Saturn Rings Closeup (2D)

What the 2D view shows

A top-down view of the ring plane (same particle swarm and integrator as the 3D page) and a local shearing box, a small patch of ring that follows Hill's equations. The box shows what the 3D view cannot: individual ring particles colliding while Keplerian shear drags neighbours past each other.

Equations

• Kepler: Ω = √(GM/r3), shear |r dΩ/dr| = (3/2) Ω.
• Hill's equations (rotating frame): x'' = 2Ω y' + 3Ω2 x, y'' = -2Ω x'; the exact epicycle solution is used between collisions, so σy/σx = κ/2Ω = 1/2 for free particles.
• Collisions: hard spheres, vn' = -ε vn; optical depth τ = N π a2/(Lx Ly).
• Resonance: a = am ((m-1)/m)2/3; Roche limit 2.456 R (ρp/ρ)1/3.

Honest simplifications

Test particles around Saturn with moons on circular orbits (no collisions or self-gravity in the ring-plane view); moon masses can be multiplied so effects show in seconds; the box is planar (no vertical motion), has no self-gravity and a fixed restitution coefficient, particle radius 1 m. The optical-depth profile is illustrative. Dissipative collisions in a real ring also damp eccentricities, which is why resonances clear gaps.

• Load Mimas 2:1 resonance: particles near 116,900 km turn orange (eccentric).
• Switch to the shearing box and lower the restitution to 0.2: the velocity dispersion collapses.
• Raise tau to 1.0: the collision rate per orbit rises sharply.
• Put the view centre at the Mimas 2:1 radius and read the orbital period (half of Mimas' period): graph 3.