Relative equilibria of four identical satellites

TL;DR

This paper rigorously proves that only three symmetric relative equilibria exist for four identical satellites in a planar 5-body Newtonian system with a small fifth body, confirming previous numerical findings.

Contribution

It establishes the uniqueness and symmetry of all relative equilibria of four identical satellites in the specified system, completing the classification.

Findings

Only three symmetric relative equilibria exist.

All equilibria possess symmetry, ruling out other configurations.

Confirmed numerical results through rigorous proof.

Abstract

We consider the Newtonian 5-body problem in the plane, where 4 bodies have the same mass m, which is small compared to the mass M of the remaining body. We consider the (normalized) relative equilibria in this system, and follow them to the limit when m/M -> 0. In some cases two small bodies will coalesce at the limit. We call the other equilibria the relative equilibria of four separate identical satellites. We prove rigorously that there are only three such equilibria, all already known after the numerical researches in [SaY]. Our main contribution is to prove that any equilibrium configuration possesses a symmetry, a statement indicated in [CLO2] as the missing key to proving that there is no other equilibrium.