Nonthermal escape

Using Planck's constant, the energy of an EUV photon with wavelength 0.05 pm is 4 • 10-18J. This is sufficient to dissociate the components of many molecules, and to knock off electrons from just about anything - a process called ionization which produces charged particles in the outer atmosphere. In fact, ionization is the principle means by which absorbed EUV heats the outer atmosphere, since ejection of an electron increases the kinetic energy of the ion left behind, as well as imparting energy to particles the electron subsquently collides with. When easily ionized species are used up, the heating is correspondingly reduced.

The energy of such a photon is somewhat in excess of the escape energy for atomic oxygen on Earth, and so if this energy can be converted into kinetic energy of an atom, it can lead to escape at rates far higher than one would get from Jeans escape. This idea underlies the subject of nonthermal escape.

The term nonthermal escape represents a whole zoo of mechanisms united only by the common theme that they are not thermal - that they rely on the strong deviations from the Maxwell-Boltzmann distribution that are possible when collisions are infrequent. The study on nonthermal escape is rather like, to paraphrase a remark of Stanislaw Ulam, the study of the physiology of non-elephants. We will discuss a few examples to give the general flavor of the issues involved, but the reader should be aware that we are barely scratching the surface of this difficult and interesting subject.

The Maxwell-Boltzmann distribution corresponding to a given temperature is maintained through frequent collisions, which redistribute energy amongst the molecules making up the gas.

If an individual particle in the gas acquires some new kinetic energy as a result of absorption of an EUV photon or a chemical reaction which releases energy, the energy may initially be much greater than the typical energy kT characteristic of the temperature of the gas. It is only after several collisions that the extra energy is equitably redistributed amongst the other particles - a process known as thermalization. If collisions are infrequent, the thermalization time can be very long, leading to a large population of particles that have anomalously large energy. Indeed in such a case, the energy distribution deviates greatly from the Maxwell-Boltzmann distribution, and the gas is no longer characterized by a temperature in the usual sense of equilibrium thermodynamics. The anomalously energetic atoms or molecules are often referred to as the "hot" population, as in "hot oxygen" or "hot hydrogen". The hot atoms may have enough energy to escape before they thermalize, or they may remain gravitationally bound but later impart their energy to lighter species which can escape. This is the general idea of nonthermal escape, and the reader can probably understand already why there are many ways in which it can happen.

When the nonthermal escape is an indirect result of the accumulation of gravitationally bound hot atoms, the calculation of escape rate is very complicated, since one needs first to model the number of hot atoms, and the distribution of their energy. Ultimately, the energy is supplied by EUV absorption (or perhaps solar wind interactions, to be discussed separately), so that the flux of EUV provides an upper limit to the rate at which any constituent can escape. However, the actual rate depends on how the delivered energy is spread amongst the escaping constituents. If the energy is concentrated in relatively few particles, than an escape flux can be sustained, whereas if the energy is spread too thinly or is wasted on heavy particles, there may be little escape. Another important consideration is that only EUV delivered above the exobase can lead to nonthermal escape; energy deposited at lower altitudes will instead thermalize through collisions. Determining the proportion of EUV which is deposited above the exobase requires consideration of the number of particles above the exobase and the EUV absorption cross section of the species making up the exosphere.

Photons have little momentum, so the absorption of an EUV photon cannot directly increase the kinetic energy of the molecule or atom absorbing it to any great degree. Rather, the increase of kinetic energy takes place through ionization or dissociation. It the first case, a fast electron is ejected in one direction while the heavier positive ion moves more slowly in the opposite direction, with such a speed as to satisfy the momentum balance. It takes energy (the ionization energy, which differs according to species) to pry loose an electron, so the total kinetic energy supplied to the electron and ion is the energy of the photon minus the ionization energy. Dissociation is similar, except that the molecule breaks apart into neutral or charged heavy constituents instead. Dissociation also requires energy, so that the energy available to increase the kinetic energy of the fragments is the energy of the photon diminished by the dissociation energy.

As a concrete and relatively simple example, let's consider photodissociation of N2 on Mars, which has been suggested as a possible means of losing the nitrogen in a hypothetical nitrogen-rich primordial Martian atmosphere. This problem is of interest because Mars at present has little N2 in its atmosphere, and we'd like to know whether this means that Mars must have formed with very little nitrogen (unlike Earth or Venus), or whether the smaller size of the planet allowed its initial nitrogen endowment to escape.

Assuming the Martian exobase to be reasonably close to the ground, an N atom requires 2.92 • 10-19J of energy to escape. This is much greater than the typical thermal energy kT = 4. • 10-21 J. Now, the dissociation energy of N2 is about 1.94 • 10-18J, so assuming the excess absorbed energy to be equally distributed between the two dissociated atoms, a photon with energy in excess of 1.94 • 10-18 J + 2 • 2.92 • 10-19 J, i.e. 2.52 • 10-18 J. Using Planck's constant, this corresponds to EUV photons with wavelengths shorter than .078 pm. Next, we must determine the rate at which such photons collide with N2 molecules and cause dissociation. For simplicity, we'll assume a pure N2 atmosphere, so we need not worry about collisions with other molecules. If the exospehere is optically thin in the EUV, then the proportion of incident photons which are absorbed is the product of the absorption cross section with the number of particles per square meter in the exobase. The latter is approximately the product of the exobase density with the scale height, i.e. 1/x%/2, where x is the molecular collisional cross section area. We are left with the rather tidy result that the proportion of photons that are absorbed in the exosphere is the ratio of the EUV cross section to the molecular collision cross-section. In the relevant part of the EUV the absorption cross section for N2 is about 3 • 10-21m2, so the proportion of incident photons which are absorbed in the N2 exosphere is 4.6%. Note that this fraction is independent of the total mass of the atmosphere, so that we cannot increase the rate of escape by increasing the total amount of N2 in the atmosphere. In general, this is one of the chief factors limiting the effectiveness of nonthermal escape due to photodissociation.

Based on satellite observations the flux of sufficiently energetic EUV photons at Earth's orbit is currently about 8 • 1014/m2s, which scales to about 4 • 1014/m2s at the orbit of Mars. Allowing for the proportion which are absorbed in the exosphere, this leads to an escape of 4.6 • 1012 N atoms per square meter of the planet's surface per second or 1.45 1029 atoms per square meter of surface in a billion years. One bar of N2 on Mars contains 1.16 • 1030 atoms per square meter, so we conclude that EUV induced nonthermal escape has the potential to remove about a half a bar of N2 from Mars over the lifetime of the Solar system, or more if the EUV flux were higher in the early Solar system.

The above calculation is sufficient to show that escape from EUV-induced dissociation is a potentially important factor worthy of more sophisticated study, but it is highly oversimplified and leaves out many considerations that could substantially reduce the escape. First, the dissociated N atoms are not always in the lowest energy electron configuration (the ground state). The energy that goes into excited electron states is not available to feed kinetic energy sustaining escape. To crudely take this into account, in the above calculation we used the energy for dissociation into the most favored excited electron configuration; the dissociation energy into ground-state N atoms would be only 1.57-10-18 J, though such dissociations are believed to be rare. Second, photons with energy higher than 2.5 • 10-18 J can cause ionization of the N2 molecule rather than dissociation. Only a small proportion of the molecules will directly dissociate. Some of the N+ ions will later dissociate and lead to escape when they recombine with the free electrons released by ionization -a process called dissociative recombination - but calculation of the proportion that do so is quite involved. Third, in an atmosphere that is not pure N2 there is a stew of other dissociation products, both neutral and ionized, that need to be considered, as well as the reactions between them.

On the other hand, dissociation due to EUV photons with energy less than 2.52 • 10-18 J leads to a population of gravitationally bound N atoms which have typical energies much larger than the typical thermal energy. Such populations of energetic particles are referred to as "hot", as in "hot nitrogen" or "hot oxygen", even though they are not in thermodynamic equilibrium and their temperature is not strictly defined. Sometimes a "temperature" is defined for such populations just as a means of characterizing the energy. For example, if a photon with energy 2.3 • 10-18J causes dissociation, it leaves behind two N atoms with an energy of 0.17 • 10-18 J each. Setting kT equal to this energy yields a "temperature" of over 12000K. This is merely a way of summarizing how the energy of the hot population compares the the much lower background thermal energy (characterized by a temperature of around 300K for the Martian exosphere). The hot population does not escape itself, but it represents a reservoir of energy that can be transferred to other species (especially lighter ones), and which can lead to the escape of those. In this mode, the outer atmosphere acts like a storage-beam particle accelerator, accumulating energetic particles for later use.

Other dissociations can also lead to escape or accumulation of hot atoms. N2 has an unusually large dissociation energy. The reactions O2 + hv ^ O + O, CO2 + hv ^ CO + O, H2O + hv ^ H + OH and OH + hv ^ O + H all have dissociation energies of around 0.8 • 10-18 J. Dissociation of these by EUV leaves more energy left over to feed escape.

Dissociative recombination, particularly involving oxygen, is an important indirect means by which EUV absorption can lead to escape or accumulation of hot atoms. For example, when the recombination of the O+ ion with an electron leads to the dissociation of the result into two O atoms in their ground states, then 1.12 • 10-18 J are released, or 0.56 • 10-18 per atom. At present, such reactions take place on Venus and Mars as a consequence of the photodissociation of CO2, but in earlier epochs on Venus water vapor could also have been involved. The escape energy for O on Venus is 1.43 • 10-18J, so dissociative recombination does not directly lead to escape of O on Venus, but rather to the accumulation of hot oxygen that can later allow a smaller part of the oxygen (or a greater part of a lighter species) to escape. Significant amounts of oxygen are indeed observed escaping at present from Venus, but it is not thought that the nonthermal mechanisms could have gotten rid of any significant part of the oxygen in a primordial Venusian ocean. On Mars, however, dissociative recombination directly leads to the escape of an oxygen atom, since the escape energy is only 0.33 • 10-18 J.

A further set of complications arises from the fact that charged particles interact with the planet's magnetic field, if it has one. Charged particles tend to spiral tightly along magnetic field lines, and so they can escape easily only when they are on the relatively limited proportion of field lines that are open - which have one end leading out into outer space. When the magnetic field is important, there is thus a great premium in generation not just of energetic particles in general, but energetic neutral particles. For this reason, a great deal of attention in work on nonthermal escape has been lavished on processes that allow energetic ions to deliver their energy to neutral particles.

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