From 3 PN order, the effects of the and integrals appearing in the 3 PN field in contribute to the 3 PN equations of motion. given in Equation (162) 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 (164), we add the corresponding acceleration given by Equation (166). Furthermore, our choice of the representative points of the stars makes appear independently of in the field, and hence affects the 3 PN equations of motion. In summary, the , and contributions to the 3 PN field can be written as
Collecting these contributions mentioned above, we obtain the 3 PN equations of motion. However, we found that logarithmic terms having the arbitrary constants in their arguments survive,. The “” stands for the terms that do not include any logarithms.
Since this equation contains two arbitrary constants, the body zone radii , 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 from our equations of motion. There, we show the explicit form of the 3 PN equations of motion we obtained.
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