2022ApJ...934...44M

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 implicit analytical formulae for the maximal eccentricity, ${e}_{\max }$, of the inner binary undergoing ZLK oscillations, where both the test-particle limit (parameterized by the inner-to-outer angular momentum ratio η) and the double-averaging approximation (parameterized by the period ratio, ϵ_SA) are relaxed, for circular outer orbits. We recover known results in both limiting cases (either η or ϵ_SA → 0) and verify the validity of our model using numerical simulations. We test our results with two accurate numerical N-body codes, REBOUND for Newtonian dynamics and TSUNAMI for general-relativistic dynamics, and find excellent correspondence. We discuss the implications of our results for stellar triples and both stellar and supermassive triple black hole mergers.