Initial velocity distribution and consequent spatial distribution of fragments

Recent work by the authors established a means by which an orbital fragmentation can be simulated as a spatial number density of objects given a distribution of initial fragmentation velocity that is added to the orbital motion of the source object. In this study, we develop a measure for the degree to which the causative initial velocity distribution can be inferred from the resultant evolved spatial density of fragments. Those properties of the initial velocity distribution that have low inferability will be unknown if only the spatial density is known; those with higher inferability might be inferred given complete knowledge of the later spatial density. Such inferability is a function of the elapsed time. A necessary property for inferability is distinguishability, that is, the degree to which two different initial velocity distributions result in distinguishably different spatial distributions. Using a multivariate Gaussian fragmentation velocity distribution, we propose the root mean square metric for comparing distributions and finding the approximate level of distinguishability for the standard deviation of velocity and for the modal velocity, or degree of anisotropy. This level is higher for the standard deviation of velocity than for the modal velocity.