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.

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

The contraction of Mercury

Hi there! Today’s post deals with the early evolution of Mercury, in particular its cooling. At the beginning of its life, a planet experiences variations of temperature, and then cooling, and while cooling, it contracts. The surface may present some signature of this contraction, and this is the object of the paper I present you today. It is entitled Timing and rate of global contraction of Mercury, by Kelsey T. Crane and Christian Klimczak, from the University of Georgia, and it has been recently accepted for publication in Geophysical Research Letters. The idea is to infer the history of the contraction from the observation of the craters and the faults.

Mercury’s facts

Mercury is the innermost planet of the Solar System, with a mean distance to the Sun which is about one third of the Sun-Earth distance. It has an eccentric orbit, with an eccentricity of 0.206, and orbits the Sun in 88 days while the planet rotates around itself in 58 days. This is very long when compared to the terrestrial day, but it also means that there is a ratio 1.5 between the spin and the orbital frequencies. This is called a 3:2 spin-orbit resonance, which is a dynamical equilibrium favored by the proximity of the Sun and the orbital eccentricity.

Mercury seen by MESSENGER (Credit: NASA)

An interesting fact is the high density of Mercury, i.e. Mercury is too dense for a terrestrial planet. Usually, a large enough body is expected to have a stratified structure, in which the heaviest elements are concentrated in the core. Mercury is so dense than it is thought to be the core of a former and larger proto-Mercury.

Mercury’s early life

There is no agreement on the way Mercury lost its mantle of lighter elements. You can find the following scenarios in the literature:

  1. Slow volatilization of the mantle by the solar wind,
  2. Very large impact,
  3. Loss of the light elements by photophoresis,
  4. Magnetic erosion.

The scenario of the large impact was very popular until the arrival of MESSENGER, in particular because the models of formation of the Solar System and the observation of the surface of Mercury suggest that Mercury has been heavily impacted in its early life. However, the detection of volatiles elements, in particular potassium, on the surface of Mercury, is interpreted by some planetary scientists as inconsistent with the large impact scenario. The large impact would have induced extreme heating of the planet, and for some scientists the potassium would not have survived this episode. The other scenarios involve much slower processes, and less heating.

This raises the question: how hot was the early Mercury? We still do not know, but this is related to the study I present here.

The exploration of Mercury

The proximity of Mercury to the Sun makes it difficult to explore, because of the large gravitational action of the Sun which significantly perturbs the orbit of a spacecraft, and more importantly because of the large temperatures in this area of the Solar System.

Contrarily to Venus and Mars, which regularly host space programs, Mercury has been and will be the target of only 3 space missions so far:

  1. Mariner 10 (NASA): It has been launched in November 1973 to make flybys of Venus and Mercury. Three flybys of Mercury have been realized between March 1974 and March 1975. This mission gave us the first images of the surface of the planet, covering some 45% of it. It also discovered the magnetic field of Mercury.
  2. MESSENGER (Mercury Surface, Space Environment, Geochemistry, and Ranging) (NASA): This was the first human-made object to orbit Mercury. It was launched in August 2004 from Cape Canaveral and has been inserted around Mercury in March 2011, after one flyby of the Earth, two flybys of Venus, and three flybys of Mercury. These flybys permitted to use the gravity of the planets to reduce the velocity of the spacecraft, which was necessary for the orbital insertion. MESSENGER gave us invaluable data, like the gravity field of Mercury, a complete cartography with topographical features (craters, plains, faults,…), new information on the gravity field, it supplemented Earth-based radar measurements of the rotation, it revealed the chemical composition of the surface… The mission stopped in April 2015.
  3. Bepi-Colombo (ESA / JAXA): This is a joint mission of the European and Japanese space agencies, which is composed of two elements: the Mercury Magnetospheric Orbiter (MMO, JAXA), and the Mercury Planetary Orbiter (MPO, ESA). It should be launched in October 2018 and inserted into orbit in December 2025, after one flyby of the Earth, two flybys of Venus, and 6 flybys of Mercury. Beside the acquisition of new data on the planet with a better accuracy than MESSENGER, it will also perform a test of the theory of the general relativity, in giving new measurements of the post-newtonian parameters β and γ. β is associated with the non-linearities of the gravity field, while γ is related with the curvature of the spacetime. In the theory of the general relativity, these two parameters should be strictly equal to 1.

This paper

The idea of the paper is based on the competition between two processes for altering the surface of Mercury:

  1. Impacts, which are violent, rapid phenomena, creating craters,
  2. Tides, which is a much slower process that creates faults, appearing while the planet is contracting. The local stress tensor can be inferred from the direction of the faults.

Dating a crater is possible, from its preservation. And when a crater and a fault are located at the same place, there are two possibilities:

  1. either the fault cuts the crater (see Enheduanna, just below), or
  2. the crater interrupts the fault.

In the first case, the fault appeared after the impact, while in the second case, the fault was already present before Mercury was impacted. So, if you can constrain the age of the crater, you can constrain the apparition of the fault, and the contraction of the planet. From a global analysis of the age of the faults, the authors deduced the variation of the contraction rate over the ages.

A close up of Enheduanna Crater. Credit: IAU

The authors used a database of 3,112 craters ranging from 20 to 2,000 km, which were classified into 5 classes, depending on their degree of preservation. And the result are given below.

Class Name Age Craters Cut Superpose
1+2 Pre-Tolstojan + Tolstojan >3.9 Gy 2,310 1,192 4
3 Calorian 3.9 – 3.5 Gy 536 266 104
4 Mansurian 3.5 – 1 Gy 244 49 55
5 Kuiperian < 1 Gy 22 0 3

We can see that the eldest craters are very unlikely to superpose a fault, while the bombardment was very intense at that time. However, the authors have detected more superposition after. They deduced the following contraction rates:

Time Contraction (radius)
Pre-Tolstojan + Tolstojan 4.0 ± 1.6 km
Calorian 0.90 ± 0.35 km
Mansurian 0.17 ± 0.07 km
Kuiperian 0

This means that the contraction rate has decreased over the ages, which is not surprising, since the temperature of Mercury has slowly reached an equilibrium.

A perspective : constraining the early days of Mercury

In my opinion, such a study could permit to constrain the evolution of the temperature of Mercury over the ages, and thus date its stratification. Maybe this would also give new clues on the way Mercury lost its light elements (impact or not?).

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