The linear stability of the post-Newtonian triangular equilibrium in the three-body problem

Abstract

Continuing work initiated in an earlier publication [Yamada, Tsuchiya, and Asada, Phys. Rev. D 91, 124016 (2015)], we reexamine the linear stability of the triangular solution in the relativistic three-body problem for general masses by the standard linear algebraic analysis. In this paper, we start with the Einstein-Infeld-Hoffman form of equations of motion for $N$-body systems in the uniformly rotating frame. As an extension of the previous work, we consider general perturbations to the equilibrium, i.e. we take account of perturbations orthogonal to the orbital plane, as well as perturbations lying on it. It is found that the orthogonal perturbations depend on each other by the first post-Newtonian (1PN) three-body interactions, though these are independent of the lying ones likewise the Newtonian case. We also show that the orthogonal perturbations do not affect the condition of stability. This is because these always precess with two frequency modes; the same with the orbital frequency and the slightly different one by the 1PN effect. The same condition of stability with the previous one, which is valid even for the general perturbations, is obtained from the lying perturbations.

21 pages, 2 figures

References in corpus (11)

• Advanced Virgo: a 2nd generation interferometric gravitational wave detector
• GW151226: Observation of Gravitational Waves from a 22-Solar-Mass Binary Black Hole Coalescence
• GW170104: Observation of a 50-Solar-Mass Binary Black Hole Coalescence at Redshift 0.2
• Collinear solution to the general relativistic three-body problem
• Gravitational waveforms for 2- and 3-body gravitating systems
• Post-Newtonian effects on Lagrange's equilateral triangular solution for the three-body problem
• Post-Newtonian effects on the stability of the triangular solution in the three-body problem for general masses
• Uniqueness of collinear solutions for the relativistic three-body problem
• Gravitational Waves from Periodic Three-Body Systems
• Can N-body systems generate periodic gravitational waves?
• Post-Newtonian effects of planetary gravity on the perihelion shift

Cited by in corpus (2)

• Gravitational wave forms, polarizations, response functions and energy losses of triple systems in Einstein-Aether theory
• Effect of a second compact object on stable circular orbits