The proliferation of orbiting fragments: A simple mathematical model

Paolo Farinella, Alessandro Cordelli, "The proliferation of orbiting fragments: A simple mathematical model," Science & Global Security, 2, no. 4, (1991): 365-378.
We present and discuss a simple mathematical model (two coupled, non-linear, first-order differential equations) for the future proliferation in low earth orbit of space debris that is created by high-velocity destructive collisions of small objects with artificial satellites or other "big" orbiting bodies. The model predicts that such collisional generation of fragments will be the dominant source of debris in a few decades. Subsequently, if the satellite launch rate remains comparable with the current one, a quasi-exponential growth of potential projectiles will cause the number of satellites to reach a peak (of the order of 104) in about 150 years and then to rapidly decline by about a factor of 10. Some alternative choices of model parameters (for example, to account for uncertainty in the projectile-to-target mass ratio required for breakup and the rate of future injection of material into orbit and intentional generation of debris) show that this evolution may be anticipated or delayed, but not qualitatively modified, unless ad hoc measures to avoid or limit collisions are adopted.

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