Three-Body Binary Formation in Clusters: Analytical Theory

TL;DR

The paper tackles how binaries form dynamically in dense clusters via three-body resonant encounters with positive total energy $E>0$, presenting an analytical probabilistic framework to derive the joint orbital-parameter distribution of the remnant binary under energy and angular-momentum conservation. It combines a conditional distribution for binary parameters given fixed $(E,\mathbf{J})$ with a rate calculation for cluster environments, yielding scaling relations and explicit expressions for both soft and hard binaries. The key findings are that soft three-body binaries are highly favored and exhibit a super-thermal eccentricity distribution, while hard binaries are rarer and tend toward a thermal distribution; the analytic results reproduce qualitative trends seen in simulations and align with recent numerical studies. This framework provides a practical tool for predicting binary populations in globular clusters and offers insights into observed eccentric wide binaries, clarifying the role of angular momentum conservation in dynamical binary formation.

Abstract

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.

Figures (9)

• Figure 1: The marginal energy distribution of the remnant binary, for equal masses. Top: $R_0 = d$, with $d = 150 GM\mu/E$. Bottom: the case $R_0 = \min\left\{R_E,d\right\}$.
• Figure 2: The marginal eccentricity distribution of the remnant binary, given that the remnant binary is bound, for equal masses. This is calculated by integrating equation \ref{['eqn:E_b and S joint distribution']} over the domain $E_b<0$ (and normalising). Top: $R_0 = d$, with $d = 150 GM\mu/E$; middle: the case $R_0 = \min\left\{R_E,d\right\}$; bottom: same as the top panel, but for hard binaries only, i.e. those with $\left\vert E_b\right\vert \geq E$.
• Figure 3: The joint energy-eccentricity distribution of the remnant binary, i.e. equation \ref{['eqn:E_b and S joint distribution']}, given that the remnant binary is bound, for equal masses. The plots in the top row show the distribution for different values of the parameter $\kappa = \frac{J^2\left\vert E\right\vert}{G^2M^2\mu^3}$. The plots in the bottom row are the probability distributions for the dimension-less angular momentum $s \equiv \sqrt{1-e_b^2}$, given$z$. The black line is the line $e_b = 1-R_0/a_b$ -- values of $s$ above that threshold cannot form.
• Figure 4: The binary formation probability, as a function of the angular momentum, made dimensionless, for equal masses. Top: the case $R_0 = d$, with $d = 150\,GM\mu/E$; bottom: the case $R_0 = R_E$. As hard binaries require significant energy exchanges to form, essentially all hard binaries form for $R_0 \leq R_E$, so that their relative fraction is lower when $R_0$ can be larger -- because only soft binaries form at $R_E \leq R_0 \leq d$.
• Figure 5: The binary formation probability, as a function of $\chi_1 = 27/(2\zeta)$, for equal masses. This assumes a density $n = 10^5 ~\textrm{pc}^{-3}$, $m = 1~M_\odot$, and a velocity dispersion $\sigma = 10 \textrm{km s}^{-1}$. Data from the numerical experiments of Atallahetal2024 are over-plotted for comparison., and re-scaled according to their equation (20). Details are in the text and appendix \ref{['appendix: rate calculation']}.