

These p r o j e c t i o n s do n o t have an e q u i v a l e n c e o f a n g l e s and a
similarity of figures.

19


4.

Arbitrary projections

P r o j e c t i o n s o f t h i s g r o u p d o n o t s a t i s f y a n y of t h e c o n d i t i o n s
mentioned above.
H o w e v e r , t h e y a r e a l s o u s e d when c o m p a r a t i v e l y
s m a l l p o r t i o n s o f t h e E a r t h ' s surface are projected onto a plane
where t h e d i s t o r t i o n s o f t h e a n g l e s and t h e s c a l e s f o r t h e p r i n c i p a l
d i r e c t i o n s a n d a l o n g t h e e n t i r e map f i e l d a r e i n s i g n i f i c a n t a n d t h e
s i m i l a r i t y o f f i g u r e s and a r e a s which s a t i s f y t h e n e e d s o f t h e i r
This group o f p r o j e c t i o n s i n ­
practical application is preserved.
c l u d e s a b a s i c f l i g h t map o n a s c a l e o f 1:1,000,000, w h i c h i s c o n ­
s t r u c t e d a c c o r d i n g t o a s p e c i a l l a w a n d w h i c h h a s b e e n a c c e p t e d by
i n t e r n a t i o n a l agreement.

For t h e p u r p o s e s o f a i r c r a f t n a v i g a t i o n , t h e m o s t n e c e s s a r y
c o n d i t i o n s a r e ( o b v i o u s l y ) i s o g o n a l i t y and e q u a l s c a l e o f t h e maps.
E q u a l l y l a r g e a n d e q u a l l y s p a c e d p r o j e c t i o n s o f maps a r e u s e d i n
a i r c r a f t n a v i g a t i o n o n l y a s s u r v e y maps f o r s p e c i a l a p p l i c a t i o n s .
They i n c l u d e maps o f h o u r z o n e s , m a g n e t i c d e c l i n a t i o n s , c o m p o s i t e
d i a g r a m s o f t o p o g r a p h i c a l map s h e e t s , c l i m a t o l o g i c a l a n d m e t e o r o l o ­
g i c a l maps, e t c .

D i v i s i o n of P r o j e c t i o n s A c c o r d i n g t o t h e Method of C o n s t r u c t i o n
( A c c o r d i n g t o t h e A p p e a r a n c e of t h e Normal G r i d )
Depending on t h e method o f c o n s t r u c t i o n , c a r t o g r a p h i c p r o j e c ­
t i o n s a r e d i v i d e d i n t o s e v e r a l g r o u p s , t h e b a s e s o f which a r e t h e
following:
(a)
group o f c y l i n d r i c a l p r o j e c t i o n s ;
(b)
group o f c o n i c p r o j e c t i o n s and t h e i r v a r i a n t s ,
projections;
( c ) group of azimuthal p r o j e c t i o n s ;
(d)
group of s p e c i a l p r o j e c t i o n s .

polyconic

Each of t h e s e p r o j e c t i o n s i s d i v i d e d i n t u r n i n t o t h e f o l l o w i n g
categories:
normal, i f t h e E a r t h ' s a x i s c o n c i d e s w i t h t h e a x i s o f
t h e f i g u r e o n t o which t h e E a r t h ' s s u r f a c e i s p r o j e c t e d ; t r a n s v e r s e ,
if t h e E a r t h ' s a x i s f o r m s a n a n g l e o f 90° w i t h t h e a x i s o f t h e f i g ­
u r e , and o b l i q u e , i f t h e a x i s of t h e E a r t h d o e s n o t c o i n c i d e w i t h
t h e a x i s of t h e f i g u r e and i n t e r s e c t s it a t an a n g l e which i s n o t
e q u a l t o 90°.
Cy1 i n d r i c a l

Projections

Normal ( e q u i v a l e n t ) cy1 i n d r i c a l p r o j e c t i o n
A l l c y l i n d r i c a l p r o j e c t i o n s a r e f o r m e d by means o f t h e i m a g i n ­
a r y t r a n s f e r o f t h e E a r t h ' s s u r f a c e ( g l o b e ) t o a t a n g e n t i a l or i n ­
tersecting cylinder, with subsequent unrolling.
I n F i g u r e 1 . 1 4 , a simple normal c y l i n d r i c a l p r o j e c t i o n i s
g i v e n , i . e . , a p r o j e c t i o n o f t h e E a r t h on a t a n g e n t i a l c y l i n d e r ,

/25
-

t h e a x i s of which c o i n c i d e s w i t h t h e a x i s * o f t h e E a r t h ( g l o b e ) ,
while the height of t h e cylinder is proportional t o the length of
the axis.

PS

Fig.

Fig.

1.14.

1.15.

Normal

(Equivalent) Cylindrical Projection

Simple E q u a l l y S p a c e d C y l i n d r i c a l P r o j e c t i o n

I n t h i s p r o j e c t i o n , t h e meridians are compressed while t h e
p a r a l l e l s are extended t o a d e g r e e which i n c r e a s e s w i t h l a t i t u d e .
The p r o j e c t i o n i n c l u d e s a c a t e g o r y o f e q u a l l y l a r g e a n d e q u i v a l e n t
p r o j e c t i o n s , s i n c e it satisfies t h e condition o f an equivalence of
areas.
I t s equation can be w r i t t e n i n t h e following form:
X = R sin v; 2 = RS,

(1.16)

where X r e p r e s e n t s t h e c o o r d i n a t e s o f a p o i n t a l o n g t h e m e r i d i a n ;
Z r e p r e s e n t s t h e coordinates of a point along t h e equator; and R
is the Earth's radius.
L e t u s d e t e r m i n e what t h e s p e c i a l s c a l e s f o r t h e d i r e c t i o n s
are equal t o i n t h i s p r o j e c t i o n :

21


(1.17)

(1.18)

where m i s a p a r t i a l scale a l o n g a m e r i d i a n ; n i s a p a r t i a l scale
along a parallel;
i s a n i n c r e a s e i n d i s t a n c e on t h e map;
d S g l o b e i s a n i n c r e a s e i n d i s t a n c e on t h e g l o b e .

asmap

The p r o d u c t o f t h e s p e c i a l s c a l e s i s
1

mn -cos

'p

or m = d1 , w h i l e n=; 1

=1

n

cos 'p

Therefore, the given projection is equal.

Since

m # n ; m # 1 and

n # 1 i n t h e p r i n c i p a l d i r e c t i o n s ( m e r i d i a n s and p a r a l l e l s ) it i s

n o t i s o g o n a l and n o t e q u a l l y spaced.
Only i n t h e e q u a t o r i a l band,
i n t h e l i m i t s f r o m 0 t o f 5 O a l o n g i t s l a t i t u d e , i s it p r a c t i c a l l y
p o s s i b l e t o c o n s i d e r i t i s o g o n a l and e q u a l l y spaced.

S i m p l e equally spaced c y l i n d r i c a l p r o j e c t i o n
If w e t a k e t h e h e i g h t of a c y l i n d e r t o be p r o p o r t i o n a l n o t t o
the length of the Earth's axis, but t o t h e length of a meridian,
and i n s t e a d o f simply p r o j e c t i n g w e u n f o l d t h e m e r i d i a n s t o t h e
c y l i n d e r w a l l s , a s shown i n F i g . 1 . 1 5 , t h e n a s i m p l e , e q u a l l y s p a c e d
I t i s r e g a r d e d as normal s i n c e
cylindrical projection is obtained.
t h e axis of t h e globe coincides with t h e a x i s o f t h e cylinder.
I n t h i s p r o j e c t i o n , t h e meridians w i l l be transformed t o t h e i r
f u l l s i z e d u r i n g t h e i r t r a n s f e r f r o m t h e g l o b e ' s s u r f a c e t o a map
( i . e . , m = l), a n d t h e e q u a t o r a l s o w i l l b e t r a n s f o r m e d t o full
s i z e ( a t t h e e q u a t o r , n = 11, w h i l e t h e p a r a l l e l s w i l l b e e x t e n d e d
The m a g n i t u d e o f t h e
j u s t as i n a normal ( e q u i v a l e n t ) p r o j e c t i o n .
effect increases with l a t i t u d e .
T h e c o o r d i n a t e g r i d o f t h e map o f t h i s p r o j e c t i o n h a s t h e a p ­
pearance of a uniform r e c t a n g u l a r r u l i n g .
I t s e q u a t i o n s have t h e
form :
X=R'p; 2 - R A .
The s p e c i a l s c a l e s a r e e q u a l t o :

(1.19)


along the p a r a l l e l

22

n=

1
= sec
cos
'p

'p.

(1.20)

/27
-

S i n c e m = 1, t h e p r o j e c t i o n i s e q u a l l y s p a c e d a l o n g t h e m e r i ­
d i a n s and a l s o a l o n g t h e e q u a t o r .
S i n c e m # n a n d mn # 1, t h e p r o ­
j e c t i o n i s n o t i s o g o n a l and n o t e q u a l l y l a r g e , e x c e p t f o r t h e equa­
t o r i a l band i n t h e l i m i t s from 0 t o + 5 O a l o n g t h e l a t i t u d e , where
it i s p r a c t i c a l l y p o s s i b l e t o c o n s i d e r i t i s o g o n a l and e q u a l l y l a r g e .
Maps i n n o r m a l ( e q u i v a l e n t ) a n d s i m p l e , e q u a l l y d i s t a n t c y l i n ­
maps
d r i c a l p r o j e c t i o n s are u s e d i n a v i a t i o n o n l y as r e f e r e n c e s :
o f h o u r z o n e s , maps o f n a t u r a l l i g h t , e t c .

I s o g o n a l cy1 i n d r i c a l p r o j e c t i o n
An i s o g o n a l c y l i n d r i c a l p r o j e c t i o n ( M e r c a t o r p r o j e c t i o n ) i s
t h e most v a l u a b l e of a l l t h e c y l i n d r i c a l p r o j e c t i o n s f o r n a v i g a t i o n .
I t i s o b t a i n e d from a s i m p l e , e q u a l l y spaced c y l i n d r i c a l p r o j e c t i o n
by a r t i f i c i a l l y e x t e n d i n g t h e s c a l e a l o n g t h e l a t i t u d e ( l e n g t h e n i n g
t h e m e r i d i a n s ) , p r o p o r t i o n a l t o t h e change i n s c a l e a l o n g t h e l o n g i ­
tude.
T h e c o o r d i n a t e g r i d o f t h e map o f t h i s p r o j e c t i o n i s s h o w n
i n Figure 1.16.
The r e a s o n f o r i t s u s e i s t h e f a c t t h a t t h e a n g l e s m e a s u r e d o n
t h e map a r e e q u a l t o t h e c o r r e s p o n d i n g a n g l e s a t t h e l o c a t i o n , % . e . ,
m = n = s e c 6.
Le
t i o n of
along a
dinate)
f i n d m:

t us w r i t e an equa­
t h i s map p r o j e c t i o n
meridian (X-coor­
f o r which w e can

m=

dS
dS

-­ dX
globe

Rdv '

w h e r e dS i s a n i n c r e a s e o f
distance along the meridian
on t h e map; a n d R d + i s a n
increase i n distance along
t h e meridian at the loca­
tion.
We m u s t h a v e
m = sec 'p,
We s h a l l t h e n e q u a t e t h e
r i g h t - h a n d s i d e s of t h e s e
equations:

Fig. 1.16. Coordinate Grid of an Isogonal Cylindrical Projection.

.sec +,'w h e n c e
R 4
dX

e:

dX-

Rd'p
cos a(l.21)
'p

23

/28

After i n t e g r a t i n g (1.211,
along the meridian:

we w i l l obtain t h e X-coordinate

X = R I n t g 45
(

while the Z-coordinate
ple equation:

+-

O

i ),

(1.22)

a l o n g t h e p a r a l l e l i s d e t e r m i n e d by t h e s i m ­

z= Rh.

(1.22a)

Since m = n , the projection is isogonal but not equally spaced
a n d n # 1) and n o t e q u a l l y l a r g e ( m n # 1 ) .

(m # 1

T h e b a s i c a d v a n t a g e o f maps i n a n i s o g o n a l c y l i n d r i c a l p r o j e c ­
t i o n i s t h e s i m p l i c i t y o f t h e i r u s e w i t h m a g n e t i c c o m p a s s e s for
moving f r o m o n e p o i n t on t h e E a r t h t o a n o t h e r , s i n c e t h e loxodrome
Therefore,
i n t h i s p r o j e c t i o n has t h e appearance of a s t r a i g h t l i n e .
t h e isogonal c y l i n d r i c a l projection has been used widely, primarily
i n m a r i n e n a v i g a t i o n d u r i n g t h e c o m p i l a t i o n o f n a v a l maps.
The c h a n g e i n s c a l e w i t h l a t i t u d e i s a d i s a d v a n t a g e o f n o r m a l
cylindrical projections.
Here, i n normal ( e q u i v a l e n t ) and s i m p l e ,
e q u a l l y s p a c e d c y l i n d r i c a l p r o j e c t i o n s , t h e map s c a l e i s n o t i d e n t i ­
c a l i n t h e p r i n c i p a l d i r e c t i o n s ( n o r t h - s o u t h and e a s t - w e s t ) , s o
t h a t t h e d i s t a n c e b e t w e e n two p o i n t s i n d i r e c t i o n s n o t p a r a l l e l t o
t h e l i n e s o f t h e g r a t i n g c a n b e d e t e r m i n e d o n l y by c a l c u l a t i o n .
I n a n i s o g o n a l c y l i n d r i c a l p r o j e c t i o n , t h e map s c a l e a l o n g t h e
l a t i t u d e i s a l s o v a r i a b l e , b u t a t a n y p o i n t o n t h e map i t i s i d e n t i ­
cal in the principal directions.
T h i s makes it p o s s i b l e t o m e a s u r e
d i s t a n c e s b y m e a n s of c o m p a s s e s , for w h i c h a s c a l e ( v a r y i n g w i t h
t h e l a t i t u d e ) i s drawn on t h e w e s t e r n an d e a s t e r n e d g e s o f t h e map.
Means for m e a s u r i n g d i s t a n c e s o n maps w i t h s u c h a p r o j e c t i o n a r e
i n d i c a t e d i n m a n u a l s for m a r i n e n a v i g a t i o n .

Isogonal oblique cylindrical projections

T h e b a s i s f o r c r e a t i n g maps i n a n i s o g o n a l c y l i n d r i c a l p r o j e c ­
its isogonality.
t i o n i s a property of t h e Mercator projection:
S u c h p r o j e c t i o n s a r e u s e d i n t h e p r e p a r a t i o n o f s p e c i a l f l i g h t maps
and 1:4,000,000
which a r e
o n s c a l e s o f 1:1,000,000, 1 : 2 , 0 0 0 , 0 0 0 ,
used i n c i v i l a v i a t i o n .
T h e t a n g e n t i a l ( F i g . 1.17) or i n t e r s e c t i n g ( F i g . 1.18) c y l i n d e r
is s i t u a t e d a t such an angle t o t h e axis of t h e globe t h a t t h e tan­
g e n t o f t h e c y l i n d e r ' s s u r f a c e t o t h e g l o b e or t h e i n t e r s e c t i o n r u n s
along t h e f l i g h t path.
Usually the s t r i p along t h e tangent does not
e x t e n d m o r e t h a n 5 0 0 - 6 0 0 km t o e i t h e r s i d e o f t h e r o u t e (or t h e m i d ­
d l e l i n e o f t h e r o u t e , i f i t h a s d i s c o n t i n u i t i e s ) , w h i l e on t h e i n ­
t e r s e c t i n g c o n e i t d o e s n o t e x t e n d m o r e t h a n 1 0 0 0 - 1 4 0 0 km t o e i t h e r
s i d e of t h e given middle l i n e of t h e r o u t e s .
24

I n p r a c t i c e , s u c h f l i g h t maps a r e i s o g o n a l , e q u a l l y s p a c e d ,
and e q u a l l y l a r g e ; however, s i n c e t h e c y l i n d e r i s i n c o n t a c t w i t h
t h e g l o b e a l o n g t h e a r c o f a g r e a t c i r c l e or c u t s t h e g l o b e c o m p a r ­
a t i v e l y c l o s e t o t h e a r c o f a g r e a t c i r c l e , t h e o r t h o d r o m e on t h e s e
maps w i l l i n p r a c t i c e b e r e p r e s e n t e d b y a s t r a i g h t l i n e .
T h e d i s t o r t i o n s o f l e n g t h s o n f l i g h t maps o f o b l i q u e t a n g e n t i a l
p r o j e c t i o n s do n o t exceed 0.5%; f o r i n t e r s e c t i n g p r o j e c t i o n s t h e y
do n o t .exceed 0.8%-1.2%.

Fig. 1.17. Isogonal Oblique
(Tangential) Cylindrical
Projection.

Isogonal Oblique
Fig. 1.18.
(Intersecting) Cylindrical
Projection.

I s o g o n a l t r a n s v e r s e and c y l i n d r i c a l G a u s s i a n p r o j e c t i o n
The a x i s o f t h e c y l i n d e r i n G a u s s i a n p r o j e c t i o n s i s p e r p e n d i c u ­
l a r t o t h e a x i s of r o t a t i o n o f t h e E a r t h ( g l o b e ) .
The c o n s t r u c t i o n
o f maps w i t h t h i s p r o j e c t i o n i s s i m i l a r t o t h e c o n s t r u c t i o n o f maps
with oblique cylindrical projections.
F o r e x a m p l e , a f l i g h t map
o n a s c a l e o f 1:1,000,000 f o r L e n i n g r a d - K i e v h a s b e e n c o m p i l e d o n
such a projection.
H o w e v e r , on t h e w h o l e , i s o g o n a l t r a n s v e r s e c y l ­
i n d r i c a l G a u s s i a n p r o j e c t i o n i s u s e d f o r c o m p i l i n g maps o n a l a r g e
s c a l e , where t h e s p e c i a l p r i n c i p l e s o f c o n s t r u c t i o n are used.
A s p h e r o i d ( E a r t h ' s e l l i p s o i d ) i s t a k e n as t h e f i g u r e from
which t h e E a r t h ' s s u r f a c e i s p r o j e c t e d , w h i l e t h e t a n g e n t i a l c y l i n ­
d e r on w h i c h t h e E a r t h ' s s u r f a c e i s p r o j e c t e d h a s a n e l l i p t i c a l b a s e
a c c o r d i n g t o t h e form of t h e E a r t h ' s e l l i p s o i d .

The e n t i r e E a r t h ' s s u r f a c e i s d i v i d e d b y m e r i d i a n s i n t o z o n e s ,
e a c h o f w h i c h h a s a l a t i t u d e o f 6 O a n d i s p r o j e c t e d o n t o i t s own
c y l i n d e r which i s t a n g e n t i a l t o t h e E a r t h ' s s u r f a c e a l o n g t h e mid­
d l e meridian of t h e g i v e n zone.
Thus, i n o r d e r t o p r o j e c t t h e whole s u r f a c e o f t h e E a r t h , it
is necessary t o t u r n t h e e l l i p t i c a l c y l i n d e r mentally around t h e
25

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a x i s of t h e E a r t k ' s e l l i p s o i d t h r o u g h 6O a t a t i m e ,
In Figure 1.19,
a , t h e p r o j e c t i o n o f o n l y one zone f o r 6 O l o n g i t u d e i s shown, w h i l e
i n Figure 1.19, b y the unrolling of a semicylinder after its rotat i o n around t h e E a r t h ' s a x i s i n o r d e r t o p r o j e c t several zones i s
shown.
W i t h s u c h a p r o j e c t i o n , a l l maps a r e c o n s t r u c t e d o n t h e
scales:
1:500,000, 1:200,000, 1:100,000, 1:50,000, and 1:25,000.
The l a t t e r a r e e s s e n t i a l l y c h a r t s .

Fig.

1.19.

Isogonal Transverse-Cylindrical Gaussian Projectioh.

E a c h z o n e o n maps w i t h a s c a l e o f 1:200,000 a n d l a r g e r h a s i t s
own s p e c i a l X a n d Y(Z) r e c t a n g u l a r c o o r d i n a t e s y s t e m , w h i c h i s
c a l l e d t h e Gaussian kilometer system.
M e r i d i a n s a n d p a r a l l e l s on
maps o f t h i s p r o j e c t i o n a r e c u r v e d l i n e s a n d d o n o t c o i n c i d e w i t h
t h e Gaussian system.
The v e r t i c a l l i n e s o f t h e r e c t a n g u l a r Gaus­
s i a n system a r e p a r a l l e l t o t h e c e n t r a l m e r i d i a n o f t h e zone and do
n o t c o i n c i d e with o t h e r meridians o f t h e zone.
The a n g l e b e t w e e n t h e v e r t i c a l l i n e X o f t h e G a u s s i a n s y s t e m
and t h e l i n e t o t h e o b j e c t ( p o i n t ) i s c a l l e d t h e d i r e c t i o n a l angle.
I n order t o o b t a i n t h e t r u e o r magnetic d i r e c t i o n ( a n g l e ) , t h e
a n g l e s o f t h e convergence o f t h e system w i t h t h e t r u e and magnetic
In addition, the
m e r i d i a n s a r e i n d i c a t e d o n t h e e d g e o f t h e map.
v e r t i c a l s e c t i o n o f a map ( f r a m e ) a l w a y s r u n s i n t h e d i r e c t i o n o f
the true meridian,
By m e a n s o f t h e G a u s s i a n s y s t e m a n d f i g u r e s i n t h e f r a m e s o f
t h e maps, it i s p o s s i b l e t o determine t h e d i s t a n c e from t h e e q u a t o r
and from t h e c e n t r a l meridian o f t h e zone t o t h e o b j e c t ( p o i n t ) .
D i s t o r t i o n s o f l e n g t h s o n t h e s e maps a r e i n s i g n i f i c a n t a n d d o n o t
exceed 0.14% a l o n g t h e edges o f t h e zone i n t h e l a t i t u d e which i s
e q u a l t o z e r o (140 m a t 1 0 0 k m ) .
Maps o n a n i s o g o n a l t r a n s v e r s e - c y l i n d r i c a l G a u s s i a n p r o j e c t i o n
are used b o t h i n a v i a t i o n f o r a d e t a i l e d o r i e n t a t i o n and l o c a t i o n

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o f t a r g e t s , a n d i n many b r a n c h e s o f t h e n a t i o n a l e c o n o m y f o r l i n k i n g
p r o j e c t s , equipment, and r a d i o engineering f a c i l i t i e s i n a l o c a t i o n , 1 3 2
f o r determining geodesic r e f e r e n c e p o i n t s , and f o r a c c u r a t e geodesic
c a l c u l a t i o n s o f d i s t a n c e s and d i r e c t i o n s , e t c .

­

Conic P r o j e c t i o n s

Conic p r o j e c t i o n s a r e c o n s t r u c t e d by p r o j e c t i n g t h e s u r f a c e o f
t h e E a r t h ' s spheroid (globe) on a tangent o r i n t e r s e c t i n g cone, w i t h
i t s subsequent u n r o l l i n g t o form a p l a n e s u r f a c e ( F i g . 1 . 2 0 , a ) .

( a ) Tangent ( i n t e r ­
Construction of Conic P r o j e c t i o n s :
F i g . 1 . 2 0 .
s e c t i n g ) c o n e ; ( b ) U n r o l l i n g o f t h e C o n e t o Form a P l a n e .

A c c o r d i n g t o t h e p o s i t i o n s of t h e axes o f t h e g l o b e a n d c o n e ,
c o n i c p r o j e c t i o n s can be normal, t r a n s v e r s e , and oblique.
However,
i n o u r p u b l i c a t i o n s n o r m a l p r o j e c t i o n s a r e g e n e r a l l y u s e d when t h e
a x j s o f t h e cone c o i n c i d e s w i t h t h e a x i s of t h e g l o b e .
I n a normal c o n i c p r o j e c t i o n , m e r i d i a n s a r e r e p r e s e n t e d by
s t r a i g h t l i n e s , w h i l e p a r a l l e l s a r e r e p r e s e n t e d by a r c s o f c o n c e n ­
t r i c circles (Fig. 1.20, b).
From F i g u r e 1 . 2 0 , a , i t i s e a s y t o s e e t h a t t h e r a d i u s o f a
p a r a l l e l of tangency (p,)
c a n b e e x p r e s s e d by t h e E a r t h ' s r a d i u s :

where R i s t h e r a d i u s o f t h e E a r t h ( g l o b e ) and $ 0 i s t h e l a t i t u d e
of t h e p a r a l l e l o f tangency.
The e q u a t i o n o f t h i s p r o j e c t i o n i s w r i t t e n i n t h e f o l l o w i n g
form:
8r.aA;

-

Translator's note:

P'= Po

+ R (cpo 3- cp);

(1.23)

ctg = cot,
27

n


x

where 6 and p are t h e p r i n c i p a l d i r e c t i o n s i n t h e p o l a r c o o r d i n a t e
s y s t e m a l o n g t h e p a r a l l e l a n d m e r i d i a n , r e s p e c t i v e l y , a n d 01 i s t h e
c o e f f i c i e n t f o r t h e a n g l e o f convergence of t h e m e r i d i a n s .

S i m p l e normal c o n i c p r o j e c t i o n
A simple normal c o n i c p r o j e c t i o n i s c o n s t r u c t e d w i t h t h e con­
s i d e r a t i o n t h a t t h e m e r i d i a n s o n t h e w h o l e map a n d t h e p a r a l l e l o f
tangency b e t r a n s f e r r e d from a g l o b e w i t h o u t d i s t o r t i o n s t o t h e i r
n a t u r a l v a l u e ( i . e . , in = l), w h i l e f o r t h e p a r a l l e l s o f t a n g e n c y
($0)

m = n = 1.

Such a p r o j e c t i o n forms t h e b a s i s o f t h e improved i n t e r s e c t i n g
conic Kavrayskiy p r o j e c t i o n (Fig. 1.21, a ) .
I t is equally spaced,
s i n c e m = 1, w h i l e o n t h e i n t e r s e c t i n g p a r a l l e l s i t i s i s o g o n a l a n d
e q u a l l y l a r g e ( F i g . 1 . 2 1 , b).

intersection para1 l e l s

@i - J - @
-b

-----

Fig.

1.21.

S i m p l e Normal C o n i c P r o j e c t i o n .
( a ) I n t e r s e c t i n g Cone;
( b ) U n f o l d i n g o f t h e Cone t o a P l a n e .

Many a i r c r a f t maps w i t h s c a l e s o f 1 : 2 , 5 0 0 , 0 0 0 , 1 : 2 , 0 0 0 , 0 0 0 ,
a n d e v e n 1 : 1 , 5 0 0 , 0 0 0 , w h i c h a r e u s e d i n a i r c r a f t n a v i g a t i o n for
g e n e r a l o r i e n t a t i o n and t h e approximate d e t e r m i n a t i o n o f t h e p o s i ­
t i o n o f a n a i r c r a f t by means o f r a d i o e n g i n e e r i n g f a c i l i t i e s ( a i r ­
c r a f t r a d i o compasses, ground r a d i o g o n i o m e t e r s , e t c . ) , have been
published.
T h e i r p o s i t i v e f e a t u r e i f t h e i n s i g n i f i c a n t d i s t o r t i o n of
l e n g t h s i n t h e s t r i p *So from t h e i n t e r s e c t i n g p a r a l l e l s , which
d o e s n o t e x c e e d 0 . 3 4 % ( 3 4 0 m for 1 0 0 k m ) .
Their disadvantage is
t h e d i s t o r t i o n of d i r e c t i o n s , which i n c r e a s e s w i t h d i s t a n c e from
the intersecting parallels.

28

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Isogonal conic projection

By a n a l o g y w i t h t h e c o n s t r u c t i o n o f a n i s o g o n a l c y l i n d r i c a l
Mercator p r o j e c t i o n , d e s t r o y i n g t h e e q u a l s p a c i n g , a simple normal
c o n i c p r o j e c t i o n i s t r a n s f o r m e d i n t o an i s o g o n a l p r o j e c t i o n by r e ­
ducing (equating) t h e scale along t h e meridians t o t h e s c a l e along
t h e p a r a l l e l s ( m = n).
T h i s i s more v a l u a b l e f o r u s e i n a v i a t i o n .
A i r c r a f t maps w i t h a s c a l e o f 1 : 2 , 0 0 0 , 0 0 0
a n d s u r v e y maps on
s c a l e s o f 1:3,000,000, 1 : 4 , 0 0 0 , 0 0 0 ,
and 1:5,000,000
are published
w i t h a normal i s o g o n a l c o n i c p r o j e c t i o n f o r a v i a t i o n .
Maps w i t h a s c a l e o f 1 : 2 , 0 0 0 , 0 0 0 i n t h i s p r o j e c t i o n , b e s i d e s
h a v i n g t h e b a s i c advantage of i s o g o n a l i t y , a l s o have d i s t o r t i o n s of
l e n g t h which a r e p e r m i s s i b l e i n t h e p r a c t i c e o f a i r c r a f t n a v i g a t i o n .
On a n i n t e r s e c t i n g c o n e i n a s t r i p f r o m 4 0 ° t o 7 0 ° i n l a t i t u d e , t h e
maximum l e n g t h d i s t o r t i o n s d o n o t e x c e e d 51.8 km f o r 1 0 0 k m .

Fig. 1 . 2 2 .
Projection:

Angle o f Convergence o f t h e M e r i d i a n s o f a Tangent Conic
( a ) A r c o f 9 P a r a l l e l on a G l o b e ; ( b ) A r c o f a P a r a l l e l
o n a Map.

T h e o r t h o d r o m e on m a p s o f a n i s o g o n a l c o n i c p r o j e c t i o n f o r d i s ­
t a n c e s u p t o 1 2 0 0 km a p p e a r s a s a p r a c t i c a l l y s t r a i g h t l i n e .
This
v a l u a b l e q u a l i t y i s u s e d d u r i n g f l i g h t s on c i v i l a v i a t i o n a i r l i n e s
o f a v e r a g e l e n g t h by u s i n g g y r o s c o p i c and a s t r o n o m i c a l c o m p a s s e s
A t g r e a t d i s t a n c e s , t h e orthodrome
f o r following t h e orthodrome.
( a s a r e s u l t o f a c h a n g e i n s c a l e ) i s b e n t by a b u l g e t e n d i n g t o ­
ward a l a r g e r s c a l e .
The

loxodrome i s

r e p r e s e n t e d by an a r c o f a l o g a r i t h m i c s p i r a l .

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T h i s c r e a t e s d i f f i c u l t i e s . i n a i r c r a f t n a v i g a t i o n by means o f m a g n e t i c
compasses.
I n t h e s e i n s t a n c e s , for d i s t a n c e s u p t o 5 0 0 - 8 0 0 km i n
d i r e c t i o n s which i n t e r s e c t t h e m e r i d i a n s on a map, a s t r a i g h t l i n e
i s c o n s t r u c t e d , w h i l e measurement o f t h e f l i g h t a n g l e i s c a r r i e d
o u t a l o n g t h e c e n t r a l m e r i d i a n o f t h e r o u t e which i s maintained i n
f l i g h t by means o f a m a g n e t i c compass.

I t i s also p o s s i b l e t o c o n s t r u c t ( c o n t i n u e ) t h e l o x o d r o m e a l o n g
a n a n g l e measured i n t h e middle o f t h e s t r a i g h t l i n e j o i n i n g t h e
/35
c o n t r o l ( r o t a t i n g ) landmarks of t h e r o u t e .

-

The d i s a d v a n t a g e o f a l l maps w i t h c o n i c p r o j e c t i o n s i s t h e p r e ­
s e n c e o f an a n g l e o f convergence o f t h e m e r i d i a n s from t h e p a r a l l e l s
It is neces­
of t a n g e n c y ( p a r a l l e l s o f i n t e r s e c t i o n ) t o t h e p o l e .
s a r y t o c o n s i d e r t h i s a n g l e when d e t e r m i n i n g d i r e c t i o n s ( f l i g h t
a n g l e s ) or t h e l o c a t i o n o f t h e a i r c r a f t b y m e a n s o f a i r c r a f t r a d i o
compasses.
I n a d d i t i o n , d e p e n d i n g on t h e p a r a l l e l s o f i n t e r s e c t i o n
or t a n g e n c y , t h e a n g l e o f c o n v e r g e n c e o f t h e m e r i d i a n s w i l l b e d i f ­
ferent.

Convergence a n g l e o f t h e m e r i d i a n s
The p r i n c i p a l s c a l e o f c o n i c p r o j e c t i o n s i s t a k e n a l o n g t h e
There­
m e r i d i a n s a n d p a r a l l e l s o f t a n g e n c y or i n t e r s e c t i o n ( $ 0 ) .
f o r e , t h e a r c MN i s e q u a l t o t h e a r c M I N I ( F i g . 1 . 2 2 ) .
I t i s known
t h a t on a g l o b e ( s p h e r o i d ) ( F i g . 1 . 2 2 , a ) , t h e a r c M N = r A A , where
r i s t h e r a d i u s o f t h e p a r a l l e l . On a map o f a c o n i c p r o j e c t i o n
(Fig. 1 . 2 2 , b) t h e arc M l N l = poA6; then
rAA = poA8.

But r = R cos $ 0 and P O = R c o t $ 0 ,
projection (1.23), A6 = aAA.

and from t h e e q u a t i o n of a c o n i c

S u b s t i t u t i n g t h e v a l u e s o f r , P O , and A 6
out the necessary reductions, we obtain:
a=Sinyo.

(1.24)

in

(1.24)

and c a r r y i n g

(1.25)

O b v i o u s l y , on t h e e q u a t o r t h e c o e f f i c i e n t o f c o n v e r g e n c e o f
t h e m e r i d i a n s a = 0 , s i n c e s i n O o = 0 ; a t t h e p o l e s c1 = 1, s i n c e
s i n 9 0 ° = 1, a n d i n t h e g e n e r a l c a s e f o r c e n t r a l l a t i t u d e s , O < a < l .
Knowing t h e c o e f f i c i e n t a , i t i s n o t d i f f i c u l t t o d e t e r m i n e
any a n g l e of convergence o f t h e m e r i d i a n s 6 a l o n g a p a r a l l e l o f
t a n g e n c y or i n t e r s e c t i o n :
8 =A h ,

w h e r e A?,

(1.26)

is t h e d i f f e r e n c e i n l o n g i t u d e between t h e given meridians.

A t any o t h e r l a t i t u d e ,

t h e c o e f f i c i e n t a w i l l b e d i f f e r e n t from

30


I

t h e c o e f f i c i e n t a a t a l a t i t u d e of tangency ( i n t e r s e c t i o n ) .
There­
f o r e , f o r approximate c a l c u l a t i o n s i n t h e p r a c t i c e of aircraft navi­
g a t i o n d u r i n g t h e d e t e r m i n a t i o n o f f l i g h t a n g l e s or l o c a t i o n o f t h e
a i r c r a f t , t h e mean l a t i t u d e o f t h e r o u t e , p a r t o f t h e r o u t e , or t h e
d i s t a n c e between t h e a i r c r a f t and t h e r a d i o s t a t i o n , i s t a k e n as

w h e r e A2 a n d A 1 a r e t h e l o n g i t u d e s o f t h e f i n a l a n d i n i t i a l p o i n t s ,
A, a n d h a a r e t h e l o n g i t u d e s o f t h e r a d i o s t a t i o n a n d a i r c r a f t , r e ­
s p e c t i v e l y , a n d $mid i s t h e m i d d l e l a t i t u d e b e t w e e n t h e i n d i c a t e d
points (places).
I n some c a s e s , f o r a p p r o x i m a t e d e t e r m i n a t i o n s o f t h e l o c a t i o n
o f an a i r c r a f t o r t h e f l i g h t a n g l e s , t h e c o e f f i c i e n t a i s assumed
c o n s t a n t f o r a g i v e n map o f a c o n i c p r o j e c t i o n .
Thus, f o r example,
f o r a map w i t h a s c a l e o f 1 : 2 , 0 0 0 , 0 0 0
and a normal i s o g o n a l c o n i c
/36
p r o j e c t i o n , it i s p o s s i b l e t o l e t a m 0 . 8 , which c o r r e s p o n d s t o t h e
s i n e o f t h e l a t i t u d e o f t h e middle p a r a l l e l between t h e i n t e r s e c t i o n
p a r a l l e l s , w h e r e t h e map s c a l e w i l l b e minimum.

­

Polyconic projections
Polyconic (multiconic) projections are t h e g r e a t e s t perfection
o f c o n i c p r o j e c t i o n s for t h e p u r p o s e o f d e c r e a s i n g d i s t o r t i o n s o f
l e n g t h s and a n g l e s i n p r o j e c t i n g t h e E a r t h ' s s u r f a c e o n t o a p l a n e .
The p r i n c i p l e o f c o n s t r u c t i o n o f s u c h p r o j e c t i o n s i s shown i n
Figure 1.23, a.
The c e n t r a l m e r i d i a n o f t h e p r o j e c t i o n s i s a

Fig.

1.23.

Polyconic Projection:
( a ) I n t e r s e c t i n g Cones on t h e
G l o b e ; (b) U n r o l l i n g o f Cones on a P l a n e .

31

s t r a i g h t l i n e , w h i l e meridians i n t h e form of curved l i n e s are s i t ­
u a t e d t o t h e w e s t and e a s t o f it.
The p a r a l l e l s a r e c o n c e n t r i c
c i r c l e s w i t h d i f f e r e n t c e n t e r s , l y i n g on t h e c e n t r a l m e r i d i a n ( F i g .
1.23, b).
A s a r e s u l t of t h e i n c r e a s e i n s c a l e i n proportion to
t h e d i s t a n c e from t h e c e n t r a l meridian t o t h e w e s t and e a s t , such
p r o j e c t i o n s a r e used o n l y t o r e p r e s e n t t h e E a r t h ' s s u r f a c e i n coun­
t i r e s extended along a meridian.

International projection
I n terms o f t h e method o f c o n s t r u c t i o n , a n i n t e r n a t i o n a l p r o ­
j e c t i o n i s r e l a t e d t o a m o d i f i e d p o l y c o n i c p r o j e c t i o n ; i n terms o f
t h e nature of t h e d i s t o r t i o n s , it is r e l a t e d t o an a r b i t r a r y pro­
jection.
This projection w a s accepted a t an i n t e r n a t i o n a l geophysical
c o n f e r e n c e i n London i n 1 9 0 9 a n d was c a l l e d I r A P r o j e c t i o n o f a n
I n t e r n a t i o n a l Map o f t h e W o r l d , w i t h a S c a l e o f 1:1,000,000; C o r ­
r e c t i o n s by t h e R u s s i a n G e o d e s i s t S h c h e t k i n " .
This projection,
which i s t h e most widely d i s t r i b u t e d p r o j e c t i o n a t t h e p r e s e n t time
i n t h e S o v i e t U n i o n , i s u s e d f o r t h e p u b l i c a t i o n o f f l i g h t maps w i t h
s c a l e s o f 1:1,000,000, 1 : 2 , 0 0 0 , 0 0 0 ,
and 1:4,000,000.
E v e r y s h e e t o f map w i t h a s c a l e 1:1,000,000, w h i c h e n c o m p a s s e s
o f l a t i t u d e a n d 6 O of l o n g i t u d e ( i n a r a n g e o f l a t i t u d e s f r o m 0
t o 6 4 O ) , i s c o n s t r u c t e d a c c o r d i n g t o i t s own l a w , w h i c h i s g e n e r a l
for a l l t h e s h e e t s o f a g i v e n l a t i t u d i n a l s t r i p .
On e a c h s h e e t ,
t h e p r i n c i p a l s c a l e i s given along t h e o u t e r p a r a l l e l s of t h e s h e e t
a s a r e s u l t o f t h e i n t e r s e c t i o n o f t h e g l o b e by a c o n e a l o n g t h e s e
p a r a l l e l s ( w h e r e n = 1) a n d a l o n g t h e m e r i d i a n s , s e p a r a t e d f r o m t h e
c e n t r a l m e r i d i a n o f t h e s h e e t by 2 O t o t h e w e s t and e a s t , where m =
1 (Fig. 1.24, a ) .
I n t h e r a n g e o f l a t i t u d e s f r o m 64O t o 8 0 ° , e a c h
The p r i n ­
s h e e t o c c u p i e s 1 2 O o f l o n g i t u d e , a n d f r o m 8 0 t o 88O, 24O.
c i p a l scale of these sheets is also given along the outer p a r a l l e l s
o f t h e s h e e t ( t h r o u g h 4O) a n d a l o n g t h e m e r i d i a n s w h i c h a r e d i s t a n t
4O

Fig.

32

1.24.

International Projection:
( a ) Construction of t h e Sheet;
(b) B r e a k s i n t h e S p l i c i n g o f S h e e t s .

/37

from t h e c e n t r a l m e r i d i a n o f t h e s h e e t by 4 and 8 O , r e s p e c t i v e l y .
The r e g i o n s o f t h e p o l e s a r e p r o j e c t e d o n t o s e p a r a t e s h e e t s i n a
central (polar) projection.
The m e r i d i a n s i n t h i s p r o j e c t i o n a r e r e p r e s e n t e d by s t r a i g h t
l i n e s which have an a n g l e o f convergence t o t h e p o l e s , s i m i l a r t o
t h e c o n i c p r o j e c t i o n s , w h i l e t h e p a r a l l e l s are curved l i n e s which
a r e constructed according t o a s p e c i a l mathematical l a w .
The c e n t e r s
o f t h e c i r c l e - p a r a l l e l s a r e s i t u a t e d on t h e c e n t r a l m e r i d i a n o f a
g i v e n sample of s h e e t s , w h i l e t h e i r r a d i i are p r o p o r t i o n a l t o t h e
cotangents of the intersection l a t i t u d e s :
R~ = ctg

R~ = c t g 'p2 e t c'..,

A c c o r d i n g t o s t u d i e s b y L i m n i t s k i y , d i s t o r t i o n s o f l e n g t h s on
maps w i t h a s c a l e of 1 : 1 , 0 0 0 , 0 0 0
w i t h s u c h a p r o j e c t i o n , i n t h e mid­
d l e l a t i t u d e s does n o t exceed 0.076% ( 7 6 m i n 1 0 0 km), w h i l e d i s t o r ­
t i o n of d i r e c t i o n s i s 5 ' .
The g r e a t e s t d i s t o r t i o n s a r i s e i n t h e
d i s t o r t i o n o f l e n g t h s up t o 0 . 1 4 % , a n g l e s
region of t h e equator:
up t o 7 ' .
I n s i g n i f i c a n t d i s t o r t i o n s make i t p o s s i b l e t o c o n s i d e r t h e map
as a p r a c t i c a l l y i s o g o n a l , e q u a l l y s p a c e d , and w i t h e q u a l l y l a r g e
projection.
A c c o r d i n g t o t h i s p r i n c i p l e , s h e e t s o f maps w i t h s c a l e s o f
1:2,000,000 a n d 1 : 4 , 0 0 0 , 0 0 0 a r e c o n s t r u c t e d .
In a range of l a t i ­
t u d e s f r o m 0 t o 6 4 O , t h e s h e e t o f a map w i t h a s c a l e o f 1 : 2 , 0 0 0 , 0 0 0
o c c u p i e s 12O of l a t i t u d e a n d 18O o f l o n g i t u d e ( n i n e s h e e t s o f a
m i l l i o n t h , 3 x 3), w h i l e a map w i t h a s c a l e o f 1:4,000,000 o c c u p i e s
24 and 3 6 O , r e s p e c t i v e l y .
T h e p r i n c i p a l s c a l e o f a 1 : 2 , 0 0 0 , 0 0 0 map
i s g i v e n a l o n g t h e o u t e r p a r a l l e l s o f t h e s h e e t and t h e m e r i d i a n s
w h i c h a r e d i s t a n t f r o m t h e c e n t r a l m e d i a n o f t h e s h e e t by 6 O t o t h e
w e s t a n d e a s t ( F i g . 1 . 2 4 , b ) , w h i l e o n a map w i t h a s c a l e o f
1:4,000,000
t h e p a r a l l e l s which a r e d i s t a n t by 8 O 5 0 ' t o t h e n o r t h
a n d 8O10' t o t h e s o u t h a r e g i v e n w i t h o u t d i s t o r t i o n s , a n d t h e m e r i ­
d i a n s a r e 12O t o t h e w e s t a n d e a s t f r o m t h e c e n t r a l p a r a l l e l a n d
the central meridian, respectively.
The d i s t o r t i o n o f l e n g t h s i n t h e m i d d l e l a t i t u d e s on maps w i t h
a s c a l e o f 1 : 2 , 0 ~ 0 , ~ 0 r0e a c h e s 0 . 5 % , a n d t h e d i s t o r t i o n o f t h e a n ­
g l e s i s 30'; o n 1 : 4 , 0 0 0 , 0 0 0
m a p s , d i s t o r t i o n o f l e n g t h s r e a c h e s l.5%,
t h a t o f a n g l e s , 1O30'.
A d i s a d v a n t a g e o f maps i n t h e i n t e r n a t i o n a l p r o j e c t i o n o n a l l
s c a l e s i s t h e p r e s e n c e of d i s c o n t i n u i t i e s i n t h e s p l i c i n g o f s e v e r a l
Sheets of
s h e e t s , as a r e s u l t of t h e f e a t u r e s of i t s c o n s t r u c t i o n .
m a p s o f o n l y o n e s t r i p or o n e c o l u m n a r e s p l i c e d w i t h o u t b r e a k s .
D u r i n g s p l i c i n g o f n i n e s h e e t s o f m a p s o n a s c a l e o f 1:1,000,000
( 3 x 3 ) , t h e d i s c o n t i n u i t i e s which arise are p a r t i a l l y evened o u t
b y d e f o r m a t i o n o f t h e p a p e r , a n d t h e u s e o f s u c h a map d o e s n o t r e ­
s u l t i n p e r c e p t i b l e d i s t o r t i o n s of l e n g t h s a n d a n g l e s .
S p l i c i n g of
a l a r g e number o f s h e e t s i s n o t recommended.

33

138
­

I t i s e v e n i m p o s s i b l e t o s p l i c e a map w i t h a s c a l e o f 1 : 2 , 0 0 0 ,
000 from f o u r s h e e t s ( 2 x . 2 ) without a break.
A t a l a t i t u d e of 60°,
t h e d i s c o n t i n u i t y o f t h e s p l i c e d s h e e t s r e a c h e s 1 . 8 cm, ;.e.,
36 km
(see Fig. 1.24, b ) .
T h e r e f o r e , it i s p o s s i b l e t o s p l i c e o n l y o n e
s t r i p or o n e c o l u m n o f t h e s e m a p s .
T h e o r t h o d r o m e w i t h a l e n g t h u p t o 1 2 0 0 km o n , m a p s w i t h a
and 1:2,000,000 ( w i t h i n t h e l i m i t s o f one
s h e e t ) a p p e a r s i n p r a c t i c e as a s t r a i g h t l i n e , w h i l e t h e loxodrome
i s t h e arc of a logarithmic s p i r a l .
U s u a l l y , i n d i r e c t i o n s which
i n t e r s e c t t h e m e r i d i a n s , t h e l o x o d r o m e s e c t i o n s w i t h a l e n g t h up t o
6 0 0 km a r e l i k e w i s e c o n s t r u c t e d i n t h e f o r m o f a s t r a i g h t l i n e , w h i l e
t h e f l i g h t a n g l e i s measured i n t h e middle o f a p a r t o f a r o u t e i n
o r d e r t o l e s s e n by a f a c t o r o f 2 t h e e r r o r o f t h e measured a n g l e
d u r i n g f l i g h t w i t h t h e u s e o f a magnetic compass.

scale o f 1:1,000,000

During t h e d e t e r m i n a t i o n o f t h e p o s i t i o n o f a n a i r c r a f t by
means o f r a d i o c o m p a s s e s , a c o r r e c t i o n i s a l l o w e d f o r t h e c o n v e r g e n c e
o f t h e m e r i d i a n s j u s t a s i n maps o f c o n i c p r o j e c t i o n s , w i t h a n a p ­
proximate formula
6 = (A,,--

ha\ sin p.

mi d

where Xr i s t h e l o n g i t u d e o f t h e r a d i o s t a t i o n ; Xa i s t h e l o n g i t u d e
o f t h e a i r ' c r a f t ; CPmid i s t h e mean l a t i t u d e b e t w e e n t h e r a d i o s t a t i o n
a n d a i r c r a f t , or t h e mean l a t i t u d e o f t h e s h e e t ( s h e e t s ) i f t h e ' a p ­
p r o x i m a t e p o s i t i o n o f t h e a i r c r a f t i s unknown.
I n c i v i l a v i a t i o n , maps w i t h a s c a l e o f 1 : 1 , 0 0 0 , 0 0 0
a n d 1:2,
0 0 0 , 0 0 0 on a n i n t e r n a t i o n a l p r o j e c t i o n a r e u s e d a s f l i g h t maps, p r i m a r i l y on p i s t o n - e n g i n e a i r c r a f t and h e l i c o p t e r s , and secondly on
aircraft with gas-turbine engines.
Maps w i t h a s c a l e o f 1 : 4 , 0 0 0 ,
0 0 0 a r e u s e d a s a i r c r a f t maps f o r g e n e r a l o r i e n t a t i o n a n d a p p r o x i ­
mate d e t e r m i n a t i o n o f t h e l o c a t i o n o f a n a i r c r a f t by means o f r a d i o engineering f a c i l i t i e s .
Azimuthal

(Perspective) Projections

Azimuthal ( p e r s p e c t i v e ) p r o j e c t i o n s are c o n s t r u c t e d according
t o t h e laws o f a s i m p l e g e o m e t r i c p e r s p e c t i v e ; t h e r e f o r e , t h e y a r e
often called perspective projections.
According t o t h e p o s i t i o n of t h e p l a n e o f t h e f i g u r e , azimuthal
p r o j e c t i o n s a r e d i v i d e d i n t o n o r m a l or p o l a r ( F i g . 1 . 2 5 , a ) , t r a n s ­
v e r s e or e q u a t o r i a l ( F i g . 1 . 2 5 , b), a n d o b l i q u e or h o r i z o n t a l ( F i g .
1 . 2 5 , c ) ; d e p e n d i n g on t h e p o s i t i o n o f t h e c e n t e r o f t h e p r o j e c t i o n
r e l a t i v e t o t h e plane of t h e f i g u r e , they can be of t h e following
types (Fig. 1.26):

a ) C e n t r a l or gnomonic, when t h e c e n t e r o f t h e p r o j e c t i o n c o ­
incides with the center of t h e Earth (globe):
p o i n t A;

34

/39
-

'

b) Steriographic, when t h e c e n t e r o f t h e p r o j e c t i o n i s s e p a ­
r a t e d from t h e p o i n t of c o n t a c t w i t h t h e p l a n e o f t h e f i g u r e by a
p o i n t B;
distance equal t o the diameter of the Earth (globe):
c)
Orthographic, when t h e c e n t e r o f t h e p r o j e c t i o n i s i n f i ­
n i t e l y s e p a r a t e d from t h e p l a n e o f t h e f i g u r e :
p o i n t C;
d)
External, when t h e c e n t e r o f t h e p r o j e c t i o n i s l o c a t e d
above t h e p l a n e o f t h e f i g u r e :
p o i n t D.

Fig.

1.25.

Azimuthal P r o j e c t i o n s :
( a ) Normal;
( c ) Oblique.

D

1
1

(b) T r a n s v e r s e ;

A s i s e v i d e n t from F i g . 1 . 2 6 ,
on s u c h p r o j e c t i o n s p o i n t s M and N
on t h e E a r t h ' s s u r f a c e w i l l b e p r o ­
j e c t e d a t a d i f f e r e n t d i s t a n c e from
t h e point of tangency of t h e plane
o f t h e f i g u r e w i t h t h e E a r t h ' s sur­
face.
Meridians i n azimuthal ( p o l a r )
p r o j e c t i o n s a r e r e p r e s e n t e d by
s t r a i g h t l i n e s which converge t o a
p o l e a t an a n g l e e q u a l t o t h e d i f ­
ference i n longitude:
6 = AA.
P a r a l l e l s a r e r e p r e s e n t e d by
concentric circles, the r a d i i of
which depend on t h e c e n t e r of t h e
p r o j e c t i o n and t h e l a t i t u d e o f t h e
position.

Position of the
Fig. 1.26.
Centers of Projection i n
Azimuthal P r o j e c t i o n s .

I n a v i a t i o n , c e n t r a l p o l a r and
s t e r e o g r a p h i c p o l a r p r o j e c t i o n s are
generally used.
35

fi

Central polar (gnomonic projection)

The c e n t e r o f p r o j e c t i o n i n t h i s p r o j e c t i o n c o i n c i d e s w i t h t h e
center of the Earth (globe) a t the point 0 (Fig. 1.27, a ) .
From F i g u r e 1 . 2 7 ,
projection:

it i s p o s s i b l e t o w r i t e t h e e q u a t i o n of t h i s
6 = A;
p = R ctg 'p.

I n order t o have a complete i d e a of t h e p r o j e c t i o n , l e t u s f i n d
t h e s p e c i a l s c a l e s ( m , n ) for t h e p r i n c i p a l d i r e c t i o n s ( m e r i d i a n s
and p a r a l l e l s ) :

w h e r e dp i s t h e i n c r e a s e i n t h e r a d i u s of t h e u n r o l l i n g , i . e . , a

p o s i t i v e i n c r e a s e i n l a t i t u d e ( $ ) c o r r e s p o n d s t o a n e g a t i v e i n c r e a s e /41
in the radius (p).
Integrating the latter, we obtain:
c

m=

or

+

Rd'p
Rdv sec2 'p

-

1
L=ec2,9

m coset2 'p;
(1.27)

Here, r = R cos $ i s t h e r a d i u s of t h e p a r a l l e l , i . e . ,
n = cosec 'p..

(1.28)

a > P l a n e o f
the Figure

90'

Fig. 1.27.
C e n t r a l P o l a r (Gnomonic) P r o j e c t i o n :
( a ) Position of
t h e P l a n e o f t h e F i g u r e ; ( b ) Appearance of t h e P r o j e c t i o n .

Translator's note:
36

cosec = c s c .

_

Therefore, t h e p r o j e c t i o n i s n o t isogonal (m # n), n o t e q u a l l y
s p a c e d ( m # 1 a n d n # 11, a n d n o t e q u a l l y l a r g e ( m n # 1 ) .
Although t h e p r o j e c t i o n i s n o t i s o g o n a l , t h e orthodrome on it
i s r e p r e s e n t e d by a s t r a i g h t l i n e .
T h i s remarkable p r o p e r t y i s ex­
p l a i n e d by t h e f a c t t h a t t h e p l a n e o f t h e c i r c u m f e r e n c e o f a g r e a t
c i r c l e ( p l a n e of t h e o r t h o d r o m e ) a l w a y s p a s s e s t h r o u g h t h e c e n t e r
o f t h e E a r t h , which i n t h i s case a p p e a r s q s t h e c e n t e r of t h e p r o ­
jection, while the intersection of the plane of a great circle with
t h e plane of t h e f i g u r e is a s t r a i g h t l i n e .
S i n c e t h e p r o j e c t i o n i s n o t i s o g o n a l , t h e moving a z i m u t h o f
t h e o r t h o d r o m e o n i t , i f i t i s n o t e q u a l t o 0 , 1 8 0 , 9 0 , or 2 7 0 ° ,
d o e s n o t c o r r e s p o n d t o t h e a z i m u t h on t h e E a r t h ' s s u r f a c e .

­
/42

D i s t o r t i o n o f d i r e c t i o n s o n t h e map w i l l b e e q u a l :
(1.29)
while it i s p o s s i b l e t o c a l c u l a t e t h e a c t u a l d i r e c t i o n of t h e ortho­
drome a t t h e l o c a t i o n a n a l o g o u s l y w i t h t h e a i d o f t h e m e a s u r e d a n g l e
o n t h e map:
(1.30)


w h e r e B i s t h e m e a s u r e d a n g l e o n t h e map o f a g i v e n p r o j e c t i o n , c1
i s t h e c o r r e s p o n d i n g a n g l e i n a l o c a t i o n , and @ i s t h e l a t i t u d e of
t h e f i n a l point of t h e orthodrome.
The d i s t o r t i o n s o f d i r e c t i o n s a n d d i s t a n c e s on t h i s p r o j e c t i o n
are g r e a t .
I n t h i s c o n n e c t i o n , it i s i m p o s s i b l e t o u s e a p r o t r a c t o r
t o m e a s u r e t h e d i r e c t i o n s a n d a s c a l e t o m e a s u r e t h e d i s t a n c e s on
t h e map w i t h o u t c o r r e s p o n d i n g c o r r e c t i o n s .
A c e n t r a l p o l a r p r o j e c t i o n i s u s e d for c o n s t r u c t i n g g n o m o n i c
systems, while the regularity i n the distortion of directions is
u s e d for c a l c u l a t i n g t h e n o m o g r a m s o f t h e o r t h o d r o m e d i r e c t i o n .

T h e g n o m o n i c s y s t e m a n d t h e nomogram o f t h e o r t h o d r o m e d i r e c ­
t i o n can be used f o r t h e g r a p h i c (approximate) c a l c u l a t i o n of t h e
length of t h e orthodrome, t h e coordinates of i t s intermediate p o i n t s ,
and t h e d i r e c t i o n .
The l o x o d r o m e a n d o t h e r l i n e s o f p o s i t i o n o f
t h e g i v e n p r o j e c t i o n a r e r e p r e s e n t e d by complex c u r v e s .
The p r o p e r t y o f o r t h o d r o m i c i t y o f a c e n t r a l p o l a r p r o j e c t i o n
h a s been used f o r t h e p u b l i s h i n g o f o b l i q u e c e n t r a l p r o j e c t i o n s
which have been used a t r a d i o g o n i o m e t r i c p o i n t s i n c i v i l a v i a t i o n .
The m i d d l e o f t h e base ( t h e m i d d l e o f t h e o r t h o d r o m e d i s t a n c e b e ­
t w e e n two r a d i o g o n i o m e t e r s ) w a s t a k e n as t h e p o i n t o f t a n g e n c y o f
t h e p l a n e o f t h e f i g u r e o f s u c h maps.
I n t h i s case, t h e c o o r d i n a t e s

37

o f t h e p o s i t i o n o f t h e a i r c r a f t a r e v e r y e a s i l y d e f i n e d as t h e i n ­
t e r s e c t i o n o f t w o s t r a i g h t orthodrome b e a r i n g s ( l i n e s ) extended
from t h e radiogoniometers.
Maps o f t h e d i f f e r e n t i a l r a n g e f i n d i n g ( h y p e r b o l i c ) s y s t e m o f
l o n g - r a n g e n a v i g a t i o n ( D S L N - 1 ) a r e made o n s u c h a p r o j e c t i o n , s i n c e
t h e s p h e r i c a l h y p e r b o l a on t h e p r o j e c t i o n i s a l s o e x p r e s s e d by a
hyperbola.

Equally spaced azimuthal ( c e n t r a l ) p r o j e c t i o n
T h i s p r o j e c t i o n i s c o n s t r u c t e d by c a l c u l a t i n g and t r a n s f o r m i n g
conventional meridians ( r a d i i ) t o f u l l s i z e , equal t o the principal
scale t r a n s f e r r e d from t h e g l o b e .
The p r o j e c t i o n i s u s e d o n l y f o r
t h e p u b l i c a t i o n o f s p e c i a l s m a l l - s c a l e maps ( 1 : 4 0 , 0 0 0 , 0 0 0 ) , w h i c h
a r e u s e d a s r e f e r e n c e maps f o r m e a s u r i n g d i s t a n c e s f r o m a c e n t r a l
p o i n t o n t h e map.
U s u a l l y a l a r g e a d m i n i s t r a t i v e or a v i a t i o n c e n t e r , f r o m w h i c h
i t i s n e c e s s a r y t o know t h e s h o r t e s t o r t h o d r o m e d i s t a n c e i n a n y a i ­
r e c t i o n t o a g i v e n p o i n t o n a map, i s c h o s e n as t h e p o i n t o f t a n ­

­
/43

gency o f t h e p l a n e o f t h e f i g u r e o f t h e p r o j e c t i o n .
I n such a pro­
j e c t i o n , f o r e x a m p l e , a map i s c o n s t r u c t e d w i t h t h e p o i n t o f t a n ­
g e n c y a t Vnukovo A i r p o r t , w i t h c i r c l e s p l o t t e d a t e q u a l d i s t a n c e s
from t h e a i r p o r t .
The g e o g r a p h i c m e r i d i a n s a n d p a r a l l e l s a r e r e p r e ­
s e n t e d by complex c u r v e s .
This does not allow t h e d i r e c t i o n s t o be
measured.

Stereographic polar projection
The c e n t e r o f p r o j e c t i o n i n a s t e r e o g r a p h i c p o l a r p r o j e c t i o n

Fig.

1.28.
Stereographic Polar Projection:
( a ) Position of the
P o i n t o f P r o j e c t i o n ; ( b ) Appearance o f t h e P r o j e c t i o n .

38

---I.-=-11=-

..1111.--111-111

I I III

111111 1 1 1 1 1 1 1 1 . 1 1 1 1

11-11

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 II”

-

I


i s s e p a r a t e d from t h e p o i n t of tangency o f t h e p l a n e of t h e f i g u r e
by two r a d i i o f t h e g l o b e a t t h e p o i n t B ( F i g . 1 . 2 8 , a ) .
H e r e t h e a n g l e 8 = 90° - 9 , w h i l e t h e a n g l e

An e q u a t i o n o f t h e p r o j e c t i o n c a n b e d e r i v e d f r o m e q u a t i o n s o f
t h e e l e m e n t s shown i n F i g u r e 1 . 2 8 .
ti = A;

p

e
= 2R tg -

2 '

The m e r i d i a n s i n t h e p r o j e c t i o n a r e s t r a i g h t l i n e s w h i c h d i ­
v e r g e r a d i a l l y from t h e p o l e ( F i g . 1 . 2 8 , b ) , and from t h e p o i n t o f
tangency of t h e plane o f t h e paper a t an angle equal t o t h e d i f f e r ­
ence i n longitude:
6 = Ax.
The p a r a l l e l s a r e c o n c e n t r i c c i r c l e s , whose r a d i i a r e p r o p o r t i o n a l t o t h e tangent of t h e l a t i t u d e .

/44

The s p e c i a l s c a l e a l o n g t h e p a r a l l e l i s d e t e r m i n e d by t h e e q u a ­
tion

H e r e C$ = 9 0 °

-

8 ,

while after integration
m=

1
cos2

e

2

= sec

26
2 -

(1.30)

The s p e c i a l s c a l e a l o n g t h e p a r a l l e l i s d e t e r m i n e d by t h e e q u a ­
tion
2R tgodA

but cos 9 = cos (90 - 9)

= s i n 8 , so t h a t
2 tg-

e

e
2
n = -= SecZsin 6
2
i.e.

,

a
2

e

m = n= s e d - = secz
2

(+)90-

,

(1.31)

.
39

ip

I 1111111 II I.
1
I III.IIIIIIIIII.~111111111111111

11111 1111

11111111 I 1111 111.1111-~-.~-I

-1

T h e r e f o r e , t h i s p r o j e c t i o n i s i s o g o n a l ( m = n), b u t n o t e q u a l l y
s p a c e d ( m # 1 a n d n # 1) or e q u a l l y l a r g e ( m n # 1 ) .
On maps o f a s t e r e o g r a p h i c p r o j e c t i o n , a c i r c l e d r a w n o n t h e
g l o b e i s r e p r e s e n t e d by a c i r c l e on t h e p l a n e ( m a p ) ; h o w e v e r , t h e
c e n t e r of t h i s c i r c l e does not coincide with t h e projection of t h e
c e n t e r o f t h e c i r c l e on t h e g l o b e .
T h i s makes t h e p r o j e c t i o n i n ­
e f f e c t i v e f o r use i n rangefinding systems, since l i n e s of equal
l e n g t h w i l l b e r e p r e s e n t e d by e c c e n t r i c c i r c l e s .
The b a s i c a d v a n t a g e o f t h e p r o j e c t i o n i s i t s i s o g o n a l i t y a n d
i n s i g n i f i c a n t d i s t o r t i o n of lengths i n the p o l a r regions.
There­
f o r e , i t h a s b e e n u s e d f o r p u b l i s h i n g maps o f t h e p o l a r r e g i o n s
with s c a l e s of 1:2,000,000,
1:3,000,000,
and 1:4,000,000,
which a r e
u s e d f o r g e n e r a l o r i e n t a t i o n and a p p r o x i m a t e d e t e r m i n a t i o n o f t h e
p o s i t i o n o f a n a i r c r a f t by m e a n s o f r a d i o d e v i c e s .
The maximum d i s t o r t i o n o f l e n g t h s a t 7 0 ° l a t i t u d e d o e s n o t e x ­
c e e d 3 % ( 3 km i n 1 0 0 k m ) , w h e r e a s i f t h e p l a n e o f t h e f i g u r e i s i n ­
t e r s e c t e d ( f o r e x a m p l e , a t 70° l a t i t u d e ) , t h e d i s t o r t i o n o f t h e
l e n g t h s a t t h e p o l e s d o e s n o t e x c e e d 3% ( a n d a t 60° l a t i t u d e , 4 % ) .
T h e o r t h o d r o m e on maps o f a s t e r e o g r a p h i c p r o j e c t i o n h a s a n
i n s i g n i f i c a n t bend toward t h e e q u a t o r and i s c o n s t r u c t e d i n p r a c ­
t i c e as a s t r a i g h t l i n e .
The l o x o d r o m e i s r e p r e s e n t e d by a l o g a r i t h m i c s p i r a l .
It is
p o s s i b l e t o c o n t i n u e it ( j u s t as i n c o n i c p r o j e c t i o n s ) a l o n g t h e
f l i g h t a n g l e , which i s measured i n t h e middle o f t h e p a r t o f t h e
s t r a i g h t l i n e j o i n i n g t h e c o n t r o l ( r o t a t i n g ) p o i n t s of t h e f l i g h t
path.
I n d e t e r m i n i n g t h e p o s i t i o n o f a n a i r c r a f t by means o f an a i r ­
c r a f t r a d i o compass, a c o r r e c t i o n f o r t h e a n g l e of convergence o f
the meridians i s allowed according t o t h e formula

w h e r e A, a n d A, a r e t h e l o n g i t u d e o f t h e r a d i o s t a t i o n a n d t h e a i r ­
craft , respectively.
On maps o f a s t e r e o g r a p h i c p r o j e c t i o n , i n o r d e r t o f a c i l i t a t e
determining d i r e c t i o n s i n t h e p o l a r regions according t o a sugges­
t i o n by V . I . A k k u r a t o v , a s u p p l e m e n t a r y s y s t e m o f " a r b i t r a r y " mer­
i d i a n s ( F i g . 1 . 2 8 , b ) p a r a l l e l t o t h e Greenwich m e r i d i a n ( A = O o )
and perpendicular t o it ( A = 90°) i s p l o t t e d .
Then t h e t r u e G r e e n ­
wich f l i g h t a n g l e w i l l e q u a l :

where TFAar i s a n a r b i t r a r y f l i g h t a n g l e m e a s u r e d f r o m a d i r e c t i o n
p e r p e n d i c u l a r t o t h e G r e e n w i c h m e r i d i a n ( A = g o o ) ; A e = 90° i s t h e
40

/45
-

l o c a t i o n of t h e r o u t e ( p a r t o f t h e p a t h ) t o t h e e a s t o f t h e Green­
wich m e r i d i a n ; and Xw = 270° i s t h e l o c a t i o n o f t h e r o u t e ( p a r t o f
t h e p a t h ) t o t h e w e s t o f t h e Greenwich m e r i d i a n .

Nomenclature o f Maps
A t t h e p r e s e n t t i m e , a map w i t h a s c a l e o f 1:1,000,000 (1 c m =
1 km) w h i c h i s e x e c u t e d i n a n i n t e r n a t i o n a l p r o j e c t i o n i s c o n s i d e r e d
t h e b a s i c t o p o g r a p h i c a l map o f t h e w o r l d .
As d e s c r i b e d a b o v e , e a c h
s h e e t o f t h i s map e n c o m p a s s e s a t e r r i t o r y w i t h i n t h e l i m i t s o f 4 O
o f l a t i t u d e and 6 O o f l o n g i t u d e .
T h i s h a s made i t p o s s i b l e t o com­
p i l e an i n t e r n a t i o n a l d e s i g n a t i o n f o r t h e s h e e t s o f maps.

For t h e p u r p o s e o f q u i c k l y c h o o s i n g a g i v e n s h e e t o f a m a p ,
e a c h o f them b e a r s a d e s i g n a t i o n o f i t s r a n k i n a d e f i n i t e s y s t e m .
T h i s d e s i g n a t i o n i s c a l l e d i n t e r n a t i o n a l map nomenclature.
The s h e e t s a r e s i t u a t e d i n r o w s a l o n g p a r a l l e l s w h i c h r u n f r o m
t h e e q u a t o r t o a l a t i t u d e o f 84O.
There a r e a t o t a l o f 2 1 rows i n
each hemisphere.
E a c h row i s d e s i g n a t e d by a l e t t e r i n t h e L a t i n
alphabet:
A , B , C , D , E , F, G , H , I , J , K , L , M , N , 0 , P , Q , R , S ,
T , U (maps f o r l a t i t u d e s g r e a t e r t h a n 84O a r e c o n s t r u c t e d i n p e r ­
spective projections).
E a c h s h e e t o f a row h a s a n o r d i n a l
i n g o f t h e s h e e t s b e g i n s from t h e 180th
w e s t t o east.
T h e map s h e e t s r e f e r r i n g
meridian from t h e east have t h e o r d i n a l
o f map n u m b e r s a r e o b t a i n e d .

number f r o m 1 t o 6 0 . Countmeridian and proceeds from
t o t h e prime (Greenwich)
number 31.
Thus, columns

To c h o o s e t h e n e c e s s a r y map s h e e t , i t i s n e c e s s a r y t o know t h e
a p p r o x i m a t e c o o r d i n a t e s o f t h e p o i n t f o r whose r e g i o n t h e s h e e t i s
selected.

For e x a m p l e :

the point

c o o r d i n a t e s l a t i t u d e 50°

N,

longitude

69O E .

L e t u s d i v i d e t h e l a t i t u d e o f t h e p o i n t by 4 , and w e w i l l ob­
t a i n t h e n e c e s s a r y row o f map s h e e t s :
50 +
> 12.
Therefore, the
map s h e e t i s l o c a t e d i n t h e t h i r t e e n t h r o w , w h i c h i s d e s i g n a t e d b y
t h e l e t t e r M.
J+

L e t u s d i v i d e t h e l o n g i t u d e o f t h e p o i n t by 6 , a n d w e w i l l o b ­
tain:
6 9 + 6 > 11. T h e o r d i n a l n u m b e r o f t h e s h e e t w i l l t h e n b e
30 + 1 2 = 42.

For c o n v e n i e n c e i n s e l e c t i n g map s h e e t s , c o m p o s i t e t a b l e s h a v e
been c o n s t r u c t e d .
T h e s e t a b l e s a r e e x e c u t e d on s m a l l - s c a l e maps
w i t h a s t r a i g h t , e q u a l l y s p a c e d c y l i n d r i c a l p r o j e c t i o n by r u l i n g
t h e i n d i c a t e d map e v e r y 4 d e g r e e s i n l a t i t u d e a n d e v e r y 6 d e g r e e s
i n l o n g i t u d e , w i t h c o r r e s p o n d i n g d e s i g n a t i o n s showing t h e rows and
c o l u m n s o f o r d i n a l n u m b e r s o f t h e maps ( S u p p l e m e n t 1 ) .
41

““4,

i

/46
-

I n a d d i t i o n , o n t h ' e f a c e o f e a c h map s h e e t i s a d i a g r a m s h o w i n g
The
how t h e g i v e n s h e e t f i t s . t o t h e a d j o i n i n g o n e ( F i g ... 1 . 2 9 ) .
s h e e t on which t h i s d i a g r a m i s drawn f i t s i n t h e m i d d l e and i s
shaded.
S h e e t s o f maps w i t h l a r g e r s c a l e s
have s t a n d a r d schemes q f arrangement with­
i n t h e l i m i t s o f a sheet with a scale o f
1:1,000,000.

F o r e x a m p l e , a map s h e e t w i t h a s c a l e
o f 1:1,000,000 c o n t a i n s 4 map s h e e t s w i t h
a scale o f 1:500,000, which are d e s i g n a t e d
by l e t t e r s o f t h e R u s s i a n a l p h a b e t :
A,
B y C y and D .
By a n a n a l o g o u s m e t h o d , t h e d i v i s i o n
o f a map s h e e t w i t h a s c a l e o f 1:1,000,
0 0 0 i n t o s h e e t s w i t h l a r g e r scales i s carried out.
Roman a n d A r a b i c n u m e r a l s a r e
u s e d for t h e i r d e s i g n a t i o n .
H e r e t h e map
nomenclature r e t a i n s t h e designation of t h e s h e e t s i n t h e i n i t i a l
d i v i s i o n , b e g i n n i n g w i t h a s c a l e o f 1:1,000,000 a n d u p ( F i g . 1 . 3 0 ) .
F i g . 1 . 2 9 . Scheme f o r
S p l i c i n g Map S h e e t s
with an I n t e r n a t i o n a l
Projection.

T h e n o m e n c l a t u r e o f map s h e e t s w i t h s m a l l s c a l e s ( 1 : 2 , 0 0 0 , 0 0 0 ,
1 : 2 , 5 0 0 , 0 0 0 , a n d 1:4,000,000) i s n o t i n t e r n a t i o n a l a n d i s e s t a b l i s h e d
when t h e y a r e p r i n t e d i n a c c o r d a n c e w i t h t h e r e g i o n s f o r w h i c h t h e y
a r e p u b l i s h e d a n d i n a c c o r d a n c e w i t h t h e d i m e n s i o n s o f t h e map s h e e t s .
Maps U s e d f o r A i r c r a f t

Navigation

'

/47
-

D e p e n d i n g on t h e n a t u r e o f t h e t a s k s t o b e f u l f i l l e d , i t i s p o s ­
s i b l e t o d i v i d e maps i n t o s e v e r a l g r o u p s a c c o r d i n g t o t h e i r s c a l e s .

1) Maps w i t h d e t a i l e d o r i e n t a t i o n , w i t h s c a l e s o f 1 : 5 0 0 , 0 0 0
a n d u p , a r e u s e d i n c i v i l a v i a t i o n d u r i n g f l i g h t s for s p e c i a l p u r ­
p o s e s ( g e o m a g n e t i c mapping and p h o t o g r a p h y , c h e m i c a l t r e a t m e n t o f
areas, searching f o r small objects i n t h e execution of s p e c i a l t a s k s ,
" j o i n i n g " o f r a d i o e n g i n e e r i n g p r o j e c t s i n a i r p o r t r e g i o n s , compi­
l a t i o n o f diagrams f o r p i e r c i n g c l o u d s , and f o r o t h e r p u r p o s e s ) .
2)
F l i g h t maps w i t h s c a l e s o f 1 : 2 , 0 0 0 , 0 0 0 , 1 : 1 , 0 0 0 , 0 0 0 ,
and
1:500,000 a r e u s e d i n c i v i l a v i a t i o n a s b a s i c f l i g h t m a p s .
Crews
o f l i g h t a i r c r a f t and h e l i c o p t e r s a t c o m p a r a t i v e l y low s p e e d s u s e
w h i l e crews o f h i g h maps w i t h s c a l e s o f 1:1,000,000 a n d 1 : 5 0 0 , 0 0 0 ,
s p e e d a i r c r a f t u s e maps w i t h s c a l e s o f 1 : 2 , 0 0 0 , 0 0 0
a n d 1:1,000,000.
3)
A i r c r a f t maps w i t h s c a l e s o f 1:4,000,000, 1:3,000,000,
1 : 2 , 5 0 0 , 0 0 0 , a n d 1:2,000,000 a r e u s e d i n c i v i l a v i a t i o n for g e n e r a l
o r i e n t a t i o n and p l o t t i n g o f p o s i t i o n l i n e s w i t h t h e a i d o f r a d i o For t h e s e p u r p o s e s , c r e w s
e n g i n e e r i n g and a s t r o n o m i c a l f a c i l i t i e s .
o f l i g h t a i r c r a f t a t l o w s p e e d s a n d h e l i c o p t e r s u s e maps w i t h o n l y
t h e l a s t two s c a l e s .
42

­
/48

S p e c i a l maps w i t h s c a l e s o f 1 : 4 0 , 0 0 0 , 0 0 0 a n d u p ( t o
w i t h s p e c i a l e m p h a s i s on d i f f e r e n t p u r p o s e s o f a p p l i ­
cation:
l i n e s of e q u a l d i s t a n c e from d e f i n i t e p o i n t s , a h y p e r b o l i c
system, azimuths from r a d i o - e n g i n e e r i n g i n s t a l l a t i o n s , etc. are used.
T h e s e i n c l u d e maps w i t h r e f e r e n c e m a t e r i a l s o f s m a l l e r s c a l e s : maps
w i t h t i m e z o n e s , m a g n e t i c d e c l i n a t i o n s , c o m p o s i t e t a b l e s o f map
sheets, etc.
4)

1:2,000,000),

Fig.

1.30.

Scheme for D i v i d i n g a Map S h e e t w i t h a n I n t e r n a t i o n a l
Projection.

A l s o , s p e c i a l f l i g h t maps w i t h s c a l e s o f 1:1,000,000 a n d
w i t h p l o t t e d and marked f l i g h t r o u t e s a r e p u b l i s h e d f o r
c i v i l aviation.
A s a r u l e , t h e y a r e c o m p i l e d on o b l i q u e c y l i n d r i c a l
or o b l i q u e c o n i c p r o j e c t i o n s , w i t h t h e l e a s t d i s t o r t i o n s o f a n g l e s
and l e n g t h s along t h e r o u t e .
T h e o r t h o d r o m e o n s u c h maps i s p r a c ­
tically a straight line.
1:2,000,000

T h e c o n t e n t s o f a map d e p e n d o n i t s s c a l e , t h e a e r o g r a p h i c f e a ­
t u r e s of t h e r e g i o n s f o r which it i s compiled, and t h e purpose o f
t h e map.

43

On maps o f a l l s c a l e s , t h e f o l l o w i n g a r e d r a w n i n some k i n d o f
detail:

(a) relief;

(b)
hydrography ( s e a s , r i v e r s , l a k e s ) ;

(c)
populated p o i n t s ;

(d)
network o f r a i l r o a d s , highways, and country r o a d s ;

(e)
v e g e t a t i o n or g r o u n d c o v e r ( l a r g e f o r e s t s , meadow, swamp,

sand, desert, etc.);
(f)
i s o l i n e s o f m a g n e t i c d e c l i n a t i o n s and m a g n e t i c a n o m a l i e s .
The l e g e n d s o f t h e i n d i c a t e d e l e m e n t s a r e u s u a l l y e x e c u t e d on
t h e maps a t t h e l o w e r e d g e o f t h e s h e e t .
On m a p s ,

a relief

i s e x p r e s s e d by t h r e e m e t h o d s :

1) I t i s e x p r e s s e d b y i s o l i n e s o f e q u a l h e i g h t on t h e s u r f a c e
o f t h e r e l i e f ( h o r i z o n t a l s ) , i . e . , l i n e s formed a t t h e i n t e r s e c t i o n
of a r e l i e f w i t h h o r i z o n t a l p l a n e s which are s i t u a t e d one above t h e
o t h e r , w i t h h e i g h t i n t e r v a l s d e p e n d i n g on t h e s c a l e o f t h e map; t h e
h e i g h t o f t h e h o r i z o n t a l s a b o v e s e a l e v e l i s d e s i g n a t e d by numbeps.
2)
I t i s e x p r e s s e d by l a y e r e d c o l o r i n g ; a s p e c i a l c o l o r d e s i g ­
n a t e d on a s p e c i a l ( h y p s o m e t r i c ) s c a l e o n t h e l o w e r e d g e o f t h e map
i s a s s i g n e d t o each i n t e r v a l of r e l i e f h e i g h t .
3)
I t i s e x p r e s s e d by brown s h a d i n g , ; . e . ,
by s p e c i a l c o l o r i n g
w i t h t h i c k e n i n g of b r o w n i n t h e h i g h e s t a r e a s o f t h e r e l i e f a n d t h e
steepest slopes.
This use of color gives a n a t u r a l , volumetric idea
of t h e nature of t h e r e l i e f .

I n a d d i t i o n t o t h e a b o v e methods o f r e p r e s e n t i n g r e l i e f on maps,
m a r k s o f command h e i g h t s ( w h i c h e x c e e d n e i g h b o r i n g h e i g h t s ) , w i t h a n
i n d i c a t i o n o f t h e h e i g h t o f t h e s e p o i n t s above s e a l e v e l , a r e shown.
H y d r o g r a p h y i s shown on m a p s b y a b l u e
p e n d s on t h e s c a l e a n d p u r p o s e o f t h e map.

color.

I t s d e t a i l de­

P o p u l a t e d p o i n t s , d e p e n d i n g o n t h e s c a l e o f t h e map a n d t h e
a r e a l d i m e n s i o n s o f t h e p o i n t s , a r e r e p r e s e n t e d b y c o n t o u r s or c o n ­
v e n t i o n a l s y m b o l s i n a c c o r d a n c e w i t h t h e p o i n t ' s d i m e n s i o n s or i t s
population.
In l i g h t l y populated areas, a l l populated points a r e designated.
On s m a l l - s c a l e maps o f d e n s e l y p o p u l a t e d a r e a s , some o f t h e p o i n t s
a r e omitted.
T h e n u m b e r o f p o i n t s d r a w n d e p e n d s on t h e s c a l e o f
t h e map a n d t h e p o p u l a t i o n d e n s i t y o f t h e a r e a .
The d e t a i l o f t h e h i g h w a y n e t w o r k d e p e n d s o n i t s d e n s i t y , t h e
v e g e t a t i v e or g r o u n d c o v e r , a n d t h e s c a l e o f t h e map a n d i t s p u r p o s e .
B e s i d e s t h e above g e n e r a l c o n t e n t s o f maps,

44

specially prepared

/49

f l i g h t maps r e p r e s e n t a n a v i g a t i o n a l s i t u a t i o n , ; . e . ,
t h e arrange­
ment o f r a d i o - e n g i n e e r i n g f a c i l i t i e s f o r a i r c r a f t n a v i g a t i o n , p o s i ­
t i o n l i n e s o f a i r c r a f t , a n d s p e c i a l m a r k i n g s f o r n a v i g a t i o n a l meas­
u r e m e n t s and c a l c u l a t i o n s a r e shown.
On s o m e f o r m s o f s p e c i a l l y p r e p a r e d m a p s ( m a p - d i a g r a m s ) , s o m e
t h e e l e m e n t s o f t h e g e n e r a l c o n t e n t s a r e o m i t t e d or s i m p l i f i e d
for t h e p u r p o s e o f a m o r e d e t a i l e d a n d g r a p h i c r e p r e s e n t a t i o n o f
the navigational situation.
of

6.

M e a s u r i n g D i r e c t i o n s and D i s t a n c e s on t h e E a r t h ' s S u r f a c e
Orthodrome on t h e E a r t h ' s S u r f a c e

In the practice of aircraft navigation at the present t i m e ,
o r t h o d r o m e d i r e c t i o n i s t h e main and most w i d e s p r e a d d i r e c t i o n .

an

I n o r d e r t o e x p l a i n a l l t h e problems connected with measuring
moving a n g l e s , d i s t a n c e s , and c o o r d i n a t e s i n f l i g h t a l o n g a n o r t h o ­
d r o m e , l e t u s e x a m i n e a n o r t h o d r o m e on t h e E a r t h ' s s u r f a c e ( F i g .
1.31).

An o r t h o d r o m e , i n g e n e r a l , l i e s a t a n a n g l e t o t h e E a r t h ' s
e q u a t o r and i n t e r s e c t s it a t two p o i n t s , t h e d i s t a n c e between which
( a l o n g t h e a r c o f t h e e q u a t o r ) i s e q u a l t o 180O.
Only t h e e q u a t o r ,
which l i k e w i s e a p p e a r s as an o r t h o d r o m e , i s a n e x c e p t i o n .
I n F i g u r e 1 . 3 1 , a a n d b , l i n e X O M ~i s t h e a r c of t h e e q u a t o r ,
l i n e X O M i s t h e o r t h o d r o m e e x a m i n e d b y u s ; p o i n t s 10 a n d X O + 1 8 0
are t h e p o i n t s of i n t e r s e c t i o n of t h e orthodrome with t h e equator;
PNXOP, i s t h e m e r i d i a n o f t h e p o i n t o f i n t e r s e c t i o n o f t h e o r t h o ­
drome w i t h t h e e q u a t o r ; P H Y M I P s i s t h e m e r i d i a n o f t h e p o i n t M on

Fig. 1.31.
Orthodrome on t h e E a r t h ' s S u r f a c e :
(a) P o s i t i o n of t h e
Orthodrome on a S p h e r e ; ( b ) R e l a t i o n s h i p b e t w e e n L o n g i t u d e s a n d L a t ­
i t u d e s o f P o i n t s on t h e Orthodrome.
45

I

/50
-

t h e o r t h o d r o m e ; 90° - a0 i s t h e a n g l e b e t w e e n t h e p l a n e o f t h e e q u a ­
t o r and t h e p l a n e of t h e o r t h o d r o m e ; X i s t h e 1ondigu.de o f t h e p o i n t
M ; (I i s t h e l a t i t u d e o f t h e p o i n t M ; - B , + B a r e p o i n t s o n t h e o r t h o ­
d r o m e o f maximum l a t i t u d e , w h i c h a r e c a l l e d v e r t e x p o i n t s .

L e t us e r e c t a normal t o t h e p l a n e
of t h e equator a t p o i n t M1 ( s e e Fig.
1 . 3 1 , b ) and e x t e n d it t o an i n t e r s e c t i o n
w i t h t h e v e r t i c a l o f p o i n t M o n t h e or­
thodrome ( p o i n t M2).
I t is obvious t h a t
t h e t r i a n g l e 0 M1M2 w i l l be a r i g h t t r i ­
angle.
H e r e MlM, w i l l b e t h e t a n g e n t
line of the angle 4 .

Xo

Fig. 1.32. Determining
D i s t a n c e on a n O r t h o ­

drome.

L e t us d r o p from p o i n t s MI and M 2 ,
perpendiculars t o t h e aperture axis of
t h e o r t h o d r o m e w i t h t h e e q u a t o r 1 0 0 . One
o f them w i l l l i e i n t h e p l a n e o f t h e
equator, t h e second i n t h e plane of t h e
orthodrome; b o t h w i l l converge a t one
p o i n t on t h e a p e r t u r e a x i s ( p o i n t N ) .

I t i s obvious t h a t l i n e M1N w i l l be

a l i n e of t h e s i n e of angle A , w h i l e an­
g l e M1NM2 w i l l b e t h e a p e r t u r e a n g l e o f t h e p l a n e o f t h e e q u a t o r
Here t h e t r i a n g l e
w i t h t h e p l a n e o f t h e o r t h o d r o m e (90° - a o ) .
NM1M2 w i l l a l s o b e a r i g h t t r i a n g l e .

T h u s , for p o i n t M a n d for a n y p o i n t o n t h e o r t h o d r o m e ,
following equation w i l l be v a l i d :

the

­
sln A

tg(90"-uo) = '8 'p

-

(1.32)
F o r m u l a ( 1 . 3 2 ) i s v a l i d o n l y f o r c a s e s when t h e p o i n t A 0 i s
the point of origin of the longitude.
When t h e l o n g i t u d e o f t h e
p o i n t i s n o t e q u a l t o z e r o , t h e l o n g i t u d e o f t h e p o i n t A0 must be
s u b t r a c t e d from t h e l o n g i t u d e o f t h e p o i n t M(AM), ? . e . , t h e r e f e r ­
ence system o f l o n g i t u d e s must be r e d u c e d t o t h i s p o i n t .
Then

I n t h e f u t u r e , f o r t h e s a k e o f s i m p l i c i t y , we w i l l c o n s i d e r
t h e l o n g i t u d e o f A0 e q u a l t o z e r o .
I t i s p o s s i b l e t o d e t e r m i n e t h e moving a z i m u t h a c c o r d i n g t o
t h e formula
tg a -- tgq, sec A sec '9,
(1.33)

46

C o n s i d e r i n g t h a t t g a0 =
t h e form:

sin A
, it
tg 4
tg a =,tg

A cosec 'p

ctg a = ctg h sifi y.

or
of

is possible t o reduce (1.33) t o

Formulas (1.33)
(1.32).
Since the r a t i o

are o b t a i n e d by d i f f e r e n t i a t i o n

and (1.33a)

sin A
tg 4

(1.33a)

= t g a0 = c o n s t r e m a i n s v a l i d f o r e v e r y

l e n g t h o f an o r t h o d r o m e , i t i s o b v i o u s t h a t t h e e l e m e n t a r y d i f f e r ­
ence q u o t i e n t s i n X and t g
w i l l a l s o be constant f o r every length
of an orthodrome and w i l l e q u a l :

+

sin h
=
d

tg a0 = const.

d t g 'p

Therefore,

it i s p o s s i b l e t o w r i t e ( 1 . 3 2 ) i n t h e form:
dsinh
dh

_-- dk - tg 4,
d'p
I

dT

wh e n c e

cos1
-,-­
sec2~

dh
dy

- tgao

or

On t h e E a r t h ' s s u r f a c e , t h e l i n e a r s c a l e o f l o n g i t u d e i s e q u a l
t o t h e l i n e a r s c a l e of l a t i t u d e m u l t i p l i e d by t h e c o s i n e o f l a t i ­
tude.
T h e r e f o r e , t h e t a n g e n t of t h e moving a z i m u t h of t h e o r t h o ­
d h , d i v i d e d by t h e co­
drome w i l l b e e x p r e s s e d b y t h e d e r i v a t i v e
3
s i n e of t h e l a t i t u d e :
dh

-

tga=-

d?
COS 'p

or, c o n s i d e r i n g t h a t

tre a r r i v e a t ( 1 . 3 3 a )

=o
- COS tga COS^
'p

sin h
tg%=-

:

tgcp

tg a = tg h cosec y.

I n t h e p r a c t i c e o f a i r c r a f t n a v i g a t i o n , it i s u s u a l l y n e c e s ­
s a r y t o d e a l w i t h two p o i n t s on t h e E a r t h ' s s u r f a c e .
With t h e ex­
c e p t i o n o f s p e c i a l c a s e s , n e i t h e r o f t h e m i s on t h e e q u a t o r .
F o r m u l a s ( 1 . 3 2 ) a n d ( 1 . 3 3 ) c a n b e u s e d o n l y i n t h o s e c a s e s when
t h e p o i n t of i n t e r s e c t i o n o f a n o r t h o d r o m e w i t h t h e e q u a t o r ( i . e . ,

47

t h e l o n g i t u d e o f a p o i n t on t h e o r t h o d r o m e ,
e q u a l t o z e r o ) i s known.

t h e w i d t h of which i s

Elements of a S p h e r i c a l T r i a n g l e .
( a ) T r i a n g l e on a
Fig. 1.33.
S p h e r e ; ( b ) R e l a t i o n s h i p o f Angles and S i d e s of a S p h e r i c a l T r i a n g l e .

L e t u s d e r i v e a n e q u a t i o n which makes i t p o s s i b l e t o d e t e r m i n e
t h e c o o r d i n a t e X o f a g i v e n p o i n t on an o r t h o d r o m e on t h e b a s i s o f
t h e c o o r d i n a t e s o f t w o known p o i n t s o n i t .

L e t u s assume t h a t w e have two a r b i t r a r y p o i n t s on t h e E a r t h ' s
We w i l l t a k e t h e d i f f e r ­
s u r f a c e w i t h c o o r d i n a t e s 4111 a n d 4 2 1 2 .
Then,
e n c e of t h e l o n g i t u d e s o f t h e s e p o i n t s a s A A ( A A = A2 - A I ) .
according t o (1.32a),

Transforming t h e right-hand
sin (?I

-4) -

sln (Al

s i d e of t h i s e q u a t i o n , w e o b t a i n :

- ),A

k 'PI

cos AX

+ cos ((Ai-

A,) sin AX

tg '92

-

A,)

sin A),

--­

D i v i d i n g b o t h s i d e s o f t h e e q u a t i o n by s i n ( A 1
i n g by t g 4 2 , w e w i l l have:
tg h -tg 'PI

from which

ctg (AI

- b)COS AA + COS (A, -4)
sin (Al - A,)
= cos Ah 4- ctg (Al - A,) sin LA,

sif(hi

- no) = tg 'pz ctg

cosec AA

-ctgdk.

and m u l t i p l y ­

(1.34)

E q u a t i o n (1.341, w h i c h m a k e s i t p o s s i b l e t o d e t e r m i n e t h e
l o n g i t u d e of t h e p o i n t of i n t e r s e c t i o n of an orthodrome with t h e
is very important.
Knowledge o f t h i s c o o r d i n a t e
equator (A,),
makes it p o s s i b l e t o c a l c u l a t e e a s i l y a l l t h e r e m a i n i n g e l e m e n t s
of t h e orthodrome.

/53
-

Having s u b s t i t u t e d t h e v a l u e X i n (1.34) f o r t h e v a l u e ( X I as b e f o r e , and s u b s t i t u t i n g i n t o ( 1 . 3 3 a ) t h e v a l u e c t g X f r o m
( 1 . 3 4 1 , we o b t a i n t h e f o l l o w i n g e q u a t i o n f o r a p o i n t w i t h t h e c o o r ­
d i n a t e s 1$1X1:

XO),

ctg a = t g '92 cos '91 cosec Ah

-ctg Ah sin 71.

(1.35)

F o r m u l a ( 1 . 3 5 ) i s u s u a l l y u s e d for c a l c u l a t i n g t h e a z i m u t h o f
an orthodrome a t t h e i n i t i a l p o i n t of t h e s t r a i g h t - l i n e segment o f
t h e p a t h when t h e r e i s n o n e c e s s i t y f o r d e t e r m i n i n g t h e r e m a i n i n g
elements of t h e orthodrome.
I n g e n e r a l , i t i s b e t t e r t o s o l v e ( 1 . 3 4 ) i n d e p e n d e n t l y , and
t h e n f i n d t h e s o l u t i o n by s u b s t i t u t i n g X i n t o ( 1 . 3 2 ) and ( 1 . 3 3 ) .
Simple t r a n s f o r m a t i o n s o f

(1.32)

r e d u c e t o f o r m u l a s w h i c h make

it p o s s i b l e t o d e t e r m i n e t h e c o o r d i n a t e s o f i n t e r m e d i a t e p o i n t s o n
an orthodrome :
tg 'p = sin h ctguo,
sin A = tg 'p t g a p

(1.36)
(l.36a)

Given t h e a r b i t r a r y v a l u e o f a p o i n t c o o r d i n a t e on t h e o r t h o ­
d r o m e $ or A , i t i s p o s s i b l e t o o b t a i n t h e v a l u e o f t h e s e c o n d c o o r ­
d i n a t e o f t h i s p o i n t on t h e b a s i s o f t h e s e f o r m u l a s .
T h e f o r m u l a s f r o m ( 1 . 3 2 ) t h r o u g h ( 1 . 3 6 1 , g i v e n b y u s , make i t
p o s s i b l e t o d e t e r m i n e t h e i n i t i a l and moving a z i m u t h s o f t h e o r t h o ­
drome, and a l s o t h e c o o r d i n a t e s o f i t s i n t e r m e d i a t e p o i n t s .
I n o r d e r t o d e t e r m i n e t h e l e n g t h o f t h e o r t h o d r o m e or d i s t a n c e s
a l o n g i t (S) l e t u s d e r i v e e q u a t i o n s w h i c h c o n n e c t t h e c o o r d i n a t e s
o f t h e p o i n t s of t h e orthodrome with i t s l e n g t h .
I n F i g u r e 1 . 3 2 , t h e t r i a n g l e s ONMl and ON1Mz are s i m i l a r .
The
s t r a i g h t l i n e ON i s a l i n e o f t h e c o s i n e o f t h e a r c A , w h i l e ON' i s
a l i n e o f t h e c o s i n e of a r c 5'.
The h y p o t e n u s e of t r i a n g l e ONMl i s e q u a l t o t h e r a d i u s o f t h e
E a r t h , w h i l e t h e h y p o t e n u s e o f t r i a n g l e ON1M2 i s t h e l i n e of t h e
cosines of arc 4.
Therefore

,

cos s,=cos A cds ,'p.

(1.37)

E q u a t i o n ( 1 . 3 7 ) makes i t p o s s i b l e t o d e t e r m i n e t h e d i s t a n c e
from t h e s t a r t i n g p o i n t o f t h e orthodrome t o any o f i t s p o i n t s w i t h
known c o o r d i n a t e s .

If t h e i n i t i a l p o i n t o f t h e o r t h o d r o m e a n d t h e c o o r d i n a t e s o f
a n y two p o i n t s a l o n g it are known, t h e d i s t a n c e ( S ) b e t w e e n t h e
l a t t e r i s d e t e r m i n e d as t h e d i f f e r e n c e b e t w e e n t h e d i s t a n c e s t o
the i n i t i a l point:
49

­
/54

SI,*= SZ - s1.
I f 2 h e c o o r d i n a t e s o f t h e s t a r t i n g p o i n t a r e n o t known, a n d
t h e n e c e s s i t y f o r determining t h e o t h e r elements of t h e orthodrome
( b e s i d e s t h e d i s t a n c e between t h e two p o i n t s ) i s l a c k i n g , t h e n t h e
i n d i c a t e d d i s t a n c e can b e d e t e r m i n e d by t h e f o r m u l a
cos S = sin 'pl sin 'pz

+ cos 'pl cos 'p2 cos Ak.

(1.38)

F o r m u l a (1.38) i s n o t d e r i v e d f r o m s i m p l e g e o m e t r i c r a t i o s .
it is necessary t o use t h e s p h e r i c a l t r i a n g l e
(PNMaMb) ( F i g . 1 . 3 3 a ) .

For i t s d e r i v a t i o n ,

L e t u s j o i n p o i n t s P N M a a n d Mb b y v e r t i c a l s w i t h t h e c e n t e r o f
t h e E a r t h 0.
L e t u s d r a w t a n g e n t s t o t h e a r c s PNMa a n d PNMb a t t h e
p o i n t P N up t o t h e i n t e r s e c t i o n w i t h t h e i n d i c a t e d v e r t i c a l s a t
p o i n t s Ma1 a n d Mbl ( F i g . 1 . 3 3 , b ) .
We w i l l o b t a i n t w o p l a n e t r i ­
a n g l e s PNMalMbl a c d OMalMbl w i t h t h e common s i d e M a l M b l .
Obviously,
Ma1Mbl=(PNMal) 2 f ( P N M b l ) '-2PNMalPNMb liCOSMalPNMb

A t t h e same t i m e ,

Ma 1 Mb = (OM,

I

+ ( 0 Mb ,I

- 20Ma 1OMb

COS

Ma

OMb 1

(1.39)

S i n c e M a l Mb1 i s t h e common s i d e o f t h e t r i a n g l e , t h e l e f t hand s i d e o f t h e f i r s t e q u a t i o n i s e q u a l t o t h e r i g h t - h a n d s i d e o f
t h e second.
T a k i n g t h e r a d i u s o f t h e E a r t h a s e q u a l t o 1, f r o m t h e r i g h t
t r i a n g l e s O P N M , ~ a n d OPNMbl we f i n d :

PNM,,= tg& Pqqb tg a; OM*, = sec b;
OMb1 = sec a; L MR1
PNMb TP; L Malo% 7p .
S u b s t i t u t i n g t h e i n d i c a t e d v a l u e s i n t o (1.39), w e o b t a i n :
tg2 b+ tg2 a - 2tg b tg a cos P = secz a + secz'b- 2sec a sec b cos p ;
seda=l+t@a;

Therefore,

seSb=l+tgzb

2 tg a tg bcos P = 2-2 Seca secb-cosp.

cos a cos b
2
s i n a sin.bcos P = cos a cos b - cosp

Multiplying both s i d e s of

(1.40) b y

(1.40)
we o b t a i n :

Formula ( 1 . 4 1 ) i s t h e f i r s t b a s i c formula of s p h e r i c a l t r i g o ­
nometry and i s widely used i n aircraft n a v i g a t i o n with t h e use of
astronomical facilities ( t h e remaining formulas of s p h e r i c a l trigo­
nometry a r e given i n Supplement 2 ) .
50

'

I n o u r case,

.

i.e.,

(1.41)

h a s t h e form:
cos S = sin y1 sin yz

i

:

+ cos

'pl

cos 'p2 cos M.

When d e t e r m i n i n g p o i n t c o o r d i n a t e s o f t h e o r t h o d r o m e , t h e r e i s
t h e same n e c e s s i t y t o s o l v e t h e i n v e r s e p r o b l e m s a c c o r d i n g t o t h e
known o r t h o c r o m i c d i s t a n c e ( S ) .
F o r t h i s l e t u s r e t u r n t o F i g u r e 1 . 3 2 , i n which it i s o b v i o u s
t h a t t h e l i n e M N 1 .is t h e l i n e o f t h e s i n e o f t h e a r c S , w h i l e l i n e
MM2 i s e q u a l t o M N 1 c o s " 0 .
A t t h e same t i m e , MM i s t h e l i n e o f
t h e s i n e s f o r arc $.
Therefore,
sln y = sin Scos a,

(1.42)

Formula ( 1 . 4 2 ) makes i t p o s s i b l e t o d e t e r m i n e t h e c o o r d i n a t e
i n i t i a l point.
The c o o r d i n a t e X i n t h i s c a s e i s d e t e r m i n e d a c c o r d i n g t o ( 1 . 3 6 a ) .
$ a l o n g t h e t r a v e r s e d orthodrome d i s t a n c e from t h e

sin h = tg 'p tg ao.

l

I
[

Thus, w e have an a n a l y t i c a l form o f a l l t h e n e c e s s a r y t r a n s ­
f o r m a t i o n s f o r d e t e r m i n i n g t h e e l e m e n t s o f t h e o r t h o d r o m e on t h e
Earth's surface.
However, i n t h e p r a c t i c e o f a i r c r a f t n a v i g a t i o n
it i s s o m e t i m e s more c o n v e n i e n t t o a p p l y o t h e r f o r m u l a s w h i c h d e ­
termine s e p a r a t e elements of t h e orthodrome.
F o r e x a m p l e , i f t h e c o o r d i n a t e s o f t w o p o i n t s on t h e E a r t h ' s
s u r f a c e a n d t h e o r t h o d r o m e d i s t a n c e b e t w e e n t h e m S a r e known, t h e
azimuth of t h e orthodrome ( a ) a t t h e s t a r t i n g p o i n t can be d e t e r mined by t h e f o r m u l a
sin a =

cos 'p2 sin Ah
SihS

*

(1.43)

Formula ( 1 . 4 3 ) can be t r a n s f o r m e d t o determine t h e d i s t a n c e
b e t w e e n p o i n t s a t a known a z i m u t h :
sin S =--

cos 'p2 sin M
sirwr

(1.43a)

I t i s obvious t h a t b o t h formulas are o b t a i n e d from t h e e q u a t i o n
sin S sin 01 = cos 'p2 sln Ah,

i

which i n t u r n i s d e r i v e d by means o f F i g u r e 1 . 3 4 , where l i n e B B ,
i s a p e r p e n d i c u l a r dropped from p o i n t B t o t h e p l a n e of t h e equat o r , and i s a l i n e of t h e s i n e o f t h e l a t i t u d e o f t h i s p o i n t , w h i l e

I

51


/56
­

l i n e B1A2 i s a p e r p e n d i c u l a r dropped from p o i n t B1 t o t h e p l a n e of
t h e m e r i d i a n which p a s s e s t h r o u g h p o i n t A .
Obviously,
A2B1 = cos Q~ sin dA.

Let us e r e c t another perpendicular t o t h e plane of t h e equator
a t p o i n t A 2 ; we w i l l t h e n o b t a i n p l a n e A l A 2 B l B p e r p e n d i c u l a r t o t h e
p l a n e o f t h e e q u a t o r and t h e p l a n e o f t h e m e r i d i a n o f p o i n t A .

Fig. 1.34. Determining S p e c i a l
Elements o f an Orthodrome.

Fig. 1.35. Determining t h e I n i t i a l Azimuth o r t h e V e r t e x o f
an Orthodrome.

I f we r o t a t e t h e i n d i c a t e d p l a n e a r o u n d t h e - l i n e B A 1 i n s u c h a
way t h a t i t r e m a i n s p e r p e n d i c u l a r t o t h e p l a n e o f t h e m e r i d i a n o f
p o i n t A u p t o t h e moment when l i n e A 2 A 1 b e c o m e s p e r p e n d i c u l a r t o
In this
t h e v e r t i c a l o f p o i n t A , t h e d i s t a n c e BA1 w i l l n o t change.
c a s e , s t r a i g h t l i n e BA1 w i l l be t h e l i n e o f t h e s i n e o f t h e a r c A B ,
w h i l e i t s l e n g t h w i l l b e d e t e r m i n e d by t h e f o r m u l a
A1B=

A2B1
­

sina

'

from which it f o l l o w s t h a t
sin S sin a = cos y2 sin dA.

By a s i m i l a r m e t h o d , t h e i n i t i a l a n g l e o f t h e o r t h o d r o m e or
t h e l a t i t u d e of t h e v e r t e x p o i n t i s determined i f t h e azimuth o f
t h e o r t h o d r o m e a t a n y p o i n t on t h e E a r t h ' s s u r f a c e i s known.
I n F i g u r e 1 . 3 5 , a r c A O A B i s t h e e q u a t o r ; a r c AoMB i s t h e o r t h o ­
d r o m e ; l i n e OPN i s t h e a x i s o f t h e E a r t h ; U P 0 i s t h e a x i s o f t h e
o r t h o d r o m e ; M i s a p o i n t on t h e E a r t h ' s s u r f a c e ; and B i s t h e v e r ­
tex point.

52

j

L e t us e r e c t a p e r p e n d i c u l a r PMO t o t h e v e r t i c a l o f p o i n t M
from t h e c e n t e r of t h e E a r t h s o t h a t it i s l o c a t e d i n t h e p l a n e o f
t h e m e r i d i a n o f p o i n t M.
L e t u s a l s o draw a p l a n e p a r a l l e l t o t h e
p l a n e of t h e e q u a t o r t h r o u g h t h e p o l e s P N , Po, and P M . The a n g l e s
P o P ~ Oa n d P M P N O w i l l b e r i g h t a n g l e s , s i n c e l i n e s P,PN a n d P M P N l i e
i n a p l a n e p e r p e n d i c u l a r t o PNO.

­
/57

Angle P o P ~ Oi s a l s o a r i g h t a n g l e , since t h e p l a n e o f t r i a n g l e
h a s a s l o p e t o t h e a x i s p e r p e n d i c u l a r t o PMO a n d p a r a l l e l t o
It i s o b v i o u s t h a t a n g l e P o O P ~i s e q u a l t o t h e l a t i t u d e o f
t h e v e r t e x p o i n t , w h i l e a n g l e PMOPN i s e q u a l t o t h e l a t i t u d e o f
p o i n t M.
Therefore,

POP@
PoPld.
'

opN=op6 cos (PB

oP&fcos TM;
OFB= OP,sin a ,

whence

cos (pB =sin a, = cos yM sin a.

(1.44)

Formula (1.44) is used t o f i n d t h e l a t i t u d e o f t h e v e r t e x p o i n t ,
which a l s o a p p e a r s as a complement o f t h e i n i t i a l a n g l e o f t h e o r ­
The l o n g i t u d e o f p o i n t M r e l a t i v e t o . t h e s t a r t ­
t h o d r o m e u p t o 90°.
i n g p o i n t o f t h e o r t h o d r o m e i n t h i s case c a n b e d e t e r m i n e d b y ( 1 . 3 6 a ) .
W i t h a known a z i m u t h o f t h e o r t h o d r o m e a t a n y p o i n t o n t h e
E a r t h ' s s u r f a c e , t h e c o o r d i n a t e s o f i t s s t a r t i n g p o i n t can be ob­
t a i n e d d i r e c t l y a c c o r d i n g t o ( 1 . 3 3 a ) , from which it f o l l o w s t h a t
tg h = sin? tg a ,

T h e n it i s n o t d i f f i c u l t
thodrome.

(1.45)

t o d e t e r m i n e t h e i n i t i a l a n g l e o f t h e or­

I n some c a s e s , i n o r d e r t o c a l c u l a t e t h e e l e m e n t s o f . t h e o r t h o ­
d r o m e , c o o r d i n a t e s o f t h e v e r t e x p o i n t r a t h e r t h a n of t h e s t a r t i n g
p o i n t a r e used.
I n t h e s e c a s e s , t h e f u n c t i o n s o f t h e a0 a n g l e a r e
r e p l a c e d by i n v e r s e f u n c t i o n s of t h e l a t i t u d e o f t h e v e r t e x p o i n t
which are e q u a l t o them, j u s t as f u n c t i o n s o f l o n g i t u d e are, s i n c e
t h e s e a n g l e s d i f f e r b y 90°.

For e x a m p l e , ( 1 . 3 6 a ) h a s t h e f o r m :
cos AB = tg ? ctg ?E,

w h i l e (1.45) h a s t h e f o r m :
ctg AB = sin 'p tg u.

To e x p l a i n t h e procedure f o r d e t e r m i n i n g a l l t h e elements o f
a n o r t h o d r o m e , l e t u s e x a m i n e ( a s an e x a m p l e ) a n o r t h o d r o m e w h i c h
p a s s e s t h r o u g h two p o i n t s on t h e E a r t h ' s s u r f a c e w i t h t h e s e c o o r ­
M I = l a t i t u d e 60° N , l o n g i t u d e 30° E ; M2 = l a t i t u d e 8 0 ° N ,
dinates:
l o n g i t u d e 40° E .
53

F i r s t w e s h a l l c a r r y o u t t h e g e n e r a l s o l u t i o n of t h e p r o b l e m
o f f i n d i n g t h e e l e m e n t s 'of a n o r t h o d r o m e .
For t h i s w e s h a l l u s e
(1.34).
S u b s t i t u t i n g i n t o t h i s equation t h e f u n c t i o n s o f t h e coord i n a t e s of t h e p o i n t s M I and M2, w e o b t a i n :

-A,,) = tg 80"ctg W."cosec10"- ctg 10";
54671'0*5774--fj,&l = 13,228;
ctg (A, - A,,)
ctg (A,

='

0,1736

The i n i t i a l a z i m u t h o f a n o r t h o d r o m e , a c c o r d i n g t o ( 1 . 3 2 ) ,
w i l l be:

tg cr, = sin A ctg 7.­
L e t us d e t e r m i n e it on t h e b a s i s o f t h e c o o r d i n a t e s o f p o i n t
tg cr, = sin 4"18'.ctg 60"= 0,57A.0,075 ='0.0433;
~ r=, 299'.

According t o ( 1 . 3 3 a ) ,

t h e moving a z i m u t h o f t h e orthodrome ( a )

for p o i n t , M I i s e q u a l t o
ctgu = ctg4°18'.sin600= 13,228.0.868

= i4.455;

u = 5".
The d i s t a n c e f r o m t h e s t a r t i n g p o i n t o f t h e o r t h o d r o m e t o
p o i n t M I , a c c o r d i n g t o (1.371, e q u a l s
COS

Si ='COS 4 " 1 8 ' * ~60'~ = 0,9972.0,5 = 0,4986;
S, = 60"4',

w h i l e t h e d i s t a n c e t o p o i n t M 2 , a c c o r d i n g t o t h e same f o r m u l a , i s
COSS~=COS

14"18'*~0~80"=0,969*0,1736=0,1682;
S2 = 80'17'.

T h e d i s t a n c e b e t w e e n M I a n d M, i s t h e n d e f i n e d a s t h e d i f f e r ­
ence between t h e d i s t a n c e s t o t h e s t a r t i n g p o i n t :
S = S, - Si = 80'17'

-60'04'

=: 20'13'.

C o o r d i n a t e s o f any i n t e r m e d i a t e p o i n t can be determined accord­
ing t o (1.36) or (1.36a).

For e x a m p l e , t h e l o n g i t u d e o f p o i n t M 2 a c c o r d i n g t o i t s l a t i ­
tude is

sin 12 =: tg 80"Ig 2"29' = 5,671.0,043 = 0,246;

A=

54

(A2-

-0)

14'15';

A2 = 39'57'.

/58
­

Thus, a l l t h e necessary elements o f an orthodrome are e a s i l y
determined.
L e t u s now a s s u m e t h a t w e h a d t o d e t e r m i n e o n l y t h e a z i m u t h o f
t h e o r t h o d r o m e a t p o i n t M I . For t h i s w e w i l l u s e ( 1 . 3 5 ) :
ctg a =

0.5-5.671
-0.866.5.671
0,1736
a = 5".

= 11,42251

Knowing t h e a z i m u t h o f t h e o r t h o d r o m e a t o n e o f i t s p o i n t s
makes. i t p o s s i b l e t o d e t e r m i n e t h e d i s t a n c e t o a n y p o i n t b y u s i n g
( l . 4 3 a ) , or i n o u r e x a m p l e :
0,1736*0,1736
sin S
= 0.3456;
0,0872
S = 20'13'.
I

Using t h e azimuth o f t h e orthodrome a t one p o i n t , it is p o s s i ­
b l e t o d e t e r m i n e t h e l a t i t u d e o f t h e v e r t e x p o i n t or t h e i n i t i a l
azimuth of t h e orthodrome according t o (1.44).

For our o r t h o d r o m e , u s i n g p o i n t M I , w e o b t a i n :
sin cr, = cos 60"sin 5"'= 0.5*0,0872=Q;0436;
~ r==,

After t h i s ,
determined.

2'30'.

t h e intermediate p o i n t s o f an orthodrome are e a s i l y /59

Thus, it i s p o s s i b l e t o determine t h e elements of an orthodrome
b e g i n n i n g w i t h t h e d i s t a n c e b e t w e e n t w o p o i n t s a c c o r d i n g t o (1.38),
c h a n g i n g t o t h e m o v i n g a z i m u t h a c c o r d i n g t o (1.431, a n d t h e n t o t h e
l a t i t u d e of t h e vertex point according t o (1.44).
O r t h o d r o m e on T o p o g r a p h i c a l

Maps o f

Different

Projections

L e t u s e x a m i n e a n o r t h o d r o m e o n maps o f a s i m p l e e q u a l l y s p a c e d
c y l i n d r i c a l p r o j e c t i o n , which e s s e n ­
t i a l l y r e p r e s e n t s a geographical coord i n a t e s y s t e m on a s c a l e o f a n g l e s .
A A sinv

/ -

AA

Fig. 1.36. Elementary
Segment o f a n Orthodrome
o n a Map of a C y l i n d r i ­

cal Projection.


draw
tary
shal
tion

T o e x p l a i n t h i s , however, l e t u s
on t h e E a r t h ' s s u r f a c e a n e l e m e n ­
n o r m a l c o n e a t some l a t i t u d e ; w e
l examine it on t h e above p r o j e c (Fig. 1.36).

A s i s a l r e a d y known, t h e r a d i u s

o f t h i s c o n e when u n r o l l e d e q u a l s :


55

or, t a k i n g t h e r a d i u s o f t h e E a r t h a s 1,
Po = ctg 'Po.

According t o
l e l is equal to:

(1.20)

t h e s c a l e of t h e p r o j e c t i o n along t h e p a r a l ­

n=-.


1
cos q

I n o r d e r t o d r a w our u n r o l l e d c o n e o n a c y l i n d r i c a l s u r f a c e ,
it i s n e c e s s a r y t o s t r a i g h t e n t h e cone f i r s t and t h e n e x t e n d i t .
Obviously, t h e segments of t h e meridians remain s t r a i g h t l i n e s dur­
i n g s t r a i g h t e n i n g of t h e cone, b u t t h e y must be u n r o l l e d t o g e t h e r
w i t h t h e s u r f a c e e l e m e n t s t o an a n g l e e q u a l t o Ah s i n $.
L e t u s now d r a w a s t r a i g h t l i n e A B i n t h e e a s t - w e s t
on a n e l e m e n t a r y c o n e .

direction

During s t r a i g h t e n i n g of t h e cone, t h e i n d i c a t e d s t r a i g h t l i n e

w i l l a c q u i r e a c u r v a t u r e , t h e r a d i u s of which w i l l be e q u a l t o t h e
r a d i u s of t h e u n r o l l e d cone (r), b u t curved i n t h e o p p o s i t e d i r e c ­
tion.

Therefore,
P = ~ A ~ B ctg
,

'Pa

D u r i n g e x t e n s i o n o f our c o n e a l o n g a p a r a l l e l t o a s c a l e n z ,
e a c h o f i t s e l e m e n t s ( i n c l u d i n g e l e m e n t s o f our s t r a i g h t l i n e ) w i l l
1
Therefore, t h e radius of t h e
undergo an e x t e n s i o n e q u a l t o cos $
s t r a i g h t - l i n e element w i l l i n c r e a s e and w i l l equal:

.

ctg
r ~ l=~Coscp
l 'p - cosec 'p.

A s is e v i d e n t , t h e s t r a i g h t - l i n e e l e m e n t , s i t u a t e d a l o n g t h e
p a r a l l e l ( i n g e n e r a l , i n a d i r e c t i o n perpendicular t o t h e a x i s o f
The s t r a i g h t - l i n e e l e m e n t s i t ­
the cylinder) acquires a curvature.
uated i n t h e d i r e c t i o n of t h e axis of t h e cylinder does not acquire
a curvature.
Therefore, i f t h e s t r a i g h t - l i n e element is s i t u a t e d
a t an a n g l e t o t h e a x i s o f t h e c y l i n d e r , i t s r a d i u s of c u r v a t u r e
w i l l equal:
ctg '9
'AB, = cos 'p sin a

=1

cosec 'p cosec a..

(1.46)

I n geometry, t h e c u r v a t u r e of a curve i s c o n s i d e r e d t o be a
value inverse t o t h e radius of the curvature.
Therefore, t h e curv­
a t u r e o f our e l e m e n t w i l l e q u a l :
1

=sin
r ~ , ~ ,

56

'p sin

a.

(1.47)

/60

An o r t h o d r o m e o n t h e E a r t h ' s s u r f a c e d o e s n o t h a v e i t s own
c u r v a t u r e o f e a c h e l e m e n t o f a n o r t h o d r o m e o n a map i n a n o r m a l ,
e q u a l l y s p a c e d c y l i n d r i c a l p r o j e c t i o n w i l l b e e x p r e s s e d b y (1.471,
f r o m w h i c h i t i s o b v i o u s t h a t t h e maximum c u r v a t u r e o f t h e o r t h o ­
drome w i l l b e o b s e r v e d a t i t s v e r t e x p o i n t s , w h i l e t h e s t a r t i n g
p o i n t s , ;.e., t h e p o i n t s o f i n t e r s e c t i o n o f t h e orthodrome w i t h t h e
equator (Fig. 1 . 3 7 ) , w i l l appear as p o i n t s o f i n f l e c t i o n .
T h u s , t h e o r t h o d r o m e o n a map
of an equally spaced c y l i n d r i c a l
p r o j e c t i o n h a s a form r e m i n i s c e n t
of a s i n e curve.
This curve is
t h e graph of t h e r a t i o of t h e coor­
d i n a t e s of t h e orthodrome with a
known i n i t i a l a z i m u t h ( a ) .

ILLIT

"I

gbTA

F i g . 1 . 3 7 . Graph o f an Ortho­
drome i n a C y l i n d r i c a l Projection.

A s a r e s u l t of t h e nonisogon­
a l i t y of an equally spaced projec­
tion, the slope of a tangent t o
t h e c u r v e o f t h e orthodrome d o e s n o t r e f l e c t i t s d i r e c t l y moving
a z i m u t h , w i t h t h e e x c e p t i o n of t h e a z i m u t h a t s t a r t i n g p o i n t s .
The m o v i n g a z i m u t h o f a n o r t h o d r o m e a l o n g a c u r v e c a n b e d e t e r ­
mined i f w e c o n s i d e r t h e r e l a t i o n s h i p between t h e s c a l e s m and n a t
the investigated points.
With e q u a l s c a l e s , t h e d i p a n g l e o f t h e
t a n g e n t t o t h e c u r v e i s d e t e r m i n e d by t h e f o r m u l a
tga-

I

d'h
­

dy

'

I n o u r c a s e , t h e s c a l e n h = no s e c 4 or n h = n 4 s e c 4 .
Theref o r e , t h e a c t u a l l y moving a z i m u t h o f t h e orthodrome i n an e q u a l l y
s p a c e d c y l i n d r i c a l p r o j e c t i o n i s d e t e r m i n e d by t h e f o r m u l a
tga=-

d)l

cos 'p.

d'p

I t i s obvious t h a t i n an i s o g o n a l normal c y l i n d r i c a l p r o j e c ­
t i o n , t h e orthodrome w i l l a l s o have a shape r e m i n i s c e n t of a s i n e
curve.
However, a s a r e s u l t o f t h e e x t e n s i o n o f t h e s c a l e a l o n g
t h e l a t i t u d e ( n = no s e c I$), t h e a m p l i t u d e o f t h i s c u r v e w i l l b e
increased.
The more it i s i n c r e a s e d , t h e s m a l l e r t h e i n i t i a l a z i ­
muth o f t h e o r t h o d r o m e w i l l b e , a n d t h e g r e a t e r t h e l a t i t u d e o f t h e
vertex points.

I n c o n t r a s t t o an e q u a l l y spaced p r o j e c t i o n , i n t h i s p r o j e c t i o n
t h e d i p angle of t h e tangent t o t h e curve w i l l correspond t o t h e
moving a z i m u t h o f t h e orthodrome a t any p o i n t , . s i n c e t h e s c a l e s
along t h e l o n g i t u d e and l a t i t u d e are e q u a l t o :
m = It = sec 'p.

57

/61

Thus, t h e orthodrome i n a c y l i n d r i c a l p r o j e c t i o n h a s t h e form
of a c u r v e which i s convex i n t h e d i r e c t i o n o f t h e i n c r e a s e i n t h e
T h i s f e a t u r e o f t h e o r t h o d r o m e i s common
s c a l e of t h e p r o j e c t i o n .
t o a l l p r o j e c t i o n s which have an i n c r e a s e i n one d i r e c t i o n .
Let us c i t e a b r i e f a n a l y s i s o f t h e bend o f t h e orthodrome with
a v a r y i n g map s c a l e , i n a c c o r d a n c e w i t h t h e g e n e & a l c a s e .
L e t u s a s s u m e t h a t we h a v e a s p h e r i c a l t r a p e z o i d w h i c h i s r e p ­
r e s e n t e d on a map i n t h e f o r m o f a r e c t a n g l e ( F i g . 1 . 3 8 ) .
The l e n g t h
o f a n y p a r a l l e l ( L A ) on t h e t r a p e z o i d i s e q u a l t o i t s l e n g t h on a
r e c t a n g l e d i v i d e d by t h e s c a l e o f i t s r e p r e s e n t a t i o n .
The s c a l e o f
r e p r e s e n t a t i o n o f t h e m e r i d i a n s i n any p a r t o f t h e r e c t a n g l e i s
e q u a l t o one.
1
LA&-;

L,=1.

R

During extension of t h e t r a p e z o i d i n t o a r e c t a n g l e , each
s t r a i g h t - l i n e e l e m e n t on i t s s u r f a c e a c q u i r e s a c u r v a t u r e .

Therefore,

f o r an e q u a l l y spaced c y l i n d r i c a l p r o j e c t i o n ,
1

a cos 'p

f-A

drq

---=_
-

Since

1
=
PA

0,

sin 7.

f o r a s t r a i g h t l i n e p a s s i n g a t an a n g l e t o t h e

m e r i d i a n we w i l l h a v e
1
-_r

sin rq sin a.

A minus s i g n shows t h a t b e n d i n g o c c u r s i n t h e d i r e c t i o n o f a
I

d e c r e a s e i n t h e v a l u e o f - or i n t h e d i r e c t i o n o f a n i n c r e a s e i n
n
the scale.

a

-I

I__-

+LA

---

Li"

Fig. 1.38. Conical Trapezoid
R e p r e s e n t e d i n t h e Form o f a
Rectangle.

58

F i g . 1 . 3 9 . Bending o f an O r thodrome i n t h e D i r e c t i o n s
of Scale Increases.

L e t u s now a s s u m e t h a t w e h a v e a p r o j e c t i o n , a c h a n g e i n t h e
s c a l e o f w h i c h o c c u r s i n t w o p r i n c i p a l d i r e c t i o n s [for e x a m p l e ,
s i m u l t a n e o u s l y i n t h e n o r t h a n d e a s t ( F i g . 1.39)].
It is obvious t h a t a s t r a i g h t l i n e AB, passing a t an angle t o
t h e m e r i d i a n w i t h a change i n t h e scales i n two p r i n c i p a l d i r e c t i o n s ,
w i l l simultaneously undergo bending i n o p p o s i t e d i r e c t i o n s , i . e . ,
i t s component c u r v a t u r e s w i l l b e s u b t r a c t e d :

a-

1

il­

r
drp

sin a -

a­ * 1
"p
dA

cosa

For t h e g e n e r a l case,

a - n1
1
--r
dz

-

sln a

a­ mz1

-cos a.
dx

T h e r e f o r e , t h e o r t h o d r o m e o n maps c o n s t r u c t e d w i t h t a n g e n t i a l
c y l i n d r i c a l p r o j e c t i o n s w i l l have convexity:

a t l a t i t u d e s g r e a t e r t h a n t h e l a t i t u d e of a p a r a l l e l which
a)
is tangent t o a geographic pole;
a t l a t i t u d e s lower t h a n t h e l a t i t u d e o f a p a r a l l e l which
b)
i s t a n g e n t t o t h e e q u a t o r ( F i g . 1.40).

Conic s t r a i g h t
--Orthodrome

line

In intersecting conic /63
projections , the ortho­
d r o m e h a s t h e same f o r m a s
i n tangential projections.
Here, i t s p o i n t of i n f l e c ­
t i o n i s s i t u a t e d on t h e
middle p a r a l l e l between
the parallels of intersection.

.

O f special interest
i s t h e o r t h o d r o m e o n maps
i n an i n t e r n a t i o n a l pro­
jection.
A s i s known,
t h e s c a l e a l o n g t h e l a t i t u d e i n t h e s e maps r e m a i n s p r a c t i c a l l y c o n ­
s t a n t , w h i l e t h e scale along t h e l o n g i t u d e h a s s m a l l changes i n t h e
l i m i t s o f e a c h map s h e e t .
T h e r e f o r e , a n o r t h o d r o m e drawn a t a n an­
g l e t o t h e m e r i d i a n w i t h i n t h e l i m i t s o f o n e s h e e t o n t h i s map w i l l
b e a wavy l i n e a r o u n d a s t r a i g h t p r i n c i p a l d i r e c t i o n .
However, on
maps w i t h a s c a l e o f 1 : 1 , 0 0 0 , 0 0 0 ,
d e v i a t i o n s from t h e p r i n c i p a l d i ­
r e c t i o n w i l l be s o i n s i g n i f i c a n t t h a t t h e y are p r a c t i c a l l y u n n o t i c e ­
a b l e , w h i l e i n t h e p l a c e s w h e r e s e p a r a t e map s h e e t s a r e s p l i c e d ,
b o t h p a r a l l e l s and t h e orthodrome w i l l have b r e a k s .

F i g . 1.40. O r t h o d r o m e o n a Map o f a
T a n g e n t i a l Conic P r o j e c t i o n .

59

A s w e have a l r e a d y shown, t h e orthodrome i n a c e n t r a l p o l a r
However, i t s moving
p r o j e c t i o n i s e x p r e s s e d by a s t r a i g h t l i n e .
a z i m u t h , w i t h t h e e x c e p t i o n o f t h e d i r e c t i o n s 0 , 9 0 , 1 8 0 , and 270°
c a n n o t b e d e t e r m i n e d b y s i m p l e m e a s u r e m e n t s o n a map, b u t demand
t h e i n t r o d u c t i o n of c o r r e c t i o n s a c c o r d i n g t o ( 1 . 2 9 ) and ( 1 . 3 0 ) .
I n a p o l a r s t e r e o g r a p h i c p r o j e c t i o n , t h e orthodrome i s a l s o a
nearly straight line.
However, t o d e t e r m i n e i t s a z i m u t h , it i s
n e c e s s a r y t o u s e g e n e r a l e q u a t i o n s o f a n o r t h o d r o m e on t h e E a r t h ' s
surface.
L o x o d r o m e on t h e E a r t h ' s

Surface

The l o x o d r o m e d i r e c t i o n a t t h e p r e s e n t t i m e i s u s e d o n l y t o
d e t e r m i n e t h e mean p a t h a n g l e o f f l i g h t o n s h o r t s e g m e n t s o f a p a t h
by t h e u s e o f m a g n e t i c compasses.
' W i t h t h e u s e o f m a g n e t i c com­
p a s s e s , n o t a geographic b u t a magnetic loxodrome d i r e c t i o n i s used.
T h i s l e a d s t o a bending o f t h e f l i g h t p a t h which does n o t l e n d i t ­
self t o precise analytical descriptions.
The u s e o f maps i n a n i n t e r n a t i o n a l p r o j e c t i o n , on w h i c h a n
o r t h o d r o m e w i t h a ! e n g t h u p t o 1 2 0 0 km i s p r a c t i c a l l y r e p r e s e n t e d
by a s t r a i g h t l i n e , as w e l l as t h e u s e o f r a d i o - e n g i n e e r i n g f a c i ­
l i t i e s f o r a i r c r a f t n a v i g a t i o n by o r t h o d r o m e b e a r i n g s , l e d t o t h e
p o s i t i o n t h a t t h e orthodrome l i n e of t h e p a t h o f an a i r c r a f t i s
p r a c t i c a l l y m a i n t a i n e d w i t h o u t b e i n g d e p e n d e n t o n a s y s t e m o f meas­
u r i n g d i r e c t i o n s on t h e E a r t h ' s s u r f a c e .
I n t h i s case, a s t r i c t l y
l o x o d r o m i c l i n e i s n o t t a k e n as t h e loxodrome, b u t r a t h e r an o r t h o ­
d r o m e s e c t i o n w i t h a n i n d i c a t i o n o f t h e mean p a t h a n g l e .
A p p l i c a t i o n o f p r e c i s e g y r o s c o p i c and a s t r o n o m i c a l n a v i g a t i o n a l
d e v i c e s g e n e r a l l y e l i m i n a t e s t h e n e c e s s i t y o f u s i n g t h e loxodrome
Therefore, i n studying the properties of a
d i r e c t i o n of f l i g h t .
l o x o d r o m e on t h e E a r t h ' s s u r f a c e , w e a r e l i m i t e d o n l y by t h e s m a l l
amount o f a v a i l a b l e g e o m e t r i c i n f o r m a t i o n .

A s w e a l r e a d y know, a loxodrome i s a l i n e on t h e E a r t h ' s s u r face which j o i n s two p o i n t s and i n t e r s e c t s t h e m e r i d i a n s a t a con­
s t an t angle.
I n g e n e r a l , a loxodrome i s a s p i r a l l i n e which goes t o t h e
A s a r e s u l t of t h i s , it has curvature not only i n
Earth's poles.
a v e r t i c a l p l a n e , b u t i n a h o r i z o n t a l p l a n e as w e l l .
Meridians,
t h e e q u a t o r , and p a r a l l e l s which are a l s o loxodrome l i n e s , e x p r e s s e d
i n t h e f i r s t t w o c a s e s by a g r e a t c i r c l e a n d i n t h e l a s t case b y a
s m a l l c i r c l e on t h e E a r t h ' s s u r f a c e , a r e t h e e x c e p t i o n .
The c u r v a t u r e o f a l o x o d r o m e i n a h o r i z o n t a l p l a n e i n c r e a s e s
s h a r p l y w i t h an approach t o t h e E a r t h ' s p o l e s .
A s a r e s u l t , it i s
n o t u s e d a t a l l for f l i g h t s i n p o l a r l a t i t u d e s .

L e t u s determine t h e c u r v a t u r e of a loxodrome, i t s e x t e n s i o n ,

60

/64

and i t s d e f l e c t i o n as compared t o t h e orthodrome d i r e c t i o n a t a
given l a t i t u d e 9.
T h e maximum c u r v a ­
t u r e of a loxodrome a t
a given f l i g h t a l t i t u d e
w i l l o c c u r when t h e
f l i g h t i s i n an e a s t e r l y
or w e s t e r l y d i r e c t i o n ,
and it w i i l v a n i s h i n a
flight t o the north or
south.
L e t u s assume t h a t
a f l i g h t a t a l t i t u d e $I
F i g . 1 . 4 1 . L o x o d r o m e o n a Map o f a C o n i c
occurs i n an e a s t e r l y
Projection.
direction.
In t h i s
case, t h e angle of t u r n
o f t h e loxodrome from p o i n t A t o p o i n t B w i l l b e e q u a l t o t h e a n g l e
of c o n v e r g e n c e o f t h e m e r i d i a n s ( 6 ) b e t w e e n t h e s e p o i n t s ( F i g . 1 . 4 1 ) .
8=-

Its length

( S ) from p o i n t A

(AB

- AA) sin 'PI

t o B w i l l be
S = (AB

- AA) COS 'p,

w h e r e A A a n d A B a r e t h e l o n g i t u d e s o f p o i n t s A a n d B a n d $I
mean l a t i t u d e b e t w e e n p o i n t s A a n d B .

is the

The r a d i u s o f c u r v a t u r e o f t h e l o x o d r o m e ( r ~ )
can be determined
as t h e r a t i o o f t h e l e n g t h o f p a r t ( S ) t o t h e angle o f t u r n ( 6 ) .
I f we t a k e t h e r a d i u s o f t h e E a r t h a s 1, t h e n
S
ri=-

8

=ctgrp.

(1.48)

The p a r t o f t h e l o x o d r o m e w h i c h r u n s a l o n g t h e m e r i d i a n d o e s
T h e r e f o r e , i f t h e loxodrome p a s s e s
n o t have a h o r i z o n t a l c u r v a t u r e .
a t an angle t o t h e meridian, t h e r a d i u s of i t s c u r v a t u r e a t any
point w i l l equal:
r = r, COSec a = clg 'p cosec a.
(1.49)

Example:
Determine t h e r a d i u s o f c u r v a t u r e o f a loxodrome
p a s s i n g a t a n a n g l e o f 30° t o t h e m e r i d i a n a t a l a t i t u d e o f 4 5 O .

S o l u t i on :
r = R, ctg 45" cosec 30"= ZR, = 12742

km

w h e r e R 3 i s t h e r a d i u s of t h e E a r t h .

61

/65

The c u r v a t u r e o f t h e , l o x o d r o m e i n a h o r i z o n t a l p l a n e c r e a t e s
some l e n g t h e n i n g o f t h e s t r a i g h t - l i n e p a r t s o f t h e p a t h .
The l a t e r ­
a l d e v i a t i o n s f r o m t h e l i n e o f t h e o r t h o d r o m e d i r e c t i o n may t u r n
out t o be very s i g n i f i c a n t here.
I n Figure 1 . 4 2 , t h e s t r a i g h t l i n e AB i s t h e orthodrome; arc AB
i s t h e loxodrome; 6 i s t h e a n g l e o f t u r n of t h e lokodrome from p o i n t
A t o point B.
The l e n g t h o f t h e s t r a i g h t l i n e i s

w h i l e t h e l e n g t h o f t h e a r c is
AB = RB.

Lengthening o f t h e p a t h a l o n g t h e loxodrome ( A S )
by t h e f o r m u l a
&3 = Rs -2R sin

6
­
2 .

i s determined

(1.50)

Example: Determine t h e l e n g t h e n i n g of t h e p a t h a l o n g t h e loxo­
drome p a s s i n g t h r o u g h p o i n t s A a n d B on t h e E a r t h ' s s u r f a c e , w i t h
A:
l a t i t u d e 55O N , l o n g i t u d e 3 8 O E ;
the following coordinates:
B:
l a t i t u d e 55O N , l o n g i t u d e 6 8 O E .
S i n c e t h e l a t i t u d e of t h e s t a r t i n g and end p o i n t s i s t h e same,
t h e d i r e c t i o n o f t h e loxodrome c o i n c i d e s w i t h t h e E a r t h ' s p a r a l l e l
at a l a t i t u d e of 5 5 O .
The r a d i u s o f c u r v a t u r e o f t h e l o x o d r o m e w i l l b e :
r=r;,=Rgctg55"=6371.0,7002=4461

The a n g l e o f t u r n of
-b=((h2-A1)slncp;

km

t h e l o x o d r o m e i s d e t e r m i n e d by t h e f o r m u l a
b=-3O0sin55"=

-3O0-O,8192=24.576",

S u b s t i t u t i n g t h e v a l u e of t h e r a d i u s of c u r v a t u r e and t h e an­
we obtain:
g l e of t u r n of t h e loxodrome i n t o ( 1 . 5 0 ) ,

As=

24*576'4461 -2.4461 4,2127 &5,6
57.3

km

From t h i s e x a m p l e , i t i s o b v i o u s t h a t a t m i d d l e l a t i t u d e s ,
w i t h f l i g h t p a t h s u p t o 2 , 0 0 0 - 3 , 0 0 0 km l o n g , t h e c u r v a t u r e o f t h e
loxodrome creates r e l a t i v e l y s m a l l l e n g t h e n i n g s o f t h e p a t h ( i n o u r

62

/66

e x a m p l e , l e s s t h a n 1 % )h;o w e v e r , i n a p p r o a c h i n g t h e p o l a r l a t i t u d e s ,
lengthening of the path w i l l increase, together with a decrease i n
t h e r a d i u s o f c u r v a t u r e o f t h e loxodrome.
S i g n i f i c a n t l e n g t h e n i n g s o f t h e p a t h a l o n g t h e loxodrome o c c u r
a t m i d d l e l a t i t u d e s w i t h v e r y l o n g d i s t a n c e s b e t w e e n p o i n t s on t h e
Earth's surface.
F o r e x a m p l e , a t a l a t i t u d e o f 40°, w i t h a d i s t a n c e
o f 1 1 , 0 0 0 km b e t w e e n p o i n t s , l e n g t h e n i n g o f t h e p a t h a l o n g t h e l o x o ­
d r o m e c a n e x c e e d 4 , 0 0 0 km, ; . e . , m o r e t h a n 3 0 % .
I n Figure 1 . 4 2 , it i s obvious t h a t
w i t h a c o n s t a n t r a d i u s of c u r v a t u r e o f t h e
loxodrome, i t s g r e a t e s t discrepancy w i t h
r e s p e c t t o t h e orthodrome ( d e f l e c t i o n ) w i l l
be observed a t h a l f t h e p a t h between p o i n t s
A and B .

0 r t h o d rome
A

A2

8

Loxodrome
F i g . 1 . 4 2 . R a d i u s of
C u r v a t u r e of a ~ 0 x 0 ­
drome.

Here

,

A2

- R COS­

A 2 = R (1 -cos

or

?i

2

$).

(1.51)


I n t h e example a n a l y z e d by u s ,
A 2 = 4461 (1 -0,9771)

= 102,6

km

Thus, t h e d i s c r e p a n c y between t h e loxodrome l i n e o f t h e p a t h
and t h e o r t h o d r o m e , e v e n a t c o m p a r a t i v e l y s m a l l d i s t a n c e s b e t w e e n
This is
p o i n t s on t h e E a r t h ' s s u r f a c e , w i l l b e v e r y s u b s t a n t i a l .
t h e b a s i c c a u s e of t h e l i m i t a t i o n of t h e l e n g t h o f t h e loxodrome
segments of t h e p a t h .
I n t h e p r a c t i c e o f a i r c r a f t n a v i g a t i o n , s i n c e t h e loxodrome
d i r e c t i o n of f l i g h t i s used only i n l i m i t e d path segments, t h e a z i ­
muth o f t h e o r t h o d r o m e ( a ) , m e a s u r e d on t h e c e n t r a l m e r i d i a n b e t w e e n
t h e s t a r t i n g and end p o i n t s o f t h e segment i s t a k e n as t h e loxodrome
d i r e c t i o n of t h e f l i g h t .
T h i s a n g l e c a n a l s o b e d e t e r m i n e d on t h e b a s i s o f t h e a p p r o x i m a t e formula
(1.52)

The l e n g t h o f t h e l o x o d r o m e s e g m e n t o f t h e p a t h
m i n e d by t h e f o r m u l a

(S) is deter­

(1.53)

or

(1.54)

63

/67

I

1111I1111IIIIIlII.

Ill

Formulas (1.54) and (1.52)
geometrical interpretation.
Formula ( 1 . 5 3 )

I,

11,

I

I I I

are approximate and have a simple

is derived analytically.

Considering t h a t t h e loxodrome i n t e r s e c t s t h e meridians a t a
constant angle, t h e r a t i o remains constant:

from which

In t h e majority of cases, i n c a l c u l a t i n g t h e distance along
t h e l o x o d r o m e , i t i s more a d v a n t a g e o u s t o a p p l y ( 1 . 5 3 ) .
However,
w i t h l o x o d r o m e d i r e c t i o n s c l o s e t o 9 0 or 2 7 0 ° , t h e v a l u e s $ 2 - $ 1
and cos a s i m u l t a n e o u s l y approach z e r o .
This leads t o large a r i t h ­
m e t i c e r r o r s i n c a l c u l a t i o n and u l t i m a t e l y t o a n a m b i g u i t y i n t h e
solution.
I n t h e s e c a s e s , i t i s more a d v a n t a g e o u s t o u s e ( 1 . 5 4 ) ,
t h e e r r o r s i n which w i l l be n e g l i g i b l y s m a l l , s i n c e a s m a l l d i f f e r ­
e n c e i n t h e l a t i t u d e s b e t w e e n t h e p o i n t s m e a n s t h a t t h e mean c o s i n e
o f t h e l a t i t u d e b e c o m e s p r a c t i c a l l y e q u a l t o t h e c o s i n e o f t h e mean
latitude.

Example:
Determine t h e loxodrome d i r e c t i o n and t h e d i s t a n c e
b e t w e e n p o i n t s A a n d B on t h e E a r t h ' s s u r f a c e , t h e c o o r d i n a t e s o f
A : l a t i t u d e 56O N , l o n g i t u d e 38O E ; B : l a t i t u d e 68O N ,
which a r e :
l o n g i t u d e 47O E .
Solution:

According t o

(1.381,

l e t us find t h e d i r e c t i o n of

t h e loxodrome:
tga=- 47

-38

68-33

COS 62" =

0,3521; a = 1994'.

Let us d e t e r m i n e t h e loxodrome d i s t a n c e a c c o r d i n g t o ( 1 . 5 3 ) :

S = l l l , l - 68-56. =I413 km
cos 1924'

Loxodrome on M a p s o f

Different Projections

A l o x o d r o m e h a s t h e a p p e a r a n c e o f a s t r a i g h t l i n e o n l y on maps
of a normal i s o g o n a l c y l i n d r i c a l p r o j e c t i o n .
O n maps o f n o r m a l i s o g o n a l c o n i c a n d a z i m u t h a l p r o j e c t i o n s ,
t h e l o x o d r o m e is a c u r v e d l i n e i n t e r s ' e c t i n g t h e m e r i d i a n s a t a c o n ­
T h e r e f o r e , knowing t h e d i r e c t i o n o f t h e loxodrome
s t a n t angle a.
i n o r d e r t o d r a w i t o n a map i t i s s u f f i c i e n t a t t h e s t a r t i n g p o i n t

64

.

t o p l o t t h i s d i r e c t i o n up t o t h e i n t e r s e c t i o n w i t h t h e n e x t m e r i ­
d i a n , where t h e i n d i c a t e d d i r e c t i o n must b e e x t e n d e d t o t h e n e x t
meridian i n l i n e .
Continuing our p l o t t i n g t o t h e f i n a l p o i n t , we
w i l l obtain a broken l i n e very c l o s e t o t h e loxodrode.
On maps w i t h n o n i s o g o n a l p r o j e c t i o n s , t h e l o x o d r o m e w i l l h a v e
a v a r i a b l e a n g l e t o t h e m e r i d i a n s , which depends on t h e r a t i o o f
t h e scales
t g a m = tga-

n
m

(1.55)

where a i s t h e a n g l e o f i n t e r s e c t i o n of t h e loxodrome w i t h t h e m e r i ­
d i a n a t a l o c a t i o n ; C ~ M i s t h e a n g l e of i n t e r s e c t i o n o f t h e l o x o ­
d r o m e w i t h t h e m e r i d i a n o n a m a p ; n a n d m a r e t h e s c a l e s o f a map
a t a g i v e n p o i n t a l o n g t h e p r i n c i p a l d i r e c t i o n s east-west and n o r t h south, respectively.

For e x a m p l e , o n m a p s w i t h a n e q u a l l y s p a c e d n o r m a l c y l i n d r i c a l
= sec @,
p r o j e c t i o n , where

m

tg am= tg a sec '4,

; . e . , t h e loxodrome w i l l have a c u r v a t u r e i n t h e d i r e c t i o n of a
p o l e , whereas it h a s a n a t u r a l c u r v a t u r e i n t h e d i r e c t i o n of t h e
equator.
General

Recommendations f o r Measuring D i r e c t i o n s and D i s t a n c e s

O r t h o d r o m e d i r e c t i o n s a n d d i s t a n c e s for s t r a i g h t - l i n e s e g m e n t s
o f a p a t h o f m o r e t h a n 1 2 0 0 - 1 5 0 0 km i n a l l c a s e s m u s t b e d e t e r m i n e d
b y a n a l y t i c a l m e a n s , i n d e p e n d e n t l y o f t h e s c a l e s a n d map p r o j e c t i o n s
used.
W i t h a l e n g t h o f t h e p a t h s e g m e n t s o f more t h a n 2 0 0 0 k m , t h e
i n t e r m e d i a t e p o i n t s of t h e orthodrome must a l s o be d e t e r m i n e d i n
s u c h a way t h a t t h e d i s t a n c e b e t w e e n t h e m d o e s n o t e x c e e d 8 0 0 - 1 0 0 0
km.
On s h o r t p a t h s e g m e n t s ( u p t o 1 2 0 0 - 1 5 0 0 k m ) , t h e m e t h o d s o f
d e t e r m i n i n g d i r e c t i o n s and d i s t a n c e s depend on t h e s c a l e and p r o ­
j e c t i o n o f t h e m a p s , a s w e l l a s on t h e means a n d m e t h o d s o f a i r ­
craft navigation used.
For e x a m p l e , i n u s i n g p r e c i s e a u t o m a t i c
n a v i g a t i o n a l d e v i c e s , it i s always advantageous t o use a n a l y t i c a l
forms t o s o i v e t h e s e problems.
I t i s p o s s i b l e t o c a r r y o u t d i r e c t measurement o f d i s t a n c e s
a n d d i r e c t i o n s o n maps b y m e a n s o f a s c a l e a n d p r o t r a c t o r , w i t h t h e
l e n g t h o f t h e p a t h s e g m e n t s b e i n g n o t m o r e t h a n 1 5 0 0 km i f t h e s e
maps a r e e x e c u t e d o n a n i n t e r n a t i o n a l p o l y c o n i c p r o j e c t i o n a n d h a v e
a s c a l e o f 1:1,000,000 or 1 : 2 , 0 0 0 , 0 0 0 ( t h e l a t t e r w i t h i n t h e l i m i t s
o f o n e (or, i n e x t r e m e c a s e s , t w o ) a d j o i n i n g ' s h e e t s ) .

65


­
/69

fl

We m u s t n o t e t h a t g o o d r e s u l t s i n m e a s u r i n g d i r e c t i o n s a n d d i s ­
t a n c e s c a n b e o b t a i n e d ' o n r o u t e maps c o n s t r u c t e d o n o b l i q u e c y l i n ­
d r i c a l or o b l i q u e c o n i c p r o j e c t i o n s when t h e f l i g h t d i r e c t i o n c o i n ­
How­
c i d e s w i t h or i s l o c a t e d c l o s e t o t h e a x i s o f t h e r o u t e map.
e v e r , i n d i r e c t i o n s a t a n a n g l e t o t h e a x i s of t h e r o u t e map, t h e
r e s u l t s o f m e a s u r e m e n t s a r e s i g n i f i c a n t l y w o r s e e h a n on maps w i t h
an i n t e r n a t i o n a l p o l y c o n i c p r o j e c t i o n .
I n u s i n g maps c o n s t r u c t e d w i t h a l l o t h e r p r o j e c t i o n s , o n l y t h e
a n a l y t i c a l form of d e t e r m i n i n g d i s t a n c e s and d i r e c t i o n s , w i t h c a l c u ­
l a t i o n o f i n t e r m e d i a t e p o i n t s a l o n g t h e orthodrome a f t e r e v e r y 200­
3 0 0 km o f t h e p a t h , m u s t b e a p p l i e d .
The l o x o d r o m i c f l i g h t d i r e c t i o n c a n b e
on maps w i t h a n i s o g o n a l n o r m a l c y l i n d r i c a l
m e n t s o f d i s t a n c e s u p t o 3 0 0 - 4 0 0 km o n t h i s
u r e d by means o f a v a r y i n g s c a l e l o c a t e d on

measured d i r e c t l y o n l y
Here, seg­
projection.
p r o j e c t i o n c a n b e meas­
t h e e d g e o f t h e map.

On maps i n o t h e r p r o j e c t i o n s , g e n e r a l l y s p e a k i n g , t h e r e i s n o
need t o m e a s u r e and p l o t t h e loxodrome l i n e o f t h e p a t h i n p a r t s of
m3rie t h a n 3 0 0 - 4 0 0 km.
Since t h e loxodromic f l i g h t d i r e c t i o n i n s h o r t p a t h segments
i s u s e d a s t h e mean o r t h o d r o m e d i r e c t i o n , i t i s c o n s i d e r e d e q u a l t o
t h e o r t h o d r o m e a s i n d i c a t e d b y t h e mean m e r i d i a n b e t w e e n t h e s t a r t ­
i n g and end p o i n t s o f t h e p a t h segment.
I n view o f t h e f a c t t h a t i n s h o r t s e g m e n t s o f t h e p a t h t h e lox-.
odrome l i n e d o e s n o t show s i g n i f i c a n t d e v i a t i o n f r o m t h e o r t h o d r o m e
a s a r u l e , i t i s n o t p l o t t e d o n maps b u t i s c o n s i d e r e d c o i n c i d e n t
w i t h t h e d i r e c t i o n of t h e orthodrome.

7.

S p e c i a l C o o r d i n a t e S y s t e m s on t h e E a r t h ' s S u r f a c e

I n t h e p r a c t i c e o f a i r c r a f t n a v i g a t i o n , r e c t a n g u l a r and geo­
g r a p h i c c o o r d i n a t e systems a r e i n s u f f i c i e n t , and it i s n e c e s s a r y t o
u s e a t l e a s t t h r e e or f o u r c o o r d i n a t e s y s t e m s s i m u l t a n e o u s l y .
A c t u a l l y , e l e m e n t s o f a i r c r a f t m o v e m e n t a r e e x a m i n e d i n a moving rectangular coordinate system.
The c e n t e r o f a r e c t a n g u l a r
s y s t e m moves i n o n e o f t h e s u r f a c e c o o r d i n a t e s y s t e m s w h i c h i s c o n ­
n e c t e d w i t h t h e g i v e n f l i g h t p a t h , which i n t u r n i s determined i n a
geographic coordinate system.
T h e i n d i c a t e d o r d e r o f t h e c o n n e c t i o n of t h e c o o r d i n a t e s y s t e m s
i s minimal.
For some p u r p o s e s , i t i s a d v a n t a g e o u s t o e x a m i n e a i r ­
c r a f t movement r e l a t i v e t o t h e a i r s p a c e , i . e . , a s u p p l e m e n t a r y c o ­
o r d i n a t e s y s t e m whose c e n t e r s h i f t s i n t h e moving r e c t a n g u l a r s y s t e m .

ones,
tem,

66

With t h e u s e of gyroscopic d e v i c e s a s w e l l as a s t r o n o m i c a l
it is necessary t o use a u n i v e r s a l ( s t e l l a r ) coordinate sys­
The u s e o f r a d i o - e n g i n e e r i n g n a v i g a t i o n a l f a c i l i t i e s i s c o n ­

~~

/70

I

n e c t e d w i t h t h e u s e of a whole s e r i e s o f s p e c i a l t y p e s o f s u r f a c e
c o o r d i n a t e s by w h i c h t h e p o s i t i o n o f t h e a i r c r a f t o n t h e E a r t h ‘ s
surface is determined.
L e t u s examine t h e most i m p o r t a n t
used i n aircraft navigation.

surface coordinate systems

Orthodromi c C o o r d i n a t e System

.The orthodromic c o o r d i n a t e system f o r c a l c u l a t i n g t h e p a t h o f
a n a i r c r a f t i s t h e one most w i d e l y u s e d a t t h e p r e s e n t t i m e .
I n t h i s system, t h e d i r e c t i o n o f t h e s t r a i g h t - l i n e p a t h segment
The l i n e p e r p e n d i c u l a r t o
( F i g . 1 . 4 3 ) i s t a k e n as t h e main a x i s X.
t h e X-axis and a l s o s i t u a t e d i n t h e p l a n e o f t h e h o r i z o n i s t h e sec­
o n d a x i s , Z.

Fig.

1.43.

Orthodromic Coordinate System.

I n F i g u r e 1 . 4 3 , a n g l e s 0.1 a n d “2 a r e t h e d i r e c t i o n s o f t h e
f i r s t and second s t r a i g h t - l i n e segments of t h e p a t h , measured from
t h e meridians of t h e i r s t a r t i n g p o i n t s .
P o i n t s 01 a n d 0 2 a r e t h e
s t a r t i n g p o i n t s of t h e segments, t h e c o o r d i n a t e s of which a r e de­
termined i n t h e geographic coordinate system.
The o r t h o d r o m e d i s t a n c e s 0102 a n d 0 2 0 3 a r e t h e l e n g t h s o f t h e s t r a i g h t - l i n e s e g m e n t s ;
t h e a n g l e T A l i s t h e a n g l e o f t u r n of t h e orthodromic c o o r d i n a t e
system a t point 0 2 .
S i n c e an a i r c r a f t moving above t h e E a r t h ‘ s s u r f a c e i n a g i v e n
d i r e c t i o n h a s o n l y s m a l l random d e v i a t i o n s from t h e g i v e n f l i g h t
p a t h ( a s a r u l e , n o t more t h a n 2 0 - 3 0 km), i t i s p o s s i b l e t o t a k e
t h e s p h e r i c a l s u r f a c e of t h e Earth within t h e a r e a of t h e p o s s i b l e
d e v i a t i o n s o f t h e a i r c r a f t from t h e X-axis o f t h e orthodrome system
as a c y l i n d r i c a l s u r f a c e .
Then t h e u n r o l l i n g o f t h e c y l i n d e r g i v e s
u s a r e c t a n g u l a r s y s t e m XZ on a p l a n e .
L e t u s a s s u m e t h a t a n a i r c r a f t m o v e s f r o m p o i n t 01 a t a s m a l l
a n g l e t o t h e 01x1 a x i s e q u a l t o J, - a l , a n d c o v e r s a d i s t a n c e S .

67

/71

111 ...
1.111...111111111--1...--..........II
,I

.
I

I...__.._.

... .....
,.,

.

- ... .

._

The c o o r d i n a t e s o f t h e a i r c r a f t a t p o i n t
t h e equations’:

M,,

a r e d e t e r m i n e d by

Xa= S cos (JI - al);

1

Za= S sin (JI - a,).

(1.56)

M e a s u r i n g t h e Xa c o o r d i n a t e c o n s t i t u t e s c h e c k i n g o f t h e p a t h
o f t h e a i r c r a f t a c c o r d i n g t o d i s t a n c e , w h i l e measuring t h e Za coor­
d i n a t e c o n s t i t u t e s checking of t h e path according t o direction.
P e r i o d i c m e a s u r e m e n t o f t h e X a a n d Z a c o o r d i n a t e s make i t p o s ­
s i b l e t o d e t e r m i n e a l l t h e b a s i c e l e m e n t s o f a i r c r a f t movement; f o r
example :
D i r e c t i o n o f a i r c r a f t movement

a)

Jr = arctg

($):

Za2-Zai
..

+

a,

(1.57)

Xa2-Xa1

where X a l , Zal are c o o r d i n a t e s of t h e a i r c r a f t a t t h e f i r s t p o i n t .
X a 2 , Za2 a r e c o o r d i n a t e s of t h e a i r c r a f t a t t h e s e c o n d p o i n t ;

b)

Speed of

a i r c r a f t movement a l o n g a g i v e n f l i g h t p a t h ( W )
(1.58)

where

t i s t h e f l y i n g t i m e o f t h e a i r c r a f t between p o i n t s Xal and

Xa2 ;
c)

Remaining f l y i n g t i m e t o p o i n t 0 2
Xrem



‘re%?

w h e r e Xrem
d)

(1.59)

= 0102 - X a .

N e c e s s a r y f l i g h t d i r e c t i o n for a r r i v a l a t p o i n t 0 2 :

+ = 01-

arctg

z
Xrek

(1.60)

Formulas ( 1 . 5 5 ) t o ( 1 . 6 0 ) a r e e n t i r e l y o b v i o u s and do n o t r e ­
q u i r e s p e c i a l d e r i v a t i o n s or p r o o f s .
To r e f i n e t h e c o o r d i n a t e s o f t h e a i r c r a f t i n t h e o r t h o d r o m e
s y s t e m , w e c a n u s e c o r r e c t i o n p o i n t s ( C P ) , v i s u a l or r a d a r l a n d m a r k s on t h e E a r t h ’ s s u r f a c e , l o c a t i o n s o f g r o u n d r a d i o f a c i l i t i e s ,
etc. (Fig. 1.44).

T r a n s l a t o r ’ s note:
68

a r c t g = cot’’.

/72
-

If t h e c o r r e c t i o n p o i n t i s o b s e r v e d from a n a i r c r a f t a t a n
d i s t a n c e from t h e a i r c r a f t e q u a l
angle 0 t o the given route, a t a
t o R , the coordinates of the aircraft
w i l l be d e t e r m i n e d by t h e f o r m u l a s :

X,= Xc p- R COS 8;
Za= X c p - R sine.

(1.61)


During f l i g h t o v e r t h e c o r r e c ­
t i o n p o i n t , ? . e . , when t h i s p o i n t i s
o b s e r v e d a t a n a n g l e e q u a l t o 90° t o
t h e f l i g h t path, (1.61) is simplified
and t a k e s t h e form:
Fig. 1.44. Determining t h e
Orthodromic Coordinates of
an A i r c r a f t from a Correc­

tion Point.


xa= X C P




z,= Z c p - R .


Fig. 1.45.
T r a n s f e r of t h e Next Stage i n a Course t o an Orthodromic
Coordinate System:
( a > w i t h t h e A i r c r a f t P o s i t i o n on t h e P a t h o f
t h e G i v e n C o u r s e ; (b) w i t h D e v i a t i o n o f t h e A i r c r a f t f r o m t h e P a t h
of a Given Course.

The s i m p l i c i t y o f t h e g e o m e t r i c a l t r a n s f o r m a t i o n s a n d t h e n a ­
t u r a l p e r c e p t i o n of t h e c o o r d i n a t e s of t h e a i r c r a f t i n an orthodrome
s y s t e m , b o t h of t h e p a t h c o v e r e d by t h e a i r c r a f t and o f t h e d e v i a ­
t i o n a l l o w e d f r o m t h e g i v e n p a t h , make i t t h e m o s t a c c e p t a b l e c o o r ­
d i n a t e system f o r a given f l i g h t path.
I n high-speed aircraft ( a s a r e s u l t of a l a r g e t u r n i n g r a d i u s ) ,
i n o r d e r t o e m e r g e w i t h o u t d e v i a t i o n a t t h e n e x t s t a g e o f t h e or­
thodrome p a t h , it i s n e c e s s a r y t o c o n s i d e r t h e l i n e a r advance t o
the angle of turn ( T A ) .
This t r a n s f e r i s connected with transforma t i o n s o f t h e c o o r d i n a t e s of t h e a i r c r a f t from t h e orthodrome s y s ­
t e m of t h e preceding s t a g e t o t h e system of t h e following stage.
I n F i g u r e 1 . 4 5 , a , p o i n t M l o c a t e d on t h e f l i g h t p a t h o f t h e
preceding stage of f l i g h t is t h e point of the beginning of turn f o r
Obviously, t h e
a r r i v a l at the flight path of the following stage.
coordinates of t h i s p o i n t i n t h e system of t h e following s t a g e w i l l
be e q u a l t o
69


/73
-

(1.62)

I n g e n e r a l , when t h e c o o r d i n a t e Z a t t h e b e g i n n i n g o f t h e t u r n
if t h e aircraft is not located s t r i c t l y
o n t h e g i v e n f l i g h t p a t h when b e g i n n i n g t h e t u r n , t h e t r a n s f o r m a ­
t i o n of t h e c o o r d i n a t e s m u s t b e c a r r i e d o u t a c c o r d s n g t o t h e f o l ­
lowing formula (Fig. 1.45, b ) :

is n o t e q u a l t o z e r o , i . e . ,

x,= z sin ~n - x,re m cos T A;
z2= z yn + x r e m . sin T A.
COS



(1.63)


I n t h e p r o c e s s of t u r n i n g , t h e c o o r d i n a t e s o f t h e a i r c r a f t are
m e a s u r e d i n t h e s y s t e m o f t h e f o l l o w i n g p a r t of t h e f l i g h t i n w h i c h
their calculation after turning is carried out.
The o r t h o d r o m e s y s t e m examined
by us i s sometimes c a l l e d t h e stage
In
orthodromic coordinate system.
some i n s t a n c e s , a r e c t a n g u l a r c o ­
o r d i n a t e s y s t e m i s u s e d for f l i g h t
o v e r a n a r e a , e . g . , for m a n e u v e r ­
i n g of an aircraft i n t h e region
o f a n a i r p o r t a n d for s p e c i a l - . p u r ­
I n t h e s e cases,
pose f l i g h t s , etc.
t h e d i r e c t i o n of t h e meridian at
F i g . 1 . 4 6 . R e c t a n g u l a r Coort h e p o i n t o f o r i g i n of t h e coor­
d i n a t e S y s t e m for F l i g h t
d i n a t e s or some o t h e r d i r e c t i o n
[for e x a m p l e , t h e d i r e c t i o n of t h e
o v e r a n Area.
t a k e - o f f - l a n d i n g zone a t an a i r ­
p o r t ( F i g . 1 . 4 6 1 1 i s t a k e n as t h e X - a x i s , and a r e c t a n g u l a r c o o r ­
d i n a t e system i s c o n s t r u c t e d from t h i s .
The f l i g h t i s c a r r i e d o u t a l o n g t h e g i v e n c o o r d i n a t e s o f t h e
p o i n t s o f t h e r o u t e [for e x a m p l e , a l o n g t h e c o o r d i n a t e s o f t h e b e ­
g i n n i n g of e a c h o f t h e four t u r n s i n t h e r e c t a n g u l a r m a n e u v e r o f
m a k i n g a n a p p r o a c h t o l a n d a t a n a i r p o r t ( X , Z , ) , (X,Z,), ( X 3 Z 3 ) ,

(X4Z4)1.
The l i m i t s o f a p p l i c a b i l i t y o f a n a r e a l r e c t a n g u l a r c o o r d i n a t e
s y s t e m a r e l i m i t e d b y t h e e f f e c t o f t h e s p h e r i c i t y of t h e E a r t h o n
t h e p r e c i s i o n of measurements.
In practice, without noticeable dis­
t o r t i o n s , s u c h a s y s t e m c a n b e u s e d w i t h i n a r a d i u s o f 3 0 0 - 4 0 0 km
from t h e p o i n t of o r i g i n of t h e c o o r d i n a t e s .
I t i s a l s o a p p l i e d w i t h t h e u s e of n a v i g a t i o n a l i n d i c a t o r s i n
f l i g h t , when t h e o r t h o d r o m e d i r e c t i o n o f p a r t o f t h e c o u r s e i s t a k e n
as t h e X a x i s .

70

A r b i t r a r y (Ob1 i q u e and T r a n s v e r s e ) S p h e r i c a l
and P o l a r C o o r d i n a t e S y s t e m s

I n t h e s o l u t i o n o f n a v i g a t i o n a l problems w i t h a geographical
coordinate system i n p o l a r regions, very s i g n i f i c a n t e r r o r s arise.
A s p e c i a l c h a p t e r i s devoted t o problems o f accuracy i n air­
craft navigation.
I n t h e present s e c t i o n , f o r t h e purpose of i l l u s ­
t r a t i o n , o n l y ( 1 . 3 6 a ) i s examined.
I t i s obvious t h a t w i t h t h e approach of t h e a i r c r a f t
t u d e e q u a l t o 90°, t h e t a n g e n t 4 w i l l a p p r o a c h i n f i n i t y .
s m a l l e r r o r s i n m e a s u r i n g t h e l a t i t u d e of t h e l o c a t i o n o f
craft w i l l cause t h e e r r o r s i n c a l c u l a t i n g t h e longitude t
indefinitely.

t o a lati­
Therefore,
the air­
o grow

To a v o i d a l o s s o f a c c u r a c y i n s o l v i n g n a v i g a t i o n a l p r o b l e m s ,
e s p e c i a l l y by a u t o m a t i c n a v i g a t i o n a l d e v i c e s , r a n d o m s p h e r i c a l c o ­
o r d i n a t e systems are employed.
P

A

Fig. 1.47. Transformation
of S p h e r i c a l C o o r d i n a t e s

on t h e E a r t h ' s S u r f a c e .

Arbitrary s p h e r i c a l systems d i f f e r from a g e o g r a p h i c a l s y s t e m by t h e
fact t h a t t h e poles of t h e s e systems
do n o t c o i n c i d e w i t h t h e g e o g r a p h i c
poles.
Therefore, i n t h e s e systems
a l l the analytical transformations of
d i s t a n c e s and d i r e c t i o n s which a r e
c a r r i e d o u t for a g e o g r a p h i c c o o r d i nate system a r e j u s t i f i e d .
For t r a n s f e r r i n g from a geograph­
i c a l c o o r d i n a t e system t o an a r b i t r a r y
s p h e r i c a l s y s t e m , or v i c e v e r s a , i t
i s n e c e s s a r y t o d e r i v e s p e c i a l equations:
l e t us examine F i g u r e 1 . 4 7 .

In Figure 1.47, a cross section

Here
o f t h e E a r t h ' s s p h e r e i s shown.
t h e p l a n e o f t h e c r o s s s e c t i o n i s c h o s e n i n s u c h a way a s t o p a s s
through t h e c e n t e r o f t h e E a r t h and t h e p o l e s o f t h e geographic s y s ­
t e m and t h e a r b i t r a r y c o o r d i n a t e s y s t e m s , i . e . , s o a s t o appear as
t h e p l a n e o f t h e m e r i d i a n i n both t h e geographic and a r b i t r a r y s y s ­
t e m s simultaneously.
I t i s obvious t h a t such a plane e x i s t s with
any d i s t r i b u t i o n o f t h e p o l e s o f an a r b i t r a r y s p h e r i c a l system.
L e t us a g r e e t h a t a r e a d i n g o f t h e l o n g i t u d e b o t h i n t h e geo­
g r a p h i c a l and a r b i t r a r y systems w i l l run from t h e i n d i c a t e d p l a n e
of i n t e r s e c t i o n .
L i n e s AB and A l B l i n F i g u r e 1 . 4 7 , and t h e l i n e s
p a r a l l e l t o t h e m , a p p e a r as l i n e s of i n t e r s e c t i o n w i t h t h e p l a n e s
o f t h e e q u a t o r and t h e p a r a l l e l s i n t h e g e o g r a p h i c a l and a r b i t r a r y
systems.
P o i n t P i s t h e p o l e of t h e geographic c o o r d i n a t e system;
P1 i s t h e p o l e o f t h e a r b i t r a r y system; angle e i s a combination of
t h e a x e s o f t h e g e o g r a p h i c a l a n d random s y s t e m s .

71


/75
-

L e t u s c h o o s e p o i n t M ( 9 1 x 1 ) on t h e E a r t h ' s s u r f a c e a n d p r o ­
j e c t i t o n t o t h e p l a n e o f t h e c r o s s s e c t i o n ( p o i n t Mi).
I t is ob­
v i o u s t h a t OL w i l l a p p e a r a s t h e l i n e o f t h e s i n e s o f t h e l a t i t u d e
o f p o i n t M i n a n a r b i t r a r y s y s t e m , w h i l e LM1 w i l l a p p e a r a s t h e
l i n e of t h e c o s i n e s o f t h e l o n g i t u d e o f p o i n t M of t h i s system i n
the plane of i t s p a r a l l e l , i.e.,
LM1=

The l a t i t u d e of p o i n t

or

COS A1 COS ~ 1 .

M i n t h e geographical system w i l l equal:
sin 'p = Ot cob e - LM,sin e

sin 'p = sin

cos 0 -cos

A, cos yl sin 0.

(1.64)

I t i s obvious t h a t a p e r p e n d i c u l a r dropped from p o i n t M t o t h e
p l a n e o f i n t e r s e c t i o n ( p o i n t M i ) w i l l b e t h e l i n e o f t h e s i n e of
t h e l o n g i t u d e i n t h e a r b i t r a r y s y s t e m i n t h e p l a n e of t h e p a r a l l e l
o f t h i s p o i n t ; a t t h e same t i m e , t h e l i n e o f t h e s i n e o f t h e l o n g i ­
t u d e i n t h e p l a n e o f t h e p a r a l l e l of t h e p o i n t i n t h e g e o g r a p h i c
c o o r d i n a t e system w i l l be
sin A = sin AI cos 'pl sec y.

from which it f o l l o w s t h a t
MMI -=sin A, cos y1 =sin A COS 'p,

(1.65)

F o r m u l a s ( 1 . 6 4 ) a n d ( 1 . 6 5 ) make i t p o s s i b l e t o d e t e r m i n e t h e
coordinates of a p o i n t i n a geographical coordinate system accord­
i n g t o i t s c o o r d i n a t e s , known i n t h e a r b i t r a r y s y s t e m u n d e r t h e
condition t h a t t h e plane coinciding with t h e axes of both systems
i s t a k e n as t h e i n i t i a l m e r i d i a n .
A f t e r s o l v i n g t h e problems ac­
cording t o (1.64) and ( 1 . 6 5 ) , it i s n e c e s s a r y t o i n t r o d u c e a cor­
r e c t i o n i n t o t h e A c o o r d i n a t e e q u a l t o t h e l o n g i t u d e of t h e P I
pole i n t h e geographic coordinate system.
S i n c e t h e p r i n c i p l e s o f c o n s t r u c t i o n o f s p h e r i c a l and geograph­
i c coordinate systems are i d e n t i c a l , f o r t h e s o l u t i o n of t h e r e v e r s e
t a s k ( t r a n s f e r r i n g from t h e geographic system t o t h e a r b i t r a r y o n e ) ,
i t i s s u f f i c i e n t t o d r o p t h e s u b s c r i p t s i n t h e f u n c t i o n s of t h e co­
o r d i n a t e s o f ( 1 . 6 4 ) and ( 1 . 6 5 ) w h e r e e v e r t h e y o c c u r and t o add them
where t h e y are a b s e n t :

-

-

sin TI -sin 'p COS 0 COS k COS 'p sin'$;
sin AI sin k cos 'p sec ql.

F o r m u l a s ( 1 . 6 4 ) a n d ( 1 . 6 5 ) were g i v e n w i t h a c o n s i d e r a t i o n o f t h e
f l a t t e n i n g o f t h e E a r t h a t t h e p o l e s , ? . e . , t h e E a r t h was t a k e n a s
a s p h e r e w i t h a mean r a d i u s .

72

176
-

Position L i n e s of

a n A i r c r a f t on t h e E a r t h ' s

Surface

Thus f a r , w e h a v e examined c o o r d i n a t e s y s t e m s on t h e E a r t h ' s
s u r f a c e as s y s t e m s which c o n n e c t t h e p o s i t i o n o f a n a i r c r a f t w i t h
t h e E a r t h ' s s u r f a c e d u r i n g i t s movement i n a g i v e n d i r e c t i o n .
I n aircraft n a v i g a t i o n , it i s o f t e n necessary t o determine t h e
e l e m e n t s of a i r c r a f t movement a c c o r d i n g t o c o n s e c u t i v e c o o r d i n a t e s .
I t i s o b v i o u s t h a t means and methods f o r m e a s u r i n g t h e c o o r d i n a t e s
of a n a i r c r a f t a r e n e c e s s a r y f o r t h i s p u r p o s e .
Usually t h e two-dimensional s u r f a c e c o o r d i n a t e s of an aircraft
a r e d e t e r m i n e d s e p a r a t e l y a c c o r d i n g t o two l i n e s o f t h e a i r c r a f t ' s
p o s i t i o n m e a s u r e d a t d i f f e r e n t t i m e s or a c c o r d i n g t o t w o l i n e s
measured s i m u l t a n e o u s l y .
I n some c a s e s , i t i s s u f f i c i e n t t o d e t e r ­
mine one l i n e o f t h e a i r c r a f t ' s p o s i t i o n .
T h e g e o m e t r i c l o c u s o f p o i n t s of t h e p r o b a b l e l o c a t i o n o f a n
a i r c r a f t o n t h e E a r t h ' s s u r f a c e i s c a l l e d t h e p o s i t i o n Z i n e of an
aircraft.
Similar groups of a i r c r a f t p o s i t i o n l i n e s are c a l l e d a
f a m i Zy of p o s i t i o n Z i n e s .
For example, i f t h e l a t i t u d e of t h e l o c a t i o n of an a i r c r a f t
i s d e t e r m i n e d by a s t r o n o m i c means b a s e d on t h e e l e v a t i o n o f P o l a r i s ,
t h e p a r a l l e l on w h i c h t h e a i r c r a f t i s l o c a t e d w i l l b e a p o s i t i o n
l i n e of t h e second family.

L e t u s a s s u m e t h a t t h e l o n g i t u d e of a n a i r c r a f t w a s d e t e r m i n e d
s i m u l t a n e o u s l y on t h e b a s i s o f t h e a l t i t u d e o f a s t a r , t h e a z i m u t h
o f w h i c h i s e q u a l t o 9 0 or 2 7 0 O .
The l o n g i t u d e o b t a i n e d by s u c h a
method i s a p o s i t i o n l i n e o f t h e s e c o n d f a m i l y .
Direct measurement o f t h e geographic c o o r d i n a t e s o f an a i r c r a f t
i s p o s s i b l e o n l y by a s t r o n o m i c m e a n s , a n d n o t i n a l l c a s e s .

I n d e t e r m i n i n g t h e l o c a t i o n o f a n a i r c r a f t b y o p t i c a l or r a d i o ­
m e t r i c means, t h e f a m i l i e s o f p o s i t i o n l i n e s g e n e r a l l y do n o t c o i n ­
c i d e e i t h e r w i t h t h e g r i d o f g e o g r a p h i c c o o r d i n a t e s or w i t h t h e
given f l i g h t d i r e c t i s n .
A t t h e present t i m e , t h e r e are several types of coordinate
systems which are used as f a m i l i e s o f a i r c r a f t p o s i t i o n l i n e s i n
t h e a p p l i c a t i o n o f r a d i o - e n g i n e e r i n g and a s t r o n o m i c f a c i l i t i e s o f
aircraft navigation.
They i n c l u d e t h e f o l l o w i n g :

1) A two-poZe a z i m u t h a Z s y s t e m , i n w h i c h t h e r a d i a l l i n e s
( b e a r i n g s ) d i v e r g i n g f r o m two p o i n t s on t h e E a r t h ' s s u r f a c e w i t h
known c o o r d i n a t e s a r e f a m i l i e s o f p o s i t i o n l i n e s .
2)
P o l a r o r azimuthaZ r a n g e - f i n d i n g s y s t e m , i n which t h e b e a r ­
i n g s f r o m a p o i n t o n t h e E a r t h ' s s u r f a c e w i t h known c o o r d i n a t e s a r e
t h e first f a m i l y o f p o s i t i o n l i n e s of t h i s system, and c o n c e n t r i c

73

/77
-

.II

.
.
I
111 1
I

I 1 1 I ~ I I . ~ - 1 1 1 1 1 1 . 1

II 1111111111

1111 I I1111 Im-1111

.
I
.
.I.
I I

11.111

II

.

11.1

I,

. .1
1

,.,...,,...,.

c i r c l e s a t e q u a l d i s t a n c e s from t h e i n d i c a t e d p o i n t
family

.

,

,

. . , . .., .

._ .._

are t h e second

3)
L i n e s of e q u a l a z i m u t h s ( L E A ) , w h i c h a r e p o s i t i o n l i n e s
r e l a t i v e t o known p o i n t s o n t h e E a r t h ' s s u r f a c e , a t e a c h o f w h i c h
t h e a z i m u t h o f a known p o i n t r e t a i n s a c o n s t a n t v a l u e .
4)
Difference-rangefinding (hyperbolic system), i n which each
family of p o s i t i o n l i n e s i s b i p o l a r ; a constant difference of dis­
t a n c e s t o t h e p o l e s of t h e s y s t e m i s p r e s e r v e d on e a c h p o s i t i o n
line.
5)
Over-aZZ r a n g e f i n d i n g ( e Z Z i p t i c a Z ) s y s t e m , i n w h i c h t h e
f a m i l y o f p o s i t i o n l i n e s i s b i p o l a r ; a c o n s t a n t sum o f t h e d i s t a n c e s
t o t h e p o l e s o f t h e s y s t e m i s p r e s e r v e d on t h e p o s i t i o n l i n e s .
6)
c o n f o c a z h y p e r b o Z i c - e Z Z i p t i c a Z s y s t e m , i n w h i c h t h e fam­
i l i e s of p o s i t i o n l i n e s a r e e l l i p s e s and hyperbolas confocal with
them.

From t h e a b o v e l i s t o f c o o r d i n a t e s y s t e m s , i t i s e v i d e n t t h a t
each has a r i s e n from t h e n a t u r e of t h e n a v i g a t i o n a l v a l u e s measured
by t h e d e v i c e s u s e d .
The i n d i c a t e d v a l u e s a r e c a l l e d n a v i g a t i o n a Z

parameters.
For.example,
f o r a hyperbolic system t h e difference i n d i s ­
t a n c e s s e r v e s as a n a v i g a t i o n a l p a r a m e t e r , and i n an a z i m u t h a l s y s ­
tem ( o r f o r l i n e s o f e q u a l a z i m u t h s ) t h e a z i m u t h s e r v e s 'as a n a v i ­
gational parameter, e t c .
I n e v a l u a t i o n s of t h e accuracy of n a v i g a t i o n a l measurements,
considering t h a t t h e i n t e r s e c t i n g segments of t h e p o s i t i o n l i n e s
of any s y s t e m can be assumed t o be s t r a i g h t - l i n e segments i n t h e
region of t h e l o c a t i o n of t h e a i r c r a f t , t h e concept of a u n i f i e d
coordinate system i s sometimes i n t r o d u c e d f o r t h e purpose of study­
i n g t h e g e n e r a l p r o p e r t i e s o f a l l t h e above systems, i n c l u d i n g t h e
g e o g r a p h i c and orthodrome s y s t e m s .
I n s t u d y i n g t h e s e c o o r d i n a t e s y s t e m s , it i s n e c e s s a r y t o con­
n e c t e a c h o f them w i t h t h e g e o g r a p h i c s y s t e m f o r l o c a t i n g t h e i n ­
t e r m e d i a t e p o i n t s o f t h e p o s i t i o n l i n e s , i n o r d e r t o p l o t t h e m on a
map.
I n a d d i t i o n , i t i s n e c e s s a r y t o know t h e a n a l y t i c a l f o r m f o r
d e t e r m i n i n g t h e c o o r d i n a t e s o f a n a i r c r a f t i n a g e o g r a p h i c or o r t h o ­
d r o m e s y s t e m o n t h e b a s i s o f known p a r a m e t e r s o f n a v i g a t i o n a l s y s ­
t e m s w i t h o u t p l o t t i n g p o s i t i o n l i n e s on t h e map, as i s done i n a u t o ­
matic navigational devices.
B i p o l a r Azimuthal

C o o r d i n a t e System

B e a r i n g s f o r an a i r c r a f t , i . e . , orthodrome l i n e s d i v e r g i n g
f r o m t w o p o i n t s o n t h e E a r t h ' s s u r f a c e w i t h known c o o r d i n a t e s , a r e
p o s i t i o n l i n e s i n t h e azimuthal coordinate system.

74

I


L e t u s a s s u m e t h a t w e h a v e t w o p o i n t s 01 a n d 0 2 o n t h e E a r t h ' s
s u r f a c e ( F i g . 1.48).
I f t h e map b e i n g u s e d h a s b e e n e x e c u t e d o n a p r o j e c t i o n h a v i n g
t h e p r o p e r t i e s o f i s o g o n a l i t y a n d o r t h o d r o m i c i t y , e . g . , on a n i n t e r n a t i o n a l p r o j e c t i o n , t h e i n d i c a t e d p o s i t i o n l i n e s o n t h e map c a n b e
t a k e n a s s t r a i g h t l i n e s o r i g i n a t i n g a t p o i n t s 01 a n d 0 2 .

However, s a t i s f a c t o r y a c c u r a c y
i n determining t h e coordinates of
an aircraft a t t h e i n t e r s e c t i o n of
t h e b e a r i n g s a s s t r a i g h t l i n e s on
a map i s p r e s e r v e d a t c o m p a r a t i v e l y
s m a l l d i s t a n c e s a n d o n l y on maps
with an i n t e r n a t i o n a l p r o j e c t i o n .
In general, f o r the precise
p l o t t i n g o f p o s i t i o n l i n e s on a
map, l e t u s c o n s i d e r t h e p o i n t s 01
F i g . 1.48. B i p o l a r A z i m u t h a l
and 0 2 a s p o l e s o f an a r b i t r a r y
Let
Coordinate System.
s p h e r i c a l coordinate system.
us c o n s i d e r t h e d i s t a n c e s S from
t h e s e p o i n t s t o a n y p o i n t on t h e E a r t h ' s s u r f a c e M a s c o m p l e m e n t s
o f t h e l a t i t u d e o f p o i n t M i n t h e s e c o o r d i n a t e s y s t e m s , up t o 9 0 ° :
S1 = 01M = 90'

- '11;

S2 = OzM = 90"- 'p2-

I n t h i s case, t h e c o o r d i n a t e s of p o i n t M i n t h e g e o g r a p h i c a l
s y s t e m a r e d e t e r m i n e d a c c o r d i n g t o (1.64) a n d ( 1 . 6 5 ) .
T a k i n g t h e m e r i d i a n o f p o i n t 01 a s t h e p r i m e m e r i d i a n o f t h e
g e o g r a p h i c s y s t e m , 0 1 a s t h e a z i m u t h a for t h e l o n g i t u d e i n t h e
s p h e r i c a l s y s t e m , a n d a v a l u e 0 f . 9 0 ~f o r SI a s t h e l a t i t u d e i n t h i s
system, l e t us obtain ( i n t h e geographical coordinate system)
sin 'p = cos S1cos 0 - sin a1sin SIsin 0,

where
0 = 90"- 'Po,;

sin A = sin al sin S1sec 'p.

G i v e n t h e d e f i n i t e v a l u e SI a n d s u b s t i t u t i n g d i f f e r e n t v a l u e s
f o r cil, e . g . , g r e a t e r t h a n l o , f o r m u l a s l e t u s u s e t h e g i v e n f o r m u ­
las t o find the coordinates of the points of intersection of the
a z i m u t h a l l i n e s w i t h t h e c i r c l e o f e q u a l d i s t a n c e SI i n t h e g e o g r a p h ­
i c system.
G i v e n a n o t h e r v a l u e �or S 1 a n d h a v i n g c a r r i e d o u t t h e same
o p e r a t i o n s w i t h "1, w e w i l l o b t a i n t h e c o o r d i n a t e s o f t h e p o i n t s o f
intersection of the azimuthal l i n e s with a circle of equal distance
having t h i s radius.
C o n t i n u i n g t o i n c r e a s e S1 t o a f u l l r a d i u s o f

operation of a
75

/78

n a v i g a t i o n a l d e v i c e , l e t us o b t a i n t h e c o o r d i n a t e s o f t h e interme­
d i a t e p o i n t s o f t h e a z i m u t h a l p o s i t i o n l i n e s , r u n n i n g f r o m p o l e 01
i n t h e g e o g r a p h i c a l c o o r d i n a t e system, w i t h t h e l o n g i t u d e changed
Introducing a correction i n the values of the
t o the value hol.
l o n g i t u d e s o f t h e i n t e r m e d i a t e p o i n t s f o r t h e i n d i c a t e d v a l u e A,,
l e t us o b t a i n t h e l o n g i t u d e s of t h e s e p o i n t s from t h e prime meridian
of t h e geographic system.
On a map o f a n y p r o j e c t i o n , b y j o i n i n g
t h e p o i n t s o b t a i n e d b y l i n e s r u n n i n g f r o m p o i n t 01, we w i l l o b t a i n
position l i n e s of t h e first family.
I n t h i s way, it i s p o s s i b l e t o o b t a i n t h e f a m i l y o f p o s i t i o n
l i n e s from p o i n t 0 2 , t a k i n g it as t h e p o l e of t h e second a r b i t r a r y
s p h e r i c a l coordinate system.
L e t u s now d e t e r m i n e t h e c o o r d i n a t e s o f p o i n t M i n t h e g e o ­
g r a p h i c a l s y s t e m , b a s e d o n known a z i m u t h s m e a s u r e d a t p o i n t s 0 1 a n d
0 2 , without recourse t o t h e p l o t t i n g of position l i n e s .
Let us
f i r s t s o l v e t h i s problem i n t h e s p h e r i c a l system o f one o f t h e p o l e s
of a navigational device, e.g., 02 (see Fig. 1.481, taking t h e azi­
m u t h o f p o i n t 01 a s t h e p r i m e m e r i d i a n .
According t o ( 1 . 6 4 ) and ( 1 . 6 5 ) , t h e c o o r d i n a t e s of p o i n t M i n
t h i s system a r e d e t e r m i n e d by t h e e q u a t i o n s :
sin 'pl = sin 9' , cos 8

where

8 is the

-cus A2 cos cpz sin 0,

a n g u l a r d i s t a n c e o f 0102;
s i n 11 = s i n A, cos 'p2 sec:cpl,

w h e r e A l h 2 a r e r e s p e c t i v e l y cil, " 2 .
From ( 1 . 6 5 )

it i s e v i d e n t t h a t
sinkl
--sinh,

- cos%

cos'pl

Substituting i n t o (1.52),
i n g t o ( 1 . 6 4 ) , we o b t a i n :

sink2
or---.
,sinkl

­

cosyl
coscpz

(1.66)

i n s t e a d o f c o s $1 i t s v a l u e a c c o r d ­

sin A2

-- tgcpz cos e - cos A2 s i n e

s i n AI

or
tg y2 = sin A, cosec hl sec 0

+ COS A,

tg 0.

(1.67)

Since t h e azimuth of p o i n t M i n t h e 02 system i s considered
k n o w n , we o b t a i n e d b o t h c o o r d i n a t e s o f p o i n t M i n t h i s s y s t e m .

For t r a n s f e r r i n g t o t h e g e o g r a p h i c a l c o o r d i n a t e s y s t e m , i t i s
a g a i n p o s s i b l e t o u s e ( 1 . 6 4 ) and ( 1 . 6 5 ) , c o n s i d e r i n g as a n g l e 0
t h e v a l u e $ 1 ~ 2 , a n d as t h e p r i m e m e r i d i a n t h e l o n g i t u d e o f t h e p o i n t
02 '

76

-.

..__..

/79
-

I t is obvious t h a t here it is necessary t o introduce a correc­
t i o n i n t o t h e A 2 c o o r d i n a t e f o r t h e v a l u e o f t h e a z i m u t h of p o i n t
0 1 f r o m p o i n t 0 2 , i . e . , t h e c o r r e c t e d v a l u e o f A2 w i l l e q u a l :

(1.68)
I t i s a l s o obvious t h a t after transforming t h e coordinates i n t o ­
/80
g e o g r a p h i c a l o n e s , it i s n e c e s s a r y t o i n t r o d u c e a c o r r e c t i o n i n t o
c o o r d i n a t e A2 f o r t h e l o n g i t u d e o f p o i n t 0 2 :
A,=

(1.69)

A -I.,,A

F o r m u l a s ( 1 . 6 4 ) a n d ( 1 . 6 5 ) a l s o make i t p o s s i b l e t o i m p l e m e n t
a t r a n s f e r from a s p h e r i c a l system w i t h p o l e 0 2 t o t h e orthodrome
system.
This i s n e c e s s a r y f o r determining t h e p o s i t i o n of an air­
c r a f t relative t o a given f l i g h t path.
A c t u a l l y , it is p o s s i b l e t o
c o n s i d e r t h e orthodrome system as a s p h e r i c a l system i f w e measure
t h e X- and Z - c o o r d i n a t e s n o t a s l i n e a r b u t as a n g u l a r m e a s u r e s ,
i . e . , w e t a k e t h e X - c o o r d i n a t e as A and t h e Z a s $.
I n t h i s i n s t a n c e , it i s advantageous t o t a k e t h e X-coordinate
o f p o i n t 0 2 a s t h e p r i m e m e r i d i a n a n d t h e v a l u e 90° - Z o f t h i s
p o i n t as t h e a n g l e 0
The c o o r d i n a t e s o f p o i n t M i n t h e o r t h o ­
drome s y s t e m w i l l t h e n e q u a l :

.

sin X = sin 'p2 COS B

- COS A,cos 'p2 sin 8;

?In Z = sin A, cos 'p2 sec X .

If t h e X o - c o o r d i n a t e , n o t e q u a l t o X o 2 , i s t a k e n a s t h e prime
meridian, then after transforming the coordinates according t o
( 1 . 6 4 ) and ( 1 . 6 5 ) a c o r r e c t i o n e q u a l t o Xo2 i s i n t r o d u c e d i n t h e
XM c o o r d i n a t e .

Goniometric Range-Finding

Coordinate System

The g o n i o m e t r i c r a n g e - f i n d i n g s y s t e m i s t h e m o s t c o n v e n i e n t
s y s t e m f o r c o n v e r s i o n t o t h e g e o g r a p h i c or o r t h o d r o m e s y s t e m .
Since d i r e c t i o n and d i s t a n c e are measured s i m u l t a n e o u s l y i n
t h i s system, f o r conversion t o t h e geographical system it i s s u f f i ­
c i e n t t o u s e ( 1 . 6 4 ) a n d ( 1 . 6 5 ) , t a k i n g t h e v a l u e 90° - $ o l for
a n g l e 0 and A o l f o r t h e prime m e r i d i a n .
I n t h i s i n s t a n c e t h e v a l u e 90° - S i s c o n s i d e r e d t h e l a t i t u d e
o f t h e p o i n t M i n t h e c o o r d i n a t e s y s t e m w i t h p o l e 01, w h i l e t h e a z i ­
muth o f p o i n t M i s c o n s i d e r e d as t h e l o n g i t u d e .
In the geographical
system, t h e coordinates of t h e point M w i l l equal:
sln 'p = sln '4, cos B -cos A1 cos 'pl sin 8;
sin A = sin k1 cos 'pl'sec'p.

A f t e r t r a n s f o r m a t i o n , it i s n e c e s s a r y t o i n t r o d u c e a c o r r e c t i o n
77

t o t h e c o o r d i n a t e A f o r t h e l o n g i t u d e o f p o i n t 01:

(1.70)

The c o n v e r s i o n t o t h e o r t h o d r o m i c c o o r d i n a t e s y s t e m i s i m p l e ­
m e n t e d i n t h e same m a n n e r a s w a s d o n e i n t h e b i p o l a r a z i m u t h a l s y s ­
t e m after solving (1.67).
If t h e r a d i u s o f a c t i o n o f t h e b o n i o m e t e r r a n g e - f i n d i n g c o o r d i n a t e system i s s m a l l (on t h e o r d e r of 300-400 km), it i s p o s s i b l e
t o d i s r e g a r d t h e s p h e r i c i t y o f t h e E a r t h i n c o n v e r t i n g t o t h e or­
thodrome system and t h e problem of t r a n s f e r i s c o n s i d e r a b l y s i m p l i ­
f i e d (Fig. 1.49).

i t i s e v i d e n t t h a t w i t h known v a l u e s o f R a n d
system, t h e coordinates of point
M i n t h e o r t h o d r o m e s y s t e m c a n b e d e t e r m i n e d a c c o r d i n g t o t h e fol­
lowing formulas:
In Figure 1.49,

a i n t h e goniometer range-finding

X ='Xoi $. R COS (U
Z = Zol

- +);

+ R sin (a-+),

(1.71)
(1.71a)

+

i s t h e d i r e c t i o n of t h e orthodrome segment of t h e p a t h r e l a ­
where
t i v e t o p o i n t 01.
Bipolar Range-Finding

(Circular) Coordinate System

I n a b i p o l a r r a n g e - f i n d i n g s y s t e m ( F i g . 1.50), t h e d i s t a n c e t o
t w o p o i n t s o n t h e E a r t h ' s s u r f a c e w i t h known c o o r d i n a t e s i s a meas­
ured navigational parameter.

Fig. 1.49. Conversion of Polar
( G o n i o m e t e r - R a n g e - F i n d i n g ) Coo r d i n a t e s t o Orthodromic Coor­
dinates.

F i g . 1 . 5 0 . e i p o l a r Range-Find­
i n g Coordinate System.

The i n d i c a t e d d i s t a n c e i s u s u a l l y d e t e r m i n e d a c c o r d i n g t o t h e
t i m e o f passage o f r a d i o s i g n a l s from t h e a i r c r a f t t o t h e ground
r a d i o - r e l a y equipment and back t o t h e a i r c r a f t .

78

/81
-

In Figure 1.50, it is evident t h a t t h e task of determining t h e
coordinates of an aircraft i n a c i r c u l a r system i s double-valued.
The p o i n t o f i n t e r s e c t i o n o f t h e c i r c l e s o f e q u a l d i s t a n c e t o t h e
p o l e s 01 a n d 02 i s c o n s i d e r e d t h e l o c a t i o n o f - t h e a i r c r a f t .
Since
t h e r e a r e t w o s u c h p o i n t s for a n y p a i r o f c i r c l e s , a d d i t i o n a l s i g n s
a r e u s e d for c h o o s i n g t h e a c t u a l p o i n t , e . g . :
a)

Provisional

a i r c r a f t p o s i t i o n a t t h e moment o f m e a s u r e m e n t .

b)
Tendency toward a change i n d i s t a n c e d u r i n g f l i g h t i n a
definite direction.
I n Figure 1.50, i t i s obvious t h a t i n f l y i n g from n o r t h t o
s o u t h , t h e d i s t a n c e s R1 a n d R2 w i l l d e c r e a s e a t p o i n t M a n d i n c r e a s e
a t point MI.
On maps w i t h d i f f e r e n t p r o j e c t i o n s , c i r c u l a r p o s i t i o n l i n e s
w i l l have a d i f f e r e n t appearance.
U s u a l l y t h e y a r e p l o t t e d o n maps
on a n o b l i q u e c e n t r a l . a n d i n t e r n a t i o n a l p r o j e c t i o n o n w h i c h t h e a p ­
proximate form o f t h e c i r c l e s i s p r e s e r v e d .
F o r p l o t t i n g t h e i n d i c a t e d l i n e s on a map w i t h a n y p r o j e c t i o n ,
it i s necessary t o determine t h e coordinates of t h e i r intermediate
points.
T h i s p r o b l e m i s s o l v e d i n t h e s a m e way a s f o r b i p o l a r a z i ­
muthal systems, with t h e s o l e difference being t h a t a f t e r determin­
ing the coordinates of intermediate points, the l a t t e r are not
j o i n e d by r a d i a l p o s i t i o n l i n e s , b u t by c i r c u l a r l i n e s .

I n c o n v e r t i n g f r o m a c i r c u l a r t o a g e o g r a p h i c or o r t h o d r o m e
s y s t e m , it i s n e c e s s a r y f i r s t t o d e t e r m i n e t h e c o o r d i n a t e s o f p o i n t
M i n t h e s p h e r i c a l s y s t e m r e l a t i v e t o one o f t h e p o l e s o f t h e c i r ­
c u l a r system.
C o n s i d e r i n g t h e l i n e 0102 a s t h e i n i t i a l m e r i d i a n o f t h i s s y s ­
t e m , t h e l a t i t u d e of p o i n t M i n t h e s y s t e m 01 a c c o r d i n g t o ( 1 . 6 4 )
w i l l be
sin.yl = sin y2 cos 0

- cos A,

cos yz sin 0 ,

w h e r e 9 i s t h e a n g u l a r d i s t a n c e b e t w e e n p o i n t s 01 a n d 0 2 ; $ 1 ; $ 2
a r e 9 0 ° - R1 a n d 90° - R 2 , r e s p e c t i v e l y .
Carrying out simple transformations,
cos A, =

sin y, cos 0.- sin il
cos 7, sin 0

we obtain:
(1.72)

'

-.
Formula ( 1 . 7 2 ) makes it p o s s i b l e t o d e t e r m i n e t h e A - c o o r d i n a t e
Since t h e $-coordinate i n t h i s system i s deter­
i n t h e 0 2 system.
m i n e d d i r e c t l y a s 90° - R 2 , i t i s p o s s i b l e t o c o n s i d e r t h e p r o b l e m
solved.

79

/82

T h e c o n v e r s i o n t o t h e g e o g r a p h i c or o r t h o d r o m e s y s t e m i s i m ­
p l e m e n t e d b y t h e same means a s i n t h e a z i m u t h a l a n d g o n i o m e t e r
range-finding systems.
Lines o f Equal Azimuths

L i n e s o f e q u a l a z i m u t h s (LEA) a r e a f a m i l y o f a i r c r a f t p o s i t i o n
l i n e s w h i c h c o n v e r g e a t one p o i n t on t h e E a r t h ‘ s s u r f a c e , on e a c h
o f w h i c h t h e a z i m u t h o f t h e known p o i n t r e t a i n s a c o n s t a n t v a l u e
(Fig. 1.51).
F o r f i n d i n g t h e l o c a t i o n o f a n a i r c r a f t , it i s u s e d a l o n g one
l i n e of e q u a l a z i m u t h s o f t w o f a m i l i e s , a s i s d o n e a l o n g two b e a r ­
i n g s i n an a z i m u t h a l b i p o l a r s y s t e m .
L i n e s o f e q u a l a z i m u t h s w e r e w i d e l y u s e d i n t h e p e r i o d when t h e
radiocompass ( a i r c r a f t r a d i o g o n i o m e t e r ) , measuring t h e d i s t a n c e from
t h e a i r c r a f t t o t h e ground r a d i o s t a t i o n , w a s t h e most r e f i n e d nav­
igational facility.
Along w i t h l i n e s of e q u a l a z i m u t h s , a method o f d e t e r m i n i n g
t h e c o o r d i n a t e s o f an a i r c r a f t by p l o t t i n g b e a r i n g s from a r a d i o
s t a t i o n t o an aircraft ( t a k i n g account of t h e convergence of t h e
m e r i d i a n s b e t w e e n t h e m ) h a s become w i d e s p r e a d .
An a d v a n + a g e o f t h e l i n e s o f e q u a l a z i m u t h s , i n c o m p a r i s o n w i t h
b e a r i n g s f o r a n a i r c r a f t , i s t h e f a c t t h a t t h e s o l u t i o n of t h e p r o b ­
l e m of determining an a i r c r a f t ’ s coordinates i s independent of i t s
l o c a t i o n , w h e r e a s i n o r d e r t o p l o t b e a r i n g s i t i s n e c e s s a r y t o know
t h e a p p r o x i m a t e c o o r d i n a t e s o f t h e a i r c r a f t for c a l c u l a t i n g t h e c o n ­
vergence of t h e meridians.
I n e x a m i n i n g l i n e s o f e q u a l a z i m u t h s , t h e r e i s no s e n s e i n de­
r i v i n g an a n a l y t i c a l form o f t r a n s f o r m a t i o n s f o r c o n v e r t i n g t o t h e

F i g . 1 . 5 1 . Line of Equal Azim u t h s (LEA).

80


Fig. 1.52. Determining t h e
Coordinates of Intermediate
P o i n t s o f a n LEA.

­
/83


g e o g r a p h i c or o r t h o d r o m i c c o o r d i n a t e s y s t e m .
L e t us l i m i t o u r s e l v e s
t o an e x a m i n a t i o n of t h e means of c a l c u l a t i n g i n t e r m e d i a t e p o i n t s
f o r p l o t t i n g t h e m o n a map i n o r d e r t o make i t p o s s i b l e t o d e t e r m i n e
t h e c o o r d i n a t e s of an a i r c r a f t a c c o r d i n g t o t h e i n t e r s e c t i o n o f t h e
l i n e s o f e q u a l a z i m u t h s o f t w o f a m i l i e s o n a map w i t h a n y p r o j e c t i o n .
I n F i g u r e 1 . 5 2 , one o f t h e l i n e s o f e q u a l a z i m u t h s o f a f a m i l y
c o n v e r g i n g a t p o > i n t M i s shown.
A t t h i s p o i n t , t h e orthodromes i n ­
t e r s e c t i n g t h e equator a t d i f f e r e n t angles a o l , ao2, e t c . converge.
A c c o r d i n g t o ( 1 . 3 2 ) , i t i s p o s s i b l e t o f i n d t h e l o n g i t u d e A0
of t h e p o i n t s of i n t e r s e c t i o n of a f a m i l y o f orthodromes with t h e
e q u a t o r , given t h e v a l u e s of t h e i n i t i a l a n g l e s "0.
A c c o r d i n g t o (1.44),
cos vs = cos 'p nn a.

Or s i n c e c o s c$B = s i n a o ,
cos 'p =

sin a,,
sin a .

(1.73)

Formula ( 1 . 7 3 ) makes it p o s s i b l e t o d e t e r m i n e t h e l a t i t u d e o f
a p o i n t on any l i n e of t h e f a m i l y o f o r t h o d r o m e s which c o n v e r g e a t
p o i n t M, where t h e azimuth o f p o i n t M i s e q u a l t o t h e given v a l u e
of a.
The l o n g i t u d e o f t h e i n d i c a t e d p o i n t c a n b e d e t e r m i n e d a c c o r d ­
i n g t o ( 1 . 3 6 a ) by s u b s t i t u t i n g i n t o i t t h e g i v e n i n i t i a l a n g l e o f
t h e orthodrome and t h e l a t i t u d e o b t a i n e d from ( 1 . 7 3 ) .
I t i s obvious
t h a t t h e l o n g i t u d e o b t a i n e d w i l l be measured from t h e s t a r t i n g
p o i n t s o f t h e family of orthodromes.
Therefore, t o reduce it t o
t h e geographic system, it i s necessary t o introduce a correction
for t h e l o n g i t u d e of t h e i n d i c a t e d i n i t i a l p o i n t s .
Having s o l v e d t h i s problem f o r e v e r y v a l u e o f a0 w i t h g i v e n
v a l u e s of a , l e t u s o b t a i n t h e i n t e r m e d i a t e p o i n t s of t h e f a m i l y
of l i n e s of e q u a l azimuths.
The p r o b l e m o f d e t e r m i n i n g t h e c o o r d i n a t e s o f i n t e r m e d i a t e
p o i n t s on l i n e s o f e q u a l a z i m u t h s o f t h e s e c o n d f a m i l y , w h o s e p l o t ­
t i n g on a m a p ' y i e l d s a g r i d o f i n t e r s e c t i n g a i r c r a f t p o s i t i o n l i n e s ,
is solved analogously.
Difference-Range-Finding

( H y p e r b o l i c ) C o o r d i n a t e System

The c i r c u l a r r a n g e - f i n d i n g s y s t e m o f a i r c r a f t p o s i t i o n l i n e s
examined e a r l i e r i s used w i t h c o m p a r a t i v e l y s m a l l d i s t a n c e s from
t h e ground r a d i o - e n g i n e e r i n g equipment t o t h e a i r c r a f t , s i n c e t h e
s e n d i n g and r a d i o - r e l a y i n g of r a d i o s i g n a l s t o t h e a i r c r a f t o v e r
great distances involves technical d i f f i c u l t i e s .
81

­
/84

The t e c h n i c a l s o l u t i o n o f t h e p r o b l e m i s g r e a t l y s i m p l i f i e d i f ,
instead of relaying aircraft r a d i o s i g n a l s , we send simultaneous
r a d i o s i g n a l s from two ground r a d i o - e n g i n e e r i n g i n s t a l l a t i o n s , w i t h
t h e i r s u b s e q u e n t r e c e p t i o n by t h e a i r c r a f t .
However, i n t h i s i n s t a n c e i t i s a d v a n t a g e o u s t o m e a s u r e n o t
t h e a b s o l u t e d i s t a n c e s from t h e ground i n s t a l l a t i o r i s t o t h e aircraft,
b u t o n l y t h e d i f f e r e n c e .in d i s t a n c e s t o t h e m .
The s y s t e m of p o s i t i o n l i n e s f o r t h e d i f f e r e n c e i n t h e d i s t a n c e s
t o two p o i n t s on t h e E a r t h ' s s u r f a c e i s c a l l e d t h e d i f f e r e n e e - r a n g e ­

finding or hyperbolic system.
The g e o m e t r i c l o c u s o f p o i n t s , t h e d i f f e r e n c e i n whose d i s t a n c e s
t o t w o g i v e n p o i n t s ( f o c i ) i s a c o n s t a n t v a l u e e q u a l t o 2a ( F i g .
1 . 5 3 1 , i s c a l l e d a hyperbola.
The d i s t a n c e a l o n g t h e f o c a l a x i s f r o m t h e p o i n t o f i n t e r s e c ­
t i o n of t h e f o c a l and c o n j u g a t e a x e s t o t h e peak of t h e h y p e r b o l a
i s t h e v a l u e "a1'. I t i s p o s s i b l e t o d e s i g n a t e h y p e r b o l i c a i r c r a f t
p o s i t i o n l i n e s o n a map b y d o u b l i n g t h e v a l u e o f
as an o r d i n a l
/85
number.
The d i s t a n c e a l o n g t h e f o c a l a x i s f r o m t h e f o c u s t o t h e
i n t e r s e c t i o n w i t h t h e c o n j u g a t e a x i s i s d e s i g n a t e d by t h e v a l u e r r e ' f .

-

To d e t e r m i n e t h e p o s i t i o n o f a n a i r c r a f t , t w o f a m i l i e s o f
h y p e r b o l i c p o s i t i o n l i n e s c o n s t r u c t e d from t h r e e p o i n t s forming
A t each of these points,
p a i r s of f o c a l axes are usually used.
ground r a d i o - e n g i n e e r i n g i n s t a l l a t i o n s f o r synchronous t r a n s m i s s i o n
of radio signals a r e established.
I n o r d e r t o p l o t h y p e r b o l i c p o s i t i o n l i n e s on
p r o j e c t i o n , t h e i n t e r m e d i a t e p o i n t s of h y p e r b o l a s i
s y s t e m o f one o f i t s f o c i , e . g . Fi, a r e d e t e r m i n e d
d i r e c t i o n F1F2 i s t a k e n a s t h e i n i t i a l m e r i d i a n o f

a map w i t h a n y
n the spherical
f i r s t . Here t h e
t h i s system.

B e a r i n g i n mind t h e f a c t
t h a t t h e l a t i t u d e of any p o i n t
i n t h e F 1 s y s t e m e q u a l s 9 0 ° - SI
and 90°
S2 i n t h e F2 s y s t e m ,
t h e v a l u e S2,
5'1 t 2 a , a n d
t a k i n g t h e d i s t a n c e FlF2 as t h e
a n g l e 8, i t i s p o s s i b l e t o w r i t e
(1.64) i n t h e f o l l o w i n g f o r m :

-

+

cos (SI 2a) = cos SI cos 2c

- cos A1 sin S1 sin 2c,

m

E

H

Fig. 1.53.
Difference-RangeFinding (Hyperbolic) Coordinate
System.
82

Hence
COS A1 =

,

- cos (3, + 2a)
sin s1sfn 2~

cos s,cos 2c

(1.74)

Given the definite values of S as circles of equal radius and

changing the values of 2a, it is possible to determine the value

of X of all the hyperbolas of the family at points of intersection

with the indicated circles.

F o r conversion t o the geographical coordinate system, the
intermediate points are recalculated according to (1.64) and (1.651,
after which they are plotted on a map and joined by smooth lines.

Hyperbolic coordinate systems are usually used in the applica­

tion of radionavigational devices with a large effective radius.

Therefore, the automatic conversion of the hyperbolic coordinates

t o geographical or orthodromic coordinates is advantageous.

The problem indicated is solved comparatively easily when all

three foci of the hyperbolic system are situated on one orthodrome

line (Fig. 1.54).

According to (1.74),

cos A, =

cos SI cos 2 q - cos
~

+

(SI 201)

~

s i n &sin 2cl

--

+

- cos (SI 2%)
sin SIs i n 2c2

cos S
I cos 2c2
~

Expanding the value of the cosines of the sum of the angles

and carrying out a reduction, we obtain:

COS 2cl

- cos 2 q + tg S I s i n 2al -

cos 22---cos
-

.
.

s i n 2cl

2% $-

tg S1 s i n 2u2

sin 2 2

Multiplying both sides o f the equation by sin 2cl.sin 2c2 and
rearranging the terms, we obtain:
cos 2c1 s i n 2 2 - cos 2c2 sin 2 c1- s i n 2 c c~os 2 a l + s l n 2EI ~ 0 ~ 2 %
= tg SI ( s i n 2cl s i n 2a2 - s i n 2 2 s i n
h

or

tg SI=

sin 2cz (cos 21-COS 2 q )
sin 2 1 Sirria2

s i n 2cl

(cos 2c2 - cos 2%)

- sin 2c2 s i n 2u1

(1.75)

The task is simplified even more if the distances FF1 and FF2
which are chosen are identical, i.e. 2cl = 2c2. In this case
tgq=

cos 2+ -,cos 2al
s i n %l- sim%1 '

(1.75a)

Formulas (1.75) are used for determining the coordinates of an

83


­
/86


aircraft in a spherical system with the pole at point F , bearing
in mind that (I = 90° - Si.
The X-coordinate with a known value of Si is easily determined

on the basis of (1.74). For conversion t o the geographic or ortho­

drome system, the same formulas (1.64) and (1.65) are used.

The problem of conversion t o the spherical (and consequently,

t o the geographic coordinate system) if the foci of the hyperbolic

system are not located on one orthodrome line (Fig. 1.551, is much

more complicated to solve.

It is obvious that (1.74) can be reduced t o the form:

cos AI =

.

COS

S1 cos 2cl~­ cos S1 cos 2a1

sin S1sin 2

+ sin S1 sin !2u1
­

1

Carrying out simple transformations, we obtain:

COS 11 == ctg S1 ctg 2 1

or
ctgS1=

- ctg S1cos 2al cosec 2 1 + sin %zl cosec %I
-

cos A,
sin 2al cosec
~. 21
ctg 2 1 - cos 2 1 cosec 2 ~ 1

*

(1.76)

In Fig. 1.55 it is evident that i n the FF2 system A M = X I t f3;
therefore, the following equation is valid:


ctgq=

-

cos (AI -tp)
sin 2 e cogec-- 2c2
ctg 2, cos 242, cose42c2

-

/87
-

(1.76a)

Designating the second terms of the numerators of (1.76a) by


X and the denominators by Y and reducing t o a common denominator,

we obtain:


M

'.
Fig. 1.54.
Conversion of Hyper­

bolic Coordinates t o Spherical

Coordinates (Special Case)

84




Fig. 1.55.
Conversion of Hyper­

bolic Coordinates t o Spherical

Coordinates (General Case).


Rearranging the terms, replacing sin A , by d 1 - cosLA, and
squaring both sides o f the equation we obtain:
c o s ~ ~ ( Y ~ - c o s p Y l ) - X l Y 2 +X2Y1= - ~ l - c o s 2 A 1 slnPY1

or

Thus, the coordinate A 1 is determined by the solution of a
quadratic equation

f

1/

(XZY, - x,Y2)2( Y 2 - cos pulp ­
- (Y:-2Y1 Y2 cos $ + Y;)[(X,Y, - x,Yz)2-si112 pv:]

Y:: -2Y,Y,cosp

where


X2,

(1.77)


+ Y2,

Xl = s i n 2al cosec 2cl;

X2 = sin 2a2cosec 2c2;

Y , = ctg 2cl- COS 2al cosec 2c,;

Y2 = ctg 2cz - cos 2a2cosec 2cz.


There is no sense in substituting the indicated values X I ,
Y 1 , and Y2 into (1.771, since this hampers its solution greatly.

In practice, it is easier t o determine the numerical values of these
magnitudes first, on the basis of the known values of 2 a 1 , 2 a 2 ,
2c1, and 2 c 2 , and t o substitute them into (1.77).
Knowing the coordinate A makes it possible t o determine easily
the coordinate $ 1 , e.g., according t o ( 1 . 7 6 1 , keeping in mind the
fact that $ 1 = 90° - Si, and then t o convert t o the geographic or
orthodrome system using (1.64) and ( 1 . 6 5 ) .
Overall-Range-Finding

/88

(Elliptical) Coordinate System

Hyperbolic navigational systems are the most easily implemented

technologically of a l l the range-finding systems. However, from


85

I

I 1


t h e p o i n t of view o f u s e i n f l i g h t ,
tageous.

t h e y are g e o m e t r i c a l l y disadvan­

Both f a m i l i e s o f p o s i t i o n l i n e s a r e d i v e r g e n t , and a t d i s t a n c e s
e x c e e d i n g 2c from t h e c e n t e r of t h e s y s t e m are p r a c t i c a l l y d i r e c t e d
This leads t o an increase i n e r r o r
a l o n g t h e r a d i i of t h i s c e n t e r .
i n determining aircraft coordinates with an increqse i n t h e distance
from t h e c e n t e r .
In a d d i t i o n , with an increase i n d i s t a n c e , t h e angle of i n t e r ­
s e c t i o n of t h e h y p e r b o l i c l i n e s of t h e two f a m i l i e s d e c r e a s e s .
T h i s a l s o l o w e r s t h e a c c u r a c y of d e t e r m i n i n g t h e a i r c r a f t c o o r d i n a t e s .
Combination of h y p e r b o l i c p o s i t i o n l i n e s with e l l i p t i c a l l i n e s
t u r n s o u t t o b e more a d v a n t a g e o u s ( F i g . 1 . 5 6 ) .

I t i s known t h a t t h e g e o m e t r i c a l p l a c e o f p o i n t s , t h e s u m o f
whose d i s t a n c e s t o two g i v e n p o i n t s ( f o c i ) i s a c o n s t a n t m a g n i t u d e
The d i s t a n c e a l o n g t h e m a j o r
e q u a l t o 2a, i s c a l l e d a n e z z i p s e .
a x i s from i t s i n t e r s e c t i o n w i t h t h e minor a x i s t o t h e t o p of t h e
e l l i p s e , i . e . , i t s s e m i - a x i s , i s c o n s i d e r e d t o be t h e v a l u e llall,
i n t h i s case.
The d i s t a n c e f r o m t h e i n t e r s e c t i o n o f t h e a x e s t o
t h e f o c i i s c o n s i d e r e d t o be t h e ItcI1 v a l u e .
I f , i’n a d d i t i o n t o t h e d i f f e r e n c e i n d i s t a n c e s t o t h e f o c i ,
t h e d i s t a n c e t o one o f them i s m e a s u r e d , it i s e a s y t o i m p l e m e n t
t h e h y p e r b o l i c - e l l i p t i c a l system of p o s i t i o n l i n e s .
A c t u a l l y , i f o n e d i s t a n c e Si i s known a n d t h e d i f f e r e n c e i n
t h e d i s t a n c e s A S = 2ah, obviously t h e d i s t a n c e t o t h e second f o c u s

is

2a, = 2S1 t 2 a h
where a e i s t h e m a j o r s e m i - a x i s
t e r a of t h e hyperbola.

of t h e e l l i p s e and ah i s t h e parame­

Obvious advantages of t h e
h y p e r b o l i c - e l l i p t i c a l system
include the following:
(a)
There i s an absence
of divergence i n t h e second
family of p o s i t i o n l i n e s .
The
elliptical position lines are
c l o s e d , s o t h a t t h e a c c u r a c y of
determining the a i r c r a f t ’ s
p o s i t i o n a l o n g them d o e s n o t
decrease with an i n c r e a s e i n
distance.
Hyperbolic-Ellipti­
Fig. 1.56.
cal Coordinate System.
86

( b ) Orthogonality of t h e

/89
­

position lines appears at any point of the system. In a confocal

hyperbolic-elliptical system, the position lines intersect only at

a right angle.

(c) Two foci, instead of the three f o r a hyperbolic system,

are sufficient for the construction of a confocal hyperbolic-

elliptical system. This leads t o a simplification of the transfor­

mations during conversion t o a geographic or orthodrome system.

However, in spite of the advantages of a hyperbolic-elliptical

system indicated above, a wide distribution was not obtained. This

was 'connected with great technical difficulties in measuring distance

t o points on the Earth's surface at distances exceeding straight-

line geometric visibility of the object from flight altitude.

The above problem is solved by keeping on board the aircraft

a reference frequency (quartz-crystal clock) which permits syn­

chronization of the transmission of radio signals from the ground

with reference signals on board the plane.
It is therefore possible

to determine the travel time of the signals.

Hence, strict stabilization of the reference frequency on

board the aircraft is the main task for the technical implementation

of hyperbolic-elliptical systems.

To plot elliptical lines on a map with any projection, the
intermediate points are determined according t o the same formulas
as the family of hyperbolas. For example , in ( 1 . 7 4 ) , considering
the second distance in it ( S 2 ) not as the sum SI + 2ah, but as the
difference 2ae
SI,

-

COB hi

cos

= -___-

s,cos 2c - cos (2a- SI)
,
sin s1sin 2~

(1.78)

where X 1 is an angle with the vertex at point F 1 , measured from the

major axis of the ellipse.

Given the different values of Si and determining the values of
for each of them with a constant value of 2 a , we shall obtain
intermediate points of an ellipse in a spherical system F 1 .
A 1

Changing the value of 2a and performing these operations with
Si, we obtain intermediate points of the next elliptical position
line, etc. Recalculation of intermediate points is implemented in
the geographic system according t o ( 1 . 6 4 ) and (1.65) as in other
cases analyzed by us.
In the hyperbolic-elliptical system, the conversion t o the

geographic or orthodromic coordinate system is very simple.

In fact, the value of Si and the parameter 2a in this system

87

a r e measured.
T h e r e f o r e , (1.78) i s u s e f u l f o r t h e p r o b l e m o f
c a l c u l a t i n g t h e s p h e r i c a l c o o r d i n a t e s of an a i r c r a f t along measured
p a r a m e t e r s a n d f o r a s u b s e q u e n t t r a n s f e r t o t h e g e o g r a p h i c or
orthodrome system.

8.

Elements o f A i r c r a f t N a v i g a t i o n

Aircraft f l i g h t s a r e c a r r i e d out i n airspace.
The p h y s i c a l
c o m p o s i t i o n o f a i r s p a c e , a s w e l l as t h e s p e e d a n d d i r e c t i o n o f i t s
s h i f t relative t o the Earth's surface, exert a substantial influence
on t h e t r a j e c t o r y of a i r c r a f t m o v e m e n t i n a g e o g r a p h i c or o r t h o d r o m i c
coordinate system.
U n t i l r e c e n t l y , t h e d i r e c t measurement o f t h e s p e e d and d i r e c ­
t i o n o f a i r c r a f t movement r e l a t i v e t o t h e E a r t h ' s s u r f a c e w a s a
problem.
A t t h e p r e s e n t t i m e , t h i s problem h a s been solved.
How­
e v e r , it i s n o t a d v i s a b l e t o i n s t a l l t h e complex and e x p e n s i v e
e q u i p m e n t w h i c h m e a s u r e s t h e i n d i c a t e d p a r a m e t e r s on a l l a i r c r a f t .
I n s o l v i n g n a v i g a t i o n a l p r o b l e m s , p a r a m e t e r s o f a i r c r a f t move­
ment r e l a t i v e t o t h e a i r s p a c e a r e u s u a l l y m e a s u r e d t o t h e g r e a t e s t
e x t e n t p o s s i b l e , a n d t h e n a d d i t i o n a l p a r a m e t e r s o f t h e movement o f
t h e a i r c r a f t w h i c h are c o n n e c t e d w i t h m o v e m e r t i n a i r s p a c e a r e
found.
Summarizing t h e m e a s u r e d p a r a m e t e r s o f a i r c r a f t movement, t h e
v a l u e and d i r e c t i o n of t h e speed v e c t o r of t h e a i r c r a f t r e l a t i v e t o
the Earth's surface a r e found.
The p a r a m e t e r s o f a i r c r a f t m o v e m e n t w i t h w h i c h we m u s t c o n c e r n
o u r s e l v e s i n c a r r y i n g o u t a i r c r a f t n a v i g a t i o n a r e c a l l e d ezements

of a i r c r a f t navigation.
Elements of a i r c r a f t n a v i g a t i o n a r e divided i n t o t h r e e groups
which d e t e r m i n e t h e d i r e c t i o n , s p e e d , and a l t i t u d e of f l i g h t .

E l e m e n t s w h i c h d e t e r m i n e F l i g h t Direction

The b a s i c e l e m e n t w h i c h d e t e r m i n e s t h e d i r e c t i o n o f a i r c r a f t
movement i n a i r s p a c e i s c a l l e d t h e a i r c r a f t c o u r s e .
The a i r c r a f t course ( g e n e r a l l y d e s i g n a t e d b y y ) i s t h e a n g l e
b e t w e e n t h e d i r e c t i o n o f a m e r i d i a n on t h e E a r t h ' s s u r f a c e a n d t h e
d i r e c t i o n of t h e l o n g i t u d i n a l a x i s of t h e a i r c r a f t i n a h o r i z o n t a l
plane.
U s u a l l y it i s c o n s i d e r e d t h a t t h e a i r s p e e d v e c t o r o f an a i r ­
c r a f t i n t h e p l a n e of t h e h o r i z o n c o i n c i d e s w i t h t h e d i r e c t i o n of
t h e longitudinal a x i s of t h e a i r c r a f t , although t h i s is a c t u a l l y
not entirely true.
Therefore, an understanding of t h e course o f t e n
c o i n c i d e s with an u n d e r s t a n d i n g of t h e d i r e c t i o n of t h e f l i g h t a i r ­
speed vector.
88

Depending on t h e reference system chosen, t h e f o l l o w i n g s p e c i a l
v a r i e t i e s o f a i r c r a f t c o u r s e s can b e d i s t i n g u i s h e d :
(a)
A true course ( T C ) i s m e a s u r e d f r o m t h e n o r t h e r n e n d o f
a geographic m e r i d i a n which p a s s e s through t h e p o i n t o f i n t e r s e c t i o n
of t h e Earth's surface with t h e v e r t i c a l of t h e aircraft.
The l a t ­
t e r i s u s u a l l y c a l l e d t h e p o s i t i o n p o i n t o f t h e a i r c r a f t (PA):
(b)
The o r t h o d r o m e c o u r s e ( O C ) i s m e a s u r e d f r o m t h e n o r t h e r n
end of a geographic meridian of t h e s t a r t i n g p o i n t of a r e c t i l i n e a r
(orthodrome) segment of t h e p a t h or from a n o t h e r c o n d i t i o n a l l y
chosen ( r e f e r e n c e ) m e r i d i a n a l o n g which t h e z e r o p o i n t o f t h e
course-reading scale i s established.

/91

(c)
The m a g n e t i c c o u r s e ( M C ) i s r e a d f r o m t h e n o r t h e r n e n d
of t h e magnetic m e r i d i a n which p a s s e s through p o i n t PA.

In addition t o these varieties of a i r c r a f t courses, there is
a n o t h e r c o n c e p t , t h e compass course ( C C ) , ? . e . , a c o u r s e b a s e d on
t h e r e s p o n s e s o f a compass.
I n t e x t b o o k s on a i r c r a f t n a v i g a t i o n ,
t h e c o n c e p t o f compass c o u r s e h a s i n c l u d e d o n l y m a g n e t i c compasses,
b u t we h a v e b r o a d e n e d t h i s c o n c e p t t o i n c l u d e a l l m e t h o d s o f
measuring an a i r c r a f t course with an i n s t r u m e n t .
A i r c r a f t c o u r s e s a r e m e a s u r e d by t h r e e d i f f e r e n t m e t h o d s , % . e . ,
s t a b i l i z a t i o n o f t h e z e r o r e a d i n g o f t h e compass a l o n g t h e m e r i d i a n s :

M a g n e t i c c o u r s e , by means o f m a g n e t i c s y s t e m s .
True c o u r s e , by means of a s t r o n o m i c a l s y s t e m s .
R e f e r e n c e c o u r s e , by means of

gyroscopic devices.

A l l o f t h e s e m e t h o d s h a v e i n s t r u m e n t a l e r r o r s or a deviation
d e s i g n a t e d b y A,.
I n d i v i d u a l components of e r r o r s i n t h e c o u r s e
d e v i c e s a r e components of t h e d e v i a t i o n .

Any o f t h e t h r e e t y p e s o f a i r c r a f t c o u r s e s c a n b e o b t a i n e d
from r e s p o n s e s o f a c o u r s e d e v i c e , a l l o w i n g for i t s d e v i a t i o n , e . g . ,

(1.79)


I n t h e g e n e r a l case,

Y

= c c +nc

A s a c o r r e c t i o n f o r any measurement, t h e v a l u e Ac i s c o n s i d e r e d
p o s i t i v e when t h e c o m p a s s u n d e r e s t i m a t e s t h e v a l u e o f t h e m e a s u r e d
m a g n i t u d e , a n d n e g a t i v e when t h e c o m p a s s r e a d i n g s a r e t o o h i g h .
89

I n t h e f u t u r e , when w e s t u d y t h e r e l a t i o n s h i p among t h e t h r e e
t y p e s of a i r c r a f t c o u r s e s q w e w i l l c o n s i d e r t h a t t h e v a l u e of each
h a s b e e n c o r r e c t e d for t h e d e v i a t i o n o f t h e d e v i c e .
The i n t e r r e l a t i o n s h i p b e t w e e n m a g n e t i c a n d t r u e f l i g h t c o u r s e s
s i n c e theqe c o u r s e s are
m e a s u r e d f r o m t h e m e r i d i a n s w h i c h p a s s t h r o u g h p o i n t PA ( F i g . 1 . 5 7 a ) .

is established with the least d i f f i c u l t y ,

I n F i g . 1 . 5 7 a t h e n o r t h e r n g e o g r a p h i c m e r i d i a n is d e s i g n a t e d
b y P n , t h e d i r e c t i o n o f t h e m a g n e t i c m e r i d i a n b y Pw.
Since the magnetic meridian is s h i f t e d t o t h e l e f t r e l a t i v e
t o t h e geographic meridian, t h e magnetic d e c l i n a t i o n a t the given
With a p o s i t i v e d e v i a t i o n , t h e m a g n e t i c m e r i d point is negative.
ian is s h i f t e d t o the r i g h t r e l a t i v e t o the geographic meridian.
From t h e f i g u r e ,

it i s e v i d e n t t h a t

MC =TC - AM;

TC =MC +.,A

(1.80)


In t h e case of AM, t h e value i s negative; t h e r e f o r e , t h e abso­
l u t e v a l u e of t h e t r u e c o u r s e t u r n s o u t t o be l e s s t h a n t h e magnetic
course.
C o n v e r t i n g f r o m t h e t r u e or m a g n e t i c c o u r s e t o t h e o r t h o d r o m e
c o u r s e (or v i c e v e r s a ) i s m o r e c o m p l e x ( F i g . 1 . 5 7 , b ) .

Fig.

1.57.

I n t e r r e l a t i o n s h i p of A i r c r a f t Courses;
and True; b:
True and Orthodrome.

a:

Magnetic

The d i r e c t i o n o f t h e g e o g r a p h i c m e r i d i a n p a s s i n g t h r o u g h p o i n t
PA i s d e s i g n a t e d by P n ; t h e d i r e c t i o n o f t h e r e f e r e n c e m e r i d i a n
P r a m , i s shown by a d o t t e d l i n e w h i c h i n t e r s e c t s t h e g e o g r a p h i c
Therefore
meridian a t angle 6.

(1.81)


90

­
/92

The v a l u e 6, t h e a n g l e o f c o n v e r g e n c e o f t h e m e r i d i a n s , i s
c o n s i d e r e d p o s i t i v e when t h e d i r e c t i o n o f t h e g e o g r a p h i c m e r i d i a n
a t p o i n t PA i s p r o p o r t i o n a l t o t h e r e f e r e n c e m e r i d i a n e x t e n d e d
c l o c k w i s e ( t o t h e r i g h t ) a n d n e g a t i v e when t h e g e o g r a p h i c m e r i d i a n
is shifted t o the left.
On s m a l l p a t h s e g m e n t s ( 5 0 0 - 6 0 0 k m ) , t h e a n g l e o f c o n v e r g e n c e
of t h e meridians is approximately equal t o :
(1.82)
In g e n e r a l , f o r c o n v e r t i n g from t h e orthodrome course t o a
t r u e c o u r s e or v i c e v e r s a , i t i s n e c e s s a r y t o d e t e r m i n e t h e l o n g i ­
tude of t h e s t a r t i n g p o i n t of t h e orthodrome of each r e c t i l i n e a r
p a t h s e g m e n t , e . g . , o n t h e b a s i s o f ( 1 . 3 4 1 , or, i f t h e a z i m u t h o f
t h e o r t h o d r o m e i s known a t t h e s t a r t i n g p o i n t o f t h e p a t h s e g m e n t ,
on t h e b a s i s o f ( 1 . 3 3 a ) .
The a n g l e o f c o n v e r g e n c e o f t h e m e r i d i a n s b e t w e e n t h e s t a r t i n g
p o i n t o f t h e p a t h s e c t i o n Mi a n d a n y m o v i n g p o i n t M on a s e c t i o n ,
a c c o r d i n g t o ( 1 . 3 3 a ) , w i l l be e q u a l t o
8 = a - a1 = arctg (tg A cosec '4)

- arctg (tg A,

cosec TI),

(1.83)

where A and A 1 a r e measured from t h e s t a r t i n g p o i n t o f t h e o r t h o ­
drome.
If t h e l o n g i t u d e o f a n y o t h e r p o i n t on t h e E a r t h ' s s u r f a c e ,
e . g . , t h e p o i n t of t a k e - o f f of t h e a i r c r a f t , i s t a k e n as t h e r e f e r ­
ence m e r i d i a n , t h e a n g l e of convergence of t h e m e r i d i a n s w i l l be
d e t e r m i n e d a s t h e sum:

a=s,+$+

...s

_..

where 6 1 ; 6 2
a r e t h e angles of convergence of t h e meridians
between t h e s t a r t i n g and end p o i n t s o f t h e p r e c e d i n g p a t h segments
d e t e r m i n e d on t h e b a s i s o f ( 1 . 8 3 ) ; 6, i s t h e a n g l e o f c o n v e r g e n c e
o f t h e m e r i d i a n s f r o m t h e s t a r t i n g p o i n t t o t h e moving p o i n t o f t h e
l a s t p a t h s e g m e n t ( o n t h e b a s i s o f t h e same f o r m u l a ) .
The a n g l e o f c o n v e r g e n c e o f t h e m e r i d i a n s c a l c u l a t e d i n t h i s
way a l l o w s c o n v e r s i o n f r o m a t r u e c o u r s e t o a n o r t h o d r o m e c o u r s e
a n d v i c e v e r s a a t a n y f l i g h t d i s t a n c e w i t h a n y number o f b r e a k s i n
the path.

For c o n v e r s i o n f r o m a n o r t h o d r o m e c o u r s e t o a m a g n e t i c c o u r s e
and v i c e v e r s a , ( 1 . 8 0 ) and ( 1 . 8 1 ) are u s e d , from which it f o l l o w s
that

91


/93
-

(1.84)

The sum o f t h e m a g n i t u d e s A M + 6 i s t a k e n a s t h e o v e r a l l c o r ­
r e c t i o n f o r conversion from a magnetic t o a n orthodrome course and
Then ( 1 . 8 4 ) a s s u m e s t h e
v i c e v e r s a , a n d i s d e s i g n a t e d by A . .
form :

..

OC = MCfA;
MC = O C - A .

The d r i f t a n g l e i s t h e s e c o n d e l e m e n t d e t e r m i n i n g t h e d i r e c t i o n
o f a i r c r a f t movement.
I n an a i r c r a f t , t h e a n g l e between t h e a i r s p e e d v e c t o r and t h e
groundspeed v e c t o r i n a h o r i z o n t a l plane is c a l l e d t h e d r i f t angle
(Fig. 1.58).
I n g e n e r a l , t h e d r i f t a n g l e i s d e s i g n a t e d by t h e
I n t h o s e i n s t a n c e s when s p e c i a l d e s i g n a t i o n s for
Latin l e t t e r a.
c o u r s e s a r e used i n t h e s o l u t i o n s of n a v i g a t i o n a l problems t h e
d r i f t a n g l e i s d e s i g n a t e d by t h e R u s s i a n l e t t e r s f o r D A .

In Fig. 1.58.
OPN i s t h e d i r e c t i o n o f t h e m e r i d i a n a t p o i n t
P A ; OV i s t h e d i r e c t i o n of t h e a i r s p e e d v e c t o r and t h e l o n g i t u d i n a l
a x i s o f t h e a i r c r a f t ; OW i s t h e d i r e c t i o n of t h e groundspeed v e c t o r
r e l a t i v e t o t h e E a r t h ’ s s u r f a c e ; u i s t h e wind s p e e d v e c t o r .
The d r i f t a n g l e o f a n a i r c r a f t i s c o n s i d e r e d p o s i t i v e when t h e
g r o u n d s p e e d v e c t o r ( v e c t o r o f a i r c r a f t movement r e l a t i v e t o t h e
E a r t h ’ s s u r f a c e ) i s f u r t h e r t o t h e r i g h t of t h e l o n g i t u d i n a l a x i s
of t h e a i r c r a f t and n e g a t i v e i f it i s f u r t h e r t o t h e l e f t .
The a n g l e b e t w e e n t h e n o r t h e r n e n d o f t h e m e r i d i a n a n d t h e
g r o u n d s p e e d v e c t o r or t h e v e c t o r o f t h e s p e e d o f t h e a i r c r a f t r e l a ­
t i v e t o t h e E a r t h ’ s s u r f a c e i s c a l l e d t h e f l i g h t angle (FA).
The
g e n e r a l d e s i g n a t i o n f o r t h e f l i g h t a n g l e i s $.
The f l i g h t a n g l e , l i k e t h e c o u r s e o f
t h e a i r c r a f t , can be measured from t h e
r e f e r e n c e meridian, t h e geographic meridian,
and a m a g n e t i c m e r i d i a n p a s s i n g t h r o u g h
p o i n t PA.
S p e c i a l v a l u e s of t h e f l i g h t a n g l e s
have t h e f o l l o w i n g d e s i g n a t i o n s :

The P a t h
Fig. 1.58.
Angle o f F l i g h t .
92

( a > The o r t h o d r o m e f l i g h t a n g l e i s
O F A , w i t h o b l i g a t o r y i n d i c a t i o n by a s u b s c r i p t of t h e l o n g i t u d e of t h e r e f e r e n c e
meridian.
For example, O F A 4 0 = 96O.

/94

(b)

The t r u e f l i g h t a n g l e i s T F A .


(c)

T h e m a g n e t i c f l i g h t a n g l e i s MFA.


I n Figure 1.58,

it i s e v i d e n t t h a t t h e f l i g h t a n g l e is g e n e r a l l y


or i n s p e c i a l c a s e s :
O F A r OC
TFA L TC
M F A = MC

+ DAI

+ DA;
+ DA;


(1.85)


The i n t e r r e l a t i o n s h i p b e t w e e n t h e s p e c i a l v a l u e s o f t h e f l i g h t
a n g l e s and t h e method o f c o n v e r s i o n from one s p e c i a l v a l u e t o
a n o t h e r c o r r e s p o n d s c o m p l e t e l y t o t h e i n t e r r e l a t i o n s h i p between
special values of a i r c r a f t courses:
OFA

TFA

TPA = O F A
MFA = O F A

+ A;
+ A,;
A = TFP - .,A

+ 6

-

=MFA

6 = MFA

I n d e t e r m i n i n g t h e d i r e c t i o n o f a i r c r a f t movement r e l a t i v e t o
­
/95
t h e E a r t h ’ s s u r f a c e , i t i s s u f f i c i e n t t o know t h e c o u r s e o f t h e a i r ­
c r a f t a s t h e a n g l e between t h e d i r e c t i o n of t h e m e r i d i a n and t h e
l a t e r a l a x i s of t h e a i r c r a f t , and t h e d r i f t a n g l e as t h e a n g l e be­
tween t h e l a t e r a l a x i s of t h e a i r c r a f t and t h e d i r e c t i o n of i t s
movement.
These e l e m e n t s , t o g e t h e r w i t h elements o f f l i g h t s p e e d ,
make i t p o s s i b l e t o d e t e r m i n e a p p r o x i m a t e l y t h e s p e e d a n d d i r e c t i o n
o f t h e wind a t f l i g h t a l t i t u d e .

For p r e c i s e l y d e t e r m i n i n g t h e w i n d a t f l i g h t a l t i t u d e , i t i s
necessary t o s e p a r a t e t h e p a r t of t h e d r i f t angle of an a i r c r a f t
c a u s e d by t h e w i n d .
I t i s obvious t h a t t o do t h i s it i s necessary

n

Fig. 1.59.
Moment a n d C o n t r o l

Force w i t h Assymetry o f Engine
Thrust.

Lateral Glide

Fig. 1.60.
w i t h T r a n s v e r s e Roll.

93


t o determine t h e direation of t h e airspeed vector of t h e aircraft
as t h e c o u r s e and g l i d e o f dynamic o r i g i n , a r i s i n g i n f l i g h t .
There are s e v e r a l c a u s e s of l a t e r a l g l i d e i n a i r c r a f t d u r i n g
flight.
The b a s i c c a u s e s a r e t h e f o l l o w i n g .
1.

Assymetry

o f

the Engine t h r u s t or A i r c r a f t

D r a g (Fig.

1.59)

L e t u s assume t h a t w i t h s y m m e t r i c a l d r a g , o n e of t h e e n g i n e s
h a s a somewhat g r e a t e r t h r u s t t h a n t h e o t h e r . The d i f f e r e n c e i n
t h r u s t AP w i l l p r o d u c e t o r q u e i n t h e a i r c r a f t r e l a t i v e t o t h e v e r ­
t i c a l a x i s , i . e . , t h e course of t h e aircraft w i l l change.

For s t a b i l i z i n g t h e f l i g h t d i r e c t i o n , a moment m u s t b e a p p l i e d
t o t h e empennage o f t h e a i r c r a f t w h i c h i s e q u a l i n m a g n i t u d e and
o p p o s i t e i n d i r e c t i o n t o t h e moment o f t h r u s t , i . e .

w h e r e AP is t h e a s s y m e t r y o f t h e t h r u s t ; FZ i s t h e c o n t r o l f o r c e ;
LT i s t h e a r m of t h r u s t a s s y m e t r y ( f r o m t h e a x i s o f t h e e n g i n e t o
t h e a x i s of t h e a i r c r a f t ) ; a n d L , i s t h e a r m o f c o n t r o l ( f r o m t h e
c e n t e r of t h e empennage a r e a t o t h e c e n t e r o f g r a v i t y o f t h e a i r ­
craft)
The l a t e r a l f o r c e w h i c h c a u s e s g l i d i n g o f t h e a i r c r a f t w i l l
be:

APL

F * = d .

(1.86)

‘X

I n an analogous manner, t h e f o r c e which c a u s e s g l i d i n g o f an
aircraft with assymetry of drag arises.
I n t h i s i n s t a n c e , t h e moment o f r o t a t i o n o f a n a i r c r a f t c a u s e s
e x c e s s d r a g on o n e wing o f t h e a i r c r a f t .
The d i s t a n c e f r o m t h e
l a t e r a l a x i s of t h e a i r c r a f t t o t h e c e n t e r o f i t s a p p l i c a t i o n i s
the a r m of t h i s force.

2.

Allowable L a t e r a l B a n k i n g o f an A i r c r a f t
Horizontal

With a l l o w a b l e l a t e r a l b a n k i n g
component o f t h e l i f t w i l l a p p e a r :

in

Flight.

(Fig.

1.60), t h e h o r i z o n t a l

w h e r e G i s t h e w e i g h t of t h e a i r c r a f t a n d B i s t h e a n g l e o f
banking.

lateral

For e x a m p l e , w i t h a f l y i n g w e i g h t o f t h e a i r c r a f t o f 7 5 t , t h e
a l l o w a b l e b a n k i n g i n h o r i z o n t a l f l i g h t , e q u a l t o lo, c a u s e s a
94

----.-._

..._..

I

­
/96

l a t e r a l component o f l i f t

=

3.

1.3 t.

Coriolis

Force

During f l i g h t i n t h e E a r t h ' s atmosphere, as a r e s u l t o f t h e
d i u r n a l r o t a t i o n o f t h e E a r t h ' s s u r f a c e , a l a t e r a l Coriolis f o r c e
a c t s on t h e a i r c r a f t :

F,

= 20,

W m sin 7,

(1.87)

where Oe i s t h e a n g u l a r v e l o c i t y of t h e E a r t h ' s r o t a t i o n ; W i s t h e
speed of t h e aircraft r e l a t i v e t o t h e E a r t h ' s s u r f a c e ; m i s t h e
mass o f t h e a i r c r a f t ; a n d $ i s t h e l a t i t u d e o f t h e p o i n t P A .

4.

Two-dimensional

Fluctuations

i n t h e A i r c r a f t Course

D u r i n g t w o - d i m e n s i o n a l rolls ( w i t h o u t b a n k i n g ) , a n a i r c r a f t
(as a r e s u l t of i n e r t i a ) t r i e s t o maintain t h e i n i t i a l d i r e c t i o n of
movement.
This causes l a t e r a l g l i d i n g of t h e a i r c r a f t V z , e q u a l t o :

w h e r e V i s t h e s p e e d o f t h e a i r c r a f t r e l a t i v e t o che a i r s p a c e ,
Ay i s t h e m a g n i t u d e o f t h e c h a n g e i n t h e a i r c r a f t ' s c o u r s e .

and

The i n d i c a t e d l a t e r a l g l i d i n g o f a n a i r c r a f t g r a d u a l l y d i e s
down a s a r e s u l t o f a c c e l e r a t i o n c a u s e d b y t h e l a t e r a l a i r f l o w
over i t s surface.

5 . G l i d i n g During Changes
Component a t

i n t h e Lateral
Flight Altitude

Wind

Speed

This type of g l i d i n g arises a s a r e s u l t of t h e i n e r t i a of t h e
aircraft.
F i r s t , l a t e r a l a i r f l o w o v e r t h e a i r c r a f t or s o m e t h i n g
s i m i l a r ( g l i d i n g i n a i r s p a c e ) w i l l a p p e a r , f o l l o w e d by a change i n
t h e d i r e c t i o n o f a i r c r a f t movement.
From t h e a b o v e f i v e e x a m p l e s o f l a t e r a l a i r c r a f t g l i d i n g ,
constant l a t e r a l f o r c e s are t h e causes of t h e g l i d i n g i n t h e first
t h r e e cases, w h i l e a b r u p t b e g i n n i n g and g r a d u a l d i m i n u t i o n o f
g l i d i n g a r e t h e c a u s e s i n t h e l a s t two c a s e s .
The m a g n i t u d e o f s t a b l e g l i d i n g w i t h a c o n s t a n t l y a c t i n g
l a t e r a l f o r c e can be determined according t o t h e formula
W
z = czs P
-

or

2

(1.89)

­
vz=f+
95

L


where Z i s t h e o p e r a t i v e l a t e r a l f o r c e ; cz i s t h e c o e f f i c i e n t o f
l a t e r a l drag of t h e a i r c r a f t ; s i s t h e area of t h e l o n g i t u d i n a l
s e c t i o n o f a n a i r c r a f t w i t h a v e r t i c a l p l a n e ; a n d p i s t h e mass
density of the a i r f l i g h t a l t i t u d e .
To c a l c u l a t e g l i d i n g i n f l i g h t , i t m u s t b e i n t e g r a t e d a n d
converted t o angular g l i d e ( a g l ) :

where V z i s t h e l a t e r a l component o f t h e a i r s p e e d and V,
l o n g i t u d i n a l component of l o n g i t u d i n a l s p e e d .
The d i r e c t i o n o f t h e a i r s p e e d v e c t o r
formula
TI,

is the

i s d e t e r m i n e d by t h e

=r+ag2

(1.91)

The d r i f t a n g l e o f a n a i r c r a f t , w h o s e c a u s e i s t h e a c t i o n o f
t h e wind a t f l i g h t a l t i t u d e , w i l l b e :
a = 9- 7" =

+ - 7 -a g l

(1.92)

A s w e have a l r e a d y s a i d , determining t h e g l i d i n g of a n a i r ­
c r a f t i n a i r s p a c e i s n e c e s s a r y only f o r p r e c i s e measurements of
For t h e p u r p o s e s o f
wind s p e e d a n d d i r e c t i o n a t f l i g h t a l t i t u d e .
a i r c r a f t n a v i g a t i o n , t h e r e i s no need t o s e p a r a t e o u t t h e c a u s e s o f
l a t e r a l a i r c r a f t movement.
E l e m e n t s Which C h a r a c t e r i z e t h e F l i g h t S p e e d o f

an A i r c r a f t

The f l i g h t s p e e d o f a n a i r c r a f t i s m e a s u r e d b o t h r e l a t i v e t o
t h e a i r s p a c e surrounding t h e a i r c r a f t and r e l a t i v e t o t h e E a r t h ' s
surface.
M e a s u r i n g t h e s p e e d o f a i r c r a f t movement r e l a t i v e t o t h e a i r space i s s i g n i f i c a n t both from t h e p o i n t o f view o f f l i g h t a e r o ­
dynamics ( s t a b i l i t y and c o n t r o l o f t h e a i r c r a f t ) and from t h e p o i n t
ov view o f a i r c r a f t n a v i g a t i o n .

I t i s known t h a t t h e l i f t o f a w i n g , t h e d r a g o f a n a i r c r a f t ,
a n d t h e s t a b i l i t y a n d c o n t r o l l a b i l i t y o f a n a i r c r a f t d e p e n d on t h e
square of t h e airspeed.

F o r example, a t f l i g h t speeds which a r e s i g n i f i c a n t l y l e s s
t h a n t h e s p e e d o f s o u n d , t h e d r a g o f a n a i r c r a f t i s d e t e r m i n e d by
t h e formula

96

/98

w h e r e Q i s t h e l a t e r a l d r a g o f a n a i r c r a f t , ex i s t h e d r a g c o e f ­
f i c i e n t , S i s t h e maximum a r e a o f t h e l a t e r a l c r o s s s e c t i o n of a n
a i r c r a f t , a n d p i s t h e mass a i r d e n s i t y a t f l i g h t a l t i t u d e .
T h e v a l u e . -p v 2
2

c h a r a c t e r i z e s t h e aerodynamic p r e s s u r e of t h e

a t m o s p h e r e on t h e s u r f a c e o f a n a i r c r a f t .

A 1 1 t h e a e r o d y n a m i c c h a r a c t e r i s t i c s of
mined r e l a t i v e t o t h i s v a l u e .

an a i r c r a f t are d e t e r ­

I n determining t h e aerodynamic c h a r a c t e r i s t i c s of an a i r c r a f t ,
the, aerodynamic p r e s s u r e (and t h e r e f o r e t h e speed of f l i g h t ) reduce
t o conditions i n a standard atmosphere, i . e . , t o f l i g h t conditions
n e a r t h e E a r t h ' s s u r f a c e , w i t h a n a t m o s p h e r i c p r e s s u r e o f 7 6 0 m m Hg
and an ambient a i r t e m p e r a t u r e of 1 5 O C .
Therefore, speed i n d i c a t o r s
which m e a s u r e a i r s p e e d on t h e b a s i s o f a e r o d y n a m i c p r e s s u r e a r e
c a l i b r a t e d a c c o r d i n g t o t h e p a r a m e t e r s of a s t a n d a r d a t m o s p h e r e .
With a n i n c r e a s e i n f l i g h t a l t i t u d e , a i r d e n s i t y d e c r e a s e s .
T o preserve aerodynamic p r e s s u r e a t f l i g h t a l t i t u d e , it i s necessary
t o increase f l i g h t airspeed, although responses of t h e airspeed
i n d i c a t o r which m e a s u r e a i r s p e e d on t h e b a s i s o f a e r o d y n a m i c p r e s ­
s u r e remain constant.
F l i g h t a i r s p e e d which i s m e a s u r e d on t h e b a s i s o f a e r o d y n a m i c
p r e s s u r e and which i n f l u e n c e s t h e a e r o d y n a m i c s of t h e f l i g h t o f t h e
a i r c r a f t i s c a l l e d aerodynamic speed (Vaer).
I t i s n e c e s s a r y t o c o n s i d e r , however, t h a t w i t h an i n c r e a s e i n
f l i g h t speed, e s p e c i a l l y i n approaching t h e speed of sound, aero­
dynamic s p e e d d o e s n o t c o m p l e t e l y c o r r e s p o n d t o t h e aerodynamic
c h a r a c t e r i s t i c s of a n a i r c r a f t which a r e d e t e r m i n e d u n d e r t h e c o n d i ­
t i o n s o f a s t a n d a r d a t m o s p h e r e and which a r e i n h e r e n t i n f l i g h t
speed.
This i s becuase t h e f a c t o r of a i r c o m p r e s s i b i l i t y begins t o
e x e r t an influence.
To e n s u r e s a f e p i l o t a g e o f t h e a i r c r a f t i n
these instances, a corresponding correction is introduced i n t o t h e
i n d i c a t i o n s of aerodynamic speed.

For t h e p u r p o s e s o f a i r c r a f t n a v i g a t i o n , i t i s n e c e s s a r y t o
know t h e a c t u a l s p e e d o f a n a i r c r a f t i n s p a c e .

/99

The a c t u a l a i r s p e e d , which w e s h a l l c a l l s i m p l y a i r s p e e d ( V ) ,
c a n b e o b t a i n e d from a e r o d y n a m i c s p e e d by i n t r o d u c i n g c o r r e c t i o n s
for t h e c h a n g e i n a i r d e n s i t y w i t h f l i g h t a l t i t u d e a n d t e m p e r a t u r e :

97

w h e r e Vaer
i s t h e a e r o d y n a m i c s p e e d ; AVH i s t h e c o r r e c t i o n f o r
s p e e d a s a r e s u l t o f f l i g h t a l t i t u d e s ; AV$ i s t h e c o r r e c t i o n f o r
is the correction
s p e e d a s a r e s u l t o f a i ' r t e m p e r a t u r e ; a n d AVa.cmp.
for s p e e d a s a r e s u l t o f a i r c o m p r e s s i b i l i t y
C o r r e c t i o n f o r f l i g h t a l t i t u d e is of b a s i c jmportance.
Cor­
r e c t i o n f o r a i r t e m p e r a t u r e i s s i g n i f i c a n t l y smaller and i s i n t r o ­
d u c e d o n l y i n t h o s e c a s e s when t h e a i r t e m p e r a t u r e a t f l i g h t a l t i ­
t u d e i s s i g n i f i c a n t l y d i f f e r e n t from t h e temperature c a l c u l a t e d f o r
this altitude.

A t t h e p r e s e n t t i m e , t h e r e a r e d e v i c e s which i n d i c a t e f l i g h t
airspeed directly, taking a ltitude i n t o account.
C o r r e c t i o n s must
be i n t r o d u c e d i n t h e r e s p o n s e 6 o f t h e s e d e v i c e s o n l y f o r i n s t r u ­
m e n t a l e r r o r s of t h e d e v i c e s and ( i n i n d i v i d u a l c a s e s ) f o r a d i s ­
crepancy between t h e a c t u a l a i r temperature and t h e c a l c u l a t e d
temperature a t a given a l t i t u d e .

I n p u b l i s h e d t e x t b o o k s on a i r c r a f t n a v i g a t i o n , a i r s p e e d h a s
b e e n c l a s s i f i e d as " i n d i c a t e d " ( m e a s u r e d on t h e b a s i s o f a e r o d y n a m i c
p r e s s u r e ) b u t t r u e , and as " i n d i c a t e d , c o r r e c t e d f o r methodological
and i n s t r u m e n t e r r o r s . "
S i n c e t h e r e a r e now d e v i c e s w h i c h m e a s u r e b o t h t h e s e s p e e d s ,
In addition, increasing airspeeds
e a c h of them i s " i n d i c a t e d " .
have r e q u i r e d t h e i n t r o d u c t i o n of c o r r e c t i o n s i n aerodynamic
flight
speed.
T h i s h a s c a u s e d a new c l a s s i f i c a t i o n o f a i r s p e e d s .
The s p e e d o f a i r c r a f t movement r e l a t i v e t o t h e E a r t h ' s
i s c a l l e d f l i g h t . groundspeed ( W ) .

surface

F l i g h t g r o u n d s p e e d c a n b e m e a s u r e d d i r e c t l y by means o f D o p p l e r
or i n e r t i a l s y s t e m s , d e t e r m i n e d b y s i g h t i n g a l o n g a s e r i e s o f l a n d ­
m a r k s on t h e E a r t h ' s s u r f a c e , a n d a l s o c a l c u l a t e d on t h e b a s i s o f
f l y i n g t i m e b e t w e e n t w o l a n d m a r k s on t h e E a r t h ' s s u r f a c e .
In
a d d i t i o n , g r o u n d s p e e d c a n b e d e t e r m i n e d by a d d i n g t h e a i r s p e e d a n d
wind v e c t o r s , i f t h e wind s p e e d a n d d i r e c t i o n a t f l i g h t a l t i t u d e
a r e known.
Navigational

Speed Triangle

The i n t e r r e l a t i o n s h i p o f t h e e l e m e n t s o f f l i g h t d i r e c t i o n a n d
s p e e d i n t h e c h o s e n frame of r e f e r e n c e of a i r c r a f t c o u r s e s i s
c l e a r l y i l l u s t r a t e d by a n a v i g a t i o n a l s p e e d t r i a n g l e .

I n F i g . 1 . 6 1 a n a v i g a t i o n a l s p e e d t r i a n g l e i s shown f o r a
g e n e r a l c a s e , i . e . , i n d e p e n d e n t l y o f t h e m e r i d i a n which i s u s e d as
t h e b a s i s for m e a s u r i n g a n a i r c r a f t c o u r s e .
S t r a i g h t l i n e s OPN a n d O ~ P Ni n t h e f i g u r e s h o w t h e d i r e c t i o n
o f t h e m e r i d i a n a t p o i n t PA; V i s t h e a i r s p e e d v e c t o r ; W i s t h e
g r o u n d s p e e d v e c t o r ; y is t h e c o u r s e o f t h e a i r c r a f t ( C ) , a i s t h e

98


/1

L

d r i f t a n g l e ( D A ) ; J, i s t h e f l i g h t a n g l e ( F A ) , 6 i s t h e d i r e c t i o n
o f t h e wind v e c t o r r e l a t i v e t o t h e m e r i d i a n f o r r e a d i n g t h e a i r ­
c r a f t c o u r s e ; & $ i s t h e f l i g h t a n g l e o f t h e w i n d (WA) r e a d f r o m t h e
g i v e n l i n e o f t h e p a t h ; a n d 6 y = a t 6 $ i s t h e c o u r s e wind a n g l e
(CWA), r e a d f r o m t h e l o n g i t u d i n a l a x i s o f t h e a i r c r a f t .

-

A s p e e d t r i a n g l e c a n be s o l v e d g r a p h i c a l l y b y c o n s t r u c t i o n o f
v e c t o r s o n p a p e r or b y a m e c h a n i c a l a p p a r a t u s , u s i n g a s p e c i a l d e ­
v i c e ( a wind-speed i n d i c a t o r which i s a combination o f r u l e s , d i a l s ,
a n d h i n g e s w i t h movable a n d immovable j o i n t s ) .
A s p e e d t r i a n g l e i s s o l v e d a n a l y t i c a l l y on t h e o a s i s o f a
known s i n e t h e o r e m .
From F i g u r e 1 . 6 3 , i t i s c l e a r t h a t i n t h e g i v e n
case t h e s i n e theorem w i l l have t h e form:

(1.93)

From (1.93) t h e v a l u e of t h e d r i f t a n g l e and t h e f l i g h t ground­
s p e e d a r e e a s i l y d e t e r m i n e d o n t h e b a s i s o f known v a l u e s o f t h e a i r ­
c r a f t c o u r s e , a i r s p e e d , and t h e speed and d i r e c t i o n of t h e wind.

Now,

l e t u s d e f i n e t h e p a t h a n g l e of t h e wind:
b+ = b

- 9.

(1.94)

The d r i f t a n g l e o f a n a i r c r a f t a c c o r d i n g t o ( 1 . 9 3 ) i s d e t e r ­
mined from t h e f o r m u l a
sin a =

U
sid 6+.
V

(1.95)

The v a l u e o f t h e f l i g h t g r o u n d s p e e d i s t h e n e a s i l y d e t e r m i n e d :

V sin 6

w=
P

4.

(1.96)

These problems a r e e s p e c i a l l y
simple t o solve with s l i d e r u l e s having
a combination of s i n e logarithms with
a l o g a r i t h m s c a l e of l i n e a r v a l u e s .
1

Fig. 1.61.
Navigational
Speed T r i a n g l e

I n t h i s case, combining t h e loga r i t h m o f t h e s i n e o f t h e wind a n g l e
with the logarithm of the airspeed, we
obtain directly:

99

I-

I

/101'

lg sin u - ig u = Ig sin b+- lg

v = lg sin 8,

- igW

or o n s c a l e s o f n a v i g a t i o n a l r u l e r s w i t h t h e d e s i g n a t i o n s u s e d w i t h
special values f o r aircraft courses.

u

v

w

(1.97)

T o d e t e r m i n e wind s p e e d s a n d d i r e c t i o n s a t f l i g h t a l t i t u d e on
t h e b a s i s o f known v a l u e s o f a i r s p e e d , g r o u n d s p e e d , a n d d r i f t a n g l e ,
l e t us use Figure 1.62.

From t h e f i g u r e , i t f o l l o w s
that
O D = VCOSDA;
01D = Vsin DA;
DM=OM-OD=

Fig. 1.62.
Determining t h e

A n g l e a n d S p e e d o f t h e Wind
w i t h Known V a l u e s o f t h e
Groundspeed and D r i f t Angle
of t h e A i r c r a f t .

Therefore

W-VCOS DA.

,


V s i n._--.
DA
tg W A = %D = DM
W - VCOSDA

(1.98)

The f l i g h t a n g l e o f t h e w i n d d e t e r m i n e d i n t h i s way p e r m i t s
t h e f u r t h e r s o l u t i o n of problems on t h e b a s i s of t h e s i n e theorem
[Equation (1.9311.

i.e.,

W i t h s m a l l d r i f t a n g l e s ( p r a c t i c a l l y u p t o loo), c o s D A e 1,
it i s p o s s i b l e t o c o n s i d e r i n approximation t h a t
tgWA=

Vsin D A

w-v

F o r s o l u t i o n on a s l i d e r u l e ,

Translator's note:
1 00

lg = log.

(1.99)

( 1 . 9 9 ) i s reduced t o t h e form:

or o n a s l i d e r u l e ,
DA.

WA

w-v v

I-igsinlgtg I
llgspl
.-

A f t e r f i n d i n g t h e f l i g h t a n g l e of t h e wind, t h e v a l u e o f t h e
w i n d s p e e d i s d e t e r m i n e d o n t h e b a s i s of t h e s i n e t h e o r e m .

E l e m e n t s Which D e t e r m i n e F l i g h t A l t i t u d e
The f l i g h t
special initial
for m e a s u r i n g f l
f o r which it i s

a l t i t u d e o f a n a i r c r a f t (HI i s measured f r o m a
l e v e l of t h e Earth’s surface.
The i n i t i a l l e v e l
i g h t a l t i t u d e i s c h o s e n d e p e n d i n g on t h e p u r p o s e s
measured.

For example, i n o r d e r t o d i s t r i b u t e t h e c o u n t e r and i n c i d e n t a l
movements o f a i r c r a f t i n a i r s p a c e ( f l i g h t e c h e l o n s ) , t h e i n i t i a l
l e v e l f o r m e a s u r i n g t h e a l t i t u d e on e a c h a i r c r a f t m u s t b e g e n e r a l .
To e n s u r e t h e s a f e t y o f f l i g h t s o f i n d i v i d u a l a i r c r a f t a t l o w a l t i ­
t u d e s , it i s d e s i r a b l e t h a t t h e f l i g h t a l t i t u d e be measured from
t h e s u r f a c e o f t h e r e l i e f o v e r which t h e a i r c r a f t i s f l y i n g .
In
m a k i n g a n a p p r o a c h t o l a n d a t a n a i r p o r t , f l i g h t a l t i t u d e i s meas­
ured from t h e l e v e l of t h e l a n d i n g p o i n t .
U s u a l l y 2 or 3 k i n d s o f a l t i t u d e s a r e m e a s u r e d a t t h e same
time.
T h e r e f o r e , it i s n e c e s s a r y t o c l a s s i f y them a n d t o e s t a b l i s h
a r e l a t i o n s h i p between them.
A t the present t i m e ,
tinguished (Fig. 1.63):

Nabs

+Fig.

1.63.

t h e following kinds of a l t i t u d e s are d i s ­

I

L e v e l / r = 7 6 0 mm H g

I n t e r r e l a t i o n s h i p of D i f f e r e n t Systems f o r Measuring
Flight Altitude

101


4


/lo2

( a ) A b s o l u t e f l i g h t a l t i t u d e ( H a b s > i s m e a s u r e d f r o m t h e mean
l e v e l o f t h e B a l t i c S e a i n t h e same way a s t h e h e i g h t o f a r e l i e f
on t h e E a r t h ' s s u r f a c e .
(b)
R e l a t i v e f l i g h t a l t i t u d e ( H r e l ) i s measured from t h e
l e v e l o f t h e t a k e - o f f or l a n d i n g a i r p o r t .
(c)
True f l i g h t a l t i t u d e
''Htr" i s measured from t h e s u r f a c e
of t h e r e l i e f o v e r which t h e a i r c r a f t i s f l y i n g .

(d)
Conventional barometric a l t i t u d e "H7601t i s measured from
t h e c o n v e n t i o n a l b a r o m e t r i c l e v e l on t h e E a r t h ' s s u r f a c e , w h e r e t h e
a t m o s p h e r i c p r e s s u r e i s e q u a l t o 7 6 0 mm Hg.
Absolute, r e l a t i v e , and t r u e f l i g h t a l t i t u d e s are determined
by b a r o m e t r i c a l t i m e t e r s w i t h c o r r e c t i o n o f t h e i r r e a d i n g s f o r
i n s t r u m e n t a l and methodological e r r o r s .
The l a t t e r c a n a l s o b e
m e a s u r e d b y r a d i o a l t i m e t e r s a n d a i r c r a f t r a d a r e q u i p m e n t or d e t e r ­
There is a r e l a t i o n s h i p be­
mined by a i r c r a , f t s i g h t i n g d e v i c e s .
t w e e n t h e t h r e e i n d i c a t e d a l t i t u d e s which makes it p o s s i b l e t o
s w i t c h from one k i n d of a l t i t u d e t o a n o t h e r .
C o n v e n t i o n a l b a r o m e t r i c a l t i t u d e i s m e a s u r e d by b a r o m e t r i c
altimeters without considering methodological e r r o r s ,
Therefore,
i t h a s no d i r e c t c o n n e c t i o n w i t h t h e f i r s t t h r e e k i n d s o f a l t i t u d e s ,
and a t a high f l i g h t a l t i t u d e it can be d i s t i n g u i s h e d from t h e abso­
l u t e a l t i t u d e c l o s e s t t o it b y 9 0 0 - 1 0 0 0 m .
The m a i n a d v a n t a g e o f a c o n v e n t i o n a l b a r o m e t r i c a l t i t u d e i s t h e
c o n v e n i e n c e o f u s i n g it f o r e c h e l o n i n g f l i g h t s a c c o r d i n g t o a l t i t u d e s
when t h e i m p o r t a n t t h i n g i s n o t t h e p r e c i s e m e a s u r i n g o f a l t i t u d e
b u t only t h e p r e s e r v a t i o n of s a f e a l t i t u d e i n t e r v a l s between
neighboring echelons.
The l a t t e r c o n d i t i o n i s s a t i s f i e d , s i n c e i f
we p e r m i t t w o a i r c r a f t t o m e e t i n o n e r e g i o n a n d a t o n e a l t i t u d e ,
t h e m e t h o d o l o g i c a l c o r r e c t i o n s i n t h e s e a i r c r a f t w i l l be i d e n t i c a l .
Therefore, such a meeting cannot occur i f a i r c r a f t maintain d i f ­
f e r e n t a l t i t u d e s b a s e d on i n s t r u m e n t s .
From F i g u r e 1 . 6 3 i t i s e v i d e n t t h a t t r u e f l i g h t a l t i t u d e i s
d i s t i n g u i s h e d from a b s o l u t e f l i g h t a l t i t u d e by t h e h e i g h t o f t h e
r e l i e f o v e r which t h e a i r c r a f t i s f l y i n g , and from r e l a t i v e a l t i ­
t u d e by t h e h e i g h t o f t h e r e l i e f above t h e a i r p o r t l e v e l from which
r e l a t i v e a l t i t u d e i s measured:

(1.100)


w h e r e H r i s t h e a l t i t u d e o f t h e r e l i e f a b o v e s e a l e v e l ; AHr
h e i g h t of t h e r e l i e f above t h e l e v e l of t h e a i r p o r t .

10 2

is the

R e l a t i v e a l t i t u d e i s d i s t i n g u i s h e d from t r u e a l t i t u d e by t h e
h e i g h t o f t h e r e l i e f , w h i l e it i s d i s t i n g u i s h e d f r o m a b s o l u t e a l t i ­
t u d e by t h e h e i g h t o f t h e a i r p o r t above sea l e v e l :

(1.101)

F i n a l l y , a b s o l u t e f l i g h t a l t i t u d e c a n b e d e t e r m i n e d on t h e
b a s i s o f t h e v a l u e s o f t r u e or r e l a t i v e f l i g h t a l t i t u d e :

/lo4

(1.102)

C a l c u l a t i n g F l i g h t A l t i t u d e i n Determining Distances
on t h e E a r t h ' s S u r f a c e
I n m e a s u r i n g d i r e c t i o n s on t h e E a r t h ' s s u r f a c e , f l i g h t a l t i t u d e
d o e s n o t e x e r t a d i r e c t i n f l u e n c e on t h e v a l u e o f t h e m e a s u r e d
a n g l e s or o n t h e a c c u r a c y o f t h e m e a s u r e m e n t s .
A c t u a l l y , b y d i r e c t i o n o n t h e E a r t h ' s s u r f a c e w e mean d i r e c t i o n
o f t h e l i n e of i n t e r s e c t i o n o f t h e h o r i z o n p l a n e w i t h t h e p l a n e o f
a g r e a t c i r c l e ( o r t h o d r o m e ) which j o i n s two p o i n t s on t h e E a r t h ' s
surface.
S i n c e t h e v e r t i c a l a t a n y o f t h e s e p o i n t s on t h e i n d i c a t e d
l i n e l i e s i n t h e plane of a g r e a t circle, f l i g h t a l t i t u d e does not
e x e r t a n i n f l u e n c e on t h e d i r e c t i o n o f t h e o r t h o d r o m e and t h e r e f o r e
on d i r e c t i o n on t h e E a r t h ' s s u r f a c e .

--

I n m e a s u r i n g d i s t a n c e s on t h e
E a r t h ' s s u r f a c e , f l i g h t a l t i t u d e can
p l a y a n i m p o r t a n t r o l e and can l e a d t o
l a r g e measurement e r r o r s i f w e do n o t
allow f o r errors i n f l i g h t altitude
(Fig. 1.64).
I n t h e f i g u r e , s t r a i g h t l i n e s OAl
a n d OB1 a r e v e r t i c a l s o f t h e p o s i t i o n
o f a n a i r c r a f t a t p o i n t s A and B .
Obviously t h e d i s t a n c e S between
p o i n t s A 1 and B1 a t f l i g h t a l t i t u d e i s
g r e a t e r t h a n d i s t a n c e S between p o i n t s
A a n d B on t h e E a r t h ' s s u r f a c e :

Fig. 1.64.
Calculating
Flight Altitude i n
Determining Distances.
1 03

4.


whence

(1.104)


w h e r e R e i s t h e r a d i u s o f t h e E a r t h ( e q u a l t o 6 3 7 1 km) a n d H i s t h e
flight altitude.
Each k i l o m e t e r o f f l i g h t a l t i t u d e l e n g t h e n s t h e p a t h between
p o i n t s on t h e E a r t h ’ s s u r f a c e by a v a l u e e x p r e s s e d i n p e r c e n t :

‘ *‘0° = 0.016%.
6371

For e x a m p l e , a t a f l i g h t a l t i t u d e o f 1 0 km, a d i s t a n c e o n t h e
E a r t h ’ s s u r f a c e e q u a l t o 3 0 0 0 km l e n g t h e n s t o t h e v a l u e
3000~10~0,016
=4,8
6371

UA,

The i n d i c a t e d l e n g t h e n i n g o f t h e p a t h o f t h e a i r c r a f t d o e s n o t
e x e r t a s u b s t a n t i a l i n f l u e n c e on t h e t i m e of t h e a i r c r a f t f l i g h t
along the path.
The i n f l u e n c e of f l i g h t a l t i t u d e on d e t e r m i n a t i o n
o f t h e p o s i t i o n o f t h e a i r c r a f t by r a n g e f i n d i n g a n d , e s p e c i a l l y ,
h y p e r b o l i c d e v i c e s t u r n s o u t t o b e more s u b s t a n t i a l .
L e t u s assume t h a t a r a n g e f i n d i n g d e v i c e i s l o c a t e d a t p o i n t
A on t h e E a r t h ’ s s u r f a c e , w h i l e t h e a i r c r a f t i s l o c a t e d a t p o i n t
B1 , a t f l i g h t a l t i t u d e .

A s i s e v i d e n t from F i g u r e 1 . 6 6 , t h e d i s t a n c e from t h e ground
radio-engineering apparatus t o the a i r c r a f t R along a straight l i n e
w i l l equal A B I , while the distance along t h e Earth’s surface S is
equal t o AB.
L e t u s d r o p a p e r p e n d i c u l a r from p o i n t A o n t h e E a r t h ‘ s s u r f a c e
t o p o i n t D on t h e v e r t i c a l O B I . O b v i o u s l y ,
AB: = AD2 +- OB;,

since

10 4

/lo5

(1.105)

With s m a l l a n g u l a r d i s t a n c e s S ( u p . t o 6 O a l o n g t h e a r c of t h e o r t h o ­
d r o m e ) , when R s i n S M S , w h i l e c o s S M 1, (1.105) t a k e s t h e f o r m :

F i g u r e 1 . 6 6 c a n l i k e w i s e b e u s e d f o r d e t e r m i n i n g t h e maximum
d i s t a n c e o f g e o m e t r i c a l v i s i b i l i t y of o b j e c t s o n t h e g r o u n d f r o m o n
b o a r d t h e a i r c r a f t , or o f a n a i r c r a f t f r o m t h e E a r t h ' s s u r f a c e .

I t i s o b v i o u s t h a t w i t h maximum v i s i b i l i t y , l i n e A B 1 m u s t b e
tangent t o the Earth's surface, ;.e.,
it i s l o c a t e d i n t h e plane of
the horizon.
I n t h i s c a s e , a n g l e OABl w i l l be a r i g h t a n g l e .
Therefore

,
OA2 + A B : = OB:

or


Expanding t h e r i g h t - h a n d

side of the equation, we obtain:

C o n s i d e r i n g t h a t a t d i s t a n c e s up t o 6 0 0 - 7 0 0 k m , R t S = S , a n d
e.g
d i s r e g a r d i n g t h e v a l u e H 2 as n e g l i g i b l y s m a l l i n c o m p a r i s o n w i t h
2ReH, we o b t a i n t h e a p p r o x i m a t e f o r m u l a

s= v2q.
S u b s t i t u t i n g i n (1.107) t h e v a l u e o f

(1.107)
t h e r a d i u s of t h e E a r t h

( 6 3 7 1 km) w e o b t a i n :

-

S=1/12742H= 113fi.
B e a r i n g i n mind t h a t a s a r e s u l t of t h e r e f r a c t i o n o f l i g h t
or r a d i o w a v e s i n a v e r t i c a l p l a n e , t h e d i s t a n c e o f g e o m e t r i c a l
v i s i b i l i t y i n c r e a s e s a p p r o x i m a t e l y by 8%, t h e p r a c t i c a l r e s u l t w i l l
be :

1 05

svis=122 VE

(1.108)

F o r m u l a (1.108) d e t e r m i n e s t h e l i m i t s o f a p p l i c a b i l i t y of
(1.104) or ( 1 . 1 0 6 ) .
S i n c e w e have a g r e e d t o c o n s i d e r c o s S = 1 and
R s i n S = S u p t o S = 6O, w h i c h o n t h e E a r t h ' s s u r f a c e c o r r e s p o n d s
t o 6 6 6 km, i t i s o b v i o u s t h a t a t f l i g h t a l t i t u d e s u p t o 2 5 km i t
is always p o s s i b l e t o use (1.106).

It i s necessary t o use t h e precise.formula (1.104) a t d i s t a n c e s
This is possible a t f l i g h t a l t i t u d e s exceeding
o f m o r e t h a n 7 0 0 km.
2 5 km.

Fig.

1.65. Calculating F l i g h t A l t i t u d e i n Determining t h e Path
L e n g t h o f E l e c t r o m a g n e t i c Wave P r o p a g a t i o n

L e t u s p a u s e now t o d i s c u s s t h e i n f l u e n c e o f f l i g h t a l t i t u d e
on t h e a c c u r a c y o f measuring d i s t a n c e s a t v e r y s m a l l r a n g e s , i . e . ,
i n c a s e s when l o n g r a d i o w a v e s c a p a b l e o f t r a v e l i n g a r o u n d t h e
E a r t h ' s surface are used (Fig. 1.65).
I n t h e f i g u r e , ground r a d i o e n g i n e e r i n g equipment i s l o c a t e d
a t p o i n t A on t h e E a r t h ' s s u r f a c e ; t h e a i r c r a f t i s a t p o i n t B a t
a l t i t u d e H.
L i n e A B i s t h e c u r v e o f p r o p a g a t i o n o f a r a d i o wave
front.

I f w e c o n d i t i o n a l l y move t h e E a r t h ' s s u r f a c e t o t h e r i g h t b y
a v a l u e e q u a l t o H / 2 , t h e n t h e l i n e o f r a d i o wave p r o p a g a t i o n b e ­
comes c o n c e n t r i c w i t h t h e E a r t h ' s s u r f a c e a n d w i l l h a v e a r a d i u s o f
c u r v a t u r e R 1 = Re + H/2.
Therefore, t h e i n c r e a s e i n d i s t a n c e from p o i n t A t o p o i n t B
c a n b e c o n s i d e r e d as a l e n g t h e n i n g o f t h e o r t h o d r o m e a t a f l i g h t
a l t i t u d e e q u a l t o H/2, i . e . ,

1 06

/.lo7

E l e m e n t s o f A i r c r a f t Roll
I t i s known t h a t t h e r a d i u s o f a i r c r a f t r o l l i n a i r s p a c e a t a
given banking B equals:

If a f l i g h t i s c a r r i e d o u t w i t h a c o u n t e r or i n c i d e n t a l w i n d ,
r o l l i n g o f a n a i r c r a f t t h r o u g h a n a n g l e o f 9 0 ° i n v o l v e s a n i n c r e a s e
or d e c r e a s e i n t h e mean r a d i u s o f
r o l l of an aircraft r e l a t i v e t o t h e
Earth's surface (Fig. 1.66).

.+

F)
uI 9-250 k m / h r

u1 =O; V = 600 k m / h r

u, = t250k m /h r

Fig. 1.66.
Deformation of
the R o l l Trajectory i n the
P r e s e n c e o f Wind.

I n f a c t , f o r a change i n t h e
d i r e c t i o n of t h e groundspeed v e c t o r
o f a n a i r c r a f t b y 90°, w i t h a s h i f t
f r o m t h e p l a n e o f i n c i d e n t wind t o
a l a t e r a l plane, it i s necessary t o
execute a r o l l of an a i r c r a f t t o t h e
r i g h t or l e f t t h r o u g h a n a n g l e o f
90° t D A Y and i n c h a n g i n g from t h e
p l a n e o f i n c i d e n t wind t o a l a t e r a l
wind t o a l a t e r a l p l a n e t h r o u g h a n
a n g l e of 90° - D A .

During r o l l i n g of an a i r c r a f t
i n airspace, i n the first instance
t h e r e w i l l be a d e v i a t i o n o f t h e a i r ­
c r a f t from t h e o r i g i n a l f l i g h t d i r e c t i o n ; i n t h e second c a s e , t h e r e / l o 8
w i l l be a deviation opposite t o the o r i g i n a l .

Example.
L e t u s e x a m i n e t h e roll o f a n a i r c r a f t t h r o u g h 9 0 ° ,
w i t h a f l i g h t a i r s p e e d o f 6 0 0 km/h a n d w i t h a c o u n t e r a n d i n c i d e n t
w i n d s p e e d o f 2 5 0 km/h ( 7 0 m / s e c ) .
A c c o r d i n g t o (1.6), t h e r a d i u s o f t h e a i r c r a f t i n a i r s p a c e
with banking of 1 5 O w i l l be

R==

1672
= 10500
9,81 tg 15"

m

The d r i f t a n g l e a t t h e e n d o f r o l l i n g t h r o u g h 9 0 °
the following value:

w i l l have

--

250
DA- arctg 6oo - 23".

107

T h e r e f o r e , i n t h e f i r s t case it w i l l b e n e c e s s a r y t o t u r n t h e a i r ­
c r a f t t h r o u g h 1 1 3 O , a n d i n t h e s e c o n d c a s e t h r o u g h 67O.

= 6 0 0 km/h

T h e a n g u l a r v e l o c i t y o f roll a t
R = 10,500 m w i l l be

(167 m/sec)

and

V.57 3 167.57.3
W = L =
=0,9deg/sec
R
10 500

L e t u s d e t e r m i n e t h e a d d i t i o n a l s h i f t o f t h e a i r c r a f t as a
r e s u l t o f wind d u r i n g r o l l i n g i n t h e f i r s t i n s t a n c e :
70 m/sec.l13O
AR, =
~9 km
0.9 d e g / s e c

and i n t h e second i n s t a n c e :
70 * 67

ARx == -" 5
0.9

km

O b v i o u s l y , d u r i n g r o l l ( i n t h e f i r s t c a s e t h r o u g h 113O a n d i n
t h e s e c o n d c a s e t h r o u g h 67') t h e movement o f t h e a i r c r a f t i n d i r e c ­
t i o n X w i l l not be i d e n t i c a l , since
B, = R sin Y P .

Then t h e g e n e r a l p a t h o f a n a i r c r a f t i n d i r e c t i o n
I n t h e first c a s e ,

Rx = 10,5 sin 113"

X w i l l equal:

+ 9 = 18,5 km

I n t h e second case,
R, = 10,5 sin 67"-5 = 4,5 k m

L e t u s now d e t e r m i n e t h e l a t e r a l s h i f t o f t h e a i r c r a f t R ,
d u r i n g roll:
I n t h e first case,
Rt= R f R s i r 1 2 3 ~ - 14,s km
and i n t h e second case

Rt=R-Rsin2$=6km.

For c o m p a r i s o n , l e t u s e x a m i n e t h e r o l l o f a n a i r c r a f t t h r o u g h
w i t h a r a d i u s c a l c u l a t e d n o t on t h e b a s i s o f a i r s p e e d , b u t on
t h e b a s i s of groundspeed:
90°,

w10 8

Vf

us.

I n t h e f i r s t case, t h e r a d i u s of r o l l i n g is

R=

2372
=21 km
9,81 tg 15"

R=

972
=3,5
9,81 tg 1_59

and i n t h e second c a s e
km

Let us compile a t a b l e with t h e r e s u l t s obtained:
.

Roll
parameters

u,=o
-

RY

R,

X

Z

~

103
10,5
10,5
10,5

10,5
21
18,5
14,5

I

10,5
3.5
4,5
6

From t h e t a b l e , i t i s e v i d e n t t h a t t h e r e s u l t s o f t h e c a l c u ­
l a t i o n s c a r r i e d o u t on t h e b a s i s o f t h e g r o u n d s p e e d a r e much c l o s e r
t o t h e a c t u a l r e s u l t s t h a n c a l c u l a t i o n s p n t h e b a s i s of a i r s p e e d .

;
7
*

----- - - _

_-----

/

C a l c u l a t i o n s o f roll on t h e
b a s i s of groundspeed with winds a s
h i g h a s 2 0 0 - 3 0 0 k m / h , when e n t e r i n g
a new l i n e o f f l i g h t , w i l l b e c a r ­
r i e d o u t w i t h an a c c u r a c y o f 1 - 2 . 5
km.
Some i n a c c u r a c i e s a r i s e , b u t
only i n the l a t e r a l d i r e c t i o n .
How­
e v e r , t h i s i s n o t of p r a c t i c a l s i g ­
n i f i c a n c e , since the d i r e c t i o n of
d e v i a t i o n c o i n c i d e s w i t h t h e new
l i n e of f l i g h t .

Fig. 1.67.
Approach o f a n
A i r c r a f t t o a Given Line
w i t h t h e P r e s e n c e o f a n Approach Angle.

With a d e c r e a s e i n t h e a n g l e
o f roll, t h e t r a j e c t o r y c a l c u l a t e d
a c c o r d i n g t o t h e g r o u n d s p e e d comes
c l o s e r t o the a c t u a l t r a j e c t o r y of
Therefore, i n t h e
aircraft roll.
f u t u r e w e w i l l p r o c e e d f r o m f l i g h t g r o u n d s p e e d i n c a l c u l a t i n g roll.
I n aircraft n a v i g a t i o n , i n c l u d i n g maneuvering before l a n d i n g ,
it i s n e c e s s a r y t o s o l v e t h r e e t y p e s o f p r o b l e m s , t a k i n e ; i n t o ac­
c o u n t t h e roll t r a j e c t o r y .

10 9

1.

Roll w i t h a S t r a i g h t L i n e

Combinati'on of

L e t u s assume t h a t an a i r c r a f t i s a p p r o a c h i n g a g i v e n l i n e of
f l i g h t at a definite angle (Fig. 1.67).
I t is o b v i o u s t h a t t h e a n g l e o f r o l l o f t h e a i r c r a f t f o r f o l ­
l o w i n g a l o n g t h e g i v e n l i n e i s e q u a l t o t h e a p p r o a c h a n g l e (a).
L e t u s d e t e r m i n e t h e d i s t a n c e ( Z ) from t h e g i v e n l i n e on which i t
i s n e c e s s a r y t o b e g i n t h e roll s o t h a t t h e r o l l t r a j e c t o r y w i l l b e
joined with t h e given l i n e .
it i s e v i d e n t t h a t t h i s d i s t a n c e i s e q u a l t o :

In Figure 1.67,

Z=R-Ucosa
= R (1

or

-

(1.109)
COS a).

Example.
An a i r c r a f t a p p r o a c h e s a g i v e n l i n e o f f l i g h t w i t h
a g r o u n d s p e e d o f 9 0 0 km/h a t a 2 5 O a n g l e .
Determine t h e l a t e r a l
d i s t a n c e from t h e l i n e of f l i g h t a t which it i s n e c e s s a r y t o b e g i n
a roll f o r a s m o o t h a p p r o a c h t o t h e l i n e .
S o l u t i on.
R---

2502

9.81 tg 15"

Z = 26,5(1-

= 2 6 , 5 km

cos 25O) = 2,46 knl

C o m b i n a t i o n of t w o rolls

2.

I f , d u r i n g f l i g h t a l o n g a given f l i g h t l i n e , a d e v i a t i o n from
i t o c c u r s a n d i t i s n e c e s s a r y t o a p p r o a c h t h e g i v e n l i n e by t h e
s h o r t e s t t r a j e c t o r y , a n a p p r o a c h maneuver i s u s e d which i s a combi­
n a t i o n o f two r o l l s ( F i g . 1 . 6 8 ) .

>
-------

L+

_--­

CLZ

4

Fig. 1.68.
Approach of
a n A i r c r a f t t o a Given
F l i g h t Line with a Paral­
l e l Flight Line.
110

S i n c e t h e v a l u e of Z i n t h i s c a s e
i s c o n s i d e r e d known, w h i l e t h e r a d i u s
o f roll i s d e t e r m i n e d on t h e b a s i s o f
t h e groundspeed and t h e given banking
i n t h e roll, i t i s n e c e s s a r y t o d e t e r ­
m i n e t h e v a l u e o f t h e a n g l e s ai = a~
o f t h e c o m b i n e d rolls.

It i s obvious t h a t i n t h i s case,
i n e a c h o f t h e t w o c o m b i n e d rolls,
t h e aircraft approaches t h e f l i g h t
p a t h by a v a l u e 2 / 2 ; t h e r e f o r e ,

/110

Z

-2 -R(1-

cosa),

whence

(1.110)


For e x a m p l e , l e t u s s a y t h a t a n a i r c r a f t h a v i n g a g r o u n d s p e e d
o f 9 0 0 km/h h a s d e v i a t e d f r o m a g i v e n f l i g h t p a t h b y 5 km; t o make
t h e a p p r o a c h , i t i s n e c e s s a r y t o e x e c u t e t w o c o m b i n e d rolls w i t h
b a n k i n g of 1 5 O t o a n g l e s u p t o 25O.

3.

Linear p r e d i c t i o n o f roll

(LPR)

L e t u s examine two s o l u t i o n s t o problems, w i t h a c o n s i d e r a t i o n
o f t h e roll t r a j e c t o r y o f a n a i r c r a f t w h i c h i n c l u d e s o n e r e c t i l i n e a r
p a r t of the path.

/111
L i n e a r p r e d i c t i o n o f roll i s c a l c u l a t e d i n i n s t a n c e s o f a
break i n t h e f l i g h t path a t turning points i n t h e route (Fig. 1.69).
I n t h e f i g u r e , TPR i s t h e t u r n i n g p o i n t i n t h e r o u t e a n d TA i s
t h e t u r n a n g l e o f t h e f l i g h t p a t h e q u a l t o t h e roll a n g l e o f a n
aircraft (RA).
A s i s clear from Figure 1 . 6 9 , t h e r a d i u s of r o l l of an a i r ­
c r a f t , a t i t s beginning and end, i s d i r e c t e d perpendicular t o t h e
preceding and following orthodrome segments of t h e p a t h .
The l i n e s
0-TPR f o r m t h e b i s e c t o r o f t h e a n g l e o f roll.

Thus, w e have two i d e n t i c a l r e c t a n g u l a r t r i a n g l e s w i t h v e r t e x
T h e l i n e a r p r e d i c t i o n o f roll ( L P R ) i s t h e
a n g l e s e q u a l t o RA/2.
l i n e o f t a n g e n c y o f t h e roll a n g l e , d i v i d e d i n h a l f :

Fig. 1.69.
Linear Prediction
o f R o l l o f a n A i r c r a f t (LPR).

L i n e a r Lag of A i r Fig. 1.70.
c r a f t R o l l (LLR).

111


LPR =Rtg

RA

-.2

(1.111)


E x a m p l e : D e t e r m i n e L P R w i t h a f l i g h t g r o u n d s p e e d o f 9 0 0 km/h
a n d a n a n g l e o f t u r n t o t h e new f l i g h t p a t h o f 40° f o r b a n k i n g i n
a roll o f 1 5 O .

S o l u t i on.
R=

­

2502
=26,5 km
9.81 tg 15"

L P R = 26.5. tg 20" = 9,6 k m

Linear predictions
i n T a b l e 1.1.

w i t h roll a n g l e s f r o m 0 t o 1 5 0 ° , a r e g i v e n
-

Wl
km/hr

R, M

'roll
t o goo
450 60" 750 900 105" 120" 135" 1500 s e c

15"
~

400

500
600
700
800
900

4600

7'w
10~600
14 700
18500
23 500

/11:
­

P r e d i c t i o n w i t h roll a n g l e s
mO t o 1 5 0 ° , k r
- 3-

I I I I -l--l--i
" I

__

.... ..

2,O
3,1
4,2
6,O
7,7
9,7

2,7 3,5 4,6 6,O 8,01!l,0115,0
4,3. 5,71 7,3 9,712,8,18,0,27,5
6,lI 8,210,613,818,3,25,5140,0
8,3,11,0:14,4 19,0,25,0,35,052,0
10,7,14,018.524,032,0142,070,0,
13,5 18,023,530,0140 058 087 0
1 'I 'I ' I
I
I

'65
82
100

116

132
148

I n some c a s e s , t h e n e c e s s i t y f o r f l i g h t a b o v e t h e T P R w i t h t h e
f l i g h t a n g l e o f t h e f o l l o w i n g p a r t o f t h e p a t h c a n a r i s e ( F i g . 1.70),
e . g . , i n f l i g h t s of d i f f e r e n t k i n d s f o r t e s t i n g a i r c r a f t and ground
navigational equipment.
In t h e s e cases, i n s t e a d of l i n e a r predic­
t i o n , l i n e a r l a g o f roll ( L L R ) i s c a l c u l a t e d , w h i l e t h e r o l l i s
c a r r i e d o u t i n t h e d i r e c t i o n o p p o s i t e t o t h e t u r n o f t h e new f l i g h t
p a t h by t h e a n g l e

RA=3600-

TA

I n F i g u r e 1 . 7 2 , it i s c l e a r t h a t t h e LLR i s a l i n e o f t h e
t a n g e n t s of t h e t u r n angle of t h e f l i g h t p a t h d i v i d e d i n h a l f , i . e . ,
w i t h t h e s a m e t u r n a n g l e s , t h e f o r m u l a for t h e L L R r e m a i n s t h e same
as f o r t h e l i n e a r p r e d i c t i o n o f roll:

112


CHAPTER TWO

AIRCRAFT NAVIGATION U S I N G MISCELLANEOUS D E V I C E S
1.

G e o t e c h n i c a l Means o f A i r c r a f t N a v i g a t i o n

G e o t e c h n i c a l means o f a i r c r a f t n a v i g a t i o n c o n s t i t u t e a p o r t i o n o f t h e n a v i g a t i o n a l equipment o f a n a i r c r a f t which h a s a n
autonomous c h a r a c t e r and i s u s e d u n d e r a l l f l i g h t c o n d i t i o n s ,
i n d e p e n d e n t l y o f t h e u s e o f o t h e r s p e c i a l d e v i c e s s u c h as t h o s e
e m p l o y i n g r a d i o e n g i n e e r i n g or a s t r o n o m y , f o r e x a m p l e .

/113

Such d e v i c e s i n c l u d e t h o s e which measure t h e a i r c r a f t c o u r s e ,
a i r s p e e d , and f l i g h t a l t i t u d e , as w e l l as d e v i c e s f o r a u t o m a t i c
s o l u t i o n of n a v i g a t i o n a l problems.
G e o t e c h n i c a l d e v i c e s f o r a i r c r a f t n a v i g a t i o n a r e b a s e d on
h i g h l y d i v e r s e p h y s i c a l p r i n c i p l e s f o r t h e u s e of n a t u r a l geo­
physical f i e l d s of t h e Earth (magnetic, g r a v i t a t i o n a l , pressure,
t h e f i e l d of e l e c t r o m a g n e t i c o s c i l l a t i o n s i n t h e o p t i c a l and i n f r a ­
red range, etc.).
The o p e r a t i o n o f t h e s e d e v i c e s i s more d e p e n ­
d e n t o n t h e p h y s i c a l p a r a m e t e r s o f t h e medium i n w h i c h t h e f l i g h t
i s being c a r r i e d out t h a n i s t h e case f o r devices employing r a d i o
T h e r e f o r e , t h e y have a complex math­
e n g i n e e r i n g or a s t r o n o m y .
ematical basis f o r t h e i r regulation, especially with regard t o
t h e s y s t e m f o r making c o r r e c t i o n s t o i n s t r u m e n t r e a d i n g s .
Aircraft navigation using only geotechnical devices can be
c a r r i e d o u t i n c a s e s when i t i s p o s s i b l e t o c h e c k t h e n a v i g a t i o n ­
a l c a l c u l a t i o n s ( e v e n p e r i o d i c a l l y ) by d e t e r m i n i n g t h e l o c u s o f
t h e a i r c r a f t b y o t h e r m e a n s or v i s u a l l y .

H i s t o r i c a l l y speaking, t h e development of radio-engineer­
i n g and a s t r o n o m i c a l means f o r a i r c r a f t n a v i g a t i o n h a s b e e n d i r e c t e d
t o w a r d a s o l u t i o n of o n l y o n e p r o b l e m , n a m e l y , t h e d e t e r m i n a t i o n
o f t h e a i r c r a f t c o o r d i n a t e s on t h e E a r t h ' s s u r f a c e , w h i c h p r o v e d
a n e c e s s a r y a d j u n c t t o t h e g e o t e c h n i c a l means o f a i r c r a f t n a v i ­
g a t i o n i n f l i g h t u n d e r c o n d i t i o n s when t h e g r o u n d w a s n o t v i s i b l e .
I n r e c e n t y e a r s , t h e r e has been a development of t h e r a d i o e n g i n e e r i n g , a s t r o n o m i c a l , and a s t r o - i n e r t i a . 1 systems f o r s o l v i n g
problems which a r e i n h c r e n t i n g e o t e c h n i c a l d e v i c e s f o r a i r c r a f t
n a v i g a t i o n , i . e . , measurement of t h e a i r c r a f t c o u r s e , a i r s p e e d ,
turn angle, altitude, etc.

/114

113


2.

Course Instruments a n d Systems

Course i n s t r u m e n t s are i n t e n d e d f o r determining t h e p o s i t i o n
o f t h e l o n g i t u d i n a l a x i s o f a n a i r c r a f t i n t h e p l a n e of t h e h o r ­
i z o n or ( w h a t a m o u n t s t o t h e same t h i n g ) for m e a s u r i n g t h e c o u r s e
of t h e aircraft.
I t i s n e c e s s a r y t o know t h e a i r c r a f t c o u r s e i n o r d e r t o d e t e r m i n e
both t h e f l i g h t d i r e c t i o n and t h e p o s i t i o n of t h e a i r c r a f t r e l a t i v e
t o o r i e n t a t i o n p o i n t s on t h e g r o u n d .

A s w e have mentioned above, t h e r e are s e v e r a l systems f o r
c a l c u l a t i n g t h e a i r c r a f t c o u r s e , and t h e s e l e c t i o n of t h e system
o f c a l c u l a t i o n i s g o v e r n e d b o t h by t h e r e q u i r e m e n t s o f a i r c r a f t
n a v i g a t i o n and by t h e t e c h n i c a l p o s s i b i l i t i e s f o r e q u i p p i n g t h e
aircraft with the corresponding instruments.
A t t h e p r e s e n t t i m e , t h e r e are no c o u r s e i n s t r u m e n t s which
completely s a t i s f y t h e requirements of aircraft c o n t r o l under a l l
conditions.
Therefore, aircraft usually are f i t t e d with several
d i f f e r e n t c o u r s e i n s t r u m e n t s o p e r a t i n g on d i f f e r e n t p r i n c i p l e s
and u s i n g d i f f e r e n t s y s t e m s o f c a l c u l a t i o n ; e a c h of them i s u s e d
I n some
under t h e c o n d i t i o n s which are most f a v o r a b l e f o r i t .
cases, thes'e i n s t r u m e n t s a r e combined i n t o complexes, c a l l e d c o u r s e
s y s t e m s , w h e r e t h e o p e r a t i o n of t h e i n d i v i d u a l i n s t r u m e n t s i s c l o s e l y
related.
T h i s makes i t p o s s i b l e t o e x p l o i t t h e p o s i t i v e q u a l i t i e s
o f e a c h o f them i n a c t u a l o p e r a t i o n .
Methods o f U s i n g

t h e Magnetic F i e l d o f t h e Earth t o Determine

Direction
D i r e c t i o n s on t h e E a r t h ' s s u r f a c e c a n b e m e a s u r e d m o s t ac­
c u r a t e l y by a s t r o n o m i c a l m e t h o d s .
However, t h i s r e q u i r e s o p t i ­
c a l v i s i b i l i t y o f t h e s k y , complex and a c c u r a t e a p p a r a t u s , and
tedious calculation.
D i r e c t i o n s on t h e E a r t h ' s s u r f a c e c a n b e
d e t e r m i n e d m o r e s i m p l y a n d i n many c a s e s q u i t e r e l i a b l y b y u s i n g
t h e magnetic f i e l d of %he Earth.
The m a g n e t i c f i e l d of t h e E a r t h ( F i g . 2 . 1 ) i s c h a r a c t e r i z e d
by t h e f o l l o w i n g p a r a m e t e r s a t e v e r y p o i n t on i t s s u r f a c e :

(a)

D i r e c t i o n a l i t y of

t h e h o r i z o n t a l component o f %he f i e l d

(b)

D i r e c t i o n a l i t y of

t h e v e r t i c a l component o f t h e f i e l d

(5;

(5;

­

-

( c ) The d i r e c t i o n o f t h e p l a n e i n w h i c h t h e v e c t o r s H a n d
Z l i e r e l a t i v e t o t h e geographic meridian a t t h e given p o i n t ,

The p l a n e i n w h i c h t h e v e c t o r s
t h e pZane of t h e m a g n e t i c m e r i d i a n .

114

3

2

and
are located i s c a l l e d
The a n g l e between t h e p l a n e s

/115

r


of t h e magnetic and g e o g r a p h i c meridians i s c a l l e d t h e magnetic

d e c l i n a t i o n and i s r e p r e s e n t e d by AM.
The p o i n t s on t h e E a r t h ' s s u r f a c e a t w h i c h t h e m a g n e t i c m e r ­
i d i a n s i n t e r s e c t are c a l l e d magnetic poles.
Obviously , t h e h o r i ­
z o n t a l component o f t h e m a g n e t i c f i e l d i s l a c k i n g a t t h e m a g n e t i c
p o l e s , w h i l e t h e i n t e n s i t y o f t h e v e r t i c a l component r e a c h e s i t s
maximum v a l u e .
The m a g n e t i c g o l e s o f t h e E a r t h
do not coincide with t h e geographic
ones.
The c o o r d i n a t e s o f t h e N o r t h
M a g n e t i c P o l e a r e 74ON a n d 1 O O O W ;
t h o s e of t h e South Magnetic Bole
a r e 6 8 O S , 143OE ( a s o f 1 9 5 2 ) .

n


The d e v i c e o f a f r e e l y r o t a t i n g
m a g n e t i c p o i n t e r mounted i n t h e p l a n e
of t h e magnetic meridian is used
t o d e t e r m i n e d i r e c t i o n on t h e E a r t h ' s
surface.
Therefore , a t every point
on t h e E a r t h ' s s u r f a c e t h e r e w i l l
be a r e l i a b l e i n d i c a t i o n of t h e t h r e e
p a r a m e t e r s which c h a r a c t e r i z e t h e
magnetic f i e l d of t h e Earth.

Magnetic F i e l d
Fig. 2.1.
of t h e E a r t h .

The t o t a l i n t e n s i t y o f t h e m a , g
n e t j c f i e l d of t h e Earth (vector T )
i s t h e r e s u l t a n t v e c t o r of H and 2 .
Consequently,

-

(2.1)


The o e r s t e d ( O e ) i s t h e u n i t o f m e a s u r e m e n t f o r t h e t o t a l
i n t e n s i t y of t h e m a g n e t i c f i e l d , as w e l l as t h e i n t e n s i t y of i t s
components; i n o t h e r w o r d s , i t i s t h e i n t e n s i t y o f a f i e l d which
i n t e r a c t s w i t h a u n i t m a g n e t i c p o l e w i t h a f o r c e o f one d y n e .
The l i m i t s o f c h a n g e i n t h e i n t e n s i t y o f t h e co mp o n en t s
t h e magnetic f i e l d of t h e Earth a r e t h e following:

i n

(a)
Horizontal:
from z e r o i n t h e v i c i n i t y of t h e magnetic
p o l e s t o a maximum a t t h e m a g n e t i c e q u a t o r ( 0 . 4 o e r s t e d s i n t h e
v i c i n i t y of Indonesia);

(b)
Vertical: from z e r o a t t h e magnetic e q u a t o r t o 0 . 6
s t e d s i n t h e v i c i n i t y of t h e magnetic p o l e s .

oer­

A s m a l l e r u n i t o f i n t e n s i t y , t h e gamma ( y ) , i s u s e d f o r v e r y
p r e c i s e m a g n e t i c measurements; it i s e q u a l t o one h u n d r e d t h o u ­
s a n d t h o f an o e r s t e d .

The a n g l e w h i c h

characterizes the inclination of the vector
115


of t o t a l i n t e n s i t y of t h e magnetic f i e l d of t h e Earth t o t h e plane
of t h e t r u e horizon is c a l l e d t h e magnetic incZination "0".
0 arctg

2
=H

(2.2)

Charts of t h e magnetic f i e l d s are prepared f o r convenience
i n using t h e magnetic f i e l d of t h e Earth t o determine d i r e c t i o n s
on t h e E a r t h ' s s u r f a c e .
A c h a r t of magnetic i n c l i n a t i o n s i s extremely important f o r
aircraft navigation.
L i n e s j o i n i n g p o i n t s on t h e E a r t h ' s s u r f a c e
w h i c h h a v e t h e same m a g n e t i c d e c l i n a t i o n a r e c a l l e d i s o g o n i c s .
They a r e p r i n t e d d i r e c t l y on f l i g h t a n d l a r g e - s c a l e g e o g r a p h i c
maps.

To d e t e r m i n e t h e t r u e c o u r s e , t h e m a g n e t i c d e c l i n a t i o n d e t e r ­
mined from t h e c h a r t a t t h e l o c u s o f t h e a i r c r a f t ( w i t h i t s s i g n ,
as a c o r r e c t i o n ) i s e n t e r e d i n t h e r e a d i n g s o f t h e m a g n e t i c c o m p a s s .
F i g u r e 2 . 2 s h o w s a map o f t h e
t i o n s e n t e r e d on i t ; t h e i s o g o n i c s
The p o s i t i v e
the Earth's surface.
by s o l i d l i n e s , w h i l e t h e n e g a t i v e

World w i t h t h e m a g n e t i c d e c l i n a ­
a r e s h o w n a s t h e y a p p e a r on
i s o g o n i c s on t h e c h a r t a r e m a r k e d
o n e s a r e marked by d a s h e d l i n e s .

All o f t h e i s o g o n i c s m e e t a t t h e m a g n e t i c p o l e s o f t h e E a r t h ,
a n d t h e compass r e a d i n g s ( a n d c o n s e q u e n t l y t h e m a g n e t i c i n c l i n ­
a t i o n ) c h a n g e b y 180° when p a s s i n g t h r o u g h t h e m a g n e t i c p o l e .
I n a d d i t i o n , t h e i s o g o n i e s a l s o meet a t t h e g e o g r a p h i c p o l e s ,
s i n c e t h e d i r e c t i o n s of t h e magnetic and geographic meridians a r e
opposite between t h e magnetic and geographic p o l e s , b u t coincide
a f t e r p a s s i n g t h r o u g h t h e p o l e , i . e . , t h e d e c l i n a t i o n c h a n g e s by
180O.
T h e map o f t h e W o r l d s h o w i n g t h e m a g n e t i c d e c l i n a t i o n s h a s
t h e isogonics only f o r t h e normal magnetic f i e l d of t h e Earth.
I n a d d i t i o n t o t h i s normal f i e l d , t h e r e i s a l s o an anomalous f i e l d ,
c a u s e d by t h e m a g n e t i z a t i o n o f t h e s o i l i n t h e u p p e r l a y e r s o f
the Earth.
Regions and areas of changes i n t h e d e c l i n a t i o n i n
s u c h r e g i o n s a r e marked on l a r g e - s c a l e c h a r t s .
The r e l i a b i l i t y o f o p e r a t i o n o f m a g n e t i c compasses a n d t h e
m a g n i t u d e o f t h e e r r o r s i n t h e i r r e a d i n g s d e p e n d on t h e i n t e n s i t y
o f t h e h o r i z o n t a l component of t h e m a g n e t i c f i e l d o f t h e E a r t h .
E r r o r s i n t h e r e a d i n g s of c o m p a s s e s , p a r t i c u l a r l y when t h e a i r ­
c r a f t i s r o l l i n g , d e p e n d o n l y on t h e i n t e n s i t y o f t h e v e r t i c a l
comonent

.

The l i n e s on t h e E a r t h ' s s u r f a c e w h i c h c o n n e c t p o i n t s w i t h
t h e same i n t e n s i t y o f t h e h o r i z o n t a l or v e r t i c a l c o m p o n e n t s o f
t h e magnetic f i e l d are c a l l e d isodynamic l i n e s .

116


1116

--

F i g u r e 2 . 3 s h o w s a map o f t h e W o r l d w i t h t h e i s o d y n a m i c l i n e s
f o r t h e h o r i z o n t a l c o m p o n e n t of t h e E a r t h ' s m a g n e t i c f i e l d , w h i l e
F i g u r e 2 . 4 shows t h o s e f o r t h e v e r t i c a l component.

Fig.

2.2.

W o r l d C h a r t of M a g n e t i c D e c l i n a t i o n s .

117


/118


Fig.

118


2 . 3 . W o r l d C h a r t o f I s o d y n a m i c L i n e s f o r t h e H o r i z o n t a l Com­
ponent of t h e E a r t h ’ s Magnetic F i e l d .

/119


I

Fig.

2.4.

W o r l d C h a r t o f I s o d y n a m i c L i n e s f o r t h e V e r t i c a l Compo­
n e n t of t h e E a r t h ' s Ma.gnetic F i e l d .

Only g e n e r a l ( o u t l i n e ) c h a r t s of i s o d y n a m i c l i n e s a r e u s e d
/120
i n aircraft navigation.
T h e s e l i n e s d o n o t a p p e a r on f l i g h t c h a r t s .
L i n e s on t h e E a r t h ' s s u r f a c e w h i c h c o n n e c t p o i n t s w i t h t h e
s a m e d e c l i n a t i o n o f t h e m a g n e t i c f i e l d a r e c a l l e d i s o c l i n e s . Form­
e r l y , o u t l i n e maps o f i s o c l i n e s w e r e u s e d j o i n t l y w i t h c h a r t s o f
i s o d y n a m i c l i n e s s h o w i n g t h e t o t a l i n t e n s i t y of t h e m a g n e t i c f i e l d
t o determine t h e e r r o r s of magnetic compasses.
A t the present
t i m e , t h e s e c h a r t s are no l v n g e r u s e d , s i n c e i t i s b e t t e r t o use
t h e isodynamic l i n e s of t h e h o r i z o n t a l and v e r t i c a l components
of t h e m a g n e t i c f i e l d .
Variations

and O s c i l l a t i o n s

i n the Earth's Magnetic F i e l d

T h e r e a r e s e v e r a l h y p o t h e s e s r e g a r d i n g t h e o r i g i n o f t h e mag­
n e t i c f i e l d o f t h e E a r t h , b u t n o n e of them h a s b e e n a d e q u a t e l y
p r o v e n as o f t h e p r e s e n t t i m e .
Possible f a c t o r s i n t h e formation
of t h e m a g n e t i c f i e l d a r e t h e s u b s u r f a c e a n d i o n o s p h e r i c e l e c t r i c a l
c u r r e n t s , as w e l l as t h e m a g n e t i c i n d u c t i o n a n d m a g n e t i c h y s t e r e s i s

119


of t h e s o i l ,

c o m p o s i n g t h e s t r u c t u r e of

the Earth's

sphere.

Even i f t h e s e f a c t o r s a r e n o t p r i m a r i l y r e s p o n s i b l e f o r t h e
f o r m a t i o n of t h e m a g n e t i c f i e l d of t h e E a r t h , t h e y are i n any c a s e
i m p o r t a n t i n f l u e n c e s on i t s s t r u c t u r e a n d s t a b i l i t y .
An a n a l y s i s . o f t h e i s o l i n e s o f t h e i n t e n s i t y o f t h e c o m p o n e n t s
i n t h e magnetic f i e l d of t h e Earth and t h e magnetic d e c l i n a t i o n s
r e v e a l s t h a t t h e i r c o n f i g u r a t i o n i s d e t e r m i n e d b o t h by g e n e r a l
l a w s o f t h e d i s t r i b u t i o n o f m a g n e t i c f o r c e s i n t h e f i e l d o f a mag­
n e t i z e d s p h e r e , as w e l l a s by l o c a l d i s t u r b a n c e s i n t h e g e n e r a l
s t r u c t u r e of t h e f i e l d .
Therefore, t h e s t a t i o n a r y magnetic f i e l d
o f t h e E a r t h i s a s s u m e d t o c o n s i s t o f a sum o f f i e l d s :
(a)

The f i e l d o f t h e u n i f o r m m a g n e t i z e d s p h e r e ;

(b)
The c o n t i n e n t a l f i e l d , r e l a t e d t o t h e n o n u n i f o r m i t y o f
t h e r e l i e f and t h e s t r u c t u r e of t h e i n t e r n a l l a y e r s of t h e E a r t h ;

( c ) The a n o m a l o u s f i e l d , r e l a t e d t o t h e e x i s t e n c e o f d e p o s ­
i t s of m a g n e t i c m a t e r i a l i n t h e u p p e r l a y e r s o f t h e E a r t h ' s c o r e .
A s s y s t e m a t i c o b s e r v a t i o n s of t h e s t r u c t u r e o f t h e m a g n e t i c
f i e l d of t h e E a r t h have shown, i t does n o t remain s t r i c t l y s t a ­
tionary but undergoes constant changes.
C h a n g e s or v a r i a t i o n s
i n t h e m a g n e t i c f i e l d of t h e E a r t h have a d i v e r s e n a t u r e .
A n n u a l or c o n s t a n t c h a n g e s i n t h e m a g n e t i c f i e l d o f t h e E a r t h
These v a r i a t i o n s c o n s t i t u t e t h e
a r e c a l l e d secuzar v a r i a t i o n .
d i f f e r e n c e between t h e average annual v a l u e s f o r t h e elements of
t h e E a r t h ' s magnetism.
T h e c a u s e s for t h e a n n u a l v a r i a t i o n s a r e
changes i n t h e components of t h e s t a t i o n a r y f i e l d w i t h t i m e , i . e . ,
t h e m a g n e t i c moment o f t h e E a r t h a n d t h e c o n t i n e n t a l f i e l d .
The a n n u a l v a r i a t i o n s i n t h e d e c l i n a t i o n a t m i d d l e l a t i t u d e s
r e a c h e d 10-12', a n d u p t o 4 0 ' a t h i g h l a t i t u d e s ; t h e r e f o r e , when
u s i n g c h a r t s o f m a g n e t i c d e c l i n a t i o n s , or i s o g o n i c s on f l i g h t c h a r t s ,
i t i s n e c e s s a r y t o c o n s i d e r t h e p e r i o d when t h e y w e r e m a d e .
If
t h e c h a r t o f m a g n e t i c d e c l i n a t i o n s i s o b s o l e t e , c h a n g e s must be
/121
made when u s i n g i t f o r t h e v a r i a t i o n i n t h e d e c l i n a t i o n d u r i n g
t h e t i m e w h i c h h a s p a s s e d s i n c e t h e c h a r t w a s made.
The d e s i r e d
c o r r e c t i o n i s determined from s p e c i a l c h a r t s of t h e s e c u l a r v a r i ­
The i s o l i n e s o f e q u a l
a t i o n s of t h e magnetic f i e l d of t h e E a r t h .
s e c u l a r v a r i a t i o n s i n d e c l i n a t i o n on a c h a r t a r e c a l l e d i s o p o r s .
I n a d d i t i o n t o t h e slow s y s t e m a t i c changes i n t h e magnetic
f i e l d of t h e E a r t h , t h e r e a r e a l s o p e r i o d i c a n d e v e n c h a o t i c c h a n g e s
which a r e r e l a t e d t o t h e s o - c a l l e d i n t e r n a l f i e l d of t h e E a r t h ,
These a r e e s t i ­
t h e main c a u s e o f w h i c h i s i o n o s p h e r i c c u r r e n t s .
m a t e d p e r i o d i c a l l y or a r e d i s r e g a r d e d e n t i r e l y .

12 0

M a g n e t i c Compasses

cases
rate.

The m a g n e t i c c o m p a s s i s t h e s i m p l e s t c o u r s e d e v i c e ; i n m o s t
, i t i s s u f f i c i e n t l y r e l i a b l e t h o u g h n o t s u f f i c i e n t l y accu­

H o w e v e r , a s i m p l e m a g n e t i c c o m p a s s w i t h a f r e e l y t u r n i n g mag­
n e t i c n e e d l e i s n o t s u i t a b l e for u s e o n b o a r d a n a i r c r a f t , s i n c e
i t s r e a d i n g s would b e i n a c c u r a t e and u n s t a b l e .
Various kinds of
i n t e r f e r e n c e would i n f l u e n c e t h e o p e r a t i o n o f t h e compass d u r i n g
f l i g h t , including:
(a)

Movements o f t h e a i r c r a f t r e l a t i v e t o i t s a x i s ;

(b)
V i b r a t i o n s p r o d u c e d by t h e o p e r a t i o n o f t h e e n g i n e s and
b y t h e movement o f t h e a i r c r a f t t h r o u g h t h e a i r ;
(c)
The e f f e c t o f t h e m a g n e t i c f i e l d o f t h e a i r c r a f t , w h i c h
would c a u s e d e f l e c t i o n s o f t h e m a g n e t i c n e e d l e from t h e p l a n e o f
t h e magnetic meridian, ;.e.,
compass d e v i a t i o n .
O b v i o u s l y , a m a g n e t i c c o m p a s s w h i c h i s i n t e n d e d f o r u s e on
an a i r c r a f t must have d e v i c e s f o r compensating t h e i n t e r f e r e n c e
mentioned above.
The s i m p l e s t f o r m o f a n a i r c r a f t m a g n e t i c co mp as s i s t h e i n t e ­
g r a t e d compass, i . e . , one i n which t h e c o u r s e t r a n s m i t t e r ( s e n s i ­
t i v e e l e m e n t ) and t h e i n d i c a t o r a r e combined i n a s i n g l e h o u s i n g .
O f t h e l a r g e number of t y p e s o f m a g n e t i c compasses which h a v e
b e e n d e v i s e d a s of t h e p r e s e n t t i m e , t h e o n e m o s t u s e d n o w a d a y s
i s t h e “KI” ( a n a b b r e v i a t i o n f o r t h e h i s t o r i c name o f t h e m a g n e t i c
compasses which were d e v i s e d i n t h e p a s t f o r f i g h t e r a i r c r a f t ) .
C o m p a s s e s u s i n g o t h e r s y s t e m s a r e c a 1 l e d d i s t a n c e - m a g n e t i c or g y r o magnetic compasses.

Any i n t e g r a t e d a v i a t i o n a l m a g n e t i c c o m p a s s c o n s i s t s o f t h e
f o l l o w i n g main p a r t s ( F i g . 2 . 5 ) :
T h e b o w l or c o n t a i n e r 1 o f t h e c o m p a s s , f i l l e d w i t h a damp­
i n g f l u i d t o d e c r e a s e t h e o s c i l l a t i o n s , u s u a l l y l i q r o i n ; on t h e
b o t t o m of t h e bowl i s a p i v o t s u p p o r t f o r t h e movable p a r t o f t h e
c o m p a s s , w i t h a d a m p i n g s p r i n g a n d p i v o t b e a r i n g made o f a g a t e ;
The m o v a b l e p a r t o f t h e c o m p a s s , c o n s i s t i n g o f a c a r d 2 , w h i c h
i s a c o m b i n a t i o n of a m a g n e t i c s y s t e m (H-shaped m a g n e t ) , a f l o a t
t o r e d u c e t h e w e i g h t o f t h e c a r d a n d r e d u c e t h e f r i c t i o n on t h e
b e a r i n g , a n e e d l e p i v o t , a n d a r o t a t i n g s c a l e f’or t h e r e a d i n g s ,
mounted on t h e m a g n e t i c s y s t e m ;
Chamber 3 ,

compensating f o r thermal expansion and contracti.on

121

/122

o f t h e damping f l u i d ; t h e e x p a n s i o n chamber i s l o c a t e d a b o v e t h e
This
bowl and i s connected t o i t by h o l e s o f v e r y small d i a m e t e r .
a l l o w s a i r b u b b l e s t o e s c a p e from t h e bowl i n t o t h e chamber and
p e r m i t s t h e f l u i d t o f l o w b a c k a n d f o r t h w i t h e x p a n s i o n a n d con­
I t a l s o p r e v e n t s i t from s p l a s h i n g i n t h e bowl as t h e
traction.
a i r p l a n e moves ;
A d e v i c e 4 for g e t t i n g r i d o f d e v i a t i o n s o f t h e c o m p a s s , w h i c h
c o n t a i n s s e v e r a l b a r m a g n e t s p r e s s e d i n t o drums which r o t a t e i n mutu­
a l l y p e r p e n d i c u l a r p l a n e s w i t h t h e a i d o f screws.
Rotation of
t h e drums p e r m i t s them t o b e s e t t o a p o s i t i o n w h e r e t h e m a g n e t i c
f i e l d of t h e b a r magnets compensates f o r t h e magnetic f i e l d of
t h e a i r c r a f t a c t i n g on t h e c o m p a s s c a r d .

4

Fig.

2.5.

3

Combined M a g n e t i c Compass:
Section; (b) External View.

( a ) Cross

The d e s i g n o f t h e m a g n e t i c c o m p a s s d e s c r i b e d a b o v e r e d u c e s
t h e effect of i n t e r f e r e n c e with i t s o p e r a t i o n t o a c o n s i d e r a b l e
d e g r e e and t h e compass r e a d i n g s a r e q u i t e s t a b l e .
Nevertheless,
m a g n e t i c compasses ( e s p e c i a l l y i n t e g r a t e d o n e s ) h a v e a number o f
shortcomings which p r e v e n t t h e c o u r s e from b e i n g c a l c u l a t e d under
certain conditions.
The m o s t i m p o r t a n t o f t h e s e s h o r t c o m i n g s a r e
the following:
(1) A l i m i t a t i o n o f t h e c h o i c e o f m o u n t i n g l o c a t i o n for t h e
compass a b o a r d t h e a i r c r a f t ;
t h e i n t e g r a t e d compass must b e l o ­
c a t e d i n a p l a c e which i s s u i t a b l e f o r d e t e r m i n i n g t h e c o u r s e ,
a n d t h e r e f o r e c l o s e t o o t h e r i n s t r u m e n t s a n d m o v i n g p a r t s for c o n ­
t r o l l i n g t h e a i r c r a f t , which produce a l a r g e and v a r y i n g d e v i a ­
t i o n of t h e compass;
(2)
T h e i m p o s s i b i l i t y o f u s i n g t h e c o m p a s s when t h e a i r c r a f t / 1 2 3
is turning.
When t h e a i r c r a f t m a k e s a t u r n , s e v e r a l f a c t o r s a c t
the
o n t h e c o m p a s s c a r d t o move i t f r o m i t s c u s t o m a r y p o s i t i o n :
p r e s s u r e o f t h e d a m p i n g f l u i d on t h e c a r d , t h e a c t i o n o f c e n t r i ­
f u g a l f o r c e o n t h e s o u t h e r n , s o m e w h a t e l o n g a t e d p o r t i o n of t h e
c a r d , as w e l l as a c h a n g e i n t h e s t r u c t u r e o f t h e m a g n e t i c f i e l d
The d e v i a t i o n s o f t h e compass c a r d
of t h e aircraft while turning.

122


f r o m t h e p l a n e o f t h e m a g n e t i c m e r i d i a n when t h e a i r c r a f t i s t u r n ­
i n g a r e p a r t i c u l a r l y n o t i c e a b l e when t h e a i r c r a f t c o u r s e c r o s s e s
t h e n o r t h e r n and s o u t h e r n d i r e c t i o n s .
These d e v i a t i o n s are c a l l e d
t h e n o r t h e r n and s o u t h e r n t u r n i n g e r r o r s .
The i n s t a b i l i t y o f t h e s t r u c t u r e o f t h e m a g n e t i c f i e l d o f
t h e E a r t h a t t h e l o c u s o f t h e a i r c r a f t and i t s changes w i t h t i m e
a r e t h e m a j o r s h o r t c o m i n g s of u s i n g m a g n e t i c c o m p a s s e s o f a l l t y p e s .
D e v i a t i o n o f M a g n e t i c Compasses and

i t s Compensation

The c a u s e o f m a g n e t i c c o m p a s s d e v i a t i o n i s t h e p r e s e n c e o f
p a r t s o n b o a r d t h e a i r c r a f t w h i c h a r e made o f m a t e r i a l s e x h i b i t i n g
magnetic p r o p e r t i e s .
Some o f t h e s e p a r t s h a v e a c o n s t a n t m a g n e t i c
field.
P a r t s o f t h i s k i n d a r e c a l l e d hard magnetic i r o n .
Another
g r o u p o f p a r t s a r e m a g n i t i z e d u n d e r t h e e f f e c t of t h e m a g n e t i c
f i e l d o f t h e E a r t h a n d a r e c a l l e d soft m a g n e t i c i r o n .
A c c o r d i n g t o C o u l o m b ’ s l a w , t h e f o r c e ( F ) of t h e i n t e r a c t i o n
o f m a g n e t i c masses ( m ) i s i n v e r s e l y p r o p o r t i o n a l t o t h e d i s t a n c e
b e t w e e n them (r).

T h e r e f o r e , t h e d e v i a t i o n o f t h e m a g n e t i c compass i n c r e a s e s
v e r y s h a r p l y w i t h t h e a p p r o a c h of i t s s e n s i t i v e e l e m e n t t o p a r t s
which have h i g h m a g n e t i z a t i o n .
A c c o r d i n g t o t h e p r i n c i p l e of i n d e p e n d e n c e of t h e a c t i o n o f
f o r c e s a t a g i v e n p o i n t i n t h e a i r c r a f t , i t i s p o s s i b l e t o sum
t h e m a g n e t i c f i e l d s coming f r o m i n d i v i d u a l p a r t s of t h e a i r c r a f t
and t o s u b j e c t them t o t h e e q u i v a l e n t e f f e c t o f a s i n g l e m a g n e t i z e d
bar located a t a certain point.
H o w e v e r , i f we t a k e i n t o a c c o u n t
t h e d i v e r s e n a t u r e of t h e a c t i o n o f t h e h a r d and s o f t magnetized
i r o n on d i f f e r e n t c o u r s e s a n d d u r i n g d i f f e r e n t m o t i o n s o f t h e a i r ­
craft, it is better t o subject t h i s f i e l d t o the equivalent action
of b a r s w h i c h h a v e a c o n s t a n t a n d v a r y i n g m a g n e t i z a t i o n .
L e t us assume t h a t t h e e q u i v a l e n t b a r of h a r d magnetized i r o n
is l o c a t e d h o r i z o n t a l l y and c o i n c i d e s with t h e d i r e c t i o n of t h e
longitudinal axis of the aircraft (Fig. 2.6).

WiZh a m a g n e t i c c o u r s e o f t h e a i r c r a f t e q u a l t o z e r o , t h e
v e c t o r F o f t h e f i e l d i n t e n s i t y of t h e b a r c o i n c i d e s i n d i r e c t i o n
w i t h t h e h o r i z o n t a l component of t h e m a g n e t i c f i e l d o f t h e E a r t h
H, which does n o t p r o d u c e any d e v i a t i o n o f t h e compass c a r d from
t h e plane of t h e magnetic meridian.

tor

F

/124

I n t h e c a s e o f a i r c r a f t c o u r s e s e q u a l t o 9 0 or 2 7 0 ° , t h e v e c ­
o f t h e f i e l d i n t e n s i t y of t h e b a r i s l o c a t e d a t r i g h t a n g l e s

123


-


t o t h e v e c t o r H , p r o d u c i n g maximum d e v i a t i o n o f t h e c a r d f r o m t h e
plane of t h e magnetic meridian.
H e n c e , when t h e a i r c r a f t i s t u r n i n g a r o u n d i t s v e r t i c a l a x i s
through 360°, t h e r e s u l t a n t v e c t o r ( P t ) of t h e hard magnetic i r o n
and t h e m a g n e t i c f i e l d o f t h e E a r t h w i l l c o i n c i d e a t two p o i n t s
w i t h t h e d i r e c t i o n o f t h e m a g n e t i c m e r i d i a n , a n d w i l l b e a t a maxi­
mum d i s t a n c e f r o m i t a t t w o o t h e r p o i n t s . D e v i a t i o n o f t h i s k i n d
is caZZed s e m i c i r c u l a r , i . e . , i t h a s z e r o v a l u e w i t h e v e r y 180°
r o t a t i o n of t h e a i r c r a f t ( F i g . 2 . 7 ) .
F i n a l l y , it cannot
be expected t h a t i n
t h e g e n e r a l case t h e
equivalent bar of hard
magnetic i r o n w i l l
I
coincide i n direction
with the longitudi­
n a l a x i s of t h e a i r craft.
However, t h i s
does not a l t e r the
n a t u r e of t h e semi­
circular deviation,
but only shifts the
Fig. 2.6.
D e v i a t i o n o f Compass C a r d b y
g r a p h of d e v i a t i o n
a B a r o f Hard M a g n e t i c I r o n .
r e l a t i v e t o the course
scale of t h e a i r c r a f t
by a n a n g l e which i s e q u a l t o t h a t between t h e a x i s o f t h e a i r ­
c r a f t and t h e a x i s of t h e e q u i v a l e n t b a r .
S e m i c i r c u l a r d e v i a t i o n o f a m a g n e t i c c o m p a s s c a n b e compen­
sated easily.
To d o t h i s , i t i s s u f f i c i e n t t o make a b a r o f h a r d
m a g n e t i c i r o n and p l a c e i t n e a r t h e compass i n s t a l l a t i o n i n s u c h
a way t h a t i t s f i e l d i s o p p o s i t e t o t h e d i r e c t i o n o f t h e f i e l d
of t h e equivalent b a r of hard magnetic i r o n .
L e t u s now a s s u m e t h a t t h e r e i s n o h a r d m a g n e t i c i r o n a b o a r d
t h e aircraft, but a f i e l d of s o f t magnetic i r o n is located hori­
z o n t a l l y and c o n t r i b u t e s t o t h e a c t i o n of t h e e q u i v a l e n t b a r , coin­
ciding i n direction with the longitudinal axis of the aircraft.

T h e e s s e n c e o f t h e e f f e c t o f t h e s o f t m a g n e t i c i r o n on t h e
compass r e a d i n g s c o n s i s t s i n t h e f a c t t h a t t h e b a r , which i s l o ­
cated i n a c e r t a i n p o s i t i o n r e l a t i v e t o t h e magnetic f i e l d of t h e
Earth, i s not magnetized i n t h e d i r e c t i o n of t h e f i e l d b u t along
t h e length of t h e b a r .
T h e m a g n e t i z a t i o n o f t h e b a r c a n b e e x p r e s s e d by t h e f o r m ­

ula
B = pHcos a ,

124

(2.4)

/125

w h e r e B i s t h e m a g n e t i c i n d u c t i o n , l~ i s t h e m a g n e t i c p e r m e a b i l ­
i t y o f t h e b a r , H i s t h e i n t e n s i t y o f t h e m a g n e t i c f i e l d , a n d ct
is
t h e a n g l e between t h e d i r e c t i o n of t h e i n t e n s i t y v e c t o r of t h e f i e l d
a n d t h e d i r e c t i o n of t h e b a r .
On c o u r s e s 0 a n d 180°,

i n t h i s case, t h e d i r e c t i o n
of t h e equivalent b a r coin­
cides with the direction
o f t h e h o r i z o n t a l compo­
nent of t h e vector of inten­
s i t y of t h e magnetic f i e l d
of the Earth (a = 0 ) ; although
t h e magnetic induction of
t h e b a r i s maximum, t h e r e
w i l l b e no compass d e v i ­
ation.
I n changing t h e aircraft
c o u r s e f r o m 0 t o 9 0 ° or
from 0 t o 270°, t h e magnetic
F i g . 2 . 7 . G r a p h of S e m i c i r c u l a r
Deviation.
induction of the b a r w i l l
decrease, but the angle
between t h e v e c t o r s H and B
w i l l
increase.
I t i s obvious t h a t t h e deviation w i l l then reach
i t s maximum a t a c o u r s e o f 4 5 or 3 1 5 O a n d w i l l r e a c h z e r o o n c e a g a i n
on c o u r s e s o f 9 0 a n d 2 7 0 ° .
A s i m i l a r change i n d e v i a t i o n w i l l V C C U ~
i n t h e f l i g h t s e c t o r s f r o m 9 0 t o 1 8 0 ° a n d f r o m 1 8 0 t o 270° ( F i g .
2.8).
I t i s c l e a r i n t h e f i g u r e t h a t t h e d e v i a t i o n from t h e s o f t
m a g n e t i c i r o n d u r i n g o n e c o m p l e t e t u r n of t h e a i r c r a f t a r o u n d t h e
v e r t i c a l a x i s passes through zero f o u r t i m e s , i . e . , it has a quar­
ternary nature.

The a c t i o n o f o n e m a g n e t i c b a r o f s o f t i r o n c l e a r l y i l l u s t r a t e s
t h e quarternary nature of t h e a l t e r n a t i n g magnetic f i e l d of t h e
aircraft.
I n p r a c t i c e , however, w i t h t h e e x c l u s i o n of r a r e cases,
t h e a l t e r n a t i n g m a g n e t i c f i e l d o f t h e a i r c r a f t c a n n o t amount t o
t h e e f f e c t o f one b a r o f s o f t m a g n e t i c i r o n .
I n f a c t , i f w e t a k e two b a r s o f s o f t i r o n and l o c a t e them a t
t o one a n o t h e r , t h e r e s u l t a n t v e c t o r of i n d u c t i o n o f t h e b a r s
w i l l c o i n c i d e w i t h t h e b i s e c t r i x between them ( B 1 = B 2 ) o n l y i n
t h e c a s e when t h e i n t e n s i t y v e c t o r o f t h e m a g n e t i c f i e l d ( H I ) c o i n ­
c i d e s w i t h t h e b i s e c t r i x of t h e a n g l e b e t w e e n t h e b a r s ( F i g . 2 . 9 ) .
I n a l l o t h e r cases, t h e i n d u c t i o n v e c t o r w i l l approach t h e a x i s
of t h e b a r which i s c l o s e r t o t h e i n t e n s i t y v e c t o r of t h e magnetic
field.
90°

If w e c o n s i d e r t h e a c t i o n o f one b a r , t h e v e c t o r o f m a g n e t i c
i n d u c t i o n w i l l change i n v a l u e b u t w i l l always c o i n c i d e w i t h t h e

125

D

/126

axis of t h e bar.
T h i s e s s e n t i a l l y e x p l a i n s t h e e x i s t e n c e on t h e
a i r c r a f t o f b o t h s e m i c i r c u l a r a n d q u a r t e r n a r y d e v i a t i o n as w e l l as
deviations o f higher order.

H

Fig.

2.8.

Fig.

2.9.

Fig.

2.8.

Graph o f Q u a r t e r n a r y D e v i a t i o n f r o m S o f t M a g n e t i c I r o n .

Fig.

2.9.

M a g n e t i c I n d u c t i o n o f C r o s s e d Bars o f

Soft Iron.

I n a d d i t i o n , if w e d i s r e g a r d t h e d e v i a t i o n o f h i g h e r o r d e r ,
t h e d e v i a t i o n f r o m s o f t m a g n e t i c i r o n c a n n o t b e e l i m i n a t e d by u s i n g
a s u i t a b l e b a r of s o f t i r o n , s i n c e i t w i l l a l s o be magnetized l i k e
a l l o t h e r p a r t s of t h e a i r c r a f t and w i l l n o t l e a d t o a r e d u c t i o n
b u t r a t h e r t o an increase of t h e deviation.

EquaZizing t h e Magnetic FieZd o f t h e A i r c r a f t
T h e c a u s e of m a g n e t i c c o m p a s s d e v i a t i o n o n b o a r d a n a i r c r a f t
i s g e n e r a l l y a l a c k of c o i n c i d e n c e between t h e r e s u l t a n t components
of t h e m a g n e t i c f i e l d o f t h e a i r c r a f t w i t h t h e v e c t o r o f i n t e n s i t y
of t h e E a r t h ' s magnetic f i e l d .
When t h e a i r c r a f t r o t a t e s a r o u n d i t s a x i s , t h e a l t e r n a t i n g
/127
magnetic f i e l d of t h e aircraft not only r o t a t e s along with i t , b u t
simultaneously changes i n magnitude and s i g n .
Therefore, i n order
t o d e t e r m i n e t h e magnitude and s i g n o f t h e d e v i a t i o n f o r v a r i o u s
a i r c r a f t c o u r s e s , it i s a d v i s a b l e t o e x p r e s s i t s f i e l d components
i n t h e f o r m of f o r c e s a c t i n g a l o n g t h e a x e s o f t h e a i r c r a f t .
Obviously, t h e magnitude of t h e s e f o r c e s (with t h e exception
o f t h e c o m p o n e n t s made o f h a r d m a g n e t i c i r o n ) w i l l v a r y w i t h c h a n g e s
i n t h e m a g n e t i c c o u r s e of t h e a i r c r a f t ( Y M ) .
D e p e n d i n g on t h e n a t u r e a n d c h a r a c t e r o f t h e a c t i o n o f t h e
components o f t h e m a g n e t i c f i e l d on t h e s e n s i t i v e e l e m e n t o f t h e
compass, w e c a n d i v i d e them i n t o t h r e e g r o u p s :

126

(1)
Components o f t h e m a g n e t i c f i e l d o f t h e E a r t h a l o n g t h e
axes of t h e aircraft; t h e i r designations coincide with t h e desig­
nations f o r t h e aircraft axes X, Y, 2.
The r e s u l t a n t v e c t o r o f
t h e s e components i s T .
( 2 )
T h e c o m p o n e n t s o f t h e m a g n e t i c f i e l d o f t h e a i r c r a f t made
of hard magnetic i r o n have t h e designations:
P along t h e X-axis
of t h e a i r c r a f t ; Q a l o n g t h e Y - a x i s , a n d R a l o n g t h e Z - a x i s .

(3)
Components o f s o f t m a g n e t i c i r o n o f t h e a i r c r a f t .
As
f o l l o w s f r o m w h a t h a s b e e n s a i d a b o v e , t h e y c a n n o t b e v i e w e d as
a simple p a r t of t h e r e s u l t a n t vector along t h e axes of t h e air­
craft.

For c o n v e n i e n c e i n m a t h e m a t i c a l o p e r a t i o n s , t h e s e c o m p o n e n t s
l e a d t o an e q u i v a l e n t effect of n i n e b a r s of s o f t magnetic i r o n ,
o f which t h r e e b a r s c o i n c i d e w i t h e a c h of t h e a x e s of t h e a i r c r a f t .
T h i s means t h a t e a c h o f t h e t h r e e b a r s which c o i n c i d e w i t h a g i v e n
a x i s o f t h e a i r c r a f t i s m a g n e t i z e d b y a component o f t h e m a g n e t i c
f i e l d of t h e E a r t h w h i c h i s l o c a t e d o n l y a l o n g some o n e a x i s o f
the. aircraft.
Equivalent b a r s a , b, e a r e l o c a t e d along t h e X-axis of t h e
a i r c r a f t ; b a r a i s m a g n e t i z e d by t h e component of t h e m a g n e t i c f i e l d
of t h e E a r t h X , b a r b by component Y , and b a r e by component 2 .
E q u i v a l e n t b a r s d, e , f a r e l o c a t e d a l o n g t h e Y - a x i s , a n d b a r s
g, h , k a r e l o c a t e d a l o n g t h e Z - a x i s ; t h e y a r e m a g n e t i z e d b y t h e
same c o m p o n e n t s o f t h e v e c t o r ?;.
The c o n t r i b u t i o n o f t h e m a g n e t i c f i e l d o f t h e s o f t i r o n i n
t h e a i r c r a f t t o t h e e q u i v a l e n t e f f e c t of n i n e b a r s a c q u i r e s phys­
i c a l s i g n i f i c a n c e i n - s u m m i n g t h e m a g n e t i c i n d u c t i o n o f t h e compo­
nents of t h e v e c t o r T , along t h e axes of t h e a i r c r a f t .

For e x a m p l e , t h e X - c o m p o n e n t o f t h e m a g n e t i c f i e l d o f t h e E a r t h
a c t s on b a r s a , d , g , a n d t h e r e s u l t a n t i n d u c t i o n f r o m t h e s e t h r e e
b a r s s h o w s how t h e v e c t o r o f t h e m a g n e t i c f i e l d f r o m t h e s o f t mag­
n e t i c i r o n o f t h e a i r c r a f t ZX w o u l d b e l o c a t e d i f t h e c o m p o n e n t s
of t h e magnetic f i e l d of t h e E a r t h Y and Z w e r e e q u a l t o zero.
I n o t h e r words, t h e e q u i v a l e n t b a r s are e q u i v a l e n t t o t h e vec­
t o r s of d i v i s i o n o f t h e m a g n e t i c i n d u c t i o n f r o m t h e components o f
t h e m a g n e t i c f i e l d o f t h e E a r t h a l o n g t h e a x e s of t h e a i r c r a f t ( T a b l e
2.1).

I n summing t h e m a g n e t i c f o r c e s a l o n g t h e a x e s , w e o b t a i n t h e
e q u a t i o n s for t h e m a g n e t i c f i e l d o f t h e a i r c r a f t :

127


/128

T h e s e f i e l d s w i l l b e u s e d a s a b a s i s for d e r i v i n g f o r m u l a s
f o r t h e d e v i a t i o n o f m a g n e t i c c o m p a s s e s on a n a i r c r a f t .
TABLE 2 . 1

axis of
the
aircraft

I

1

T

'

1

E

I
I

I

ox
OY
oz

R e s u l t a n t forces

lx

I

X
Y
Z

P

aX

Q

dX
gx

R

x',

1

,.I

?

my
'

I

.

nZ

1

CZ
fZl

6Y
eY
hY

kZ

-

T h e sum o f t h e v e c t o r s
Y ' a n d Z' g i v e s a t o t a l v e c t o r
a c t i n g on t h e s e n s i t i v e e l e m e n t o f t h e c o m p a s s .
,

?;I

De v i u t i o n Fo rmu Zas
I n t h e e q u a t i o n s of t h e m a g n e t i c f i e l d of t h e a i r c r a f t , t h e
c o n s t a n t t e r m s a r e o n l y t h e c o m p o n e n t s of t h e f i e l d o f t h e h a r d
m a g n e t i c i r o n , P , Q, R .
However, t o c a l c u l a t e t h e d e v i a t i o n i n h o r i ­
z o n t a l f l i g k t , we c a n c o n s i d e r t h a t t h e m a g n e t i c i n d u c t i o n Z f r o m
the vector T is constant along the v e r t i c d l axis of the a i r c r a f t
( t e r m s c Z , fZ, k Z ) .
In addition, horizontal f l i g h t w i l l not involve the t h i r d
equation i n (2.5),
determining Z'.
I f we a l s o c o n s i d e r t h a t t h e sum o f t h e v e c t o r s X a n d Y c o n ­
s t i t u t e s t h e h o r i z o n t a l c o m p o n e n t of t h e m a g n e t i c f i e l d o f t h e E a r t h

H,

X=HCQS~;
U = HsinT,

t h e f i r s t two e q u a t i o n s
lows :

i n (2.5)

+

c a n b e r e w r i t t e n t o r e a d as f o l ­

-

x' = H cos 7 aH cos r bH sin 7 + CL + P ;
U'=Hsiny+dHcosr-eHsinr+
fZ+Q,
where y

I

i s t h e magnetic course of t h e a i r c r a f t .

a r e t h e components of t h e magnetic f i e l d
The v e c t o r s XI, Y'
along t h e l o n g i t u d i n a l and t r a n s v e r s e axes of t h e a i r c r a f t a t t h e
l o c u s of t h e c o m p a s s .
The m a g n e t i c c o m p a s s d e v i a t i o n ( 6 ) i s e x p r e s s e d b y t h e a n g l e
between t h e d i r e c t i o n of t h e h o r i z o n t a l component of t h e m a g n e t i c
f i e l d o f t h e E a r t h H a n d t h e h o r i z o n t a l c o m p o n e n t o f t h e t o t a l magn e t i c f i e l d on t h e a i r c r a f t H ' ( F i g . 2 . 1 0 ) .

128


/129

O b v i o u s l y , t g 6 i s e q u a l t o t h e r a t i o of t h e p r o j e c t i o n o f v e c ­
t o r HI i n a d i r e c t i o n p e r p e n d i c u l a r t o t h e m a g n e t i c m e r i d i a n H " ,
t o i t s p r o j e c t i o n on t h e magnetic m e r i d i a n H " ' :
H"
H"

tg8=-=

X' sin 7 f Y' cos 7
X' cog 7 - Y' sin 7'
~

If w e s u b s t i t u t e i n t o E q u a t i o n ( 2 . 7 ) t h e v a l u e s o f X' a n d I"
from E q u a t i o n ( 2 . 6 1 ,
and a l s o r e d u c e s i m i l a r t e r m s , r e p l a c i n g t h e
v a l u e s s i n y c o s y , s i n 2 y and cos2y by t h e i r obvious homologues 4 s i n 2 y ,
?i ( l - c o s 2 y ) a n d % ( l t c o s 2 y ) , w e w i l l h a v e :

(2.8)

The t e r m s i n E q u a t i o n ( 2 . 8 1 , w i t h a c o e f f i c i e n t e q u a l t o u n i t y ,
have a c o n s t a n t c h a r a c t e r , i . e . , t h e y are independent o f t h e a i r ­
craft course at a given magnetic l a t i t u d e .
The terms which h a v e
t h e c o e f f i c i e n t s 2 s i n y and 2 cos y have a q u a r t e r n a r y c h a r a c t e r .
The t e r m s w i t h c o e f f i c i e n t s s i n y a n d c o s y h a v e a s e m i c i r c u l a r
character.

All of t h e f o r c e s d e s i g n a t e d by v a l u e s l o c a t e d i n t h e n u m e r a t o r
of E q u a t i o n ( 2 . 8 ) a r e d i r e c t e d a t a n a n g l e o f 90° t o t h e m a g n e t i c
meridian while those i n t h e denominator coincide with it.
The f o r c e

~

-2

H i s i n d e p e n d e n t of

the a i r c r a f t course;

it i s

p r o p o r t i o n a l t o t h e h o r i z o n t a l c o m p o n e n t of t h e m a g n e t i c f i e l d o f
t h e E a r t h a n d i s d i r e c t e d a t a n a n g l e of 9 0 ° t o t h e m a g n e t i c m e r i d ­
ian.
T h i s f o r c e i s r e l a t e d t o t h e m a g n e t i z a t i o n of t h e s o f t i r o n
o f t h e a i r c r a f t b y tFle m a g n e t i c f i e l d o f t h e E a r t h a n d v a r i e s a s
a f u n c t i o n o f t h e magnetic l a t i t u d e of
t h e l o c u s of t h e a i r c r a f t .
We w i l l d e s i g ­
n a t e t h i s f o r c e by A o X H .
T h e f o r c e cZtP i s d i r e c t e d a l o n g t h e
l o n g i t u d i n a l a x i s of t h e a i r c r a f t ; it i s
t h e r e s u l t o f t h e l o n g i t u d i n a l component
o f t h e f i e l d from t h e h a r d magnetic i r o n
P and t h e i n d u c t i o n from t h e v e r t i c a l
Fig.

2.10.

G e v i a t i o n o f M a g n e t i c Compass
Aboard an A i r c r a f t .

129


.component o f t h e m a g n e t i c f i
n a t e d BQXH and changes w i t h
l o c a t i o n only i n accordgnce
o f t h e f o r c e on t h e n o r m a l t
t o t h e s i n e of t h e m a g n e t i c

e l d of the Earth.
This force is desig­
t h e magnetic l a t i t u d e of t h e aircraft
/130
with t h e first t e r m .
The p r o j e c t i o n
o t h e magnetic meridian is proportional
c o u r s e of t h e a i r c r a f t .

T h e f o r c e fZtQ i s d e s i g n a t e d b y C Q A H , a n d i s ' a n a l o g o u s i n t h e
n a t u r e and c h a r a c t e r of i t s changes t o t h e f o r c e BQAH, b u t i s d i r e c t e d
Consequently, i t s pro­
a l o n g t h e t r a n s v e r s e a x i s of t h e a i r c r a f t .
j e c t i o n on t h e n o r m a l t o t h e m a g n e t i c m e r i d i a n i s p r o p o r t i o n a l t o
t h e cosine of t h e magnetic course of t h e aircraft.
The f o r c e s

'
a - b
a n d -a r e
2
2

r e l a t e d t o t h e s o f t magnetic

i r o n on t h e a i r c r a f t , m a g n e t i z e d by t h e m a g n e t i c f i e l d o f t h e E a r t h .
The f o r m e r i s d e s i g n a t e d D Q A H a n d c o i n c i d e s w i t h t h e d i r e c t i o n o f
t h e double c o u r s e o f t h e a i r c r a f t ; t h e l a t t e r i s E Q X H and i s perpen­
d i c u l a r t o t h e double course of t h e a i r c r a f t .
T h e f o r c e H+- a'b H i s d e s i g n a t e d AH.
In Equation (2.8), it
i s i n t h e denominagor, and t h e r e f o r e c o i n c i d e s w i t h t h e d i r e c t i o n
of t h e magnetic meridian.
If w e s u b s t i t u t e i n t o E q u a t i o n ( 2 . 8 ) t h e s e d e s i g n a t i o n s f o r
t h e f o r c e s a n d d i v i d e t h e n u m e r a t o r a n d d e n o m i n a t o r b y AH, t h e L a t t e r
w i l l g i v e us

Expression ( 2 . 9 ) i s c a l l e d t h e p o i n t - d e v i a t i o n f o r m u l a , and
t h e c o e f f i c i e n t s AQ, BQ, CQ, D Q and E Q a r e t h e p o i n t c o e f f i c i e n t s
of deviation.
The p o i n t - d e v i a t i o n f o r m u l a i s i n c o n v e n i e n t t o u s e ,
been s i m p l i f i e d f o r p r a c t i c a l purposes.

s o it has

S i n c e i t i s a l m o s t a l w a y s n e c e s s a r y t o s e l e c t a p l a c e f o r mount­
i n g t h e c o m p a s s on t h e a i r c r a f t w h e r e t h e d e v i a t i o n d o e s n o t e x c e e d
8-10°, w e can l e t t g 6 = 6 .
The d e n o m i n a t o r o f F o r m u l a ( 2 . 9 )
of a binomial:
[1

can b e e x p r e s s e d i n t h e form

+ (Bo cos 7 - Cosin 7 + Do cos 27 -Eosin 27)]-1=

We know t h a t w i t h
the converging series:

a

< 1, t h e

(1

+ a)-'.

expansion of t h e binomial gives

(1 + a ) - ~ = l - U + a 2 - a 3 . . .

For p r a c t i c a l p u r p o s e s , w e c a n l i m i t o u r s e l v e s t o t h e f i r s t

130


two terms o f t h e series
(1 +a)-* x 1 - 0 ,

so t h a t E q u a t i o n ( 2 . 9 )

assumes t h e form:

+ Do

8 = (Ao f Bo sin 7 f Cocos 1

SIR

27 f Eo COI 27)(1 -Bo cos 7 +

+ C, sln 7 -Do cos 27 + EOsln q).

(2.10)

1131
Having c a r r i e d o u t t h e m u l t i p l i c a t i o n of t h e m u l t i p l i e r s , reduced t h e s i m i l a r t e r m s , and c a r r i e d o u t s i m p l e t r i g o n o m e t r i c conver­
s i o n s , Equation ( 2 . 1 0 ) assumes t h e form:
8=A

+ E sln 7 + C C O S+~D sin27 + ~ c 0 ~ +2 7Fsln37 + Gcos3y +
+ H sin 47 + K c o s 4 7 . ..

Here t h e c o e f f i c i e n t s A , B ,
than i n the point-deviation

(2.11)

C, D, E h a v e a s o m e w h a t d i f f e r e n t v a l u e
formula:

The c o e f f i c i e n t s o f d e v i a t i o n o f h i g h e r o r d e r s , i , e . , p r o p o r ­
t i o n a l t o t h e s i n e s and c o s i n e s 3 y , 4y,
can be d i s r e g a r d e d ,
s i n c e t h e y a r e much s m a l l e r t h a n a n y o f t h e f i r s t f i v e c o e f f i c i e n t s .
Then F o r m u l a ( 2 . 1 1 ) a s s u m e s t h e f o r m :

...,

8 =A

+ B sin 7 + C cos 7 + D sin 27 + E cos 27,

(2.12)

where A i s t h e c o e f f i c i e n t of c o n s t a n t d e v i a t i o n , B , C are t h e coef­
f i c i e n t s o f s e m i c i r c u l a r d e v i a t i o n , a n d D,E a r e t h e c o e f f i c i e n t s
of quarternary deviation.
Formula ( 2 . 1 2 ) i s c a l l e d t h e approximate formula o f d e v i a t i o n ,
and i t s c o e f f i c i e n t s are t h e approximate d e v i a t i o n c o e f f i c i e n t s .
However, i t i s c o m p l e t e l y s a t i s f a c t o r y f o r p r a c t i c a l a p p l i c a t i o n s ,
e s p e c i a l l y i f w e r e c a l l t h a t o t h e r f a c t o r s a r e a c t i n g on t h e compass
which are very d i f f i c u l t t o allow f o r .

C a Z c u Z a t i o n of A p p r o x i m a t e D e v i a t i o n C o e f f i c i e n t s
We w i l l a s s u m e t h a t w e know t h e d e v i a t i o n o f a m a g n e t i c com­
pass at e i g h t symmetrical points:
0 , 4 5 , 9 0 , 1 3 5 , 1 8 9 , 2 2 5 , 270
a n d 315O.

According t o Equation
must have t h e v a l u e s :
%=A+

(2.12),

the deviation at these points

BsinOO+CcosO"+DslnOO+EcosOO.

131


t
Since sin00 = 0 ,

cosOo
=A

-

1, t h e n 6 0 = A f

C f E;

+ B sin 45" + Ccos 45' + D sin 90"-I-E cos 90"

or, i f w e c o n s i d e r t h e v a l u e s s i n 9 0 O = 1, c o s 9 0 ° = 0 ,
ti4, = A

+ Bsin45"+

Ccos45" f

Dd

S i m i l a r l y , w e can o b t a i n a system of e q u a t i o n s f o r t h e devia t i o n of t h e e i g h t points:

/132

+

= A iC E;
A f B sin45" -I-Ccos 45" + D;
tm=A+B-E:
&&=A'+ B sin 45" Ccos 45" D;
o!

b5=

-

-

(2.13)


b , w = A - C + E;

= A - B sin 45"- Ccos 45" D;

%i-o=A-B-E;

8316 = A - B sin 45" f CCOS

45"- D.

+

%26

Summing E q u a t i o n

(2.13), w e o b t a i n :

+ 8185 -k

-i-b-kb

+gci

f %~IJ

= 8A

or

consequently

,

To f i n d t h e a p p r o x i m a t e d e v i a t i o n c o e f f i c i e n t B , w e m u l t i p l y
e a c h o f t h e E q u a t i o n s ( 2 . 1 3 ) by t h e c o e f f i c i e n t a t B , depending
on t h e a i r c r a f t c o u r s e .
T h e n , k e e p i n g i n mind t h e f a c t t h a t sin45O
= c o s 4 5 O , t h e e q u a t i o n s f o r 6 0 a n d 6 1 8 0 become z e r o a n d t h e r e m a i n ­
d e r s assume t h e form:
6,s sin 45" = A sin 45"

8 ~ 2 A f B - 4
sin 45"
- 8 ~ 2 sin
~ 45" = A sin 45"
-8270 = A + B + E;
-$I5
sin 45" =A sin 45"
6135 sin 45" = A

+ B sin2 45" + C sln2 45" + D sin 45";
+ B sin2 45"- C sin2 45"-D sin 45";
+ B sin2 45" + C sin2 45"-D sin 45";

+ B sin2 45"- C sin2 45" + D sin 45".

I n s u m m i n g t h e s i x r e m a i n i n g e q u a t i o n s , t h e sum o f t h e t e r m s
c o n t a i n i n g c o e f f i c i e n t A becomes z e r o , s i n c e t h r e e o f them h a v e
a p l u s s i g n and t h e remaining t h r e e , symmetrical t o t h e f i r s t , have
a minus s i g n .
The sum o f t h e t e r m s c o n t a i n i n g c o e f f i c i e n t B i s e q u a l t o
132

2B

+

4B s i n 2 45O, b u t s i n c e s i n 2 45O =(?)fi

= 1; t h i s sum w i l l b e
2,

e q u a l t o 4B.
T h e sum o f t h e t e r m s c o n t a i n i n g c o e f f i c i e n t C , a s w e l l a s t h e
sum o f t h e t e r m s c o n t a i n i n g c o e f f i c i e n t D , i s e q u a l t o z e r o .
Consequently,

/133

8

2 b i l l = 4B
1=0

or

S i m i l a r l y , we c a n f i n d t h e f o r m u l a s f o r d e t e r m i n i n g t h e r e ­
maining t h r e e c o e f f i c i e n t s :

D=

8

C
4
1

61 COS 21i;

i-0
8


(2.14)

E=
i=O

Change i n D e v i a t i o n of M a g n e t i c Compasses a s a F u n c t i o n o f
t h e M a g n e t i c L a t i t u d e o f t h e L o c u s of t h e A i r c r a f t
The d e v i a t i o n o f a m a g n e t i c c o m p a s s d e t e r m i n e d f o r a g i v e n
p o i n t on t h e E a r t h ' s s u r f a c e , d o e s n o t r e m a i n f i x e d f o r o t h e r p o i n t s ,
b u t c h a n g e s d e p e n d i n g on t h e m a g n e t i c l a t i t u d e o f t h e l o c u s o f t h e
aircraft.
~ ~ v i o u s l ya , c h a n g e i n d e v i a t i o n c a n n o t t a k e p l a c e a s a r e s u l t
o f c h a n g e s i n t h e m a g n e t i c i n d u c t i o n of s o f t m a g n e t i c i r o n f r o m
t h e h o r i z o n t a l component o f t h e m a g n e t i c f i e l d o f t h e E a r t h .

the
the
the
the

By t h e same t o k e n , t h e i n d u c t i o n f r o m t h e c o m p o n e n t p r o d u c i n g
d e v i a t i o n w i l l change i n t h e s a m e p r o p o r t i o n with a change i n
h o r i z o n t a l component o f t h e m a g n e t i c f i e l d o f t h e E a r t h , as
p r i n c i p a l d i r e c t i o n a l p o s i t i o n o f t h e compass c a r d .
Consequently,
compass d e v i a t i o n r e m a i n s c o n s t a n t .

T h e d e v i a t i o n f r o m m a g n e t i c i n d u c t i o n o f t h e h o r i z o n t a l com­
p o n e n t of t h e f i e l d o f t h e E a r t h h a s a c o n s t a n t a n d q u a r t e r n a r y
character:

133


ary

Hence, w e r e a c h t h e c o n c l u s i o n t h a t t h e c o n s t a n t and q u a r t e r n ­
deriation a t various magnetic l a t i t u d e s remains constant.

E s s e n t i a l l y , t h e change i n t h e d e v i a t i o n with a change i n t h e
/134
magnetic l a t i t u d e i s t h e r e s u l t of t h e influence of hard magnetic
i r o n a n d p a r t i a l l y as a r e s u l t o f i n d u c t i o n w i t h s o f t m a g n e t i c i r o n
f r o m t h e v e r t i c a l component of t h e E a r t h ' s m a g n e t i c f i e l d .
This
remains constant
t a k e s p l a c e b e c a u s e t h e m a g n i t u d e o f t h e v e c t o r s P,
w i t h a change i n t h e d i r e c t i o n a l v e c t o r H .
Consequently, with an
i n c r e a s e i n t h e magnetic l a t i t u d e , t h e s e m i c i r c u l a r d e v i a t i o n must
increase.

8

I n a d d i t i o n , with an i n c r e a s e i n t h e magnetic l a t i t u d e , t h e
i n d u c t i o n o f t h e s o f t m a g n e t i c i r o n f r o m t h e v e r t i c a l component
of t h e E a r t h ' s f i e l d i n c r e a s e s with a simultaneous decrease i n t h e
directional force H .
However, i f w e c o n s i d e r t h e p r e d o m i n a n t i n f l u ­
e n c e on t h e a i r c r a f t p r o d u c e d b y t h e h a r d m a g n e t i c i r o n , we c a n
consider i n approximation t h a t t h e semicircular deviation is inversely
p r o p o r t i o n a l t o t h e h o r i z o n t a l component o f t h e m a g n e t i c f i e l d of
the Earth.

which g i v e s t h e f o l l o w i n g f o r t h e a p p r o x i m a t e c o e f f i c i e n t s o f de­
v i a t i o n B and C :

(2.15)

w h e r e B 1 , C1, H I a r e t h e a p p r o x i m a t e c o e f f i c i e n t s a n d t h e h o r i z o n t a l
component of t h e E a r t h ' s f i e l d a t t h e p o i n t where t h e d e v i a t i o n
i s m e a s u r e d ; B,,
C2, H2 a r e t h e s a m e v a l u e s a t a p o i n t w i t h a d i f ­
f e r e n t magnetic l a t i t u d e .
W i t h known c o e f f i c i e n t s B + C , t h e s e m i c i r c u l a r d e v i a t i o n a t
a g i v e n p o i n t on t h e E a r t h ' s s u r f a c e c a n b e d e t e r m i n e d b y t h e f o r m ­
ula

E l i m i n a t i o n o f D e v i a t i o n i n t h e M a g n e t i c Compasses
Modern m a g n e t i c c o m p a s s e s a r e f i t t e d w i t h a d e v i c e f o r com­
p e n s a t i n g o n l y s e m i c i r c u l a r d e v i a t i o n , r e s u l t i n g from h a r d magnetic
iron.

134


I n a d d i t i o n , by a s u i t a b l e r o t a t i o n o f t h e compass h o u s i n g
i n i t s m o u n t i n g s , w e can c o m p e n s a t e f o r t h e c o n s t a n t component of
d e v i a t i o n a l o n g w i t h t h e a d j u s t m e n t e r r o r o f t h e compass.
E l i m i n a t i o n o f q u a r t e r n a r y d e v i a t i o n b y m a g n e t i c means e n c o u n ­
ters considerable technical difficulty.
Therefore , i f w e keep i n
mind t h e r e l a t i v e l y low v a l u e o f t h e q u a r t e r n a r y d e v i a t i o n r e l a t i v e
t o t h e s e m i c i r c u l a r d e v i a t i o n , as w e l l as i t s c o n s t a n t v a l u e a t
various latitudes, w e w i l l not be able t o get r i d of the latter
b u t w i l l e n t e r i t on s p e c i a l g r a p h s f o r co mp as s c o r r e c t i o n .
Modern r e m o t e c o n t r o l m a g n e t i c c o m p a s s e s h a v e d e v i c e s f o r m e c h a n i c a l compensation of d e v i a t i o n of a l l o r d e r s .

1135

The d e v i c e f o r c o m p e n s a t i n g s e m i c i r c u l a r d e v i a t i o n c o n s i s t s
of a s y s t e m o f f o u r c y l i n d e r s m o u n t e d i n p a i r s , w i t h p e r m a n e n t m a g n e t s
i n s t a l l e d i n them ( F i g . 2 . 1 1 ) .
The c y l i n d e r s , i n t e n d e d f o r c o m p e n s a t i n g f o r d e v i a t i o n i n 0
a n d 180° c o u r s e s , i n w h i c h t h e f i e l d o f h a r d m a g n e t i c i r o n p r o d u c e s
t h e deviation, are arranged along t h e t r a n s v e r s e a x i s of t h e air­
c r a f t , and p l a c e d p a r a l l e l t o t h e a x i s o f t h e a i r c r a f t i n such a
way t h a t w h e n t h e y a r e r o t a t e d , t h e s m a l l m a g n e t s c a n t u r n f r o m
a v e r t i c a l p o s i t i o n t o one which i s c o i n c i d e n t w i t h t h e t r a n s v e r s e
a x i s of t h e a i r c r a f t . W i t h a v e r t i c a l p o s i t i o n o f t h e m a g n e t s , t h e i r
f i e l d d o e s n o t h a v e a n y i n f l u e n c e on t h e p o s i t i o n o f t h e compass
card (neutral position, Fig. 2.11, a ) .
4

N

S

Fig. 2.11.
D e v i c e f o r C o r r e c t i n g S e m i c i r c u l a r Devi­
(1) F r a m e ;
( 2 ) Transverse
a t i o n o f t h e Compass.
Cylinders;
(3) Longitudinal Cylinders;
(4) A c t u a ­
t i n g Cylinders;
( 5 ) Embedded M a g n e t s .
When t h e m a g n e t s a r e t i l t e d ( F i g . 2 . 1 1 , c ) , t h e h o r i z o n t a l
component o f t h e i r f i e l d a p p e a r s , and c a n b e s e t s o t h a t i t i s e q u a l
b u t d i r e c t e d opposite t o t h e magnetic f i e l d of t h e aircraft (hori­
z o n t a l component), l o c a t e d along i t s t r a n s v e r s e a x i s .
The m a x i ­
mum e f f e c t o f t h e s m a l l m a g n e t s w i l l b e o b s e r v e d w h e n t h e y a r e i n
the horizontal position (Fig. 2.11, b ) .
135

The c y l i n d e r s f o r c o m p e n s a t i n g d e v i a t i o n a t c o u r s e s o f 9 0 a n d
are mounted i n t h e t r a n s v e r s e a x i s o f t h e a i r c r a f t i n s u c h
a way t h a t t h e s m a l l m a g n e t s c a n b e u s e d t o c o m p e n s a t e f o r t h e compo­
n e n t o f t h e m a g n e t i c f i e l d of t h e a i r c r a f t which i s d i r e c t e d a l o n g
its longitudinal axis.
270°

The r o t a t i o n o f t h e l o n g i t u d i n a l a n d t r a n s v e r s e c y l i n d e r s i s
a c c o m p l i s h e d b y m e a n s o f s p e c i a l h a n d l e s made o f d i a m a g n e t i c m a t e r ­
ial.
To d e t e r m i n e a n d g e t r i d o f d e v i a t i o n s , t h e a i r c r a f t i s p l a c e d
o n a s p e c i a l l y p r e p a r e d s t a n d , made o f c o n c r e t e ( f o r h e a v y a i r c r a f t ) / l 3 6
b u t without a metal c o r e .
The s t a n d m u s t b e o f s u f f i c i e n t s i z e s o t h a t a i r c r a f t o f a n y
k i n d can be r o t a t e d i n a c i r c l e and t h e d i s t a n c e from t h e s t a n d
t o o t h e r a i r c r a f t and metal s t r u c t u r e s i s a t l e a s t 200 m.
The a c c u r a c y o f t h e s e t t i n g o f t h e a i r c r a f t on a g i v e n c o u r s e
for d e t e r m i n i n g a n d g e t t i n g r i d o f d e v i a t i o n c a n b e c h e c k e d i n o n e
of t h e f o l l o w i n g two ways:

1. D i r e c t i o n f i n d i n g o f l a n d m a r k s f r o m o n b o a r d t h e a i r c r a f t .
I n t h e c e n t e r of t h e area where t h e a i r c r a f t i s t o t u r n , a magnetic
d i r e c t i o n f i n d e r or t h e o d o l i t e i s m o u n t e d o n a s t a n d s o t h a t t h e
i n d i c a t i n g d i a l i s l o c a t e d e x a c t l y i n a h o r i z o n t a l p o s i t i o n , and
t h e z e r o r e a d i n g on t h e d i a l c o i n c i d e s w i t h t h e d i r e c t i o n o f t h e
magnetic meridian.
For t h i s p u r p o s e , t h e s e i n s t r u m e n t s a r e f i t t e d
wi.th a b u b b l e l e v e l a n d o r i e n t i n g m a g n e t i c n e e d l e .
T h e n t w o or t h r e e d i s t i n c t a n d p r o m i n e n t l a n d m a r k s on t h e h o r ­
i z o n are s e l e c t e d ( t o w e r s and chimneys a r e b e s t f o r t h i s p u r p o s e ) ,
a n d t h e i r m a g n e t i c b e a r i n g s (MB) a r e d e t e r m i n e d w i t h t h e a i d o f
a s i g h t , r o t a t i n g o n t h e d i a l u s e d for d e t e r m i n i n g t h e b e a r i n g s .
T h e l a n d m a r k s s h o u l d b e l o c a t e d as f a r as p o s s i b l e f r o m t h e
area s o t h a t t h e s h i f t i n g o f t h e a i r c r a f t from i t s c e n t e r d u r i n g
r o t a t i o n w i l l n o t produce any n o t i c e a b l e changes i n t h e b e a r i n g s
of t h e landmarks.
For l i g h t a i r c r a f t , t h i s d i s t a n c e s h o u l d b e a t
l e a s t 2 - 3 km; f o r l a r g e r a i r c r a f t w i t h a g r e a t e r r a d i u s o f t u r n
o n t h e g r o u n d , i t s h o u l d b e a t l e a s t 5 - 6 km.
A f t e r d e t e r m i n i n g and r e c o r d i n g t h e magnetic b e a r i n g s o f t h e
The d i r e c t i o n
l a n d m a r k s , t h e a i r c r a f t i s m o u n t e d on t h e s t a n d .
f i n d e r i s p l a c e d i n f r o n t o f or b e h i n d t h e a i r c r a f t a t a d i s t a n c e
o f 2 0 - 1 0 0 m , d e p e n d i n g on t h e l e n g t h o f t h e a i r c r a f t , e x a c t l y a l o n g
i t s l o n g i t u d i n a l a x i s s o t h a t t h e f o r w a r d a n d r e a r p o i n t s on t h e
a x i s o f t h e a i r c r a f t w i l l b e p r o j e c t e d on t h e s i g h t , e . g . , t h e c e n t e r s
of t h e nose and k e e l .
Then t h e d i a l on t h e d i r e c t i o n f i n d e r i s
s e t t o t h e magnetic m e r i d i a n , and t h e d i r e c t i o n of t h e l o n g i t u d i n a l
a x i s of t h e a i r c r a f t i s m e a s u r e d , a n d i t s i n i t i a l c o u r s e i s s e t .

136

I t i s n e c e s s a r y t o r e c a l l t h a t t h e minimum d i s t a n c e f o r t h e
d i r e c t i o n f i n d e r from t h e a i r c r a f t i s l i m i t e d by t h e e f f e c t o f t h e
a i r c r a f t on t h e m a g n e t i c n e e d l e o f t h e d e v i a t i o n d i r e c t i o n f i n d e r ,
a n d t h e maximum d i s t a n c e i s s e t b y t h e l e n g t h o f t h e a i r c r a f t , s i n c e
a t a d i s t a n c e o f more t h a n 1 0 0 m y w i t h a n a i r c r a f t which i s n o t
v e r y l o n g , t h i s method w i l l b e i n s u f f i c i e n t l y p r e c i s e .
A f t e r t h e d i r e c t i o n f i n d e r h a s b e e n moved t o t h e a i r c r a f t ,
t h e magnetic n e e d l e i s f i x e d and s e t s o t h a t one of t h e s e l e c t e d
l a n d m a r k s ( F i g . 2 . 1 2 ) a p p e a r s a t a c o u r s e a n g l e (CA) e q u a l t o

CA = MBL

-

(2.17)

MC

w h e r e MBL e q u a l s t h e m a g n e t i c b e a r i n g o f t h e l a n d m a r k a n d MC i s
t h e i n i t i a l magnetic course of t h e aircraft.
I f t h e above c o n d i t i o n i s s a t i s f i e d , t h e z e r o p o i n t on t h e
direction finder d i a l w i l l coincide exactly with t h e longitudinal
a x i s of t h e aircraft.
To s e t t h e a i r c r a f t on d e f i n i t e c o u r s e s , a t a b l e o f c o u r s e
a n g l e s for landmarks f o r e a c h a i r c r a f t c o u r s e i s c o m p i l e d .

F o r e x a m p l e , if t h e d e v i a t i o n h a s b e e n d e t e r m i n e d a t e i g h t
p o i n t s , b u t t h e s e l e c t e d landmarks have magnetic b e a r i n g s of 115
a n d 3 2 8 O , t h e n t h e c o u r s e a n g l e s �or t h e c o u r s e s w h i c h w e r e q u i r e
w i l l have t h e v a l u e s shown i n T a b l e 2 . 2 .
. -

MC
0
45
90
135
I80
225
270
315

115
70
25
340

295
250
205
160

326

283
238
19 3
148

358
42

1-2

+2

91

-1

133
178

+2
t-2
+I

103

224

58

271

13

313

-1
+2

When u s i n g t h i s t a b l e , t h e s i g h t o f t h e d i r e c t i o n f i n d e r i s
s e t t o a g i v e n c o u r s e a n g l e f o r a landmark and t h e a i r c r a f t is t h e n
t u r n e d u n t i l t h e a x i s o f t h e s i g h t l i n e s up w i t h t h e d i r e c t i o n o f
It is clear t h a t t h e aircraft is then set
t h e s e l e c t e d landmark.
p r e c i s e l y on t h e d e s i r e d c o u r s e .
The s e c o n d l a n d m a r k i s a n e x t r a o n e i n c a s e t h e f i r s t i s ob­
s t r u c t e d b y some p a r t o f t h e a i r c r a f t s u c h a s t h e e m p e n n a g e o r w i n g .

137

/137

The m e t h o d o f s e t t i n g a n a i r c r a f t on c o u r s e b y t h e m e t h o d d e ­
s c r i b e d above f o r o b t a i n i n g t h e c o u r s e a n g l e s of landmarks i s t h e
most p r e c i s e and r e l i a b l e one,
e s p e c i a l l y s i n c e a f i x e d area
c a n b e s e t u p a t a n a i r p o r t for
c o r r e c t i n g deviaFions and doing
o t h e r work t o s e t t h e b e a r i n g s
of landmarks and compiling t a b l e s
o f c o u r s e a n g l e s for g i v e n a i r ­
craft courses.
However, t h i s method i s n o t
always p r a c t i c a b l e .
I n some c a s e s ,
i t may b e i m p o s s i b l e t o s e l e c t
s u i t a b l e landmarks, and i n o t h e r
c a s e s t h e visibility may b e i n a d Fig. 2.12.
D e t e r m i n a t i o n of
e q u a t e f o r them t o b e s e e n .
In
Aircra'ft Course by t h e Course
some a i r c r a f t , t h e r e may b e d i f f i culty i n fastening the direction
/138
Angle o f a Landmark.
f i n d e r on b o a r d t h e a i r c r a f t i n
a p l a c e where t h e r e would be a c l e a r f i e l d o f v i s i o n f o r o b s e r v ­
ing the landmarks.

2.
D i r e c t i o n f i n d i n g o f a n a i r c r a f t from t h e nose o r t a i l .
T h i s m e t h o d i s u s e d i n c a s e s when i t i s i m p o s s i b l e t o s e t t h e a i r ­
c r a f t on c o u r s e s o f 0 , 4 5 , 9 0 ° , e t c . , b y t h e m e t h o d d e s c r i b e d a b o v e .
In t h i s case, t h e aircraft is s e t each time(e.g., according
t o t h e r e a d i n g s of t h e magnetic compass) t o a given c o u r s e .
Then
t h e d i r e c t i o n f i n d e r is l o c a t e d along t h e extension of t h e longi­
t u d i n a l a x i s a t a d i s t a n c e of 20-100 m from t h e a i r c r a f t , depend­
i n g on t h e t y p e o f t h e l a t t e r ; t h e c o r r e c n e s s o f t h e s e t t i n g o f
t h e a i r c r a f t on c o u r s e i s t h e n d e t e r m i n e d a s i n t h e f i r s t c a s e b e f o r e
m o u n t i n g t h e d i r e c t i o n f i n d e r on b o a r d t h e a i r c r a f t .
I t may b e
necessary t o t u r n t h e a i r c r a f t f o r a secondary check.
This method i s l e s s c o n v e n i e n t t h a n t h e f i r s t , s i n c e it i s
necessary t o s h i f t t h e d i r e c t i o n f i n d e r f o r each course, s e t it
exactly along t h e extension of t h e a i r c r a f t a x i s , a d j u s t the zero
on t h e d i a l a l o n g t h e m a g n e t i c m e r i d i a n , a n d m a k e t h e d i a l l e v e l ,
Under u n f a v ­
i n addition t o measuring t h e distance t o t h e aircraft.
o r a b l e c o n d i t i o n s a b o a r d t h e a i r c r a f t , t h i s o p e r a t i o n may h a v e t o
The a d v a n t a g e o f t h i s m e t h o d
b e r e p e a t e d a f t e r moving t h e a i r c r a f t .
i s i t s independence of t h e e x i s t e n c e of landmarks, meteorological
v i s i b i l i t y , and p e c u l i a r i t i e s of a i r c r a f t d e s i g n .
S e m i c i r c u l a r d e v i a t i o n of m a g n e t i c c o m p a s s e s i s c o r r e c t e d a n d
eliminated at four basic points:
0 , 180 , 90 a n d 2 7 0 ° .

I t i s c l e a r from ( 2 . 1 3 ) t h a t s e m i c i r c u l a r d e v i a t i o n a t t h e
0 a n d 1 8 0 ° p o i n t s i s e q u a l i n v a l u e , b u t o p p o s i t e i n s i g n , a n d ex.Deviation from
p r e s s e d b y t h e maximum v a l u e o f c o e f f i c i e n t C .

138

c o e f f i c i e n t B i s e q u a l t o z e r o on t h e s e c o u r s e s .
However, a l l of t h e s e c o u r s e s a r e s u b j e c t t o t h e a c t i o n o f
a c o n s t a n t d e v i a t i o n i n t h e c o e f f i c i e n t A and q u a r t e r n a r y d e v i a t i o n
E i n addition t o the semicircular deviation.
T h i s means t h a t t h e
v a l u e s of t h e c o n s t a n t a n d q u a r t e r n a r y d e v i a t i o n a r e e q u a l i n v a l u e
and s i g n .
C o n s e q u e n t l y , i f t h e d e v i a t i o n on c o u r s e O o i s s e t t o z e r o
by t u r n i n g t h e c y l i n d e r o f t h e d e v i a t i o n - c o r r e c t i n g a p p a r a t u s w i t h
t h e m a r k i n g IrN-S" , t h e s e m i c i r c u l a r d e v i a t i o n w i l l b e c o m p e n s a t e d
f o r and t h e c o n s t a n t and q u a r t e r n a r y d e v i a t i o n w i l l s i m u l t a n e o u s l y
b e compensated f o r .
I t w i l l c h a n g e w i t h t h e same s i g n t o a c o u r s e
Therefore, after s e t t i n g the
o f 180°, w h e r e i t s v a l u e d o u b l e s .
a i r c r a f t t o a c o u r s e o f 180°, i t i s n e c e s s a r y t o s e t t h e d e v i a t i o n
n o t t o zero, b u t t o h a l f t h e r o t a t i o n of t h a t c y l i n d e r , and i n t h e
reverse direction.
Hence , t h e s e m i c i r c u l a r d e v i a t i o n f r o m c o e f f i c i e n t C c a n b e
e l i m i n a t e d completely and p r e c i s e l y w i t h o u t d i s t u r b i n g t h e c o n s t a n t
and q u a r t e r n a r y d e v i a t i o n s .
A n a l o g o u s l y , by t u r n i n g t h e c y l i n d e r o f t h e d e v i a t i o n - c o r r e c t ­
i n g a p p a r a t u s w i t h t h e m a r k i n g "E-W", i t i s p o s s i b l e t o r e d u c e t h e
d e v i a t i o n t o zerio f o r a 90° c o u r s e a n d by h a l f f o r a 270° c o u r s e ,
/139
which c o m p l e t e l y g e t s r i d o f t h e s e m i c i r c u l a r d e v i a t i o n from c o e f f i ­
c i e n t B without d i s t u r b i n g t h e c o n s t a n t and q u a r t e r n a r y d e v i a t i o n s .

TABLE 2 3
MC

12

0
180

0
+2
0
-1

+4
+7
-2

90
270

+5

+S

a

-50

Fig'.

0

45

YO

135

180

225

270

315

0 -5

2 . 1 3 . Graph o f D e v i a t i o n o f a M a g n e t i c
Compass.
139

L.

.

The o p e r a t i o n w i t h s e m i c i r c u l a r d e v i a t i o n i s d e s c r i b e d i n a
s p e c i a l t a b l e (Table 2.3).
Obviously, t h e remaining deviation a t t h e s e p o i n t s w i l l be
e q u a l t o t 2 O for c o u r s e s o f 0 a n d 180° a n d -lo f o r c o u r s e s o f 9 0
a n d 270°.

A f t e r g e t t i n g r i d of t h e s e m i c i r c u l a r d e v i a t i o n , t h e a i r c r a f t
i s s e t t o c o u r s e s a t 45O i n t e r v a l s a n d t h e r e m a i n i n g d e v i a t i o n i s
An e x a m p l e o f t h e r e c o r d i n g i s s h o w n i n T a b l e 2 . 2 .
measured.
A f t e r summing t h e r e m a i n i n g d e v i a t i o n f o r e i g h t c o u r s e s ( G r a p h
5 , T a b l e 2 . 2 ) a n d d i v i d i n g t h e sum b y e i g h t , w e o b t a i n t h e v a l u e
of t h e constant deviation
A=

2

+ 3- 1 + 2 + 2 + 1 - 1 + 2

=+

8

The b o w l o f t h e c o m p a s s m u s t b e s e t i n i t s m o u n t i n g t o t h i s
If w e d i s r e g a r d t h e v a l u e of 0.25O produced by t u r n i n g t h e
value.
bowl o f t h e compass t h r o u g h lo, t h e r e m a i n i n g d e v i a t i o n f o r t h e
e i g h t c o u r s e s w i l l h a v e a v a l u e o f t1, t 2 , - 2 , t1, t 1 , 0 , - 2 , t 1
so t h a t t h e graph o f t h e c o r r e c t i o n s c a n b e compared w i t h t h e r e a d ­
i n g s o f t h e compass ( F i g . 2.13).
If t h e a i r c r a f t i s i n t e n d e d f o r u s e on f l i g h t s a t m a g n e t i c
l a t i t u d e s w h e r e t h e r e w i l l o n l y b e s m a l l c h a n g e s , t h i s w i l l mark
t h e e n d o f t h e work w i t h d e v i a t i o n .
I n p r e p a r i n g f o r long d i s t a n c e f l i g h t s , w i t h c o n s i d e r a b l e changes
/140
i n t h e magnetic l a t i t u d e s , t h e c o e f f i c i e n t s of t h e s e m i c i r c u l a r
d e v i a t i o n B and C must a l s o b e found w i t h d e t e r m i n a t i o n of t h e i r
changes w i t h magnetic l a t i t u d e .
I n t h i s case, t h e c o e f f i c i e n t B w i l l b e e q u a l t o :

+ I sin'0 + 2 sin 450- 2 sin + I sin + I sin 1800 +
+ 0-2 sin 2x0' + 1 sin 315"
--­
900

B-

1.3~0



4

--0
-

+ 1,4-2+0,7 + 0 + 0 + 2-0.7
--__­- 0.35.
4

and c o e f f i c i e n t C w i l l be

C=

140


1+1,4-0-0,7+

-1 + 0 --0 + 0 , 7

4


= 0.85.

G y r o s c o p i c C o u r s e Devices
Regardless o f t h e f a c t t h a t measures have been employed f o r
a long p e r i o d of t i m e which are d i r e c t e d toward i n c r e a s i n g t h e accur­
a c y o f r e a d i n g s a n d t h e s t a b i l i t y o f o p e r a t i o n o f i n t e g r a t e d mag­
n e t i c compasses, t h e i r shortcomings have n o t been completely over­
come.
I n a d d i t i o n , magnetic course devices are d i f f i c u l t t o use i n
a f l i g h t a l o n g a n orthodrome f o r l o n g d i s t a n c e s , due t o t h e complex­
i t y o f t h e c a l c u l a t i o n o f t h e m a g n e t i c d e c l i n a t i o n as i t changes
along the route.
A l l o f t h i s h a s made i t n e c e s s a r y t o s e e k new w a y s o f d e v i s ­
i n g c o u r s e i n s t r u m e n t s and systems which w i l l s a t i s f y t h e r e q u i r e ­
ments of a i r c r a f t n a v i g a t i o n a t a l l s t a g e s and a l l c o n d i t i o n s o f
flight.

The f i r s t s t e p s i n t h i s d i r e c t i o n w e r e made b y t h e r e m o t e c o n ­
t r o l magnetic compasses, c o n t a i n i n g a magnetic t r a n s m f t t e r ( a s e n s i ­
t i v e element) l o c a t e d a t any convenient p o i n t i n t h e a i r c r a f t ,
whose r e a d i n g s w e r e t r a n s m i t t e d b y means o f s p e c i a l p o t e n t i o m e t r i c
t r a n s m i t t e r s t o d i a l s mounted i n t h e c o c k p i t .
T h i s made i t p o s s i b l e t o m o u n t t h e c o m p a s s i n t h e p i l o t ' s
f i e l d o f v i s i o n a n d e n s u r e optimum c o n d i t i o n s f o r o p e r a t i o n o f t h e
compass f r o m t h e s t a n d p o i n t o f d e v i a t i o n .
However, t h e r e were s t i l l
c o n s i d e r a b l e shortcomings i n t h e o p e r a t i o n of t h e compass, such
as i n s t a b i l i t y o f t h e r e a d i n g s w i t h m o v e m e n t of t h e a i r c r a f t a n d
t h e i m p o s s i b i l i t y o f u s i n g i t when t h e a i r c r a f t w a s t u r n i n g .
I n a d d i t i o n , t h e r e l i a b i l i t y o f o p e r a t i o n o f t h e compass de­
creased, since t h e potentiometric connection with r e l i a b l e contacts
produced an a d d i t i o n a l d e l a y i n t h e t u r n i n g of t h e s e n s o r c a r d t o
a s i g n i f i c a n t l y g r e a t e r d e g r e e t h a n w a s t h e case f o r t h e r o t a t i o n
o f a f r e e l y m o v i n g c a r d on i t s b e a r i n g i n a n i n t e g r a t e d c o m p a s s .
The n e x t s t e p s i n i n c r e a s i n g t h e a c c u r a c y a n d r e l i a b i l i t y o f
/141
o p e r a t i o n o f c o u r s e d e v i c e s w a s made b y t h e g y r o s c o p i c s e m i c o m p a s s e s
and magnetic c o u r s e s e n s o r s l i n k e d w i t h g y r o s c o p i c dampers.
This
made i t p o s s i b l e t o u s e t h e c o u r s e i n s t r u m e n t s w h i l e t h e a i r c r a f t
w a s t u r n i n g and t o a c h i e v e s t a b i l i t y o f c o u r s e r e a d i n g s under any
f l i g h t conditions.
Analysis of t h e induction course sensors, free
of f r i c t i o n d u r i n g t u r n i n g o f a n a i r c r a f t , s i g n i f i c a n t l y i n c r e a s e d
t h e r e l i a b i l i t y of magnetic compasses.
H o w e v e r , t h e g r e a t e s t r e l i a b i l i t y a n d a c c u r a c y i n c o u r s e meas­
urements f o r a i r c r a f t h a s b e e n a c h i e v e d by t h e b u i l d i n g of complexes
of c o u r s e i n s t r u m e n t s ( c o u r s e s y s t e m s ) , c o m b i n i n g t h e o p e r a t i o n
of gyroscopic, magnetic, and astronomic s e n s o r s .
The p r i n c i p l e o f
t h e s e systems is a s t a b l e and prolongad maintenance o f t h e system

141

4


f o r estimating the course. w i t h a gyroscopic assembly having p e r i ­
o d i c c o r r e c t i o n ?f t h e r e a d i n g s b y m e a n s o f a m a g n e t i c or a s t r o ­
n o m i c a l s e n s o r , or i n p u t o f c o r r e c t i o n s m a n u a l l y a s d e s i r e d b y t h e
crew.

P r i n c i p Z e o f O p e r a t i o n o f Gyroscopic I n s t r u m e n t s
The g y r o s c o p e i s a m a s s i v e b a l a n c e d b o d y ,
a x i s of symmetry a t a h i g h a n g u l a r v e l o c i t y .

r o t a t i n g around i t s

G y r o s c o p e s a r e u s u a l l y made i n a f o r m s u c h t h a t t h e y h a v e r e l ­
a t i v e l y low w e i g h t a n d s m a l l s i z e , y e t h a v e a maximum i n e r t i a l moment
which i s r e a c h e d r e l a t i v e t o t h e b a s i c m a s s of t h e g y r o s c o p e as
f a r as p o s s i b l e f r o m t h e c e n t e r o f r o t a t i o n w i t h i n t h e g i v e n dimen­
s i o n s of t h e g y r o s c o p e .
L e t u s r e c a l l t h a t t h e i n e r t i a l moment J i n m e c h a n i c s i s t h e
p r o d u c t of t h e m a s s t i m e s t h e s q u a r e o f t h e d i s t a n c e t o t h e a x i s
o f r o t a t i on :

J = mr;",


(2.18)

w h e r e r l i s t h e d i s t a n c e f r o m t h e mass t o t h e a x i s o f r o t a t i o n .

For a . c o m p l e t e c y l i n d e r , w h i c h c o n s t i t u t e s t h e b a s i c mass o f
a g y r o s c o p e ( F i g . 2 . 1 4 ) , t h e i n e r t i a l moment i s
m (ri - r;")

J=

2

(2.19)

The g y r o s c o p e h a s t w o i n t e r e s t i n g p r o p e r t i e s w h i c h a r e u s e d
i n a n u m b e r o f d e v i c e s for p i l o t a g e a n d n a v i g a t i o n :

(1) A x i a Z s t a b i l i t y , i . e . , t h e a b i l i t y t o m a i n t a i n t h e d i ­
r e c t i o n o f i t s a x i s o f r o t a t i o n i n s p a c e i n t h e a b s e n c e o f moments
of e x t e r n a l f o r c e s t e n d i n g t o c h a n g e t h i s d i r e c t i o n ;
( 2 ) A x i a Z p r e c e s s i o n of r o t a t i o n u n d e r t h e i n f l u e n c e o f mo­
ments of e x t e r n a l f o r c e , i . e . , a slow r o t a t i o n of t h e a x i s i n a
p l a n e which i s p e r p e n d i c u l a r t o t h e a p p l i e d f o r c e , w i t h maintenance
of t h e d i r e c t i o n i n t h e plane of t h e a p p l i c a t i o n of t h e f o r c e .
The f i r s t p r o p e r t y of t h e g y r o s c o p e i s u s u a l l y u s e d f o r s t a b i l i z i n g t h e d i r e c t i o n s of t h e axes of t h e c o o r d i n a t e s f o r d e t e r ­
mining t h e r e q u i r e d v a l u e s , t h e banking of t h e a i r c r a f t , t h e a n g l e
of p i t c h , and t h e c o u r s e .
The s e c o n d p r o p e r t y i s u s e d t o s e t t h e
a x i s of t h e gyroscope i n t h e d e s i r e d p o s i t i o n , e . g . , t o t h e v e r t ­
i c a l of t h e locus of t h e a i r c r a f t , t o t h e plane of t h e t r u e h o r i ­
z o n , for c o m p e n s a t i o n o f t h e a p p a r e n t r o t a t i o n o f t h e a x i s d u e t o
the diurnal r o t a t i o n of the Earth, e t c .
In addition, the property
of p r e c e s s i o n i s s o m e t i m e s e m p l o y e d i n d e v i c e s w h i c h i n t e g r a t e t h e

142

/142

a c t i o n of t h e f o r c e s with t i m e ,
navigation devices.

e.g.,

i n t h e construction of i n e r t i a l

T o e x p l a i n t h e p r i n c i p l e s of o p e r a t i o n o f g y r o s c o p i c d e v i c e s
c o n s i d e r t h e p h y s i c a l s i g n i f i c a n c e o f t h e two p r o p e r t i e s
of a gyroscope mentioned above.

,

l e t us

For t h e s a k e o f s i m p l i c i t y , w e s h a l l
a s s u m e t h a t t h e mass o f t h e g y r o s c o p e
i s located along t h e circumference around
and
t h e a x i s of r o t a t i o n ( F i g . 2.15),
w e s h a l l select an element of t h i s m a s s
a t some p o i n t on t h e c i r c u m f e r e n c e .
L e t u s assume t h a t under t h e i n f l u e n c e
of a f o r c e F, t h e a x i s of t h e gyroscope
has been t i l t e d t o an a n g l e A + .
Fig.

2.14.

Gyroscope
Rotor.

Obviously, t h e d i r e c t i o n of r o t a t i o n
o f t h e e l e m e n t o f mass o f t h e g y r o s c o p e
d o e s n o t c h a n g e when i t p a s s e s t h r o u g h p o i n t s A a n d B , s i n c e t h e
motion of t h e element a t p o i n t s A , A 1 and B y B1 t a n g e n t t o t h e circum­
ference remain p a r a l l e l .
The t a n g e n t s t o t h e d i r e c t i o n o f m o t i o n
a t t h e p o i n t C and d i a m e t r i c a l l y o p p o s i t e t o it are a t an a n g l e
equal t o A+.
Consequently, a t these p o i n t s t h e r e arises a difference i n
the velocities

-

AV = V s i n A + .

(2.20)

The g r e a t e r t h e a n g u l a r v e l o c i t y of r o t a t i o n of t h e g y r o s c o p e
and t h e r a d i u s of t h e r i n g , t h e g r e a t e r w i l l b e t h e c i r c u m f e r e p t i a l
s p e e d o f t h e e l e m e n t o f mass a n d t h e m a g n i t u d e o f t h e v e c t o r A V .
O b v i o u s l y , t h e r e a c t i o n o f t h e mass o f t h e g y r o s c o p e must p r o ­
duce r e s i s t a n c e t o t h e v e c t o r of v e l o c i t y change a t t h e p o i n t s C
and C1, i . e . , t h e f o r c e s F
and Fp
arise at these points, directed
P
1
opposite t o v e c t o r AV and producing t h e p r e c e s s i o n of t h e gyroscope
axis.
Hence, t h e i n e r t i a of t h e m a s s of t h e g y r o s c o p e w i l l c a u s e p r e ­
c e s s i o n of t h e gyroscope a x i s .
The f o r c e s p r o d u c i n g t h e p r e c e s ­
s i o n w i l l i n t u r n c a u s e a t i l t i n g of t h e a x i s i n a p l a n e perpen­
d i c u l a r t o t h e a c t i o n of t h e e x t e r n a l f o r c e , thus generating iner­
t i a l forces analogous t o t h e precession f o r c e s b u t directed a g a i n s t
the external force.
It i s easy t o see t h a
the external force w i l l be
no r o t a t i o n o f t h e a x i s o f
D f the external force w i l l

/143
t the inertial forces directed against
exactly equal t o the latter, so that
t h e gyroscope i n t h e p l a n e of t h e a c t i o n
be observed.
143

The p r e c e s s i o n r a t e of t h e g y r o s c o p e c a n b e d e t e r m i n e d e a s i l y
i f we know t h e moment of i n e r t i a o f t h e r o t o r a n d t h e moment o f
the applied external force.
A c h a n g e i n t h e moment of i n e r t i a o f t h e g y r o s c o p e w i t h t i m e
w i l l b e p r o p o r t i o n a l t o t h e moment o f t h e e x t e r n a l f o r c e

(2.21)

whence
0
1
'

M
-,
JW

(2.22)

w h e r e M e q u a l s t h e moment of t h e e x t e r n a l f o r c e , J e q u a l s t h e mo­
ment o f i n e r t i a o f t h e g y r o s c o p e , w i s t h e a n g u l a r v e l o c i t y o f g y r o - '
scope r o t a t i o n , and w 1 i s t h e a n g u l a r v e l o c i t y of p r e c e s s i o n .
M

By t h e c h a n g e i n t h e moment
of i n e r t i a of t h e gyroscope,
we mean h e r e t h e c h a n g e i n t h e
d i r e c t i o n of t h e v e c t o r of i n e r ­

tia.
A t t h e same t i m e , t h e r o t a t i o n
o f t h e a x i s of t h e g y r o s c o p e
t h r o u g h 180° p r o d u c e s a n o p p o s i t e
m o t i o n o f a l l p o i n t s on t h e r o t o r ,
which amounts t o a b r a k i n g of
t h e gyroscope from i t s i n i t i a l
angular velocity t o zero, with
a s u b s e q u e n t s p e e d i n g up i n t h e
o p p o s i t e d i r e c t i o n t o t h e same
angular velocity.

dV

Fig.

2.15.

P r e c e s s i o n of a
Gyroscope Axis.

D e g r e e of Freedom of t h e G y r o s c o p e
By d e g r e e s o f f r e e d o m i n m e c h a n i c s , we mean t h e d i r e c t i o n s
o f f r e e m o t i o n o f a body which i s n o t l i m i t e d b y c o n n e c t i o n s o f
any s o r t .
For example, an o b j e c t s l i d i n g along a given l i n e ( r a i l )
h a s one d e g r e e of f r e e d o m ; a n o b j e c t moving i n any d i r e c t i o n i n
a p l a n e h a s t w o d e g r e e s o f f r e e d o m , a n d an o b j e c t w h i c h i s m o v i n g
i n t h r e e dimensional space has t h r e e degrees of freedom.
B e s i d e s t h e d e g r e e s of freedom of l i n e a r m o t i o n , t h e r e a r e
a l s o d e g r e e s of f r e e d o m o f r o t a t i o n a l m o t i o n o f a body a r o u n d i t s
three axes.
Hence,

144


a c o m p l e t e l y f r e e body h a s s i x d e g r e e s of f r e e d o m .

/144

The r o t o r s o f g y r o s c o p e s i n n a v i g a t i o n a l a n d p i l o t a g e i n s t r u ­
ments have s u p p o r t s which l i m i t t h e i r l i n e a r motion i n a c e r t a i n
d i r e c t i o n r e l a t i v e t o t h e a x e s o f t h e a i r c r a f t , s o t h a t when w e
a r e t a l k i n g a b o u t t h e d e g r e e s of f r e e d o m o f a g y r o s c o p e w e a r e r e +
f e r r i n g only t o t h e degrees of r o t a t i o n a l motion.
A gyroscope is considered t o be f r e e i f a l l t h r e e degrees of
r o t a t i o n a l motion are free ( F i g . 2.16).

C

The f i r s t d e g r e e o f f r e e d o m
of a gyroscope is t h e r o t a t i o n
o f its r o t o r a r o u n d t h e a x i s i n
b e a r i n g s A,A,.
If t h e s e b e a r i n g s
a r e t i g h t l y f a s t e n e d t o t h e body
o f t h e machine, as i s done f o r
example f o r t h e f l y w h e e l s i n machin­
e r y , t h e gyroscope w i l l have only
one d e g r e e o f f r e e d o m .
However,
i f t h e s e b e a r i n g s c a n move a r o u n d
an a x i s perpendicular t o A,A1 (bear­
ings B , B 1 ) , then t h e r e w i l l be
two d e g r e e s o f f r e e d o m .

If b e a r i n g s B , B 1 c a n a l s o
h a v e t h e f r e e d o m t o move a r o u n d
s t i l l another ( t h i r d ) a x i s , per­
Fig. 2.16.
Gyroscope w i t h
pendicular t o B,B1 (bearings C,C1),
Three Degrees of R o t a t i o n a l
t h e gyroscope w i l l have t h r e e degrees
Freedom.
of freedom and i t s a x i s can b e
s e t r e a d i l y t o any d i r e c t i o n i n s p a c e .
CI

A s w e c a n s e e f r o m F i g u r e 2 . 1 6 : t h e d e g r e e s of f r e e d o m o f t h e
g y r o s c o p e a r e e n s u r e d by p a i r s o f b e a r i n g s and ( w i t h t h e e x c l u s i o n
of t h e f i r s t ) r o t a t i n g frames.
A g y r o s c o p e u s u a l l y h a s two r o t a t i n g f r a m e s , i n t e r n a l and ex­
ternal.
In course gyroscopic instruments, t h e i n t e r n a l frame, to­
g e t h e r with t h e r o t o r and t h e bearings of t h e gyroscope, serves
The
t o s e t t h e gyroscope a x i s i n t h e plane of t h e t r u e horizon.
same f r a m e c o n t a i n s a s e n s i t i v e e l e m e n t f o r c o r r e c t i n g t h e g y r o ­
scope a x i s f o r t h i s plane.
The i n t e r n a l frame o f t h e g y r o s c o p e
along with t h e r o t o r and s e n s i t i v e element f o r correction are c a l l e d
t h e gyro assembly.

The e x t e r n a l f r a m e e n s u r e s f r e e m o t i o n o f t h e a x i s o f t h e g y r o ­
s c o p e i n t h e p l a n e of t h e h o r i z o n ; f r o m i t s p o s i t i o n i n t h e u n i t ,
w e c a n g e t a n i d e a of t h e d i r e c t i o n o f t h e g y r o s c o p e a x i s r e l a t i v e
/l45
t o t h e a x i s o f t h e a i r c r a f t , or v i c e v e r s a , t h u s m a k i n g i t p o s s i b l e
t o determine t h e aircraft course.

145

D i r e c t i o n o f P r e c e s s i o n of t h e G y r o s c o p e A x i s
The d i r e c t i o n o f t h e c p r e c e s s i o n o f t h e g y r o s c o p e a x i s u n d e r
t h e i n f l u e n c e o f t h e moment o f e x t e r n a l f o r c e s c a n b e s e e n i n F i g u r e
2.15.

For a r a p i d a n d e r r o r - f r e e d e t e r m i n a t i o n o f t h e d i r e c t i o n o f
t h e p r e c e s s i o n o f t h e gyroscope a x i s , w e u s e t h e concepts o f Ifpole
o f t h e g y r o s c o p e " a n d " p o l e of t h e e x t e r n a l f o r c e " , a n d u s e t h e
r u l e of t h e r i g h t - h a n d screw.

For e x a m p l e , i n o b s e r v i n g t h e r o t a t i o n o f a g y r o s c o p e w h i c h
i s t u r n i n g c l o c k w i s e as v i e w e d f r o m t h e t o p ( t u r n i n g t h e screw i n ­
w a r d ) , t h e p o l e o f t h e g y r o s c o p e w i l l b e c o n s i d e r e d as b e i n g l o c a t e d
a t t h e l o w e r e n d o f i t s a x i s ( P o i n t s P a n d P I ) ; w i t h l e f t - h a n d ro­
t a t i o n o f t h e g y r o s c o p e , a t t h e u p p e r e n d of t h e a x i s .
Analogously,
w i t h a r i g h t - h a n d d i r . e c t i o n of t h e moment o f e x t e r n a l f o r c e , t h e
p o l e o f t h e moment i s c o n s i d e r e d a s b e i n g d i r e c t e d a l o n g t h e s c r e w ,
i n i t s r e a r p o r t i o n as shown i n o u r d i a g r a m ( P o i n t (21).
With a
l e f t - h a n d d i r e c t i o n o f t h e moment o f e x t e r n a l f o r c e , i t s p o l e i s
l o c a t e d i n t h e f r o n t p a r t o f t h e p i c t u r e ( P o i n t (2).
The p r e c e s s i o n o f t h e g y r o s c o p e i s a l w a y s d i r e c t e d i n s u c h
a manner t h a t t h e p o l e of t h e g y r o s c o p e a t t e m p t s t o r e a c h t h e p o l e
o f t h e e x t e r n a l f o r c e by t h e s h o r t e s t p a t h .
I n our diagram, t h e lower end o f t h e gyroscope a x i s w i l l tilt
b a c k w a r d , a n d t h e u p p e r o n e f o r w a r d , i . e . , if w e l o o k a t t h e draw­
i n g from l e f t t o r i g h t , t h e a x i s of t h e g y r o s c o p e w i l l r o t a t e c l o c k ­
wise.

A p p a r e n t R o t a t i o n of G y r o s c o p e A x i s o n t h e E a r t h ' s S u r f a c e
A f r e e l y moving g y r o s c o p e , w i t h a n i d e a l l y s t a b i l i z e d e x t e r n a l
and i n t e r n a l s u p p o r t and t h e l a c k o f n o t i c e a b l e f r i c t i o n i n t h e
b e a r i n g s , t e n d s t o keep t h e p o s i t i o n o f t h e a x i s of r o t a t i o n of
the rotor i n space.
On t h e E a r t h ' s s u r f a c e , h o w e v e r , d u e t o t h e d i u r n a l r o t a t i o n
o f t h e E a r t h a n d p a r t i a l l y d u e t o t h e c u r v i l i n e a r i t y of i t s m o t i o n
a r o u n d t h e S u n , t h e r e a r i s e s a n a p p a r e n t r o t a t i o n of t h e g y r o s c o p e
a x i s i n t h e v e r t i c a l and h o r i z o n t a l p l a n e s .
The a p p a r e n t r o t a t i o n o f t h e g y r o s c o p e d u e t o t h e m o t i o n o f
t h e E a r t h a r o u n d t h e Sun i s e x p r e s s e d a s a s l i g h t d e v i a t i o n o f t h e
r o t a t i o n of t h e g y r o s c o p e a x i s f r o m t h e a p p a r e n t d i u r n a l r o t a t i o n
o f t h e E a r t h , a s a r e s u l t o f t h e f a c t t h a t t h e E a r t h m a k e s a com­
p l e t e r o t a t i o n a r o u n d t h e Sun a l o n g i t s o r b i t i n t h e c o u r s e o f a
year.
This c o n d i t i o n a l r o t a t i o n amounts t o a t o t a l of about 1/365
o f t h e a p p a r e n t r o t a t i o n of t h e g y r o s c o p e d u e t o t h e d i u r n a l r o t a ­
t i o n of the Earth.
Hence, t h i s v a l u e w i l l n o t be c o n s i d e r e d i n
future.
146

L e t u s c o n s i d e r t h e a p p a r e n t r o t a t i o n of t h e gyroscope a x i s
a t v a r i o u s p o i n t s on t h e E a r t h ' s s u r f a c e , w h i c h a p p e a r s as a r e s u l t
of t h e r o t a t i o n of t h e Earth around i t s a x i s .
We w i l l a s s u m e t h a t
/146
w e h a v e a f r e e l y m o u n t e d g y r o s c o p e , w h o s e a x i s a t t h e i n i t i a l moment
coincides with the v e r t i c a l of t h e locus (Fig. 2.17, a).

O b v i o u s l y , i f s u c h a g y r o s c o p e i s p l a c e d on a p o l e o f t h e E a r t h ,
t h e axis of i t s r o t a t i o n w i l l coincide with t h e axis of r o t a t i o n
of t h e E a r t h and t h e r e w i l l b e no a p p a r e n t r o t a t i o n of t h e gyro­
scope a x i s (position A i n t h e diagram).
I f t h e g y r o s c o p e w i t h a v e r t i c a l a x i s i s p l a c e d on some l a t ­
i t u d e tp ( p o s i t i o n B i n thse d i a g r a m ) , i t s a x i s w i l l b e a t a n a n g l e
t o t h e a x i s o f r o t a t i o n o f t h e E a r t h , e q u a l t o 90°-4.
As w e c a n
see from t h e d i a g r a m , t h e a p p a r e n t r o t a t i o n of t h e gyroscope a x i s
w i l l d e s c r i b e a cone w i t h an a p e r t u r e a n g l e a t t h e v e r t e x e q u a l
t o 2 (90-4).

I n t h e c a s e when t h e l a t i t u d e o f t h e l o c u s i s e q u a l t o z e r o
( p o s i t i o n C i n t h e d i a g r a m ) , t h e a p e r t u r e a n g l e of t h e cone w i l l
b e e q u a l t o 180°, i . e . , i t w i l l t u r n i n t h e p l a n e o f r o t a t i o n .
Now l e t u s e x a m i n e t h e c a s e when t h e a x i s of t h e g y r o s c o p e
a t t h e i n i t i a l moment i s l o c a t e d h o r i z o n t a l l y a t v a r i o u s p o i n t s
on t h e E a r t h ' s s u r f a c e ( F i g . 2 . 1 7 , b ) a n d c o i n c i d e s i n d i r e c t i o n
w i t h t h e m e r i d i a n of t h e E a r t h .
I t i s o b v i o u s t h a t t h e a x i s o f t h e g y r o s c o p e l o c a t e d on t h e
p o l e ( p o s i t i o n A ) w i l l remain h o r i z o n t a l and w i l l r o t a t e i n t h e
p l a n e o f t h e h o r i z o n w i t h t h e a n g u l a r v e l o c i t y of t h e E a r t h .
The
a x i s o f a g y r o s c o p e l o c a t e d a t some l a t i t u d e ( p o s i t i o n B ) w i l l de.­
s c r i b e a cone w i t h an a p e r t u r e a n g l e e q u a l t o 2$.
The a x i s o f t h e
g y r o s c o p e l o c a t e d on t h e E q u a t o r w i l l r e m a i n h o r i z o n t a l a n d w i l l
h a v e no a p p a r e n t d i u r n a l r o t a t i o n .
I t i s i m p o r t a n t t o n o t e i n t h i s r e g a r d t h a t i f t h e r e i s any
k i n d o f c o r r e c t i n g f o r c e w h i c h a c t s c o n s t a n t l y on t h e g y r o s c o p e
a x i s i n t h e p l a n e of t h e t r u e h o r i z o n , t h e a n g u l a r v e l o c i t y of t h e
r o t a t i o n of t h e g y r o s c o p e a x i s i n t h e p l a n e o f t h e h o r i z o n w i l l
be equal t o (Fig. 2.17, c ) :
a t t h e pole, t h e angular velocity of
r o t a t i o n of t h e E a r t h ; a t t h e E q u a t o r , z e r o ; a t any o t h e r p o i n t ,
w

= 52 s i n $ ,

(2.23)

where R i s t h e a n g u l a r v e l o c i t y of t h e E a r t h ' s r o t a t i o n and w i s
t h e a n g u l a r v e l o c i t y of t h e a p p a r e n t r o t a t i o n of t h e gyroscope a x i s .
From t h e e x a m p l e s w h i c h w e h a v e s e e n , i t i s c l e a r t h a t a f r e e l y
moving g y r o s c o p e c a n b e u s e d t o d e t e r m i n e t h e p o s i t i o n o f t h e a i r ­
c r a f t a x i s o n l y i n t h e f o l l o w i n g cases:
(a)

To d e t e r m i n e t h e p o s i t i o n o f t h e v e r t i c a l a x i s

(banking,

147

pitch)

only a t t h e poles;

(b)
To d e t e r m i n e t h e d i r e c t i o n o f t h c l o n g i t u d i n a l a x i s
of t h e a i r c r a f t ) only a t t h e Equator.

(course

I n o r d e r t o r e n d e r t h e g y r o s c o p e u s e f u l for d e t e r m i n i n g t h e
p o s i t i o n o f t h e a i r c r a f t a x i s a t a n y o t h e r p o i n t on t h e E a r t h ' s
s u r f a c e , w e used d e v i c e s which compensate f o r t h e a p p a r e n t r o t a t i o n o f t h e a x i s of t h e g y r o s c o p e d u e t o t h e d i u r n a l r o t a t i o n o f
t h e E a r t h , a s w e l l a s i t s own d r i f t , w h i c h a r i s e s a s a r e s u l t o f
imperfect balance, f r i c t i o n i n the bearings, etc.

Fig. 2.17.
A p p a r e n t R o t a t i o n of a G y r o s c o p e on t h e
( b ) With
Earth's Surface:
( a ) With V e r t i c a l A x i s ;
Horizontal Axis;
( c ) With C o n s t a n t C o r r e c t i o n of t h e
Axis i n t h e H o r i z o n t a l Plane.
To k e e p t h e a x i s of a g y r o s c o p e c o n s t a n t l y i n t h e v e r t i c a l
p o s i t i o n , p i l o t a g e d e v i c e s ( g y r o h o r i z o n , g y r o v e r t i c a l ) , or i n t h e
h o r i z o n t a l p o s i t i o n i n t h e case o f c o u r s e i n s t r u m e n t s , a r e u s u a l l y
f i t t e d w i t h pendulum d e v i c e s which a c t as s e n s i t i v e e l e m e n t s r e a c t ­
i n g t o a n y d e v i a t i o n s w h i c h may a r i s e .

The s i g n a l s f r o m t h e s e d e v i c e s a r e c o n v e r t e d t o a i r c u r r e n t s
i n p n e u m a t i c d e v i c e s a n d t o m o m e n t s of s p e c i a l e l e c t r i c m o t o r s i n
electrical devices.

148


/147

E l e c t r o l y t i c g r a v i t a t i o n a l c o r r e c t i o n ( F i g . 2.18) i s most widely
T h i s d e v i c e c o n s i s t s of a b u b b l e l e v e l
used at t h e present t i m e .
a t t a c h e d t o t h e lower p a r t of t h e gyro assembly.
U n l i k e a conven­
t i o n a l l e v e l , i t s chamber i s f i l l e d w i t h a n e l e c t r i c a l l y c o n d u c t i v e
l i q u i d ( e l e c t r o l y t e ) , w h i l e on t h e t o p of t h e s p h e r i c a l s u r f a c e
are mounted f o u r c u r r e n t - c a r r y i n g c o n t a c t s .
When t h e g y r o a s s e m b l y i s i n a v e r t i c a l p o s i t i o n ( F i g . 2 . 1 8 ,
a ) , t h e b u b b l e l e v e l i s l o c a t e d s o t h a t a l l f o u r c o n t a c t s a r e cov­
e r e d h a l f - w a y b y e l e c t r o l y t e , s o t h a t t h e moment a p p l i e d t o t h e
/148
frame o f t h e g y r o a s s e m b l y b y t h e c o r r e c t i n g m o t o r i s e q u a l t o z e r o .
If �or s o m e r e a s o n t h e g y r o a s ­
sembly v a r i e s from t h e v e r t i c a l p o s i t i o n ,
the current-carrying contacts w i l l
n o t b e u n i f o r m l y c o v e r e d by t h e f l u i d
(Fig. 2.18, b ) , r e s u l t i n g i n a s u i t ­
able distribution of currents t o the
w i n d i n g s of a s m a l l m o t o r a n d i n a
moment w h i c h i s a p p l i e d t o t h e a x i s
o f t h e g y r o s c o p e i n s u c h a way t h a t
t h e p r e c e s s i o n which i s p r o d u c e d b r i n g s
t h e gyro assembly t o a given v e r t ­
Fig. 2.18.
Electrolytic
ical position.
For c o u r s e d e v i c e s
Gravitational Correction.
which have a v e r t i c a l e x t e r n a l frame
and a h o r i z o n t a l l y l o c a t e d axis of t h e gyroscope, i n order t o c o r r e c t
t h e l a t t e r t o t h e p l a n e of t h e h o r i z o n , i t i s s u f f i c i e n t t o h a v e
one p a i r o f c u r r e n t - c a r r y i n g c o n t a c t s w i t h a g r a v i t a t i o n a l l e v e l ,
i n o r d e r t o r e g u l a t e t h e moment o f t h e f o r c e s a c t i n g o n t h e e x t e r n a l
frame

..

O b v i o u s l y , f o r t h o s e d e v i c e s w h i c h m e a s u r e d i r e c t i o n on t h e
E a r t h ' s s u r f a c e , i n a d d i t i o n t o d e v i c e s for c o r r e c t i n g t h e a x i s
of t h e g y r o s c o p e i n t h e p l a n e of t h e t r u e h o r i z o n , t h e r e m u s t a l s o
b e o t h e r d e v i c e s w h i c h c o m p e n s a t e for t h e a p p a r e n t r o t a t i o n o f t h e
a x i s of t h e gyroscope i n t h e h o r i z o n t a l p l a n e due t o t h e d i u r n a l
r o t a t i o n of t h e Earth.
G y r o s c o p i c Semicompass

I n p r i n c i p l e o f o p e r a t i o n , t h e g y r o s e m i c o m p a s s (GSC) i s a g y r o ­
s c o p e w i t h t h r e e d e g r e e s of f r e e d o m a n d i t s a x i s o f r o t a t i o n l o c a t e d
i n t h e h o r i z o n t a l , a v e r t i c a l e x t e r n a l frame, and a f l u i d g r a v i t a ­
t i o n a l c o r r e c t o r , a t t a c h e d t o t h e gyro assembly.
The r o t a t i o n o f
t h e g y r o s c o p e r o t o r i s p r o d u c e d by a l t e r n a t i n g t h r e e - p h a s e c u r r e n t ,
while t h e c o r r e c t i o n of t h e a x i s i n t h e h o r i z o n t a l p o s i t i o n i s
a c h i e v e d b y a n e l e c t r o m a g n e t i c moment a p p l i e d t o t h e e x t e r n a l f r a m e .
The g y r o c o m p a s s h a s h a s a v e r y s e n s i t i v e b a l a n c e a n d low f r i c ­
t i o n i n t h e a x e s o f t h e s u p p o r t s , which e n s u r e s a low i n t r i n s i c
s h i f t of t h e g y r o s c o p e ( c a l l e d t ' d r i f t t t ) . I n a d d i t i o n , i n o r d e r
t o compensate f o r t h i s " d r i f t " , t h e gyroscope is f i t t e d i n t h e

149


h o r i z o n t a l p l a n e w i t h a ' s p e c i a l b a l a n c i n g p o t e n t i o m e t e r and motor,
w h i c h a p p l y a moment t o t h e e x t e r n a l f r a m e o f t h e g y r o s c o p e i n t h e
v e r t i c a l plane.
T h i s same m o t o r i s u s e d f o r c o m p e n s a t i n g t h e a p p a r e n t d i u r n a l
r o t a t i o n o f t h e a x i s of t h e g y r o s c o p e , a n d i s t h e r e f o r e f i t t e d w i t h
a s p e c i a l l a t i t u d i n a l p o t e n t i o m e t e r , w h i c h r e g u l a t e s t h e moment
o f t h e m o t o r i n s u c h a way t h a t t h e r a t e o f p r e c e b s i o n o f t h e g y r o ­
scope a x i s is e q u a l t o and c o i n c i d e s i n d i r e c t i o n w i t h t h e rate
/ 149
o f r o t a t i o n o f t h e E a r t h ' s m e r i d i a n i n t h e p l a n e of t h e t r u e h o r i ­
zon a t t h e g i v e n l a t i t u d e .
By c o m p a r i n g t h e f o r m u l a f o r t h e p r e c e s s i o n o f t h e g y r o s c o p e
a x i s (2.22) and t h e formula f o r t h e angular v e l o c i t y of r o t a t i o n
o f t h e E a r t h ' s m e r i d i a n ( 2 . 2 3 ) , w e c a n d e t e r m i n e t h e moment w h i c h
i s required t o be a p p l i e d t o t h e gyroscope a x i s t o compensate f o r
t h e d i u r n a l r o t a t i o n of t h e Earth

M = QJw s i n 9 ,

(2.24)

w h e r e M i s t h e moment a p p l i e d t o t h e g y r o s c o p e a x i s , Si i s t h e a n g ­
u l a r r o t a t i o n a l v e l o c i t y of t h e E a r t h , J i s t h e i n e r t i a l moment
o f t h e r o t o r o f t h e g y r o s c o p e i n t h e p l a n e of i t s r o t a t i o n , w i s
t h e a n g u l a r v e l o c i t y of r o t a t i o n o f t h e r o t o r , a n d 9 i s t h e l a t i ­
tude of t h e aircraft's location.
W i t h a c o n s t a n t r a t e of r o t a t i o n
of t h e r o t o r o f t h e gyroscope, a l l
of t h e c o e f f i c i e n t s which e n t e r i n t o
t h e r i g h t - h a n d s i d e of ( 2 . 2 4 ) , w i t h
t h e e x c e p t i o n o f s i n $, are c o n s t a n t s .
The l a t t e r m u s t b e r e g u l a t e d i n f l i g h t .
T h e r e f o r e , t h e p o t e n t i o m e t e r which
r e g u l a t e s t h e moment a c c o r d i n g t o
t h e l a t i t u d e of t h e a i r c r a f t , as
w e l l as t h e b a l a n c i n g p o t e n t i o m e t e r ,
a r e m o u n t e d on t h e c o n t r o l p a n e l
of t h e gyrocompass ( F i g . 2.19).

.

F i g . 2.19.
Control Panel
o f KPK-52 G y r o s e m i c o m p a s s .

The e x t e r n a l frame o f t h e g y r o ­
scope is f i t t e d with a scale f o r estimatj-ng t h e gyroscopic course
and a s e l s y n - t r a n s m i t t e r f o r t r a n s m i t t i n g t h e c o u r s e t o t h e i n d i ­
cators.
The i n d i c a t i n g d i a l a n d t h e s e l s y n - t r a n s m i t t e r a r e f r e e t o
r o t a t e along w i t h t h e e x t e r n a l frame and can a l s o be s e t with t h e
The s e t t i n g
a i d o f a motor t o any a n g l e r e l a t i v e t o t h e frame.
of t h e i n d i c a t o r d i a l t o t h e zero p o s i t i o n i s accomplished manually
b y t u r n i n g a s p e c i a l h a n d l e on t h e c o n t r o l p a n e l m a r k e d r r L - R ' l ( l e f t r i g h t ) , see F i g u r e 2.19.
Hence,

15 0


t h e gyrocompass i s a s o r t o f

"keeper"

f o r the course

c a l c u l a t i o n s e t by hand:
t h e d i r e c t i o n of t h e zero s e t t i n g of t h e
c o u r s e on t h e GSC r e m a i n s c o n s t a n t i n t h e p l a n e of t h e h o r i z o n , s o
t h a t t h e gyrocompass i s a n orthodromic course d e v i c e , and i s cap­

a b l e of g u i d i n g a f l i g h t a l o n g a n orthodrome o v e r any d i s t a n c e .

The a d v a n t a g e o f a g y r o c o m p a s s i s i t s i n d e p e n d e n c e of o p e r a t i o n

from t h e magnetic f i e l d of t h e E a r t h , and c o n s e q u e n t l y t h e f i x e d
a c c u r a c y a n d s t a b i l i t y i n o p e r a t i o n a t a n y p o i n t on t h e E a r t h ' s

s u r f a c e , as w e l l as t h e ease o f d e t e r m i n i n g t h e c o u r s e w i t h o u t a n y
k i n d of m e t h o d o l o g i c a l c o r r e c t i o n s ; t h i s i s p a r t i c u l a r l y i m p o r t a n t
f o r automatic navigational devices.







/150






However , t h e g y r o s e m i c o m p a s s i s n o t a m e a s u r i n g d e v i c e , b u t
one which r e t a i n s t h e c o u r s e s e t t i n g ( t h i s i s where i t g e t s i t s
name o f s e m i c o m p a s s ) ; t h e r e f o r e , i t c a n n o t b e u s e d a l o n e w i t h o u t
o t h e r course sensors.
Nevertheless, it does not reduce t h e value
of t h e g y r o s e m i c o m p a s s , s i n c e t h e u s e of o t h e r c o u r s e s e n s o r s b e c o m e s
necessary only i n t h e i n i t i a l s e t t i n g of t h e readings of t h e GSC
a n d a t v a r i o u s p o i n t s t o make c o r r e c t i o n s f o r t h e a c c u m u l a t e d e r r o r s
i n its operation.
It is relatively easy t o eliminate errors i n the operation
o f t h e GSC, w h i c h a r i s e i n t h e f o r m o f " d r i f t " .
For t h i s p u r p o s e ,
t h e o p e r a t i o n o f t h e GSC i s t e s t e d on t h e g r o u n d f o r a p e r i o d o f
o n e t o t w o h o u r s w i t h a n a t t e m p t b e i n g made t o u s e t h e r o t a t i o n
o f t h e b a l a n c i n g p o t e n t i o m e t e r t o s e t t h e minimum e x c u r s i o n s o f
t h e needle with t i m e from t h e t r u e s e t t i n g s .
If a c o n s i d e r a b l e d e v i a t i o n o f t h e n e e d l e f r o m t h e c o r r e c t
r e a d i n g s of t h e gyroscope i s n o t i c e d d u r i n g f l i g h t , t h i s can be
c o r r e c t e d by s h i f t i n g t h e l a t i t u d e s c a l e on t h e c o n t r o l p a n e l r e l a ­
t i v e t o t h e a v e r a g e l a t i t u d e of t h e g i v e n p a t h s e g m e n t .
T h i s means
t h a t t h e d e g r e e by which t h e s c a l e i s s h i f t e d f o r e a c h d e g r e e a t
t h e t i m e t h a t the d r i f t occurs w i l l be t h e following a t various
flight latitudes :

Range o f L a t i t u d e s ,
Degrees
0 - 32
32 - 42
42 - 6 0
60 70 -

70
90

Magnitude of Scale
Deviation , Degrees
4

5

6

10

20

The l a t i t u d e on t h e s c a l e m u s t b e i n c r e a s e d i f t h e t e n d e n c y
o f t h e GSC i s d i r e c t e d t o w a r d a r e d u c t i o n o f t h e r e a d i n g s f o r t h e
c o u r s e w i t h t i m e , and i t must b e reduced i f t h e c o u r s e r e a d i n g s
increase with t i m e .
I t should be mentioned t h a t a l l s h i f t i n g mentioned above with
r e g a r d t o t h e gyrosemicompass i s i n r e f e r e n c e t o n o r t h e r n l a t i t u d e s .
I n southern l a t i t u d e s , t h e l a t i t u d i n a l compensations f o r the apparent

151


r o t a t i o n o f t h e a x i s of t h e g y r o s c o p e m u s t b e r e v e r s e d , s i n c e t h e
rotation of the meridian takes place i n t h e opposite direction rela­
tive t o the northern latitudes.
I n a d d i t i o n , t h e system f o r i n t r o ­
d u c i n g c o r r e c t i o n s t o t h e movement of t h e n e e d l e of t h e g y r o s e m i ­
compass must a l s o b e s h i f t e d t o t h e o p p o s i t e d i r e c t i o n .
Shortcomings o f t h e gyrosemicompass i n c l u d e t h e f a c t t h a t it
i s necessary t o s e t i t s readings manually a t t h e beginning of a
f l i g h t a n d t o make c o r r e c t i o n s e n r o u t e .
During f l i g h t , e s p e c i a l l y
i n rough a i r , t h i s i n v o l v e s a c e r t a i n amount o f d i f f i c u l t y , s i n c e
i t i s i m p o s s i b l e t o s e p a r a t e t h e movement o f t h e i n d i c a t o r n e e d l e
due t o c o u r s e v a r i a t i o n s from t h o s e motions which a r e caused by
s e t t i n g t h e c o u r s e m a n u a l l y , i . e . , t h e v a l u e o f t h e c o u r s e t o w h i c h /151
t h e GSC must b e s e t becomes v a r i a b l e .
In addition,

t h e GSC i s s u b j e c t t o Cardan e r r o r s d u r i n g t u r n s .

The e s s e n c e o f t h e C a r d a n e r r o r s i s t h e s h i f t i n t h e r e a d i n g
of t h e i n d i c a t o r d i a l during banking.
When t h e a i r c r a f t i s b a n k i n g
less t h a n 8 O , t h e s e e r r o r s do n o t h a v e a n y p r a c t i c a l s i g n i f i c a n c e ,
b u t t h e y r a p i d l y i n c r e a s e w i t h t h e d e g r e e of b a n k i n g a n d c a n r e a c h
6-8O.

The C a r d a n e r r o r s . h a v e a q u a t e r n a r y n a t u r e .
They a r e e q u a l
t o z e r o i n b a n k i n g i n t h e p l a n e o f r o t a t i o n o f t h e rotor o f t h e
g y r o s c o p e a n d i n t h e p l a n e of t h e p o s i t i o n o f t h e a x i s o f i t s r o t a ­
tion.
Maximum e r r o r s a r i s e when t h e g y r o s c o p e a x i s i s t h e n a t a n
angle of 45O t o t h e plane of t h e banking.
T h e r e f o r e , t h e a x i s of t h e g y r o s c o p e c a n - a s s u m e a n y p o s i t i o n
r e l a t i v e t o t h e a x e s of t h e a i r c r a f t , a n d a l s o w i t h r e s p e c t t o t h e
z e r o p o i n t on t h e c o u r s e i n d i c a t o r s c a l e , a n d t h e g r a p h o f t h e b a n k ­
i n g e r r o r i s " f l o a t i n g " , i . e . , i t s maxima a n d m i n i m a c a n a s s u m e
a n y p o s i t i o n on t h e i n d i c a t o r d i a l w h i l e r e t a i n i n g t h e v a l u e s a n d
p e r i o d i c i t y of t h e e r r o r s .
T h e s e e r r o r s a u t o m a t i c a l l y d i s a p p e a r when t h e a i r c r a f t comes
o u t o f t h e t u r n ; however, t h e y do c o n s t i t u t e c e r t a i n s h o r t c o m i n g s
i n t h e p i l o t a g e of an aircraft, i . e . , they d i s t u r b t h e c o r r e c t e s t i ­
m a t i o n o f t h e moment w h e n t h e a i r c r a f t b e g i n s t o s t o p b a n k i n g i n
making a t u r n .
D i s t a n c e G y r o m a g n e t i c Compass
The d i s t a n c e g y r o m a g n e t i c compass ( D G M C ) h a s s i g n i f i c a n t ad ­
vantages over t h e i n t e g r a t e d and d i s t a n c e magnetic compasses , s i n c e
i t i s s u i t a b l e f o r u s e when t h e a i r c r a f t i s b a n k i n g a t a c e r t a i n
a n g l e a n d c o m p l e t e l y damps t h e o s c i l l a t i o n s o f t h e m a g n e t i c c a r d
i n f l i g h t i n a t u r b u l e n t atmosphere.
The g y r o m a g n e t i c c o m p a s s i s a c o m b i n a t i o n o f m a g n e t i c a n d g y r o ­
s c o p i c c o u r s e d e v i c e s , i n which t h e r o l e o f t h e c o u r s e s e n s o r i s

152

p l a y e d by t h e m a g n e t i c t r a n s m i t t e r and t h e r o l e o f t h e s t a b i l i z e r
o f t h e r e a d i n g s i s p l a y e d by t h e g y r o a s s e m b l y .
L e t u s c o n s i d e r t h e combined s y s t e m which i s p r e s e n t l y u s e d
f o r d i s t a n c e g y r o m a g n e t i c c o m p a s s e s , e . g . , t h e DGMC-7 ( F i g . 2 . 2 0 ) .

The b a s i c p a r t s o f t h e d i s t a n c e g y r o m a g n e t i c compass a r e t h e
magnetic s e n s o r , t h e gyro assembly, and t h e main c o u r s e c o r r e c t o r .
I n a d d i t i o n t o t h e s e main p a r t s , t h e compass must b e f i t t e d
w i t h a p o w e r s u p p l y ( n o t shown i n t h e d i a g r a m ) , as w e l l a s compen­
s a t i n g and r e g u l a t i n g devices:
(a)

Compensating mechanism (combined w i t h t h e g y r o a s s e m b l y ) ;

(b)

Rapid compensation b u t t o n ;

(c)
A mechanism f o r c o m p e n s a t i n g t h e r e m a i n i n g d e v i a t i o n (com-/152
b i n e d w i t h t h e main c o u r s e i n d i c a t o r ) ;
(d)

Outputs f o r c o u r s e r e p e a t e r s and o t h e r i n d i c a t o r s ;

(e)

Two-channel

amplifier.

The m a g n e t i c t r a n s m i t t e r o f t h e compass h a s a c a r d whose a x i s
c a r r i e s a d i a l f o r s h o w i n g t h e c o u r s e d i r e c t l y on t h e t r a n s m i t t e r
( i t c a n b e u s e d t o g e t r i d o f s e m i c i r c u l a r d e v i a t i o n ) , as w e l l a s
t h e b r u s h e s f o r t h e w i r e s l e a d i n g t o t h e p o t e n t i o m e t e r on t h e t r a n s ­
m i t t e r .
The t r a n s m i t t e r p o t e n t i o m e t e r h a s a t h r e e - w i r e c i r c u i t con­
n e c t i n g i t t o t h e gyro-assembly p o t e n t i o m e t e r , through which it
r e c e i v e s a l t e r n a t i n g c u r r e n t from t h e power s u p p l y .
The t r a n s m i t t e r i n t h e damping s u s p e n s i o n i s mo u n t ed i n t h e
a i r c r a f t a t a l o c a t i o n w h e r e t h e r e i s a minimum i n f l u e n c e on t h e
c a r d s of t h e magnetic and e l e c t r o m a g n e t i c f i e l d s of t h e a i r c r a f t .
Compensation

L-------_r

Fig.

2 . 2 0 . F u n c t i o n a l D i a g r a m o f D i s t a n c e G y r o m a g n e t i c Compass
( D G M C 1.

153


The t r a n s m i t t e r h o u s i n g carries a d e v i c e f o r c o r r e c t i n g s e m i ­
circular deviation.
If t h e s e m i c i r c u l a r d e v i a t i o n a t . t h e p o i n t
where t h e magnetic s e n s o r i s mounted does n o t exceed 1-2O, t h e devia­
t i o n d e v i c e i s n o t u s e d , s i n c e i n t h i s case i t would n o t improve
b u t would r a t h e r d e t r a c t from t h e o p e r a t i n g c o n d i t i o n s of t h e t r a n s ­
mitter.
The g y r o a s s e m b l y c o n s i s t s of t h e g y r o s c o p e w i t h a h o r i z o n ­
t a l a x i s and a Cardan s u p p o r t , which e n s u r e s t h r e e d e g r e e s of free­
dom f o r t h e g y r o s c o p e r o t a t i o n .
T h e e x t e r n a l frame o f t h e g y r o
assembly r o t a t e s around t h e v e r t i c a l a x i s .
The g y r o s c o p e i s s e t i n m o t i o n b y means o f a t h r e e - p h a s e m o t o r ,
w h o s e s t a t o r i s m o u n t e d o n t h e i n t e r n a l f r a m e o f t h e g y r o assem­
b l y and whose s h o r t - c i r c u i t e d r o t o r i s t h e r o t o r o f t h e g y r o s c o p e .

For c o r r e c t i o n o f t h e g y r o s c o p e a x i s i n t h e h o r i z o n t a l p o s i ­
t i o n , t h e l o w e r p a r t o f t h e g y r o a s s e m b l y i s f i t t e d w i t h a twoc o n t a c t g r a v i t a t i o n a l c o r r e c t o r , whose a c t i v a t i n g mechanism i s a
m o t o r w h i c h p r o d u c e s a moment o f f o r c e t h a t
i s a p p l i e d t o t h e ext e r n a l frame o f t h e g y r o s c o p e and a c t s i n t h e h o r i z o n t a l p l a n e .
I f f o r some r e a s o n t h e a x i s o f t h e g y r o s c o p e v a r i e s f r o m t h e
plane of t h e t r u e h o r i z o n , t h e c o n t a c t s of t h e c o r r e c t o r w i l l be
c o v e r e d n o n u n i f o r m l y by t h e s h i f t i n g c o n d u c t i n g f l u i d , t h u s r e s u l t ­
ing i n a d i s t r i b u t i o n of c u r r e n t s passing through t h e c o r r e c t o r .
T h i s i n t u r n t r a n s m i t s a s i g n a l f o r a c o r r e c t i n g moment o f f o r c e
A s a r e s u l t of t h e preces­
t o be a p p l i e d t o t h e e x t e r n a l frame.
s i o n of t h e gyroscope a x i s , i t i s s h i f t e d t o a h o r i z o n t a l p o s i t i o n .

The e x t e r n a l f r a m e o f t h e g y r o a s s e m b l y c a r r i e s a m a s t e r
s e l s y n �or c o n n e c t i n g t o t h e p r i n c i p a l i n d i c a t o r o f t h e c o m p a s s
( t h e p i l o t ' s i n d i c a t o r , P I ) and a t h r e e - c o n d u c t o r c o r d f o r connec­
t i o n t o t h e magnetic t r a n s m i t t e r .
The m a s t e r s e l s y n a n d c a b l e a r e c o n n e c t e d c l o s e l y t o g e t h e r
a n d c a n r o t a t e t o g e t h e r w i t h t h e e x t e r n a l f r a m e o f t h e g y r o assem­
bly.
H o w e v e r , t h e y c a n a l s o r o t a t e r e l a t i v e t o t h e e x t e r n a l frame
by m e a n s o f a s p e c i a l c o o r d i n a t i o n m e c h a n i s m .
The c o o r d i n a t i o n mechanism c o n s i s t s o f a s m a l l m o t o r w i t h a
r e d u c t i o n g e a r f o r t h e s l o w - c o o r d i n a t i o n regime, i n which t h e r a t e
o f r o t a t i o n of t h e s e l s y n i s 1 - 4 O p e r m i n u t e .
When i t i s n e c e s s a r y t o c a r r y o u t a r a p i d c o o r d i n a t i o n , t h e
m o t o r i s s w i t c h e d t o r e d u c e d r e d u c t i o n b y means of t h e r a p i d - c o o r d ­
The r a t e o f r o t a t i o n of t h e
i n a t i o n button and a s p e c i a l r e l a y .
s e l s y n i n t h i s c a s e i s r a i s e d t o 15-16O p e r s e c o n d .
The p o t e n t i o m e t e r o f t h e g y r o a s s e m b l y i s f i r m l y f a s t e n e d t o
t h e housing.

15 4

/153

T h e c o o r d i n a t i o n o f t h e m a g n e t i c t r a n s m i t t e r w i t h a g y r o assem­
b l y i s a c c o m p l i s h e d as f o l l o w s ( F i g . 2 . 2 1 ) .
The a l t e r n a t i n g c u r r e n t p a s s e s t h r o u g h c o n t a c t s A
and B t o
r e a c h t h e p o t e n t i o m e t e r o f t h e g y r o assembly and i s p i c k e d up by
p i c k u p s 1, 2 , 3 m o u n t e d o n t h e e x t e r n a l f r a m e o f t h e g y r o assem­
b l y , from w h i c h i t p a s s e s t o t h e p i c k u p s on t h e t r a n s m i t t e r p o t e n ­
tiometer, l a , 2a, 3a.
I t is clear from t h e f i g u r e t h a t i f t h e p o s i t i o n of t h e brushes
o f t h e c u r r e n t p i c k u p s on t h e t r a n s m i t t e r A 1 , B l r e l a t i v e t o t h e
c u r r e n t l e a d s of t h e p o t e n t i o m e t e r l a , 2 a , 3a d i f f e r s from t h e p o s i ­
t i o n of t h 8 c u r r e n t p i c k u p s o f p o t e n t i o m e t e r A , B r e l a t i v e t o t h e i r
c u r r e n t c o n n e c t i o n s 1, 2 , 3 b y 90°, t h e r e w i l l b e a c u r r e n t i n t h e
pickups of t h e t r a n s m i t t e r .
A t t h e s a m e t i m e , between t h e c u r r e n t connection A and t h e
c u r r e n t pickup A 1 i n t h i s case, t h e r e w i l l be a p o r t i o n of t h e poten­
t i o m e t e r i n t'he g y r o a s s e m b l y A - 1 a n d a p o r t i o n o f t h e t r a n s m i t t e r
p o t e n t i o m e t e r l a - A 1 , r e p r e s e n t e d a s a sum o f t h e f o u r c i r c u m f e r ­
ences.
Such a l e n g t h o f w i n d i n g of p o t e n t i o m e t e r w i l l b e p l a c e d
b e t w e e n c u r r e n t c o n n e c t i o n A a n d c u r r e n t p i c k u p B 1 ( s e g m e n t s A­
2 and 2a-Bl).
Consequently, a p o t e n t i a l difference w i l l develop
between p o i n t s A 1 and B 1 .

We c a n r e a c h a n a n a l o g o u s c o n c l u s i o n i f w e c o n s i d e r t h e p a t h
o f t h e c u r r e n t from c o n n e c t i o n B t o p i c k u p s A 1 and B 1 .
If t h e p o s i t i o n o f t h e b r u s h e s o f c u r r e n t p i c k u p s A 1 a n d B 1
d i f f e r s from t h e p o s i t i o n o f c o n n e c t o r s A a n d B by an a n g l e which
i s n o t 90° ( c o n s i d e r i n g t h e i r r e l a t i o n s h i p t o t h e p o t e n t i o m e t e r
/
s e c t i o n s ) , t h e r e w i l l b e a c u r r e n t i n pickups A 1 and B 1 .
T h i s cur-...
rent is fed t o the first
channel of t h e a m p l i f i e r ,
and t h e n t o t h e motor of
t h e c o o r d i n a t i o n mechanism.
The p o t e n t i o m e t e r b r u s h e s
i n t h e gyro assembly, along
with t h e selsyn-transmitter,
begin t o r o t a t e a t a very
low s p e e d u n t i l t h e r e i s
an e q u i l i b r i u m of t h e cur­
r e n t s on p i c k u p s A 1 a n d B 1 .

Fig.
m i t t

less
tion

Thus, t h e p o s i t i o n of
t h e m a s t e r o f t h e g y r o assembly constantly shifts t o
2.21.
Potentiometric Transagree with the position of
e r of Position Signal.
t h e transmitter card, regard­
of t h e a p p a r e n t r o t a t i o n o f t h e g y r o s c o p e a x i s due t o t h e r o t a ­
of t h e E a r t h and t h e n a t u r a l changes i n t h e gyroscope a x i s .

155


,

I n a s m u c h as t h e a g r e e m e n t of t h e r e a d i n g s o f t h e s e l s y n s o f
t h e t r a n s m i t t e r and gyro assembly t a k e s p l a c e a t an angular veloc­
i t y which d o e s n o t e x c e e d 4 O p e r m i n u t e , t h e r e a d i n g s o f t h e g y r o
a s s e m b l y c a n n o t show t h e i n f l u e n c e o f r a p i d c h a n g e s i n t h e p o s i ­
t i o n of t h e t r a n s m i t t e r c a r d , ;.e.,
t h e mechanism f o r c o o r d i n a t i o n
i s a damper which a v e r a g e s o u t t h e r e a d i n g s o f t h e compass f o r a n
a v e r a g e p o s i t i o n of t h e c a r d .
I n o r d e r t h a t no t r a n s m i t t e r e r r o r s b e t r a n s m i t t e d t o t h e g y r o
a s s e m b l y when t h e a i r c r a f t i s m a k i n g a t u r n , t h e DGMC c o m p l e x i n c l u d e s
a c o r r e c t i o n s w i t c h which a u t o m a t i c a l l y s h u t s o f f t h e c o r r e c t i o n
mechanism o f t h e g y r o a s s e m b l y f r o m t h e c o m p a s s c a r d when t h e a i r ­
craft is turning.
E s t i m a t i o n of t h e r e a d i n g s o f t h e a i r c r a f t ‘ s
c o u r s e d u r i n g t u r n s i s made w i t h a p u r e l y g y r o s c o p i c o p e r a t i o n r e g i m e
o f t h e DGMC.
Inasmuch as t h e a p p a r e n t d i u r n a l r o t a t i o n o f t h e g y r o s c o p e
a x i s c a n n o t e x c e e d lo i n f o u r m i n u t e s o f t u r n , w h i l e t h e t u r n i n g
t i m e o f t h e a i r c r a f t a t a n a n g l e up t o 90° as a r u l e d o e s n o t e x c e e d
1 - 3 m i n u t e s , n o g r e a t e r r o r s i n t h e compass r e a d i n g s are p r o d u c e d
d u r i n g t h e t u r n a n d t h e g y r o m a g n e t i c compass c a n b e u s e d s u c c e s s ­
f u l l y f o r t u r n i n g an aircraft a t a d e s i r e d angle.
Agreement of t h e g y r o a s s e m b l y w i t h t h e b a s i c c o u r s e i n d i c a t o r
i s a c c o m p l i s h e d by means o f a m a s t e r s e l s y n ( F i g . 2 . 2 2 ) .
Winding AB r o t a t e s i n s i d e t h e h o u s i n g o f t h e master s e l s y n ,
a l l o w i n g a l t e r n a t i n g c u r r e n t t o f l o w i n t h e w i n d i n g s of t h e s e l s y n
0 - 1 , 0-2, 0-3.
C u r r e n t s which a r e s y m m e t r i c a l i n p h a s e a l s o a r i s e
i n t h e windings of t h e s l a v e s e l s y n 01-11, 01-21, 01-31.
Hence,
t h e magnetic f i e l d of t h e r e s u l t a n t c u r r e n t s of t h e s l a v e s e l s y n
w i l l be p a r a l l e l t o t h e magnetic f i e l d o f t h e supply winding AB.
T h e r e f o r e , i f winding A l B l of t h e s l a v e s e l s y n o c c u p i e s a p o s i t i o n
which i s p e r p e n d i c u l a r t o t h e s u p p l y w i n d i n g A B , t h e c u r r e n t i n
it w i l l be e q u a l t o zero.

J,

21

a motor which t u r n s winding
with an i n d i c a t o r

A1B1,

156


/155

o f r e a d i n g s a n d t r a n s m i t t h e m w i t h h i g h m e c h a n i c a l moments a n d good
damping.
This p e r m i t s us n o t only t o o b t a i n p r e c i s e and s t a b l e
r e a d i n g s w i t h t h e compass, b u t a l s o t o apply a n a d d i t i o n a l stress
t o t h e c o u r s e i n d i c a t o r s or t h e i n t e r m e d i a t e l i n k s .
For e x a m p l e ,
t h e y c a n b e u s e d t o s e t t h e m e c h a n i c a l c o m p e n s a t o r s for d e v i a t i o n ,
a n d t o t a k e r e a d i n g s f r o m o t h e r i n d i c a t o r s or d e v i c e s w h i c h u s e
course signals.
The d e v i c e f o r m e c h a n i c a l c o m p e n s a t i o n o f t h e r e s i d u a l d e v i ­
a t i o n c o n s i s t s of a c i r c u l a r c u r v e d s t r i p w i t h s p e c i a l b e n d s , w h i c h
o p e r a t e s by means o f a l e v e r a n d p i n i o n t o p r o d u c e a n a d d i t i o n a l
t u r n i n g o f t h e n e e d l e on t h e s c a l e f o r s h o w i n g t h e m a g n e t i c c o u r s e .
The a d j u s t m e n t s c r e w s a r e m o u n t e d a l o n g th.e e d g e o f t h e s t r i p , u s u a l l y
a t every 15O, t h u s making i t p o s s i b l e t o compensate f o r t h e r e s i d ­
u a l d e v i a t i o n p r a c t i c a l l y down t o z e r o .
However, i t i s n o t recommended t h a t r e s i d u a l d e v i a t i o n g r e a t e r
t h a n 2-3O b e c o m p e n s a t e d , if i t i s p o s s i b l e t o g e t r i d o f i t by a
deviation device with a magnetic t r a n s m i t t e r , f o r t h e following
reasons:
(a)
Not g e t t i n g r i d o f , b u t c o m p e n s a t i n g f o r , s e m i c i r c u l a r
d e v i a t i o n l e a d s t o c o n s i d e r a b l e c h a n g e s i n i t , d e p e n d i n g on t h e
magnetic l a t i t u d e of t h e l o c u s of t h e a i r c r a f t .

/156

(b)
When t h e a i r c r a f t i s t u r n i n g a n d t h e m a g n e t i c c o r r e c t i o n
is s w i t c h e d o f f w h i l e t h e compass i s o p e r a t i n g i n a regime o f gyro­
s c o p i c s t a b i l i z a t i o n , t h e mechanical compensation f o r d e v i a t i o n
( i f i t i s shown on t h e i n d i c a t o r ) c a u s e s e r r o r s i n t h e c o u r s e r e a d i n g s
i n t h e form of o v e r s h o o t i n g and l a g g i n g , e q u a l t o t h e v a l u e of t h e
c o m p e n s a t e d d e v i a t i o n , t h u s m a k i n g i t more d i f f i c u l t t o t u r n t h e
aircraft a t a given angle.
I n a d d i t i o n t o t h e mechanical compensator f o r t h e r e s i d u a l
d e v i a t i o n , t h e main i n d i c a t o r h a s a d e c l i n a t i o n s c a l e whose r e v o l u ­
t i o n t o t h e v a l u e of t h e m a g n e t i c d e c l i n a t i o n of t h e l o c u s of t h e
a i r c r a f t c o n v e r t s t h e c o m p a s s r e a d i n g s from m a g n e t i c t o t r u e .
T o l i n k it w i t h o t h e r d e v i c e s , t h e main i n d i c a t o r h a s b o t h
a master and a s l a v e s e l s y n , whose i n d i c a t i o n s can b e t r a n s m i t t e d
e i t h e r w i t h t h e a i d o f t h e a c t i v a t i n g m o t o r s or b y a d i r e c t s e l s y n
connection.

I n t h e case of d i r e c t s e l s y n c o n n e c t i o n , t h e windings o f t h e
s e l s y n i n t r a n s m i t t e r AB and t h e s e l s y n of t h e i n d i c a t o r A l B l are
connected i n p a r a l l e l with t h e a l t e r n a t i n g current source.
In this
case, t h e winding A B
of the slave selsyn attempts t o set itself
a c c o r d i n g t o t h e r e g u l a t i o n of t h e m a g n e t i c f i e l d , p r o d u c e d b y wind;
i n g s 0111, 0 1 2 1 , 0 1 3 1 , i . e . , i t a u t o m a t i c a l l y a s s u m e s t h e p o s i t i o n
of t h e p o w e r w i n d i n g A B o f t h e m a s t e r s e l s y n .
The d i r e c t s e l s y n c o n n e c t i o n h a s a l o w e r s e n s i t i v i t y for t h e

157

I 1 I I


m a t c h i n g o f t h e s e l s y n s a n d a s m a l l e r w o r k i n g moment,
i s a r e d u c e d a c c u r a c y of t r a n s m i s s i o n .
Hence, i t i s
m i s s i o n s where t h e r e a r e n o p a r t i c u l a r l y h i g h demands
acy, e.g., f o r p i l o t a g e course repeaters connected t o
cator.

so that there
used f o r t r a n s ­
made on a c c u r ­
t h e main i n d i ­

Gyroinduction Compass

In t h e preceding paragraph, it w a s mentioned t h a t t h e d i s t a n c e
g y r o m a g n e t i c compass h a s c o n s i d e r a b l e a d v a n t a g e s o v e r t h e i n t e g r a t e d
compass.
However, t h e m a g n e t i c t r a n s m i t t e r o f t h i s compass h a s
a s e r i o u s shortcoming.
T h e f a c t i s , t h a t t h e m a g n e t i c moment w h i c h m o v e s t h e t r a n s ­
m i t t e r c a r d t o t h e p l a n e of t h e m a g n e t i c m e r i d i a n i s i t s e l f v e r y
s m a l l , and w h i l e i t i s s u f f i c i e n t f o r t u r n i n g t h e f l o a t i n g c a r d ,
it i s f r e q u e n t l y i n s u f f i c i e n t f o r overcoming t h e f r i c t i o n of t h e
b r u s h e s o n t h e c u r r e n t p i c k u p s , e s p e c i a l l y i n f l i g h t a t h i g h mag­
netic latitudes.
Therefore, t h i s t r a n s m i t t e r is unstable i n oper­
a t i o n and f r e q u e n t l y goes o u t of o r d e r .
To o v e r c o m e t h i s s h o r t c o m i n g , new t y p e s o f i n d u c t i o n m a g n e t i c
t r a n s m i t t e r s have b e e n d e v e l o p e d ; i n a d d i t i o n t o h a v i n g an i n c r e a s e d
t h r e s h o l d o f s e n s i t i v i t y , t h e y d o n o t h a v e t h e a b i l i t y t o move i n
/157
t h e h o r i z o n t a l p l a n e ( i n t h e a z i m u t h ) ; c o n s e q u e n t l y t h e r e a r e no.
e r r o r s d u e t o s p l a s h i n g o f t h e f l u i d o v e r t h e s e n s i t i v e e l e m e n t or
o b s t r u c t i o n ; t h e y a r e l e s s s e n s i t i v e t o t h e i n f l u e n c e of accelera­
t i o n s when t h e a i r c r a f t i s y a w i n g , a n d t h e s i z e o f t h e t r a n s m i t ­
t e r is smaller.
The o p e r a t i n g p r i n c i p l e of t h e i n d u c t i o n - t y p e s e n s i t i v e e l e ­
ment i s t h e d e p e n d e n c e o f t h e v a l u e o f t h e a l t e r n a t i n g m a g n e t i c
i n d u c t i o n of t h e c o r e upon t h e p r e s e n c e o f i t s c o n s t a n t c o m p o n e n t ,
e x e r t e d i n t h e c o r e by t h e h o r i z o n t a l component o f t h e t e r r e s t r i a l
magnetism.

For e x a m p l e , i f t h e c o r e h a s a c o n s t a n t c o m p o n e n t o f m a g n e t i c
i n d u c t i o n i n t h e d i r e c t i o n of t h e v e c t o r O A ( F i g . 2 . 2 3 , a ) , t h e n
i n o r d e r t o b r i n g i t u p t o c o m p l e t e s a t u r a t i o n i n t h i s same d i r e c ­
The c h a n g e i n i n d u c ­
t i o n we w i l l r e q u i r e an a d d i t i o n a l v e c t o r AB.
a)

.

B

l7

I

-.f

00 - UA

J
6

Fig. 2.23.
I n d u c t i o n S a t u r a t i o n of t h e Core of t h e
S e n s i t i v e Element:
( a ) Induction Vector Coincides
with Saturation Vector;
( b ) I n d u c t i o n Vector and
S a t u r a t i o n Vector are i n Opposite Directions.

158

t i o n i n t h i s case i s e x p r e s s e d by t h e d i f f e r e n c e b e t w e e n t h e v e c t o r s
OB-OA.
A s w e s e e f r o m F i g u r e 2 . 2 3 , b , when t h e m a g n e t i c i n d u c t i o n
i s b r o u g h t up t o f u l l s a t u r a t i o n , t h e change i n i n d u c t i o n i n t h e
o p p o s i t e d i r e c t i o n w i l l b e e q u a l t o t h e sum o f t h e v e c t o r s O A +
OB.

The t r a n s m i t t e r o f a n i n d u c t i o n compass h a s t h r e e s e n s i t i v e
e l e m e n t s , e a c h o f which i s made as f o l l o w s :
Two p a r a l l e l m a g n e t i c
c o r e s ' m a d e o f p e r m a l l o y ( a m a t e r i a l w i t h a h i g h m a g n e t i c permea­
b i l i t y a n d a v e r y low v a l u e o f m a g n e t i c h y s t e r e s i s ) h a v e s e p a r a t e
p r i m a r y w i n d i n g s , c o n n e c t e d i n o p p o s i t e p h a s e , a n d a common s e c o n ­
dary winding around both cores (Fig. 2.24, a ) .
A l t e r n a t i n g c u r r e n t /158
flows through t h e primary windings of t h e cores.
O b v i o u s l y , i f t h e c o n s t a n t component of t h e m a g n e t i c i n d u c ­
t i o n o f t h e c o r e s f r o m t h e h o r i z o n t a l c o m p o n e n t o f t h e E a r t h ' s mag­
n e t i c f i e l d i s z e r o , t h e v e c t o r s of i t s c h a n g e w i t h p a s s a g e o f a n
a l t e r n a t i n g c u r r e n t t h r o u g h t h e w i n d i n g w i l l b e t h e same i n b o t h
Cores, b u t i n o p p o s i t e d i r e c t i o n s , and t h e r e w i l l b e no a l t e r n a t i n g
current i n t h e secondary winding.
If t h e c o r e s h a v e a c o n s t a n t c o m p o n e n t o f m a g n e t i c i n d u c t i o n ,
t h e v e c t o r of t h e change i n magnetic i n d u c t i o n w i l l be g r e a t e r i n
one and s m a l l e r i n t h e o t h e r ; t h i s w i l l p r o d u c e p u l s e s o f a l t e r ­
n a t i n g c u r r e n t as shown i n t h e g r a p h i n F i g u r e 2 . 2 4 , b .
T h e mag­
n i t u d e of t h e c u r r e n t p u l s e s w i l l b e p r o p o r t i o n a l t o t w i c e t h e v a l u e
of t h e c o n s t a n t component o f t h e m a g n e t i c i n d u c t i o n o f t h e c o r e s .

The s e n s i t i v e e l e m e n t s i n t h e t r a n s m i t t e r a r e a r r a n g e d i n t h e
form of a t r i a n g l e and t h e i r secondary windings form a s o r t of master
selsyn (Fig. 2.25).
T h e r o t a t i n g w i n d i n g ot t h e s l a v e s e l s y n i s
c o n n e c t e d t o t h e a m p l i f i e r and mounted i n a p o s i t i o n p e r p e n d i c u l a r
t o t h e r e s u l t a n t v e c t o r of t h e e l e c t r o m a g n e t i c f i e l d of t h e s l a v e
s e l s y n b y means o f a n a c t i v a t i n g m o t o r w i t h r e d u c t i o n g e a r i n g .
The p r i m a r y w i n d i n g o f t h i s t r a n s m i t t e r i s mo u n t ed i n a n i n t e r ­
mediate element between t h e t r a n s m i t t e r and t h e gyro assembly i n
a c o r r e c t i o n mechanism which h a s a d e v i c e f o r mechan c o m p e n s a t i o n
of
r e s i d u a l d e v i a t i o n a n d i s u s e d as a c o r r e c t i o n mechanism f o r
t h e following system.

Fig.

2.24. S e n s i t i v e Element of Induction Transmitter:
ing;
( b ) Graph o f C u r r e n t .

( a ) Wind­

159

The i n d u c t i o n t r a n s m i t t e r s f o r t h e c o u r s e a r e r e l i a b l e a n d
s t a b l e i n o p e r a t i o n , b u t t h e i r a c c u r a c y o f o p e r a t i o n d r o p s when
the transmitter is t i l t e d t o a sufficiently greater degree than
i s t h e case f o r m a g n e t i c t r a n s m i t t e r s .
A t t h e same t i m e , i f t h e t i l t i n g o f t h e t r a n s m i t t e r t a k e s p l a c e
i n t h e plane perpendicular t o t h e magnetic meridian, t h e v e r t i c a l
component o f t h e m a g n e t i c f i e l d o f t h e E a r t h , p r o j e c t e d on t h e p l a n e
of t h e s e n s i t i v e element, forms a magnetic induction normal t o t h e
magnetic meridian; t h e banking deviation w i l l then be determined
by t h e f o r m u l a
(2.25)

where i i s t h e b a n k i n g of t h e t r a n s m i t t e r , 0 i s t h e a n g l e between
t h e plane of t h e magnetic meridian and t h e banking plane of t h e
t r a n s m i t t e r , and Z,H are t h e v e r t i c a l and h o r i z o n t a l components
of t h e E a r t h ’ s f i e l d , r e s p e c t i v e l y .

Z

For e x a m p l e , w i t h t h e r a t i o = 3 a n d t h e a n g l e 0 = 90°, e a c h
H
banking r a d i u s of t h e t r a n s m i t t e r w i l l produce a n e r r o r of approx­
i m a t e l y 3O i n t h e o p e r a t i o n of t h e compass.
L

The r a t i o - = 3 c o r r e s p o n d s

H

(e.g.1

t o t h e l a t i t u d e o f Moscow

/159

and i n c r e a s e s r a p i d l y w i t h an approach t o t h e p o l a r r e g i o n s .
There­
f o r e , t h e b a n k i n g e r r o r s i n t h e i n d u c t i o n t r a n s m i t t e r c a n t a k e on
very s i g n i f i c a n t values.
In order t o reduce t h e e r r o r s i n
i t s s e n s i t i v e e l e m e n t i s mounted on a
support.
The body of t h e t r a n s m i t t e r
t h e p r e s s u r e on t h e a x i s o f t h e frame

Fig. 2.25.
D i a g r a m S h o w i n g Conn e c t i o n o f E l e m e n t s i n S e n s o r of
G y r o i n d u c t i o n Compass.

16 0

the induction transmitter,
f l o a t mounted i n a Cardan
is f i l l e d with f l u i d t o reduce
of t h e Cardan suspension
( a mixture of l i g r o i n and
methylvinylpyridine o i l ) .
The Cardan s u s p e n s i o n e n s u r e s
t h e h o r i z o n t a l p o s i t i o n of
t h e s e n s i t i v e element during
banking and p i t c h i n g t o w i t h i n
17O.

The i n d u c t i o n t r a n s m i t t e r ,
l i k e t h e magnetic one, i s
mounted a b o a r d t h e a i r c r a f t
i n a p o s i t i o n such t h a t it
i s exposed t o t h e smallest
magnetic f i e l d of t h e air­
c r a f t a n d one w h i c h i s as
c o n s t a n t as p o s s i b l e ; a
d e v i a t i o n mechanism i s mounted
on i t t o r e c o r d t h e semi­

circular deviation of t h e t r a n s m i t t e r .
However, t h e c u r v i l i n e a r t r a j e c t o r y o f f l i g h t ( a l t h o u g h t h e
r a d i u s of c u r v a t u r e i s v e r y g r e a t ) , i n a d d i t i o n t o t h e accelera­
t i o n p r o d u c e d by C o r i o l i s f o r c e s , p r o d u c e s a c o n s t a n t t i l t i n g o f
t h e s e n s i t i v e e l e m e n t of t h e t r a n s m i t t e r , t h e d e v i a t i o n from which
i s t r a n s m i t t e d t o t h e main i n d i c a t o r a n d i t s r e p e a t e r s .

For e x a m p l e , a t t h e l a t i t u d e o f Moscow a n d a n
km/hr, t h e t i l t i n g of t h e s e n s i t i v e e l e m e n t o f t h e
t o t h e acceleration of t h e Coriolis forces w i l l be
i m a t e l y 2 0 ' , w h i c h u n d e r g o e s d e v i a t i o n e q u a l t o lo
t h e n o r t h e r l y and s o u t h e r l y ' d i r e c t i o n s .

a i r s p e e d o f 800
t r a n s m i t t e r due
e q u a l t o approx­
in a flight in

T h e g y r o s c o p i c i n d u c t i o n c o m p a s s ( w i t h t h e e x c e p t i o n of t h e
i n d u c t i o n t r a n s m i t t e r ) i s b u i l t i n a m a n n e r s i m i l a r t o t h a t of t h e
d i s t a n c e magnetic compass.
Its p r i n c i p a l components a r e t h e i n d u c t i o n t r a n s m i t t e r , t h e
gyro assembly and t h e course i n d i c a t o r .

I n a d d i t i o n t o t h e p r i n c i p a l u n i t s , t h e r e i s a l s o a power s u p p l y ,
a m p l i f i e r s , c o r r e c t i o n mechanism w i t h a c u r v e d d e v i c e f o r g e t t i n g
/160
r i d of r e s i d u a l d e v i a t i o n , a c o n n e c t i n g chamber, a b u t t o n w i t h a
mechanism for r a p i d c o o r d i n a t i o n , a c o r r e c t i o n s w i t c h , a n d r e p e a t e r s
from t h e main c o u r s e i n d i c a t o r .
The c o r r e c t i o n m e c h a n i s m i s t h e i n t e r m e d i a t e l i n k b e t w e e n t h e
i n d u c t i o n t r a n s m i t t e r and t h e gyro assembly.
The c o n n e c t i o n b e t w e e n
t h e i n d u c t i o n t r a n s m i t t e r a n d t h e c o r r e c t i o n m e c h a n i s m i s made w i t h
a s e l s y n , w h i l e t h e c o n n e c t i o n b e t w e e n t h e c o r r e c t i o n mechanism
a n d t h e g y r o a s s e m b l y , t h e g y r o a s s e m b l y w i t h t h e main i n d i c a t o r ,
and t h e main i n d i c a t o r w i t h t h e
r e p e a t e r s i s made b y p o t e n t i o m ­
eters.
T h e m a i n i n d i c a t o r a l s o h a s a c u r v e d d e v i c e for g e t t i n g r i d
of e r r o r s i n t h e d i s t a n c e t r a n s m i s s i o n o f t h e c o u r s e i n d i c a t i o n s
from t h e g y r o assembly t o t h e i n d i c a t o r a t t h e f a c t o r y .
The c o r r e c t i o n s w i t c h i s a t w o - s t a g e g y r o s c o p e w h i c h s e r v e s
f o r a u t o m a t i c a l l y d i s c o n n e c t i n g t h e gyro assembly from t h e c o r r e c t i o n
mechanism; t h i s d i s c o n n e c t s t h e c i r c u i t f o r a z i m u t h c o r r e c t i o n from
t h e i n d u c t i o n t r a n s m i t t e r and d i s c o n n e c t s t h e c o r r e c t i o n of t h e
h o r i z o n t a l p o s i t i o n o f t h e a x i s o f t h e g y r o s c o p e r o t o r when t h e
a i r c r a f t i s m a k i n g t u r n s w i t h a n a n g u l a r v e l o c i t y g r e a t e r t h a n 36
deg/min.
Disconnecting t h e induction t r a n s m i t t e r during t u r n s g e t s r i d
o f t h e c o n s i d e r a b l e errors w h i c h a r i s e d u e t o t h e i n f l u e l i c e o f t h e
v e r t i c a l component of t h e E a r t h ' s m a g n e t i c f i e l a 2 .
In order t o
ensure t h a t t h e gyroscope c o r r e c t i o n w i l l n o t be disconnected i n
a t u r b u l e n t a t m o s p h e r e when t h e a i r c r a f t i s b u mp i n g a n d y a w i n g ,

16 1


t h e c o r r e c t i o n s w i t c h h a s a d e l a y mechanism which d i s c o n n e c t s t h e
c o r r e c t i o n o n l y a f t e r 5 - 1 5 s e c h a v e e l a p s e d f o l l o w i n g t h e moment
when t h e a i r c r a f t r e a c h e s a n a n g u l a r v e l o c i t y o f 3 6 ' d e g / m i n .
The c o u r s e r e p e a t e r s a r e s i m p l e i n d e s i g n a n d c o n s i s t o f t h r e e p h a s e m a g n e t o e l e c t r i c l a g o m e t e r s whose a c c u r a c y f o r d e t e r m i n i n g
t h e c o u r s e i s l o w e r t h a n t h a t of t h e main i n d i c a t o r .
D e s p i t e t h e numerous a d v a n t a g e s of d i s t a n c e g y r o m a g n e t i c and
g r y o i n d u c t i o n c o m p a s s e s o v e r i n t e g r a t e d c o m p a s s e s , t h e y d o n o t com­
p l e t e l y s a t i s f y t h e requirements of a i r c r a f t navigation, p a r t i c ­
u l a r l y with regard t o automation of i t s processes, since the follow­
i n g s h o r t c o m i n g s of c o m p a s s e s s t i l l p e r s i s t :
(a)
The d e p e n d e n c e o f t h e a c c u r a c y w i t h w h i c h t h e c o u r s e i s
m e a s u r e d upon t h e m a g n e t i c l a t i t u d e a n d t h e i m p o s s i b i l i t y o f u s i n g
the instrument a t high magnetic l a t i t u d e s .
( b ) . The d i f f i c u l t y o f m a i n t a i n i n g a n o r t h o d r o m i c d i r e c t i o n
of f l i g h t , s i n c e t h e magnetic f l i g h t a n g l e s which a r e t h e n o b t a i n e d
vary.
(c)
The m a g n e t i c l o x o d r o m e a l o n g w h i c h a f l i g h t c a n b e c a r ­
r i e d o u t w i t h a c o n s t a n t m a g n e t i c f l i g h t a n g l e i s a complex c u r v e ,
s i n c e i t - d e p e n d s on t h e i n t e r s e c t i o n o f m e r i d i a n s a n d m a g n e t i c d e c l i n ­
a t i o n s , which l i m i t t h e l e n g t h of t h e s t r a i g h t - l i n e f l i g h t s e g m e n t s , / l 6 1
a l o n g which t h e f l i g h t a n g l e c a n b e assumed c o n s t a n t .
(d)
R e g a r d l e s s of a l l t h e m e a s u r e s w h i c h h a v e b e e n t a k e n t o
g e t r i d of a n d c o r r e c t f o r d e v i a t i o n s , as w e l l as t h e c o n s i d e r a t i o n .
of m a g n e t i c d e c l i n a t i o n s , t h e a c c u r a c y o f t h e measurements o f t h e
magnetic course s t i l l remain low(within t h e l i m i t s of 2 - 3 O ) .
The m a j o r i t y o f t h e s e s h o r t c o m i n g s c a n b e o v e r c o m e b y u s i n g
g y r o s c o p i c s e m i c o m p a s s e s w i t h h i g h a c c u r a c y , or c o u r s e s y s t e m s w h i c h
make i t p o s s i b l e t o f l y i n a r e g i m e u s i n g h i g h l y s e n s i t i v e g y r o ­
semicompasses ( t h e GSC r e g i m e ) .
Details

of

D e v i a t i o n O p e r a t i o n s on D i s t a n c e G y r o m a g n e t i c
and G y r o i n d u c t i o n Compasses

D e v i a t i o n o p e r a t i o n s on d i s t a n c e c o m p a s s e s a r e c a r r i e d o u t
u s i n g t h e same m e t h o d a s f o r i n t e g r a t e d c o m p a s s e s , w i t h c e r t a i n
c h a n g e s n e c e s s i t a t e d by f e a t u r e s o f t h e d e s i g n a n d m o u n t i n g o f t h e s e
compasses.
I n s e v e r a l types of a i r c r a f t , t h e s e m i c i r c u l a r d e v i a t i o n a t
t h e p o i n t where t h e t r a n s m i t t e r s a r e mounted can b e v e r y low.
In
t h e s e cases, t h e d e v i a t i o n d e v i c e s must b e removed from t h e t r a n s ­
m i t t e r s a n d a l l f o r m s o f d e v i a t i o n a r e c o m p e n s a t e d f o r by a mechan­
i c a l c o m p e n s a t o r o n t h e m a i n c o u r s e i n d i c a t o r or on t h e c o r r e c t i o n
mechanism.

162

I

I

The c o m p e n s a t i o n f o r t h e r e s i d u a l d e v i a t i o n , u s i n g a mechan­
i c a l c o m p e n s a t o r , i s c a r r i e d o u t on 24 c o u r s e s :
0 , 15, 30,
,
345O, i n w h i c h t h e a i r c r a f t i s s e t t o t h e d e s i r e d c o u r s e s , a n d a
screw i s t u r n e d ( c o r r e s p o n d i n g t o t h e c o u r s e of t h e a i r c r a f t ) i n
order t o bring the remaining deviation t o zero.
The g r a p h o f t h e
r e m a i n i n g d e v i a t i o n on t h e main c o u r s e i n d i c a t o r i s n o t p l o t t e d .
However, i f d i f f e r e n c e s i n r e a d i n g s b e t w e e n t h e main i n d i c a t o r a n d
i t s r e p e a t e r s are n o t i c e d , it i s necessary t o p l o t a graph of t h e
c o r r e c t i o n s f o r t h e r e a d i n g s on t h e r e p e a t e r s .

. ..

A f t e r e a c h two i n t e r m e d i a t e s e t t i n g s o f t h e a i r c r a f t on c o u r s e
( a t t h e p o i n t s 0 , 4 5 , 9 0 , 1 3 5 , 1 8 0 , 225, 270 a n d 315O), i t i s n e c e s ­
s a r y t o mark t h e r e a d i n g s o f t h e c o m p a s s t r a n s m i t t e r on t h e s c a l e
o f t h e c o m p a s s c o u r s e on t h e m a i n i n d i c a t o r ( f o r i n d u c t i o n t r a n s ­
m i t t e r s , on t h e s c a l e o f t h e c o r r e c t i o n m e c h a n i s m ) , a n d u s e t h i s
t o d e t e r m i n e t h e c o e f f i c i e n t s o f s e m i c i r c u l a r d e v i a t i o n B a n d C.
The f o r m shown i n T a b l e 2 . 4 i s r e c o m m e n d e d for c o n v e n i e n c e i n d e t e r ­
mining these c o e f f i c i e n t s .

The c o e f f i c i e n t s a r e c a l c u l a t e d a c c o r d i n g t o t h e f o r m u l a s :

where 8

i i s t h e c o m p a s s d e v i a t i o n on i n d i v i d u a l c o u r s e s .
a0

MC
0

45

90

135

180

225

270

315


I

/162


TABLE 2 . 4 .
-

sin MC

8 sin

MC

0

0.7

1

0.7

0'

-0.7

-1

-0.7


cos

MC

8cos

MC

1

0.7

0

-0.7

-1

-0.7

0

0.7


The c a l c u l a t e d c o e f f i c i e n t s m u s t b e i n t h e f o r m o f t a b l e s ,
a t t a c h e d t o t h e i n s t r u m e n t p a n e l a l o n g w i t h t h e main c o u r s e i n d i ­
cator.
I n a d d i t i o n t o t h e c o e f f i c i e n t s on t h e t a b l e , i t i s a l s o
n e c e s s a r y t o show t h e p l a c e w h e r e t h e d e v i a t i o n s w e r e c o r r e c t e d
or t h e h o r i z o n t a l c o m p o n e n t o f t h e m a g n e t i c f i e l d of t h e E a r t h a t
t h e p o i n t where t h e c o r r e c t i o n w a s c a r r i e d o u t .
Since the semicircular deviation,

as w e l l a s a l l i t s o t h e r

1 63

f o r m s , c a n b e made b y a m e c h a n i c a l c o m p e n s a t o r a t t h e m a g n e t i c l a t i t u d e o f t h e p o i n t where t h e c o r r e c t i o n w a s made, Formula ( 2 . 1 6 )
for c a l c u l a t i n g t h e d e v i a t i o n f o r o t h e r m a g n e t i c l a t i t u d e s a s s u m e s
t h e form
(2.26)

Course

Systems

The m o s t c o m p l e t e d e v i c e s f o r m e a s u r i n g t h e c o u r s e o f a n a i r craft are t h e course systems.
C o u r s e s y s t e m s a r e c o m b i n a t i o n s or
c o m p l e x e s o f v a r i o u s c o u r s e t r a n s m i t t e r s m o u n t e d on t h e a i r c r a f t ,
w i t h t h e i r r e a d i n g s d i s p l a y e d on g e n e r a l i n d i c a t o r s .
Such t r a n s ­
mitters include the following:
M a g n e t i c i n d u c t i o n (MC r e g i m e ) ;
A s t r o n o m i c a l (AC r e g i m e ) ;
G y r o s c o p i c (GSC r e g i m e ) .
I n p r i n c i p l e , t h e course system c o n s i s t s of a combination of
t h e d e s i g n f e a t u r e s of a g y r o i n d u c t i o n compass, gyrosemicompass
and a s t r o n o m i c a l c o u r s e t r a n s m i t t e r , whose o p e r a t i n g p r i n c i p l e w i l l
b e d i s c u s s e d i n t h e c h a p t e r d e v o t e d t o a s t r o n o m i c a l means o f a i r ­
craft navigation.
The p r i m a r y f e a t u r e of t h e d e s i g n o f t h e g y r o s c o p i c p o r t i o n
o f t h e c o u r s e s y s t e m i s t h e p r e s e n c e o f a t h i r d f r a m e �or t h e g y r o ­
scope with a h o r i z o n t a l a x i s , coinciding with t h e longitudinal a x i s
The p u r p o s e o f t h e t h i r d frame i s t o s e l e c t t h e
of t h e aircraft.
C a r d a n e r r o r s i n t h e r e a d i n g s o f t h e g y r o s e m i c o m p a s s when t h e a i r - / 1 6 3
craft is turning.
The u s e o f t h i s t h i r d f r a m e c o m p l e t e l y e x c l u d e s C a r d a n e r r o r s
from t h e t r a n s v e r s e r o l l i n g o f t h e a i r c r a f t , s i n c e t h e s e c o n d f r a m e
of t h e gyroscope ( w i t h a master s e l s y n ) w i l l always be i n a v e r t ­
ical position.
The s e t t i n g o f t h e s e c o n d f r a m e o f t h e g y r o s c o p e i n a v e r t ­
i c a l p o s i t i o n i s a c c o m p l i s h e d by means o f a n e l e c t r i c a l c i r c u i t
and a mechanical device f o r matching it w i t h t h e s o - c a l l e d gyrov e r t i c a l , m o u n t e d on a i r c r a f t f o r p i l o t a g e p u r p o s e s .
The s e c o n d f e a t u r e of c o u r s e s y s t e m s i s t h e u s e ( a s a r u l e )
o f two g y r o a s s e m b l i e s , a main one a n d a s t a n d b y , which improve
t h e r e l i a b i l i t y of t h e s y s t e m a n d e n s u r e r e c i p r o c a l c o n t r o l o f t h e
readings.
F i g u r e 2.26 shows t h e c o n t r o l p a n e l a n d t h e i n d i c a t o r o f t h e
course system.
The c o u r s e s y s t e m o p e r a t e s on t h e main i n d i c a t o r
i n a r e g i m e i n w h i c h t h e s w i t c h for t h e o p e r a t i n g r e g i m e i s s e t
a t t h e t o p p a r t o f t h e p a n e l ( M C , A C , or GSC).

1 64

r

When s w i t c h i n g t h e c o u r s e s y s t e m f r o m t h e G S C r e g i m e t o t h e
i n order t o c o r r e c t t h e readings, it is neces­
sary t o press the button f o r rapid correlation i n order t o adjust
t h e r e a d i n g s of t h e gyro assembly t o
t h e readings of these t r a n s m i t t e r s .
A f t e r correlation, the switch is again
r e t u r n e d t o t h e GSC p o s i t i o n .

MC or A C r e g i m e s ,

The p u s h b u t t o n c o u r s e c o n t r o l s e r v e s
f o r manual s e t t i n g o f t h e v a l u e s f o r
t h e course system only i n t h e GSC regime.
The s w i t c h on t h e l e f t - h a n d s i d e o f
t h e p a n e l , m a r k e d "N-S", i s u s e d t o
switch t h e p o l a r i t y of t h e l a t i t u d i n a l
p o t e n t i o m e t e r i n o r d e r t o compensate
f o r t h e r o t a t i o n of t h e Earth i n t h e
N o r t h e r n or S o u t h e r n H e m i s p h e r e .
The
c o v e r s a t t h e b o t t o m o f t h e p a n e l , marked
"main" a n d " s t a n d b y " , c o v e r a d j u s t m e n t s
f o r t h e balancing potentiometers of
t h e main and s t a n d b y g y r o a s s e m b l i e s .
Methods o f U s i n g C o u r s e D e v i c e s
f o r Purposes o f A i r c r a f t Navi­
gation

/164

The m e t h o d s of u s i n g c o u r s e e q u i p m e n t
d e p e n d upon t h e r e s o l v i n g p o w e r s o f
t h e c o m p l e x of c o u r s e d e v i c e s m o u n t e d
on t h e a i r c r a f t , t h e p r e s e n c e o f o t h e r
equipment f o r purposes of a i r c r a f t navi­
g a t i o n , a n d a l s o on t h e d i s t a n c e , g e o g r a p h ­
i c and m e t e o r o l o g i c a l c o n d i t i o n s of
flight.
While t h e m e t e o r o l o g i c a l f l i g h t
conditions along a given route (path)
change i n t h e c o u r s e of t i m e a n d - c a n
v a r y d e p e n d i n g on a l t i t u d e a n d d i s t a n c e
of f l i g h t , t h e remaining c o n d i t i o n s f o r a g i v e n t y p e of a i r c r a f t
and a given r o u t e ( a i r r o u t e ) remain c o n s t a n t .

Fig. 2.26.
Control
P a n e l of Course System.

I n d i s c u s s i n g t h e methods of u s i n g c o u r s e d e v i c e s i n f l i g h t ,
t h e c o n s t a n t c o n d i t i o n s l i s t e d above can b e d i v i d e d i n t o t h r e e groups.
(1)
The a i r c r a f t i s e q u i p p e d w i t h a n i n t e g r a t e d or d i s t a n c e
gyromagnetic ( i n d u c t i o n ) compass.
Flights are c a r r i e d out over
l o n g or medium d i s t a n c e s w i t h o u t s i g n i f i c a n t c h a n g e s i n m a g n e t i c
latitude.
The e q u i p m e n t f o r c o n s t a n t m e a s u r e m e n t o f t h e a i r s p e e d ,
d r i f t a n g l e , a n d a u t o m a t i c c a l c u l a t i o n of t h e p a t h a r e l a c k i n g on
the aircraft.

16 5


P

(2)
T h e a i r c r a f t i s f i t t e d w i t h a d i s t a n c e g y r o m a g n e t i c or
g y r o i n d u c t i o n c o m p a s s a n d a g y r o s e m i c o m p a s s or c o u r s e s y s t e m o f
average accuracy.
F l i g h t s a r e made o v e r l o n g d i s t a n c e s w i t h c o n s i d ­
e r a b l e changes i n magnetic l a t i t u d e .
There is no equipment f o r
a u t o m a t i c m e a s u r e m e n t o f t h e d r i f t a n g l e or a i r s p e e d , o r c a l c u l a t ­
i n g t h e f l i g h t a c c o r d i n g t o t h e s e p a r a m e t e r s on b o a r d t h e a i r c r a f t .
(3)
The a i r c r a f t i s f i t t e d w i t h a c o u r s e s y s t e m o f h i g h a c c u r ­
a c y , as w e l l as d e v i c e s f o r a u t o m a t i c a l l y m e a s u r i n g t h e d r i f t a n g l e ,
t h e a i r s p e e d , and c a l c u l a t i n g t h e p a t h .
F l i g h t s a r e made a t a n y
g e o g r a p h i c a l l a t i t u d e and f o r any d i s t a n c e .

Methods of U s i n g C o u r s e D e v i c e s Under C o n d i t i o n s IncZu­
ded i n t h e F i r s t Group
Under t h e c o n d i t i o n s i n t h e f i r s t g r o u p , ; . e . ,
when f l i g h t s
a r e b e i n g made o v e r s h o r t d i s t a n c e s i n a i r c r a f t w h i c h h a v e s i m p l e
n a v i g a t i o n equipment, t h e f o l l o w i n g methods a r e used t o p r e p a r e
t h e c a l c u l a t e d d a t a and use t h e course d e v i c e s i n f l i g h t .
I n p r e p a r i n g f o r a f l i g h t , t h e r o u t e o f t h e f l i g h t t o b e made
i s e n t e r e d on a f l i g h t c h a r t .
If t h e f l i g h t c h a r t i s one which
i s i n a n i n t e r n a t i o n a l or d i a g o n a l c y l i n d r i c a l p r o j e c t i o n , t h e
s t r a i g h t - l i n e p o r t i o n s of t h e f l i g h t b e t w e e n t h e t u r n i n g p o i n t s
a l o n g t h e r o u t e a r e p l o t t e d as s t r a i g h t l i n e s b y means o f a r u l e r .
When u s i n g c h a r t s w h i c h a r e p l o t t e d w i t h a n i s o g o n a l c y l i n d r i c a l
p r o j e c t i o n (Mercator), t h e s t r a i g h t - l i n e p o r t i o n s of a f l i g h t which/l65
i s v e r y l o n g a r e p l o t t e d a s a c u r v e d l i n e on t h e b a s i s o f t h e i n t e r ­
mediate p o i n t s a l o n g t h e orthodrome, c a l c u l a t e d by a n a l y t i c a l means.
S i n c e t h e m a g n e t i c compass i s a l o x o d r o m i c c o u r s e - m e a s u r i n g
d e v i c e , a n d t h e p a r t s of t h e r o u t e s w h i c h h a v e b e e n p l o t t e d a r e
very nearly orthodromic, i n order t o avoid overly high deflections
i n t h e loxodrome from t h e given l i n e o f f l i g h t , t h e l e n g t h o f t h e
f l i g h t segments with a c o n s t a n t f l i g h t p a t h a n g l e are s e l e c t e d s o
t h a t t h e i n i t i a l and f i n a l f l i g h t p a t h a n g l e s under c o n d i t i o n s of
f o l l o w i n g a n o r t h o d r o m e do n o t d i f f e r by more t h a n 2 - 3 O , ? . e . , s o
t h a t t h e t o t a l c o r r e c t i o n f o r t h e r e a d i n g s o f t h e m a g n e t i c compass
a t t h e end of t h e segment r e l a t i v e t o i t s r e a d i n g s a t t h e begin­
n i n g of t h e s e g m e n t i s n o more t h a n 3 O :

A = (1,

- A,)

sin tm 4-(AMr

-),A < 3".

If t h e i n d i c a t e d c o r r e c t i o n i s m o r e t h a n 3 O i n t h e s t r a i g h t l i n e p o r t i o n o f t h e f l i g h t , t h i s segment i s d i v i d e d i n t o two, t h r e e
or m o r e p a r t s a n d t h e f l i g h t p a t h a n g l e i s d e t e r m i n e d f o r e a c h .
T h i s i s u s u a l l y n o t done by s i m p l e d i v i s i o n of a s t r a i g h t l i n e i n t o
e q u a l p a r t s , b u t by s e l e c t i n g c h a r a c t e r i s t i c o r i e n t a t i o n p o i n t s
a l o n g t h e s e c t i o n o f t h e r o u t e , t h e f l i g h t between which can be
made a t t h e c o n s t a n t f l i g h t p a t h a n g l e .
If w e

166

c o n s i d e r t h e low a c c u r a c y o f t h e i n d i c a t i o n s o f t h e

m a g n e t i c compasses i n a r e l a t i v e l y s h o r t l e n g t h of f l i g h t segment
f o r a f l i g h t with a given f l i g h t path angle, t h e l a t t e r are deter­
mined n o t by a n a l y t i c a l means, b u t by s i m p l e measurement o f t h e
d i r e c t i o n of t h e s e g m e n t on t h e c h a r t b y means o f a p r o t r a c t o r .
M e a s u r e m e n t o f t h e l o x o d r o m i c f l i g h t p a t h a n g l e c a n b e made
r e l a t i v e t o t h e m e r i d i a n which i n t e r s e c t s t h e segment a t a p o i n t
which i s c l o s e s t t o i t s c e n t e r , c o n s i d e r i n g t h e m a g n e t i c d e c l i n ­
a t i o n of t h i s p o i n t .
H o w e v e r , t o i n c r e a s e t h e a c c u r a c y o f t h e meas­
u r e m e n t s , i t i s recommended t h a t i t b e done a t two p o i n t s , a t t h e
beginning and end of t h e segment, considering t h e average d e c l i n ­
a t i o n of t h e s e p o i n t s .
O b v i o u s l y , i n t h e f i r s t case t h e m a g n e t i c f l i g h t a n g l e o f t h e
segment w i l l be

w h i l e i n t h e s e c o n d case
MFA =

ab

f

Cte

- AMb
__

~

2

-

‘Me
Y

w h e r e a b , Ctm, a, a r e t h e a z i m u t h s o f t h e o r t h o d r o m e a t t h e b e g i n ­
ning, t h e middle, and end, r e s p e c t i v e l y .
An a d v a n t a g e o f t h e s e c o n d m e t h o d i s t h e d o u b l e m e a s u r e m e n t
of t h e angles and t h e averaging of t h e d e c l i n a t i o n s , s i n c e t h e accur­
a c y o f t w o m e a s u r e m e n t s a n d t h e a v e r a g i n g o f t h e i r r e s u l t i s a l w a y s 1166
h i g h e r t h a n t h e a c c u r a c y of a s i n g l e measurement.
F o r t h e f i r s t g r o u p o f c o n d i t i o n s , i t i s p o s s i b l e t o h a v e some
s i m p l i f i e d p r e p a r a t i o n for t h e c o u r s e e q u i p m e n t o f t h e a i r c r a f t
for the flight.
S i n c e t h e f l i g h t s a r e made w i t h r e l a t i v e l y l o w
measurements of magnetic l a t i t u d e , t h e r e i s no need t o d e t e r m i n e
t h e c o e f f i c i e n t s o f s e m i c i r c u l a r d e v i a t i o n B a n d C or t o c o n s i d e r
t h e i r changes during t h e f l i g h t .
I f t h e d e v i a t i o n i s c o m p e n s a t e d by a m e c h a n i c a l c o m p e n s a t o r ,
i t i s assumed t o b e z e r o d u r i n g t h e f l i g h t .
In considering the
r e s i d u a l d e v i a t i o n , a v a l u e i s a s s i g n e d t o i t as s h o w n on t h e g r a p h .

During t h e f l i g h t , t h e course of t h e a i r c r a f t i s checked s o
t h a t i t s value t o g e t h e r with t h e d r i f t angle of t h e aircraft w i l l
b e e q u a l t o a g i v e n m a g n e t i c f l i g h t p a t h a n g l e o f t h e f l i g h t seg­
ment.
MFA,

= M C t US = MFAg.

On t h e o t h e r h a n d , s i n c e t h e m a g n e t i c c o u r s e o f t h e a i r c r a f t i s
e q u a l t o t h e compass c o u r s e , i t i s n e c e s s a r y t o a d d t h e compass
deviation:
16 7

I

I l l 1


MFA,

= C C t A,

t

U S = MFAg.

Prob Zems
1. T h e d i r e c t i o n o f a f l i g h t s e g m e n t m e a s u r e d a l o n g t h e a v e r ­
age meridian is e q u a l t o 4 8 O ; t h e magnetic d e c l i n a t i o n i n t h e middle
o f t h e segment is t 7 O .
Determine t h e g i v e n magnetic f l i g h t p a t h
angle of t h e segment.

Answer:

MFAg = 4 1 O .

2.
The d i r e c t i o n o f a f l i g h t s e g m e n t m e a s u r e d a l o n g t h e i n ­
i t i a l m e r i d i a n i s e q u a l t o 1 3 6 O , 132O a t t h e f i n a l m e r i d i a n , w i t h
a n i n i t i a l m a g n e t i c d e c l i n a t i o n o f t 7 O a n d a f i n a l one o f t 5 O .
Deter­
m i n e t h e MFAg.

Answer:

MFAg

= 128O.

3.
The g i v e n m a g n e t i c p a t h f l i g h t a n g l e o f a s e g m e n t i s e q u a l
t o 84O, t h e d r i f t a n g l e w a s e q u a l t o - 6 O ,
t h e d e v i a t i o n o f t h e mag­
n e t i c compass i s t 4 O .
Determine t h e r e q u i r e d compass c o u r s e f o r
following the f l i g h t lines.

Answer:

CC

=

86O.

4.
The c o m p a s s c o u r s e o f a n a i r c r a f t i s e q u a l t o 5 4 O , t h e
Determine t h e
compass d e v i a t i o n i s t 3 O , t h e d r i f t a n g l e i s + 6 O .
actual f l i g h t path angle.

Answer:

MFA,

= 63O.

Methods of U s i n g Course D e v i c e s Under C o n d i t i o n s o f
t h e S e c o n d Group
When f l i g h t s a r e b e i n g made o v e r l o n g d i s t a n c e s u s i n g d i s t a n c e
g y r o m a g n e t i c a n d g y r o s e m i c o m p a s s e s or c o u r s e s y s t e m s , b u t w i t h o u t
a n y a u t o m a t i c c o u r s e c a l c u l a t i o n , t h e u s e of c o u r s e i n s t r u m e n t s
i n f l i g h t a n d p r e p a r a t i o n of c h a r t s f o r a f l i g h t i s a c c o m p l i s h e d
by d e v i c e s which a r e somewhat d i f f e r e n t f r o m t h o s e which a r e recom­
m e n d e d f o r t h e c o n d i t i o n s of t h e f i r s t g r o u p .
The m o s t i m p o r t a n t o f t h e s e d e v i c e s i s t h e p l o t t i n g o f t h e
orthodromic course along t h e s t r a i g h t - l i n e segments of t h e f l i g h t
w i t h a g y r o s e m i c o m p a s s or a c o u r s e s y s t e m i n t h e I r G S C " r e g i m e , w i t h
p e r i o d i c c o r r e c t i o n o f t h e g y r o s c o p e c o u r s e by means o f a m a g n e t i c
or a s t r o n o m i c t r a n s m i t t e r .

A s a r u l e , i n f l i g h t s over long distances, t h e f l i g h t chart
/167
i s one w i t h a s c a l e of .1:2,000,000
on t h e i n t e r n a t i o n a l p r o j e c t i o n .
If a s t r a i g h t l i n e w i t h i n t h e l i m i t s o f o n e s h e e t o f t h i s m a p , w i t h
d i s t a n c e s q t o 1 2 0 0 - 1 5 0 0 km, c a n b e a s s u m e d w i t h i n s i g n i f i c a n t
e r r o r t o b e a n o r t h o d r o m e , t h e n when t w o o r m o r e s h e e t s a r e c o m b i n e d

1 68

..

s,

I

a n d t h e r o u t e d o e s n o t r u n a l o n g a m e r i d i a n or w h e n s h e e t s o f t h i s
c h a r t are used s e p a r a t e l y a t g r e a t d i s t a n c e s , t h e orthodrome must
be l o c a t e d a l o n g p o i n t s which are determined by c a l c u l a t i o n .
When
s p l i c i n g two a d j a c e n t s h e e t s a l o n g t h e m e r i d i a n , t h e o r t h o d r o m e
h a s a s i g n i f i c a n t b r e a k i n i t , and i n t h i s case (when i t c r o s s e s
t h e a d j a c e n t s h e e t s ) a s t r a i g h t l i n e cannot be t a k e n as t h e o r t h o ­
drome.
On t h e c h a r t s o f a l l o t h e r p r o j e c t i o n s , e x c e p t t h e c e n t r a l
p o l a r a n d s p e c i a l r o u t e maps i n a d i a g o n a l , c y l i n d r i c a l p r o j e c t i o n ,
when t h e l i n e o f t h e t a n g e n t ( c r o s s - s e c t i o n a l s t r i p ) o f t h e c y l i n ­
d e r c o i n c i d e s w i t h t h e a x i s of t h e r o u t e , t h e o r t h o d r o m e i s c a l c u ­
l a t e d a n a l y t i c a l l y a n d p l o t t e d on t h e c h a r t a c c o r d i n g t o t h e c a l c u ­
lated intermediate points.
The d i s t a n c e s f o r t h e s e c t i o n s o f t h e
orthodrome a r e a l s o d e t e r m i n e d by a n a l y t i c a l means.
The o r t h o d r o m i c f l i g h t p a t h a n g l e s o f t h e r o u t e s e g m e n t s u n d e r
t h e s e c o n d i t i o n s a r e m e a s u r e d or c a l c u l a t e d a n a l y t i c a l l y r e l a t i v e
If t h e s t r a i g h t t o t h e i n i t i a l meridian of each f l i g h t segment.
l i n e segments of t h e f l i g h t have a very s h o r t l e n g t h , t h e f l i g h t
p a t h a n g l e s c a l c u l a t e d from t h e i n i t i a l m e r i d i a n s of t h e segments
can be a p p l i e d t o t h e system r e l a t i v e t o t h e s e l e c t e d r e f e r e n c e
meridian (Fig. 2.27) according t o t h e following formula:

where 6 i s t h e a n g l e of convergence between t h e r e f e r e n c e
meridians of t h e segment.

and i n i t i a l

Since
t h e c o n d i t i o n f o r t h e second group assumes f l i g h t s over
long d i s t a n c e s with c o n s i d e r a b l e changes i n t h e magnetic l a t i t u d e s ,
t h e p r e p a r a t i o n o f t h e m a g n e t i c c o m p a s s e s m u s t b e made w i t h a c o n s i d ­
e r a t i o n of d e t e r m i n a t i o n o f t h e c h a n g e s i n t h e s e m i c i r c u l a r d e v i ­
ation during the f l i g h t .
C o u r s e d e v i c e s i n t e n d e d for f l i g h t s u n d e r c o n d i t i o n s o f t h e
second group
have d e v i c e s f o r mechanical compensation of t h e r e s i d ­
ual deviation.
T h e r e f o r e , t h e g r a p h o f t h e d e v i a t i o n f o r them i s
not plotted.
However, i n g e t t i n g r i d of t h e d e v i a t i o n , i t i s n e c e s ­
s a r y t o d e t e r m i n e a n d w r i t e down t h e c o e f f i c i e n t s o f d e v i a t i o n B
a n d C:

I t i s t h e n n e c e s s a r y t o w r i t e down t h e i n t e n s i t y o f t h e h o r i z o n /168
t a l component o f t h e E a r t h ' s m a g n e t i c f i e l d a t t h e p o i n t where t h e
d e v i a t i o n s were c o r r e c t e d .

16 9

4


To c a l c u l a t e t h e c h a n g e s i n t h e s e m i c i r c u l a r d e v i a t i o n d u r i n g
f l i g h t , t h e corrections f o r t h e magnetic course a t d i f f e r e n t seg­
m e n t s o� t h e r o u t e m u s t b e d e t e r m i n e d w h e n p r e p a r i n g f o r a f l i g h t .
They a r e d e t e r m i n e d f o r a number o f p o i n t s a l o n g t h e f l i g h t p a t h ,
o n t h e b a s i s o f t h e m a g n e t i c f l i g h t a n g l e s of t h e r o u t e a t t h e s e
p o i n t s w i t h a f r e q u e n c y s u c h t h a t t h e d i f f e r e n c e Fetween two a d j a ­
c e n t c o r r e c t i o n s a l o n g a s t r a i g h t l i n e p a t h d o e s n o t e x c e e d lo a n d
a f t e r e a c h t u r n i n g p o i n t on t h e r o u t e .

#/\
A init

'ref

I

I n f a c t , t h e changes i n t h e s e m i ­
circular
deviation a t correspond­
ing points along the route w i l l d i f f e r
only s l i g h t l y from t h e c a l c u l a t e d cor­
r e c t i o n s , s i n c e t h e c o u r s e which i s
followed w i l l b e prepared with a consid­
e r a t i o n of t h e d r i f t a n g l e o f t h e a i r ­
craft.
However, t h e e r r o r s which arise
i n t h i s p r o c e s s w i l l be small and can
be disregarded.

During t h e f l i g h t , t h e gyrosemi­
c o m p a s s o r t h e c o u r s e s y s t e m i s corr e c t e d f o r t h e m a g n e t i c or a s t r o n o m i c a l
Fig. 2.27.
Calculation
t r a n s m i t t e r when f l y i n g a l o n g t h e r e f e r of F l i g h t P a t h A n g l e s
e n c e m e r i d i a n s or t h e t u r n i n g p o i n t s
from Reference Meridian.
If t h e c o r r e c ­
o f t h e r o u t e (TPR).
t i o n i s made on t h e b a s i s o f t h e mag­
n e t i c t r a n s m i t t e r , t h e n t h e main i n d i c a t o r w i l l h a v e t h e r e q u i r e d
This correction is equal t o the
c o r r e c t i o n e n t e r e d on i t s d i a l .
sum o f t h e m a g n e t i c d e c l i n a t i o n a n d t h e c h a n g e i n t h e s e m i c i r c u l a r
deviation along the magnetic l a t i t u d e .
I

For c o r r e c t i o n , t h e course system i s switched t o t h e IIMC" r e ­
The s y s t e m
gime and t h e b u t t o n i s p u s h e d t o match t h e r e a d i n g s .
o p e r a t e s f o r a p e r i o d o f 1 - 2 min i n t h e s l o w c o o r d i n a t i o n r e g i m e
and i s t h e n s w i t c h e d t o t h e llGSC" r e g i m e .
I n t h i s m a n n e r , t h e s y s t e m s a r e c o r r e c t e d for t h e a s t r o n o m ­
ical transmitter.
H a v i n g d e t e r m i n e d t h e l a t t e r on t h e b a s i s o f
t h e c o o r d i n a t e s of a s t a r a n d t h e l o c u s o f t h e a i r c r a f t , t h e s y s t e m
i s s w i t c h e d t o t h e "AC" r e g i m e , t h e c o o r d i n a t i o n i s c a r r i e d o u t ,
T h i s means t h a t a t
and t h e n s w i t c h e d back t o t h e "GSC" regime.
t h e t u r n i n g p o i n t s o f t h e r o u t e , n o c o r r e c t i o n s a r e r e q u i r e d on
t h e s c a l e of t h e d e c l i n a t i o n s .
The c o r r e c t i o n o f t h e g y r o c o m p a s s i s made i n t h e s a m e m a n n e r ,
e x c e p t t h a t t h e c o u r s e i s s e t on t h e g y r o s e m i c o m p a s s n o t by com­
p a r i n g t h e r e a d i n g s of t h e t r a n s m i t t e r s , b u t by m a n u a l s e t t i n g on
t h e b a s i s o f t h e r e a d i n g s o f t h e m a g n e t i c or a s t r o n o m i c a l t r a n s ­
mitters.
After correction,

170


t h e f l i g h t i s c a r r i e d out with an orthodromic

c o u r s e u p t o t h e n e x t t u r n i n g p o i n t o f t h e r o u t e or r e f e r e n c e m e r i d ­

ian.
When i t i s n e c e s s a r y t o make a c o r r e c t i o n for t h e o r t h o d r o m i c c o u r s e between two r e f e r e n c e m e r i d i a n s , t h e c o r r e c t i o n i s s e t
on t h e main i n d i c a t o r a n d i s e q u a l t o :

/16g

f o r t h e magnetic transmitter,
A = A

M

t

sin

+m y

for t h e a s t r o n o m i c a l t r a n s m i t t e r

Then t h e r e a d i n g s a r e m a t c h e d i n t h e man n er d e s c r i b e d a b o v e .

Prob I ems

1. The e a s t l o n g i t u d e o f t h e r e f e r e n c e m e r i d i a n i s 40°,
the
The c o o r d i n a t e s o f
n o r t h l a t i t u d e o f t h e r e f e r e n c e p o i n t i s 52O.
t h e s e t t i n g p o i n t of t h e r o u t e are:
l o n g i t u d e 43O, l a t i t u d e 54O.
The t r u e f l i g h t p a t h a n g l e o f t h e s e g m e n t a t t h e s t a r t i n g p o i n t
i s 67O.
Determine t h e orthodromic f l i g h t p a t h a n g l e c a l c u l a t e d
from t h e r e f e r e n c e m e r i d i a n .

Answer:

64.5O.

2.
The i n t e n s i t y o f t h e h o r i z o n t a l c o m p o n e n t o f t h e E a r t h ' s
magnetic f i e l d a t t h e p o i n t where t h e d e v i a t i o n s are c o r r e c t e d i s
0.24 o e r s t e d s , w h i l e a t a c e r t a i n p o i n t a l o n g t h e f l i g h t r o u t e i t
is 0.08 oersteds.
Determine t h e c o r r e c t i o n s f o r t h e magnetic c o u r s e
of t h e aircraft a t t h i s p o i n t , i f t h e magnetic f l i g h t path angle
o f t h e f l i g h t s e g m e n t i s e q u a l t o 60°, c o e f f i c i e n t B = t 1 . 5 , a n d
coefficient C = tO.9.

Answer:

t3.50.

3.
The e a s t l o n g i t u d e o f t h e r e f e r e n c e m e r i d i a n i s e q u a l t o
The a i r ­
70°, t h e n o r t h l a t i t u d e of t h e r e f e r e n c e p o i n t i s 5 8 O .
c r a f t i s l o c a t e d a t t h e p o i n t A = 76O,
= 60°; t h e magnetic declin­
a t i o n o f t h e l o c a t i o n o f t h e a i r c r a f t i s e q u a l t o tilo, w h i l e t h e
c o r r e c t i o n f o r t h e c h a n g e i n t h e s e m i c i r c u l a r d e v i a t i o n ABC = t 2 O .
Determine t h e c o r r e c t i o n f o r t h e r e a d i n g s o f t h e m a g n e t i c compass
f o r correction of t h e orthodromic course.

+

Answer:

A = t8O.

171

Methods of U s i n g Course D e v i c e s Under t h e C o n d i t i o n s o f
t h e T h i r d Group
The t h i r d g r o u p o f c o n d i t i o n s f o r u s i n g c o u r s e d e v i c e s r e f e r s
t o f l i g h t s i n a i r c r a f t which are f i t t e d w i t h p r e c i s e c o u r s e s y s ­
t e m s , a p p a r a t u s f o r a u t o m a t i c measurement o f t h e a i r s p e e d of t h e
a i r c r a f t , t h e d r i f t a n g l e , and automatic c a l c u l a t i o n of t h e f l i g h t
path of t h e aircraft.
The c o n d i t i o n s of t h e t h i r d g r o u p a s s u m e a p r o l o n g e d a u t o n o m i c
a i r c r a f t n a v i g a t i o n w i t h no v i s i b i l i t y of t h e ground o r o v e r water,
w i t h c o r r e c t i o n of t h e a i r c r a f t c o o r d i n a t e s o n l y a t i n d i v i d u a l p o i n t s
located significant distances apart.
This places particularly strict
r e q u i r e m e n t s on t h e a c c u r a c y o f t h e p l o t t i n g o f t h e o r t h o d r o m e o n
t h e c h a r t s , t h e d e t e r m i n a t i o n of t h e f l i g h t p a t h a n g l e s , and t h e
r e t e n t i o n of systems f o r c a l c u l a t i n g t h e aircraft course, s i n c e
t h e c o u r s e i s a b a s i s f o r t h e a u t o m a t i c c a l c u l a t i o n of t h e f l i g h t
i n t e r m s of d i r e c t i o n .
From t h e t h e o r e t i c a l s t a n d p o i n t , a m o r e p r e c i s e a n d c o n v e n /170
i e n t form f o r using t h e course devices under c o n d i t i o n s of t h e t h i r d
group i s t h e following:

I n preparing t h e f l i g h t c h a r t s f o r each orthodrome s e c t i o n
t h e f l i g h t b e t w e e n t h e t u r n i n g p o i n t s on t h e r o u t e , r e g a r d l e s s
of t h e i r l e n g t h , w e d e t e r m i n e t h e c o n d i t i o n a l s h i f t i n t h e l o n g i t u d e
(A,),
i . e . , t h e d i f f e r e n c e between t h e l o n g i t u d e c a l c u l a t e d from
t h e p o i n t w h e r e t h e g i v e n o r t h o d r o m e i n t e r s e c t s t h e E q u a t o r (1,)
and t h e g e o g r a p h i c a l l o n g i t u d e ( A ) :
of

As

=

A0

-

A.

Here, t h e orthodromic longitude A0
i s d e t e r m i n e d for t h e s t a r t ­
1
i n g p o i n t o f e a c h s e g m e n t by t h e f o r m u l a
ctg A,, = tgy2 ctg+l cosec AA

- ctg Ah.

A f t e r d e t e r m i n i n g t h e change i n t h e l o n g i t u d e , t h e l o n g i t u d e
of any p o i n t a l o n g t h e r o u t e can b e c o n v e r t e d e a s i l y t o t h e o r t h o ­
d r o m i c s y s t e m , t h u s making it p o s s i b l e t o d e t e r m i n e r e l a t i v e l y e a s i l y
a l l o f t h e r e q u i r e d e l e m e n t s o f t h e o r t h o d r o m e for t h e s e p o i n t s :
(a)
The a z i m u t h o f t h e p o i n t o f i n t e r s e c t i o n o f t h e o r t h o d r o m e
with t h e Equator (ao)

(b)
The c o o r d i n a t e s o f i n t e r t e r m e d i a t e p o i n t s f o r p l o t t i n g
t h e o r t h o d r o m e o n t h e map:

172

(c)

The i n i t i a l ,

i n t e r m e d i a t e , and f i n a l azimuths of t h e o r t h o ­

drome
tg
tgq=-'.

sin

vi '

(d)
The d i s t a n c e t o a n y p o i n t a l o n g t h e o r t h o d r o m e (Si) f r o m
t h e p o i n t of i t s i n t e r s e c t i o n w i t h t h e Equator
cos

si

= c o s A0

i

cos

0i'­

(e)
The d i s t a n c e b e t w e e n a n y t w o p o i n t s a l o n g t h e o r t h o d r o m e
as a d i f f e r e n c e i n t h e d i s t a n c e f r o m t h e p o i n t o f i n t e r s e c t i o n w i t h
t h e Equator

Considering t h e necessity of precisely calculating t h e course
f o r a u t o m a t i c computation of t h e p a t h i n t e r m s of t h e d i r e c t i o n ,
and t h e d i f f i c u l t y of an e x a c t s e t t i n g of t h e c o u r s e i n f l i g h t rela­
t i v e t o t h e new r e f e r e n c e m e r i d i a n s , i t i s d e s i r a b l e f o r t h e c o n d i ­
t i o n s i n t h e t h i r d group t o r e t a i n a s i n g l e system f o r c a l c u l a t ­
i n g t h e c o u r s e s o v e r t h e e n t i r e l e n g t h of t h e f l i g h t f r o m t a k e o f f
t o landing.

/171
I n t h i s c a s e , t h e p a t h a n g l e of t h e f i r s t orthodromic f l i g h t
segment i s c o n s i d e r e d t o b e e q u a l t o t h e azimuth of t h i s segment
r e l a t i v e t o t h e m e r i d i a n of t h e a i r p o r t from which t h e a i r c r a f t
took o f f .
The p a t h a n g l e s o f a l l s u b s e q u e n t s e g m e n t s a r e o b t a i n e d
by c o m b i n i n g t h e o r t h o d r o m i c f l i g h t a n g l e ( O F A ) o f t h e p r e v i o u s
s e c t i o n with t h e t u r n angle (TA) of t h e l i n e of f l i g h t a t t h e turning
points along the route (Fig. 2.28):

T h e t u r n a n g l e s a l o n g t h e l i n e of f l i g h t a r e f o u n d a s t h e d i f ­
ferences of t h e azimuths of t h e orthodrome, i n t e r s e c t i n g a t t h e
t u r n i n g p o i n t s of t h e r o u t e , determined a c c o r d i n g t o t h e formula

O b v i o u s l y , t h e l a t i t u d e o f t h e t u r n i n g p o i n t s w i l l b e common
f o r t h e two o r t h o d r o m e s ; f o r one i t w i l l b e f ; n a l ,
for t h e o t h e r

1 73

it w i l l be i n i t i a l .
A s f a r as t h e l o n g i t u d e i s c o n c e r n e d , i t i s
d e t e r m i n e d on t h e b a s i s o f t h e g e o g r a p h i c a l l o n g i t u d e o f t h e t u r n i n g
p o i n t o f t h e r o u t e , c o n s i d e r i n g t h e s h i f t i n l o n g i t u d e of t h e p r e v ­
i o u s and subsequent segments.
When f l y i n g a b . o v e a c o n t i n e n t ,
t h e b e s t method of c o r r e c t i o n
f o r t h e orthodromic course under
conditions of t h e t h i r d group
is t o introduce corrections i n t o
t h e c o u r s e as a r e s u l t of calcu­
l a t i o n s of t h e a i r c r a f t p a t h .

Fig. 2.28.
System f o r Calcul a t i n g P a t h A n g l e s b y Combining t h e Turn Angles along
the Flight Path.

For e x a m p l e , i f t h e r e a d i n g s
o f t h e c a l c u l a t i n g d e v i c e s on
board t h e aircraft at both t h e
i n i t i a l and f i n a l p o i n t s i n d i c a t e
t h a t i t i s on t h e l i n e o f f l i g h t ,
b u t has undergone a l a t e r a l devia t i o n AZ during t h e f l i g h t , then

ob v i ous l y

tg AT =

AZ

-

S ’

w h e r e Ay e q u a l s t h e e r r o r i n t h e r e a d i n g s o f t h e o r t h o d r o m i c c o u r s e ,
a n d S i s t h e l e n g t h o f t h e c o n t r o l s e c t i o n of t h e f l i g h t .
The s i g n o f t h e c o r r e c t i o n t o t h e c o m p a s s r e a d i n g c o i n c i d e s
with t h e sign of A z .
With p o s i t i v e v a l u e s o f A Z , ( a s h i f t f r o m t h e l i n e o f t h e de­
s i r e d f l i g h t t o t h e r i g h t ) , t h e r e a d i n g s o f t h e compass w i l l b e
r e d u c e d and t h e c o r r e c t i o n must be p o s i t i v e ; i n t h e c a s e o f d e v i ­
a t i o n t o t h e l e f t , t h e c o r r e c t i o n m u s t b e made w i t h a m i n u s s i g n .
I n f l i g h t s o v e r w a t e r , when d e t e r m i n a t i o n o f t h e c o r r e c t n e s s
/172
o f t h e c a l c u l a t i o n o f t h e a i r c r a f t p a t h i n t e r m s of d i r e c t i o n i s
m o r e d i f f i c u l t , t h e c o r r e c t i o n o f t h e g y r o s c o p e c o u r s e m u s t b e made
by a s t r o n o m i c a l methods.
T h i s means t h a t t h e d i f f e r e n c e b e t w e e n
t h e orthodromic and t r u e c o u r s e s a t any p o i n t w i l l b e e q u a l t o t h e
d i f f e r e n c e between t h e orthodromic p a t h a n g l e of t h e segment and
t h e running azimuth of t h e orthodrome a t a given p o i n t :

If

t h e p o s i t i v e d i f f e r e n c e of t h e c o u r s e s t u r n s o u t t o be g r e a t e r

(or i f i t i s n e g a t i v e , t u r n s o u t t o b e s m a l l e r ) t h a n t h e d i f f e r ­
ence between t h e p a t h a n g l e s , t h e r e a d i n g f o r t h e orthodromic course
w i l l be i n c r e a s e d and it w i l l be n e c e s s a r y t o r e d u c e it manually
by t h e c o u r s e d e t e c t o r .
When t h e r e a d i n g s of t h e o r t h o d r o m i c c o u r s e
are low, it must b e i n c r e a s e d .

174

.-

..

.- .. -.....-_.

..

.,

II

I .

r

I n t h i s manner, b u t w i t h reduced accuracv,
course can be c o r r e c t e d magnetically:
OC

-

(MC+ M ) = OFA

-

t h e orthodromic

~ 1 .

For t h e c o n d i t i o n s o f t h e t h i r d g r o u p , t h e p r e p a r a t i o n o f t h e
magnetic compasses must b e c a r r i e d o u t a c c o r d i n g t o t h e r u l e s g i v e n
above f o r t h e c o n d i t i o n s of t h e second group.
However, t h e u s e
of m a g n e t i c t r a n s m i t t e r s f o r c o r r e c t i o n o f o r t h o d r o m i c c o u r s e d u r i n g
f l i g h t i s l i m i t e d t o cases when t h e r e a d i n g s of t h e o r t h o d r o m i c
c o u r s e c a n n o t b e c h e c k e d on t h e b a s i s of t h e r e s u l t s o f c a l c u l a ­
t i o n s of t h e p a t h or b y m e a n s of a s t r o n o m i c a l c o u r s e t r a n s m i t t e r s .
The m e t e o r o l o g i c a l c o n d i t i o n s o f a p l a n n e d f l i g h t , e s p e c i a l l y
o v e r l o n g d i s t a n c e s , c a l l for c a r e f u l p r e p a r a t i o n o f a l l c o u r s e
e q u i p m e n t on t h e p l a n e , s i n c e i t may b e c o m e n e c e s s a r y t o u s e d e v i c e s
f o r measuring t h e c o u r s e s which b e l o n g t o a l l t h r e e groups of condi­
tions.

3.

Barometric Altimeters

The p r i n c i p a l m e t h o d o f m e a s u r i n g f l i g h t a l t i t u d e f o r n a v i g a ­
t i o n a l purposes i s t h e b a r o m e t r i c method.
I t i s b a s e d o n t h e meas­
urement of t h e atmospheric p r e s s u r e a t t h e f l i g h t l e v e l of t h e a i r ­
craft

.

For s p e c i a l p u r p o s e s , s u c h a s a e r i a l p h o t o g r a p h y or a e r i a l
g e o d e s i c s t u d i e s , as w e l l as f o r s i g n a l i n g dangerous a p p r o a c h e s
t o t h e l o c a l r e l i e f when c o m i n g i n f o r a l a n d i n g u n d e r d i f f i c u l t
meteorological c o n d i t i o n s , e l e c t r o n i c devices f o r measuring a l t i ­
t u d e a r e u s e d , which a r e more a c c u r a t e i n p r i n c i p l e t h a n t h e b a r o ­
metric method.
However, t h e y a r e n o t w i d e l y employed f o r n a v i g a t i o n a l
purposes because they a r e used only f o r measuring t h e t r u e f l i g h t
altitude.
On t h e b a s i s o f t h e b a r o m e t r i c m e t h o d o f m e a s u r i n g a l t i ­
t u d e , it is t h e l a w of change of atmospheric p r e s s u r e with i n c r e a s e
i n h e i g h t which means t h a t t h e c a l i b r a t i o n o f t h e a l t i m e t e r d i a l
m u s t b e made o n t h e b a s i s o f t h e c o n d i t i o n s o f t h e i n t e r n a t i o n a l
/173
standard atmosphere.
The c o n d i t i o n s o f
(a)
kg/cm2.

t h e s t a n d a r d atmosphere are as f o l l o w s :

The p r e s s u r e a t s e a l e v e l i s e q u a l t o 7 6 0 mm Hg,

or 1 . 0 3 3 3

(b)
T h e a i r t e m p e r a t u r e a t s e a l e v e l i s +15O C w i t h a l i n ­
e a r d e c r e a s e for f l i g h t a l t i t u d e s u p t o 1 1 , 0 0 0 m o f 6.5O for e a c h
1000 m of a l t i t u d e .
Beginning a t 11,000 m y t h e a i r temperature
i s c o n s i d e r e d c o n s t a n t a n d e q u a l t o -56.5O.
To u n d e r s t a n d t h e o p e r a t i n g p r i n c i p l e of t h e b a r o m e t r i c a l ­
t i m e t e r , l e t us r e c a l l t h e f a m i l i a r e q u a t i o n s from p h y s i c s which

175

d e s c r i b e t h i s s t a t e of

g a s e s and t h e c o n d i t i o n s o f t h e i r change.

Thus , a c c o r d i n g t o t h e B o y l e - M a r i o t t e l a w , w i t h i s o t h e r m a l
c o m p r e s s i o n ( i . e . , f i x e d t e m p e r a t u r e ) , t h e p r e s s u r e o f a gas c h a n g e s
i n i n v e r s e p r o p o r t i o n t o i t s volume S O t h a t t h e p r o d u c t of t h e volume
t i m e s t h e pressure remains constant:

pv = c o n s t ,
w h e r e p i s t h e p r e s s u r e of
at temperature t .

t h e gas and

v i s t h e volume o f t h e g a s

A c c o r d i n g t o t h e G a y - L u s s a c l a w , h e a t i n g a g a s b y lo C a t c o n ­
s t a n t p r e s s u r e c a u s e s t h e g a s t o e x p a n d t o 1 / 2 7 3 . 1 o f t h e volume
which it occupied a t z e r o t e m p e r a t u r e :
v-vo=­

VO

273.1 t s

w h e r e 0 0 i s t h e v o l u m e a t z e r o t e m p e r a t u r e a n d t h e same p r e s s u r e .
By c o m b i n i n g t h e B o y l e - M a r i o t t e
t h e state equation of a gas:

and Gay-Lussac

laws

,

we obtain

- + 273.11.

p v = POVO ( t
273. I

T h i s e q u a t i o n i s known a s t h e C l a p e y r o n e q u a t i o n .
The t e m p e r ­
a t u r e ( t t 2 7 3 . l 0 C > i s c a l l e d t h e absolute temperature ( T I , i . e . ,
c a l c u l a t e d r e l a t i v e t o a b s o l u t e z e r o ( - 2 7 3 . 1 O C)l, a n d t h e c o n s t a n t
v a l u e of-

P V

273.1

i s c a l l e d t h e gas c o n s t a n t .

A gram m o l e c u l e o f a n y g a s ( g r a m mole , o r s i m p l y m o l e ) , i . e . ,
t h e number o f grams o f a g a s which i s e q u a l t o i t s m o l e c u l a r w e i g h t ,
a l w a y s o c c u p i e s e x a c t l y t h e same v o l u m e ( 2 2 . 4 1 l i t e r s ) a t z e r o t e m p e r ­
a t u r e and a p r e s s u r e of 1 a t m .
T h e g a s c o n s t a n t for o n e m o l e o f g a s i s c a l l e d t h e u n i v e r s a l
g a s c o n s t a n t (R):

With P

= 1atm, v

= 22.41 l i t e r s .

T h e C l a p e y r o n e q u a t i o n for o n e m o l e o f
sumes t h e f o r m
~~

g a s i n t h i s case as­

-

~

1 T h i s v a l u e i s u s u a l l y a s s u m e d t o b a a p p r o x i m a t e l y 273O i n c a l ­
culations.
176

L7'!

pv
The n u m e r i c a l v a l u e of

= RT.

t h e universal gas constant is

R = 1*033*n410=84,8k g / c m ( d e g r e e s / m o l e ) .
273.1

I n t e c h n i c a l c a l c u l a t i o n s , t h e w e i g h t of t h e g a s i s u s u a l l y
T h e r e f o r e , w e do n o t u s e t h e u n i v e r s a l
expressed i n kilograms.
gas constant b u t r a t h e r t h e c h a r a c t e r i s t i c gas constant

where M i s t h e number o f grams o f g a s p e r m o l e ,
weight.

or i t s m o l e c u l a r

Then

pt,

=

BT.

The c o n s t a n t B f o r a i r i s 2 9 . 2 7 m / d e g r e e .
By u s i n g t h e g a s c o n s t a n t , we c a n f i n d t h e w e i g h t d e n s i t y o f
a i r ( y ) a t a given p r e s s u r e p and absolute temperature T .

L e t u s d e f i n e a n a r e a on t h e E a r t h ' s s u r f a c e m e a s u r i n g 1 c m 2 ,
a n d e r e c t a v e r t i c a l c o l u m n on i t w h i c h e x t e n d s u p w a r d t o t h e l i m i t s
of t h e E a r t h ' s atmosphere (Fig. 2 . 2 9 ) .

Obviously, t h e drop i n pressure with increased a l t i t u d e t o
t h e d i s t a n c e AH a t a c e r t a i n h e i g h t w i l l b e e q u a l t o :
Ap = rAH=

BT

AH

or


-

Ap

=

AH
-4

BT

(2.27)

By u s i n g E q u a t i o n ( 2 . 2 7 ) a n d t h e a l t i t u d e t e m p e r a t u r e g r a d i e n t , / l 7 5
w e obtain t h e so-called barometric formula

(2.28)
w h e r e T O i s t h e t e m p e r a t u r e on t h e g r o u n d u n d e r s t a n d a r d c o n d i t i o n s
e q u a l t o 281O K , a n d t g r i s t h e v e r t i c a l t e m p e r a t u r e g r a d i e n t .

1 77

I1

I

I1

I


Formula ( 2 . 2 8 ) i s o b t a i n e d from Formula ( 2 . 2 7 ) ,
i n f i n i t e l y s m a l l values:

switching t o

2 =d­
H

p

Integrating (2.27a)

BT'

(2.27a)

a n d k e e p i ng i n mind t h a t

TH = T O - g r H ,

we obtain:
PH

H

PO

0

or

Fig.

2.29.

Column o f A i r o n t h e
Earth's Surface.

Solving Equation ( 2 . 2 8 ) f o r H , w e o b t a i n t h e s t a n d a r d hypso­
metric formula f o r t h e troposphere:
(2.29)

S u b s t i t u t i n g i n t o Formula ( 2 . 2 9 )
obtain:

t h e n u m e r i c a l v a l u e s of TO,

t g ra n d B , w e

(2.30)

We c a n u s e F o r m u l a ( 2 . 3 0 ) t o c a l c u l a t e t h e h y p s o m e t r i c t a b l e s
w h i c h r e l a t e t h e f l i g h t a l t i t u d e up t o 1 1 , 0 0 0 m t o t h e a t m o s p h e r i c
p r e s s u r e ; t h e s e t a b l e s a r e used t o a d j u s t and c o r r e c t altimeters.

Under t h e c o n d i t i o n s o f a s t a n d a r d a t m o s p h e r e , t h e a i r t e m ­
perature at a l t i t u d e s g r e a t e r than 11,000 m i s considered t o be
c o n s t a n t , s o t h a t t h e b a r o m e t r i c f o r m u l a for t h e s e a l t i t u d e s c a n
b e w r i t t e n as f o l l o w s :
(2.31)
We o b t a i n F o r m u l a ( 2 . 3 1 ) b y i n t e g r a t i n g E q u a t i o n
1 1 , 0 0 0 m a n d c o n s i d e r T H e q u a l t o Til:

178

(2.27) f o r

/176

or

Solving Equation (2.31) f o r H, w e o b t a i n t h e s t a n d a r d s t a t e
f o r m u l a o f t h e h y p s o m e t r i c t a b l e ( T a b l e 2 . 5 ) �or a l t i t u d e s g r e a t e r
t h a n 11,000 m.
H = 11 OOO

H,

M



-500
0
500
1000
I 500
2000
2500
3000
3500
4000
4500
5000
5500
6000
6 500
7000
7500
8000
8500
9000
9500
I0000

+ BTI1In PI1
PH
-a

(2.32)

TABLE 2 . 5 .
~

PHI

PHI

uN Hg

806.
Z

760.2

716.0
674.j
034.2
596.2
560.1
525.8

493.2

462.2
432.9
405,l
378.7
353.8
330.2
307.8
286.8
266.9
248.1
230.5

213.8

198.2


Hg
183.40
169,60
144.87
123.72
105.67
90.24
77.07
65.82
56.21
48,Ol
41.00
35.02
29.90
25.54
21.81
18.63
15.91
13,59
11.60
9.91
8.46

219,25

216.50

216.50

216.50

216.50

216.50

216.50

216.50

216.50

216.50

216.50

216.50

216.50

216.50

216.50

216.50

216.50

216.50

216.50

216.50

216,50


uM

~~

291.25
228.00
284.75
281.50
278.25
275.00
271.75
268.50
265.25
262.00
258.75
255.50
252.25
249.00
245.75
242.50
239.25
236.00
232.75
229.50
226.25
223.00

342.1
340.2
338.3
336.4
334.4
332,5
330.5
328.5
326.5
324.5
322.5
320.5
318.4
316,3
314.3
312.2
310.1
308.0
305.9
303.7
301.6
299.4


10500
11 OOO
12000
13000
14000
15000
16000
17000
18000
19000
20000
21 000
22000
23000
24000
25000
26000
27000
28000

29000

30000


~~

TH,"K

a,


m_
/sec ­
297.2

295.0

295.0

295.0

295.0

295.0

295,0

295.0

295.0

295,0

295.0

295.0

295.0

295.0

295.0

295.0

295.0

295.0

295.0

295.0

295.0


Note:
T h e t a b l e for a d j u s t i n g a n d c o r r e c t i n g t h e b a r o m e t r i c
a l t i m e t e r s is given i n abbreviated form.
The v a l u e a r e p r e s e n t s
t h e speed of sound a t f l i g h t a l t i t u d e under s t a n d a r d c o n d i t i o n s ,
g i v e n i n t h e f o u r t h column o f t h e t a b l e .
= 216.5O, a n d s h i f t i n g
S u b s t i t u t i n g t h e value of B and T
t o t h e l o g t e n ( I n N = 2.30259 I g N ) , t h i s f o r m u l a assumes t h e form:

179

I.'

F o r m u l a s ( 2 . 3 0 ) a n d ( 2 . 3 3 1 , s u i t a b l e for c o m p i l i n g h y p s o m e t - /177
ri c t a b l e s and c a l i b r a t i n g a l t i m e t e r s , are n o t completely s u i t a b l e
f or c a l c u l a t i n g t h e m e t h o d o l o g i c a l e r r o r s i n t h e a l t i m e t e r , r e l a t e d
t o a f a i l u r e o f t h e a c t u a l a i r t e m p e r a t u r e a t h e i g h t s from z e r o
t o t h e f l i g h t a l t i t u d e of t h e a i r c r a f t t o a g r e e w i t h t h e c o n d i t i o n s
of t h e standard atmosphere.
S i n c e t h e accur,acy o f a l t i t u d e measurement i s a f f e c t e d by t h e
air temperature not only at the f l i g h t a l t i t u d e but a t a l l inter­
m e d i a t e l a y e r s f r o m t h e o n e on t h e g r o u n d up t o t h a t a t t h e f l i g h t
a l t i t u d e , it i s b e t t e r t o u s e t h e formula which r e l a t e s t h e f l i g h t
a l t i t u d e n o t t o t h e t e m p e r a t u r e g r a d i e n t , b u t t o t h e average temper­
a t u r e o f t h e column o f a i r which w e h a v e s e l e c t e d , and t o u s e t h i s
t o c a l c u l a t e a hypsometric t a b l e f o r a d j u s t i n g and c o r r e c t i n g baro­
metric altimeters.
This formula has t h e form:
H-BT

in-Po

(2.34)

av PH

Formula ( 2 . 3 4 ) i s o b t a i n e d by i n t e g r a t i n g E q u a t i o n ( 2 . 2 7 a )
a t a constant average temperature:

whence

If w e c o n s i d e r t h a t

GV=273 + LV=273 1 + - ,

(

z,)

and t h e v a l u e B = 29.27, by u s i n g t h e c o e f f i c i e n t f o r t r a n s i t i o n
from n a t u r a l l o g a r i t h m s t o t h e l o g 1 0 , Formula ( 2 . 3 4 ) assumes t h e
form :

T h i s f o r m u l a i s known a s t h e L a p l a c e f o r m u l a .
D e s c r i p t i o n of a B a r o m e t r i c A l t i m e t e r

T h e s e n s i t i v e e l e m e n t i n t h e b a r o m e t r i c a l t i m e t e r i s a cor­
r u g a t e d m a n o m e t r i c ( a n e r o i d ) b o x 1 ( F i g . 2 . 3 0 1 , made o f b r a s s .
The
box h a s two r i g i d p o i n t s ( o n t h e t o p a n d b o t t o m c o r r u g a t e d s u r f a c e s ) ,
o n e o f w h i c h i s f i x e d or t i g h t l y f a s t e n e d t o t h e c a s i n g o f t h e a p p a r ­
a t u s , w h i l e t h e o t h e r i s movable.

180


-_

--I1111

I I 1111I 1 1 1 1 I1111II I I 1.11111111

I I

111111.1111 I

.

In principle,
with a gas.

t h e a n e r o i d b o x c a n b e e i t h e r e v a c u a t e d or f i l l e d / 1 7 8

U s u a l l y , t h e s p a c e w i t h i n t h e box i s f i l l e d w i t h a gas t o a
p r e s s u r e s u c h t h a t when t h e b o x i s h e a t e d , t h e t h e r m a l l o s s e s o f
i t s e l a s t i c p r o p e r t i e s w i l l b e r o u g h l y compensated by a n i n c r e a s e
i n g a s p r e s s u r e w i t h i n t h e b o x when i t i s h e a t e d .
The c a s i n g o f t h e a l t i m e t e r i s h e r m e t i c a l l y s e a l e d a n d c o n ­
n e c t e d by a n i p p l e t o a s e n s o r o f t h e a t m o s p h e r i c ( s t a t i c ) p r e s ­
sure.

Fig.,.

2.30.

S c h e m a t i c Diagram of
Altimeter.

Barometric

When t h e a i r c r a f t i s l o c a t e d a t s e a l e v e l , t h e a n e r o i d b o x
i s c o m p r e s s e d t o t h e maximum d e g r e e , s i n c e t h e a t m o s p h e r i c p r e s ­
s u r e a c t i n g on i t h a s a maximum v a l u e .
W i t h a g a i n i n a l t i t u d e , t h e a t m o s p h e r i c p r e s s u r e i n t h e cham­
b e r d e c r e a s e s and t h e a n e r o i d box expands due t o i t s e l a s t i c prop­
e r t i e s , s h i f t i n g i t s m o v a b l e c e n t e r ( w i t h b i m e t a l l i c s h a f t 2 ) up­
ward.
A s it moves, t h e c e n t e r d i s p l a c e s r o d 3 , which i n t u r n a c t s
through a l e v e r 4 t o convey a r o t a r y motion t o s h a f t 5 .

Shaft 5 carries a toothed s e c t o r 6 with a counterweight, f i t t e d
w i t h a c o g w h e e l 7 , w h i c h t r a n s m i t s t h e movement t o t h e p o i n t e r
through another gear.
Thus, t h e motion of t h e c e n t e r of t h e box i s used t o i n d i c a t e
t h e f l i g h t a l t i t u d e on t h e s c a l e o f t h e i n s t r u m e n t .
I n a d d i t i o n t o t h e p a r t s l i s t e d above, t h e k i n e m a t i c p o r t i o n
of t h e i n s t r u m e n t i n c l u d e s elements i n t e n d e d f o r r e g u l a t i n g t h e

181


i n s t r u m e n t a n d a d j u s t i n g t h e b a c k l a s h i n t h e t r a n s m i s s i o n mechan­

ism.

1.
Zero-point b i m e t a l l i c compensator,
This device is intended/l79
f o r compensating t h e temperature changes i n t h e elastic p r o p e r t i e s
If t h e a t m o s p h e r i c p r e s s u r e i n t h e
of t h e box f o r z e r o a l t i t u d e .
c a s i n g of t h e instrument i s s e t t o z e r o a l t i t u d e , b u t t h e temper­
a t u r e o f t h e b o x i n c r e a s e s , t h e loss o f e l a s t i c p r o p e r t i e s o f t h e
material i n t h e box creates a d d i t i o n a l compression, causing t h e
i n d i c a t o r n e e d l e t o s h i f t from t h e z e r o a l t i t u d e r e a d i n g .
The b i ­
m e t a l l i c s t r i p b e n d s as t h e t e m p e r a t u r e c h a n g e s , due t o d i f f e r e n t
c o e f f i c i e n t s o f l i n e a r e x p a n s i o n f o r t h e two m a t e r i a l s o f which
i t i s made.
By r o t a t i n g t h e s t r i p i n i t s s o c k e t , i t i s p o s s i b l e
t o s e t it i n a p o s i t i o n such t h a t t h e d e f l e c t i o n i n t h e d i r e c t i o n
of t h e s h i f t of t h e c e n t e r of t h e box w i l l e x a c t l y correspond t o
the additional t r a v e l of t h i s center, but i n t h e opposite direc­
tion.
Then r o d 3 r e m a i n s i n p l a c e a n d t h e i n d i c a t o r n e e d l e w i l l
n o t move f r o m t h e z e r o p o s i t i o n .
2.
T h e r e g u l a t i n g mechanism o f
of s t r i p 4 and a n adjustment screw.

t h e device.

This consists

T u r n i n g t h e s c r e w p u s h e s t h e s t r i p away f r o m r o d 5 , c h a n g i n g
This i s used t o r e g u l a t e t h e angular veloc­
t h e a r m of t h e l e v e r .
i t y of r o t a t i o n of t h e s h a f t , i . e . , t h e transmission r a t i o of t h e
apparatus.
The t r a n s m i s s i o n r a t i o o f t h e r o t a t i o n o f t h e s h a f t
i s s e t s o t h a t t h e r e a d i n g s of t h e n e e d l e c o r r e s p o n d t o t h e atmo­
s p h e r i c p r e s s u r e i n t h e casing of t h e a p p a r a t u . ~ .

3 . H i g h t e m p e r a t u r e c o m p e n s a t o r . When t h e e l a s t i c p r o p e r t i e s
o f t h e box change due t o t h e e f f e c t o f t e m p e r a t u r e , t h i s n o t o n l y
causes an a d d i t i o n a l compression a t z e r o a l t i t u d e b u t a l s o changes
t h e amount b y w h i c h i t s c e n t e r moves w i t h a c h a n g e i n a l t i t u d e .
For c o m p e n s a t i o n o f t h i s e r r o r , s t r i p 4 i s o f b i m e t a l l i c c o n s t r u c ­
tion.
When t h e i n s t r u m e n t is h e a t e d , a n d t h e t r a v e l o f t h e c e n t e r
o f t h e b o x i n c r e a s e s , t h e e n d of t h e s t r i p b e n d s away f r o m t h e s h a f t ,
t h u s r e d u c i n g t h e t r a n s m i s s i o n r a t i o f o r r o t a t i n g s h a f t 5 a n d compen­
s a t i n g f o r t h e i n c r e a s e i n s e n s i t i v i t y of t h e box.
I t is important not t o confuse t h e instrumental temperature
e r r o r s o f t h e i n s t r u m e n t , which a r e compensated by t h e z e r o a n d
a l t i t u d e b i m e t a l l i c compensators, with t h e methodological temper­
ature errors i n the altimeter.
The i n s t r u m e n t a l e r r o r s a r e r e l a t e d t o t h e t e m p e r a t u r e i n t h e
c a s i n g o f t h e i n s t r u m e n t , a c t i n g on t h e p r o p e r t i e s o f t h e mater­
i a l from which t h e s e n s i t i v e e l e m e n t i s made, a n d c a n b e overcome
by c o m p e n s a t o r s

.

The m e t h o d o l o g i c a l e r r o r s w h i c h a r e r e l a t e d t o t h e n a t u r e o f
t h e changes i n p r e s s u r e with f l i g h t a l t i t u d e can only b e c o r r e c t e d
by s p e c i a l f o r m u l a s .
The b u i l d i n g o f a c o m p e n s a t o r f o r methodo­

182


l o g i c a l e r r o r s i s i m p o s s i b l e , s i n c e i n t h e g e n e r a l case t h e temper­
a t u r e of t h e casing is not equal t o t h e average a i r temperature
f r o m z e r o a l t i t u d e up t o t h e f l i g h t a l t i t u d e o f t h e a i r c r a f t .
I n o r d e r t o i n c r e a s e t h e a c c u r a c y of t h e a l t i t u d e r e a d i n g s ,
/180
a l t i m e t e r s a r e made w i t h t w o p o i n t e r s .
T h i s means t h a t t h e a n e r o i d
b o x i s made d o u b l e , i n c r e a s i n g t h e t r a v e l o f t h e m o v a b l e c e n t e r
by a f a c t o r o f two.
Between t h e t o o t h e d s e c t o r and t h e a x i s o f
t h e p o i n t e r , t h e r e are a d d i t i o n a l g e a r s which i n c r e a s e t h e t r a n s ­
m i s s i o n r a t i o o f t h e mechanism s e v e r a l t i m e s .
The main p o i n t e r
o f t h e i n s t r u m e n t makes s e v e r a l r e v o l u t i o n s ; t h e number o f r e v o l u ­
t i o n s of t h e p o i n t e r i s e q u a l t o t h e change i n a l t i t u d e i n thou­
sands of meters.
I n addition, t h e r e is a pressure scale 8 f o r s e t t i n g t h e al­
timeter readings re l a t i v e t o a desired level.
The a l t i m e t e r mechanism, a l o n g w i t h t h e a x i s o f t h e main p o i n t e r ,
i s r o t a t e d w i t h i n t h e h o u s i n g by means o f a r a c k a n d p i n i o n 9 , c o n s i s t ­
i n g o f a d r i v i n g g e a r 1 0 a n d a d r i v e n g e a r 11. T h u s , t h e m a i n p o i n t e r
o f t h e i n s t r u m e n t c a n b e s e t t o any d i v i s i o n on t h e s c a l e .
S i m u l t a n e o u s l y , by means o f d r i v i n g g e a r 1 0 , t h e p r e s s u r e s c a l e
which can b e u s e d i n c o n j u n c t i o n w i t h t h e main
s c a l e t o d e t e r m i n e t h e p r e s s u r e a t t h e l e v e l a t which t h e f l i g h t
altitude is calculated.
8 is s e t i n motion,

I n t h e VD-10 a n d VD-20 a l t i m e t e r s , a m o v a b l e r i n g i s m o u n t e d
a r o u n d t h e main s c a l e ; i t i s r o t a t e d by means of a r a c k and p i n i o n
and d r i v i n g g e a r 10 a t an angular v e l o c i t y e q u a l t o t h e r a t e of
t u r n o f t h e mechanism.
I t i s u s e d f o r s h i f t i n g a movable i n d e x
a l o n g t h e c i r c u l a r s c a l e of t h e i n s t r u m e n t , and can b e s e t t o t h e
b a r o m e t r i c a l t i t u d e o f t h e a i r p o r t where t h e l a n d i n g i s t o b e made.
T h i s s e r v e s t h e same p u r p o s e as t h e p r e s s u r e s c a l e .
However, t h e
l a t t e r can o n l y be used o v e r a r a n g e o f p r e s s u r e s from 6 7 0 t o 790
m m Hg, w h i l e t h e m o v a b l e i n d e x c a n b e s e t t o a n y a i r p o r t a l t i t u d e .
I n c a s e s when p r e s s u r e s c a l e s a r e n o t s u f f i c i e n t f o r a i r p o r t s
located a t high a l t i t u d e s , t h e pressure a t t h e l e v e l of t h e air­
p o r t is n o t measured aboard t h e aircraft, b u t r a t h e r t h e baromet­
r i c a l t i t u d e o f t h e a i r c r a f t i s u s e d f o r s e t t i n g t h e movable i n d e x
of t h e a l t i m e t e r .
Errors

i n Measuring A l t i t u d e w i t h a Barometric Altimeter

The e r r o r s i n m e a s u r i n g t h e f l i g h t a l t i t u d e w i t h b a r o m e t r i c
altimeters can be divided i n t o instrumental and methodological e r r o r s :
These are r e l a t e d t o i n c o r r e c t adjustment
. I r ~ s t r u m e n t a le r r o r s .
o f t h e a l t i m e t e r , f r i c t i o n ( w e a r ) i n t h e t r a n s m i s s i o n mechanism,
as w e l l as t e m p e r a t u r e e f f e c t s on t h e m a t e r i a l o f t h e s e n s i t i v e
element.
The e r r o r s f r o m s o - c a l l e d h y s t e r e s i s a r e p a r t i c u l a r l y

183


I

i m p o r t a n t , i . e . , t h e r e s i d u a l deformation o f t h e s e n s i t i v e box w i t h
changes i n f l i g h t a l t i t u d e of t h e aircraft over wide l i m i t s .
In addition, instrumental errors include errors i n sensing
t h e s t a t i c p r e s s u r e , r e l a t e d t o dynamic f l i g h t of t h e a i r c r a f t .

/181

Methodological e r r o r s .
I n t h e b a r o m e t r i c method o f measur­
i n g a l t i t u d e , t h e s e include e r r o r s i n correspondence of t h e i n i t i a l
a t m o s p h e r i c p r e s s u r e , t h e p r e s s u r e a l o n g t h e f l i g h t r o u t e , and t h e
average air temperature with t h e calculated data.
Under f l i g h t c o n d i t i o n s e n c o u n t e r e d i n c i v i l a i r c r a f t , method­
o l o g i c a l e r r o r s i n measuring a l t i t u d e i n approaching a i r c r a f t are
e x t r e m e l y r a r e , s o t h a t t h e s e e r r o r s do n o t d i s t u r b t h e m u t u a l p o s i ­
t i o n of t h e a i r c r a f t a n d a r e n o t t a k e n i n t o a c c o u n t .
However, t h e y
do have s i g n i f i c a n t value i n determining t h e safe f l i g h t a l t i t u d e
a b o v e t h e r e l i e f , as w e l l as i n making s p e c i a l f l i g h t s ( f o r p u r ­
poses of aerial photography, e.g.1.
I n p r a c t i c e , t h e b a r i c s t a g e a t low f l i g h t a l t i t u d e s ( t h e d i f ­
f e r e n c e i n a l t i t u d e which corresponds t o a drop i n p r e s s u r e of 1
m m Hg) i s c o n s i d e r e d r o u g h l y e q u a l t o 11 m .
However, a t f l i g h t
a l t i t u d e s of 20,000 m y t h e b a r i c s t a g e i s e q u a l t o 1 5 5 m , i . e . ,
1 4 t i m e s g r e a t e r t h a n on t h e g r o u n d .
T h e i n c r e a s e i n t h e b a r i c s t a g e w i t h f l i g h t a l t i t u d e , as w e l l
as t h e e r r o r s i n m e a s u r i n g s t a t i c p r e s s u r e due t o a e r o d y n a m i c p r o c ­
esses, c o m p l i c a t e a p r e c i s e measurement of t h e b a r o m e t r i c a l t i t u d e
a t great a l t i t u d e s and high speeds.
In a f l i g h t according t o a t a b l e of c o r r e c t i o n s , it i s rela­
t i v e l y easy t o compensate f o r i n s t r u m e n t a l e r r o r s i n t h e a p p a r a t u s ,
r e l a t e d only t o i t s r e g u l a t i o n .
Consideration of a l l o t h e r i n s t r u ­
mental e r r o r s p r e s e n t s g r e a t e r d i f f i c u l t y , s o t h a t a l l measures
a r e u s u a l l y t a k e n t o r e d u c e t h e m t o a minimum b y c a r e f u l l y p r e p a r ­
i n g t h e a p p a r a t u s , s e l e c t i n g t h e p o i n t o f c a l i b r a t i o n , and d e s i g n ­
ing the static pressure sensor.
M e t h o d o l o g i c a l e r r o r s i n a l t i m e t e r s a r e e s t i m a t e d by d e t e r ­
mining t h e t r u e a l t i t u d e of t h e a i r c r a f t above t h e r e l i e f f o r s p e c i a l
purposes and i n c a l c u l a t i n g safe f l i g h t a l t i t u d e s above t h e r e l i e f .
Changes i n a t m o s p h e r i c p r e s s u r e a l o n g t h e f l i g h t r o u t e , r e l a t i v e
t o sea l e v e l , are c a l c u l a t e d i n b a r i c s t a g e s , s o t h a t t h e lowest
f l i g h t a l t i t u d e o s c i l l a t e s as f o l l o w s :

For e x a m p l e , i f t h e p r e s s u r e m e a s u r e d a t s e a l e v e l a t t h e p o i n t
w h e r e t h e a l t i t u d e i s m e a s u r e d d i f f e r s f r o m 7 6 0 mm Hg t o 1 5 m m , t h e
m e t h o d o l o g i c a l e r r o r i n measuring t h e a l t i t u d e from t h e l e v e l of
7 6 0 mm w i l l b e 1 5 - 1 1 = 1 6 5 m .

184


H e n c e , i f t h e c o r r e c t e d p r e s s u r e i s g r e a t e r t h a n 7 6 0 mm H g ,
i . e . , e q u a l t o 7 7 5 mm i n our e x a m p l e , t h e r e a d i n g s o f t h e a l t i m e t e r w i l l b e reduced and t h e c o r r e c t i o n w i l l have t o c a r r y a p l u s
s i g n , while i f t h e corrected pressure is lower than t h e c a l c u l a t e d
p r e s s u r e , it w i l l have a minus s i g n .

/182

Methodological e r r o r s i n t h e a l t i m e t e r , which a r i s e due t o
a f a i l u r e o f t h e a c t u a l mean a i r t e m p e r a t u r e t o c o i n c i d e w i t h t h e
c a l c u l a t e d t e m p e r a t u r e , a r e a c c o u n t e d f o r by means o f a n a v i g a t i o n a l
s l i d e r u l e , a d e s c r i p t i o n of which i s g i v e n below.
Proceeding from
t h e f a c t t h a t t h e i n s t r u m e n t i n d i c a t e s a f l i g h t a l t i t u d e on t h e
b a s i s o f t h e c a l c u l a t e d mean t e m p e r a t u p e o f t h e a i r , a n d t h e c o r r e c t e d
a l t i t u d e m u s t b e d e t e r m i n e d on t h e b a s i s of t h e a c t u a l a l t i t u d e ,
t h e e q u a t i o n r e a d s as f o l l o w s :
Hinst

Hcorr

= BT

= BT

P

1n-j-Q ;

av.c.

av.a.

Po
p

I n -,

where T a v a c i s t h e a v e r a g e c a l c u l a t e d t e m p e r a t u r e and TaVsa. i s
t h e average a c t u a l temperature.
Whence

H

Theref o r e

c o rr

- -H i n s t
T

1’

.

h v .

av. c.

a.

,

(2.35)

w h e r e To a n d TH a r e t h e t e m p e r a t u r e s on t h e g r o u n d a n d a t f l i g h t
altitude , respectively.
By u s i n g F o r m u l a ( 2 . 3 5 ) , w e c a n c a l c u l a t e t h e s c a l e s o f t h e
n a v i g a t i o n a l s l i d e r u l e NL-10 f o r making c o r r e c t i o n s i n t h e r e a d i n g s
of a l t i m e t e r s f o r a i r t e m p e r a t u r e u p t o a l t i t u d e s o f 1 2 , 0 0 0 m .

For a l t i t u d e s a b o v e 1 2 , 0 0 0 m y t h e c o r r e c t e d a l t i t u d e i s f o u n d
by t h e f o r m u l a
m

H

corr
w h e r e TH,

a n d TH

C

- 11,000 =

a
‘Hinst
TH

- ll,OOO>,

C

are t h e a c t u a l and c a l c u l a t e d temperatures a t

the altitude.
185

The n a v i g a t i o n a l s l i d e r u l e f o r t h e s e a l t i t u d e s i s a l s o p r o ­
v i d e d w i t h l o g a r i t h m i c scales a c c o r d i n g t o t h e f o r m u l a

(

co r r

-11,000)

+

= lgTH

lg

Hinst

a

4.

-11,000
(2.36)

216.5O

Airspeed I n d i c a t o r s

/183

T h e f l i g h t o f a n a i r c r a f t t a k e s p l a c e i n t h e medium o f a i r ,
s o t h a t a s i m p l e s t a n d e a s i e s t method f r o m t h e t e c h n i c a l s t a n d p o i n t
f o r measuring a i r s p e e d would b e t o measure t h e aerodynamic p r e s ­
s u r e or s o - c a l l e d v e l o c i t y h e a d o f t h e i n c i d e n t a i r f l o w .

For p u r p o s e s o f a i r c r a f t n a v i g a t i o n ,
t h e s p e e d of t h e a i r c r a f t r e l a t i v e t o t h e
s i n c e t h e a i r m a s s p r a c t i c a l l y always has
tive t o the latter.
A t the present t i m e ,
t i a l methods of m e a s u r i n g t h e s p e e d r e l a t i
t h e measurement of a i r s p e e d does n o t l o s e
i n t h e presence of such equipment.

i t i s b e t t e r t o measure
s u r f a c e of t h e ground,
i t s own m o v e m e n t r e l a ­
t h e r e a r e r a d i a l a n d i n e r­
v e t o t h e ground, but
i t s s i g n i f i c a n c e even

The f a c t i s t h a t t h e s t a b i l i t y a n d m a n e u v e r a b i l i t y o f a n a i r ­
c r a f t depen.ds on t h e a i r s p e e d .
In a d d i t i o n , t h e o p e r a t i o n a l regime
o f t h e m o t o r s on t h e a i r c r a f t a n d t h e f u e l c o n s u m p t i o n d e p e n d on
the airspeed.
The o p e r a t i n g p r i n c i p l e of a i r s p e e d i n d i c a t o r s i s b a s e d on
a m e a s u r e m e n t of t h e a e r o d y n a m i c p r e s s u r e of t h e i n c i d e n t a i r f l o w .
The r e l a t i o n s h i p b e t w e e n t h e r a t e o f m o t i o n o f a l i q u i d or
g a s and i t s dynamic a n d s t a t i c p r e s s u r e w a s f i r s t e s t a b l i s h e d by
t h e S t . P e t e r s b u r g A c a d e m i c i a n D a n i e l B e r n o u l l i (1738), w o r k i n g
w i t h i n c o m p r e s s i b l e l i q u i d s or g a s e s ( F i g . 2 . 3 1 ) .
According t o t h e p r i n c i p l e of i n s e p a r a b i l i t y of flow, t h e prod­
u c t o f t h e s p e e d of a n a i r c u r r e n t ( V ) m u l t i p l i e d b y t h e c r a s s s e c ­
t i o n a l area of a t u b e ( S ) must b e uniform everywhere w i t h i n i t s
cross section.
Consequently, i n a narrow p a r t of t h e t u b e , t h e
speed of t h e flow must b e g r e a t e r t h a n i n a wide s e c t i o n .
I n t h e g e n e r a l c a s e , i f t h e t u b e i s n o t h o r i z o n t a l , a mass
o f g a s m e n t e r s t h e t u b e d u r i n g a t i m e At w h i c h i n t r o d u c e s a n e n e r g y
c o n s i s t i n g of t h r e e components:
t h e p o t e n t i a l energy of t h e gas

t h e k i n e t i c energy

--.2V:
ni



a n d t h e work o f i n f l u x i n t o t h e t u b e

18 6

I

I 1I

1

where g i s t h e a c c e l e r a t i o n due t o t h e E a r t h ' s g r a v i t y , h i s t h e
d i f f e r e n c e i n t h e g a s l e v e l s , a n d p i s t h e gas p r e s s u r e i n s i d e t h e
tube.
These components d e t e r m i n e t h e e n e r g y o f t h e g a s f l o w i n g o u t
of t h e tube.
Therefore

The p r o d u c t S V A t i s t h e volume of f l u i d f l o w i n g t h r o u g h t h e
cross section of t h e tube i n a t i m e A t .
Therefore, dividing t h e
mass i n t o t h e v o l u m e g i v e s u s t h e d e n s i t y ( p ) , w h i c h i s

Fig.

tal,

2.31.

Flow i n a Tube w i t h V a r y i n g Cross
Section.

If t h e t u b e t h r o u g h which t h e c u r r e n t i s f l o w i n g i s h o r i z o n ­
hl = h2, therefore

(2.37)

i . e . , t h e sum o f t h e d y n a m i c a n d s t a t i c p r e s s u r e s a t a n y p o i n t i n
t h e t u b e r e m a i n s c o n s t a n t , s i n c e t h e dynamic component i s p r o p o r ­
t i o n a l t o t h e gas d e n s i t y ( f l u i d d e n s i t y ) a n d t h e s q u a r e o f t h e
speed of flow.

For a d i a b a t i c c o m p r e s s i o n , i . e . , w h e n t h e p r o c e s s t a k e s p l a c e
with compression of t h e gas ( a i r ) without exchange of h e a t energy
w i t h t h e s u r r o u n d i n g medium, which a l m o s t a l w a y s c a n b e c o n s i d e r e d
v a l i d f o r h i g h s p e e d e v e n t s , t h i s e q u a t i o n t a k e s t h e form:
(2.38)

w h e r e y i s t h e u n i t w e i g h t ( w e i g h t d e n s i t y ) of t h e g a s , V i s t h e
i n t e r n a l (thermal) energy of t h e g a s , and E is t h e p o t e n t i a l energy
of t h e g a s .

187


/184

T h e r e f o r e , a change i n t h e r a t e of a i r f l o w d u r i n g f l i g h t due
t o t h e f l o w b e i n g r e t a r d e d i s u s u a l l y n e g l i g i b l e ; t h e component E
c a n b e c o n s i d e r e d c o n s t a n t a n d may b e o m i t t e d f r o m t h e e q u a t i o n .
Then e a c h o f t h e r e m a i n i n g t e r m s of t h e e q u a t i o n , i f w e m u l t i p l y
them by m g , w i l l c h a r a c t e r i z e t h e component e n e r g y i n c l u d e d i n a
u n i t mass o f g a s f l o w :
V2/2g e q u a l s t h e k i n e t i c e n e r g y o f t h e f l o w
( f o r a u n i t mass m V 2 / 2 ) , p / y i s t h e e n e r g y o f t h e p r e s s u r e , a n d
U i s t h e thermal energy.

For m e a s u r e m e n t s o f a i r s p e e d , w e c a n u s e s e n s o r s w h i c h a l l o w
us t o s e p a r a t e t h e dynamic a i r p r e s s u r e from t h e s t a t i c p r e s s u r e .
F i g u r e 2.32
tube).

shows t h e o p e r a t i o n of a n a i r p r e s s u r e s e n s o r ( P i t o t

I n t h e cross s e c t i o n of t h e a i r f l o w , t h e speed V I w i l l cor­
respond t o t h e a i r s p e e d , and t h e p r e s s u r e p 1 w i l l correspond t o
t h e static pressure of t h e a i r at f l i g h t a l t i t u d e .

I’
Air-Pressure
Fig. 2.32.
(1) S t a t i c P r e s s u r e p s t ;

Sensor ( P i l o t Tube).
( 2 ) T o t a l P r e s s u r e ptotal.

Within t h e l i m i t s of t h e opening i n t h e s e n s o r f o r t o t a l pres­
s u r e , t h e r a t e of flow w i l l be equal t o zero ( t h e c r i t i c a l current
or c u r r e n t o f c o m p l e t e b r a k i n g ) .
Obviously, a t t h i s p o i n t t h e pressure p 2 w i l l correspond t o
t h e t o t a l p r e s s u r e ( t h e v e l o c i t y head p l u s t h e s t a t i c p r e s s u r e ) ,
a n d E q u a t i o n ( 2 . 3 8 ) a c q u i r e s t h e f o l l o w i n g f o r m for t h i s c a s e :

(2.39)

L e t u s c o n s i d e r t h a t f o r a i r s p e e d s u p t o 4 0 0 k m / h r t h e com­
p r e s s i o n of t h e a i r can b e d i s r e g a r d e d , i . e . , t h e v a l u e s y and U
are constants.
Then E q u a t i o n ( 2 . 3 9 ) assumes t h e form:

v2

- PtotalPst

29-

(2.40)

YH

H is t h e u n i t weight of gas a t a given a l t i t u d e .
S i n c e y H = p H g ( w h e r e p H i s t h e mass d e n s i t y ) , t h e d i f f e r e n c e

where y

188

/185

between t h e t o t a l and s t a t i c p r e s s u r e s
to

( v e l o c i t y head) w i l l be equal

whence
(2.41)


The t o t a l p r e s s u r e a l o n g t h e t u b e i s a d m i t t e d t o t h e i n t e r ­
i o r of a f l e x i b l e b o x .
The s t a t i c p r e s s u r e r e a c h e s t h e h e r m e t i c a l l y
s e a l e d c h a m b e r o f t h e i n d i c a t o r t h r o u g h a n o p e n i n g made i n t h e s i d e
of t h e pressure s e n s o r and through a n i p p l e .
A s a result, there
w i l l b e a p r e s s u r e d r o p b e t w e e n t h e i n t e r n a l s p a c e o f t h e box and
/186
t h e medium s u r r o u n d i n g t h e b o x , w h i c h w i l l b e e q u a l t o t h e v e l o c i t y
head.
T h i s d r o p c a u s e s movement o f t h e t o p o f t h e b o x , w h i c h c a n
t r a n s m i t i t s movement by means o f a s y s t e m o f g e a r s s i m i l a r t o t h e
mechanism i n a s i n g l e - p o i n t e r a l t i m e t e r , e v e n t u a l l y moving a p o i n t e r
on a n a x i s t o s h o w t h e a i r s p e e d on a s c a l e w h i c h i s g r a d u a t e d i n
kilometers per hr.
Formula

(2.41)

can be used t o d e s c r i b e a i r s p e e d i n d i c a t o r s
I f we i n t r o d u c e t h e w e i g h t
d e n s i t y t o t h i s formula i n t h e form

for l o w s p e e d s , s u c h a s t h e U S - 3 5 0 .

whence

v =

(2.42)

I t i s c l e a r from t h i s formula t h a t i n o r d e r t o determine
t h e t r u e a i r s p e e d , i t i s n e c e s s a r y t o know n o t o n l y t h e v a l u e o f
t h e v e l o c i t y head, b u t a l s o t h e a t m o s p h e r i c p r e s s u r e and t h e temper­
ature of the a i r at f l i g h t a l t i t u d e .

T h e a i r s p e e d , w h i c h i s m e a s u r e d o n l y on t h e b a s i s o f t h e v e l o c ­
I n view
i t y h e a d , i s c a l l e d t h e a e r o d y n a m i c or i n d i c a t e d s p e e d .
o f t h e f a c t t h a t c a l i b r a t i o n o f t h e s p e e d i n d i c a t o r i s made f o r
f l i g h t c o n d i t i o n s a t sea l e v e l a t s t a n d a r d t e m p e r a t u r e and a i r p r e s ­
s u r e , during f l i g h t under these conditions t h e indicated speed w i l l
be equal t o t h e t r u e airspeed.
Under o t h e r c o n d i t i o n s , however,
t h e i n d i c a t e d s p e e d must b e c o n v e r t e d t o t h e t r u e a i r s p e e d .
A t high a l t i t u d e s and speeds, t h e d i f f e r e n c e between t h e a i r ­

189


s p e e d and t h e i n d i c a t e d s p e e d becomes s o s i g n i f i c a n t t h a t it becomes
d i f f i c u l t t o use t h e l a t t e r f o r navigational purposes.
In addi­
t i o n , f o r a i r s p e e d s above 400 km/hr, it becomes n e c e s s a r y t o t a k e
Therefore, f o r
t h e c o m p r e s s i o n o f t h e a i r i n t o a c c o u n t as w e l l .
a i r c r a f t o p e r a t i n g a t h i g h a l t i t u d e s and s p e e d s , a combined s p e e d
i n d i c a t o r "CSI" h a s been developed, which measures both t h e i n d i ­
c a t e d and t r u e a i r s p e e d .
I n terms o f i t s d e s i g n , t h i s i n d i c a t o r d i f f e r s f r o m t h e u s u a l
s p e e d i n d i c a t o r s i n t h a t t h e s p e e d i s measured i n two ways:
( a ) T h e f i r s t m e t h o d c o n s i s t s o f t h e c o n v e n t i o n a l s y s t e m for
i n d i c a t i n g speed and i s used t o measure t h e i n d i c a t e d a i r s p e e d ( t h e
l a r g e p o i n t e r on t h e d i a l ) ;
(b)
The s e c o n d s y s t e m i n c o r p o r a t e s a s p e c i a l c o m p e n s a t o r f o r
c h a n g e s i n a i r d e n s i t y w i t h a l t i t u d e b y m e a n s o f a s y s t e m o f g e a r s /187
a n d i s u s e d t o measllre t h e a i r s p e e d .
The c o m p e n s a t o r i s a n a n e r o i d b o x , w h i c h c h a n g e s t h e l e n g t h
of t h e a r m of a c o n t r o l l e v e r , increasing t h e l a t t e r ' s mechanical
a d v a n t a g e when t h e a t m o s p h e r i c p r e s s u r e ( a s w e l l a s t h e d e n s i t y
of t h e a i r a t f l i g h t a l t i t u d e ) i s reduced, and v i c e v e r s a .

I t s h o u l d b e mentioned t h a t i n t h e case of h i g h s p e e d a i r c r a f t ,
t h e s e n s o r f o r t o t a l p r e s s u r e i s u s u a l l y s e p a r a t e d from t h e s t a t i c
p r e s s u r e i n d i c a t o r , s o t h a t i t i s p o s s i b l e t o s e l e c t t h e most s u i t ­
T h i s means t h a t
a b l e p o s i t i o n f o r m o u n t i n g them on t h e a i r c r a f t .
t h e r o l e o f t h e s t a t i c p r e s s u r e i n d i c a t o r i s p l a y e d by o p e n i n g s
w h i c h a r e made o n t h e l a t e r a l s u r f a c e o f t h e f u s e l a g e o f t h e a i r ­
c r a f t and are l i n k e d t o t h e i n s t r u m e n t i t s e l f by t u b i n g .
In
ulation
sion of
tor for

addition t o t h e d e t a i l s of design described above, t h e reg­
o f t h e s y s t e m s i n t h e C S I a r e made b y t a k i n g t h e c o m p r e s ­
t h e a i r i n t o a c c o u n t when t h e f l o w i s r e t a r d e d i n t h e d e t e c ­
t o t a l pressure.

T h e r e f o r e , c o m p r e s s i o n o f t h e a i r on b r a k i n g w i l l b e accom­
p a n i e d by h e a t i n g , and t h e r e f o r e by an i n c r e a s e i n i t s i n t e r n a l
energy.
The r e l a t i o n s h i p b e t w e e n t h e i n t e r n a l e n e r g y o f t h e g a s ,
p r e s s u r e , and w e i g h t d e n s i t i e s i s e x p r e s s e d by t h e
formula:

its

(2.43)

3

i s t h e r a t i o o f t h e s p e c i f i c heats r e t h e g a s when i t
bV
i s h e a t e d , w i t h r e t e n t i o n o f c o n s t a n t p r e s s u r e and c o n s t a n t volume.
where K =

190


For a i r , t h i s c o e f f i c i e n t i s K = 1.4.
By s u b s t i t u t i n g t h e v a l u e U i n t o F o r m u l a ( 2 . 3 9 1 , w e c a n c h a n g e
i t t o r e a d as f o l l o w s :

or

(2.44)
A f t e r m a k i n g some s i m p l e c o n v e r s i o n s ,

taking pst/yl

out of t h e parentheses, w e w i l l have:
(2.45)

For t h e a d i a b a t i c p r o c e s s , t h e r e i s a n e q u a t i o n w h i c h i s k n o w n / l 8 8
as t h e M e n d e l e y e v - C l a p e y r o n e q u a t i o n :

from which w e o b t a i n f o r o u r c a s e
1

S u b s t i t u t i n g t h e v a l u e y1/y2 i n F o r m u l a

Assuming t h a t y 1 = Y H , s o t h a t P s t / y l
equation i n t h e form:

(2.45),

we o b t a i n

= BTH, w e can r e w r i t e t h i s

and f i n a l l y o b t a i n t h e f o r m u l a which c a n b e u s e d t o c a l i b r a t e t h e
combined s p e e d i n d i c a t o r by t h e a i r s p e e d i n t h e c h a n n e l f o r s u b s o n i c
airspeeds:

V=

K-

1

(2.46)

19 1


I


The t e m p e r a t u r e a t f l i g h t a l t i t u d e ( T H ) i s assumed t o b e s t a n ­
d a r d a c c o r d i n g t o t h e f l i g h t a l t i t u d e ( o r p s t ) , i . e . , up t o 1 1 , 0 0 0
m y T H = 288O-6.5OHY a n d a b o v e 11,000 m T H = 216.5O K ( - 5 6 . 5 O C).
To c a l i b r a t e t h e a i r s p e e d i n d i c a t o r , i t i s n e c e s s a r y t o know
t h e p r e s s u r e i n i t s manometric box and i n t h e housing o f t h e appar­
a t u s , corresponding t o t h e p r e s s u r e i n t h e s e n s o r s of t o t a l and
s t a t i c pressure under t h e given f l i g h t conditions.
Therefore, (2.46)
i s s o l v e d r e l a t i v e t o t h e p r e s s u r e s and assumes t h e form:

(2.47)

or, i f w e i n s e r t t h e n u m e r i c a l v a l u e s o f K ,

9, B,

A s w e have already pointed o u t , (2.46) i s v a l i d f o r subsonic
A t s p e e d s which exceed t h e s p e e d o f sound, t h e flow
airspeeds.
of t h e p a r t i c l e s d i f f e r s from t h e i r flow a t s u b s o n i c s p e e d .

The l o c a l c o m p r e s s i o n p r o d u c e d i n t h e a i r b y t h e a i r c r a f t c a n ­
n o t propagate i t s e l f i n t h e atmosphere faster t h a n t h e speed of
sound.
Therefore, at supersonic speeds, l o c a l interruptions i n
d e n s i t y are produced, i n which t h e r a t e of f l o w d e c r e a s e s s h a r p l y
while the pressure increases sharply.

We know t h a t t h e r a t e a t w h i c h s o u n d t r a v e l s ( a ) i n a i r d e ­
p e n d s o n l y o n t h e t e m p e r a t u r e o f t h e medium a n d i s e x p r e s s e d b y
t h e formula

a = JKqBT.
I n o t h e r words, i f g = 9 . 8 1 m/sec2, and t h e c o e f f i c i e n t
is e q u a l t o 1 . 4 , w h i l e B = 29.27 m/degree,
= ) / m T = 20.3

for a i r

1/T m / s e c .

T h e r a t i o o f t h e a i r s p e e d t o t h e r a t e o f p r o p a g a t i o n of

sound

i n a i r i s c a l l e d t h e Mach number:

M = -V
a'
If w e r e p l a c e K g B T i n F o r m u l a ( 2 . 4 6 ) b y a 2 , w e w i l l h a v e t h e
e x p r e s s i o n f o r M (Mach n u m b e r ) f o r s u b s o n i c a i r s p e e d s :

(2.48)

19 2

/189

The l a t t e r f o r m u l a i n d i c a t e s t h a t i n o r d e r t o d e t e r m i n e t h e
Mach n u m b e r , i t i s n e c e s s a r y t o know o n l y t h e v e l o c i t y h e a d a n d
There i s no n e c e s s i t y
the static pressure at f l i g h t altitudes.
t o measure a i r t e m p e r a t u r e f o r t h i s purpose.
F or s u b s o n i c a i r s p e e d s , t h e r e l a t i o n s h i p b e t w e e n t h e t o t a l
p r e s s u r e , t h e s t a t i c p r e s s u r e , a n d t h e Mach n u m b e r i s e XP r e s s e d
as f o llows :
2K

-P total-

'st
P st

M- K-

-.

1
~

1
-

- 1.
'

(2.49)

I n t h i s f o r m u l a , i f w e r e p l a c e M 2 by i t s v a l u e as o b t a i n e d
i n E q u a t i o n ( 2 . 4 8 ) , w e w i l l o b t a i n t h e f o r m u l a for c a l i b r a t i n g t h e
airspeed indicator f o r supersonic airspeeds:

Pst

1.4,

(2.50)

If w e s u b s t i t u t e t h e n u m e r i c a l v a l u e s o f K f o r a i r , e q u a l t o
i n Equation ( 2 . 5 0 ) , w e can c o n v e r t i t t o t h e s i m p l e r form:

Errors

/190

i n Measuring A i r s p e e d

Errors i n measuring airspeed, l i k e those involved i n measuring
f l i g h t a l t i t u d e , c a n b e d i v i d e d i n t o i n s t r u m e n t a l a n d me-thodolog­
i c a l ones.
I n s t r u m e n t a l e r r o r s i n c l u d e t h o s e which a r e r e l a t e d
t o i m p r o p e r a d j u s t m e n t of t h e a p p a r a t u s a n d i n s t a b i l i t y of i t s o p e r ­
a t i o n w i t h c h a n g e s i n t h e t e m p e r a t u r e o f t h e mechanism i n t h e d e v i c e .
In addition, instrumental errors also include errors i n sensing
dynamic and e s p e c i a l l y s t a t i c p r e s s u r e s w i t h s e n s o r s which depend
on t h e m o u n t i n g l o c a t i o n o n t h e a i r c r a f t .
I n s t r u m e n t a l e r r o r s are c o r r e c t e d by c o r r e c t i o n c h a r t s , which
a r e c o m p i l e d wh.en t h e a p p a r a t u s i s t e s t e d , t a k i n g i n t o a c c o u n t t h e
errors i n indicating the static pressure f o r a given type of air­
craft.
M e t h o d o l o g i c a l errors i n c l u d e t h o s e i n v o ' l v i n g f a i l u r e o f t h e
a c t u a l a i r temperature a t f l i g h t a l t i t u d e t o correspond with t h e

193

c a l c u l a t e d t e m p e r a t u r e f o r combined i n d i c a t o r s of s p e e d , and w i t h
t h e temperature and p r e s s u r e f o r o t h e r speed i n d i c a t o r s .
S t r i c t l y s p e a k i n g , t h e methodological c o r r e c t i o n s which must
be taken i n t o account i n converting t h e indicated speed t o t h e air+
soeed, are n o t i n s t r u m e n t e r r o r s , s i n c e t h e i n d i c a t e d speed has
i t s own i n d e p e n d e n t v a l u e .
However, f r o m t h e n a v i g q t i o n a l s t a n d ­
p o i n t , it i s convenient t o c o n s i d e r them methodological e r r o r s .
I n aircraft n a v i g a t i o n , it i s p o s s i b l e t o use both t h e s i n g l e
p o i n t e r d i a l f o r i n d i c a t e d s p e e d ( T y p e U S - 3 5 0 or US-700), a s w e l l
as t h e combined i n d i c a t o r (Type CSI-1200) and o t h e r s , s o t h a t t h e
methods of c a l c u l a t i n g t h e methodological e r r o r s can be viewed sepa­
rately.
I t s h o u l d b e m e n t i o n e d f i r s t of a l l t h a t t h e d i a l s o f s p e e d
i n d i c a t o r s are c a l i b r a t e d t o t a k e i n t o account t h e compressibil­
i t y of t h e a i r f o r a t r u e a i r s p e e d e q u a l t o t h e i n d i c a t e d speed,

I n f a c t , a t h i g h a l t i t u d e s , t h e t r u e a i r s p e e d i s almost always
much g r e a t e r t h a n t h e i n d i c a t e d s p e e d , s o t h a t i t i s n e c e s s a r y t o
c o n s i d e r t h a t t h e r e i s a n e r r o r i n t h e d i f f e r e n c e b e t w e e n t h e com­
p r e s s i o n of t h e a i r a t t h e a c t u a l a n d c a l c u l a t e d a i r s p e e d s :
"comp

-

- "c0mp.a.

-AV

c0mp.c.

There are s p e c i a l , p r e c i s e formulas f o r determining t h e cor­
r e c t i o n s f o r AVcomp f o r u s e w i t h i n d i c a t o r s o f i n s t r u m e n t s p e e d
a t s u b s o n i c a n d s u p e r s o n i c a i r s p e e d s , a n d t h e y a r e u s e d t o draw
u p a t a b l e of c o r r e c t i o n s ( F i g . 2 . 3 3 ) ; w e w i l l l i m i t o u r s e l v e s
discussing only t h e simple approximate formula
(2
w h e r e Vir?d i s t h e i n d i c a t e d a i r s p e e d a n d A V c o m p
f o r t h e indicated speed.

is the correction

I n t h e a p p r o x i m a t e f o r m u l a g i v e n a b o v e , a s w e l l as i n t h e g r a p h
w h i c h i s t h e r e s u l t o f a c c u r a t e f o r m u l a s , t h e c o r r e c t i o n s f o r com­
It
p r e s s i o n a r e d e t e r m i n e d o n l y as a f u n c t i o n o f f l i g h t a l t i t u d e .
i s c l e a r t h a t t h i s i n v o l v e s an e r r o r which i s r e l a t e d t o a f a i l ­
u r e of t h e a c t u a l a i r t e m p e r a t u r e a t t h e g i v e n a l t i t u d e t o c o r r e ­
spond t o s t a n d a r d t e m p e r a t u r e .
However, if w e r e c a l l t h a t l a r g e
v a r i a t i o n s i n t e m p e r a t u r e u s u a l l y o c c u r o n l y a t low a l t i t u d e s , w h e r e
high-speed a i r c r a f t p r a c t i c a l l y n e v e r f l y , t h e s e e r r o r s can be d i s ­
regarded.
A f t e r making t h e c o r r e c t i o n s i n t h e i n d i c a t e d speed f o r t h e
compression of t h e a i r , conversion o f t h e l a t t e r i n t o a i r s p e e d s
i s d o n e on a n n a v i g a t i o n a l s l i d e r u l e .

19 4

S i n c e t h e d i a l of t h e i n d i c a t e d a i r s p e e d i s c a l i b r a t e d by t h e
formula

and t h e a i r s p e e d is

t h e n i f w e d i v i d e t h e s e c o n d f o r m u l a by t h e f i r s t w e w i l l o b t a i n :
(2.52)

+gr

L e t u s s u b s t i t u t e i n t o Formula ( 2 . 2 8 ) t h e f o l l o w i n g v a l u e s :
T o = 288O K , a n d B = 2 9 . 2 7 .
We w i l l t h e n o b t a i n :

= 6 . 5 deg/km,

PH =Po ( 1

-0 . 0 2 2 6 ~ p ~ ,

and i f w e l e t t h e v a l u e p H b e s u b s t i t u t e d i n t o Formula ( 2 . 5 2 ) ,
w i l l obtain:

we

1
(1 -0,0226H$6628

or

'true-

-

Ig Vinst+-1g(273+
1
2

fH)-- 1 Ig288-2.628lg(l-O,0226H).

2

(2.53)

A c c o r d i n g t o F o r m u l a ( 2 . 5 3 ) we c a n c o n v e r t t h e l o g a r i t h m i c
s c a l e s of n a v i g a t i o n a l s l i d e r u l e s f o r c o n v e r t i n g t h e i n d i c a t e d
airspeed into the true airspeed.
C a l i b r a t i o n of t h e combined s p e e d i n d i c a t o r on t h e b a s i s o f
t h e t r u e a i r s p e e d i s p e r f o r m e d by t a k i n g i n t o a c c o u n t t h e compres­
s i b i l i t y of t h e a i r over t h e e n t i r e range of t h e scale.
The methodo­
l o g i c a l e r r o r i n t h e reading is r e l a t e d only t o t h e differences
between t h e a c t u a l a i r temperature and t h e c a l c u l a t e d temperature
at the flight altitude.
1192
S i n c e t h e a i r s p e e d , as shown b y a c o m b i n e d s p e e d i n d i c a t o r
u n d e r s t a n d a r d t e m p e r a t u r e c o n d i t i o n s , i s e x p r e s s e d b y t h e formula­

195

I1

I


a n d t h e c o r r e c t e d v a l u e f o r t h e a i r s p e e d a t f l i g h t a l t i t u d e i n ac­
cordance with t h e actual temperature i s
[(Ptotal-Pst

+ i fl
) ~- 1 1 ,

p st
if w e d i v i d e t h e s e c o n d f o r m u l a i n t o t h e f i r s t , w e w i l l h a v e :
V

corr

=V

CSI

1/

g1
P

or

(2.54)

After looking up t h e logarithm of t h e l a t t e r , w e w i l l o b t a i n a
f o r m u l a w h i c h c a n b e u s e d t o c o n s t r u c t t h e l o g a r i t h m i c s c a l e on
t h e N L - 1 O M for a c o m b i n e d s p e e d i n d i c a t o r :
(2.54a)
Relationship Between Errors i n Speed
F1 i g h t A 1 t i t u d e

I n d i c a t o r s and

I n d e s c r i b i n g t h e e r r o r s i n b a r o m e t r i c a l t i m e t e r s and a i r s p e e d
i n d i c a t o r s , i n s t r u m e n t a l e r r o r s of aerodynamic o r i g i n are found,
which a r e r e l a t e d t o e r r o r s i n r e c o r d i n g t h e s t a t i c p r e s s u r e by
the air pressure sensors.
E x p e r i e n c e h a s shown t h a t a e r o d y n a m i c e r r o r s i n t h e s p e e d i n ­
d i c a t o r s due t o i n c o r r e c t r e c o r d i n g of t h e dynamic p r e s s u r e are
n e g l i g i b l y small by comparison w i t h t h e e r r o r s i n i n c o r r e c t re­
cording of s t a t i c p r e s s u r e .
T h i s i s e x p l a i n e d by t h e f a c t t h a t it
it immensely e a s i e r t o measure t h e p r e s s u r e of a r e t a r d e d a i r f l o w
w i t h a s e n s o r t h a t i s aimed i n t o t h e a i r f l o w , t h a n i t i s t o s e l e c t
a l o c a t i o n on a n a i r c r a f t f o r a s t a t i c - p r e s s u r e s e n s o r , s u c h t h a t
t h e l a t t e r w i l l n o t b e d i s t o r t e d by t h e a i r f l o w o v e r t h e body o f
the aircraft.

I n connection w i t h t h e f a c t t h a t t h e s t a t i c p r e s s u r e from
t h e s e n s o r i s t r a n s m i t t e d s i m u l t a n e o u s l y t o t h e h e r m e t i c chambers
of t h e s p e e d a n d a l t i t u d e i n d i c a t o r s , t h e r e m u s t b e a m u t u a l r e l a ­

19 6

t i o n s h i p between t h e e r r o r s i n t h e measurement of a l t i t u d e and
s p e e d owing t o e r r o r s i n r e c o r d i n g t h e p r e s s u r e .

/193

A t t h e same t i m e , t h e v e l o c i t y h e a d a c c o r d i n g t o w h i c h t h e
d i a l of t h e speed i n d i c a t o r i s c a l i b r a t e d i s equal t o
(2.55)
Since t h e e r r o r s i n measuring t h e v e l o c i t y head are e q u a l t o
t h e e r r o r s i n measuring t h e s t a t i c p r e s s u r e , then
(2.56)
The s t a t i c
Under s t a n d a r d c o n d i t i o n s , P O = 0 . 1 2 5 k g / s e c 2 / m 4 .
The s p e c i f i c g r a v i t y o f m e r ­
p r e s s u r e i s u s u a l l y g i v e n i n m m Hg.
c u r y i s 1 3 . 6 , s o t h a t t h e p r e s s u r e of 1 k g / c m 2 w o u l d e q u a l l O , O O O /
1 3 . 6 = 7 3 5 m m Hg.
On t h e o t h e r h a n d , s i n c e t h e p a r a m e t e r p h a s m 4 i n t h e d e n o m i ­
n a t o r , t h e p r e s s u r e e x p r e s s e d by ( 2 . 5 6 ) , r e l a t i v e t o a n a r e a o f
1 m 2 , must be d i v i d e d by 1 0 , 0 0 0 t o d e t e r m i n e t h e v a l u e f o r 1 c m 2 ,
SO t h a t w e f i n a l l y obtain
APst=

735.0.125
2.10000

A ( W) -0.0048 A ( V z ) .

Example:

A t a n i n d i c a t e d s p e e d o f 396 k m / h r ( 1 1 0 m / s e c ) , a t a
f l i g h t a l t i t u d e of 5000 m y t h e aerodynamic c o r r e c t i o n f o r t h e speed
Find t h e aerodynamic e r r o r i n
i n d i c a t o r i s 36 k m / h r , ( 1 0 m / s e c ) .
the altimeter.

Sol u t i on:
ApP,t=0.0048(11M- IO@) =0.0048.2100=

10.8 mm Hg

.

According t o t h e hypsometric t a b l e , t h e b a r i c s t a g e a t a
f l i g h t a l t i t u d e o f 5 0 0 0 m i s e q u a l t o 1 8 . 5 mm Hg; h e n c e , t h e a e r o ­
dynamic component i n t h e a l t i m e t e r e r r o r i s

AH = - 1 0 . 8 . 1 8 . 5

= - 200 m.

Formula ( 2 . 5 6 ) i s an approximate one, b u t it y i e l d s s u f f i c i e n t l y
a c c u r a t e r e s u l t s u p t o a n i n d i c a t e d a i r s p e e d of 4 0 0 k m / h r .
The
a l t i m e t e r error c a n b e d e t e r m i n e d m o r e p r e c i s e l y i f w e know t h e
dynamic p r e s s u r e and t a k e i n t o account t h e compression o f t h e a i r
at different instrument readings.
T a b l e 2.6 shows t h e v e l o c i t y head a t v a r i b u s i n d i c a t e d a i r ­
s p e e d s , and can be used t o d e t e r m i n e t h e aerodynamic c o r r e c t i o n s
of t h e altimeter.
T h e t h i r d c o l u m n i n T a b l e 2 . 6 s h o w s t h e mano­

19 7

I

I

I l l 1

I l l

I 1

r


m e t r i c s t a g e , i . e . , t h e change i n p r e s s u r e w i t h change i n a i r s p e e d
If w e m u l t i p l y t h e a e r o d y n a m i c c o r r e c t i o n o f t h e s p e e d
by 1 km/hr.
i n d i c a t o r by t h e manometric s t a g e and t h e n u s e t h e h y p s o m e t r i c
t a b l e , it w i l l be easy t o determine t h e aerodynamic c o r r e c t i o n f o r
t h e altimeter f o r a given f l i g h t a l t i t u d e .

/ 19
-

TABLE 2.6.

4

'inst

1 km/hr

_.
I

50
100
150
200
250
300

350
400
450

500
550
600

0.89
3.57
8
14.3
22.37
32.4
44.27
58.25
74.23
92.35
112.7
135.7

0.054
0,089
0,126
0,162
0*2
0.24
0.28
0.32
0.36
0,41
0.46
0.53

700
800
900
loo0
1100
1200
1300
1400
1500
1600
1700
1800

188.3
252
322
418
522.8
645.8
787.2
947.2
1125,4
1317.6
1525.7
1748.8

0.65
0,s

0,96
1.04
1.23
1.42
1.6
1.78
1,92
2,08
2.23
2.4

inst
F i g . 2.33.

19 8

G r a p h of C o r r e c t i o n s for A i r C o m p r e s

sion.

5.

Measurement o f t h e Temperature o f t h e Outside Air

Measurement o f t h e t e m p e r a t u r e o f t h e o u t s i d e a i r d u r i n g f l i g h t
is necessary first of a l l f o r determining t h e t r u e values of t h e
a i r s p e e d and f l i g h t a l t i t u d e .
/195
The t h e r m o m e t e r f o r m e a s u r i n g t h e o u t s i d e a i r t e m p e r a t u r e i s
a r e m o t e - c o n t r o l l e d i n s t r u m e n t , i . e . , i t s s e n s i t i v e e l e m e n t i s mounted
o u t s i d e t h e c a b i n o f t h e a i r c r a f t a n d i:s e x p o s e d t o t h e a i r f l o w , w h i l e
t h e i n d i c a t o r i s mounted on t h e i n s t r u m e n t p a n e l i n t h e c o c k p i t .

A t t h e p r e s e n t t i m e , e l e c t r i c t h e r m o m e t e r s a r e u s e d f o r meas­
u r i n g t h e o u t s i d e a i r temperature, and t h e i r o p e r a t i n g p r i n c i p l e
i s b a s e d on t h e c h a n g e s i n e l e c t r i c a l c o n d u c t i v i t y of m a t e r i a l s
d e p e n d i n g on t h e i r temp e r a t u r e

.

2.34

A s c h e m a t i c d i a g r a m o f s u c h a t h e r m o m e t e r i s shown i n F i g u r e
a n d c o n s i s t s o f a n e l e c t r i c a l b r i d g e made o f r e s i s t o r s .

I f t h e arms o f t h e b r i d g e 1 a n d 1 1 , a s w e l l a s 2 a n d 2 1 , h a v e
. t h e same r e s i s t a n c e w h e n c o n n e c t e d i n p a i r s , n o s u p p l y v o l t a g e w i l l
flow t h r o u g h b r i d g e AB and c o n s e q u e n t l y t h r o u g h t h e t e m p e r a t u r e
indicator.

One a r m o f t h e b r i d g e ( 2 1 )
i s made o f a m a t e r i a l w h i c h h a s
a high thermoelectric coefficient,
a n d i s m o u n t e d on t h e s u r f a c e
of t h e a i r c r a f t t o be exposed
t o airflow.
D e p e n d i n g on t h e t e m p e r a t u r e
of a r m 2 1 , i t s r e s i s t a n c e c h a n g e s ,
t h u s a f f e c t i n g t h e amount o f c u r r e n t
which p a s s e s through b r i d g e A B
with t h e temperature indicator
connected t o it.
Fig. 2.34.
S c h e m a t i c Diagram
of E l e c t r i c Thermometer.

Thermometers of t h i s k i n d ,
when u s e d a t low a i r s p e e d s , i n d i ­
c a t e t h e t e m p e r a t u r e w i t h a n a c c u r a c y o f 2-3O.
However, a t h i g h
a i r s p e e d s , due t o d r a g a n d a d i a b a t i c c o m p r e s s i o n o f t h e a i r f l o w
on t h e f o r w a r d s e c t i o n o f t h e s e n s o r , t h e l a t t e r i s s u b j e c t e d t o
l o c a l h e a t i n g t h a t c r e a t e s m e t h o d o l o g i c a l errors i n m e a s u r i n g t e m p e r ­
ature.

For an e x a c t determination of t h e methodological e r r o r s of
t h i s thermometer, w e w i l l r e q u i r e a s e n s o r with complete braking
of t h e a i r f l o w , as i s t h e c a s e i n s e n s o r s u s e d t o m e a s u r e t h e t o t a l
pressure i n airspeed indicators.
If w e k e e p i n mind t h a t y = p / B T ,

(2.44)

can b e changed t o

19 9

r e a d as f o l l o w s :

w h e r e TT i s t h e t e m p e r a t u r e o f

t h e retarded flow.

Therefore

/196
(2.57)

If w e s u b s t i t u t e i n ( 2 . 5 7 )
w i l l obtain

the values

K =

1 . 4 and B = 29.27,

we

v2

AT= A t = 2000 ’

where V i s t h e v e l o c i t y ,

e x p r e s s e d i n m/sec.

Fig. 2.35.
S e n s o r s for E l e c t r i c T h e r m o m e t e r f o r
Measuring Outside A i r Temperature.
( a ) TUE; ( b ) TNV.

km/hr

S i n c e t h e c o n v e r s i o n c o e f f i c i e n t f o r c h a n g i n g from m / s e c t o
i s 3 . 6 , f o r a s p e e d e x p r e s s e d i n km/hr
(2.57a)

P r a c t i c a l l y s p e a k i n g , i t i s h i g h l y u n s u i t a b l e t o u s e thermom­
e t e r s f o r measuring o u t s i d e a i r t e m p e r a t u r e which have complete
r e t a r d a t i o n o f a i r f l o w , s i n c e i n t h i s case t h e s e n s o r w i l l n o t b e
exposed t o t h e flow and t h i s w i l l r e s u l t i n a high thermal i n e r t i a
of t h e thermometer, ?.e. , r a p i d changes i n temperature during f l i g h t ,
which c o u l d t a k e p l a c e a t h i g h f l i g h t s p e e d s , would n o t be d e t e c t e d
by t h e t h e r m o m e t e r .

For s e n s o r s w h i c h a r e e x p o s e d t o t h e a i r f l o w , t h e c o e f f i c i e n t
of d r a g i s w i t h i n t h e l i m i t s o f 0 . 5 t o 0 . 8 5 .
The TUE a n d T N V t h e r ­
mometers i n u s e a t t h e p r e s e n t t i m e h a v e c o e f f i c i e n t s o f d r a g which
a r e n e a r l y t h e same ( a p p r o x i m a t e l y 0 . 7 ) .
The s c a l e of c o r r e c t i o n s
f o r t h e thermometer f o r measuring o u t s i d e a i r temperature (TUE),
l o c a t e d on t h e n a v i g a t i o n a l s l i d e r u l e , c a n b e u s e d w i t h s u f f i c i e n t
a c c u r a c y f o r t h e TNV thermometers as w e l l .
The s e n s o r o f t h e TUE t h e r m o m e t e r i s in t h e s h a p e of a r o d

200

w i t h a w i n d i n g on t h e s u r f a c e , c o v e r e d b y a c y l i n d r i c a l h o u s i n g
(Fig. 2.35, a ) .
When a f l o w o f a i r p a s s e s t h r o u g h s u c h a s e n s o r ,
i t i s h e a t e d on one s i d e .

/19 7

T h e s e n 5 o r o f t h e T N V t h e r m o m e t e r i s made i n t h e f o r m o f a
de Lava1 nozzle.
The s e n s i t i v e e l e m e n t i s l o c a t e d i n t h e n a r r o w e s t
p o r t i o n of t h e nozzle (Fig. 2.35, b ) and t h e a i r flows symmetrically
over it.
Therefore, t h i s s e n s o r has less thermal i n e r t i a and gives
more a c c u r a t e r e a d i n g s i n d i f f e r e n t f l i g h t r e g i m e s .

6.

A v i a t i o n Clocks

The m e a s u r e m e n t o f t i m e p l a y s a n e x t r e m e l y i m p o r t a n t r o l e i n
aircraft navigation, s i n c e t h e c a l c u l a t i o n of t h e path of t h e air­
c r a f t on t h e b a s i s o f t h e component a i r s p e e d a n d t i m e i s i n v o l v e d
i n almost a l l navigational equations.
T h i s means t h a t a n i n c r e a s e
i n t h e a i r s p e e d p l a c e s i n c r e a s e d demands on t h e a c c u r a c y o f t h e
measurement o f t i m e .
It i s e s p e c i a l l y important t o have an exact
d e t e r m i n a t i o n o f t h e moments o f p a s s a g e o v e r c o n t r o l c h e c k p o i n t s ,
;.e.,
i n t h i s c a s e , t h e exact measurement n o t of e l a p s e d t i m e b u t
o f t i m e s e g m e n t s b e t w e e n t h e m o m e n t s when t h e a i r c r a f t i s p a s s i n g
over landmarks.
T h e r e a r e a l s o f a c t o r s w h i c h demand h i g h a c c u r a c y i n d e t e r ­
For
mining t h e time and t h e e x a c t o p e r a t i o n of a v i a t i o n c l o c k s .
example, t h e coincidence of t h e f l i g h t p l a n s of i n d i v i d u a l a i r c r a f t ,
communication w i t h t h e tower , and e s p e c i a l l y i n a s t r o n o m i c a l c a l c u ­
l a t i o n s , w h e r e a n e r r o r i n c a l c u l a t i n g t h e e l a p s e d t i m e o f 1 min
could produce an e r r o r i n determining t h e a i r c r a f t coordinates of
2 7 km.
T h e o p e r a t i n g p r i n c i p l e o f a l l e x i s t i n g d e v i c e s for m e a s u r i n g
t i m e i s t h e i r c o m p a r i s o n w i t h t h e t i m e r e q u i r e d for some s t a n d a r d
event t o occur.
I n t h i s c a s e , t h e s t a n d a r d e v e n t is t h e p e r i o d
of o s c i l l a t i o n o f t h e b a l a n c e w h e e l o f a c l o c k ( a c i r c u l a r pendu­
lum).
A l l o f t h e r e m a i n i n g mecLdnism of t h e c l o c k a c t s m a i n l y as
a m e c h a n i c a l c o u n t e r o f t h e number o f o s c i l l a t i o n s o f t h e pendu­
lum.
H o w e v e r , i t e x e r t s a c o n s i d e r a b l e i n f l u e n c e on t h e a c c u r a c y
o f o p e r a t i o n o f t h e c l o c k ; w h e n t h e m a i n s p r i n g o f a c l o c k i s wound
c o m p l e t e l y , t h e c l o c k r u n s s o m e w h a t f a s t e r , a n d when t h e s p r i n g
h a s r u n down t h e c l o c k r u n s s l o w e r .
The most i m p o r t a n t r o l e i n
m e a s u r i n g t i m e i s p l a y e d by t h e a c c u r a c y o f a d j u s t m e n t o f t h e a c t u a l
p e r i o d o f o s c i l l a t i o n o f t h e pendulum.
We know t h a t t h e p e r i o d o f o s c i l l a t i o n o f a b o d y a r o u n d i t s
a x i s ( t o r s i o n a l o s c i l l a t i o n ) is r e l a t e d t o t h e deformation of t h e
body as d e t e r m i n e d by t h e f o r m u l a

201


w h e r e T i s t h e p e r i o d of o s c i l l a t i o n o f t h e b o d y a r o u n d t h e a x i s ,
J i s t h e moment of i n e r t i a o f t h e b o d y , a n d D i s t h e m o d u l u s o f
torsion.
The p r o d u c t of t h e modulus of t o r s i o n t i m e s t h e a n g l e t h r o u g h
w h i c h t h e b o d y r o t a t e s ( 4 ) i s t h e t o r s i o n a l moment:

M = D4.
The p e r i o d o f o s c i l l a t i o n o f a b a l a n c e c a n b e a d j u s t e d b o t h
/198
by c h a n g i n g i t s moment o f i n e r t i a ( f o r w h i c h p u r p o s e a d j u s t i n g s c r e w s
a r e l o c a t e d a l o n g i t s o u t e r c i r c u m f e r e n c e ) , or b y c h a n g i n g t h e mod­
u l u s of t o r s i o n .
T h e moment o f i n e r t i a o f t h e b a l a n c e w h e e l i s c h a n g e d b y s c r e w ­
i n g t h e a d j u s t i n g s c r e w s s y m m e t r i c a l l y i n or o u t a l o n g t h e e n t i r e
c i r c u m f e r e n c e , i n o r d e r n o t t o d i s t u r b t h e b a l a n c e of t h e pendulum.
T h i s m e a n s t h a t a p o r t i o n o f t h e mass i s b r o u g h t c l o s e r t o or moved
f u r t h e r away f r o m t h e c e n t e r o f r o t a t i o n o f t h e b a l a n c e .
The m o d u l u s o f t o r s i o n i s a d j u s t e d by means o f a h a i r s p r i n g ;
t h e b a l a n c e w h e e l i s a d j u s t e d by c h a n g i n g t h e f r e e l e n g t h o f t h e
h a i r s p r i n g , f o r w h i c h p u r p o s e a movable s t o p , w h i c h a c t s as a r e g u ­
l a t o r , is' m o u n t e d n e a r t h e p o i n t w h e r e t h e h a i r s p r i n g i s f a s t e n e d .
I t s h o u l d b e m e n t i o n e d t h a t many f a c t o r s a f f e c t t h e p r e c i s ­
i o n w i t h which a clock o p e r a t e s , b u t t h e most i m p o r t a n t ones are
temperature and magnetic eff6cts.
T h e r e f o r e , a number o f m e a s u r e s
are taken t o exclude t h e s e f a c t o r s .

T h e b a l a n c e w h e e l o f a n a c c u r a t e c l o c k i s u s u a l l y made o f b i ­
metallic m a t e r i a l and d i v i d e d a l o n g t h e p l a n e of t h e d i a m e t e r .
When t h e t e m p e r a t u r e f a l l s a n d t h e f l e x i b i l i t y o f t h e h a i r ­
s p r i n g i n c r e a s e s ( t h e m o d u l u s D i n c r e a s e s ) , o n e - h a l f of t h e b a l a n c e
e x p a n d s a n d i t s e n d s move f u r t h e r away f r o m t h e c e n t e r o f r o t a t i o n ,
t h u s compensating f o r t h e temperature e r r o r i n t h e clock.
The h a r m f u l e f f e c t of m a g n e t i c f i e l d s on t h e a c c u r a c y o f c l o c k s
c a n u s u a l l y b e overcome by u s i n g d i a m a g n e t i c p a r t s i n t h e b a l a n c e
w h e e l , h a i r s p r i n g a n d e s c a p e m e n t , or e l s e t h e e n t i r e c l o c k m e c h a n ­
i s m i s p l a c e d w i t h i n a s h i e l d e d h o u s i n g made o f i r o n a l l o y .
Special Requirements f o r Aviation Clocks
I n a d d i t i o n t o t h e g e n e r a l r e q u i r e m e n t s for c l o c k m e c h a n i s m s
( h i g h a c c u r a c y , compensation f o r t e m p e r a t u r e and magnetic e f f e c t s ) ,
a v i a t i o n c l o c k s h a v e a d d i t i o n a l r e q u i r e m e n t s p l a c e d upon t h e m :
(a)
P r o t e c t i o n a g a i n s t v i b r a t i o n and shock, s o t h a t t h e clocks
on a n a i r c r a f t m u s t b e m o u n t e d i n s p e c i a l s h o c k m o u n t i n g s .

202

(b) Ensuring r e l i a b l e o p e r a t i o n u n d e r c o n d i t i o n s of l o w t e m ­
p e r a t u r e ; f o r t h i s purpose, a v i a t i o n clocks are usually f i t t e d with
electric heaters.
(c)
R e l i a b i l i t y a n d a c c u r a c y o f o p e r a t i o n u n d e r v a r i o u s con­
ditions.
The h a n d s , n u m e r a l s , a n d p r i n c i p a l s c a l e d i v i s i o n s a r e
made l a r g e r a n d c o v e r e d w i t h a l u m i n o u s m a t e r i a l t o p e r m i t t h e i r
use during night f l i g h t s .
(d)
The p o s s i b i l i t y o f m e a s u r i n g s i m u l t a n e o u s l y s e v e r a l t i m e
parameters.
T h i s means t h a t s e v e r a l d i a l s a r e u s u a l l y d r i v e n b y
t h e mechanism.
A v i a t i o n c l o c k s o f t h e ACCH t y p e ( a v i a t i o n c l o c k - c h r o n o m e t e r
w i t h h e a t e r ) a r e made t o s a t i s f y a l l t h e c o n d i t i o n s l i s t e d a b o v e .

/199

The e l a p s e d t i m e i s i n d i c a t e d on t h e s e c l o c k s b y a main d i a l
T o c a l c u l a t e t h e t o t a l f l i g h t t i m e or t h e
with a central pointer.
f l i g h t time over i n d i v i d u a l s t a g e s , t h e r e is an a d d i t i o n a l scale
i n t h e upper p a r t of t h e clock.
The s t a r t o f t h e c l o c k h a n d s i s
m a r k e d on t h i s s c a l e , w h i l e t h e t i m e when t h e y s t o p a s w e l l a s t h e
r e s e t t i n g t o z e r o a r e a c c o m p l i s h e d b y p u s h i n g a b u t t o n on t h e l e f t hand s i d e of t h e c l o c k h o u s i n g .
T h i s same b u t t o n , when p u l l e d o u t ,
i s u s e d t o wind t h e main s p r i n g o f t h e c l o c k .
Below t h e " f l i g h t t i m e " s c a l e , t h e r e i s a p i l o t l i g h t w h i c h
i s u s e d t o s i g n a l t h e f o l l o w i n g by means o f a s p e c i a l s h u t t e r :
(a)

S t a r t o f mechanism:

(b)

S t o p mechanism:

(c)

Pause:

red light.

t h e l i g h t i s h a l f r e d and h a l f w h i t e .

white l i g h t .

T o measure s h o r t t i m e e v e n t s , t h e c l o c k i s f i t t e d w i t h a sweep
hand ( t h i n c e n t r a l p o i n t e r ) and an a d d i t i o n a l s c a l e a t t h e bottom
of t h e a p p a r a t u s where t h e m i n u t e s a r e c o u n t e d .
The sweep h a n d
i s s t a r t e d , s t o p p e d a n d h e l d by p r e s s i n g a b u t t o n on t h e r i g h t hand s i d e of t h e h o u s i n g .

I n a d d i t i o n t o t h e A C C H , t h e a v i a t i o n c h r o n o m e t e r 1 3 ChP i s
currently i n use.
I t employs a p o t e n t i o m e t r i c c i r c u i t ; t h e v e r s i o n
f i t t e d w i t h i n d i c a t o r s i s t h e 2 0 ChP.
This chronometer, e s p e c i a l l y
intended f o r purposes of astronomical o r i e n t a t i o n , i s operated by
remote c o n t r o l and c o n s i s t s of t h r e e main i n d i c a t o r s :
(a)
An e l a p s e d - t i m e i n d i c a t o r w h o s e r e a d i n g s a r e a l w a y s l i n k e d
t o t h e chronometer a t t h e t r a n s m i t t e r .
(b)
Two t i m e i n d i c a t o r s f o r m e a s u r i n g t h e a l t i t u d e o f l u m i n ­
aries;
t h e i r readings are a l s o connected t o t h e chronometer a t t h e
t r a n s m i t t e r , b u t a t t h e moment o f m e a s u r e m e n t o f t h e a l t i t u d e o f

20 3

t h e l u m i n a r y by means of a s e x t a n t , a s t o p s i g n a l i s s e n t t o o n e
of them a n d t h e t i m e o f m e a s u r e m e n t i s n o t e d .

A f t e r t h e r e a d i n g i s made, t h e m i n u t e h a n d i s s e t t o t h e e l a p s e d
t i m e a c c o r d i n g t o t h e r e a d i n g s of t h e f i r s t d i a l by p u s h i n g t h e
button.
Each t i m e t h e b u t t o n i s p r e s s e d , t h e h a n d moves f o r w a r d
one minute.
The s w e e p s e c o n d h a n d l i n e s u p w i t h t h e r e a d i n g s o f
t h e t r a n s m i t t e r immediately a f t e r t h e i n d i c a t o r i s switched on.
These d i a l s do n o t have any hour hands.
The t i m e i n h o u r s
i s d e t e r m i n e d by r e a d i n g s f r o m a Type ACCH c l o c k .

7.

Navigational Sights

A t t h e p r e s e n t t i m e , n a v y g a t i o n a l s i g h t s a r e 11.sed o n l y f o r
s p e c i a l p u r p o s e s s u c h as a e r i a l p h o t o g r a p h y .
They a r e n o t u s e d
i n passenger aircraft.
T h e r e a r e s e v e r a l t y p e s of n a v i g a t i o n a l s i g h t s , w h i c h d i f f e r
/200
i n t h e i r design.
However, a l l a r e i n t e n d e d f o r m e a s u r i n g t h e c o u r s e
a n g l e s o f l a n d m a r k s ( C A L I a n d t h e i r v e r t i c a l a n g l e s (VA).
The c o u r s e a n g l e
g i t u d i n a l a x i s of t h e
The v e r t i c a l a n g l e i s
where t h e a i r c r a f t i s

of a Zandmark i s t h e a n g l e b e t w e e n t h e l o n ­
a i r c r a f t and t h e d i r e c t i o n of t h e landmark.
t h e angle between t h e v e r t i c a l a t t h e p o i n t
l o c a t e d and t h e d i r e c t i o n of t h e landmark.

T h e s i g h t c a n b e u s e d t o s o l v e a g r e a t many n a v i g a t i o n a l p r o b ­
l e m s r e l a t e d t o d e t e r m i n a t i o n of t h e l o c u s of t h e a i r c r a f t and t h e
parameters of i t s motion.

Fig.
ing;

2.36.
D e t e r m i n i n g t h e V a l u e ( a ) o f A i r c r a f t Bera­
( b ) o f t h e D i s t a n c e f r o m a Landmark t o t h e A i r c r a f t
Vertical.

1.
D e t e r m i n a t i o n of t h e locus of t h e a i r c r a f t i n terms o f
t h e c o u r s e and v e r t i c a l a n g l e s o f t h e landmark ( F i g . 2 . 3 6 ) .
In
t h i s c s e , t h e t r u e b e a r i n g from t h e landmark t o t h e a i r c r a f t i s
( F i g . 2.36, a )
T B A = T C t C A L 2 180°,
204

while t h e d i s t a n c e from t h e landmark t o t h e v e r t i c a l of t h e a i r ­
craft (Fig. 2.36, b ) is

S = H t g VA,
where TBA i s t h e t r u e b e a r i n g f r o m t h e landmark t o t h e a i r c r a f t ,
TC i s t h e t r u e c o u r s e o f t h e a i r c r a f t , CAL i s t h e c o u r s e a n g l e o f
t h e l a n d m a r k , H i s t h e f l i g h t a l t i t u d e , a n d VA i s t h e v e r t i c a l a n g l e
of t h e landmark.
Obviously, i f t h e a i r c r a f t course i s determined by a magnetic
compass, i n o r d e r t o s o l v e t h i s problem w e must a l s o add t o t h e
r e a d i n g s o f t h e compass t h e c o r r e c t i o n s f o r t h e d e v i a t i o n o f t h e
compass o f t h e m a g n e t i c d e c l i n a t i o n of t h e l o c u s o f t h e a i r c r a f t .
TC = CC

+

A

C

+

AM.

T h e c o r r e ' c t i o n for t h e d e v i a t i o n of t h e m e r i d i a n s b e t w e e n l a n d ­
m a r k s a n d t h e l o c u s of t h e a i r c r a f t i n t h i s c a s e i s n o t t a k e n i n t o
a c c o u n t , s i n c e t h e m e a s u r e m e n t o f t h e v e r t i c a l a n g l e s c a n b e made
s a t i s f a c t o r i l y up t o 7 0 - 7 5 O , i . e . , a t d i s t a n c e s w h i ch d o n o t e x c e e d / 2 0 1
three t o four times the f l i g h t a l t i t u d e .
I n s o l v i n g t h i s p r o b l e m , i t i s p a r t i c u l a r l y i m p o r t a n t t o know
t h e t r u e f l i g h t a l t i t u d e a b o v e t h e l e v e l of t h e v i s i b l e l a n d m a r k ,
since e r r o r s i n determining t h e distance w i l l be proportional t o
t h e e r r o r s i n measuring t h e f l i g h t a l t i t u d e .
Therefore, the readings
o f t h e a l t i m e t e r must b e s u b j e c t e d t o c o r r e c t i o n s f o r t h e i n s t r u ­
mental and m e t h o d o l o g i c a l e r r o r s and t h e ele.vation of t h e landmark
above s e a l e v e l must a l s o b e t a k e n i n t o a c c o u n t if measurements
a r e n o t b e i n g made i n a l e v e l l o c a t i o n .

2.
D e t e r m i n a t i o n o f t h e l o c a t i o n o f an a i r c r a f t i n terms of
t h e b e a r i n g s f r o m two l a n d m a r k s ( F i g . 2 . 3 7 ) .
In this case,
IPS1 = TC
IPS2 = TC

+
+

CALI

f

180O;

C A L 2 f 180'.

T h e p o s i t i o n of t h e a i r c r a f t i s d e t e r m i n e d b y t h e i n t e r s e c ­
t i o n o f b e a r i n g s I P S 1 a n d I P S 2 on t h e map.
If t h e d i r e c t i o n f i n d i n g
i s made o v e r g r e a t d i s t a n c e s , e s p e c i a l l y i n t h e p o l a r r e g i o n s , t h e
measurements o f t h e b e a r i n g s must i n c l u d e a c o r r e c t i o n f o r t h e d i s ­
placement of t h e meridians.
An a d v a n t a g e o f t h i s m e t h o d i s i t s i n d e p e n d e n c e o f f l i g h t a l ­
t i t u d e , and c o n s e q u e n t l y , of t h e n a t u r e of t h e l o c a l r e l i e f .
However, t h i s method r e q u i r e s c a r e f u l measurement o f t h e c o u r s e
- ? g l e o f t h e s e c o n d l a n d m a r k , s i n c e t h e a i r c r a f t may move c o n s i d ­
e r a b l y away f r o m t h e l i n e o f t h e f i r s t b e a r i n g d u r i n g a p r o l o n g e d
measurement.

20 5

3.
D e t e r m i n a t i o n of t h e d r i f t a n g l e of t h e a i r c r a f t a c c o r d ­
ing t o visual points.
T o d e t e r m i n e t h e d r i f t a n g l e by t h i s means,
t h e s i g h t i s s e t - a t a c o u r s e a n g l e o f 180° a n d a z e r o v e r t i c a l a n g l e .
With a n e x a c t m a i n t e n a n c e
o f t h e course, by t h e p i l o t ,
o b s e r v i n g . t h e d i r e c t i o n s of
v i s u a l p o i n t s and t u r n i n g t h e
s i g h t t o keep it p a r a l l e l t o
t h e course c h a r t , the s i g h t
i s s e t i n t h e d i r e c t i o n i n which
The
t h e a i r c r a f t i s moving.
. d r i f t angle of t h e aircraft
i s t h e n c a l c u l a t e d on a s p e c i a l
scale.

Fig. 2.37.
D e t e r m i n i n g t h e POs i t i o n L i n e o f a n A i r c r a f t by
Two o f i t s B e a r i n g s .

T h i s method i s u s e d f o r
low f l i g h t a l t i t u d e s , i . e . ,
with r a p i d l y changing v i s u a l
l'andmarks.

4.
Determination of the
d r i f t a n g l e of an a i r c r a f t b y
using a backsight.
T h e e s s e n c e o f t h i s m e t h o d l i e s i n t h e meas/202
urement o f t h e c o u r s e a n g l e a t which v i s u a l p o i n t s r e c e d e from t h e
a i r c r a f t . A f t e r s e t t i n g t h e s i g h t , as i n m e a s u r i n g t h e d r i f t a n g l e
i n t e r m s o f t h e l o c a t i o n o f v i s u a l p o i n t s (CAL = 180°, V A = 0 ) ,
t h e p i l o t waits u n t i l t h e c h a r a c t e r i s t i c v i s u a l p o i n t appears i n
t h e cross h a i r s of t h e s i g h t a t t h e p o s i t i o n of t h e bubble l e v e l
i n these cross hairs.
T h e n , k e e p i n g t h e a i r c r a f t s t r i c t l y on c o u r s e ,
t h e p i l o t w a i t s u n t i l t h e landmark l e a v e s t h e c r o s s h a i r s i n t h e
v e r t i c a l p l a n e a t a n a n g l e o f 4 0 - 5 0 ° a t a v e r a g e a l t i t u d e s or 1 5 ­
20° a t h i g h a l t i t u d e s .
Then, by t u r n i n g t h e s i g h t , h e matches t h e
v i s u a l p o i n t with t h e course marking and c a l c u l a t e s t h e d r i f t angle.

5.
D e t e r m i n a t i o n o f t h e d r i f t a n g l e o f an a i r c r a f t by s i g h t ­
ing forward.
I n m e a s u r i n g t h e d r i f t a n g l e by s i g h t i n g f o r w a r d ,
t h e s i g h t is s e t t o t h e zero course angle and a v i s u a l p o i n t i s
s e l e c t e d on t h e c o u r s e c h a r t , wh i ch p r e f e r a b l y l i e s a t a v e r t i c a l
a n g l e o f 4 5 or 2 6 . 5 O .
I n t h i s c a s e , w i t h VA = 45O, t h e d i s t a n c e
t o t h e l a n d m a r k w i l l b e e q u a l t o t h e a l t i t u d e , w h i l e a t V A = 26.5O
it w i l l equal half the altitude:
S = H t g VA.

Then, k e e p i n g t h e a i r c r a f t s t r i c t l y on c o u r s e , t h e s i g h t i s
s e t t o z e r o on t h e s c a l e o f v e r t i c a l a n g l e s a n d t h e p i l o t w a i t s
u n t i l t h e v i s u a l p o i n t c r o s s e s t h e t r a n s v e r s e l i n e on t h e c r o s s
h a i r s on t h e s i g h t ( t r a v e r s e of t h e l a n d m a r k ) .
Noting t h e l a t e r a l
d e v i a t i o n of t h e landmark i n d e g r e e s , i t i s p o s s i b l e t o d e t e r m i n e
t h e l i n e a r l a t e r a l d e v i a t i o n as f o l l o w s :

206

Sg = H t g V A .

of

The d r i f t a n g l e o f t h e a i r c r a f t i s d e t e r m i n e d a s t h e r a t i o
its i n i t i a l distance t o its f i n a l distance:

A t d r i f t a n g l e s o n t h e o r d e r o f l o o , t h e t a n g e n t US c a n b e
r e p l a c e d by i t s v a l u e , w h i l e t h e t a n g e n t VU2 can be r e p l a c e d by
t h e v a l u e o f t h e l a t e r a l d e v i a t i o n (LD):

or, w i t h a n i n i t i a l v a l u e o f V A 1 = 4 5 O ,
V A 1 = 2 6 . 5 O Y US = 2 L D .

US = L D ;

with an i n i t i a l

A l l t h r e e o f t h e s e m e t h o d s d e s c r i b e d a b o v e for d e t e r m i n i n g
t h e ' d r i f t a n g l e a r e u s e d i n l o c a t i o n s w h i c h h a v e many l a n d m a r k s ,
i . e . , where i t i s e a s y t o p i c k o u t a v i s u a l landmark a t t h e d e s i r e d
visual angle.

6 . D e t e r m i n a t i o n o f t h e g r o u n d s p e e d o f t h e a i r c r a f t b y means
a backsight.
To d e t e r m i n e t h e g r o u n d s p e e d b y t h i s m e t h o d , r h e
s i g h t i s s e t o n t h e c o u r s e a n g l e s c a l e t o 180°, a n d t o z e r o o n t h e
v e r t i cal angle scale.
The b u b b l e i n t h e l e v e l i s s e t a t t h e i n t e r ­
s e c t i o n of t h e c r o s s h a i r s .

of

Having s e l e c t e d t h e c h a r a c t e r i s t i c p o i n t as i t p a s s e s t h r o u g h / 2 0 3
t h e i n t e r s e c t i o n of t h e s i g h t , t h e s w e e p s e c o n d h a n d i s s t a r t e d
a n d t h e p i l o t w a i t s u n t i l t h i s p o i n t h a s moved t o a v e r t i c a l a n g l e
O f

35-40°.

T h e n t h e c o u r s e m a r k i n g o f t h e s i g h t i s made t o c o i n c i d e w i t h
t h e v i s u a l p o i n t a n d t h e s i g h t i s s e t t o V A = 45O. T h e moment w h e n
t h e v i s u a l p o i n t a g a i n c r o s s e s t h e i n t e r s e c t i o n of t h e c r o s s h a i r s
I n t h i s case, t h e
i n t h e s i g h t , t h e sweep second hand i s s t o p p e d .
p a t h t r a v e l e d by t h e a i r c r a f t w i l l be e q u a l t o t h e f l i g h t a l t i t u d e .
C o n s e q u e n t l y , t h e ground s p e e d c a n b e d e t e r m i n e d by t h e s i m p l e form­
ula

w h e r e H i s t h e f l i g h t a l t i t u d e a n d t i s t h e t i m e m e a s u r e d by t h e
sweep s e c o n d hand.

7.
D e t e r m i n a t i o n of t h e d r i f t a n g l e and t h e ground s p e e d of
t h e a i r c r a f t from a landmark l o c a t e d t o t h e s i d e .
T h i s method i s
u s e d i n t h e c a s e when i t i s d e s i r e d t o m e a s u r e t h e d r i f t a n g l e a n d
t h e ground speed and t h e p i l o t h a s o n l y one landmark a t h i s d i s p o s a l ,
20 7

which i s n o t l o c a t e d a l o n g t h e l i n e
c a r e f u l t o keep t h e aircraft s t r i c t l
t h e s i g h t a t t h e landmark and waits
t o 4 5 or 315O, d e p e n d i n g o n w h e t h e r
t h e f l i g h t path of t h e aircraft.

of f l i g h t o f t h e a i r c r a f t .
Being
y on c o u r s e , h e l o o k s t h r o u g h
u n t i l its course angle is equal
i t i s t o t h e l e f t or r i g h t o f

A t a c o u r s e a n g l e f o r t h e l a n d m a r k o f 4 5 or 3 1 5 O , t h e v e r t ­
i c a l a n g l e of t h e landmark i s measured and t h e sweep second hand
is started.
Leaving t h e s e t t i n g of t h e
v e r t i c a l a n g l e i n t h e same p o s i ­
tion, the sight is rotated t o follow
t h e motion of t h e landmark, n o t i n g
i t s f i x e d p o s i t i o n on t h e c o u r s e
A t t h e beginning (up t o
chart.
CAL = 90° + U S ) , t h e d i s t a n c e t o
t h e landmark w i l l d e c r e a s e , b u t
C3n­
then w i l l increase again.
s e q u e n t l y , t h e landmark w i l l a t
f i r s t move away t o o n e s i d e f r o m
t h e i n t e r s e c t i o n of t h e cross h a i r s
a n d w i l l t h e n a g a i n b e g i n t.0 a p p r o a c h
i t . A t t h e moment when t h e l a n d ­
mark i s a t t h e i n t e r s e c t i o n of
t h e c r o s s h a i r s , t h e sweep s e c o n d
hand i s stopped and t h e course
a n g l e o f t h e landmark i s calculated.

---

-\

-/--

F i g . 2. 3 8 .
Determining t h e
D r i f t A n g l e a n d Ground Speed
b y a Landmark L o c a t e d t o t h e
Side.

If C A L I = 4 5 O , t h e b i s e c t r i x o f t h e t r i a n g l e OAB ( F i g .
w i l l be l o c a t e d a t t h e c o u r s e a n g l e , which i s e q u a l t o :
45O+CAL2
2

-

CALbis -

2.38)

Y

w h i l e t h e d r i f t a n g l e of t h e a i r c r a f t w i l l b e e q u a l t o CALbis-900,
so that

us

=

CAL2-135O.
2

I f C A L I = 315O,
CALbis

+

-- 315O

US = C A L

bis

-270°

or

us
208

=

CAL7 - 2 25O
2

CALz

2

/204

A t p o i n t s 1 and 2 t h e d i s t a n c e from t h e a i r c r a f t t o t h e land­
mark i s e q u a l t o

Consequently, t h e d i s t a n c e between p o i n t s 1 and 2 i s d e t e r ­
mined by t h e f o r m u l a
51-2

= 2H t g V A s i n

C A L 1+CALL
2

C l e a r l y , t h e r e a s o n f o r t h e change i n t h e c o u r s e a n g l e of t h e
landmark from C A L I t o C A L 2 , w a s t h e s h i f t of t h e a i r c r a f t from p o i n t
0 t o p o i n t 01, s o t h a t

Consequently, t h e ground speed i s

The m a j o r i t y o f n a v i g a t i o n a l p r o b l e m s w h i ch w e h a v e d i s c u s s e d ,
w h i c h a r e s o l v e d b y m e a n s o f m e c h a n i c a l or o p t i c a l s i g h t s , c a n b e
s o l v e d u s i n g t h e r a d i o d e v i c e s which a r e i n s t a l l e d nowadays a b o a r d
modern t u r b o p r o p a n d j e t a i r c r a f t , which w i l l b e d e s c r i b e d i n t h e
next chapter.

20 9

I-


8.

Automatic Navigation Instruments

I n S e c t i o n 2 of C h a p t e r I , i t w a s m e n t i o n e d t h a t i n t h e
g e n e r a l c a s e , a l l t h e e l e m e n t s of a f l i g h t r e g i m e a r e n o t s t r i c t l y
f i x e d , with t h e exception of t h e extreme p o i n t s of d e v i a t i o n from
a given trajectory.
T h e r e f o r e , t h e crew o f a n a i r c r a f t m u s t c o n ­
s t a n t l y d e a l with average values of measured n a v i g a t i o n a l elements
(average course, average speed, average wind, e t c . ) .
If a l l t h e e l e m e n t s which h a v e b e e n m e n t i o n e d h a d a c o n s t a n t
/205
given v a l u e , t h e p r a c t i c a l problems of a i r c r a f t n a v i g a t i o n could
b e s o l v e d q u i t e simply and t h e q u e s t i o n o f automating t h e p r o c e s s e s
o f a i r c r a f t n a v i g a t i o n would b e s u p e r f l u o u s .

Without u s i n g automatic n a v i g a t i o n d e v i c e s , t h e p i l o t of an
a i r c r a f t must s y s t e m a t i c a l l y c a r r y o u t o b s e r v a t i o n s u s i n g a l l t h e
n a v i g a t i o n a l i n s t r u m e n t s , a v e r a g e them f o r t i m e i n t e r v a l s c o v e r i n g
t h e o b s e r v a t i o n t i m e , and r e c o r d t h e t i m e of change i n average v a l u e s
of t h e measured parameters.
This introduces considerable tedious
However, i f t h e p i l o t
work i n t o t h e t a s k o f a i r c r a f t n a v i g a t i o n .
d i d n o t d e v o t e s u f f i c i e n t a t t e n t i o n t o c h a n g e s i n t h e e l e m e n t s of
a i r c r a f t n a v i g a t i o n , i t w o u l d s o o n h a v e a n e f f e c t on t h e p r e c i s i o n
of aircraft navigation.
T h e s i m p l e s t d e v i c e u s e d for a u t o m a t i n g t h e c o m p u t a t i o n o f t h e
a i r c r a f t p a t h i n terms o f t h e c h a n g i n g v a l u e s of n a v i g a t i o n a l param­
e t e r s and times i s t h e a u t o m a t i c n a v i g a t i o n a l d e v i c e , which h a s been
d e v i s e d on t h e b a s i s o f t h e g e n e r a l f e a t u r e s o f a i r c r a f t n a v i g a t i o n .
A t t h e p r e s e n t t i m e , t h e n a v i g a t i o n i n d i c a t o r Type N I - S O B , i s
widely used.
We s h a l l now d i s c u s s i t s d e s i g n a n d t h e m e t h o d o f i t s
application.
The NI-50B n a v i g a t i o n i n d i c a t o r i s a n a u t o m a t i c n a v i g a t i o n d e v i c e
w h i c h c a l c u l a t e s t h e p a t h o f t h e a i r c r a f t on t h e b a s i s o f s i g n a l s
f r o m s e n s o r s for t h e c o u r s e a n d a i r s p e e d , t a k i n g i n t o a c c o u n t t h e
measured wind s p e e d d u r i n g f l i g h t .
I n a d d i t i o n , t h e i n d i c a t o r can
b e u s e d t o d e t e r m i n e t h e wind p a r a m e t e r s a t t h e f l i g h t a l t i t u d e .
C a l c u l a t i o n o f t h e p a t h of t h e a i r c r a f t w i t h t h e u s e of t h e
N I - S O B c a n b e p e r f o r m e d b o t h on t h e b a s i s o f o r t h o d r o m i c s y s t e m s
of c o o r d i n a t e s f o r s t r a i g h t - l i n e f l i g h t s e g m e n t s , as w e l l as i n a
r e c t a n g u l a r s y s t e m of c o o r d i n a t e s w i t h a n y o r i e n t a t i o n o f i t s a x e s .
W i t h o u t g o i n g i n t o t h e d e t a i l s of t h e d e s i g n o f t h e i n s t r u m e n t ,
l e t us examine i t s s c h e m a t i c diagram, p u r p o s e , and o p e r a t i n g p r i n ­
c i p l e s o f t h e i n d i v i d u a l p a r t s , as w e l l as t h e ways i n w h i c h t h e
s y s t e m as a w h o l e c a n b e e m p l o y e d .
autoThe n a v i g a t i o n i n d i c a t o r c o n s i s t s of t h e f o l l o w i n g p a r t s :
matic speed i n d i c a t o r , c o n t r o l u n i t , automatic course-setting device,
wind i n d i c a t o r , and d e v i c e f o r c a l c u l a t i n g t h e a i r c r a f t c o o r d i n a t e s
(Fig. 2.39).

210


­

The a u t o m a t i c s p e e d c o n t r o l c o n s i s t s o f a d e v i c e w h i c h c o n v e r t s
t h e p r e s s u r e from t h e s e n s o r s of t o t a l and s t a t i c p r e s s u r e i n t o elec­
trical s i g n a l s , corresponding i n value t o t h e airspeed of t h e air­
c r a f t , a c c o r d i n g t o Formula ( 2 . 4 7 a )

2060 T

The a u t o m a t i c s p e e d c o n t r o l h a s t wo h o r i z o n t a l m a n o m e t r i c b o x e s .
One o f t h e m ( a n e r o i d 1) i s u s e d t o m e a s u r e t h e s t a t i c p r e s s u r e , w h i l e
t h e o t h e r i s u s e d t o measure t h e a e r o d y n a m i c p r e s s u r e 2 as t h e d i f f e r ­
ence between p t o t a l and pst.
/206
Both b o x e s a r e c o n n e c t e d by means o f l i n k i n g mechanisms t o po­
t e n t i o m e t e r s 3, which r e g u l a t e t h e c u r r e n t r a t i o i n t h e b a l a n c i n g
c i r c u i t , a c c o r d i n g t o t h e r a t i o o f t h e dynamic p r e s s u r e t o t h e s t a t i c
pressure.
I t i s c l e a r f r o m F o r m u l a ( 2 . 4 7 , a ) t h a t t h e r a t i o o f t h e dy­
namic p r e s s u r e t o t h e s t a t i c p r e s s u r e i s n o t l i n e a r l y r e l a t e d t o
the airspeed of t h e aircraft.
In order t o develop e l e c t r i c a l s i g n a l s
which a r e p r o p o r t i o n a l t o t h e a i r s p e e d , t h e c o n t r o l u n i t c o n t a i n s
an a u t o m a t i c s p e e d c o n t r o l mechanism.
T h i s mechanism c o n s i s t s o f
a m a g n e t i c s i g n a l a m p l i f i e r 4 , coming from t h e a u t o m a t i c s p e e d con­
t r o l , a c t i v a t i n g motor 5 , and a p o t e n t i o m e t e r 6 w i t h a s p e c i a l pro­
f i l e , which l e v e l s o u t t h e n o n l i n e a r i t y of t h e s i g n a l s from t h e a u t o ­
m a t i c airspeed control.
T h u s , t h e t u r n a n g l e of t h e a x i s o f t h e
p o t e n t i o m e t e r of t h e a n a l y z i n g mechanism becomes p r o p o r t i o n a l t o
the airspeed.

S

amp l i f i e r

coor

automatic
course control
Fig.

2.39.

S c h e m a t i c Diagram o f N a v i g a t i o n a l I n d i c a t o r .

211

By m e a n s of a s e c o n d p o t e n t i o m e t e r , c o n n e c t e d b y i t s a x i s o f
r o t a t i o n t o t h e a c t i v a t i n g mechanism, s e n d s o u t e l e c t r i c a l s i g n a l s
which are p r o p o r t i o n a l t o t h e a i r s p e e d , i n t h e form o f a DC v o l t ­
age.
The a u t o m a t i c c o u r s e c o n t r o l i s i n t e n d e d t o d i s t r i b u t e t h e s i g ­
n a l s which are p r o p o r t i o n a l t o t h e a i r s p e e d , a l o n g t h e axes of t h e
c o o r d i n a t e s for c a l c u l a t i n g t h e p a t h .

L e t us assume t h a t w e must make a f l i g h t o v e r a p a t h segment
with t h e orthodromic f l i g h t angle $ (Fig. 2.40).
I f t h e a i r c r a f t i s now t o f l y w i t h a n o r t h o d r o m i c c o u r s e y ,
t h e a i r s p e e d must b e d i v i d e d i n t o two c o m p o n e n t s :

V, = V cos (7- Jy);
Vz = V sin (1

-+).

I t i s clear t h a t i f t h e r e i s no wind a t t h e f l i g h t a l t i t u d e ,
t h e s e c o m p o n e n t s of t h e a i r s p e e d m u s t b e m u l t i p l i e d b y t h e f l i g h t
t i m e t o g i v e us t h e change i n t h e a i r c r a f t c o o r d i n a t e s d u r i n g t h i s
time:
A$ = V&. A2 = V&.

Fig.

2.40.

Fig.

2.41.

Fig.

2.40.

D i s t r i b u t i o n of t h e Airspeed Vector a l o n g t h e Coordi­
n a t e Axes.

Fig.

2.41.

Sine-Cosine

Distributor.

T h e d i v i s i o n of t h e c o u r s e s i g n a l s b y t h e a x e s o f t h e c o o r d ­
i n a t e s i n t h e a u t o m a t i c c o u r s e c o n t r o l i s a c c o m p l i s h e d by m e a n s o f
a sine-cosine potentiometer (Fig. 2.41).
T h e s i n e - c o s i n e p o t e n t i o m e t e r c o n s i s t s of a c i r c u l a r w i n d i n g
w i t h power s u p p l i e d t o it a t two d i a m e t r i c a l l y o p p o s i t e p o i n t s .

212


/207

Two p a i r s o f p i c k u p s s l i d e a l o n g t h e c o i l s ; t h e y a r e l o c a t e d a t r i g h t
a n g l e s t o one a n o t h e r .
Obviously, i f w e say t h a t t h e zero p o s i t i o n of t h e pickups is
t h e one i n which one p a i r ( c o s i n e ) c o i n c i d e s w i t h t h e s u p p l y l e a d s
a n d t h e s e c o n d ( s i n e ) w i l l b e l o c a t e d a t a n a n g l e of 90° t o t h e m ,
t h e n t h e maximum c u r r e n t w i l l f l o w t h r o u g h t h e f i r s t p a i r o f p i c k u p s
while t h a t through t h e second p a i r w i l l be zero.
By t u r n i n g t h e
pickups from z e r o t o 90°, t h e c u r r e n t i n t h e c o s i n e p i c k u p s w i l l
d r o p f r o m maximum t o z e r o a n d t h a t i n t h e s i n e p i c k u p s w i l l i n c r e a s e
f r o m z e r o t o t h e maximum.
However, t h e c h a n g e i n t h e c u r r e n t i n
t h e pickups w i l l n o t t a k e p l a c e according t o t h e s i n e and cosine
l a w s , b u t p r o p o r t i o n a t e l y t o t h e a n g l e of r o t a t i o n of t h e p i c k u p s .
I n o r d e r f o r t h e l a w of change of c u r r e n t s t o approach t h e s i n e c o s i n e , t h e w i n d i n g o f t h e p o t e n t i o m e t e r i s g i v e n a p r o f i l e or i s
/208
f i t t e d with special regulating shunt r e s i s t o r s .
Rotation of t h e pick-up shoes of t h e potentiometer i s involved
i n f i g u r i n g t h e c o u r s e of t h e a i r c r a f t which i s a r r i v i n g from a c o u r s e
s y s t e m or o t h e r c o u r s e i n s t r u m e n t .
I n o r d e r t o a p p l y t h e components of t h e a i r s p e e d t o t h e r e c e i v ­
i n g system f o r c a l c u l a t i n g t h e a i r c r a f t coordinates, t h e c i r c u l a r
w i n d i n g o f t h e p o t e n t i o m e t e r i s made m o v a b l e a n d c a n b e m o u n t e d i n
any p o s i t i o n by means o f a r a c k a n d p i n i o n , l o c a t e d i n t h e a u t o m a t i c
c o u r s e c o n t r o l , and a s p e c i a l scale f o r c a l c u l a t i n g i t s p o s i t i o n .
T h e a n g l e for s t u d y i n g t h e s y s t e m o f c o o r d i n a t e s f o r c a l c u l a t ­
i n g t h e p a t h r e l a t i v e t o t h e m e r i d i a n from which t h e a i r c r a f t c o u r s e
is measured i s c a l l e d t h e chart angle.
I n t h e majority of cases,
t h e c h a r t a n g l e i s made e q u a l t o t h e o r t h o d r o m i c p a t h a n g l e o f t h e
path segment.
Hence, by a p p l y i n g t o t h e w i n d i n g of t h e s i n e - c o s i n e p o t e n t i o m ­
e t e r a v o l t a g e which i s p r o p o r t i o n a l t o t h e a i r s p e e d , w e o b t a i n s i g n a l s
a t t h e o u t p u t s of t h e p o t e n t i o m e t e r w h i c h a r e p r o p o r t i o n a l t o t h e
component o f t h e a i r s p e e d a l o n g t h e a x e s of t h e c o o r d i n a t e s Vx a n d

vz *
For a p r e c i s e r e g u l a t i o n o f t h e n a v i g a t i o n a l i n d i c a t o r a s a
w h o l e , t h e s e s i g n a l s a r e c a l i b r a t e d m a n u a l l y by means o f a p o t e n ­
t i o m e t e r ( s e e Fig. 2.39, P o s i t i o n 81, l o c a t e d i n t h e c o n t r o l u n i t .
The w i n d sensor h a s a s c h e m a t i c s i m i l a r t o t h a t f o u n d i n t h e
a u t o m a t i c c o u r s e c o n t r o l , w i t h t h e e x c e p t i o n t h a t t h e v o l t a g e which
i s p r o p o r t i o n a l t o t h e windspeed i s analyzed d i r e c t l y a t t h e s e n s o r
by means o f a p o t e n t i o m e t e r ( s e e F i g . 2 . 3 9 , P o s i t i o n 9 ) and i s s e t
by m a n u a l l y t u r n i n g knob "u" s o t h a t t h e s e t t i n g of t h e p i c k - u p s h o e s
on t h e s i n e - c o s i n e p o t e n t i o m e t e r a g r e e s w i t h t h e wind d i r e c t i o n .
T h u s , w e h a v e t h r e e s e t p a r a m e t e r s on t h e w i n d s e n s o r :

the

213

wind s p e e d (u), t h e wind ' d i r e c t i o n

(61,

and t h e c h a r t angle (+).

+

It i s c l e a r t h a t t h e d i f f e r e n c e between a n g l e s 6 and
gives
A s a result, we obtain signals at the
t h e p a t h a n g l e of t h e wind.
o u t p u t of t h e s i n e - c o s i n e p o t e n t i o m e t e r which a r e p r o p o r t i o n a l t o
t h e component o f t h e wind s p e e d a l o n g t h e a x e s o f t h e c o o r d i n a t e s
f o r calculating the path.
The o u t p u t s of t h e s i n e - c o s i n e p o t e n t i o m e t e r s o f t h e a u t o m a t i c
c o u r s e c o n t r o l and t h e wind s e n s o r a r e c o n n e c t e d i n s e r i e s , s o t h a t
w e o b t a i n s i g n a l s a t t h e i r common o u t p u t s w h i c h a r e p r o p o r t i o n a l
as f o l l l o w s
V , + U ~ =V C o s ( ~ - + ) + u j o s A W
V z + u z = V s i n ( y - + ) + u s i n AW

i . e . , s i g n a l s which m a k e i t p o s s i b l e t o c a l c u l a t e t h e p a t h of t h e
a i r c r a f t w i t h t i m e , c o n s i d e r i n g t h e manual s e t t i n g o f t h e wind v a l u e
for the flight altitude.
The c o o r d i n a t e caZcuZator c o n s i s t s o f t w o i n t e g r a t i n g m o t o r s
/209
t h a t w o r k on d i r e c t c u r r e n t ( s e e F i g . 2 . 3 9 , P o s i t i o n lo), w h o s e s p e e d
of r o t a t i o n s t r i c t l y c o r r e s p o n d s t o t h e m a g n i t u d e o f t h e s i g n a l s
coming f r o m t h e a u t o m a t i c c o u r s e c o n t r o l a n d t h e wind s e n s o r .
The
revolutions of
t h e m o t o r s a r e summed b y t w o c o u n t e r s , w h o s e r e a d i n g s
a r e shown on a s c a l e which i s g r a d u a t e d i n k i l o m e t e r s o f p a t h cov­
e r e d by t h e a i r c r a f t a l o n g t h e c o r r e s p o n d i n g a x e s .
A p o i n t e r m a r k e d "N" s h o w s t h e p a t h o f t h e a i r c r a f t a l o n g t h e
X - a x i s , ; . e . , a l o n g t h e o r t h o d r o m e , w h i l e a p o i n t e r m a r k e d "E" s h o w s
t h e t r a v e l a l o n g t h e Z - a x i s , or t h e l a t e r a l d e v i a t i o n f r o m t h e d e s i r e d
l i n e of f l i g h t .
T h e n a m e s o f t h e p o i n t e r s ("N" a n d " E " ) w e r e g i v e n b e c a u s e a t
a c h a r t a n g l e e q u a l t o z e r o , t h e p o i n t e r "N" w i l l s h o w t h e p a t h
t r a v e l e d b y t h e a i r c r a f t i n a n o r t h e r l y d i r e c t i o n from t h e s t a r t ­
i n g p o i n t w h i l e t h e p o i n t e r "E" shows t r a v e l i n a n e a s t e r l y d i r e c ­
tion.
To s e t t h e p o i n t e r s o f t h e c o u n t e r t o z e r o ( a t t h e s t a r t i n g
p o i n t o f a r o u t e ) or t o t h e a c t u a l c o o r d i n a t e s o f t h e a i r c r a f t w h e n
c o r r e c t i n g i t s c o o r d i n a t e s , t h e r e i s a s p e c i a l r a c k and p i n i o n which
i s u s e d t o t u r n t h e "N" p o i n t e r w h e n i t i s p u s h e d i n w a r d a n d t o t u r n
t h e ''E" p o i n t e r w h e n i t i s p u l l e d o u t .

9 . P r a c t i c a l Methods o f A i r c r a f t N a v i g a t i o n Using
Geotechni c a l Devices
F l i g h t e x p e r i e n c e shows t h a t i n a d d i t i o n t o a knowledge o f t h e
devices f o r determining each of t h e elements of aircraft navigation,
s u c c e s s f u l c o m p l e t i o n o f a f l i g h t , means t h a t i t i s n e c e s s a r y t o

214

o b t a i n and use t h e measured v a l u e s , ? . e . , t o master t h e d e v i c e s used
f o r aircraft navigation p r i o r t o automation.
T h e s e d e v i c e s d o n o t d e p e n d on s y s t e m s o f m e a s u r i n g f l i g h t a n g l e s
and a i r c r a f t c o u r s e s , s i n c e t h e y have l i m i t e d f i e l d s of a p p l i c a t i o n .
I n a d d i t i o n , i n d e s c r i b i n g them, i t i s necessary t o r e c a l l t h a t t h e
readings of navigational devices contain a l l necessary corrections.
T h e r e f o r e , i n t h e formulas which have been found t o b e n e c e s s a r y ,
w e h a v e u s e d t h e common d e s i g n a t i o n s f o r n a v i g a t i o n a l p a r a m e t e r s .
U n d e r p r a c t i c a l c o n d i t i o n s of a i r c r a f t n a v i g a t i o n , a n i m p o r ­
t a n t r o l e i s p l a y e d by t h e p i l o t s ' c a l c u l a t i n g a n d m e a s u r i n g i n s t r u ­
ments.
H o w e v e r , i n many c a s e s , i n s t e a d o f u s i n g t h e s e i n s t r u m e n t s ,
approximate c a l c u l a t i o n s are performed m e n t a l l y .
Approximate m e n t a l
e x t i m a t e s c a n b e u s e d t o a d v a n t a g e i n a l l c a s e s when t h e p r o b l e m
c a n b e s o l v e d more p r e c i s e l y b y means o f c a l c u l a t i n g i n s t r u m e n t s
i n o r d e r t o a v o i d any chance g r o s s e r r o r s .
Methods of a p p r o x i m a t e ( y e t s u f f i c i e n t l y a c c u r a t e f o r p r a c t i c a l
p u r p o s e s ) e s t i m a t i o n of n a v i g a t i o n a l e l e m e n t s i n f l i g h t w i t h o u t t h e
use of c a l c u l a t i n g and measuring i n s t r u m e n t s a r e c a l l e d p i l o t s ' vis­
ual estimates.
The r u l e s f o r p i l o t s ' v i s u a l e s t i m a t e s w i l l b e g i v e n
l a t e r on i n t h e d e s c r i p t i o n o f t h e s u i t a b l e m e t h o d s o f a i r c r a f t n a v i g a ­
tion.
Takeoff

of

t h e A i r c r a f t a t t h e S t a r t i n g Point of

t h e Route

/210

T h e s t a r t i n g p o i n t of t h e r o u t e ( S P R ) i s t h e f i r s t c o n t r o l l a n d ­
mark a l o n g t h e f l i g h t p a t h f r o m w h i c h t h e a i r c r a f t w i l l t r a v e l a l o n g
t h e r o u t e a t a given path angle $.
T h e f i n a l p o i n t on t h e r o u t e ( F P R ) i s t h e l a s t c o n t r o l l a n d ­
m a r k a l o n g t h e r o u t e , from which t h e maneuver t o l a n d t h e a i r c r a f t
begins.
Regardless of t h e f a c t t h a t t h e p a t h a n g l e of t h e f l i g h t i s
u s u a l l y r e c k o n e d from t h e a i r p o r t from which t h e a i r c r a f t t o o k o f f
u p t o t h e S P R Y a s w e l l a s f r o m t h e FPR t o t h e a i r p o r t w h e r e i t i s
t o l a n d , t h e s e values have s i g n i f i c a n c e only f o r g e n e r a l o r i e n t a t i o n
i n t h e v i c i n i t y of t h e a i r p o r t s .
I n connection with t h e fact t h a t t h e f i r s t t u r n of t h e aircraft
a f t e r t a k e o f f i s made a f t e r t h e a i r c r a f t r e a c h e s a c e r t a i n a l t i t u d e
( 2 0 0 m y e . g . 1 a n d t h a t many f a c t o r s i n f l u e n c e t a k e o f f c o n d i t i o n s
( s u c h as a t m o s p h e r i c p r e s s u r e , w i n d s p e e d a n d d i r e c t i o n , f l y i n g w e i g h t
o f t h e a i r c r a f t , e t c . ) , a n e x a c t d e t e r m i n a t i o n of t h e l o c a t i o n o f
t h e b e g i n n i n g and end of a t u r n i s u s u a l l y d i f f i c u l t .
Therefore,
t h e p a t h a n g l e and t h e d i s t a n c e from t h e f i r s t t u r n t o t h e SPR h a s
a v a r i a b l e n a t u r e and cannot be determined e x a c t l y .
Methods o f b r i n g i n g t h e a i r c r a f t t o t h e i n i t i a l p o i n t on t h e
r o u t e d i f f e r somewhat f r o m t h e g e n e r a l methods o f a i r c r a f t n a v i g a t i o n
along t h e f l i g h t route.

215


I

.I

The b a s i c d i f f e r e n c e b e t w e e n t h e m e t h o d s o f a i r c r a f t n a v i g a t i o n
i n v o l v e d i n b r i n g i n g an a i r c r a f t t o t h e SPRY and t h e a i r c r a f t navi­
g a t i o n a l o n g t h e r o u t e , i s t h a t i n t h e f i r s t case w e do n o t h a v e
a s t r i c t l y determined p a t h a n g l e f o r t h e f l i g h t and can r e a c h t h e
g i v e n p o i n t from a n y d i r e c t i o n , i . e . , i n t h e g i v e n case t h e n a v i ­
g a t i o n i s made i n a p o l a r s y s t e m o f c o o r d i n a t e s .
I n t h e second case,
w e h a v e a g i v e n l i n e of f l i g h t , a n d t h e a i r c r a f t n a v i g a t i o n t a k e s
p l a c e along a s t r a i g h t - l i n e orthodromic system of c o o r d i n a t e s .
I n Figure 2.42, a , w e see t h a t t h e f l i g h t p a t h a n g l e from t h e
c e n t e r o f a n a i r p o r t i n t h e d i r e c t i o n a l o n g t h e SPR a n d t h e s h o r t ­
e s t l i n e f o r t h e a i r c r a f t ' s p a t h t o t h e SPR a f t e r t a k e o f f a n d g a i n ­
i n g a l t i t u d e u n t i l t h e first t u r n are a t r i g h t angles.
S i n c e t h e l i n e o f f l i g h t i s n o t c o n s t a n t when t h e a i r c r a f t r e a c h e s
t h e SPR, t h e problem i n v o l v e s b r i n g i n g t h e a i r c r a f t t o a g i v e n p o i n t
w i t h t h e minimum n u m b e r o f c h a n g e s i n t h e c o u r s e , o r ( i n o t h e r w o r d s )
along the shortest path.
P r a c t i c a l l y speaking, v i s u a l c o n t r o l of an a i r c r a f t t o b r i n g
i t t o t h e SPR i s d o n e a s f o l l o w s .
If t h e c o u r s e l e a d i n g f r o m t h e a i r p o r t t o t h e SPR d i f f e r s f r o m
t h e t a k e o f f c o u r s e by l e s s t h a n 90°, a f t e r t a k e o f f a n d g a i n i n g t h e
d e s i r e d a l t i t u d e , t h e a i r c r a f t makes a r i g h t a n g l e t u r n t o t h e s t a r t ­
i n g p o i n t o f t h e r o u t e , s o t h a t when t h e w i n d i n t h e v i c i n i t y o f
t h e a i r p o r t c a u s e s d r i f t i n g of t h e a i r c r a f t t o t h e r i g h t , t h e l a n d ­
mark w h i c h i s d e s i g n a t e d a s t h e SPR m u s t r e m a i n t o t h e r i g h t o f t h e
a i r c r a f t c o u r s e b y 5 - 1 0 ° , d e p e n d i n g on t h e d i r e c t i o n a n d s p e e d o f
/211
t h e wind.
I n t h e case of a l e f t - h a n d d r i f t , t h e S P R m u s t r e m a i n
a t t h e same a n g l e t o t h e l e f t o f t h e a i r c r a f t c o u r s e .
W i t h t h e p r o p e r s e l e c t i o n o f t h e c o u r s e t o t h e S P R Y ; . e . , when
t h e l e a d angle (LA) i s e q u a l i n v a l u e t o t h e d r i f t angle of t h e a i r ­
c r a f t , t h e landmark w i l l be observed a t a c o n s t a n t a n g l e t o t h e axis
o f t h e a i r c r a f t , CAL = c o n s t ( F i g . 2.42, b ) .
I n t h i s case, it i s necessary t o c o n t i n u e t h e f l i g h t along t h e
p r e v i o u s c o u r s e u n t i l t h e SPR i s p a s s e d or ( i n h i g h - s p e e d a i r c r a f t )
u n t i l t h e r e i s a l i n e a r l e a d on t h e t u r n .
If t h e d r i f t a n g l e t u r n s o u t t o b e l e s s t h a n t h e l e a d w h i c h
has been t a k e n ( F i g . 2.42, c ) , a s l i p p i n g o f t h e landmark w i l l be
observed from t h e d i r e c t i o n of t h e l o n g i t u d i n a l a x i s of t h e a i r c r a f t .
I n t h i s case, t h e a i r c r a f t must b e s h i f t e d i n t h e d i r e c t i o n o f t h e
landmark s o t h a t . i t s c o u r s e a n g l e t u r n s o u t t o b e l e s s t h a n t h e i n i t i a l
one.
The s l i p p i n g of t h e l a n d m a r k i n t h e
i n a l a x i s o f t h e a i r c r a f t (Fig'. 2 . 4 3 , d )
which has been t a k e n is l e s s t h a n t h e d r
m u s t b e t u r n e d away f r o m t h e l a n d m a r k s o
g r e a t e r than t h e i n i t i a l one.
216

d i r e c t i o n of t h e l o n g i t u d ­
indicates that the lead
i f t a n g l e , and t h e a i r c r a f t
t h a t its course angle is

T h u s , t h e c o u r s e t o b e f o l l o w e d by t h e a i r c r a f t i s s e t v i s u a l l y
This problem i s
when t h e S P R i s l o c a t e d a l o n g a s t r a i g h t l i n e .

Fig. 2.42.
L i n i n g Up a n A i r c r a f t w i t h t h e SPR: ( a ) P a t h
Angle ( $ 1 and S h o r t e s t D i s t a n c e ( S I ;
( b ) Aircraft Course
Chosen C o r r e c t l y ;
( c ) A i r c r a f t Course must b e I n c r e a s e d ;
(d)
A i r c r a f t C o u r s e Must b e D e c r e a s e d .
b e s t s o l v e d when t h e r e i s a n a v i g a t i o n l e v e l on b o a r d , b y u s i n g t h e ./-212
s o - c a l l e d method of h a l f c o r r e c t i o n s .
T h i s method i n v o l v e s t h e f o l ­
i f t h e l e a d which h a s b e e n t a k e n t u r n s o u t t o b e g r e a t e r
lowing:
or l e s s t h a n t h e r e q u i r e d o n e , i t t h e n c h a n g e s i n t h e r e q u i r e d d i r e c ­
If t h i s t u r n s
t i o n by h a l f of t h e i n i t i a l l e a d which w a s t a k e n .
o u t t o b e i n s u f f i c i e n t , i t i s c h a n g e d a g a i n by h a l f of t h e i n i t i a l
v a l u e u n t i l t h e c o u r s e a n g l e b e c o m e s s t a b l e or t h e s i g n o f t h e c o r ­
r e c t i o n must be changed t o t h e o p p o s i t e .
R e v e r s e c o r r e c t i o n i s made b y o n e - f o u r t h o f t h e i n i t i a l l e a d ,
a n d i f t h i s i s i n s u f f i c i e n t or t o o m u c h , a c o r r e c t i o n i s made w h i c h
i s e q u a l t o o n e - e i g h t h of t h e i n i t i a l l e a d .
I t is not usually neces­
s a r y t o b r e a k down t h e c o r r e c t i o n s m o r e t h a n e i g h t t i m e s , s i n c e t h e
v a l u e o f t h e c o r r e c t i o n w i l l t h e n b e n o more t h a n 1-1.5O , w h i c h i s
no l o n g e r of p r a c t i c a l i m p o r t a n c e f o r v i s u a l a i r c r a f t n a v i g a t i o n .
I n t h e absence of a s i g h t aboard t h e a i r c r a f t , t h e c o u r s e a n g l e s
for t h e SPR a r e d e t e r m i n e d b y v i s u a l o b s e r v a t i o n ; t o s o l v e t h i s p r o b ­
lem, t h e p i l o t r e q u i r e s a c e r t a i n d e g r e e of e x p e r i e n c e which i s
gained i n t h e course of t h e t r a i n i n g of f l i g h t c r u i s e i n a c t u a l
f l i g h t or i n s p e c i a l t r a i n i n g d e v i c e s , a s w e l l a s i n p r a c t i c e f l i g h t s .

217


l i-

S e l e c t i n g the C o u r s e to be F o l l o w e d f o r the F l i g h t Route
The c o u r s e t o b e f o l l o w e d b y t h e a i r c r a f t a l o n g t h e f l i g h t
r o u t e n o t o n l y must b e s e t s o t h e air'craft p a s s e s o v e r c e r t a i n c o n t r o l
landmarks i n t h e p r o p e r o r d e r , b u t must a l s o e n s u r e t h a t t h e f l i g h t
t a k e s p l a c e e x a c t l y a c c o r d i n g t o t h e g i v e n l i n e of f l i g h t .
There are t h r e e p r i n c i p a l methods o f s e l e c t i n g t h e c o u r s e t o
be followed:
(a)
When d e v i a t i o n s o c c u r f r o m t h e l i n e o f a g i v e n p a t h
during the f l i g h t ,
(b)

A t a landmark a l o n g t h e l i n e ,

(c)

I n t h e d i r e c t i o n o f t h e landmark p o i n t s ,

(LGP)

The m o s t u n i v e r s a l a n d w i d e l y u s e d m e t h o d i s t h e f i r s t o n e .
a f t e r f l y i n g over a c e r t a i n
T h i s method i n v o l v e s t h e f o l l o w i n g :
control point, the calculated course t o be followed along the given
l i n e of f l i g h t i s d e t e r m i n e d as f o l l o w s
y

=

$

-

a

calc'

which t h e a i r c r a f t f o l l o w s u n t i l t h e f i r s t c h a r a c t e r i s t i c p o i n t
along the f l i g h t path.
I f , a t t h e moment t h a t i t i s f l y i n g o v e r t h i s p o i n t , t h e a i r ­
c r a f t t u r n s o u t t o b e on t h e g i v e n l i n e o f f l i g h t , t h e c o u r s e i s
then considered t o be sufficiently correct.
I f t h e a i r c r a f t h a s u n d e r g o n e some s h i f t t o t h e r i g h t w h e n
it p a s s e s o v e r t h i s p o i n t , t h e l i n e a r l a t e r a l d e v i a t i o n from t h e
d e s i r e d l i n e of f l i g h t i s determined and t h e r e q u i r e d c o r r e c t i o n
i s f o u n d for t h e c o u r s e o f t h e a i r c r a f t :

t g Ay =

LLD
­

sc

,

where LLD i s t h e l i n e a r l a t e r a l d e v i a t i o n and Sc i s t h e d i s t a n c e
covered.

/213

E x a m p l e : An a i r c r a f t h a s f l o w n f r o m a c o n t r o l l a n d m a r k f o r
a d i s t a n c e o f 36 km a n d h a s d e v i a t e d 3 km t o t h e r i g h t o f t h e d e s i r e d
path.
Determine t h e r e q u i r e d c o r r e c t i o n i n t h e c o u r s e ( F i g . 2 . 4 3 ) :

S o l u t i on.

218


To r e a c h t h e d e s i r e d l i n e of f l i g h t , i t i s u s u a l l y p e c e s s a r y
f i r s t o f a l l t o make a d o u b l e c o u r s e c o r r e c t i o n ( i n o u r c a s e , loo),
a n d t h e n (when t h e a i r c r a f t h a s c o v e r e d a d i s t a n c e e q u a l t o t h e
b a s e o f t h e m e a s u r e m e n t , or i s t r a v e l i n g a l o n g t h e l i n e o f t h e d e s i r e d
p a t h ) t h e l e a d i n t h e c o u r s e i s r e d u c e d by a f a c t o r o f two, l e a v ­
i n g a c o r r e c t i o n i n t h e course which i s e q u a l t o t h e s e t a n g l e of
drift.
If t h e c l o s e s t t u r n i n g p o i n t i n t h e r o u t e ( C T R ) i s l o c a t e d
a t a d i s t a n c e which i s s m a l l e r t h a n t h e b a s e of m e a s u r e m e n t , t h e n
i n o r d e r t o a t t a i n i t , c o r r e c t i o n m u s t b e made i n t h e c o u r s e for
t h e d i s t a n c e covered f o r t h e t r a v e l p a r a l l e l t o t h e l i n e of t h e
d e s i r e d p a t h and o v e r t h e d i s t a n c e covered, i n o r d e r t o r e a c h t h e
d e s i r e d p a t h a t t h e moment w h e n t h e n e x t c o n t r o l l a n d m a r k i s b e i n g
passed.

L e t us s a y t h a t i n o u r example t h e d i s t a n c e t o t h e n e x t land­
m a r k i s s t i l l 3 0 km; t h e c o r r e c t i o n f o r t h e r e m a i n i n g d i s t a n c e w i l l
be equal t o :
lg AyremtAI ==

3
30

==

1
10

- 6".

S i n c e t h e c o r r e c t i o n for t h e
distance covered w a s equal t o So, the t o t a l correction for the
course i n order t o get the air­
c r a f t t o t h e CTR must b e e q u a l
t o -110.

Fig. 2.43.
Determination of
C o r r e c t i o n s i n C o u r s e to b e
Followed.

The p r o b l e m i s s o l v e d s i m i l a r l y
when t h e a i r c r a f t h a s w a n d e r e d
t o t h e l e f t of t h e desired path,
but with the difference that the
correction i n the course t o be
followed i s p o s i t i v e i n t h i s case.

I n s o l v i n g problems i n determining t h e d e s i r e d c o r r e c t i o n s
i n t h e c o u r s e t o b e f o l l o w e d , we p r e f e r a b l y u s e m e t h o d s i n v o l v i n g
v i s u a l o b s e r v a t i o n by t h e p i l o t w i t h o u t t h e u s e of a n y c a l c u l a t ­
i n g i n s t r u m e n t s or t a b l e s .
In the opposite case, while the p i l o t
i s solving t h e problems, t h e a i r c r a f t w i l l cover a considerable
d i s t a n c e , t h u s c o m p l i c a t i n g t h e r e a l i z a t i o n of t h e d e s i r e d s o l u tions.
The f i r s t m e t h o d o f p i l o t ' s v i s u a l e s t i m a t i o n i n t h i s c a s e
w i l l be t h e v i s u a l e s t i m a t i o n of t h e l a t e r a l d r i f t from t h e l i n e
of f l i g h t .

2 19

/214

If a n a r r c r a f t i s t r a v e l i n g t o t h e s i d e o f t h e a b o v e m e n t i o n e d
c h a r a c t e r i s t i c p o i n t , t h e d i s t a n c e from it by f l i g h t along t h e tra­
v e r s e i s determined by t h e v e r t i c a l a n g l e .
When t h e v e r t i c a l a n g l e
o f t h e p o i n t i s c l o s e t o 26.5O, t h e a i r c r a f t i s l o c a t e d a t a d i s ­
t a n c e f r o m t h e p o i n t which i s e q u a l t o h a l f t h e f l i g h t a l t i t u d e ;
w i t h a v e r t i c a l a l t i t u d e o f 45O, t h e d i s t a n c e i s e q u a l t o t h e f l i g h t
a l t i t u d e , w h i l e a t a,n a n g l e o f 63.5O i t i s t w i c e t h e f l i g h t a l t i ­
tude.
These a n g l e s are u s u a l l y determined by v i s u a l observation.
Intermediate values of v e r t i c a l angles and distances are determined
For e x a m p l e , i f t h e v e r t ­
by v i s u a l o b s e r v a t i o n a n d i n t e r p o l a t i o n .
i c a l a n g l e i s r o u g h l y e q u a l t o 55O, t h e n t h e d i s t a n c e t o t h e p o i n t
is approximately equal t o 1 . 5 f l i g h t a l t i t u d e s .

This method, with s u f f i c i e n t t r a i n i n g , g i v e s a very high
accuracy f o r determining t h e l o c a t i o n of t h e aircraft r e l a t i v e
t o a given p o i n t along t h e r o u t e , and consequently, with r e s p e c t
t o t h e l i n e of f l i g h t (on t h e o r d e r of 0 . 1 H ) a t v e r t i c a l angles
u p t o 65O.
A t v e r y l a r g e a n g l e s ( g r a t e r t h a n 65O) f r o m t h e v e r t ­
i c a l of t h e a i r c r a f t , t h e e r r o r s i n d i s t a n c e w i l l b e g r e a t e r
a n d t h i s method c a n n o t b e u s e d .
The s e c o n d method o f v i s u a l e s t i m a t i o n by t h e p i l o t w h i c h
i s used i n s o l v i n g t h i s problem i s t h e mental c a l c u l a t i o n of t h e
required course corrections following l i n e a r l a t e r a l deviation

(LLD).

For c o n v e n i e n c e i n m e t a l c a l c u l a t i o n , o n e r a d i a n i s a s s u m e d
t o b e 60' r a t h e r t h a n 5 7 . 3 , b u t t h i s d o e s n o t i n t r o d u c e a n y c o n s i d ­
e r a b l e e r r o r s ( t h e maximum e r r o r i n a n g l e s u p t o 20° d o e s n o t e x c e e d
10 )

.

T h i s a l l o w s t h e r e q u i r e d c o r r e c t i o n t o b e made i n t h e c o u r s e
i n terms of t h e approximate r a t i o of t h e l a t e r a l d e v i a t i o n t o t h e
d i s t a n c e covered:

1/60
1/40
1/30
1
1/15

/a

1
1.5
2

3
4

1/12.
1/10
118
1/7

5

6
7
8

1I6
1/5
114
113

10
12
15

20

T h e s e r a t i o s a r e e a s y t o r e m e m b e r i f w e know t h a t i n o r d e r
t o o b t a i n t h e i r r e q u i r e d c o r r e c t i o n it i s adequate t o divide t h e
number 60 i n t o t h e d i s t a n c e c o v e r e d , when t h e l a t e r a l d e v i a t i o n
i s t a k e n p e r u n i t of measurement.
O b v i o u s l y , i f t h i s method f o r c o u r s e c o r r e c t i o n i s employed
and t h e a i r c r a f t does n o t reach t h e d e s i r e d p o i n t along t h e l i n e
220

o f f l i g h t , s o t h a t t h e r e i s s t i l l some l a t e r a l d e v i a t i o n , t h e l a t e r a l
/215
d e vi a t i o n a n d t h e d i s t a n c e f r o m t h e p o i n t a t w h i c h t h e c o u r s e w a s
l a st changed can b e used t o c o r r e c t t h e c o u r s e .
Selection of t h e course t o be followed according t o a land­
mark a l o n g t h e r o u t e c a n b e u s e d i n t h e c a s e when t h e f l i g h t t a k e s
p l a c e a l o n g a s t r a i g h t - l i n e p o r t i o n o f a r a i l w a y or h i g h w a y a n d
means t h a t t h e c r e w m u s t c h a n g e t h e c o u r s e of t h e a i r c r a f t s o t h a t
it follows t h i s l i n e a r landmark.
A f t e r c h a n g i n g t h e c o u r s e by
a n a d d i t i o n a l t u r n i n g of t h e a i r c r a f t , t h e crew r e t u r n s t o t h e
d e s i r e d c o u r s e and t r a v e l s i n t h e d e s i r e d d i r e c t i o n once a g a i n .
T h e s e l e c t i o n o f t h e c o u r s e t o b e f o l l o w e d on t h e b a s i s o f
o r i e n t a t i o n landmarks i s a v a r i e t y o f t h e l a t t e r method.
I n t h i s case, t h e course i s s e l e c t e d s o t h a t t h e c l o s e r of
two s e l e c t e d l a n d m a r k s a l o n g t h e l i n e o f f l i g h t c o n s t a n t l y ( u p
t o t h e moment t h a t t h e a i r c r a f t f l i e s o v e r i t ) r e m a i n s i n a l i n e
with t h e f u r t h e r landmark.
A f t e r p a s s i n g by t h e c l o s e r l a n d m a r k ,
t h e a i r c r a f t f o l l o w s t h e d e s i r e d c o u r s e or c h o s e s t h e n e x t l a n d ­
m a r k , l o c a t e d beyond t h e second one, and c o n t i n u e s i t s f l i g h t a l o n g
this line.
Change

i n Navigational

Elements During F l i g h t

The m a j o r i t y o f n a v i g a t i o n a l e l e m e n t s ( c o u r s e , a l t i t u d e , s p e e d )
a r e d e t e r m i n e d i n f l i g h t on t h e b a s i s of i n d i c a t i o n s o f t h e c o r r e ­
sponding instruments, with i n t r o d u c t i o n of c o r r e c t i o n s f o r i n s t r u ­
mental and methodological e r r o r s .
A u t o m a t i c r a d i o d e v i c e s , b a s e d o n t h e D o p p l e r p r i n c i p l e , make
it p o s s i b l e t o make measurements d i r e c t l y ( d u r i n g f l i g h t ) of s u c h
e l e m e n t s as t h e d r i f t a n g l e and t h e ground s p e e d .
O t h e r methods o f a i r c r a f t n a v i g a t i o n do n o t p e r m i t d i r e c t
measurement o f t h e l a t t e r two e l e m e n t s , s o t h a t i n o r d e r t o de­
t e r m i n e them i t i s n e c e s s a r y t o u s e v a r i o u s p i l o t a g e t e c h n i q u e s .
In t h e absence of s i g h t s , t h e d r i f t angle of t h e aircraft
can b e d e t e r m i n e d as f o l l o w s .
L e t us suppose t h a t w e are t r a v e l i n g along a given r o u t e and
t h a t a c o n t r o l landmark on t h i s r o u t e h a s b e e n p a s s e d .
A f t e r 15­
20 min o f f l y i n g t i m e , w e s e l e c t a n o t h e r l a n d m a r k by w h i c h w e t e s t
t h e c o r r e c t n e s s o f t h e c o u r s e which h a s b e e n s e l e c t e d .
If no l a t ­
e r a l d e v i a t i o n o f t h e a i r c r a f t o c c u r s on t h i s s e g m e n t , i t means
t h a t t h e aircraft course has been properly s e t , i . e . , t h e d r i f t
a n g l e i s e q u a l i n v a l u e t o t h e p r e v i o u s c o u r s e , b u t h a s t h e oppo­
site sign

221

w h e r e a i s t h e d r i f t a n g l e of t h e a i r c r a f t ,
and $ i s t h e g i v e n f l i g h t p a t h angle.

It i s n o t always p o s s i b l e ,
t o be followed.

y is the aircraft

course,

however, t o c o r r e c t l y set t h e course

If a l a t e r a l d e v i a t i o n o f t h e a i r c r a f t f r o m t p e l i n e o f t h e
d e s i r e d path arises i n our f l i g h t segment, t h e course t o be f o l ­
lowed w i l l b e i n c o r r e c t and t h e a c t u a l f l i g h t a n g l e w i l l b e
+Q = q3 -I-arctg

AZ
S

w h e r e A Z e q u a l s t h e d e v i a t i o n of t h e a i r c r a f t f r o m t h e L G F , a n d S
i s t h e l e n g t h o f t h e s e g m e n t o v e r w h i c h t h e d r i f t a n g l e w a s meas­
ured.
T h e a n g l e o f d e v i a t i o n of t h e a i r c r a f t f r o m t h e l i n e o f t h e
d e s i r e d f l i g h t p a t h a r c t g AZ/S i s c o n s i d e r e d t o b e n e g a t i v e i f
t h e a i r c r a f t d e v i a t e s from it t o t h e l e f t , and p o s i t i v e i f it devi­
ates t o t h e r i g h t .
A s i n t h e method o f s e l e c t i n g t h e c o u r s e , t h i s
a n g l e i s d e t e r m i n e d by methods o f v i s u a l e s t i m a t i o n by t h e p i l o t .
I n t h e c a s e of i m p r o p e r s e l e c t i o n o f t h e c o u r s e t o b e f o l ­
lowed, t h e . l a t t e r c a n b e d e t e r m i n e d as t h e d i f f e r e n c e between t h e
a c t u a l f l i g h t angle and t h e course being followed:

I t i s much e a s i e r i n f l i g h t t o d e t e r m i n e t h e g r o u n d s p e e d
of an aircraft:
t h e same l a n d m a r k s a r e u s e d f o r t h i s p u r p o s e as
those used f o r determining t h e d r i f t angle of t h e aircraft.
To
d o t h i s , i t i s s u f f i c i e n t t o d e t e r m i n e t h e t i m e s when t h e a i r c r a f t
f l i e s o v e r t h e f i r s t and second landmarks, a f t e r which t h e ground
speed i s determined by t h e formula

where S i s t h e d i s t a n c e between t h e landmarks and t i s t h e f l y i n g
t i m e between t h e landmarks,
T h e d i v i s i o n S/t i s d o n e a s a r u l e o n s c a l e s 1 a n d 2 o f a
n a v i g a t i o n a l s l i d e r u l e ( F i g . 2.44), w i t h t h e e x c e p t i o n of t h o s e
For example, 6 ,
c a s e s when t h e f l y i n g t i m e i s l e s s t h a n 6 0 m i n .
1 0 , 1 2 , 1 5 , 2 0 a n d 3 0 , or e v e n 4 0 a n d 4 8 m i n a r e p o s s i b l e .
In
t h e s e c a s e s , t h e g r o u n d s p e e d w i l l b e e q u a l t o l O S , 6 S , 5 5 , 4 5 , 3S,
2 5 , 1 . 5 5 a n d 1 . 2 5 S , r e s p e c t i v e l y , a n d i s e a s i l y d e t e r m i n e d men­
t a l l y by m u l t i p l y i n g t h e d i s t a n c e b e t w e e n t h e l a n d m a r k s by one
o f t h e numbers g i v e n a b o v e .

222

/216

Fig.

h)

h)

W


2.44.

S c a l e s on N a v i g a t i o n a l S l i d e Rule N L - 1 O M .

To m e a s u r e t h e g r o u n d s p e e d a s w e l l a s t h e d r i f t a n g l e , i t
i s d e s i r a b l e t o s e l e c t d i s t a n c e s between landmarks which are no
l e s s t h a n 5 0 - 7 0 km a p a r t .
Over s h o r t d i s t a n c e s , i n o r d e r t o a v o i d
g r o s s e r r o r s , i t i s n e c e s s a r y t o d e t e r m i n e a n d m a r k down v e r y e x a c t l y
t h e t i m e t h a t t h e aircraft passes over t h e c o n t r o l landmarks.

Measuring the Wind at Flight Altitude and Calculating
Navigational Elements a t Successive Stages
The p r i n c i p a l f a c t o r w h i c h c o m p l i c a t e s t h e p r o c e s s e s o f a i r ­
c r a f t n a v i g a t i o n a t f l i g h t a l t i t u d e i s t h e wind.
With a v a i l a b i l ­
i t y of exact d a t a r e g a r d i n g i t s d i r e c t i o n and s p e e d , a l l problems
o f a i r c r a f t n a v i g a t i o n c a n b e s o l v e d by a c o m b i n a t i o n of g e n e r a l
m e t h o d s o f a i r c r a f t n a v i g a t i o n i n d e p e n d e n t l y of t h e v i s i b i l i t y
of t e r r e s t r i a l landmarks.
When t h e a i r c r a f t h a s o n b o a r d o n l y t h e m o s t g e n e r a l d e v i c e s
f o r a i r c r a f t n a v i g a t i o n , t h e problem of d e t e r m i n i n g t h e wind a t
t h e f l i g h t a l t i t u d e a s w e l l as t h e d r i f t a n g l e and t h e ground s p e e d
can b e s o l v e d i f t e r r e s t r i a l landmarks a r e v i s i b l e .
The w i n d a t f l i g h t a l t i t u d e d o e s n o t r e m a i n c o n s t a n t b u t i s
c o n s t a n t l y changing w i t h t i m e and e s p e c i a l l y w i t h d i s t a n c e .
In
order t o be able t o prepare t h e navigational data f o r t h e next
s t a g e of f l i g h t , it i s n e c e s s a r y t o d e t e r m i n e t h e wind a t t h e v e r y
e n d o f t h e p r e c e d i n g s t a g e a n d e v e n i n t h i s c a s e , t h e d a t a on t h e
wind which a r e o b t a i n e d a r e o b s o l e t e t o a c e r t a i n degree and a r e
not completely s a t i s f a c t o r y f o r t h e needs of calculating.
U n d e r t h e c o n d i t i o n s when a n a i r c r a f t i s f l y i n g a l o n g a n a i r
r o u t e , t h e r e are t h r e e n a v i g a t i o n a l p a r a m e t e r s which b a s i c a l l y
determine t h e s p e e d and d i r e c t i o n of t h e wind a t f l i g h t a l t i t u d e :
t h e a i r s p e e d ( V ) , ground speed ( W ) , and t h e d r i f t a n g l e f o r a given
course.
T h e w i n d c a l c u l a t e d on t h e b a s i s o f t h e s e p a r a m e t e r s w i l l
n o t b e r e c k o n e d f r o m t h e m e r i d i a n o f t h e l o c u s of t h e a i r c r a f t
( L A ) b u t from t h e l i n e of f l i g h t of t h e a i r c r a f t .
T h e c a l c u l a t i o n o f t h e p a t h a n g l e o f t h e w i n d (AW) i s c a r ­
r i e d o u t on t h e n a v i g a t i o n a l s l i d e r u l e b y means o f a k e y ( F i g .
2.45, a ) .

Example: W = 360 k m / h r ;
Determine t h e wind a n g l e .

Solution:

Answer:

(Fig.

2.45,

V = 320 k m / h r ;

d r i f t a n g l e = +8O.

b).

AW = 48O.

I f w e know t h e w i n d , i t i s e a s y t o d e t e r m i n e i t s s p e e d b y
For our
means o f a k e y w h i c h i s m a r k e d on t h e r u l e ( F i g . 2 . 4 6 , a ) .

224

/218

e x a m p l e , see F i g u r e 2 . 4 6 ,

b.

u = 60 k m / h r .

Answer:

The d i r e c t i o n o f t h e w i n d r e l a t i v e t o t h e m e r i d i a n o f t h e
l o c u s o f t h e a i r c r a f t ( L A ) i s d e t e r m i n e d by t h e f o r m u l a
6

= AW t I
).

I f t h e f l i g h t i s made w i t h m a g n e t i c f l i g h t a n g l e s , t h e w i n d
d i r e c t i o n i s o b t a i n e d r e l a t i v e t o t h e magnetic meridian of t h e
LA.
This d i r e c t i o n is a l s o used t o c a l c u l a t e t h e navigational
elements i n t h e next s t a g e of t h e f l i g h t .

I n f o r m a t i o n on t h e s p e e d o f t h e w i n d a n d i t s d i r e c t i o n i s t r a n s ­
m i t t e d from t h e a i r c r a f t t o ground s t a t i o n s , a l s o r e l a t i v e t o t h e
magnetic m e r i d i a n of t h e L A , and i s used f o r c o n t r o l l i n g t h e f l i g h t
of t h e aircraft.
T h e a n g l e o f t h e w i n d �or t h e n e x t s t a g e o f
AW = 6

-

the flight is

/219

+,

w h e r e 6 i s t h e w i n d d i r e c t i o n a n d J, i s t h e f l i g h t p a t h a n g l e o f
t h e n e x t s t a g e of t h e f l i g h t .

b)


a)

0 s1n us
@ w-"

1

Y

0

8'

48'

@

40

20

0

Fig. 2.45.
C a l c u l a t i o n o f t h e P a t h A n g l e o f t h e Wind o n t h e
( a ) Key for D e t e r m i n i n g t h e Wind
Navigational S l i d e Rule:
Angle;
(b) D e t e r m i n i n g t h e Wind A n g l e .
T h e v a l u e s for t h e g r o u n d s p e e d a n d d r i f t a n g l e o f t h e a i r ­
c r a f t for t h e n e x t s t a g e o f t h e f l i g h t a r e c a l c u l a t e d o n t h e n a v i ­
g a t i o n a l s l i d e r u l e by m e a n s o f a k e y ( F i g . 2 . 4 7 , a ) .
L e t u s a s s u m e t h a t t h e f l i g h t i n t h e p r e c e d i n g s t a g e w a s made
w i t h a MFA = 38O, i n t h e n e x t s t a g e w i t h a n M F A = 5 6 O , a n d w i t h
a n a i r s p e e d o f 320 k m / h r .
The d a t a o b t a i n e d on t h e w i n d a t t h e
p r e c e d i n g s t a g e a r e AW = 48O, u = 6 0 k m / h r .

Fig.
.46.
C a l c u l a t i o n o f t h e Wind S p e e d o n t h e N a v i g a t - o n a l
(b) D e t e r ­
S l i d e Rule:
( a ) Key f o r D e t e r m i n i n g t h e S p e e d ;
mination of t h e Speed.

225

II I I I

11111

.

The d i r e c t i o n o f t h e wind r e l a t i v e t o t h e m e r i d i a n o f t h e
LA i s

6 = 48

+

38 = 86O,

w h i l e t h e a n g l e o f t h e w i n d f o r t h e n e x t s t a g e o f t,he f l i g h t i s
AW = 8 6

-

a)

0us:
0"

5 6 = 30'.

b)

-A$

'US+AW

V

W

@

$5'

@

60

30'
320

3JS'
370.

Fig. 2.47.
C a l c u l a t i o n o f t h e D r i f t Angle a n d Ground Speed
on t h e N a v i g a t i o n a l A l i d e R u l e :
( a ) Key for D e t e r m i n i n g t h e
D r i f t Angle a n d Ground S p e e d ;
(b) Determination of t h e
D r i f t Angle a n d Ground S p e e d .

The v a l u e o f t h e g r o u n d s p e e d a n d t h e d r i f t a n g l e f o r t h e n e x t
s t a g e o f t h e f l i g h t a r e a l s o d e t e r m i n e d by means o f t h e n a v i g a ­
t i o n a l s l i d e r u l e (Fig. 2.47, b ) , ?.e.,

W = 370 k m / h r ;

US = + 5 . 5 O .

The v a l u e s o f t h e d r i f t a n g l e c a n b e u s e d t o d e t e r m i n e t h e
c a l c u l a t e d c o u r s e t o b e f o l l o w e d i n t h e n e x t s t a g e of t h e f l i g h t .
I n our case,
y

= 58

-

/220

5 . 5 = 52.5O.

I f t h e f l i g h t i s made w i t h o r t h o d r o m i c f l i g h t a n g l e s , t h e n
i n order t o c a l c u l a t e t h e navigational elements f o r t h e next stage
o f t h e f l i g h t it i s u n n e c e s s a r y t o c o n v e r t t h e wind a n g l e t o i t s
d i r e c t i o n r e l a t i v e t o t h e m e r i d i a n of t h e LA.
In t h i s case, the
wind a n g l e f o r t h e n e x t s t a g e of t h e f l i g h t i s d e t e r m i n e d as t h e
d i f f e r e n c e b e t w e e n t h e w i n d a n g l e o f t h e p r e c e d i n g s t a g e of t h e
f l i g h t and t h e angle of t u r n i n t h e r o u t e ( F i g . 2.48):
AW2

I n o u r e x a m p l e , AW1
1 8 = 30°.

= AW1 - TA.

= 48O, T A = 5 6 - 3 8 = 1 8 O , a n d A W 2 = 4 8 ­

However, i n o r d e r t o t r a n s m i t i n f o r m a t i o n r e g a r d i n g t h e wind
t o ground s t a t i o n s , i t i s n e c e s s a r y t o d e t e r m i n e t h e wind d i r e c ­
t i o n r e l a t i v e t o t h e m e r i d i a n of t h e LA.
O b v i o u s l y , t h e t r u e w i n d d i r e c t i o n a t t h e p o i n t LA i s

226

....

... .. .

___

w h e r e ct i s t h e a z i m u t h o f t h e o r t h o d r o m e a t t h e p o i n t L A ;
n e t i c d i r e c t i o n of t h e wind i s

Consequently,

i f t h e c a l c u l a t i o n of

t h e mag­

t h e orthodromic path angles

i s made f r o m t h e r e f e r e n c e m e r i d i a n , t h e n

-

6 M - AW + ( A L A - X r e f ) s i n $ a v - A M .

ExampZs:

,

,

A ~ = - ~ ~o 1 = 3 a o

SOlutiOn:

‘true

Aor=700

ALA=850 , @ a v = 5 2 0,

,

A W = ~ ~ O .

The t r u e wind d i r e c t i o n i s

= 48

+

38

+

15-0.8

= 980,

and t h e m a g n e t i c wind d i r e c t i o n i s
8~

= 48 + 3 8

+

15.0.8

+

5 = 1030.

Fig. 2.48.
Determination
o f t h e Wind A n g l e i n a
Successive Flight Stage.

C a l c u l a t i o n o f t h e P a t h o f t h e A i r c r a f t and M o n i t o r i n g
A i r c r a f t N a v i g a t i o n i n Terms o f D i s t a n c e and D i r e c t i o n

I n t h e p r e c e d i n g p a r a g r a p h s , we h a v e d i s c u s s e d t h e m e t h o d s
o f p l a c i n g t h e a i r c r a f t on c o u r s e , d e t e r m i n i n g t h e n a v i g a t i o n a l
e l e m e n t s d u r i n g f l i g h t , and c a l c u l a t i n g them f o r t h e f o l l o w i n g
s t a g e s of t h e f l i g h t .
Under c o n d i t i o n s o f c o n t i n u o u s v i s u a l o r i e n t a t i o n , t h e s e s methods
c o m p l e t e l y e n s u r e r e l i a b l e a i r c r a f t n a v i g a t i o n w i t h r e s p e c t t o d i s - /221
t a n c e and d i r e c t i o n , and no a d d i t i o n a l c a l c u l a t i o n s a r e r e q u i r e d .
H o w e v e r , v e r y o f t e n i t may n o t b e p o s s i b l e t o d e t e r m i n e t h e
l o c a t i o n of t h e a i r c r a f t c o n t i n u o u s l y r e l a t i v e t o a g i v e n p a t h .
For example, i n l o c a t i o n s where t h e r e a r e no d i s t i n g u i s h i n g f e a t u r e s
(steppe, d e s e r t , t a i g a , bodies of w a t e r , e t c ) , v i s u a l o r i e n t a t i o n
i s o n l y p o s s i b l e o v e r i n d i v i d u a l s e c t i o n s of t h e r o u t e .
T h e c o n d i t i o n s o f m e t e o r o l o g i c a l v i s i b i l i t y may n o t allow
u s e of landmarks which h a v e b e e n s e l e c t e d a l o n g t h e r o u t e .
T h e r e f o r e , i t becomes n e c e s s a r y t o u s e c o n t i n u o u s c a l c u l a ­
t i o n of t h e a i r c r a f t path i n t e r m s of t i m e a t c e r t a i n p e r i o d s ,
when i t b e c o m e s n e c e s s a r y t o c h e c k t h e a i r c r a f t p a t h w i t h r e s p e c t
t o d i s t a n c e and d i r e c t i o n .
C a l c u l a t i o n of t h e a i r c r a f t p a t h i s a l w a y s done w i t h p r e v ­
i o u s l y c a l c u l a t e d p a r a m e t e r s ( t h e c a l c u l a t e d c o u r s e and g r o u n d s p e e d ,
227

calculated t i m e ) .
A t t h e same t i m e , a l l t h e v a l u e s a n d m o m e n t s
o f c h a n g e i n t h e a i r c r a f t c o u r s e a r e d e t e r m i n e d , w h i c h make i t
p o s s i b l e t o d e t e r m i n e t h e c a l c u l a t e d p o s i t i o n o f t h e a i r c r a f t by
p l o t t i n g and t h u s t o determine t h e a d d i t i o n a l e r r o r s i n a i r c r a f t
navigation.
C a l c u l a t i o n of t h e p a t h o f t h e a i r c r a f t means t h a t a f t e r t h e
l a s t i d e n t i f i e d l a n d m a r k h a s b e e n l e f t b e h i n d , t h e crew aims t h e
a i r c r a f t toward t h e n e x t landmark d u r i n g a c e r t a i n p e r i o d o f t i m e
which i s used t o f i x a l l t h e v a l u e s of t h e a c t u a l c o u r s e of t h e
aircraft.
I f t h e p r o p e r l a n d m a r k h a s n o t b e e n s i g h t e d when t h e s c h e d ­
u l e d t i m e h a s e l a p s e d , due t o m e t e o r o l o g i c a l c o n d i t i o n s , t h e c a l c u ­
l a t e d t i m e f o r f l y i n g o v e r t h i s landmark i s determined, and t h e
a i r c r a f t i s s e t t o t h e n e x t p h a s e o f c a l c u l a t e d f l i g h t on t h e b a s i s
of t h e p r e v i o u s v a l u e s f o r d i r e c t i o n and v e l o c i t y of t h e wind.

T h u s , c a l c u l a t i o n o f t h e p a t h ( f l i g h t on t h e b a s i s o f p r e v ­
i o u s l y determined d a t a ) can continue u n t i l t h e conditions f o r v i s u a l
o r i e n t a t i o n improve.
However, i t i s n e c e s s a r y t o k e e p i n mind
t h a t the accuracy of aircraft navigation then decreases contin­
u o u s l y due t o t h e accummulation o f e r r o r s w i t h t i m e , a s w e l l as
i n c o n n e c t i o n w i t h t h e o b s o l e s c e n c e o f t h e d a t a o n t h e w i n d , meas­
ured p r i o r t o t h e l a s t r e l i a b l y s i g h t e d landmark.
When t h e c o n d i t i o n s f o r v i s u a l o r i e n t a t i o n i m p r o v e , t h e c r e w
t a k e s measures t o check t h e p a t h of t h e a i r c r a f t i n t e r m s o f d i s ­
t a n c e and d i r e c t i o n .
T o check t h e p a t h i n terms of d i s t a n c e , l i n e a r landmarks a r e
u s u a l l y employed, which i n t e r s e c t t h e r o u t e of t h e f l i g h t a t an
angle c l o s e t o 90°.

Five t o t e n minutes before t h e c a l c u l a t e d time f o r f l y i n g
o v e r t h e s e l a n d m a r k s , d e p e n d i n g on t h e f l y i n g t i m e a c c o r d i n g t o
t h e p r e v i o u s l y c a l c u l a t e d d a t a and t h e s p e e d of t h e a i r c r a f t , t h e
p i l o t c a r e f u l l y b e g i n s t o examine t h e l a n d s c a p e , l o o k i n g f o r t h e
l a n d m a r k ; a t t h e moment t h a t h e f l i e s o v e r i t , t h e a p p r o x i m a t e
p o s i t i o n o f t h e a i r c r a f t i s determined r e l a t i v e t o d i s t a n c e and
time

.

/222

When f l y i n g o v e r a c o n t r o l l a n d m a r k , t h e p i l o t a l s o t r i e s
t o determine t h e l a t e r a l d e v i a t i o n o f t h e a i r c r a f t from t h e d e s i r e d
p a t h on t h e b a s i s o f a d d i t i o n a l f e a t u r e s of t h e landmark ( c u r v e s
i n r i v e r s , t r i b u t a r i e s , r o a d j u n c t i o n s , populated areas , f o r e s t
outlines, etc.).
Having determined t h e p o i n t of i n t e r s e c t i o n of t h e landmark,
the p i l o t projects it along t h e l i n e of the desired path, fixing
t h e p o s i t i o n o f t h e a i r c r a f t ( i n t e r m s o f d i s t a n c e a t t h e moment
t h a t it f l i e s o v e r t h e 1andmark)and t h e d i r e c t i o n .

228

I

I

A t t h e present t i m e , aircraft used f o r long distance f l i g h t s
when t h e g r o u n d i s n o t v i s i b l e a r e f i t t e d w i t h s p e c i a l r a d i o n a v i g a ­
t i o n a l equipment.
L i g h t p l a n e s ( w h i c h f l y a t low a l t i t u d e s ) a r e
o c c a s i o n a l l y r e q u i r e d t o make
l o n g d i s t a n c e f l i g h t s when
the conditions f o r visual
o r i e n t a t i o n are poor.
Never­
t h e l e s s , when s u c h c a s e s d o
o c c u r , t h e f l i g h t c a n b e made
on t h e b a s i s o f a s l i g h t l e a d
i n t h e course and t h e d i r e c ­
t i o n of an acute angle of
intersection with the route
Fig. 2.49.
Lead Toward t h e A c u t e
by a l i n e a r landmark ( F i g .
2.49).
In t h i s case, t h e
Angle of t h e T r a v e r s e o f a Landlandmark must a p p e a r i n t h e
mark.
f i e l d o f view somewhat e a r l i e r
After t h i s , t h e air­
than t h e calculated time f o r f l y i n g p a s t it.
c r a f t c a n b e aimed a t a c o n t r o l l a n d m a r k w h i c h l i n e s up w i t h t h e
l i n e a r landmark.

I n i n d i v i d u a l c a s e s , when t h e p i l o t d o e s n o t r e c o g n i z e t h e
t e r r a i n o v e r which t h e a i r c r a f t must f l y , a f t e r t h e c o n d i t i o n s
f o r v i s u a l o b s e r v a t i o n have i m p r o v e d , t h e crew s e t s t h e a i r c r a f t
on c o u r s e t o f l y t o w a r d t h e n e x t c o n t r o l l a n d m a r k , a n d t h e p i l o t
m a k e s a n e s t i m a t e on t h e c h a r t o f t h e a i r c r a f t f l i g h t i n t e r m s
of a i r s p e e d , f i x e d c o u r s e , and f l y i n g t i m e w i t h t h e s e c o u r s e s from
t h e l a s t recognized landmark.
The p o i n t o b t a i n e d h a s a c a l c u l a t e d wi n d v e c t o r d u r i n g t h e
f l i g h t t i m e , a f t e r which t h e p i l o t compares t h e c h a r t w i t h t h e
l o c a t i o n i n t h e f o l l o w i n g manner:
(a)
I n t h e r e g i o n o f t h e e n d o f t h e wind v e c t o r ( t h e most
probable p o s i t i o n of t h e a i r c r a f t ) ;
(b)

I n t h e v i c i n i t y of

a calm p o i n t ;

(c)
I n terms o f t h e wind v e c t o r d i r e c t i o n from i t s b e g i n ­
n i n g t o e n d , w i t h a c o n t i n u a t i o n o f t h e wind v e c t o r 1 . 5 t o 2 t i m e s
and t u r n i n g i t t o t h e l e f t and r i g h t a t a n g l e s up t o 90° f r o m t h e
calculated direction;
(d)
T u r n i n g t h e wind v e c t o r ( e x t e n d e d 1 . 5 t o 2 t i m e s ) i n
t h e remaining semicircle.
N a t u r a l l y , t h e s e o p e r a t i o n s must b e c a r r i e d o u t w i t h c o n s t a n t
c h a n g e o f t h e c a l m p o i n t , d e p e n d i n g o n t h e d i r e c t i o n o f t h e move­
ment o f t h e a i r c r a f t .
If t h e l o c a t i o n o f t h e a i r c r a f t c a n n o t b e d e t e r m i n e d i n t h i s
m a n n e r , o t h e r m e a s u r e s m u s t b e t a k e n t o f i n d l a n d m a r k s , s u c h as

229

/223

t h e l o c a t i o n o f a c h a r a c t e r i s t i c l i n e a r or l a r g e - a r e a l a n d m a r k
( l a k e ; s e a ) , and a l s o by making i n q u i r i e s from t h e g r o u n d , e t c .
Use o f A u t o m a t i c N a v i g a t i o n a l D e v i c e s f o r C a l c u l a t i n g t h e
A i r c r a f t P a t h and M e a s u r i n g t h e Wind P a r a m e t e r s

To a c o n s i d e r a b l e d e g r e e , a u t o m a t i c n a v i g a t 5 o n a l d e v i c e s s i m ­
p l i f y t h e work o f t h e p i l o t i n c a l c u l a t i n g t h e p a t h o f t h e a i r ­
c r a f t and i n m e a s u r i n g t h e wind p a r a m e t e r s a t f l i g h t a l t i t u d e .
T h e s e d e v i c e s a r e mounted on h i g h - s p e e d p a s s e n g e r a i r c r a f t
which have complete r a d i o n a v i g a t i o n a l equipment, t h u s c o n s i d e r a b l y
i n c r e a s i n g t h e e f f e c t i v e n e s s of t h e i r use.
Such d e v i c e s , w h i c h a r e b a s e d on t h e g e n e r a l m e t h o d s o f a i r ­
craft n a v i g a t i o n , can b e used i n s t r a i g h t - l i n e systems of coord­
i n a t e s a t any o r i e n t a t i o n of t h e i r a x e s .
The d i r e c t i o n o f t h e a x e s of t h e c o o r d i n a t e s i s s e l e c t e d by
t h e p i l o t d e p e n d i n g o n t h e c o n d i t i o n s for w h i c h t h e s y s t e m i s b e i n g
used.
For e x a m p l e , f o r f l y i n g a l o n g a r o u t e , i t i s m o s t a d v a n ­
t a g e o u s t o c o m b i n e t h e a x i s o f t h e s y s t e m OX w i t h t h e d i r e c t i o n s
o f t h e s t r a i g h t - l i n e s e g m e n t s of t h e f l i g h t , i . e . , t o c a l c u l a t e
t h e p a t h i n an orthodromic system of c o o r d i n a t e s i n s t a g e s .
To c a r r y o u t s p e c i a l o p e r a t i o n s i n t h i s r e g i o n , e . g . , a t t e s t
s i t e s f o r r a d i o n a v i g a t i o n a l s y s t e m s for s h o r t - r a n g e o p e r a t i o n ,
t h e a x i s OX i s combined w i t h t h e a v e r a g e m e r i d i a n o f t h e f l i g h t
a r e a ( m a g n e t i c or t r u e ) , d e p e n d i n g o n w h i c h s y s t e m for c a l c u l a t i n g
t h e f l i g h t a n g l e s i s b e i n g u s e d t o make t h e f l i g h t .
I n p r e p a r i n g t o l a n d and maneuvering i n t h e v i c i n i t y of t h e
a i r p o r t , t h e a x i s OX c o i n c i d e s w i t h t h e a x i s o f t h e l a n d i n g s t r i p
at the airport, etc.
I n a l l c a s e s when a n a u t o m a t i c n a v i g a t i o n a l d e v i c e i s b e i n g
u s e d , a r e c t a n g u l a r system of c o o r d i n a t e s s h o u l d b e a p p l i e d t o
t h e f l i g h t c h a r t i n t h e given r e g i o n , p a r a l l e l t o t h e axes of t h e
s y s t e m OX a n d O Z .

P a r a l l e l l i n e s a r e d r a w n a t 2 0 mm i n t e r v a l s , s o t h a t on c h a r t s
w i t h a s c a l e o f 1:1,000,000 t h i s c o r r e s p o n d s t o 2 0 k m , w h i l e o n
t h o s e w i t h a s c a l e of 1 : 2 , 0 0 0 , 0 0 0
i t i s 40 km, e t c .
For t h i s p u r p o s e ,
a s p e c i a l s t e n c i l is included i n t h e set of navigational instru­
m e n t s f o r t h e NI-50B i n d i c a t o r .
In using an automatic navigational device with orthodromic
c o o r d i n a t e s i n s t a g e s , no a d d i t i o n a l d e v i c e s a r e needed o t h e r t h a n
t h e general navigational d i v i s i o n s of t h e c h a r t .
During f l i g h t , t h e apparatus i s connected t o a source o f d i r e c t
c u r r e n t , a n d t h e c h a r t a n g l e on t h e a u t o m a t i c c o u r s e c o n t r o l i s s e t

230

i n accordance with t h e s e l e c t e d system f o r c a l c u l a t i n g t h e air­

craft t o coordinates.
T h e w i n d s p e e d a n d d i r e c t i o n a r e s e t on t h e
wind s e n s o r on t h e b a s i s o f t h e r e s u l t s o f m e a s u r e m e n t s d u r i n g

t h e preceding f l i g h t segment.


/224


If t h e n a v i g a t i o n a l i n d i c a t o r i s u s e d w i t h an o r t h o d r o m i c
s y s t e m of c o o r d i n a t e s i n s e c t i o n s , t h e s e t t i n g o f t h e c h a r t a n g l e
a n d w i n d i s made a t t h e e n d o f t h e p r e c e d i n g s t a g e o f t h e f l i g h t
b e f o r e f l y i n g o v e r a t u r n i n g p o i n t i n t h e r o u t e (TPR).
On t h e
c o o r d i n a t e c a l c u l a t o r i n t h i s c a s e , t h e p o i n t e r IIN" i s s e t t o a
v a l u e e q u a l t o t h e l i n e a r l e a d f o r t h e t u r n (LLT) a n d p o i n t e r r f E T f
is set t o zero.
A t t h e moment when t h e a i r c r a f t
e m e r g e s f r o m t h e t u r n o n ' t h e new
course (Fig. 2.50), t h e a l t e r n a t i n g
current i s connected t o t h e i n s t r u ­
ment and t h e i n d i c a t o r b e g i n s
t o calculate the f l i g h t path.
A t s m a l l turn angles i n the
l i n e of f l i g h t (up t o 3 0 ° ) , t h e
t u r n t r a j e c t o r y of t h e a i r c r a f t
i s very c l o s e t o TPR.
In this
Fig. 2.50.
Transition t o an
c a s e , t h e t w o p o i n t e r s on t h e
O r t h o d r o m i c System o f Coorindicator should be s e t t o zero,
dinates i n a Successive
a n d t h e mechanism s w i t c h e d on
Flight Stage.
when t h e T P R i s p a s s e d a s t h e a i r ­
craft is turning.
A t t h e beginning
o f t h e s t r a i g h t - l i n e s e g m e n t o f f l i g h t , if p o s s i b l e , i t i s n e c e s ­
s a r y t o m a r k t h e e s t a b l i s h e d c o o r d i n a t e s o f t h e a i r c r a f t on t h e
computer as t h e a i r c r a f t p a s s e s o v e r a g i v e n landmark.

During f l i g h t along t h e s t r a i g h t - l i n e segment, t h e p o i n t e r s
o f i n d i c a t o r s "N" a n d " E " w i l l a l w a y s s h o w t h e d i s t a n c e c o v e r e d
by t h e a i r c r a f t from t h e l a s t landmark a l o n g t h e l i n e o f t h e g i v e n
p a t h and t h e l a t e r a l d e v i a t i o n from t h e l a t t e r i f i t t a k e s p l a c e ,
e . g . , due t o i n a c c u r a t e m a i n t e n a n c e of t h e c o u r s e , i m p r o p e r s t u d y ­
i n g o f t h e d a t a r e g a r d i n g t h e w i n d , or i n c a s e o f d a n g e r o u s m e t e o r o ­
l o g i c a l phenomena.
C o n s t a n t knowledge of t h e a i r c r a f t c o o r d i n a t e s f a c i l i t a t e s
b o t h v i s u a l and r a d i a l o r i e n t a t i o n .
However, a i r c r a f t c o o r d i n a t e s
o b t a i n e d on t h e b a s i s of a c o m p u t e r w i l l n o t a l w a y s c o r r e s p o n d
p r e c i s e l y with t h e a c t u a l c o o r d i n a t e s , s i n c e t h e speed and d i r e c ­
t i o n o f t h e wind d u r i n g f l i g h t change o v e r t h e d i s t a n c e c o v e r e d .
The n a v i g a t i o n a l i n d i c a t o r a l s o makes i t e a s i e r t o d e t e r m i n e
t h e wind p a r a m e t e r s a t f l i g h t a l t i t u d e .
T h i s i s done as f o l l o w s :
A t t h e end of a s t a g e i n t h e f l i g h t , t h e a i r c r a f t c o o r d i n a t e s
are recorded with t h e computer ( P o i n t B i n Figure 2.51, a ) and t h e

231

a c t u a l l o c a t i o n o f t h e a i r c r a f t i s d e t e r m i n e d v i s u a l l y or b y m e a n s
of r a d i o n a v i g a t i o n a l d e v i c e s ( P o i n t B1).
These P o i n t s BB1 d e t e r ­
mine t h e v e c t o r o f t h e c h a n g e i n t h e w i n d a t f l i g h t a l t i t u d e f o r
t h e f l i g h t t i m e of a given s t a g e of f l i g h t .
The p r o b l e m o f d e t e r m i n i n g t h e w i n d v e c t o r i n t h i s case c a n
/225
b e s o l v e d e a s i l y on a f l i g h t c h a r t .
To d o t h i s , a r e v e r s e l i n e
must b e drawn from P o i n t B and t h e l e n g t h o f t h e wind v e c t o r i s s e t
on t h e s e n s o r ( u p a t ) d u r i n g t h e f l y i n g t i m e f r o m P o i n t A t o P o i n t
B ( P o i n t 0). T h e n t h e v e c t o r o f O B w i l l c o n s t i t u t e t h e v e c t o r
of t h e c a l c u l a t e d wind (and O B I , t h e a c t u a l wind) a t f l i g h t a l t i ­
tude.

M e a s u r e m e n t o f t h e Wind b y Means o f a N a v i ­
Fig. 2.51.
gational Indicator:
( a ) W i n d - C h a n g e V e c t o r ; ( b ) Wind V e c t o r .
I n o r d e r t o o b t a i n t h e v a l u e of t h e wind i n km/hr, i t i s s u f ­
f i c i e n t t o d i v i d e t h e l e n g t h o f t h e v e c t o r O B 1 by t h e f l y i n g t ' i m e
between P o i n t s A and B y e x p r e s s e d i n h o u r s .
The p r o b l e m o f m e a s u r i n g t h e w i n d c a n b e s i m p l i f i e d i f w e
c o n s i d e r t h a t t h e w i n d a t t h e s e n s o r i s z e r o for t h e f l i g h t s t a g e ,
i . e . , w e i n t r o d u c e t h e v a l u e o f AW = 0 , u = 0 i n t o t h e w i n d s e n s o r .
Then P o i n t B w i l l b e t h e i n d i c a t i o n o f t h e c o o r d i n a t e s o f t h e a i r ­
c r a f t a t t h e end o f t h e f l i g h t s t a g e , w h i l e P o i n t B 1 w i l l r e p r e ­
Consequently, vector
sent the actual coordinates (Fig. 2.51, b ) .
B B 1 w i l l be t h e wind v e c t o r f o r t h e f l y i n g t i m e i n t h i s s t a g e .
The u s e o f t h e n a v i g a t i o n a l i n d i c a t o r i n r e c t i l i n e a r c o o r d ­
inates f o r f l i g h t s i n a given region is not different i n principle
However,
from u s i n g i t i n orthodromic c o o r d i n a t e s and s t a g e s .
t h e i m p o r t a n t a d v a n t a g e of t h e o r t h o d r o m i c s y s t e m o f c o o r d i n a t e s
is then l o s t , ;.e., t h e r e l a t i o n s h i p of t h e coordinates t o t h e
Therefore, the
c h e c k i n g o f t h e p a t h for d i s t a n c e a n d d i r e c t i o n .
p o s i t i o n of t h e a i r c r a f t i n t h i s c a s e c a n b e d e t e r m i n e d o n l y i n
t e r m s o f t h e c o o r d i n a t e s of t h e n e t w o r k s u p e r i m p o s e d o n t h e c h a r t .
The r e c t a n g u l a r s y s t e m o f c o o r d i n a t e s c a n b e e x t e n d e d o v e r
a r e l a t i v e l y s m a l l a r e a (on t h e o r d e r o f 400 x 400 k m ) , s i n c e t h e
e f f e c t o f t h e s p h e r i c i t y of t h e E a r t h b e g i n s t o show up i n l a r g e
areas.
I n c o n j u n c t i o n w i t h t h i s , i n t h e case o f f l i g h t s by a c o o r d ­
i n a t e s y s t e m , i t i s n o t n e c e s s a r y t o s e t a new c h a r t a n g l e f o r e a c h

232

c h a n g e i n t h e l i n e o f f l i g h t a n d -to d e s c r i b e t h e c o o r d i n a t e s o f
t h e a i r c r a f t i n a new s y s t e m f o r c a l c u l a t i o n , w h i c h t o a c o n s i d ­
e r a b l e degree compensates f o r t h e l o s s o f t h o s e advantages which
w e have i n t h e orthodromic system of coordinates i n s t a g e s .
Details o f A i r c r a f t Navigation Using Geotechnical
i n Various F l i g h t Conditions

Methods

/226

The c o n d i t i o n s f o r a i r c r a f t n a v i g a t i o n u s i n g g e o t e c h n i c a l
d e v i c e s are d e t e r m i n e d p r i m a r i l y by t h e p r e s e n c e and n a t u r e o f
l a n d m a r k s , as w e l l as b y t h e i r c o n t r a s t r e l a t i v e t o t h e s u r r o u n d ­
ing terrain.
The b e s t l a n d m a r k s f o r v i s u a l a i r c r a f t n a v i g a t i o n a r e l i n ­
ear ones ( l a r g e r i v e r s , r a i l w a y s and highways, t h e shores of l a r g e
bodies of water).
L a k e s , l a r g e and s m a l l populated areas, char­
a c t e r i s t i c m o u n t a i n p e a k s , e t c . , a r e a l s o good l a n d m a r k s , w h i l e
grain elevators, water tanks, churches, i n d u s t r i a l enterprises,
e t c . , c a n b e u s e d f o r f l i g h t s a t low a l t i t u d e s .

For a i r c r a f t n a v i g a t i o n i n a n a r e a w h i c h i s p o o r i n l a n d m a r k s ,
w e c a n u s e s e p a r a t e s i g h t i n g p o i n t s on t h e E a r t h ' s s u r f a c e i n t h e
form of s p o t s , i n d i v i d u a l t r e e s , foam on t h e s u r f a c e o f t h e w a t e r ,
etc.
Such p o i n t s a r e n o t l a n d m a r k s , s i n c e i t i s i m p o s s i b l e t o
d e t e r m i n e t h e i r l o c a t i o n on a f l i g h t c h a r t , b u t t h e y c a n b e u s e d
t o m e a s u r e t h e d r i f t a n g l e a n d t h e g r o u n d s p e e d when t h e r e i s a
s i g h t o n b o a r d a n d a l s o make i t p o s s i b l e t o i n c r e a s e t h e a c c u r ­
a c y of a i r c r a f t n a v i g a t i o n d u r i n g f l i g h t b e t w e e n c o n t r o l l a n d m a r k s .
The c o n t r a s t o f l a n d m a r k s i s d e t e r m i n e d by w e a t h e r c o n d i t i o n s ,
i n which t h e v i s u a l f l i g h t s m u s t b e made.
With good c o n t r a s t ,
e . g . , i n t h e presence of l a r g e populated a r e a s , r i v e r v a l l e y s ,
l o c a t e d i n f o r e s t e d t e r r a i n , a i r c r a f t n a v i g a t i o n b o t h summer a n d
w i n t e r c a n b e c a r r i e d o u t w i t h a h o r i z o n t a l v i s i b i l i t y on t h e o r d e r
of 1 . 5 km.
A t p l a c e s where t h e r e i s l i t t l e v e g e t a t i o n , a v i s i ­
b i l i t y o n t h e o r d e r o f 1 0 km i s r e q u i r e d f o r a i r c r a f t n a v i g a t i o n
i n winter.
The v i s i b i l i t y o f a l l l a n d m a r k s , w i t h t h e e x c e p t i o n o f il­
luminated populated areas , i s c o n s i d e r a b l y decreased a t n i g h t ,
e s p e c i a l l y when t h e Moon i s n o t o u t .
Therefore, populated areas
are t h e p r i n c i p a l landmarks a t n i g h t ; t h e i r appearance a t n i g h t
can d i f f e r from t h e i r appearance i n t h e day.

An i m p o r t a n t f a c t o r w h i c h d e t e r m i n e s t h e c o n d i t i o n s for a i r ­
c r a f t n a v i g a t i o n i s t h e s t a b i l i t y o f o p e r a t i o n o f m a g n e t i c com­
passes.
C o n d i t i o n s of a i r c r a f t n a v i g a t i o n w i t h o u t t h e u s e of gyro­
s c o p i c compasses are u n f a v o r a b l e i n t h e p o l a r r e g i o n s , as w e l l
as low a l t i t u d e s i n t h e v i c i n i t y of t h e m a g n e t i c a n o m a l i e s .
The f l i g h t a l t i t u d e a l s o h a s a s i g n i f i c a n t i n f l u e n c e on t h e
aircraft navigation conditions.
I n c l e a r w e a t h e r , optimum c o n d i t i o n s

233


I-


for v i s u a l o r i e n t a t i o n e x i s t a t h e i g h t s o n t h e o r d e r o f 1000-1500
m , s i n c e a t t h i s a l t i t u d e t h e a n g u l a r v e l o c i t y a t which t h e land­
marks g o by i s s m a l l , a l l of t h e i r d e t a i l s c a n b e s e e n c l e a r l y ,
a n d t h e f i e l d o f v i e w of t h e crew c o v e r s a v e r y l a r g e a r e a , w h i c h
is important i n comparing t h e c h a r t s with t h e landscape.

/227

H o w e v e r , t h e s e a l t i t u d e s c a n o n l y b e u s e d whGen t h e r e i s a
s m a l l amount o f c l o u d s a l o n g t h e f l i g h t r o u t e .
I n cloudy weather,
f l i g h t s a r e made a t l o w e r a l t i t u d e s , a s l o w a s t h e r e l i e f o f t h e
t e r r a i n w i l l allow.
A t l o w a l t i t u d e s , t h e c o n d i t i o n s for v i s u a l o r i e n t a t i o n a r e
w o r s e , s i n c e t h e a n g u l a r v e l o c i t y w i t h which t h e landmarks go by
i n c r e a s e s a n d t h e a r e a w h i c h t h e c r e w of t h e a i r c r a f t c a n s c a n
i s reduced.
An i n c r e a s e i n t h e f l i g h t a l t i t u d e ( a b o v e 1 . 5 km) i n c l e a r
w e a t h e r h a v e a s m a l l i n f l u e n c e on t h e c o n d i t i o n s o f v i s u a l o r i e n ­
t a t i o n , b u t a t g r e a t h e i g h t s t h e v i s u a l v i s i b i l i t y of landmarks
( d e p e n d i n g on w e a t h e r c o n d i t i o n s ) i s much w o r s e t h a n a t l o w a n d
medium a l t i t u d e s .
The s e l e c t i o n o f s c a l e s a n d c h a r t p r o j e c t i o n s f o r mak i n g a
f l i g h t d e p e n d p r i m a r i l y on t h e a l t i t u d e a n d s p e e d of t h e f l i g h t .
A t low a l t i t u d e s , i t i s b e s t t o u s e c h a r t s w i t h a l a r g e s c a l e o f
1:500,000
or 1:1,000,000. A t h i g h a l t i t u d e s a n d h i g h s p e e d s , i t
i s b e s t t o u s e c h a r t s w i t h scales of 1:2,000,000
a n d 1:4,000,000.

For f l i g h t s a l o n g r o u t e s w h i c h a r e v e r y l o n g , c h a r t s a r e u s e d
w h i c h a r e made u p o f p r o j e c t i o n s s h o w i n g t h e p r o p e r t i e s o f o r t h o ­
d r o m i c i t y ( t h e o r t h o d r o m e on t h e c h a r t h a s a s h a p e c l o s e t o a s t r a i g h t
l i n e ) , i . e . , c h a r t s i n t h e i n t e r n a t i o n a l or t r a n s v e r s e c y l i n d r i c a l
projection.
For t h e p o l a r r e g i o n s , c h a r t s w i t h t a n g e n t s t e r e o graphic projection a r e used.

10.

C a l c u l a t i n g and Measuring P i l o t a g e Instruments

Purpose of

C a l c u l a t i n g and M e a s u r i n g P i l o t a g e

Instruments

P i l o t a g e c a l c u l a t i n g and measuring instruments are intended
f o r t h e following:
(a)

Measuring d i s t a n c e s

a n d d i r e c t i o n s on f l i g h t c h a r t s .

(b)
Calculating navigational elements both i n preparing f o r
f l i g h t a n d when c o m p l e t i n g i t .
(c)
Calculating methodological e r r o r s i n t h e readings of
navigational instruments ( t h e readings of t h e airspeed, a l t i m e t e r ,
and o u t s i d e a i r thermometer).
(d)

C a l c u l a t i n g t h e elements of a i r c r a f t maneuvering.

234

I

M e a s u r e m e n t o f d i s t a n c e s o n f l i g h t c h a r t s i s made b y m e a n s
of a s p e c i a l navigational s l i d e r u l e .
A f e a t u r e which d i s t i n g u i s h e s
t h i s s l i d e r u l e from conventional s l i d e r u l e s i s t h e presence of
s e v e r a l s c a l e s f o r m e a s u r i n g d i s t a n c e s on c h a r t s w i t h d i f f e r e n t
/228
scales.
M e a s u r e m e n t o f d i r e c t i o n s o n f l i g h t c h a r t s i s made b y m e a n s
o f n a v i g a t i o n a l p r o t r a c t o r s , made o f t r a n s p a r e n t m a t e r i a l .
The p r o t r a c t o r s a r e s i m u l t a n e o u s l y u s e d as t r i a n g l e s , w h i c h
make i t p o s s i b l e t o m a k e c e r t a i n c o n s t r u c t i o n s o n f l i g h t c h a r t s
and diagrams ( l a y i n g o u t t h e t r a v e r s e s of landmarks, p a r a l l e l s h i f t
of l i n e s , e t c . ) .
C a l c u l a t i o n s o f n a v i g a t i o n a l e l e m e n t s , c o r r e c t i o n s t o nav­
i g a t i o n a l d e v i c e s , and e l e m e n t s o f maneuvering are p r e s e n t l y car­
r i e d o u t w i t h t h e a i d of n a v i g a t i o n a l l o g a r i t h m i c s l i d e r u l e s ,
t h e b e s t m o d i f i c a t i o n of which i s t h e n a v i g a t i o n a l c a l c u l a t i n g
s l i d e r u l e NL-1OM.
In addition, t o calculate c e r t a i n navigational elements, w e
c a n u s e s p e c i a l d e v i c e s f o r s e t t i n g up t h e s p e e d t r i a n g l e (windspeed i n d i c a t o r s ) .
However, due t o t h e improvements i n n a v i g a t i o n a l
c a l c u l a t i n g s l i d e r u l e s , they have a very l i m i t e d a p p l i c a t i o n .

Navigational

Slide Rule NL-1OM


T h e p r i n c i p l e of t h e c o n s t r u c t i o n o f n a v i g a t i o n a l s l i d e r u l e s
i s t h e p r o p e r t y o f l o g a r i t h m s t o make i t p o s s i b l e . t o r e p l a c e t h e
p r o c e s s e s of m u l t i p l i c a t i o n , d i v i s i o n , r a i s i n g t o a p o w e r , e x t r a c t ­
i n g t h e s q u a r e r o o t , as w e l l as m u l t i p l i c a t i o n and d i v i s i o n o f
t r i g o n o m e t r i c f u n c t i o n s , by o p e r a t i o n s i n v o l v i n g t h e a d d i t i o n and
subtraction of t h e logarithms of these values.
Thus, t h e o p e r a t i o n s d e s c r i b e d above i n v o l v i n g numbers c a n
b e a p p l i e d t o t h e summing o f t h e s e g m e n t s o f a s c a l e on t h e r u l e r ,
which s i m p l i f i e s c a l c u l a t i o n t o a c o n s i d e r a b l e d e g r e e .
The s c a l e s o f t h e n a v i g a t i o n a l s l i d e r u l e N L - 1 O M ( s e e F i g .
2.44) are grouped s o t h a t one s i d e i s used f o r s o l v i n g problems
i n d e t e r m i n i n g n a v i g a t i o n a l e l e m e n t s o f f l i g h t a s w e l l a s maneu­
vering elements, while t h e o t h e r s i d e i s used f o r c a l c u l a t i n g t h e
c o r r e c t i o n s for t h e r e a d i n g s o f n a v i g a t i o n a l i n s t r u m e n t s .
I n a d d i t i o n , t h e upper beveled edge of t h e r u l e r ( P o s i t i o n
1 7 ) c a r r i e s a scale d i v i d e d i n t o m i l l i m e t e r s , which can b e used
t o m e a s u r e d i s t a n c e s on t h e map.
The s c a l e s on t h e r u l e r 1 a n d 2 a r e i n t e n d e d t o d e t e r m i n e
t h e g r o u n d s p e e d f r o m a known d i s t a n c e c o v e r e d i n a g i v e n t i m e ,
or f r o m a g i v e n d i s t a n c e a t a known g r o u n d s p e e d a n d t i m e .

235

Therefore

w = -

t

S = Wt,

and

so t h a t
1gW = 1 g S - l g t

and

1gS = 1gW t

lgt.

S c a l e 1 i s t h e s.cale o f l o g a r i t h m s o f d i s t a n c e s i n k i l o m e t e r s /229
s c a l e 2 i s a s c a l e o f l o g a r i t h m s of
f l y i n g t i m e i n m i n or s e c u p t o t h e r e c t a n g u l a r i n d e x m a r k e d 1 0 0 a n d
b e y o n d , i n h r s or m i n .

or f l i g h t s p e e d s i n k m / h r ;

The p r i n c i p l e o f s o l v i n g p r o b l e m s by d e t e r m i n i n g t h e a i r s p e e d
o v e r a g i v e n d i s t a n c e a t a g i v e n t i m e i s as f o l l o w s :

L e t u s s a y t h a t a n a i r c r a f t h a s c o v e r e d a d i s t a n c e of 1 6 5
km i n 1 2 m i n a n d t h a t w e m u s t d e t e r m i n e t h e g r o u n d s p e e d i n k m / h r .
We s e t t h e m a r k i n g o n t h e s l i d e r t o t h e 1 6 5 p o s i t i o n o n t h e
d i s t a n c e s c a l e ; by moving t h e a d j u s t a b l e s c a l e 2 , w e s e t d i v i s i o n
We c a n t h e n r e a d
1 2 on i t o p p o s i t e t h e m a r k i n g on t h e s l i d e r .
o f f t h e d i s t a n c e c o v e r e d b y t h e a i r c r a f t i n m i n u t e s o f f l i g h t oppo­
s i t e t h e number 1 a t t h e b e g i n n i n g o f t h e s c a l e :

1gW k m / m i n = l g 1 6 5 - l g 1 2 = l g 1 3 . 8 .
However,
equal t o

s i n c e 1 h r i s 60 m i n ,

1gW km/h

t h e s p e e d i n km/hr would b e

= l g 1 3 . 8 t l g 60 = l g 825

or

W = 825 km/h.
By c o m b i n i n g t h e f i r s t a n d s e c o n d e f f e c t s , w e o b t a i n
1gW = l g 1 6 5

-

lg 12

+

l g 60,

i . e . , i n order t o s o l v e t h e problem, it i s s u f f i c i e n t t o set t h e
n u m b e r 1 2 o n s c a l e 2 o p p o s i t e t h e n u m b e r 1 6 5 on s c a l e 1 a n d o p p o ­
s i t e n u m b e r 60 o n s c a l e 2 , w h i c h i s m a r k e d w i t h a t r i a n g u l a r m a r k ­
i n g , and t h e n t o c a l c u l a t e t h e ground speed from scale 1 ( F i g .
2.52, a ) .
T h e p r o b l e m i s s o l v e d a n a l o g o u s l y i f t h e f l i g h t t i m e i s meas­
u r e d i n sec.
I n o u r e x a m p l e , i t w i l l b e 720 s e e :
1gW = l g 1 6 5 - l g 7 2 0 t l g 3 6 0 0 = l g 8 2 5 ;

W = 825 km/h.
236

T h e g r o u n d s p e e d i n t h i s c a s e is c a l c u l a t e d f r o m s c a l e 1 o p p o ­
s i t e t h e number 3600 on s c a l e 2 ( t h e number o f s e c o n d s i n 1 h r ) ,
marked w i t h a c i r c u l a r i n d e x .
To d e t e r m i n e t h e d i s t a n c e a t a g i v e n g r o u n d s p e e d a n d f l y i n g
t i m e ( S = Wt), t h e l o g a r i t h m s o f t h e s e numbers a r e a d d e d :
1 g S = 1gW

+

lgt.

On t h e r u l e , t h e t r i a n g u l a r or c i r c u l a r i n d e x on t h e m o v a b l e
s c a l e 2 i s s e t o p p o s i t e t h e known g r o u n d s p e e d o n s c a l e 1. T h e
i n d e x m a r k i n g on t h e s l i d e r i s s e t o p p o s i t e t h e g i v e n f l y i n g t i m e
on s c a l e 2 , a f t e r which t h e p o s i t i o n o f t h e i n d i c a t o r on s c a l e
1 shows t h e d i s t a n c e c o v e r e d i n t h i s t i m e .

Example.

W = 750 k m / h r ,

t = 1 h r a n d 36 m i n .

Find t h e dis­

tance covered.
See Figure 2.52, b .
Ig s = Ig 750 + Ig 1h 36 m i l l = 1200;
S = 1 2 0 0 km.

SOlUtiOn.

Answer.

a)

0

iss

825

I2

“ 0

b?

@

750

IZUU

0-36
1 h

min

Fig. 2.52.
C a l c u l a t i o n on t h e N L - 1 O M :
( a ) o f t h e Ground
S p e e d ; (b) o f t h e D i s t a n c e C o v e r e d o n t h e B a s i s o f G r o u n d
Speed and Time.
L e t u s a p p l y t h e k e y s t o N L - 1 0 for s o l v i n g p r o b l e m s i n d e t e r ­
m i n i n g t h e g r o u n d s p e e d a n d d i s t a n c e c o v e r e d on s c a l e s 1 a n d 2 :
(a)
To d e t e r m i n e t h e g r o u n d s p e e d f o r a d i s t a n c e c o v e r e d
i n a known t i m e ( f i g . 2 . 5 3 , a or 2 . 5 4 , a ) .

(b)
and t i m e

To d e t e r m i n e t h e d i s t a n c e c o v e r e d f r o m t h e g r o u n d s p e e d
( F i g . 2 . 5 3 , b or 2 . 5 4 , b).

Fig. 2.53.
t h e NL-lOM,

Keys f o r D e t e r m i n i n g t h e Ground S p e e d o n
on t h e Basis of t h e D i s t a n c e Covered and
the Time.

237

/230

Movable s c a l e 3 w i t h t h e s i g n s o f t h e l o g a r i t h m s , w h i c h i s
t h e same ( u p t o 5O) a s s c a l e 4 f o r t h e l o g a r i t h m s o f t h e t a n g e n t s
and i s a l s o d i v i d e d i n t o scales 3 and 4, a l o n g w i t h t h e f i x e d scale
o f d i s t a n c e s or a l t i t u d e s 5 , w h i c h e s s e n t i a l l y r e p e a t s s c a l e 1,
are a l l i n t e n d e d f o r working w i t h t r i g o n o m e t r i c f u n c t i o n s .
The m a j o r i t y o f p r o b l e m s w h i c h a r e s o l v e d on t h e s e s c a l e s
a r e b a s e d on t h e p r o p e r t i e s of a r i g h t t r i a n g l e , s o t h a t t h e v a l u e
o f t h e s i n e o f 9 0 ° a n d t a n g e n t 45O ( s c a l e s 3 a n d 4 ) , w h o s e l o g a ­
r i t h m s a r e e q u a l t o z e r o , a r e marked on t h e r u l e by a t r i a n g u l a r
index.
I f t h e p r o b l e m i s s o l v e d f r o m a known l e g , e . g . , d e t e r m i n ­
i n g t h e e r r o r i n t h e c o u r s e on t h e b a s i s o f t h e d i s t a n c e c o v e r e d
and t h e l i n e a r l a t e r a l d e v i a t i o n ( F i g . 2 . 5 5 1 , w e u s e s c a l e s 4 and
5 on t h e r u l e .
Z
tg A7 = - or Ig tg A7 9 lg 2 -.lg X.

X

The key t o s o l v i n g t h i s p r o b l e m i s shown i n F i g u r e 2 . 5 6 .
I n t h e c a s e when t h e h y p o t e n u s e of t h e t r i a n g l e i s k n o w n ,
F o r example, sup­
t h e problems are s o l v e d by u s i n g scales 3 and 5 .
pose w e wish t o determine t h e l o c a t i o n of t h e a i r c r a f t i n orthod r o m i c c o o r d i n a t e s ( X a , Z a > o n t h e b a s i s o f known c o o r d i n a t e s o f
a l a n d m a r k ( X l y Z,),
t h e d i s t a n c e and d i r e c t i o n o f which have been
d e t e r m i n e d by means o f a r a d a r l o c a t e d on b o a r d t h e a i r c r a f t ( F i g .
2.57).
I t i s clear from t h e f i g u r e t h a t t h e orthodromic c o o r d i n a t e s
of t h e a i r c r a f t w i l l be equal t o

Xa

= Xi - R

Za

=

COS

8 ;

Z1 - R s i n

8,

where R i s t h e d i s t a n c e t o t h e landmark and 8 i s t h e p a t h b e a r ­
i n g of t h e landmark ( t h e a n g l e between t h e g i v e n l i n e of f l i g h t
and t h e d i r e c t i o n of t h e landmark).

Fig. 2.54.
Keys f o r D e t e r m i n i n g t h e D i s t a n c e C o v e r e d on
t h e B a s i s of t h e Ground S p e e d a n d T i m e , U s i n g t h e N L - 1 O M .
The d i f f e r e n c e i n t h e c o o r d i n a t e s of
are r e p r e s e n t e d by X and 2 , r e s p e c t i v e l y ,
l o g a r i t h m i c r u l e (Fj-g. 2 . 5 8 , a , b ) .
238

t h e landmark i n a i r c r a f t
a n d a r e f o u n d on t h e

/231

I n a i r c r a f t n a v i g a t i o n , a number of p r o b l e m s are s o l v e d which
are connected w i t h t h e d i s t a n c e s and d i r e c t i o n s ( e . g . ) , t h e check­
i n g of a course i n t e r m s of t h e d i s t a n c e covered, determination
of t h e p o s i t i o n o f t h e a i r c r a f t by u s i n g methods o f v i s u a l a n d
r a d a r m e a s u r e m e n t s , a n d many o t h e r s .
The e s s e n c e o f t h e s o l u t i o n
of t h e s e problems is obvious from t h e examples given.

Fig.

2.55.

Fig.

2.56.

Fig.

2 . 5 5 . D e t e r m i n a t i o n o f t h e C o u r s e Error f r o m t h e C h a n g e i n
the Lateral Coordinate.

Fig.

2.56.

Key o n t h e N L - 1 O M
Error.

for D e t e r m i n i n g t h e A i r c r a f t C o u r s e

For c a s e s w h e n t h e a n g l e s m e a s u r e d a r e g r e a t e r t h a n r i g h t
a n g l e s , t h e s i n e s c a l e 3 i s numbered b a c k w a r d s , s o t h a t s i n 180-a
= s i n a , f o r example:
l g s i n 135O = l g s i n 45O.
S c a l e s 3 , 4 a n d 5 c a n b e use.d t o s o l v e s p e c i a l p r o b l e m s o f
oblique-angled t r i a n g l e s , e.g., t h e solving of speed t r i a n g l e s .
T h e k e y f o r s o l v i n g t h i s k i n d o f p r o b l e m i s g i v e n on t h e r i g h t h a n d s i d e of t h e s c a l e 3 .

/232

T h e t h e o r e m o f s i g n s , w e l l known f r o m t r i g o n o m e t r y , d e t e r ­
mines t h e r e l a t i o n s h i p between t h e a n g l e s a n d l e n g t h s o f t h e s i d e s
I n t h e c a s e where t h e speed t r i a n g l e
of oblique-angled t r i a n g l e s .
i s used ( F i g . 2.59), t h i s theorem has t h e form:
s i n US - s i n AW -- s i n ( A W + U S )
U
V
W

(2.58)

It is obvious t h a t t h e r e l a t i o n ­
s h i p of E q u a t i o n ( 2 . 5 8 ) i s e q u i v a ­
l e n t t o the following:

1gsinUS - l g u = IgsinAW-lgV

=

l g s i n ( A W + U S ) - IgW,
F i g . 2.57.
Determination
o f t h e O r t h o d r o m i c Coord i n a t e s of t h e Aircraft.

which i s e x p r e s s e d by t h e key on
t h e navigational r u l e (see Fig. 2.60,
,
a).

239

R

Fig. 2.58.
Keys f o r D e t e r m i n i n g t h e A i r c r a f t C o o r d i n a t e s
on t h e N L - 1 O M ;
( a ) X-coordinates;
( b ) Z-coordinates.

Exampbe.
MFA
= 35O, V t s u e
= 4 0 0 k m / h r , 6 = 85O, u = 6 0 k m / h r .
F i n d t h e d r i f t a n g f e of t h e a i r c r a f t a n d t h e g r o u n d s p e e d .
SOlUtiOn.

I n o u r example, t h e wind a n g l e i s
AW =

- MFAg = 8 5 - 3 5 = 5 0 ° .

Having s e t t h e s l i d e r i n d i c a t o r t o t h e d i v i s i o n r e p r e s e n t ­
i n g 4 0 0 k m / h r on s c a l e 5 , a n d a l s o h a v i n g l i n e d u p t h e 50° d i v i ­
s i o n o n t h e l o g a r i t h m s i n e s c a l e 3 w i t h t h e same s l i d e r i n d i c a t o r ,
w e o b t a i n t h e d r i f t a n g l e e q u a l t o 6.5O , a n d a g r o u n d s p e e d o f
440 km/hr ( F i g . 2 . 6 0 , b ) .
T h i s k e y for s o l v i n g s p e e d t r i a n g l e s i s s u i t a b l e for d e t e r ­
m i n i n g s p e e d a n d d r i f t a n g l e o f a n a i r c r a f t a t known w i n d p a r a m ­
eters.
However, i t i s n o t s u i t a b l e f o r d e t e r m i n i n g wind param­
e t e r s ‘ i n m e a s u r i n g t h e d r i f t a n g l e .and t h e g r o u n d s p e e d .
T h i s p r o b l e m c a n b e s o l v e d as f o l l o w s .

L e t u s s a y t h a t o n t h e b a s i s o f m e a s u r e m e n t s , w e know t h e
a i r s p e e d of an a i r c r a f t , t h e ground speed and t h e d r i f t a n g l e ,
./.2 3 3
and w e want t o f i n d t h e s p e e d and d i r e c t i o n o f t h e wind ( u ) a t f l i g h t
a l t i t u d e (see Fig. 2.59).
I t i s c l e a r from t h e diagram t h a t t h e r u n n i n g component o f
t h e wind a t f l i g h t a l t i t u d e i s

ux

=

w - v

cos

us,

(2.59)

w h i l e t h e l a t e r a l component i s
u z = V s i n US = ( W - V

cos US)

t g AW.

(2.60)

If w e c o n s i d e r t h a t t h e d r i f t a n g l e of t h e a i r c r a f t r a r e l y
e x c e e d s 15O, a n d t h e c o s i n e of t h e a n g l e o f d r i f t i s p r a c t i c a l l y
always c l o s e t o u n i t y , Formula ( 2 . 6 0 ) can b e w r i t t e n a s f o l l o w s :

t g AN =
However,

240

since

V s i n US

w - v

*

V s i n U S = ( W - V ) t g AW,
then w e have t h e r a t i o
s i n US - t g AW

w - v -

v

*

which c a n be u s e d as a k e y on t h e s l i d e r u l e ( F i g .

2.61,

a).

In practice, within the l i m i t s
o f th.e d r i f t a n g l e s which a r e encoun­
t e r e d i n f l i g h t , t h e scales 3 and
4 on t h e r u l e c o i n c i d e ( t h e s i n e e q u a l s
t h e t a n g e n t ) , a n d s o t h a t we w i l l
n o t have t o d e a l with t h r e e b u t with
two s c a l e s , w e o f t e n u s e a key which
i s shown i n F i g u r e 2 . 6 1 , b .

0

E x a m p l e . V = 450 k m / h r ; W =
5 2 0 k m / h r - ; US = + l o o . F i n d t h e w i n d
angle.

Fig. 2.59.
Navigational
Speed T r i a n g l e .

The d i f f e r e n c e b e ­
i s e q u a l t o 70 km/hr.

SOlUtiOtl.

tween t h e ground speed and a i r s p e e d ( W - V )
If w e s e t t h i s
2.61, c ) , w e w i l l f i
speed i s found with
theorem (Fig. 2.61,

v a l u e on s c a l e 5 o p p o s i t e l o o on s c a l e 4 ( F i g .
n d t h e wind angle t o be e q u a l t o 48O.
The w i n d
t h e a i d of a k ey wh i ch i s d e s c r i b e d i n t h e s i n e
d).

Answer. u = 105 km/hr.
The f i x e d s c a l e on t h e r u l e r 6 , l i k e s c a l e 5 , i s a s c a l e o f
l o g a r i t h m s o f l i n e a r v a l u e s , b u t t h e s c a l e i s t w i c e t h a t u s e d on
T h e r e f o r e , when c o m p a r i n g a n y o f t h e f i r s t
t h e first f i v e scales.
f i v e s c a l e s t o t h e f i x e d s c a l e , a n u m b e r i s o b t a i n e d on t h e l a t t e r
whose l o g a r i t h m i s e q u a l t o h a l f t h e l o g a r i t h m of t h e numbers o f
t h e first five scales.

Example.
I n s e t t i n g t h e m a r k e r o f t h e s l i d e r t o t h e number
4 0 0 o n s c a l e 5 or 1, t h i s m a r k e r s h o w s h a l f t h e l o g o f 4 0 0 on t h e
s i x t h s c a l e , w h i c h c o r r e s p o n d s t o t h e s q u a r e r o o t o f 4 0 0 or 2 0 .
I f t h e d e s i r e d n u m b e r i s s e t o n s c a l e 6 , w e w i l l o b t a i n num­
b e r s on s c a l e s 5 and 1 whose l o g a r i t h m s a r e e q u a l t o t w i c e t h e l o g a ­
r i t h m o f t h e g i v e n number, t h u s c o r r e s p o n d i n g t o t h a t number r a i s e d
t o a power o f t w o .

(p),

The t u r n r a d i u s of t h e a i r c r a f t w i t h a g i v e n b a n k i n g a n g l e
a s w e know, i s d e t e r m i n e d f r o m F o r m u l a ( 1 . 6 ) .

R = -

V
g tgB

/234

­
241

T h e r e f o r e , t h e problem of d e t e r m i n i n g t h e t u r n r a d i u s i s s o l v e d
by means o f s c a l e s 4 , 5 a n d 6 :
1 g R = 21gV

-

lgg - l g tg8.

Therefore, i n s o l v i n g t h i s problem, it i s necessary t o have
t h e l o g a r i t h m of t h e s q u a r e o f t h e s p e e d a n d t o s e t i t o n s c a l e
6.
The l o g a r i t h m of t h e t a n g e n t o f t h e b a n k i n g a n g l e i s c a l c u l a t e d
with t h e a i d of scale 4.

Fig. 2.60.
C a l c u l a t i o n on t h e N L - 1 O M :
( a ) Key f o r S o l v ­
i n g t h e N a v i g a t i o n a l Speed T r i a n g l e ;
( b ) S o l u t i o n of
N a v i g a t i o n a l Speed T r i a n g l e .
If w e c o n s i d e r t h a t i n o r d e r t o d e t e r m i n e t h e t u r n i n g r a d i u s ,
t h e a i r s p e e d o f t h e a i r c r a f t must b e e x p r e s s e d i n m/sec and n o t
i n k m / h r , as w e d i d on s c a l e 6 , a n d a l s o t h a t i t i s n e c e s s a r y t o
t a k e i n t o a c c o u n t t h e a c c e l e r a t i o n due t o g r a v i t y g , w e have a marking
R on s c a l e 4 w h i c h c o r r e s p o n d s t o t h e l o g a r i t h m o f t h e number
1

3,@. 9,81

i.e.,

Formula ( 1 . 6 )

= 0,00787,

assumes t h e form:
0,00787 V2
R=
tg P

w h i c h c o r r e s p o n d s t o t h e k e y for t h e n a v i g a t i o n a l s l i d e r u l e w h i c h
w a s shown i n F i g u r e 2 . 6 2 , a n d wh i ch i s f o u n d a t t h e b e g i n n i n g o f
t h e t h i r d scale of s i n e s .
The l a s t s c a l e on t h e s l i d e r u l e N L - 1 O M i s t h e s c a l e l a , wh i ch
i s i n t e n d e d t o determine t h e t u r n i n g t i m e (t
of t h e a i r c r a f t a t
P
a g i v e n a n g l e ( U T ) a t a known t u r n i n g r a d i u s ( R ) a n d f l i g h t s p e e d
( V ) . T h i s s c a l e i s a s c a l e o f l o g a r i t h m s for t h e a r c o f t h e c i r c u m ­
f e r e n c e , r e l a t i v e t o t h e r a d i u s of t u r n of t h e a i r c r a f t .
Obviously, t h e t u r n i n g t i m e o f t h e a i r c r a f t a t t h e g i v e n a n g l e /235
w i l l be
2xR * - UT
t -_
p-

v

360

(2.61)
242

I n t h i s f o r m u l a , t h e v a l u e 2 1 ~ / 3 6 0is a c o n s t a n t m u l t i p l i e r .
I n order not t o have t o c a l c u l a t e it each t i m e , scale l a i s set
t o t h e value of t h e logarithm of t h i s m u l t i p l i e r at t h e left-hand
side.
After dropping t h i s m u l t i p l i e r ,

Formula

(2.61) a s s u m e s t h e

form :

%

c)

0

44'

IO'

48'

IO'

Fig. 2.61.
C a l c u l a t i o n on t h e N L - 1 O M :
( a , b ) : Keys f o r
D e t e r m i n i n g t h e Wind A n g l e ;
( c , d ) : Determining t h e
A n g l e a n d S p e e d o f t h e Wind.
w h i c h c a n b e e x p r e s s e d on t h e r u l e s c a l e s b y a k e y shown i n F i g .
2.62, b .

Example. R = 4.5 km,
ing time of t h e aircraft.
Solution.

See F i g .

V = 400 km/hr,

2.62,

c.

UT = 90°.

Answer.

Find t h e turn­

tp = 64 sec.

On t h e b a c k o f t h e r u l e a r e s c a l e s for m a k i n g m e t h o d o l o g i c a l
corrections i n t h e readings of navigational instruments ( a l t i m e t e r s ,
airspeed indicators, outside-air thermometers).
A d j u s t a b l e s c a l e 7 , w i t h a movable diamond-shaped i n d e x a n d
t h e a d j a c e n t s c a l e s ( f i x e d s c a l e 8 a n d movable s c a l e 9 ) a r e i n t e n d e d
f o r making c o r r e c t i o n s i n t h e r e a d i n g s of b a r o m e t r i c a l t i m e t e r s
i n c a s e t h e a c t u a l mean a i r t e m p e r a t u r e d o e s n o t a g r e e w i t h t h e
c a l c u l a t e d t e m p e r a t u r e o b a t i n e d when a d j u s t i n g t h e a p p a r a t u s .
These
c o r r e c t i o n s c a n b e made w i t h t h e f o r m u l a

'gH c orr

= l

TO+TH
g
2

+

lg

Hinst
r
.
av.c.

243

According t o t h i s formula, t h e a d j u s t a b l e scale 7 i s a scale
of l o g a r i t h m s To+TH/2.
For convenience i n use, t h e logarithms
The a r i t h m e t i c e f f e c t s o f
T o + T H / ~o n t h e r u l e a r e m a r k e d t o + t .
c o n v e r t i n g t e m p e r a t u r e s from t h e c e n t i g r a d e scale t o t h e a b s o l u t e
scale and t h e i r d i v i s i o n i n h a l f are t a k e n i n t o c o n s i d e r a t i o n i n
t h e d e s i g n o f t h e s c a l e s i n s u c h a way t h a t i t i s n o t n e c e s s a r y
t o make t h e m e a c h t i m e d u r i n g t h e f l i g h t .

b)

a)

@ @

..

B
- # ­

v 4 a @-

0.
3

ViIp

UT @

R'

(P

c),

@

40

@

4.5

. .@

90

64

Fig. 2.62.
C a l c u l a t i o n on N L - 1 O M :
( a ) Key for D e t e r m i n ­
(c)
i n g Turn Radius.
( b ) Key f o r D e t e r m i n i n g T u r n T i m e .
Determination of Turn T i m e .
Fixed s c a l e 8 ( c o r r e c t e d a l t i t u d e ) i s simply a scale of log­
a r i t h m s o f a l t i t u d e , w h i l e t h e movable s c a l e 9 ( i n s t r u m e n t a l a l t i ­
t u d e ) i s a s c a l e of l o g a r i t h m s o f a l t i t u d e , d i v i d e d b y t h e a v e r ­
age c a l c u l a t e d temperature o b t a i n e d f o r each a l t i t u d e , i . e . ,

The key f o r s o l v i n g p r o b l e m s by i n t r o d u c i n g m e t h o d o l o g i c a l c o r r e c ­
t i o n s t o t h e r e a d i n g s o f t h e a l t i m e t e r a r e shown i n F i g u r e 2 . 6 3 ,

a.

ExampZe. The f l i g h t a l t i t u d e a c c o r d i n g t o t h e i n s t r u m e n t i s
H i n s t = 6000 m ; t H = -35O.
Find t h e f l i g h t a l t i t u d e corrected f o r
t h e methodological e r r o r .
Solution.
T h e a c t u a l t e m p e r a t u r e for a z e r o a l t i t u t d e
t e r m i n e d f r o m t h e t e m p e r a t u r e g r a d i e n t e q u a l t o 6 . 5 deg/km:

t o = tH +
S O

b),

6.Skm = - 3 5

6.5.6

= +4O,

t h a t t o + t H = -31O.

I f w e s e t t h i s t e m p e r a t u r e v a l u e on t h e s l i d e r u l e ( F i g .
w e w i l l o b t a i n H C O r r = 5 . 7 4 km.

To i n t r o d u c e c o r r e c t i o n s i n
f l i g h t a l t i t u d e s a b o v e 1 2 km, w e
c e n t f i x e d diamond-shaped i n d e x ,
f i x e d scale 14 f o r t h e corrected

244

+

i s de­

2.63,

t h e r e a d i n g s of t h e altimeter a t
u s e movable s c a l e 1 0 w i t h t h e a d j a ­
as w e l l as t h e a d j a c e n t s c a l e s :
a l t i t u d e and s p e e d , and f i x e d

/236

-

s c a l e 1 5 for t h e i n s t r u m e n t a l a l t i t u d e a n d s p e e d .
C o r r e c t i o n s t o t h e r e a d i n g s of t h e a l t i m e t e r a t f l i g h t a l t i ­
t u d e s a b o v e 12 km a r e made b y F o r m u l a (2.36).

/237

E x p r e s s i n g t h e a l t i t u d e i n km, t h i s f o r m u l a c a n b e w r i t t e n
as f o l l o w s :

lg(Hcorr

-11) = l g T H
- l g 216.5 t l g ( H i n s t -11).
av

(2.62)

Adjustable scale 1 0 i s a scale of logarithms (lgTHav-lg216.5).
S c a l e s 14 a n d 1 5 a r e s c a l e s o f l o g a r i t h m s ( H - 1 1 k m ) , s o t h a t t h e y
a r e s i m p l e , u n i q u e l o g a r i t h m i c s c a l e s -on w h i c h w e c a n c a r r y o u t
m u l t i p l i c a t i o n and d i v i s i o n o f numbers, b u t w i t h a d d i t i o n a l numbers
w h i c h a r e s h i f t e d b y 11 km t o c a l c u l a t e a l t i t u d e .

F i g . 2.63.
C a l c u l a t i o n o n NL-1OM:
( a ) Key f o r I n t r o d u c i n g
Methodological Correction i n A l t i m e t e r Reading.
( b ) Determ i n a t i o n o f C o r r e c t i o n f o r Measured F l i g h t A l t i t u d e .

I n a c c o r d a n c e w i t h F o r m u l a (2.62), t h e k e y f o r i n t r o d u c i n g
corrections i n t h e readings of t h e altimeter a t f l i g h t a l t i t u d e s
a b o v e 12 km i s s h o w n i n F i g u r e 2.64 a .

ExampZe. H i n s t

Solution.

= 1 4 km;

tH = -5OO.

S e e F i g u r e 2.64, b .

Find Hcorr.

Answer:

Hcorr

= 14,400 m .

Note.
Since t h e a l t i t u d e of t h e tropopause a t middle l a t i t u d e s
i s n o t e x a c t l y a t a n a l t i t u d e o f 11 k m , b u t c a n c h a n g e w i t h i n l i m i t s
o f 9-13 k m , a f t e r s o l v i n g t h e p r o b l e m b y m e a n s o f t h e k e y s h o w n
i n F i g u r e 2.63, b , t h e f l i g h t a l t i t u d e m u s t b e c o r r e c t e d f o r t h e
a d d i t i o n a l c o r r e c t i o n AH = 9 O O t 2 0 ( t O + t H ) w h i c h i s s h o w n o n t h e
r u l e a t t h e r i g h t - h a n d s i d e b e l o w s c a l e 14.

F i g . 2.64.
C a l c u l a t i o n on N L - 1 O M :
( a ) Key f o r I n t r o ­
d u c i n g C o r r e c t i o n i n F l i g h t A l t i t u d e s a b o v e 12,000 m ;
( b ) Determination of Correction f o r F l i g h t A l t i t u d e
above 12,000 m.

245


The m e t h o d o l o g i c a l c o r r e c t i o n s d u e t o t h e f a i l u r e o f a g r e e ­
m e n t o f t h e a c t u a l a i r t e m p e r a t u r e w i t h a c a l c u l a t e d v a l u e a r e made
) i t h t h e a i d o f Formula
t o c a l i b r a t e t h e s p e e d i n d i c a t o r ( t y p e l l U S 1 lw
(2.53).
I n a c c o r d a n c e w i t h t h i s f o r m u l a , t h e s c a l e 1 4 on t h e r u l e r f o r
logVtrue a n d s c a l e 1 5 f o r l o g V i n s t a r e p u r e l y l o g a r i t h m i c s c a l e s
of l i n e a r v a l u e s .
A d j u s t a b l e s c a l e 11 ( t e m p e r a t u r e f o r s p e e d ) i s
a scale of logarithms
1
Ig (273" t ~ ) s
2

-

+

while a d j a c e n t t o it i s f i x e d scale 1 2 (instrument a l t i t u d e a l t i t u d e i n km) w i t h a s c a l e o f l o g a r i t h m s
1

-lg288+
2

2,6281g(1 + 0 , 0 2 2 6 H ) .

The k e y f o r i n t r o d u c i n g c o r r e c t i o n s i n t h e r e a d i n g s o f t h e s p e e d
i n d i c a t o r "US" i s s h o w n i n F i g u r e 2 . 6 5 , a .

Example.

t H = -30°,

Hinst

= 7 km;

vinst

= 450 k m / h r .

Find

the airspeed.

Solution:

See F i g u r e 2 . 6 5 ,

b.

Answer:

638 km/hr.

( a ) Key f o r I n t r o d u c i n g
Fig. 2.65.
C a l c u l a t i o n on N L - 1 O M :
(b)
C o r r e c t i o n i n R e a d i n g s o f T y p e "LIS" S p e e d I n d i c a t o r ;
D e t e r m i n a t i o n o f C o r r e c t i o n f o r R e a d i n g o f T y p e "US" S p e e d
Indicator.
For speed i n d i c a t o r s of t y p e llCSI1l, t h e c o r r e c t i o n s g i v e n above
a r e found by Formula ( 2 . 5 4 a ) .

I t i s c l e a r f r o m t h i s f o r m u l a t h a t f i x e d s c a l e 11 a n d m o v a b l e
s c a l e 1 5 (for V i n s t ) a n d 1 4 (for V c o r r ) w i l l b e t h e s a m e f o r t h e
speed i n d i c a t o r s o f t y p e s rlUSfl and "CSI".
I n s t e a d o f f i x e d s c a l e 1 2 , w e c a n s c a l e 1 3 on s p e e d i n d i c a t o r s
o f t y p e rlCSI1l,which i s a s c a l e o f l o g a r i t h m s

1

lg(288 - 0 . 0 0 6 5 H i n s t
2

1.

The k e y f o r i n t r o d u c i n g c o r r e c t i o n s i n t h e r e a d i n g s of t h e s e
i n d i c a t o r s i s shown i n F i g u r e 2 . 6 6 , a .

246

/238

ExampZe.
H = 1 0 km,
corrected airspeed.

Solution:

tH =

See Figure 2.66,

VCSI

-45O,

b.

= 800 k m / h r .

Answer:

Find t h e

808 km/hr.

R u l e s c a l e 1 6 i s s e t up a c c o r d i n g t o t h e f o r m u l a

At=K-

112.

26 000

and is used f o r i n t r o d u c i n g c o r r e c t i o n s i n t o t h e readings of t h e
t h e r m o m e t e r f o r t h e o u t s i d e a i r , t y p e "TUE".
T h i s same s c a l e c a n
be used a t subsonic a i r s p e e d s , and t h e e r r o r w i l l n o t be g r e a t e r
t h a n 1 - 2 O f o r t h e t y p e I'TNV".
I n p r a c t i c e , t h e f r o n t s i d e of t h e navigational s l i d e r u l e
NL-1OM c a n b e u s e d t o s o l v e a number of o t h e r p r o b l e m s , t h e k e y s
f o r w h o s e s o l u t i o n a r e d i r e c t l y d e p e n d e n t on t h e n a t u r e o f t h e p r o b ­
lem.
One e x a m p l e o f s u c h a p r o b l e m i s t h e d e t e r m i n a t i o n o f t h e d e v i a t i o n a n g l e of t h e m e r i d i a n s b e t w e e n t w o p o i n t s o n t h e E a r t h ' s
surface.
The a n g l e of
Formula ( 1 . 8 2 ) .

d e v i a t i o n of t h e m e r i d i a n s c a n b e d e t e r m i n e d by

O b v i o u s l y , t h i s p r o b l e m c a n b e s o l v e d on a r u l e r by means o f
a key shown i n F i g u r e 2 . 6 7 , a .

Fig. 2.66.
C a l c u l a t i o n on N L - 1 O M :
( a ) Key for I n t r o d u c i n g
(b)
C o r r e c t i o n i n R e a d i n g o f Type "CSI" Speed I n d i c a t o r ;
D e t e r m i n a t i o n o f C o r r e c t i o n f o r R e a d i n g of Type "CSI" Speed
Indicator.
The s c a l e s o n t h e b a c k o f t h e r u l e r c a n b e u s e d t o s o l v e some
o t h e r problems.
For e x a m p l e , m o v a b l e s c a l e s 1 4 a n d 1 5 a r e t h e o n e s
m o s t s u i t a b l e for m u l t i p l i c a t i o n a n d d i v i s i o n o f n u m b e r s .
S c a l e 1 4 i s marked o f f w i t h t h e f o l l o w i n g v a l u e s :
AM ( A m e r i ­
c a n s t a t u t e m i l e , e q u a l t o 1 . 6 3 k m ) ; NM ( n a u t i c a l m i l e , e q u a l t o
1 . 8 5 2 km), a n d f o o t ( e q u a l s 3 2 . 8 cm).
T h e s e m a r k i n g s a r e u s e d for
r a p i d c o n v e r s i o n o f measurements from one s y s t e m t o a n o t h e r .

247

/239

Fig. 2.67.
C a l c u l a t i o n on N 1 - 1 O M :
( a ) Key f o r D e t e r ­
m i n i n g Angle of D e v i a t i o n o f M e r i d i a n s ;
( b ) Conver­
s i o n o f t h e Len.gth of t h e A r c o f t h e O r t h o d r o m e i n t o
Kilometers.

Example.

C o n v e r t t h e l e n g t h o f t h e a r c o f t h e o r t h o d r o m e 5O16'

t o kilometers.

Solution.

5O16'

= 3 1 6 NM ( n a u t i c a l m i l e s ) .

Having s e t d i v i s i o n 1 0 0 on s c a l e 1 4 on t h e n a v i g a t i o n a l s l i d e
r u l e o p p o s i t e 316 on s c a l e 1 5 ( F i g . 2 . 6 7 , b ) , w e o b t a i n t h e a n s w e r
( 5 8 5 km).
On s c a l e s 1 4 a n d 1 5 , b y
1 2 , w e can s o l v e problems i n
a i r s p e e d and a i r temperature
t h e a i r s p e e d a t a g i v e n Mach
Therefore,

u s i n g t h e s e t t i n g s o f s c a l e s 11 a n d
d e t e r m i n i n g t h e Mach n u m b e r a t a known
a t a f l i g h t a l t i t u d e , or d e t e r m i n e
number a n d a i r t e m p e r a t u r e .

t h e s p e e d o f s o u n d i n a i r i s f o u n d by t h e f o r m u l a
M=-- V,t r u e

-

a

Vtrue
20,3 1/273"

+ tH

or


S c a l e s 1 4 a n d 1 5 a r e s c a l e s o f l o g V , f i x e d s c a l e 11 i s t h e
s c a l e o f 1 / 2 l o g ( 2 7 3 O + t ~ ) ,a n d f i x e d s c a l e 1 2 i s a s c a l e o f 2 . 6 2 8
l o g ( 1 - 0 . 0 2 2 6 H ) , w h i c h i s m o v a b l e r e l a t i v e t o s c a l e 11 t o t h e v a l u e
1/21og288.

Fig.

248

2.68.

D e t e r m i n a t i o n o f Mach Number o n N L - 1 O M .

I n o r d e r t o g e t log20.3 from t h e v a l u e 2.628 l o g (1-0.0226H),
i t i s i m p o r t a n t t o r e p l a c e H b y a v a l u e o f 3 . 2 5 km.
Therefore,
i f w e f i n d a n a l t i t u d e o f 3 . 2 5 km a n d s e t i t o n f i x e d s c a l e 1 2 ,
w e w i l l o b t a i n t h e key f o r s o l v i n g t h e problem with a c e r t a i n M
number ( F i g . 2 . 6 8 ) .
Obviously, t h e value M = 1 corresponds t o t h e airspeed ( i n
km/hr) which is e q u a l t o t h e s p e e d o f s o u n d .
To d e t e r m i n e t h e s p e e d o f s o u n d i n m / s e c , i t i s n e c e s s a r y t o
s e t t h e v a l u e o f 0 . 2 7 7 5 (1/36) on s c a l e 1 5 .
If w e u s e t h e r e c t a n g ­
u l a r i n d e x w i t h t h e m a r k i n g o f 1 0 0 0 f o r ' M = 1, t h e n d i v i s i o n 0 . 2 7 7 5
w i l l c o r r e s p o n d t o t h e number 277.5.

Note.
I n general, f o r converting zero a l t i t u d e , correspond­
i n g t o 1 / 2 log288, t o t h e value of log20.3, it i s necessary t o s h i f t
it t o t h e r i g h t t o t h e v a l u e 2.51 km, and t h e f u n c t i o n s of s c a l e s
1 4 a n d 1 5 i n t h e key shown i n F i g u r e 2.100 w i l l change p l a c e s .
Then
Formula ( 2 . 6 3 ) w i l l b e v a l i d .
H o w e v e r , t h i s w i l l c a l l f o r u s i n g f i x e d s c a l e 11 ( t e m p e r a t u r e
for s p e e d ) i n r e v e r s e o r d e r .
T h e r e f o r e , i t w a s f o u n d t o b e more
c o n v e n i e n t f o r s o l v i n g t h e s e p r o b l e m s r e l a t e d t o t h a t Mach n u m b e r
t o i n t e r c h a n g e t h e l o c a t i o n s o f t h e v a l u e s on s c a l e s 1 4 a n d 1 5 ,
and t o s h i f t t h e a l t i t u d e t o t h e l e f t , t o t h e d i v i s i o n e q u i v a l e n t
t o 3 . 2 5 km, a l t h o u g h t h e o r e t i c a l l y F o r m u l a ( 2 . 6 3 ) i s n o t t h e n v a l i d ,
The f a c t t h a t t h e numbers 2 . 5 1 ( w i t h a s h i f t t o t h e r i g h t ) a n d
3.25 ( w i t h a s h i f t t o t h e l e f t ) are n o t e q u a l is e x p l a i n e d by t h e
f a c t t h a t z e r o a l t i t u d e under s t a n d a r d c o n d i t i o n s does not corre­
s p o n d t o z e r o t e m p e r a t u r e b u t t o +15O.
T h e r e f o r e , t o make z e r o
t e m p e r a t u r e m a t c h d i v i s i o n H = 3 . 2 5 , t h e s c a l e m u s t b e moved b y
a n a m o u n t s u c h t h a t i t l i n e s u p w i t h t h e m a r k i n g H = - 2 . 5 1 km.

249

IIIIII I I l l 1 I

I I


CHAPTER THREE

AIRCRAFT NAVIGATION USING RADIO-ENGINEERING DEVICES

1. Pri nci p l e s o f t h e T h e o r y of Radi onavi g a t i o n a l
Instruments
G e o t e c h n i c a l methods o f a i r c r a f t n a v i g a t i o n , a l t h o u g h t h e y
f o r m t h e b a s i s o f t h e c o m p l e x o f n a v i g a t i o n a l e q u i p m e n t on a n a i r ­
c r a f t , do n o t p e r m i t a c o m p l e t e s o l u t i o n o f t h e problems o f a i r ­
c r a f t n a v i g a t i o n w h e n t h e r e a r e n o t e r r e s t r i a l l a n d m a r k s or w h e n
the latter are invisible.

/241

The p r i n c i p a l r e a s o n f o r t h i s i s t h e v a r i a t i o n o f t h e w i n d
a t f l i g h t a l t i t u d e , which means t h a t t h e f l i g h t c a n n o t b e maint.ained
f o r a s i g n i f i c a n t p e r i o d of t i m e w i t h o u t checking t h e d i s t a n c e and
d i r e c t i o n of t h e path being followed.
Astronomical means, however, a r e n o t always h e l p f u l i n d e t e r ­
mining t h e l o c a t i o n of t h e a i r c r a f t , s i n c e t h e heavenly bodies are
j u s t as i n v i s i b l e as t e r r e s t r i a l l a n d m a r k s when f l y i n g i n c l o u d s
or b e t w e e n c l o u d l a y e r s .
I n a d d i t i o n , i n order t o determine t h e
l o c a t i o n o f t h e a i r c r a f t , i t i s n e c e s s a r y t o s e e a t l e a s t two l u m i n a r ­
i e s i n t h e sky s i m u l t a n e o u s l y , which i s n o t always p o s s i b l e under
normal f l i g h t c o n d i t i o n s .
T h i s h a s made i t n e c e s s a r y t o s e e k new m e t h o d s o f r e l i a b l y
c a r r y i n g o u t a i r c r a f t n a v i g a t i o n under any p h y s i c a l and geograph­
i c a l c o n d i t i o n s e i t h e r d a y or n i g h t , w i t h o u t d e p e n d e n c e u p o n m e t e o r ­
o l o g i c a l c o n d i t i o n s , and has l e d t o t h e development of radio-engin­
eering devices f o r aircraft navigation.

All r a d i o - e n g i n e e r i n g d e v i c e s f o r a i r c r a f t n a v i g a t i o n u s e t h e
p r o p e r t i e s of t h e p r o p a g a t i o n o f e l e c t r o m a g n e t i c waves i n t h e E a r t h I s
atmosphere t o v a r y i n g degree,s

.

We know t h a t t h e p h a s e v e l o c i t y o f t h e p r o p a g a t i o n o f w a v e
energy i n d i e l e c t r i c media i s
c1 =

250

C

VF'

'

w h e r e c1 i s t h e r a t e o f p r o p a g a t i o n o f e l e c t r o m a g n e t i c w a v e s i n
t h e m e d i u m , c i s t h e r a t e of p r o p a g a t i o n o f e l e c t r o m a g n e t i c w a v e s
i n a v a c u u m , 1.1 i s t h e m a g n e t i c p e r m e a b i l i t y , a n d E i s t h e d i e l e c t r i c constant.
F o r a v a c u u m , U = E = 1.

/242

I n a d d i t i o n t o t h e p h a s e p r o p a g a t i o n r a t e of e l e c t r o m a g n e t i c
w a v e s , t h e r e i s a l s o a g r o u p p r o p a g a t i o n r a t e of e l e c t r o m a g n e t i c
energy.
I n a vacuum, t h e p h a s e a n d g r o u p p r o p a g a t i o n r a t e s f o r e l e c ­
t r o m a g n e t i c waves a r e t h e same i n a l l c a s e s .
'In d i e l e c t r i c m e d i a , e s p e c i a l l y i n s o l i d s , l i q u i d s , a n d ( t o
a much s m a l l e r d e g r e e ) g a s e s , t h e p h a s e p r o p a g a t i o n r a t e d e p e n d s
on t h e f r e q u e n c y o f t h e o s c i l l a t o r y p r o c e s s .
This is explained
by t h e i n e r t i a o f t h e d i e l e c t r i c medium, ? . e . , t h e d i e l e c t r i c perme­
a b i l i t y o f t h e medium d e p e n d s o n t h e o s c i l l a t i o n f r e q u e n c y .
The d e p e n d e n c e o f t h e p h a s e p r o p a g a t i o n r a t e upon t h e o s c i l ­
If t h e waves p r o p a g a t e i n
l a t i o n frequency i s c a l l e d dispersion.
a n e l e c t r o m a g n e t i c medium w i t h d i f f e r e n t f r e q u e n c i e s , t h e i r p h a s e
r a t e may n o t b e t h e s a m e .
I n t h i s case, t h e t o t a l energy of t h e
w a v e s w i l l b e maximum a t t h o s e p o i n t s i n s p a c e w h e r e t h e p h a s e s
of t h e waves a r e c l o s e s t t o c o i n c i d e n c e .
In addition, there w i l l
b e p o i n t s where t h e t o t a l e n e r g y of a l l t h e waves w i l l b e e q u a l
t o z e r o , i . e . , where t h e p o s i t i v e p h a s e s o f t h e waves w i l l b e b a l ­
a n c e d by t h e n e g a t i v e o n e s .
T h e p o i n t s w i t h maximum t o t a l e n e r g y a r e c a l l e d c e n t e r s o f
wave e n e r g y .
T h e r a t e a t w h i c h t h e c e n t e r s o f w a v e e n e r g y move
i n space i s t h e group r a t e of t h e waves.
The g r o u p r a t e o f p r o p a g a t i o n

o f e l e c t r o m a g n e t i c waves i n s p a c e

is
c:

'rg'

where e
and el

!?E

c*-w--J-

dc
do

'

i s t h e group r a t e , w i s t h e average s p e c t r a l frequency,
t h e average phase rate of t h e spectrum.

I t i s c l e a r from t h e f o r m u l a t h a t w i t h p o s i t i v e d i s p e r s i o n ,
t h e g r o u p r a t e of t h e waves e x c e e d s t h e p h a s e r a t e o f t h e i r p r o p ­
agation.

Wave P o l a r i z a t i o n
Figure 3.1 is a graphic representation of a propagating elec­
t r o m a g n e t i c wave i n t h e h o r i z o n t a l p l a n e as a f u n c t i o n o f t h e v e r t i c a l
open c i r c u i t .

251

I n t h i s case, t h e v e c t o r o f t h e e l e c t r i c a l f i e l d , and t h e r e ­
f o r e t h e displacement currents , w i l l coincide with t h e direction
of t h e d i p o l e of t h e c i r c u i t ( d i p o l e open a n t e n n a ) .
The p l a n e o f
t h e vector of t h e magnetic f i e l d coincides with t h e h o r i z o n t a l plane,
O b v i o u s l y , t h e e l e c t r o m a g n e t i c wave i s a t r a n s v e r s e wave , i . e . , / 2 4 3
t h e a m p l i t u d e s of t h e o s c i l l a t i o n s o f t h e e l e c t r i c and magnetic
f i e l d s a r e l o c a t e d a t r i g h t a n g l e s t o t h e d i r e c t i o n o f wave p r o p a ­
gation.
The d i r e c t i o n o f t h e p l a n e o f o s c i l l a t i o n o f t h e e l e c t r i c f i e l d
i s c a l l e d t h e v e c t o r o f wave p o l a r i z a t i o n .
I n o u r s k e t c h , w e have
e l e c t r o m a g n e t i c waves w i t h a v e r t i c a l p o l a r i z a t i o n v e c t o r .
I n r e c e i v i n g electromagnetic waves, it i s important t o b e s u r e
t h a t t h e d i r e c t i o n of t h e dipole of t h e receiving c i r c u i t coincides
w i t h t h e v e c t o r o f wave p o l a r i z a t i o n .
I n t h i s case, t h e o s c i l l a t i o n s
o f t h e e l e c t r i c f i e l d a n d t h e a x i s of r o t a t i o n o f t h e m a g n e t i c f i e l d
coincide with t h e d i r e c t i o n of t h e d i p o l e , and both of t h e s e f a c t o r s
w i l l b r i n g t h e e l e c t r o m a g n e t i c f o r c e ( a n d c o n s e q u e n t l y t h e conduc­
t i v i t y currents) t o the receiving antenna.

i s o p h a s a l circles

If t h e waves a r e v e r t i c a l l y
p o l a r i z e d and t h e r e c e i v i n g antenna
is located i n a horizontal position,
no e m f w i l l be produced i n t h e
dipole.

With t h e d i p o l e i n a h o r i z o n t a l
p o s i t i o n , t h e e l e c t r o m a g n e t i c waves
reaching t h e antenna w i l l have
a h o r i z o n t a l v e c t o r of polarization.
I n t h i s case, t h e r e c e i v ­
Propagation of an
Fig. 3.1.
i n g a n t e n n a must be h o r i z o n t a l ;
E l e c t r o m a g n e t i c Wave f r o m a
i n a d d i t i o n , t h e d i r e c t i o n of t h e
Verticle Dipole.
antenna i n the horizontal plane
must b e p e r p e n d i c u l a r t o t h e l i n e
t o t h e t r a n s m i t t i n g a n t e n n a , i . e . , i t must c o i n c i d e w i t h t h e p o l a r ­
i z a t i o n v e c t o r of t h e waves.
The c i r c l e s i n F i g u r e 3 . 1 j o i n p o i n t s i n t h e h o r i z o n t a l p l a n e
which have i d e n t i c a l phases f o r t h e e l e c t r o m a g n e t i c waves.
These
circles are c a l l e d isophasal.
From t h e v i e w p o i n t o f t h e r e c e i v i n g a n t e n n a , t h e i s o p h a s a l
c i r c l e s (and t h e isophasal spheres i n t h e propagation area) are
t h e d i r e c t i o n s o f t h e wave f r o n t .

252

P r o p a g a t i o n o f E l e c t r o m a g n e t i c O s c i l l a t i o n s
Homogeneous Media

in

I n o r d e r t o make u s e o f t h e p r i n c i p l e s o f d e s i g n o f v a r i o u s
t r a n s m i t t i n g and r e c e i v i n g r a d i o n a v i g a t i o n a l i n s t r u m e n t s , it i s
n e c e s s a r y t o become a c q u a i n t e d w i t h t h e c h a r a c t e r i s t i c s o f t h e p r o p ­
a g a t i o n o f e l e c t r o m a g n e t i c o s c i l l a t i o n s i n inhomogeneous c o n d u c t ­
i n g and nonconducting media.
/24&
E l e c t r o m a g n e t i c wave p r o c e s s e s i n d i e l e c t r i c s c o n s t i t u t e t h e
c o n v e r s i o n of t h e p o t e n t i a l energy o f ' t h e e l e c t r i c a l l y deformed
medium t o t h e k i n e t i c e n e r g y o f d i s p l a c e m e n t c u r r e n t s a n d v i c e v e r s a
( t h e k i n e t i c energy of t h e f i e l d i n t o t h e p o t e n t i a l deformation
o f t h e medium).

For t h e m a j o r i t y o f d i e l e c t r i c m a t e r i a l s , p o l a r i z a t i o n i s n o t
r e l a t e d t o a b s o r p t i o n o f wave e n e r g y , b e c a u s e wave e n e r g y i s p r o p ­
agated practically without losses i n a l l directions.
A decrease
i n t h e o s c i l l a t i o n power w i t h d i s t a n c e t a k e s p l a c e due t o t h e f a c t
t h a t t h e wave e n e r g y f i l l s a n i n c r e a s i n g l y l a r g e v o l u m e , w h i c h ( a s
we k n o w ) i s p r o p o r t i o n a l t o t h e c u b e o f t h e r a d i u s o f t h e s p h e r e
which i t f i l l s .
S i g n i f i c a n t l o s s e s i n wave e n e r g y c a n o c c u r i n s o l i d d i e l e c ­
trics with polar molecules.
I n t h i s case, t h e p o l a r i z a t i o n i s n o t
r e l a t e d t o e l a s t i c d e f o r m a t i o n b u t t o t h e motion o f m o l e c u l e s , which
c a u s e s a c o n v e r s i o n o f wave e n e r g y i n t o h e a t .
I n c o n d u c t i n g m e d i a , t h e e l e c t r o m a g n e t i c waves c a r r y a l t e r ­
nating conductivity currents.
T h i s means t h a t c o n d u c t o r s a l w a y s
u n d e r g o a b s o r p t i o n o f wave e n e r g y a n d i t s c o n v e r s i o n t o h e a t .
T h u s , t h e p r o p a g a t i o n o f wave e n e r g y i n m e d i a , e x h i b i t i n g b o t h
e l e c t r o n i c and i o n i c c o n d u c t i v i t y , i s p r a c t i c a l l y p o s s i b l e t o a
s l i g h t d e p t h w h i c h d e p e n d s o n t h e c o n d u c t i v i t y o f t h e medium a n d
t h e frequency of t h e o s c i l l a t i o n s .
The h i g h e r t h e c o n d u c t i v i t y

o f t h e medium and t h e g r e a t e r t h e f r e q u e n c y of o s c i l l a t i o n , t h e
shallower t h e depths t o which t h e o s c i l Z a t i o n s w i Z Z propagate.
S i n c e t h e p r o p a g a t i o n r a t e of e l e c t r o m a g n e t i c waves d e p e n d s
on t h e d i e l e c t r i c a n d m a g n e t i c p e r m e a b i l i t y of t h e medium, a n d t h e
e l e c t r o n i c or i o n i c c o n d u c t i v i t y o f m e d i a c a n b e a s s u m e d t o b e a
very high (approaching i n f i n i t y ) d i e l e c t r i c permeability, t h e concept
of o p t i c a l d e n s i t y of media has been introduced.
The m i n i m a l o p t i c a l d e n s i t y ( e q u a l t o o n e ) i s p o s s e s s e d b y
a vacuum ( w h e r e t h e p r o p a g a t i o n r a t e of t h e waves i s e q u a l t o e ) .
The o p t i c a l d e n s i t y o f a l l o t h e r d i e l e c t r i c s i s g r e a t e r t h a n u n i t y .
In i d e a l conductors, t h e o p t i c a l density is equal t o i n f i n i t y ( t h e
p r o p a g a t i o n r a t e of e l e c t r o m a g n e t i c waves i s e q u a l t o z e r o ) .
I n p o r t i o n s o f a medium w i t h v a r y i n g o p t i c a l d e n s i t y , e l e c t r o ­

253
5

m a g n e t i c o s c i l l a t i o n s change t h e d i r e c t i o n of t h e i r p r o p a g a t i o n .
T h e c h a n g e i n d i r e c t i o n of p r o p a g a t i o n o f e l e c t r o m a g n e t i c w a v e s
o n t h e s u r f a c e s o f p a r t i c l e s o f t h e medium w i t h d i f f e r e n t o p t i c a l
d e n s i t y i s c a l l e d r e f r a c t i o n of r a y s .
I n a d d i t i o n , under c e r t a i n
c o n d i t i o n s , t h e r e i s r e f l e c t i o n o f waves f r o m t h e s u r f a c e s o f t h e
sections.
The c o e f f i c i e n t of r e f l e c t i o n d e p e n d s on t h e d i f f e r e n c e
b e t w e e n t h e o p t i c a l d e n s i t i e s of t h e m e d i a , t h e f r e q u e n c y o f t h e
o s c i l l a t i o n s , a n d t h e a n g l e of i n c i d e n c e o f t h e wave.
/245
When t h e p a t h o f a w a v e ( p r o p a g a t i o n d i r e c t i o n ) r u n s f r o m a
l e s s d e n s e medium t o a m o r e d e n s e o n e , w i t h a c e r t a i n a n g l e o f i n c i ­
d e n c e t o t h e s u r f a c e , t h e r e may b e n o s e p a r a t i o n o f t h e r e f l e c t e d ,'
wave.
Such a n a n g l e i s c a l l e d t h e a n g l e o f t o t a l i n t e r n a l r e f l e c ­
t i o n o f t h e d e n s e r medium.
I f t h e medium w i t h t h e g r e a t e r o p t i c a l
d e n s i t y i s a c o n d u c t o r , i r r e v e r s i b l e a b s o r p t i o n o f w a v e e n e r g y may
t a k e p l a c e i n i t ( c o n v e r s i o n o f wave e n e r g y i n t o h e a t ) .

With t h e g r a d u a l c h a n g e i n t h e o p t i c a l d e n s i t y o f t h e medium,
t h e r e i s a c o n t i n u o u s r e f r a c t i o n ( b e n d i n g ) of t h e l i n e o f p r o p a g a ­
t i o n , c a l l e d radiorefraction.
The o p t i c a l i n h o m o g e n e i t y o f a medium c h a r a c t e r i z e s t h e p r o p ­
a g a t i o n c h a r a c t e r i s t i c s of waves o f d i f f e r e n t f r e q u e n c i e s i n t h e
Earth's atmosphere.

A l l h a r m o n i c o s c i l l a t i o n s i n a medium a r e c h a r a c t e r i z e d b y
an o s c i l l a t i o n frequency ( w ) and an amplitude o s c i l l a t i o n ( E ) .
I f w e s a y t h a t t h e a m p l i t u d e o s c i l l a t i o n i s t h e maximum v a l u e
o f t h e i n t e n s i t y of t h e e l e c t r i c a l f i e l d , t h e n a t a n y f i x e d p o i n t
t h e o s c i l l a t i o n process w i l l s a t i s f y the expression:

E = E,
where

sin(wtt$),

i s t h e i n i t i a l phase of o s c i l l a t i o n .

The d e r i v a t i v e o f t h e f i e l d i n t e n s i t y w i t h t i m e w i l l c h a r a c ­
t e r i z e t h e m a g n i t u d e of t h e d i s p l a c e m e n t c u r r e n t

Idis

dE =

E-

dt

while t h e second d e r i v a t i v e w i l l express t h e a c c e l e r a t i o n of t h e
displacement current

The d i s t a n c e b e t w e e n t h e t w o c l o s e s t p o i n t s i n s p a c e w h i c h
l i e a l o n g t h e l i n e of p r o p a g a t i o n of t h e wave f r o n t , i n w h i c h t h e
wave p h a s e i s i d e n t i c a l , a r e c a l l e d t h e w a v e l e n g t h ( A ) , w h i c h i s
e q u a l t o cl/w

254

I


E l e c t r o m a g n e t i c w a v e s c a n b e s u b d i v i d e d i n t o f o u r g r o u p s on
t h e b a s i s o f t h e i r p r o p a g a t i o n c h a r a c t e r i s t i c s i n t h e E a r t h ' s atmo­
sphere.
1. L o n g w a v e s , f r o m 3 0 , 0 0 0 t o 3 0 0 0 m ( 1 0 - 1 0 0 k H z ) .
These
waves h a v e a s u r f a c e t y p e o f p r o p a g a t i o n .
Conducting media such
as t h e E a r t h ' s s u r f a c e a n d t h e u p p e r i o n i z e d l a y e r s o f t h e atmo­
s p h e r e h a v e a d e f l e c t i n g e f f e c t upon them.
2.
Medium Waves, f r o m 3 0 0 0 t o 2 0 0 m ( 1 0 0 - 1 5 0 0 k H z ) h a v e a
complex t y p e of p r o p a g a t i o n .
I n t h e d a y , when t h e i o n i z e d l a y e r s
of t h e a t m o s p h e r e a r e l o w e r , t h e t y p e o f p r o p a g a t i o n i s s u p e r f i c i a l
a s i n t h e c a s e o f l o n g w a v e s ; a t n i g h t , t h e medium w a v e s h a v e b o t h
a s u r f a c e and s p a t i a l t y p e of propagation.

3.
S h o r t waves, from 2 0 0 t o 1 0 m (1500-30,000
s p a t i a l type of propagation.

/246

kHz) h a v e a

4.
UZtra-short waves, l e s s t h a n 1 0 m , h a v e a r a d i a l t y p e o f
propagation.
They c a n b e r e f l e c t e d f r o m c o n d u c t i n g l a y e r s on t h e
E a r t h ' s s u r f a c e , but only under c e r t a i n conditions can they be re­
f l e c t e d from t h e i o n i z e d l a y e r s of t h e atmosphere.
Therefore, it
i s t h e s e waves which a r e u s e d w i t h i n t h e l i m i t s o f g e o m e t r i c v i s i ­
b i l i t y of o b j e c t s .
T h e r e s i s t a n c e t o t h e s e w a v e s on t h e E a r t h ' s
surface is insignificant.
From t h e p o i n t o f v i e w o f e l e c t r i c a l c o n d u c t i v i t y a n d r e l i e f ,
t h e E a r t h ' s s u r f a c e h a s a c o m p l e x n a t u r e w h i c h d e p e n d s on t h e t i m e
of y e a r and weather conditions.
The i o n i z e d l a y e r s o f t h e a t m o ­
s p h e r e also h a v e a v a r y i n g n a t u r e .
The i o n i z e d D l a y e r , which i s c l o s e s t t o t h e E a r t h ' s s u r f a c e ,
i s o n l y o b s e r v e d i n t h e d a y t i m e a n d d e p e n d s on t h e t i m e o f y e a r ,
t i m e o f d a y , a n d g e o g r a p h i c a l l a t i t u d e ; i t may a p p e a r a t h e i g h t s
from 60-90 km.
T h i s l a y e r h a s a n e f f e c t on t h e p r o p a g a t i o n o f l o n g
a n d medium w a v e s .
T h e c r i t i c a l f r e q u e n c y o f t h e l a y e r i s 0 . 4 MHz
( 7 5 0 m).
Waves w i t h f r e q u e n c i e s h i g h e r t h a n t h e c r i t i c a l a r e n o t
r e f l e c t e d from t h e l a y e r .
Above t h i s i s t h e E l a y e r , w h o s e i o n i z a t i o n maximum i s r e a c h e d
a t a h e i g h t o f 120-130 km.
T h i s l a y e r i s t h e most s t a b l e one and
r e t a i n s i t s e f f e c t b o t h day and n i g h t .
The c r i t i c a l f r e q u e n c y o f
t h i s l a y e r w i t h m a x i m u m i l l u m i n a t i o n i s 4 . 5 MHz; a t n i g h t i t d r o p s
t h i s l a y e r h a s a maximum e f f e c t o n t h e p r o p a g a t i o n
t o 0 . 9 MHz.
o f medium a n d i n t e r m e d i a t e w a v e s ( t h e s h o r t w a v e s a t t h e s p e c t r a l
b o u n d a r y w i t h t h e medium w a v e s ) .
During t h e e v e n i n g and morning
h o u r s , t h e l a y e r c h a n g e s i t s p a r a m e t e r s s o t h a t t h e s u r f a c e of maxi­
mum i o n i z a t i o n d e c r e a s e s .
This leads t o e r r o r s i n radionavigation
m e a s u r e m e n t s , s i n c e i t r e v e r s e s t h e v e c t o r o f t h e wave p o l a r i z a t i o n
a n d c o n s e q u e n t l y t h e d i r e c t i o n o f p r o p a g a t i o n o f t h e wave f r o n t
i n the horizontal plane.

255

The t h i r d i o n i z e d l a y e r ( F ) i s t h e m o s t u n s t a b l e o n e b o t h i n
t e r m s o f t i m e o f d a y as w e l l as s e a s o n of t h e y e a r .
Its average
D u r i n g t h e d a y t i m e i n summer, t h i s l a y e r
h e i g h t i s 2 7 0 - 3 0 0 km.
I n a d d i t i o n , t h e F l a y e r shows
d i v i d e s i n t o two p a r t s ( F I a n d F2).
some s h i f t i n g i n h o m o g e n e i t i e s , w h i c h m a k e i t d i f f i c u l t t o p r e d i c t
t h e propagation c o n d i t i o n s f o r e l e c t r o m a g n e t i c waves.
The F l a y e r
h a s a n i n f l u e n c e on t h e p r o p a g a t i o n o f s h o r t w a v e s .
I t s h o u l d b e m e n t i o n e d t h a t t h e medium a n d s h o r t w a v e s a r e
r e f l e c t e d b o t h from t h e i o n i z e d l a y e r s o f t h e a t m o s p h e r e as w e l l
a s f r o m t h e E a r t h ' s s u r f a c e , s o t h a t t h e y may u n d e r g o m u l t i p l e r e f l e c ­
tion.
A l l of t h i s combines t o g i v e u s t h e complex p i c t u r e of t h e
p r o p a g a t i o n o f e l e c t r o m a g n e t i c w a v e s i n t h e E a r t h Is a t m o s p h e r e ,
which must be t a k e n i n t o a c c o u n t i n r a d i o n a v i g a t i o n a l measurements.

The p e c u l i a r i t i e s o f p r o p a g a t i o n o f e l e c t r o m a g n e t i c o s c i l l a t i o n s
i n a conducting f e e d e r channel i n r e c e i v e r s and t r a n s m i t t e r s include/247
the following.
Unlike c o n s t a n t and low-frequency a l t e r n a t i n g c u r r e n t s , high
frequency c u r r e n t s propagate mainly along t h e s u r f a c e of a conductor,
s i n c e t h e r e a c t i o n of t h e m a g n e t i c f i e l d w i t h i n t h e c o n d u c t o r i s
g r e a t e r t h a n on i t s s u r f a c e ( t h e s k i n e f f e c t ) .
This causes a l l
high frequency conductors t o b e c o n s t r u c t e d w i t h an eye toward i n c r e a s ­
i n g t h e s u r f a c e , e . g . , t u b u l a r and multiple-filament (stranded w i r e ) .
However, t h e s e m e a s u r e s a r e i n s u f f i c i e n t f o r waves i n t h e cen­
I t i s much b e t t e r t o u s e h o l l o w c o n d u c t o r s f o r t h e s e
timeter range.
w a v e s , c a l l e d wave g u i d e s ( F i g . 3 . 2 ) .

pma

I n a homogeneous medium, t h e
propagation r a t e of t h e f i e l d along
t h e v e c t o r of t h e wave p o l a r i z a t i o n ,
4%
i n t h e d i r e c t i o n of t h e perpendicular
vector of p o l a r i z a t i o n , i n t h e plane
o f t h e wave f r o n t , a n d i n t h e p r o p a Fig. 3.2.
Propagation of
E l e c t r o m a g n e t i c Waves
g a t i o n d i r e c t i o n o f t h e wave f r o n t
a r e t h e same.
T h e r e f o r e , i n a boxA l o n g a Wave G u i d e .
t y p e wave g u i d e , t h e d i s t a n c e b e t w e e n
t h e s u r f a c e s o f which i s measured i n whole numbers o f h a l f waves,
t h e v e r t i c a l l y p o l a r i z e d wave ( s t r i k i n g t h e t o p a n d b o t t o m i n t e r n a l
w a l l s of t h e box a n d b e i n g r e f l e c t e d f r o m t h e m ) w i l l s t r i k e t h e
o p p o s i t e w a l l a l s o w i t h a whole number o f h a l f w a v e s .
Consequently,
r e f l e c t i o n of t h e waves w i l l t a k e p l a c e i n r e s o n a n c e w i t h i t s o s c i l ­
lations.
I n t h i s c a s e , t h e b o x - t y p e wave g u i d e w i l l a c t as a c h a n n e l
a l o n g which t h e waves w i l l p r o p a g a t e t h e m s e l v e s p r a c t i c a l l y w i t h o u t
any r e s i s t a n c e .
If t h e d i s t a n c e b e t w e e n t h e w a l l s i s e q u a l t o a w h o l e , odd
number w h i c h i s o n e - q u a r t e r of t h e w a v e l e n g t h , t h e n ( a s i s e a s i l y

256

s e e n ) t h e r e f l e c t i o n s f r o m t h e w a l l s o f t h e wave g u i d e w i l l t a k e
In this
place each t i m e i n opposite phase with t h e o s c i l l a t i o n s .
c a s e t h e w a v e g u i d e w i l l h a v e i n f i n i t e r e s i s t a n c e , a n d t h e wave
energy w i l l not be propagated i n it.
In Figure 3.2, the vector
o f w a v e p o l a r i z a t i o n m u s t b e t u r n e d 90° t o a c c o m p l i s h t h i s .

P r i n c i p l e s o f S u p e r p o s i t i o n and

I n t e r f e r e n c e o f R a d i o Waves

The p r i n c i p l e of s u p e r p o s i t i o n i s a p p l i e d t o wave p r o c e s s e s ,
i . e . , e a c h o f t h e wave p r o c e s s e s a c t s i n d e p e n d e n t l y o f o t h e r p r o c e s s e s
w h i c h a r e t a k i n g p l a c e i n t h e medium or c i r c u i t s .
A t t h e same t i m e , t h e r e s u l t s of d i f f e r e n t p r o c e s s e s c a n b e
summed b y m e a n s o f a s i m p l e s u p e r p o s i t i o n o f o s c i l l a t i o n v e c t o r s .
If t h e v e c t o r s o f two c o h e r e n t ( c o i n c i d i n g i n f r e q u e n c y ) p r o c e s s e s
s u c h a s t h e o s c i l l a t i o n s i n t h e i n t e n s i t y o f a f i e l d or d i s p l a c e ment c u r r e n t s , a r e e q u a l i n a m p l i t u d e a n d c o i n c i d e i n p h a s e , t h e
Under t h e s e
t o t a l amplitude of t h e o s c i l l a t i o n s w i l l be doubled.
conditions, i f the oscillations are i n opposite phase, the t o t a l
a m p l i t u d e o f t h e o s c i l l a t i o n s w i l l b e e q u a l t o z e r o and no method
w i l l s u f f i c e t o d e t e c t t h e p r e s e n c e o f t h e wave p r o c e s s e s i n v o l v e d .

/248

Summing o f t h e r e s u l t s o f t h e p r o c e s s e s i n o p p o s i t e p h a s e i s
c a l l e d wave i n t e r f e r e n c e .
The c a s e i n w h i c h t h e r e s u l t o f summing
of t h e o s c i l l a t i o n s is e q u a l t o zero i s c a l l e d t o t a l interference.
The p r o p e r t i e s o f i n t e r f e r e n c e o f r a d i o w a v e s a r e w i d e l y em­
ployed and r a d i o n a v i g a t i o n a l d e v i c e s b o t h i n r e c e i v e r s and t r a n s ­
m i t t e r s , e s p e c i a l l y i n measuring t h e d i r e c t i o n of an o b j e c t .

P r i n c i p l e C h a r a c t e r i s t i cs o f

Rad i o n a v i g a t i o n a 1

The p r i n c i p l e c h a r a c t e r i s t i c s o f
instruments are the following:

(1) The r a d i a t e d p o w e r ,
of t h e system.

I n s t rumen t s

transmitting radionavigational

characterizing t h e operating range

(2)
A c c u r a c y a n d s t a b i l i t y of t h e f r e q u e n c y s t r u c t u r e , a s
w e l l as s y n c h r o n i z a t i o n o f s p e c i a l n a v i g a t i o n a l s i g n a l s .

A s f a r as t h e a n t e n n a a r r a y s a r e c o n c e r n e d , w h i c h i n c o r p o r a t e
c e r t a i n c h a r a c t e r i s t i c s for r a d i a t i o n o f s i g n a l s , w e w i l l d i s c u s s
them u n d e r t h e h e a d i n g o f " P r i n c i p l e s o f O p e r a t i o n o f C o n c r e t e Naviga­
t i o n a l Instruments".

Receiving navigational instruments are a l s o combinations of
t u n i n g c i r c u i t s a n d a m p l i f i e r s w h i c h u s e v a c u u m t u b e s or s e m i c o n ­
ductors.
In the majority of receivers, special generators (hetero­
d y n e s ) a r e u s e d , whose c o n s t r u c t i o n d i f f e r s f o r t h e p a r t i c u l a r d i f ­
ference i n frequency with respect t o t h e received s i g n a l .
In special
mixing t u b e s , t h e f r e q u e n c i e s of t h e t r a n s m i t t e r and heterodyne

257

s i g n a l s are combined and an i n t e r m e d i a t e f r e q u e n c y i s produced which
is e q u a l t o t h e d i f f e r e n c e b e t w e e n t h e f r e q u e n c i e s g i v e n a b o v e .
I n such devices, f u r t h e r amplification of t h e s i g n a l is c a r r i e d out
w i t h a c o n s t a n t , l o w e r f r e q u e n c y , w h i c h makes i t p o s s i b l e t o u s e
amplifier devices with very high c o e f f i c i e n t s of amplification,
as w e l l a s t o e n s u r e a h i g h s e l e c t i v i t y o f t h e r e c e i v e r .
U s u a l l y , r e c e i v i n g r a d i o n a v i g a t i o n a l i n s t r u m e n t s f u l f i l l two
functions:
(a)
r e c e p t i o n and a m p l i f i c a t i o n o f t h e s i g n a l s from
a t r a n s m i t t e r ( b ) s e p a r a t i o n and i n d i c a t i o n o f measured n a v i g a t i o n a l
parameters.
The b a s i c c h a r a c t e r i s t i c s o f r e c e i v e r n a v i g a t i o n a l i n s t r u m e n t s
are t h e following:
(1) S e n s i t i v i t y o f t h e r e c e i v e r w h i c h c h a r a c t e r i z e s t h e p o s ­
s i b l e r e c e i v i n g r a n g e f o r s i g n a l s from a t r a n s m i t t e r .
(2)
S e l e c t i v i t y of t h e r e c e p t i o n ; t h i s p a r a m e t e r i s u s u a l l y
o b t a i n e d by n a r r o w i n g t h e f r e q u e n c y b a n d w h i c h t h e r e c e i v e r w i l l
p a s s , which u s u a l l y c h a r a c t e r i z e s t h e freedom from n o i s e of t h e
receiver.
(3)
The a c c u r a c y w i t h w h i c h t h e n a v i g a t i o n a l p a r a m e t e r s
s e l e c t e d and r e c o r d e d .

Operating

P r i n c i p l e s of

Radionavigational

are

Instruments

I n a c c o r d a n c e w i t h t h e l a w s of p r o p a g a t i o n o f e l e c t r o m a g n e t i c
waves i n s p a c e , i t i s p o s s i b l e i n p r i n c i p l e t o measure t h e f o l l o w ­
i n g parameters o f e l e c t r o m a g n e t i c waves:
amplitude, phase , fre­
q u e n c y , a n d t r a n s m i s s i o n t i m e of t h e s i g n a l .
A c c o r d i n g t o t h e p r i n c i p l e of t e c h n i c a l o p e r a t i o n , r a d i o n a v ­
i g a t i o n a l d e v i c e s a r e d i v i d e d i n t o a m p l i t u d e , p h a s e , frequency and
t i m e devices.
I n a d d i t i o n , w i t h a mutual exchange of r a d i o s i g n a l s between
o b j e c t s which have r e l a t i v e motion t o one a n o t h e r , changes i n t h e
f r e q u e n c y c h a r a c t e r i s t i c s o f t h e s i g n a l s o c c u r w h i c h a r e known a s
t h e Doppler e f f e c t , which i s used t o b u i l d a u t o m a t i c a i r s p e e d i n d i ­
c a t o r s and d e v i c e s f o r measuring t h e d r i f t a n g l e of an a i r c r a f t .
Measurements of t h e parameters l i s t e d above f o r electromag­
n e t i c waves f r o m t h e n a v i g a t i o n a l s t a n d p o i n t m a k e i t p o s s i b l e t o
determine t h e following navigational elements:

(a)
tems;

The d i r e c t i o n of

(b)
tems ;

The d i s t a n c e t o a n o b j e c t , b y m e a n s o f r a n g e f i n d i n g s y s ­

258

t h e o b j e c t , by means o f g o n i o m e t r i c s y s ­

/249

(c)
The d i f f e r e n c e o r sum o f t h e d i s t a n c e s t o t h e o b j e c t :
h y p e r b o l i c or e l i p t i c a l s y s t e m s ;
(d)
S p e e d a n d d i r e c t i o n o f movement o f t h e a i r c r a f t : a u t o ­
m a t i c Doppler meters f o r ground speed and d r i f t a n g l e .

For c o n v e n i e n c e o f a p p l i c a t i o n , i n many c a s e s t h e n a v i g a t i o n a l
s y s t e m s a r e c o m p e n s a t e d for m e a s u r i n g t w o n a v i g a t i o n a l p a r a m e t e r s
simultaneously.
For e x a m p l e , t h e r e a r e t h e g o n i o m e t r i c - r a n g e f i n d i n g
systems, difference-rangefinding instruments, etc.
T h e p a n o r a m i c r a d a r l o c a t e d o n t h e g r o u n d a n d on t h e a i r c r a f t
are goniometric-range f i n d i n g devices with a s i n g l e u n i t of navi­
g a t i o n a l equipment.
I n s t u d y i n g methods o f a p p l y i n g r a d i o n a v i g a t i o n a l s y s t e m s ,
i t i s a good i d e a t o c l a s s i f y them a c c o r d i n g t o t h e p r i n c i p l e s by
Therefore, the
which t h e n a v i g a t i o n a l p a r a m e t e r s a r e measured.
f u r t h e r s u b d i v i s i o n o f t h e m a t e r i a l w i l l b e made on t h e b a s i s o f
these principles.
Radionavigational devices can a l s o be subdivided i n t o auto­
matic and non-automatic.
N o n - a u t o m a t i c d e v i c e s , when t h e y c o n s i s t
of systems of ground c o n t r o l and apparatus aboard t h e a i r c r a f t ,
a r e c a l l e d navigational systems.
A u t o m a t i c d e v i c e s a r e c a l l e d auto­
m a t i c n a v i g a t i o n a z s y s t e m s when t h e o p e r a t i o n o f s e v e r a l t y p e s o f
n a v i g a t i o n a l d e v i c e s i s combined o r g a n i c a l l y on b o a r d t h e a i r c r a f t .
For example, t h e automatic Doppler system f o r aircraft navigation,
/250
which c o n s i s t s o f a Doppler meter f o r t h e d r i f t a n g l e and t h e
ground speed, course d e v i c e s , and t h e automatic n a v i g a t i o n a l i n s t r u ­
ments.

2.

GONIOMETRIC A N D GONIOMETRIC-RANGEFINDING SYSTEMS


The g o n i o m e t r i c r a d i o n a v i g a t i o n a l s y s t e m s a r e t h e s i m p l e s t
o n e s f r o m t h e s t a n d p o i n t of t e c h n i c a l r e q u i r e m e n t s , a n d a r e t h e r e ­
f o r e t h o s e which a r e most w i d e l y employed a t t h e p r e s e n t t i m e .
I n t h e m a j o r i t y of t h e s e s y s t e m s ,
t h e a m p l i t u d e method o f measurement i s
e m p l o y e d , b a s e d on t h e i n t e r f e r e n c e o f
e l e c t r o m a g n e t i c waves.
This principle
s e r v e s as t h e b a s i s o f t h e o p e r a t i o n o f
ground and aircraft-mounted r a d i o d i r e c ­
t i o n f i n d e r s , which a r e a l s o c a l l e d r a d i o
compasses.
Fig. 3.3.
Reception
of Electromagnetic
Waves b y a F r a m e
Antenna.

L e t us imagine a frame-type r e c e i v i n g
antenna, l o c a t e d i n a f i e l d of outwardly
d i r e c t e d r a d i o waves ( F i g . 3 . 3 ) .
If t h e
frame a n t e n n a i s l o c a t e d r e l a t i v e t o t h e
t r a n s m i t t e r s o t h a t t h e directiqn,--;of t h e
,

+',-"'
.,%
."
.'. ­
2.

2::

259

I


p r o p a g a t i n g waves w i l l b e p e r p e n d i c u l a r t o t h e p l a n e o f t h e frame,
t h e l e f t a n d r i g h t v e r t i c a l s i d e s o f t h e f r a m e w i l l b e o n t h e same
isophasal circle.
I n t h i s case, t h e high-fi-equency c u r r e n t s which
a r e c o n d u c t e d i n t h e s i d e s of t h e frame w i l l a g r e e i n p h a s e a n d
w i l l c o n s e q u e n t l y b e d i r e c t e d t o w a r d one a n o t h e r .
This gives complete
i n t e r f e r e n c e of t h e o s c i l l a t i o n s of t h e c u r r e n t s i n t h e frame, and
t h e r e w i l l no r e c e p t i o n of s i g n a l s from t h e t r a n s m i t t i n g s t a t i o n .
If t h e frame i s t u r n e d a r o u n d t h e v e r t i c a l a x i s t h r o u g h 90°,
s o t h a t t h e d i r e c t i o n o f t h e p l a n e o f t h e frame c o i n c i d e s w i t h t h e
d i r e c t i o n o f t h e t r a n s m i t t i n g s t a t i o n , t h e s i d e s o f t h e frame w i l l
b e a t d i f f e r e n t i s o p h a s a l c i r c l e s , m a x i m a l l y d i s t a n t for t h e g i v e n
device.
Thus, t h e c u r r e n t s i n t h e v e r t i c a l s i d e s of t h e frame w i l l
u n d e r g o a c e r t a i n p h a s e s h i f t w h i c h w i l l g i v e maximum r e c e p t i o n
of t h e s i g n a l s from t h e s t a t i o n .
T h e maximum e f f e c t o f t h e f r a m e
w i l l b e o b s e r v e d i n t h e c a s e when t h e d i s t a n c e b e t w e e n i t s s i d e s
i s equal t o h a l f t h e wavelength.
Then t h e c u r r e n t s i n t h e v e r t ­
i c a l s i d e s o f t h e frame w i l l b e i n o p p o s i t e p h a s e a n d t h e i r a m p l i ­
However, t h i s r e q u i r e m e n t ( i n t h e m a j o r i t y
tudes w i l l be added.
o f c a s e s ) c a n n o t b e f u l f i l l e d , s i n c e t h e a n t e n n a d e v i c e becomes
t o o u n w i e l d y ; t h e r e f o r e , w e u s e t h a t p a r t of t h e e f f e c t which i s
obtained with a phase shi-ft through a s m a l l (frequently very small) /251
angle.
I n t h e s e c a s e s , t h e r e c e i v i n g f r a m e i s s u p p l i e d w i t h many
t u r n s and a r a d i o r e c e i v e r w i t h v e r y h i g h s e n s i t i v i t y i s employed.
The v e c t o r d i a g r a m o f t h e r e c e p t i o n d i r e c t i o n a l i t y o f t h e frame
w i l l h a v e t h e f o r m of a f i g u r e e i g h t ( F i g . 3 . 4 ) .
The g r e a t e s t a c c u r a c y i n r a n g e f i n d i n g i s o b t a i n e d w i t h min­
imum r e c e p t i o n , w h i l e a t t h e maximum t h e c h a n g e i n a m p l i t u d e o f
t h e r e c e i v e d s i g n a l i s o b t a i n e d b y t u r n i n g t h e frame a t a s l i g h t
angle.
T h e r e f o r e , r a n g e f i n d i n g b y means o f a frame i s a l w a y s d o n e
w i t h minimum r e c e p t i o n or a u d i b i l i t y o f t h e s i g n a l .

Fig.

3.4.

Fig.

3.5.

Fig.

3.4.

Diagram o f R e c e p t i o n o f a Frame Antenna.

Fig.

3.5.

Edcock-Type

260

Antenna.

The r e c e i v i n g frame a n t e n n a h a s t h e s h o r t c o m i n g t h a t when e l e c ­
t r o m a g n e t i c waves a r e b e i n g p r o p a g a t e d t h r o u g h s p a c e , i t p i c k s up
n o t o n l y t h e v e r t i c a l l y p o l a r i z e d w a v e b u t a l s o t h e h o r i z o n t a l compo­
n e n t of t h e p o l a r i z a t i o n v e c t o r i n t h e t o p a n d b o t t o m s i d e s of t h e
frame.
I n a d d i t i o n , t h e frame a n t e n n a w i t h i t s l a r g e d i m e n s i o n s
and mechanical r o t a t i o n i s inconvenient t o use.
T h e r e f o r e , groundb a s e d r a d i o r a n g e f i n d i n g i n s t a l l a t i o n s u s e s p e c i a l a n t e n n a s which
a r e e q u i v a l e n t t o a frame t y p e i n t h e c h a r a c t e r i s t i c s o f r e c e p t i o n
f o r v e r t i c a l l y p o l a r i z e d waves, b u t are f r e e of t h e shortcomings
m e n t i o n e d a b o v e ; t h e y a r e c a l l e d Edcock a n t e n n a s ( F i g . 3 . 5 ) .
The p i c t u r e s h o w s o n e p a i r o f E d c o c k d i p o l e s w i t h t h e c o i l
O b v i o u s l y , i n open d i p o l e s , no i n t e r ­
of a g o n i o m e t e r b e t w e e n t h e m .
f e r e n c e w i l l b e o b s e r v e d when t h e y a r e l o c a t e d on o n e i s o p h a s a l
circle.
However, t h e d i f f e r e n c e i n p o t e n t i a l a t t h e e n d s o f t h e
goniometric c o i l w i l l b e equal t o zero, since they are connected
t o s y m m e t r i c a l p o i n t s on t h e d i p o l e .
If t h e d i p o l e s a r e l o c a t e d
on d i f f e r e n t i s o p h a s a l c i r c l e s , t h e n t h e p h a s e s h i f t w i l l d i s t u r b
t h e p o t e n t i a l e q u i l i b r i u m a t t h e ends of t h e c o i l and a high-fre­
quency c u r r e n t w i l l p a s s through i t .
A s i m i l a r p a i r o f d i p o l e s i s mounted i n t h e p l a n e p e r p e n d i c ­
ular t o the first pair.

The h i g h - f r e q u e n c y c u r r e n t i n t h e g o n i o m e t e r c o i l s w i l l d e p e n d / 2 5 2
on t h e d i r e c t i o n o f t h e t r a n s m i t t i n g s t a t i o n r e l a t i v e t o t h e c r o s s e d
dipoles.
I n a g o n i o m e t r i c i n s t r u m e n t , i n a d d i t i o n t o t h e two f i x e d d i p o l e
c o i l s mounted a t a n a n g l e of 90°, t h e r e i s a movable s e a r c h i n g c o i l ,
c o n n e c t e d t o t h e i n p u t c i r c u i t of t h e r e c e i v e r .
If t h e s e a r c h i n g c o i l i s p l a c e d i n t h e r e s u l t a n t f i e l d o f t h e
f i x e d c o i l s , t h e r e c e p t i o n w i l l b e maximum; w h e n a c o i l i s p l a c e d
a t a n a n g l e o f 9 0 ° t o t h e r e s u l t a n t f i e l d , r e c e p t i o n w i l l b e min­
imal.

Fig. 3.6.
Inclusion of an
Open A n t e n n a f o r S o l v i n g
Ambiguity of Reception.

The h o r i z o n t a l w i r e s c o n n e c t i n g
t h e a n t e n n a d i p o l e s a r e l o c a t e d as
c l o s e as p o s s i b l e t o one a n o t h e r ,
s o t h a t t h e e l e c t r o m o t i v e f o r c e con­
d u c t e d i n them f r o m t h e h o r i z o n t a l
component v e c t o r o f p o l a r i z a t i o n
w i l l be i n t h e s a m e phase, and t h e i r
t o t a l interference w i l l appear a t
t h e inputs i n t h e goniometer c o i l s .
Therefore, t h e antenna does not pick
up component waves w i t h h o r i z o n t a l
polarization , thus considerably reducing
t h e r a n g e f i n d i n g e r r o r f o r waves
i n space.

261

I


The r e c e p t i o n c h a r a c t e r i s t i c s o f t h e frame a n t e n n a ( i n c l u d ­
i n g t h e E d c o c k t y p e ) h a v e t w o s i g n s , i . e . , w e h a v e t w o maxima a n d
two minima of a u d i b i l i t y , s o t h a t i t c a n b e u s e d t o d e t e r m i n e t h e
d i r e c t i o n l i n e on which t h e t r a n s m i t t i n g a n d r e c e i v i n g o b j e c t s a r e
l o c a t e d , b u t does n o t s o l v e t h e problem of t h e s i d e s of t h e mutual
p o s i t i o n of t h e o b j e c t s (see Fig. 3.4).
To s o l v e t h e a m b i g u i t y o f r e c e p t i o n w i t h r a d i o r a n g e f i n d i n g
i n s t r u m e n t s , an open antenna with an e x t e r n a l l y d i r e c t e d ( c i r c u l a r )
r e c e p t i o n c h a r a c t e r i s t i c i s u s e d i n a d d i t i o n t o t h e frame a n t e n n a
(Fig. 3.6).
The p h a s e o f t h e h i g h - f r e q u e n c y c u r r e n t i n t h e o p e n a n t e n n a ,
d e p e n d i n g on t h e r e c e p t i o n d i r e c t i o n , w i l l c o i n c i d e w i t h t h e p h a s e
o f o n e of t h e s i d e s of t h e frame r e c e i v e r a n d w i l l b e i n o p p o s i t e
As
phase with t h e c u r r e n t s i n t h e second s i d e of t h e receiver.
a r e s u l t , t h e c u r r e n t amplitudes of an open antenna w i l l be added
t o o n e - h a l f o f t h e f i g u r e e i g h t of t h e frame a n t e n n a a n d w i l l i n t e r ­
f e r e with t h e o t h e r h a l f of t h e f i g u r e e i g h t (Fig. 3.7, a ) .
I n combining t h e c h a r a c t e r i s t i c s o f t h e frame and open a n t e n n a s ,
w e o b t a i n a t o t a l c h a r a c t e r i s t i c which h a s t h e form o f a c a r d i o i d .
If w e c o n n e c t t h e open a n t e n n a a n d t u r n t h e f r a m e a n t e n n a t h r o u g h
9 0 ° c l o c k w i s e , t h e maximum r e c e p t i o n s h o w n i n F i g u r e 3 . 7 , b w i l l
s h i f t t o t h e u p p e r p a r t of t h e p i c t u r e w h i l e t h e minimum w i l l s t i i f t
t o t h e lower p a r t (Fig. 3.7, c ) .
T h i s c o r r e s p o n d s t o one s i d e o f
t h e minimum o f t h e f r a m e r e c e i v e r b e i n g t r a n s f e r r e d t o t h e maximum, / 2 5 3
a n d t h e s e c o n d r e m a i n i n g minimum.

Fig.

3 . 7 . D i a g r a m o f D i r e c t i o n a l i t y o f a Frame A n t e n n a
C o m b i n e d w i t h a n Open A n t e n n a .

L e t u s s u p p o s e t h a t w e h a v e d e f i n e d a l i n e ( b e a r i n g ) on which
t h e t r a n s m i t t i n g a n d r e c e i v i n g p o i n t s a r e l o c a t e d a t minimum a u d ­
ibility.
A f t e r c o n n e c t i n g t h e open a n t e n n a and t u r n i n g t h e g o n i ­
ometer c o i l through 90°, w e can determine t h e d i r e c t i o n of t h e t r a n s ­
mitter.
If t h e a u d i b i l i t y o f t h e s i g n a l s i n c r e a s e s s h a r p l y , t h e
If i t
t r a n s m i t t e r is located i n t h e d i r e c t i o n of t h e upper p a r t .
r e m a i n s a s b e f o r e or c h a n g e s s l i g h t l y , t h e t r a n s m i t t e r i s l o c a t e d
at the opposite side.

26 2

The p r i n c i p l e s d e s c r i b e d a b o v e f o r f i n d i n g t h e d i r e c t i o n o f a
t r a n s m i t t e r are used i n ground r a d i o d i r e c t i o n - f i n d i n g i n s t a l l a t i o n s .
I n t h i s c a s e , t h e t r a n s m i t t e r i s t h e r a d i o on b o a r d t h e a i r c r a f t .
Ground r a d i o d i r e c t i o n - f i n d e r s i n p r i n c i p l e c a n o p e r a t e a t
a l l wavelengths.
The most w i d e l y u s e d r a d i o r a n g e f i n d e r s o p e r a t e
on s h o r t a n d u l t r a - s h o r t w a v e s .
T h e p o s i t i o n o f t h e a i r c r a f t c a n b e d e t e r m i n e d b y m e a n s of
t h e g r o u n d r a d i o r a n g e f i n d e r i n t e r m s o f t h e minimum a u d i b i l i t y
o f t h e s i g n a l f r o m t h e t r a n s m i t t e r l o c a t e d on b o a r d .
In addition,
v i s u a l ’ i n d i c a t o r s a r e m o u n t e d o n t h e u l t r a s h o r t w a v e (USW) r a n g e f i n d e r s , s u c h as c a t h o d e - r a y t u b e s .
I n t h i s c a s e , t h e f r a m e o f t h e d i r e c t i o n - f i n d e r or t h e g o n ­
i o m e t e r c o i l i s s e t t o r o t a t i n g r a p i d l y , and t h e s c a n o f t h e cathoderay tube is synchronized with it.
The a m p l i t u d e o f t h e s c a n i s
r e l a t e d i n magnitude t o t h e amplitude of t h e r e c e i v e d s i g n a l s i n
s u c h a way t h a t a t minimum r e c e p t i o n t h e maximum a m p l i t u d e o f t h e
scan is observed.
Then, on a s c a l e which i s marked a l o n g t h e p e r i p h e r y
o f t h e t u b e f a c e , we c a n d e t e r m i n e t h e d i r e c t i o n o f t h e a i r c r a f t
i n t e r m s o f t h e p o s i t i o n o f t h e maximum d e f l e c t i o n o f t h e s c a n .
With a r e l a t i v e l y low d e n s i t y o f a i r m o t i o n , t h e g r o u n d r a d i o
r a n g e f i n d e r s a r e a s u f f i c i e n t l y e f f e c t i v e and p r e c i s e method o f
aircraft navigation.
An a d v a n t a g e o f g r o u n d r a d i o r a n g e f i n d e r s
i s t h e l a c k o f a n e e d t o m o u n t s p e c i a l r a d i o e q u i p m e n t on t h e a i r ­
craft.
The r a d i o r a n g e f i n d e r s a n d r e c e i v e r s w h i c h a r e u s e d f o r
/254
r e c e i v i n g s i g n a l s from ground d i r e c t i o n - f i n d i n g p o i n t s mainly have
o t h e r purposes, and t h e i r use f o r n a v i g a t i o n a l purposes i s n o t r e l a t e d
t o t h e i n c r e a s e d c o m p l e x i t y a n d w e i g h t o f t h e e q u i p m e n t on b o a r d .
A t t h e same t i m e , h o w e v e r , t h e u s e o f g r o u n d r a d i o d i r e c t i o n f i n d e r s h a s a number o f s e r i o u s s h o r t c o m i n g s , which h a v e l e d t o
a s e a r c h t o f i n d new w a y s o f r a d i o n a v i g a t i o n a l c o n t r o l o f f l i g h t .

The most i m p o r t a n t o f t h e s e s h o r t c o m i n g s a r e :
L a c k o f a v i s u a l i n d i c a t o r on b o a r d t h e a i r c r a f t t o show
(a)
i t s p o s i t i o n , thus reducing t h e ease of a i r c r a f t navigation.

(b)
A small capacity f o r t h e ground i n s t a l l a t i o n s ; a t t h e
same t i m e , t h e r a d i o d i r e c t i o n - f i n d e r c a n o n l y o p e r a t e w i t h o n e
a i r c r a f t , w h i c h i s c l e a r l y i n a d e q u a t e w h e n t h e r e a r e a g r e a t many
flights.
A i r c r a f t N a v i g a t i o n U s i n g Ground-Based

Radio D i r e c t i o n - F i n d e r s

The u s e o f g r o u n d - b a s e d r a d i o d i r e c t i o n - f i n d e r s
t o solve t h e following n a v i g a t i o n a l problems:
(a>

can be used

S e l e c t i o n of t h e course t o be followed and f l i g h t along

26 3

t h e s r r a i g h t - l i n e s e g m e n t s of
which r a d i o d i r e c t i o n - f i n d e r s
(b)

C o n t r o l of

a r o u t e , a t t h e b e g i n n i n g or e n d o f
are l o c a t e d .

t h e a i r c r a f t p a t h i n terms o f d i s t a n c e .

(c)
D e t e r m i n a t i o n of t h e a i r c r a f t l o c a t i o n on t h e b a s i s o f
b e a r i n g s o b t a i n e d from two ground-based r a d i o d i r e c t i o n - f i n d e r s .
(d)
D e t e r m i n a t i o n of t h e g r o u n d s p e e d o f t h e a i r c r a f t , as
w e l l as t h e d r i f t a n g l e , d i r e c t i o n a n d s p e e d o f t h e wind a t f l i g h t
altitude.
Usually, t h e i n t e r n a t i o n a l ltShchll-code i s used f o
The crew
i n g t h e b e a r i n g s f r o m on b o a r d t h e a i r c r a f t .
c r a f t r e p o r t s i t s p o s i t i o n , g i v e s t h e r e q u i r e d code f o
t i o n , and p r e s s e s t h e t e l e g r a p h key o f t h e t r a n s m i t t e r
of 20 sec.

r determin­
of t h e air­
r i t s posi­
for a p e r i o d

I n r e c e n t y e a r s , both s t a t e and l o c a l c i v i l a i r l i n e s have adopted
USW d i r e c t i o n f i n d e r s , w i t h v i s u a l i n d i c a t o r s .
They a r e o r i e n t e d
a c c o r d i n g t o t h e m a g n e t i c m e r i d i a n o f t h e l o c a t i o n o f t h e USW d i r e c ­
t i o n - f i n d e r , a n d ( d e p e n d i n g on t h e f l i g h t a l t i t u d e ) a r e u s e d i n
a r a d i u s o f 1 0 0 - 2 0 0 km a s a f o r m o f t r a c e d i r e c t i o n - f i n d e r , r e p o r t ­
i n g on b o a r d t h e a i r c r a f t t h e " f o r w a r d " ( a w a y ) a n d " b a c k " ( r e t u r n )
m a g n e t i c b e a r i n g s of t h e a i r c r a f t .
I f t h e crew o f t h e a i r c r a f t
r e q u e s t s t h e f o r w a r d t r u e b e a r i n g , t h e o p e r a t o r o f t h e USW d i r e c ­
tion-finder (supervisor) c a l c u l a t e s t h e magnetic declination of
t h e l o c a t i o n of t h e d i s t a n c e f i n d e r and r e p o r t s t h e forward t r u e
bearing t o the aircraft.
D i s t a n c e f i n d i n g b y m e a n s o f USW d i s t a n c e f i n d e r s w i t h a v i s i b l e i n d i c a t o r i s used i n t h e c o u r s e of communication w i t h an a i r ­
c r a f t , i . e . , w i t h a d e p r e s s e d t a n g e n t o f t h e c o n n e c t e d USW t r a n s ­
m i t t e r on b o a r d t h e a i r c r a f t .
The o p e r a t o r o f t h e g r o u n d r a d i o d i r e c t i o n - f i n d e r , a f t e r t h e
required measurements, gives t h e c a l l l e t t e r s of t h e a i r c r a f t , t h e
c o d e e x p r e s s i o n f o r t h e b e a r i n g a s r e q u e s t e d b y t h e a i r c r a f t or
u s e d f o r USW c o m m u n i c a t i o n , a n d g i v e s t h e m a g n i t u d e o f t h e b e a r i n g
i n degrees.
The c o d e e x p r e s s i o n s f o r t h e b e a r i n g s i n t h e i n t e r n a t i o n a l
Shch code h a v e t h e f o l l o w i n g meanings ( F i g . 3 . 8 ) :
ShchDR:

magnetic b e a r i n g from d i s t a n c e - f i n d e r

t o the aircraft,

or f o r w a r d b e a r i n g .
ShchDM:
magnetic b e a r i n g from t h e a i r c r a f t t o t h e d i s t a n c e f i n d e r (measured r e l a t i v e t o t h e l o c a l meridian of t h e l o c a t i o n
o f t h e d i s t a n c e - f i n d e r ) , or r e v e r s e b e a r i n g .
ShchTE:

t r u e b e a r i n g from t h e d i s t a n c e - f i n d e r t o t h e a i r c r a f t ,

or t h e f o r w a r d t r u e b e a r i n g .
264

/255

ShchGE:
a z i m u t h o f t h e a i r c r a f t a t a d i s t a n c e from t h e con­
t r o l distance-finding station.
ShchTF:
location of t h e
a i r c r a f t ( c o o r d i n a t e s or l i n k ) .
Due t o t h e s m a l l effective
r a d i u s o f t h e USW r a n g e f i n d e r s ,
they are n o t grouped i n c o d i s ­
tance-finding n e t s l i k e long
O D medium-wave s t a t i o n s , b u t
o p e r a t e i n d e p e n d e n t l y , and do
not give t h e location of t h e
aircraft.

Fig. 3.8.
Code E x p r e s s i o n s f o r
Bearings i n t h e Shch-code.

S e Z e c t i o n o f t h e Course t o
b e FoZZowed and ControZ o f F Z i g h t
D i r e e t i on

The s e l e c t i o n o f t h e c o u r s e
t o b e f o l l o w e d and f l i g h t a l o n g a s t r a i g h t - l i n e p a t h segment are
a c c o m p l i s h e d by means o f p e r i o d i c i n q u i r i e s a n d d e t e r m i n a t i o n s o f
t h e f o r w a r d or r e v e r s e b e a r i n g s o f t h e a i r c r a f t (ShchDR or S h c h D M ) .
If t h e r a d i o d i s t a n c e - f i n d e r i s l o c a t e d a t t h e s t a r t i n g p o i n t
of a f l i g h t segment ( f l i g h t from t h e d i s t a n c e - f i n d e r ) , t h e n t h e
When t h e a i r c r a f t i s p a s s i n g p r e ­
ShchDR b e a r i n g s a r e r e q u e s t e d .
c i s e l y over t h e ground r a d i o d i s t a n c e d e t e c t o r and follows a c o n s t a n t
c o u r s e f o r a c e r t a i n p e r i o d of t i m e , t h e f i r s t b e a r i n g o f t h e a i r ­
c r a f t a f t e r passing over t h e distance-finder can be used t o deter­
U s u a l l y i n t h i s c a s e t h e ShchDR
mine t h e d r i f t a n g l e ( F i g . 3 . 9 ) .
so that
b e a r i n g w i l l b e e q u a l t o MFA,,
US = ShchDR - M C .

I f t h e ShchDR d o e s n o t c o r r e s p o n d t o t h e g i v e n f l i g h t p a t h
/256
a n g l e f o r t h e p a t h s e g m e n t , t h e n t h e a i r c r a f t i s p u t on t h e d e s i r e d
l i n e of f l i g h t a f t e r determining t h e d r i f t angle and t h e course
t o be followed i s s e t s o t h a t t h e t o t a l of t h e course and t h e d r i f t
angle of t h e aircraft w i l l equal t h e given path angle.

I t s h o u l d b e k e p t i n m i n d t h a t i n t h e g e n e r a l c a s e ShchDR i s
n o t e q u a l t o MFA , s i n c e t h e f o r m e r i s t h e o r t h o d r o m i c b e a r i n g meas­
u r e d a t t h e s t a r f i n g p o i n t o f t h e s e g m e n t a n d t h e MFA i s t h e l o x o d r o m i c
p a t h a n g l e m e a s u r e d r e l a t i v e t o t h e mean m a g n e t i c m e r i d i a n :

ShchDR

-

MFA = A

av

­
- A ~ l 6avy

where AMav i s t h e m a g n e t i c d e c l i n a t i o n a t t h e m i d p o i n t o f t h e s e g ­
is t h e magnetic declination a t t h e location of t h e r a d i o
ment,
d i s t a n c e - f i n d e r , and
i s t h e d e v i a t i o n of t h e meridians between

265

t h e i n i t i a l a n d m i d d l e p o i n t s on t h e p a t h s e g m e n t .

Fig.

3.9.

Fig.

3.10.

Fig.

3 . 9 . D e t e r m i n a t i o n of t h e D r i f t A n g l e A f t e r F l y i n g O v er a
Radio Distance-Finding S t a t i o n .

Fig.

3.10.

P a t h S e g m e n t B e t w e e n Two R a d i o D i s t a n c e - F i n d i n g

Stations.

I n p r i n c i p l e , orthodromic c o n t r o l of t h e path f o r a loxodromic
f l i g h t is i n c o n s i s t e n t , because i n p r a c t i c e t h e course t o be f o l ­
lowed i n a l o x o d r o m i c s y s t e m of p a t h a n g l e s i s s e l e c t e d s o t h a t
f l i g h t t a k e s place along t h e orthodrome.
In order t o maintain t h e given f l i g h t d i r e c t i o n over t h e path
segment with s u f f i c i e n t accuracy, i t i s necessary t o n o t e t h a t a t
e a c h b e a r i n g (ShchDR or ShchDM) t h e a i r c r a f t w i l l b e l o c a t e d o n
an orthodromic l i n e of t h e given p a t h and w i l l t h e r e f o r e maintain
t h i s bearing.
L e t u s e x p l a i n t h i s by a c o n c r e t e e x a m p l e .

We w i l l a s s u m e t h a t we m u s t make a f l i g h t f r o m a p o i n t A(A =
1 0 5 O , A M = -1") t o a p o i n t B(A = 1 1 5 O , A M = - 7 O ) a n d r e t u r n ( F i g .
3.10).
T h e m a g n e t i c f l i g h t a n g l e o f t h e s e g m e n t i s 9 5 or 2 7 5 O ,
w h i l e t h e a v e r a g e l a t i t u d e o f t h e s e g m e n t i s 52O.
Obviously, f o r f l i g h t i n an e a s t e r l y d i r e c t i o n from t h e d i s ­
tance-finder, located a t point A
ShchDR = M F A + A M

-AM1-6av

= 95-4+1-4

= 88O.

av
F o r f l i g h t i n a w e s t e r l y d i r e c t i o n , t h e ShchDM f r o m t h i s d i s ­
t a n c e - f i n d e r m u s t b e e q u a l t o 268O.
For a f l i g h t i n an e a s t e r l y d i r e c t i o n , t h e i n i t i a l course of
t h e a i r c r a f t m u s t b e s e t n o t o n t h e b a s i s o f MFA = 9 5 O , b u t f r o m
I n t h e o p p o s i t e case, t h e a i r c r a f t slowly begins
ShchDR = 8 8 O .
t o d e v i a t e f r o m t h e l i n e o f t h e d e s i r e d b e a r i n g a t a n a n g l e o f 7O.

266

/257

A n a l o g o u s l y , f o r t h e p o i n t B ( S h c h D R = 2 8 2 O , ShchDM = 1 0 2 O ) ,
t h e i n i t i a l c o u r s e must b e s e t 7 O g r e a t e r t h a n one would c o n c l u d e
on t h e b a s i s o f t h e MEA.
O f c o u r s e , i t i s i m p o s s i b l e t o make f l i g h t s w i t h a c o n s t a n t
MFA a t d i s t a n c e s a t w h i c h t h e m a g n e t i c d i r e c t i o n o f t h e f l i g h t c h a n g e s
b y 14O.
T h i s example i s g i v e n o n l y t o i l l u s t r a t e t h e g e o m e t r y of
the process.
I t would b e more a c c u r a t e t o d i v i d e t h i s segment i n t o
f o u r p a r t s 1 5 0 km l o n g w i t h t h e f o l l o w i n g f l i g h t a n g l e s :
MFAl =
90°,
MFA2 = 9 3 O , MFA3 = 9 7 O , a n d MFA4 = l o o o .
I n t h e f i r s t two
s e g m e n t s , w e must u s e a d i s t a n c e f i n d e r which i s l o c a t e d a t P o i n t
A (ShchDR = 8 8 O ) , w h i l e f o r t h e l a t t e r w e m u s t u s e t h e d i s t a n c e
f i n d e r ' a t P o i n t B (ShchDM = 102O).

T h i s d i v i s i o n of t h e f l i g h t segment i n t o p a r t s f o r t h e case
of a f l i g h t a c c o r d i n g t o a g r o u n d d i s t a n c e f i n d e r i s an approxima­
t i o n o f t h e i n i t i a l MFA t o t h e ShchDR o f t h e i n i t i a l d i s t a n c e f i n d e r ,
w h i l e t h e l a t t e r i s a p p r o a c h i n g t h e ShchDM o f t h e r a n g e f i n d e r l o c a t e d
a t t h e terminus of t h e f l i g h t .
In t h e orthodromic system of c a l c u l a t i n g f l i g h t angles, t h e
d i s t a n c e b e t w e e n t h e O F A a n d b e a r i n g s ShchDR a n d ShchDM f r o m o n e
of t h e d i s t a n c e f i n d e r s w i l l b e c o n s t a n t i n v a l u e a n d w i l l depend
o n i y on t h e m e r i d i a n s e l e c t e d f o r c a l c u l a t i n g t h e p a t h a n g l e s .
In
t h e s p e c i a l c a s e when t h e r e f e r e n c e m e r i d i a n c o i n c i d e s w i t h t h e
m e r i d i a n where t h e r a n g e f i n d e r i s l o c a t e d , OFA w i l l d i f f e r from
ShchDR o n l y i n t h e m a g n i t u d e o f m a g n e t i c d e c l i n a t i o n f o r t h e l o c a ­
t i o n of t h e distance f i n d e r :
O F A = ShchDR

+

AM.

T h e r e f o r e , i n an o r t h o d r o m i c s y s t e m o f c a l c u l a t i n g f l i g h t a n g l e s ,
t h e c o u r s e t o b e f o l l o w e d by t h e a i r c r a f t c h a n g e s more r a r e l y a n d
t o a much l e s s e r d e g r e e t h a n i n a l o x o d r o m i c s y s t e m , b u t a l l e l e m e n t s
o f a i r c r a f t n a v i g a t i o n , i n c l u d i n g t h e s p e e d and wind d i r e c t i o n ,
a r e d e t e r m i n e d more a c c u r a t e l y .
For s e l e c t i n g a course and maintaining t h e f l i g h t d i r e c t i o n
o f a n a i r c r a f t i n t e r m s o f g r o u n d r a d i o d i s t a n c e - f i n d e r s , t h e method
of h a l f c o r r e c t i o n s i s u s e d , which c o n s i s t s of t h e f o l l o w i n g :
L e t u s s a y t h a t a t a p o i n t p o s i t i o n o f t h e a i r c r a f t on t h e
l i n e o f a g i v e n p a t h , t h e l a t t e r i s on c o u r s e w i t h a c e r t a i n a n t i ­
c i p a t i o n of d r i f t .

A f t e r a c e r t a i n p e r i o d o f t i m e , on t h e b a s i s o f t h e b e a r i n g
o b t a i n e d from t h e d i s t a n c e f i n d e r , it i s found t h a t t h e a i r c r a f t
i s s h i f t i n g from t h e l i n e of f l i g h t toward t h e d i r e c t i o n of t h e
wind v e c t o r .
T h i s i n d i c a t e s t h a t t h e c o r r e c t i o n i n t h e c o u r s e which
has been taken is i n s u f f i c i e n t .
Therefore, it is necessary t o r e t u r n
t h e a i r c r a f t a t a n a n g l e o f 10-15O t o t h e g i v e n l i n e o f f l i g h t ,
and t h e p r e v i o u s l y employed l e a d i n t h e c o u r s e t o b e f o l l o w e d i s
doubled.
267

If i n t h i s c a s e t h e a i r c r a f t b e g i n s t o s h i f t f r o m t h e l i n e
of f l i g h t toward t h e s i d e opposite t h e wind v e c t o r , t h e n a f t e r t h e
second aiming of t h e aircraft along t h e given l i n e of f l i g h t , it
i s n e c e s s a r y t o make a c o r r e c t i o n i n t h e c o u r s e w h i c h i s h a l f w a y
If t h e d e v i a t i o n t a k e s p l a c e
between t h e l a t t e r and t h e former.
a l o n g t h e d i r e c t i o n of t h e wind v e c t o r , t h e c o r r e c t i o n i n t h e c o u r s e
must be i n c r e a s e d .
I n a d d i t i o n , i f t h e d e v i a t i o n of t h e a i r c r a f t from t h e l i n e
of d e s i r e d f l i g h t t a k e s p l a c e , t h e d i f f e r e n c e between t h e l a t t e r
and t h e former c o r r e c t i o n s i s d i v i d e d and added t o t h e course with
a p o s i t i v e or n e g a t i v e s i g n , d e p e n d i n g o n t h e d i r e c t i o n o f t h e a i r ­
craft deviation.
T h e p l a c i n g o f t h e a i r c r a f t on t h e d e s i r e d l i n e o f f l i g h t b y
s e l e c t i n g t h e course with a l l t h e d e v i a t i o n s mentioned is oblig­
a t o r y o n l y i n a f l i g h t from t h e d i s t a n c e - f i n d e r a l o n g a f o r w a r d
b e a r i n g (ShchDR).
I n a f l i g h t toward a r a d i o distance-finder along
a r e v e r s e b e a r i n g (ShchDM), t h e a i r c r a f t m u s t f o l l o w t h e l i n e o f
t h e d e s i r e d p a t h o n l y i n t h e c a s e when i t i s g o i n g b e y o n d t h e l i m i t s
of t h e e s t a b l i s h e d t r a c e .
With s m a l l d e v i a t i o n s ( b y d i s t a n c e s f r o m
t h e r a d i o d i s t a n c e - f i n d e r o f u p t o 2 0 0 km w i t h i n t h e l i m i t s o f 1­
2 O ) , i t i s s u f f i c i e n t t o s e l e c t t h e c o u r s e t o b e f o l l o w e d by t h e
same m e t h o d o f h a l f c o r r e c t i o n s r e l a t i v e t o t h e l a s t ShchDM ( r e v e r s e
b e a r i n g ) , w i t h o u t g o i n g t o t h e d e s i r e d l i n e of f l i g h t e a c h t i m e .
The m e t h o d o f h a l f c o r r e c t i o n s i s t h e g e n e r a l o n e u s e d f o r
However,
f l i g h t t o w a r d t h e r a d i o d i s t a n c e - f i n d e r a n d away f r o m i t .
i n p r a c t i c a l use, t h e r e are c o n s i d e r a b l e d i f f e r e n c e s between f l i g h t
t o w a r d t h e d i s t a n c e - f i n d e r a n d away f r o m i t :
(1) I n a f l i g h t f r o m t h e r a d i o d i s t a n c e - f i n d e r , t h e d r i f t
angle can be measured a t t h e beginning of t h e segment, while i n
a f l i g h t toward t h e distance-finder it can be determined only a f t e r
s e l e c t i n g t h e c o u r s e t o b e f o l l o w e d w i t h a s t a b l e ShchDM.
(2)
I n a f l i g h t from a r a d i o d i s t a n c e - f i n d e r , t h e c o u r s e t o
b e f o l l o w e d by t h e a i r c r a f t must change i n t h e d i r e c t i o n o p p o s i t e
t h e change of t h e b e a r i n g :
ShchDR i n c r e a s e s , a n d t h e c o u r s e m u s t
a l s o decrease, and vice versa.
I n a f l i g h t toward a r a d i o distancef i n d e r , t h e change i n t h e c o u r s e must t a k e p l a c e i n t h e d i r e c t i o n
of t h e change i n b e a r i n g :
ShchDM i n c r e a s e s , t h e c o u r s e m u s t b e
i n c r e a s e d , and v i c e v e r s a .

(3)
A s we
distance-finder
bearing, while i
certain limits)
lowed a c c o r d i n g

268

have already pointed o u t , a
i n a l l c a s e s m u s t b e made s t
n t h e f l i g h t toward a r a d i o
it is permissible t o select
t o the last stable bearing.

f l i g h t away f r o m a r a d i o
r i c t l y along the given
distance-finder (within
t h e f l i g h t t o be f o l ­

/258

P a t h C o n t r o Z i n Terms

o f D i s t a n c e and D e t e r m i n a t i o n o f
t h e A i r c r a f t ' s Location
For t h e purposes o f c o n t r o l l i n g t h e p a t h i n terms of d i s t a n c e ,
as w e l l as d e t e r m i n i n g t h e l o c a t i o n o f t h e a i r c r a f t , w e c a n u s e
t h e t r u e b e a r i n g s from t h e ground r a d i o d i s t a n c e - f i n d e r t o t h e a i r ­
c r a f t ( SchTE ) .
F o r c h e c k i n g a f l i g h t i n t e r m s of d i s t a n c e , w e u s u a l l y s e l e c t
t h e c o n t r o l landmarks along t h e f l i g h t r o u t e and determine t h e i r
p r e c a l c u l a t e d b e a r i n g s from t h e r a d i o d i s t a n c e - f i n d e r l o c a t e d t o
t h e s i d e of t h e aircraft route (Fig. 3.11).
Three t o f i v e minutes before t h e aircraft reaches t h e c o n t r o l
l a n d m a r k , a s e r i e s of " f o r w a r d t r u e " b e a r i n g s a r e r e q u e s t e d ( S h c h T E ) .
When t h e b e a r i n g o f t h e a i r c r a f t b e c o m e s e q u a l t o t h e c a l c u l a t e d
/259
one, t h e passage of t h e c o n t r o l landmark i s n o t e d .
By u s i n g l o n g - a n d m e d i u m - w a v e r a d i o d i r e c t i o n - f i n d e r s , t h e
l o c a t i o n o f t h e a i r c r a f t i s d e t e r m i n e d f r o m b e a r i n g s o f t w o or t h r e e
m u t u a l l y r e l a t e d g r o u n d r a d i o d i r e c t i o n - f i n d e r s , one o f which i s
t h e command s t a t i o n .
Upon r e q u e s t f r o m t h e c r e w
of an a i r c r a f t , with regard t o
t h e azimuth and d i s t a n c e from
t h e command d i s t a n c e - f i n d i n g s t a ­
t i o n ( ShchGE) , t h e a i r c r a f t m e a s u r e s
i t s d i s t a n c e s i m u l t a n e o u s l y from
two ( t h r e e ) d i s t a n c e measuring
s t a t i o n s , while auxiliary distance
f i n d e r s r e p o r t t h e measured b e a r ­
i n g t o t h e command d i s t a n c e s t a t i o n .
T h e o p e r a t o r o f t h e command
radio distance-finding station
uses a s p e c i a l p l o t t i n g board
t o determine t h e t r u e bearings
o f t h e a i r c r a f t w i t h t h e a i d o f movable r u l e r s w i t h t h e i r c e n t e r s
of r o t a t i o n a t t h e p o i n t s where t h e r a d i o d i s t a n c e - f i n d i n g s t a t i o n s
a r e l o c a t e d ; having measured t h e d i s t a n c e t o t h e a i r c r a f t ( t h e p o i n t s
o f i n t e r s e c t i o n of t h e b e a r i n g s ) , t h e o p e r a t o r t r a n s m i t s t o t h e
crew o f t h e a i r c r a f t i t s p o s i t i o n ( t h e t r u e d i r e c t i o n and d i s t a n c e
f r o m t h e command r a d i o d i s t a n c e - f i n d i n g s t a t i o n ) .

Fig. 3.11.
Previously Calcu­
l a t e d B e a r i n g of a Landmark.

1




If t h e c r e w of t h e a i r c r a f t d e s i r e s t o o b t a i n d a t a r e g a r d i n g
t h e l o c a t i o n of t h e a i r c r a f t i n d i f f e r e n t forms ( e . g . , geograph­
i c a l c o o r d i n a t e s or r e l a t i o n s h i p t o s o m e l a n d m a r k ) , t h e y m u s t a s k
f o r t h e S h c h T F b e a r i n g f r o m t h e command r a d i o d i s t a n c e - f i n d i n g s t a ­
tion.

269

D e t e r m i n a t i o n of t h e Ground S p e e d , D r i f t A n g l e , and Wind
The g r o u n d s p e e d of a n
r a d i o d i s t a n c e - f i n d e r s as w
t i o n a l devices during f l i g h
by t h e a i r c r a f t b e t w e e n two
(LA):

a i r c r a f t c a n b e f o u n d by u s i n g g r o u n d
e l l as o t h e r n o n - a u t o m a t i c r a d i o n a v i g a ­
t on t h e b a s i s o f t h e d i s t a n c e c o v e r e d
s u c c e s s i v e i n d i c a t i o n s of i t s p o s i t i o n

The s u c c e s s i v e l a n d m a r k s f o r t h e LA a r e t h e p o i n t s a t w h i c h t h e
aircraft passes over previously calculated bearings along the route
or l o c a t i o n s f o r t h e a i r c r a f t m a r k e d o n a map w h i c h w e r e o b t a i n e d
f r o m t h e command d i s t a n c e - f i n d e r s u p o n r e q u e s t o f b e a r i n g s ShchGE
or S h c h T F .
The d r i f t a n g l e c a n b e d e t e r m i n e d i n t h r e e ways w i t h t h e a i d
of ground r a d i o distance-finders:
(1) The d i f f e r e n c e b e t w e e n t h e " f o r w a r d " b e a r i n g (ShchDR)
and t h e course of t h e a i r c r a f t a f t e r passing over t h e r a d i o d i s ­
tance-finding station:
US = ShchDR

/260

- MC;

(2)
By t h e d i f f e r e n c e b e t w e e n t h e p a t h a n g l e o f t h e f l i g h t
and t h e c o u r s e o f t h e a i r c r a f t a f t e r s e l e c t i n g a s t a b l e "forward1'
b e a r i n g ( S h c h D R ) o r " r e v e r s e " ShchDM:

+

where a i s t h e d r i f t a n g l e of t h e a i r c r a f t ,
is the path angle
of t h e f l i g h t , and y i s t h e course of t h e a i r c r a f t .
(3)
On t h e b a s i s o f t h e p a t h a n g l e a n d t h e mean c o u r s e o f
t h e a i r c r a f t b e t w e e n s u c c e s s i v e i n d i c a t i o n s o f t h e PA ( A a n d B ) :

a

=

av

I n t h e magnetic loxodromic system, of e s t i m a t i n g path a n g l e s ,
The
t h e b e s t way for d e t e r m i n i n g t h e d r i f t a n g l e i s t h e f i r s t .
second and t h i r d methods g i v e e x a c t r e s u l t s o n l y i n t h e middle p a r t
o f t h e p a t h s e g m e n t , i . e . , when c r o s s i n g t h e m e r i d i a n , r e l a t i v e
A t t h e begin­
t o which t h e p a t h a n g l e o f t h e segment i s measured.
n i n g a n d e n d o f t h e s e g m e n t , t h e e r r o r s a r e maximum.
\

I n t h e o r t h o d r o m i c system of c a l c u l a t i n g p a t h a n g l e s and c o u r s e s ,
t h e accuracy of determining t h e d r i f t angle is approximately t h e
same f o r a l l t h r e e m e t h o d s .
The s p e e d a n d d i r e c t i o n o f t h e w i n d a t f l i g h t a l t i t u d e i s

270

determined with t h e a i d of ground r a d i o distance-finders
ways:

i n two

(1)
According t o t h e ground speea of t h e a i r c r a f t , t h e a i r ­
s p e e d , and t h e d r i f t a n g l e .
T o solve t h i s problem, w e can use a
k e y cn t h e n a v i g a t i o n a l s l i d e r u l e f o r d e t e r m i n i n g t h e w i n d a n g l e
( F i g . 3.12, a ) and f o r d e t e r m i n i n g t h e wind s p e e d ( F i g . 3 . 1 2 , b ) .

(2)
By t h e d i f f e r e n c e b e t w e e n t h e a c t u a l a n d c a l m c o o r d i n a t e s
o f t h e a i r c r a f t on t h e f l i g h t c h a r t .
T h i s method means t h a t t h e
f i r s t l q c a t i o n o f t h e a i r c r a f t o n t h e b a s i s o f t h e ShchGE or S h c h T F
b e a r i n g i s marked on t h e f l i g h t c h a r t .
During t h e t i m e t h a t t h e
a i r c r a f t is f l y i n g from t h e f i r s t l o c a t i o n , t h e c a l m p a t h calcu­
l a t i o n i s made ( a c c o r d i n g t o t h e a v e r a g e c o u r s e , a i r s p e e d a n d t i m e ) ,
t h e calm p o s i t i o n o f t h e a i r c r a f t i s d e t e r m i n e d , a n d a l s o e n t e r e d
on t h e c h a r t w i t h s i m u l t a n e o u s r e q u e s t o f t h e s e c o n d p o s i t i o n o f
The v e c t o r b e t w e e n
t h e a i r c r a f t i n t e r m s o f t h e ShchGE or S h c h T F .
t h e calm p o i n t a n d t h e s e c o n d p o s i t i o n o f t h e a i r c r a f t , d e t e r m i n e d
o n t h e b a s i s o f t h e ShchGE b e a r i n g , i s t h e w i n d v e c t o r f o r t h e f l i g h t
time over a given path segment.
L e t u s c o n s i d e r t h e f o l l o w i n g ex­
ample:
A f t e r 24 min o f f l i g h t b e t w e e n t w o s u c c e s s i v e l o c a t i o n s , t h e
w i n d v e c t o r i s e q u a l t o 140° i n d i r e c t i o n a n d 30 km i n m a g n i t u d e .
If w e d i v i d e t h e modulus o f t h e wind v e c t o r by t h e f l i g h t t i m e / 2 6 1
i n h o u r s ( 0 . 4 ) , w e w i l l g e t t h e wind s p e e d
u = 30:0.4

= 75 k m / h .

The f i r s t m e t h o d o f d e t e r m i n i n g t h e w i n d i s t h e o n e m o s t w i d e l y
employed.
However , on l a r g e p a s s e n g e r a i r c r a f t w i t h a u t o m a t i c n a v i ­
g a t i o n a l i n d i c a t o r s on b o a r d ( e . g . , N I - S O ) , by means of wh i ch a u t o ­
matic q u i e t c a l c u l a t i o n of t h e a i r c r a f t path can be c a r r i e d o u t ,
When t h i s i s
t h e s e c o n d method i s t h e most s u i t a b l e a n d p r e c i s e .
d o n e , i t i s n o l o n g e r n e c e s s a r y t o p l o t t h e w i n d v e c t o r on t h e f l i g h t
chart.

Fig.
Fig.

3.12.

I

Fig.

3.13.

3 . 1 2 . K e y s f o r D e t e r m i n i n g t h e ( a ) Wind A n g l e a n d ( b ) Wind
Speed on t h e N L - 1 O M .

Fig. 3.13.
D e t e r m i n a t i o n o f t h e Wind b y t h e D i f f e r e n c e i n t h e C o o r ­
d i n a t e s of t h e C a l m Point and t h e Location of t h e Aircraft.
271

I t i s clear from Figure 3.13,
tgAW =

AZ
AX '

that

AZ
sinAW '

ut =

where

A Z = Z
A X = X

LA

LA

- Z
- X

*

P'
'P

.

I f w e know t h e d i s t a n c e o f t h e o r t h o d r o m i c c o o r d i n a t e s o f t h e
l o c a t i o n o f t h e a i r c r a f t a n d t h e calm p o i n t , t h i s problem i s e a s i l y
s o l v e d on t h e n a v i g a t i o n a l s l i d e r u l e u s i n g t h e f o l l o w i n g k e y :
F o r d e t e r m i n g AW
3.14,

(Fig.

3.14,

a ) , a n d for d e t e r m i n i n g ut ( F i g .

b)

Fig.

3.14. D e t e r m i n a t i o n of
on t h e N L - 1 O M .

( a ) Wind A n g l e a n d ( b ) Wind S p e e d

ExampZe:
1 5 min.

MFA = 110O; A M = - 7 O ; AX = 4 0 k m ; A Z = 2 0 km; t =
Find t h e d i r e c t i o n and wind s p e e d a t f l i g h t a l t i t u d e .

Solution:

AW = 2 6 O

(Fig.

3.15,

a)

ut = 45 km ( F i g . 3 . 1 5 , b ) .

T o d e t e r m i n e t h e wind d i r e c t i o n r e l a t i v e t o t h e m e r i d i a n of
t h e a i r c r a f t ' s l o c a t i o n , t h e c a l c u l a t i o n of t h e g i v e n p a t h a n g l e
o f f l i g h t s h o u l d b e a p p l i e d t o t h a t m e r i d i a n , and t h e n t h e wind
angle should be added.

Fig.

3 . 1 5 . D e t e r m i n a t i o n o f ( a ) Wind A n g l e a n d ( b )
Wind S p e e d o n t h e N L - 1 O M .

I n a f l i g h t with magnetic p a t h a n g l e s , w e w i l l have approx­
i m a t e ly

272

/262

fiM = MEA

+

AW

or i n o u r c a s e
6M = 110
6

+

26 = 1 3 6 O ;

= 6 +A
= 136-7
M
M

= 129O

.

Automatic A i r c r a f t Radio D i s t a n c e - F i n d e r s

(Radiocompasses)

Automatic a i r c r a f t r a d i o d i r e c t i o n - f i n d e r s (radiocompasses)
are v e r y widely employed.
A i r c r a f t w i t h p i s t o n e n g i n e s u s e them
as a r e l i a b l e , o p e r a t i v e , a n d h i g h l y p r e c i s e method o f a i r c r a f t
navigation.
L a r g e p a s s e n g e r a i r c r a f t w i t h j e t e n g i n e s , f o r a number
o f r e a s o n s , c a n n o t make s u c h e f f e c t i v e u s e o f r a d i o c o m p a s s e s , b u t
t h e y c o n t i n u e t o u s e t h e m s u c c e s s f u l l y a l o n g w i t h o t h e r more p r e ­
c i s e means o f a i r c r a f t n a v i g a t i o n .
T h e o p e r a t i n g p r i n c i p l e o f r a d i o c o m p a s s e s i s i n n o way d i f ­
f e r e n t from t h e p r i n c i p l e of o p e r a t i o n of ground r a d i o d i s t a n c e finders.
However, r a d i o c o m p a s s e s a r e more o p e r a t i v e a n d s u i t a b l e
f o r t h e p u r p o s e s o f a i r c r a f t n a v i g a t i o n , s i n c e t h e y a l l o w t h e crew
of t h e a i r c r a f t t o have a c o n s t a n t v i s u a l i n d i c a t i o n o f t h e p o s i ­
t i o n of t h e aircraft.
The a c c u r a c y o f d i s t a n c e - f i n d i n g f o r g r o u n d
w i t h t h e a i d o f r a d i o c o m p a s s e s i s somewhat l o w e r
acy of d i s t a n c e f i n d i n g f o r a i r c r a f t with ground
f i n d e r s , which c a n b e e x p l a i n e d by t h e f o l l o w i n g

radio stations
than t h e accur­
radio distancethree reasons:

(1) S t a t i o n a r y r a d i o d i s t a n c e - f i n d e r s c a n h a v e s p e c i a l a n ­
t e n n a s which a r e e q u i v a l e n t t o f r a m e - t y p e a n t e n n a s b u t a r e f r e e
o f t h e e f f e c t r e l a t e d t o t h e h o r i z o n t a l s i d e s o f t h e f r a m e ; on a i r ­
c r a f t , i t i s n o t p o s s i b l e t o i n s t a l l s u c h a n t e n n a s due t o t h e i r
unwieldiness.
(2)
The b e a r i n g o f a n a i r c r a f t i s measured w i t h t h e a i d o f
ground r a d i o d i s t a n c e - f i n d e r s d i r e c t l y from t h e d i r e c t i o n of t h e
m a g n e t i c or t r u e m e r i d i a n , p a s s i n g t h r o u g h t h e r a d i o d i s t a n c e - f i n d e r
a t a f i x e d s e t t i n g of t h e antenna system r e l a t i v e t o t h e v e r t i c a l ;
/263
i n d i s t a n c e - f i n d i n g w i t h g r o u n d r a d i o s t a t i o n s by r a d i o c o m p a s s e s
l o c a t e d on b o a r d a i r c r a f t , t h e e r r o r i n t h e b e a r i n g i n c l u d e s t h e
e r r o r s i n measuring t h e a i r c r a f t course; i n a d d i t i o n , t h e accur­
a c y o f d i s t a n c e measurement i s r e d u c e d due t o t h e l o n g i t u d i n a l and
transverse banking of t h e aircraft.

(3)
Errors i n d i s t a n c e - f i n d i n g d u e t o t h e e f f e c t on t h e p r o p ­
a g a t i o n of e l e c t r o m a g n e t i c waves o v e r t h e r e l i e f o f t h e s u r r o u n d ­
i n g medium t o a c e r t a i n d e g r e e i s t a k e n i n t o a c c o u n t i n m e a s u r i n g
t h e d i s t a n c e o f a i r c r a f t w i t h t h e a i d of g r o u n d r a d i o d i s t a n c e - f i n d e r s
( b y means o f p r e l i m i n a r y t e s t f l i g h t s a n d t h e r e c o r d i n g o f a c u r v e
of radio deviation).
273

The c o n s i d e r a b l e d i f f e r e n c e

i n f l i g h t conditions does not permit

u s t o s o l v e t h i s p r o b l e m f o r r a d i o c o m p a s s e s l o c a t e d on b o a r d a n
aircraft.
On t h e a v e r a g e ( w i t h a p r o b a b i l i t y o f 95%), t h e e r r o r s
i n l o c a t i n g aircraft w i t h ground r a d i o d i s t a n c e - f i n d e r s i n f l a t
c o u n t r y i s 1 - 2 O , a n d 3-5O i n t h e m o u n t a i n s .
T h e e r r o r s i n meas­
u r i n g t h e d i s t a n c e s w i t h t h e a i d o f radiocompasses i n f l a t areas
i s 3-5O, a n d c a n r e a c h 10-15O i n m o u n t a i n o u s a r e a s , e s p e c i a l l y a t
low flight altitudes.
Accordingly, t h e p r a c t i c a l o p e r a t i n g r a n g e of a ground r a d i o
d i s t a n c e - f i n d e r w i t h s a t i s f a c t o r y r e s u l t s o f t r a c k i n g i s 300-400
km ( e x c e p t f o r t h o s e w h i c h w o r k o n U S W , w h e r e t h e o p e r a t i n g r a n g e
i s d e t e r m i n e d by t h e s t r a i g h t - l i n e g e o m e t r i c v i s i b i l i t y ) .
A s a t i s f a c t o r y accuracy i n determining t h e bearings of r a d i o
s t a t i o n s w i t h t h e a i d of on-board radiocompasses i s o b t a i n e d a t
d i s t a n c e s up t o 1 8 0 - 2 0 0 km.
N e v e r t h e l e s s , radiocompasses have found
increasingly broad a p p l i c a t i o n f o r purposes of aircraft navigation,
a n d a r e more p o p u l a r t h a n t h e g r o u n d r a d i o d i s t a n c e - f i n d e r s d u e
t o t h e i r c o n s i d e r a b l e autonomousness and t h e ease w i t h which t h e y
can be employed.

For p u r p o s e s o f i n c r e a s i n g o p e r a t i v e n e s s , a s w e l l a s f o r m i n g
a r e s e r v e and e n s u r i n g r e l i a b l e o p e r a t i o n o f r a d i o c o m p a s s e s , two
s e t s o f them a r e u s e d i n m o s t a i r c r a f t .
The b a s i c c o n t r o l s y s t e m f o r a n o n - b o a r d
sists of the following:

r a d i o compass con­

( a ) Frame a n t e n n a w i t h mechanical d e v i c e f o r r o t a t i n g i t and
a mechanism f o r c o m p e n s a t i n g r a d i o d e v i a t i o n .
(b)

Open a n t e n n a .

(c)
S u p e r h e t e r o d y n e r e c e i v e r w i t h a d e v i c e for c o m m u t a t i o n
o f t h e p h a s e o f t h e frame a n t e n n a a n d a n e l e c t r i c a l d e v i c e f o r t u r n i n g
t h e frame a n t e n n a ( t r a c k i n g s y s t e m ) .
(d)

I n d i c a t o r of c o u r s e a n g l e s o f r a d i o s t a t i o n s .

(e>

A s h i e l d for t h e r e m o t e c o n t r o l o f t h e r a d i o c o m p a s s .

T h e s i z e o f t h e a n t e n n a f o r t h e r a d i o c o m p a s s l o c a t e d on b o a r d
a n a i r c r a f t i s many t i m e s s m a l l e r t h a n h a l f t h e l e n g t h o f t h e r e c e i v e d
w a v e , a n d t h e r e f o r e many t u r n s a r e made i n t h e f r a m e t o i n c r e a s e
the effectiveness.
I n a d d i t i o n , t h e i n t e r n a l s p a c e between t h e
c o i l s i s f i l l e d w i t h a m a t e r i a l w i t h a v e r y h i g h magnetic and d i e l e c ­
tric permeability (ferro-dielectric).
T h i s p r o d u c e s a s h a r p de/264
c r e a s e i n t h e p r o p a g a t i o n r a t e of e l e c t r o m a g n e t i c o s c i l l a t i o n s b e t w e e n
t h e s i d e s o f t h e frame, s o t h a t
c1 =

274

c
VF ’

w h i c h i s e q u i v a l e n t t o a d r o p i n t h e w a v e l e n g t h of t h e r e c e i v e d
s i g n a l and t h e r e f o r e an i n c r e a s e i n t h e phase s h i f t between t h e
s i d e s o f t h e frame.
To increase t h e magnetic and d i e l e c t r i c per­
m e a b i l i t y o f t h e medium, t h e e f f e c t of t h e frame w i l l i n c r e a s e .

f r e q u e n c yg e n e r a t o r

Diagram o f A m p l i t u d e M o d u l a t o r a t t h e O u t p u t o f
Fig. 3.16.
t h e Radiocompass R e c e i v e r .
I n c o n t r a s t t o t h e amplitude ground r a d i o d i r e c t i o n - f i n d e r s
which w e h a v e d i s c u s s e d t h u s f a r and which b e l o n g t o t h e llE1t t y p e
( c a r r i e r - w a v e a m p l i t u d e ) , r a d i o c o m p a s s e s p r e s e n t l y u s e t h e method
of amplitude modulation of t h e received s i g n a l s (direction-finder
t y p e "M" )

.

The e s s e n c e o f t h e m e t h o d i s t h a t - r e c e p t i o n o f t h e s i g n a l s
t a k e s p l a c e s i m u l t a n e o u s l y w i t h an open and a frame a n t e n n a , w i t h
t h e phase o f t h e frame a n t e n n a b e i n g c o n s t a n t l y s w i t c h e d by t h e
low-frequency g e n e r a t o r .
T h i s means t h a t a n a m p l i t u d e - m o d u l a t e d
s i g n a l i s obtained a t t h e input of t h e r e c e i v e r .
A s i m p l i f i e d d i a g r a m of t h e a m p l i t u d e m o d u l a t o r a t t h e i n p u t
o f t h e r e c e i v e r i s shown i n F i g u r e 3 . 1 6 .

The c o n t r o l g r i d s o f L 1 a n d L 2 r e c e i v e a n e g a t i v e v o l t a g e u g 0 ,
s o t h a t when t h e l o w - f r e q u e n c y g e n e r a t o r i s t u r n e d o f f , t h e s e t u b e s
w i l l be c l o s e d and t h e s i g n a l s from t h e frame a n t e n n a w i l l n o t b e
passed.
When t h e l o w - f r e q u e n c y g e n e r a t o r i s t u r n e d o n , t u b e s L 1 a n d
L 2 open a l t e r n a t e l y , a n d t h e s i g n a l from t h e frame a n t e n n a r e a c h e s
t h e i n p u t o f t h e r e c e i v e r i n p h a s e s w h i c h a r e s e p a r a t e d b y 180°,
a n d when t h e s e a r e c o m b i n e d w i t h t h e s i g n a l s f r o m t h e o p e n a n t e n n a ,
they undergo amplitude modulation.

275

/265

O b v i o u s l y , d e p e n d i n g on t h e d i r e c t i o n o f t h e r a d i o s t a t i o n
( F i g . 3 . 1 7 ) , w i t h a f i x e d r a d i o c o m p a s s f r a m e , t h e a m p l i t u d e modu­
l a t i o n c a n b e p o s i t i v e ( P o s i t i o n l), z e r o ( P o s i t i o n 2 1 , a n d n e g a ­
tive (Position 3).
T h e t r a c k i n g s y s t e m a t t h e o u t p u t of t h e r e c e i v e r i s d e s i g n e d
s o t h a t t h e frame of t h e r a d i o c o m p a s s r o t a t e s i n t h e d i r e c t i o n w h i c h
w i l l produce a zero modulation of t h e s i g n a l .
A diagram of t h e output s e c t i o n
o f t h e r e c e i v e r i s shown i n F i g ­
u r e 3.18.

A r e f e r e n c e v o l t a g e on t h e a n o d e s
o f t u b e s L 1 a n d L2, f o r m e d p r e v i o u s l y
i n t h e p o s i t i v e half-periods of
the rectangular pulse, is supplied
t o t h e switching c i r c u i t of t h e
frame a n t e n n a a t t h e i n p u t o f t h e
If t h e i n p u t s i g n a l i s
receiver.
m o d u l a t e d by t h e frame s i g n a l , t h e
a v e r a g e anode c u r r e n t o f one o f
t h e tubes w i l l be g r e a t e r than t h a t
F i g . 3.17.
Zero, P o s i t i v e ,
of t h e o t h e r .
This produces a disturband Negative Modulation.
ance of t h e b a l a n c e of t h e b r i d g e
c i r c u i t i n t h e magnetic a m p l i f i e r ,
made o f p e r m a l l o y c o r e s , a n d a c u r r e n t p a s s e s t h r o u g h t h e r o t o r
The s t a t o r w i n d i n g o f t h e m o t o r i s con­
winding of a s m a l l motor.
s t a n t l y s u p p l i e d w i t h a v o l t a g e which i s s h i f t e d 90° i n p h a s e by
c a p a c i i t o r C , f r o m a l o w - f r e q u e n c y g e n e r a t o r wh'ich s u p p l i e s t h e
bridge circuit.
The m o t o r w i l l c o n t i n u e t o r o t a t e u n t i l t h e d i r e c t i o n o f t h e
r a d i o s t a t i o n i s no l o n g e r p e r p e n d i c u l a r t o t h e frame of t h e r a d i o compass, and t h e m o d u l a t i o n o f t h e s i g n a l o f t h e open a n t e n n a by
t h e frame becomes z e r o .
I n t h e case when t h e frame a n t e n n a i s t u r n e d t o w a r d t h e r a d i o
s t a t i o n i n t h e o p p o s i t e p l a n e , t h e p h a s e of t h e frame c h a n g e s by
180O.
I n t h i s case, i n t h e presence of modulation, t h e r o t a t i o n
o f t h e frame w i l l t a k e p l a c e n o t i n t h e d i r e c t i o n o f r e d u c t i o n ,
but i n i t i a l l y i n t h e d i r e c t i o n of increase of modulation, thus causing
t h e f r a m e t o t u r n t h r o u g h 180°.
I n t h i s manner, t h e readings from
t h e r a d i o c o m p a s s a r e a l l g i v e n t h e same s i g n .

A b l o c k d i a g r a m o f t h e r a d i o c o m p a s s i s shown i n F i g u r e 3 . 1 9 .
The c o n t r o l s f o r t h e r a d i o c o m p a s s a r e mo u n t ed on a s p e c i a l
control panel.
Usually, t h e radiocompass has t h r e e operating regimes
( b e s i d e s t h e " o f f " p o s i t i o n ) , s o t h a t a s e l e c t o r s w i t c h i s mounted
on t h e p a n e l .

276

I .
Tuning.
I n t h i s r e g i m e , o n l y t h e open a n t e n n a o f t h e r a d i o / 2 6 6
compass i s c o n n e c t e d .
A s p e c i a l v e r n i e r on t h e c o n t r o l p a n e l i s
used t o tune t h e device t o t h e frequency of t h e ground r a d i o sta­
t i o n , e i t h e r b y e a r or b y a v i s u a l t u n i n g i n d i c a t o r .
When t u n i n g
by e a r , r e c e p t i o n t a k e s p l a c e i n t h e " t e l e g r a p h " r e g i m e , i . e . , t h e
s e c o n d h e t e r o d y n e o f t h e r e c e i v e r i s t u r n e d on t o c o n v e r t t h e i n t e r ­
mediate frequency of t h e r e c e i v e r t o sound.
I n t h e t e l e g-~
raph regime,
t h e c a l l l e t t e r s o f t h e r a d i o s t a t i o n are a l s o h e a r d , i f t h e s t a t i o n
i s t r a n s m i t t i n g on a n o n - m o d u l a t e d f r e q u e n c y .

Fig.

3.18.

D i a g r a m o f O u t p u t S e c t i o n o f R a d i ocompas s
Receiver.

I I .
Compass r e g i m e .
I n t h i s r e g i m e , b o t h t h e open and frame
antennas of t h e radiocompass are connected.
I n t h i s case, t h e track­
i n g s y s t e m o f t h e r e c e i v e r t u r n s t h e f r a m e a n t e n n a d e p e n d i n g on
t h e d i r e c t i o n of t h e r a d i o s t a t i o n a n d t h e d i r e c t i o n o f t h e r a d i o
s t a t i o n i s s h o w n o n a n i n d i c a t o r ( c o u r s e a n g l e or b e a r i n g ) .
I l l .
Frame r e g i m e .
I n t h i s r e g i m e , o n l y t h e frame a n t e n n a
o f t h e radiocompass i s connected, and t h e b e a r i n g of t h e r a d i o s t a ­
t i o n c a n b e d e t e r m i n e d w i t h minimum a u d i b i l i t y o f i t s s i g n a l s i n
t h e telegraph regime.
The r o t a t i o n o f t h e frame i s c a r r i e d o u t
b y means o f a s p e c i a l p u s h b u t t o n s w i t c h on t h e c o n t r o l p a n e l w i t h
the label "left-right".
The r e a d i n g of t h e b e a r i n g i n t h i s c a s e
h a s two s i g n s .
/267

R e c e n t models o f t h e A R K - 1 1 a u t o m a t i c r a d i o c o m p a s s d o n o t differ--i n t h e i r p r i n c i p l e of o p e r a t i o n from t h e o p e r a t i n g p r i n c i p l e d e s c r i b e d
a b o v e f o r t h e ARK-5 r a d i o c o m p a s s , b u t t h e y h a v e s e v e r a l d e s i g n f e a t u r e s
and advantages :

27 7

I


(a)

Complete e l e c t r i c a l remote c o n t r o l .

(b)
Possibility of setting the apparatus t o nine previously
s e l e c t e d c h a n n e l s ( f r e q u e n c i e s ) i n t h e r a n g e f r o m 1 2 0 t o 1 3 4 0 kHz
a n d s w i t c h i n g f r o m one r e c e i v e r c h a n n e l t o a n o t h e r by means o f a n
a u t o m a t i c p u s h b u t t o n s w i t c h , l o c a t e d on t h e c o n t r o l l p a n e l .
There
i s a l s o a p r o v i s i o n f o r . s m o o t h manual s e t t i n g o v e r t h e e n t i r e oper­
a t i n g range of t h e radiocompass (with t h e t e n t h button depressed).
(c)

I n c r e a s e d n o i s e s t a b i l i t y of t h e r e c e i v e r .

(d)
P o s s i b i l i t y of o p e r a t i o n i n c o m b i n a t i o n w i t h a n o n -co n ­
t r o l l e d a n t e n n a o f o p e n t y p e w i t h a low a e r o d y n a m i c r e s i s t a n c e and
a low c p e r a t i n g a l t i t u d e ( o n t h e o r d e r o f 2 0 cm).

f

7

Fig.

3.19.

F u n c t i o n a l Diagram o f Radiocompass.

The c o n t r o l p a n e l o f t h e ARK-11 d i f f e r s i n d e s i g n f r o m t h a t
of t h e ARK-5.
I n a d d i t i o n t o t h e Itoff" p o s i t i o n , t h e r e a r e f o u r
operating regimes.
T h e f i r s t t h r e e r e g i m e s a r e t h e same a s d e s c r i b e d
above.
T h e f o u r t h r e g i m e " C o m p a s s 11" i s a s p a r e a n d i s u s e d i n
t h e c a s e o f i n t e n s e e l e c t r o s t a t i c n o i s e s when t h e u s u a l d i s t a n c e f i n d i n g m e t h o d s b e come u n s t a b l e .
I n t h e " C o m p a s s 11" r e g i m e , i n s t e a d o f t h e o p e n a n t e n n a , a
s e c o n d f r a m e a n t e n n a i s u s e d , m o u n t e d o n a common f r a m e - a n t e n n a
b l o c k , p e r p e n d i c u l a r t o t h e b a s i c frame a n d f o r m i n g a u n i t w i t h
t h e b a s i c frame.
The r e f e r e n c e s i g n a l i n t h i s c a s e r e a c h e s t h e i n p u t o f t h e
r e c e i v e r n o t from t h e open b u t from t h e a d d i t i o n a l frame a n t e n n a ,
w h i c h i s less s e n s i t i v e t o n o i s e .
However, t h e a d d i t i o n a l frame
a n t e n n a which h a s t h e same p r o p e r t i e s as t h e main a n t e n n a , c h a n g e s
t h e p h a s e o f t h e r e f e r e n c e s i g n a l b y a f u r t h e r 180° when i t i s
t u r n e d t h r o u g h 180°, s o t h a t b o t h p o s i t i o n s o f zero r e c e p t i o n o f
t h e main frame a n t e n n a w i l l b e p o s i t i o n s o f s t a b l e e q u i l i b r i u m ,
and consequently it i s p o s s i b l e t o have an e r r o r i n determining
t h e c o u r s e a n g l e o f t h e r a d i o s t a t i o n o f 180O.

The c o n t r o l p a n e l o f t h e A R K - 1 1 h a s a t o g g l e s w i t c h f o r n a r row a n d w i d e f r e q u e n c y b a n d p a s s : " w i d e - n a r r o w " .
I n t h e "narrow"
p o s i t i o n , t h e extraneous noises i n t h e earphones are reduced and
t h e d e s i r e d r a d i o s t a t i o n c a n b e h e a r d more c l e a r l y .

/268

O t h e r c o n t r o l u n i t s on t h e A R K - 1 1 p a n e l ( s u b r a n g e s w i t c h , k n o b s
f o r coarse and f i n e s e t t i n g , toggle switches and b u t t o n s ) have t h e
same m a r k i n g s a s i n t h e ARK-5.

Radiocompass Deviation

C o n d i t i o n s f o r d i r e c t i o n a l r e c e p t i o n of e l e c t r o m a g n e t i c waves
on a n a i r c r a f t a r e n o t f a v o r a b l e a n d d e p e n d on t h e d i r e c t i o n o f
p r o p a g a t i o n o f t h e wave f r o n t i n b o t h t h e h o r i z o n t a l a n d v e r t i c a l
planes.
If t h e r e c e p t i o n o f s i g n a l s from a ground r a d i o s t a t i o n i s
b e i n g made a t c o n s i d e r a b l e d i s t a n c e s w h i c h e x c e e d 5-6 t i m e s t h e
f l i g h t a l t i t u d e , t h e v e r t i c a l component of t h e v e c t o r o f p r o p a g a t i o n
o f t h e w a v e f r o n t h a s l e s s o f a n e f f e c t on t h e r e c e p t i o n c o n d i t i o n s .
I n t h i s c a s e , we c a n u s e a c o m p e n s a t e d c u r v e o f r a d i o d e v i a t i o n ,
whi'ch i s a f u n c t i o n o n l y o f t h e c o u r s e a n g l e s o f t h e r a d i o s t a t i o n .

The r e a s o n f o r t h e r a d i o d e v i a t i o n i s a r e f l e c t i o n o f e l e c ­
t r o m a g n e t i c w a v e s f r o m t h e s u r f a c e o f t h e a i r c r a f t or t h e i r r e Since the r a d i o
r e f l e c t i o n from i n d i v i d u a l p a r t s of t h e a i r c r a f t .
compass frame i s mounted i n t h e p l a n e o f symmetry o f t h e a i r c r a f t
X - Z , t h e d e v i a t i o n a t c o u r s e a n g l e s z e r o a n d 180° i s c l o s e t o z e r o .
The t r a n s v e r s e p l a n e o f t h e a i r c r a f t Y-Z i s a l s o c l o s e t o t h e
p l a n e of symmetry, s o t h a t t h e d e v i a t i o n a t c o u r s e a n g l e s 90 and
270° i s n o t g r e a t and p a s s e s t h r o u g h z e r o a t c o u r s e a n g l e s c l o s e
t o it.
T h e maximum a s y m m e t r y o f t h e a i r c r a f t t a k e s p l a c e r e l a t i v e
t o t h e d i r e c t i o n s 4 5 , 1 3 5 , 2 2 5 a n d 315O.
Therefore, t h e r a d i o devi­
a t i o n a t t h e s e c o u r s e a n g l e s r e a c h e s a maximum.
Hence, t h e c u r v e of r a d i o d e v i a t i o n h a s a q u a r t e r n a r y a p p e a r ­
+ 1 2 t o 25O d e p e n d i n g
a n c e ( F i g . 3 . 2 0 ) w i t h e x t r e m e v a l u e s AP = on t h e t y p e o f a i r c r a f t .
R a d i o d e v i a t i o n i s c o m p e n s a t e d by a m e c h a n i c a l c o m p e n s a t o r
The compen­
l o c a t e d on t h e a x i s o f r o t a t i o n o f t h e frame a n t e n n a .
s a t o r h a s a c o n t r o l s t r i p which produces a n a d d i t i o n a l r e v o l u t i o n
o f t h e a x i s o f t h e m a s t e r s e l s y n by means of a s p e c i a l t r a n s m i s ­
sion.
The r e q u i r e d s h a p e i s g i v e n t o t h e c o n t r o l s t r i p b y means
of 2 4 c o m p e n s a t i n g s c r e w s t o s e t t h e r e a d i n g s f o r t h e r a d i o c o m p a s s
a t 15O i n t e r v a l s on t h e s c a l e f r o m z e r o t o 360O.
B e f o r e t h e f i r s t d e t e r m i n a t i o n of
pensator is usually neutralized, <.e.,

r a d i o d e v i a t i o n , t h e com­
each of t h e screws i s unscrewed

279

t o such a p o s i t i o n t h a t t h e c o n t r o l s t r i p has a shape with t h e
c o r r e c t c u r v a t u r e and t h e a d d i t i o n a l r - + a t i o n o f t h e a x i s o f t h e
m a s t e r s e l s y n i s e q u a l t o z e r o a t a l l cou;-se a n g l e s .

/269

T o d e t e r m i n e r a d i o d e v i a t i o n , a g r o u n d r a d i o s t a t i o n i s se­
l e c t e d ( p r e f e r a b l y a t a d i s t a n c e o f 5 0 - 1 0 0 km f r o m t h e a i r p o r t )
a n d t h e t r u e b e a r i n g i s m e a s u r e d as a c c u r a t e l y as p o s s i b l e on a
l a r g e - s c a l e c h a r t ( u s u a l l y 1:500,000),
and t h e n t h e magnetic b e a r i n g
o f t h i s r a d i o s t a t i o n (MBR) i s d e t e r m i n e d .

Fig.

3.20.

Graph o f R a d i o D e v i a t i o n .

By m e a n s o f a d e v i a t i o n d i s t a n c e f i n d e r , m a g n e t i c b e a r i n g s
o f o n e or t w o s e p a r a t e l a n d m a r k s (MBL) a r e m e a s u r e d f r o m t h e c e n t e r
o f t h e a r e a w h e r e t h e r a d i o d e v i a t i o n w i l l b e p l o t t e d , i n t h e way
w h i c h w a s d e s c r i b e d i n C h a p t e r 11, w i t h a d e s c r i p t i o n o f t h e i r d e v i ­
I f t h e a r e a �or t h e d e v i a t i o n
a t i o n of t h e m a g n e t i c c o m p a s s e s .
o p e r a t i o n s a t t h e a e r o d r o m e i s c o n s t a n t , t h e MBL w i l l b e known e a r l i e r .
The a i r c r a f t i s t h e n r o l l e d o u t o n t h e r u n w a y .
The d e v i a t i o n
d i s t a n c e f i n d e r i s i n s t a l l e d i n t h e a i r c r a f t i n a l i n e 0-180° e x a c t l y
a l o n g i t s l o n g i t u d i n a l a x i s , and t h e c o u r s e a n g l e o f t h e landmark
(CAL) i s c a l c u l a t e d t o g e t r i d o f i n s t a l l a t i o n e r r o r s i n t h e r a d i o compass.
The c o r r e s p o n d i n g C A R = 0 :
C A L = MBL

-

MBR.

I n t h e d e v i a t i o n d i s t a n c e - f i n d e r , t h e l e v e l l i n e is s e t t o t h e cal­
c u l a t e d CAL, and t h e a i r c r a f t i s t u r n e d u n t i l t h e s i g h t l i n e of
t h e d i s t a n c e - f i n d e r c o i n c i d e s w i t h t h e d i r e c t i o n of t h e s e l e c t e d
landmark.
I n t h i s case, t h e l o n g i t u d i n a l a x i s of t h e a i r c r a f t w i l l
b e l i n e d up e x a c t l y w i t h t h e r a d i o s t a t i o n (CAR = O ) , a n d t h e mag­
n e t i c course of t h e a i r c r a f t w i l l be e q u a l t o t h e magnetic b e a r i n g
o f t h e r a d i o s t a t i o n a s m e a s u r e d o n t h e c h a r t (MBR).
T u r n i n g on t h e r a d i o compass a n d s e t t i n g i t t o t h e d e s i r e d
r a d i o s t a t i o n , t h e reading of t h e radiocompass i s taken (RRC).
If
RRC i s n o t e q u a l t o z e r o , w e w i l l h a v e t h e i n s t a l l a t i o n e r r o r o f
t h e frame:

280

I

I I II I

1111111IIII

11111 I111111111111 I I II II

'est


= CAR - RRC.


Then, w i t h o u t t u r n i n g o f � t h e r a d i o c o m p a s s , it i s n e c e s s a r y
t o l o o s e n t h e f a s t e n i n g screws which h o l d t h e frame t o t h e f u s e ­
l a g e and t h e n (by t u r n i n g t h e b a s e of t h e frame) a d j u s t it u n t i l
the indicator points t o
RRC = CAR = 0 ,

a f t e r w h i c h t h e frame i s r e - f a s t e n e d

to the fuselage.

The r e m a i n i n g i n s t a l l a t i o n e r r o r , i f R R C i s n o t e q u a l t o z e r o
a f t e r t h e f r a m e h a s b e e n f a s t e n e d d o w n , c a n b e c o m p e n s a t e d �or i m m e ­
d i a t e l y e i t h e r b y t h e n a v i g a t o r or t h e p i l o t b y t u r n i n g t h e b o d y
of t h e s e l s y n r e l a t i v e t o t h e i n d i c a t o r scale.
A f t e r c o m p e n s a t i n g f o r t h e i n s t a l l a t i o n e r r o r , t h e r a d i o de­
v i a t i o n is determined successively a t 24 RRC's a t 15O i n t e r v a l s .
To d o t h i s , i t i s n e c e s s a r y t o s e t t h e s i g h t l i n e o f t h e d e v i a t i o n
distance-finder along t h e l o n g i t u d i n a l a x i s of t h e a i r c r a f t t o 0­
180°, l o o s e n t h e d i a l o f t h e d e v i a t i o n d i s t a n c e - f i n d e r a n d move
i t s o t h a t t h e l i n e o f s i g h t 0-180° p a s s e s t h r o u g h t h e s e l e c t e d
landmark, and t h e n f a s t e n t h e s c a l e of t h e d i s t a n c e - f i n d e r once
again.
I n t h i s case
C A R = 0 a n d MBR = 0 .

The d e v i a t i o n d i s t a n c e - f i n d e r m o u n t e d i n t h i s m a n n e r makes
it p o s s i b l e t o c a l c u l a t e t h e c o u r s e a n g l e s of t h e r a d i o s t a t i o n
(CAR) o n t h e s c a l e d i a l b y t u r n i n g t h e a i r c r a f t t o a n y a n g l e .
Consequently, i f w e t u r n t h e aircraft according t o t h e indi­
c a t i o n s of t h e r a d i o c o m p a s s t o a RRC = 1 5 O , a n d t h e n t o 3 0 , 4 5 ,
60°, e t c . , successively ( s e t t i n g t h e s i g h t system of t h e deviation
d i s t a n c e - f i n d e r each t i m e t o a s e l e c t e d landmark) , w e can calcu­
l a t e t h e C A R i m m e d i a t e l y f r o m t h e s c a l e on t h e d i a l .
T h u s , i n e a c h r e a d i n g o f t h e r a d i o c o m p a s s , we d e t e r m i n e t h e
a c t u a l c o u r s e a n g l e of t h e r a d i o s t a t i o n a n d c a n w r i t e t h e r a d i o
d e v i a t i o n as f o l l o w s :

'r

= CAR - R R C .

Compensation of r a d i o d e v i a t i o n i s p e r f o r m e d a f t e r i t h a s b e e n
determined.
To d o t h i s , t h e g r a p h o f r a d i o d e v i a t i o n i s p l o t t e d
and t h e extreme values of t h e graph a r e d i v i d e d i n t o t h r e e e q u a l
p a r t s t o a v o i d s h a r p b e n d s i n t h e s t r i p , a f t e r which two i n t e r m e d ­
iate graphs of radio deviation are plotted.
The c o m p e n s a t o r i s t h e n r e m o v e d f r o m t h e a x i s o f t h e f r a m e ;
by t u r n i n g t h e p r o p e r s c r e w s , c o m p e n s a t i o n i s made f o r t h e r a d i o

281


/210

d e v i a t i o n i n terms of t h e f i r s t i n t e r m e d i a t e graph, c a l c u l a t i n g
t h e c o r r e c t i o n made i n t h e s e l e c t e d p o r t i o n o f t h e r a d i o c o m p a s s
Then t h e d e v i ­
b y m e a n s of a s p e c i a l p o i n t e r o n t h e c o m p e n s a t o r .
a t i o n i s compensated by t h e second i n t e r m e d i a t e g r a p h , and f i n a l l y
b y t h e c u r v e of r a d i o d e v i a t i o n .
Compensation f o r r a d i o d e v i a t i o n by a l l t h r e e g r a p h s i s p e r ­
formed i n an o r d e r such t h a t a f t e r each i n t r o d u c t i o n of a p o s i t i v e
c o r r e c t i o n t h e r e i s a c o r r e c t i o n of e q u a l magnitude b u t n e g a t i v e ,
i . e . , w i t h a m i r r o r image of t h e c o u r s e a n g l e s .
Usually, the order
0 , 1 5 , 345, 30, 330, 45,
o f c o m p e n s a t i o n i s s e l e c t e d as f o l l o w s :
315, 6 0 , 300, 75, 285, 90, 270, 105, 255, 120, 240, 135, 225, 150,
/271
2 1 0 , 1 6 5 , 1 9 5 a n d 180O.
A f t e r compensation f o r r a d i o d e v i a t i o n , t h e compensator is
m o u n t e d on t h e m e c h a n i s m o f t h e frame; t h e a i r c r a f t i s t u r n e d a n d
t h e d e v i a t i o n d i s t a n c e - f i n d e r i s used t o check t h e c o r r e c t n e s s of
t h e o p e r a t i o n s which have been c a r r i e d o u t .
If a n y e r r o r s i n compen­
s a t i o n a r e d i s c o v e r e d , t h e r a d i o d e v i a t i o n i s compensated once a g a i n
by a n a d d i t i o n a l t u r n i n g of t h e screws c o r r e s p o n d i n g t o t h e r e a d i n g s
of t h e radiocompass.
I n a d d i t i o n t o t h e m e t h o d d e s c r i b e d a b o v e for c o r r e c t i n g r a d i o
d e v i a t i o n s on t h e g r o u n d , t h e r e a r e o t h e r s .
For e x a m p l e :
(a)
D e t e r m i n a t i o n of t h e m a g n e t i c c o u r s e o f a n a i r c r a f t b y
d i s t a n c e - f i n d i n g a t t h e t a i l ( n o s e ) , a s d e s c r i b e d i n C h a p t e r 11,
a n d t h e c a l c u l a t i o n o f c o u r s e a n g l e s o f t h e r a d i o s t a t i o n on t h e
b a s i s of it.
(b)
R a n g e f i n d i n g of a r a d i o s t a t i o n w h i c h i s v i s i b l e f r o m
t h e a i r p o r t ( e . g . , a d i s t a n t power r a d i o s t a t i o n ) .
I n a i r c r a f t w h e r e t h e frame a n t e n n a o f t h e r a d i o compass i s
m o u n t e d b e l o w t h e f u s e l a g e , d e t e r m i n a t i o n o f r a d i o d e v i a t i o n on
t h e ground i s i m p r a c t i c a l , s i n c e t h e r e f l e c t i o n of e l e c t r o m a g n e t i c
waves from t h e s u r f a c e o f t h e ground c a u s e s a d i s t o r t i o n of t h e
electromagnetic f i e l d .
In these aircraft, the radio deviation is
determined i n f l i g h t .
To d e t e r m i n e r a d i o d e v i a t i o n i n f l i g h t , a r a d i o s t a t i o n l o ­
c a t e d 2 0 0 - 3 0 0 km a w a y f r o m t h e f l i g h t a r e a i s s e l e c t e d .
The f l i g h t
i s c a r r i e d o u t i n s u c h a way t h a t t h e a i r c r a f t c r o s s e s t h e l i n e
o f t h e g i v e n b e a r i n g a t e a c h s e g m e n t of t h e f l i g h t a n g l e o f t h e
radio station.
Usually, t h e order of t h e course angles i s then
s e l e c t e d s o t h a t t h e f o l l o w i n g c o m p e n s a t i o n mechanism i s employed
0 , 1 5 , 345, 30, e t c . , approach­
i n compensating f o r r a d i o d e v i a t i o n :
i n g t h e r a d i o s t a t i o n , a n d 1 0 5 , 2 5 5 , 1 2 0 , 2 4 0 , e t c . , up t o 1 8 0 ° ,
g o i n g away f r o m i t .
To s a v e t i m e , t h e f l i g h t c a n b e c a r r i e d o u t o v e r a 2 4 - a n g l e
r o u t e , i . e . , p r a c t i c a l l y a l o n g a c o u r s e which c r o s s e s t h e s t r a i g h t ­

2 82

l i n e f l i g h t f o r 20-30 s e c f o r e a c h r e c o r d i n g of t h e r e a d i n g s o f
t h e r a d i o compass a n d c o u r s e .
However, i n t h i s c a s e , it i s neces­
s a r y t o determine t h e l o c a t i o n of t h e aircraft a t each p o i n t being
m e a s u r e d a n d t o e n t e r i t on a c h a r t s o t h a t when t h e d a t a i s a n a l y z e d
i t w i l l b e p o s s i b l e t o d e t e r m i n e t h e b e a r i n g of t h e r a d i o s t a t i o n
from t h e p o i n t a t which t h e r e a d i n g w a s t a k e n .
I n f a c t , t h e c o u r s e a n g l e o f t h e r a d i o s t a t i o n (CAR) a t t h e
moment w h e n t h e r e c o r d i n g s a r e made i s d e t e r m i n e d b y t h e f o r m u l a
CAR = TBR

-

TK,

a n d t h e ' r a d i o d e v i a t i o n o f t h e r a d i o compass i s d e t e r m i n e d as t h e
difference:
Ar

= CAR - RRC.

C o m p e n s a t i o n f o r r a d i o d e v i a t i o n i s made a f t e r t h e a i r c r a f t
l a n d s i n t h e same way a s a f t e r d e t e r m i n i n g i t o n t h e g r o u n d , b u t
w i t h o u t c h e c k i n g t h e a c c u r a c y of t h e work which h a s b e e n c a r r i e d
o u t , s i n c e t h i s would r e q u i r e r e p e t i t i o n of t h e f l i g h t .

A?:rcraft N a v i g a t i o n Using R a d i o c o m p a s s e s on B o a r d t h e Aircraft

/272

R a d i o c o m p a s s e s on b o a r d t h e a i r c r a f t m a k e i t p o s s i b l e t o s o l v e
t h e same n a v i g a t i o n a l p r o b l e m s as g r o u n d r a d i o d i s t a n c e - f i n d e r s .
(a)
Path c o n t r o l i n t e r m s of d i r e c t i o n and s e l e c t i o n of t h e
course t o be followed during f l i g h t toward t h e r a d i o s t a t i o n and
away f r o m i t .
(b)
s t a tion.

Measurement o f t h e d r i f t a n g l e a f t e r f l y i n g o v e r t h e r a d i o

(c)
Checking t h e p a t h f o r d i s t a n c e by measuring t h e d i s t a n c e
t o a radio station located t o the side.
(d)
D e t e r m i n a t i o n o f t h e p o s i t i o n o f t h e a i r c r a f t by o b t a i n ­
i n g b e a r i n g s from two r a d i o s t a t i o n s .
(e)
Determining t h e d r i f t angle and t h e groundspeed from suc­
c e s s i v e p o s i t i o n s o f t h e a i r c r a f t , as w e l l a s t h e w i n d p a r a m e t e r s
at flight altitude.
The s o l u t i o n o f t h e s e p r o b l e m s by means o f a r a d i o c o m p a s s mounted
on b o a r d t h e a i r c r a f t i s v e r y s i m i l a r i n p r i n c i p l e o f s o l u t i o n t o
t h e ground r a d i o d i s t a n c e - f i n d e r s , e s p e c i a l l y if t h e i n d i c a t o r f o r
t h e c o u r s e a n g l e s o f t h e r a d i o compass i s combined w i t h t h e c o u r s e
i n d i c a t o r o f t h e a i r c r a f t a n d t h u s shows t h e r e a d i n g f o r t h e b e a r ­
ing (Fig. 3.21).
The f i g u r e s h o w s t h e c o u r s e i n d i c a t o r f o r t h e n a v i g a t o r ,

283

li


c o m b i n e d w i t h t h e b e a r i n g i n d i c a t o r s o f t w o r a d i o c o m p a s s e s (UShM).
The c o u r s e o f t h e a i r c r a f t i s m e a s u r e d o n t h e i n n e r , m o v a b l e
s c a l e o f t h i s i n d i c a t o r ( r e l a t i v e t o a t r i a n g u l a r m a r k on t h e o u t e r
s c a l e ) , w h i l e t h e c o u r s e a n g l e s of t h e r a d i o s t a t i o n s a r e i n d i c a t e d
on t h e o u t e r f i x e d s c a l e a c c o r d i n g t o t h e p o s i t i o n o f t h e p o i n t ­
ers of t h e radiocompasses.
On t h e i n n e r , m o v a b l e s c a l e , o p p o s i t e
t h e arrows, it is possible t o c a l c u l a t e t h e bearings of t h e r a d i o
s t a t i o n s , w h i l e t h e o t h e r e n d s o f t h e p o i n t e r s c a n b e u s e d t o show
t h e bearings of t h e aircraft.
However, t h i s method i s p r a c t i c a l o n l y f o r u s e w i t h one r a d i o c o m p a s s , s i n c e t h e t o t a l c o r r e c t i o n i s t h e n shown on t h e s c a l e o f
d e v i a t i o n s , and i s e f f e c t i v e o n l y f o r one r a d i o s t a t i o n
A = A

M

+ 6 ,

w h e r e 6 i s t h e d i f f e r e n c e b e t w e e n t h e m e r i d i a n of t h e a i r c r a f t l o c a ­
t i o n and t h e m e r i d i a n of t h e r a d i o s t a t i o n , a n d AM i s t h e m a g n e t i c
d e c l i n a t i o n of t h e l o c a t i o n of t h e aircraft.
Obviously, i n t h e g e n e r a l case t h i s c o r r e c t i o n w i l l be d i f ­
ferent for different radio stations.
When i t i s n e c e s s a r y t o o b t a i n t h e t r u e b e a r i n g s o f a n a i r ­
c r a f t s i m u l t a n e o u s l y from two r a d i o s t a t i o n s , o n l y t h e m a g n e t i c
d e c l i n a t i o n o f t h e p o s i t i o n o f t h e a i r c r a f t i s s e t on t h e d e c l i n ­
a t i o n scale, and a f t e r c a l c u l a t i n g t h e approximate b e a r i n g s of t h e
a i r c r a f t by t h e o p p o s i t e e n d s of t h e p o i n t e r s o f t h e r a d i o c o m p a s s e s ,
c o r r e c t i o n s f o r d e v i a t i o n o f t h e m e r i d i a n s a r e made i n t h e s e r e a d ­
ings.
T h e n e c e s s i t y t o make c o r r e c t i o n s for t h e d e v i a t i o n o f t h e
m e r i d i a n s i s one o f t h e p r i n c i p a l s h o r t c o m i n g s o f r a d i o c o m p a s s e s .
T h i s s h o r t c o m i n g t o a c e r t a i n d e g r e e c a n b e r e d u c e d by u s i n g an
orthodromic system for, e s t i m a t i n g t h e p a t h a n g l e s and courses of
the aircraft.
I n t h i s case, t h e need t o introduce c o r r e c t i o n s i s
n o l o n g e r a p p l i c a b l e , i f t h e r a d i o s t a t i o n i s l o c a t e d on t h e r e f e r ­
e n c e m e r i d i a n for c o m p u t i n g t h e p a t h a n g l e s , a n d i n a n y c a s e t h e
c o r r e c t i o n r e m a i n s c o n s t a n t if t h i s c o n d i t i o n i s n o t o b s e r v e d .

/273

The u s e o f combined i n d i c a t o r s c o n s i d e r a b l y s i m p l i f i e s t h e
o p e r a t i o n s r e l a t e d t o t h e u s e o f r a d i o c o m p a s s e s mounted on b o a r d
aircraft.
T h e r e f o r e , t h e methods of u s i n g them f o r n a v i g a t i o n a l
purposes must b e viewed as non-recorded i n d i c a t o r s of c o u r s e a n g l e s
of r a d i o s t a t i o n s , a s s u v i n g t h a t i n t h e combined i n d i c a t o r s , t h e
a d d i t i o n a n d s u b t r a c t i o n o f t h e a n g l e s a c c o r d i n g t o t h o s e same r u l e s
is carried out automatically.
I t i s c l e a r from F i g u r e 3.22 t h a t t h e magnetic b e a r i n g of t h e
r a d i o s t a t i o n (MBR) a n d t h e t r u e b e a r i n g o f t h e r a d i o s t a t i o n (TBR)
a r e added from t h e c o u r s e of t h e a i r c r a f t i n c o r r e s p o n d i n g s y s t e m s
o f c a l c u l a t i o n a n d t h e c o u r s e a n g l e of t h e r a d i o s t a t i o n :

2 84

1111

I .I11

Ill 111

Fig.

3 . 2 1 . Combined I n d i c a t o r f o r C o u r s e a n d C o u r s e
Angles of a Radio S t a t i o n .

Similarly,

TBR = TC

+

CAR;

MBR = MC

+

CAR.

/274

i n t h e o r t h o d r o m i c s y s t e m of
OBR = OC

+

calculating courses

CAR.

where OBR i s t h e o r t h o d r o m i c b e a r i n g o f t h e r a d i o s t a t i o n and OC
i s t h e orthodromic course.

I n determi- i n g t h e l o c a t i o n of t h e a i r c r a f t , t h e t r u e b e a r ­
i n g s of t h e a i r c r a f t ( T B A ) a r e p l o t t e d on t h e f l i g h t c h a r t f r o m
grBound r a d i o s t a t i o n s .
TBA = TBR

where 6 is t h e a n g l e of

+

6

+ l8Oo,

convergence of t h e meridians.

I n c a l c u l a t i n g t h e t r u e b e a r i n g of t h e a i r c r a f t , i S O o a r e d d d e d
if t h e TBA h a s a n u m e r i c a l v a l u e l e s s t h a n 1 8 0 ° a n d s u b t r a c t e d when
t h e v a l u e of t h e b e a r i n g exceeds 180°.

of

Consequently, i n c a l c u l a t i n g c o u r s e s from t h e t r u e m e r i d i a n
t h e a i r c r a f t ' s l o c a t i o n , where
285

+

TBA = TC

CAR

+

6

+
-

180°,

and i n c a l c u l a t i n g t h e course from t h e magnetic meridian

+

T B A = MC

AM

+

CAR

+

6

+

18'0O.

I t i s a s s u m e d t h a t t h e L A i s d e t e r m i n e d w i t h t h e a i d o f mag­
n e t i c compass d e v i a t i o n .
I n t h e orthodromic system of c a l c u l a t i n g courses,
TBA = OC

+

CAR

+

6

r.m.r.s.

+

-

180°,

w h e r e 6,.,.,,,.
i s t h e a n g l e of c o n v e r g e n c e of t h e m e r i d i a n s ( r e f ­
e r e n c e a n d r a d i o s t a t i o n ) w h i c h i s e q u a l t o (Xr,s.-X,,m.)sin~av.
L e t u s c o n s i d e r t h e means of s o l v i n g p r o b l e m s by means o f r a d i o c o m p a s s e s l o c a t e d on b o a r d a i r c r a f t , t a k i n g t h e r u l e s m e n t i o n e d
above i n t o account.

6
,


/$a+*
2$c

/

sv

CAR

must t a k e p l a c e w i t h a c o n s t a n t
t r u e b e a r i n g of t h e a i r c r a f t .
In
a f l i g h t from a r a d i o s t a t i o n ,
t h i s bearing is equal t o the azi­
muth o f t h e o r t h o d r o m e r e l a t i v e
t o t h e m e r i d i a n of t h e r a d i o sta-

For e x a m p l e , i n c a l c u l a t i n g t h e c o u r s e f r o m t h e m a g n e t i c m e r i d ­
i a n of t h e a i r c r a f t ' s l o c a t i o n , w e must s a t i s f y t h e e q u a t i o n :

286

c1

c1

ref

init
= MC

= MC

+

+

AM t 6 t CAR;

AM t 6

+

CAR

+ 180°,

where c 1 i n i t i s t h e t r u e b e a r i n g of t h e a i r c r a f t and “ref
bearing of t h e r a d i o s t a t i o n .

is the true

I n u s i n g a c o m b i n e d i n d i c a t o r for t h e b e a r i n g , t h e t o t a l c o r ­
r e c t i o n A = A M t 6 c a n b e e n t e r e d on t h e s c a l e of d e c l i n a t i o n .
Then
i t i s n e c e s s a r y t o s a t i s f y t h e c o n d i t i o n t h a t “ i n i t = TBA and t h a t
“ref = TBA t 180’.
The t r u e b e a r i n g o f t h e a i r c r a f t i s c a l c u l a t e d f r o m t h e o t h e r
e n d of t h e i n d i c a t o r p o i n t e r , i . e . , i n f l i g h t t o w a r d a r a d i o s t a t i o n ,
t h e d i r e c t r e a d i n g o f t h e p o i n t e r on t h e r a d i o c o m p a s s must b e e q u a l
t o t h e f i n a l a z i m u t h o f t h e o r t h o d r o m e , w h i l e i n f l i g h t away f r o m
a r a d i o s t a t i o n t h e o t h e r e n d o f t h e p o i n t e r m u s t show a r e a d i n g
equal t o t h e i n i t i a l azimuth of t h e orthodrome.
Inasmuch as t h e t o t a l c o r r e c t i o n ( A > o v e r t h e l e n g t h o f t h e
p a t h segment w i l l change c o n s t a n t l y , i t i s n e c e s s a r y t o d e t e r m i n e
t h i s c o r r e c t i o n i n each measurement i n o r d e r t o t a k e i n t o c o n s i d e r ­
a t i o n o t h e r i n d i c a t o r s or e n t e r i n g t h e d e c l i n a t i o n s o f t h e c o m b i n e d
This poses considerable d i f f i c u l t y i n using
i n d i c a t o r on t h e s c a l e .
t h e radiocompass i n f l i g h t .
The p r o b l e m i s s i m p l i f i e d c o n s i d e r a b l y i n a n o r t h o d r o m i c s y s ­
t e m of calculations f o r t h e aircraft course.
I n t h i s case
TBA = O B A

+

6r,m.r,s

where O B A i s t h e o r t h o d r o m i c b e a r i n g of

,

the aircraft.

The c o r r e c t i o n 6r.m.r.s. i s c o n s t a n t f o r a l l s t r a i g h t - l i n e
p a t h s e g m e n t s ; a f t e r s e t t i n g i t on t h e s c a l e o f d e v i a t i o n s , t h e
T B A c a n b e r e a d o f f i m m e d i a t e l y on t h e i n d i c a t o r o v e r t h e e n t i r e
s t r a i g h t - l i n e path segment.

Note.
If a r a d i o s t a t i o n i s l o c a t e d a t t h e s t a r t i n g p o i n t
of t h e r o u t e (SPR) with a r e f e r e n c e meridian, t h e f l i g h t can be
made d i r e c t l y a l o n g t h e o r t h o d r o m i c b e a r i n g o f t h e a i r c r a f t w i t h
a z e r o c o r r e c t i o n on t h e d e c l i n a t i o n s c a l e .
Correction f o r devi­
a t i o n o f m e r i d i a n s i s o n l y v a l u a b l e when t h e r a d i o s t a t i o n i s l o c a t e d
t o t h e s i d e o f t h e f l i g h t r o u t e or t h e m e r i d i a n o f t h e r a d i o s t a ­
t i o n does not coincide with t h e r e f e r e n c e meridian.
A s i n t h e case o f u s i n g a ground r a d i o d i r e c t i o n - f i n d e r i n
s e l e c t i n g t h e course t o be followed along s t r a i g h t - l i n e segments
of a p a t h b y means o f a r a d i o c o m p a s s , i t i s n e c e s s a r y t o b e g u i d e d
by t h e f o l l o w i n g r u l e s :

287

/276

(a)
I n a f l i g h t from t h e r a d i o s t a t i o n , w i t h a d r i f t of t h e
a i r c r a f t t o t h e r i g h t , t h e b e a r i n g i s i n c r e a s e d and t h e c o u r s e t o
be followed must b e reduced; with a d e c r e a s e i n t h e b e a r i n g , t h e
c o u r s e must be i n c r e a s e d .
(b)
I n t h e case o f a f l i g h t t o w a r d t h e r a d i o s t a t i o n , t h e
c o u r s e m u s t b e i n c r e a s e d when t h e b e a r i n g i n c r e a s e s a n d d e c r e a s e d
i f t h e aircraft bearing decreases.
I n a f l i g h t along a course determined by a r a d i o d i r e c t i o n
f i n d e r , i f t h e c o u r s e t o b e f o l l o w e d i s s e l e c t e d on t h e b a s i s o f
s t a b l e b e a r i n g s ShchDR or ShchDM, t h e c o u r s e i s c o n s i d e r e d t o h a v e
b e e n s e l e c t e d b y u s i n g t h e r a d i o c o m p a s s on b o a r d i f t h e c o u r s e a n g l e
of t h e r a d i o s t a t i o n remains c o n s t a n t .

For e x a m p l e , i n a f l i g h t t o w a r d a r a d i o s t a t i o n a t a c o n s t a n t
course of t h e a i r c r a f t , an i n c r e a s e i n t h e course angle of t h e r a d i o
s t a t i o n corresponds t o a d r i f t of t h e aircraft t o t h e l e f t ( t h e
TBA i n c r e a s e s ) .
In order f o r the a i r c r a f t t o follow a constant
When t h e
b e a r i n g , t h e c o u r s e o f t h e a i r c r a f t must b e i n c r e a s e d .
lead i n t h e course is equal i n value t o t h e d r i f t angle of t h e air­
c r a f t , t h e c o u r s e a n g l e of t h e r a d i o s t a t i o n w i l l r e m a i n c o n s t a n t
and e q u a l t o t h e d r i f t a n g l e b o t h i n v a l u e and i n s i g n .
I n s e l e c t i n g a c o u r s e , t h e same m e t h o d o f h a l f

correc tion is

used.

Example.
I t i s n e c e s s a r y t o make a f l i g h t t o w a r d a r a d i o s t a ­
t i o n w i t h a o r t h o d r o m i c p a t h a n g l e o f 82O.
L e t us assume t h a t t h e
d r i f t of t h e aircraft w i l l b e t o t h e l e f t w i t h i n l i m i t s of approx­
i m a t e l y loo. S e l e c t a c o u r s e t o b e f o l l o w e d b y u s i n g t h e m e t h o d
of h a l f c o r r e c t i o n .
In t h i s case, after f l y i n g over t h e turning point i n t h e r o u t e ,
it i s n e c e s s a r y t o assume an orthodromic c o u r s e o f 9 2 O .
The c o u r s e
a n g l e t h e of r a d i o s t a t i o n w i l l t h e n b e e q u a l t o 350°, which i s
e q u i v a l e n t t o a n u m e r i c a l v a l u e of - l o o .
I n a f l i g h t w i t h a c o u r s e of 9 2 " , i f t h e c o u r s e a n g l e o f t h e
r a d i o s t a t i o n i s i n c r e a s e d ( ? . e . , if w e a c q u i r e t h e v a l u e s o f 351,
3 5 2 , 353O i n s u c c e s s i o n ) , i t i s n e c e s s a r y t o p l a c e t h e a i r c r a f t
o n t h e l i n e o f f l i g h t a n d t o t a k e a l e a d o f 15O ( c o u r s e e q u a l s 9 7 O ,
C A R e q u a l s 345O).
L e t us assume t h a t t h i s l e a d t u r n s o u t t o be t o o g r e a t ; t h e n
t h e C A R b e g i n s t o d e c r e a s e , t a k i n g o n v a l u e s o f 3 4 4 , 3 4 3 , a n d 342O
T h e n , a f t e r a s e c o n d p l a c i n g of t h e a i r c r a f t on t h e p a t h , i t i s
n e c e s s a r y t o t a k e a n i n t e r m e d i a t e l e a d i n t h e c o u r s e o f 12-13O (CAR
If t h e CAR i s t o b e s t a b l e , i t i s n e c e s s a r y t o
e q u a l s 348-347O).
ensure t h a t t h e orthodromic bearing of t h e r a d i o s t a t i o n is equal
t o t h e orthodromic p a t h a n g l e and t o c o n t i n u e t h e f l i g h t w i t h t h e
selected course.

288

I

The c o u r s e t o f o l l o w e d i n a f l i g h t away f r o m a r a d i o s t a t i o n
i s s e l e c t e d i n t h e same m a n n e r , t h e o n l y d i f f e r e n c e b e i n g t h a t when
t h e CAR i n c r e a s e s , i t s h o u l d n o t b e r e d u c e d b u t i n c r e a s e d f u r t h e r .
I n using o t h e r i n d i c a t o r s , t h e s e l e c t i o n of t h e course is a l s o
a c c o m p l i s h e d by means o f s t a b l e p a t h a n g l e s o f r a d i o s t a t i o n s .
How-:
ever, i n order t o c o n t r o l t h e path of t h e aircraft i n t e r m s of direc­
t i o n , i t i s n e c e s s a r y t o d e t e r m i n e on e a c h o c c a s i o n t h e b e a r i n g
o f t h e a i r c r a f t or t h e r a d i o s t a t i o n b y s u m m i n g t h e c o u r s e a n d c o u r s e
angles of t h e r a d i o s t a t i o n , taking i n t o account t h e deviation of
t h e m e r i d i a n s o f t h e r a d i o s t a t i o n and a i r c r a f t and t h e magnetic
d e c l i n a t i o n a t t h e p o i n t where t h e a i r c r a f t i s l o c a t e d .
I t should be mentioned once again t h a t i n a f l i g h t toward a
r a d i o s t a t i o n , t h e s e l e c t e d s t a b l e course angle of t h e r a d i o sta­
t i o n i s always e q u a l t o t h e d r i f t angle of t h e a i r c r a f t , regard­

less of whether t h e aircraft i s
f l i g h t or a s a s l i g h t d e v i a t i o n
TAR = 350° c o r r e s p o n d s t o a d r i
from t h e r a d i o s t a t i o n , t h e d r i
C A R m i n u s 180°.

/277

l o c a t e d on t h e l i n e o f t h e d e s i r e d
from i t .
For e x a m p l e , a s t a b l e
f t a n g l e o f -loo.
I n f l i g h t away
f t angle i s always e q u a l t o a s t a b l e

The d r i f t a n g l e o f t h e a i r c r a f t c a n b e m e a s u r e d d i r e c t l y a f t e r
f l y i n g over t h e radio s t a t i o n .
A t t h e same t i m e , a f t e r f l y i n g o v e r t h e r a d i o s t a t i o n w i t h
any c o n s t a n t c o u r s e , t h e c o u r s e a n g l e s of t h e r a d i o s t a t i o n w i l l
be s t a b l e , s o t h a t
US = C A R

- 180'.

However, i n t h e m a j o r i t y o f c a s e s t h e d r i f t a n g l e i s d e t e r ­
m i n e d d u r i n g f l i g h t away f r o m a r a d i o s t a t i o n a s a d i f f e r e n c e b e t w e e n
t h e m a g n e t i c ( t r u e or o r t h o d r o m i c ) b e a r i n g o f t h e a i r c r a f t a n d t h e
m a g n e t i c ( t r u e or o r t h o d r o m i c ) c o u r s e o f t h e a i r c r a f t :
U S = MBA

-

MC,

U S = OBA

-

OC,

or

or

where OBA i s t h e o r t h o d r o m i c b e a r i n g o f t h e a i r c r a f t and t h e OC
is t h e orthodromic course of t h e aircraft.
I n t h i s case, w e can s i m u l t a n e o u s l y d e t e r m i n e t h e s i d e t o which
t h e a i r c r a f t d e v i a t e s ( l e f t or r i g h t ) b y c o m p a r i n g t h e g i v e n p a t h
a n g l e a n d t h e d e t e r m i n e d r a n g e of t h e a i r c r a f t i n t h e s y s t e m o f
2 89

coordinates being used;'the aircraft acquires t h e given l i n e of
f l i g h t according t o t h e c a l c u l a t e d course angle of t h e r a d i o s t a t i o n ,
w h i l e on t h e l i n e o f f l i g h t c o r r e c t i o n s a r e made t o t h e c o u r s e w h i c h
a r e & q u a l t o t h e a v e r a g e . a n g l e of d r i f t .
The m o n i t o r i n g of t h e a i r c r a f t p a t h i n t e r m s o f d i s t a n c e b y
means o f t h e r a d i o c o m p a s s i s a c c o m p l i s h e d w i t h p r ' e v i o u s l y c a l c u ­
lated bearings of the lateral radio station.
To d o t h i s , c o n t r o l l a n d m a r k s a r e m a r k e d o n a map a n d t h e b e a r ­
For
i n g s of r a d i o s t a t i o n s a r e d e t e r m i n e d f r o m t h e s e l a n d m a r k s .
t h e s a k e of convenience i n c a l c u l a t i o n , t h e bearings of t h e r a d i o
s t a t i o n s a r e d e t e r m i n e d i n t h e same s y s t e m i n w h i c h t h e c o u r s e o f
For e x a m p l e , f o r f l i g h t w i t h m a g n e t i c
t h e a i r c r a f t i s measured.
p a t h a n g l e s i t i s MBR a n d w i t h t r u e f l i g h t a n g l e s i t i s T B R , w h i l e
w i t h orthodromic a n g l e s i t i s OBR.
Then t h e b e a r i n g o f t h e r a d i o
s t a t i o n w i l l c o n s i s t of o n l y two c o m p o n e n t s , t h e a i r c r a f t c o u r s e
a n d t h e c o u r s e a n g l e o f t h e r a d i o s t a t i o n ; on t h e o t h e r i n d i c a t o r s ,
it can be e s t i m a t e d d i r e c t l y w i t h o u t any c o r r e c t i o n s from t h e scale
of t h e instrument.
In approaching a c o n t r o l landmark, t h e readings f o r t h e course
A t the
a n d t h e c o u r s e a n g l e of t h e r a d i o s t a t i o n a r e o b s e r v e d .
moment when t h e sum o f t h e a i r c r a f t c o u r s e a n d t h e c o u r s e a n g l e
o f t h e r a d i o s t a t i o n become e q u a l t o t h e p r e v i o u s l y c a l c u l a t e d b e a r ­
i n g ( i n c o m b i n e d i n d i c a t o r s , t h e b e a r i n g s o f r a d i o s t a t i o n s become
e q u a l t o t h e p r e v i o u s l y c a l c u l a t e d v a l u e ) , t h e moment f o r f l y i n g
/278
o v e r t h e landmark i s determined.
With t h e a i d o f a r a d i o c o m p a s s l o c a t e d on b o a r d , i t i s a l s o
p o s s i b l e t o d e t e r m i n e t h e l o c a t i o n o f t h e a i r c r a f t on t h e b a s i s
o f t r u e b e a r i n g s from two r a d i o s t a t i o n s .
However, t h e a c c u r a c y
o f d e t e r m i n i n g t h e a i r c r a f t l o c a t i o n by t h i s method, i n v o l v i n g c o n s i d ­
erable difficulty i n the process, is insufficiently higt.
There­
f o r e , t h e method i s n o t w i d e l y employed i n a i r c r a f t n a v i g a t i o n ,
being used only f o r determining approximate a i r c r a f t coordinates
i n f i n d i n g l o s t landmarks.
when t w o r a d i o The e s s e n c e o f t h e m e t h o d i s t h e f o l l o w i n g :
c o m p a s s e s a r e on b o a r d , one i s s e t t o t h e f r e q u e n c i e s o f two g r o u n d
r a d i o s t a t i o n s , l o c a t e d n o m o r e t h a n 1 8 0 - 2 0 0 km f r o m t h e a i r c r a f t .
I t i s d e s i r a b l e when d o i n g t h i s t o e n s u r e t h a t t h e b e a r i n g s o f t h e s e
r a d i o s t a t i o n s cross a t an angle close t o 90°.
If t h e i n d i c a t o r s o f t h e r a d i o c o m p a s s e s d o n o t a g r e e , t h e c o u r s e
, o f t h e a i r c r a f t , t h e c o u r s e a n g l e s o f t h e two r a d i o s t a t i o n s , and
t h e d i s t a n c e - f i n d i n g t i m e must a l l b e d e s c r i b e d s i m u l t a n e o u s l y f o r
a g i v e n moment o f t i m e .
Then t h e a p p r o x i m a t e t r u e b e a r i n g s o f t h e
aircraft are determined:

= MC

+

AM

T B A 2 = MC

+

AM

TBAl

+
+

CAR1
CAR2

+
t

180°;

180O.

29 0

1 1 1 1

I 11111111111.

I

Ill

I1

I

T h e b e a r i n g s w h i c h h a v e b e e n o b t a i n e d a r e p l o t t e d on t h e f l i g h t
c h a r t from t h e m e r i d i a n s o f s e l e c t e d r a d i o s t a t i o n s by means of
a p r o t r a c t o r and scale r u l e .
Having t h u s d e t e r m i n e d t h e a p p r o x i m a t e p o s i t i o n o f t h e a i r ­
c r a f t , w e c a n f i n d i t s t r u e b e a r i n g by i n t r o d u c i n g t h e p r e c i s e v a l u e
of t h e m a g n e t i c d e c l i n a t i o n a n d m a k i n g c o r r e c t i o n s t o t h e d e v i a t i o n
a n g l e s of t h e meridians o f t h e r a d i o s t a t i o n and t h e aircraft loca­
tion.
T h e s e c o r r e c t e d b e a r i n g s a r e a g a i n p l o t t e d on t h e c h a r t t o
g i v e a p r e c i s e p o s i t i o n o f t h e a i r c r a f t a t t h e moment o f d i r e c t i o n
finding.
If o n l y o n e r a d i o c o m p a s s i s m o u n t e d on b o a r d t h e a i r c r a f t , i t
i s n e c e s s a r y t o c o n s i d e r i t s p a t h when d e t e r m i n i n g t h e l o c a t i o n
o f t h e a i r c r a f t f o r t h e t i m e b e t w e e n t h e moments o f d i r e c t i o n f i n d i n g ,
and t h i s i s done as f o l l o w s ( F i g . 3 . 2 3 ) .

A f t e r d e t e r m i n i n g t h e a v e r a g e b e a r i n g s of t h e a i r c r a f t , t h e
l a t t e r a r e p l o t t e d on a c h a r t , a n d t h e n t h e f l i g h t p a t h o f t h e a i r ­
c r a f t i s o b t a i n e d f r o m t h e p o i n t o f l o c a t i o n of t h e f i r s t r a d i o
s t a t i o n f o r t h e time between t h e measurements of t h e c o u r s e a n g l e s
of t h e r a d i o s t a t i o n i n a d i r e c t i o n which c o i n c i d e s w i t h t h e c o u r s e
of t h e a i r c r a f t .
A l i n e i s drawn t h r o u g h t h e p o i n t which i s o b t a i n e d ,
p a r a l l e l t o t h e f i r s t b e a r i n g up t o t h e i n t e r s e c t i o n w i t h t h e l i n e
of t h e second bearing, defining t h e p o s i t i o n of t h e aircraft a t
t h e moment o f d i r e c t i o n f i n d i n g f o r t h e s e c o n d r a d i o s t a t i o n .
In
a d d i t i o n , t h e t r u e b e a r i n g s o f t h e a i r c r a f t a r e f o u n d i n t h e same
way a s i n t h e c a s e o f t w o r a d i o c o m p a s s e s .

The l a b o r i o u s n e s s o f t h e p r o c e s s o f d e t e r m i n i n g t h e p o s i t i o n
o f t h e a i r c r a f t i s c o n s i d e r a b l y r e l i e v e d i f t h e f l i g h t i s made w i t h
o r t h o d r o m i c c o u r s e s , b u t t h e i n d i c a t o r s of t h e r a d i o compasses must
/279
match.
I n t h i s c a s e , t h e a n g l e s of convergence of t h e meridians
o f t h e r a d i o s t a t i o n s w i t h t h e r e f e r e n c e m e r i d i a n for c a l c u l a t i n g
t h e course a r e determined beforehand.
A t t h e t i m e of measurement, t h e a n g l e of d e v i a t i o n o f t h e merid­
i a n s f o r t h e f i r s t and t h e n t h e second r a d i o s t a t i o n a r e e n t e r e d
on t h e s c a l e o f d e v i a t i o n s
of t h e i n d i c a t o r i n s u c c e s s i o n .
They a r e c a l c u l a t e d f r o m t h e
r e a d i n g s of t h e o p p o s i t e e n d s
of t h e p o i n t e r s o f t h e r a d i o
compasses and are d e s i g n a t e d
as T B S l and TBS2.
The b e a r ­
i n g s o b t a i n e d are f i n a l and
no c o r r e c t i o n s a r e r e q u i r e d .

F i g . 3.23.
Diagram f o r L o c a t i n g
t h e P o s i t i o n of an Aircraft from
t h e B e a r i n g s o f Two R a d i o S t a ­
stions.

A s we have already mentioned,
when u s i n g r a d i o c o m p a s s e s m o u n t e d on b o a r d a i r c r a f t , t h e d r i f t
angle of t h e a i r c r a f t is

291

d e t e r m i n e d f r o m t h e s t a b l e c o u r s e a n g l e s o f t h e r a d i o s t a t i o n s or
measured a f t e r f l y i n g o v e r a r a d i o s t a t i o n .
The g r o u n d s p e e d o f t h e a i r c r a f t i s d e t e r m i n e d b y c h e c k i n g
t h e f l i g h t i n t e r m s o f d i s t a n c e b y means o f r a d i o s t a t i o n s l o c a t e d
t o t h e s i d e o f , t h e c o u r s e or f r o m t h e m o m e n t s when t h e a i r c r a f t p a s s e s
over radio stations.
The l a t t e r m e t h o d i s n o t a c c u r a t e , e s p e c i a l l y
when f l y i n g a t h i g h a l t i t u d e s , d u e t o t h e e r r o r i n t h e r e a d i n g s
o f r a d i o c o m p a s s e s when f l y i n g o v e r r a d i o s t a t i o n s .
The d e t e r m i n a t i o n o f t h e w i n d a t f l i g h t a l t i t u d e i s accomp­
l i s h e d on t h e b a s i s o f t h e g r o u n d s p e e d of t h e a i r c r a f t , t h e a i r ­
s p e e d , a n d t h e d r i f t a n g l e b y t h e same m e t h o d s a s f o r g r o u n d r a d i o
distance-finders.
The method o f d e t e r m i n i n g t h e wi n d b y u s i n g t h e
s u c c e s s i v e p o s i t i o n s o f t h e a i r c r a f t , o b t a i n e d by d i s t a n c e m e a s u r e - '
ment f r o m two r a d i o s t a t i o n s , u s u a l l y i s n o t u s e d d u e t o t h e i n a d ­
equate precision of t h e determination of t h e aircraft location.

S p e c i a l F e a t u r e s o f U s i n g R a d i o c o m p a s s e s on Board A i r c r a f t
a t High A l t i t u d e s and F l i g h t S p e e d s
High a l t i t u d e s a n d f l i g h t s p e e d s c a u s e d e t e r i o r a t i o n o f t h e
c o n d i t i o n s for u s i n g r a d i o c o m p a s s e s a b o a r d a i r c r a f t f o r p u r p o s e s
of aircraft navigation.
The u s e o f

radiocompasses and t h e o b s e r v a t i o n of a l l r u l e s

for r e t a i n i n g a c c u r a c y o f d i s t a n c e f i n d i n g i s a l a b o r i o u s p r o c e s s ,
s o t h a t t h e i n c r e a s e i n f l i g h t s p e e d , c a l l i n g f'or o p e r a t i v e n e s s
of navigational c a l c u l a t i o n s , creates d i f f i c u l t i e s i n using radioc o m p a s s e s on b o a r d t h e a i r c r a f t .
This shortcoming can b e l a r g e l y overcome by u s i n g combinations
/280
of b e a r i n g i n d i c a t o r s , e s p e c i a l l y i n t h e orthodromic system of calcu­
lating aircraft courses.
I n a d d i t i o n , another shortcoming of aircraft radiocompasses
w h i c h o p e r a t e o n medium a n d s h o r t w a v e s , d u e t o t h e i n c r e a s e d s p e e d
o f f l i g h t , i s t h e e f f e c t o f e l e c t r o s t a t i c n o i s e on t h e i r o p e r a t i o n .
A t high a i r s p e e d s , e s p e c i a l l y i n clouds and i n p r e c i p i t a t i o n ,
a c o n s i d e r a b l e e l e c t r i f i c a t i o n of t h e a i r c r a f t s u r f a c e s o c c u r s .
Static e l e c t r i c i t y , emitted a t pointed portions of the aircraft
( i n c l u d i n g open a n t e n n a s ) creates n o i s e and r a d i o i n t e r f e r e n c e i n
t h e frequency r a n g e a t which radiocompasses o p e r a t e .
Despite the
m e a s u r e s w h i c h a r e t a k e n t o p r e v e n t t h e c h a r g e s f r o m f l o w i n g by
u s i n g s p e c i a l d i s c h a r g e d e v i c e s , as w e l l as s h i e l d i n g t h e open a n t e n n a s
o f t h e r a d i o c o m p a s s e s , t h i s s h o r t c o m i n g c a n b e overcome o n l y p a r t i a l l y
and m a n i f e s t s i t s e l f i n very d i f f i c u l t f l i g h t c o n d i t i o n s .

High f l i g h t a l t i t u d e s h a v e a n e f f e c t m a i n l y on t h e a c c u r a c y
o f o p e r a t i o n of r a d i o c o m p a s s e s a n d e s p e c i a l l y on t h e a c c u r a c y of
29 2

I


d e t e r m i n i n g t h e moment when t h e a i r ­
craft f l i e s over a r a d i o s t a t i o n .

.

yb('!3-_
1q
\
b

a

ca,

br

1

i
I
II
I

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1 ;
I

I
\
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/"\

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'.-A'

,
.
\

I

/

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Fig. 3.24.
O p e r a t i o n of
a n Open A n t e n n a When F l y i n g P a s t a Radio S t a t i o n .

The d e c r e a s e i n t h e a c c u r a c y
o f o p e r a t i o n t a k e s p l a c e due t o t h e
change i n t h e n a t u r e of r a d i o d e v i a t i o n
a t d i f f e r e n t a n g l e s o f d e v i a t i o n of
t h e p r o p a g a t i o n v e c t o r o f r a d i o waves.
The l a t t e r c h a n g e s w i t h i n w i d e l i m i t s
when t h e a i r c r a f t a p p r o a c h e s t h e l o c a ­
t i o n of a r a d i o s t a t i o n .
A d i a g r a m of t h e a p p e a r a n c e o f
e r r o r s i n d e t e r m i n i n g t h e moment w h e n
t h e a i r c r a f t f l i e s over a ground r a d i o
s t a t i o n i s shown i n F i g u r e 3 . 2 4 w h e r e
there is a picture of the electrical
f i e l d r a d i a t e d by a n open v e r t i c a l
a n t e n n a on a g r o u n d r a d i o s t a t i o n .

A t l a r g e d i s t a n c e s from t h e r a d i o s t a t i o n , t h e e l e c t r o m a g n e t i c
wave i s v e r t i c a l l y p o l a r i z e d .
However, t h e r e i s a s p a c e n e a r t h e
r a d i o s t a t i o n and above it w h e r e t h e p o l a r i z a t i o n s h i f t s t o t h e
h o r i z o n t a 1 , t h e n back t o t h e v e r t i c a l b u t i n opposite phase.
L e t us assume t h a t a n open a n t e n n a o f t h e r a d i o c o m p a s s i s t i l t e d
backward ( p o s i t i o n a , F i g . 3.24) and t h e a i r c r a f t i s approaching
t h e r a d i o s t a t i o n a t a high f l i g h t a l t i t u d e i n t h e d i r e c t i o n of
v e c t o r iT.

Obviously, a t p o s i t i o n a t h e antenna w i l l have zero r e c e p t i o n .
The r e c e p t i o n o f t h e a n t e n n a w i l l t h e n i n c r e a s e , b u t i n a p h a s e
This leads
/281
which i s o p p o s i t e t o t h e r e c e p t i o n up t o t h e p o i n t a .
t o a r o t a t i o n o f t h e r a d i o c o m p a s s f r a m e b y 180° u n t i l t h e a i r c r a f t
passes a radio st a t i o n .
Then, a f t e r p a s s i n g t h e s t a t i o n , t h e phases
of b o t h t h e frame a n d open a n t e n n a s c h a n g e a l m o s t s i m u l t a n e o u s l y
( a t t h e p o i n t all.
T h u s , t h e c h a n g e i n t h e r e a d i n g s o f t h e r a d i o c o m p a s s b y 180°
t a k e s p l a c e u n t i l t h e moment w h e n t h e a i r c r a f t p a s s e s o v e r t h e r a d i o
s t a t i o n ( a t point a ) and only t h e o s c i l l a t i o n of t h e needle w i l l
b e o b s e r v e d from t h e n on.

Fig. 3.25.
Equivalent of an
Open A n t e n n a o n B o a r d a n A i r ­
craft.

2 9.3

When t h e a n t e n n a i s ' t i l t e d f o r w a r d ( p o s i t i o n c ) , t h e o s c i l ­
l a t i o n s of t h e radiocompass n e e d l e w i l l b e g i n a t p o i n t c , w h i l e
t h e passage by t h e r a d i o s t a t i o n with r o t a t i o n of t h e needle through
180° w i l l b e n o t e d a t p o i n t c l , i . e . , t h e r e w i l l b e a d e l a y i n m a r k i n g
t h e passage.
F o r a s t r i c t l y v e r t i c a l a n t e n n a ( p o s i t i o n b ) , ' t h e movement
Then t h e
o f t h e n e e d l e t h r o u g h 180° c a n t a k e p l a c e p r e m a t u r e l y .
p o i n t e r c a n make a r e v e r s e t u r n a n d a g a i n s h o w t h e p a s s a g e b y t h e
radio station at point bl.

I t s h o u l d be m e n t i o n e d t h a t an e q u i v a l e n t t o t h e open a n t e n n a
o f t h e r a d i o c o m p a s s i n terms of i t s i n c l i n a t i o n i n t h e v e r t i c a l
p l a n e i s t h e r e s u l t a n t c o m b i n i n g t h e u p p e r or l o w e r p o i n t s o f t h e
antenna with t h e e l e c t r i c a l center of t h e aircraft, constituting
i t s g r o u n d i n g or c o u n t e r w e i g h t ( F i g . 3 . 2 5 ) .
I n t h i s f i g u r e , p o i n t 1 i s t h e t o p of t h e open a n t e n n a , p o i n t
2 i s t h e r e c e i v e r and p o i n t 3 i s t h e e l e c t r i c a l c e n t e r of t h e a i r ­
craft.
Obviously, s t r a i g h t l i n e 1-3 is t h e equivalent
a t i o n o f t h e open a n t e n n a , which i s f o r w a r d i n t h i s
t h e s e t t i n g o f t h e open a n t e n n a above t h e f u s e l a g e i
s e c t i o n c a u s e s a d e l a y i n t h e r e a d i n g o f t h e moment
s t a t i o n is passed.

of t h e i n c l i n ­
case.
Thus,
n i t s forward
when t h e r a d i o

M o u n t i n g o f t h e a n t e n n a i n t h e same p o s i t i o n w i t h r e s p e c t t o
t h e c e n t e r b u t below t h e f u s e l a g e l e a d s t o a p r e l i m i n a r y r e a d i n g
o f t h e moment w h e n t h e a i r c r a f t p a s s e s t h e r a d i o s t a t i o n .
The oppo­
s i t e p i c t u r e i s o b s e r v e d when t h e a n t e n n a i s mo u n t ed b e h i n d t h e
e l e c t r i c a l c e n t e r of t h e a i r c r a f t .
The b e s t p l a c e t o m o u n t t h e a n t e n n a i s a b o v e or b e l o w t h e e l e c ­
t r i c a l c e n t e r o f t h e a i r c r a f t , b u t i n t h i s c a s e a n a d v a n c e or d e l a y
i n t h e r e a d i n g s i s o b s e r v e d ; i n some c a s e s , t h e s e d e v i a t i o n s d e p e n d
on t h e h e i g h t a n d s p e e d of f l i g h t , b u t t h e r e c a n a l s o b e a d o u b l e
r e a d i n g involving both an advance and delayed i n d i c a t i o n .
This system f o r t h e c r e a t i o n of e r r o r s i n measuring a f l i g h t
only approximately r e f l e c t s t h e reasons f o r these e r r o r s .
In prac­
t i c e , t h e y w i l l d e p e n d b o t h on t h e a n g l e of p i t c h on t h e a i r c r a f t
a n d on t h e a c c u r a c y w i t h w h i c h t h e p a s s a g e of t h e a i r c r a f t o v e r
t h e r a d i o s t a t i o n is determined.

For e x a m p l e , i f t h e a i r c r a f t i s p a s s i n g a r a d i o s t a t i o n t o
t h e s i d e , t h e n o b v i o u s l y t h e r e will n o t b e a n i n d i c a t i o n o f p a s s ­
a g e w i t h movement o f t h e n e e d l e t h r o u g h 1 8 0 ° , b u t a d e t e r i o r a t i o n
i n t h e passage over t h e r a d i o s t a t i o n , i . e . , e r r o r s i n determini n g t h e passage of t h e t r a v e r s e of t h e r a d i o s t a t i o n w i l l be prac­
t i c a l l y non-existent.

29 4

/282

Usually, t h e exact passage of an aircraft over a r a d i o sta­
t i o n occurs only i n s p e c i a l and e x c e p t i o n a l c o n d i t i o n s .
Therefore,
i n p r a c t i c e t h e r e i s always a c o n s i d e r a t i o n of t h e effect of passage
w i t h t h e e f f e c t o f e r r o r , w h i c h d o e s n o t make i t p o s s i b l e t o c o n ­
s i d e r t h e m a g n i t u d e of t h e d e l a y a d v a n c e i n m a r k i n g t h e p a s s a g e .
D e p e n d i n g on t h e t y p e o f a i r c r a f t a n d t h e f l i g h t c o n d i t i o n s ,
these e r r o r s can occur w i t h i n l i m i t s e q u a l t o 1-3 f l i g h t a l t i t u d e s ,
e x c l u d i n g t h e case o f e x a c t d e t e r m i n a t i o n .
However, e x a c t d e t e r ­
m i n a t i o n c a n o c c u r a t d i s t a n c e s which e x c e e d t h e f l i g h t a l t i t u d e
of t h e a ' i r c r a f t , beyond t h e l i m i t s of a zone w i t h h o r i z o n t a l p o l a r ­
i z a t i o n , i . e . , a t v e r y c o n s i d e r a b l e d e v i a t i o n s of t h e a i r c r a f t f r o m
t h e g i v e n l i n e of f l i g h t .

D e t a i Z s o f U s i n g R a d i o c o m p a s s e s i n Making Maneuvers i n
t h e V i c i n i t y o f t h e A i r p o r t a t Which a L a n d i n g is
t o b e Made
The m a n e u v e r o f a p p r o a c h i n g f o r a l a n d i n g u s u a l l y b e g i n s a t
a r e l a t i v e l y l o w f l i g h t a l t i t u d e ( 1 2 0 0 - 4 0 0 0 m) w i t h a g r a d u a l r e d u c ­
t i o n of t h e a i r s p e e d .
Therefore, the e f f e c t s related t o height
and f l i g h t speed i n t h i s c a s e are c o n s i d e r a b l y reduced.
U n l i k e a f l i g h t a l o n g t h e r o u t e , where t h e main o p e r a t i o n i n
a i r c r a f t n a v i g a t i o n i n v o l v i n g t h e u s e o f r a d i o c o m p a s s e s i s meas­
u r i n g o f b e a r i n g s , t h e work i s s h i f t e d when m a n e u v e r i n g t h e a i r ­
craft mainly t o measuring course angles of t h e r a d i o s t a t i o n s a t
i n d i v i d u a l p o i n t s a l o n g t h e maneuver.
This i s p r o f i t a b l e i n t h i s
respect:
r e g a r d l e s s o f t h e o r i e n t a t i o n of t h e l a n d i n g s t r i p , and
consequently of t h e course of t h e aircraft a t d i f f e r e n t s t a g e s of
t h e m a n e u v e r , t h e s y s t e m f o r u s i n g r a d i o c o m p a s s e s i s b a s e d on two
or t h r e e s t a n d a r d s w h i c h a r e u s e d a t a l l a i r p o r t s .
O f course, i f t h e r e i s a tendency t o d r i f t i n t h e a i r c r a f t
c o u r s e , t h e c o r r e s p o n d i n g c o r r e c t i o n s m u s t b e made i n t h e r e a d i n g s
of t h e c o u r s e a n g l e o f t h e r a d i o s t a t i o n wh i ch a r e e q u a l i n magni­
t u d e and s i g n t o t h e l e a d which h a s b e e n t a k e n .

The e f f e c t i v e n e s s o f u s i n g r a d i o c o m p a s s e s i n t h e v i c i n i t y o f
a i r p o r t s w h e r e l a n d i n g s a r e t o made i s i n c r e a s e d a l s o b y t h e f a c t
t h a t t h e f l i g h t i s made a t s h o r t d i s t a n c e s f r o m g r o u n d r a d i o s t a ­
t i o n s , which g i v e s r e l a t i v e l y s m a l l l i n e a r e r r o r s i n d e t e r m i n i n g
t h e p o s i t i o n of t h e a i r c r a f t i n view o f t h e e r r o r s a l r e a d y committed
i n measuring t h e course angles of t h e r a d i o s t a t i o n .
In addition,
it i s no l o n g e r n e c e s s a r y t o c a l c u l a t e t h e magnetic d e c l i n a t i o n
a n d t h e d e v i a t i o n a n g l e s of t h e m e r i d i a n s .
The a c c u r a c y o f a i r ­
c r a f t n a v i g a t i o n i n t h e v i c i n i t y o f t h e aerodrome, u s i n g radiocom­
passes, i s considered t o be q u i t e s a t i s f a c t o r y i n a l l s t a g e s of
t h e maneuver w i t h t h e f o l l o w i n g e x c e p t i o n s :
(a>

D e t e r m i n a t i o n o f t h e s t a r t i n g p o i n t o f t h e maneuver by

29 5

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F

i

t

f l y i n g p a s t t h e power r a d i o s t a t i o n , if t h e maneuver i s b e g i n n i n g
a t a high a l t i t u d e .
(b)
On a l a n d i n g s t r i p , w h e r e i t i s n e c e s s a r y t o h a v e v e r y
h i g h a c c u r a c y of f l i g h t a l o n g a g i v e n t r a j e c t o r y f o r b r i n g i n g t h e
aircraft i n f o r a landing.
U 1 t r a - S h o r t w a v e Gon i ome t r i c a n d Gon i ome t r i c - R a n g e F i n d i n g
Systems

A s w e have mentioned, radiocompasses have s i g n i f i c a n t advan­
t a g e s o v e r ground r a d i o d i s t a n c e - f i n d e r s with r e s p e c t t o u n i n t e r ­
r u p t e d v i s u a l i n f o r m a t i o n on b o a r d t h e a i r c r a f t r e g a r d i n g i t s p o s i ­
tion.
T h i s means t h a t t h e y h a v e b e e n v e r y w i d e l y e m p l o y e d a n d a r e
i n s t a l l e d i n p r a c t i c a l l y a l l t y p e s o f a i r c r a f t as a r u l e i n a d o u b l e
set.
I n a d d i t i o n , t h e r e a r e a number o f i m p o r t a n t s h o r t c o m i n g s
f o r r a d i o c o m p a s s e s m o u n t e d on b o a r d a i r c r a f t , w h i c h r e d u c e t h e a c c u r ­
acy and f e a s i b i l i t y of a i r c r a f t n a v i g a t i o n .
I n a d d i t i o n t o t h e e r r o r s c a u s e d by t h e e f f e c t o f t h e l o c a l
r e l i e f , which a f f e c t a l l systems f o r s h o r t - r a n g e n a v i g a t i o n , r a d i o compasses have t h e f o l l o w i n g shortcomings:
(a)
Unfavorable c o n d i t i o n s f o r d i r e c t i o n a l r e c e p t i o n of elec­
t r o m a g n e t i c waves on b o a r d t h e a i r c r a f t ( r a d i o d e v i a t i o n o f a n u n s t a b l e
nature) ;
(b)
An i n c r e a s e i n t h e e r r o r s i n d i s t a n c e f i n d i n g d u e t o i n a c ­
c u r a t e measurements of t h e a i r c r a f t c o u r s e ;
(c)
The n e c e s s i t y t o c o n s i d e r t h e d e v i a t i o n o f t h e m e r i d i a n s
a n d m a g n e t i c d e c l i n a t i o n s when u s i n g m a g n e t i c c o m p a s s e s t o d e t e r ­
mine b e a r i n g s ;
(d)
The e f f e c t o f s t a t i c n o i s e i n t h e r a n g e o f r e c e i v e d r a d i o
frequencies a t high airspeeds ;

( e ) T h e e f f e c t o f f l i g h t a l t i t u d e o n t h e a c c u r a c y o f meas­
u r i n g t h e r a n g e a n d d e t e r m i n i n g t h e moment o f f l y i n g o v e r t h e r a d i o
stations.
I n a d d i t i o n t o t h e s e shortcomings, radiocompasses are s u b j e c t
t h e need t o p l o t
t o a g e n e r a l d i s a d v a n t a g e of g o n i o m e t r i c s y s t e m s :
b e a r i n g s on t h e c h a r t f r o m t w o g r o u n d p o i n t s t o d e t e r m i n e t h e l o c a ­
t i o n of t h e aircraft.
T h e r e f o r e i t seems n a t u r a l t o t r y t o b u i l d d e v i c e s f o r s h o r t r a n g e r a d i o n a v i g a t i o n which would have t h e advantages of t h e r a d i o c o m p a s s e s m o u n t e d on a i r c r a f t b u t w o u l d n o t h a v e t h e s h o r t c o m i n g s
from which t h e y s u f f e r .
Such d e v i c e s a r e t h e g o n i o m e t r i c a n d g o n i o m e t r i c - r a n g e f i n d ­
i n g s y s t e m s w h i c h o p e r a t e on u l t r a - s h o r t w a v e s .
296

A common f e a t u r e o f t h e s e s y s t e m s i s t h e d i r e c t i o n a l r a d i a t i o n
o f e l e c t r o m a g n e t i c waves by g r o u n d i n s t r u m e n t s and t h e i r d i r e c t i o n a l
This f e a t u r e , together with t h e
r e c e p t i o n on b o a r d t h e a i r c r a f t .
r a n g e o f waves employed, g i v e s t h r e e v e r y i m p o r t a n t a d v a n t a g e s f o r
navigational systems:
(1)

I t f r e e s t h e s y s t e m f r o m r a d i o d e v i a t i o n s on b o a r d ;

(2)
The b e a r i n g o f t h e a i r c r a f t becomes i n d e p e n d e n t o f t h e
aircraft course, magnetic d e c l i n a t i o n , and d e v i a t i o n of meridians;
(3)
I t sharply i n c r e a s e s t h e freedom of t h e system
and atmospheric i n t e r f e r e n c e .

/284

from s t a t i c

There are s e v e r a l t y p e s of goniometric d i r e c t i o n a l r a d i o bea­
cons and r e c e i v i n g d e v i c e s t o c a r r y aboard a i r c r a f t , which o p e r a t e
on u l t r a - s h o r t w a v e s .
The u s u a l p r i n c i p l e o f o p e r a t i o n f o r t h e s e s y s t e m s i s t h e r o t a ­
t i o n of the d i r e c t i o n a l c h a r a c t e r i s t i c of t h e r a d i a t i o n of t h e trans­
m i t t i n g a n t e n n a of a g r o u n d i n s t a l l a t i o n w i t h i t s r e c e p t i o n a b o a r d
t h e a i r c r a f t by a n open ( o m n i d i r e c t i o n a l ) a n t e n n a f o r a t r a n s m i t t e r
of r e f e r e n c e s i g n a l s , r e l a t e d t o t h e p a s s a g e of t h e c h a r a c t e r i s t i c
of t h e a n t e n n a t h r o u g h t h e s t a r t i n g p o i n t f o r m e a s u r i n g b e a r i n g s ,
e . g . , t h r o u g h t h e n o r t h e r n d i r e c t i o n o f t h e m a g n e t i c or t r u e m e r i d ­
i a n of t h e b e a c o n .
F i g u r e 3 . 2 6 shows t h e s c h e m a t i c d i a g r a m of a r a d i o b e a c o n w i t h
a r o t a t i n g d i r e c t i o n a l antenna.
The g e n e r a t o r of low f r e q u e n c i e s
p r o d u c e s a f r e q u e n c y which i s s y n c h r o n i z e d w i t h t h e r o t a t i o n o f
t h e d i r e c t i o n a l antenna.
On t h e a x i s o f r o t a t i o n o f t h e a n t e n n a
i s a s p e c i a l d i s k , which g e n e r a t e s t h e r e f e r e n c e s i g n a l r e l a t e d
t o t h e p o s i t i o n of t h e r o t a t i n g antenna.
The r e f e r e n c e s i g n a l p a s s e s
t h r o u g h t h e m o d u l a t o r and t r a n s m i t t e r t o r e a c h t h e open a n t e n n a
of t h e r a d i o b e a c o n .

­

trans
mitter
1

modu­
call
-Eiato+
signal

Fig. 3.26.
S c h e m a t i c Diagram o f a R a d i o Beacon
With R o t a t i n g D i r e c t i o n a l A n t e n n a .
29 7

From t h e t r a n s m i t t e r , t h e s i g n a l r e a c h e s t h e r o t a t i n g a n t e n n a
t h r o u g h a m o d u l a t i o n s u p p r e s s o r , s o t h a t t h e a m p l i t u d e of t h e s i g n a l
r a d i a t e d b y t h e a n t e n n a i n a n y g i v e n d i r e c t i o n d e p e n d s o n l y on t h e
position of the antenna r e l a t i v e t o this. direction.
Consequently,
t h e s i g n a l from a d i r e c t i o n a l a n t e n n a is modulated by a l o w fre­
q u e n c y w h o s e p h a s e r e l a t i v e t o t h e r e l a t i v e s i g n a l i’s s h i f t e d t h r o u g h
a n a n g l e e q u a l t o t h e a z i m u t h of t h e a i r c r a f t .
T h u s , two s i g n a l s r e a c h t h e a i r c r a f t on t h e c a r r i e r f r e q u e n c y
i n addition t o the call-letter signals:

(1)

Reference s i g n a l f o r beginning t h e reading.

/285

(2)
The s i g n a l f r o m t h e d i r e c t i o n a l a n t e n n a , whose a m p l i t u d e
maximum c o i n c i d e s w i t h t h e moment when t h i s a n t e n n a c r o s s e s t h e
l i n e t o the aircraft.
The r e c e i v e r o n t h e a i r c r a f t h a s t h r e e c h a n n e l s ( F i g .

(1)
(earphones

3.27):

C h a n n e l f o r p i c k i n g up t h e c a l l l e t t e r s f r o m t h e b e a c o n
;

(2)

Reference-channel;

(3)

Azimuthal v o l t a g e channel.

earphone
channel

I

refer-.
vo 1ta g e
channel

receiver

azimuthal
II
azimuthal-’
voltage

E

I



channel
I

phase
di sc r i m anator

I

Fig. 3.27.
Apparatus f o r Goniometric System
Aboard a n A i r c r a f t .
The i n d i c a t o r m e c h a n i s m i s u s u a l l y b a s e d o n m e a s u r e m e n t o f
t h e p h a s e r a t i o o f t h e r e f e r e n c e a n d a z i m u t h a l s i g n a l s w i t h low
f r e q u e n c y by t h e c o m p e n s a t i o n m e t h o d , i . e . , t h e p h a s e o f t h e r e f e r ­
e n c e s i g n a l i s changed by a n a u t o m a t i c p h a s e s h i f t e r s o t h a t i t
298

c o i n c i d e s w i t h t h e p h a s e of t h e a z i m u t h a l s i g n a l .
I n t h i s case,
t h e s i g n a l on t h e p h a s e d i s c r i m i n a t o r w i l l b e e q u a l t o z e r o w h i l e
t h e b e a r i n g i n d i c a t o r on t h e a i r c r a f t w i l l
a c t as t h e p o i n t e r o f
t h e phase s h i f t e r .

to a
(and
only
line

For f l i g h t a l o n g a g i v e n b e a r i n g , t h e p h a s e s h i f t e r i s s e t
g i v e n p o s i t i o n , s o t h a t t h e s i g n a l on t h e p h a s e d i s c r i m i n a t o r
t h e r e f o r e on t h e z e r o i n d i c a t o r d e v i c e ) w i l l b e e q u a l t o z e r o
i n t h e case when t h e a i r c r a f t i s l o c a t e d e x a c t l y a l o n g t h e
of the desired bearing.

The g o n i o m e t r i c s y s t e m f o r s h o r t - r a n g e n a v i g a t i o n , o p e r a t i n g
o n u l t r a s o n i c w a v e s , i n t h e c a s e when t h e c h a r a c t e r i s t i c o f t h e
r o t a t i n g d i r e c t i o n a l a n t e n n a h a s a s h a r p l y p r o n o u n c e d maximum ( w h i c h
u s u a l l y i s a c h i e v e d b y u s i n g r e f l e x r e f l e c t o r s ) , c a n b e b u i l t on
t h e p r i n c i p l e of t i m e r a t i o s r a t h e r than phase r a t i o s .
I n t h i s c a s e , t h e r e f e r e n c e s i g n a l , when t h e r o t a t i n g d i r e c ­
t i o n a l antenna passes through zero reading, has a pulsed character,
a n d t h e e q u i p m e n t on b o a r d must i n c l u d e a g e n e r a t o r o f a r e f e r e n c e
f r e q u e n c y as w e l l as s p e c i a l d e l a y d e v i c e s t o d e t e r m i n e t h e b e a r i n g
o f t h e a i r c r a f t i n t i m e b e t w e e n t h e m o m e n t s when t h e r e f e r e n c e a n d
azimu'thal s i g n a l s are r e c e i v e d .
The g e o m e t r y o f n a v i g a t i o n a l a p p l i c a t i o n s o f U S W b e a c o n s w i t h
d i r e c t i o n a l r a d i a t i o n i s e x a c t l y t h e same a s t h e u s e o f g r o u n d r a d i o
All t h e problems of a i r c r a f t n a v i g a t i o n such as
distancefinders.
s e l e c t i o n of t h e course t o be followed, monitoring of t h e path f o r
d i s t a n c e and d i r e c t i o n , determination of t h e l o c a t i o n of t h e a i r ­
c r a f t f r o m two b e a c o n s , measurement o f t h e d r i f t a n g l e and g r o u n d
s p e e d , measurement o f t h e wind a t f l i g h t a l t i t u d e , e t c . , a r e s o l v e d
In
i n e x a c t l y t h e same way a s f o r g r o u n d r a d i o d i s t a n c e - f i n d e r s .
f l i g h t away f r o m a r a d i o b e a c o n , t h e r u l e s for f l i g h t a l o n g t h e
ShchDR b e a r i n g a r e o b s e r v e d , w h i l e i n f l i g h t t o w a r d a r a d i o b e a c o n
i t i s t h e r u l e s f o r b e a r i n g ShchDM w h i c h a r e f o l l o w e d .
I f t h e f l i g h t i s made u s i n g a z e r o - i n d i c a t o r i n s t r u m e n t , t h e
method o f s e l e c t i n g t h e c o u r s e i n f l i g h t from t h e b e a c o n and t o w a r d
t h e b e a c o n w i t h a c o r r e s p o n d i n g s w i t c h i n t h e mode o f o p e r a t i o n
o f t h e r e c e i v e r l e a d s us t o o n l y one t y p e :
t h e p o i n t e r of t h e z e r o
i n d i c a t o r shows t h e d i r e c t i o n of d e v i a t i o n o f t h e a i r c r a f t from
t h e LGF.

The m a i n d i f f e r e n c e b e t w e e n u s i n g U S W b e a c o n s a n d g r o u n d r a d i o
distance-finders i s only t h a t i n order t o determine t h e location
o f t h e a i r c r a f t from two b e a r i n g s u s i n g a r a d i o d i r e c t i o n - f i n d e r ,
t h e p l o t t i n g o f t h e b e a r i n g s i s d o n e b y t h e o p e r a t o r o f t h e com­
mand d i s t a n c e f i n d i n g s t a t i o n , w h i l e i n t h e c a s e o f U S W b e a c o n s
i t i s d o n e b y t h e crew o f t h e a i r c r a f t .
N e v e r t h e l e s s , g o n i o m e t r i c U S W b e a c o n s h a v e a much w i d e r r a n g e
of a p p l i c a t i o n than ground r a d i o - d i s t a n c e - f i n d e r s and a i r c r a f t radio­

29 9

/286

c o m p a s s e s , t h a n k s t o t h e c o n s t a n t i n d i c a t i o n o f b e a r i n g s on b o a r d
the aircraft.
T h e i n s t r u m e n t a l a c c u r a c y o f g o n i o m e t r i c USW n a v i g a t i o n a l s y s ­
t e m s i s higher than for ground r a d i o d i s t a n c e - f i n d e r s .
The p r a c ­
t i c a l a c c u r a c y u n d e r a v e r a g e c o n d i t i o n s o f a p p l i c a t i o n i s a l s o somewhat
h i g h e r or e q u a l t o t h e a c c u r a c y o f d i s t a n c e - f i n d e r s .
However, i n
using r a d i o d i s t a n c e - f i n d e r s , it i s p o s s i b l e t o consider t o a c e r t a i n
e x t e n t t h e i n f l u e n c e of t h e l o c a l r e l i e f on t h e r a d i u s of a p p l i ­
c a t i o n , w h i c h c a n n o t b e d o n e f o r USW b e a c o n s .
In this respect,
t h e USW b e a c o n s h a v e l e s s f a v o r a b l e o p e r a t i n g c o n d i t i o n s t h a n g r o u n d
r a d i o distance-measuring s t a t i o n s .

s

T h e o p e r a t i n g r a n g e o f a USW s y s t e m
i s l i m i t e d by t h e l i m i t s
o f d i r e c t g e o m e t r i c v i s i b i l i t y f r o m t h e g r o u n d b e a c o n t o t h e air­
c r a f t w i t h an i n s i g n i f i c a n t i n c r e a s e c a u s e d by r a d i o r e f r a c t i o n .
I t i s d e t e r m i n e d by t h e a p p r o x i m a t e f o r m u l a

s =

122

JK.

H o w e v e r , i f t h e r e a r e some o b s t a c l e s a l o n g t h e p a t h o f t h e
p r o p a g a t i o n o f t h e r a d i o waves ( e . g . , m o u n t a i n p e a k s ) , t h e y w i l l
a p p e a r i n s u r m o u n t a b l e f o r USW.
From t h e s t a n d p o i n t o f n a v i g a t i o n a l a p p l i c a t i o n s , i t i s v e r y
a d v a n t a g e o u s t o c o m b i n e t h e o p e r a t i o n o f a g o n i o m e t r i c USW s y s t e m
with range f i n d e r s .

type

R a n g e f i n d i n g USW n a v i g a t i o n a l s y s t e m s a r e u s u a l l y o f i m p u l s e
(Fig. 3.28).

The a i i a c r a f t t r a n s m i t t e r s e n d s o u t i m p u l s e s o f u l t r a s h o r t waves
w h i c h r e a c h t h e r e c e i v e r a b o a r d t h e a i r c r a f t a t t h e same t i m e a s
a reference signal.

Fig. 3.28.
System.

300

/287
-

Diagram o f Long-Range

Navigational

,

The g r o u n d r e c e i v e r r e c e i v e s p u l s e s o f wave e n e r g y e m i t t e d
by t h e a i r c r a f t , a m p l i f i e s them a n d s e n d s them o u t a g a i n t h r o u g h
a t r a n s m i t t e r i n t o t h e e t h e r , t o b e r e c e i v e d by t h e a i r c r a f t .
The r a n g e i n d i c a t o r a b o a r d t h e a i r c r a f t h a s a g e n e r a t o r o f
s t a n d a r d f r e q u e n c i e s , a f r e q u e n c y - d i v i d e r c i r c u i t , and a d e l a y l i n e
f o r t h e r e f e r e n c e p u l s e t o measure t h e t i m e r e q u i r e d f o r t h e s i g n a l
t o p a s s from t h e a i r c r a f t t o t h e ground beacon and back t o t h e a i r ­
craft.
While t h e s i g n a l i s t r a v e l i n g , t h e d u r a t i o n o f t h e d e l a y i n
t h e reference p u i s e p r i o r t o i t s combination with t h e received s i g n a l
determines t h e d i s t a n c e t o t h e ground beacon, which i s u s u a l l y used
as a v i s u a l i n d i c a t o r o f t h e a z i m u t h o f t h e a i r c r a f t ( t h e d i r e c t r e a d i n g i n s t r u m e n t f o r d i s t a n c e and azimuth, D R I D A ) .
The c o m b i n a t i o n o f a z i m u t h a n d d i s t a n c e r e a d i n g s m a k e s i t v e r y
easy t o s o l v e t h e problems of a i r c r a f t n a v i g a t i o n , e s p e c i a l l y i f
t h e b e a c o n i s m o u n t e d a t t h e s t a r t i n g or e n d p o i n t o f a s t r a i g h t l i n e f l i g h t segment.
I n t h e l a t t e r c a s e , t h e crew o f t h e a i r c r a f t
has a constant supply of d i r e c t data regarding t h e p o s i t i o n of t h e
a i r c r a f t r e l a t i v e t o t h e l i n e o f f l i g h t i n t e r m s o f d i r e c t i o n and
distance.
When t h e g r o u n d b e a c o n i s l o c a t e d t o t h e s i d e o f t h e p a t h t o
b e c o v e r e d by t h e a i r c r a f t , t h e problem o f d e t e r m i n i n g t h e a i r c r a f t
c o o r d i n a t e s i s s o l v e d a n a l y t i c a l l y , or v e r y s i m p l e c a l c u l a t i n g d e v i c e s
are used t o convert t h e p o l a r system of c o o r d i n a t e s f o r t h e posi­
t i o n of t h e a i r c r a f t i n t o t h e o r t h o d r o m i c s y s t e m .
One t y p e o f s u c h d e v i c e i s t h e c o m p u t e r w h i c h i s i n s t a l l e d
f o r z e r o i n d i c a t i o n o f t h e p o s i t i o n of t h e a i r c r a f t on t h e l i n e
of t h e p a t h t o b e t r a v e l e d d u r i n g f l i g h t i n a g i v e n d i r e c t i o n .
L e t u s assume t h a t w e have a s t r a i g h t - l i n e
point A t o point B (Fig. 3.29).

p a t h segment from

I f we a r e g i v e n t h e p a t h a n g l e o f t h e s e g m e n t ( $ 1 , m e a s u r e d
r e l a t i v e t o t h e meridian f o r c a l c u l a t i n g t h e bearings (magnetic
or t r u e m e r i d i a n o f t h e l o c a t i o n o f t h e g r o u n d b e a c o n ) , a n d w e know
t h e a z i m u t h o f t h e e n d p o i n t o f t h e s e g m e n t ( A f i n ) a s w e l l as t h e
d i f f e r e n c e from i t t o t h e beacon ( R f i n ) , t h e s h o r t e s t d i s t a n c e from
t h e beacon a l o n g t h e l i n e of f l i g h t (Rs) ( d i s r e g a r d i n g t h e spher­
i c i t y o f t h e E a r t h ) , c a n b e d e t e r m i n e d by t h e f o r m u l a

or f o r a n y p o i n t l y i n g o n t h e l i n e o f f l i g h t
Rs
I n o t h e r words,

- Ri

s i n ( 9 - Ai).

t h e g i v e n l i n e of f l i g h t i s t h e g e o m e t r i c l o c u s

30 1


/288

o f p o i n t s f o r which
R.sin($-Ai)
2

= const =

Rfin

sin.( $-Afin).

F i g . 3.29.
Diagram Showing O p e r a t i o n o f Computer
f o r Zero Indication of Path Line.
on t h e
Thus, having s e t t h e p a t h angle of t h e segment ($1
c a l c u l a t o r , t h e d i s t a n c e from t h e beacon t o t h e end p o i n t of t h e
segment ( R f i n ) a n d t h e a z i m u t h o f t h e e n d p o i n t ( A f i n ) , w e can f.ind
t h e n a v i g a t i o n a l p a r a m e t e r Rs which c o r r e s p o n d s t o t h e p o s i t i o n
o f t h e a i r c r a f t e x a c t l y on t h e l i n e o f f l i g h t .
If i t t u r n s o u t i n t h e c o u r s e o f t h e f l i g h t t h a t Rs i s g r e a t e r
t h a n t h e g i v e n v a l u e , t h e n i n t h e example shown i n F i g u r e 3.29 t h e
a i r c r a f t d e v i a t e s f r o m t h e LGF t o t h e l e f t , a n d t h e p o i n t e r o f t h e
With a d e v i a t i o n f r o m t h e
z e r o i n d i c a t o r a l s o moves t o t h e l e f t .
L G F t o t h e r i g h t , t h e a r r o w o f t h e z e r o i n d i c a t o r w i l l move t o t h e
right.

When a f l i g h t i s made i n a d i r e c t i o n w h i c h i s o p p o s i t e t o t h a t
shown i n F i g u r e 3 . 4 0 , t h e v a l u e s i n ( $ A i ) a n d c o n s e q u e n t l y R s , w i l l
h a v e a n e g a t i v e s i g n , s o t h a t when t h e a i r c r a f t m o v e s t o t h e l e f t
o f t h e l i n e of f l i g h t t h e p o i n t e r of t h e z e r o i n d i c a t o r w i l l a l s o
move t o t h e l e f t r e g a r d l e s s o f t h e f a c t t h a t t h e a b s o l u t e d i s t a n c e
Rs i n t h i s c a s e d e c r e a s e s , a n d v i c e v e r s a .
The c a l c u l a t i n g d e v i c e f o r z e r o i n d i c a t i o n o f t h e g i v e n l i n e
o f f l i g h t i s v e r y s i m p l e a n d makes i t p o s s i b l e t o s o l v e o n l y o n e
p r o b l e m , :.e. , t o s e l e c t t h e c o u r s e t o b e f o l l o w e d by t h e a i r c r a f t
f o r a f l i g h t a l o n g a g i v e n l i n e o f f l i g h t , i n a manner s i m i l a r t o
t h a t f o r a f l i g h t f r o m a r a d i o b e a c o n or a l o n g t h e ShchDR b e a r i n g s ,
u s i n g t h e method of h a l f c o r r e c t i o n .

I t i s b e t t e r t o s o l v e t h e p r o b l e m or t o u s e c o m p u t o r s t o s o l v e
it using t h e computation of numerical v a l u e s of orthodromic coord­
i n a t e s o f t h e a i r c r a f t , w o r k i n g on t h e b a s i s o f t h e i n d i c a t i o n s
from t h e D R I D A :
302

/28'

I n t h i s case, t h e angle of s h i f t of t h e a i r c r a f t r e l a t i v e t o
t h e l i n e of f l i g h t i s d e t e r m i n e d v e r y s i m p l y as t h e r a t i o o f t h e
change i n t h e c o o r d i n a t e Z t o t h e d i s t a n c e covered between t h e p o i n t s
of two m e a s u r e m e n t s (Xcov):
tg A+ -.I

AZ

XC&

Example.
The d i s t a n c e c o v e r e d b e t w e e n meas u remen t p o i n t s i s
e q u a l t o 6 0 km.
T h e c o o r d i n a t e Z v a r i e s f r o m z e r o t o + 4 km.
Find
t h e required correction i n t h e course f o r traveling p a r a l l e l t o
t h e l i n e of f l i g h t .

&.

S o l u t i on
A+ = arctg

of'fl
must
(and
w i l l

We w i l l
ight fo
be -E0,
if this
have t o

assume t h a t i t i s
r a n o t h e r 6 0 km s o
b u t a t t h e moment
t a k e s p l a c e a s we
i n c r e a s e d by 4 O .

= 4".
60

necessary t o t r a v e l along t h i s l i n e
t h a t t h e correction i n t h e course
when t h e c o o r d i n a t e Z b e c o m e s z e r o
h a v e c a l c u l a t e d a t 6 0 km) t h e c o u r s e

I f t h e j u n c t u r e w i t h t h i s c o u r s e t a k e s p l a c e e a r l i e r or l a t e r ,
t h e n i t i s n e c e s s a r y once a g a i n t o determine t h e a n g l e of s h i f t
o f t h e a i r c r a f t (A$) a n d t o move t h e a i r c r a f t t o t h e r i g h t b y t h i s
angle.
F o r e x a m p l e , i f we h a v e a n i n i t i a l s h i f t f r o m t h e d e s i r e d
p a t h o f 4 k m , t h e a i r c r a f t w i l l r e a c h t h e l i n e o f f l i g h t i n 80 k m ,
so that
4
A+ = arctg -= 3".
80

I n t h e orthodromic system, t h e problem of checking t h e path
for d i s t a n c e a n d d e t e r m i n i n g t h e g r o u n d s p e e d i s s o l v e d s i m p l y

For l a c k o f a c a l c u l a t o r , t h e p r o b l e m s i n f i n d i n g t h e a n g l e
of s h i f t of t h e a i r c r a f t and checking t h e p a t h f o r d i s t a n c e and
/290
d i r e c t i o n , c a n b e s o l v e d a n a l y t i c a l l y by means o f a n a v i g a t i o n a l
slide rule.
I n a d d i t i o n , t h e s e same p r o b l e m s c a n b e s o l v e d by p l o t ­
t i n g on t h e c h a r t t h e i n d i c a t i o n s of t h e a z i m u t h a n d d i s t a n c e o f
t h e a i r c r a f t as o b t a i n e d from t h e b e a c o n .
On a f l i g h t c h a r t w h i c h h a s a n i n d i c a t i o n o f t h e g i v e n l i n e
o f f l i g h t , t w o p o i n t s b a s e d on t h e b e a r i n g s a n d d i s t a n c e s f r o m a
ground beacon are p l o t t e d every 15-20 min.
On t h e b a s i s o f t h e

30 3

p o s i t i o n s of t h e s e p o i n t s r e l a t i v e t o t h e l i n e o f f l i g h t , w e can
It is then easy
determine t h e i r orthodromic c o o r d i n a t e s X and Z .
t o s o l v e t h e p r o b l e m s i n d e t e r m i n i n g t h e a n g l e o f s h i f t of t h e a i r ­
craft (A$),
and a l s o t h e d r i f t angle and t h e r e q u i r e d angle f o r
t u r n i n g t h e a i r c r a f t , t h e g r o u n d s p e e d as w e l l as t h e w i n d param­
eters a t f l i g h t a l t i t u d e .

D e t a i l s of U s i n g G o n i o m e t r i c - R a n g e F i n d i n g S y s t e m s a t D i f f e r ­
ent Flight AZtitudes
A s p e c i a l f e a t u r e o f u l t r a s h o r t waves i s t h e i r a b i l i t y t o b e
r e f l e c t e d from t h e i n t e r f a c e s of media with d i f f e r e n t o p t i c a l densi­
t i e s , a n d e s p e c i a l l y f r o m c o n d u c t i n g m e d i a i n a more s h a r p l y p r o ­
n o u n c e d f o r m t h a n i s t h e c a s e for w a v e s o f s h o r t e r f r e q u e n c i e s .
I n a d d i t i o n , a t s h o r t w a v e l e n g t h s , t h e i n t e r f e r e n c e which a r i s e s
w i t h c o m b i n a t i o n o f o s c i l l a t i o n s shows up more r a r e l y t h a n i n t h e
case o f l o n g waves, s i n c e t h e s m a l l d i f f e r e n c e i n t h e p a t h o f t h e
c o h e r e n t waves i n t h e case o f s h o r t w a v e l e n g t h s g i v e s a c o n s i d e r a b l e
s h i f t i n t h e i r phase.
Let us say t h a t the antenna
o f a g r o u n d t r a n s m i t t e r o f a gon­
i o m e t r i c or r a n g e - f i n d i n g s y s t e m
i s mounted a i a c e r t a i n a l t i t u d e
above t h e s u r f a c e of t h e ground ( p o i n t
A i n Fig. 3.30).
The e l e c t r o m a g n e t i c wave a t
t h e r e c e i v e r p o i n t B w i l l be prop­
a g a t e d a l o n g two p a t h s :
Fig. 3.30.
D i a g r a m Forma­
t i o n o f L o b e s o f Maximum
Radiation.

(a)

AB

Along t h e s t r a i g h t l i n e

3

(b)
Along t h e b r o k e n l i n e ACB w i t h r e f l e c t i o n a t p o i n t C o f f
the Earth's surface.

I t i s clear i n t h e diagram t h a t s t r a i g h t l i n e A1B i s equal
t o t h e b r o k e n l i n e A C B , s i n c e t h e a n g l e of i n c i d e n c e o f t h e wave
i s e q u a l t o t h e a n g l e of r e f l e c t i o n .
L e t u s d r a w l i n e A A 7 i n s u c h a way t h a t t r i a n g l e A B A 2 i s a n
isosceles triangle.
Obviously, l i n e A1A2 w i l l r e p r e s e n t t h e path d i f f e r e n c e of
the rays
i n t h e s t r a i g h t and r e f l e c t e d waves.
The r e f l e c t i o n of r a d i o waves i n v o l v e s a p h a s e s h i f t i n t h e
w a v e w h i c h d e p e n d s o n t h e o p t i c a l p r o p e r t i e s o f t h e r e f l e c t i n g med­
ium.
A p u r e l y m i r r o r r e f l e c t i o n c h a n g e s t h e p h a s e o f a wave by
180O.
With a s m a l l d i f f e r e n e i n o p t i c a l d e n s i t i e s o f t h e m e d i a ,
30 4

/291

when t h e p r o p a g a t i o n o f t h e r e f l e c t e d w a v e t a k e s p l a c e a l o n g a c u r v e
w i t h a d i p i n t h e r e f l e c t i n g medium, t h e p h a s e s h i f t c a n t a k e p l a c e
differently.
L e t u s s a y t h a t u p o n r e f l e c t i o n , t h e p h a s e o f a wave
remains fixed.
Then t h e r e s u l t a n t o f t h e d i r e c t and r e f l e c t e d s i g n a l s
a t t h e r e c e i v i n g p o i n t B w i l l h a v e a maximum w h e n t h e p a t h d i f f e r ­
e n c e of t h e beams h a s a v a l u e w h i c h i s a n e v e n w h o l e m u l t i p l e o f
t h e h a l f wave:
A
AS=2%-~ = 0 , 2 , 4 , . .2n

.

2 '

a n d a minimum i f

K

i s a n odd m u l t i p l e o f t h e h a l f - w a v e :
x=

1,3,5.

. .(2n-

1).

Thus, t h e r e w i l l b e an i n t e r f e r e n c e p a t t e r n f o r t h e propaga­
t i o n o f r a d i o w a v e s i n t h e v e r t i c a l p l a n e w i t h maxima a n d m i n i m a
of d i r e c t i o n a l i t y of t h e r a d i a t i o n c h a r a c t e r i s t i c (Fig. 3.31).

Fig. 3.31.
Multilobe Radia t i o n C h a r a c t e r i s t i c of
E l e c t r o m a g n e t i c Waves.

A change i n t h e phase of t h e
wave w i t h r e f l e c t i o n f r o m t h e E a r t h ' s
s u r f a c e causes corresponding changes
i n t h e d i s t r i b u t i o n o f t h e maxima
a n d minima o f t h e c h a r a c t e r i s t i c
of directionality, but the t o t a l
s t r u c t u r e of t h e i n t e r f e r e n c e p a t t e r n
w i l l b e s i m i l a r t o t h a t shown i n
t h e diagram.

The i n t e r f e r e n c e p a t t e r n o f
shading i n t h e d i r e c t i o n s of radi­
of r a d i o waves b y o b j e c t s on t h e E a r t h ' s s u r f a c e , as w e l l as t h e
a l t i t u d e a t which t h ? a n t e n n a i s mounted above t h e E a r t h ' s s u r f a c e ,
introduce considerable corrections i n t h e possible range of recep­
t i o n of u l t r a s h o r t waves.
T h e o p e r a t i n g r a n g e of a s y s t e m , e x p r e s s e d b y t h e a p p r o x i m a t e
f o r m u l a S = 122&,
i s maximum a t a s u f f i c i e n t p o w e r o f t h e t r a n s ­
m i t t e r a n d s e n s i t i v i t y o f t h e r e c e i v e r , if t h e a i r c r a f t i s l o c a t e d
i n t h e l o b e o f t h e maximum o f d i r e c t i o n a l i t y .
However, a t c e r t a i n
h e i g h t s a n d d i s t a n c e s , t h e r e c a n b e r ' d i p s r ' i n a u d i b l i t y , when t h e
a i r c r a f t p a s s e s t h r o u g h r e g i o n s o f r a d i o s h a d o w or i n t e r f e r e n c e
minima.
I n a d d i t i o n , s p e c i a l f e a t u r e s u s i n g USW g o n i o m e t r i c - r a n g e
f i n d i n g d e v i c e s a t h i g h f l i g h t a l t i t u d e s art: r e l a t e d t o t h e i r r a n g e finding sections.
Rangefinding instruments can b e used t o measure n o t only t h e
horizontal but a l s o t h e sloping distance from t h e aircraft t o i t s
r a d i o beacon (Fig. 3.32).
Therefore,

sh=SH cos e
or

30 5

/292

I n t h e s p e c i a l case when t h e a i r c r a f t i s p a s s i n g a b o v e t h e
r a d i o beacon

L e t us suppose t h a t an a i r c r a f t
i s f l y i n g a l o n g a g i v e n r o u t e w i t h Rs
= 1 0 km, a t a n a l t i t u d e which i s a l s o
e q u a l t o 1 0 km w i t h t h e u s e o f a t y p e

Fig. 3.32.
Sloping and
Horizontal Distance t o
Radio Beacon.

Risin($-Ai) = const calculating device.
W i t h $-A = 9 0 ° , d i s t a n c e R m u s t b e e q u a l
t o 1 0 km, i . e . , t h e a i r c r a f t m u s t d e v i a t e
from t h e given course and pass over
t h e r a d i o beacon.
The h e i g h t e r r o r s
i n goniometric-rangefinding devices
h a v e some i m p o r t a n t s h o r t c o m i n g s i n
t h e i r use i n the shortrange applications
and e s p e c i a l l y i n maneuverings i n t h e
v i c i n i t y of an a i r p o r t .

C o n s i d e r a t i o n o f a l t i t u d e e r r o r s i s v e r y i m p o r t a n t due t o t h e
r a p i d i t y w i t h w h i c h t h e a i r c r a f t p a s s e s o v e r t h e b e a c o n , when t h e
e r r o r s i n m e a s u r i q g t h e d i s t a n c e c h a n g e s o r a p i d l y t h a t i t becomes
impossible t o e n t e r corrections without using s p e c i a l calculating
devices.
Therefore, t h e use of goniometric-range finding instruments
f o r n a v i g a t i o n a l measurements u s u a l l y l i m i t s t h e d i s t a n c e from t h e
b e a c o n t o 3-4 f l i g h t a l t i t u d e s , i . e . , i t d e f i n e s a n e f f e c t i v e zone
around t h e beacon with t h i s r a d i u s .

For e x a m p l e , a t a f l i g h t a l t i t u d e o f 1 2 km, t h e r a d i u s o f t h e
i n o p e r a t i v e zone t h u s d e f i n e d must b e e q u a l t o a p p r o x i m a t e l y 50
km.
Fan-Shaped

G o n i o m e t r i c R a d i o Beacons

The p o s s i b i l i t i e s o f a i r c r a f t r a d i o compasses a r e i n c r e a s e d
c o n s i d e r a b l y by u s i n g f a n - s h a p e d g o n i o m e t r i c b e a c o n s ( F i g . 3 . 3 3 ) .
The p i c t u r e shows t h e s c h e m a t i c d i a g r a m o f a r a d i o b e a c o n .
The t w o o u t e r m o s t a n t e n n a s a r e s e t t o some w a v e l e n g t h a n d t h e power
f o r them i s i n o p p o s i t e p h a s e .
The t o t a l c h a r a c t e r i s t i c o f t h e
t h r e e a n t e n n a s g i v e s t h e m u l t i l o b e p i c t u r e of r a d i a t i o n as s e e n
i n Figure 3.34.
The number o f l o b e s d e p e n d s on t h e r a t i o of t h e
/293
l e n g t h of t h e b a s e l i n e between t h e end antennas t o t h e wavelength,
a n d t h e i r d i r e c t i o n d e p e n d s on t h e . r a t i o of t h e p h a s e s i n t h e o u t e r
and i n n e r antennas of t h e r a d i o beacon.

306

With a c h a n g e i n t h e p h a s e o f t h e m i d d l e a n t e n n a by 1 8 0 ° , t h e
p o s i t i o n s o f t h e l o b e s s h i f t t o t h e i r m i r r o r images ( t h e s o l i d and
d o t t e d l o b e s i n F i g . 3.34), w h i l e t h e p o i n t s w h e r e t h e d o t t e d a n d
s o l i d l o b e s i n t e r s e c t become axes o f e q u a l s i g n a l s .

ain phas
~-s h i f t e r

Fig.

3.33.

Fig.

3.34.

Fig:

3.33.

Fan-Shaped

Radio Beacon.

Fig.

3.34.

R a d i a t i o n C h a r a c t e r i s t i c of a Fan-Shaped

Radio Beacon.

D u r i n g t h e p e r i o d s b e t w e e n c o m m u t a t i o n s , if w e t r a n s m i t s h o r t
and long s i g n a l s i n t h e forms of d o t s and dashes i n an overlapping
p a t t e r n , s i g n a l s of o n l y one t y p e w i l l b e h e a r d w i t h i n t h e e d g e s
of t h e s o l i d l o b e s ( e . g . , l o n g s i g n a l s ) , w h i l e w i t h i n t h e l i m i t s
of t h e dotted lobes, only s h o r t s i g n a l s w i l l be heard.
I n zones
o f e q u a l s i g n a l s ( n e a r t h e a x e s of i n t e r s e c t i o n o f t h e l o b e s ) , one
w i l l hear a continuous tone.
I f we t h e n s m o o t h l y c h a n g e t h e p h a s e
r a t i o i n t h e end antennas, t h e l o b e s w i l l begin t o r o t a t e , e . g . ,
t o t h e r i g h t , and t h e phase r a t i o s w i l l change i n t h e r e v e r s e d i r e c
tion:
each of t h e s o l i d l o b e s w i l l change p l a c e s with t h e d o t t e d
l o b e t o t h e r i g h t of i t , a n d e a c h d o t t e d l o b e w i l l change p l a c e
with t h e s o l i d lobe t o t h e r i g h t of i t .
L e t u s assume t h a t
3.341, ;.e.,
within the
hand l i m i t of t h e s o l i d
beginning after a pause

an a i r c r a f t i s l o c a t e d a t P o i n t B ( s e e Fig.
l i m i t s of a d o t t e d l o b e , n e a r t h e r i g h t l o b e , with each o p e r a t i n g cycle of t h e beacon
i n radiation.

I n t h i s case, a t t h e b e g i n n i n g of a c y c l e and a f t e r t h e p a u s e ,
s e v e r a l f a d i n g d o t s w i l l b e h e a r d , t h e n a continuous s i g n a l , and
f i n a l l y a long series of dashes.
If t h e a i r c r a f t i s l o c a t e d i n t h e
of d o t s w i l l be e q u a l i n l e n g t h t o t h e
which i s c l o s e t o t h e r i g h t - h a n d l i m i t
of d o t s w i l l b e l o n g e r t h a n t h e s e r i e s

middle of t h e l o b e , t h e series
s e r i e s of dashes.
A t a point/294
of t h e d o t t e d l o b e , t h e s e r i e s
of dashes.

30 7

A similar picture f o r t h e
the aircraft is located within
t h e sole difference being t h a t
d a s h e s would b e h e a r d , a n d t h e
continuous tone.

a u d i b i l i t y w o u l d b e o b t a i n e d when
t h e l i m i t s of t h e s o l i d lobe with
a t t h e beginning of the cycle t h e
d o t s would b e h e a r d o n l y a f t e r t h e

Thus, t o o b t a i n t h e b e a r i n g
o f an a i r c r a f t , it i s s u f f i c i e n t
t o p l o t t h e orthodromic l i n e s
on a c h a r t a c c o r d i n g t o t h e
l o c a t i o n of t h e axes of equal
signals i n order t o obtain the
b e a r i n g of t h e a i r c r a f t , a n d
a l s o t o d i v i d e t h e angles between
them by o r t h o d r o m e s as w e l l
as t h i n l i n e s i n a r a t i o w h i c h .
i s a m u l t i p l e o f t h e number
of signals i n the cycle.
With a s e c t o r w i d t h b e t w e e n
t h e axes of equal signals of
15O and 60 s i g n a l s p e r c y c l e ,
each s i g n a l w i l l correspond
t o 1 5 min o f a n g l e .
If t h e
Fig. 3.35.
Grid of Position
s e c t o r between t h e axes of e q u a l
L i n e s f o r a n A i r c r a f t on t h e
signals is then divided into
Basis o f Fan-Shaped Radio
f i v e p a r t s , t h e t h i n orthodromic
Beacons.
l i n e s w i l l diverge a t angles of
3O.
T h i s n a r r o w s e c t o r w i l l c o n t a i n 1 2 s i g n a l s o f t h e same t y p e
(Fig. 3.35).

For e x a m p l e , i f a n a i r c r a f t i s l o c a t e d a t t h e s e c t o r o f p o i n t s
on t h e f i r s t t h i n l i n e t o t h e r i g h t of t h e a x i s o f e q u a l s i g n a l s ,
t h e n 1 2 d o t s w i l l b e h e a r d which w i l l f a d e i n t o a c o n t i n u o u s t o n e ,
a f t e r which t h e r e w i l l b e 48 d a s h e s .
On t h e s e c o n d l i n e t h e r e w i l l
b e 2 4 d o t s a n d 36 d a s h e s , 36 d o t s a n d 2 4 d a s h e s o n t h e t h i r d , e t c ,
A t t h e l i m i t of t h e s e c t o r ( t h e a x i s of e q u a l s i g n a l s ) , a t o t a l
o f 60 d o t s and 60 dashes w i l l b e h e a r d .

Note.
P r a c t i c a l l y s p e a k i n g , i f w e c o n s i d e r t h a t p a r t of t h e
s i g n a l s ( d o t s and d a s h e s ) are mixed w i t h t h e c o n t i n u o u s t o n e , t h e
number o f a u d i b l e s i g n a l s w i l l b e l e s s t h a n 6 0 , s o t h a t a f t e r c o u n t i n g
them t h e number of a u d i b l e s i g n a l s s h o u l d b e t a k e n s u b t r a c t e d from
6 0 , t h e n d i v i d e d i n h a l f a n d a d d e d t o t h e number o f s i g n a l s o f b o t h
t y p e s t h a t were h e a r d .
If t h e a i r c r a f t i s l o c a t e d b e t w e e n t h e t h i n o r t h o d r o m i c l i n e s
p l o t t e d on t h e c h a r t , t h e n t h e l i n e of t h e b e a r i n g o f t h e a i r c r a f t
can e a s i l y be found by i n t e r p o l a t i o n of t h e d i s t a n c e between t h e
plotted lines.
Fan-shaped

30 8

/295

b e a c o n s make i t p o s s i b l e t o d e t e r m i n e v e r y a c c u r a t e l y

t h e p o s i t i o n l i n e s of an aircraft.
T o do t h i s , with t h e a i d of
a r a d i o c o m p a s s or b y g e n e r a l l y c a l c u l a t i n g t h e p a t h o f t h e a i r c r a f t ,
it is necessary t o determine t h e approximate p o s i t i o n of t h e air­
c r a f t w i t h an e r r o r which i s n o g r e a t e r t h a n t h e w i d t h o f one s e c t o r .
Then, h a v i n g l i s t e n e d t o t h e o p e r a t i n g c y c l e of t h e beacon w i t h
t h e r a d i o c o m p a s s , or w i t h t h e c o h e r e n t r e c e i v e r , w e c a n d e t e r m i n e
t h e position of t h e aircraft i n t h e s e c t o r .
A s i m i l a r method i s u s e d t o d e t e r m i n e t h e s e c o n d l i n e o f p o s i ­
t i o n o f t h e a i r c r a f t , u s i n g t h e s e c o n d f a n - s h a p e d b e a c o n , whose
f a m i l y of p o s i t i o n l i n e s i n t e r s e c t s t h e l i n e s o f t h e f i r s t b e a c o n .
I n order not t o t a k e i n t o account t h e . s h i f t of t h e aircraft during
t h e t i m e between t h e t a k i n g o f b e a r i n g s from t h e two b e a c o n s , i t
i s d e s i r a b l e t o l i s t e n t o t h e o p e r a t i n g c y c l e s o f t h e two b e a c o n s
s i m u l t a n e o u s l y u s i n g t w o m e m b e r s o f t h e c r e w who a r e u s i n g t w o r a d i o c o m p a s s e s or o n e r a d i o c o m p a s s a n d t h e c o h e r e n t r a d i o r e c e i v e r .
The a c c u r a c y o f d i s t a n c e f i n d i n g w i t h t h e a i d o f f a n - s h a p e d
Under t h e
b e a c o n s d u r i n g t h e d a y t i m e i s n o w o r s e t h a n 0.1-0.3O.
most u n f a v o r a b l e c o n d i t i o n s f o r d i s t a n c e measurement ( i n t w i l i g h t
when w o r k i n g w i t h t h e s p a c e w a v e , or a t t h e b o u n d a r y f o r t h e u s e
of s u r f a c e waves), t h e e r r o r s can reach 3 and sometimes 5 O .
In
a f u r t h e r zone o f d i s t a n c e measurement, and a l s o t h e s h o r t - r a n g e
z o n e , w i t h o p e r a t i o n on a s u r f a c e wav e, t h e e r r o r s d o n o t e x c e e d
0 . 5-1°.
The o p e r a t i n g r a n g e o f a f a n - s h a p e d b e a c o n d u r i n g t h e d a y t i m e
r e a c h e s 1 3 5 0 km o n d r y l a n d a n d 1 7 5 0 km a b o v e t h e s e a .
A t night
a b o v e d r y l a n d , t h i s f i g u r e i s 7 4 0 km a n d a b o v e t h e s e a , 9 5 0 km.
An i m p o r t a n t a d v a n t a g e o f f a n - s h a p e d b e a c o n s i s t h e i n d e p e n ­
dence of d i s t a n c e measurement from t h e a i r c r a f t c o u r s e , magnetic
d e c l i n a t i o n , d e v i a t i o n of meridians , and r a d i o d e v i a t i o n s aboard
t h e a i r c r a f t , as w e l l as t h e s i m p l i c i t y of o b t a i n i n g p o s i t i o n l i n e s
of t h e aircraft during f l i g h t .
One s h o r t c o m i n g t h a t s h o u l d b e men­
tioned i s t h e l a c k of a continuous indication of a navigational
p a r a m e t e r , s u c h as w e h a v e i n t h e c o n v e n t i o n a l d i s t a n c e measure­
ment w i t h g r o u n d r a d i o s t a t i o n s .
Unlike t h e r a d i o beacons with n o n - d i r e c t i o n a l and omnidirec­
t i o n a l o p e r a t i o n , which a r e mounted as a r u l e a t t h e t u r n i n g p o i n t s
of a i r r o u t e s , f l i g h t a l o n g t h e b e a r i n g l i n e of a f a n - s h a p e d b e a c o n
i s only a very r a r e case.
T h e r e f o r e , t h e p r i n c i p a l method o f a i r ­
craft navigation using fan-shaped beacons is determining a l l navi­
g a t i o n a l e l e m e n t s i n c l u d i n g t h e wind p a r a m e t e r s a t f l i g h t a l t i t u d e
by s u c c e s s i v e m e a s u r e m e n t s of t h e L A .
T h i s method i s t h e most s u i t a b l e one f o r f a n - s h a p e d b e a c o n s
b e c a u s e t h e l o c a t i o n o f t h e a i r c r a f t c a n b e d e t e r m i n e d i n t h i s manner
with a s u f f i c i e n t l y high accuracy.
It is often desirable t o carry out aircraft navigation during

30 9

/296

f l i g h t using fan-shaped beacons with conventional distance-finding
For e x a m p l e , i n a f l i g h t t o w a r d a r a d i o s t a ­
from r a d i o s t a t i o n s .
t i o n or away f r o m a r a d i o s t a t i o n , i t i s d e s i r a b l e t o u s e b e a r i n g s
f r o m f a n - s h a p e d b e a c o n s for c h e c k i n g t h e p a t h f o r d i s t a n c e a n d d e t e r ­
mining t h e ground speed.

3. D I F F E R E N C E - R A N G E F I N D I N G ( H Y P E R B O L I C ) < N A V I G A T I O N A L

SYSTEMS

The b e s t n a v i g a t i o n a l d e v i c e s f r o m t h e s t a n d p o i n t o f g e o m e t r y
are t h e goniometric-range f i n d i n g systems, s i n c e t h e l i n e s of posi­
t i o n of an aircraft i n t h e s e systems always c r o s s a t . r i g h t angles.
However, t h e t e c h n i c a l r e q u i r e m e n t s o f s u c h s y s t e m s , which s a t i s f y
t h e requirements of accuracy i n a i r c r a f t n a v i g a t i o n and o p e r a t e
o u t i s d e t h e l i m i t s of d i r e c t geometric v i s i b i l i t y , are connected
with very high technical d i f f i c u l t i e s .
The a z i m u t h l i n e s o f p o s i t i o n a r e d i v e r g e n t b e c a u s e as t h e
range of operation of a system increases, increasingly high require­
m e n t s a r e i m p o s e d on t h e m e a s u r e m e n t a c c u r a c y , w h i l e b e y o n d t h e
l i m i t s of d i r e c t geometric v i s i b i l i t y it is very d i f f i c u l t t o r e t a i n
d i r e c t i o n a l i t y o f t r a n s m i s s i o n or r e c e p t i o n d u e t o t h e e f f e c t o f
l o c a l r e l i e f a n d e s p e c i a l l y t h e i o n i z e d l a y e r s of t h e a t m o s p h e r e .
The s i t u a t i o n i s somewhat b e t t e r as f a r a s t h e c i r c u l a r p.osi­
t i o n l i n e s are concerned.
C i r c u l a r l i n e s do n o t d i v e r g e , s o t h a t
t h e r e q u i r e m e n t f o r a c c u r a c y i n d e t e r m i n i n g them r e m a i n s c o n s t a n t
at a l l distances.
In a d d i t i o n , t h e l i n e a r e r r o r i n determining t h e p o s i t i o n of
t h e a i r c r a f t i n a.goniometric system i s proportional t o t h e s i n e s
of t h e angles of t h e propagation e r r o r s :

A2 = S sin Ad.
I n range finding systems, these e r r o r s a r e proportional to t h e cosines
o f t h e a n g l e s of t h e p r o p a g a t i o n e r r o r s :

AS = S ( I - COS PA).
A t s m a l l a n g l e s , o n t h e o r d e r o f 6O , t h e c o s i n e i s p r a c t i c a l l y
equal t o unity.
Therefore, t h e e r r o r s i n determining t h e distance
a r e u s u a l l y many t i m e s l e s s t h a n t h e e r r o r s i n t h e a z i m u t h a l s h i f t
(Fig. 3.36).
We c a n s e e f r o m t h e f i g u r e t h a t t h e l i n e a r e r r o r i n d e t e r m i n ­
i n g t h e d i r e c t i o n C C l = SsinAA, and t h e l i n e a r e r r o r i n d i s t a n c e
i s A S = ABC - A C
S(1-cosAA).

However, t h e t e c h n i c a l a c h i e v e m e n t i n m e a s u r i n g d i s t a n c e o v e r
l o n g d i s t a n c e s i s much m o r e c o m p l e x t h a n t h a t i n m e a s u r e m e n t o f
t h e azimuth.

310

.11.11.1...1.,,111.III

-

I

I 1111 .
I I1 1111111.

I .I

II 1I

I

1

A s w e s a w i n t h e c a s e o f USW s y s t e m s , d i s t a n c e i s d e t e r m i n e d
/297
by r e t r a n s l a t i o n o f s i g n a l s f r o m on b o a r d t h e a i r c r a f t by a g r o u n d
This method,
b e a c o n a n d t h e i r r e c e p t i o n b a c k on b o a r d t h e a i r c r a f t .
which i s r e l a t i v e l y e a s i l y a c c o m p l i s h e d a t s h o r t d i s t a n c e s , t u r n s
o u t t o b e u n s a t i s f a c t o r y o v e r l o n g d i s t a n c e s f o r u s e on medium a n d
long waves.

Fig. 3.36.
Errors i n M e a s u r i n g
B e a r i n g a n d Range w i t h R e f l e c t i o n o f E l e c t r o m a g n e t i c Waves
from Obstacles:
A:
Location
o f Ground R a d i o B e a c o n s ;
B:
Point of Mirror Reflection of
R a d i o Waves;
C : Location of
t h e A i r c r a f t ( A c t u a l ) ; C1:
M e a s u r e d P o s i t i o n of t h e A i r -
craft;
L A : A n g u l a r Error i n
Propagation.

The b e s t method o f m e a s u r i n g
long distances at the present
t i m e i s t h e maintenance of a
c a l i b r a t i o n f r e q u e n c y on b o a r d
The g e n e r a t o r
the aircraft.
f o r t h e c a l i b r a t i o n frequency
is set a t t h e frequency of a
ground t r a n s m i t t e r and r e t a i n s
a given frequency f o r long periods
o f t i m e by means o f s p e c i a l
s t a b i l i z i n g elements.

By m e a n s o f t h e s e s p e c i a l
timing devices , t h e calibrat i o n frequency can be converted
t o a l o w e r f r e q u e n c y which i s
synchronous with t h e s i g n a l s
of t h e g r o u n d b e a c o n .
I f we
t a k e t h e s i g n a l s from t h e ground
s t a t i o n s a n d compare them i n
t i m e with t h e s i g n a l s from t h e g e n e r a t o r of t h e c a l i b r a t i o n fre­
quency, w e can determine t h e d i s t a n c e t o t h e s e r a d i o s t a t i o n s .
However, t h i s method h a s n o t b e e n w i d e l y employed due t o t h e
complexity involved i n keeping a highly s t a b l e r e f e r e n c e frequency
on b o a r d t h e a i r c r a f t , a l t h o u g h i t o f f e r s c o n s i d e r a b l e p r o m i s e i n
future.

I t i s s i m p l e r t o s o l v e t h e problem of d e t e r m i n i q g t h e p o s i ­
t i o n l i n e o f t h e a i r c r a f t on t h e b a s i s o f t h e d i s t a n c e b e t w e e n t h e
d i s t a n c e s t o two g r o u n d r a d i o s t a t i o n s .
In t h i s case, there is
no n e c e s s i t y f o r a s t r i c t s y n c h r o n i z a t i o n o f t h e o p e r a t i o n o f t h e
g r o u n d i n s t a l l a t i o n s w i t h t h o s e on b o a r d .
Only t h e t r a n s m i s s i o n
of s i g n a l s from t h e ground s t a t i o n s must b e s y n c h r o n i z e d .
The a i r ­
craft generator f o r t h e c a l i b r a t i o n frequency i n t h i s p a r t i c u l a r
case a c t s o n l y as a c e n t r a l m e a s u r i n g gauge t o d e t e r m i n e t h e t i m e
i n t e r v a l s b e t w e e n t h e moments o f r e c e p t i o n o f t h e s i g n a l s f r o m t h e
two r a d i o s t a t i o n s .

S y n c h r o n i z a t i o n of t h e o p e r a t i o n o f t h e a p p a r a t u s a t ground
r a d i o s t a t i o n s c a n b e a c h i e v e d i n c o m p a r a b l y more e a s i l y t h a n s y n ­
c h r o n i z a t i o n o f a g r o u n d a p p a r a t u s w i t h one a b o a r d an a i r c r a f t ,
s i n c e t h e d i s t a n c e between ground s t a t i o n s remains c o n s t a n t , thus
a l l o w i n g u s t o u s e a s y n c h r o n i z i n g d e v i c e f o r t w o or t h r e e s t a t i o n s
together.
I n a d d i t i o n , ground i n s t a l l a t i o n s are n o t l i m i t e d by

311


s i z e and weight r e s t r i c t i o n s ,

n o t t o m e n t i o n t h e a p p a r a t u s on b o a r d .

The m e t h o d s of m e a s u r i n g t h e d i f f e r e n c e i n d i s t a n c e t o g r o u n d
r a d i o s t a t i o n s c a n i n v o l v e e i t h e r t i m e ( p u l s e ) s y s t e m s or p h a s e
systems.
E a c h o f t h e s e m e t h o d s h a s i t s own a d v a n t a g e s a n d d i s a d ­
vantages.

/298

An a d v a n t a g e o f t h e p h a s e m e t h o d s i s t h e h i g h e r i n s t r u m e n t a l
a c c u r a c y o f t h e m e a s u r e m e n t s , b u t i n t h i s c a s e t h e r e s u l t o f meas­
u r i n g i s o b t a i n e d a m b i g u o u s l y , i . e . , t h e r e may b e s e v e r a l i s o p h a s a l
p a t h s simultaneously with d i f f e r e n t d i s t a n c e s t o t h e ground r a d i o
s t a t i o n s , which d i f f e r i n magnitude and a r e m u l t i p l e s of t h e l e n g t h
o f t h e measured wave.
On e a c h o f t h e s e p a t h s , t h e r e s u l t o f meas,­
u r e m e n t i s t h e same a n d m u s t b e u s e d a s a m e a s u r e f o r d e t e r m i n i n g
t h e pathway a l o n g which t h e a i r c r a f t i s t r a v e l i n g .
The p u l s e m e t h o d s o f m e a s u r i n g d i s t a n c e h a v e somewhat l e s s
i n s t r u m e n t a l a c c u r a c y , b u t t h e i r r e s u l t s a r e more d e f i n i t e .
O f c o u r s e . it should b e mentioned t h a t f o r long-range navi­
g a t i o n a l s y s t e m s , t h e i n s t r u m e n t a l a c c u r a c y o f measurement which
c a n b e a t t a i n e d a t t h e p r e s e n t t i m e b o t h by t h e p u l s e a n d p h a s e
m e t h o d s i s s u f f i c i e n t l y h i g h S O t h a t t h e i r e r r o r s a r e many t i m e s
l e s s t h a n o t h e r s y s t e m a t i c e r r o r s which a r e r e l a t e d t o c o n d i t i o n s
o f p r o p a g a t i o n of e l e c t r o m a g n e t i c e n e r g y .
Since t h e e r r o r s i n oper­
a t i o n of t h e s y s t e m s u n d e r p r o p a g a t i o n c o n d i t i o n s o f r a d i o waves
a r e p r a c t i c a l l y t h e s a m e for b o t h p u l s e d a n d p h a s e s y s t e m s , t h e
a d v a n t a g e o f p h a s e m e t h o d s o f m e a s u r e m e n t may b e r e s t r i c t e d o n l y
t o s h o r t d i s t a n c e s from ground r a d i o s t a t i o n s ( i n t h e short-range
zone of e f f e c t i v e n e s s ) .

Operating P r i n c i p l e s of

Differential

Rangefinding Systems

D i f f e r e n t i a l rangefinding systems of a i r c r a f t navigation c o n s i s t
o f two p a i r s o f s y n c h r o n o u s l y o p e r a t i n g ground r a d i o s t a t i o n s and
a receiving-indicating apparatus aboard an aircraft.
For p u r p o s e
o f r e d u c i n g t h e a m o u n t o f g r o u n d e q u i p m e n t for t h e s y s t e m , o n e o f
t h e t r a n s m i t t i n g r a d i o s t a t i o n s ( t h e m a s t e r ) i s made common f o r
two p a i r s s o t h a t t h e s y s t e m c a n i n c l u d e t h r e e ground r a d i o s t a ­
tions.
The o p e r a t i o n o f t h e t w o s l a v e s t a t i o n s i s s y n c h r o n i z e d w i t h
t h e master s t a t i o n b y s y n c h r o n i z i n g s i g n a l s s e n t o u t by t h e m a s t e r
station.

L e t u s b e g i n b y e x a m i n i n g t h e g e o m e t r y of
one p a i r o f ground r a d i o s t a t i o n s ( F i g . 3 . 3 7 ) .

t h e o p e r a t i o n of

Two g r o u n d r a d i o s t a t i o n s a r e l o c a t e d a t p o i n t s F1 a n d F2.
The l i n e c o n n e c t i n g p o i n t s F 1 a n d F p w i l l b e c o n s i d e r e d as t h e f o c a l
l i n e o f t h e b a s e , w h i l e t h e p o i n t s F1 a n d F 2 a r e t h e f o c i o f t h e
system.

312


L e t u s assume t h a t a t p o i n t M t h e r e i s a n a i r c r a f t which i s
r e c e i v i n g s i g n a l s f r o m r a d i o s t a t i o n s F1 a n d F 2 .
A t the beginning,
t h e a i r c r a f t w i l l r e c e i v e a s i g n a l from t h e f i r s t r a d i o s t a t i o n
/299
and t h e n from t h e s e c o n d .
The d i f f e r e n c e i n t h e d i s t a n c e s f r o m
t h e a i r c r a f t t o t h e s e r a d i o s t a t i o n s i s d e t e r m i n e d by t h e d i f f e r ­
ence i n t i m e between t h e a r r i v a l of t h e s i g n a l s i n t h e p u l s e system
o r b y t h e d i f f e r e n c e i n m o d u l a t i o n of t h e p h a s e s i n t h e w a v e s r e ­
c e i v e d from t h e two r a d i o s t a t i o n s i n t h e p h a s e s y s t e m .

Fig. 3.37.
Hyperbolic System o f P o s i t i o n L i n e s .

We know t h a t t h e l i n e w h i c h
is t h e geometrical locus of p o i n t s ,
t h e d i f f e r e n c e i n whose d i s t a n c e
t o two g i v e n p o i n t s i s a c o n s t a n t
v a l u e , i s c a l l e d a hyperbola.
The
g i v e n p o i n t s , t o which t h e d i s t a n c e s
are measured, are c a l l e d t h e f o c i
of t h e h y p e r b o l a .
Consequently,
knowing t h e d i f f e r e n c e o f t h e d i s ­
t a n c e s t o t h e two r a d i o s t a t i o n s ,
we c a n a l w a y s p l o t t h e h y p e r b o l i c
l i n e o f t h e p o s i t i o n where t h e a i r ­
craft is located.

The h y p e r b o l i c l i n e w i t h a d i f f e r e n c e i n d i s t a n c e s e q u a l t o
z e r o becomes a s t r a i g h t l i n e p e r p e n d i c u l a r t o t h e f o c a l a x i s and
d i v i d i n g t h e d i s t a n c e between t h e f o c i of t h e system i n h a l f ( s e e
Fig. 3.37).
T h i s l i n e i s c a l l e d t h e i m a g i n a r y axis of t h e h y p e r ­

bola.
The d i s t a n c e a l o n g t h e f o c a l a x i s o f t h e f a m i l y o f h y p e r b o l a s
from t h e f o c i t o t h e imaginary a x i s i s c a l l e d t h e p a r a m e t e r e .
I t i s obvious t h a t t h e d i f f e r e n c e i n d i s t a n c e s from t h e
o f t h e h y p e r b o l a t o any p o i n t a l o n g i t s b r a n c h e s i s e q u a l t o
t h e d i s t a n c e along t h e f o c a l a x i s from t h e imaginary a x i s t o
This distance is c a l l e d parameter a .
peak o f t h e h y p e r b o l a .
i n g l y , t h e d i f f e r e n c e i n d i s t a n c e s from any p o i n t t o t h e f o c
t h e hyperbola i s always e q u a l t o 2a.

foci
t w i c e
the
Accord­
i of

T h e maximum d e n s i t y o f h y p e r b o l i c l i n e s o f p o s i t i o n i s f o u n d
a l o n g t h e f o c a l a x i s b e t w e e n t h e f o c i of t h e s y s t e m , where t h e d i s ­
t a n c e between t h e peaks of t h e hyperbola is e q u a l t o t h e d i f f e r ­
ence i n parameters a .
T h e m a g n i t u d e o f t h e v a l u e 2a i s m e a s u r e d b y n a v i g a t i o n a l p a r a m ­
eters of t h e system s o t h a t t h e accuracy i n determining t h e l i n e s
o f p o s i t i o n o f t h e a i r c r a f t d e p e n d s on t h e a c c u r a c y w i t h which t h i s
parameter i s measured.
Consequently, an e r r o r i n determining t h e
p o s i t i o n l i n e o f t h e a i r c r a f t on t h e f o c a l a x i s i s e q u a l t o t h e e r r o r
i n m e a s u r i n g p a r a m e t e r 2a, d i v i d e d i n h a l f .
A s w e see from F i g u r e 3 . 3 7 ,

t h e family of hyperbolas is divided

313


by a family o f p o s i t i o n l i n e s .
A t d i s t a n c e s from t h e c e n t e r o f
t h e s y s t e m w h i c h e x c e e d 2c, t h e h y p e r b o l a s p r a c t i c a l l y become s t r a i g h t
l i n e s , whose d i r e c t i o n c o i n c i d e s w i t h t h e d i r e c t i o n o f t h e r a d i i
Thus, t h e h y p e r b o l i c system
e x t e n d e d from t h e c e n t e r o f t h e s y s t e m .
i s converted i n t o a goniometric one.
/300
H o w e v e r , t h e d e n s i t y of t h e l i n e s o f p o s i t i o n , i n t h i s c a s e
w i l l n o t b e e q u a l a l o n g t h e c i r c u m f e r e n c e , as i s t h e case i n p u r e l y
goniometric systems.
A t a given d i s t a n c e from t h e c e n t e r of t h e
s y s t e m , t h e maximum d e n s i t y of p o s i t i o n l i n e s w i l l b e f o u n d a t t h e
imaginary a x i s of t h e hyperbola, gradually decreasing along t h e
The d e n s i t y o f p o s t
c i r c u m f e r e n c e as t h e y a p p r o a c h t h e f o c a l a x i s .
t i o n l i n e s a t a d i s t a n c e g r e a t e r t h a n e on t h e f o c a l a x i s becomes
s o s m a l l t h a t t h e s y s t e m becomes u n s u i t a b l e f o r d e t e r m i n i n g t h e
l o c a t i o n of t h e a i r c r a f t .

Fig.

3.38.

E f f e c t i v e Area of H y p e r b o l i c
N a v i g a t i o n a l System.

The m a s t e r s t a t i o n o f t h e s e c o n d h y p e r b o l i c p a i r c a n b e l o ­
c a t e d a l o n g t h e e x t e n s i o n of t h e f o c a l a x i s o f t h e f i r s t p a i r .
In
t h i s case, t h e a n g l e of f r a c t u r e of t h e b a s e ( 8 ) i s e q u a l t o z e r o .
If t h e m a s t e r s t a t i o n of t h e s e c o n d p a i r i s n o t l o c a t e d on
t h e f o c a l axis of t h e first p a i r , t h e r e is a d e f i n i t e f r a c t u r e of
the base (Fig. 3.38).

The a n g l e o f f r a c t u r e o f t h e b a s e c r e a t e s a more f a v o r a b l e
condition f o r i n t e r s e c t i o n of t h e p o s i t i o n l i n e i n t h a t region of
a p p l i c a t i o n of t h e s y s t e m toward which i t i s d i r e c t e d , s i n c e t h e
a n g l e o f i n t e r s e c t i o n o f t h e h y p e r b o l a s i n t h i s case a p p r o a c h e s
a r i g h t angle and t h e r e f o r e t h e accuracy i n determining t h e l o c u s
o f t h e a i r c r a f t i s i n c r e a s e d when t w o p o s i t i o n l i n e s i n t e r s e c t .
However,

t h i s involves a decrease i n t h e q u a l i t y of t h e condi­

314

I

t i o n s f o r n a v i g a t i o n a l e x p l o i t a t i o n , as w e l l as a n a r r o w i n g o f t h e
range of a p p l i c a t i o n f o r t h e system from t h e opposite s i d e of t h e
I n a d d i t i o n , t h e a n a l y t i c a l s o l u t i o n of problems r e l a t e d
focal axis.
t o n a v i g a t i o n a l a p p l i c a t i o n of t h e s y s t e m i s c o m p l i c a t e d t o a s i g ­
n i f i c a n t degree by t h e f r a c t u r e of t h e b a s e , e . g . , i n converting
hyperbolic coordinates t o geographical orthodromic ones ( s e e Chapter
I , Section 7 ) .
The c o m p l e x o f e q u i p m e n t f o r a h y p e r b o l i c n a v i g a t i o n a l s y s t e m aboard an aircraft usually c o n s i s t s of t h e following:
a non­
d i r e c t i o n a l r e c e i v i n g antenna, a matching block f o r t h e antenna
with a r e c e i v i n g device, a r e c e i v e r , and an i n d i c a t o r .

/301

The m a t c h i n g b l o c k s e r v e s t o p r o d u c e p a r a m e t e r s o f t h e r e c e i v ­
i n g a n t e n n a s when s i g n a l s a r e r e c e i v e d f r o m g r o u n d r a d i o s t a t i o n s .
S i g n a l s r e c e i v e d by t h e a n t e n n a a r e t r a n s m i t t e d t o t h e i n d i ­
c a t o r for m e a s u r e m e n t o f t h e n a v i g a t i o n a l p a r a m e t e r .
The i n d i c a t o r
has a g e n e r a t o r f o r a c a l i b r a t i o n f r e q u e n c y , which produces s t a n ­
d a r d s i g n a l s f o r p u r p o s e s o f m e a s u r e m e n t , a n d a number o f f r e q u e n c y
d i v i d e r s w h i c h a r e r e q u i r e d f o r f o r m i n g e l e c t r o n i c m a r k i n g s on t h e
r e a d i n g s c a l e s , a s w e l l a s r e p e t i t i o n f r e q u e n c i e s f o r t h e s c a n on
a c a t h o d e r a y t u b e , s y n c h r o n i z e d w i t h t h e t r a n s m i s s i o n s of s i g n a l s
from ground r a d i o s t a t i o n s .
The s i g n a l s w h i c h a r e r e c e i v e d p a s s t o t h e s c a n o f t h e c a t h ­
ode r a y t u b e , where t h e o p e r a t o r c o n t r o l s t h e i r s i z e a c c o r d i n g t o
t h e amplification of t h e receiver.
The s y n c h r o n i z a t i o n o f t h e s c a n
on t h e s c r e e n i s t h e n r e g u l a t e d w i t h t h e f r e q u e n c y o f t h e r e c e i v e d
p u l s e s s o t h a t t h e l a t t e r r e m a i n f i x e d on t h e s c r e e n .
The o p e r a t o r t h e n m i x e s a r e f e r e n c e ( s e l e c t i n g ) s i g n a l f r o m
t h e g e n e r a t o r w i t h t h e s i g n a l from t h e m a s t e r s t a t i o n , which i s
a c h i e v e d by t h e i n t e r m i t t e n t i n t r o d u c t i o n of s m a l l d i s t o r t i o n s i n
t h e generator f o r the c a l i b r a t i o n frequency, s o t h a t t h e pulses
of t h e s i g n a l s b e g i n t o move a c r o s s t h e s c r e e n .
The m o t i o n o f t h e
p u l s e s s t o p s when t h e s i g n a l o f t h e m a s t e r s t a t i o n c o i n c i d e s w i t h
t h e reference s i g n a l of t h e generator (usually a rectangular base
a t t h e beginning of t h e scan).
To m e a s u r e t h e t i m e d i f f e r e n c e b e t w e e n t h e a r r i v a l o f t h e s i g n a l s
and t h e s i g n a l from t h e s l a v e s t a t i o n , a s e l e c t i n g p u l s e i s g i v e n
which i s r e l a t e d t o t h e d e l a y i n s c a n n i n g o f t h e r e f e r e n c e s i g n a l ,
a f t e r which t h e i n d i c a t o r i s s w i t c h e d t o t h e r e f e r e n c e regime and
t h e r e a d i n g i s t a k e n on t h e e l e c t r o n i c s c a l e .
I n some t y p e s of r e c e i v e r i n d i c a t o r s , t h e r e c o r d i n g o f t h e
r e a d i n g i s made o n a d i a l w i t h t w o or t h r e e s c a l e s ( f o r d i f f e r e n t
scanning r a t e s ) , f o r example, beginning with thousands of micro­
s e c o n d s , t h e n h u n d r e d s a n d f i n a l l y t e n s , w i t h i n t e r p o l a t i o n up t o
u n i t s of m i c r o s e c o n d s .
This provides increased accuracy of readings
due t o t h e many-fold i n c r e a s e i n t h e scale of t h e i n d i c a t o r .

315


I I I1


I n systems w i t h a u t o m a t i c t r a c k i n g o f t h e s i g n a l s from ground
s t a t i o n s , t h e t i m e i n t e r v a l s b e t w e e n t h e moments o f a r r i v a l o f t h e
s i g n a l s a r e c a l c u l a t e d on m e c h a n i c a l c o u n t i n g d i a l s , whose r o t a ­
t i o n i s r e l a t e d t o t h e d e l a y mechanisms f o r t h e s e l e c t i n g p u l s e .
The r e f e r e n c e s i g n a l f r o m t h e g e n e r a t o r i s t h e n r e i n f o r c e d , t o g e t h e r
w i t h t h e s i g n a l f r o m t h e master s t a t i o n , by a n a u t o m a t i c f r e q u e n c y
adjustment of t h e c a l i b r a t i o n generator.
Thus, t h e r e i s an automatic t r a c k i n g of t h e s i g n a l s
r a d i o s t a t i o n and a c o n s t a n t numerical i n d i c a t i o n of t h e
navigational parameter of t h e system, and t h e difference
t a n c e s from t h e a i r c r a f t t o t h e ground r a d i o s t a t i o n s i s
i n m i c r o s e c o n d s o f r a d i o wave p r o p a g a t i o n .

from t h e
output
i n dis­
expessed

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I n p h a s e s y s t e m s , b y means o f d i s t r i b u t i n g e l e m e n t s i n t h e
c a l i b r a t i o n g e n e r a t o r , i t s phase i s matched w i t h t h e phase of t h e
s i g n a l s from t h e master r a d i o s t a t i o n , a f t e r which a phasometer
i s used t o measure t h e phase d i f f e r e n c e between t h e c a l i b r a t i o n
generator and t h e s l a v e r a d i o s t a t i o n , and t h e p o s i t i o n l i n e of
t h e a i r c r a f t i s determined from t h i s d i f f e r e n c e .

A s w e have already pointed o u t , i f t h e d i f f e r e n c e i n d i s t a n c e s
t o t h e r a d i o s t a t i o n s includes s e v e r a l periods of t h e modulating
f r e q u e n c y o f t h e g r o u n d s t a t i o n s , t h e d e t e r m i n a t i o n w i l l b e ambig­
uous.
The s o l u t i o n o f t h e a m b i g u i t y of
p l i s h e d by s e v e r a l methods.

t h i s e s t i m a t e c a n b e accom­

(1) An i n i t i a l s e t t i n g o f t h e c o o r d i n a t e s o f t h e a i r c r a f t
with automatic t r a c k i n g of t h e r a d i o s t a t i o n s i g n a l s .
In t h i s case,
u s i n g known c o o r d i n a t e s o f t h e a i r c r a f t ( e . g . , on t h e b a s i s o f t h e
v i s u a l determination of t h e aircraft l o c a t i o n ) , t h e i n d i c a t o r is
s e t b y h a n d t o show t h e i s o p h a s a l l i n e on w h i c h t h e a i r c r a f t i s
located.
If c o n s t a n t t r a c k i n g o f t h e r a d i o s t a t i o n s i g n a l s i s t h e n
c a r r i e d o u t , c o m p l e t e l y r e l i a b l e r e a d i n g s of t h e p o s i t i o n l i n e w i l l
be obtained.
A shortcoming of t h i s system is t h e n e c e s s i t y t o r e l a t e t h e
a i r c r a f t t o t h e l o c a l t e r r a i n o n t h e b a s i s of t h e i n i t i a l r e a d i n g
of t h e hyperbolic coordinates.
In addition, during f l i g h t , there
may b e r e a d i n g s o f o t h e r i s o p h a s a l l i n e s , d u e t o i n t e r f e r e n c e , w h i c h
c a n be d e t e r m i n e d a n d c o r r e c t e d o n l y by a r e p e a t e d r e l a t i o n o f t h e
a i r c r a f t t o t h e l o c a l t e r r a i n by means of o t h e r m e t h o d s .

(2)
By m o d u l a t i o n o f t h e c a r r i e r f r e q u e n c y o f t h e g r o u n d r a d i o
s t a t i o n s a t v e r y low f r e q u e n c i e s ( w i t h l o n g m o d u l a t i n g w a v e s , c o n s i d ­
e r a b l y i n c r e a s i n g t h e p o s s i b l e d i f f e r e n c e i n d i s t a n c e s from t h e
aircraft t o the radio stations).
I n t h i s case, a t a low f r e q u e n c y
p h a s e , t h e r o u g h p o s i t i o n of t h e i s o p h a s a l l i n e o f a c a r r i e r f r e ­
q u e n c y or t h e f r e q u e n c y o f t h e s e c o n d m o d u l a t i o n w i t h a s m a l l , l o n g
period can be determined.

3 16


(3)
By u s i n g s e v e r a l c a r r i e r f r e q u e n c i e s f o r t h e g r o u n d r a d i o
s t a t i o n s , t h e isophasal l i n e can be considered t o be determined
i f it i s s i m u l t a n e o u s l y on t h e i s o p h a s a l l i n e s f o r a l l f r e q u e n c i e s
a t which t h e measure ment i s c a r r i e d o u t ( u s u a l l y t h r e e f r e q u e n ­
c i e s , s i n c e t w o w i l l b e i n a d e q u a t e i n some c a s e s ) .
On a d j a c e n t
isophasal l i n e s , f o r each frequency used, t h e isophasal l i n e s of
o t h e r f r e q u e n c i e s w i l l n o t c o i n c i d e with t h e readings of t h e phaso­
meter.

Navigational Applications o f Differential-Rangefinding Systems
Differential-rangefinding navigational systems, l i k e fan-type
beacons, are intended primarily f o r determining t h e locus of t h e
a i r c r a f t on two p o s i t i o n l i n e s .
T h e r e f o r e , t h e p r i n c i p a l method
of aircraft navigation using these systems is t h e determination
o f t h e n a v i g a t i o n a l e l e m e n t s on t h e b a s i s o f a s e r i e s o f d e t e r m i n ­
a t i o n s of t h e LA.
By r e c o r d i n g a n d p l o t t i n g on a c h a r t a s e r i e s o f p o i n t s f o r
t h e l o c u s of t h e a i r c r a f t , r e c o r d i n g t h e time a t which t h e y were
passed, and using a s c a l e r u l e r and p r o t r a c t o r t o measure t h e d i s ­
t a n c e s b e t w e e n t h e m o n t h e c h a r t , a s w e l l a s t h e d i s t a n c e Erom t h e
f i r s t r e c o r d i n g of t h e LA t o t h e s e c o n d , i t i s e a s y l t o d e t e r m i n e
t h e speed and f l i g h t angle of t h e a i r c r a f t .

4J = a1,2;

where a1.2 i s t h e azimuth of t h e second r e c o r d i n g of t h e LA from
t h e f i r s t and S1.2 is t h e d i s t a n c e b e t w e e n t h e r e c o r d i n g s o f t h e
LA.
The d r i f t a n g l e o f t h e a i r c r a f t i s d e t e r m i n e d a s t h e d i s t a n c e
between t h e a c t u a l f l i g h t p a t h a n g l e and t h e average c o u r s e of t h e
a i r c r a f t o v e r t h e segment b e t w e e n two s u c c e s s i v e r e c o r d i n g s o f t h e
LA:

W i t h a known g r o u n d s p e e d a n d d r i f t a n g l e , t a k i n g t h e a i r s p e e d
i n t o a c c o u n t as w e l l as t h e c o u r s e t o b e f o l l o w e d , t h e wind param­
eters a t f l i g h t a l t i t u d e can be determined with t h e a i d of a naviga­
tional slide rule.
I n s p e c i a l c o n d i t i o n s , when t h e f l i g h t d i r e c t i o n c o i n c i d e s
w i t h one of t h e b r a n c h e s of t h e h y p e r b o l i c f l i g h t l i n e s , t h e f l i g h t
To d o t h i s , i t i s s u f f i c i e n t t o m a i n ­
c a n b e made a l o n g t h e l a t t e r .
t a i n a c o n s t a n t r e a d i n g f o r t h e c a l c u l a t o r of h y p e r b o l i c c o o r d i n a t e s
of one p a i r .
The f a m i l y o f p o s i t i o n l i n e s f o r t h e s e c o n d p a i r i n
t h i s case i s u s e d t o m o n i t o r t h e p a t h f o r d i s t a n c e .

317


/303

M o n i t o r i n g t h e p a t h f o r d i s t a n c e b y means o f t h e r e a d i n g s o f
o n e o f t h e c o u n t e r s c a n b e u s e d i n t h e c a s e when t h e a i r c r a f t n a v i ­
g a t i o n i n t e r m s o f d i r e c t i o n i s c a r r i e d o u t u s i n g two d e v i c e s , e . g . ,
t h e U S W b e a r i n g o f a g o n i o m e t r i c s y s t e m or a f a n - t y p e b e a c o n .
To increase the f e a s i b i l i t y of using differenfial-rangefind­
ing systems, t h e hyperbolic coordinates can be converted t o ortho­
d r o m i c or g e o g r a p h i c a l o n e s ( s e e C h a p t e r I , S e c t i o n 7 ) .

I n some h y p e r b o l i c s y s t e m s o f a i r c r a f t n a v i g a t i o n , e . g . , t h a t
o f Decca a n d D e c t r a ( E n g l a n d ) , s i m p l i f i e d m e t h o d s o f a u t o m a t i c p l o t ­
t i n g o f t h e a i r c r a f t c o u r s e on a s p e c i a l c h a r t u s e t h e movement
For t h i s p u r p o s e ,
of a pen i n mutually perpendicular d i r e c t i o n s .
s p e c i a l c h a r t s a r e made on w h i c h t h e h y p e r b o l i c l i n e s o f t h e f i r s t
and second family a r e l a i d out a t r i g h t a n g l e s .
Naturally t h i s
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r e s u l t s i n d i s t o r t i o n o f t h e c o n t o u r s o f t h e t e r r a i n on t h e c h a r t ,
as w e l l as t h e s c a l e a n d g e o g r a p h i c g r i d , a n d t h e l i n e o f f l i g h t
of t h e aircraft i s a l s o bent.
Such a method o f r e c o r d i n g h a s a number o f s h o r t c o m i n g s ( e . g . ,
i n r e l a t i o n t o t h e c a l c u l a t i o n of o r t h o d r o m i c c o o r d i n a t e s f o r t h e
a i r c r a f t ) , b u t it i s v e r y e a s y t o a c h i e v e from t h e t e c h n i c a l s t a n d ­
p o i n t and i t s shortcomings are considerably reduced i f t h e p a t h
of t h e a i r c r a f t has markings f o r d i s t a n c e .
Methods o f

Improving D i f f e r e n t i a l R a n g e f i n d i n g
Navigational Systems

The d e s i g n o f h y p e r b o l i c s y s t e m s c o n t a i n s e l e m e n t s w h o s e i m ­
provement l e a d s t o a c o n v e r s i o n of t h e system t o a h y p e r b o l i c - r a n g e ­
f i n d i n g or h y p e r b o l i c - e l l i p t i c a l s y s t e m .
Such e l e m e n t s i n c l u d e t h e s t a n d a r d f r e q u e n c y g e n e r a t o r s a b o a r d
When t h e s e g e n e r a t o r s o p e r a t e i n a h i g h l y s t a b l e
the aircraft.
r e g i m e , t h e r e f e r e n c e s i g n a l s from t h e s e g e n e r a t o r s c a n b e k e p t
s o p r e c i s e t h a t i t becomes p o s s i b l e t o m e a s u r e d i s t a n c e s t o one
of t h e ground r a d i o s t a t i o n s .
T o do t h i s , i t i s s u f f i c i e n t t o combine
t h e p h a s e s o f t h e f r e q u e n c i e s of t h e g e n e r a t o r a b o a r d t h e a i r c r a f t
and t h e ground r a d i o s t a t i o n w i t h an i n i t i a l d i s t a n c e s e t t i n g ( e . g . ,
t h e takeoff point of t h e a i r c r a f t ) .
F u r t h e r changes i n d i s t a n c e
c a n be d e t e r m i n e d by t h e d e v i a t i o n o f t h e p h a s e s o f t h e s e f r e q u e n c i e s
or b y t h e d e v i a t i o n o f t h e p u l s e s i g n a l s , i f t h e s y s t e m i s o p e r ­
a t i n g i n a pulse regime.
Measurement of d i s t a n c e i n c o n n e c t i o n w i t h one p a i r o f h y p e r b o l i c
p o s i t i o n l i n e s makes it p o s s i b l e t o c o n s i d e r a b l y improve t h e accur­
acy w i t h which t h e l o c u s o f t h e a i r c r a f t i s d e t e r m i n e d o v e r l o n g
d i s t a n c e s , a n d one p a i r o f g?ound r a d i o s t a t i o n s w i l l s u f f i c e f o r
measurements.
However, t h e c o n d i t i o n s f o r measurement b e t w e e n t h e
f o c i of t h e system near t h e f o c a l a x i s w i l l remain unfavorable (Fig.
3.39).

318


I t i s more a d v a n t a g e o u s i n t h i s case t o u s e t h e h y p e r b o l i c
network of p o s i t i o n l i n e s ( s e e Chapter I , S e c t i o n 7).

H o w e v e r , s i n c e w e know t h e d i f f e r e n c e b e t w e e n t h e d i s t a n c e s
t o t h e two r a d i o s t a t i o n s as w e l l a s t h e d i s t a n c e t o one o f them,
i t i s e a s y t o d e t e r m i n e t h e sum o f t h e d i s t a n c e s t o t h e s e r a d i o
stations, e.g., if
S2 > SI a n d AS=s2-s,,

so that

and

Sz = Si

SI

+ AS

+ s, = 2s1 -I-p3.

S i m i l a r l y , � o r t h e c a s e when 5 2 < S I ,

T h e r e f o r e , i n o r d e r t o o b t a i n t h e number o f t h e h y p e r b o l a ,
it is s u f f i c i e n t t o use t h e difference i n distances, while t o obtain
t h e n u m b e r o f t h e e l l i p s e , we m u s t d o u b l e t h e d i s t a n c e t o o n e o f
t h e r a d i o s t a t i o n s and add t h e d i f f e r e n c e i n d i s t a n c e s with t h e
/305
corresponding sign.
One g r e a t a d v a n t a g e o f t h e h y p e r b o l i c - e l l i p t i c a l n e t w o r k i s
t h e o r t h o g o n a l i t y of t h e i n t e r s e c t i o n of t h e p o s i t i o n l i n e s a t any
p o i n t i n t h e f i e l d which i s i n v o l v e d .
On i n d i v i d u a l s h e e t s o f t h e
c h a r t , t h e h y p e r b o l i c - e l l i p t i c a l network has t h e appearance of a
nearly rectangular g r i d with noticeable curvature of t h e p o s i t i o n
l i n e s o n l y i n t h e v i c i n i t y of t h e f o c i of t h e system.
When u s i n g a h y p e r b o l i c - r a n g e ­
f i n d i n g (and e s p e c i a l l y a hyper­
b o l i c - e l l i p t i c a l system, a system
of p o s i t i o n l i n e s ) t h e accuracy
w i t h which t h e c o o r d i n a t e s o f t h e
aircraft are determined over long
distances is increased several fold,
s o t h a t t h e p r a c t i c a l range of appli­
cation of t h e system is a l s o consid­
erably increased, with t h e use of
o n l y one p a i r o f ground r a d i o s t a ­
tions.
Fig. 3.39.
Combination of
H y p e r b o l i c and R a n g e f i n g i n g
Navigational Systems.

It should be mentioned, however,
t h a t a serious obstacle t o t h e devel­
opment
of systems of long-range
n a v i g a t i o n f o r u s e on h i g h - s p e e d
a i r c r a f t i s t h e low n o i s e s t a b i l i t y

3 19

o f o p e r a t i o n o f t h e s e s y s t e m s , s i n c e o n l y v e r y l o n g waves c a n b e
used f o r navigation over long distances.

4.

AUTONOMOUS RADIO-NAVIGATIONAL INSTRUMENTS


I n r e c e n t y e a r s , t h e r e has been a considerable i n c r e a s e i n
t h e use of r a d i o n a v i g a t i o n a l i n s t r u m e n t s which are housed completely
aboard t h e a i r c r a f t and o p e r a t e without t h e need f o r ground f a c i l ­
ities.
Such i n s t r u m e n t s a r e c a l l e d autonomous r a d i o - n a v i g a t i o n a l
i n s t r u m e n t s or, i f t h e i r o p e r a t i o n i s c o m b i n e d w i t h s o m e o t h e r n a v i g a ­
t i o n a l e q u i p m e n t a b o a r d t h e a i r c r a f t , autonomous n a v i g a t i o n a l s y s ­
tems.
These i n c l u d e a i r c r a f t n a v i g a t i o n a l r a d a r , Doppler systems
f o r a i r c r a f t navigation, and r a d i o altimeters.

All a u t o n o m o u s r a d i o - n a v i g a t i o n a l i n s t r u m e n t s o p e r a t e o n u l t r a ­
s h o r t waves, s i n c e they have a very high ( p r a c t i c a l l y complete) free­
dom f r o m i n t e r f e r e n c e d u r i n g o p e r a t i o n ( n o t c o u n t i n g a r t i f i c i a l
interference).
Doppler m e t e r s f o r measuring t h e ground s p e e d and d r i f t a n g l e
of t h e a i r c r a f t m e a s u r e t h e m o t i o n p a r a m e t e r s o f t h e a i r c r a f t d i r e c t l y
r e l a t i v e t o t h e E a r t h ' s s u r f a c e , which c l e a r l y d i f f e r e n t i a t e s them
from a l l e x i s t i n g forms of n a v i g a t i o n a l equipment, e s p e c i a l l y w i t h
/306
r e g a r d t o problems of automation of a i r c r a f t n a v i g a t i o n an? p i l o t age of aircraft.

A i rcraft Navigational

Radar

A i r c r a f t n a v i g a t i o n a l r a d a r i s a v e r y f l e x i b l e and e f f e c t i v e
m e t h o d o f a i r c r a f t n a v i g a t i o n d u r i n g f l i g h t o v e r l a n d or s e a c l o s e
t o coastal regions.

I n terms o f t h e geometry of t h e i r u s e , a i r c r a f t r a d a r d e v i c e s
However,
c a n b e i n c l u d e d among t h e goniometric-rangefinding s y s t e m s .
i n c o m p a r i s o n t o t h e goniometric-rangefinding n a v i g a t i o n a l s y s t e m s ,
t h e y h a v e a number o f t a c t i c a l a d v a n t a g e s :

(1) T h e h i g h s a t u r a t i o n o f g r o u n d l a n d m a r k s m a k e s i t p o s s i b l e
t o s e l e c t t h e most s u i t a b l e ones f o r measurement i n n a v i g a t i o n .
(2)
The l a c k o f e r r o r s i n d e t e r m i n i n g t h e b e a r i n g s o f l a n d ­
marks from t h e r a d i o d e v i a t i o n o f b o t h t h e a i r c r a f t i t s e l f and t h e
l o c a l r e l i e f , s o m e t h i n g which a f f e c t s a l l non-autonomous n a v i g a ­
t i o n a l systems.
(3)
The p o s s i b i l i t y o f v i s u a l i z j - n g g r o u n d l a n d m a r k s w i t h t h e
purposes of determining ground speed and d r i f t a n g l e t o a b e t t e r
degree t h a n with o p t i c a l methods.

(4)
The p o s s i b i l i t y o f i d e n t i f y i n g d a n g e r o u s m e t e o r o l o g i c a l
c o n d i t i o n s i n f l i g h t ( t h u n d e r s t o r m s , p o w e r f u l cumulus and cumulonim­
bus clouds).
320

... -

___ __ .

...

. ..

.... .. ... .. - . ..

.

.

.
.

... ..

. .

.-.- .. ...

(5)
The h i g h a c c u r a c y a n d e a s e o f t h e m e a s u r e m e n t s u s i n g o n l y
one o p e r a t i o n a l f r e q u e n c y .

A t t h e same t i m e ,
s e v e r a l shortcomings:

t h e navigational use of aircraft r a d a r has

(a)
The b e a r i n g o f t h e a i r c r a f t c a n b e u s e d o n l y a s a b a s i s
f o r measuring t h e aircraft course, t h u s lowering t h e accuracy of
distance findings.
A c e r t a i n amount o f e x p e r i e n c e i s n e e d e d f o r c o r r e c t r e c o g ­
(b)
n i t i o n of ground landmarks and t h e p o s s i b i l i t y of e r r o r s i n d e t e r ­
mining a landmark, s i n c e they are n o t l a b e l l e d .

The o p e r a t i n g p r i n c i p l e o f r a d a r i s b a s e d on t h e a b i l i t y o f
e l e c t r o m a g n e t i c waves a t h i g h f r e q u e n c i e s t o b e r e f l e c t e d from o b j e c t s
l o c a t e d along t h e i r propagation path (from t h e i n t e r f a c e between
media w i t h d i f f e r e n t o p t i c a l d e n s i t i e s ) .
T o o b t a i n a p a n o r a m i c i m a g e o f t h e t e r r a i n , a r o t a t i n g or s c a n ­
ning antenna is used t o cover a c e r t a i n s e c t o r , s o t h a t i t s posi­
t i o n m u s t b e s y n c h r o n i z e d w i t h t h e p o s i t i o n o f t h e s c k n n i n g beam
on t h e s c r e e n o f a c a t h o d e r a y t u b e .
In addition t o synchronizing
t h e d i r e c t i o n of t h e antenna with t h e scanning d i r e c t i o n of t h e
beam, i t i s a l s o n e c e s s a r y t o e n s u r e t h a t t h e b e g i n n i n g o f t h e s c a n
i s s y n c h r o n i z e d w i t h t h e moment when t h e U S W p u l s e s a r e o m i t t e d f r o m
t h e antenna transmitter

.

The r a d a r t r a n s m i t t e r e m i t s h i g h - f r e q u e n c y p u l s e s whose p r o p ­
a g a t i o n d i r e c t i o n d e p e n d s on t h e p o s i t i o n of t h e r o t a t i n g a n t e n n a
/307
a t t h e moment o f e m i s s i o n .
T h e s c a n n i n g o f t h e i n d i c a t o r beam b e g i n s
s i m u l t a n e o u s l y w i t h t h e e m i s s i o n o f t h e p u l s e , moving from t h e c e n t e r
o f t h e s c r e e n t o w a r d t h e p e r i p h e r y ; t h e d i r e c t i o n i n w h i c h t h e beam
moves c o i n c i d e s w i t h t h e movement o f t h e a n t e n n a .
S i n c e t h e p r o p a g a t i o n r a t e of e l e c t r o m a g n e t i c waves i s v e r y
h i g h , and t h e r o t a t i o n a l s p e e d of t h e a n t e n n a and t h e s c a n r a t e
a r e s l o w , t h e p u l s e o f wave e n e r g y c a n c o v e r t h e d i s t a n c e t o t h e
i r r a d i a t e d o b j e c t a n d r e t u r n i n a p e r i o d of t i m e w h i c h i s s u f f i ­
c i e n t l y s h o r t s o t h a t t h e a n t e n n a h a s n o t y e t moved t h r o u g h a n y
noticeable angle.
Therefore, t h e d i r e c t i o n of t h e antenna a t t h e
moment o f r e c e i v i n g t h e s i g n a l c o i n c i d e s w i t h i t s d i r e c t i o n a t t h e
moment o f e m i s s i o n .
The r e c e i v e d r e f l e c t e d s i g n a l i s a m p l i f i e d
i n t h e r e c e i v e r a n d p a s s e s t o a n i n d i c a t o r , where i t c o n t r o l s t h e
b r i g h t n e s s o f t h e s c a n n i n g beam.
Thus, t h e r a d a r s c r e e n shows t h e f o l l o w i n g :
(a)
The d i r e c t i o n o f t h e o b j e c t on t h e b a s i s o f t h e a n t e n n a
p o s i t i o n a t t h e moment o f e m i s s i o n a n d r e c e p t i o n o f t h e s i g n a l .

(b)

The d i s t a n c e t o t h e o b j e c t on t h e b a s i s o f t h e t i m e r e ­

321


q u i r e d for t h e s i g n a l t o t r a v e l b e t w e e n t h e moment when i t i s e m i t ­
t e d t o t h e moment w h e n i t i s r e c e i v e d .
(c)
The n a t u r e o f t h e o b j e c t , on t h e b a s i s o f t h e b r i g h t n e s s
o f t h e s c a n n i n g beam a t t h e p o i n t w h e r e t h e r e f l e c t e d wave i s r e c e i v e d .

u
s t a n d a . r d f r ea ue'n CY
and dividers
-

Fig.

3.40.

Diagram o f A i r c r a f t R a d a r .

T h e r a d a r s c r e e n h a s a l o n g a f t e r g l o w s o t h a t when t h e a n t e n n a
h a s made a c o m p l e t e r e v o l u t i o n , t h e s c r e e n s t i l l s h o w s a t r a c e o f
a l l t h e i r r a d i a t e d o b j e c t s on t h e E a r t h ' s s u r f a c e wh i ch a r e l o c a t e d
i n t h e f i e l d s c a n n e d by t h e r a d a r .
T h e m a i n s e c t i o n o f t h e r a d a r , c o n t r o l l i n g t h e pera at ion o f
t h e e n t i r e system, i s t h e standard-frequency generator with fre­
quency d i v i d e r s f o r forming d i s t a n c e markings and a s i g n a l - t r a n s ­
m i s s i o n f r e q u e n c y s y n c h r o n i z e d w i t h t h e s a w - t o o t h s c a n n i n g image
on t h e s c r e e n ( F i g . 3 . 4 0 ) .
The s i g n a l s from t h e s t a n d a r d - f r e q u e n c y g e n e r a t o r r e a c h t h e
modulator, where t h e y a r e c o n v e r t e d t o h i g h - v o l t a g e r e c t a n g u l a r
T h e h i g h - v o l t a g e p u l s e s from
o s c i l l a t i o n s of a s p e c i a l l e n g t h .
t h e m o d u l a t o r p a s s t o t h e t r a n s m i t t e r m a g n e t r o n , where h i g h - f r e ­
quency groups a r e g e n e r a t e d a c c o r d i n g t o t h e p u l s e l e n g t h .

1308

The h i g h f r e q u e n c y r e a c h e s t h e a n t e n n a t h r o u g h a w a v e g u i d e
and i s r a d i a t e d i n t o space.
A t t h e same t i m e , i n s y n c h r o n i z a t i o n w i t h t h e p u l s e s o f h i g h
v o l t a g e which are s e n t t o t h e t r a n s m i t t e r , t h e s c a n n i n g g e n e r a t o r .
forms a s a w - t o o t h v o l t a g e w h i c h c o n t r o l s t h e s c a n n i n g b e a m on t h e
screen.
The s c a n n i n g r a t e d e p e n d s on t h e s t e e p n e s s o f t h e s l o p e
for t h e s a w - t o o t h w a v e s .
A t a low s c a n n i n g r a t e , a f i n e image s c a l e
i s o b t a i n e d as w e l l as l o n g - d i s t a n c e d e t e c t i o n o f o b j e c t s .
When
t h e s c a n n i n g r a t e i s i n c r e a s e d , t h e s c a l e o f t h e image d e c r e a s e s
p r o p o r t i o n a t e l y w i t h t h e d i s t a n c e c o v e r e d by t h e r a d i u s o f t h e s c r e e n .
The c o n t r o l o f t h e s c a n n i n g r a t e i s a c h i e v e d w i t h t h e a i d o f a s w i t c h
on t h e c o n t r o l p a n e l .

322

The d u r a t i o n o f t h e h i g h - f r e q u e n c y p u l s e s c o n t r o l s t h e r e s o l v ­
For e x a m p l e , a p u l s e d u r a t i o n o f 2 m i c r o ­
i n g power o f t h e r a d a r .
s e c o n d s c o r r e s p o n d s t o a p r o p a g a t i o n d i s t a n c e o f t h e wave ( b a c k
a n d f o r t h ) o f 300 m .
Hence, i f two r e f l e c t i n g o b j e c t s are l o c a t e d
a t a d i s t a n c e o f l e s s t h a n 300 m a l o n g t h e p r o p a g a t i o n d i r e c t i o n
o f t h e wave, t h e y w i l l a p p e a r as one o b j e c t .
T h e r e f o r e , when w o r k ­
i n g a t c o u r s e s c a l e s , t h e m o d u l a t o r i s s e t for t h e f o r m a t i o n of
s h o r t e r pulses with an increase i n t h e frequency of t h e i r trans­
mission.
The r e s o l v i n g p o w e r o f t h e r a d a r i n t e r m s o f a z i m u t h s i s a
f u n c t i o n o f t h e s h a r p n e s s o f t h e d i r e c t i o n a l i t y o f t h e a n t e n n a beam.
The a z i m u t h s o f g r o u n d l a n d m a r k s c a n b e d e t e r m i n e d i m m e d i a t e l y
by t h e p o s i t i o n o f t h e a n t e n n a ( a n d t h e r e f o r e by t h e s c a n n i n g l i n e
on t h e s c r e e n ) , a n d t h e a n t e n n a mechanism i s f i t t e d w i t h a s e l s y n
mechanism f o r t i l t i n g t h e i n d i c a t o r .
T h e a z i m u t h r e a d i n g i s made
on a s c a l e l o c a t e d a l o n g t h e p e r i p h e r y o f t h e s c r e e n .
To m e a s u r e t h e d i s t a n c e t o a l a n d m a r k , p u l s e s f r o m t h e f r e ­
quency d i v i d e r a r e s e n t t o t h e r e c e i v e r (and t h e r e f o r e t o t h e scan­
n i n g beam).
T h e s e p u l s e s i n c r e a s e t h e b r i g h t n e s s o f t h e beam a t
c e r t a i n d i s t a n c e s from t h e c e n t e r o f t h e s c r e e n , f o r m i n g c i r c u l a r
distance markings.
When u s i n g t h e r a d a r on d i f f e r e n t s c a l e s t h e d i s t a n c e m a r k ­
For example,
ings are shifted t o different distance intervals.
w i t h a s c a l e o f 1 0 km f o r t h e r a d i u s o f t h e s c r e e n , t h e m a r k i n g s
a r e u s u a l l y 2 km a p a r t ; when u s i n g s c a l e s f r o m 1 0 t o 1 0 0 km, t h e
m a r k i n g s a r e 1 0 km a p a r t ; a t a s c a l e o f 2 0 0 km, t h e y a r e 2 0 or 4 0
km a p a r t .
Now l e t u s f o l l o w t h e p a t h o f t h e h i g h - f r e q u e n c y p u l s e s f r o m
t h e t r a n s m i t t e r t o t h e o b j e c t a n d b a c k a g a i n , a n d s e e how t h e y c o n t r o l
t h e b r i g h t n e s s o f t h e s c a n n i n g beam.
The h i g h - f r e q u e n c y p u l s e p a s s e s t h r o u g h t h e wave g u i d e t o t h e
r a d i a t i n g h o r n o f t h e a n t e n n a , a f t e r which i t i s s h a p e d i n t o t h e
r e q u i r e d d i r e c t i o n a l d i a g r a m f o r r a d i a t i o n by means o f a r e f l e c ­
tor.
Usually, t h e d i r e c t i o n a l i t y of t h e antenna i n t h e h o r i z o n t a l
T o do t h i s , i t i s n e c e s s a r y
p l a n e i s made a s s h a r p a s p o s s i b l e .
f o r t h e p h a s e o f t h e beam when e m e r g i n g f r o m t h e a n t e n n a t o r e m a i n
constant over its e n t i r e perpendicular cross section (Fig. 3.41),
i . e . , t h e r e f l e c t i o n i n t h i s p l a n e must have a shape such t h a t t h e
wave p a t h from t h e h o r n r a d i a t o r t o t h e s u r f a c e o f t h e r e f l e c t o r
and a l o n g i t s chord of emergence i s uniform.
The c h a r a c t e r i s t i c o f d i r e c t i o n a l i t y o f t h e r a d i a t i o n i n t h e
v e r t i c a l p l a n e m u s t b e s u c h t h a t t h e i l l u m i n a t i o n ot t h e t e r r a i n
from t h e v e r t i c a l o f t h e a i r c r a f t i s as u n i f o r m as p o s s i b l e o v e r
t h e e n t i r e e f f e c t i v e r a d i u s of t h e r a d a r .
T o do t h i s , i t i s n e c e s ­
s a r y t o h a v e t h e maximum a m o u n t o f w a v e e n e r g y t r a n s m i t t e d a t s m a l l

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a n g l e s t o t h e p l a n e o f t h e h o r i z o n , i . e . , o v e r t h e maximum r a n g e ,
and t o h a v e t h e s m a l l e s t amount of e n e r g y r a d i a t e d a l o n g t h e v e r t ­
ical of t h e aircraft.
S u c h a c h a r a c t e r < s t i c is c a l l e d t h e c o s e c a n t square, i . e . , t h e r e f l e c t o r s i n t h e vert­
i c a l plane are given a shape such t h a t
t h e a m o u n t of e n e r g y r a d i a t e d i n t o s p a c e
is roughly proportional t o t h e square
of t h e cosecant of t h e angle of t h e plane
of the horizon t o the propagation direc­
tion.

4;

I n some t y p e s o f r a d a r , a n a c i c u l a r
c h a r a c t e r i s t i c o f d i r e c t i o n a l i t y i s emplQyed,
:.e.,
one which i s s h a r p e s t i n b o t h t h e
h o r i z o n t a l a n d v e r t i c a l p l a n e s , combin­
ing it with t h e cosesant-square i n the
v e r t i c a l p l a n e , e . g . , by a s c a n n i n g c y c l e .
Fig. 3.41.
R a d a r Ant e n n a f o r Use A b o a r d
This i s achieved by u s i n g s p e c i a l l y shaped
Aircraft.
reflectors with a telescoping deflector
or b y s e n d i n g e n e r g y t o t h e a n t e n n a b y
d i f f e r e n t wave g u i d e s f o r t h e a c i c u l a r a n d c o s e c a n t - s q u a r e a n t e n n a
c h a r a c t e r i s t i c s af t h e r a d a r .
A n t e n n a s w i t h c o s e c a n t - s q u a r e c h a r a c t e r i s t i c s a r e u s e d �or
c i r c u l a r - s c a n r a d a r , mounted below t h e f u s e l a g e o f t h e a i r c r a f t .
Antennas w i t h combined r a d i a t i o n a r e u s e d f o r s e c t o r - s c a n r a d a r s
and a r e mounted i n t h e n o s e o f t h e f u s e l a g e t o c o v e r o n l y t h e a r e a
ahead of t h e aircraft.
I n t h i s c a s e , t h e r a d a r s c r e e n i s made w i t h
t h e c e n t e r d i s p l a c e d s o t h a t t h e maximum a r e a o f t h e s c r e e n c a n
be used.
Usually, t h e t i l t i n g of t h e
t h e r e f o r e t h e c h a r a c t e r i s t i c s of
i s a d j u s t e d m a n u a l l y by means o f
a s w i t c h on t h e c o n t r o l p a n e l o f

antenna i n t h e v e r t i c a l plane (and
d i r e c t i o n a l i t y of t h e r a d i a t i o n )
a s p e c i a l e l e c t r i c a l d e v i c e and
the radar.

The t r a n s m i t t i n g a n t e n n a o f t h e r a d a r a c t s s i m u l t a n e o u s l y a s
a receiving antenna, since t h e d i r e c t i o n a l c h a r a c t e r i s t i c s of t h e
a n t e n n a a r e r e v e r s e d , i . e . , u s e d b o t h for e m i t t i n g a n d r e c e i v i n g
t h e wave e n e r g y .
I n o r d e r f o r t h e p u l s e s o f wave e n e r g y e m i t t e d f r o m t h e t r a n s ­
m i t t e r n o t t o r e t u r n i m m e d i a t e l y t o t h e wave g u i d e o f t h e r e c e i v e r ,
s p e c i a l a r r e s t e r s a r e u s e d w h i c h b l o c k t h e wave e n e r g y f r o m e n t e r ­
i n g t h e r e c e i v e r a t t h e moment w h e n t h e t r a n s m i t t e r i s o p e r a t i n g .
The t r a n s m i s s i o n f r e q u e n c y o f t h e p u l s e s o f wave e n e r g y f r o m
t h e t r a n s m i t t e r i s s e t s o t h a t t h e t i m e i n t e r v a l s b e t w e e n them a r e
not s h o r t e r than those required f o r propagation of electromagnetic
w a v e s t o t h e m o s t d i s t a n t o b j e c t a t a g i v e n o p e r a t i n g r a n g e for
t h e r a d a r and f o r i t s r e t u r n t o t h e a i r c r a f t .
When u s i n g t h e r a d a r
a t large-scale settings, the decrease i n t h e pulse duration is

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accompanied
serving the
t i o n of t h e
between t h e

by a n i n c r e a s e i n t h e t r a n s m i s s i o n f r e q u e n c y , t h u s p r e ­
a v e r a g e power of t h e t r a n s m i t t e r .
Hence, t h e r e c e p ­
reflected signals takes place i n the t i m e intervals
p u l s e s o f wave e n e r g y e m i t t e d b y t h e t r a n s m i t t e r .

The r a d a r r e c e i v e r s h a v e s p e c i a l vacuum d e v i c e s ( k l y s t r o n s
f o r g e n e r a t i n g h i g h f r e q u e n c y ) w h i c h p l a y t h e same r o l e a s h e t e r ­
odynes i n c o n v e n t i o n a l r e c e i v e r s .
The s i g n a l s r e c e i v e d b y t h e a n t e n n a a r e mi x ed w i t h t h e f r e ­
quency o f t h e k l y s t r o n ; a n i n t e r m e d i i i t , e f r e q u e n c y i s p r o d u c e d which
t h e n g o e s on ( a f t e r d e t e c t i o n a n d a m p 1 i f i c a t i o n ) t o c o n t r o l t h e b r i g h t ­
n e s s o f t h e s c a n n i n g beam.
I n a d d i t i o n t o t h e s p e c i a l f e a t u r e s of t h e r a d a r which w e have
d i s c u s s e d above, t h e r e c e i v e r h a s a d d i t i o n a l c i r c u i t s and c o n t r o l
units.
I n p a r t i c u l a r , t o a l l o w t h e f r e q u e n c y of t h e k l y s t r o n t o
b e c h a n g e d , t h e r e i s a n a u t o m a t i c f r e q u e n c y a d j u s t e r (AFA), e t c .
F o r i m p r o v e d c o n t r a s t o f t h e i m a g e on t h e s c r e e n , i n a d d i t i o n
t o t h e devices f o r a d j u s t i n g t h e o v e r a l l amplification of t h e r e c e i v e r ,
t h e o p e r a t o r o f t h e r a d a r c a n u s e a s e p a r a t e s i g n a l a m p l i f i e r which
This makes it p o s s i b l e t o d i s ­
o p e r a t e s a t h i g h a n d low l e v e l s .
t i n g u i s h s h a d e d or i l l u m i n a t e d o b j e c t s o n t h e E a r t h ' s s u r f a c e a s
desired.
For example, t o examine p o p u l a t e d a r e a s , t h e h i g h - l e v e l
s i g n a l s a r e i n c r e a s e d and t h e low-level s i g n a l s a r e reduced (by
decreasing t h e b r i g h t n e s s of t h e background of t h e s c r e e n ) .
To
pick out r i v e r s and l a k e s , t h e low-level s i g n a l s are i n c r e a s e d ,
t h u s improving t h e v i s i b i l i t y of shaded o b j e c t s a g a i n s t a b r i g h t e r
g e n e r a l background.
I t should be mentioned t h a t f o r t h e formation of high-frequency
p u l s e s by t h e t r a n s m i t t e r , v e r y h i g h v o l t a g e s m u s t b e p r o d u c e d i n
t h e m o d u l a t o r ; t h i s means t h a t a t h i g h a l t i t u d e s ( i . e . , a t l o w atmo­
s p h e r i c p r e s s u r e ) , t h e r e may b e f l a s h o v e r s i n t h e w i r i n g o f t h e s e
units.
T h e r e f o r e , t h e s e u n i t s ( i n c l u d i n g t h e wave g u i d e s o f t h e
/311
t r a n s m i t t e r ) a r e hermetically s e a l e d and t h e r e q u i r e d p r e s s u r e i s
m a i n t a i n e d i n t h e m b y a s p e c i a l pump or b y s y s t e m s f o r p r e s s u r i z i n g
the aircraft cabin.

I n d i c a t o r s of A i r c r a f t N a v i g a t i o n a E R a d a r s
The a i r c r a f t r a d a r i s a n a u t o n o m o u s g o n i o m e t r i c - r a n g e - f i n d ­
i n g a n d s i g h t i n g d e v i c e , s o t h a t i t s i n d i c a t o r m u s t b e made s o t h a t
a l l r e q u i r e d n a v i g a t i o n a l measurements can be performed satisfac­
t o r i l y with it.
C i r c u l a r i n d i c a t o r s a r e t h e o n e s which a r e o f g r e a t e s t i n t e r ­
e s t from t h e n a v i g a t i o n a l s t a n d p o i n t ( F i g . 3 . 4 2 ) .
The c e n t e r o f t h i s i n d i c a t o r , m a r k i n g t h e ' p o s i t i o n o f t h e a i r ­
c r a f t a g a i n s t t h e panorama o f t h e f i e l d of v i s i o n , c o i n c i d e s w i t h

325

the center of the screen.
Around t h e e d g e o f t h i s s c r e e n i s a s c a l e
of b e a r i n g s , which c a n b e r o t a t e d manually; i n t h e upper p a r t of
i t i s a c o u r s e m a r k i n g which shows t h e p o s i t i o n of t h e l o n g i t u d ­
The s c a l e of b e a r i n g s i s s e t t o i t s
i n a l axis of t h e aircraft.
own d i v i s i o n s b y m e a n s o f a " c o u r s e ' l . r a c k a n d p i n i o n d e v i c e , f o r
s e t t i n g t h e course o f t h e aircraft by t h e c o u r s e markings, accord­
i n g t o t h e r e a d i n g s of t h e c o u r s e i n s t r u m e n t s .
The s i g h t i n g l i n e s o f t h e i n d i c a t o r a r e m a r k e d on t h e p r o t e c ­
t i v e g l a s s o f t h e s c r e e n , which c a n b e r o t a t e d by means o f " s i g h t "
rack and p i n i o n .
F o r c o n v e n i e n c e i n s i g h t i n g , t h r e e movable p o i n t s
f o r l o n g i t u d i n a l s i g h t i n g l i n e s a r e p r o v i d e d , and one . t r a n s v e r s e
s i g h t i n g l i n e i s p r o v i d e d f o r i n d i c a t i n g t r a v e r s e s when f l y i n g o v e r
landmarks.
When t h e r a d a r i s o p e r a t i n g ,
c i r c u l a r d i s t a n c e markings appear
on t h e s c r e e n , a n d t h e d e f l e c t i o n
of t h e luminous course l i n e s of t h e
a i r c r a f t may a l s o b e i n c l u d e d .
I n t h e lower p a r t of t h e i n d i c a t o r
u n i t , i n a d d i t i o n t o t h e "course"
and " s i g h t " adjustments , t h e r e are
other controls:
"scale illumination",
"beam s c a n f o c u s " , "beam b r i g h t n e s s
a d j u s t m e n t " , a n d some t y p e s o f i n d i ­
c a t o r s a l s o have a " v e r t i c a l and h o r i ­
zontal centering of scan".
Thus, t h e c i r c u l a r s c r e e n of
t h e r a d a r can b e used t o measure b e a r ­
Fig. 3.42.
Indicator for
i n g s p r e c i s e l y or d e t e r m i n e t h e c o u r s e
Radar w i t h
Screen'
angle of a landmark, i t s d i s t a n c e ,
as w e l l as t h e p r o v i s i o n a l l i n e o f
motion of t h e landmark f o r purposes of determining t h e d r i f t angle
/312
a n d t h e g r o u n d s p e e d on t h e b a s i s of t h e t r a v e r s e of t h e f l i g h t
over t h e landmark.

(Fig.

Sector-type
3.43).

r a d a r s c r e e n s h a v e somewhat fewer p o s s i b i l i t i e s

Since t h e c e n t e r of r o t a t i o n ( t h e p o s i t i o n of t h e aircraft
o n t h e s c r e e n ) i s s h i f t e d f r o m t h e c e n t e r o f t h e s c r e e n on t h i s
t y p e o f i n d i c a t o r , t h e r e i s n o p o s s i b i l i t y for u s i n g a r o t a t i n g
s c a l e o f b e a r i n g s or s i g h t i n g l i n e s , t h u s c o m p l i c a t i n g t h e d e t e r ­
m i n a t i o n o f b e a r i n g s a n d t h e m o t i o n o f l a n d m a r k s on t h e s c r e e n .
Instead, t h e screen i s f i t t e d with a system of divergent l i n e s
f o r t h e c o u r s e a n g l e of t h e landmark ( C A L ) .
The d e t e r m i n a t i o n o f
t h e b e a r i n g s i n t h i s c a s e i s made b y a d d i n g t h e c o u r s e a n g l e o f
t h e landmark t o t h e c o u r s e o f t h e a i r c r a f t by t h e f o r m u l a :

326

TBL = TC
TBA = T C

+

CAL

+

CAL;

+ 180°
-

+

6.

I t i s very d i f f i c u l t and n o t always
p o s s i b l e on t h e s e i n d i c a t o r s t o d e t e r m i n e
t h e moment o f f l y i n g o v e r t h e t r a v e r s e s
of landmarks.

I n s t e a d o f v i s u a l i z i n g t h e movement
o f landmarks, t h e s o l u t i o n of n a v i g a t i o n a l
p r o b l e m s on t h e s e i n d i c a t o r s i s more
o f t e n accomplished by a succession of
m e a s u r e m e n t s o f t h e LA on a c h a r t .
An
e x c e p t i o n t o t h i s i s c o n s t i t u t e d by l a n d ­
m a r k s w h i c h move a c r o s s t h e s c r e e n i n
t h e immediate v i c i n i t y of t h e course
Fig. 3.43.
Indicator
m a r k i n g , and c a n b e u s e d t o d e t e r m i n e
f o r Sector-Type Radar.
t h e d r i f t a n g l e by t h e p a r a l l e l i s m of
t h e i r s h i f t i n g , using t h e l i n e s of t h e
c o u r s e a n g l e s a n d t h e g r o u n d s p e e d when p a s s i n g o v e r t h e d i s t a n c e
m a r k i n g s on t h e s c r e e n .

N a t u r e of t h e V i s i b i Z i t y of Landmarks on t h e S c r e e n of a n
A i r c r a f t Radar
For purposes of a i r c r a f t navigation using aircraft r a d a r ,
following landmarks can be used:

the

1.
Large populated a r e a s and i n d u s t r i a l e n t e r p r i s e s .
The
v i s i b i l i t y a n d o u t l i n e s o f t h e s e l a n d m a r k s d e p e n d on t h e n u m b e r
and l o c a t i o n of m e t a l s t r u c t u r e s and c o v e r i n g s i n t h e o b j e c t .
Popu­
l a t e d a r e a s a n d i n d u s t r i a l e n t e r p r i s e s a p p e a r as b r i g h t s p o t s on
t h e s c r e e n , as a r u l e , w i t h s h a r p l y bounded o u t l i n e s .
T h i s means
t h a t t h e o u t l i n e s of t h e landmarks c o i n c i d e c l o s e l y w i t h t h e i r o u t ­
l i n e s o n a c h a r t or a s t h e y a r e s e e n b y v i s u a l o b s e r v a t i o n , a s g r o u p s
o f s t r u c t u r e s w i t h n o n - m e t a l l i c c o v e r i n g s s h o w u p much l e s s c l e a r l y / 3 1 3
and a r e v i s i b l e from s h o r t e r d i s t a n c e s t h a n m e t a l s t r u c t u r e s and
coverings.
P o p u l a t e d a r e a s s h o w u p m o s t c l e a r l y w i t h maximum a m p l i f i c a ­
t i o n o f t h e h i g h - l e v e l s i g n a l s a n d a minimum a m p l i f i c a t i o n o f t h e l o w level signals.
2.
R i v e r s and l a k e s .
D u r i n g t h e summer, t h e s e l a n d m a r k s a r e
v i s i b l e as d a r k a r e a s a n d s p o t s whose o u t l i n e s m a t c h t h o s e o f t h e
landmarks a g a i n s t t h e a l i g h t e r background of t h e surrounding ter­
rain.
I n t h e w i n t e r , when t h e s e b o d i e s o f w a t e r a r e c o v e r e d b y
a smooth l a y e r o f i c e , o n l y t h e r i v e r v a l l e y s a r e s e e n , e s p e c i a l l y
a g a i n s t f o r e s t e d areas.
I c e p a c k s on r i v e r s c a n b e s e e n i n t h e
f o r m o f b r i g h t s p o t s a g a i n s t a d a r k e r b a c k g r o u n d o f snow c o v e r e d
banks.
R i v e r s and l a k e s c a n b e d i s t i n g u i s h e d by a m p l i f y i n g t h e
low-level s i g n a l s t o i n c r e a s e t h e b r i g h t n e s s of t h e e n t i r e background
327

of t h e screen.
Then t h e d a r k o b j e c t s w i l l b e o b s e r v e d as s t i l l
d a r k e r areas a g a i n s t t h e l i g h t background.

3.

Mountains.

T h e s e l a n d m a r k s a p p e a r on t h e r a d a r s c r e e n
, as t h e y
appear t o v i s u a l observation.
Mountains c a n b e d i s t i n g u i s h e d by
a s u i t a b l e s e l e c t i o n o f s i g n a l a m p l i f i c a t i o n a t b o t h h i g h a n d low
levels.

i n a form which i s v e r y c l o s e t o t h e i r n a t u r a l one, i . e .

4.
Forested areas.
Landmarks o f t h i s t y p e c a n o n l y be s e e n
c l e a r l y i n w i n t e r , a g a i n s t a g e n e r a l background of snow-covered
s u r f a c e , b y a m p l i f y i n g t h e l o w - l e v e l s i g n a l s ; i n summer, a g a i n s t
a background of v e g e t a t i o n and c u l t i v a t e d areas, f o r e s t s a r e seen
v e r y dimly and c a n n o t b e u s e d as l a n d m a r k s .
5.
Highway and r a i l w a y b r i d g e s .
T h e s e l a n d m a r k s show up e s p e c ­
The r a i l w a y s
i a l l y w e l l a g a i n s t t h e background of l a r g e r i v e r s .
t h e m s e l v e s s h o w u p c l e a r l y o n l y when t h e r e a r e e m b a n k m e n t s or s t e e l
s t r u c t u r e s for s u p p o r t i n g c a t e n a r i e s f o r e l e c t r i f i e d r a i l w a y s .
I n summer, t h e d e v e l o p m e n t o f p o w e r f u l cumulus a n d cumulonim­
b u s c l o u d s shows up v e r y c l e a r l y on r a d a r s c r e e n s .
Areas w h i c h
are dangerous f o r f l i g h t (with a l a r g e - d r o p l e t s t r u c t u r e , and t h e r e ­
f o r e w i t h i n t e n s e t u r b u l e n c e and h i g h i n t e n s i t y of e l e c t r i c a l f i e l d s )
a p p e a r on t h e s c r e e n i n t h e f o r m o f b r i g h t s p o t s w i t h d i f f u s e e d g e s .
T h e s e s t o r m s c a n b e d i s t i n g u i s h e d v e r y w e l l w i t h maximum a m p l i ­
f i c a t i o n o f h i g h - l e v e l s i g n a l s a n d minimum a m p l i f i c a t i o n o f l o w level signals.
A m p l i f i c a t i o n o f l o w - l e v e l s i g n a l s r e d u c e s t h e con­
t r a s t of t h e images o f t h e s e dangerous s t o r m s , b u t areas of r a d a r
shadows b e g i n t o a p p e a r , which a r e v e r y c l e a r on t h e s c r e e n a n d
are c h a r a c t e r i s t i c s i g n s of storm clouds.
I n o b s e r v i n g t e r r e s t r i a l landmarks a n d c l o u d s i n which t h e r e
is thunderstorm a c t i v i t y , it i s necessary (besides adjusting t h e
a m p l i f i c a t i o n l e v e l of t h e r e c e i v e r ) t o choose t h e p r o p e r i n c l i n ­
A s a r u l e , landmarks which a r e l o c a t e d
a t i o n of t h e r a d a r antenna.
c l o s e t o t h e aircraft are observed with an increased i n c l i n a t i o n
o f t h e a n t e n n a downward, w h i l e t h o s e f u r t h e r away ( a n d s t o r m c l o u d s ) / 3 1 4
a r e v i e w e d w i t h a s l i g h t i n c l i n a t i o n d o w n w a r d or w i t h t h e a n t e n n a
aimed upward, d e p e n d i n g on t h e f l i g h t a l t i t u d e a n d t h e v i e w i n g r a n g e .
The i n c l i n a t i o n o f t h e a n t e n n a c a n b e s e l e c t e d t o p r o v i d e t h e
optimum c l a r i t y o f t h e i m a g e s of t h e l a n d m a r k s on t h e s c r e e n .

Use o f A i r c r a f t Radar f o r P u r p o s e s of A i r c r a f t N a v i g a ­
t i o n and A v o i d a n c e o f Dangerous M e t e o r o l o g i c a l Phenomena
Aircraft r a d a r can b e used t o s o l v e a l l problems of aircraft
n a v i g a t i o n , b e g i n n i n g w i t h t h e r e c o g n i t i o n of l a n d m a r k s o v e r which
t h e a i r c r a f t i s f l y i n g and e n d i n g w i t h measurement of a l l b a s i c
elements of aircraft navigation.

328

For r e c o g n i t i o n o f t e r r e s t r i a l l a n d m a r k s , i t i s d e s i r a b l e t o
u s e o p e r a t i n g s c a l e s of t h e r a d a r which c o i n c i d e w i t h t h e s c a l e s
of f l i g h t c h a r t s .
W i t h a n i n d i c a t o r s c r e e n r a d i u s o f 5 5 m m , a n i m a g e s c a l e of
1:1,000,000 p r o d u c e s a r a n g e o f 5 5 km o n t h e s c r e e n .
This oper­
a t i n g s c a l e f o r t h e r a d a r i s m o s t s u i t a b l e when u s i n g m a p s w i t h
a s c a l e o f 1 0 km t o 1 c m .
H e n c e , when u s i n g c h a r t s w i t h a s c a l e o f 1 : 2 , 0 0 0 , 0 0 0 ,
one must
u s e a r a d a r s c a l e o f 1 0 0 km; 1 1 0 km i s p o s s i b l e , i f t h e d e s i g n o f
t h e radar permits.
The s h a r p e s t d i s t i n c t i o n o f r a d a r l a n d m a r k s i s o b t a i n e d by
u s i n g t h e p r o p e r s e l e c t i o n of c o n t r a s t i n t h e i mag e b y u s i n g v a r ­
i o u s a m p l i f i c a t i o n s of t h e s i g n a l s a t h i g h and low l e v e l s , a d j u s t ­
i n g t h e i n c l i n a t i o n of t h e a n t e n n a t o t h e p r o p e r a n g l e , and s e t t i n g
t h e beam b r i g h t n e s s o n t h e s c r e e n .
The l o c a t i o n o f t h e a i r c r a f t c a n b e d e t e r m i n e d v e r y a c c u r a t e l y
i n terms of t h e b e a r i n g and d i r e c t i o n from a p o i n t landmark.
Point
l a n d m a r k s i n t h i s c a s e c a n b e t h e c e n t e r s of p o p u l a t e d a r e a s , c h a r a c ­
t e r i s t i c f e a t u r e s of t h e s h o r e s o f r i v e r s and l a k e s , i n d i v i d u a l
mountain peaks , e t c .
When u s i n g c i r c u l a r s c a n r a d a r , w i t h a r o t a t i n g s c a l e o f b e a r ­
i n g s , t h e l o c u s o f t h e a i r c r a f t i s d e t e r m i n e d b y t h e same m e t h o d
used f o r goniometric-rangefinding n a v i g a t i o n a l USW systems, i . e . ,
p l o t t i n g t h e b e a r i n g and d i s t a n c e from a landmark t o t h e a i r c r a f t
on a c h a r t .
However, i t i s n e c e s s a r y t o t a k e i n t o c o n s i d e r a t i o n
t h e correction f o r t h e deviation of t h e meridians, i f t h e d i f f e r ­
ence i n t h e l a t i t u d e s of t h e landmark and l o c u s of t h e a i r c r a f t
is significant.
In using sector-type radars, the bearing of t h e aircraft is
o b t a i n e d by a d d i n g t h e a i r c r a f t c o u r s e and t h e c o u r s e a n g l e o f t h e
l a n d m a r k , a s i s d o n e when u s i n g a i r c r a f t r a d i o c o m p a s s e s w i t h n o n ­
integrated indicators.
A s i n t h e c a s e w h e n USW r a n g e f i n d i n g s y s t e m s a r e u s e d , t h e
measurement o f d i s t a n c e s w i t h a n a i r c r a f t r a d a r means t h a t t h e r a d a r / 3 1 5
m e a s u r e s n o t t h e h o r i z o n t a l b u t t h e o b l i q u e d i s t a n c e ( O D ) of t h e
landmark.
T h e r e f o r e , when m e a s u r i n g d i s t a n c e s t o l a n d m a r k s , w h i c h
a r e l e s s t h a n f i v e t i m e s t h e f l i g h t a l t i t u d e (H), t h e measurement
must i n c l u d e a c o r r e c t i o n A R , which always h a s a n e g a t i v e s i g n :

AR = -

( v o m- R);

R=OD-AR,
where OD i s t h e o b l i q u e d i s t a n c e , H i s t h e f l i g h t a l t i t u d e , and R
is the horizontal distance.

329

If t h e o b l i q u e d i s t a n c e t o t h e l a n d m a r k i s e q u a l t o t h e f l i g h t
a l t i t u d e ( t h e c o r r e c t i o n f o r t h e f l i g h t a l t i t u d e becomes e q u a l t o
the oblique distance), the horizontal distance w i l l be equal t o
zero.
T h i s i s a l s o r e f l e c t e d i n t h e p a n o r a m a o f t h e i m a g e , when
a dark s p o t appears i n t h e middle of t h e i n d i c a t o r s c r e e n with a
s h a r p l i m i t for t h e b e g i n n i n g o f i m a g e f o r m a t i o n .
The b e g i n n i n g
of image f o r m a t i o n i s s e p a r a t e d from t h e c e n t e r o f t h e s c r e e n by
a d i s t a n c e w h i c h i s e q u a l t o t h e f l i g h t a l t i t u d e on t h e s c a n n i n g
scale.
T h i s s p o t i s caZZed t h e a Z t i m e t r a Z a n d i s u s e d for meas­
u r i n g t h e t r u e a l t i t u d e of f l i g h t a b o v e t h e l o c a l r e l i e f .

TABLE 3.1.

Flight1 a l t i t u d e , ~

.u

- -

1 1 2 1 3 i 4 1 5 1 6 1 7 1 8 1 9 _­
1 1 0 1 1 1 ) 12
K.u

c ) r r e c t i onsr,

5
10
15
20
25
30
35
40
45
50

5.0
1,5
1.0
0.5

0,o
0,o
0,o
0'0
0,o
0,o

2.0
.1,5
1,0

0.0
0.0
0,o
0,o

0.0
0.0

KX

1.

- - - - - 3,O

2.0
1.5
0,5
0.0

4,O
2,5

660 10,O *3,O 4,O 5,O

2.0

2,5
1,5
1,0
0,5

1,0
0.5
0.0

0.0
040 0,O 0,O
O,? 0.0 0.0
0,B 0.0 0,o

3,O 3.5
240 2.5
1,5 2,O
I t 0 1.5
015 1.0
0,O 0,5
0.0 0,o

­

630
430
3.0
2,5
250
!,5
1.0

0,o

I

For m a k i n g c o r r e c t i o n s i n t h e m e a s u r e d d i s t a n c e s f o r t h e f l i g h t
a l t i t u d e , w e can use Table 3.1.
The l o c a t i o n o f t h e a i r c r a f t c a n b e d e t e r m i n e d b y m e a n s o f
a i r c r a f t r a d a r and d i r e c t l y i n s t a g e s of orthodromic c o o r d i n a t e s .
T o d o t h i s , t h e s c a l e o f b e a r i n g s on t h e i n d i c a t o r m u s t b e s e t n o t
t o t h e c o u r s e of t h e a i r c r a f t , b u t t o t h e l e a d a n g l e (LA) o n t h e
course of t h e a i r c r a f t r e l a t i v e t o a g i v e n orthodromic path a n g l e
o f f l i g h t or d r i f t a n g l e .

/ 316

The s i g h t i n g d e v i c e c a n t h e n b e u s e d t o d e t e r m i n e t h e p a t h
For e x a m p l e , w i t h LA = y-JI = - l o o ,
b e a r i n g o f t h e l a n d m a r k (PBL).
t h e b e a r i n g s c a l e must b e s e t t o 350° o p p o s i t e t h e c o u r s e m a r k i n g ;
w i t h a c o u r s e a n g l e o f 4 0 ° , t h e p a t h b e a r i n g o f t h e l a n d m a r k (PBL)
w i l l b e e q u a l t o 30°; w i t h a n e g a t i v e d r i f t a n g l e , a n d t h e r e f o r e
a p o s i t i v e l e a d a n g l e , s u c h a s l o o , e . g . , t h e b e a r i n g s c a l e must
b e s e t t o loo o p p o s i t e t h e c o u r s e m a r k i n g .
Knowing t h e p a t h b e a r i n g a n d t h e d i s t a n c e t o a l a n d m a r k ( R ) ,
w e can v e r y s i m p l y d e t e r m i n e t h e o r t h o d r o m i c c o o r d i n a t e s o f t h e
aircraft:

330


I

X = X

-

1

RcosPBL = X

2 = 2

1

-

-

1
RsinPBL.

Rsin(9OO-PBL);

These formulas are d i f f e r e n t from ( 1 . 7 1 ) and ( 1 . 7 1 a ) only i n
t h e s i g n o f t h e s e c o n d t e r m s on t h e r i g h t - h a n d s i d e .
This is explained
b y t h e f a c t t h a t when w e a r e u s i n g g o n i o m e t r i c - r a n g e f i n d i n g s y s t e m s ,
t h e d i r e c t i o n i s reckoned from a ground beacon t o t h e a i r c r a f t ,
w h i l e i n t h i s case i t i s r e c k o n e d from t h e a i r c r a f t t o a ground
landmark.

Example.
The r a d a r l a n d m a r k h a s o r t h o d r o m i c c o o r d i n a t e s X1
= 2 5 0 km; 2 1 = 8 0 km a n d i s o b s e r v e d w i t h a p a t h b e a r i n g o f 40°
a s a d i s t a n c e o f 1 2 5 km.
and 2 .

Find t h e coordinates of t h e aircraft

X

Solution:
X = 250 - 125~sin50"= 250- 96 = 156 K A .
Z=80-~25*~0~40~=80-82=--!2~~~.

T h u s we h a v e f o u n d t h a t t h e a i r c r a f t i s l o c a t e d a t a d i s t a n c e
o f 1 5 6 km f r o m t h e l a s t P B L , 2 km t o t h e l e f t o f t h e L G F , w i t h o u t
r e s o r t i n g t o a p l o t t i n g o f t h e b e a r i n g s on t h e f l i g h t c h a r t .

0

40"

5

@

82

96

V
125

n

I n s o l v i n g t h i s problem, it
i s very convenient t o use t h e cal­
culating navigational slide rule.

To d o t h i s , t h e t r i a n g u l a r
i n d e x on s c a l e 4 i s s e t t o t h e
d i s t a n c e of t h e landmark a l o n g
scale 5.
The s l i d e r i n d i c a t o r
i s t h e n s e t on s c a l e 3 t o t h e mark­
i n g which corresponds t o 90° -PBL and P B L , w h i l e t h e v a l u e s R s i n
( 9 0 ° - P B L ) and R s i n P B L a r e s e t on s c a l e 5 ( F i g . 3 . 4 4 ) .

Fig. 3.44.
Determination of
Orthodromic Coordinates of
a n A i r c r a f t on t h e N L - 1 O M .

A f t e r t h i s , t h e r e remains only t h e c a l c u l a t i o n of these values
from t h e c o o r d i n a t e s of t h e landmark and t h e d e t e r m i n a t i o n of t h e
aircraft coordinates.

T h e p r o b l e m i s c o n s i d e r a b l y s i m p l i f i e d when t h e p a t h b e a r i n g
of t h e landmark i s e q u a l t o 90° ( f l i g h t o v e r t h e t r a v e r s e of t h e
landmark).
Then

I t should be mentioned t h a t t h e d e t e r m i n a t i o n of t h e a i r c r a f t
c o o r d i n a t e s when f l y i n g o v e r t h e t r a v e r s e o f a l a n d m a r k i s a d v a n ­
t a g e o u s , s i n c e i n t h i s case t h e e r r o r s i n measuring t h e p a t h b e a r ­
i n g o f t h e l a n d m a r k h a v e a b s o l u t e l y n o e f f e c t on t h e a c c u r a c y o f
d e t e r m i n a t i o n o f t h e l a t e r a l d e v i a t i o n of t h e a i r c r a f t f r o m t h e

331


/317

l i n e of f l i g h t .
This is very u s e f u l f o r monitoring t h e path i n
t e r m s o f d i r e c t i o n a n d c o r r e c t i n g t h e c o u r s e o f t h e a i r c r a f t by
u s i n g a u t o n o m o u s D o p p l e r m e a s u r e m e n t s of t h e g r o u n d s p e e d a n d d r i f t
angle.
T h i s m e t h o d o f d e t e r m i n i n g t h e o r t h o d r o m i c c o o r d i n a t e s of a n
In this
aircraft is a l s o suitable f o r use with sector-type radars.
c a s e , t h e p a t h b e a r i n g o f t h e landmark i s d e t e r m i n e d by t h e f o r m u l a

PBL = C A L

+

LA.

T h i s p r o b l e m c a n t h e n b e s o l v e d i n t h e same way a s f o r c i r c ­
ular-screen radars.
However , on s e c t o r - t y p e s c r e e n s as a r u l e ,
it i s n o t possible t o determine t h e markings of t h e t r a v e r s e of
f l i g h t s over landmarks.
Therefore, f o r an accurate c o n t r o l of t h e
path, taking i n t o account t h e reduced accuracy of d i r e c t i o n finding,
due t o t h e l a c k o f s i g h t i n g l i n e s , i t i s n e c e s s a r y t o image t h e
landmarks a t c o u r s e a n g l e s which a r e as l a r g e as p o s s i b l e .
The g r o u n d s p e e d o f t h e a i r c r a f t a n d d r i f t a n g l e c a n b e d e t e r ­
m i n e d m o s t e a s i l y w i t h t h e a i d of a i r c r a f t r a d a r by u s i n g s u c c e s s i v e
m e a s u r e m e n t s o f t h e L A , ( l o c u s of t h e a i r c r a f t ) , e s p e c i a l l y i n o r t h o ­
d r o m i c c o o r d i n a t e s , when i t i s n o t n e c e s s a r y t o p l o t t h e l o c u s o f
The e s s e n c e o f t h i s met h o d h a s b e e n d e­
t h e a i r c r a f t on a c h a r t .
s c r i b e d above.
However, t h e method u s e d f o r m e a s u r i n g t h e g r o u n d
s p e e d o n t h e b a s i s o f s u c c e s s i v e m e a s u r e m e n t s of t h e l o c u s o f t h e
a i r c r a f t , i s i n s u f f i c i e n t l y p r a c t i c a l f o r measurements of t h e d r i f t
The f a c t i s t h a t f o r a n a c c u r a t e m e a s u r e ­
angle of an a i r c r a f t .
m e n t t o b e made a l o n g a g i v e n p a t h , i t i s n e c e s s a r y t o d e t e r m i n e
t h e d r i f t a n g l e q u i t e f r e q u e n t l y a n d r a p i d l y , s o t h a t t h e method
of s u c c e s s i v e measurements of t h e l o c u s of t h e a i r c r a f t r e q u i r e s
a l a r g e b a s e for m e a s u r e m e n t s .
I n some c a s e s , i t i s a d v i s a b l e t o u s e o t h e r m e t h o d s f o r d e t e r ­
mining t h e ground speed ( e . g . , i f v i s u a l p o i n t s l i e i n t h e f i e l d
o f v i s i o n of t h e r a d a r which do n o t a l l o w t h e p o s i t i o n o f t h e a i r ­
c r a f t t o b e d e t e r m i n e d ) s i n c e t h e y do n o t a p p e a r on t h e c h a r t .
How­
e v e r , t h e y a r e s u i t a b l e f o r d e t e r m i n i n g t h e d r i f t a n g l e and t h e
g r o u n d s p e e d by v i s u a l methods.
T h e r e a r e s e v e r a l methods o f d e t e r m i n i n g t h e d r i f t a n g l e and
t h e g r o u n d s p e e d by v i s u a l means.
L e t u s d i s c u s s s e v e r a l of them
which a r e most o f t e n employed:

1.
Measurement o f t h e d r i f t a n g l e o f an a i r c r a f t on t h e b a s i s
o f t h e secondary Doppler e f f e c t .
The d i r e c t i o n a l i t y o f t h e c h a r a c ­
t e r i s t i c of r a d i a t i o n from an a i r c r a f t r a d a r i n t h e h o r i z o n t a l p l a n e
i s made a s n a r r o w a s p o s s i b l e .
The n a r r o w e r t h e beam f o r t h e p r o p ­
a g a t i o n of e l e c t r o m a g n e t i c w a v e s , t h e b e t t e r t h e r e s o l v i n g p o w e r
of t h e radar i n a tangential direction (perpendicular t o the radius
/318
of t h e scan).
However, i n o r d e r t o p r o d u c e a v e r y n a r r o w c h a r a c t e r i s t i c o f r a d i a t i o n , w e m u s t u s e a n a n t e n n a r e f l e c t o r on t h e r a d a r
which h a s v e r y l a r g e d i m e n s i o n s .
Therefore, t h e p r a c t i c a l width
o f t h e beam i s 2-3O.
332


The w i d e n i n g o f t h e c h a r a c t e r i s t i c o f d i r e c t i o n a l i t y w i t h i n
these l i m i t s is undesirable i n principle f o r surveying the terrain,
b u t can be used very advantageously f o r measuring t h e d r i f t angle
by t h e s o - c a l l e d s e c o n d a r y D o p p l e r e f f e c t .
The e s s e n c e o f t h i s
method i s t h e f o l l o w i n g .
L e t u s s a y t h a t w e h a v e s t o p p e d t h e r o t a t i o n o f t h e r a d a r an­
tenna a t a c e r t a i n angle t o t h e d i r e c t i o n of t h e a i r c r a f t ' s motion
(Fig. 3.45).

F
L

' R

F i g . 3.45.
C r e a t i o n of t h e
Secondary Doppler E f f e c t .

I n t h e p i c t u r e , w e can see
t h e r e f l e c t i o n of t h e electromag­
n e t i c waves f r o m t h e e l e m e n t a r y
a r e a S which w e have s e l e c t e d .
The h i g h f r e q u e n c y r e f l e c t e d
f r o m t h e E a r t h ' s s u r f a c e , when r e ­
ceived aboard t h e aircraft, w i l l
n o t be equal t o t h e frequency r a d i a t e d by t h e r a d a r , b u t w i l l h a v e
a c e r t a i n p o s i t i v e or n e g a t i v e f r e ­
quency s h i f t which i s c a l l e d t h e
Doppler e f f e c t .

L e t us a l s o n o t e t h a t t h e Doppler e f f e c t i s p r o p o r t i o n a l t o
t h e cosine of t h e angle between t h e d i r e c t i o n of t h e a i r c r a f t ' s
m o t i o n a n d t h e d i r e c t i o n o f t h e wave p r o p a g a t i o n ( i . e . , f3-a).
Angle
f3 h e r e r e p r e s e n t s t h e c o u r s e a n g l e o f t h e a n t e n n a p o s i t i o n o f t h e
r a d a r , while t h e angle a r e p r e s e n t s t h e d r i f t angle of t h e aircraft.

For t h e s a k e o f s i m p l i c i t y , l e t u s c o n s i d e r t h e D o p p l e r e f f e c t
o n l y f o r t w o e x t r e m e l i m i t s o f t h e b e a m w i t h a common c h a r a c t e r ­
i s t i c o f r a d i a t i o n d i r e c t i o n a l i t y , t h e l e f t - h a n d beam i s m a r k e d
L a n d t h e r i g h t - h a n d beam R i n o u r d i a g r a m .
The s o l u t i o n o f t h e c h a r a c t e r i s t i c w i l l b e r e p r e s e n t e d b y t h e
angle 6 , so t h a t

f D R -cos(P-u+-

where f D

2

') ',

i s t h e Doppler frequency.

T h u s , w e s e e t h a t t h e D o p p l e r e f f e c t on t h e l e f t - h a n d e d g e
o f t h e beam i s g r e a t e r t h a n on t h e r i g h t - h a n d s i d e , s o t h a t t h e
frequency r e c e i v e d by t h e a n t e n n a from t h e l e f t - h a n d s i d e of t h e
beam w i l l b e s o m e w h a t h i g h e r t h a n t h a t f r o m t h e r i g h t .
The f r e q u e n c i e s o f t h e l e f t ( L ) a n d r i g h t ( R ) b o u n d a r i e s o f
t h e beam w i l l b e c o m b i n e d i n t h e r e c e i v e r a n d p r o d u c e a n i n t e r m e d i a t e
333

/319

f r e q u e n c y as f o l l o w s

which w i l l amount t o a m p l i t u d e m o d u l a t i o n o f t h e r e c e i v e d s i g n a l .
Now l e t u s s a y t h a t t h e d i r e c t i o n o f t h e a n t e n n a c o i n c i d e s
w i t h t h e d i r e c t i o n i n w h i c h t h e a i r c r a f t i s m o v i n g , i . e . , B=a.
Then
t h e D o p p l e r f r e q u e n c i e s o f t h e l e f t a n d r i g h t s i d e s o f t h e beam
w i l l be uniform i n value and p r o p o r t i o n a l according t o t h e cosines
6/2:

fDL

- +cos -;8

fDR--c0s-

8

2 .’

a n d t h e a m p l i t u d e m o d u l a t i o n f r o m t h e e d g e s o f t h e beam w i l l b e
absent.
I n a c t u a l i t y , t h e r e w i l l be a very low-frequency amplitude
m o d u l a t i o n owing t o t h e d i f f e r e n c e i n t h e D o p p l e r f r e q u e n c i e s of
t h e e d g e s o f t h e beam r e l a t i v e t o t h e e f f e c t o f t h e c e n t e r o f t h e
beam ( t h e b i s e c t r i x o f t h e r a d i a t i o n c h a r a c t e r i s t i c ) , b u t d u e t o
t h e very s m a l l d i f f e r e n c e between t h e c o s i n e s of t h e a n g l e s , t h e
b e a t f r e q u e n c y w i l l b e v e r y low ( e x p r e s s e d i n H e r t z ) , w h i l e t h e
v i s u a l e f f e c t o f t h e b e a t i s maximum.
With c i r c u l a r r o t a t i o n o f t h e a n t e n n a , t h e b e a t i n g o f t h e f r e ­
quencies is n o t n o t i c e a b l e t o t h e e y e , s i n c e each of t h e luminous
p o i n t s i s r a p i d l y c r o s s e d b y t h e s c a n n i n g beam a n d a p p e a r s on t h i s
s c r e e n as an i n d i v i d u a l p o i n t w i t h s u b s e q u e n t a f t e r g l o w .
A s l i g h t impression remains of t h e secondary Doppler e f f e c t
i n a f i x e d a n t e n n a , when i t s d i r e c t i o n d i f f e r s c o n s i d e r a b l y f r o m
t h e d i r e c t i o n i n which t h e a i r c r a f t i s moving, s i n c e t h e f l i c k e r ­
i n g of t h e p o i n t s i n t h i s c a s e t a k e s p l a c e a t h i g h f r e q u e n c i e s and
i s b l u r r e d by t h e a f t e r g l o w on t h e s c r e e n .
If t h e d i r e c t i o n of t h e a n t e n n a s l o w l y a p p r o a c h e s t h e d i r e c ­
t i o n i n which t h e a i r c r a f t i s moving, t h e luminous p o i n t s a l l b e g i n
t o f l a s h a t a reduced frequency and increased amplitude.
A slow
b u t b r i g h t f l a s h i n g o f t h e l u m i n o u s p o i n t s on t h e s c r e e n i n d i c a t e s
a c o i n c i d e n c e of t h e d i r e c t i o n o f t h e a n t e n n a w i t h t h e d i r e c t i o n
The d r i f t a n g l e o f t h e a i r c r a f t
i n which t h e a i r c r a f t i s moving.
i s d e t e r m i n e d b y t h e p o s i t i o n of t h e s c a n n i n g l i n e s on t h e s c r e e n
w i t h naximum s e c o n d a r y D o p p l e r e f f e c t .
Measurement i s performed b e s t o f a l l w i t h a l a r g e - s c a l e o p e r ­
a t i o n o f t h e r a d a r ( 2 0 km f o r t h e s c r e e n r a d i u s ) , u s i n g a s c a n n i n g
d e l a y o f 2 0 km.
I t i s t h e n necessary t o make a corresponding ampli­
f i c a t i o n i n t h e r e c e i v e r for t h e common a m p l i f i c a t i o n c h a n n e l , i n
b o t h t h e h i g h a n d low s i g n a l l e v e l s , a s w e l l as t h e c o r r e s p o n d i n g
i n c l i n a t i o n of t h e antenna.
334

I


One a d v a n t a g e o f t h e m e t h o d o f d e t e r m i n i n g t h e d r i f t a n g l e
of t h e a i r c r a f t according t o t h e secondary Doppler effect i s i t s
high accuracy.
With a l i t t l e e x p e r i e n c e i n s e l e c t i n g t h e r e c e i v e r
a m p l i f i c a t i o n and t h e a n g l e f o r t i l t i n g t h e a n t e n n a , measurements
c a n b e made l i t e r a l l y w i t h i n s e v e r a l s e c o n d s .
S e v e r a l t y p e s o f s e c t o r - t y p e r a d a r s have a s p e c i a l o p e r a t i n g
regime and an a d d i t i o n a l i n d i c a t o r f o r measuring t h e d r i f t a n g l e
according t o t h e secondary Doppler e f f e c t .

2.
Measurement o f t h e
d r i f t angle and ground speed
by s i g h t i n g p o i n t s near t h e
course.
If a c l e a r l y v i s i b l e
point is located near the line
o f f l i g h t of t h e a i r c r a f t on
t h e r a d a r s c r e e n , t h e ground
speed and t h e d r i f t a n g l e of
t h e a i r c r a f t can be measured by
t h e movement o f t h i s p o i n t .

0

Fig. 3.46.
Change i n t h e D r i f t
A n g l e a n d Ground S p e e d o f a
P o i n t Near t h e C o u r s e I n d i c a t o r
on a R a d a r S c r e e n .
set with the zero division opposite
s i g h t i s moved s o t h a t t h e l a n d m a r k
lines (Fig. 3.46).
When t h e o b j e c t
t i m e r i s switched off and t h e f l i g h

To a v o i d g r o s s e r r o r s i n
measurement due t o a l t i t u d e
e r r o r s , t h e s i g h t i n g of t h e
p o i n t s m u s t b e made a t d i s t a n c e s
f r o m 6 0 t o 30 km.
A t t h e moment
when t h e p o i n t b e i n g o b s e r v e d
c r o s s e s t h e 6 0 km m a r k i n g , t h e
timer i s switched on.
The b e a r ­
i n g s c a l e of t h e r a d a r i s t h e n
t h e c o u r s e i n d i c a t i o n , and t h e
is located along its parallel
c r o s s e s t h e 30 km m a r k i n g , t h e
t t i m e of t h e b a s e i s c a l c u l a t e d .

The d r i f t a n g l e i s c a l c u l a t e d d i r e c t l y f r o m t h e b e a r i n g s c a l e ,
w i t h n e g a t i v e d r i f t a n g l e s b e i n g c a l c u l a t e d a s a d d e d t o 360O.
To
determine t h e ground speed, t h e c o r r e c t i o n f o r f l i g h t a l t i t u d e f o r
a d i s t a n c e o f 30 km i s a d d e d t o t h e l e n g t h o f t h e b a s e , t a k i n g i n t o
a c c o u n t t h e c o r r e c t i o n f o r a d i s t a n c e o f 6 0 km a s e q u a l t o z e r o .
T h u s , a t f l i g h t a l t i t u d e s o f 8 - 1 0 km, t h e l e n g t h o f t h e b a s e t u r n s
out t o be equal t o :
A t a h e i g h t of 8 km, 30.5
a h e i g h t o f 1 0 km, 3 1 . 5 km.

km;

a t a h e i g h t o f 9 km,

The g r o u n d s p e e d i s d e t e r m i n e d b y means o f
r u l e (Fig. 3.47, a ) .

3 1 km;

at

a navigational slide

L e t u s s a y t h a t a t a f l i g h t a l t i t u d e o f 1 0 km t h e t i m e r e q u i r e d
t o f l y a l o n g t h e b a s e b e t w e e n t h e 60 a n d 30 km m a r k i n g s i s 2 m i n
The g r o u n d s p e e d i n t h i s case i s 840
and 1 5 sec ( F i g . 3.47, b ) .
km/hr.

335

/320

T h i s method can b e used w i t h s u f f i c i e n t a c c u r a c y f o r measuring
The a c c u r a c y of d e t e r m i n a t i o n
t h e d r i f t angle of t h e aircraft.
o f t h e g r o u n d s p e e d i s o b t a i n e d w i t h a low a n d t h e r e f o r e v e r y s m a l l /321
measurement b a s e .
Thus, e . g . , a t an a i r s p e e d o f 900 km/hr, t h e
e r r o r i n m e a s u r i n g t h e f l i g h t t i m e on t h e b a s e ( w h i c h amounts t o
4 s e c ) produces an e r r o r i n measuring t h e ground speed of about
30 k m / h r .
I n a d d i t i o n , a t l a r g e d r i f t a n g l e s , when t h e v e c t o r o f
t h e motion of t h e t a r g e t p o i n t does n o t agree with t h e r a d i u s of
t h e s c r e e n , e r r o r s a r i s e i n d e t e r m i n i n g t h e measurement b a s e from
t h e d i s t a n c e m a r k i n g s on t h e s c r e e n .

Fig.

3.47.

F i g . 3.47.
D e t e r m i n a t i o n o f Ground Speed
of a P o i n t Near t h e Course I n d i c a t o r on a
Radar Screen.
F i g . 3.48.
D e t e r m i n a t i o n of D r i f t Angle
a n d Ground S p e e d b y Means o f a R i g h t
Triangle.
Fig.

3 . Determination of t h e d r i f t angle of t h e a i r c r a
ground s p e e d b y means o f a r i g h t t r i a n g l e .
T h i s method
convenient and p r e c i s e i n comparison t o t h e s i g h t i n g of
of a landmark n e a r t h e c o u r s e .
I n addition, t h e use of
t r i a n g l e method makes i t p o s s i b l e t o s e l e c t more f r e e l y
m a r k s on t h e r a d a r s c r e e n i n o r d e r t o t r a c k them.

3.48.

f t and t h e
i s more
t h e motion
the right
t h e land­

The b e a r i n g s c a l e o f t h e r a d a r i s s e t t o z e r o o p p o s i t e t h e
c o u r s e m a r k i n g , a f t e r which t h e c o u r s e a n g l e of t h e l a n d m a r k i s
m e a s u r e d w i t h t h e s i g h t , i t s d i s t a n c e on t h e c i r c u l a r m a r k i n g s i s
observed, and t h e timer i s switched on.
Leaving t h e s i g h t i n g i n s t r u ­
ment i n a f i x e d p o s i t i o n , t h e o p e r a t o r o b s e r v e s t h e m o t i o n o f t h e
A t t h e moment w h e n i t c r o s s e s t h e p e r ­
landmark a c r o s s t h e s c r e e n .
p e n d i c u l a r l i n e on t h e s i g h t ( F i g . 3 . 4 8 ) , t h e t i m e r i s s w i t c h e d
o f f , t h e d i s t a n c e t o t h e landmark i s d e t e r m i n e d by t h e c i r c u l a r
markings, and t h e f l i g h t t i m e along t h e b a s e i s c a l c u l a t e d .
C o r r e c t i o n s a r e t h e n made i n t h e f i r s t a n d s e c o n d m e a s u r e m e n t s
o f t h e o b l i q u e d i s t a n c e f o r t h e f l i g h t a l t i t u d e ; a n g l e c1 b e t w e e n
t h e p o s i t i o n o f t h e s i g h t i n g l i n e a n d t h e d i r e c t i o n o f t h e move­
ment of t h e landmark i s t h e n d e t e r m i n e d as f o l l o w s :
$a=-,

R2


4

and t h e l e n g t h o f t h e measurement b a s e i s :

336

.

,,.. .

.

T h i s p r o b l e m i s e a s i l y s o l v e d on a n a v i g a t i o n a l s l i d e r u l e
(Fig.

/322

3.49).

F i g . 3.49.
Keys f o r D e t e r m i n i n g t h e ( a ) A c u t e An g l e o f t h e
T r i a n g l e a n d ( b ) Measurement Base on t h e N L - 1 O M .
The d r i f t a n g l e i s d e t e r m i n e d as t h e d i f f e r e n c e b e t w e e n t h e
f i r s t c o u r s e a n g l e o f t h e l a n d m a r k a n d t h e a n g l e c1 ( F i g . 3 . 4 9 , a ) ,
w h i l e t h e g r o u n d s p e e d i s d e t e r m i n e d a s t h e l e n g t h of t h e b a s e r e l a ­
t i v e t o t h e t i m e required t o cover t h e distance (Fig. 3.49, b ) .

Example.

a f l i g h t a l t i t u d e o f 1 0 k m , t h e c o u r s e a n g l e of
i n i t i a l l y e q u a l t o 8O a t O D 1 = 6 0 km.
The o b l i q u e
d i s t a n c e a t t h e moment w h e n t h e l a n d m a r k c r o s s e s t h e t r a n s v e r s e
The f l i g h t t i m e a l o n g t h e b a s e w a s
l i n e i n t h e s i g h t w a s 2 3 km.
5 min a n d 35 s e c .
F i n d t h e d r i f t a n g l e of t h e a i r c r a f t a n d t h e
ground speed.
A t

a l a n d m a r k was

S o l u t i o n . T h e c o r r e c t i o n for t h e f l i g h t a l t i t u d e f o r t h e f i r s t
d i s t a n c e w i l l be c o n s i d e r e d as e q u a l t o z e r o .
The c o r r e c t i o n f o r
t h e s e c o n d d i s t a n c e (OD = 2 3 k m , H = 1 0 km) i s e q u a l t o 3 km, S O
t h a t t h e h o r i z o n t a l d i s t a n c e i s HD2 = 2 0 km.
On t h e n a v i g a t i o n a l s l i d e r u l e , w e f i n d t h e a n g l e c1 = 1 8 . 5 O
( F i g . 3 . 5 0 , a ) and t h e l e n g t h of t h e measurement b a s e i s S = 6 0
km ( F i g . 3 . 5 0 , b ) .

Fig.

3.50.
angle,

D e t e r m i n a t i o n o f ( a ) t h e Acute Angle o f a T r i ­
( b ) t h e B a s e a n d ( c ) t h e Ground Speed on t h e
NL-1OM.

T h e r e f ore

U S = 8 - 18.5O

= -10.5O.

337

I

3.50,

The g r o u n d s p e e d ( W ) i s t h e r e f o r e e q u a l t o 830 km/hr
c).

(Fig.

4.
D e t e r m i n a t i o n o f t h e g r o u n d s p e e d a n d d r i f t a n g l e of a n
a i r c r a f t by d o u b l e d i s t a n c e f i n d i n g u s i n g a s i g h t i n g p o i n t w i t h
equal oblique distances.
T h i s method i s t h e most p r e c i s e of t h e methods which w e have
d i s c u s s e d w h i c h u s e s i g h t i n g of l a n d m a r k s .
However, it c a l l s f o r
t h e maximum t i m e f o r m e a s u r e m e n t a n d c a l c u l a t i o n .
When a h i g h l y v i s i b l e p o i n t s h o w s u p i n t h e f o r w a r d p a r t o f
t h e s c r e e n , t h e crew w a i t s u n t i l i t r e a c h e s o n e o f t h e c i r c u l a r
A t t h e moment when t h i s p o i n t c r o s s e s
d i s t a n c e markings ( F i g . 3.51).
t h e d i s t a n c e m a r k i n g , t h e t i m e r i s s w i t c h e d on a n d t h e c o u r s e a n g l e
of t h i s p o i n t i s m e a s u r e d .
The crew t h e n w a i t s u n t i l t h i s p o i n t
m o v e s a c r o s s t h e s c r e e n a n d c r o s s e s t h e same c i r c u l a r d i s t a n c e m a r k ­
i n g a t t h e r e a r cf t h e s c r e e n .
A t t h e moment w h e n i t c r o s s e s i t ,
t h e t i m e r i s s w i t c h e d o f f a n d t h e c o u r s e a n g l e o f t h e p o i n t i s meas­
u r e d once a g a i n .
S i n c e i n t h i s c a s e HD1 = HD2, t h e l i n e o f m o t i o n o f t h e p o i n t
( f r o m A t o B) i s p e r p e n d i c u l a r t o t h e b i s e c t r i x b e t w e e n CALl a n d
CAL2, ; . e . , i f t h e p o i n t moves t o t h e r i g h t o f t h e c o u r s e l i n e o f
t h e a i r c r a f t , t h e d r i f t l i n e o f t h e a i r c r a f t i s d e t e r m i n e d by .the
formula

us

/323

= CAL1tCAL2 - 9 0 0 ,
2

a n d if t h e p o i n t moves t o t h e l e f t of t h e c o u r s e l i n e

us --

CALltCALp
2


+

360°

,

or

us

=

CAL1tCAL7

-

2700,

2

To d e t e r m i n e t h e g r o u n d s p e e d , a c o r r e c t i o n f o r f l i g h t a l t i ­
t u d e i s made i n t h e o b l i q u e d i s t a n c e a t p o i n t s A a n d B a n d t h e l e n g t h
of t h e measurement b a s e i s determined by t h e formula

S = 2Rsin

CAL~-CALL

2

If H D 1 = HD2 e x c e e d s f i v e t i m e s t h e f l i g h t a l t i t u d e ,
rection f o r a l t i t u d e is considered t o be zero.

the cor­

Example.
H = 1 0 km, HD1 = HD2 = 60 km; CALI = 32O; CAL2 =
152O; t h e f l i g h t t i m e a l o n g t h e b a s e i s 8 min a n d 1 5 s e c .
Find
t h e d r i f t a n g l e and t h e ground speed.
338


Solution.
320+15i"
'-90"e + 20;
2
152- 32
S = 2.60 ah--- 120 sin 60°;
2
'

us

=

'

B y u s i n g a n a v i g a t i o n a l s l i d e r u l e , w e can s o l v e t h e l a t t e r
e q u a t i o n and f i n d t h e ground speed (Fig. 3.52).
Sg105K-q

wt=765

km /hr

.

a

Fig. 3.52.
Fig. 3.52.
Determination of ( a ) t h e
M e a s u r e m e n t B a s e a n d (b) t h e G r o u n d
Speed on t h e N L - 1 O M .

CAL
Fig.

3.51.

Fig. 3.51.
D e t e r m i n a t i o n o f t h e Ground
Speed and D r i f t Angle by Double D i s ­
t a n c e F i n d i n g o f a Landmark a t E q u a l
Oblique Distances.

We s h o u l d m e n t i o n t h a t i n s o l v i n g p r o b l e m s i n d e t e r m i n i n g t h e
/324
d r i f t a n g l e o f an a i r c r a f t by t h e f o u r methods e n u m e r a t e d a b o v e ,
t h e b e a r i n g s c a l e o f t h e r a d a r may b e s e t t o t h e a i r c r a f t c o u r s e
r a t h e r than zero, e . g . , according t o t h e orthodrome.
Then t h e c o u r s e
angles i n a l l t h e formulas w i l l b e r e p l a c e d by t h e b e a r i n g s of t h e
landmarks, and t h e r e s u l t of t h e s o l u t i o n w i l l n o t be t h e d r i f t
a n g l e b u t t h e a c t u a l f l i g h t a n g l e of t h e a i r c r a f t .

Autonomous

Doppler Meters f o r D r i f t A n g l e and G r o u n d S p e e d

Autonomous m e t e r s f o r g r o u n d s p e e d a n d d r i f t a n g l e o f a n a i r ­
c r a f t , b a s e d on t h e D o p p l e r e f f e c t , o f f e r b r o a d p e r s p e c t i v e s f o r
automation of t h e processes of aircraft navigation and p i l o t a g e
of aircraft.

i
F i g . 3.53.
D i a g r a m of F o r m a t i o n o f D o p p l e r F r e ­
q u e n c y With a Moving O b j e c t .
339

Continuous measurement of t h e motion p a r a m e t e r s of a n a i r c r a f t
makes i t p o s s i b l e t o use simple i n t e g r a t i n g devices f o r a p r e c i s e
In addition,
automatic c a l c u l a t i o n of t h e aircraft path i n t i m e .
a c o n s t a n t knowledge o f t h e s e p a r a m e t e r s makes i t p o s s i b l e t o r e g u ­
l a t e t h e m i n s u c h a way t h a t t h e a i r c r a f t f o l l o w s a g i v e n f l i g h t
t r a j e c t o r y w i t h a minimum n u m b e r o f d e v i a t i o n s .

A l l o t h e r r a d i o d e v i c e s f o r a i r c r a f t n a v i g a t i o n make i t p o s s i b l e
The m o t i o n p a r a m e t e r s
t o determine only t h e locus of t h e aircraft.
o f a n a i r c r a f t c a n b e d e t e r m i n e d o n l y d i s c r e t e l y �or i n d i v i d u a l
path segments , using t h e n a v i g a t i o n a l devices described above.
A s w e p o i n t e d o u t a t t h e b e g i n n i n g o f C h a p t e r One, t h e f l i g h t
regimes of an aircraft are almost never s t a b l e , with t h e exception
A strictly
o f t h e e n d p o i n t s of c u r v e s a l o n g s e p a r a t e p a r a m e t e r s .
s t a b l e f l i g h t regime f o r a l l parameters simultaneously is never
encountered.
T h e r e f o r e , a u t o m a t i c or s e m i - a u t o m a t i c c a l c u l a t i o n
o f t h e p a t h on t h e b a s i s o f m o t i o n p a r a m e t e r s m e a s u r e d o v e r i n d i v i d ­
u a l segments i s a v e r y approximate and u n r e l i a b l e method.

The o p e r a t i n g p r i n c i p l e of

Doppler meters is t h e following.

L e t u s s a y t h a t w e h a v e a moving s o u r c e of e l e c t r o m a g n e t i c
o s c i l l a t i o n s a t a h i g h f r e q u e n c y A and a f i x e d o b j e c t B which r e f l e c t s
these o s c i l l a t i o n s (Fig. 3.53).
If t h e s o u r c e A r e m a i n s f i x e d r e l a t i v e t o o b j e c t B y t h e n a f t e r
a p e r i o d o f t i m e which i s r e q u i r e d f o r t h e e l e c t r o m a g n e t i c waves
t o t r a v e l from p o i n t A t o p o i n t B , e l e c t r o m a g n e t i c o s c i l l a t i o n s
w i l l b e s e t u p i n t h e l a t t e r a t t h e same f r e q u e n c y a s t h o s e e m i t ­
t e d by t h e s o u r c e .

When t h e s o u r c e o f o s c i l l a t i o n s m o v e s t o w a r d p o i n t B , e a c h
s u c c e s s i v e c y c l e o f o s c i l l a t i o n s i s e m i t t e d somewhat c l o s e r t o t h i s
p o i n t ; i t s p r o p a g a t i o n t i m e t o r e a c h p o i n t B i s somewhat l e s s t h a n
i n t h e p r e c e d i n g c y c l e , s o t h a t t h e moments a t w h i c h t h e o s c i l l a t i o n s
a r r i v e a t p o i n t B can b e compared.
L e t us c a l l t h e wavelength of t h e source
With a
t i o n r a t e of e l e c t r o m a g n e t i c waves e .
f r e q u e n c y o f t h e o s c i l l a t i o n s (f) b o t h a t t h e
a n d a t t h e p o i n t of r e f l e c t i o n o f t h e w av es w

A , and t h e propaga­
fixed source, the
p o i n t of emission
i l l be equal t o

f = ,e.


With a movable s o u r c e , t h e number o f o s c i l l a t i o n s r e a c h i n g
p o i n t B p e r u n i t t i m e i s i n c r e a s e d by t h e number of w a v e l e n g t h s
c o n t a i n e d i n t h e d i s t a n c e c o v e r e d b y t h e a i r c r a f t i n t h a t same u n i t
time, i.e.,
c+w
c
w
I=--A

340

-

A

+Tr

/325

The i n c r e a s e i n t h e f r e q u e n c y W / X , p r o d u c e d by t h e m o t i o n o f
t h e s o u r c e , i s c a l l e d t h e Doppler frequency (fD).
S i m i l a r l y , t h e o s c i l l a t i o n frequency a t t h e p o i n t of reflec­
t i o n w i l l d e c r e a s e i f t h e s o u r c e r e c e d e s from t h e r e f l e c t i o n p o i n t
f o r t h e e l e c t r o m a g n e t i c waves.
D o p p l e r m e t e r s w o r k o n t h e same p r i n c i p l e o f s i g n a l t r a n s m i s ­
s i o n as a i r c r a f t r a d a r s , i . e . , f r e q u e n c i e s a r e r e c e i v e d t h a t h a v e
b e e n e m i t t e d by a i r c r a f t s o u r c e s a f t e r t h e i r r e f l e c t i o n from t h e
Earth's surface.
Therefore , a double Doppler frequency i s r e c e i v e d
which a r i s e s a l o n g t h e p a t h o f e l e c t r o m a g n e t i c waves, from t h e a i r ­
c r a f t t o t h e r e f l e c t i n g s u r f a c e and a l o n g t h e r e v e r s e r o u t e from
t h e r e f l e c t i n g s u r f a c e b a c k t o t h e a p p r o a c h i n g or r e c e d i n g a i r c r a f t
T h e r e a r e t h r e e ways o f s e p a r a t i n g t h e D o p p l e r f r e q u e n c y i n
receiving s i g n a l s aboard an aircraft:

(1) T h e i n t e r n a l c o h e r e n c e o f t h e s i g n a l s , w h e n t h e r e c e i v e d
f r e q u e n c y i s combined w i t h i n t h e r e c e i v e r w i t h a f r e q u e n c y r a d i ­
a t e d by t h e s o u r c e , a s a r e s u l t o f w h i c h t h e r e i s a b e a t i n g o f t h e
double Doppler frequency;
(2)
E x t e r n a l c o h e r e n c e , when t h e r e c e i v i n g a n t e n n a p i c k s u p
s i g n a l s which have b e e n r e f l e c t e d from t h e ground as w e l l as s i g ­
n a l s r a d i a t e d b y t h e t r a n s m i t t i n g a n t e n n a t h r o u g h t h e e x t e r n a l med­
ium ;
(3)
Autocoherence of t h e s i g n a l s ; i n t h i s case, t h e frequen/326
c i e s of s i g n a l s r e f l e c t e d from t h e E a r t h ' s s u r f a c e i n t h e forward
and backward r a d i a t i o n o f t h e r e c e i v i n g - t r a n s m i t t i n g a n t e n n a a r e
combined i n t h e r e c e i v e r w i t h o u t t h e f r e q u e n c y r a d i a t e d by t h e a n t e n n a .
S i n c e t h e o s c i l l a t i o n f r e q u e n c y i s i n c r e a s e d by 2 f D r e l a t i v e t o
t h e p r e c e d i n g beam, a n d d e c r e a s e d by t h e same v a l u e f o r t h e f o l l o w ­
i n g b e a m , t h e b e a t f r e q u e n c y w i l l b e e q u a l t o f o u r t i m e s t h e Dop­
p l e r frequency.
I f we a g r e e t o c a l l t h e D o p p l e r f r e q u e n c y t h e b e a t f r e q u e n c y
s e p a r a t e d i n t h e r e c e i v e r as a r e s u l t o f s u p e r p o s i t i o n o f t h e s i g n a l s ,
t h e n f o r t h e cases o f i n t e r n a l and e x t e r n a l c o h e r e n c e w e w i l l have

and f o r t h e case o f

autocoherence w e w i l l have:

I n p r i n c i p l e , Doppler meters w i t h i n t e r n a l and e x t e r n a l coher­
e n c e c a n b e made w i t h a s i n g l e - b e a m a n t e n n a , b u t w i t h a u t o c o h e r ­
e n c e a minimum o f t w o b e a m s i s r e q u i r e d .
I n p r a c t i c e , as w e w i l l
s e e l a t e r o n , i t i s c o n v e n i e n t t o u s e a n t e n n a s w i t h t h r e e or f o u r
beams.
R e c e n t l y , t h e most w i d e l y employed t y p e i s t h e Doppler meter
w i t h four-beam a n t e n n a s .

341


Since t h e characteristics
o f d i r e c t i o n a l i t y of t h e a n t e n n a s
o f D o p p l e r meters i n t h e g e n e r a l
case do n o t c o i n c i d e w i t h t h e
v e c t o r of t h e ground speed of
t h e aircraft, it is necessary
t o c o n s i d e r t h e ' a c t u a l Doppler
frequencies separated i n the receivers.

Fig. 3.54.
Projection of the
Ground-Speed V e c t o r on t h e
Direction of t h e Radiation
of E l e c t r o m a g n e t i c Waves.

Usually, the slope of t h e
a n t e n n a beams o f t h e meter i s
s e l e c t e d s o t h a t t h e areas o f
t h e i r r e f l e c t i o n from t h e E a r t h ' s
s u r f a c e are n o t t o o f a r from t h e
a i r c r a f t , i . e . , t h e power o f t h e
t r a n s m i t t e r i s used most e f f e c t i v e l y .
The s l o p e a n g l e o f t h e beam r e l a ­
t i v e t o the horizontal plane is
called t h e angle 8 (Fig. 3.54).

O b v i o u s l y , w h e n t h e beam i s i n c l i n e d r e l a t i v e t o t h e p l a n e
of t h e horizon, t h e Doppler frequency w i l l b e p r o p o r t i o n a l not t o
t h e modulus of t h e g r o u n d s p e e d v e c t o r , b u t t o i t s p r o j e c t i o n i n
t h e d i r e c t i o n o f t h e a n t e n n a beam.
For example, f o r a meter with
i n t e r n a l coherence,

On t h e o t h e r h a n d , t h e g r o u n d s p e e d v e c t o r o f t h e a i r c r a f t
can b e d i v i d e d i n t o two v e c t o r components:

.I327

T h e v e c t o r WlW i s d i r e c t e d p e r p e n d i c u l a r t o t h e a n t e n n a b e a m ,
The v e c t o r
and t h e r e f o r e t h e Doppler e f f e c t i s n o t produced.

O W ]= owcos
is effective.
I n a d d i t i o n t o t h e f a c t t h a t t h e a n t e n n a beam i s s e t a t a c e r ­
t a i n a n g l e t o t h e v e r t i c a l p l a n e , t h e a n t e n n a beam i s u s u a l l y d i r e c t e d
at a c e r t a i n angle t o t h e longitudinal a x i s of t h e aircraft i n t h e
horizontal plane.
For example, w i t h a four-beam a n t e n n a , t h e l o n g i ­
t u d i n a l a x i s of t h e a i r c r a f t i s t h e b i s e c t r i x of t h e a n g l e s betweeh
t h e d i r e c t i o n s o f t h e f o r w a r d a n d r e a r beams o f t h e a n t e n n a ( F i g .
3.55).
The a n g l e b e t w e e n t h e l o n g i t u d i n a l a x i s o f t h e a i r c r a f t
a n d t h e d i r e c t i o n o f t h e a n t e n n a beam i n t h e h o r i z o n t a l p l a n e i s
called the angle 8 .
Hence, i n r e c e i v e r s w i t h i n t e r n a l and e x t e r n a l c o h e r e n c e , t h e
separated Doppler frequency

342

I I II11111 I I I I

where a i s t h e d r i f t a n g l e o f t h e
aircraft.

I n t h e s p e c i a l case where t h e
d r i f t a n g l e o f t h e a i r c r a f t i s ab­
s e n t , t h e Doppler frequency f o r
e a c h a n t e n n a b e a m w i l l b e t h e same:

Fig. 3.55.
Diagram o f t h e

P o s i t i o n s o f t h e B e a m s from
a n Antenna on a D o p p l e r
Meter.

Three- and four-beam a n t e n n a s

a r e d e s i r a b l e b e c a u s e t h e y make

it p o s s i b l e t o compensate a u t o m a t i c a l l y

f o r e r r o r s i n measurements which
a r i s e with l o n g i t u d i n a l and t r a n s ­

verse r o l l i n g of t h e aircraft.
A t t h e same t i m e , i n c a s e s when s i n g l e b e a m or t w o - b e a m a n ­
t e n n a s a r e u s e d , t h e y m u s t b e p l a c e d on g y r o s t a b i l i z i n g d e v i c e s .
I n t h e o p p o s i t e c a s e , l o n g i t u d i n a l or t r a n s v e r s e r o l l i n g o f t h e
a i r c r a f t w i l l change t h e s l o p e a n g l e of t h e a n t e n n a 0 , t h u s l e a d i n g
t o a change i n t h e Doppler f r e q u e n c y .

I n t h e c a s e o f a f o u r - b e a m a n t e n n a , t h e l o n g i t u d i n a l or t r a n s ­
v e r s e r o l l i n g of t h e a i r c r a f t produces a change i n t h e s l o p e a n g l e
o f one p a i r o f beams i n a p o s i t i v e d i r e c t i o n a n d c h a n g e s t h e oppo­
s i t e p a i r i n t h e n e g a t i v e d i r e c t i o n by t h e same m a g n i t u d e .
If a n g l e s
8 a r e t h e n l o c a t e d on a n a p p r o x i m a t e l y l i n e a r s e c t i o n o f t h e c o s i n e / 3 2 8
c u r v e , t h e f r e q u e n c y s h i f t o f t h e o p p o s i t e a n t e n n a beams w i l l b e
o p p o s i t e i n s i g n b u t a p p r o x i m a t e l y t h e same i n m a g n i t u d e , w h i c h
c a n a l s o b e u s e d for c o m p e n s a t i n g roll e r r o r s i n t h e s y s t e m ( F i g .
3.56, a ) .
a


Fig. 3.56.
S h i f t s i n t h e Doppler Frequency with
T i l t i n g of t h e System:
(a)
Change i n t h e C o s i n e s
of t h e Angles;
( b ) Frequency S h i f t .
343


For e x a m p l e , i n t
t h e Doppler frequency
d e c r e a s e s b y t h e same
b e a t f r e q u e n c y of one

h e c a s e of r e c e i v e r s w i t h a u t o c o h e r e n c e , when
i n c r e a s e s i n t h e f r o n t r i g h t - h a n d beam a n d
m a g n i t u d e i n t h e r e a r l e f t - h a n d beam, t h e
p a i r w i l l simply be r e t a i n e d .

I n systems w i t h i n t e r n a l and e x t e r n a l coherence, t u r n i n g of
t h e a n t e n n a l e a d s t o d o u b l i n g o f t h e f r e q u e n c y s p e c t r u m o f t h e oppo­
s i t e beams ( F i g . 3 . 5 6 , b ) .
With a h o r i z o n t a l p o s i t i o n o f t h e l o n g i t u d i n a l a n d t r a n s v e r s e
a x e s o f t h e a i r c r a f t , t h e D o p p l e r f r e q u e n c y i n t h e f o r w a r d a n d oppo­
s i t e r e a r beams w i l l b e t h e same.
T i l t i n g t h e system s h i f t s t h e
f r e q u e n c y s p e c t r u m o f o n e of t h e beams f o r w a r d , a n d t h a t o f t h e
However, if w e add up
o p p o s i t e b e a m b a c k w a r d t o t h e same e x t e n t .
t h e s e f r e q u e n c i e s w i t h t i m e and d i v i d e them by t h e measurement time',
t h e average frequency w i l l t u r n out t o be equal t o t h e frequency
of t h e h o r i z o n t a l p o s i t i o n of t h e a x i s of t h e a i r c r a f t .
Doppler meters which a r e p r e s e n t l y i n u s e c a n b e d i v i d e d i n t o
f o u r t y p e s , d e p e n d i n g on t h e r e g i m e of e m i s s i o n a n d r e c e p t i o n o f
signals:

1.
P u l s e meters.
I n t r a n s m i t t e r s , t h e s e meters produce highf r e q u e n c y p u l s e s i n t h e same m a n n e r a s i s d o n e i n a i r c r a f t r a d a r s .
Reception of r e f l e c t e d s i g n a l s takes place i n t h e i n t e r v a l s between
pulse emission.
I n o r d e r t o s e p a r a t e t h e Doppler frequency, autoc o h e r e n c e b y beam p a i r s i s e m p l o y e d .
A s h o r t c o m i n g o f t h i s method i s t h e p r e s e n c e o f "dead" a l t i ­
t u d e s , ; . e . , w h e n t h e r e f l e c t e d s i g n a l s a r r i v e a t t h e moment c o i n /329
c i d i n g with t h e emission of p u l s e s , and n o t i n t h e i n t e r v a l s between
them.
I n a d d i t i o n , when f l y i n g o v e r m o u n t a i n o u s t e r r a i n , t h e d i s t a n c e
t o t h e E a r t h ' s s u r f a c e a c c o r d i n g t o o p p o s i t e beams o f t h e a n t e n n a
may n o t b e t h e s a m e , t h u s l e a d i n g t o a f a i l u r e o f t h e a r r i v a l o f
r e f l e c t e d s i g n a l s t o c o i n c i d e f o r t h e s e beams a n d p r o d u c i n g a d i s t u r b ­
ance of t h e i r c o h e r e n c e .

Another shortcoming of p u l s e meters i s t h e need f o r high v o l t ­
ages t o d r i v e t h e magnetron i n analyzing t h e high-frequency p u l s e s ,
t h u s n e c e s s i t a t i n g a hermetic s e a l i n g of t h e t r a n s m i t t e r u n i t s and
s u b j e c t i n g them t o a c e r t a i n p r e s s u r e .

2.
Meters u s i n g continuous r a d i a t i o n of h i g h frequency.
In
t h i s c a s e , t h e h i g h f r e q u e n c y i s r a d i a t e d c o n t i n u o u s l y by t h e t r a n s ­
mitter.
Reception of s i g n a l s is accomplished with a separate antenna,
having a c e r t a i n by-pass c o e f f i c i e n t with t h e t r a n s m i t t i n g antenna.
The r e f l e c t e d s i g n a l s a r e co mb i n ed i n t h e r e c e i v i n g a n t e n n a
w i t h t h e f r e q u e n c y p r o d u c e d by t h e t r a n s m i t t i n g a n t e n n a , s o t h a t
t h e b e a t frequency i s separated out i n an e x t e r n a l coherence system.
The a d v a n t a g e s of

344

t h i s method a r e t h e i n d e p e n d e n c e o f t h e re­

c e p t i o n c o n d i t i o n s f o r t h e s i g n a l s of f l i g h t a l t i t u d e a n d t h e l o c a l
relief.
I n a d d i t i o n , d e v i c e s of t h i s t y p e w o rk ed a t r e l a t i v e l y
low p o w e r s i n t h e r e c e i v e r m e c h a n is m.
A s h o r t c o m i n g o f t h i s method i s t h e n e e d t o have s e p a r a t e an­
tennas f o r transmission and r e c e p t i o n of t h e s i g n a l s .

3.
Meters w i t h c o n t i n u o u s l y p u l s e d r a d i a t i o n .
These meters
employ a c o n s t a n t r e g i m e o f g e n e r a t i o n a n d t r a n s m i s s i o n w i t h p u l s e d
e m i s s i o n o f a p o r t i o n o f t h e h i g h f r e q u e n c y i n t o t h e a n t e n n a s by
means o f c o m m u t a t i n g d e v i c e s .
The r e f l e c t e d s i g n a l s a r e combined
w i t h t h e f r e q u e n c y d e v e l o p e d by t h e t r a n s m i t t e r i n t h e i n t e r v a l s
b e t w e e n t h e moments when t h e h i g h - f r e q u e n c y s e g m e n t s a r e e m i t t e d
( i n t e r n a l coherence).
T h i s means t h a t i t becomes p o s s i b l e t o u s e
p a r t of t h e advantages of continuous emission (operation a t rela­
t i v e l y low v o l t a g e s i n t h e t r a n s m i t t e r c i r c u i t s ) a n d t h e p u l s e s y s t e m s
( r e c e p t i o n and t r a n s m i s s i o n w i t h a s i n g l e a n t e n n a ) .
However, t h e s h o r t c o m i n g s s t i l l r e m a i n which a f f l i c t p u l s e d
meters, i . e . , t h e p r e s e n c e of "dead" a l t i t u d e s and t h e e f f e c t o f
t h e r e l i e f on r e c e p t i o n c o n d i t i o n s .
In addition, there are also
d i f f i c u l t i e s i n u s i n g t h e s e m e t e r s a t low f l i g h t a l t i t u d e s , s i n c e
a t a v e r y s h o r t s i g n a l p a t h , t h e moments o f t r a n s m i s s i o n a n d r e c e p ­
t i o n are p r a c t i c a l l y impossible t o separate.

4.
Meters w i t h f r e q u e n c y m o d u l a t i o n o f s i g n a l t r a n s m i s s i o n .
Frequency modulation of s i g n a l t r a n s m i s s i o n can be used e i t h e r i n
a p u l s e d or c o n t i n u o u s - p u l s e d r e g i m e o f o p e r a t i o n f o r t h e m e t e r .
If t h e t r a n s m i s s i o n of h i g h - f r e q u e n c y p u l s e s a t a c o n s t a n t
f r e q u e n c y a t low a l t i t u d e s m a k e s i t p o s s i b l e t o s u p e r p o s e t h e moments
o f r e c e p t i o n o f s i g n a l s o n t h e m o m e n t s o f e m i s s i o n , w h i l e i n moun/330
t a i n o u s r e g i o n s t h e r e may b e d i s r u p t i o n s i n t h e c o h e r e n c e o f t h e
s i g n a l s , when t h e r e i s a c h a n g e i n t h e f r e q u e n c y o f t h e t r a n s m i s ­
s i o n o f t h e s i g n a l s , and o n l y a p o r t i o n of them w i l l c o n t r i b u t e
t o t h e c o m b i n a t i o n w i t h t h e r a d i a t i o n moments.
The m a j o r i t y o f
s i g n a l s w i l l be r e c e i v e d i n t h e i n t e r v a l s b e t w e e n t h e moments o f
e m i s s i o n , s i n c e t h e d u r a t i o n o f t h e i n t e r v a l s i s made s u f f i c i e n t l y
To a certain degree, i n
g r e a t e r than t h e duration of t h e pulses.
t h i s c a s e , t h e e f f e c t o f d i s r u p t i o n o f c o h e r e n c e i s r e d u c e d i n moun­
tainous regions.

The b e s t p r o p e r t i e s a r e e x h i b i t e d b y c o n t i n u o u s - p u l s e d m e t e r s
with frequency modulation of s i g n a l transmission, s i n c e i n t h i s
case a l l p o s i t i v e q u a l i t i e s o f t h e c o n t i n u o u s and p u l s e d systems
a r e employed.
However, t h e s h o r t c o m i n g s of t h e p u l s e d s y s t e m s r e m a i n ,
i n c l u d i n g d i f f i c u l t y i n making m e a s u r e m e n t s a t v e r y low f l i g h t a l t i ­
tudes.
O f t h e t y p e s o f D o p p l e r meters which w e have d i s c u s s e d , t h e
o n e s which a r e c u r r e n t l y u s e d m o s t w i d e l y a r e t h e meters w i t h con­
t i n u o u s r a d i a t i o n a n d c o n t i n u o u s - p u l s e d m e t e r s w i t h f r e q u e n c y modu­
l a t i o n of s i g n a l transmission.
345

I..

a.


Fig. 3.57.
Doppler-Meter Antenna:
( a ) Waveguide
L a t t i c e ; ( b ) Diagram o f B e a m F o r m a t i o n .

I n t h e f i r s t tyj?es of D o p p l e r m e t e r s , b e g i n n i n g w i t h t h e s i n g l e
beam v e r s i o n s , r e f l e c t o r - t y p e a n t e n n a s w e r e u s e d w i t h a n i s o s c e l e s
directional characteristic.
R e c e n t l y , a n t e n n a s of t h e " w a v e g u i d e
n e t w o r k " t y p e h a v e come i n t o u s e .
The p r i n c i p l e o f o p e r a t i o n of
i n g ( F i g . 3.57, a ) .

t h e s e antennas i s t h e follow­

I m a g i n e a r e c t a n g u l a r l a t t i c e , made u p o f w a v e g u i d e s , t o o n e
c o r n e r o f w h i c h a n e l e c t r o m a g n e t i c wave o f h i g h f r e q u e n c y is a p p l i e d .
On t h e u p p e r w a l l s o f t h e t r a n s v e r s e w a v e g u i d e i n t h i s l a t ­
t i c e , t h e r e a r e s l o t s f o r e m i s s i o n of wave e n e r g y i n t o s p a c e .
The wave e n e r g y , p r o p a g a t e d a l o n g a t r a n s v e r s e w a v e g u i d e , e m e r g e s
/331
t h r o u g h t h e s l i t s w i t h a c e r t a i n s h i f t i n t i m e f r o m one s l i t t o
t h e n e x t , s o t h a t t h e r e i s a n i n t e r f f e r e n c e of t h e waves e m e r g i n g
from t h e s l i t s , as i s t h e case i n s p a c e d a n t e n n a s ( F i g . 3.57, b ) .
T h e d i r e c t i o n o f t h e r a d i a t i o n maximum a n d t h e i s o p h a s a l l i n e s
a r e l o c a t e d a t a n a n g l e t o t h e s u r f a c e of t h e w a v e g u i d e .
Since t h e electromagnetic energy propagated along t h e longitud­
i n a l wave g u i d e r e a c h e s t h e n e x t t r a n s v e r s e w a v e g u i d e a l s o w i t h
a s h i f t i n t i m e , a similar picture of interference with a t i l t i n g
of t h e i s o p h a s a l l i n e a l s o t a k e s p l a c e a l o n g t h e waveguide l a t t i c e .
A s a r e s u l t , an i s o p h a s a l s u r f a c e i s formed above t h e waveguide
l a t t i c e , having a s l o p e i n t h e d i r e c t i o n of i t s diagonal toward
t h e c o r n e r o p p o s i t e t h e c o r n e r a t which t h e e l e c t r o m a g n e t i c waves
C o n s e q u e n t l y , one of t h e beams w i l l b e f o r m e d
enter the lattice.
along t h e diagonal of t h e antenna.
If wave e n e r g y i s a l s o t r a n s m i t t e d f r o m t h e d i a g o n a l l y o p p o ­
s i t e c o r n e r o f t h e l a t t i c e , two o p p o s i t e l y d i r e c t e d a n t e n n a beams
can b e formed s i m u l t a n e o u s l y .

346


F l a t , m u l t i - b e a m a n t e n n a s , e s p e c i a l l y when f i x e d i n p o s i t i o n ,
are v e r y u s e f u l , s i n c e t h e y can b e p l a c e d below t h e r a d i o - t r a n s ­
p a r e n t h o u s i n g f l u s h w i t h t h e s k i n o f t h e a i r c r a f t and do n o t p r o d u c e
any a d d i t i o n a l aerodynamic r e s i s t a n c e d u r i n g f l i g h t .

S c h e m a t i c Diagram of t h e O p e r a t i o n of a M e t e r w i t h C o n t i n ­
uous R a d i a t i o n Regime
The h i g h f r e q u e n c y p r o c e s s e d b y t h e t r a n s m i t t e r p a s s e s t h r o u g h
a commutation d e v i c e t o t h e t r a n s m i t t i n g a n t e n n a , where t h e beams
The
f o r p r o p a g a t i o n of e l e c t r o m a g n e t i c waves a r e f o r m e d i n p a i r s .
commutating d e v i c e i s connected t o t h e c o u n t e r o f t h e meter, t o
s e p a r a t e t h e f r e q u e n c i e s of t h e f i r s t a n d s e c o n d p a i r s o f beams
(Fig. 3.58).
A p o r t i o n o f t h e wave e n e r g y r a d i a t e d b y t h e t r a n s m i t t e r r e a c h e s
t h e r e c e i v i n g a n t e n n a , where i t i s combined w i t h t h e r e c e i v e d s i g n a l s
r e f l e c t e d from t h e E a r t h ' s s u r f a c e , s o t h a t t h e Doppler f r e q u e n c y
of t h e g i v e n p a i r of beams c a n b e s e p a r a t e d .

The s e p a r a t e d D o p p l e r f r e q u e n c y , a f t e r a m p l i f i c a t i o n , p a s s e s
t o t h e c a l c u l a t i n g d e v i c e , a t whose o u t p u t i s a n i n d i c a t o r f o r t h e
ground speed and d r i f t angle of t h e a i r c r a f t .
A s w e a l r e a d y know, f o r t h e four-beam a n t e n n a o f a D o p p l e r
m e t e r w i t h i n t e r n a l c o h e r e n c e , t h e s e p a r a t e d f r e q u e n c y by beam p a i r s
w i l l be equal to:

(a)

For t h e f i r s t p a i r ,
fDl =

(b)

7cos0 cos (p + a);

For t h e s e c o n d p a i r ,
fD

=

/332

2w
COS 0 COS (p - a).

For a D o p p l e r m e t e r , t h e s i g n o f t h e a n g l e i s n o t i m p o r t a n t ,
but its absolute value is.
T h e r e f o r e , w e c a n s i m p l y assume t h a t
with a positive d r i f t angle, t h e d r i f t angle i n t h e right-hand p a i r
o f beams w i l l b e c a l c u l a t e d f r o m t h e a n g l e 8 , w h i l e i n t h e l e f t hand p a i r t h e s e a n g l e s w i l l b e added.
With a n e g a t i v e d r i f t a n g l e ,
t h e c a l c u l a t i o n of t h e angles w i l l be performed i n t h e left-hand
p a i r o f beams, and combined i n t h e r i g h t .
T h e r e f o r e , t h e l a s t two
f o r m u l a s g i v e n above can b e w r i t t e n i n t h e form

347

where t h e s i g n
shows t h a t t h e f o r m u l a s c h a n g e p l a c e s f o r t h e
l e f t a n d r i g h t - h a n d p a i r s o f beams when t h e s i g n o f t h e d r i f t a n g l e
changes.

c o u n t e r X, 2

i n d i c a t o r YP

mitter w

receiver
.device.
r eac eni v it n ge n n a L - F y n

tarnatnesnm
n iat t i n g

F u n c t i o n a l Diagram of A Doppler

Fig. 3.58.
Meter.

Note.
S i n c e t h e p a i r s o f beams a r e d i a g o n a l t o t h e wave g u i d e
l a t t i c e , a n d e a c h o f t h e m c o n t a i n s a l e f t a n d r i g h t - h a n d beam r e l a t i v e
t o t h e l o n g i t u d i n a l a x i s o f t h e a i r c r a f t , t h e l e f t or r i g h t - h a n d
p a i r o f beams h e r e i s r e f e r r e d t o a s a p a i r w h o s e l e a d i n g beam i s
d i r e c t e d t o t h e l e f t or r i g h t o f t h e l o n g i t u d i n a l a x i s o f t h e a i r ­
craft.
O b v i o u s l y , i n t h e c a s e o f a f i x e d a n t e n n a on t h e a i r c r a f t ( F i g .
3.59), t h e f i r s t problem f o r t h e c a l c u l a t i n g device of t h e Doppler
/333
meter i s t h e d e t e r m i n a t i o n o f a n a n g l e a t which
c


Since t h e angle 6 i s a c o n s t a n t v a l u e , and t h e f r e q u e n c i e s f D Y
a
re v a r i a b l e , t h e s o l u t i o n of a problem of t h i s kind does n o t
fD2
p r e s e n t any s i g n i f i c a n t d i f f i c u l t i e s .
The d e s i r e d a n g l e a i s t h e
d r i f t angle of t h e aircraft.
The s e c o n d p r o b l e m f o r t h e c o m p u t i n g d e v i c e i s t h e d e t e r m i n ­
a t i o n o f t h e g r o u n d s p e e d ( W ) w i t h a p r e v i o u s l y known d r i f t a n g l e

348


and Doppler f r e q u e n c y f o r a p a i r o f beams:
fD,

fD,X

The c a l c u l a t e d d r i f t a n g l e o f
t h e a i r c r a f t and t h e ground speed are
transmitted t o the visual indicator
of t h e s e p a r a m e t e r s a n d a l s o t o t h e
automatic navigational device f o r inte­
g r a t i o n of t h e aircraft path i n t i m e .
The p r o b l e m o f t h e c a l c u l a t i n g
d e v i c e of t h e D o p p l e r m e t e r i s s i g ­
n i f i c a n t l y s i m p l i f i e d by m o u n t i n g a
movable a n t e n n a on t h e a i r c r a f t .
In
t h i s case, t h e d i r e c t i o n of t h e antenna

i s s e t s o t h a t t h e Doppler frequen­

c i e s o f b o t h a n t e n n a s fD
1 and
b e t h e same, i . e . , t h e b i s e c t r i x o f
t h e beams w i l l c o i n c i d e w i t h t h e d i r e c ­

Fig. 3.59.
Differece i n
Doppler Frequencies of

t h e Right- and Left-Hand
B e a m s of a n Antenna.

%

t i o n of t h e a i r c r a f t motion:

Then t h e d r i f t a n g l e o f t h e a i r c r a f t i s d e t e r m i n e d by t h e c o u r s e
a n g l e o f t h e a n t e n n a s e t t i n g , a n d t h e g r o u n d s p e e d i s f o u n d by t h e
formula :
(a)

With i n t e r n a l a n d e x t e r n a l c o h e r e n c e

w=
2

(b)

fDX
.- .
e COS p ’

With a u t o c o h e r e r l c e

fDx
W=

4 COS e COS B’

T h i s means t h a t a l l c o e f f i c i e n t s e n t e r e d i n t o t h e f o r m u l a s ( w i t h
t h e e x c e p t i o n o f fD) a r e c o n s t a n t s w h i l e f D i s a v a r i a b l e q u a n t i t y .
We s h o u l d m e n t i o n t h a t d u r i n g f l i g h t a b o v e t h e o c e a n , D o p p l e r
f r e q u e n c i e s f r o m p a i r s o f beams i n a D o p p l e r m e t e r a r e somewhat
l o w e r t h a n a b o v e d r y l a n d a t t h e same a i r s p e e d s .
This is caused
by p e c u l i a r f e a t u r e s of t h e r e f l e c t i o n of e l e c t r o m a g n e t i c waves
from t h e s u r f a c e o f t h e w a t e r .

/334

I n f l i g h t a b o v e d r y l a n d , i f t h e c o n d i t i o n s for d i f f u s e r e f l e c ­
t i o n o f t h e w a v e s a r e a p p r o x i m a t e l y t h e same o v e r a l l a r e a s i n c o n t a c t
w i t h t h e E a r t h ’ s s u r f a c e , a n d t h e maximum a m p l i t u d e c o i n c i d e s w i t h

349


I*


I1 I l l

t h e c e n t e r o f t h e beam a t t h e maximum o f t h e r a d i a t i o n c h a r a c t e r ­
i s t i c , t h e n t h e r e f l e c t i o n c o n d i t i o n s above a watery s u r f a c e w i l l
depend t o a c o n s i d e r a b l e e x t e n t upon t h e a n g l e o f i n c i d e n c e of t h e
beam.
T h e r e f o r e , t h e l e a d i n g e d g e o f t h e beam w i l l h a v e a s h a r p e r
a n g l e o f i n c i d e n c e ( a n d t h e r e f o r e a l o w e r s i g n a l amp'litude), w h i l e
t h e t r a i l i n g e d g e o f t h e beam w i l l s t r i k e more o b l i q u e l y a n d h a v e
somewhat g r e a t e r a m p l i t u d e .
C o n s e q u e n t l y , t h e maximum a m p l i t u d e
of t h e s i g n a l s s h i f t s from t h e c e n t e r t o a r e g i o n of lower Doppler
frequency (see Fig. 3.54).
To c o m p e n s a t e f o r e r r o r s i n t h e o p e r a t i o n o f t h e m e t e r a b o v e
water, t h e c i r c u i t i s designed t o i n c l u d e a c a l i b r a t i o n element
w h i c h i s s w i t c h e d on f r o m t h e c o n t r o l p a n e l b y t u r n i n g a s w i t c h
from t h e " l a n d " p o s i t i o n t o t h e "sea" p o s i t i o n .
Over a smooth w a t e r y s u r f a c e ( w i t h a s w e l l l e s s t h a n a s c a l e
v a l u e o f o n e ) , t h e p o t e n t i a l o f t h e r e f l e c t e d s i g n a l s becomes i n a d ­
equate t o ensure o p e r a t i o n of t h e meter, and t h e l a t t e r t h e n i s
t u r n e d o f f b y s w i t c h i n g t h e a u t o m a t i c n a v i g a t i o n a l d e v i c e t o memory
operation.
The c h a n n e l o f t h e D o p p l e r f r e q u e n c y r e c e i v e r i s f i t t e d w i t h
a f i l t e r i n t e n d e d t o damp o u t a l l p a r a s i t i c f r e q u e n c i e s p r o d u c e d
by o t h e r e l e c t r o n i c d e v i c e s m o u n t e d a b o a r d t h e a i r c r a f t w h i c h c o u l d
d i s t u r b r e c e p t i o n of r e f l e c t e d s i g n a l s from t h e E a r t h ' s s u r f a c e .
T h e f i l t e r must h a v e a n a r r o w p a s s b a n d w i t h i n t h e r e g i o n o f D o p p l e r
frequencies of t h e received s i g n a l s .
If t h e f r e q u e n c y o f a c a r e f u l l y a d j u s t e d f i l t e r d i f f e r s c o n ­
s i d e r a b l y from t h e midpoint of t h e r a n g e o f Doppler f r e q u e n c i e s
b e i n g employed, it b e g i n s t o i n t r o d u c e e r r o r s i n t h e measurement
of t h e Doppler frequency, s h i f t i n g it toward t h e p o i n t o f f i n e t u n i n g
Therefore, f i l t e r s a r e used with automatic tuning
of t h e f i l t e r .
f o r t h e frequency o f t h e s i g n a l s employed.

Use of D o p p l e r M e t e r s f o r P u r p o s e s of A i r c r a f t N a v i g a t i o n
Doppler meters f o r ground speed and d r i f t a n g l e are v e r y effec­
tive i n aircraft navigation.
The f o l l o w i n g p r o b l e m s c a n b e s o l v e d
d i r e c t l y by u s i n g a D o p p l e r m e t e r :
(a)
M a i n t a i n a n c e o f a g i v e n d i r e c t i o n of f l i
d r o m e or l o x o d r o m e , a u t o m a t i c a l l y i f d e s i r e d .
TO
o n l y n e c e s s a r y t h a t t h e sum o f t h e c o u r s e ( y ) a n d
of t h e a i r c r a f t b e c o n s t a n t l y e q u a l t o a g i v e n f l i

g h t a l o n g an ortho­
do t h i s , it i s
d r i f t (a> angles
/335
g h t path angle ($):

$ = y t a ;

(b)
The c a l c u l a t i o n o f t h e p a t h o f t h e a i r c r a f t i n t e r m s o f
d i s t a n c e c a n b e s o l v e d on t h e b a s i s o f t h e g r o u n d s p e e d and t i m e :

s
350

= Wt.

The r e s u l t s o f s o l v i n g t h e s e p r o b l e m s by u s i n g D o p p l e r m e t e r s
a r e much m o r e a c c u r a t e t h a n t h o s e o b t a i n e d w h e n o t h e r t y p e s o f e l e c ­
t r o n i c d e v i c e s a r e u s e d , a n d t h e r e i s l e s s work i n v o l v e d .
H o w e v e r , a c c u r a t e a i r c r a f t n a v i g a t i o n r e q u i r e s c o n s t a n t mon­
i t o r i n g of a l l v a r i a b l e f a c t o r s i n t h e ground speed, d r i f t a n g l e ,
and a i r c r a f t c o u r s e , which a r e t e d i o u s f o r t h e crew.
However, i f
we use individual, discrete values f o r calculating t h e d r i f t angle,
c o u r s e , a n d g r o u n d s p e e d , ( e . g . , e v e r y 1 5 m i n o f f l i g h t t i m e ) , as
i s done w i t h a i r c r a f t r a d a r , one o f t h e b a s i c a d v a n t a g e s o f Doppler
meters w i l l b e l o s t :
the constant supply of information regard­
i n g t h e motion parameters of t h e a i r c r a f t .
I n view o f t h e a b o v e , as w e l l as t h e r e l a t i v e s i m p l i c i t y o f
a u t o m a t i n g a i r c r a f t n a v i g a t i o n on t h e b a s i s o f D o p p l e r m e a s u r e m e n t s ,
t h e l a t t e r a r e p r a c t i c a l l y i m p o s s i b l e t o u s e w i t h o u t c o m b i n i n g them
with automatic navigational instruments.
Automatic n a v i g a t i o n a l i n s t r u m e n t s connected t o Doppler meters
c a l c u l a t e t h e a i r c r a f t p a t h w i t h t i m e i n a n o r t h o d r o m i c or g e o g r a p h i c
system of c o o r d i n a t e s .
To c a l c u l a t e t h e p a t h o f t h e a i r c r a f t i n a n o r t h o d r o m i c s y s ­
t e m of coordinates, t h e navigational devices a r e connected t o trans­
m i t t e r s of t h e o r t h o d r o m i c c o u r s e ( a gyro assembly f o r t h e c o u r s e
s y s t e m , o p e r a t i n g i n t h e GSC r e g i m e ) .
The a u t o m a t i c s y s t e m i n c l u d e s
a t r a n s m i t t e r of t h e f l i g h t a n g l e or ( a s i t i s u s u a l l y c a l l e d ) t h e
g i v e n c h a r t a n g l e (GCA)

.

The s i g n a l s f o r t h e d r i f t a n g l e o f t h e a i r c r a f t , o b t a i n e d f r o m
t h e m e t e r , and t h e c o u r s e s i g n a l s o f t h e a i r c r a f t , o b t a i n e d from
t h e c o u r s e s y s t e m , a r e c o m b i n e d a n d t h e i r sum c o m p a r e d w i t h a g i v e n
path angle fed i n t o t h e transmitter.
I f t h e sum o f t h e c o u r s e a n d t h e d r i f t a n g l e o f t h e a i r c r a f t
is e q u a l t o t h e g i v e n p a t h a n g l e of t h e f l i g h t $ = y t a , t h e ground
s p e e d i s d i r e c t e d a l o n g t h e X - a x i s : W = W,; W, = 0 .

If t h i s e q u a t i o n i s n o t s a t i s f i e d ,
s p e e d i s d i v i d e d i n t o two components:

t h e v e c t o r of t h e ground

wx'= w cos (7 + - 49;
W,= Wsin (1 +

- +I-

The v e c t o r c o m p o n e n t s o b t a i n e d for t h e g r o u n d s p e e d a l o n g t h e
axes of t h e c o o r d i n a t e s are i n t e g r a t e d over t i m e and c a l c u l a t o r s
a r e u s e d t o f i n d t h e r u n n i n g v a l u e s of t h e a i r c r a f t c o o r d i n a t e s X
and 2 .
C a l c u l a t i o n of t h e a i r c r a f t p a t h a n d g e o g r a p h i c c o o r d i n a t e s
c a n a l s o b e d o n e d i r e c t l y o n t h e b a s i s o f t h e s i g n a l s f r o m t h e Dop­
However, t o do t h i s i t i s
p l e r meter and t h e course c a l c u l a t o r .
n e c e s s a r y t o h a v e a n exact knowledge of t h e t r u e c o u r s e of t h e a i r ­
351

/336

craft and t o express t h e d i v i s i o n of t h e ground-speed
t h e aircraft according t o t h e formulas:

*w
dt

=

cos (7

vector of

+ a);

T o ensure operation of t h e gyroscopic t r a n s m i t t e r i n a regime
o f t r u e c o u r s e , i n a d d i t i o n t o t h e moment w h i c h c o m p e n s a t e s for
t h e d i u r n a l r o t a t i o n of t h e E a r t h
= 52 sin 'p,

i t i s n e c e s s a r y t o a d d t h e moment w h i c h c o m p e n s a t e s f o r t h e c h a n g e
i n t h e t r u e c o u r s e w i t h t i m e d u e t o t h e e a s t e r n or w e s t e r n compo­
nent of t h e ground-speed v e c t o r of t h e a i r c r a f t :

However, c a l c u l a t i o n of t h e a i r c r a f t c o u r s e b y t h i s s y s t e m
c a n n o t b e c o n s i d e r e d a d e q u a t e for t h r e e r e a s o n s :

(1) T h e p a t h d u r i n g f l i g h t w i t h a c o n s t a n t t r u e c o u r s e i s
l o x o d r o m i c , b u t t h i s c o m p l i c a t e s t h e p r e p a r a t i o n s a n d makes i t more
d i f f i c u l t t o u s e t h e r a d i o - e n g i n e e r i n g and a s t r o n o m i c a l methods
There­
during f l i g h t f o r c o r r e c t i n g t h e coordinates of t h e aircraft.
f o r e , i n a d d i t i o n t o t h e m a g n e t i c ( t r u e ) compass on t h e a i r c r a f t ,
t h e r e must a l s o b e an orthodromic couse d e v i c e .
(2)
T h e c o n s t a n t d e p e n d e n c e o f t h e o p e r a t i o n of t h e c o u r s e
s y s t e m o n t h e o p e r a t i o n of t h e D o p p l e r m e t e r a n d a c a l c u l a t i n g d e v i c e
introduces inaccuracies i n t o t h e aircraft navigational elements.
For e x a m p l e , w h e n t h e E a r t h is n o t v i s i b l e , a f l i g h t c a n b e made
over dry land; however, i f t h e a i r c r a f t t h e n b e g i n s t o t r a v e l over
a smooth w a t e r y s u r f a c e , t h e r e f l e c t e d D o p p l e r s i g n a l s w i l l n o t
o n l y i n t r o d u c e e r r o r s i n t o t h e a c c u r a c y w i t h which t h e p a t h i s calcu­
l a t e d with time, but w i l l a l s o incorporate e r r o r s i n the operation
of t h e c o u r s e s y s t e m .

(3)
T h e e r r o r s w h i c h a p p e a r i n t h e c a l c u l a t i o n of t h e a i r ­
c r a f t c o u r s e a t t h e p o i n t s of c o r r e c t i o n o f i t s c o o r d i n a t e s c a n n o t
b e u s e d d i r e c t l y f o r c o r r e c t i o n o f t h e a i r c r a f t c o u r s e , as can e a s i l y
be done i n a n o r t h o d r o m i c s y s t e m o f c o o r d i n a t e s .

A more l o g i c a l c a l c u l a t i o n o f t h e g e o g r a p h i c c o o r d i n a t e s o f
t h e a i r c r a f t would i n v o l v e t h e o r t h o d r o m i c s y s t e m of a i r c r a f t n a v i g a ­
t i o n , b a s e d o n a c o n s t a n t c o n v e r s i o n o f t h e o r t h o d r o m i c c o u r s e of
t h e a i r c r a f t t o t h e t r u e c o u r s e on t h e b a s i s of t h e r u n n i n g c o o r d ­
i n a t e s of t h e aircraft:

352

where

The t r u e c o u r s e f o r t h e a i r c r a f t o b t a i n e d i n t h i s man n er c a n
be u s e d t o c a l c u l a t e t h e g e o g r a p h i c c o o r d i n a t e s o f a n a i r c r a f t as
w a s shown e a r l i e r ; i t c a n a l s o b e u s e d f o r c o r r e c t i n g t h e o r t h o ­
dromic c o u r s e by a s t r o n o m i c a l means.
1337
The a d v a n t a g e s o f a m e t h o d o f t h i s k i n d a r e t h e i n d e p e n d e n c e
of t h e t r u e c o u r s e f r o m t h e g r o u n d s p e e d a n d i t s a u t o m a t i c c o r r e c t i o n
a l o n g w i t h t h e c o r r e c t i o n of t h e a i r c r a f t c o o r d i n a t e s .

However, w e s h o u l d m e n t i o n t h a t t h e c a l c u l a t i o n o f t h e a i r ­
c r a f t course and geographic c o o r d i n a t e s should r e a l l y be r e p l a c e d
by a c o n s t a n t c o n v e r s i o n o f t h e r u n n i n g o r t h o d r o m i c c o o r d i n a t e s
i n t o g e o g r a p h i c o n e s , e . g . , by F o r m u l a s ( 1 . 6 4 ) a n d ( 1 . 6 5 ) :
sin+
= sin+
cose-cosh
sine;
geog
ort
ort
sinX

Jeog

= sinh

ort

cos+

ort

sec+
geog'

I n t h i s c a s e , t h e geographic coordinates w i l l always
s t r i c t l y with t h e orthodromic ones, so t h a t t h e r e w i l l be
p a r a m e t e r s from o n l y one i n t e g r a t i n g d e v i c e and a u t o m a t i c
t i o n i n t h e second system w i t h c o r r e c t i o n of c o o r d i n a t e s i
of t h e m .

agree
output
correc­
n one

I n g e n e r a l , t h e g e o g r a p h i c c o o r d i n a t e s a r e n o t o f much i n t e r ­
e s t as f a r as a i r c r a f t n a v i g a t i o n i s concerned.
However, t h e y a r e
i m p o r t a n t for e n s u r i n g a c c u r a t e o p e r a t i o n o f n a v i g a t i o n a l t r a n s ­
m i t t e r s ( l a t i t u d i n a l c o r r e c t i o n of course systems, a n a l y s i s of gyrov e r t i c a l s , r e l i a b l e operation of i n e r t i a l navigational systems,
etc.).
I n a d d i t i o n , t h e geographic coordinates can be conveniently
used i n t h e p r e s e n c e of a s t r o n o m i c a l methods of a i r c r a f t n a v i g a t i o n
and f o r i n t r o d u c i n g t h e c o o r d i n a t e s o f r e f e r e n c e p o i n t s i n t o t h e
c a l c u l a t i n g device, used f o r c o r r e c t i o n of aircraft coordinates.
For purposes of a i r c r a f t navigation, automatic n a v i g a t i o n a l
d e v i c e s a r e much m o r e d e p e n d a b l e f o r c a l c u l a t i n g t h e p a t h o f t h e
a i r c r a f t i n orthodromic coordinates.
I n a d d i t i o n t o t h e b a s i c r e g i m e o f o p e r a t i o n by s i g n a l s f r o m
a Doppler m e t e r , a u t o m a t i c n a v i g a t i o n a l d e v i c e s as a r u l e have an
o p e r a t i n g r e g i m e w i t h "memorized" n a v i g a t i o n a l p a r a m e t e r s .
T h e r e g i m e f o r o p e r a t i n g b y "memory"
one o f t h e f o l l o w i n g two v e r s i o n s .
1.
By "memorizing" t h e
d r i f t angle o f the a i r c r a f t .

can be incorporated i n

l a s t v a l u e s o f t h e g r o u n d speed and
I n t h i s v e r s i o n , i n t h e case when

353


t h e r e i s a n i n t e r r u p t i o n i n t h e a r r i v a l o f D o p p l e r s i g n a l s f o r some
r e a s o n , ( e . g . , when t h e r e a r e n o w a v e s i n a f l i g h t o v e r w a t e r ) ,
t h e p a t h c a n b e c a l c u l a t e d b y "memory" f o r a p e r i o d o f 1 5 - 2 0 m i n ,
only under t h e c o n d i t i o n t h a t t h e f l i g h t d i r e c t i o n and a i r s p e e d
have been recorded.
With a c h a n g i n g f l i g h t r e g i m e f o r t h e a i r c r a f t ,
c a l c u l a t i o n b y "memory" l e a d s t o l a r g e e r r o r s , s i n c e t h e g r o u n d
s p e e d a n d d r i f t a n g l e c h a n g e o n a new c o u r s e or w i t h a c h a n g e i n
other parameters.

2.
By " m e m o r i z i n g " w i n d p a r a m e t e r s a t f l i g h t a l t i t u d e .
In
t h i s v a r i e t y , t h e c a l c u l a t i n g d e v i c e i s p r o v i d e d w i t h s p e c i a l "memory"
p o t e n t i o m e t e r s , which c o n s t a n t l y s e t t h e v a l u e o f t h e wind param­
eters:
ux = w c o s (7 a - 4 )
v c o s (7- $);

+

u,=

Wsin(r+a-+)-

-

Vsln(r-+).

Then, i f t h e s i g n a l s s h o u l d n o t b e r e c e i v e d from t h e Doppler meter,

t h e p a t h of t h e a i r c r a f t c a n b e c a l c u l a t e d b y c o m p a r i n g t h e v e c t o r

o f t h e w i n d s p e e d a l o n g t h e a x i s of t h e s y s t e m o f c o o r d i n a t e s w i t h
-~
/33:
t h e wind v e c t o r components added t o i t .
If t h e given path angle
o f t h e f l i g h t t h e n c h a n g e s , t h e components of t h e wind v e c t o r are
r e d i s t r i b u t e d among t h e c o o r d i n a t e a x e s a n d t h e c a l c u l a t i o n r e g i m e
is not disturbed.
However, i n b o t h t h e f i r s t a n d s e c o n d methods o f "memorizing"
n a v i g a t i o n a l p a r a m e t e r s , n o p r o v i s i o n i s made f o r a n e x a c t c a l c u ­
l a t i o n of t h e aircraft path during a long period of t i m e , s i n c e
t h e wind p a r a m e t e r s change w i t h d i s t a n c e .
I n t h e s e cases, t h e navi­
g a t i o n a l mechanism i s u s e d f o r c a l c u l a t i n g t h e p a t h o f t h e a i r c r a f t
on t h e b a s i s o f d i s c r e t e d a t a o b t a i n e d b y m e a s u r i n g t h e g r o u n d s p e e d
a n d d r i f t a n g l e , e . g . , b y m e a n s o f a a i r c r a f t r a d a r or some o t h e r
d e v i c e , a s i s d o n e ( e . g . 1 when u s i n g t h e n a v i g a t i o n a l i n d i c a t o r
NI-SOB.
I n some t y p e s o f n a v i g a t i o n a l i n s t r u m e n t s , i n e r t i a l or a s t r o ­
i n e r t i a l i n s t r u m e n t s a r e u s e d a s memory d e v i c e s .
The p r o b l e m d o e s n o t i n v o l v e a d e t a i l e d s t u d y o f i n e r t i a l n a v i ­
g a t i o n a l d e v i c e s , because t h e l a t t e r have n o t y e t found wide a p p l i ­
cation i n c i v i l aviation.
I n a d d i t i o n , t h e problem of t h e a d v i s ­
a b i l i t y o f i n s t a l l i n g them i s s t i l l n o t s u f f i c i e n t l y c l e a r , s i n c e
t h e c o n s i d e r a b l e c o m p l e x i t y a n d s t a b i l i t y o f t h e s e i n s t r u m e n t s means
t h a t t h e range o f problems which t h e y can s o l v e i s s t i l l extremely
narrow.
T h e r e f o r e , w e w i l l c o n t e n t o u r s e l v e s with a b r i e f mention
of t h e o p e r a t i n g p r i n c i p l e o f t h e s e i n s t r u m e n t s .
I n e r t i a l navigational devices are gyrostabilized platforms
on which a c c e l e r o m e t e r s a n d s p e c i a l g y r o s c o p e s a r e mounted which
i n t e g r a t e t h e a c c e l e r a t i o n s of t h e a i r c r a f t w i t h t i m e a l o n g t h e
axes of t h e reference system.
I n t h e case when t h e m o t i o n of

354

t h e a i r c r a f t a l o n g o n e or t w o

I


a x e s t a k e s p l a c e w i t h a c c e l e r a t i o n , a moment i s a p p l i e d t o t h e a x e s
o f t h e g y r o s c o p e which i s p r o p o r t i o n a l t o t h e s e a c c e l e r a t i o n s , s o
t h a t precession of t h e gyroscope axes takes place, i.e. , t h e r e i s
integration of accelerations with t i m e .
Since

s

t

w, =

cr,(f)dt

0

and

s

w, = 0

az(t)df,

w h e r e ax a n d a , a r e t h e a c c e l e r a t i o n s a l o n g t h e c o r r e s p o n d i n g a x e s ,
we can use t h e p o s i t i o n of t h e gyroscope axes t o g e t an i d e a of
t h e components of t h e a i r c r a f t s p e e d a l o n g t h e axes of t h e coord­
inates.
T h e c o m p o n e n t s o f t h e g r o u n d s p e e d a l o n g t h e a x e s of t h e r e f ­
e r e n c e s y s t e m c a n b e i n t e g r a t e d i n t u r n w i t h t i m e by means o f a
navigational instrument.
I n an o p e r a t i n g Doppler meter, t h e p o s i t i o n of t h e axes of
t h e i n t e g r a t i n g g y r o s c o p e s c a n b e c o r r e c t e d by s i g n a l s from t h i s
meter.
I n t h e c a s e when t h e D o p p l e r i n f o r m a t i o n d o e s n o t a r r i v e ,
t h e i n e r t i a l device can be used f o r a long period of t i m e t o r e t a i n
"remembered" v a l u e s of t h e components o f t h e s p e e d a l o n g t h e a x e s
of t h e c o o r d i n a t e s , c o r r e c t i n g t h e m for a n y a c c e l e r a t i o n s t h a t a r i s e
i n t h e way o f w i n d c h a n g e s , a s w e l l a s i n c h a n g e s i n t h e f l i g h t
regime.
A i r c r a f t n a v i g a t i o n u s i n g Doppler m e t e r s and a u t o m a t i c nav­
i g a t i o n a l i n s t r u m e n t s becomes e x t r e m e l y s i m p l e and p r a c t i c a l , b u t
v e r y c a r e f u l p r e p a r a t i o n s f o r f l i g h t and e x a c t measurements of t h e
c o o r d i n a t e s of t h e a i r c r a f t a t t h e c o r r e c t i o n p o i n t s a r e r e q u i r e d .
An e x a c t m e a s u r e m e n t o f t h e a i r c r a f t c o u r s e i s e x t r e m e l y i m p o r t a n t
i n t h i s regard.

/339

On t h e o t h e r h a n d , t h e f a c t t h a t t h e c r e w i s c o n s t a n t l y a w a r e
o f t h e g r o u n d s p e e d , t h e d r i f t a n g l e of t h e a i r c r a f t , a n d i t s c o o r d ­
i n a t e s makes i t p o s s i b l e t o m a i n t a i n a g i v e n f l i g h t t r a j e c t o r y f o r
long periods of time according t o t h e i n d i c a t i o n s of t h e i n s t r u ­
ments.
To d o t h i s , i t i s s u f f i c i e n t t h a t t h e sum o f t h e a i r c r a f t
course and t h e d r i f t angle b e c o n s t a n t l y e q u a l t o t h e given path
a n g l e , a n d t h a t t h e Z - c o o r d i n a t e of t h e a i r c r a f t b e e q u a l t o z e r o .

I-t i s p a r t i c u l a r l y e a s y t o s o l v e p r o b l e m s i n a i r c r a f t n a v i g a t i o n
i f che readings of t h e a i r c r a f t course and t h e d r i f t angle are obtained
f r o m t h e i n d i c a t o r i n t h e f o r m of a sum, : . e . ,
as t h e a c t u a l p a t h
a n g l e of t h e a i r c r a f t f l i g h t .
It is then sufficient t o p i l o t the
aircraft so t h a t with Z equal t o zero, t h e f l i g h t angle w i l l actually
be equal t o t h e given one.
355

I n a case when t h e p a t h a n g l e o f t h e f l i g h t i s n o t m a i n t a i n e d
p r e c i s e l y and t h e Z-coordinate of t h e a i r c r a f t i s n o t e q u a l t o z e r o ,
or, i f t h e i m p r o p e r o p e r a t i o n o f a s y s t e m h a s c a u s e d t h e a i r c r a f t t o
d e v i a t e from t h e g i v e n f l i g h t p a t h as r e v e a l e d by c o r r e c t i o n o f
i t s coordinates, t h e path angle of t h e f l i g h t is set so t h a t t h e
aircraft approaches t h e given l i n e of f l i g h t a t an angle of 3-5O.
When t h e Z - c o o r d i n a t e d e c r e a s e s t o z e r o , t h e p a t h a n g l e o f t h e f l i g h t
becomes e q u a l t o t h e g i v e n v a l u e .
The a i r c r a f t c a n b e p l a c e d on t h e g i v e n l i n e of f l i g h t b y u s i n g
For t h i s p u r p o s e t h e r e m u s t b e a c a l c u l a t i n g u n i t
the autopilot.
a b o a r d t h e a i r c r a f t for r e l a t i n g t h e D o p p l e r m e t e r w i t h t h e a u t o ­
matic n a v i g a t i o n a l d e v i c e and an a u t o p i l o t which s o l v e s t h e s i m p l e
problem :
AZ+kA$ = 0 ,
w h e r e A Z i s t h e l a t e r a l d e v i a t i o n f r o m t h e l i n e o f f l i g h t , A$ i s
t h e angle of approach t o t h e l i n e of f l i g h t , and
k is the selected
coupling f a c t o r .
The a i r c r a f t i s t h e n s t e e r e d s o t h a t a l e a d i n t h e p a t h a n g l e
o f t h e f l i g h t i s t a k e n when t h e a i r c r a f t d e v i a t e s t o a c e r t a i n d e g r e e
from t h e g i v e n l i n e of f l i g h t w i t h a c e r t a i n c o e f f i c i e n t .
Then,
i n t h e presence of l a t e r a l d e v i a t i o n , t h e a i r c r a f t w i l l automat­
i c a l l y move i n t o t h e l i n e o f f l i g h t , d e c r e a s i n g i t s l e a d a s i t a p ­
proaches t h e l a t t e r .
C e r t a i n d i f f i c u l t i e s i n a i r c r a f t n a v i g a t i o n when u s i n g Dop­
p l e r meters with automatic navigational devices are encountered
i n converting t h e computer t o c a l c u l a t e t h e p a t h i n orthodromic
coordinates of t h e previous s t a g e , a t t h e t u r n i n g points along t h e
route.
T h e m e t h o d s o f c o n v e r s i o n t o t h e new s y s t e m o f c a l c u l a t i o n
o f c o o r d i n a t e s i s s h o w n i n C h a p t e r 11, S e c t i o n 9 .
H o w e v e r , when
u s i n g Doppler m e t e r s , i t i s b e t t e r t o s e t t h e a i r c r a f t coordinates
t o t h e reference system of t h e previous s t a g e before beginning t h e
t u r n of t h e aircraft.
For e x a m p l e , w i t h 21 = 0 , Xi = - L L T .
X2 = -LLT

COS

/340

TA;

22 = L L T s i n T A .

For p u r p o s e s o f s i m p l i f y i n g t h e c o n v e r s i o n o f t h e p a t h c a l ­
c u l a t i o n i n t o t h e s y s t e m of c o o r d i n a t e s o f t h e n e x t s t a g e , d o u b l e
c o o r d i n a t e c a l c u l a t o r s a r e u s e d , l i n k e d m u t u a l l y w i t h one a n o t h e r .
I n t h i s c a s e , t h e c a l c u l a t i o n of t h e a i r c r a f t p a t h i s p e r f o r m e d
by one o f t h e c a l c u l a t o r s i n t h e s y s t e m o f c o o r d i n a t e s o f t h e f l i g h t
The s e c o n d c a l c u l a t o r
segment i n which t h e f l i g h t i s b e i n g made.
a d j u s t s i t s e l f according t o t h e path angle of t h e next path seg­
ment, and it c a n b e used t o c a l c u l a t e t h e a i r c r a f t c o o r d i n a t e s i n
t h e r e f e r e n c e system of t h i s segment.

356

The t r a n s i t i o n o f t h e a i r c r a f t t o t h e n e x t o r t h o d r o m i c s e g ­
m e n t of t h e p a t h i s a c c o m p l i s h e d b y t h e i n d i c a t i o n s o f t h e s e c o n d
c a l c u l a t o r , a f t e r which t h e f i r s t c a l c u l a t o r i s c l e a r e d and s e t
f o r t h e next path segment.
A s w e have a l r e a d y p o i n t e d o u t , i n t h e case o f double c a l c u ­
l a t o r s , t h e i r r e a d i n g s a r e m u t u a l l y r e l a t e d , i . e . , t h e y a r e con­
verted according t o t h e formulas:
X2

= XlcosTA-21sin

22

= X1sinTAtZlcosTA.

TA;

T h e r e f o r e , i n c o r r e c t i n g t h e c o o r d i n a t e s of t h e a i r c r a f t on one
o f t h e s e c o m p u t e r s , a c o r r e c t i o n i s a u t o m a t i c a l l y made i n t h e a i r ­
craft coordinate i n t h e reference system of t h e next stage.
T h u s , a t e a c h t u r n i n g p o i n t a l o n g t h e r o u t e , t h e a i r c r a f t makes
a t u r n i n a previously prepared and c o r r e c t e d system of c o o r d i n a t e s
for t h e n e x t s t a g e o f f l i g h t , t h u s c o m p l e t e l y g e t t i n g r i d o f a n y
u n d e s i r a b l e f e a t u r e s of t h e t r a n s i t i o n which m i g h t o c c u r i f o n l y
one c a l c u l a t o r were u s e d .

P r e p a r a t i o n f o r F Z i g h t and C o r r e c t i o n of E r r o r s i n A i r c r a f t
N a v i g a t i o n b y U s i n g DoppZer M e t e r s
A i r c r a f t n a v i g a t i o n u s i n g Doppler meters f o r measuring t h e ground
s p e e d and d r i f t a n g l e o f a n a i r c r a f t c a n b e done v e r y s i m p l y and
rapidly.
However, t h e r e q u i r e d a c c u r a c y f o r a i r c r a f t n a v i g a t i o n
when u s i n g t h e s e d e v i c e s c a n o n l y b e a c h i e v e d w i t h v e r y c a r e f u l
p r e p a r a t i o n f o r f l i g h t , as w e l l as c a r e f u l c o r r e c t i o n f o r e r r o r s
i n a i r c r a f t n a v i g a t i o n which a r i s e d u r i n g f l i g h t .
When u s i n g D o p p l e r m e t e r s , t h e r e may b e e r r o r s i n m e a s u r i n g
t h e f o l l o w i n g e l e m e n t s i n a i r c r a f t n a v i g a t i o n due t o e r r o r s i n t h e
transmitters:
(a)

M e a s u r e m e n t of t h e a i r c r a f t c o u r s e ;

(b)

M e a s u r e m e n t of t h e d r i f t a n g l e a n d g r o u n d s p e e d ;

(c)
I n t h e p r o g r a m m i n g of t h e g i v e n p a t h a n g l e a n d t h e d i s t a n c e of t h e f l i g h t s t a g e s ;
(d)
I n t h e integration of t h e aircraft f l i g h t along t h e axes
o f t h e c o o r d i n a t e s by t h e a u t o m a t i c n a v i g a t i o n a l d e v i c e .
The a c c u r a c y w i t h w h i c h t h e a i r c r a f t c o u r s e i s m e a s u r e d i s
o f e x t r e m e i m p o r t a n c e f o r a i r c r a f t n a v i g a t i o n when u s i n g D o p p l e r
meters and i s c l o s e l y r e l a t e d t o t h e p r o p e r programming o f p a t h
angles f o r each f l i g h t s t a g e .
T h i s i s e x p l a i n e d by t h e v e r y h i g h

357

/341

requirements f o r accuracy i n determining path angles i n preparing
for flight.
P r e p a r a t i o n for f l i g h t u s i n g D o p p l e r m e t e r s m u s t b e c a r r i e d
o u t properly according t o t h e t h i r d group of conditions i n Chapter
Two, S e c t i o n 2 .

For e a c h f l i g h t s e g m e n t , a l l p a r a m e t e r s o f t h e o r t h o d r o m e m u s t
:
be determined, b e g i n n i n g w i t h X
dis

G )bl=

tg 'p2 c t g 'pr c o s e c ~b

- ctg AL

It i s then necessary t o determine t h e o r i g i n a l azimuth of t h e
o r t h o d r o m e a0 b y t h e f o r m u l a

a n d t h e n t h e c o o r d i n a t e s of t h e i n t e r m e d i a t e p o i n t s on t h e o r t h o ­
drome f o r p l o t t i n g t h e m on t h e c h a r t :

-.

sin h

tg7/=

01

tg a0

The d i s t a n c e b e t w e e n t h e t u r n i n g p o i n t s a l o n g t h e r o u t e a l o n g
t h e orthodrome can b e determined by t h e f o r m u l a
COS

Si = COS A

01

COS 'pi.

If w e i n t r o d u c e i n t o t h i s f o r m u l a t h e c o o r d i n a t e s o f t h e i n i t i a l
and f i n a l p o i n t s o f t h e f l i g h t s e g m e n t , and (when n e c e s s a r y ) any
i n t e r m e d i a t e p o i n t s , w e can f i n d t h e d i s t a n c e t o t h o s e p o i n t s from
t h e s t a r t i n g p o i n t of t h e orthodrome.
The d i s t a n c e s b e t w e e n t h e
p o i n t s a r e d e t e r m i n e d by c a l c u l a t i n g t h e d i s t a n c e s from t h e s t a r t ­
i n g p o i n t of t h e orthodrome t o them.

For p r o g r a m m i n g t h e f l i g h t p a t h a n g l e w e d e t e r m i n e t h e a z i ­
muths o f t h e orthodrome a t t h e b e g i n n i n g and end o f e a c h segment
according t o t h e formula

If i t i s p r o p o s e d t h a t w e u s e a s t r o n o m i c a l m e t h o d s for c o r ­
r e c t i n g t h e a i r c r a f t course i n f l i g h t ( e . g . , i n f l i g h t over water
or t e r r a i n w h i c h h a s n o i d e n t i f y i n g l a n d m a r k s ) , t h e c o u r s e c o r r e c ­
t i o n p o i n t s are marked and t h e azimuths of t h e orthodromes a t t h e
c o r r e c t i o n p o i n t s a r e d e t e r m i n e d by t h i s f o r m u l a .
Reference points are s e l e c t e d f o r correcting t h e coordinates
of t h e a i r c r a f t d u r i n g f l i g h t .
Usually t h e s e a r e landmarks which
s h o w u p c l e a r l y o n r a d a r or p l a c e s w h e r e g o n i o m e t r i c - r a n g e f i n d i n g
Then t h e o r t h o d r o m i c c o o r d i n a t e s o f
i n s t a l l a t i o n s are l o c a t e d .

358

1342

t h e s e p o i n t s are determined, and t h e ground goniometric-rangefinding
instruments are used t o determine t h e azimuths of t h e orthodromic
s e g m e n t s o f t h e p a t h on w h i c h t h e s e d e v i c e s w i l l b e u s e d , r e l a t i v e
t o t h e m e r i d i a n s on which t h e g r o u n d b e a c o n s a r e l o c a t e d .
The g i v e n p a t h a n g l e f o r t h e f i r s t f l i g h t s e g m e n t i s c o n s i d e r e d
e q u a l t o t h e ' a z i m u t h of t h e o r t h o d r o m e a t t h e s t a r t i n g p o i n t o f
t h i s segment.
The p a t h a n g l e s of a l l s u b s e q u e n t p a t h s e g m e n t s a r e
c o n s i d e r e d t o b e e q u a l t o t h e sum o f t h e p a t h a n g l e o f t h e p r e v ­
i o u s s e g m e n t p l u s t h e a n g l e of t u r n i n t h e p a t h a t t h e t u r n i n g p o i n t
on t h e r o u t e .
D o p p l e r m e t e r s h a v e r e l a t i v e l y low e r r o r s i n m e a s u r i n g t h e
d r i f t a n g l e of an a i r c r a f t , s o t h a t t h e y can b e compensated f o r
i n t h e t o t a l by t h e e r r o r s i n a i r c r a f t c o u r s e .
In g e n e r a l , besides t h e e r r o r s i n measuring t h e d r i f t angle,
d e p e n d i n g on t h e o p e r a t i n g r e g i m e o f t h e m e t e r , t h e h e i g h t a n d s p e e d
o f f l i g h t , w h i c h h a v e a more or l e s s c o n s t a n t c h a r a c t e r , t h e r e a r e
e r r o r s which h a v e a f l u c t u a t i n g n a t u r e ( o s c i l l a t i o n s i n t h e m e t e r
r e a d i n g s from t h e a v e r a g e v a l u e ) .
The p r i n c i p a l r e a s o n f o r f l u c ­
t u a t i o n s i s t h e v a r y i n g c o n d i t i o n s of r e f l e c t i o n of e l e c t r o m a g n e t i c
waves from t h e E a r t h ' s s u r f a c e .
When s o m e p o i n t i s e n c o u n t e r e d w h i c h r e f l e c t s e l e c t r o m a g n e t i c
waves w e l l , i n t h e e l l i p s e o f r e f l e c t i o n from t h e E a r t h ' s s u r f a c e ,
t h e maximum o f t h e a m p l i t u d e o f D o p p l e r f r e q u e n c y i s f i r s t d i s p l a c e d
f o r w a r d ( f o r a r e a r beam, b a c k w a r d ) ; t h e n , as t h e p o i n t p a s s e s t h r o u g h
t h e e l l i p s e o f r e f l e c t i o n , t h e maximum o f t h e a m p l i t u d e s h i f t s t o w a r d
the average Doppler frequency and then backward, i n t o a region of
lower frequencies.
Thus, t h e r e i s f i r s t a p o s i t i v e " f i r i n g " of t h e Doppler f r e ­
quency, then a l e v e l i n g o f f , and f i n a l l y a negative " f i r i n g " .
For
t h e r e a r beam, t h e " f i r i n g s " of f r e q u e n c i e s t a k e p l a c e i n r e v e r s e
order.
The p e r i o d s o f f l u c t u a t i n g o s c i l l a t i o n s a r e s h o r t a n d d e p e n d
on t h e t i m e r e q u i r e d f o r t h e r e f l e c t i n g p o i n t s t o p a s s t h r o u g h t h e
e l l i p s e of r e f l e c t i o n .
P r a c t i c a l l y speaking, they are l o c a t e d w i t h i n
t h e l i m i t s of 3-6 s e c , s o t h a t t h e y c a n b e s m o o t h e d o u t t o a c o n s i d ­
e r a b l e d e g r e e by s e l e c t i n g t h e p r o p e r r a t e o f a n a l y s i s f o r t h e r e a d ­
ings of t h e d r i f t angle and ground speed.
A s f a r as t h e c a l c u l a t i o n s of t h e a i r c r a f t p a t h f o r d i s t a n c e
and d i r e c t i o n a r e c o n c e r n e d , t h e f l u c t u a t i n g o s c i l l a t i o n s do n o t
h a v e a n y n o t i c e a b l e e f f e c t on i t , s i n c e a f t e r 3-5 min o f f l i g h t
t h e i n t e g r a l v a l u e of t h e p o s i t i v e f l u c t u a t i o n s becomes e q u a l t o
t h e i n t e g r a l value of t h e negative f l u c t u a t i o n s .

T h e p r o c e s s o f n a v i g a t i o n a l e x p l o i t a t i o n o f a u t o n o m o u s Dop/343
p l e r s y s t e m s f o r a i r c r a f t n a v i g a t i o n c a n b e employed f o r a d j u s t ­
ing t h e system i t s e l f , i - e . , i n correcting t h e aircraft coordinates
359

m a n u a l l y or a u t o m a t i c a l l y i t i s p o s s i b l e t o d e t e r m i n e a n d c o m p e n ­
sate simultaneously the systematic errors i n t h e operation of the
s y s t e m as a whole.
In fact, i f t h e aircraft ( a t t h e s t a r t i n g point of a f l i g h t
s e g m e n t ) i s l o c a t e d p r e c i s e l y on t h e d e s i r e d f l i g h t l i n e , b u t i t s
Z-coordinate i s e q u a l t o z e r o , and w e keep t h e c o o r d i n a t e Z e q u a l
t o z e r o d u r i n g a11 s u b s e q u e n t s t a g e s o f t h e f l i g h t , t h e a i r c r a f t
w i l l h a v e t o r e m a i n on t h i s l i n e c o n s t a n t l y .
If t h i s i s n o t t h e
c a s e , a n e r r o r w i l l c r o p up i n t h e c a l c u l a t i o n o f t h e a i r c r a f t p a t h
w i t h r e s p e c t t o d i r e c t i o n , i . e . , a c e r t a i n a n g l e w i l l develop between
t h e g i v e n a n d a c t u a l f l i g h t p a t h a n g l e s of t h e a i r c r a f t .
I t i s most l i k e l y t h a t under t h e c o n d i t i o n s of p r e c i s e deter.­
m i n a t i o n and s e t t i n g o f a g i v e n p a t h a n g l e on t h e t r a n s m i t t e r , a n
e r r o r i n c a l c u l a t i o n w i l l a r i s e as a r e s u l t o f i m p r o p e r measure­
ment o f t h e a i r c r a f t c o u r s e , s i n c e g y r o s c o p i c d e v i c e s c a n show d r i f t
i n t h e i r readings with t i m e .
T h e r e f o r e , t h e t o t a l c o r r e c t i o n which
i s r e q u i r e d for p r o p e r c a l c u l a t i o n o f t h e p a t h s h o u l d m o s t l o g i c a l l y
b e made i n t h e r e a d i n g s o f t h e c o u r s e i n s t r u m e n t .
If a p o r t i o n o f t h e e r r o r s i n c a l c u l a t i n g i s n o t r e l a t e d t o
t h e o p e r a t i o n of t h e c o u r s e i n s t r u m e n t , t h e n t h e i r c o n t r i b u t i o n
t o t h e c o u r s e e r r o r s w i l l n o t make t h e a c c u r a c y o f a i r c r a f t n a v i g a ­
t i o n any w o r s e .

The l a t t e r s t a t e m e n t i s v a l i d f o r a c o m p l e x o f i n s t r u m e n t s
w h i c h p e r m i t c a l c u l a t i o n of t h e p a t h i n t e r m s of d i r e c t i o n , b u t
it is t h e o r e t i c a l l y n o t completely v a l i d f o r instruments intended
f o r d i s t a n c e f i n d i n g o f landmarks f o r t h e p u r p o s e o f making c o r r e c ­
tions i n the aircraft coordinates.
N e v e r t h e l e s s , if w e c o n s i d e r t h a t t h e t o t a l e r r o r i n measuring
t h e d r i f t a n g l e and c a l c u l a t i n g t h e p a t h s i n terms of d i r e c t i o n
w i t h a n a u t o m a t i c a p p a r a t u s i s n o m o r e t h a n 0 . 2 t o 0.3O a s a r u l e ,
w e must r e c o g n i z e t h a t c o r r e c t i o n o f t h e a i r c r a f t c o u r s e by t h e
r e s u l t s o f c a l c u l a t i n g t h e p a t h i s much m o r e a c c u r a t e t h a n c o r r e c t ­
i n g i t by any o t h e r methods, i n c l u d i n g a s t r o n o m i c a l o n e s .
During f l i g h t , t h e a c t u a l a i r c r a f t c o o r d i n a t e s are determined
by t h e d i s t a n c e s R and t h e p a t h b e a r i n g s o f t h e landmarks (PBL),
s e l e c t e d f o r t h i s purpose by t h e formulas:
(a)

I n t h e m e a s u r e m e n t of

X

=

X -R
7,

2 = 2 -R

2

(b)

I n t h e measurement of

aircraft radars
COS

PBL;

s i n PBL.
goniometric-rangefinding systems:

X = X u + Rcos(A-+M);

Z = $4

+ R sin ( A -h).

360

I1 I111

I

P


where
con.

XM a n d ZM a r e t h e o r t h o d r o m i c c o o r d i n a t e s o f a g r o u n d b e a ­

/344
O b v i o u s l y , t h e f o r m u l a s f o r t h e a i r c r a f t r a d a r a n d t h e goniometric-rangefinding systems are i n v a r i a b l e .
The d i f f e r e n c e i n
t h e s i g n s o f t h e s e c o n d t e r m s on t h e r i g h t - h a n d s i d e s i s e x p l a i n e d
by t h e f a c t t h a t t h e b e a r i n g o f a landmark i s o b t a i n e d w i t h t h e
a i d of a n a i r c r a f t r a d a r b u t t h e b e a r i n g of a n a i r c r a f t r e l a t i v e
t o a g r o u n d b e a c o n i s o b t a i n e d w i t h t h e a i d of a g o n i o m e t r i c - r a n g e ­
finding system.

A t t h e moment when t h e d i s t a n c e a n d p a t h b e a r i n g o f a l a n d ­
m a r k or a i r c r a f t a r e d e t e r m i n e d f r o m a g r o u n d b e a c o n , t h e i n d i c a t o r
A f t e r deter­
readings f o r t h e aircraft coordinates are recorded.
m i n i n g t h e a c t u a l c o o r d i n a t e s o f t h e a i r c r a f t by means o f a n a v i g a ­
t i o n a l s l i d e r u l e , t h e y a r e compared w i t h t h e c o o r d i n a t e s on t h e
i n d i c a t o r r e c o r d e d a t t h e moment o f d i s t a n c e f i n d i n g , a n d t h e e r r o r s
i n c a l c u l a t i n g t h e coordinates are found:

AX = X

act

A2 = Z

act

-

'talc;

'talc'

where Xact and Zact
a r e t h e c o o r d i n a t e s o f t h e a i r c r a f t on t h e b a s i s
are the coordinates
of t h e measurement r e s u l t s and X c a l c and Z c a l c
o f t h e a i r c r a f t a c c o r d i n g t o t h e r e a d i n g s on t h e c a l c u l a t o r .
The c o r r e s p o n d i n g c o r r e c t i o n s a r e t h e n e n t e r e d i n t h e r e a d ­
i n g s o f t h e r u n n i n g o r t h o d r o m i c c o o r d i n a t e s o f t h e a i r c r a f t on t h e
calculator.
The c h a v a c t e r i s t i c f e a t u r e o f t h e s o l u t i o n o f t h e s e p r o b l e m s
i s t h e l a c k of a need t o f i x t h e t i m e of measurement
of t h e a i r ­
c r a f t c o o r d i n a t e s a n d t h e i n t r o d u c t i o n of c o r r e c t i o n s i n t h e r e a d i n g s
o f t h e c a l c u l a t o r s w h e n t h e y c h a n g e , a s i s n e c e s s a r y when u s i n g
a l l other radionavigational instruments.
This f e a t u r e i s completely c h a r a c t e r i s t i c f o r Doppler systems.
The r e l a t i o n s h i p t o t i m e h e r e i s m a i n t a i n e d o n l y w i t h s e l e c t i o n
of regimes of speed f o r reaching checkpoints a t a given t i m e .
In
measuring t h e a i r c r a f t c o o r d i n a t e s and a l l o t h e r elements of a i r ­
c r a f t navigation, t h e time need not be taken i n t o account.
L e t us examine f u r t h e r t h e methods of g e t t i n g r i d of s y s t e ­
matic e r r o r s i n c a l c u l a t i n g t h e a i r c r a f t path and p r i m a r i l y t h e
measurements of t h e a i r c r a f t c o u r s e w i t h t h e u s e of Doppler meters.

For a p r e c i s e d e t e r m i n a t i o n o f t h e e r r o r s i n m e a s u r i n g t h e
aircraft course, w e need t o determine t h e a c t u a l coordinates of
t h e a i r c r a f t a t a t l e a s t two s u c c e s s i v e p o i n t s , w%th t h e measure­
m e n t b a s e on t h e o r d e r o f 2 0 0 - 3 0 0 km.

361


A t t h e first p o i n t , t h e actual coordinates of t h e aircraft
are determined and t h e readings of t h e c a l c u l a t o r a r e c o r r e c t e d .
A t t h e s e c o n d p o i n t , t h e a c t u a l c o o r d i n a t e s of t h e a i r c r a f t a r e
determined once a g a i n and t h e e r r o r i n c a l c u l a t i o n i s found, which
has been accumulated d u r i n g t h e f l i g h t time a l o n g t h e base from
t h e f i r s t t o t h e second measurement p o i n t .

I f we c o n s i d e r t h e e r r o r i n t h e r e a d i n g s

of

the calculator

a t t h e f i r s t p o i n t t o be e q u a l t o z e r o ( s i n c e t h e y have been cor­
r e c t e d ) , t h e e r r o r i n m e a s u r i n g t h e c o u r s e i s d e t e r m i n e d by t h e
formula

where AZ, i s t h e e r r o r i n c a l c u l a t i n g t h e a i r c r a f t c o o r d i n a t e s a t
t h e second p o i n t , and X1,2 i s t h e l e n g t h o f t h e measurement b a s e
between p o i n t s 1 and 2 .

(Fig.

T h i s p r o b l e m i s e a s i l y s o l v e d on a n a v i g a t i o n a l s l i d e r u l e
3.60).

By u s i n g a D o p p l e r m e t e r , i t i s p o s s i b l e t o f i n d n o t o n l y t h e
e r r o r s i n m e a s u r i n g t h e a i r c r a f t c o u r s e , b u t also t h e n a t u r e o f
t h e i r accumulation with time.

Fig.

3.60.

Fig.

3.61.

Fig.

3 . 6 0 . D e t e r m i n a t i o n o f t h e Error i n M e a s u r i n g t h e C o u r s e o n
t h e NL-1OM.

Fig.

3 . 6 1 . D e t e r m i n a t i o n of G y r o s c o p e D e v i a t i o n on t h e S e c o n d
Measurement Base.

a

b


Fig. 3.62.
Use o f t h e N L - 1 O M t o D e t e r m i n e ( a ) D e g r e e o f D e v i a t i o n
of t h e Gyroscope and ( b ) Orthodromic C o o r d i n a t e s of t h e A i r c r a f t
f r o m Ground R a d i o B e a c o n s .

362

/345

A s w e a l r e a d y know, t h e d e v i a t i o n o f a g y r o s c o p e w i t h t i m e
c a n b e c o m p e n s a t e d b y a s u i t a b l e s h i f t o f t h e l a t i t u d e on t h e c o m p e n ­
s a t o r f o r t h e d i u r n a l r o t a t i o n of t h e Earth.
If t h e d e v i a t i o n o f
t h e gyroscope i s s i g n i f i c a n t (2-3 d e g / h r ) , it can b e determined
by c h a n g e s i n t h e e r r o r s i n c a l c u l a t i n g t h e p a t h on t w o a d j a c e n t
b a s e s , p r e f e r a b l y o f t h e same l e n g t h ( F i g . 3 . 6 1 ) .
With c o n s i d e r a b l e d e v i a t i o n s o f t h e g y r o s c o p e a x i s , t h e p a t h
o f t h e a i r c r a f t t u r n s o u t t o b e c u r v i l i n e a r if t h e Z - c o o r d i n a t e
r e c o r d e d on t h e c a l c u l a t o r i s e q u a l t o z e r o , a n d t h e f i n a l e r r o r
i n measuring t h e course w i l l be g r e a t e r than t h i s average e r r o r
which a p p e a r s on t h e f i r s t b a s e a t t h e i n i t i a l v a l u e (Ayd) d i v i d e d
in half.
Therefore, after introducing the corrections i n the readings
o f t h e c o u r s e i n s t r u m e n t , a n e r r o r remains i n t h e measurement of
t h e c o u r s e which i s e q u a l t o t h i s v a l u e .
I f t h e m e a s u r e m e n t i s r e p e a t e d on t h e a d j a c e n t b a s e , o f a p p r o x ­
i m a t e l y the s a m e l e n g t h , t h e e r r o r found i n t h e measurement of t h e
c o u r s e w i l l c o n s i s t o f two v a l u e s :

(a)

The e r r o r i n t h e i n i t i a l s e t t i n g ,

e q u a l t o Ayd/2;

(b)
The a v e r a g e e r r o r d u e t o t h e d e v i a t i o n o f t h e g y r o s c o p e
on t h e s e c o n d b a s e , a l s o e q u a l t o A y d / 2 .
Thus, t h e e r r o r which i s found w i l l c o n s t i t u t e t h e magnitude
of t h e g y r o s c o p e d e v i a t i o n d u r i n g t h e f l i g h t t i m e a l o n g t h e s e c o n d
base.

/346

I n o r d e r t o d e t e r m i n e t h e m a g n i t u d e of g y r o s c o p e d e v i a t i o n
p e r h o u r of f l i g h t , i t i s s u f f i c i e n t t o d i v i d e t h e e r r o r which h a s
b e e n f o u n d i n t o t h e f l i g h t t i m e on t h e s e c o n d b a s e .

Example.
The f l i g h t t i m e o f a n a i r c r a f t on t h e f i r s t a n d s e c ­
ond b a s e s i s 2 0 min e a c h .
On t h e f i r s t b a s e , a n e r r o r i n m e a s u r i n g
t h e a i r c r a f t c o u r s e w a s found and compensated f o r .
H o w e v e r , on
t h e second base t h e e r r o r i n measuring t h e course turned out t o
b e e q u a l t o lo. F i n d t h e m a g n i t u d e of g y r o s c o p e d r i f t p e r h o u r
of f l i g h t .

Solution:
"d

--

10
= 3 degrees/hr.
0.33 h r

I f t h e d e v i a t i o n s o f t h e g y r o s c o p e a r e s m a l l (0.5-lo), t h e y
However, i n
c a n n o t be found by measurement from a s e c o n d b a s e .
t h i s case, it i s n o t n e c e s s a r y t o s h i f t t h e l a t i t u d i n a l p o t e n t i o m e t e r
i n making c o m p e n s a t i o n .
It is sufficient t o correct the readings
of t h e c o u r s e p e r i o d i c a l l y ( a l o n g w i t h t h e c o r r e c t i o n o f t h e a i r ­
c r a f t c o o r d i n a t e s ) by t h e r e s u l t s of t h e m e a s u r e m e n t s on o n e b a s e .

363

When n e c e s s a r y , a D o p p l e r m e t e r c a n b e u s e d t o d e t e r m i n e t h e
small-scale v a r i a t i o n s i n t h e gyroscope (0.5-2 d e g / h r ) .
T o do t h i s ,
both t h e a i r c r a f t c o o r d i n a t e s and e r r o r i n measuring t h e course
a r e d e t e r m i n e d on t h e f i r s t b a s e .
On s u b s e q u e n t b a s e s , o n l y t h e a i r c r a f t c o o r d i n a t e s a r e d e t e r ­
mined and c o r r e c t e d .
On t h e l a s t b a s e , t h e e r r o r i n t h e a i r c r a f t
course i s again determined.
The e r r o r w h i c h i s f o u n d w i l l c o n s t i ­
t u t e t h e d e v i a t i o n o f t h e g y r o s c o p e f r o m t h e moment o f t h e e n d o f
t h e first t o t h e end of t h e last base.
A t t h e same t i m e , t h e r e m a i n i n g e r r o r a t t h e e n d o f t h e f i r s t
b a s e i s e q u a l t o Ayd/2, w h i l e t h e e r r o r f o u n d a t t h e end o f t h e
last base,(e.g.) the fourth, is equal to:

i.e., if the last base is equal t o the first, the
b e f o u n d by m e a s u r i n g t h e c o u r s e w i l l b e e q u a l t o
of t h e gyroscope i n t h e second, t h i r d , and f o u r t h
t h e f l i g h t t i m e w i l l b e s u f f i c i e n t f o r s h o w i n g up
deviations of t h e gyroscope p e r hour of f l i g h t .

e r r o r which w i l l
the deviation
bases.
Thus,
even small-scale

Since t h e n a v i g a t i o n a l u s e o f Doppler meters does n o t pose
any d i f f i c u l t i e s , b u t t h e d e t e c t i o n of e r r o r s i n c a l c u l a t i n g t h e
p a t h a n d m e a s u r i n g t h e a i r c r a f t c o u r s e a r e much m o r e d i f f i c u l t ,
w e w o u l d l i k e t o c o n c l u d e b y p r o v i d i n g s e v e r a l e x a m p l e s o f how t o
determine these e r r o r s .
1. The l a s t c o r r e c t i o n o f a i r c r a f t c o o r d i n a t e s w a s made a t
t h e p o i n t X = 1 5 6 km, w h e r e t h e e r r o r i n t h e Z - c o o r d i n a t e w a s f o u n d
t o be z e r o .
The f l i g h t t h e n c o n t i n u e d w i t h m a i n t a i n a n c e o f t h e Z - c o o r d ­
A t t h e p o i n t X = 330 km, a c c o r d ­
i n a t e on t h e c o m p u t e r e q u a l t o z e r o .
i n g t o t h e r e a d i n g s of t h e c o m p u t e r , t h e a c t u a l c o o r d i n a t e s o f t h e
a i r c r a f t w e r e d e t e r m i n e d on t h e b a s i s o f a r a d a r l a n d m a r k , h a v i n g
the coordinates
Xi= 3 7 5 km, Zi = 6 1 km. T h e p o l a r c o o r d i n a t e s
/347
o f t h e landmark were as f o l l o w s :
P B L = 5 4 O , R = 72 km.
Find t h e
errors i n c a l c u l a t i n g t h e coordinates i n measuring t h e aircraft
course.
SOlUtiOn:
following:

by u s i n g a n a v i g a t i o n a l s l i d e r u l e , w e f i n d t h e

R c o s P B L = 4 2 . 5 km;
R s i n P B L = 5 8 km.
C o n s e q u e n t l y , t h e a c t u a l c o o r d i n a t e s o f t h e a i r c r a f t a r e as
follows: X
= 3 7 5 - 4 2 . 5 = 3 3 2 . 5 km; Z = 6 1 - 5 8 = 3 km, w h i l e t h e e r r o r s
i n calculating t h e coordinates are:
A X = t 2 . 5 O km, A Z = + 3 km.
364

The e r r o r i n m e a s u r i n g t h e a i r c r a f t c o u r s e i s
67 = arctg

'

3
332.5-156

==

0'58'.

I n t h i s case, t h e a i r c r a f t d e v i a t e d t o t h e r i g h t from t h e g i v e n
path, s o t h a t t h e readings of t h e course instrument w e r e reduced
a n / i t w a s n e c e s s a r y t o m a k e a c o r r e c t i o n e q u a l t o tOO58' o r a p p r o x ­
imately tlO.
2.
A f t e r correcting the coordinates i n the aircraft course,
considered i n t h e first example, t h e aircraft t r a v e l e d along a base
e q u a l t o 1 8 0 km w i t h a g r o u n d s p e e d o f 8 5 0 k m / h r .
A second check of t h e a i r c r a f t c o o r d i n a t e s r e v e a l e d t h a t t h e
T h e f l i g h t w a s made w i t h i n
e r r o r i n measuring t h e course w a s + O 0 3 5 ' .
l a t i t u d i n a l l i m i t s o f 50-60°.
Find t h e degree of d e v i a t i o n of t h e
gyroscope p e r hour of f l i g h t and t h e required s h i f t i n t h e l a t i ­
t u d i n a l compensator t o g e t r i d of it.

S o l u t i o n : The f l i g h t t i m e o f t h e a i r c r a f t a l o n g t h e b a s e i s
e q u a l t o 1 3 . 5 min.
The l e n g t h o f t h e s e c o n d b a s e i s a p p r o x i m a t e l y
equal t o t h e f i r s t base, s o t h a t t h e deviation of t h e gyroscope
a l o n g t h e s e c o n d b a s e i s e q u a l t o t h e e r r o r f o u n d by m e a s u r i n g t h e
course.
The d e v i a t i o n o f t h e g y r o s c o p e w a s f o u n d b y m e a n s o f a n a v i ­
gational s l i d e r u l e (Fig. 3.62, a ) .
Answer:
t h e d e v i a t i o n of t h e gyroscope p e r hour of f l i g h t
a m o u n t s t o 1 5 4 ' or 2 O 3 4 ' .
A t l a t i t u d e s of 50-60°, f o r each d e g r e e p e r hour of d e v i a t i o n
i n t h e g y r o s c o p e , i t i s n e c e s s a r y t o s h i f t t h e l a t i t u d i n a l compen­
s a t o r by 6 O .
In our example, t h e gyroscope deviated i n t h e direc­
t i o n of a reduction of t h e course indication s o t h a t t h e l a t i t u d e
6 O x 2 . 6 = 15.6O.
on t h e c o m p e n s a t o r h a d t o b e s e t t o t h e v a l u e :
A f t e r t h e d e s i r e d change i n t h e s e t t i n g of t h e l a t i t u d i n a l poten­
t i o m e t e r i s m a d e , t h e d e v i a t i o n o f t h e g y r o s c o p e s h o u l d c e a s e com­
pletely.

For c o r r e c t i n g t h e a i r c r a f t c o o r d i n a t e s , a g o n i o m e t r i c ­
3.
r a n g e f i n d i n g system i s employed.
The o r t h o d r o m i c c o o r d i n a t e s o f
a ground r a d i o beacon are:

x,

187 U S ;

z~= 142 K J l .

The f l i g h t a n g l e o f a n o r t h o d r o m e s e g m e n t , m e a s u r e d r e l a t i v e
t o t h e m e r i d i a n of t h e p o i n t where t h e beacon i s e s t a b l i s h e d , i s
equal t o 64O.
Find t h e orthodromic coordinates, of t h e aircraft
i f i t s a z i m u t h ( A ) i s e q u a l t o 24O a n d R = 2 2 5 km.

365

Solution:

+,,

A - = 320";
cos 3200 = COS 40"=sin 50";
sin 3%" = - sin 40".

By u s i n g a n a v i g a t i o n a l s l i d e r u l e , w e f i n d ( F i g .

i.e.

,

3.62,

b),

R cos 320"= 170 #A;
R sin 3%"= - I45 U.U.
C o n s e q u e n t l y , t h e o r t h o d r o m i c c o o r d i n a t e s of

t h e aircraft are

+

X = 187 170 = 357 K X ;
Z = 142-145=*3~A.

5.

PRINCIPLES O F COMBINING NAVIGATIONAL INSTRUMENTS

/348

I n C h a p t e r s Two a n d T h r e e o f t h e p r e s e n t work, w e d i s c u s s e d
t h e c o m p l e x e s o f n a v i g a t i o n a l i n s t r u m e n t s , w h i c h make i t p o s s i b l e
i n o n e way or a n o t h e r t o a u t o m a t e t h e p r o c e s s e s o f a i r c r a f t n a v i g a ­
t i o n or m e a s u r e m e n t o f i n d i v i d u a l n a v i g a t i o n a l p a r a m e t e r s .
The f i r s t n a v i g a t i o n a l co mp l ex i s t h e c o u r s e s y s t e m .
The b a s i c p r i n c i p l e s of c o m b i n i n g i n d i v i d u a l t r a n s m i t t e r s i n t o
a c o u r s e s y s t e m i s t h e c o m b i n a t i o n o f t h e r e a d i n g s for p u r p o s e s
o f a u t o m a t i c m u t u a l c o r r e c t i o n ( t h e M C , A C , GSC r e g i m e s ) , a n d a l s o
t o combine t h e r e a d i n g s of i n d i v i d u a l i n s t r u m e n t s t o improve t h e
n a v i g a t i o n a l v a l u e s , c o n s t i t u t i n g t h e sum o f i n d i v i d u a l e l e m e n t s .
For e x a m p l e :
OBR = O C t C A R .
The s e c o n d c o m p l e x i s t h e n a v i g a t i o n a l i n d i c a t o r N I - S O B , i n
which t h e r e i s a c o u r s e t r a n s m i t t e r , a t r a n s m i t t e r of t h e a i r s p e e d ,
and a m a n u a l l y - s e t wind t r a n s m i t t e r .
The m o s t c o m p l e t e o f t h e s e c o m p l e x e s i s t h e a u t o n o m o u s D o p p l e r
s y s t e m o f a i r c r a f t n a v i g a t i o n , which works i n c o n j u n c t i o n w i t h c o u r s e
t r a n s m i t t e r s and an automatic n a v i g a t i o n a l device.
Thus, t h e b a s i c reasons f o r combining n a v i g a t i o n a l instruments
are t h e following:
(a)

C o m p a r i s o n o f r e a d i n g s for p u r p o s e s o f m u t u a l c o r r e c t i o n ,

(b)

C o m b i n a t i o n o f r e a d i n g s f o r p u r p o s e s of

a u t o m a t i c summation.

Combination of i n d i v i d u a l t r a n s m i t t e r s i n t o n a v i g a t i o n a l sys­
tems n o t o n l y m a k e s i t p o s s i b l e t o s o l v e n a v i g a t i o n a l p r o b l e m s a u t o ­
m a t i c a l l y or s e m i - a u t o m a t i c a l l y , b u t a l s o m a k e s i t p o s s i b l e t o r e a l i z e
t h e i r s o l u t i o n f o r automatic p i l o t a g e of an aircraft along t h e given
trajectory.
An e x a m p l e o f s u c h a r e a l i z a t i o n i s t h e a u t o m a t i c p i l o t ­
366

...

a g e o f a n a i r c r a f t on t h e b a s i s o f s i g n a l s from D o p p l e r meters w i t h
automatic navigational instruments.
These complexes g e n e r a l l y i n v o l v e autonomous n a v i g a t i o n a l i n ­
struments:
t r a n s m i t t e r s f o r t h e c o u r s e , a i r s p e e d , and d r i f t angle.
The o n l y e x c e p t i o n i s t h e a i r c r a f t r a d i o c o m p a s s , whose r e a d i n g s a r e
combined w i t h t h e r e a d i n g s o f c o u r s e i n s t r u m e n t s t o o b t a i n h e a r i n g s .
However, t h i s i s a r e s u l t o f a p e c u l i a r f e a t u r e on t h e u s e o f r a d i o compasses ( f o r o b t a i n i n g t h e b e a r i n g it i s n e c e s s a r y t o add t h e
course angle of t h e r a d i o s t a t i o n t o t h e a i r c r a f t course).
The u s e
o f s i m p l e c o m b i n a t i o n s o f n a v i g a t i o n a l s y s t e m s s u c h as g r o u n d ra­
d a r s , r a d i o d i s t a n c e - f i n d e r s , e x t e r n a l l y d i r e c t e d g o n i o m e t r i c and
goniometric-rangefinding systems, fan-type beacons, and h y p e r b o l i c
s y s t e m s c a n n o t b e combined s a t i s f a c t o r i l y .
Nevertheless, it i s d e s i r a b l e t h a t t h e s e r e l a t e d n a v i g a t i o n a l /349
s y s t e m s , as w e l l as t h e a i r c r a f t r a d a r a n d a s t r o n o m i c a l n a v i g a t i o n a l
i n s t r u m e n t s , s h o u l d a l s o b e combined i n t o n a v i g a t i o n a l complexes f o r
t h e purposes of automatic c o r r e c t i o n of aircraft coordinates.
How­
e v e r , t h e p r i n c i p l e o f combining t h e s e i n s t r u m e n t s i n t o g e n e r a l nav­
i g a t i o n a l systems must be of a q u i t e d i f f e r e n t n a t u r e t h a n i s t h e
c a s e f o r t h e c o m p l e x e s w h i c h we h a v e d i s c u s s e d .
The f i r s t c h a r a c t e r i s t i c of co mb i n ed n a v i g a t i o n a l s y s t e m s an d
a v i a t i o n a l s e x t a n t s i s t h a t they are intended only f o r determining
Therefore, they can be
d i s c r e t e values of aircraft coordinates.
u s e d i n n a v i g a t i o n a l complexes as s o u r c e s o f i n f o r m a t i o n which dup­
l i c a t e t h e r e s u l t s of automatic c a l c u l a t o r s of t h e a i r c r a f t p a t h ,
i . e . , o n l y f o r p u r p o s e s of c o r r e c t i n g p r e v i o u s l y o b t a i n e d n a v i g a t i o n a l
parameters.
The s e c o n d f e a t u r e o f t h e s e d e v i c e s i s t h a t w i t h a r e l a t i v e l y
h i g h a c c u r a c y of c o o r d i n a t e measurement f o r t h e a i r c r a f t , t h e y can­
n o t be used t o d e t e r m i n e t h e f i r s t d e r i v a t i v e s of t h e s e c o o r d i n a t e s
with t i m e .
Let us i l l u s t r a t e t h i s with a concrete example.
L e t u s s a y t h a t some n a v i g a t i o n a l i n s t r u m e n t , t a k i n g i t s i n s t r u ­
mental e r r o r s i n t o account ( f o r e l e c t r o n i c devices, considering t h e
c o n d i t i o n s f o r p r o p a g a t i o n of e l e c t r o m a g n e t i c waves, and f o r a s t r o n ­
omical o n e s , t h e a c c e l e r a t i o n s of t h e l e v e l of t h e a i r c r a f t ) make i t
possible t o determine t h e successive coordinates of an aircraft with
a n e r r o r w h i c h d o e s n o t e x c e e d 1 km, s o t h a t t h e e r r o r i n m e a s u r e ­
ment c a n c h a n g e i n v a l u e a n d s i g n .

I n t h i s c a s e , t h e e r r o r i n d e t e r m i n i n g t h e d i r e c t i o n of t h e
a i r c r a f t m o t i o n o n t h e b a s i s o f t w o s u c c e s s i v e m e a s u r e m e n t s may h a v e
a maximum v a l u e o f
AZ
2
---

s -s.


W i t h a m e a s u r e m e n t b a s e o f 3 0 km ( a p p r o x i m a t e l y 2 m i n o f f l i g h t
i n a j e t a i r c r a f t ) , t h e a n g u l a r error i n t h e m e a s u r e m e n t s c a n r e a c h

367

-2_-30

1

15

z 4".

I f t h e m e a s u r e m e n t s a r e made m o r e f r e q u e n t l y , ( e . g . 1 o f t e n e r
than each minute o f f l i g h t , t h e e r r o r i n measuring t h e d i r e c t i o n can
reach 8 O .
With c o n t i n u o u s m e a s u r e m e n t of t h e a i r c r a f t c o o r d i n a t e s , t h e
n u m e r a t o r i n our e x a m p l e c a n r e t a i n i t s v a l u e , b u t t h e d e n o m i n a t o r
w i l l tend toward z e r o , i . e . , t h e e r r o r i n determining t h e d i r e c t i o n
o f f l i g h t or ( w h a t a m o u n t s t o t h e s a m e t h i n g ) t h e f i r s t d e r i v a t i v e
of t h e Z-coordinate with t i m e , w i l l be equal t o i n f i n i t y .
We c a n r e a c h a n a n a l o g o u s c o n c l u s i o n f o r t h e c a s e o f d e t e r m i n - / 3 5 0
i n g t h e g r o u n d s p e e d o f a n a i r c r a f t ( t h e f i r s t d e r i v a t i v e o f t h e X­
c o o r d i n a t e w i t h t i m e ) by c o n t i n u o u s measurement o f i t a t a s u c c e s s i o n
of p o i n t s w h e r e t h e L A i s m e a s u r e d .
T h i s example shows t h a t communication a n d a s t r o n o m i c a l n a v i g a ­
t i o n a l systems can only provide a rough p i l o t a g e of t h e a i r c r a f t
along a given trajectory.
With a v e r y p r e c i s e measurement o f t h e
a i r c r a f t c o o r d i n a t e s a l o n g t h e r o u t e ( w i t h e r r o r s no g r e a t e r t h a n
2 0 0 - 3 0 0 m) a n d a v e r y c a r e f u l d a m p i n g o f t h e r e a d i n g s ( a v e r a g i n g
f o r t i m e ) , automatic p i l o t a g e w i l l t a k e place with v a r i a t i o n s of
t h e c o u r s e w i t h i n l i m i t s of 5 - 6 O , i . e . , 5-10 times g r e a t e r t h a n
would b e o b t a i n e d by t h e r e s u l t s o f m e a s u r i n g t h e d r i f t a n g l e b y
a Doppler meter.
The o n l y e x c e p t i o n t o t h i s i s t h e p i l o t a g e o f a n a i r c r a f t
u s i n g s t r i c t l y s t a b i l i z e d zones of l a n d i n g b e a c o n s , where t h e e r r o r s
i n d e t e r m i n i n g t h e d e v i a t i o n s from a g i v e n t r a j e c t o r y are measured
i n several meters.
Under t h e s e c o n d i t i o n s , t h e p i l o t a g e o f a n a i r ­
c r a f t c a n t a k e p l a c e w i t h v a r i a t i o n of t h e c o u r s e w i t h i n l i m i t s of
1 - 2 O with a very p r e c i s e maintainance of t h e g e n e r a l d i r e c t i o n of
flight.
The t h i r d f e a t u r e o f t h e s e m e t h o d s i s t h e c o n s i d e r a b l e d i v e r ­
s i t y of s p e c i a l coordinate systems used (Chapter I , Section 7 ) .
T o i n t r o d u c e t h e s e methods i n t o n a v i g a t i o n a l complexes, it i s nec­
c e s s a r y t o have an automatic conversion of s p e c i a l c o o r d i n a t e s i n t o
o r t h o d r o m i c c o o r d i n a t e s , c a l l i n g for t h e a v a i l a b i l i t y a b o a r d t h e
a i r c r a f t of v e r y complex and p r e c i s e m a t h e m a t i c a l i n s t r u m e n t s , t o
make t h e s p h e r i c a l c o n v e r s i o n s .

I t i s somewhat s i m p l e r i n t h i s r e g a r d t o u s e g o n i o m e t r i c - r a n g e ­
f i n d i n g methods f o r s h o r t - r a n g e n a v i g a t i o n and a i r c r a f t r a d a r s .
Due t o t h e l i m i t e d r a d i u s o f t h e i r o p e r a t i o n , t h e p o l a r c o o r d i n a t e s
of t h e s e d e v i c e s c a n b e c o n v e r t e d i n t o o r t h o d r o m i c o n e s b y s o l v i n g
simple equations f o r plane representations, using simple calcula­
t i n g devices with l o w accuracy.

368

.-11.11

I1111.111.11111.1.1.11

111

I. 1.1,

I .I1

I1

I 1 I I

I111 I

I I 111

1111

I

I

I

Combined n a v i g a t i o n a l s y s t e m s and a s t r o n o m i c a l methods c a n b e
c o m b i n e d i n t o n a v i g a t i o n a l c o m p l e x e s o n l y f o r p u r p o s e s of c o r r e c t i n g
t h e aircraft coordinates a t d i s c r e t e p o i n t s , maintaining t h e regime
of c o r r e c t i o n by t h e a i r c r a f t crew.
The a d v i s a b i l i t y o f i n c l u d i n g
e a c h o f t h e s e d e v i c e s i n t h e t o t a l c o m p l e x or i t s i n d e p e n d e n t u s e
f o r c o r r e c t i o n of t h e c o o r d i n a t e s o f t h e a i r c r a f t b y t h e c r e w i s
determined by t h e t a c t i c a l c h a r a c t e r i s t i c s of t h e system i n t h e
c o n d i t i o n s for f l i g h t o f t h e p a r t i c u l a r t y p e o f a i r c r a f t i n v o l v e d .

369

CHAPTER FOUR

D E V I C E S A N D METHODS F O R M A K I N G A N I N S T R U M E N T L A N D I N G
S Y S T E M S FOR M A K I N G A N I N S T R U M E N T L A N D I N G
Landing an a i r c r a f t under c o n d i t i o n s of l i m i t e d c e i l i n g and
meteorological v i s i b i l i t y i n t h e l a y e r of t h e atmosphere n e a r t h e
g r o u n d i s t h e most: c o m p l i c a t e d a n d d i f f i c u l t s t a g e o f t h e f l i g h t .
Even u n d e r f a v o r a b l e m e t e o r o l o g i c a l c o n d i t i o n s , a p r o p e r l a n d i n g
o f t h e a i r c r a f t r e q u i r e s c o n s i d e r a b l e a t t e n t i o n a n d e x p e r i e n c e on
t h e p a r t o f t h e crew.

Fig. 4.1.
Setting the
Aircraft Course f o r
L i n i n g Up w i t h t h e
Runway.

/351

E x p e r i e n c e h a s shown t h a t i n o r d e r
t o l a n d any k i n d of a i r c r a f t , it i s neces­
s a r y t h a t i t b e l o c a t e d e x a c t l y on t h e
l a n d i n g p a t h a t a c e r t a i n d i s t a n c e from
It is
t h e touchdown p o i n t ( F i g . 4 . 1 ) .
a l s o necessary t h a t t h e course followed
by t h e a i r c r a f t b e s e l e c t e d s o t h a t t h e
v e c t o r of t h e ground speed is d i r e c t e d
along t h e a x i s of t h e landing and take­
off s t r i p (LTS).

However, i t i s n o t d e s i r a b l e t o l a n d a n a i r c r a f t w i t h a l e a d
i n t h e course being followed i n o r d e r t o compensate f o r t h e d r i f t
a n g l e , s i n c e t h i s c a u s e s c o n s i d e r a b l e l a t e r a l s t r e s s e s on t h e a i r ­
c r a f t u n d e r c a r r i a g e when i t b e g i n s t o t a x i a l o n g t h e r u n w a y .
There­
f o r e , t h e l o n g i t u d i n a l a x i s m u s t b e l i n e d u p w i t h t h e LTS i m m e d i a t e l y
b e f o r e l a n d i n g , by making a f l a t t u r n w i t h o u t b a n k i n g .
Then ( s i n c e
t h e t u r n w a s f l a t ) t h e aircraft w i l l keep t h e desired d i r e c t i o n
of motion r e l a t i v e t o t h e Earth’s s u r f a c e f o r a s h o r t period of
t i m e , s h o w i n g l a t e r a l d e v i a t i o n r e l a t i v e t o t h e a i r mass f l o w i n g
over it.
This s h i f t gradually dies o u t , eventually turning i n t o a d r i f t
Therefore, the
a n g l e r e l a t i v e t o t h e new c o u r s e o f t h e a i r c r a f t .
s e l e c t i o n o f t h e a p p r o a c h a n g l e m u s t b e made s e v e r a l s e c o n d s ( n o
more t h a n 5 o r 7 ) b e f o r e l a n d i n g t h e a i r c r a f t .
It should be mentioned t h a t t h e c o r r e c t s e l e c t i o n of an air­
c r a f t c o u r s e w h i l e k e e p i n g i t s i m u l t a n e o u s l y on t h e g i v e n t r a j e c t o r y f o r l a n d i n g p o s e s c o n s i d e r a b l e d i f f i c u l t i e s f o r t h e crew i n
preparing t o land.

370

/352

I n c a s e s when t h e a i r c r a f t i s n o t l i n e d u p w i t h t h e r u n w a y ,
i t i s n e c e s s a r y t o c a r r y o u t a maneuver w h i c h w i l l b r i n g i t on t o
t h e a x i s o f t h e LTS, a n d w h i c h i n v o l v e s a c o n s i d e r a b l e loss o f t i m e
a n d a l s o a loss o f d i s t a n c e a l o n g a x i s LTS ( F i g . 4 . 2 ) .
L e t us say t h a t i n a f l i g h t
a l o n g t h e LTS a x i s , t h e c r e w h a s
r e a c h e d a p o i n t a t which t h e i r l a t e r a l
deviation is equal t o 2.

Obviously, i n order t o l i n e
u p t h e a i r c r a f t w i t h t h e LTS a x i s
i n t h e most economical f a s h i o n and
w i t h o u t any remaining d e v i a t i o n r e l a ­
t i v e t o t h e LTS a x i s , i t i s n e c e s ­
s a r y t o t u r n t h e aircraft toward
Fig. 4.2.
S - S h a p e d Maneut h e r u n w a y t h r o u g h a t u r n a n g l e (TA)
v e r f o r L i n i n g Up t h e A i r of a m a g n i t u d e s u c h t h a t t h e l a t e r a l
c r a f t w i t h t h e Runway.
d e v i a t i o n o f t h e a i r c r a f t from t h e
LTS a x i s i s r e d u c e d by a f a c t o r o f t w o .
Then t h e a i r c r a f t must
b e t u r n e d by t h e same amount i n t h e o p p o s i t e d i r e c t i o n b u t t h r o u g h
an a n g l e s u c h t h a t t h e t r a j e c t o r y a l o n g which t h e a i r c r a f t i s t r a ­
v e l i n g when i t e m e r g e s f r o m t h e t u r n c o i n c i d e s w i t h t h e LTS a x i s .
I n o r d e r t o a v o i d loss o f t h e s e l e c t e d d i r e c t i o n o f t h e g r o u n d
s p e e d v e c t o r w h i l e making t h e t u r n s , i . e . , s h i f t i n g t h e a i r c r a f t
a f t e r p l a c i n g i t on t h e l a n d i n g c o u r s e , t h e t u r n s made b y t h e a i r ­
c r a f t m u s t b e c o o r d i n a t e d a s much a s p o s s i b l e .
I t i s o b v i o u s from F i g u r e 4 . 2 t h a t t h e m a g n i t u d e of e a c h o f
t h e t w o c o o r d i n a t e d t u r n s f o r b r i n g i n g t h e a i r c r a f t on t o t h e r u n w a y
c a n b e d e t e r m i n e d by t h e f o r m u l a
2 = 2(R-

COSTA) = ~ R ( ~ - c o s T A ) ,

whence
COSTA

2

= 1 - 2R

,

where R i s t h e t u r n i n g r a d i u s o f t h e a i r c r a f t w i t h a g i v e n bank­
i n g and a i r s p e e d d u r i n g t h e t u r n .
O b v i o u s l y , w h i l e t h e a i r c r a f t i s making t h e maneuver t o l a n d ,
i t m u s t t r a v e l t h r o u g h a p a t h a l o n g a x i s LTS

X = 2RsinTA.
ExampZe.
When a n a i r c r a f t i s d e s c e n d i n g a n d i s l i n e d u p w i t h
t h e runway on t h e d e s i r e d c o u r s e a n d w i t h a h o r i z o n t a l g r o u n d s p e e d
o f 2 8 0 k m / h r , t h e r e i s a l a t e r a l d e v i a t i o n f r o m t h e LTS a x i s e q u a l
t o 60 m .

371


F i n d t h e a n g l e s o f t h e combined t u r n s o f t h e a i r c r a f t w i t h
/353
a g i v e n b a n k i n g o f 8O a n d t h e p a t h o f t h e a i r c r a f t a l o n g t h e d e s c e n t
p a t h d u r i n g t h e completion of t h e maneuver.

Solution.
T h e r a d i u s o f t . h e t u r n s made b y t h e a i r c r a f t a r e
f o u n d by u s i n g a n a v i g a t i o n a l s l i d e r u l e ( F i g . 4 . 3 ) , which g i v e s
t h e a n s w e r 4500 m.
COSTA = 1

60

- = 0.9867;
4500

TA = 9O20';

sinTA = 0.1625;
ANSWER:

R = 4500 m ;

X = 2 4500 0 . 1 6 2 5 = 1 4 6 3 m .

TA = 9O20';

X = 1463 m.

However, w e must t a k e i n t o a c c o u n t t h e f a c t t h a t t h e d e s i r e d
a i r c r a f t p a t h i n l i n i n g u p w i t h t h e r u n w a y m u s t b e c h o s e n on t h e
b a s i s o f t h e a s s u m p t i o n t h a t t h e t u r n s a r e made w i t h a c o n s t a n t
A t t h e same t i m e ,
banking angle, i . e . , with a s t a b l e t u r n regime.
t h e r e i s a d e l a y i n t h e maneuver p r o d u c e d by t h e r e a c t i o n o f t h e
c r e w a n d m a i n l y d u e t o t h e i n e r t i a o f t h e a i r c r a f t when e n t e r i n g
and emerging from t h e t u r n s .

8'

a

4500

280


Fig.

4.3.

Fig.

4.3. Using t h e NL-1OM
Aircraft.

Fig.

4.4.

Fig.

4.4.

t o Determine t h e Turning Radius of an

Landing P r o f i l e f o r a J e t A i r c r a f t .

A s s p e c i a l t e s t s have shown, t h e d e l a y i n t h e maneuver o c c u r s
p r i m a r i l y a l o n g t h e d e s c e n t p a t h and h a s p r a c t i c a l l y no i n f l u e n c e
on t h e d e s i r e d m a g n i t u d e o f t h e a n g l e s o f t h e c o m b i n e d t u r n s .
This
i s e x p l a i n e d b y t h e f a c t t h a t when t h e a i r c r a f t i s e n t e r i n g a n d
e m e r g i n g from a bank a t t h e b e g i n n i n g a n d e n d o f t h e maneuver, t h e
a x i s of t h e aircraft p r a c t i c a l l y coincides with t h e a x i s of t h e
LTS a n d t h e a i r c r a f t h a s p r a c t i c a l l y n o l a t e r a l v e l o c i t y a t t h e s e
points.
A s f a r as t h e m o v e m e n t o f t h e c o n t r o l s u r f a c e s when m a k i n g
t h e t u r n s i s c o n c e r n e d , t h e t i m e r e q u i r e d t o move t h e m i s a p p r o x ­
i m a t e l y t w o t i m e s l e s s t h a n t h e t i m e r e q u i r e d for t h e a i r c r a f t t o
e n t e r and l e a v e t h e t u r n , s o t h a t t h e l a t e r a l component o f t h e a i r ­

372

c r a f t s p e e d a t t u r n a n g l e s up t o 1 2 O h a s a magnitude l e s s t h a n onef i f t h of t h e l o n g i t u d i n a l v e l o c i t y .

The d e l a y t i m e i n t h e m a n e u v e r d e p e n d s on t h e s q u a r e o f t h e
A t g l i d e s p e e d s o f 280 k m / h r ,
horizontal velocity of t h e aircraft.
t h e d e l a y t i m e i s e q u a l t o 4 . 5 s e c o f f l i g h t t i m e on t h e a v e r a g e ,
or 3 5 0 m o f t h e a i r c r a f t ' s f l i g h t a l o n g t h e LTS a x i s .
T h i s means
t h a t i n our example, t h e r e q u i r e d t r a v e l of t h e a i r c r a f t i n l i n i n g
up w i t h t h e runway i s e q u a l t o a p p r o x i m a t e l y 1800 m .
A t t h e same t i m e t h a t t h e c o u r s e i s b . e i n g s e l e c t e d w h i c h m u s t
b e f o l l o w e d i n o r d e r t o make t h e l a n d i n g , t h e c r e w m u s t b e g i n s o m e
d i s t a n c e away f r o m t h e l a n d i n g p o i n t t o s e t u p t h e d e s i r e d d e s c e n t / 3 5 4
trajectory i n the v e r t i c a l plane (Fig. 4.4).

I n F i g u r e 4 . 4 , P o i n t A i s t h e p o i n t of t r a n s i t i o n from hor­
i z o n t a l f l i g h t along t h e landing path t o t h e descent regime of t h e
aircraft.
P o i n t B i s t h e p o i n t where t h e l a n d i n g d i s t a n c e b e g i n s , which
i s a l s o c a l l e d t h e c r i t i c a l p o i n t f o r s a f e t r a n s i t i o n t o making
another pass.
After t h i s point has been passed, a second attempt
a t l a n d i n g c a n n o t b e m a d e , s o t h a t t h e a i r c r a f t m u s t make a f i n a l
s e l e c t i o n of t h e a i r c r a f t c o u r s e b e f o r e t h i s p o i n t i s r e a c h e d and
t h e d e v i a t i o n of t h e a i r c r a f t from t h e g i v e n t r a j e c t o r y (upward
and downward) must n o t e x c e e d c e r t a i n l i m i t s .
Before t h i s point
i s r e a c h e d , a d e c i s i o n m u s t b e made e i t h e r t o make t h e l a n d i n g or
c i r c l e around t h e a i r p o r t once a g a i n .
After t h e s t a r t i n g point f o r t h e landing distance has been
p a s s e d , t h e crew c a r e f u l l y o b s e r v e s t h e a l t i t u d e .
To d o t h i s , a
l e v e l i n g point C i s s e l e c t e d along t h e approach t o t h e a i r p o r t ( t h i s
i s a c o n d i t i o n a l d e s i g n a t i o n f o r t h e p o i n t where t h e d e s c e n t t r a ­
j e c t o r y of t h e a i r c r a f t c r o s s e s t h e E a r t h ' s s u r f a c e ) , toward which
t h e f u r t h e r descent of t h e a i r c r a f t i s aimed.
With p r o p e r d e s c e n t and a c o n s t a n t p i t c h a n g l e o f t h e a i r c r a f t ,
t h i s p o i n t i s p r o j e c t e d a t a c o n s t a n t l e v e l on t h e c o c k p i t w i n d o w .
I f t h e a p p r o a c h i s b e i n g made t o o r a p i d l y , t h i s p o i n t s h i f t s u p w a r d
on t h e g l a s s , a n d i f t h e a i r c r a f t i s c o m i n g i n t o o s l o w l y i t m o v e s
downward.
B e f o r e r e a c h i n g P o i n t C ( a t a n a l t i t u d e o f 8-15 m , d e p e n d i n g
on t h e t y p e o f a i r c r a f t ) t h e a i r c r a f t l e v e l s o f f a n d t h e n l a n d s
a t Point D.
The d e s c e n t t r a j e c t o r y o f t h e a i r c r a f t i n t h e v e r t i c a l p l a n e
is c a l l e d t h e gZide path.
The a i r c r a f t i s k e p t on a f i x e d g l i d e
p a t h by s e l e c t i n g t h e p r o p e r a n g l e o f p i t c h f o r t h e a i r c r a f t a n d
t h e c o r r e c t amount o f power t o t h e e n g i n e s .
T h i s p r o c e s s i s much
s i m p l e r i n p r i n c i p l e t h a n t h e s e l e c t i o n of t h e c o u r s e t o b e f o l ­
lowed by t h e a i r c r a f t , s i n c e i t d o e s n o t r e q u i r e m a n e u v e r i n g b u t

373

I


o n l y t h e p r o p e r s e t t i n g of t h e p i t c h a n g l e a n d t h e l e v e r s w h i c h
c o n t r o l t h e motors.
However, i t c o m p l i c a t e s l a n d i n g as a whole
because b o t h p r o c e s s e s must b e c a r r i e d o u t s i m u l t a n e o u s l y w h i l e
a given horizontal airspeed is being maintained.
Unlike a l l o t h e r n a v i g a t i o n a l devices, t h e systems used i n
making an i n s t r u m e n t l a n d i n g are i n t e n d e d s p e c i a l l y f o r k e e p i n g
t h e a i r c r a f t on a g i v e n d e s c e n t t r a j e c t o r y b e f o r e l a n d i n g i n t h e
h o r i z o n t a l and v e r t i c a l p l a n e s .
The p r o p e r o p e r a t i o n o f t h e s e d e v i c e s a n d t h e m a n e u v e r a b i l ­
i t y o f t h e a i r c r a f t d e t e r m i n e t h e minimum p e r m i s s i b l e d i s t a n c e f r o m
t h e LTS a t w h i c h t h e a i r c r a f t c a n b e p i l o t e d b y i n s t r u m e n t s or b y
i n s t r u c t i o n s from t h e ground, w i t h c o r r e c t i o n o f any e r r o r s t h a t
The more p r e c i s e l y
may o c c u r a f t e r c h a n g e o v e r t o v i s u a l f l i g h t .
t h e d e s i r e d t r a j e c t o r y i s m a i n t a i n e d by i n s t r u m e n t s , t h e c l o s e r
t h e t r a n s i t i o n t o v i s u a l f l i g h t w i l l l i e t o t h e l a n d i n g p o i n t and
t h e lower t h e a l t i t u d e a t t h a t p o i n t .
The l i m i t s w i t h i n w h i c h a n a i r c r a f t c a n b e p i l o t e d b y i n s t r u - / 3 5 5
ments w i t h o u t t h e a i r p o r t b e i n g v i s i b l e and w i t h no t e r r e s t r i a l
l a n d m a r k s i n s i g h t which c o u l d show a p p r o a c h e s t o t h e a i r p o r t i s
c a l l e d t h e w e a t h e r m i n i m u m for l a n d i n g t h e a i r c r a f t .
A t t h e present t i m e , t h e r e a r e t h r e e p r i n c i p a l types of sys­
tems f o r making i n s t r u m e n t l a n d i n g s :

(a)
A s i m p l i f i e d l a n d i n g s y s t e m w h i c h i n v o l v e s l i n i n g up t h e
aircraft with r a d i o s t a t i o n s .
(b)

A course-glide

landing system.

(c)

A r a d a r landing sytem.

A n e c e s s a r y complement t o e a c h of t h e s e s y s t e m s i s t h e s y s ­
t e m of landing l i g h t s a t t h e a i r p o r t .

S i m p l i f i e d System f o r Making an

I n s t r u m e n t Landing

The c o m p l e x of d e v i c e s i n t h e s i m p l i f i e d s y s t e m f o r mak i n g
a n i n s t r u m e n t l a n d i n g on t h e b a s i s o f i n f o r m a t i o n f r o m two m a s t e r
radio s t a t i o n s includes the following:
(1) Two m a s t e r r a d i o b e a c o n s , l o c a t e d o n t h e LTS a x i s , w h o s e
s t a n d a r d d e s i g n a t i o n i s t h e s h o r t - r a n g e m a s t e r s t a t i o n (SRMS), l o c a t e d
1 0 0 0 m from t h e e n d of t h e L T S , a n d t h e l o n g - r a n g e master s t a t i o n
( L R M S ) , l o c a t e d 4 0 0 0 m f r o m t h e e n d o f t h e LTS.
(2)
Two USW m a r k e r b e a c o n s w i t h a n a r r o w v e r t i c a l p r o p a g a ­
t i o n c h a r a c t e r i s t i c f o r e l e c t r o m a g n e t i c w a v e s , l o c a t e d on t h e same
s i t e s a s t h e LRMS a n d SRMS.

374


(3)

The l i g h t i n g o f t h e a p p r o a c h e s t o t h e LTS a n d i t s o u t ­

line.
(4)
The complex o f a i r c r a f t r a d i o n a v i g a t i o n a l a n d p i l o t a g e n a v i g a t i o n a l equipment as a w h o l e .
This includes :

(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)

One or t w o r a d i o c o m p a s s e s ,

A marker r e c e i v e r ,

Course c o n t r o l of t h e a i r c r a f t ,

A barometric altimeter,

A r a d i o a l t i m e t e r f o r low a l t i t u d e s ,

An a i r s p e e d i n d i c a t o r ,

A gyrohorizon,

A v e r t i c a l speed indicator (variometer).


We h a v e a l r e a d y d i s c u s s e d i n g r e a t d e t a i l ( i n C h a p t e r s T w o
and T h r e e ) most o f t h e g r o u n d and a i r c r a f t equipment which i s i n c l u d e d
i n t h e s i m p l i f i e d s y s t e m f o r making i n s t r u m e n t l a n d i n g s .
In this
c h a p t e r , w e w i l l p r o v i d e o n l y a b r i e f d e s c r i p t i o n of t h e o p e r a t i n g
p r i n c i p l e s of t h e p i l o t a g e and s p e c i a l l a n d i n g equipment , which
were n o t d i s c u s s e d i n t h e o t h e r c h a p t e r s , s i n c e t h i s e q u i p m e n t h a s
a v e r y l i m i t e d a p p l i c a t i o n f o r p u r p o s e s of a i r c r a f t n a v i g a t i o n a n d
i t s u s e i s v e r y s i m p l e f r o m t h e s t a n d p o i n t of t h e m e t h o d o l o g i c a l
/356
e r r o r s which m u s t be t a k e n i n t o a c c o u n t .

I n p a r t i c u l a r , w e s h a l l acquaint ourselves with the operating
p r i n c i p l e s of t h e following pieces of equipment:
marker devices ,
r a d i o a l t i m e t e r s f o r low a l t i t u d e s , t h e g y r o h o r i z o n a n d v a r i o m e t e r .

Marker Devices
I n o r d e r t o make a landing with t h e s i m p l i f i e d system, it i s
v e r y i m p o r t a n t t o know ( a d m i t t e d l y , a t s e p a r a t e p o i n t s ) t h e d i s ­
t a n c e remaining u n t i l t h e end of t h e runway.
A s w e know, a i r c r a f t r a d i o c o m p a s s e s d o n o t p e r m i t a p r e c i s e
d e t e r m i n a t i o n o f t h e moment when a n a i r c r a f t f l i e s o v e r t h e c o n t r o l
r a d i o s t a t i o n ; t h i s i s due t o t h e s p e c i a l c h a r a c t e r i s t i c s o f t h e
To solve t h i s
o p e r a t i o n of t h e open a n t e n n a a b o a r d t h e a i r c r a f t .
problem, marker beacons and a i r c r a f t m a r k e r r e c e i v e r s have been
d e v ised

.

Marker r a d i o b e a c o n s a r e t r a n s m i t t e r s w i t h a d i r e c t i o n a l t r a n s ­
mission c h a r a c t e r i s t i c v e r t i c a l l y upward, sometimes with a s l i g h t
d e v i a t i o n t o w a r d t h e LTS s o t h a t t h e l i m i t o f t h e d i r e c t i o n a l c h a r ­
a c t e r i s t i c o f t h e r a d i a t i o n i s l o c a t e d t o o n e s i d e o f t h e LTS a n d
as d o s e as p o s s i b l e t o t h e v e r t i c a l .
I n t h i s case, an aircraft
w h i c h i s f l y i n g o v e r t h e b e a c o n t o w a r d s t h e LTS w i l l r e c e i v e t h e
s i g n a l s f r o m t h e m a r k e r t r a n s m i t t e r a t t h e moment when i t is e x a c t l y
above t h e b e a c o n .

For p u r p o s e s of

r e c o g n i t i o n , t h e t r a n s m i s s i o n from t h e m a r k e r

375

I

d

b e a c o n i s n o t c o n t i n u o u s b u t i n t h e form o f f r e q u e n t s h o r t p u l s e s
(SRMS) or l o n g e r , l e s s f r e q u e n t s i g n a l s (LRMS).
These s i g n a l s are
h e a r d a b o a r d t h e a i r c r a f t f o r a p e r i o d o f 3-6 s e c a f t e r i t h a s f l o w n
o v e r t h e v e r t i c a l l i m i t of t h e r a d i a t i o n c h a r a c t e r i s t i c and b e f o r e
it c r o s s e s t h e second, d e f l e c t e d l i m i t of t h e c h a r a c t e r i s t i c .

A s t i l l simpler device i s t h e aircraft marker r e c e i v e r .
It
i s s e t t o o n e f r e q u e n c y w h i c h i s t h e same f o r a l l b e a c o n s .
There­
f o r e , it i s v e r y s i m p l e i n d e s i g n , h a s s m a l l dimensions, and r e q u i r e s
no a t t e n t i o n f o r u s e e x c e p t t o b e s w i t c h e d on a n d o f f .
When u s e d i n a c o m p l e x t o g e t h e r w i t h c o u r s e - g l i d e d e v i c e s ,
t h e m a r k e r r e c e i v e r i s t u r n e d on by a s w i t c h w h i c h i s combined w i t h
t h e c o u r s e - g l i d e e q u i p m e n t , s o t h a t t h e crew d o e s n o t h a v e t o i n t e r ­
I n many c a s e s , t h e m a r k e r r e c e i v e r
fere i n its operation at a l l .
i s combined w i t h t h e s w i t c h f o r t h e r a d i o c o m p a s s e s , t h e p u r p o s e
b e i n g t o e n s u r e a low c o n s u m p t i o n o f e l e c t r i c a l e n e r g y , a n d a l l o w
s t a b i l i t y and h i g h r e l i a b i l i t y i n t h e o p e r a t i o n of t h i s r e c e i v e r .
The m a r k e r r e c e i v e r i s c o n n e c t e d t o a l i g h t s i g n a l
on t h e i n s t r u m e n t p a n e l i n t h e c o c k p i t marked “ m a r k e r “ )
d e v i c e which g i v e s a s i m u l t a n e o u s s o u n d s i g n a l by means
T h u s , when t h e a i r c r a f t f l i e s o v e r t h e m a r k e r , t h e l a m p
and a s e r i e s of s h o r t r i n g s i s h e a r d .

(a red light
and t o a
of a b e l l .
flashes

Low-Altitude Radio A Ztimeters

/357

A t t h e p r e s e n t t i m e , l o w - a l t i t u d e r a d i o a l t i m e t e r s b a s e d on
t h e p r i n c i p l e of f r e q u e n c y m o d u l a t i o n a r e t h e o n e s most w i d e l y e m ­
ployed.

A s c h e m a t i c d i a g r a m o f s u c h a r a d i o a l t i m e t e r i s shown i n F i g u r e
4.5.

indicator

Fig.

376

4.5.

Diagram o f L o w - A l t i t u d e R a d i o a l t i m e t e r .

The r a d i o a l t i m e t e r t r a n s m i t t e r h a s a m o d u l a t i n g d e v i c e which
For t h i s Purpose, w e can use ( e . g . >
p r o d u c e s a s a w - t o o t h wave.
a v a r i a b l e membrane c a p a c i t o r w i t h m e c h a n i c a l o s c i l l a t i o n of t h e
membrane.

I

1.

t

Fig. 4.6.
Frequency Char­
acteristic of Radioaltimeter.

The f r e q u e n c y o f t h e s i g n a l s
r e f l e c t e d from t h e ground and
p i c k e d up by t h e r e c e i v i n g a n t e n n a
h a s t h e same s a w - t o o t h c h a r a c t e r ­
istic, but is shifted i n t i m e
by a v a l u e T , r e q u i r e d f o r t h e
e l e c t r o m a-g n e t i c w a v e s t o t r a v e 1
from t h e t r a n s m i t t i n g a n t e n n a
t o t h e ground and back a g a i n t o
t h e receiving antenna (Fig. 4.6).

I t i s c l e a r from t h e f i g u r e t h a t t h e f r e q u e n c y d i f f e r e n c e be­
t w e e n t h e e m i t t e d a n d r e c e i v e d w a v e s a t a n y moment i n t i m e ( w i t h
t h e exception of t h e segments between t h e extreme values of t h e
frequency c h a r a c t e r i s t i c ) w i l l be s t r i c t l y l i n e a r with respect t o
the flight altitude.
For a complete r e t e n t i o n of t h e l i n e a r i t y ,
t h e s e s e g m e n t s c a n b e c u t o u t by c u t t i n g o f f t h e r e c e i v i n g s e c t i o n
w i t h a II-shaped v o l t a g e a t t h e end p o i n t s o f t h e e m i t t e d f r e q u e n c y .

The e m i t t e d a n d r e c e i v e d f r e q u e n c i e s a r e co mb i n ed i n t h e b a l ­
a n c i n g d e t e c t o r , w h e r e a low f r e q u e n c y i s f o r m e d w h i c h i s p r o p o r ­
tional t o the flight altitude.
/358
F o l l o w i n g a m p l i f i c a t i o n , t h e low f r e q u e n c y i s c o n v e r t e d t o
r e c t a n g u l a r o s c i l l a t i o n s which a r e c a l i b r a t e d b o t h i n t e r m s of ampli­
tude and d u r a t i o n .
Thus, t h e counting c i r c u i t w i l l receive pulses
which a r e o f u n i f o r m m a g n i t u d e , and whose number p e r u n i t t i m e w i l l
d e p e n d on t h e f l i g h t a l t i t u d e .
T h e n u m b e r o f c a l i b r a t e d p u l s e s i s summed a n d f e d i n t h e f o r m
o f a d i r e c t c u r r e n t t o t h e i n d i c a t o r , whose p o i n t e r shows t h e a l t i ­
t u d e i n meters.
In t h e s i m p l i f i e d landing system, t h e r a d i o a l t i m e t e r plays
o n l y an a u x i l i a r y r o l e as a n i n d i c a t o r o f a d a n g e r o u s a p p r o a c h t o
t h e g r o u n d , s i n c e i t s r e a d i n g s d e p e n d upon t h e n a t u r e o f t h e r e l i e f
and cannot be used f o r checking t h e r a t e of d e s c e n t .
T o s e t up
t h e descent t r a j e c t o r y of t h e a i r c r a f t , barometric altimeters are
used.
I n more c o m p l e t e l a n d i n g s y s t e m s , t h e r a d i o a l t i m e t e r c a n b e
u s e d t o g i v e a t r a j e c t o r y v a l u e as w e l l , b u t o n l y i n t h e l a s t s t a g e
of descent b e f o r e landing above a given f i n a l area of s a f e t y adjoin­
i n g t h e LTS.
S i n c e t h o s e l a n d i n g s y s t e m s which e n s u r e d e s c e n t o f t h e a i r ­
c r a f t by i n s t r u m e n t s u n t i l t h e p o i n t where t h e l a n d i n g d i s t a n c e
377

b e g i n s use t h e r a d i o altimeter only t o s i g n a l a dangerous approach
t o t h e g r o u n d , w e c a n e x c l u d e them f o r c o n v e n i e n c e f r o m t h e g r o u p
of basic pilotage instruments located i n t h e center of the f i e l d
o f v i s i o n of t h e p i l o t , and u s e a u d i b l e s i g n a l s .
If a n a i r c r a f t
i s making a d e s c e n t and r e a c h e s t h e l i m i t of p e r m i s s i b l e a l t i t u d e
a b o v e t h e g r o u n d , t h e a u d i b l e s i g n a l w a r n s t h e crew o f t h e n e c e s ­
s i t y t o terminate descent.

Gy ro h or iz o n
The a r t i f i c i a l i n d i c a t o r of t h e p o s i t i o n o f t h e h o r i z o n r e l a ­
t i v e t o t h e a x i s o f t h e a i r c r a f t ( g y r o h o r i z o n ) i s a common p i l o t ­
a g e i n s t r u m e n t , i n t e n d e d f o r p i l o t i n g t h e a i r c r a f t when t h e t r u e
horizon is not v i si b l e .
However, i t i s v e r y i m p o r t a n t i n g u i d i n g
t h e a i r c r a f t a l o n g a l a n d i n g t r a j e c t o r y , where i t i s u s e d f o r main­
taining a desired landing trajectory.
In p r i n c i p l e , t h e design of t h e gyrohorizon is simpler than
t h a t of t h e gyrosemicompass , e .g. , u n l i k e t h e l a t t e r , t h e gyrohor­
i z o n h a s a v e r t i c a l a x i s o f r o t a t i o n for t h e g y r o s c o p e , a n d a g r a v ­
i t a t i o n a l c o r r e c t i o n d e v i c e suspended from t h e bottom of t h e gyro
assembly.
This s e r v e s t o keep t h e gyroscope a x i s c o n s t a n t l y v e r t i c a l
in the aircraft.
The e x t e r n a l f r a m e o f t h e g y r o h o r i z o n i s l o c a t e d h o r i z o n t a l l y ,
while i t s axis of r o t a t i o n coincides with t h e longitudinal axis
T h e r e f o r e , w e can immediately determine t h e e x i s t ­
of t h e aircraft.
e n c e and magnitude o f a l a t e r a l r o l l i n g o f t h e a i r c r a f t by t h e p o s i ­
t i o n o f t h e e x t e r n a l frame r e l a t i v e t o t h e a x i s o f t h e a i r c r a f t .

For t h i s p u r p o s e , a s i l h o u e t t e o f t h e a i r c r a f t h a s b e e n p a s t e d
on t h e g l a s s which c o v e r s t h e d i a l , a n d a h o r i z o n t a l s t r i p which
moves u p a n d down i m i t a t e s t h e p o s i t i o n o f t h e v i s i b l e h o r i z o n .
F i g u r e 4 . 7 s h o w s t h e s c h e m a t i c d i a g r a m of t h e g y r o h o r i z o n .

Fig.
378

4.7.

D i a g r a m s of

Gyrohorizon:

( a ) Kinematics;

(b) Indicator.

G y r o a s s e m b l y 1, w i t h a v e r t i c a l a x i s o f t h e g y r o s c o p e r o t o r
and t h e g r a v i t a t i o n a l c o r r e c t i o n d e v i c e 2 mounted a t t h e b o t t o m ,
a r e s u s p e n d e d i n t h e h o r i z o n t a l e x t e r n a l f r a m e 3 3 n t h e g y r o assem­
b l y b e a r i n g s 4.
The c a r r i e r f o r t h e h o r i z o n l i n e 5 i s f a s t e n e d
t o t h e c a s i n g o f t h e g y r o a s s e m b l y a n d d i s p l a c e d somewhat f o r w a r d
r e l a t i v e t o t h e h o r i z o n t a l axis of t h e gyro assembly (along t h e
d i r e c t i o n of t h e a i r c r a f t ' s f l i g h t from t h e forward p a r t o f t h e
instrument).
The a x i s o f t h e l i n e i s f a s t e n e d a t t h e f r o n t t o t h e
e x t e r n a l frame, a l s o along t h e f l i g h t d i r e c t i o n of t h e aircraft.
T h e r e f o r e , when r e d u c i n g t h e a n g l e of p i t c h o f t h e a i r c r a f t , t h e
s t r i p o f t h e g y r o h o r i z o n 7 moves u p w a r d , r e m a i n i n g p a r a l l e l t o t h e
h o r i z o n t a l a x i s of t h e g y r o assembly.
When t h e p i t c h a n g l e i n c r e a s e s ,
t h e h o r i z o n l i n e moves downward as t h e t r u e h o r i z o n d o e s .
During l a t e r a l r o l l i n g of t h e a i r c r a f t , t h e c a s i n g of t h e gyroh o r i z o n ( a l o n g w i t h t h e s i l h o u e t t e of t h e a i r c r a f t ) r o t a t e s r e l a ­
t i v e t o t h e b e a r i n g s of t h e e x t e r n a l f r a m e 9 i n t h e d i r e c t i o n i n
which t h e a i r c r a f t i s r o l l i n g , which p r o v i d e s a n i n d i c a t i o n o f t h e
For e s t i ­
r o l l i n g of t h e aircraft r e l a t i v e t o t h e h o r i z o n t a l s t r i p .
mating and m a i n t a i n i n g g i v e n l o n g i t u d i n a l and l a t e r a l r o l l i n g of
t h e a i r c r a f t , a s c a l e i s l o c a t e d b e t w e e n t h e o u t e r frame a n d t h e
h o r i z o n l i n e a n d shows s c a l e d i v i s i o n s f o r e s t i m a t i n g t h e m a g n i t u d e
of t h e r o l l i n g i n d e g r e e s .
The g y r o h o r i z o n , f i t t e d w i t h t h e k i n e m a t i c s y s t e m d e s c r i b e d
a b o v e , c a n be u s e d w i t h i n l i m i t e d d e g r e e s of l o n g i t u d i n a l and t r a n s ­
verse r o l l i n g of t h e aircraft.
Obviously, t h e r e a r bearing of t h e
o u t e r frame, i . e . , t h e one l o c a t e d b e t w e e n t h e o u t e r frame a n d t h e
s c a l e , m u s t b e m o u n t e d on a s u p p o r t i n t h e u n i t .
This support acts/360
as a p i v o t f o r t h e l e v e r s u p p o r t i n g t h e h o r i z o n l i n e , e . g . , when
t h e a i r c r a f t rolls o v e r o n o n e w i n g .
I n t h e case o f c o n s i d e r a b l e changes i n t h e p i t c h i n g a n g l e of
t h e aircraft (e.g., i n a Nestrov loop), a support w i l l hold t h e
l e v e r f o r t h e s t r i p i n a n o t c h on t h e o u t e r f r a m e .
The p r o j e c t i o n o f one o f t h e s e s u p p o r t s l i m i t s t h e d e g r e e o f
f r e e d o m of t h e g y r o s c o p e , t h u s l e a d i n g t o a " d i s - l o c a t i o n " o f i t s
i n d i c a t i o n s , and a v e r y l o n g p e r i o d of t i m e i s r e q u i r e d t o r e a d j u s t
t h e m by g r a v i t a t i o n a l c o r r e c t i o n .
To e n s u r e " n o n d i s l o c a t i o n " o f t h e o p e r a t i o n o f t h e g y r o h o r ­
izon, t h e gyroscopic section protrudes outside t h e housing of t h e
instrument, i.e., constitutes a separate gyroscopic instrument,
a g y r o c o m p a s s w i t h o u t a l i m i t e d d e g r e e of f r e e d o m .
The r e a d i n g s
of t h e g y r o v e r t i c a l are t r a n s m i t t e d t o t h e horizon i n d i c a t o r by
means o f m a s t e r a n d s l a v e s e l s y n s .
We s h o u l d a l s o n o t e t h a t g y r o h o r i z o n s or g y r o v e r t i c a l s a r e
t r a n s m i t t e r s w h i c h i n d i c a t e l o n g i t u d i n a l a n d l a t e r a l r o l l i n g for
t h e o p e r a t i o n of a u t o p i l o t s , a c t i n g as t r a n s m i t t e r s of t u r n angles
of t h e a i r c r a f t i n t h e h o r i z o n t a l p l a n e , i n w h i c h g y r o s c o p i c s e m i compasses are u s e d .

379
H


V a r i ome t e r
A uar<ometer i s a d e v i c e which m e a s u r e s t h e r a t e o f v e r t i c a l
d e s c e n t or c l i m b o f a n a i r c r a f t .
T h e o p e r a t i n g p r i n c i p l e o f a v a r i o m e t e r i s b a s e d on t h e d e c e l ­
e r a t i o n o f a c u r r e n t o f a i r which e q u a l i z e s t h e p r e s s u r e i n s i d e
t h e body o f t h e u n i t w i t h t h e e x t e r n a l s t a t i c p r e s s u r e .
T h i s means
t h a t when v e r t i c a l m o v e m e n t o c c u r s , a p r e s s u r e d r o p d e v e l o p s w i t h i n
t h e body o f t h e u n i t and i n t h e s t a t i c t u b e ( F i g . 4 . 8 ) .

t h e m a n o m e t r i c chamber o f t h e i n s t r u ment.
W i t h i n t h e body o f t h e i n s t r u ­
ment, t h i s p r e s s u r e passes through a

capillary
E=

To measure t h i s drop, t h e variometer i s f i t t e d with a t r a n s m i t ­
t e r mechanism, s i m i l a r i n p r i n c i p l e t o t h e mechanism o f t h e a l t i m ­
e t e r or s p e e d i n d i c a t o r .
The i n d i c a t o r s c a l e i s g r a d u a t e d d i r e c t l y
i n t e r m s o f v e r t i c a l s p e e d , as e x p r e s s e d m e t e r s / s e c .
A n g l e o f Slope

/ 361

for Aircraft G1 ide

The p r o p e r s e l e c t i o n o f a n a n g l e o f s l o p e for g l i d i n g i s v e r y
important f o r a l l instrument landing systems, and e s p e c i a l l y f o r
t h e s i m p l i f i e d s y s t e m s g u i d e d by m a s t e r r a d i o s t a t i o n s , b o t h f r o m
t h e s t a n d p o i n t of making a s a f e l a n d i n g a n d t h e m e t e o r o l o g i c a l
minimum a t w h i c h a l a n d i n g c a n b e m a d e .
When m a k i n g a n a p p r o a c h t o l a n d , i t i s v e r y i m p o r t a n t t h a t t h e
t o the
f l i g h t a l t i t u d e (H) correspond t o t h e remaining d i s t a n c e
p o i n t w h e r e t h e a i r c r a f t t o u c h e s down:

(s)

H

= S

rem

tg0

where 8 i s t h e g l i d e a n g l e .
A s i m p l i f i e d s y s t e m o f i n s t r u m e n t l a n d i n g makes i t p o s s i b l e
t o d e t e r m i n e t h e r e m a i n i n g d i s t a n c e t o t h e l a n d i n g p o i n t o n l y when
p a s s i n g o v e r t h e LRMS a n d SRMS.
The p o i n t a t w h i c h t h e a i r c r a f t b e g i n s t o d e s c e n d f r o m t h e

3 80

a l t i t u d e e s t a b l i s h e d f o r c i r c l i n g above t h e f i e l d i s d e t e r m i n e d by
c a l c u l a t i n g t h e t i m e , and i s t h e r e f o r e i n s u f f i c i e n t l y e x a c t .
A
d e s c e n t b e t w e e n t h e LRMS a n d SRMS i s a l s o made b y c a l c u l a t i n g t h e
path of the aircraft with t i m e , but t h i s calculation takes only a
s h o r t p e r i o d of t i m e and i s performed a f t e r a c e r t a i n p o i n t has
b e e n p a s s e d ; it i s t h e r e f o r e more a c c u r a t e .
A c c o r d i n g t o t h e s t a n d a r d s a d o p t e d i n t h e USSR, t h e f l i g h t
a l t i t u d e f o r c i r c l i n g o v e r an a i r p o r t ( f o r a i r c r a f t w i t h g a s t u r ­
b i n e e n g i n e s ) h a s been s e t a t 400 m; f o r p i s t o n - e n g i n e a i r c r a f t ,
i t is 300 m . I n b o t h cases, however, t h e t r u e f l i g h t a l t i t u d e above
t h e l o c a l t e r r a i n s u r r o u n d i n g t h e a i r p o r t must b e n o l e s s t h a n
200 m .
T h i s a l t i t u d e r e s e r v e i s r e t a i n e d e v e n when c o m i n g s t r a i g h t
i n f o r a landing, u n t i l t h e beginning of descent i n t h e designated
glide pattern.
From t h e moment when d e s c e n t b e g i n s i n a g l i d e , a n d u n t i l t h e
a i r c r a f t p a s s e s o v e r a c e r t a i n m a r k e r (LRMS), t h e a l t i t u d e r e s e r v e
a b o v e t h e r e l i e f i s k e p t a t a minumum o f 1 5 0 m .
After flying over
t h e L R M S , a n d b e f o r e r e a c h i n g t h e SRMS, t h e h e i g h t o f t h e a i r c r a f t
above t h e t e r r a i n i s r e d u c e d from 1 5 0 t o 5 0 m .
During t h i s maneuver,
h o w e v e r , i t i s n e c e s s a r y t o k e e p i n m i n d t h e f a c t t h a t t h e r e may b e
a p o s s i b l e p r e m a t u r e l o s s o f a l t i t u d e , i n c a s e o f an u n e x p e c t e d
s t r o n g head wind.
F o r t h i s r e a s o n , i t i s c o n s i d e r e d t h a t t h e minimum
f l i g h t a l t i t u d e b e t w e e n t h e LRMS a n d t h e SRMS ( f l i g h t a l t i t u d e
a b o v e t h e SRMS) m u s t b e a t l e a s t 5 0 m e t e r s a b o v e t h e h e i g h e s t p o i n t
i n t h e v i c i n i t y , b e g i n n i n g a t h a l f t h e d i s t a n c e b e t w e e n t h e LRMS
a n d SRMS a n d e x t e n d i n g t o t h e p o i n t w h e r e t h e SRMS i s l o c a t e d .
T h e s e s a m e a l t i t u d e r e s e r v e s a r e m a i n t a i n e d e v e n when u s i n g
more c o m p l e t e l a n d i n g s y s t e m s , a l t h o u g h i n t h i s c a s e t h e g i v e n g l i d e
p a t h f o r t h e a i r c r a f t i s d e f i n e d i n s p a c e a n d t h e p r o b a b i l i t y of a
premature descent is sharply reduced.
I n t h i s case, however, t h e
/362
b a s i c m e t h o d �or c h e c k i n g t h e p r o p e r d e s c e n t i s t h e m e a s u r e m e n t o f
t h e b a r o m e t r i c a l t i t u d e when f l y i n g o v e r t h e m a r k e r p o i n t s , t h u s
guaranteeing safety of f l i g h t i n case the landing instruments
a b o a r d t h e a i r c r a f t o r on t h e g r o u n d s h o u l d m a l f u n c t i o n .
I n c a s e s when t h e a p p r o a c h e s t o a n a i r p o r t a r e f r e e o f o b ­
structions, the angle of slope i n the glide path is set equal t o
2'40'.
The f l i g h t a l t i t u d e r e l a t i v e t o t h e l e v e l o f t h e a i r p o r t i n
t h i s c a s e i s s e t a t 2 0 0 m a b o v e t h e LRMS a n d 60 m a b o v e t h e SRMS.
Typical

Maneuvers

i n Landing

an A i r c r a f t

S i m p l i f i e d s y s t e m s f o r b r i n g i n g an a i r c r a f t i n f o r a l a n d i n g
a r e u s e d a t a i r p o r t s w i t h a low t r a f f i c d e n s e , wh ere t h e i n s t a l l a ­
t i o n o f complex l a n d i n g s y s t e m s would n o t b e j u s t i f i e d .
Conse­
q u e n t l y , i t i s d i f f i c u l t t o know i n a d v a n c e w h e t h e r t h e s e a i r p o r t s
w i l l h a v e p r o v i s i o n f o r r a d a r c o n t r o l , t o s e t up t h e a p p r o a c h a n d
l a n d i n g p a t t e r n on command f r o m t h e g r o u n d .
Hence, t h e a p p r o a c h
f o r l a n d i n g i s made w i t h t h e same d e v i c e s w h i c h a r e u s e d i n l a n d i n g
381


t h e aircraft along a s t r a i g h t line.
For t h i s r e a s o n , a s u c c e s s f u l
a c c o m p l i s h m e n t o f t h e maneuv,er u n d e r t h e s e c o n d i t i o n s w i l l b e a s s u r e d
i f t h e s t a r t i n g p o i n t f o r t h e maneuver i s one o f t h e marker p o i n t s
of t h e system.
U s u a l l y a LRMS i s u s e d f o r
t h i s purpose, Bince a t t h e major­
i t y o f a i r p o r t s , it i s t h e main
control f a c i l i t y at the airport.
There are t h e n t h r e e p o s s i b l e
ways t o b r i n g t h e a i r c r a f t t o
the starting point f o r the ma­
neuver:
&
.
(1) An a p p r o a c h o f t h e a i r ­
H 3900-42011
c r a f t t o t h e LRMS, w i t h a p a t h
angle close t o the landing course.
Fig. 4.9. Large and Small
( 2 ) An a p p r o a c h t o t h e LRMS,
Rectangular Landing P a t t e r n s .
with a path angle n e a r l y perpen­
d i c u l a r t o t h e landing course.
( 3 ) An a p p r o a c h o f t h e a i r c r a f t t o t h e LRMS, w i t h a p a t h a n g l e
nearly the reverse o f the landing course.
I n d i r e c t i n g t h e a i r c r a f t t o w a r d t h e LRMS a t p a t h a n g l e c l o s e
t o t h e l a n d i n g c o u r s e , t h e a p p r o a c h �or l a n d i n g c a n b e made a l o n g
a m o r e or l e s s s t r a i g h t - l i n e c o u r s e ( F i g . 4 . 9 ) .
A l a r g e r e c t a n g u l a r r o u t e i s c o v e r e d i n t h i s case, i f t h e
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aircraft approaches t h e a i r p o r t a t a g r e a t a l t i t u d e ( f o r aircraft
w i t h g a s t u r b i n e e n g i n e s , t h i s i s 3 9 0 0 t o 4200 m), and an a d d i t i o n a l .
l e n g t h o f t i m e i s r e q u i r e d for t h e a i r c r a f t t o d e s c e n d b e f o r e l a n d ­
ing.
I n t h i s c a s e , i n m a k i n g t h e a p p r o a c h t o t h e LRMS, t h e a i r c r a f t
makes a t u r n t o t h e p a t h a n g l e f o r l a n d i n g ( i n t h e f o l l o w i n g , t h e
p a t h a n g l e s w ' i l l b e r e f e r r e d t o as m a g n e t i c ) , a t w h i c h t h e a i r c r a f t
d e s c e n d s t o 2800 m ( r e l a t i v e t o t h e p r e s s u r e a t t h e l e v e l o f t h e
a i r p o r t where i t i s l a n d i n g ) .
A t an a l t i t u d e o f 2800 m , t h e d o u b l e t u r n b e g i n s ( f i r s t a n d
second t u r n s w i t h o u t a s t r a i g h t l i n e between them) a t 180° w i t h a
d e s c e n t t o 1200 m.
F l i g h t then continues with a magnetic path
a n g l e (MPA) o p p o s i t e t o t h e l a n d i n g a n g l e , w i t h d e s c e n t t o t h e
a l t i t u d e set f o r c i r c l i n g over the a i r p o r t .
I n a i r c r a f t with gas t u r b i n e engines, l i m i t s have been s e t
f o r t h e h o r i z o n t a l a i r s p e e d with t h e undercarriage lowered.
There­
f o r e , i n a f l i g h t w i t h a MPA o p p o s i t e t o t h e l a n d i n g a n g l e , t h e
f l i g h t a l t i t u d e f o r c i r c l i n g t h e f i e l d i s m a i n t a i n e d f o r 5 t o 6 km
u n t i l t h e LRMS i s p a s s e d , s o t h a t a t t h e moment when i t a c t u a l l y
i s p a s s e d , t h e s p e e d o f t h e a i r c r a f t i n h o r i z o n t a l f l i g h t can b e
cut t o the speed e s t a b l i s h e d f o r lowering t h e undercarriage.

3 82

A f t e r p a s s i n g o v e r t h e t r a v e r s e of t h e
continues opposite t o t h e landing direction
s t a r t i n g t h e t h i r d turn ( u s u a l l y a t a f l i g h
up t o C A R = 1 2 0 ° t o t h e r i g h t a n d up t o C A R
straight-line paths).
The u n d e r c a r r i a g e i s
segment.

LRMS, t h e f l i g h t
f o r 70 s e c , p r i o r t o
t a l t i t u d e o f 400 m ,
= 2 4 0 ° on t h e l e f t
lowered i n t h i s path

A f t e r a p e r i o d o f 7 0 s e c f l y i n g t i m e f r o m t h e moment when t h e
t r a v e r s e o f t h e LRMS i s p a s s e d or u n t i l CAR-120 ( 2 4 0 O ) i s r e a c h e d ,
t h e t h i r d t u r n i s made.
Since t h e horizontal airspeed i n t h e
v i c i n i t y o f t h e t h i r d t u r n i s much l e s s t h a n i n t h e v i c i n i t y o f t h e
doubling of t h e first and second t u r n s , t h e r a d i i o f t h e t h i r d
a n d f o u r t h t u r n s ( w i t h a b a n k i n g a n g l e o f 1 5 t o 1 7 O ) a r e t h e n much
less than t h e r a d i u s o f t h e double t u r n .
T h e r e f o r e , between t h e
t h i r d and f o u r t h t u r n s t h e r e i s a p e r i o d of s t r a i g h t - l i n e f l i g h t
This s t r a i g h t - l i n e segment is used f o r
which l a s t s 50 t o 5 5 s e c .
p r e l i m i n a r y l o w e r i n g o f t h e wing f l a p s b e f o r e l a n d i n g , and a l s o
a c t s as a " b u f f e r t ' , w h i c h c o m p e n s a t e s f o r e r r o r s i n a i r c r a f t n a v ­
i g a t i o n i n c a s e s when t h e e f f e c t o f a s i d e w i n d i n m a k i n g t h e m a ­
n e u v e r from t h e s t a r t i n g p o i n t u n t i l t h e end o f t h e t h i r d t u r n h a s
n o t been e s t i m a t e d s u f f i c i e n t l y p r e c i s e l y .

I n t h e s e c a s e s , t h e l l b u f f e r " l i n e c a n b e e x t e n d e d or s h o r t e n e d
s o m e w h a t , b u t t h e l a s t ( f o u r t h ) t u r n m u s t b e a l w a y s made on t i m e .
A t a i r p o r t s w h e r e t h e n a t u r e o f t h e l o c a l t e r r a i n or c o m p l e x
wind c o n d i t i o n s r e n d e r f l i g h t a l o n g a s t r a i g h t l i n e a t 400 m i m ­
possible ( f o r aircraft with gas t u r b i n e engines), b u t t h e estab­
l i s h e d f l i g h t a l t i t u d e i s 6 0 0 or 9 0 0 m , t h e d u r a t i o n o f t h e f l i g h t
f r o m t h e t r a v e r s e o f t h e LRMS t o t h e b e g i n n i n g o f t h e t h i r d t u r m
i s i n c r e a s e d , s o t h a t a f t e r t h e a i r c r a f t emerges from t h e f o u r t h
t u r n it i s l o c a t e d below t h e g l i d e p a t h e s t a b l i s h e d f o r a g i v e n
approach d i r e c t i o n and h a s a segment o f h o r i z o n t a l f l i g h t t o t h e
This t i m e
e n d o f t h e g l i d e p a t h w h i c h i s o n l y 2J t o 30 s e c l o n g .
i s n e e d e d t o p r e p a r e t h e crew f o r l a n d i n g and f o r e x t e n d i n g t h e
flaps fully.

/364

For e x a m p l e , i f t h e f l i g h t a l t i t u d e a l o n g a s t r a i g h t - l i n e
course is s e t a t 600 m, and t h e s l o p e angle o f t h e g l i d e p a t h is
2 O 4 ' , t h e f o u r t h t u r n m u s t b e e x e c u t e d n o c l o s e r t h a n 1 5 km f r o m
t h e e n d o f t h e LTS, s i n c e t h e a i r c r a f t
( a t a n a l t i t u d e o f 6 0 0 m)
e n t e r s t h e g l i d e p a t h a t a d i s t a n c e o f 1 3 km f r o m t h e e n d o f t h e
LTS, a n d 2 km a r e r e q u i r e d f o r t h e h o r i z o n t a l f l i g h t s e g m e n t b e f o r e
entering the glide path.
C o n s e q u e n t l y , t h e s t a r t of t h e t h i r d t u r n u n d e r c a l m c o n d i ­
t i o n s , a f t e r p a s s i n g t h e t r a v e r s e o f t h e LRMS, l a s t s 2 m i n u t e s
a n d 30 s e c o n d s o f f l y i n g t i m e ( a t V = 3 5 0 k m / h r ) , w i t h C A R a p p r o x ­
i m a t e l y e q u a l t o 135O ( 2 2 5 O ) .
If t h e f l i g h t a l t i t u d e along t h e
s t r a i g h t - l i n e p a t h i s s e t a t 900 m, t h e f l y i n g t i m e from t h e
t r a v e r s e o f t h e LRMS t o t h e b e g i n n i n g o f t h e f o u r t h t u r n i s i n c r e a s e d
t o 3 m i n u t e s a n d 30 s e c o n d s , s o t h a t i t i s a d v i s a b l e t o i n c r e a s e

383

t h e g l i d e p a t h u p t o 4O f o r t h e p u r p o s e o f s h o r t e n i n g t h e t i m e i n ­
volved i n making t h e d e s c e n t .
I n c a s e s when a n a i r c r a f t i s a p p r o a c h i n p a n a i r p o r t w i t h a p a t h
a n g l e c l o s e t o t h e l a n d i n g a n g l e , a t a n a l t i t u d e o f 1 5 0 0 m or l e s s ,
t h e d o u b l e f i r s t a n d s e c o n d t u r n s a r e made i m m e d i a t e l y a f t e r p a s ­
The d e s c e n t t o c i r c l i n g a l t i t u d e a n d r e d u c t i o n o f
s i n g t h e LRMS.
s p e e d t o l o w e r t h e u n d e r c a r r i a g e i n t h i s case a r e p e r f o r m e d i n t h e
designated turn.
Hence, t h e l a r g e r e c t a n g u l a r f l i g h t p a t t e r n i n
converted t o a s m a l l one, and t h e maneuvering t i m e i s s h o r t e n e d t o
a b o u t 4.5 min.

If t h e a i r c r a f t approaches t h e a i r p o r t a t t h e a l t i t u d e e s t a b ­
l i s h e d f o r c i r c l i n g t h e f i e l d , t h e r a d i i o f a l l f o u r t u r n s a r e made
approximately t h e same, s o t h a t i n o r d e r t o create t h e “buffer1’
l i n e between t h e t h i r d and f o u r t h t u r n s , t h e f i r s t and second t u r n s
of t h e a i r c r a f t a r e e x e c u t e d i n s u c c e s s i o n w i t h a t i m e i n t e r v a l
between t h e end o f t h e f i r s t and t h e b e g i n n i n g o f t h e second t u r n
which e q u a l s 40 sec.
The f o u r t h t u r n on t h e l a r g e a n d s m a l l r e c t a n g u l a r p a t t e r n s
i s made a l o n g t h e c o u r s e a n g l e .
In aircraft with gas turbine
e n g i n e s , t h e C A R a t t h e b e g i n n i n g o f t h e f o u r t h t u r n (when t u r n i n g
t o t h e r i g h t ) m u s t b e e q u a l t o 7 0 ° ( a n d - 2 9 0 ° when t u r n i n g t o
the left).
The l a n d i n g a p p r o a c h f o r a i r c r a f t w i t h p i s t o n e n g i n e s , i s made
according t o t h e small rectangular p a t t e r n , with d i f f e r e n t first
a n d s e c o n d t u r n s , a n d t h e same t i m e p a r a m e t e r s b e t w e e n t h e f i r s t
a n d s e c o n d t u r n s ( 4 0 s e c ) , f r o m t h e t r a v e r s e o f t h e LRMS t o t h e
beginning of the t h i r d turn (70 s e c ) ( a t a f l i g h t a l t i t u d e of
3 0 0 m).
T h e f o u r t h t u r n for t h e s e a i r c r a f t b e g i n s a t C A R = 75 or
285O.
However, due t o t h e l o w e r a i r s p e e d a l o n g t h e s t r a i g h t - l i n e
segments and t h e smaller t u r n i n g r a d i i , t h e l i n e a r dimensions o f
t h e m a n e u v e r f o r a i r c r a f t w i t h p i s t o n e n g i n e s a r e much l e s s t h a n
I n a d d i t i o n , due t o t h e
for aircraft with gas turbine engines.
s h o r t e r t i m e f o r each t u r n , t h e t o t a l t i m e f o r executing the ma­
neuver f o r a i r c r a f t with p i s t o n engines i s s h o r t e r ( f o r example)
by 1 m i n u t e .

F i g . 4 . 1 0 . L a n d i n g M a n e u v e r When
Approaching t h e LTS Axis a t a
90° Angle.

384

..

..

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When a n a i r c r a f t i s a p p r o a c h i n g an a i r p o r t a t a n MPA w h i c h i s
perpendicular t o the landing angle , the landing altitude f o r air­
c r a f t with gas t u r b i n e e n g i n e s i s u s u a l l y s e t a t 1 2 0 0 m above t h e
l e v e l of t h e ai rp o rt (Fig. 4.10).
A f t e r p a s s i n g t h e LRMS, t h e a i r ­
c r a f t c o n t i n u e s on a c o u r s e w h i c h l a s t s f o r 40 s e c u n t i l d e s c e n t ,
a n d t h e s e c o n d t u r n i s a l s o e x e c u t e d w i t h loss o f a l t i t u d e .
A f t e r c o m p l e t i n g t h e s e c o n d t u r n , t h e f l i g h t l a s t s 30 s e c u n t i l
t h e b e g i n n i n g o f t h e t h i r d t u r n , when t h e u n d e r c a r r i a g e i s l o w e r e d .

A s i m i l a r m a n e u v e r i s e x e c u t e d by ' p i s t o n - e n g i n e a i r c r a f t ,
w i t h t h e s o l e d i f f e r e n c e t h a t t h e f l y i n g t i m e f r o m t h e LRMS t o t h e
beginning o f t h e second t u r n is s e t a t no less than 1 min, s i n c e
t h e a i r s p e e d o f t h e s e a i r c r a f t i n a l l s t a g e s o f t h e landing approach
u n t i l e m e r g e n c e f r o m t h e f o u r t h t u r n i s r o u g h l y t h e same.
I f a n a i r c r a f t a p p r o a c h e s a n a i r p o r t w i t h a n MPA w h i c h i s c l o s e
t o t h e r e v e r s e o f t h e l a n d i n g a n g l e , t h e crew o f a g a s t u r b i n e
aircraft travels along a s m a l l rectangular pattern with different
s i d e s f o r t h e f i r s t and second t u r n s (Fig. 4.11,a).
I n t h e case
of aircraft with piston engines, t h e so-called standard turn is
e x e c u t e d i n t h i s i n s t a n c e ( F i g . 4 . 1 1 , b ) on t h e l a n d i n g c o u r s e .

These maneuvers a g r e e i n t e r m s o f t h e magnitude and d i r e c t i o n
o f t h e t u r n s ; i n t h e f o r m e r , however, t h e r e i s a s t r a i g h t - l i n e
segment between t h e second and thkd t u r n s f o r lowering t h e under­
c a r r i a g e , w h i l e t h e r e i s a " b u f f e r " l i n e between t h e t h i r d and
fourth turns.
I n a d d i t i o n , i f t h e aircraft approaches t h e a i r p o r t
a t t h e f l i g h t a l t i t u d e f o r c i r c l i n g t h e f i e l d , and a l l t h e t u r n s
o f t h e a i r c r a f t a r e made w i t h o u t loss o f a l t i t u d e ( t h e r a d i i o f
a l l t h e t u r n s being t h e same), then between t h e f i r s t and second
t u r n s t h e r e w i l l a l s o be a p e r i o d of f l i g h t along a s t r a i g h t l i n e
f o r a p e r i o d o f 40 s e c .
I n t h e c a s e o f a s t a n d a r d t u r n , a l l f o u r t u r n s w i l l b e made i n /366
s u c c e s s i o n w i t h o u t t h e r e b e i n g any s t r a i g h t - l i n e s e g m e n t s between
turns.

F i g . 4 . 1 1 . Landing Maneuver w i t h a Course O p p o s i t e t o t h e Landing
C o u r s e . ( a > Along a S t r a i g h t - L i n e P a t h ; ( b ) S t a n d a r d T u r n .

Analogous maneuvers

f o r approaching t o make a l a n d i n g can a l s o

385

b e made b y u s i n g m o r e c o m p l e t e l a n d i n g s y s t e m s .
However, a i r ­
p o r t s t h a t h a v e s u c h s y s t e m s , as a r u l e , a r e a l s o e q u i p p e d w i t h
r a d a r devices t o monitor t h e a i r c r a f t maneuvering i n t h e v i c i n i t y
of the airport.
T h e r e f o r e , t h e b e g i n n i n g o f t h e l a n d i n g maneuver
n e e d n o t n e c e s s a r i l y b e made a t t h e m a r k e r p o i n t o n t h e L T S a x i s ,
t h u s m a k i n g i t p o s s i b l e t o come i n f o r a l a n d i n g a l o n g t h e s h o r t e s t
p a t h from any d i r e c t i o n .
A s m a l l or l a r g e r e c t a n g u l a r p a t t e r n i s u s u a l l y u s e d as t h e
b a s i s f o r s e t t i n g up a l a n d i n g a p p r o a c h a l o n g t h e s h o r t e s t p a t h .
However, i t i s n o t g e n e r a l l y c o m p l e t e d , u s u a l l y b e g i n n i n g a t t h e
p o i n t o f t a n g e n c y of t h e e n t r a n c e i n t o t h e m a n e u v e r t o o n e o f i t s
turns.

C a l c u l a t i o n o f Landing Approach P a r a m e t e r s
for a Simplified System
I n t h e p r e c e J i n g s e c t i o n , we d i s c u s s e d t h e t y p i c a l m a n e u v e r s
f o r l a n d i n g a n a i r c r a f t when a p p r o a c h i n g t h e a i r p o r t f r o m a n y d i ­
rection.
The e x e c u t i o n o f t h e s e m a n e u v e r s d o e s n o t p o s e g r e a t
d i f f i c u l t y f o r t h e c r e w o f t h e a i r c r a f t , s i n c e t h e f l i g h t i s made
w i t h a s u f f i c i e n t a l t i t u d e r e s e r v e and s u f f i c i e n t s p e e d , w h i l e t h e
d e m a n d s on t h e a c c u r a c y o f m a k i n g t h e m a n e u v e r a r e n o t v e r y h i g h .
The m a i n d i f f i c u l t y l i e s i n f l y i n g a l o n g a g i v e n d e s c e n t t r a ­
j e c t o r y i n t h e g l i d e p a t h , d u e t o t h e v e r y h i g h demands on t h e
maintenance o f f l i g h t d i r e c t i o n , a l t i t u d e , and h o r i z o n t a l g l i d e
s p e e d , d e p e n d i n g on t h e r e m a i n i n g d i s t a n c e t o t h e t o u c h d o w n p o i n t .
I n t h e case o f a i r c r a f t w i t h g a s t u r b i n e e n g i n e s , t h e r e i s t h e
a d d i t i o n a l n e e d t o r e d u c e t h e a i r s p e e d g r a d u a l l y as t h e a j r p o r t i s
approached

.

I n o r d e r t o f a c i l i t a t e t h e t a s k of d e s c e n d i n g a l o n g a g i v e n
t r a j e c t o r y t o a c e r t a i n d e g r e e , as w e l l a s t o a v o i d s e r i o u s e r r o r s
i n f l i g h t a l o n g t h e l a n d i n g p a t h , some p r e l i m i n a r y c a l c u l a t i o n s
are made, o f which t h e f o l l o w i n g i s t h e most i m p o r t a n t .

I f t h e l a n d i n g a p p r o a c h i s made i n a d e a d calm, t h e g e o m e t r i c
dimensions of t h e maneuver ( a n d c o n s e q u e n t l y , t h e p o i n t where t h e
d e s c e n t b e g i n s a l o n g t h e l a n d i n g p a t h ) are d e t e r m i n e d by s i m p l e
r e l a t i o n s h i p s between t h e a i r s p e e d , t i m e , t u r n r a d i i , f l i g h t
a l t i t u d e i n c i r c l i n g t h e f i e l d , and e s t a b l i s h e d steepness of t h e
glide path.
T h e c a l c u l a t e d d a t a f o r m a k i n g a l a n d i n g i n a calm a r e u s u a l l y
p l o t t e d on s p e c i a l l a n d i n g p a t t e r n s , d e v i s e d f o r e a c h a i r p o r t .
Under a c t u a l c o n d i t i o n s , however, i t i s n e c e s s a r y t o t a k e i n t o ac­
count t h e head wind and s i d e wind components ( f o r the l a n d i n g
c o u r s e ) , w h i c h c a n h a v e a v e r y g r e a t e f f e c t on t h e m a k i n g o f a
landing.

3 86

.. .

_.

/367

C a Z c u Z a t i o n of C o r r e c t i o n s for t h e Time for
B e g i n n i n g t h e T h i r d Turn
In preparing t o land, especially with the a i d of a simplified
system, it i s n e c e s s a r y t o e n s u r e t h a t t h e a i r c r a f t emerges from
t h e f o u r t h t u r n o n t o t h e l a n d i n g a p p r o a c h a l w a y s a t t h e same d i s t a n c e
f r o m t h e LTS.
Obviously, i n o r d e r t o s o l v e t h i s problem, i t is
n e c e s s a r y t o c o n s i d e r o n l y t h e head-wind component f o r t h e l a n d i n g
course.
I n making an a p p r o a c h t o l a n d a l o n g a r e c t a n g u l a r p a t t e r n , t h e
last r e l i a b l e point f o r determining t h e X-coordinate of t h e air­
craft (the distance along the axis of the direction of the a i r p o r t )
i s t h e t r a v e r s e o f t h e LRMS, w h i l e i n a s t a n d a r d t u r n i t i s t h e
p a s s a g e o v e r t h e LRMS w i t h a n MPA o p p o s i t e t o t h e l a n d i n g a n g l e .
I f w e d o n o t t a k e t h e w i n d i n t o a c c o u n t when c o m i n g i n f o r a
landing, the aircraft w i l l enter the landing path a t a distance
f r o m t h e L T S w h i c h e x c e e d s t h e d i s t a n c e f o r c a l m c o n d i t i o n s by
t h e value

w h e r e t i s t h e f l y i n g t i m e f r o m t h e t r a v e r s e o f t h e LRMS t o t h e
e m e r g e n c e f r o m t h e f o u r t h t u r n , or f r o m t h e moment when t h e air­
c r a f t p a s s e s o v e r t h e LRMS u n t i l i t e m e r g e s f r o m t h e s t a n d a r d t u r n .

Example: The f l y i n g t i m e f r o m t h e t r a v e r s e o f t h e LRMS t o t h e
e m e r g e n c e f r o m t h e f o u r t h t u r n i n a calm i s 4 m i n , d i v i d e d i n t o
these stages:
T r a v e r s e o f t h e LRMS t o b e g i n n i n g o f t h i r d t u r n . . . 7 0 s e c
Third turn...
60 sec
Buffer l i n e
50 s e c
Fourth t u r n .
60 sec

.....................................
.......................................
......................................

The s p e e d o f

t h e head-wind

c o m p o n e n t on t h e l a n d i n g c o u r s e i s

ux = 1 5 m/sec.
F i n d t h e v a l u e o f A X for e m e r g e n c e f r o m t h e f o u r t h t u r n .

Solution:
AX

= 240

x

1 5 = 360p m.

I n o r d e r f o r t h e d i s t a n c e f o r emergence from t h e f o u r t h t u r n
t o r e m a i n t h e same a s i n a c a l m , i t i s n e c e s s a r y t o s h o r t e n t h e
f l y i n g t i m e f r o m t h e t r a v e r s e o f t h e LRMS t o t h e b e g i n n i n g o f t h e
t h i r d t u r n by a v a l u e
3600
At

=

V+UX

387

/368

I n o u r e x a m p l e , f o r a n a i r s p e e d o f 4 0 0 k m / h r (110 m / s e c ) a n d
a course opposite t o the landing course,
3600
3600 - . 2 9 s e c
At = l l O + 15 - 125
I-­

T h u s , a f l i g h t f r o m t h e t r a v e r s e o f t h e LRMS t o t h e s t a r t o f
t h e t h i r d t u r n w o u l d l a s t 41 s e c i n s t e a d o f 7 0 s e c .
By c o m b i n i n g t h e f o r m u l a s for o b t a i n i n g t h e v a l u e s �or A X a n d
A t , we f i n a l l y obtain t h e formula f o r determing t h e value A t :
At=

tux

Vfu,

For our e x a m p l e ,
At==

240.15

l l O + 15

=29sec

The p r o b l e m f o r a s t a n d a r d t u r n i s s o l v e d i n t h e same w a y .
I n t h i s case, t h e t i m e t i s t h e t i m e from t h e p a s s a g e o v e r t h e
LRMS i n a c o u r s e o p p o s i t e t o t h e l a n d i n g c o u r s e , t o t h e e n d o f t h e
s t a n d a r d t u r n ; t h e v a l u e o f A t i s c a l c u l a t e d f r o m t h a t f o r a calm
i n f l y i n g f r o m t h e LRMS t o t h e s t a r t o f t h e s t a n d a r d t u r n .

C a Z c u Z a t i o n of t h e C o r r e c t i o n f o r t h e Time
of S t a r t i n g t h e F o u r t h T u r n
The b e g i n n i n g o f t h e f o u r t h t u r n i n c o m i n g i n f o r a l a n d i n g i s
u s u a l l y d e t e r m i n e d f r o m t h e c o u r s e a n g l e o f t h e LRMS.
For e x a m p l e ,
when e x e c u t i n g a m a n e u v e r t o t h e r i g h t :
R
ctgCAR= -

x

*’

w h e r e R i s t h e r a d i u s o f t h e t u r n made b y t h e a i r c r a f t , a n d X i s
t h e d i s t a n c e a l o n g t h e LTS a x i s f r o m t h e LRMS t o t h e s t a r t i n g po3n.t.
of t h e f o u r t h t u r n .
Under t h e i n f l u e n c e o f a s i d e w i n d , t h e f o u r t h t u r n i s begun
e a r l i e r i f t h e l a t e r a l c o m p o n e n t o f t h e w i n d on t h e l a n d i n g c ~ u r s e
is favorable between t h e t h i r d and f o u r t h t u r n s , l a t e r i f t h i s
component i s u n f a v o r a b l e .
O b v i o u s l y , if w e t a k e t h e wind i n t o a c c o u n t :
ctg C A R =

R+
X
t*uz

w h e r e t i s t h e t i m e o f t h e f o u r t h t u r n a n d u Z i s t h e l a t e r a l com­
ponent o f t h e wind speed.

For e x a m p l e , w i t h a t u r n i n g r a d i u s o f 4 5 0 0 m a n d X = 12.5 km,
388

/ 369

CtgCAl?,

e--

4500

12500 '


If t h e l a t e r a l component o f t h e w i n d a p p e a r s on t h e " b u f f e r "
l i n e a n d i s f a v o r a b l e , w i t h a s p e e d o f 1 0 m/sec, t h e n f o r a t u r n i n g
t i m e o f 60 sec w e w i l l have:

ctg CAR=

4500+60*10

12'500

.
'

CAR= 68".

i.e.,

t h e t u r n must b e g i n 2 O e a r l i e r t h a n under calm c o n d i t i o n s .

C a Z c u Z a t i o n of t h e Moment f o r B e g i n n i n g D e s c e n t
Along t h e Landing Course
Under calm c o n d i t i o n s , t h e distance a t w h i c h t h e a i r c r a f t
e m e r g e s f r o m t h e f o u r t h t u r n i s d e t e r m i n e d by t h e f l y i n g t i m e f r o m
t h e LRMS t o t h e s t a r t o f t h e t h i r d t u r n .

For e x a m p l e , w i t h V = 111 m / s e c ( 4 0 0 k m / h r ) ,

t h i s distance

w i l l be :
X---:70*111x 8 K A c t o L m S

The f o u r t h t u r n w i l l b e c o m p l e t e d a t a p p r o x i m a t e l y t h i s d i s ­
t a n c e i f a c o r r e c t i o n f o r t h e e f f e c t o f t h e w i n d i s made i n t h e
time f o r s t a r t i n g the t h i r d t u r n .
Consequently, with a standard
l o c a t i o n o f t h e LRMS, t h e d i s t a n c e f r o m t h e p o i n t w h e r e t h e a i r ­
c r a f t comes o u t o f t h e f o u r t h t u r n t o t h e touchdown p o i n t i s 1 2 km.
The d i s t a n c e f o r ; b e g i n n i n g t h e d e s c e n t a l o n g t h e g l i d e p a t h
i s d e t e r m i n e d by t h e f o r m u l a
xd =tic,ctge,
where X
i s t h e d i s t a n c e f o r beginning t h e descent and H
is the
C
flight daltitude for circling the field.

For e x a m p l e , a t a c i r c l i n g a l t i t u d e o f H c
angle f o r the glide path 8 = 2040f:

= 400 m and a s l o p e

x d = 400.ctg2Q40'= 8500.u.
Thus, a f t e r coming o u t o f t h e f o u r t h t u r n a t a d i s t a n c e o f 1 2
km f r o m t h e L T S , t h e a i r c r a f t m u s t f o l l o w t h e l a n d i n g p a t h w i t h o u t
losing altitude for a period of t i m e

3 89

/370

"h =

Xe - x d
'

v-uu,

where t h is t h e t i m e o f h o r i z o n t a l f l i g h t a l o n g t h e l a n d i n g p a t h .

For example, i f t h e h o r i z o n t a l a i r s p e e d a f t e r coming o u t o f a
t u r n i s 360 km/hr ( 1 0 0 m / s e c ) , and t h e h e a d w i n d ' i s moving a t 1 5
m/sec, t h e t i m e f o r h o r i z o n t a l f l i g h t i n o u r case w i l l b e

fh =

12000-8500

100-

15

--3500
85

= 4 1 sec

U n d e r calm c o n d i t i o n s , t h e t i m e f o r h o r i z o n t a l f l i g h t i n t h i s
case w i l l b e :

P r a c t i c a l l y s p e a k i n g , t h e d e s c e n t o f t h e a i r c r a f t must begin
5 t o 6 seconds before t h i s time has a c t u a l l y elapsed, since a
c e r t a i n period of t i m e i s required t o guide t h e aircraft i n t o i t s
landing regime.

C a l c u l a t i o n of t h e V e r t i c a l R a t e of D e s c e n t
Along t h e G l i d e Path
The v e r t i c a l r a t e o f d e s c e n t o f a n a i r c r a f t a l o n g t h e g l i d e
p a t h i s d e t e r m i n e d by t h e s i m p l e f o r m u l a

vy= wxtgo = ( V x -

a x ) tgo.

For e x a m p l e , w i t h a mean h o r i z o n t a l r a t e o f d e s c e n t o f 2 9 0
km/hr (80 m/sec), a head wind o f 1 5 m/sec, and a s l o p e a n g l e i n
t h e g l i d e p a t h o f 2O40':
Vy = 65. tg 2'40' = 3 W / sec
The c a l c u l a t i o n o f t h e v e r t i c a l r a t e o f d e s c e n t i s o f p a r t i c ­
u l a r i n t e r e s t for p i s t o n - e n g i n e a i r c r a f t , w h o s e h o r i z o n t a l g l i d e
i s a b o u t 50 m/sec.
S i n c e t h e h e a d w i n d c a n b e as h i g h as 2 5 m / s e c on l a n d i n g ,
t h e v e r t i c a l g l i d e s p e e d f o r t h e s e a i r c r a f t can change by a f a c t o r
from 2 . 3 t o 1 . 1 5 m/sec.
o f 2 , :.e.,
I n t h e case of a i r c r a f t w i t h g a s t u r b i n e e n g i n e s , t h e r a t i o
o f t h e maximum r a t e o f d e s c e n t t o t h e minimum r a t e , w i t h t h e same
steepness of glide, is 1.5.

390

D e t e r m i n a t i o n of t h e Lead AngZe for t h e
Landing Path

/371

A knowledge o f t h e approximate v a l u e o f t h e d r i f t a n g l e , and
c o n s e q u e n t l y t h e n e c e s s a r y l e a d a n g l e f o r t h e l a n d i n g p a t h o f an
aircraft, considerably facilitates t h e choice of t h e course t o be
followed along a given descent t r a j e c t o r y .
The v a l u e o f t h e d r i f t a n g l e a l o n g t h e l a n d i n g p a t h c a n b e
d e t e r m i n e d by t h e a p p r o x i m a t e f o r m u l a

I n f l i g h t along a given descent t r a j e c t o r y , however, t h e hor­
i z o n t a l a i r s p e e d , a l t i t u d e , and wind are v a r i a b l e s , s o t h a t it i s
s u f f i c i e n t t o use t h e following r u l e i n finding the d r i f t angle:
( a ) For a i r c r a f t w i t h g a s t u r b i n e e n g i n e s , a t g l i d e s p e e d s o f
2 7 0 - 2 9 0 k m / h r , t h e l e a d a n g l e i s c o n s i d e r e d t o b e e q u a l t o 0.7O
f o r each 1 m/sec of s i d e wind.
( b ) For a i r c r a f t w i t h p i s t o n e n g i n e s , ( g l i d e s p e e d s o f 1 8 0 - 2 0 0
km/hr), t h e l e a d a n g l e i s c o n s i d e r e d t o be lo f o r each 1 m/sec o f
s i d e wind.

For e x a m p l e , w i t h a
8 mjsec, coming from t h e
- f o r aircraft with
- f o r aircraft with

s i d e wind a l o n g t h e l a n d i n g p a t h o f
r i g h t , t h e l e a d angle w i l l be:
gas t u r b i n e engines, 5 . 5 O t o t h e l e f t ;
p i s t o n e n g i n e s , 8O t o t h e l e f t .

The c a l c u l a t i o n s g i v e n a b o v e for t h e t i m e o f s t a r t i n g t h e
t h i r d t u r n , the course angle f o r beginning the fourth t u r n , the
time f o r beginning t h e d e s c e n t , t h e v e r t i c a l rate of d e s c e n t , and
t h e l e a d a n g l e for t h e l a n d i n g p a t h , m u s t a l l b e made b y t h e c r e w
o f t h e a i r c r a f t b e f o r e a p p r o a c h i n g t h e a i r p o r t on t h e b a s i s o f
landing-condition information.
A l l c a l c u l a t i o n s must b e c o m p l e t e
b e f o r e t h e l a n d i n g maneuver b e g i n s .
L a n d i n g t h e A i r c r a f t o n t h e Runway a n d F l i g h t
a l o n g a Given T r a j e c t o r y w i t h a S i m p l i f i e d Landing System

W h i l e m a k i n g p r e p a r a t i o n s for l a n d i n g , t h e c r e w m u s t p r e p a r e
t h e course t o be followed by t h e aircraft along a l l t h e s t r a i g h t l i n e segments of t h e approach p a t t e r n , with t h e exception of t h e l i n e
between t h e t h i r d and f o u r t h t u r n s , beginning with a c a l c u l a t i o n
of the d r i f t angle.
T h e r a d i o c o m p a s s m u s t b e s e t b y t h e LRMS; i f t h e r e a r e t w o
s e t s o f r a d i o c o m p a s s e s , t h e s e c o n d must b e s e t by t h e SRMS.
Along t h e l i n e b e t w e e n t h e t h i r d a n d f o u r t h t u r n s , t h e c o u r s e
t o b e f o l l o w e d i s a l w a y s e q u a l t o t h e MPA o f t h e " b u f f e r " s e g m e n t ,

391


s o t h a t t h e s t a r t o f t h e f o u r t h t u r n w i l l b e determined by t h e CAR.
The s l i g h t d r i f t o f t h e a i r c r a f t w h i c h o c c u r s a t t h i s t i m e , a s w e
h a v e s e e n , i s c o m p e n s a t e d b y r e d e f i n i n g t h e t i m e for s t a r t i n g t h e
third turn.
When t h e c o u r s e a n g l e o f t h e LRMS b e c o m e s e q u a l t o t h e c a l ­
culated value, t h e fourth t u r n is executed with a banking angle o f
15O b e f o r e a c q u i r i n g t h e c a l c u l a t e d l a n d i n g p a t h .
I f a l l t h e c a l c u l a t e d d a t a a r e c o r r e c t , t h e a i r c r a f t w i l l come
o u t o f t h e t u r n p r e c i s e l y on t h e l a n d i n g p a t h w i t h t h e d e s i r e d
At t h e moment when t h e a i r c r a f t e m e r g e s f r o m t h e f o u r t h
course.
t u r n , t h e t i m e r i s s w i t c h e d on t o d e t e r m i n e t h e t i m e f o r b e g i n n i n g
descent i n the glide path.
I n t h e m a j o r i t y o f cases, however, due t o e r r o r s i n t h e oper­
a t i o n o f t h e radiocompass, improper maintenance o f t h e c o u r s e and
a i r s p e e d o f t h e a i r c r a f t , e r r o r s i n d e t e r m i n i n g t h e s i d e - w i n d com­
p o n e n t , a n d f a i l u r e t o b a n k ;It t h e p r o p e r a n g l e when t u r n i n g , t h e
a c q u i s i t i o n o f t h e g l i d e p a t h by t h e a i r c r a f t i s n o t a c c u r a t e .
The a c c u r a c y w i t h w h i c h t h e a i r c r a f t a c q u i r e s t h e l a n d i n g
p a t h i s d e t e r m i n e d by a comparison o f t h e m a g n e t i c b e a r i n g o f t h e
LRMS w i t h t h e MPA f o r l a n d i n g .
IF MC + C A R = MPA1, b u t t h e com­
the aircraft w i l l
b i n e d r e a d i n g o f t h e r a d i o c o m p a s s i s M B R = MPA1,
b e e x a c t l y on t h e a x i s o f t h e L T S .
I f MBR i s g r e a t e r t h a n MPA1,
t h e aircraft w i l l be t o the l e f t
of t h e g i v e n l a n d i n g p a t h .
W i t h MBR s m a l l e r t h a n MPA1, t h e a i r c r a f t
w i l l be t o t h e r i g h t o f t h e given landing path.
The d i f f e r e n c e b e t w e e n M P A l a n d MBR i s c a l l e d t h e a c q u i s i t i o n
e r r o r a.

Example: MPAl = 68O, w i t h a c a l c u l a t e d d r i f t a n g l e o f t 3 O ;
t h e a i r c r a f t e m e r g e d f r o m t h e f o u r t h t u r n w i t h MC = 6 5 O , t h e c o u r s e
a n g l e f o r t h e t u r n o v e r t h e LRMS w a s 358O; f i n d t h e a c q u i s i t i o n
error.

Solution:
4%

a =68 - (65 + 358) = 5",
i.e.,the

acquisition error is 5O t o the right.

For l i n i n g u p t h e a i r c r a f t w i t h t h e l a n d i n g p a t h , t h e c o u r s e
f o l l o w e d by t h e a i r c r a f t i s u s u a l l y c h a n g e d b y d o u b l i n g t h e a c q u i s i ­
tion error.
I n our e x a m p l e , t h e c o u r s e t o b e f o l l o w e d m u s t b e r e ­
d u c e d l o o , s o t h a t t h e C A R o f t h e LRMS b e c o m e s 8O; t h e f l i g h t i s
continued at t h i s course u n t i l the value of the course angle in­
creases t o t h e m a g n i t u d e o f t h e a c q u i s i t i o n e r r o r , ; . e . , becomes
13O.
39 2

/372

When t h e p o i n t e r o f t h e r a d i o c o m p a s s i s o n t h e 1 3 O m a r k ( o n a
combined i n d i c a t o r , a b e a r i n g o f 6 8 O ) , w i t h a s l i g h t l e a d ( n o more
t h a n 1 t o 2 O ) , t h e a i r c r a f t makes a n o t h e r t u r n t o t h e c a l c u l a t e d
l a n d i n g p a t h , a n d t h e C A R o f t h e LRMS b e c o m e s e q u a l t o t h e c a l c u ­
l a t e d d r i f t angle o f t h e aircraft (3O i n t h e example).
A s t h e a i r c r a f t c o n t i n u e s t o f o l l o w t h e l a n d i n g p a t h on t h e
c a l c u l a t e d c o u r s e , t h e CAR w i l l remain e q u a l t o t h e c a l c u l a t e d
d r i f t angle i f t h e course o f t h e aircraft has been properly s e l e c t e d .
If t h e CAR i s i n c r e a s e d , t h e a i r c r a f t w i l l d r i f t t o t h e l e f t o f
t h e LTS a x i s , a n d t h e p a t h b e i n g f o l l o w e d w i l l h a v e t o b e i n c r e a s e d
f o r a c q u i s i t i o n of t h e d e s i r e d l i n e o f f l i g h t , and decreased l a t e r
o n , a l t h o u g h i t w i l l r e m a i n somewhat g r e a t e r t h a n t h e c a l c u l a t e d
v a l u e ( t h e CAR i s t h e n l e s s t h a n t h e c a l c u l a t e d d r i f t a n g l e ) .
If
/373
t h e CAR i s t h e n t o remain c o n s t a n t , t h e c o u r s e t o b e f o l l o w e d must
be s e l e c t e d properly.

S i m i l a r o p e r a t i o n s i n s e l e c t i n g a c o u r s e a r e c a r r i e d o u t when
the aircraft deviates t o the r i g h t of the desired l i n e of f l i g h t .
T h e s e o p e r a t i o n s w i l l h a v e t h e f o r m of a m i r r o r image o f t h e o p e r ­
a t i o n s d e s c r i b e d a b o v e , : . e . , when t h e C A R i s r e d u c e d , i t i s a l s o
n e c e s s a r y t o reduce t h e c o u r s e t o be f o l l o w e d i n a c q u i r i n g t h e de­
s i r e d l i n e o f f l i g h t , t h e n i n c r e a s e i t somewhat, b u t s t i l l k e e p it
below t h e c a l c u l a t e d v a l u e .
I n t h e c a s e when t h e c o u r s e a n g l e o f t h e LRMS c o n t i n u e s t o
change, a f t e r t h e f i r s t o p e r a t i o n t o c o r r e c t t h e course by acquir­
i n g t h e l i n e of t h e g i v e n c o u r s e , t h e o p e r a t i o n s are r e p e a t e d u s i n g
t h e f a m i l i a r method o f h a l f c o r r e c t i o n s .
Thus, t h e r e a d i n g s o f t h e radiocompasses , b e g i n n i n g w i t h t h e
LRMS a n d t h e n t h e SRMS, a r e u s e d t o m a i n t a i n t h e g i v e n d i r e c t i o n
of the descent trajectory.
When t h e a i r c r a f t is c a l c u l a t e d t o h a v e r e a c h e d t h e p o i n t f o r
b e g i n n i n g i t s d e s c e n t , it i s s h i f t e d t o a d e s c e n t regime w i t h a
calculated r a t e of descent.
The v e r t i c a l r a t e o f d e s c e n t i s m a i n ­
t a i n e d by o b s e r v i n g t h e v a r i o m e t e r r e a d i n g s a n d t h o s e o f t h e g y r o horizon , while maintaining t h e e s t a b l i s h e d regime of h o r i z o n t a l
a i r s p e e d on t h e b a s i s o f t h e i n s t r u m e n t - s p e e d i n d i c a t o r .
The g y r o h o r i z o n m u s t b e u s e d t o m a i n t a i n t h e v e r t i c a l r a t e o f
descent, because t h e readings o f t h e variometer are less s t a b l e than
those of the angle of p i t c h of t h e aircraft obtained with the a i d
of t h e gyrohorizon i n d i c a t o r .
The r e a d i n g s o f t h e v a r i o m e t e r m u s t
be averaged o v e r t h e t i m e .
In addition, t h e variometer has s l i g h t delays i n the readings
w i t h a change i n t h e a n g l e o f p i t c h o f t h e a i r c r a f t .
Therefore,
t h e gyrohorizon i s employed t o s e l e c t t h e a n g l e o f p i t c h f o r t h e
a i r c r a f t a t which t h e a v e r a g e r e a d i n g s o f t h e v a r i o m e t e r are e q u a l
393


t o the calculated values,
on t h e g y r o h o r i z o n .

If t h e
tive t o the
the engines
t o maintain

a n d t h i s a n g l e i s m a i n t a i n e d by t h e r e a d i n g s

h o r i z o n t a l a i r s p e e d i s t h e n i n c r e a s e d or d e c r e a s e d r e l a ­
g i v e n v a l u e , it i s r e g u l a t e d by changing t h e t h r u s t o f
and s i m u l t a n e o u s l y c h a n g i n g t h e aogle o f p i t c h s l i g h t l y
t h e c a l c u l a t e d r a t e of d e s c e n t .

A f a i l u r e t o maintain t h e c a l c u l a t e d settirgi f o r t h e g l i d e
p a t h , or e r r o r s i n c a l c u l a t i o n s , may c a u s e t h e a i r c r a f t t o p a s s
o v e r t h e LRMS e a r l i e r a t t h e r e q u i r e d a l t i t u d e , s o t h a t t h e d e s c e n t
of t h e a i r c r a f t i s t e r m i n a t e d and t h e a i r c r a f t i s once a g a i n p l a c e d
i n t h e r e g i m e o f d e s c e n t a t t h e moment i t p a s s e s o v e r t h e LRMS.
H o w e v e r , i f t h e g i v e n a l t i t u d e h a s n o t b e e n a t t a i n e d when p a s s i n g
o v e r t h e LRMS, t h e v e r t i c a l r a t e o f d e s c e n t i s i n c r e a s e d a t t h e
s t a g e o f t h e f l i g h t b e t w e e n t h e LRMS a n d t h e SRMS.

Similarly, t h e descent of t h e aircraft is terminated i f it
r e a c h e s t h e a l t i t u d e s e t f o r p a s s i n g o v e r t h e SRMS b e f o r e t h e s o u n d
/374
o f t h e SRMS i s h e a r d , m a r k i n g t h e l o c a t i o n o f t h e l a t t e r .
The minimum w e a t h e r f o r t h e c e i l i n g when l a n d i n g w i t h a s i m p l i ­
f i e d s y s t e m , i n t h e case o f a i r c r a f t w i t h p i s t o n e n g i n e s , i s n o t
s e t a n y l o w e r t h a n t h e a l t i t u d e for p a s s i n g o v e r t h e SRMS; i n t h e
case o f a i r c r a f t w i t h g a s t u r b i n e e n g i n e s , i t i s s i g n i f i c a n t l y
higher.
T h e r e f o r e , t h e a i r c r a f t can be allowed t o descend only i n
t h e c a s e when t h e c r e w o f t h e a i r c r a f t c a n s e e t h e l i g h t s o f t h e
a p p r o a c h e s t o t h e LTS a n d t h e e n d o f t h e r u n w a y .

Course-Glide Landing Systems

The s i m p l i f i e d s y s t e m f o r l a n d i n g a n a i r c r a f t as d e s c r i b e d i n
t h e p r e c e d i n g s e c t i o n , u s i n g t h e master r a d i o s t a t i o n s , h a s a
number o f i m p o r t a n t d e f i c i e n c i e s :
( a ) The m e a s u r e m e n t a c c u r a c y o f t h e a i r c r a f t b e a r i n g , u s i n g
an a i r c r a f t radiocompass and course meter, i s v e r y low, s o t h a t
i t d o e s n o t make i t p o s s i b l e t o l a n d t h e a i r c r a f t ( e s p e c i a l l y
t h o s e w i t h g a s - t u r b i n e e n g i n e s ) w i t h low w e a t h e r minima.
( b ) The o p e r a t i o n o f r a d i o c o m p a s s e s d u r i n g f l i g h t i n c l o u d s
and p r e c i p i t a t i o n i s h i g h l y s u b j e c t t o a t m o s p h e r i c d i s t u r b a n c e s ,
t h u s c o m p l i c a t i n g a l a n d i n g w i t h t h e s e d e v i c e s as g u i d e s .
( c ) The s i m p l i f i e d s y s t e m r e q u i r e s c o n s t a n t c h e c k i n g o f t h e
p o s i t i o n o f t h e a i r c r a f t a l o n g a given d e s c e n t t r a j e c t o r y i n terms
of d i r e c t i o n o n l y ; t h e d e s c e n t o f t h e a i r c r a f t i n a g i v e n g l i d e
path i s accomplished by maintaining t h e v e r t i c a l rate of descent
of t h e a i r c r a f t and c a l c u l a t i n g t h e t i m e , t h u s c o m p l i c a t i n g t h e
l a n d i n g procedure and n o t e n s u r i n g safe d e s c e n t under e s p e c i a l l y
d i f f i c u l t conditions

.

394

If w e c o n s i d e r t h a t t h e p e r i o d o f l a n d i n g t h e a i r c r a f t w i t h
low c e i l i n g a n d low m e t e o r o l o g i c a l v i s i b i l i t y i s t h e mo s t d i f f i c u l t
a n d d a n g e r o u s s t a g e o f t h e f l i g h t , it i s n e c e s s a r y t o d e v i s e more
complete systems o f instrument l a n d i n g .
One s u c h s y s t e m i s t h e
course-glide landing system.
The g e o m e t r i c e s s e n c e o f c o u r s e - g l i d e s y s t e m s i s t h e u s e o f
r a d i o - e n g i n e e r i n g methods t o d e f i n e t w o m u t u a l l y p e r p e n d i c u l a r
planes i n space ( F i g . 4.12):
(.a> A v e r t i c a l . p l a n e which i n t e r s e c t s t h e E a r t h ’ s s u r f a c e
a l o n g t h e LTS a x i s .
( b ) An i n c l i n e d p l a n e w h i c h r e p r e s e n t s t h e g l i d e p a t h o f t h e

aircraft.
I f t h e a i r c r a f t i s i n one o f t h e s e two p l a n e s , t h e r e a d i n g s
o f t h e c o r r e s p o n d i n g p o i n t e r o n t h e i n d i c a t o r ( d i r e c t i o n or g l i d e )
must be e q u a l t o z e r o .

When t h e a i r c r a f t m o v e s o u t o f o n e o f t h e s e p l a n e s , t h e
corresponding p o i n t e r s h i f t s from z e r o .
The s h i f t o f t h e p o i n t e r
proportional t o the
must be l i n e a r w i t h i n c e r t a i n l i m i t s ( ; . e . ,
d e v i a t i o n o f t h e a i r c r a f t from t h e given p l a n e ) .
Obviously, t h e given t r a j e c t o r y f o r t h e descent of t h e airc r a f t i s t h e l i n e o f i n t e r s e c t i o n o f t h e s e two p l a n e s .
When t h e
a i r c r a f t i s on t h e g i v e n t r a j e c t o r y , b o t h i n d i c a t o r p o i n t e r s m u s t
p o i n t t o z e r o on t h e i n d i c a t o r .

Fig.

4.12.

Radio-Signal Planes o f a Course-Glide
Landing
System.

For t h e b e s t v i s u a l d e t e r m i n a t i o n o f t h e p o s i t i o n o f t h e a i r ­
c r a f t r e l a t i v e t o a g i v e n d e s c e n t t r a j e c t o r y , t h e p o i n t e r s on t h e
i n d i c a t o r a r e made i n t h e f o r m o f s t r i p s , o n e h o r i z o n t a l for g l i d e
and one v e r t i c a l . f o r d i r e c t i o n .
The movement o f t h e s t r i p s t h e n
o c c u r s i n a d i r e c t i o n which i s o p p o s i t e t o t h e d e v i a t i o n o f t h e
aircraft from a g i v e n t r a j e c t o r y ( F i g . 4.13).

395

/375

The c e n t e r o f t h e i n s t r u m e n t , w i t h a s i l h o u e t t e o f a n a i r ­
c r a f t shown on t h e s c a l e , shows t h e p o s i t i o n ' o f t h e a i r c r a f t r e l ­
a t i v e t o t h e course plane and t h e g l i d e plane.
Thus, f o r example,
i n Fig. 4.13 t h e aircraft i s l o c a t e d below t h e given g l i d e p a t h
To s e t t h e a i r c r a f t o n t h e d e ­
a n d t o t h e l e f t o f t h e LTS a x i s .
s i r e d t r a j e c t o r y , it must b e t u r n e d i n t h e d i r e c t i o n o f t h e p l a n e s ,
;.e.,
upward ( t o i n c r e a s e t h e a n g l e o f p i t c h ) a n d t o t h e r i g h t .
The i n d i c a t o r f o r t h e d i r e c t i o n a n d g l i d e h a s t h e t r a d i t i o n a l
name o f " L a n d i n g S y s t e m A p p a r a t u s " , or LAS for s h o r t .

Ground ControZ o f Course-GZide S y s t e m s

The p r i n c i p a l p i e c e s o f e q u i p m e n t i n a c o u r s e - g l i d e l a n d i n g
s y s t e m a r e two g r o u n d b e a c o n s which f o r m t h e c o u r s e zone a n d t h e
g l i d e zone marking t h e g i v e n t r a j e c t o r y f o r t h e d e s c e n t of t h e
aircraft.
B o t h b e a c o n s o p e r a t e on m e t e r or c e n t i m e t e r w a v e l e n g t h s .
The a n t e n n a s o f t h e b e a c o n s t h a t u s e m e t e r w a v e s a r e c r o s s e d
h o r i z o n t a l dipoles ( h o r i z o n t a l frames) i n course beacons and

h o r i z o n t a l dipoles i n g l i d e beacons.
T h u s , t h e e l e c t r o m a g n e t i c waves f r o m t h e b e a c o n s are h o r i z o n ­
t a l l y p o l a r i z e d , w h i c h t o a c e r t a i n d e g r e e r e d u c e s t h e i r e f f e c t on
the d i r e c t i o n a l ~
c h a r a c t e r i s t i c s o f t h e a n t e n n a s on t h e g r o u n d
control facilities at the airport.

1
Fig.

4.13.
Course-Glide

-

to transmitting device

Fig.

Fig.

4.13.

I n d i c a t o r of

Fig.

4.14.

Diagram o f L o c a t i o n o f Antennas

4.14.

Landing System.
o f Course Radio Beacon.

However, t h e E a r t h ' s s u r f a c e p l a y s a r o l e i n t h e f o r m a t i o n o f
t h e c o u r s e zone a n d t h e g l i d e zone by t h e s e b e a c o n s .
The c o u r s e
zone t h e n becomes m u l t i l o b e d i n t h e v e r t i c a l p l a n e , w i t h t h e
major l o b e b e i n g t h e working l o b e , which h a s a g l i d e a n g l e o f t h e
b i s e c t r i x which c o r r e s p o n d s r o u g h l y t o t h e s l o p e a n g l e o f t h e g l i d e
The E a r t h ' s s u r f a c e i s o f s t i l l g r e a t e r
plane of the aircraft.
396

/376


I

i m p o r t a n c e f o r t h e f o r m a t i o n o f t h e g l i d e z o n e , whose s l o p e a n g l e
d e p e n d s on t h e h e i g h t of t h e a n t e n n a ab o v e t h e g r o u n d .
The i n v o l v e m e n t o f t h e E a r t h ' s s u r f a c e i n t h e f o r m a t i o n o f t h e
b e a c o n z o n e s imposes l i m i t a t i o n s on t h e p o s s i b i l i t i e s o f t h e
beacons i n t e r m s of e n s u r i n g t h e a c c u r a c y w i t h which t h e a i r c r a f t
can b e landed.
T h i s i s e s p e c i a l l y t r u e f o r t h e g l i d e z o n e , whose
l o c a t i o n c a n c h a n g e w i t h t h e s t a t e o f t h e E a r t h ' s s u r f a c e ( w e t or
dry ground, g r a s s cover, snow).
The a c c u r a c y o f t h e l o c a t i o n o f
t h e c o u r s e zone i s s u b j e c t t o t h e i n f l u e n c e o f t h e l o c a l r e l i e f
and equipment l o c a t e d w i t h i n t h e l i m i t s o f t h e d i r e c t i o n a l charac­
teristic of the antennas.
The m o s t i m p o r t a n t o f t h e s e s h o r t c o m i n g s c a n b e o v e r c o m e t o a
g r e a t e x t e n t by e m p l o y i n g b e a c o n s w h i c h o p e r a t e on t h e c e n t i m e t e r
w a v e l e n g t h , u s i n g r e f l e c t i n g a n t e n n a s t o form v e r y n a r r o w d i r e c t i o n a l
characteristics

.

A t t h e p r e s e n t t i m e , however, t h e s e beacons have n o t been
a d o p t e d s u f f i c i e n t l y w i d e l y and a r e n o t u s e d i n enough l o c a t i o n s .
Therefore, w e s h a l l give a b r i e f description of t h e course-glide
s y s t e m s o n l y for t h e m e t e r w a v e l e n g t h s .

I n a d d i t i o n t o t h e b e a c o n s , which form t h e c o u r s e and g l i d e
z o n e s , t h e c o u r s e - g l i d e s y s t e m for l a n d i n g a l s o i n c l u d e s m a r k e r
d e v i c e s , whose l o c a t i o n s c a n c o i n c i d e w i t h t h o s e f o r t h e m a r k e r s
i n a s i m p l i f i e d landing system.
In foreign practice,

the first (long-range)

marker i s l o c a t e d

7 km f r o m t h e e n d o f t h e L T S ; a t a s l o p e a n g l e f o r t h e g l i d e p a t h
o f 2O30' and a c i r c l i n g a l t i t u d e o f 300 m , t h i s marks t h e p o i n t a t

which t h e a i r c r a f t b e g i n s t o d e s c e n t i n a g l i d e .
However, n o
s i g n i f i c a n t a d v a n t a g e s a r e g a i n e d by p l a c i n g t h e m a r k e r a t t h i s
s p o t , s i n c e t h e f l i g h t a l t i t u d e o f t h e a i r c r a f t when c i r c l i n g t h e
f i e l d d e p e n d s on t h e t y p e o f a i r c r a f t , w h i l e t h e s l o p e a n g l e for
t h e g l i d e p a t h d e p e n d s on t h e n a t u r e o f t h e s u r r o u n d i n g t e r r a i n .
T h i s means t h a t t h e p o i n t f o r b e g i n n i n g t h e g l i d e d o e s n o t always
c o i n c i d e w i t h t h e s t a n d a r d l o c a t i o n o f t h e a i r c r a f t ( 7 km).

For p u r p o s e s o f c h e c k i n g f o r t h e c o r r e c t n e s s o f t h e l o c a t i o n
o f t h e g l i d e z o n e , i t i s b e t t e r t o c h o o s e a m a r k e r l o c a t e d 4 km
f r o m t h e e n d o f t h e LTS, s i n c e a t t h i s p o i n t t h e a i r c r a f t w i l l a l ­
r e a d y have t h e s e l e c t e d r a t e of d e s c e n t f o r f o l l o w i n g t h e g l i d e
p a t h , a n d t h e a l t i t u d e o f i t s l o c a t i o n w i l l b e d e t e r m i n e d more
precisely.
An i n h e r e n t p a r t o f t h e c o u r s e - g l i d e l a n d i n g s y s t e m i s a l s o
t h e l i g h t i n g s y s t e m f o r t h e a p p r o a c h e s t o t h e runways a n d a l o n g
t h e e d g e s o f t h e runway i t s e l f .
T h e c o u r s e beacon i s a t r a n s m i t t i n g d e v i c e w i t h a n a n t e n n a
s y s t e m which c o n s i s t s ( a s a r u l e ) of f i v e o r s e v e n h o r i z o n t a l
antennas (Fig. 4.14).
397


/377

Antenna A h a s a r a d i a t i o n c h a r a c t e r i s t i c which i s d i r e c t e d
e x t e r n a l l y i n t h e h o r i z o n t a l p l a n e , a n d is p o w e r e d b y a t r a n s m i t t e r
o p e r a t i n g w i t h o u t m o d u l a t i o n on t h e m e t e r w a v e l e n g t h .
Antennas A1 and A 2 r e c e i v e amplitude-modulated f r e q u e n c i e s
t h e t r a n s m i t t e r , o n e a t 9 0 Hz a n d t h e o t h e r a t 1 5 0 'Hz.

from

A n t e n n a s A 3 a n d A 4 ( a s w e l l as A 5 a n d A g , i n s o m e t y p e s o f
beacons) s e r v e t o r e g u l a t e t h e d i r e c t i o n a l i t y of t h e r a d i a t i o n
c h a r a c t e r i s t i c , as w e l l as t h e d i r e c t i o n o f t h e r a d i o - s i g n a l zone
of the e n t i r e system.
The c o m b i n e d r e s u l t o f t h e e l e c t r o m a g n e t i c o s c i l l a t i o n s o f t h e
e n t i r e antenna system forms t h e d i r e c t i o n a l r a d i a t i o n c h a r a c t e r i s t i c
o f t h e e l e c t r o m a g n e t i c waves i n t h e h o r i z o n t a l p l a n e ; an example o f
t h i s i s shown i n F i g . 4 . 1 5 .
F i g u r e 4 . 1 5 . a , shows t h e s h a p e of t h e
radiation c h a r a c t e r i s t i c i n the horizontal plane ; the l e f t s i d e
i s ihe o n e m o d u l a t e d ' b y t h e 1 5 0 Hz f r e q u e n c y , w h i l e t h e r i g h t s i d e
i s m o d u l a t e d by t h e 9 0 Hz f r e q u e n c y .
Along a x i s AB , w h e r e t h e r a d i a t i o n c h a r a c t e r i s t i c s i n t e r s e c t
t h e m o d u l a t i o n f r e q u e n c i e s o f 1 5 0 and 90 Hz, t h e m o d u l a t i o n d e p t h
o f t h e c a r r i e r f r e q u e n c y by b o t h l o w f r e q u e n c i e s i s t h e s a m e ( ; . e .
t h e difference i n modulation depth is zero).

,

When t h e a i r c r a f t m o v e s t o t h e l e f t o f a x i s A B , t h e d e p t h o f
t h e m o d u l a t i o n w i t h t h e 9 0 Hz f r e q u e n c y i n c r e a s e s a n d t h a t w i t h t h e
1 5 0 Hz f r e q u e n c y d e c r e a s e s .
The p i c t u r e i s r e v e r s e d when t h e a i r ­
c r a f t moves t o t h e r i g h t o f t h e a x i s .
The d o t t e d l i n e s i n F i g . 4 . 1 5 s h o w t h e p r o j e c t i o n s o f t h e
r a d i a t i o n l o b e s o f t h e e l e c t r o m a g n e t i c waves i n t h e v e r t i c a l
p l a n e ( a s s h o w n i n F i g . 4 . 1 5 , b ) on t h e h o r i z o n t a l p l a n e .
L i n e A B i s t h e common a x i s w i t h a d i f f e r e n c e i n m o d u l a t i o n
d e p t h s which i s e q u a l t o z e r o f o r a l l l o b e s .
However, t h e p r i n ­
c i p a l operating lobes are the first ones, located n e a r e s t t o the
ground.

a)


/378

b)


Fig. 4.15. Radlatrsn Characteri'sti'cs
of Course Radio Beacon: ( a ) i n t h e
Horizontal Plane; (b) i n the Verti­
c a l Plane.

398

The r a d i a t i o n c h a r a c t e r i s t i c o f t h e c o u r s e b e a c o n i s r e g u l a t e d
i n s u c h a way t h a t t h e a x i s o f t h e z e r o d i f f e r e n c e i n m o d u l a t i o n
d e p t h c o i n c i d e s e x a c t l y w i t h t h e a x i s o f t h e LTS.
Hence, it i s
necessary t h a t t h e d i f f e r e n c e i n modulation depths within l i m i t s o f
3-4O o f t h e e q u a l - s i g n a l a x i s i n c r e a s e l i n e a r l y w i t h t h e l a t e r a l
deviation.
W i t h f u r t h e r d e v i a t i o n f r o m t h e LTS a x i s ( w i t h i n l i m i t s
up t o l o o ) , t h e d i f f e r e n c e i n m o d u l a t i o n d e p t h s must a l s o i n c r e a s e ,
b u t n o t i n p r o p o r t i o n t o t h e l a t e r a l d e v i a t i o n ; it can m a i n t a i n i t s
v a l u e or d e c r e a s e , b u t w i t h o u t c h a n g i n g s i g n i n t h e e n t i r e h e m i s p h e r e
t o t h e l e f t or r i g h t o f t h e r a d i o - s i g n a l a x i s .
The d i s t a n c e f o r p o s s i b l e r e c e p t i o n o f t h e b e a c o n s i g n a l s i n
t h e s e c t o r l o o from t h e e q u a l - s i g n a l a x i s i n t h e working l o b e of
t h e z o n e m u s t b e w i t h i n t h e l i m i t s o f 45 t o 70 km.
T h e g l i d e beacon i s a l s o a t r a n s m i t t i n g d e v i c e , o p e r a t i n g i n
t h e m e t e r wavelength, b u t a t a frequency d i f f e r e n t from t h a t o f t h e
course beacon.
The a n t e n n a s y s t e m o f t h e g l i d e b e a c o n c o n s i s t s o f o n l y t w o
The
a n t e n n a s ( a n u p p e r a n d a l o w e r ) , m o u n t e d o n a common m a s t .
u p p e r a n t e n n a i s d o u b l e , as s h o w n i n F i g . 4 . 1 6 , a .
Both t h e u p p e r a n d l o w e r a n t e n n a s r e c e i v e an a m p l i t u d e - m o d u l a ­
t e d frequency, but with d i f f e r e n t modulation frequencies ( f o r
example, 90 and 150 Hz).
Each o f t h e a n t e n n a s , t o g e t h e r w i t h t h e g r o u n d forms a n i n d e ­
p e n d e n t w o r k i n g l o b e w i t h i t s own m o d u l a t i o n f r e q u e n c y ( F i g . 4 . 1 6 , b ) .
The p o i n t s o f i n t e r s e c t i o n o f t h e w o r k i n g l o b e s i n t h e v e r t i c a l
p l a n e a l s o f o r m a r a d i o - s i g n a l a x i s AB w i t h a z e r o d i f f e r e n c e i n
t h e modulation depth.
S i n c e t h e c h a r a c t e r i s t i c of t h e a n t e n n a d i r e c t i o n a l i t y i n t h e
/379
horizontal plane is r a t h e r broad, the surface with a zero difference
i n m o d u l a t i o n d e p t h i s c o n i c a l , w i t h AB as t h e g e n e r a t r i x .
There­
f o r e , t h e g l i d e p a t h can b e an i d e a l s t r a i g h t l i n e o n l y i n t h e case
when t h e a n t e n n a s y s t e m o f t h e b e a c o n i s l o c a t e d a t t h e p o i n t w h e r e
t h e a i r c r a f t t o u c h e s down on t h e r u n w a y .

A

A

Fig.

4.16.

F i g . 4.16. G l i d e R a d i o Beacon:
(b) Radiation Characteristic.
Fig.

4.17.

Fig. 4.17
( a ) Diagram o f A n t e n n a L o c a t i o n ;

Hyperbolic Trajectory f o r Glide Plane.
39 9

However,

t h e g l i d e b e a c o n c a n n o t b e l o c a t e d on t h e LTS a x i s

or e v e n i n t h e i m m e d i a t e v i c i n i t y o f t h e LTS, s i n c e i t w o u l d c o n ­
T h e r e f o r e , t h e i n t e r s e c t i o n o f t h e cone
s t i t u t e a f l i g h t hazard.
with t h e zero d i f f e r e n c e i n t h e modulation depth f o r t h e g l i d e
beacon o f t h e e q u a l - s i g n a l plane o f a course beacon g i v e s a hyper­
b o l i c t r a j e c t o r y which does n o t touch t h e ground ( F i g . 4.17).

A s t h e aircraft approaches along t h e landing path toward t h e
t r a v e r s e of t h e g l i d e beacon, t h e g l i d e path begins t o "float"
above t h e g r o u n d , moving upward a f t e r p a s s i n g o v e r t h e b e a c o n .
Since t h e l o c a t i o n and shape o f t h e d i r e c t i o n a l c h a r a c t e r i s t i c
o f t h e t w o a n t e n n a s o f t h e g l i d e b e a c o n d e p e n d s on t h e h e i g h t o f
t h e a n t e n n a s above t h e ground, t h e c h a r a c t e r i s t i c and t h e p o s i t i o n
of t h e l i n e of t h e i r i n t e r s e c t i o n i n t h e v e r t i c a l plane i s regu­
l a t e d by t h e c h a n g e i n t h e h e i g h t o f t h e u p p e r a n d l o w e r a n t e n n a s
above t h e ground.
A s in t h e c a s e o f t h e c o u r s e b e a c o n , t h e i n c r e a s e i n t h e
d i f f e r e n c e o f t h e modulation depth w i t h d e v i a t i o n from t h e g l i d e
s u r f a c e u p w a r d or d o w n w a r d m u s t b e l i n e a r w i t h t h i s d e v i a t i o n .
However, t h e c u r v a t u r e o f t h e c u r v e o f t h e c h a n g e i n t h e d i f f e r e n c e
i n m o d u l a t i o n d e p t h w i l l n o t b e s y m m e t r i c i n t h i s c a s e , as i t i s
f o r t h e course beacon.
A s t e e p e r curve f o r t h e change i n t h e d i f ­
f e r e n c e o f modulation depth i s found above t h e g l i d e s u r f a c e , and
a l e s s s t e e p c u r v e i s f o u n d below t h e s u r f a c e .
+8O
The o p e r a t i n g r a n g e f o r a g l i d e b e a c o n i n a s e c t o r d f ­
t h e LTS a x i s m u s t b e a t l e a s t 1 8 t o 2 5 km.

from

A i r c r a f t - M o u n t e d E q u i p m e n t for t h e C o u r s e - G l i d e
Landing S y s t e m
T h e f o l l o w i n g u n i t s m a k e up t h e a i r c r a f t - m o u n t e d e q u i p m e r l t
t h e course-glide landing system:
( a ) Antenna and r e c e i v e r f o r course-beacon s i g n a l s .
( b ) Antenna and r e c e i v e r f o r glide-beacon s i g n a l s .
( c > Control panel.
( d ) L a n d i n g - s y s t e m a p p a r a t u s (LSA)

/380

for

The r e c e i v e r s o f s i g n a l s f r o m t h e c o u r s e a n d g l i d e b e a c o n s
c o n t a i n e s s e n t i a l l y t h e same e l e m e n t s , w i t h t h e e x c e p t i o n o f t h e
AAC ( a u t o m a t i c a m p l i f i c a t i o n c o n t r o l ) , w h i c h i s n o t shown i n t h e
figure

.

Fig.
400

4.18.

Diagram o f A i r c r a f t - M o u n t e d

G l i d e Radio Beacon.

The g l i d e - b e a c o n r e c e i v e r u s e s t h e c i r c u i t f o r t h e r e i n f o r c e d
AAC.
The l a t t e r i s n o t e m p l o y e d i n c o u r s e - b e a c o n r e c e i v e r s , s i n c e
i t w o u l d a d d t h e m i c r o p h o n e commands r e l a y e d v i a t h i s b e a c o n t o t h e
a i r c r a f t when t h e c o m m u n i c a t i o n s r e c e i v e r s a r e o u t o f o r d e r .
The s i g n a l s f r o m t h e c o u r s e a n d g l i d e b e a c o n s a r e p i c k e d up
by t h e a n t e n n a s a n d a m p l i f i e d b y t h e H F A .
The s e l e c t i o n o f t h e
f r e q u e n c y c h a n n e l i s made o n t h e b a s i s o f t h e f i r s t i n t e r m e d i a t e
f r e q u e n c y by q u a r t z r e t u r n i n g o f t h e h e t e r o d y n e fyom t h e c o n t r o l
panel.
The s i g n a l s a r e t h e n a m p l i f i e d b y t h e I F A a n d LFA c h a n n e l s ,
s o t h a t t h e s i g n a l s p a s s t h r o u g h 9 0 Hz a n d 1 5 0 Hz f i l t e r s t o t h e
r e c t i f i e r s , t h e n t o t h e emergency b l i n k e r , and f i n a l l y t o t h e
r e c e i v e r ground.
The i n d i c a t o r for t h e c o u r s e or g l i d e z o n e i s
c o n n e c t e d i n a b r i d g e c i r c u i t between t h e r e c t i f i e r s f o r t h e 90
a n d 1 5 0 Hz s i g n a l s .
I f t h e s i g n a l s d o n o t r e a c h t h e r e c e i v e r or t h e r e i s s o m e
m a l f u n c t i o n i n t h e r e c e i v e r b l o c k s somewhere a h e a d o f t h e 9 0 a n d
1 5 0 Hz f i l t e r s , t h e r e a d i n g s o f t h e L S A i n d i c a t o r on t h a t c h a n n e l
w i l l b e z e r o ; i f t h e e q u i p m e n t i s o p e r a t i n g p r o p e r l y , i t means
t h a t the a i r c r a f t i s l o c a t e d p r e c i s e l y i n t h e corresponding zone.
T h e r e f o r e , t h e LSA s y s t e m i n c l u d e s t h e emergency b l i n k e r s .
When
n o c u r r e n t i s f l o w i n g i n t h e 9 0 a n d 1 5 0 Hz r e c t i f i e r s , t h e c u r r e n t
t h r o u g h t h e emergency b l i n k e r w i n d i n g s w i l l n o t f l o w , a n d a s i g n a l
i n d i c a t i n g t h a t t h e a p p a r a t u s i s malfunctioning w i l l be d i s p l a y e d
o n ihe i n d i c a t o r .

/381
The d e s i g n c i r c u i t f o r t h e r e c e i v e r s o f t h e s i g n a l s f r o m t h e
g l i d e a n d c o u r s e b e a c o n s i n c l u d e s p o t e n t i o m e t e r s for e l e c t r l c a l
Each o f t h e r e c t i f i e r s r e c e i v e s
b a l a n c e o f t h e LSA i n d i c a t o r s .
s i g n a l s w h i c h d o n o t p a s s t h r o u g h t h e 9 0 a n d 1 5 0 Hz f i l t e r s .
The i n ­
dicator pointer should then point t o zero.
If t h e balance of t h e
c u r r e n t s i n t h e r e c t i f i e r i s u p s e t , it causes t h e r e g u l a t i n g poten­
tiometer t o rotate.
The b a l a n c i n g p o t e n t i o m e t e r o f t h e r e c e i v e r for t h e s i g n a l s
f r o m t h e g l i d e b e a c o n i s u s u a l l y m o u n t e d on t h e r e c e i v e r h o u s i n g ,
w h i l e t h e r e c e i v e r f o r s i g n a l s from t h e c o u r s e b e a c o n i s mounted
on t h e c o n t r o l p a n e l .
For smoothing t h e s h o r t - p e r i o d o s c i l l a t i o n s o f t h e c o u r s e and
g l i d e i n d i c a t o r o f t h e LAS, d u e t o l o c a l d i s t u r b a n c e s i n t h e
r a d i o - s i g n a l zone , t h e i n d i c a t o r c i r c u i t c o n t a i n s a s p e c i a l s e a l e d
u n i t damping c a p a c i t o r s i n t h e c i r c u i t f o r t u r n i n g on t h e a p p a r a t u s .

L o c a e i o n and P a r a m e t e r s f o r R e g u l a t i n g t h e

E q u i p m e n t f o r t h e C o u r s e - G l i d e LandZng S y s t e m

The r a d i o b e a c o n f o r t h e c o u r s e z o n e o f t h e c o u r s e - g l i d e
s y s t e m f o r l a n d i n g an a i r c r a f t i s mounted a t a d i s t a n c e o f 600 t o
1000 m f r o m t h e e n d o f t h e r u n w a y , a l o n g a n e x t e n s i o n o f t h e a x i s
o f t h e LTS.
401


.
I

...
11
.1

I II

I, .I I 1 1 1 111.1.1.1.

The b e a c o n f o r t h e g l i d e zo n e i s mo u n t ed t o t h e s i d e of t h e
LTS ( a s a r u l e , t o t h e l e f t o f t h e l a n d i n g p a t h ) , a t a d i s t a n c e o f
glide beacon

-mnu

c o u r s e beacon

s h o r t r a n g e imarke:r
Fig.

4.19.

Diagram o f L o c a t i o n o f Ground-Based Equipment f o r C o u r s e Glide System.

1 5 0 t o 200 m f r o m i t s a x i s a n d 2 5 0 - 2 7 5

m from t h e e n d o f t h e runway.

The a x i s o f t h e z o n e o f t h e c o u r s e b e a c o n c o i n c i d e s w i t h t h e
LTS a x i s .
A c o n t r o l p o i n t is chosen f o r measuring t h e parameters
f o r r e g u l a t i n g t h e s y s t e m on t h e LTS a x i s .
T h e c o n t r o l p o i n t i s s e l e c t e d as a p o i n t w h e r e t h e a n t e n n a
r e c e i v i n g s i g n a l s from t h e g l i d e beacon aboard t h e a i r c r a f t w i l l be
l o c a t e d a t t h e moment when t h e a i r c r a f t t o u c h e s down on t h e r u n w a y .
I t i s c o n s i d e r e d t h a t t h i s p o i n t i s l o c a t e d a t an a l t i t u d e o f 6 m
a b o v e t h e s u r f a c e o f t h e LTS, a n d i s p l o t t e d f r o m t h e l o c a t i o n o f
t h e g l i d e b e a c o n , 7 5 m t o w a r d t h e e n d o f t h e LTS ( i . e . , t h e d i s ­
t a n c e from t h e e n d o f t h e runway t o t h e c o n t r o l p o i n t i s 1 8 0 t o
2 0 0 m).
The s l o p e a n g l e f o r t h e g l i d e p a t h i s c a l c u l a t e d f r o m a t h e o r e t i c a l p l a n e l o c a t e d 6 m a b o v e t h e s u r f a c e o f t h e LTS.
The v e r ­
t e x o f t h e s l o p e a n g l e o f t h e g l i d e p a t h i s the c o n t r o l p o i n t ( C P ) .
The w i d t h o f t h e z o n e o f t h e c o u r s e a n d g l i d e b e a c o n s i s r e c k ­
oned
from t h e a n g l e s o f d e v i a t i o n from t h e g i v e n d e s c e n t t r a j e c ­
t o r y , c a l c u l a t e d r e s p e c t i v e l y from t h e p o i n t where t h e c o u r s e
beacon i s l o c a t e d and from t h e c o n t r o l p o i n t , w i t h i n t h e l i m i t s o f
which t h e s t r i p s o f t h e landing-system a p p a r a t u s d e v i a t e from t h e
z e r o p o s i t i o n t o t h e l i m i t s of t h e scale.
O b v i o u s l y , t h e a n g l e o f d e v i a t i o n o f t h e LSA s t r i p d e p e n d s o n
t h e d i f f e r e n c e i n t h e m o d u l a t i o n d e p t h s i n t h e b e a c o n z o n e s , as
w e l l as on t h e s e n s i t i v i t y o f t h e r e c e i v e r a b o a r d t h e a i r c r a f t .
Therefore, t h e a n g u l a r width o f t h e zones o f t h e course and g l i d e
b e a c o n s i s r e g u l a t e d by t h e s e n s i t i v i t y o f t h e r e c e i v e r s mounted
a b o a r d t h e a i r c r a f t , which a r e used as s t a n d a r d s .
The s t a n d a r d s f o r t h e w i d t h o f t h e c o u r s e - b e a c o n
s e t as f o l l o w s :

zone a r e

( a > The a n g u l a r w i d t h o f h a l f t h e z o n e mu s t b e l o c a t e d w i t h i n
2 t o 3 O o f t h e LTS a x i s .

402

/382

( b ) The l i n e a r w i d t h o f h a l f t h e z o n e a t a d i s t a n c e o f 1 3 5 0
m from t h e c o n t r o l p o i n t (1150 m t o t h e end of t h e runway) must
b e e q u a l t o 150 m.
An e x p a n s i o n o f t h e z o n e f r o m t h e n o m i n a l v a l u e
t o 45 m a n d a n a r r o w i n g t o 30 m i s c o n s i d e r e d p e r m i s s i b l e .
The h o r i z o n t a l s c a l e o f t h e LSA ( s e e F i g . 4 . 1 3 ) f r o m t h e c e n t e r
t o the scale s t o p has 6 divisions.
The f i r s t d i v i s i o n i s t h e w h i t e
c i r c l e on t h e s i l h o u e t t e o f t h e a i r c r a f t , t h e s e c o n d i s t h e e n d o f
t h e v a n e , t h e t h i r d , f o u r t h , a n d f i f t h a r e p o i n t s on t h e h o r i z o n t a l
a x i s o f t h e s c a l e , while t h e s i x t h is t h e scale s t o p .
The v e r t i c a l s c a l e a l s o h a s s i x d i v i s i o n s , o f w h i c h t h e s e c o n d
d i v i s i o n h e r e i s t h e f i r s t p o i n t on t h e v e r t i c a l a x i s o f t h e
apparatus.
E a c h d i v i s i o n o f t h e h o r i z o n t a l s c a l e o f t h e LSA c o r r e s p o n d s t o
a d e v i a t i o n o f t h e a i r c r a f t f r o m t h e LSA a x i s ( r e l a t i v e t o t h e p o i n t
w h e r e t h e c o u r s e b e a c o n i s l o c a t e d ) w i t h i n l i m i t s o f 2 0 t o 3 0 ' or
2 5 m ( t 7 , - 5 m) f r o m t h e LTS a x i s a t a d i s t a n c e o f 1 3 5 0 m f r o m t h e
control point.
The a n g l e w i d t h o f t h e z o n e o f t h e g l i d e b e a c o n i s l i n k e d b
t h e s l o p e a n g l e o f t h e g l i d e p a t h , which i s d e t e r m i n e d by t h e con­
d i t i o n s o f t h e f o r m a t i o n of t h e z o n e .
The w i d t h o f t h e zone b e ­
n e a t h t h e g l i d e p a t h i s t h e n somewhat g r e a t e r t h a n above t h e g l i d e
path.
The s t a n d a r d s f o r r e g u l a t i n g t h e g l i d e z o n e a r e t h e f o l l o w i n g :
( a ) The p o s i t i o n o f t h e u p p e r l i m i t a t a n a n g l e t o t h e a x i s
o f t h e zone w i t h i n t h e l i m i t s from 0 . 1 9 t o 0 . 2 1 8 , ? . e . , a p p r o x i ­
mately 1/5 of t h e slope angle f o r t h e g l i d e path.
( b ) The l o c a t i o n o f t h e l o w e r l i m i t a t a n a n g l e t o t h e a x i s
o f t h e zone w i t h i n l i m i t s from 0 . 2 9 t o 0 . 3 1 8 (somewhat l e s s t h a n
1 / 3 of the slope angle f o r the glide path).
A c c o r d i n g l y , one d i v i s i o n o f t h e v e r t i c a l s c a l e o f t h e LSA i n
t h e upper p a r t i s e q u a l t o about 0.030 8 , while i n t h e lower p a r t
it i s about 0.05 8 , where 0 is t h e s l o p e a n g l e o f t h e g l i d e p a t h .

L a n d i n g an A i r c r a f t w i t h t h e C o u r s e - G Z i d e S y s t e m

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S e t t i n g up t h e m a n e u v e r f o r a n a i r c r a f t a p p r o a c h i n g a n a i r p o r t
t o descend with t h e use of t h e course-glide system i s performed
a c c o r d i n g t o t h e s a m e r u l e s as i n t h e s i m p l i f i e d s y s t e m for l a n d i n g
an a i r c r a f t .
The c o m p l e m e n t
l a n d i n g an a i r c r a f t
radio s t a t i o n s with
simplified landing,

o f e q u i p m e n t f o r t h e c o u r s e - g l i d e s y s t e m for
i s u s u a l l y s u p p l e m e n t e d b y o n e or t w o m a s t e r
marker beacons, located i n t h e system f o r
w h i c h i s u s e d f o r s e t t i n g up t h e m a n e u v e r f o r

403

bringing t h e aircraft i n f o r a landing and t o a c e r t a i n degree
r e s e r v e s t h e c o u r s e - g l i d e s y s t e m f o r cases o f m a l f u n c t i o n o f t h e
g r o u n d or a i r b o r n e e q u i p m e n t , as w e l l as d u r i n g t i m e s w h e n e q u i p ­
m e n t i s b e i n g r e p a i r e d or a d j u s t e d .
If i n a d d i t i o n t o t h e c o u r s e - g l i d e and master beacons, t h e air­
p o r t i s equipped w i t h r a d a r f o r o b s e r v i n g t h e a i r c r a f t , t h e maneuver
f o r l a n d i n g i n minimum w e a t h e r c a n b e made a l o n g t h e s h o r t e s t p a t h
f o r each l a n d i n g d i r e c t i o n and t a k e o f f d i r e c t i o n .
By t h e same r u l e s w h i c h g o v e r n t h e s i m p l i f i e d l a n d i n g s y s t e m ,
p r e l i m i n a r y c a l c u l a t i o n s a r e c a r r i e d o u t which e n s u r e a s i m p l e r and
more e x a c t a c t i o n o f t h e crew i n f l i g h t a l o n g a g i v e n d e s c e n t t r a ­
jectory.
A p o r t i o n o f t h e p r e l i m i n a r y c a l c u l a t i o n s , s u c h a s (for e x a m p l e )
t h e d e t e r m i n a t i o n o f t h e moment f o r s t a r t i n g t h e d e s c e n t i n a g l i d e ,
c a n n o t be done i n t h i s case i f w e k e e p i n mind t h e f a c t t h a t t h e
The c a l c u l a t i o n s o f t h e
given glide path is defined i n space.
d r i f t angle o f the aircraft and t h e v e r t i c a l rate of descent along
t h e l a n d i n g p a t h a r e o f somewhat l e s s i m p o r t a n c e i n t h i s c a s e .
When t h e m a n e u v e r f o r m a k i n g a l a n d i n g i s made on command
f r o m t h e g r o u n d , t h e n e e d f o r s u c h c a l c u l a t i o n s as t h e d e t e r m i n a t i o n
o f t h e moment f o r m a k i n g t h e t h i r d t u r n n o l o n g e r e x i s t s .
However,
t h e moment �or b e g i n n i n g t h e f o u r t h t u r n m u s t i n a l l c a s e s b e
d e t e r m i n e d b y t h e c r e w o f t h e a i r c r a f t , w i t h t h e maximum a c c u r a c y
possible.
I n s e t t i n g up t h e m a n e u v e r f o r l a n d i n g , t h e s t r i p s o f t h e L S A
c a n b e l o c a t e d on a n y d i v i s i o n s o f t h e s c a l e a n d n o a t t e n t i o n n e e d
b e p a i d t o t h e i r r e a d i n g s ; h o w e v e r , when a p p r o a c h i n g t h e f o u r t h
t u r n , b o t h s t r i p s m u s t b e l o c a t e d on t h e s c a l e s t o p s .
The s t r i p
f o r t h e c o u r s e z o n e r e s t s o n t h e s t o p on t h e s i d e o p p o s i t e t h e
d i r e c t i o n o f t h e m a n e u v e r , t h e s t r i p for t h e g l i d e z o n e r e s t s o n
The e m e r g e n c y b l i n k e r s m u s t t h e n b e o f f .
the stop at the top.
T h e s t r i p f o r t h e c o u r s e z o n e m u s t move a w a y f r o m t h e s c a l e
stop during the fourth turn.
T h e m o v e m e n t o f t h e s t r i p away f r o m
the stop is called deflection.
When t h e f o u r t h t u r n i s made c o r r e c t l y , d e f l e c t i o n o f t h e s t r i p
f o r t h e c o u r s e z o n e o c c u r s a t t h e moment w h e n t h e t u r n a n g l e i s h e l d
u n t i l the aircraft acquires the calculated landing course (Fig.
For a i r c r a f t w i t h p i s t o n e n g i n e s , t h i s t u r n a i s l e is a b o u t
4.20,a).
45O; f o r a i r c r a f t w i t h t u r b o j e t or t u r b o p r o p e n g i n e s , i t i s a b o u t
/384
30°.
W i t h a r e s i d u a l t u r n a n g l e o f 45O for a i r c r a f t w i t h p i s t o n
engines (30° f o r a i r c r a f t with gas t u r b i n e e n g i n e s ) , i f d e f l e c t i o n
o f t h e c o u r s e - z o n e s t r i p d o e s n o t o c c u r , it means t h a t t h e f o u r t h
t u r n i s b e i n g made w i t h a l e a d .
404

I n t h i s case, it is d e s i r a b l e t o s i g n i f i c a n t l y reduce t h e
b a n k i n g a n g l e d u r i n g t h e t u r n or e v e n t o s t o p t u r n i n g a n d f o l l o w
t h e L T S a x i s a t t h e r e s i d u a l t u r n a n g l e u n t i l t h e LSA s t r i p d e f l e c t s .

F i g . 4 . 2 0 . A c q u i s i t i o n o f t h e L a n d i n g P a t h by a n A i r c r a f t :
P r o p e r T u r n ; ( b ) With Turn Begun L a t e .

( a ) With

When t h e c o u r s e - z o n e s t r i p d e f l e c t s , t h e t u r n m u s t b e c o n t i n u e d
u n t i l the landing course is acquired.
When t h e l a n d i n g c o u r s e
i s a c q u i r e d , t h e c o u r s e - z o n e s t r i p m u s t b e l o c a t e d n e a r t h e zero
marking ( c e n t e r of t h e s c a l e ) .
I n c a s e s when t h e f o u r t h t u r n i s made w i t h a d e l a y ( F i g . 4 . 2 0 , b ) ,
t h e d e f l e c t i o n o f t h e LSA s t r i p t a k e s p l a c e e a r l i e r t h a n 45 or 3 0 °
b e f o r e a c q u i s i t i o n of t h e l a n d i n g c o u r s e .
In t h i s case, t h e t u r n
must l a s t u n t i l t h e l a n d i n g c o u r s e and beyond, a t a l a n d i n g a n g l e
o p p o s i t e t o t h e LTS a x i s , d e p e n d i n g on t h e m a g n i t u d e o f t h e t r a n s i ­
t i o n o f t h e course s t r i p through t h e center of t h e scale.
For example, i f t h e d e s c a l i n g occurs a t t h e very beginning
/385
of t h e fourth t u r n , i t i s necessary t o increase the banking angle
i n t h e t u r n up t o 2 0 ° , a n d t h e a i r c r a f t w i l l c o n t i n u e t o t u r n t o
t h e opposite angle f o r l a n d i n g (20° i n aircraft with p i s t o n engines
a n d 30° f o r a i r c r a f t w i t h g a s t u r b i n e e n g i n e s ) .
With l e s s d e l a y i n t u r n i n g , t h e o p p o s i t e a n g l e f o r a p p r o a c h
can b e w i t h i n t h e l i m i t s o f 5 t o 20°.
With r e v e r s e d e f l e c t i o n of t h e c o u r s e - z o n e s t r i p , t h e a i r c r a f t
makes a reverse turn onto t h e landing course, with a simultaneous
After the aircraft has acquired t h e
f l a t t u r n onto t h e LTS a x i s .

405

LTS a x i s , t h e f l i g h t c o n t i n u e s f o r a t i m e u n t i l d e f l e c t i o n o f t h e
glide-zone s t r i p takes place a t a constant a l t i t u d e .
A t t h e moment w h e n t h e g l i d e - z o n e s t r i p m o v e s a w a y f r o m t h e
u p p e r s t o p , t h e a i r c r a f t s h i f t s t o a d e s c e n t regime w i t h a smooth
a c q u i s i t i o n o f t h e d e s i r e d g l i d e p a t h downward.

D i r e c t i o n a Z P r o p e r t i e s o f t h e Landing S y s t e m A p p a r a t u s
The s e l e c t i o n o f t h e d e s i r e d c o u r s e a n d t h e v e r t i c a l r a t e of
d e s c e n t a r e s o u r c e s o f c o n s i d e r a b l e d i f f i c u l t y f o r t h e crew a n d
require a c e r t a i n degree of training.
However, t h e s e d i f f i c u l t i e s
do n o t a r i s e from p r i n c i p l e s o f p i l o t i n g t h e a i r c r a f t a l o n g t h e
LSA, b u t r a t h e r f r o m t h e n e c e s s i t y o f s i m u l t a n e o u s l y o b s e r v i n g
s e v e r a l d e v i c e s and i n s t r u m e n t s and s e l e c t i n g a f l i g h t regime i n
t h e v e r t i c a l and h o r i z o n t a l planes simultaneously.
N e v e r t h e l e s s , w i t h a p r o p e r r e a c t i o n o f t h e crew t o a c h a n g e
i n t h e p o s i t i o n s o f t h e s t r i p s on t h e LAS, t h e l a n d i n g m a n e u v e r s h o u l d
b e s u c c e s s f u l i n a l l cases and n o t v e r y d i f f i c u l t .
I n p i l o t i n g t h e a i r c r a f t b y t h e LSA,two o f i t s p r i n c i p a l c h a r ­
a c t e r i s t i c s must b e employed:
( 1 ) The i n d i c a t i n g c h a r a c t e r i s t i c , ; . e . ,
the indication of the
position of t h e aircraft r e l a t i v e t o a given descent t r a j e c t o r y .
( 2 ) The command c h a r a c t e r i s t i c , ; . e . ,
the a b i l i t y t o predeter­
s e l e c t i n g t h e f l i g h t regime.
m i n e t h e a c t i o n s o f t h e crew
I n a s m u c h a s t h e f i r s t p r o p e r t y o f t h e LSA i s o b v i o u s ,
examine t h e second.

l e t us

The c o u r s e a n d g l i d e z o n e s a r e r a t h e r n a r r o w i n s p a c e , s u f f i ­
c i e n t l y s o t h a t t h e l i m i t s of t h e s e zones can be c o n s i d e r e d
p a r a l l e l over s h o r t segments o f t h e t r a j e c t o r y .
L e t u s s a y t h a t a n a i r c r a f t a t a g i v e n moment i s l o c a t e d t o
t h e s i d e o f t h e LTS a x i s , a n d t h e g r o u n d s p e e d v e c t o r o f t h e a i r ­
craft does n o t coincide w i t h t h e d i r e c t i o n of t h i s a x i s ( F i g . 4 . 2 1 ) .
O b v i o u s l y , t h e g r o u n d s p e e d v e c t o r o f t h e a i r c r a f t c a n be d i v i d e d
i n t o t w o c o m p o n e n t s : a l o n g i t u d i n a l o n e W, a n d a l a t e r a l o n e W,.
T h e l o n g i t u d i n a l c o m p o n e n t W,
is not involved i n the selection
of t h e c o u r s e t o b e f o l l o w e d .
The p r i n c i p a l r o l e i s p l a y e d b y t h e
l a t e r a l or t r a n s v e r s e c o m p o n e n t ,

wz

*

T h e c o m p o n e n t W, d e t e r m i n e s t h e r a t e o f m o t i o n o f a n L S A s t r i p / 3 8 6
along t h e h o r i z o n t a l scale of t h e apparatus.
With t h e s t r i p f i x e d
a t a n y s c a l e d i v i s i o n , t h e c o m p o n e n t W, i s e q u a l t o z e r o , w h i c h
agrees precisely with t h e selected a i r c r a f t course, i . e . , its path
i s p r a c t i c a l l y p a r a l l e l t o t h e a x i s LTS.

406

The r e g u l a t i o n o f t h e L S A i s s e t s o t h a t t h e c h a n g e i n t h e
c o u r s e o f t h e a i r c r a f t ( 1 . 5 t o 2 0 ) makes t h e m o t i o n o f t h e v e r t i c a l

F i g . 4 . 2 1 . D i v i s i o n o f Ground Speed V e c t o r i n t o L o n g i t u d i n a l and
L a t e r a l Components a l o n g t h e L a n d i n g P a t h .

For e x a m p l e , i n a i r c r a f t w i t h p i s t o n
s t r i p LSA v i s i b l e t o t h e e y e .
e n g i n e s , a change i n t h e c o u r s e by 2 O produces a l a t e r a l s h i f t o f
t h e a i r c r a f t o f 2 m/sec.
T h i s means t h a t t h e most d a n g e r o u s r e g i o n
o f f l i g h t ( 1 2 0 0 t o 1 5 0 0 m t o t h e e n d o f t h e r u n w a y ) , t h e LSA s t r i p
c r o s s e s e a c h s c a l e d i v i s i o n i n 11 t o 1 2 s e c , i . e . , a s u f f i c i e n t l y
noticeable value equal t o h a l f a s c a l e division a f t e r each 5 t o 6
s e c of f l i g h t .
I f a t u r n i s made i n t h e d i r e c t i o n o f t h e m o t i o n
o f t h e LSA s t r i p b y 2 O , i t s m o t i o n c a n b e h a l t e d a t a n y d i v i s i o n
on t h e s c a l e .
On t h i s b a s i s , t h e p r i n c i p l e o f s e l e c t i n g t h e c o u r s e f o r t h e
a i r c r a f t b y t h e LSA m u s t b e t h e f o l l o w i n g :
If t h e v e r t i c a l s t r i p i s l o c a t e d a t a
t h e c e n t e r of t h e instrument (on t h e t h i r d
i s t h e n n e c e s s a r y t h a t t h e r a t e of i t s s h i
instrument be s i g n i f i c a n t .
To d o t h i s , i t
t h e a i r c r a f t i n t h e d i r e c t i o n i n which t h e
to 6O.

s i g n i f i c a n t d i s t a n c e from
or f o u r t h d i v i s i o n ) , i t
f t t o the center of the
is sufficient t o turn
s t r i p i s moving, by 4

A s t h e s t r i p a p p r o a c h e s t h e c e n t e r of t h e a p p a r a t u s , i t s r a t e
of m o t i o n m u s t b e a r r e s t e d b y t u r n i n g t h e a i r c r a f t 1 t o 2 O i n t h e
d i r e c t i o n shown b y t h e a r r o w .
A t t h e moment when t h e s t r i p r e a c h e s
t h e c e n t e r o f t h e i n s t r u m e n t , i t s m o t i o n i s a r r e s t e d by a f i n a l
t u r n o f t h e a i r c r a f t b y 1 t o 2 O , a n d t h e a i r c r a f t w i l l b e s e t on
t h e LTS a x i s , w i t h t h e c o u r s e a l r e a d y s e l e c t e d .

T h i s method r e q u i r e s a v e r y p r e c i s e n i g h t o f t h e a i r c r a f t
along t h e a x i s of t h e course zone, with p e r i o d i c changes i n t h e
course within t h e l i m i t s of 1 t o 2 O .
An a n a l o g o u s m e t h o d i s e m p l o y e d t o s e t t h e v e r t i c a l r a t e o f
descent of t h e a i r c r a f t , w i t h simultaneous a c q u i s i t i o n of t h e
d e s i r e d g l i d e plane and subsequent f l i g h t along i t .
M a i n t e n a n c e of t h e d e s c e n t r e g i m e o f t h e a i r c r a f t a l o n g a
g i v e n t r a j e c t o r y b y t h e r e a d i n g s o f t h e LSA c o n t i n u e s up t o t h e
moment when t h e a i r c r a f t e m e r g e s f r o m t h e c l o u d s a n d m a k e s a

407

t r a n s i t i o n t o v i s u a l f l i g h t , a f t e r which a v i s u a l estimate o f a l t i t u d e i s made a n d t h e a i r c r a f t t o u c h e s down o n t h e r u n w a y .

/387

D i r e c t i o n a l D e v i c e s f o r Landing A i r c r a f t
Determination o f t h e rate of s h i f t o f t h e s t r i p s calls f o r in­
c r e a s e d v i g i l a n c e i n o b s e r v i n g each of them.
In addition, local
i r r e g u l a r i t i e s i n th.e c o u r s e z o n e a n d g l i d e z o n e a t i n d i v i d u a l
p o i n t s d i s t u r b t h e r e g u l a r i t y o f t h e p r o c e s s ; t h i s must b e t a k e n i n t o
c o n s i d e r a t i o n by t h e c r e w and c a r e f u l l y s e p a r a t e d f r o m t h e g e n e r a l l y
e s t ab l i s h e d t e n d e n c y .
A l l o f t h i s r e q u i r e s c o n s i d e r a b l e c a u t h n a n d t r a i n i n g on t h e
p a r t o f t h e crew f o r m a k i n g a d e s c e n t a l o n g a g i v e n t r a j e c t o r y .

R e c e n t l y , s p e c i a l d i r e c t i o n a l d e v i c e s f o r p i l o t i n g an a i r c r a f t
i n t h e c o u r s e and g l i d e zones have begun t o b e employed w i d e l y .
Unlike t h e LSA, t h e d i r e c t i o n a l p r o p e r t i e s o f t h e s e d e v i c e s are
n o t e x p r e s s e d b y t h e d e r i v a t i v e s o f t h e p o s i t i o n s o f t h e s t r i p s on
t h e i n s t r u m e n t w i t h t i m e , b u t d i r e c t l y by t h e p o s i t i o n s o f t h e s e
strips.
The m o s t w i d e l y e m p l o y e d d i r e c t i o n a l d e v i c e s a t t h e p r e s e n t
t i m e a r e t h o s e w h i c h a r e b a s e d on v a r i o u s l a w s o f c o n t r o l , w i t h a n
i n d i c a t i o n which i s l i n k e d t o t h e b a n k i n g o f t h e a i r c r a f t d u r i n g a
c o o r d i n a t e d s t a b l e t u r n , or t o t h e a n g l e o f p i t c h a t a s e t r a t e o f
descent.
In pilotage of the aircraft in the horizontal plane, these
laws r e p r e s e n t a d e f i n i t e l i n k b e t w e e n t h e c o u r s e a n d t h e b a n k i n g
o f t h e a i r c r a f t i n a t u r n , w i t h l a t e r a l d e v i a t i o n from t h e r a d i o s i g n a l p l a n e o f t h e c o u r s e zone a n d t h e f i r s t d e r i v a t i v e of t h i s
deviation with time.

w h e r e Ay i s t h e a n g l e o f a p p r o a c h t o t h e l a n d i n g p a t h , B i s t h e
banking angle of t h e aircraft i n t h e t u r n , Z i s t h e lateral devi­
a t i o n o f t h e a i r c r a f t f r o m t h e z o n e a x i s , V z i s t h e r a t e of l a t e r a l
s h i f t o f t h e a i r c r a f t , and K are t h e c o e f f i c i e n t s f o r t h e c o r r e s ­
ponding parameters.
A s i m i l a r l a w i s employed f o r p i l o t i n g an a i r c r a f t i n t h e
vertical plane:
Kau

+ KyY f K

"Y

v y = 0,

where V i s t h e a n g l e o f p i t c h o f t h e a i r c r a f t , Y i s t h e d e v i a t i o n
o f t h e a i r c r a f t from t h e g l i d e p a t h i n t h e v e r t i c a l p l a n e , and V
Y
i s t h e rate o f v e r t i c a l motion o f t h e a i r c r a f t .
40 8

S i n c e t h e l i n e a r v a l u e s Z and Y and t h e i r f i r s t d e r i v a t i v e s
/388
cannot be measured d i r e c t l y i n p o l a r systems, t h e i r v a l u e s are re­
p l a c e d by a n g l e v a l u e s ( a a n d A O ) a n d t h e i r d e r i v a t i v e s .
The
v a l u e s a a n d A 0 a n d t h e i r d e r i v a t i v e s a a n d A0 a r e m e a s u r e d by t h e
d i f f e r e n c e s i n modulation depths and t h e i r d e r i v a t i v e s i n t h e zones
of t h e course and g l i d e beacons.
O b v i o u s l y , by s e l e c t i n g t h e p r o p e r b a n k i n g a n g l e a n d p i t c h
a n g l e f o r t h e a i r c r a f t , t h e a i r c r a f t can be p o s i t i o n e d s o t h a t b o t h
s t r i p s on t h e d i r e c t i o n a l i n d i c a t o r a r e l o c a t e d on z e r o .
The c o e f f i c i e n t s f o r t h e c o n v e r t e d p o s i t i o n p a r a m e t e r s f o r t h e
aircraft axes are s e l e c t e d s o t h a t whatever deviations t h e aircraft
may m a k e f r o m t h e g i v e n t r a j e c t o r y ( i f t h e i n d i c a t o r s t r i p s r e m a i n
on z e r o ) , t h e a i r c r a f t w i l l s t i l l t r a v e l a l o n g t h e g i v e n l a n d i n g
and g l i d e p a t h w i t h a p r e d e t e r m i n e d t r a j e c t o r y (whose c o u r s e depends
upon t h e c o e f f i c i e n t s s e l e c t e d ) .
T h i s means t h a t t h e l a n d i n g
c o u r s e and v e r t i c a l s p e e d must b e s e l e c t e d s i m u l t a n e o u s l y , s i n c e
they are r e q u i r e d f o r f l y i n g t h e a i r c r a f t along a given t r a j e c t o r y .
Hence, i n s t e a d o f a d j u s t i n g t h e r a t e o f motion o f t h e s t r i p s
i n accordance with t h e i r motion toward t h e c e n t e r o f t h e instrument
as i n a n o r m a l LSA, i n d i r e c t i o n a l i n s t r u m e n t s t h e crew n e e d o n l y
b r i n g t h e i n d i c a t o r s t r i p s t o t h e c e n t e r o f t h e i n s t r u m e n t by c h a n g ­
i n g t h e b a n k i n g a n g l e o f t h e a i r c r a f t as w e l l as i t s a n g l e o f p i t c h ;
t h i s s i g n i f i c a n t l y f a c i l i t a t e s t h e t a s k o f p i l o t i n g an a i r c r a f t .
To f u r t h e r r e d u c e t h e w o r k o f t h e c r e w , d i r e c t i o n a l i n s t r u m e n t s
are u s u a l l y combined w i t h a g y r o h o r i z o n i n d i c a t o r .
In t h i s case,
t h e e n t i r e a t t e n t i o n o f t h e p i l o t i s c o n c e n t r a t e d p r a c t i c a l l y on
t h e r e a d i n g s o f o n l y one i n s t r u m e n t .
However, d i r e c t i o n a l i n s t r u ­
m e n t s b a s e d on t h e r u l e s s t a t e d a b o v e h a v e s o m e i m p o r t a n t s h o r t ­
comings, which t o a c e r t a i n d e g r e e r e d u c e t h e a c c u r a c y o f p i l o t i n g
an a i r c r a f t r e l a t i v e t o p i l o t i n g by t h e i n d i c a t i o n s o f an L S A .
'The p r o p e r s e l e c t i o n o f c o e f f i c i e n t s f o r m a k i n g a t u r n a n d
t h e a n g l e o f p i t c h o f t h e a i r c r a f t c a n b e made o n l y a t a c e r t a i n
d i s t a n c e o f t h e a i r c r a f t from t h e ground beacons.
During measure­
ment of t h e d i s t a n c e , t h e l i n e a r w i d t h o f t h e c o u r s e and g l i d e z o n e s
c h a n g e s , t h u s l e a d i n g t o a f a i l u r e o f t h e s y s t e m r e g u l a t i o n param­
e t e r s t o a g r e e w i t h t h e dynamic f l i g h t t r a j e c t o r y o f t h e a i r c r a f t .
T h i s s h o r t c o m i n g can b e c o m p l e t e l y overcome i f t h e s y s t e m i s r e g u ­
l a t e d n o t o n l y by t h e a n g u l a r d e v i a t i o n o f t h e a i r c r a f t from t h e
r a d i o - s i g n a l a x i s , b u t by c a l c u l a t i o n o f t h e d i s t a n c e r e m a i n i n g t o
the ground r a d i o beacons:

Z = L,tga;
Y = L tg AO,
g
where Lc and Lg a r e t h e d i s t a n c e s
beacons.

t o t h e c0urs.e a n d g l i d e r a d i o

409

Control can t h e n be e f f e c t e d i n a r e c t a n g u l a r system of coord i n a t e s , and t h e r e f o r e with constant agreement of t h e r e g u l a t i o n
o f t h e s y s t e m w i t h t h e dynamic t r a j e c t o r y o f t h e a i r c r a f t ' s f l i g h t .

/389

In p o l a r coordinates, shortcomings i n t h e operation o f t h e
d i r e c t i o n a l s y s t e m c a n b e e l i m i n a t e d by a s p e c i a l s e l e c t i o n o f
converted s i g n a l coefficients (not proportional t o the values of
t h e s i g n a l s i n various s e c t i o n s of t h e t r a j e c t o r y ) i n accordance
with t h e tactical c h a r a c t e r i s t i c s o f aircraft o f various types.
It should a l s o be mentioned t h a t i n d i r e c t i o n a l systems, t h e
i n d i c a t i o n of t h e p o s i t i o n o f t h e a i r c r a f t r e l a t i v e t o a g i v e n
T h i s m e a n s t h a t on b o a r d t h e a i r c r a f t ,
descent trajectory is l o s t .
i n a d d i t i o n t o t h e d i r e c t i o n a l d e v i c e s , t h e r e must s t i l l be a
c o n v e n t i o n a l LSA i n d i c a t o r , which i s u s e d as a s t a n d a r d t o c h e c k t h e
accuracy of p i l o t a g e according t u t h e d i r e c t i o n a l i n d i c a t o r .
So-called p a r a v i s u a l d i r e c t i o n a l instruments are a l s o beginning
t o b e u s e d nowadays ; i n p r i n c i p l e , t h e y r e p r e s e n t a r e i n f o r c e m e n t
o f t h e d i r e c t i o n a l p r o p e r t i e s o f t h e LSA.
I n t h i s c a s e , t h e u s u a l LSA i n d i c a t o r s a r e l o c a t e d i n t h e c e n ­
t e r of t h e f i e l d o f t h e p i l o t ' s vision, while a t t h e periphery o f
h i s v i s i o n t h e r e are i m i t a t o r s o f t h e motion of t h e s t r i p s according
t o t h e f i r s t d e r i v a t i v e s c1 a n d A B , w h i c h l i n k t h e i n d i c a t i o n s h o w n
with t h e l o n g i t u d i n a l and l a t e r a l r o l l i n g of t h e a i r c r a f t according
t o the l a w s of t h e design of directional instruments.
Radar Landing Systems

From t h e t a c t i c a l s t a n d p o i n t , r a d a r l a n d i n g s y s t e m s h a v e n o
s p e c i a l a d v a n t a g e s o v e r c o u r s e - g l i d e s y s t e m s ; on t h e c o n t r a r y ,
t h e i r u s e i s l e s s c o n v e n i e n t , s i n c e t h e r e a r e n o i n s t r u m e n t s a­
board t h e a i r c r a f t f o r i n d i c a t i n g t h e p o s i t i o n of t h e a i r c r a f t and
n o commands f o r p i l o t i n g i t r e l a t i v e t o a g i v e n d e s c e n t t r a j e c t o r y .
The a c c u r a c y w i t h w h i c h a n a i r c r a f t c a n b e l a n d e d by means o f
r a d a r landing systems is roughly equal t o t h a t o f landing it with
course-glide systems.
Nevertheless , r a d a r landing systems are
widely employed, a l o n g with course-glide systems.
The p r i m a r y r e a s o n why r a d a r l a n d i n g s y s t e m s h a v e b e e n e m p l o y e d
s o w i d e l y i s t h e n e e d f o r a c o n s t a n t c h e c k on a i r c r a f t making t h e i r
l a n d i n g a p p r o a c h e s b y c o u r s e - g l i d e s y s t e m s , for t h e p u r p o s e s o f
p o i n t i n g o u t e r r o r s made b y t h e c r e w a n d p r e v e n t i n g t h e v e r y d a n g e r ­
ous c o n s e q u e n c e s o f e r r o r .
The s e c o n d r e a s o n i s t h e n e e d t o g i v e t h e c r e w a s s i s t a n c e i n
l a n d i n g t h e a i r c r a f t i f t h e y s h o u l d r e q u e s t i t , i f for s o m e r e a s o n
t h e course-glide system cannot be used.
T h e same r e a s o n i n g a p p l i e s / 3 9 0
i n r e t a i n i n g t h e c o u r s e - g l i d e s y s t e m i n c a s e t h e g r o u n d c o n t r o l is
not functioning.

410

- - - . . - . - = l l l - ~ - ~ - 1 1 1 1 1 1 1

I I111 I 111-=1
.1=11.1111

11111111 11111 111111I111 I I I 11111111 I I I II II I 111111111111I I

I l l

Ill

The r a d a r l a n d i n g s y s t e m c o n s i s t s o f a c o m p l e x o f d e v i c e s
f o r observing t h e f l i g h t o f approaching aircraft ( r a d a r screen,
USW r a d i o d i s t a n c e - f i n d e r ) a n d t h o s e a c t u a l l y making a l a n d i n g
(landing radar).
I n a d d i t i o n , t h e system i n c l u d e s communication
a p p a r a t u s f o r t r a n s m i t t i n g i n f o r m a t i o n a n d n e c e s s a r y commands t o
the aircraft.
The l a n d i n g r a d a r i s t h e h e a r t o f t h e r a d a r l a n d i n g s y s t e m ,
s o w e s h a l l pause t o examine t h e p r i n c i p l e s o f i t s o p e r a t i o n .
Unlike ground r a d a r i n st a l l a t i o n s with c i r c u l a r s c r e e n s , t h e
landing r a d a r s have a s e c t o r s c r e e n , i . e . , t h e r e i s n o r o t a t i n g
d i r e c t i o n a l c h a r a c t e r i s t i c of t h e antenna, b u t one which s c a n s
(oscillates) in a certain sector.
Accordingly, t h e scanning l i n e
on t h e r a d a r s c r e e n a l s o o s c i l l a t e s .
The l a n d i n g r a d a r h a s t w o a n t e n n a s :
( a ) The c o u r s e - s e c t o r a n t e n n a , w i t h a w i d e c h a r a c t e r i s t i c i n
t h e v e r t i c a l p l a n e a n d a n a r r o w one i n t h e h o r i z o n t a l .
( b ) The g l i d e - s e c t o r a n t e n n a , w i t h a w i d e c h a r a c t e r i s t i c i n
t h e h o r i z o n t a l p l a n e and a n a r r o w one i n t h e v e r t i c a l ; t h e s c a n n i n g
of the c h a r a c t e r i s t i c of t h i s antenna takes place i n t h e v e r t i c a l
plane.
The s c a n n i n g o f t h e d i r e c t i o n a l c h a r a c t e r i s t i c s o f t h e l a n d i n g r a d a r a n t e n n a c a n b e a c h i e v e d e i t h e r by m e c h a n i c a l o s c i l l a t i o n o f
t h e a n t e n n a r e f l e c t o r or b y s p e c i a l d e v i c e s w h i c h c h a n g e t h e p h a s e
o f t h e wave a l o n g t h e c h o r d o f t h e a n t e n n a r e f l e c t o r , t h u s c a u s i n g
t h e p l a n e o f t h e wave f r o n t t o o s c i l l a t e ( s o t h a t a l l t h e wavepropagation c h a r a c t e r i s t i c s a l s o o s c i l l a t e ) .
The s c a n n i n g s e c t o r s o f t h e d i r e c t i o n a l c h a r a c t e r i s t i c s o f t h e
a n t e n n a a r e made n a r r o w ;
( a > For a c o u r s e s e c t o r o f 15O: t o e i t h e r s i d e o f t h e LTS a x i s .
( b ) For a g l i d e s e c t o r , 9 O w i d e : + 8 O u p w a r d a n d - l o d o w n w a r d
from t h e p l a n e o f t h e h o r i z o n .
A peculiar feature of the landing radar i s t h e s p e c i a l design
o f t h e s c a n n i n g on t h e c o u r s e a n d g l i d e s c r e e n s .
Thus, i n s t e a d of
t h e c i r c u l a r d i s t a n c e m a r k s on c o n v e n t i o n a l c i r c u l a r r a d a r s c r e e n s ,
t h e d i s t a n c e m a r k s on l a n d i n g r a d a r s a r e s t r a i g h t l i n e s , ; . e . , t h e
d e l a y i n t h e d i s t a n c e m a r k s i s made p r o p o r t i o n a l n o t t o R , b u t t o
R / c o s a , w h e r e c1 i s t h e a n g l e o f d e v i a t i o n o f t h e s c a n n i n g l i n e f r o m
Hence, a r e c t a n g u l a r system of
t h e a x i s of t h e scanning s e c t o r .
c o o r d i n a t e s i s f o r m e d on t h e s c r e e n f r o m t h e p o l a r s y s t e m of
coordinates f o r the aircraft.
In addition, t h e r a d a r screen has a t r a n s v e r s e scale t h r e e
t i m e s l a r g e r t h a n t h e d i s t a n c e scale f o r t h e course s e c t o r and
five t i m e s larger than that for the glide sector.
T h i s means t h a t
t h e r e i s a corresponding r e l a t i o n s h i p between t h e i n c r e a s e i n t h e
scale i n d i c a t i n g the p o s i t i o n of t h e aircraft r e l a t i v e t o t h e given
t r a j e c t o r y f o r t h e same s c r e e n r a d i u s .

411


A g e n e r a l view o f t h e s c r e e n s o f t h e c o u r s e and g l i d e s e c t o r s
i s shown i n F i g . 4 . 2 2 .

g l i d e path s e c t o r

course sector

Fig.
Fig. 4.22.
Sector.
F i g . 4.23.

/391

4.22

Landing-Radar Screen:

Fig.

4.23

( a ) Glide S e c t o r ; ( b ) Course

P a t t e r n on C o u r s e S c r e e n o f L a n d i n g R a d a r .

The l a n d i n g r a d a r i s m o u n t e d on t h e t r a v e r s e o f t h e c e n t e r o f
t h e LTS, a t a d i s t a n c e o f 1 0 0 t o 1 5 0 m t o t h e s i d e , s o t h a t t h e
c o n d i t i o n s for u s i n g i t when l a n d i n g a t e i t h e r e n d o f t h e r u n w a y
w i l l b e t h e same.
I n t h e immediate v i c i n i t y of t h e l a n d i n g r a d a r , t h e r e is a
c i r c u l a r - s c a n r a d a r f o r o b s e r v i n g a i r c r a f t n e a r and f a r from t h e
airport

.

I n s e t t i n g up t h e l a n d i n g m a n e u v e r , i m m e d i a t e l y b e f o r e c o m p l e t i n g
t h e f o u r t h t u r n , t h e short-range r a d a r approach system i s used, a l s o
I t s s c r e e n can b e u s e d t o show
c a l l e d c o n t r o l - t o w e r r a d a r (CTR).
l a n d i n g m a n e u v e r s for a i r c r a f t a p p r o a c h i n g f r o m a l l d i r e c t i o n s .
A l l t u r n s o f t h e a i r c r a f t a r e made on command f r o m t h e f l i g h t
s u p e r v i s o r , as a r e t h e c o u r s e c o r r e c t i o n s on t h e s t r a i g h t - l i n e
segments between t h e t u r n s , i f t h e given f l i g h t d i r e c t i o n s a r e n o t
maintained s u f f i c i e n t l y accurately.
O b s e r v a t i o n o f an a i r c r a f t w i t h t h e l a n d i n g r a d a r b e g i n s
w h i l e i t i s making the f o u r t h t u r n , u s i n g o n l y t h e c o u r s e s e c t o r
screen

.

I n o r d e r t o e n s u r e t h a t t h e a i r c r a f t l a n d s p r e c i s e l y on a
g i v e n d e s c e n t t r a j e c t o r y , t h e r e q u i r e d p a t t e r n i s s u p e r p o s e d on t h e
landing radar screen.
T h i s p a t t e r n on t h e s c r e e n s e r v e s t h r e e
purposes:
(1) T o show t h e g i v e n t r a j e c t o r y f o r t h e a i r c r a f t ' s d e s c e n t .

412

( 2 ) To p r o v i d e a u x i l i a r y l i n e s f o r g i v i n g commands t o t h e
c r e w of the aircraft.
( 3 ) To s h o w t h e b o u n d a r y l i n e s for s a f e f l i g h t a l t i t u d e a n d
t h e p e r m i s s i b l e zones f o r landing t h e aircraft.

/392

S i n c e t h e l a n d i n g r a d a r i s u s u a l l y u s e d for t w o d i r e c t i o n s o f
l a n d i n g a n d t a k e o f f , a n d c a n e v e n b e u s e d f o r t h r e e or f o u r i s i f
o t h e r runways i n t e r s e c t , t h e p a t t e r n s f o r t h e s c r e e n s are p r i n t e d
on r e m o v a b l e c e l l u l o i d s h e e t s w h i c h c a n b e c h a n g e d when s h i f t i n g
t h e l a n d i n g r a d a r t o a new l a n d i n g d i r e c t i o n .
The s c r e e n f o r t h e c o u r s e s e c t o r o f a l a n d i n g r a d a r ( F i g .
u s u a l l y shows t h e f o l l o w i n g :

4.23)

1. T h e g i v e n l a n d i n g p a t h ( a x i s o f L T S ) , b e g i n n i n g a t t h e e n d
t h e runway and e x t e n d i n g t o t h e l i m i t o f t h e s c r e e n .
The f o l l o w ­
i n g p o i n t s a r e m a r k e d on t h i s l i n e : t h e b e g i n n i n g o f d e s c e n t a l o n g
a s e t g l i d e p a t h a n d t h e l o c a t i o n s o f t h e LRMS a n d SRMS l a n d i n g
T h e SRMS
systems w i t h i n t h e r a n g e o f t h e master r a d i o s t a t i o n s .
i s u s u a l l y f i t t e d w i t h a c o r n e r r e f l e c t o r , which p r o d u c e s a b r i g h t
s p o t on t h e s c r e e n a n d i s u s e d i n s e t t i n g t h e r a d a r f o r t h e g i v e n
l a n d i n g d i r e c t i o n a n d as a c o n t r o l t o c h e c k t h e a c c u r a c y o f t h e
s e t t i n g of t h e r a d a r a f t e r i t i s turned around.
of

2 . T h e l i n e s d e l i m i t i n g t h e zone o f
T h e s e l i n e s a r e d e f i n e d on t h e b a s i s o f
c r a f t , b e i n g on a c o u r s e c l o s e t o t h a t f
up w i t h t h e LTS a x i s p r i o r t o t h e s t a r t
o n l y i n t h e c a s e when
x > 2R sin UT’

where

YP = arccos

possible a i r c r a f t landings.
t h e assumption t h a t t h e a i r ­
o r landing, can be l i n e d
of the landing distance

( -2
3
1

where X i s t h e r e m a i n i n g d i s t a n c e t o t h e s t a r t o f t h e l a n d i n g d i s ­
t a n c e , Z i s t h e l a t e r a l d e v i a t i o n from t h e l a n d i n g p a t h , and R i s
t h e t u r n i n g r a d i u s w i t h a b a n k i n g a n g l e o f ioo.
The o r d e r i n w h i c h t h e s e l i n e s a r e p l o t t e d i s t h e f o l l o w j n g :
( a > S e v e r a l p o i n t s o f d e v i a t i o n o f t h e a i r c r a f t from t h e
l a n d i n g p a t h a r e g i v e n ( e . g . , 30, 1 0 0 , 2 0 0 , 500, 1000, 2000, and
4 0 0 0 m) a n d t h e r e q u i r e d t u r n a n g l e s t o c o r r e c t t h e s e d e v i a t i o n s
are determined:
Z

COSUT= 1 - -;
2R

( b ) T h e r e q u i r e d c o u r s e f o r l i n i n g up t h e a i r c r a f t w i t h t h e
LTS a x i s i s d e t e r m i n e d :
X = 2R sinUT.

413

/393

To t h i s p a t h , w e a d d t h e d i s t a n c e t r a v e l e d b y t h e a i r c r a f t
( i n 4 sec f o r piston-engine aircraft, 7 sec f o r gas turbine a i r c r a f t ) ,
r e q u i r e d f o r r e c e i v i n g commands a n d c a r r y i n g o u t t h e m a n e u v e r t o
l i n e up t h e a i r c r a f t w i t h t h e r u n w a y .
( c ) The p a t h o b t a i n e d f o r t h e a i r c r a f t i s m e a s u r e d f r o m t h e s t a r t ­
i n g p o i n t o f t h e l a n d i n g d i s t a n c e ( a s r u l e , f r o m t h e SRMS), a n d w e
o b t a i n t h e minimum a t t a i n a b l e d i s t a n c e s o f t h e s e l e c t e d p o i n t s f o r
the lateral deviations of the aircraft.
By c o n n e c t i n g t h e p o i n t s b y a s m o o t h c u r v e , w e o b t a i n t h e l i m i t
o f t h e p o s s i b l e l a n d i n g zone o f an a i r c r a f t , w i t h p e r m i s s i b l e l a t e r a l
deviations.
I n t h e c o u r s e o f l a n d i n g a n a i r c r a f t , i f i t s h o w s up o u t s i d e
t h e i n d i c a t e d l i m i t s , t h e l a n d i n g c a n n o t b e a l l o w e d a n d t h e command
i s g i v e n t o make a x o t h e r p a s s a t t h e f i e l d .
The b o u n d a r y l i n e s a r e u s u a l l y p l o t t e d f o r t w o t y p i c a l g l i d e
speeds of aircraft:
for t h o s e w i t h p i s t o n - e n g i n e s , 2 0 0 k m / h r ;
for t h o s e w i t h g a s t u r b i n e e n g i n e s , 2 8 0 k m / h r .
The t u r n r a d i u s i s c a l c u l a t e d f o r a c o o r d i n a t e d t u r n w i t h a
banking a n g l e o f loo, w i t h t h e l i n e s f o r s t a r t i n g t h e t u r n p l o t t e d
f o r making a l a n d i n g a t a p p r o a c h a n g l e s o f 1 0 a n d 30°.

I f t h e a i r c r a f t h a s a s i g n i f i c a n t d e v i a t i o n f r o m t h e LTS a x i s
a f t e r emerging from t h e f o u r t h t u r n , w e can i n p r i n c i p l e use any
a n g l e o f a p p r o a c h t o t h e L T S a x i s which makes i t p o s s i b l e t o l i n e
up t h e a i r c r a f t w i t h t h e l a n d i n g p a t h b e f o r e t h e l a n d i n g d i s t a n c e
is reached.
H o w e v e r , a s e x p e r i e n c e h a s s h o w n , i t i s s i m p l e s t t o l i n e up t h e
a i r c r a f t w i t h t h e l a n d i n g p a t h by u s i n g o n l y two v a l u e s f o r t h e
approach a n g l e s : l o o i f t h e d e v i a t i o n o f t h e a i r c r a f t from t h e
g i v e n l i n e p a t h i s l e s s t h a n 5 0 0 m y a n d 30 m f o r d e v i a t i o n s e x c e e d ­
Then t h e l a n d i n g - r a d a r s c r e e n c a n b e b o u n d e d b y a
i n g 500 m.
t o t a l o f two a u x i l i a r y l i n e s f o r b e g i n n i n g t h e t u r n o n t o t h e l a n d i n g
path.
I n t h i s c a s e , t h e d i s t a n c e f r o m t h e LTS a x i s t o t h e a u x i l i a r y
l i n e can b e d e t e r m i n e d by t h e f o r m u l a
2 = R (1

-

cos UT$

However, e x p e r i m e n t a l d a t a show t h a t t h e r e i s a n a p p r e c i a b l e
d e l a y i n t h e a i r c r a f t ’ s a c q u i r i n g t h e l a n d i n g p a t h , due t o t h e
t i m e i n v o l v e d i n t r a n s m i t t i n g commands a n d d u e t o t h e r e a c t i o n o f
Therefore, it is b e t t e r
t h e a i r c r a f t a n d crew i n making t h e t u r n .
t o p l o t t h e s e l i n e s on t h e b a s i s o f s t a t i s t i c a l d a t a o b t a i n e d f r o m
e x p e r i e n c e , as d e t e r m i n e d f r o m a l a r g e n u m b e r o f a i r c r a f t l a n d i n g s .

414


/394

According t o t h e s e d a t a , t h e t u r n t o t h e l a n d i n g c o u r s e must
begin :
( a ) For a n a p p r o a c h a n g l e o f l o o , i n a i r c r a f t w i t h p i s t o n
e n g i n e s , 1 5 0 m f r o m t h e LTS a x i s ( 5 mm on t h e s c r e e n s c a l e ) ; f o r
a i r c r a f t w i t h g a s t u r b i n e e n g i n e s , i t i s 2 5 0 m f r o m t h e LTS a x i s
( 8 mm o n t h e s c r e e n s c a l e ) .
( b ) W i t h a n a p p r o a c h a n g l e o f 30°, t h e s e d i s t a n c e s a r e 4 5 0 a n d
7 5 0 m , r e s p e c t i v e l y ( 1 5 a n d 2 5 mm on t h e s c r e e n s c a l e ) .
T h e m a r k i n g s on t h e g l i d e s c r e e n o f t h e l a n d i n g r a d a r a r e
shown i n F i g . 4 . 2 4 .
I n t h i s case, t h e d e s c e n t
trajectory f o r the glide path is
set a t the airport.
Above t h i s
g l i d e p a t h a r e two b o u n d a r y l i n e s
f o r landing the aircraft; f o r
aircraft with gas t u r b i n e engines
i t i s 4O, a n d f o r a i r c r a f t w i t h
p i s t o n engines it i s 5 O .

F i g . 4 . 2 4 . P a t t e r n on G l i d e
Screen o f Landing Radar.

If t h e b l i p r e p r e s e n t i n g an
a i r c r a f t a p p e a r s above t h e boun­
dary l i n e designated f o r a given
type of a i r c r a f t , the landing of
t h e aircraft w i l l be complicated.
T h e r e f o r e , when c o n t r o l l i n g t h e
l a n d i n g o f an a i r c r a f t , it s h o u l d
n o t be allowed t o go beyond t h e

l i m i t s of t h e s e boundary l i n e s .
Below t h e e s t a b l i s h e d g l i d e p a t h , t h e r e a r e b o u n d a r y l i n e s f o r
p e r m i s s i b l e d e s c e n t o f t h e a i r c r a f t below t h e g l i d e p a t h , i . e . , t h e
l i n e s l i m i t i n g t h e f l i g h t a l t i t u d e a b o v e t h e l o c a l t e r r a i n : 200 m
p r i o r t o beginning descent i n a g l i d e , 150 m before passing over
t h e LRMS, a n d 5 0 m b e f o r e p a s s i n g o v e r t h e SRMS.
In addition, there
may a l s o b e f l i g h t a l t i t u d e s f o r c i r c l i n g t h e f i e l d , s e t a t 3 0 0 , 4 0 0
a n d 500m.
T h e s e l i n e s a r e u s e d for a i r c r a f t c o m i n g i n f o r a l a n d i n g
according t o t h e CGS (course-glide system).
I n t h e case w h e r e t h e b l i p m a r k i n g an a i r c r a f t i n t e r s e c t s o n e
o f t h e s e l i n e s , f u r t h e r d e s c e n t of t h e a i r c r a f t i s t o b e c o n s i d e r e d
d a n g e r o u s and t h e i n t e r v e n t i o n of t h e f l i g h t s u p e r v i s o r o p e r a t i n g
the landing radar is required.

B r i n g i n g an A i r c r a f t In f o r a L a n d i n g w i t h L a n d i n g R a d a r

/395

The m e t h o d o f b r i n g i n g a n a i r c r a f t i n f o r a l a n d i n g w i t h a
landing r a d a r i s very simple and q u i t e e f f e c t i v e a t t h e p r e s e n t
time

.

415

The s e t t i n g up of t h e l a n d i n g m a n e u v e r a n d t h e c a l c u l a t i o n s o f
t h e e l e m e n t s o f t h e d e s c e n t i s made b y t h e same r u l e s a s i n u s i n g
t h e s i m p l i f i e d or c o u r s e - g l i d e l a n d i n g s y s t e m s .
T h e moment f o r s t a r t i n g t h e f o u r t h t u r n i s d e t e r m i n e d o n t h e
b a s i s o f t h e b l i p r e p r e s e n t i n g t h e a i r c r a f t on t h e f l i g h t s u p e r ­
No commands a r e g i v e n t o t h e c r e w d u r i n g t h e
visor's screen.
fourth turn.
A f t e r t h e a i r c r a f t emerges from t h e f o u r t h t u r n , t h e c a l c u l a t e d
l a n d i n g p a t h m u s t b e f o l l o w e d for 1 0 t o 1 5 s e c .
If t h e b l i p o n t h e
l a n d i n g - r a d a r s c r e e n i s p a r a l l e l t o t h e LTS a x i s , t h e c a l c u l a t e d
course o f t h e a i r c r a f t i s e q u a l t o t h e l a n d i n g course and t h e a i r ­
c r a f t n e e d m e r e l y b e l i n e d up w i t h t h e l a n d i n g p a t h .
If t h e b l i p
i s a t a n a n g l e t o t h e LTS a x i s , t h e c a l c u l a t e d c o u r s e o f t h e a i r ­
c r a f t i s n o t e q u a l t o t h e l a n d i n g c o u r s e , b u t it i s v e r y e a s y t o
d e t e r m i n e t h e d e s i r e d c o u r s e c o r r e c t i o n by v i s u a l i n s p e c t i o n , s i n c e
t h e angle of t h e b l i p i s equal t o t h r e e t i m e s t h e angle o f t h e
For e x a m p l e , w i t h a b l i p a n g l e o f l o o , t h e c o u r s e
course e r r o r .
c o r r e c t i o n must be 3 O .
Having t h u s d e t e r m i n e d t h e r e q u i r e d c o r r e c t i o n i n t h e c o u r s e
t o b e f o l l o w e d , t h e s u p e r v i s o r g i v e s a command t o t h e c r e w , t e l l i n g
them t o a c q u i r e t h e d e s i r e d l a n d i n g p a t h a t an a n g l e o f 1 0 o r 3 0 ° ,
thus s e t t i n g t h e course t o be followed.
A t t h e moment when t h e b l i p c r o s s e s t h e c o r r e s p o n d i n g a u x i l i a r y
l i n e , a command i s g i v e n t o t u r n t h e a i r c r a f t o n t o t h e l a n d i n g c o u r s e ,
considering the correction given.
I n t h e m a j o r i t y o f c a s e s , when t h e s e t w o commands a r e g i v e n ,
i t i s s u f f i c i e n t t o l i n e up t h e a i r c r a f t w i t h t h e l a n d i n g p a t h on
If a t e n d e n c y i s o b s e r v e d d u r i n g f l i g h t a l o n g
the desired course.
t h e l a n d i n g p a t h f o r t h e a i r c r a f t t o s h i f t l a t e r a l l y , i t can b e
c o r r e c t e d b y commands for s m a l l c h a n g e s i n t h e a i r c r a f t c o u r s e
( b y 2 or 3 O ) , w i t h i n d i c a t i o n e a c h t i m e o f t h e c o u r s e w h i c h m u s t b e
followed.
When t h e b l i p a p p r o a c h e s t h e p o i n t w h e r e t h e a i r c r a f t i s t o
b e g i n i t s d e s c e n t i n a g l i d e , a command i s g i v e n t o d e s c e n d a t a
calculated v e r t i c a l speed.
If it t h e n develops t h a t t h e a i r c r a f t
i s d e v i a t i n g f r o m t h e g i v e n g l i d e p a t h ( e i t h e r u p w a r d or d o w n w a r d ) ,
t h e f l i g h t s u p e r v i s o r c o r r e c t s t h e v e r t i c a l s p e e d , g i v i n g new
v a l u e s f o r it and e n s u r i n g t h a t t h e a i r c r a f t t r a v e l s e x a c t l y a l o n g
t h e given path.
An a d v a n t a g e o f t h e r a d a r l a n d i n g s y s t e m i s t h e r e l a t i v e s i m p l i ­
c i t y of the supervisor's task i n directing the aircraft t o a landing
a n d t h e u n c o m p l i c a t e d a c t i o n s o f t h e crew i n c a r r y i n g o u t t h e s u p e r ­
v i s o r ' s commands, w i t h n o p r e v i o u s t r a i n i n g r e q u i r e d .
These advan­
t a g e s a r e a l s o r e i n f o r c e d by t h e f a c t t h a t t h e f l i g h t s u p e r v i s o r ,
/396
who c o n s t a n t l y w a t c h e s o v e r s e v e r a l a i r c r a f t c o m i n g i n f o r a l a n d i n g

416


and g i v e s them i n s t r u c t i o n s , a c q u i r e s a v e r y g r e a t amount o f e x p e r ­
i e n c e i n t h e c o u r s e o f h i s work, a g r e a t d e a l more t h a n t h a t which
t h e c r e w c a n a c q u i r e f r o m t h e l a n d i n g s o f t h e i r own a i r c r a f t a l o n e .
I n a d d i t i o n , t h e s u p e r v i s o r , i n t h e c o u r s e o f h i s work i n g u i d i n g
one a i r c r a f t a f t e r a n o t h e r t o a s a f e l a n d i n g , a c q u i r e s a p e c u l i a r
" f e e l " f o r e s t i m a t i n g t h e n a v i g a t i o n a l d i f f i c u l t i e s on a g i v e n d a y
( s e l e c t i o n o f t h e r e q u i r e d v e r t i c a l s p e e d a n d l a n d i n g c o u r s e on t h e
b a s i s o f h i s experience with aircraft t h a t have landed e a r l i e r ) .
T h e r e f o r e , i n p r a c t i c e , t h e a c c u r a c y o f l a n d i n g an a i r c r a f t
w i t h a r a d a r system i s no worse t h a n w i t h a c o u r s e - g l i d e system.
N e v e r t h e l e s s , t h e main s h o r t c o m i n g of t h e s y s t e m ( a l a c k o f i n d i ­
c a t i o n f o r t h e c r e w a s t o t h e p o s i t i o n o f t h e a i r c r a f t on a g i v e n
descent t r a j e c t o r y ) creates a c e r t a i n degree of inaccuracy i n
making t h e l a n d i n g , a n d i n t h i s r e s p e c t t h e r a d a r l a n d i n g s y s t e m i s
i n f e r i o r t o t h e course-glide system.

417


CHAPTER F I V E

AVIATION ASTRONOMY]

1.

The C e l e s t i a l S p h e r e

The s k y a p p e a r s t o t h e o b s e r v e r a s a n immense h e m i s p h e r e .
The c e Z e s t i a Z s p h e r e i s a n i m a g i n a r y s p h e r e o f a r b i t r a r y r a d i u s , /397
whose c e n t e r i s t h e e y e o f t h e o b s e r v e r ( F i g . 5 . 1 ) .
An o b s e r v e r o n t h e E a r t h ’ s s u r f a c e c a n s e e o n l y t h e h a l f o f t h e
c e l e s t i a l s p h e r e which i s l o c a t e d above t h e h o r i z o n , s i n c e t h e
o t h e r hemisphere i s l o c a t e d below t h e h o r i z o n .
If t h e E a r t h were t r a n s p a r e n t , a n o b s e r v e r l o c a t e d a t a n y p o i n t
on i t s s u r f a c e w o u l d s e e n o t o n e b u t t w o domes w h i c h t o g e t h e r f o r m
the celestial sphere.
Special

Points,

Planes,

and C i r c l e s

i n the Celestial

Sphere

If a l i n e i s p l o t t e d p e r p e n d i c u l a r t o t h e
Z e n i t h and n a d i r .
l o c a t i o n o f t h e o b s e r v e r ( t h r o u g h t h e c e n t e r of t h e c e l e s t i a l s p h e r e ) ,
it w i l l i n t e r s e c t t h e imaginary l i m i t s o f t h e c e l e s t i a l s p h e r e a t
two p o i n t s ( s e e F i g . 5 . 1 ) .
The p o i n t w h i c h i s l o c a t e d a b o v e t h e
T h e o p p o s i t e p o i n t i s t h e n a d i r (Z’).
observer is the zenith (Z).
True horizon.
If a p l a n e i s d e f i n e d t h r o u g h t h e c e n t e r o f t h e
c e l e s t i a l s p h e r e a n d i s p e r p e n d i c u l a r t o t h e v e r t i c a l l i n e Z Z ’ , we
/398
c a n c a l l i t t h e p l a n e of t h e h o r i z o n .

I


Fig.

5.1.

The p l a n e o f
sects the celesti
t h e circumference
( t h e p o i n t s NESW)
t h e true horizon.

C e l e s t i a l Sphere
~.

’ T h i s c h a p t e r w a s w r i t t e n by M . I .
418

t h e horizon i n t e r ­
a l sphere along
of a g r e a t c i r c l e
which i s c a l l e d

World a x i s .
The i m a g i n a r y
l i n e P P ’ , around which t h e a p p a r e n t
r o t a t i o n of t h e c e l e s t i a l s p h e r e
t a k e s p l a c e , i s c a l l e d t h e world
It passes through the point
axis.
of t h e observer, located a t t h e
c e n t e r of t h e c e l e s t i a l s p h e r e ,
and i n t e r s e c t s t h e a r b i t r a r y l i m i t s
of t h e c e l e s t i a l s p h e r e a t two
__

Gurevich.

I'


d i a m e t r i c a l l y opposed p o i n t s PP'.
The w o r l d a x i s i s i n c l i n e d t o
t h e h o r i z o n a t a n a n g l e w h i c h d e p e n d s on t h e l a t i t u d e of t h e ob­
server.
Z

N

S


Fig. 5.2.
V e r t i c a l and
Almucantar.

Fig. 5.3.
C e l e s t i a l Meridian,
Hour C i r c l e , a n d C e l e s t i a l P a r a l l e l .

Celestial poles.
The p o i n t s w h e r e t h e i m a g i n a r y w o r l d a x i s
i n t e r s e c t s t h e a r b i t r a r y l i m i t of t h e c e l e s t i a l sphere a r e c a l l e d
the celestial poles.
Point P i s c a l l e d t h e superior (north) celes­
t i a l p o l e , and t h e o p p o s i t e p o i n t P ' i s c a l l e d t h e i n f e r i o r ( s o u t h )
ceZestiaZ pole.
Only t h e n o r t h c e l e s t i a l p o l e i s v i s i b l e i n t h e
Northern Hemisphere, and only t h e south c e l e s t i a l p o l e i s v i s i b l e
i n t h e Southern Hemisphere.
Celestial equator.
The p l a n e w h i ch p a s s e s t h r o u g h t h e c e n t e r
o f t h e c e l e s t i a l s p h e r e a n d i s p e r p e n d i c u l a r t o t h e w o r l d a x i s is
c a l l e d t h e p l a n e of t h e c e l e s t i a l e q u a t o r .
The g r e a t c i r c l e QEQ'W,
a l o n g which t h e p l a n e of t h e c e l e s t i a l e q u a t o r i n t e r s e c t s t h e c e l e s ­
t i a l sphere, i s c a l l e d t h e ceZestiaZ equator.
The c e l e s t i a l e q u a t o r d i v i d e s t h e c e l e s t i a l s p h e r e i n t o
n o r t h e r n (QPQ ' 1 a n d s o u t h e r n (Q ' P 'Q) p a r t s .
The p l a n e o f t h e c e l e s t i a l e q u a t o r i s i n c l i n e d t o t h e p l a n e
of t h e t r u e h o r i z o n a t a n a n g l e which a l s o d e p e n d s on t h e l a t i t u d e
of t h e o b s e r v e r .
Vertical.
The g r e a t c i r c l e on t h e c e l e s t i a l s p h e r e whose
plane passes through t h e v e r t i c a l l i n e i s called t h e v e r t i c a l .
Every v e r t i c a l p a s s e s t h r o u g h t h e z e n i t h Z and t h e n a d i r Z ' .
The
p l a n e of t h e v e r t i c a l i s p e r p e n d i c u l a r t o t h e p l a n e of t h e t r u e
horizon (Fig. 5.2).
The v e r t i c a l w h i c h p a s s e s t h r o u g h t h e e a s t a n d w e s t p o i n t s ( E
and W , r e s p e c t i v e l y ) i s c a l l e d t h e primary v e r t i c a Z .

419


,


/399

It

The g r e a t c i r c l e ZMZ’ o f t h e c e l e s t i a l s p h e r e , w h i c h p a s s e s
t h r o u g h t h e z e n i t h of t h e o b s e r v e r a n d a c e r t a i n s t a r ( P o i n t M ,
F i g . 5.2), i s c a l l e d t h e v e r t i c a l o f t h a t s t a r .

Almucantar.
The s m a l l c i r c l e D M D ’ on t h e c e l e s t i a l s p h e r e ,
whose p l a n e i s p a r a l l e l t o t h e p l a n e o f t h e t r u e h o r i z o n , i s c a l l e d
t h e almucantar.
The a l m u c a n t a r w h i c h p a s s e s t h r o u g h a g i v e n s t a r i s c a l l e d t h e

almucantar o f t h a t s t a r .
Hour c i r c l e . T h e g r e a t c i r c l e PMP’ o f t h e c e l e s t i a l s p h e r e ,
w h o s e p l a n e p a s s e s t h r o u g h t h e w o r l d a x i s , i s c a l l e d t h e c i r c l e of
declination (Fig. 5.3).
Since t h e world a x i s i s perpendicular t o
t h e c e l e s t i a l equator, t h e plane of t h e hour circle is a l s o per­
pendicular t o the equator.
T h e h o u r c i r c l e w h i c h p a s s e s t h r o u g h a g i v e n s t a r i s t h e hour

circZe of that star.
Celestial m e r i d i a n .
The v e r t i c a l P Z P ’ Z ’ , w h i c h p a s s e s t h r o u g h
t h e celestial poles, i s c a l l e d t h e ceZestiaZ.meridian (since i t s
plane coincides with t h e plane of t h e meridian of t h e observer).
The c e l e s t i a l m e r i d i a n d i v i d e s t h e c e l e s t i a l s p h e r e i n t o t h e e a s t e r n
and w e s t e r n hemispheres.
T h e north point N a n d south point S.
c r o s s e s t h e t r u e h o r i z o n a t two p o i n t s ,

The c e l e s t i a l m e r i d i a n
c a l l e d t h e n o r t h and s o u t h

points.
Meridian line. T h e p l a n e of t h e c e l e s t i a l m e r i d i a n c r o s s e s
Obviously,
t h e plane of t h e t r u e horizon t o form t h e meridian l i n e .
t h e e n d s o f t h e m e r i d i a n l i n e c o i n c i d e w i t h t h e n o r t h and s o u t h
(N a n d S , r e s p e c t i v e l y ) .
T h i s l i n e i s c a l l e d t h e “noon
points.
l i n e ” i n R u s s i a n b e c a u s e t h e shadows o f v e r t i c a l o b j e c t s f a l l a l o n g
t h i s l i n e a t noon.
T h e east point E and west point W .
If w e p l o t a s t r a i g h t l i n e
i n t h e p l a n e of t h e h o r i z o n p e r p e n d i c u l a r t o t h e m e r i d i a n l i n e ( s e e
F i g . 5 . 3 ) and f a c e n o r t h , t h e e a s t p o i n t E w i l l l i e on t h e r i g h t a t
t h e p o i n t where t h e p l a n e i n t e r s e c t s t h e c i r c u m f e r e n c e of t h e t r u e
h o r i z o n , w h i l e t h e w e s t p o i n t w i l l b e l o c a t e d on t h e l e f t .
A s t h e f i g u r e shows, t h e e a s t and w e s t p o i n t s a r e 90° d i s t a n t
from t h e n o r t h and s o u t h p o i n t s .
T h e same f i g u r e a l s o s h o w s t h a t
t h e e a s t a n d w e s t p o i n t s ( E a n d W , r e s p e c t i v e l y ) mark t h e p o i n t s
of i n t e r s e c t i o n o f t h e c e l e s t i a l e q u a t o r w i t h t h e t r u e h o r i z o n .

Celestial parallel.
The s m a l l c i r c l e on t h e c e l e s t i a l s p h e r e , / 4 o o
whose p l a n e i s p a r a l l e l t o t h e p l a n e o f t h e c e l e s t i a l e q u a t o r , i s
c a l l e d t h e ceZestiaZ parallel ( s i m i l a r t o t h e t e r r e s t r i a l p a r a l l e l s ) .

420

Diurnal c i r c l e o f a s t a r .
The s m a l l c i r c l e o n t h e c e l e s t i a l
s p h e r e , drawn t h r o u g h a s t a r p a r a l l e l t o t h e c e l e s t i a l e q u a t o r , i s
c a l l e d t h e d i u r n a l c i r c l e of t h e s t a r .
Astronomical coordinates.
A s w e know, i n o r d e r t o d e t e r m i n e
t h e l o c a t i o n o f a n y p o i n t on t h e E a r t h ’ s s u r f a c e , i t i s s u f f i c i e n t
t o know t h e t w o a n g u l a r c o o r d i n a t e s o f t h i s p o i n t , t h e l a t i t u d e
and l o n g i t u d e .

1

I n a s t r o n o m y , t h e l o c a t i o n o f s t a r s on t h e s p h e r e i s accom­
p l i s h e d by means o f two a n g u l a r s y s t e m s of c e l e s t i a l c o o r d i n a t e s :
t h e a p p a r e n t s y s t e m of c o o r d i n a t e s and t h e e q u a t o r i a l s y s t e m o f
coordinates.
I n e a c h o f t h e s e s y s t e m s , t h e p o s i t i o n o f a p o i n t ( s t a r ) on
t h e c e l e s t i a l s p h e r e i s d e t e r m i n e d by two c e l e s t i a l c o o r d i n a t e s .
L e t u s examine t h e s y s t e m s o f c e l e s t i a l c o o r d i n a t e s i n d i v i d u a l l y .

S y s t e m s of

Coordinates

A p p a r e n t S y s t e m of C o o r d i n a t e s
The main c i r c l e s r e l a t i v e t o wh i ch c o o r d i n a t e s a r e d e t e r m i n e d
i n t h i s system ( F i g . 5 . 4 ) a r e t h e t r u e h o r i z o n and t h e m e r i d i a n of
the observer.
The c o o r d i n a t e s t h e m s e l v e s a r e c a l l e d t h e a l t i t u d e
o f t h e s t a r (h) a n d t h e a z i m u t h o f t h e s t a r ( A ) .
Altitude of a star.
The a n g l e b e t w e e n t h e p l a n e o f t h e t r u e
h o r i z o n and a l i n e from t h e c e n t e r of t h e s p h e r e t o t h e s t a r ( a n g l e
M ’ O M , F i g . 5 . 4 ) i s c a l l e d t h e a z t i t u d e of t h e s t a r .
The a l t i t u d e
o f a s t a r c a n a l s o b e m e a s u r e d by t h e a r c o f t h e v e r t i c a l from t h e
t r u e h o r i z o n t o t h e l o c a t i o n o f t h e g i v e n s t a r (M’M).

The a l t i t u d e o f t h e s t a r i s m
easured from 0 t o 90° ( p o s i t i v e
v a l u e s toward t h e z e n i t h from t h e
h o r i z o n , n e g a t i v e v a l u e s from t h e
horizon toward t h e n a d i r ) .
Z e n i t h distance.
Instead of
t h e s t a r , w e can a l s o use t h e soc a l l e d zenith distance of t h e star
as a c o o r d i n a t e , measured a l o n g
t h e a r c ZM.

Fig. 5.4. Horizontal
of C o o r d i n a t e s .

System

A s we c a n s e e f r o m F i g u r e 5 . 4 ,
t h e z e n i t h d i s t a n c e i s t h e a r c from
t h e z e n i t h t o t h e l o c a t i o n of t h e
I t i s e a s y t o s e t up
given star.
a formula t o express t h e r e l a t i o n - /401
shiD between t h e a l t i t u d e and t h e
zenith distance of a s t a r , since
t h e two add up t o 90°:
h + Z = 90°,

421

h = 9 0 0 - Z, Z = 9 0 ° - h .
Obviously, t h e value of t h e zenith d i s ­
t a n c e w i l l b e s o m e w h e r e b e t w e e n 0 a n d 180'.
Azimuth.
The s e c o n d c o o r d i n a t e i n t h e a p p a r e n t s y s t e m of c o ­
ordinates i s t h e azimuth of t h e star.
T h e a z i m u t h of a s t a r i s t h e
s p h e r i c a l a n g l e b e t w e e n t h e p l a n e of t h e m e r i d i a n 3f t h e o b s e r v e r
and t h e p l a n e of t h e c i r c l e of t h e v e r t i c a l of t h e g i v e n s t a r .

The a z i m u t h i s c a l c u l a t e d d i f f e r e n t l y i n d i f f e r e n t a r e a s o f
astronomy:
f r o m t h e s o u t h p o i n t or f r o m t h e n o r t h p o i n t t o w a r d t h e
e a s t and w e s t .
I n a v i a t i o n astronomy, t h e azimuth i s always calcu­
l a t e d from t h e n o r t h p o i n t along t h e horizon i n an e a s t e r l y d i r e c ­
t i o n (clockwise) from 0 t o 360'.
We c a n t h e r e f o r e d e f i n e t h e
azimuth i n a v i a t i o n astronomy as t h e a n g l e measured a l o n g t h e a r c
NSM' o f t h e t r u e h o r i z o n from t h e n o r t h p o i n t through t h e e a s t ( t h e
e a s t p o i n t ) t o t h e v e r t i c a l o f t h e s t a r ( s e e F i g . 5 . 4 ) , from 0 t o
360'.

Hence, t h e f i r s t system of c o o r d i n a t e s f o r c e l e s t i a l l u m i n a r i e s
i s c a l l e d t h e apparent system.
The c o o r d i n a t e s o f t h i s s y s t e m a r e
t h e a l t i t u d e o f t h e s t a r ( h ) and t h e azimuth o f t h e s t a r ( A ) .
The a l t i t u d e a n d a z i m u t h w i l l s u f f i c e c o m p l e t e l y t o d e t e r m i n e
For e x a m p l e , t h e
t h e l o c a t i o n o f a s t a r on t h e c e l e s t i a l s p h e r e .
s t a r M , w i t h h = 60'
a n d A = 2 4 0 ° , i s i n d i c a t e d on t h e s p h e r e ( s e e
Fig. 5.4).

E q u a t o r i a l S y s t e m of

Coordinates

T h e e q u a t o r i a l s y s t e m o f c o o r d i n a t e s is t h e s e c o n d s y s t e m o f
c o o r d i n a t e s which i s u s e d t o d e t e r m i n e t h e l o c a t i o n o f a s t a r on
The m a i n c i r c l e s r e l a t i v e t o w h i c h c a l c u l a t i o n s
the celestial sphere.
a r e made i n t h i s s y s t e m a r e t h e c e l e s t i a l m e r i d i a n a n d t h e c e l e s t i a l
equator.
The c o o r d i n a t e s i n t h i s s y s t e m a r e t h e d e c l i n a t i o n o f t h e s t a r
( 6 ) and t h e hour a n g l e of t h e s t a r ( t ) ; s e e F i g u r e 5.5.
Declination of t h e s t a r .
The a r c o f t h e c i r c l e m a r k i n g t h e
d i s t a n c e f r o m t h e e q u a t o r t o t h e l o c a t i o n o f t h e g i v e n s t a r , or t h e
a n g l e between t h e p l a n e of t h e e q u a t o r znd a l i n e from t h e c e n t e r
o f t h e s p h e r e t o t h e s t a r , c a l l e d t h e d e c l i n a t i o n of t h e s t a r .
D e c l i n a t i o n i s m e a s u r e d by t h e a r c o f a c i r c l e w h i c h m a r k s t h e
d i s t a n c e from t h e e q u a t o r t o t h e l o c a t i o n of t h e g i v e n s t a r , from
0 t o _C 9 0 ' .
If t h e s t a r i s l o c a t e d i n t h e N o r t h e r n H e m i s p h e r e , i t s
d e c l i n a t i o n i s c o n s i d e r e d p o s i t i v e , w h i l e i f it i s i n t h e S o u t h e r n
Hemisphere, it i s considered n e g a t i v e .

I t i s c l e a r f r o m F i g u r e 5 . 5 t h a t i f t h e s t a r i s l o c a t e d on t h e
e q u a t o r , i t s d e c l i n a t i o n w i l l be e q u a l t o z e r o , while t h e d e c l i n a t i o n
o f t h e n o r t h c e l e s t i a l p o l e i s + 90' a n d t h a t o f t h e s o u t h c e l e s t i a l
422

I


p o l e i s -goo.
Polar distance.
O c c a s i o n a l l y , i n s t e a d of t h e d e c l i n a t i o n , t h e
p o l a r d i s t a n c e i s u s e d a s a c o o r d i n a t e , m e a s u r e d a l o n g t h e a r c PM. j 4 0 2
The p o Z a r d i s t a n c e i s t h e a r c o f t h e c i r c l e w h i c h m a r k s t h e d i s ­
t a n c e from t h e n o r t h c e l e s t i a l p o l e t o t h e l o c a t i o n of t h e s t a r .
The r e l a t i o n s h i p b e t w e e n t h e d e c l i n a t i o n a n d t h e p o l a r d i s ­
t a n c e i s e x p r e s s e d by t h e f o r m u l a

B

or

+ PM = 90"

PM

or

==

-6 ,

90"

B = 90"- P M ,
i . e . , t h e d e c l i n a t i o n and p o l a r d i s t a n c e t o g e t h e r add up t o 90°.
T h e r e f o r e t h e p o i n t of t h e s o u t h p o l e h a s a p o l a r d i s t a n c e e q u a l t o
180O.
Hour a n g l e o f a s t a r .
T h e a r c o f t h e c e l e s t i a l e q u a t o r Q'M'
( F i g . 5 . 5 ) b e t w e e n t h e s o u t h p o i n t of t h e e q u a t o r a n d t h e h o u r
c i r c l e o f a g i v e n s t a r i s c a l l e d t h e hour a n g l e of a s t a r ( t ) .
I n a v i a t i o n astronomy, t h e hour angle i s measured from t h e
s o u t h p o i n t o f t h e e q u a t o r a l o n g t h e e q u a t o r i n t h e e a s t e r l y and
w e s t e r l y d i r e c t i o n s f r o m 0 t o 180O.
The w e s t e r n h o u r a n g l e i s r e p r e s e n t e d by t h e l e t t e r W , f o r
e x a m p l e , t = 135' W ; t h e e a s t e r n h o u r a n g l e i s r e p r e s e n t e d by t h e
l e t t e r E , for e x a m p l e , t = 6 0 ° E .
I n making c a l c u l a t i o n s , t h e
w e s t e r n h o u r a n g l e m u s t s o m e t i m e s b e c a l c u l a t e d f r o m 0 t o 360O.
I f t h e w e s t e r n h o u r a n g l e i s f o u n d t o b e g r e a t e r t h a n 180°, i t i s
r e l a t e d t o 360°, b u t i n t h i s case
t h e r e s u l t i s given as an e a s t e r n
P
For e x a m p l e , t = 265O
hour angle.
W or t = 360' - 265' = 95OE.

P

P'

P'

Fig. 5.5.
E q u a t o r i a l System
of Coordinates.

R i g h t ascension o f a s t a r .
I n s t e a d of t h e hour a n g l e , it i s
s o m e t i m e s more c o n v e n i e n t t o u s e
another coordinate , t h e r i g h t as­
c e n s i o n o f t h e s t a r ( a ) . The r i g h t
a s c e n s i o n of a s t a r i s t h e a n g l e
as measured a l o n g t h e e q u a t o r from
t h e p o i n t of t h e v e r n a l equinox
( y ) t o t h e hour c i r c l e of t h e given
star (see Fig. 5.5).

T h e p o i n t of t h e v e r n a l e q u i n o x
is t h e imaginary point of t h e

423

i n t e r s e c t i o n o f t h e e c l i p t i c w i t h t h e c e l e s t i a l e q u a t o r , when t h e
S u n p a. s s e s f r o m t h e S o u t h e r n H e m i s p h e r e i n t o t h e N o r t h e r n H e m i s p h e r e .
T h e o p p o s i t e p o i n t o n t h e e c l i p t i c i s c a l l e d t h e p o i n t of t h e
autumn.aZ equinox
(a).
I n a n c i e n t G r e e c e , t h e s t a r s were u s e d t o r e c k o n t i m e .
The
/403
c o n s t e l l a t i o n A r i e s w a s l o c a t e d a t t h e p o i n t of t h e v e r n a l e q u i n o x ,
Due t o t h e p r e c e s s i o n o f t h e
and w a s r e p r e s e n t e d by t h e symbol ( y ) .
E a r t h , A r i e s h a s now moved a w a y f r o m t h e p o i n t o f t h e v e r n a l e q u i n o x .
T h i s p o i n t h a s r e m a i n e d u n m a r k e d , t h o u g h i t s name h a s b e e n r e t a i n e d ,
a n d i t s p o s i t i o n i n t h e s k y i s d e t e r m i n e d by u s i n g some o t h e r s t a r
which i s a f i x e d d i s t a n c e from t h e p o i n t of t h e v e r n a l equinox.
Right ascension i s c a l c u l a t e d from t h e p o i n t of t h e v e r n a l
e q u i n o x a l o n g t h e e q u a t o r up t o t h e h o u r c i r c l e of a g i v e n s t a r i n
a c l o c k w i s e d i r e c t i o n ( a s s e e n f r o m t h e n o r t h c e l e s t i a l p o l e ) , from
0 t o 360O.
Like t h e hour angle of a s t a r , t h e r i g h t ascension of a star
c a n b e r e c k o n e d i n e i t h e r d e g r e e s or h o u r s , m i n u t e s , a n d s e c o n d s .
T h i s i s b e c a u s e b o t h of t h e s e c o o r d i n a t e s ( e s p e c i a l l y t h e h o u r
a n g l e ) a r e c l o s e l y r e l a t e d t o t h e measurement of t i m e .

T h u s , t h e e q u a t o r i a l s y s t e m of c o o r d i n a t e s c a n b e u s e d t o
d e t e r m i n e t h e l o c a t i o n o f a s t a r on t h e c e l e s t i a l s p h e r e .
I f w e know t h e d e c l i n a t i o n a n d t h e h o u r a n g l e or t h e r i g h t
a s c e n s i o n , w e c a n d e t e r m i n e t h e l o c a t i o n o f a s t a r on t h e s p h e r e .
For e x a m p l e , t h e s t a r M , w i t h 6 = + 5 0 ° ,
= 4 5 O , i s shown on t h e
sphere (see Fig. 5.5).

G r a p h i c R e p r e s e n t a t i o n of

t h e Celestial

Sphere

I n solving textbook problems i n a v i a t i o n astronomy, it i s o f t e n
n e c e s s a r y t o s k e t c h t h e c e l e s t i a l s p h e r e a n d p l o t t h e s t a r s on i t
according t o t h e i r coordinates.
Let u s u s e a c o n c r e t e example t o
s t u d y t h e o r d e r i n which t h e c e l e s t i a l s p h e r e i s s k e t c h e d .

Example.
1. T h e l a t i t u d e o f t h e o b s e r v e r i s 4 = 6 0 ° N ,
a l t i t u d e of t h e s t a r h = 7 0 ° , and i t s azimuth A = 240°.
it.

the

D r a w t h e c e l e s t i a l s p h e r e a n d p l o t t h e p o s i t i o n o f t h e s t a r on
(Fig. 5.6,a).

Solution.
( 1 ) Use a c o m p a s s t o d r a w t h e c e l e s t i a l m e r i d i a n
i n t h e form of a c i r c l e of a r b i t r a r y r a d i u s .
(2)
D r a w a v e r t i c a l d i a m e t e r ( p e r p e n d i c u l a r l i n e ) and mark
t h e z e n i t h and n a d i r ( Z and Z ' , r e s p e c t i v e l y ) a t t h e p o i n t s where
it c r o s s e s t h e circumference.
(3)

424

Perpendicular t o the v e r t i c a l l i n e ,

through t h e c e n t e r of

t h e s p h e r e , draw a l a r g e c i r c l e which w i l l b e t h e t r u e h o r i z o n of
the observer.
D r a w t h e world
(4)
plane of the horizon w i l
:.e.,
@ = 6 0 ° N ; mark t h e
circumference ( t h e north
pole P').

a x i s s u c h t h a t t h e a n g l e it forms w i t h t h e
l be equal t o t h e l a t i t u d e of t h e observer,
p o i n t s where t h e world a x i s c r o s s e s t h e
c e l e s t i a l p o l e P and t h e s o u t h c e l e s t i a l

( 5 )
A t t h e p o i n t s where t h e t r u e hori.zon i n t e r s e c t s t h e m e r i d ­
i a n of t h e o b s e r v e r , m a r k t h e n o r t h p o i n t N ( c l o s e t o t h e n o r t h
c e l e s t i a l p o l e ) and t h e south p o i n t S ( c l o s e t o t h e south c e l e s t i a l
pole )

.

( 6 )
P e r p e n d i c u l a r t o t h e p o i n t of i n t e r s e c t i o n of t h e c e l e s t i a l
e q u a t o r w i t h t h e t r u e h o r i z o n , mark t h e e a s t p o i n t E ( o n t h e r i g h t ,
a s v i e w e d by s o m e o n e f a c i n g n o r t h ) a n d t h e w e s t p o i n t W ( o n t h e
left).

We h a v e
This completes t h e sketching of t h e c e l e s t i a l sphere.
y e t t o p l o t t h e p o s i t i o n o f t h e s t a r o n t h e s p h e r e on t h e b a s i s o f
i t s coordinate d a t a , as follows:

(1) From t h e n o r t h p o i n t N , p l o t t h e a z i m u t h o f t h e s t a r
( e q u a l t o 240°) a l o n g t h e c i r c u m f e r e n c e of t h e h o r i z o n , j u d g i n g
t h e a n g l e by e y e .
of

(2)
Through t h i s p o i n t M ' ,
the vertical.

draw t h e c i r c l e

/404

(semicircumference)

(3)
Along t h e c i r c l e o f t h e v e r t i c a l , from t h e p l a n e o f t h e
horizon, p l o t t h e a l t i t u d e of t h e s t a r , equal t o 70°, judging t h e
d i s t a n c e by e y e .

N

E

Fig. 5.6.
Examples of G r a p h i c C o n s t r u c t i o n o f t h e C e l e s t i a l S p h e r e ;
( a ) A t a L a t i t u d e o f 6 0 ° ; (b) A t a L a t i t u d e o f 50°.
425

The r e s u l t o f t h i s c o n s t r u c t i o n w i l l b e t h e c e l e s t i a l s p h e r e
a s s e e n b y a n o b s e r v e r a t 60°N a n d t h e p o s i t i o n o f a s t a r o n t h e
sphere according t o i t s apparent coordinates.

Example. 2 .
The o b s e r v e r i s l o c a t e d a t a l a t i t u d e o f 5 0 ° .
S k e t c h t h e c e l e s t i a l s p h e r e for t h i s o b s e r v e r a n d p l o t o n i t t h e
hour
position of a star with t h e following equatorial boordinates:
a n g l e t = 130°, d e c l i n a t i o n 6 = + 4 0 ° .

Solution.
(1) S k e t c h t h e c e l e s t i a l s p h e r e i n t h e same o r d e r
o u t l i n e d i n E x a m p l e 1.
(2)
From t h e s o u t h p o i n t o n t h e e q u a t o r Q ’ , p r o c e e d i n g a l o n g
t h e c i r c u m f e r e n c e of t h e e q u a t o r i n a w e s t e r l y d i r e c t i o n , p l o t a n
h o u r a n g l e t = 130° b y e y e ( F i g . 5 . 6 , b ) .

( 3 )

Through t h i s p o i n t

draw t h e h o u r c i r c l e

(M’),

(PM’P’).

From t h e p l a n e o f t h e e q u a t o r , a l o n g t h e h o u r c i r c l e , m e a s u r e
o f f t h e d e c l i n a t i o n 6 = +40° a n d mark t h e l o c a t i o n of t h e s t a r on
t h e sphere (point M ) .
The r e s u l t o f t h i s c o n s t r u c t i o n i s t h e h o u r c i r c l e f o r a n o b ­
s e r v e r l o e a t e d a t a l a t i t u d e of @ = 5 0 ° N ; t h e s t a r h a s b e e n p l o t t e d
on t h e s p h e r e on t h e b a s i s o f i t s e q u a t o r i a l c o o r d i n a t e s .

2.

D i u r n a l Motion o f t h e S t a r s

If one o b s e r v e s t h e h e a v e n l y b o d i e s , e v e n i n t h e c o u r s e o f a
s i n g l e n i g h t , he w i l l see t h a t t h e appearance of t h e sky changes
S e v e r a l c o n s t e l l a t i o n s which were l o ­
with t h e passage of t i m e .
c a t e d n e a r t h e e a s t e r n p o r t i o n of t h e h o r i z o n g r a d u a l l y r i s e h i g h e r
a b o v e t h e h o r i z o n , w h i l e o t h e r s , which were l o c a t e d c l o s e r t o t h e
Some s t a r s w h i c h w e r e v i s i b l e i n
western horizon, gradually set.
t h e s k y s e t i n t h e w e s t , w h i l e o t h e r s , which were n o t v i s i b l e be­
T h e e n t i r e s k y s e e m s t o move c o n s t a n t l y
1405,
fore, rise i n the east.
during t h e course of t h e n i g h t from e a s t t o west.
A t t h e same
t i m e , one c a n see t h a t t h e mutual p o s i t i o n s of t h e c o n s t e l l a t i o n s
and stars do n o t change i n t h e course of t h e motion of t h e e n t i r e
sky ( s e e Supplement 3 ) .
The r e a s o n f o r t h i s a p p a r e n t m o t i o n o f t h e s t a r s (or o f t h e
s k y ) i s t h e d i u r n a l r o t a t i o n o f t h e E a r t h on i t s a x i s from w e s t t o
east.
I n order t o f a c i l i t a t e a study of t h e d i u r n a l r o t a t i o n of t h e
s t a r s , we w i l l a s s u m e f o r t h e s a k e o f d i s c u s s i o n t h a t t h e E a r t h i s
f i x e d a n d t h e c e l e s t i a l s p h e r e r o t a t e s on t h e w o r l d a x i s a t t h e
same r a t e t h a t t h e E a r t h a c t u a l l y r o t a t e s o n i t s a x i s , b u t i n t h e
o p p o s i t e d i r e c t i o n , from e a s t t o w e s t ( i n o t h e r w o r d s , t h e way i t
actually looks t o us).
Since t h e e n t i r e celestial sphere r o t a t e s
on t h e w o r l d a x i s , a l l t h e p o i n t s ( s t a r s ) l o c a t e d on t h e s p h e r e
426

I

I1 111 I I 111 I1 I I I1II

I

1.1

I

II 1111I I
.

111111

I

I

Id

w i l l t u r n a l o n g w i t h i t , i . e . , i t is c l e a r t h a t e a c h s t a r d e s c r i b e s
a s o r t of circle around t h e world axis.
Diurnal p a r a l l e l of a s t a r .
All o f t h e s t a r s r o t a t e t o g e t h e r
with t h e celestial sphere around t h e world axis.
From t h i s i t i s
c l e a r t h a t every s t a r , f i x e d permanently i n t h e sky, d e s c r i b e s a
c i r c l e o f s o m e s i z e i n t h e c o u r s e of 24 h o u r s .
The c i r c l e d e s c r i b e d b y a s t a r i n 2 4 h o u r s i n t h e c o u r s e o f
i t s m o v e m e n t a r o u n d t h e w o r l d a x i s i s c a l l e d t h e d i u r n a l c i r c l e of
the star.
T h i s c i r c l e i s a l s o c a l l e d t h e c e z e s t i a z paraZZeZ.
Since t h e e n t i r e c e l e s t i a l sphere r o t a t e s around t h e world
a x i s , it i s e a s y t o see ( a n d i m p o r t a n t t o remember) t h a t t h e d i ­
u r n a l r o t a t i o n of t h e heavenly bodies takes place p a r a l l e l t o t h e
c e l e s t i a l e q u a t o r , i . e . , t h e d i u r n a l p a r a l l e l of t h e s t a r ( t h e p a t h
o f t h e s t a r a r o u n d t h e w o r l d a x i s i n 24 h o u r s ) i s a l w a y s l o c a t e d
parallel t o the celestial equator.
The m a g n i t u d e o f t h e d i u r n a l p a r a l l e l o f t h e s t a r d e p e n d s on
t h e l o c a t i o n of t h e s t a r i n t h e s k y .
O b v i o u s l y , s t a r s which a r e
l o c a t e d c l o s e r t o t h e c e l e s t i a l p o l e s (and have h i g h e r d e c l i n a t i o n
v a l u e s ) have a small d i u r n a l c i r c l e .
The c l o s e r a s t a r i s l o c a t e d
relative t o the c e l e s t i a l equator (the smaller its declination),
t h e l a r g e r its diurnal circle w i l l be.
The l a r g e s t d i u r n a l c i r c l e
b e l o n g s t o t h o s e s t a r s w h i c h a r e l o c a t e d on t h e c e l e s t i a l e q u a t o r ,
and whose d e c l i n a t i o n i s z e r o .
Motion o f

t h e Stars a t Different Latitudes

If w e observe t h e d i u r n a l motion of t h e s t a r s a t d i f f e r e n t
l a t i t u d e s , w e w i l l see t h a t t h e sky and s t a r s t u r n r e l a t i v e t o t h e
observer's horizon at different angles.
T h i s phenomenon becomes
understandable i f we r e c a l l t h e location of the world a x i s r e l a t i v e
t o the horizon at d i ffe re n t latitudes.

The w o r l d a x i s i s l o c a t e d r e l a t i v e t o t h e h o r i z o n a t a n a n g l e 1 4 0 6
which i s e q u a l t o t h e l a t i t u d e o f t h e l o c a t i o n .
From t h i s i t f o l ­
lows t h a t t h e h i g h e r t h e l a t i t u d e of a . l o c a t i o n , t h e c l o s e r t h e
c e l e s t i a l p o l e s P P ' w i l l be l o c a t e d t o t h e z e n i t h Z and t h e n a d i r
Z ' , and t h e s m a l l e r t h e a n g l e w i l l be between t h e t r u e h o r i z o n and
the celestial equator.
Conversely, t h e lower t h e l a t i t u d e of t h e
l o c a t i o n , t h e f u r t h e r t h e c e l e s t i a l p o l e s w i l l be from t h e z e n i t h
and n a d i r , and t h e a n g l e between t h e t r u e h o r i z o n and t h e c e l e s t i a l
equator w i l l be l a r g e r .
F i g u r e 5 . 7 , a shows t h e a n g l e b e t w e e n t h e t r u e h o r i z o n a n d t h e
c e l e s t i a l e q u a t o r for a n o b s e r v e r l o c a t e d a t a m i d d l e l a t i t u d e , e . g .
50° ( a n g l e 900 - @ = 4 0 O ).
F i g u r e 5 . 7 , b shows t h e a n g l e b e t w e e n
t h e t r u e h o r i z o n and t h e c e l e s t i a l e q u a t o r f o r an o b s e r v e r l o c a t e d
o n t h e E q u a t o r ( a n g l e 9 0 ° - @ = g o o ) , w h i l e F i g u r e 5 . 7 , ~s h o w s t h e
a n g l e between t h e t r u e h o r i z o n and t h e c e l e s t i a l e q u a t o r f o r an

427

o b s e r v e r l o c a t e d a t t h e N o r t h or S o u t h P o l e .
(The a n g l e 90° - 4 =
0 , the t r u e horizon is p a r a l l e l t o the celestial equator, the zenith
/407
p o i n t Z c o i n c i d e s w i t h t h e n o r t h c e l e s t i a l p o l e P , and t h e n a d i r
Z' coincides with the south celestial pole P ' ) .

I t i s clear i n a l l t h r e e f i g u r e s t h a t t h e a n g l e between t h e
t r u e h o r i z o n of t h e o b s e r v e r and t h e c e l e s t i a l e q u a t o r i s always
e q u a l t o 90° m i n u s t h e l o c a l l a t i t u d e ( 9 0 - $1.
We c a n d r a w t h e f o l l o w i n g c o n c l u s i o n f r o m t h e a b o v e :
the slope
o f t h e d i u r n a l p a r a l l e l of s t a r s r e l a t i v e t o t h e t r u e h o r i z o n o f
t h e o b s e r v e r d e p e n d s on t h e l a t i t u d e o f t h e o b s e r v e r .
The h i g h e r
t h e l a t i t u d e of t h e observer, t h e smaller t h e slope of t h e d i u r n a l
p a r a l l e l s of t h e stars r e l a t i v e t o t h e horizon; t h e lower t h e
latitude, the greater the slope.

Rising and S e t t i n g , N e v e r - R i s i n g and N e v e r - S e t t i n g S t a r s

If w e know t h a t t h e p o s i t i o n o f t h e c e l e s t i a l e q u a t o r ( a n d
c o n s e q u e n t l y t h e d i u r n a l p a r a l l e l s of t h e s t a r s ) r e l a t i v e t o t h e
t r u e h o r i z o n of t h e o b s e r v e r depends on t h e l a t i t u d e o f t h e o b s e r v e r ,
i t w i l l b e c l e a r why s o m e s t a r s a t a c e r t a i n l a t i t u d e r i s e a n d s e t

Fig. 5.7.
A n g l e s Between t h e T r u e
Horizon and t h e Celestial Equator;
( a ) A t a L a t i t u d e o f 5 0 ° ; ( b ) On
t h e Equator; ( c ) A t the Poles.

428

at the horizon,

o t h e r s never set, and s t i l l o t h e r s never r i s e .

A s t a r n e v e r s e t s i f i t s d e c l i n a t i o n i s g r e a t e r t h a n 90° m i n u s
t h e l a t i t u d e of t h e l o c a t i o n , i . e . , i f 6 > 90° - $ .

For e x a m p l e , s e e F i g u r e 5 . 8 , a .
Given t h e l a t i t u d e o f t h e ob­
s e r v e r $ = 6 0 ° , t h e d e c l i n a t i o n o f t h e s t a r 6 = +45O.
From t h i s i t
i s c l e a r t h a t 9 0 ° - $ = 90 - 6 0 ° = 3 0 ° .
Since t h e declination 6
= +45O, i . e . , g r e a t e r t h a n 9 0 ° - $ , i t i s c l e a r t h a t t h e s t a r c a n /408
n o t s e t below t h e h o r i z o n of t h e o b s e r v e r .
I n Figure 5.8,a w e have
s k e t c h e d t h e c e l e s t i a l s p h e r e for a n o b s e r v e r l o c a t e d a t a l a t i t u d e
of 60°.
We m a r k o f f t h e d e c l i n a t i o n o f t h e s t a r a l o n g t h e m e r i d i a n
of t h e observer (;.e.,
t h e h o u r c i r c l e ) S O t h a t 6 = +45O, a n d t h e n
l a y out t h e d i u r n a l c i r c l e ( d i u r n a l p a r a l l e l ) of t h e star p a r a l l e l

b)

L

7

L

Fig. 5.8.
E x a m p l e s o f N e v e r - S e t t i n g S t a r s ; ( a ) The S t a r N e v e r S e t s
Below t h e H o r i z o n ; ( b ) The S t a r T o u c h e s t h e H o r i z o n .

Fig. 5.9.
Examples o f S t a r s t h a t
( b ) The S t a r d o e s n o t R i s e .

Set;

( a ) The S t a r R i s e s and S e t s ;

429

t o the celestial equator.
A s w e can see from t h e f i g u r e , t h i s
c i r c l e i s l o c a t e d above t h e h o r i z o n of t h e o b s e r v e r , and s o a s t a r
w h i c h moves a l o n g t h i s c i r c l e i n t h e c o u r s e of 24 h o u r s w i l l n e v e r
s e t below t h e o b s e r v e r ’ s horizon.
The s t a r t o u c h e s t h e h o r i z o n , b u t d o e s n o t g o b e l o w i t , i n
t h e c a s e when i t s d e c l i n a t i o n i s e q u a l t o 9 0 ° m i n u s t h e l a t i t u d e
i f 6 = 90 - 4 .
of t h e observer, ;.e.,
T a k e F i g u r e 5 . 8 , b for e x a m p l e .
The
t h e d e c l i n a t i o n of t h e s t a r 6 =
c l e a r t h a t 9 0 ° - C$ = 9 0 - 6 0 ° = 3 0 ° .
In
h a v e s a i d , i f 6 = 90° - 4 , t h e s t a r w i l l
h o r i z o n b u t w i l l n o t s e t below i t .

4 = 60°,

l a t i t u d e of t h e observer
t3Oo.
From t h i s i t i s
a c c o r d a n c e w i t h what w e
touch the observer’s

I n Figure 5.8,b w e have sketched t h e c e l e s t i a l sphere f o r an
o b s e r v e r l o c a t e d a t a. l a t i t u d e o f 6 0 ° .
Along t h e m e r i d i a n o f t h e
observer ( i . e . , t h e hour c i r c l e ) , we have p l o t t e d t h e declinati.on
of a s t a r 6 = t 3 0 ° , a n d h a v e t h e n d r a w n t h e d i u r n a l p a r a l l e l o f
t h i s star parallel t o the c e l e s t i a l equator.
A s we c a n s e e , t h e
d i u r n a l p a r a l l e l of t h e s t a r t o u c h e s t h e o b s e r v e r ’ s h o r i z o n , b u t
d o e s n o t c r o s s i t , ; . e . , a s t a r moving a l o n g i t s d i u r n a l p a r a l l e l
i n t h e c o u r s e o f 2 4 h o u r s g o e s down t o t h e h o r i z o n a n d t h e n r i s e s
again i n t h e course of its d i u r n a l journey.
A s t a r r i s e s a n d s e t s when i t s d e c l i n a t i o n ( i n t e r m s o f a b s o l u t e v a l u e ) i s l e s s t h a n 90° minus t h e l a t i t u d e o f t h e l o c a t i o n ,
i . e . , i f 6 < 90° - 4 .

L e t u s c o n s i d e r t h e following example:
The l a t i t u d e o f t h e
In
o b s e r v e r i s C$ = 3 0 ° , t h e d e c l i n a t i o n o f t h e s t a r i s 6 = t 4 O o .
F i g u r e 5 . 9 , a we h a v e s k e t c h e d t h e c e l e s t i a l s p h e r e for a n o b s e r v e r
located a t a l a t i t u d e of 30°.
Along t h e m e r i d i a n ( h o u r c i r c l e )
we h a v e m a r k e d o f f t h e d e c l i n a t i o n o f t h e s t a r 6 = t 4 0 ° a n d h a v e
drawn t h e d i u r n a l p a r a l l e l of t h e s t a r p a r a l l e l t o t h e e q u a t o r ( q ­
q’).
A s w e c a n s e e f r o m t h e d i a g r a m , a s t a r w h i c h moves i n t h e
c o u r s e of 24 h o u r s a l o n g i t s d i u r n a l p a r a l l e l w i l l be l o c a t e d below
t h e horizon f o r a c e r t a i n time ( t h e shaded p a r t of t h e d i u r n a l
p a r a l l e l ) , and w i l l be above t h e h o r i z o n
t h e rest of t h e t i m e .

A star never rises if i t s declina­
t i o n i s e q u a l t o or g r e a t e r t h a n 9 0 °
minus t h e l a t i t u d e o f t h e o b s e r v e r and
h a s a s i g n which d i f f e r s w i t h l a t i t u d e
( t h e l a t i t u d e i s p o s i t i v e and t h e

I’

430

Fig. 5.10
Division of t h e C e l e s t i a l
Sphere i n t o R e g i o n s w i t h R i s i n g and
S e t t i n g , Never-Setting and Never-Rising
Stars.

/409

d e c l i n a t i o n i s n e g a t i v e , or v i c e v e r s a ) , ; . e . ,
i f - 6 > 90° - 4 ,
o r 6 = 4 - 90°.
For e x a m p l e , t h e l a t i t u d e o f t h e o b s e r v e r 4 = 60°N,
t h e d e c l i n a t i o n of t h e s t a r 6 = - 3 O O .
I n F i g u r e 5 . 9 , b w e h a v e s k e t c h e d t h e c e l e s t i a l s p h e r e for a n
o b s e r v e r a t a l a t i t u d e of 4 = 6 0 ° N .
Along t h e m e r i d i a n o f t h e
o b s e r v e r ( h o u r c i r c l e ) w e h a v e m a r k e d t h e d e c l i n a t i o n of t h e s t a r
6 = -30° ( b e l o w t h e e q u a t o r ) a n d t h e d i u r n a l p a r a l l e l o f t h e s t a r
parallel t o the equator.
A s w e can see from t h e diagram, a star
which moves a l o n g i t s d i u r n a l p a r a l l e l w i l l a l w a y s b e b e l o w t h e
hori,zon and w i l l never r i s e .
This is completely understandable,
since the declination of the star is negative.
If t h e s t a r h a d a
negative d e c l i n a t i o n s t i l l g r e a t e r than 90 - 4 , i t s d i u r n a l c i r c l e
would b e l o c a t e d s t i l l f u r t h e r below t h e h o r i z o n .
Consequently, t h e e n t i r e c e l e s t i a l s p h e r e of an o b s e r v e r l o ­
c a t e d a t a given l a t i t u d e can be d i v i d e d i n t o t h r e e p a r t s :
(1)

The p o r t i o n o f t h e c e l e s t i a l s p h e r e w i t h s t a r s t h a t n e v e r

set.
( 2 )
The p o r t i o n o f t h e c e l e s t i a l s p h e r e w i t h s e t t i n g a n d
r i s i n g stars.

(3)

The p o r t i o n o f t h e c e l e s t i a l s p h e r e w i t h s t a r s t h a t n e v e r

rise.
A l l t h r e e p o r t i o n s o f t h e c e l e s t i a l s p h e r e a r e shown i n F i g u r e

5.10.for a n o b s e r v e r a t a l a t i t u d e o f 6 0 ° N .
The c i r c u m f e r e n c e i s
t h e p l a n e o f t h e c e l e s t i a l m e r i d i a n , ZZ' i s t h e v e r t i c a l l i n e o f
t h e o b s e r v e r , PP' i s t h e w o r l d a x i s .
The s t r a i g h t l i n e B B ' i s t h e
s e c t i o n o f t h e p l a n e o f t h e c e l e s t i a l m e r i d i a n a s c u t by t h e d i u r n a l
c i r c l e o f t h e s t a r , t o u c h i n g t h e h o r i z o n of t h e o b s e r v e r b u t n o t
/410
p a s s i n g b e l o w i t ( a s t a r w h o s e 6 = 90° - + ) .
This i s t h e boundary
o f t h e r e g i o n of stars t h a t n e v e r s e t w i t h t h a t o f t h e o n e s w h i c h
r i s e a n d s e t for a g i v e n l a t i t u d e of t h e o b s e r v e r .
The s t r a i g h t
l i n e DD' i s t h e s e c t i o n of t h e plane of t h e c e l e s t i a l meridian as
c u t by t h e h o u r c i r c l e o f t h e s t a r i n t h e S o u t h e r n H e m i s p h e r e ,
t o u c h i n g t h e h o r i z o n b u t n o t g o i n g below i t ( a s t a r whose d e c l i n a ­
t i o n i s - 6 = 90° - 4 ) .
T h i s i s t h e b o u n d a r y o f t h e r e g i o n of s t a r s
t h a t n e v e r r i s e w i t h t h a t o f t h e o n e s which r i s e and s e t f o r a
given l a t i t u d e of t h e observer.
Motion o f

Stars a t

t h e T e r r e s t r i a l Poles

In order t o g e t a b e t t e r i d e a of t h e nature of t h e d i u r n a l
motion of s t a r s a t t h e t e r r e s t r i a l p o l e s , l e t u s c o n s t r u c t a form
of c e l e s t i a l s p h e r e f o r a n o b s e r v e r l o c a t e d a t t h e N o r t h P o l e .
In
t h i s c a s e , t h e a l t i t u d e of t h e P o l e a b o v e t h e h o r i z o n w i l l b e e q u a l
t o the l a t i t u d e of the observer.
Since t h e observer is located a t
t h e N o r t h P o l e , 4 = 90°N a n d c o n s e q u e n t l y t h e a l t i t u d e o f t h e P o l e
a b o v e t h e h o r i z o n w i l l b e 9 0 ° ( F i g . 5.11).

431

_-

I


The w o r l d a x i s c o i n c i d e s w i t h t h e v e r t i c a l l i n e , i . e . , t h e
n o r t h c e l e s t i a l p o l e P c o i n c i d e s with t h e z e n i t h Z and t h e south
celestial pole P' c o i n c i d e s with t h e n a d i r Z ' , while t h e plane of
t h e c e l e s t i a l e q u a t o r w i l l c o i n c i d e w i t h t h e p l a n e of t h e h o r i z o n .
T h i s means t h a t a l l s t a r s , d e p e n d i n g on t h e i r d i u r n a l r o t a t i o n ,
w i l l move p a r a l l e l t o t h e h o r i z o n a n d t h e i r a l t i t u d e s w i l l n o t
change as t h e c e l e s t i a l s p h e r e r o t a t e s .
All s t a r s l o c a t e d i n t h e
N o r t h e r n H e m i s p h e r e ( h a v i n g p o s i t i v e d e c l i n a t i o n s ) w i l l move a b o v e
t h e h o r i z o n , w h i l e t h e s t a r s which a r e l o c a t e d i n t h e S o u t h e r n
H e m i s p h e r e ( a n d h a v e n e g a t i v e d e c l i n a t i o n s ) w i l l move b e l o w t h e
horizon, i . e . , a l l stars with 6 > Oo w i l l never s e t and a l l those
with 6 < Oo w i l l never r i s e .

M o t i o n of S t a r s at M i d d l e L a t i t u d e s

L e t u s examine t h e n a t u r e o f t h e d i u r n a l m o t i o n of t h e s t a r s
a t m i d d l e l a t i t u d e s , when 0 < 4 < 9 0 ° .

Parallel-\

diurnal

h o r i z o n and.eguato.rs(Ql

P 72 '

F i g u r e 5 . 1 2 shows t h e a p ­
p e a r a n c e of t h e c e l e s t i a l s p h e r e
a t a l a t i t u d e c l o s e t o 45O.
Due
t o t h e i n c l i n a t i o n of t h e world
a x i s , a l l s t a r s move a t a n a n g l e
t o the horizon (parallel t o the
celestial equator).
A t middle
l a t i t u d e s , a c o n s i d e r a b l e number
of s t a r s r i s e and s e t (between
/411
t h e p a r a l l e l s NK a n d DS).
Stars
which are l o c a t e d f a r t h e r t h a n
t h e s e p a r a l l e l s from t h e c e l e s t i a l
e q u a t o r e i t h e r n e v e r r i s e or
never set.

Fig. 5.11.
Motion of t h e
Stars a t the T e r r e s t i a l Poles.
reeioq of

Fig. 5.12.
Motion of t h e S t a r s
a t Middle L a t i t u d e s .
432

.-

..

Fig. 5.13.
Motion o f t h e
S t a r s at t h e Equator.

Motion o f

S t a r s a t t h e Equator

S i n c e t h e l a t i t u d e o f a n o b s e r v e r l o c a t e d on t h e E q u a t o r i s
equal t o zero, it i s clear t h a t t h e world axis l i e s i n t h e plane of
t h e h o r i z o n and c o i n c i d e s w i t h t h e m e r i d i a n l i n e on t h e p l a n e o f
t h e horizon, while t h e terrestrial poles PP' coincide with the
n o r t h and s o u t h p o i n t s N a n d S , r e s p e c t i v e l y .
Hence, t h e p l a n e of t h e c e l e s t i a l e q u a t o r , as w e l l as t h e p l a n e s
of t h e c e l e s t i a l p a r a l l e l s , are l o c a t e d p e r p e n d i c u l a r t o t h e p l a n e
of t h e observer's horizon.
H o w e v e r , s i n c e we know t h a t t h e d i u r n a l
motion of t h e s t a r s o c c u r s p a r a l l e l t o t h e c e l e s t i a l e q u a t o r , w e
can see from Figure 5.13 t h a t t h e r e a r e no stars v i s i b l e a t t h e
All s t a r s a r e l o c a t e d above
E q u a t o r w h i c h n e v e r r i s e or n e v e r s e t .
t h e h o r i z o n f o r 1 2 h o u r s and below t h e h o r i z o n f o r 1 2 h o u r s , s i n c e
a l l t h e p a r a l l e l s o f t h e s t a r s a r e c u t i n h a l f by t h e h o r i z o n .
Culmination of

Stars

The d i u r n a l p a r a l l e l o f a s t a r c r o s s e s t h e c e l e s t i a l m s r i d i a n
a t two p o i n t s ( F i g . 5 . 1 4 , a ) .
These p o i n t s a r e c a l l e d t h e c u l m i n a t i o n
points.
T h e moment o f p a s s a g e o f a g i v e n s t a r t h r o u g h t h e c e l e s t i a l
m e r i d i a n i s c a l l e d t h e moment of c u l m i n a t i o n , or i t i s s a i d t h a t
the star culminates.
The u p p e r c u l m i n a t i o n of a s t a r i s t h e moment when t h e s t a r i s
a t i t s g r e a t e s t a l t i t u d e above t h e horizon.
The l o w e r c u l m i n a t i o n
of a s t a r i s t h e moment when t h e s t a r i s a t i t s l o w e s t a l t i t u d e
above t h e h o r i z o n .
In t h e case of stars t h a t s e t , t h e lower cul1412
m i n a t i o n t a k e s p l a c e below t h e h o r i z o n .
Upper c u l m i n a t i o n o f a s t a r c a n t a k e p l a c e on t h e s o u t h e r n
p o r t i o n of a m e r i d i a n (between t h e s c u t h p o i n t and t h e z e n i t h ) , and
on t h e s o u t h e r n p o r t i o n of t h e m e r i d i a n ( b e t w e e n t h e z e n i t h a n d t h e
n o r t h c e l e s t i a l p o l e ) , d e p e n d i n g on t h e r e l a t i o n s h i p b e t w e e n t h e

culmination

S


Fig. 5.14.
Meridian:

C u l m i n a t i o n o f S t a r s on t h e S o u t h e r n S e c t i o n o f t h e
( a ) i n t h e A p p a r e n t System of C o o r d i n a t e s ; ( b ) i n t h e
E q u a t o r i a l System.
433

l a t i t u d e of t h e o b s e r v e r and t h e d e c l i n a t i o n of t h e s t a r .

A s t a r c u l m i n a t e s on t h e s o u t h e r n p a r t of t h e m e r i d i a n (between
t h e s o u t h p o i n t a n d t h e z e n i t h ) when t h e l a t i t u d e o f t h e o b s e r v e r
i s g r e a t e r t h a n t h e d e c l i n a t i o n o f t h e s t a r , ? . e . , i f (9 > 6 .
A s t a r c u l m i n a t e s on t h e n o r t h e r n p a r t of t h e m e r i d i a n ( b e t w e e n
t h e z e n i t h a n d t h e n o r t h c e l e s t i a l p o l e ) when t h e l a t i t u d e o f t h e
o b s e r v e r i s l e s s t h a n t h e d e c l i n a t i o n o f t h e s t a r , i . e . , i f (9 < 6 .
I n Figure 5.14,b t h e c e l e s t i a l sphere has been sketched i n
s i m p l i f i e d f a s h i o n , i . e . , t h e c i r c l e s o f t h e h o r i z o n , e q u a t o r , and
p a r a l l e l s a r e n o t r e p r e s e n t e d as c i r c l e s b u t as d i a m e t e r s and
chords.
A s w e c a n s e e f r o m t h e d i a g r a m , t h e l a t i t u d e o f t h e ob­
v

v

s e r v e r i s g r e a t e r than t h e declination of t h e star:
NP > Q M , i . e . ,
(9 > 6 , s o t h a t t h e u p p e r c u l m i n a t i o n o f t h e s t a r ( p o i n t M') l i e s
on t h e s o u t h e r n p a r t o f t h e m e r i d i a n ( b e t w e e n t h e z e n i t h Z a n d t h e
south point S ) .
L e t us determine t h e a l t i t u d e of t h e s t a r i n t h i s case.
T h e a l t i t u d e o f t h e s t a r ( h ) i s t h e a r c SM',
c/

U

= SQ'

-

((9

t

Q'Z

v

- M'Z,

3

a n d SQ'

= 90° - ( 9 ;

c/

Q'Z

b u t t h e a r c SM'
0

= $I a n d M ' Z

= $I - 6.

By s u b s t i t u t i n g t h e s e v a l u e s , w e w i l l o b t a i n h = 90° - (9 t (9
- 6 ) or h = 90° - $I t 6 .

1413
I n Figure 5.15,a t h e c e l e s t i a l sphere has a l s o been sketched
i n a s i m p l i f i e d form.
H e r e t h e l a t i t u d e o f t h e o b s e r v e r (NP) i s
l e s s t h a n t h e d e c l i n a t i o n o f t h e s t a r ( Q ' M ' ) , ? . e . , (9 < 6 , s o t h a t
t h e u p p e r c u l m i n a t i o n o f t h e s t a r ( p o i n t M ' o c c u r s on t h e n o r t h e r n
p a r t of t h e m e r i d i a n ( b e t w e e n t h e z e n i t h a n d t h e p o i n t o f t h e n o r t h
celestial pole).
L e t u s d e t e r m i n e t h e a l t i t u d e of t h e s t a r i n t h i s

N

I'

Fig. 5.15.
Meridian:

434

C u l m i n a t i o n o f a S t a r on t h e N o r t h e r n S e c t i o n o f t h e
( a ) C o o r d i n a t e s o f Upper C u l m i n a t i o n ; ( b ) C o o r d i n a t e s
o f Lower C u l m i n a t i o n .

The a l t i t u d e o f t h e s t a r ( h ) i s N M ' Y b u t

case.

LJ

U

NM'

= 180"

-M'Q'

- Q'S
U

h = 180°-6-(90-~),

or

h = 90"- 6

i.e.,

+ 'p.

If t h e s t a r d o e s n o t s e t , w e w i l l s o m e t i m e s b e i n t e r e s t e d i n
i t s a l t i t ' u d e a t t h e moment o f l o w e r c u l m i n a t i o n .
A s w e s e e from Figure 5.15,b,

U

U

MQ = QN

+

0

NM,

u

but MQ = 6;

d

QN =

'd

90 - $ ;

NM = h .

By s u b s t i t u t i n g t h e v a l u e s o f t h e s e a r c s , w e w i l l o b t a i n 6 =
90° - $ + h , s o t h a t h = $ + 6 - 90°, ; . e . , t h e a l t i t u d e of t h e
s t a r a t t h e moment o f l o w e r c u l m i n a t i o n i s e q u a l t o t h e l a t i t u d e
o f t h e o b s e r v e r p l u s t h e d e c l i n a t i o n o f t h e s t a r minus 90°.

ProbZems a n d Exercises
1. What m u s t b e t h e d e c l i n a t i o n o f a s t a r i f a t t h e l a t i t u d e
Moscow ( 9 = 5 5 O 4 8 ' ) ( a ) i t n e v e r s e t s , ( b ) i t r i s e s a n d s e t s ,
or ( c ) i t n e v e r r i s e s ?

of

Solution.
( a ) I n o r d e r f o r a s t a r n e v e r t o s e t , we m u s t h a v e
6 > 90 - $.
I f w e s u b s t i t u t e t h e v a l u e o f t h e l a t i t u d e o f Moscow
6 must be g r e a t e r
( 5 S 0 4 8 ' ) , we w i l l o b t a i n 6 > 9 0 - 5 5 O 4 8 ' , ; . e . ,
t h a n +34O12'.
C o n s e q u e n t l y , a l l s t a r s which have a d e c l i n a t i o n
g r e a t e r t h a n + 3 4 O 1 2 ' w i l l n e v e r s e t d t t h e l a t i t u d e o f Moscow.
T y p i c a l s t a r s i n t h i s c a t e g o r y a r e C a p e l l a , A l i o t h , Vega, Deneb,
and P o l a r i s .

/414

( a ) S t a r s r i s e and s e t , a s w e know, i f t h e a b s o l u t e v a l u e o f
In
t h e i r d e c l i n a t i o n i s l e s s t h a n 90° - $, i . e . , 6 < 90° - $1.
o u r e x a m p l e , 6 < 34O12'.
Stars i n t h i s category f o r the latitude
o f Moscow a r e R e g u l u s , A r c t u r u s , A l t a i r , e t c .
( c ) I n o r d e r f o r a s t a r n e v e r t o r i s e , i t s d e c l i n a t i o n must
b e e q u a l t o or g r e a t e r t h a n 9 0 ° - $ a n d v a r i e s w i t h t h e l a t i t u d e o f
t h e observer, i . e . , -6
90° - 9 .
I n o u r e x a m p l e , 6 must b e e q u a l
t o or g r e a t e r t h a n 3 4 O 1 2 ' .
I n a d d i t i o n , i t must a l s o b e n e g a t i v e
(inasmuch as w e are t a l k i n g about n o r t h l a t i t u d e ) .

(2)
C a l c u l a t e which o f t h e f o l l o w i n g s t a r s :
Aldebaran,
Alpherants, Capella, S i r i u s , Procyon, Arcturus) w i l l never r i s e ,
r i s e a n d s e t , and n e v e r s e t a t t h e l a t i t u d e o f L e n i n g r a d ( $ = 59O59'N).

435

., ,

.

...._.. ....... .._._._
_.. . . . . .....

...

... .... . . ... . .

... .

a---_.-_-.

I
3.
55O48’)

C a l c u l a t e t h e a l t i t u d e o f t h e s t a r D u b k h e a t Moscow ( 4 =
a t t h e moment o f u p p e r c u l m i n a t i o n .

4.
A t what a l t i t u d e d o e s t h e s t a r S i r i u s c u l m i n a t e . ( u p p e r
culmination) a t Leningrad?
5.
Show m a t h e m a t i c a l l y t h a t a l l s t a r s w h o s e 6 > 0 d o n o t s e t
a t t h e P o l e s , w h i l e t h o s e which have 6 < 0 n e v e r r i s e .

3.

The Motion of the Sun


The A n n u a l

M o t i o n of

the

Sun

The Sun p a r t i c i p a t e s i n t h e d i u r n a l m o t i o n a l o n g w i t h a l l
other stars.
The a p p a r e n t d i u r n a l m o t i o n o f t h e Sun i s a l s o
r e s u l t o f t h e d i u r n a l m o t i o n o f t h e E a r t h i n r o t a t i n g on i t s
H o w e v e r , t h e S u n a l s o h a s i t s own s o - c a l l e d i n t r i n s i c m o t i o n
t h e c o u r s e o f a y e a r , c a l l e d t h e annuaz m o t i o n of t h e Sun.

the
the
axis.
in

The a n n u a l m o t i o n o f t h e Sun i s d i f f i c u l t t o o b s e r v e d i r e c t l y .
However, i f t h e s t a r s w e r e v i s i b l e i n t h e d a y t i m e , a n d w e w e r e t o
o b s e r v e t h e m u t u a l p o s i t i o n s o f t h e Sun a n d s t a r s f o r a c e r t a i n
p e r i o d o f t i m e , we w o u l d s e e t h a t t h e m u t u a l p o s i t i o n s o f t h e s e
b o d i e s would c h a n g e i n t h e c o u r s e o f t i m e , w h i l e t h e m u t u a l p o s i t i o n s
o f t h e s t a r s and c o n s t e l l a t i o n s i n t h e s k y would n o t c h a n g e .
The d i r e c t i o n o f t h i s i n t r i n s i c a n n u a l m o t i o n o f t h e Sun i s
o p p o s i t e t o t h e d i u r n a l motion of t h e s t a r s , ; . e . ,
from w e s t t o e a s t .
The a n n u a l m o t i o n o f t h e Sun i s a p p a r e n t ( a s i s t h e d i u r n a l
motion), and occurs as t h e r e s u l t of t h e annual r o t a t i o n of t h e
E a r t h a r o u n d t h e Sun.
A s we d i d i n d e s c r i b i n g t h e d i u r n a l m o t i o n o f t h e s k y a n d
s t a r s , w e w i l l c o n s i d e r t h a t t h e Sun i s m o v i n g a n d t h e E a r t h s t a n d s
still.

Due t o t h e e x i s t e n c e o f s o - c a l l e d a n n u a l m o t i o n o f t h e S u n , t h e
d i u r n a l m o t i o n of t h e Sun h a s some u n u s u a l a s p e c t s , s u c h a s :
(a)
The i n t e r v a l o f t i m e b e t w e e n t h e r i s i n g a n d s e t t i n g o f
Sun v a r i e s c o n t i n u o u s l y i n t h e c o u r s e o f a y e a r .
(b)
The a z i m u t h o f t h e p o i n t s o f t h e r i s i n g a n d s e t t i n g o f
t h e S u n c h a n g e s c o n s t a n t l y d u r i n g t h e y e a r , a n d t h e p o i n t s wher:e
t h e S u n r i s e s a n d s e t s move a b o u t , s h i f t i n g f r o m o n e q u a r t e r o f t h e
horizon t o another.
For e x a m p l e , i n summer t h e S u n r i s e s i n t h e
n o r t h e a s t , and s e t s i n t h e n o r t h w e s t .
In winter, it r i s e s i n the
s o u t h e a s t and s e t s i n t h e southwest.
(c)
The m e r i d i o n a l a l t i t u d e o f t h e Sun c h a n g e s c o n s t a n t l y i n
t h e course of a y e a r .
436

/415

Ecliptic.
I n t h e c o u r s e o f i t s i n t r i n s i c m o t i o n , t h e c e n t e r of
t h e Sun moves a l o n g a g r e a t c i r c l e of a s p h e r e c a l l e d t h e e c l i p t i c
(Fig. 5.16,a).
The p l a n e o f t h e e c l i p t i c i n t e r s e c t s t h e p l a n e o f
t h e c e l e s t i a l e q u a t o r a t a n a n g l e o f 23O27' a t t w o p o i n t s :
at the
p o i n t of t h e v e r n a l e q u i n o x ( y ) and t h e p o i n t o f t h e a u t u m n a l
e q u i n o x (52).
Tropic year.
The Sun c o m p l e t e s a j o u r n e y a r o u n d t h e e c l i p t i c
( t h r o u g h 3 6 0 O ) i n 3 6 5 . 2 4 2 2 mean d a y s .

The i n t e r v a l o f t i m e b e t w e e n s u c c e s s i v e p a s s a g e s o f t h e c e n t e r
o f t h e Sun t h r o u g h t h e p o i n t of t h e v e r n a l e q u i n o x i s c a l l e d t h e

t r o p i c year.
Sidereal year.
I n t h e c o u r s e o f i t s a n n u a l m o t i o n , t h e Sun
makes a f u l l r o t a t i o n r e l a t i v e t o t h e s t a r s i n a p e r i o d of t i m e
somewhat l o n g e r t h a n t h e t r o p i c y e a r ( i . e . , i n 365.25636 d a y s ) .
T h i s t i m e i n t e r v a l , e q u a l t o t h e p e r i o d of t i m e r e q u i r e d f o r t h e
A f t e r
Earth t o r o t a t e around t h e Sun, i s c a l l e d t h e sidereal year.
t h i s i n t e r v a l , t h e Sun w i l l h a v e r e t u r n e d t o i t s o r i g i n a l p o s i t i o n
among t h e s t a r s .

Motion o f t h e Sun Along t h e E c Z i p t i c
On M a r c h 2 1 , i n t h e c o u r s e o f i t s a n n u a l m o t i o n , t h e S u n
crosses t h e c e l e s t i a l equator a t t h e point of t h e vernal equinox.
This d a t e i s c a l l e d t h e date o f t h e vernal equinox.
When t h e S u n
p a s s e s t h r o u g h t h e p o i n t y , i t s d e c l i n a t i o n and r i g h t a s c e n s i o n a r e
A f t e r March 2 1 , t h e Sun c o n t i n u e s i t s m o t i o n a n d
equal t o zero.
p a s s e s i n t o t h e Northern Hemisphere, and i t s d e c l i n a t i o n b e g i n s t o
increase (;.e.,
becomes p o s i t i v e ) .
Thus, a f t e r t h r e e months (on
J u n e 2 2 ) t h e Sun i s a t t h e p o i n t K ( s e e F i g . 5 . 1 6 , a ) , w h i c h i s
c a l l e d t h e p o i n t of t h e summer s o l s t i c e .
A t t h i s p o i n t , t h e Sun
i s a t i t s h i g h e s t p o s i t i o n above t h e c e l e s t i a l e q u a t o r .
The
a=180'

S e p t 22

a=9 P

J u n e 2;

0'

CC=OO

M a r 21

Fig.

5.16.

Annual Motion of t h e Sun:

( a ) Motion a l o n g t h e E c l i p t i c ;

(b) C o o r d i n a t e s o n t h e D a t e s o f t h e E q u i n o x e s a n d S o l s t i c e s .
437


d e c l i n a t i o n o f t h e Sun a t t h i s p o i n t i s +23O27', a n d i t s r i g h t
For s e v e r a l d a y s , i t s a l t i t u d e a t
a s c e n s i o n i s 90° o r 6 h o u r s .
noon r e m a i n s n e a r l y c o n s t a n t , i . e . , +23O27', s o t h a t t h e p o i n t K ,
w h i c h c o r r e s p o n d s t o t h e c o n s t e l l a t i o n C a p r i c o r n , h a s b e e n named
The p o i n t s w h e r e t h e S u n r i s e s
t h e d a t e of t h e summer s o t s t i c e .
a n d s e t s o n t h i s d a t e w i l l b e a t t h e i r maximum d i s t a n c e s f r o m t h e
e a s t a n d w e s t p o i n t s on t h e h o r i z o n .
A f t e r t h e d a t e o f t h e summer
s o l s t i c e , t h e Sun b e g i n s t o a p p r o a c h t h e c e l e s t i a l e q u a t o r , i t s
d e c l i n a t i o n b e g i n s t o d e c r e a s e , a n d by S e p t e m b e r 23 i t a g a i n i n t e r ­
s e c t s t h e c e l e s t i a l e q u a t o r a t t h e p o i n t o f t h e v e r n a l e q u i n o x (&,
i n the constellation Libra).
When t h e S u n p a s s e s t h r o u g h t h e p o i n t o f t h e v e r n a l e q u i n o x
d e c l i n a t i o n becomes e q u a l t o z e r o , w h i l e i t s r i g h t a s c e n s i o n
b e c o m e s 180° or 1 2 h o u r s .

( c ) ,i t s

TABLE 5 . 1
I'

I

Date

~

O c c u r s on

Coordinates

I

Declination

(6)IRightA
scensbn ( a )
-~

i

Vernal equinox
Summer s o l s t i c e
Autumnal e q u i n o x
Winter s o l s t i c e

00

+23'27
00

-23O27

'
'

00

or 6 h o u r s
180° or 1 2 h o u r s
2 7 0 ° or 1 8 h o u r s
90°

S e p t e m b e r 2 3 i s c a l l e d t h e d a t e of t h e autumnaZ e q u i n o x .
A l l
of t h e e v e n t s o f t h e d a t e o f t h e v e r n a l e q u i n o x a r e r e p e a t e d on
t h i s date.
A f t e r S e p t e m b e r 2 3 , t h e Sun p a s s e s i n t o t h e S o u t h e r n Hemisphere
and i t s d e c l i n a t i o n becomes n e g a t i v e .
On D e c e m b e r 2 2 , t h e S u n i s
a t i t s lowest p o s i t i o n r e l a t i v e t o t h e c e l e s t i a l e q u a t o r and i s a t
/417
t h e point of the winter so l st ic e (the point L, i n the constellation
On t h e
Leo).
T h i s d a t e i s c a l l e d t h e d a t e of t h e w i n t e r s o Z s t i c e .
d a t e o f t h e w i n t e r s o l s t i c e , t h e Sun h a s a d e c l i n a t i o n o f - 2 3 O 2 7 ' ,
The p o i n t s w h e r e
w h i l e i t s r i g h t a s c e n s i o n i s 2 7 0 ° or 1 8 h o u r s .
t h e Sun r i s e s a n d s e t s o n t h i s d a t e a r e f a r t h e s t s o u t h f r o m t h e
e a s t and w e s t p o i n t s on t h e h o r i z o n .

A f t e r December 2 2 , t h e Sun b e g i n s i t s r i s e a l o n g t h e e c l i p t i c ,
a n d on March 2 1 i t h a s a g a i n r i s e n t o t h e p o i n t o f t h e v e r n a l
e q u i n o x , w h e r e i t s d e c l i n a t i o n a n d r i g h t a s c e n s i o n a r e o n c e more
equal t o zero.
T h u s , we c a n d r a w u p a s p e c i a l t a b l e f o r t h e a n n u a l m o t i o n o f
t h e Sun a l o n g t h e e c l i p t i c , s h o w i n g i t s c o o r d i n a t e s ( T a b l e 5 . 1 ;
Fig. 5.16,b).

438

I n t h e c o u r s e o f a y e a r , a s i t moves t h r o u g h t h e s k y (among
t h e s t a r s ) , t h e Sun p a s s e s t h r o u g h 1 2 c o n s t e l l a t i o n s , c a l l e d t h e
signs of t h e zodiac.
T h e y h a v e r e c e i v e d t h i s name b e c a u s e t h e
m a j o r i t y o f them b e a r t h e names o f a n i m a l s ( A r i e s , t h e R a m ; T a u r u s ,
t h e B u l l , e t c . ) , and t h e word z o a n i n Greek means " a n i m a l " .
A s the
S u n m o v e s among t h e s t a r s i n t h e c o u r s e of a y e a r , i t i s i n t h e
following positions:
on t h e d a t e o f t h e v e r n a l e q u i n o x (March 211,
i n t h e c o n s t e l l a t i o n P i s c e s ( t h e F i s h e s ) ; o n t h e d a t e o f t h e summer
s o l s t i c e J u n e 2 2 , i n t h e c o n s t e l l a t i o n Gemini ( t h e T w i n s ) ; on t h e
d a t e of t h e a u t u m n a l e q u i n o x (September 231, i n t h e c o n s t e l l a t i o n
V i r g o ( t h e V i r g i n ) , a n d on t h e d a t e of t h e w i n t e r s o l s t i c e (December
2 2 1 , i n t'he c o n s t e l l a t i o n S a g i t t a r i u s ( t h e A r c h e r ) .

Diurnal Motion o f the Sun

T h e M o t i o n of t h e S u n at t h e North P o l e
Due t o t h e c o n s t a n t c h a n g e i n t h e d e c l i n a t i o n o f t h e S u n , i t s
a l t i t u d e a t t h e t e r r e s t r i a l poles a l s o changes.
I t d o e s n o t move
p a r a l l e l t o the horizon, but along a s p i r a l path.
Since t h e declin­
a t i o n o f t h e Sun w i l l b e p o s i t i v e f o r s i x mo n t h s ( a n d t h e c e l e s t i a l
equator f o r an observer a t t h e North Pole w i l l coincide with t h e
h o r i z o n ) , t h e c e n t e r o f t h e Sun w i l l a c t l i k e a s t a r t h a t n e v e r s e t s
a t t h e N o r t h P o l e f r o m March 2 1 t o S e p t e m b e r 2 3 , < . e . , i t w i l l
remain above t h e h o r i z o n .
D u r i n g t h e o t h e r h a l f o f t h e y e a r , when t h e S u n h a s a n e g a t i v e
d e c l i n a t i o n , it w i l l be below t h e h o r i z o n as s e e n from t h e North
Pole.
T h e r e f o r e , t h e r e a r e s i x m o n t h s of d a y a n d s i x m o n t h s o f
night at the terrestrial poles.
T h e S u n r e a c h e s i t s maximum a l t i t u d e a b o v e t h e h o r i z o n a t t h e
N o r t h P o l e o n t h e d a t e o f t h e summer s o l s t i c e , < . e . , o n J u n e 2 2 .
The a l t i t u d e o f t h e S u n o n t h a t d a t e i s e q u a l t o i t s maximum d e c l i n ­
a t i o n , i . e . , 23O27'.
A t t h e S o u t h P o l e , t h e Sun r e a c h e s i t s m a x i ­
mum a l t i t u d e o n t h e d a t e o f o u r w i n t e r s o l s t i c e , ; . e . , D e c e m b e r 2 2 .

M o t i o n of t h e S u n b e t w e e n t h e N o r t h P o l e a n d t h e A r c t i c CircZe
If an o b s e r v e r i s beyond t h e A r c t i c C i r c l e :.e.,
f a r t h e r northI41*
t h a n t h e l a t i t u d e o f 66O33' ( t h e l a t i t u d e o f t h e A r c t i c C i r c l e ) ,
t h e S u n d u r i n g i t s d i u r n a l m o t i o n w i l l b e a n e v e r - s e t t i n g s t a r for
p a r t of t h e y e a r , a r i s i n g and s e t t i n g s t a r p a r t of t h e y e a r and a
never-rising star f o r p a r t of t h e year.
In order t o understand
t h i s p h e n o m e n o n , l e t U S r e c a l l t h e c o n d i t i o n s �or t h e n o n r i s i n g ,
s e t t i n g , r i s i n g and n o n s e t t i n g o f heavenly b o d i e s .

Heavenly b o d i e s do n o t s e t i f t h e i r d e c l i n a t i o n i s e q u a l t o
or m o r e t h a n 9 0 ° m i n u s t h e l a t i t u d e o f t h e o b s e r v e r , i . e . i f 6 >
9 0 ° - $ I . T h i s s i t u a t i o n a l s o a p p l i e s t o t h e d i u r n a l m o t i o n o f the
Sun.
I f , f o r example, t h e observer i s standing a t a l a t i t u d e of
76ON ( b e t w e e n t h e N o r t h P o l e a n d t h e A r c t i c C i r c l e ) , t h e n a c c o r d i n g
439

t o t h e c o n d i t i o n s e t f o r t h above t h e Sun w i l l n o t s e t a f t e r t h e
d a t e when i t s d e c l i n a t i o n i s e q u a l t o or more t h a n 9 0 ° - 4 , i . e .
m o r e t h a n 90-76O = + 1 4 O .
T h i s phenomenon ( t o c i t e a s p e c i f i c
A f t e r A p r i l 26 t h e S u n w i l l r i s e
e x a m p l e ) b e g i n s on A p r i l 26.
h i g h e r and h i g h e r above t h e horizon.
T h e S u n r e a c h e s a maximum a l t i t u d e a b o v e t h e h o r i z o n o n t h e
A f t e r J u n e 2 2 , t h e Sun w i l l d i p
d a t e o f t h e summer s o l s t i c e .
When i t s d e c l i n a t i o n
toward t h e horizon b u t w i l l s t i l l not s e t .
i s a g a i n e q u a l t o 900 - 4 , t h e Sun w i l l t o u c h t h e h o r i z o n .
A f t e r August 1 9 , t h e S u n ' s d e c l i n a t i o n w i l l b e l e s s t h a n 90°
i . e . 6 < 90° - $ and f o r a f i x e d t i m e i t w i l l a p p e a r t o a n
o b s e r v e r l o c a t e d a t t h i s l a t i t u d e as a r i s i n g and s e t t i n g s t a r .

-

$,

T h i s phenomenon w i l l c o n t i n u e u n t i l t h e d e c l i n a t i o n o f t h e
6 > 90° - $ and w i l l
S u n i s n o t e q u a l t o or m o r e t h a n 9 0 ° - 4 , : . e .
have a s i g n o p p o s i t e t o t h a t of t h e l a t i t u d e ( i . e . t h e l a t i t u d e
i s p o s i t i v e and t h e d e c l i n a t i o n i s n e g a t i v e ) .
For a n o b s e r v e r
B e g i n n i n g on
l o c a t e d a t 76ON t h i s p h e n o m e n o n b e g i n s o n N o v e m b e r 3 .
N o v e m b e r 3 , for a n o b s e r v e r l o c a t e d a t a l a t i t u d e o f 7 6 O , t h e S u n
w i l l not s e t , s i n c e -6 > 90° - $.
The Sun w i l l n o t r i s e u n t i l i t s n e g a t i v e d e c l i n a t i o n i s e q u a l
t o or l e s s t h a n 9 0 ° - $ ( - 6 < 9 0 ° - $ 1 , ; . e . , b e g i n n i n g o n t h e d a t e
t h e S u n ' s d e c l i n a t i o n h a s a-value
o f 14O a n d l e s s .
I n our example,
t h i s phenomenon s t a r t s on F e b r u a r y 9 .
A f t e r t h i s d a t e , t h e Sun
w i l l r i s e a n d s e t e v e r y d a y a n d t h e p e r i o d of d a y l i g h t w i l l g r a d u a l l y
increase.
F i n a l l y , when t h e S u n ' s d e c l i n a t i o n i s e q u a l t o or m o r e
b e g i n n i n g on A p r i l 2 6 , ' t h e Sun a g a i n w i l l n o t
t h a n 90° - $, ; . e . ,
set.
/419

TABLE 5 . 2
.~
'

Spring

tLhaet i p
t uods ei t ioofn ' B e g .
i n degrees
ldate
68
70
72
74
76
78
80
82
84
86
88

90

44 0

4.1
17.1
26.1
3.11
9.11
15.11
22.11
27.11
4.111
9.111
14.111
19.111

1

]

Summer

27.V
17.V
9.v
2.v
26.1V
20.IV
14.IV
9.IV
4.lV
30.111
25.111
19.111-.

53
72
88
I 02
115
127
139
150
159
169
179
189
-____

_ _

Autumn

Dumtim B e g . lDuration]".g.
i n . d a y s u d __
a y s ,date
143
120
103
88
76
64
51
41
31
2!
11
0

-___

~

I

~

144
121
104
90
76
63
52
41
31
21
10

0

.

-~

Winter

j h r a t i o n [Beg.
i n d ay s -~d a t e

19.VII
28.VII
5.VIII
12.v111
I9.VIII
25.VllI
31.VIII
6.IX
1O.IX
15.1X
20.IX
25.1x. J
-.

I

..

1o:xr I
26.XI
17.XI
1O.XI
3.XI
28.X
22.x
17.X
ll.x
5.x
30.IX
25.IX

Duration
i n d- -a­
ys
&5
52
70
85
98
111
123
133
r44
I55
165
176

T h u s , l e t u s sum u p t h e d i u r n a l m o t i o n o f t h e S u n i n t h e c o u r s e
o f a y e a r f o r a n o b s e r v e r l o c a t e d a t a l a t i t u d e o f 76O ( b e t w e e n t h e
North P o l e and t h e Arctic C i r c l e ) .
( 1 ) From A p r i l 26 t o A u g u s t 1 9 , i . e . , f o r 1 1 5 d a y s , t h e S u n
T h i s p e r i o d o f t i m e i s c a l l e d t h e p o z a r summer.
does not set.

(2)
From A u g u s t 1 9 t o N o v e m b e r 3 , i . e . , for 76 d a y s , t h e S u n
w i l l r i s e and s e t d a i l y a n d t h e p e r i o d o f d a y l i g h t w i l l d i m i n i s h
T h i s p e r i o d o f t i m e i s c a l l e d t h e p o l a r autumn.
each day.
(3)
From N o v e m b e r 3 t o F e b r u a r y 9 , i . e . , f o r 9 8 d a y s , t h e
This period is c a l l e d t h e
Sun w i l l n o t r i s e f o r t h e o b s e r v e r .

p o Zar winter.
(4)
From F e b r u a r y 9 t o A p r i l 26 i . e . , f o r 7 6 d a y s t h e S u n w i l l
r i s e and s e t d a i l y and t h e p e r i o d o f d a y l i g h t w i l l i n c r e a s e e a c h
day.
T h i s p e r i o d o f t i m e i s c a l l e d t h e p o l a r spring.

The d a t e s f o r t h e s t a r t o f t h e s e a s o n s d e p e n d on t h e l a t i t u d e
of t h e observer.
T h e d a t e s g i v e n i n o u r e x a m p l e a r e f o r 76ON.
We
have provided a t a b l e f o r t h e seasons as a f u n c t i o n of t h e l a t i t u d e
of t h e observer (Table 5.2).

M o t i o n of t h e S u n a b o v e t h e Arctic CircZe

A s w e a l r e a d y know, t h e l a t i t u d e of t h e A r c t i c C i r c l e i s e q u a l
t o $ = 6 6 O 3 3 ' , or t h e c o m p l e m e n t o f i t s l a t i t u d e t o 90° i s 2 3 O 2 7 ' .

90°
r$
-

-

T h e r e f o r e , o n t h e d a t e o f t h e summer s o l s t i c e ( J u n e 2 2 ) , 6 =
$ , a n d on t h e d a t e o f t h e w i n t e r s o l s t i c e ( D e c e m b e r 2 2 ) , 6 =

900.

On t h e s e d a t e s a t t h e A r c t i c C i r c l e , t h e c e n t e r o f t h e S u n
t o u c h e s t h e h o r i z o n o n J u n e 2 2 a t t h e n o r t h p o i n t a t t h e moment o f
l o w e r c u l m i n a t i o n ( p o i n t N , F i g . 5 . 1 7 ) , a n d on December 2 2 a t t h e
s o u t h p o i n t a t t h e moment o f u p p e r c u l m i n a t i o n ( p o i n t S ) .
During
t h e r e s t o f t h e y e a r , t h e Sun w i l l r i s e a n d s e t d a i l y .
The p e r i o d
o f d a y l i g h t w i l l i n c r e a s e d a i l y f r o m December 2 2 t o J u n e 2 2 ; a f t e r
June 2 2 , i t w i l l d e c r e a s e .
The S u n r e a c h e s i t s maximum a l t i t u d e a b o v e t h e h o r i z o n o n
June 2 2 .
I t w i l l b e : h = 9 0 - 66O33' + 23O27' f i 4 6 O 5 4 ' .

M o t i o n of t h e S u n a t MiddZe L a t i t u d e s

K n o w i n g t h e maximum d e c l i n a t i o n o f t h e S u n ( e q u a l t o 2 3 O 2 7 ' ) ,
i t i s n o t d i f f i c u l t t o c a l c u l a t e t h e l a t i t u d e s o f t h e o b s e r v e r on
t h e E a r t h ' s s u r f a c e w h e r e t h e Sun w i l l b e a r i s i n g a n d a s e t t i n g
h e a v e n l y body d u r i n g t h e y e a r .
From t h e c o n d i t i o n s f o r t h e r i s i n g a n d s e t t i n g o f h e a v e n l y

/420

b o d i e s , it f o l l o w s t h a t t h e d e c l i n a t i o n must b e l e s s t h a n t h e comple­
m e n t o f t h e l a t i t u d e t o 9 0 ° , or t h e c o m p l e m e n t o f t h e l a t i t u d e t o
90° m u s t b e m o r e t h a n t h e d e c l i n a t i o n o f t h e S u n , i . e . , 9 0 - @ > 6 .
T h i s m e a n s t h a t b e t w e e n 66O33'N a n d 66O33'SY t h e S u n w i l l r i s e a n d
s e t every day of t h e y e a r ( s e e Fig. 5.17).

M o t i o n o f t h e Sun a t t h e T e r r e s t r i a l Equator.
I n o r d e r t o b e t t e r understand t h e n a t u r e of t h e d i u r n a l motion
Here w e
o f t h e Sun a t t h e E q u a t o r , l e t u s a n a l y z e F i g u r e 5 . 1 8 .
have sketched schematically t h e d i u r n a l t r a j e c t o r i e s ( c i r c l e s ) of
/421
AS w e see from t h e d r a w i n g , a l l t h e d i u r n a l
t h e Sun a t t h e e q u a t o r .
c i r c l e s a r e d i v i d e d i n two by t h e h o r i z o n l i n e .
This i s under­
s t a n d a b l e , s i n c e w e a l r e a d y know t h a t t h e t r u e h o r i z o n o f a n o b ­
s e r v e r l o c a t e d a t o n e o f t h e p o l e s i s s i t u a t e d a t a n a n g l e o f 90°
t o the equator.
S i n c e t h e d a i l y c i r c l e s o f t h e Sun a r e d i v i d e d i n
h a l f by t h e h o r i z o n , t h e Sun w i l l b e l o c a t e d a b o v e t h e h o r i z o n h a l f
t h e day and below t h e h o r i z o n h a l f t h e d a y , i . e . d u r i n g t h e whole
y e a r t h e Sun r i s e s a n d s e t s , a n d t h e d a y s a n d n i g h t s a r e e q u a l i n
length.

T w i c e a y e a r , o n t h e d a t e s o f t h e e q u i n o x e s , when t h e S u n m o v e s
along t h e c e l e s t i a l equator (the equator itself i s t h e diurnal
c i r c l e ) , it p a s s e s t h r o u g h t h e z e n i t h o f t h e o b s e r v e r and i t s h e i g h t
i s 90°.
D u r i n g t h e r e s t o f t h e y e a r , t h e Sun c u l m i n a t e s t o t h e
n o r t h or s o u t h o f t h e z e n i t h a n d i s a t i t s g r e a t e s t d i s t a n c e f r o m
it (23O27') on t h e d a t e s o f t h e s o l s t i c e s .

4.

M o t i o n o f t h e Moon

I n t r i n s i c Motion of

t h e Moon

T h e Moon, p a r t i c i p a t i n g w i t h a l l t h e h e a v e n l y b o d i e s i n t h e
s k y i n t h e d i u r n a l r o t a t i o n o f t h e c e l e s t i a l s p h e r e , h a s i t s own
i n t r i n s i c motion.

SP'

PN

2'

Fig. 5.17.
D i u r n a l Motion

of t h e S u n A b o v e t h e A r c t i c
Circle.
442

Fig. 5.18.
D i u r n a l Motion o f

t h e Sun a t t h e E q u a t o r .


If xe o b s e r v e t h e Moon for o n e n i g h t , w e c a n e a s i l y s e e t h a t
i t t r a v e l s l i k e t h e Sun i n t h e s k y ( r e l a t i v e t o t h e s t a r s ) .
The
a p p a r e n t m o t i o n o f t h e S u n i s t h e r e s u l t of t h e E a r t h ' s m o t i o n
a r o u n d t h e S u n ; t h e Moon a c t u a l l y m o v e s a r o u n d t h e E a r t h .

D i r e c t i o n and R a t e o f t h e Moon's M o t i o n
T h e Moon m o v e s a l o n g t h e c e l e s t i a l s p h e r e f r o m w e s t t o e a s t ,
i . e . , i n a d i r e c t i o n o p p o s i t e t h e d i u r n a l motion of t h e c e l e s t i a l
sphere.
T h e g r e a t c i r c l e a l o n g w h i c h t h e Moon c o m p l e t e s i t s m o t i o n
around t h e E a r t h h a s t h e shape of an e l l i p s e and i s c a l l e d t h e
Moon's o r b i t .
The Moon's o r b i t i s i n t e r s e c t e d b y t h e s o l a r e c l i p t i c
a t a n a n g l e o f 5O08' ( F i g . 5 . 1 9 ) .
The two d i a m e t r i c a l l y o p p o s e d
p o i n t s a t which t h e Moon's o r b i t i s i n t e r s e c t e d by t h e s o l a r
e c l i p t i c a r e c a l l e d t h e n o d e s of t h e o r b i t .
T h e Moon c o m p l e t e s a f u l l r e v o l u t i o n a l o n g i t s o r b i t
t o t h e s t a r s i n 27.32 d a y s .
This t i m e i n t e r v a l is c a l l e d
s i d e r e a l ( s t e Z Z a r l month.
Thus, it i s easy t o c a l c u l a t e t
one d a y i t moves 1 3 . 2 O .
Its hourly s h i f t relative t o the
a p p r o x i m a t e l y 0.5O.

relative
the
h a t during
stars is

T h e m o t i o n o f t h e Moon i n i t s o r b i t may b e s t u d i e d i n r e l a t i o n
t o t h e S u n ( w h i c h h a s , a s we k n o w , i t s own m o t i o n ) .
T h e p e r i o d o f r e v o l u t i o n r e q u i r e d f o r t h e Moon t o r e t u r n t o a
p r e v i o u s p o s i t i o n i n r e l a t i o n t o t h e Sun i s c a l l e d t h e s y n o d i c
I t i s a p p r o x i m a t e l y 2 9 . 5 mean s o l a r d a y s .
month.

/422

The Moon, r e v o l v i n g a r o u n d t h e E a r t h , a c c o m p a n i e s i t i n i t s
m o t i o n around t h e Sun.
T h e Moon i s 3 5 6 , 0 0 0 km f r o m t h e E a r t h a t
p e r i g e e ( t h e c l o s e s t p o i n t t o t h e E a r t h ) a n d 407,100 km f r o m t h e
E a r t h a t apogee ( t h e most d i s t a n t
P
p o i n t from t h e E a r t h ) .

P h a s e s of t h e Moon

Fig.

5.19.

O r b i t o f t h e Moon.

T h e Moon, l i k e o u r E a r t h ,
i s a n opaque body which i l l u m i n a t e s
the Earth's surface with reflected
sunlight.
The i l l u m i n a t i o n o f
t h e E a r t h ' s s u r f a c e b y t h e Moon,
as w e can see, i s not always t h e
same.
A t different t i m e s the
Moon i s v i s i b l e i n t h e f o r m o f
a luminous d i s k , i n t h e form o f
a l u m i n o u s h a l f - d i s k , or a c r e s ­
cent.
T h e r e i s a t i m e when t h e
The
Moon i s e n t i r e l y i n v i s i b l e .
Moon h a s v a r i o u s p h a s e s .
The
443

p e r i o d i c a l l y r e p e a t e d c h a n g e i n t h e s h a p e o f t h e Moon i s c a l l e d t h e

change i n l u n a r p h a s e s .
In order t o explain the
Figure 5.20.
A t Point 0 (in
P o i n t s 1 - 8 w e s h o w t h e Moon i
around t h e Earth.
Below t h e
i n those eight positions f o r
surface.

cause of l u n a r phases, l e t u s look a t
the orbit's center) is our Earth; at
n various positions i n its revolution
f i g u r e w e s h o w t h e s h a p e o f t h e Moon
a n o b s e r v e r l o c a t e d on t h e E a r t h ' s

An o b ­
When t h e Moon i s i n P o s i t i o n 1, i t s p h a s e i s new moon.
s e r v e r on t h e E a r t h ' s s u r f a c e d u r i n g t h i s p h a s e d o e s n o t s e e t h e
Moon, s i n c e i t s n o n i l l u m i n a t e d s i d e i s t u r n e d t o t h e E a r t h a n d i s
l o c a t e d a t a n a n g u l a r d i s t a n c e o f n o t m o r e t h a n 5O f r o m t h e S u n .
I n P o s i t i o n 2 , t h e Moon i s v i s i b l e on t h e E a r t h i n t h e f o r m o f a
narrow c r e s c e n t only during evening hours.
When t h e Moon i s i n
P o s i t i o n 3 , i t s phase i s c a l l e d t h e f i r s t q u a r t e r and an observer
This phase is
s e e s it i n t h e s h a p e o f h a l f a n i l l u m i n a t e d d i s k .
c a l l e d t h e first q u a r t e r because a t t h i s t i m e a q u a r t e r of t h e
entire lunar surface is visible.
D u r i n g f i r s t q u a r t e r , t h e Moon i s v i s i b l e i n t h e e a s t a t n o o n ,
i n t h e s o u t h a b o u t 1 8 0 0 , and i n t h e w e s t a t m i d n i g h t .
After first
q u a r t e r , t h e i l l u m i n a t e d p a r t of t h e l u n a r d i s k begins t o i n c r e a s e
(Position 4); i n Position 5, the entire lunar disk is illuminated.
T h i s i s t h e half-moon p h a s e .
I n t h i s p h a s e , t h e Moon i s v i s i b l e
a l l night.
A f t e r h a l f moon, t h e i l l u m i n a t e d p a r t o f t h e l u n a r d i s k b e g i n s / 4 2 3
t o d e c r e a s e from t h e r i g h t s i d e o f t h e d i s k ( i n P o s i t i o n 6 ) , and i n

I

new
moon
Fig.
444

2

4

3

first
quarter
5.20.

5

moon

6

7

last
quarter

P h a s e s o f t h e Moon.

8

P o s i t i o n 7 it r e a c h e s t h e p h a s e of l a s t q u a r t e r .
In last quarter,
t h e Moon i s v i s i b l e i n t h e e a s t a t m i d n i g h t , i n t h e s o u t h a b o u t
0 6 0 0 , and i n t h e w e s t a t noon.
A f t e r last quarter, the illuminated
p a r t o f t h e l u n a r d i s k d i m i n i s h e s e v e n more and i n P o s i t i o n 8 i t i s
a l r e a d y v i s i b l e i n t h e form of a narrow c r e s c e n t (on t h e l e f t s i d e
of t h e d i s k ) ; it t h e n becomes i n v i s i b l e a g a i n , i . e . , t h e p h a s e o f
new moon a g a i n s e t s i n .
Nature o f

t h e M o t i o n o f t h e Moon a r o u n d t h e E a r t h

I n v i e w o f t h e f a c t t h a t t h e Moon, i n i t s m o t i o n a r o u n d t h e
E a r t h , i s s o m e t i m e s c l o s e r t o t h e Sun t h a n t h e E a r t h i s a n d some­
t i m e s f a r t h e r a w a y , i t r e c e i v e s a c c e l e r a t i o n f r o m t h e Sun w h i c h i s
sometimes more and s o m e t i m e s l e s s t h a n t h a t o f t h e E a r t h .
As a
r e s u l t , t h e m o t i o n o f t h e Moon a r o u n d t h e E a r t h i s c o m p l e x , s i n c e
t h e above f a c t o r s n o t o n l y change t h e shape and dimensions o f t h e
lunar o r b i t , but its position i n space.
A s a r e s u l t of t h e change i n p o s i t i o n of t h e plane of t h e l u n a r
o r b i t , t h e angle of i n c l i n a t i o n of t h i s plane t o t h e plane of t h e
c e l e s t i a l e q u a t o r c h a n g e s c o n t i n u o u s l y from 28.5 t o 18.5O.
Because
o f t h i s , t h e d e c l i n a t i o n o f t h e Moon a l s o c h a n g e s i n t h e r a n g e
f r o m - 2 8 . 5 t o +28.5O a n d i n t h e r a n g e f r o m -18.5O t o +18.5O.
In
a l l c a s e s , however, t h e p l a n e of t h e l u n a r o r b i t i s i n c l i n e d t o t h e
p l a n e o f t h e e c l i p t i c a t 5O08'.
A s a r e s u l t of t h e change i n t h e
p o s i t i o n of t h e plane of t h e l u n a r o r b i t , t h e nodes of t h e l u n a r
o r b i t a l s o t r a v e l a l o n g t h e e c l i p t i c a g a i n s t t h e annual motion o f
Thus, t h e nodes of t h e
t h e S u n ( o r Moon) b y 19.3O e v e r y y e a r .
l u n a r o r b i t complete one r e v o l u t i o n a l o n g it i n 18.6 y e a r s .

/45!

T h e Moon, c o m p l e t i n g o n e r e v o l u t i o n i n i t s o r b i t , i n t e r s e c t s
t h e p l a n e o f t h e c e l e s t i a l e q u a t o r t w i c e ; i n t h e c o u r s e o f one
s i d e r e a l m o n t h , i t s d e c l i n a t i o n c h a n g e s f r o m a maximum p o s i t i v e
v a l u e t o a maximum n e g a t i v e o n e .
T h e maximum d e c l i n a t i o n o f t h e Moon i n a p e r i o d o f o n e m o n t h
may b e + 2 8 O 2 7 ' a n d t h e minimum may b e - 2 8 2 7 ' .
Location o f

t h e Moon A b o v e t h e H o r i z o n

T h e l o c a t i o n o f t h e Moon a b o v e t h e h o r i z o n d e p e n d s o n t h e s o c a l l e d w a x i n g o f t h e Moon ( t i m e e l a p s e d a f t e r new m o o n ) a n d t h e
season.
I n t h e N o r t h e r n H e m i s p h e r e , d u r i n g t h e summer a n d t h e p h a s e o f
f u l l m o o n , t h e Moon i s l o c a t e d c o m p a r a t i v e l y l o w a n d f o r a s h o r t
t i m e a b o v e t h e h o r i z o n ; i n t h e w i n t e r , d u r i n g t h e f u l l moon, i t i s
v i s i b l e a l l n i g h t and r i s e s r a t h e r high above t h e horizon.
This
d e p e n d s o n t h e d e c l i n a t i o n o f t h e Moon.
I n winter i n t h e Northern
H e m i s p h e r e , t h e f u l l moons o c c u r w i t h a p o s i t i v e . d e c l i n a t i o n , w h i l e
t h e f u l l m o o n s i n summer o c c u r w i t h n e g a t i v e d e c l i n a t i o n .

445

5.

Measurement o f Time

Essence o f Calculating Time
Measurement o f t i m e i s t h e r e s u l t of t h e p r e s e n c e o f m o t i o n i n
T i m e a n d m o t i o n a r e synonymous.
I'f m o t i o n c e a s e d
universal space.
i n Nature and i n t h e Universe, t h e r e could be no discussion of t i m e .
I t i s e n t i r e l y u n d e r s t a n d a b l e t h a t t o m e a s u r e t i m e , some c o n s t a n t
and uniform motion must b e u s e d .
T h e r o t a t i o n o f t h e E a r t h o n i t s a x i s (or, a s a r e s u l t o f t h i s ,
t h e a p p a r e n t r o t a t i o n of t h e c e l e s t i a l s p h e r e around t h e world a x i s )
could be such a motion.
S p e c i a l o b s e r v a t i o n s o v e r a v e r y l o n g t i m e i n t e r v a l h a v e shown
t h a t t h e duration of t h e E a r t h ' s r o t a t i o n around i t s a x i s has not
This characterizes very
c h a n g e d e v e n by a f r a c t i o n o f a s e c o n d .
w e l l t h e c o n s t a n c y and u n i f o r m i t y of t h e E a r t h ' s r o t a t i o n .
I n t h e p r a c t i c e o f a v i a t i o n , t h e f o l l o w i n g k i n d s of t i m e must
be s t u d i e d and used:
s i d e r e a l , t r u e s o l a r , mean s o l a r , G r e e n w i c h ,
l o c a l , zone a n d s t a n d a r d t i m e .

Sidereal T i m e

Time m e a s u r e d o n t h e b a s i s o f t h e a p p a r e n t m o t i o n o f h e a v e n l y
T h i s t i m e may b e m e a s u r e d
bodies ( s t a r s ) is called sidereal time.
by t h e h o u r a n g l e o f some h e a v e n l y b o d y w i t h r e s p e c t t o t h e m e r i d i a n
of t h e observer.
However, f o r c o n v e n i e n c e , it i s a d v a n t a g e o u s t o
t a k e t h e h o u r a n g l e n o t o f some s t a r b u t o f t h e p o i n t o f t h e v e r n a l
equinox from which t h e r i g h t a s c e n s i o n i s r e a d .
T h e w e s t e r n h o u r a n g l e of t h e p o i n t o f t h e v e r n a l e q u i n o x t Y
i s c a l l e d s i d e r e a l t i m e a n d i s r e p r e s e n t e d b y t h e l e t t e r S (S = t y )
S i d e r e a l t i m e i s e q u a l t o t h e sum of t h e h o u r
i n Figure 5.21.
angle and t h e r i g h t ascension of a
S = a
+ t.
star:

Y
Sidereal days.

Z'

Fig. 5.21.
H o u r A n g l e of
t h e Point of t h e Vernal
Equinox.
446

The t i m e i n t e r ­
v a l between two s u c c e s s i v e u p p e r
c u l m i n a t i o n s of t h e p o i n t o f t h e
v e r n a l equinox i s c a l l e d a sidereal
day.
T h e moment o f t h e u p p e r c u l ­
mination of t h e v e r n a l equinox point
w a s t a k e n as t h e beginning of s i d e ­
real days.
A t t h i s moment, s i d e r e a l
Sidereal
time i s e q u a l t o 0O:OO:OO.
d a y s a r e d i v i d e d i n t o 24 s i d e r e a l
hours, each hour i s divided i n t o 60
s i d e r e a l minutes, and each minute
i n t o 60 s i d e r e a l seconds.

/425

S i d e r e a l t i m e does n o t have a d a t e and t h e r e f o r e i n c a l c u l a t i o n s
when t h e s i d e r e a l t i m e i s m o r e t h a n 24:00, t h e r e i s o n l y a s u r p l u s
o f t i m e a b o v e 24:OO.

Exumple.
I n c a l c u l a t i o n s , t h e r e is a s i d e r e a l t i m e of S =
29:20. D i s c a r d i n g 24:00, we o b t a i n a s i d e r e a l t i m e o f 5 : 2 0 .
The practical application of sidereal time.
Sidereal t i m e is
not used i n everyday l i f e , but as t h e b a s i s of t i m e s i g n a l s .
In
e v e r y a s t r o n o m i c a l o b s e r v a t o r y , t h e r e a r e s p e c i a l c l o c k s which r u n
I n a v i a t i o n , s i d e r e a l t i m e must b e
according t o si d e re a l t i m e .
used i n observing stars f o r determining t h e p o s i t i o n of t h e aircraft
or t h e p o s i t i o n l i n e o f t h e a i r c r a f t .
I n t h e a v i a t i o n astronomical yearbook, t h e s i d e r e a l t i m e i s
g i v e n f o r e a c h d a t e a n d e a c h h o u r o f G r e e n w i c h mean t i m e .
Therefore,
t h e s i d e r e a l t i m e f o r a n y moment a t a n y p o i n t o n E a r t h may b e d e t e r ­
mined by means o f t h e a v i a t i o n a s t r o n o m i c a l y e a r b o o k on t h e b a s i s
/426
of Greenwich t i m e .
I n everyday l i f e , s o l a r t i m e r a t h e r than side­
r e a l t i m e i s u s e d , s i n c e on t h e whole man's a c t i v i t y o c c u r s i n t h e
daytime hours.
True Solar T i m e

True s o l a r t i m e t n i s t h e t i m e measured on t h e b a s i s o f t h e
d i u r n a l motion of t h e t r u e Sun.
of
T r u e s o l a r t i m e i s m e a s u r e d by t h e w e s t e r n h o u r a n g l e ( t w >
t h e c e n t e r o f t h e t r u e Sun.

True s o l a r days a r e t h e t i m e i n t e r v a l s b e t w e e n two s u c c e s s i v e
T h e moment o f
upper c u l m i n a t i o n s of t h e c e n t e r of t h e t r u e Sun.
t h e u p p e r c u l m i n a t i o n o f t h e c e n t e r of t h e t r u e Sun i s t a k e n a s t h e
beginning of t r u e s o l a r days.
A t t h e moment o f u p p e r c u l m i n a t i o n , when t h e h o u r a n g l e o f t h e
Sun i s z e r o , t h e t r u e s o l a r t i m e i s 0 O : O O : O O .

I n p r o p o r t i o n t o t h e d i u r n a l motion of t h e Sun,
i n c r e a s e s ; thse t r u e s o l a r t i m e a l s o i n c r e a s e s .

i t s hour angle

A t t h e moment o f t h e l o w e r c u l m i n a t i o n o f t h e S u n , t h e t r u e
s o l a r t i m e i s 12:OO; when t h e c e n t e r o f t h e S u n i s a g a i n i n t h e
p o s i t i o n o f u p p e r c u l m i n a t i o n , t h e t r u e s o l a r t i m e i s 24:OO. A f t e r
t h i s , new d a y s b e g i n .

T h e d u r a t i o n o f t r u e s o l a r days changes i n t h e course o f a
year.
T h i s o c c u r s b e c a u s e t h e t r u e Sun moves d u r i n g t h e y e a r a l o n g
t h e e c l i p t i c , which i s i n c l i n e d t o t h e c e l e s t i a l e q u a t o r a t a n
a n g l e o f 23O27' a n d i s n o t a c i r c l e b u t a n e l l i p s e .
For t h i s reason,
t h e d a i l y s h i f t o f t h e Sun t o t h e e a s t i s d i f f e r e n t on d i f f e r e n t
days of t h e year.
T h i s s h i f t i s a t a maximum n e a r t h e s o l s t i c e s ,
447

d

when t h e S u n m o v e s p a r a l l e l t o t h e e q u a t o r .
On t h e o t h e r h a n d ,
near t h e equinoxes t h e s h i f t t o t h e east i s smallest,
The S u n l a g s b e h i n d t h e m o t i o n o f t h e s t a r s b y e i t h e r a v e r y
g r e a t or a v e r y s m a l l v a l u e ; t h e d u r a t i o n o f t r u e d a y s c h a n g e s a l l
the time.
I n u s i n g t r u e s o l a r t i m e i n e v e r y d a y l i f e , it would a l m o s t b e
n e c e s s a r y (as a r e s u l t of i t s n o n u n i f o r m i t y ) t o r e g u l a t e c l o c k s
e v e r y d a y , moving them a h e a d and b a c k .
T h i s would be e x t r e m e l y
inconvenient.
I n view o f t h e i n c o n v e n i e n c e o f c a l c u l a t i n g t i m e on t h e b a s i s
o f t h e t r u e S u n , t i m e i n e v e r y d a y l i f e i s c a l c u l a t e d on t h e b a s i s
o f t h e s o - c a l l e d mean S u n .
T h e mean Sun i s t h e i m a g i n a r y or r e a l S u n m o v i n g u n i f o r m l y
a l o n g t h e c e l e s t i a l e q u a t o r i n t h e same d i r e c t i o n t h a t t h e t r u e S u n
i n a direction opposite t o the
moves a l o n g t h e e c l i p t i c , : . e . ,
d i u r n a l motion of t h e c e l e s t i a l sphere.
Mean S o l a r T i m e

T i m e c a l c u l a t e d o n t h e b a s i s o f t h e d i u r n a l m o t i o n o f t h e mean
( i m a g i n a r y ) Sun i s c a l l e d t h e mean soZar t i m e ( m ) .
Mean s o l a r t i m e i s m e a s u r e d b y t h e w e s t e r n h o u r a n g l e ( t , )
of
t h e mean ( i m a g i n a r y ) S u n .
Mean s o l a r d a y s a r e t h e b a s i c u n i t o f
mean s o l a r t i m e .

Mean s o Z a r d a y s a r e t h e t i m e i n t e r v a l s b e t w e e n t w o s u c c e s s i v e
u p p e r c u l m i n a t i o n s o f t h e mean S u n .
T h e y a r e d i v i d e d i n t o 2 4 mean
h o u r s , each hour i s d i v i d e d i n t o 60 minutes and each minute i n t o
60 seconds.
The d u r a t i o n o f mean s o l a r d a y s i s c o n s t a n t .
Mean c i v i l t i m e ( m c ) .
S i n c e a t t h e moment o f t h e u p p e r c u l ­
m i n a t i o n o f t h e mean S u n i t s h o u r a n g l e w i l l b e z e r o , t h e mean
s o l a r t i m e a t t h i s moment ( m e a n n o o n ) w i l l a l s o b e z e r o .
I n e v e r y d a y l i f e , w i t h a 24-hour r e c k o n i n g of t i m e , t h i s i s
v e r y i n c o n v e n i e n t s i n c e i n t h i s c a s e t h e b e g i n n i n g o f mean d a y s
comes a t n o o n .
T h e r e f o r e , i n everyday l i f e , s o - c a l l e d c i v i l time (which i s
d i f f e r e n t f r o m mean solar t i m e b y 1 2 h r s . ) i s u s e d a s a v a r i e t y o f
mean s o l a r t i m e .
Mean m i d n i g h t , i . e . t h e mean t i m e e q u a l t o t h e w e s t e r n h o u r
a n g l e o f t h e mean S u n p l u s 1 2 h r s , i s t a k e n a s t h e b e g i n n i n g o f
mean c i v i l d a y s :

448

w h e r e mc

i s t h e mean c i v i l t i m e ,

a n d m i s t h e mean s o l a r t i m e .

T h e p l u s s i g n i s u s e d when t h e mean t i m e i s l e s s t h a n 1 2 h r s
a n d t h e m i n u s s i g n i s u s e d i f t h e mean t i m e i s more t h a n 1 2 h r s .

ExampZe.
D e t e r m i n e t h e mean c i v i l t i m e i f t h e w e s t e r n h o u r
a n g l e o f t h e mean S u n ( m e a n t i m e ) i s 6 : O O .
Solution.

mc

= m

+

12:OO

= 6:OO

+

12:OO = 1 8 : O O

Local C i v i l T i m e

S i d e r e a l t i m e , t r u e s o l a r a n d mean s o l a r t i m e a r e m e a s u r e d b y
t h e h o u r a n g l e o f a h e a v e n l y b o d y or t h e p o i n t o f t h e v e r n a l e q u i n o x .
But t h e h o u r a n g l e ( t ) o f a h e a v e n l y body c a l c u l a t e d a t one p h y s i c a l
moment f o r o n e h e a v e n l y b o d y f r o m t h e m e r i d i a n s o f v a r i o u s p o i n t s
on t h e E a r t h ' s s u r f a c e v a r i e s i n v a l u e .
F o r some p o i n t s on t h e
E a r t h ' s s u r f a c e it w i l l b e l a r g e and f o r o t h e r s it w i l l b e s m a l l .
I n F i g u r e 5.22 w e s h o w a c e l e s t i a l s p h e r e w h o s e e q u a t o r l i e s i n t h e
b e t h e p o s i t i o n o f t h e mean
plane of t h e drawing.
L e t p o i n t Ma,
R a d i u s PnA i s t h e m e r i d i a n o f o n e p o i n t
/428
Sun a t a c e r t a i n moment:
on t h e E a r t h ' s s u r f a c e a n d r a d i u s P n B i s t h e m e r i d i a n o f a s e c o n d
p o i n t on t h e E a r t h ' s s u r f a c e .
From t h e f i g u r e , i t i s a p p a r e n t t h a t
t h e h o u r a n g l e o f t h e mean S u n for t h e f i r s t p o i n t ( t l ) i s e x p r e s s e d
b y t h e a n g l e APnMaV a n d t h e h o u r a n g l e f o r t h e s e c o n d p o i n t ( t 2 )
a t t h i s moment i s e x p r e s s e d b y t h e a n g l e BPnMav.
From t h i s f i g u r e
i t i s o b v i o u s t h a t t h e a n g l e APnMav i s g r e a t e r t h a n t h e a n g l e BPnMaV.
T h e r e f o r e , a t t h e s a m e p h y s i c a l moment t h e t i m e a t d i f f e r e n t
meridians w i l l be d i f f e r e n t .
T i m e c a l c u l a t e d r e l a t i v e t o t h e m e r i d i a n o f m i d n i g h t o f some
p o i n t o n t h e E a r t h ' s s u r f a c e i s c a l l e d t h e ZocaZ c i v i l t i m e a n d i s
L o c a l t i m e may b e s i d e r e a l or s o l a r .
r e p r e s e n t e d b y T1.
It w i l l
b e t h e same f o r a l l p o i n t s l y i n g on
ElS)
. .
one m e r i d i a n ( ; . e .
h a v i n g t h e same
geographic longitude).

Greenwich Time

Local c i v i l time calculated
from t h e Greenwich m e r i d i a n i s
c a l l e d Greenwich t i m e and i s r e p r e ­
s e n t e d by T G ~ .

. .

Determining
Fig. 5.22.
L o c a l T i m e on t h e Basis o f

t h e Mean S u n .

In astronomical yearbooks, t h e
t i m e s o f some c e l e s t i a l phenomena
are g i v e n as w e l l as t h e a s t r o n o m i c a l
values necessary f o r practical
c a l c u l a t i o n s on t h e b a s i s of Greenwich t i m e .


T h e relation between local

449

civil time a n d Greenwich time. With a knowledge o f the Greenwich
time and longitude o f a place expressed in time units, it is simple
t o determine the local civil time and vice versa.
The local civil time is equal t o the Greenwich time plus or

minus the longitude o f the place, :.e.


Here the plus sign is used if the longitude of the place is east

and the minus sign is used if the longitude is west.


Example.
(AE

for M O S C O W

1. Greenwich time is 14:15.
= 2:28).

Find the local time

Solution.
Substituting the values f o r T G a
~nd A E in the formu­
la TI = TGr t A E , we obtain the local time T1 = 14:15 t 2:28 = 16.43.

Example.
2.
The Greenwich time is 10:42.
time for Washington ( A w = 05:08).
Solution.

T~ = T G ~
- A W = 10:42

-

Find the local

5:08 = 5:34.

When solving practical problems in astronomy, it is often
necessary t o change local time t o Greenwich time.


(Aw)

The plus sign is used i f the longitude of the place is west
and minus if the longitude o f the place is east ( A E ) .

Example.
1. The local time in Ryazan ( A ,
Determine the Greenwich time.

= 2:39) is 1430.

Solution.
Substituting the values f o r T 1 and A E into the
= T 1 - A E , we obtain TGr = 14:30 - 2:39 = 11:51.
formula T G ~

Example.
is 1520.

2 .
The local time in San Francisco ( A w
Determine the Greenwich time.

= 8:09:44)

Solution.
Substituting the values �or Ti and A W into the
formula T G =~ Ti t A w , we obtain T G ~
= 1520 t 8:09:44 = 23:09:44.
Time difference on t w o meridians. It is easy to imagine that
the difference in local time on any two meridians is equal to the
difference in their longitudes.

In Figure 5.23 the diameter EQ is the Greenwich meridian, the

diameter BA is the meridian of a point on the Earth's surface, the

450

/429


d i a m e t e r DC i s t h e m e r i d i a n o f a s e c o n d p o i n t on t h e E a r t h ' s s u r f a c e ,
Ma,
i s t h e l o c a t i o n o f t h e m e a n S u n a t a c e r t a i n moment a n d t h e
d i a m e t e r KMav i s t h e c i r c l e o f t h e S u n ' s d e c l i n a t i o n .

I n F i g u r e 5.23 i t i s e v i d e n t t h a t t h e h o u r a n g l e s o f t h e mean
Sun ( t i p a v a n d t 2 0 a v ) m e a s u r e d f r o m b o t h l o c a l m e r i d i a n s a r e d i f ­
f e r e n t f r o m one a n o t h e r by a v a l u e e q u a l t o t h e d i f f e r e n c e i n t h e
l o n g i t u d e s o f t h e s e m e r i d i a n s , s i n c e t i - t 2 = A2 - h i .
Hence it f o l l o w s t h a t t h e l o c a l t i m e on t h e s e m e r i d i a n s w i l l
d i f f e r by t h e d i f f e r e n c e i n t h e l o n g i t u d e s o f t h e s e two m e r i d i a n s .

Example.
L e t t h e l o c a l t i m e b e Ti = 1204 a t a p o i n t h a v i n g a n
e a s t l o n g i t u d e o f A E = 2:35.
What i s t h e l o c a l t i m e a t t h i s moment
a t a p o i n t h a v i n g a n e a s t l o n g i t u d e A E = 4:35?
Solution.
L e t us f i n d t h e difference i n t h e longitudes of
t h e s e t w o p o i n t s A A = 4:35 - 2:35 = 2:OO.
L e t us f i n d t h e l o c a l t i m e f o r t h e point having an east longi­
T 1 = 12:04 + 2:OO = 14:04.
t u d e XE = 4:35.

I n s o l v i n g s u c h p r o b l e m s , i t i s i m p o r t a n t t o r e m e m b e r t h a t t h e /430
l a r g e r t h e e a s t l o n g i t u d e of t h e p o i n t , t h e l a r g e r w i l l be t h e l o c a l
t i m e a t t h i s p o i n t ; t h e smaller t h e east longitude of t h e p o i n t ,
the smaller the local t i m e .
Zone T i m e

We h a v e a l r e a d y s e e n t h a t e a c h m e r i d i a n o f t h e E a r t h ' s s u r f a c e
I f we t a k e K h a b a r o v s k , w h o s e l o n g i t u d e i s 135O5'
h a s i t s own t i m e .
(9:00:20) i t s l o c a l t i m e i s 6:29:48 d i f f e r e n t f r o m t h e l o c a l t i m e
o f Moscow, w h i c h h a s a l o n g i t u d e o f 37'38'
(2:30:32).
E (5)

4 (NJ
F i g . 5.23.
Time Difference
o n Two M e r i d i a n s

S i n c e e a c h p o i n t on t h e E a r t h ' s
s u r f a c e h a s i t s own ( l o c a l ) t i m e ,
it i s t o o inconvenient t o use l o c a l
t i m e i n everyday l i f e .
For e x a m p l e ,
when m o v i n g f r o m w e s t t o e a s t , i t
would be n e c e s s a r y t o c o n t i n u o u s l y
move t h e h o u r h a n d s 4 m i n a h e a d
f o r e v e r y d e g r e e o f l o n g i t u d e or
1 h r f o r e v e r y 15O o f l o n g i t u d e .
On t h e o t h e r h a n d , in m o v i n g f r o m
e a s t t o w e s t , t h e h o u r h a n d s would
h a v e t o b e moved b a c k c o n s t a n t l y .
Therefore, since t h e middle of t h e
last century, t h e countries of
Europe have begun t o i n t r o d u c e a
single t i m e i n their territories.
T h i s t i m e i s measured from t h e
meridians of t h e p r i n c i p a l observa­

451


I

..

t o r i e s of t h e s e c o u n t r i e s .
I n France so-called " P a r i s t i m e " , w a s introduced,
"Rome t i m e " ,
and i n England "Greenwich t i m e " , e t c .

i n Italy

The i n t r o d u c t i o n o f a s i n g l e c o n v e n t i o n a l t i m e i n t h e s e c o u n t r i e s
d i d n o t create great d i f f i c u l t i e s , s i n c e t h e l o c a l t i m e o f any
meridian o f t h e s e c o u n t r i e s ( i n view o f t h e i r s m a l l area r e l a t i v e
t o t h e m e r i d i a n of t h e c o n v e n t i o n a l t i m e i n t r o d u c e d ) d i f f e r e d i n s i g ­
If w e t a k e E n g l a n d a n d
n i f i c a n t l y (only several minutes i n a l l ) .
France f o r example, t h e i r outermost populated p o i n t s ( t o t h e east
a n d w e s t ) a r e s i t u a t e d i n a r a n g e o f 7-8O f r o m t h e i r r e s p e c t i v e
m e r i d i a n s (Greenwich and P a r i s ) :
t h e t i m e d i f f e r e n c e amounts t o a
t o t a l o f 30 m i n .
If we t a k e s u c h a c o u n t r y a s t h e USSR, w e know
t h a t t h e d i f f e r e n c e i n t h e l o n g i t u d e s of i t s e a s t e r n and w e s t e r n
However, i n p r e b o u n d a r i e s a m o u n t s t o more t h a n 1 0 h r s i n t i m e .
R e v o l u t i o n a r y R u s s i a , a common P e t e r s b u r g t i m e ( t h e l o c a l t i m e o f
t h e Pulkovo Observatory meridian) w a s introduced only f o r r a i l r o a d s .
T h i s t i m e w a s 0 0 : 2 8 : 5 8 b e h i n d Moscow l o c a l t i m e ( l o c a l t i m e o f t h e
Moscow U n i v e r s i t y O b s e r v a t o r y m e r i d i a n ) .
The i n t r o d u c t i o n o f a common t i m e i n i n d i v i d u a l c o u n t r i e s
p a r t i a l l y f a c i l i t a t e d i t s c a l c u l a t i o n w i t h i n each country, b u t it
d i d n o t s o l v e t h e p r o b l e m on a n i n t e r n a t i o n a l s c a l e .
The p r o b l e m
o f c a l c u l a t i n g t i m e was s o l v e d m o s t s u c c e s s f u l l y a f t e r t h e i n t r o ­
duction of a zone t i m e system.
I n some c o u n t r i e s , t h i s s y s t e m w a s i n t r o d u c e d a t t h e e n d o f t h e / 4 3 1
1 9 t h and beginning of t h e 20th c e n t u r y .
I n R u s s i a , t h e zone t i m e
s y s t e m w a s i n t r o d u c e d o n l y a f t e r t h e R e v o l u t i o n , o n J u l y 1, 1 9 1 9 b y
a s p e c i a l d e c r e e of t h e S o v i e t Government.
E s s e n c e o f t h e zone time s y s t e m .
The e n t i r e E a r t h i s d i v i d e d
i n t o 24 h o u r z o n e s .
The o u t e r m e r i d i a n s ( b o u n d a r i e s ) o f e a c h b a n d
a r e 15O o f l o n g i t u d e (1 h r i n t i m e ) a p a r t f r o m o n e a n o t h e r .
The z o n e s a r e n u m b e r e d f r o m w e s t t o e a s t f r o m t h e z e r o z o n e
t o t h e 23rd zone, i n c l u s i v e .
The z o n e i n c l u d e d b e t w e e n t h e m e r i d i a n s
7O30'N a n d 7 O 3 0 ' E i s t a k e n a s t h e z e r o z o n e , i . e . t h e i n i t i a l z o n e .
T h e G r e e n w i c h m e r i d i a n , w h i c h h a s a l o n g i t u d e o f O o , i s t h e mean
meridian of t h i s zone.
Obviously, t h e f i r s t zone w i l l be l o c a t e d
b e t w e e n t h e m e r i d i a n s A = 7O30'E a n d X = 2 2 O 3 0 ' E a n d t h e mean
m e r i d i a n o f t h i s zone h a s a l o n g i t u d e o f 15O; t h e s e c o n d zone i s
l o c a t e d b e t w e e n t h e m e r i d i a n s X = 22O30'E a n d X = 37O30'E a n d t h e
mean m e r i d i a n o f t h i s z o n e h a s a l o n g i t u d e o f 3 0 ° , e t c .
A t all
t h e common t
t i m e of t h e
ventional t i

t h e p o i n t s l o c a t e d w i t h i n t h e l i m i t s o f t h e same z o n e ,
i m e ( t i m e of t h e g i v e n z o n e ) which r e p r e s e n t s t h e l o c a l
mean m e r i d i a n o f t h e g i v e n z o n e i s t a k e n .
Such a con­
m e i s c a l l e d t h e zone t i m e ( T z ) .

I n t h e zero zone,
452

t h e t i m e i s c a l c u l a t e d on t h e b a s i s of t h e

l o c a l t i m e of t h e z e r o (Greenwich) meridian.
I n t h e first zone,
t h e t i m e i s c a l c u l a t e d on t h e b a s i s o f t h e l o c a l t i m e o f t h e m e r i d i a n
h a v i n g a l o n g i t u d e o f 15OE.
I n t h e second z o n e , it i s c a l c u l a t e d
on t h e b a s i s o f t h e l o c a l t i m e of t h e m e r i d i a n h a v i n g a l o n g i t u d e
S i n c e t h e mean m e r i d i a n s o f t w o a d j a c e n t z o n e s a r e
o f 30°E, e t c .
15O o f l o n g i t u d e a p a r t f r o m o n e a n o t h e r , t h e d i f f e r e n c e b e t w e e n
t h e zone t i m e o f a d j a c e n t z o n e s i s one h o u r .
T h e n u m b e r o f e a c h z o n e s h o w n b y how many h o u r s t h e t i m e i n
For e x a m p l e , i n t h e c a s e o f
t h i s zone i s a h e a d of Greenwich t i m e .
t h e t i m e o f t h e f o u r t h z o n e , t h i s means t h a t i t s t i m e i s 4 h r s
a h e a d o f Greenwich t i m e (Supplement 4 ) .
The i n t r o d u c t i o n o f z o n e t i m e g r e a t l y f a c i l i t a t e d t h e c a l c u l a ­
t i o n of t i m e on a n i n t e r n a t i o n a l s c a l e , s i n c e t h e m i n u t e a n d s e c o n d
h a n d s i n a l l t h e z o n e s i n d i c a t e t h e s a m e number o f m i n u t e s and
seconds.
The h o u r h a n d s m u s t b e moved a w h o l e h o u r o n l y when m o v i n g
from one zone t o a n o t h e r .
When t h e b o u n d a r i e s o f t h e h o u r z o n e s
were d e t e r m i n e d , t h e b o u n d a r i e s o f s t a t e s , r e g i o n s and c i t i e s a s
w e l l as n a t u r a l b o u n d a r i e s ( e . g . r i v e r s , e t c ) were t a k e n i n t o ac­
count.
If t h e b o u n d a r i e s o f t h e hour zones had been d e t e r m i n e d
s t r i c t l y a c c o r d i n g t o t h e m e r i d i a n s , c a l c u l a t i n g t h e zone t i m e ( e . g .
i n MOSCOW,
which i s l o c a t e d on t h e b o u n d a r y b e t w e e n t h e s e c o n d a n d
t h i r d z o n e s ) would h a v e t o b e done on t h e b a s i s o f two z o n e s :
in
t h e w e s t e r n p a r t o f t h e c i t y on t h e b a s i s o f t h e s e c o n d z o n e , a n d
i n t h e e a s t e r n p a r t o f t h e c i t y on t h e b a s i s o f t h e t h i r d z o n e .
1 . e . t h e t i m e d i f f e r e n c e i n t h e two p a r t s o f t h e c i t y would b e one
h o u r and i n c r o s s i n g t h e b o u n d a r y t h e h o u r h a n d s would have t o b e
1432
moved o n e h o u r .
I n view o f t h i s , t h e d i f f e r e n c e between t h e l o c a l t i m e o f t h e
o u t e r p o i n t s o f t h i s z o n e may b e s o m e w h a t m o r e or l e s s t h a n 3 0 m i n
r e l a t i v e t o t h e zone t i m e .
Standard Time

I n t h e S o v i e t U n i o n , on t h e b a s i s o f a government d e c r e e ,
c l o c k s w e r e moved a h e a d o n e h o u r b e g i n n i n g w i t h t h e summer o f 1 9 3 0 .
S i n c e t h i s t i m e , t h e e n t i r e USSR r e c k o n s t i m e on t h e b a s i s o f s o called standard time.
T h u s , t h e zone t i m e w a s s h i f t e d a h e a d l h r ,
i . e . e a c h zone l i v e s n o t on t h e b a s i s o f i t s zone t i m e b u t r a t h e r
on t h e b a s i s o f t h e t i m e o f t h e a d j a c e n t e a s t e r n z o n e .
For e x a m p l e ,
MOSCOW,
which i s l o c a t e d i n t h e s e c o n d z o n e , l i v e s a c c o r d i n g t o
t h e t i m e of t h e t h i r d zone.
The t i m e r u n n i n g 3 h r s a h e a d o f G r e e n w i c h t i m e
3 r d z o n e ) i s c a l l e d Moscow t i m e .

(time of t h e

All t h e r a i l r o a d , w a t e r , and a i r r o u t e s o f communication
t h e S o v i e t U n i o n o p e r a t e a c c o r d i n g t o Moscow t i m e .

in

453

Relation between G r e e n w i c h , Local and Z o n e ( S t a n d a r d ) T i m e

When s o l v i n g p r a c t i c a l p r o b l e m s o f a i r c r a f t n a v i g a t i o n o n t h e
g r o u n d a n d i n t h e a i r , it is v e r y o f t e n n e c e s s a r y t o c o n v e r t f r o m
one form of t i m e t o a n o t h e r .
T h e s e p r o b l e m s may b e s o l v e d c o r r e c t l y o n l y i f ' t h e c r e w
c o n s c i e n t i o u s l y m a s t e r s t h e e s s e n c e of c a l c u l a t i n g t i m e .
To f a c i l i ­
t a t e t h e w o r k o f t h e crew i n s o l v i n g s u c h p r o b l e m s , t h e r e a r e
f o r m u l a s f o r c o n v e r t i n g t i m e from one form t o a n o t h e r .
Zone t i m e ( T , )
( T G ~ )p l u s t h e n u m b e r o f t h e z o n e :

C o n v e r t i n g G r e e n w i c h t i m e t o Mean time.
e q u a l t o Greenwich t i m e

Tz= T

is

+ N.

GR

I n s o l v i n g p r o b l e m s f o r t h e U S S R , t h e number o f t h e zone i s
taking i n t o account t h e standard t i m e , ;.e. 1 h r . later.

used,

Example.
What i s t h e z o n e
c l o c k s i n G r e e n w i c h show 1 2 : 0 0 ?

( s t a n d a r d ) t i m e i n N o v o s i b i r s k when

SOlUtiOn.
On t h e b a s i s o f a map of h o u r z o n e s or o n t h e b a s i s
of a l i s t o f t h e most i m p o r t a n t p o p u l a t e d
points, l e t us find the
Taking i n t o
number o f t h e zone i n which N o v o s i b i r s k i s l o c a t e d .
account t h a t t h e standard hour i n Novosibirsk, t h e clocks run ac­
c o r d i n g t o t h e t i m e o f t h e 7 t h zone ( 6 t h zone t 1 h r ) , l e t u s f i n d
t h e z o n e t i m e on t h e b a s i s o f t h e f o r m u l a

Greenwich t i m e i s
e q u a l t o t h e zone ( s t a n d a r d ) t i m e minus t h e number o f t h e zone
(taking i n t o account t h e standard time):

Converting Z o n e T i m e t o G r e e n w i c h T i m e .

Example.

/433

What i s t h e G r e e n w i c h t i m e when t h e c l o c k s s h o w 17:OO

i n Krasnoyarsk?

S o l u t i o n . On t h e b a s i s o f a map o f h o u r z o n e s or o n t h e b a s i s
o f a l i s t of p o p u l a t e d p o i n t s , l e t u s f i n d t h e n u m b e r o f t h e z o n e
Taking i n t o account t h e s t a n d a r d
where Krasnoyarsk i s l o c a t e d .
t i m e i n K r a s n o y a r s k , t h e c l o c k s r u n o n t h e b a s i s of t h e t i m e o f t h e
7 t h zone ( 6 t h zone t 1 h r ) .
Let us f i n d t h e Greenwich t i m e .

T G r = Tz-N

= 17:OO - 7:OO =

1o:oo.
454

.
. .. ..

..

..

I

C o n v e r t i n g Z o n e ( s t a n d a r d t i m e ) t i m e t o local t i m e .
Local
t i m e i s e q u a l t o t h e z o n e T i m e m i n u s t h e n u m b e r o f t h e z o n e p l u s or
m i n u s t h e l o n g i t u d e of t h e p l a c e ( p l u s 5 s u s e d w h e n t h e l o n g i t u d e
i s e a s t , m i n u s when t h e l o n g i t u d e i s w e s t ) :

This formula assumes t h a t t h e hour zones are counted from 0 t o
24 e a s t w a r d f r o m G r e e n w i c h .

Example.
What i s t h e l o c a l t i m e i n Omsk w h e n t h e z o n e ( s t a n d a r d )
t i m e t h e r e i s 16:00?

S o l u t i o n . U s i n g a map o f h o u r z o n e s or a l i s t o f t h e m o s t
i m p o r t a n t p o p u l a t e d p o i n t s , l e t u s f i n d t h e number o f t h e zone
( t a k i n g i n t o a c c o u n t t h e s t a n d a r d t i m e ) w h e r e Omsk i s l o c a t e d a n d
its longitude.
Taking i n t o account t h e standard t i m e i n O m s k ,
t h e c l o c k s r u n a c c o r d i n g t o t h e t i m e of t h e 6 t h z o n e ( 5 t h z o n e t
1 h r ) ; t h e l o n g i t u d e o f Omsk A = 73O24’E ( 4 : 5 3 : 3 6 ) .
L e t us f i n d t h e l o c a l t i m e T 1 = Tz
4:53:36 = 14:53:36.

- N

t

X E = 16:OO

-

6:OO

t

C o n v e r t i n g l o c a l t i m e to z o n e t i m e ( s t a n d a r d t i m e ) .
Zone
( s t a n d a r d ) t i m e i s e q u a l t o t h e l o c a l t i m e p l u s t h e number o f t h e
z o n e ( t a k i n g i n t o a c c o u n t t h e s t a n d a r d t i m e ) p l u s or m i n u s t h e
l o n g i t u d e o f t h e p l a c e ( p l u s i s u s e d when t h e l o n g i t u d e i s w e s t
a n d m i n u s when t h e l o n g i t u d e i s e a s t ) .

Example.
What i s t h e z o n e
l o c a l t i m e t h e r e i s 18:00?

( s t a n d a r d ) t i m e i n I r k u t s k when t h e

Solution.
Using a l i s t o f t h e most i m p o r t a n t p o p u l a t e d p o i n t s ,
l e t u s f i n d t h e number o f t h e zone ( t a k i n g i n t o a c c o u n t t h e s t a n d a r d
t i m e ) and t h e l o n g i t u d e of I r k u t s k .
Taking i n t o account t h e s t a n d a r d t i m e , I r k u t s k i s l o c a t e d i n
t h e 8 t h z o n e ( 7 t h z o n e + 1 h r ) ; t h e l o n g i t u d e o f I r k u t s k A = 104O
18’E ( 6 : 5 7 : 1 2 ) .

8:OO

L e t u s f i n d t h e zone s t a n d a r d t i m e Tz
- 6:57:12 = 19:02:48.

Measuring Angles

= TI

t N

- A E = 18:OO

+

i n Time Units

S i n c e t h e v a l u e s of hour a n g l e s and r i g h t a s c e n s i o n s a r e used
i t i s o f t e n more c o n v e n i e n t t o e x p r e s s t h e s e

for m e a s u r i n g t i m e ,

455

Also i t i s o f t e n
values i n t i m e units rather than i n degrees.
necessary t o express t h e longitude of a place i n t i m e units.
T o c o n v e r t h o u r a n g l e s a n d r i g h t a s c e n s i o n s as w e l l as l o n g i t u d e from degrees t o hours and back a g a i n , t h e f o l l o w i n g e q u a t i o n s
must be used:
2 4 h r = 3 6 0 O ; .1 h r = 15O or lo = 4 m i n ;
s e c ; 1 s e c = 1 5 " or 1" = 1 / 1 5 s e c .

1 m i n = 1 5 ' o r 1' = 4

These e q u a t i o n s a r e b a s e d on t h e f a c t t h a t t h e c e l e s t i a l
s p h e r e (or t h e E a r t h o n i t s a x i s ) m a k e s a c o m p l e t e r e v o l u t i o n i n
24 h r s , w h i c h c o r r e s p o n d s t o 3 6 0 O .

To convert hour a n g l e s , r i g h t a s c e n s i o n s and t h e l o n g i t u d e of
a p l a c e from d e g r e e s t o h o u r s , t h e f o l l o w i n g must b e done:

(1)

D i v i d e t h e number o f d e g r e e s by 1 5 a n d o b t a i n whole h o u r s .

( 2 )
M u l t i p l y t h e r e m a i n d e r f r o m d i v i d i n g t h e d e g r e e s and ob­
t a i n minutes of t i m e .

(3)
D i v i d e t h e number o f m i n u t e s o f a r c by 1 5 t o o b t a i n whole
minutes of t i m e , which must be added t o t h e m i n u t e s of t i m e a l r e a d y
o b t a i n e d , and o b t a i n t h e t o t a l number o f m i n u t e s o f t i m e .
(4)
M u l t i p l y t h e r e m a i n d e r f r o m d i v i d i n g t h e m i n u t e s by 4 and
obtain seconds of time.
(5)
D i v i d e t h e s e c o n d s of a r c by 1 5 and o b t a i n a n a d d i t i o n a l
number o f s e c o n d s .
Add t h e s e s e c o n d s t o t h e p r e c e d i n g o n e s a n d
o b t a i n a t o t a l number o f s e c o n d s .
(6)
D i s c a r d t h e r e m a i n d e r o f s e c o n d s o f a r c when i t i s l e s s
t h a n 8 ; i f it i s g r e a t e r t h a n 8 , c o n s i d e r it as 1 s e c of t i m e .

Example.

1.

E x p r e s s t h e h o u r a n g l e 163O57'35"

Solution.

i500=
13O =
45' =
12' =
00'30" =
Total:

i n hours.

10:oo
0:52
0:03
0:00:48
0:00:02

10:55:50

T o c o n v e r t h o u r a n g l e s , r i g h t a s c e n s i o n s a n d l o n g i t u d e from
h o u r s t o d e g r e e s , t h e f o l l o w i n g must b e d o n e :

456

(1)

M u l t i p l y t h e h o u r s by 1 5 and o b t a i n t h e r a d i i .

(2)

D i v i d e t h e m i n u t e s by 4 a n d s e p a r a t e o u t t h e whole d e g r e e s .

/434

( 3 ) Multiply the remainder of the minutes of time by 15 and

obtain minutes of arc.


(4) Divide the seconds of time by 4 and separate out minutes

of arc.


(5) Multiply the remainder of the seconds of time by 15 and

obtain seconds of arc.


Example.

Solution.

2.

Express an hour angle of 11:27:15 in degrees.

11 hr
24 min
3 min
1 2 sec
3 sec
Total:

= 165O

=
=
=
=

6O
45'
3'

45'
171O48 '45''
/435

Time Signals

Accurate time in astronomical observatories is determined by

means of astronomical observations of the culmination of heavenly

bodies.

Special transit instruments are used for this purpose. A

transit is mounted in such a way that its main part (the terrestrial

telescope) is always located in the plane of the meridian. Such a

location of the terrestrial telescope permits the observation of a

heavenly body only at the moment when it crosses the meridian, i.e.

at the moment of culmination. Since the Sun is at the point of

upper culmination (crosses the southern part of the meridian) at

true noon, it can be observed only when the hour angle of the true

Sun is zero. Therefore, when the Sun passes through the terrestrial

telescope of the transit instrument, the moment of true noon on the

given meridian will be recorded.

Knowing the precise time of the moment of true noon and com­

paring it with the actual indication of the clock at the moment of

observation, it is possible to check the clock and determine its

error. In the field of vision of a transit, vertical lines are

drawn which permit more accurate determination of the moment that

stars cross the meridian.
In astronomical observatories, the

accurate time is determined on the basis of observing the passage

of stars is based on the fact that at the moment that any star is

at the meridian (we already know), the sidereal time will be equal

t o the right ascension of the star.

Thus, these observations make it possible t o determine the

exact sidereal time. The time obtained is set on special clocks,

which run according t o sidereal time. They differ from normal

clocks in that (by the means of a special cont.ro1) they run 3 min

and 56 sec ahead of normal clocks per day.

457


I


Once the exact sidereal time is obtained, the mean s o l a r time
is calculated at astronomical observatories and set on mean solar
astronomical clocks. The time obtained will be the local mean s o l a r
time on the meridian of the observatory.
When necessary it is always possible t o coqvert this time t o
zone and standard time. The necessary accuracy in determining the
true time at astronomical observatories is computed in hundredths / 4 3 6
of a second, and therefore the process of determining accurate time

is much more complex than we have described here.


-

For accurately determining the moments of the passage of the

stars through the meridian, the moments of the culmination o f the

stars are automatically recorded at astronomical observatories.

With these methods, the determination of the true time is accurate

t o 2 or 3 hundredths of a second. At every observatory, there are

accurate astronomical clocks which are manufactured on special

order. Special care is required for these clocks, since the

continuous changes in temperature and atmospheric pressure strongly

affect the steadiness of the oscillation period of the clock balance

wheels. Therefore, astronomical clocks are kept in a special room

where a constant temperature is maintained, and they are placed

under a hermetically sealed bell jar where a constant atmospheric

pressure is maintained.

In order that the astronomical clocks will run smoothly, they

are protected from vibrations and their hands are very rarely moved.

The clock corrections obtained after calculation are recorded in a

special log, but the c3ock's hands are not moved. This makes it

possible t o know the accurate clock correction at any moment of

time. At astronomical observatories, the clocks are kept running

fast or slow t o the same degree, ?.e. uniformities in the operation

of the clocks are obtained. In modern astronomical clocks, the

change in the readings of such clocks amounts t o hundredths of a

second p e r day. On the basis of the precise time data available at

astronomical observatories, radio stations supply special radio

signals for checking the time.


O r g a n i z a t i o n of T i m e S i g n a l s in A v i a t i o n

Time signals in aviation are organized t o ensure accuracy of

aircraft navigation. The basic task of the time signals in aviation

is the systematic checking of clocks and the guarantee of knowing

the accurate time a t any time of day. The presence of accurate

time is especially important when using astronomical means of air­

craft navigation. This necessitates knowing the accurate time not

only on the ground, but in flight.

In order for the crew t o know the correct time at any time o f

day there must be master clocks with a constant daily speed. They

are installed in the cockpit, at the weather station, or in other

special places.

458

Master c l o c k s a r e u s e d t o d e t e r m i n e t h e c o r r e c t i o n o f o t h e r
c l o c k s i n t h e p e r i o d s between t h e t r a n s m i s s i o n s of a c c u r a t e t i m e
signals.
Master c l o c k s a r e c h e c k e d a n d s e t a c c o r d i n g t o t h e c o r r e c t
t i m e on t h e b a s i s o f a c c u r a t e t i m e r a d i o s i g n a l s t r a n s m i t t e d b y
The c o r r e c t i o n o f t h e c l o c k s
b r o a d c a s t i n g s t a t i o n s o f t h e USSR.
a n d i t s v e r i f i c a t i o n on t h e b a s i s o f s i g n a l s i s r e c o r d e d i n a
special log.

/437

A B r i e f History o f T i m e Reckoning.

S o m e t i m e s we h e a r t h e t e r m s

"Old

S t y l e " and "New

Style".

Systems f o r measuring and c a l c u l a t i n g l a r g e t i m e i n t e r v a l s a r e
c a l l e d calendars.
The b a s i s f o r t i m e r e c k o n i n g i s t h e t r o p i c a l y e a r , ; . e . ,
the
t i m e i n t e r v a l b e t w e e n t w o s u c c e s s i v e p a s s a g e s o f t h e Sun t h r o u g h
t h e point of t h e v e r n a l equinox.
The l e n g t h o f t h e t r o p i c a l y e a r
i s a p p r o x i m a t e l y e q u a l t o 3 6 5 d a y s , 5 h o u r s , 48 m i n u t e s a n d 46
seconds, but it i s very inconvenient t o use t h e t r o p i c a l y e a r f o r
t i m e r e c k o n i n g , s i n c e i t d o e s n o t c o n t a i n a whole number o f d a y s .
T h u s , f o r e x a m p l e , i f w e t a k e m i d n i g h t on J a n u a r y 1 a s t h e b e ­
g i n n i n g o f one y e a r , t h e s e c o n d y e a r w i l l b e g i n n o t a t m i d n i g h t o f
J a n u a r y 1 b u t on J a n u a r y 1 a t 5 : 4 8 : 4 6 A M , t h e t h i r d y e a r a t 1 1 : 3 7 : 3 2
AM, e t c .
We may c o n c l u d e t h a t e a c h y e a r t h e b e g i n n i n g o f t h e new
y e a r would b e s h i f t e d by 5 : 4 8 : 4 6 .
Old S t y l e (Julian) Calendar.
F o r Romans, t h e y e a r w a s o r i g i n a l ­
The l e n g t h o f t h e l u n a r
l y l u n a r and c o n s i s t e d o f 1 2 l u n a r months.
y e a r w a s 355 d a y s , i . e . , t h e i r y e a r w a s 1 0 d a y s s h o r t e r t h a n t h e
accepted year a t t h e present time.
With s u c h a t i m e r e c k o n i n g , t h e
b e g i n n i n g o f t h e new y e a r s h i f t e d r a t h e r q u i c k l y f r o m o n e m o n t h t o
another.
I f , f o r e x a m p l e , we t a k e a t i m e i n t e r v a l o f 1 0 y e a r s ,
t h e n 1 0 0 d a y s were a c c u m u l a t e d d u r i n g t h i s p e r i o d , ; . e . ,
t h e begin­
n i n g o f t h e new y e a r s h i f t e d b y m o r e t h a n t h r e e m o n t h s .

To e l i m i n a t e t h e s e i n c o n v e n i e n c e s , a p p r o x i m a t e l y e v e r y t h r e e
y e a r s t h e y e a r w a s l e n g t h e n e d by one month, : . e . , t h i s y e a r had
13 r a t h e r t h a n 1 2 months.
The Roman d i c t a t o r , J u l i u s C a e s a r , i n t r o d u c e d a c a l e n d a r r e f o r m i n 46 B . C .
This c a l e n d a r w a s c a l l e d t h e J u l i a n c a l e n d a r and
The e s s e n c e o f i t w a s t h a t t h e
we now c a l l i t t h e " O l d S t y l e " .
d u r a t i o n o f o n e y e a r w a s c o n s i d e r e d t o b e 365 d a y s r a t h e r t h a n 355
days.
I n a d d i t i o n , i n � e b r u a r y (which w a s t h e n c o n s i d e r e d t h e l a s t
month) an e x t r a day w a s included every f o u r t h y e a r , i . e . , t h a t y e a r
h a d 366 d a y s .
The a d d i t i o n o f t h e e x t r a d a y o n c e e v e r y 4 y e a r s
n e a r l y compensated f o r t h e d i f f e r e n c e accumulated i n 4 y e a r s (about
6 h r p e r y e a r ) and t h u s t h e constancy of t h e d a t e of t h e v e r n a l
459

e q u i n o x (March 2 1 ) w a s p r e s e r v e d .
u n t i l the present t i m e .

This p r i n c i p l e has been r e t a i n e d

T h e e x t r a d a y a s w e k n o w , i s now a d d e d i n F e b r u a r y :
instead
o f 28 d a y s t h e r e a r e 2 9 d a y s i n a l l t h e y e a r s w h i c h a r e d i v i s i b l e
by 4 , e . g . 1 9 5 6 , 1 9 6 0 , 1 9 6 4 , e t c .
These y e a r s have been c a l l e d
Zeap y e a r s u p u n t i l t h e p r e s e n t t i m e .
New S t y l e ( G r e g o r i a n ) C a l e n d a r .
The l e n g t h o f t h e s o - c a l l e d
t r o p i c a l y e a r i s , a s w e know, 365 d a y s , 5 h o u r s , 48 m i n u t e s a n d Q6
seconds.
The J u l i a n y e a r ( o n t h e a v e r a g e ) i s e q u a l t o 365 d a y s an d
6 hours.
T h u s , t h e J u l i a n y e a r i s 11 m i n a n d 1 4 s e c l o n g e r t h a n
the tropical year.
Although t h i s d i f f e r e n c e i s small, over a l a r g e
i n t e r v a l o f t i m e i t may a l s o c a u s e t h e b e g i n n i n g o f t h e y e a r t o
It is not d i f f i c u l t t o calculate that i n order t o s h i f t the
shift.
d a t e o f t h e v e r n a l e q u i n o x one c a l e n d a r d a t e (one d a y ) , a l m o s t 128
y e a r s ( 2 4 h r or 1 4 4 0 m i n , ' d i v i d e d b y 11 m i n 1 4 s e c ) a r e r e q u i r e d .
The g r a d u a l l y a c c u m u l a t i n g d i f f e r e n c e i n t h e s e c o n d h a l f o f t h e
A s a r e s u l t , t h e d a t e of t h e
1 6 t h c e n t u r y amounted t o 1 0 d a y s .
v e r n a l e q u i n o x came ( a c c o r d i n g t o t h e c a l e n d a r ) n o t o n t h e 2 1 s t ,
b u t on t h e 1 1 t h o f March.

A s a r e s u l t of t h e s h i f t of t h e b e g i n n i n g of s p r i n g from 2 1 t o
11 M a r c h , t h e h o l i d a y o f E a s t e r ( w h i c h m u s t b e c l o s e t o s p r i n g )
g r a d u a l l l y moved t o w a r d t h e s u m m e r .
This g r e a t l y disturbed t h e
c l e r y , who d i d n o t w a n t t o d e p a r t f r o m t h e i r r u l e s .

T h e Roman P o p e G r e g o r y X I 1 1 i n t r o d u c e d a new c a l e n d a r r e f o r m
i n 1 5 8 2 by d e c r e e .
The e s s e n c e o f t h i s r e f o r m , ; . e . , t h e t r a n s f e r
t o a new s t y l e o f c a l e n d a r , w a s t h e f o l l o w i n g :
a f t e r October 4 , 1582,
he ordered everyone t o c o n s i d e r t h e d a t e t o be October 1 5 , r a t h e r
t h a n October 5 , i . e . , he o r d e r e d t h e 1 0 days accumulated o v e r 1 2 0 0 /438
y e a r s t o be dropped s o as t o r e t u r n t h e d a t e of t h e v e r n a l equinox
t o March 2 1 .
I n o r d e r t o avoid t h e accumulation of an e r r o r i n
f u t u r e , i t w a s d e c i d e d t h a t e v e r y 400 y e a r s t h o s e t h r e e d a y s which
d i f f e r e n t i a t e t h e J u l i a n y e a r from t h e t r o p i c a l y e a r be dropped.
To d o t h i s , i t w a s d e c i d e d t h a t t h r e e l e a p y e a r s i n e v e r y 4 0 0 y e a r s
b e c o n s i d e r e d r e g u l a r y e a r s , ; . e , n o t o n t h e b a s i s o f 366 d a y s b u t
r a t h e r on t h e b a s i s o f 365 d a y s .
I n o r d e r t o remember t h i s more
e a s i l y , t h e c e n t u r i a l y e a r s i n which t h e numbers o f t h e c e n t u r y were
d i v i s i b l e by 4 were t a k e n a s l e a p y e a r s ( f o r e x a m p l e , o f t h e c e n t u r i a l
y e a r s 1 6 0 0 , 1 7 0 0 , 1800 and 1 9 0 0 , o n l y t h e y e a r 1600 r e m a i n e d a l e a p
year.
The o t h e r s became r e g u l a r y e a r s , s i n c e o n l y 1 6 i s d i v i s i b l e
by 4 ) .
O f t h e c e n t u r i a l y e a r s , t h e next l e a p y e a r w i l l be t h e year
2000.

-

A s regards the other years (besides the centurial years), the
c a l c u l a t i o n o f t h e l e a p y e a r s r e m a i n e d t h e same a s i n t h e J u l i a n
calendar.
The G r e g o r i a n c a l e n d a r w a s g r a d u a l l y i n t r o d u c e d i n a l l c i v i l i z e d
countries.
I n T s a r i s t R u s s i a , t h e i n t r o d u c t i o n o f t h e new c a l e n d a r

460

..

_.

,

II II

..,-..

(New S t y l e ) m e t w i t h g r e a t o p p o s i t i o n .
Only a f t e r t h e Great
O c t o b e r S o c i a l i s t R e v o l u t i o n o n F e b r u a r y 1, 1 9 1 8 w a s t h e new s t y l e
quickly introduced.
Then t h e d i f f e r e n c e b e t w e e n t h e Old a n d N e w
Styles has already 13 days.
N e i t h e r t h e Old n o r t h e N e w S t y l e i s
a b s o l u t e l y a c c u r a t e , b u t t h e G r e g o r i a n (New S t y l e ) c a l e n d a r h a s
l e s s e r r o r (1 day i n 3000 y e a r s ) .

6.

Use o f A s t r o n o m i c a l D e v i c e s

A s t r o n o m i c a l means o f a i r c r a f t n a v i g a t i o n p e r m i t t h e d e t e r ­
m i n a t i o n of f l i g h t d i r e c t i o n and t h e p o s i t i o n l i n e of t h e a i r c r a f t
on t h e b a s i s of t h e s t a r s .
The a d v a n t . a g e o f a s t r o n o m i c a l d e v i c e s
i s t h e i r autonomy.
T h e i r u s e i n f l i g h t i s n o t r e l a t e d t o any
ground equipment and t h e i r accuracy i s not a f u n c t i o n of f l i g h t
distance.
The u s e o f a s t r o n o m i c a l d e v i c e s i s b a s e d o n t h e p r i n c i p l e
o f m e a s u r i n g t h e a z i m u t h s or t h e a l t i t u d e s o f h e a v e n l y b o d i e s .
The p o s i t i o n o f t h e a i r c r a f t i s d e t e r m i n e d by a s t r o n o m i c a l
i n s t r u m e n t s from t h e i n t e r s e c t i o n o f two a s t r o n o m i c a l p o s i t i o n l i n e s
( A P L ) or ( a s t h e y a r e s t i l l c a l l e d ) l i n e s of e q u a l a l t i t u d e .

The a s t r o n o m i c a l p o s i t i o n l i n e ( l i n e o f e q u a l a 1 t i t u d e ) i s t h e
s t r a i g h t e n e d a r c o f t h e c i r c l e o f e q u a l a l t i t u d e whose c e n t e r i s
t h e geographic p o s i t i o n of t h e star.
The g e o g r a p h i c p o s i t i o n of a s t a r ( G P S ) i s t h e p o i n t o n t h e
E a r t h ' s s u r f a c e a t which t h e g i v e n s t a r i s observed a t t h e z e n i t h ,
or t h e p r o j e c t i o n o f t h e s t a r o n t o t h e s u r f a c e o f t h e E a r t h .
The
c o o r d i n a t e s of t h e g e o g r a p h i c p o s i t i o n of t h e s t a r r e p r e s e n t t h e
equatorial coordinates of t h e given star, ? . e . , t h e l a t i t u d e is
equal t o t h e d e c l i n a t i o n of t h e s t a r ( 4 = 6 ) and t h e longitude i s
e q u a l t o t h e Greenwich hour a n g l e of t h e s t a r ( A = t G r ) .
This is
evident i n Figure 5.24.

C i r c l e of e q u a l a l t i t u d e .
L e t a n o b s e r v e r a t p o i n t A on t h e
E a r t h ' s s u r f a c e ( F i g . 5 . 2 5 ) a t a n y moment o f t i m e m e a s u r e t h e a l t i ­
tude of t h e star M .
On t h e c e l e s t i a l s p h e r e , i f we d r a w a c i r c l e
from p o i n t M w i t h a s p h e r i c a l r a d i u s e q u a l t o t h e z e n i t h d i s t a n c e ,
i t w i l l b e c a l l e d t h e c i r c l e of e q u a l z e n i t h d i s t a n c e s , s i n c e t h e
d i s t a n c e from a n y p o i n t on t h i s c i r c l e t o t h e s t a r M i s e q u a l t o
the zenith distance (radius).
The p r o j e c t i o n o f t h e c i r c l e o f e q u a l z e n i t h d i s t a n c e s ( e a c h
p o i n t o n t h e c i r c l e ) o n t o t h e E a r t h i s c a l l e d t h e c i r c l e of e q u a l
a l t i t u d e (AA') a n d t h e c e n t e r o f t h i s c i r c l e i s t h e g e o g r a p h i c
p o s i t i o n of t h e s t a r M on t h e E a r t h (GPS).
It is called the circle
o f e q u a l a l t i t u d e b e c a u s e a t any p o i n t on t h i s c i r c l e , t h e s t a r M
w i l l h a v e t h e same a l t i t u d e , i . e . i t i s o b s e r v e d a t a n a n g l e f o r t h e
same a l t i t u d e o f t h e s t a r .
T h i s may b e p r o v e d r a t h e r s i m p l y i f w e
r e c a l l t h e r e l a t i o n between t h e a l t i t u d e of a s t a r ( h ) and i t s
zenith distance (Z):
h = 9 0 ° - Z. B u t s i n c e t h e z e n i t h d i s t a n c e
f o r a n y o b s e r v e r l o c a t e d o n t h e c i r c l e o f e q u a l ' a l t i t u d e i s t h e same

461


/439

(R of the circle = Z)
t h e same.

then t h e h e i g h t ( h ) f o r a l l observers w i l l be

K n o w i n g t h i s p r i n c i p l e , w e may m a k e t h e r e v e r s e c o n c l u s i o n :
i f a n o b s e r v e r , by m e a s u r i n g , d e t e r m i n e s t h e a l t i t u d e o f a s t a r a t
a c e r t a i n moment o f t i m e , t h e n b y p l o t t i n g t h e g e o g r a p h i c p o s i t i o n
of t h e star (GPS) w i t h a r a d i u s e q u a l t o t h e z e n i t h d i s t a n c e of t h e
s t a r ( Z = 90° - h ) , t h e c i r c l e o f e q u a l a l t i t u d e may b e d r a w n .
Ob­
v i o u s l y , a t t h e moment o f m e a s u r i n g t h e a l t i t u d e o f t h e s t a r t h e
o b s e r v e r ( a i r c r a f t ) i s l o c a t e d on t h e c i r c u m f e r e n c e of t h e c i r c l e
of equal altitude.
Therefore, the c i r c l e of equal a l t i t u d e is the
circle of t h e p o s i t i o n of t h e aircraft.
I n p r a c t i c e , t h e a l t i t u d e ( h ) a n d a z i m u t h (A) o f a s t a r a r e
d e t e r m i n e d o n t h e b a s i s o f t a b l e s a t a moment o f t i m e p l a n n e d
b e f o r e h a n d f o r t h e p o i n t b e i n g c a l c u l a t e d whose c o o r d i n a t e s a p p r o x i ­
/ 49 u
mately coincide with t h e location of t h e aircraft a t t h e given
moment.
On t h e m a p , a s t r a i g h t - l i n e s e g m e n t e q u a l t o Ah i s d r a w n
from t h e p o i n t being c a l c u l a t e d i n t h e d i r e c t i o n of t h e azimuth
of the star.
The s e g m e n t APL i s drawn p e r p e n d i c u l a r t o t h e s t r a i g h t
l i n e s through i t s end.
A t t h e moment f o r w h i c h t h e c a l c u l a t i o n o f t h e A P L o f t h e p o i n t
b e i n g c a l c u l a t e d w a s made, t h e a l t i t u d e o f t h e s t a r h i s measured
I n g e n e r a l , t h e measured a l t i t u d e does n o t
by means o f a s e x t a n t .
coincide with t h e a l t i t u d e of t h e star a t t h e point being calcu­
lated.
The d i f f e r e n c e b e t w e e n
these altitudes is equal t o the
zenith
d i f f e r e n c e between t h e r a d i i
z distance
of t h e circles of equal a l t i ­
tude of t h e c a l c u l a t e d and a c t u a l
p o i n t s of t h e aircraft's posi­
tion.
Since a l l t h e circles of

P


D'

Fig. 5.24.
of a S t a r .
462

Geographic P o s i t i o n

Fig. 5.25.
Altitude.

C i r c l e of Equal

e q u a l a l t i t u d e o f t h e same s t a r a t t h e same moment o f t i m e a r e c o n ­
c e n t r i c , t h e APL r e p r e s e n t i n g v a r i o u s a l t i t u d e s o f t h e s t a r a r e
p a r a l l e l t o one a n o t h e r .
A s t r o n o m i c a l p o s i t i o n l i n e s which r e p r e s e n t h i g h a l t i t u d e s
are s i t u a t e d c l o s e r t o t h e geographic p o s i t i o n of t h e s t a r , and
vice versa.
T h e r e f o r e , if t h e m e a s u r e d a l t i t u d e (hm) i s g r e a t e r
t h a n t h e c a l c u l a t e d a l t i t u d e ( h c ) , t h i s means t h a t . t h e a i r c r a f t a t
t h e moment o f m e a s u r i n g w a s l o c a t e d n o t o n t h e A P L o f t h e p o i n t
b e i n g c a l c u l a t e d b u t o n t h e A P L p a r a l l e l t o i t moved i n t h e d i r e c ­
t i o n o f t h e s t a r b y a v a l u e Ah = hm - h c .
If t h e m e a s u r e d a l t i t u d e i s l e s s t h a n t h e c a l c u l a t e d a l t i t u d e ,
t h e A P L i s moved i n a d i r e c t i o n o p p o s i t e t h e d i r e c t i o n o f t h e s t a r .
To c o n s t r u c t t h e A P L o n a m a p ,

i t i s n e c e s s a r y t o know:

(a)
The a p p r o x i m a t e c o o r d i n a t e s o f t h e a i r c r a f t ' s p o s i t i o n
( t h e point being c a l c u l a t e d ) $, A .
(b)

The a z i m u t h o f t h e s t a r ( A ) f o r t h e p o i n t b e i n g c a l c u l a t e d :

(c)
The d i s t a n c e b e t w e e n t h e m e a s u r e d a n d c a l c u l a t e d a l t i t u d e s
of t h e star (Ah).

Determining t h e astronomical p o s i t i o n l i n e s .
Before beginning
t h e measurements, t h e e l e c t r i c a l supply and t h e s e x t a n t l i g h t are
s w i t c h e d o n , t h e a v e r a g i n g m e c h a n i s m i s wound u p , a n d t h e r e p e a t e r s
of t h e c h r o n o m e t e r a r e m a t c h e d w i t h t h e i n d i c a t o r o f t h e c u r r e n t
time.
On t h e b a s i s o f a map o f t h e s t e l l a r s k y or t a b l e s o f a l t i t u d e s
and a z i m u t h s , t h e azimuth and t h e n t h e c o u r s e a n g l e of t h e s t a r a r e
determined roughly.
By r o t a t i n g t h e c o u r s e - a n g l e d r u m , t h e s e x t a n t
i s s e t t o t h e c o u r s e a n g l e of t h e s t a r .
By r o t a t i n g t h e a l t i t u d e d r u m , we b r i n g t h e s t a r i n t o t h e s e x ­
t a n t ' s f i e l d of v i s i o n .
The s e x t a n t i s s e t o n a l e v e l b a s e ; by
r o t a t i n g t h e c o u r s e - a n g l e drum, t h e s t a r i s l i n e d up w i t h t h e b u b b l e
level.
By r o t a t i n g t h e a l t i t u d e d r u m , t h e s t a r i s made t o c o i n c i d e
w i t h t h e b u b b l e l e v e l and t h e a v e r a g i n g mechanism i s c o n n e c t e d .
Once t h e p i l o t h a s b e e n n o t i f i e d b e f o r e h a n d a b o u t t h e b e g i n n i n g o f
m e a s u r e m e n t , a n d t h e s t a r h a s b e e n a c c u r a t e l y s u p e r p o s e d on t h e
b u b b l e l e v e l , t h e a v e r a g i n g mechanism i s s w i t c h e d o n .
When t h e
a v e r a g e r h a s completed i t s work, a r e a d i n g i s t a k e n .
On t h e b a s i s o f t h e m e a s u r e d a l t i t u d e s o f t h e s t a r s , p r o b l e m s
i n determining t h e following are solved:
(a)
One a s t r o n o m i c a l p o s i t i o n l i n e o n t h e b a s i s o f t h e S u n
t o check t h e p a t h with r e s p e c t t o d i s t a n c e and d i r e c t i o n .

463


/441

(b)
Two p o s i t i o n l i n e s o n t h e b a s i s o f t w o n a v i g a t i o n a l s t a r s
or o n t h e b a s i s o f o n e n a v i g a t i o n a l s t a r a n d P o l a r i s t o d e t e r m i n e
t h e p o s i t i o n of t h e a i r c r a f t
To c h e c k t h e p a t h w i t h r e s p e c t t o d i s t a n c e b y m e a n s o f o n e
A P L , s t a r s a r e u s e d whose d i r e c t i o n s a r e s i m i l a r t o t h e d i r e c t i o n
of t h e l i n e of t h e given path.
To c h e c k t h e p a t h w i t h r e s p e c t t o
d i r e c t i o n , s t a r s a r e used which a r e s i t u a t e d a t r i g h t a n g l e s t o t h e
l i n e of the given path.
I n d e t e r m i n i n g t h e p o s i t i o n of a n a i r c r a f t o n t h e b a s i s o f t w o
APL's, s t a r s must b e c h o s e n s o t h a t t h e d i r e c t i o n s t o them d i f f e r
b y a n a n g l e c l o s e t o 90°.
In calculating the astronomical position l i n e s , the following
auxiliary t a b l e s a r e used:
(a)
Detached pages from an a v i a t i o n astronomical yearbook
(AAY) for t h e f l i g h t d a t e ;
(b)
T a b l e s o f t h e a l t i t u d e s a n d a z i m u t h s o f t h e S u n , Moon a n d
planets (TAA):
(c)

Tables of t h e a l t i t u d e s and azimuths of t h e stars

(TAAS).

The a s t r o n o m i c a l p o s i t i o n l i n e i s c a l c u l a t e d o n a f o r m l i k e
t h e following:
- .. -. -_
-

Order of
Ooeration

Date
GPA,

W_ _ _ 2

1
1

5
6

9

4

10
3
7

13
2
8

464

x

tl(S1)
$J

6
FB
hnl
P

14
15

S

16

0

17
11
18
19
12

h
hC
Ah

-r

ah^^

A

-.

C
a l c u l a t i .n. g t h e .~A P L
.Name of
the star

__
____

-

3

Name o f
_ _t h e

star
4

-~

Calculation of t h e astronomical p o s i t i o n l i n e with r e s p e c t t o
t h e S u n , Moon or s o m e p l a n e t i s d o n e i n t h e o r d e r i n d i c a t e d i n t h e
l e f t - h a n d column o f t h e f o r m :
(1)
recorded
(2)

point

T h e Moscow t i m e o f m e a s u r i n g t h e a l t i t u d e o f t h e s t a r i s
(TMOSCOW).
The m e a s u r e d a l t i t u d e o f t h e s t a r i s r e c o r d e d

(hm).

( 3 ) and ( 4 ) .
The l a t i t u d e a n d l o n g i t u d e o f t h e c a l c u l a t e d
(r$
and A ) a r e r e c o r d e d .

(5)
The G r e e n w i c h t i m e o f m e a s u r i n g i s d e t e r m i n e d on t h e b a s i s
of t h e formula

w h e r e (N, + 1 ) i s t h e n u m b e r o f t h e h o u r z o n e p l u s t h e s t a n d a r d
h o u r a n d i s r e c o r d e d on t h e f o r m .
( 6 ) a n a ( 7 ) The d e c l i n a t i o n o f t h e s t a r ( 6 ) a n d i t s h o u r a n g l e
( t G r ) for t h e w h o l e h o u r c o r r e s p o n d i n g t o t h e t i m e o f m e a s u r e m e n t
are copied out of t h e AAY.
(When u s i n g t h e Moon, t h e t G r i s w r i t t e n
f o r whole t e n s o f m i n u t e s . )
( 8 )
I n m e a s u r i n g t h e a l t i t u d e of t h e Moon,
i s copied from t h e AAY.

the parallax (P)

(9).
On t h e b a s i s o f t h e i n t e r p o l a t i o n t a b l e a v a i l a b l e i n
t h e T A A or t h e A A Y , t h e c o r r e c t i o n ( A t G r ) f o r T G i~ n m i n u t e s a n d
s e c o n d s i s found and r e c o r d e d .

(10) The l o c a l h o u r a n g l e o f t h e s t a r ( t i ) i s d e t e r m i n e d by
adding t G r , h
t a n~d A ~.
I n c r e a s i n g or d e c r e a s i n g X , i t i s n e c e s ­
s a r y t h a t t l b e e x p r e s s e d by a whole e v e n number o f d e g r e e s .
If
t h e w e s t e r n h o u r a n g l e i s m o r e t h a n 180°, i t s c o m p l e m e n t t o 3 6 0 °
is taken.
The v a l u e f o u n d f o r t l i s c o n s i d e r e d t h e e a s t e r n h o u r
a n g l e and i s r e c o r d e d i n t h e form.
The v a l u e f o r t h e l o n g i t u d e o f t h e c a l c u l a t e d p o i n t X , w r i t t e n
e a r l i e r on t h e f o r m , i s r e f i n e d i n a c c o r d a n c e w i t h t h e c h a n g e
introduced with t h e s e l e c t i o n of tl.
(11) a n d ( 1 2 )
From t h e T A A , t h e v a l u e f o r t h e a l t i t u d e o f t h e
s t a r a t t h e c a l c u l a t e d p o i n t (h,) i s w r i t t e n , t a k i n g i n t o a c c o u n t
t h e c o r r e c t i o n f o r m i n u t e s of d e c l i n a t i o n , and t h e azimuth A i s
recorded.
If t h e h o u r a n g l e i s w e s t e r n , t h e n t h e complement o f i t s
t a b u l a r v a l u e up t o 360° i s t a k e n a s t h e a z i m u t h .

(13)
The p a t h b e a r i n g ( P B ) o f t h e s t a r ( p a t h a n g l e ) i s d e t e r ­
m i n e d o n t h e b a s i s o f t h e f o r m u l a PB = A - GPA a n d i s w r i t t e n o n
465

/442

t h e form.

(14) ( 1 5 ) a n d (16) C o r r e c t i o n s a r e w r i t t e n o n t h e f o r m f r o m
s e x t a n t (SI, f o r t h e r e f r a c t i o n ( - r ) a n d
t h e pertinent tables:
f o r the Earth's rotation ( 6 ) .
(17) The m e a s u r e d a l t i t u d e o f t h e s t a r ( h ) i s a d j u s t e d f o r
c o r r e c t i o n s N o s . 8 , 14, 15, 16.
(18)
The d i f f e r e n c e b e t w e e n t h e c o r r e c t e d v a l u e o f t h e
measured a l t i t u d e (hm) and t h e a l t i t u d e of t h e s t a r a t t h e c a l c u ­
l a t e d p o i n t (h,) i s c a l c u l a t e d on t h e b a s i s o f t h e f o r m u l a
Ah = h m ­
hC'

(19) A n o t h e r v a l u e f o r Ah i s r e c a l c u l a t e d i n k i l o m e t e r s .
A f t e r t h e c a l c u l a t i o n s , A P L i s p l o t t e d on t h e c h a r t a s shown a b o v e .

/443

The A P L o n t h e b a s i s o f s t a r s i s c a l c u l a t e d i n t h e s a m e o r d e r
a s o n t h e b a s i s o f t h e S u n , Moon a n d p l a n e t s .
T a b l e s o f t h e TAAS
and AAY a r e u s e d .
I n a d d i t i o n , i n s t e a d o f t h e Greenwich and l o c a l
hour a n g l e s , t h e Greenwich and l o c a l s i d e r e a l t i m e a r e determined.
The s i d e r e a l G r e e n w i c h t i m e ( S G ~ i) s t a k e n f r o m t h e t a b l e
e n t i t l e d " S t a r s " i n t h e A A Y f o r t h e moment T G ~ . T h e l o c a l s i d e r e a l
t i m e i s d e t e r m i n e d on t h e b a s i s o f t h e f o r m u l a

where X i s t h e l o n g i t u d e of t h e c a l c u l a t e d p o i n t r e f i n e d w i t h t h i s
c a l c u l a t i o n s o t h a t S 1 i s e q u a l t o a whole number o f d e g r e e s .
When d e t e r m i n i n g t h e p o s i t i o n o f t h e a i r c r a f t o n t h e b a s i s o f
t h e i n t e r s e c t i o n o f t h e APL from two n a v i g a t i o n a l s t a r s , t h e meas­
u r i n g and r e c o r d i n g o f t h e t i m e o f t h e r e a d i n g s a r e done s u c c e s s i v e ­
l y with t h e s h o r t e s t possible time i n t e r v a l .
When p l o t t i n g o n t h e
c h a r t , t h e f i r s t APL i s s h i f t e d p a r a l l e l t o i t s e l f i n t h e d i r e c t i o n
of t h e vector of t h e f l i g h t speed f o r t h e distance traversed i n
t h i s interval of t i m e .
The c o r r e c t i o n f o r t h e movement o f t h e a i r c r a f t b e t w e e n t h e
moments o f t h e f i r s t a n d s e c o n d m e a s u r e m e n t s i s a l s o d e t e r m i n e d b y
means o f a s p e c i a l t a b l e a p p l i e d t o t h e t a b l e s o f t h e a l t i t u d e s
and azimuths o f s t a r s .
The c o r r e c t i o n f o r t h e r o t a t i o n o f t h e E a r t h ( 0 ) i s i n t r o d u c e d
It i s not necessary
i n t h e a l t i t u d e o f t h e s t a r measured f i r s t .
t o s h i f t t h e f i r s t APL d e t e r m i n e d , t a k i n g i n t o a c c o u n t t h e c o r r e c t i o n
o f 6 by t h e movement o f t h e a i r c r a f t i n t h i s c a s e .
When d e t e r m i n i n g t h e p o s i t i o n o f a n a i r c r a f t o n t h e b a s i s o f
s t a r s i n t h e N o r t h e r n Hemisphere, P o l a r i s a n d one o f t h e n a v i g a t i o n a l

466

,.e--;-

s t a r s s i t u a t e d i n a w e s t e r l y or e a s t e r l y d i r e c t i o n a r e u s e d .
Polaris
i s a p p r o x i m a t e l y lo f r o m t h e n o r t h c e l e s t i a l p o l e a n d t h e r e f o r e
i t s h e i g h t above t h e horizon i s always roughly e q u a l t o t h e l a t i t u d e
of t h e p o s i t i o n .
This s i m p l i f i e s t h e c a l c u l a t i n g and p l o t t i n g of
APL's.

The a c c u r a t e l a t i t u d e o f t h e p o s i t i o n o f a n a i r c r a f t on t h e
b a s i s o f P o l a r i s i s d e t e r m i n e d by s i m p l e a d d i t i o n :
I t s measured a l t i t u d e hm; t h e c o r r e c t i o n of
c o r r e c t i o n f o r r e f r a c t i o n -r; t h e c o r r e c t i o n f o r
o and t h e c o r r e c t i o n f o r t h e a l t i t u d e of P o l a r i s
A + i s g i v e n i n TAAS o n t h e b a s i s o f t h e v a l u e o f
t i m e S1.

the sextant S; the
the Earth's rotation
A+.
The c o r r e c t i o n
the local sidereal

The a l t i t u d e o f P o l a r i s i s m e a s u r e d l a t e r t h a n t h e a l t i t u d e
of t h e n a v i g a t i o n a l s t a r and t h e r e f o r e t h e p a r a l l e l corresponding
t o t h e l a t i t u d e found i s s h i f t e d i n t h e d i r e c t i o n of t h e f l i g h t speed v e c t o r f o r t h e segment of t h e p a t h t r a v e r s e d during t h e time
i n t e r v a l between t h e f i r s t and s e c o n d m e a s u r e m e n t s .
The c o r r e c t i o n d u e t o t h e t r a v e l o f t h e a i r c r a f t i s a l s o
i n t r o d u c e d d i r e c t l y i n t h e c a l c u l a t e d l a t i t u d e by means o f a t a b l e
o f c o r r e c t i o n s I c D " i n t h e TAAS.

Astronomical

Compasses


Modern a s t r o n o m i c a l c o m p a s s e s a r e a u t o m a t i c d e v i c e s f o r d e t e r ­
m i n i n g t h e t r u e c o u r s e o f t h e a i r c r a f t by t h e d i r e c t i o n - f i n d i n g o f
t h e S u n or o t h e r s t a r s .
Astronomical compasses of t h e t y p e DAK-DB

a r e u s e d on a i r c r a f t .

T h e s e a s t r o c o m p a s s e s a r e m a i n l y i n t e n d e d for:
(a)
I n c i d e n t a l d e t e r m i n a t i o n o f t h e t r u e c o u r s e on t h e b a s i s
of t h e Sun;
(b)
Continuous measurement of t h e c o u r s e i n f l i g h t a l o n g t h e
o r t h o d r o m e on t h e b a s i s o f t h e Sun .
Astrocompasses o f t h e DAK-DB t y p e c a n t r a n s m i t t h e v a l u e s o f
t h e t r u e c o u r s e t o c o u r s e system i n d i c a t o r s , and t h e y can a l s o
p e r m i t t h e t r u e c o u r s e t o b e d e t e r m i n e d on t h e b a s i s o f s t a r s a t
n i g h t by means o f a p e r i s c o p e s e x t a n t .
A s t r o c o m p a s s e s o f D A K - D B t y p e may b e u s e d i n t h e r a n g e o f
l a t i t u d e s from t h e North P o l e t o 10°S.
Astrocompasses of a s p e c i a l
t y p e are i n t e n d e d f o r u s e i n t h e Southern Hemisphere as w e l l .
They
c a n o p e r a t e when t h e S u n i s n o t m o r e t h a n 7 0 ° a b o v e t h e h o r i z o n .
Here t h e p e r m i s s i b l e e r r o r i n d e t e r m i n i n g t h e t r u e c o u r s e must n o t
exceed +2O.
467

/444

An astrocompass automatically solves problems o f determining

the true course of an aircraft according t o the equation:

TC

A - CA

where A is the azimuth o f the heavenly body and CA is the course

angle of the heavenly body.

The course angle of the Sun is determined automatically by

means of a course-angle data transmitter (CAD).

The photoelectric head is situated in a transparent case in

the fuselage of the aircraft; by means of an electronic system,

it is automatically oriented in the direction of the Sun and sup­

plies an electrical signal representing the course angle (CA) t o

a computer device.

The azimuth of the star is determined by a special computer
/445
whose basis is a spatial computer mechanism (spherant).
When es­

tablishing the equatorial coordinates on computers, the hour angle

and declination of the star as well as the latitude and longitude

of the position, the azimuth of the star, i.e. the horizontal co­

ordinate, is given at the output in the form of electrical signals.

The table for the Greenwich hour angles of the Sun is given in

Supplement 5 .

star


1x

\

A signal representing the dif­

ference in the azimuth and course

angle, ?.e. the value of the true

course, is fed t o the indicator

of the astrocompass.


T

L

\
\

\

the horizon


4
I
I

mz3

I

3-

I
I

63

I
I

6

b

light

source
Fig. 5.26. Optical Diagram

o f an Aviational Sextant.

468

When using the astrocompass

t o determine and retain the ortho­

drome course, coordinates pertaining

t o the initial point of the ortho­

drome path line are fed into the­

astrocompass. During flight along

the orthodrome, the course angles

at the initial point of the route.

To preserve a constant value of

the true course relative t o the

reference meridian of the beginning

of the path, a correction on the

basis of the flight correction

method is automatically fed in.

This method entails the following:

The axis of rotation of the head

of the CAD is vertical at the be­

ginning of the path. Later with

movement o f the aircraft along the


o r t h o d r o m e , it s l o p e s back t o w a r d t h e t a i l o f t h e a i r c r a f t by a n
angle e q u a l t o t h e arc o f t h e t r a v e r s e d p a r t of t h e orthodrome a t
t h e s a m e t i m e remaining p a r a l l e l t o t h e o r i g i n a l position.
The
automatic c a l c u l a t i o n of t h e angle proportional t o t h e arc of t h e
t r a v e r s e d segment of t h e o r t h o d r o m e i s p e r f o r m e d by t h e f l i g h t
c o r r e c t o r , w i t h a manual s e t t i n g of t h e a i r s p e e d o f t h e a i r c r a f t .
Astronomical

Sextants

A v i a t i o n a l a s t r o n o m i c a l s e x t a n t s a r e i n t e n d e d for m e a s u r i n g
t h e a l t i t u d e s of stars t o determine t h e astronomical p o s i t i o n l i n e s
and t h e p o s i t i o n of t h e a i r c r a f t , as w e l l as f o r measuring t h e c o u r s e
angles of stars.
A t t h e p r e s e n t t i m e , p e r i s c o p e s e x t a n t s ( P S ) which are a d a p t e d
f o r m o u n t i n g on a i r c r a f t w i t h h e r m e t i c f u s e l a g e s a r e t h e most
common v a r i e t y .

The o p t i c a l s y s t e m o f t h e P S s e x t a n t ( F i g . 5 . 2 6 ) i n c l u d e s a
c u b i c p r i s m 1 for s i g h t i n g s t a r s .
The c u b i c p r i s m t u r n s i n a v e r t i ­
c a l p l a n e f r o m 0 t o 8 5 O , w i t h a g o n i o m e t e r drum t o i n d i c a t e a l t i ­
tude of a star.
I n t h e h o r i z o n t a l p l a n e , i t t u r n s t h r o u g h 360° w i t h t h e g o n i o ­
m e t e r drum f o r c o u r s e a n g l e s .
The c u b i c p r i s m p r o j e c t s a n i m a g e o f
Between t h e n o d e s
t h e s t a r a l o n g t h e o p t i c a l a x i s of t h e s e x t a n t .
o f t h e t r a n s f o r m a t i o n s y s t e m , a s p e c i a l p r i s m 2 i s s i t u a t e d on t h e
optical axis.
I n t h i s p r i s m , t h e image of t h e s t a r i s matched w i t h
t h e bubble l e v e l .
T h e i m a g e o f t h e b u b b l e i s p r o j e c t e d o n t o p r i s m /446;
2 by means o f r o t a t i n g p r i s m 4 .
The combined i m a g e s o f t h e s t a r a n d
t h e b u b b l e , by means o f r o t a t i n g p r i s m 5 , a r e d i r e c t e d t o t h e
o c u l a r 6 , where t h e y a r e o b s e r v e d by t h e e y e .
The s e x t a n t h a s a c h r o n o m e t e r w i t h t w o i n d e p e n d e n t r e p e a t e r s ,
t h e c l o c k mechanism of t h e a v e r a g e r o f t h e r e a d i n g s a n d t h e c o u r s e angle transmitting selsyn.

469

CHAPTER S I X
A C C U R A C Y I N AIRCRAFT N A V I G A T I O N

1. A c c u r a c y i n M e a s u r i n g N a v i g a t i o n a l E l e m e n t s a n d i n
A i r c r a f t N a v i g a t i o n as a Whole
The p r o c e s s o f a i r c r a f t n a v i g a t i o n i s d i r e c t e d t o w a r d a c r e w ' s
m a i n t a i n i n g g i v e n t r a j e c t o r i e s o f a i r c r a f t movement w i t h r e s p e c t t o
d i r e c t i o n , a l t i t u d e , d i s t a n c e , and t i m e .
S i n c e t h e c o o r d i n a t e s of a n a i r c r a f t and t h e p a r a m e t e r s o f i t s
s p e e d a l o n g t h e a x e s o f c o o r d i n a t e s o f a c h o s e n frame o f r e f e r e n c e
a r e measured w i t h d e f i n i t e e r r o r s , i t i s n a t u r a l t h a t a g i v e n
t r a j e c t o r y o f a i r c r a f t movement w i l l l i k e w i s e b e m a i n t a i n e d w i t h
some e r r o r s .
By a c c u r a c y of a i r c r a f t n a v i g a t i o n i s m e a n t t h e l i m i t s w i t h i n
which t h e e r r o r s o f a n y f l i g h t - t r a j e c t o r y p a r a m e t e r a r e i n c l u d e d
with a definite probability.
I n c o n t r a s t t o t h e a c c u r a c y o f n a v i g a t i o n a l d e v i c e s , which
c h a r a c t e r i z e s ( i n t h e majority of c a s e s ) t h e e r r o r s i n measuring
o n e c o o r d i n a t e or t w o a i r c r a f t c o o r d i n a t e s s i m u l t a n e o u s l y , t h e
a c c u r a c y of a i r c r a f t n a v i g a t i o n d e p e n d s o n t h e c o n d i t i o n s o f i m p l e ­
m e n t i n g i n d i c a t e d m e a s u r e m e n t s a n d , i n some c a s e s , o n t h e d y n a m i c s
of a i r c r a f t f l i g h t .
L e t u s assume t h a t a n a i r c r a f t i s moving i n a f i e l d o f c o n s t a n t
The d i r e c t i o n o f f l i g h t i s m a i n ­
w i n d or u n d e r c o n d i t i o n s o f c a l m .
t a i n e d on t h e b a s i s o f r e s u l t s o f m e a s u r i n g t h e l a t e r a l d e v i a t i o n
of t h e a i r c r a f t ( 2 ) f r o m t h e l i n e o f t h e g i v e n p a t h a t d e s i g n a t e d
points (Fig. 6.1).
P o i n t s A and B i n t h e f i g u r e correspond t o t h e a c t u a l coordi­
n a t e s of t h e a i r c r a f t , w h i l e p o i n t s A 1 a n d B 1 c o r r e s p o n d t o m e a s u r e d
coordinates.

I t i s o b v i o u s t h a t on t h e b a s i s o f r e s u l t s o f m e a s u r e m e n t s
( A 1 a n d B l ) , t h e a i r c r a f t crew d o e s n o t o b t a i n a n a c c u r a t e n o t i o n
c o n c e r n i n g t h e d i r e c t i o n of movement, i . e . t h e r e i s a n e r r o r i n
determining t h e actual angle of f l i g h t A$.
470

/447

I n g e n e r a l , e r r o r s i n measuring t h e Z-coordinate (and, t h e r e ­
f o r e , $ > w i l l e x e r t t h e same i n f l u e n c e o n t h e a c c u r a c y o f a i r c r a f t
navigation with r e s p e c t t o d i r e c t i o n , independently of whether t h e
a c t u a l t r a j e c t o r y o f a i r c r a f t movement w i l l c o i n c i d e w i t h t h e g i v e n
t r a j e c t o r y or w h e t h e r i t i s s i t u a t e d a t s o m e s l i g h t a n g l e t o i t .

Fig.

6.1

D i a g r a m o f t h e O c c u r r e n c e o f Errors i n A i r c r a f t N a v i g a t i o n
with Respect t o Direction.

However, f o r s i m p l i c i t y o f a r g u m e n t , w e w i l l c o n s i d e r t h a t on
segment AB t h e a c t u a l p a t h l i n e of t h e a i r c r a f t a c c i d e n t a l l y t u r n e d
out t o correspond s t r i c t l y with t h e given l i n e .
In t h i s case, angle
A$ a n d c o o r d i n a t e B 1 w i l l b e m a g n i t u d e s o f m i s i n f o r m a t i o n f o r t h e
c r e w which, i n t h e i r graphic form, determine e r r o r s i n t h e crew’s
a c t i o n s i n t h e f l i g h t s e g m e n t BC.
A c t u a l l y , a crew l o c a t e d a t p o i n t B p r e c i s e l y on t h e g i v e n
p a t h w i l l assume t h a t t h e a i r c r a f t i s l o c a t e d a t p o i n t B 1 .
There­
f o r e , f o r a n a p p r o a c h t o p o i n t C i t w i l l b e o b l i g e d t o make a n a d ­
vance i n t h e c o u r s e :
BB,

A: = arctg __

BC

.

I n a d d i t i o n , t h e crew w i l l assume t h a t a n a i r c r a f t on segment
A B d i d no? t r a v e l p a r a l l e l t o t h e g i v e n p a t h l i n e , b u t a t a n a n g l e
A$, e q u a l t o

A+i - nrctg

Therefore,

An,
--,--

+

B.5,
AB.

*

t h e t o t a l i n c o r r e c t advance i n t h e course
A y t o t a l =L+~=A+,+I;.

T h e r e f o r e , i f t h e d i s t a n c e BC i s a p p r o x i m a t e l y e q u a l t o A B ,
t h e a i r c r a f t must g o n o t t o p o i n t C b u t t o p o i n t C2, s i t u a t e d t h e
f o l l o w i n g d i s t a n c e from p o i n t C:

/448(

where AA1

= A Z l ; BB1 = A Z 2 .

Under a c t u a l f l i g h t c o n d i t i o n s , it i s d i f f i c u l t t o e x p e c t t h a t
There­
t h e w i n d i n s e g m e n t BC w i l l b e t h e s a m e a s i n s e g m e n t AB.
f o r e , i f a f l i g h t i s made o v e r BC b y m a i n t a i n i n g t h e c o n d i t i o n s e ­
l e c t e d i n s e g m e n t AB, t h e a i r c r a f t w i l l n o t a p p e a r a t p o i n t C2, b u t
a t p o i n t C3, d i s p l a c e d f r o m p o i n t C2 b y t h e v a l u e o f t h e c h a n g e i n
t h e w i n d v e c t o r i n s e g m e n t BC w i t h r e s p e c t t o s e g m e n t AB r e l a t i v e
t o t h e f l y i n g t i m e BC.
For a i r c r a f t n a v i g a t i o n with r e s p e c t t o d i r e c t i o n , only t h e
l a t e r a l c o m p o n e n t o f t h e w i n d c h a n g e v e c t o r ATz w i l l h a v e a n y s i g ­
Thus, t h e general e r r o r i n a i r c r a f t navigation with
nificance.
r e s p e c t t o d i r e c t i o n i n s e g m e n t BC i s :
(6.1)

I t i s p o s s i b l e t o come t o a n a n a l o g o u s c o n c l u s i o n b y e x a m i n i n g
t h e accuracy of aircraft navigation with r e s p e c t t o distance i f t h e
c o n d i t i o n of s p e e d i s c h o s e n on t h e b a s i s o f r e s u l t s o f m e a s u r i n g
t h e X - c o o r d i n a t e a t p o i n t s A and B:

Ax~=
c AX1

+ 2AX2 + AITxt.

(6.2)

F o r m u l a s (6.1) a n d ( 6 . 2 ) d e t e r m i n e t h e a b s o l u t e e r r o r s i n a i r ­
craft n a v i g a t i o n w i t h r e s p e c t i o n t o d i r e c t i o n a n d d i s t a n c e .
In

(Am

t h e s e f o r m u l a s , o n l y t h e t h i r d t e r m on t h e r i g h t - h a n d s i d e
a n d A v ) i s a v a l u e w h i c h d e p e n d s o n t h e l e n g t h 'of t h e s t a g e o f t h e
p a t h a n d t h e r e f o r e , on f l i g h t t i m e .
Therefore, the absolute error
grows smoothly w i t h an i n c r e a s e i n t h e l e n g t h of t h e s t a g e i n t h e
p a t h between t h e c o n t r o l p o i n t s .
The r a t i o o f t h e a b s o l u t e e r r o r o f a g i v e n
length of t h e stage i n t h e path of t h e a i r c r a f t
a r i s e s i s c a l l e d t h e r e l a t i v e e r r o r of a i r c r a f t
f o r e , t h e r e l a t i v e e r r o r e x e r t s a n i n f l u e n c e on
t h e f l i g h t c o n d i t i o n s of t h e a i r c r a f t .
Let us i
a s p e c i f i c example.

parameter t o the
i n which t h i s e r r o r
navigation.
There­
t h e s t a b i l i t y of
llustrate this with

L e t u s assume t h a t a t t h e c o n t r o l s t a g e i n t h e p a t h of an a i r ­
c r a f t , w i t h a l e n g t h o f 2 0 0 km, a n e r r o r o f a i r c r a f t n a v i g a t i o n o f
5 km i n d i s t a n c e a n d 4 km i n d i r e c t i o n h a s a c c u m u l a t e d .
The r e l a t i v e e r r o r i n a i r c r a f t n a v i g a t i o n w i t h r e s p e c t t o d i s ­
t a n c e and d i r e c t i o n w i l l be:

AX

-

X
A2

X

472

5
200

-

-

1

40

4 .-

200

- 2,596;

50

-2%.

/449

The r e l a t i v e e r r o r w i t h r e s p e c t t o d i r e c t i o n c h a r a c t e r i z e s t h e
conditional errors of aircraft navigation:
A+ = arctg'

A2

.
X

I n t h e f o l l o w i n g s t a g e o f f l i g h t o f e q u a l l e n g t h ( 2 0 0 km) i n
o r d e r t o b a l a n c e t h e e r r o r s o f a i r c r a f t n a v i g a t i o n which were accumu­
l a t e d i n t h e preceding s t a g e , it is necessary:
t o introduce a c o r r e c t i o n i n 'the aircraft course equal t o
(a)
t h e e r r o r A $ , i n o u r case a r c t g 1 / 5 0
lo, a n d
(b)

t o change t h e a i r s p e e d ,

i n our e x a m p l e b y 2 . 5 % .

L e t u s a s s u m e now t h a t t h e s a m e e r r o r i n a i r c r a f t n a v i g a t i o n
a r o s e a t a s t a g e i n t h e p a t h a b o u t 5 0 km l o n g .
Then

-AX

X

-

sa.

-1096;

I n t h i s case it would b e n e c e s s a r y f o r u s t o change t h e a i r ­
c r a f t c o u r s e b y 4 O a n d t h e a i r s p e e d b y 1 0 % f o r e v e r y 5 0 km o f t h e
p a t h , i . e . i n modern a i r c r a f t , e v e r y 3 - 4 min o f f l i g h t .
Considering t h a t t h e e r r o r i n a i r c r a f t navigation increases
w i t h r e s p e c t t o t i m e o n l y a s a r e s u l t of a c h a n g e i n t h e wind v e c t o r ,
it becomes e n t i r e l y o b v i o u s t h a t i t i s a d v a n t a g e o u s t o c h o o s e
c o n t r o l s t a g e s o f f l i g h t which a r e v e r y l o n g , b o t h from t h e p o i n t
of view of t h e f r e q u e n c y o f i n t r o d u c i n g c o r r e c t i o n s i n t h e a i r c r a f t
f l i g h t c o n d i t i o n and i n t h e v a l u e s of t h e c o r r e c t i o n s b e i n g i n t r o ­
duced.
However, w i t h e x c e s s i v e l y l o n g f l i g h t s t a g e s b e t w e e n c o n t r o l
p o i n t s , an a b s o l u t e e r r o r can be accumulated i n a i r c r a f t n a v i g a t i o n ,
b i t h r e s p e c t t o b o t h d i s t a n c e and d i r e c t i o n ) which exceeds t h e
p e r m i s s i b l e l i m i t as a r e s u l t of t h e t h i r d t e r m s of ( 6 . 1 ) and ( 6 . 2 ) .
T h e r e f o r e t h e r e a r e c o n d i t i o n s f o r c h o o s i n g a n optimum l e n g t h o f
t h e c o n t r o l f l i g h t s t a g e a t which t h e r e s u l t s of a i r c r a f t n a v i g a t i o n
a r e optimum, b o t h f r o m t h e p o i n t o f v i e w of m a i n t a i n i n g a g i v e n
t r a j e c t o r y with allowable l i m i t s of d e v i a t i o n and s t a b i l i t y of a i r ­
craft f l i g h t conditions.
The n e c e s s a r y a c c u r a c y o f a i r c r a f t n a v i g a t i o n w i t h r e s p e c t t o
d i r e c t i o n o f t h e f l i g h t p a t h i s d e t e r m i n e d by t h e s e t w i d t h o f a i r
r o u t e s and approach p a t h s t o a i r p o r t s , as w e l l as n a t i o n a l
473

b


/450)

boundaries.

However, it is necessary t o consider that at turning points on

the paths, with significant turn angles for the route, the errors

of aircraft navigation with respect t o distance become errors with

respect t o direction, and vice versa.

The accuracy of aircraft navigation during the approach of an

aircraft landing on instruments acquires a special significance.

The necessary length of the path of an aircraft's approach to a

given trajectory, after changing t o visual flight, depends on the

magnitude of the aircraft's deviation from the given descent trajec­

tory during an instrument approach for landing, and therefore on

/451

the weather conditions during which a landing can be made.
With automatic or semiautomatic approach t o landing by air­

craft up to low altitudes (for example, up t o leveling off.or landing)

the accuracy of aircraft navigation must be such that the landing

of the aircraft in all cases will be ensured with the execution of

safe deviation norms with respect to the landing position and

direction of the aircraft vector in the path.


2.

Methods of

E v a l u a t i n g t h e Accuracy o f A i r c r a f t Navigation

In special books on the study of the accuracy of aircraft n.avi­

gation with the application of navigational systems, the methods of

probability theory (Laws of the distribution of random variables)

are used.

To evaluate the accuracy of aircraft navigation under practical

conditions, it is sufficient to use only the basic conclusions of

probability theory. Since the study of probability theory as a

science is not the purpose of this textbook, in the majority of

cases these conclusions will be given without proofs.

In probability theory, variables which cannot be determined

in advance by classical methods of mathematics, or are determined

by methods so complex that they cannot be used for practical pur­

poses, are considered t o be random variables.

In connection with problems of
gation or the accuracy of measuring
of navigational systems, the errors
some of the navigational parameters

the accuracy of aircraft navi­

aircraft coordinates by means

in measuring or maintaining

will be random variables.


Let us assume that the value of some navigational flight param­

eter (on the basis of some especially precise control device) is

known exactly. However, in carrying out a number of measurements

by the usual means, we always obtain new values for the parameter

which differ from its precise value.

The precise value of a measured parameter will be called its


474

mathematical expectation. If a series of measurements is suffi­
ciently great, then in all probability we will obtain many values
for the measured parameter, with both positive and negative errors.
Here the mean arithmetic value of all the measurements will ap­
proach (depending on the increase in their number) the mathematical
expectation of the measured value. Therefore, t o raise the accuracy
of aircraft navigation, in many cases measurements are carried out
repeatedly and the arithmetic mean of the series of measurements is
found

.

The arithmetic mean of a measured parameter cannot characterize / 4 5 2
the probable accuracy of carrying out individual measurements.
Therefore, probability theory includes a concept of mean square
deviation from the precise value.
Let us designate the precise value of a measured quantity by
2, 3

a , and its measured values by x i , where i = 1,

...

Let us call the value ( x i - a ) the m e a s u r e m e n t e r r o r .
The value obtained by extracting the square root from the sum

of the squares of the errors divided by the number of measurements

is considered the mean square error of measurement:


(6.3)


According to (6.31, the mean square error of measurement is
determined when the precise value of magnitude a is known.

If the value of the measured magnitude is determined as an
arithmetic mean from a series of observations, it is considered
that one of the measured magnitudes coincides with or very closely
approaches the arithmetic mean. The error of this measurement is
considered t o be zero, resulting in an increase in the sum in the
numerator under the root of (6.3) equal t o zero. Therefore, in
order to avoid decreasing the value of the mean square error,
especially with a short series of measurements, the denominator of
(6.3) reduces t o 1. Then this formula assumes the form:

The mean square error characterizes the accuracy of the meas­

urements in a rather definite way. With the raising of each of the

errors to a square, its sign always becomes positive. Therefore,

475

in determining mean square errors, only the absolute value of each

plays a role.

It is considered that the mean square e r r o r does not have a
sign.

If we examine only one of a series of measurements, with a
probability equal t o 1 (complete probability), it is possible t o
say that the magnitude being measured will undoubtedly have some
value. However the probability that the magnitude being measured
will have a strict and absolutely precise value is practically
equal t o zero, except in cases when it can assume only a,discrete
value. Therefore, in determining the probability of an error of
measurement it is not the precise value of the error, but the limits / 4 5 5
in which it must be found, which are given (for example, the proba­
bility of error in the range from 500-600 m or from 2 t o 2.5 k m ,
etc. 1.
All the measured navigational magnitudes are (to a certain

degree) calibrated magnitudes, ;.e., they have errors limited by

certain boundaries. These boundaries depend on the allowances in

the regulation of the measuring apparatus and on the maximum pos­

sible distortions of the measured magnitudes as a result of the in­

fluence of external factors (electromagnetic wave propagation, the

physical composition of the airspace, variations in the Earth's

magnetic field, etc.).

Allowances in the regulation o f measuring apparatus are known

quantities. Century-old observations permit the .determination of

the limit of change in the parameters of the environment. There

are ways of evaluating the maximum influence and other factors on

the accuracy of measurements. Therefore, it is always possible to

predetermine the maximum errors of some kind of measurements.

The quantitative characteristics of the distribution of errors

from their zero t o maximum values, in the majority of cases, are

subject to the normal law of random variable distribution.

If in some cases the law of e.rror distribution is not normal,

it will be close in any case.

Considering that devices of probability theory are used not
in calculating measurement errors, but only in evaluating limits
and the probability of possible measurement errors within these
limits, it is considered permissible in a l l cases to use the
normal law of distribution of random variables.
The normal law of random variable distribution (Gauss formula)
characterizes the probability density of a random variable, in our
case of the measurement errors ( X - a ) , depending on its value:

476


.- .- .. .._..._. . .

p ( x -a ) =

1

-

E

G e

(X-LZ)'

29

(6.5)

~

2

w h e r e @(x - a ) i s t h e p r o b a b i l i t y d e n s i t y o f e r r o r s o f a g i v e n
m a g n i t u d e , (T i s t h e m e a n s q u a r e e r r o r o f a s e r i e s o f mea u r e m e t s ,
e i s a N a p i e r number e q u a l t o 2 . 7 1 8 2 8 , and a i s t h e p r e c i s e v a l u e
of t h e magnitude being measured.
I t i s o b v i o u s t h a t t h e p r o b a b i l i t y of f i n d i n g t h e r e s u l t o f
m e a s u r i n g (x) i n t h e r a n g e o f v a l u e s f r o m a t o x c a n b e d e t e r m i n e d
b y i n t e g r a t i n g ( 6 . 5 ) o v e r x:

(6.6)

The g r a p h o f t h e p r o b a b i l i t y o f r a n d o m v a r i a b l e s s u b o r d i n a t e
t o t h e n o r m a l d i s t r i b u t i o n l a w i s shown i n F i g u r e 6 . 2 .
The
variable
values.
negative

c u r v e on t h e g r a p h shows t h e p r o b a b i l i t y d e n s i t y o f random
d e v i a t i o n s f r o m z e r o t o maximum p o s i t i v e a n d n e g a t i v e
The l e f t s i d e o f t h e g r a p h c o r r e s p o n d s t o e r r o r s w i t h a
sign, the r i g h t t o errors with a positive sign.

Since t h e absolute p r o b a b i l i t y of obtaining any value of t h e
measured magnitude i s e q u a l t o one, t h e p r o b a b i l i t y t h a t t h e v a l u e
o f t h e m a g n i t u d e w i l l b e n e g a t i v e or p o s i t i v e i s 0 . 5 .
L e t u s n o t e t h a t on t h e a b c i s s a o f t h e g r a p h t h e r e a r e two
v a l u e s o f a r a n d o m v a r i a b l e , xi a n d x , .
The a r e a bounded by t h e
s e g m e n t ~ 1 x 2 ,b y t h e o r d i n a t e s P,,, P,,, a n d b y t h e c u r v e i s t h e
p r o b a b i l i t y of f i n d i n g t h e r e s u l t of measurement i n t h e l i m i t s be­
t w e e n xi a n d x2.

A

With t h e c o n v e r g e n c e o f
p o i n t s xi a n d x2 a t o n e p o i n t ,
t h e p r o b a b i l i t y of f i n d i n g an
e r r o r of measurement between
t h e s e p o i n t s w i l l d i m i n i s h and
converge t o zero.

@(z2-u)-@[x,-a)

5

(W-o)<O

-

.
.-

i~

xz

x1

a

-c

(x-a);

Fig. 6.2
Graph of t h e Probab i l i t y of Random V a r i a b l e s U n d e r
t h e Normal L a w o f D i s t r i b u t i o n .

T h e a n a l o g o u s p r o b l e m for
determining p r o b a b i l i t y can be
s o l v e d on t h e b a s i s of t h e
r i g h t s i d e of t h e g r a p h f o r
e r r o r s of measurement which
have a p o s i t i v e s i g n .

47 7

/454

Without s t o p p i n g a t t h e methods of s o l v i n g a n i n t e g r a l ( 6 . 6 ) ,

>

l e t us indicate t h a t the overall probability of finding positive
and n e g a t i v e e r r o r s of measurements is 68.3% i n t h e range from 0 t o
0 , 95% from 0 t o 2 0 ,
and 99.7% from 0 t o 30.
A table of values of
t h e f u n c t i o n @(x - a ) f o r ( z - a ) f r o m 0 t o 5 0 i s g i v e n i n S u p p l e ­
ment 6 .

For e x a m p l e , i f t h e mean s q u a r e e r r o r o f m e a s u r i n g t h e d r i f t
a n g l e w i t h a Doppler meter i s e q u a l t o 1 5 ' , t h e n w i t h a p r o b a b i l i t y
o f 95% i t i s p o s s i b l e t o e x p e c t t h a t t h e measurement e r r o r w i l l n o t
exceed 3 0 ' , and w i t h a p r a c t i c a l l y complete p r o b a b i l i t y ( 9 9 . 7 % ) , 45'.
T h e v a l u e o f t h e mean s q u a r e e r r o r o f m e a s u r i n g a g i v e n k i n d
of parameter permits evaluation of t h e accuracy of o t h e r parameters
which h a v e a f u n c t i o n a l d e p e n d e n c e on t h e f i r s t .

Example:
T h e mean s q u a r e e r r o r o f t h e d i r e c t i o n - f i n d i n g o f a n
a i r c r a f t by means o f a g r o u n d d i r e c t i o n f i n d e r 0 = lo. D e t e r m i n e
t h e l i m i t s of l i n e a r e r r o r i n d e t e r m i n i n g t h e l a t e r a l d e v i a t i o n o f
a n a i r c r a f t f r o m t h e l i n e o f a g i v e n p a t h w i t h a p r o b a b i l i t y o f 95%
i f t h e a i r c r a f t i s l o c a t e d a t a d i s t a n c e o f 3 0 0 km f r o m t h e d i r e c ­
tion finder.

Solution: With a p r o b a b i l i t y o f 9 5 % , t h e a n g u l a r e r r o r o f a
d i r e c t i o n f i n d e r i n measuring does not exceed 2 O .
T h e r e f o r e , AZ(P = 9 5 % ) = 3 0 0 t g 2 O w 1 0 km.
S o l v i n g t h e same
/455
p r o b l e m f o r a p r a c t i c a l p r o b a b i l i t y o f 1 0 0 % (more p r e c i s e l y , 9 9 . 7 % ) ,
AZ(P = 1 0 0 % ) = 3 0 0 t g 3O = 1 5 k m .
L e t u s now a s s u m e t h a t w e m u s t s o l v e t h e r e v e r s e p r o b l e m , ; . e .
d e t e r m i n e t h e n e c e s s a r y a c c u r a c y o f a d i r e c t i o n f i n d e r which en­
s u r e s t h e given accuracy of measurements of l a t e r a l d e v i a t i o n s .

Example:
AZma,(P
= 1 0 0 % ) = 1 0 km.
Determine t h e necessary
a c c u r a c y o f a d i r e c t i o n f i n d e r f o r d i s t a n c e s u p t o 3 0 0 km.

Solution:
3.

10

30d = a r c t g 300

20.

Therefore,

ad = 0 . 7 0 .

Linear a n d Two-Dimensional Problems o f Probability Theory

The n o r m a l l a w o f r a n d o m v a r i a b l e d i s t r i b u t i o n e x a m i n e d i n t h e
preceding paragraph i n c l u d e s t h e l i n e a r (one-dimensional) problem
o f p r o b a b i l i t y t h e o r y f o r one p a r a m e t e r o f measurement.
I n aircraft navigation, it i s o f t e n necessary t o d e a l with
For e x a m p l e , i n c a l c u l a t i n g t h e
s e v e r a l measurement p a r a m e t e r s .
p a t h o f a n a i r c r a f t w i t h r e s p e c t t o d i r e c t i o n by a u t o m a t i c n a v i ­
g a t i o n a l d e v i c e s , on t h e b a s i s o f r e s u l t s o f m e a s u r i n g t h e d r i f t
a n g l e and groundspeed of t h e a i r c r a f t w i t h a Doppler meter, t h e
f o l l o w i n g e r r o r s e x e r t a n i n f l u e n c e on t h e a c c u r a c y o f c a l c u l a t i n g
t h i s parameter:
errors i n calculating the given f l i g h t angle;
478

e r r o r s i n measuring t h e c o u r s e , d r i f t a n g l e , and groundspeed;
e r r o r s i n t h e operation of an i n t e g r a t i n g device,
Each o f t h e s e f a c t o r s s e p a r a t e l y w i l l c r e a t e t h e f o l l o w i n g
e r r o r components i n c a l c u l a t i n g t h e p a t h w i t h r e s p e c t t o d i r e c t i o n :
A Z + = x sin A+ = Wt sin Arb;
AZ1 = Wtsin Ay;
AZn =. Wt sin Aa;


.


AZw = AlVt sin (++ - +3);


If t h e i n d i c a t e d c o m p o n e n t s h a d t h e same s i g n a n d h a d a m a x i ­
mum v a l u e w i t h i n t h e c a l i b r a t i o n l i m i t s o f e a c h o f t h e p a r a m e t e r s ,
t h e g e n e r a l e r r o r w o u l d b e e q u a l t o t h e a r i t h m e t i c sum o f t h e s e
components.
However, a c c o r d i n g t o t h e l a w o f n o r m a l random v a r i a b l e
d i s t r i b u t i o n , e v e n when m e a s u r i n g o n e p a r a m e t e r , t h e maximum e r r o r
i s encountered r a t h e r r a r e l y .
The p r o b a b i l i t y t h a t a l l t h e e r r o r s
/456
w i l l t a k e o n a maximum v a l u e , a n d e v e n o n e s i g n , w i l l b e e x t r a ­
o r d i n a r i l y low.

+


I n s p i t e o f t h e f a c t t h a t we m u s t d e a l s i m u l t a n e o u s l y w i t h
many m e a s u r e d p a r a m e t e r s , t h e s o l u t i o n o f t h e a b o v e e x a m p l e i n c l u d e s
a l i n e a r p r o b l e m o f p r o b a b i l i t y t h e o r y , s i n c e t h e random v a r i a b l e s
a r e summed a l o n g o n e a x i s o f t h e c h o s e n f r a m e o f r e f e r e n c e o f t h e i r
coordinates.
To s o l v e s i m i l a r p r o b l e m s , t h e c o n c e p t o f t h e d i s p e r s i o n o f
random v a r i a b l e s cr2 i s i n t r o d u c e d i n t o p r o b a b i l i t y t h e o r y .
I t i s known t h a t t h e l a w o f r a n d o m v a r i a b l e d i s t r i b u t i o n , o b ­
t a i n e d by a d d i n g o t h e r random v a r i a b l e s which a r e s u b j e c t t o t h e
Here,
normal d i s t r i b u t i o n l a w , i s a l s o a normal d i s t r i b u t i o n l a w .
t h e s c a t t e r o f a n o v e r a l l r a n d o m v a r i a b l e i s t h e sum o f t h e s c a t t e r s
of t h e values being added.

I n o u r example o f c a l c u l a t i n g t h e p a t h o f a n a i r c r a f t by means
of a u t o m a t i c n a v i g a t i o n a l d e v i c e s , t h e v a l u e u 2 i s t h e s c a t t e r o f
2
t h e sum.

(6.7)

The mean s q u a r e e r r o r o f t h e m e a s u r e m e n t i s e q u a l t o t h e s q u a r e
r o o t of t h e scatter:
u =
T h e r e f o r e , t h e mean s q u a r e e r r o r o f
t h e t o t a l v a l u e w i l l e q u a l t h e s q u a r e r o o t o f t h e sum o f t h e s c a t ­
ters.
For o u r e x a m p l e ,
479

The v a l u e a 2

i n (6.8)

zw

i s a small second-order value:
Az,t = A W s i n A+.

Therefore,

it i s n e c e s s a r y t o d i s r e g a r d t h i s v a l u e .

L e t u s assume t h a t t h e r e m a i n i n g v a l u e s i n c l u d e d i n ( 6 . 8 ) have
b e e n mean s q u a r e e r r o r s a s f o l l o w s :

= 20’;

(i+,

(i

7

= 20’;

(ia

= 15’;



as,

=0,5%of

x.

Since t h e first t h r e e values are s m a l l , t h e i r s i n c e s can be
r e p l a c e d by a n g l e v a l u e s .
T h e n , c o n s i d e r i n g lo e q u a l t o 0 . 0 1 7 b y
1 . 7 % X, t h e i r v a l u e c a n be e x p r e s s e d i n p e r c e n t o f t h e d i s t a n c e
traversed:
a+, = 0,56%X;

where

aT = 0,56%X;

aa = 0,42%X;

-O , ~ % X ,

“Jz ­

X = Wt.

T h e r e f o r e , t h e mean s q u a r e e r r o r i n c a l c u l a t i n g t h e p a t h w i t h
respect t o direction is

/457

-

H e n c e , it i s p o s s i b l e t o c o n s i d e r t h a t t h e mean s q u a r e e r r o r
1%
i n c a l c u l a t i n g t h e p a t h w i t h r e s p e c t t o d i r e c t i o n amounts t o
of t h e distance traversed.
L e t u s assume t h a t w e h a v e s e t o u r s e l v e s t h e g o a l o f m a i n t a i n i n g
a n a i r c r a f t w i t h i n t h e l i m i t s o f a n a i r r o u t e w i t h a w i d t h o f 2 0 km
( u p t o 1 0 km f r o m LGP) w i t h a p r o b a b i l i t y o f 9 5 % . Here t h e mean
square e r r o r i n determining t h e i n i t i a l coordinates of t h e aircraft
e q u a l s 2 km.

For a p r o b a b i l i t y o f 9 5 % , t h e e r r o r i n t h e i n i t i a l f o r m u l a t i o n
o f t h e a i r c r a f t ’ s c o o r d i n a t e s m u s t b e t a k e n a s 4 km, w h i l e t h e
a c c u r a c y o f c a l c u l a t i n g t h e p a t h w i t h r e s p e c t t o d i r e c t i o n must be
t a k e n a s 2 % . T h e maximum e r r o r i n c a l c u l a t i n g t h e p a t h w i t h r e s p e c t
t o d i s t a n c e must n o t e x c e e d

T h e v a l u e 9 km m u s t a m o u n t t o 2 % o f t h e d i s t a n c e c o v e r e d .
480

Therefore, the allowable length of the stage of the path between

the control points (5') must be not more than


If we set ourselves the goal of maintaining an aircraft within
the limits of a route with a probability of 9 9 . 7 % , the accuracy of
the initial display of coordinates and the calculation of the path
of the aircraft would have to be t a k e n , a s 3 0 or a reading accuracy
equal to 6 k m and an accuracy for calculating the path equal t o
3% X.
Then

If we take the limits o f calibrating each parameter as 30,

then by adding the errors on the basis of the calibration rules we

would obtain the value


or, in our example,

AZmax:--;. 1,7+ 1 , 7 + 1,5+ 1 , 2 = 6 , 1 % ,

i.e., in the case when all errors have a maximum value and the same
sign, the error of calculating the path can reach 6%. Since it
reaches 2% with a probability of 9 5 % , with a practically complete
probability of 99.7% it reaches 3%.
The probability that calculating the path will occur with
errors within the range of an overall calibration of the system is
expressed by in hundred millionths of a percent. Therefore, when
there is no threat of disturbing the safety of a flight, it is not
necessary for practical purposes to take the limits of overall
calibration into consideration.

ai

.

e
.

Fig. 6.3. Diagram of the
Occurrence of E r r o r s in
Determining the Position of
an Aircraft: (a) with Multilateral. E r r o r s in Rearings;
(b) With Unilateral E r r o r s .
481


/458

I n t h e m a j o r i t y of cases, t h e r e i s s u f f i c i e n t e r r o r i n c a l c u ­
l a t i n g t h e p a t h t o c a l c u l a t e w i t h a p r o b a b i l i t y o f 95%, and o n l y i n
e s p e c i a l l y r e s p o n s i b l e cases, w i t h 99.7%.
The m a j o r i t y o f p r o b l e m s i n d e t e r m i n i n g t h e a c c u r a c y o f a i r ­
c r a f t n a v i g a t i o n or m e a s u r i n g i t s s e p a r a t e p a r a m e t e r s w i t h t h e u s e
o f some m e t h o d r e d u c e s d i r e c t l y t o l i n e a r p r o b l e m s p f p r o b a b i l i t y
theory.
The f i n a l r e s u l t o f s o l v i n g a l l t h e p r o b l e m s o f a i r c r a f t
n a v i g a t i o n must be one-dimensional, s i n c e t h e g o a l s and r e q u i r e m e n t s
f o r accuracy of aircraft navigation with respect t o distance,
d i r e c t i o n , and f l i g h t a l t i t u d e are d i f f e r e n t .

A t t h e p r e s e n t t i m e , t h e r e a r e no n a v i g a t i o n a l systems which
determine t h e p o s i t i o n of an aircraft i n three-dimensional space.
Therefore, t h e n e c e s s i t y f o r s o l v i n g v o l u m e t r i c problems i n proba­
b i l i t y theory is superfluous.
However, a number o f n a v i g a t i o n a l
s y s t e m s s u c h a s a h y p e r b o l i c , t w o - p o l e g o n i o m e t e r , or g o n i o m e t r i c
r a n g e f i n d e r ( i f o n l y t h e l o c a t i o n o f a g r o u n d b e a c o n i s known f o r
t h e LGP o f a g i v e n s e g m e n t ) , p e r m i t t h e s o l u t i o n o f a p r o b l e m i n
determining an a i r c r a f t ' s coordinates i n two-dimensional space.
An e v a l u a t i o n o f t h e a c c u r a c y o f a i r c r a f t n a v i g a t i o n a l o n g
each of t h e axes of t h e c o o r d i n a t e system chosen f o r a i r c r a f t n a v i ­
g a t i o n , i n t h i s case, can be c a r r i e d o u t o n l y a f t e r s o l v i n g a onedimensional' problem i n p r o b a b i l i t y t h e o r y .

L e t u s assume t h a t w e have an o b l i q u e - a n g l e d s u r f a c e system of
a i r c r a f t p o s i t i o n l i n e s , e a c h of w h i c h d o e s n o t c o i n c i d e w i t h t h e
given f l i g h t path (Fig. 6.3, a,b).
The l i n e a r e r r o r i n d e t e r m i n i n g t h e f i r s t ( r l ) a n d s e c o n d ( r 2 )
a i r c r a f t p o s i t i o n l i n e s d e p e n d s on b o t h t h e a c c u r a c y o f m e a s u r i n g
t h e n a v i g a t i o n a l p a r a m e t e r and i t s g r a d i e n t .

The g r a d i e n t of a n a v i g a t i o n a 2 p a r a m e t e r i s t h e r a t i o o f i t s
/455
i n c r e a s e t o t h e movement o f a n a i r c r a f t i n a d i r e c t i o n p e r p e n d i c u l a r
t o t h e p o s i t i o n l i n e s of t h e operating region of t h e system
g =

da
-dr

(6.9)

'

where g i s t h e g r a d i e n t o f t h e n a v i g a t i o n a l p a r a m e t e r , and a i s t h e
n a v i g a t i o n a l parameter being measured.

For e x a m p l e , i f t h e n a v i g a t i o n a l s y s t e m i s a g o n i o m e t e r , t h e n
dU

-

dA

1

­
g = d r . - d=-­S '
r

where A i s t h e a z i m u t h o f t h e a i r c r a f t and S i s t h e d i s t a n c e from
482

t h e ground beacon t o p o i n t PA.

I n t h i s case,

With t h e i n t r o d u c t i o n o f t h e c o n c e p t of t h e g r a d i e n t of a
navigational parameter, a l l t h e e x i s t i n g coordinate systems reduce
t o a g e n e r a l i z e d s y s t e m , i . e . , t h e p r o b l e m s of d e t e r m i n i n g t h e
a c c u r a c y o f n a v i g a t i o n a l m e a s u r e m e n t s a r e s o l v e d on t h e b a s i s o f a
g e n e r a l scheme, i n d e p e n d e n t of t h e geometry o f a p p l i c a t i o n of t h e
navigational device.
I n F i g . 6 . 3 a , b two p o s s i b l e c a s e s of t h e a p p e a r a n c e o f e r r o r s
i n d e t e r m i n i n g t h e p o s i t i o n o f a n a i r c r a f t on t h e b a s i s o f t h e
i n t e r s e c t i o n o f p o s i t i o n l i n e s a r e shown:
! a > E r r o r s i n r l a n d r2 h a v e d i f f e r e n t s i g n s ; i n t h i s c a s e ,
t h e measured p o s i t i o n o f t h e a i r c r a f t l i e s i n an a c u t e a n g l e between
the actual position lines.
This leads t o l a r g e r e r r o r s of deter­
mination.

Errors i n r l a n d 2-2 h a v e i d e n t i c a l s i g n s ; t h e m e a s u r e d
(b)
p o s i t i o n of t h e a i r c r a f t l i e s i n an obtuse angle between t h e p o s i t i o n
lines.
The e r r o r s i n d e t e r m i n i n g t h e p o s i t i o n o f t h e a i r c r a f t i n
t h i s c a s e a r e c l o s e t o t h e l i n e a r e r r o r s o f one b e a r i n g .
I t i s necessary t o n o t e t h a t t h e p r o b a b i l i t y of e r r o r s i n rl
a n d r 2 w i t h t h e same s i g n i n t h e m a j o r i t y o f c a s e s i s more t h a n t h e
p r o b a b i l i t y of e r r o r s with d i f f e r e n t signs.
For e x a m p l e , i n t a k i n g
b e a r i n g s w i t h a r a d i o c o m p a s s , t h e e r r o r component as a r e s u l t o f
an e r r o r i n m e a s u r i n g t h e a i r c r a f t c o u r s e w i l l be g e n e r a l f o r two
If t h e a n g l e b e t w e e n t h e b e a r i n g s i s s u f f i c i e n t l y
measurements.
a c u t e , t h e r a d i o d e v i a t i o n w i l l h a v e e i t h e r o n e s i g n or d i f f e r e n t
s i g n s , b u t a s m a l l v a l u e i n any c a s e .
A s i m i l a r r e l a t i o n s h i p between measurement e r r o r s i n p r o b a b i l i t y
theory i s c a l l e d correzation ( p ) .

The g e n e r a l e r r o r i n d e t e r m i n i n g t h e p o s i t i o n o f t h e a i r c r a f t
i n our c a s e w i l l be ( F i g . 6 . 4 ) :

or

(6.10)
Formula

(6.10)

c h a r a c t e r i z e s only t h e magnitude of e r r o r i n
483

/460
-

determining the position of an aircraft on the basis of two position
lines with known errors in measurement f o r each of them. However,
it does not give us an idea of the nature o f the distribution of
the indicated errors around the point of the actual position of the
aircraft (center of scatter).
In contrast t o a linear problem, where the probability of an

overall error in several measurements is examined, in a two-dimen­

sional problem it is. necessary t o examine the products of the

probabilities of these errors.

For simplicity of argument, let u s assume that we have a

rectangular coordinate system (Fig. 6.5); let us set ourselves the

goal of limiting the area within which the aircraft is located with

a probability of 95%. Here, the mean square errors of measuring

the two position lines will be considered identical.

Let us examine a certain large number of measurements (e.g.,
10,000) and let us see what will be the probability that the

measured position of the aircraft will be i n an exterior angle at
a distance from the center of the area of scatter which exceeds
the diagonal of a square constructed with errors 20, and 2 a 2 .
Since the probability of an error in the first measurement
exceeding 2 0 equals 5 % , then 500 of 10,000 measurements must be be­
yond the limits o f a side of the indicated square. The remaining
9,500 measurements lie in the range from zero to 2 0 , and there is
no need to calculate them during common measurements with the
second position line.
It is obvious that of the 500 remaining measurements, where an
aircraft will be located a distance more than 2 0 , from the first
position line, the errors in the second bearing will exceed the
value 202 only in 5% of the cases, and thus there will be only 25
cases (or 0.25%) simultaneously exceeding the errors of the values /461
2 0 , and 202.

\
'\

Fig. 6.4. Total E r r o r in
Determining the Position of
an Aircraft.
484


Fig. 6.5. Probability of the
Simultaneous Yield of E r r o r s Be­
yond the Limits of the Given Values.

The e x a m p l e e x a m i n e d shows c l e a r l y t h a t t h e p r o b a b i l i t y d e n s i t y
of e r r o r s d i r e c t e d toward t h e i n d i c a t e d angle diminishes sharply.
I t i s p r a c t i c a l l y p o s s i b l e t o c o n s i d e r t h a t a l a r g e number o f t h e
common m e a s u r e m e n t s o f t h e f i r s t a n d s e c o n d p o s i t i o n l i n e s l i e i n
a c i r c l e , t h e r a d i u s o f w h i c h e q u a l s 2al = 202, w h i l e t h e l i m i t o f
equal p r o b a b i l i t y of d e v i a t i o n s from t h e c e n t e r of s c a t t e r i n g w i l l
be a c i r c l e .
I n g e n e r a l , when t h e e r r o r s i n t h e f i r s t b e a r i n g a r e
n o t e q u a l t o t h e e r r o r s i n t h e second b e a r i n g , t h i s boundary has
t h e shape of an e l l i p s e .
If w e examine a number o f c a s e s i n , w h i c h a n a i r c r a f t i s w i t h i n
t h e l i m i t s o f a n e l l i p s e w i t h a x e s e q u a l t o 2a, i t t u r n s o u t t o b e
s i g n i f i c a n t l y l e s s t h a n 95%, s i n c e e v e n i n r e c t a n g l e s c o n s t r u c t e d
w i t h s i d e s e q u a l t o 2 a t h e r e w i l l b e 95% o f 95%, or 90.025%.

H o w e v e r , t h i s i s o f v a l u e o n l y when t h e p r o b a b i l i t y o f a n
a i r c r a f t ' s e n t e r i n g a g i v e n area i s examined.
From t h e p o i n t o f
view o f a i r c r a f t n a v i g a t i o n , it i s n o t t h e l o c a t i o n of an a i r c r a f t
i n a g i v e n a r e a , b u t t h e d e v i a t i o n from t h e g i v e n p a t h t r a j e c t o r y
and t h e r e t a i n i n g of f l i g h t d i s t a n c e w i t h r e s p e c t t o t i m e which
Therefore, t h e r e s u l t s of adding t h e p r o b a b i l i t i e s
play a role.
are again d i s t r i b u t e d according t o d i r e c t i o n .
This again raises
t h e p r o b a b i l i t y o f e a c h o f t h e m t o 95% ( F i g . 6.6).
I n t h e f i g u r e , an e l l i p s e of measurement e r r o r s , l o c a t e d i n a
c e r t a i n p o s i t i o n r e l a t i v e t o t h e path l i n e of an aircraft, i s
shown.
I n t h e above case, t h e p o s s i b l e e r r o r s i n measuring t h e
coordinates of an aircraft with a given p r o b a b i l i t y are determined
by t a n g e n t s , p a r a l l e l s , and p e r p e n d i c u l a r s t o t h e l i n e o f t h e g i v e n
The d i s t a n c e
p a t h ( t h e orthodrome c o o r d i n a t e s are k e p t i n mind).
o f t h e t a n g e n t s from p o i n t PA, as w e l l as t h e m a i n t e n a n c e o f c o r r e ­
l a t i o n dependence between t h e e r r o r s i n t h e maintenance of t h e p a t h
w i t h r e s p e c t t o d i s t a n c e and d i r e c t i o n , w i l l depend on t h e s h a p e
d i m e n s i o n s , and o r i e n t a t i o n o f t h e e l l i p s e o f e r r o r s .
The c o r r e ­
l a t i o n dependence p l a y s a r o l e only a t t u r n i n g p o i n t s i n t h e r o u t e ,
when e r r o r s o f a i r c r a f t n a v i g a t i o n w i t h r e s p e c t t o d i s t a n c e d e f i ­
n i t e l y become e r r o r s o f a i r c r a f t n a v i g a t i o n w i t h r e s p e c t t o d i r e c ­
t i o n , and v i c e v e r s a .

I@
t


'J

F i g . 6.6. E l l i p s e o f E r r o r s
i n t h e Aircraft P o s i t i o n .

The p h y s i c a l s e n s e o f t h e
above a r g u m e n t s becomes c l e a r i f
w e assume t h a t t h e e l l i p s e o f e r ­
rors c o n s t r u c t e d f r o m mean s q u a r e
measurement e r r o r s can be a r r a n g e d
with t h e major a x i s both i n t h e
d i r e c t i o n of t h e p a t h l i n e and
/462

perpendicular t o it.
Then t h e
accuracy of aircraft navigation
w i t h re.spect t o d i s t a n c e and
d i r e c t i o n c a n b e d e t e r m i n e d by
mean s q u a r e v a l . u e s o f t h e a x i s o f
the ellipse.
Obviously t h e

o r i e n t a t i o n o f t h e a x i s o f t h e e l l i p s e a t a n a n g l e t o t h e l i n e of
t h e given p a t h occupies an i n t e r m e d i a t e p o s i t i o n between t h o s e i n
the figures.
The a c c u r a c y o f m a i n t e n a n c e o f t h e p a t h a l o n g t h e
a x e s o f t h e c o o r d i n a t e s i n t h i s c a s e i s d e t e r m i n e d i n t h e same way
as f o r t h e case shown i n ( F i g . 6 . 7 a , b ) .
However, a c o r r e l a t i o n
dependence between t h e s e measurements w i l l be seen.
I n t h e above examples of a c i r c l e and e l l i p s e of e r r o r s , w e
assumed t h a t t h e a i r c r a f t p o s i t i o n l i n e s were s i t u a t e d a t r i g h t
a n g l e s t o one a n o t h e r , a l t h o u g h a n a n g l e t o t h e g i v e n p a t h l i n e i s
p o s s i b l e , and t h a t t h e c o r r e l a t i o n dependence between t h e measure­
ments of t h e p o s i t i o n l i n e s i s a b s e n t .
L e t u s now p r e s e n t , w i t h o u t d e r i v a t i o n , t h e f o r m u l a s w h i c h c a n
be used as a b a s i s f o r d e t e r m i n i n g t h e dimensions and o r i e n t a t i o n
o f t h e a x e s o f a n e l l i p s e f o r a s i t u a t i o n when t h e p o s i t i o n l i n e s
are s i t u a t e d a t an angle (not a r i g h t a n g l e ) with an independent
a c c u r a c y o f m e a s u r c m e n t f o r e a c h of t h e m ( c o r r e l a t i o n c o e f f i c i e n t
equal t o zero):

(6.11)

sin 20

tg 2.1 =­
,c;

$ .;,cos 26)

'


(6.12)

where a i s t h e m a j o r s e m i a x i s o f t h e e l l i p s e ; b i s t h e minor semia x i s of t h e e l l i p s e ; w i s t h e angle between t h e p o s i t i o n l i n e s ; e
i s t h e p a r a m e t e r o f t h e e l l i p s e chosen f o r a g i v e n p r o b a b i l i t y ; and
c1 i s t h e a n g l e b e t w e e n t h e b i s e c t o r o f t h e a n g l e o f i n t e r s e c t i o n o f
t h e p o s i t i o n l i n e s and t h e m a j o r a x i s o f t h e e l l i p s e ( i t i s p l o t t e d
i n t h e d i r e c t i o n of t h e p o s i t i o n l i n e with t h e smallest e r r o r ) .
These f o r m u l a s a r e used i n t h e m a j o r i t y of cases, s i n c e t h e
a c c u r a c y o f d e t e r m i n i n g p o s i t i o n l i n e s i s u s u a l l y i n d e p e n d e n t or
t h e c o r r e l a t i o n c o e f f i c i e n t i s unknown.
Only i n i n d i v i d u a l c a s e s , e . g . , i n d e t e r m i n i n g p o s i t i o n l i n e s
with an a i r c r a f t r a d i o compass, i s t h e c o r r e l a t i o n c o e f f i c i e n t
d e t e r m i n e d r a t h e r s i m p l y , s i n c e p a r t o f t h e e r r o r of m e a s u r i n g t h e
For
b e a r i n g ( d e p e n d i n g on t h e a i r c r a f t c o u r s e ) w i l l be g e n e r a l .
example:
(scoUL1se
= 3O; 0
= 2O.

Y

The g e n e r a l mean s q u a r e e r r o r i n m e a s u r i n g a b e a r i n g w i l l b e :

48 6

G A=

I n m e a s u r i n g two b e a r i n g s on one a i r c r a f t c o u r s e , i t i s n e c e s ­
s a r y t o e x p e c t t h a t b o t h e r r o r s w i l l b e s h i f t e d i n one d i r e c t i o n by
t h e mean s q u a r e e r r o r o f m e a s u r i n g t h e a i r c r a f t c o u r s e , i n o u r c a s e
by 2 O , which amounts t o 0.55 o f t h e t o t a l e r r o r .
Therefore, the
c o r r e l a t i o n c o e f f i c i e n t P = 0.55.
A s shown a b o v e ( s e e F i g . 6 . 3 1 , t h e p r e s e n c e o f a t o t a l compo­
n e n t ‘in t h e e r r o r s o f m e a s u r i n g p o s i t i o n l i n e s r a i s e s t h e a c c u r a c y
of determining t h e p o s i t i o n of an aircraft.
According t o (6.10),
s i n c e t h e t h i r d t e r m under t h e r a d i c a l i n t h e numerator can be
given both p o s i t i v e and n e g a t i v e v a l u e s , f o r independent measure­
ments it i s p o s s i b l e t o w r i t e

ar

= --

sin w

(6.13)

With d e p e n d e n t m e a s u r e m e n t s , t h e t h i r d t e r m u n d e r t h e r a d i c a l
must b e m u l t i p l i e d by t h e c o r r e l a t i o n c o e f f i c i e n t p and ( 6 . 1 3 ) t a k e s
t h e form:

(6.14)

With a c o r r e l a t i o n c o e f f i c i e n t e q u a l t o z e r o ,
g o o , (6.14) is transformed i n t o (6.13).

or w i t h a n g l e

w equal t o

I n t h e p r e s e n c e o f c o r r e l a t i o n , t h e d i m e n s i o n s and o r i e n t a t i o n
of t h e a x e s of t h e e l l i p s e are determined a c c o r d i n g t o t h e f o r m u l a s :

(6.15)

(6.16)

The l e n g t h o f t h e m i n o r a x i s o f a n e l l i p s e i s a l s o d e t e r m i n e d
on t h e b a s i s o f ( 6 . 1 6 ) , w i t h r e p l a c e m e n t o f t h e p o s i t i v e s i g n i n
f r o n t of t h e r a d i c a l by a n e g a t i v e s i g n .

487

/463

I n p r o b a b i l i t y t h e o r y , t h e p r o b a b i l i t y of l o c a t i n g a n o b j e c t
within t h e l i m i t s of t h e indicated e l l i p s e of e r r o r s (given d e f i n i t e
v a l u e s of t h e m a g n i t u d e e ) i s e x a m i n e d .
Since w e have a g r e e d t o examine t h e a c c u r a c y o f a i r c r a f t n a v i ­
g a t i o n s e p a r a t e l y w i t h r e s p e c t t o d i s t a n c e and d i r e c t i o n , t h i s prob­
l e m w i l l not interest us.
We a r e u s i n g t h e e l l i p s e for a n e v a l u a ­
t i o n of t h e accuracy.of a i r c r a f t navigation with r e s p e c t t o both
d i s t a n c e and d i r e c t i o n .
L e t U S a s s u m e t h a t w e know t h e o r i e n t a t i o n o f a n a i r c r a f t ' s
p o s i t i o n l i n e s a n d a g i v e n p a t h l i n e on a map, a n d h a v e d e t e r m i n e d ,
o n t h e b a s i s o f ( 6 . 1 1 ) a n d ( 6 . 1 2 ) or ( 6 . 1 5 ) a n d ( 6 . 1 6 ) , t h e l e n g t h s
o f t h e a x e s o f t h e e l l i p s e 2a a n d 2 b , a s w e l l a s t h e o r i e n t a t i o n
of t h e m a j o r a x i s of t h e e l l i p s e w i t h p a r a m e t e r e = 1 (mean s q u a r e
e l l i p s e 1.

I n t h i s c a s e , t h e mean s q u a r e e r r o r i n t h e m a i n t e n a n c e o f t h e
p a t h w i t h r e s p e c t t o d i s t a n c e and d i r e c t i o n i s d e t e r m i n e d by t a n g e n t s
t o t h e e l l i p s e a t p o i n t s Xo and Z o , p e r p e n d i c u l a r t o t h e p a t h l i n e ,
a n d a t p o i n t s Xi a n d Z 1 , p a r a l l e l t o i t ( F i g . 6 . 7 ) .
I t i s p o s s i b l e t o show t h a t t h e mean s q u a r e e r r o r s i n m a i n ­
in
t a i n i n g t h e p a t h ( w i t h r e s p e c t t o d i s t a n c e ox a n d d i r e c t i o n 0 , )
t h i s case a r e

(6.17)

w h e r e c1 i s t h e a n g l e b e t w e e n t h e l i n e o f t h e g i v e n p a t h a n d t h e
major a x i s of t h e e l l i p s e .
I n F i g . 6 . 7 , i t i s o b v i o u s t h a t i n a g e n e r a l c a s e , when t h e
/464
major a x i s of an e l l i p s e does n o t c o i n c i d e w i t h t h e g i v e n p a t h l i n e
or t h e l i n e p e r p e n d i c u l a r t o i t , t h e r e i s a c o r r e l a t i o n d e p e n d e n c e
between t h e e r r o r s i n t h e maintenance of t h e p a t h w i t h r e s p e c t t o
d i s t a n c e and d i r e c t i o n .
A c t u a l l y , i f w e have a p o s i t i v e e r r o r i n determining t h e X
c o o r d i n a t e , t h e measured p o s i t i o n of t h e a i r c r a f t i s l o c a t e d i n t h e
The m a t h e m a t i c a l e x p e c t a t i o n o f
r i g h t - h a n d s i d e of t h e e l l i p s e .
t h e v a l u e of t h e Z - c o o r d i n a t e i n t h i s case w i l l be found i n t h e
m i d d l e of t h e c h o r d of t h e e l l i p s e , p a r a l l e l t o O Z a n d i n t e r s e c t i n g
t h e X-axis a t a p o i n t c o r r e s p o n d i n g t o A X .
The d i a m e t e r o f t h e e l l i p s e ,

48 8

d i v i d i n g i t s c h o r d s (which are

p e r p e n d i c u l a r t o some o t h e r d i a m e t e r ) i n h a l f ,

is called the

conjugate diameter.

The d i r e c t i o n o f t h e c o n j u ­
g a t e d i a m e t e r i s d e t e r m i n e d ac­
cording t o t h e formula

-2

tgp =
X

Fig. 6.7.
C o r r e l a t i o n Depende n c e o f t h e Errors i n t h e Cont r o l Path with Respect t o D i s t a n c e and D i r e c t i o n .

bz
a (tg90-u)'

(6.18 )

I n p r o b a b i l i t y theory , l i n eS
which d e t e r m i n e t h e dependence
b e t w e e n random v a r i a b l e s a r e
s i m i l a r t o t h e c o n j u g a t e diameters
of an e l l i p s e of e r r o r s .
In the
case d e s c r i b e d by u s , t h e y a r e
c a l l e d r e g r e s s i o n lines, w h i l e
t h e a n g u l a r c o e f f i c i e n t s o f t h e se
l i n e s (tangents of t h e angles t o
t h e a x e s o f t h e frame o f r e f e r ­
e n c e ) a r e c a l l e d angular c o e f ­

ficients of regression.
I n our c a s e , t h e d i r e c t i o n o f t h e c o n j u g a t e d i a m e t e r c o n n e c t i n g
t h e e r r o r s o f m e a s u r e m e n t o f t h e Z - c o o r d i n a t e w i t h e r r o r s for t h e
X - c o o r d i n a t e i s d e t e r m i n e d o n t h e b a s i s o f (6.18), w h e r e a i s t h e
r o t a t i o n a n g l e o f t h e m a j o r a x i s o f t h e e l l i p s e r e l a t i v e t o t h e Xaxis.
The d i r e c t i o n o f t h e s e c o n d c o n j u g a t e d i a m e t e r , w h i c h c o n n e c t s
t h e measurement e r r o r s of t h e X-coordinate w i t h t h e e r r o r s of t h e
Z-coordinate, i s determined according t o t h e formula
tg

Here,

62

=-

a2 tg a

(6.19)

t h e angular c o e f f i c i e n t s of regression w i l l be:

(6.20)

489

4.

C o m b i n a t i o n o f M e t h o d s o f M a t h e m a t i c a l A n a l y s i s and
M a t h e m a t i c a l S t a t i s t i c s in E v a l u a t i n g t h e A c c u r a c y o f
Navigational Measurements

/465

In the preceding paragraphs, we examined methods of mathematical

statistics (probability theory) used t o evaluate the' accuracy of

navigational measurements. As a group, these devices permit the

solution of any two-dimensional or linear problem encountered in

aircraft navigation.

However, in examining the methods of probability theory, we

assumed that the accuracy (in general, mean square error) of

measurements of separate parameters was known.

Actually, the accuracy of measurements of navigational parame­

ters has a functional dependence on other physical or geometric

values connected with the principles of measuring or determining a

navigational parameter.

Since this functional dependence is always known, the simplest

(and presently most universal) method of determining the accuracy

of a navigational parameter is that of the variation of independent

variables included in the equations of formulas which determine a

navigational parameter within the limits in which the indicated

variations are encountered in the practice of aircraft navigation.

F o r example, the basic equation for an orthodrome which deter­
mines its shift in a geographic coordinate system has the form:
c t g hoi = tg Y Z c t g y1 cosec Ak - c t g ~ h .

Let U S determine the accuracy of solving the above equation,
assuming that the measurement accuracy of each of the parameters
included in the equation is known.
The dependence of the accuracy of the solution of the equation

on the accuracy of measuring the coordinates of $2 is expressed by

the equation:

(6.21)


The final result of solving (6.21) would have t o be viewed as

an arc tangent of the right-hand side. However, from the point of

view of a mathematical solution, this would lead to a significant

complication of the given problem. It is advisable to use the

following method:

dctgA0,
-=-.-

dctgho,

diol

4 2

diol

dF2

490

I

I

I

II I

or

cosec2 Aol.

Therefore,

(6.22)

T h u s , i f t h e mean s q u a r e e r r o r i n m e a s u r i n g t h e @ 2 c o o r d i n a t e
e q u a l s ~ $ 2 ,i t c a u s e s a n e r r o r i n d e t e r m i n i n g 1 0 1 :
ahOlp, =

-a
91

c t g ' p ~cosec Ah sec2 y z
cosecz Lol

(6.23)

The d e p e n d e n c e o f t h e a c c u r a c y i n d e t e r m i n g A 0 1 on t h e a c c u r a c y / 4 6 6
of coordinate $1 can be obtained analogously:
tg y2 cosec A L cosec2 'p1
cosecz hol
For t h e p a r a m e t e r A A
a b p , = GP1

ahOlAA= GAL

,.

t g v z c.t g.p­l c t g Ahcosec Ah-coseczdh
cosecz h,l

(6.24)

(6.25)

The t o t a l e r r o r i n s o l v i n g t h e e q u a t i o n w i l l b e :

S i n c e we h a v e e x a m i n e d a s a n e x a m p l e t h e a c c u r a c y o f s o l v i n g
t h e b a s i c e q u a t i o n f o r an orthodrome, it i s a p p r o p r i a t e t o examine
t h e a c c u r a c y o f s o l v i n g a l l t h e s p e c i a l e q u a t i o n s which d e t e r m i n e
i t s parameters:
(a)

I n i t i a l azimuth of an orthodrome:

Using t h i s method o f t r a n s i t i o n from t h e a r c t a n g e n t of t h e a n g l e
t o i t s v a l u e , as i n t h e preceding example, w e o b t a i n :

491

(6.26)


The moving azimuth of an orthodrome:


t g a i = - . t g hoi
s i n yi '
aai

=- ( c t g 'pi cosec y i t g Aoi cos2 a i )a'pi;

'pi

(6.27)

= ( s e d hoi cosec yi cos2 ai)d 0 i ;

aa.
(A01

aai = 1/.zaiYi

+ a2aiioi :

Coordinates of intermediate points:


(6.28)


Distance along the orthodrome from its source:

cos

0s.

Qoi

asi
'pi

si = cos k0i cos pi;

= sin hoi cos ' p i cosec

SiohOi;

= COS bi sin y i cosec S i ~ y i ;

(6.29)


above formulas (6.23) t o (6.29) have the following practi­

cal significance.

Let us assume that in solving the basic and special equations

of an orthodrome, we use trigonometric tables to five decimal

places or a computer with 18 binary digit bits (which are also

492


equivalent t o 5 decimal places).
In the first case, the error in
the vaiue of each independent variable wili have a magnitude of
from 0 - 5 units of the sixth sign, and (in the second case) from
0 - 10 units of the sixth sign. Substituting the values of the
possible errors of each independent variable into these formulas,
we will obtain the possible errors in the solution of the equations
Thus, it is possible t o determine the necessary accuracy of the
tables (number of signs) o r the computers (number of orders) for
obtaining a satisfactory result in solving equations within the
given value limits of the independent variables.
Calculations show that for geographic latitudes from 0 t o 8 0 ° ,
while solving equations f o r an orthodrome, it is necessary t o use
tables with 6 decimal points or computers with 21-22 binary digit
bits.

5 . I n f l u e n c e o f t h e Geometry o f a N a v i g a t i o n a l System on t h e
Accuracy o f Determining A i r c r a f t C o o r d i n a t e s
The accuracy of determining the coordinates of an aircraft by

means of navigational systems depends both on the accuracy of

measuring a navigational parameter and on the geometry of the navi­

gational system being used.

Means of solving one-dimensional problems of probability

theory for a generalized oblique-angled coordinate system were

examined above. The azimuth coordinate system was given as an

example for determining the gradient of a navigational parameter.

Since it is necessary to know the value and direction of the
gradient vector ( g ) of the navigational parameter to solve problems
in a generalized coordinate system, only the reduction of different
coordinate systems t o a generalized system is examined in this
section.
Two-pole goniometric, two-pole circular and one-pole range-

finding are most simply reduced t o a generalized coordinate system.

As has already been indicated, for an azimuthal system at

distances on the order of up to 3,000 k m , we can consider


-.1

g =' S

(6.30)

where S is the distance from the aircraft t o the ground radio beacon.

F o r greater distances, we must consider the convergence of the
/468,
position lines as a result of the spher'icity of the Earth's surface,
and ( 6 . 3 0 ) assumes the form:

493

1

g = RsinS

'

(6.31)

where R i s t h e r a d i u s of t h e E a r t h and
dr = dAR s i n S ;

where d r i s t h e i n c r e a s e i n l i n e a r e r r o r ; d A i s t h e i n c r e a s e i n
azimuth.
The d i r e c t i o n s o f t h e p o s i t i o n l i n e s i n t h i s c a s e c a n b e d e t e r ­
mined a s t h e moving a z i m u t h s of t h e o r t h o d r o m e s a t a g i v e n p o i n t M ,
which i n t e r s e c t f o c i o f t h e s y s t e m s A 1 and A 2 .
We m u s t t a k e p o i n t
i n t h i s case,

M as t h e s t a r t i n g p o i n t of b o t h orthodromes;

T h e p r o b l e m o f f i n d i n g a z i m u t h s o f t h e p o s i t i o n l i n e s for a
t w' 0 - p o l e c i r c u l a r s y s t e m i s s o l v e d a n a l o g o u s l y , w i t h t h e s o l e d i f ­
f er e n c e b e i n g t h a t t h e v e c t o r - g r a d i e n t w i l l n o t be d i r e c t e d p e r P en d i c u l a r t o t h e a z i m u t h s o f t h e o r t h o d r o m e s , b u t a l o n g t h e s e
o rthodromes; a c c o r d i n g l y , t h e f o r m u l a s f o r d e t e r m i n i n g t h e a z i m u t h s
of t h e p o s i t i o n l i n e s t a k e t h e f o r m :

S i n c e t h e d e n s i t y o f c i r c u l a r p o s i t i o n l i n e s l/g d o e s n o t d e ­
pend on d i s t a n c e ,
g = 1: dr = d R u Ar = h R ,

w h e r e AR i s t h e e r r o r i n m e a s u r i n g d i s t a n c e .
494

In goniometric range-finding systems, the task of finding the

density and position of the azimuthal position lines is solved in

the same way as for goniometric systems. In the case of circular

position lines at point M , their direction will differ from the

azimuthal lines by 90°.

The axes of the ellipses of errors in this case will coincide

with the position lines. Here, at short distances from the focus

of the system (without taking into account the convergence of the

azimuthal position lines), the minor axis of the ellipse coincides

with the position line which is determined most accurately (usually

the circular line).
At great distances, we must also consider the

convergence of the azimuthal position lines according to (6.31).

The problem of conversion to the generalized coordinate system
from the hyperbolic or hyperbolic-elliptical system is somewhat
more complicated. Let us use (1.74) f o r this purpose:
cos h , =

cos

SI cos 2c - cos (SI- 2a)
sin SIs i n 2c

Developing cos(S1 - 2 a ) , we can present this formula in the
form (Fig. 6.8a):
cos A1

0


c

--

cos SIcos 2c - cos SI cos 2a

- s i n SIs i n 2a

Sin s1s i h c




‘I

Fig. 6.8.

Determining Hyperbolic Position Lines:
(b) Distance.


(a) Direction;


The direction of the position lines at point M can be deter­

mined after differentiating (1.74) on the basis of s:


-

- cos SI s i n 2c (cos SI cos 2c - s i n SIcos 2a s i n SIs i n 2a)
sin2 SI sin2 2c s i n 11
dS1
- s i n S1 s i n 2c (sin SIcos 2 ~ - sin SIcos 2a - cos SI sin 2a)
sin2 SIsin22c s i n A,

dh,
--

(6.32)

495


/469

The h y p e r b o l i c p o s i t i o n l i n e i n t e r s e c t s t h e a z i m u t h l i n e o f
p o i n t M, drawn f r o m t h e f o c u s F 1 , a t a n a n g l e
a = arctg-

dS

s.

(6.33)

L e t u s determine t h e density of t h e hyperbolic p o s i t i o n l i n e s
a f t e r d i f f e r e n t i a t i n g ( 1 . 7 4 ) on t h e b a s i s o f t h e p a r a m e t e r and w i t h
a constant S (Fig. 6.8, b):

- dhl
_
da

Then,

-

sin SI sin 2c (sin SI cos 2a - cos SI sin 2a)
*
sin2 SI sin2 2c sin A,
dh
--da
da
dr

s cos a .

(6.34)
(6.35)

For c o n v e r s i o n a g e n e r a l i z e d c o o r d i n a t e s y s t e m f r o m a h y p e r ­
b o l i c - e l l i p t i c a l s y s t e m , i t i s s u f f i c i e n t t o s o l v e t h e p r o b l e m for
hyperbolic position l i n e s.
The d i r e c t i o n s o f t h e e l l i p t i c a l p o s i t i o n l i n e s a r e t h e n e a s i l y
The d e n s i t y o f
determined as b e i n g normal t o t h e h y p e r b o l i c ones.
t h e e l l i p t i c a l l i n e s i s c o n s t a n t f o r t h e whole a r e a of t h e a c t i v i t y
/470
of t h e s y s t e m , s i n c e t h e s e l i n e s do n o t d i v e r g e .
The r e d u c t i o n o f s p e c i a l c o o r d i n a t e s y s t e m s o f r a d i o - e n g i n e e r ­
i n g devices t o a generalized system permits t h e c o n s t r u c t i o n of
e l l i p s e s of e r r o r s i n d e t e r m i n i n g t h e c o o r d i n a t e s o f an a i r c r a f t
by means o f t h e s e d e v i c e s and t h e e s t i m a t i o n o f t h e m a i n t e n a n c e o f
t h e a i r c r a f t p a t h w i t h r e s p e c t t o d i s t a n c e and d i r e c t i o n .
I n some c a s e s , t h e b o u n d a r i e s o f t h e o p e r a t i o n a l a r e a o f t h e
system a r e d e s i g n a t e d , w i t h i n t h e l i m i t s of which t h e dimensions of
t h e a x e s of t h e e l l i p s e s of e r r o r s do n o t exceed t h e g i v e n v a l u e s .
The o p e r a t i o n a l a r e a s o f t h e s y s t e m c a n b e c o n s t r u c t e d on t h e b a s i s
o f o t h e r f a c t o r s ( f o r e x a m p l e , on t h e b a s i s o f t h e a c c e p t e d a c c u r a c y
o f t h e m a i n t e n a n c e o f t h e p a t h w i t h r e s p e c t t o d i r e c t i o n a l o n e or
with respect t o distance alone.
The e x a m i n e d m e a n s o f e v a l u a t i n g t h e a c c u r a c y o f t h e n a v i ­
g a t i o n a l m e a s u r e m e n t s i n c l u d e , on t h e w h o l e , r a d i o - e n g i n e e r i n g
devices.
However, t h e a c c u r a c y o f t h e c a l c u l a t i o n o f a n a i r c r a f t
p a t h by means o f g e o t e c h n i c a l means o f a i r c r a f t n a v i g a t i o n c a n b e
The a c c u r a c y of
examined w i t h an a z i m u t h a l r a n g e - f i n d i n g system.
d e t e r m i n i n g a i r c r a f t c o o r d i n a t e s by a s t r o n o m i c a l means c a n b e
examined by a c i r c u l a r s y s t e m .

496

6.

Evaluation of t h e Accuracy o f Measuring a Navigational Parameter

In the preceding paragraph, the effect of the geometry of a

navigational system on the accuracy of determining aircraft coordi­

nates (assuming that the accuracy of measuring a navigational param­

eter is known) was examined.

By measured navigational parameter of a system, we mean the

value being measured at the output of a navigational device: azi­

muth, course angle, distance, difference in distances, or sum of

distances t o objects on the ground.

In addition t o measured navigational parameters, there are

parameters which are determined by calculating the path of an air­

craft on the basis of its speed and time components; for example,

the orthodrome or geographic coordinates of the aircraft.

Let us examine briefly the reasons for errors and the methods

of evaluating the accuracy of measurements or determinations of the

indicated parameters.

During visual aircraft navigation with the use of geotechnical

devices and with the use of astronomical and nonautonomous radio

devices, the calculation of the aircraft path at each succeeding

stage is carried out on the basis of the results of measuring the

parameters of aircraft movement in the preceding stage. In this

case, two factors will influence the accuracy of aircraft naviga­

tion:

(a) The accuracy of juncture (determination of the location

of the aircraft) at the beginning and end of the control stage of

the path.

(b)

Wind variation at flight altitude from stage t o stage.


The accuracy of visual junctures depends on flight altitude

and on methods of measuring both the vertical and course angles o f

reference points.

Visual methods of aircraft navigation are usually used at low

flight altitudes, in conjunction with closely spaced reference

points. Therefore, the errors of juncture are very small, and do
/471

not have values comparable t o the wind variation at flight altitude.

The accuracy of measuring a navigational parameter with a
radio-navigational system depends on the principle of operation,
the frequency range used, the distance of the measurements, the
effect of a relief or the ionizing layers of the atmosphere,and
also (partially) on the physical state of the atmosphere (optical
density).
In a number of cases, the accuracy of measuring a navi­
gational parameter of a radio-navigational system is evaluated
statistically on the basis of the results of tests of the system
497

_-

u'nder d i f f e r e n t o p e r a t i n g c o n d i t i o n s .
The a c c u r a c y o f n a v i g a t i o n a l j u n c t u r e s o f a n a i r c r a f t b y means
o f n a v i g a t i o n a l s y s t e m s i s e v a l u a t e d by t h e means s e t f o r t h i n t h e
p r e c e d i n g s e c t i o n , p r o c e e d i n g f r o m t h e mean s q u a r e e r r o r i n m e a s u r i n g
a n a v i g a t i o n a l p a r a m e t e r , c o n s i d e r i n g t h e g e o m e t r y 05 t h e s y s t e m i n
t h i s area of a p p l i c a t i o n .
The a c c u r a c y o f m e a s u r i n g a n a v i g a t i o n a l p a r a m e t e r b y a s t r o n o m ­
i c a l means ( a s a r u l e , t h e a l t i t u d e o f a s t a r ) i s d e t e r m i n e d , i n
t h e f i r s t p l a c e , by t h e a c c u r a c y o f t h e i n s t a l l a t i o n a t t h e l e v e l
from which t h e measurements are c a r r i e d o u t .
The b u b b l e l e v e l s o f a v i a t i o n s e x t a n t s , w h i c h a r e s u b j e c t t o
c o n s t a n t a c c e l e r a t i o n under t h e e f f e c t o f C o r i o l i s f o r c e s , which
c a n b e t a k e n i n t o a c c o u n t i n f l i g h t if t h e g r o u n d s p e e d i s known,
t o a s t i l l g r e a t e r d e g r e e , t h e y can be s u b j e c t e d t o varying acceler­
a t i o n s by u n s t a b l e f l i g h t c o n d i t i o n s .
Gyroscopic l e v e l s are a l s o s u b j e c t t o C o r i o l i s a c c e l e r a t i o n s
and a l s o t o l o n g - p e r i o d f l u c t u a t i o n s i n t h e shape of a p r e c e s s i o n
cone, connected w i t h t h e e r r o r s o f b a l a n c i n g a gyroscope, where
calculation is practically impossible.
However, g y r o s c o p i c l e v e l s a r e f r e e f r o m t h e s h o r t - p e r i o d
i n t e r f e r e n c e s which a r e connected w i t h d i s t u r b a n c e of t h e a i r c r a f t ' s
f l i g h t condition.
I f , as a r e s u l t o f t h e a s t r o n o m i c a l o b s e r v a t i o n s , w e i n t r o d u c e
c o r r e c t i o n s f o r t h e i n s t r u m e n t e r r o r of t h e s e x t a n t , t h e r e f r a c t i o n
of t h e atmosphere and t h e C o r i o l i s a c c e l e r a t i o n s of t h e l e v e l , t h e n
t h e e r r o r s as a r e s u l t of s h o r t - p e r i o d f l u c t u a t i o n s of t h e bubble
l e v e l o r long-period ( g y r o s c o p i c ) f l u c t u a t i o n s w i l l be predominant.
The m a g n i t u d e o f t h e e r r o r s i n b u b b l e l e v e l s d e p e n d s on t h e
s t a t e of t h e atmosphere and t h e f l i g h t speed w h i l e t h e e r r o r s i n
t h e g y r o s c o p i c l e v e l s d e p e n d on t h e m a n u f a c t u r i n g a c c u r a c y o f t h e
level (precession cone).
The a c c u r a c y o f a s t r o n o m i c a l m e a s u r e m e n t s i s d e t e r m i n e d b y
s t a t i s t i c a l methods.
The e r r o r o f m e a s u r e m e n t s w i t h b u b b l e l e v e l s
( 2 0 ) i s w i t h i n t h e l i m i t s o f 5 - 6 ' i n a clam a t m o s p h e r e a t a low
f l i g h t s p e e d ( u p t o 1 5 - 2 0 l f o r h i g h - s p e e d a i r c r a f t ) , a n d a t low
f l i g h t speeds with t h e presence of an atmospheric disturbance.
The a c c u r a c y o f g y r o s c o p i c l e v e l s i s w i t h i n t h e l i m i t s o f 1 0 ­
15'.

498

7 . C a l c u l a t i o n o f t h e Wind w i t h an E v a l u a t i o n o f t h e A c c u r a c y

of A i r c r a f t Navigation

/4722)

I n a l l c a s e s when t h e p a r a m e t e r s o f a i r c r a f t movement a r e
d e t e r m i n e d o n t h e b a s i s o f s u c c e s s i v e i n d i c a t i o n s o f PA or b y t h e
s i g h t i n g o f g r o u n d r e f e r e n c e p o i n t s a t s e p a r a t e p o i n t s , t h e wind
variation a t f l i g h t a l t i t u d e with respect t o the distance traversed
e x e r t s t h e g r e a t e s t i n f l u e n c e on t h e a c c u r a c y of a i r c r a f t n a v i g a t i o n .

It i s only with t h e use of Doppler o r i n e r t i a l guidance d e v i c e s
t h a t t h e wind v a r i a t i o n d o e s n o t e x e r t a d i r e c t i n f l u e n c e on t h e
a c c u r a c y o f a i r c r a f t n a v i g a t i o n , w i t h t h e e x c e p t i o n o f c a s e s when
t h e r a n g e o f c r u i s i n g s p e e d s o f t h e a i r c r a f t d o e s n o t make i t p o s ­
s i b l e t o compensate completely f o r t h e changes of t h e groundspeed
during f l i g h t with r e s p e c t t o a given groundspeed.
The a c c u r a c y o f a i r c r a f t n a v i g a t i o n a t a n y s t a g e i n f l i g h t ,
w i t h r e s p e c t t o d i s t a n c e and d i r e c t i o n , c a n b e e x p r e s s e d by t h e
formulas :
(6.36)
(6.37)

where ux

1

a n d crz

1

a r e t h e e r r o r s i n m e a s u r i n g t h e X- a n d Z - c o o r d i ­

n a t e s of an a i r c r a f t a t t h e beginning of a s t a g e ;

and u

0

w1

are
$1

t h e e r r o r s i n d e t e r m i n i n g t h e groundspeed and f l i g h t a n g l e i n a
p r e c e d i n g s t a g e ; a n d uu
a n d uu
a r e t h e v a r i a t i o n s of t h e longiXS

ZS

t u d i n a l and l a t e r a l components of
path t o the next.

t h e wind f r o m one s t a g e o f t h e

A s has a l r e a d y been s a i d , t h e accuracy of measuring t h e ground­
s p e e d and t h e a c t u a l f l i g h t a n g l e d e p e n d s on t h e a c c u r a c y o f t h e
i n i t i a l and f i n a l " j u n c t u r e s " of t h e a i r c r a f t t o t h e preceding
s t a g e ; i f t h e length of t h e chosen c o n t r o l f l i g h t s t a g e s i s t h e
same, t h e n w i t h t h e same a c c u r a c y o f t h e i n i t i a l a n d f i n a l " j u n c t u r e s "
o f t h e a i r c r a f t it c a n b e e x p r e s s e d by t h e f o r m u l a s :

(6.38)

To e v a l u a t e t h e w i n d v a r i a t i o n a t f l i g h t a l t i t u d e w i t h r e s p e c t
t o d i s t a n c e , t h e following formula i s used:

499

The c o e f f i c i e n t o f wind v a r i a t i o n w i t h r e s p e c t t o d i s t a n c e K 3
d e p e n d s on t h e f l i g h t a l t i t u d e a n d t h e t i m e o f y e a r ( F i g . 6 . 9 a ) .
The c o m p o n e n t s o f t h e w i n d v a r i a t i o n v e c t o r , o n t h e b a s i s o f
t h e c o o r d i n a t e a x e s , a r e c o n s i d e r e d e q u a l t o i t s m o d u l u s d i v i d e d by
t h e r o o t of t h e two:

(6.40)

To c h a r a c t e r i z e t h e p o s s i b l e e r r o r s i n a i r c r a f t n a v i g a t i o n
f o r a f o l l o w i n g a i r c r a f t on t h e b a s i s o f r e s u l t s o f wind m e a s u r e ­
ments f o r t h e a i r c r a f t ahead, t h e f o l l o w i n g formula i s used:
UUf

where

=K f

vr.

(6.41)

Kt i s t h e c o e f f i c i e n t o f wind v a r i a t i o n w i t h r e s p e c t t o t i m e .

Kt on t h e f l i g h t a l t i t u d e i s

The d e p e n d e n c e o f t h e c o e f f i c i e n t
shown i n F i g . 6 . 9 , b .
a)

b)
H.

12

9
6

3


Fig.

6.9.
D e p e n d e n c e o f Wind V a r i a t i o n o n F l i g h t A l t i t u d e :
With R e s p e c t t o D i s t a n c e ; ( b ) With R e s p e c t t o Time.

(a)

F o r t h e c a s e o f wind v a r i a t i o n w i t h r e s p e c t t o d i s t a n c e , t h e
c o m p o n e n t s of wind v a r i a t i o n a l o n g t h e c o o r d i n a t e a x e s a r e c o n s i d e r e d
equal:

(6.42)

500

/473

8.

Consideration o f t h e Polar Flattening o f t h e E a r t h in the
Determination o f D i r e c t i o n s a n d D i s t a n c e s on t h e
E a r t h’ s Surface

The p o l a r f l a t t e n i n g o f t h e E a r t h i s e x p l a i n e d b y i t s d i u r n a l
r o tation.
In fact, each point of t h e Earth’s surface i s subject t o t h e
a c t i o n o f two f o r c e s :
(a)
Earth.

A f o r c e of g r a v i t a t i o n , direc,ted toward t h e c e n t e r of t h e

A c e n t r i f u g a l force, directed along t h e radius of t h e
(b)
p a r a l l e l of t h e p o i n t .
I n F i g . 6 . 1 0 , 0 i s t h e p o i n t o f s u s p e n s i o n o f a p e n d u l u m ; OD
i s t h e v e c t o r of t h e f o r c e of g r a v i t a t i o n , and DFc i s t h e v e c t o r of
the centrifugal force.
A s a r e s u l t , t h e d i r e c t i o n o f t h e p e n d u l u m c o r d (or, i n o t h e r
words, t h e d i r e c t i o n of t h e v e r t i c a l of t h e l o c u s ) coincides with
It is n a t u r a l t h a t t h e plane of t h e
t h e r e s u l t a n t of t h e s e f o r c e s .
t r u e h o r i z o n must be p e r p e n d i c u l a r t o t h e v e r t i c a l o f t h e l o c u s ,
e s p e c i a l l y s i n c e t h e m a j o r i t y of t h e E a r t h ‘ s s u r f a c e i s covered
w i t h w a t e r , w h i l e t h e l e v e l of t h e d r y l a n d d i f f e r s from sea l e v e l
s i g n i f i c a n t l y i n o n l y a few p l a c e s .

T h e r e f o r e , t h e E a r t h h a s assumed a s h a p e c l o s e t o a n e l l i p s o i d
of r o t a t i o n .
The a n g l e b e t w e e n t h e p l a n e o f t h e e q u a t o r a n d t h e v e r t i c a l
o f t h e l o c u s i s c o n s i d e r e d t h e g e o g r a p h i c l a t i t u d e of t h e l o c u s .
The a n g l e b e t w e e n t h e p l a n e
of t h e e q u a t o r and t h e g e o c e n t r i c
v e r t i c a l o f t h e l o c u s ( l i n e con­
necting a point at the Earth’s
center with t h e c e n t e r of t h e
Earth) i s c a l l e d t h e geocentric

l a t i t u d e of t h e Zocus ( $ 1 1 .
The a n g l e b e t w e e n t h e p l a n e
of t h e e q u a t o r and t h e g e o c e n t r i c
v e r t i c a l o f s u c h a p o i n t on t h e
E a r t h ’ s s u r f a c e ( t a k e n as a s p h e r e ) ,
t h e r a d i u s of whose p a r a l l e l i s
e q u a l t o t h e r a d i u s of t h e p a r a l l e l

G e o ’ g r a p h i c a n d Geo­
Fig. 6.10.
centric Verticals.

501


I..


/474

o f our point on the ellipsoid of rotation, is called the c o r r e c t e d

Z a t i t u d e ( u1.
The indicated latitudes are
connected by the following
relationships:
5

tg u =

or

tg Y1

vCi2 '

(6.45)

9 s­

where e is a parameter equal to
e
c is a parameter equal t o
a'
Ja2 - b 2 , and a is the large

Geographic, GeoFig. 6.11.
centric and Reduced Latitudes.

semiaxis of the ellipse.

From Fig. 6.11, it is obvious that the ratio of the Earth's

radius at any point t o the major semiaxis of the ellipse can be

expressed by the formula

tg 'p1
cos arc tg r

where R is the radius of the Earth.
Since

.arc tg

tg

VG2

= arc cos

1

fi+(

QYl

VI-

j

e2

Then,

(6.46)


F o r the Earth's surface, the polar flattening is
a--6 - 1
--a
298

502

/475

'

where b i s t h e minor semiaxis o f t h e e l l i p s e .
I f we t a k e t h e m a j o r s e m i a x i s a s e q u a l t o 1, t h e n t h e m i n o r
semiaxis w i l l be:
6

1
1-i
298 = 0,99665.

Therefore, t h e parameter e has t h e value:
e=

$1

l/ac2--bz ..= 0,006689.
a

The l e n g t h o f t h e a r c o f t h e E a r t h ' s m e r i d i a n f r o m l a t i t u d e
t o $2 w i l l equal:

(6.47)

I n p r a c t i c e , t h e maximum d i f f e r e n c e $ - $ 1 c a n n o t e x c e e d 12',
t h e r e f o r e , t h e c o s i n e i n t h e denominator of ( 6 . 4 7 ) can be t a k e n as
one ( f i r s t a p p r o x i m a t i o n ) .
Then ( 6 . 4 7 ) h a s t h e f o r m :

(6.47a)
Y1


However,

a c c o r d i n g t o (6.461,

--

Therefore,

( 6 . 4 7 a ) can be reduced t o t h e form:

(6.47b)

T h e i n t e g r a l ( 6 . 4 7 b ) i s n o t s o l v e d i n f i n a l f o r m or i s s o l v e d
H o w e v e r , c o n s i d e r i n g t h a t t h e v a l u e e' c o s 2 $ 1 < < 1,
very complexly.
with a very s m a l l e r r o r w e can consider (second approximation) t h a t

503

/476

I-

Then t h e i n t e g r a l ( 6 . 4 7 b )

t a k e s t h e form:

(6.47~)

The i n t e g r a l ( 6 . 4 7 ~ )c a n b e f u r t h e r s i m p l i f i e d i f w e t a k e 1

e* as X and t h e v a l u e e 2 cos2$1 as

­

E

Since E <<X, it i s p o s s i b l e t o consider approximately
approximat i o n ) :

(third

or i n o u r e x a m p l e

VI - e2 + e2 cos2 VI

-

=

e2cos2y1
2

+V G 2 .

(6.48)

Taking t h e t h i r d approximation i n t o a c c o u n t , t h e i n t e g r a l
( 6 . 4 7 ~ )t a k e s t h e f o r m :

dpl + a
'p1

The i n t e g r a l

e2
2

i

cos2 yldyl.

(6.47d)

'PI

(6.47d) has t h e simple solution:

'PI

Considering t h a t

a 1/G2=b,

(6.49)

or

504

. ...

,

Formula ( 6 . 4 9 ) i s t h e s i m p l e s t one f o r f i n d i n g t h e d i s t a n c e s
a l o n g t h e a r c of t h e m e r i d i a n a f t e r r e d u c i n g t h e g e o g r a p h i c
l a t i t u d e s t o geocentric ones.
S i n c e t h e f o r m u l a i s o b t a i n e d by t h r e e s u c c e s s i v e a p p r o x i m a t i o n s ,
w e must i n d i c a t e t h e d e g r e e of i t s a c c u r a c y .

I n t h e f i r s t a p p r o x i m a t i o n , w e s u b s t i t u t e d 1 f o r COS ( 4 - 4 1 ) .
On t h e E a r t h ' s s u r f a c e , t h e d i f f e r e n c e $ 1 - $ 2 c h a n g e s f r o m z e r o a t
t h e p o l e or e q u a t o r t o 12' a t m i d d l e l a t i t u d e s .
Therefore, cos
41 - 4 1 c a n n o t b e l e s s t h a n 0 . 9 , 9 9 9 , 9 3 9 , w h i l e i t s m e a n v a l u e i s
0.9,999,985.
T h e maximum e r r o r a s a r e s u l t o f t h e f i r s t a p p r o x i m a t i o n i s
e q u a l t o 0 . 0 0 0 , 0 0 6 o f t h e m e a s u r e d v a l u e , w h i l e i t s mean e r r o r i s
0.0,030,015 of i t s value.
T h i s g i v e s a t o t a l o f 30 m f o r t h e
d i s t a n c e from t h e North t o t h e South Pole.
The s e c o n d a p p r o x i m a t i o n w i t h r e s p e c t t o t h e p a r a m e t e r o f t h e
f l a t t e n i n g of t h e E a r t h a l s o r e s u l t s i n an underestimation of t h e
c a l c u l a t e d v a l u e a t a maximum o f 0 . 0 0 0 , 0 0 2 o f i t s v a l u e .
T h e t h i r d a p p r o x i m a t i o n , a t a maximum o f u p t o 0.000,Oll o f t h e
measured v a l u e , causes t h e g r e a t e s t e r r o r .
However, t h i s e r r o r ( a s
opposed t o t h e f i r s t two a p p r o x i m a t i o n s ) t e n d s toward a n o v e r e s t i ­
mation of t h e value being c a l c u l a t e d .
T h u s , t h e maximum e r r o r o f t h e t h r e e a p p r o x i m a t i o n s d o e s n o t
e x c e e d 0.000,OOl o t t h e c a l c u l a t e d v a l u e , w h i l e t h e mean e r r o r i s
0.000,005
of i t s v a l u e (around 1 0 0 m f o r t h e d i s t a n c e from t h e
North t o t h e South P o l e ) .
L e t u s now e x a m i n e t h e m e a n s o f c a l c u l a t i n g t h e f l a t t e n i n g o f
t h e E a r t h w i t h m e a s u r e m e n t s o f d i s t a n c e s a l o n g t h e o r t h o d r o m e for
any f l i g h t d i r e c t i o n .

O b v i o u s l y , j u s t as t h e E a r t h ' s m e r i d i a n , a n o r t h o d r o m e drawn
i n any d i r e c t i o n i s a n e l l i p s e , t h e d e g r e e o f f l a t t e n i n g o f which
d e p e n d s on t h e d i p a n g l e o f t h e p l a n e o f t h e o r t h o d r o m e t o t h e
plane of t h e equator ( l a t i t u d e of t h e vertex).

I n a s p e c i a l c a s e , when t h e l a t i t u d e o f t h e v e r t e x i s 9 0 ° , t h e
o r t h o d r o m e c o i n c i d e s w i t h t h e m e r i d i a n a n d i t s m i n o r a x i s is e q u a l
t o t h e minor a x i s o f t h e e l l i p s o i d of r o t a t i o n of t h e E a r t h .
When
t h e l a t i t u d e of t h e v e r t e x i s O o , t h e orthodrome c o i n c i d e s with
t h e e q u a t o r and i s c o n v e r t e d t o a c i r c l e .
I n t h e g e n e r a l case, t h e minor semiaxis of t h e e l l i p s e of t h e
orthodrome i s e q u a l t o t h e r a d i u s o f t h e E a r t h a t t h e p o i n t of i t s
v e r t e x , ?.e., according t o 16.48):

505

(6.50)

where a = 6,378,245; b = 6,356, 863; and e * = 0.00669.
Having thus determined the parameter b for the orthodrome,
the distance along its arc can be determined according t o (6.49),
by substitution as follows:
(a)

parameter b for

berth;

(b) geocentric latitudes for distances along the orthodrome
from the point of intersection with the equator expressed in angular
measurement ;
(c)

parameter e for the value

Since the flattening of the ellipse of an orthodrome is always

less than the flattening of an ellipse of a meridian, except when

they coincide, the calculation errors resulting from the three

approximations introduced by us for the arc of an ellipse will be

significantly less in the general case than for the arc of the

meridian.


To determine directions on the Earth's surface, taking into

account the flattening of the Earth, it is sufficient t o convert

the geographic longitudes of the starting and end points of the

orthodrome segment o f the path into geocentric longitudes on the

basis of (6.43).
The azimuth of the orthodrome at any point is

then determined on the basis of those formulas, just as for the

spherical shape of the Earth.


506


CHAPTER S E V E N
F L I G H T PREPARATION
1.

Goals a n d P r o b l e m s o f F l i g h t P r e p a r a t i o n

Flight preparation plays an exceptionally important r o l e i n
e n s u r i n g a c c u r a c y and r e l i a b i l i t y i n a i r c r a f t n a v i g a t i o n .
The i m ­
provement o f n a v i g a t i o n a l equipment f o r a i r c r a f t i s making h i g h e r
demands on f l i g h t p r e p a r a t i o n and i s a l s o c h a n g i n g i t s n a t u r e .
D u r i n g p r e p a r a t i o n f o r v i s u a l f l i g h t a t low a l t i t u d e s and f o r
s h o r t d i s t a n c e s , a t t e n t i o n must be p a i d t o t h e n a t u r e of t h e l o c a l
r e l i e f , b o t h from t h e p o i n t of view of t h e o r i e n t a t i o n c o n d i t i o n s
and t h e p o i n t o f view o f f l i g h t s a f e t y .
I n t h i s case it i s permis­
s i b l e t o measure t h e n e c e s s a r y n a v i g a t i o n a l magnitudes ( a n g l e s and
d i s t a n c e s ) d i r e c t l y on a f l i g h t c h a r t w i t h a s c a l e and p r o t r a c t o r .
Detailed study of t h e r e l i e f plays a s i g n i f i c a n t l y smaller
r o l e i n a u t o m a t i c or s e m i a u t o m a t i c n a v i g a t i o n a t h i g h a l t i t u d e s a n d
airspeeds.
However, t h e a c c u r a t e c a l c u l a t i o n of n a v i g a t i o n a l
c o u r s e m a g n i t u d e s , e s p e c i a l l y f l i g h t a n g l e s and d i s t a n c e s , becomes
more i m p o r t a n t .
More a t t e n t i o n m u s t b e p a i d t o p r o b l e m s o f f u e l
consumption during f l i g h t , maneuvering i n a i r p o r t landing areas,
etc.
F l i g h t p r e p a r a t i o n i s u s u a l l y d i v i d e d i n t o two s t a g e s :
l i m i n a r y and p r e - f l i g h t .

pre­

All t h e b a s i c and t i m e - c o n s u m i n g p r o b l e m s a r e s o l v e d i n t h e
p r e l i m i n a r y p r e p a r a t i o n which can be e x e c u t e d b e f o r e h a n d , t h e day
For e x a m p l e :
b e f o r e f l i g h t or e a r l i e r .
(a)
S e l e c t i n g and p r e p a r i n g f l i g h t c h a r t s , p l o t t i n g t h e r o u t e ,
c a l c u l a t i n g t h e p a t h a n g l e s and d i s t a n c e s , marking t h e c h a r t s .
(b)

Studying t h e r o u t e ,

c a l c u l a t i n g a safe f l i g h t a l t i t u d e .

(c)
S p e c i a l p r e p a r a t i o n of c h a r t s and a i d s f o r t h e u s e of
r a d i o - e n g i n e e r i n g and a s t r o n o m i c a l d e v i c e s d u r i n g f l i g h t .
(d)

S e l e c t i n g , studying and r e f i n i n g t h e maneuvering diagrams

507

/478

i n a i r p o r t areas and t h e o p e r a t i n g r u l e s f o r r a d i o - e n g i n e e r i n g
vices.
(e)
flight.

de­

P r e l i m i n a r i l y c a l c u l a t i n g t h e f u e l consumption during

(f)
Checking n a v i g a t i o n a l equipment and e l i m i n a t i n g compass
deviations, etc.
I f t h e c r e w h a s made r e p e a t e d f l i g h t s a l o n g t h e same r o u t e ,
most and p e r h a p s a l l o f t h e problems o f p r e l i m i n a r y p r e p a r a t i o n
I n a d d i t i o n , each pre­
have been solved d u r i n g a previous f l i g h t .
ceding f l i g h t is t h e b e s t preparation f o r t h e subsequent one.
There­
f o r e , i n c a s e s o f r e p e a t e d f l i g h t s t h e volume o f p r e l i m i n a r y p r e p a r ­
a t i o n c a n b e s u b s t a n t i a l l y r e d u c e d or e x c l u d e d e n t i r e l y i f o n l y a
s h o r t t i m e has e l a p s e d s i n c e completing t h e previous f l i g h t and i f
t h e r e h a v e b e e n n o s u b s t a n t i a l c h a n g e s i n r o u t e s or f l i g h t c o n d i ­
t i o n s and r a d i o - e n g i n e e r i n g e q u i p m e n t , e t c . , a l o n g t h e r o u t e .
During p r e - f l i g h t p r e p a r a t i o n , only t h o s e problems a r e solved
which c o u l d n o t be s o l v e d b e f o r e h a n d , : . e . :
(a)
Studying t h e meteorological s i t u a t i o n and compiling a
navigational f l i g h t plan.
(b)
Refining t h e c a l c u l a t i o n of f u e l consumption during
f l i g h t and t h e p o i n t s of c l o s e s t approach t o r e s e r v e a i r p o r t s .
(c)
Introducing changes i n t h e operating r u l e s f o r radioe n g i n e e r i n g d e v i c e s a n d i n t h e m a n e u v e r i n g d i a g r a m s , w h i c h may o c ­
cur directly before f l i g h t .
(d)
Surveying and i n s p e c t i n g t h e o p e r a t i n g c o n d i t i o n of t h e
aircraft’s n a v i g a t i o n a l equipment.

2.

Preparing F l i g h t Charts and Marking t h e Route

The s e l e c t i o n o f a s c a l e a n d t h e met h o d f o r p r e p a r i n g f l i g h t
c h a r t s d e p e n d on t h e t a c t i c a l d a t a o f t h e a i r c r a f t ( a l t i t u d e , s p e e d ,
type of n a v i g a t i o n a l equipment).

or 1:1,000,000 may b e u s e d o n
C h a r t s w i t h a scale of 1:500,000
a i r c r a f t w i t h l o w s p e e d s and f l i g h t a l t i t u d e s ( s p e e d up t o 1 5 0 - 2 5 0
k m / h r , a l t i t u d e up t o 2 0 0 0 - 3 0 0 0 m).
A s c a l e o f 1 : 1 , 0 0 0 , 0 0 0 i s m o s t s u i t a b l e for f l i g h t s p e e d s o f
3 0 0 - 6 0 0 km/hr a n d i n a number o f c a s e s i s d e s i r a b l e f o r s p e e d s o f
800 - 1000 km/hr.
However, c h a r t s w i t h a s c a l e o f 1 : 2 , 0 0 0 , 0 0 0
are
u s e d m a i n l y a t f l i g h t s p e e d s of m o r e t h a n 6 0 0 k m / h r , e s p e c i a l l y f o r
g r e a t f l i g h t d i s t a n c e s , s i n c e a s c a l e o f 1:1,000,000 b e c o m e s v e r y
unwieldy f o r long-distance f l i g h t s .

508

/479

C h a r t s must b e s e l e c t e d on t h e b a s i s of s p e c i a l c o m p o s i t e
tables.
C h a r t s w i t h s c a l e s o f 1:1,000,000, 1 : 5 0 0 , 0 0 0 a n d l a r g e r
w h i c h h a v e c o n v e n t i o n a l s t a n d a r d i n t e r n a t i o n a l , s y m b o l s may a l s o b e
s e l e c t e d on t h e b a s i s of t h e c o o r d i n a t e s o f t h e i n i t i a l , f i n a l a n d
i n t e r m e d i a t e p o i n t s o f t h e p a t h and a c c o r d i n g t o t h e s t r u c t u r a l
p r i n c i p l e of t h e c o n v e n t i o n a l symbols and t h e d i v i s i o n o f c h a r t
sheets.
/480
A c c u r a t e c u t t i n g of t h e s h e e t s must be done from t h e b o t t o m
and r i g h t o f t h e s h e e t ( s o u t h and e a s t ) w i t h a l l o w a n c e of s p a c e f o r
s p l i c i n g from t h e t o p and l e f t .
T h i s makes i t p o s s i b l e t o draw
l i n e s on t h e c h a r t , e s p e c i a l l y w i t h I n d i a i n k .
If t h e s h e e t s a r e
s p l i c e d t h e o t h e r way r o u n d , e a c h o f t h e seams w i l l p o s e a n o b s t a c l e
t o a p e n c i l or d r a w i n g p e n .

C h a r t s o f l a r g e a r e a s must f i r s t be s p l i c e d by columns and n o t
A f t e r t h i s , t h e a r r a n g e d columns a r e s p l i c e d .
by r o w s .
A f t e r t h e s h e e t s a r e s p l i c e d on f l i g h t c h a r t s , t h e r o u t e o f t h e
f o r t h c o m i n g f l i g h t i s c o n s t r u c t e d , p a t h a n g l e s and d i s t a n c e s a r e
c a l c u l a t e d or c h a n g e d , a n d t h e f l i g h t p a t h i s d r a w n .
A s we a l r e a d y know ( C h a p t e r T w o ) , t h e r e f e r e n c e s y s t e m o f t h e
p a t h a n g l e s d e p e n d s on t h e f l i g h t d i s t a n c e and t h e n a v i g a t i o n a l
equipment used.

For s h o r t - d i s t a n c e f l i g h t s , p a t h a n g l e s a n d d i s t a n c e s may b e
m e a s u r e d d i r e c t l y on a f l i g h t c h a r t w i t h t h e u s e o f m a g n e t i c
compasses.
I n a d d i t i o n , t h e p a t h a n g l e s and d i s t a n c e s f o r f l i g h t s o v e r a
g r e a t e r d i s t a n c e may b e m e a s u r e d v i a g i v e n p o i n t s o n a c h a r t w i t h
t h e use of course instruments of average accuracy, without automatic
c a l c u l a t i o n of t h e p a t h .
However, i f t h e m a g n e t i c l a t i t u d e c h a n g e s
t o a s i g n i f i c a n t degree, during f l i g h t preparation t h e deviation of
t h e m a g n e t i c compass must be c a l c u l a t e d f o r e a c h s t r a i g h t segment
of t h e p a t h , t a k i n g i n t o account t h e change i n t h e h o r i z o n t a l
If t h e s t r a i g h t s e g m e n t
component o f t h e E a r t h ' s m a g n e t i c f i e l d .
o f t h e p a t h i s v e r y l o n g , i t must be d i v i d e d i n t o s e v e r a l s e g m e n t s
n o t m o r e t h a n 1 0 0 0 - 1 2 0 0 km i n l e n g t n .
The m o s t t i m e - c o n s u m i n g a n d c o m p l i c a t e d p r o c e s s i s c a l c u l a t i n g
t h e p a t h a n g l e s , d i s t a n c e s , i n t e r m e d i a t e p o i n t s of t h e p a t h and
o r t h o d r o m e a z i m u t h s f o r l o n g - d i s t a n c e f l i g h t s by means o f a c c u r a t e
c o u r s e d e v i c e s and a u t o m a t i c means o f a i r c r a f t n a v i g a t i o n .
Below w e g i v e a s e r i e s o f t h e mo s t c o m p r e h e n s i v e s y s t e m a t i c
c a l c u l a t i o n of t h e orthodrome elements f o r a f l i g h t with t h e use of
t h e above means.

I t i s advantageous t o c a l c u l a t e t h e orthodrome e l e m e n t s i n two
s t a g e s w i t h t h e u s e of one t a b l e which s y s t e m a t i z e s t h e c a l c u l a t i o n

509

sequence a t each s t a g e .
I n t h e first stage, t h e s h i f t of t h e orthodrome r e l a t i v e t o
t h e Greenwich m e r i d i a n and t h e a z i m u t h o f t h e o r i g i n a t i t s s o u r c e
are determined, u s i n g t h e following formulas and t h e form of Table
7.1.
ctgAo1 = tg 92 ctg yl cosec AA - ctg Ak

where $ 1 and $ 2 r e p r e s e n t t h e g e o g r a p h i c l a t i t u d e s o f t h e i n i t i a l
and f i n a l p o i n t s ; AI i s t h e d i f f e r e n c e i n l o n g i t u d e s o f t h e i n i t i a l
and f i n a l p o i n t s ; A 0 1 i s t h e l o n g i t u d e o f t h e i n i t i a l p o i n t measured
from t h e p o i n t of i n t e r s e c t i o n of t h e orthodrome with t h e Equator;
Xcm i s t h e s h i f t o f t h e o r t h o d r o m e f r o m t h e G.reenwich m e r i d i a n ( d i f ­
f e r e n c e i n g e o g r a p h i c a n d o r t h o d r o m i c l o n g i t u d e ) ; a n d a0 i s t h e
azimuth of t h e orthodrome a t i t s o r i g i n .
TABLE 7 . 1
Argui e n t o r

V a l u e of t h e
arguement

or

function __
1

91

92

Ah

2

3

4

Angular
magnitude
sin

­

cos

rg
ctg
cosec

-

Columns 1 t h r o u g h 4 o f T a b l e 7 . 1 . a r e f i l l e d d i r e c t l y on t h e
b a s i s of t h e v a l u e s o f t h e t r i g o n o m e t r i c f u n c t i o n s o f t h e c o o r d i n a t e s
of t h e i n i t i a l and f i n a l p o i n t s of t h e orthodrome, w i t h a c c u r a c y t o
t h e s i x t h decimal place.
Then t h e f i g u r e s from t h e t a b l e a r e t r a n s f e r r e d t o t h e f o r m u l a
The r e s u l t o f t h e s o l u t i o n i s g i v e n i n Column 5 .
t o determine X01.

A f t e r t h i s , t h e f o r m u l a f o r d e t e r m i n i n g a0 i s s o l v e d and t h e
r e s u l t o f t h e s o l u t i o n i s g i v e n i n Column 6 .
F i n a l l y t h e v a l u e o f A,,

510


i s d e t e r m i n e d a n d w r i t t e n i n Column 7 .

/481,

I t i s c o n v e n i e n t t o a p p l y t h e a b o v e o r d e r o f s o l u t i o n when
u s i n g a computer w i t h a keyboard.

When u s i n g t a b l e s i n s t e a d o f t r i g o n o m e t r i c f u n c t i o n s , i t i s
more c o n v e n i e n t t o r e c o r d t h e i r l o g a r i t h m s w i t h t h e e x c e p t i o n o f t h e
m a g n i t u d e s c t g AA and c t g A 0 1 which ( w i t h r e s p e c t t o t h e c h a r a c t e r ­
i s t i c s o f t h e s o l u t i o n ) must b e r e p r e s e n t e d by t r i g o n o m e t r i c v a l u e s .
I n t h e second s t a g e , a l l t h e o t h e r s p e c i a l values of t h e ortho­
drome e l e m e n t s a r e d e t e r m i n e d on t h e b a s i s o f t h e f o r m u l a s :
tg 'Ti =

s i n hoi
,tgao

t g at .= t g lor ,. cos sol = cos h,i cos 'pi;
,
sin pi
A

w h e r e + i ,A o i , S o i a r e t h e c o o r d i n a t e s o f a n y p o i n t o n t h e o r t h o ­
d r o m e ( A o i a n d So< a r e m e a s u r e d f r o m t h e p o i n t o f i n t e r s e c t i o n o f
t h e orthodrome with t h e Equator).
/482
COS

__
-70'00'
-60'00'
-51"OO'

9'49'
1 9"49'
28"49'

0,17049
0,33901
0,48201

A,,

0,98536
0,94078
0,87617

tgloi

0,17303
0,36035
0,55013

I

tgyi

0,37057
0,73653
1,04764

1

Pi

20"20'
3G023'
46'20'

The i n i t i a l d a t a f o r s o l v i n g t h e s e p r o b l e m s a n d t h e r e s u l t s o f
t h e s o l u t i o n a r e w r i t t e n i n Table 7 . 2 ( s e e t h e example of t h e
s o l u t i o n ) using d a t a from Table 7 . 1

ExampZe.
Determine t h e orthodrome parameters f o r a f l i g h t
$ 1 = 2 0 ° 2 0 ' t o p o i n t A2 = - 5 1 O ; + 2 = 4 6 O 2 0 ' .
from p o i n t s A 1 = - 7 0 ° 0 0 ' ;
C a l c u l a t e t h e i n t e r m e d i a t e p o i n t of t h e orthodrome f o r t h e given
longitude 60°.
Solution.
P r o c e e d i n g from c o o r d i n a t e s of t h e i n i t i a l and
f i n a l p o i n t s o f t h e o r t h o d r o m e , l e t u s f i l l i n Columns 1 - 4 i n
Table 7.1.
S u b s t i t u t i n g t h e a b o v e v a l u e s i n t h e f o r m u l a f o r A 0 1 a n d cxo,
l e t u s f i n d t h e v a l u e s t o f i l l Columns 5 - 7 .
Using t h e d a t a from Table 7 . 1 and t h e f o r m u l a s f o r Table 7 . 2 ,
t h i s table i n succession.

l e t u s f i l l i n t h e columns o f

After f i l l i n g i n Table 7 . 2 , l e t u s w r i t e o u t t h e r e s u l t s from
it which a r e n e c e s s a r y f o r e x e c u t i n g t h e f l i g h t .

511


Knowing t h e e l e m e n t s o f t h e o r t h o d r o m e i s n e c e s s a r y w i t h a n y
frame o f r e f e r e n c e o f p a t h a n g l e s a n d a i r c r a f t c o u r s e s .
If c o u r s e
i n s t r u m e n t s o f h i g h a c c u r a c y a r e u s e d , t h e m o s t a d v a n t a g e o u s frame
o f r e f e r e n c e f o r f l i g h t a n g l e s i s t h e s y s t e m o f t h e summation o f
t h e t u r n a n g l e s of t h e p a t h l i n e .
I n t h i s case, t h e path angle of
t h e f i r s t s e c t i o n i s assumed t o be e q u a l t o t h e azimuth o f t h e
orthodrome of t h e i n i t i a l p o i n t of t h e r o u t e ( f o r example, a t t h e
point of t h e aircraft's take-off).
The p a t h a n g l e o f e a c h s u c c e s ­
s i v e segment w i l l be

The t u r n a n g l e s o f t h e r o u t e a r e d e t e r m i n e d a s t h e d i f f e r e n c e
of t h e azimuths of t h e i n t e r s e c t i n g orthodromes a t t h e intermediate
points of t h e r o u t e .
The i n t e r m e d i a t e p o i n t s o f t h e o r t h o d r o m e s a r e d r a w n o n t h e
flight chart.
On t h e b a s i s o f t h e s e p o i n t s , t h e o r t h o d r o m e l i n e s
o f a p a t h o f g r e a t l e n g t h a r e d r a w n on t h e f l i g h t c h a r t .

All t h e i n f o r m a t i o n a b o u t t h e o r t h o d r o m e r o u t e m u s t b e t a b u ­
l a t e d i n a s p e c i a l t a b l e (Table 7.3).
Some o f t h e i n f o r m a t i o n m u s t
b e drawn d i r e c t l y on t h e f l i g h t c h a r t .
In addition t o information
a b o u t t h e o r t h o d r o m e , T a b l e 7 . 3 must i n c l u d e d a t a on t h e c o n t r o l
r e f e r e n c e p o i n t s or o t h e r c o r r e c t i n g p o i n t s (CP) a l o n g t h e s e g m e n t s
of t h e r o u t e .

0,34748
0,59318
0,72337

512

0,9376s
0,80507
0,69046

0,49796
0,60749
0,76050

26'25'
31"17'
37'15'

0,92395
0,75739
0,60496

22'29'
46'46'
52"46'

­

2032
1333

T h e o r t h o d r o m e c o o r d i n a t e s o f t h e c o r r e c t i n g p o i n t s may b e
determined a c c u r a t e l y on t h e b a s i s of (1.64 and 1 . 6 5 ) .
However,
taking i n t o account t h e fact t h a t t h e aircraftls coordinates with
r e s p e c t t o t h e s e p o i n t s a r e c o r r e c t e d a t d i s t a n c e s o f n o t more t h a n
3 0 0 - 4 0 0 km, t h e o r t h o d r o m e c o o r d i n a t e s o f t h e c o r r e c t i n g p o i n t s may
a l s o b e m e a s u r e d on t h e f l i g h t c h a r t .
To d o t h i s , t h e c o r r e c t i n g
p o i n t must b e p r o j e c t e d a c c u r a t e l y t o t h e p a t h l i n e by means o f a
protractor (Fig. 7.1).
It i s p r a c t i c a l l y always p e r m i s s i b l e t o measure t h e Z C a p .
o r d i n a t e o n t h e map b y m e a n s o f a s c a l e .
A s f o r t h e Xc.p. c o o r d i ­
n. a t e , i t c a n b e m e a s u r e d o n l y w h e n i + , i s s i t u a t e d n o t f a r f r o m t h e
turning point of t h e r o u t e .
I n o t h e r c a s e s , it must be c a l c u l a t e d
on t h e b a s i s o f t h e f o r m u l a a s t h e d i s t a n c e a l o n g t h e o r t h o d r o m e
t o the traverse of the correcting point.
Co

The a z i m u t h o f t h e o r t h o d r o m e r e l a t i v e t o t h e m e r i d i a n o f t h e
c o r r e c t i n g p o i n t i s d e t e r m i n e d when t h i s p o i n t i s a g o n i o m e t r i c o r
goniometric-rangefinding device and t h e azimuth of t h e a i r c r a f t i s
measured from i t .
The a b o v e a n g l e i s c a l c u l a t e d on t h e b a s i s o f t h e c o o r d i n a t e s
of t h e p o i n t of i n t e r s e c t i o n of t h e orthodrome with t h e meridian of
If t h e f l i g h t i s e x e c u t e d p a r a l l e l t o t h e
the correcting point.
m e r i d i a n o f t h e c o r r e c t i n g p o i n t or a t a s m a l l a n g l e t o t h e m e r i d i a n ,
t h e a z i m u t h o f t h e o r t h o d r o m e i s f i r s t d e t e r m i n e d on t h e b a s i s o f
t h e c o o r d i n a t e s o f t h e p o i n t o f t h e o r t h o d r o m e on t h e t r a v e r s e o f
/484
t h e c o r r e c t i n g p o i n t , and t h e convergence of t h e meridians between
t h e p o i n t of t h e orthodrome and t h e c o r r e c t i n g p o i n t i s taken i n t o
account.
W i t h l o n g o r t h o d r o m e s e c t i o n s o f t h e r o u t e , i t may b e n e c e s s a r y
t o c o r r e c t t h e a i r c r a f t ' s c o o r d i n a t e s w i t h r e s p e c t t o more t h a n one
correcting point.
I n t h i s case, t h e s e c t i o n of t h e r o u t e i s divided
i n t o t w o or t h r e e a n d s o m e t i m e s m o r e s e c t i o n s , o n t h e b a s i s o f t h e
i n t e r m e d i a t e p o i n t s of t h e orthodrome
being calculated; a separate l i n e of
Table 7.3 i s f i l l e d f o r each section.
Therefore, the turning angle of the
path l i n e a t t h e beginning of these
LGF'
Q
s e c t i o n s w i l l be z e r o and t h e f l i g h t
a n g l e i s common for a l l t h e s e c t i o n s .

Fig. 7.1.
Orthodromic
C o o r d i n a t e s o f t h e Correcting Point

From t h e i n f o r m a t i o n i n T a b l e 7 . 3 ,
i t i s a d v i s a b l e t o draw on t h e c h a r t
t h e numbers o f t h e s e c t i o n s o f t h e
r o u t e , and t h e orthodrome f l i g h t a n g l e s
and d i s t a n c e s .
I n a d d i t i o n , it i s
d e s i r a b l e t o draw on t h e c h a r t t h e
loxodromic magnetic f l i g h t angles
w i t h r e s p e c t t o t h e s e c t i o n s of t h e
r o u t e and t o mark.the p a t h l i n e with
513

dashes:
5 0 km.

n o r m a l o n e s w i t h n u m b e r i n g e v e r y 1 0 0 km a n d s h o r t o n e s e v e r y

S i n c e r e a d i n g c o o r d i n a t e s i n f l i g h t i s done from t h e i n i t i a l
p o i n t of e a c h s t r a i g h t s e c t i o n of t h e p a t h , t h e m a r k i n g w i t h r e s p e c t
t o d i s t a n c e m u s t a l s o b e g i n f r o m t h e t u r n i n g p o i n t ‘of t h e r o u t e
i n d e p e n d e n t l y f o r each s e c t i o n of t h e p a t h i n t h e forward and r e ­
verse directions.

3.

S t u d y i n g t h e Route a n d C a l c u l a t i n g a S a f e F l i g h t A l t i t u d e

J u s t as i n t h e p r e p a r a t i o n of f l i g h t c h a r t s , t h e procedure f o r
s t u d y i n g t h e f l i g h t r o u t e d e p e n d s on t h e n a t u r e o f t h e f l i g h t t o be
e x e c u t e d a n d t h e a i r c r a f t n a v i g a t i o n means b e i n g u s e d .

For s h o r t - d i s t a n c e v i s u a l f l i g h t s , t h e f o l l o w i n g m u s t b e c a r e ­
fully studied:
t h e r e l i e f , t h e presence of l i n e a r and areal r e f e r ­
ence p o i n t s and t h e i r a d d i t i o n a l c h a r a c t e r i s t i c s , t h e procedure f o r
r e - e s t a b l i s h i n g o r i e n t a t i o n when i t i s l o s t , t h e e q u i p m e n t a n d
c o n d i t i o n s f o r a p p r o a c h i n g t h e a i r p o r t s and l a n d i n g a r e a s , and s i t e s
w h i c h a r e s u i t a b l e for a f o r c e d l a n d i n g o u t s i d e a n a i r p o r t .
The s a f e f l i g h t a l t i t u d e i n t h i s c a s e i s c a l c u l a t e d r e l a t i v e
t o t h e l e v e l of t h e take-off a i r p o r t , separately f o r each s e c t i o n
of t h e r o u t e .
I n a d d i t i o n , s a f e f l i g h t a l t i t u d e s must be c a l c u l a t e d a l o n g t h e s e c t i o n s o f t h e r o u t e f o r c a s e s when d e v i a t i o n f r o m
t h e route is necessary i n order t o re-establish orientation.
I n c a l c u l a t i n g t h e safe f l i g h t a l t i t u d e , temperature c o r r e c t i o n s
f o r a l t i m e t e r r e s p o n s e s and t h e n e c e s s a r y a l t i t u d e a l l o w a n c e s above
o b s t a c l e s ( e s t a b l i s h e d f o r a g i v e n t y p e o f r e l i e f by t h e i n s t r u c ­
t i o n s on e x e c u t i n g f l i g h t s ) a r e t a k e n i n t o a c c o u n t .
I n d e t e r m i n i n g t h e s a f e f l i g h t a l t i t u d e on t h e b a s i s o f a n
i n s t r u m e n t , t h e f o l l o w i n g f o r m u l a may b e u s e d

Hs i n s t r = Hr

+

Hi

- H a i r + AHt

w h e r e Hs i n s t r i s t h e s a f e f l i g h t a l t i t u d e o n t h e b a s i s o f a n i n ­
s t r u m e n t ; Hr i s t h e maximum h e i g h t o f a r e l i e f o n t h e f l i g h t p a t h ;
H i i s t h e p e r m i s s i b l e f l i g h t a l t i t u d e above t h e r e l i e f i n accordance
with the requirement of t h e instructions f o r executing f l i g h t s ;
H a i r i s t h e a l t i t u d e o f t h e t a k e - o f f a i r p o r t above sea l e v e l ; and
AH$ r e p r e s e n t s t h e m e t h o d o l o g i c a l e r r o r i n m e a s u r i n g t h e a l t i t u d e
as a r e s u l t of t h e d i s c r e p a n c y between t h e a c t u a l a i r t e m p e r a t u r e
and t h e s t a n d a r d c o n d i t i o n s .
The r e d u c e d a t m o s p h e r i c p r e s s u r e a l o n g t h e s e c t i o n s o f t h e
r o u t e is not taken i n t o account i n t h i s case since v i s u a l f l i g h t s
a t low a l t i t u d e s a r e u s u a l l y e x e c u t e d o v e r a s m a l l d i s t a n c e and t h e
A s regards t h e reduced pres­
pressure changes w i t h i n small l i m i t s .
s u r e i n t h e v i c i n i t y of t h e t a k e - o f f a i r p o r t , it i s a u t o m a t i c a l l y
514

/48E

taken into account by setting the pointers of the altimeters t o

zero before the aircraft's take-off.

The methodological errors in measuring altitude as a result
o f the discrepancy between the actual air temperature and the stand­
ard conditions are taken into account by means of a navigational
slide rule, for example, the N S - 1 O M .

ExampZe.
The height o f the take-off airport above sea level
is 270 m. The air kmperature at the airport is + 1 8 O ; the maximum
height of the relief on the flight path is 680 m ; the permissible
flight altitude above the relief is 300 m and the temperature at
flight altitude is +12O. Determine the safe flight altitude on the
basis of an instrument.

Solution.

The safe flight altitude relative t o the take-off


airport is

Hs instr

= 680

+

300 - 2 7 0

The total air temperature to

+

+ AHt

= 710 m

tH = 1 8

+ 12

+ AHt

= 30°.

On the N S - 1 O M ruler let us find the safe flight altitude on
= 660 m.
the basis of an instrument (Fig. 7.2).
Answer:
H s instr
The nature of studying the flight path for aircraft navigation
with instruments at average flight altitudes changes significantly.
In this case, most of the attention i s devoted t o the use of air­
craft navigational devices on the path and also for the aircraft's
approach to the airport region and the approach for landing. The
relief and the visual reference point conditions on the flight path
are a l s o studied in great detail.
/486

The safe flight altitude in this case is calculated with
reference to standard conditions with an atmospheric pressure of

760 mm H g , taking into account the reduced atmospheric pressure and

the air temperature along sections of the path:


H

s760

where H

Hi + Hr

+

(760 - p

min

1-11 + AHt,

is the safe flight altitude on the basis of an instru­

s760

ment, established for a pressure of 760 mm Hg; Hi is the permissible


Pig. 7.2. Determining the
Safe Flight Altitude by
Instrument on the N S - 1 O M .

Fig. 7.3.
Determining the
Safe Flight Echelon on the
NS-1OM.

515

I


safe f l i g h t a l t i t u d e with instruments i n accordance with t h e require­
m e n t s o f t h e i n s t r u c t i o n s f o r e x e c u t i n g f l i g h t s ; a n d Hmin i s t h e
minimum a t m o s p h e r i c p r e s s u r e o n a s e c t i o n o f t h e f l i g h t p a t h .
I n t h i s case, moreover, t h e a l t i t u d e of t h e lowest safe f l i g h t
echelon i s determined, t a k i n g i n t o account t h e e s t a b l i s h e d system
of f l i g h t echeloning with respect t o a l t i t u d e s , ?.e., the reverse
problem i s solved.

Example.
T h e minimum a t m o s p h e r i c p r e s s u r e r e d u c e d t o s e a
l e v e l o n t h e f l i g h t p a t h is 7 5 0 mm Hg.
The h e i g h t o f t h e r e l i e f o n
t h e p a t h i s 1 3 0 0 m ; t h e a i r t e m p e r a t u r e o n t h e g r o u n d i s 35O, a n d
t h e a i r t e m p e r a t u r e a t f l i g h t a l t i t u d e i s 45O.
Determine whether an a l t i t u d e of 2100 m i s a safe a l t i t u d e f o r
t h e lowest f l i g h t e c h e l o n i f t h e t r u e f l i g h t a l t i t u d e above t h e
r e l i e f must be n o t l e s s t h a n 6 0 0 m .

Solution.
1. L e t u s d e t e r m i n e o n t h e n a v i g a t i o n a l s l i d e r u l e ,
t h e f l i g h t a l t i t u d e r e l a t i v e t o t h e s t a n d a r d l e v e l o f 7 6 0 mm Hg a t
H i n s t r = 2100 m y t a k i n g i n t o account t h e a i r temperature (Fig. 7.3).
Answer:
1750 m.
2.
L e t u s d e t e r m i n e t h e t r u e f l i g h t a l t i t u d e above t h e r e l i e f
t a k i n g i n t o a c c o u n t t h e minimum r e d u c e d a t m o s p h e r i c p r e s s u r e ( p m i n
- 7 6 0 ) - 1 1 = -110 m and t h e h e i g h t of t h e r e l i e f a l o n g t h e p a t h .

Htr

= 1750 - 1 1 0 - 1 3 0 0 = 340 m .

Thus, Hinstr
= 2100 m w i l l n o t be a safe a l t i t u d e f o r t h e
lowest f l i g h t e c h e l o n , s i n c e t h e t r u e a l t i t u d e above t h e r e l i e f i s
e q u a l t o o n l y 340 m .
The s a f e a l t i t u d e of t h e l o w e s t f l i g h t e c h e l o n
= 2400 m.
i n t h i s c a s e w i l l be H i n s t r
The s t u d y o f t h e r o u t e s f o r f l i g h t s o n h i g h - s p e e d
i s of an e n t i r e l y d i f f e r e n t n a t u r e .

j e t aircraft

I n t h i s case, t h e f l i g h t a l t i t u d e i s chosen e x c l u s i v e l y f o r
r e a s o n s o f f l i g h t economy a n d t h e r e l a t i v e l o c a t i o n o f a i r c r a f t
/48i
moving i n t h e o t h e r d i r e c t i o n , t a k i n g i n t o a c c o u n t t h e m e t e o r o l o g i c a l
situation with respect t o a l t i t u d e .
The s a f e f l i g h t a l t i t u d e a b o v e t h e r e l i e f i s s i g n i f i c a n t o n l y
u n d e r c o n d i t i o n s of t h e a i r c r a f t ' s c l i m b a n d d e s c e n t .

For example, t h e p o s s i b i l i t y of overcoming o b s t a c l e s d u r i n g
climb i s determined, t a k i n g i n t o account t h e a i r s p e e d , t h e climbing
r a t e o f t h e a i r c r a f t , a n d t h e c o u n t e r or i n c i d e n t a l c o m p o n e n t o f
t h e wind s p e e d .
If t h e r e s e r v e a l t i t u d e a b o v e t h e r e l i e f i s i n s u f ­
f i c i e n t , p a r t o f t h e f l i g h t a l t i t u d e must be g a i n e d a f t e r t a k e o f f
by c i r c l i n g above t h e a i r p o r t .

516

The distance and vertical speed of descent from the given flight

echelon are determined analogously, taking into account the obstacles

on the approaches t o the landing airport. When the vertical rate

of descent is more than is permissible, the approach t o the airport

is executed at a certain safe altitude which is later lost in the

landing approach maneuver.

In studying the flight path f o r jet aircraft, special attention

must be focused on the possibility of using radio-engineering de­

vices for aircraft navigation, the typical radar reference points,

conditions for take-off and climb along the path, conditions of the

approaches t o the landing airports and.maneuvering before landing.


4. S p e c i a l P r e p a r a t i o n o f C h a r t s and Aids f o r U s i n g
V a r i o u s N a v i g a t i o n a l Devices i n F l i g h t .
The special preparation of charts and aids depends on the nature

of the navigational devices which are supposed t o be used in the

flight.

In using goniometric and goniometric-rangefinding devices, it

is sufficient t o note on the chart the points where these devices

are located, and the calculated directions and distances from them

t o the most typical points on the flight path (turning points of

the path, control reference points, points of the beginning of the

descent, and points of the entrance and exit corridors, etc.).

In using hyperbolic navigational systems, it is necessary to

have special charts with hyperbolic position lines plotted on them.

Before flight, the path of the forthcoming flight is drawn.

In using autonomous Doppler aircraft navigational systems with

automatic navigational devices, several things must be prepared

carefully: the flight angles and distances, as well as the ortho­

drome coordinates of the visual and radar reference points, the

azimuth-rangefinding devices and the azimuths of the orthodromes

relative to the meridians of the locations of the orthodromes.

If astronomical devices are supposed t o be used in flight,
special astronomical tables must be prepared: detached sheets from
the astronomical yearbook for the flight date, tables of azimuths

and altitudes of stars, special calculating tables for recording

and calculating astronomical parameters.


/488

The navigational equipment of the aircraft must be prepared

and checked accordingly.

During the inspection of this equipment, attention is focused

on its general condition, efficiency, regulating parameters of

currents and voltage potentials, as well as the calibration of

measured parameters according t o instructions on the use of specific

types of equipment.

517


I n u s i n g a s t r o n o m i c a l d e v i c e s , s p e c i a l a t t e n t i o n must a l s o be
f o c u s e d on i n s p e c t i n g t h e a i r c r a f t ' s c l o c k s a n d s p e c i a l c h r o n o m e t e r s .

5.

C a l c u l a t i n g t h e D i s t a n c e and D u r a t i o n o f F l i g h t

The d i s t a n c e a n d d u r a t i o n o f t h e f l i g h t a r e ' c a l c u l a t e d i n o r d e r
t o determine t h e n e c e s s a r y f u e l s u p p l y , as w e l l as monitoring i t s
c o n s u m p t i o n i n f l i g h t a n d d e t e r m i n i n g t h e maximum d i s t a n c e o f t h e
p o i n t o f p o s s i b l e r e t u r n t o t h e t a k e - o f f a i r p o r t or t h e a l t e r n a t e
a i r p o r t s when t h e w e a t h e r i s c h a n g i n g a t t h e l a n d i n g a i r p o r t .
Depending on t h e t y p e s of a i r c r a f t , a n d p r i m a r i l y on t h e t y p e
o f p r o p u l s i o n and a l t i t u d e o f t h e a i r c r a f t , one of t h r e e methods
f o r c a l c u l a t i n g t h e d i s t a n c e and d u r a t i o n of f l i g h t i s used:
(a)
For a i r c r a f t with low-altitude p i s t o n engines, with re­
s p e c t t o t h e hourly f u e l consumption a t c r u i s i n g speed.
(b)
For aircraft with high-altitude p i s t o n engines (with
s u p e r c h a r g e r s ) , with r e s p e c t t o t h e hourly f u e l consumption under
t h e m o s t a d v a n t a g e o u s s e l e c t e d c o n d i t j . o n s o f s u p e r c h a r g i n g a n d rpm
of t h e e n g i n e s a t a g i v e n a l t i t u d e and a i r s p e e d .
(c)
For a i r c r a f t w i t h j e t and t u r b o p r o p e n g i n e s , w i t h r e s p e c t
t o t h e c o n s u m p t i o n o f f u e l p e r km a t d i f f e r e n t v a l u e s o f t h e f l y i n g
weight of t h e a i r c r a f t and t h e a l t i t u d e and speed of f l i g h t .
C a l c u l a t i n g t h e F u e l S u p p l y f o r F l i g h t on A i r c r a f t
w i t h Low-Altitude P i s t o n E n g i n e s
On a i r c r a f t w i t h l o w - a l t i t u d e p i s t o n e n g i n e s a n d f i x e d p i t c h
of t h e p r o p e l l e r i n f l i g h t , t h e c o n t r o l o f t h e e n g i n e s i s e x e c u t e d
o n l y by t h e f u e l - c o n t r o l l e v e r .
F l i g h t s on t h e s e a i r c r a f t a r e
e x e c u t e d a t low a l t i t u d e s , which makes i t p o s s i b l e i n c a l c u l a t i o n
not t o t a k e i n t o account t h e change i n f u e l consumption with t h e
/489
flight altitude.
The h o u r l y f u e l c o n s u m p t i o n ( Q , k g / h r ) i s a p p r o x i ­
m a t e l y p r o p o r t i o n a l t o t h e c u b e of t h e a i r s p e e d ; t h e f u e l consump­
t i o n p e r k i l o m e t e r o f a i r s p e e d ( 9 , kg/km) i s a p p r o x i m a t e l y p r o p o r ­
t i o n a l t o the square o f the airspeed.
S i n c e t h e f u e l c o n s u m p t i o n p e r km
s h a r p l y d e p e n d i n g on a i r s p e e d , t a k i n g i
o f t h e wind s p e e d t o t h e a i r s p e e d , t h e
customarily calculated with respect t o
given airspeed.
I n t h i s case,

w

=
G

518


vtr +

req

uz;

= Q*t

+

on t h e s e a i r c r a f t c h a n g e s
n t o account t h e large r a t i o
necessary f u e l supply is
t h e hourly consumption a t a

S

t = w
Gn,r

,

where W i s t h e g r o u n d s p e e d of t h e a i r c r a f t ; Vtr
is the airspeed
( a s a r u l e i t i s assumed t o b e e q u a l t o t h e s p e e d on t h e b a s i s o f
an i n s t r u m e n t ) ; ux i s t h e i n c i d e n t a l (encountered w i t h a minus s i g n )
component of t h e wind s p e e d ; S i s t h e d i s t a n c e a l o n g t h e r o u t e ; t i s
t h e f l y i n g t i m e ; Greq
i s t h e necessary f u e l supply; and G n . r .
is the
e s t a b l i s h e d n a v i g a t i o n a l supply ( r e s e r v e over t h e c a l c u l a t e d amount).
Combining t h e above f o r m u l a s , w e o b t a i n :

ExampZe.
The a i r s p e e d i s 1 5 0 k m / h r ; t h e d i s t a n c e a l o n g t h e
r o u t e i s 4 8 0 km; t h e i n c i d e n t a l w i n d c o m p o n e n t o n t h e r o u t e i s 3 0
km/hr; t h e h o u r l y f u e l consumption i s 30 k g / h r which i s t h e e s t a b ­
l i s h e d n a v i g a t i o n a l s u p p l y for 1 h r o f f l i g h t .
D e t e r m i n e t h e r e q u i r e d amount o f f u e l t o e x e c u t e t h e f l i g h t .

Solution.

It i s c h a r a c t e r i s t i c f o r aircraft of t h i s type t h a t t h e addi­
t i o n a l f u e l consumption due t o climb i s n o t t a k e n i n t o a c c o u n t .

Obviously, proceeding from t h e energy e q u a t i o n s ,
f u e l c o n s u m p t i o n a s a r e s u l t o f c l i m b GH must b e :

the additional

GH = q K H
where q i s t h e f u e l c o n s u m p t i o n p e r k i l o m e t e r of t h e p a t h i n h o r i ­
z o n t a l f l i g h t ; K i s t h e aerodynamic q u a l i t y of t h e a i r c r a f t under
c o n d i t i o n s o f c l i m b K = cy/cx, w h e r e c y i s t h e c o e f f i c i e n t o f l i f t
and cxis t h e c o e f f i c i e n t of d r a g ) ; and H i s t h e g i v e n f l i g h t
altitude.

For e x a m p l e , i n o u r c a s e , i f t h e g i v e n f l i g h t a l t i t u d e i s 3 0 0
m a n d t h e q u a l i t y of t h e a i r c r a f t i s 1 5 , w e w i l l h a v e :

w h i c h r e p r e s e n t s t h e f u e l c o n s u m p t i o n for 4 . 5
i n our case, 0.6% of t h e t o t a l f u e l supply.

km o f t h e a i r p a t h ,

or

However, t a k i n g i n t o a c c o u n t t h e f a c t t h a t t h e a i r c r a f t ’ s
d e s c e n t i n v o l v e s a s a v i n g o f f u e l , t h o u g h somewhat l e s s t h a n t h e

519


/490

Ln

10

0


“760

m
T
­
0

Fig. 7.4.

1000
I

7000

3000

4000

5000

6000u

E n g l n e r Ash-82FN

Graph f o r Determining t h e Operating Conditions of Engines w h e n Cruising.


loss during climb ( a s a r e s u l t of t h e lower q u a l i t y of t h e aircraft
under descent c o n d i t i o n s ) , t h e t o t a l overexpenditure of f u e l under
c l i m b and d e s c e n t c o n d i t i o n s i s i n s i g n i f i c a n t on t h e whole and i s
disregarded.
Calculating

t h e Fuel Supply for Flight i n Aircraft w i t h
High-Altitude Piston Engines

The means o f c a l c u l a t i n g t h e n e c e s s a r y f u e l s u p p l y f o r f l i g h t s
on a i r c r a f t w i t h h i g h - a l t i t u d e p i s t o n e n g i n e s c o i n c i d e s i n p r i n c i p l e
w i t h t h e means f o r c a l c u l a t i n g i t f o r a i r c r a f t w i t h l o w - a l t i t u d e
engines.
The d i s t i n g u i s h i n g f e a t u r e of t h e c a l c u l a t i o n i s t h e
n e c e s s i t y f o r a p r e l i m i n a r y estimate o f t h e most advantageous oper­
a t i n g c o n d i t i o n s o f t h e e n g i n e s a t a g i v e n a l t i t u d e and f l i g h t
speed ( s u p e r c h a r g i n g , t u r n s ) and a l s o a d e t e r m i n a t i o n of t h e h o u r l y
f u e l consumption under t h e s e c o n d i t i o n s .
I n F i g u r e 7 . 4 , we h a v e p r o v i d e d a g r a p h f o r d e t e r m i n i n g t h e
optimum o p e r a t i n g c o n d i t i o n s f o r e n g i n e s on t h e 1 1 - 1 2 a i r c r a f t .
Usually, i n a d d i t i o n t o graphs f o r each type of a i r c r a f t , t a b u l a r
d a t a f o r t h e optimum o p e r a t i n g c o n d i t i o n s o f e n g i n e s and f o r f u e l
consumption a t v a r i o u s a l t i t u d e s and f l i g h t s p e e d s a r e g i v e n .
A f t e r determining t h e hourly f u e l consumption f o r a given a l t i ­
t u d e a n d s p e e d on t h e b a s i s o f t h e g r a p h , t h e f u e l s u p p l y i s c a l c u ­
l a t e d ( j u s t as f o r a i r c r a f t with low-altitude engines).
The a d d i ­
t i o n a l t o t a l overconsumption of f u e l due t o climb and d e s c e n t
conditions i s disregarded, s i n c e it i s i n s i g n i f i c a n t i n t h i s case.
Calculating

t h e F u e l S u p p l y f o r F l i g h t on A i r c r a f t w i t h
Gas T u r b i n e E n g i n e s

The m e a n s f o r c a l c u l a t i n g t h e n e c e s s a r y f u e l s u p p l y f o r f l i g h t s
on a i r c r a f t w i t h g a s t u r b i n e e n g i n e s d i f f e r somewhat f r o m t h e means
used f o r a i r c r a f t with p i s t o n engines.
With an i n c r e a s e i n t h e a i r s p e e d i n a i r c r a f t w i t h g a s t u r b i n e
e n g i n e s , t h e i r o p e r a t i n g c o n d i t i o n s a r e improved ( t h e v o l u m e t r i c
efficiency increases).
I n a d d i t i o n , an increase i n t h e f l i g h t
s p e e d i s d u e t o t h e d e c r e a s e i n t h e a n g l e of a t t a c k o f t h e wing
( a n d t h e r e f o r e t h e c o e f f i c i e n t o f d r a g ex).
T h e s e f a c t o r s a r e o f g r e a t s i g n i f i c a n c e on a i r c r a f t w i t h g a s
t u r b i n e e n g i n e s ; up t o c e r t a i n l i m i t s , t h e y n o t o n l y c o m p e n s a t e f o r
t h e n a t u r a l i n c r e a s e i n t h e f u e l c o n s u m p t i o n on t h e b a s i s o f t h e
square of t h e a i r s p e e d , but exceed i t .
A S a r e s u l t , on t h e s e a i r ­
c r a f t ( w i t h a c o n s t a n t f l y i n g w e i g h t ) t h e f u e l c o n s u m p t i o n p e r km
over t h e e n t i r e range of c r u i s i n g speeds remains p r a c t i c a l l y constant
(with an accuracy of 2 t o 3%).
Beyond t h e l i m i t s o f t h e r a n g e o f c r u i s i n g s p e e d s , e s p e c i a l l y
i f t h e maximum s p e e d o f t h e a i r c r a f t i s c l o s e t o t h e s p e e d o f

521


/492

s o u n d , t h e f u e l c o n s u m p t i o n p e r km i n c r e a s e s s h a r p l y .
T h e minimum f u e l c o n s u m p t i o n p e r km o n a i r c r a f t w i t h g a s t u r b i n e
e n g i n e s c o r r e s p o n d s t o t h e mean c r u i s i n g s p e e d a n d i n c r e a s e s s l o w l y
w i t h a change i n t h e f l i g h t c o n d i t i o n s , b o t h i n t h e d i r e c t i o n s of
an i n c r e a s e and a decrease of speed ( w i t h i n t h e l i m i t s of t h e c r u i s ­
ing speed).

If w e a s s u m e t h a t t h e f u e l c o n s u m p t i o n p e r km w i t h i n t h e l i m i t s
of t h e r a n g e o f c r u i s i n g s p e e d s of t h e a i r c r a f t i s p r a c t i c a l l y
c o n s t a n t , t h e n t h e h o u r l y f u e l consumption w i l l change i n p r o p o r t i o n
t o t h e f l i g h t speed.
The c o m p a r a t i v e l y c l e a r l y e x p r e s s e d s t e a d i n e s s o f t h e f u e l
c o n s u m p t i o n p e r km o n a i r c r a f t w i t h g a s t u r b i n e e n g i n e s , o v e r t h e
e n t i r e r a n g e o f c r u i s i n g s p e e d s w i t h a v a r i a b l e h o u r l y f u e l con­
s u m p t i o n , makes i t n e c e s s a r y t o c a l c u l a t e t h e f u e l s u p p l y on t h e
b a s i s of t h e consumption p e r hour r a t h e r than p e r kilometer.
This
m u s t b e d o n e on t h e b a s i s o f o t h e r c o n s i d e r a t i o n s .
In contrast t o aircraft with piston engines, t h e f u e l supply
on g a s t u r b i n e a i r c r a f t r e p r e s e n t s a l a r g e p a r t o f t h e f l y i n g w e i g h t
o f t h e a i r c r a f t ( o n i n d i v i d u a l t y p e s o f a i r c r a f t , i t may b e c l o s e
t o half t h e flying weight).
This causes g r e a t changes i n t h e f l y i n g
weight and t h e r e f o r e i n t h e f u e l consumption w i t h r e s p e c t t o t h e
aircraft's flying t i m e .
While i t i s c o m p a r a t i v e l y s i m p l e t o c a l c u l a t e t h e f u e l on t h e
b a s i s o f t h e v a r i a b l e ( d e p e n d i n g on f l y i n g w e i g h t ) f u e l c o n s u m p t i o n
p e r km, i t i s v e r y d i f f i c u l t t o c a l c u l a t e it on t h e b a s i s o f t h e
h o u r l y e x p e n d i t u r e , which v a r i e s b o t h on t h e b a s i s o f t h e f l y i n g
weight and t h e a i r s p e e d .
I n c o n s t r u c t i n g t h e g r a p h o f t h e f u e l c o n s u m p t i o n p e r km o n
a i r c r a f t w i t h g a s t u r b i n e e n g i n e s as a f u n c t i o n o f t h e a l t i t u d e and
a i r s p e e d , t h e r e d u c e d w e i g h t of a n a i r c r a f t (Gre+) which h a s t h e
following p h y s i c a l s t r u c t u r e i s s e l e c t e d as t h e i n i t i a l argument.
L e t u s assume t h a t t h e f l i g h t i s e x e c u t e d a t s p e e d s a t which
t h e aerodynamic d r a g of t h e a i r c r a f t s a t i s f i e s t h e Bernoulli l a w ,
then
/493

P v*
x=c,s---.
2 '

Y = c y S - -P -VT2,
2

w h e r e X i s t h e a i r c r a f t ' s d r a g ; Y i s t h e l i f t o f t h e w i n g ; ex a n d
cy a r e t h e c o e f f i c i e n t s o f d r a g a n d l i f t ; S r e p r e s e n t s t h e c r o s s ­

522

I111111II

I

I

I 1

I

s e c t i o n a l areas of t h e a i r c r a f t which r e p r e s e n t p l a n e s ; and p i s
t h e d e n s i t y of t h e a i r a t f l i g h t a l t i t u d e .
Obviously, t h e h o r i z o n t a l f l i g h t of t h e aircraft a t various
a l t i t u d e s , w i t h t h e same v a l u e s o f f l i g h t s p e e d a n d a n g l e o f a t t a c k
o f t h e a i r c r a f t ( e = c o n s t and V = c o n s t ) w i l l be p o s s i b l e o n l y
when t h e f l y i n g w e i g h t o f t h e a i r c r a f t a n d t h e a i r d e n s i t y a r e
changed i n e q u a l p r o p o r t i o n .

For e x a m p l e , w i t h t h e g i v e n v a l u e s o f t h e s p e e d a n d
a t t a c k , if h o r i z o n t a l f l i g h t n e a r t h e ground i s p o s s i b l e
f l y i ' n g w e i g h t o f 6 0 t o n s , t h e n a t a n a l t i t u d e w h e r e p~ =
h o r i z o n t a l f l i g h t i s p o s s i b l e w i t h a f l y i n g w e i g h t o f 30

angle of
with a
0 . 5 ~ 0 ,

tons.

H o w e v e r , a t t h e f l i g h t a l t i t u d e i n our e x a m p l e , t h e d r a g i s
2 times less t h a n near t h e ground because t h e f u e l consumption f o r
each t o n of f l y i n g weight remains constant.
Thus, i n o r d e r t o reduce t h e f l y i n g weight of t h e a i r c r a f t t o
standard atmospheric conditions i n order t o determine t h e f u e l
consumption p e r k i l o m e t e r f o r each t o n of f l y i n g weight ( a c t u a l ,
n o t r e d u c e d ) it i s s u f f i c i e n t t o m u l t i p l y t h e a c t u a l f l y i n g weight
by t h e p r o p o r t i o n p o / p
H'
I n o u r example, t h e a c t u a l weight of t h e a i r c r a f t a t an a l t i ­
t u d e i s 30 t o n s ; t h e r a t i o p 0 / p H = 2 ; t h e r e f o r e , Gred = 3 0 . 2 = 6 0
tons.
I f , w i t h a w e i g h t o f 6 0 t o n s t h e f u e l c o n s u m p t i o n a t a mean
c r u i s i n g s p e e d i s 0 . 2 kg p e r t o n o f a c t u a l f l y i n g w e i g h t , t h i s means
t h a t n e a r t h e g r o u n d t h e c o n s u m p t i o n p e r km i s 0 . 2 0 6 0 = 1 2 k g / k m ,
a n d a t a n a l t i t u d e w h e r e p~ = 0 . 5 p o i t w i l l b e 6 k g / k m .
Therefore,
f o r a n g l e s o f a t t a c k o f a n a i r c r a f t which a r e s m a l l e r t h a n t h e
angle of t h e g r e a t e s t q u a l i t y , i . e . , over t h e e n t i r e range of t h e
a i r c r a f t ' s c r u i s i n g speed with an i n c r e a s e i n t h e reduced weight,
t h e f u e l consumption p e r t o n of a c t u a l weight w i l l d e c r e a s e .

A c t u a l l y , t h e c o e f f i c i e n t Cy, with an i n c r e a s e i n t h e a n g l e of
a t t a c k w i t h i n t h e a b o v e l i m i t s , w i l l g r o w much m o r e r a p i d l y t h a n
Cx a n d t h e s i g n i f i c a n t i n c r e a s e i n t h e w e i g h t o f t h e a i r c r a f t l e a d s
t o a r e l a t i v e l y smaller increase i n t h e drag and, t h e r e f o r e , i n t h e
f u e l consumption p e r k i l o m e t e r .
Therefore, with an increase i n
f l i g h t a l t i t u d e , when t h e r e d u c e d w e i g h t g r o w s w i t h o u t a n i n c r e a s e
/494
i n t h e a c t u a l w e i g h t o f t h e a i r c r a f t , t h e f u e l c o n s u m p t i o n p e r km
decreases significantly.
I n a d d i t i o n , t h e Mach n u m b e r i s u s e d i n c o n s t r u c t i n g g r a p h s o f
t h e f u e l c o n s u m p t i o n p e r km o n a i r c r a f t w i t h g a s t u r b i n e e n g i n e s .
The method f o r c a l c u l a t i n g t h e f u e l c o n s u m p t i o n on t h e b a s i s
o f t h e r e d u c e d f l y i n g w e i g h t a n d t h e Mach n u m b e r i s v e r y a d v a n t a g e o u s ,
s i n c e it p e r m i t s a l l t h e c h a r a c t e r i s t i c s o f f u e l consumption a l o n g

523

I


t h e a i r p a t h over t h e e n t i r e range of speeds, a l t i t u d e s , and f l y i n g
w e i g h t s o f t h e a i r c r a f t t o b e r e p r e s e n t e d on a t w o - d i m e n s i o n a l
g r a p h or t a b l e .
W i t h o u t t h e r e d u c e d w e i g h t , a t h r e e - d i m e n s i o n a l g r a p h or t a b l e
w o u l d h a v e b e e n t h e minimum r e q u i r e m e n t f o r t h i s .
Since such a
g r a p h must b e v o l u m e t r i c r a t h e r t h a n f l a t , it c a n n o t b e r e p r e s e n t e d
on p a p e r a n d i t must b e r e p l a c e d by a s e r i e s o f g r a p h s wh i ch h a v e
been c a l c u l a t e d , f o r example, f o r each constant value of f l i g h t
a l t i t u d e w i t h v a r i a b l e f l y i n g w e i g h t s a n d s p e e d s or o n t h e b a s i s o f
any o t h e r c o n s t a n t v a l u e w i t h two o t h e r v a r i a b l e m a g n i t u d e s .
T h i s i s e x a c t l y what i s done i n c o m p i l i n g o p e r a t i n g t a b l e s f o r
c a l c u l a t i n g t h e d i s t a n c e a n d d u r a t i o n o f f l i g h t on a i r c r a f t w i t h
gas turbine engines.
Using t h e g e n e r a l g r a p h c o n s t r u c t e d on t h e b a s i s o f t h e r e d u c e d
f l y i n g w e i g h t a n d t h e Mach n u m b e r a s t h e i n i t i a l g e n e r a l i z e d s o u r c e / 4 9 5
t a b l e s o f t h e f u e l consumption as a f u n c t i o n of t h e f l y i n g weight
and t h e f l i g h t a l t i t u d e with a given a i r speed are compiled.
Here it i s
consumption f o r
consumption p e r
a l m o s t t h e same

often s u f f i c i e n t t o c a l c u l a t e t h e t a b l e of f u e l
one (mean) f l i g h t s p e e d , b e a r i n g i n mind t h a t t h e
km o v e r t h e e n t i r e r a n g e o f c r u i s i n g s p e e d s i s
a s t h e f u e l c o n s u m p t i o n a t t h e mean s p e e d .

I n any case, f o r a i r c r a f t w i t h g a s t u r b i n e e n g i n e s and w i t h
g r e a t r a n g e , it i s s u f f i c i e n t t o compile t a b l e s of f u e l consumption
F.or t h e o t h e r
f o r n o t more t h a n t h r e e f l i g h t c r u i s i n g s p e e d s ,

G
instr

Fig.

524

7.5.

Graph o f t h e F u e l Consumption/km

on a J e t A i r c r a f t

s p e e d s , t h e f u e l c o n s u m p t i o n may b e i n t e r p o l a t e d when n e c e s s a r y .
The f u e l c o n s u m p t i o n f o r t h e g i v e n a i r s p e e d i s d e t e r m i n e d i n
c o m p i l i n g t h e t a b l e s o n t h e b a s i s o f t h e Mach n u m b e r f o r t h i s s p e e d ,
which c o r r e s p o n d s t o t h e s t a n d a r d t e m p e r a t u r e a t f l i g h t a l t i t u d e .
In practice, a t high f l i g h t a l t i t u d e s t h e a c t u a l a i r temperature
d i f f e r s s h a r p l y from t h e s t a n d a r d t e m p e r a t u r e ( b y more t h a n 10-15O).
I n t h e above r a n g e s o f t e m p e r a t u r e c h a n g e , t h e h o u r l y f u e l consump­
t i o n c h a n g e s somewhat ( w i t h a n i n c r e a s e i n t e m p e r a t u r e , i t i n c r e a s e s ;
with a decrease i n temperature, it decreases).
However, t h e t r u e
a i r s p e e d o f f l i g h t a t a c o n s t a n t Mach n u m b e r c h a n g e s i n a p p r o x i m a t e l y
t h e same p r o p o r t i o n .
This causes a r e l a t i v e steadiness i n the f u e l
consumption p e r k i l o m e t e r .
In Figure 7.5, w e have p l o t t e d a g e n e r a l graph of t h e f u e l
consumption p e r k i l o m e t e r of a c i v i l a i r c r a f t .
Tables 7.4 and 7.5
r e p r e s e n t t h e t o t a l f u e l c o n s u m p t i o n p e r f l i g h t f o r t h e same a i r ­
c r a f t and t h e f u e l consumption w i t h r e s p e c t t o s t a g e s of t h e f l i g h t .
We c o m p i l e d t h e s e t a b l e s f o r a mean c r u i s i n g s p e e d o f 8 0 0 k m / h r a n d
a mean f l y i n g w e i g h t o f 6 2 t o n s , b e a r i n g i n m i n d t h a t t h e f u e l c o n ­
s u m p t i o n p e r km f o r a n a i r c r a f t o v e r t h e e n t i r e r a n g e o f c r u i s i n g
s p e e d s d i f f e r s b y n o t m o r e t h a n 2 % f r o m t h e mean a n d t h e f l y i n g
w e i g h t a t d i f f e r e n t s t a g e s o f h o r i z o n t a l f l i g h t d o e s n o t d i f f e r by
n o t more t h a n 5 t o n s .

Example.
T h e d i s t a n c e a l o n g t h e p a t h i s 2 2 0 0 km;
the flight
a l t i t u d e is 10,000 m ; t h e adjusted navigational f u e l supply is 5.4
t o n s , and t h e wind component a t t h e f l i g h t a l t i t u d e i s minus 6 0
km/hr.
D e t e r m i n e t h e amount o f f u e l n e c e s s a r y t o e x e c u t e t h e f l i g h t
and t h e c a l c u l a t e d f u e l s u r p l u s a t t h e f o l l o w i n g s t a g e s of f l i g h t :
a t t h e end o f t h e c l i m b , a t d i s t a n c e s from t h e t a k e - o f f p o i n t o f
8 0 0 a n d 1 6 0 0 km, a t t h e b e g i n n i n g o f d e s c e n t , a n d d u r i n g t h e l a n d i n g
of t h e a i r c r a f t .
SOlUtiOn.
On t h e b a s i s o f T a b l e 7 . 4 , f o r a d i s t a n c e o f 2 2 0 0
km, a f l i g h t a l t i t u d e of 1 0 , 0 0 0 m and a head wind, l e t u s f i n d t h e
amount o f f u e l r e q u i r e d (Greq) t o e x e c u t e t h e f l i g h t , a d d i n g t o i t
t h e adjusted navigational supply of 5.4 t o n s :
15.5 t (0.62.2) t
5 . 4 = 22.14 t o n s .

On t h e b a s i s o f T a b l e 7 . 5 f o r t h e same c o n d i t i o n s , l e t u s f i n d :
(a)
253 km).
(b)

The a i r c r a f t ' s p a t h d u r i n g t h e c l i m b ( S

cl

The f u e l c o n s u m p t i o n d u r i n g t h e c l i m b ( &

cl

= 275 - 22 =

= 3.95 t o n s ) .

525

TABLE 7.4
F l i g h t altitude, m

~

path
lengtl

_ . ,7000

I

SO00

I

9000

1

/496
~

1

10000

11000

F u e l c nsumption for t h e e x e c u t i n g f l i g h t

k$

- --

Grec A, T
T i

T

A, 7

?re


A, T
.

so0

7, z

900
1000
1100
1200
1300

8.E
9,:
10,c

1300

1500
1GOO
1700
1so0
1900
2000
2100
2200
2300
2400
2500
2G00
2700
2soo
2900
3000

12.1
12,s
13.5
14.2

0.2;
0,3C
0.34
0.37
0.41
0.44
0.47
0.51
3.34
3.55

14,s

3.61

15.5
1G.2
16.9
17,G
1s.3
19.0
19.7
20.4
21.1
21.7
22.4
23.1

3, G4
1, GS
1.71
1.75
). 7s
),S1
),S5
),SS
), 92
). 95

10.7
11.4

7.5

0.2:
0.28
0.31
0.34
0.37
0.41
0.44
12.2 0.47
12,s 0.50
13.5 0.53
14.1 0.57
14-7 0. GO
15.4 0.63
8.2
9. C
9, E
10.3
10.9
11.5

16.0

16.7
17.3
7.9
8.6

9.2

),98

9.9
!O, 5
!1,1

.02

!I ,S

0,GG
O.G9

0.73
0.76
0.79
3.s2
3.85
3,SO
3.92
3.95

7, E
5.1

s 17

- 9.3
9.9
10.5
11.1

11.7
12.3
12.9
13.5
14,l
14.7
15.3
15.9
16.5
17.1
17.7
1s.3
1S.9
19.5
20.1
20.7

0.2:
0.26
0.29
0.32
0.35
0.38
0.41
0.4.1
0.47

7,:
8. C
8.6
9.2
9,s
10.3
10.9
11.5
12.0
12.6
13.2

0.21
0.24
0.2i
0.30
0.33
0.33
0.39
0,42
0,45
0.45

14.3

7.4
8.0

8.5
9.1
9.6

0.53
0,5G
0.59
0.62

0.50
13,s 0,53

10.2
10.7
11.3
11.8
12.4
12.9
13.5

O,G5
0.GS
si71

5,5
6.0
6.6
7.2
7,s
5.3

0.5G
0.59
0.62

14.0
466
5.1

0.65

5.7
6.2
6.8
7.3
7.9
8.4
9.0
9*5

0.50

3.74
3.77

3.so

3.83
3,SG
1,SD

4.9

8.9

9,'5
0.0

3,GS
3.71
3.74
3.77
3,79
1.82
1.85

0.20
0,23
0.26
0.29

0.32
0.34
0.37
0.40
0,43
0,4G
0.45
0,51
0.54

0.57
0,g0
3.G2

3-65
3. GS
3.71
1.74
),76
1.79
1.52

Note.
I n t h e T a b l e , A i s t h e i n d i c a t e d c o r r e c t i o n for t h e
t o t a l f u e l c o n s u m p t i o n f o r e v e r y 30 km/hr o f t h e wind component:
f a v o r a b l e w i t h a minus s i g n and c o n t r a r y w i t h a p l u s s i g n .
(c)

P a t h o f t h e a i r c r a f t d u r i n g d e s c e n t S d e s = 2 2 0 - 20 = 2 0 0

(d)

F u e l consumption f o r d e s c e n t &des = 0.8

km;
tons.

The f u e l c o n s u m p t i o n on t h e s e c t i o n s o f h o r i z o n t a l f l i g h t i s :

0.58)

(a)

From 2 5 3 t o 8 0 0 k m , & = 5 5 3 ( 5 . 7 2

(b)

From 8 0 0 t o

+

1600 k m , Q = 8 0 0 ( 5 . 7 2

0.58)
t

= 3.44

0.58)

tons.

= 5.04

(c)
From 1 6 0 0 t o t h e b e g i n n i n g o f d e s c e n t Q = 4 0 0 ( 5 . 7 2
= 2.52 t o n s .
The f u e l c o n s u m p t i o n f o r t h e l a n d i n g a p p r o a c h m a n e u v e r Q

= 1 ton.

tons;
t

a PP

L e t u s t a b u l a t e t h e r e s u l t of t h e s o l u t i o n i n T a b l e 7 . 6 .

The a b o v e c a l c u l a t i o n p r o c e d u r e i s a c c e p t a b l e f o r a i r c r a f t w i t h
a f l i g h t d u r a t i o n o f 3 - 3 . 5 h r when t h e f u e l c o n s u m p t i o n i n h o r i ­
z o n t a l f l i g h t does n o t exceed 1 0 - 1 2 t o n s .

Note.
I n t h e Table, N i s t h e i n d i c a t e d f u e l consumption, path /497
t r a v e r s e d and f l i g h t t i m e f o r a s t a g e .
I n t h e Table, A is t h e in­
c r e a s e i n t h e p a t h a n d t h e f u e l c o n s u m p t i o n p e r km f o r e v e r y 30 km/
h r o f a . h e a d o r t a i l rrind.

T a k e - o f f and c l i m b f u e l consump­
tion, tons
t i m e , min
p a t h , km

2.7
14
145

Horizontal f l i g h t

f u e l c o n s u m p t i o n kg/km

6.9 0.346.4

.....................
......................
.......................

Descent
f u e l consumption, t o n s
t i m e , min
p a t h , km

.........

7

I

I

.........

-

-

I

-.
lso 3.0
16.5

I

I

0,35,720.295.5
I

I

I

0,42 - 0.55 - 0.6s - 0.S
11 - 14 - 17 - 20
110 5 140
G 180 s 220

......................
.......................

Landing approach maneuver
f u e l consumption, t o n s

.........

1

t

-

1

-

1

-


-

4.5
3.95
30
23
10 275 11 375

G

0.32

I

-

3.4
19
220

-

1

-

12


0,2&


I'

I


i

0.95 -

93 -

10 250 12


-

-

1

­

TABLE 7 . 6

Sections of t h e r o u t e

Before t a k e - o f f . . . . . . . . . . . . .
T a k e - o f f a n d Climb
2 5 3 - 8 0 0 km
8 0 0 - 1 6 0 0 km
1 6 0 0 - b e g i n n i n g of d e s c e n t .
Descent
Landing

.........
..............
.............

....................
....................

- -~
253
547
so0
400
200

-

3.95
3.44
5.032.52

0.so
1 .oo

22.14

IS. 19

14.75

9.71

7.19
6.39
5.39

527


PILOT NAVIGATOR GRAPH

Flight Route Moscow-Khabarovsk, 196 g

Aircraft No:
Navigator

Crew Commander
Aircraft Engineer


Fig. 7.6.

Pilot-Navigator Graph


For aircraft operating over long distances, calculating the

fuel consumption on the basis o f one mean flying weight of the air­

craft becomes entirely insufficient.

For such aircraft, it is advantageous t o compile a pilot-navi- / 4 9 9
gator table for checking the fuel consumption in flight. To do
this, a form of Table 7.7 is used. Data for the fuel consumption
at various altitudes and airspeeds as a function of the aircraft's
flying weight are entered into the table.
TABLE 7.7

_ .

Flight
Limits of
Vtr 750 km/hr
altitude,
flying

km
.weights, T

3

1-

4

1-

Vtr 800 km/hr

5

_L__

i - G-.__I__
I-

Vtr 850 km/hr


-7. .--_8

!GO-150

G

:50-140
140-130
130-120
120-1 10
I -.

I__._

-.--_­

lG0-150
150-140
140-1 30
130- 1 20
120-110

7

etc


I

Usually it is sufficient t o have a table with the altitude
indicated for every 1 k m , and the flying weight is indicated for
every 10 tons for the mean and for two other cruising speeds of the
flight. This is given in the table when the weight of the aircraft
changes in horizontal flight within the limits from 160 t o 110 tons
and the cruising speed changes from 750 - 850 km/hr.
In addition t o Table 7.7, to compile a pilot-navigator table,

data for the aircraft's path and the fuel consumption at stages of

the climb, descent and pre-landing maneuvering are necessary.

The most convenient form of the pilot-navigator table for

compilation and usage is given in Figure 7.6.

First, the curve of the change in flying weight of the air­

craft along the flight path is plotted on the graph:

(a)

During climb.


529


(b)
From t h e moment o f a p p r o a c h i n g t h e g i v e n a l t i t u d e t o t h e
c l o s e s t f l y i n g w e i g h t g i v e n i n Column 2 o f T a b l e 7 . 7 .
(c)
Later, after every 10 t o n s of f l y i n g weight along t h e
d i s t a n c e t r a v e r s e d , w i t h t h e i r development i n aFcordance with
Columns 4 , 6 a n d 8 i n T a b l e 7 . 7 ;
(d)

During t h e t i m e of descent and pre-landing

maneuvering.

After t h i s , a c u r v e p a r a l l e l t o t h e c u r v e of t h e change i n
f l y i n g w e i g h t a l o n g t h e f l i g h t p a t h i s p l o t t e d o n t h e g r a p h i n s u c h /500
a way t h a t t h e f u e l s u r p l u s ( i n m u l t i p l e s o f 1 0 t o n s ) c o i n c i d e s
with t h e thickened h o r i z o n t a l l i n e s of t h e graph g r i d .
This creates
conditions f o r conveniently measuring t h e f u e l s u r p l u s , with i t s
measurement a t 1 0 t o n i n t e r v a l s d i r e c t l y on t h e b a s i s o f t h e c u r v e .
For example. i f t h e f l y i n g weight of t h e a i r c r a f t of 1 4 0 t o n s
c o r r e s p o n d s t o a f u e l s u r p l u s e q u a l t o 36 t o n s , i t i s c o n v e n i e n t
t o p l a c e t h e c u r v e o f f u e l c o n s u m p t i o n 6 t o n s h i g h e r or 4 t o n s
lower t h a n t h e f l y i n g weight curve.
I n t h i s case, a l l t h e p o i n t s
f o r t h e f u e l surplus i n multiples of 1 0 t o n s , coincide with t h e
t h i c k e n e d l i n e s on t h e g r a p h g i v e n e v e r y 5 o t 1 0 t o n s .
I n a d d i t i o n t o t h e c u r v e s o f t h e change i n f l y i n g w e i g h t and
f u e l consumption along t h e f l i g h t p a t h of t h e a i r c r a f t , t h e f o l ­
l o w i n g h a v e b e e n p l o t t e d on t h e g r a p h :
t h e p r o f i l e of t h e f l i g h t
a n d t h e s l o p i n g g r i d of t h e wind a t f l i g h t a l t i t u d e f o r t r a n s f e r r i n g
t h e a i r p a t h o f t h e a i r c r a f t t o t h e a c t u a l p a t h , a s w e h a v e shown
i n Figure 7.6.
F o r c h e c k i n g t h e f u e l c o n s u m p t i o n i n f l i g h t , P o i n t 1 i s marked
on t h e t i m e s c a l e ; s i g h t i n g u p w a r d , t h e f o l l o w i n g c a n b e o b t a i n e d
i n succession:
t h e a i r p a t h t r a v e r s e d (2), t h e c a l c u l a t e d f u e l
surplus ( 3 1 , t h e c a l c u l a t e d w e i g h t o f t h e a i r c r a f t ( 4 ) a n d t h e
p o i n t o f i n t e r s e c t i o n w i t h t h e s l o p e l i n e o f t h e mean w i n d ( 5 ) .
T h i s p o i n t may b e s i g h t e d a l o n g t h e h o r i z o n t a l t o t h e l e f t t o
p o i n t 6 , showing t h e c a l c u l a t e d p a t h of t h e a i r c r a f t r e l a t i v e t o
t h e ground.
The a b o v e c a l c u l a t e d d a t a a r e c o m p a r e d w i t h t h e a c t u a l d a t a
Checking t h e
f o r t h e a i r c r a f t ' s p a t h and f o r t h e f u e l s u r p l u s .
f u e l consumption i n f l i g h t p r o p e r l y i n c l u d e s t h i s procedure.
Calculating t h e Greatest Distance o f the A i r c r a f t ' s Point
o f C l o s e s t Approach t o a R e s e r v e A i r p o r t
I n order t o guarantee t h e r e g u l a r i t y of f l i g h t s ,
sometimes p e r m i t t e d t o f l y w i t h o u t complete a s s u r a n c e
w i l l land at t h e designated a i r p o r t .
However, s u c h a
accepted only with t h e f u l l guarantee t h a t (depending
weather conditions) t h e p o s s i b i l i t y of t h e a i r c r a f t ' s
one o f t h e a l t e r n a t e a i r p o r t s ( i n c l u d i n g t h e t a k e - o f f
530

aircraft are
t h a t they
decision is
on t h e
landing a t
airport) is

r e l i a b l y ensured.
In these instances, the distance of the f l i g h t t o a point
w h e r e i t i s s t i l l p o s s i b l e t o make a f i n a l d e c i s i o n t o l a n d a t t h e
d e s i g n a t e d a i r p o r t or t o r e t u r n t o a n a l t e r n a t e a i r p o r t i s c a l c u ­
lated.
T h i s p o i n t i s c a l l e d t h e p o i n t of c Z o s e s t a p p r o a c h .
The p o i n t o f c l o s e s t a p p r o a c h mu s t b e l o c a t e d a t t h e t a k e o f f
a i r p o r t , a t such a d i s t a n c e from it t h a t t h e a i r p a t h of t h e a i r ­
c r a f t t o , t h i s p o i n t and back t o t h e t a k e o f f a i r p o r t does n o t exceed
t h e maximum f l i g h t d i s t a n c e d u r i n g a c a l m w i t h t h e s u r p l u s a d j u s t e d
n a v i g a t i o n a l f u e l s u p p l y on b o a r d .
L e t u s r e p r e s e n t t h e maximum a i r p a t h o f t h e a i r c r a f t b y S a p ,
and l e t u s assume t h a t t h e a i r c r a f t w i l l t r a v e l t o t h e p o i n t o f
c l o s e s t approach w i t h a t a i l wind, and back a g a i n w i t h a head wind.
Without t a k i n g i n t o account t h e a i r c r a f t ’ s p a t h during a t u r n t o
t h e r e v e r s e c o u r s e , �or c a l m f l i g h t c o n d i t i o n s t h e f o l l o w i n g e q u a ­
t i o n w i l l hold:

s

aP

=

+

where V i s t h e a i r s p e e d , t i i s t h e f l i g h t t i m e up t o t h e p o i n t o f
c l o s e s t a p p r o a c h , and t 2 i s t h e f l i g h t t i m e i n t h e r e v e r s e d i r e c ­
tion.
Since

where S p . c . a . i s t h e d i s t a n c e o f t h e p o i n t o f c l o s e s t a p p r o a c h ;

a n d uz i s t h e w i n d c o m p o n e n t a l o n g t h e p a t h l i n e .

We c a n w r i t e t h e f o r m u l a f o r S
i n t h e form:

aP

s a p =Ispca (\
= Spca

(

v 4 v-ux
I/
)=
v*- u x V t v2 $- u,v
V f U ,

v2 - u;

)

= SPCa(-)

2 v2



whence

Taking i n t o account t h e p a t h of t h e a i r c r a f t i n t h e t u r n t o
t h e reverse course ( L r ) , we f i n a l l y obtain:

531


/501

Obviously, t h e c o e f f i c i e n t 1

-

U2X

- and, t h e r e f o r e , t h e distance
V2

o f t h e p o i n t o f c l o s e s t a p p r o a c h w i l l b e maximum a t ux = 0 a n d w i l l
d i m i n i s h w i t h a n i n c r e a s e i n t h e wind component a l o n g t h e p a t h l i n e ,
independent of i t s s i g n .
It i s important t o note t h a t t h e distance of t h e point of
c l o s e s t approach f o r aircraft with gas t u r b i n e engines depends l i t t l e
on t h e a i r s p e e d w i t h i n t h e l i m i t s o f t h e r a n g e o f c r u i s i n g s p e e d s
o f t h e a i r c r a f t , s i n c e t h e d i s t a n c e o f t h e f l i g h t i n a calm c h a n g e s
insignificantly.
I n a d d i t i o n , w i t h a n i n c r e a s e i n t h e a i r s p e e d and
w i t h some i n c r e a s e i n t h e f u e l c o n s u m p t i o n p e r k i l o m e t e r of a i r

the coefficient 1 -

path,

5

i n c r e a s e s somewhat.

This a l s o

TI

s t a b i l i z e s t h e d i s t a n c e of t h e p o i n t of c l o s e s t approach.

For a s p e c i f i c a i r s p e e d , t h e c o e f f i c i e n t 1 -

.;
1

0
50
100
150
200
250

100,o
99. G
9s.5
96.5
94.0
91 .o

-

may b e e x V2
This simplifies t h e calculations s i g n i f i c a n t l y .

pressed i n percent.

ux, k m/hr

ULX

Multiplication

%

8 0 0 km/hr

it w i l l be:

f o r an a l t e r n a t e a i r p o r t
s i t u a t e d on t h e f l i g h t p a t h ,
t h e c a l c u l a t i o n i s performed
0.940
o n t h e b a s i s o f t h e same f o r m u 0,910
la.
Instead of a completely
calm d i s t a n c e , i n t h i s case
t h e calm d i s t a n c e f r o m t h e
a l t e r n a t e a i r p o r t i s t a k e n , t a k i n g i n t o a c c o u n t t h e amount of f u e l
remaining u n t i l t h e a l t e r n a t e a i r p o r t is reached,

6.

0.996
0,983
0,965

I

P r e - f l i g h t P r e p a r a t i o n and F l i g h t C a l c u l a t i o n

P r e - f l i g h t p r e p a r a t i o n i n p r a c t i c e i s always necessary.
How­
e v e r , i t s volume w i l l b e l e s s a s more p r o b l e m s a r e s o l v e d i n t h e
preliminary preparation.
Pre-flight
(a)

p r e p a r a t i o n i s c a u s e d by t h e n e e d t o :

Analyze t h e meteorological s i t u a t i o n p r i o r t o t a k e o f f ;

532


~.
... ...

. . . -.

..

/ 5 0~.
2

(b)
Introduce p o s s i b l e changes i n t h e operating r u l e s f o r
radio-engineering devices or f l i g h t conditions along t h e a i r route
and i n t h e v i c i n i t y of a i r p o r t s .
I n a d d i t i o n , b e f o r e t a k e - o f f t h e n a v i g a t i o n a l e q u i p m e n t on
b o a r d t h e a i r c r a f t m u s t b e s w i t c h e d on a n d i n s p e c t e d .
In studying t h e meteorological s i t u a t i o n along t h e f l i g h t path
a n d a t t h e l a n d i n g a i r p o r t , a t t e n t i o n must b e f o c u s e d on t h e c o r ­
r e s p o n d e n c e o f t h e a c t u a l or p r e d i c t e d w e a t h e r w i t h t h e e s t a b l i s h e d
minimum, t h e d i s t r i b u t i o n o f t h e w i n d s a t f l i g h t a l t i t u d e , t h e
l o c a t i o n o f z o n e s w i t h d a n g e r o u s w e a t h e r phenomena and c o n d i t i o n s
f o r avoiding them, as w e l l as t h e s t a t e of t h e weather a t t h e a l t e r ­
nate airports.
Calculating t h e f l i g h t consists of taking i n t o account t h e
wind d i s t r i b u t i o n a t f l i g h t a l t i t u d e , t h e l o c a t i o n o f z o n e s w i t h
d a n g e r o u s m e t e o r o l o g i c a l phenomena, a s w e l l a s t a k i n g i n t o a c c o u n t
t h e s i t u a t i o n of t h e a i r on t h a t d a y .
In c a l c u l a t i n g t h e f l i g h t , it i s necessary t o average t h e
e q u i v a l e n t wind i n o r d e r t o d e t e r m i n e t h e n e c e s s a r y a i r s p e e d t o
maintain the f l i g h t according t o schedule.
T h e p r e c i s e v a l u e o f t h e e q u i v a l e n t w i n d ( u e > may b e d e t e r m i n e d / 5 0 3
on t h e b a s i s o f t h e f o r m u l a

or, o n t h e b a s i s o f t h e r o u g h f o r m u l a ,
u3 = 1i C O S

u:!'

Y B - - - sin:! Y B
21.1

.

I n p r a c t i c e , however, with a s u f f i c i e n t degree of accuracy,
t h e h e a d - w i n d or t a i l - w i n d c o m p o n e n t o f t h e e q u i v a l e n t w i n d may b e
t a k e n as t h e e q u i v a l e n t wind:

u e z u c o s WA
Averaging t h e e q u i v a l e n t wind a l o n g t h e e n t i r e r o u t e i s done
on t h e b a s i s o f t h e f o r m u l a :

533

The n e c e s s a r y c o n s t a n t a i r s p e e d f o r t r a v e l i n g a c c o r d i n g t o
schedule is then e a s i l y determined:
'sched

- 'sched

- 2 4

e. av.

After t h e n e c e s s a r y a i r s p e e d f o r f l i g h t a l o n g $ t h e p a t h h a s been
obtained tentatively, the elements of the f l i g h t along sections of
t h e f l i g h t p a t h are i n c l u d e d i n t h e c a l c u l a t i o n .

For e a c h s e c t i o n o f t h e p a t h , t h e f o l l o w i n g m u s t b e d e t e r m i n e d :
(a)
The p a t h o f t r a v e l a n d t h e a i r s p e e d , p r o c e e d i n g f r o m t h e
f l i g h t a n g l e o f a s e c t i o n of t h e p a t h , t h e a i r s p e e d d e t e r m i n e d f o r
t h e f l i g h t a c c o r d i n g t o t h e s c h e d u l e , and t h e wind a t f l i g h t a l t i ­
tude.
(b)
The f l y i n g t i m e p r o c e e d i n g f r o m t h e l e n g t h o f t h e s e c t i o n
and t h e f l i g h t s p e e d .
(c)

The f u e l s u r p l u s a t t h e e n d o f t h e s e c t i o n .

The r e s u l t s o f t h e c a l c u l a t i o n a r e e n t e r e d i n a p r e l i m i n a r y
f l i g h t c a l c u l a t i o n t a b l e i n a l o g on t h e a i r c r a f t , w h i c h h a s t h e
following form:

the route

I

beginningc
fhn

c n c f i n .

The f l i g h t a n g l e s a n d c a l c u l a t e d t r a v e l p a t h s a r e d e t e r m i n e d
a n d r e c o r d e d i n a c c o r d a n c e w i t h a s e l e c t e d frame o f r e f e r e n c e
( o r t h o d r o m i c , t r u e or m a g n e t i c ) .
I n some c a s e s , d u r i n g f l i g h t p r e p a r a t i o n t h e t i m e a n d p l a c e o f
t h e a i r c r a f t ' s e n c o u n t e r i n g d a r k n e s s must be determined, as w e l l a s
t h e l o c a t i o n f o r e n c o u n t e r i n g or o v e r t a k i n g o t h e r a i r c r a f t .
In
solving t h i s problem, t h e p o s i t i o n of a given a i r c r a f t a t t h e
moment when d a r k n e s s or a n o t h e r a i r c r a f t p a s s e s a d e t e r m i n e d p o i n t
on t h e r o u t e ( f o r e x a m p l e , t h e t e r m i n a l a i r p o r t ) i s d e t e r m i n e d .
T h e s p e e d o f t h e a p p r o a c h ( a s t h e sum or d i f f e r e n c e o f t h e s p e e d s )
and t h e l o c a t i o n of t h e e n c o u n t e r are t h e n determined.

Example.
An a i r c r a f t i s f l y i n g f r o m p o i n t A t o p o i n t B a t
2 0 1 5 Moscow t i m e .
The mean a i r s p e e d i s 3 5 0 k m / h r .
The d i s t a n c e
Determine t h e t i m e of t h e encounter
b e t w e e n t h e p o i n t s i s 1 1 0 0 km.
o f t h e a i r c r a f t w i t h d a r k n e s s i f d a r k n e s s r e a c h e s p o i n t B a t 2103
and p o i n t A a t 2218.
534

/SO4

Solution.
B

The rate of movement of darkness from point

(a)

t o point A is

1100
- 1 hr 1 5 min = 880 km/hr.

d'

(b) At the moment of the arrival of darkness at point B y the
aircraft will be at the following distance from point A :
'instr

= Wa

(2103

-

2015) = 35000.8 = 280 k m

and the following distance from point B :
= 1 1 0 0 - 280 = 820 k m ;

'rem
(c)

The speed of the approach of the aircraft to darkness is


W
(d)

aPP

= 350

+

880 = 1 2 3 0 km/hr

The aircraft's encounter with darkness occurs at:


'enc

820 k m
= 2143
= 2103 + 1230 km/hr

at a distance from point A
'enc

= 280

+

350 km/hr*0040 = 5 1 4 k m

Pre-flight preparation is completed on board the aircraft,

where the following events occur: a general survey of the equip­

ment, switching on the equipment and displaying the aircraft course,

adjusting and inspecting the radio-engineering equipment, reading

the pressure on the altimeters, and other operations in accordance

with the manual on flight operation and the instructions on air­

craft navigation for a given type of aircraft.


535


C H A P T E R EIGHT
G E N E R A L P R O C E D U R E F O R AIRCRAFT NAVIGATION
1.

G e n e r a l Methods o f A i r c r a f t N a v i g a t i o n a l o n g Air R o u t e s

The a c c u r a c y a n d r e l i a b i l i t y o f a i r c r a f t n a v i g a t i o n a l o n g a i r
r o u t e s may b e e n s u r e d o n l y when t h e c r e w c o r r e c t l y u s e s e a c h i n ­
d i v i d u a l type of a i r c r a f t n a v a g a t i o n a l equipment and s k i l l f u l l y
c o m b i n e s t h e o p e r a t i o n o f t h e e n t i r e c o m p l e x of t h i s e q u i p m e n t .

/505

By t o t a l u t i l i z a t i o n o f n a v i g a t i o n a l e q u i p m e n t , w e mean t h e
c o r r e c t combination of operating t h e n a v i g a t i o n a l means'at t h e d i s ­
p o s a l o f t h e crew i n o r d e r t o

(1) S o l v e t h e n a v i g a t i o n a l p r o b l e m s w h o s e e l e m e n t s , m e a s u r e d
by v a r i o u s n a v i g a t i o n a l means, a r e t h e p a r e n t e l e m e n t s ;
(2)
Compensate f o r t h e o p e r a t i o n of i n d i v i d u a l t y p e s o f
n a v i g a t i o n a l e q u i p m e n t by means o f o t h e r t y p e s o f e q u i p m e n t ;
(3)
S e l e c t t h e most s u i t a b l e means, u n d e r t h e g i v e n c o n d i t i o n s ,
and d u p l i c a t e them by o t h e r means.

When s e v e r a l n a v i g a t i o n a l e l e m e n t s m u s t b e m e a s u r e d s i m u l t a n e o u s ­
l y ( f o r e x a m p l e , when t h e a i r c r a f t ' s b e a r i n g i s d e t e r m i n e d o n t h e
b a s i s o f t h e r e s p o n s e s of a c o u r s e i n s t r u m e n t a n d a r a d i o c o m p a s s ,
a n d t h e d i r e c t i o n - f i n d i n g t i m e i s r e a d on t h e b a s i s o f a c l o c k on
b o a r d ) t h e p r o c e d u r e f o r t a k i n g r e a d i n g s must b e s u c h a s t o e n s u r e
t h e r a p i d r e c o r d i n g of t h e most u n s t a b l e e l e m e n t s .
In addition,
t h e moments o f r e c o r d i n g s i m u l t a n e o u s l y c h a n g i n g i n d i c a t i o n s mu s t
be as c l o s e t o g e t h e r as p o s s i b l e .
I n our e x a m p l e , t h e a i r c r a f t ' s c o u r s e m u s t f i r s t b e r e c o r d e d ;
t h e n it i s d e s i r a b l e t o r e c o r d t h e c o u r s e a n g l e o f t h e r a d i o
s t a t i o n w i t h t h e s m a l l e s t l a p s e o f t i m e , s i n c e t h e s e e l e m e n t s may
change q u i c k l y and s i m u l t a n e o u s l y due t o t h e f l u c t u a t i o n s i n t h e
longitudinal a x i s of t h e aircraft.
A s r e g a r d s t h e d i r e c t i o n - f i n d i n g t i m e , i t may b e r e c o r d e d
a f t e r t h e f i r s t t w o e l e m e n t s by t a k i n g i n t o a c c o u n t t h e a p p r o x i m a t e
c o r r e c t i o n for t i m e e x p e n d e d o n t h e r e a d i n g a n d r e c o r d i n g o f t h e

536

first elements.

/506

If t h e r a d i o c o m p a s s h a s a n i n d i c a t o r combined w i t h t h e c o u r s e
instrument, t h e b e a r i n g r e a d i n g s do n o t change with f l u c t u a t i o n s i n
the aircraft’s longitudinal axis.
I n such c a s e s , only t h e accuracy
o f m e a s u r i n g t h e t i m e i n t e r v a l s b e t w e e n t h e moments o f r e c o r d i n g
t h e bearings r a t h e r t h a n t h e sequence of recording t h e elements i s
important.

A d d i t i o n a l v a l u e s o b t a i n e d by c a l c u l a t i o n s o r on t h e b a s i s o f
t a b l e s ( t h e c o n v e r g e n c e o f m e r i d i a n s , d e v i a t i o n s o f t h e compass and
radiocompass) are recorded l a t e r , since t h e i r values f o r t h e b a s i c
recorded elements remain unchanged.
I n a l l c a s e s when p r e c a l c u l a t e d or d i s c r e t e r e a d i n g s a r e r e ­
c o r d e d ( e . g . , a p r e c a l c u l a t e d b e a r i n g , t h e moment when a s e l e c t e d
l a n d m a r k p a s s e s a c i r c u l a r d i s t a n c e mark on a r a d a r s c r e e n , e t c . ) ,
t h e moment o f t i m e o f t h e r e a d i n g i s r e c o r d e d f i r s t a n d t h e m e a s u r e d
e l e m e n t s e c o n d , s i n c e t h e p i l o t remembers i t .
Recordings of measured e l e m e n t s are a r r a n g e d i n a sequence which
i s most c o n v e n i e n t f o r a d d i n g i n n a v i g a t i o n a l c a l c u l a t i o n s , u s u a l l y
on t h e b a s i s o f a f o r m o f a i r c r a f t l o g e s t a b l i s h e d f o r a i r c r a f t t y p e d a t a or o n t h e b a s i s o f a s p e c i a l c a l c u l a t i n g f o r m .
If one o f t h e t y p e s o f n a v i g a t i o n a l e q u i p m e n t i s u s e d t o c o r ­
r e c t r e a d i n g s by a n a v i g a t i o n a l d e v i c e o f a n o t h e r t y p e ( e . g . , t o
c o r r e c t t h e a i r c r a f t ’ s c o o r d i n a t e s ) w h i c h a r e c a l c u l a t e d by a n
a u t o m a t i c n a v i g a t i o n a l d e v i c e or b y m e a n s o f a i r c r a f t r a d a r , a t t e n ­
t i o n must b e f o c u s e d on t h e a c c u r a c y o f s e l e c t i n g t h e c o r r e c t i o n
l o c a t i o n where t h e g e o m e t r y o f t h e s o l u t i o n t o t h e problem w i l l b e
most a d v a n t a g e o u s .

I n our example, it i s advantageous t o c o r r e c t t h e X-coordinate
o f t h e a i r c r a f t when t h e l a n d m a r k m a r k l o c a t e d o n t h e l i n e o f t h e
g i v e n p a t h p a s s e s t h r o u g h t h e c i r c u l a r d i s t a n c e m a r k on t h e r a d a r
s c r e e n , a n d t o c o r r e c t t h e Z - c o o r d i n a t e on t h e b a s i s o f d i s t a n c e
m a r k s with p a t h b e a r i n g s of landmarks 90 o r 2 7 0 O .
D u p l i c a t i n g t h e n a v i g a t i o n a l measurements by d i f f e r e n t n a v i ­
g a t i o n a l means i s o b l i g a t o r y i n o r d e r t o a v o i d g r o s s e r r o r s i n
d e t e r m i n i n g t h e l o c a t i o n o f t h e a i r c r a f t or t o r e s o l v e a m b i g u i t y i n
t h e instrument readings.
I n i n d i v i d u a l c a s e s when t h e a c c u r a c y o f m e a s u r i n g n a v i g a t i o n a l
e l e m e n t s by d i f f e r e n t means i s a p p r o x i m a t e l y t h e same, t h e d u p l i ­
c a t i o n o f m e a s u r e m e n t s may b e u s e f u l f o r r a i s i n g t h e i r a c c u r a c y .
I n t h e s e c a s e s t h e mean v a l u e o f t h e r e s u l t s o f t w o m e a s u r e m e n t s b y
two d i f f e r e n t means i s assumed t o b e t h e measured v a l u e .
I n a d d i t i o n , t h e d u p l i c a t i o n o f n a v i g a t i o n a l measurements by
d i f f e r e n t means i s a n emergency a r r a n g e m e n t i n c a s e o f breakdowns

537

I


in the operation of individual elements o f the equipment on board

the aircraft.


Examples.
1. In determining the location of an aircraft
visually or by means of aircraft radar, the crew may commit an
error in identifying a landmark. Even if the locat'ion of an air­
craft is determined approximately by means of a n electronic navi­
gational indicator or by astronomical means, the probability of a
gross error in determining the aircraft's position is practically
excluded.
2. In determining the bearing of an aircraft on the basis of

a fan-type beacon or a hyperbolic position line by phase measure­

ments, the readings are ambiguous. The ambiguity is resolved by

an additional determination of the sector or phase path by less

accurate means.

3.
The accuracy in determining the aircraft's location by
means of the goniometric rangefinding system and by means of air­
craft radar is the same under normal conditions. Duplicating the
measurements permits an increase in the accuracy of determining the
aircraft's location as a mean based on the results of two measure­
ments.

The above examples are characteristic only of the general

principles of the total utilization of aircraft navigational equip­

ment in executing a flight.

The great variety of flight conditions, navigational equipment

on air routes, and aircraft equipment has determined the various

methods of their total utilization.

Specific recommendations for the total utilization of navi­

gational means over individual sections of air routes are usually

given in descriptions and instructions on executing flights along

air routes.

On flights not along air routes, the procedure for using navi­

gational devices is determined during the flight preparation after

studying the navigational situation along the flight route.


2.

S t a g e s in E x e c u t i n g t h e F l i g h t

Executing flight with the use of any aircraft navigational

device may be divided into stages with the following characteristic

features: ( a > take-off and climb; (b) flight along the route;

(c) descent and approach t o the airport; and (d) maneuvering and

landing approach.

The stages of climb, descent and maneuvering are executed at

varying altitudes and airspeeds, and flight along the route is

executed under more stable conditions.


/507

The m o s t c o m p l e x s t a g e f o r t h e a i r c r a f t crew i s u s u a l l y t h e
maneuvering s t a g e n e a r t h e a i r p o r t , combined w i t h t h e l a n d i n g ap­
p r o a c h , e s p e c i a l l y under complex m e t e o r o l o g i c a l c o n d i t i o n s .
Despite t h e general f e a t u r e s of each f l i g h t s t a g e , t h e airc r a f t n a v i g a t i o n a l p r o c e d u r e i n them i s n e v e r t h e l e s s v e r y d i f f e r e n ?
i f v a r i o u s a i r c r a f t n a v i g a t i o n a l d e v i c e s are employed.

.-/ 5 0 8

The p r i n c i p a l d i f f e r e n c e s i n t h e a i r c r a f t n a v i g a t i o n a l m e t h o d s
o c c u r when a n y n a v i g a t i o n a l d e v i c e ( g e o t e c h n i c a l , r a d i o - e n g i n e e r i n g
and a s t r o n o m i c a l ) i s used w i t h o u t a u t o m a t i c measurement o f t h e a i r ­
s p e e d 'components a l o n g t h e c o o r d i n a t e a x e s , w i t h o u t c a l c u l a t i n g
t h e a i r c r a f t ' s p a t h , a n d i n o t h e r c a s e s when t h e r e a r e d e v i c e s o n
board f o r solving t h e s e problems.
T h e r e f o r e , i n d e s c r i b i n g -the
general procedure of aircraft navigation a t d i f f e r e n t f l i g h t s t a g e s ,
w e s h a l l discuss i t s f e a t u r e s with t h e a p p l i c a t i o n of automatic
n a v i g a t i o n a l d e v i c e s and w i t h o u t them.
Take-Off

and Climb

During f l i g h t p r e p a r a t i o n , t h e s e c t i o n of t h e p a t h from t h e
t a k e o f f a i r p o r t t o t h e f i r s t c o n t r o l landmark on t h e r o u t e i s
p l o t t e d on t h e map b y a s t r a i g h t l i n e a n d t h e d i s t a n c e a n d d i r e c t i o n
t o t h e landmark from t h e c e n t e r o f t h e a i r p o r t i s i n d i c a t e d .
However, i n p r a c t i c e t h e c o u r s e o f t h e t a k e o f f a l m o s t n e v e r
coincides with t h e indicated f l i g h t direction.
In addition, the
t a k e o f f d i r e c t i o n c h a n g e s d e p e n d i n g on t h e wind d i r e c t i o n , a n d t h e
i n i t i a l p o i n t o f t h e f i r s t t u r n a f t e r t a k e o f f may c h a n g e t h e d i s ­
t a n c e , d e p e n d i n g on t h e t a k e o f f c o n d i t i o n s ( p r e s s u r e , t e m p e r a t u r e ,
Therefore, the
s p e e d , wind d i r e c t i o n and a i r c r a f t f l i g h t w e i g h t ) .
a i r c r a f t ' s a p p r o a c h t o t h e f i r s t c o n t r o l l a n d m a r k may n o t b e e x e ­
c u t e d on t h e b a s i s o f t h o s e r u l e s wh i ch a r e u s e d on t h e f l i g h t
chart.
Without u s i n g a u t o m a t i c d e v i c e s , t h e crew a t t h i s s t a g e of
f l i g h t i s obliged t o perform continuous v i s u a l o r i e n t a t i o n or t o
c a l c u l a t e t h e path of t h e aircraft i n accordance with t h e d i r e c t i o n s
of t h e s t r a i g h t l i n e s between t u r n s and t h e t r a j e c t o r i e s of t h e
t u r n s made t o g a i n a l t i t u d e a b o v e t h e a i r p o r t , r e f i n i n g t h e a i r c r a f t ' s
p o s i t i o n by r a d i o - e n g i n e e r i n g means.
After t h e maneuvering i s completed, t h e a i r c r a f t ' s course t o
t h e f i r s t landmark of t h e f l i g h t r o u t e i s t a k e n i n accordance w i t h
the a i r c r a f t ' s position before acquiring t h i s course.
The c o u r s e
i s t h e n c o r r e c t e d , depending on t h e a i r c r a f t ' s f l i g h t a l o n g t h e
indicated arrival trajectory.

Before a r r i v a l a t t h e landmark, i n t h e case o f a i r c r a f t w i t h
gas turbine engines, t h e necessary value of t h e l i n e a r lead f o r t h e
t u r n w h i c h m u s t b e made t o a c q u i r e t h e g i v e n r o u t e i s d e t e r m i n e d .

539

For f u r t h e r f l i g h t a l o n g t h e r o u t e w i t h c l i m b , t h e c a l c u l a t e d
This course, p r i o r t o a r r i v a l a t the
course i s followed first.
g i v e n f l i g h t e c h e l o n , i s r e f i n e d t w o or t h r e e t i m e s b y m e a s u r i n g
t h e a i r c r a f t ’ s d r i f t a n g l e o r on t h e b a s i s o f s u c c e s s i v e r e c o r d i n g s
/509
of t h e LA.
The a i r c r a f t ’ s a p p r o a c h t o t h e f i r s t l a n d m a r k i s s i m p l i f i e d
c o n s i d e r a b l y when a ’ u t o m a t i c a i r c r a f t n a v i g a t i o n a l d e v i c e s a r e u s e d .
I n t h i s case, t h e automatic device i s a d j u s t e d f o r c a l c u l a t i n g t h e
p a t h i n a c o o r d i n a t e system f o r t h e s e c t i o n o f t h e p a t h from t h e
c e n t e r of t h e a i r p o r t t o t h e f i r s t landmark.
After t a k e o f f a t any
p o i n t , t h e a i r c r a f t ’ s c o o r d i n a t e s are g i v e n i n t h i s system whenever
possible.
After maneuvering i s completed, t h e course t o t h e f i r s t land­
mark i s f o l l o w e d i n a c c o r d a n c e w i t h t h e f l i g h t a n g l e o f t h e a r r i v a l
c o o r d i n a t e system and i n accordance with t h e a n g l e of d e p a r t u r e
f o r t h e f i n a l point of t h e stage.

where y i s t h e a i r c r a f t ’ s c o u r s e i n t h e s e l e c t e d frame o f r e f e r e n c e ;
$ i s t h e f l i g h t a n g l e o f t h e s t a g e i n t h e same f r a m e o f r e f e r e n c e ;
a n d c1 i s t h e a n g l e o f d e p a r t u r e d e t e r m i n e d o n t h e b a s i s o f t h e
formula

where Z i s t h e l a t e r a l c o o r d i n a t e a t t h e end o f m a n e u v e r i n g ; X i s
t h e l o n g i t u d i n a l c o o r d i n a t e a t t h e end of maneuvering; and S i s t h e
d i s t a n c e from t h e c e n t e r of t h e a i r c r a f t t o t h e landmark.

After t h i s , t h e aircraft’s course i s r e f i n e d i n accordance with
t h e d r i f t angle of t h e aircraft.
B e f o r e a r r i v a l a t t h e l a n d m a r k , t h e Z - c o o r d i n a t e must g r a d u a l l y
d e c r e a s e t o z e r o and t h e X - c o o r d i n a t e must a p p r o a c h S.
The t u r n f o r m o v i n g a l o n g t h e r o u t e i s e x e c u t e d , t a k i n g i n t o
account t h e l i n e a r lead.
Further f l i g h t along t h e route is executed
with t h e given f l i g h t path angle, taking i n t o account t h e d r i f t
a n g l e on t h e b a s i s o f t h e r e a d i n g s o f t h e D o p p l e r m e t e r .

Executing

.J

F l i g h t A l o n g a Route

The g e n e r a l a i r c r a f t n a v i g a t i o n a l p r o c e d u r e a l o n g a r o u t e
w i t h o u t t h e u s e o f a u t o m a t i c d e v i c e s i s g o v e r n e d by t h e p r e s e n c e of
navigational devices of aircraft navigation.
I n t h e beginning of t h e first s t a g e of h o r i z o n t a l fligh-t; t h e
calculated course is followed.
I t i s l a t e r r e f i n e d on t h e b a s i s o f
540

t h e r e s u l t s o f m e a s u r i n g t h e d r i f t a n g l e or o n t h e b a s i s o f s u b s e ­
quent r e c o r d i n g s o f t h e LA.
If t h e l a n d m a r k s a l o n g t h e r o u t e a r e m a r k e d b y r a d i o b e a c o n s ,
In t h e ab­
t h e a r r i v a l of t h e aircraft a t t h e s e p o i n t s i s noted.
s e n c e o f s u c h r a d i o d e v i c e s , t h e moment o f p a s s i n g a l a n d m a r k i s
d e t e r m i n e d v i s u a l l y o n t h e b a s i s o f t h e f l i g h t t i m e or b y m e a n s o f
aircraft radar.

The a i r s p e e d i s d e t e r m i n e d a n d c h e c k e d o n t h e b a s i s o f t h e
moments o f p a s s i n g t h e l a n d m a r k s .
When n e c e s s a r y , t h e a i r s p e e d f o r
maintaining t h e f l i g h t t i m e along t h e given p a t h i s changed.
On t h e b a s i s o f t h e m e a s u r e d v a l u e s o f t h e d r i f t a n g l e a n d t h e
a i r s p e e d , t h e s p e e d and d i r e c t i o n o f t h e wind a r e d e t e r m i n e d and
t h e d i r e c t i o n o f movement f o r t h e f l i g h t i n t h e n e x t s e c t i o n o f t h e
r o u t e i s c a l c u l a t e d on t h i s b a s i s .
Turrsalong t h e f l i g h t route ( i n aircraft with gas turbine
e n g i n e s ) a r e e x e c u t e d t a k i n g t h e l i n e a r l e a d i n t o a c c o u n t (LLT).
A t t h e c o n t r o l p o i n t s o f t h e r o u t e where it h a s been s p e c i f i e d
by p l a n , t h e f u e l c o n s u m p t i o n i s c h e c k e d by c o m p a r i n g t h e a c t u a l
fuel surplus t o the calculated surplus.

With a p i l o t - n a v i g a t o r c h a r t , c h e c k i n g o f t h e f u e l c o n s u m p t i o n
may b e e x e c u t e d a t a n y p o i n t o n t h e r o u t e w h e r e i t i s c o n v e n i e n t or
after equal intervals of f l i g h t .
If a i r c r a f t n a v i g a t i o n i s e x e c u t e d by means o f a u t o m a t i c n a v i ­
g a t i o n a l d e v i c e s , t h e d i r e c t i o n o f movement i n a l l c a s e s i s d e t e r ­
mined by m e a s u r i n g t h e d r i f t a n g l e on t h e b a s i s o f t h e D o p p l e r
m e t e r r e a d i n g s ; t h e a i r s p e e d i s s e l e c t e d i n s u c h a way a s t o m a i n ­
t a i n a given f l i g h t speed along t h e r o u t e .

When t h e l a n d m a r k s a r e p a s s e d , t h e a i r c r a f t ' s c o o r d i n a t e s a r e
r e f i n e d and t h e r e a d i n g s of t h e c o o r d i n a t e m e t e r s are c o r r e c t e d .
A t t h e same t i m e , s y s t e m a t i c e r r o r s i n t h e e n t i r e c o m p l e x o f n a v i ­
g a t i o n a l equipment occur.
They a r e e l i m i n a t e d by i n t r o d u c i n g
p e r t i n e n t corrections i n t o t h e readings of t h e course device.
A t t h e t u r n i n g p o i n t s on t h e r o u t e , t h e frame o f r e f e r e n c e o f
t h e a i r c r a f t ' s c o o r d i n a t e s i s c h a n g e d b y e s t a b l i s h i n g a new v a l u e
f o r t h e f l i g h t p a t h a n g l e o n t h e a n g l e s e t t e r o f t h e map a n d r e ­
c a l c u l a t i n g t h e a i r c r a f t ' s c o o r d i n a t e s i n a new f r a m e o f r e f e r e n c e .
Checking t h e f u e l consumption i n f l i g h t , u s i n g automatic navi­
g a t i o n a l d e v i c e s , i s d o n e o n t h e b a s i s o f t h e same r u l e s a s a r e
employed w i t h o u t s u c h d e v i c e s .
I n h o r i z o n t a l f l i g h t along t h e r o u t e , j u s t . a s under conditions
of climb and de.scent, t h e c r e w i s o b l i g e d t o f o l l o w t h e meteoro­

541

I-1111. ­

1111 111

I111

111 111 I 11111111 11111.111.111111.

I111 I I 1 1 1 1 1 1111.11111..

1111111.111

I1

/510

l o g i c a l s i t u a t i o n c o n t i n u o u s l y and t o t a k e measures t o circumvent
d a n g e r o u s w e a t h e r p h e n o m e n a w h i c h a r e o b s e r v e d v i s u a l l y or b y m e a n s
of aircraft radar.
When p a s s i n g l a n d m a r k s or a t o t h e r p o i n t s o f t h e r o u t e , t h e
crew t r a n s m i t s d a t a on t h e f l i g h t c o n d i t i o n s t o t h e g r o u n d p e r s o n ­
nel:
t h e measured v a l u e of t h e wind p a r a m e t e r s , t h e a i r t e m p e r a t u r e ,
c l o u d i n e s s , m e t e o r o l o g i c a l v i s i b i l i t y , bumpy a i r , i c e d e p o s i t s , e t c .
These d a t a are necessary f o r s u p e r v i s i n g t h e f l i g h t s , planning
/511
t h e t a k e o f f s o f o t h e r a i r c r a f t , and r e f i n i n g t h e s i t u a t i o n f o r
forecasting t h e weather along t h e a i r r o u t e s a t later times.
I n approaching t h e calculated points f o r t h e closest possible
a p p r o a c h of t h e a i r c r a f t t o a l t e r n a t e a i r p o r t s , t h e crew must ob­
t a i n d a t a on t h e s t a t e o f t h e w e a t h e r a t t h e l a n d i n g a i r p o r t a n d
m u s t make t h e f i n a l d e c i s i o n o n w h e t h e r t o c o n t i n u e t h e f l i g h t or
return t o the alternate airport.
A t t h e e n d o f t h e f l i g h t r o u t e , t h e crew m u s t o b t a i n f r o m t h e
d i s p a t c h i n g p e r s o n n e l o f t h e l a n d i n g a i r p o r t , d a t a on t h e c o n d i t i o n s
of t h e approach t o t h e a i r p o r t and t h e p a t t e r n of t h e l a n d i n g
approach maneuver.
They must t h e n c a l c u l a t e t h e i n i t i a l p o i n t o f
descent from t h e a l t i t u d e of t h e given f l i g h t echelon, proceeding
f r o m t h e g i v e n v e r t i c a l s p e e d a n d mean a i r s p e e d a t t h e d e s c e n t
stage.
The c r e w m u s t a l s o a s k p e r m i s s i o n o f t h e d i s p a t c h e r ( f l i g h t
supervisor) for the aircraft's descent.
Descent and Entrance t o t h e Region o f
Landing A i r p o r t by an A i r c r a f t

the

The d i s t a n c e o f t h e i n i t i a l p o i n t o f d e s c e n t b e f o r e a r r i v a l i n
t h e v i c i n i t y o f t h e a i r p o r t i s u s u a l l y d e t e r m i n e d on t h e b a s i s o f
t h e d i f f e r e n c e i n t h e a l t i t u d e of t h e f l i g h t e c h e l o n a l o n g t h e
r o u t e and t h e given a l t i t u d e f o r approaching t h e landing a i r p o r t .
Here t h e d e s c e n t t i m e

where Hech i s t h e f l i g h t a l t i t u d e a l o n g t h e r o u t e ;
is the
g i v e n a l t i t u d e o f t h e a p p r o a c h t o t h e a i r p o r t ; a n d Fzp?s t h e g i v e n
v e r t i c a l rate of the aircraft's descent.
T h e mean a i r s p e e d f o r t h e a i r c r a f t ' s d e s c e n t i s d e t e r m i n e d
w i t h s u f f i c i e n t a c c u r a c y i f t h e t a i l - w i n d c o m p o n e n t i s known a t
t h e beginning and end of t h e a i r c r a f t ' s descent s t a g e :
UXech
'av
542

= 'av

+

+

2

Uxapp
Y

w h e r e Vav i s t h e m e a n a i r s p e e d i n t h e d e s c e n t s t a g e o f t h e a i r c r a f t
a n d ux i s t h e t a i l - w i n d c o m p o n e n t a t t h e b e g i n n i n g a n d e n d o f t h e
descent.
If t h e t a i l - w i n d component a t t h e end o f t h e d e s c e n t i s n o t
known b y t h e c r e w , t h e c a l c u l a t i o n i s p e r f o r m e d o n t h e b a s i s of
h a l f t h e v a l u e of t h e w i n d v e c t o r a t f l i g h t a l t i t u d e , t a k i n g i n t o
account t h e aircraft’s f l i g h t d i r e c t i o n under descent conditions.

When t h e a p p r o a c h e s t o t h e a i r p o r t a r e n o t l i m i t e d b y t h e l o c a u 5 1 2 ,
t e r r a i n , it i s a d v a n t a g e o u s t o b e g i n t h e a i r c r a f t ’ s d e s c e n t 1 - 1 . 5
min p r i o r t o p a s s i n g t h e c a l c u l a t e d p o i n t i n o r d e r t h a t t h e g i v e n
approach a l t i t u d e t o t h e a i r p o r t be a c h i e v e d b e f o r e h a n d and h o r i ­
z o n t a l f l i g h t be e x e c u t e d a t t h i s altj.:clde f o r a s h o r t l e n g t h o f
time.
Otherwise, t h e aircraft approaches t h e a i r p o r t a t an a l t i t u d e
above t h a t g i v e n .
This s l i g h t l y complicated t h e maneuvering before
landing.

I n t h e d e s c e n t s t a g e ( j u s t a s d u r i n g c l i m b ) i n a i r c r a f t which
do not have automatic n a v i g a t i o n a l d e v i c e s , t h e d r i f t a n g l e of t h e
a i r c r a f t m u s t b e d e t e r m i n e d more o f t e n a n d t h e c o u r s e o f movement
must b e r e f i n e d .
With a u t o m a t i c d e v i c e s , a i r c r a f t n a v i g a t i o n i n
t h e d e s c e n t s t a g e i s p r a c t i c a l l y t h e same a s a i r c r a f t n a v i g a t i o n
i n horizontal flight.
We m u s t b e a r i n m i n d t h a t o n j e t a n d t u r b o p r o p a i r c r a f t ,
d a n g e r o u s m e t e o r o l o g i c a l phenomena ( t h u n d e r s t o r m s , i c e d e p o s i t s ,
a n d v i o l e n t b u m p s ) a r e more o f t e n e n c o u n t e r e d u n d e r c l i m b a n d
descent conditions than i n given echelons of horizontal f l i g h t .
T h i s r e q u i r e s more a t t e n t i o n by t h e a i r c r a f t c r e w t o d e t e c t i n g and
c i r c u m v e n t i n g t h e s e m e t e o r o l o g i c a l phenomena.

Maneuvering

i n t h e V i c i n i t y o f t h e A i r p o r t and t h e
Landing Approach

The l a n d i n g a p p r o a c h i s t h e m o s t c o m p l e x a n d i m p o r t a n t f l i g h t
s t a g e , e s p e c i a l l y a t a i r p o r t s w i t h a m o u n t a i n o u s r e l i e f , u n d e r com­
plex meteorological conditions.
I n t h i s stage of f l i g h t , t h e air­
craft approaches t h e ground; f l i g h t i s executed with a changing
a l t i t u d e and a i r s p e e d , w i t h a n u n s t e a d y wind.
Under complex
m e t e o r o l o g i c a l c o n d i t i o n s , i n t e r f e r e n c e i n r a d i o r e c e p t i o n i s most
probable.
This complicates t h e use of radiocompasses.
In addi­
t i o n , t h e landing approach i s t h e f i n a l s t a g e of f l i g h t and t h e
All t h i s r e q u i r e s c a r e f u l p r e p a r a t i o n by
crew i s most f a t i g u e d .
t h e crew i n e x e c u t i n g t h e a i r c r a f t ’ s l a n d i n g a p p r o a c h maneuver.
B e f o r e a p p r o a c h i n g t h e a i r p o r t , t h e crew must o b t a i n t h e
n e c e s s a r y d a t a f o r t h e c o n d i t i o n s of maneuvering and t h e Landing
approach.
The c r e w m u s t e s t a b l i s h a l t i m e t e r p r e s s u r e s c a l e s on
t h e b a s i s of t h e p r e s s u r e a t t h e l a n d i n g a i r p o r t l e v e l and must

543

r e f r e s h i n t h e i r memory t h e a p p r o a c h s c h e m e s p e c i f i e d f o r t h e g i v e n
a i r p o r t i n accordance with t h e landing directions.
T h e crew m u s t
also p e r f o r m p r e l i m i n a r y c a l c u l a t i o n s f o r t h e a p p r o a c h .
I n e x e c u t i n g t h e m a n e u v e r , t h e crew m u s t c o n t r o l t h e m o t i o n
p a r a m e t e r s o f t h e a i r c r a f t by a l l a v a i l a b l e m e a n s a n d m u s t f o l l o w
t h e commands o f t h e g r o u n d p e r s o n n e l who a r e c o n t r o l l i n g t h e e x e cution o f t h e landing approach.
The a i r c r a f t ' s d e s c e n t a l o n g t h e g i v e n t r a j e c t o r y on t h e
l a n d i n g p a t h when t h e g r o u n d i s n o t v i s i b l e m u s t b e e x e c u t e d o n l y
u p t o t h e d e t e r m i n e d minimum a l t i t u d e f o r t h e g i v e n a i r p o r t .
If
t h e w e a t h e r i s l o w e r t h a n t h e e s t a b l i s h e d minimum a n d t h e a i r c r a f t
does not t r a n s f e r t o v i s u a l f l i g h t a t t h e e s t a b l i s h e d a l t i t u d e , t h e
crew must c e a s e f u r t h e r d e s c e n t and must a c t a c c o r d i n g t o t h e i n ­
s t r u c t i o n s of t h e d i s p a t c h e r ( f l i g h t s u p e r v i s o r ) .
I n t r a n s f e r r i n g t o v i s u a l f l i g h t , t h e crew must e v a l u a t e t h e
a v a i l a b l e d e v i a t i o n s from t h e g i v e n f l i g h t t r a j e c t o r y ; i f t h e y do
n o t exceed t h e p e r m i s s i b l e l i m i t s , t h e crew must v i s u a l l y d i r e c t
t h e a i r c r a f t t o t h e g i v e n t r a j e c t o r y and e x e c u t e t h e l a n d i n g .
Otherwise, t h e crew must c i r c l e a g a i n and a c t a c c o r d i n g t o t h e
d i r e c t i o n s of t h e d i s p a t c h e r ( f l i g h t s u p e r v i s o r ) .

544

/513

I I I111

Supplement 1
Composite C h a r t o f Topographical

Ma p s

/514

545

Composite C h a r t of

Topographical

Maps

(con't)

/515

Supplement 2
Spherical Trigonometry Formulas

/516

In solving special navigational problems, especially problems

of aviational astronomy, formulas of spherical trigonometry are used.

Spherical trigonometry formulas establish the interdependence

between the sides and angles of a spherical triangle.

The sides of a spherical triangle a , b , C
central angles of a sphere a ; the angles A , B y
spherical angles whose sides are the verticals
B and C. Therefore, both the sides and angles
expressed in the same way (by angular degrees).

are the arcs of the
C are the surface
from the points A ,
of the triangle are

Formulas known as the law of cosines (the first group of formulas

for spherical trigonometry) are the most widely used in navigation:

cos a = cos b cos c + s i n b s i n c cos A ;
c o s d = cos B cos C

+s i n B sin Ccos a.

These formulas may be written in three ways (for sides a , b y
c and angles A , B y C).
The second formula of spherical trigonometry is the law of

sines:


-s_i n_a _-_s_i n _6 ­
sind

sinB

sinc
sinC

*

The third group of formulas is used much less often:


+

s i n a cos 6 = cos a sin 6 cos C
s i n c cos B;
sin a ctg 6 = ctg B s i n C cos a cos C;

+

sin A cos B = cos 6 s i n c - cos c s i n B cos A;
s i n d c t g B= ctg 6 s i n c - cos c cos A .

547

i

E a c h o f t h e s e f o r m u l a s may b e w r i t t e n i n s i x w a y s ( i n t w o w a y s
f o r s i n e s of t h e s i d e s a , b , e and a n g l e s A , B , C>.
I f o n e o f t h e s u r f a c e a n g l e s (for e x a m p l e , C )
t h e Napier r u l e i s used t o solve t h e t r i a n g l e .

is a s t r a i g h t l i n e ,

Five elements of t h e t r i a n g l e (excluding t h e r i g h t angle C)
a r e a r r a n g e d i n a c i r c l e , a s shown i n t h e F i g u r e , t h u s d e s i g n a t i n g
t h e a d j a c e n t and n o n a d j a c e n t e l e m e n t s of t h e t r i a n g l e .

F i r s t Principle.
The c o s i n e o f a n y e l e m e n t o f a r i g h t t r i a n g l e
i s e q u a l t o t h e p r o d u c t of t h e c o t a n g e n t s of t h e a d j a c e n t e l e m e n t s .
For e x a m p l e :
cosC=ctgActgB.

F i v e s u c h f o r m u l a s may b e w r i t t e n f o r e l e m e n t s A ,
B a n d e.

90-b,

90-a,

S e c o n d Principle.
T h e c o s i n e o f e a c h e l e m e n t is e q u a l t o t h e
p r o d u c t of t h e s i n e s of t h e n o n a d j a c e n t e l e m e n t s .
For example:

cos A =: s i n (90 - a) sin’ B.

F i v e o f t h e s e f o r m u l a s may b e w r i t t e n ,

548

o n e for e a c h e l e m e n t .

/517

Supplement 3
Map of t h e H e a v e n s

549


I


Supplement 4

-..................

-

_c

.....

0

.......
-z

c

B

.................................

....

0

............... .......

IOf.Zl.1

.......
0

-2

C


.
-,lI

0

.......

B


v)


a
c

0
N

a

.- E

o




P

c




-l

ll­

0
Q

m

E

C

c

a

L
c

0

Supplement 5
-

-

T a b l e of G r e e n w i c h Hour A n g l e s of the Sun a n d Chart o f
T h e i r C o r r e c t i o n s for the F1igh.t D a t e
Hour
Angle,
deg

Moscow
Time

hr, m i n

I

0.00
12
24
36
48
1 .oo
12
24
36
48
2.00
12
24
36
48
3.00
12
24
36
48
4.00
12
24
36
48
5,OO
12
24
36
4y

II

135
138
141
144
147
150
153
156
159
162
165
168
171
174
177
180
183
186
189
192
195
198
20 1
204
207
210
213
216
219
222

Moscow
Time
h r , min
- ._

6.00
12
24
36
48
7.00
12
24
36
48
8.00
12
24
36
A8
9.00
12
24
36
48
10.00
12
24
36
48
11.00
12
24
36
48

Hour
Angle,
deg
_.

225
228
231
234
237
240
243
246
249
252
255
258
261
264
267
270
273
276
279
282
285
288
29 1
294
297
300
303
306
309
312

Moscow
Time ,
h r , min

Hour
Moscow
Angle
Time,
h r , min
deg
12.00
12
24
?6
48
13.00
12
24
36
48
14.00
12
24
36
48
15.00
12.
24
36
48
16.00
12
24
36
4s
17.00
12
24
36
48

315
318
321
324
327
330
333
336
339
342
345
348
35 1
354
337
360
03
06
09
12
15
18
21
21
27
30
33
36
39
42

/5 19

I

18.00
12
24
36
48
19.00
12
24
36
48
20.00
12
24
36
48
21 -00
12
24
36
48
22.00
12
24
36
48
23.00
12
24
36
48

45
48
51
54
57

GO
63
GG
69
72
75
78
81
84
87
90
93
96
99
102
105
108
111
114
117
120
123
126
129
132

551

Supplement 6


-

T a b l e of

x-a

­

Values of
x-a

t h e F u n c t i o n CP (x


@(x-a

x-a

04.r

-a

x-a

-

U(X-Q)

-

-

0,oo
00
.1

0,02
0,03
O"O4

o;ooso

010120
0,0160

0,05

0,0199

0.06

0,0239
0,0279
0,0319
0,0359
0,0395
0,0435
0,0478
0,0517
0,0557
Oc059G
0 p636
0,0675
0,0714
0,0753
0,0793
0,0532
08871
0,09 10
0,0945
0,0987
0,1026
0, IO64
0,1103
0,1141
0,1179
0,1217
0, I255
0,1293
0,1331
0,1368
0,1406
0,1443
0,1480
0,1517
Oil554

0.07
0.0s
0.09
0,lO
0'1 1
0.12
0.13
0'14
0,15
Oar6
0.17

0.1s
ob 19

0.20
0.2 1
0,22
0'23
0,24
0,25
0.26
0.27
0.25
0.29
0,30
0,31
0,32
0,33
0.34
0,35
0-36
0,37
0.38
0.39
0.40

552

O,,bOOO
0,0040

0, 66
0,67
0,GS
0,69
0'70
0,71
0172
0,73
0- 74
0'75
0.76
0.77
0,75
Oa79
0.80
0,81
0,82
0,83
06 84
0.85
0&6

0,87

0,88
0.89
0490
0.91
0,92
0.93
0,94
0,95
0*96
0.97
0'98
0*99
1z o o
1 col
I ,02
1a03

1,04
1.05

0,2454
0,2456
42517
Os2549
0,2550
0,2611
0,2642
0,2673
0,2703
0,2734
0,2764
0,2794
0,2823
0,2852
0,2551
0,29 10
0,2?39
0,2967
012995
0,3023
0,3051
0.3078
0,3 106
02133
0,3 159
0,3156
0,3212
0,3238
0,3264
0,3259
0,3315
0,3340
0,3365
0,3389
0,3413
0,3438
0,346 1
033485
0,3508
0,353 1

1'31
1,32
1-33
1,34
1.35
1.36
1,37
1338
1,39
1.40
1041
1,42
1,43
1.44
1,45
1.46
10.47
1,'48
I, 49
L50
1A1
1.52
I, 53
1-54
1.55
136
Is 57
19.58
169
1G
,O

1661
1962
1,63
1,64
1,65
1,66
1.67
1,68
1,,69
1e70

0,4049
0,4066
0,4082
014099
0,4115
0,4131
0,4147
0,4 162
0,4177
0,4192
0,4207
0,4222
0,4236
094251
0,4265
0,4279
0,4292
0,4306
0,4319
0.4332
0,4345
0,4357
014370
014352
094394
0,4406
0,4418
0,4429
0,4441
0,4452
Ob4463
0,4474
0,4484
0.4495
0.4505
014515
0.4525
0,4535
0.4545
014554

/521
­

a)

J,96
le97

1,98
1,99
2,OO
2d02
2,04
2,OG
2. os

5 10
2,12
2,14
29 16
25 1s
22.0
2.22
2,24

2,2G
2*2s
2.30
2*32
2.34
2,36
2,35
2.40
2'42
2,44
2,4G.
2.48
2,50
2,52
2,54
2.56
2.58
2,60
2.62
2.64
2.66
2'68
2.70

0,4750
0,4756
0,476 1
0,4767
0,4772
0,4733
004793
0,4803
0,4S 12
0,4821
094530
0,4838
004S46
004854
0.4561
0,4SGS
0,4875
0.4S5'1
0,4557
0,4593
0,4898
0,4904
0,4909
0,49!3
0,4915
0,4922
0,4927
0,493 I
0,4934
0,4935
0,494 I
0,4945
0,4945
0,4951
0,4953
0,4956
0,4959
0,4961
0,4963
0,4965

/521

T a b l e o f Values o f the Function @ ( x - a ) , ( c o n l t )
.

_ _

-..

X - U

P

(x - a )

x-a

­

-

0.41
0,42
0.43
0,44
0.45
0.46
0,47
0148
0 #49
0.50
0,51
0452
0,53
0.54
0.55
Om56
0157
0.58
0-59
os60
0,61
0662
0863
0064
0.65

0,1591
0,1628
0, b664
0,1700
0,1736
0,1772
0,1808
0,1844
0,1879
0,1915
0,1950
01. 985
0.20 19
0,2054
0,2088
0,2123
042 157
Oa2190
0,2224
0,2257
0,229 1
042324
0,2357
0,2389
0,2422

1.06
1,07
1.08
1609
1.10
1,11
1,12
1613
1,14
1,15
I, 16
1.17
1,18
1.19
1.20
1.21
1.22
1a23
1.24
1,25
1626
1,27
1 &8
1'29
1.30

0,3554
0,3577
0,3599
0,3621
0,3643
0,3665
0,3686
0,3708
0,3729
0,3749
0,3770
0,3790
0,3810
0,3830
02849
0,3869
0,3SSS
0,3907
Od925
0.3944
0,3962
08980
03997
0,40 15
0,4032

.

1.71
1.72
1.73
1,74
1*75
1,'76
1,77
la78
1.79
1,SO
1.81
1.82
I,S3
1,84
1 $85
1,86
1.87
18
,8

1.89
1.90
1.91
1,92
1,93
1094
1*95

0,4564
0,4573
0,4582
0,4591
Oe4599
0,4608

0,4616
0.4625
0,4633
014641
0,4649
0,4656
0,4664
0,4671
0,4678
0,4686
0,4693
0,4699
0,4706
0.4713
0,4719
0,4726
0,4732
0,4738
0#4744

2, i 2
2,74
2,76
2.78
2,80
2.82
2,84
2,86
2,88
2-90
2,92
2,94
2E96
26 98
3,OO
3.20
3u40
3,60
3-80
4.00
4.50
5.00

0,4967
0,4969
0,497 1
0,4973
0,4974
0,4976
0.4977
0,4979
Oa4980
0,498 1
Oa 4982
0,4954
0,4935
0,4986
0149SG5
0,49931
06499G6
01499841
0,499928
0,499968
0,499997
0,49999997

553

Supplement 7
U n i t s o f t e n E n c o u n t e r e d in A i r c r a f t N a v i g a t i o n

and Thei r Va 1 ues


-- - .

-.

-

R a t i o o f t h e c i r c u m f e r e n c e of a c i r c l e t o i t s
diameter

n

L e n g t h of a n a r c o f a c i r c l e ,
r a d i u s of t h e c i r c l e

Radian

I

10


1'


equal t o the
57.30
0.01U53 r a d

A n g u l a r u n i t or l e n g t h o f a r c

1/60°
0.030291 rd
11

1 I'

3.1416

I'

It

11

It

I1

1/60
O.OOOoc6

The mean v a l u e o f t h e l e n g t h o f t h e a r c o f a

g r e a t c i r c l e on t h e E a r t h ' s s u r f a c e a t lo

10


t°C

T e m p e r a t u r e on t h e C e l s i u s s c a l e

T°K

Absolute temperature

R


30.9 m

to(? + 273O

(Kelvin s c a l e )

Radius of t h e E a r t h ' s e l l i p s o i d reduced t o
a sphere

a

6 3 7 1 km
6378.245 km

Semimajor a x i s of t h e E a r t h ' s e l l i p s o i d

I

e

e

!

554

1.8523km

Length o f t h e a r c o f a g r e a t c i r c l e

1' I

b

111.1 km


L e n g t h of t h e a r c o f a g r e a t c i r c l e ( n a u t i ­

cal mile)

1'


Semiminor a x i s of t h e E a r t h ' s e l l i p s o i d

',

rad

1

6356.863 km

E c c e n t r i c i t y o f t h e e l l i p s e formed by t h e
i n t e r s e c t i o n of t h e E a r t h by a p l a n e o f
t h e meridian

0.006689

N a p i e r number ( f o u n d a t i o n of n a t u r a l
logarithms)

2.718282

Modulus o f t h e t r a n s f e r f r o m n a t u r a l
l o g a r i t h m s t o common l o g a r i t h m s

0.43429

/522


Symbol
of the

'

II

w1

2.30259

B

Characteristic gas constant for air

29.27 d d e g

K

Specific heat ratio for air Cp/Cv

P

Mass density o f the air near the ground
under standard atmospheric conditions

Y

a

1

1 mm Hg
~

Numerical
Value and

Modulus of the transfer from common logarithms t o natural logarithms

MI =

-

i

Geometric or Physical Meaning

~-

-

1.4

0.125 k
sec2/m

E'

Weight density of the air near the ground
under standard atmospheric conditions

1.226 w m 3

Temperature gradient to an altitude of
11,000 m (standard atmospheric conditions)

6.5 deg/km

Speed of sound near the ground under
standard atmospheric conditions

340.2 m/sec

Acceleration of gravity

9.81 & e c 2

Unit of measurement of atmospheric pressure
.

-~

~

_-

-.

I

~~T

1.36 g/cm
~

__

CONVERSION FROM ENGLISH AND AMERICAN UNITS TO METRIC UNITS

1.609 k m

Statue mile
Foot

-

_ .

Inch, Hg
Millibar

0.3048 m
254 mm Hg

-

__

_ _

0.7501 mm Hg

Gallon (Imperial)

4.546 1

Gallon (American)-

3.785 1

T r a n s l a t e d f o r t h e N a t i o n a l A e r o n a u t i c s a n d S p a c e A d m i n i s t r a t i o n by:

Aztec School of Languages, Inc.

Research Translation Division (152)

Acton, Massachusetts.

NASW-1692


NASA-Langley, 1967

-2 1

555

NATIONAL
AERONAUTICS
AND SPACE ADMINISTRATION
WASHINGTON,
D. C. 20546

SPACE ADMINISTRATION

FIRST CLASS MAIL

OFFICIAL BUSINESS

.

.
I

I
POSTMASTER:

If Undeliverable (Section 158
Postal Manual) D o N o t Returi

' T h e aeronautical and space activities of the United States shall be
coadacted so as t o contribute . . . t o the expansion of human knowl­
edge of phenoiiiena in the ntmosphere and space. T h e Administration
shall provide folr the widest practicable aizd appropriate disseminatjon
of inf oriiiation concerning its activities and the resdts thereof."

-NATIONALAERONAUTICS
AND SPACE ACT OF 1958

NASA SCIENTIFIC AND TECHNICAL PUBLICATIONS
TECHNICAL REPORTS: Scientific and
technical information considered important,
complete, and a lasting contribution to existing
knowledge:
TECHNICAL NOTES': Information less broad
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contribution to existing knowledge.
TECHNICAL MEMORANDUMS :
Information receiving limited distribution
because of preliminary data, security classifica­
tion, or other reasons.
CONTRACTOR REPORTS: Scientific and
technical information generated under a NASA
contract or grant and considered an important
contribution to existing knowledge.

TECHNICAL TRANSLATIONS: Information
published in a foreign language considered
to merit NASA distribution in English.
SPECIAL PUBLICATIONS: Information
derived from or of value to NASA activities.
Publications include conference proceedings,
monographs, data compilations, handbooks,
sourcebooks, and special bibliographies.
TECHNOLOGY UTILIZATION
PUBLICATIONS: Information on technology
used by NASA that may be of particular
interest in commercial and other non-aerospace
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Technology Utilization Reports and Notes,
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Details on the availability of these publications may be obtained from:

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D.C. 20546

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