Monthly Archives: April 2017

Asteroids: Main Belt

Identifying an asteroid family

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Hi there! Today’s post deals with an asteroid family, more precisely the Datura family. The related study is New members of Datura family, by A. Rosaev and E. Plávalová, it has recently been accepted for publication in Planetary and Space Science. The Datura family is a pretty recent one, with only 7 known members when that study started. The authors suggest that 3 other bodies are also members of this family.

Some elements of the dynamics of asteroids

Detailing the dynamics of asteroids would require more than a classical post, here I just aim at giving a few hints.
Asteroids can be found at almost any location in the Solar System, but the combination of the gravitational effects of the planets, of thermal effects, and of the formation of the Solar System, result in preferred locations. Most of the asteroids are in the Main Belt, which lies between the orbits of Mars and Jupiter. And most of these bodies have semimajor axes between 2.1 and 3.2 astronomical units (AU), i.e. between 315 and 480 millions of km. Among these bodies can be found interesting dynamical phenomena, such as:

  • Mean motion resonances (MMR) with planets, especially Jupiter. These resonances can excite the eccentricities of the asteroids until ejecting them, creating gaps known as Kirkwood gaps. At these locations, there are much less asteroids than nearby.
  • Stable chaos. Basically, a chaotic dynamics means that you cannot predict the orbit at a given accuracy over more than a given timespan, because the orbit is too sensitive to uncertainties on its initial conditions, i.e. initial location and velocity of the asteroid. Sometimes chaos is associated with instability, and the asteroid is ejected. But not always. Stable chaos means that the asteroid is confined in a given zone. You cannot know accurately where the asteroid will be at a given time, but you know that it will be in this zone. Such a phenomenon can be due to the overlap of two mean-motion resonances (Chirikov’s criterion).

Anyway, when an asteroid will or will not be under the influence of such an effect, it will strongly be under the influence of the planets, especially the largest ones. This is why it is more significant to describe their dynamics with proper elements.

Proper elements

Usually, an elliptical orbit is described with orbital elements, which are the semimajor axis a, the eccentricity e, the ascending node Ω, the pericentre ω, the inclination I, and the mean longitude λ. Other quantities can be used, like the mean motion n, which is the orbital frequency.

Because of the large influence of the major planets, these elements present quasiperiodic variations, i.e. sums of periodic (sinusoidal) oscillations. Since it is more significant to give one number, the oscillations which are due to the gravitational perturbers are removed, yielding mean elements, called proper elements. These proper elements are convenient to characterize the dynamics of asteroids.

Asteroid families

Most of the asteroids are thought to result from the disruption (for instance because of a collision) of a pretty large body. The ejecta resulting from this disruption form a family, they share common properties, regarding their orbital dynamics and their composition. A way to guess the membership of an asteroid to a family is to compare its proper elements with others’. This guess can then be enforced by numerical simulations of the orbital motion of these bodies over the ages.

Usually a family is named from its largest member. In 2015, 122 confirmed families and 19 candidates were identified (source: Nesvorný et al. in Asteroids IV, The University of Arizona Press, 2015). Many of these families are very old, i.e. more than 1 Gyr, which complicates their identification in the sense that their orbital elements are more likely to have scattered.
The Datura family is thought to be very young, i.e. some 500 kyr old.

A funny memory: in 2005 David Nesvorný received the Urey Prize of the Division of Planetary Sciences of the American Astronomical Society. This prize was given to him at the annual meeting of the Division, that year in Cambridge, UK. He then gave a lecture on the asteroid families, and presented the “Nesvorný family”, i.e. his father, his wife, and so on.

Datura’s facts

The asteroid (1270) Datura has been discovered in 1930. It orbits the Sun in 3.34 years, and has a semimajor axis of 2.23 AU. As such, it is a member of the inner Main Belt. Its orbit is highly elongated, between 1.77 and 2.70 AU, with an orbital eccentricity of 0.209. It rotates very fast, i.e. in 3.4 hours. Its diameter is about 8.2 km.

It is an S-type asteroid, i.e. it is mainly composed of iron- and magnesium-silicates.

This study

After having identified 10 potential family members from their proper elements, the authors ran backward numerical simulations of them, cloning each asteroid 10 times to account for the uncertainties on their locations. The simulations were ran over 800 kyr, the family being supposed to be younger than that. The simulations first included the 8 planets of the Solar System, and Pluto. The numerical tool is a famous code, Mercury, by John Chambers.

The 10 asteroids identified by the authors include the 7 already known ones, and 3 new ones: (338309) 2002 VR17, 2002 RH291, and 2014 OE206. These are all sub-kilometric bodies. The authors point out that these bodies share a linear correlation between their node and their pericentre.

This study also shows that 2014 OE206 has a chaotic resonant orbit, because of the proximity of the 9:16 MMR with Mars. This resonance also affects 2001 VN36, but this was known before (Nesvorný et al., 2006). The authors also find that this chaotic dynamics can be significantly enhanced by the gravitational perturbations of Ceres and Vesta. Finally, they say that close encounters might happen between (1270) Datura and two of its members: 2003 SQ168 and 2001 VN36.

Another study

Now, to be honest, I must mention another study, The young Datura asteroid family: Spins, shapes, and population estimate, by David Vokrouhlický et al., which was published in Astronomy and Astrophysics in February 2017. That study goes further, in considering the 3 new family members found by Rosaev and Plávalová, and in including other ones, updating the Datura family to 17 members.

This seems to be a kind of anachronism: how could a study be followed by another one, which is published before? In fact, Rosaev and Plávalová announced their results during a conference in 2015, this is why they could be cited by Vokrouhlický et al. Of course, their study should have been published earlier. Those things happen. I do not know the specific case of this study, but sometimes this can be due to a delayed reviewing process, another possibility could be that the authors did not manage to finish the paper earlier… Something that can be noticed is that the study by Vokrouhlický is signed by a team of 13 authors, which is expected to be more efficient than a team of two. But the very truth is that I do not know why they published before. This is anyway awkward.

A perspective

I notice something which could reveal a rich dynamics: the authors show (their Figure 7) a periodic variation of the distance between (1270) Datura and 2003 SQ168, from almost zero to about twice the semimajor axis… This suggests me a horseshoe orbit, i.e. a 1:1 mean-motion resonance, the two bodies sharing the same orbit, but with large variations of their distance. If you look at the orbit of the smallest of these two bodies (here 2003 SQ168) in a reference frame which moves with (1270) Datura, you would see a horseshoe-shaped trajectory. To the best of my knowledge, such a configuration has been detected in the satellites of Saturn between Janus and Epimetheus, suggested for exoplanetary systems, maybe detected between a planet and an asteroid, but never between two asteroids…

By the way, 2003 SQ168 is the asteroid, which has the closest semimajor axis to the one of (1270) Datura, in Rosaev and Plávalová’s paper. Now, when I look at Vokrouhlický et al.’s paper, I see that 2013 ST71 has an even closer semimajor axis. I am then tempted to speculate that these two very small bodies are coorbital to (1270) Datura. Maybe a young family favors such a configuration, which would become unstable over millions of years… Speculation, not fact.

Update

This is actually not an horsehoe orbit. The large variation of the distance is due to the fact that 2003 SQ168 is on a orbit, which is close to the one of (1270) Datura, with a slightly different orbital frequency. Regarding 2013 ST71, a numerical simulation by myself suggests the possibility of a temporary (i.e. unstable) capture in a 1:1 MMR.

To know more…

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Asteroids: Near-Earth Objects, Comets

The activity of the comet C/2015 ER61

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Hi there! Today’s post is on the comet C/2015 ER61. Behind this weird name is a small object orbiting the Sun on a highly elongated orbit, which currently shows us a tail. The associated study is Beginning of activity in long-period comet C/2015 ER61 (PANSTARRS), by Karen J. Meech, Charles A. Schambeau, Kya Sorli, Jan T. Kleyna, Marco Micheli, James Bauer, Larry Denneau, Jacqueline V. Keane, Elizabeth Toller, Richard Wainscoat, Olivier Hainaut, Bhuwan Bhatt, Devendra Sahu, Bin Yang, Emily Kramer and Gene Magnier. It has recently been published in The Astronomical Journal.

C/2015 ER61‘s facts

This comet was discovered two years ago, in March 2015, by the telescope Pan-STARRS 1, located on the Haleakalā, Hawai’i. Its distance to the Sun was then 8.44 Astronomical Units, its absolute magnitude about 12, and no tail was visible. As such, it was supposed to be a Manx object, a Manx being a tailless cat. A Manx object would be a comet, which had no activity anymore, as if the lighter elements had already gone.

From its magnitude, it was guessed that its radius was about 10 km. Its apparent lack of activity triggered enough interest for the object to be followed, this in particularly permitted to determine its orbit, and showed that it had a huge eccentricity, i.e. some 0.998. When the eccentricity reaches 1, then the orbit is parabolic, so the orbit of C/2015 ER61 is almost parabolic. Further observations showed the beginning of a period of activity, proving that C/2015 ER61 (I would appreciate a funnier nickname…) is actually not a Manx. This period is not done yet, and the activity is actually increasing, as the comet is approaching the Sun. At its smallest distance, i.e. the perihelion, its distance to the Sun is 1.04 AU, i.e. it almost crosses the orbit of the Earth (don’t worry, I said “almost”). So, observing this comet today reveals a tail.

We are actually pretty lucky to be able to observe it, since its orbital period is some 10,000 years. This comet is considered to belong to the Oort cloud, which is a reservoir of comets at the edge of our Solar System.

Cometary outgassing

Since the comet model by Fred L. Whipple, published between 1950 and 1955, a comet is seen as a kind of dirty snowball, with a nucleus, and icy elements, which tend to sublimate when approaching the Sun, because of the elevation of the temperature. This hypothesis was confirmed in 1986 when we were visited by the well-known comet 1P/Halley (you know, Halley’s comet).
The idea is this: you have some water ice, some CO, some CO2, trapped on the comet. When it is warm enough, it sublimates.

But the intensity of the sublimation depends on several parameters:

  • the thermal inertia of the comet: how does the temperature elevate?
  • its albedo: which fraction of the incident Solar light flux is reflected?
  • its density
  • the quantity of elements, which are likely to be sublimated
  • their depth: if they are not at the surface, the heat needs to be conducted deep enough for them to sublimate
  • the distance to the Sun (of course)
  • etc.

This means that observing and measuring this outgassing gives some physical properties of the comet.

The observation facilities

To conduct this study, several observation facilities were used:

  • Pan-STARRS1 (PS1): This stands for Panoramic Survey Telescope and Rapid Response System. This is a 1.8m wide-field telescope,
  • Gemini North: this is a 8.19 m telescope, which is based in Hawai’i. It has a twin brother, Gemini South, which is based in Chile,
  • Canada-France-Hawai’i Telescope (CFHT): this 3.58m telescope is part of the Mauna Kea Observatory. For this study, the MegaPrime/Megacam wide-field imager was used, which gives of fied of view of 1°,
  • ATLAS: (for Asteroid Terrestrial-impact Last Alert System). This will be a network of two 0.5m-telescopes, both based in Hawai’i. At this time, only the ATLAS-Haleakalā has begun full operation,
  • Himalayan Chandra Telescope (HCT): this is a 2.01 m optical-infrared telescope, which is part of the Indian Astronomical Observatory, which stands on Mount Saraswati, Digpa-ratsa Ri, Hanle, India,
  • Wide-field Infrared Survey Explorer (WISE): this is an infrared space telescope, on a Sun-synchronous polar orbit. It is used in the program NEOWISE, NEO standing for Near-Earth Objects.

The diversity of observation facilities explains the numbers of authors signing this study. The observations span from February 2014 to February 2017, which means that there are pre-discovery observations. It is always easier to find an object when you know where it is, which permitted to find C/2015 ER61 on images, which were taken before its discovery.

Results

These observations (see the Figure) has shown a variation of the magnitude, which could be expected since the comet approached the Earth, but too large to be explained by its trajectory. Actually, it is enhanced by the activity of the comet, more precisely by the sublimation of CO and CO2, starting in early 2015.

The measured apparent magnitude of the comet, with respect to the date and the distance to the Sun. We can see that the comet is brighter when closer to the Sun, because of the outgassing. The measurements have some uncertainties, which are not represented here. This figure is drawn for the Tab.1 Observation Log of the paper.

The authors modeled the warming of the comet and the sublimation of the elements, in using the well-known heat equation. The observed tail suggests a radius of the nucleus of about 9 km, which is consistent with previous guesses. Moreover, they suggest that the CO2 is present at a depth of about 0.4 m. If it were present at the surface, then sublimation would have been observed even when the comet was 20 AU away from the Sun.

The closest approach of the comet with the Earth was on April 4, and with the Sun on May 10, which would result in a peak of activity… probably with some delay, please give the comet a chance to warm!

To know more

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Satellites of Jupiter

Periodic volcanism on Io

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Hi there! Today’s post addresses the volcanic activity of Io, you know, this very active large satellite of Jupiter. It appears from long-term observations that this activity is somehow periodic. This is not truly a new result, but the study I present you enriches the database of observations to refine the measurement of the relevant period. This study is entitled Three decades of Loki Patera observations, by I. de Pater, K. de Kleer, A.G. Davies and M. Ádámkovics, and has been recently accepted for publication in Icarus.

Io’s facts

Io is one the Galilean satellites of Jupiter. it was discovered in 1610 by Galileo Galilei, when he pointed its telescope to Jupiter. It is the innermost of them, with a semimajor axis of 422,000 km, and a orbital period of 1 day and 18 hours. Its mean radius is 1,822 km.

Io has been visited by the spacecrafts Pioneer 10 and 11, Voyager 1 and 2, Galileo, Cassini and New Horizons, Galileo being the only one of these missions to have orbited Jupiter. The first images of the surface of Io are due to Voyager 1. As most of the natural satellites in our Solar System, it rotates synchronously, permanently showing the same face to a fictitious jovian observer.

Its orbital dynamics in interesting, since it is locked in a 1:2:4 three-body mean motion resonance (MMR), with Europa and Ganymede. This means that during 4 orbits of Ganymede, Europa makes exactly two, and Io 4. While two-body MMR are ubiquitous in the Solar System, this is the only known occurrence of a three-body MMR, which is favored by the significant masses of these three bodies.

Three full disk views of Io, taken by Galileo in June 1996. Loki Patera is the small black spot appearing in the northern hemisphere of the central image. The large red spot on the right is Pele. Credit: NASA.

Such a resonance is supposed to raise the orbital eccentricity, elongating the orbit. Nevertheless, it appears that the eccentricity of Io is small, i.e. 0.0041, on average. How can this be possible? Because there is a huge dissipation of energy in Io.

Volcanoes on Io

This energy dissipation appears as many volcanoes, which activities can now be monitored from the Earth. When active, they appear as hot spots on infrared images. More than 150 volcanoes have been identified so far, among them are Loki, Pele, Prometheus, Tvashtar…

This dissipation has been anticipated by the late Stanton J. Peale, who compared the expected eccentricity from the MMR with Europa and Ganymede with the measured one. This way, he predicted dissipation in Io a few days before the arrival of Voyager 1, which detected plumes. This discovery is narrated in the following video (credit: David Rothery).

Dissipation induces geological activity, which another signature is tectonics. Tectonics create mountains, and actually Io has some, with a maximum height of 17.5 km.

But back to the volcanoes. We are here interested in Loki. The Loki volcano is the source of Loki Patera, which is a 200-km diameter lava lake. This feature appears to be actually very active, representing 9% of the apparent energy dissipation of Io.

The observation facilities

This study uses about 30 years of observations, from

  • the Keck Telescopes: these are two 10-m telescopes, which constitute the W.M. Keck Observatory, based on the Mauna Kea, Hawaii. This study enriches the database of observations thanks to Keck data taken between 1998 and 2016.
  • Gemini: the Gemini Observatory is constituted of two 8.19-m telescopes, Gemini North and Gemini South, which are based in Hawaii and in Chile, respectively.
  • Galileo NIMS: the Galileo spacecraft was a space mission which was sent in 1989 to Jupiter. It has been inserted into orbit in December 1995 and has been deorbited in 2003. NIMS was the Near-Infrared Mapping Spectrometer.
  • the Wyoming Infrared Observatory (WIRO): this is a 2.3-m infrared telescope operating since 1977 on Jelm Mountain, Wyoming.
  • the Infrared Telescope Facility (IRTF): this is a 3-m infrared telescope based on the Mauna Kea, Hawaii.
  • the European Southern Observatory (ESO) La Silla Observatory: a 3.6-m telescope based in Chile.

All of these facilities permit infrared observations, i.e. to observe the heat. In this study, the most relevant observations have wavelengths between 3.5 and 3.8 μm. Some of these observations benefited from adaptive optics, which somehow compensates the atmospheric distortion.

Results

And here are the results:

Periodicity

The authors notice a periodicity in the activity of Loki Patera. More particularly, they find a period between 420 and 480 days between 2009 and 2016, while a period of about 540 days was estimated for the activity before 2002. Moreover, Loki Patera appears to have been pretty inactive between 2002 and 2009, and the propagation direction of the eruptions seems to have reversed from one of these periods of activity to the other one.

Temperature

The authors show variations of temperature of the Loki Patera, in estimating it from the infrared photometry, assuming the surface to be a black body, i.e. which emission would only depend on its temperature. They analyzed in particular a brightening event, which occurred in 1999. They showed that it consisted in the emergence of hot magma, at a temperature of 600 K.
On the whole dataset, temperatures up to 1,475 K have been observed, which correspond to the melting temperature of basalt.

Resurfacing rate

This production of magma renews the surface. The observations of such events by different authors suggest a resurfacing rate between 1,160 and 2,100 m2/s, while the surface of Loki Patera is about 21,500 km2, which means that the surface can be renewed in between 118 and 215 days. At this rate, we would be very lucky to observe impact craters on Io… we actually observe none.

A perspective

The authors briefly mention the variation of activity of Pele, Gilbil, Janus Patera, and Kanehekili Fluctus. The intensity of the events affecting Loki Patera makes it easier to study, but similar studies on the other volcanoes would probably permit a better understanding of the phenomenon. They would reveal in particular whether the cause is local or global, i.e. whether the same periods can be detected for other volcanoes, or not.

To know more

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Satellites of Neptune

Where is Triton?

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Hi there! Today’s post is on the location of Triton. Triton is the largest satellite of Neptune and, of course, we know where it is. The paper I present you, entitled Precise CCD positions of Triton in 2014-2016 using the newest Gaia DR1 star catalog, aims at assisting an accurate modeling of its motion. This is a Chinese study, by Na Wang, Qing-Yu Peng, Huan-Wen Peng, H.J. Xie, S. Ma and Q.F. Zhang, which presents observations made at the Yunnan Observatory. This study has recently been published in The Monthly Notices of the Royal Astronomical Society.

Triton’s facts

Triton is by far the largest satellite of Neptune, with a mean radius of 1,350 km. It was discovered almost simultaneously with Neptune, i.e. 17 days later, in 1846. It orbits Neptune in a little less than 6 days, and rotates synchronously.

Surprisingly, its orbit is very inclined with respect to the equator of Neptune (some 24°), and it is retrograde… which means that the inclination should not be given as 24° but as 180-24 = 156°. Usually the satellites have a very small inclination, since they are supposed to have been formed in a nebula which gave birth to the planet. Such a large inclination means that Triton has probably not been formed in situ but was an asteroid, which has been trapped by Neptune. Since then, it loses some orbital energy, which has resulted in a circularization of its orbit and a fall on Neptune.

Triton has been visited by the Voyager 2 spacecraft in 1989, which covered about 40% of its surface. It revealed an atmosphere of nitrogen and evidences of melting, which indicate a geological activity, probably resulting in a differentiated body. It could even harbor or have harbored a subsurface ocean.

Astrometry in the Solar System

The goal of the paper I present today is to give accurate positions of Triton. This is called astrometry. The idea is this: measuring accurately the position of a body at a given date requires to take a picture of the satellite. Stars must be present in the field since the satellite will be positioned according to them. The result will then be compared to the predictions given by dynamical models, called ephemerides. Since we observe only in 2 dimensions, i.e. on the celestial sphere, which is a surface, then an astrometric position consists of two coordinates: the right ascension and the declination.

The positions of the stars surrounding the satellite, actually its image projected on the celestial sphere, since the stars are much further, are known thanks to systematic surveys. The satellite Gaia is currently conducting the most accurate of such surveys ever performed, and the first data were released in 2016, in the Gaia Data Release 1 (DR1, see this post, from October 2016). The accuracy of Gaia lets us hope very accurate future astrometry, and even past, since old observations could be retreated in using Gaia’s catalog.

But why making astrometry? For improving the ephemerides, which would give us a better knowledge of the orbital motion of the satellite. And why improving the ephemerides? I see at least two reasons:

  1. To help future space missions,
  2. To have a better knowledge of the physical properties of the bodies. Some of these properties, like the mass and the energy dissipation (tides), affect the orbital motion.

Potential difficulties

This study presents Earth-based observations, which are affected by:

  1. Diffraction on the CCD chip. When you observe a point as a light-source, you actually see a diffraction disk, and you have to decide which point on the disk is the position of your object. You can partly limit the size of the diffraction disk in limiting the exposure time, this prevents the chip from being saturated. If this results in a too faint object, then you can add several images.
  2. The anisotropy of the light scattering by the surface of the body (here Triton). When we see Triton, we actually see the Solar light, which is reflected by the surface of Triton. Since the surface could be pretty rough, since the limb is darker than the center because of a different incidence angle, and since Triton is not seen as a whole disk (remember the lunar crescent), then the center of the diffraction disk, i.e. the photocentre, is not exactly the location of the body.
  3. The refraction by the atmosphere. The atmosphere distorts the image, which makes the satellite-based images more accurate than the Earth-based ones. This distortion depends on the location of the observatory, and the weather. Some systems of adaptive optics exist, which partly overcome this problem.
  4. The aberration. The relative velocity of the observed object with the observer (the Earth is moving, remember?) alters the apparent position of the objects.
  5. The seeing. When you have some wind, the locations of the stars present some erratic variations.
  6. The inhomogeneity of the CCD chip. An homogeneous lightning will not result in a homogeneous response, because of the positions of the pixel on the image (you have a better sensitivity in the center than close to an edge), and some technical differences between the pixels. A way to overcome this problem is to normalize the light measurement by a flat image, which is the response to a homogeneous lightning. This is usually obtained in taking a picture in the dome, or from the averaging of many images.

This paper

This paper presents 775 new observations, i.e. 775 new positions of Triton at given dates, between 2014 and 2016. The images were taken with a 1-meter refractor at the Yunnan National Observatory, Kunming, Yunnan, China. The residuals, i.e. observed minus predicted positions, are obtained from the ephemerides made by the Jet Propulsion Laboratory in California, USA. The authors obtain mean residuals of a few tens of milli-arcsec, i.e. some thousands of kilometers. Something interesting is the dispersion of these residuals: the authors show that when the stars are positioned with Gaia DR1, the residuals are much less dispersed than with an older catalog. The authors used the catalog UCAC4, released in August 2012 by the US Naval Observatory, for comparison.

These new observations will enrich the databases and permit the future improvements of ephemerides.

Some links

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Theoretical celestial mechanics

Mathematics of the spin-orbit resonance

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Hi there! Today things are a little bit different. The paper I present you is not published in a journal of astronomy, nor of planetary sciences, but of mathematics. It is entitled Hamiltonian formulation of the spin-orbit model with time-varying non-conservative forces, by Ioannis Gkolias, Christos Efthymiopoulos, Giuseppe Pucacco and Alessandra Celletti, and it has been recently published in Communications in Nonlinear Science and Numerical Simulation. It deals with a mathematical way to express and solve the spin-orbit problem. This mathematical way is the Hamiltonian formulation.

The spin-orbit problem

It is something I already discussed on this blog, but never mind. Imagine you have a triaxial body orbiting a largest one… e.g. the Moon orbiting the Earth… or a satellite orbiting a giant planet. Usually the satellite always show the same face to the planet, which is a consequence of a synchronous rotation, which you can call 1:1 spin-orbit resonance. It can be shown that this synchronous resonance is a dynamical equilibrium, i.e. the fact that the angular momentum of the satellite is almost orthogonal to its orbit, and the long axis always points to the parent planet, is a stable position. This is makes the synchronous rotation ubiquitous in the Solar System. Initially the satellite had some rotation, which could have had any spin and orientation. And then, the dissipations of energy, mostly tides raised by the planet, have damped the rotation until reaching the synchronous rotation. At this point, the energy given by the gravitational torque of the planet is large enough to compensate the tides. Since it is a stable equilibrium, then the system stays there, i.e. the rotation remains synchronous.

Hamiltonian formulation

Let us start from conservative mechanics, i.e. in the absence of dissipation. Neglecting the dissipation might be a priori surprising, but this approximation is used since centuries. In planetary systems, dissipation can be easily seen from geysers, volcanoes…, but its effects on the orbital and rotational dynamics are very small, and hence difficult to measure. Lunar Laser Ranging have shown us that the Earth-Moon distance is increasing by some 3.9 cm / yr, as a consequence of the dissipation. We have measurements of such an effect in the system of Jupiter since 2009, and in the system of Saturn since 2011. Moreover, if we assume that the equilibrium has been reached, then we can consider that the loss of energy is compensated by the energy exchanges between the parent planet and the satellite. This is why neglecting the dissipation is sometimes allowed… even if the paper I present you does not neglect it.

So, in conservative mechanics, the total energy of the system is conserved. The total energy of the system is the sum of the kinetic and potential energies of all of the bodies involved. This total energy depends on the variables of the system, i.e. the orbital and rotational variables. It can be shown that convenient sets of variables exist, i.e. canonical variables, which time derivatives are the partial derivatives of the total energy, written with this set of variables, which respect to their conjugate variables. In that case, the formulation of the total energy is called Hamiltonian of the system, and the ensuing equations are the Hamilton equations.

The Hamiltonian formulation is very convenient from a mathematical point of view. Its properties make the dynamics of the system easier to interpret. For instance, in manipulating the Hamiltonian, you can determine its equilibrium, their stability, and the small oscillations (librations) around it. This mathematical structure can also be used to construct dedicated numerical integrators, called symplectic integrators, which solve the equations numerically. Symplectic integrators are reputed for their numerical stability.

Viscoelasticity and tides

Let us talk now on the dissipation. The main source of dissipation is the tides raised by the parent planet. Since its gravitational torque felt by the satellite is not homogeneous over its volume, as distance-dependent, then the satellite experiences stress and strains which alter its shape and induces energy loss. So, the tides have two consequences: loss of energy and variation of the shape. The paper proposes a way to consider these effects in a Hamiltonian formalism.

This paper

As the authors honestly admit, it is somehow inaccurate to speak of Hamiltonian formulation when you have dissipation. Their paper deals with the dissipative spin-orbit problem, so their “Hamiltonian” function is not an Hamiltonian strictly speaking, but the ensuing equations have a symplectic structure.

They assume that the dissipation is contained in a function F, which depends on the time t, and discuss the resolution of the problem with respect to the form of F: either a constant dissipation, or a quasi-periodic one, or the sum of a constant and a quasi-periodic one.

Of course, this paper is very technical, and I do not want to go too deep into the details. I would like to mention their treatment of the quasi-periodic case. Quasi-periodic means that the function F, i.e. the dissipation, can be written under a sum of sines and cosines, i.e. oscillations, of different frequencies. This is physically realistic, in the sense that the material constituting the satellite has a different response with respect to the excitation frequency, and the time evolution of the distance planet-satellite and a pretty wide spectrum itself.
In that case, the dissipation function F depends on the time, which is a problem. But it is classically by-passed in assuming the time to be a new variable of the problem, and in adding to the Hamiltonian a dummy conjugate variable. This is a way to transform a non-autonomous (time-dependent) Hamiltonian into an autonomous one, with an additional degree of freedom.
Once this is done, the resolution of the problem is made with a perturbative approach. It is assumed, which is physically realistic, that the amplitudes of the oscillations which constitute the F function are of different orders of magnitudes. This allows to classify them from the most important to the less important ones, with the help of a virtual book-keeping parameter λ. This is a small parameter, and the amplitude of the oscillations will be normalized by λq, q being an integer power. The largest is q, the smallest is the amplitude of the oscillations. The resolution process is iterative, and each iteration multiplies the accuracy by λ.

It is to be noted that such algorithms are usually written as formal processes, but their convergence is not guaranteed, because of potential resonances between the different involved frequencies. When two frequencies become too close to each other, the process might be destabilized. But usually, this does not happen before a reasonable order, i.e. before a reasonable number of iterations, and this is why such methods can be used. The authors provide numerical tests, which prove the robustness of their algorithm.

Potential applications

Such a study is timely, since dissipation can now be observed. For instance, the variations of the shapes of planetary bodies have been observed by measurements of variations of their gravity fields, which give the tidal Love number k2. k2 has been measured for Mercury, Venus, the Earth, the Moon, Mars, Saturn, and Titan, thanks to space missions. Moreover, its dissipative counterpart, i.e. k2/Q, has been measured for the Earth, Mars, Jupiter and Saturn. This means that conservations models for the spin-orbit problem are not sufficient anymore.

To know more

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