Numerical confirmation of the existence of triple collision orbits inside the domain of the free-fall three-body problem

Abstract

We performed numerical orbit integrations of the equal-mass free-fall three-body problem. We provided numerical evidence of the existence of triple collision orbits concentrating on the shape of triangles formed by the three bodies. In the process, in the negative side, we confirmed that the shape of the triangle of non-triple collision orbits deviates from equilateral as the orbit approaches the instant of supposed triple collision. In the positive side, the orbit integrations have been done along the three binary collision curves on which the orbit perform binary collision. These curves meet at the triple collision point (TCP). Three binary collisions have the three base edges, whose direction defines the edges of the triple collision orbit. The triangle defined by these three edges precisely coincides with the equilateral triangles of the triple collision orbit.