Transition regions between stable periodic solutions of the general three-body problem

Abstract

The behavior of triple systems in the transition zones located between regions of stability is studied in the framework of the general three-body problem with equal masses and zero angular momentum. It is well known that there are exist three stable periodic orbits, namely, the Schubart orbit for the rectilinear problem, the Broucke orbit for the isosceles problem, and the Moore eight-figure orbit. In the space of the initial conditions, these orbits are surrounded by sets of points where bounded motions are observed over substantial time intervals. The transition zones between the Schubart and Moore orbits and the Moore and Broucke orbits are studied. It is shown that the boundaries of the regions of stability can be either smooth and sharp or diffuse. Beyond these boundary regions, the dynamical evolution of triple systems results in a distant ejection of one of the components, or the decay of the system. The distribution of the times when the stability is lost is constructed, and obeys a power law for long time intervals. Three stages in the evolution of an unstable triple system are identified.