Category Archives: Planets

Ice on Mercury

Hi there! You know Mercury, the innermost planet of the Solar System. It has recently been explored during 4 years by the American spacecraft MESSENGER, which gave us invaluable data on its surface, its magnetic field, its interior…
Today I present you a study on the ice on Mercury. It is entitled Constraining the thickness of polar ice deposits on Mercury using the Mercury Laser Altimeter and small craters in permanently shadowed regions, by Ariel N. Deutsch, James W. Head, Nancy L. Chabot & Gregory A. Neumann, and has recently been accepted for publication in Icarus.
We know that there is some ice at the surface of Mercury, and the study wonders how much. Since Mercury is close to the Sun, its surface is usually hot enough to sublimate the ice… except in permanently shadowed regions, i.e. in craters. For that, the authors compared the measured depth of small craters, and compared it with the expected depth from the excavation of material by an impactor. The difference is supposed to be ice deposit.

Mercury and MESSENGER

The planet Mercury is known at least since the 14th century BC. It was named after the Roman messenger god Mercurius, or Hermes in Greek, since the messengers saw it at dawn when they left, and at dusk when they arrived. The reason is that Mercury is in fact pretty close to the Sun, i.e. three times closer than our Earth. So, usually the Sun is so bright that it prevents us from observing it. Unless it is below the horizon, which happens at dawn and at dusk.

Mercury makes a full revolution around the Sun in 88 days, and a full rotation in 58 days. This 2/3 ratio is a dynamical equilibrium, named 3:2 spin-orbit resonance, which has been reached after slow despinning over the ages. This despinning is indeed a loss of energy, which has been favored by the tidal (gravitational) action of the Sun. This resulting spin-orbit resonant configuration is a unique case in the Solar System. A consequence is that the Solar day on Mercury lasts 176 days, i.e. if you live on Mercury, the apparent course of the Sun in the sky lasts 176 days.

The proximity of the Sun makes Mercury a challenge for exploration. Mariner 10 made 3 fly-bys of it in 1974-1975, mapping 45% of its surface, and measuring a tiny magnetic field. We had to wait until 2011 for the US spacecraft MESSENGER (MErcury Surface, Space ENvironment, GEochemistry and Ranging) to be the first human-made object inserted into orbit around Mercury. The orbital phase lasted 4 years, and gave us a full map of the planet, gravity data, accurate measurements of its rotation, a list of craters, measurements of the magnetic field,…

The instrument of interest today is named MLA, for Mercury Laser Altimeter. This instrument used an infrared laser (wavelength: 1,064 nanometers) to estimate the height of the surface from the reflection of the laser: you send a laser signal, you get it back some time later, and from the time you have the distance, since you know the velocity, which is the velocity of the light. And in applying this technique all along the orbit, you produce a map of the whole planet. This permits for instance to estimate the size and depth of the craters.

The Mercury Laser Altimeter (MLA).
The Mercury Laser Altimeter (MLA).

Ice on Mercury

The discovery of ice at the poles of Mercury was announced in 1992. It was permitted by Earth-based radar imagery made at Goldstone Deep Space Communications Complex in the Mojave desert, in California (USA). Ice is pretty easy to uncover, because of its high reflectivity. But this raises some questions:

  1. How can ice survive on Mercury?
  2. How much ice is there?
  3. How did it arrive?
The Goldstone facilities in 2018. © Google
The Goldstone facilities in 2018. © Google

The first question is not really a mystery. Because of its long Solar day and its absence of atmosphere (actually Mercury has a very tenuous exosphere, but we can forget it), Mercury experiences huge variations of temperature between day and night, i.e. from 100K to 700K, or -173°C to 427°C, or -279°F to 801°F (it is in fact not accurate at the 1°F level…). So, when a region is illuminated, the water ice is definitely not stable. However, there are regions, especially at the poles, which are never illuminated. There ice can survive.

The last two questions are answered by this study.

Ice is still present in craters

For not being illuminated, it helps to be close to a pole, but the topography can be helpful as well. The surface of Mercury is heavily cratered, and the bottoms of some of these craters are always hidden from the Sun. This is where the authors looked for ice. More precisely, they investigated 10 small craters within 10 degrees of the north pole. And for each of them, they estimated the expected depth from the diameter, and compared it with the measured depth. If it does not match, then you have water ice at the bottom. Easy, isn’t it?

The Carolan crater, one of the craters studied. © NASA
The Carolan crater, one of the craters studied. © NASA

Well, it is not actually that easy. The question is: did the water ice arrive after or before the excavation of the crater? If it arrived before, then the impactor just excavated some ice, and the measurements do not tell you anything.

Another challenge is to deal with the uncertainties. MLA was a wonderful instrument, with an accuracy smaller than the meter. Very well. But you are not that accurate if you want to predict the depth of a crater from its diameter. The authors used an empirical formula proposed by another study: d=(0.17±0.04)D0.96±0.11, where d is the depth, and D the diameter. The problem is the ±, i.e. that formula is not exact. This uncertainty is physically relevant, since the depth of the crater might depend on the incidence angle of the impact, which you don’t know, or on the material at the exact location of the impact… and this is a problem, since you cannot be that accurate on the theoretical depth of the crater. The authors provide a numerical example: a 400-m diameter crater has an expected depth between 21.2 and 127.7 m… So, there is a risk that the thickness of ice that you would measure would be so uncertain that actual detection would be unsure. And this is what happens in almost of all the craters. But the detection is secured by the fact that several craters are involved: the more data you have, the lower the uncertainties. And the ice thickness derived from several craters is more accurate than the one derived from a single crater.

Results: how much ice?

And the result is: the ice thickness is 41+30-14m. The uncertainty is large, but the number remains positive anyway, which means that the detection is positive! Moreover, it is consistent with previous studies, from the detection of polar ice with Goldstone facilities, to similar studies on other regions of Mercury. So, there is ice on Mercury.
An extrapolation of this result suggests that the total mass of water ice on the surface of Mercury is “1014-1015 kg, which is equivalent to ~100-1,000 km3 ice in volume, assuming pure water ice with no porosity” (quoted from the study).

The origin of ice

Mercury is a dense planet, i.e. too dense for such a small planet. It is widely accepted that Mercury as we see it constituted a core of a proto-Mercury, which has been stripped from its mantle of lighter elements. Anyway, Mercury is too dense for the water ice to originate from it. It should come from outside, i.e. it has been brought by impactors. The authors cite studies stating that such a quantity could have been brought by micrometeorites, by Jupiter-family comets, and even by a single impactor.

Another spacecraft soon

Such a study does not only exploit the MESSENGER data, but is also a way to anticipate the future measurements by Bepi-Colombo. This mission will be constituted of two orbiters, one supervised by the European Space Agency (ESA), and the other one by the Japanese agency JAXA. Bepi-Colombo should be launched in October 2018 from Kourou (French Guiana), and inserted into orbit around Mercury in April 2026. Its accuracy is expected to be 10 times better than the one of MESSENGER, and the studies inferring results from MESSENGER data can be seen as predictions for Bepi-Colombo.

The study and its authors

And that’s it for today! Please do not forget to comment. You can also subscribe to the RSS feed, and follow me on Twitter, Facebook, Instagram, and (NEW) Pinterest.

The origin of giant impactors

Hi there! You may know that our Solar System has had a catastrophic youth, destructive impacts playing an inescapable role in sculpting its structure. The paper I present you today, Constraints on the pre-impact orbits of Solar System giant impactors, by Alan P. Jackson, Travis S.J. Gabriel & Erik I. Asphaug, proposes an efficient way to determine the orbit of a giant impactor before the impact. This tells us where this impactor could have come from. The study has recently been published in The Monthly Notices of the Royal Astronomical Society.

Giant impacts in the Solar System

In its early history, the Solar System was composed of many small bodies, i.e. many much than now, most of them having been cleared since then. They have been cleared because the gravitational perturbation of the planets made their orbits unstable. They may also have been destroyed by collisions. Before the clearing was completed, the presence of so many bodies favored intense bombardments. You know for instance the Late Heavy Bombardment (LHB), which probably happened 4 Byr ago, during 200 Myr.
The violence and the outcome of an impact depend on the relative sizes of the target and the impactor, and their relative velocities. Here, the relative velocity should be seen as a vector, i.e. not only the velocity itself is important (the norm of the vector), but its orientation as well, since it directly affects the incidence angle. The craters on the surface of telluric planets, asteroids, and planetary satellites tell us about the history of the bodies, and are the signature of such bombardment. They have been excavated by impactors of moderate size. But now, imagine that the impactor has the size of a planetary body. This is what the authors address as giant impactors.
Giant impactors could have been responsible for

  • the formation of the Moon,
  • the removal of light elements on Mercury,
  • the formation of the two satellites of Mars, Phobos and Deimos,
  • the tilt of Uranus,
  • the disruption of dwarf planets, creating asteroids families,
  • the rings of Saturn,

and many more.

2 kinds of massive impacts

When two bodies of pretty similar size collide, they could

  • either be destroyed, or just one of them be destroyed,
  • survive.

The last case is known as hit-and-run. It happens when the impact is tangential, like between two billiard balls. But it is a little more complicated, of course. This last decade has seen the publication of many Smooth Particle Hydrodynamics (SPH) simulations, in which the impactor and the target are modeled as aggregates of particles. Their mutual interactions are of course considered. Such simulations permit to model the differentiated composition of the two involved bodies, i.e. heavy elements constitute the core, while lighter ones make the mantle, and to trace the outcome of the different geochimical components during and after the impact. This way, the results can be compared with our knowledge of the composition of the bodies. We can evaluate which fraction of the material of the impactor is finally reaccreted on the target, and we can also determine the consequences of hit-and-run collisions. It appears that these collisions do not leave the two bodies intact, but they may strip them from their outer layers.

Why determining their origin

It appeared from the simulations and from the observed compositions of planetary bodies, like the Earth-Moon pair, that impacts do affect the composition of the resulting bodies, and that the difference of composition between the target and the impactor may result in variations of composition after the process is completed, i.e. not only the collision, but also the coalescence of the dust cloud and the reaccretion of the debris. Determining the composition of an impacted planetary body can tell us something on the composition of the impactor.
The composition of planetary bodies depend on their location in the Solar System. The distance with the Sun affects the temperature, which itself affects the viscoelastic properties of the material. Moreover, the initial protoplanetary nebula which gave birth to the Solar System had probably a radial dependent composition, which affected the composition of the resulting planetary bodies. If we could know where an impactor came from, that would tell us something on the primordial Solar System.

The impact velocity gives the orbit of the impactor

In 2014 the first author, i.e. Alan P. Jackson, while he was in UK, lead a study in which the orbital elements of the impacted bodies were determined, in modeling the impact as an impulsive velocity kick. In the present study, the authors invert the formulae, to get the pre-impact elements from the post-impact ones, which are observed, and from the impact velocity, which is estimated by other studies.

This seems easy, but actually is not. One of the problem is that the uncertainties on the impact velocity translates into a family of possible pre-impact elements. Anyway, they give constraints on the semimajor axis, eccentricity and inclination of the impactor. Something very interesting with this method is that the inversion of analytical formulae is very fast with a computer, i.e. you can have the result in a few seconds, maybe less, while the classical method would consist to run N-body simulations during days, where you model the motions of thousands of candidate-impactors (of which you know nothing), until you observe a collision… or not.

In determining the possible pre-impact orbital elements, the authors can assess whether the impactors are likely to fall on the Sun, or to cross the orbit of another planet. In particular, a Sun-grazing solution should obviously be rejected. Moreover, the authors consider that an impactor which would have crossed the orbit of Jupiter would probably have been ejected from the Solar System. As a consequence, such a solution has only a low probability.

And now, let us have a look at the results! The authors applied their method on the formation of the Moon, of Mercury, and on the northern hemisphere of Mars.

A slow impact formed the Moon

It is widely accepted that a giant impactor, nicknamed Theia, has split the proto-Earth between the Earth and the Moon. The literature proposes us three scenarios:

  1. A canonical scenario, in which Theia is a Mars-sized object,
  2. A hit-and-run scenario, in which only part of Theia constitutes the protolunar disk,
  3. A violent scenario, which assumes that the Earth spun very fast at the time of the impact, and that the total angular momentum of the Earth-Moon system was twice the present one. This last scenario requires high impact velocities.

It appears from the results that the last of these scenarios, which requires high velocities, is much less probable than the others, because high velocities translate into highly eccentric orbits. Highly eccentric orbits are the less stable ones, in particular many of them cross the orbit of Jupiter, which could eject them from the Solar System.
So, a conclusion of this study is that the formation of the Moon probably results from a low velocity impact between the proto-Earth and Theia.

Stripping Mercury from its light elements

The composition of Mercury is intriguing, since it is anomalously dense with respect to its size. It is as if the observed Mercury would only be the core of a planet. A proposed explanation is that the proto-Mercury was composed of that core and a mantle of lighter elements (an alternative one is a depletion of lighter elements in the protoplanetary nebula in that region of the Solar System). And of course, it has been imagined that the removal of the mantle is due to a giant impact.

Two scenarios are present in the literature:

  1. A violent impact on the proto-Mercury, which would have removed the mantle,
  2. A succession of hit-and-run collisions in which the proto-Mercury would have been the impactor on the proto-Earth and / or the proto-Venus, and which would have been progressively enriched in iron.

The authors consider the multiple hit-and-run scenario as the most probable one, since it is the one involving the smallest velocities, and limits the possibility of gravitational scattering by Jupiter.

A giant impact on Mars

The North Polar Basin of Mars, or Borealis Basin, covers 40% of the surface of Mars. It may be the largest impact basin in the Solar System, and it creates a dichotomy between the northern and the southern hemispheres.

Topography of Mars, from the instrument MOLA. Borealis Basin is the large blue region in the north. © USGS
Topography of Mars, from the instrument MOLA. Borealis Basin is the large blue region in the north. © USGS

The authors stress that the exact location of Mars at the date of the impact does affect the results, in particular Earth-crossing orbits are allowed only is Mars was close to its pericentre (the location on its orbit, where it is the closest to the Sun). Anyway, they find that the impactor should have had an orbit close to the one of Mars, and suggest that its semimajor axis could have been between 1.2 and 2.2 astronomical unit (the one of Mars being 1.52 AU).

The study and its authors

And that’s it for today! Please do not forget to comment. You can also subscribe to the RSS feed, and follow me on Twitter and Facebook.

The lowlands of Mars

Hi there! Today I will give you the composition of the subsurface of the lowlands of Mars. This is the opportunity for me to present you The stratigraphy and history of Mars’ northern lowlands through mineralogy of impact craters: A comprehensive survey, by Lu Pan, Bethany L. Ehlmann, John Carter & Carolyn M. Ernst, which has recently been accepted for publication in Journal of Geophysical Research: Planets.

Low- and Highlands

Topography of Mars. We can see lowlands in the North, and highlands in the South. © USGS
Topography of Mars. We can see lowlands in the North, and highlands in the South. © USGS

As you can see on this image, the topography of Mars can be divided into the Northern and the Southern hemispheres, the Northern one (actually about one third of the surface) being essentially constituted of plains, while the Southern one is made of mountains. The difference of elevation between these two hemispheres is between 1 to 3 km. Another difference is the fact that the Southern hemisphere is heavily cratered, even if craters exist in the lowlands. This Martian dichotomy is very difficult to explain, some explanations have been proposed, e.g., the lowlands could result from a single, giant impact, or the difference could be due to internal (tectonic) processes, which would have acted differentially, renewing the Northern hemisphere only… Anyway, whatever the cause, there is a dichotomy in the Martian topography. This study examines the lowlands.

The lowlands are separated into: Acidalia Planitia (for plain), Arcadia Planitia, Amazonis Planitia, Chryse Planitia, Isidis Planitia, Scandia Cavi (the polar region), Utopia Planitia, Vatistas Borealis,…

Plains also exist in the Southern hemisphere, like the Hellas and the Argyre Planitiae, which are probably impact basins. But this region is mostly known for Olympus Mons, which is the highest known mountain is the Solar System (altitude: 22 km), and the Tharsis Montes, which are 3 volcanoes in the Tharsis region.

To know the subsurface of a region, and its chemical composition, the easiest way is to dig… at least on Earth. On Mars, you are not supposed to affect the nature… Fortunately, the nature did the job for us, in bombarding the surface. This bombardment was particularly intense during the Noachian era, which correspond to the Late Heavy Bombardment, between 4.1 to 3.7 Gyr ago. The impacts excavated some material, that you just have to analyze with a spectrometer, provided the crater is preserved enough. This should then give you clues on the past of the region. Some say the lowlands might have supported a global ocean once.

The past ocean hypothesis

Liquid water seems to have existed at the surface of Mars, until some 3.5 Gyr ago. There are evidences of gullies and channels in the lowlands. This would have required the atmosphere of Mars to be much hotter, and probably thicker, than it is now. The hypothesis that the lowlands were entirely covered by an ocean has been proposed in 1987, and been supported by several data and studies since then, even if it is still controversial. Some features seem to be former shorelines, and evidences of two past tsunamis have been published in 2016. These evidences are channels created by former rivers, which flowed from down to the top. These tsunamis would have been the consequences of impacts, one of them being responsible for the crater Lomonosov.

The fate of this ocean is not clear. Part of it would have been evaporated in the atmosphere, and then lost in the space, part of it would have hydrated the subsurface, before freezing… This is how the study of this subsurface may participate in the debate.

The CRISM instrument

To study the chemical composition of the material excavated by the impacts, the authors used CRISM data. CRISM, for Compact Reconnaissance Imaging Spectrometer for Mars, is an instrument of Mars Reconnaissance Orbiter (MRO). MRO is a NASA spacecraft, which orbits Mars since 2006.
CRISM is an imaging spectrometer, which can observe both in the visible and in the infrared ranges, which requires a rigorous cooling of the instrument. These multi-wavelengths observations permit to identify the different chemical elements composing the surface. The CRISM team summarizes its scientific goals by follow the water. Studying the chemical composition would permit to characterize the geology of Mars, and give clues on the past presence of liquid water, on the evolution of the Martian climate,…

In this study, the authors used CRISM data of 1,045 craters larger than 1 km, in the lowlands. They particularly focused on wavelengths between 1 and 2.6μm, which is convenient to identify hydrated minerals.

Hydrated vs. mafic minerals

The authors investigated different parts of the craters: the central peak, which might be constituted of the deepest material, the wall, the floor… The CRISM images should be treated, i.e. denoised before analysis. This requires to perform a photometric, then an atmospheric correction, to remove spikes, to eliminate dead pixels…

And after this treatment, the authors identified two kinds of minerals: mafic and hydrated ones. Mafic minerals are silicate minerals, in particular olivine and pyroxenes, which are rich in magnesium and iron, while hydrated minerals contain water. They in particular found a correlation between the size of the crater and the ratio mafic / hydrated, in the sense that mafic detections are less dependent on crater size. Which means that mafic minerals seem to be ubiquitous, while the larger the crater, the likelier the detection of hydrated minerals. Since larger craters result from more violent impacts, this suggests that hydrated minerals have a deeper origin. Moreover, no hydrated material has been found in the Arcadia Planitia, despite the analysis of 85 craters. They also noticed that less degraded craters have a higher probability of mineral detection, whatever the mineral.

However, the authors did not find evidence of concentrated salt deposits, which would have supported the past ocean hypothesis.

The study and the authors

That’s it for today! Please do not forget to comment. You can also subscribe to the RSS feed, and follow me on Twitter and Facebook.

How rough is Mercury?

Hi there! Today I will tell you on the smoothness of the surface of Mercury. This is the opportunity for me to present The surface roughness of Mercury from the Mercury Laser Altimeter: Investigating the effects of volcanism, tectonism, and impact cratering by H.C.M. Susorney, O.S. Barnouin, C.M. Ernst and P.K. Byrne, which has recently been published in Journal of Geophysical Research: Planets. This paper uses laser altimeter data provided by the MESSENGER spacecraft, to measure the regularity of the surface in the northern hemisphere.

The surface of Mercury

I already had the opportunity to present Mercury on this blog. This is the innermost planet of the Solar System, about 3 times closer to the Sun than our Earth. This proximity makes space missions difficult, since they have to comply with the gravitational action of the Sun and with the heat of the environment. This is why Mercury has been visited only by 2 space missions: Mariner 10, which made 3 fly-bys in 1974-1975, and MESSENGER, which orbited Mercury during 4 years, between 2011 and 2015. The study of MESSENGER data is still on-going, the paper I present you today is part of this process.

Very few was known from Mercury before Mariner 10, in particular we just had no image of its surface. The 3 fly-bys of Mariner 10 gave us almost a full hemisphere, as you can see below. Only a small stripe was unknown.

Mercury seen by Mariner 10. © NASA.
Mercury seen by Mariner 10. © NASA.

And we see on this image many craters! The details have different resolutions, since this depends on the distance between Mercury and the spacecraft when a given image was taken. This map is actually a mosaic.
MESSENGER gave us full maps of Mercury (see below).

Mercury seen by MESSENGER. © USGS
Mercury seen by MESSENGER. © USGS

Something that may be not obvious on the image is a non-uniform distribution of the craters. So, Mercury is composed of cratered terrains and smooth plains, which have different roughnesses (you will understand before the end of this article).
Craters permit to date a terrain (see here), i.e. when you see an impact basin, this means that the surface has not been renewed since the impact. You can even be more accurate in dating the impact from the relaxation of the crater. However, volcanism brings new material at the surface, which covers and hides the craters.

This study focuses on the North Pole, i.e. latitudes between 45 and 90°N. This is enough to have the two kinds of terrains.

Three major geological processes

Three processes affect the surface of Mercury:

  1. Impact cratering: The early Solar System was very dangerous from this point of view, having several episodes of intense bombardments in its history. Mercury was particularly impacted because the Sun, as a big mass, tends to focus the impactors in its vicinity. It tends to rough the surface.
  2. Volcanism: In bringing new and hot material, it smoothes the surface,
  3. Tectonism: Deformation of the crust.

If Mercury had an atmosphere, then erosion would have tended to smooth the surface, as on Earth. Irrelevant here.

To measure the roughness, the authors used data from the Mercury Laser Altimeter (MLA), one of the instruments of MESSENGER.

The Mercury Laser Altimeter (MLA) instrument

This instrument measured the distance between the spacecraft and the surface of Mercury from the travel time of light emitted by MLA and reflected by the surface. Data acquired on the whole surface permitted to provide a complete topographic map of Mercury, i.e. to know the variations of its radius, detect basins and mountains,… The accuracy and the resolution of the measurements depend on the distance between the spacecraft and the surface, which had large variations, i.e. between 200 and 10,300 km. The most accurate altimeter data were for the North Pole, this is why the authors focused on it.

Roughness indicators

You need at least an indicator to quantify the roughness, i.e. a number. For that, the authors work on a given baseline on which they had data, removed a slope, and calculated the RMS (root mean square) deviation, i.e. the average squared deviation to a constant altitude, after removal of a slope. When you are on an inclined plane, then your altitude is not constant, but the plane is smooth anyway. This is why you remove the slope.

But wait a minute: if you are climbing a hill, and you calculate the slope over 10 meters, you have the slope you are climbing… But if you calculate it over 10 km, then you will go past the summit, and the slope will not be the same, while the summit will affect the RMS deviation, i.e. the roughness. This means that the roughness depends on the length of your baseline.

This is something interesting, which should be quantified as well. For this, the authors used the Hurst exponent H, such that ν(L) = ν0LH, where L is the length of the baseline, and ν the standard deviation. Of course, the data show that this relation is not exact, but we can say it works pretty well. H is determined in fitting the relation to the data.


To summarize the results:

  • Smooth plains: H = 0.88±0.01,
  • Cratered terrains: H = 0.95±0.01.

The authors allowed the baseline to vary between 500 m and 250 km. The definition of the Hurst exponent works well for baselines up to 1.5 km. But for any baseline, the results show a bimodal distribution, i.e. two kinds of terrains, which are smooth plains and cratered terrains.

It is tempting to compare Mercury to the Moon, and actually the results are consistent for cratered terrains. However, the lunar Maria seem to have a slightly smaller Hurst exponent.

To know more

That’s it for today! The next mission to Mercury will be Bepi-Colombo, scheduled for launch in 2018 and for orbital insertion in 2025. Meanwhile, please do not forget to comment. You can also subscribe to the RSS feed, and follow me on Twitter and Facebook.

Measuring the tides of Mercury

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

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