The use of renormalization group techniques on fragmentation problems is examined. The equations which represent fractals and the size-frequency distributions of fragments are presented. Method for calculating the size distributions of asteriods and meteorites are described; the frequency-mass distribution for these interplanetary objects are due to fragmentation. The application of two renormalization group models to fragmentation is analyzed. It is observed that the models yield a fractal behavior for fragmentation; however, different values for the fractal dimension are produced . It is concluded that fragmentation is a scale invariant process and that the fractal dimension is a measure of the fragility of the fragmented material.

Fractals and fragmentation

Hydrothermal breccias in vein-type ore deposits: A review of mechanisms, morphology and size distribution

The present study is focused on the distributions in particle size produced in dynamic fragmentation processes. Previous work on this subject is reviewed. We then examine the one‐dimensional fragmentation problem as a random Poisson process and provide comparisons with expanding ring fragmentation data. Next we explore the two‐dimensional (area) and, less extensively, the three‐dimensional (volume) fragmentation problem. Mott’s theory of random area fragmentation is developed, and we propose an alternative application of Poisson statistics which leads to an exponential distribution in fragment size. Both theoretical distributions are compared with analytic and computer studies of random area geometric fragmentation problems, including those suggested by Mott, the Voronoi construction, a variation of the Johnson–Mehl construction, and several methods of our own. We find that size distributions from random geometric fragmentation are construction dependent, and that a conclusive choice between the two distributions cannot be made. A tentative application of the maximum entropy principle to fragmentation is discussed. The statistical theory is extended to include a concept of statistical heterogeneity in the fragmentation process. Finally, comparisons are made with various, dynamic fragmentation data.

Geometric statistics and dynamic fragmentation

10 – dynamic rock fragmentation

Fractal geometry applications in description and analysis of patch patterns and patch dynamics

We have tested the hypothesis that the universe underwent a single fragmentation event, separating into protogalactic volumes at a relatively early stage after the Big Bang. Assuming that the present total luminosity of each galaxy is proportional to its mass, we plotted the number and mass distribution of a thousand nearby galaxies just as we would in an analysis of fragments from a laboratory high-explosive experiment. The results are consistent with the single-fragmentation hypothesis.

Fragmentation of the universe

We have further developed Brown's model of solar system formation. In this model, each fragment of an ejected supernova shell evolves into a separate solar system. Specifically, we have formulated the reverse-flow hypothesis that may be responsible for the inner, earthlike planets. We have written a computer program with which it is possible to calculate mass distributions within a solar nebula. We have found mass distributions similar to our solar system over a wide range of the model parameters.

The supernova fragmentation model of solar system formation

A one-parameter theory of sequential fragmentation is formulated, validated, and compared with the Initial Mass Function of stars. The agreement is excellent.

Comparison of a Theory of Sequential Fragmentation with the Initial Mass Function of Stars

The problem of evolution of viscous protogalactic disks is examined. A modification of traditional turbulent viscosity theory is proposed which is based on the premise that the eddies are so small and numerous that each eddy can be considered to be large molecule. The radial flow term is omitted from the viscosity equation to make it tractable, and the radial flow is later reintroduced via the requirement of gravitational balance within the disk. The main result is a family of rotation curves that, as the system evolves, serially reproduces most types of observed rotation curves. With the use of Seiden's star formation theory, the present model produces an exponential-like luminosity profile whenever stars form and the viscous action ceases.

Approximate rotation curve solutions for the evolution of a viscous protogalactic disk

Galactic mass distributions produced by the fragmentation model (in addition to those published previously) are examined. This is done by allowing the cubical fragment to assume any orientation in space. All distributions examined closely resemble observed galactic luminosity profiles, and, among the variety produced, was found the special logI versusr1/4 behavior of de Vaucouleurs' empirical law as well as the exponential behavior of spirals. Because of the apparent success of the model, the initial conditions and analytical methods are re-examined in detail.

Wilbur K. Brown

Papers

Fragmentation of the universe

The supernova fragmentation model of solar system formation

Comparison of a Theory of Sequential Fragmentation with the Initial Mass Function of Stars

Approximate rotation curve solutions for the evolution of a viscous protogalactic disk

A model of the formation of spherical galaxies