Existence and stability of Lagrangian points in the relativistic restricted three body problem

TL;DR

This paper analyzes the stability and existence of Lagrangian point L4 in the relativistic restricted three-body problem, comparing different models and confirming L4's existence in the Sun-Earth system.

Contribution

It provides an exact expression for the oscillation period at L4, compares relativistic and non-relativistic models, and proves L4's existence in the Sun-Earth system.

Findings

Relativistic corrections have minimal impact on the characteristic polynomial.

Previous relativistic models show larger errors compared to non-relativistic calculations.

L4 exists in the Sun-Earth system as proven by the Poincare Miranda theorem.

Abstract

It is well known that objects can oscillate around the Lagrangian point L4. In this manuscript we compute the period of these oscillations by computing the exact expression of the characteristic polynomial of the matrix that determined the stability of the Lagrangian point. When we use relativity theory we are supposed to get a small improvement of this period. This paper compares the values of the the characteristic polynomials computed in the following way: 1. Using relativity with an error smaller than 10^{-30}. 2. The characteristic polynomial coming from a paper written in 2002, 3. The characteristic polynomial coming from a paper written in 2006 and 4. The characteristic polynomial obtain without using relativity. We show that the polynomial 1 is closer to the polynomial 4 than it is to the polynomials 2 and 3. This is, the error in both polynomials that dealt with relativity coming from previous papers is bigger than the error coming from the characteristic polynomial that do not use relativity at all. We also prove the existence of the Lagrangian point L4 in the Sun-Earth system using the Poincare Miranda theorem.