Tag Archives: numerical methods

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…

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.

A polar resonant asteroid

Hi there! Did you know that an asteroid could be resonant and in polar orbit? Yes? No? Anyway, one of them has been confirmed as such, i.e. this body was already discovered, known to be on a polar orbit, but it was not known to be in mean-motion resonance with Neptune until now. This is the opportunity for me to present you First transneptunian object in polar resonance with Neptune, by M.H.M. Morais and F. Namouni. This study has recently been accepted for publication in The Monthly Notices of the Royal Astronomical Society.

Polar asteroids

The planets of the Solar System orbit roughly in the same plane. In other words, they have small mutual inclinations. However, asteroids are much more scattered, and can have any inclination with respect to the ecliptic, i.e. the orbital plane of the Earth, even if low inclinations are favored.

Two angles are needed to orientate an orbit:

  • the ascending node, which varies between 0 and 360°, and which is the angle between a reference and the intersection between the ecliptic and the orbital plane,
  • the inclination, which is the angle between the ecliptic and the orbital plane. It varies between 0° and 180°.

So, an almost planar orbit means an inclination close to 0° or close to 180°. Orbits close to 0° are prograde, while orbits close to 180° are retrograde. However, when your inclination is close to 90°, then you have a polar orbit. There are prograde and retrograde polar orbits, whether the inclination is smaller (prograde) or larger (retrograde) than 90°.

There are 7 known Trans-Neptunian Objects with an eccentricity smaller than 0.86 and inclination between 65 and 115°, hence 7 known polar TNOs. You can find them below:

Semimajor axis Eccentricity Inclination Ascending node Period
(471325) 2011 KT19 (Niku) 35.58 AU 0.33 110.12° 243.76° 212.25 y
2008 KV42 (Drac) 41.44 AU 0.49 103.41° 260.89° 266.75 y
2014 TZ33 38.32 AU 0.75 86.00° 171.79° 237.20 y
2015 KZ120 46.07 AU 0.82 85.55° 249.98° 312.70 y
(127546)2002 XU93 67.47 AU 0.69 77.95° 90.39° 554.18 y
2010 WG9 52.90 AU 0.65 70.33° 92.07° 384.77 y
2017 CX33 73.97 AU 0.86 72.01° 315.88° 636.21 y

These bodies carry in their names their year of discovery. As you can see, the first of them has been discovered only 15 years ago. We should keep in mind that TNOs orbit very far from the Earth, this is why they are so difficult to discover, polar or not.

The last of them, 2017 CX33, is so recent that the authors did not study it. A recent discovery induces a pretty large uncertainty on the orbital elements, so waiting permits to stay on the safe side. Among the 6 remaining, 4 (Niku, Drac, 2002 XU93 and 2010 WG9) share (very) roughly the same orbit, 2 of them being prograde, while the others two are retrograde. This happened very unlikely by chance, but the reason for this rough alignment is still a mystery.

Orbits of the polar TNOs, in the x-y plane.
Orbits of the polar TNOs, in the x-y plane.
Orbits of the polar TNOs, in the y-z plane.
Orbits of the polar TNOs, in the y-z plane.

The study I present you today investigated the current dynamics of these bodies, and found a resonant behavior for one of them (Niku).

Behavior of the resonant asteroids

By resonant behavior, I mean that an asteroid is affected by a mean-motion resonance with a planet. This means that it makes a given (integer) number of revolutions around the Sun, while the planet makes another number of revolutions. Many outcomes are possible. It can slowly enhance the eccentricity and / or the inclination, which could eventually lead to a chaotic behavior, instability, collision… it could also protect the body from close encounters…

It usually translates into an integer combination of the fundamental frequencies of the system (orbital frequencies, frequencies of precession of the nodes and pericentres), which is null, and this results in an integer combination of angles positioning the asteroid of the planet, which oscillates around a given number instead of circulating. In other words, this angle is bounded.

Another point of interest is how the asteroid has been trapped into the resonance. A resonance is between two interacting bodies, but the mass ratio between an asteroid and a planet implies that the planet is insensitive to the gravitational action of the asteroid, and so the asteroid is trapped by the planet. The fundamental frequencies of the orbital motion are controlled by the semimajor axes of the two bodies, so a trapping into a resonance results from a variation of the semimajor axes. Models of formation of the Solar System suggest that the planets have migrated, this could be a cause. Another cause is close encounters between planets and asteroids, which result in abrupt changes in the trajectory of the asteroid. And this is probably the case here: Niku got trapped after a close encounter.

Numerical and analytical study

The authors used both numerical and analytical methods to get, understand, and secure their results.

Numerical study

The authors ran long-term numerical simulations of the orbital motion of the 6 relevant asteroids, perturbed by the planets. They ran 3 kinds of simulations: 2 with different integrators (algorithms) over 400 kyr and 100 Myr and 8 planets, and one over 400 Myr and the four giant planets. With less planets, you go faster. Moreover, since the inner planets have shorter orbital periods, removing them allows you to increase the time-step, and thus go further in time, inward and backward. In each of these simulations, the authors cloned the asteroids to take into consideration the uncertainty on the orbital elements. They used for that a well-known devoted code, MERCURY.

Analytical study

Numerical studies give you an idea of the possible dynamical states of a system, but you need to write down equations to fully understand it. Beside these numerical simulations, the authors wrote a dynamical theory of resonant polar orbits, in another paper (or here).

This consists in reducing the equations to the only terms, which are useful to reproduce the resonant dynamics. For that, you keep the secular variations, i.e. precessions of the nodes and pericentres, and the term involving the resonant argument. This is a kind of averaged dynamics, in which all of the small oscillations of the orbital elements have been dropped. To improve the relevance of the model, the authors used orbital elements which are based on the barycenter (center of mass) of the whole Solar System instead on the Sun only. This is a small correction, since the barycenter is at the edge of the Sun, but the authors mention that it improves their results.


Niku, i.e. (471325) 2011 KT19, is trapped into a 7:9 mean-motion resonance with Neptune. In other words, it makes 7 revolutions around the Sun (sorry: the barycenter of the Solar System) while Neptune makes 9. More precisely, its resonant argument is φ=9λ-7λN-4ϖ+2Ω, where λ and λN are the mean longitudes of the asteroid and of Neptune, respectively, ϖ is the longitude of its pericenter, and Ω is the one of its ascending node. Plotting this argument shows a libration around 180°. Niku has been trapped in this resonance after a close encounter with Neptune, and should leave this resonance in 16±11 Myr. This means that all of the numerical simulations involving Niku show a resonant object, however they disagree on the duration of the resonance.
Their might be another resonant object: a few simulations suggest that Drac, i.e. 2008 KV42 is in a 8:13 mean-motion resonance with Neptune.

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.

Our water comes from far away

Hi there! Can you imagine that our water does not originally come from the Earth, but from the outer Solar System? The study I present you today explains us how it came to us. This is Origin of water in the inner Solar System: Planetesimals scattered inward during Jupiter and Saturn’s rapid gas accretion by Sean Raymond and Andre Izidoro, which has recently been published in Icarus.

From the planetary nebula to the Solar System

There are several competing scenarios, which describe a possible path followed by the Solar System from its early state to its current one. But all agree that there was originally a protoplanetary disk, orbiting our Sun. It was constituted of small particles and gas. Some of the small particles accreted to form the giant planets, first as a massive core, then in accreting some gas around. The proto-Jupiter cleared a ring-shaped gap around its orbit in the disk, Saturn formed as well, the planets migrated, in interacting with the gas. How fast did they migrate? Inward? Outward? Both? Scenarios diverge. Anyway, the gas was eventually ejected, and the protoplanetary disk was essentially cleared, except when it is not. There remains the telluric planets, the giant planets, and the asteroids, many of them constituting the Main Belt, which lies between the orbits of Mars and Jupiter.
If you want to elaborate a fully consistent scenario of formation / evolution of the Solar System, you should match the observations as much as possible. This means matching the orbits of the existing objects, but not only. If you can match their chemistry as well, that is better.

No water below this line!

The origin of water is a mystery. You know that we have water on Earth. It seems that this water comes from the so-called C-type asteroids. These are carbonaceous asteroids, which contain a significant proportion of water, usually between 5 and 20%. This is somehow the same water as on Earth. In particular, it is consistent with the ratios D/H and 15N/14N present in our water. D is the deuterium, it is an isotope of hydrogen (H), while 15N and 14N are two isotopes of nitrogen (N).

These asteroids are mostly present close to the outer boundary of the Main Belt, i.e. around 3.5 AU. An important parameter of a planetary system is the snow line: below a given radius, the water cannot condensate into ice. That makes sense: the central star (in our case, the Sun) is pretty hot (usually more than pretty, actually…), and ice cannot survive in a hot environment. So, you have to take some distance. And the snow line of the Solar System is currently lose to 3.5 AU, where we can find these C-type asteroids. Very well, there is no problem…

But there is one: the location of the snow line changes during the formation of the Solar System, since it depends on the dynamical structure of the disk, i.e. eccentricity of the particles constituting it, turbulence in the gas, etc. in addition to the evolution of the central star, of course. To be honest with you, I have gone through some literature and I cannot tell you where the snow line was at a given date, it seems to me that this is still an open question. But the authors of this study, who are world experts of the question, say that the snow line was further than that when these C-types asteroids formed. I trust them.

And this raises an issue: the C-types asteroids, composed of at least 5% of water, have formed further than they are. This study explains us how they migrated inward, from their original location to their present one.

Planet encounter and gas drag populate the Asteroid Belt

The authors ran intensive numerical simulations, in which the asteroids are massless particles, but with a given radius. This seems weird, but this just means that the authors neglected the gravitational action of the asteroids on the giant planets. The reason why they gave them a size in that it influences the way the gas drag (remember: the early Solar System was full of gas) affects their orbits. This size actually proved to be a key parameter. So, these asteroids were affected by the gas and the giant planets, but in the state they were at that time, i.e. initially Jupiter and Saturn were just slowly accreting cores, and when these cores of solid material reached a critical size, then they were coated by a pretty rapid (over a few hundreds of kyr) accretion of gas. The authors considered only Jupiter in their first simulations, then Jupiter and Saturn, and finally the four giant planets. Their different parameters were:

  • the size of the asteroids (planetesimals),
  • the accretion velocity of the gas around Jupiter and Saturn,
  • the evolution scenario of the early Solar System. In particular, the way the giant planets migrated.

Simulating the formation of the planet actually affects the orbital evolution of the planetesimals, since the mass of the planets is increasing. The more massive the planet, the most deviated the asteroid.

And the authors succeed in putting C-type asteroids with this mechanism: when a planetesimal encounters a proto-planet (usually the proto-Jupiter), its eccentricity reaches high numbers, which threatens its orbital stability around the Sun. But the gas drag damps this eccentricity. So, these two effects compete, and when ideally balanced this results in asteroids in the Main-Belt, on low eccentric orbits. And the authors show that this works best for mid-sized asteroids, i.e. of the order of a few hundreds of km. Below, Jupiter ejects them very fast. Beyond, the gas drag is not efficient enough to damp the eccentricity. And this is consistent with the current observations, i.e. there is only one C-type asteroid larger than 1,000 km, this is the well-known Ceres.

However, the scenarios of evolution of the Solar System do not significantly affect this mechanism. So, it does not tell us how the giant planets migrated.

Once the water ice has reached the main asteroid belt, other mechanism (meteorites) carry it to the Earth, where it can survive thanks to our atmosphere.

Making the exoplanets habitable

This study proposes a mechanism of water delivery, which could be adapted to any planetary system. In particular, it tells us a way to make exoplanetary planets habitable. Probably more to come in the future.

To know more…

  • The study, presented by the first author (Sean N. Raymond) on his own blog,
  • The website of Sean N. Raymond,
  • The IAU page of Andre Izidoro.
  • And I would like to mention Pixabay, which provides free images, in particular the one of Cape Canaveral you see today. Is this shuttle going to fetch some water somewhere?

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 fate of the Alkyonides

Hello everybody! Today, I will tell you on the dynamics of the Alkyonides. You know the Alkyonides? No? OK… There are very small satellites of Saturn, i.e. kilometer-sized, which orbit pretty close to the rings, but outside. These very small bodies are known to us thanks to the Cassini spacecraft, and a recent study, which I present you today, has investigated their long-term evolution, in particular their stability. Are they doomed or not? How long can they survive? You will know this and more after reading this presentation of Long-term evolution and stability of Saturnian small satellites: Aegaeon, Methone, Anthe, and Pallene, by Marco Muñoz-Gutiérrez and Silvia Giuliatti Winter. This study has recently been accepted for publication in The Monthly Notices of the Royal Astronomical Society.

The Alkyonides

As usually in planetary sciences, bodies are named after the Greek mythology, which is the case of the four satellites discussed today. But I must admit that I cheat a little: I present them as Alkyonides, while Aegeon is actually a Hecatoncheires. The Alkyonides are the 7 daughters of Alcyoneus, among them are Anthe, Pallene, and Methone.

Here are some of there characteristics:

Methone Pallene Anthe Aegaeon
Semimajor axis 194,402 km 212,282 km 196,888 km 167,425 km
Eccentricity 0 0.004 0.0011 0.0002
Inclination 0.013° 0.001° 0.015° 0.001°
Diameter 2.9 km 4.4 km 2 km 0.66 km
Orbital period 24h14m 27h42m 24h52m 19h24m
Discovery 2004 2004 2007 2009

For comparison, Mimas orbits Saturn at 185,000 km, and the outer edge of the A Ring, i.e. of the main rings of Saturn, is at 137,000 km. So, we are in the close system of Saturn, but exterior to the rings.

Discovery of Anthe, aka S/2007 S4. Copyright: NASA.
Discovery of Anthe, aka S/2007 S4. Copyright: NASA.

These bodies are in mean-motion resonances with main satellites of Saturn, more specifically:

  • Methone orbits near the 15:14 MMR with Mimas,
  • Pallene is close to the 19:16 MMR with Enceladus,
  • Anthe orbits near the 11:10 MMR with Mimas,
  • Aegaeon is in the 7:6 MMR with Mimas.

As we will see, these resonances have a critical influence on the long-term stability.

Rings and arcs

Beside the main and well-known rings of Saturn, rings and arcs of dusty material orbit at other locations, but mostly in the inner system (with the exception of the Phoebe ring). In particular, the G Ring is a 9,000 km wide faint ring, which inner edge is at 166,000 km… Yep, you got it: Aegaeon is inside. Some even consider it is a G Ring object.

Methone and Anthe have dusty arcs associated with them. The difference between an arc and a ring is that an arc is longitudinally bounded, i.e. it is not extended enough to constitute a ring. The Methone arc extends over some 10°, against 20° for the Anthe arc. The material composing them is assumed to be ejecta from Methone and Anthe, respectively.

However, Pallene has a whole ring, constituted from ejecta as well.

Why sometimes a ring, and sometimes an arc? Well, it tell us something on the orbital stability of small particles in these areas. Imagine you are a particle: you are kicked from home, i.e. your satellite, but you remain close to it… for some time. Actually you drift slowly. While you drift, you are somehow shaken by the gravitational action of the other satellites, which disturb your Keplerian orbit around the planet. If you are shaken enough, then you may leave the system of Saturn. If you are not, then you can finally be anywhere on the orbit of your satellite, and since you are not the only one to have been ejected (you feel better, don’t you?), then you and your colleagues will constitute a whole ring. If you are lucky enough, you can end up on the satellite.

The longer the arc (a ring is a 360° arc), the more stable the region.

Frequency diffusion

The authors studied

  1. the stability of the dusty particles over 18 years
  2. the stability of the satellites in the system of Saturn over several hundreds of kilo-years (kyr).

For the stability of the particles, they computed the frequency diffusion index. It consists in:

  1. Simulating the motion of the particles over 18 years,
  2. Determining the main frequency of the dynamics over the first 9 years, and over the last 9 ones,
  3. Comparing these two numbers. The smaller the difference, the more stable you are.

The numerical simulations is something I have addressed in previous posts: you use a numerical integrator to simulate the motion of the particle, in considering an oblate Saturn, the oblateness being mostly due to the rings, and several satellites. Our four guys, and Janus, Epimetheus, Mimas, Enceladus, and Tethys.

How resonances destabilize an orbit

When a planetary body is trapped in a mean-motion resonance, there is an angle, which is an integer combination of angles present in its dynamics and in the dynamics of the other body, which librates. An example is the MMR Aegaeon-Mimas, which causes the angle 7λMimas-6λAegaeonMimas to librate. λ is the mean longitude, and ϖ is the longitude of the pericentre. Such a resonance is supposed to affect the dynamics of the two satellites but, given their huge mass ratio (Mimas is between 300 and 500 millions times heavier than Aegaeon), only Aegaeon is affected. The resonance is at a given location, and Aegaeon stays there.
But a given resonance has some width, and several resonant angles (we say arguments) are associated with a resonance ratio. As a consequence, several resonances may overlap, and in that case … my my my…
The small body is shaken between different locations, its eccentricity and / or inclination can be raised, until being dynamically unstable…
And in this particular region of the system of Saturn, there are many resonances, which means that the stability of the discovered body is not obvious. This is why the authors studied it.


Stability of the dusty particles

The authors find that Pallene cannot clear its ring efficiently, despite its size. Actually, this zone is the most stable, wrt the dynamical environments of Anthe, Methone and Aegaeon. However, 25% of the particles constituting the G Ring should collide with Aegaeon in 18 years. This probably means that there is a mechanism, which refills the G Ring.

Stability of the satellites

From long-term numerical simulations over 400 kyr, i.e. more than one hundred millions of orbits, these 4 satellites are stable. For Pallene, the authors guarantee its stability over 64 Myr. Among the 4, this is the furthest satellite from Saturn, which makes it less affected by the resonances.

A perspective

The authors mention as a possible perspective the action of the non-gravitational forces, such as the solar radiation pressure and the plasma drag, which could affect the dynamics of such small bodies. I would like to add another one: the secular tides with Saturn, and the pull of the rings. They would induce drifts of the satellites, and of the resonances associated. The expected order of magnitude of these drifts would be an expansion of the orbits of a few km / tens of km per Myr. This seems pretty small, but not that small if we keep in mind that two resonances affecting Methone are separated by 4 km only.

This means that further results are to be expected in the upcoming years. The Cassini mission is close to its end, scheduled for 15 Sep 2017, but we are not done with exploiting its results!

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.

On the interior of Mimas, aka the Death Star

Hi there! Today I will tell you on the interior of Mimas. You know, Mimas, this pretty small, actually the smallest of the mid-sized, satellite of Saturn, which has a big crater, like Star Wars’ Death Star. Despite an inactive appearance, it presents confusing orbital quantities, which could suggest interesting characteristics. This is the topic of the study I present you today, by Marc Neveu and Alyssa Rhoden, entitled The origin and evolution of a differentiated Mimas, which has recently been published in Icarus.

Mimas’ facts

The system of Saturn is composed of different groups of satellites. You have

  • Very small satellites embedded into the rings,
  • Mid-sized satellites orbiting between the rings and the orbit of Titan
  • The well-known Titan, which is very large,
  • Small irregular satellites, which orbit very far from Saturn and are probably former asteroids, which had been trapped by Saturn,
  • Others (to make sure I do not forget anybody, including the coorbital satellites of Tethys and Dione, Hyperion, the Alkyonides, Phoebe…).

Discovered in 1789 by William Herschel, Mimas is the innermost of the mid-sized satellites of Saturn. It orbits it in less than one day, and has strong interactions with the rings.

Semimajor axis 185,520 km
Eccentricity 0.0196
Inclination 1.57°
Diameter 396.4 km
Orbital period 22 h 36 min

As we can see, Mimas has a significant eccentricity and a significant inclination. This inclination could be explained by a mean-motion resonance with Tethys (see here). However, we see no obvious cause for its present eccentricity. It could be due to a past gravitational excitation by another satellite.

Mimas, seen by Cassini. We can the crater Herschel, which makes Mimas look like Star Wars' Death Star. Credit: NASA
Mimas, seen by Cassini. We can the crater Herschel, which makes Mimas look like Star Wars' Death Star. Credit: NASA

The literature is not unanimous on the formation of Mimas. It was long thought that the satellites of Saturn formed simultaneously with the planet and the rings, in the proto-Saturn nebula. The Cassini space mission changed our view of this system, and other scenarios were proposed. For instance, the mid-sized satellites of Saturn could form from the collisions between 4 big progenitors, Titan being the last survivor of them. The most popular explanation seems to be that a very large body impacted Saturn, its debris coalesced into the rings, and then particles in the rings accreted, forming satellites which then migrated outward… these satellites being the mid-sized satellites, i.e. Rhea, Dione, Tethys, Enceladus, and Mimas. This scenario would mean that Mimas would be the youngest of them, and that it formed differentiated, i.e. that the proto-Mimas was made of pretty heavy elements, on which lighter elements accreted. Combining observations of Mimas with theoretical studies of its long-term evolution could help to determine which of these scenarios is the right one… if there is a right one. Such studies can of course involve other satellites, but this one is essentially on Mimas, with a discussion on Enceladus at the end.

The rotation of Mimas

As most of the natural satellites of the giant planets, Mimas is synchronous, i.e. it shows the same face to Saturn, its rotational (spin) period being on average equal to its orbital one. “On average” means that there are some variations. These are actually a sum of periodic oscillations, which are due to the variations of the distance Mimas-Saturn. And from the amplitude and phase of these variations, you can deduce something on the interior, i.e. how the mass is distributed. This could for instance reveal an internal ocean, or something else…

This rotation has been measured in 2014 (see this press release). The mean rotation is indeed synchronous, and here are its oscillations:

Period Measured
amplitude (arcmin)
amplitude (arcmin)
70.56 y 2,616.6 2,631.6±3.0
23.52 y 43.26 44.5±1.1
22.4 h 26.07 50.3±1.0
225.04 d 7.82 7.5±0.8
227.02 d 3.65 2.9±0.9
223.09 d 3.53 3.3±0.8

The most striking discrepancy is at the period 22.4 h, which is the orbital period of Mimas. These oscillations are named diurnal librations, and their amplitude is very sensitive to the interior. Moreover, the amplitude associated is twice the predicted one. This means that the interior, which was hypothesized for the theoretical study, is not a right one, and this detection of an error is a scientific information. It means that Mimas is not exactly how we believed it is.

The authors of the 2014 study, led by Radwan Tajeddine, investigated 5 interior models, which could explain this high amplitude. One of these models considered the influence of the large impact crater Herschel. In all of these models, only 2 could explain this high amplitude: either an internal ocean, or an elongated core of pretty heavy elements. Herschel is not responsible for anything in this amplitude.

The presence of an elongated core would support the formation from the rings. However, the internal ocean would need a source of heating to survive.

Heating Mimas

There are at least three main to heat a planetary body:

  1. hit it to heat it, i.e. an impact could partly melt Mimas, but that would be a very intense and short heating, which would have renewed the surface…nope
  2. decay of radiogenic elements. This would require Mimas to be young enough
  3. tides: i.e. internal friction due to the differential attraction of Saturn. This would be enforced by the variations of the distance Saturn-Mimas, i.e. the eccentricity.

And this is how we arrive to the study: the authors simulated the evolution of the composition of Mimas under radiogenic and tidal heating, in also considering the variations of the orbital elements. Because when a satellite heats, its eccentricity diminishes. Its semimajor axis varies as well, balanced between the dissipation in the satellite and the one in Saturn.

The problems

For a study to be trusted by the scientific community, it should reproduce the observations. This means that the resulting Mimas should be the Mimas we observe. The authors gave themselves 3 observational constraints, i.e. Mimas must

  1. have the right orbital eccentricity,
  2. have the right amplitude of diurnal librations,
  3. keep a cold surface.

and they modeled the time evolution of the structure and the orbital elements using a numerical code, IcyDwarf, which simulates the evolution of the differentiation, i.e. separation between rock and water, porosity, heating, freezing of the ocean if it exists…


The authors show that in any case, the ocean cannot survive. If there would be a source of heating sustaining it, then the eccentricity of Mimas would have damped. In other words, you cannot have the ocean and the eccentricity simultaneously. Depending on the past (unknown) eccentricity of Mimas and the dissipation in Saturn, which is barely known, an ocean could have existed, but not anymore.
As a consequence, Mimas must have an elongated core, coated by an icy shell. The eccentricity could be sustained by the interaction with Saturn. This elongated core could have two origins: either a very early formation of Mimas, which would have given enough time for the differentiation, or a formation from the rings, which would have formed Mimas differentiated.

Finally the authors say that there model does not explain the internal ocean of Enceladus, but Marc Neveu announces on his blog that they have found another explanation, which should be published pretty soon. Stay tuned!

Another mystery

The 2014 study measured a phase shift of 6° in the diurnal librations. This is barely mentioned in the literature, probably because it bothers many people… This is huge, and could be more easily, or less hardly, explained with an internal ocean. I do not mean that Mimas has an internal ocean, because the doubts regarding its survival persist. So, this does not put the conclusions of the authors into question. Anyway, if one day an explanation would be given for this phase lag, that would be warmly welcome!

To know more…

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