Category Archives: Planets

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.

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

I hope you liked it! Please do not forget to comment. You can also subscribe to the RSS feed, and follow me on Twitter.

The surface of Mars is fractal

Hi there! Today’s post is a pretty much different than usual. I will present you a mathematical analysis of planetary features. More precisely, a paper investigating the fractal structure of the surface of Mars. This is a paper entitled Mars topography investigated through the wavelet method: A multidimensional study of its fractal structure, by Adrien Deliège, Thomas Kleyntssens and Samuel Nicolay, which has been recently published in Planetary and Space Science. This study has been conducted at the University of Liège (Belgium).

The surface of Mars

The Mars Orbiter Laser Altimeter (MOLA), as instrument of Mars Global Surveyor, provided us a very accurate map of the whole surface of Mars, which is far from boring. It has for instance an hemispheric asymmetry, the Northern hemispheric being composed of pretty flat, new terrains, which the Southern one is very cratered (several thousands of craters). The northern new terrains are made of lava, which is a fingerprint of past geophysical activity. Moreover, Mars has two icy polar caps.

Among the remarkable features are:

  • Olympus Mons, which is the highest known mountain in the Solar System. This is a former volcano, which rises 22 km above the surrounding volcanic plains.
  • The Tharsis region, which contains many volcanoes.
  • Hellas Planitia, which is a huge impact basin (diameter: 2300 km, depth: 7 km), located in the Southern hemisphere.

You can find below an annotated map, please click!

The topology of Mars. Credit: USGS Astrogeology Science Center

The mission Mars Global Surveyor

The missions Mars Global Surveyor (MGS) is a NASA mission, which has been launched in November 1996, and has been inserted into orbit around Mars 10 months later, i.e. September 1997. It became silent in November 2006 after 3 extensions of the nominal mission, and gave us invaluable data during almost 10 years. It embarked 5 scientific instruments:

  • the Mars Orbiter Camera (MOC), a wide angle camera which gave us images of the surface and of the two satellites of Mars Phobos and Deimos,
  • the Mars Orbiter Laser Altimeter (MOLA), which gave us the most accurate topographic measurements of Mars. The study I present today uses its data,
  • the Thermal Emission Spectrometer (TES), which studied the atmosphere of Mars, and the thermal emission of the surface. This instrument observed in the infrared band,
  • the magnetometer, which studied the magnetic field of Mars,
  • and the radio-science, which measured the gravity field of the planet.

Mars Global Surveyor was of great help to prepare the further missions. It allowed in particular to identify landing sites for rovers.

The rich topography of Mars has encouraged many scientists to characterize it with a fractal structure.

Fractals and multifractals

A fractal is a mathematical set that exhibits a repeating pattern displayed at every scale, see the following figure, which shows the well-known Mandelbrot set.

The Mandelbrot set, plotted by myself after an inspiration from Rosetta Code. The zoom on the right shows the same structure than on the left, with a larger scale.

It is tempting to quantify the “fractality” of such a set. A convenient indicator is the Hausdorff dimension, which is an extension of the dimension of a space. A line is a space of dimension 1, a plane is of dimension 2, and a volume of dimension 3. Now, if you look at the Mandelbrot set, for instance, its contour is a line of infinite length (actually depending on the resolution of the plot), which tends to fill the plane, but does not fill it entirely. So, it makes sense that its dimension should be a real number larger than 1 and smaller than 2. The Hausdorff dimension quantifies how a fractal set fills the space. The Hausdorff dimension of the Mandelbrot set is 2, the one of the coastline of Great-Britain is 1.25, and the one of the coastline of Norway is 1.52.

For a natural object, things are not necessarily that easy, in the sense that some parts of the objects could look like a fractal, and some not, or look like another fractal. Then the object is said multifractal.

The Hausdorff dimension is not the only possible measure of a fractal object. In the paper I present today, the authors use the Hölder exponent, which represents how continuous the function is. Here, the function is the height of a terrain, it depends on its coordinates, i.e. longitude and latitude, on the surface of Mars. The Hölder exponent is usually more appropriate for sets of numerical data.

The wavelet transforms

The wavelet transform is a mathematical transform, which aims at measuring the periodicity of a phenomenon, and gives the amplitude of a periodic contribution, at a given period. In our case, the idea is to measure periodic patterns in the spatial evolution of the height of the surface of Mars.
For that, the authors use more specifically the wavelet leaders methods, which will in particular give them the Hölder exponent, and tell them how (mono)fractal / multifractal the topography of Mars is.

Results

The “fractality” actually depends on the scale you are considering. The authors disposed of MOLA data, with a resolution of 0.463 km. They analyzed them twice, once in performing 1-D analyses, in considering the longitude and the latitude independently, and once in a 2-dimensional analysis, which is probably new in this context. And here are their results:

  • The surface of Mars is monofractal if you look at it at scales smaller than 15 km.
  • It is multifractal for scales larger than 60 km (the authors considered that the range 15-60 km is a transtition zone).
  • The “monofractality” is better in longitude than in latitude. This could be due to the hemispherical asymmetry of Mars, to the polar caps, and / or to the fact that the representation surface is just a planar projection, which necessarily alters it.
  • Some features can be detected from the variations of the Hölder exponent, especially the plains. However, this technique seems to fail for the volcanoes.

Some links

That’s it for today! I hope you enjoyed this post. I particularly like the idea to give a mathematical representation of a natural object. Please feel free to comment! You can also subscribe to the Twitter @planetmechanix and to the RSS feed.

A periodic variation in the atmosphere of Venus

Hi there! Today’ post will be my first on Venus. More precisely, it deals with its atmosphere. As you may know, the planet Venus is known for its very thick atmosphere, which precludes optical observations of its surface. The study I present today is entitled “Discovery of a 150 day period in the Venus condensational clouds”, by Kevin McGouldrick and Constantine Tsang, who work in the city of Boulder, CO (I love this place). This study has been recently accepted for publication in Icarus.

Some Venus facts

Venus is the second innermost planet of the Solar System, which means that its orbit is interior to the one of the Earth. It is sometimes said to be a twin sister of the Earth because its diameter is 95% the diameter of the Earth. However, the meteorological conditions make it a very hostile place for life. The surface pressure is ~93 times the one of the Earth, the temperature is about 470˚C, and the atmosphere is essentially made of carbon dioxide.

Its rotation is very interesting, since it rotates very slowly, and in the retrograde direction. It has a rotation period of 245 days, while its orbital period around the Sun is only 225 days. This means that a Venusian day is longer than a Venusian year. This peculiar rotational state could result from the atmospheric tides, i.e. the way the dense atmosphere interacts with the gravitational forcing of the Sun, loses some energy, and also interacts with the surface. However, the atmosphere moves much faster, with a period of about 4.2 days.

The exploration of Venus

As a putative twin sister of the Earth and a nearby planet, Venus has been a priority target of the Space Race. This is why several American and Soviet probes reached it between 1962 and 1984, allowing major progress in our knowledge of the planet. Here are the probes:

  • 1962: Mariner 2 (USA). This probe was the first one to perform successfully a flyby of another planet than the Earth. It proved that the surface was hot, detected no magnetic field, and it improved our knowledge of the mass of the planet. Beside these results of Venus, it made measurements of the Solar wind and allowed many technological improvements in space navigation and telecommunication.
  • 1965: Venera 4 (USSR), Mariner 5 (USA). Venera 4 crashed on Venus after a fall in the atmosphere with a parachute, permitting the first in situ measurements of its chemical composition, and detection of a weak magnetic field, which Mariner 2 could not have detected. Mariner 5 made a flyby of Venus and analyzed its outer atmosphere.
  • 1969: Venera 5 & 6 (USSR) were technologically similar to Venera 4, but with specific improvements of the analysis of the atmosphere, based on the results of Venera 4.
  • 1970: Venera 7 (USSR) was the first probe to land on another planet than the Earth and to transmit data from the surface. It made the first accurate measurement of the temperature and the pressure at the surface.
  • 1972: Venera 8 (USSR) showed that the atmosphere of Venus was pretty clear below 50 km, meaning that the clouds had a higher altitude.
  • 1975: Venera 9 & 10 (USSR). These two probes were the first ones to send images of the surface of another planet than the Earth. Moreover, Venera 10 measured the velocity of the wind.
  • 1978: Venera 11 & 12 (USSR), Pioneer Venus Multiprobe (USA). Venera 11 & 12 made more accurate measurements of the composition of the atmosphere, and detected lightning and thunder. Pioneer Venus Multiprobe launched 4 probes to the surface of the planet, to analyse the atmosphere during their fall. One of these probes survived the impact, but did not have any imaging instrument. These probes identified 3 layers of clouds in the atmosphere.
  • 1978-1992: Pioneer Venus Orbiter (USA). This spacecraft was the companion of Pioneer Venus Multiprobe, and was inserted into orbit on Dec 4th 1978. Its orbit was very eccentric (0.8), and it contained 17 instruments, allowing to study the magnetic field of Venus, its gravity field, its atmosphere… It also monitored the water loss of the Halley’s comet in 1986.
  • 1981: Venera 13 & 14 (USSR) were landers, they made measurements of the atmosphere during the fall and took images of the surface.
  • 1983: Venera 15 & 16 (USSR). These probes were orbiters equipped with radars. They mapped ~25% of the surface.
  • 1984: Vega 1 & 2 (USSR + Europa). These two probes made a flyby of Venus to launch a lander devoted to make measurements of the atmosphere. After the flyby, the probes approached Halley’s comet and took ~1,500 images of it.
  • 1990: Flyby by Galileo (USA). Galileo was sent to Jupiter, but used the gravitational assistance of Venus on its way. This was the opportunity to study the composition of the clouds of Venus, in comparing the measurements at 1.74 and 2.30 μm, i.e. in the infrared. These two bandwidths correspond to minimal absorption by carbon dioxide and by water, so they can be used not only to detect a signal from the surface of Venus, i.e. the Solar light reflected by the surface, but also to estimate the temporal evolution and the composition of the clouds.
  • 1990-1994: Magellan (USA). This orbiter studied the gravity field of the planet, and also provided a detailed map. It particularly revealed the presence of many volcanoes, few impact craters and large lava plains, meaning that the surface is geologically young, and evidence of some tectonic activity, which is pretty different than the terrestrial one. It was revealed by low domical structures called coronae, produced by the upwelling and subsidence of magma from the mantle.
  • 1998-1999: 2 flybys by Cassini (USA), on its way to Saturn.
  • 2006-2015: Venus Express (Europa), see next paragraph.
  • Since 2015: Akatsuki (Japan). This spacecraft should have orbited Venus since 2010, but that maneuver failed. It then orbited the Sun during 5 years in safe mode before succeeding another orbital insertion in December 2015. This spacecraft essentially studies the dynamics of the atmosphere of Venus during a 2 year regular scientific mission, which has started in May 2016.

Venus Express (VEX)

This ESA spacecraft has been launched in November 2005, and was inserted in orbit in April 2006, originally for a 2-year mission… which was completed 9 years later! The main objective of that mission was to understand the dynamics of the atmosphere of Venus, with the hope of a better understanding of the atmospheric evolution in general. It contained 7 instruments, 3 of them being devoted to spectrometry (VIRTIS, SPICAV and PFS), one to radioscience (VeRa, for Venus Radioscience), one was a magnometer (MAG), one for imaging (VMC, for Venus Monitoring Camera), and the last one, ASPERA-4, investigated the interaction between the Solar wind and the Venusian atmosphere. We are today particularly interested by VIRTIS, for Visible and Infrared Thermal Imaging Spectrometer, which measured the emitted radiance in 1.74 μm and 2.30 μm of the night-side of Venus.
Venus Express had a polar and highly eccentric orbit. Its high eccentricity resulted in a large variation of the distance between the probe and the planet, i.e. from 460 to 63,000 km, with a period of 24 hours. As a consequence, the field of view and resolution of the measurements experienced large variations.
An interesting thing to notice is the fact that Venus Express reused some technologies designed for Mars Express and Rosetta.

This paper

The authors analyzed the emitted radiance in the infrared at different latitudes, for the two wavelengths 1.74 μm and 2.30 μm. Unfortunately, they do not have measurements later than 2008 October 27, because of the failure of the instrument’s cooling system (keep in mind that infrared is very sensitive to the temperature). Moreover, they used only data taken at a distance larger than 10,000 km. The variation of this radiance characterizes the dynamics of the lower region of the clouds, at an altitude between 50 and 55 km. Observing at these two wavelengths permits to draw conclusions on the size of the particles constituting the clouds. Actually, 4 sizes of particles are expected in the clouds of Venus, and in this specific region:

  • Mode 1 particles: they have an average diameter of 0.6 μm, and are expected in the upper region,
  • Mode 2 particles: they have an average diameter of 2 μm, and are expected in the upper region as well,
  • Mode 2′ particles: they have an average diameter of 3 μm, and are expected in the lower and middle regions,
  • Mode 3 particles, with a diameter of 7 μm, are expectd in the lower region.

So, for our lower clouds, we expect only Mode 2′ and Mode 3 particles.

The authors used VIRTIS data, and after denoising they averaged the measurements over 7 days, since they are interested only in the long-term dynamics. Since the atmosphere is rotating, the authors could thus only detect variations in time and in latitude, but not in longitude.

And the results are these: the radiance steadily increases at mid-latitudes, while it decreases near the poles, which could reveal a circulation of clouds over a very-long term. This long-term variation should be a periodic effect, which future measurements by Akatsuki should help to understand.
Moreover, the authors noticed a 150-day periodic variation in the cloud coverage, especially in the 1.74 μm radiance data, at mid-latitude. This is an unexpected result, which had already been hinted by the same authors 4 years before, with less data. The cause of this periodicity still needs to be elucidated. The authors notice that this period is almost two thirds of the rotation period of Venus, but this may be by chance. This could be the manifestation of a Hadley-like circulation, i.e. a kind of circular motion of the atmosphere driven by variations of its temperature, itself controlled by the latitude and the altitude.

Some links

That’s it for today! As usual, I am interested in your feedback. Let me know what you think about this article, what kind of articles you are interested in, if you have specific questions on the science behind…
 

Thanks!

Hinting for Planet Nine in the orbits of Trans-Neptunian Objects

Hi There! Today I will present you a paper by Matthew Holman and Matthew Payne, entitled Observational constraints on Planet Nine: Astrometry of Pluto and other Trans-Neptunian Objects, which aims to derive constraints on the hypothetical Planet Nine from the orbits of small bodies, which orbit beyond the orbit of Neptune. For that, the authors investigate how an unknown, distant and massive planet, could improve the ephemerides of the known Trans-Neptunian Objects (TNOs). This study has recently been accepted for publication in The Astronomical Journal.

The quest for Planet Nine

Here is a longstanding pending question: is there a ninth planet on the Solar System? Some will answer: Yes, and its name is Pluto. But as you may know, Pluto has been reclassified in 2006 as a dwarf planet by the International Astronomical Union. So, is there another ninth planet, still to be discovered? In January 2016, Konstantin Batygin and Michael Brown, answered “probably yes” to this question, from the orbits of TNOs. They discovered that the clustering of their orbits could hardly be due to chance, and so there should be a cause, which has a gravity action. Since this study, several groups try to constrain its orbit and mass, while observers try to detect it.

The purpose of this post is to discuss the study of one of these groups. Let me briefly cite other ones (sorry for oblivion):

  • In 2014, Chad Trujillo and Scott Sheppard discovered a TNO, 2012VP113, whose apparent orbit seemed to be too difficult to explain with the known planets only. This made a case for the existence of the Planet Nine.
  • In 2015, a team led by the Brazilian astronomer Rodney Gomes, showed that a Planet Nine could explain an excess of bright object in the population of the most distant TNOs.
  • In January 2016, Batygin and Brown published their result, which triggered a bunch of other studies.
  • Hervé Beust, from Grenoble (France), showed from a statistical analysis that resonant effects with Neptune could explain the observed clustering,
  • Renu Malhotra, Kat Volk and Xianyu Wang, from the University of Arizona, considered that the largest TNOs could be in mean-motion resonance with the Planet Nine, i.e. that their orbital periods could be commensurate with the one of the Planet Nine. Such a configuration has a dynamical implication on the stability of these bodies. In such a case, the TNO Sedna would be in a 3:2 resonance with the Planet Nine.
  • A team led by Agnès Fienga, from the Observatoire de la Côte d’Azur (France), has suggested that a signature of the Planet Nine could be found in the deviation of the Cassini spacecraft, which currently orbits Saturn. The JPL (Jet Propulsion Laboratory, NASA) does not seem to believe in this option, and indicates that the spacecraft does not present any anomaly in its motion.
  • Gongjie Li and Fred Adams, based respectively at the Harvard-Smithsonian Center for Astrophysics, and at the University of Michigan, show that the orbit of the Planet Nine is pretty unlikely to be stable, because of passing stars close to the outer Solar System, which should have ejected it.
  • de la Fuente Marcos and de la Fuente Marcos, from Spain, reexamined the statistics, and concluded that there should be at least two massive perturbers beyond the orbit of Pluto
  • Matthew Holman and Matthew Payne, from the Harvard-Smithsonian Center for Astrophysics, tried to constrain the orbit of the Planet Nine from the orbits of the TNOs.

All this should result in the present architecture for the Solar System (AU stand for Astronomical Unit, i.e. ≈150 million km:

  • 1 AU: the Earth,
  • 5.2 AU: Jupiter,
  • 9.55 AU: Saturn,
  • 19.2 AU: Uranus,
  • 30.1 AU: Neptune,
  • 39.5 – 48 AU: the Kuiper Belt,
  • 39.5 AU: Pluto,
  • >50 AU: the scattered disk,
  • 67.8 AU: Eris
  • 259.3 AU: 2012VP113
  • 526.2 AU: Sedna,
  • 300 – 1500 AU: the Planet Nine,
  • 50,000 AU: the Oort Cloud,
  • 268,000 AU: Proxima Centauri, which is the closest known star beside the Sun.

Astrometry

The astrometry consists to measure the position of an object in the sky. Seen by a terrestrial observer, the sky is a spherical surface. You can determine two angles which will give the direction of the object, but no distance. These two angles are the right ascension and the declination.

Determining the right ascension and the declination of an object you observe is not that easy. It involves for example to have good reference points on the sky, whose positions are accurately known, with respect to which you will position your object. These reference points are usually stars, and their positions are gathered in catalogs. You should also consider the fact that an object is more than a dot, it appears on your image as a kind of a circle. To be accurate, you should determine the location of the center of the object from its light circle, due to light diffraction. You should in particular consider the fact that the center of the light is not necessarily the center of this object.

When all this is done, you have a right ascension and a declination with uncertainties, at a given date. This date is corrected from the light travel time, i.e. the position of an object we observe was the position of the object when the Solar light was refracted on its surface, not when we observe it. Gathering several observations permits to fit ephemerides of the considered body, i.e. a theory which gives its orbit at any time. These ephemerides are very convenient to re-observe this object, and to send a spacecraft to it…

Fitting an orbit

Ephemerides give you the orbit of a given body. Basically, the orbit of a Solar System body is an ellipse, on which the body is moving. For that, a set of 6 independent orbital elements shall be defined. The following set is an example:

  1. the semimajor axis,
  2. the eccentricity (a null eccentricity means that the orbit is circular; an elliptical orbit means that the eccentricity is smaller than 1),
  3. the inclination, usually with respect to the ecliptic, i.e. the orbital plane of the Earth,
  4. the pericentre, at which the distance Sun-body is the smallest,
  5. the ascending node, locating the intersection between the orbital plane and the ecliptic,
  6. the longitude, which locates the body on its orbit.

The first 5 of these elements are constant if you have only the Sun and an asteroid; in practice they have a time dependence due to the gravitational perturbations of the other bodies, in particular the giant planets, i.e. Jupiter, Saturn, Uranus and Neptune. This study aims at identifying the gravitational influence of the Planet Nine.

A numerical simulation gives you the orbit of an asteroid perturbed by the Sun and the giant planets. But for that, you need to know initial conditions, i.e. the location of the body at a given date. The initial conditions are derived from astrometric positions. Since the astrometry does not give exact positions but positions with some uncertainty, you may have many solutions to the problem. The best fit is the solution which minimizes what we call the residuals, or the O-C, for Observed Minus Calculated. All the O-C are gathered under a statistical quantity known as χ2. The best fit minimizes the χ2.

This study

The purpose of this study is to use 42,323 astrometric positions of TNOs with a semimajor axis larger than 30 AU, 6,677 of them involving Pluto. For that, the fitting algorithm not only includes the gravitational influence of the giant planets, but also of 10 large TNOs, and of the hypothetical Planet Nine, in considering two models: either the Planet Nine is moving on a circular orbit, or it is a fixed point-mass. Its expected orbital period, i.e. several thousands of years, is so large that no significant difference between the two models is expected, given the time span covered by the observations.

Indeed, the two models give pretty the same result. The authors split the sky into several tiles, to check the preferred locations for the Planet Nine, and it appears that for some locations the fit is better, while it is worse for some others.

They also find that if the Planet Nine has a mass of 10 Earth masses, then the distance of the Planet Nine to the Sun should be between 300 and 1,000 AU, while Batygin and Brown found it to be between 400 and 1,500 AU. This discrepancy could be explained by the presence of an another planet at a distance of 60 to 100 AU. In addition to that, the node of the Planet Nine seems to be aligned with the one of Pluto, which had already been noticed by other authors. This could reveal an enhanced dynamical interaction between them.

Finally the authors acknowledge that the astrometric positions have some inaccuracy, and that further observations could affect the results.

The quest for Planet Nine is very exciting, and I am pretty sure that new results will come in a next future!

To know more…

  • The study, made freely available by the authors here, thanks to them for sharing!
  • The webpage of Matthew Holman
  • The profile of Matthew Payne on ResearchGate
  • The press release relating the likely existence of the Planet Nine
  • The study by Trujillo and Sheppard
  • The study by Gomes et al.
  • The study by Batygin and Brown, freely available here
  • The study by Beust, also freely available here
  • The study by Malhotra et al., also freely available here
  • The study by Fienga et al., also freely available here
  • The study by de la Fuente Marcos and de la Fuente Marcos, also freely available here

 

Don’t forget to leave comments!

New clues on the interior of Mercury

Hi there! Thanks for coming on the Planetary Mechanics Blog.

Today I will tell you about new results on the interior of the planet Mercury, by Ashok Kumar Verma and Jean-Luc Margot.
Mercury has been orbited during 4 years by the spacecraft MESSENGER, and gravity data have been derived from the deviations of the spacecraft. These data tell us how the mass is distributed in the planet.

 

Planet Mercury facts

Mercury is the innermost planet of the Solar System. Its radius is about one third of the one of the Earth, and its closeness to the Sun associated with the absence of an atmosphere induces large temperature variations between the day and the night. Another consequence is its very slow rotation, i.e. a Hermean (Mercurian) day lasts 58 terrestrial days, while its revolution around the Sun lasts 88 days, which is exactly 50% longer! This phenomenon is called a 3:2 spin-orbit resonance state, it is a unique case in the Solar System but is somehow analogous to the spin-orbit synchronization of our Moon. It is a consequence of the Solar tides, which despin the planet.

A last interesting fact I would like to mention is that Mercury is too dense for a such a small planet. This suggests that in the early ages of the Solar System, the proto-Mercury was much bigger, and differentiated between a core of pretty heavy elements and a less dense mantle. And then, Mercury has been stripped from this mantle, either slowly, or because of a catastrophic event, i.e. an impact.

 

The missions to Mercury

Sending a spacecraft to Mercury is a challenge, once more because of the proximity of the Sun. Not only the spacecraft should be protected from the Solar radiations, heat,… but it also tends to fall on the Sun instead of visiting the planet. For these reasons, only two spacecrafts have visited the Mercury up to now:

  • the US spacecraft Mariner 10 made 3 flybys of Mercury in 1974-1975. It mapped 45% of the surface and measured a magnetic field,
  • the US spacecraft MESSENGER orbited Mercury during 4 years between March 2011 and April 2015. It gave us invaluable information on the planet, including the ones presented here,
  • and let me mention the European-Japanese mission Bepi-Colombo, which should be launched to Mercury in April 2018.

 

The rotation of Mercury

The rotation of Mercury is in a resonant state, known as 3:2 spin-orbit resonance. This is a dynamical equilibrium reached after dissipation of its rotational energy, in which

  • Mercury rotates about one axis,
  • this axis is nearly perpendicular to its orbit, the deviation, named obliquity, being a signature of the interior,
  • the rotation and orbital periods are commensurate, here with a ratio 3/2. Around this exact commensurability are small librations, due to the periodic variations of the Solar gravitational torque acting on Mercury. The main period of these librations is the orbital one, i.e. 88 days, which is a direct consequence of Mercury’s eccentric orbit. They are supplemented by smaller oscillations, at harmonics of the orbital period (44 d, 29 d, 22 d, etc…), and at the periods of the other planets, meaning that they result from the planetary perturbations on the orbit of Mercury. The largest of these perturbations is expected to be due to Jupiter, but it has not been measured yet.

 

What the rotation can tell us

An issue in the pre-MESSENGER era was: does Mercury have an at least partially molten (outer) core? We now know that it has, thanks to Peale’s experiment, due to the late Stan Peale. The idea was this: the viscous core responds like a fluid to short-period excitations, and like a rigid body for long-period (secular) excitations. And the good thing is that the librations (called longitudinal physical librations) are due to a 88 d-oscillations, while the obliquity is due to a secular one (actually an oscillation which is some 200 kyr periodic, i.e. the rotation of the orbital plane of Mercury). So, in measuring these 2 quantities, one should be able to invert for the size of the core. This was achieved in 2007 thanks to radar measurements of the rotation of Mercury, and confirmed from additional Earth-based measurements, and MESSENGER data, since.

We now know that Mercury has a large molten core, which does not rule out the presence of a solid inner core. For that, additional investigations should be conducted.

 

The gravity field

The most basic model of gravity is the point-mass, which just gives us a mean density of the planet. This can be obtained from planetary ephemerides, i.e. in studying how Mercury affects the motion of the other planets, and with more accuracy from the deviations of the spacecraft. We know since Mariner 10 that Mercury has a density of 5.43 g/cm3, while 1g/cm3 is expected for ice, 3.3 g/cm3 for silicates, and 8 g/cm3 for iron.

A more accurate model is to see Mercury has a triaxial ellipsoid. This requires to add two parameters in the gravity field: J2 and C22, also know as Stokes coefficients. A positive J2 means that the body is flattened at its poles, while C22 represents the equatorial ellipticity of the planet. A positive polar flattening is expected as a consequence of the rotation of the planet, while the equatorial ellipticity can result from differential gravitational action of the Sun, i.e. the tides.

Knowing these two Stokes coefficients is possible from gravity data, and this would give us the triaxility of the mass distribution in Mercury. But something is missing: we do not know its radial distribution, i.e. heavier elements are expected to be in the core. For that, we need the polar momentum C, which could be derived from the obliquity, knowing the Stokes coefficient.

For a spherical homogeneous body, C=2/5 MR2, M being the mass and R the radius, and is smaller when heavier elements are concentrated in the core.

 

The tidal Love coefficient k2

The tides tend to alter the shape of the planet. In addition to a mean shape, there are periodic variations, which are due to the variations of the distance between Mercury and the Sun.

The amplitude of these variations depend on the Love parameter k2, which characterizes the response of the material to the periodic excitations. Actually, k2 depends on the frequency of excitation, in the specific case of Mercury k2@88d and k2@44d affect the gravity field. But distinguishing these two quantities requires a too high accuracy in the data, this is why k2 is often mentioned without precising the frequency involved.

If Mercury were spherical and fluid, k2 would be 1.5, while it would be null if Mercury were fully rigid. Actually, all the natural bodies are somewhere between these two end-members.

The frequency-dependence of the tides is based on the assumption that if you impose a slow deformation of a viscous body, it will not loose any internal energy and slowly recover its shape after (elastic deformation). However, rapid solicitations induce permanent deformations. The numbers associated with these two different regimes depend on the interior of the planet.

 

In this paper

This study, Mercury’s gravity tides, and spin from MESSENGER radio data, by A.K. Verma and J.-L. Margot, has been accepted for publication in Journal of Geophysical Research – Planets. It presents

  • an updated gravity field for Mercury,
  • an updated Love number,
  • an updated spin orientation.

These results are based on measurements of the instantaneous gravity field of Mercury. This is particularly interesting for the determination of the spin, since classical methods are based on the observation of the surface, while the gravity field is ruled by the whole planet. This means that here, the rotation of the whole planet is observed, not just its surface. This allows to constrain the possible differential rotation between the surface and the core.

For the gravity field of Mercury, a 40th order solution is considered, because Mercury is something more complicated than a triaxial ellipsoid. The second order Stokes coefficients are consistent with previous studies, which were also based on MESSENGER data. Some higher-order coefficients are identified as well.

This is the second determination of the Love number k2 = 0.464, which implies than the mantle of Mercury is pretty hot.

 

Some perspectives

We are some years away from the orbital insertion of the European / Japanese mission Bepi-Colombo, which is expected to be ten times more accurate than MESSENGER. So, results like the ones presented here are in some sense preparing the Bepi-Colombo’s measurements. This mission will also secure the results, and providing independent determinations.

Knowing Mercury is also a way to understand planetary formation. There are many discoveries of exoplanets, which orbit close to their parent star, but are so far from us that we cannot hope to send spacecrafts orbiting them. So, understand the way Mercury has been formed helps understanding the other planetary systems.

I hope that one day we will be able to measure the frequency-dependence of the Love numbers, this would be very helpful to constrain the tidal models.

 

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