Category Archives: Asteroids: Main Belt

Breaking an asteroid

Hi there! Asteroids, these small bodies in the Solar System, are fascinating by the diversity of their shapes. This is a consequence of their small sizes. Another consequence is their weakness, which itself helps to split some of them into different parts, sometimes creating binary objects, asteroids families… The study I present you today, Internal gravity, self-energy, and disruption of comets and asteroids, by Anthony R. Dobrovolskis and Donald G. Korycansky, proposes an accurate computation of the required energy to provoke this break-up, at any place of the asteroid, i.e. you are more efficient when you hit at a given location. This study has recently been accepted for publication in Icarus.

Shapes of asteroids

Please allow me, in this context, to call asteroid a comet, a comet being a small body, i.e. like an asteroid, but with a cometary activity. The important thing is that the involved bodies are small enough.

Beyond a given size, i.e. a diameter of ~400 km, a planetary body is roughly spheroidal, i.e. it is an ellipsoid with it two equatorial axes almost equal and the polar one smaller, because of its rotation. For a tidally despun body, like the Moon, or a satellite of a giant planet, the shape is more triaxial, since the tidal (gravitational) action of the parent planet tends to elongate the equatorial plane. The same phenomenon affects Mercury.

However, for smaller bodies, the self-gravitation is not strong enough to make the body look more or less like a sphere. As a consequence, you can have almost any shape, some bodies are bilobate, some are contact binaries, i.e. two bodies which permanently touch together, some others are rubble piles, i.e. are weak aggregates of rocks, with many voids.

These configurations make these bodies likely to undergo or have undergone break-up. This can be quantified by the required energy to extract some material from the asteroid.

The energies involved

For that, an energy budget must be performed. The relevant energies to consider are:

  • The impact disruption energy: the minimum kinetic energy of an impactor, to shatter the asteroid and remove at least half of its mass,
  • The shattering energy: the minimum energy needed to shatter the asteroid into many small pieces. It is part of the impact disruption energy. This energy is roughly proportional to the mass of the asteroid. It represents the cohesion between the adjacent pieces.
  • The binding energy: this energy binds the pieces constituting the asteroid. In other words, once you have broken an asteroid (don’t try this at home!), you have to make sure the pieces will not re-aggregate… because of the binding energy. For that, you have to bring enough energy to disperse the fragments.
  • The self-gravitational energy: due to the mutual gravitational interaction between the blocks constituting the asteroids. Bodies smaller than 1 km are strength-dominated, i.e. they exist thanks to the cohesion between the blocks, which is the shatter energy. However, larger bodies are gravity-dominated.
  • The kinetic energy of rotation: the spin of these bodies tends to enlarge the equatorial section. In that sense, it assists the break-up process.

This study addresses bodies, which are far enough from the Sun. This is the reason why I do not mention its influences, i.e. the tides and the thermic effects, which could be relevant for Near-Earth Objects. In particular, the YORP effect is responsible for the fission of some of them. I do not mention the orbital kinetic energy of the asteroid either. Actually the orbital motion is part of the input energy brought by an impact, since the relative velocity of the impactor with respect to the target is relevant in this calculation.

I now focus on the two cases studied by the authors to illustrate their theory: the asteroid Kleopatra and the comet 67P/Churyumov-Gerasimenko.

2 peculiar cases: Kleopatra and Churyumov-Gerasimenko

216 Kleopatra is a Main-Belt asteroid. Adaptive optics observations have shown that is is constituted of two masses bound by material, giving a ham-bone shaped. As such, it can be considered as a contact binary. It is probably a rubble pile. Interestingly, observations have also shown that Kleopatra has 2 small satellites, Alexhelios and Cleoselene, which were discovered in 2008.

Reconstruction of the shape of Kleopatra. © NASA
Reconstruction of the shape of Kleopatra. © NASA

However, 67P Churyumov-Gerasimenko is a Jupiter-family comet, i.e. its aphelion is close to the orbit of Jupiter, while its perihelion is close to the one of the Earth. It has an orbital period of 6.45 years, and was the target of the Rosetta mission, which consisted of an orbiter and a lander, Philae. Rosetta orbited Churyumov-Gerasimenko between 2014 and 2016. The shape of this comet is sometimes described as rubber ducky, with two dominant masses, a torso and a head, bound together by some material, i.e. a neck.

Churyumov-Gerasimenko seen by Rosetta. © ESA
Churyumov-Gerasimenko seen by Rosetta. © ESA
216 Kleopatra 67P/Churyumov-Gerasimenko
Semimajor axis 2.794 AU 3.465 AU
Eccentricity 0.251 0.641
Inclination 13.11° 7.04°
Spin period 5.385 h 12.761 h
Mean radius 62 km 2.2 km
Magnitude 7.30 11.30
Discovery 1880 1969

The irregular shapes of these two bodies make them interesting targets for a study addressing the gravitation of any object. Let us see now how the authors addressed the problem.

Numerical modeling

Several models exist in the literature to address the gravity field of planetary bodies. The first approximation is to consider them as spheres, then you can refine in seeing them as triaxial ellipsoids. For highly irregular bodies you can try to model them as cuboids, and then as polyhedrons. Another way is to see them as duplexes, this allows to consider the inhomogeneities dues to the two masses constituting bilobate objects. The existence of previous studies allow a validation of the model proposed by the authors.

And their model is a finite-element numerical modeling. The idea is to split the surface of the asteroid into small triangular planar facets, which should be very close to the actual surface. The model is all the more accurate with many small facets, but this has the drawback of a longer computation time. The facets delimit the volume over which the equations are integrated, these equations giving the local self-gravitational and the impact disruption energies. The authors also introduce the energy rebate, which is a residual energy, due to the fact that you can remove material without removing half of it. This means that the impact disruption energy, as it is defined in the literature, is probably a too strong condition to have extrusion of material.
The useful physical quantities, which are the gravitational potential, the attraction, and the surface slope, are propagated all along the body thanks to a numerical scheme, which accuracy is characterized by an order. This order quantifies the numerical approximation which is made at each integration step. A higher order is more accurate, but is computationally more expensive.

Once the code has been run on test cases, the authors applied it on Kleopatra and Churyumov-Gerasimenko, for which the shape is pretty well known. They used meshes of 4,094 and 5,786 faces, respectively.


The validation phase is successful. The authors show that with a 3rd order numerical scheme, they recover the results present in the literature for the test cases with an accuracy of ~0.1%, which is much better than the accuracy of the shape models for the real asteroids. Regarding Kleopatra and Churyumov-Gerasimenko, they get the gravity field at any location, showing in particular excesses of gravity at the two lobes.

Such a study is particularly interesting for further missions, which would determine the gravity field of asteroids, which would then be compared with the theoretical determination by this code. Other applications are envisaged, the authors mentioning asteroid mining.

The study and its authors

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

And Merry Christmas!

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 rotation of Fagus

Hi there! Today I will tell you on the rotation of the asteroid (9021) Fagus. The first determination of its spin period is given in Rotation period determination for asteroid 9021 Fagus, by G. Apostolovska, A. Kostov, Z. Donchev, and E. Vchkova Bebekovska. This study has recently been published in the Bulgarian Astronomical Journal.

(9021) Fagus’s facts

(9021) Fagus is a small, Main Belt asteroid. You can find below some of its characteristics:

Semimajor axis 2.58 AU, i.e. 386 millions km
Eccentricity 0.173
Inclination 13.3°
Orbital period 4.14 y
Diameter 13.1 km
Absolute magnitude 12.4
Discovery February 14, 1988

Its small magnitude explains that its discovery was acknowledged only in 1988. Once identified, it was found on older photographic plates, providing observations from 1973 (yes, you can observe an object before it was discovered… you just do not know that you observed it). This body is so small, that the authors of this study observed it by accident: in 2013, they observed in fact (901) Brunsia during two nights, which is brighter (absolute magnitude: 11.35), but Fagus was in the field. The collected photometric data were supplemented in March 2017 by two other nights of observations, which permitted the authors to determine the spin (rotation) period with enough confidence.

Measuring the rotation

I address the measurement of the rotation of an asteroid here. Such a small body may have an irregular shape, and tumble. But since it is very difficult to get accurate data for such a small body, it is commonly assumed that the body rotates around one principal axis, this hypothesis being confronted with the observations. In other words, if you can explain the observations with a rotation around one axis, then you have won.

The irregularity of the shape makes that the light flux you record presents temporal variations, i.e. the surface elements you face is changing, so the reflection of the incident Solar light is changing, which means that these variations are correlated with the rotational dynamics. If these variations are dominated by a constant period of oscillation, then you have the rotation period of the asteroid. Typically, the rotation period of the Main-Belt asteroids are a few hours. These numbers are strongly affected by the original dynamics of the planetary nebula, the despinning of the asteroids being very slow. This is a major difference with the planetary satellites, which rotates in a few days since they are locked by the tides raised by their parent planet. For comparison, the spin period of the Moon is 28 days.

Photometric observations

Detecting the photometric variations of the incident light of such a small body requires to be very accurate. The overall signal is very faint, its variations are even fainter. To avoid errors, the observer should consider:

  • The weather. A bright sky is always better, preferably with no wind, which induces some seeing, i.e. apparent scintillation of the observed object.
  • The anthropogenic light pollution.
  • The variations of the thickness of the atmosphere during the observation. If your object is at the zenith, then it is pretty good. If it is low in the sky, then its course during the night will involve variations of the thickness of the atmosphere during the observations.
  • Instrumental problems. Usually you use a chip of CCD sensors, these sensors do not have exactly the same response. A way to compensate this is to measure a flat, i.e. the response of the chip to a homogeneous incident light flux.

The observation conditions can be optimized, for instance in observing from a mountain area. The observer should also be disciplined, for instance many professional observatories forbid to smoke under the domes. In the past, this caused wrong detections. A good way to secure the photometric results is to have several objects in the fields, and to detect the correlations between their variations of flux. Intrinsic properties of an object would emerge from light variations, which would be detected for this object only.

The observation facilities

The observations were made at Rozhen Observatory, also known as Bulgarian National Astronomical Observatory. It is located close to Chepelare, Bulgaria, at an altitude of 1,759 m. It consists of 4 telescopes.

The 2013 observations were made with a 50/70 cm Schmidt telescope, and the 2017 ones with a 2m-Ritchey-Chrétien-Coude telescope. In both cases, the observations were made through a red filter. The faintness of the asteroid required exposure times between 5 and 6 minutes.

The Schmidt telescope used for the 2013 observations. Copyright: P. Markishky
The Schmidt telescope used for the 2013 observations. Copyright: P. Markishky
The 2m telescope, used for the 2017 observations. Copyright: P. Markishky
The 2m telescope, used for the 2017 observations. Copyright: P. Markishky

The softwares

The authors used two softwares in their study: CCDPHOT, and MPO Canopus. CCDPHOT is a software running under IDL, which is another software, commonly used to treat astrophysical data, and not only. With CCDPHOT, the authors get the photometric measurements. MPO Canopus could give these measurements as well, but the authors used it for another functionality: it fits a period to the lightcurve, in proving an uncertainty. This is based on a Fourier transform, i.e. a spectral decomposition of the signal. In other words, the lightcurves, with are recorded as a set of pairs (time, lightflux), are transformed into a triplet of (amplitude, frequency, phase), i.e. it is written as a sum of sinusoidal oscillations. If one of these oscillations clearly dominates the signal, then its period is the rotation period of the asteroid.


And the result is this: the rotation period of (9021)Fagus is 5.065±0.002 hours. In practice, being accurate on such a number requires to collect data over several times this interval. An ideal night of observation would permit to measure during about 2 periods. Here, data have been collected over 4 nights.
Up to now, we had no measurement of the spin period of Fagus, which makes this result original. It not only helps to understand the specific Fagus, but it is also a new data in the catalog of the rotational periods of Main-Belt asteroids.

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.

An asteroid pair

Hi there! Today I present you the study of an asteroid pair. Not a binary, a pair. A binary asteroid is a couple of asteroids which are gravitationally bound, while in pair, the asteroids are just neighbors, they do not live together… but have. The study is entitled Detailed analysis of the asteroid pair (6070) Rheinland and (54287) 2001 NQ8, by Vokrouhlický et al., and it has recently been published in The Astronomical Journal.

Asteroid pairs

I have presented asteroid families in a previous post. These are groups of asteroids which present common dynamical and physical properties. They can be in particular identified from the clustering of their proper elements, i.e. you express their orbital elements (semimajor axis, eccentricity, inclination, pericentre, …), you treat them properly so as to get rid off the gravitational disturbance of the planets, and you see that some of these bodies tend to group. This suggests that they constitute a collisional family, i.e. they were a unique body in the past, which has been destroyed by collisions.
An asteroid pair is something slightly different, since these are two bodies which present dynamical similarities in their osculating elements, i.e. before denoising them from the gravitational attraction of the planets. Of course, they would present similarities in their proper elements as well, but the fact that similarities can be detected in the osculating elements means that they are even closer than a family, i.e. the separation occurred later. Families younger than 1 Myr (1 million of years) are considered to be very young; the pair I present you today is much younger than that. How much? You have to read me before.
A pair suggests that only two bodies are involved. This suggests a non-collisional origin, more particularly an asteroid fission.

Asteroid fission

Imagine an asteroid with a very fast rotation. A rotation so fast that it would split the asteroid. We would then have two components, which would be gravitationally bound, and evolving… Depending on the energy involved, it could remain a stable binary asteroid, a secondary fission might occur, the two or three components may migrate away from each other… and in that case we would pair asteroid with very close elements of their heliocentric orbits.
It is thought that the YORP (Yarkovsky – O’Keefe – Radzievskii – Paddack) could trigger this rotational fission. This is a thermic effect which alter the rotation, and in some cases, in particular when the satellite has an irregular shape, it could accelerate it. Until fission.
Thermic effects are particularly efficient when the Sun is close, which means that NEA (Near Earth Asteroids) are more likely to be destroyed by this process than Main Belt asteroids. Here, we deal with Main Belt asteroids.

The pair 6070-54827 (Rheinland – 2001 NQ8)

The following table present properties of Rheinland and 2001 NQ8. The orbital elements are at Epoch 2458000.5, i.e. September 4th 2017. They come from the JPL Small-Body Database Browser.

(6070) Rheinland (54827) 2001 NQ8
Semimajor axis (AU) 2.3874015732216 2.387149297807496
Eccentricity 0.2114524962733347 0.211262507795103
Inclination 3.129675305535938° 3.128927421642917°
Node 83.94746016534368° 83.97704257098502°
Pericentre 292.7043398319871° 292.4915004062336°
Orbital period 1347.369277588708 d (3.69 y) 1347.155719572348 d (3.69 y)
Magnitude 13.8 15.5
Discovery 1991 2001

Beside their magnitudes, i.e. Rheinland is much brighter than 2001 NQ8, this is why it was discovered 10 years earlier, we can see that all the slow orbital elements (i.e. all of them, except the longitude) are very close, which strongly suggests they shared the same orbit. Not only their orbits have the same shape, but they also have the same orientation.

Shapes and rotations from lightcurves

A useful tool for determining the rotation and shape of an asteroid is the lightcurve. The object reflects the incident Solar light, and the way it reflects it will tell us something on its location, its shape, and its orientation. You can imagine that the surfaces of these bodies are not exclusively composed of smooth terrain, and irregularities (impact basins, mountains,…) will result in a different Solar flux, which also depends on the phase, i.e. the angle between the normale of the surface and the asteroid – Sun direction… i.e. depends whether you see the Sun at the zenith or close to the horizon. This is why recording the light from the asteroid at different dates tell us something. You can see below an example of lightcurve for 2001 NQ8.

Example of lightcurve for 2001 NQ8, observed by Vokrouhlický et al.

Recording such a lightcurve is not an easy task, since the photometric measurements should be denoised, otherwise you cannot compare them and interpret the lightcurve. You have to compensate for the variations of the luminosity of the sky during the observation (how far is the Moon?), of the thickness of the atmosphere (are we close to the horizon?), of the heterogeneity of the CCD sensors (you can compensate that in measuring the response of a uniform surface). And the weather should be good enough.

Once you have done that, you get a lightcurve alike the one above. We can see 3 maxima and 2 minima. Then the whole set of lightcurves is put into a computational machinery which will give you the parameters that best match the observations, i.e. periods of rotation, orientation of the spin pole at a given date, and shape… or at least a diameter. In this study, the authors already had the informations for Rheinland but confirmed them with new observations, and produced the diameter and rotation parameters for 2001 NQ8. And here are the results:

Spin pole(124°,-87°)(72°,-49°) or (242°,-46°)

(6070) Rheinland (54827) 2001 NQ8
Diameter (km) 4.4 ± 0.6 2.2 ± 0.3
Spin period (h) 4.2737137 ± 0.0000005 5.877186 ± 0.000002

We can see rapid rotation periods, as it is often the case for asteroids. The locations of the poles mean that their rotations
are retrograde, with respect to their orbital motions. Moreover, two solutions best match the pole of 2001 NQ8.

Dating the fission

The other aspect of this study is a numerical simulation of the orbital motion of these two objects, backward in time, to date their separation. Actually, the authors considered 5,000 clones of each of the two objects, to make their results statistically relevant.
They not only considered the gravitational interactions with other objects of the Solar System, but also the Yarkovsky effect, i.e. a thermal pull due to the Sun, which depends on the reflectivity of the asteroids, and favors their separation. For that, they propose new equations implementing this effect. They also simulated the variations of the spin pole orientation, since it affects the thermal acceleration.

And here is the result: the fission probably occurred 16,340 ± 40 years ago.


Why doing that? Because what we see is the outcome of an asteroid fission, which occurred recently. The authors honestly admit that this result could be refined in the future, depending on

  • Possible future measurements of the Yarkovsky acceleration of one or two of these bodies,
  • The consideration of the mutual interactions between Rheinland and 2001 NQ8,
  • Refinements of the presented measurements,
  • Discovery of a third member?

To date the fission, they dated a close approach between these two bodies. They also investigated the possibility that that
close approach, some 16,000 years from now, could have not been the right one, and that the fission could have been much older. For that, they ran long-term simulations, which suggest that older close approaches should have been less close: if the pair were older, Yarkovsky would have separated it more.

To know more

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

Chaotic dynamics of asteroids

Hi there! Today’s post deals with the fate of an asteroid family. You remember Datura? Now you have Hungaria! Datura is a very young family (< 500 kyr), now you have a very old one, i.e. probably more than 1 Gyr, and you will see that such a long time leaves room for many uncertainties… The paper I present is entitled Planetary chaos and the (In)stability of Hungaria asteroids, by Matija Ćuk and David Nesvorný, it has recently been accepted for publication in Icarus.

The Hungaria asteroids

Usually an asteroid family is a cluster of asteroids in the space of the orbital elements (semimajor axis, eccentricity, inclination), which share, or a supposed to share, a common origin. This suggests that they would originate from the same large body, which would have been destroyed by a collision, its fragments then constituting an asteroid family. Identifying an asteroid family is not an easy task, because once you have identified a cluster, then you must make sure that the asteroids share common physical properties, i.e. composition. You can get this information from spectroscopy, i.e. in comparing their magnitudes in different wavelengths.

The following plot gives the semimajor axis / eccentricity repartition of the asteroids in the inner Solar System, with a magnitude smaller than 15.5. We can clearly see gaps and clusters. Remember that the Earth is at 1 UA, Mars at 1.5 UA, and Jupiter at 5.2. The group of asteroids sharing the orbit of Jupiter constitute the Trojan population. Hungaria is the one on the left, between 1.8 and 2 AU, named after the asteroid 434 Hungaria. The gap at its right corresponds to the 4:1 mean-motion resonance with Jupiter.

Distribution of the asteroids in the inner Solar System, with absolute magnitude < 15.5. Reproduced from the data of The Asteroidal Elements Database. Copyright:

If we look closer at the orbital elements of this Hungaria population, we also see a clustering on the eccentricity / inclination plot (just below).

Eccentricity / Inclination of the asteroids present in the Hungaria zone. Copyright:

This prompted Anne Lemaître (University of Namur, Belgium) to suggest in 1994 that Hungaria constituted an asteroid family. At that time, only 26 of these bodies were identified. We now know more than 4,000 of them.

The origin of this family can be questioned. The point is that these asteroids have different compositions, which would mean that they do not all come from the same body. In other words, only some of them constitute a family. Several dynamics studies, including the one I present today, have been conducted, which suggest that these bodies are very old (> 1 Gyr), and that their orbits might be pretty unstable over Gyrs… which suggests that it is currently emptying.

This raises two questions:

  1. What is the origin of the original Hungaria population?
  2. What is the fate of these bodies?

Beside the possible collisional origin, which is not satisfying for all of these bodies since they do not share the same composition, it has been proposed that they are the remnants of the E-Belt, which in some models of formation of the Solar System was a large population of asteroid, which have essentially been destabilized. Another possibility could be that asteroids might pass by and eventually be trapped in this zone, feeding the population.

Regarding the fate, the leaving asteroids could hit other bodies, or become Trojan of Jupiter, or… who knows? Many options seem possible.

The difficulty of giving a simple answer to these questions comes partly from the fact that these bodies have a chaotic dynamics… but what does that mean?

Chaos, predictability, hyperbolicity, frequency diffusion, stability,… in celestial dynamics

Chaos is a pretty complicated mathematical and physical notion, which has several definitions. A popular one is made by the American mathematician Robert L. Devaney, who said that a system is chaotic if it has sensitive dependence on initial conditions, it is topologically transitive (for any two open sets, some points from one set will eventually hit the other set), and its periodic orbits form a dense set.

Let us make things a little simpler: in celestial mechanics, you assume to have chaos when you are sensitive to the initial conditions, i.e. if you try to simulate the motion of an object with a given uncertainty on its initial conditions, the uncertainties on its future will grow exponentially, making predictions impossible beyond a certain time, which is related to the Lyapunov time. But to be rigorous, this is the definition of hyperbolicity, not of chaos… but never mind.

A chaotic orbit is often thought to be unstable. This is sometimes true, especially if the eccentricity of your object becomes large… but this is not always the same. Contrarily, you can have stable chaos, in which you know that your object is not lost, it is in a given bounded zone… but you cannot be more accurate than that.

Chaos can also be related to the KAM theory (for Kolmogorov-Arnold-Moser), which says that when you are chaotic, you have no tores in the dynamics, i.e. periodic orbits. When your orbit is periodic, its orbital frequency is constant. If this frequency varies, then you can suspect chaos… but this is actually frequency diffusion.

And now, since I have confused you enough with the theory, comes another question: what is responsible for chaos? The gravitational action of the other bodies, of course! But this is not a satisfying answer, since a gravitational system is not always chaotic. There are actually many configurations in which a gravitational system could be chaotic. An obvious one is when you have a close encounter with a massive object. An other one is when your object is under the influence of several overlapping mean-motion resonances (Chirikov criterion).

This study is related to the chaos induced by the gravitational action of Mars.

The orbit of Mars

Mars orbits the Sun in 687 days (1.88 year), with an inclination of 1.85° with respect to the ecliptic (the orbit of the Earth), and an eccentricity of 0.0934. This is a pretty large number, which means that the distance Mars – Sun experiences some high amplitude variations. All this is valid for now.

But since the Hungaria asteroids are thought to be present for more than 1 Gyr, a study of their dynamics should consider the variations of the orbit of Mars over such a very long time-span. And this is actually a problem, since the chaos in the inner Solar System prevents you from being accurate enough over such a duration. Recent backward numerical simulations of the orbits of the planets of the Solar System by J. Laskar (Paris Observatory), in which many close initial conditions were considered, led to a statistical description of the past eccentricity of Mars. Some 500 Myr ago, the eccentricity of Mars was most probably close to the current one, but it could also have been close to 0, or close to 0.15… actually it could have taken any number between 0 and 0.15.

The uncertainty on the past eccentricity of Mars leads uncertainty on the past orbital behavior of Solar System objects, including the stability of asteroids. At least two destabilizing processes should be considered: possible close encounters with Mars, and resonances.

Among the resonances likely to destabilize the asteroids over the long term are the gi (i between 1 and 10) and the fj modes. These are secular resonances, i.e. involving the pericentres (g-modes) and the nodes (f-modes) of the planets, the g-modes being doped by the eccentricities, and the f-modes by the inclinations. These modes were originally derived by Brouwer and van Woerkom in 1950, from a secular theory of the eight planets of the Solar System, Pluto having been neglected at that time.

The eccentricity of Mars particularly affects the g4 mode.

This paper

This paper consists of numerical integrations of clones of known asteroids in the Hungaria region. By clones I mean that the motion of each asteroid is simulated several times (21 in this study), with slightly different initial conditions, over 1 Gyr. The authors wanted in particular to test the effect of the uncertainty on the past eccentricity of Mars. For that, they considered two cases: HIGH and LOW.

And the conclusion is this: in the HIGH case, i.e. past high eccentricity of Mars (up to 0.142), less asteroids survive, but only if they experienced close encounters with Mars. In other words, no effect of the secular resonance was detected. This somehow contradicts previous studies, which concluded that the Hungaria population is currently decaying. An explanation for that is that in such phenomena, you often have a remaining tail of stable objects. And it seems make sense to suppose that the currently present objects are this tail, so they are the most stable objects of the original population.

Anyway, this study adds conclusions to previous ones, without unveiling the origin of the Hungaria population. It is pretty frustrating to have no definitive conclusion, but we must keep in mind that we cannot be accurate over 1 Gyr, and that there are several competing models of the evolution of the primordial Solar System, which do not affect the asteroid population in the same way. So, we must admit that we will not know everything.

To know more

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