Tag Archives: stability

Triton: a cuckoo around Neptune

Hi there! Did you know that Neptune had a prominent satellite, i.e. Triton, on a retrograde orbit? This is so unusual that it is thought to have been trapped, i.e. was originally an asteroid, and has not been formed in the protoneptunian nebula. The study I present you today, Triton’s evolution with a primordial Neptunian satellite system, by Raluca Rufu and Robin M. Canup, explains how Triton ejected the primordial satellites of Neptune. This study has recently been published in The Astronomical Journal.

The satellites of Uranus and Neptune

We are tempted to see the two planets Uranus and Neptune as kinds of twins. They are pretty similar in size, are the two outermost known planets in the Solar System, and are gaseous. A favorable orbital configuration made their visitation possible by the spacecraft Voyager 2 in 1986 and 1989, respectively.

Among their differences are the high obliquity of Uranus, the presence of rings around Uranus while Neptune displays arcs, and different configurations in their system of satellites. See for Uranus:

Semimajor axis Eccentricity Inclination Radius
Miranda 5.12 Ru 0.001 4.338° 235.8 km
Ariel 7.53 Ru 0.001 0.041° 578.9 km
Umbriel 10.49 Ru 0.004 0.128° 584.7 km
Titania 17.20 Ru 0.001 0.079° 788.9 km
Oberon 23.01 Ru 0.001 0.068° 761.4 km
Puck 3.39 Ru 0 0.319° 81 km
Sycorax 480.22 Ru 0.522 159.420° 75 km

I show on this table the main satellites of Uranus, and we can see that the major ones are at a reasonable distance (in Uranian radius Ru) of the planet, and orbit almost in the same plane. The orbit of Miranda is tilted thanks to a past mean-motion resonance with Umbriel, which means that it was originally in the same plane. So, we can infer that these satellites were formed classically, i.e. the same way as the satellites of Jupiter, from a protoplanetary nebula, in which the planet and the satellites accreted. An exception is Sycorax, which is very far, highly inclined, and highly eccentric. As an irregular satellite, it has probably been formed somewhere else, as an asteroid, and been trapped by the gravitational attraction of Uranus.

Now let us have a look at the system of Neptune:

Semimajor axis Eccentricity Inclination Radius
Triton 14.41 Rn 0 156.865° 1353.4 km
Nereid 223.94 Rn 0.751 7.090° 170 km
Proteus 4.78 Rn 0 0.075° 210 km
Larissa 2.99 Rn 0.001 0.205° 97 km

Yes, the main satellite seems to be an irregular one! It does not orbit that far, its orbit is (almost) circular, but its inclination is definitely inconsistent with an in situ formation, i.e. it has been trapped, which has been confirmed by several studies. Nereid is much further, but with a so eccentric orbit that it regularly enters the zone, which is dynamically perturbed by Triton. You can also notice the absence of known satellites between 4.78 and 14.41 Neptunian radii. This suggests that this zone may have been cleared by the gravitational perturbation of a massive body… which is Triton. The study I present you simulates what could have happened.

A focus on Triton

Before that, let us look at Triton. The system of Neptune has been visited by the spacecraft Voyager 2 in August 1989, which mapped 40% of the surface of Triton. Surprisingly, it showed a limited number of impact craters, which means that the surface has been renewed, maybe some 1 hundred of millions of years ago. Renewing the surface requires an activity, probably cryovolcanism as on the satellite of Saturn Enceladus, which should has been sustained by heating. Triton was on the action of the tides raised by Neptune, but probably not only, since tides are not considered as efficient enough to have circularized the orbit. The tides have probably been supplemented by gravitational interactions with the primordial system of Neptune, i.e. satellites and / or disk debris. If there had been collisions, then they would have themselves triggered heating. As a consequence of this heating, we can expect a differentiated structure.

Moreover, Triton orbits around Neptune in 5.877 days, on a retrograde orbit, while the rotation of Neptune is prograde. This configuration, associated with the tidal interaction between Triton and Neptune, makes Triton spiral very slowly inward. In other words, it will one day be so close to Neptune that it will be destroyed, and probably create a ring. But we will not witness it.

A numerical study with SyMBA

This study is essentially numerical. It aimed at modeling the orbital evolution of Triton, in the presence of Nereid and the putative primordial satellites of Neptune. The authors assumed that there were 4 primordial satellites, with different initial conditions, and considered 3 total masses for them: 0.3, 1, and 3 total masses of the satellites of Uranus. For each of these 3 masses, they ran 200 numerical simulations.

The simulations were conducted with the integrator (numerical code) SyMBA, i.e. Symplectic Massive Body Algorithm. The word symplectic refers to a mathematical property of the equations as they are written, which guarantee a robustness of the results over very long timescales, i.e. there may be an error, but which does not diverge. It may be not convenient if you make short-term accurate simulations, for instance if you want to design the trajectory of a spacecraft, but it is the right tool for simulating a system over hundreds of Myrs (millions of years). This code also handles close encounters, but not the consequences of impacts. The authors bypassed this problem in treating the impacts separately, determining if there were disrupting, and in that case estimated the timescales of reaccretion.

Results

The authors found, consistently with previous studies, that the interaction between Triton and the primordial system could explain its presently circular orbit, i.e. it damped the eccentricity more efficiently than the tides. Moreover, the interaction with Triton caused collisions between the primordial moons, but usually without disruption (hit-and-run impacts). In case of disruption, the authors argue that the reaccretion would be fast with respect of the time evolution of the orbit of Triton, which means that we could lay aside the existence of a debris disk.

Moreover, they found that the total mass of the primordial system had a critical role: for the heaviest one, i.e. 3 masses of the Uranian system, Triton had only small chances to survive, while it had reasonable chances in the other two cases.

Something frustrating when you try to simulate something that happened a few hundreds of Myrs ago is that you can at the best be probabilistic. The study shows that a light primordial system is likelier to have existed than a heavy one, but there are simulations with a heavy system, in which Triton survives. So, a heavy system is not prohibited.

The study and its authors

  • The study, which is available as free article. The authors probably paid extra fees for that, many thanks to them! You can also look at it on arXiv.
  • A conference paper on the same study,
  • The ResearchGate profile of Raluca Rufu,
  • The Homepage of Robin M. Canup.

Before closing this post, I need to mention that the title has been borrowed from Matija Ćuk (SETI, Mountain View, CA), who works on this problem as well (see these two conference abstracts here and here).

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

An interstellar asteroid

Hi there! You may have heard this week of our Solar System visited by an asteroid probably formed in another planetary system. This is why I have decided to speak about it, so this article will not be based on a peer-reviewed scientific publication, but on good science anyway. The name of this visitor is for now A/2017 U1.

History of the discovery

Discovering a new object usually consists in

  1. Taking a picture of a part of a sky. Usually these are parts of the order of the degree, maybe much less… so, small parts. And this also requires to treat the image, to correct for atmospheric (brightness of the sky, wind,…) and instrumental (dead pixels…) effects,
  2. Comparing in with the objects, which are known to be in that field.

If there is an unexpected object, then it could be a discovery. Here is the history of the discovery of A/2017 U1:

  • Oct. 19, 2017: Robert Weryk, a researcher of the University of Hawaii, discovers a new object while searching for Near-Earth Asteroids with the Pan-STARRS 1 telescope. An examination of images archives revealed that the object had already been photographed the day before.
  • Oct. 25, 2017: The Minor Planet Center (Circular MPEC 2017-U181) gives orbital elements for this new object, from 34 observations over 6 days, from Oct. 18 to 24. Surprisingly, an eccentricity bigger than 1 (1.1897018) is announced, which means that the trajectory follows a hyperbola. This means that if this object would be affected only by the Sun, then it would come from an infinite distance, and would leave us for infinity. In other words, this object would not be fated to remain in our Solar System. That day, the object was thought to be a comet, and named C/2017 U1. 10 observation sites were involved (once an object has been detected and located, it is easier to re-observe it, even with a smaller telescope).
  • Oct. 26, 2017: Update by the Minor Planet Center (Circular MPEC 2017-U185), using 47 observations from Oct. 14 on. The object is renamed A/2017 U1, i.e. from comet “C” to asteroid “A”, since no cometary activity has been detected. Same day: the press release announcing the first confirmed discovery of an interstellar object. New estimation of the eccentricity: e = 1.1937160.
  • Oct. 27, 2017: Update by the Minor Planet Center (Circular MPEC 2017-U234), using 68 observations. New estimation of the eccentricity: e = 1.1978499.

And this is our object! It has an absolute magnitude of 22.2 and a diameter probably smaller than 400 meters. These days, spectroscopic observations have revealed a red object, alike the KBOs (Kuiper Belt Objects). It approached our Earth as close as 15 millions km (0.1 astronomical unit), i.e. one tenth of the Sun-Earth distance.

The trajectory of A/2017 U1.
The trajectory of A/2017 U1.

What are these objects?

The existence of such objects is predicted since more than 40 years, in particular by Fred Whipple and Viktor Safronov. This is how they come to us:

  1. A protoplanetary disk creates a star, planets, and small objects,
  2. The small objects are very sensitive to the gravitational perturbations of the planets. As a consequence, they may be ejected from their planetary system, and become interstellar objects,
  3. They visit us.

Calculations indicate that A/2017 U1 comes roughly from the constellation Lyra, in which the star Vega is (only…) at 25 lightyears from our Sun. It is tempting to assume that A/2017 U1 was formed around Vega, but that would be only speculation, since many perturbations could have altered its trajectory. Several studies will undoubtedly address this problem within next year.

Maybe not the first one

Here we have an eccentricity, which is significantly larger (some 20%) than 1. Moreover, our object has a very inclined orbit, which means that we can neglect the perturbations of its orbit by the giant planets. In other words, it entered the Solar System on the trajectory we see now. But a Solar System object can get a hyperbolic orbit, and eventually be ejected. This means that when we detect an object with a very high eccentricity, like a long-period comet, it does not necessary mean that it is an interstellar object. In the past, some known objects have been proposed to be possible interstellar ones. This is for example the case for the comet C/2007 W1 (Boattini), which eccentricity is estimated to be 1.000191841611794±0.000041198 at the date May 26, 2008. It could be an IC (Interstellar Comet), but could also be an Oort cloud object, put on a hyperbolic orbit by the giant planets.

Detecting interstellar objects

A/2017 U1 object has been detected by the Pan-STARRS (for Panoramic Survey Telescope and Rapid Response System) 1 telescope, which is located at Haleakala Observatory, Hawaii. Pan-STARRS is constituted of two 1.8 m Ritchey–Chrétien telescopes, with a field-of-view of 3°. This is very large compared with classical instruments, and it is suitable for detection of bodies. It operates since 2010.

Detections could be expected from the future Large Synoptic Survey Telescope (LSST), which should operate from 2022 on. This facility will be a 8.4-meter telescope based in Chile, and will conduct surveys with a field-of-view of 3.5°. A recent study by Nathaniel Cook et al. suggests that LSST could detect between 0.001 and 10 interstellar comets during its nominal 10 year lifetime. Of course, 0.001 detection should be understood as the result of a formula. The authors give a range of 4 orders of magnitude in their estimation, which reflects how barely constrained the theoretical models are. This also means that we could be just lucky to have detected one.

What Pan-STARRS can do, LSST should be able to do. In a few years, i.e. in the late 2020s, the number or absence of new discoveries will tell us something on the efficiency of creation of interstellar objects in the nearby stars. Meanwhile, let us enjoy this exciting discovery!

The press release and its authors

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 dynamics of the Quasi-Satellites

Hi there! After reading this post, you will know all you need to know on the dynamics of quasi-satellites. This is the opportunity to present you On the co-orbital motion in the planar restricted three-body problem: the quasi-satellite motion revisited, by Alexandre Pousse, Philippe Robutel and Alain Vienne. This study has recently been published in Celestial Mechanics and Dynamical Astronomy.

The 1:1 mean-motion resonance at small eccentricity

(see also here)

Imagine a pretty simple case: the Sun, a planet with a keplerian motion around (remember: its orbit is a static ellipse), and a very small third body. So small that you can neglect its mass, i.e. it does not affect the motion of the Sun and the planet. You know that the planet has no orbital eccentricity, i.e. the static ellipse serving as an orbit is actually a circle, and that the third body (let us call it the particle) has none either. Moreover, we want the particle to orbit in the same plane than the planet, and to have the same revolution period around the Sun. These are many conditions.
Under these circumstances, mathematics (you can call that celestial mechanics) show us that, in the reference frame which is rotating with the planet, there are two stable equilibriums 60° ahead and astern the planet. These two points are called L4 and L5 respectively. But that does not mean that the particle is necessary there. It can have small oscillations, called librations around these points, the resulting orbits being called tadpole orbits. It is even possible to have orbits enshrouding L4 and L5, this results in large librations orbits, called horseshoe orbits.

All of these configurations are stable. But remember: the planet is much less massive than the Sun, the particle is massless, the orbits are planar and circular… Things become tougher when we relax one of these assumptions. And the authors assumed that the particle had a significant eccentricity.

At high eccentricities: Quasi-satellites

Usually, increasing the eccentricity destabilizes you. This is still true here, i.e. co-orbital orbits are less stable when eccentric. But increasing the eccentricity also affects the dynamical structure of your problem in such a way that other dynamical configurations may appear. And this is the case here: you have an equilibrium where your planet lies.

Ugh, what does that mean? If you are circular, then your particle is at the center of your planet… Nope, impossible. But wait a minute: if you oscillate around this position without being there… yes, that looks like a satellite of the planet. But a satellite is under the influence of the planet, not of the star… To be dominated by the star, you should be far enough from the planet.

I feel the picture is coming… yes, you have a particle on an eccentric orbit around the star, the planet being in the orbit. And from the star, this looks like a satellite. Funny, isn’t it? And such bodies exist in the Solar System.

Orbit of a quasi-satellite. It follows the planet, but orbits the star.
Orbit of a quasi-satellite. It follows the planet, but orbits the star.

Known quasi-satellites

Venus has one known quasi-satellite, 2002 VE68. This is a 0.4-km body, which has been discovered in 2002. Like Venus, it orbits the Sun in 225 days, but has an orbital eccentricity of 0.41, while the one of Venus is 0.007. It is thought to be a quasi-satellite of Venus since 7,000 years, and should leave this configuration in some 500 years.

The Earth currently has several known quasi-satellites, see the following table:

(277810) 2006 FV350.387.1°10,000 y2013 LX280.4550°40,000 y2014 OL3390.4610.2°1,000 y(469219) 2016 HO30.107.8°400 y

Known quasi-satellites of the Earth
Name Eccentricity Inclination Stability
(164207) 2004 GU9 0.14 13.6° 1,000 y

These bodies are all smaller than 500 meters. Because of their significant eccentricities, they might encounter a planet, which would then affect their orbits in such a way that the co-orbital resonance would be destabilized. However, significant inclinations limit the risk of encounters. Some bodies switch between quasi-satellite and horseshoe configurations.

Here are the known quasi-satellites of Jupiter:

Known quasi-satellites of Jupiter
Name Eccentricity Inclination Stability
2001 QQ199 0.43 42.5° > 12,000 y
2004 AE9 0.65 1.6° > 12,000 y
329P/LINEAR-Catalina 0.68 21.5° > 500 y
295P/LINEAR 0.61 21.1° > 2,000 y

329P/LINEAR-Catalina and 295P/LINEAR being comets.

Moreover, Saturn and Neptune both have a confirmed quasi-satellite. For Saturn, 2001 BL41 should leave this orbit in about 130 years. It has an eccentricity of 0.29 and an inclination of 12.5°. For Neptune, (309239) 2007 RW10 is in this state since about 12,500 years, and should stay in it for the same duration. It has an orbital eccentricity of 0.3, an inclination of 36°, and a diameter of 250 km.

Understanding the dynamics

Unveiling the dynamical/mathematical structure which makes the presence of quasi-satellites possible is the challenge accepted by the authors. And they succeeded. This is based on mathematical calculation, in which you write down the equations of the problem, you expand them to retain only what is relevant, in making sure that you do not skip something significant, and you manipulate what you have kept…

The averaging process

The first step is to write the Hamiltonian of the restricted planar 3-body problem, i.e. the total energy of a system constituted by the Sun, the planet, and the massless particle. The dynamics is described by so-called Hamiltonian variables, which allow interesting mathematical properties…
Then you expand and keep what you need. One of the pillars of this process is the averaging process. When things go easy, i.e. when your system is not chaotic, you can describe the dynamics of the system as a sum of sinusoidal contributions. This is straightforward to figure out if you remember that the motions of the planets are somehow periodic. Somehow means that these motions are not exactly sinusoidal, but close to it. So, you expand it in series, in which other sinusoids (harmonics) appear. And you are particularly interested in the one involving λ-λ’, i.e. the difference between the mean longitude of the planet and the particle. This makes sense since they are in the co-orbital configuration, that particular angle should librate with pretty small oscillations around a given value, which is 60° for tadpole orbits, 180° for horseshoes, and 0° for quasi-satellites. Beside this, you have many small oscillations, in which you are not interested. Usually you can drop them in truncating your series, but actually you just average them, since they average to 0. This is why you can drop them.
To expand in series, you should do it among a small parameter, which is usually the eccentricity. This means that your orbit looks pretty like a circle, and the other terms of the series represent the difference with the circle. But here there is a problem: to get quasi-satellite orbits, your eccentricity should be large enough, which makes the analytical calculation tougher. In particular, it is difficult to guarantee their convergence. The authors by-passed this problem in making numerical averaging, i.e. they computed numerically the integrals of the variables of the motion over an orbital period.

Once they have done this, they get a simplified system, based on one degree-of-freedom only. This is a pair of action-angle variables, which will characterize your quasi-satellite orbit. This study also requires to identify the equilibriums of the system, i.e. to identify the existing stable orbits.

Perspectives

So, this study is full of mathematical calculations, aiming at revisiting this problem. The authors mention as possible perspective the study of resonances between the planets, which disturb the system, and the proper frequency of the quasi-satellite orbit. This is the oscillating frequency of the angle characterizing the orbit, and if it is equal to a frequency already present in the system, it could have an even more interesting dynamics, e.g. transit between different states (quasi-satellite / horsehoe,…).

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.

Results

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.

(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.