Second-order relative motion equations

An approximate solution of second-order relative motion equations is presented. The equations of motion for a Keplerian orbit in spherical coordinates are expanded in Taylor series form using reference conditions consistent with that of a circular orbit. Only terms that are linear or quadratic in state variables are kept in the expansion. The method of multiple scales is employed to obtain an approximate solution of the resulting nonlinear differential equations, which are free of false secular terms. This new solution is compared with the previously known solution of the linear case to show improvement and with numerical integration of the quadratic differential equation to understand the error incurred by the approximation. In all cases, the comparison is made by computing the difference of the approximate state (analytical or numerical) from numerical integration of the full nonlinear Keplerian equations of motion. The results of two test cases show two orders of magnitude improvement in the second-order analytical solution compared with the previous linear solution over one period of the reference orbit.