Attempts to deflect a near-Earth object (NEO) from an impact trajectory using high-energy methods, such as nuclear explosions, can plausibly fragment the NEO rather than deflect it. This paper addresses the orbital dispersion problem of a fragmented NEO using the asteroid 99942 Apophis as a model, and presents a physical description of the relative motion phenomena of a fragmented NEO. The overall stochastic model is approximated using a quasi-random process in which the randomness of fragmentation is associated only with the initial condition. Categories of initial probability distributions are analyzed to obtain relationships between initial conditions and eventual percentage of impacting bodies. A Direct Simulation Monte Carlo approach is used to further investigate the sample initial conditions for the Apophis example previously studied in the literature. Consequences of ignoring mutual gravitation of the fragmented system or relying on the linearized Clohessy-Wiltshire-Hill model for large time scales are discussed.
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