Perturbation solutions to the nonlinear motion of displaced orbits

This paper analytically describes the nonlinear motion of displaced orbits obtained by low-thrust propulsion, and focuses on the first-order analytical derivation of perturbation solutions of osculating Keplerian element (OKE) time histories. A virtual Earth (VE) model is developed to transform non-Keplerian orbits above the Earth into Keplerian orbits around the VE by treating the thrust and coordinate transformation effects as perturbations. The numerical OKEs computed from the Cartesian position and velocity exhibit secular and periodic variations. The short-period perturbations of the orbital elements are derived by the conventional quasi-mean element method. The differential equations of the secular and long-period perturbations are linearized for an analytical solution. A comparison between the analytical OKEs and numerical OKEs of a specific displaced orbit validates the correctness and good accuracy of the analytical derivation. One contribution of this paper is that the analytical OKEs can be applied for a fast calculation of a displaced orbit in the preliminary mission design phase, which has more significance than accurate but complicated time-consuming calculations by complete variational equations. Another contribution is revealing the connections between the amplitude as well as frequency of perturbation solutions and displaced orbits from a physical perspective, from which the topology and geometrical size of a displaced orbit can be easily inferred from the time history of OKEs.