Stochastic Dynamics of and Collision Prediction for Low Altitude Earth Satellites

Abstract

Air drag B^∗ factors from Earth satellite element sets often show the characteristic near Gaussian distribution and autocorrelation exponential decay typical of a Gauss-Markov process. Assuming the “most current” set of orbital elements are correct, earlier elements can be used to construct covariance matrices as a function of prediction time into the future. If resolved in cylindrical orbit frame coordinates, these are remarkably structured, essentially showing only in-track error growth. Often the in-track position covariance element growth follows a fourth power in time rule, and is definitely forced by the uncertainty in the air drag factor. This observation is confirmed both by perturbation theory and by modeling stochastic state covariance propagation. Realizing that almost all error growth under the SGP4 model is in track, the Cosmos 2251/Iridium 33 event is reexamined. While a collision prediction from the last elements shows a minimum miss distance of about 700 m, those same elements show a closest approach distance of the orbits of only 32 m. Given large in-track uncertainty, minimum orbit separation may be a much more reliable metric for maneuver decisions.