New State Transition Matrices for Spacecraft Relative Motion in Perturbed Orbits

This paper presents new state transition matrices that model the relative motion of two spacecraft in arbitrarily eccentric orbits perturbed by and differential drag for three state definitions based on relative orbital elements. Both density-model-specific and density-model-free formulations of the effects of differential drag are included. The state transition matrices are derived by first performing a Taylor expansion on the equations of relative motion and subsequently computing an exact closed-form solution of the resulting linear differential equations. The resulting state transition matrices are used to generalize the geometric interpretation of the effects of and differential drag on relative motion in near-circular orbits provided in previous works to arbitrarily eccentric orbits. Additionally, this paper harmonizes current literature by demonstrating that a number of state transition matrices derived by previous authors using various techniques can be found by subjecting the models presented in this paper to more restrictive assumptions. Finally, the presented state transition matrices are validated through comparison with a high-fidelity numerical orbit propagator. It is found that these models are able to match or exceed the accuracy of comparable models in the literature over a broad range of orbit scenarios.