A semi-analytical model for secular dynamics of test particles in hierarchical triple systems

Abstract

ABSTRACT In this work, a semi-analytical model is formulated up to an arbitrary order in the semimajor axis ratio of the inner and outer binaries to describe the long-term (secular) dynamics of test particles in hierarchical triple systems. The third-body disturbing function is expressed as a Fourier series, where the harmonic arguments are linear combinations of the perturber’s mean anomaly, and the test particle’s mean anomaly, longitude of the ascending node, and argument of pericentre. Based on the series expansion, it is straightforward to arrive at the secular equations of motion by directly eliminating those terms that are irrelevant to the long-term dynamics. When the perturbations are so strong that the system’s hierarchy is no longer high, the conventional double-averaged model fails to predict the long-term behaviours of test particles. To overcome the difficulty, we develop a corrected double-averaged model by taking into account the short-term effects within the orbital periods of the inner and outer binaries. The resulting averaged model is applied to Jupiter’s irregular satellites, and simulation results show that the corrected model can reproduce the behaviours on time-scales much longer than the orbital periods. Moreover, we retrieve a triple-averaged model and discuss the associated dynamics in the phase space. It is found that the Kozai resonance in the corrected model occurs at a higher inclination than that in the conventional model.