Nonlinear dynamic modeling of satellite relative motion with differential $\pmb{J}_{2}$ and drag

A systematic approach to modeling the relative motion of artificial satellites in the presence of perturbations is presented. The relative motion is described using relative position and velocities as states. The modeling here is restricted to low Earth orbit (LEO) satellites and therefore includes the differential J2 and drag effects. In this paper we expand on the modeling approach that makes use of the Reference Satellite Variables for the chief's orbit using simple Newtonian mechanics to systematically derive the exact nonlinear relative motion model with differential J2 and drag. These equations are exact for eccentric reference orbits as well as equatorial. This intuitive modeling approach shall establish a framework to incorporate other kinds of differential perturbations for higher fidelity models based on the significance of application. Simulation results of the developed nonlinear relative motion model show the effect of differential J2 and drag captured by the equations for a LEO leader-follower formation with large intersatellite distances. The propagation errors of the model are studied for varying initial conditions and reference orbits. A subsequent analysis gives further insight into how the model developed is particularly free from singularities in the special case of J2 and drag disturbances alone.