Now we obtain from the Newtonian momentum-velocity relation:
We evaluate the surface integrals in the evolution equation for at 1 PN order. The result for star is
Then we have to calculate the 1 PN order . The result is . As depends on , we ignore it (see Section 4.5). As a result we obtain the momentum-velocity relation at 1 PN order, from Equation (86).
Now as for , we first calculate the surface integrals , , and from Equations (91) and (92). We then find that they depend on , hence we ignore them and obtain
To derive and , we have to evaluate non-compact support integrals for and , and the term in Equation (76) for :. Equation (101) for each integrand then becomes
Using the above results and the Newtonian equations of motion for the twice retardation expansion term, we finally obtain
Evaluating the surface integrals in Equation (111) as in the Newtonian case, we obtain the 1 PN equations of motion,
Finally let us give a summary of our procedure (see Figure 4). With the PN order equations of motion and in hand, we first derive the PN evolution equation for . Then we solve it functionally and obtain the mass-energy relation at PN order. Next we calculate at PN order and derive the momentum-velocity relation at PN order. Then we calculate and . With the PN mass-energy relation, the PN momentum-velocity relation, , and , we next derive the PN deviation field . Finally we evaluate the surface integrals which appear in the right hand side of Equation (111) and obtain the PN equations of motion. In the above calculations we use the PN order equations of motion to reduce the order of the equations of motion whenever an acceleration appears in the right hand of the resulting equations of motion. For instance, when we meet in the right hand side of the equations motion and we have to evaluate this up to , then using the Newtonian equations of motion, we replace it by . Basically we shall derive the 3 PN equations of motion with the procedure as described above.
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