Existence and Stability the Lagrangian point $L_4$ for the Earth-Sun system under a relativistic framework

TL;DR

This paper extends the classical analysis of the stability of the Lagrangian point L4 in the Sun-Earth system by incorporating relativistic effects, providing a formula for eigenvalues and demonstrating stability under relativistic dynamics.

Contribution

It introduces a formula to compute eigenvalues for stability analysis and shows L4 remains stable in a relativistic framework for the Sun-Earth system and similar systems.

Findings

L4 is stable in the relativistic Sun-Earth system.

A new formula for eigenvalues of stability matrices.

Stability extends to systems with similar mass ratios.

Abstract

It is well known that, from the Newtonian point of view, the Lagrangian point $L_4$ in the circular restricted three body is stable if $\mu< \frac{1}{18}(9-\sqrt{19})\approx 0.03852$. In this paper we will provide a formula that allows us to compute the eigenvalues of the matrix that determines the stability of the equilibrium points of a family of ordinary differential equations. As an application we will show that, under the relativistic framework, the Lagrangian point $L_4$ is also stable for the Sun-Earth system. Similar arguments show the stability for $L_4$ not only for the Sun-Earth system but for systems coming from a range of values for $\mu$ similar to those in the Newtonian restricted three body problem.