Planetary Dynamics / Celestial MechanicsJovian Resonance Lab

Jupiter’s moons keep time. Which physics keeps them together?

Current research status: product v1.0.0 preserves the original protocol-frozen thirty-year model ablation and adds a second protocol-frozen temporal replication on 3,653 non-overlapping JUP365 days from 2031–2040. The model ordering, numerical-convergence, and reference-adequacy gates all passed. Because the second interval extends beyond the 2026 release date, its JUP365 values are ephemeris predictions—not future observations.

The question

Io, Europa, and Ganymede orbit Jupiter in periods close to a 4:2:1 ratio. The deeper relationship is three-body: their mean longitudes combine into the Laplace angle `φL = λIo − 3λEuropa + 2λGanymede`, which remains close to 180 degrees instead of circulating around the full circle.

Which minimum dynamical ingredients can reproduce that angle over a fixed thirty-year interval? Jovian Resonance Lab compares four models against the same JPL reference rather than asking whether one visually convincing simulation “looks right.”

What was fixed before the result

The protocol was committed before the complete scored interval was downloaded or any confirmatory model was evaluated. It fixed 2001-01-01 through 2030-12-31 TDB, one daily sample, circular RMSE as the primary endpoint, and four models:

  • K2: independent Jupiter–moon pairs, without moon–moon forces or oblateness;
  • G3: Jupiter, Io, Europa, and Ganymede as mutually interacting point masses;
  • G4: G3 plus Callisto;
  • G4J2: G4 plus Jupiter's axisymmetric `J2` oblateness term.

A disclosed two-date feasibility check from 2000 sits outside the scored interval. No solar force, higher Jovian harmonic, relativity, tide, or pole-precession term was added after seeing results.

The evidence

NASA/JPL Horizons supplied JUP365 osculating elements for 10,957 aligned daily epochs and initial Cartesian states for all four Galilean moons. Every scored response identified its source as `JUP365_MERGED`. The normalized reference registry has a published SHA-256 identity, while the full query mappings, retrieval time, row counts, states, and model traces remain machine-readable.

That provenance matters, but JUP365 is still a fitted and propagated ephemeris—not raw astrometric measurements and not an uncertainty distribution. The experiment measures model adequacy relative to that reference product.

What we found

The JUP365 Laplace angle had a circular mean of 180.0036 degrees. Every daily reference sample was within two degrees of 180 degrees, and the unwrapped half peak-to-peak range was 0.4169 degrees. That two-degree band is a descriptive containment check, not a confidence interval or universal definition of resonance.

The four frozen circular RMSE values were:

  • K2: 103.5936°;
  • G3: 28.2122°;
  • G4: 28.3195°;
  • G4J2: 11.2678°.

G4J2 was lower than K2, G3, and G4, so all three prespecified comparisons passed. Mutual satellite gravity recovered much of the phase relationship, while Jupiter's oblateness improved the primary metric further. Adding Callisto without `J2` did not improve this endpoint over G3. That does not make Callisto dynamically irrelevant; it is a result about this model family, interval, and angle.

The result that prevents overclaiming

The nominal integration used velocity Verlet with a 0.01-day step. The protocol required a 0.005-day sensitivity run. Those two G4J2 traces differed by 8.1342° RMSE across thirty years—too large to call the nominal absolute trajectory numerically converged.

After that result was known, two smaller steps were added as an explicitly exploratory convergence study. Successive trace differences fell from 8.1342° to 2.1487° and then 0.5417°, close to the fourfold reduction expected when halving the step of a second-order method. Against JUP365, the finest 0.00125-day run reached 0.4641° RMSE.

That was scientifically interesting: numerical phase error likely explained much of the nominal 11.27-degree discrepancy. It did not provide permission to replace the frozen primary endpoint. Instead, it motivated a second protocol with the step sizes and pass thresholds committed before new reference data were retrieved.

The v1 temporal replication

The new protocol fixed 2031-01-01 through 2040-12-31 TDB, 3,653 daily epochs, a fresh JUP365 initial state at the start of the interval, and a common 0.0025-day step for all four models. G4J2 again had the lowest circular RMSE:

  • K2: 97.9578°;
  • G3: 25.1741°;
  • G4: 24.7109°;
  • G4J2: 1.0434°.

All three frozen ordering comparisons passed. The protocol also required G4J2 runs at 0.005 and 0.00125 day. The 0.0025-versus-0.00125-day traces differed by 0.5735° RMSE, below the fixed 0.75-degree ceiling. Their estimated convergence order was 1.984, inside the fixed 1.5–2.5 interval. The finest trace reached 0.4699° RMSE against JUP365, passing the fixed one-degree reference-adequacy gate.

This makes the numerical result confirmatory for the new interval rather than a favorable post-result refinement. It still does not prove convergence to exact Solar System dynamics or attach an uncertainty distribution to JUP365.

Other checks

The frozen 2011 reference segment sampled every twelve hours ranged only 0.2273 degrees and showed no concealed circulation. Recalculating that segment in the ICRF/J2000 element frame changed the Laplace angle by 0.2019 degrees RMS and at most 0.2037 degrees. The frame effect is small relative to model RMSE, but substantial relative to the narrow reference libration range.

Point-mass energy drift remained `−2.57 × 10⁻⁷` for G3 and `−9.72 × 10⁻⁹` for G4. Interacting-model barycenter displacement stayed below `6 × 10⁻⁸` km. K2 deliberately fixes Jupiter, so its represented collection does not provide the same barycenter-conservation diagnostic.

Try it yourself

The interactive Jovian Resonance Lab animates a sixty-day JUP365 vector extract, lets you scrub through the moon positions, compares each model's validation residual, and lets you switch among the three frozen G4J2 step sizes. The angular timing comes from JPL; radial spacing and body sizes are visibly rescaled to keep every moon on screen.

What this cannot establish

Agreement with a fitted ephemeris cannot determine how the Laplace relation formed, whether it survives for billions of years, or how tides alter the system. The validation restarts from JUP365 in 2031, so it is not an uninterrupted forty-year forecast. It says nothing direct about tidal quality factors, internal heating, subsurface oceans, or habitability. Paita, Celletti, and Pucacco provide the closest published model context; Lainey and colleagues constrain the idea of an immutable exact clock; Lari, Saillenfest, and Fenucci show why later tidal evolution involving Callisto remains sensitive to uncertain parameters.

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