Analysis of fidelities of linearized orbital models using least squares

Satellites orbiting in Low Earth Orbit (LEO) are accelerated by Earth gravity and dominant orbital perturbations due to Earth oblateness and atmospheric drag. The equations of motion describing such a motion are highly nonlinear in nature. Linearized orbital models only approximate these nonlinear dynamics. The difference between a linearized model and full nonlinear dynamical equations of motion is termed as process noise. To determine the accuracy of these approximate models, we need to compare these with numerical integration of the full non-linear dynamical equations. Linearized solution propagations are characterized by a set of initial conditions which determine orbital evolution. The question arises on how to choose the initial conditions of the analytical approximation appropriate to a given choice of initial conditions for the numerically propagated orbit such that the process noise is minimized. An algorithm is developed, based upon the statistical method of nonlinear least squares to compare linearized orbital models for relative and absolute satellite dynamical motion with numerically propagated orbits to evaluate their accuracy. Due to recent interest in formation flying missions a comparison of accuracies of linearized relative orbital models i.e., Hill-Clohessy-Wiltshire (HCW) equations, J₂ Modified Hills equations by Schweighart-Sedwick (SS), and for absolute orbital models described by analytical equations for Kepler´s problem and Epicycle Model by Hashida and Palmer have been carried out.