Thermospheric density uncertainty propagation based on linearized relative orbital dynamics

Thermospheric density uncertainty is the dominant source of uncertainty for orbit prediction in Low Earth Orbit (LEO) and Very Low Earth Orbit (VLEO). While analytical uncertainty propagation methods based on mean orbital elements accurately capture long-term secular drift, they fail to represent short-term dynamics, leading to a significant overestimation of along-track uncertainty for prediction horizons below one orbital period. These short-term dynamics are characterized by a zero-crossing of the along-track error caused by the initial differential drag acceleration counteracting the secular drift known as the drag paradox. This limitation is addressed in this paper in the form of an uncertainty propagation model based on linearized relative orbit mechanics using the Hill-Clohessy-Wiltshire (HCW) equations. An analytical solution is derived to characterize the zero-crossing phenomenon and evaluate the effect on the along-track uncertainty. To account for time-varying dynamics and temporally correlated atmospheric density errors, the model is extended using Linear Covariance Propagation with a First-Order Gauss–Markov Process (GMP). Validation with Monte Carlo simulations demonstrates that the numerical HCW model accurately reproduces the empirical uncertainty and the zero-crossing effect that the mean element approaches miss. A sensitivity analysis confirms the model’s robustness as well as the general zero-crossing behavior across varying altitudes, correlation times and solar activities while identifying valid propagation timeframes and the limiting effects of orbital eccentricity.