Stability and secondary resonances in the spatial restricted three-body problem for small mass ratios

TL;DR

This study examines the stability of L4 in the spatial restricted three-body problem for small mass ratios, identifying secondary resonances and their effects on stability through numerical analysis.

Contribution

It provides new insights into secondary resonances and stability regions for small mass ratios relevant to Solar system configurations.

Findings

High order secondary resonances are present at various inclinations.

Stability varies around secondary resonances, affecting Trojan orbit stability.

Secondary resonances are identified using Rabe's equation and frequency analysis.

Abstract

This paper is devoted to the study of secondary resonances and the stability of the Lagrangian point L4 in the spatial restricted three-body problem for moderate mass ratios (mu), meaning that mu is smaller than 0.0045. However, we concentrated our investigations on small mass ratios for mu smaller than 0.001, which represent the mass ratios for stable configurations of tadpole orbits in the Solar system. The stability is investigated by numerical methods, computing stability maps in different parameter planes. We started investigating the mass of the secondary; from Earth-mass bodies up to Jupiter-mass bodies. In addition we changed the orbital elements (eccentricity and inclination) of the secondary and Trojan body. For this parameter space we found high order secondary resonances, which are present for various inclinations. To determine secondary resonances we used Rabe's equation and the frequency analysis. In addition we investigated the stability in and around these secondary resonances.