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

Abstract

This paper is devoted to the study of secondary resonances and the stability of the Lagrangian point L-4 in the spatial restricted three-body problem for moderate mass ratios mu, meaning that mu <= 0.0045. However, we concentrated our investigations on small mass ratios mu <= 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.