Librating Kozai-Lidov Cycles with a Precessing Quadrupole Potential are Analytically Approximately Solved

TL;DR

This paper extends an analytical solution for the long-term evolution of librating Kozai-Lidov cycles in a hierarchical three-body system with a precessing quadrupole potential, covering a wider range of precession rates and initial conditions.

Contribution

It generalizes previous solutions to include a broader spectrum of precession rates, accurately describing dynamics far from the fixed point.

Findings

Analytic solution effectively models a wide range of initial conditions.

Rich dynamical structures emerge at precession rates near the Kozai-Lidov timescale.

Solution provides insights into long-term orbital evolution in hierarchical systems.

Abstract

The very long-term evolution of the hierarchical restricted three-body problem with a slightly aligned precessing quadrupole potential is investigated analytically for librating Kozai-Lidov cycles (KLCs). \citet{klein2023} presented an analytic solution for the approximate dynamics on a very long timescale developed in the neighborhood of the KLCs fixed point where the eccentricity vector is close to unity and aligned (or anti aligned) with the quadrupole axis and for a precession rate equal to the angular frequency of the secular Kozai-Lidov Equations around this fixed point. In this Letter, we generalize the analytic solution to encompass a wider range of precession rates. We show that the analytic solution approximately describes the quantitative dynamics for systems with librating KLCs for a wide range of initial conditions, including values that are far from the fixed point which is somewhat unexpected. In particular, using the analytic solution we map the strikingly rich structures that arise for precession rates similar to the Kozai-Lidov timescale (ratio of a few).