Predicting the chemical composition of (4)Vesta

Hi there! Today I present you a study entitled Chlorine and hydrogen degassing in Vesta’s magma ocean, by Adam R. Sarafian, Timm John, Julia Roszjár and Martin J. Whitehouse. This study has recently been published in Earth and Planetary Science Letters. The goal here is, from the chemical analysis of meteorites which are supposed to come from Vesta, understand the evolution of its chemical evolution. In particular, how the degassing of its magma ocean impacts its chemical evolution.


I have presented the small planet (4)Vesta in that post. Basically, it is one of the largest Main-Belt asteroids, with a mean radius of some 500 km. The craters at its surface and the dynamical models of the early Solar System show that Vesta has been intensively bombarded. The largest of these impacts were energetic enough to melt Vesta and trigger its differentiation between a pretty dense core, a shallow magma ocean and a thin crust.

Despite having been visited by the spacecraft Dawn, the magma ocean has not been detected. Its presence is actually confirmed by the analyses of meteorites which fell on Earth.

The HED meteorites

Every day, about 6 tons of material hit the surface of the Earth, after having survived the atmospheric entry. Mineralogists split these meteorites into several groups. 5% of these meteorites are HEDs, for Howardite-Eucrite-Diogenites. These are achondritic basaltic meteorites, which are supposed to present similarities with Vesta. This hypothesis has been proposed in 1970 after comparison of the spectrum of Vesta and the one of these meteorites, and enforced since by the observations and theoretical works. So, it is now accepted that these meteorites come from Vesta or bodies similar to it, and studying them is a way to study the chemical composition of Vesta.
In this study, only the Eucrites will be addressed. They represent most of the HEDs, and contain 2 phosphates: the merrillite and the apatite. Moreover, they are systematically depleted in volatile elements, compared to carbonaceous chondrites and the Earth.

Chemical analysis

The authors have analyzed the chemical composition of 7 samples of eucrites, which were found on Earth. They present a variety of thermal alteration. Comparing them would be like watching a movie of the process of evolution of the material during the degassing in the magma ocean. The analyses were conducted on two sites: the Natural History Museum Vienna, in Austria, and the Woods Hole Oceanographic Institution (MA, USA). The involved technology is the scanning electron microscopy, which consists in obtaining images from the interaction of the sample with a focused bean of electrons, supplemented with an energy-dispersive X-ray spectrometer. This spectrometer gives the spectral signature of the interactions of the electrons with the rock sample, and so reveals the elements which constitute it.

The authors were particularly interested in measuring the concentrations of halogen (fluorine, chlorine, bromine and iodine), of stable isotopes of the chlorine, isotopes of hydrogen, and water. Comparing the relative concentration of these elements in the seven samples would give information on their volatilization during the outgassing process of the magma ocean, in conditions that do not exist on Earth.


The samples show different compositions in volatile elements (H2, H20, and metal chlorides), which show that there is some outgassing in Vesta’s magma ocean. The authors show in particular a large variability in the ratio [Cl]/[K], i.e. chlorite with respect to potassium. This means that not only the thermal evolution tends to reject volatile elements, but also that they are effectively ejected. This might be a concern since the ocean cannot be seen at the surface of Vesta. Anyway, this does not preclude outgassing, either through the crust, which is supposed to be thin, and/or with the assistance of giant impacts, which created craters deep enough to reach the ocean.

This way, we have a signature of the history of a planetary body in material found on the Earth. These results might have implications beyond Vesta, i.e. could be extended to other dwarf planets, and so give us information on the chemical evolution of the Solar System.

I hope you enjoyed this article. As usual, I am interested in your feed-back. So please, leave me some comments, share it, and happy new year!

To know more…

  • The study, which can also be found on ResearchGate, thanks to the authors for sharing!
  • The webpage of Adam Robert Sarafian, grad student at the Woods Hole Oceanographic Institution (USA)
  • The webpage of Timm John, Freie Universität Berlin, Germany
  • The webpage of Julia Roszjár, Natural History Museum, Vienna, Austria
  • The webpage of Martin Whitehouse, Swedish Museum of Natural History, Stockholm, Sweden

The dynamics of Saturn’s F Ring

Hi there! Today: a new post on the rings of Saturn. I will more specifically discuss the F Ring, in presenting you the study A simple model for the location of Saturn’s F ring, by Luis Benet and Àngel Jorba, which has recently been accepted for publication in Icarus.

The F Ring

The F Ring of Saturn is a narrow ring of particles. It orbits close to the Roche limit, which is the limit below which the satellites are not supposed to accrete because the differential gravitational action of Saturn on different parts of it prevents it. This is also the theoretical limit of the existence of the rings.

The F Ring seen by Cassini (Credit: NASA)
The F Ring seen by Cassini (Credit: NASA)

Its mean distance from the center of Saturn is 140,180 km, and its extent is some hundreds of kilometers. It is composed of a core ring, which width is some 50 km, and some particles which seem to be ejected in spiral strands.

Orbiting nearby are the two satellites Prometheus (inside) and Pandora (outside), which proximity involves strong gravitational perturbations, even if they are small.

The images of the F Ring, and in particular of its structures, are sometimes seen as an example of observed chaos in the Solar System. This motivates many planetary scientists to investigate its dynamics.

Mean-motion resonances in the rings

Imagine a planar configuration, in which we have a big planet (Saturn), a small particle orbiting around (the rings are composed of particles), and a third body which is very large with respect to the particle, but very small with respect to the planet (a satellite). The orbit of the particle is essentially an ellipse (Keplerian motion), but is also perturbed by the gravitational action of the satellite. This usually results in oscillating, periodic variations of its orbital elements, in particular the semimajor axis… except in some specific configurations: the mean motion resonances.

When the orbital periods of the particle and of the satellite are commensurate, i.e. when you can write the ratio of their orbital frequencies as a fraction of integers, then you have part of the gravitational action of the satellite on the particle which accumulates during the orbital history of the two bodies, instead of cancelling out. In such a case, you have a resonant interaction, which usually produces the most interesting effects in planetary systems.

There are resonances among planetary satellites as well, but here I will stick to the rings-satellites interactions, for which a specific formalism has been developed, itself inspired from the galactic dynamics. Actually, 4 angles should be considered, which are

  • the mean longitude of the particle λp, which locates the particle on its orbit,
  • the mean longitude of the satellite λs
  • the longitude of the pericentre of the particle ϖp, which locates the point of the orbit which is the closest to Saturn,


  • the longitude of the pericentre of the satellite ϖs.

The situation is a little more complicated when the orbits are not planar, please allow me to dismiss that question for this post.

You have a mean-motion resonance when you can write <pλp-(p+q)λs+q1ϖp+q2ϖs>=0, <> meaning on average. p, q, q1 and q2 are integer coefficients verifying q1+q2=q. The sum of the integer coefficients present in the resonant argument is null. This rule is sometimes called d’Alembert rule, and is justified by the fact that you do not change the physics of a system if you change the reference frame in which you describe it. The only way to preserve the resonant argument from a rotation of an angle α and axis z is that the sum of the coefficients is null.

It can be shown that the strongest resonances happen with |q|=1, meaning either |q1|=1 and q2=0, or
|q2|=1 and q1=0.

In the first case, pλp-(p+1)λsp is the argument of a Lindblad resonance, which pumps the eccentricity of the particle, while pλp-(p+1)λss is a corotation resonance, which is doped by the eccentricity of the satellite. Here I supposed a positive q, which means that the orbit of the satellite is exterior to the one of the particle. This is the case for the configurations F Ring – Pandora and F Ring – Titan. However, when the satellite is interior to the particle, like in the configuration F Ring Prometheus, then the argument of the Lindblad resonance should read pλs-(p+1)λpp, and the one of the corotation resonance is pλs-(p+1)λps.

As I said, these resonances have cumulative effects on the orbits. This means that we could expect that something happens, this something being possibly anything: a Lindblad resonance should pump the eccentricity of a particle and favor its ejection, but this also means that particle which would orbit nearby without being affected by the resonance would be more stable… chaotic effects might happen, which would be favored by the accumulation of resonances, the consideration of higher-order ones, the presence of several perturbers… This is basically what is observed in the F Ring.

The method: numerical integrations

The authors address this problem in running intensive numerical simulations of the behavior of the particles under the gravitational action of Saturn and some satellites. Let me specify that, usually, the rings are seen as clouds of interacting particles. They interact in colliding. In that specific study, the collisions are neglected. This allows the authors to simulate the trajectory of any individual particle, considered as independent of the other ones.

They considered that the particles are perturbed by the oblateness of Saturn expanded until the order 2 (actually this has been measured with a good accuracy until the order 6), Prometheus, Pandora, and Titan. Why these bodies? Because they wanted to consider the most significant ones on the dynamics of the F Ring. When you model so many particles (2.5 millions) over such a long time span (10,000 years), you are limited by the computation time. A way to reduce it is to remove negligible effects. Prometheus and Pandora are the two closest ones and Titan the largest one. The authors have detected that Titan slightly shifts the location of the resonances. However, they admit that they did not test the influence of Mimas, which is the closest of the mid-sized satellites, and which is known for having a strong influence on the main rings.

A critical point when you run numerical integrations, especially over long durations, is the accuracy, because you do not want to propagate errors. The authors use a symplectic scheme, based on a Hamiltonian formulation, i.e. on the conservation of the total energy, which can be expanded up to the order 28. The conservation of the total energy makes sense as long as the dissipation is neglected, which is the case here. The internal accuracy of the integrator was set to 10-21, which translated into a relative error on the angular momentum of Titan below 2.10-14 throughout the whole integration.

Measuring the stability

It might be tough to determine from a numerical integration whether a particle has a stable orbit or not. If you simulate its ejection, then you know, but if you do not see its ejection, you have to decide from the simulated trajectory whether the particle will be ejected one day or not, and possibly when.
For this, two kinds of indicator exists in the literature. The first kind addresses the chaos, or most specifically the hyperbolicity of the trajectory, while the second one addresses the variability of the fundamental frequencies of the system. From a rigorous mathematical point of view, these two notions are different. Anyway, the ensuing indicators are convenient ways to characterize non-periodic trajectories, and their use are commonly accepted as indicators of stability.
A hyperbolic point is an unstable equilibrium. For instance a rigid pendulum has a stable equilibrium down (when you perturb it, it will return down), but an unstable one up (it stays up until you perturb it). The up position is hyperbolic, while the down one is elliptic. The hyperbolicity of a trajectory implies a significant dependency on the initial conditions of the system: a slightly different initial position or different initial velocity will give you a very different trajectory. In systems having some complexity, this strongly suggests a chaotic behavior. The hyperbolicity can be measured with Lyapunov exponents. Different definitions of these exponents exist in the literature, but the idea is to measure the evolution of the norm of the vector which is tangent to the trajectory. Is this norm has an exponential growth, then you strongly depend on the initial conditions, i.e. you are hyperbolic, i.e. you are likely chaotic. Some indicators of stability are thus based on the evolution of the tangent vector.
The other way to estimate the stability is to focus on the fundamental frequencies of the trajectory. Each of the two angles which characterize the trajectory of the particle, i.e. its mean longitude λp and the longitude of its pericentre ϖp can be associated with a frequency of the problem. It is actually a little more complicated than just a time derivative of the relevant angle, because in that case you would have a contribution of the dynamics of the satellite. A more proper determination is made with a frequency analysis of the orbital elements, kind of Fourier. You are very stable when these frequencies do not drift with time. Here, the authors used first the relative variations of the orbital frequency as indicator of the stability. The most stable particles are the ones which present the smallest relative variations. In order to speed up the calculations, they also used the variations of the semimajor axis as an indicator, and considered that a particle was stable when the variations were smaller than 1.5 km.


A study of stability necessarily focuses on the core of the rings, because the spiral strands are supposed to be doomed. And the authors get very confined zones of stability. A comparison between these zones of stability shows that several mean-motion resonances with Prometheus, Pandora and Titan are associated with them. This could be seen as consistent with the global aspect of the F Ring, but neither with the measured width of the core ring, nor with its exact location.

This problem emphasizes the difficulty to get accurate results with such a complex system. The study manages, with a simplified system of an oblate Saturn and 3 satellites, to render the qualitative dynamics of the F Ring, but this is not accurate enough to predict the future of the observed structures.

Some links

  • The study, also made freely available by the authors on arXiv. Thanks to them for sharing!
  • The web page of Luis Benet (UNAM, Mexico).
  • The web page of Àngel Jorba (University of Barcelona, Spain).

Thanks for having read all this. I wish you a Merry Christmas, and please feel free to share and comment!

Measuring an asteroid from its shadow

Hi there! Today’s post is on the paper Results from the 2014 November 15th multi-chord stellar occultation by the TNO (229762) 2007 UK126, by Gustavo Benedetti-Rossi and 28 colleagues, which has recently been published in The Astronomical Journal. It explains us what 22 simultaneous observations of the same event, i.e. the occultation of a star by an asteroid, tell us about this asteroid.

The asteroid (229762) 2007 UK126

(229762) 2007 UK126 is a Trans-Neptunian Object, which was discovered in October 2007. Its highly eccentricity orbit (0.49) makes it a probable scattered disc object, i.e. its eccentricity should have been pumped by the planets, in particular Neptune. Its estimated rotation period is 11.05 hours. The Hubble Space Telescope has revealed the presence of an orbital companion.

Even at its perihelion, this object is further than Neptune, which makes it difficult to observe. The stellar occultations permit to bypass this problem.

The strategy of observation

The idea is this: while a pretty dark object passes just between you and a star, you do no see the star anymore. The dark object occults it. This occultation contains information.

This is the reason why some planetary scientists try to predict occultations from simulations of the motion of asteroids in the sky, maintaining lists of such events. These predictions suffer from uncertainties on the orbit of the asteroid, this motivates the need to refine the predictions just before the predicted event. For that, astrometric observations of the object are performed, to better constrain its orbital ephemerides.

Once the occultation is predicted with enough accuracy, the observers are informed of the date and the places from where to observe. Multiple observations of the same event, at different locations, represent a set of data which will then be inverted to get information on the asteroid. For these observations to be conducted, amateur astronomers are solicited. They usually constitute networks, which efficiency is doped by their enthusiasm.

The observation of a stellar occultation consists to measure the light flux received from the star during a time interval which includes the predicted event, and when the occultation happens, then a flux drop should be registered. For an observation to be useful, the observer should take care to have an accurate time reference. Moreover, a clear sky, preferably with no wind, makes the measurements more accurate. Some flux drops could be actually due to clouds passing by!

What can these observations tell us?

The first information we get from these occultations addresses the motion of the asteroid: the date and length of the occultation is an information, because we know where the asteroid was on the celestial sphere when this happened. When no occultation is detected, this is an information as well, even if it is frustrating.
Observing at different places permits to observe the occultation of the star by different parts of the asteroid. This is called a multi-chord occultation. From the duration of the event, we can deduce the size of the object with a much better accuracy than direct observation. Such a technique could also detect companions, as it might have been the case for the Main-Belt asteroid (146)Lucina in 1982.

A compelling information on a planetary body is its mass. The best way to measure its mass is by observing the orbit of a companion, if there is one. If there is none, or if its orbit cannot be observed, then we can combine the different measurements of its radius with the measurement of its rotation period and the assumption that its shape is at an hydrostatic equilibrium, i.e. a balance between its own gravity, its rotation, and possibly the gravitational (tidal) attraction of a planetary companion. In the absence of a companion, the equilibrium figure is an oblate, MacLaurin spheroid, which has a circular equatorial section, and a rotation axis which is smaller than the two other ones. If a companion is involved, then the object could be a Jacobi ellipsoid, i.e. an ellipsoid with 3 different principal axes.

This study

This study gathers the results of 20 observations of the stellar occultation of the star UCAC4 448-006503 by the TNO (229762) 2007 UK126 in November 2014, all over the United States, and 2 negative observations, i.e. no flux drop measured, in Mexico. One of the difficulties is is to be accurate on the exact times of the beginning (ingress) and the end (egress) of the event, i.e. the star disappearance and reappearance. This is the reason why the authors of the study split into two teams, which treated the same data separately, with their own techniques (denoted GBR and MWB, since conducted by Gustavo Benedetti-Rossi and Marc W. Buie, respectively).

And here are some of their results:

Before GBR MWB
Longest radius (km) 339+15-10 340+12-8
Equivalent radius (km) 299.5±38.9 319+14-7 319+12-6
Circular fit radius (km) 324+27-23 328+26-21
Apparent oblateness 0.106+0.050-0.040 0.118+0.055-0.048
Density (kg/m3) <1740 <1620

This table tells us that, before the occultation, only a mean radius was known, and with a much larger uncertainty than now. It also tells us that assuming the asteroid to be circular instead of elliptical gives a larger uncertainty. Wait… why circular and not spherical? Why elliptical and not ellipsoidal? Because the occultation is ruled by the projection of the shape of the asteroid on the celestial sphere, which is a 2D surface. So, we observe a surface, and not a volume, even if our assumptions on the shape (remember, the MacLaurin spheroid) give us a 3D information.
This is why the oblateness is just an apparent oblateness. It is actually biased by the projection on the celestial sphere. This oblateness is defined by the quantity (a-b)/a, where a and b are the two axes of the projected asteroid, with a > b.

To know more…

You can find the study on the web site of The Astronomical Journal. It has also been freely made available by the authors on arXiv. Thanks to the authors for sharing! Here is the webpage of the RECON and IOTA networks, which were of great help for the observations.

The authors

Here are their web pages or research profiles:

I hope you enjoyed this post. As usual, please let me know what you think about it. Happy holidays to everybody, and see you soon!

A quest for sources of meteor showers

Hi there! Today I will present you a study entitled Dynamical modeling validation of parent bodies associated with newly discovered CMN meteor showers, by D. Šegon, J. Vaubaillon, P. Gural, D. Vida, Ž. Andreić, K. Korlević & I. Skokić, which has recently been accepted for publication in Astronomy and Astrophysics. It addresses the following question: when you see meteors, where do they come from?

The meteor showers

Imagine you have a comet, i.e. a small body, which wanders in the Solar System with a large eccentricity. This means that it orbits around the Sun, but with large variations of its distance with the Sun. The consequence is that it experiences large variations of temperature during its journey. In particular, when it reaches the perihelion, i.e. when its distance to the Sun is the smallest, the temperature is so hot that it outgasses. The result is the ejection of a cloud of small particles, which itself wanders in the Solar System, on its own orbit.

When the Earth meets it, then these particles are burnt in our atmosphere. This results in meteor showers. Such showers can be sporadic, or happen every year if the cloud is pretty static with respect to the orbit of the Earth. The body from which the particles originate is called the parent body. The study I present you today aims at identifying the parent body of some of these meteor showers.

How to observe them

Understanding the meteor showers is an issue for the safety of the Earth environment, particularly our artificial satellites. Some meteors can even impact the surface of the Earth. This is why numerous observation programs exist, and for that amateurs are very helpful!

The first way to observe meteor showers is visually. When you know that meteor showers are likely to happen, you look at the sky and take note of the meteors you see: when you saw it, from where, where it came from, its magnitude (~its brightness), etc. The point from where the meteor seems to come is called the radiant. It is written as a set of two angles α and δ, i.e. right ascension and declination, which localize it on the celestial sphere.

For unpredicted showers, we can use cameras, which continually observe and record the sky. Then, algorithms of image processing can detect the meteor. Meteors can also be detected in the radio wavelengths.

Dynamical modeling

If you want to simulate the orbit of a particle, you have to consider:

  • the location of the parent body when the particle was ejected (initial position),
  • the ejection velocity,
  • the ejection time, likely when the parent body was close to its perihelion. The question how close? cannot be accurately answered,
  • the gravitational action of the Sun and the planets of the Solar System,
  • the non-gravitational forces, which might have a strong effect on such small bodies.

These non-gravitational forces, here the Poynting-Robertson drag, are due to the Solar radiation, which causes a loss of angular momentum of the particle during its orbital journey around the Sun. It is significant for particles smaller than the centimeter, which is often the case for such ejecta.

You cannot simulate the orbit of a specific particle that you would have identified before, just because they are too small to be observed as individuals. However, you can simulate a cloud, composed of a synthetic population of fictitious particles, with various sizes, ejection times, initial velocities… in such a way that your resulting cloud will have global properties which are close to the real cloud of ejecta. Then you can simulate the evolution of the cloud with time, and in particular determine the time, duration, direction, and intensity of a meteor shower.

Simulating such a cloud reveals interesting dynamical features. It presents an initial size, because of the variations in the ejection times of the particles. But it also widens with time, since the particles present different ejection velocities. This usually (but not always!) results in a kind of a tire which enshrouds the whole orbit of the parent body. Unfortunately, it can be observed only when the Earth crosses it. So, simulating the behavior of the cloud will tell you when the Earth crosses it, how long the crossing lasts, and the density of the cloud during the crossing.
It should be kept in mind that a cloud is composed of a hyue number of particles. For this reason, dedicated computation means are required.

This study

This study aims at identifying the parent body of meteor showers, which were detected by the Croatian Meteor Network (CMN in the title). For that, the first step is to make sure that a shower is a shower.
The detected meteors should resemble enough, which can be measured with the D-criteria, that are a measurement of a distance, in a given space, between the orbits of two objects. Once a meteor shower is identified, the same D-criteria can be used to try to identify the parent body, from its orbit. The parent bodies are comets or asteroids, they are usually known enough for candidates to be determined. And once candidates are identified, then their outgassing is simulated, to predict the meteor showers associated. If a calculated meteor shower is close enough to an observed one, then it is considered that the parent body has been successfully identified. This last close enough is related to the time and duration of the showers, and the location of the radiants.

The authors analyzed 13 meteor showers, and successfully identified the parent body for 7 of them. Here is their list, the showers are identified under their IAU denominations:

  • #549FAN – 49 Andromedids comes from the comet 2001 W2 Batters,
  • #533 JXA – July ξ Arietids comes from the comet 1964 N1 Ikeya,
  • #539 ACP – α Cepheids comes from the comet 255P Levy,
  • #541 SSD – 66 Draconids comes from the asteroid 2001 XQ,
  • #751 KCE – κ Cepheids comes from the asteroid 2009 SG18,
  • #753 NED – November Draconids comes from the asteroid 2009 WN25,
  • #754 POD – ψ Draconids comes from the asteroid 2008 GV.

For this last stream, the authors acknowledge that another candidate parent body has not been investigated: the asteroid 2015 FA118.

For the 6 other cases, either the identification of a parent body is speculated but not assessed enough, or just no candidate has been hinted, possibly because it is an asteroid or a comet which has not discovered yet, and / or because data are missing on the meteor shower.

Some links

That’s it for today! As usual, I accept any comment, feel free to post!