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 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.

Empirical stability boundary for hierarchical triples

“…The SB also peaks around inclination 50 • more smoothly in the q out = 1 case, while the decay is more abrupt in extreme q out cases and occurs close to 60 • . The increase of the critical inclination for the onset of LK oscillations for triples with mild hierarchy (close to their stability limit) has been observed by Grishin et al (2017), was later derived analytically by Grishin, Perets, & Fragione (2018) for small q out , and was recently extended to any mass ratio by Mangipudi et al (2022). These results are consistent with the analytic expectations.…”

“…Right panels: V+22's threshold for instability (either separation varies by >10%). Hamers et al 2021;Grishin & Perets 2022), or semi-analytical (Muñoz et al 2016;Mangipudi et al 2022). However, even numerical population studies generally require secular approximations to reduce computational costs, and therefore rely on dynamical stability checks during the evolution.…”

“…The impulse from the outer object is greatest at its pericentre. However, the eccentricity of the inner orbit is determined by the three-body interactions rather than by its initial value for q out < 1 (e.g., the maximum eccentricity in the Lidov-Kozai regime is set by the masses, separations and the mutual inclination Grishin et al 2017;Mangipudi et al 2022).…”

Lagrange versus Lyapunov Stability of Hierarchical Triple Systems: Dependence on the Mutual Inclination between Inner and Outer Orbits

“…This is not expected from the quadrupole ZKL interaction mechanism alone, because it should be symmetric around 90°. The broken symmetry might arise due to semisecular effects that excite the eccentricity of the inner binary more efficiently than the simple quadrupole interaction mechanism (Grishin et al 2018;Mangipudi et al 2022), and thus increase the energy diffusion between inner and outer orbits (Paper I).…”

Ultrawide Black Hole—Neutron Star Binaries as a Possible Source for Gravitational Waves and Short Gamma-Ray Bursts

“…(4) Mergers from ultrawide systems in the field: Recently, it was shown that isolated ultrawide systems (with a semimajor axis (SMA) > 100 au), either binaries or triples, in the field of the host galaxy, may be driven to extremely high eccentricities, either by flyby stars or galactic tides, to allow strong interaction between the two binary objects (Michaely & Perets 2016Michaely 2021;Michaely & Shara 2021;Mangipudi et al 2022;Grishin & Perets 2022). For ultrawide systems, the field is a collisional environment due to consecutive frequent flyby interactions with random field stars.…”