Three-Body Binary Formation in Clusters: Analytical Theory

Binary formation in clusters through triple encounters between three unbound stars, 'three-body' binary formation, is one of the main dynamical formation processes of binary systems in dense environments. In this paper, we use an analytical probabilistic approach to study the process for the equal mass case and calculate a probability distribution for the orbital parameters of three-body-formed binaries, as well as their formation rate. For the first time, we give closed-form analytical expressions to the full orbital parameter distribution, accounting for both energy and angular momentum conservation. This calculation relies on the sensitive dependence of the outcomes of three-body scatterings on the initial conditions: here we compute the rate of three-body binaries from ergodic interactions, which allow for an analytical derivation of the distribution of orbital parameters of the binaries thus created. We find that soft binaries are highly favoured in this process and that these binaries have a super-thermal eccentricity distribution, while the few hard three-body binaries have an eccentricity distribution much closer to thermal. The analytical results predict and reproduce simulation results of three-body scattering experiments in the literature well.