Laplace Resonance Simulator

Laplace Resonance

Simulation speed 6 d/s

Elapsed time

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Resonant angle φ

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T(Europa)/T(Io)

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T(Ganymede)/T(Europa)

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How it works

Io, Europa and Ganymede orbit Jupiter with periods in an almost exact 1 : 2 : 4 ratio — Kepler's third law ties each period to its orbital radius, T² ∝ a³, and mutual gravitational tugs locked the three moons into this chain over billions of years of tidal migration.

T_Io : T_Europa : T_Ganymede = 1 : 2 : 4
φ = λ_Io − 3·λ_Europa + 2·λ_Ganymede

φ is the resonant angle. If the ratio were perfectly exact and static, φ would freeze at a constant value. In reality — and in this simulation — it librates: it oscillates back and forth around 180° instead of drifting through all 360°. That bounded libration (amplitude exaggerated here for visibility) is the mathematical fingerprint of the lock, and it guarantees the three moons can never all align with Jupiter at the same instant.

Simulation speed — how many simulated days pass per real second.
Orbit lines — toggle the three Keplerian orbit paths on or off.
Resonance angle — toggle the φ indicator arc drawn out from Jupiter.

Real-world relevance: this same tidal flexing forced by the resonance is what powers Io's 400+ active volcanoes and is thought to help keep Europa's subsurface ocean liquid.

Io (T ≈ 1.77 d) Europa (T ≈ 3.55 d) Ganymede (T ≈ 7.16 d) Resonant angle φ

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