Tag Archives: Topography

Fracturing the crust of an icy satellite

Hi there! You may know that the space missions to the systems of giant planets have revealed that the surface of several of theirs satellites are fractured. We dispose of images of such structures on Jupiter’s Europa and Ganymede, Saturn’s Enceladus (the famous tiger stripes at its South Pole), and even on Uranus’ Miranda, which has been visited by Voyager II. These satellites are thought to be icy bodies, with an icy crust enshrouding a subsurface, global ocean (maybe not for Miranda, but certainly true for the other guys).

The study I present you today, Experimental constraints on the fatigue of icy satellite lithospheres by tidal forces, by Noah P. Hammond, Amy C. Barr, Reid F. Cooper, Tess E. Caswell, and Greg Hirth, has recently been accepted for publication in Journal of Geophysical Research: Planets. The authors particularly tried to produce in labs the process of fatigue, which would weaken a material after a certain number of solicitations, i.e. it would become easier to break.

Cycloids on Europa

The Galilean satellite of Jupiter Europa may be the most interesting satellite to focus on, since it is the most fractured, at least to the best of our knowledge. The observation of the surface of Europa, first by Voyager I and II in 1979, and after by Galileo between 1995 and 2003, revealed many structures, like lineae, i.e. cracks, due to the geophysical activity of the satellite. This body is so active that only few craters are visible, the surface having been intensively renewed since the impacts. Something particularly appealing on Europa is that some of these lineae present a cycloidal pattern, which would reveal a very small drift of the orientation of the surface. Some interpret it has an evidence of super-synchronous rotation of Europa, i.e. its rotation would not be exactly synchronous with its orbital motion around Jupiter.

Cycloids on Europa, seen by the spacecraft Galileo. © NASA
Cycloids on Europa, seen by the spacecraft Galileo. © NASA

Beside Europa, fractures have also been observed on Ganymede, but with less frequency. For having such fractures, you need the surface to be brittle enough, so that stress will fracture it. This is a way to indirectly detect a subsurface ocean. But you also need the stress. And this is where tides intervene.

Fractures on Ganymede. © Paul M. Schenk
Fractures on Ganymede. © Paul M. Schenk

Tides can stress the surface

You can imagine that Jupiter exerts a huge gravitational action on Europa. But Europa is not that small, and its finite size results in a difference of Jovian attraction between the point which is the closest to Jupiter, and the furthest one. The result of this differential attraction is stress and strain in the satellite. The response of the satellite will depend on its structure.

A problem is that calculations suggest that the tidal stress may be too weak to generate alone the observed fractures. This is why the authors suggest the assistance of another phenomenon: fatigue crack growth.

The phenomenon of fatigue crack growth

The picture is pretty intuitive: if you want to break something… let’s say a spoon. You twist it, bend it, wring it… once, twice, thrice, more… Pretty uneasy, but you do not give up, because you see that the material is weakening. And finally it breaks. Yes you did it! But what happened? You slowly created microcracks in the spoon, which weakened it, the cracks grew… until the spoon broke.

For geophysical materials, it works pretty much the same: we should imagine that the tides, which vary over an orbit since the eccentricity of the orbit induces variations of the Jupiter-Europa distance, slowly create microcracks, which then grow, until the cracks are visible. To test this scenario, the authors ran lab experiments.

Lab experiments

The lab experiments consisted of Brazil Tests, i.e. compression of circular disks of ice along their diameter between curved steel plates. The resulting stress was computed everywhere in the disk thanks to a finite-element software named Abaqus, and the result was analyzed with acoustic emissions, which reflections would reveal the presence of absence of microcracks in the disk. The authors ran two types of tests: both with cyclic loading, i.e. oscillating loading, but one with constant amplitude, and the other one with increasing amplitude, i.e. a maximum loading becoming stronger and stronger.

But wait: how to reproduce the conditions of the real ice of these satellites? Well, there are things you cannot do in the lab. Among the problems are: the exact composition of the ice, the temperature, and the excitation frequency.

The authors conducted the experiments in assuming pure water ice. The temperature could be below 150 K (-123°C, or -189°F), which is very challenging in a lab, and the main period of excitation is the orbital one, i.e. 3.5 day… If you want to reproduce 100,000 loading cycles, you should wait some 1,000 years… unfeasible…

The authors bypassed these two problems in constraining the product frequency times viscosity to be valid, the viscosity itself depending on the temperature. This resulted in an excitation period of 1 second, and temperatures between 198 and 233 K (-75 to -40°C, or -103 to -40°F). The temperature was maintained thanks to a liquid nitrogen-cooled, ethanol bath cryostat.

And now the results!

No fatigue observed

Indeed, the authors observed no fatigue, i.e. no significant microcracks were detected, which would have altered the material enough, to weaken it. This prompted the authors to discuss the application of their experiments for understanding the crust of the real satellites, and they argue that fatigue could be possible anyway.

Why fatigue may still be possible

As the authors recall, these experiments are not the first ones. Other authors have had a negative result with pure water ice. However, fatigue has been detected on sea ice, which could mean that the presence of salt favors fatigue. And the water ice of icy satellite may not be pure. Salt and other chemical elements may be present. So, even if these experiments did not reveal fatigue, there may be some anyway.

But the motivation for investigating fatigue is that a process was needed to assist the tides to crack the surface. Why necessarily fatigue? Actually, other processes may weaken the material.

How to fracture without fatigue

The explanation is like the most (just a matter of taste) is impacts: when you impact the surface, you break it, which necessarily weakens it. And we know that impacts are ubiquitous in the Solar System. In case of an impact, a megaregolith is created, which is more likely to get fractured. The authors also suggest that the tides may be assisted, at least for Europa, by the super-synchronous rotation possibly suggested by the geometry of the lineae (remember, the cycloids). Another possibility is the large scale inhomogeneities in the surface, which could weaken it at some points.

Anyway, it is a fact that these surfaces are fractured, and the exact explanation for that is still in debate!

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.

Ice on Mercury

Hi there! You know Mercury, the innermost planet of the Solar System. It has recently been explored during 4 years by the American spacecraft MESSENGER, which gave us invaluable data on its surface, its magnetic field, its interior…
Today I present you a study on the ice on Mercury. It is entitled Constraining the thickness of polar ice deposits on Mercury using the Mercury Laser Altimeter and small craters in permanently shadowed regions, by Ariel N. Deutsch, James W. Head, Nancy L. Chabot & Gregory A. Neumann, and has recently been accepted for publication in Icarus.
We know that there is some ice at the surface of Mercury, and the study wonders how much. Since Mercury is close to the Sun, its surface is usually hot enough to sublimate the ice… except in permanently shadowed regions, i.e. in craters. For that, the authors compared the measured depth of small craters, and compared it with the expected depth from the excavation of material by an impactor. The difference is supposed to be ice deposit.

Mercury and MESSENGER

The planet Mercury is known at least since the 14th century BC. It was named after the Roman messenger god Mercurius, or Hermes in Greek, since the messengers saw it at dawn when they left, and at dusk when they arrived. The reason is that Mercury is in fact pretty close to the Sun, i.e. three times closer than our Earth. So, usually the Sun is so bright that it prevents us from observing it. Unless it is below the horizon, which happens at dawn and at dusk.

Mercury makes a full revolution around the Sun in 88 days, and a full rotation in 58 days. This 2/3 ratio is a dynamical equilibrium, named 3:2 spin-orbit resonance, which has been reached after slow despinning over the ages. This despinning is indeed a loss of energy, which has been favored by the tidal (gravitational) action of the Sun. This resulting spin-orbit resonant configuration is a unique case in the Solar System. A consequence is that the Solar day on Mercury lasts 176 days, i.e. if you live on Mercury, the apparent course of the Sun in the sky lasts 176 days.

The proximity of the Sun makes Mercury a challenge for exploration. Mariner 10 made 3 fly-bys of it in 1974-1975, mapping 45% of its surface, and measuring a tiny magnetic field. We had to wait until 2011 for the US spacecraft MESSENGER (MErcury Surface, Space ENvironment, GEochemistry and Ranging) to be the first human-made object inserted into orbit around Mercury. The orbital phase lasted 4 years, and gave us a full map of the planet, gravity data, accurate measurements of its rotation, a list of craters, measurements of the magnetic field,…

The instrument of interest today is named MLA, for Mercury Laser Altimeter. This instrument used an infrared laser (wavelength: 1,064 nanometers) to estimate the height of the surface from the reflection of the laser: you send a laser signal, you get it back some time later, and from the time you have the distance, since you know the velocity, which is the velocity of the light. And in applying this technique all along the orbit, you produce a map of the whole planet. This permits for instance to estimate the size and depth of the craters.

The Mercury Laser Altimeter (MLA).
The Mercury Laser Altimeter (MLA).

Ice on Mercury

The discovery of ice at the poles of Mercury was announced in 1992. It was permitted by Earth-based radar imagery made at Goldstone Deep Space Communications Complex in the Mojave desert, in California (USA). Ice is pretty easy to uncover, because of its high reflectivity. But this raises some questions:

  1. How can ice survive on Mercury?
  2. How much ice is there?
  3. How did it arrive?
The Goldstone facilities in 2018. © Google
The Goldstone facilities in 2018. © Google

The first question is not really a mystery. Because of its long Solar day and its absence of atmosphere (actually Mercury has a very tenuous exosphere, but we can forget it), Mercury experiences huge variations of temperature between day and night, i.e. from 100K to 700K, or -173°C to 427°C, or -279°F to 801°F (it is in fact not accurate at the 1°F level…). So, when a region is illuminated, the water ice is definitely not stable. However, there are regions, especially at the poles, which are never illuminated. There ice can survive.

The last two questions are answered by this study.

Ice is still present in craters

For not being illuminated, it helps to be close to a pole, but the topography can be helpful as well. The surface of Mercury is heavily cratered, and the bottoms of some of these craters are always hidden from the Sun. This is where the authors looked for ice. More precisely, they investigated 10 small craters within 10 degrees of the north pole. And for each of them, they estimated the expected depth from the diameter, and compared it with the measured depth. If it does not match, then you have water ice at the bottom. Easy, isn’t it?

The Carolan crater, one of the craters studied. © NASA
The Carolan crater, one of the craters studied. © NASA

Well, it is not actually that easy. The question is: did the water ice arrive after or before the excavation of the crater? If it arrived before, then the impactor just excavated some ice, and the measurements do not tell you anything.

Another challenge is to deal with the uncertainties. MLA was a wonderful instrument, with an accuracy smaller than the meter. Very well. But you are not that accurate if you want to predict the depth of a crater from its diameter. The authors used an empirical formula proposed by another study: d=(0.17±0.04)D0.96±0.11, where d is the depth, and D the diameter. The problem is the ±, i.e. that formula is not exact. This uncertainty is physically relevant, since the depth of the crater might depend on the incidence angle of the impact, which you don’t know, or on the material at the exact location of the impact… and this is a problem, since you cannot be that accurate on the theoretical depth of the crater. The authors provide a numerical example: a 400-m diameter crater has an expected depth between 21.2 and 127.7 m… So, there is a risk that the thickness of ice that you would measure would be so uncertain that actual detection would be unsure. And this is what happens in almost of all the craters. But the detection is secured by the fact that several craters are involved: the more data you have, the lower the uncertainties. And the ice thickness derived from several craters is more accurate than the one derived from a single crater.

Results: how much ice?

And the result is: the ice thickness is 41+30-14m. The uncertainty is large, but the number remains positive anyway, which means that the detection is positive! Moreover, it is consistent with previous studies, from the detection of polar ice with Goldstone facilities, to similar studies on other regions of Mercury. So, there is ice on Mercury.
An extrapolation of this result suggests that the total mass of water ice on the surface of Mercury is “1014-1015 kg, which is equivalent to ~100-1,000 km3 ice in volume, assuming pure water ice with no porosity” (quoted from the study).

The origin of ice

Mercury is a dense planet, i.e. too dense for such a small planet. It is widely accepted that Mercury as we see it constituted a core of a proto-Mercury, which has been stripped from its mantle of lighter elements. Anyway, Mercury is too dense for the water ice to originate from it. It should come from outside, i.e. it has been brought by impactors. The authors cite studies stating that such a quantity could have been brought by micrometeorites, by Jupiter-family comets, and even by a single impactor.

Another spacecraft soon

Such a study does not only exploit the MESSENGER data, but is also a way to anticipate the future measurements by Bepi-Colombo. This mission will be constituted of two orbiters, one supervised by the European Space Agency (ESA), and the other one by the Japanese agency JAXA. Bepi-Colombo should be launched in October 2018 from Kourou (French Guiana), and inserted into orbit around Mercury in April 2026. Its accuracy is expected to be 10 times better than the one of MESSENGER, and the studies inferring results from MESSENGER data can be seen as predictions for Bepi-Colombo.

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 (NEW) Pinterest.

The Brazil Nut Effect on asteroids

Hi there! You know these large nuts called Brazil nuts? Don’t worry, I will not make you think that they grow on asteroids. No they don’t. But when you put nuts in a pot, or in a glass, have you ever noticed that the biggest nuts remain at the top? That seems obvious, since we are used to that. But let us think about it… these are the heaviest nuts, and they don’t sink! WTF!!! And you have the same kind of effect on small bodies, asteroids, planetesimals, comets… I present you today a Japanese study about that, entitled Categorization of Brazil nut effect and its reverse under less-convective conditions for microgravity geology by Toshihiro Chujo, Osamu Mori, Jun’ichiro Kawaguchi, and Hajime Yano. This study has recently been published in The Monthly Notices of the Royal Astronomical Society.

Brazil Nut and Reverse Brazil Nut effects

The idea is easy to figure out. If you have a pot full of different nuts, then the smallest ones will be naturally closer to the bottom, since they are small enough to fill the voids between the largest ones. For the same reason, if you fill a bucket first with stones and then with sand, the sand will naturally reach the bottom, flowing around the stones. Flowing is important here, since the sand pretty much behaves as a fluid. And of course, if you put the sand in the bucket first, and then the stones, the stones will naturally be closer to the top. Well, this is the Brazil Nut Effect.

OK, now let us make the story go one step further… You have an empty bucket, and you put sand inside… a third of it, or a half… this results as a flat structure. You put stones, which then cover the sand, lying on its surface… and you shake. You shake the bucket, many times… what happen? the sand is moving, and makes some room for the stones, or just some of them, which migrate deeper… if you shake enough, then some of them can even reach the bottom. This is the Reverse Brazil Nut Effect.

And the funny thing is that you can find this effect on planetary bodies! Wait, we may have a problem… when the body is large enough, then the material tends to melt, the heaviest one migrating to the core. So, the body has to be small enough for its interior being ruled by the Brazil Nut Effect, or its reversed version. If the body is small enough, then we are in conditions of microgravity. The authors give the examples of the Near-Earth Asteroid (433)Eros, its largest diameter being 34.4 km, the comet 67P/Churyumov-Gerasimenko, which is ten times smaller in length, and the asteroid (25143)Itokawa, its largest length being 535 meters. All of these bodies are in conditions of microgravity, and were visited by spacecraft, i.e. NEAR Shoemaker for Eros in 2001, Rosetta for Churyumov-Gerasimenko in 2014, and Hayabusa for Itokawa in 2003. And all of these space missions have revealed pebbles and boulders at the surface, which motivated the study of planetary terrains in conditions of microgravity.

Eros seen by NEAR Shoemaker. © NASA/JPL-Caltech/JHUAPL
Eros seen by NEAR Shoemaker. © NASA/JPL-Caltech/JHUAPL

I mentioned the necessity to shake the bucket to give a chance to Reverse Brazil Nut Effect. How to shake these small bodies? With impact, of course. You have impactors everywhere in the Solar System, and small bodies do not need impactors to be large to be shaken enough. Moreover, this shaking could come from cometary activity, in case of a comet, which is true for Churyumov-Gerasimenko.

The authors studied this process both with numerical simulations, and lab experiments.

Numerical simulations

The numerical simulations were conducted with a DEM code, for Discrete Element Modeling. It consisted to simulate the motion of particle which touch each others, or touch the wall of the container. These particles are spheres, and you have interactions when contact. These interactions are modeled with a mixture of spring (elastic interaction, i.e. without dissipation of energy) and dashpot (or damper, which induces a loss of energy at each contact). These two effects are mixed together in using the so-called Voigt rheology.

In every simulation, the authors had 10,224 small particles (the sand), and a large one, named intruder, which is the stone trying to make its way through the sand.

The simulations differed by

  • the density of the intruder (light as acryl, moderately dense as glass, or heavy as high-carbon chromium steel),
  • the frequency of the shaking, modeled as a sinusoidal oscillation over 50 cycles,
  • the restitution coefficient between the sand of the intruder. If it is null, then you dissipate all the energy when contact between the intruder and the sand, and when it is equal to unity then the interaction is purely elastic, i.e. you have no energy loss.

Allowing those parameters to vary will result in different outcomes of the simulations. This way, the influence of each of those parameters is being studied.

A drawback of some simulations is the computation time, since you need to simulate the behavior of each of the particles simultaneously. This is why the authors also explored another way: lab experiments.

Lab experiments

You just put sand in a container, you put an intruder, you shake, and you observe what is going on. Well, said that way, it seems to be easy. It is actually more complicated than that if you want to make proper job.

The recipient was an acryl cylinder, put on a vibration test machine. This machine was controlled by a device, which guaranteed the accuracy of the sinusoidal shaking, i.e. its amplitude, its frequency, and the total duration of the experiment. The intruder was initially put in the middle of the sand, i.e. half way between the bottom of the recipient and the surface of the sand. If it reached the bottom before 30,000 oscillation cycles, then the conclusion was RBNE, and if it raised from the surface the conclusion was BNE. Otherwise, these two effects were considered to be somehow roughly balanced.

But wait: the goal is to model the surface of small bodies, i.e. in conditions of microgravity. The authors did the experiment on Earth, so…? There are ways to reproduce microgravity conditions, like in a parabolic flight, or on board the International Space Station, but this was not the case here. The authors worked in a lab, submitted to our terrestrial gravity. The difficulty is to draw conclusions for the asteroids from Earth-based lab experiments.

At this point, the theory assists the experimentation. If you write down the equations ensuing from the physics (I don’t do it… feel free to do so if you want), these equations ruling the DEM code for instance, you will be able to manipulate them (yes you will) so as to make them depend on dimensionless parameters. For instance: your size is in meters (or in feet). It has the physical dimension of a length. But if you divide your size with the one of your neighbor, you should get something close to unity, but this will be a dimensionless quantity, as the ratio between your size and your neighbor’s. The size of your neighbor is now your reference (let him know, I am sure he would be delighted), and if your size if larger than 1, it means that you are taller than your neighbor (are you?). In the case of our Brazil Nut experiment, the equations give you a gravity, which you can divide by the local one, i.e. either the gravity of your lab, or the microgravity of an asteroid. The result of your simulation will be expressed with respect to this ratio, which you can then re-express with respect to the microgravity of your asteroid. So, all this is a matter of scale. These scaling laws are ubiquitous in lab experiments, and they permit to work in many other contexts.

Triggering the Reverse Brazil Nut effect

And here are the results:

  • The outcomes of the experiments match the ones of the numerical simulations.
  • The authors saw practically no granular convection, i.e. the sand initially at the bottom does not migrate to the top. This is here an analogy with fluid mechanics, in which water at the bottom can raise to the top, especially when it warms (warm water is less dense than cold one).
  • Densest intruders are the likeliest to migrate to the bottom.
  • The authors identified 3 distinct behaviors for the particles, depending on a dimensionless acceleration Γ.

These behaviors are:

  1. Slow Brazil Nut Effect,
  2. Fast BNE, for which the intruder requires less oscillation cycles to raise,
  3. Fluid motion, which may induce RBNE. This is favored by rapid oscillations of the shaking.

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, and (NEW) Instagram.

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.