The formation of periodic three-body orbits for Newtonian systems

TL;DR

This study investigates the formation and stability of periodic three-body braids within Newtonian four-body systems by reverse-engineering braid formation: a predefined braid is bombarded by a fourth object and the resulting outcomes are catalogued. Using a 4th-order Hermite integrator in N-body units, the authors quantify formation channels, stability, and the dependence on initial conditions, including planar and non-planar encounters. They find that braid formation occurs mainly through binary–binary and triple–single interactions, with several braids remaining long-lived and linearly stable, while one braid is unstable; the angular distribution of successful formations is anisotropic and exhibits fractal-like structure. The results imply braids could be more common as transient configurations in shallow gravitational potentials such as the Oort cloud or Galactic halo, and, if composed of compact objects, may serve as potential gravitational-wave sources.

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.

Figures (9)

• Figure 1: The four 3-body braids adopted in this study. The classic Figure-8 (top panel) problem, and the three more complicated braids, the unequal-mass problem, $I.A.^{i.c.}_4(0.5)$, and $I.A.^{i.c.}_{68}(0.5)$ for the bottom panel. Integrated with a time-step parameter of $\eta = 0.01$, a time step of $dt = 0.001$, and integrated for one periodic orbit. The bottom panel shows $I.A.^{i.c.}_{68}(0.5)$, which is a complex orbit for which the period exceeds 83 time units, much more than any of the others. The orbit closes only after many orbits, giving rise to the filled-in surfaces. For the Figure-8, only one line color is visible, because all three objects have the same orbit and therefore overlap.
• Figure 2: Two representations of braid $I.A.^{i.c.}_4(0.5)$ after 100 time units. The top panel shows the unperturbed solution integrated for 10 N-body time units. The bottom panel shows the result of the perturbed solution in the same time frame. The introduced perturbation was $10^{-5}$ in the $x$-coordinate. Both solutions have clearly deviated from the original braid, and from each other. By this time, the phase space distance between the two solutions is on the order of unity. The orbits are unstable, and in the next 100 crossing time, one particle is ejected.
• Figure 3: Braid model A after $10^6$ N-body time units. By this time, an initial perturbation of $10^{-5}$ has grown to O($10^{-2}$). The braid still resembles the original, but the entire orbit has been precessing.
• Figure 4: Three examples of how the braid $I.A.^{i.c.}_4(0.5)$ forms from an encounter between a binary (a,b) and two single stars (c) and (d), between two binaries (a,b), and (c,d), and a triple ((a,b),c) encountering a single (d).
• Figure 5: Model A distributions of orbital elements, semi-major axis and eccentricity for interaction Model A, resulting in a binary-binary with $f_{\rm BB} = 0.058$ (see \ref{['tab:Lyapunov_timescale']}). The top panel shows the semi-major axis of the two binaries, where we present the tighter binary along the x-axis and the wider binary along the y-axis (explaining the empty lower right corner bordered by the dashed line). The bottom panel shows the eccentricity distributions of various orbits. The Kolmogorov-Smirnov test indicates that for model A, the tighter and wider distributions for eccentricity are indistinguishable. For the Figure-8 ($p=0.0064$) and $I.A.^{i.c.}_{68}(0.5)$ ($p=0.034$) the eccentricity distribution of the tighter binaries appears not to be randomly sampled from a single parent distribution.