All posts by Terryl Coron

The fate of the Alkyonides

Hello everybody! Today, I will tell you on the dynamics of the Alkyonides. You know the Alkyonides? No? OK… There are very small satellites of Saturn, i.e. kilometer-sized, which orbit pretty close to the rings, but outside. These very small bodies are known to us thanks to the Cassini spacecraft, and a recent study, which I present you today, has investigated their long-term evolution, in particular their stability. Are they doomed or not? How long can they survive? You will know this and more after reading this presentation of Long-term evolution and stability of Saturnian small satellites: Aegaeon, Methone, Anthe, and Pallene, by Marco Muñoz-Gutiérrez and Silvia Giuliatti Winter. This study has recently been accepted for publication in The Monthly Notices of the Royal Astronomical Society.

The Alkyonides

As usually in planetary sciences, bodies are named after the Greek mythology, which is the case of the four satellites discussed today. But I must admit that I cheat a little: I present them as Alkyonides, while Aegeon is actually a Hecatoncheires. The Alkyonides are the 7 daughters of Alcyoneus, among them are Anthe, Pallene, and Methone.

Here are some of there characteristics:

Methone Pallene Anthe Aegaeon
Semimajor axis 194,402 km 212,282 km 196,888 km 167,425 km
Eccentricity 0 0.004 0.0011 0.0002
Inclination 0.013° 0.001° 0.015° 0.001°
Diameter 2.9 km 4.4 km 2 km 0.66 km
Orbital period 24h14m 27h42m 24h52m 19h24m
Discovery 2004 2004 2007 2009

For comparison, Mimas orbits Saturn at 185,000 km, and the outer edge of the A Ring, i.e. of the main rings of Saturn, is at 137,000 km. So, we are in the close system of Saturn, but exterior to the rings.

Discovery of Anthe, aka S/2007 S4. Copyright: NASA.
Discovery of Anthe, aka S/2007 S4. Copyright: NASA.

These bodies are in mean-motion resonances with main satellites of Saturn, more specifically:

  • Methone orbits near the 15:14 MMR with Mimas,
  • Pallene is close to the 19:16 MMR with Enceladus,
  • Anthe orbits near the 11:10 MMR with Mimas,
  • Aegaeon is in the 7:6 MMR with Mimas.

As we will see, these resonances have a critical influence on the long-term stability.

Rings and arcs

Beside the main and well-known rings of Saturn, rings and arcs of dusty material orbit at other locations, but mostly in the inner system (with the exception of the Phoebe ring). In particular, the G Ring is a 9,000 km wide faint ring, which inner edge is at 166,000 km… Yep, you got it: Aegaeon is inside. Some even consider it is a G Ring object.

Methone and Anthe have dusty arcs associated with them. The difference between an arc and a ring is that an arc is longitudinally bounded, i.e. it is not extended enough to constitute a ring. The Methone arc extends over some 10°, against 20° for the Anthe arc. The material composing them is assumed to be ejecta from Methone and Anthe, respectively.

However, Pallene has a whole ring, constituted from ejecta as well.

Why sometimes a ring, and sometimes an arc? Well, it tell us something on the orbital stability of small particles in these areas. Imagine you are a particle: you are kicked from home, i.e. your satellite, but you remain close to it… for some time. Actually you drift slowly. While you drift, you are somehow shaken by the gravitational action of the other satellites, which disturb your Keplerian orbit around the planet. If you are shaken enough, then you may leave the system of Saturn. If you are not, then you can finally be anywhere on the orbit of your satellite, and since you are not the only one to have been ejected (you feel better, don’t you?), then you and your colleagues will constitute a whole ring. If you are lucky enough, you can end up on the satellite.

The longer the arc (a ring is a 360° arc), the more stable the region.

Frequency diffusion

The authors studied

  1. the stability of the dusty particles over 18 years
  2. the stability of the satellites in the system of Saturn over several hundreds of kilo-years (kyr).

For the stability of the particles, they computed the frequency diffusion index. It consists in:

  1. Simulating the motion of the particles over 18 years,
  2. Determining the main frequency of the dynamics over the first 9 years, and over the last 9 ones,
  3. Comparing these two numbers. The smaller the difference, the more stable you are.

The numerical simulations is something I have addressed in previous posts: you use a numerical integrator to simulate the motion of the particle, in considering an oblate Saturn, the oblateness being mostly due to the rings, and several satellites. Our four guys, and Janus, Epimetheus, Mimas, Enceladus, and Tethys.

How resonances destabilize an orbit

When a planetary body is trapped in a mean-motion resonance, there is an angle, which is an integer combination of angles present in its dynamics and in the dynamics of the other body, which librates. An example is the MMR Aegaeon-Mimas, which causes the angle 7λMimas-6λAegaeonMimas to librate. λ is the mean longitude, and ϖ is the longitude of the pericentre. Such a resonance is supposed to affect the dynamics of the two satellites but, given their huge mass ratio (Mimas is between 300 and 500 millions times heavier than Aegaeon), only Aegaeon is affected. The resonance is at a given location, and Aegaeon stays there.
But a given resonance has some width, and several resonant angles (we say arguments) are associated with a resonance ratio. As a consequence, several resonances may overlap, and in that case … my my my…
The small body is shaken between different locations, its eccentricity and / or inclination can be raised, until being dynamically unstable…
And in this particular region of the system of Saturn, there are many resonances, which means that the stability of the discovered body is not obvious. This is why the authors studied it.

Results

Stability of the dusty particles

The authors find that Pallene cannot clear its ring efficiently, despite its size. Actually, this zone is the most stable, wrt the dynamical environments of Anthe, Methone and Aegaeon. However, 25% of the particles constituting the G Ring should collide with Aegaeon in 18 years. This probably means that there is a mechanism, which refills the G Ring.

Stability of the satellites

From long-term numerical simulations over 400 kyr, i.e. more than one hundred millions of orbits, these 4 satellites are stable. For Pallene, the authors guarantee its stability over 64 Myr. Among the 4, this is the furthest satellite from Saturn, which makes it less affected by the resonances.

A perspective

The authors mention as a possible perspective the action of the non-gravitational forces, such as the solar radiation pressure and the plasma drag, which could affect the dynamics of such small bodies. I would like to add another one: the secular tides with Saturn, and the pull of the rings. They would induce drifts of the satellites, and of the resonances associated. The expected order of magnitude of these drifts would be an expansion of the orbits of a few km / tens of km per Myr. This seems pretty small, but not that small if we keep in mind that two resonances affecting Methone are separated by 4 km only.

This means that further results are to be expected in the upcoming years. The Cassini mission is close to its end, scheduled for 15 Sep 2017, but we are not done with exploiting its results!

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.

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…

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

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.

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.

Venus Earth
Semimajor axis (AU) 0.723332 1.000001
Eccentricity 0.0068 0.0167
Inclination 3.86° 7.155°
Obliquity 177.36° 23.439°
Orbital period 224.701 d 365.256 d
Spin period 243.025 d 0.997 d
Surface pressure 92 bar 1.01 bar
Magnetic field (none) 25-65 μT
Mean density 5,243 kg/m3 5,514 kg/m3

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.

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.

Tilting and retilting our Moon

Hi there! The Moon is so close and so familiar to us, but I realize this is my first post on it. Today I present you a paper entitled South Pole Aitken Basin magnetic anomalies: Evidence for the true polar wander of Moon and a lunar dynamo reversal, by Jafar Arkani-Hamed and Daniel Boutin, which will be published soon in Journal of Geophysical Research: Planets. The idea is to track the variations of the magnetic field of the Moon along its history, as a signature of the motion of its rotation pole, i.e. of a polar wander.

Our Moon’s facts

The Moon is a fascinating object, as it is the only known natural satellite of the Earth, and we see it as large as the Sun in our sky. It orbits around the Earth at a distance of almost 400,000 km in 27.3 days. It shows us always the same face, as a result of a tidal locking of its rotation, making it synchronous, i.e. its spin period is equal to its orbital period.

Moonset over Paris, France. Copyright: Josselin Desmars.

Something interesting is its pretty large size, i.e. its radius is one fourth of the one of the Earth. It is widely admitted that the Moon and the Earth have a common origin, i.e. either a proto-Earth has been impacted by a Mars-sized impactor, which split it between the Earth and the Moon, or the Earth-Moon system results from the collision of two objects of almost the same size. In both cases, the Earth and the Moon would have been pretty hot just after the impact, which also means active… and this has implications for the magnetic field.

A very weak magnetic field has been detected for the Moon, but which is very different from the Earth’s. The magnetic field of the Earth, or geomagnetic field, has the signature of a dipolar one, in the sense that it has a clear orientation. This happens when the rotating core acts as a dynamo. The north magnetic pole is some 10° shifted from the spin pole of the Earth, and has an amplitude between 25 and 65 μT (micro-teslas). However, the magnetic field of the Moon, measured at its surface, does not present a clear orientation, and never reaches 1 μT. Its origin is thus not obvious, even if we could imagine that the early Moon was active enough to harbor a dynamo, from which the measured magnetic field would be a signature… But the absence of preferred orientation is confusing.

The core dynamo

The core of the Earth spins, it is surrounded by liquid iron, which is conductive, and there is convection in this fluid layer, which is driven by the heat flux diffusing from the core to the surface of the Earth. This process creates and maintains a magnetic field.

For the Earth, the core dynamo is assumed to account for 80 to 90% of the total magnetic field. This results in a preferred orientation. Other processes that could create a magnetic field are a global asymmetry of the electric charges of the planet, or the presence of an external magnetic field, for instance due to a star.

A dynamo could be expected for many planetary objects, which would be large enough to harbor a global fluid layer. It is usually thought that the detection of a magnetic field is a clue for the presence of a global ocean. Such a magnetic field has been detected for Jupiter’s moon Ganymede, which is probably due to an outer liquid layer coating its iron core.

The Moon has probably no dynamo, but could have had one in the past. The measured magnetic field could be its signature. A question is: what could have driven this dynamo? The early Moon was hotter than the current one, so a magnetic field existed at that time. And after that, the Moon experienced intense episodes of bombardment, like the Late Heavy Bombardment. The resulting impacts affected the orientation of the Moon, its shape, and also its temperature. This could have itself triggered a revival of the magnetic field, particularly for the biggest impact.

The study I present today deals with measurements of the magnetic field in the South Pole-Aitken Basin, not to be confused with the Aitken crater, which is present in its region. The South Pole-Aitken Basin is one of the largest known impact crater in the Solar System, with a diameter of 2,500 km and a depth of 13 km. This basin contains other craters, which means that it is older than all of them, its age is estimated to be 4.1 Gyr (gigayears, i.e. billions of years). Measurements of the magnetic field in each of these craters could give its evolution over the ages. But why is it possible?

The magnetic field as a signature of the history

When a material is surrounded by a magnetic field, it can become magnetic itself. This phenomenon is known as induced magnetization, and depends on the magnetic susceptibility of the material, i.e. the efficiency of this process depends on the material. Once the surrounding magnetic field has disappeared, the material might remain magnetic anyway, i.e. have its remanent magnetic field. This is what has been measured by the Lunar Prospector mission, whose data originated this study.
An issue is the temperature. The impact should be hot enough to trigger the magnetic field, which implies that the material would be hot, but it cannot be magnetized if it is too hot. Below a Curie temperature, the process of induced magnetization just does not work. You can even demagnetize a material in heating it. For the magnetite, which is a mineral containing iron and present on the Moon, the Curie temperature is 860 K, i.e. 587°C, or 1089°F.

Lunar Prospector

This study uses data of the Lunar Prospector mission. This NASA mission has been launched in January 1998 from Cape Canaveral and has orbited the Moon on a polar orbit during 18 months, until July 1999. It made a full orbit in a little less than 2 hours, at a mean altitude of 100 km (60 miles). This allowed to cover the whole surface of the Moon, and to make measurements with 6 instruments, related to gamma rays, electrons, neutrons, gravity… and the magnetic field.

Results of this study

This study essentially consists of two parts: a theoretical study of the temperature evolution of the Moon over its early ages, including after impacts, and the interpretation of the magnetic field data. These data are 14 magnetic anomalies in the South Pole-Aitken Basin, which the theoretical study helps to date. And the data show two orientations of the magnetic field in the magnetic in the past, giving an excursion of more than 100° over the ages.

Now, if we consider that in the presence of a core dynamo, the magnetic field should be nearly aligned with the spin pole, this means that the Moon has experienced a polar wander of more than 100° in its early life. More precisely, the two orientations are temporally separated by the creation of the Imbrium basin, 3.9 Gyr ago. In other words, the Moon has been tilted. This is not the only case in the Solar System, see e.g. Enceladus.

To know more

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