Roche limit

Move a moon closer to its planet and it is first stretched into an ellipsoid. Cross a critical distance and it is torn apart into a ring of debris.

The limit comes from setting tidal acceleration equal to the moon’s own surface gravity: d_R = 2.44 Rₚ (ρₚ/ρₛ)^(1/3). The density ratio decides everything, which is why an ice moon disintegrates farther out than a rock one.

The demo uses a linearised tidal field and a uniform-density sphere with no cohesion, so it shows the fluid Roche limit. Real rubble piles can hold together closer in thanks to material strength.