Open Access

ARTICLE

Solution of Liouville's Equation for Uncertainty Characterization of the Main Problem in Satellite Theory

by Ryan Weisman³, Manoranjan Majji⁴, Kyle T. Alfriend⁵

1 Presented in Part at the 2015 International Conference on Computational and Experimental Engineering and Sciences (ICCES) Modeling in Engineering and Sciences Mini-Symposium: Computational Methods in Celestial Mechanics as paper ICCES 1520-150422180.
2 This work was prepared as part of official Government duties and it not subject to U.S. copyright protection in the United States.
3 Research Aerospace Engineer, Air Force Research Laboratory, AFRL/RVSV 3550 Aberdeen Ave. SE, Kirtland AFB, 87117
4 Assistant Professor, Mechanical and Aerospace Engineering Department, University at Buffalo, 318 Jarvis Hall, Buffalo, NY, 14260.
5 TEES Research Professor, Aerospace Engineering Department, Texas A& M University, College Station, TX, 77843-3141.

Computer Modeling in Engineering & Sciences 2016, 111(3), 269-304. https://doi.org/10.3970/cmes.2016.111.269

Abstract

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.

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