Tag Archives: tides

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.

Tides in the lakes of Titan

Hi there! The satellite of Saturn Titan has hydrocarbon seas, i.e. lakes made of liquid ethane and methane. When you have a sea, or a lake, you may have tides, and this is what this study is about. I present you A numerical study of tides in Titan’s northern seas, Kraken and Ligeia Maria, by David Vincent, Özgür Karatekin, Jonathan Lambrechts, Ralph D. Lorenz, Véronique Dehant, and Éric Deleersnijder, which has recently been accepted for publication in Icarus.

The lakes of Titan

The presence of hydrocarbons in such a thick atmosphere as the one of Titan has suggested since the spacecraft Voyager 1 than methane and ethane could exist in the liquid state on the surface of Titan. There could even be a cycle of methane, as there is a hydrological cycle on Earth, in which the liquid methane on the surface feeds the clouds of gaseous methane in the atmosphere, and conversely.

The spacecraft Cassini has detected dark smooth features, which revealed to be these hydrocarbon seas. Here is a list of the largest ones:

Location Diameter
Kraken Mare 68.0°N 310.0°W 1,170 km
Ligeia Mare 79.0°N 248.0°W 500 km
Punga Mare 85.1°N 339.7°W 380 km
Jingpo Lacus 73.0°N 336.0°W 240 km
Ontario Lacus 72.0°S 183.0°W 235 km
Mackay Lacus 78.32°N 97.53°W 180 km
Bolsena Lacus 75.75°N 10.28°W 101 km

I present you only the detected lakes with a diameter larger than 100 km, but some have been detected with a diameter as small as 6 km. It appears that these lakes are located at high latitudes, i.e. in the polar regions. Moreover, there is an obvious North-South asymmetry, i.e. there are much more lakes in the Northern hemisphere than in the Southern one. This could be due to the circulation of clouds of Titan: they would form near the equator, from the evaporation of liquid hydrocarbons, and migrate to the poles, where they would precipitate (i.e. rain) into lakes. Let us now focus on the largest two seas, i.e. Kraken and Ligeia Maria.

Kraken and Ligeia Maria

Kraken and Ligeia Maria are two adjacent seas, which are connected by a strait, named Trevize Fretum, which permit liquid exchanges. Kraken is composed of two basins, named Kraken 1 (north) and Kraken 2 (south), which are connected by a strait named Seldon Fretum, which dimensions are similar to the strait of Gibraltar, between Morocco and Spain.

Kraken and Ligeia Maria. © NASA
Kraken and Ligeia Maria. © NASA

Alike the Moon and Sun which raise tides on our seas, Saturn raises tides on the lakes. These tides cannot be measured yet, but they can be simulated, and this is what the authors did. In a previous study, they had simulated the tides on Ontario Lacus.

They honestly admit that the tides on Kraken and Ligeia Maria have already been simulated by other authors. Here, they use a more efficient technique, i.e. which uses less computational resources, and get consistent results.

Numerical modeling with SLIM

Computational fluid dynamics, often referred as CFD, is far from an easy task. The reason is that the dynamics of fluids in ruled by non-linear partial derivative equations like the famous Navier-Stokes, i.e. equations which depend on several variables, like the time, the temperature, the location (i.e. where are you exactly on the lake?), etc. Moreover, they depend on several parameters, some of them being barely constrained. We accurately know the gravitational tidal torque due to Saturn, however we have many uncertainties on the elasticity of the crust of Titan, on the geometry of the coast, on the bathymetry, i.e. the bottom of the seas. So, several sets of parameters have to be considered, for which numerical simulations should be run.

It is classical to use a finite element method for problems of CFD (Computational Fluid Dynamics, remember?). This consists to model the seas not as continuous domains, but as a mesh of finite elements, here triangular, on which the equations are defined.
The structure of the mesh is critical. A first, maybe intuitive, approach would be to consider finite elements of equal size, but it appears that this way of integrating the equations is computationally expensive and could be optimized. Actually, the behavior of the fluid is very sensitive to the location close to the coasts, but much less in the middle of the seas. In other words, the mesh needs to be tighter at the coasts. The authors built an appropriate mesh, which is unstructured and follow the so-called Galerkin method, which adapts the mesh to the equations.

The authors then integrated the equations with their homemade SLIM software, for Second-generation Louvain-la-Neuve Ice-ocean Model. The city of Louvain-la-Neuve hosts the French speaking Belgian University Université Catholique de Louvain, where most of this study has been conducted. The model SLIM has been originally built for hydrology, to model the behavior of fluids on Earth, and its simulations have been successfully confronted to terrain measurements. It thus makes sense to use it for modeling the behavior of liquid hydrocarbons on Titan.

In this study, the authors used the 2-dimensional shallow water equations, which are depth-integrated. In other words, they directly simulated the surface rather than the whole volume of the seas, which of course requires much less computation time.
Let us now see their results.

Low diurnal tides

The authors simulated the tides over 150 Titan days. A Titan day is 15.95 days long, which is the orbital period of Titan around Saturn. During this period, the distance Titan-Saturn varies between 1,186,680 and 1,257,060 km because the orbit of Titan is eccentric, and so does the intensity of the tidal torque. This intensity also varies because of the obliquity of Titan, i.e. the tilt of its rotation axis, which is 0.3°. Because of these two quantities, we have a period of variation of 15.95 days, and its harmonics, i.e. half the period, a third of the period, etc.

It appeared from the simulations that the 15.95-d response is by far the dominant one, except at some specific locations where the tides cancel out (amphidromic points). The highest tides are 0.29 m and 0.14 m in Kraken and Ligeia, respectively.

Higher responses could have been expected in case of resonances between eigenmodes of the fluids, i.e. natural frequencies of oscillations, and the excitation frequencies due to the gravitational action of Saturn. It actually appeared that the eigenmodes, which have been computed by SLIM, have much shorter periods than the Titan day, which prevents any significant resonance. The author did not consider the whole motion of Titan around Saturn, in particular the neglected planetary perturbations, which would have induced additional exciting modes. Anyway, the corresponding periods would have been much longer than the Titan day, and would not have excited any resonance. They would just have given the annual variations of tides, with a period of 29.4 years, which is the orbital period of Saturn around the Sun.

Fluid exchanges between the lakes

SLIM permits to trace fluid particles, which reveals the fluid exchanges between the basins. Because of their narrow geometry, the straits are places where the currents are the strongest, i.e. 0.3 m/s in Seldon Fretum.
The volumetric exchanges are 3 times stronger between Kraken 1 and Kraken 2 than between Kraken and Ligeia. These exchanges behave as an oscillator, i.e. they are periodic with respect to the Titan day. As a consequence, there is a strong correlation between the volume of Kraken 1, and the one of Kraken 2. Anyway, these exchanges are weak with respect to the volume of the basins.

The attenuation is critical

The authors studied the influence of the response with respect to different parameters: the bathymetry of the seas (i.e., the geometry of the bottom), the influence of bottom friction, the depth of Trevize Fretum, and the attenuation factor γ2, which represents the viscoelastic response of the surface of Titan to the tidal excitation. It appears that γ2 plays a key role. Actually, the maximum tidal range is an increasing function of the attenuation, and in Seldon and Trevize Fretum, the maximum velocities behave as a square root of γ2. It thus affects the fluid exchanges. Moreover, these exchanges are also affected by the depth of Trevize Fretum, which is barely constrained.

Another mission to Titan is needed to better constrain these parameters!

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 and Facebook. And let me wish you a healthy and happy year 2018.

Water-ice boundary on Titan

Hi there! Titan may be the most famous satellite in the Solar System, I realize that I never devoted a post to it. It is high time to make it so. I present you Does Titan’s long-wavelength topography contain information about subsurface ocean dynamics? by Jakub Kvorka, Ondřej Čadek, Gabriel Tobie & Gaël Choblet, which has recently been accepted for publication in Icarus. This paper tries to understand the mechanisms responsible for the location of the boundary between the icy crust and the subsurface ocean. This affects the thickness of the crust, which itself affects the topography of Titan.


The existence of Titan is known since 1655 thanks to the Dutch astronomer Christiaan Huygens. It was the only known satellite of Saturn until the discovery of Iapetus in 1671. It is the second largest natural satellite of the Solar System (mean radius: 2,575 km), and it orbits Saturn in almost 16 days, on a 3% eccentric and almost equatorial orbit (actually, a small tilt is due to the gravitational influence of the Sun).

It has interesting physical characteristics:

  • A thick atmosphere (pressure at the surface: 1.5 bar) mainly composed of nitrogen, with clouds of methane and ethane.
  • A complex surface. We can find hydrocarbon seas, i.e. lakes of methane and ethane (Kraken Mare, Ontario Lacus…), we also have a mountain chain, which features have been named after Tolkien’s Lords of the Rings (Gandalf Colles, Erebor Mons,…). There are some impact craters as well, but not that many, which suggests a geologically young surface. There is probably cryovolcanism on Titan, i.e. eruptions of volatile elements. The surface and the atmosphere interact, i.e. there are exchange between the liquid methane and ethane of the lakes and the gaseous ones present in the atmosphere, and the atmosphere is responsible for erosion of the surface, for winds which are likely to create dunes, and for heat exchanges.
  • A global subsurface ocean, lying under the icy crust.
Map of Titan.
Map of Titan.

The quest for the internal ocean

An internal, water ocean is considered to be of high interest for habitability, i.e. we cannot exclude the presence of bacteriological life in such an environment. This makes Titan one of the priority targets for future investigations.

The presence of the ocean was hinted long ago, from the consideration that, at some depth, the water ice covering the surface would be in such conditions of temperature and pressure that it should not be solid anymore, but liquid. The detection of this ocean has been hoped from the Cassini-Huygens mission, and this was a success. More precisely:

  • The rotation of the surface of Titan is synchronous, i.e. Titan shows on average the same face to Saturn, like our Moon, but with a significant obliquity (0.3°), which could reveal the presence of a global ocean which would decouple the rotation of the crust from the one of the core.
  • A Schumann resonance, i.e. an electromagnetic signal, has been detected by the lander Huygens in the atmosphere of Titan, during its fall. This could be due to an excitation of a magnetic field by a global conductive layer, i.e. a global subsurface ocean.
  • The gravitational Love number k2, which gives the amplitude of the response of the gravity field of Titan to the variations of the gravitational attraction of Saturn, is too large to be explained by a fully solid Titan.

All of these clues have convinced almost all of the scientific community that Titan has a global subsurface ocean. Determining its depth, thickness, composition,… is another story. In the study I present you today, the authors tried to elucidate the connection between its depth and the surface topography.

Modeling the ice-water boundary

The authors tried to determine the depth of the melting point of the water ice with respect to the latitude and longitude. This phase boundary is the thickness of the icy crust. For that, they wrote down the equations governing the viscoelastic deformation of the crust, its thermal evolution, and the motion of the boundary.

The viscoelastic deformation, i.e. deformation with dissipation, is due to the varying tidal action of Saturn, and the response depends on the properties of the material, i.e. rigidity, viscosity… The law ruling the behavior of the ice is here the Andrade law… basically it is a Maxwell rheology at low frequencies, i.e. elastic behavior for very slow deformations, viscoelastic behavior when the deformations gets faster… and for very fast excitation frequencies (tidal frequencies), the Maxwell model, which is based on one parameter (the Maxwell time, which gives an idea of the period of excitation at the transition between elastic and viscoelastic behavior), underestimates the dissipation. This is where the more complex Andrade model is useful. The excitation frequencies are taken in the variations of the distance Titan-Saturn, which are ruled by the gravitational perturbations of the Sun, of the rings, of the other satellites…

These deformations and excitations are responsible for variations of the temperature, which are also ruled by physical properties of the material (density, thermal conductivity), and which will determine whether the water should be solid or liquid. As a consequence, they will induce a motion of the phase change boundary.

Resolution by spectral decomposition

The equations ruling the variables of the problem are complex, in particular because they are coupled. Moreover, we should not forget that the density, thickness, temperature, resulting heat flows… not only depend on time, but also on where you are on the surface of Titan, i.e. the latitude and the longitude. To consider the couplings between the different surface elements, the authors did not use a finite-element modeling, but a spectral method instead.

The idea is to consider that the deformation of the crust is the sum of periodic deformations, with respect to the longitude and latitude. The basic shape is a sphere (order 0). If you want to be a little more accurate, you say that Titan is triaxial (order 2). And if you want to be more accurate, you introduce higher orders, which would induce bulges at non equatorial latitudes, north-south asymmetries for odd orders, etc. It is classical to decompose the tidal potential under a spectral form, and the authors succeeded to solve the equations of the problem in writing down the variables as sums of spherical harmonics.

The role of the grain size

And the main result is that the grain size of the ice plays a major role. In particular, the comparison between the resulting topography and the one measured by the Cassini mission up to the 3rd order shows that we need grains larger than 10 mm to be consistent with the observations. The authors reached an equilibrium in at the most 10 Myr, i.e. the crust is shaped in a few million years. They also addressed the influence of other parameters, like the rigidity of the ice, but with much less significant outcomes. Basically, the location of the melting / crystallization boundary is ruled by the grain size.

In the future

Every new study is another step forward. Others will follow. At least two directions can be expected.

Refinements of the theory

The authors honestly admit that the presence of other compounds in the ocean, like ammonia, is not considered here. Adding such compounds could affect the behavior of the ocean and the phase boundary. This would require at least one additional parameter, i.e. the fraction of ammonia. But the methodology presented here would still be valid, and additional studies would be incremental improvements of this one.
A possible implication of these results is the ocean dynamics, which is pretty difficult to model.

More data?

Another step forward could come from new data. Recently the mission proposal Dragonfly has been selected as a finalist by the NASA’s New Frontiers program. It would be a rotorcraft lander on Titan. Being selected as a finalist is a financial encouragement to refine and mature the concept within the year 2018, before final decision in July 2019 (see video below).

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 and Facebook.

On the interior of Mimas, aka the Death Star

Hi there! Today I will tell you on the interior of Mimas. You know, Mimas, this pretty small, actually the smallest of the mid-sized, satellite of Saturn, which has a big crater, like Star Wars’ Death Star. Despite an inactive appearance, it presents confusing orbital quantities, which could suggest interesting characteristics. This is the topic of the study I present you today, by Marc Neveu and Alyssa Rhoden, entitled The origin and evolution of a differentiated Mimas, which has recently been published in Icarus.

Mimas’ facts

The system of Saturn is composed of different groups of satellites. You have

  • Very small satellites embedded into the rings,
  • Mid-sized satellites orbiting between the rings and the orbit of Titan
  • The well-known Titan, which is very large,
  • Small irregular satellites, which orbit very far from Saturn and are probably former asteroids, which had been trapped by Saturn,
  • Others (to make sure I do not forget anybody, including the coorbital satellites of Tethys and Dione, Hyperion, the Alkyonides, Phoebe…).

Discovered in 1789 by William Herschel, Mimas is the innermost of the mid-sized satellites of Saturn. It orbits it in less than one day, and has strong interactions with the rings.

Semimajor axis 185,520 km
Eccentricity 0.0196
Inclination 1.57°
Diameter 396.4 km
Orbital period 22 h 36 min

As we can see, Mimas has a significant eccentricity and a significant inclination. This inclination could be explained by a mean-motion resonance with Tethys (see here). However, we see no obvious cause for its present eccentricity. It could be due to a past gravitational excitation by another satellite.

Mimas, seen by Cassini. We can the crater Herschel, which makes Mimas look like Star Wars' Death Star. Credit: NASA
Mimas, seen by Cassini. We can the crater Herschel, which makes Mimas look like Star Wars’ Death Star. Credit: NASA

The literature is not unanimous on the formation of Mimas. It was long thought that the satellites of Saturn formed simultaneously with the planet and the rings, in the proto-Saturn nebula. The Cassini space mission changed our view of this system, and other scenarios were proposed. For instance, the mid-sized satellites of Saturn could form from the collisions between 4 big progenitors, Titan being the last survivor of them. The most popular explanation seems to be that a very large body impacted Saturn, its debris coalesced into the rings, and then particles in the rings accreted, forming satellites which then migrated outward… these satellites being the mid-sized satellites, i.e. Rhea, Dione, Tethys, Enceladus, and Mimas. This scenario would mean that Mimas would be the youngest of them, and that it formed differentiated, i.e. that the proto-Mimas was made of pretty heavy elements, on which lighter elements accreted. Combining observations of Mimas with theoretical studies of its long-term evolution could help to determine which of these scenarios is the right one… if there is a right one. Such studies can of course involve other satellites, but this one is essentially on Mimas, with a discussion on Enceladus at the end.

The rotation of Mimas

As most of the natural satellites of the giant planets, Mimas is synchronous, i.e. it shows the same face to Saturn, its rotational (spin) period being on average equal to its orbital one. “On average” means that there are some variations. These are actually a sum of periodic oscillations, which are due to the variations of the distance Mimas-Saturn. And from the amplitude and phase of these variations, you can deduce something on the interior, i.e. how the mass is distributed. This could for instance reveal an internal ocean, or something else…

This rotation has been measured in 2014 (see this press release). The mean rotation is indeed synchronous, and here are its oscillations:

Period Measured
amplitude (arcmin)
amplitude (arcmin)
70.56 y 2,616.6 2,631.6±3.0
23.52 y 43.26 44.5±1.1
22.4 h 26.07 50.3±1.0
225.04 d 7.82 7.5±0.8
227.02 d 3.65 2.9±0.9
223.09 d 3.53 3.3±0.8

The most striking discrepancy is at the period 22.4 h, which is the orbital period of Mimas. These oscillations are named diurnal librations, and their amplitude is very sensitive to the interior. Moreover, the amplitude associated is twice the predicted one. This means that the interior, which was hypothesized for the theoretical study, is not a right one, and this detection of an error is a scientific information. It means that Mimas is not exactly how we believed it is.

The authors of the 2014 study, led by Radwan Tajeddine, investigated 5 interior models, which could explain this high amplitude. One of these models considered the influence of the large impact crater Herschel. In all of these models, only 2 could explain this high amplitude: either an internal ocean, or an elongated core of pretty heavy elements. Herschel is not responsible for anything in this amplitude.

The presence of an elongated core would support the formation from the rings. However, the internal ocean would need a source of heating to survive.

Heating Mimas

There are at least three main to heat a planetary body:

  1. hit it to heat it, i.e. an impact could partly melt Mimas, but that would be a very intense and short heating, which would have renewed the surface…nope
  2. decay of radiogenic elements. This would require Mimas to be young enough
  3. tides: i.e. internal friction due to the differential attraction of Saturn. This would be enforced by the variations of the distance Saturn-Mimas, i.e. the eccentricity.

And this is how we arrive to the study: the authors simulated the evolution of the composition of Mimas under radiogenic and tidal heating, in also considering the variations of the orbital elements. Because when a satellite heats, its eccentricity diminishes. Its semimajor axis varies as well, balanced between the dissipation in the satellite and the one in Saturn.

The problems

For a study to be trusted by the scientific community, it should reproduce the observations. This means that the resulting Mimas should be the Mimas we observe. The authors gave themselves 3 observational constraints, i.e. Mimas must

  1. have the right orbital eccentricity,
  2. have the right amplitude of diurnal librations,
  3. keep a cold surface.

and they modeled the time evolution of the structure and the orbital elements using a numerical code, IcyDwarf, which simulates the evolution of the differentiation, i.e. separation between rock and water, porosity, heating, freezing of the ocean if it exists…


The authors show that in any case, the ocean cannot survive. If there would be a source of heating sustaining it, then the eccentricity of Mimas would have damped. In other words, you cannot have the ocean and the eccentricity simultaneously. Depending on the past (unknown) eccentricity of Mimas and the dissipation in Saturn, which is barely known, an ocean could have existed, but not anymore.
As a consequence, Mimas must have an elongated core, coated by an icy shell. The eccentricity could be sustained by the interaction with Saturn. This elongated core could have two origins: either a very early formation of Mimas, which would have given enough time for the differentiation, or a formation from the rings, which would have formed Mimas differentiated.

Finally the authors say that there model does not explain the internal ocean of Enceladus, but Marc Neveu announces on his blog that they have found another explanation, which should be published pretty soon. Stay tuned!

Another mystery

The 2014 study measured a phase shift of 6° in the diurnal librations. This is barely mentioned in the literature, probably because it bothers many people… This is huge, and could be more easily, or less hardly, explained with an internal ocean. I do not mean that Mimas has an internal ocean, because the doubts regarding its survival persist. So, this does not put the conclusions of the authors into question. Anyway, if one day an explanation would be given for this phase lag, that would be warmly welcome!

To know more…

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.

Measuring the tides of Mercury

Hi there! I have already told you about the tides. If you follow me, you know that the tides are the deformations of a planet from the gravitational action of its parent star (the Sun for Mercury), and that a good way to detect them is to measure the variations of the gravity field of a planet from the deviations of a spacecraft orbiting it. From periodic variations we should infer a coefficient k2, known as the potential Love number, which represents the response of the planet to the tides…

That’s all for today! Please feel free to comment… blablabla…

Just kidding!

Today, I will tell you about another way to measure the tides, from the rotation of Mercury. For this, I will present you a study entitled Periodic and quasi-periodic attractors for the spin-orbit evolution of Mercury with a realistic tidal torque, which was recently published in The Monthly Notices of the Royal Astronomical Society. This is a collaboration between English and Italian mathematicians, i.e. Michele Bartuccelli, Jonathan Deane, and Guido Gentile. In planetary sciences mathematics can lead to new discoveries. In this case, the idea is: tides slow down the rotation of a planetary body, which eventually reaches an equilibrium rotation (or spin). For the Moon, the equilibrium is the synchronous rotation, while for Mercury it is the 3:2 spin-orbit resonance. Very well. A very good way to describe this final state is to describe the equilibrium rotation, i.e. in considering that the tides do not affect the spin anymore. But this is just an approximation. The tides are actually still active, and they affect the final state. In considering it, the authors show that the variations of the spin rate of Mercury should be composed of at least two sinusoids, i.e. two periodic effects, the superimposition of these two periods being quasi-periodic… you now understand the title.

The rotation of Mercury

I have already presented you Mercury here. Mercury is the innermost planet of the Solar System, with a semimajor axis which is about one third of the one of the Earth, i.e. some 58 million km, and a surprisingly large orbital eccentricity, which is 0.206. These two elements favor a spin-orbit resonance, i.e. the rotation rate of Mercury is commensurate with its orbital rate. Their ratio is 3/2, Mercury performing a revolution about the Sun in 88 days, while a rotation period is 58 days. You can notice a 3/2 ratio between these two numbers.

The 3:2 spin-orbit resonance of Mercury
The 3:2 spin-orbit resonance of Mercury

Why is this configuration possible as an equilibrium state? If you neglect the dissipation (the authors do not) and the obliquity (the authors do, and they are probably right to do it), you can write down a second-degree ODE (ordinary differential equation), which rules the spin. In this equation, the triaxiality of Mercury plays a major role, i.e. Mercury spins the way it spins because it is triaxial. Another reason is its orbital eccentricity. This ODE has equilibriums, i.e. stable spin rates, among them is the 3:2 spin-orbit resonance.

And what about the obliquity? It is actually an equilibrium as well, known as Cassini State 1, in which the angular momentum of Mercury is tilted from the normal to its orbit by 2 arcminutes. This tilt is a response to the slow precessing motion (period: 300,000 years) of the orbit of Mercury around the Sun.

Let us forget the obliquity. There are several possible spin-orbit ratios for Mercury.

Possible rotation states

If you went back to the ODE which rules the spin-rate of Mercury, you would see that there are actually several equilibrium spin rates, which correspond to p/2 spin-orbit resonances, p being an integer. Among them are the famous synchronous resonance 1:1 (p=2), the present resonance of Mercury (p=3), and other ones, which have never been observed yet.

If we imagine that Mercury initially rotated pretty fast, then it slowed down, and crossed several resonances, e.g. the 4:1, the 7:2, 3:1, 5:2, 2:1… and was trapped in none of them, before eventually being trapped in the present 3:2 one. Or we can imagine that Mercury has been trapped for instance in the 2:1 resonance, and that something (an impact?) destabilized the resonance…
And what if Mercury had been initially retrograde? Why not? Venus is retrograde… In that case, the tides would have accelerated Mercury, which would have been trapped in the synchronous resonance, which is the strongest one. This would mean that this synchronous resonance would have been destabilized, to allow trapping into the 3:2 resonance. Any worthwhile scenario of the spin evolution of Mercury must end up in the 3:2 resonance, since it is the current state. The scenario of an initially retrograde Mercury has been proposed to explain the hemispheric repartition of the observed impacts, which could be a signature of a past synchronous rotation. Could be, but is not necessarily. Another explanation is that the geophysical activity of Mercury would have renewed the surface of only one hemisphere, making the craters visible only on the other part.

Anyway, whatever the past of Mercury, it needed a dissipative process to end up in an equilibrium state. This dissipative process is the tides, assisted or not by core-mantle friction.

The tides

Because of the differential attraction of the Sun on Mercury, you have internal friction, i.e. stress and strains, which dissipate energy, and slow down the rotation. This dissipation is enforced by the orbital eccentricity (0.206), which induces periodic variations of the Sun-Mercury distance.
An interesting question is: how does the material constituting Mercury react to the tides? A critical parameter is the tidal frequency, i.e. the way you dissipates depends on the frequency you shake. A derivation of the tidal torque raised by the Sun proves to be a sum of periodic excitations, one of them being dominant in the vicinity of a resonance. This results in an enforcement of all the spin-orbit resonances, which means that a proper tidal model is critical for accurate simulations of the spin evolution.
A pretty common way to model the tides is the Maxwell model: you define a Maxwell time, which is to be compared with the period of the tidal excitation (the shaking). If your excitation is slow enough, then you will have an elastic deformation, i.e. Mercury will have the ability to recover its shape without loss of energy. However, a more rapid excitation will be dissipative. Then this model can be improved, or refined, in considering more dissipation at high frequencies (Andrade model), or grain-boundary slip (Burgers model)… There are several models in the literature, which are supported by theoretical considerations and lab experiments. Choosing the appropriate one depends on the material you consider, under which conditions, i.e. pressure and temperature, and the excitation frequencies. But in any case, these physically realistic tidal models will enforce the spin-orbit resonances.

Considering only the tides assumes that your body is (almost) homogeneous. Mercury has actually an at least partially molten outer core, i.e. a global fluid layer somewhere in its interior. This induces fluid-solid boundaries, the outer one being called CMB, for core-mantle boundary, and you can have friction there. The authors assumed that the CMB was formed after the trapping of Mercury into its present 3:2 spin-orbit resonance, which is supported by some studies. This is why they neglected the core-mantle friction.

This paper

This paper is part of a long-term study on the process of spin-orbit resonance. The authors studied the probabilities of capture (when you slow down until reaching a spin-orbit resonance, will you stay inside or leave it, still slowing down?), proposed numerical integrators adapted to this problem…
In this specific paper, they write down the ODE ruling the dynamics in considering the frequency-dependent tides (which they call realistic), and solve it analytically with a perturbation method, i.e. first in neglecting a perturbation, that they add incrementally, to eventually converge to the real solution. They checked their results with numerical integrations, and they also studied the stability of the solutions (the stable solutions being attractors), and the probabilities of capture.

In my opinion, the main result is: the stable attractor is not periodic but quasi-periodic. Fine, but what does that mean?

If we neglect the influence of the other planets, then the variations of the spin rate of Mercury is expected to be a periodic signal, with a period of 88 days. This is due to the periodic variations of the Sun-Mercury distance, because of the eccentricity. This results in longitudinal librations, which are analogous to the librations of the Moon (we do not see 50% of the surface of the Moon, but 59%, thanks to these librations). The authors say that this solution is not stable. However, a stable solution is the superimposition of these librations with a sinusoid, which period is close to 15 years, and an amplitude of a few arcminutes (to be compared to 15 arcminutes, which is the expected amplitude of the 88-d signal). So, it is not negligible, and this 15-y period is the one of the free (or proper) oscillations of Mercury. A pendulum has a natural frequency of oscillations, here this is exactly the same. But contrarily to a pendulum, the amplitude of these oscillations does not tend to 0. So, we could hope to detect it, which would be a direct observation of the tidal dissipation.

Measuring the rotation

What can we observe? We should first keep in mind that the authors addressed the early Mercury, when being trapped into the 3:2 spin-orbit resonance, which was pretty homogeneous. The current Mercury has a global fluid layer, which means a larger (about twice) amplitude of the 88-d signal, and a different dissipative process, the tides being assisted by core-mantle friction. As a consequence, there is no guarantee that the 15-y oscillation (actually a little shorter, some 12 years, because of the fluid core) would still exist, and that would require a dedicated study. But measuring it would be an information anyway.

How to measure it? The first observations of the rotation of Mercury in 1965 and of the librations in 2007 were Earth-based, radar observations, which are sensitive to the velocity. This means that they are more likely to detect a rapid oscillation (88 d, e.g.) than a slow one (12 years…). Observations of the surface of Mercury by the spacecraft MESSENGER confirmed those measurements. In 2018 the ESA/JAXA (Europe / Japan) joint mission Bepi-Colombo will be sent to Mercury, for orbital insertion in 2025 and hopefully a 2-y mission, with a better accuracy than MESSENGER. So, we could hope a refinement of the measurements of the longitudinal motion.

Purple: The 88-d oscillation. Green: Superimposed with the 15-y one. Keep in mind that Bepi-Colombo will orbit Mercury during some 2 years.
Purple: The 88-d oscillation. Green: Superimposed with the 15-y one. Keep in mind that Bepi-Colombo will orbit Mercury during some 2 years.

To know more

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