All posts by Terryl Coron

The mass of Cressida from Uranus’ rings

Hi there! Today I will present you a new way to weigh an inner satellite of a giant planet. This is the opportunity for me to present you Weighing Uranus’ moon Cressida with the η Ring by Robert O. Chancia, Matthew M. Hedman & Richard G. French. This study has recently been accepted for publication in The Astronomical Journal.

The inner system of Uranus

Uranus is known to be the third planet of the Solar System by its radius, the 4th by its mass, and the 7th by its distance to the Sun. It is also known to be highly tilted, its polar axis almost being in its orbital plane. You may also know that it has 5 major satellites (Ariel, Umbriel, Titania, Oberon, and Miranda), and that it has been visited by the spacecraft Voyager 2 in 1986. But here, we are interested in its inner system. If we traveled from the center of Uranus to the orbit of the innermost of its major satellites, i.e. Miranda, we would encounter:

  • At 25,559 km: the location where the atmosphere reaches the pressure 1 bar. This is considered to be the radius of the planet.
  • Between 37,850 and 41,350 km: the ζ Ring,
  • At 41,837 km: the 6 Ring,
  • At 42,234 km: the 5 Ring,
  • At 42,570 km: the 4 Ring,
  • At 44,718 km: the α Ring,
  • At 45,661 km: the β Ring,
  • At 47,175 km: the η Ring,
  • At 47,627 km: the γ Ring,
  • At 48,300 km: the δ Ring,
  • At 49,770 km: the satellite Cordelia (radius: 20 km),
  • At 50,023 km: the λ Ring,
  • At 51,149 km: the ε Ring
  • At 53,790 km: the satellite Ophelia (radius: 22 km),
  • At 59,170 km: the satellite Bianca (radius: 26 km),
  • At 61,780 km: the satellite Cressida (radius: 40 km),
  • At 62,680 km: the satellite Desdemona (radius: 34 km),
  • At 64,350 km: the satellite Juliet (radius: 47 km),
  • At 66,090 km: the satellite Portia (radius: 68 km),
  • Between 66,100 and 69,900 km: the ν Ring,
  • At 69,940 km: the satellite Rosalind (radius: 36 km),
  • At 74,800 km: the satellite Cupid (radius: 9 km),
  • At 75,260 km: the satellite Belinda (radius: 45 km),
  • At 76,400 km: the satellite Perdita (radius: 15 km),
  • At 86,010 km: the satellite Puck (radius: 81 km),
  • Between 86,000 and 103,000 km: the μ Ring,
  • In the μ Ring, at 97,700 km: the satellite Mab (radius: 13 km)
  • At 129,390 km: the satellite Miranda (radius: 236 km).

The rings of Uranus are being discovered since 1977. Originally it was from star occultations observed from the Earth. Then Voyager 2 visited Uranus in 1986, which revealed other rings, and more recently the Hubble Space Telescope imaged some of them, permitting other discoveries.. Most of them have a width of ≈1 km.
All of the inner moons have been discovered on Voyager 2 images, except Cupid and Mab, which have been discovered in 2003, once more thanks to Hubble. On the contrary, the major moons have been discovered between 1787 and 1948.

Today we will focus only on

  • At 47,175 km: the η Ring,
  • At 61,780 km: the satellite Cressida (radius: 40 km).

The η Ring is very close to the 3:2 mean-motion resonance (MMR) with Cressida, which means that any particle of the η Ring makes 3 revolutions around Uranus while Cressida makes 2. As a consequence, Cressida has a strong gravitational action on the η Ring.

Gravitational interactions

How do we know the mass of planetary bodies? When we send a spacecraft close enough, the spacecraft is deviated, and from the deviation we have the gravity field, or at least the mass. If we cannot send a spacecraft, then we can invert, i.e. analyze, the interactions between different bodies. We know the mass of the Sun thanks to the orbits of the planets, we know the mass of Jupiter thanks to the orbits of its satellites, and the deviations of the spacecraft. We can also use MMR. For instance, in the system of Saturn, the mass ratios between Mimas and Tethys, between Enceladus and Dione, and between Janus and Epimetheus, were accurately known before the arrival of Cassini, thanks to resonant relations.

We can have resonant interactions between a satellite and a ring, as well. A ring is actually a cloud of small particles, and the way their motion is affected reveals the gravitational interaction with something. When you have a MMR, then the ring exhibits streamlines, which give a pattern with equally spaced corners. From the number of these corners you can determine the MMR involved, and from the size of the pattern you get the mass of the disturbing satellite. This is exactly what happens here, i.e. 3:2 MMR with Cressida affects the η Ring in such a way that you can read the mass of Cressida from the shape of this ring. But for that, you need to be accurate enough on the location of the ring.

The data

The authors used 49 observations, including 3 Voyager 2 ones, the other ones being star occultations by rings. Such an observation should be anticipated, i.e. the relative position of Uranus with respect to thousands of stars is calculated, then the star has to be observed where possible, i.e. in a place where it will be high enough in the sky, and of course at night. You measure the light flux coming from the star, which should be pretty constant… and is not because of the variability of the atmospheric thickness since the star is moving in the sky (remember: the Earth rotates in one day), so you have to compensate with other stars… and if you detect a flux drop, then this means that something is occulting the star. Possibly a ring.
Most of the observations were made in the K band, i.e. at an infrared wavelength of 2.2 μm, where Uranus is fainter than its rings. These observations have been made between 1977 and 1996. Since then, the opening of the rings is too small, i.e. we see Uranus by the edge, which reduces the chances to occult a star.


The authors made a least-square fit. This means that they fitted their corpus of observations with a shape of the ring as R-A cos (mθ), where R is a constant radius, A is an amplitude of distortion of the ring, θ is the angle (a longitude), and m is a factor giving the frequency of the distortion, which could be related to its cause, i.e. the orbital motion of the satellite affecting the ring. You fit R, A and m, i.e. you adjust them so as to reduce the difference (the error, which is mathematically seen as a distance) between your model and the observations. From R you have a ring (and you can check whether there should be a ring there), from A you have the mass of the satellite, and from m and have its frequency (and you can check whether a known satellite has this frequency).
The authors show that the highest effect of the inner satellites on the rings should be the effect of Cressida on the η Ring, thanks to the 3:2 MMR.


The authors find that Cressida should have a density of 0.86±0.16, which is lighter than water. Usually these bodies are supposed to be kind of porous dirty ice, which would mean this kind of density. This is the first measurement of the density of an inner satellite of Uranus. A comparison with other systems shows that this is much denser than the inner satellites of Saturn. However, the inner satellite of Jupiter Amalthea has a pretty similar density.

Finally the authors say that they used this method on other rings, and that additional results should be expected, so we stay tuned. They also say that a spacecraft orbiting Uranus would help knowing these satellites. I cannot agree more. Some years ago, a space mission named Uranus Pathfinder has been proposed to ESA, and another one, named Uranus orbiter and probe, has been proposed to NASA.

The study and the authors

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 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

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.

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.


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

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.

How the Planet Nine would affect the furthest asteroids

Hi there! You have heard of the hypothetical Planet Nine, which could be the explanation for an observed clustering of the pericentres of the furthest asteroids, known as eTNOS for extreme Trans-Neptunian Objects. I present you today a theoretical study investigating in-depth this mechanism, in being focused on the influence of the inclination of this Planet Nine. I present you Non-resonant secular dynamics of trans-Neptunian objects perturbed by a distant super-Earth by Melaine Saillenfest, Marc Fouchard, Giacomo Tommei and Giovanni B. Valsecchi. This study has recently been accepted for publication in Celestial Mechanics and Dynamical Astronomy.

Is there a Planet Nine?

An still undiscovered Solar System planet has always been dreamed, and sometimes even hinted. We called it Tyche, Thelisto, Planet X (“X” for mystery, unknown, but also for 10, Pluto having been the ninth planet until 2006). Since 2015, this quest has been renewed after the observation of clustering in the pericentres of extreme TNOS. Further investigations concluded that at least 5 observed dynamical features of the Solar System could be explained by an additional planet, now called Planet Nine:

  1. the clustering of the pericentres of the eTNOs,
  2. the significant presence of retrograde orbits among the TNOs,
  3. the 6° obliquity of the Sun,
  4. the presence of highly inclined Centaurs,
  5. the dynamical detachment of the pericentres of TNOs from Neptune.

The combination of all of these elements tends to rule out a random process. It appears that this Planet Nine would be pretty like Neptune, i.e. 10 times heavier than our Earth, that its pericentre would be at 200 AU (while Neptune is at 30 AU only!), and its apocentre between 500 AU and 1200 AU. This would indeed be a very distant object, which would orbit the Sun in several thousands of years!

Astronomers (Konstantin Batygin and Michael Brown) are currently trying to detect this Planet Nine, unsuccessfully up to now. You can follow their blog here, from which I took some inspiration. The study I present today investigates the secular dynamics that this Planet Nine would induce.

The secular dynamics of an asteroid

The secular dynamics is the one involving the pericentre and the ascending node of an object, without involving its longitude. To make things clear, you know that a planetary object orbiting the Sun wanders on an eccentric, inclined orbit, which is an ellipse. When you are interested in the secular dynamics, you care of the orientation of this ellipse, but not of where the object is on this ellipse. The clustering of pericentres of eTNOs is a feature of the secular dynamics.

This is a different aspect from the dynamics due to mean-motion resonances, in which you are interested in objects, which orbital periods around the Sun are commensurate with the one of the Planet Nine. Some studies address this issue, since many small objects are in mean-motion resonance with a planet. Not this study.

The Kozai-Lidov mechanism

A notable secular effect is the Kozai-Lidov resonance. Discovered in 1961 by Michael Lidov in USSR and Yoshihide Kozai in Japan, this mechanism says that there exists a dynamical equilibrium at high inclination (63°) for eccentric orbits, in the presence of a perturber. So, you have the central body (the Sun), a perturber (the planet), and your asteroid, which could have its inclination pushed by this effect. This induces a libration of the orientation of its orbit, i.e. the difference between its pericentre and its ascending node would librate around 90° or 270°.

This process is even more interesting when the perturber has a significant eccentricity, since the so-called eccentric Kozai-Lidov mechanism generates retrograde orbits, i.e. orbits with an inclination larger than 90°. At 117°, you have another equilibrium.

Now, when you observe a small body which dynamics suggests to be affected by Kozai-Lidov, this means you should have a perturber… you see what I mean?

Of course, this perturber can be Neptune, but only sometimes. Other times, the dynamics would rather be explained by an outer perturber… which could be the Planet Nine, or a passing star (who knows?)


Before mentioning the results of this study I must briefly mention the methodology. The authors made what I would call a semi-analytical study, i.e. they manipulated equations, but with the assistance of a computer. They wrote down the Hamiltonian of the restricted 3-body problem, i.e. the expression of the whole energy of the problem with respect to the orbital elements of the perturber and the TNO. This energy should be constant, since no dissipation is involved, and the way this Hamiltonian is written has convenient mathematical properties, which allow to derive the whole dynamics. Then this Hamiltonian is averaged over the mean longitudes, since we are not interested in them, we want only the secular dynamics.

A common way to do this is to expand the Hamiltonian following small parameters, i.e. the eccentricity, the inclination… But not here! You cannot do this since the eccentricity of the Planet Nine (0.6) and its inclination are not supposed to be small. So, the authors average the Hamiltonian numerically. This permits them to keep the whole secular dynamics due to the eccentricity and the inclination.

Once they did this, they looked for equilibriums, which would be preferential dynamical states for the TNOs. They also detected chaotic zones in the phase space, i.e. ranges of orbital elements, for which the trajectory of the TNOs would be difficult to predict, and thus potentially unstable. They detected these zones in plotting so-called Poincaré sections, which give a picture of the trajectories in a two-dimensional plane that reduces the number of degrees-of-freedom.


And the authors find that the two Kozai-Lidov mechanisms, i.e. the one due to Neptune, and the one due to the Planet Nine, conflict for a semimajor axis larger than 150 AU, where orbital flips become possible. The equilibriums due to Neptune would disappear beyond 200 AU, being submerged by chaos. However, other equilibriums appear.

For the future, I see two ways to better constrain the Planet Nine:

  1. observe it,
  2. discover more eTNOs, which would provide more accurate constraints.

Will Gaia be useful for that? Anyway, this is a very exciting quest. My advice: stay tuned!

To know more…

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.

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.


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.


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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