Tag Archives: Saturn

Planets

Thunderstorms on Saturn

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Hi there! You know the thunderstorms on our Earth. In fact, you can have thunderstorms once you have an atmosphere. And you have many atmospheres in our Solar System, particularly on the giant planets. Today we describe a thunderstorm on Saturn, which happened between November 2007 and July 2008, and was observed by Cassini. This thunderstorm is described in Analysis of a long-lived, two-cell lightning storm on Saturn, by G. Fischer et al. This study will be published soon in Astronomy and Astrophysics.

Outline

Physics of a thunderstorm
The atmosphere of Saturn
The detected episodes
Radio and optical observations
The storm F
The study and its authors

Physics of a thunderstorm

Basically, a thunderstorm results from the encounter between cool and hot air. For instance, after a hot summer day, you have hot air in the low atmosphere, while colder air is brought by the wind. Then the hot air, which is lighter, gains altitude. This convective motion induces displacements of electric charges, and so a difference of electrostatic potential between the ground and the top atmosphere. This difference in electrostatic potential creates electric lightning, which actually balances the charges between the sky and the ground. All this results in unstable weather conditions, in particular rain and strong wind. The rain is due to the moist contained in the hot air, which coalesced as clouds while gaining altitude. Thunderstorms are among the most dangerous natural phenomena.

As I said, you can have thunderstorms on any planet with an atmosphere. Today, we are on Saturn.

The atmosphere of Saturn

The radius of Saturn is about 60,000 km, which corresponds to the distance to the center, where the atmospheric pressure reaches 1 bar. At its center Saturn has probably a rocky core, which radius is about 25,000 km. This leaves room for a very thick atmosphere, i.e. what I would call the Saturnian air, mainly composed of molecular hydrogen and helium. Interestingly for us, there are clouds in the atmosphere of Saturn, which composition depend on the altitude, itself correlated with the pressure. The less dense clouds (up to 2 bars), in the upper atmosphere, mainly consist of ammonia ice, while denser clouds contain water ice. The densest clouds, which pressure exceeds 9.5 bars, contain water droplets with ammonia in aqueous solution.

The winds on Saturn are very strong, i.e. up to 1,800 km/h, or 1,120 mph, which of course facilitates the encounters between different air masses (with different temperatures). Moreover, the atmosphere of Saturn is organized into parallel bands, as is the atmosphere of Jupiter. These bands rotate at slightly different rates, which prompted the International Astronomical Union to define 3 reference systems for the rotation of Saturn:

  • System I: spin period of 10 h14 min for the equatorial bands,
  • System II: spin period of 10 h 39 min 24 s, at the other latitudes,
  • System III: spin period of 10 h 39 min 22.3 s, for the radio emissions.

The detected episodes

To be honest with you, I did not manage to get an exhaustive list of the detected events. By the way, if you have some information, I would be glad to get it. You can comment at the end of this article.

You can find below a list of thunderstorms, which have been detected by the Cassini spacecraft between 2004 and 2010. The study we discuss today is on the Storm F.

  • Storm 0: May 26–31, 2004
  • Storm A: July 13–27, 2004
  • Storm B: August 3–15, 2004
  • Storm C: Sept. 4–28, 2004
  • Storm D: June 8–15, 2005
  • Storm E: Jan. 23 – Feb. 23, 2006
  • Storm F: Nov. 27, 2007 – July 15, 2008
  • Storm G: Nov. 19 – Dec. 11, 2008
  • Storm H: Jan. 14 – Dec. 13, 2009
  • Storm I: Feb. 7 – July 14, 2010

These events were identified in detecting radio emissions, due to Saturn electrostatic discharges (SEDs for short). Before that, the Voyager spacecrafts have detected SEDs in 1980 and 1981, but attributed their origins to impacts in the rings. Since then, other events have been detected. In particular, Great White Spot events, i.e. huge disturbance encircling the planets, can be seen from the Earth. They seem to appear roughly every 30 years, which could be correlated with the duration of Saturn’s year (29.46 years). The last Great White Spot has been observed in 2010-11.

The Great White Spot observed by Cassini in February 2011. Credit: NASA/JPL-Caltech/Space Science Institute
The Great White Spot observed by Cassini in February 2011. Credit: NASA/JPL-Caltech/Space Science Institute

Radio and optical observations

As I said, these events are usually detected thanks to their radio emissions. For that, Cassini disposed of the Radio and Plasma Wave Science (RPWS) instrument, equipped with a High Frequency Receiver.
This receiver listened to Saturn in 3 different modes alternatively, allowing to cover a pretty wide range between 325 and 16025 kHz.
These radio measurements were supplemented by optical observations by the Cassini ISS (Imaging Science Subsystem), by optical observations from Earth, and even by Earth-based radiotelescopes, for the strongest discharges.

The detection of such events strongly depends on the location of the spacecraft with respect to the storm. When the spacecraft is opposite to the storm, you detect almost nothing. Almost, because measuring radio emissions permits over-the-horizon detection, especially when the SED storm is located on the night side (opposite the Sun) and Cassini on the day side. This could be due to a temporary trapping of the radio waves below Saturn’s ionosphere before they are released.

So, Cassini’s RPWS detects the discharges, ISS and the Earth-based telescopes see the storms… Let us see the results for the Storm F (November 2007 to July 2008).

The Storm F

RPWS detected about 277,000 SEDs related to this Storm F. But the analysis of the images revealed two phases.

One or two events?

From November 2007 to March 2008, ISS saw one convection cell, at the latitude of ~35° south. And in March 2008 a second cell appeared, at roughly the same latitude, and separated from the first cell by about 25° in longitude. These two cells drifted both of about 0.35° per day. The presence of these two cells with a correlated motion makes this event a very interesting one… and the authors also detected dark ovals.

Dark ovals

A storm appears as a a bright spot, while a dark oval is a dark one. Several dark ovals were seen, the largest one, nicknamed S3 drifted by 0.92° per day, i.e. much faster than the storms. These dark ovals have probably no SED activity. Several explanations have been proposed to explain these features. They could either be clouds of carbon soot particles, produced by the dissociation of methane in the lightning channels, or remnants of convection cells, in which the ammonia particles have fallen deeper into the atmosphere, leaving darker spots.

Features related to the Storm F. The rectangle focuses on the so-called Storm Alley. This image was taken by Cassini ISS on 23 April 2008. © NASA
Features related to the Storm F. The rectangle focuses on the so-called Storm Alley. This image was taken by Cassini ISS on 23 April 2008. © NASA

So, this paper describes the event. The physics behind still needs some clarification, so you can be sure that devoted papers will follow. Stay tuned!

The study and its authors

You can find the study here. And now, the authors:

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, Facebook, Instagram, and Pinterest.

Rings of Saturn

A constantly renewed ring of Saturn

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Hi there! The outstanding Cassini mission ended last September with its Grand Finale, and it gave us invaluable data, which will still be studied for many years. Today I present you a study which has recently been published in The Astrophysical Journal: Particles co-orbital to Janus and Epimetheus: A firefly planetary ring, by a Brazilian team composed of Othon C. Winter, Alexandre P.S. Souza, Rafael Sfair, Silvia M. Giuliatti Winter, Daniela C. Mourão, and Dietmar W. Foryta. This study tells us how the authors characterized a dusty ring in the system of Saturn, studied its stability, and investigated its origin.

Outline

The rings of Saturn
Janus and Epimetheus
Images of a new ring
Dynamical stability
Renewing the ring
A firefly behavior
The study and its authors

The rings of Saturn

As you may know, Saturn is the ringed planet, its rings being visible from Earth-based amateur telescopes. Actually, the 4 major planets of our Solar System have rings, and some dwarf planets as well, i.e. Chariklo, Haumea, and possibly Chiron. But Saturn is the only one with so dense rings. I summarize below the main relevant structures and distances, from the center of Saturn:

Distance Structure
60,268 km The atmospheric pressure of Saturn reaches 1 bar.
This is considered as the equatorial radius of Saturn.
66,900 – 74,510 km D Ring
74,658 – 92,000 km C Ring
92,000 – 117,580 km B Ring
117,580 – 122,170 km Cassini Division
122,170 – 136,775 km A Ring
133,589 km Encke Gap
140,180 km F Ring
151,500 km Orbits of Janus and Epimetheus
189,000 km Orbit of Mimas
1,222,000 km Orbit of Titan

The A and B Rings are the densest ones. They are separated by the Cassini Division, which appears as a lack of material. It actually contains some, arranged as ringlets, but they are very faint. The Encke Gap is a depletion of material as well, in which the small satellite Pan confines the boundaries. Here we are interested in a dusty ring enshrouding the orbits of Janus and Epimetheus, i.e. outside the dense rings. The discovery of this ring had been announced in 2006, this study reveals its characteristics.

The rings of Saturn seen by Cassini. From right to left: the A Ring with the Encke Gap, the Cassini Division, the B Ring, the C Ring, and the D Ring. © NASA
The rings of Saturn seen by Cassini. From right to left: the A Ring with the Encke Gap, the Cassini Division, the B Ring, the C Ring, and the D Ring. © NASA

Janus and Epimetheus

The two coorbital satellites Janus and Epimetheus are a unique case in the Solar System, since these are two bodies with roughly the same size (diameters: ~180 and ~120 km, respectively), which share the same orbit around Saturn. More precisely, they both orbit Saturn in 16 hours, i.e. at the same mean orbital frequency. This is a case of 1:1 mean-motion resonance, involving peculiar mutual gravitational interactions, which prevent them from colliding. They swap their orbits every four years, i.e. the innermost of the two satellites becoming the outermost. The amplitudes of these swaps (26 km for Janus and 95 for Epimetheus) have permitted to know accurately the mass ratio between them, which is 3.56, Janus being the heaviest one.

Interestingly, Epimetheus is the first among the satellites of Saturn for which longitudinal librations have been detected. As many natural satellites, Janus and Epimetheus have a synchronous rotation, showing the same face to a fictitious observer at the surface of Saturn. For Epimetheus, large librations have been detected around this direction, which are a consequence of its elongated shape, and could reveal some mass inhomogeneities, maybe due to variations of porosity, and/or to its pretty irregular shape.

Janus and Epimetheus seen by Cassini (mosaic of 2 images). © NASA
Janus and Epimetheus seen by Cassini (mosaic of 2 images). © NASA

Images of a new ring

So, Cassini images have revealed a dusty ring in that zone. To characterize it, the authors have first extracted images likely to contain it. Such images are made publicly available on NASA’s Planetary Data System. Since that ring had been announced to have been observed on Sept 15th 2006 (see the original press release), the authors restricted to 2 days before and after that date. The data they used were acquired by the ISS (Imaging Science Subsystem) instrument of Cassini, more precisely the NAC and WAC (Narrow- and Wide-Angle-Camera). They finally found 17 images showing the ring.

The images are given as raw data. The authors needed to calibrate their luminosity with a tool (a software) provided by the Cassini team, and sometimes to smooth them, to remove cosmic rays. Moreover, they needed to consider the position of the spacecraft, to be able to precisely locate the structures they would see.

One of the Cassini images used by the authors. I have added red stars at the location of the ring. © NASA / Ciclops
One of the Cassini images used by the authors. I have added red stars at the location of the ring. © NASA / Ciclops

It appears that the ring presents no longitudinal brightness variation. In other words, not only this is a whole ring and not just an arc, but no density variation is obvious. However, it presents radial brightness variations, over a width of 7,500 km, which is wider than the 5,000 km announced in the 2006 press release.

The next step is to understand the dynamics of this ring, i.e. its stability, its origin, the properties of the particles constituting it… Let us start with the stability.

The ring is removed in a few decades

The authors ran N-body simulations, i.e. numerical integrations of the equations ruling the motion of a ring particle, which would be gravitationally perturbed by the surrounding bodies, i.e. Saturn, and the Janus, Epimetheus, Mimas, Enceladus, Tethys, Dione, and Titan. Moreover, for a reason that I will tell you at the end of this article, the authors knew that the particles were smaller than 13 μm. The motions of such small particles are affected by the radiation pressure of the Sun, in other words the Solar light pushes the particles outward.

The authors simulated 14 times the motion of 18,000 particles equally distributed in the rings. Why 14 times? To consider different particle sizes, i.e. one set with 100 μm-sized particles, and the other sets with sizes varying from 1μm to 13μm.
And it appears that these particles collide with something in a few decades, mostly Janus or Epimetheus. This leaves two possibilities: either we were very lucky to be able to take images of the ring while it existed, or a process constantly feeds the ring. The latter option is the most probable one. Let us now discuss this feeding process.

Renewing the ring

The likeliest sources of material for the rings are ejecta from Janus and Epimetheus. The question is: how were these ejecta produced? By impacts, probably. This study show that Janus and Epimetheus are impacted by the particles constituting the rings, but the impact velocities would not permit to produce ejecta. This is why the authors propose a model, in which interplanetary particles collide with the satellites, generating ejecta.

A firefly behavior

And let us finish with something funny: the ring seems to behave like a firefly, i.e. sometimes bright, and sometimes dark, which means undetectable while present.
To understand what happens, figure out how the light would cross a cloud of particles. If the cloud is dense enough, then it would reflect the light, and not be crossed. But for dust, the light would be refracted, i.e. change its direction. This depends on the incidence angle of the Solar light, i.e. on the geometrical configuration of the Sun-Saturn-ring system. The Solar incidence angle is also called phase. And this phase changes with the orbit of Saturn, which results in huge brightness variations of the ring. Sometimes it can be detected, but most of the time it cannot. This can be explained and numerically estimated by the Mie theory, which gives the diffusion of light by small particles. This theory also explains the creation of rainbows, the Solar light being diffracted by droplets of water.

The study and its authors

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Satellites of Saturn

The fate of the Alkyonides

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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.

Outline

The Alkyonides
The G and Pallene rings
Frequency diffusion
How resonances destabilize an orbit
Results
A perspective
To know more…

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-6λAegaeon-ϖMimas 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…

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Planets

Energy dissipation in Saturn

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Hi there! I will tell you today about the letter Frequency-dependent tidal dissipation in a viscoelastic Saturnian core and expansion of Mimas’ semi-major axis, by Daigo Shoji and Hauke Hussmann, both working at the DLR in Berlin, Germany. This paper has recently been published in Astronomy and Astrophysics.

Saturn’s facts

Do I need to introduce Saturn? Saturn is the sixth planet of the Solar System by its distance to the Sun, and the second by its size. It orbits the Sun at a mean distance of 1.5 billions of km, in 29.4 years. It has more than 200 satellites, which comprises small moons embedded in the rings, mid-sized icy satellites, a large one, i.e. Titan, and very far small moons which are probably trapped objects. Which means that the other bodies are expected to have formed while orbiting around Saturn, or formed from the same protoplanetary disk.
Saturn is particularly known for its large rings, which can be observed from the Earth with almost any telescope. Moreover this planet is on average less dense than the water, which is due to a large atmosphere enshrouding a core. The total radius of Saturn is about 60,000 km, which actually corresponds to a pressure of 1 bar in the atmosphere, while the radius of the core is about 13,000 km. The paper I present today is particularly focused on the core.

A new view of the formation of the satellites of Saturn

The spacecraft Cassini orbits Saturn since 2004, and has given us invaluable information on the planet, the rings, and the satellites. Some of these information pushed the French planetologist Sébastien Charnoz, assisted by French and US colleagues, to propose a new model of formation of the satellites from the rings: this model states that instead of having formed with Saturn, the rings are pretty recent, i.e. less than 1 Gyr, and are due to the disintegration of an impactor.
Once the debris rearranged as a disk, reaccretion of material would have created the satellites, which would then have migrated outward, because of the tidal interaction with the planet… This means that it is crucial to understand the tidal interaction.

Tidal dissipation in the planets

I have already discussed of tides in this blog. Basically: when you are a satellite (you dream of that, don’t you?) orbiting Saturn, you are massive enough (sorry) to alter the shape of the planet, and raise a bulge which would almost be aligned with you… Almost because while the material constituting the planet responds, you have moved, but actually the bulge is in advance because the planet rotates faster than you orbit around it (you still follow me?). As a consequence, you generate a torque which tends to slow down the spin of the planet, and this is compensated by an outward migration of the satellite (of you, since you are supposed to be the satellite). This compensation comes from the conservation of the angular momentum. You can imagine that the planet also raises a tidal bulge on the satellite, but this does not deal with our paper. So, not today.

A consequence of tides is the secular migration of the planetary satellites. Lunar Laser Ranging measurements have detected an outward migration of the Moon at a rate of 3 cm/y. It is not that easy to measure the migration of the satellites of Saturn. An initial estimation, based on the pre-Cassini assumption that the satellites were as old as the Solar System, considered that the satellite Mimas would have at the most migrated from the synchronous orbit to its present one, in 4.5 Gyr. The relevant quantity is the dissipation function Q, and this condition would have meant Q>18,000, in neglecting dissipation in Mimas. Recent measurements based on Cassini observations suggest Q ≈ 2,600, which would be another invalidation of the assumption of primordial satellites.

Several models of dissipation

To make things a little more technical: we are interested in the way the material responds to an external, gravitational sollicitation. This sollicitation is quasi-periodic, i.e. it can be expressed as a sum of periodic, sinusoidal terms. With each of these terms is associated a frequency, on which the response of the material depends. This affects the quantity k2/Q, k2 being a Love number and Q the dissipation function I have just presented. Splitting these two quantities is sometimes useless, since they appear as this ratio in the equations ruling the orbital evolution of the satellites.

Tides in a solid body

By solid body, I mean a body with some elasticity. Its shape can be altered, but not that much. An elastic response would not dissipate any energy, while a viscoelastic one would, and would be responsible for the migration of the orbits of the satellites.
It was long considered that the tidal dissipation did not depend on the excitation frequency, which is physically irrelevant and could lead to non-physical conclusions, e.g. the belief in a stable super-synchronous rotation for planetary satellites.
We now consider that the response of the material is pretty elastic for slow excitations, and viscoelastic for rapid ones. If you do not shake the material too much, then you have a chance to not alter it. If you are brutal, then forget it.
For that, a pretty simple tidal model rendering this behavior is the Maxwell model, based on one parameter which is the Maxwell time. It is defined as the ratio between the viscosity and the rigidity of the material, and it somehow represents the limit between the elastic and the viscoelastic responses.
A refining model for icy satellites is the Andrade model, which considers a higher dissipation at high frequencies.

Tides in a gaseous planet

If the planet is a ball of gas, a fortiori a fluid, then the behavior is different, actually much more complicated. You should consider Coriolis forces in the gas, turbulent behaviors, which can be highly non-linear.
A recent model has been presented by Jim Fuller, in which he considers the possibility of resonant interactions between the fluid and the satellites, which would result in a high dissipation at the exact orbital frequency of the satellite, and the resonant condition would induce that this high dissipation would survive the migration of the satellite. You can see here an explanation of this phenomenon, drawn by James T. Keane.

This paper

This paper aims at checking whether a dissipation of the planet, which would be essentially viscoelastic, could be consistent with the recent measurements of tides. For that, the authors modeled Saturn as an end-member, in neglecting every dissipation in the atmosphere. They considered different plausible numbers for the viscosity and rigidity in the core Saturn, in assuming it has no internal fluid layer, and numerically integrated the migration of Mimas, the variation of its orbital frequency in the expression of tides being taken into account.

And the result is that the viscosity should be of the order of 1013-1014 Pa.s. Smaller and higher numbers would be inconsistent with the measured dissipation.
Moreover, some of these viscosities are found to be consistent with the assumption of a primordial Mimas, i.e. with an inward migration from the synchronous orbit in 4.5 Gyr.

Perspectives

This letter probably presents a preliminary study, the whole study requiring to consider additional effects, like the pull of the rings, the influence of the atmosphere, and the mean-motion resonances between the satellites (see this post), which themselves alter the rate of migration. And this is why this letter does not invalidate Charnoz’s model of formation, nor Fuller’s tides, but just says that other explanations are possible.

Useful links

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Rings of Saturn

The dynamics of Saturn’s F Ring

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

The F Ring

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

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

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

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

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

Mean-motion resonances in the rings

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

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

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

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

and

  • the longitude of the pericentre of the satellite ϖs.

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

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

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

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

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

The method: numerical integrations

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

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

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

Measuring the stability

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

Results

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

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

Some links

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

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