TY - JOUR

T1 - Regularized phase-space volume for the three-body problem

AU - Dandekar, Yogesh

AU - Kol, Barak

AU - Lederer, Lior

AU - Mazumdar, Subhajit

N1 - Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Nature B.V.

PY - 2022/12

Y1 - 2022/12

N2 - 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.

AB - 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.

KW - Chaos

KW - Phase-space volume

KW - Statistical mechanics

KW - Three-body problem

UR - http://www.scopus.com/inward/record.url?scp=85142638965&partnerID=8YFLogxK

U2 - 10.1007/s10569-022-10108-1

DO - 10.1007/s10569-022-10108-1

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AN - SCOPUS:85142638965

SN - 0923-2958

VL - 134

JO - Celestial Mechanics and Dynamical Astronomy

JF - Celestial Mechanics and Dynamical Astronomy

IS - 6

M1 - 55

ER -