C H A P T E R 15
G E N E R A L R E L A T I V I S T I C DYNAMICAL
MODEL
T h e r e l a t i v i s t i c t r e a t m e n t of t h e n e a r - E a r t h s a t e l l i t e orbit
d e t e r m i n a t i o n p r o b l e m i n c l u d e s c o r r e c t i o n t o t h e e q u a t i o n s of
m o t i o n , t h e t i m e t r a n s f o r m a t i o n s , and t h e m e a s u r e m e n t m o d e l .
The
t w o c o o r d i n a t e S y s t e m s g e n e r a l l y used w h e n including r e l a t i v i t y in
n e a r - E a r t h o r b i t d e t e r m i n a t i o n S o l u t i o n s a r e t h e s o l a r system
b a r y c e n t r i c frame of r e f e r e n c e and t h e g e o c e n t r i c or E a r t h - c e n t e r e d
frame of r e f e r e n c e .
A s h b y and B e r t o t t i (1986) c o n s t r u c t e d a locally inertial frame
in t h e n e i g h b o r h o o d of t h e g r a v i t a t i n g E a r t h and d e m o n s t r a t e d t h a t
t h e g r a v i t a t i o n a l e f f e c t s of t h e S u n , M o o n , and o t h e r p l a n e t s are
basically
reduced
to
their
tidal
forces,
with
very
small
relativistic corrections.
T h u s t h e m a i n r e l a t i v i s t i c e f f e c t s on
a n e a r - E a r t h s a t e l l i t e a r e t h o s e d e s c r i b e d by t h e S c h w a r z s c h i l d
field of t h e E a r t h i t s e l f . T h i s r e s u l t m a k e s t h e g e o c e n t r i c frame
m o r e s u i t a b l e for d e s c r i b i n g t h e m o t i o n of a n e a r - E a r t h s a t e l l i t e
(Ries, et a l . , 1 9 8 8 ) .
T h e t i m e c o o r d i n a t e in t h e inertial
Dynamical T i m e ( T D T ) . T h i s t i m e c o o r d i n a t e
by I n t e r n a t i o n a l A t o m i c T i m e ( T A I ) , w h o s e
a t o m i c second in t h e I n t e r n a t i o n a l System
E - f r a m e is T e r r e s t r i a l
is r e a l i z e d in p r a c t i c e
r a t e is d e f i n e d by t h e
of U n i t s ( S I ) .
E Q U A T I O N S O F M O T I O N FOR A N A R T I F I C I A L E A R T H
The correction
s a t e l l i t e Aa is
Aa = - - ^ %
c r
\[2(ß
to
+
7)
the
S B
^
acceleration
- 7V 2 ]? +
r
of
[2(1 +
SATELLITE
an
7)
artificial
(r-v)v]
Earth
(1)
where
c = speed of l i g h t ,
ß , 7 = PPN p a r a m e t e r s equal to 1 in G e n e r a l
Relativity,
r,v,a = g e o c e n t r i c s a t e l l i t e p o s i t i o n , v e l o c i t y ,
acceleration, respectively,
and
GMe = g r a v i t a t i o n a l p a r a m e t e r of t h e E a r t h .
T h e v a l u e o f GM® in t h e N e w t o n i a n t w o - b o d y a c c e l e r a t i o n and in Eq,
66
(1) s h o u l d b e t h e g e o c e n t r i c v a l u e b u t t h e d i f f e r e n c e b e t w e e n t h e
b a r y c e n t r i c and g e o c e n t r i c m a s s of t h e S u n , M o o n , and p l a n e t s is
not important when Computing the indirect Newtonian pertubations.
T h e e f f e c t s of L e n s e - T h i r r i n g p r e c e s s i o n (frame-dragging) , g e o d e s i c
(de S i t t e r ) p r e c e s s i o n , and t h e r e l a t i v i s t i c e f f e c t s of t h e E a r t h ' s
oblateness have been negelcted.
E O U A T I O N S O F M O T I O N IN T H E B A R Y C E N T R I C
FRAME
T h e n-body e q u a t i o n s of m o t i o n for t h e s o l a r s y s t e m frame of
r e f e r e n c e (the i s o t r o p i c P a r a m e t e r i z e d P o s t - N e w t o n i a n system w i t h
T D B as t h e t i m e c o o r d i n a t e ) a r e required t o d e s c r i b e t h e d y n a m i c s
of t h e s o l a r system and a r t i f i c i a l p r o b e s m o v i n g a b o u t t h e s o l a r
s y s t e m (for e x a m p l e , s e e M o y e r , 1 9 7 1 ) .
These are the equations
a p p l i e d t o t h e M o o n ' s m o t i o n for LLR (Newhall, W i l l i a m s , and
D i c k e y , 1987) . In a d d i t i o n , r e l a t i v i s t i c c o r r e c t i o n s t o t h e laser
r ä n g e m e a s u r e m e n t , t h e d a t a t i m i n g , and t h e S t a t i o n c o o r d i n a t e s a r e
r e q u i r e d (see C h a p t e r 1 6 ) .
SCALE EFFECT AND CHOICE OF TIME
COORDINATE
B e c a u s e t h e IAU d e f i n i t i o n of t h e t i m e c o o r d i n a t e in t h e
b a r y c e n t r i c frame r e q u i r e s t h a t only p e r i o d i c d i f f e r e n c e s e x i s t
b e t w e e n T D B and T D T (Kaplan, 1 9 8 1 ) , t h e s p a t i a l c o o r d i n a t e s in t h e
b a r y c e n t r i c frame h a v e e f f e c t i v e l y b e e n r e s c a l e d t o k e e p t h e speed
of l i g h t u n c h a n g e d b e t w e e n t h e b a r y c e n t r i c and t h e g e o c e n t r i c
frames (Misner, 1 9 8 2 ; H e l l i n g s , 1 9 8 6 ) .
T h u s , w h e n b a r y c e n t r i c (or
T D B ) u n i t s of length are compared to g e o c e n t r i c (or T D T ) u n i t s of
l e n g t h , a s c a l e d i f f e r e n c e , L, a p p e a r s .
Noting that the mass
p a r a m e t e r G M / c 2 or G m / c 2 h a s u n i t s of l e n g t h , t h e v a l u e for t h e m a s s
p a r a m e t e r of a b o d y in T D B u n i t s , GM, is r e l a t e d to its v a l u e in
T D T u n i t s , Gm, a c c o r d i n g to GM = (1-L) Gm.
It can b e s h o w n t h a t the v a l u e of t h e s c a l e d i f f e r e n c e d o e s
n o t i n c l u d e t h e c o n t r i b u t i o n of t h e g r a v i t a t i o n a l and r o t a t i o n a l
p o t e n t i a l of t h e E a r t h (Guinot and S e i d e l m a n n , 1 9 8 8 ; H u a n g , et a l . ,
1989) so t h a t t h e v a l u e of t h e scale d i f f e r e n c e b e t w e e n t h e t w o
frames is L = 1.4808 x 10" 8 (Fukushima, et a l . , 1 9 8 6 ) .
REFERENCES
A s h b y , N . and B e r t o t t i , B . , 1 9 8 6 , " R e l a t i v i s t i c E f f e c t s
I n e r t i a l F r a m e s , " P h y s . R e v . , D34 ( 8 ) , 2 2 4 6 .
in
Local
F u k u s h i m a , T . , F u j i m o t a , M. K., K i n o s h i t a , H . , and A o k i , S., 1 9 8 6 ,
"A S y s t e m of A s t r o n o m i c a l C o n s t a n t s
in t h e
Relativistic
Framework," Celest. Mech., 38, pp. 215-230.
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G u i n o t , B. and S e i d e l m a n n , P. K. , 1 9 8 8 , "Time S c a l e s : T h e i r
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H e l l i n g s , R. W. , 1 9 8 6 , " R e l a t i v i s t i c E f f e c t s in A s t r o n o m i c a l T i m i n g
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H u a n g , C. , R i e s , J. C , T a p l e y , B. D. , W a t k i n s , M . M. , 1 9 8 9 ,
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Kaplan,
G.
H.,
1981,
"The
IAU
Resolutions
on
Astronomical
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M i s n e r , C. W . , 1 9 8 2 , "Scale F a c t o r s for R e l a t i v i s t i c E p h e m e r i s
Coordinates,"
NASA
Contract
NAS5-25885,
Report,
EG&G,
Washington Analytical Services Center, Inc.
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32-1527.
N e w h a l l , X. X., W i l l i a m s , J. G., D i c k e y , J. O., 1 9 8 7 , " R e l a t i v i t y
Modelling
in
Lunar
Laser
Ranging
Data
Analysis,"
in
Proceedings
of t h e I n t e r n a t i o n a l A s s o c i a t i o n
of
Geodesy
Symposia, Vancouver, pp. 78-82.
R i e s , J. C , H u a n g , C. , and W a t k i n s , M. M. , 1 9 8 8 , "The E f f e c t of
G e n e r a l R e l a t i v i t y on N e a r - E a r t h S a t e l l i t e s in t h e S o l a r
S y s t e m B a r y c e n t r i c and G e o c e n t r i c R e f e r e n c e F r a m e s , " P h y s .
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68