Probability of collision during the early evolution of debris clouds

It has been a common practice in the literature to use the kinetic theory of gases, in order to obtain an estimate of the probability of collision posed to an orbital asset by the background debris population. This has been advantageous, because it features the use of the Poisson distribution to model encounters between an asset and background objects. This model yields a very simple method for computing probability of collision. It is fairly accurate in earth orbital studies over periods of several years in certain orbital regimes. It also has been the practice to use the Poisson model for debris clouds while they still are in their early evolutionary phase. However, newly formed debris clouds resulting from orbital fragmentations are characterized by fragments with relative motion that is highly correlated by the central gravitational field, thereby eliminating any resemblance to a gas. While the use of the Poisson model in this context has been criticized, it generally has been used anyway due to the lack of a well-known and accepted alternative model. A precise probabilistic assessment generally involves Monte Carlo analysis. This method is effective but often is computationally burdensome. By making some simple assumptions that hold in the vast majority of scenarios, it is shown that collision hazard for short-term debris cloud evolution can in fact be described by a Poisson model. These assumptions concern the way in which a fragmentation process is modeled and the orbital geometry between assets and a debris cloud. The derivation of this result is quite different from that used in kinetic gas theory but is nonetheless a direct application of standard probability theory. The ramification for short term debris cloud modeling is a theoretical substantiation of formulations in software like program DEBRIS. The purpose of this paper is to present the derivation and substantiation of this result.