Tidal Evolution into the Laplace Resonance and the Resurfacing of Ganymede

Abstract

We use the numerical model of R. Malhotra (1991, Icarus94, 399–412) to explore the orbital history of Io, Europa, and Ganymede for a large range of parameters and initial conditions near the Laplace resonance. We identify two new Laplace-like resonances which pump Ganymede's eccentricity and may help explain the resurfacing of Ganymede. Near the Laplace resonance, the Io–Europa conjunction drifts at a mean angular velocity ω 1≡ 2 n 2− n 1, while the Europa-Ganymede conjunction drifts at a rate ω 2≡ 2 n 3− n 2, where n 1, n 2, and n 3are the mean motions of Io, Europa, and Ganymede. We find that Laplace-like resonances characterized by ω 1/ω 2≈ 3/2 and ω 1/ω 2≈ 2 can pump Ganymede's eccentricity to ∼0.07, producing tidal heating several hundred times higher than at the present epoch and 2 to 30 times greater than that occurring in the ω 1/ω 2≈ 1/2 resonance identified previously by Malhotra. The evolution of ω 1and ω 2prior to capture is strongly affected by Q′ Io/ Q′ J. (Here, Q′ = Q/ kis the ratio of the tidal dissipation function to second-degree Love number; the subscript Jis for Jupiter.) We find that capture into ω 1/ω 2≈ 3/2 or 2 occurs over a large range of possible initial satellite orbits if Q′ Io/ Q′ J≤ 4 × 10 −4, but cannot occur for values ≥ 8 × 10 −4. (The latter is approximately two-thirds the value required to maintain Io's current eccentricity in steady state.) For constant Q/ k, the system, once captured, remains trapped in these resonances. We show, however, that they can be disrupted by rapid changes in the tidal dissipation rate in Io or Europa during the course of the evolution; the satellites subsequently evolve into the Laplace resonance (ω 1= ω 2) with high probability. Because the higher dissipation in these resonances increases the likelihood of internal activity within Ganymede, we favor the ω 1/ω 2≈ 3/2 and 2 resonances over ω 1/ω 2≈ 1/2 for the evolutionary path taken by the Galilean satellites before their capture into the Laplace resonance. In addition to its surface appearance, Ganymede's large free eccentricity (0.0015) has long been a puzzle. We find that the ω 1/ω 2≈ 3/2 and ω 1/ω 2≈ 2 resonances can pump Ganymede's free eccentricity up to ∼10 −3independent of Q′ Gany/ Q′ J. We also show that Ganymede's free eccentricity cannot have been produced by impact with a large asteroid or comet.