Tag Archives: asteroids

The system of (107) Camilla

Hi there! I will present you today a fascinating paper. It aims at a comprehensive understanding of the system composed of an asteroid, (107) Camilla, and its two satellites. For that, the authors acquired, processed and used 5 different types of observations, from all over the world. A consequence is that this paper has many authors, i.e. 27. Its title is Physical, spectral, and dynamical properties of asteroid (107) Camilla and its satellites, by Myriam Pajuelo and 26 colleagues, and it has recently been published in Icarus. This paper gives us the shape of Camilla and its main satellites, their orbits, the mass of Camilla, its composition, its spin period,… I give you these results below.

The system of Camilla

The asteroid (107) Camilla has been discovered in 1868 by Norman Pogson at Madras Observatory, India. It is located in the
outer Main-Belt, and more precisely it is a member of the Cybele group. This is a group of asteroids, named after the largest of them (65) Cybele, which is thought to have a common origin. They probably originate from the disruption of a single progenitor. I show you below some Camilla’s facts, taken from the JPL Small-Body Database Browser:

Discovery 1868
Semimajor axis 3.49 AU
Eccentricity 0.066
Perihelion 3.26 AU
Inclination 10.0°
Orbital period 6.52 yr

We have of course other data, which have been improved by the present study. Please by a little patient.

In 2001 the Hubble Space Telescope revealed a satellite of Camilla, S1, while the second satellite, S2, and has been discovered in 2016 from images acquired by the Very Large Telescope of Cerro Paranal, Chile. This makes (107) Camilla a ternary system. Interesting fact, there is at least another ternary system in the Cybele group: the one formed by (87) Sylvia, and its two satellites Romulus and Remus.

Since their discoveries, these bodies have been re-observed when possible. This resulted in a accumulation of different data, all of them having been used in this study.

5 different types of data

The authors acquired and used:

  • optical lightcurves,
  • high-angular-resolution images,
  • high-angular-resolution spectrum,
  • stellar occultations,
  • near-infrared spectroscopy.

You record optical lightcurves in measuring the variations of the solar flux, which is reflected by the object. This results in a curve exhibiting periodic variations. You can link their period to the spin period of the asteroid, and their amplitudes to its shape. I show you an example of lightcurve here.

High-angular-resolution imaging requires high-performance facilities. The authors used data from the Hubble Space Telescope (HST), and of 3 ground-based telescopes, equipped with adaptive optics: Gemini North, European Southern Observatory Very Large Telescope (VLT), and Keck. Adaptive optics permits to correct the images from atmospheric distortion, while the HST, as a space telescope, is not hampered by our atmosphere. In other words, our atmosphere bothers the accurate observations of such small objects.

A spectrum is the amplitude of the reflected Solar light, with respect to its wavelength. This permits to infer the composition of the surface of the body. The high-angular-resolution spectrum were made at the VLT, the resulting data also permitting astrometry of the smallest of the satellites, S2. These spectrum were supplemented by near-infrared spectroscopy, made with a dedicated facility, i.e. the SpeX spectrograph of the NASA InfraRed Telescope Facility (IRTF), based on Mauna Kea, Hawaii. Infrared is very sensitive to the temperature, this is why their observations require dedicated instruments, which need a dedicated cooling system.

Finally, stellar occultations consist to record the light of a star, which as some point is occulted by the asteroid you study. This is particularly interesting for a faint body, which you cannot directly observe. Such observations can be made by volunteers, who use their own telescopes. You can deduce clues on the shape, and sometimes on the presence of a satellite, from the duration of the occultation. In comparing the durations of the same occultation, recorded at different locations, you may even reconstruct the shape (actually a 2-D shape, which is projected on the celestial sphere). See here.

And from all this, you can infer the orbits of the satellites, and the composition of the primary (Camilla) and its main satellite (S1), and the spin and shape of Camilla.

The orbits of the satellites

All of these observations permit astrometry, i.e. they give you the relative location of the satellites with respect to Camilla, at given dates. From all of these observations, you fit orbits, i.e. you numerically determine the orbits, which have the smallest distances (residuals), with the data.

This is a very tough task, given the uncertainty of the recorded positions. For that, the authors used their own genetic-based algorithm, Genoid, for GENetic Orbit IDentification, which relies on a metaheuristic method to minimize the residuals. Many trajectories are challenged in this iterative approach, and only the best ones are kept. These remaining trajectories, designed as parents, are used to generate new trajectories which improve the residuals. This algorithm has proven its efficiency for other systems, like the binary asteroid (22) Kalliope-Linus. In such cases, the observations lack of accuracy and many parameters are involved.

You can find the results below.

S/2001 (107) 1
Semimajor axis 1247.8±3.8 km
Eccentricity <0.013
Inclination (16.0±2.3)°
Orbital period 3.71234±0.00004 d
S/2016 (107) 2
Semimajor axis 643.8±3.9 km
Eccentricity ~0.18 (<0.23)
Inclination (27.7±21.8)°
Orbital period 1.376±0.016 d

You can deduce the mass of (107) Camilla from these numbers, i.e. (1.12±0.01)x1019 kg. The ratio of two orbital periods probably rule out any significant mean-motion resonance between these two satellites.

Spin and shape

The authors used their homemade algorithm KOALA (Knitted Occultation, Adaptive-optics, and Lightcurve Analysis) to determine the best-fit solution (once more, minimization of the residuals) for spin period, orientation of the rotation pole, and 3-D shape model, from lightcurves, adaptive optics images, and stellar occultations. And you can find the solution below:

Camilla
Diameter 254±36 km
a 340±36 km
b 249±36 km
c 197±36 km
Spin period 4.843927±0.00004 h

This table gives two solutions for the shape: a spherical one, and an ellipsoid. In this last solution, a, b, and c are the three diameters. We can see in particular that Camilla is highly elongated. Actually a comparison between the data and this ellipsoid, named the reference ellipsoid, revealed two deep and circular basins at the surface of Camilla.

Moreover, a comparison of the relative magnitudes of Camilla and its two satellites, and the use of the diameter of Camilla as a reference, give an estimation of the diameters of the two satellites. These are 12.7±3.5 km for S1 and 4.0±1.2 km for S2. These numbers assume that S1 and S2 have the same albedo. This assumption is supported for S1 by the comparison of its spectrum from the one of Camilla.

The composition of these objects

In combining the shape of Camilla with its mass, the authors deduce its density, which is 1,280±130 kg/m3. This is slightly larger than water, while silicates should dominate the composition. As the authors point out, there might be some water ice in Camilla, but this pretty small density is probably due to the porosity of the asteroid.

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.

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.

‘Oumuamua

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

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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

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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.

Results

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

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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.

Yarkovsky

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.

YORP

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.

Validation

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.

Perspectives

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

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