Linearized SDE for Propagating Density Model Uncertainty

摘要

Orbital uncertainty or covariance plays an important role in both Space Objects (SO) catalogue maintenance and applications of a SO catalogue. Uncertainty within the estimated orbital elements, initial uncertainty, depends on the method of orbit determination and quality of the observations. The uncertainties in propagated orbital states also depend on the quality of the models employed in representing an orbit and the chosen uncertainty propagation technique. In low Earth orbiters, drag from upper atmosphere is the second most prominent force acting on them, after the perturbing force caused by the oblateness of Earth. Dedicated missions to develop detailed gravity models have reduced uncertainty in gravitational perturbations. The upper atmosphere densities are both spatially and temporally correlated, making it hard to accurately capture within a physical model. The nature of the atmosphere makes it hard to understand the uncertainty within the existing models, and to study its influence on propagating orbital uncertainty. In this paper, we address the problem of covariance propagation due to uncertainty in atmospheric density model. We establish a stochastic model of linear Clohessy-Wiltshire-Hill (CWH) equations of relative motions. The growth of uncertainty in satellite position due to model uncertainty is then obtained by deriving the time variation of covariance of the established Stochastic Differential Equations (SDE). To this end, a simplified exponential model is established, which can be calibrated with real observations. We use it to evaluate the density model induced uncertainties in the propagated states without using an orbit propagator.