Tag Archives: tides

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)
Theoretical
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…

Results

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…

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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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Inferring the interior of Venus from the tides

Hi there! Today’s post presents you the study Tidal constraints on the interior of Venus, by Caroline Dumoulin, Gabriel Tobie, Olivier Verhoeven, Pascal Rosenblatt, and Nicolas Rambaux. This study has recently been accepted for publication in Journal of Geophysical Research: Planets. The idea is: because of its varying distance to the Sun, Venus experiences periodic variations. What could their measurements tell us on the interior?

Venus vs. the Earth

Venus is sometimes called the twin sister of the Earth, because of its proximity and its size. However, their physical properties show crucial differences, the most crucial one being the atmosphere.

Inclination3.86°7.155°Obliquity177.36°23.439°Orbital period224.701 d365.256 dSpin period243.025 d0.997 dSurface pressure92 bar1.01 barMagnetic field(none)25-65 μTMean density5,243 kg/m35,514 kg/m3

Venus Earth
Semimajor axis (AU) 0.723332 1.000001
Eccentricity 0.0068 0.0167

As you can see:

  • Venus has a retrograde and very slow rotation,
  • it has a very thick and dense atmosphere,
  • it has no magnetic field.

For a magnetic field to exist, you need a rotating solid core, a global conductive fluid layer, and convection, which is triggered by heat transfers from the core to the mantle. The absence of magnetic field means that at least one of these conditions is not fulfilled. Given the size of Venus and its measured k2 by the spacecraft Magellan (explanations in the next section), it has probably a fluid global layer. However, it seems plausible that the heat transfer is missing. Has the core cooled enough? Is the surface hot enough so that the temperature has reached an equilibrium? Possible.

Probing the interior of Venus is not an easy task; an idea is to measure the time variations of its gravity field.

Tidal deformations

The orbital eccentricity of Venus induces variations of its distance to the Sun, and variations of the gravitational torque exerted by it. Since Venus is not strictly rigid, it experiences periodic deformations, which frequencies are known as tidal frequencies. These deformations can be expressed with the potential Love number k2, which gives you the amplitude of the variations of the gravity field. Since the gravity field can be measured from deviations of the spacecraft orbiting the planet, we dispose of a measurement, i.e. k2 = 0.295 ± 0.066. It has been published in 1996 from Magellan data (see here a review on the past exploration of Venus). You can note the significant uncertainty on this number. Actually k2 should be decorrelated from the other parameters affecting the trajectory of the spacecraft, e.g. the flattening of the planet, the atmosphere, which is very dense, motor impulses of the spacecraft… This is why it was impossible to be more accurate.

Other parameters can be used to quantify the tides. Among them are

  • the topographic Love number h2, which quantified the deformations of the surface. Observing the surface of Venus is a task strongly complicated by the atmosphere. Magellan provided a detailed map thanks to a laser altimeter. Mountains have been detected. But these data do not permit to measure h2.
  • The dissipation function Q. If I consider that the deformations of the gravity field are periodic and represented by k2 only, I mean that Venus is elastic. That mean that it does not dissipate any energy, it has an instantaneous response to the tidal solicitations, and the resulting tidal bulge always points exactly to the Sun. Actually there is some dissipation, which results as a phase lag between the tidal bulge and the Venus – Sun direction. Measuring this phase lag would give k2/Q, and that information would help to constrain the interior.

A 3-layer-Venus

Such a large body is expected to be denser in the core than at the surface, and is usually modeled with 3 layers: a core, a mantle, and a crust. Venus also have an atmosphere, but this is not a very big deal in this specific case. These are not necessarily homogeneous layers, the mantle and the core are sometimes assumed to have a global outer fluid layer. If this would happen for the core, then we would have a solid (rigid) inner core, and a fluid (molten) outer core. This interior must be modeled to estimate the tidal quantities. More precisely, you need to know the radial evolution of the density, and of the velocities of the longitudinal (P) and transverse (S) seismic waves. These two velocities tell us about the viscosity of the material.

Possible interior of Venus (not to scale). Copyright: The Planetary Mechanics Blog

Modeling the core from PREM

PREM is the Preliminary Reference Earth Model. It was published in 1981, and elaborated from thousands of seismic observations. Their inversions gave the radial distribution of the density, dissipation function, and elastic properties for the Earth. It is now used as a standard Earth model.

The lack of data regarding the core of Venus prompted the authors, and many of their predecessors, to rescale PREM from the Earth to Venus.

Modeling the mantle from Perple_X

The properties of the mantle of Venus depend on its composition and the radial distribution of its temperature, its composition itself depending on the formation of the planet. The authors identified 5 different models of formation of Venus in the literature, which affect 5 variables: mass of the core, abundance of uranium (U), K/U ratio (K: potassium), Tl/U ratio (Tl: thallium), and FeO/(FeO+MgO) ratio (FeO: iron oxide, MgO: magnesium oxide). Only 3 of these 5 models were kept, two being end-members, and the third one being pretty close to the Earth. These 3 models were associated with two end-members for temperature profiles, which can be found in the scientific literature. This then resulted in 6 models, and their properties, i.e. density and velocities of the P- and S-waves, were obtained thanks to the Perple_X code. This code gives phase diagrams in a geodynamic context, i.e. under which conditions (pressure and temperature) you can have a solid, liquid, and / or gaseous phase(s) (they sometimes coexist) in a planetary body.

Numerical modeling of the tidal parameters

Once the core and the mantle have been modeled, a 60-km-thick crust have been added on the top, and then the tidal quantities have been calculated. For that, the authors used a numerical algorithm elaborated in Japan in 1974, using 6 radial functions y. y1 and y3 are associated with the radial and tangential displacements, y2 and y4 are related to the radial and tangential stresses, y5 is associated with the gravitational potential, while y6 guarantees a property of the continuity of the gravitational force in the structure. These functions will then give the tidal quantities.

Results

The results essentially consist of a description of the possible interiors and elastic properties of Venus for different values of k2, which are consistent with the Magellan measurements. But the main information is this: Venus may have a solid inner core. Previous studied had discarded this possibility, arguing that k2 should have been 0.17 at the most. However, the authors show that considering viscoelastic properties of the mantle, i.e. dissipation, would result in a smaller pressure in the core, i.e. <300 GPa, for a k2 consistent with Magellan. This does not mean that Venus has a solid inner core, this just means that it is possible. Actually, the authors also get interior models with a fully fluid core.
The atmosphere would alter k2 by only 3 to 4%.

The authors claim that the uncertainty on k2 is too large to have an accurate knowledge on the interior, and they hope that future measurements of k2 and of k2/Q, which has never been measured yet, would give better constraints.

The forthcoming and proposed missions to Venus

For this hope to be fulfilled, we should send spacecrafts to Venus in the future. The authors mention EnVision, which applies to the ESA M5 call (M for middle-class). This is a very competitive call, and we should know the finalists very soon. If selected, EnVision would consist of an orbiter on a low and circular orbit, which would focus on geology and geochimical cycles. It should also measure k2 with an accuracy of 3%, and give us a first measurement of k2/Q.

In America, two missions to Venus have been proposed for the Discovery program of NASA: VERITAS (Venus Emissivity, Radio Science, InSAR, Topography, and Spectroscopy) and DAVINCI (Deep Atmosphere Venus Investigation of Noble gases, Chemistry, and Imaging). They have both been rejected.

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Energy dissipation in Saturn

Hi there! I will tell you today about the letter Frequency-dependent tidal dissipation in a viscoelastic Saturnian core and expansion of Mimas’ semi-major axis, by Daigo Shoji and Hauke Hussmann, both working at the DLR in Berlin, Germany. This paper has recently been published in Astronomy and Astrophysics.

Saturn’s facts

Do I need to introduce Saturn? Saturn is the sixth planet of the Solar System by its distance to the Sun, and the second by its size. It orbits the Sun at a mean distance of 1.5 billions of km, in 29.4 years. It has more than 200 satellites, which comprises small moons embedded in the rings, mid-sized icy satellites, a large one, i.e. Titan, and very far small moons which are probably trapped objects. Which means that the other bodies are expected to have formed while orbiting around Saturn, or formed from the same protoplanetary disk.
Saturn is particularly known for its large rings, which can be observed from the Earth with almost any telescope. Moreover this planet is on average less dense than the water, which is due to a large atmosphere enshrouding a core. The total radius of Saturn is about 60,000 km, which actually corresponds to a pressure of 1 bar in the atmosphere, while the radius of the core is about 13,000 km. The paper I present today is particularly focused on the core.

A new view of the formation of the satellites of Saturn

The spacecraft Cassini orbits Saturn since 2004, and has given us invaluable information on the planet, the rings, and the satellites. Some of these information pushed the French planetologist Sébastien Charnoz, assisted by French and US colleagues, to propose a new model of formation of the satellites from the rings: this model states that instead of having formed with Saturn, the rings are pretty recent, i.e. less than 1 Gyr, and are due to the disintegration of an impactor.
Once the debris rearranged as a disk, reaccretion of material would have created the satellites, which would then have migrated outward, because of the tidal interaction with the planet… This means that it is crucial to understand the tidal interaction.

Tidal dissipation in the planets

I have already discussed of tides in this blog. Basically: when you are a satellite (you dream of that, don’t you?) orbiting Saturn, you are massive enough (sorry) to alter the shape of the planet, and raise a bulge which would almost be aligned with you… Almost because while the material constituting the planet responds, you have moved, but actually the bulge is in advance because the planet rotates faster than you orbit around it (you still follow me?). As a consequence, you generate a torque which tends to slow down the spin of the planet, and this is compensated by an outward migration of the satellite (of you, since you are supposed to be the satellite). This compensation comes from the conservation of the angular momentum. You can imagine that the planet also raises a tidal bulge on the satellite, but this does not deal with our paper. So, not today.

A consequence of tides is the secular migration of the planetary satellites. Lunar Laser Ranging measurements have detected an outward migration of the Moon at a rate of 3 cm/y. It is not that easy to measure the migration of the satellites of Saturn. An initial estimation, based on the pre-Cassini assumption that the satellites were as old as the Solar System, considered that the satellite Mimas would have at the most migrated from the synchronous orbit to its present one, in 4.5 Gyr. The relevant quantity is the dissipation function Q, and this condition would have meant Q>18,000, in neglecting dissipation in Mimas. Recent measurements based on Cassini observations suggest Q ≈ 2,600, which would be another invalidation of the assumption of primordial satellites.

Several models of dissipation

To make things a little more technical: we are interested in the way the material responds to an external, gravitational sollicitation. This sollicitation is quasi-periodic, i.e. it can be expressed as a sum of periodic, sinusoidal terms. With each of these terms is associated a frequency, on which the response of the material depends. This affects the quantity k2/Q, k2 being a Love number and Q the dissipation function I have just presented. Splitting these two quantities is sometimes useless, since they appear as this ratio in the equations ruling the orbital evolution of the satellites.

Tides in a solid body

By solid body, I mean a body with some elasticity. Its shape can be altered, but not that much. An elastic response would not dissipate any energy, while a viscoelastic one would, and would be responsible for the migration of the orbits of the satellites.
It was long considered that the tidal dissipation did not depend on the excitation frequency, which is physically irrelevant and could lead to non-physical conclusions, e.g. the belief in a stable super-synchronous rotation for planetary satellites.
We now consider that the response of the material is pretty elastic for slow excitations, and viscoelastic for rapid ones. If you do not shake the material too much, then you have a chance to not alter it. If you are brutal, then forget it.
For that, a pretty simple tidal model rendering this behavior is the Maxwell model, based on one parameter which is the Maxwell time. It is defined as the ratio between the viscosity and the rigidity of the material, and it somehow represents the limit between the elastic and the viscoelastic responses.
A refining model for icy satellites is the Andrade model, which considers a higher dissipation at high frequencies.

Tides in a gaseous planet

If the planet is a ball of gas, a fortiori a fluid, then the behavior is different, actually much more complicated. You should consider Coriolis forces in the gas, turbulent behaviors, which can be highly non-linear.
A recent model has been presented by Jim Fuller, in which he considers the possibility of resonant interactions between the fluid and the satellites, which would result in a high dissipation at the exact orbital frequency of the satellite, and the resonant condition would induce that this high dissipation would survive the migration of the satellite. You can see here an explanation of this phenomenon, drawn by James T. Keane.

This paper

This paper aims at checking whether a dissipation of the planet, which would be essentially viscoelastic, could be consistent with the recent measurements of tides. For that, the authors modeled Saturn as an end-member, in neglecting every dissipation in the atmosphere. They considered different plausible numbers for the viscosity and rigidity in the core Saturn, in assuming it has no internal fluid layer, and numerically integrated the migration of Mimas, the variation of its orbital frequency in the expression of tides being taken into account.

And the result is that the viscosity should be of the order of 1013-1014 Pa.s. Smaller and higher numbers would be inconsistent with the measured dissipation.
Moreover, some of these viscosities are found to be consistent with the assumption of a primordial Mimas, i.e. with an inward migration from the synchronous orbit in 4.5 Gyr.

Perspectives

This letter probably presents a preliminary study, the whole study requiring to consider additional effects, like the pull of the rings, the influence of the atmosphere, and the mean-motion resonances between the satellites (see this post), which themselves alter the rate of migration. And this is why this letter does not invalidate Charnoz’s model of formation, nor Fuller’s tides, but just says that other explanations are possible.

Useful links

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Hinting the interior of planetary satellites from energy dissipation

Hi there! Today I will present you a paper that has recently been accepted for publication in Celestial Mechanics and Dynamical Astronomy, entitled Constraints on dissipation in the deep interiors of Ganymede and Europa from tidal phase-lags. This study has been conducted in Germany, at the DLR, by Hauke Hussmann.

The idea is here to get some clues on the interior of the satellites of Jupiter Ganymede and Europa, from two different signatures of the tides raised by Jupiter.

The tidal Love numbers h2 and k2

I have recently presented the tidal Love number k2 in a post on Mercury. In a nutshell: it represents the amplitude of variation of the gravity field of the satellite, at the orbital frequency. Please note that contrary to Mercury, only the orbital frequency is to be considered in the periodic variations of the gravity field. The reason for that is in the rotational dynamics: the main satellites of Jupiter rotate synchronously, showing the same face to their planet like our Moon, while Mercury is in a 3:2 spin-orbit resonance.
The tidal Love number h2 represents the amplitude of the tidal deformation of the topography of the satellite. Something remarkable on these 2 numbers is that h2 is mostly sensitive to the surface, while k2 is the response of the whole body. The idea of this study is to compare the two numbers, to get clues on the interior.

The satellites of Jupiter

At this time, 67 natural satellites are known for Jupiter. They can be classified into 3 groups:

  • The inner satellites Metis, Adrastea, Amalthea and Thebe. These are small bodies, their mean radii being between 8 and 85 km. They orbit at less than 3 Jupiter radii.
  • The Galilean satellites Io, Europa, Ganymede and Callisto. These are pretty large bodies, which were discovered in 1610 by Galileo Galileo. They orbit between 6 and 25 Jupiter radii. They contain almost of the mass of the satellites of Jupiter, which make them particularly interesting. For instance, their large masses is responsible for an interesting 3-bodies mean-motion resonance involving Io, Europa, and Ganymede. Basically, Io makes 4 revolutions around Jupiter while Europa makes 2 and Ganymede exactly one. This configuration is known as Laplacian resonance. Moreover the sizes of the 4 Galilean satellites, combined with the tides raised by Jupiter, are also responsible for internal differentiation. In particular, these 4 bodies are all considered to harbor global internal fluid layers.
  • The irregular satellites. These are small bodies orbiting far much further from Jupiter. They are probably former asteroids which were trapped by the gravity field of Jupiter. Contrary to the two other groups, which have pretty circular and coplanar orbits, the irregular satellites can have highly eccentric and inclined orbits. Some of them are even retrograde.

The next space missions JUICE and Europa Multiple Flyby

Ganymede and Europa are the main targets of the next two missions to the system of Jupiter. These two missions are the ESA mission JUICE, and the NASA Europa Mission.

JUICE, for JUpiter ICy moons Explorer, is planned to be launched in 2022 and to orbit Jupiter in 2030. Then, it will make flybys of Europa and Callisto, before becoming a satellite of Ganymede. Ganymede is thus the main target. Among the 11 instruments constituting JUICE, let us focus on two of them: GALA and 3GM.

GALA, for GAnymede Laser Altimeter, will measure the topography of the planet, while 3GM, for Gravity and Geophysics of jupiter and the Galilean Moons, is the radioscience experiment. It will in particular measure the gravity field of the body. The connection with the study I am presenting you is that h2 is expected from GALA, while k2 is expected from 3GM. Another connection is that Hauke Hussmann is both the first author of this study, and the principal investigator of GALA.

The NASA Europa Mission, also known as Europa Multiple-Flyby Mission, and previously Europa Clipper, will obviously target Europa. It should be launched in the 2020’s, and the nominal mission plans to perform 45 flybys of Europa.

One of the motivations to explore these bodies is the search for extraterrestrial life. Europa and Ganymede are known to harbor a subsurface ocean, and we wonder whether these oceans contain the ingredients for bacteriological life. These two missions will give us more information on the interior, from gravity data, analysis of the topography, imagery of the surface, measurements of the magnetic field… bringing new constraints on the oceans, like their depths, density, or viscosity…

This study

The idea of these studies is to compare the Love number h2, from the topography, and k2, from the gravity field, to constrain the interior. For that, the authors have considered several models of interior of Europa and Ganymede, and simulated the resulting Love numbers.

These interior models have to be realistic, which means being consistent with our current knowledge of these bodies, i.e. their total mass and their shapes, and being physically relevant. This implies that their densities increase radially, from the surface to the center. So, the surface is assumed to be made of ice coating a water ocean. Below the ocean is another ice layer, which itself surrounds a denser core. The ocean tends to decouple the icy shell from the action of the interior.

The authors particularly focus on the phase difference between h2 and k2. Basically, the Love numbers are complex numbers, the imaginary part representing the dissipation, while the real part is related to a purely elastic tide. From their simulations, they show that these phase differences should be of several degrees. Their possible measurements should constrain the viscosity of the ice shell coating the core of Ganymede, and the temperature of the mantle of Europa.

Some perspectives

Of course, the most interesting perspective is the future measurements of these phase differences by JUICE and NASA Europa Mission. The information they will provide will be supplemented by better constraints on the gravity field, on the magnetic field, on the rotation…

The authors assumed the rotations of these satellites to be synchronous, as suggested by the theory. But features at the surface of Europa suggest that the rotation of its surface could be actually slightly super-synchronous. This is something that the dynamical theories still need to understand, but this would probably affect the tidal action of Jupiter on Europa.

 

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