Io, Europa and Ganymede — three of Jupiter's four large Galilean moons — orbit with periods locked almost exactly in a 1:2:4 ratio. This is the Laplace resonance, a three-body mean-motion resonance first analyzed mathematically by Pierre-Simon Laplace in 1784. It is not a coincidence: tidal friction over billions of years pushed the moons into this configuration and now holds them there.
ω = √(GM/r³) (Kepler's third law), so Ganymede's period is fixed at very close to 4× Io's and 2× Europa's.φ = λ_Io − 3λ_Europa + 2λ_Ganymede, where λ is each moon's orbital longitude. In a perfect lock this angle would sit exactly at 180°; in the real resonance it librates (oscillates) around 180° instead of drifting through all values.Because of the resonance, Io, Europa and Ganymede can never all line up with Jupiter at the same time — whenever Io and Europa are in conjunction, Ganymede is always on the opposite side. This forced non-alignment is exactly what keeps pumping their orbital eccentricities and, in turn, keeps Io's volcanoes erupting.
Io, Europa and Ganymede orbit Jupiter with real Keplerian angular speeds locked in an almost exact 1:2:4 ratio — watch the resonant angle librate around 180°, mark conjunctions, and detune Ganymede's period to see the lock break.
A genuine three-body mean-motion resonance: the resonant angle φ = λ_Io − 3λ_Europa + 2λ_Ganymede stays bounded (librates) instead of drifting through all values, which is the mathematical signature of a resonance lock.
Watch the libration graph and resonant-angle readout while the moons orbit. Drag the Ganymede period-offset slider to detune the resonance and see φ start to circulate instead of librate.
The resonance forces Io, Europa and Ganymede to never all align with Jupiter at once — that forced misalignment pumps Io's orbital eccentricity and powers its extreme volcanism.