Abstract
The first-order solution of the problem of relative motion of a spacecraft in near-circular orbit is known to degrade in accuracy, when compared to the numerically integrated exact solution, at greater distances from the origin of the rotating reference frame. These solutions have been developed to study the problem of the terminal rendezvous guidance where an active spacecraft at several hundred km from its rendezvous target centered at the rotating frame, must maneuver to intercept the target in a given time. In References [1, 2], the relative motion technique was used for a different purpose, namely to describe the future motion of a spacecraft relative to the rotating frame, as it is perturbed by the Earth zonal harmonics J2 and J3, and by the luni-solar gravity effects. Although these perturbations have a small effect on the spacecraft motion which would not wander to great distances from the origin of the frame, it is felt that because of the presence of initial non-zero velocities, the subsequent motion may drift to considerable distances from the origin of the frame, thereby, degrading the accuracy of the analytic first-order solution of the equations of motion. The initial velocities exist because the orbit determination-generated osculating orbit is necessarily elliptical in nature with small eccentricity, such that at time zero, or epoch, a reference circular orbit having the same radius as the radial distance of the actual spacecraft is assumed, to describe the future motion of the vehicle itself, which, unlike the frame, experiences the various perturbations just mentioned. Second-order corrections to the linear solution of Reference [3] have been obtained in References [4, 5] to extend the region of accuracy of the analytic solutions at greater distances from the origin. This chapter rederives the second-order solutions, resolving the errors of Reference [4], and the typographical errors of Reference [5], by adopting the nomenclature of these two references in defining the coordinates. The radial coordinate is depicted as y, and the “tangential” coordinate by x, although x is pointing in the opposite direction of motion. The out-of-plane or z coordinate, is along the orbital angular momentum vector. In References [1, 2], x is along the radial direction, y along the “tangential” direction in the direction of motion, and z along the orbital angular momentum vector. When the second-order expressions developed here are added to the first-order solutions of the perturbed motion of References [1, 2], the differences in the coordinates must be properly accounted for.