### 8.2 Arbitrary constant

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 and are typical the mass and length scale in the orbital motion, respectively, and
is a positive integer. For instance consider as an integrand. Then we find
Actually, we could remove the 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 removes
the 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 instead of ). Then we have
Comparing the above equations with Equation (171), we easily conclude that the representative point
of star 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 is defined by the value of , and in turn we
have a freedom to assign to any value as we like (though it may be natural to set the value
of such that resides inside star ). We also note that we define 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
radii.
It is well-known that choosing different values of dipole moments corresponds to a coordinate
transformation.