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.



How Ceres and Vesta shape the asteroid belt

Hi! Today I will tell you about a recent study made in Serbia on the dynamical influence of the small planets Ceres and Vesta on the Asteroid Belt. This study, Secular resonances with Ceres and Vesta by G. Tsirvoulis and B. Novaković, has been accepted for publication in Icarus.

The Asteroid Main Belt

There are many small bodies in the Solar System, here we just focus on the so-called Main Belt, i.e. a zone “full” of asteroids, which lies between the orbits of Mars and Jupiter. The word “full” should be taken with care, since there are many asteroids populating it, but if we cross it, we would be very unlikely to meet one. This zone is essentially void. It is estimated that the total mass of these asteroids is only 4% of the mass of the Moon.

It is called “Main Belt” since the first asteroids were discovered in this zone, and it was long thought that most of them were in this Main Belt. At this time, hundreds of thousands of them have been identified, but the Kuiper Belt, which lies behind the orbit of Neptune, might be even more populated.

The dynamics of these bodies is very interesting. It could contain clues on the early ages of the Solar System. Moreover, they are perturbed by the planets of the Solar System, especially the giant planets.

As a consequence, they have pretty complex dynamics. Their orbits can be approximated with ellipses, but these are not constant ellipses. They are precessing, i.e. their pericentres and nodes are moving, but their semi-major axes, eccentricities and inclinations are time-dependent as well. To represent their dynamics, so-called proper elements are used, which are kind of mean values of these orbital elements, and which are properties of these bodies.

Ceres and Vesta

Ceres and Vesta, or more precisely 1 Ceres and 4 Vesta, are the two largest objects of the Main Belt, with mean radii of 476 and 263 km, respectively. So large objects could present complex interior structures, this is one motivation for the US space mission Dawn, which has orbited Vesta between July 2011 and September 2012, and is currently in orbit around Ceres, since March 2015.

This space mission has given, and is still giving, us invaluable data on these two bodies, like a cartography of the craters of Vesta, and the recent proof that Ceres is differentiated, from the analysis of its gravity field.

The orbital resonances

The asteroids are so small bodies than they are subjected to the gravitational influence of the planets, in particular Jupiter. The most interesting dynamical effect is the orbital resonances, which occur when a proper frequency of the orbit of the asteroid (for instance its orbital frequency, or the frequency of precession of its orbital plane, known as nodal precession) is commensurate with a proper frequency of a planet. In such a case, orbital parameters are excited. In particular, an excitation of the eccentricity results in a destabilization of the orbit, since the asteroid is more likely to collide with another body, and/or to be finally ejected from the Main Belt.

This results in gaps in the Main Belt. The most famous of them are the Kirkwood Gaps, which correspond to mean-motion resonances between the asteroids and Jupiter. When the orbital frequency of the asteroid is exactly three times the one of Jupiter, i.e. when its orbital period is exactly one third of the one of Jupiter, then the asteroid is at the 3:1 resonance, its eccentricity is excited, and its orbit is less stable. We thus observe depletions of asteroids at the resonances 3:1, 5:2, 7:3, and 2:1.

Another type of resonance are the secular resonances, which involve the precession of the pericentres and / or of the node (precession of the orbital plane) of the asteroid. In such a case, this is a much slower phenomenon, since the periods involved are of the order of millions of years, while the orbital period of Jupiter is 11.86 years.

The asteroid families

An analysis of the dynamics (proper elements) and the physical properties of the asteroids shows that it is possible to classify them into families. The asteroids of these families are thought to originate from the same body, which has been destroyed by a collision. They are usually named among the largest of these bodies, for instance Vesta is also the name of a family.

This study

In this study, the authors investigate the dynamical influence of Vesta and Ceres on the Main Belt. They particularly focus on the secular resonances, in identifying four of them, i.e. resonances with the precessions of the pericentres and nodes of these two bodies.

For that, they perform numerical integrations of the motion of 20 test particles over 50 Myr, perturbed by the 4 giant planets, with and without Ceres and Vesta, and show significant influence of these bodies for some of the particles.

Finally, they show that some asteroid families do cross these resonances, like the Hoffmeister family.

Some links

  • The study, Secular resonances with Ceres and Vesta by G. Tsirvoulis and B. Novakovic, accepted for publication in Icarus, and made freely available by the authors on arXiv (thanks to them for sharing)
  • The web site of Georgios Tsirvoulis
  • The web site of Bojan Novaković
  • The mission DAWN

Surviving as a Trojan of Neptune

Hi! Today I will tell you about a study recently accepted for publication in Astronomy & Astrophysics, by Rodney Gomes and David Nesvorný, on the survival of the asteroids which precede and follow Neptune on its orbit.

The coorbital resonance

In the Solar System, the mean motion resonances are ubiquitous. When the orbital frequencies of two bodies are commensurate, interesting phenomena might happen: they could have a more stable orbit, or they could experience a permanent forcing which raise their eccentricity and / or inclination, and in some cases could result in an ejection. A resonance has particularly strong effects on a small body which orbit resonates with the one of a large planet. This is for instance how the giant planets shaped the asteroid belt.

Here, we deal with the coorbital resonance, which is a very specific and interesting case. This happens between two bodies which have on average the same orbital frequencies, and the perturbations associated result in some zones of stability. In particular, there are five equilibrium positions for the coorbital restricted 3-body problem, i.e. if we consider the Sun, a planet, here Neptune, and a small body. These equilibriums are known as Lagrangian points, and the most remarkable of them are denoted L4 and L5. They precede and follow the planet at an angular distance of 60°, and are stable equilibriums. As a consequence, they are likely to accumulate several small bodies, and this is verified by the observations, which have detected asteroids which coorbit with Jupiter, Uranus and Neptune.

At this time, 6,288 of these objects have been detected for Jupiter, 1 for Uranus, and 17 for Neptune.

The planetary migration

Since 2004 and the first version of the Nice model, the giant planets are assumed to have formed closer to the Sun than they are now, and have migrated to their current orbit. The reason for this migration is that they were form in a large proto-planetary disk, full of planetesimals which drove migration. The asteroids are some of these planetesimals. This raises the following question: could the coorbital (or Lagrangian) asteroids survive this migration?

Long-term numerical integrations

Addressing this problem requires long-term and intensive numerical simulations. The issue is this: you need to simulate the evolution of the Solar System over 4.5 Gyr. For that, you write down the gravity equations ruling the motion of the planets and the planetesimals (these are many objects… the authors considered 60,000 of them), and you propagate them numerically.

To propagate them, you start from a given position and velocity of each of your bodies (initial conditions), and the equations give you the time-derivative at this point. You then use it to extrapolate the trajectory in the time, and you reiterate…

Of course, this algorithm does not give exact results. To lower the error, you should take a small time-step, but a too small time-step requires more iterations, and at each iteration you add an error due to the internal accuracy of the computer. To make your life easier, numerical integrators have been developed to improve the accuracy for a given time-step. In this study, the authors use two very well-known tools, SWIFT and MERCURY, dedicated to the integration of the motion of the planets and asteroids.

In this paper

The authors show it is difficult to get Trojans of Neptune that survive over the lifetime of the Solar System. In a first numerical integration, they do get captures, but none of them survive. Then they consider planetesimals which are very close to the observed Trojan, and they get some captures.

Something interesting is that they show that the orbital inclination of these Trojans can be excited during the migration process. For that, the migration should be slow enough, i.e. over 150 Myr, while previous studies, which assumed a migration ten times faster, did not excite the inclinations up to observed values.

Some perspectives

Even if it is now accepted that the planets have migrated, several competing scenarios exist (Nice, Nice 2, Grand Tack,…) and some are probably to come, just because there are many ranges of initial conditions which are possible, many possible assumptions on the initial state of the proto-planetary nebula… and these scenarios should of course impact the capture of Trojans of the giant planets.


New clues on the interior of Mercury

Hi there! Thanks for coming on the Planetary Mechanics Blog.

Today I will tell you about new results on the interior of the planet Mercury, by Ashok Kumar Verma and Jean-Luc Margot.
Mercury has been orbited during 4 years by the spacecraft MESSENGER, and gravity data have been derived from the deviations of the spacecraft. These data tell us how the mass is distributed in the planet.


Planet Mercury facts

Mercury is the innermost planet of the Solar System. Its radius is about one third of the one of the Earth, and its closeness to the Sun associated with the absence of an atmosphere induces large temperature variations between the day and the night. Another consequence is its very slow rotation, i.e. a Hermean (Mercurian) day lasts 58 terrestrial days, while its revolution around the Sun lasts 88 days, which is exactly 50% longer! This phenomenon is called a 3:2 spin-orbit resonance state, it is a unique case in the Solar System but is somehow analogous to the spin-orbit synchronization of our Moon. It is a consequence of the Solar tides, which despin the planet.

A last interesting fact I would like to mention is that Mercury is too dense for a such a small planet. This suggests that in the early ages of the Solar System, the proto-Mercury was much bigger, and differentiated between a core of pretty heavy elements and a less dense mantle. And then, Mercury has been stripped from this mantle, either slowly, or because of a catastrophic event, i.e. an impact.


The missions to Mercury

Sending a spacecraft to Mercury is a challenge, once more because of the proximity of the Sun. Not only the spacecraft should be protected from the Solar radiations, heat,… but it also tends to fall on the Sun instead of visiting the planet. For these reasons, only two spacecrafts have visited the Mercury up to now:

  • the US spacecraft Mariner 10 made 3 flybys of Mercury in 1974-1975. It mapped 45% of the surface and measured a magnetic field,
  • the US spacecraft MESSENGER orbited Mercury during 4 years between March 2011 and April 2015. It gave us invaluable information on the planet, including the ones presented here,
  • and let me mention the European-Japanese mission Bepi-Colombo, which should be launched to Mercury in April 2018.


The rotation of Mercury

The rotation of Mercury is in a resonant state, known as 3:2 spin-orbit resonance. This is a dynamical equilibrium reached after dissipation of its rotational energy, in which

  • Mercury rotates about one axis,
  • this axis is nearly perpendicular to its orbit, the deviation, named obliquity, being a signature of the interior,
  • the rotation and orbital periods are commensurate, here with a ratio 3/2. Around this exact commensurability are small librations, due to the periodic variations of the Solar gravitational torque acting on Mercury. The main period of these librations is the orbital one, i.e. 88 days, which is a direct consequence of Mercury’s eccentric orbit. They are supplemented by smaller oscillations, at harmonics of the orbital period (44 d, 29 d, 22 d, etc…), and at the periods of the other planets, meaning that they result from the planetary perturbations on the orbit of Mercury. The largest of these perturbations is expected to be due to Jupiter, but it has not been measured yet.


What the rotation can tell us

An issue in the pre-MESSENGER era was: does Mercury have an at least partially molten (outer) core? We now know that it has, thanks to Peale’s experiment, due to the late Stan Peale. The idea was this: the viscous core responds like a fluid to short-period excitations, and like a rigid body for long-period (secular) excitations. And the good thing is that the librations (called longitudinal physical librations) are due to a 88 d-oscillations, while the obliquity is due to a secular one (actually an oscillation which is some 200 kyr periodic, i.e. the rotation of the orbital plane of Mercury). So, in measuring these 2 quantities, one should be able to invert for the size of the core. This was achieved in 2007 thanks to radar measurements of the rotation of Mercury, and confirmed from additional Earth-based measurements, and MESSENGER data, since.

We now know that Mercury has a large molten core, which does not rule out the presence of a solid inner core. For that, additional investigations should be conducted.


The gravity field

The most basic model of gravity is the point-mass, which just gives us a mean density of the planet. This can be obtained from planetary ephemerides, i.e. in studying how Mercury affects the motion of the other planets, and with more accuracy from the deviations of the spacecraft. We know since Mariner 10 that Mercury has a density of 5.43 g/cm3, while 1g/cm3 is expected for ice, 3.3 g/cm3 for silicates, and 8 g/cm3 for iron.

A more accurate model is to see Mercury has a triaxial ellipsoid. This requires to add two parameters in the gravity field: J2 and C22, also know as Stokes coefficients. A positive J2 means that the body is flattened at its poles, while C22 represents the equatorial ellipticity of the planet. A positive polar flattening is expected as a consequence of the rotation of the planet, while the equatorial ellipticity can result from differential gravitational action of the Sun, i.e. the tides.

Knowing these two Stokes coefficients is possible from gravity data, and this would give us the triaxility of the mass distribution in Mercury. But something is missing: we do not know its radial distribution, i.e. heavier elements are expected to be in the core. For that, we need the polar momentum C, which could be derived from the obliquity, knowing the Stokes coefficient.

For a spherical homogeneous body, C=2/5 MR2, M being the mass and R the radius, and is smaller when heavier elements are concentrated in the core.


The tidal Love coefficient k2

The tides tend to alter the shape of the planet. In addition to a mean shape, there are periodic variations, which are due to the variations of the distance between Mercury and the Sun.

The amplitude of these variations depend on the Love parameter k2, which characterizes the response of the material to the periodic excitations. Actually, k2 depends on the frequency of excitation, in the specific case of Mercury k2@88d and k2@44d affect the gravity field. But distinguishing these two quantities requires a too high accuracy in the data, this is why k2 is often mentioned without precising the frequency involved.

If Mercury were spherical and fluid, k2 would be 1.5, while it would be null if Mercury were fully rigid. Actually, all the natural bodies are somewhere between these two end-members.

The frequency-dependence of the tides is based on the assumption that if you impose a slow deformation of a viscous body, it will not loose any internal energy and slowly recover its shape after (elastic deformation). However, rapid solicitations induce permanent deformations. The numbers associated with these two different regimes depend on the interior of the planet.


In this paper

This study, Mercury’s gravity tides, and spin from MESSENGER radio data, by A.K. Verma and J.-L. Margot, has been accepted for publication in Journal of Geophysical Research – Planets. It presents

  • an updated gravity field for Mercury,
  • an updated Love number,
  • an updated spin orientation.

These results are based on measurements of the instantaneous gravity field of Mercury. This is particularly interesting for the determination of the spin, since classical methods are based on the observation of the surface, while the gravity field is ruled by the whole planet. This means that here, the rotation of the whole planet is observed, not just its surface. This allows to constrain the possible differential rotation between the surface and the core.

For the gravity field of Mercury, a 40th order solution is considered, because Mercury is something more complicated than a triaxial ellipsoid. The second order Stokes coefficients are consistent with previous studies, which were also based on MESSENGER data. Some higher-order coefficients are identified as well.

This is the second determination of the Love number k2 = 0.464, which implies than the mantle of Mercury is pretty hot.


Some perspectives

We are some years away from the orbital insertion of the European / Japanese mission Bepi-Colombo, which is expected to be ten times more accurate than MESSENGER. So, results like the ones presented here are in some sense preparing the Bepi-Colombo’s measurements. This mission will also secure the results, and providing independent determinations.

Knowing Mercury is also a way to understand planetary formation. There are many discoveries of exoplanets, which orbit close to their parent star, but are so far from us that we cannot hope to send spacecrafts orbiting them. So, understand the way Mercury has been formed helps understanding the other planetary systems.

I hope that one day we will be able to measure the frequency-dependence of the Love numbers, this would be very helpful to constrain the tidal models.


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