Principal Periodic Orbits of the Keplerian Dumbbell System

We derive the exact equations of motion of the Keplerian dumbbell (KD) system, which consists of two point masses connected by a rigid massless rod, moving under the gravitational influence of a third point mass. We find several classes of continuous families of periodic orbits (principal periodic orbits) of this system, and we analyze their stability. In particular, these principal periodic orbits are stable both when the dumbbell arm points to the primary body and for a large circular orbital radius. At all times, this class of stable orbits has one of the dumbbell masses closer to the primary body, implying that axisymmetric bodies captured in these orbits have a hidden face relative to the primary body. This is an example of synchronous rotation without tidal locking. Moreover, we show that periodic points of the Poincare map of the equations of motion do not necessarily correspond to periodic orbits of the dumbbell, which is a consequence of one of the symmetries of the Hamiltonian of the KD system.