Monthly Archives: July 2017

Asteroids: Extreme Trans-Neptunian Objects

Discovery of 6 new Extreme Trans-Neptunian Objects

Leave a comment

Hi there! You know the Trans-Neptunian Objects, these bodies which orbit beyond the orbit of Neptune… There are the furthest known objects in the Solar System. Today I will particularly tell you on the most distant of them, which have a semimajor axis larger than 150 AU, while Neptune is at 30 AU… Yes, we can observe some of them. 6 have been recently discovered by the OSSOS survey, in OSSOS. VI. Striking biases in the detection of large semimajor axis Trans-Neptunian Objects, by Cory Shankman and 11 collaborators (full list at the end). This paper has recently been published in The Astronomical Journal. In this study, the authors particularly focus on the possible observational biases, and discuss the Planet Nine hypothesis.

Outline

The OSSOS survey
The Canada-France-Hawaii Telescope
Observational biases
Statistical tests
This study
What about the Planet Nine?
To know more…

The OSSOS survey

I should probably write OSSOSurvey instead, since it stands for Outer Solar System Origins Survey. It is a systematic observation program that ran on the Canada-France-Hawaii Telescope (CFHT) between 2013 and 2017, devoted to the discovery and orbit determination of Trans-Neptunian Objects. For that, the program used an imager with a field of 1×1 degree, to image 21 square degree fields, in different parts of the sky. During the 4 years, these fields were regularly re-observed to follow the motion of the discovered objects. 16 months of astrometric observations are required to obtain an accurate orbit.
The authors announce that OSSOS permitted the detection of more than 830 TNOs, with a “40% detection efficiency at r(ed)-band magnitude 24.4-24.5”. OSSOS followed another survey, CFEPS, for Canada-France-Hawaii Ecliptic Plane Survey, which discovered some 200 Kuiper Belt Objects, i.e. Trans-Neptunian Objects, which are not as far as the objects we discuss today. This makes more than 1,000 small objects discovered by the CFHT.

Some TNOs detected by CFEPD and OSSOS. Replotted from the public data. Copyright: The Planetary Mechanics Blog.
Some TNOs detected by CFEPD and OSSOS. Replotted from the public data. Copyright: The Planetary Mechanics Blog.

The Canada-France-Hawaii Telescope

The Canada-France-Hawaii Telescope is a joint facility of the University of Hawaii, the French Centre National de la Recherche Scientifique, and the Canadian National Research Council. It has also partnerships with institutions based in the two Chinas, South Korea, and Brazil. It has a 3.58-m telescope, which is functional since 1979.
It is ideally situated, close to the summit of the Mauna Kea mountain, Hawaii (altitude: 4,204 m). It is equipped of different instruments, to observe in the visible to infrared bands. One of them, the wide field imager MegaCam, was used for OSSOSurvey.

Observational biases

If you are looking for stars to the West, you will find some. But only on the West, and brighter than a given magnitude. Does that mean that there are no fainter stars, and no stars in the opposite direction? Of course not. You have found only those stars because your observation means and protocol precluded from discovering other stars. This is an observational bias.

This is a very important issue for understanding surveys, i.e. how to extrapolate the catalog of discovered objects to the existing but unknown ones? Observational biases can be due to:

  • The direction in which you observe. Since our sky is moving, this is strongly correlated to the observation date.
  • The weather. Hard to see something behind a cloud.
  • Your field of view. Is there something behind this tree?
  • The limitations of your instrument.
  • The albedo of your object. How efficiently does it reflect the incident Solar light?
  • …

There is something very significant in the name of CFEPS… E stands for ecliptic, which is the orbital plane of the Earth. The Solar System is roughly planar (with many exceptions of course), and it made sense to look for objects with a small orbital inclination. Consequence: most of the objects discovered by CFEPS have a low inclination… observational bias, which was in fact a way to optimize the chances to discover objects. But it would be wrong to conclude from these discoveries a lack of objects with a small inclination.

OSSOS had observational biases as well, mostly due to the absence of observations in the direction of the Galactic Plane, and to the allocated observation time. The Galactic Plane is full of stars, which complicates the observations of faint objects. This is why the authors maximized their chances in avoiding that part of the sky. As a consequence, OSSOS could not detect objects with an ascending node (the point where the orbit of the object crosses the ecliptic) between -120° and -20°, and had a poor sensitivity between 115° and 165°.

In the specific case of extreme TNOs, there is another bias due to their dynamics: a small object orbiting at a distance of 150 AU has no chance to be detected from the Earth. So, the only detected objects came close enough, which means that their orbits is highly elongated, i.e. highly eccentric. The 8 objects considered in this study, i.e. 6 newly discovered and 2 already known, have a pericentric distance between 31 and 50 AU, which involves an eccentricity between 0.727 and 0.932 (the eccentricity of the Earth is some 0.016). Among the extreme TNOs, only the highly eccentric ones can be detected. This does not mean that they are all highly eccentric.

The reason why the scientific community became excited about the Planet Nine is that a clustering of the orbits of the extreme TNOs was identified in other data, in particular a clustering of the pericentres of the objects. It was then concluded that this clustering was the dynamical signature of the Planet Nine, proving its existence. OSSOS gives independent data, are they clustered?

Answering such a question is not straightforward when the data are scattered. Looking at them with the naked eye is not enough, there are mathematical tools which can measure the statistical relevance of an hypothesis. In particular, the Planet Nine hypothesis should be compared with the null hypothesis, i.e. an equal distribution of the pericentres of the extreme TNOs.

Statistical tests

A common tool is the Kolmogorov-Smirnov test, or KS-test. The idea is to determine a distance between your sample and the one that a given law would give you. If the distance is small enough, then it makes sense to conclude that your sample obeys the law you tested.
This test has been refined as Kuiper’s test, which is insensitive to cyclic transformations of the variables. Cyclic phenomena are everywhere in orbital dynamics.

This study

The following table presents you the 8 eTNOs presented in this study.

Name Semimajor axis Eccentricity Inclination Magnitude
2013 GP136 150.2 AU 0.727 33.5° 23.1
2013 SY99 735 AU 0.932 4.2° 24.8
2013 UT15 200 AU 0.780 10.7° 24.1
2015 KH163 153 AU 0.739 27.1° 24.7
2015 RY245 226 AU 0.861 6° 24.6
2015 GT50 312 AU 0.877 8.8° 24.5
2015 RX245 430 AU 0.894 12.1° 24.1
2015 KG163 680 AU 0.940 14° 24.3

All of them have been discovered by OSSOS, the first two ones being known before that study. The 6 other ones are the 6 newly discovered. We can see that all have huge semimajor axes and eccentricities. You can see the high relative magnitudes in the red band during their discoveries. Their discoveries were made possible by their high eccentricities, which reduce significantly the minimal distances to the Sun and to the Earth.

And now their orbits are drawn!

Projection of the orbits of the 8 eTNOs on the ecliptic. The orbit of Neptune is embedded into the small circle delimited by the orbits. Copyright: The Planetary Mechanics Blog, after inspiration from OSSOS.
Projection of the orbits of the 8 eTNOs on the ecliptic. The orbit of Neptune is embedded into the small circle delimited by the orbits. Copyright: The Planetary Mechanics Blog, after inspiration from OSSOS.

Do they look clustered?

What about the Planet Nine?

The Kuiper’s test used by the authors say that the orbital elements of the detected eTNOS are statistically consistent with a uniform repartition. We must be careful with words. This means that there is no evidence of clustering in this sample. That does not mean that there is no Planet Nine. We should keep in mind that 8 objects do not constitute a statistically relevant sample.

My feeling is that if you were skeptical about the existence of the Planet Nine, you remain skeptical. However, if you believed in it, there is still room for belief. The fact is that this study does not comfort the existence of the Planet Nine.

To know more…

  • The study, made freely available by the authors on arXiv. You can also find a presentation of the study by the authors themselves here.
  • A presentation by Nature.
  • The website of the OSSOS survey, and its Twitter.
  • The quest for the Planet Nine.

And the authors of the study:

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.

Asteroids: Main Belt

The rotation of Fagus

Leave a comment

Hi there! Today I will tell you on the rotation of the asteroid (9021) Fagus. The first determination of its spin period is given in Rotation period determination for asteroid 9021 Fagus, by G. Apostolovska, A. Kostov, Z. Donchev, and E. Vchkova Bebekovska. This study has recently been published in the Bulgarian Astronomical Journal.

Outline

(9021) Fagus’s facts
Measuring the rotation
Photometric observations
The observation facilities
The softwares
Result
To know more…

(9021) Fagus’s facts

(9021) Fagus is a small, Main Belt asteroid. You can find below some of its characteristics:

Semimajor axis 2.58 AU, i.e. 386 millions km
Eccentricity 0.173
Inclination 13.3°
Orbital period 4.14 y
Diameter 13.1 km
Absolute magnitude 12.4
Discovery February 14, 1988

Its small magnitude explains that its discovery was acknowledged only in 1988. Once identified, it was found on older photographic plates, providing observations from 1973 (yes, you can observe an object before it was discovered… you just do not know that you observed it). This body is so small, that the authors of this study observed it by accident: in 2013, they observed in fact (901) Brunsia during two nights, which is brighter (absolute magnitude: 11.35), but Fagus was in the field. The collected photometric data were supplemented in March 2017 by two other nights of observations, which permitted the authors to determine the spin (rotation) period with enough confidence.

Measuring the rotation

I address the measurement of the rotation of an asteroid here. Such a small body may have an irregular shape, and tumble. But since it is very difficult to get accurate data for such a small body, it is commonly assumed that the body rotates around one principal axis, this hypothesis being confronted with the observations. In other words, if you can explain the observations with a rotation around one axis, then you have won.

The irregularity of the shape makes that the light flux you record presents temporal variations, i.e. the surface elements you face is changing, so the reflection of the incident Solar light is changing, which means that these variations are correlated with the rotational dynamics. If these variations are dominated by a constant period of oscillation, then you have the rotation period of the asteroid. Typically, the rotation period of the Main-Belt asteroids are a few hours. These numbers are strongly affected by the original dynamics of the planetary nebula, the despinning of the asteroids being very slow. This is a major difference with the planetary satellites, which rotates in a few days since they are locked by the tides raised by their parent planet. For comparison, the spin period of the Moon is 28 days.

Photometric observations

Detecting the photometric variations of the incident light of such a small body requires to be very accurate. The overall signal is very faint, its variations are even fainter. To avoid errors, the observer should consider:

  • The weather. A bright sky is always better, preferably with no wind, which induces some seeing, i.e. apparent scintillation of the observed object.
  • The anthropogenic light pollution.
  • The variations of the thickness of the atmosphere during the observation. If your object is at the zenith, then it is pretty good. If it is low in the sky, then its course during the night will involve variations of the thickness of the atmosphere during the observations.
  • Instrumental problems. Usually you use a chip of CCD sensors, these sensors do not have exactly the same response. A way to compensate this is to measure a flat, i.e. the response of the chip to a homogeneous incident light flux.

The observation conditions can be optimized, for instance in observing from a mountain area. The observer should also be disciplined, for instance many professional observatories forbid to smoke under the domes. In the past, this caused wrong detections. A good way to secure the photometric results is to have several objects in the fields, and to detect the correlations between their variations of flux. Intrinsic properties of an object would emerge from light variations, which would be detected for this object only.

The observation facilities

The observations were made at Rozhen Observatory, also known as Bulgarian National Astronomical Observatory. It is located close to Chepelare, Bulgaria, at an altitude of 1,759 m. It consists of 4 telescopes.

The 2013 observations were made with a 50/70 cm Schmidt telescope, and the 2017 ones with a 2m-Ritchey-Chrétien-Coude telescope. In both cases, the observations were made through a red filter. The faintness of the asteroid required exposure times between 5 and 6 minutes.

The Schmidt telescope used for the 2013 observations. Copyright: P. Markishky
The Schmidt telescope used for the 2013 observations. Copyright: P. Markishky
The 2m telescope, used for the 2017 observations. Copyright: P. Markishky
The 2m telescope, used for the 2017 observations. Copyright: P. Markishky

The softwares

The authors used two softwares in their study: CCDPHOT, and MPO Canopus. CCDPHOT is a software running under IDL, which is another software, commonly used to treat astrophysical data, and not only. With CCDPHOT, the authors get the photometric measurements. MPO Canopus could give these measurements as well, but the authors used it for another functionality: it fits a period to the lightcurve, in proving an uncertainty. This is based on a Fourier transform, i.e. a spectral decomposition of the signal. In other words, the lightcurves, with are recorded as a set of pairs (time, lightflux), are transformed into a triplet of (amplitude, frequency, phase), i.e. it is written as a sum of sinusoidal oscillations. If one of these oscillations clearly dominates the signal, then its period is the rotation period of the asteroid.

Result

And the result is this: the rotation period of (9021)Fagus is 5.065±0.002 hours. In practice, being accurate on such a number requires to collect data over several times this interval. An ideal night of observation would permit to measure during about 2 periods. Here, data have been collected over 4 nights.
Up to now, we had no measurement of the spin period of Fagus, which makes this result original. It not only helps to understand the specific Fagus, but it is also a new data in the catalog of the rotational periods of Main-Belt asteroids.

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.

Satellites of Saturn

The fate of the Alkyonides

Leave a comment

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…

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.

Satellites of Saturn

On the interior of Mimas, aka the Death Star

Leave a comment

Hi there! Today I will tell you on the interior of Mimas. You know, Mimas, this pretty small, actually the smallest of the mid-sized, satellite of Saturn, which has a big crater, like Star Wars’ Death Star. Despite an inactive appearance, it presents confusing orbital quantities, which could suggest interesting characteristics. This is the topic of the study I present you today, by Marc Neveu and Alyssa Rhoden, entitled The origin and evolution of a differentiated Mimas, which has recently been published in Icarus.

Outline

Mimas’ facts
The rotation of Mimas
Heating Mimas
The problems
Results
Another mystery
To know more…

Mimas’ facts

The system of Saturn is composed of different groups of satellites. You have

  • Very small satellites embedded into the rings,
  • Mid-sized satellites orbiting between the rings and the orbit of Titan
  • The well-known Titan, which is very large,
  • Small irregular satellites, which orbit very far from Saturn and are probably former asteroids, which had been trapped by Saturn,
  • Others (to make sure I do not forget anybody, including the coorbital satellites of Tethys and Dione, Hyperion, the Alkyonides, Phoebe…).

Discovered in 1789 by William Herschel, Mimas is the innermost of the mid-sized satellites of Saturn. It orbits it in less than one day, and has strong interactions with the rings.

Semimajor axis 185,520 km
Eccentricity 0.0196
Inclination 1.57°
Diameter 396.4 km
Orbital period 22 h 36 min

As we can see, Mimas has a significant eccentricity and a significant inclination. This inclination could be explained by a mean-motion resonance with Tethys (see here). However, we see no obvious cause for its present eccentricity. It could be due to a past gravitational excitation by another satellite.

Mimas, seen by Cassini. We can the crater Herschel, which makes Mimas look like Star Wars' Death Star. Credit: NASA
Mimas, seen by Cassini. We can the crater Herschel, which makes Mimas look like Star Wars’ Death Star. Credit: NASA

The literature is not unanimous on the formation of Mimas. It was long thought that the satellites of Saturn formed simultaneously with the planet and the rings, in the proto-Saturn nebula. The Cassini space mission changed our view of this system, and other scenarios were proposed. For instance, the mid-sized satellites of Saturn could form from the collisions between 4 big progenitors, Titan being the last survivor of them. The most popular explanation seems to be that a very large body impacted Saturn, its debris coalesced into the rings, and then particles in the rings accreted, forming satellites which then migrated outward… these satellites being the mid-sized satellites, i.e. Rhea, Dione, Tethys, Enceladus, and Mimas. This scenario would mean that Mimas would be the youngest of them, and that it formed differentiated, i.e. that the proto-Mimas was made of pretty heavy elements, on which lighter elements accreted. Combining observations of Mimas with theoretical studies of its long-term evolution could help to determine which of these scenarios is the right one… if there is a right one. Such studies can of course involve other satellites, but this one is essentially on Mimas, with a discussion on Enceladus at the end.

The rotation of Mimas

As most of the natural satellites of the giant planets, Mimas is synchronous, i.e. it shows the same face to Saturn, its rotational (spin) period being on average equal to its orbital one. “On average” means that there are some variations. These are actually a sum of periodic oscillations, which are due to the variations of the distance Mimas-Saturn. And from the amplitude and phase of these variations, you can deduce something on the interior, i.e. how the mass is distributed. This could for instance reveal an internal ocean, or something else…

This rotation has been measured in 2014 (see this press release). The mean rotation is indeed synchronous, and here are its oscillations:

Period Measured
amplitude (arcmin)		 Theoretical
amplitude (arcmin)
70.56 y 2,616.6 2,631.6±3.0
23.52 y 43.26 44.5±1.1
22.4 h 26.07 50.3±1.0
225.04 d 7.82 7.5±0.8
227.02 d 3.65 2.9±0.9
223.09 d 3.53 3.3±0.8

The most striking discrepancy is at the period 22.4 h, which is the orbital period of Mimas. These oscillations are named diurnal librations, and their amplitude is very sensitive to the interior. Moreover, the amplitude associated is twice the predicted one. This means that the interior, which was hypothesized for the theoretical study, is not a right one, and this detection of an error is a scientific information. It means that Mimas is not exactly how we believed it is.

The authors of the 2014 study, led by Radwan Tajeddine, investigated 5 interior models, which could explain this high amplitude. One of these models considered the influence of the large impact crater Herschel. In all of these models, only 2 could explain this high amplitude: either an internal ocean, or an elongated core of pretty heavy elements. Herschel is not responsible for anything in this amplitude.

The presence of an elongated core would support the formation from the rings. However, the internal ocean would need a source of heating to survive.

Heating Mimas

There are at least three main to heat a planetary body:

  1. hit it to heat it, i.e. an impact could partly melt Mimas, but that would be a very intense and short heating, which would have renewed the surface…nope
  2. decay of radiogenic elements. This would require Mimas to be young enough
  3. tides: i.e. internal friction due to the differential attraction of Saturn. This would be enforced by the variations of the distance Saturn-Mimas, i.e. the eccentricity.

And this is how we arrive to the study: the authors simulated the evolution of the composition of Mimas under radiogenic and tidal heating, in also considering the variations of the orbital elements. Because when a satellite heats, its eccentricity diminishes. Its semimajor axis varies as well, balanced between the dissipation in the satellite and the one in Saturn.

The problems

For a study to be trusted by the scientific community, it should reproduce the observations. This means that the resulting Mimas should be the Mimas we observe. The authors gave themselves 3 observational constraints, i.e. Mimas must

  1. have the right orbital eccentricity,
  2. have the right amplitude of diurnal librations,
  3. keep a cold surface.

and they modeled the time evolution of the structure and the orbital elements using a numerical code, IcyDwarf, which simulates the evolution of the differentiation, i.e. separation between rock and water, porosity, heating, freezing of the ocean if it exists…

Results

The authors show that in any case, the ocean cannot survive. If there would be a source of heating sustaining it, then the eccentricity of Mimas would have damped. In other words, you cannot have the ocean and the eccentricity simultaneously. Depending on the past (unknown) eccentricity of Mimas and the dissipation in Saturn, which is barely known, an ocean could have existed, but not anymore.
As a consequence, Mimas must have an elongated core, coated by an icy shell. The eccentricity could be sustained by the interaction with Saturn. This elongated core could have two origins: either a very early formation of Mimas, which would have given enough time for the differentiation, or a formation from the rings, which would have formed Mimas differentiated.

Finally the authors say that there model does not explain the internal ocean of Enceladus, but Marc Neveu announces on his blog that they have found another explanation, which should be published pretty soon. Stay tuned!

Another mystery

The 2014 study measured a phase shift of 6° in the diurnal librations. This is barely mentioned in the literature, probably because it bothers many people… This is huge, and could be more easily, or less hardly, explained with an internal ocean. I do not mean that Mimas has an internal ocean, because the doubts regarding its survival persist. So, this does not put the conclusions of the authors into question. Anyway, if one day an explanation would be given for this phase lag, that would be warmly welcome!

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.

Planets

Measuring the tides of Mercury

Leave a comment

Hi there! I have already told you about the tides. If you follow me, you know that the tides are the deformations of a planet from the gravitational action of its parent star (the Sun for Mercury), and that a good way to detect them is to measure the variations of the gravity field of a planet from the deviations of a spacecraft orbiting it. From periodic variations we should infer a coefficient k2, known as the potential Love number, which represents the response of the planet to the tides…

That’s all for today! Please feel free to comment… blablabla…

Just kidding!

Today, I will tell you about another way to measure the tides, from the rotation of Mercury. For this, I will present you a study entitled Periodic and quasi-periodic attractors for the spin-orbit evolution of Mercury with a realistic tidal torque, which was recently published in The Monthly Notices of the Royal Astronomical Society. This is a collaboration between English and Italian mathematicians, i.e. Michele Bartuccelli, Jonathan Deane, and Guido Gentile. In planetary sciences mathematics can lead to new discoveries. In this case, the idea is: tides slow down the rotation of a planetary body, which eventually reaches an equilibrium rotation (or spin). For the Moon, the equilibrium is the synchronous rotation, while for Mercury it is the 3:2 spin-orbit resonance. Very well. A very good way to describe this final state is to describe the equilibrium rotation, i.e. in considering that the tides do not affect the spin anymore. But this is just an approximation. The tides are actually still active, and they affect the final state. In considering it, the authors show that the variations of the spin rate of Mercury should be composed of at least two sinusoids, i.e. two periodic effects, the superimposition of these two periods being quasi-periodic… you now understand the title.

The rotation of Mercury

I have already presented you Mercury here. Mercury is the innermost planet of the Solar System, with a semimajor axis which is about one third of the one of the Earth, i.e. some 58 million km, and a surprisingly large orbital eccentricity, which is 0.206. These two elements favor a spin-orbit resonance, i.e. the rotation rate of Mercury is commensurate with its orbital rate. Their ratio is 3/2, Mercury performing a revolution about the Sun in 88 days, while a rotation period is 58 days. You can notice a 3/2 ratio between these two numbers.

The 3:2 spin-orbit resonance of Mercury
The 3:2 spin-orbit resonance of Mercury

Why is this configuration possible as an equilibrium state? If you neglect the dissipation (the authors do not) and the obliquity (the authors do, and they are probably right to do it), you can write down a second-degree ODE (ordinary differential equation), which rules the spin. In this equation, the triaxiality of Mercury plays a major role, i.e. Mercury spins the way it spins because it is triaxial. Another reason is its orbital eccentricity. This ODE has equilibriums, i.e. stable spin rates, among them is the 3:2 spin-orbit resonance.

And what about the obliquity? It is actually an equilibrium as well, known as Cassini State 1, in which the angular momentum of Mercury is tilted from the normal to its orbit by 2 arcminutes. This tilt is a response to the slow precessing motion (period: 300,000 years) of the orbit of Mercury around the Sun.

Let us forget the obliquity. There are several possible spin-orbit ratios for Mercury.

Possible rotation states

If you went back to the ODE which rules the spin-rate of Mercury, you would see that there are actually several equilibrium spin rates, which correspond to p/2 spin-orbit resonances, p being an integer. Among them are the famous synchronous resonance 1:1 (p=2), the present resonance of Mercury (p=3), and other ones, which have never been observed yet.

If we imagine that Mercury initially rotated pretty fast, then it slowed down, and crossed several resonances, e.g. the 4:1, the 7:2, 3:1, 5:2, 2:1… and was trapped in none of them, before eventually being trapped in the present 3:2 one. Or we can imagine that Mercury has been trapped for instance in the 2:1 resonance, and that something (an impact?) destabilized the resonance…
And what if Mercury had been initially retrograde? Why not? Venus is retrograde… In that case, the tides would have accelerated Mercury, which would have been trapped in the synchronous resonance, which is the strongest one. This would mean that this synchronous resonance would have been destabilized, to allow trapping into the 3:2 resonance. Any worthwhile scenario of the spin evolution of Mercury must end up in the 3:2 resonance, since it is the current state. The scenario of an initially retrograde Mercury has been proposed to explain the hemispheric repartition of the observed impacts, which could be a signature of a past synchronous rotation. Could be, but is not necessarily. Another explanation is that the geophysical activity of Mercury would have renewed the surface of only one hemisphere, making the craters visible only on the other part.

Anyway, whatever the past of Mercury, it needed a dissipative process to end up in an equilibrium state. This dissipative process is the tides, assisted or not by core-mantle friction.

The tides

Because of the differential attraction of the Sun on Mercury, you have internal friction, i.e. stress and strains, which dissipate energy, and slow down the rotation. This dissipation is enforced by the orbital eccentricity (0.206), which induces periodic variations of the Sun-Mercury distance.
An interesting question is: how does the material constituting Mercury react to the tides? A critical parameter is the tidal frequency, i.e. the way you dissipates depends on the frequency you shake. A derivation of the tidal torque raised by the Sun proves to be a sum of periodic excitations, one of them being dominant in the vicinity of a resonance. This results in an enforcement of all the spin-orbit resonances, which means that a proper tidal model is critical for accurate simulations of the spin evolution.
A pretty common way to model the tides is the Maxwell model: you define a Maxwell time, which is to be compared with the period of the tidal excitation (the shaking). If your excitation is slow enough, then you will have an elastic deformation, i.e. Mercury will have the ability to recover its shape without loss of energy. However, a more rapid excitation will be dissipative. Then this model can be improved, or refined, in considering more dissipation at high frequencies (Andrade model), or grain-boundary slip (Burgers model)… There are several models in the literature, which are supported by theoretical considerations and lab experiments. Choosing the appropriate one depends on the material you consider, under which conditions, i.e. pressure and temperature, and the excitation frequencies. But in any case, these physically realistic tidal models will enforce the spin-orbit resonances.

Considering only the tides assumes that your body is (almost) homogeneous. Mercury has actually an at least partially molten outer core, i.e. a global fluid layer somewhere in its interior. This induces fluid-solid boundaries, the outer one being called CMB, for core-mantle boundary, and you can have friction there. The authors assumed that the CMB was formed after the trapping of Mercury into its present 3:2 spin-orbit resonance, which is supported by some studies. This is why they neglected the core-mantle friction.

This paper

This paper is part of a long-term study on the process of spin-orbit resonance. The authors studied the probabilities of capture (when you slow down until reaching a spin-orbit resonance, will you stay inside or leave it, still slowing down?), proposed numerical integrators adapted to this problem…
In this specific paper, they write down the ODE ruling the dynamics in considering the frequency-dependent tides (which they call realistic), and solve it analytically with a perturbation method, i.e. first in neglecting a perturbation, that they add incrementally, to eventually converge to the real solution. They checked their results with numerical integrations, and they also studied the stability of the solutions (the stable solutions being attractors), and the probabilities of capture.

In my opinion, the main result is: the stable attractor is not periodic but quasi-periodic. Fine, but what does that mean?

If we neglect the influence of the other planets, then the variations of the spin rate of Mercury is expected to be a periodic signal, with a period of 88 days. This is due to the periodic variations of the Sun-Mercury distance, because of the eccentricity. This results in longitudinal librations, which are analogous to the librations of the Moon (we do not see 50% of the surface of the Moon, but 59%, thanks to these librations). The authors say that this solution is not stable. However, a stable solution is the superimposition of these librations with a sinusoid, which period is close to 15 years, and an amplitude of a few arcminutes (to be compared to 15 arcminutes, which is the expected amplitude of the 88-d signal). So, it is not negligible, and this 15-y period is the one of the free (or proper) oscillations of Mercury. A pendulum has a natural frequency of oscillations, here this is exactly the same. But contrarily to a pendulum, the amplitude of these oscillations does not tend to 0. So, we could hope to detect it, which would be a direct observation of the tidal dissipation.

Measuring the rotation

What can we observe? We should first keep in mind that the authors addressed the early Mercury, when being trapped into the 3:2 spin-orbit resonance, which was pretty homogeneous. The current Mercury has a global fluid layer, which means a larger (about twice) amplitude of the 88-d signal, and a different dissipative process, the tides being assisted by core-mantle friction. As a consequence, there is no guarantee that the 15-y oscillation (actually a little shorter, some 12 years, because of the fluid core) would still exist, and that would require a dedicated study. But measuring it would be an information anyway.

How to measure it? The first observations of the rotation of Mercury in 1965 and of the librations in 2007 were Earth-based, radar observations, which are sensitive to the velocity. This means that they are more likely to detect a rapid oscillation (88 d, e.g.) than a slow one (12 years…). Observations of the surface of Mercury by the spacecraft MESSENGER confirmed those measurements. In 2018 the ESA/JAXA (Europe / Japan) joint mission Bepi-Colombo will be sent to Mercury, for orbital insertion in 2025 and hopefully a 2-y mission, with a better accuracy than MESSENGER. So, we could hope a refinement of the measurements of the longitudinal motion.

Purple: The 88-d oscillation. Green: Superimposed with the 15-y one. Keep in mind that Bepi-Colombo will orbit Mercury during some 2 years.
Purple: The 88-d oscillation. Green: Superimposed with the 15-y one. Keep in mind that Bepi-Colombo will orbit Mercury during some 2 years.

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.