Extreme eccentricities of triple systems: Analytic results

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, $e_{\max }$, 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, ${\epsilon }_{\mathrm{SA}}$) are relaxed, for circular outer orbits. We recover known results in both limiting cases (either $\eta$ or ${\epsilon }_{\mathrm{SA}}\to 0$) and verify the validity of our model using numerical simulations. We test our results with two accurate numerical N-body codes, $\text{Rebound}$ for Newtonian dynamics and $\text{Tsunami}$ 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.

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