A comet nucleus is a few kilometers in diameter and consists of a very porous mixture of ice, organics and a fine rocky powder. They often move on a highly elliptic orbit around the Sun, which means that they spend most of their time in extreme cold at large distances from the Sun. The ice is then so cold that it is stable, even when it is directly illuminated by the Sun and exposed to the vacuum of space. But periodically the comet passes its perihelion, its smallest possible distance to the Sun, and on its way there, the surface layer of the comet is heated sufficiently to vaporize the ice. The gases rush out from the comet nucleus, drags dust grains along and form the coma, a vast temporary atmosphere, along with two tails consisting of gas and dust, respectively. The basic properties of comets have been described in an earlier post.
But how close does a comet nucleus need to get to the Sun, in order to send gas and dust into space? How do the production rates of gas and dust vary with the distance to the Sun? Are all parts of the comet nucleus emitting the same amount of gas and dust, or are some regions more productive than others? In order to answer questions of this type astronomers use a tool called thermophysical modeling. The purpose of this modeling is to calculate the temperature of the comet nucleus at different depths below the surface, and how these temperatures are changing with time, as the Sun rises and sets in the sky as seen from a specific point on the comet surface, due to the rotation of the nucleus. These temperatures are depending on a number of different things, like the intensity of solar light, the capability of the surface material to absorb solar radiation, its heat conductivity, its porosity, the abundance of ice, and the physical properties of water as a chemical species. The final product of these calculations is an estimate of the number of water molecules that leaves a square meter of comet surface every second. By adding contributions from different parts of the comet nucleus, that are illuminated by the Sun in different ways at any given moment, it is possible to calculate the total production of gas of the nucleus.
My research has focused on thermophysical modeling for many years, and in a number of posts I will describe the thermophysical model I developed together with Dr. Yuri Skorov, a Russian scientist who presently works at Technische Universität Braunschweig in Germany. The model has several parts that are joined to form a whole – I here describe the first part, that deals with the way the surface material of comet nuclei absorb sunlight. Such research can provide clues as to why comets are so dark – they reflect merely a few percent of the sunlight – and why they are so red. As a fact, they reflect red light much more efficiently than blue light.
What happens to the solar radiation at the surface of the comet?
In order to describe the outgassing of a comet one needs to solve several sub-problems. One of these deals with the sunlight as it hits the surface of the comet. What fraction of this radiation is reflected back into space? What fraction of the radiation is absorbed, so that its energy can be used to vaporize ice? Where does this absorption take place – how deep are the solar rays penetrating?
When I started my PhD studies at Uppsala university in 1998, most thermophysical models used by scientists worldwide, assumed that solar radiation was absorbed at the very surface of the comet, within an infinitely thin layer. The reason for this assumption was very practical – there are so many other things to take into account, that the details of solar light absorption were overlooked in favor of other mechanisms considered more important. However, we envision the comet surface material as very porous – the material consists of loosely bound grains of ice and rock, surrounded by cavities and empty spaces since the gravitational force of the little nucleus is incapable of compacting the material. The solar radiation should be able to penetrate fairly deep into such a porous structure before being absorbed, especially since many of the grains consist of ice, which is rather transparent. Since ice at a depth of a few millimeters or centimeters below the surface therefore may be illuminated directly by sunlight, and if large amounts of vapor therefore are produced at depth, the outgassing of a comet would perhaps behave differently, compared to an object were all solar energy indeed is absorbed within the uppermost layer of tiny grains.
On a related note – the space probe Voyager 2 had flown by Jupiter in July 1979, and observed its ice-rich satellite Europa with an infrared spectrometer, that measures the heat radiated from the surface of Europa. Based on these measurements it is possible to calculate the temperature that prevails on different parts of the surface of Europa. When these temperatures, valid at different local times (night, early morning, noon), were compared to calculations, it was found that thermophysical models failed to reproduce the observations if they assumed that all radiation was absorbed at the very surface. Only when the model allowed radiation to be absorbed gradually within a thicker layer of ice near the surface was it possible to match the calculations with the measurements.
If all radiation is absorbed at the very surface, a sunlit spot on the dayside of Europa will have its highest temperature at the very surface, and the temperature decreases with depth below the surface. However, if the solar radiation is absorbed gradually within a thicker layer, the surface becomes rather cold. Now the temperature increases with depth and reaches a maximum a few millimeters or centimeters below the surface, and is then falling at even larger depths. This phenomena is called a solid-state greenhouse effect and is due to the fact that radiation easily can enter, but has difficulties getting back out. This is because sunlight primarily consists of visible light, and at such wavelengths ice is rather transparent. But the heated ice below the surface tries to get rid of this heat by emitting infrared radiation, at wavelengths where ice no longer is transparent. The energy cannot get out, but accumulates – this causes the temperature peak just below the surface – and it can only be removed by heat conduction or by consumption through ice sublimation.
Nobody had seriously considered solid-state greenhouse effects for comet nuclei, and since I needed a relatively poorly studied topic for my doctoral thesis, I thought I could work on this problem. But where to start?
First one needs to have an idea of how the grains look like in the surface region of the comet – what is the grain size, what minerals and substances are present, how are these materials arranged within the grains, how much void space surrounds the grains, and are the grains homogeneously distributed or lumped together into little clusters?
We know that comets contain grains consisting of silicates, sulfides and metals, typically having a size of a tenth of a micrometer, that is to say, ten thousand times smaller than a millimeter. This is because oxygen, silicon, magnesium, iron and sulphur are some of the most common chemical elements in the interstellar medium from which the Solar System formed (see the post about Voyager 1 and the interstellar medium), and since these elements team up to form silicates, sulfides, and metals – the mixture we call rock. We know that most of the grains have a size of a micrometer or smaller, because they are expelled in large numbers by active comets to form their huge dust tails. The sizes of these grains can be inferred from their motion – the dynamics is determined by the gravity and radiation pressure of the Sun, in a way that makes it possible to estimate the sizes of the grains.
It is likely that such small cores of rock are covered by a mantle of organic substances. This is because carbon is even more common in the interstellar medium than the previously mentioned elements, and it must be present in the comet in one form or another. This is because organic substances have a much lower condensation temperature than silicates, sulfides and metals, and require very cold environments to solidify from the gas phase. Therefore, organic species are rare in the inner Solar System (for Mercury, Venus, Earth, Mars and the inner parts of the asteroid belt), but much more common in the outer Solar System – for example in the comets. The inner Solar System was too warm for organic species to condense in large numbers, but in the outer Solar System this process could proceed undisturbed. In this way, mantles of organics likely formed on the pre-existing grains of rock. Such a grain can be seen in panel A in the figure below.
Another species that was present in huge quantities was water – which has an even lower condensation temperature than the organic species. Therefore, essentially all water in the inner Solar System was in the form of vapor, while the substance gradually could condense and freeze in the outer parts of the Solar System. Thus thick crusts of ice were formed on the grains, on top of the layer of organics. This ice could either be rather pure (panel B) or possibly mixed with organic substances and rather dirty (panel C).
About 4.57 billion years ago such free-flying grains started to stick to each other as they collided at low speed, and with time this fluffy ball of rock, organics and ice grew to a size of several kilometers – this is the comet nucleus. We have reasons to believe that these grains are not strongly compressed within the nucleus, but form a loosely bound structure where grains barely touch, and where cavities of different sizes surrounds threads or membranes of loosely bound grains, which makes the material very porous and fragile. The force of gravity is too weak to squeeze the material into a more compact dirty snowball – it may have a structure resembling cotton sugar. We often see that comet nuclei fragment, split or are pulverized completely, which means that they are extremely fragile. We know that comets have a very low density and high porosity since we can measure their masses and volumes. Their masses are very low compared to what we could expect, given their volumes. It is necessary to take this very high porosity into account when trying to calculate the penetration depth of the solar radiation.
We do not know if the grains are homogeneously distributed, or if they form lumps here and there – something we call clusters. Panels D and E show examples of such clusters – in the former case the cluster consists of type A grains (they only contain rock and organics), while clusters of type E consist of type B grains (containing ice in addition to rock and organics). We can also imagine that ice evaporates but that the vapor freezes anew. If this happens over and over again, larger chunks of more compact ice could form, where particles of rock and organics (type A grains) are finely distributed within the chunk – such a dirty ice particle is seen in panel F.
The interaction of sunlight with single grains and clusters
Light, also called electromagnetic radiation, consists of an electrical field and a magnetic field that oscillate about each other. When light from the Sun penetrates the surface layer of the comet, these electric and magnetic fields start to interact with the electrically charged particles – atomic nuclei and electrons – that the grains consist of. As a result of such interaction some of the solar energy is absorbed – this gives the atoms higher vibration speed since they have absorbed the energy of the light. The parameter we call “temperature” is a direct measure of these vibration velocities, so the absorption of sunlight of the grains simply increase their temperature. But a part of the radiation will not be absorbed, but re-emitted by the grain in various directions – we say that the radiation is being scattered.
If we want to know how solar radiation behaves as it enters the surface layer of the comet we therefore first need to understand how the solar radiation interacts with a single grain, or a cluster of such grains. Exactly how much radiation is absorbed and how much is scattered? Exactly how much radiation is scattered into a specific direction, for example, perpendicularly to the original direction of propagation of the light? How do these things depend on the particle size, its chemical composition, and internal structure? How are these dependencies changing depending on the wavelength of the radiation?
There is a whole branch of science dealing with questions of this sort – light scattering theory. The first step was therefore to use techniques that had been developed in this research field. For the simplest grains (type A, B, and C), consisting of spheres or concentric shells with different composition, it is possible to use Mie theory. Mie theory is based on Maxwell’s equations – a group of equations that describe the properties of the electric and magnetic fields that light consists of. By using the Maxwell equations, Mie theory calculates exactly what happens to the electromagnetic wave as it encounters a spherical particle. The result is an analytical solution – a mathematical formula were one can enter the particle size, the thickness of the different layers, the wavelength of the radiation, and parameters that describes the optical properties of the materials (these differ strongly between various types of silicates, sulfides, organics and ice and are measured in the laboratory). The analytical solution then tells us how much radiation is being absorbed or scattered, and in what way.
However, it is not possible to apply Mie theory for the clusters seen in panels D and E. These grains have an extremely complex geometry – matter with very different optical properties are mixed in an elaborated manner, and there are a lot of cavities and spaces where radiation may bounce around and change direction. If a ray of light falls on such a complex particle, how much is absorbed and how much is scattered?
There are many techniques available to calculate the behavior of the electromagnetic wave when it interacts with such a strange grain. In our work, we used a technique called the Discrete Dipole Approximation or DDA. A dipole consists of a particle with positive electrical charge and a particle with negative electrical charge, separated by some distance. If the dipole is illuminated, the electrical and magnetic fields of the light will set the charged particles in motion. An oscillation takes place, so that the distance between the two parts of the dipole periodically is decreasing and increasing. Electrical charges that perform acceleration and deceleration will emit light themselves, into completely different directions compared to the light that initially set the dipole in motion. The energy theft of the dipole is the very reason for the light absorption of the material, while its emission is the source of the scattering phenomenon.
In DDA, the grains are represented by tens of thousands of individual dipoles, that each have their specific position. The dipoles located in a region consisting of silicates are given properties that are typical of silicate. The dipoles located in a region consisting of organics or ice are given properties consistent with those substances. Cavities between grains are not given any dipoles at all. A computer code then calculates what happens to every single dipole, when it is exposed to the combination of sunlight and radiation produced by all other dipoles within the grain. In this way, DDA can calculate very accurately how much sunlight is being absorbed or scattered by the cluster as a whole.
For grains of type F we used a third technique called geometrical optics. That is the type of physics used when calculating what happens to radiation when it passes through large objects like lenses (in telescopes, glasses, etc) or large drops of water. Geometrical optics assumes that the objects is much larger than the wavelength of the light. It is possible to calculate how much radiation that is reflected at the surface of the grain, and how much that enters the grain. For the transmitted radiation, the gradual absorption can be calculated, as well as the scattering, especially considering the presence of small particles of rock and organics that are baked into the grain. This calculation provides the same type of information as Mie theory and DDA – how much radiation is absorbed, how much is scattered, into what directions is the scattered radiation going, and how does these things depend on the size of the chunk and its content of type A grains?
Thanks to these calculations we had come quite a way. We now had a fairly good idea of how individual grains, or clusters of such grains, interact with sunlight. The next step was to figure out how an entire medium of such grains behave – that is to say, large quantities of such grains or clusters that are spread out within a volume, with empty space between them in a porous yet coherent structure. Basically – what happens to sunlight as it shines on a dirty and very fluffy snowball consisting of millions of little grains?
The first step was to cut out small cubes of material and figure out how such cubes would absorb and scatter radiation, accounting for the fact that only a fraction of their interior consisted of grains or clusters, while most of their volumes were empty space. Through this calculation we could account for the porosity of the comet surface material.
The next problem was far more complicated. The surface region of the comet will consist of an entire network of such cubes. Each cube will receive some radiation from the Sun, keep some for itself, and pass on the rest into different directions. But since each cube is emitting radiation like this, it means that the Sun is not the only source of light. In reality, each cube will be illuminated both by the Sun and by all other cubes, and it is this combined radiation field each cube needs to deal with. Luckily it is possible to describe this complex network of interdependencies of cubes with the aid of a powerful mathematical tool – the so called equation of radiative transfer.
In fact, all of our work had aimed at realistically describing the terms for absorption, degree of scattering and type of scattering that need to be known before one even can attempt to solve this equation. However, the equation of radiative transfer is far too complex to be solved analytically, why we used the computers again. By doing so we obtained numerical solutions that finally allowed us to achieve our goal. We could tell, in a fairly realistic manner, what happens when the Sun shines on a comet nucleus.
Reflectivity and color
By observing comet nuclei we know that they are very dark – they reflect only a few percent of the sunlight that illuminates them. For example, Comet 1P/Halley only reflects 4% of the sunlight. We also know that they are rather red, in the sense that they reflect light at long wavelengths (red) more efficiently than light at short wavelengths (blue). We also know that most comets produce very little water vapor compared to the available nucleus surface area, which means that large parts of this area is ice-free and inactive. The inactive areas, that dominates the surface of comets, could perhaps consist of grains similar to type A or D, that lack ice.
Therefore it is interesting to compare our theoretical calculations to the actual observations. As it turns out, a porous medium consisting of 0.1 micrometer type A grains with 70% porosity, will reflect 9% of the sunlight if they are not forming clusters. This is a bit too much, but we also found that the reflectivity is strongly dependent on the size of the grains – if the grain size is cut in half, the reflectivity drops to 2%. This is very close to the reflectance of real comets. However, there is a problem – our modeled medium is strongly blue – it reflects the short-wavelength light more effectively than the long-wavelength light, which is not the case for real comets.
If the grains are allowed to form small clusters (like in panel D), something interesting happens – the reflected radiation becomes very red, and is similar to that of real comets. A medium consisting of small clusters of type D do reflect a bit too much radiation, about 13%. However, this is possibly due to the porosity of the medium. We had assumed a very high porosity in our calculations, about 76%, but laboratory experiments show that this type of media may be even more porous, reaching values in excess of 90%. Such a high porosity makes it even easier for radiation to enter, and if a reflection takes place, the radiation most likely does not manage to exit, but illuminates some other part of the medium, where it may get absorbed. Therefore, if the porosity is increased further, the reflectivity could be substantially lowered.
It is therefore possible that the inactive parts of cometary surfaces are dominated by extremely small grains of rock and organics, that often form little clusters, and that this medium is extremely porous. In fact, such a medium would be as dark and as red as real comet nuclei.
How would a spot of dirty ice look like? According to our calculations, a porous medium consisting of type E clusters would not look much different than a porous medium of ice-free clusters. It is still very red, but reflects a bit more – about 20% of the sunlight if the ice is clean, but only 13% if the ice is dirty. These clusters contain three times as much ice as rock. An icy spot therefore does not necessarily look white and shiny like newly fallen snow, but can be very dark. The reflectivity depends on the way ice, organics and rocks are mixed with each other, and the size of pieces of “pure” rock and organics compared with the wavelength of the light that illuminates them.
If the grains look like those in panel F, the calculations show that the material may become very dark – it only reflects 3% of the light and the color is fairly neutral. There is not a big difference in reflectivity at long and short wavelengths.
How deep does the radiation penetrate?
The single most important property of the medium that decides where and how sunlight is absorbed is the size of the cavities between the grains. If a medium consists of 0.1-1 micrometer grains or clusters that are homogeneously distributed, the cavities will also have sizes of a few micrometers. On its way through the medium, the radiation is “forced” to pass through grains, where it rapidly becomes absorbed.
If the grains look like those in panel B, about 90% of the radiation will be absorbed at a depth of 10 micrometers, while 99% of the radiation is absorbed in the upper 45 micrometers. This is very deep in a relative sense – the grains are assumed to have a radius of merely 0.16 micrometers – but is very short in an absolute sense. For such a medium, an assumption of immediate absorption at the surface is probably accurate.
If the grains look like those in panel F, and if they have a size of 100 micrometers (i.e., a tenth of a millimeter) due to repeated cycles of sublimation and condensation, about 1% of the sunlight is still present at a depth of two millimeters. If the grains instead have a size of a centimeter, there may be unabsorbed sunlight even at a depth of 20 centimeters. This is because the cavities have sizes of 0.01-1 centimeters in these media, which allows radiation to reach rather large depths.
It is therefore clear that the properties of the surface material – not only its composition and porosity, but also its purely geometrical arrangement – has a potentially important effect on the absorption of sunlight, and hence the heating and outgassing of comets. Many other factors play a role as well – but that is for another post!
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