On a circular orbit the planet's pull on the moon never changes strength, so the permanent tidal bulge it raises never changes shape either — no flexing, no friction, no heat. Give the orbit eccentricity and the moon's distance swings between a close periapsis and a far apoapsis every revolution (Kepler's equal-areas law also makes it swing fastest right at periapsis). Because the tidal bulge scales with 1/r³, it grows and shrinks every orbit, kneading the moon's interior like a bent paperclip and depositing frictional heat — a rate that rises roughly with e².
r(E) = a(1 − e·cosE), E − e·sinE = M(t)
bulge(t) ∝ 1/r(t)³ − 1/a³
tidal heating rate ⟨Ė⟩ ∝ e²
- Eccentricity — how elongated the orbit is; 0 is a perfect circle, higher values swing the moon closer at periapsis and farther at apoapsis each lap.
- Flex gauge — the instantaneous rate of shape change, spiking as the moon rushes through periapsis and relaxing near apoapsis.
- Geological time — fast-forward to let the ice shell / ocean layer relax toward the equilibrium its current heating rate can sustain, over millions of simulated years.
- Ocean thickness — grows when tidal heating outruns heat lost to space through the ice shell, and shrinks back to zero (frozen solid) when it can't.
This is the mechanism thought to keep a liquid layer under the ice of several outer-solar-system moons — Saturn's Enceladus, Jupiter's Europa, and to a lesser degree bodies like Rhea — even though sunlight this far out is far too weak and the moons too small to stay warm from their own formation heat alone.