Regularized phase-space volume for the three-body problem

Yogesh Dandekar, Barak Kol, Lior Lederer, Subhajit Mazumdar

Research output: Contribution to journalArticlepeer-review


The micro-canonical phase-space volume for the three-body problem is an elementary quantity of intrinsic interest, and within the flux-based statistical theory, it sets the scale of the disintegration time. While the bare phase-volume diverges, we show that a regularized version can be defined by subtracting a reference phase-volume, which is associated with hierarchical configurations. The reference quantity, also known as a counter-term, can be chosen from a 1-parameter class. The regularized phase-volume of a given (negative) total energy, σ¯ (E) , is evaluated. First, it is reduced to a function of the masses only, which is sensitive to the choice of a regularization scheme only through an additive constant. Then, analytic integration is used to reduce the integration to a sphere, known as shape sphere. Finally, the remaining integral is evaluated numerically and presented by a contour plot in parameter space. Regularized phase-volumes are presented for both the planar three-body system and the full 3d system. In the test mass limit, the regularized phase-volume is found to become negative, thereby signaling the breakdown of the non-hierarchical statistical theory. This work opens the road to the evaluation of σ¯ (E, L) , where L is the total angular momentum, and in turn to comparison with simulation determined disintegration times.

Original languageAmerican English
Article number55
JournalCelestial Mechanics and Dynamical Astronomy
Issue number6
StatePublished - Dec 2022


  • Chaos
  • Phase-space volume
  • Statistical mechanics
  • Three-body problem

All Science Journal Classification (ASJC) codes

  • Modelling and Simulation
  • Mathematical Physics
  • Astronomy and Astrophysics
  • Space and Planetary Science
  • Computational Mathematics
  • Applied Mathematics


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