Collision probability due to space debris clouds through a continuum approach

As the debris population increases, the probability of collisions in space grows. Because of the high level of released energy, even collisions with small objects may produce thousands of fragments. Propagating the trajectories of all the objects produced by a breakup could be computationally expensive. Therefore, in this work, debris clouds are modeled as a fluid, whose spatial density varies with time under the effect of atmospheric drag. By introducing some simplifying assumptions, such as an exponential model of the atmosphere, an analytical expression for the cloud density evolution in time is derived. The proposed approach enables the analysis of many potential fragmentation scenarios that would be time limited with current numerical methods that rely on the integration of all the fragments' trajectories. In particular, the proposed analytical method is applied to evaluate the consequences of some recent breakups on a list of target objects. In addition, collision scenarios with different initial conditions are simulated to identify which parameters have the largest effect on the resulting collision probability. Finally, the proposed model is used to study the mutual influence among a set of high-risk targets, analyzing how a fragmentation starting from one spacecraft affects the collision probability of the others.