Trojan resonant dynamics, stability, and chaotic diffusion, for parameters relevant to exoplanetary systems

TL;DR

This paper develops a Hamiltonian formalism to analyze the resonant dynamics, stability, and chaotic diffusion of trojan exoplanets, considering both the elliptic restricted three-body problem and multi-planet effects, with numerical demonstrations of chaotic diffusion processes.

Contribution

It introduces a unified Hamiltonian approach to study resonant and secular effects on trojan exoplanets in both simplified and multi-planet systems, highlighting the diffusion mechanism.

Findings

Chaotic diffusion occurs via modulational diffusion in both ERTBP and RMPP.

Resonant web survey reveals secondary resonances from 1:5 to 1:12.

Power-law tails observed in the distribution of escape times for chaotic orbits.

Abstract

We investigate the dynamics of small trojan exoplanets in domains of secondary resonances within the tadpole domain of motion. We consider the limit of a massless trojan companion of a giant planet. Without other planets, this is a case of the elliptic restricted three body problem (ERTBP). The presence of more planets (the restricted multi-planet problem, RMPP) induces new direct and indirect secular effects on the trojan's dynamics. In the theoretical part of this paper, we develop a Hamiltonian formalism in action-angle variables, which allows to treat in a unified way resonant dynamics and secular effects on the trojan body in both the ERTBP or the RMPP. Our formalism leads to a decomposition of the Hamiltonian in two parts, $H=H_b+H_{sec}$. $H_b$, called the basic model, describes resonant dynamics in the short-period (epicyclic) and synodic (libration) degrees of freedom. $H_{sec}$ contains only terms depending on slow (secular) angles. $H_b$ is formally identical in the ERTBP and the RMPP, apart from a re-definition of angular variables. An important physical consequence is that the slow chaotic diffusion proceeds in both the ERTBP and the RMPP by a qualitatively similar dynamical mechanism better approximated by the paradigm of `modulational diffusion'. In the numerical part, we focus on the ERTBP for making a numerical demonstration of the chaotic diffusion process along resonances. Using color stability maps, we provide a survey of the resonant web for characteristic mass parameters of the primary, in which the secondary resonances from 1:5 to 1:12 (ratio of the short over the synodic period) and their resonant multiplets appear. We give numerical examples of diffusion of weakly chaotic orbits in the resonant web. We make a statistics of the escaping times in the resonant domain, and find power-law tails of the distribution of escaping times for slowly diffusing chaotic orbits.