Arbitrary constant εRA

8.2 Arbitrary constant $εRA$

The reason logarithms appear in the 3 PN equations of motion (171

) 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

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

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 (171 ):

Note that this redefinition of the center of mass does not affect the existence of the energy conservation as was shown by Equation (167

). We can examine the effect of this redefinition on the equations of motion using Equation (166

) (using $i δA ln$ instead of $δA Θ$ ). Then we have

i i 2 i m1ai1&#x007C;&#x03B4;A ln = &#x2212; &#x03B5;63m1-&#x03B4;2ln-n&#x27E8;1ik2&#x27E9;+ &#x03B5;6 3m2-&#x03B4;1lnn &#x27E8;i12k&#x27E9;&#x2212; &#x03B5;6d-&#x03B4;1ln r312 r312 d &#x03C4;2 44m3 m2 &#x0028; r12 &#x0029; 44m2 m3 &#x0028; r12 &#x0029; = &#x2212; ----15--2ni12 ln ---- + -----15-2 ni12ln ---- 3r12 &#x03B5;R1 3r12 &#x03B5;R2 &#x0028; &#x0029; 22m31m2- &#x0028; &#x20D7; 2 i 2 i &#x20D7; i&#x0029; -r12 &#x2212; r4 5&#x0028;&#x20D7;n12 &#x22C5;V &#x0029;n 12 &#x2212; V n12 &#x2212; 2&#x0028;&#x20D7;n12 &#x22C5;V &#x0029;V ln &#x03B5;R 132 &#x0028; 1 &#x0029; + 22m-1m2- m1-ni + m2-ni &#x2212; V2ni + 8&#x0028;&#x20D7;n &#x22C5; &#x20D7;V &#x0029;2ni &#x2212; 2&#x0028;&#x20D7;n &#x22C5; &#x20D7;V &#x0029;2Vi . &#x0028;175 &#x0029; 3r412 r12 12 r12 12 12 12 12 12

Comparing the above equations with Equation (171

), we easily conclude that the representative point $i zA$ of star $A$ defined by

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 [27 ] 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.