Saturn rings: fractal structure and random field model

TL;DR

This paper investigates the fractal radial structure of Saturn's rings using non-integer dimensional calculus and models the rings' kinematics as a random field of angular velocities, providing a mathematical framework for their statistical properties.

Contribution

It introduces a novel mathematical model of Saturn's rings' fractal structure and kinematics using non-integer dimensional calculus and group representation theory.

Findings

Fractal structure of rings supported by non-integer dimensional calculus.

Kinematic model of rings as a random field of angular velocities.

Complete determination of the velocity field by positive-definite matrix functions.

Abstract

This study is motivated by the observation, based on photographs from the Cassini mission, that Saturn's rings have a fractal structure in radial direction. Accordingly, two questions are considered: (1) What Newtonian mechanics argument in support of that fractal structure is possible? (2) What kinematics model of such fractal rings can be formulated? Both challenges are based on taking Saturn's rings' spatial structure as being statistically stationarity in time and statistically isotropic in space, but statistically non-stationary in space. An answer to the first challenge is given through the calculus in non-integer dimensional spaces and basic mechanics arguments (Tarasov (2006) \textit{Celest. Mech. Dyn. Astron.} \textbf{94}). The second issue is approached in Section~3 by taking the random field of angular velocity vector of a rotating particle of the ring as a random section of a special vector bundle. Using the theory of group representations, we prove that such a field is completely determined by a sequence of continuous positive-definite matrix-valued functions defined on the Cartesian square $F^{2}$ of the radial cross-section $F$ of the rings, where $F$ is a fat fractal.