Category Archives: Asteroids: Centaurs

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.

(10199)Chariklo

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.

Forming the rings of Chariklo

Hi there! Today’s article is on the rings of the small planet Chariklo. Their origin is being discussed in Assessment of different formation scenarios for the ring system of (10199) Chariklo, by Mario Melita, René Duffard, Jose-Luis Ortiz and Adriano Campo Bagatin, which has recently been accepted for publication in Astronomy and Astrophysics.

The Centaur (10199)Chariklo

As a Centaur, (10199)Chariklo orbits around the Sun, between the orbits of Saturn and Uranus. It has been discovered in February 1997 thanks to the Spacewatch program, which was a systematic survey conducted at Kitt Peak National Observatory in
Arizona, USA. It orbits about the Sun in 63 years, has an orbital inclination of 23°, and an eccentricity of 0.17, which results in significant variations of its distance to the Sun. Moreover, it orbits close to the 4:3 mean-motion resonance with Uranus, which means that it performs 4 revolutions around the Sun while Uranus performs almost 3.

(10199)Chariklo is considered to be possibly a dwarf planet. A dwarf planet is not a planet, since the International Astronomical Union reserved this appellation for only 8 objects, but looks like one. As such, it is large enough to have a pretty spherical shape, with a mean radius of 151 km. It has a pretty fast rotation, with a period of 7 hours. Something unusual to notice: its equatorial section is almost circular (no problem), but its polar axis is the longest one, while it should be the shortest if Chariklo had been shaped by its rotational deformation.

The planetary rings

Everybody knows the massive rings of Saturn, which can be seen from the Earth with any telescope. These rings are composed of particles, which typical radius ranges from the centimeter to some meters. These particles are mostly water ice, with few contamination by silicates.
The spacecrafts Voyager have revealed us the presence of a tiny ring around Jupiter, mainly composed of dust. Moreover, Earth-based observations of Uranus and Neptune revealed rings in 1978 and 1984, respectively. We now know 13 rings for Uranus, which should be composed of submillimetric particles, and 5 rings for Neptune. Interestingly, one of the rings of Neptune, Adams, is composed of 5 arcs, i.e. 5 zones of surdensity, which seem to be pretty stable.

It is usually assumed that rings around a planet originate from the disruption of a small body, possibly an impactor. A question is : why do these rings not reaccrete into a new planetary body, which could eventually become a satellite of a planet? Because its orbit is above the Roche limit.

The Roche limit

The Roche limit is named after the French astronomer and mathematician Édouard Albert Roche who discovered that when a body was too close from a massive object, it could just not survive. This allowed him to say that the distance Mars-Phobos which was originally announced when Phobos was discovered was wrong, and he was right.

Imagine a pretty small object orbiting around a massive planet. Since the object has a finite dimension, the gravitational force exerted by the planet has some variation over the volume of the object. More precisely, it decreases with the square of the distance to the planet. If the internal cohesion in the body is smaller than the variations of the gravitational attraction which affect the body, then it just cannot survive, and is tidally disrupted.

It was long thought than you need a very massive central object to get rings around. This is why the announcement of the discovery of rings around Chariklo, in 2014, was a shock.

The rings of Chariklo

The discovery of these rings has been announced in March 2014, and was the consequence of the observation of the occultation by Chariklo of the star UCAC4 248-108672 in June 2013 by 13 instruments, in South America. This was a multichord observation mostly aiming at characterizing a stellar occultation observed from different sites, to infer clues on the shape of the occulting body, and possibly discover a satellite (see this related post). In the case of Chariklo, short occultations before AND after the main one have been measured, which meant a ring system around Chariklo. The following video, made by the European South Observatory, illustrates the light flux drops due to the rings and to Chariklo itself.

Actually two rings were discovered, which are now named Oiapoque and Chuí. They have both a radius close to 400 km, Oiapoque being the inner one. These two rings are separated by a gap of about 9 km. Photometric measurements suggest there are essentially composed of water ice.

This study

This study investigates and discusses different possible causes for the formation of the rings of Chariklo.

Tidal disruption of a small body: REJECTED

It can be shown that, for a satellite which orbits beyond the Roche limit, i.e. which should not be tidally disrupted, the tides induce a secular migration of its orbit: if the satellite orbits faster than the central body (here, Chariklo) rotates around its polar axis, then the satellites migrates inward, i.e. gets closer to the satellite. In that case, it would eventually reach the Roche limit and be disrupted; this is the expected fate of the satellite of Mars Phobos. However, if it orbits above the synchronous orbit, which means that its orbital angular velocity is smaller than the rotation of Chariklo, then it would migrate outward.
In the case of Chariklo, the synchronous orbit is closer than the Roche limit. The rotation period of Chariklo is 7 hours, while the rings’ one is 20 hours. As a consequence, tidal inward migration until disruption is impossible. It would have needed Chariklo to have spun much slower in the past, while a faster rotation is to be expected because of the loss of rotational energy over the ages.

Collision between a former satellite of Chariklo and another body: VERY UNLIKELY

If the rings are the remnants of a former satellite of Chariklo, then models of formation suggest that this satellite should have had a radius of about 3 km. The total mass of the rings is estimated to be the one of a satellite of 1 km, but only part of the material would have stayed in orbit around Chariklo.
The occurence of such an impact is almost precluded by the statistics.

Collision between Chariklo and another body: UNLIKELY

We could imagine that the rings are ejectas of an impact on Chariklo. The authors estimate that this impact would have left a crater with a diameter between 20 and 50 km. Once more, the statistics almost preclude it.

Three-body encounter: POSSIBLE

Imagine an encounter between an unringed Chariklo and another small planet, which itself has a satellite. In that case, favorable conditions could result in the trapping of the satellite in the gravitational field of Chariklo, and its eventual disruption if it is below the Roche limit. The author estimate that it would require the largest body to have a radius of about 6.5 km, and its (former) satellite a radius of 330 meters.

The authors favor this scenario, but I do not see how a satellite of a radius of 330 m could generate a ring, which material should correspond to a 1 km-radius body.

Beyond Chariklo

The quest for rings is not done. Since 2015, another Centaur, (2060)Chiron, is suspected to harbor a system of rings. This could mean that rings are not to be searched around large bodies, as long thought, but in a specific region of the Solar System. Matt Hedman has proposed that the weakness of ice at 70K, which is its temperature in that region of the Solar System, favors the formation and the stability of rings.

To know more

That’s all for today! I hope you liked it. As usual, you are free to comment. You can also subscribe to the RSS feed, and follow me on Twitter.

On the dynamics of small bodies beyond Neptune

Hi there! Today I will present you a study on the possible dynamics of some Trans-Neptunian Objects (TNOs). This study, Study and application of the resonant secular dynamics beyond Neptune by M. Saillenfest, M. Fouchard, G. Tommei and G.B. Valsecchi, has recently been accepted for publication in Celestial Mechanics and Dynamical Astronomy.
This is a theoretical study, which presents some features of the dynamics that could one day be observed. This manuscript follows another one by the same authors, in which a theory of the “resonant secular dynamics” is presented. Here it is applied to small bodies, which are thought to be in mean-motion resonances with Neptune. This study results from a French-Italian collaboration.

The Kozai-Lidov mechanism

The dynamics that is presented here uses the so-called Kozai-Lidov mechanism. This is a mechanism which has been simultaneously and independently discovered in Russia (by Lidov) and in Japan (by Kozai), and which considers the following configuration: a massive central body, another massive one called the perturber, and a test-particle, i.e. a massless body, which orbits the central one. This problem is called the Restricted 3-body problem. Originally, the central body was the Earth, the perturber the Moon, and the test-particle an artificial satellite of the Earth. In such a case, the orbit of the test-particle is an ellipse, which is perturbed by the perturber; this results in variation of the elliptical elements, i.e. eccentricity, inclination… moreover, the orientation of the ellipse is moving…

To describe the problem, I need to introduce the following orbital elements:

  • The semimajor axis a, which is half the long axis of the orbit,
  • the mean anomaly M, which locates the satellite on the ellipse,
  • the eccentricity e, which is positive and smaller than 1. It tells us how eccentric the orbit is (e=0 means that the orbit is circular),
  • the pericentre ω, which is the point of the orbit which is the closest to the central body (undefined if the orbit is circular),
  • the inclination I, which is the angle between the orbital plane and the reference plane,
  • the ascending node Ω, which locates the intersection between the orbital plane and the reference plane.

The Kozai-Lidov mechanism allows a confinement of the pericentre with respect to the ascending node, and it can be shown that it results in a raise of the eccentricity of the inclination. Exploiting such a mechanism gives frozen orbits, i.e. configurations for which the orbit of an artificial orbiter, even inclined and eccentric, will keep the same spatial orientation.

These recent years, this mechanism has been extended for designing space missions around other objects than the Earth, but also to explain the dynamics of some exoplanetary systems, of small distant satellites of the giant planets, and of Trans-Neptunian Objects, as it is the case here. In this last problem, the central body is the Sun, the perturber is a giant planet (more specifically here, it is Neptune), and the test-particle is a TNO, with the hope to explain the inclined and eccentric orbit of some of them. A notable difference with the original Kozai-Lidov problem is that here, the test-particle orbits exterior to the perturber. Another difference is that its dynamics is also resonant.

Resonant and secular dynamics

The authors do not speak of resonant secular dynamics, but of dynamics that is both resonant and secular. The difference is that the involved resonance is not a secular one. Let me explain.

The authors consider that the TNO is in a mean-motion resonance with Neptune. This implies an integer commensurability between its orbital period around the Sun and the one of Neptune, with results in large variations of its semi-major axis. If we look at the orbital elements, this affects the mean anomaly M, while, when a resonance is secular, M is not affected.

So, these objects are in a mean-motion resonance with Neptune. Moreover, they have an interested secular dynamics. By secular, I mean that the mean anomaly is not affected, but something interesting involves the node and/or the pericentre. And this is where comes Kozai-Lidov. The paper studies the objects which are trapped into a mean-motion resonance with Neptune, and which are likely to present a confinement of the pericentre ω, which could explain a significant eccentricity and a high inclination.

For that, they make an analytical study, which theory had been developed in the first paper, and which is applied here.

Why an analytical study?

The modern computing facilities allow to simulate the motion of millions of test-particles over the age of the Solar System, in considering the gravitational interaction of the planets, the galactic tide, a star passing by… and this results in clusters of populations of fictitious TNOs. Very well. But when you do that, you do not know why this particular object behaves like that. However, an analytical study will give you zones of stability for the orbits, which are preferred final states. It will tell you: there will probably be some objects in this state, BECAUSE… and in the case of this study, the because has something to do with the Kozai-Lidov mechanism. Moreover, the because also gives you some confidence in your results, since you have an explanation why you get what you get.

To make things short, a numerical study shows you many things, while an analytical one proves you a few things. A comprehensive study of the problem requires combining the two approaches.

This paper

This paper specifically deals with fictitious objects, which are in mean-motion with Neptune, and are likely to be affected by the Kozai-Lidov mechanism. After many calculations presented in the first paper, the authors show that the problem can be reduced to one degree of freedom, in a Hamiltonian formalism.

The Hamiltonian formalism is a common and widely used way to treat problems of celestial mechanics. It consists in expressing the total energy of the problem, i.e. kinetic + potential energy, and transform it so that trajectories can be described. These trajectories conserve the total energy, which may seem weird for a physical problem. Actually there is some dissipation in the dynamics of TNOs, but so small that it can be neglected in many problems. The most recent numerical studies in this topic consider the migration of the planets, which is not a conservative process. In the paper I present you today, this migration is not considered. This is one of the approximations required by the analytical study.

The remaining degree of freedom is the one relevant to the Kozai-Lidov mechanism. The one associated with the mean-motion resonance is considered to be constant. For that it involves the area enshrouded by the libration of the resonant argument, which is constant (hypothesis of the adiabatic invariant). So, the authors get a one degree-of-freedom Hamiltonian, for which they draw phase spaces, showing the trajectory in the plane q vs. ω, q=a(1-e) being the distance between the Sun and the pericentre of the TNO, i.e. its closest distance to the Sun. These phase portraits depend on other parameters, like the mean-motion resonance with Neptune that is considered, and a parameter η, which combines the inclination and the eccentricity.

The results are a catalog of possible trajectories, some of them presenting a confinement of the pericentre ω. For a large cloud of objects, this would result in an accumulation of pericentres in a constrained zone. The authors try to find confirmation of their results with existing objects, but their limited number and the inaccuracy on their location make this comparison inconclusive. They also point out that the orbits of Sedna and 2012VP113 cannot be explained by this mechanism.

Perspectives

The future observations of TNOs will give us access to more objects and more accurate trajectories, and it is to be hoped that some of them will fit into the trajectories found by the authors. That would be a great success for that, and that would be deserved regarding the effort necessary to achieve such an analytical study.

As I said, such a problem needs analytical and numerical studies, but some of the authors (Marc Fouchard and Giovanni Valsecchi) are also involved in such a numerical exploration, which starts from a fictitious Oort cloud and simulates the excitation of the eccentricity and inclination of some of the objects.

For the two studies to meet, it should also be investigated how the planetary migration, which results from models of formation and evolution of the Solar System, affects the zones of stability due to the Kozai-Lidov mechanism.

Finally, we should not forget the quest for the Planet Nine. As the authors honestly point out, an additional planet could break down some of the conclusions.

To know more

Feel interested? Please leave a comment!

Surviving as a Trojan of Neptune

Hi! Today I will tell you about a study recently accepted for publication in Astronomy & Astrophysics, by Rodney Gomes and David Nesvorný, on the survival of the asteroids which precede and follow Neptune on its orbit.

The coorbital resonance

In the Solar System, the mean motion resonances are ubiquitous. When the orbital frequencies of two bodies are commensurate, interesting phenomena might happen: they could have a more stable orbit, or they could experience a permanent forcing which raise their eccentricity and / or inclination, and in some cases could result in an ejection. A resonance has particularly strong effects on a small body which orbit resonates with the one of a large planet. This is for instance how the giant planets shaped the asteroid belt.

Here, we deal with the coorbital resonance, which is a very specific and interesting case. This happens between two bodies which have on average the same orbital frequencies, and the perturbations associated result in some zones of stability. In particular, there are five equilibrium positions for the coorbital restricted 3-body problem, i.e. if we consider the Sun, a planet, here Neptune, and a small body. These equilibriums are known as Lagrangian points, and the most remarkable of them are denoted L4 and L5. They precede and follow the planet at an angular distance of 60°, and are stable equilibriums. As a consequence, they are likely to accumulate several small bodies, and this is verified by the observations, which have detected asteroids which coorbit with Jupiter, Uranus and Neptune.

At this time, 6,288 of these objects have been detected for Jupiter, 1 for Uranus, and 17 for Neptune.

The planetary migration

Since 2004 and the first version of the Nice model, the giant planets are assumed to have formed closer to the Sun than they are now, and have migrated to their current orbit. The reason for this migration is that they were form in a large proto-planetary disk, full of planetesimals which drove migration. The asteroids are some of these planetesimals. This raises the following question: could the coorbital (or Lagrangian) asteroids survive this migration?

Long-term numerical integrations

Addressing this problem requires long-term and intensive numerical simulations. The issue is this: you need to simulate the evolution of the Solar System over 4.5 Gyr. For that, you write down the gravity equations ruling the motion of the planets and the planetesimals (these are many objects… the authors considered 60,000 of them), and you propagate them numerically.

To propagate them, you start from a given position and velocity of each of your bodies (initial conditions), and the equations give you the time-derivative at this point. You then use it to extrapolate the trajectory in the time, and you reiterate…

Of course, this algorithm does not give exact results. To lower the error, you should take a small time-step, but a too small time-step requires more iterations, and at each iteration you add an error due to the internal accuracy of the computer. To make your life easier, numerical integrators have been developed to improve the accuracy for a given time-step. In this study, the authors use two very well-known tools, SWIFT and MERCURY, dedicated to the integration of the motion of the planets and asteroids.

In this paper

The authors show it is difficult to get Trojans of Neptune that survive over the lifetime of the Solar System. In a first numerical integration, they do get captures, but none of them survive. Then they consider planetesimals which are very close to the observed Trojan, and they get some captures.

Something interesting is that they show that the orbital inclination of these Trojans can be excited during the migration process. For that, the migration should be slow enough, i.e. over 150 Myr, while previous studies, which assumed a migration ten times faster, did not excite the inclinations up to observed values.

Some perspectives

Even if it is now accepted that the planets have migrated, several competing scenarios exist (Nice, Nice 2, Grand Tack,…) and some are probably to come, just because there are many ranges of initial conditions which are possible, many possible assumptions on the initial state of the proto-planetary nebula… and these scenarios should of course impact the capture of Trojans of the giant planets.

Links