The goal of this paper is to show how traditional covariance matrix propagation is not always fit for the purpose of forecasting either the distribution of space debris or (which turns out to be equivalent) the probability of finding a body drifting on a gravitational field with incomplete knowledge of its initial conditions. The main limitation of covariance matrix propagation comes from the fact that the debris density function is poorly described by the two first moments alone, even if the initial density function is spherical. Given enough time, the chaotic nature of the motion under gravity stretches and bends the initial debris distribution into distorted and growing shapes ( called by us "bananoids") that the two lowest moments can no longer model it. To illustrate this fact we analyze a very simple model composed of an almost massless body under the action of a central gravitational force to show that, even under such favorable conditions (no atmospheric effects, no non-gravitational forces, no random forces), covariance propagation fails miserably. To overcome the limitations of the traditional approach we turn to the differential equations of the motion and its topological and measure properties. The partial differential equation that describes the time history of the debris distribution is of the same mathematical nature of the Lagrangian or material derivative of the fluid mechanics. If we add random forces the motion becomes a diffusion process. Its distribution is thus governed by the well known Kolmogorov-Fokker-Planck partial differential equation. Covariance matrix propagation is indeed an approximate solution to the KFP equation. We propose that more elaborate approximations be used, either including higher moments or moving to a new set of base functions.
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