Tag Archives: Dawn

Rotational stability of Ceres and Vesta

Hi there! The question we address today is: how stable are the rotations of Ceres and Vesta? Do you remember these two guys? These are the largest two asteroids in the Main Belt, and the spacecraft Dawn visited them recently. It gave us invaluable information, like the maps of these bodies, their shapes, their gravity fields, their rotational states…
The study I present you today, Long-term orbital and rotational motions of Ceres and Vesta, by T. Vaillant, J. Laskar, N. Rambaux, and M. Gastineau, wonders how permanent the observed rotational state is. This French study has recently been accepted for publication by Astronomy & Astrophysics.

Ceres and Vesta

I already told you about these two bodies. (1)Ceres (“(1)” because it was the first asteroid to be discovered) is known since January 1801. It has been discovered by the Italian astronomer Giuseppe Piazzi at Palermo Astronomical Observatory. The spacecraft Dawn orbits it since April 2015, but is now inoperative since November 1st, 2018. We see Ceres as a body with a rocky core and an icy mantle, possibly with an internal ocean.

Before visiting Ceres, Dawn orbited Vesta, between July 2011 and September 2012. (4)Vesta has been discovered 6 years after Ceres, in 1807, by the German astronomer Heinrich Olbers. This is a differentiated body, probably made of a metallic core, a rocky mantle, and a crust. It has been heavily bombarded, showing in particular two large craters, Rheasilvia and Veneneia. Vesta is the source of the HED (Howardite Eucrite Diogenite) meteorites, which study is an invaluable source of information on Vesta (see here).

The surface of Vesta (detail). © NASA/JPL-Caltech/UCLA/MPS/DLR/IDA
The surface of Vesta (detail). © NASA/JPL-Caltech/UCLA/MPS/DLR/IDA

You can find below some numbers regarding Ceres and Vesta.

(1) Ceres (4) Vesta
Discovery 1801 1807
Semimajor axis 2.77 AU 2.36 AU
Eccentricity 0.116 0.099
Inclination 9.65° 6.39°
Orbital period 4.604 yr 3.629 yr
Spin period 9.07 h 5.34 h
Obliquity 4.00° 27.47°
Shape (965.2 × 961.2 × 891.2) km (572.6 × 557.2 × 446.4) km
Density 2.08 g/cm3 3.47 g/cm3

As you can see, Vesta is the closest one. It is also the most elongated of these bodies, i.e. you definitely cannot consider it as spherical. Both have significant orbital eccentricities, which means significant variations of the distance to the Sun (this will be important, wait a little). You can also see that these are fast rotators, i.e. they spin in a few hours, while their revolution periods around the Sun are of the order of 4 years. By the way, Vesta rotates twice faster than Ceres. Such numbers are pretty classical for asteroids.
You can also notice that Vesta is denser than Ceres, which is consistent with a metallic core.
Finally, the obliquities. The obliquity is the angle between the angular momentum (somehow the rotation axis… this is not exactly the same, but not too far) and the normal to the Sun. In other words, a null obliquity means that the body rotates along its orbit. An obliquity of 90° means that the body rolls on its orbit. An obliquity of 180° means that the body rotates along its orbit… but its rotation is retrograde (while it is prograde with a null obliquity).
Here, you can see that the obliquity of Ceres is close to 0, while the one of Vesta is 27°, which is significant. It is actually close to the obliquity of the Earth, this induces yearly variations of the insolation, and the seasons. On bodies like Ceres and Vesta, the obliquity would affect the survival of ice in deep craters, i.e. if the obliquity and the size of the crater prevents the Sun to illuminate it, then it would survive as ice.
From these data, the authors simulated the rotational motion of Ceres and Vesta.

Simulating their rotation

Simulating the rotation consists in predicting the time variations of the angles, which represent the rotational state of the bodies. For that, you must start from the initial conditions (what is the current rotational state?), and the physical equations, which rule the rotational motion.
For rigid bodies, rotation is essentially ruled by gravity. The gravitational perturbation of the Sun (mostly) and the planets affects the rotation. You quantify this perturbation with the masses of the perturbers, and the distances between your bodies (Ceres and Vesta), and these perturbers. To make things simple, just take Ceres and the Sun. You know the Solar perturbation on Ceres from the mass of the Sun, and the orbit of Ceres around it. This is where the eccentricity intervenes. Once you have the perturbation, you also need to determine the response of Ceres, and you have it from its shape. Since Vesta is more triaxial than Ceres, then its sensitivity to a gravitational should be stronger. It mostly is, but you may have some resonances (see later), which would enhance the rotational response.

The rotational stability

The question of the rotational stability is: you know, the numbers I gave you on the rotation… how much would they vary over the ages? This is an interesting question, if you want to know the variations of temperature on the surface. Would the ice survive? Would the surface melt? Would that create an atmosphere? For how long? Etc.
For instance, the same team showed that the obliquity of the Earth is very stable, and we owe it to our Moon, which stabilized the rotation axis of the Earth. This is probably a condition for the habitability of a telluric planet.

Let us go back to Ceres and Vesta. The authors focused on the obliquity, not on the spin period. In fact, they considered that the body rotated so fast, that the spin period would not have any significant effect. This permitted them to average the equations over the spin period, and resulted in a rotational dynamics, which moves much slower. And this allows to simulate it over a much longer time span.

A symplectic integrator for a long-term study

A numerical integration of the equations of the rotational motion, even averaged over the fast angle (I mean, the rotation period), may suffer from numerical problems over time. If you propagate the dynamics over millions of years, then the resulting dynamics may diverge significantly from the real one, because of an accumulation of numerical errors all along the process of propagation.

For that, use symplectic integrators. These are numerical schemes, which preserve the global energy of the dynamics, if you have no dissipation of course. But there are many problems of planetary dynamics, which permit you to neglect the dissipation.

When you can neglect the dissipation, your system is conservative. In that case, you can use the mathematical properties of the Hamiltonian systems, which preserve the total energy. That way, your solution does not diverge.

But how to determine whether your dynamics is stable or not? There are many tools for that (Lyapunov exponents, alignment indexes…) Here, the authors determined the diffusion of the fundamental frequencies of the system.

Diffusion of the fundamental frequencies

Imagine you orbit around the Sun, at a given period… actually the period depends on your semimajor axis, so, if it remains constant, then the orbital period remains constant. If your orbit is also disturbed by another perturber, you will see periodic variations in your orbital elements, which correspond to the period of the perturber. Very well. So, analyzing the frequencies which are present in your motion should give you constant numbers…

But what happens if your bodies drift? Then your frequencies will drift as well. In detecting these variations, which result from the so-called diffusion of the fundamental frequencies of the system, you detect some chaos in the system. I took the example of the orbital dynamics, but the same works for the rotation. For instance, the orbital frequencies appear in the time evolution of the rotational variables, since the orbit affects the rotation. But you also have proper frequencies of the rotational motion, for instance the period at which the angular momentum precesses around the normale to the orbit, and this period may drift as well…

The diffusion of the fundamental frequencies is one indicator of the stability. The authors also checked the variations of the obliquity of Ceres and Vesta, along their trajectories. They simulated the motion over 40 Myr (million years), in considering different possible numbers for the interior, and different initial obliquities.

Let us see now the results.

Obliquity variations up to 20 degrees

If you consider different possibilities, i.e. we do not know how these bodies were 40 Myr ago, then we see that it is theoretically possible for them to have been highly influenced by a resonance. This means that one fundamental frequency of the rotation would have been commensurable with periodic contributions of the orbital motion, and this would have resulted in a high response of the obliquity. For the present trajectories, the author estimate that the obliquity of Ceres could have varied between 2 and 20° these last 20 Myr, and the obliquity of Vesta between 21 and 45°.

To be honest, this is only a part of a huge study, which also investigates the stability of the orbital motions of Ceres and Vesta. Actually, these bodies are on chaotic orbits. This does not mean that they will be ejected one day, but that their orbits becomes uncertain, or inaccurate, after some tens of Myr.

The study and its authors

  • You can find the study here. The authors made it also freely available on arXiv, many thanks to them for sharing! And now the authors
  • Unfortunately I did not find any webpage for the first author Timothée Vaillant. You can find here the one of Jacques Laskar, second author of the study,
  • and the IAU page of Mickaël Gastineau.

And that’s it for today! Please do not forget to comment. You can also subscribe to the RSS feed, and follow me on Twitter, Facebook, Instagram, and Pinterest.

Big impact on Ceres

Hi there! Today: Ceres. (1) Ceres is the largest object of the main asteroid belt, so large that the International Astronomical Union (IAU) classified it as a dwarf planet in 2006. As many planetary bodies, it is craterized, the largest crater being named Kerwan. This crater has a diameter of 280 km. But this is not the most remarkable one. The crater Occator, which diameter almost reaches 100 km, is particularly interesting since it exhibits bright spots, which are probably the signature of past hydrothermal activity. This raises the interest of the scientific community, since it could reveal a geophysical activity and water below the surface.
The study I present, The various ages of Occator crater, Ceres: Results of a comprehensive synthesis approach, by A. Neesemann et al., tries to be as accurate as possible on the age of Occator, in summarizing the previous studies and in using as many data as possible. These are actually data provided by the spacecraft Dawn. This paper will be published in Icarus soon.

The dwarf planet (1)Ceres


The quest for an object between the orbits of Mars and Jupiter was initially motivated by the Titius-Bode law. This empirical law, which is now proven to be absolutely wrong, noticed a arithmetic progression between the orbital radii of the known planets, and was confirmed by the discovery of Uranus in 1781 (however, it is inconsistent with the presence of Neptune). Anyway, this convinced former astronomers that something was there, and it revealed to be true. A group led by Franz Xaver von Zach looked for an object with a semimajor axis close to 2.8 AU (astronomical units, remember that 1 AU is 150 million kilometers, which is the orbital radius of our Earth). But that group did not discover Ceres.

Ceres has been serendipitously discovered in 1801 by the Italian astronomer Giuseppe Piazzi in Palermo, Sicilia. He wanted to observe the star HR 1110, but saw a slowly moving object instead. He noticed that it looked somehow like a comet, but that it was probably better than that. Ceres was found!

Giuseppe Piazzi (1746-1826) pointing at Ceres. © Palermo Observatory
Giuseppe Piazzi (1746-1826) pointing at Ceres. © Palermo Observatory

Later, the group led by von Zach discovered many asteroids. One of them, Heinrich Olbers, is credited for the discoveries of Pallas, Vesta, and the periodic comet 13P/Olbers. He also gave his name to the Olbers paradox, which wonders why the night is so dark while we are surrounded by so many stars.


You can find below some of the orbital and physical properties of Ceres.

Semimajor axis 2.77 AU
Eccentricity 0.075
Inclination 10.6°
Revolution 4.60 yr
Rotation 9 h 4 min
Diameters (965.2 × 961.2 × 891.2) km
Density 2.161 g/cm3

These orbital elements and its size make it the largest object of the main asteroid belt. You can see a small eccentricity, and a pretty fast rotation period with respect to its orbital one (i.e., the revolution). Moreover, its equatorial section is pretty circular, i.e. if you look at its 3 diameters, the two largest ones of them are very close, and in fact the uncertainties on the measurements are even consistent with a strict equality. However, the polar diameter is much smaller. This is a consequence of its rotation, which flattens the body.

You can also notice a density, which is between the one of the water (1) and the one of silicates (3.3). This means that its composition should be a mixture of both, i.e. silicates and water ice.

Ceres seen by Dawn © NASA/JPL-Caltech/UCLA/MPS/DLR/IDA
Ceres seen by Dawn © NASA/JPL-Caltech/UCLA/MPS/DLR/IDA

The physical properties and the image above are due to the spacecraft Dawn. This mission is close to its end.

Dawn at Ceres

The spacecraft Dawn has been launched from Cape Canaveral, Florida, in September 2007, and reached the asteroid Vesta in 2011. After a little more than one year in orbit around Vesta, it left it and has been trapped by the gravity field of Ceres in March 2015. This mission will be completed soon.

Dawn consists of three instruments:

  • the Gamma Ray and Neutron Detector (GRaND) Instrument,
  • the Visible and Infrared Spectrometer (VIR) Instrument,
  • and the Framing Camera (FC).

Dawn is essentially an American mission, even if Germany provided the Framing Camera. The German study we discuss today uses FC data.

The orbital journey of Dawn around Ceres consists of several phases, which are different orbits. This results in variable resolutions of the images. The prime mission considered two mapping orbits, the HAMO (High Altitude Mapping Orbit) and the LAMO (for Low Altitude), at distances of 1,470 and 375 km of the surface, respectively. Since then, the mission has been extended, and the spacecraft is now at only 50 kilometers of the surface. High resolution expected.

This mapping orbits permitted to map comprehensively the surface of Ceres. Unsurprisingly, that survey revealed many craters.
We are today interested in Occator, which is not the largest one, but contains bright spots, possibly signatures of a recent hydrothermal activity.

Occator crater

Occator crater is located in the northern hemisphere of Ceres. Its diameter is some 90 km, which does not make it the largest one, but it is particularly interesting for the bright spots it shows. To be honest, there are bright spots at other locations of Ceres, but anyway Occator is remarkable for that. The spot in the center is a dome called Cerealia Facula, while the small spots are called the Vinalia Faculae. You can see them below, on these high-resolution images due to the extended mission.

Occator Crater on Ceres, with its central bright area called Cerealia Facula. © NASA/JPL-Caltech/UCLA/MPS/DLR/IDA/PSI
Occator Crater on Ceres, with its central bright area called Cerealia Facula. © NASA/JPL-Caltech/UCLA/MPS/DLR/IDA/PSI
Vinalia Faculae © NASA/JPL-Caltech/UCLA/MPS/DLR/IDA/PSI
Vinalia Faculae © NASA/JPL-Caltech/UCLA/MPS/DLR/IDA/PSI

Yes, there are domes, due to hydrothermal activity! You can find interesting analogies with Earth features here. But basically, the two possible explanations are for now:

  1. either the heat from the impact that formed the crater caused briny liquid or mushy ice to push up on the surface,
  2. or the heat from the impact could have enhanced activity related to pre-existing liquid reservoirs just below the surface.

Anyway, this reveals water! And this makes Ceres and its crater Occator fascinating.

How a crater evolves

This study wants to estimate the age of Occator, or in other words, date the impact that created it. For that, you examine its current state, and guess how long it took from the impact to the observed state.

Because of the elasticity of the surface, after some time (hundreds of millions years, maybe a little more…) the surface relaxes. The consequence is that the crater gets less deep, and its slopes get gentler. A 3-D terrain model will give you the numbers. But the dynamics of the relaxation process is barely constrained.

Another evolution is that the crater is covered by something else. This something could be other, more recent craters. If the new crater is larger than the older one, then the oldest disappears. However, if the new crater is small with respect to the old one, then you see both, and by counting the small craters, you can say “it took this time to get so many craters, so the age is at least…”. OK

But how to constrain this? You calibrate your models with better known bodies, like the Moon, and / or dynamical models of the bombardments. Previous studies have used Lunar Derived Models and Asteroid-flux Models… of course with different outcomes.

In the specific case of Occator, the hydrothermal activity revealed by the bright spots has generated ejecta blankets, as lobate deposits.

Previous estimations of Occator’s age

The quest for the age of Occator crater began with the first data on Ceres, i.e. in 2015. Here are the already published numbers

  • Nathues et al. 2015: 78 ± 5 Ma (million years). This measurement is based on crater counting, and only HAMO data. In particular, the more accurate low-altitude data were missing at that time,
  • Nathues et al. 2016: 6.9 Ma, based on the interior lobate deposits,
  • Jaumann et al. 2016: between 100 and 200 Ma, depending on how you calibrate the dating from craters,
  • Nathues et al. 2017: 34 ± 2 Ma, from the creation of the central dome, i.e. Cerealia Facula,
  • Nathues et al. 2018, stated that the dispersed bright deposits Vinalia Faculae were younger than 2 Ma, in using low-altitude high-resolution images.

The study we now discuss uses almost all of the data, and so should be more accurate.

A young crater anyway

It is interesting that a study points out all of the possible numbers, given the models, the data, and the physical process considered (crater counting, age of ejecta,…). In particular, if the hydrothermal activity has been triggered by the impact which created Occator, then dating the ejecta should tell us something accurate.

The authors find an age of 21.9 ± 0.7 Ma for the crater in using the Lunar Derived Model, and between 1 and 64 Ma in using the Asteroid-flux Derived Model. You see, lots of uncertainties… as they say, the model ages are a matter of perspective. But anyway, this is a very young and interesting crater!

The study and its authors

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