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The three classical resonance angles (Io, Europa with Io, Europa with Ganymede); φL = ψ2 − ψ3
Phase plane: Laplace angle against its rate (closed loop = libration, open line = circulation)
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The science
The Laplace resonance. Io, Europa and Ganymede have mean motions that obey (Laplace, 1784)
n1 − 3 n2 + 2 n3 = 0, φL = λ1 − 3 λ2 + 2 λ3 = 180°
(1 = Io, 2 = Europa, 3 = Ganymede, λ = mean longitude, n = mean motion). Neither pair is exactly 2:1 on its own: n1 − 2 n2 = 0.7395 deg/day and n2 − 2 n3 = 0.7395 deg/day, the same number, which is what the first equation says. Each pair is locked by its own classical resonance angle ψ1 = λ1 − 2λ2 + ϖ1, ψ2 = λ1 − 2λ2 + ϖ2, ψ3 = λ2 − 2λ3 + ϖ2 (ϖ = longitude of pericentre); subtracting gives the identity φL = ψ2 − ψ3, in which the pericentres cancel. The observed libration of φL about 180° has an amplitude of only 0.064° and a period of about 2071 days (5.7 years).
Geometry that follows. If φL = 180° and Io and Europa are in conjunction (λ1 = λ2), then 2(λ3 − λ1) = 180°: Ganymede is 90° away. At a Europa-Ganymede conjunction, λ1 − λ2 = 180°: Io is on the opposite side of Jupiter. A triple conjunction would need φL = 0, so it can never happen. Both facts are measured in the page (conjunction counters and checks); an older statement that Ganymede is “opposite” at an Io-Europa conjunction is wrong, it is at right angles.
Equations of motion (what is integrated). Planar, Jupiter-centred coordinates; the moons feel Jupiter's point mass plus its equatorial J2 and J4 terms, and each other (direct and indirect terms):
ai = −(GMJ+GMi)/r2 [1 + (3/2) J2 (RJ/r)2 − (15/8) J4 (RJ/r)4] r̂i + Σj≠i GMj [ (rj − ri)/|rj − ri|3 − rj/|rj|3 ]
with GMJ = 126 686 535 km³/s², RJ = 71 492 km, J2 = 0.014736, J4 = −0.000587 and moon masses from the published GM values. The oblateness matters: it makes pericentres precess by about 0.13 deg/day for Io, comparable to the mean-motion mismatch, so it sets where the resonance sits. The equations are advanced with a fourth-order symplectic composition (Yoshida 1990) at a fixed step of 0.01 day; angular momentum is conserved to rounding of the scheme and the energy error stays bounded.
Libration as a pendulum. Close to the lock, the Laplace angle behaves like a pendulum, φ̈ ≈ −Ω2 sin(φL − 180°), with Ω set by the moon masses (the small-amplitude libration period in this model is about 2110 days; heavier moons give faster swings). The gauge in the corner draws exactly this picture: a ball in the cosine well. A large kick makes the swing slower; beyond about 100° the ball goes over the top (the separatrix) and the angle circulates.
Tidal heating. The resonance keeps the eccentricities from decaying. Dissipation inside a synchronously rotating moon of radius R on an orbit with semi-major axis a, mean motion n and eccentricity e is (Peale, Cassen and Reynolds 1979)
P = (21/2) (k2/Q) G MJ2 R5 n e2 / a6, τe = (2/21) (Q/k2) (m/MJ) (a/R)5 / n
τe is how fast e would die out if nothing forced it: about 150 000 years for Io, tiny compared with 4.5 billion. With k2/Q = 0.015 and e = 0.0041 the formula gives 9.3e13 W for Io, 2.2 W/m² averaged over the surface, the measured value. k2/Q is the calibrated quantity: the page uses the same value for all three moons, so the Europa and Ganymede powers are order-of-magnitude upper estimates (their interiors are much less dissipative than Io's).
Initial conditions. The semi-major axes follow from the observed mean motions in the J2+J4 field; the eccentricities (0.0041, 0.0094, 0.0013) and pericentres follow the conjunction lines. Phases and the three eccentricity vectors were then refined once with a numerical search (Nelder-Mead on 10 parameters) so that the lock starts near its libration centre; this is a model setup, not a measurement.
Honest simplifications. (1) Planar: the real inclinations (0.04°, 0.47°, 0.2°) are ignored. (2) Callisto is not in the resonance and is not integrated (it is drawn on a circular orbit for contrast; its period ratio to Ganymede, 2.33, is not a ratio of small integers). (3) No solar or Saturnian perturbations, no Jupiter recoil, no relativity. (4) Tidal dissipation is computed from the orbit but not fed back into it, so there is no secular evolution of the orbits. (5) The libration amplitude of this N-body model is about 0.3° instead of the observed 0.064°, because it has no Callisto, Sun, higher gravity terms or inclinations to damp the start-up. (6) Moons are drawn enlarged by the slider; Jupiter's spin is decorative.
Checks run in the page. (a) Io alone on a circular orbit against the analytic rate sqrt(Fr/r); (b) the Io-Europa synodic period; (c) Ganymede's 90° offset at Io-Europa conjunctions; (d) Io's 180° offset at Europa-Ganymede conjunctions; (e) the mean-motion relation n1 − 3n2 + 2n3 = 0; (f) angular-momentum conservation; (g) the libration period against 2071 days; (h) Io's heat flux against 2.24 W/m².