Hierarchical Three-Body Problem at High Eccentricities = Simple Pendulum

TL;DR

This paper simplifies the analytical understanding of high-eccentricity orbital flips in the hierarchical three-body problem by reducing the complex dynamics to a simple pendulum model, clarifying previous derivation errors.

Contribution

It introduces a simplified pendulum-based model for analyzing orbital flips, improving the analytical approach to high-eccentricity dynamics in the three-body problem.

Findings

The simplified model accurately predicts orbital flips for most initial conditions.

The new flip criterion is significantly simpler than previous criteria.

An earlier logical error was identified but shown not to affect the main results.

Abstract

The gradual evolution of the restricted hierarchical three body problem is analyzed analytically, focusing on conditions of Kozai-Lidov Cycles that may lead to orbital flips from prograde to retrograde motion due to the octupole (third order) term which are associated with extremely high eccentricities. We revisit the approach described by Katz, Dong and Malhotra (\href{https://doi.org/10.1103/PhysRevLett.107.181101}{Phys. Rev. Lett. 107, 181101 (2011)}) and show that for most initial conditions, to an excellent approximation, the analytic derivation can be greatly simplified and reduces to a simple pendulum model allowing an explicit flip criterion. The resulting flip criterion is much simpler than the previous one but the latter is still needed in a small fraction of phase space. We identify a logical error in the earlier derivation but clarify why it does not affect the final results.