Hierarchical Three-Body Problem at High Eccentricities = Simple Pendulum III: Precessing Quadrupole

TL;DR

This paper analytically investigates the long-term evolution of the hierarchical three-body problem with a precessing quadrupole potential, revealing resonant dynamics and similarities to a simple pendulum model at high eccentricities.

Contribution

It introduces an analytical solution for the resonant dynamics of high-eccentricity Kozai-Lidov cycles with precession, demonstrating their equivalence to a simple pendulum model.

Findings

Resonant dynamics occur when KLC frequency matches precession rate.

Librating and rotating KLCs show striking similarity under certain conditions.

High-eccentricity KLCs can be modeled as a simple pendulum.

Abstract

The very long-term evolution of the hierarchical restricted three-body problem with a slightly aligned precessing quadrupole potential is investigated analytically and solved for both rotating and librating Kozai-Lidov cycles (KLCs) with high eccentricities. We describe the finding of a striking similarity between librating and rotating KLCs for some range of precession rates. We show that the main effect occurs in both categories when the KLC frequency is equal to the precession rate of the perturbing potential. We solve the resonant dynamics analytically and show that it is equivalent to a simple pendulum model allowing us to map the strikingly rich structures that arise for precession rates similar to the Kozai-Lidov timescale (ratio of a few) and explain the similarity and when it vanishes. Additionally, we show that the regular KLCs at high eccentricities can also be described using a simple pendulum.