Uniqueness of collinear solutions for the relativistic three-body problem

Journal Article · Sat Jan 15 04:00:00 EST 2011 · Physical Review. D, Particles Fields

DOI:https://doi.org/10.1103/PHYSREVD.83.024040· OSTI ID:21513076

Yamada, Kei; Asada, Hideki ^[1]

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1. Faculty of Science and Technology, Hirosaki University, Hirosaki 036-8561 (Japan)

Continuing work initiated in an earlier publication [Yamada, Asada, Phys. Rev. D 82, 104019 (2010)], we investigate collinear solutions to the general relativistic three-body problem. We prove the uniqueness of the configuration for given system parameters (the masses and the end-to-end length). First, we show that the equation determining the distance ratio among the three masses, which has been obtained as a seventh-order polynomial in the previous paper, has at most three positive roots, which apparently provide three cases of the distance ratio. It is found, however, that, even for such cases, there exists one physically reasonable root and only one, because the remaining two positive roots do not satisfy the slow-motion assumption in the post-Newtonian approximation and are thus discarded. This means that, especially for the restricted three-body problem, exactly three positions of a third body are true even at the post-Newtonian order. They are relativistic counterparts of the Newtonian Lagrange points L{sub 1}, L{sub 2}, and L{sub 3}. We show also that, for the same masses and full length, the angular velocity of the post-Newtonian collinear configuration is smaller than that for the Newtonian case. Provided that the masses and angular rate are fixed, the relativistic end-to-end length is shorter than the Newtonian one.

• OSTI ID: 21513076
• Journal Information: Physical Review. D, Particles Fields, Journal Name: Physical Review. D, Particles Fields Journal Issue: 2 Vol. 83; ISSN PRVDAQ; ISSN 0556-2821
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ANGULAR VELOCITY
APPROXIMATIONS
CALCULATION METHODS
CONFIGURATION
ENERGY RANGE
MANY-BODY PROBLEM
MASS
MATHEMATICAL SOLUTIONS
RELATIVISTIC RANGE
THREE-BODY PROBLEM
VELOCITY