Tag Archives: chaos

Asteroids: Main Belt

Chaotic dynamics of asteroids

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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: planetary-mechanics.com

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: planetary-mechanics.com

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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Asteroids: Main Belt

Identifying an asteroid family

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Hi there! Today’s post deals with an asteroid family, more precisely the Datura family. The related study is New members of Datura family, by A. Rosaev and E. Plávalová, it has recently been accepted for publication in Planetary and Space Science. The Datura family is a pretty recent one, with only 7 known members when that study started. The authors suggest that 3 other bodies are also members of this family.

Some elements of the dynamics of asteroids

Detailing the dynamics of asteroids would require more than a classical post, here I just aim at giving a few hints.
Asteroids can be found at almost any location in the Solar System, but the combination of the gravitational effects of the planets, of thermal effects, and of the formation of the Solar System, result in preferred locations. Most of the asteroids are in the Main Belt, which lies between the orbits of Mars and Jupiter. And most of these bodies have semimajor axes between 2.1 and 3.2 astronomical units (AU), i.e. between 315 and 480 millions of km. Among these bodies can be found interesting dynamical phenomena, such as:

  • Mean motion resonances (MMR) with planets, especially Jupiter. These resonances can excite the eccentricities of the asteroids until ejecting them, creating gaps known as Kirkwood gaps. At these locations, there are much less asteroids than nearby.
  • Stable chaos. Basically, a chaotic dynamics means that you cannot predict the orbit at a given accuracy over more than a given timespan, because the orbit is too sensitive to uncertainties on its initial conditions, i.e. initial location and velocity of the asteroid. Sometimes chaos is associated with instability, and the asteroid is ejected. But not always. Stable chaos means that the asteroid is confined in a given zone. You cannot know accurately where the asteroid will be at a given time, but you know that it will be in this zone. Such a phenomenon can be due to the overlap of two mean-motion resonances (Chirikov’s criterion).

Anyway, when an asteroid will or will not be under the influence of such an effect, it will strongly be under the influence of the planets, especially the largest ones. This is why it is more significant to describe their dynamics with proper elements.

Proper elements

Usually, an elliptical orbit is described with orbital elements, which are the semimajor axis a, the eccentricity e, the ascending node Ω, the pericentre ω, the inclination I, and the mean longitude λ. Other quantities can be used, like the mean motion n, which is the orbital frequency.

Because of the large influence of the major planets, these elements present quasiperiodic variations, i.e. sums of periodic (sinusoidal) oscillations. Since it is more significant to give one number, the oscillations which are due to the gravitational perturbers are removed, yielding mean elements, called proper elements. These proper elements are convenient to characterize the dynamics of asteroids.

Asteroid families

Most of the asteroids are thought to result from the disruption (for instance because of a collision) of a pretty large body. The ejecta resulting from this disruption form a family, they share common properties, regarding their orbital dynamics and their composition. A way to guess the membership of an asteroid to a family is to compare its proper elements with others’. This guess can then be enforced by numerical simulations of the orbital motion of these bodies over the ages.

Usually a family is named from its largest member. In 2015, 122 confirmed families and 19 candidates were identified (source: Nesvorný et al. in Asteroids IV, The University of Arizona Press, 2015). Many of these families are very old, i.e. more than 1 Gyr, which complicates their identification in the sense that their orbital elements are more likely to have scattered.
The Datura family is thought to be very young, i.e. some 500 kyr old.

A funny memory: in 2005 David Nesvorný received the Urey Prize of the Division of Planetary Sciences of the American Astronomical Society. This prize was given to him at the annual meeting of the Division, that year in Cambridge, UK. He then gave a lecture on the asteroid families, and presented the “Nesvorný family”, i.e. his father, his wife, and so on.

Datura’s facts

The asteroid (1270) Datura has been discovered in 1930. It orbits the Sun in 3.34 years, and has a semimajor axis of 2.23 AU. As such, it is a member of the inner Main Belt. Its orbit is highly elongated, between 1.77 and 2.70 AU, with an orbital eccentricity of 0.209. It rotates very fast, i.e. in 3.4 hours. Its diameter is about 8.2 km.

It is an S-type asteroid, i.e. it is mainly composed of iron- and magnesium-silicates.

This study

After having identified 10 potential family members from their proper elements, the authors ran backward numerical simulations of them, cloning each asteroid 10 times to account for the uncertainties on their locations. The simulations were ran over 800 kyr, the family being supposed to be younger than that. The simulations first included the 8 planets of the Solar System, and Pluto. The numerical tool is a famous code, Mercury, by John Chambers.

The 10 asteroids identified by the authors include the 7 already known ones, and 3 new ones: (338309) 2002 VR17, 2002 RH291, and 2014 OE206. These are all sub-kilometric bodies. The authors point out that these bodies share a linear correlation between their node and their pericentre.

This study also shows that 2014 OE206 has a chaotic resonant orbit, because of the proximity of the 9:16 MMR with Mars. This resonance also affects 2001 VN36, but this was known before (Nesvorný et al., 2006). The authors also find that this chaotic dynamics can be significantly enhanced by the gravitational perturbations of Ceres and Vesta. Finally, they say that close encounters might happen between (1270) Datura and two of its members: 2003 SQ168 and 2001 VN36.

Another study

Now, to be honest, I must mention another study, The young Datura asteroid family: Spins, shapes, and population estimate, by David Vokrouhlický et al., which was published in Astronomy and Astrophysics in February 2017. That study goes further, in considering the 3 new family members found by Rosaev and Plávalová, and in including other ones, updating the Datura family to 17 members.

This seems to be a kind of anachronism: how could a study be followed by another one, which is published before? In fact, Rosaev and Plávalová announced their results during a conference in 2015, this is why they could be cited by Vokrouhlický et al. Of course, their study should have been published earlier. Those things happen. I do not know the specific case of this study, but sometimes this can be due to a delayed reviewing process, another possibility could be that the authors did not manage to finish the paper earlier… Something that can be noticed is that the study by Vokrouhlický is signed by a team of 13 authors, which is expected to be more efficient than a team of two. But the very truth is that I do not know why they published before. This is anyway awkward.

A perspective

I notice something which could reveal a rich dynamics: the authors show (their Figure 7) a periodic variation of the distance between (1270) Datura and 2003 SQ168, from almost zero to about twice the semimajor axis… This suggests me a horseshoe orbit, i.e. a 1:1 mean-motion resonance, the two bodies sharing the same orbit, but with large variations of their distance. If you look at the orbit of the smallest of these two bodies (here 2003 SQ168) in a reference frame which moves with (1270) Datura, you would see a horseshoe-shaped trajectory. To the best of my knowledge, such a configuration has been detected in the satellites of Saturn between Janus and Epimetheus, suggested for exoplanetary systems, maybe detected between a planet and an asteroid, but never between two asteroids…

By the way, 2003 SQ168 is the asteroid, which has the closest semimajor axis to the one of (1270) Datura, in Rosaev and Plávalová’s paper. Now, when I look at Vokrouhlický et al.’s paper, I see that 2013 ST71 has an even closer semimajor axis. I am then tempted to speculate that these two very small bodies are coorbital to (1270) Datura. Maybe a young family favors such a configuration, which would become unstable over millions of years… Speculation, not fact.

Update

This is actually not an horsehoe orbit. The large variation of the distance is due to the fact that 2003 SQ168 is on a orbit, which is close to the one of (1270) Datura, with a slightly different orbital frequency. Regarding 2013 ST71, a numerical simulation by myself suggests the possibility of a temporary (i.e. unstable) capture in a 1:1 MMR.

To know more…

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Rings of Saturn

The dynamics of Saturn’s F Ring

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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,

and

  • 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)λs+ϖp is the argument of a Lindblad resonance, which pumps the eccentricity of the particle, while pλp-(p+1)λs+ϖs 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)λp+ϖp, and the one of the corotation resonance is pλs-(p+1)λp+ϖs.

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.

Results

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!