Post-Newtonian effects on Lagrange's equilateral triangular solution for the three-body problem

TL;DR

This paper explores how post-Newtonian corrections influence Lagrange's equilateral triangle solutions in the three-body problem, revealing conditions for their existence and differences from Newtonian predictions.

Contribution

It demonstrates that the post-Newtonian equilateral configuration exists only for equal masses or specific test mass cases, and compares angular velocities with Newtonian results.

Findings

Equilateral configuration exists only for equal masses in full three-body case.

Post-Newtonian angular velocity is always smaller than Newtonian.

Configurations with test masses are possible under specific mass conditions.

Abstract

Continuing work initiated in earlier publications [Yamada, Asada, Phys. Rev. D 82, 104019 (2010), 83, 024040 (2011)], we investigate the post-Newtonian effects on Lagrange's equilateral triangular solution for the three-body problem. For three finite masses, it is found that the equilateral triangular configuration satisfies the post-Newtonian equation of motion in general relativity, if and only if all three masses are equal. When a test mass is included, the equilateral configuration is possible for two cases: (1) one mass is finite and the other two are zero, or (2) two of the masses are finite and equal, and the third one is zero, namely a symmetric binary with a test mass. The angular velocity of the post-Newtonian equilateral triangular configuration is always smaller than the Newtonian one, provided that the masses and the side length are the same.