Roche Limit Calculator

The Roche limit is the distance from a primary body inside which the tidal forces it exerts on an orbiting satellite become stronger than the satellite’s own self-gravity. Inside this distance, a fluid satellite is torn apart. Outside it, moons can survive.

Formula (for rigid satellite):

d = 1.26 × R_M × (ρ_M / ρ_m)^(1/3)

For fluid satellite:

d = 2.455 × R_M × (ρ_M / ρ_m)^(1/3)

The gap between those two coefficients is the whole point, and it is nearly a factor of two. A rigid body is held together by its own material strength, so it survives much closer in. A fluid body has nothing but self-gravity, so the tide deforms it into an ellipsoid, which makes the near side stick out further, which makes the tide stronger, and the whole thing runs away. Real moons sit between the two: a rubble pile behaves closer to fluid, a solid iron fragment closer to rigid. Use the rigid answer as the optimistic bound and the fluid answer as the pessimistic one.

Where:

• R_M = radius of the primary body
• ρ_M = density of the primary body
• ρ_m = density of the satellite

Saturn’s rings: Put Saturn’s numbers into the calculator (radius 58,232 km, density 687 kg/m³) with ice at 900 kg/m³ and you get a fluid limit of about 130,700 km. That is why the rings exist: inside it, particles cannot pull themselves together into a moon no matter how long you wait.

Compare that to what is actually up there. The A ring, the outermost of the bright rings, runs from 122,170 km out to 136,775 km, so the calculated limit lands squarely inside it. Just past the A ring the shepherd moons Prometheus and Pandora hold the narrow F ring in place, and beyond them the real moons begin. Two tiny moonlets do orbit inside the A ring, Pan in the Encke gap and Daphnis in the Keeler gap, and both are visibly deformed by the tide, squashed into the ravioli shapes Cassini photographed in 2017. For a formula with one coefficient and two densities, landing within a few percent of the boundary between rings and moons is about as well as it has any right to do.

Shoemaker-Levy 9: In July 1992, Comet Shoemaker-Levy 9 passed about 91,000 km from Jupiter’s center and was torn into 21 fragments. A loose comet nucleus at roughly 600 kg/m³ has a fluid Roche limit near 224,000 km around Jupiter, so the comet went more than twice as deep as it needed to. The fragments came back and hit the planet in July 1994 over six days, leaving dark scars bigger than Earth.

Earth-Moon Roche limit: For a rocky moon (ρ_m = 3,000 kg/m³) orbiting Earth, the rigid limit is about 9,830 km and the fluid limit about 19,160 km. Both sit above Earth’s own 6,371 km radius, so a rocky body really would break up if it drifted that close. For an icy body (ρ_m = 900 kg/m³) the limits are roughly 14,690 km rigid and 28,620 km fluid. Our own Moon has a density of 3,342 kg/m³, near enough to the rocky row, and its fluid limit works out at 18,480 km. It orbits at 384,400 km, twenty times further out than that, so nothing about it is in any danger. Do not read the icy figures as applying to the Moon; they are there for the kind of body that ends up as a ring.

Densities to type in, if you do not have them handy

Use the mean radius with the mean density, and do not mix in an equatorial radius. The two always travel together: the formula can be rewritten to depend on the primary’s mass alone, and R and ρ only appear as the combination that reproduces it. Pair a mean density with an equatorial radius and you have silently invented a heavier planet.

For satellite density, ice is about 900 kg/m³, rock 3,000, a loose comet nucleus 500 to 600, and a rubble-pile asteroid anywhere from 1,200 to 2,500 depending on how much empty space it contains.

The Roche limit is also relevant for accretion disks around black holes and neutron stars.