An evaluation and comparison of relative motion theories

A modeling error index is introduced for evaluating and comparing the accuracy of various theories of relative the motion of satellites. The derived index does not require linearization of the equations of motion so nonlinear theories can also be evaluated. The approximate methods include Hill's equation, small eccentricity state transition, matrix, non-J(2) state transition matrix, Gim-Alfriend state transition matrix and a nonlinear method. The two key parameters in the evaluation are the eccentricity of the reference orbit and the size measure of the relative motion orbit. The numerical results show the sequence of the index from high to low should be Hill's equation, non-J(2), small eccentricity, Gim-Alfriend state transition matrix and nonlinear method index. A size measure of the relative motion has a major effect on Gim-Alfriend, non-J(2) and small eccentricity index, whereas the eccentricity of the Chief orbit is the parameter which has the most effect on the Hill's equation and small eccentricity index.