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Around a rocky planet or small gas giant, could there be a system of rings that follow an elliptical orbit? I have used a gravity simulator (this one) to test it, but the way it handles its particles doesn't seem like it would work. My question is, can a planet have an elliptical ring system?

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It's unlikely, on the whole, for a ring system to maintain a high eccentricity on significant timescales. Dissipative collisions tend to circularize the orbits of individual particles, even if the original constituent body traveled on a fairly eccentric orbit. Therefore, you need some external perturbation keeping the particles on substantially elliptical orbits.

This typically involves another orbiting body - say, a moon - creating, for instance, a Lindblad resonance with a subset of the particles in the ring that have the proper orbital period. These particles will be forced into slightly more eccentric orbits. While this has not occurred stably on a large scale in the present-day Solar System, it might occur in some circumstellar disks around other stars. The star Fomalhaut has a ring of dust of eccentricity $e\simeq0.11\pm0.01$ orbiting at a distance of $a\simeq133\text{ AU}$. The star is a couple hundred million years old, but the circularization timescale is only $\tau\sim10^6$ years. It has been suggested that the culprit is a planet orbiting at $a_p\simeq119\text{ AU}$, with an eccentricity similar to that of the disk (Quillen 2006). The theory holds up, but observations have not confirmed such an object.

In short, yes, it's quite possible for there to be a ring system that maintains a fairly eccentric shape for long periods of time, but there needs to be a perturbing body (a high-mass moon, for instance) in the proper orbit to force the ring particles to maintain their eccentricities. Otherwise, collisions will very quickly circularize the ring system.

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