# 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.

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.

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

And Merry Christmas!

# The chemistry of Pluto

Hi there! The famous dwarf planet Pluto is better known to us since the flyby of the spacecraft New Horizons in 2015. Today, I tell you about its chemistry. I present you Solid-phase equilibria on Pluto’s surface, by Sugata P. Tan & Jeffrey S. Kargel, which has recently been accepted for publication in The Monthly Notices of the Royal Astronomical Society.

## The atmosphere of Pluto

I do not want here to recall everything about Pluto. This is a dwarf planet, which has been discovered by Clyde Tombaugh in 1930. It orbits most of the time outside the orbit of Neptune, but with such an eccentricity that it is sometimes inside. It was discovered in 1978 that Pluto has a large satellite, Charon, so large that the system Pluto-Charon can be seen as a binary object. This binary has at least 4 small satellites, which were discovered thanks to the Hubble Space Telescope.

Pluto has a tenuous atmosphere. It was discovered from the Earth in 1985 in analyzing a stellar occultation: when a faint, atmosphereless object is aligned between a star and a observer, the observer does not see the star anymore. However, when the object has an atmosphere, the light emitted by the star is deviated, and can even be focused by the atmosphere, resulting in a peak of luminosity.

Several occultations have permitted to constrain the atmosphere. It has been calculated that its pressure is about 15 μbar (the one of the Earth being close to 1 bar, so it is very tenuous), and that it endured seasonal variations. By seasonal I mean the same as for the Earth: because of the variations of the Sun-Pluto distance and the obliquity of Pluto, which induces that every surface area has a time-dependent insolation, thermic effects affect the atmosphere. This can be direct effects, i.e. the Sun heats the atmosphere, but also indirect ones, in which the Sun heats the surface, triggering ice sublimation, which itself feeds the atmosphere. The seasonal cycle, i.e. the plutonian (or hadean) year lasts 248 years.

Observations have shown that this atmosphere is hotter at its top than at the surface, i.e. the temperature goes down from 110 K to about 45 K (very cold anyway). This atmosphere is mainly composed of nitrogen N2, methane CH4, and carbon monoxide CO.

## The surface of Pluto

The surface is known to us thanks to New Horizons. Let me particularly focus on two regions:

• Sputnik Planitia: this is the heart that can be seen on a map of Pluto. It is directed to Charon, and is covered by volatile ice, essentially made of nitrogen N2,
• Cthulhu Regio: a large, dark reddish macula, on which the volatile ice is absent.

The reason why I particularly focus on these two regions is that they have two different albedos, i.e. the bright Sputnik Planitia is very efficient to reflect the incident Solar light, while Cthulhu Regio is much less efficient. This also affects the temperature: on Sputnik Planitia, the temperature never rises above 37 K, while it never goes below 42.5 K in Cthulhu Regio. We will see below that it affects the composition of the surface.

## An Equation Of State

The three main components, i.e. nitrogen, methane, and carbon monoxide, have different sublimation temperatures at 11μbar, which are 36.9 K, 53 K, and 40.8 K, respectively (sublimation: direct transition from the solid to the gaseous state. No liquid phase.). A mixture of them will then be a coexistence of solid and gaseous phases, which depends on the temperature, the pressure, and the respective abundances of these 3 chemical components. The pressure is set to 11μbar, since it was the pressure measured by New Horizons, but several temperatures should be considered, since it is not homogeneous. The authors considered temperatures between 36.5 K and 41.5 K. Since the atmosphere has seasonal variations, a pressure of 11μbar should be considered as a snapshot at the closest encounter with New Horizons (July 14, 2015), but not as a mean value.

The goal of the authors is to build an Equation Of State giving the phases of a given mixture, under conditions of temperature and pressure relevant for Pluto. The surface is thus seen as a multicomponent solid solution. For that, they develop a model, CRYOCHEM for CRYOgenic CHEMistry, which aims at predicting the phase equilibrium under cryogenic conditions. The paper I present you today is part of this development. Any system is supposed to evolve to a minimum of energy, which is an equilibrium, and the composition of the surface of Pluto is assumed to be in thermodynamic equilibrium with the atmosphere. The energy which should be minimized, i.e. the Helmholtz energy, is related to the interactions between the molecules. A hard-sphere model is considered, i.e. a minimal distance between two adjacent particles should be maintained, and for that the geometry of the crystalline structure is considered. Finally, the results are compared with the observations by New Horizons.

Such a model requires many parameters. Not only the pressure and temperature, but also the relative fraction of the 3 components, and the parameters related to the energies involved. These parameters are deduced from extrapolations of lab experiments.

## Results

The predicted coexistence of states predicted by this study is consistent with the observations. Moreover, it shows that the small fraction of carbon monoxide can be neglected, as the behavior of the ternary mixture of N2/CH4/CO is very close to the one of the binary N2/CH4. This results in either a nitrogren-rich solid phase, for the coolest regions (the bright Sputnik Planitia, e.g.), and a methane-rich solid phase for the warmest ones, like Cthulhu Regio.

Developing such a model has broad implications for predicting the composition of bodies’s surfaces, for which we lack of data. The authors give the example of the satellite of Neptune Triton, which size and distance to the Sun present some similarities with Pluto. They also invite the reader to stay tuned, as an application of CRYOCHEM to Titan, which is anyway very different from Pluto, is expected for publication pretty soon.

# Resurfacing Ganymede

Hi there! After Europa last week, I tell you today on the next Galilean satellite, which is Ganymede. It is the largest planetary satellite in the Solar System, and it presents an interesting surface, i.e. with different terrains showing evidence of past activity. This is the opportunity for me to present you Viscous relaxation as a prerequisite for tectonic resurfacing on Ganymede: Insights from numerical models of lithospheric extension, by Michael T. Bland and William B. McKinnon. This study has recently been accepted for publication in Icarus.

## The satellite Ganymede

Ganymede is the third, by its distance to the planet, of the 4 Galilean satellites of Jupiter. It was discovered with the 3 other ones in January 1610 by Galileo Galilei. These are indeed large bodies, which means that they could host planetary activity. Io is known for its volcanoes, and Europa and Ganymede (maybe Callisto as well) are thought to harbour a global, subsurfacic ocean. The table below lists their size and orbital properties, which you can compare with the 5th satellite, Amalthea.

J-1 Io 5.90 Rj 0.0041 0.036° 1821.6 km
J-2 Europa 9.39 Rj 0.0094 0.466° 1560.8 km
J-3 Ganymede 14.97 Rj 0.0013 0.177° 2631.2 km
J-4 Callisto 26.33 Rj 0.0074 0.192° 2410.3 km
J-5 Amalthea 2.54 Rj 0.0032 0.380° 83.45 km

We have images of the surface of Ganymede thanks to the spacecraft Voyager 1 & 2, and Galileo. These missions have revealed different types of terrains, darker and bright, some impacted, some pretty smooth, some showing grooves… “pretty smooth” should be taken with care, since the feeling of smoothness depends on the resolution of the images, which itself depends on the distance between the spacecraft and the surface, when this specific surface element was directed to the spacecraft.

A good way to date a terrain is to count the craters. It appears that the dark terrains are probably older than the bright ones, which means that a process renewed the surface. The question this paper addresses is: which one(s)?

## Resurfacing a terrain

These four mechanisms permit to renew a terrain from inside:

• Band formation: The lithosphere, i.e. the surface, is fractured, and material from inside takes its place. This phenomenon is widely present on Europa, and probably exists on Ganymede.
• Viscoelastic relaxation: When the crust has some elasticity, it naturally smooths. As a consequence, craters tend to disappear. Of course, this phenomenon is a long-term process. It requires the material to be hot enough.
• Cryovolcanism: It is like volcanism, but with the difference that the ejected material is mainly composed of water, instead of molten rock. Part of the ejected material falls on the surface.
• Tectonics: Extensional of compressional deformations of the lithosphere. This is the phenomenon, which is studied here.

Beside these processes, I did not mention the impacts on the surface, and the erosion, which is expected to be negligible on Ganymede.

The question the authors addressed is: could tectonic resurfacing be responsible for some of the actually observed terrains on Ganymede?

## Numerical simulations

To answer this question, the authors used the numerical tool, more precisely the 2-D code Tekton. 2-D means that the deformations below the surface are not explicitly simulated. Tekton is a viscoelastic-plastic finite element code, which means that the surface is divided into small areas (finite elements), and their locations are simulated with respect to the time, under the influence of a deforming cause, here an extensional deformation.

The authors used two kinds of data, that we would call initial conditions for numerical simulations: simulated terrains, and real ones.
The simulated terrains are fictitious topographies, varying by the amplitude and frequency of deformation. The deformations are seen as waves, the wavelength being the distance between two peaks. A smooth terrain can be described by long-wavelength topography, while a rough one will have short wavelength.
The real terrains are Digital Terrain Models, extracted from spacecraft data.

The authors also considered different properties of the material, like the elasticity, or the cohesion.

## A new scenario of resurfacing

It results from the simulations that the authors can reproduce smooth terrains with grooves, starting from already smooth terrains without grooves. However, extensional tectonics alone cannot remove the craters. In other words, if you can identify craters at the surface of Ganymede, after millions of years of extensional tectonics you will still observe them. To make smooth terrains, you need the assistance of another process, the viscoelastic relaxation of the lithosphere being an interesting candidate.

This pushed the authors to elaborate a new scenario of resurfacing of Ganymede, involving different processes.
They consider that the dark terrains are actually the eldest ones, having remaining intact. However, there was indeed tectonic resurfacism of the bright terrains, which formed grooved. But the deformation of the lithosphere was accompanied by an elevation of the temperature (which is not simulated by Tekton), which itself made the terrain more elastic. This elasticity itself relaxed the craters.

Anyway, you need elasticity (viscoelasticity is actually more accurate, since you have energy dissipation), and for that you need an elevation of the local temperature. This may have been assisted by heating due to internal processes.

## In the future

Ganymede is the main target of the ESA mission JUICE, which should orbit it 2030. We expect a big step in our knowledge of Ganymede. For this specific problem, we will have a much better resolution of the whole surface, the gravity field of the body (which is related to the interior), maybe a magnetic field, which would constrain the subsurface ocean and the depth of the crust enshrouding it, and the Love number, which indicates the deformation of the gravity field by the tidal excitation of Jupiter. This last quantity contains information on the interior, but it is related to the whole body, not specifically to the structure. I doubt that we would have an accurate knowledge of the viscoelasticity of the crust. Moreover, the material properties which created the current terrains may be not the current ones; in particular the temperature of Ganymede is likely to have varied over the ages. We know for example that this temperature is partly due to the decay of radiogenic elements shortly after the formation of the satellite. During this heating, the satellite stratifies, which alters the tidal response to the gravitational excitation of Jupiter, and which itself heats the satellite. This tidal response is also affected by the obliquity of Ganymede, by its eccentricity, which is now damped… So, the temperature is neither constant, nor homogeneous. There will still be room for theoretical studies and new models.

# Plate tectonics on Europa?

Hi there! Jupiter has 4 large satellites, known as Galilean satellites since they were discovered by Galileo Galilei in 1610. Among them is Europa, which ocean is a priority target for the search for extraterrestrial life. Many clues have given us the certainty that this satellite has a global ocean under its icy surface, and it should be the target of a future NASA mission, Europa Clipper. Meanwhile it will also be visited by the European mission JUICE, before orbital insertion around Ganymede. Since Europa presents evidences of tectonic activity, the study I present you today, i.e. Porosity and salt content determine if subduction can occur in Europa’s ice shell, by Brandon Johnson et al., wonders whether subduction is possible when two plates meet. This study has been conducted at Brown University, Providence, RI (USA).

## Subduction on Earth

I guess you know about place tectonics on Earth. The crust of the Earth is made of several blocks, which drift. As a consequence, they collide, and this may be responsible for the creation of mountains, for earthquakes… Subduction is a peculiar kind of collision, in which one plate goes under the one it meets, just because their densities are significantly different. The lighter plate goes up, while the heavier one goes down. This is what happens on the west coast of South America, where the subduction of the oceanic Nazca Plate and the Antarctic Plate have created the Andean mountains on the South America plate, which is a continental one.

Even if our Earth is unique in the Solar System by many aspects, it is highly tempting to use our knowledge of it to try to understand the other bodies. This is why the authors simulated the conditions favorable to subduction on Europa.

## The satellite Europa

Europa is the smallest of the four Galilean satellites of Jupiter. It orbits Jupiter in 3.55 days at a mean distance of 670,000 km, on an almost circular and planar orbit. It has been visited by the spacecraft Pioneer 10 & 11 in 1973-1974, then by Voyager 1 & 2 in 1979. But our knowledge of Europa is mostly due to the spacecraft Galileo, which orbited Jupiter between 1995 and 2003. It revealed long, linear cracks and ridges, interrupted by disrupted terrains. The presence of these structures indicates a weakness of the surface, and argues for the presence of a subsurface ocean below the icy crust. Another argument is the tidal heating of Jupiter, which means that Europa should be hot enough to sustain this ocean.
This active surface shows extensional tectonic feature, which suggests plate motion, and raises the question: is subduction possible?

## Numerical simulations of the phenomenon

To determine whether subduction is possible, the authors performed one-dimensional finite-elements simulations of the evolution of a subducted slab, to determine whether it would remain below another plate or not. The equation is: would the ocean be buoyant? If yes, then the slab cannot subduct, because it would be too light for that.

The author considered the time and spatial evolution of the slab, i.e. over its length and over the ages. They tested the effect of

1. The porosity: Planetary ices are porous material, but we do not know to what extent. In particular, at some depth the material is more compressed, i.e. less porous than at the surface, but it is not easy to put numbers behind this phenomenon. Which means that the porosity is a parameter. The porosity is defined as a fraction of the volume of voids over the total volume investigated. Here, total volume should not be understood as the total volume of Europa, but as a volume of material enshrouding the material element you consider. This allows you to define a local porosity, which thus varies in Europa. Only the porosity of the icy crust is addressed here.
2. The salt content: the subsurface ocean and the icy crust are not pure ice, but are salty, which affects their densities. The authors assumed that the salt of Europa is mostly natron, which is a mixture essentially made of sodium carbonate decahydrate and sodium bicarbonate. Importantly, the icy shell has probably some lateral density variations, i.e. the fraction of salt is probably not homogeneous, which gives room for local phenomenons.
3. The crust thickness: barely constrained, it could be larger than 100 km.
4. The viscosity: how does the material react to a subducting slab? This behavior depends on the temperature, which is modeled here with the Fourier law of heat,
5. The spreading rate, i.e. the velocity of the phenomenon,
6. The geometry of the slab, in particular the bending radius, and the dip angle.

And once you have modeled and simulated all this, the computer tells you under which conditions subduction is possible.

## Yes, it is possible

The first result is that the two critical parameters are the porosity and the salt content, which means that the conditions for subduction can be expressed with respect to these two quantities.
Regarding the conditions for subduction, let me quote the abstract of the paper: If salt contents are laterally homogeneous, and Europa has a reasonable surface porosity of 0.1, the conductive portion of Europa’s shell must have salt contents exceeding ~22% for subduction to occur. However, if salt contents are laterally heterogeneous, with salt contents varying by a few percent, subduction may occur for a surface porosity of 0.1 and overall salt contents of ~5%.

A possible subduction does not mean that subduction happens. For that, you need a cause, which would trigger activity in the satellite.

## Triggering the subduction

The authors propose the following two causes for subduction to happen:

1. Tidal interaction with Jupiter, enhanced by non-synchronous rotation: Surface features revealed by Galileo are consistent with a crust which would not rotate synchronously, as expected for the natural satellites, but slightly faster, the departure from supersynchronicity inducing a full rotation with respect to the Jupiter-Europa direction between 12,000 and 250,000 years… to be compared with an orbital period of 3.55 days. So, this is a very small departure, which would enhance the tidal torque of Jupiter, and trigger some activity. This interpretation of the surface features as a super-synchronous rotation is controversial.
2. Convection, i.e. fluid motion in the ocean, due to the variations of temperature.

No doubt Europa Clipper and maybe JUICE will tell us more!

# 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.