Three-body capture, ejection, and the demographics of bound objects in binary systems

We study the capture of light objects of arbitrary velocity by binary systems. Extending results for the capture of comets in the solar system, we develop a simple geometric characterization of the capture cross section, leading directly to the distribution of orbital parameters of captured objects. We use the same framework to study the lifetimes of these bound orbits prior to ejection, and find that a simplified version of the \"Opik--Arnold approach readily yields a closed-form estimate for the ejection rate that agrees well with numerical experiments. Without any detailed-balance assumptions, our results make manifest the characteristics of close encounters leading to capture and ejection. As an application of our results, we demonstrate the estimation of the equilibrium population of captured dark matter particles in a binary system.