Surprises in the recent orbital evolution of the Galilean moons

Introduction

The orbital configuration of the Galilean moons of Jupiter is the result of billions of years of evolution. Despite the extensive studies on this system, many orbital features are still unexplained or debated. A prominent example is the formation age of the Laplace resonance between Io, Europa and Ganymede, which could be either primordial (Peale and Lee 2002) or the product of tidal orbital migration of the moons (Yoder and Peale 1981). Callisto is currently not involved in any resonance, but it is very close to the 7:3 commensurability with Ganymede (De Haerdtl 1892, Noyelles and Vienne 2007).

Other orbital features that still need to be explained are the free eccentricities of Ganymede and Callisto (Malhotra 1991, Downey et al. 2020). Furthermore, the amplitude of the free libration of the Laplace resonance angle is 0.066° (Lieske 1980). Since this amplitude decreases with time because of tidal dissipation, its current value was used to date the formation of the Laplace resonance (Yoder and Peale 1981, Henrard 1983), assuming a smooth damping from an initial value of 360°.

The dynamical evolution of the Galilean moon system is mainly driven by the tides acting between Jupiter and Io (Peale et al. 1979), which cause a secular change in the satellite's semi-major axis. The Laplace resonance allows the exchange of angular momentum between the three inner moons, so that Ganymede results to be the moon that migrates faster. From astrometric and Juno data (Lainey et al. 2009, Park et al. 2025), it was possible to estimate the tidal parameters of Jupiter and Io, and the consequent migration rates of the satellites. In particular, Ganymede moves away from Jupiter at a rate of about 10 cm/yr. Such a value implies that Ganymede encountered the 7:3 resonance with Callisto about two million years ago.

Methods

In order to investigate the effects caused by the encounter with the 7:3 resonance, we ran dedicated numerical simulations of the orbital evolution of the satellites. We considered a pre-resonance orbital configuration of the moons almost identical to the current one, with the three inner satellites slightly closer to Jupiter and locked in the Laplace resonance. We took different initial values of the eccentricities of Ganymede and Callisto, considering both the case their free eccentricities were smaller or slightly larger than today's.
 
Setting tidal parameters close to the estimated ones (Lainey et al. 2009), we propagated the system starting just before the encounter with the 7:3 resonance and ending at J2000 epoch. We ran hundreds of numerical simulations and we compared the resulting orbital elements in order to determine the dynamical pathway that best matches the current configuration of the Galilean moon system.

Results

Starting from eccentricities of Ganymede and Callisto slightly larger than today's, we obtained that the most probable outcome is that the two moons crossed the resonance without being captured. At the resonance crossing, their eccentricities experienced a downward kick that decreased their values of about 15% (Ganymede) and 6% (Callisto). By tuning the pre-resonance values of the eccentricities, it was possible to perfectly match their current values at the end of the integration. This kind of evolution is the one that best reproduces the current orbital elements of the Galilean moons.

Although the probability of capture into the 7:3 resonance was significant, the consequent evolution is generally not compatible with the current orbital configuration of the system. In fact, in almost all simulations for which there was capture, the 7:3 resonance persisted for many millions of years, forcing the eccentricities of Ganymede and Callisto to increase well above their current values. Once the resonance eventually broke down, there was not enough time for tidal dissipation to damp these values.

From our simulations, we found that in the case the initial free eccentricity of Ganymede had been null, the capture into the 7:3 resonance would have been extremely probable. Since this kind of evolution must be avoided, it is necessary that Ganymede conserved part of its free eccentricity before the 7:3 resonance crossing. This requires a low tidal dissipation within Ganymede, so that the damping timescale in eccentricity is at least a few hundreds of millions of years.

One of the effect of the the passage through the 7:3 resonance was to excite the amplitude of the free libration of the Laplace resonance angle. In our simulations, we started with an amplitude much smaller than today's, and we observed a jump of its value right at the resonance crossing. In many simulations, we obtained a final amplitude close to the current value of 0.066°. Therefore, the observed libration amplitude of the Laplace resonance angle is the result of the recent encounter with the 7:3 resonance, and the estimates of the age of the Laplace resonance based on its value are not valid.

Finally, the numerical simulations revealed the effect of an extremely recent dynamical excitation occurred just 20 thousand years ago. This was due to the crossing of a three-body resonance between the three outer Galilean moons, and its main effect was to increase the oscillation amplitude of the eccentricity of Europa.

This study provides an accurate reconstruction of the orbital evolution of the Galilean moons over the last millions of years. Although this is a relatively short period of time, the dynamical evolution of the moon system was very rich and could have taken different pathways, including the formation of a four-body resonant chain. The results presented in this work could be useful for future studies of the evolution of the system over larger timescales.

 

Acknowledgments:

This research was developed under the ASI/CRAS agreement no. 2022-16-HH.0.

References:

De Haerdtl (1892). Bull. Astron. 9, 212–215.
Downey et al. (2020). Mon. Not. R. Astron. Soc. 499, 40–51.
Henrard (1983). Icarus 53, 55–67.
Lainey et al. (2009). Nature 459, 957–959.
Lieske (1980). Astron. Astrophys. 82, 340–348.
Malhotra (1991). Icarus 94, 399–412.
Noyelles and Vienne (2007). Icarus 190, 594–607.
Park et al. (2025). Nature 638, 69–73.
Peale et al. (1979). Science 203, 892–894.
Peale and Lee (2002). Science 298, 593–597.
Yoder and Peale (1981). Icarus 47, 1–35.