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 [26] 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 like

where and . When and , this integral is perfectly well-defined (recall that the finite part deals with the bound at infinity). When or , our basic ansatz is that we apply the definition of the Hadamard partie 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 ) We denote for example ; the constant is the one pertaining to the finite-part process (see Eq. (36)). One example where the integral diverges at the location of the particle 1 is where is the Hadamard-regularization constant introduced in Eq. (119)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 [26]

Also needed are the 1PN mass -pole, 1PN current -pole (octupole), Newtonian mass -pole and Newtonian current -pole: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 time derivatives of the source moments (157, 158, 159, 160) are computed by means of the circular-orbit equations of motion given by Eq. (146, 147); then we substitute them into Eq. (161) (for circular orbits there is no difference at this order between , and , ). The net result is The Newtonian approximation, , is the prediction of the Einstein quadrupole formula (4), as computed by Landau and Lifchitz [97]. 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).http://www.livingreviews.org/lrr-2002-3 |
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