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Imagine if the Moon had its own moon. A moon (likely an asteroid) that lies within the Moon's Hill Sphere. I have read elsewhere about the fate of such a moon. Over time due to the Moon being tidally locked to the Earth, tidal forces would make as such that this small moon of the Moon would see its orbit shrink. This tiny object would potentially cross the Roche Limit and be ripped apart forming a ring or would simply come crashing down on the Moon.

In a scenario I am imagining, his tiny moon could have circled around our satellite for around (or more than) 100 million years before being discovered. Is it possible for this moon to survive more than 100 million years in orbit around our natural satellite before being disintegrated, or a a duration between around 4 billion years to 100 million years, assuming that this object was naturally captured by the Moon sometime in the past?

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    $\begingroup$ Not quite related to the question, but you say it has orbited the Moon a hundred million years until discovery? If it were asteroid-sized, we likely would have discovered it in ancient times, or up until the use of the telescope for astronomy. We were able to find the small moons of Mars over a century ago. $\endgroup$ – B.fox Jul 18 '18 at 12:14
  • $\begingroup$ @B.fox ten thousand years of uncertainty on the dates is four nines over 100 million years. $\endgroup$ – user25818 Jul 18 '18 at 18:57
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This is complicated because the main effect producing instability depends on where the satellite's orbit is and how big the satellite is.

Close to the Moon there are two main perturbations: lumps on the Lunar gravity field and tidal friction. Lumps in the Moon's own gravitational field are caused by non-uniform distribution of mass (Mascons, mostly) within the Moon. It can have a surprisingly large effect -- close-in artificial satellites need to be tended and their orbits adjusted if they're to remain stable for thousands of orbits. These effects are independent of the mass of the satellite. I'm having difficulty finding papers I can link to, but here's an abstract of one.

Tidal friction is slower, but very, very certain and will cause larger masses to spiral in.

Higher altitude circular Lunar orbits in general are not stable. For example:

"High-altitude circular orbits around the Moon are unstable," says Todd A. Ely, senior engineer for guidance, navigation, and control at NASA's Jet Propulsion Laboratory. "Put a satellite into a circular lunar orbit above an altitude of about 750 miles (1200 km) and it'll either crash into the lunar surface or it'll be flung away from the Moon altogether in a hyperbolic orbit." Depending on the specific orbit, this can happen fast: within tens of days.

And here's a link to a great presentation which notes the effects: Main perturbations:

  • Lunar mass concentrations
  • Earth/Sun third body effects:
    • Main contribution over 700 km
    • Significant between 400-700 km
    • Small between 100-400 km
    • Almost negligible below 100 km
  • Solar radiation pressure

OTOH, some papers have suggested that some elliptical orbits which are far from the plane of the Moon's orbit may be moderately stable.

But it appears that nothing can give you stability in the hundreds of years or longer range.

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This is a good question. Chances are that the Moon being tidally locked would cause the satellite to eventually deorbit, this would happen faster if the orbit was retrograde but even a prograde orbit would still have it very slowly slow down to deorbiting.

Roche Limit

If the satellite was fairly substantial, made of rock, not a collection of dust and ice as some asteroids are then providing it was not too big then it shouldn't break apart. The larger the satellite, the higher the stress that can be exerted upon it by the gravity of two bodies. So a fairly small, maybe 20-40km asteroid should be fine. I'm just a space fan, not an astrophysicist, so if I'm wrong on that then anyone please correct me

However

A way to limit its deceleration, and eventually de-orbit, would be to have the satellite in a polar orbit, it would still slow down, but at a much much slower rate, and the tidal locking wouldn't have as much of an effect.

Victorbrine Cassini, I wouldn't accept this as an answer unless you don't get any other answers, Hopefully, someone more qualified would be able to answer your question with some maths.

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Your satellite needs to be small, persistent, and evade detection from Earth. High satellites are unstable and might be noticed from earth.
Here is a scheme of mine from the Halfbakery: the Moon VLO (for Very Low Orbit) device. The halfbakery is sadly incommunicado these past weeks, so link is via the Wayback Machine.

http://web.archive.org/web/20171015065017/http://www.halfbakery.com/idea/Moon_20VLO_20Device#1331956156

Stuff in orbit is always falling down toward the planet, but always missing on account of its sideways motion. The closer the orbiter is to the planet, the harder it is to miss it as it is falling. But it is still possible, provided there is enough sideways motion. I propose that a VLO {very low orbit) device be constructed and set into orbit around the moon. I choose the moon for its lack of atmosphere, which otherwise would slow the VLO and make it crash. The VLO would be set into orbit about 2 feet (or 1 meter) off the ground. A path would need to be carefully cleared and marked, as a rock in the way would have tragic consequences. The device would need to be very shiny, to reflect rays which might push it down toward the surface.

A railgun might be used to start the VLO. The railgun would need to be quickly moved out of the way as the VLO completed its first orbit. The VLO would be good for all the usual sort of things very low lunar satellites are good for, except it would be better, because it is lower.

— bungston, Sep 11 2003

from the comments

The thing would be going at about 6000 km/h and complete a circle every hour and 50 minutes. It would take a big rail gun...

— AO, Sep 11 2003

So too your satellite. No-one has noticed because it is so small. Nonuniform gravity is not important because its path goes thru regions such that there is a uniform pull along the entire course.

If your characters visit the moon and find a funny little path cut through a hill, have them stand off to the side a ways while they do their wondering. They will figure it out within an hour and 50 minutes.

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