Tag Archives: chaos

How the Planet Nine would affect the furthest asteroids

Hi there! You have heard of the hypothetical Planet Nine, which could be the explanation for an observed clustering of the pericentres of the furthest asteroids, known as eTNOS for extreme Trans-Neptunian Objects. I present you today a theoretical study investigating in-depth this mechanism, in being focused on the influence of the inclination of this Planet Nine. I present you Non-resonant secular dynamics of trans-Neptunian objects perturbed by a distant super-Earth by Melaine Saillenfest, Marc Fouchard, Giacomo Tommei and Giovanni B. Valsecchi. This study has recently been accepted for publication in Celestial Mechanics and Dynamical Astronomy.

Is there a Planet Nine?

An still undiscovered Solar System planet has always been dreamed, and sometimes even hinted. We called it Tyche, Thelisto, Planet X (“X” for mystery, unknown, but also for 10, Pluto having been the ninth planet until 2006). Since 2015, this quest has been renewed after the observation of clustering in the pericentres of extreme TNOS. Further investigations concluded that at least 5 observed dynamical features of the Solar System could be explained by an additional planet, now called Planet Nine:

  1. the clustering of the pericentres of the eTNOs,
  2. the significant presence of retrograde orbits among the TNOs,
  3. the 6° obliquity of the Sun,
  4. the presence of highly inclined Centaurs,
  5. the dynamical detachment of the pericentres of TNOs from Neptune.

The combination of all of these elements tends to rule out a random process. It appears that this Planet Nine would be pretty like Neptune, i.e. 10 times heavier than our Earth, that its pericentre would be at 200 AU (while Neptune is at 30 AU only!), and its apocentre between 500 AU and 1200 AU. This would indeed be a very distant object, which would orbit the Sun in several thousands of years!

Astronomers (Konstantin Batygin and Michael Brown) are currently trying to detect this Planet Nine, unsuccessfully up to now. You can follow their blog here, from which I took some inspiration. The study I present today investigates the secular dynamics that this Planet Nine would induce.

The secular dynamics of an asteroid

The secular dynamics is the one involving the pericentre and the ascending node of an object, without involving its longitude. To make things clear, you know that a planetary object orbiting the Sun wanders on an eccentric, inclined orbit, which is an ellipse. When you are interested in the secular dynamics, you care of the orientation of this ellipse, but not of where the object is on this ellipse. The clustering of pericentres of eTNOs is a feature of the secular dynamics.

This is a different aspect from the dynamics due to mean-motion resonances, in which you are interested in objects, which orbital periods around the Sun are commensurate with the one of the Planet Nine. Some studies address this issue, since many small objects are in mean-motion resonance with a planet. Not this study.

The Kozai-Lidov mechanism

A notable secular effect is the Kozai-Lidov resonance. Discovered in 1961 by Michael Lidov in USSR and Yoshihide Kozai in Japan, this mechanism says that there exists a dynamical equilibrium at high inclination (63°) for eccentric orbits, in the presence of a perturber. So, you have the central body (the Sun), a perturber (the planet), and your asteroid, which could have its inclination pushed by this effect. This induces a libration of the orientation of its orbit, i.e. the difference between its pericentre and its ascending node would librate around 90° or 270°.

This process is even more interesting when the perturber has a significant eccentricity, since the so-called eccentric Kozai-Lidov mechanism generates retrograde orbits, i.e. orbits with an inclination larger than 90°. At 117°, you have another equilibrium.

Now, when you observe a small body which dynamics suggests to be affected by Kozai-Lidov, this means you should have a perturber… you see what I mean?

Of course, this perturber can be Neptune, but only sometimes. Other times, the dynamics would rather be explained by an outer perturber… which could be the Planet Nine, or a passing star (who knows?)


Before mentioning the results of this study I must briefly mention the methodology. The authors made what I would call a semi-analytical study, i.e. they manipulated equations, but with the assistance of a computer. They wrote down the Hamiltonian of the restricted 3-body problem, i.e. the expression of the whole energy of the problem with respect to the orbital elements of the perturber and the TNO. This energy should be constant, since no dissipation is involved, and the way this Hamiltonian is written has convenient mathematical properties, which allow to derive the whole dynamics. Then this Hamiltonian is averaged over the mean longitudes, since we are not interested in them, we want only the secular dynamics.

A common way to do this is to expand the Hamiltonian following small parameters, i.e. the eccentricity, the inclination… But not here! You cannot do this since the eccentricity of the Planet Nine (0.6) and its inclination are not supposed to be small. So, the authors average the Hamiltonian numerically. This permits them to keep the whole secular dynamics due to the eccentricity and the inclination.

Once they did this, they looked for equilibriums, which would be preferential dynamical states for the TNOs. They also detected chaotic zones in the phase space, i.e. ranges of orbital elements, for which the trajectory of the TNOs would be difficult to predict, and thus potentially unstable. They detected these zones in plotting so-called Poincaré sections, which give a picture of the trajectories in a two-dimensional plane that reduces the number of degrees-of-freedom.


And the authors find that the two Kozai-Lidov mechanisms, i.e. the one due to Neptune, and the one due to the Planet Nine, conflict for a semimajor axis larger than 150 AU, where orbital flips become possible. The equilibriums due to Neptune would disappear beyond 200 AU, being submerged by chaos. However, other equilibriums appear.

For the future, I see two ways to better constrain the Planet Nine:

  1. observe it,
  2. discover more eTNOs, which would provide more accurate constraints.

Will Gaia be useful for that? Anyway, this is a very exciting quest. My advice: stay tuned!

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.

The fate of the Alkyonides

Hello everybody! Today, I will tell you on the dynamics of the Alkyonides. You know the Alkyonides? No? OK… There are very small satellites of Saturn, i.e. kilometer-sized, which orbit pretty close to the rings, but outside. These very small bodies are known to us thanks to the Cassini spacecraft, and a recent study, which I present you today, has investigated their long-term evolution, in particular their stability. Are they doomed or not? How long can they survive? You will know this and more after reading this presentation of Long-term evolution and stability of Saturnian small satellites: Aegaeon, Methone, Anthe, and Pallene, by Marco Muñoz-Gutiérrez and Silvia Giuliatti Winter. This study has recently been accepted for publication in The Monthly Notices of the Royal Astronomical Society.

The Alkyonides

As usually in planetary sciences, bodies are named after the Greek mythology, which is the case of the four satellites discussed today. But I must admit that I cheat a little: I present them as Alkyonides, while Aegeon is actually a Hecatoncheires. The Alkyonides are the 7 daughters of Alcyoneus, among them are Anthe, Pallene, and Methone.

Here are some of there characteristics:

Methone Pallene Anthe Aegaeon
Semimajor axis 194,402 km 212,282 km 196,888 km 167,425 km
Eccentricity 0 0.004 0.0011 0.0002
Inclination 0.013° 0.001° 0.015° 0.001°
Diameter 2.9 km 4.4 km 2 km 0.66 km
Orbital period 24h14m 27h42m 24h52m 19h24m
Discovery 2004 2004 2007 2009

For comparison, Mimas orbits Saturn at 185,000 km, and the outer edge of the A Ring, i.e. of the main rings of Saturn, is at 137,000 km. So, we are in the close system of Saturn, but exterior to the rings.

Discovery of Anthe, aka S/2007 S4. Copyright: NASA.
Discovery of Anthe, aka S/2007 S4. Copyright: NASA.

These bodies are in mean-motion resonances with main satellites of Saturn, more specifically:

  • Methone orbits near the 15:14 MMR with Mimas,
  • Pallene is close to the 19:16 MMR with Enceladus,
  • Anthe orbits near the 11:10 MMR with Mimas,
  • Aegaeon is in the 7:6 MMR with Mimas.

As we will see, these resonances have a critical influence on the long-term stability.

Rings and arcs

Beside the main and well-known rings of Saturn, rings and arcs of dusty material orbit at other locations, but mostly in the inner system (with the exception of the Phoebe ring). In particular, the G Ring is a 9,000 km wide faint ring, which inner edge is at 166,000 km… Yep, you got it: Aegaeon is inside. Some even consider it is a G Ring object.

Methone and Anthe have dusty arcs associated with them. The difference between an arc and a ring is that an arc is longitudinally bounded, i.e. it is not extended enough to constitute a ring. The Methone arc extends over some 10°, against 20° for the Anthe arc. The material composing them is assumed to be ejecta from Methone and Anthe, respectively.

However, Pallene has a whole ring, constituted from ejecta as well.

Why sometimes a ring, and sometimes an arc? Well, it tell us something on the orbital stability of small particles in these areas. Imagine you are a particle: you are kicked from home, i.e. your satellite, but you remain close to it… for some time. Actually you drift slowly. While you drift, you are somehow shaken by the gravitational action of the other satellites, which disturb your Keplerian orbit around the planet. If you are shaken enough, then you may leave the system of Saturn. If you are not, then you can finally be anywhere on the orbit of your satellite, and since you are not the only one to have been ejected (you feel better, don’t you?), then you and your colleagues will constitute a whole ring. If you are lucky enough, you can end up on the satellite.

The longer the arc (a ring is a 360° arc), the more stable the region.

Frequency diffusion

The authors studied

  1. the stability of the dusty particles over 18 years
  2. the stability of the satellites in the system of Saturn over several hundreds of kilo-years (kyr).

For the stability of the particles, they computed the frequency diffusion index. It consists in:

  1. Simulating the motion of the particles over 18 years,
  2. Determining the main frequency of the dynamics over the first 9 years, and over the last 9 ones,
  3. Comparing these two numbers. The smaller the difference, the more stable you are.

The numerical simulations is something I have addressed in previous posts: you use a numerical integrator to simulate the motion of the particle, in considering an oblate Saturn, the oblateness being mostly due to the rings, and several satellites. Our four guys, and Janus, Epimetheus, Mimas, Enceladus, and Tethys.

How resonances destabilize an orbit

When a planetary body is trapped in a mean-motion resonance, there is an angle, which is an integer combination of angles present in its dynamics and in the dynamics of the other body, which librates. An example is the MMR Aegaeon-Mimas, which causes the angle 7λMimas-6λAegaeonMimas to librate. λ is the mean longitude, and ϖ is the longitude of the pericentre. Such a resonance is supposed to affect the dynamics of the two satellites but, given their huge mass ratio (Mimas is between 300 and 500 millions times heavier than Aegaeon), only Aegaeon is affected. The resonance is at a given location, and Aegaeon stays there.
But a given resonance has some width, and several resonant angles (we say arguments) are associated with a resonance ratio. As a consequence, several resonances may overlap, and in that case … my my my…
The small body is shaken between different locations, its eccentricity and / or inclination can be raised, until being dynamically unstable…
And in this particular region of the system of Saturn, there are many resonances, which means that the stability of the discovered body is not obvious. This is why the authors studied it.


Stability of the dusty particles

The authors find that Pallene cannot clear its ring efficiently, despite its size. Actually, this zone is the most stable, wrt the dynamical environments of Anthe, Methone and Aegaeon. However, 25% of the particles constituting the G Ring should collide with Aegaeon in 18 years. This probably means that there is a mechanism, which refills the G Ring.

Stability of the satellites

From long-term numerical simulations over 400 kyr, i.e. more than one hundred millions of orbits, these 4 satellites are stable. For Pallene, the authors guarantee its stability over 64 Myr. Among the 4, this is the furthest satellite from Saturn, which makes it less affected by the resonances.

A perspective

The authors mention as a possible perspective the action of the non-gravitational forces, such as the solar radiation pressure and the plasma drag, which could affect the dynamics of such small bodies. I would like to add another one: the secular tides with Saturn, and the pull of the rings. They would induce drifts of the satellites, and of the resonances associated. The expected order of magnitude of these drifts would be an expansion of the orbits of a few km / tens of km per Myr. This seems pretty small, but not that small if we keep in mind that two resonances affecting Methone are separated by 4 km only.

This means that further results are to be expected in the upcoming years. The Cassini mission is close to its end, scheduled for 15 Sep 2017, but we are not done with exploiting its results!

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.

On the stability of Chariklo

Hi there! Do you remember Chariklo? You know, this asteroid with rings (see this post on their formation). Today, we will not speak on the formation of the rings, but of the asteroid itself. I present you the paper entitled The dynamical history of Chariklo and its rings, by J. Wood, J. Horner, T. Hinse and S. Marsden, which has recently been published in The Astronomical Journal. It deals with the dynamical stability of the asteroid Chariklo as a Centaur, i.e. when Chariklo became a Centaur, and for how long.


Chariklo is a large asteroid orbiting between the orbits of Saturn and Uranus, i.e. it is a Centaur. It is the largest known of them, with a diameter of ~250 km. It orbits the Sun on an elliptic orbit, with an eccentricity of 0.18, inducing variations of its distance to the Sun between 13.08 (perihelion) and 18.06 au (aphelion), au being the astronomical unit, close to 150 millions km.
But the main reason why people are interested in Chariklo is the confirmed presence of rings around it, while the scientific community expected rings only around large planets. These rings were discovered during a stellar occultation, i.e. Chariklo occulting a distant star. From the multiple observations of this occultation in different locations of the Earth’s surface, 2 rings were detected, and announced in 2014. Since then, rings have been hinted around Chiron, which is the second largest one Centaur, but this detection is still doubtful.
Anyway, Chariklo contributes to the popularity of the Centaurs, and this study is focused on it.

Small bodies populations in the Solar System

The best known location of asteroids in the Solar System is the Main Belt, which is located between the orbits of Mars and Jupiter. Actually, there are small bodies almost everywhere in the Solar System, some of them almost intersecting the orbit of the Earth. Among the other populations are:

  • the Trojan asteroids, which share the orbit of Jupiter,
  • the Centaurs, which orbit between Saturn and Uranus,
  • the Trans-Neptunian Objects (TNOs), which orbit beyond the orbit of Neptune. They can be split into the Kuiper Belt Objects (KBOs), which have pretty regular orbits, some of them being stabilized by a resonant interaction with Neptune, and the Scattered Disc Objects (SDOs), which have larger semimajor axes and high eccentricities
  • the Oort cloud, which was theoretically predicted as a cloud of objects orbiting near the cosmological boundary of our Solar System. It may be a reservoir of comets, these small bodies with an eccentricity close to 1, which can sometimes visit our Earth.

The Centaurs are interesting from a dynamical point of view, since their orbits are not that stable, i.e. it is estimated that they remain in the Centaur zone in about 10 Myr. Since this is very small compared to the age of our Solar System (some 4.5 Gyr), the fact that Centaurs are present mean that the remaining objects are not primordial, and that there is at least one mechanism feeding this Centaur zone. In other words, the Centaurs we observe were somewhere else before, and they will one day leave this zone, but some other guys will replace them.

There are tools, indicators, helpful for studying and quantifying this (in)stability.

Stability, Lyapunov time, and MEGNO

Usually, an orbiting object is considered as “stable” (actually, we should say that its orbit is stable) if it orbits around its parent body for ever. Reasons for instability could be close encounters with other orbiting objects, these close encounters being likely to be favored by a high eccentricity, which could itself result from gravitational interactions with perturbing objects.
To study the stability, it is common to study chaos instead. And to study chaos, it is common to actually study the dependency on initial conditions, i.e. the hyperbolicity. If you hold a broom vertically on your finger, it lies in a hyperbolic equilibrium, i.e. a small deviation will dramatically change the way it will fall… but trust me, it will fall anyway.
And a good indicator of the hyperbolicity is the Lyapunov time, which is a timescale beyond which the trajectory is so much sensitive on the initial conditions that you cannot accurately predict it anymore. It will not necessarily become unstable: in some cases, known as stable chaos, you will have your orbit confined in a given zone, you do not know where it is in this zone. The Centaur zone has some kind of stable chaos (over a given timescale), which partly explains why some bodies are present there anyway.
To estimate the Lyapunov time, you have to integrate the differential equations ruling the motion of the body, and the ones ruling its tangent vector, i.e. tangent to its trajectory, which will give you the sensitivity to the initial conditions. If you are hyperbolic, then the norm of this tangent vector will grow exponentially, and from its growth rate you will have the Lyapunov time. Easy, isn’t it? Not that much. Actually this exponential growth is an asymptotic behavior, i.e. when time goes to infinity… i.e. when it is large enough. And you have to integrate over a verrrrry loooooooong time…
Fortunately, the MEGNO (Mean Exponential Growth of Nearby Orbits) indicator was invented, which converges much faster, and from which you can determine the Lyapunov time. If you are hyperbolic, the Lyapunov time is contained in the growth rate of the MEGNO, and if not, the MEGNO tends to 2, except for pretty simple systems (like the rotation of synchronous bodies), where it tends to zero.

We have now indicators, which permit to quantify the instability of the orbits. As I said, these instabilities are usually physically due to close encounters with large bodies, especially Uranus for Centaurs. This requires to define the Hill and the Roche limits.

Hill and Roche limits

First the Roche limit: where an extended body orbits too close to a massive object, the difference of attraction it feels between its different parts is stronger than its cohesion forces, and it explodes. As a consequence, satellites of giant planets survive only as rings below the Roche limit. And the outer boundary of Saturn’s rings is inner and very close to the Roche limit.

Now the Hill limit: it is the limit beyond which you feel more the attraction of the body you meet than the parent star you both orbit. This may result in being trapped around the large object (a giant planet), or more probably a strong deviation of your orbit. You could then become hyperbolic, and be ejected from the Solar System.

This paper

This study consists in backward numerical integrations of clones of Chariklo, i.e. you start with many fictitious particles (the authors had 35,937 of them) which do not interact with each others, but interact with the giant planets, and which are currently very close to the real Chariklo. Numerical integration over such a long timespan requires accurate numerical integrators, the authors used a symplectic one, i.e. which presents mathematical properties limiting the risk of divergence over long times. Why 1 Gyr? The mean timescale of survival (called here half-life, i.e. during which you lose half of your population) is estimated to be 10 Myr, so 1 Gyr is 100 half-lives. They simulated the orbits and also drew MEGNO maps, i.e. estimated the Lyapunov time with respect to the initial orbital elements of the particle. Not surprisingly, the lower the eccentricity, the more stable the orbit.

And the result is: Chariklo is in a zone of pretty stable chaos. Moreover, it is probably a Centaur since less than 20 Myr, and was a Trans-Neptunian Object before. This means that it was exterior to Neptune, while it is now interior. In a few simulations, Chariklo finds its origin in the inner Solar System, i.e. the Main Belt, which could have favored a cometary activity (when you are closer to the Sun, you are warmer, and your ice may sublimate), which could explain the origin of the rings. But the authors do not seem to privilege this scenario, as it supported by only few simulations.

What about the rings?

The authors wondered if the rings would have survived a planetary encounter, which could be a way to date them in case of no. But actually it is a yes: they found that the distance of close encounter was large enough with respect to the Hill and Roche limits to not affect the rings. So, this does not preclude an ancient origin for the rings… But a specific study of the dynamics of the rings would be required to address this issue, i.e. how stable are they around Chariklo?

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: 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

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.

Identifying an asteroid family

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.


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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