The previous article showed that tides come from the difference in the pull, and that difference grows as the inverse cube of the distance.
What holds a moon together, meanwhile, is its own gravity, which has nothing to do with how far it is from anything else.
Come close enough and the two exchange places. Past the point where the stretching exceeds the holding, a moon cannot remain one body.
That boundary is the Roche limit, calculated by Roche in the middle of the nineteenth century.
Setting the two against each other fixes a distance: , with the radius of the central body and the densities.
No mass appears. What matters is not weight but how tightly packed each body is.
At equal densities the limit is times the radius of the central body. The lighter and fluffier the moon, the further out the limit reaches.
The formula is for a body held together by nothing but its own gravity. That qualification matters in the last scene.
The largest example is the rings of Saturn.
The rings lie inside the Roche limit and the moons lie outside it. The division is not a coincidence.
Inside the limit, anything that begins to gather is pulled apart again, so nothing large can form. The rings are the fine ice that was never allowed to assemble.
Jupiter, Uranus and Neptune have rings too, all in the same place. A ring is a shape that exists only inside the limit.
In the comet Shoemaker-Levy 9 passed inside the Roche limit of Jupiter.
Being a loosely bound lump of ice held by its own gravity alone, it simply came apart. Twenty-one pieces, strung out along the same orbit like beads.
Two years later, in July , those pieces fell into Jupiter one after another. Dark marks the size of the Earth appeared in its clouds, one at a time.
It was the first collision between bodies ever observed in the solar system, and a rare case of a quantity from inside a calculation becoming visible through a telescope.
The Roche limit concerns bodies held together by gravity alone. Rock has strength.
A small enough body can resist the tide on strength alone. Gravity grows with size while strength does not, so the smaller the body, the better strength fares.
Phobos, a moon of Mars, orbits inside the Roche limit of Mars and has not broken. At about km across, the strength of rock is holding it. Grooves run across its surface that are thought to be tidal.
Phobos is nonetheless drawing in by about m a century. In thirty to fifty million years, strength will no longer be enough. Mars may acquire a ring.