There is an explanation for why rings flatten out here. The general mechanism is that particles collide, and gets a very uniform momentum. Thus, any set-up giving unusually thick rings is in essence "cheating".
Here are some ways:
Moons can cause spiral waves in the rings, giving them more of a structure in the z direction. The ones known in Saturn's rings has a modes amplitude of just 10-100 m, but larger Moons can easily increase that.
Another way is simply having massive rings. Then they can not get more flattened, as there are no more empty space to remove.
A tilted ring relative to the Planets orbit around the star is going to experience tidal forces, as long as the radius of the rings is some notable fraction of the planet's orbital radius. From the context that sparked the question, that is not a suitable mechanism though, along with the possibility of having a so low density that particle collisions are rare.
However, more promising:
The halo ring of Jupiter is estimated to be around 12500 km thick (about the same as the diameter of the Earth), and are very fine dust kept from condensing into a disc by both the magnetic fields of Jupiter, and by iterations with the Galilean Moons.
We have four planets with rings in the solar system, so the sample size is quite small. Applying some small-sample-size statistical methodology, in this case an unusual application of the German Tank Problem, we can give a rough but realistic maximum thickness of a ring:
$$N approx m+frac{m}{k}-1$$
Where $m$ is the highest observed value, and $k$ the sample size.
Modified slightly to get a non-integer version that makes some sense, we get:
$$max_{thickness} approx 12500km+frac{12500km}{4} approx 16000km$$
By no means a very certain limit, but at least about what can obtain from what we know.
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