Hong Life CalculatorsUnderstanding the Roche limit calculator — concept and principle
The Roche limit is the closest distance at which a moon held together only by its own gravity can orbit a planet without being torn apart by tides. This calculator gives two classic estimates, one for a rigid spherical satellite and one for a fluid satellite that can deform, from the planet's radius and the densities of both bodies.
How it is calculated
- Rigid body: d = 1.26 × R × (ρ_M ÷ ρ_m)^(1/3)
- Fluid body: d = 2.44 × R × (ρ_M ÷ ρ_m)^(1/3)
R is the primary's radius (km), ρ_M the primary's density and ρ_m the satellite's density (kg/m³). Distances are measured from the planet's centre.
The principle
A planet pulls harder on the near side of a moon than on the far side; this difference is the tidal force, and it grows as 1/d³. The moon's own gravity holds it together. Setting the two equal for a small rigid sphere gives d = R(2ρ_M/ρ_m)^(1/3), and 2^(1/3) ≈ 1.26. A fluid moon is stretched into an elongated shape by the tide, which makes it easier to pull apart; Édouard Roche's 1848 analysis of such a body gives the larger coefficient of about 2.44. Expressing masses through density and radius makes the planet's mass and the moon's size cancel, so only the density ratio matters.
Worked example
- Defaults (Earth–Moon): R = 6,371 km, ρ_M = 5,514 kg/m³, ρ_m = 3,344 kg/m³.
- Density ratio: 5,514 ÷ 3,344 = 1.648923; cube root = 1.181409.
- Rigid: 1.26 × 6,371 × 1.181409 = 9,483.711075 km.
- Fluid: 2.44 × 6,371 × 1.181409 = 18,365.28176 km.
- The Moon actually orbits at about 384,400 km, roughly 20 times farther than even the fluid limit, so it is in no danger.
| Primary radius (km) | Rigid body (km) |
|---|---|
| 1,593 | 2,370.927769 |
| 3,186 | 4,741.855537 |
| 4,778 | 7,112.783306 |
| 6,371 | 9,483.711075 |
| 7,964 | 11,854.63884 |
| 9,556 | 14,225.56661 |
| 12,742 | 18,967.42215 |
How to read the result
Real bodies fall between the two values: rocky moons with internal strength can survive inside the fluid limit, while loose rubble piles behave closer to fluids. Saturn's main rings lie largely inside its Roche limit, which is consistent with material that cannot gather into a single moon. Comets that pass inside a planet's limit can break up, as Shoemaker-Levy 9 did near Jupiter before its 1994 impacts.
Common mistakes
- Measuring distance from the planet's surface rather than its centre.
- Using the satellite's radius in place of the primary's radius.
- Applying the limit to small objects held together by material strength, such as spacecraft or boulders.
Frequently asked questions
Can an artificial satellite orbit inside the Roche limit?
Yes. It is held together by its structure, not gravity, so the limit does not apply; the ISS orbits well inside it.
Why is the fluid coefficient sometimes given as 2.455?
More detailed calculations of a deformed fluid body give values of about 2.44 to 2.46 depending on the assumptions.
Does Phobos face this problem?
Phobos is spiralling slowly towards Mars and is expected eventually to break up or crash, which is often discussed in terms of tidal limits.
Sources: Murray C.D. and Dermott S.F., Solar System Dynamics, Cambridge University Press, 1999 · Chandrasekhar S., Ellipsoidal Figures of Equilibrium, Yale University Press, 1969