Relative Orbit Geometry Through Classical Orbit Element Differences

The relative orbit geometry of a spacecraft formation can be elegantly described in terms of a set of orbit element differences relative to a common chief orbit. For the nonperturbed orbit motion, these orbit element differences remain constant if the anomaly difference is expressed in terms of the mean anomaly and all semimajor axes are equal. A general method is presented to estimate the linearized relative orbit geometry for both circular and elliptic chief reference orbits. The relative orbit is described purely through relative orbit element differences, not through the classical method of using Cartesian initial conditions. Analytical solutions of the relative motion are provided in terms of the true anomaly angle. When this angle is swept from 0 to 2π, it is trivial to estimate the along-track, out-of-plane, and orbit radial dimensions. The orbit element-based relative motion predictions are valid for both the osculating element space and the mean element space. The main assumption being made in the linearization is that the relative orbit radius is small compared to the inertial orbit radius relative to Earth. The resulting linearized relative motion solution can be used to assist in selecting the orbit element differences that yield a natural desired relative orbit geometry. Linearized analytical solutions are provided that show what secular drift is caused by having a nonzero semimajor axis difference, or what influence the J 2 gravitational perturbation will have on the mean relative motion.