The Laplace resonance

Talking points

• NASA Jovian Satellite Fact Sheet, orbital periods: Io 1.769138, Europa 3.551181, Ganymede 7.154553, Callisto 16.689017 days; semi-major axes 421.8, 671.1, 1070.4 and 1882.7 thousand km.
• The ratios are close to, and not exactly, 2: P₂/P₁ = 2.00729 and P₃/P₂ = 2.01470, so P₃/P₁ = 4.04409. The departures matter: they are what the Laplace relation ties together.
• In the viewer each moon turns at 360°/P with P from its data, the fact sheet's 1.769138, 3.551181 and 7.154553 days, starting from a random angle picked when the page loads. The periods are real; where each moon is on a given date is not.
• This is the artistic scale at spread 1: the moons are drawn at 18, 24 and 32 units, ratios 1 : 1.33 : 1.78, against the real 1 : 1.59 : 2.54.

Talking points

• Peale and Lee (2002): the 2:1 resonance variables θ₁ = λ₁ − 2λ₂ + ϖ₁ and θ₃ = λ₂ − 2λ₃ + ϖ₂ librate about 0° and θ₂ = λ₁ − 2λ₂ + ϖ₂ about 180°, all with small amplitude. Since θ₂ − θ₃ = λ₁ − 3λ₂ + 2λ₃, that angle stays near 180° and n₁ − 3n₂ + 2n₃ = 0. "This last libration condition is called the Laplace relation because Laplace first understood its stability."
• Peale and Lee on the geometry: conjunctions of Io and Europa occur with Io near its periapse and Europa near its apoapse; conjunctions of Europa and Ganymede occur with Europa near its periapse, and Ganymede can be anywhere in its orbit then because θ₄ = λ₂ − 2λ₃ + ϖ₃ circulates.
• Origin: Peale and Lee contrast assembly by differential tidal expansion of the orbits (tides raised on Jupiter, mainly by Io, with libration damped by dissipation in the satellites) with their primordial assembly by migration in a circumjovian disk. The resonances sustain the tidal heating behind Io's volcanism.
• Units: n = 360°/P in degrees per day, P from the fact sheet.

Talking points

• JPL Planetary Satellite Mean Elements (ephemeris JUP365, Laplace-plane frame, epoch 2000-01-01.5 TDB), in degrees, node, argument of periapsis ω, mean anomaly M: Io 0.0, 49.1, 330.9; Europa 184.0, 45.0, 345.4; Ganymede 58.5, 198.3, 324.8. The mean longitude is λ = node + ω + M.
• Each moon's elements are referred to its own Laplace plane, with the node counted from that plane's node on the ICRF equator. The three poles (R.A., Dec.) are 268.1°, 64.5°; 268.1°, 64.5°; 268.2°, 64.6°, so the origins agree to about a tenth of a degree.
• JPL: mean elements are "simply the elements of a precessing ellipse which has been fit in a least squares sense to the numerically integrated orbit", "not intended for ephemeris computation". Each λ is the sum of three angles given to 0.1°, so good to 0.15°, and the coefficients of λ₁ − 3λ₂ + 2λ₃ add up to 6 in size: the Laplace angle from these elements is good to 0.9° at worst.
• The same elements give θ₁ = 0.3°, θ₂ = 180.2° and θ₃ = 0.2° (with ϖ = node + ω), the libration centres of the previous step.
• Libration of the Laplace angle: Lari (2018) finds a period of 2047.85 days in his semi-analytical model and quotes 2059.62 days from Lainey and colleagues' frequency analysis. Lari and Rossi (2026, Icarus) give the free libration, from the JUP365 and NOE-5 ephemerides, a period of 2060 days and an amplitude of 0.061°.

Talking points

• With λ₁ = λ₂ = λ₃ = λ, λ₁ − 3λ₂ + 2λ₃ = λ(1 − 3 + 2) = 0. Peale and Lee: the angle librates about 180° with small amplitude, so that configuration never occurs.
• Rearranged, the Laplace relation reads n₁ − n₂ = 2(n₂ − n₃): Europa-Ganymede conjunctions come exactly half as often as Io-Europa ones. From the fact sheet periods the synodic periods are 3.525464 and 7.050927 days, a ratio of 1.999999.
• In the viewer the three moons start from random angles, so its Laplace angle rests at whatever value the page load gave it (step 6). Three moons within ε of each other need that angle within 3ε of 0°, so when it happens to start near 0° the viewer shows close line-ups that the real system never makes.
• NASA lists the inclinations as 0.04°, 0.47° and 0.18°, so the three orbits are close to coplanar.

Talking points

• From the fact sheet periods, n₁ − 2n₂ = 0.73948 and n₂ − 2n₃ = 0.73951 degrees per day. Lari (2018) gives 486.89 days for the circulation related to ν = n₁ − 2n₂ = n₂ − 2n₃, and 486.81 days from Lainey and colleagues' frequency analysis.
• θ₁ = λ₁ − 2λ₂ + ϖ₁ librating about 0° means ϖ̇₁ = 2n₂ − n₁ = −ν: Io's periapse regresses at ν. So does Europa's (from θ₂). Io's periapse-to-periapse period is then 360°/(n₁ + ν) = 1.762732 days, and Europa's 360°/(n₂ + ν) = 3.525464 days, which is exactly the Io-Europa synodic period, since n₂ + ν = n₁ − n₂. That is why the conjunctions always fall at the same point of Europa's orbit.
• A trap in JPL's mean-element table: the period column, which the table's key calls the sidereal period, lists Io 1.762732 and Europa 3.525463 days, the periapse-to-periapse values above. Ganymede's 7.155588 and Callisto's 16.690440 also equal 360° over the mean anomaly's rate built from JPL's own apsidal and nodal periods. For mean motions use the fact sheet's orbital periods.
• The clock in this view runs at a day a second.

Talking points

• In src/main.js each moon's angle is its starting angle plus (360°/P) × JD, with P parsed from the orbitalPeriod strings in src/data/celestialData.js: Io 1.769138, Europa 3.551181, Ganymede 7.154553 days, the fact sheet's values. createMoon picks the starting angle with Math.random(), so it differs every time the page loads.
• With those periods n₁ − 3n₂ + 2n₃ is −0.00003 degrees per day (step 2), so the viewer's Laplace angle moves about a degree in a century of simulated time, from wherever it happened to start.
• The relation is sensitive to the third decimal place of a day. Rounded to two decimals (1.77, 3.55 and 7.15 days), the same periods would turn the Laplace angle once round in about 2,640 days, 7.2 years.
• What the viewer is good for here: the 1:2:4 rhythm (Io makes 4.044 laps per Ganymede lap), the real periods and the moons in Jupiter's equatorial plane. What it does not have: real positions on a date, the 180° lock, or the small eccentricities (NASA: 0.004, 0.009, 0.001), since it draws each orbit as a circle.
• At a week a second and 60 frames a second, Io moves about 24° between frames, so the rhythm strobes; the earlier steps run slower.

Talking points

• At six hours a second Ganymede's lap takes 7.154553 / 0.25 = 28.6 seconds and Io's 7.1 seconds.
• The viewer runs the real ratio, 7.154553 / 1.769138 = 4.044, so Io finishes its fourth lap just before Ganymede finishes its first.