Third post-Newtonian equations of motion with logarithmic terms

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 (111

) 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 (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 $Di Aχ$ 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

where “ $...$ ” are other contributions. On the other hand, $6QiA Θ$ and $δiAΘ$ affect the equations of motion through the momentum-velocity relation in Equation (111

but they cancel each other out, since we choose Equation (164

). Then these contributions to a 3 PN acceleration can be summarized into

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,

&#x0028; dvi&#x0029;with log &#x0028; dvi &#x0029; m1 --1- = m1 --1- d&#x03C4; d&#x03C4; &#x2264;2.5PN m2 m &#x005B; 44m2 &#x0028; r &#x0029; 44m2 &#x0028; r &#x0029; + &#x03B5;6--13--2 --2-1ni12ln --12 &#x2212; ---22ni12 ln -12- r12 3r12 &#x03B5;R1 3r12 &#x03B5;R2 &#x0028; &#x0029; &#x005D; 22m1 &#x0028; 2 i 2 i i&#x0029; r12 + -r---- 5&#x0028;&#x20D7;n12 &#x22C5; &#x20D7;V &#x0029;n 12 &#x2212; V n12 &#x2212; 2&#x0028;&#x20D7;n12 &#x22C5; &#x20D7;V&#x0029;V ln &#x03B5;R--- 12 1 + &#x22C5;&#x22C5;&#x22C5; + &#x1D4AA;&#x0028;&#x03B5;7&#x0029;, &#x0028;171 &#x0029;

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 [95

]. 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.