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8.2 Arbitrary constant εRA

The reason logarithms appear in the 3 PN equations of motion (171View Equation) is in a sense easy to understand. Since the post-Newtonian approximation is a weak field expansion, at some level of iteration in the evaluation of field, we have inevitably logarithms in the field through volume integrals such as
∫ 3 ( )n --d-y-- m- , (172 ) N∕B |⃗x − ⃗y| y
where m and y are typical the mass and length scale in the orbital motion, respectively, and n is a positive integer. For instance consider 3 5 m 1(⃗r1 ⋅⃗a1)∕r1 as an integrand. Then we find
∫ ( ) --d3y--m31(⃗y1 ⋅⃗a1) m31(⃗r1-⋅⃗a1)- -r1- N∕B |⃗x − ⃗y| y51 = 3r31 ln εR1 + .... (173 )

Actually, we could remove the εRA dependence in our 3 PN equations of motion via an alternative choice of the center of mass. The following alternative choice of the representative point of star A removes the εRA dependence in Equation (171View Equation):

22 ( r12) DiA Θ,new(τ) = ε4δiAΘ (τ ) − ε4--m3AaiA ln ---- ≡ ε4δiAΘ(τ) + ε4δiAln(τ) ≡ ε4δiA(τ). (174 ) 3 εRA
Note that this redefinition of the center of mass does not affect the existence of the energy conservation as was shown by Equation (167View Equation). We can examine the effect of this redefinition on the equations of motion using Equation (166View Equation) (using i δA ln instead of δA Θ). Then we have
i i 2 i m1ai1|δA ln = − ε63m1-δ2ln-n⟨1ik2⟩+ ε6 3m2-δ1lnn ⟨i12k⟩− ε6d-δ1ln r312 r312 d τ2 44m3 m2 ( r12 ) 44m2 m3 ( r12 ) = − ----15--2ni12 ln ---- + -----15-2 ni12ln ---- 3r12 εR1 3r12 εR2 ( ) 22m31m2- ( ⃗ 2 i 2 i ⃗ i) -r12 − r4 5(⃗n12 ⋅V )n 12 − V n12 − 2(⃗n12 ⋅V )V ln εR 132 ( 1 ) + 22m-1m2- m1-ni + m2-ni − V2ni + 8(⃗n ⋅ ⃗V )2ni − 2(⃗n ⋅ ⃗V )2Vi . (175 ) 3r412 r12 12 r12 12 12 12 12 12
Comparing the above equations with Equation (171View Equation), we easily conclude that the representative point i zA of star A defined by
∫ i − 6 3 i i ττ k 4 i D AΘ,new(τ) = ε d y (y − zA(τ)) ΘN (τ,y ) = ε δA(τ) (176 ) BA
obeys the equations of motion free from any logarithmic term and hence free from any ambiguity up to 3 PN order inclusively. We note that in our formalism ziA is defined by the value of DiA, and in turn we have a freedom to assign to Di A any value as we like (though it may be natural to set the value of i D A such that i zA resides inside star A). We also note that we define i zA order by order.

We mention here that Blanchet and Faye [27Jump To The Next Citation Point] have already noticed that in their 3 PN equations of motion a suitable coordinate transformation removes (parts of) the logarithmic dependence of arbitrary parameters corresponding (roughly) to our body zone radii14. It is well-known that choosing different values of dipole moments corresponds to a coordinate transformation.

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