High-Order Analytical Solutions of Hill's Equations

G. Gomez and M. Marcote

Abstract. The purpose of this paper is the semi-analytical computation of the
bounded orbits of Hill's equations, describing the relative motion of two
particles in their Keplerian motion around a central body. We have considered
the case in which one of the particles moves along a circular reference orbit.
The solutions obtained are the generalisation of the usual periodic orbits
obtained for the linearised equations and also of the third-order solution
computed by D.L. Richardson and J.W. Mitchell (2003). With the algorithm
presented, those orbits can be computed in a fast and efficient way up to a
high-order in the in-plane and out-of-plane amplitudes.