Lagrangian Relative Equilibria in a Modified Three-Body Problem with a Rotationally Symmetric Ellipsoid

We consider the gravitational interaction of two point masses (or two perfect spheres) and a symmetric spheroid whose axis of symmetry is perpendicular to the plane of the centers of mass. Under appropriate approximations, the system formed by the three centres of mass decouples and may be considered as a separate three point mass problem with modified gravitational interaction. We find all the relative equilibria for the centres of mass system, and show that they can have isosceles or scalene configurations. The linear stability is studied by using the phase space splittings provided by the Reduced Energy Momentum Method. In our stability calculations, we are able to reduce the number of parameters by observing that the stability type is invariant under certain rescalings. We generalise these rescaling symmetries and prove some theoretical invariance results applicable to Lie-symmetric Hamiltonian systems. We comment the results in light of physically realistic situations and note that the scalene relative equilibria are present in models which validate the truncation of the gravitational series expansion within O(10).