Are Saturn's Rings Caused by Fractal Structures?

[Jimster41]

It's very recent, but it discusses the question of cause of and stability of Saturn's and other ring systems. I came to this site wondering about them. I got the impression from answers received here they were well understood? This paper seems to suggest there are a number of big questions, and has some interesting (if a bit exotic) ideas about what's going on with them. Anyone heard of the guy?

I'm interested in other references to the topic.

The fractal theory of the Saturn Ring
Mikhail Zelikin
(Submitted on 9 Jun 2015)
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.
Comments: 19 pages, 1 figure
Subjects: Dynamical Systems (math.DS); Mathematical Physics (math-ph)
Cite as: arXiv:1506.02908 [math.DS]
(or arXiv:1506.02908v1 [math.DS] for this version)

http://arxiv.org/abs/1506.02908

 

[Dr. Courtney]

https://www.researchgate.net/profile/Ralf_Srama/publication/42345171_An_evolving_view_of_Saturn%27s_dynamic_rings/links/00b4951d46424cb9d3000000.pdf

 

[tech99]

It's very recent, but it discusses the question of cause of and stability of Saturn's and other ring systems. I came to this site wondering about them. I got the impression from answers received here they were well understood? This paper seems to suggest there are a number of big questions, and has some interesting (if a bit exotic) ideas about what's going on with them. Anyone heard of the guy?

I'm interested in other references to the topic.

The fractal theory of the Saturn Ring
Mikhail Zelikin
(Submitted on 9 Jun 2015)
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.
Comments: 19 pages, 1 figure
Subjects: Dynamical Systems (math.DS); Mathematical Physics (math-ph)
Cite as: arXiv:1506.02908 [math.DS]
(or arXiv:1506.02908v1 [math.DS] for this version)

http://arxiv.org/abs/1506.02908

James Clerk Maxwell worked on the structure and stability of the rings in 1857 and won the Adam's Prize at Cambridge for his work. His public paper was published in 1859 "On the Stability of the Motion of Saturn's Rings". He also made a mechanical model showing the particles having four modes of vibration. Sorry, that's all I know.

 

[Jimster41]

James Clerk Maxwell worked on the structure and stability of the rings in 1857 and won the Adam's Prize at Cambridge for his work. His public paper was published in 1859 "On the Stability of the Motion of Saturn's Rings". He also made a mechanical model showing the particles having four modes of vibration. Sorry, that's all I know.

Yeah, the paper refs LaPlace, Boltzmann and that very Maxwell paper, he then works through a Boltzmann model, applies the stationary (non-time dependent) Maxwell solution, then "proves" the solutions cannot explain observation.

P.12 "Hence the steady state surface density in two-dimensional model B cannot be smooth relative to the space variables"

I couldn't follow his Diff EQ but I got some of it. I'd be interested in hearing a real explanation and critique of it.

The paper DrCourtney linked is from 2010, but does reference the "as yet undetected satellite resonance influences" something like that. Which seems to be where he's coming from.

The gist of his theory has to do with the polyhedron of satellites and their effect, and solutions to the Boltzmann model related to the "Fuller Problem" and "Pontryagin type Hamiltonian Problems" page 15 is totally Greek to me. Wish I understood it. The "Concluding remarks" section is wild. He quotes Maxwell at the end. It seems like a wild paper, but I want to understand how far our understanding is advanced, what is and isn't well understood.