Chaos in the vicinity of a singularity in the Three-Body Problem: The equilateral triangle experiment in the zero angular momentum limit

We present numerical simulations for the three-body problem, in which three particles lie at rest at the vertex of a perturbed equilateral triangle. In the unperturbed problem, the three particles fall towards the center of mass of the system to form a three-body collision, or singularity, where the particles overlap in space and time. By perturbing the initial positions of the particles, we are able to study chaos in the vicinity of the singularity. Here we cover the full range in parameter space for binary formation due to three-body interactions of isolated single stars, covering the singular region corresponding to an equilateral triangle and extending to sufficiently deformed triangles that we enter the binary-single scattering regime (i.e., one side of the triangle is very short and the other two are very long). We make phase space plots to study the regular and ergodic subsets of our simulations independently and derive the expected properties of the left-over binaries from three-body binary formation in isotropic cluster environments. We further provide fits to the ergodic subset to characterize the properties of the left-over binaries. We identify the discrepancy between the statistical theory and the simulations to the regular subset of interactions, which exhibit only weak chaos. As we decrease the scale of the perturbations in the initial positions, the phase space becomes entirely dominated by regular interactions, according to our metric for chaos.