Relative Motion of Satellites Exploiting the Super-Integrability of Kepler’s Problem

This paper builds upon the work of Palmer and Imre exploring the relative motion of satellites on neighbouring Keplerian orbits. We make use of a general geometrical setting from Hamiltonian systems theory to obtain analytical solutions of the variational Kepler equations in an Earth centred inertial coordinate frame in terms of the relevant conserved quantities: relative energy, relative angular momentum and the relative eccentricity vector. The paper extends the work on relative satellite motion by providing solutions about any elliptic, parabolic or hyperbolic reference trajectory, including the zero angular momentum case. The geometrical framework assists the design of complex formation flying trajectories. This is demonstrated by the construction of a tetrahedral formation, described through the relevant conserved quantities, for which the satellites are on highly eccentric orbits around the sun to visit the Kuiper belt.