Spacecraft Formation Flying

7.3 The Gim–Alfriend Geometric Method

The purpose of this section is to derive the state transition matrices for both osculating and mean elements for the relative motion of two neighboring satellites when the reference satellite (chief) is in an elliptic orbit, and both satellites are subjected to the $J_2$ perturbation. The linear differential equations of relative motion for this system are formidable. A perturbation solution in powers of $J_2$ could probably be obtained, but there is an easier approach. Rather than solving the equations of motion, the ...