The Binary Returns!

Consider the spatial Newtonian three body problem at fixed negative energy and fixed angular momentum. The moment of inertia $I$ provides a measure of the overall size of a three-body system. We will prove that there is a positive number $I_0$ depending on the energy and angular momentum levels as well as the masses such that every solution at these levels passes through $I\leq I_0$ at some instant of time. Motivation for this result comes from trying to prove the impossibility of realizing a certain syzygy sequence in the zero angular momentum problem.