Tag Archives: internal structure

Planets

Evolution of Venus’ crust

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Hi there! Of course, you know Venus. This planet is sometimes nicknamed the twin sister of the Earth, but beside its size, it does not look like the Earth. Venus is closer to the Sun than us, and it has a very thick atmosphere, which is essentially composed of carbon dioxide. This atmosphere has a pressure of 93 bar at the surface of the planet, to be compared with 1 bar for the Earth, and the temperature reaches there 470°C. Definitely hostile.

Anyway, I do not speak of the atmosphere today, but of the surface. I present Inferences on the mantle viscosity structure and the post-overturn evolutionary state of Venus, by T. Rolf and collaborators, which has recently been published in Icarus.

Outline

The interior of Venus
Gravity and topography
Modeling the crustal evolution
Stagnant-lid vs. overturn
Overturn should have happened
The study and its authors

The interior of Venus

Given its size, i.e. a diameter of 12,000 km, which is 95% of the one of the Earth, Venus must be differentiated. It has a crust, a mantle, and core, with increasing densities when you go deeper below the surface. We think the crust to be essentially basaltic, while the core must contain heavy elements. Surprisingly, the space missions did not detect any magnetic field, which means that the core may be not solid, or may be not cooling…

The outer part of the mantle should be fluid, which means that a fluid layer separates the core from the mantle. We know very few of the thicknesses and the compositions of these different layers. Actually, these could only be guessed from the measurements we dispose on, which are the gravity and the topography (see just below). Once you know the gravity field of Venus and its topography, you can elaborate interior models, which would be consistent with your data.

Gravity and topography

First, gravity. When a small body, like an artificial satellite, orbits a spherical planetary body, the gravitational perturbation affecting its motion depends only on the distance between the satellite and the planet. Now, if the planet is not spherical, and has mass anomalies, then the perturbation will not only depend on the distance, but also on the direction planet-satellite. You can determine the gravity field from the orbital deviation of your spacecraft.

It is convenient to write the gravity field as a sum of spherical harmonics. The first term (order 0) is a spherical one, then the order 2 (you have no order 1 if the center of your reference frame is the center of mass) represents the triaxiality of the planet, i.e. the planet seen as a triaxial ellipsoid. And the higher order terms will represent anomalies, with increasing resolutions. These resolutions are modeled as spatial periods. Such a representation has usually an efficient convergence, except for highly elongated bodies (see here).

We use such a representation for the topography as well. The difference is that the result is not the gravity field in any direction, but the altitude of the surface for a given point, i.e. a latitude and a longitude. The spacecraft measure the topography with a laser, which echo gives you the distance between the spacecraft and the surface. The altitude is directly deduced from this information.

Topography of Venus. The altitude variations are about 13 km with respect to a reference ellipsoid. © Calvin Hamilton, Johns Hopkins University Applied Physics Laboratory
Topography of Venus. The altitude variations are about 13 km with respect to a reference ellipsoid. © Calvin Hamilton, Johns Hopkins University Applied Physics Laboratory

The best representations we dispose on for Venus come from the American spacecraft Magellan, which orbited Venus between 1990 and 1994. These representations go to the order 180.

Modeling the crustal evolution

In this study, the authors simulated possible evolutionary paths for the crust of Venus, and compared their results with the present Venus, i.e. the gravity and topography as we know them.

For that, they simulated the thermochemical evolution of Venus in using a numerical code, StagYY. This is a 3D-code, which models convection in the mantle, i.e. internal motions. This code is based on finite elements, i.e. the interior of Venus is split into small elements. This splitting is made following a so-called Yin-Yang grid, which is appropriate for spherical geometries. This code includes several features like phase transition (i.e. from solid to fluid, and conversely), compositional variations, partial melting and melt migration. Moreover, it is implemented for parallel computing.

In other words, these are huge calculations. The authors started with 10 simulations in which the crust was modeled as a single plate, i.e. a stagnant lid. The simulations differed by the modeling of the viscosity, and by the radiogenic heating rate. This is the heating of Venus by the decay of the radiogenic elements, which was most effective in the early Solar System.

Once these 10 simulations have run, the authors kept the one, which resulted in the closest Venus to the actual one, and introduced episodic overturns in it.

Stagnant-lid vs. overturn

Venus does not present any tectonic activity. Did it have some in the past? This is a question this study tried to answer.

An overturn is a sudden peak in the heat transfer from the core to the crust through the mantle, due to a too strong difference of temperature, i.e. when the mantle gets colder. Such an episodic phenomenon is triggered by a too thick crust, and results in a melting of this crust, in heating it. In other words, it regulates the thickness of the crust.

Overturns should have happened

And here are the results: the best stagnant-lid scenario, called S2 in the study, presents some discrepancy between the simulated present Venus and the observed one. These discrepancies are present in the topography, in the gravity field, and in the age of the surface. The surface is estimated to be between 0.3 and 1 Gyr old, while the best stagnant-lid scenario predicts that the most probable age is 0.25 Gyr… a little too young.

However, episodic overturns give a surface, which is 0.6 Gyr old. Moreover, the gravity and topography are much better fit. The only remaining problem is that this scenario should result in much detections of plumes than actually detected.

As the authors honestly recall, some physical phenomena were not considered, in particular the influence of the dense atmosphere, and intrusive volcanism. Anyway, this study strongly suggests that episodic overturn happened.

Further data will improve our understanding of Venus. Recently, the European Space Agency (ESA) has pre-selected 3 potential future space missions, including EnVision, i.e. an orbiter around Venus. The final decision is expected in 2021.

The study and its authors

And that’s it for today! Please do not forget to comment. You can also subscribe to the RSS feed, and follow me on Twitter, Facebook, Instagram, and Pinterest.

Asteroids: Main Belt

The system of (107) Camilla

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Hi there! I will present you today a fascinating paper. It aims at a comprehensive understanding of the system composed of an asteroid, (107) Camilla, and its two satellites. For that, the authors acquired, processed and used 5 different types of observations, from all over the world. A consequence is that this paper has many authors, i.e. 27. Its title is Physical, spectral, and dynamical properties of asteroid (107) Camilla and its satellites, by Myriam Pajuelo and 26 colleagues, and it has recently been published in Icarus. This paper gives us the shape of Camilla and its main satellites, their orbits, the mass of Camilla, its composition, its spin period,… I give you these results below.

Outline

The system of Camilla
5 different types of data
The orbit of the satellites
Spin and shape
The composition of these objects
The study and its authors

The system of Camilla

The asteroid (107) Camilla has been discovered in 1868 by Norman Pogson at Madras Observatory, India. It is located in the
outer Main-Belt, and more precisely it is a member of the Cybele group. This is a group of asteroids, named after the largest of them (65) Cybele, which is thought to have a common origin. They probably originate from the disruption of a single progenitor. I show you below some Camilla’s facts, taken from the JPL Small-Body Database Browser:

Discovery 1868
Semimajor axis 3.49 AU
Eccentricity 0.066
Perihelion 3.26 AU
Inclination 10.0°
Orbital period 6.52 yr

We have of course other data, which have been improved by the present study. Please by a little patient.

In 2001 the Hubble Space Telescope revealed a satellite of Camilla, S1, while the second satellite, S2, and has been discovered in 2016 from images acquired by the Very Large Telescope of Cerro Paranal, Chile. This makes (107) Camilla a ternary system. Interesting fact, there is at least another ternary system in the Cybele group: the one formed by (87) Sylvia, and its two satellites Romulus and Remus.

Since their discoveries, these bodies have been re-observed when possible. This resulted in a accumulation of different data, all of them having been used in this study.

5 different types of data

The authors acquired and used:

  • optical lightcurves,
  • high-angular-resolution images,
  • high-angular-resolution spectrum,
  • stellar occultations,
  • near-infrared spectroscopy.

You record optical lightcurves in measuring the variations of the solar flux, which is reflected by the object. This results in a curve exhibiting periodic variations. You can link their period to the spin period of the asteroid, and their amplitudes to its shape. I show you an example of lightcurve here.

High-angular-resolution imaging requires high-performance facilities. The authors used data from the Hubble Space Telescope (HST), and of 3 ground-based telescopes, equipped with adaptive optics: Gemini North, European Southern Observatory Very Large Telescope (VLT), and Keck. Adaptive optics permits to correct the images from atmospheric distortion, while the HST, as a space telescope, is not hampered by our atmosphere. In other words, our atmosphere bothers the accurate observations of such small objects.

A spectrum is the amplitude of the reflected Solar light, with respect to its wavelength. This permits to infer the composition of the surface of the body. The high-angular-resolution spectrum were made at the VLT, the resulting data also permitting astrometry of the smallest of the satellites, S2. These spectrum were supplemented by near-infrared spectroscopy, made with a dedicated facility, i.e. the SpeX spectrograph of the NASA InfraRed Telescope Facility (IRTF), based on Mauna Kea, Hawaii. Infrared is very sensitive to the temperature, this is why their observations require dedicated instruments, which need a dedicated cooling system.

Finally, stellar occultations consist to record the light of a star, which as some point is occulted by the asteroid you study. This is particularly interesting for a faint body, which you cannot directly observe. Such observations can be made by volunteers, who use their own telescopes. You can deduce clues on the shape, and sometimes on the presence of a satellite, from the duration of the occultation. In comparing the durations of the same occultation, recorded at different locations, you may even reconstruct the shape (actually a 2-D shape, which is projected on the celestial sphere). See here.

And from all this, you can infer the orbits of the satellites, and the composition of the primary (Camilla) and its main satellite (S1), and the spin and shape of Camilla.

The orbits of the satellites

All of these observations permit astrometry, i.e. they give you the relative location of the satellites with respect to Camilla, at given dates. From all of these observations, you fit orbits, i.e. you numerically determine the orbits, which have the smallest distances (residuals), with the data.

This is a very tough task, given the uncertainty of the recorded positions. For that, the authors used their own genetic-based algorithm, Genoid, for GENetic Orbit IDentification, which relies on a metaheuristic method to minimize the residuals. Many trajectories are challenged in this iterative approach, and only the best ones are kept. These remaining trajectories, designed as parents, are used to generate new trajectories which improve the residuals. This algorithm has proven its efficiency for other systems, like the binary asteroid (22) Kalliope-Linus. In such cases, the observations lack of accuracy and many parameters are involved.

You can find the results below.

S/2001 (107) 1
Semimajor axis 1247.8±3.8 km
Eccentricity <0.013
Inclination (16.0±2.3)°
Orbital period 3.71234±0.00004 d
S/2016 (107) 2
Semimajor axis 643.8±3.9 km
Eccentricity ~0.18 (<0.23)
Inclination (27.7±21.8)°
Orbital period 1.376±0.016 d

You can deduce the mass of (107) Camilla from these numbers, i.e. (1.12±0.01)x1019 kg. The ratio of two orbital periods probably rule out any significant mean-motion resonance between these two satellites.

Spin and shape

The authors used their homemade algorithm KOALA (Knitted Occultation, Adaptive-optics, and Lightcurve Analysis) to determine the best-fit solution (once more, minimization of the residuals) for spin period, orientation of the rotation pole, and 3-D shape model, from lightcurves, adaptive optics images, and stellar occultations. And you can find the solution below:

Camilla
Diameter 254±36 km
a 340±36 km
b 249±36 km
c 197±36 km
Spin period 4.843927±0.00004 h

This table gives two solutions for the shape: a spherical one, and an ellipsoid. In this last solution, a, b, and c are the three diameters. We can see in particular that Camilla is highly elongated. Actually a comparison between the data and this ellipsoid, named the reference ellipsoid, revealed two deep and circular basins at the surface of Camilla.

Moreover, a comparison of the relative magnitudes of Camilla and its two satellites, and the use of the diameter of Camilla as a reference, give an estimation of the diameters of the two satellites. These are 12.7±3.5 km for S1 and 4.0±1.2 km for S2. These numbers assume that S1 and S2 have the same albedo. This assumption is supported for S1 by the comparison of its spectrum from the one of Camilla.

The composition of these objects

In combining the shape of Camilla with its mass, the authors deduce its density, which is 1,280±130 kg/m3. This is slightly larger than water, while silicates should dominate the composition. As the authors point out, there might be some water ice in Camilla, but this pretty small density is probably due to the porosity of the asteroid.

The study and its authors

And that’s it for today! Please do not forget to comment. You can also subscribe to the RSS feed, and follow me on Twitter, Facebook, Instagram, and Pinterest.

Moon

The lunar history

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(Alternative title: The search for the origin of the Late Heavy Bombardment)

Hi there! It is a pleasure for me to present you today a multi-disciplinary study, which mixes celestial mechanics with geochemistry. If you want to know the past of a planetary body, you must go backward: you start from the body as you observe it nowadays, and from this you infer the processes which made it evolve from its formation to its present state. In The timeline of the Lunar bombardment – revisited, by A. Morbidelli, D. Nesvorný, V. Laurenz, S. Marchi, D.C. Rubie, L. Elkins-Tanton, M. Wieczorek and S. Jacobson, the authors exploit our observations of the craters and the chemistry of the Moon, and simulations of the motion of asteroids in the early Solar System, to give new constraints on the bombardment of the Moon between 3.9 and 3.7 Gyr (billions of years) ago, which is famous as the Late Heavy Bombardment (LHB). We will see that the results have implications for Mars. This study has recently been accepted for publication in Icarus.

Outline

The Lunar basins
Origin of the LHB: cataclysm or accretion tail?
What siderophile elements tell us
Simulating the bombardment
Cataclysm possible, accretion tail preferred
Tungsten is another marker
Implications for Mars
Predictions
The study and its authors

The Lunar basins

Let us start from what we observe: the Lunar surface. This is a heavily cratered surface. Actually, the absence of atmosphere preserves it from erosion, and the small size of the Moon limits its heating, as a consequence the craters neither erode nor relax. Hence, the surface of the Moon is a signature of the activity in the early Solar System.

Let us focus on the largest structures, i.e. the maria and the basins. The maria are lava plains, which result from a volcanic activity of the early Moon. However, the basins are the largest impact craters. You can find below the largest ones, of course many smaller craters exist.

Basin Diameter (km)
South Pole-Aitken 2,600
Imbrium 1,100
Orientale 930
Serenitatis 920
Australe 880
Nectaris 860
Smythii 740
Crisium 740
Tranquillitatis 700
Tsiolkovsky-Stark 700
Fecunditatis 690
Mutus-Vlacq 690
Nubium 690

The early Moon was hot, because of the impact which created it. As a hot body, it stratified into a fluid core, a mantle and a crust, while cooling. The visible impact craters are younger than the crust, i.e. they are younger than 3.9 Gyr, and were created at least 600 Myr after the formation of the Moon… pretty late, hence due to the Late Heavy Bombardment.

Orientale Basin. © NASA
Orientale Basin. © NASA

Origin of the LHB: cataclysm or accretion tail?

Late Heavy Bombardment means that the inner Solar System have been intensively bombarded late after its genesis. But how did that happen? Two scenarios can be found in the literature:

  1. Cataclysm: the very young Solar System was very active, i.e. composed of many small bodies which collided, partly accreting… and became pretty quiet during some hundreds of Myr… before suddenly, a new phase of bombardment occurred.
  2. Accretion tail: the Solar System had a slowly decreasing activity, and the craters on the Moon are just the signature of the last 200 Myrs. The previous impacts were not recorded, since the surface was still molten.

The second scenario could be preferred, as the simplest one. The first one needs a cause which would trigger this second phase of bombardment. Anyway, many numerical simulations of the early Solar System got such an activity, as a dynamical phenomenon destabilizing the orbits of a group of small bodies, which themselves entered the inner Solar System and collided with the planets, accreting on them. The giant planets Jupiter and Saturn have a dominant dynamical influence on the small bodies of the Solar System, and could have triggered such an instability. One of the theories existing in the literature is the E-Belt, for extended belt. It would have been an internal extension of the Main Belt of asteroids, which would have been destabilized by a secular resonance with Saturn, and has finished as the impactors of the LHB. Why not, this is a theory.

When you model phenomena having occurred several billions years ago, you have so many uncertainties that you cannot be certain that your solution is the right one. This is why the literature proposes several scenarios. Further studies test them, and sometimes (this is the case here) give additional constraints, which refine them.

Thanks to the Apollo mission, samples of the Moon have been analyzed on Earth, and geochemistry can tell us many things on the history of a body. For the Moon, focus has been put on siderophile elements.

What siderophile elements tell us

A siderophile element is a chemical element which has affinity with iron. Among these elements are iron, iridium, palladium, platinum, rubidium… When a planetary body is hot, it tends to differentiate, and its heaviest elements, i.e. iron, migrate to the core. This results in a depletion of highly siderophile elements (HSE). Since a very small abundance of these elements has been observed, then we have no problem, thank you…

NO NO NO there is actually a problem, since these siderophile elements should be present in the impactors, which are supposed to have accreted on the Moon AFTER its stratification… yes we have a problem.

But some of the authors have shown recently that on Earth, another phenomenon could remove the HSEs from the crust, well after the formation of the core: the exsolution and segregation of iron sulfide. In other words, the bombardment could have brought more HSEs than currently recorded. And this motivates to revisite the history of the Lunar bombardment.

Simulating the bombardment

So, the observations are: the craters, and the HSEs. The craters are not only the basins, but also the smaller ones, with diameters larger than 1 km. Even smaller craters could be used, but the data are considered to be reliable, i.e. exhaustive, for craters larger than 1 km. From that size to the large basins, we can fit a function of distribution, i.e. number of craters vs. diameter. Since there is an obvious correlation between the size of a crater and the one of the impactor, a population of craters corresponds to a population of impactors.

The authors dispose of statistics of collisions, which could be seen as mass accretion, between the Moon and small bodies during the early ages of the Solar System. These statistics result from numerical simulations conducted by some of them, and they can be fine-tuned to fit the crater distribution, their estimated ages, and the abundance of highly siderophile elements. Fine-tuning the statistics consist in artificially moving the parameters of the simulation, for instance the initial number of small bodies, or the date of the instability provoking the cataclysm, in the cataclysm scenario.

Cataclysm possible, accretion tail preferred

And here is the result: if the HSEs are only due to the mass accretion after the cooling of the Lunar crust, then the observations can only be explained by the cataclysm, i.e. the LHB would be due to a late instability. This instability would have resulted in a mass accretion from comets, and this raises another problem: this accretion seems to lack of primitive, carbonaceous material, while the comets contain some.

However, if the HSEs have been removed after the cooling of the crust, then the accretion tail scenario is possible.

We should accept that for this kind of study, the solution is not unique. A way to tend to the unicity of the solution is to discuss further implications, in examining other clues. And the authors mention the tungsten.

Tungsten is another marker

Tungsten is rather a lithophile than a siderophile element, at least in the presence of iron sulfide. In other words, even if it does not dislike iron, it prefers lithium (I like this way of discussing chemistry). Something puzzling is a significant difference in the ratios of two isotopes of tungsten (182W and 184W) between the Moon and the Earth. This difference could be primordial, as brought by the projectile which is supposed to have splitted the proto-Earth into the Earth and the Moon (nickname of the projectile: Theia), or it could be due to the post-formation mass accumulation. In that case, that would be another constraint on the LHB.

Implications for Mars

The LHB has affected the whole inner Solar System. So, if a scenario is valid for the Moon, it must be valid for Mars.
This is why the authors did the job for Mars as well. A notable difference is that Mars would be less impacted by comets than the Moon, and this would affect the composition of the accreted material. More precisely, a cataclysmic LHB would be a mixture of asteroids and comets, while an accretion tail one would essentially consist of leftover planetesimals. It appears that this last scenario, i.e. the accretion tail one, can match the distribution of craters and the abundance of HSEs. However, the cataclysmic scenario would not bring enough HSEs on Mars.

Predictions

This study tells us that the accretion tail scenario is possible. The authors show that it would imply that

  1. The quantity of remaining HSEs on the Moon is correlated with the crystallization of the Lunar magma ocean, which itself regulates the age of the earliest Lunar crust.
  2. For Mars, the Noachian era would have started 200 Myr earlier than currently thought, i.e. 4.3 Gyr instead of 4.1 Gyr. That period is characterized by high rates of meteorite and asteroid impacts and the possible presence of abundant surface water. Moreover, the Borealis formation, i.e. the northern hemisphere of Mars, which seems to be a very large impact basin, should have been formed 4.37 Gyr ago.

Further studies, explorations, space missions, lab experiments,… should give us new data, which would challenge these implications and refine these scenarios. So, the wording prediction can seem weird for past phenomena, but the prediction is for new clues.

The study and its authors

And that’s it for today! Please do not forget to comment. You can also subscribe to the RSS feed, and follow me on Twitter and Facebook.

Satellites of Saturn

Water-ice boundary on Titan

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

Outline

Titan
The quest for the internal ocean
Modeling the ice-water boundary
Resolution by spectral decomposition
The role of the grain size
In the future
The study and its authors

Titan

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

It has interesting physical characteristics:

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

The quest for the internal ocean

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

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

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

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

Modeling the ice-water boundary

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

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

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

Resolution by spectral decomposition

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

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

The role of the grain size

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

In the future

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

Refinements of the theory

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

More data?

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

The study and its authors

And that’s it for today! Please do not forget to comment. You can also subscribe to the RSS feed, and follow me on Twitter and Facebook.

Asteroids: Main Belt, Comets

Breaking an asteroid

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Hi there! Asteroids, these small bodies in the Solar System, are fascinating by the diversity of their shapes. This is a consequence of their small sizes. Another consequence is their weakness, which itself helps to split some of them into different parts, sometimes creating binary objects, asteroids families… The study I present you today, Internal gravity, self-energy, and disruption of comets and asteroids, by Anthony R. Dobrovolskis and Donald G. Korycansky, proposes an accurate computation of the required energy to provoke this break-up, at any place of the asteroid, i.e. you are more efficient when you hit at a given location. This study has recently been accepted for publication in Icarus.

Outline

Shapes of asteroids
The energies involved
Kleopatra and Churyumov-Gerasimenko
Numerical modeling
Results
The study and its authors

Shapes of asteroids

Please allow me, in this context, to call asteroid a comet, a comet being a small body, i.e. like an asteroid, but with a cometary activity. The important thing is that the involved bodies are small enough.

Beyond a given size, i.e. a diameter of ~400 km, a planetary body is roughly spheroidal, i.e. it is an ellipsoid with it two equatorial axes almost equal and the polar one smaller, because of its rotation. For a tidally despun body, like the Moon, or a satellite of a giant planet, the shape is more triaxial, since the tidal (gravitational) action of the parent planet tends to elongate the equatorial plane. The same phenomenon affects Mercury.

However, for smaller bodies, the self-gravitation is not strong enough to make the body look more or less like a sphere. As a consequence, you can have almost any shape, some bodies are bilobate, some are contact binaries, i.e. two bodies which permanently touch together, some others are rubble piles, i.e. are weak aggregates of rocks, with many voids.

These configurations make these bodies likely to undergo or have undergone break-up. This can be quantified by the required energy to extract some material from the asteroid.

The energies involved

For that, an energy budget must be performed. The relevant energies to consider are:

  • The impact disruption energy: the minimum kinetic energy of an impactor, to shatter the asteroid and remove at least half of its mass,
  • The shattering energy: the minimum energy needed to shatter the asteroid into many small pieces. It is part of the impact disruption energy. This energy is roughly proportional to the mass of the asteroid. It represents the cohesion between the adjacent pieces.
  • The binding energy: this energy binds the pieces constituting the asteroid. In other words, once you have broken an asteroid (don’t try this at home!), you have to make sure the pieces will not re-aggregate… because of the binding energy. For that, you have to bring enough energy to disperse the fragments.
  • The self-gravitational energy: due to the mutual gravitational interaction between the blocks constituting the asteroids. Bodies smaller than 1 km are strength-dominated, i.e. they exist thanks to the cohesion between the blocks, which is the shatter energy. However, larger bodies are gravity-dominated.
  • The kinetic energy of rotation: the spin of these bodies tends to enlarge the equatorial section. In that sense, it assists the break-up process.

This study addresses bodies, which are far enough from the Sun. This is the reason why I do not mention its influences, i.e. the tides and the thermic effects, which could be relevant for Near-Earth Objects. In particular, the YORP effect is responsible for the fission of some of them. I do not mention the orbital kinetic energy of the asteroid either. Actually the orbital motion is part of the input energy brought by an impact, since the relative velocity of the impactor with respect to the target is relevant in this calculation.

I now focus on the two cases studied by the authors to illustrate their theory: the asteroid Kleopatra and the comet 67P/Churyumov-Gerasimenko.

2 peculiar cases: Kleopatra and Churyumov-Gerasimenko

216 Kleopatra is a Main-Belt asteroid. Adaptive optics observations have shown that is is constituted of two masses bound by material, giving a ham-bone shaped. As such, it can be considered as a contact binary. It is probably a rubble pile. Interestingly, observations have also shown that Kleopatra has 2 small satellites, Alexhelios and Cleoselene, which were discovered in 2008.

Reconstruction of the shape of Kleopatra. © NASA
Reconstruction of the shape of Kleopatra. © NASA

However, 67P Churyumov-Gerasimenko is a Jupiter-family comet, i.e. its aphelion is close to the orbit of Jupiter, while its perihelion is close to the one of the Earth. It has an orbital period of 6.45 years, and was the target of the Rosetta mission, which consisted of an orbiter and a lander, Philae. Rosetta orbited Churyumov-Gerasimenko between 2014 and 2016. The shape of this comet is sometimes described as rubber ducky, with two dominant masses, a torso and a head, bound together by some material, i.e. a neck.

Churyumov-Gerasimenko seen by Rosetta. © ESA
Churyumov-Gerasimenko seen by Rosetta. © ESA
216 Kleopatra 67P/Churyumov-Gerasimenko
Semimajor axis 2.794 AU 3.465 AU
Eccentricity 0.251 0.641
Inclination 13.11° 7.04°
Spin period 5.385 h 12.761 h
Mean radius 62 km 2.2 km
Magnitude 7.30 11.30
Discovery 1880 1969

The irregular shapes of these two bodies make them interesting targets for a study addressing the gravitation of any object. Let us see now how the authors addressed the problem.

Numerical modeling

Several models exist in the literature to address the gravity field of planetary bodies. The first approximation is to consider them as spheres, then you can refine in seeing them as triaxial ellipsoids. For highly irregular bodies you can try to model them as cuboids, and then as polyhedrons. Another way is to see them as duplexes, this allows to consider the inhomogeneities dues to the two masses constituting bilobate objects. The existence of previous studies allow a validation of the model proposed by the authors.

And their model is a finite-element numerical modeling. The idea is to split the surface of the asteroid into small triangular planar facets, which should be very close to the actual surface. The model is all the more accurate with many small facets, but this has the drawback of a longer computation time. The facets delimit the volume over which the equations are integrated, these equations giving the local self-gravitational and the impact disruption energies. The authors also introduce the energy rebate, which is a residual energy, due to the fact that you can remove material without removing half of it. This means that the impact disruption energy, as it is defined in the literature, is probably a too strong condition to have extrusion of material.
The useful physical quantities, which are the gravitational potential, the attraction, and the surface slope, are propagated all along the body thanks to a numerical scheme, which accuracy is characterized by an order. This order quantifies the numerical approximation which is made at each integration step. A higher order is more accurate, but is computationally more expensive.

Once the code has been run on test cases, the authors applied it on Kleopatra and Churyumov-Gerasimenko, for which the shape is pretty well known. They used meshes of 4,094 and 5,786 faces, respectively.

Results

The validation phase is successful. The authors show that with a 3rd order numerical scheme, they recover the results present in the literature for the test cases with an accuracy of ~0.1%, which is much better than the accuracy of the shape models for the real asteroids. Regarding Kleopatra and Churyumov-Gerasimenko, they get the gravity field at any location, showing in particular excesses of gravity at the two lobes.

Such a study is particularly interesting for further missions, which would determine the gravity field of asteroids, which would then be compared with the theoretical determination by this code. Other applications are envisaged, the authors mentioning asteroid mining.

The study and its authors

And that’s it for today! Please do not forget to comment. You can also subscribe to the RSS feed, and follow me on Twitter and Facebook.

And Merry Christmas!