Laplace Resonance of Jupiter Moons - 2D Conjunction Map and Resonance Angle

The 1:2:4 lock

Io, Europa and Ganymede have orbital periods of 1.769, 3.551 and 7.155 days, close to 1 : 2 : 4. Their mean motions satisfy n₁ - 3 n₂ + 2 n₃ = 0, and the corresponding combination of mean longitudes, the Laplace angle φ_L = λ₁ - 3 λ₂ + 2 λ₃, is held near 180° (it librates by only about 0.064° with a period near 2.07 years). Consequently when Io overtakes Europa, Ganymede is always about 90° away, and the three moons are never in a triple conjunction. The mutual pulls at each conjunction keep Io's orbit slightly eccentric, which drives the tidal heating behind Io's volcanism.

What the page computes

Mean longitudes advance linearly, λ_i = λ_i,0 + n_i t, with the libration added to Io as a prescribed small sinusoid. Conjunctions are found by detecting when λ₁ - λ₂ crosses zero; their count is compared with the synodic prediction.

Honest simplifications: this is a kinematic model of mean longitudes, not an N-body integration; the libration is imposed rather than produced by the dynamics, Callisto and eccentricities are ignored, and the detune slider is a thought experiment.

• Run the Real system and watch the yellow dots: successive Io-Europa conjunctions happen at slowly shifting longitudes, always with Ganymede a quarter turn away.
• Pick the Exaggerated libration preset to see the Laplace angle oscillate visibly.
• Pick Broken resonance: the angle drifts away from 180 degrees and the third graph scatters.
• Raise the days-per-second rate to see years of motion in seconds.