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Planet and moon
Real planets and moons; density and size set the Roche limit in km.
Orbit and pile
Distance is given relative to the fluid Roche limit of the chosen pair.
Experiments
Layers
Model checks
The science
Roche limit
A fluid satellite held together only by its own gravity is torn apart inside d = 2.44 R_p (ρ_p / ρ_m)^(1/3); a rigid body survives down to about 1.26 R_p (ρ_p / ρ_m)^(1/3). The planet's tide across a moon of radius R is a_tide = G M_p [1/(d-R)^2 - 1/d^2] ≈ 2 G M_p R / d^3, and the moon holds itself with g = G M_m / R^2. Both are plotted against distance.
The simulation
The moon is a pile of soft discs integrated in the frame that co-rotates with the orbit (Hill's equations): x'' = 3Ω^2 x + 2Ω y' + a_self, y'' = -2Ω x' + a_self, with Ω^2 = G M_p / d^3 = (ρ_p/ρ_m) (R_p/d)^3 in units where G M_m / R^3 = 1. Discs attract each other with softened Newtonian gravity, repel on contact with a damped spring, and can be glued by breakable bonds (cohesion). Only the ratio d / d_Roche and the density ratio matter, which is why the planet choice changes the distance in km but not the physics of the pile.
Honest simplifications
This is a two-dimensional disc model in the orbital plane. A flat pile of discs has less self-gravity per unit mass than a sphere, so its measured disruption distance differs from the 3D formula; the sweep graph shows this difference instead of hiding it. Tidal acceleration is the leading-order Hill approximation, the planet is a point mass, and no heating or spin-up of the planet is modelled.
What to try
- Press "Inspiral" and watch the pile stretch into a bar, then a stream, as the distance crosses the Roche limit.
- Raise cohesion to 1 (iron-rich asteroid) and compare: bonds delay the break-up until well inside the fluid limit.
- Run the disruption sweep to see the bound fraction collapse near d / d_Roche of order one.
- Change the planet: the distance in km moves, but the dimensionless behaviour stays the same.