Dynamics of the post-Newtonian circular restricted three-body problem with compact objects

Abstract

By applying scaling transformations to distance and time, we obtain the first post-Newtonian equations of motion for a relativistic circular restricted three-body problem, where the Newtonian terms do not depend on the separation of a parent binary, though the post-Newtonian terms do. The postNewtonian contributions consist of the relativistic effects from the circular orbital frequencies between the primaries and those from the primaries to a third body. When the former post-Newtonian contribution and the nonrelativistic terms are considered, the post-Newtonian dynamics are qualitatively different from the Newtonian dynamics if the separation between the two primaries is insufficiently large. When the latter post-Newtonian contribution is also included, some orbits become unstable. By scanning the dependence of the dynamics on the separation with fast Lyapunov indicators, the separation is classified into three domains for dynamically unstable, bounded chaotic, and bounded regular dynamics.

Full three-body problem in effective-field-theory models of gravity

“…(4.15), which depend on a parameter ρ, possess for ρ ¼ 0 a periodic solution whose characteristic exponents (see Appendix B) are all nonvanishing, they have again a periodic solution for small values of ρ. In our case, the small parameter ρ is the Planck length l P , and when ρ ¼ 0 we revert to the three-body problem in post-Newtonian mechanics, for which, in the circular restricted case, one knows from recent work [13] that orbits may be unstable, or bounded chaotic, or bounded regular. In the case of Newtonian mechanics instead, Chenciner and Montgomery [14] have found a class of solutions where three bodies of equal mass move periodically on the plane along the same curve.…”

“…The desired periodic solutions, whose existence is a special rather than generic property [1,2,13], can be written in the form…”

“…At finite values of t, there exist intervals of this time variable where neither (7.5) nor (7.6) converges in Newtonian physics [1,2]. One should notice that the actual evaluation of periodic solutions of the full three-body problem within the framework of parametrized post-Newtonian formalism is still in its infancy, since, to the best of our knowledge, only results for the circular restricted three-body problem are available so far [13], unlike the case of Newtonian theory, where, after centuries of efforts, some periodic solutions of the full three-body problem are explicitly known by now [14]. The years to come will hopefully tell us whether the scheme described by our Sec.…”

Unveiling the basins of convergence in the pseudo-Newtonian planar circular restricted four-body problem

“…A series of research papers (e.g., [23]; [12]; [14]) are available on the 1 st order post-Newtonian equations of motion for the restricted problem of three bodies which are deduced by using the Einstein-Infeld-Hoffmann theory (e.g., [19], [18]). [17] studied the dynamics of the planar circular restricted problem of three-bodies in the context of a pseudo-Newtonian approximation by using the Fodor-Hoenselaers-Perjé procedure, while [20] examined the influence of the separation between the primaries. It is concluded that the post-Newtonian dynamics substantially differ from the corresponding classical Newtonian dynamics provided the distance between the primaries is sufficiently small.…”

“…When the value of the transition parameter ∈ (0, 0.67752839], the libration points L 11,12,13,14 , L 15,16,17,18 and L 19,20,21,22 move away from the centers of the primaries P 1 , P 2 , and P 3 , respectively, while on the other hand the libration points L 2,7,8 move towards the centers of the primaries P 1 , P 2 , and P 3 , respectively. In particular, L 1,3,5,6,9 and L 10 , move towards the central libration point L 4 .…”