Triple collision orbits with nonzero initial velocities

Triple collision orbits of the three-body problem with nonzero initial velocities have been systematically surveyed. For this purpose, we have formulated the sufficient conditions of the velocities of three bodies so that the total angular momentum of the triple system is zero. The velocity conditions are parameterized by two parameters α _2 and β _1 in the initial condition plane. We introduce the characteristic as the curve in the initial condition plane of the non-free-fall triple collision points (TCPs) parameterized by one parameter, say, α _2/β _1 . The velocity conditions with full parameters suggest that non-free-fall TCPs (nonff-TCPs) occupy two-dimensional areas in the initial condition plane. We plotted five characteristics passing through eight ff-TCPs, one of which forms a closed circuit. Two ff-TCPs on a characteristics are called a twin. This gives us a criterion of classification of TCPs in addition to the one obtained in Tanikawa and Mikkola (Cel Mech Dyn Astron 133:52, 2001) which connects the directions of the initial and final triangles formed by three bodies. We find that the twin ff-TCP TSM-1 and TSM-2 are connected by all the characteristics, which pass through one of them. A neighborhood of ff-TCP TSM-2 has been confirmed to be occupied by nonff-TCPs. We expect that the same is true with a neighborhood of any ff-TCP. Further, we expect that this is true with a neighborhood of any nonff-TCP. We do not continue the characteristics until its end because the continuation seems endless.