Debris Cloud Evolution: Mathematical Modeling and Application to Satellite Constellation Design

Orbital break-ups produce a large number of fragments, which constitute an obvious hazard for other satellites in nearby orbits. Of these fragments, many are too small to be detected by ground-based facilities: this leads to the need for mathematical modelling as a tool for adequate risk analysis. In this paper an average spatial density model is presented. It is based on the Gauss analogy and, for unperturbed Keplerian orbits, it matches the asymptotic density model developed by other authors. Risk analysis for satellite constellations is an interesting application of debris cloud evolution models: the survivability of a constellation as a whole following the break-up of one of its satellites is obviously of primary concern in the constellation design. Risk analysis is conducted over a number of traditional configurations in order to achieve an additional constraint on the design parameters. Results indicate the remarkable influence of the fragmentation point position along the orbit; moreover, the higher risk for low orbit and the advantage of placing more satellites on a limited number of planes are assessed.