Orbital distribution arbitrarily close to the homothetic equilateral triple collision in the free-fall three-body problem with equal masses (Q1976129)

The existence of escape and non-escape orbits arbitrarily close to the homothetic equilateral triple-collision orbit is proved analytically in the three-body problem with zero initial velocities and equal masses. The orbital distribution in a small neighborhood of a particular triple-collision orbit is obtained analytically, taking into account some numerical results. The paper consists of five sections. In the first section the authors present a short review of the literature on the title problem. In section two numerical results on orbital distributions are obtained, and a definition of the free-fall three-body problem is introduced. The dynamical evolution is described by Newtonian equations of motion. The authors investigate the case of bodies with equal masses, and discuss triple-collision points, hyperbolic-elliptic escape, and parabolic-elliptic escape. Theorems on the behavior of solutions near homothetic equilateral solutions are topics of section three, whereas section four deals with the escape in planar isosceles subsystem, and formulates an escape criterion. The final section five analyzes the distribution of hyperbolic-elliptic regions around the homothetic equilateral point. In any small neighborhood of the triple-collision orbit corresponding to the homothetic equilateral solution, all types of orbits exist, so that the system can lead to hyperbolic-elliptic escape, parabolic-elliptic escape, and to the non-escape, respectively, after the first triple encounter.