Tag Archives: asteroids

9 interstellar asteroids?

Hi there! You may have recently heard of 1I/’Oumuamua, initially known as C/2017 U1, then A/2017 U1 (see here), where C stands for comet, A for asteroid, and I for interstellar object. This small body visited us last fall on a hyperbolic orbit, i.e. it came very fast from very far away, flew us by, and then left… and we shall never see it again. ‘Oumuamua has probably been formed in another planetary system, and its visit has motivated numerous studies. Some observed it to determine its shape, its composition, its rotation… and some conducted theoretical studies to understand its origin, its orbit… The study I present you today, Where the Solar system meets the solar neighbourhood: patterns in the distribution of radiants of observed hyperbolic minor bodies, by Carlos and Raúl de la Fuente Marcos, and Sverre J. Aarseth, is a theoretical one, but with a broader scope. This study examines the orbits of 339 objects on hyperbolic orbits, to try to determine their origin, in particular which of them might be true interstellar interlopers. This study has recently been accepted for publication in The Monthly Notices of the Royal Astronomical Society.


I detail the discovery of ‘Oumuamua there. Since that post, we know that ‘Oumuamua is a red dark object, probably dense. It is tumbling, i.e. does not rotate around a single rotation axis, in about 8 hours. The uncertainties on the rotation period are pretty important, because of this tumbling motion. Something really unexpected is huge variations of brightness, which should reveal either a cigar-shaped object, or an object with extreme variations of albedo, i.e. bright regions alternating with dark ones… but that would be inconsistent with the spectroscopy, revealing a reddish object. This is why the dimensions of ‘Oumuamua are estimated to be 230 × 35 × 35 meters.

Artist's impression of 'Oumuamua. © ESO/M. Kornmesser
Artist’s impression of ‘Oumuamua. © ESO/M. Kornmesser

One wonders where ‘Oumuamua comes from. An extrapolation of its orbit shows that it comes from the current direction of the star Vega, in constellation Lyra… but when it was there, the star was not there, since it moved… We cannot actually determine around which star, and when, ‘Oumuamua has been formed.

Anyway, it was a breakthrough discovery, as the first certain interstellar object, with an eccentricity of 1.2. But other bodies have eccentricities larger than 1, which make them unstable in the Solar System, i.e. gravitationally unbound to the Sun… Could some of them be interstellar interlopers? This is the question addressed by the study. If you want to understand what I mean by eccentricity, hyperbolic orbit… just read the next section.

Hyperbolic orbits

The simplest orbit you can find is a circular one: the Sun is at the center, and the planetary object moves on a circle around the Sun. In such a case, the eccentricity of the orbit is 0. Now, if you get a little more eccentric, the trajectory becomes elliptical, and you will have periodic variations of the distance between the Sun and the object. And the Sun will not be at the center of the trajectory anymore, but at a focus. The eccentricity of the Earth is 0.017, which induces a closest distance of 147 millions km, and a largest one of 152 millions km… these variations are pretty limited. However, Halley’s comet has an eccentricity of 0.97. And if you exceed 1, then the trajectory will not be an ellipse anymore, but a branch of hyperbola. In such a case, the object can just make a fly-by of the Sun, before going back to the interstellar space.

Wait, it is a little more complicated than that. In the last paragraph, I assumed that the eccentricity, and more generally the orbital elements, were constant. This is true if you have only the Sun and your object (2-body, or Kepler, problem). But you have the gravitational perturbations of planets, stars,… and the consequence is that these orbital elements vary with time. You so may have a hyperbolic orbit becoming elliptical, in which case an interstellar interloper gets trapped, or conversely a Solar System object might be ejected, its eccentricity getting larger than 1.

The authors listed three known mechanisms, likely to eject a Solar System object:

  1. Close encounter with a planet,
  2. Secular interaction with the Galactic disk (in other words, long term effects due to the cumulative interactions with the stars constituting our Milky Way),
  3. Close encounter with a star.

339 hyperbolic objects

The authors identified 339 objects, which had an eccentricity larger than 1 on 2018 January 18. The objects were identified thanks to the Jet Propulsion Laboratory’s Small-Body Database, and the Minor Planet Center database. The former is due to NASA, and the latter to the International Astronomical Union.

Once the authors got their inputs, they numerically integrated their orbits backward, over 100 kyr. These integrations were made thanks to a dedicated N-body code, powerful and optimized for long-term integration. Such algorithm is far from trivial. It consists in numerically integrating the equations of the motion of all of these 339 objects, perturbed by the Sun, the eight planets, the system Pluto-Charon, and the largest asteroids, in paying attention to the numerical errors at each iteration. This step is critical, to guarantee the validity of the results.

Some perturbed by another star

And here is the result: the authors have found that some of these objects had an elliptical orbit 100 kyr ago, meaning that they probably formed around the Sun, and are on the way to be expelled. The authors also computed the radiants of the hyperbolic objects, i.e. the direction from where they came, and they found an anisotropic distribution, i.e. there are preferred directions. Such a result has been obtained in comparing the resulting radiants from the ones given by a random process, and the distance between these 2 results is estimated to be statistically significant enough to conclude an anisotropic distribution. So, this result in not based on a pattern detected by the human eye, but on statistical calculations.

In particular, the authors noted an excess of radiants in the direction of the binary star WISE J072003.20-084651.2, also known as Scholz’s star, which is currently considered as the star having had the last closest approach to our Solar System, some 70 kilo years ago. In other words, the objects having a radiant in that direction are probably Solar System objects, and more precisely Oort cloud objects, which are being expelled because of the gravitational kick given by that star.

8 candidate interlopers

So, there is a preferred direction for the radiants, but ‘Oumuamua, which is so eccentric that it is the certain interstellar object, is an outlier in this radiant distribution, i.e. its radiant is not in the direction of Scholz’s star, and so cannot be associated with this process. Moreover, its asymptotical velocity, i.e. when far enough from the Sun, is too large to be bound to the Sun. And this happens for 8 other objects, which the authors identify as candidate interstellar interlopers. These 8 objects are

  • C/1853 RA (Brunhs),
  • C/1997 P2 (Spacewatch),
  • C/1999 U2 (SOHO),
  • C/2002 A3 (LINEAR),
  • C/2008 J4 (McNaught),
  • C/2012 C2 (Bruenjes),
  • C/2012 S1 (ISON),
  • C/2017 D3 (ATLAS).

Do we know just one, or 9 interstellar objects? Or between 1 and 9? Or more than 9? This is actually an important question, because that would constrain the number of detections to be expected in the future, and have implications for planetary formation in our Galaxy. And if these objects are interstellar ones, then we should try to investigate their physical properties (pretty difficult since they are very small and escaping, but we did it for ‘Oumuamua… maybe too late for the 8 other guys).

Anyway, more will be known in the years to come. More visitors from other systems will probably be discovered, and we will also know more on the motion of the stars passing by, thanks to the astrometric satellite Gaia. Stay tuned!

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 Pinterest.

The Brazil Nut Effect on asteroids

Hi there! You know these large nuts called Brazil nuts? Don’t worry, I will not make you think that they grow on asteroids. No they don’t. But when you put nuts in a pot, or in a glass, have you ever noticed that the biggest nuts remain at the top? That seems obvious, since we are used to that. But let us think about it… these are the heaviest nuts, and they don’t sink! WTF!!! And you have the same kind of effect on small bodies, asteroids, planetesimals, comets… I present you today a Japanese study about that, entitled Categorization of Brazil nut effect and its reverse under less-convective conditions for microgravity geology by Toshihiro Chujo, Osamu Mori, Jun’ichiro Kawaguchi, and Hajime Yano. This study has recently been published in The Monthly Notices of the Royal Astronomical Society.

Brazil Nut and Reverse Brazil Nut effects

The idea is easy to figure out. If you have a pot full of different nuts, then the smallest ones will be naturally closer to the bottom, since they are small enough to fill the voids between the largest ones. For the same reason, if you fill a bucket first with stones and then with sand, the sand will naturally reach the bottom, flowing around the stones. Flowing is important here, since the sand pretty much behaves as a fluid. And of course, if you put the sand in the bucket first, and then the stones, the stones will naturally be closer to the top. Well, this is the Brazil Nut Effect.

OK, now let us make the story go one step further… You have an empty bucket, and you put sand inside… a third of it, or a half… this results as a flat structure. You put stones, which then cover the sand, lying on its surface… and you shake. You shake the bucket, many times… what happen? the sand is moving, and makes some room for the stones, or just some of them, which migrate deeper… if you shake enough, then some of them can even reach the bottom. This is the Reverse Brazil Nut Effect.

And the funny thing is that you can find this effect on planetary bodies! Wait, we may have a problem… when the body is large enough, then the material tends to melt, the heaviest one migrating to the core. So, the body has to be small enough for its interior being ruled by the Brazil Nut Effect, or its reversed version. If the body is small enough, then we are in conditions of microgravity. The authors give the examples of the Near-Earth Asteroid (433)Eros, its largest diameter being 34.4 km, the comet 67P/Churyumov-Gerasimenko, which is ten times smaller in length, and the asteroid (25143)Itokawa, its largest length being 535 meters. All of these bodies are in conditions of microgravity, and were visited by spacecraft, i.e. NEAR Shoemaker for Eros in 2001, Rosetta for Churyumov-Gerasimenko in 2014, and Hayabusa for Itokawa in 2003. And all of these space missions have revealed pebbles and boulders at the surface, which motivated the study of planetary terrains in conditions of microgravity.

Eros seen by NEAR Shoemaker. © NASA/JPL-Caltech/JHUAPL
Eros seen by NEAR Shoemaker. © NASA/JPL-Caltech/JHUAPL

I mentioned the necessity to shake the bucket to give a chance to Reverse Brazil Nut Effect. How to shake these small bodies? With impact, of course. You have impactors everywhere in the Solar System, and small bodies do not need impactors to be large to be shaken enough. Moreover, this shaking could come from cometary activity, in case of a comet, which is true for Churyumov-Gerasimenko.

The authors studied this process both with numerical simulations, and lab experiments.

Numerical simulations

The numerical simulations were conducted with a DEM code, for Discrete Element Modeling. It consisted to simulate the motion of particle which touch each others, or touch the wall of the container. These particles are spheres, and you have interactions when contact. These interactions are modeled with a mixture of spring (elastic interaction, i.e. without dissipation of energy) and dashpot (or damper, which induces a loss of energy at each contact). These two effects are mixed together in using the so-called Voigt rheology.

In every simulation, the authors had 10,224 small particles (the sand), and a large one, named intruder, which is the stone trying to make its way through the sand.

The simulations differed by

  • the density of the intruder (light as acryl, moderately dense as glass, or heavy as high-carbon chromium steel),
  • the frequency of the shaking, modeled as a sinusoidal oscillation over 50 cycles,
  • the restitution coefficient between the sand of the intruder. If it is null, then you dissipate all the energy when contact between the intruder and the sand, and when it is equal to unity then the interaction is purely elastic, i.e. you have no energy loss.

Allowing those parameters to vary will result in different outcomes of the simulations. This way, the influence of each of those parameters is being studied.

A drawback of some simulations is the computation time, since you need to simulate the behavior of each of the particles simultaneously. This is why the authors also explored another way: lab experiments.

Lab experiments

You just put sand in a container, you put an intruder, you shake, and you observe what is going on. Well, said that way, it seems to be easy. It is actually more complicated than that if you want to make proper job.

The recipient was an acryl cylinder, put on a vibration test machine. This machine was controlled by a device, which guaranteed the accuracy of the sinusoidal shaking, i.e. its amplitude, its frequency, and the total duration of the experiment. The intruder was initially put in the middle of the sand, i.e. half way between the bottom of the recipient and the surface of the sand. If it reached the bottom before 30,000 oscillation cycles, then the conclusion was RBNE, and if it raised from the surface the conclusion was BNE. Otherwise, these two effects were considered to be somehow roughly balanced.

But wait: the goal is to model the surface of small bodies, i.e. in conditions of microgravity. The authors did the experiment on Earth, so…? There are ways to reproduce microgravity conditions, like in a parabolic flight, or on board the International Space Station, but this was not the case here. The authors worked in a lab, submitted to our terrestrial gravity. The difficulty is to draw conclusions for the asteroids from Earth-based lab experiments.

At this point, the theory assists the experimentation. If you write down the equations ensuing from the physics (I don’t do it… feel free to do so if you want), these equations ruling the DEM code for instance, you will be able to manipulate them (yes you will) so as to make them depend on dimensionless parameters. For instance: your size is in meters (or in feet). It has the physical dimension of a length. But if you divide your size with the one of your neighbor, you should get something close to unity, but this will be a dimensionless quantity, as the ratio between your size and your neighbor’s. The size of your neighbor is now your reference (let him know, I am sure he would be delighted), and if your size if larger than 1, it means that you are taller than your neighbor (are you?). In the case of our Brazil Nut experiment, the equations give you a gravity, which you can divide by the local one, i.e. either the gravity of your lab, or the microgravity of an asteroid. The result of your simulation will be expressed with respect to this ratio, which you can then re-express with respect to the microgravity of your asteroid. So, all this is a matter of scale. These scaling laws are ubiquitous in lab experiments, and they permit to work in many other contexts.

Triggering the Reverse Brazil Nut effect

And here are the results:

  • The outcomes of the experiments match the ones of the numerical simulations.
  • The authors saw practically no granular convection, i.e. the sand initially at the bottom does not migrate to the top. This is here an analogy with fluid mechanics, in which water at the bottom can raise to the top, especially when it warms (warm water is less dense than cold one).
  • Densest intruders are the likeliest to migrate to the bottom.
  • The authors identified 3 distinct behaviors for the particles, depending on a dimensionless acceleration Γ.

These behaviors are:

  1. Slow Brazil Nut Effect,
  2. Fast BNE, for which the intruder requires less oscillation cycles to raise,
  3. Fluid motion, which may induce RBNE. This is favored by rapid oscillations of the shaking.

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, and (NEW) Instagram.

Breaking an asteroid

Hi there! Asteroids, these small bodies in the Solar System, are fascinating by the diversity of their shapes. This is a consequence of their small sizes. Another consequence is their weakness, which itself helps to split some of them into different parts, sometimes creating binary objects, asteroids families… The study I present you today, Internal gravity, self-energy, and disruption of comets and asteroids, by Anthony R. Dobrovolskis and Donald G. Korycansky, proposes an accurate computation of the required energy to provoke this break-up, at any place of the asteroid, i.e. you are more efficient when you hit at a given location. This study has recently been accepted for publication in Icarus.

Shapes of asteroids

Please allow me, in this context, to call asteroid a comet, a comet being a small body, i.e. like an asteroid, but with a cometary activity. The important thing is that the involved bodies are small enough.

Beyond a given size, i.e. a diameter of ~400 km, a planetary body is roughly spheroidal, i.e. it is an ellipsoid with it two equatorial axes almost equal and the polar one smaller, because of its rotation. For a tidally despun body, like the Moon, or a satellite of a giant planet, the shape is more triaxial, since the tidal (gravitational) action of the parent planet tends to elongate the equatorial plane. The same phenomenon affects Mercury.

However, for smaller bodies, the self-gravitation is not strong enough to make the body look more or less like a sphere. As a consequence, you can have almost any shape, some bodies are bilobate, some are contact binaries, i.e. two bodies which permanently touch together, some others are rubble piles, i.e. are weak aggregates of rocks, with many voids.

These configurations make these bodies likely to undergo or have undergone break-up. This can be quantified by the required energy to extract some material from the asteroid.

The energies involved

For that, an energy budget must be performed. The relevant energies to consider are:

  • The impact disruption energy: the minimum kinetic energy of an impactor, to shatter the asteroid and remove at least half of its mass,
  • The shattering energy: the minimum energy needed to shatter the asteroid into many small pieces. It is part of the impact disruption energy. This energy is roughly proportional to the mass of the asteroid. It represents the cohesion between the adjacent pieces.
  • The binding energy: this energy binds the pieces constituting the asteroid. In other words, once you have broken an asteroid (don’t try this at home!), you have to make sure the pieces will not re-aggregate… because of the binding energy. For that, you have to bring enough energy to disperse the fragments.
  • The self-gravitational energy: due to the mutual gravitational interaction between the blocks constituting the asteroids. Bodies smaller than 1 km are strength-dominated, i.e. they exist thanks to the cohesion between the blocks, which is the shatter energy. However, larger bodies are gravity-dominated.
  • The kinetic energy of rotation: the spin of these bodies tends to enlarge the equatorial section. In that sense, it assists the break-up process.

This study addresses bodies, which are far enough from the Sun. This is the reason why I do not mention its influences, i.e. the tides and the thermic effects, which could be relevant for Near-Earth Objects. In particular, the YORP effect is responsible for the fission of some of them. I do not mention the orbital kinetic energy of the asteroid either. Actually the orbital motion is part of the input energy brought by an impact, since the relative velocity of the impactor with respect to the target is relevant in this calculation.

I now focus on the two cases studied by the authors to illustrate their theory: the asteroid Kleopatra and the comet 67P/Churyumov-Gerasimenko.

2 peculiar cases: Kleopatra and Churyumov-Gerasimenko

216 Kleopatra is a Main-Belt asteroid. Adaptive optics observations have shown that is is constituted of two masses bound by material, giving a ham-bone shaped. As such, it can be considered as a contact binary. It is probably a rubble pile. Interestingly, observations have also shown that Kleopatra has 2 small satellites, Alexhelios and Cleoselene, which were discovered in 2008.

Reconstruction of the shape of Kleopatra. © NASA
Reconstruction of the shape of Kleopatra. © NASA

However, 67P Churyumov-Gerasimenko is a Jupiter-family comet, i.e. its aphelion is close to the orbit of Jupiter, while its perihelion is close to the one of the Earth. It has an orbital period of 6.45 years, and was the target of the Rosetta mission, which consisted of an orbiter and a lander, Philae. Rosetta orbited Churyumov-Gerasimenko between 2014 and 2016. The shape of this comet is sometimes described as rubber ducky, with two dominant masses, a torso and a head, bound together by some material, i.e. a neck.

Churyumov-Gerasimenko seen by Rosetta. © ESA
Churyumov-Gerasimenko seen by Rosetta. © ESA
216 Kleopatra 67P/Churyumov-Gerasimenko
Semimajor axis 2.794 AU 3.465 AU
Eccentricity 0.251 0.641
Inclination 13.11° 7.04°
Spin period 5.385 h 12.761 h
Mean radius 62 km 2.2 km
Magnitude 7.30 11.30
Discovery 1880 1969

The irregular shapes of these two bodies make them interesting targets for a study addressing the gravitation of any object. Let us see now how the authors addressed the problem.

Numerical modeling

Several models exist in the literature to address the gravity field of planetary bodies. The first approximation is to consider them as spheres, then you can refine in seeing them as triaxial ellipsoids. For highly irregular bodies you can try to model them as cuboids, and then as polyhedrons. Another way is to see them as duplexes, this allows to consider the inhomogeneities dues to the two masses constituting bilobate objects. The existence of previous studies allow a validation of the model proposed by the authors.

And their model is a finite-element numerical modeling. The idea is to split the surface of the asteroid into small triangular planar facets, which should be very close to the actual surface. The model is all the more accurate with many small facets, but this has the drawback of a longer computation time. The facets delimit the volume over which the equations are integrated, these equations giving the local self-gravitational and the impact disruption energies. The authors also introduce the energy rebate, which is a residual energy, due to the fact that you can remove material without removing half of it. This means that the impact disruption energy, as it is defined in the literature, is probably a too strong condition to have extrusion of material.
The useful physical quantities, which are the gravitational potential, the attraction, and the surface slope, are propagated all along the body thanks to a numerical scheme, which accuracy is characterized by an order. This order quantifies the numerical approximation which is made at each integration step. A higher order is more accurate, but is computationally more expensive.

Once the code has been run on test cases, the authors applied it on Kleopatra and Churyumov-Gerasimenko, for which the shape is pretty well known. They used meshes of 4,094 and 5,786 faces, respectively.


The validation phase is successful. The authors show that with a 3rd order numerical scheme, they recover the results present in the literature for the test cases with an accuracy of ~0.1%, which is much better than the accuracy of the shape models for the real asteroids. Regarding Kleopatra and Churyumov-Gerasimenko, they get the gravity field at any location, showing in particular excesses of gravity at the two lobes.

Such a study is particularly interesting for further missions, which would determine the gravity field of asteroids, which would then be compared with the theoretical determination by this code. Other applications are envisaged, the authors mentioning asteroid mining.

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.

And Merry Christmas!

Rough terrains spin up asteroids

Hi there! If you follow me, you have already heard of the Yarkovsky effect, or even of the YORP, which are non-gravitational forces affecting the dynamics of Near-Earth Asteroids. Today I tell you about the TYORP, i.e. the Tangential YORP. This is the opportunity for me to present you Analytic model for Tangential YORP, by Oleksiy Golubov. This study has recently been published in The Astronomical Journal. The author meets the challenge to derive an analytical formula for the thermal pressure acting on the irregular regolith of an asteroid. Doing it requires to master the physics and make some sound approximations, following him tells us many things on the Tangential YORP.

From Yarkovsky to TYORP

When we address the dynamics of Near-Earth Asteroids, we must consider the proximity of the Sun. This proximity involves thermal effects, which significantly affect the dynamics of such small bodies. In other words, the dynamics is not ruled by the gravitation only. The main effect is the Yarkovsky effects, and its derivatives.


The Sun heats the surface of the asteroid which faces it. When this surface element does not face the Sun anymore, because of the rotation of the asteroid, it cools, and radiates some energy. This effect translates into a secular drift in the orbit, which is known as the Yarkovsky effect. This Yarkovsky effect has been directly measured for some asteroids, in comparing the simulated orbit from a purely gravitational simulation, with the astrometric observations of the objects. Moreover, long-term studies have shown that the Yarkovsky effect explains the spreading of some dynamical families, i.e. asteroids originating from a single progenitor. In that sense, observing their current locations proves the reality of the Yarkovsky effect.
When the asteroid has an irregular shape, which is common, the thermal effect affects the rotation as well.


Cooling a surface element which has been previously heated by the Sun involves a loss of energy, which depends on the surface itself. This loss of energy affects the rotational dynamics, which is also affected by the heating of some surface. But for an irregular shaped body, the loss and gain of energy does not exactly balance, and the result is an asteroid which spins up, like a windmill. In some cases, it can even fission the body (see here). This effect is called YORP, for Yarkovsky-O’Keefe–Radzievskii–Paddack.

This is a large-scale effect, in the sense that it depends on the shape of the asteroid as a whole. Actually, the surface of an asteroid is regolith, it can have boulders… i.e. high-frequency irregularities, which thus will be heated differently, and contribute to YORP… This contribution is known as Tangential YORP, or TYORP.

Modeling the physics

When you heat a boulder from the Sun, you create an inhomogeneous elevation of temperature, which can be modeled numerically, with finite elements. For an analytical treatment, you cannot be that accurate. This drove the author to split the boulder into two sides, the eastern and the western sides, both being assumed to have an homogeneous temperature. Hence, two temperatures for the boulder. Then the author wrote down a heat conduction equation, which says that the total heat energy increase in a given volume is equal to the sum of the heat conduction into this volume, the direct solar heat absorbed by its open surface, and the negative heat emitted by the open surface (which radiates).

These numbers depend on

  • the heat capacity of the asteroid,
  • its density,
  • its heat conductivity,
  • its albedo, i.e. its capacity to reflect the incident Solar light,
  • its emissivity, which characterizes the radiated energy,
  • the incident Solar light,
  • the time.

The time is critical since a surface will heat as long it is exposed to the Sun. In the calculations, it involves the spin frequency. After manipulation of these equations, the author obtains an analytical formula for the TYORP pressure, which depends on these parameters.

A perturbative treatment

In the process of solving the equations, the author wrote the eastern and western temperatures as sums of periodic sinusoidal solutions. The basic assumption, which seems to make sense, is that these two quantities are periodic, the period being the rotation period, P, of the asteroid. This implicitly assumes that the asteroid rotates around only one axis, which is a reasonable assumption for a general treatment of the problem.
As a result, the author expects these two temperatures to be the sum of sines and cosines of periods P/n, P being an integer. For n=1, you have a variation of period P, i.e. a diurnal variation. For n = 2, you have a semi-diurnal one, etc.

The perturbative treatment of the problem consists in improving the solution in iterating it, first in expressing only one term, i.e. the diurnal one, then in using the result to derive the second term, etc. This assumes that these different terms have amplitudes, which efficiently converge to 0, i.e. the semi-diurnal effect is supposed to be negligible with respect to the diurnal one, but very large with respect to the third-diurnal, etc. Writing down the solution under such a form is called Fourier decomposition.

The author says honestly that he did not check this convergence while solving the equation. However, he successfully tested the validity of his obtained solution, which means that the resolution method is appropriate.


The author is active since many years on the (T)YORP issue, and has modeled it numerically in a recent past. So, validating his analytical formula consisted in confronting it with his numerical results.

He particularly confronted the two results in the cases of a wall, a half buried spherical boulder, and a wave in the regolith, with respect to physical characteristics of the material, i.e. dimension and thermic properties. Even though visible differences, the approximation is pretty good, validating the methodology.

This allowed then the author to derive an analytical formula of the TYORP pressure on a while regolith, which is composed of boulders, which sizes are distributed following a power law.


This is the first analytical formula for the TYORP, and I am impressed by the author’s achievement. We can expect in the future that this law (should we call it the Golubov law?) would be a reference to characterize the thermic properties of an asteroid. In other words, future measurements of the TYORP effect could give the thermic properties, thanks to this law. This is just a possibility, which depends on the reception of this study by the scientific community, and on future studies as well.

The study and its author

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.

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

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