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We think, more or less, that our Moon was formed from an accretion disk caused by a Mars-sized impactor. Cool.

Tidal forces can break apart a moon, causing rings.

Ashes to ashes, dust to dust: the moon came from an orbiting debris disk and may return to an orbiting debris disk someday.

What determines whether an accretion disk or ring system will form moons vs. remain as stable rings?

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I believe the two biggest things that determine whether or not a ring system forms into a moon depends on are the parent planet's Roche Limit and whether or not there are other moons that can disrupt the ring.

The roche limit of a planet is how close a large body can orbit the planet without being torn apart by the planet's gravitational pull. The article I've referenced can show how this is calculated. Small moons like Jupiter's Metis can orbit within the roche limit as they are so small, the difference between the tidal forces acting on their near and far sides is not large enough to tug the moon apart.

The presence of other moons - or in the case of stellar disks, other planets - will play a large role. Like this article explains about our asteroid belt, the gravitational perturbations of Jupiter prevented a planet forming between it and Mars. Smaller bodies cannot easily gather together when a beast like Jupiter swings by to scatter them all!

This also means though that most asteroids within the belt are confined between Mars and Jupiter. Around Saturn's F ring, there are two small moons that slow down / speed up debris within the ring so that it remains gathered in a small space. This does keep rings and disks stable, but it also stops larger bodies forming.

Long story short; if the disk if far enough away from its parent, and nothing else large is orbiting nearby, you should get a moon / planet. If it's too close to the parent and being shepherded by another large body, it's more likely to remain a stable disk.

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