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With what sort of gravitation are Saturn's Rings held together? It is very spectacular and intriguing compared to the way Earth's gravitation acts.

I do not have a basic or conceptual idea of what/how motion happens in the Rings. What makes water/ice matter to be confined to the equatorial planes and yet be bound within so many individual Rings? Does inverse square law operate on the icy plate orbiting matter in a force field ? Is there a mathematical model of relative motion of individual glacier sort of ice-block stone masses orbiting inside of the flat planetary Rings?

EDIT1:

Has an ordinary differential equation been written for dynamic motion of particles in the Rings?

A purely conceptual model is written to include strong precession. Here each particle has radial velocity along with circumferential and is bound between extreme radii. Wish to see such Ring orbits theoretically derived like Kepler/Newton planetary orbits of Earth satellites considering conservation of momentum, net forces acting as a function of radius and the Inverse squared law of gravitational attraction that operates universally.

EDIT2:

Is a particle forever confined to a Ring ? Are there no Kepler type laws or rules for Saturn? Is there an elementary method to verify calculate/ correlate observed and calculated time period of some Ring particles and some moons in most simple cases? ( For time being we leave out the moon darting across some Rings and such advanced situations.) I hope to gain clarity about satellite particle fundamentals.Please point out references about same.

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    $\begingroup$ Welcome to Space Exploration @Narasimham. While your question would be better suited to Astronomy, I'll answer briefly here. There is only one kind of gravity that we know of, so that's what holds the rings. A flat ring is the lowest energy state for many particles or bodies orbiting at similar speeds. Yes, the inverse square law holds. I'm not sure what you mean by the last qn. For much more detail, see the Wikipedia article. $\endgroup$ – user8406 Apr 21 '15 at 12:51
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    $\begingroup$ Albeit this questions is interdisciplinary, unless it's strictly off-topic, it shouldn't be voted off site. We included planetary-science in our scope long ago, and we also have astronomy, celestial-mechanics, orbital-mechanics, gravity, even rings tags. And if we want to stay strictly on-topic, as our community seems to have agreed, then I can see how explaining dynamics of a planetary ring, their stability and perturbing effects could influence planetary mission designs, too. In fact, it already did. $\endgroup$ – TildalWave Apr 21 '15 at 14:07
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    $\begingroup$ The question seems misinformed. The material of Saturn's rings is not cohesively held together. The rings consist mostly of innumerable small ice crystals in independent circular orbits. There is no "icy plate". en.wikipedia.org/wiki/Rings_of_Saturn $\endgroup$ – Russell Borogove Apr 21 '15 at 15:41
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    $\begingroup$ Your edits substantially change the question, invalidating the existing answer. I am voting to close since this question in its present form indicates a) lack of prior research, b) a hidden agenda, c) lack of motivation to stick to the original wording to make answering easier. All in all, the close vote's reason is going to be "too broad". $\endgroup$ – Deer Hunter Apr 22 '15 at 22:37
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    $\begingroup$ @DeerHunter I opted for "unclear" because, well, it is. And because the text adds a link to How to Ask. Because I can't tell any more if this is still a question, or a statement. I'm also tempted to revert to previous revision, but I'll let OP realize that might be a good thing to do considering the question already received one answer to its previous state and we aren't a discussion forum but a Q&A. $\endgroup$ – TildalWave Apr 23 '15 at 2:23
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The rings became flat over time as the trillions of particles in them collided over and over, slowly causing their vectors (direction of motion) to average out until they were all aligned in the same direction. See this short video from Minute Physics. A flat disk rotating in one direction is the only arrangement of these particles that is stable.

The divisions among the rings exist for two reasons. One, there are moonlets in the rings, too small to see from our distance (or for even the Cassini probe to catch). In fact many of these moonlets may be very short-lived, clumping together and then breaking apart due to collisions, tidal forces from Saturn, and gravitational interactions among all the many moons. These moonlets clear the particles in their orbit. Two, the gravitational fields of the moons collectively influence the rings in such a way that resonances clear the particles at certain distances from Saturn.

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  • $\begingroup$ Since a lot of mass data is available, is there a mathematical model or ordinary differential equation that represents motion of rotating ice particles,the shepherding moons satisfying gravity of ice and gravity interactions among Saturn moons ? Is there any model based video simulation? $\endgroup$ – Narasimham Apr 22 '15 at 12:36
  • $\begingroup$ @Narasimham Yes, here's one of the latest ones, where actual observation precisely matches previous simulations in a bit more dynamic setting (accretion of E ring) than I presume your question comes from (judging by your image, it appears you're only considering rings A, B and C) ciclops.org/view_event/205 You can find more on the subject if you follow up on image descriptions here saturn.jpl.nasa.gov/photos/?category=5. See videos here saturn.jpl.nasa.gov/video (e.g. saturn.jpl.nasa.gov/video/videodetails/?videoID=241) $\endgroup$ – TildalWave Apr 23 '15 at 2:18
  • $\begingroup$ Shepherd moons like Prometheus and Pandora also play a role in rings stability Prometheus Teases the F-Ring of Saturn. $\endgroup$ – mins Apr 23 '15 at 19:40

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