Hi there! You may have heard that subsurface oceans have been hinted / discovered / confirmed for some major satellites of Jupiter and Saturn. What if bacteriological life existed there? Wait a minute… it is too early to speak about that. But anyway, these oceans are interesting, and the study I present you today, i.e. Ocean tidal heating in icy satellites with solid shells, by Isamu Matsuyama et al., discusses the response of these oceans to the tidal heating, in considering the icy shell coating the oceans. This study has recently been accepted for publication in Icarus.

Ocean worlds in the Solar System

First of all, let us see how you can have a subsurface ocean. The main satellites of our giant planets are in general frozen worlds, where the heaviest elements have migrated to the center. As a consequence, the surface is essentially water ice. If you go a little deeper, i.e. some kilometers below the surface, then you increase the pressure and the temperature, and you meet conditions under which liquid water may survive. This is why large and mid-sized satellites may support a global, subsurface ocean. Let us see now the direct and indirect detections

Certain: Titan

Titan is the largest satellite of Saturn, and is hinted since at least 30 years to have a global ocean. The spacecraft Cassini-Huygens has provided enough data to confirm this assumption, i.e.

• The detection of a so-called Schumann resonance in the atmosphere of Titan, i.e. an electromagnetic resonance, which could be excited by a rotating magnetosphere, which would itself be generated by a global liquid layer, i.e. an ocean,
• the obliquity of the surface of Titan, i.e. 0.3°, is thrice too large for a body in which no ocean would decouple the surface from the core,
• the variations of the gravity field of Titan, which are contained in a so-called tidal Love number k₂, are too large for an oceanless body.

Certain: Europa

Europa has been visited by the Galileo spacecraft, which orbited Jupiter between 1995 and 2003. Galileo revealed in particular

• a fractured surface (see featured image), which means a pretty thin crust, and an ocean beneath it,
• a significant magnetic field, due to a subsurface conductive layer, i.e. an ocean.

Certain: Ganymede

Ganymede has a strong magnetic field as well. Observations by the Hubble Space Telescope revealed in 2015 that the motion of auroras on Ganymede is a signature of that magnetic field as well, i.e. the internal ocean. Theoretical studies in fact suggest that there could be several oceanic layers, which alternate with water ice.

Certain: Enceladus

We can see geysers at the surface of Enceladus, which reveal liquid water below the surface. In particular, we know that Enceladus has a diapir at its South Pole. Cassini has proven by its gravity data that the ocean is in fact global.

Suspected: Dione

A recent theoretical study, led by Mikael Beuthe who also co-authors the present one, shows that Dione could not support its present topography if there were no subsurface ocean below the crust. The same methodology applied on Enceladus gives the same conclusion. In some sense, this validated the method.

Suspected: Callisto

Measurements by Galileo suggest that the magnetic field of Jupiter does not penetrate into Callisto, which suggests a conductive layer, i.e. once more, an ocean.

Suspected: Pluto

Pluto exhibits a white heart, Sputnik Planitia, which frozen material might originate from a subsurface ocean.

Doubtful: Mimas

Mimas is the innermost of the mid-sized satellites of Saturn. It is often compared to the Death Star of Star Wars, because of its large crater, Herschel. The surface of Mimas appears old, i.e. craterized, and frozen, so no heating is to be expected to sustain an ocean. However, recent measurements of the diurnal librations of Mimas, i.e. its East-West oscillations, give too large numbers. This could be the signature of an ocean.

Other oceanic worlds may exist, in particular among the satellites of Uranus and Neptune.

Tidal heating

Tides are the heating of a body by another, massive one, due to the variations of its gravitational action. For natural satellites, the tides are almost entirely due to the parent planet. The variations of the gravitational attraction over the volume of the satellite, and their time variations, generate stress and strain which deform and heat the satellite. The time-averaged tide will generate an equilibrium shape, which is a triaxial ellipsoid, while the time variations heat it.
The time variations of the tides are due to the variations of the distance between a satellite element and the planet. And for satellites, which rotate synchronously, two elements rule these variations of distance: the orbital eccentricity, and the obliquity.

For solid layers, rheological models give laws ruling the tidal response. However, the problem is more complex for fluid layers.

Waves are generated in the ocean

In a fluid, you have waves, which transport energy. In other words, you must considerate them when you estimate the heating. The authors considered two classes of waves:

1. Gravity waves: when a body moves on its orbit, the ocean moves, but the gravity of the body acts as a restoring force. This way, it generates gravity waves.
2. Rossby-Haurwitz waves: these waves are generated by the rotation of the body, which itself is responsible for the Coriolis force.

A wave has a specific velocity, wavelength, period… and if you excite it at a period which is close to its natural period of oscillation, then you will generate a resonant amplification of the response, i.e. your wave will meet a peak of energy.

All this illustrates the complexity of resolving such a problem.

The physical model

Solving this problem requires to write down the equations ruling the dynamics of the fluid ocean. The complete equations are the Navier-Stokes equations. Here the authors used the Laplace tidal equations instead, which derive from Navier-Stokes in assuming a thin ocean. This dynamics depends on drag coefficients, which can only be estimated, and which will rule the dissipation of energy in the oceans.
Once the equations are written down, the solutions are decomposed as spectral modes, i.e. as sums of periodic contributions, which amplitudes and phases are calculated separately. This requires to model the shapes of the satellites as sums of spherical harmonics, i.e. as sums of ideal shapes, from the sphere to more and more distorted ones. And the shapes of the two boundaries of the ocean are estimated from the whole gravity of the body. As you may understand, I do not want to enter into specifics…
Let us go to the results instead.

The response of the oceans may be measured

The authors applied their model to Europa and Enceladus. They find that eccentricity tides give a higher amplitude of deformation, but the obliquity tides give a higher phase lag, because the the Rossby-Haurwitz waves, that the eccentricity tides do not produce. For instance, and here I cite the abstract of the paper If Europa’s shell and ocean are respectively 10 and 100 km thick, the tide amplitude and phase lag are 26.5 m and <1° for eccentricity forcing, and <2.5 m and <18° for obliquity forcing. The expected NASA mission Europa Clipper should be able to detect such effects. However, no space mission is currently planned for Enceladus.

I have a personal comment: for Mimas, a phase lag in libration of 6° has been measured. Could it be due an internal ocean? This probably requires a specific study.

The study and its authors

• You can find the study here. Moreover, the authors shared it on arXiv, so that you can get it for free. Many thanks to them! And now, the presentation of the authors:
• The homepage of Isamu Matsuyama,
• the one of Hamish Hay,
• the one of Francis Nimmo,
• and the one of Shunichi Kamata.

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.