Roche Limit Calculator

The Roche limit is the orbital distance inside which tidal stretching by a primary body overcomes the self-gravity holding a satellite together. Tidal acceleration across a satellite of radius r at distance d scales as 2GMr divided by d cubed, while the self-gravity at its own surface scales as its density times its radius. Setting the two equal and expressing the primary mass through its density and radius leaves a limit that depends on nothing but the primary radius and the ratio of the two densities. For a rigid satellite the result is d equal to the primary radius times the cube root of twice the density ratio. For a fluid satellite, which deforms into an elongated equilibrium figure and is therefore easier to pull apart, the coefficient rises to 2.4554, placing the boundary roughly twice as far out. The numbers follow directly. Earth, with a radius of 6,371km and a mean density of 5,514 kg per cubic metre, has a fluid Roche limit near 18,500km for a satellite of lunar density, so the Moon at 384,400km is in no danger. The main rings of Saturn, spanning roughly 74,000 to 137,000km from the planet centre, lie inside its limit for icy bodies, which is why that material has never accreted into a moon. The formulas idealize uniform density, spherical shape, synchronous rotation, and zero material strength. Bodies held by cohesion or tensile strength survive well inside the fluid limit, which is how the small shepherd moons of Saturn and the inner Martian moon Phobos, orbiting at 9,376km, persist at all.

Edouard Albert Roche, professor of mathematics at Montpellier, published the calculation in the late 1840s in the memoirs of the Academy of Sciences and Letters of Montpellier, in the course of investigating what shape a fluid satellite would take within the tidal field of a primary. His motivating question was the origin of the rings of Saturn, which were then still widely thought to be solid or liquid sheets. Roche showed that a fluid moon orbiting closer than about 2.44 primary radii could not hold itself together, which supplied a natural explanation for a broad band of debris in a region where no moon exists. The ring question was settled on independent grounds when James Clerk Maxwell won the Adams Prize in 1857 with an essay, published in 1859, proving that neither a solid nor a fluid ring could be mechanically stable and that the rings must therefore consist of countless separate particles, a conclusion confirmed spectroscopically by James Keeler in 1895. The name of Roche later attached to the related idea of the Roche lobe in close binary stars, where overflow through the inner Lagrangian point drives mass transfer and powers novae and cataclysmic variables. The most direct modern demonstration came in July 1992, when Comet Shoemaker-Levy 9 passed inside the Roche limit of Jupiter and was torn into more than twenty fragments, which struck the planet two years later in July 1994.