Gravitational radiation damping and the three-body problem.

摘要

A model of three-body motion is developed which includes the effects of gravitational radiation reaction. The radiation reaction due to the emission of gravitational waves is the only post-Newtonian effect that is included here. For simplicity, all of the motion is taken to be planar. Two of the masses are viewed as a binary system and the third mass, whose motion will be a fixed orbit around the center-of-mass of the binary system, is viewed as a perturbation. This model aims to describe the motion of a relativistic binary pulsar that is perturbed by a third mass. Numerical integration of this simplified model reveals that given the right initial conditions and parameters one can see resonances. These (m, n) resonances are defined by the resonance condition, mω = 2 nΩ, where m and n are relatively prime integers and ω and Ω are the angular frequencies of the binary orbit and third mass orbit (around the center-of-mass of the binary), respectively. The resonance condition consequently fixes a value for the semimajor axis of the binary orbit for the duration of the resonance; therefore, the binary energy remains constant on the average while its angular momentum changes during the resonance. Numerical integration of an equation of relative motion that approximates the binary gives evidence of such resonances. This paper outlines a method of averaging developed by Chicone, Mashhoon, and Retzloff which renders a nonlinear system that undergoes resonance capture into a mathematically amenable form. This method is applied to the present system and one arrives at an analytical solution that describes the average motion during resonance. Furthermore, prominent features of the full nonlinear system, such as the frequency of oscillation and antidamping, accord with their analytically derived formulae.