GRAVITATIONAL-WAVES FROM BINARY-SYSTEMS IN CIRCULAR ORBITS - CONVERGENCE OF A PARTIALLY BARE MULTIPOLE EXPANSION

The gravitational radiation originating from a compact binary system i n circular orbit is usually expressed as an infinite sum over radiativ e multipole moments. In a slow-motion approximation, each multipole mo ment is expressed as a post-Newtonian expansion in powers of upsilon/c , the ratio of the orbital velocity to the speed of light. The 'bare m ultipole truncation' of the radiation consists in keeping only the lea ding-order (Newtonian) term in the post-Newtonian expansion of each mo ment, but summing over all the multipole moments. In the case of binar y systems with small mass ratios, the bare multipole series was shown in a previous paper (Simone et al 1997 Class. Quantum Grav. 14 237) to converge for all values upsilon/c < 2/e, where e is the base of natur al logarithms. (These include all physically relevant values for circu lar inspiral.) In this paper, we extend the analysis to a 'partially b are multipole truncation' of the radiation, in which the leading-order moments are corrected with terms of relative order (upsilon/c)(2) (fi rst post-Newtonian, or 1 PN, terms) and (upsilon/c)(3) (1.5 PN terms). We find that the partially bare multipole series also converges for a ll values upsilon/c < 2/e, and that it coincides (to within 1%) with t he numerically 'exact' results for upsilon/c < 0.2. Although this mult ipole series converges, it is an unphysical approximation, and the iss ue of the convergence of the true post-Newtonian series remains open. However, our analysis shows that an eventual failure of the true post- Newtonian series to converge cannot originate from summing over the Ne wtonian, 1 PN and 1.5 PN part of all the multipole moments.