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The behavior of object density after a fragmentation has been analyzed using ���Eulerian orbit dynamics���, our term for the propagation of densities, rather than points. We previously pioneered a technique that propagates the spatial density of orbital objects exactly. In the present work, we show the effects of the J2 gravitational perturbation on the density evolution from a low-earth orbit fragmentation. The fractional change in density due to the perturbation averages on the order of a tenth of a percent., Paper AAS 21-217

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.

A debris cloud from a fragmentation on orbit may be modeled by transformation of variables from the instantaneous velocity distribution at the fragmentation time to the spatial distribution at some elapsed time later. There are no Gaussian distributions assumed and the evolution map is quite nonlinear, being derived from the solution of the Lambert, two-point boundary value, problem and the state transition matrix for unperturbed propagation, so the traditional tools of analysis that assume these qualities fail dramatically. The transformation of variables technique does not suffer from any such assumptions, and unlike the Monte Carlo method, is not subject to sampling errors or approximations. Structures and features are evident in the density maps, and these structures show promise for simplified approximation of the density map. Most prominent of the structures is the well-known pinch point at the fragmentation location in inertial space. The anti-pinch line, or wedge, is also observed. Bands on the opposite side of the fragmentation are very noticeable, and their existence may be motivated from simple orbit dynamics. These bands make the anti-pinch line actually more of a set of anti-pinch line segments. By computing these density maps over time, the evolution may be studied. There is a density generator, a density band at roughly the same altitude as the pinch point, that cycles around the earth and appears a source of the bands, with newly created bands moving radially outward and diminishing in density. Although the initial velocity distribution affects the final spatial distribution, the Lambert solutions, which are the most time consuming to compute, need only be computed once. Therefore, different initial distributions may be changed and the results recomputed with relative speed. A comparison of the effects of initial distributions is shown in this paper., AMOS-2016-presentation.pdf has two movies embedded in it, evolve.mp4 and tophat.mp4, which are available as separate files. AMOS-2016-presentation-2up.pdf is the same presentation formatted two slides per page (US letter size) without the movies embedded.

The early phases of a debris cloud from an instantaneous fragmentation on orbit are dominated by the effect of the two-body gravitational force on the delta-v imparted on each fragment, which gives an in-track spreading of the cloud. This spreading is seen to form radial bands opposite to the fragmentation point. The bands may be motivated by straightforward orbit mechanics. Accurate density maps are computed with the transformation of variables technique, which opens a world of cloud dynamics to contrast with the point dynamics of traditional astrodynamics. Structure and evolution of the bands and finer details are noted and discussed., Paper number AAS 16-430. This work has been mostly superseded by later work and is provided for historical reference.