Nonchaotic evolution of triangular configuration due to gravitational radiation reaction in the three-body problem

TL;DR

This paper analytically investigates how gravitational radiation reaction affects the evolution of a three-body system in a triangular configuration, revealing that it shrinks but remains in equilibrium under certain conditions.

Contribution

It provides a direct analysis of gravitational radiation reaction on a three-body triangular system, extending previous energy-based methods to include degrees of freedom and post-Newtonian effects.

Findings

Triangular configuration shrinks adiabatically due to radiation reaction.

System remains in equilibrium if mass ratios satisfy stability conditions.

Long-term stability with post-Newtonian corrections is discussed.

Abstract

Continuing work initiated in an earlier publication [H. Asada, Phys. Rev. D {\bf 80}, 064021 (2009)], the gravitational radiation reaction to Lagrange's equilateral triangular solution of the three-body problem is investigated in an analytic method. The previous work is based on the energy balance argument, which is sufficient for a two-body system because the number of degrees of freedom (the semimajor axis and the eccentricity in quasi-Keplerian cases, for instance) equals that of the constants of motion such as the total energy and the orbital angular momentum. In a system with three (or more) bodies, however, the number of degrees of freedom is more than that of the constants of motion. Therefore, the present paper discusses the evolution of the triangular system by directly treating the gravitational radiation reaction force to each body. The perturbed equations of motion are solved by using the Laplace transform technique. It is found that the triangular configuration is adiabatically shrinking and is kept in equilibrium by increasing the orbital frequency due to the radiation reaction if the mass ratios satisfy the Newtonian stability condition. Long-term stability involving the first post-Newtonian corrections is also discussed.