The physics of planetary rings

Fractal theory of Saturn’s ring

“…Friedman and Gorkavyj [8] explored the Boltzmann equation for the Ring. They took as a pattern the method of expanding of solution in moments of velocity distribution of particles relative to the stationary Maxwell distribution.…”

“…The real failure is the following: attempts of the strict justification of its existence in the work [8] are incorrect. The writing of variational equations as such presuppose the differentiability of the equation (8) relative to the initial data. However, it was proved in the theorem 2 that the equation (8) can not have stationary, differentiable solutions.…”

“…The Boltzmann equation in connection with the Saturn Ring was considered in works of Friedman and Gorkavyj [8]. Their predecessors sought reasons of stability of the Ring because it does not destruct almost 4 century.…”

“…This method was used by Chapman end Enskog for investigation of plasma equations. In the work [8] It was written out the kinetic Boltzmann equation for three-dimensional model of the Ring, consisted of rigid nonelastic particles executed the differential rotation in the gravitational field of Saturn. For simplification of explorations, it was considered a flat (two-dimensional) model which was an analog to the model B and the transport equations for it was written.…”

“…Conditions for instability for each of selected types of oscillations were found in the work [8] . The necessity of stratification of the Ring on the variety of thin ringlets was explained by these instabilities.…”