Gravitational waves from binary systems in circular orbits: Convergence of a dressed multipole expansion

The gravitational radiation originating from a compact binary system in circular orbit is usually expressed as an infinite sum over radiative multipole moments. In a slow-motion approximation, each multipole moment is then expressed as a post-Newtonian expansion in powers of v/c, the ratio of the orbital velocity to the speed of light. The “bare multipole truncation” of the radiation consists in keeping only the leading-order (Newtonian) term in the post-Newtonian expansion of each moment, but summing over all the multipole moments. In the case of binary 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 v/c < 2/e, where e is the base of natural logarithms. (These include all physically relevant values for circular inspiral.) In this paper, we extend the analysis to a “dressed multipole truncation” of the radiation, in which the leading-order moments are corrected with terms of relative order (v/c)2 (first post-Newtonian, or 1PN, terms) and (v/c)3 (1.5PN terms). We find that the dressed multipole series converges also for all values v/c < 2/e, and that it coincides (within 1 %) with the numerically “exact” results for v/c < 0.2. Although the dressed multipole series converges, it is an unphysical approximation, and the issue of the convergence of the true post-Newtonian series remains uncertain. However, our analysis shows that an eventual failure of the true post-Newtonian series to converge cannot originate from summing over the Newtonian, 1PN, and 1.5PN part of all the multipole moments. Pacs numbers: 0430, 0425N Typeset using REVTEX 1