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A statistical solution to the chaotic, non-hierarchical three-body problem.

Nicholas C Stone1,2,3, Nathan W C Leigh4,5

  • 1Columbia Astrophysics Laboratory, Columbia University, New York, NY, USA. nicholas.stone@mail.huji.ac.il.

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|December 20, 2019
PubMed
Summary

We present a statistical solution to the chaotic three-body problem, predicting outcome distributions for non-hierarchical systems. This work offers insights into astrophysical phenomena like black-hole mergers.

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Area of Science:

  • Astrophysics
  • Dynamical Systems
  • Statistical Mechanics

Background:

  • The three-body problem remains a centuries-old challenge in astrophysics, lacking a general analytic solution.
  • Perturbation theory and numerical integrations offer partial solutions but struggle with non-hierarchical systems and chaotic dynamics.
  • The chaotic nature prevents deterministic analytic solutions, yet suggests ergodicity.

Purpose of the Study:

  • To develop a statistical solution for the non-hierarchical three-body problem using the ergodic hypothesis.
  • To provide closed-form distributions for outcomes, such as binary orbital elements.
  • To compare theoretical predictions with numerical integrations and identify key dynamical states.

Main Methods:

  • Application of the ergodic hypothesis to the non-hierarchical three-body problem.
  • Derivation of closed-form outcome distributions based on conserved integrals of motion.
  • Comparison with large ensembles of numerical three-body integrations, focusing on resonant encounters.

Main Results:

  • Good agreement found between statistical predictions and numerical integrations for chaotic, resonant encounters.
  • Identification of 'scrambles' as the crucial dynamical state that leads to ergodicity in non-hierarchical triples.
  • Prediction of generally super-thermal distributions for survivor binary eccentricity.

Conclusions:

  • The ergodic hypothesis provides a viable statistical solution to the non-hierarchical three-body problem.
  • The findings have significant implications for understanding astrophysical scenarios, including black-hole mergers.
  • Accurate prediction of post-disintegration eccentricities is crucial for modeling gravitational wave events.