Analytical model for collision probability assessments with large satellite constellations

At a time when space debris are already a growing issue in the space sector, the deployment of large constellations, made of hundreds to thousands of satellites, is of concern from an environmental point of view. In the next decade, the space sector will undergo a considerable change as the population of active satellites is about to quintuple. This scenario will pose new challenges regarding space traffic management, generating the demand for more powerful and efficient tools. In this study, an analytical model for collision probability assessments between de-orbiting or injecting space objects and satellite constellations is presented. Considering the first to be subjected to a continuous tangential acceleration, its spiraling motion would result in a series of close approaches in the proximity of a constellation. The mathematical description of the crossing dynamics relies on the assumptions of circular orbits and independent collision probabilities, but does not require to propagate the satellites’ orbit. The statistical model presented in the current work constitutes an efficient tool for the evaluation of the mean collision probability related to this type of events. A comparison with a propagation method has been performed for validation purposes, showing that the model is capable of delivering accurate results, also in case of elliptical crossing orbits. The statistical model has been used to assess the risk connected to constellation's satellites replacement, once they have reached their programmed End-of-Life. The environmental impact of the full replacement of 12 approved constellations is analysed by means of average collision probability. In particular, it is shown that the key features for space exploitation sustainability are the maximum thruster acceleration, the size of the satellites and the true anomaly phases between constellations’ and crossing satellites. The consequences of an in-orbit collision are also investigated by assessing the collision risk generated by the formation of a debris cloud. The results corroborate the need for international standards for space traffic management as an exponentially increasing satellite population could trigger a chain reaction of collisions, making Low Earth Orbit inaccessible for decades.