The derivation of a second-order satellite trajectory predictor in terms of geometrical variables

Abstract

This paper develops a new semianalytic theory by generalization of the Method of Averaging (MoA) and the Stroboscopic Method (SM). Unlike the traditional development of the MoA, which must use the mean anomaly, this new approach can treat any orbital anomaly. In particular, a more geometrically significant position variable, the true anomaly, is considered here. This extension of the traditional methods presents significant analytical and numerical advantages. For instance, the averaging of the rate functions does not require the usual change of variable to true anomaly and the evaluation of the short-period terms does not involve the solution of the transcendental Kepler equation. The new set of differential equations is completely independent of the satellite anomalistic motion, thus allowing the propagation of the state with a relatively large (up to a few days) integration stepsize. This feature effectively increases the flexibility of the SM, which normally restricts the propagation of the state to integral multiples fo the orbital period. The transformations that relate this new set of averaged rate functions and short-period terms to the traditional ones are derived in a series of analytic expressions called Equivalence Relations. These relations allow a successful comparison of this theory with the classical results found in the literature. Further changes of independent variables to “mean time” and “mean orbit number” naturally lead to difference equations and multirevolution integration algorithms. The principles of this theory are applied to the equations that describe the motion of an earth satellite subjected to gravity ( J 2, J 3 and J 4 harmonics) and drag perturbations. The atmospheric model includes the effects of solar and geomagnetic activity, the diurnal, semiannual and seasonal-latitudinal cycles of density variations and the flattening and rotation of the atmosphere. The short-period oscillatory terms and the averaged differential equations associated with these perturbations are developed into completely analytical expressions. This feature provides insights into the dynamics of the satellite and simplifies the numerics of the prediction process. Numerical simulation of this new set of differential equations and comparison with the propagation of the unaveraged (osculating) set of differential equations show that the predictor proposed in this paper significantly increases computation speed without loss in accuracy. This theory has been used to predict the orbital lifetime of actual satellites and to conduct a parametric analysis of satellite orbital lifetime.