Come too close and a moon becomes a ring

A moon doesn't come apart because the planet pulls too hard. It comes apart because the planet pulls its near side harder than its far side, and past one particular distance that difference is stronger than the moon's own grip on itself. Its size makes no difference at all. Only what it's made of.

What you're seeing

Every part of the moon is falling around the planet on its own. The near side is closer, so it's pulled a little harder; the far side is further, so it's pulled a little less. Relative to the moon's centre — which is what "falling freely" means — the near side drifts planetward and the far side drifts away. The moon is being stretched, not dragged. That difference is the tide, and it's the same effect that raises two bulges of ocean on the Earth, one of them on the side facing away from the Moon.

Holding against it is the moon's own gravity, pulling every loose rock back toward its middle. Drag the orbit inward and watch the two bars: the stretch grows like 1/d³ while the grip doesn't change at all. Somewhere they cross. Inside that crossing the moon can't hold itself, and the rubble spreads — quickly, because debris on a slightly smaller orbit travels slightly faster and pulls ahead. That shearing is exactly how a ring is made and why rings are so thin and so flat.

Now the part worth pausing on. Set the moon's material and the planet's, and the crossing point is fixed — and nowhere in it is the moon's size. A pebble and a thousand-kilometre moon made of the same stuff fail at exactly the same distance. Make the moon bigger and it gains more self-gravity to hold with, but it also spans more distance for the tide to work across, and the two grow at precisely the same rate. What's left is a contest between two densities.

Which is why Saturn has rings and moons and almost nothing in between. Inside the limit, ice can't gather into a moon and any moon that wanders in doesn't survive; outside it, the same ice happily assembles. Press the preset: the ring system sits inside, and Mimas — the innermost round moon — sits outside.

The rule, exactly. Take a grain resting on the near face of a rigid, tidally locked moon of radius r and density ρ_m, orbiting a planet of radius R_p and density ρ_p at distance d. In the frame turning with the orbit, the grain is pulled off with 3GMr/d³ — the planet's pull plus the centrifugal term, both measured relative to the moon's centre and held on by the moon's own surface gravity Gm/r². Setting them equal, the rs and the Gs cancel: d_Roche = R_p · (3ρ_p/ρ_m)^1/3 and the stretch-to-grip ratio the bars show is simply (d_Roche/d)³. If the moon is denser than three times the planet, that distance is inside the planet — such a moon can never be torn apart at all. Verified in node (improve/verify/83-roche-limit.js, 37 checks): the proof never assumes the formula above. It locates the limit by bisection on the exact, un-expanded balance — the real 1/x² pull at the surface, no small-moon approximation — and then checks the closed form against it: the error falls monotonically as the moon shrinks (2.3% at r = R_p/10, 0.02% at R_p/1000), and the exact limit always sits slightly outside the formula, never inside, which is the honest direction for the approximation to err. It confirms the size cancellation, the cube-root density law to 12 decimal places, the sign flip either side of the root, and the Keplerian shear rate this page animates. The gate corrected me while I wrote it: I had expected a 2%-wide clump to spread twice as far per orbit as it actually does — the closed form 6πε set me straight. Five negative controls: in a uniform field of the same strength the moon never tears at any distance out to a million planet radii (so it is the gradient, not the strength); with no self-gravity it tears everywhere (so both sides are load-bearing); and the intuitive test — "the planet out-pulls the moon" — is wrong by more than tenfold and would have condemned Mimas, which has been sitting there for four billion years, because it forgets that a moon in orbit is already in free fall. This is the rigid-body result. A fluid body deforms as it is stretched and gives way further out; the coefficient there is about 2.46 rather than 3^1/3 = 1.44, quoted from the literature rather than derived here, and shown as a second dashed circle. Small bodies also hold together with material strength rather than gravity — a boulder inside any Roche limit is perfectly fine. Saturn and Earth figures use published radii and densities; what's checked is the arithmetic that follows from them.

← the cabinet · Thought Toys — a cabinet of explorable explanations. Exhibit 83.