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How to use
Pick a preset, or choose the planet, moon density and structure, then set how close the moon's closest approach is in units of the fluid Roche limit (below 1 means inside it). The orbit view shows the dashed rigid and fluid limits, the moon in orange-grey, debris on bound orbits in amber and escaping material in blue; the close-up window follows the moon with the planet on its left. Try: circular orbit at 1.6 (safe), then slide to 0.5 (ring), switch to Fluid, switch to Rigid and lower the bond strain, raise eccentricity to 0.9 for a comet-like fly-by, or add orbital decay. Use the view buttons for the Roche potential and the force balance. Space pauses, R restarts.
The science
A moon orbiting a planet feels the planet's gravity slightly stronger on its near side than on its far side. That difference is the tide-raising force. For a moon of radius Rm at distance d from a planet of mass M, the stretching acceleration at the sub-planet point is, to first order in Rm/d,
atide = GM/(d − Rm)² − GM/d² ≈ 2GM Rm/d³ versus gself = G m/Rm²
Rigid-sphere limit. Setting the two equal and writing the masses with densities gives d = Rp (2 ρp/ρm)1/3 = 1.26 Rp (ρp/ρm)1/3. The moon's own size cancels: only the density ratio and the planet's radius matter. If the moon is tidally locked, the centrifugal term adds another GM Rm/d³ and the factor 2 becomes 3, a coefficient of 31/3 = 1.44 for a loose grain on the surface.
Fluid limit. A strengthless fluid moon is not a sphere: it stretches into a prolate ellipsoid, which has less self-gravity at its tips. Roche (1850) showed that a synchronously rotating incompressible fluid has an equilibrium figure only outside d = 2.44 Rp (ρp/ρm)1/3 (Chandrasekhar's more exact value is 2.455). Inside it the moon cannot stay in one piece, whatever its size. Both coefficients are drawn as dashed circles; the page uses 1.26 and 2.44.
The Roche potential. In the frame that corotates with the orbit the effective potential of two point masses is Φ = −GM/r1 − Gm/r2 − ½Ω²r² with Ω² = G(M+m)/d³. The closed equipotential through the inner Lagrange point L1 is the critical Roche lobe: material that is pushed above it flows through L1 toward the planet. The check list compares the numerical L1 with the Hill-sphere distance d (m/3M)1/3 as the mass ratio goes to zero. The Roche potential view is drawn with the true mass ratio of the moon you set, so it shows how much of the lobe the (enlarged) moon fills.
The N-body model. The moon is N equal spheres on a randomly oriented close-packed lattice. They attract each other with softened gravity, push apart by a damped spring on contact (coefficient of restitution 0.6 for the fluid-like pile, 0.35 for the others) and the planet is a point mass. Friction (a velocity-dependent tangential force limited to 0.6 of the normal force) turns the fluid-like pile into a rubble pile; neighbouring spheres joined by breakable springs make a rigid rock. The integrator is kick-drift-kick leapfrog with an adaptive step of at most one fourteenth of the contact period and 1.2% of the local dynamical time. The moon starts tidally locked to its mean motion.
What the simulation finds. A pile of discrete spheres is granular, not a true fluid, so its breakup distances sit between the textbook values: the fluid-like pile survives at 1.1 and breaks at 0.65 of the fluid limit (a coefficient of about 1.6 to 1.8), and a frictional rubble pile lasts down to about 0.5 of it (coefficient near 1.25, the rigid value). Bonded rock holds as long as the bond strain you allow is not exceeded, and rock can be much stronger than self-gravity, which is why real small moons such as Phobos live inside the fluid limit.
Numerical checks run in the page. (1) The rigid coefficient from a bisection on the exact differential gravity for three density ratios, against 21/3; (2) the tidally locked coefficient including centrifugal terms against 31/3; (3) L1 against the Hill radius; (4) the finite-difference slope of the Roche potential at L1; (5) the engine's eccentric Kepler period and energy drift; (6) angular-momentum conservation of a frictionless pile; (7) the fluid-like pile surviving at 1.1 and breaking at 0.5 of the fluid limit.
Simplifications. The planet is a spherical point mass with no rotation, oblateness, atmosphere or other moons; the moon is enlarged against the planet so the pile is resolved (the limit does not depend on it); N is a few hundred spheres, not the billions of grains in a real rubble pile; the bond break strain is a tunable stand-in for rock strength, not a measured value; and there is no thermal or chemical evolution. Real densities are taken from NASA planetary fact sheets (Saturn 687, Jupiter 1326, Neptune 1638, Mars 3934, Earth 5514 kg/m³). Treat the model's thresholds as accurate to tens of percent.