Stability of the Rhomboidal Symmetric-Mass Orbit

TL;DR

This paper investigates the stability of a symmetric four-body orbit with rhomboidal configuration, analyzing both two- and four-degree-of-freedom models, and identifies conditions for linear stability across different mass ratios.

Contribution

It provides a comprehensive stability analysis of the rhomboidal symmetric-mass orbit in both simplified and full models, including regularization and Poincaré section methods.

Findings

Linear stability in the two-degree-of-freedom model for a wide range of mass ratios.

Failure of linear stability in the four-degree-of-freedom model except for very small mass ratios.

Regularization of binary collisions at the origin in both models.

Abstract

We study the rhomboidal symmetric-mass 4-body problem in both a two-degree-of-freedom and a four-degree-of-freedom setting. Under suitable changes of variables in both settings, isolated binary collisions at the origin are regularizable. Linear stability analysis is performed in both settings. For the two-degree-of-freedom setting, linear stability is established for a wide interval of mass ratios. A Poincar\'{e} section analysis is also performed, showing stability. In the four-degree-of-freedom setting, linear stability fails except for a very small interval of mass ratios.