Stochastic models of lunar rocks and regolith. Part I. Catastrophic splitting theory

It is assumed that a rock on the lunar surface loses mass as a result of bombardment by hypervelocity meteoroids. The mass of rock and its fragments can be modeled as a nonincreasing stochastic process with independent increments. In the case of a self-similar, one-shot splitting law, Filippov's extension of Kolmogorov's results produces asymptotic mass densities (number densities) which can be of lognormal, fractional exponential (Rosin-Rammler), or inverse-power (Pareto) types. The results are extended in three directions. A new explicit formula for the number density is obtained in the case where the splitting law is a two-term polynomial. The effect of splitting laws and splitting rates which depend on randomly varying parameters, e.g., meteoroid mass and velocity, is considered. The average number density with respect to a distribution of initial rock masses and initial rock birthdays also is studied. The asymptotic average density for an inverse-power distribution of initial masses has the same shape as the unaveraged density, but a beta (Β, 1) distribution of rock birthdays strongly alters the shape of the asymptotic number density. Research supported by Office of Naval Research, under contract NONR 4010(09) awarded to Department of Statistics, The Johns Hopkins University.