Search for periodic orbits in the general three-body problem

Abstract

An original method for searching for regions of initial conditions giving rise to close to periodic orbits is proposed in the framework of the general three-body problem with equal masses and zero angular momentum. Until recently, three stable periodic orbits were known: the Schubart orbit for the rectilinear problem, the Broucke orbit for the isosceles problem, and the Moore eight-figure orbit. Recent studies have also found new periodic orbits for this problem. The proposed method minimizes a functional that calculates the sum of squared differences between the initial and current coordinates and the velocities of the bodies. The search is applied to short-period orbits with periodsT < 10τ, where τ is the mean crossing time for the components of the triple system. Twenty one regions of initial conditions, each corresponding to a particular periodic orbit, have been found in the current study. A criterion for the reliability of the results is that the initial conditions for the previously known stable periodic orbits are located inside the regions found. The trajectories of the bodies with the corresponding initial conditions are presented. The dynamics and geometry of the orbits constructed are described.