Energy dissipation in Saturn

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.


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.

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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.