The formation of periodic three-body orbits for Newtonian systems

¹ Leiden Observatory, Leiden University PO Box 9513 2300 RA Leiden, The Netherlands
² Mathematisch instituut, Leiden University PO Box 9513 2300 RA Leiden, The Netherlands
³ Niels Bohr International Academy, University of Copenhagen Copenhagen, Denmark

^★ Corresponding author.

Received: 24 November 2025
Accepted: 28 January 2026

Abstract

Context. Braids are periodic solutions to the general N-body problem in gravitational dynamics. These solutions seem special and unique but may result from rather usual encounters between four bodies.

Aims. We attempted to learn more about the existence of braids in the Galaxy by reverse engineering the interactions in which they formed.

Methods. We carried out simulations of self-gravitating systems of N particles using fourth-order integration. We started by constructing the specific braid and subsequently bombard it with a single object. We subsequently studied how frequently the bombarded braid dissolves into four singles, a triple and a single, a binary and two singles, or two binaries. The relative proportion of these events gives us insight into how easy it is to generate a braid through the reverse process.

Results. Braids are relatively easily generated from encounters between two binaries or between a triple and a single object, so long as they meet each other, independent of the braid’s stability. We find that three of the explored braids are long-lived and linearly stable against small perturbations, whereas the fourth is unstable and short-lived. The shortest-lived braid appears to be the least stable and the most chaotic. Nonplanar encounters can also lead to braid formation, which, in our experiments, are themselves planar. The parameter space in azimuth and polar angle that leads to braid formation via binary–binary or triple–single encounters is anisotropic, and the distribution has a low fractal dimension.

Conclusions. Since ∼9% of our calculations lead to periodic three-body systems, a substantial fraction, braids may be more common than currently assumed. They could easily temporarily exist as a result of multi-body (binary–binary or triple–single) interactions. We do not expect many stable braids to exist for an extensive period of time, but they may be quite common as transients, surviving for tens to hundreds of periodic orbits. We argue that braids are particularly common in relatively shallow-potential background fields, such as the Oort cloud or the Galactic halo. The formation of braids, however, easily leads to collisions between two or more of their constituents. If composed of compact objects, they potentially form interesting targets for gravitational wave detectors.