C H A P T E R
13
GENERAL
RELATIVISTIC
MODELS
FOR
COORDINATES A N D EQUATIONS OF
TIME,
MOTION
The r e l a t i v i s t i c t r e a t m e n t of the 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 problem includes c o r r e c t i o n to the e q u a t i o n s of
m o t i o n , the time t r a n s f o r m a t i o n s , and the m e a s u r e m e n t m o d e l .
The
two c o o r d i n a t e Systems g e n e r a l l y used when including r e l a t i v i t y in
n e a r - E a r t h orbit d e t e r m i n a t i o n S o l u t i o n s are the solar
System
b a r y c e n t r i c frane of r e f e r e n c e and the 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 .
Ashby and Bertotti (1986) constructed a locally inertial Eframe in the n e i g h b o r h o o d of the g r a v i t a t i n g Earth and d e m o n s t r a t e d
that the g r a v i t a t i o n a l e f f e c t s of the Sun, M o o n , and other p l a n e t s
are basically reduced to their tidal f o r c e s , with very small
r e l a t i v i s t i c c o r r e c t i o n s . T h u s the main 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 satellite are those described by the S c h w a r z s c h i l d field
of the Earth itself.
T h i s result makes the 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 the motion of a n e a r - E a r t h satellite (Ries,
et a l . ,
1988).
The time c o o r d i n a t e in the inertial E-frame is T e r r e s t r i a l
Time (designated TT) (Guinot, 1991) which can be c o n s i d e r e d to be
e q u i v a l e n t to the p r e v i o u s l y defined T e r r e s t r i a l D y n a m i c a l T i m e
(TDT) .
T h i s time c o o r d i n a t e
(TT) is realized in p r a c t i c e by
I n t e r n a t i o n a l Atomic T i m e ( T A I ) , w h o s e rate is defined by the
a t o m i c second in the International System of U n i t s (SI) . T e r r e s trial T i m e adopted by the International A s t r o n o m i c a l U n i o n in 1991
d i f f e r s from G e o c e n t r i c C o o r d i n a t e Time (TCG) by a scaling factor:
X 10 !u X (MJD-4 3144 . 0) X 86400 s e c o n d s ,
JA *9?i
w h e r e MJD r e f e r s to the modified J u l i a n d a t e .
Figure 13.1 s h o w s
g r a p h i c a l l y the r e l a t i o n s h i p s between the time s c a l e s .
T C G - T T = 6.9693
E q u a t i o n s of M o t i o n for an Artificial E a r t h Satellite
The correction
s a t e l l i t e Aa is
Aa =
-
GM,
2-f
c2rJ
where
|[2(/3
ff
L
to
+7)
the
GM,e
—
r
acceleration
"
7V2]r
+
of
[2(1
+
an
7)
artificial
(r-v)v]
c = speed of light,
= 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,ä = g e o c e n t r i c satellite p o s i t i o n , v e l o c i t y ,
acceleration, respectively,
Earth
(1)
ß,y
123
and
G M @ = g r a v i t a t i o n a l parameter of t h e
Earth.
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 ( f r a m e - d r a g g i n g ) , g e o d e s i c
(de Sitter) p r e c e s s i o n , and t h e relativistic e f f e c t s of t h e Earth's
oblateness have been neglected.
E q 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 F r a m e
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 solar s y s t e m frame of
r e f e r e n c e (the isotropic 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
B a r y c e n t r i c C o o r d i n a t e T i m e (TCB) as t h e t i m e c o o r d i n a t e ) are
required to d e s c r i b e t h e d y n a m i c s of the solar s y s t e m and a r t i f i c i a l p r o b e s m o v i n g about t h e s o l a r system (for e x a m p l e , see M o y e r ,
1971).
T h e s e are the e q u a t i o n s applied to t h e M o o n ' s m o t i o n for
L u n a r L a s e r R a n g i n g (Newhall, W i l l i a m s , and D i c k e y , 1 9 8 7 ) .
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 to the 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 required (see
Chapter 1 4 ) .
Scale Effect and Choice of T i m e Coordinate
T h e p r e v i o u s IAU d e f i n i t i o n of the time c o o r d i n a t e in the
b a r y c e n t r i c frame required 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 B a r y c e n t r i c D y n a m i c a l T i m e (TDB) and T e r r e s t r i a l Dynamical
T i m e (TDT) (Kaplan, 1 9 8 1 ) .
A s a c o n s e q u e n c e , the s p a t i a l c o o r d i n a t e s in the b a r y c e n t r i c f r a m e had to be r e s c a l e d t o k e e p the speed
of light u n c h a n g e d b e t w e e n t h e barycentric 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 w e r e c o m p a r e d 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 scale d i f f e r e n c e , L, appeared.
T h i s is no longer
r e q u i r e d w i t h the use of t h e TCG time s c a l e .
The difference between
F u k u s h i m a et al.
(1986) as
TCB
and
TDB
is
given
T C B - T D B = 1.550505X10' 8 (±1X10* 1 4 ) X (MJD-43144 . 0)
in
seconds
by
X 86400.
T h e d i f f e r e n c e b e t w e e n Barycentric C o o r d i n a t e T i m e (TCB) and
Geocentric
Coordinate
Time
(TCG)
involves a
four-dimensional
transformation,
t
T C B - T C G = c- 2 {/[v-v/2 + Uexl(xc)]dt + ve- (x-xc) } ,
t0
w h e r e x c and vc d e n o t e the b a r y c e n t r i c p o s i t i o n and v e l o c i t y of the
Earth's c e n t e r of m a s s and x is the b a r y c e n t r i c p o s i t i o n of the
o b s e r v e r . Ucxl is the N e w t o n i a n potential of all of t h e solar system
b o d i e s a p a r t from the Earth evaluated a t the g e o c e n t e r .
t 0 is
chosen t o be c o n s i s t e n t w i t h 1977 January 1, 0 h 0 m 0 S T A I and t is
124
TCB.
An a p p r o x i m a t i o n
(1986) as
is g i v e n
in
seconds
by
Fukushima
(TCB-TCG) = 1.480813X10- 8 (±1X10' 1 4 ) X (MJD-43144 . 0)
+ c"2vc- (x-x c ) + P.
al.
et
X 86400
w i t h M J D m e a s u r e d in T A I .
For o b s e r v e r s on the E a r t h ' s s u r f a c e ,
diurnal p e r i o d i c d i f f e r e n c e s d e n o t e d by P w i t h a m a x i m u m amplitude
of 2.1 ßs also r e m a i n .
T h e s e can be e v a l u a t e d from p o s i t i o n s and
m o t i o n s of solar system b o d i e s using e x p r e s s i o n s of H i r a y a m a et
al.
(1987) .
1976
1991
RECOMMENDATION
TDT
Terrestrial Dynamical Time
RECOMMENDATION
TT
Terrestrial Time
TDT = TT « TAI + 32sl84
V
II
T
TCG
Geocentric Coordinate Time
TCG - TT = 6.9693 X 10*"10 X AT
V
4-dimensional
space
transformation
Linear transformation
1.480813 X 10"8 X AT
T
TDB
Barycentric Dynamical Time
T
TCB
Barycentric Coordinate Time
TCB = TDB + 1,550505 X 10~8 X AT
AT = (date in days - 1977 January 1,0h)TAI X86400 see
Fig 13.1
R e l a t i o n s between time s c a l e s .
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125
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Constants
in the
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126