Roche Limit: Tearing a Rubble Pile Apart at Saturn

This 2D page

It runs the same Hill-frame N-body code, the same J2 orbit integrator and all six checks as the 3D page; only the drawing differs. The left view is the tidal frame seen from above the ring plane (the pieces are the rubble-chunk model from the 3D page, rendered in Blender); the right view is the Blender ring model seen from above, with the 3D page's optical-depth profile applied as opacity. The planet's shadow, the Sun-elevation control and the ring-opacity shader are only on the 3D page.

What is modelled

Why does Saturn have rings and not a moon? A body held together only by its own gravity cannot survive too close to a planet, because the tide pulls its near and far sides apart. The page puts a self-gravitating pile of 201 spheres on a circular orbit around Saturn and integrates it in the tidal (Hill) frame. You choose the orbit radius and the bulk density, or press Spiral in and watch what happens when the pile crosses its limit.

Equations

• Orbit: mean motion of a circular equatorial orbit with the oblateness term, n² = (GM/a³)(1 + (3/2) J2 (R/a)²), GM = 3.7931 × 10¹⁶ m³/s², R = 60 268 km, J₂ = 0.016291.
• Hill (tidal) equations for each piece, in a frame that turns with the orbit (x outward, y along the orbit, z out of the ring plane): x″ = 2n y′ + 3n² x + gx, y″ = −2n x′ + gy, z″ = −n² z + gz, where g is the softened gravity of all other pieces plus a soft-sphere contact force with restitution e. Only one number controls the result: η = G Mb / (rb³ n²) = (4π/3) G ρ / n², the ratio of self gravity to the tide.
• Fluid Roche limit: d = 2.456 R (ρS/ρ)1/3, which is exactly η = 14.84 in these variables; a tidally locked fluid satellite is stretched into a teardrop below it. The continuum estimate for a particle resting on the surface is η = 3 r³/(mass inside).
• Integration: a second-order time-reversible scheme (kick, half Coriolis rotation solved exactly, drift, half rotation, kick). The pile starts as a spherical piece of a close-packed lattice with a little noise and a very small random velocity.
• Rings: the opacity shown follows 1 − exp(−τ/sin B) for normal optical depth τ and elevation B of the line of sight above the ring plane; on the side of the rings away from the Sun only the light that was transmitted, exp(−τ/sin S), is seen, which is why the dense B ring looks dark and the Cassini Division bright from there. Saturn's shadow darkens the rings behind the planet.

Checks run in the page

• A single particle in the tidal frame against the exact epicycle x = 4 − 3 cos t, y = 6 sin t − 6t.
• Hill energy conservation of a cloud with collisions switched off.
• The separation threshold of two touching spheres against its closed form.
• Mimas and Enceladus periods by RK4 integration in the J2 field against their published values.
• The Roche parameter of Mimas from published GM, radius and period against the page's density form.
• The rubble-pile bracket: torn apart at η = 3.5, intact at η = 6.5.

Honest simplifications

• The tidal (Hill) frame is valid while the pieces stay within a few per cent of the orbit radius from each other, so the run is meaningful for the first few orbits only; the pieces are drawn along a circular arc and the ring they form is not followed to a closed ring.
• 201 equal spheres in a lattice with soft contacts are a rubble pile of unusual strength: its limit (η about 4.6) is much lower than the fluid value, and real icy bodies lie between the two, depending on their strength and friction. The soft-sphere stiffness, softening length and e are numerical choices; the tide acts on a body at rest in the frame (synchronous rotation); there is no spin-up, no Saturn oblateness inside the Hill frame, no radiation and no ring particle collisions.
• The ring optical depths are representative rounded values (B ring 1.5, A ring 0.55, Cassini Division 0.12, C ring 0.11), the single-scattering light model is a sketch, and the cloud band texture of the globe is procedural, not a photograph. Ring thickness is not drawn.
• The body radius slider only sets the drawing scale; it does not change the result, which depends only on η.

• Pick Below the rubble-pile limit: the bound fraction falls and the pieces stream apart along the orbit (vertical in the left view).
• Tick Spiral in slowly at 900 kg/m³ and read the orbit radius where the pile breaks: near 1.5 Saturn radii, well inside the fluid limit near 2.2.
• Choose the dense rocky body: it survives much closer to Saturn than the snowball does.
• Raise the restitution to 0.9 and compare how long the pieces stay together.