Tag Archives: Topography

How rough is Mercury?

Hi there! Today I will tell you on the smoothness of the surface of Mercury. This is the opportunity for me to present The surface roughness of Mercury from the Mercury Laser Altimeter: Investigating the effects of volcanism, tectonism, and impact cratering by H.C.M. Susorney, O.S. Barnouin, C.M. Ernst and P.K. Byrne, which has recently been published in Journal of Geophysical Research: Planets. This paper uses laser altimeter data provided by the MESSENGER spacecraft, to measure the regularity of the surface in the northern hemisphere.

The surface of Mercury

I already had the opportunity to present Mercury on this blog. This is the innermost planet of the Solar System, about 3 times closer to the Sun than our Earth. This proximity makes space missions difficult, since they have to comply with the gravitational action of the Sun and with the heat of the environment. This is why Mercury has been visited only by 2 space missions: Mariner 10, which made 3 fly-bys in 1974-1975, and MESSENGER, which orbited Mercury during 4 years, between 2011 and 2015. The study of MESSENGER data is still on-going, the paper I present you today is part of this process.

Very few was known from Mercury before Mariner 10, in particular we just had no image of its surface. The 3 fly-bys of Mariner 10 gave us almost a full hemisphere, as you can see below. Only a small stripe was unknown.

Mercury seen by Mariner 10. © NASA.
Mercury seen by Mariner 10. © NASA.

And we see on this image many craters! The details have different resolutions, since this depends on the distance between Mercury and the spacecraft when a given image was taken. This map is actually a mosaic.
MESSENGER gave us full maps of Mercury (see below).

Mercury seen by MESSENGER. © USGS
Mercury seen by MESSENGER. © USGS

Something that may be not obvious on the image is a non-uniform distribution of the craters. So, Mercury is composed of cratered terrains and smooth plains, which have different roughnesses (you will understand before the end of this article).
Craters permit to date a terrain (see here), i.e. when you see an impact basin, this means that the surface has not been renewed since the impact. You can even be more accurate in dating the impact from the relaxation of the crater. However, volcanism brings new material at the surface, which covers and hides the craters.

This study focuses on the North Pole, i.e. latitudes between 45 and 90°N. This is enough to have the two kinds of terrains.

Three major geological processes

Three processes affect the surface of Mercury:

  1. Impact cratering: The early Solar System was very dangerous from this point of view, having several episodes of intense bombardments in its history. Mercury was particularly impacted because the Sun, as a big mass, tends to focus the impactors in its vicinity. It tends to rough the surface.
  2. Volcanism: In bringing new and hot material, it smoothes the surface,
  3. Tectonism: Deformation of the crust.

If Mercury had an atmosphere, then erosion would have tended to smooth the surface, as on Earth. Irrelevant here.

To measure the roughness, the authors used data from the Mercury Laser Altimeter (MLA), one of the instruments of MESSENGER.

The Mercury Laser Altimeter (MLA) instrument

This instrument measured the distance between the spacecraft and the surface of Mercury from the travel time of light emitted by MLA and reflected by the surface. Data acquired on the whole surface permitted to provide a complete topographic map of Mercury, i.e. to know the variations of its radius, detect basins and mountains,… The accuracy and the resolution of the measurements depend on the distance between the spacecraft and the surface, which had large variations, i.e. between 200 and 10,300 km. The most accurate altimeter data were for the North Pole, this is why the authors focused on it.

Roughness indicators

You need at least an indicator to quantify the roughness, i.e. a number. For that, the authors work on a given baseline on which they had data, removed a slope, and calculated the RMS (root mean square) deviation, i.e. the average squared deviation to a constant altitude, after removal of a slope. When you are on an inclined plane, then your altitude is not constant, but the plane is smooth anyway. This is why you remove the slope.

But wait a minute: if you are climbing a hill, and you calculate the slope over 10 meters, you have the slope you are climbing… But if you calculate it over 10 km, then you will go past the summit, and the slope will not be the same, while the summit will affect the RMS deviation, i.e. the roughness. This means that the roughness depends on the length of your baseline.

This is something interesting, which should be quantified as well. For this, the authors used the Hurst exponent H, such that ν(L) = ν0LH, where L is the length of the baseline, and ν the standard deviation. Of course, the data show that this relation is not exact, but we can say it works pretty well. H is determined in fitting the relation to the data.

Results

To summarize the results:

  • Smooth plains: H = 0.88±0.01,
  • Cratered terrains: H = 0.95±0.01.

The authors allowed the baseline to vary between 500 m and 250 km. The definition of the Hurst exponent works well for baselines up to 1.5 km. But for any baseline, the results show a bimodal distribution, i.e. two kinds of terrains, which are smooth plains and cratered terrains.

It is tempting to compare Mercury to the Moon, and actually the results are consistent for cratered terrains. However, the lunar Maria seem to have a slightly smaller Hurst exponent.

To know more

That’s it for today! The next mission to Mercury will be Bepi-Colombo, scheduled for launch in 2018 and for orbital insertion in 2025. Meanwhile, please do not forget to comment. You can also subscribe to the RSS feed, and follow me on Twitter and Facebook.

Enceladus lost its balance

Hi there! Today I will present you True polar wander of Enceladus from topographic data, by Tajeddine et al., which has recently been published in Icarus. The idea is this: Enceladus is a satellite of Saturn which has a pretty stable rotation axis. In the past, its rotation axis was already stable, but with a dramatically different orientation, i.e. 55° shifted from the present one! The authors proposed this scenario after having observed the distribution of impact basins at its surface.

Enceladus’s facts

Enceladus is one of the mid-sized satellites of Saturn, it is actually the second innermost of them. It has a mean radius of some 250 km, and orbits around Saturn in 1.37 day, at a distance of ~238,000 km. It is particularly interesting since it presents evidence of past and present geophysical activity. In particular, geysers have been observed by the Cassini spacecraft at its South Pole, and its southern hemisphere presents four pretty linear features known as tiger stripes, which are fractures.

Enceladus seen by Cassini (Credit: NASA / JPL / Space Science Institute).
Enceladus seen by Cassini (Credit: NASA / JPL / Space Science Institute).

Moreover, analyses of the gravity field of Enceladus, which is a signature of its interior, strongly suggest a global, subsurfacic ocean, and a North-South asymmetry. This asymmetry is consistent with a diapir of water at its South Pole, which would be the origin of the geysers. The presence of the global ocean has been confirmed by measurements of the amplitude of the longitudinal librations of its surface, which are consistent with a a crust, that a global ocean would have partially decoupled from the interior.

The rotation of a planetary satellite

Planetary satellites have a particularly interesting rotational dynamics. Alike our Moon, they show on average always the same face to a fictitious observer, which would observe the satellite from the surface of the parent planet (our Earth for the Moon, Saturn for Enceladus). This means that they have a synchronous rotation, i.e. a rotation which is synchronous with their orbit, but also that the orientation of their spin axis is pretty stable.
And this is the key point here: the spin axis is pretty orthogonal to the orbit (this orientation is called Cassini State 1), and it is very close to the polar axis, which is the axis of largest moment of inertia. This means that we have a condition on the orientation of the spin axis with respect to the orbit, AND with respect to the surface. The mass distribution in the satellite is not exactly spherical, actually masses tend to accumulate in the equatorial plane, more particularly in the satellite-planet direction, because of the combined actions of the rotation of the satellites and the tides raised by the parent planet. This implies a shorter polar axis. And the study I present today proposes that the polar axis has been tilted of 55° in the past. This tilt is called polar wander. This result is suggested by the distribution of the craters at the surface of Enceladus.

Relaxing a crater

The Solar System bodies are always impacted, this was especially true during the early ages of the Solar System. And the inner satellites of Saturn were more impacted than the outer ones, because the mass of Saturn tends to attract the impactors, focusing their trajectories.
As a consequence, Enceladus got heavily impacted, probably pretty homogeneously, i.e. craters were everywhere. And then, over the ages, the crust slowly went back to its original shape, relaxing the craters. The craters became then basins, and eventually some of them disappeared. Some of them, but not all of them.
The process of relaxation is all the more efficient when the material is hot. For material which properties strongly depend on the temperature, a stagnant lid can form below the surface, which would partly preserve it from the heating by convection, and could preserve the craters. This phenomenon appears preferably at equatorial latitudes.
This motivates the quest for basins. A way for that is to measure the topography of the surface.

Modeling the topography

The surface of planetary body can be written as a sum of trigonometric series, known as spherical harmonics, in which the radius would depend on 2 parameters, i.e. the latitude and the longitude. This way, you have the radius at any point of the surface. Classically, two terms are kept, which allow to represent the surface as a triaxial ellipsoid. This is the expected shape from the rotational and tidal deformations. If you want to look at mass anomalies, then you have to go further in the expansion of the formula. But to do that, you need data, i.e. measurements of the radius at given coordinates. And for that, the planetologists dispose of the Cassini spacecraft, which made several flybys of Enceladus, since 2005.
Two kinds of data have been used in this study: limb profiles, and control points.
Limb profiles are observations of the bright edge of an illuminated object, they result in very accurate measurements of limited areas. Control points are features on the surface, detected from images. They can be anywhere of the surface, and permit a global coverage. In this study, the authors used 41,780 points derived from 54 limb profiles, and 6,245 control points.
Measuring the shape is only one example of use of such data. They can also be used to measure the rotation of the body, in comparing several orientations of given features at different dates.
These data permitted the authors to model the topography up to the order 16.

The result

The authors identified a set of pretty aligned basins, which would happen for equatorial basins protected from relaxation by stagnant lid convection. But the problem is this: the orientation of this alignment would need a tilt of 55° of Enceladus to be equatorial! This is why the authors suggest that Enceladus has been tilted in the past.

The observations do not tell us anything on the cause of this tilt. Some blogs emphasize that it could be due to an impact. Why not? But less us be cautious.
Anyway, the orientation of the rotation axis is consistent with the current mass distribution, i.e. the polar axis has the largest moment of inertia. Actually, mid-sized planetary satellites like Enceladus are close to sphericity, in the sense that there is no huge difference between the moments of inertia of its principal axes. So, a redistribution of mass after a violent tilt seems to be possible.

To know more

And now the 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.

The contraction of Mercury

Hi there! Today’s post deals with the early evolution of Mercury, in particular its cooling. At the beginning of its life, a planet experiences variations of temperature, and then cooling, and while cooling, it contracts. The surface may present some signature of this contraction, and this is the object of the paper I present you today. It is entitled Timing and rate of global contraction of Mercury, by Kelsey T. Crane and Christian Klimczak, from the University of Georgia, and it has been recently accepted for publication in Geophysical Research Letters. The idea is to infer the history of the contraction from the observation of the craters and the faults.

Mercury’s facts

Mercury is the innermost planet of the Solar System, with a mean distance to the Sun which is about one third of the Sun-Earth distance. It has an eccentric orbit, with an eccentricity of 0.206, and orbits the Sun in 88 days while the planet rotates around itself in 58 days. This is very long when compared to the terrestrial day, but it also means that there is a ratio 1.5 between the spin and the orbital frequencies. This is called a 3:2 spin-orbit resonance, which is a dynamical equilibrium favored by the proximity of the Sun and the orbital eccentricity.

Mercury seen by MESSENGER (Credit: NASA)

An interesting fact is the high density of Mercury, i.e. Mercury is too dense for a terrestrial planet. Usually, a large enough body is expected to have a stratified structure, in which the heaviest elements are concentrated in the core. Mercury is so dense than it is thought to be the core of a former and larger proto-Mercury.

Mercury’s early life

There is no agreement on the way Mercury lost its mantle of lighter elements. You can find the following scenarios in the literature:

  1. Slow volatilization of the mantle by the solar wind,
  2. Very large impact,
  3. Loss of the light elements by photophoresis,
  4. Magnetic erosion.

The scenario of the large impact was very popular until the arrival of MESSENGER, in particular because the models of formation of the Solar System and the observation of the surface of Mercury suggest that Mercury has been heavily impacted in its early life. However, the detection of volatiles elements, in particular potassium, on the surface of Mercury, is interpreted by some planetary scientists as inconsistent with the large impact scenario. The large impact would have induced extreme heating of the planet, and for some scientists the potassium would not have survived this episode. The other scenarios involve much slower processes, and less heating.

This raises the question: how hot was the early Mercury? We still do not know, but this is related to the study I present here.

The exploration of Mercury

The proximity of Mercury to the Sun makes it difficult to explore, because of the large gravitational action of the Sun which significantly perturbs the orbit of a spacecraft, and more importantly because of the large temperatures in this area of the Solar System.

Contrarily to Venus and Mars, which regularly host space programs, Mercury has been and will be the target of only 3 space missions so far:

  1. Mariner 10 (NASA): It has been launched in November 1973 to make flybys of Venus and Mercury. Three flybys of Mercury have been realized between March 1974 and March 1975. This mission gave us the first images of the surface of the planet, covering some 45% of it. It also discovered the magnetic field of Mercury.
  2. MESSENGER (Mercury Surface, Space Environment, Geochemistry, and Ranging) (NASA): This was the first human-made object to orbit Mercury. It was launched in August 2004 from Cape Canaveral and has been inserted around Mercury in March 2011, after one flyby of the Earth, two flybys of Venus, and three flybys of Mercury. These flybys permitted to use the gravity of the planets to reduce the velocity of the spacecraft, which was necessary for the orbital insertion. MESSENGER gave us invaluable data, like the gravity field of Mercury, a complete cartography with topographical features (craters, plains, faults,…), new information on the gravity field, it supplemented Earth-based radar measurements of the rotation, it revealed the chemical composition of the surface… The mission stopped in April 2015.
  3. Bepi-Colombo (ESA / JAXA): This is a joint mission of the European and Japanese space agencies, which is composed of two elements: the Mercury Magnetospheric Orbiter (MMO, JAXA), and the Mercury Planetary Orbiter (MPO, ESA). It should be launched in October 2018 and inserted into orbit in December 2025, after one flyby of the Earth, two flybys of Venus, and 6 flybys of Mercury. Beside the acquisition of new data on the planet with a better accuracy than MESSENGER, it will also perform a test of the theory of the general relativity, in giving new measurements of the post-newtonian parameters β and γ. β is associated with the non-linearities of the gravity field, while γ is related with the curvature of the spacetime. In the theory of the general relativity, these two parameters should be strictly equal to 1.

This paper

The idea of the paper is based on the competition between two processes for altering the surface of Mercury:

  1. Impacts, which are violent, rapid phenomena, creating craters,
  2. Tides, which is a much slower process that creates faults, appearing while the planet is contracting. The local stress tensor can be inferred from the direction of the faults.

Dating a crater is possible, from its preservation. And when a crater and a fault are located at the same place, there are two possibilities:

  1. either the fault cuts the crater (see Enheduanna, just below), or
  2. the crater interrupts the fault.

In the first case, the fault appeared after the impact, while in the second case, the fault was already present before Mercury was impacted. So, if you can constrain the age of the crater, you can constrain the apparition of the fault, and the contraction of the planet. From a global analysis of the age of the faults, the authors deduced the variation of the contraction rate over the ages.

A close up of Enheduanna Crater. Credit: IAU

The authors used a database of 3,112 craters ranging from 20 to 2,000 km, which were classified into 5 classes, depending on their degree of preservation. And the result are given below.

Class Name Age Craters Cut Superpose
1+2 Pre-Tolstojan + Tolstojan >3.9 Gy 2,310 1,192 4
3 Calorian 3.9 – 3.5 Gy 536 266 104
4 Mansurian 3.5 – 1 Gy 244 49 55
5 Kuiperian < 1 Gy 22 0 3

We can see that the eldest craters are very unlikely to superpose a fault, while the bombardment was very intense at that time. However, the authors have detected more superposition after. They deduced the following contraction rates:

Time Contraction (radius)
Pre-Tolstojan + Tolstojan 4.0 ± 1.6 km
Calorian 0.90 ± 0.35 km
Mansurian 0.17 ± 0.07 km
Kuiperian 0

This means that the contraction rate has decreased over the ages, which is not surprising, since the temperature of Mercury has slowly reached an equilibrium.

A perspective : constraining the early days of Mercury

In my opinion, such a study could permit to constrain the evolution of the temperature of Mercury over the ages, and thus date its stratification. Maybe this would also give new clues on the way Mercury lost its light elements (impact or not?).

To know more

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

Charon has had an active youth

Hi there! The scientific review Icarus releases an issue dedicated to New Horizons, which made a fly-by of the system of Pluto-Charon in July 2015. Now all the data have been transmitted to Earth. This is the opportunity for me to present you one of the new papers, entitled Charon tectonics, by Ross Beyer et al. This papers presents evidences of an active tectonic youth of Charon, and probably a former subsurface ocean, which is now frozen.

Charon facts

Charon has been discovered in 1978, as the first known satellite of Pluto. It actually appeared that Charon is massive enough, so that Pluto-Charon should be considered as a binary system, which orbits around a common barycenter. Moreover, the gravitational interactions (one call them tidal interactions) between these two bodies are so strong that they rotate synchronously with their mutual orbit. This means that they always show the same face to each other.

Charon seen by New Horizons. Credit: NASA

The recent flyby of the space mission New Horizons gave us several details on Charon, the paper I present here addresses some of them, more specifically linked to the features observed at the surface of Charon, which are linked to a past geophysical activity. Let us now speak a little about planetary tectonics.

Planetary tectonics

The tectonics is the process that controls the shaping of the surface of the Earth. It is responsible for the apparition of mountains, for earthquakes, for the continental drifts (plate tectonics). Tectonics does not appear only on Earth, this is why we can speak of planetary tectonics.

Tectonics results from the heating of a planetary body, and the loss of this heat. This heat is responsible for melting of some elements, differentiation of the planet, and thus activity. A spectacular example in the Solar System is the intense volcanic activity of Io. This satellite of Jupiter is intensively heated by the tidal interaction with its parent planet.
Another example is the geysers on the satellite of Saturn Enceladus.
Beside this observable activity, the observation of irregular features at the surface of a planetary body is an evidence of a past tectonic activity, which is actually ubiquitous in the Solar System. Just a few examples:

  • Plains have been detected at the surface of Mercury, which means that these are renewed terrains,
  • Our Earth has many volcanoes,
  • The highest known volcano in the Solar System is Olympus Mons, on Mars (see this post),
  • The surface of the satellite of Jupiter Europa presents many ridges,
  • The satellites of Uranus Ariel and Miranda present interesting features as well.

And now Charon!

A glossary of planetary features

Some of the definitions I present above have been borrowed from the official nomenclature of the International Astronomical Union. Such a nomenclature has been established to name the planetary features actually observed.

And the terms to know are:

  • Chasma: A deep, elongated, steep-sided depression. The plural is chasmata.
  • Scarp: An escarpment, i.e. a vertical feature which separates two zones of different elevation.
  • Macula: A dark spot, which may be irregular. The plural is maculae.
  • Planum: A plateau, or a high plain.
  • Elastic thickness: This is not a topographical feature. This is the thickness that would have the crust if it were fully elastic in showing the features actually observed. This quantity helps to characterize the crust from the observation of the surface.

The spacecraft New Horizons

The spacecraft New Horizons has been launched in January 2006 and has encountered the system of Pluto in July 2015, after a Jupiter flyby in February 2007. It is composed of 7 science instruments:

  • The ultraviolet imaging spectrometer Alice, dedicated to the study of the atmosphere of Pluto,
  • The imager Ralph, actually composed of 8 imagers, in different wavelengths. It is in charge of mapping the encountered bodies,
  • The Radio Science Experiment REX, which has measured the masses of Pluto and Charon, and probed their atmospheres,
  • The Long Range Reconnaissance Imager LORRI, which gave the first images of Pluto and its satellites by New Horizons,
  • SWAP, for Solar Wind Around Pluto, dedicated to the Solar wind,
  • PEPSSI, for Pluto Energetic Particle Spectrometer Science Investigation. It studied the interactions of the atmosphere of Pluto with the Solar wind,
  • and the Venetia Burney Student Dust Counter SDC, which studied the dust in the system of Pluto. This instrument was part of a New Horizons Education and Public Outreach project, it was designed and built by students. It was named after Venetia Burney, who proposed the name of Pluto after its discovery. She was 11 then. The 250-km-wide Burney Crater, on Pluto, is named after her.

The paper I present today use mainly LORRI and LEISA data, LEISA being an infrared detector of Ralph (not to be confused with LESIA, which is a planetary lab of Paris Observatory).

This mission did not permit a global high-resolution mapping of Charon, since New Horizons did not orbit in the Pluto system. So, the highest resolution images we dispose of are limited to one hemisphere, and the way to analyze them depends on the varying Solar insolation angle. A scarp, a mountain, a crater… will appear differently if enlightened from the zenith or from the horizon.

This paper

This paper represents the main surface features that can be seen, before discussing their origin.

The most striking features are:

  • Mordor Macula, which is a polar dark spot,
  • an equatorial belt of chasmata, which splits the hemisphere into two plains: Oz Terra (North), and Vulcan Planum (South),
  • Argo Chasma, which appears at the limb,
  • many craters.

Craters give us the chronology of the tectonics. Tectonic activity tends to melt the surface, renew it, and relax the crater basins, which should then be barely visible. The fact that many craters can be seen means a very old surface. This also means that the other features are even older, i.e. they were created some 4 Gyr ago.

Let us concentrate now on the equatorial belt. The two main features are Serenity Chasma, which is 40-50 km wide and over 200 km long, and Mandjet Chasma, which is 30 km wide and at least 450 km long. These two structures have a depth of typically 5-7 km.

These chasmata suggest an elastic thickness of 2.5 km. Moreover, the structures indicate that Charon experienced a radial extension, which could be due to the freezing of a global surface ocean. So, in its early ages, Charon has had a global subsurface ocean, which is now frozen.

Creating a subsurface ocean requires some heating. The system Pluto-Charon could originate from the destruction of a progenitor by an impact, which would have induced intense heating. Moreover, this heating has probably been assisted by the tidal heating of Charon by Pluto.

The discovery of these features gives us another signature of the early ages of the Solar System, and would surely contribute to the global understanding of the formation of planetary systems.

To know more

Links to the study, the authors, and the mission

That’s all for today! I hope you liked it. As usual, you are free to comment. You can also subscribe to the RSS feed, and follow me on Twitter.

The surface of Mars is fractal

Hi there! Today’s post is a pretty much different than usual. I will present you a mathematical analysis of planetary features. More precisely, a paper investigating the fractal structure of the surface of Mars. This is a paper entitled Mars topography investigated through the wavelet method: A multidimensional study of its fractal structure, by Adrien Deliège, Thomas Kleyntssens and Samuel Nicolay, which has been recently published in Planetary and Space Science. This study has been conducted at the University of Liège (Belgium).

The surface of Mars

The Mars Orbiter Laser Altimeter (MOLA), as instrument of Mars Global Surveyor, provided us a very accurate map of the whole surface of Mars, which is far from boring. It has for instance an hemispheric asymmetry, the Northern hemispheric being composed of pretty flat, new terrains, which the Southern one is very cratered (several thousands of craters). The northern new terrains are made of lava, which is a fingerprint of past geophysical activity. Moreover, Mars has two icy polar caps.

Among the remarkable features are:

  • Olympus Mons, which is the highest known mountain in the Solar System. This is a former volcano, which rises 22 km above the surrounding volcanic plains.
  • The Tharsis region, which contains many volcanoes.
  • Hellas Planitia, which is a huge impact basin (diameter: 2300 km, depth: 7 km), located in the Southern hemisphere.

You can find below an annotated map, please click!

The topology of Mars. Credit: USGS Astrogeology Science Center

The mission Mars Global Surveyor

The missions Mars Global Surveyor (MGS) is a NASA mission, which has been launched in November 1996, and has been inserted into orbit around Mars 10 months later, i.e. September 1997. It became silent in November 2006 after 3 extensions of the nominal mission, and gave us invaluable data during almost 10 years. It embarked 5 scientific instruments:

  • the Mars Orbiter Camera (MOC), a wide angle camera which gave us images of the surface and of the two satellites of Mars Phobos and Deimos,
  • the Mars Orbiter Laser Altimeter (MOLA), which gave us the most accurate topographic measurements of Mars. The study I present today uses its data,
  • the Thermal Emission Spectrometer (TES), which studied the atmosphere of Mars, and the thermal emission of the surface. This instrument observed in the infrared band,
  • the magnetometer, which studied the magnetic field of Mars,
  • and the radio-science, which measured the gravity field of the planet.

Mars Global Surveyor was of great help to prepare the further missions. It allowed in particular to identify landing sites for rovers.

The rich topography of Mars has encouraged many scientists to characterize it with a fractal structure.

Fractals and multifractals

A fractal is a mathematical set that exhibits a repeating pattern displayed at every scale, see the following figure, which shows the well-known Mandelbrot set.

The Mandelbrot set, plotted by myself after an inspiration from Rosetta Code. The zoom on the right shows the same structure than on the left, with a larger scale.

It is tempting to quantify the “fractality” of such a set. A convenient indicator is the Hausdorff dimension, which is an extension of the dimension of a space. A line is a space of dimension 1, a plane is of dimension 2, and a volume of dimension 3. Now, if you look at the Mandelbrot set, for instance, its contour is a line of infinite length (actually depending on the resolution of the plot), which tends to fill the plane, but does not fill it entirely. So, it makes sense that its dimension should be a real number larger than 1 and smaller than 2. The Hausdorff dimension quantifies how a fractal set fills the space. The Hausdorff dimension of the Mandelbrot set is 2, the one of the coastline of Great-Britain is 1.25, and the one of the coastline of Norway is 1.52.

For a natural object, things are not necessarily that easy, in the sense that some parts of the objects could look like a fractal, and some not, or look like another fractal. Then the object is said multifractal.

The Hausdorff dimension is not the only possible measure of a fractal object. In the paper I present today, the authors use the Hölder exponent, which represents how continuous the function is. Here, the function is the height of a terrain, it depends on its coordinates, i.e. longitude and latitude, on the surface of Mars. The Hölder exponent is usually more appropriate for sets of numerical data.

The wavelet transforms

The wavelet transform is a mathematical transform, which aims at measuring the periodicity of a phenomenon, and gives the amplitude of a periodic contribution, at a given period. In our case, the idea is to measure periodic patterns in the spatial evolution of the height of the surface of Mars.
For that, the authors use more specifically the wavelet leaders methods, which will in particular give them the Hölder exponent, and tell them how (mono)fractal / multifractal the topography of Mars is.

Results

The “fractality” actually depends on the scale you are considering. The authors disposed of MOLA data, with a resolution of 0.463 km. They analyzed them twice, once in performing 1-D analyses, in considering the longitude and the latitude independently, and once in a 2-dimensional analysis, which is probably new in this context. And here are their results:

  • The surface of Mars is monofractal if you look at it at scales smaller than 15 km.
  • It is multifractal for scales larger than 60 km (the authors considered that the range 15-60 km is a transtition zone).
  • The “monofractality” is better in longitude than in latitude. This could be due to the hemispherical asymmetry of Mars, to the polar caps, and / or to the fact that the representation surface is just a planar projection, which necessarily alters it.
  • Some features can be detected from the variations of the Hölder exponent, especially the plains. However, this technique seems to fail for the volcanoes.

Some links

That’s it for today! I hope you enjoyed this post. I particularly like the idea to give a mathematical representation of a natural object. Please feel free to comment! You can also subscribe to the Twitter @planetmechanix and to the RSS feed.