Limiting Eccentricity in Restricted Hierarchical Three-Body Systems with Short-Range Forces

TL;DR

This paper studies how short-range forces like General Relativity limit the maximum eccentricity in hierarchical three-body systems, providing analytical formulas and comparing different averaging models to understand orbital flip phenomena.

Contribution

It analytically determines the limiting eccentricity in restricted hierarchical three-body systems considering short-range forces, comparing DA, Brown Hamiltonian, and SA models.

Findings

Limiting eccentricity can be calculated analytically using quadrupole Hamiltonian.

Brown Hamiltonian does not alter the limiting eccentricity.

SA model predicts higher limiting eccentricity than DA model.

Abstract

A hierarchical three-body model can be widely applied to diverse astrophysical settings, from satellite-planet-star systems to binaries around supermassive black holes. The octupole-order perturbation on the inner binary from the tertiary can induce extreme eccentricities and cause orbital flips of the binary, but short-range forces such as those due to General Relativity (GR) may suppress extreme eccentricity excitations. In this paper, we consider restricted hierarchical three-body systems, where the inner binary has a test-mass component. We investigate the maximum possible eccentricity (called "limiting eccentricity") attainable by the inner binary under the influence of the tertiary perturbations and GR effect. In systems with sufficiently high hierarchy, the double averaging (DA) model is a good approximation; we show that the orbits which can flip under the octupole-order perturbation reach the same limiting eccentricity, which can be calculated analytically using the quadrupole-order Hamiltonian. In systems with moderate hierarchy, DA breaks down and the so-called Brown Hamiltonian is often introduced as a correction term; we show that this does not change the limiting eccentricity. Finally, we employ the single averaging (SA) model and find that the limiting eccentricity in the SA model is higher than the one in the DA model. We derive an analytical scaling for the modified limiting eccentricity in the SA model.