A Model for Satellite Collisions

To many aerospace enthusiasts, the concept of space debris is no stranger. Indeed, with the exponential growth of satellites in space, space debris is becoming a pressing issue for agencies worldwide. Although there is currently no real solution available, a closely related concept is that of Space Surveillance and Tracking (SST). This field seeks to mitigate the problem by ensuring no new debris is formed and has received much attention in recent years as clients seek to protect their assets in space. As an equivalent to SST, it is possible to build simple mathematical models which encapture the key properties of the problem.

As a first step towards this goal, we could consider static random geometric graph on a spherical shell with SIS interactions between nodes. This structure allows us to model the intricate interactions between satellites, which only occur when satellites are close to each other. Using standard network techniques, such as clustering, it is then possible to find a threshold for when the so-called Kessler syndrome might occur.

It is quickly noted that this model is quite unrealistic however as it assumes satellites remain static in time. To remedy this problem, we thus incorporate node dynamics in the model, which transforms it to a temporal graph. This new model will now have more realistic results, although the field of temporal graph is still in its infancy and no well-defined concepts of clustering or epidemic threshold are yet available for such graphs. It is however still possible to simulate these dynamics and examine the computational results thus obtained. This could prove particularly useful for space companies looking to simplify their calculations and understand the underlying space debris problem.

For a more detailed account of this project, as well as some Python code, the following report might be of interest: https://www.researchgate.net/publication/361450914_Temporal_Random_Geometric_Graphs_on_Spherical_Shells_A_Model_for_Satellite_Collisions.