and contributions to the field depend on the body zone boundary . But itself does not depend on . Thus it is natural to expect that there are renormalized multipole moments which are independent of since we use nonsingular matter sources. This renormalization would absorb the dependence occuring in the computation of the field (see Section 4.8 for an example of such a renormalization). One possible practical obstacle for this expectation might be the dependence of multipole moments. Although at 3 PN order there appear such logarithmic terms, it is found that we could remove them by rechoosing the value of the dipole moment of the star.
Though we use the same symbol for the moments henceforth as before for notational simplicity, it should be understood that they are the renormalized ones. For instance, we use the symbol “” for the renormalized .
Since we compute integrals over the body zone boundary, in general the resulting equations of motion seem to depend on the size of the body zone boundary, . Actually this is not the case.
In the derivation of Equation (111), if we did not use the conservation law (67) until the final step, we have).
Along the same line, the momentum-velocity relation (86) does not depend on .
In Section 4.8 we shall explicitly show the irrelevance of the field and the equations of motion to by checking the cancellation among the dependent terms up to 0.5 PN order.
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