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Satellites of Saturn

On the orbital evolution of Saturn’s mid-sized moons

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Hi there! On the moons of Saturn today. Of course, you have heard of the Cassini mission, which toured around Saturn during 12 years. Its journey ended one year ago, after the Grand Finale, during which it was destroyed in the atmosphere of Saturn. It provided us during these 12 years a colossal amount of data, which is a chance for science. It is a chance, since it improves our knowledge of the system.

But this also gives birth to new challenges. Indeed, all of these new observations are constraints, with which the models must comply. They must explain why the satellites are where they are, AND why they present the surface features they present, AND why they have their measured gravity field, AND why they have their current shape, AND why the rings are like this, AND why Saturn is like that… You see the challenge. This is why it sparks so many studies.

Today we discuss about Orbital evolution of Saturn’s mid-sized moons and the tidal heating of Enceladus, by Ayano Nakajima, Shigeru Ida, Jun Kimura, and Ramon Brasser. This Japanese team performed numerical simulations to try to understand how the orbits of Enceladus, Tethys and Dione, evolved, with being consistent with their possible heating. The evolution is driven by the dissipation in Saturn, in the satellites, and the pull of the rings. This study has recently been accepted for publication in Icarus.

Outline

The mid-sized moons of Saturn
Interesting features, hot past
Resonances everywhere
Long-term migration of the satellites
Coping with the observational constraints
4 sets of numerical simulations
Tethys is older than Enceladus
The study and its authors

The mid-sized moons of Saturn

When we speak about the mid-sized satellites of Saturn, usually we mean Mimas, Enceladus, Tethys, Dione, and sometimes Rhea.
The inner moons orbit inner to the orbit of Mimas, and are embedded into the rings. However, Titan, Hyperion, Iapetus and Phoebe are just too far. Besides these, there are small moons which are embedded into the mid-sized system of Saturn.

Let us go back to the mid-sized. You can find below some of their characteristics.

Semi-major axis Eccentricity Inclination Orbital period Diameter
Mimas 3.19 R 0.02 1.57° 0.92 d 396 km
Enceladus 4.09 R 0.005 0.02° 1.37 d 504 km
Tethys 5.06 R ≈0 1.12° 1.89 d 1,062 km
Dione 6.48 R 0.002 0.02° 2.74 d 1,123 km
Rhea 9.05 R 0.001 0.35° 4.52 d 1,528 km

The unit “R” in the semimajor axis column is Saturn’s radius, i.e. 58,232 km. You can see that the size of the satellites increases with the distance. This has motivated the elaboration of a scenario of formation of the satellites from the rings, by Sébastien Charnoz et al. In this scenario, the rings would be initially much more massive than they are now, and the satellites would have emerged from them as droplets, removing their mass from the rings. Then they would have migrated outward. In such a scenario, the further satellites would be the older ones, and the massive ones as well. Regarding the mass, this is just true.

Craters, ridges, and internal oceans

This is what Cassini told us:

  • Mimas is known for its large crater Herschel, which diameter (139 km) is almost one-third the diameter of Mimas. It makes it look alike Star Wars’ Death Star. Its widely craterized surface suggests an inactive body. However, measurements of its east-west librations are almost inconsistent with a rigid body. It would contain an internal ocean, but explaining why this ocean is not frozen is a challenge.
    • Mimas seen by Cassini. © NASA / JPL-Caltech / Space Science Institute
    Mimas seen by Cassini. © NASA / JPL-Caltech / Space Science Institute
  • Enceladus may be the most interesting of these bodies, because its surface presents geysers, and tiger stripes, which are tectonic fractures and ridges. This proves Enceladus to be a differentiated and hot, active body. It dissipates energy, and we need to explain why.
    • The tiger stripes at the South Pole of Enceladus. © NASA
    The tiger stripes at the South Pole of Enceladus. © NASA
  • Tethys is quieter. It presents many craters, the largest one being Odysseus. Besides, it has a large valley, Ithaca Chasma. It is up to 100 km wide, 3 to 5 km deep and 2,000 km long. Its presence reveals a hot past.
    • Ithaca Chasma on Tethys © Cassini Imaging Team, SSI, JPL, ESA, NASA
    Ithaca Chasma on Tethys © Cassini Imaging Team, SSI, JPL, ESA, NASA
  • Like Tethys, Dione and Rhea present craters and evidences of past activity.

Interesting features, hot past

Enceladus, Tethys, Dione and Rhea present evidences of activity. Enceladus and Dione have global, internal oceans, while the other two may have one. Mimas presents a very quiet surface, but may have an ocean as well. All this means that these 5 moons are, or have been excited, i.e. shaken, to partly melt, crack the surface, and dissipate energy.

The primordial heat source is the decay of radiogenic elements, but this works only during the early ages of the body. After that, the dissipation is dominated by the tides raised by Saturn. Because of the variations of the distance between Saturn and the satellite, the gravitational torque changes. Its variations generate stress and strain, which are likely to dramatically affect the internal structure of the satellite. Variations of distance are due to orbital eccentricity. As you can see, some of the satellites have a significant one, with the exception of Tethys. And the eccentricity may be excited by mean-motion resonances.

Resonances everywhere

Let us go back to the orbital properties of the satellites. You can see that the orbital period of Tethys is twice the one of Mimas. Same for Enceladus and Dione. This did not happen by chance. These are mean-motion resonances. The 2:1 Enceladus-Dione one excites the eccentricity of Enceladus, and so is responsible for its currently observed activity. However, the Mimas-Tethys resonance, which is a 4:2 one (the reason why it is 4:2 and not 2:1 is pretty technical, see here), excites the inclination of Mimas, and slightly the one of Tethys as well.

As I said, this configuration did not happen by chance. The satellites have migrated since their formation, and once they encountered a resonant configuration, they actually encountered a stable location. And sometimes stable enough to stay there.

Long-term migration of the satellites

Two processes have been identified for being responsible of the long-term migration: the tides and the pull of the rings.

The tides are the result of the interaction with Saturn, the satellites being finite-size bodies. As a consequence of their size, the different parts of the satellite undergo a different torque from Saturn, and this generates stress and strain, i.e. dissipation of energy. But the satellite exerts a torque on Saturn as well. The consequence is a competition between the two processes, resulting in a variation of the orbital energy of the satellite. If the satellite gains energy, then it moves outward. However, if it dissipates energy, it moves inward. The tides also tend to circularize the orbits, i.e. damp the eccentricities.

Beside this, the rings exert a pull on the satellites. The main effect is on Mimas, because of its distance to the rings, its limited size, and the fact that it has a resonance with the rings. It has a 2:1 mean-motion resonance with the inner edge of the well-known Cassini Division, i.e. a 4,500-km wide depletion of material in the rings. At the inner edge of the Division, which is actually the outer edge of the B ring, you have an accumulation of material. This accumulation tends to push Mimas outward.

Coping with the observational constraints

The spacecraft Cassini gave us numbers. In particular

  • We have an estimation of the tidal response of Saturn,
  • we know the masses of the rings and of the satellites,
  • we can estimate the current dissipation, in particular for Enceladus,
  • we know the main geological features, in particular the impacts and the ridges, to estimate the energies which has created them.

If you want to explain something, you should better try to not violate any of these observations. A very tough task.

4 sets of numerical simulations

To elaborate an acceptable scenario for the orbital evolution of the mid-sized system, the authors ran 4 sets of intensive numerical simulations:

  1. SET 1a: Enceladus older than Tethys. This is suggested by the backward extrapolation of the orbits of Enceladus and Tethys, without mutual interaction, but migrating because of a highly dissipative Saturn… which can be allowed by the data. The consequence of such a scenario is that Tethys is originally closer to Saturn than Enceladus, and must cross its orbit to be further.
  2. SET 1b: Enceladus and Tethys starting with the same semimajor axis. Actually an end-member of the previous case.
  3. SET 2a: Tethys is older than Enceladus, and the rings affect only the semimajor axes.
  4. SET 2b: Almost the same as SET 2a, with the exception that the rings also affect the eccentricities of the satellites.

And now, the results.

Tethys is older than Enceladus

The hypothesis that Enceladus is older than Tethys should probably be discarded. Indeed, the simulations end up in collisions between the two bodies, which is inconsistent with the fact that we can actually see them.

So, this means that Tethys is older than Enceladus. However, the simulations of the sets 2a/b are not entirely satisfying, since the satellites end up in resonances, in which they are not now, which constitutes a violation of the observational data. This is particularly true if you include Dione in the simulations.

These resonances should have been encountered before the current ones. In other words, either the satellites were not trapped, but the simulations show they were, or they escaped these resonances after trapping. Some studies suggest that a catastrophic event could do that. A catastrophic event is an impact, and the surfaces of these bodies show that they underwent intense bombardments. Why not?

The study and its authors

And that’s it for today! Please do not forget to comment. You can also subscribe to the RSS feed, and follow me on Twitter, Facebook, Instagram, and Pinterest.

Mean-motion resonances

Resonances around the giant planets

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Hi there! Today the release of the paper Classification of satellite resonances in the Solar System, by Jing Luan and Peter Goldreich, is the opportunity for me to present you the mean-motion resonances in the system of satellites of the giant planets. That paper has recently been published in The Astronomical Journal, but the topic it deals with is present in the literature since more than fifty years. This is why I need to detail some of the existing works.

The mean-motion resonances (MMR)

Imagine that you have a planet orbited by two satellites. In a convenient case, their orbits will be roughly elliptical. The ellipse results from the motion of a small body around a large spherical one; deviations from the exact elliptical orbit come from the oblateness of the central body and the gravitational perturbation of the other satellite. If the orbital frequencies of the two satellites are commensurate, i.e. if Satellite A accomplishes N revolutions around the planet, while Satellite B accomplishes (almost exactly) M revolutions, i.e. M orbits, N and M being integers, then the 2 satellites will be in a configuration of mean-motion resonance. It can be shown that the perturbation of A on B (respectively of B on A) will not average to 0 but have a cumulative effect, due to the repetition, at the same place, of the smallest distance between the two bodies, the smallest distance meaning the highest gravitational torque. A consequence of a MMR is the increase of the eccentricity of one of the satellites, or of both of them, and / or their inclinations… or only the inclination of one of them. In the worst case, this could result in the ejection of one of the satellites, but it can also have less catastrophic but not less interesting consequences, like the heating of a body, and the evolution of its internal structure… We will discuss that a little later.

A mean-motion resonance can be mathematically explained using the orbital elements, which describe the orbit of a satellite. These elements are

  • The semimajor axis a,
  • the eccentricity e. e=0 means that the orbit is circular, while e<1 means that the orbit is elliptical. For planetary satellites, we usually have e<0.05. With these two elements, we know the shape of the orbit. We now need to know its orientation, which is given by 3 angles:
  • the inclination i, with respect to a given reference plane. Usually it is the equatorial plane of the parent planet at a given date, and the inclination are often small,
  • the longitude of the ascending node Ω, which orientates the intersection of the orbital plane with the reference plane,
  • the longitude of the pericentre ϖ, which gives you the pericentre, i.e. the point at which the distance planet-satellite is the smallest. With these 5 elements, you know the orbit. To know where on its orbit the satellite is, you also need
  • the mean longitude λ.

Saying that the Satellites A and B are in a MMR means that there is an integer combination of orbital elements, such as φ=pλA-(p+q)λA+q1ϖA+q2ϖB+q3ΩA+q4ΩB, which is bounded. Usually an angle is expected to be able to take any real value between 0 and 2π radians, i.e. between 0 and 360°, but not our φ. The order of the resonance q is equal to q1+q2+q3+q4, and q3+q4 must be even. Moreover, it stems from the d’Alembert rule, which I will not detail here, that a strength can be associated with this resonance, which is proportional to eAq1eBq2iAq3iBq4. This quantity also gives us the orbital elements which would be raised by the resonance.

In other words, if the orbital frequency of A is twice the one of B, then we could have the following resonances:

  • λA-2λB+ϖA (order 1), which would force eA,
  • λA-2λB+ϖB (order 1), which would force eB,
  • 2λA-4λB+ϖA+ϖB (order 2), which would force eA and eB,
  • 2λA-4λB+2ΩA (order 2), which would force iA,
  • 2λA-4λB+2ΩB (order 2), which would force iB,
  • 2λA-4λB+2ΩA+ΩB (order 2), which would force iA and iB.

Higher-order resonances could be imagined, but let us forget them for today.

The next two figures give a good illustration of the way the resonances can raise the orbital elements. All of the curves represent possible trajectories, assuming that the energy of the system is constant. The orbital element which is affected by the resonance, can be measured from the distance from the origin. And we can see that the trajectories tend to focus around points which are not at the origin. These points are the centers of libration of the resonances. This means that when the system is at the exact resonance, the orbital element relevant to it will have the value suggested by the center of libration. These plots are derived from the Second Fundamental Model of the Resonance, elaborated at the University of Namur (Belgium) in the eighties.

The Second Fundamental Model of the Resonance for order 1 resonances, for different parameters. On the right, we can see banana-shaped trajectories, for which the system is resonant. The outer zone is the external circulation zone, and the inner one is the internal circulation zone. Inspired from Henrard J. & Lemaître A., 1983, A second fundamental model for resonance, Celestial Mechanics, 30, 197-218.
The Second Fundamental of the Resonance for order 2 resonances, for different parameters. We can see two resonant zones. On the right, an internal circulation zone is present. Inspired from Lemaître A., 1984, High-order resonances in the restricted three-body problem, Celestial Mechanics, 32, 109-126.

Here, I have only mentioned resonances involving two bodies. We can find in the Solar System resonances involving three bodies… see below.

It appears, from the observations of the satellites of the giant planets, that MMR are ubiquitous in our Solar System. This means that a mechanism drives the satellite from their initial position to the MMRs.

Driving the satellites into resonances

When the satellites are not in MMR, the argument φ circulates, i.e. it can take any value between 0 and 2π. Moreover, its evolution is monotonous, i.e. either constantly increasing, or constantly decreasing. However, when the system is resonant, then φ is bounded. It appears that the resonance zones are levels of minimal energy. This means that, for the system to evolve from a circulation to a libration (or resonant zone), it should loose some energy.

The main source of energy dissipation in a system of natural satellites is the tides. The planet and the satellites are not exactly rigid bodies, but can experience some viscoelastic deformation from the gravitational perturbation of the other body. This results in a tidal bulge, which is not exactly directed to the perturber, since there is a time lag between the action of the perturber and the response of the body. This time lag translates into a dissipation of energy, due to tides. A consequence is a secular variation of the semi-major axes of the satellites (contraction or dilatation of the orbits), which can then cross resonances, and eventually get trapped. Another consequence is the heating of a satellite, which can yield the creation of a subsurface ocean, volcanism…

Capture into a resonance is actually a probabilistic process. If you cross a resonance without being trapped, then your trajectories jump from a circulation zone to another one. However, if you are trapped, you arrive in a libration zone, and the energy dissipation can make you spiral to the libration center, forcing the eccentricity and / or inclination. It can also be shown that a resonance trapping can occur only if the orbits of the two satellites converge.

The system of Jupiter

Jupiter has 4 large satellites orbiting around: J1 Io, J2 Europa, J3 Ganymede, and J4 Callisto. There are denoted Galilean satellites, since they were discovered by Galileo Galilei in 1610. The observations of their motion has shown that

  • Io and Europa are close to the 2:1 MMR,
  • Europa and Ganymede are close to the 2:1 MMR as well,
  • Ganymede and Callisto are close to the 7:3 MMR (De Haerdtl inequality)
  • Io, Europa and Ganymede are locked into the Laplace resonance. This is a 3-body MMR, which resonant argument is φ=λ1-3λ2+2λ3. It librates around π with an amplitude of 0.5°.

This Laplace resonance is a unique case in the Solar System, to the best of our current knowledge. It is favored by the masses of the satellites, which have pretty the same order of magnitude. Moreover, Io shows signs of intense dissipation, i.e. volcanism, which were predicted by Stanton Peale in 1979, before the arrival of Voyager I in the vicinity of Jupiter, from the calculation of the tidal effects.

The system of Saturn

Besides the well-known rings and a collection of small moons, Saturn has 8 major satellites, i.e.

  • S1 Mimas,
  • S2 Enceladus,
  • S3 Tethys,
  • S4 Dione,
  • S5 Rhea,
  • S6 Titan,
  • S7 Hyperion,
  • S8 Iapetus,

and resonant relations, see the following table.

Satellite 1 Satellite 2 MMR Argument φ Libration center Libration amplitude Affected quantities
S1 Mimas S3 Tethys 4:2 2λ1-4λ3+Ω1+Ω3 0 95° i1,i3
S2 Enceladus S4 Dione 2:1 λ2-2λ4+ϖ2 0 0.25° e2
S6 Titan S7 Hyperion 4:3 3λ6-4λ7+ϖ7 π 36° e7

The amplitude of the libration tells us something about the age of the resonance. Dissipation is expected to drive the system to the center of libration, where the libration amplitude is 0. However, when the system is trapped, the transition from circulation to libration of the resonant argument φ induces that the libration amplitude is close to π, i.e. 180°. So, the dissipation damps this amplitude, and the measured amplitude tells us where we are in this damping process.

This study

This study aims at reinvestigating the mean-motion resonances in the systems of Jupiter and Saturn in the light of a quantity, kcrit, which has been introduced in the context of exoplanetary systems by Goldreich & Schlichting (2014). This quantity is to be compared with a constant of the system, in the absence of dissipation, and the comparison will tell us whether an inner circulation zone appears or not. In that sense, this study gives an alternative formulation of the results given by the Second Fundamental Model of the Resonance. The conclusion is that the resonances should be classified into two groups. The first group contains Mimas-Tethys and Titan-Hyperion, which have large libration amplitudes, and for which the inner circulation zone exists (here presented as overstability). The other group contains the resonances with a small amplitude of libration, i.e. not only Enceladus-Dione, but also Io-Europa and Europa-Ganymede, seen as independent resonances.

A possible perspective

Io-Europa and Europa-Ganymede are not MMR, and they are not independent pairs. They actually constitute the Io-Europa-Ganymede resonance, which is much less documented than a 2-body resonance. An extensive study of such a resonance would undoubtedly be helpful.

Some links

  • The paper, i.e. Luan J. & Goldreich P., 2017, Classification of satellite resonances in the Solar System, The Astronomical Journal, 153:17.
  • The web page of Jing Luan at Berkeley.
  • The web page of Peter Goldreich at Princeton.
  • The Second Fundamental Model of the Resonance, for order 1 resonances and for higher orders.
  • A study made in Brazil by Nelson Callegary and Tadashi Yokoyama, on the same topic: Paper 1 Paper 2, also made available by the authors here and here, thanks to them for sharing!.