A linear model for relative motion in an elliptical orbit based on a spherical coordinate system

When compared with Cartesian coordinates, curvilinear coordinates have shown significant superior accuracy when applied to a linear model for relative motion in a circular reference orbit. Experts assume the same superiority for an elliptical reference orbit. However, the curvilinear model in an elliptical reference orbit has not been established, and the superiority has not been proved. After a strictly theoretical derivation this paper establishes a linear model with spherical coordinates for relative motion in an elliptical reference orbit. The differential equation has the same form as the Lawden or Tschauner-Hempel equation and can be easily solved with the solutions from the literature. In addition both the theoretical derivation and the numerical examples prove the model's superior accuracy in leader-follower formation and formations with a large along-track distance. The proposed model reduces the approximate limits of close relative motion, and it is expected to be applied in the propagation, guidance, and control for longer term relative motion with large along-track distances.