Interpolation and Integration of Phase Space Density for Estimation of Fragmentation Cloud Distribution

To calculate the effects of on-orbit fragmentations on current or future space missions, accurate estimates of the fragment density and its time evolution are required. Current operational tools estimate the risks involved through representative objects. Such tools, however, cannot accurately estimate the fragment density at any point in space and time. Rather, they directly calculate the number of close approaches from the representative objects. As such, they require a large number of Monte Carlo (MC) simulations to accurately find the collision risk over a large domain. Instead, the continuity equation can be applied to model the fragment density as a continuum, and propagate it forward in time. To model the evolution in any orbital region, the continuum can be propagated semi-analytically along its characteristics. The difficulty arises in estimating the density in between the cloud of samples. Here, the underlying density distribution is estimated by fitting a Gaussian Mixture Model (GMM) to the characteristics. An example of a break-up in three dimensions is given. It is shown that the model can accurately be fitted at different snapshots after the fragmentation, even with a low number of sample points. Given an analytical expression of the density enables the subsequent integration of the collision risk at any point in the phase space.