Solution of Liouville's Equation for Uncertainty Characterization of the Main Problem in Satellite Theory - Weisman, Ryan; Majji, Manoranjan; Alfriend, Kyle T (Computer modeling in engineering & sciences, 2016)
This paper presents a closed form solution to Liouville's equation governing the evolution of the probability density function associated with the motion of a body in a central force field and subject to J2. It is shown that the application of transformation of variables formula for mapping uncertainties is equivalent to the method of characteristics for computing the time evolution of the probability density function that forms the solution of the Liouville's partial differential equation. The insights derived from the nature of the solution to Liouville's equation are used to reduce the dimensionality of uncertainties in orbital element space. It is demonstrated that the uncertainty propagation is fastest in the semi-major axis and the mean anomaly phase sub-space. The results obtained for uncertainty propagation for the two body problem are applied to investigate the uncertainty propagation in the presence of the J2 perturbation using a combination of osculating and mean element perturbation theory. Analytical orbital uncertainty propagation calculations are validated using Monte-Carlo results for several representative orbits.