Extreme eccentricities of triple systems: Analytic results

TL;DR

This paper derives explicit analytical formulas for the maximum eccentricity of the inner binary in triple systems undergoing ZLK oscillations, relaxing common approximations and validating results with numerical simulations, with implications for stellar and black hole mergers.

Contribution

The authors provide new analytical expressions for the maximal eccentricity in triple systems without relying on the test particle or double-averaging approximations, validated by numerical simulations.

Findings

Analytical formulas match numerical simulations accurately.

Results applicable to stellar triples and black hole mergers.

Relaxation of common approximations enhances understanding of ZLK oscillations.

Abstract

Triple stars and compact objects are ubiquitously observed in nature. Their long-term evolution is complex; in particular, the von-Zeipel-Lidov-Kozai (ZLK) mechanism can potentially lead to highly eccentric encounters of the inner binary. Such encounters can lead to a plethora of interacting binary phenomena, as well as stellar and compact-object mergers. Here we find explicit analytical formulae for the maximal eccentricity, $e_{\rm max}$, of the inner binary undergoing ZLK oscillations, where both the test particle limit (parametrised by the inner-to-outer angular momentum ratio $\eta$) and the double-averaging approximation (parametrised by the period ratio, $\epsilon_{\rm SA}$) are relaxed, for circular outer orbits. We recover known results in both limiting cases (either $\eta$ or $\epsilon_{\rm SA} \to 0$) and verify the validity of our model using numerical simulations. We test our results with two accurate numerical N-body codes, $\texttt{Rebound}$ for Newtonian dynamics and $\texttt{Tsunami}$ for general-relativistic (GR) dynamics, and find excellent correspondence. We discuss the implications of our results for stellar triples and both stellar and supermassive triple black hole mergers.