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to Equation (171
), we obtain our 3 PN equations of motion for two spherical
compact stars whose representative points are defined by Equation (176),
in the harmonic gauge.
Now we list some features of our 3 PN equations of motion. In the test-particle limit, our 3 PN
equations of motion coincide with a geodesic equation for a test-particle in the Schwarzschild metric in
harmonic coordinates (up to 3 PN order). Suppose that star 
is a test particle, star 
is represented
by the Schwarzschild metric, and 
. Then the geodesic equation for star 
becomes
) coincides with the geodesic equation
for a test particle in the Schwarzschild metric up to 3 PN order.
With the help of the formulas developed in [28], we have checked the Lorentz invariance of
Equation (181) (in the post-Newtonian perturbative sense). Also, we have checked that our
3 PN acceleration admits a conserved energy of the binary orbital motion (modulo the 2.5 PN
radiation reaction effect). In fact, the energy 
of the binary associated with Equation (181) is
], the
relation between their 3 PN equations of motion and our result described in Section 8.5 below, and
Equation (167). (After constructing 
given as in Equation (183), we have checked that our 3 PN
equations of motion make 
to be conserved.)
We note that Equation (171) as well gives a correct geodesic equation in the test-particle limit, is
Lorentz invariant, and admits the conserved energy. These facts can be seen by the form of
, Equation (175); it is zero when 
, is Lorentz invariant up to 3 PN order,
and is the effect of the mere redefinition of the dipole moments which does not break energy
conservation.
Finally, we here mention one computational detail. We have retained during our calculation
-dependent terms with positive powers of 
or logarithms of 
. As stated below Equation (76), it
is a good computational check to show that our equations of motion do not depend on 
physically. In
fact, we found that the 
-dependent terms cancel each other out in the final result. There is no
need to employ a gauge transformation to remove such an 
dependence. As for terms with
negative powers of 
, we simply assume that those terms cancel out the 
dependent
terms from the far zone contribution. Indeed, Pati and Will [129, 130], whose method we have
adopted to compute the far zone contribution, have proved that all the 
-dependent terms
cancel out between the far zone and the near zone contributions through all post-Newtonian
orders.
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| http://www.livingreviews.org/lrr-2007-2 | 
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![dvi1 m1m2 i m1 ----= − --2---n12 dτ r12 [ ] 2m1m2-- i 2 2 3- 2 5m1- 4m2- +ε r212 n 12 − v1 − 2v2 + 4(⃗v1 ⋅⃗v2) + 2(⃗n12 ⋅⃗v2) + r12 + r12 m1m2 +ε2---2--V i[4(⃗n12 ⋅⃗v1) − 3(⃗n12 ⋅⃗v2)] r12 [ 4m1m2-- i 4 2 2 3-2 2 9- 2 2 +ε r2 n 12 − 2v2 + 4v2(⃗v1 ⋅⃗v2) − 2(⃗v1 ⋅⃗v2) + 2v1(⃗n12 ⋅⃗v2) + 2 v2(⃗n12 ⋅⃗v2) 12 2 2 − 6(⃗v ⋅⃗v )(⃗n ⋅⃗v )2 − 15(⃗n ⋅⃗v )4 − 57m-1 − 9 m-2− 69m1m2-- 1 2 12 2 8 12 2 4 r212 r212 2 r212 m ( 15 5 5 39 + --1 − ---v21 + -v22 − --(⃗v1 ⋅⃗v2) +---(⃗n12 ⋅⃗v1)2 r12 4 4 2 2 ) 17 2 − − 39(⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2) + --(⃗n12 ⋅⃗v2) 2 ] m2- ( 2 2 2) + r12 4v2− 8(⃗v1 ⋅ ⃗v2)+2 (⃗n12 ⋅⃗v1) − 4(⃗n12 ⋅ ⃗v1)(⃗n12 ⋅⃗v2)− 6(⃗n12 ⋅⃗v2) [ ( ) +ε4m1m2--V i m1- − 63(⃗n12 ⋅⃗v1) + 55(⃗n12 ⋅⃗v2) + m2- (− 2 (⃗n12 ⋅⃗v1) − 2(⃗n12 ⋅⃗v2)) r212 r12 4 4 r12 2 2 2 ⃗ + v1(⃗n12 ⋅⃗v2) + 4v2(⃗n12 ⋅⃗v1) − 5v 2(⃗n1]2 ⋅⃗v2) − 4(⃗v1 ⋅⃗v2)(⃗n12 ⋅V ) 2 9- 3 − 6(⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2) + 2(⃗n12 ⋅⃗v2) 2 [ ( ) ( ) ] +ε54m--1m2- ni (⃗n ⋅ ⃗V ) − 6 m1-+ 52m2- + 3V 2 + Vi 2m1- − 8 m2-− V 2 5r312 12 12 r12 3 r12 r12 r12 m m [35 15 15 +ε6--12-2ni12 ---(⃗n12 ⋅⃗v2)6 −---(⃗n12 ⋅⃗v2)4v21 + --(⃗n12 ⋅⃗v2)4(⃗v1 ⋅⃗v2) r12 16 8 2 2 2 15- 4 2 3- 2 2 2 + 3(⃗n12 ⋅⃗v2)(⃗v1 ⋅⃗v2) − 2 (⃗n12 ⋅⃗v2) v2 + 2(⃗n12 ⋅⃗v2)v1v2 15 − 12(⃗n12 ⋅⃗v2)2(⃗v1 ⋅⃗v2)v22 − 2(⃗v1 ⋅⃗v2)2v22 +---(⃗n12 ⋅⃗v2)2v42 4 6 2 + 4(⃗v1 ⋅⃗v2)v2 − 2v 2 m1 ( 171 171 + --- − ----(⃗n12 ⋅⃗v1)4 +----(⃗n12 ⋅⃗v1)3(⃗n12 ⋅⃗v2) r12 8 2 723- 2 2 383- 3 − 4 (⃗n12 ⋅⃗v1) (⃗n12 ⋅⃗v2) + 2 (⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2) 455 4 229 2 2 205 2 − ----(⃗n12 ⋅⃗v2) + ----(⃗n12 ⋅⃗v1) v1 −----(⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2)v1 8 4 2 + 191-(⃗n12 ⋅⃗v2)2v21 − 91-v41 − 229(⃗n12 ⋅⃗v1)2(⃗v1 ⋅⃗v2) 4 8 2 225- 2 + 244(⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2)(⃗v1 ⋅⃗v2) − 2 (⃗n12 ⋅⃗v2) (⃗v1 ⋅⃗v2) 91 2 177 2 229 2 2 + ---v1(⃗v1 ⋅⃗v2) −----(⃗v1 ⋅⃗v2) + ----(⃗n12 ⋅⃗v1) v2 2 4 4 − 283-(⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2)v22 + 259(⃗n12 ⋅⃗v2)2v22 − 91v21v22 2 ) 4 4 2 81-4 + 43(⃗v1 ⋅⃗v2)v2 − 8 v2 m ( + --2 − 6(⃗n12 ⋅⃗v1)2(⃗n12 ⋅⃗v2)2 + 12(⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2)3 r12 + 6(⃗n12 ⋅⃗v2)4 + 4(⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2)(⃗v1 ⋅⃗v2) 2 2 + 12(⃗n12 ⋅⃗v2) (⃗v1 ⋅⃗v2) + 4(⃗v1 ⋅⃗v2) ) − 4(⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2)v22 − 12(⃗n12 ⋅⃗v2)2v22 − 8(⃗v1 ⋅ ⃗v2)v22 + 4v42 2( + m-2 − (⃗n12 ⋅⃗v1)2 + 2(⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2) + 43(⃗n12 ⋅⃗v2)2 r212 2 ) + 18 (⃗v1 ⋅⃗v2) − 9v22 ( m1m2-- 415- 2 375- 1113- 2 + r2 8 (⃗n12 ⋅⃗v1) − 4 (⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2) + 8 (⃗n12 ⋅⃗v2) 12 2 2 ) − 615π--(⃗n ⋅V⃗)2+18v2 + 123-π-V 2+33 (⃗v ⋅⃗v )− 33v2 64 12 1 64 1 2 2 2 m2 ( 2069 939 + -21 − -----(⃗n12 ⋅⃗v1)2 + 543 (⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2) −--(⃗n12 ⋅ ⃗v2)2 r12 8 ) 4 471 2 357 357 2 + ---v1 − ----(⃗v1 ⋅⃗v2) +----v2 8 ( 4 ) 8 ( )] 16m32- m21m2-- 547- 41-π2 13m31- m1m22- 545- 41π2- + r3 + r3 3 − 16 − 12r3 + r3 3 − 16 [ 12 12 12 12 +ε6m1m2--V i 15(⃗n ⋅⃗v )(⃗n ⋅⃗v )4 − 45(⃗n ⋅⃗v )5 − 3(⃗n ⋅⃗v )3v2 r212 2 12 1 12 2 8 12 2 2 12 2 1 + 6(⃗n ⋅⃗v )(⃗n ⋅⃗v )2(⃗v ⋅⃗v ) − 6(⃗n ⋅⃗v )3(⃗v ⋅ ⃗v ) 12 1 12 2 1 2 12 2 1 2 − 2(⃗n12 ⋅⃗v2)(⃗v1 ⋅⃗v2)2 − 12 (⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2)2v22 + 12(⃗n12 ⋅⃗v2)3v22 + (⃗n ⋅⃗v )v2v2 − 4(⃗n ⋅⃗v )(⃗v ⋅⃗v )v2+ 8(⃗n ⋅⃗v )(⃗v ⋅⃗v )v2 12 2 142 12 1 41 2 2 12 2 1 2 2 + 4(⃗n12 ⋅⃗v1)v2 − 7(⃗n12 ⋅⃗v2)v2 m2-( 2 2 3 + r12 − 2(⃗n12 ⋅⃗v1) (⃗n12 ⋅⃗v2) + 8(⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2) + 2 (⃗n12 ⋅⃗v2) + 2(⃗n ⋅⃗v )(⃗v ⋅⃗v ) + 4(⃗n ⋅⃗v )(⃗v ⋅⃗v ) 12 1 1 2 12 ) 2 1 2 − 2(⃗n12 ⋅⃗v1)v22 − 4(⃗n12 ⋅⃗v2)v22 m ( 243 565 + --1 − ----(⃗n12 ⋅⃗v1)3 + ----(⃗n12 ⋅⃗v1)2(⃗n12 ⋅⃗v2) r12 4 4 269- 2 95- 3 207- 2 − 4 (⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2) − 12(⃗n12 ⋅⃗v2) + 8 (⃗n12 ⋅⃗v1)v1 137 27 − ----(⃗n12 ⋅⃗v2)v21 − 36(⃗n12 ⋅⃗v1)(⃗v1 ⋅⃗v2) +--(⃗n12 ⋅⃗v2)(⃗v1 ⋅⃗v2) 8 ) 4 + 81-(⃗n ⋅⃗v )v2 + 83-(⃗n ⋅⃗v )v2 8 12 1 2 8 12 2 2 2 + m-2(4(⃗n12 ⋅⃗v1) + 5(⃗n12 ⋅⃗v2)) r212 m1m2 ( 307 479 123π2 ) + --2--- − ---(⃗n12 ⋅⃗v1) + ---(⃗n12 ⋅⃗v2) + ------(⃗n12 ⋅ ⃗V ) r12( 8 8 ) ] 32](article849x.gif)
![1- 2 m1m2-- E = 2 m1v 1 − 2r [ 12 2 ( ) ] + ε2 3-m1v4 + m-1m2-+ m1m2-- 3v2− 7(⃗v1 ⋅⃗v2) − 1(⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2) 8 1 2r212 2r12 1 2 2 [ 5 m3 m 19m2 m2 + ε4 ---m1v61 − --13-2− ----13--2 16 ( 2r12 8r12 ) m21m2 2 7 2 29 2 13 2 + ---2-- − 3v1 + -v2 + ---(⃗n12 ⋅⃗v1) − ---(⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2) + (⃗n12 ⋅⃗v2) 2r12 ( 2 2 2 m1m2-- 3- 3 3- 2 2 9- 2 + 4r12 2(⃗n12 ⋅⃗v1) (⃗n12 ⋅⃗v2) + 4 (⃗n12 ⋅⃗v1) (⃗n12 ⋅⃗v2) − 2 (⃗n12 ⋅⃗v1)(⃗n12 ⋅ ⃗v2)v 1 13 21 13 − ---(⃗n12 ⋅⃗v2)2v21 +---v41 + --(⃗n12 ⋅⃗v1)2(⃗v1 ⋅⃗v2) 2 2 2 )] 55- 2 17- 2 31- 2 2 + 3(⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2)(⃗v1 ⋅⃗v2) − 2 v1(⃗v1 ⋅⃗v2) + 2 (⃗v1 ⋅⃗v2) + 4 v1v 2 [ 4 3 2 + ε6 -35-m v8+ 3m-1m2- + 469m-1m-2- 128 1 1 8r412 18r412 m2 m2 ( 547 3115 123π2 + --13-2 ----(⃗n12 ⋅⃗v1)2 −-----(⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2) −------(⃗n12 ⋅⃗v1)(⃗n12 ⋅ ⃗V) 2r12 6 24 ) 32 575 2 41π2 4429 − ----v1 + -----(V⃗ ⋅⃗v2) + ----(⃗v1 ⋅⃗v2) ( 9 32 72 m31m2- 437- 2 317- 2 301- 2 + 2r3 − 4 (⃗n12 ⋅⃗v1) + 4 (⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2) + 3(⃗n12 ⋅⃗v2) + 12 v1 12 ) − 337-(⃗v ⋅⃗v ) + 5-v2 12 1 2 2 2 m m ( 5 5 5 + --1--2 − --(⃗n12 ⋅⃗v1)5(⃗n12 ⋅⃗v2) −---(⃗n12 ⋅⃗v1)4(⃗n12 ⋅⃗v2)2 −--(⃗n12 ⋅⃗v1)3(⃗n12 ⋅⃗v2)3 r12 16 16 32 19- 3 2 15- 2 2 2 + 16 (⃗n12 ⋅⃗v1) (⃗n12 ⋅⃗v2)v 1 + 16 (⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2) v1 3 19 21 + --(⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2)3v21 +---(⃗n12 ⋅⃗v2)4v21 −---(⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2)v41 4 16 16 − 2(⃗n ⋅⃗v )2v4 + 55v6 − 19-(⃗n ⋅⃗v )4(⃗v ⋅⃗v ) − (⃗n ⋅ ⃗v )3(⃗n ⋅⃗v )(⃗v ⋅⃗v ) 12 2 1 16 1 16 12 1 1 2 12 1 12 2 1 2 15- 2 2 45- 2 2 − 32 (⃗n12 ⋅⃗v1) (⃗n12 ⋅ ⃗v2)(⃗v1 ⋅⃗v2) + 16(⃗n12 ⋅⃗v1) v1(⃗v1 ⋅⃗v2) 5 11 139 + --(⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2)v21(⃗v1 ⋅⃗v2) +--(⃗n12 ⋅⃗v2)2v21(⃗v1 ⋅⃗v2) −----v41(⃗v1 ⋅⃗v2) 4 4 16 − 3-(⃗n ⋅⃗v )2(⃗v ⋅⃗v )2 + -5-(⃗n ⋅⃗v )(⃗n ⋅ ⃗v )(⃗v ⋅⃗v )2 + 41v2(⃗v ⋅⃗v )2 4 12 1 1 2 16 12 1 12 2 1 2 8 1 1 2 -1- 3 45- 2 2 2 23- 2 2 79- 4 2 + 16 (⃗v1 ⋅⃗v2) − 16(⃗n12 ⋅⃗v1) v1v2 − 32(⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2)v1v2 + 16 v1v2 ) − 161-v21v22(⃗v1 ⋅⃗v2) 32 m2 m2 ( 49 75 187 11 + --12--- − --(⃗n12 ⋅⃗v1)4 + --(⃗n12 ⋅⃗v1)3(⃗n12 ⋅⃗v2) − ----(⃗n12 ⋅⃗v1)2(⃗n12 ⋅⃗v2)2 +---v41 r12 8 8 8 2 247- 3 49- 2 2 81- 2 + 24 (⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2) + 8 (⃗n12 ⋅⃗v1)v1 + 8 (⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2)v1 21 15 3 − ---(⃗n12 ⋅⃗v2)2v21 −---(⃗n12 ⋅⃗v1)2(⃗v1 ⋅⃗v2) −--(⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2)(⃗v1 ⋅⃗v2) 4 2 2 + 21-(⃗n12 ⋅⃗v2)2(⃗v1 ⋅⃗v2) − 27v2(⃗v1 ⋅⃗v2) + 55(⃗v1 ⋅⃗v2)2 + 49(⃗n12 ⋅⃗v1)2v2 4 1 2 4 2 27- 2 3- 2 2 55- 2 2 2 − 2 (⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2)v2 + 4(⃗n12 ⋅⃗v2)v2 + 4 v1v2 − 28(⃗v1 ⋅⃗v2)v2 135 ) ] + ----v42 16 + (1 ↔ 2) + 𝒪 (ε7). (183 )](article856x.gif)