Tag Archives: rings

The mass of Cressida from Uranus’ rings

Hi there! Today I will present you a new way to weigh an inner satellite of a giant planet. This is the opportunity for me to present you Weighing Uranus’ moon Cressida with the η Ring by Robert O. Chancia, Matthew M. Hedman & Richard G. French. This study has recently been accepted for publication in The Astronomical Journal.

The inner system of Uranus

Uranus is known to be the third planet of the Solar System by its radius, the 4th by its mass, and the 7th by its distance to the Sun. It is also known to be highly tilted, its polar axis almost being in its orbital plane. You may also know that it has 5 major satellites (Ariel, Umbriel, Titania, Oberon, and Miranda), and that it has been visited by the spacecraft Voyager 2 in 1986. But here, we are interested in its inner system. If we traveled from the center of Uranus to the orbit of the innermost of its major satellites, i.e. Miranda, we would encounter:

  • At 25,559 km: the location where the atmosphere reaches the pressure 1 bar. This is considered to be the radius of the planet.
  • Between 37,850 and 41,350 km: the ζ Ring,
  • At 41,837 km: the 6 Ring,
  • At 42,234 km: the 5 Ring,
  • At 42,570 km: the 4 Ring,
  • At 44,718 km: the α Ring,
  • At 45,661 km: the β Ring,
  • At 47,175 km: the η Ring,
  • At 47,627 km: the γ Ring,
  • At 48,300 km: the δ Ring,
  • At 49,770 km: the satellite Cordelia (radius: 20 km),
  • At 50,023 km: the λ Ring,
  • At 51,149 km: the ε Ring
  • At 53,790 km: the satellite Ophelia (radius: 22 km),
  • At 59,170 km: the satellite Bianca (radius: 26 km),
  • At 61,780 km: the satellite Cressida (radius: 40 km),
  • At 62,680 km: the satellite Desdemona (radius: 34 km),
  • At 64,350 km: the satellite Juliet (radius: 47 km),
  • At 66,090 km: the satellite Portia (radius: 68 km),
  • Between 66,100 and 69,900 km: the ν Ring,
  • At 69,940 km: the satellite Rosalind (radius: 36 km),
  • At 74,800 km: the satellite Cupid (radius: 9 km),
  • At 75,260 km: the satellite Belinda (radius: 45 km),
  • At 76,400 km: the satellite Perdita (radius: 15 km),
  • At 86,010 km: the satellite Puck (radius: 81 km),
  • Between 86,000 and 103,000 km: the μ Ring,
  • In the μ Ring, at 97,700 km: the satellite Mab (radius: 13 km)
  • At 129,390 km: the satellite Miranda (radius: 236 km).

The rings of Uranus are being discovered since 1977. Originally it was from star occultations observed from the Earth. Then Voyager 2 visited Uranus in 1986, which revealed other rings, and more recently the Hubble Space Telescope imaged some of them, permitting other discoveries.. Most of them have a width of ≈1 km.
All of the inner moons have been discovered on Voyager 2 images, except Cupid and Mab, which have been discovered in 2003, once more thanks to Hubble. On the contrary, the major moons have been discovered between 1787 and 1948.

Today we will focus only on

  • At 47,175 km: the η Ring,
  • At 61,780 km: the satellite Cressida (radius: 40 km).

The η Ring is very close to the 3:2 mean-motion resonance (MMR) with Cressida, which means that any particle of the η Ring makes 3 revolutions around Uranus while Cressida makes 2. As a consequence, Cressida has a strong gravitational action on the η Ring.

Gravitational interactions

How do we know the mass of planetary bodies? When we send a spacecraft close enough, the spacecraft is deviated, and from the deviation we have the gravity field, or at least the mass. If we cannot send a spacecraft, then we can invert, i.e. analyze, the interactions between different bodies. We know the mass of the Sun thanks to the orbits of the planets, we know the mass of Jupiter thanks to the orbits of its satellites, and the deviations of the spacecraft. We can also use MMR. For instance, in the system of Saturn, the mass ratios between Mimas and Tethys, between Enceladus and Dione, and between Janus and Epimetheus, were accurately known before the arrival of Cassini, thanks to resonant relations.

We can have resonant interactions between a satellite and a ring, as well. A ring is actually a cloud of small particles, and the way their motion is affected reveals the gravitational interaction with something. When you have a MMR, then the ring exhibits streamlines, which give a pattern with equally spaced corners. From the number of these corners you can determine the MMR involved, and from the size of the pattern you get the mass of the disturbing satellite. This is exactly what happens here, i.e. 3:2 MMR with Cressida affects the η Ring in such a way that you can read the mass of Cressida from the shape of this ring. But for that, you need to be accurate enough on the location of the ring.

The data

The authors used 49 observations, including 3 Voyager 2 ones, the other ones being star occultations by rings. Such an observation should be anticipated, i.e. the relative position of Uranus with respect to thousands of stars is calculated, then the star has to be observed where possible, i.e. in a place where it will be high enough in the sky, and of course at night. You measure the light flux coming from the star, which should be pretty constant… and is not because of the variability of the atmospheric thickness since the star is moving in the sky (remember: the Earth rotates in one day), so you have to compensate with other stars… and if you detect a flux drop, then this means that something is occulting the star. Possibly a ring.
Most of the observations were made in the K band, i.e. at an infrared wavelength of 2.2 μm, where Uranus is fainter than its rings. These observations have been made between 1977 and 1996. Since then, the opening of the rings is too small, i.e. we see Uranus by the edge, which reduces the chances to occult a star.

Methodology

The authors made a least-square fit. This means that they fitted their corpus of observations with a shape of the ring as R-A cos (mθ), where R is a constant radius, A is an amplitude of distortion of the ring, θ is the angle (a longitude), and m is a factor giving the frequency of the distortion, which could be related to its cause, i.e. the orbital motion of the satellite affecting the ring. You fit R, A and m, i.e. you adjust them so as to reduce the difference (the error, which is mathematically seen as a distance) between your model and the observations. From R you have a ring (and you can check whether there should be a ring there), from A you have the mass of the satellite, and from m and have its frequency (and you can check whether a known satellite has this frequency).
The authors show that the highest effect of the inner satellites on the rings should be the effect of Cressida on the η Ring, thanks to the 3:2 MMR.

Results

The authors find that Cressida should have a density of 0.86±0.16 g.cm-3, which is lighter than water. Usually these bodies are supposed to be kind of porous dirty ice, which would mean this kind of density. This is the first measurement of the density of an inner satellite of Uranus. A comparison with other systems shows that this is much denser than the inner satellites of Saturn. However, the inner satellite of Jupiter Amalthea has a pretty similar density.

Finally the authors say that they used this method on other rings, and that additional results should be expected, so we stay tuned. They also say that a spacecraft orbiting Uranus would help knowing these satellites. I cannot agree more. Some years ago, a space mission named Uranus Pathfinder has been proposed to ESA, and another one, named Uranus orbiter and probe, has been proposed to NASA.

The study and the 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 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.

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.

The dynamics of Saturn’s F Ring

Hi there! Today: a new post on the rings of Saturn. I will more specifically discuss the F Ring, in presenting you the study A simple model for the location of Saturn’s F ring, by Luis Benet and Àngel Jorba, which has recently been accepted for publication in Icarus.

The F Ring

The F Ring of Saturn is a narrow ring of particles. It orbits close to the Roche limit, which is the limit below which the satellites are not supposed to accrete because the differential gravitational action of Saturn on different parts of it prevents it. This is also the theoretical limit of the existence of the rings.

The F Ring seen by Cassini (Credit: NASA)
The F Ring seen by Cassini (Credit: NASA)

Its mean distance from the center of Saturn is 140,180 km, and its extent is some hundreds of kilometers. It is composed of a core ring, which width is some 50 km, and some particles which seem to be ejected in spiral strands.

Orbiting nearby are the two satellites Prometheus (inside) and Pandora (outside), which proximity involves strong gravitational perturbations, even if they are small.

The images of the F Ring, and in particular of its structures, are sometimes seen as an example of observed chaos in the Solar System. This motivates many planetary scientists to investigate its dynamics.

Mean-motion resonances in the rings

Imagine a planar configuration, in which we have a big planet (Saturn), a small particle orbiting around (the rings are composed of particles), and a third body which is very large with respect to the particle, but very small with respect to the planet (a satellite). The orbit of the particle is essentially an ellipse (Keplerian motion), but is also perturbed by the gravitational action of the satellite. This usually results in oscillating, periodic variations of its orbital elements, in particular the semimajor axis… except in some specific configurations: the mean motion resonances.

When the orbital periods of the particle and of the satellite are commensurate, i.e. when you can write the ratio of their orbital frequencies as a fraction of integers, then you have part of the gravitational action of the satellite on the particle which accumulates during the orbital history of the two bodies, instead of cancelling out. In such a case, you have a resonant interaction, which usually produces the most interesting effects in planetary systems.

There are resonances among planetary satellites as well, but here I will stick to the rings-satellites interactions, for which a specific formalism has been developed, itself inspired from the galactic dynamics. Actually, 4 angles should be considered, which are

  • the mean longitude of the particle λp, which locates the particle on its orbit,
  • the mean longitude of the satellite λs
  • the longitude of the pericentre of the particle ϖp, which locates the point of the orbit which is the closest to Saturn,

and

  • the longitude of the pericentre of the satellite ϖs.

The situation is a little more complicated when the orbits are not planar, please allow me to dismiss that question for this post.

You have a mean-motion resonance when you can write <pλp-(p+q)λs+q1ϖp+q2ϖs>=0, <> meaning on average. p, q, q1 and q2 are integer coefficients verifying q1+q2=q. The sum of the integer coefficients present in the resonant argument is null. This rule is sometimes called d’Alembert rule, and is justified by the fact that you do not change the physics of a system if you change the reference frame in which you describe it. The only way to preserve the resonant argument from a rotation of an angle α and axis z is that the sum of the coefficients is null.

It can be shown that the strongest resonances happen with |q|=1, meaning either |q1|=1 and q2=0, or
|q2|=1 and q1=0.

In the first case, pλp-(p+1)λsp is the argument of a Lindblad resonance, which pumps the eccentricity of the particle, while pλp-(p+1)λss is a corotation resonance, which is doped by the eccentricity of the satellite. Here I supposed a positive q, which means that the orbit of the satellite is exterior to the one of the particle. This is the case for the configurations F Ring – Pandora and F Ring – Titan. However, when the satellite is interior to the particle, like in the configuration F Ring Prometheus, then the argument of the Lindblad resonance should read pλs-(p+1)λpp, and the one of the corotation resonance is pλs-(p+1)λps.

As I said, these resonances have cumulative effects on the orbits. This means that we could expect that something happens, this something being possibly anything: a Lindblad resonance should pump the eccentricity of a particle and favor its ejection, but this also means that particle which would orbit nearby without being affected by the resonance would be more stable… chaotic effects might happen, which would be favored by the accumulation of resonances, the consideration of higher-order ones, the presence of several perturbers… This is basically what is observed in the F Ring.

The method: numerical integrations

The authors address this problem in running intensive numerical simulations of the behavior of the particles under the gravitational action of Saturn and some satellites. Let me specify that, usually, the rings are seen as clouds of interacting particles. They interact in colliding. In that specific study, the collisions are neglected. This allows the authors to simulate the trajectory of any individual particle, considered as independent of the other ones.

They considered that the particles are perturbed by the oblateness of Saturn expanded until the order 2 (actually this has been measured with a good accuracy until the order 6), Prometheus, Pandora, and Titan. Why these bodies? Because they wanted to consider the most significant ones on the dynamics of the F Ring. When you model so many particles (2.5 millions) over such a long time span (10,000 years), you are limited by the computation time. A way to reduce it is to remove negligible effects. Prometheus and Pandora are the two closest ones and Titan the largest one. The authors have detected that Titan slightly shifts the location of the resonances. However, they admit that they did not test the influence of Mimas, which is the closest of the mid-sized satellites, and which is known for having a strong influence on the main rings.

A critical point when you run numerical integrations, especially over long durations, is the accuracy, because you do not want to propagate errors. The authors use a symplectic scheme, based on a Hamiltonian formulation, i.e. on the conservation of the total energy, which can be expanded up to the order 28. The conservation of the total energy makes sense as long as the dissipation is neglected, which is the case here. The internal accuracy of the integrator was set to 10-21, which translated into a relative error on the angular momentum of Titan below 2.10-14 throughout the whole integration.

Measuring the stability

It might be tough to determine from a numerical integration whether a particle has a stable orbit or not. If you simulate its ejection, then you know, but if you do not see its ejection, you have to decide from the simulated trajectory whether the particle will be ejected one day or not, and possibly when.
For this, two kinds of indicator exists in the literature. The first kind addresses the chaos, or most specifically the hyperbolicity of the trajectory, while the second one addresses the variability of the fundamental frequencies of the system. From a rigorous mathematical point of view, these two notions are different. Anyway, the ensuing indicators are convenient ways to characterize non-periodic trajectories, and their use are commonly accepted as indicators of stability.
A hyperbolic point is an unstable equilibrium. For instance a rigid pendulum has a stable equilibrium down (when you perturb it, it will return down), but an unstable one up (it stays up until you perturb it). The up position is hyperbolic, while the down one is elliptic. The hyperbolicity of a trajectory implies a significant dependency on the initial conditions of the system: a slightly different initial position or different initial velocity will give you a very different trajectory. In systems having some complexity, this strongly suggests a chaotic behavior. The hyperbolicity can be measured with Lyapunov exponents. Different definitions of these exponents exist in the literature, but the idea is to measure the evolution of the norm of the vector which is tangent to the trajectory. Is this norm has an exponential growth, then you strongly depend on the initial conditions, i.e. you are hyperbolic, i.e. you are likely chaotic. Some indicators of stability are thus based on the evolution of the tangent vector.
The other way to estimate the stability is to focus on the fundamental frequencies of the trajectory. Each of the two angles which characterize the trajectory of the particle, i.e. its mean longitude λp and the longitude of its pericentre ϖp can be associated with a frequency of the problem. It is actually a little more complicated than just a time derivative of the relevant angle, because in that case you would have a contribution of the dynamics of the satellite. A more proper determination is made with a frequency analysis of the orbital elements, kind of Fourier. You are very stable when these frequencies do not drift with time. Here, the authors used first the relative variations of the orbital frequency as indicator of the stability. The most stable particles are the ones which present the smallest relative variations. In order to speed up the calculations, they also used the variations of the semimajor axis as an indicator, and considered that a particle was stable when the variations were smaller than 1.5 km.

Results

A study of stability necessarily focuses on the core of the rings, because the spiral strands are supposed to be doomed. And the authors get very confined zones of stability. A comparison between these zones of stability shows that several mean-motion resonances with Prometheus, Pandora and Titan are associated with them. This could be seen as consistent with the global aspect of the F Ring, but neither with the measured width of the core ring, nor with its exact location.

This problem emphasizes the difficulty to get accurate results with such a complex system. The study manages, with a simplified system of an oblate Saturn and 3 satellites, to render the qualitative dynamics of the F Ring, but this is not accurate enough to predict the future of the observed structures.

Some links

  • The study, also made freely available by the authors on arXiv. Thanks to them for sharing!
  • The web page of Luis Benet (UNAM, Mexico).
  • The web page of Àngel Jorba (University of Barcelona, Spain).

Thanks for having read all this. I wish you a Merry Christmas, and please feel free to share and comment!