What is the Roche Limit?
The Roche limit is the closest distance a celestial body can orbit a larger one without being torn apart by tidal forces. Named after French astronomer Edouard Roche, this boundary explains why planets like Saturn have rings: material that crosses the Roche limit is ripped apart and cannot coalesce into a single moon.
How to Calculate the Roche Limit
Here is the formula for the fluid Roche limit:
[d = 2.44 \times \left(\frac{3M}{4\pi\rho}\right)^{1/3}]
Where:
- d is the Roche limit in meters.
- M is the mass of the primary body in kilograms.
- ρ is the density of the satellite in kg/m³.
The calculator accepts satellite density in g/cm³ and converts internally (1 g/cm³ = 1,000 kg/m³).
Calculation Example
Calculate the Roche limit for a satellite with a density of 3.3 g/cm³ orbiting Earth (mass 5.97 × 10²⁴ kg).
Convert density: 3.3 g/cm³ = 3,300 kg/m³
[d = 2.44 \times \left(\frac{3 \times 5.97 \times 10^{24}}{4\pi \times 3{,}300}\right)^{1/3} \approx 18{,}442{,}000 \text{ m}]
The Roche limit is approximately 18,442 km. Any satellite of this density orbiting closer than this distance would be torn apart by tidal forces.