Illustrating chaos: a schematic discretization of the general three-body problem in Newtonian gravity

Abstract

We present a formalism for constructing schematic diagrams to depict chaotic threebody interactions in Newtonian gravity. This is done by decomposing each interaction in to a series of discrete transformations in energy-and angular momentum-space. Each time a transformation is applied, the system changes state as the particles redistribute their energy and angular momenta. These diagrams have the virtue of containing all of the quantitative information needed to fully characterize most bound or unbound interactions through time and space, including the total duration of the interaction, the initial and final stable states in addition to every intervening temporary meta-stable state. As shown via an illustrative example for the bound case, prolonged excursions of one of the particles, which by far dominates the computational cost of the simulations, are reduced to a single discrete transformation in energy-and angular momentum-space, thereby potentially mitigating any computational expense. We further generalize our formalism to sequences of (unbound) three-body interactions, as occur in dense stellar environments during binary hardening. Finally, we provide a method for dynamically evolving entire populations of binaries via three-body scattering interactions, using a purely analytic formalism. In principle, the techniques presented here are adaptable to other three-body problems that conserve energy and angular momentum.

Starfall: A heavy rain of stars in 'turning on' AGN

“…Since the orbits of retrograde stars quickly collapse into the SMBH, this suggests that the ionization of binaries due to three-body interactions (e.g. Leigh et al 2018;Leigh & Wegsman 2018;Barrera et al 2020) (and therefore the suppression of the compact object merger rate) by embedded objects on retrograde orbits is mostly a consideration in the early stages of AGN (< 0.1 Myr) (Secunda et al 2020a).…”

“…Among this population in the inner disk, strong dynamical encounters, particularly between binaries and tertiaries (e.g. Leigh et al 2018;Samsing et al 2020), can quickly fill the AGN loss cone (see §7 below) and lead to tidal disruption events (TDEs) in AGN with 𝑀 SMBH < ∼ 10 8 𝑀 around non-spinning SMBH.…”

A statistical solution to the chaotic, non-hierarchical three-body problem

“…Here, v crit is the critical velocity, defined as the relative velocity at infinity required for a total encounter energy of zero. The motivation for these choices is that, as found in previous studies 39,40 , lower relative velocities at infinity and smaller impact parameters maximize the probability of long-lived resonant interactions occurring, which is probably needed to uphold the assumption of ergodicity. All angles defining the relative configurations of the binary orbital planes and phases are sampled to ensure isotropic scattering.…”

Binaries are softer than they seem: Effects of an external potential on the scattering dynamics of binaries

“…Heggie 1975;Hills 1975;Heggie & Hut 2003). Because these interactions are chaotic, past modelling relied heavily on numerical simulations (Saslaw et al 1974;Hills 1975;Hills & Fullerton 1980;Anosova 1986;Anosova & Orlov 1986;Hills 1989Hills , 1992Heggie & Hut 1993;Hut 1993;Sigurdsson & Phinney 1993;Mikkola 1994;Samsing et al 2014;Leigh & Wegsman 2018;Manwadkar et al 2020Manwadkar et al , 2021, but they were also modelled statistically, under certain approximations (Heggie 1975;Monaghan 1976;Nash & Monaghan 1978;Valtonen & Karttunen 2006;Kol 2021).…”