Tag Archives: Enceladus

Heating the subsurface oceans

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.
Mosaic of Titan, due to Cassini. © NASA/JPL/University of Arizona/University of Idaho
Mosaic of Titan, due to Cassini. © NASA/JPL/University of Arizona/University of Idaho
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.

Ganymede seen by Galileo. © NASA / JPL / DLR
Ganymede seen by Galileo. © NASA / JPL / DLR
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.

Enceladus seen by Cassini. © NASA/JPL
Enceladus seen by Cassini. © NASA/JPL
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.

Dione seen by Cassini. © NASA
Dione seen by Cassini. © NASA
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.

Callisto seen by Galileo. © NASA
Callisto seen by Galileo. © NASA
Suspected: Pluto

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

Pluto seen by New Horizons. ©NASA/APL/SwRI
Pluto seen by New Horizons. ©NASA/APL/SwRI
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.

Mimas seen by Cassini. © NASA
Mimas seen by Cassini. © NASA

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

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.

Fracturing the crust of an icy satellite

Hi there! You may know that the space missions to the systems of giant planets have revealed that the surface of several of theirs satellites are fractured. We dispose of images of such structures on Jupiter’s Europa and Ganymede, Saturn’s Enceladus (the famous tiger stripes at its South Pole), and even on Uranus’ Miranda, which has been visited by Voyager II. These satellites are thought to be icy bodies, with an icy crust enshrouding a subsurface, global ocean (maybe not for Miranda, but certainly true for the other guys).

The study I present you today, Experimental constraints on the fatigue of icy satellite lithospheres by tidal forces, by Noah P. Hammond, Amy C. Barr, Reid F. Cooper, Tess E. Caswell, and Greg Hirth, has recently been accepted for publication in Journal of Geophysical Research: Planets. The authors particularly tried to produce in labs the process of fatigue, which would weaken a material after a certain number of solicitations, i.e. it would become easier to break.

Cycloids on Europa

The Galilean satellite of Jupiter Europa may be the most interesting satellite to focus on, since it is the most fractured, at least to the best of our knowledge. The observation of the surface of Europa, first by Voyager I and II in 1979, and after by Galileo between 1995 and 2003, revealed many structures, like lineae, i.e. cracks, due to the geophysical activity of the satellite. This body is so active that only few craters are visible, the surface having been intensively renewed since the impacts. Something particularly appealing on Europa is that some of these lineae present a cycloidal pattern, which would reveal a very small drift of the orientation of the surface. Some interpret it has an evidence of super-synchronous rotation of Europa, i.e. its rotation would not be exactly synchronous with its orbital motion around Jupiter.

Cycloids on Europa, seen by the spacecraft Galileo. © NASA
Cycloids on Europa, seen by the spacecraft Galileo. © NASA

Beside Europa, fractures have also been observed on Ganymede, but with less frequency. For having such fractures, you need the surface to be brittle enough, so that stress will fracture it. This is a way to indirectly detect a subsurface ocean. But you also need the stress. And this is where tides intervene.

Fractures on Ganymede. © Paul M. Schenk
Fractures on Ganymede. © Paul M. Schenk

Tides can stress the surface

You can imagine that Jupiter exerts a huge gravitational action on Europa. But Europa is not that small, and its finite size results in a difference of Jovian attraction between the point which is the closest to Jupiter, and the furthest one. The result of this differential attraction is stress and strain in the satellite. The response of the satellite will depend on its structure.

A problem is that calculations suggest that the tidal stress may be too weak to generate alone the observed fractures. This is why the authors suggest the assistance of another phenomenon: fatigue crack growth.

The phenomenon of fatigue crack growth

The picture is pretty intuitive: if you want to break something… let’s say a spoon. You twist it, bend it, wring it… once, twice, thrice, more… Pretty uneasy, but you do not give up, because you see that the material is weakening. And finally it breaks. Yes you did it! But what happened? You slowly created microcracks in the spoon, which weakened it, the cracks grew… until the spoon broke.

For geophysical materials, it works pretty much the same: we should imagine that the tides, which vary over an orbit since the eccentricity of the orbit induces variations of the Jupiter-Europa distance, slowly create microcracks, which then grow, until the cracks are visible. To test this scenario, the authors ran lab experiments.

Lab experiments

The lab experiments consisted of Brazil Tests, i.e. compression of circular disks of ice along their diameter between curved steel plates. The resulting stress was computed everywhere in the disk thanks to a finite-element software named Abaqus, and the result was analyzed with acoustic emissions, which reflections would reveal the presence of absence of microcracks in the disk. The authors ran two types of tests: both with cyclic loading, i.e. oscillating loading, but one with constant amplitude, and the other one with increasing amplitude, i.e. a maximum loading becoming stronger and stronger.

But wait: how to reproduce the conditions of the real ice of these satellites? Well, there are things you cannot do in the lab. Among the problems are: the exact composition of the ice, the temperature, and the excitation frequency.

The authors conducted the experiments in assuming pure water ice. The temperature could be below 150 K (-123°C, or -189°F), which is very challenging in a lab, and the main period of excitation is the orbital one, i.e. 3.5 day… If you want to reproduce 100,000 loading cycles, you should wait some 1,000 years… unfeasible…

The authors bypassed these two problems in constraining the product frequency times viscosity to be valid, the viscosity itself depending on the temperature. This resulted in an excitation period of 1 second, and temperatures between 198 and 233 K (-75 to -40°C, or -103 to -40°F). The temperature was maintained thanks to a liquid nitrogen-cooled, ethanol bath cryostat.

And now the results!

No fatigue observed

Indeed, the authors observed no fatigue, i.e. no significant microcracks were detected, which would have altered the material enough, to weaken it. This prompted the authors to discuss the application of their experiments for understanding the crust of the real satellites, and they argue that fatigue could be possible anyway.

Why fatigue may still be possible

As the authors recall, these experiments are not the first ones. Other authors have had a negative result with pure water ice. However, fatigue has been detected on sea ice, which could mean that the presence of salt favors fatigue. And the water ice of icy satellite may not be pure. Salt and other chemical elements may be present. So, even if these experiments did not reveal fatigue, there may be some anyway.

But the motivation for investigating fatigue is that a process was needed to assist the tides to crack the surface. Why necessarily fatigue? Actually, other processes may weaken the material.

How to fracture without fatigue

The explanation is like the most (just a matter of taste) is impacts: when you impact the surface, you break it, which necessarily weakens it. And we know that impacts are ubiquitous in the Solar System. In case of an impact, a megaregolith is created, which is more likely to get fractured. The authors also suggest that the tides may be assisted, at least for Europa, by the super-synchronous rotation possibly suggested by the geometry of the lineae (remember, the cycloids). Another possibility is the large scale inhomogeneities in the surface, which could weaken it at some points.

Anyway, it is a fact that these surfaces are fractured, and the exact explanation for that is still in debate!

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.

Spatial variations of Enceladus’ plumes

Hi there! I guess most of you have heard of Enceladus. This mid-sized icy satellite of Saturn arouses the interest of planetologists, because of its geological activity. Permanent eruptions of plumes, essentially made of water ice, have been detected at its South Pole, by the Cassini spacecraft. The study I present you today, Spatial variations in the dust-to-gas ratio of Enceladus’ plume, by M.M. Hedman, D. Dhingra, P.D. Nicholson, C.J. Hansen, G. Portyankina, S. Ye and Y. Dong, has recently been published in Icarus.

The South Pole of Enceladus

Enceladus orbits around Saturn in one day and 9 hours, at a mean distance of 238,000 km. It is the second of the mid-sized satellites of Saturn by its distance from the planet, and is in an orbital 2:1 resonance with Dione, i.e. Dione makes exactly one revolution around Saturn while Enceladus makes 2. This results in a slight forcing of its orbital eccentricity, which remains anyway modest, i.e. 0.005. Like our Moon and many satellites of the giant planets, Enceladus rotates synchronously.

Interestingly, the Cassini spacecraft detected geysers at the South Pole of Enceladus, and fractures, which were nicknamed tiger stripes. They were named after 4 Middle East cities: Alexandria, Cairo, Baghdad, and Damascus.

The South Pole of Enceladus. We can see from left to right the famous tiger stripes, i.e. Alexandria, Cairo, Baghdad and Damascus sulci. © NASA/JPL/Space Science Institute/DLR
The South Pole of Enceladus. We can see from left to right the famous tiger stripes, i.e. Alexandria, Cairo, Baghdad and Damascus sulci. © NASA/JPL/Space Science Institute/DLR

These 4 fractures are 2km-large and 500m-deep depressions, which extend up to 130 km. The plumes emerge from them. Interior models suggest that the source of these geysers is a diapir of water, located at the South Pole.

Analysis of these plumes require them to be illuminated, and observed with spectroscopic devices. This is where the instruments UVIS and VIMS get involved.

The instruments UVIS & VIMS of Cassini

The study I present you today presents an analysis of VIMS data, before comparing the results of the same event given by UVIS.

UVIS and VIMS are two instruments of the Cassini mission, which completed a 13-years tour in the system of Saturn in September 2017 with its Grand Finale, crashing in the atmosphere of Saturn. It was accompanied by the lander Huygens, which landed on Titan in 2005, and had 12 instruments on board. Among them were UVIS and VIMS.

And then, you wonder, dear reader, whether I will introduce you UVIS and VIMS, since I mention them since the beginning without introducing them. Yes, this is now.

UVIS stands for Ultraviolet Imaging Spectrograph, and VIMS for Visible and Infrared Mapping Spectrometer. Their functions are in their names: both analyze the incoming light, UVIS in the ultraviolet spectrum, and VIMS in the visible and infrared ones. And the combination of these two spectra is relevant in this study: the analysis in the ultraviolet tells you one thing (quantity of gas), while the analysis in the infrared gives the quantity of dust. When you compare them, you have the dust-to-gas ratio. Of course, this is not that straightforward. First you have to collect the data.

Analyzing a Solar occultation by the plumes

As I said, the plume needs to be illuminated. And for that, you have to position the spacecraft where the plumes occult the Sun. So, this could happen only during a fly-by of Enceladus, which means that it was impossible to have a permanent monitoring of these plumes. Moreover, from the geometry of the configuration, i.e. location of the plume, of the Sun, of the spacecraft,… you had the data at a given altitude. It is easy to figure out that the water is more volatile than dust, is ejected faster, and higher… In other words, the higher is the observation, the lower the dust-to-gas ratio.

The studied occultation happened on May 18, 2010, and lasted approximately 70 seconds, during which the illuminated plumes originated from different tiger stripes. This means that a temporal variation of the composition of plumes during the event means a spatial variation in the subsurface of the South Pole. The altitude was 20-30 km.

But detecting a composition is a tough task. Actually the UVIS data, i.e. detection of water, were published in 2011, and the VIMS ones (detection of dust) only in 2018, probably because the signal is very weak. The authors observed a Solar spectrum in the infrared, and at the exact date of the occultation, a slight flux drop occurred, which was the signature of a dusty plume. For it to be exploitable, the authors had to treat the signal, i.e. de-noise it.

After this treatment, the resulting signal was an optical depth in 256 spectral channels between 0.85 an 5.2 microns. You then need to compare it with a theoretical model of diffraction by micrometric particles, the Mie diffraction, to have an idea of the particle-size distribution. Because the particles do not all have the same size, of course! Actually, the distribution is close to a power law of index 4.

Spatial variations detected

And here is the results: at an altitude of around 25 km, the authors have found that the material emerging from Baghdad and Damascus are up to one order of magnitude, i.e. 10 times, more particle-rich than the ones emerging from Alexandria and Cairo sulci.

It is not straightforward to draw conclusions from this single event. Once more, a permanent monitoring of the plumes was impossible. Spatial variations of the dust-to-gas ratio at a given altitude could either mean something on the variations of the dust-to-gas ratio in the subsurface diapir, and/or something on the spatial variations of the ejection velocities of dust and gas. Once more, the ratio is expected to decline with the altitude, since the water is more volatile.

We dispose of data from other events, for instance a fly-by, named E7, which occurred in November 2009, of the South Pole at an altitude of 100 km, during which the Ion and Neutral Mass Spectrometer (INMS) analyzed the plumes. The data are pretty consistent with the ones presented here, but the altitude is very different, so be careful.

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.

Enceladus lost its balance

Hi there! Today I will present you True polar wander of Enceladus from topographic data, by Tajeddine et al., which has recently been published in Icarus. The idea is this: Enceladus is a satellite of Saturn which has a pretty stable rotation axis. In the past, its rotation axis was already stable, but with a dramatically different orientation, i.e. 55° shifted from the present one! The authors proposed this scenario after having observed the distribution of impact basins at its surface.

Enceladus’s facts

Enceladus is one of the mid-sized satellites of Saturn, it is actually the second innermost of them. It has a mean radius of some 250 km, and orbits around Saturn in 1.37 day, at a distance of ~238,000 km. It is particularly interesting since it presents evidence of past and present geophysical activity. In particular, geysers have been observed by the Cassini spacecraft at its South Pole, and its southern hemisphere presents four pretty linear features known as tiger stripes, which are fractures.

Enceladus seen by Cassini (Credit: NASA / JPL / Space Science Institute).
Enceladus seen by Cassini (Credit: NASA / JPL / Space Science Institute).

Moreover, analyses of the gravity field of Enceladus, which is a signature of its interior, strongly suggest a global, subsurfacic ocean, and a North-South asymmetry. This asymmetry is consistent with a diapir of water at its South Pole, which would be the origin of the geysers. The presence of the global ocean has been confirmed by measurements of the amplitude of the longitudinal librations of its surface, which are consistent with a a crust, that a global ocean would have partially decoupled from the interior.

The rotation of a planetary satellite

Planetary satellites have a particularly interesting rotational dynamics. Alike our Moon, they show on average always the same face to a fictitious observer, which would observe the satellite from the surface of the parent planet (our Earth for the Moon, Saturn for Enceladus). This means that they have a synchronous rotation, i.e. a rotation which is synchronous with their orbit, but also that the orientation of their spin axis is pretty stable.
And this is the key point here: the spin axis is pretty orthogonal to the orbit (this orientation is called Cassini State 1), and it is very close to the polar axis, which is the axis of largest moment of inertia. This means that we have a condition on the orientation of the spin axis with respect to the orbit, AND with respect to the surface. The mass distribution in the satellite is not exactly spherical, actually masses tend to accumulate in the equatorial plane, more particularly in the satellite-planet direction, because of the combined actions of the rotation of the satellites and the tides raised by the parent planet. This implies a shorter polar axis. And the study I present today proposes that the polar axis has been tilted of 55° in the past. This tilt is called polar wander. This result is suggested by the distribution of the craters at the surface of Enceladus.

Relaxing a crater

The Solar System bodies are always impacted, this was especially true during the early ages of the Solar System. And the inner satellites of Saturn were more impacted than the outer ones, because the mass of Saturn tends to attract the impactors, focusing their trajectories.
As a consequence, Enceladus got heavily impacted, probably pretty homogeneously, i.e. craters were everywhere. And then, over the ages, the crust slowly went back to its original shape, relaxing the craters. The craters became then basins, and eventually some of them disappeared. Some of them, but not all of them.
The process of relaxation is all the more efficient when the material is hot. For material which properties strongly depend on the temperature, a stagnant lid can form below the surface, which would partly preserve it from the heating by convection, and could preserve the craters. This phenomenon appears preferably at equatorial latitudes.
This motivates the quest for basins. A way for that is to measure the topography of the surface.

Modeling the topography

The surface of planetary body can be written as a sum of trigonometric series, known as spherical harmonics, in which the radius would depend on 2 parameters, i.e. the latitude and the longitude. This way, you have the radius at any point of the surface. Classically, two terms are kept, which allow to represent the surface as a triaxial ellipsoid. This is the expected shape from the rotational and tidal deformations. If you want to look at mass anomalies, then you have to go further in the expansion of the formula. But to do that, you need data, i.e. measurements of the radius at given coordinates. And for that, the planetologists dispose of the Cassini spacecraft, which made several flybys of Enceladus, since 2005.
Two kinds of data have been used in this study: limb profiles, and control points.
Limb profiles are observations of the bright edge of an illuminated object, they result in very accurate measurements of limited areas. Control points are features on the surface, detected from images. They can be anywhere of the surface, and permit a global coverage. In this study, the authors used 41,780 points derived from 54 limb profiles, and 6,245 control points.
Measuring the shape is only one example of use of such data. They can also be used to measure the rotation of the body, in comparing several orientations of given features at different dates.
These data permitted the authors to model the topography up to the order 16.

The result

The authors identified a set of pretty aligned basins, which would happen for equatorial basins protected from relaxation by stagnant lid convection. But the problem is this: the orientation of this alignment would need a tilt of 55° of Enceladus to be equatorial! This is why the authors suggest that Enceladus has been tilted in the past.

The observations do not tell us anything on the cause of this tilt. Some blogs emphasize that it could be due to an impact. Why not? But less us be cautious.
Anyway, the orientation of the rotation axis is consistent with the current mass distribution, i.e. the polar axis has the largest moment of inertia. Actually, mid-sized planetary satellites like Enceladus are close to sphericity, in the sense that there is no huge difference between the moments of inertia of its principal axes. So, a redistribution of mass after a violent tilt seems to be possible.

To know more

And now the 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 and Facebook.

Resonances around the giant planets

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-2λBA (order 1), which would force eA,
  • λA-2λBB (order 1), which would force eB,
  • A-4λBAB (order 2), which would force eA and eB,
  • A-4λB+2ΩA (order 2), which would force iA,
  • A-4λB+2ΩB (order 2), which would force iB,
  • A-4λB+2ΩAB (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 1-4λ313 0 95° i1,i3
S2 Enceladus S4 Dione 2:1 λ2-2λ42 0 0.25° e2
S6 Titan S7 Hyperion 4:3 6-4λ77 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!.