Can moons have moons?
We observe planets, asteroids, comets etc that are in orbit around the sun. There are satellites (moons) orbiting around the planets, and asteroid pairs and more. However, I do not recall seeing a reference to bodies (satellites-squared?) orbiting moons. Question is, do moons have satellites? But you don't get away with it that easy :-), so if not, why not? Looking forward to your response!
Due to the popularity of this question and the large number of follow-up questions I have received, I have decided to update my answer with the help of someone much more knoledgeable on the subject than I am. Thanks to Patrick "I Love Asteroids" Taylor for all of his shared wisdom.
Yes, in theory, moons can have moons. The region of space around a satellite there a sub-satellite can exist is called the Hill sphere. Outside the Hill sphere, a sub-satellite would be lost from its orbit about the satellite.
An easy example is the Sun-Earth-Moon system. Earth is a satellite of the Sun and the Moon is a sub-satellite orbiting Earth. The Moon orbits the Earth because the Moon is 370,000 km from Earth, well within Earth's Hill sphere, which has a radius of 1.5 million km (0.01 AU or 235 Earth radii). Loosely speaking, the Hill sphere defines the space where the Earth's gravity is more important than the Sun's gravity on another object. If the Moon somehow ended up outside Earth's Hill sphere, the Moon would orbit the Sun instead of the Earth just like all the other planets, asteroids, and comets. For comparison, Jupiter's Hill sphere has a radius of 0.35 AU which is much much larger than the Earth's Hill sphere. This is because Jupiter is more massive than the Earth and has a stronger gravitational pull, but more importantly because it is further from the Sun than Earth so the Sun's gravity is weaker at Jupiter than at Earth. This gives Jupiter a lot of gravitational influence on the space around it. Having such a big Hill radius could explain why Jupiter has a lot of moons and can affect the orbits of passing comets so strongly.
Can the Moon have a moon?
Yes, the Moon could have a sub-satellite. If we look at a system of the Earth, Moon, and a sub-satellite, the same idea as above applies. The Moon has its own Hill sphere with a radius of 60,000 km (1/6th of the distance between the Earth and Moon) where a sub-satellite could exist. If an object lies outside the Moon's Hill sphere, it will orbit Earth instead of the Moon. The only problem is that the sub-satellite cannot stay in orbit around the Moon indefinitely because of tides.
The Moon, like almost all other moons in the solar system, is in synchronous rotation about the Earth meaning it shows the same face to Earth at all times (its rotation period about its own axis is the same as its orbital period about the Earth), which is a result of tidal forces between the Earth and Moon. These are the same tidal forces that cause the high and low tides on Earth. In this configuration, any object within the Hill sphere of the Moon will have its orbit decay due to tides! That means the orbit of any sub-satellite of the Moon will shrink over time. In other words, the distance between the sub-satellite and the Moon will get smaller and smaller until the sub-satellite crashes into the Moon or the lunar tides rip the sub-satellite apart!
How does the Moon exist if it is a sub-satellite itself?
The reason this argument does not apply to the Sun-Earth-Moon system is that the Earth itself does not synchronously rotate (nor do any of the planets) about the Sun like the Moon and other satellites do around the planets. This allows the Moon to have a stable orbit around the Earth.
What about man-made lunar orbiters? How do they survive?
Lunar orbiters only orbit the Moon for a few years, a very short time by astronomical standards. Man-made satellites can stay in orbit around the Moon or any moon for the duration of a mission because tidal effects require thousands or millions or more years, depending on the system, to cause the loss of a sub-satellite. Because of this we can leave a man-made satellite in stable orbit around a moon for a few years using the spacecraft's rocket thrusters to correct for any changes in its orbit.
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