Closed-Form Nonlinear Covariance Prediction for Two-Body Orbits

A closed-form solution for nonlinear covariance prediction is formulated for two-body orbits. Nonlinear terms arise from errors in the nonlinear state transition matrix caused by errors in the state. With nonlinear terms included, the predicted covariance cannot become unrealistically small after long prediction intervals. Simulations demonstrate that the second moments of Monte Carlo error distributions are accurately characterized by nonlinear covariance prediction, whereas linear covariance prediction grossly under-estimates these moments. For near-circular orbits, the nonlinear terms exceed the linear terms after 1/4 revolution. The covariance solution is useful for real orbit determination and prediction because it is the zeroth-order solution of a state-dependent matrix-Riccati equation for perturbed orbits.