Resonance Behavior and Partial Averaging in a Three-Body System with Gravitational Radiation Damping

TL;DR

This paper investigates resonance phenomena in a three-body system with gravitational radiation damping, using averaging methods to derive analytical solutions that match numerical results, revealing key features of the system's behavior.

Contribution

It introduces an averaging method applied to a three-body system with gravitational radiation, providing analytical solutions for resonance behavior and validating them against numerical simulations.

Findings

Resonance conditions fix the binary's semimajor axis during resonance.

Analytical solutions accurately describe the average motion during resonance.

Frequency of oscillation and antidamping effects match derived formulas.

Abstract

In a previous investigation, a model of three-body motion was developed which included the effects of gravitational radiation reaction. The aim was to describe the motion of a relativistic binary pulsar that is perturbed by a third mass and look for resonances between the binary and third mass orbits. Numerical integration of an equation of relative motion that approximates the binary gives evidence of such resonances. These $(m:n)$ resonances are defined for the present purposes by the resonance condition, $m\omega=2n\Omega$, where $m$ and $n$ are relatively prime integers and $\omega$ and $\Omega$ are the angular frequencies of the binary orbit and third mass orbit, respectively. The resonance condition consequently fixes a value for the semimajor axis $a$ of the binary orbit for the duration of the resonance because of the Kepler relationship $\omega=a^{-3/2}$. 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.