Second-Order State Propagation for Continuous-Thrust Relative Motion in $J_{2}$-Perturbed Elliptic Orbits

The second-order analytical state propagation under an arbitrary continuous thrust is proposed for relative motion in
$J_{2}$
-perturbed elliptic chief orbits. The mean relative orbital elements are used to describe the relative motion. The second-order relative motion equations are obtained by performing the Taylor expansion of the nonlinear relative motion equations about the chief orbit. The original thrust acceleration is approximated by a truncated Fourier series in terms of the eccentric anomaly of the chief orbit. Based on the first-order state propagation and the thrust acceleration Fourier-series expansion, the second-order propagation of the mean relative orbital elements is obtained. Numerical simulations show that the state propagation errors of the second-order solution are reduced by 1–2 orders of magnitude when compared with the first-order solution.