The formation of periodic three-body orbits for Newtonian systems

Abstract

Braids are periodic solutions to the general N-body problem in gravitational dynamics. These solutions seem special and unique, but they may result from rather usual encounters between four bodies. We aim at understanding the existence of braids in the Galaxy by reverse engineering the interactions in which they formed. We simulate self-gravitating systems of N particles, starting with the constructing of a specific braid, and bombard it with a single object. We study how frequently the bombarded braid dissolves in four singles, a triple and a single, a binary and 2 singles, or 2 binaries. The relative proportion of those events gives us insight into how easy it is to generate a braid through the reverse process. It turns out that braids are easily generated from encounters between 2 binaries, or a triple with a single object, independent on the braid's stability. We find that 3 of the explored braids are linearly stable against small perturbations, whereas one is unstable and short-lived. The shortest-lived braid appears the least stable and the most chaotic. nonplanar encounters also lead to braid formation, which, in our experiments, themselves are planar. The parameter space in azimuth and polar angle that lead to braid formation via binary-binary or triple-single encounters is anisotropic, and the distribution has a low fractal dimension. Since a substantial fraction of ~9% of our calculations lead to periodic 3-body systems, braids may be more common than expected. They could form in multi-body interactions. We do not expect many to exist for time, but they may be common as transients, as they survive for tens to hundreds of periodic orbits. We argue that braids are common in relatively shallow-potential background fields, such as the Oort cloud or the Galactic halo. If composed of compact objects, they potentially form interesting targets for gravitational wave detectors.