The difficult part of the analysis is to find the closed-form expressions, fully explicit in terms of the particle’s positions and velocities, of many non-linear integrals. We refer to  for full details; nevertheless, let us give a few examples of the type of technical formulas that are employed in this calculation. Typically we have to compute some integrals likepartie finie provided by Eq. (119). Two examples of closed-form formulas that we get, which do not necessitate the Hadamard partie finie, are (quadrupole case ) 23. The crucial input of the computation of the flux at the 3PN order is the mass quadrupole moment , since this moment necessitates the full 3PN precision. The result of Ref.  for this moment (in the case of circular orbits) is via the post-Newtonian parameter given by Eq. (145). As we see, there are two types of logarithms in the moment: One type involves the length scale related by Eq. (148) to the two gauge constants and present in the 3PN equations of motion; the other type contains the different length scale coming from the general formalism of Part A - indeed, recall that there is a operator in front of the source multipole moments in Theorem 6. As we know, that is pure gauge; it will disappear from our physical results at the end. On the other hand, we have remarked that the multipole expansion outside a general post-Newtonian source is actually free of , since the ’s present in the two terms of Eq. (67) cancel out. We shall indeed find that the constants present in Eqs. (158) are compensated by similar constants coming from the non-linear wave “tails of tails”. More seriously, in addition to the harmless constants and , there are three unknown dimensionless parameters in Eqs. (158), called , and . These parameters reflect some incompleteness of the Hadamard self-field regularization (see the discussion in Section 8.2).
Besides the 3PN mass quadrupole (157, 158), we need also the mass octupole moment and current quadrupole moment , both of them at the 2PN order; these are given by 
These results permit one to control what can be called the “instantaneous” part, say , of the total energy flux, by which we mean that part of the flux that is generated solely by the source multipole moments, i.e. not counting the “non-instantaneous” tail integrals. The instantaneous flux is defined by the replacement into the general expression of given by Eq. (60) of all the radiative moments and by the corresponding (th time derivatives of the) source moments and . Actually, we prefer to define by means of the intermediate moments and . Up to the 3.5PN order we have. The self-field regularization ambiguities arising at the 3PN order are the equation-of-motion-related constant and the multipole-moment-related constant (see Section 8.2).
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