T Earth m

Given the weak variation of the factors multiplying nh (Earth) in typical cases, a good rule of thumb for use in making crude estimates of escape fluxes is to simply assume the homopause density to be the same as Earth's, though of course where observations are available it is better to use the observed value. For most escape calculations, it is not necessary to know the homopause altitude, though it can be estimated from the lower atmosphere scale height if it is desired. The homopause altitude only becomes important when it is high enough that gravity is significantly attenuated relative to surface gravity.

The homopause density provides the essential point of reference for most atmospheric escape problems involving multicomponent atmospheres. Since most of the atmospheric mass is in the troposphere, in order to understand how long it takes to lose some part of an atmosphere, we need to relate the escape flux to the tropospheric composition, which in turn requires us to relate the composition of the exobase to the tropospheric composition. This proceeds via the intermediary of determining the homopause composition. To take the simplest case first consider gases that do not undergo condensation or significant chemical sinks or sources within the lower atmosphere - for example O2 and N2 on Earth. These will be well-mixed throughout the homosphere, so if there is about 20% molar O2 in the troposphere, there will be about 20% molar O2 at the homopause. To get the O2 particle density, we multiply this ratio by the homopause density, which we know how to determine. This then gives the supply of O2 molecules, which dissociate above the homopause into atomic oxygen, allowing us to determine the atomic oxygen concentration at the exobase using the scale height for atomic oxygen alone to extrapolate from the homopause to the exobase.

3Venus at present does have a region above the exobase which is dominated by atomic oxygen, but this layer is too tenuous to affect the exobase height. The question of the circumstances in which oxygen can build up to a hot Earth-type exobase is a delicate and difficult one, which hinges on details of atmospheric chemistry and atmospheric composition. An early water-rich Venus atmosphere would have another source of oxygen through dissociation of water vapor, which could conceivably lead to an oxygen-dominated exobase. Calculations performed to date do not seem to bear out this possibility, but the situation has not been thoroughly explored and there is plenty of room for surprises

(A more precise calculation would require modelling of dissociation and reaction rates above the homopause). Things would work similarly for other gases that are nonreactive/noncondensing in the lower atmosphere.

For condensing gases, such as water vapor mixed with air on Earth, CH4 mixed with N2 on Titan, or water vapor mixed with CO2 on Early Venus, there is an additional step on the way to determining the composition of the homopause. Condensible gases do not have uniform concentrations in the homosphere, because of the limitations imposed by Clausius-Clapeyron. Let's take water vapor on Earth as an example. Water vapor makes up a few percent of the lower atmosphere at present, but most water vapor entering the stratosphere must make it through the cold tropical tropopause. The temperature there is around 200K, and the corresponding water vapor mixing ratio, by Clausius Clapeyron, is 1.6 • 10-5, given a tropopause pressure of 100mb. Though there are slight additional water vapor sources in the stratosphere from oxidation of methane, the tropopause concentration is a good estimate of the water vapor concentration that will be found at the homopause. The tropopause acts as a cold trap, dehumidifying the upper atmosphere and strongly limiting the opportunity for water vapor to escape or for hydrogen to build up in Earth's upper atmosphere through decomposition of water vapor.

The cold trap temperature is defined as the lowest temperature encountered below the homopause, and to determine it precisely, one must carry out a full radiative-convective calculation of the atmospheric structure. On an adiabat, temperature would go down indefinitely with height until absolute zero were reached; it is the interruption of temperature decay by the takeover of radiative equilibrium in the stratosphere that usually determines the cold trap temperature. In the absence of a full radiative-convective equilibrium calculation, the skin temperature of the planet often provides an adequate crude estimate of the cold trap temperature. Once the cold trap temperature is known, the maximum possible partial pressure of the condensible at the cold trap is given by Clausius-Clapeyron. However, it is the molar concentration of the condensible we need, since this is the quantity that is preserved as air is mixed up toward the homopause without further condensation. To determine the molar concentration we need the partial pressure of the non-condensible gas at the cold trap. This is obtained using the tools provided in Chapter 2. One computes the adiabat starting from a specified surface pressure, surface temperature, and surface condensible concentration - following the dry adiabat with constant condensible condensation until the atmosphere becomes saturated, and following the moist adiabat thereafter until the cold trap temperature is reached. The usual procedure for computing the moist adiabat then yields the necessary molar concentration. All other things being equal, as more non-condensible is added to the atmosphere, the cold trap concentration goes down owing to greater dilution of the condensible substance. The precise functional form of the dilution depends on the thermodynamic constants of the condensible and noncondensible substances under consideration.

As an example, let's look at the cold trap water vapor concentration that would be encountered during a dry runaway greenhouse in a CO2-H2O system. Recall that in a dry runaway the surface gets so hot that the entire ocean is evaporated into the atmosphere, and there is no liquid water at the surface; in this case, the shutoff of silicate weathering should allow any outgassed CO2 to accumulate in the atmosphere, resulting in atmospheres consisting of CO2 and water vapor in proportions determined by the abundance of these substances in the planetary composition (less whatever water may have already escaped). Results for various sizes of oceans and various CO2 abundances are given in Table 8.3, based on a cold trap temperature of 200K. The saturation vapor pressure and the adiabat were computed using the ideal gas equation of state and the idealized exponential form of Clausius-Clapeyron; these are not quantitatively accurate for the pressures and temperature under consideration, but they suffice to delineate the general behavior of the cold trap concentration. We see that, for any given inventory of water, the cold trap concentration

0bar

1bar

10bar

30bar

60 bar

90 bar

25 bar

1

.83

.23

.016

3.3-10-4

5.7-10-5

50 bar

1

.90

.42

.11

.0090

9.3-10-4

100 bar

1

.95

.61

.28

.092

.024

Table 8.3: Table of water vapor molar concentrations at a 200K cold trap, for a CO2-water atmosphere. The column headers give the partial pressure of CO2 at the surface. For each row, the mass of the ocean is held fixed at the indicated amount. The mass of the ocean is expressed as the pressure that would be exerted by the ocean if the water were condensed out into a liquid layer. For a planet with g = 10m/s2, a 100 bar ocean corresponds to a mass of 106kg/m2, or a depth of about 1 km. The 25 bar and 50 bar cases were computed with a surface temperature of 540K, while the 100 bar case was computed at 570K so as to allow for a more massive water content without bringing the surface too close to saturation. Note that, as discussed in Chapter 2, the equivalent pressure of ocean differs somewhat from the partial pressure of water at the surface, since the mixing ratio of water is not uniform above the altitude where condensation first occurs.

Table 8.3: Table of water vapor molar concentrations at a 200K cold trap, for a CO2-water atmosphere. The column headers give the partial pressure of CO2 at the surface. For each row, the mass of the ocean is held fixed at the indicated amount. The mass of the ocean is expressed as the pressure that would be exerted by the ocean if the water were condensed out into a liquid layer. For a planet with g = 10m/s2, a 100 bar ocean corresponds to a mass of 106kg/m2, or a depth of about 1 km. The 25 bar and 50 bar cases were computed with a surface temperature of 540K, while the 100 bar case was computed at 570K so as to allow for a more massive water content without bringing the surface too close to saturation. Note that, as discussed in Chapter 2, the equivalent pressure of ocean differs somewhat from the partial pressure of water at the surface, since the mixing ratio of water is not uniform above the altitude where condensation first occurs.

approaches unity (pure steam) when there is little CO2 present, but that the cold trap concentration falls to very small values as the CO2 inventory approaches values similar to that of Venus. or the CO2 equivalent of Earth's crustal carbonates. Also, for any fixed partial pressre of CO2 at the surface, the cold trap concentration increases as the water inventory increases. Still, for a 90bar ocean (about half the mass of Earth's), and with a 90bar inventory of CO2, the cold trap concentration is only 2.4%. Thus, unless the CO2 inventory on a planet is very low or the water inventory is very high, the cold trap is likely to impose a significant barrier to water loss during a dry runaway scenario. Even if the water inventory is initally high, as water is lost the cold trap becomes a progressively more severe impediment, making it hard to lose the last 90bars worth of ocean, and even harder to lose the last 50bars.

For a single-component condensing atmosphere such as a water-vapor dominated runaway atmosphere on Venus or a condensing CO2 atmosphere on Mars, one no longer has to consider the cold-trap issue, however. If there is only a single atmospheric component, then perforce knowing the total homopause density tells us the particle density of the atmospheric substance, regardless of how much condensation it has undergone in the troposphere.

Besides condensation traps, there can be chemical reactions which affect the homopause concentration. Notably, H2 has little chance to escape in the modern oxygenated Earth, because it oxidizes to the heavier, condensible H2O before it has a chance to reach the homopause.

Now let's revisit the problem of hydrogen loss from Early Earth, a runaway-state Venus, and Titan. We'll assume a mixture of hydrogen with some other gase in a known proportion at the homopause, and then use the scale heights of the two gases to compute the changing composition as the exobase is approached. This allows us to say when hydrogen dominates the exobase, and what the resulting exobase height is. An important complication is the anomalously small collision cross-section of atomic hydrogen, and we must remember to take this into account when computing the mean free path for hydrogen-dominated exospheres.

For the anoxic Earth, we wish to determine how high the homopause concentration has to be in order for the escape flux to equal the volcanic outgassing. We simplify the problem by assuming H2 to be well mixed below the homopause, but to dissociate into atomic hydrogen just above the homopause. Thus, if we know the homopause concentration of atomic hydrogen, the well-mixed tropospheric H2 density is half this value. Start by assuming the atomic hydrogen density at the homopause to be 20%, and that the balance of the atmosphere is N2. Using the scale heights for the two gases, when we compute the exobase position taking into account the varying composition with height, we find that the exobase is completely hydrogen dominated, and that the exobase has moved out to an altitude of 1853km (based on an exobase temperature of 300K). The escape flux from this extended pure hydrogen exobase is $ = 1012/m2s, which is still three orders of magnitude below the estimated volcanic outgassing rate of H2. Unless some more effective escape mechanism intervenes, hydrogen should build up to extremely high concentrations in the lower atmosphere.

For Venus, we assume an all water-vapor lower atmosphere. We take the homopause density to be 1.2 • 1019 and assume that one half of the water vapor there breaks up into atomic hydrogen and oxygen. To avoid dealing with a three-component atmosphere,we'll somewhat arbitrarily ignore the resulting oxygen (perhaps it recombines into O2 which has such a small scale height that not much of it reaches the exobase) and compute the exobase from a homopause composition consisting of one third water vapor and two thirds atomic hydrogen; in addition, we'll assume a 300K exobase temperature. The exobase is again found to be hydrogen dominated, and at the relatively high altitude of 3050 km above the surface. The escape flux is $ = 1.1 • 1014/m2s, which would remove the hydrogen in one bar of water vapor in 200 million years. This is significant, but in 2 billion years one could only remove ten bars of ocean. By this means one could get rid of an ocean only about a tenth the mass of Earth's, though one could get rid of more if one could justify using a higher exobase temperature. Assuming that the water vapor at the honopause dissociates completely into atomic oxygen and atomic hydrogen changes these numbers very little, since the exobase is still hydrogen dominated.

It should be remarked that it is hard enough to get rid of an ocean's worth of hydrogen on a runaway Venus, but getting rid of an ocean's worth of oxygen by escape to space is completely out of bounds and none of the other escape mechanisms we will consider come close to closing the gap. The only hope of getting rid of the oxygen resulting from runaway followed by hydrogen escape is to react the oxygen with crustal rocks. Even this is problematic, since a great volume of crustal rock must be made available in order to take up the oxygen from an appreciable ocean. Whether this is indeed possible is one of the outstanding Big Questions. There is no data that absolutely forces us to assume that Venus indeed started with an ocean, so it remains possible that Venus was quite dry from the very beginning.

In our earlier calculation of hydrogen loss from Titan we found that a 10% hydrogen concentration at the exobase was sufficient to sustain a large thermal escape rate. How low does the homopause concentration have to be in order to keep the exobase N2-dominated? To answer this, we again make use of the scale heights of the two gases to compute the exobase composition simultaneously with the exobase height. In this case, we find that with a 300K exobase, the homopause mixing ratio of hydrogen must be 10-6 or less in order to keep Titan's exobase N2 dominated. With that homopause concentration, the escape flux is $ = 8.8 • 1015/m2s, which is somewhat less that our previous estimate (mainly because of the different means of estimating the exobase density). The main conclusion to be drawn from this exercise is that it only takes a tiny hydrogen concentration at the homopause to sustain the large escape rates we computed earlier. If the hydrogen concentration is increased to the point that the exosphere begins to become hydrogen dominated, then the exobase in fact moves out to infinity, because of the large scale height and low gravity. In that regime, hydrogen is likely to escape hydrodynamically (Section 8.7.4) rather than thermally.

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