Transformation of Satellite Breakup Distribution for Probabilistic Orbital Collision Hazard Analysis

Fragmentation clouds from explosions or collision of payloads and rocket bodies in space pose a threat to objects in Earth orbit. Most of the fragments are too small to be tracked and can only be accounted for statistically. Here, a framework for the fully statistical treatment of a fragmentation cloud, its evolution and ramifications, without the need of simplifying assumptions, is presented. The cloud is modeled as an uncertainty around a single fragment, which can be propagated using any of the existing, nondeterministic, nonlinear orbital uncertainty propagation methods. This work is focused on providing the initial distribution and the estimation of the statistical collision probability. The NASA standard breakup model is revisited to derive a probability distribution of the initial fragment cloud. Two density transformation methods are discussed to obtain the distribution in a subset of orbital elements, suitable for mid- to long-term evolution. The fragment spatial density and the impact rates on targets in any orbit are obtained. The method is applied to show the fragment cloud distribution of a payload collision in low Earth orbit (LEO). Its collision probability with a satellite in LEO and a rocket body in the geostationary transfer orbit are estimated. The result is compared against, and shows the limitations of, sampling and methods based on finite differences.