A search for triple collision orbits inside the domain of the free-fall three-body problem

Abstract

We look for triple collision orbits which are collisionless before triple collision. We developed a procedure of fixing the positions of these orbits inside the initial condition plane of the free-fall three-body problem as a natural consequence of the use of symbol sequences. Before looking for these orbits, an error regarding the relation between triple collision points and binary collision curves is corrected, that is, we confirmed that the intersections of binary collision curves of different generations (see the text for definition) are not the initial points of triple collision orbits but of the orbits with plural binary collisions along their trajectories. Then, we numerically established that a triple collision point (i.e., a point of the initial condition plane whose orbit ends at triple collision) can be found as an intersection of three cylinders of the same generation. We do not obtain triple collision orbits with symbol sequences shorter than eight digits. We obtained 11 triple collision points inside the initial condition plane. The orbits starting from these points have finite lengths in the future and in the past since the problem is free fall. These orbits start at triple collision, expand the size until the free-fall states, and go back to triple collision. Thus, these are time symmetric with respect to the time of free fall. Two types of triple collision orbits are identified. One type of orbits starts with a positive triangle formed with three bodies and ends at triple collision also with a positive triangle. The other type starts with a positive triangle and ends with a negative triangle.