Secular resonance of inner test particles in hierarchical planetary systems

TL;DR

This study analyzes the secular resonance involving the longitude of pericentre for inner test particles in hierarchical planetary systems, providing an analytical model that describes phase-space structure and resonant dynamics.

Contribution

It formulates a high-order double-averaged Hamiltonian and derives a simplified resonant model to analytically explore the resonance's phase-space and width.

Findings

Resonant width correlates with maximum eccentricity variation.

Secular dynamics are dominated by the resonance for low-eccentricity, low-inclination particles.

Analytical model matches numerical results for resonant behavior.

Abstract

The present work studies the secular resonance associated with the critical argument $\sigma = \varpi$ ($\varpi$ is the longitude of pericentre) for inner test particles moving in low-eccentricity region with inclination $i$ smaller than $39^{\circ}$. To formulate the dynamical model, the double-averaged Hamiltonian is formulated up to an arbitrary order in the semimajor axis ratio, and then those high-order periodic terms are removed from the double-averaged Hamiltonian by means of Hori--Deprit transformation technique. The resulting Hamiltonian determines a resonant model with a single degree of freedom. Based on the resonant model, it becomes possible to explore the phase-space structure, resonant centre, and resonant width in an analytical manner. In particular, an excellent correspondence is found between the resonant width in terms of the eccentricity variation and the maximum variation of eccentricity ($\Delta e$) for test particles initially placed on quasi-circular orbits. It means that the secular dynamics in the low-eccentricity space with $i < 39^{\circ}$ is dominantly governed by the secular resonance associated with $\sigma = \varpi$.