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Symmetric Periodic Orbits in the Dipole-segment Problem for Two Equal Masses

We consider a relatively simple model to represent some types of asteroids that are very elongated and with two protuberances on their end points. A conspicuous example of this type of body is asteroid 216-Kleopatra. To model such bodies, we consider a dipole segment, which consists of a massive segment with two spherical masses at the end point of the rod and which is rotating about its center of mass with uniform velocity. In particular, we restrict our study to the case in which both massive spheres are equal. For this problem, we obtain the potential, equilibria, and stability and compute the families of symmetric periodic orbits. We also discover asymptotic points in the family map, which represents heteroclinic orbits connecting different equilibria.