The fractal theory of the Saturn Ring

TL;DR

This paper proposes a fractal-based explanation for the division of Saturn's rings into multiple sub-rings, linking it to solutions of Hamiltonian systems and analyzing their stability.

Contribution

It introduces a fractal theory for planetary rings, connecting ring structure to Hamiltonian system solutions and discussing stability issues.

Findings

Fractal structure explains ring partitioning.

Instability demonstrated for models with continuous density.

Questions raised about ring stability and future research directions.

Abstract

The true reason for partition of the Saturn ring as well as rings of other planets into great many of sub-rings is found. This reason is the theorem of Zelikin-Lokutsievskiy-Hildebrand about fractal structure of solutions to generic piece-wise smooth Hamiltonian systems. The instability of two-dimensional model of rings with continues surface density of particles distribution is proved both for Newtonian and for Boltzmann equations. We do not claim that we have solved the problem of stability of Saturn ring. We rather put questions and suggest some ideas and means for researches.