An evaluation of Monte Carlo for nonlinear initial uncertainty propagation in Keplerian mechanics

This paper evaluates the performance of traditional Monte Carlo (FMC) for the nonlinear propagation of initial uncertainty in the two-body problem: an essential task in space situational awareness. This is done in light of a newly developed Markov chain Monte Carlo (MCMC) based particle approach that combines the benefits of MCMC sampling with the method of characteristics (MOC) for solving first order partial differential equations - in this case, the stochastic Liouville equation (SLE). The resulting MCMC-MOC ensemble is by construction, equivalent in measure to the true state probability density. Our recent results on the MCMC-MOC approach indicate that for systems with divergence-free dynamics, the FMC and MCMC-MOC ensembles are statistically consistent. Unfortunately, the unperturbed two-body problem (Keplerian motion) is one such system. In this paper, we demonstrate through simulation that the traditional MC propagated ensemble is indeed equivalent in measure to MCMC-MOC ensemble, which in turn is the true system measure by construction. As a result, it is not possible to improve upon FMC and its slow convergence rate for this problem.