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

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 post-Newtonian 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.