Submoons — satellites of satellites — are theoretically possible. However, they'd suffer the same eventual fate of the things we launch into orbit of our planet: eventual decay of orbit and crash into the planet below. That's a fascinating location to set a story on.
Image credit: Lunar Farside, As Seen by Apollo 8 by NASA
In a Q&A in The New York Times, C. Clairborne Ray answers the question of whether or not the moon could have a moon. While there aren't any submoons in our solar system, it's possible that another system could have just the right conditions for one to exist. Ray's answer draws on Sabrina Stierwalt and Patrick Taylor's response on "Ask an Astronomer," which explains:
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.
... 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!
Ray points out that most moons in our solar system are likely the result of the dust and gas that circulated around the planets long ago consolidating into satellites, which could happen around a moon elsewhere in the galaxy. Less likely, Ray says, a moon could capture a passing object.
Can someone work out the distances and masses that would allow a submoon to last longer than a few years? Then we could worldbuild a society of planet dwellers and submoon dwellers. What would the history look like? Would the submoon inhabitants think the planet was the sun? Would thinking the planet orbited another large body be heresy? What about the doomday cult surrounding the day the submoon falls? Would there be attempts to leave the submoon for the planet? And would the planet reject the refugees? Let's write all the stories about the moon of a moon.
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