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(1)

1

2

2

(2)

(3)

A

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x

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X 2 , ' Χ · « 1 ,

2

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(4)

« 1

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Q

3

x i X 2

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b

3

(5)

### where the approximate laws of motion are

f i ( x i , X 2 , * · • > χ η; « ι , Α2>· .

f( X ; A , T ) = f a ( x i » X 2 , * · • > xn = « ι , Α2 )· ·

### [5]

, « 2 , · · , « 2 , · ·

### The difference between the unknown laws of nature and the formulated laws are next defined as

(X,t) F ( X;A , T ) - F(X;OT,T)

Q

Q

2

ο

(6)

[ 9 ]

2

;

;

d

f(

(7)

x i

2

( 9 χ ι d x2

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Ο 7

q

Ο

I

Ο

o

TT(

### t)

^ 1 1 ,

^ 2 1 , 7 l1 2 Π 2 2 ,

1 y

Ο 7

### coordinate

T TLK ~ ^ K ~ " ^ K' V 0

7 12 K

2 k

o

0

1

I ι

0

NK Π .

" N K' V

ό

=

o

a

(9)

7l(

b

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κ + A

7i(t)

=

(10)

K[t)

L

L

o

o

1

η(τ) άτ

1

o

1 K(t)

η{τ) άτ

β

(11)

2

n

/ Kit) *

η(τ) dr

· η(τ) άτ +

η

Χ ι

2

n

=

—• ο °

Q

Q

(12)

ο I

Q

— A X

r f

r

rf

### L

g ê 1

[30]

The matrix d i f f e r e n t i a l equation which generates the fundamental s o l u t i o n matrix with respect to reverse time i s

— M t , ) = - B( tr f) " M t . ) [31a]

dt* β g g

= I

— A

d

d

r f

A

) = I

(13)

g

g

é

g

g

f

r f

0

[ 3 7

(14)

=

Χ 1 (Τ)

2

d

0

o

τ

ο

τ

Ο

(15)

X i

2

x

f

β

9

n

(16)

(17)

- » /df.\

;

df/

d

d f !

df/ df2

df2 df2

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doL± da2

df2

### 16J

(18)

R e f s . 1 1 - 1 6 in t h e E n g l i s h l i t e r a t u r e d e m o n s t r a t e t h e d e t a i l s o f t h i s t y p e o f n u m e r i c a l i n t e g r a t i o n e r r o r a n a l y s i s , w h e r e a s t h e R u s s i a n s V . P. M y a c h i n a n d A. S . S o c h i l i n a i n R e f s . 1 7 a n d 1 8 h a v e d e m o n s t r a t e d a m o r e d e t a i l e d a n a l y s i s , w i t h n u m e r i c a l e x a m p l e s , u s i n g C o w e l l ' s i n t e g r a t i o n a l g o r i t h m a p p l i e d t o t h e e q u a t i o n s o f m o t i o n o f J u p i t e r , U r a n u s , a n d S a t u r n .

D e t e r m i n a t i o n o f D i s t u r b i n g F o r c e s

T h e d e t e r m i n a t i o n o f u n k n o w n d i s t u r b i n g f o r c e s w h i c h c a u s e a s p a c e v e h i c l e t o d e p a r t f r o m i t s r e f e r e n c e t r a j e c t o r y i n t o d i s t u r b e d m o t i o n i m p l i e s t h a t o b s e r v a t i o n s o f t h e d i s t u r b e d p a t h X ( t ) at a r b i t r a r y i n s t a n t s o f t i m e t i , t2 , tm a r e a v a i l a b l e . T h e s e o b s e r v a t i o n s m a y b e c o m p l e t e at a n y t i m e i n s t a n t o r t h e y m a y b e i n c o m p l e t e , i . e . , s o m e o r a l l o b s e r - v a t i o n s o n t h e η - c o o r d i n a t e s o f )?(t) m a y b e a v a i l a b l e . T h e n u m e r i c a l d i f f e r e n c e b e t w e e n t h e o b s e r v e d s t a t e v e c t o r Xi t ) a n d t h e r e f e r e n c e s t a t e v e c t o r at t h e s a m e t i m e i n s t a n t X ( t ) r e p r e s e n t s t h e o b s e r v e d d i s t u r b e d m o t i o n , i . e .

^ o b s ^ ' - ^ o b s ' t ' - * r e f( t)

I f o b s e r v a t i o n s a r e c o m m e n c e d at a p a r t i c u l a r i n s t a n t o f t i m e t f r o m w h i c h a c o m p l e t e r e f e r e n c e t r a j e c t o r y i s d e t e r m i n e d , t h e n it m a y b e a s s u m e d t h a t ks( to) i-s

i d e n t i c a l l y z e r o a n d r e c a l l i n g E q . 2 5 , t h e n

A* o b s( t)

t K

e

### %

w h e r e t h e G r e e n ' s f u n c t i o n o f E q . 1 4 is K(t,r ) = K{t) m

### Mr )

T h e m a t r i c e s K i t ) a n d A( t ) a r e g e n e r a t e d b y s o l u t i o n o f t h e m a t r i x d i f f e r e n t i a l E q s . 1 8 a n d 1 9 . T h e d i s t u r b i n g v e c t o r η i t ) a b o v e i s d e f i n e d b y E q . 4 2 a n d i s t h e o b j e c t o f t h e

i n v e s t i g a t i o n .

E q . 4 4 i s a h o m o g e n e o u s l i n e a r m a t r i x i n t e g r a l e q u a t i o n o f V o l t e r r a t y p e . T h e e q u a t i o n m a y b e s o l v e d n u m e r i c a l l y w i t h o u t u n d u e d i f f i c u l t y f o r η i t ) u s i n g f i n i t e d i f f e r e n c e m e t h o d s .

T h i s a p p l i c a t i o n o f t h e c a l c u l u s o f p e r t u r b a t i o n s h a s b e e n r e p o r t e d o n i n d e t a i l b y t h e R u s s i a n A. A. D e b e r d e e v i n b o t h t h e s i s w o r k ( 1 9 5 5 ) a n d m o r e r e c e n t l y in R e f . 2 0 . M e t h o d s o f s o l u t i o n o f t h e i n t e g r a l e q u a t i o n , E q . 4 4 , a r e s u m m a r i z e d a n d c o m p a r e d in R e f . 2 1 .

(19)

N E W T O N I A N E Q U A T I O N S O P M O T I O N IN T H E E A R T H - M O O N S Y S T E M N - b o d y P r o b l e m

T h e d i f f e r e n t i a l e q u a t i o n s o f m o t i o n f o r t h e N - b o d y p r o b l e m w i l l f i r s t b e d e f i n e d i n b o t h a b s o l u t e a n d r e l a t i v e f o r m a n d t h e n m o d i f i e d to i n c l u d e a t m o s p h e r i c a n d n o n p o i n t m a s s g r a v i t a t i o n a l f i e l d f o r c e s . T h e d e r i v a t i o n e s s e n t i a l l y f o l l o w s t h e e x c e l l e n t t r e a t m e n t o f t h e s u b j e c t f o u n d i n C h a p . 7 o f R e f . 1 0 .

T h e a b s o l u t e f o r m i s w i t h r e s p e c t to an a r b i t r a r y i n e r t i a l o r s p a c e s t a b l e c a r t e s i a n c o o r d i n a t e f r a m e o f r e f e r e n c e e i t h e r in u n i f o r m m o t i o n o f p u r e t r a n s l a t i o n o r at r e s t in i n e r t i a l s p a c e . T h e r e l a t i v e f o r m o f t h e e q u a t i o n s o f m o t i o n i s r e l a t i v e t o a n i n e r t i a l o r s p a c e s t a b l e c a r t e s i a n c o o r d i n a t e f r a m e o f r e f e r e n c e w h o s e o r i g i n c o i n c i d e s w i t h t h e c e n t e r o f m a s s o f o n e o f t h e N - b o d i e s . T h e r e l a t i v e f o r m o f t h e e q u a t i o n s o f m o t i o n w i l l b e t h e m o s t u s e f u l s i n c e t h e p o s i t i o n s o f t h e s u n , p l a n e t s , m o o n s , a s t e r o i d s , a n d o t h e r m a s s i v e b o d i e s i n t h e s o l a r s y s t e m a r e k n o w n t a b u l a r q u a n - t i t i e s i n o n e o r m o r e c o o r d i n a t e f r a m e s w h o s e o r i g i n i s

u s u a l l y a s s u m e d t o b e l o c a t e d at a d y n a m i c a l c e n t e r , i . e . , t h e c e n t e r o f m a s s o f t h e m o o n o r t h e p l a n e t E a r t h .

A b s o l u t e P o r m o f E q u a t i o n s o f M o t i o n

A N e w t o n i a n f r a m e o f r e f e r e n c e is a s s u m e d w h i c h i s c o m p o s e d o f a r i g h t - h a n d e d a n d o r t h o g o n a l s e t o f a x i s w i t h o r i g i n l o c a t e d i n i t i a l l y at a n y c o n v e n i e n t p o i n t i n t h r e e - d i m e n s i o n a l s p a c e a s i n d i c a t e d in P i g . 3.

I t i s a s s u m e d t h a t w i t h i n t h e N - b o d y s y s t e m , i . e . , t h e v o l u m e o f s p a c e o c c u p i e d b y t h e Ν b o d i e s , t h e r e e x i s t f o r c e s w h i c h a c t o n e a c h o f t h e b o d i e s . T h e s e f o r c e s a r e o f t w o t y p e s , i n t e r n a l a n d e x t e r n a l f o r c e s . T h e i n t e r n a l f o r c e s a r e d e f i n e d t o b e t h e N e w t o n i a n g r a v i t a t i o n a l f o r c e s c a u s e d b y t h e m u t u a l g r a v i t a t i o n a l a t t r a c t i o n o f e a c h o f t h e b o d i e s f o r e a c h o f t h e r e m a i n i n g b o d i e s . N e w t o n ' s T h i r d L a w i s a l s o a s s u m e d so t h a t t h e a t t r a c t i o n o f o n e b o d y f o r a n y o t h e r b o d y is e q u a l a n d o p p o s i t e t o t h e a t t r a c t i o n o f t h e s e c o n d b o d y f o r t h e f i r s t . N e w t o n ' s L a w o f G r a v i t a t i o n t h e n s t a t e s t h a t t h e f o r c e a c t i n g o n t h e i t h b o d y d u e t o t h e g r a v i t a t i o n a l a t t r a c t i o n o f t h e j t h b o d y i s

i n t

3

### I65

(20)

w h e r e "r^j i s t h e p o s i t i o n v e c t o r d i r e c t e d f r o m t h e i t h b o d y t o t h e j t h b o d y

k2 = t h e u n i v e r s a l o r G a u s s i a n g r a v i t a t i o n a l c o n s t a n t mi = m a s s o f t h e i t h b o d y

m- = m a s s o f t h e j t h b o d y J

rH = r — = m a g n i t u d e o f t h e v e c t o r ri i

It s h o u l d b e n o t e d t h a t t h e u n i t s o f m a s s , f o r c e , a n d d i s - p l a c e m e n t m u s t b e c h o s e n t o b e c o n s i s t e n t in t h e a b o v e e q u a t i o n .

-> i n t

T h e t o t a l i n t e r n a l g r a v i t a t i o n a l f o r c e F^ o n t h e i t h m a s s p o i n t is t h e s u m .of t h e i n d i v i d u a l g r a v i t a t i o n a l f o r c e s

a c t i n g o n t h e i t h m a s s c a u s e d b y t h e (N_l) r e m a i n i n g m a s s e s ; i . e .

Ν k2mimk

i n t

l k

### [46]

ri k k ^ i

w h e r e k is a d u m m y v a r i a b l e o f s u m m a t i o n a n d s h o u l d n o t b e c o n f u s e d w i t h k2.

e x t

T h e t o t a l e x t e r n a l f o r c e (F^) a c t i n g o n t h e i t h b o d y m a y b e t h e s u m o f s u c h f o r c e s o t h e r t h a n t h e g r a v i t a t i o n a l f o r c e s , s u c h as a t m o s p h e r i c d r a g o r l i f t f o r c e s , e l e c t r o m a g n e t i c r a d i a t i o n f o r c e s , p o w e r e d r o c k e t t h r u s t f o r c e s . T h e t o t a l f o r c e a c t i n g o n t h e i t h b o d y r ^ is t h e n

-* -* i n t -> e x t

i

i

### )

o r t h e d i f f e r e n t i a l e q u a t i o n s o f m o t i o n f o r t h e N - b o d y p r o b l e m in a b s o l u t e f o r m i s

m .

d2r; ^ k m

k2

k +

e X t

### [47]

1 l , — 1 - , 3 I K 1

d t2 =1 Fi k k

k ^ i

R e l a t i v e F o r m o f E q u a t i o n s o f M o t i o n

T h e r e l a t i v e f o r m o f t h e e q u a t i o n s o f m o t i o n s o f a s p a c e v e h i c l e u n d e r e x t e r n a l f o r c e s a n d i n t h e g r a v i t a t i o n a l f i e l d

(21)

J

2

2

2

2 lJ

2 J

2

2

i i = 2 k

2

mi,

i k3

k

k l

HL

(F.)

### 167

(22)

R e w r i t i n g E q . 5 1 g i v e s

d2 _ d t2

i k

•ik

z * e x t -» e x t

### [52]

B u t

->

τ . . = — r · · : r · · -

J i i j * J i 1J

T h e r e f o r e , E q . 5 2 m a y b e w r i t t e n a s

->

d2

d t2

^ - k2( m . + m ·

ri j =

ri j " k = l

i k

### [53]

ζ * e x t -* e x t

### [54]

w h i c h is t h e d i f f e r e n t i a l e q u a t i o n o f r e l a t i v e m o t i o n f o r t h e N - b o d y p r o b l e m o f t h e j t h b o d y ( o r s p a c e v e h i c l e ) r e l a t i v e t o a N e w t o n i a n f r a m e at t h e i t h b o d y .

It i s c o n v e n i e n t , at t i m e s , t o s e p a r a t e t h e a c c e l e r a t i o n s c o n t a i n e d in t h e r i g h t - h a n d m e m b e r o f E q . 5 4 i n t o t w o p a r t s : o n e p a r t c o r r e s p o n d i n g to t h e a c c e l e r a t i o n p r o d u c i n g K e p - l e r i a n o r t w o - b o d y m o t i o n , a n d t h e s e c o n d p a r t c o n s i d e r e d as K e p l e r i a n p e r t u r b i n g a c c e l e r a t i o n s a c t i n g o n t h e t w o - b o d y r e f e r e n c e m o t i o n ; i . e .

d2

d t2

r · · + r · ·

(23)

2

2

l

x

?L

#

Ί

•P 3 -p 3

r

r

2

m

(24)

Φ1( γ )

2

„ 3

4

### relative to the ith body

(25)

T h e g r a v i t a t i o n a l f i e l d i n t e n s i t y v e c t o r g ^ i r ) o f t h e i t h o b l a t e p l a n e t i s d e f i n e d i n c a r t e s i a n i n e r t i a l c o o r d i n a t e s a s

x

y

z

### d χ dy d ζ

T h e s t a n d a r d o b l a t e p l a n e t p o t e n t i a l f u n c t i o n a s g i v e n b y E q . 6 0 m a y t h e n b e u s e d t o o b t a i n

w h e r e c a r e m u s t b e t a k e n i n t h e n u m e r i c a l e v a l u a t i o n o f t h e a b o v e e x p r e s s i o n s f o r t h e c a s e ζ = 0 s o a s t o a v o i d n u m e r i c a l i n s t a b i l i t y .

T h e s t a n d a r d r e l a t i v e f o r m o f t h e g e n e r a l i z e d - e q u a t i o n s o f m o t i o n o f a p o i n t m a s s r e l a t i v e t o a N e w t o n i a n c a r t e s i a n r e f e r e n c e f r a m e w h o s e o r i g i n c o i n c i d e s w i t h t h e c e n t e r o f

### 1 7 1

(26)

m a s s o f t h e s u n o r a n y p l a n e t o r p l a n e t o i d in t h e s o l a r s y s t e m m a y t h e n b e w r i t t e n as

N

e x t

(F.) e x t

### [62]

w h e r e g^ i s g i v e n b y E q . 6 1 .

It s h o u l d b e n o t e d t h a t t h e a b o v e e q u a t i o n s E q s . 6 2 d o n o t c o n t a i n t e r m s c o r r e s p o n d i n g t o t h e p e r t u r b i n g g r a v i t a t i o n a l a c c e l e r a t i o n s i m p o s e d u p o n t h e s p a c e v e h i c l e ( t h e j t h b o d y ) b y E a r t h ' s t r i a x i a l m o o n w h e n t h e i t h b o d y is t a k e n to b e E a r t h o r w h e n t h e i t h b o d y ( a n d h e n c e , t h e d y n a m i c a l c e n t e r ) is t a k e n to b e t h e m o o n . T h e s e c a s e s a r e c o n s i d e r e d as e x c e p t i o n s to t h e s t a n d a r d r e l a t i v e f o r m , E q . 6 2 , a n d t h e l a s t c a s e is t r e a t e d n e x t .

T h e s t a n d a r d f o r m f o r t h e g r a v i t a t i o n a l p o t e n t i a l f u n c t i o n f o r t h e t r i a x i a l e l l i p s o i d a l m o o n i s

k2m .

+

3

1 - 3

w h e r e r

### «Δ

p o s i t i o n v e c t o r o f a p o i n t r e l a t i v e t o t h e s e l e n o g r a p h i c c o o r d i n a t e f r a m e w i t h c a r t e s i a n c o m p o n e n t s x ^ , y ^ , a n d z ^ .

m ^ = m o o n ' s m a s s

= s p h e r i c a l h a r m o n i c c o e f f i c i e n t s

(27)

T h e c o r r e s p o n d i n g g r a v i t a t i o n i n t e n s i t y v e c t o r , gy

r e l a t i v e t o t h e c a r t e s i a n ( n o n i n e r t i a l ) s e l e n o - g r a p h i c c o o r d i n a t e f r a m e i s

:

( Γ

,

(9Φ

[ 6 4 ]

### <[Δ ' ( Δ (Δ

W h e r e t h e u n i t v e c t o r t r i a d o f t h e s e l e n o g r a p h i c f r a m e i s d e n o t e d b y x ^ , y ^ , a n d z ^ a n d

[ 6 5 ]

C a r r y i n g o u t t h e i n d i c a t e d p a r t i a l d i f f e r e n t i a t i o n

5 Χ

<?Φ„

k

J

5 ^

- 5

1 - 5

r 2

r 2

[ 6 6 A]

[ 6 6B]

[ 6 6C]

### 175

(28)

I t i s a p p a r e n t f r o m t h e a b o v e e q u a t i o n s t h a t t h e p o i n t m a s s f i e l d e x p r e s s i o n s m a y b e s e p a r a t e d f r o m t h e r e m a i n d e r o f E q . 6 6 so t h a t

1671

### da

3

S i n c e t h e f o r e g o i n g e x p r e s s i o n f o r g^ r e f e r s to t h e n o n - i n e r t i a l s e l e n o g r a p h i c c o o r d i n a t e f r a m e t h a t is d e f i n e d b y t h e r o t a t i n g and l i b r a t i n g p r i n c i p a l a x i s o f s y m m e t r y o f t h e m o o n ' s f i g u r e , t h e r e l a t i v e e q u a t i o n s o f m o t i o n o f a s p a c e v e h i c l e a r e w r i t t e n r e l a t i v e t o a N e w t o n i a n f r a m e w i t h c o m m o n o r i g i n to t h e s e l e n o g r a p h i c f r a m e . T h i s c h o i c e s i m p l i f i e s t h e r e s u l t i n g e x p r e s s i o n s a n d c a l c u l a t i o n s b u t r e q u i r e s t h e i n t r o d u c t i o n o f t h e s e l e n o g r a p h i c r o t a t i o n m a t r i x 0( t ) , w h i c h is d i s c u s s e d l a t e r . T h e e q u a t i o n s o f m o t i o n o f a s p a c e v e h i c l e r e l a t i v e to an i n e r t i a l f r a m e w i t h o r i g i n at t h e m o o n ' s c e n t e r o f m a s s a n d w i t h i , a n d i c o p l a n a r t o E a r t h ' s e q u a t o r i a l p l a n e a r e

r A

3 m

### A \

w h e r e r ^ = p o s i t i o n v e c t o r f r o m m o o n c e n t e r t o v e h i c l e r_ = p o s i t i o n v e c t o r f r o m e a r t h c e n t e r t o v e h i c l e r k = p o s i t i o n v e c t o r f r o m v e h i c l e t o t h e k t h b o d y

( p l a n e t o r s u n )

r ^k = p o s i t i o n v e c t o r f r o m t h e m o o n c e n t e r t o t h e k t h b o d y ( p l a n e t o r s u n )

T h e e q u a t i o n s o f m o t i o n o f a s p a c e v e h i c l e r e l a t i v e t o an i n e r t i a l f r a m e w i t h o r i g i n at E a r t h ' s c e n t e r o f m a s s u n d e r t h e p e r t u r b i n g a c c e l e r a t i o n s o f b o t h t h e o b l a t e e a r t h g r a v i t a t i o n a l f i e l d a n d t h e t r i a x i a l m o o n g r a v i t a t i o n f i e l d a r e

(29)

### -k

2

d t2 Δ Φ

% Δ3

- Γ Θ Δ + V W +0 () ' W -t

### Ν k=l

?. e x t -» e x t

FA F0

r

3

3

w h e r e

Δ ( « Δ

=

### % k - ^ φ

Δ

A t m o s p h e r i c R e a c t i o n

A t m o s p h e r i c r e a c t i o n f o r c e s , i . e . , d r a g , s i d e , a n d l i f t f o r c e s a r e c o n s i d e r e d as e x t e r n a l f o r c e s a c t i n g o n l y o n t h e v e h i c l e a n d n o t o n a n y o f t h e r e m a i n i n g b o d i e s i n t h e N - b o d y p r o b l e m . H e n c e , t h e t e r m F ^e xV m ^ i s a s s u m e d t o b e

i d e n t i c a l l y z e r o i n t h e e q u a t i o n s o f m o t i o n d e v e l o p e d u p t o t h i s p o i n t . I f t h e s p a c e v e h i c l e , w h i c h w a s c o n s i d e r e d a s a p o i n t m a s s i n t h e p r e c e d i n g d i s c u s s i o n , i s i n r e a l i t y a

" w i n g e d " v e h i c l e , t h e n a r e l a t i v e w i n d - f i x e d c o o r d i n a t e f r a m e m u s t b e d e f i n e d a s w e l l a s t h e f o r c e l a w s .

B y c o n v e n t i o n , d r a g f o r c e s a r e a s s u m e d t o b e a c t i n g i n t h e d i r e c t i o n o f t h e r e l a t i v e w i n d w h e r e t h e r e l a t i v e w i n d v e l o c i t y v e c t o r i s s i m p l y t h e a p p a r e n t v e l o c i t y o f t h e

a t m o s p h e r e w i t h r e s p e c t t o a n a s s u m e d m o t i o n l e s s v e h i c l e . I t s h o u l d b e n o t e d t h a t o t h e r c o n v e n t i o n s e x i s t i n w h i c h d r a g f o r c e s a r e d e f i n e d a s t h e v e c t o r c o m p o n e n t o f t h e t o t a l a t m o s p h e r i c r e a c t i o n f o r c e r e s o l v 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 o f a v e h i c l e . U s i n g t h e a d o p t e d c o n v e n t i o n , t h e r e m a i n - ing t w o o r t h o g o n a l c o m p o n e n t s o f r e a c t i o n f o r c e m u s t l i e b y n e c e s s i t y i n t h e p l a n e n o r m a l t o t h e r e l a t i v e w i n d v e c t o r . T h e s e t w o c o m p o n e n t s , w h i c h w i l l b e d e f i n e d , a r e t h e l i f t f o r c e a n d s i d e f o r c e v e c t o r c o m p o n e n t s o f t h e t o t a l r e a c t i o n f o r c e .

T h e a t m o s p h e r e o f a p l a n e t i s a s s u m e d to^ b e r o t a t i n g w i t h t h e p l a n e t1s r o t a t i o n a l v e l o c i t y v e c t o r , Ω ι . H e n c e , t h e

### 175

(30)

v e l o c i t y o f t h e a t m o s p h e r e w i t h r e s p e c t to i n e r t i a l s p a c e at t h e p o s i t i o n o f a v e h i c l e is

VA(R I) = Ω . Χ Γ - J J

### [70]

T h e v e l o c i t y o f t h e a t m o s p h e r e V ^ , r e l a t i v e to w h i c h h a s an i n e r t i a l v e l o c i t y ( d / d t ) r -

t h e r e l a t i v e w i n d

a v e h i c l e i s t h e v e l o c i t y o f

VR =

d t

### [71]

In i n e r t i a l g e o c e n t r i c - e q u a t o r i a l c o o r d i n a t e s , a s s u m i n g Üi

to be d i r e c t e d a l o n g iz, E q . 7 1 m a y b e w r i t t e n

w h e r e

VR

v Y

fi

iV( VV -

i7V

### y y [72]

d t

V i + V i + V i x x y y ζ ζ

T h e p r o j e c t i o n s o f t h e t o t a l a t m o s p h e r i c r e a c t i o n f o r c e v e c t o r o n t o t h e p l a n e n o r m a l t o t h e r e l a t i v e w i n d v e l o c i t y v e c t o r i n t h e d i r e c t i o n s o f VDx ~ r a n d VDx ( VDx r ) a r e d e f i n e d

as t h e s i d e - f o r c e S a n d t h e l i f t - f o r c e v e c t o r s , L , r e s p e c - t i v e l y . T h e r e m a i n i n g c o m p o n e n t o f t h e t o t a l a t m o s p h e r i c r e a c t i o n v e c t o r is i n t h e d i r e c t i o n o f t h e r e l a t i v e w i n d v e c t o r a n d i s d e f i n e d a s t h e d r a g - f o r c e D . T h e t r i a d D , S,

a n d L f o r m s a m u t u a l l y o r t h o g o n a l s e t w i t h u n i t v e c t o r s

iD ' a n d w h e r e

R- VX - Q y >

y

+

z

### I — 1 [73]

(31)

VRx r VRr

y Vz - z Vy + Ωχζ + i y

z Vx " x Vz + ß z y

VD * r

+ ir

x Vy - y Vx - Ω( χ 2 + y2

VR * r

[74]

XL = iDx iS =

- ( Vy- O x ) x Vy- y Vx ) - Ω ( x2+ y2 jj

+ Vz [( z Vx- x Vz +O y z ]

"Vz ((yVz-zVy)+OxzJ

+ ( Vx+ ny) [ u Vy- y Vx) - n(x2 + y 2 jj

- ( Vx+ O y ) ^ z Vx- x Vz) + nyz j

+ (Vy-Ωχ) £ ( y Vz- z Vy) + O x z

/ith vR = + £ νχ+ Ω7) 3 + ( V y - Ω χ )2 +

## vz2J

r = + [ x2 + y2 + z2]

1/2 1 / 2

[75]

[76a]

[76b]

D r a g , s i d e , a n d l i f t f o r c e l a w s a r e t h e n d e f i n e d i n f o l l o w i n g m a n n e r

### D » 1/2 C u V V j ,

2

S Ξ 1/2 C g A g p VR 2

L *

### 1/2

[77 a]

[77 b]

[77c]

w i t h D = D i p , S = S is, L = L iL [78]

### 177

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