Extreme Eccentricities of Triple Systems: Analytic Results

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.