Study on Relative Orbit Geometry of Spacecraft Formations in Elliptical Reference Orbits

This paper studies the relative orbit geometry of a leader-follower spacecraft formation flying in unperturbed elliptical reference orbits. The first-order relative position equations, derived using the reference orbital element approach under the condition that the follower and leader spacecraft have equal semimajor axes, are transformed from trigonometric forms to parametric and algebraic forms. The conditions for and the number of self-intersections of the relative orbit projected onto the three coordinate planes of the leader local-vertical-local-horizontal frame are obtained. The relative orbit proves to be three-dimensional instead of planar in most cases, and may self-intersect spatially at most once. The collision between the follower and leader spacecraft corresponds to the case where the solution curve to the first-order relative motion equations passes through the origin. The conditions for collision are subsequently determined. For a nondegenerate case (in which none of the relative motion in the radial, in-track, and cross-track directions vanish), three types of relative orbit are possible. Most frequently, the relative orbit is on a one-sheet hyperboloid. Otherwise, when the relative orbit has a real or finite imaginary self-intersection, it rests on an elliptic cone. In the rest of the cases, including that with an imaginary self-intersection at infinity, the relative orbit is on an elliptic cylinder. The criteria for these three types are given, respectively, followed by examples. [PUBLISHER ABSTRACT]