Secondary resonances and the boundary of effective stability of Trojan motions

TL;DR

This paper investigates the boundary of effective stability in Trojan co-orbital motions, emphasizing the role of secondary resonances and using a Hamiltonian model to estimate stable regions, with applications to exoplanetary Trojans.

Contribution

It introduces a method using a basic Hamiltonian model and resonant normal form to accurately estimate the stability boundary in Trojan dynamics.

Findings

The inner border of low-order secondary resonances predicts the regular orbit region.

The method combines resonant normal form with an asymmetric Hamiltonian expansion.

Applications include assessing stability domains for exoplanetary Trojans.

Abstract

One of the most interesting features in the libration domain of co-orbital motions is the existence of secondary resonances. For some combinations of physical parameters, these resonances occupy a large fraction of the domain of stability and rule the dynamics within the stable tadpole region. In this work, we present an application of a recently introduced `basic Hamiltonian model' Hb for Trojan dynamics, in Paez and Efthymiopoulos (2015), Paez, Locatelli and Efthymiopoulos (2016): we show that the inner border of the secondary resonance of lowermost order, as defined by Hb, provides a good estimation of the region in phase-space for which the orbits remain regular regardless the orbital parameters of the system. The computation of this boundary is straightforward by combining a resonant normal form calculation in conjunction with an `asymmetric expansion' of the Hamiltonian around the libration points, which speeds up convergence. Applications to the determination of the effective stability domain for exoplanetary Trojans (planet-sized objects or asteroids) which may accompany giant exoplanets are discussed.