Tidal Orbital Resonance (2D)

Three moons, one rhythm

Io, Europa and Ganymede orbit Jupiter in periods that are close to 1 : 2 : 4. The page integrates the planar four-body problem (Jupiter plus the three moons, all gravitating on each other) with a symplectic leapfrog step of 1/300 Io period:

From the position and velocity of each moon relative to Jupiter the page computes the osculating orbital elements: the eccentricity vector e = [(v² − μ/r)r − (r·v)v]/μ with μ = G(M + m), the semi-major axis a = 1/(2/r − v²/μ), and the mean longitude λ = M + ϖ.

The Laplace resonance

Neighbouring moons are near a 2:1 mean-motion ratio, but the three together obey an exact relation between mean motions and a locked relation between longitudes (Laplace, 1784):

The second relation says that the three moons can never be in conjunction at the same time (when Io and Europa line up, Ganymede is on the far side of Jupiter). The mutual tugs at conjunctions always fall in the same places, which keeps the orbits slightly eccentric (forced eccentricity) instead of letting tides circularise them. The tidal heating that results powers Io's volcanoes: the dissipated power scales as P ∝ (k₂/Q) G M_J² R_Io⁵ n e² / a⁶, proportional to e².

Honest simplifications

• Point masses in a plane. Jupiter's oblateness (J₂) is not included; in reality it makes the orbits precess quickly and keeps the libration amplitude of φ_L near 0.06°. Here it librates by roughly ±12°.
• No tides: the page shows the dynamics that make tides matter but does not damp or heat anything.
• The starting orbits are near-circular at the observed size ratio, not a fit to the real ephemeris, so exact numbers (libration range, forced eccentricities) are illustrative.
• Only the three Galilean moons that take part in the resonance are included (Callisto is outside it).

Checks run in the page

(1) The measured Io period (from the unwrapped mean longitude) equals Kepler's third law with the moon's mass included. (2) The total energy of the four bodies stays constant to better than 10^-4. (3) The mean drift of φ_L stays below 10^-3 of Io's mean motion: the Laplace relation holds. Set Europa orbit size to 1.03 to break it and watch check 3 fail.

• Press Real Galilean moons and raise the speed to ~100 periods/s: φ_L (graph 1) wobbles around 180° and stays on the green arc.
• Watch graph 2: the eccentricities oscillate as the moons exchange energy at conjunctions.
• Select Europa out of resonance (+3 %): the period ratio in graph 3 leaves 2:1, φ_L circulates and check 3 fails.
• Select Moons 5x heavier: stronger tugs, larger libration and eccentricities.
• Select Nearly massless moons: with almost no mutual attraction the orbits barely change.