USNO Circular 179 - U.S. Naval Observatory

IAU RESOLUTIONS 2000 73
and admit an expansion into positive powers of X,
explicitly,
G00 = −1 + 2W
c2 2W 2
− c4 ,
G0a = − 4
c3 W a ,
Gab = δab
1 + 2
c 2 W
.
The potentials W and W a should be split according to
W (T, X) = WE(T, X) + Wext(T, X),
W a (T, X) = W a E (T, X) + W a ext(T, X).
The Earth’s potentials WE and W a E are defined in the same way as w and wi but with quantities
calculated in the GCRS with integrals taken over the whole Earth.
4. using, if accuracy requires, the full post-Newtonian coordinate transformation between the BCRS
and the GCRS as induced by the form of the corresponding metric tensors,
explicitly, for the kinematically non-rotating GCRS (T=TCG, t=TCB, ri E ≡ xi − xi E (t) and a
summation from 1 to 3 over equal indices is implied),
T = t − 1
c 2 [A(t) + v i E ri E
Xa
= δai ri E
where
+ 1
c 2
d 1
dtA(t) = 2v2 E + wext(xE),
] + 1
c 4 [B(t) + B i (t)r i E + Bij (t)r i E rj
E + C(t, x)] + O(c−5 ),
1
2vi Evj Erj E + wext(xE)ri E + ri Eaj Erj 1
E − 2aiE r2 E
d
1
dtB(t) = − 8v4 3
E − 2v2 Ewext(xE) + 4vi Ewi ext(xE) + 1
2w2 ext(xE),
B i (t) = − 1
2 v2 E vi E + 4wi ext(xE) − 3v i E wext(xE),
+ O(c −4 ),
B ij (t) = −v i E δajQ a + 2 ∂
∂x j w i ext(xE) − v i E ∂
∂x j wext(xE) + 1
2 δij ˙wext(xE),
C(t, x) = − 1
10 r2 E ( ˙ai E ri E ).
Here xi E , vi E , and aiE are the barycentric position, velocity and acceleration vectors of the Earth,
the dot stands for the total derivative with respect to t, and
Qa
= δai
.
∂
∂xi wext(xE) − a i E
The external potentials, wext and w i ext, are given by
wext =
A=E wA, w i ext =
A=E wi A ,
where E stands for the Earth and wA and w i A are determined by the expressions for w and wi with