Analysis of J ₂-Perturbed Relative Orbits for Satellite Formation Flying

Abstract

We study the concept of satellite formation flying in a geodetic context, namely as a viable alternative for future gravity field satellite missions. The feasibility of formation flight is demonstrated. In particular the stability of such a formation in a J ₂ gravity field is investigated. To this end three orbit computation approaches are compared: 1) numerical integration of Newton’s equations (nn) of motion, 2) numerical integration of Hill equations (he), and 3) a new set of nontrivial, non-homogeneous analytical solutions of he.

Hill equations provide an elementary description of relative orbital motion. In order to accommodate J ₂ gravitational perturbations we modify the he in several steps: evaluating the J ₂ disturbing force function on the nominal orbit; changing the orbital rotation rate (frequency matching), due to in-orbit J ₂ precession; as well as evaluating the time-averaged J ₂ gravity gradient tensor. The resulting he are solved analytically.

The orbit simulations show that the analytical solution of the modified he are consistent with their numerically integrated counterpart. Differences with respect to the reference nn method remain, which means that not all J ₂ effects have been captured yet in the modified he. The usefulness of HE as a formation design tool are demonstrated by simulations of circular relative motion.