Figure-eight orbits in three post-Newtonian formulations of triple black holes

Abstract

There are three types of dynamical equations for a post-Newtonian Lagrangian formulation of three black holes. They are approximate post-Newtonian Lagrangian equations of motion, coherent post-Newtonian Lagrangian equations of motion, and post-Newtonian Hamiltonian canonical equations. The energy integral is conserved approximately in the first model, but exactly in the second and third models. The three formulations are not exactly equivalent, while they are approximately related in the orbital dynamical behavior. Figure-eight orbits in the three problems may have different initial conditions. Imai, Chiba, and Asada adjusted the initial velocities satisfying the figure-eight orbits for given initial positions in the first model. The initial velocities are still suitable for the figure-eight orbits of the latter two models when the initial separations of the first body are enough large. However, they are not generally applicable to relatively small initial separations in the three systems. We can obtain the desired initial velocities by scanning the initial velocities of the third body in the three systems. The figure-eight orbits in these models exhibit sensitive dependence on the scanned initial velocities for smaller initial separations. Although the differences in the scanned initial velocities among the three problems are minor for the small initial separations, the scanned initial velocities for any one of the three systems do not yield the figure-eight orbits for the other two systems.