International Journal of Modern Physics B Vol. 31, No. 30 (2017) 1750236 (14 pages) c  World Scientific Publishing Company DOI: 10.1142/S0217979217502368 Fractal planetary rings: Energy inequalities and random field model Anatoliy Malyarenko *,‡ and Martin Ostoja-Starzewski †,§ * Division of Applied Mathematics, M¨ alardalen University, H¨ogskoleplan 1, V¨ aster˚ as, SE 721 23, Sweden † Department of Mechanical Science and Engineering, Institute for Condensed Matter Theory and Beckman Institute, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA ‡ anatoliy.malyarenko@mdh.se § martinos@illinois.edu Received 6 April 2017 Revised 26 May 2017 Accepted 29 May 2017 Published 10 July 2017 This study is motivated by a recent 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 such a fractal structure of planetary rings is possible? (2) What kinematics model of such fractal rings can be formulated? Both challenges are based on taking planetary rings’ spatial structure as being statistically stationary in time and statistically isotropic in space, but statistically nonstationary in space. An answer to the first challenge is given through an energy analysis of circular rings having a self-generated, noninteger-dimensional mass distribution [V. E. Tarasov, Int. J. Mod Phys. B 19, 4103 (2005)]. The second issue is approached 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. Keywords : Planetary rings; fractal; dynamics. PACS numbers: 96.30.Wr, 05.45.Df, 02.50.Cw, 45.50.Jf, 45.50.Pk 1. Introduction A recent study of the photographs of Saturn’s rings taken during the Cassini mission has demonstrated their fractal structure. 1 This leads us to ask these ‡ Corresponding author. 1750236-1