Post-Newtonian effects on the stability of the triangular solution in the three-body problem for general masses

TL;DR

This paper investigates how first post-Newtonian relativistic corrections affect the stability of the triangular solutions in the three-body problem with general masses, revealing that stability persists but is reduced compared to Newtonian and restricted cases.

Contribution

It derives a stability condition for the relativistic three-body problem at 1PN order for general masses, extending previous restricted case results.

Findings

Stability regions exist at 1PN order for general masses.

PN triangular configuration is less stable than Newtonian and restricted cases.

The stability condition reduces to known results in the zero-mass limit.

Abstract

Continuing work initiated in earlier publications [Ichita, Yamada and Asada, Phys. Rev. D {\bf 83}, 084026 (2011); Yamada and Asada, Phys. Rev. D {\bf 86}, 124029 (2012)], we examine the post-Newtonian (PN) effects on the stability of the triangular solution in the relativistic three-body problem for general masses. For three finite masses, a condition for stability of the triangular solution is obtained at the first post-Newtonian (1PN) order, and it recovers previous results for the PN restricted three-body problem when one mass goes to zero. The stability regions still exist even at the 1PN order, though the PN triangular configuration for general masses is less stable than the PN restricted three-body case as well as the Newtonian one.