[Paper Review] An Analytical, Statistical Solution of Dissipative and non-Dissipative Binary-Single Stellar Encounters

Yonadav Barry Ginat, Hagai B. Perets|arXiv (Cornell University)|Oct 30, 2020

Stellar, planetary, and galactic studies4 citations

TL;DR

This paper presents a statistical, random walk model that analytically solves the bound, non-hierarchical three-body problem in binary-single stellar encounters, using transitions between hierarchical and non-hierarchical phases to predict remnant binary orbital parameters. The method generalizes to include tidal and dissipative forces, replacing costly N-body simulations with an efficient, environment-agnostic framework for stellar dynamics research.

ABSTRACT

We present a statistical solution of the bound, non-hierarchical three-body problem, and extend it to a general analysis of encounters between hard binary systems and single stars. Any such encounter terminates when one of the three stars is ejected to infinity, leaving behind a remnant binary; the problem of binary-single star-scattering consists of finding the probability distribution of the orbital parameters of the remnant binary, as a function of the total energy and the total angular momentum. Here, we model the encounter as a series of close, non-hierarchical, triple approaches, interspersed with hierarchical phases, in which the system consists of an inner binary and a star that orbits it -- this turns the evolution of the entire encounter to a random walk between consecutive hierarchical phases. We use the solution of the bound, non-hierarchical three-body problem to find the walker's transition probabilities, which we generalise to situations in which tidal interactions are important. Besides tides, any dissipative process may be incorporated into the random walk model, as it is completely general. Our solution can reproduce the results of the extensive body of past numerical simulations, and can account for different environments and different dissipative effects. Therefore, this model can effectively replace the need for direct few-body integrations for the study of binary-single encounters in any environment. Furthermore, it allows for a simply inclusion of dissipative forces typically not accounted for in full N-body integration schemes.

Motivation & Objective

• To develop a statistical solution for the bound, non-hierarchical three-body problem in stellar encounters.
• To model binary-single star scattering as a random walk between hierarchical phases, enabling probabilistic prediction of remnant binary parameters.
• To generalize the framework to include tidal interactions and arbitrary dissipative processes.
• To replace computationally expensive few-body integrations with an efficient, analytically grounded model.
• To enable accurate modeling of stellar dynamics across diverse astrophysical environments without full N-body simulations.

Proposed method

• Model the encounter as a sequence of non-hierarchical triple approaches separated by hierarchical phases (inner binary + distant third star).
• Use the solution of the bound, non-hierarchical three-body problem to compute transition probabilities between hierarchical phases.
• Treat the evolution as a random walk in phase space, with transitions governed by energy and angular momentum conservation.
• Generalize the transition probabilities to include tidal forces and other dissipative effects via modified energy and momentum exchange rules.
• Construct a statistical framework that maps initial encounter conditions to the probability distribution of remnant binary orbital elements.
• Validate the model against existing numerical simulations and extend its applicability to various astrophysical environments.

Experimental results

Research questions

• RQ1 How can the statistical distribution of remnant binary orbital parameters be predicted after a binary-single stellar encounter?
• RQ2 What is the role of hierarchical vs. non-hierarchical phases in shaping the outcome of three-body stellar encounters?
• RQ3 How can tidal and dissipative forces be consistently incorporated into a statistical model of three-body scattering?
• RQ4 To what extent can this model reproduce results from full N-body simulations without direct integration?
• RQ5 Can this framework be generalized to different stellar environments and dissipation mechanisms?

Key findings

• The model successfully reproduces the statistical outcomes of extensive past numerical simulations of binary-single encounters.
• The inclusion of tidal and dissipative forces is naturally embedded in the random walk framework through modified transition probabilities.
• The method enables accurate prediction of remnant binary orbital parameters (e.g., energy, angular momentum) without requiring full few-body integration.
• The framework is general enough to be applied across diverse astrophysical environments, including dense stellar clusters and galactic nuclei.
• The model provides a computationally efficient alternative to direct N-body simulations for studying stellar dynamics in systems with binary-single interactions.
• The approach captures the stochastic nature of three-body encounters by modeling evolution as a Markov process between hierarchical phases.