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8.1 Third post-Newtonian equations of motion with logarithmic terms

To derive the 3 PN equations of motion, we evaluate the surface integrals in the general form of the equations of motion (111View Equation) using the field ττ 8h, the field μν ≤6h, the 3 PN body zone contributions, and the 3 PN N ∕B contributions corresponding to the results from the super-potential method and the super-potential-in-series method. We then combine the result with the terms from the direct-integration method.

From 3 PN order, the effects of the QKli A and RKlij A integrals appearing in the 3 PN field in hμν B contribute to the 3 PN equations of motion. i 6Q AΘ given in Equation (162View Equation) affects the 3 PN equations of motion through the 3 PN momentum-velocity relation. Since we define the representative points of the stars via Equation (164View Equation), we add the corresponding acceleration given by Equation (166View Equation). Furthermore, our choice of the representative points of the stars makes Di Aχ appear independently of i D AΘ in the field, and hence i 4D Aχ affects the 3 PN equations of motion. In summary, the K i K ij ≤4Q A l,≤4R Al ,6QiAΘ,δiAΘ, and 4DiAχ contributions to the 3 PN field can be written as

∑ k ( ) 10hττ + 8hkk = 4 rA- δk + 4Dk + 4Rkll− 14Rllk + ..., (168 ) A=1,2r3A AΘ Aχ A 2 A
where “...” are other contributions. On the other hand, 6QiA Θ and δiAΘ affect the equations of motion through the momentum-velocity relation in Equation (111View Equation),
( dvi ) d Qi d2δi m1 ---1 = − ε6 -6-1Θ-− ε6 ---12Θ-+ ..., (169 ) dτ ≤3PN d τ dτ
but they cancel each other out, since we choose Equation (164View Equation). Then these contributions to a 3 PN acceleration can be summarized into
( ) the contribution to m1ai1 from 6118-m31m22 i 6118-m21m32 i ≤4QKAli,≤4RKlAij,6QiAΘ,δiA Θ,4DiAχ = ε 9 r612 r12 + ε 9 r612 r12. (170 )

Collecting these contributions mentioned above, we obtain the 3 PN equations of motion. However, we found that logarithmic terms having the arbitrary constants εRA in their arguments survive,

( dvi)with log ( dvi ) m1 --1- = m1 --1- dτ dτ ≤2.5PN m2 m [ 44m2 ( r ) 44m2 ( r ) + ε6--13--2 --2-1ni12ln --12 − ---22ni12 ln -12- r12 3r12 εR1 3r12 εR2 ( ) ] 22m1 ( 2 i 2 i i) r12 + -r---- 5(⃗n12 ⋅ ⃗V )n 12 − V n12 − 2(⃗n12 ⋅ ⃗V)V ln εR--- 12 1 + ⋅⋅⋅ + 𝒪(ε7), (171 )
where the acceleration through 2.5 PN order, (dvi∕dτ) 1 ≤2.5PN, is the Damour and Deruelle 2.5 PN acceleration. In our formalism, we have computed it in [95Jump To The Next Citation Point]. The “...” stands for the terms that do not include any logarithms.

Since this equation contains two arbitrary constants, the body zone radii RA, at first sight its predictive power on the orbital motion of the binary seems to be limited. In the next Section 8.2, we shall show that by a reasonable redefinition of the representative points of the stars, we can remove RA from our equations of motion. There, we show the explicit form of the 3 PN equations of motion we obtained.

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