Point Process Formulation of Long-Term Collision Risk Statistics from Orbital Fragmentation

Abstract

This paper focuses on the estimation of statistics (mean and variance) on collision risks induced by on-orbit fragmentation, evaluated over a long horizon of time (several decades). Two major contributions are introduced and can be used independently from one another to address the estimation problem. First, a representation of the cloud of debris with point processes is proposed, focusing on the population statistics rather than individual pieces of debris. It allows for the limited propagation of the cloud statistics required for the computation of the risk statistics, and nothing more, reducing significantly the computational costs when compared to traditional methods propagating individual pieces of debris. Then, a novel risk function is proposed, exploiting the ergodicity of the encounter problem to estimate the number of collisions between a pieces of debris and a target, following a rigorous mathematical construction based on well-defined assumptions. These two methods are then combined and illustrated on a simulated scenario and compared with a traditional Monte Carlo approach. The results show that the point process-based estimate of the risk statistics is consistent with the Monte Carlo reference, while being faster to run by at least two orders of magnitude.