The motion of a satellite in resonance with the longitude-dependent harmonics.

The solution to the motion of a satellite in an eccentric orbit and in resonance with one or more of the longitude-dependent harmonics of the central planet is developed. The method of solution parallels the well known von Zeipel method of general perturbations. The solution consists of expressions for the variations of the Delaunay variables. These expressions are composed of the perturbations developed by Brouwer in 1959 for the motion of an artificial satellite plus first-order resonant perturbations due to longitude-dependent harmonics (in terms of Legendre normal elliptic integrals of the first and second kind).