Roche Limit Calculator (How Close a Moon Can Orbit Before Tides Tear It Apart)
Distance inside which the tides of a planet or star would pull a moon, comet or spacecraft apart, for a fluid body and for a rigid one, from the two densities.
Results
The dashed line marks the value you entered.
What this tool does
Gravity pulls harder on the near side of a moon than on the far side, and inside a certain distance that difference in the pull exceeds the moon's own gravity holding it together. The limit depends on the ratio of the two densities, not on their masses: a fluid, loosely bound body breaks up at about 2.44 times the larger body's radius times the cube root of the density ratio, while a rigid one holds on until roughly half that distance. Saturn's rings lie inside its limit, and a body held together by material strength, such as a small rocky moon, can survive well inside the fluid value. The factor 2.44 is an approximation to the exact fluid result.
Formula
d (fluid) ≈ 2.44 R (ρ_M ÷ ρ_m)^(1/3) · d (rigid) = R (2 ρ_M ÷ ρ_m)^(1/3) ≈ 1.26 R (ρ_M ÷ ρ_m)^(1/3)
Variables
| Symbol | Meaning | Unit |
|---|---|---|
rm | Radius of the larger body | km |
dp | Mean density of the larger body | kg/m³ |
ds | Mean density of the smaller body | kg/m³ |
a | Distance between their centres | km |
DF | Roche limit for a fluid body | km |
DR | Roche limit for a rigid body | km |
RR | Fluid limit in radii of the larger body | R |
OR | Orbit distance ÷ fluid limit | — |
VD | Is the smaller body safe? | — |
Worked example
- Radius of the larger body6371 km
- Mean density of the larger body5514 kg/m³
- Mean density of the smaller body3344 kg/m³
- Distance between their centres384,400 km
- Roche limit for a fluid body18,365 km
- Roche limit for a rigid body9483 km
- Fluid limit in radii of the larger body2.88 R
- Orbit distance ÷ fluid limit20.93
- Is the smaller body safe?Yes — outside both limits
Limitations
- The result is an estimate based only on the values you type. Real situations often include factors this calculator does not know about.
- The default values are typical reference figures, not measurements of your situation. Replace them with your own data whenever you have it.