Given that moons commonly orbit planets, why do we never encounter a comet orbiting a planet?
What would happen (exactly, in detail) if a large one did settle into a stable orbit around the Earth?
Sorry... just making the question a bit clearer! The accepted answer will describe where all the water ends up.
Its very unlikely for a comet to become a satellite of an inner solar system planet. Much less likely than it is for an asteroid. Most asteroids are on fairly circular orbits, and so the relative velocity between asteroids and planets is quite low. In comparison comets have very elliptical orbits, and their relative velocities to the planets are much larger.
For an asteroid to be captured it must lose momentum. This is possible, though rare. For example, a binary asteroid can be captured if it is separated by tidal forces. For an comet with much more momentum, the chance of being captured is much much lower. Asteroids are captured by the Earth moon system, but not into stable orbits, they don't stay long.
If it did occur, the comet would still be active, with a coma of gas, which would be visible just like a very nearby comet. It wouldn't be particularly bright, since the surface brightness of a comet doesn't depend on distance from the Earth.
Over time the comet would run out of volatiles and become more or less indistinguishable from a captured asteroid. If it were in the Earth's orbit it probably wouldn't last that long, as there are not many orbits that are stable in the long term around the Earth, due to perturbations from the moon.
The dust and gas, including water vapour, will initally remain in orbit, forming a faint ring. It will, over time, be disrupted, and either end up in the atmosphere, on the moon, or ejected from the system. A comet doesn't contain enough water to make a difference to the Earth's ecosystem.
Given that moons commonly orbit planets, why do we never encounter a comet orbiting a planet?
By definition, it would no longer be a comet, but rather a moon (or more properly a satellite). Comets are icy bodies that orbit the Sun, satellites are any body which orbits a larger body than itself, other than the Sun. There are many examples of asteroids and comets that have been captured and become satellites, although it is not easy to prove with 100% certainty that a satellite is indeed captured and did not form in place. Some familiar (potential) examples might be Phobos and Deimos, around Mars, or the various moons around Pluto such as Nix, Hydra, Kerberos, and Stix.
What would happen (exactly, in detail) if a large one did settle into a stable orbit around the Earth?
I have to take exception to this question by pointing out that it is impossible for a comet to "settle into a stable orbit" around the Earth. Even basic physics will tell you conservation of energy and momentum disallow capturing comets by them "settling". In order for one body to capture another, you need a third body in the system to conserve angular momentum. What are the chances that two comets come by Earth at the same time and with exactly the correct orbital properties such that they can interact and cause one of them to be captured? I'm not going to do the math but I can guarantee the chances are diminishingly small. The moon could play the role of the third body, but again, you need precise orbital characteristics which seems unlikely to me.
But we disregard that point and just say that somehow a comet has been put into orbit around the Earth, be it a nearly miraculous capture or else put there by human intervention or by some other effect. What would happen to it?
The answer to that depends on where its orbit is. It seems unlikely to me that a comet could survive for long in some orbits. If it was very close to the Earth, it would eventually fall onto the planet through friction with our atmosphere. If it was farther out, you'd have to consider how the Earth-Moon system affected its dynamics. Possibly there could be some stable resonant orbit between the Earth, Moon, and comet. It is hard to say what the long term results would be as an analytical solution to the three body problem is impossible. The only way to answer this is through a simulation.
Yes. See Split comet Shoemaker-Levy 9 became a satellite of Jupiter Ríse hvezd, Vol. 74, p. 224-225
and The Century of Space Science
comet Shoemaker-Levy 9 which was probably captured into a Jupiter-centric orbit and thus, temporarily, became a satellite of Jupiter before splitting into at least 21 piece and crashing into Jupiter's atmosphere
And according to Disintegration of fragments Q1 and Q2 of comet Shoemaker-Levy 9 during the fall to Jupiter Astronomy Letters, Volume 22, November 1996, pp.771-779:
The comet was captured by Jupiter and became its satellite in 1929 ± 9 according to Chodas and Yeomans (1995) and in 1959 according to the calculations Chemetenko and Medvedev (1994).
It is absolutely possible for a planet to 'capture' a comet, Mars has already captured two asteroids and the only difference between the asteroids and comets are what they are made of. Plus, there are theories that believe that Saturn's rings were formed by a captured comet that strayed too close to the gas giant and thus, was ripped apart into millions of pieces of rock and ice (Assuming theory is accurate). Note that this is much more likely to happen to the outer planets due to their immense gravitational influence and their distance from the sun. Distance matters because comets generally go faster when they get nearer to the sun which makes it harder for a planet to capture comets.
Although, technically it wouldn't be a comet anymore but a satellite.
