Bounded, Periodic Relative Motion using Canonical Epicyclic Orbital Elements

In this paper, we use a Hamiltonian approach to derive the equations of motion for an object relative to a circular or slightly elliptical reference orbit. By solving the Hamilton-Jacobi equation we develop constants of the relative motion called epicyclic elements. A perturbation Hamiltonian is formulated in order to derive variational equations for the “constants” via Hamilton’s equations. We use this formalism to derive bounded, periodic orbits in the presence of various perturbations. In particular, we show a simple no-drift condition that guarantees bounded orbits in the presence of J2 forces. We also derive the relative motion deviations and boundedness conditions due to eccentricity of the reference orbit and higher-order terms in the gravitational potential.