Tag Archives: Topography

The lunar history

(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.

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.

Tides in the lakes of Titan

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

The lakes of Titan

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

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

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

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

Kraken and Ligeia Maria

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

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

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

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

Numerical modeling with SLIM

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

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

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

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

Low diurnal tides

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

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

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

Fluid exchanges between the lakes

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

The attenuation is critical

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

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

The study and its authors

And that’s it for today! Please do not forget to comment. You can also subscribe to the RSS feed, and follow me on Twitter and Facebook. And let me wish you a healthy and happy year 2018.

Water-ice boundary on Titan

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

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.

Breaking an asteroid

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.

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!

Resurfacing Ganymede

Hi there! After Europa last week, I tell you today on the next Galilean satellite, which is Ganymede. It is the largest planetary satellite in the Solar System, and it presents an interesting surface, i.e. with different terrains showing evidence of past activity. This is the opportunity for me to present you Viscous relaxation as a prerequisite for tectonic resurfacing on Ganymede: Insights from numerical models of lithospheric extension, by Michael T. Bland and William B. McKinnon. This study has recently been accepted for publication in Icarus.

The satellite Ganymede

Ganymede is the third, by its distance to the planet, of the 4 Galilean satellites of Jupiter. It was discovered with the 3 other ones in January 1610 by Galileo Galilei. These are indeed large bodies, which means that they could host planetary activity. Io is known for its volcanoes, and Europa and Ganymede (maybe Callisto as well) are thought to harbour a global, subsurfacic ocean. The table below lists their size and orbital properties, which you can compare with the 5th satellite, Amalthea.

Semimajor axis Eccentricity Inclination Radius
J-1 Io 5.90 Rj 0.0041 0.036° 1821.6 km
J-2 Europa 9.39 Rj 0.0094 0.466° 1560.8 km
J-3 Ganymede 14.97 Rj 0.0013 0.177° 2631.2 km
J-4 Callisto 26.33 Rj 0.0074 0.192° 2410.3 km
J-5 Amalthea 2.54 Rj 0.0032 0.380° 83.45 km

We have images of the surface of Ganymede thanks to the spacecraft Voyager 1 & 2, and Galileo. These missions have revealed different types of terrains, darker and bright, some impacted, some pretty smooth, some showing grooves… “pretty smooth” should be taken with care, since the feeling of smoothness depends on the resolution of the images, which itself depends on the distance between the spacecraft and the surface, when this specific surface element was directed to the spacecraft.

Dark terrain in Galileo Regio. © NASA
Dark terrain in Galileo Regio. © NASA
Bright terrain with grooves and a crater. © NASA
Bright terrain with grooves and a crater. © NASA

A good way to date a terrain is to count the craters. It appears that the dark terrains are probably older than the bright ones, which means that a process renewed the surface. The question this paper addresses is: which one(s)?

Marius Regio and Nippur Sulcus. © NASA
Marius Regio and Nippur Sulcus. © NASA

Resurfacing a terrain

These four mechanisms permit to renew a terrain from inside:

  • Band formation: The lithosphere, i.e. the surface, is fractured, and material from inside takes its place. This phenomenon is widely present on Europa, and probably exists on Ganymede.
  • Viscoelastic relaxation: When the crust has some elasticity, it naturally smooths. As a consequence, craters tend to disappear. Of course, this phenomenon is a long-term process. It requires the material to be hot enough.
  • Cryovolcanism: It is like volcanism, but with the difference that the ejected material is mainly composed of water, instead of molten rock. Part of the ejected material falls on the surface.
  • Tectonics: Extensional of compressional deformations of the lithosphere. This is the phenomenon, which is studied here.

Beside these processes, I did not mention the impacts on the surface, and the erosion, which is expected to be negligible on Ganymede.

The question the authors addressed is: could tectonic resurfacing be responsible for some of the actually observed terrains on Ganymede?

Numerical simulations

To answer this question, the authors used the numerical tool, more precisely the 2-D code Tekton. 2-D means that the deformations below the surface are not explicitly simulated. Tekton is a viscoelastic-plastic finite element code, which means that the surface is divided into small areas (finite elements), and their locations are simulated with respect to the time, under the influence of a deforming cause, here an extensional deformation.

The authors used two kinds of data, that we would call initial conditions for numerical simulations: simulated terrains, and real ones.
The simulated terrains are fictitious topographies, varying by the amplitude and frequency of deformation. The deformations are seen as waves, the wavelength being the distance between two peaks. A smooth terrain can be described by long-wavelength topography, while a rough one will have short wavelength.
The real terrains are Digital Terrain Models, extracted from spacecraft data.

The authors also considered different properties of the material, like the elasticity, or the cohesion.

A new scenario of resurfacing

It results from the simulations that the authors can reproduce smooth terrains with grooves, starting from already smooth terrains without grooves. However, extensional tectonics alone cannot remove the craters. In other words, if you can identify craters at the surface of Ganymede, after millions of years of extensional tectonics you will still observe them. To make smooth terrains, you need the assistance of another process, the viscoelastic relaxation of the lithosphere being an interesting candidate.

This pushed the authors to elaborate a new scenario of resurfacing of Ganymede, involving different processes.
They consider that the dark terrains are actually the eldest ones, having remaining intact. However, there was indeed tectonic resurfacism of the bright terrains, which formed grooved. But the deformation of the lithosphere was accompanied by an elevation of the temperature (which is not simulated by Tekton), which itself made the terrain more elastic. This elasticity itself relaxed the craters.

Anyway, you need elasticity (viscoelasticity is actually more accurate, since you have energy dissipation), and for that you need an elevation of the local temperature. This may have been assisted by heating due to internal processes.

In the future

Ganymede is the main target of the ESA mission JUICE, which should orbit it 2030. We expect a big step in our knowledge of Ganymede. For this specific problem, we will have a much better resolution of the whole surface, the gravity field of the body (which is related to the interior), maybe a magnetic field, which would constrain the subsurface ocean and the depth of the crust enshrouding it, and the Love number, which indicates the deformation of the gravity field by the tidal excitation of Jupiter. This last quantity contains information on the interior, but it is related to the whole body, not specifically to the structure. I doubt that we would have an accurate knowledge of the viscoelasticity of the crust. Moreover, the material properties which created the current terrains may be not the current ones; in particular the temperature of Ganymede is likely to have varied over the ages. We know for example that this temperature is partly due to the decay of radiogenic elements shortly after the formation of the satellite. During this heating, the satellite stratifies, which alters the tidal response to the gravitational excitation of Jupiter, and which itself heats the satellite. This tidal response is also affected by the obliquity of Ganymede, by its eccentricity, which is now damped… So, the temperature is neither constant, nor homogeneous. There will still be room for theoretical studies and new models.

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.