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Roche Limit Calculator

Find how close a moon can orbit before breaking up.

How it is calculated

The engine uses the rigid-body formula d = R × (2 × ρM / ρm)1/3. R is the radius of the central body in kilometres; ρM is its mean density and ρm the satellite's mean density, both in kg/m³ (only their ratio matters, so any common unit gives the same answer). It returns two values: the distance d in whole kilometres, and the same distance as a multiple of R.

Radius and both densities must be greater than zero, otherwise there is no result. This is the rigid limit for a satellite held together only by gravity; a fluid moon would break up farther out, and a body with real material strength can survive closer.

Worked example

Take Earth (R = 6371 km, density 5514 kg/m³) and a satellite with the Moon's density, 3344 kg/m³. The density ratio is 2 × 5514 / 3344 ≈ 3.298. Its cube root is about 1.488. Multiplying by the radius gives 6371 × 1.488 ≈ 9,483 km from Earth's centre, which is 1.49 R. Measured from the centre, anything closer than that would be torn apart if held by gravity alone.

Try this example →

Common mistakes

  • Entering the primary's radius in metres or miles; the field expects kilometres, so the km result would be wrong.
  • Swapping the two densities. The primary's density goes in the numerator, and swapping gives a much smaller limit.
  • Treating the result as altitude above the surface. It is measured from the centre of the primary, so subtract R to get surface height.
  • Applying the result to a fluid or loosely bound body: the rigid formula is the lower bound, and such bodies break up farther out.
  • Using the surface or core density rather than the mean density of each body.

About this calculator

The Roche limit is the minimum distance a smaller celestial body (a moon or satellite) can orbit a larger one before the larger body's tidal forces exceed the smaller body's self-gravity and pull it apart. This calculator uses the rigid Roche limit formula, d = R · (2 · ρM/ρm)^(1/3), where R is the primary body's radius, ρM is its density and ρm is the satellite's density — it assumes the satellite holds together only by gravity, not by any internal tensile strength.

The concept, named after French astronomer Édouard Roche, explains why planets like Saturn have ring systems rather than a single large moon close in — any moon that formed or wandered inside the Roche limit would be shredded into a debris ring by tidal stress. Astronomers and astrophysics students use the Roche limit to predict whether a close-orbiting moon or asteroid is stable, to understand the origin of planetary rings, and to model tidal disruption events around stars and black holes.

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