Regularized phase-space volume for the three-body problem

Yogesh Dandekar*, Barak Kol, Lior Lederer, Subhajit Mazumdar

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


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

Bibliographical note

Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Nature B.V.


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


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