Tag Archives: Geophysics

Plate tectonics on Europa?

Hi there! Jupiter has 4 large satellites, known as Galilean satellites since they were discovered by Galileo Galilei in 1610. Among them is Europa, which ocean is a priority target for the search for extraterrestrial life. Many clues have given us the certainty that this satellite has a global ocean under its icy surface, and it should be the target of a future NASA mission, Europa Clipper. Meanwhile it will also be visited by the European mission JUICE, before orbital insertion around Ganymede. Since Europa presents evidences of tectonic activity, the study I present you today, i.e. Porosity and salt content determine if subduction can occur in Europa’s ice shell, by Brandon Johnson et al., wonders whether subduction is possible when two plates meet. This study has been conducted at Brown University, Providence, RI (USA).

Subduction on Earth

I guess you know about place tectonics on Earth. The crust of the Earth is made of several blocks, which drift. As a consequence, they collide, and this may be responsible for the creation of mountains, for earthquakes… Subduction is a peculiar kind of collision, in which one plate goes under the one it meets, just because their densities are significantly different. The lighter plate goes up, while the heavier one goes down. This is what happens on the west coast of South America, where the subduction of the oceanic Nazca Plate and the Antarctic Plate have created the Andean mountains on the South America plate, which is a continental one.

Even if our Earth is unique in the Solar System by many aspects, it is highly tempting to use our knowledge of it to try to understand the other bodies. This is why the authors simulated the conditions favorable to subduction on Europa.

The satellite Europa

Europa is the smallest of the four Galilean satellites of Jupiter. It orbits Jupiter in 3.55 days at a mean distance of 670,000 km, on an almost circular and planar orbit. It has been visited by the spacecraft Pioneer 10 & 11 in 1973-1974, then by Voyager 1 & 2 in 1979. But our knowledge of Europa is mostly due to the spacecraft Galileo, which orbited Jupiter between 1995 and 2003. It revealed long, linear cracks and ridges, interrupted by disrupted terrains. The presence of these structures indicates a weakness of the surface, and argues for the presence of a subsurface ocean below the icy crust. Another argument is the tidal heating of Jupiter, which means that Europa should be hot enough to sustain this ocean.
This active surface shows extensional tectonic feature, which suggests plate motion, and raises the question: is subduction possible?

Numerical simulations of the phenomenon

To determine whether subduction is possible, the authors performed one-dimensional finite-elements simulations of the evolution of a subducted slab, to determine whether it would remain below another plate or not. The equation is: would the ocean be buoyant? If yes, then the slab cannot subduct, because it would be too light for that.

The author considered the time and spatial evolution of the slab, i.e. over its length and over the ages. They tested the effect of

  1. The porosity: Planetary ices are porous material, but we do not know to what extent. In particular, at some depth the material is more compressed, i.e. less porous than at the surface, but it is not easy to put numbers behind this phenomenon. Which means that the porosity is a parameter. The porosity is defined as a fraction of the volume of voids over the total volume investigated. Here, total volume should not be understood as the total volume of Europa, but as a volume of material enshrouding the material element you consider. This allows you to define a local porosity, which thus varies in Europa. Only the porosity of the icy crust is addressed here.
  2. The salt content: the subsurface ocean and the icy crust are not pure ice, but are salty, which affects their densities. The authors assumed that the salt of Europa is mostly natron, which is a mixture essentially made of sodium carbonate decahydrate and sodium bicarbonate. Importantly, the icy shell has probably some lateral density variations, i.e. the fraction of salt is probably not homogeneous, which gives room for local phenomenons.
  3. The crust thickness: barely constrained, it could be larger than 100 km.
  4. The viscosity: how does the material react to a subducting slab? This behavior depends on the temperature, which is modeled here with the Fourier law of heat,
  5. The spreading rate, i.e. the velocity of the phenomenon,
  6. The geometry of the slab, in particular the bending radius, and the dip angle.

And once you have modeled and simulated all this, the computer tells you under which conditions subduction is possible.

Yes, it is possible

The first result is that the two critical parameters are the porosity and the salt content, which means that the conditions for subduction can be expressed with respect to these two quantities.
Regarding the conditions for subduction, let me quote the abstract of the paper: If salt contents are laterally homogeneous, and Europa has a reasonable surface porosity of 0.1, the conductive portion of Europa’s shell must have salt contents exceeding ~22% for subduction to occur. However, if salt contents are laterally heterogeneous, with salt contents varying by a few percent, subduction may occur for a surface porosity of 0.1 and overall salt contents of ~5%.

A possible subduction does not mean that subduction happens. For that, you need a cause, which would trigger activity in the satellite.

Triggering the subduction

The authors propose the following two causes for subduction to happen:

  1. Tidal interaction with Jupiter, enhanced by non-synchronous rotation: Surface features revealed by Galileo are consistent with a crust which would not rotate synchronously, as expected for the natural satellites, but slightly faster, the departure from supersynchronicity inducing a full rotation with respect to the Jupiter-Europa direction between 12,000 and 250,000 years… to be compared with an orbital period of 3.55 days. So, this is a very small departure, which would enhance the tidal torque of Jupiter, and trigger some activity. This interpretation of the surface features as a super-synchronous rotation is controversial.
  2. Convection, i.e. fluid motion in the ocean, due to the variations of temperature.

No doubt Europa Clipper and maybe JUICE will tell us more!

The study and its authors

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The mountainous equator of Iapetus

Hi there! You may have heard of Iapetus, a large satellite of Saturn which orbits very far. Today I will tell you about its equatorial ridge. This is the opportunity for me to present you Testing models for the formation of the equatorial ridge on Iapetus via crater counting, by Amanda L. Damptz, Andrew J. Dombard & Michelle R. Kirchoff, which has recently been accepted for publication in Icarus.

The satellite Iapetus

The satellite of Saturn Iapetus was discovered by Giovanni Domenico Cassini in October 1671. It orbited then on the western side of Saturn. During many years, i.e. until 1705, he was unable to observe it on the eastern side, since it was two magnitudes fainter. This has two implications:

  1. Iapetus has a two-tone coloration, i.e. a dark and a bright hemisphere,
  2. its rotation is synchronous. Like our Moon, it is locked in the synchronous 1:1 spin-orbit resonance, constantly showing the same face to Saturn.

Beside this, Iapetus is a large body (diameter: 1,470 km), which orbits at 3.5 millions km from Saturn (for comparison, Titan orbits almost thrice closer), with an orbital eccentricity of 0.028, very close to the one of Titan. It has an unexpectedly high orbital inclination, i.e. 15.47° with the equator of Saturn, and 8.13° with the Laplace Plane. We should imagine the proto-Saturn nebula, from which Iapetus has probably been formed. It was pretty much like a disc, but distorted by the Sun if you were far enough from Saturn, which is the case for Iapetus. What I mean is that the gravitational action of the Sun tends to shift the equilibrium orbital plane from the equatorial one, this is why we need to distinguish it, and we call it the Laplace Plane. In that case, the orbital inclination of Iapetus with respect to the Laplace Plane should be very small, but it is not. This probably contains an information on the history of Iapetus, but we do not know which one yet.

Let me go back to the rotation. Iapetus is so far from Saturn that it needs almost 80 days to complete one revolution, and 80 days for a rotation, since it is synchronous. This is the largest known spin period for a natural satellite in the Solar System.

As most of the satellites of Saturn, our knowledge of Iapetus made invaluable progress since 2004 thanks to the Cassini spacecraft, which imaged it. It confirmed the two-tone coloration, and detected a mountainous equatorial ridge.

Iapetus seen by the Cassini spacecraft. © NASA
Iapetus seen by the Cassini spacecraft. © NASA

The equatorial ridge

The Cassini images showed a 20 km-wide mountainous ridge, which is very close to the equator. So close that it is hard to believe it appeared there by chance. It is present on the dark hemisphere, while isolated equatorial mountains can be seen on the bright side. Some peaks reach 20 km.

Since its discovery late 2004, this ridge is a matter of investigation, and several competing explanations can be found in the literature:

  1. A signature of its past, fast rotation (the measured oblateness of Iapetus is consistent with a rotation period of 16 hours)
  2. A signature of a past critical spin state, i.e. close to provoke disruption of Iapetus,
  3. Upwarping of the lithosphere from below,
  4. Cryovolcanism,
  5. Planetary contraction,
  6. Material from an ancient ring system,
  7. Material from impact generated debris.

We can see that some of these scenarios propose an inner (endogenic) cause, while others propose an outer (exogenic) one. Almost all of them suggest an early formation of the ridge, except the last one.
One way to date a geological feature is to count its craters, and this is where this study intervenes. Its first product is a database of 7,748 craters ranging from 0.83 to 591 km in diameter.

Counting the craters

When an impactor reaches a planetary surface, it creates a crater. If one day geological processes are strong enough to create a tectonic feature, then it may at least alter the crater, or even hide it. If we see an uncraterized geological feature, that means that it is pretty young. We could even try to give it an age in estimating the evolution of the cratering rates over the evolution of the Solar System. By the way, the early Solar System was very intensively bombarded, with an episode of Late Heavy Bombardment occurring between 4.1 and 3.8 billions years ago. Bombardments still happen nowadays, but are much less frequent.

In this study the authors worked from Cassini and Voyager images of the surface of Iapetus, and considered different zones: central ridge, peripheral ridge, and off ridge. Moreover they classified the craters following their diameters, so as to estimate a distribution law: number of craters vs. size. They also catalogued the orientation of the deformations of the craters, since it could tell us something on the geological evolution of Iapetus (how did it alter the surface?)

This systematic search for craters was assisted by the commercial software Esri’s ArcGIS, supplemented by the dedicated add-on Crater Helper Tools.

Results

The first result is a database of 7,748 craters. But the main question is: what can we say about the ridge? The authors observe a depletion of large craters, i.e. with a diameter bigger than 16 km, in the ridge, which would be consistent with a pretty recent formation, and thus would favor the scenario of a ridge created by the debris of an impact. Nevertheless, the authors are prudent with this conclusion, they seem to suggest that the resolution of the images and the risk of saturation of small craters (when you are heavily bombarded, new craters destroyed ancient ones, and the overall number does not increase) do not permit to discard a scenario of early formation of the ridge. Further studies will probably be needed to reach an agreement on the origin of this mountainous equator.

The study and its authors

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The lowlands of Mars

Hi there! Today I will give you the composition of the subsurface of the lowlands of Mars. This is the opportunity for me to present you The stratigraphy and history of Mars’ northern lowlands through mineralogy of impact craters: A comprehensive survey, by Lu Pan, Bethany L. Ehlmann, John Carter & Carolyn M. Ernst, which has recently been accepted for publication in Journal of Geophysical Research: Planets.

Low- and Highlands

Topography of Mars. We can see lowlands in the North, and highlands in the South. © USGS
Topography of Mars. We can see lowlands in the North, and highlands in the South. © USGS

As you can see on this image, the topography of Mars can be divided into the Northern and the Southern hemispheres, the Northern one (actually about one third of the surface) being essentially constituted of plains, while the Southern one is made of mountains. The difference of elevation between these two hemispheres is between 1 to 3 km. Another difference is the fact that the Southern hemisphere is heavily cratered, even if craters exist in the lowlands. This Martian dichotomy is very difficult to explain, some explanations have been proposed, e.g., the lowlands could result from a single, giant impact, or the difference could be due to internal (tectonic) processes, which would have acted differentially, renewing the Northern hemisphere only… Anyway, whatever the cause, there is a dichotomy in the Martian topography. This study examines the lowlands.

The lowlands are separated into: Acidalia Planitia (for plain), Arcadia Planitia, Amazonis Planitia, Chryse Planitia, Isidis Planitia, Scandia Cavi (the polar region), Utopia Planitia, Vatistas Borealis,…

Plains also exist in the Southern hemisphere, like the Hellas and the Argyre Planitiae, which are probably impact basins. But this region is mostly known for Olympus Mons, which is the highest known mountain is the Solar System (altitude: 22 km), and the Tharsis Montes, which are 3 volcanoes in the Tharsis region.

To know the subsurface of a region, and its chemical composition, the easiest way is to dig… at least on Earth. On Mars, you are not supposed to affect the nature… Fortunately, the nature did the job for us, in bombarding the surface. This bombardment was particularly intense during the Noachian era, which correspond to the Late Heavy Bombardment, between 4.1 to 3.7 Gyr ago. The impacts excavated some material, that you just have to analyze with a spectrometer, provided the crater is preserved enough. This should then give you clues on the past of the region. Some say the lowlands might have supported a global ocean once.

The past ocean hypothesis

Liquid water seems to have existed at the surface of Mars, until some 3.5 Gyr ago. There are evidences of gullies and channels in the lowlands. This would have required the atmosphere of Mars to be much hotter, and probably thicker, than it is now. The hypothesis that the lowlands were entirely covered by an ocean has been proposed in 1987, and been supported by several data and studies since then, even if it is still controversial. Some features seem to be former shorelines, and evidences of two past tsunamis have been published in 2016. These evidences are channels created by former rivers, which flowed from down to the top. These tsunamis would have been the consequences of impacts, one of them being responsible for the crater Lomonosov.

The fate of this ocean is not clear. Part of it would have been evaporated in the atmosphere, and then lost in the space, part of it would have hydrated the subsurface, before freezing… This is how the study of this subsurface may participate in the debate.

The CRISM instrument

To study the chemical composition of the material excavated by the impacts, the authors used CRISM data. CRISM, for Compact Reconnaissance Imaging Spectrometer for Mars, is an instrument of Mars Reconnaissance Orbiter (MRO). MRO is a NASA spacecraft, which orbits Mars since 2006.
CRISM is an imaging spectrometer, which can observe both in the visible and in the infrared ranges, which requires a rigorous cooling of the instrument. These multi-wavelengths observations permit to identify the different chemical elements composing the surface. The CRISM team summarizes its scientific goals by follow the water. Studying the chemical composition would permit to characterize the geology of Mars, and give clues on the past presence of liquid water, on the evolution of the Martian climate,…

In this study, the authors used CRISM data of 1,045 craters larger than 1 km, in the lowlands. They particularly focused on wavelengths between 1 and 2.6μm, which is convenient to identify hydrated minerals.

Hydrated vs. mafic minerals

The authors investigated different parts of the craters: the central peak, which might be constituted of the deepest material, the wall, the floor… The CRISM images should be treated, i.e. denoised before analysis. This requires to perform a photometric, then an atmospheric correction, to remove spikes, to eliminate dead pixels…

And after this treatment, the authors identified two kinds of minerals: mafic and hydrated ones. Mafic minerals are silicate minerals, in particular olivine and pyroxenes, which are rich in magnesium and iron, while hydrated minerals contain water. They in particular found a correlation between the size of the crater and the ratio mafic / hydrated, in the sense that mafic detections are less dependent on crater size. Which means that mafic minerals seem to be ubiquitous, while the larger the crater, the likelier the detection of hydrated minerals. Since larger craters result from more violent impacts, this suggests that hydrated minerals have a deeper origin. Moreover, no hydrated material has been found in the Arcadia Planitia, despite the analysis of 85 craters. They also noticed that less degraded craters have a higher probability of mineral detection, whatever the mineral.

However, the authors did not find evidence of concentrated salt deposits, which would have supported the past ocean hypothesis.

The study and the authors

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Avalanches on the Moon

Hi there! Did you know that there could be avalanches on the Moon? Why not? You have slopes, you have boulders, so you can have avalanches! Not snow avalanches of course. This is the topic of Granular avalanches on the Moon: Mass-wasting conditions, processes and features, by B.P. Kokelaar, R.S. Bahia, K.H. Joy, S. Viroulet and J.M.N.T. Gray, which has recently been accepted for publication in Journal of Geophysical Research: Planets.

The Moon vs. the Earth

On the Moon we have

  • No atmosphere: The wind cannot trigger an avalanche. Moreover, the erosion is much slower than on Earth, since it is only due to micrometeorites bombardment. The erosion tends to flatten the terrains. When you have no erosion, an steep terrain may remain steep for ages/
  • No liquid water: This means no snow! This is why you have no snow avalanche. Another consequence of this absence of fluid is that no rain can trigger an avalanche, and the regolith involved is necessarily dry. Wet sand does not behave like dry sand.
  • Less gravity: The gravity on the Moon is about one sixth of the gravity of the Earth, and as you can imagine, gravity assists the avalanches. It appears that a smaller gravity results in slower avalanches, but the final result remains pretty the same, i.e. you cannot infer the gravity from the final result of an avalanche.

The irregularity of the Moon’s topography is mainly due to the numerous impact craters. The steep edges of the craters are where avalanches happen.

Causes of the avalanches

For an avalanche to happen, you need a favorable terrain, and a triggering event.

A favorable terrain is first a slope. If you are flat enough, then the boulders would not be inclined to roll. The required limit inclination is called the dynamic angle of repose. On Earth, the dry sand has a dynamic angle of repose of 34°, while the wet sand remains stable up to 45°. This illustrates pretty well the influence of the water.

Triggering an avalanche requires to shake the terrain enough. A way is an impact occurring far enough to not alter the slope, but close enough to shake the terrain. Another way is a seismic phenomenon, due to geophysical activity of the Moon.

Datasets

The authors focused their efforts on the Kepler crater, before investigating 6 other ones. The impact craters have to be preserved enough, in particular from micrometeorite impacts. These craters are:

Crater Diameter Slope
Kepler 31 km ~32°
Gambart B 11 km ~30°
Bessel 16 km 31.5°
Censorinus 3.8 km 32°
Riccioli CA 14.2 km 34°
Virtanen F 11.8 km 32°
Tralles A 18 km 32°

The first 4 of these craters are situated in maria, while the last three are in highlands. These means that we have different types of regolith.

Kepler seen by LROC (© NASA/GSFC/Arizona State University)
Kepler seen by LROC (© NASA/GSFC/Arizona State University)

We need high-precision data to determine the shape of the avalanches. The space mission Lunar Reconnaissance Orbiter (LRO) furnishes such data. In particular the authors used:

  • Images from the LROC, for LRO Camera. This instrument is equipped of 3 cameras, two Narrow Angle Cameras (NACs), with a resolution between 0.42 and 1.3 meter per pixel, and a WAC, for Wide Angle Camera, with a resolution of 100 m /pixel, but with a much wider field. The NAC data permitted to characterize the type of flow, while the WAC data gave their extent.
  • Digital Elevation Models (DEM), obtained from the Lunar Orbiter Laser Altimeter (LOLA), mentioned here, and from the Terrain Camera of the Japanese mission SELENE / Kaguya. Knowing the variations of the topography permitted to estimate the slopes of the craters and the volume of flowing material.

Three flow types

And from the images, the authors determined 3 types of flows:

  • Multiple Channel and Lobe (MCL): these are accumulations of multiple small-volume flows. These flows are the most common in the study, and can be found on Earth too,
  • Single-Surge Polylobate (SSP): the flows have the structure of fingers,
  • Multiple Ribbon (MR): these are very elongated flows with respect to their widths, i.e. they are typically kilometer-long and meter-wide. These flows have been predicted by lab experiments, but this is their first observation on a planetary body. In particular, they are not present on the Earth. Lab experiments suggest that they are extremely sensitive to slope changes.
Debris flows observed on the northeast inner wall Kepler. This is NOT water! © NASA/GSFC/Arizona State University
Debris flows observed on the northeast inner wall Kepler. This is NOT water! © NASA/GSFC/Arizona State University

The word flow evokes a fluid phenomenon. Of course, there is no fluid at the surface of the Moon, but granular regolith may have a kind of fluid behavior. A true fluid would have a dynamic angle of repose of 0°. Regolith has a higher angle of repose because of friction, that prevents it from flowing. But it of course depends on the nature of the regolith. In particular, fine-grained material tends to reduce friction, and consequently increases the mobility of the material. This results in extended flows.

But this extension has some limitation. On Earth, we observe flows on adverse slopes, which are thought to be facilitated by the presence of liquid water. This statement is enforced by the fact that no such flow has been observed on the Moon.

The accuracy of data we dispose on the Moon has permitted the first observations of granular flows in dry and atmosphereless conditions. Such results could probably be extrapolated to other similar bodies (Mercury? Ceres? Pluto?).

Laboratory experiments

The multiple ribbon have been predicted by lab experiments. It is fascinating to realize that we can reproduce lunar condition in a room, and with accelerated timescales. This is made possible by the normalization of physical quantities.
If we write down the equations ruling the granular flows, we have a set of 3 partial derivative equations, involving the avalanche thickness, and the concentration and velocity of the particles. Mathematical manipulations on these equations permit to emphasize quantities, which have no physical dimension. For instance, the height of a mountain divided by the radius of the planet, or the time you need to read this article divided by the time I need to write it… In acting on all the quantities involved in such adimensional numbers, we can reduce an impact crater of the Moon evolving during millions of years, to a room evolving during a few days…
In this problem, a critical number is the Froude number, which depends on the gravity, the avalanche thickness, the velocity, and the slope.

The study and the authors

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Modeling the shape of a planetary body

Hi there! Do you know the shape of the Moon? You say yes of course! But up to which accuracy? The surface of the Moon has many irregularities, which prompted Christian Hirt and Michael Kuhn to study the limits of the mathematics, in modeling the shape of the Moon. Their study, entitled Convergence and divergence in spherical harmonic series of the gravitational field generated by high-resolution planetary topography — A case study for the Moon, has recently been accepted for publication in Journal of Geophysical Research: Planets.

The shape of planetary bodies

If you look at a planetary body from far away (look at a star, look at Jupiter,…), you just see a point mass. If you get closer, you would see a sphere, if the body is not too small. Small bodies, let us say smaller than 100 km, can have any shape (may I call them potatoids?) If they are larger, the material almost arranges as a sphere, which gives the same gravity field as the point mass, provided you are out of the body. But if you look closer, you would see some polar flattening, due to the rotation of the body. And for planetary satellites, you also have an equatorial ellipticity, the longest axis pointing to the parent planet. Well, in that case, you have a triaxial ellipsoid. You can say that the sphere is a degree 0 approximation of the shape, and that the triaxial ellipsoid is a degree 2 approximation… but still an approximation.

A planetary body has some relief, mountains, basins… there are explanations for that, you can have, or have had, tectonic activity, basins may have been created by impacts, you can have mass anomalies in the interior, etc. This means that the planetary body you consider (in our example, the Moon), is not exactly a triaxial ellipsoid. Being more accurate than that becomes complicated. A way to do it is with successive approximations, in the same way I presented you: first a sphere, then a triaxial ellipsoid, then something else… but when do you stop? And can you stop, i.e. does your approximation converge? This study addresses this problem.

The Brillouin sphere

This problem is pretty easy when you are far enough from the body. You just see it as a sphere, or an ellipsoid, since you do not have enough resolution to consider the irregularities in the topography… by the way, I am tempted to make a confusion between topography and gravity. The gravity field is the way the mass of your body will affect the trajectory of the body with which it interacts, i.e. the Earth, Lunar spacecrafts… If you are close enough, you will be sensitive to the mass distribution in the body, which is of course linked to the topography. So, the two notions are correlated, but not fully, since the gravity is more sensitive to the interior.

But let us go back to this problem of distance. If you are far enough, no problem. The Moon is either a sphere, or a triaxial ellipsoid. If you get closer, you should be more accurate. And if you are too close, then you cannot be accurate enough.

This limit is given by the radius of the Brillouin sphere. Named after the French-born American physicist Léon Brillouin, this is the circumscribing sphere of the body. If your planetary body is spherical, then it exactly fills its Brillouin sphere, and this problem is trivial… If you are a potatoidal asteroid, then your volume will be only a fraction of this sphere, and you can imagine having a spacecraft inside this sphere.

The asteroid Itokawa in its Brillouin-sphere. Credit: JAXA.
The asteroid Itokawa in its Brillouin-sphere. Credit: JAXA.

The Moon is actually pretty close to a sphere, of radius 1737.4±1 km. But many mass anomalies have been detected, which makes its gravity field not that close to the one of the sphere, and you can be inside the equivalent Brillouin sphere (if we translate gravity into topography), in flying over the surface at low altitude.

Why modeling it?

Why trying to be that accurate on the gravity field / topography of a planetary object? I see at least two good reasons, please pick the ones you prefer:

  • to be able to detect the time variations of the topography and / or the gravity field. This would give you the tidal response (see here) of the body, or the evolution of its polar caps,
  • because it’s fun,
  • to be able to control the motion of low-altitude spacecrafts. This is particularly relevant for asteroids, which are somehow potatoidal (am I coining this word?)

You can object that the Moon may be not the best body to test the gravity inside the Brillouin sphere. Actually we have an invaluable amount of data on the Moon, thanks to the various missions, the Lunar Laser Ranging, which accurately measures the Earth-Moon distance… Difficult to be more accurate than on the Moon…

The goal of the paper is actually not to find something new on the Moon, but to test different models of topography and gravity fields, before using them on other bodies.

Spherical harmonics expansion

Usually the gravity field (and the topography) is described as a spherical harmonics expansion, i.e. you model your body as a sum of waves with increasing frequencies, over two angles, which are the latitude and the longitude. This is why the order 0 is the exact sphere, the order 2 is the triaxial ellipsoid… and in raising the order, you introduce more and more peaks and depressions in your shape… In summing them, you should have the gravity field of your body… if your series converge. It is usually assume that you converge outside the Brillouin sphere… It is not that clear inside.

To test the convergence, you need to measure a distance between your series and something else, that you judge relevant. It could be an alternative gravitational model, or just the next approximation of the series. And to measure the distance, a common unit is the gal, which is an acceleration of 1 cm/s2 (you agree that gravity gives acceleration?). In this paper, the authors checked differences at the level of the μgal, i.e. 1 gal divided by 1 million.

Methodology

In this study, the authors used data from two sources:

  • high-resolution shape maps from the Lunar Orbiter Laser Altimeter (LOLA),
  • gravity data from the mission GRAIL (Gravity Recovery And Interior Laboratory),

and they modeled 4 gravity fields:

  1. Topography of the surface,
  2. Positive topographic heights, i.e. for basins the mean radius was considered, while the exact topography was considered for mountains,
  3. “Brillouin-sphere”, at a mean altitude of 11 km,
  4. “GRAIL-sphere”, at a mean altitude of 23 km.

In each of these cases, the authors used series of spherical harmonics of orders between 90 (required spatial resolution: 60.6 km) and 2,160 (resolution: 2.5 km). In each case, the solution with spherical harmonics was compared with a direct integration of the potential of the body, for which the topography is discretized through an ensemble of regularly-shaped elements.

Results

And here are the results:

Not surprisingly, everything converges in the last two cases, i.e. altitudes of 11 and 23 km. However, closer to the surface the expansion in spherical harmonics fails from orders 720 (case 1) and 1,080 (case 2), respectively. This means that adding higher-order harmonics does not stabilize the global solution, which can be called divergence. The authors see from their calculations that this can be predicted from the evolution of the amplitude of the terms of the expansion, with respect to their orders. To be specific, their conclusion is summarized as follows:

A minimum in the degree variances of an external potential model foreshadows divergence of the spherical harmonic series expansions at points inside the Brillouin-sphere.

 

My feeling is that this study should be seen as a laboratory test of a mathematical method, i.e. testing the convergence of the spherical harmonics expansion, not on a piece of paper, but in modeling a real body, with real data. I wonder how the consideration of time variations of the potential would affect these calculations.

To know more…

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