Osculating Relative Orbit Elements Resulting from Chief Eccentricity and J2 Perturbing Forces

Relative orbit elements (ROEs) based on a circular chief satellite orbit are erroneous when applied to a perturbed,non-circular reference orbit. In those situations, the ROEs will encounter geometric instability and drift. To counter this, a set of time-variant ROEs have been derived to describe the relative orbit for both the unperturbed, elliptical chief, and the perturbed, circular chief. A highly coupled relationship is found that describes the relative trajectory to higher accuracy when compared to numerical integration. To show the applicability of the ROEs to formation design, methods to initialize a stationary relative orbit are detailed and an algorithm for ROE based guidance and navigation is proposed. The results provide a method to predict the relative motion, while examining time-varying parameters of the motion. Eccentricity effects are shown to induce severe time-variance to the system and introduce a level of mathematical abstraction with the current parameterization. Perturbing J2 effects are shown to introduce periodic effects and compound the secular variations to the circular ROEs.