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  • Open Access

Analytical, Statistical Approximate Solution of Dissipative and Nondissipative Binary-Single Stellar Encounters

Yonadav Barry Ginat* and Hagai B. Perets†

  • Faculty of Physics, Technion-Israel Institute of Technology, Haifa, 3200003, Israel

  • *ginat@campus.technion.ac.il
  • †hperets@physics.technion.ac.il
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Phys. Rev. X 11, 031020 – Published 23 July, 2021

DOI: https://doi.org/10.1103/PhysRevX.11.031020

Abstract

We present a statistical approximate solution of the bound, nonhierarchical 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 with 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, nonhierarchical, triple approaches, interspersed with hierarchical phases, in which the system consists of an inner binary and a star that orbits it; this series of approaches turns the evolution of the entire encounter to a random walk between consecutive hierarchical phases. We use the solution of the bound, nonhierarchical three-body problem to find the walker’s transition probabilities, which we generalize 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 approximate 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.

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Corrections

28 April, 2022

Correction: Equation (D3) contained an error and has been fixed.

Popular Summary

The three-body problem—the problem of determining the evolution of three masses under their own gravity—has been studied by physicists for centuries, ever since Newton. In the late 19th century, Poincaré proved that this problem does not have an analytical solution, which led to the creation of the field of chaotic dynamics. For this reason, three-body interactions in astrophysics have been primarily studied using numerical simulations. Here, we draw on the inherent randomness of chaos to derive a closed-form statistical prediction for the outcome of a close triple approach.

In numerical simulations of three intertwined stars, the interactions among them proceed by a sequence of close three-body approaches, interspersed with phases where one of the stars is ejected; if it has positive energy, it leaves, and if its energy is negative, it eventually returns to another close approach. Our work, for the first time, also models the intermediate steps—when the third body emerges with negative energy—while exactly accounting for both energy and angular-momentum conservation. This solution allows us to model the sequence of close approaches as a random walk, enabling us to describe the entire encounter, which we then solve explicitly.

This model can incorporate more general physical effects, such as tidal dissipation, and thereby provide accurate predictions for many diverse phenomena involving three-body interactions. We compare our results with decades’ worth of numerical simulations, and find excellent agreement.

Article Text

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  47. The two are not independent, and in the plots, we have kept ηβ constant when varying β; this affects only the low-energy cutoff of the bound case.

Outline

Information

Phys. Rev. X 11, 031020– Published 23 July, 2021
  • Received 30 October 2020
  • Accepted 3 June 2021
  • Corrected 28 April 2022
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