All posts by Terryl Coron

Triton: a cuckoo around Neptune

Hi there! Did you know that Neptune had a prominent satellite, i.e. Triton, on a retrograde orbit? This is so unusual that it is thought to have been trapped, i.e. was originally an asteroid, and has not been formed in the protoneptunian nebula. The study I present you today, Triton’s evolution with a primordial Neptunian satellite system, by Raluca Rufu and Robin M. Canup, explains how Triton ejected the primordial satellites of Neptune. This study has recently been published in The Astronomical Journal.

The satellites of Uranus and Neptune

We are tempted to see the two planets Uranus and Neptune as kinds of twins. They are pretty similar in size, are the two outermost known planets in the Solar System, and are gaseous. A favorable orbital configuration made their visitation possible by the spacecraft Voyager 2 in 1986 and 1989, respectively.

Among their differences are the high obliquity of Uranus, the presence of rings around Uranus while Neptune displays arcs, and different configurations in their system of satellites. See for Uranus:

Semimajor axis Eccentricity Inclination Radius
Miranda 5.12 Ru 0.001 4.338° 235.8 km
Ariel 7.53 Ru 0.001 0.041° 578.9 km
Umbriel 10.49 Ru 0.004 0.128° 584.7 km
Titania 17.20 Ru 0.001 0.079° 788.9 km
Oberon 23.01 Ru 0.001 0.068° 761.4 km
Puck 3.39 Ru 0 0.319° 81 km
Sycorax 480.22 Ru 0.522 159.420° 75 km

I show on this table the main satellites of Uranus, and we can see that the major ones are at a reasonable distance (in Uranian radius Ru) of the planet, and orbit almost in the same plane. The orbit of Miranda is tilted thanks to a past mean-motion resonance with Umbriel, which means that it was originally in the same plane. So, we can infer that these satellites were formed classically, i.e. the same way as the satellites of Jupiter, from a protoplanetary nebula, in which the planet and the satellites accreted. An exception is Sycorax, which is very far, highly inclined, and highly eccentric. As an irregular satellite, it has probably been formed somewhere else, as an asteroid, and been trapped by the gravitational attraction of Uranus.

Now let us have a look at the system of Neptune:

Semimajor axis Eccentricity Inclination Radius
Triton 14.41 Rn 0 156.865° 1353.4 km
Nereid 223.94 Rn 0.751 7.090° 170 km
Proteus 4.78 Rn 0 0.075° 210 km
Larissa 2.99 Rn 0.001 0.205° 97 km

Yes, the main satellite seems to be an irregular one! It does not orbit that far, its orbit is (almost) circular, but its inclination is definitely inconsistent with an in situ formation, i.e. it has been trapped, which has been confirmed by several studies. Nereid is much further, but with a so eccentric orbit that it regularly enters the zone, which is dynamically perturbed by Triton. You can also notice the absence of known satellites between 4.78 and 14.41 Neptunian radii. This suggests that this zone may have been cleared by the gravitational perturbation of a massive body… which is Triton. The study I present you simulates what could have happened.

A focus on Triton

Before that, let us look at Triton. The system of Neptune has been visited by the spacecraft Voyager 2 in August 1989, which mapped 40% of the surface of Triton. Surprisingly, it showed a limited number of impact craters, which means that the surface has been renewed, maybe some 1 hundred of millions of years ago. Renewing the surface requires an activity, probably cryovolcanism as on the satellite of Saturn Enceladus, which should has been sustained by heating. Triton was on the action of the tides raised by Neptune, but probably not only, since tides are not considered as efficient enough to have circularized the orbit. The tides have probably been supplemented by gravitational interactions with the primordial system of Neptune, i.e. satellites and / or disk debris. If there had been collisions, then they would have themselves triggered heating. As a consequence of this heating, we can expect a differentiated structure.

Moreover, Triton orbits around Neptune in 5.877 days, on a retrograde orbit, while the rotation of Neptune is prograde. This configuration, associated with the tidal interaction between Triton and Neptune, makes Triton spiral very slowly inward. In other words, it will one day be so close to Neptune that it will be destroyed, and probably create a ring. But we will not witness it.

A numerical study with SyMBA

This study is essentially numerical. It aimed at modeling the orbital evolution of Triton, in the presence of Nereid and the putative primordial satellites of Neptune. The authors assumed that there were 4 primordial satellites, with different initial conditions, and considered 3 total masses for them: 0.3, 1, and 3 total masses of the satellites of Uranus. For each of these 3 masses, they ran 200 numerical simulations.

The simulations were conducted with the integrator (numerical code) SyMBA, i.e. Symplectic Massive Body Algorithm. The word symplectic refers to a mathematical property of the equations as they are written, which guarantee a robustness of the results over very long timescales, i.e. there may be an error, but which does not diverge. It may be not convenient if you make short-term accurate simulations, for instance if you want to design the trajectory of a spacecraft, but it is the right tool for simulating a system over hundreds of Myrs (millions of years). This code also handles close encounters, but not the consequences of impacts. The authors bypassed this problem in treating the impacts separately, determining if there were disrupting, and in that case estimated the timescales of reaccretion.


The authors found, consistently with previous studies, that the interaction between Triton and the primordial system could explain its presently circular orbit, i.e. it damped the eccentricity more efficiently than the tides. Moreover, the interaction with Triton caused collisions between the primordial moons, but usually without disruption (hit-and-run impacts). In case of disruption, the authors argue that the reaccretion would be fast with respect of the time evolution of the orbit of Triton, which means that we could lay aside the existence of a debris disk.

Moreover, they found that the total mass of the primordial system had a critical role: for the heaviest one, i.e. 3 masses of the Uranian system, Triton had only small chances to survive, while it had reasonable chances in the other two cases.

Something frustrating when you try to simulate something that happened a few hundreds of Myrs ago is that you can at the best be probabilistic. The study shows that a light primordial system is likelier to have existed than a heavy one, but there are simulations with a heavy system, in which Triton survives. So, a heavy system is not prohibited.

The study and its authors

  • The study, which is available as free article. The authors probably paid extra fees for that, many thanks to them! You can also look at it on arXiv.
  • A conference paper on the same study,
  • The ResearchGate profile of Raluca Rufu,
  • The Homepage of Robin M. Canup.

Before closing this post, I need to mention that the title has been borrowed from Matija Ćuk (SETI, Mountain View, CA), who works on this problem as well (see these two conference abstracts here and here).

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.

Rotation and activity of a comet

Hi there! We, Earthians, are regularly visited by periodic comets, the most famous one being probably 1P/Halley, which will visit us in 2061. Since we cannot wait, we study others of that kind. Today I tell you about 49P / Arend-Rigaux. This is the opportunity for me to present you The rotation and other properties of Comet 49P/Arend-Rigaux, 1984 – 2012, by Nora Eisner, Matthew M. Knight and David G. Schleicher. This study has recently been published in The Astronomical Journal.

The comet 49P / Arend-Rigaux

The comet 49P / Arend-Rigaux has been discovered in February 1951 at the Royal Observatory of Belgium, by Sylvain Arend and Fernand Rigaux. It is a periodic comet of the Jupiter family, i.e. with a period smaller than 20 years. Its period is actually 6.71 years, its semimajor axis 3.55 AU (astronomical units, 1 AU being 150 millions km, i.e. the Sun-Earth distance), its eccentricity 0.6, and its orbital inclination 19°, with respect to the ecliptic. These numbers are extracted from the JPL Small-Body Database Browser, and are calculated at the date Apr 6, 2010. I have plotted below the distances Sun-comet and Earth-comet.

Distance to the Sun.
Distance to the Sun.
Distance to the Earth.
Distance to the Earth.

The distance to the Sun clearly shows the periodic variations. The orbit of the Earth is at 1 AU, the one of Mars at 1.5 AU, and the one of Jupiter at 5.2 AU. Every 6.71 years, the comet reaches its perihelion, i.e. minimizes its distance to the Sun. This proximity warms the comet and provokes an excess of cometary activity, i.e. sublimation of dirty ice. At these occasions, the distance with the Earth is minimized, but with variations due to the orbital motion of the Earth. We can see for instance that the comet gets pretty close to the Earth in 1951 (when it was discovered), in 1984, and in early 2032. These are favorable moments to observe it. The paper I present you today is mainly (but not only) based on photometric observations made between January and May 2012, at Lowell Observatory.

Observations at Lowell Observatory

Lowell Observatory is located close to Flagstaff, AZ (USA). It was founded by the famous Percival Lowell in 1894, and is the place where Clyde Tombaugh discovered Pluto, in 1930. Among its facilities is the 4.28 m Discovery Channel Telescope, but most of the data used in this study were acquired with the 1.1 m Hall telescope, which is devoted to the study of comets, asteroids, and Sun-like stars. The authors also used a 79 cm telescope. The observations were made in the R(ed) band.

The data

Besides these 33 observation nights during the first half of 2012, the authors used data acquired close to the 1984 and 2005 perihelion passages, even if the 2005 ones revealed unusable. The observations consists to measure the magnitude (somehow, the luminosity) of the comet, in correcting for atmospheric problems, so as to be able to detect the variations of this magnitude. You can find below an example of data:

Magnitude of 49P / Arend-Rigaux measured in April 2012.
Magnitude of 49P / Arend-Rigaux measured in April 2012.

Of course, the data have holes, since you cannot observe during the day. Moreover, the comet needs to be visible from Arizona, otherwise it was just impossible to observe it and make any measurements.

We can see a kind of periodicity in the magnitude, this is a signature of the rotation of the comet.

Measuring the rotation

Most of the planetary bodies are kinds of triaxial ellipsoids. Imagine we are in the equatorial plane of one of them. We see an alternation of the long and short axes of its equatorial section. If the albedo of the surface element we face depends mainly on its curvature (it depends on it, but mainly may be an overstatement), then we should see two peaks during a period. As a consequence, the period of the lightcurve we observe should be half the rotation period of the comet.

In combining all the measurements, the authors managed to derive a rotation period of 13.45 ± 0.01 hour. For that, they used two different algorithms, which gave very close results, giving the authors confidence in their conclusions. The first one, Phase Dispersion Minimization (PDM), consists to assume a given period, split the measurements into time intervals of this period, and overlap them. The resulting period gives to the best overlap. The other algorithm is named Lomb-Scargle, following its authors. It is a kind of Discrete Fourier Transform, but with the advantage of not requiring uniformly sampled data.

In addition to this rotation period, the authors detected an increasing trend in the 2012 data, as if the spin of the comet accelerated. This is in agreement with an alteration of the measured rotation from the Earth, which moves, and reveals a retrograde rotation, i.e. an obliquity close to 180°. In other words, this is an illusion due to the motion of the observer, but this illusion reveals the obliquity.

Moreover, in comparing the 2012 data with the ones of 1984, the authors managed to detect a variation in the rotation period, not larger than 54 seconds. This is possible regarding the fact that the comet is altered by each perihelion passage, since it outgasses. In this case, that would imply a change of at the most 14 seconds of the rotation period between two passages. Such variations have also been detected for at least 4 other comets (2P/Encke, 9P/Tempel 1, 10P/Tempel 2, and 103P/Hartley 2, see Samarinha and Mueller (2013)).

Comet Period (h) Variation (s)
2P/Encke 11 240
9P/Tempel 1 41 -840
10P/Tempel 2 9 16.2
103P/Hartley 2 18 7200
49P/Arend-Rigaux 13.45 -(>14)

Finally, since the lightcurve is a signature of the shape as well, the authors deduced from the amplitude of variation that the axial ratio of the nucleus, i.e. long axis / short axis, should be between 1.38 and 1.63, while an independent, previous study found 1.6.

Cometary activity

49P / Arend-Rigaux has a low activity. Anyway, the authors detected an event of impulse-type outburst, which lasted less than 2 hours. The analysis of the coma revealed an excess of cyanides with respect to the 1984 passage. Moreover, 49P / Arend-Rigaux is the first comet to show hydroxyde.

The study and its 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.

An interstellar asteroid

Hi there! You may have heard this week of our Solar System visited by an asteroid probably formed in another planetary system. This is why I have decided to speak about it, so this article will not be based on a peer-reviewed scientific publication, but on good science anyway. The name of this visitor is for now A/2017 U1.

History of the discovery

Discovering a new object usually consists in

  1. Taking a picture of a part of a sky. Usually these are parts of the order of the degree, maybe much less… so, small parts. And this also requires to treat the image, to correct for atmospheric (brightness of the sky, wind,…) and instrumental (dead pixels…) effects,
  2. Comparing in with the objects, which are known to be in that field.

If there is an unexpected object, then it could be a discovery. Here is the history of the discovery of A/2017 U1:

  • Oct. 19, 2017: Robert Weryk, a researcher of the University of Hawaii, discovers a new object while searching for Near-Earth Asteroids with the Pan-STARRS 1 telescope. An examination of images archives revealed that the object had already been photographed the day before.
  • Oct. 25, 2017: The Minor Planet Center (Circular MPEC 2017-U181) gives orbital elements for this new object, from 34 observations over 6 days, from Oct. 18 to 24. Surprisingly, an eccentricity bigger than 1 (1.1897018) is announced, which means that the trajectory follows a hyperbola. This means that if this object would be affected only by the Sun, then it would come from an infinite distance, and would leave us for infinity. In other words, this object would not be fated to remain in our Solar System. That day, the object was thought to be a comet, and named C/2017 U1. 10 observation sites were involved (once an object has been detected and located, it is easier to re-observe it, even with a smaller telescope).
  • Oct. 26, 2017: Update by the Minor Planet Center (Circular MPEC 2017-U185), using 47 observations from Oct. 14 on. The object is renamed A/2017 U1, i.e. from comet “C” to asteroid “A”, since no cometary activity has been detected. Same day: the press release announcing the first confirmed discovery of an interstellar object. New estimation of the eccentricity: e = 1.1937160.
  • Oct. 27, 2017: Update by the Minor Planet Center (Circular MPEC 2017-U234), using 68 observations. New estimation of the eccentricity: e = 1.1978499.

And this is our object! It has an absolute magnitude of 22.2 and a diameter probably smaller than 400 meters. These days, spectroscopic observations have revealed a red object, alike the KBOs (Kuiper Belt Objects). It approached our Earth as close as 15 millions km (0.1 astronomical unit), i.e. one tenth of the Sun-Earth distance.

The trajectory of A/2017 U1.
The trajectory of A/2017 U1.

What are these objects?

The existence of such objects is predicted since more than 40 years, in particular by Fred Whipple and Viktor Safronov. This is how they come to us:

  1. A protoplanetary disk creates a star, planets, and small objects,
  2. The small objects are very sensitive to the gravitational perturbations of the planets. As a consequence, they may be ejected from their planetary system, and become interstellar objects,
  3. They visit us.

Calculations indicate that A/2017 U1 comes roughly from the constellation Lyra, in which the star Vega is (only…) at 25 lightyears from our Sun. It is tempting to assume that A/2017 U1 was formed around Vega, but that would be only speculation, since many perturbations could have altered its trajectory. Several studies will undoubtedly address this problem within next year.

Maybe not the first one

Here we have an eccentricity, which is significantly larger (some 20%) than 1. Moreover, our object has a very inclined orbit, which means that we can neglect the perturbations of its orbit by the giant planets. In other words, it entered the Solar System on the trajectory we see now. But a Solar System object can get a hyperbolic orbit, and eventually be ejected. This means that when we detect an object with a very high eccentricity, like a long-period comet, it does not necessary mean that it is an interstellar object. In the past, some known objects have been proposed to be possible interstellar ones. This is for example the case for the comet C/2007 W1 (Boattini), which eccentricity is estimated to be 1.000191841611794±0.000041198 at the date May 26, 2008. It could be an IC (Interstellar Comet), but could also be an Oort cloud object, put on a hyperbolic orbit by the giant planets.

Detecting interstellar objects

A/2017 U1 object has been detected by the Pan-STARRS (for Panoramic Survey Telescope and Rapid Response System) 1 telescope, which is located at Haleakala Observatory, Hawaii. Pan-STARRS is constituted of two 1.8 m Ritchey–Chrétien telescopes, with a field-of-view of 3°. This is very large compared with classical instruments, and it is suitable for detection of bodies. It operates since 2010.

Detections could be expected from the future Large Synoptic Survey Telescope (LSST), which should operate from 2022 on. This facility will be a 8.4-meter telescope based in Chile, and will conduct surveys with a field-of-view of 3.5°. A recent study by Nathaniel Cook et al. suggests that LSST could detect between 0.001 and 10 interstellar comets during its nominal 10 year lifetime. Of course, 0.001 detection should be understood as the result of a formula. The authors give a range of 4 orders of magnitude in their estimation, which reflects how barely constrained the theoretical models are. This also means that we could be just lucky to have detected one.

What Pan-STARRS can do, LSST should be able to do. In a few years, i.e. in the late 2020s, the number or absence of new discoveries will tell us something on the efficiency of creation of interstellar objects in the nearby stars. Meanwhile, let us enjoy this exciting discovery!

The press release and its 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.

The composition of Himalia, Elara, and Carme

Hi there! Today I tell you on 3 irregular satellites of Jupiter, you know, these small bodies which orbit very far from the planet. Himalia, Elara and Carme have been observed in the Near-InfraRed (NIR), and this gave Composition of Jupiter irregular satellites sheds light on their origin, by M. Bhatt et al., which has been recently accepted for publication in Astronomy and Astrophysics.

The irregular satellites of Jupiter

Jupiter has 69 known satellites, which we can divide into 3 groups:

  1. The 4 Galilean satellites Io, Europa, Ganymede and Callisto. These are large bodies, discovered in 1610 by Galileo Galilei,
  2. The 4 inner satellites Amalthea, Metis, Adrastea, and Thebe. These are small bodies, orbiting inside the orbit of Io,
  3. The irregular satellites, which orbit very far from Jupiter. These are small bodies as well, which are usually thought to have been captured, i.e. they probably not formed in the protojovian nebula.

Contrary to the inner and the Galilean satellites, the irregular satellites have pretty eccentric and inclined orbits. Their eccentricities may exceed 0.4, and most of them are retrograde, i.e. with an inclination larger than 90°. In fact, plotting their inclination vs. their semimajor axes reveals clustering.

Semimajor axes and inclinations of the irregular satellites of Jupiter. The inclinations are given with respect to the ecliptic.
Semimajor axes and inclinations of the irregular satellites of Jupiter. The inclinations are given with respect to the ecliptic.

At least 4 dynamical groups have been defined, all of them being named after the largest of their members:

  1. The Himalia group is made of prograde bodies, with inclinations between 26.6° and 28.3°, eccentricities between 0.11 and 0.25, and semimajor axes between 159 and 176 Jupiter radii (while Callisto orbits at 27 Jupiter radii),
  2. The Ananke group is composed of bodies with inclinations between 145.7° and 154.8°, eccentricities between 0.02 and 0.28, and semimajor axes between 250 and 305 Jupiter radii,
  3. The Pasiphase group is made of bodies with inclinations between 144.5° and 158.3°, eccentricities between 0.25 and 0.43, and semimajor axes between 320 and 350 Jupiter radii,
  4. The Carme group is made of bodies with inclinations between 164.9° and 165.5°, eccentricities between 0.23 and 0.27, and semimajor axes between 329 and 338 Jupiter radii

The clustering among these bodies suggests a common origin, i.e. a group of objects would have a unique progenitor. It is also interesting to notice that some groups are more dispersed than others. In particular, the dispersion of the Carme group is very limited. This could tell us something on the date of the disruption of the progenitor. Another clue regarding a common origin is the composition of these bodies.

Before addressing our 3 objects of interest, i.e. Himalia, Elara (member of the Himalia group), and Carme, I would like to mention Themisto and Carpo, which seem to be pretty isolated, and so would not share a common origin with the other bodies. Their dynamics might be affected by the Kozai-Lidov mechanism, which induces a correlated periodic evolution of their eccentrities and inclinations.

Himalia, Elara, and Carme

These 3 bodies are the ones addressed in this study. You can find below their relevant characteristics.

Semimajor axis Eccentricity Inclination Discovery Radius Albedo
Himalia 163.9 Rj 0.16 27.50° 1904 70-80 km 0.04
Elara 167.9 Rj 0.22 26.63° 1905 43 km 0.04
Carme 334.7 Rj 0.25 164.91° 1938 23 km 0.04

These were among the first known irregular moons of Jupiter. The inclinations are given with respect to the ecliptic, i.e. the orbital plane of the Earth. As a member of the Himalia group, Elara has similar dynamical properties with Himalia. We can also notice the small albedo of these bodies, i.e. of the order of 4%, which means that only 4% of the incident Solar light is reflected by the surface! In other words, these bodies are very dark, which itself suggests a carbonaceous composition. Spectroscopic observations permit to be more accurate.

Spectroscopic observations

These bodies were observed in the near infrared, at wavelengths between 0.8 and 5.5 μm. The observations were made at the IRTF (InfraRed Telescope Facility), located on the Mauna Kea (Hawai’i), with the SpeX spectrograph, during 4 nights, in 2012 and 2013. In measuring the light flux over a specific range of the spectrum, one can infer the presence of some material, which would absorb the light at a given wavelength. For that, we need to be accurate in the measurements, while the atmospheric conditions might alter them. This difficulty is by-passed by the presence of a star in the field, which serves as a reference for the measured light flux.

Detection of minerals

Once a spectrum reflectance vs. wavelength is obtained, it needs to be interpreted. In this study, the authors assumed that the observed spectra were a mixture of the spectra given by different minerals, which have been obtained in laboratories. They disposed of a database of 30 minerals, and fitted mixtures involving 4 of them, to the obtained spectra. This is an optimization algorithm, here named Spectral Mixture Analysis, which fits the relative proportion of the minerals. 4 minerals is actually the best they could obtain, i.e. they failed to produce a significantly better fit in adding a 5th mineral.

In other words, from the absorption spectrum of such a body, you can guess its 4 main components… at least of the surface.

Himalia and Elara are alike, Carme is different

Well, the title contains the conclusion. This is not very surprising, since Himalia and Elara belong to the same group. We can say that the composition confirms the guess that they should have a common origin. Previous studies gave the same conclusions.

In this specific case, Himalia and Elara have a peak of absorption centered around 1.2 μm, and their spectra are similar to C-type, i.e. carbonaceous, asteroids (52) Europa and (24) Themis, of the outer asteroid belt. The best match for Himalia is obtained with a mixture of magnetite and ilmenite, both being iron oxides, with minnesotaite, which is a ferric phyllosilicate. Elara seems to have a similar composition, but the match is not that good. In particular, the spectrum is more dispersed than for Himalia, and a little redder.

Carme has a different spectrum, with a peak of absorption centered around 1.6 μm, and is probably composed of black carbon, minnesotaite, and ilmenite. Another study has proposed that Carme could have a low-level cometary activity, but that would require to observe it at shorter wavelengths. Out of the scope of this study.

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.

Indirect measurement of an asteroid’s pole

Hi there! Today, another paper on the Yarkovsky effect. You know, this non-gravitational force which acts on the asteroid, especially if it is close enough to the Sun. After reading this post, you will know how it can reveal us the obliquity of an asteroid. I present you Constraints on the near-Earth asteroid obliquity distribution from the Yarkovsky effect, by C. Tardioli, D. Farnocchia, B. Rozitis, D. Cotto-Figueiroa, S.R. Chesley, T.S. Statler & M. Vasile. This paper has recently been accepted for publication in Astronomy and Astronomy.

The way it works

Imagine you want to know the rotation of an asteroid… but you cannot measure it directly. However, you can measure the orbital motion of the asteroid, with enough accuracy to detect an effect (here Yarkovsky), which itself depends on the rotation… measuring Yarkovsky is measuring the rotation! Easy, isn’t it?

The rotation of an asteroid

As any planetary body, an asteroid has a rotational motion, which consists in spinning around one axis (actually 3, but you can safely neglect this fact), at a given rate. We can consider that we know its rotation when

  1. We know its spin rate, or its rotational period (let us assume it is constant),
  2. We know the orientation of its spin pole. We will call it the obliquity.

Usually the asteroids spin in a few hours, which is very fast since they need at least several months to complete one revolution around the Sun. The obliquity is between 0° and 180°. 0° means that the spin axis is orthogonal to the orbital plane, and that the rotation is prograde. However, 180° is the other extreme case, the spin axis is orthogonal, but with a retrograde rotation.

A direct measurement of these two quantities would consist in following the surface of the asteroid, to observe the rotation. Usually we cannot observe the surface, but sometimes we can measure the variations of the magnitude of the asteroid over time. This is directly due to the Solar light flux, which is reflected by the surface of the asteroid. Because the topography is irregular, the rotation of the asteroid induces variations of this reflection, and by analyzing the resulting lightcurve we can retrieve the rotational quantities.

Very well, but sometimes the photometric observations are not accurate enough to get these quantities. And other times, the measured rotational quantities present an ambiguity, i.e. 2 solutions, which would need an independent measurement to discriminate them, i.e. determine which of the two possible results is the right one.

It appears that the Yarkovsky effect, which is an alteration of the orbital motion of the body due to the inhomogeneity of its temperature, itself due to the Solar incident flux and the orientation of the body, i.e. its rotation, can sometimes be measured. When you know Yarkovsky, you know the obliquity. Well, it is a little more complicated than that.

Yarkovsky: A thermal effect

Since I have already presented you Yarkovsky with words, I give you now a formula.

The Yarkovsky effect, i.e. the thermal heating of the asteroid, induced a non-gravitational acceleration of its orbital motion. This acceleration reads A2/r2, where r is the distance to the Sun (remember that the asteroid orbits the Sun), and

A2 = 4/9(1-A)Φ(αf(θs)cos(ε)-f(θo)sin2(ε)),


  • A: albedo of the asteroid, i.e. quantity of the reflected light wrt the incident one,
  • Φ: Solar radiation,
  • α: an enhancement factor. This is a parameter…
  • ε: the obliquity (which the authors determined),
  • θs / θo: thermal parameters which depend on the spin period (s), and the orbital one (o), respectively.

If you know Yarkovsky, you know A2, since you know the distance r (you actually know where the asteroid is). If you know all the parameters except ε, then A2 gives you ε. In fact, some of the other parameters need to be estimated.

Measuring Yarkovsky

As you can see, this study is possible only for asteroids, for which you can know the Yarkovsky acceleration. Since it is a thermal effect, you can do it only for Near-Earth Asteroids, which are closer to the Sun than the Main Belt. And to measure Yarkovsky, you must simulate the orbital motion of the asteroid, which is perturbed by the main planets and Yarkovsky, with the Yarkovsky acceleration as a free parameter. A fit of the simulations to the actual astrometric observations of the asteroid gives you a number for the Yarkovsky acceleration, with a numerical uncertainty. If your number is larger than the uncertainty, then you have detected Yarkovsky. And this uncertainty mainly depends on the accuracy of your astrometric observations. It could also depend on the validity of the dynamical model, i.e. on the consideration of the forces perturbing the orbital motion, but usually the dynamical model is very accurate, since the masses and motions of the disturbing planets are very well known.
The first detection of the Yarkovsky acceleration was in 2003, when a drift of 15 km over 12 years was announced for the asteroid 6489 Golevka.

So, you have now a list of asteroids, with their Yarkovsky accelerations. The authors worked with a final dataset of 125 asteroids.

So many retrograde asteroids

The authors tried to fit a distribution of the obliquities of these asteroids. The best fit, i.e. which reduces the distance between the resulting obliquities and the Yarkovsky acceleration that they would have produced, is obtained from a quadratic model, i.e. 1.12 cos2(ε)-0.32 cos(ε)+0.13, which is represented below.

Distribution of the asteroids with respect to their obliquity.
Distribution of the asteroids with respect to their obliquity.

What you see is the number of asteroids with respect to their obliquity. The 2 maxima at 0° and 180° mean that most of the asteroids spin about an axis, which is almost orthogonal to their orbital plane. From their relative heights, it appears that there about twice more retrograde asteroids than prograde ones. This is consistent with previous studies, these obliquities actually being a consequence of the YORP effect, which is the influence of Yarkovsky on the rotation.

The study and its 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.