Analytical models for secular descents in hierarchical triple systems

TL;DR

This paper develops analytical models to describe the eccentricity evolution in hierarchical triple systems, enabling efficient estimation of close-encounter timescales and migration rates without extensive numerical simulations.

Contribution

It introduces three novel analytical approximations for the secular evolution of eccentricity in hierarchical triples, extending beyond the test particle limit.

Findings

Models accurately match numerical solutions for eccentricity evolution.

Approximations effectively estimate descent timescales and close-encounter probabilities.

Application to Hot Jupiters demonstrates practical utility.

Abstract

Triple body systems are prevalent in nature, from planetary to stellar to supermassive black hole scales. In a hierarchical triple system, oscillations of the inner orbit's eccentricity and inclination can be induced on secular timescales. Over many cycles, the octupole-level terms in the secular equations of motion can drive the system to extremely high eccentricities via the Eccentric Kozai-Lidov (EKL) mechanism. The overall decrease in the inner orbit's pericenter distance has potentially dramatic effects for realistic systems, such as tidal disruption events. We present an analytical approximation in the test particle limit to describe individual step-wise increases in eccentricity of the inner orbit. A second approximation, also in the test particle limit, is obtained by integrating the equations of motion and calibrating to numerical simulations to estimate the overall octupole-level time evolution of the eccentricity. The latter approach is then extended beyond the test particle to the general case. The three novel analytical approximations are compared to numerical solutions to show that the models accurately describe the form and timescale of the secular descent from large distances to a close-encounter distance (e.g., the Roche limit). By circumventing the need for numerical simulations to obtain the long-term behavior, these approximations can be used to readily estimate properties of close encounters and descent timescales for populations of systems. We demonstrate by calculating rates of EKL-driven migration for Hot Jupiters in stellar binaries.