Corrected closed-form solutions to the Schweighart–Sedwick satellite relative motion model including differential drag

Rendezvous maneuvers represent one of the most complex maneuvers in spaceflight. To approximate the relative motion between two objects during short time periods, linearized motion models, such as the well-known Clohessy-Wiltshire model, are commonly employed. To address the inaccuracy of these equations arising from the non-spherical gravitational potential field, Schweighart and Sedwick developed a linear relative motion model which accounts for the $J_2$ effect. Closed-form solutions to these equations including differential specific forces between the two spacecrafts have been widely applied by multiple researchers to study the methodology of differential lift and drag.

While working with these solutions, however, it became apparent that small deviations between the numerically integrated and analytical solutions result if specific differential forces are applied in the tangential direction $\delta f_y$, which in the case of differential aerodynamic forces corresponds exactly to the main influence, namely differential drag. This insight was taken as an opportunity to determine corrected closed-form solutions via the computer algebra system Maxima (Maxima, 2023), a software package for performing computer algebra calculations. In this technical note, these solutions are presented, their correctness is demonstrated and subsequently the effect of the motion in the phase plane due to perturbing forces is discussed. This note is intended to contribute to the field of research by increasing the accuracy of future studies.