Hyperbolic Solutions of the Full Two-Body Problem

Abstract

In this work, we study a system of two rigid bodies interacting through Newtonian gravitation and establish conditions for the occurrence of Hill instability. Specifically, we prove the existence of forward hyperbolic orbits for an open and unbounded set of initial conditions. These orbits are characterized by a linear growth in time of the mutual distance between the bodies. We analyze the asymptotic behavior of both the rotational and translational motions associated with such trajectories. For the rotational dynamics, our results rely on the theory of asymptotically autonomous systems. Regarding the translational motion, we derive an asymptotic formula for the relative position of the barycenters of the bodies. In the final section, we outline several open questions concerning forward hyperbolic orbits, which we believe may be of further interest.