Regularized phase-space volume for the three-body problem

• Published in: Celestial mechanics and dynamical astronomy 2022-12, Vol.134 (6), p.55, Article 55
• Main Authors: Dandekar, Yogesh, Kol, Barak, Lederer, Lior, Mazumdar, Subhajit
• Format: Article
• Language: English
• Subjects: Aerospace Technology and Astronautics; Angular momentum; Astrophysics and Astroparticles; Classical Mechanics; Disintegration; Dynamical Systems and Ergodic Theory; Geophysics/Geodesy; Original Article; Parameters; Physics; Physics and Astronomy; Regularization; Three body problem
• ISSN: 0923-2958; 1572-9478
• DOI: 10.1007/s10569-022-10108-1

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.