When tides tear a moon apart
A moon is squeezed and stretched by the gravity of the planet it orbits, because its near side is pulled harder than its far side. Far away, the moon’s own gravity easily resists these tidal forces. Closer in, there is a distance at which the tidal stretching wins and a moon held together by gravity alone breaks up. This is the Roche limit, named after the French astronomer Édouard Roche.
The formulas
With R the radius of the primary body, ρM its density and ρm the density of the satellite:
- Rigid satellite: d ≈ 1.26 × R × (ρM ÷ ρm)^(1/3)
- Fluid satellite: d ≈ 2.44 × R × (ρM ÷ ρm)^(1/3)
The rigid case treats the moon as a solid sphere that keeps its shape. The fluid case lets it deform into an elongated shape under the tides, which makes it more fragile, so its limit is about twice as far out. Real bodies, made of rubble, ice and rock with some strength, fall between the two.
Example: the Moon and the Earth
The Earth has a radius of 6,371 km and an average density of 5.51 g/cm³; the Moon has a density of 3.34 g/cm³. The density ratio is 1.65, whose cube root is 1.18:
- Rigid limit: 1.26 × 6,371 × 1.18 ≈ 9,485 km from Earth’s center
- Fluid limit: 2.44 × 6,371 × 1.18 ≈ 18,368 km, or 2.88 Earth radii
- The Moon’s average distance, 384,400 km, is about 21 times the fluid limit, so it is in no danger.
Rings and broken comets
Most of Saturn’s spectacular rings lie inside its Roche limit, where ice and rock cannot gather into a moon. In 1992, comet Shoemaker-Levy 9 passed within Jupiter’s Roche limit and was torn into a string of more than twenty fragments that struck the planet two years later. Mars’s moon Phobos is slowly spiraling inward and is expected to break up into a ring within tens of millions of years.
Limits of the model
The formulas assume a satellite with no tensile strength, orbiting in a circle. A small rocky asteroid or a spacecraft is held together by chemical bonds, not gravity, so it can orbit well inside the limit. The calculator is meant for moons, comets and large bodies, for which self-gravity dominates.