Roche Limit Calculator

Calculate how close a moon or other satellite can orbit a much more massive primary before tidal forces overcome its self-gravity, d ≈ 2.44 R_M(ρ_M/ρ_m)^(1/3) for a fluid satellite (or the rigid-body variant), in km and primary radii, compared against the actual orbital distance — with a tidal-disruption diagram and a density-dependence chart.

Roche limit calculator

Inside the Roche limit, tidal forces from the primary body stretch a satellite harder than its own gravity can hold it together. Fluid model (a satellite with little internal strength, deforming as it's pulled apart): d ≈ 2.44 R_M (ρ_M/ρ_m)^(1/3). Rigid-body model (an idealized perfectly stiff satellite): d = R_M(2ρ_M/ρ_m)^(1/3) ≈ 1.26 R_M (ρ_M/ρ_m)^(1/3) — a smaller, more optimistic limit.

Primary radius (R_M)

Primary density (ρ_M)

kg/m³

Satellite density (ρ_m)

kg/m³

Actual orbital distance — optional

Roche limit (fluid) ≈ 106992.1 km = 1.837 R_M

Fluid: 106992.1 km · Rigid: 55246.6 km

Actual distance 238000 km is 2.224× the fluid Roche limit — outside it, a stable orbit is plausible

Marker placed at the actual orbital distance you entered. Dashed circle is the Roche limit boundary.

A straight line here means d ∝ ρ_m^(−1/3) exactly — icy satellites (low density) have Roche limits much farther out than iron-rich ones, for the same primary.

Every moon orbiting a planet is in a permanent tug-of-war between two forces holding it together and pulling it apart: its own gravity trying to keep it round, and the planet’s tidal pull trying to stretch it into an ellipsoid and, eventually, shreds. Get close enough, and the tide wins outright — that boundary is the Roche limit, and it’s the reason some planets have rings instead of an extra moon.

The formula

For a satellite with little internal strength — deforming into its own tidal shape as the primary’s gravity stretches it, the more physically realistic case for icy moons and rubble-pile bodies:

d_{\rm fluid} \approx 2.44\, R_M \left(\frac{\rho_M}{\rho_m}\right)^{1/3}

For an idealized perfectly rigid satellite that stays spherical right up until it’s torn apart — an optimistic lower bound, more relevant to a strong, monolithic body:

d_{\rm rigid} = R_M\left(\frac{2\rho_M}{\rho_m}\right)^{1/3} \approx 1.26\, R_M \left(\frac{\rho_M}{\rho_m}\right)^{1/3}

R_M and ρ_M are the primary’s radius and mean density; ρ_m is the satellite’s mean density. Neither version is a hard boundary in practice — real bodies with internal cohesion or unusual shapes can survive a bit inside either idealized limit, and both are approximations to a genuinely three-dimensional problem.

Why density matters so much

Both formulas depend on ρ_m through an inverse cube root, so a lower- density (icy, porous) satellite has a larger Roche limit than a dense, iron-rich one at the same distance from the same primary. This is the whole reason composition, not just size, decides whether a close-in body survives intact — Saturn’s icy rings sit at distances that would be perfectly safe for a solid rock or metal body of the same size.

Two real examples worth knowing

Saturn’s main rings sit at distances right around the fluid Roche limit for water ice — not a coincidence. Material there can’t gravitationally clump into a single moon because tidal forces keep tearing any growing clump back apart, which is a large part of why Saturn has spectacular rings instead of one more icy moon.
Comet Shoemaker-Levy 9 passed roughly 96,000 km from Jupiter’s center in 1992 — deep inside the Roche limit for a weak, porous comet nucleus (around 3.4 Jupiter radii out) — and was torn into a string of fragments that, two years later, crashed into Jupiter one after another in one of the most-watched astronomical events of the 20th century. It’s the textbook real-world demonstration of exactly the physics this calculator computes.

Reading the two visuals

The disruption diagram draws the primary, the Roche limit as a dashed boundary, and the satellite either as an intact disk (safely outside) or a stretched, debris-shedding shape (inside) — using your actual orbital distance if you give one, or a representative just-inside-the-limit position if you don’t.
The density-dependence chart plots the Roche limit against satellite density on a log-log scale, where the inverse-cube-root relation is a straight line — landmark densities from porous ice to iron-rich rock show directly how much composition alone can shift where the boundary sits.