Planetary rings seem to generally form on the ecliptic plane because of the rotation of the planet, but I was wondering what kind of effect the magnetic field of the planet might have on the ring system. Would it help keep the ring system together, disrupt it, or have little to no effect? What if the magnetic poles were on or near the ecliptic?
(I will assume that these planets orbit a star.) Planetary rings are formed by the gravitational capture of small objects, mostly ice and dust. These particles can be magnetic and in such cases would be affected by the planetary magnetic field. It is also to be pointed out that the magnetic field is not really uniform, but let us assume that it is. The presence of a magnetic field would be important for the sustenance of a ring, but the stability of the ring is an entirely different issue. If the ring does consist of magnetically sensitive materials, then large enough fluctuations in the magnetic field can in principle cause the ring to become unstable. So depending on the answer of stability, I choose to say yes, it does depend on the magnetic field. To calculate the stability would be a complex n-body problem but can be averaged over.
There are radial structures in Saturn's rings called spokes, which are likely produced by a mechanism involving electromagnetic effects linked to Saturn's magnetic field. So there would certainly appear to be some aspects of ring phenomena that are affected by magnetic fields.
General Formation Theory
The reason rings form in the ecliptic plane of the planet's rotation is the same reason why planets are usually found in the equatorial plane of a star. During star/planet formation, collisions and pressure from particles and gasses with orbits in different directions cause debris to generally orbit in the same direction in order to conserve angular momentum. This helps material spin in the same direction. While the centrifugal force "pulls" particles outward from the orbital axis of revolution, gravity pulls everything radially inward. There's therefore a component of gravity that pulls things above the equatorial plane towards the equatorial plane, and vice-versa. Over time, with lots of collisions, particles and gas settle into the equatorial plane. In the case of a star, solar wind pushes gas away, but larger chunks of material remain. There's an astrobites article that summarizes this pretty well.
Magnetic Influences to Ring Formation
Can a planet's magnetic field affect ring formation? The short answer is that it depends on a number of factors:
- If the rings are charged (i.e. plasma), then lightweight ring particles will interact with the magnetic field. In general, the result of this interaction is to try to make the ring materials orbit with the same angular velocity as the rotating magnetic field. (In the case of Saturn, this causes plasma to speed up in the ring plane and the magnetic field to slow down.)
- If the rings have a neutral electric charge, then it can't directly interact at all with the magnetic field.
- If the magnetic field is such that it channels Van Allen-like radiation belts through the ring plane, then plasma flowing along the magnetic field lines could interact with the ring particles. Radiation and ions could collide with ring particles, knocking them from their orbit, causing chemical reactions, or causing charge exchange.
- Stellar radiation, micrometeorites and ionized plasma from other sources (e.g. a moon like Enceladus) can bombard ring particles with energy that causes ionization -- a photon splits a molecule or knocks an electron off an atom, causing it to be ionized. This then makes ring particles charged, allowing them to interact with the magnetic field.
- Ionized plasma from other sources (again, like Enceladus) will interact with the magnetic field. If the plasma is sufficiently dense, it can impart momentum or pressure to the rings like a puff of air blowing on wind chimes.
There's solid evidence that Saturn's magnetic field is interacting with its rings. In the case of Saturn, electrical currents from the ionosphere and thermosphere flow along magnetic field lines across the ring plane. These currents interact with so-called "ring plasma". It's uncertain right now what effect the ring plasma has on the rings, or what the effect of the rings on the magnetic field is. (This is in part because nobody knows how to estimate the electrical conductivity of the rings!)
My personal guess — as somebody doing active research in electrodynamics in Saturn's upper atmosphere — is that a spinning magnetic field will help (indirectly) transport angular momentum from the rotating planet to ionized rings, causing them to orbit slightly further out than they would ordinarily orbit.