Describing the hierarchical gravitational three-body problem with force and torque vectors

TL;DR

This paper presents a simplified, force and torque-based approach to analyze the long-term evolution of the hierarchical gravitational three-body problem, specifically the Kozai-Lidov mechanism, making it more accessible for educational purposes.

Contribution

It introduces a novel force and torque framework to describe the Kozai-Lidov mechanism, offering an alternative to the traditional Hamiltonian approach.

Findings

Derives the long-term evolution using elementary force and torque concepts.

Provides a more accessible method suitable for undergraduate teaching.

Highlights the binary as a dipole interacting with a gravitational field.

Abstract

We investigate the hierarchical gravitational three-body problem, in which a binary is perturbed by a distant object that orbits on a Keplerian ellipse around the binary itself. This phenomenon, known as Kozai-Lidov mechanism in the literature, results in large-amplitude oscillations of the orbital elements, like eccentricity, which may significantly influence the evolution of many astrophysical systems: from minor bodies and planets to even supermassive black holes. The standard approach found in the literature describes this phenomenon in the framework of Hamiltonian mechanics. Here we derive the long term evolution of the triple with elementary tools, treating the binary as a dipole interacting with the gravitational field of the tertiary and describe the dynamics using forces and torques instead of the usual Hamiltonian formalism. This highlights another aspect of the problem and is likely an easier way to introduce the problem at the undergraduate level.