Roche limit

Remarks

The pull of the primary on the particle exceeds its pull on the centre of the satellite by $GM/(d - r)^2 - GM/d^2 \approx 2GMr/d^3$. The particle stays when the self-gravity $Gm/r^2$ is larger, that is, when $d^3 > 2 (M/m) r^3$. The second form follows from $M = \frac{4}{3}\pi R^3 \rho_M$ and $m = \frac{4}{3}\pi r^3 \rho_m$. The limit depends only on the ratio of densities, not on the size of the satellite.

A fluid satellite deforms into an elongated shape under the tide, which lowers its self-gravity at the tips. Roche's analysis of this case gives the larger value $d \approx 2.44 \, R (\rho_M/\rho_m)^{1/3}$. Real moons have material strength, so small bodies can survive inside these limits; the limits apply to bodies held together by gravity alone. The main rings of Saturn lie inside its Roche limit for icy bodies.

Going deeper

For a student

The computation is a comparison of the tidal acceleration, the first-order term of the gravitational field across the satellite, with the satellite's surface gravity. A similar balance between the tide of the primary and the gravity of the satellite gives the radius of the Hill sphere of a satellite on a Kepler orbit. The same tidal effect explains why comets that pass close to a planet can break into pieces. Refined versions include the rotation of the satellite and its fluid deformation, and they change only the numerical factor in front of $R (\rho_M/\rho_m)^{1/3}$.