As in the treatment of the hydrostatic relation in Chapter 2, the equation is closed by using the ideal gas law, p = pRT, where R is the gas constant for the mixture making up the uppermost part of the atmosphere. In escape problems it is often more convenient to deal with particle number density rather than mass density. The particle density n(r) is obtained by dividing p by the mass of a molecule, m. If the upper atmosphere is isothermal with temperature To, then the solution expressed as number density is:

Hs r ro gs where ro is some reference distance, generally presumed to be in the upper atmosphere. From this equation, it follows that the local scale height at radius r is H(r) = (r/rs)2Hs.

Because of the attenuation of gravity, the density no longer asymptotes to zero at large distances from the planet. The limiting value is = n(ro) exp —(rs/Hs)(rs/ro). Since this formula neglects all gravity but that of the planet, it is not surprising that one can fill infinite space with a finite density and have it stay put, since there is essentially zero gravity out there. To do so would require infinite mass, though, which means that if the planet is initially endowed with a finite-mass atmosphere, it will all leak away to space, given sufficient time. The estimate of that time scale is the objective of this section. In reality, the limiting density is not achieved, because the atmosphere is trucated by particle loss from the exobase. Still, it is important that there is a nonzero limiting density, since this implies that there is a nonzero limiting mean free path. Combined with the fact that the scale height increases with r, this can remove the exobase to infinity, meaning that there is no altitude at which the atmosphere can be considered collisionless. This situation won't arise for heavy constituents on reasonably massive bodies, since is exceedingly small. Using a collision radius of 125 picom, the limiting mean free path is over 10167m for O on Earth, based on a temperature of 300K. This is well in excess of the size of the Universe. The limiting value for heavier constituents is even greater, and considerations for Venus turn out similarly. Even for Titan, the limiting mean free path for N2 is about 1044m based on a temperature of 100K. The situation for the light species H and H2 is more ambiguous. For example, if Titan had a pure H2 atmosphere with a temperature of 100K, the limiting mean free path would be only 850 m even if the surface pressure were a mere 0.1 Pa. This would certainly preclude a collisionless exosphere. Since H and H2 typically appear in the atmospheres of terrestrial-type bodies as minor constituents mixed in with a heavier gas, further discussion is best deferred until after we have considered inhomogeneous atmospheres.

Exercise 8.7.1 Show that Eq. 8.13 reduces to the conventional hydrostatic relation derived in Chapter 2 when r = rs + z with z/a < 1, rs being the radius of the planet.

To determine the exobase height, we need a model of the atmospheric structure which gives us the total number density n(r) as a function of position. This can be challenging to do precisely, since n(r) depends on the temperature and composition profile of the atmosphere; often, exobase heights for present-day Solar system planets are calculated from measured rather than theoretical density profiles. The other factor involved is the scale height at the exobase, which depends on the exosphere composition, position (through gravity) and temperature. Larger temperature increases the scale height and hence tends to move the exobase further out. The temperature of the exosphere is determined by a balance between heating by absorption of solar radiation (mainly ultraviolet), outgoing thermal infrared from deeper in the atmosphere, and infrared radiation. In some cases, there can be energy gain from collision with solar wind particles, and there can also be energy loss by outward streaming of mass in hydrodynamic escape (see Section 8.7.4). The radiative transfer in the exosphere is simplified by the fact that the atmosphere is optically thin, but is complicated by the fact that it is so tenuous that local thermodynamic equilibrium (and hence Kirchoff's laws) is not accurate. Still, when the exosphere is made of a good infrared emitter, the temperature tends to be on the order of a skin temperature, augmented a bit by solar absorption. Thus, the CO2 dominated exobase of Venus has temperature between 200K and 300 K, and that of Mars is somewhat larger. The Earth's exosphere is unusually hot, since it is dominated by atomic oxygen which comes from photodissociation of the large O2 concentration of the lower atmosphere. Atomic oxygen is a good ultraviolet absorber, but radiates infrared poorly, leading to high temperatures. Exosphere temperatures for N2 or N dominated exospheres are a delicate matter, since N2 neither absorbs nor emits well, and slight contamination by infrared emitters or good solar absorbers can make a big difference. Titan's N2 dominated exosphere has an observed temperature of about 200K.

Once the parameters of the outer atmosphere are settled, the exobase position is determined by solving i(r)/H(r) = 1 by iteration, where H(r) is the scale height at position r and the mean free path ell(r) is inversely proportional to n(r). To get some numbers on the table for discussion, let's adopt a simple model atmosphere in which n(r) is computed based on Eq. 8.13 with a uniform equivalent lower atmosphere temperature To all the way down to the planet's surface, where the surface pressure ps is specified. The exobase temperature is specified separately. Exobase altitudes for some hypothetical single-component atmospheres are given in Table 8.2. The atomic oxygen case for Earth is meant to serve as a crude representation of the oxygen-dominated exosphere of Earth. In this approximation, the atmospheric structure is computed as if all the Earth's oxygen were in the form of O, and ignores the fact that the O2 is only converted to O at altitudes above 100 m as well as ignoring the effect of other gases on the vertical structure. We'll be able to do better later when we take up mixed atmospheres, but it is interesting to note that the exobase height of 400 km in this approximation is not too far from the true exobase height (500 km) computed on the basis of the observed O density in Earth's upper atmosphere. The N2 case may

Planet |
ps (bar) |
T o |
ex |
Zex (km) |
toss (GYr) |
wj,h, m/s |

Earth,N2 |
1.0 |
300 |
300 |
221 |
7 • 10286 |
4.97 • 10-7 |

Earth, O |
0.2 |
300 |
1000 |
401 |
4 • 1042 |
7.94 |

Venus, CO2 |
90.0 |
500 |
300 |
304 |
> 10300 |
1.7 • 10-5 |

Venus, H2O |
1.0 |
400 |
300 |
542 |
7 • 10147 |
3.458 • 10-5 |

Mars, CO2 |
2.0 |
250 |
300 |
352 |
3 • 1081 |
36.41 |

Mars, O |
1.0 |
250 |
1000 |
1312 |
4000 |
808.13 |

Titan,N2 |
1.5 |
100 |
200 |
774 |
3 • 1015 |
268.83 |

Titan,N |
1.5 |
100 |
200 |
2454 |
5000 |
365.18 |

Moon, N2 |
1.0 |
260 |
300 |
6145 |
4 |
613.54 |

Table 8.2: Characteristics of the exosphere and loss rates for various hypothetical single-component atmospheres. The planet and the atmospheric composition are given in the leftmost column. ps is the surface pressure in bars, To is the effective mean temperature of the atmosphere below the exobase, Tex is the exobase temperature, zex is the altitude of the exobase above the surface (in kilometers), tioss is the time needed to lose the atmosphere by thermal escape (in billions of years), and wjh is the Jeans escape coefficient for atomic hydrogen (in m/s). The hydrogen escape coefficient assumes that hydrogen is a minor constituent at the exobase. The mean free path was computed using a fixed molecular collision radius of 125 picom in all cases.

Table 8.2: Characteristics of the exosphere and loss rates for various hypothetical single-component atmospheres. The planet and the atmospheric composition are given in the leftmost column. ps is the surface pressure in bars, To is the effective mean temperature of the atmosphere below the exobase, Tex is the exobase temperature, zex is the altitude of the exobase above the surface (in kilometers), tioss is the time needed to lose the atmosphere by thermal escape (in billions of years), and wjh is the Jeans escape coefficient for atomic hydrogen (in m/s). The hydrogen escape coefficient assumes that hydrogen is a minor constituent at the exobase. The mean free path was computed using a fixed molecular collision radius of 125 picom in all cases.

be thought of as approximating an Early Earth situation in which there is little O2 available to feed an oxygen-dominated exosphere. The two Venus cases represent approximations to the present state of Venus, and a hypothetical past near-runaway state with a pure steam atmosphere. The first Mars case assumes a dense CO2 atmosphere such as might have prevailed on Early Mars, while the second is an approximate to the situation where the exosphere is dominated by atomic oxygen arising from photodissociation of CO2. The first Titan case approximatess the present, while the second gives some indication of what would happen if the N2 were to dissociate into molecular nitrogen (a somewhat implausible situation, but one which is included to allow us later to put a generous bound on nitrogen loss from Titan). Finally, the N2 lunar atmosphere gives us an indication of what an atmosphere on Earth's Moon might have looked like if it retained or gained an atmosphere after formation. The lower atmosphere temperature approximates the temperature the Moon would have with little or no greenhouse gases in its atmosphere.

Relative to the planetary radius, the estimated exobases are all fairly close to the ground with the exception of the atomic oxygen case on Mars, the N2 case on Titan, the atomic nitrogen case on Titan, and the N2 case on the Moon. In the first two cases, the altitude of the exobase is on the order of a third of the planetary radius, but in the latter two cases the exobgse extends far out into space. The effect is particularly pronounced in the Lunar N2 case. Note that the exobase extends much farther out than in the Titan N2 case, even though the Moon has somewhat higher surface gravity than Titan. This happens because we have assumed a greater atmospheric temperature for the Lunar case, consistently with its closer proximity to the Sun. This remark underscores the importance of lower atmospheric temperature in determining the characteristics of atmospheric escape: perfectly apart from the exospheric temperature, a hotter lower atmosphere has a larger scale height, and therefore can extend further out to where the gravity is lower and the atmosphere can escape more easily. This is not much of an issue for bodies as massive as Earth or Venus but for smaller bodies it can be quite a significant effect.

Let's take stock of what we know so far. To determine the rate of escape of a consitutent, we need to know the height of the exobase, the number density of that constituent at the exobase, and the proportion of particles whose energy exceeds the escape energy computed at the exobase. The definition of the exobase involves the temperature at the exobase, through the definition of scale height, so we must know a temperature for the exobase as well. This temperature might or might not also serve to characterize the distribution of particle velocity, depending on circumstances. Note that the concept of "exobase" is itself a severe idealization. The picture this calls to mind is of a distinct surface separating lower altitudes where collisions are frequent enough to maintain thermal equilibrium and higher altitudes where particles undergo ballistic trajectories without collision. This would be nearly the case for evaporation of a liquid into a vacuum, since there is a near-discontinuity in density in that situation. For gases, the transition is gradual, and it would be better to talk in terms of an "exobase region" involving a continuous profile of collision frequency and some escape to space from each layer - more toward the top, less toward the bottom. Modern calculations of atmospheric escape do indeed employ this level of sophistication, but the refinement alters estimates based on the ideal picture only by a factor of two or so. We'll see soon that this is not a serious threat to our main conclusions.

To proceed further, we need a probability distribution for molecular energy. The simplest case is one in which the molecules near the exobase can be regarded as being in thermodynamic equilibrium. This leads to what is called Rayleigh-Jeans or thermal escape. It is by far the simplest theory of atmospheric escape, but it is also the most useless; its main utility is to show that thermal escape is not a significant means of removing atmospheric constituents with the possible exception of light species such as He or molecular hydrogen (and even those only to a limited extent and in limited circumstances). The calculation proceeds as follows. For a gas in thermodynamic equilibrium at temperature T, the probability of a molecule with mass m having speed v is proportional to exp(-1 mv2/kT). This is the Maxwell-Boltzmann distribution. Note that if the gas is a mixture of molecules with various m, this formula still applies for the velocity distribution of each species separately, with the corresponding m used in the formula. The Maxwell-Boltzmann distribution has the important property that the proportion of molecules with energy much greater than kT becomes exponentially small. To determine the escape flux, one must integrate the Maxwell-Boltzmann distribution to determine the proportion of particles that have enough energy to escape, taking into account also the fact that particles are moving isotropically in all directions and that it is only the part of energy associated with radially outward motion that contributes to escape. If nex,m is the number density at the exobase of a species whose molecules have mass m, then the flux of particles to space is wj,mnex,m, where where Ac(m) = mg(rex)rex/kT is the escape parameter defined previously. The Jeans flux coefficient wJm has the dimensions of a velocity, and consists of the typical thermal velocity at the exobase reduced by an exponential factor that accounts for the proportion of molecules whose energy exceeds the escape energy. The total escape flux from the planet, expressed as molecules per second is 4nr'2xwJ,mnex,m. For calculations of the lifetime of an atmospheric constituent, it is convenient to introduce the escape flux per unit surface area of the planet, for which we will use the notation $. Thus, $ = wj,mnex,m(rex/rs)2.

The penultimate column of Table 8.2 gives the characteristic loss time of the dominant constituent of the hypothetical atmosphere by Jeans escape. This loss time is obtained by dividing the Jeans loss rate for the dominant constituent into the total number of particles in the atmosphere. With the exception of the Lunar N2 case, all the loss times are far too long to allow significant loss over the lifetime of the Solar system. For that matter, most of the loss times are well in excess of the lifetime of the Universe, and not even the extreme assumption of total decomposition of the atmosphere into lighter atomic constituents changes this conclusion. For the most part,

the main constituents of terrestrial-type atmospheres cannot escape to any significant degree by thermal means. In particular, it is impossible to lose a primordial Venusian ocean by Jeans escape of H2O. Even if we split off the O, the Earth atomic oxygen case says that it would be impossible to lose any significant quantity of the oxygen in the water by Jeans escape. The one case in which thermal escape is of interest for a heavy constituent is the warm Lunar case. This case may seem somewhat fanciful but it is of considerable relevance ot the question of habitable moons, such as might belong extrasolar gas giants which orbit their primaries at Earthlike distances. Jeans escape is a real threat to the atmospheres of small moons if they are warm enough to support liquid water.

To get a feeling for the magnitude of the thermal escape of atomic hydrogen in the regime where H is a minor constituent of the exosphere, let's suppose that the molar concentration of H is 10% at the exobase. Later we'll learn how to relate the exobase concentration to the composition of the lower atmosphere. Let's take first the Earth case with an O dominated exosphere. Using the assumptions of Table 8.2 the total number density at the exobase, obtained by plugging the exobase position and gas constants into Eq. 8.13, is 2.4 • 1014/m3, whence the H number density is 2.4 • 1013/m3. Multiplying this by the Jeans escape coefficient from the table and normalizing to surface area, we find an H escape flux $ = 2.15 • 1014/m2s. If the H ultimately came from decomposition of sea water, then each two atoms of H that escape account for the loss of one water molecule and the generation of one atom of oxygen. Converting this to mass, we find that in four billion years you could lose about 365, 000kg of water from each square meter of the Earth's surface, equal to a depth of about 365m. You could not come close to losing an ocean on Earth or Venus this way, even with a hot exosphere; with a colder exosphere such as on Venus or the Early Earth, the loss rate would dwindle to practically nothing. If we want to get rid of a primordial Venusian ocean, we'll have to look at hydrogen-dominated exospheres, and find means other than thermal escape to pump the hydrogen into space.

Hydrogen, in the form of H2 is one of the substances commonly outgassed from volcanoes on Earth and probably on other geologically active planets. In Earth's present highly oxygenated atmosphere, this hydrogen rapidly oxidizes into water, so there is little opportunity for free hydrogen to accumulate. On the anoxic early Earth, however, the accumulation of hydrogen would be limited by the rate of escape to space. For a cold N2 dominated exosphere, the exobase density of atomic hydrogen is 6.7 • 1015/m3 if the molar concentration is 10%. Using the Jeans escape coefficient from Table 8.2, the hydrogen escape flux would be a mere $ = 3.32 • 109/m2s . It has been estimated that the volcanic outgassing rate of H2 on the Early Earth could have been on the order of 1015 molecules per second per square meter of Earth's surface, which is many orders of magnitude in excess of the Jeans escape flux. Thus, if Jeans escape were the only escape mechanism for hydrogen, hydrogen would accumulate to very high concentrations on the Early Earth. In reality, it would only accumulate to the point where the exosphere became hydrogen-dominated, whereupon other, more efficient escape mechanisms would take over. Still, we have learned from this exercise that hydrogen has the potential to build up to high values on an anoxic planet, that a cold exobase plays an important role in allowing this to happen, and that the exosphere is likely to have been hydrogen-dominated.

Another case of interest is hydrogen loss from Titan. In this case, the hydrogen is supplied by decomposition of CH4 in the atmosphere, and the loss is important because the atmospheric chemistry would change quite a bit if hydrogen stuck around in the atmosphere. Let's suppose a 10% atomic hydrogen concentration at an N2 dominated exobase on Titan. From Table 8.2 , we see that the Jeans escape coefficient for atomic hydrogen is very large in the Titan case, owing to the low gravity. The exobase particle density is about 1.3 • 1014/m3 based on the table, whence the assumed hydrogen density is 1.3 • 1013/m3 and the escape flux is $ = 6 • 1016/m2s. Assuming a CH4 molar concentration of 30% over a layer 15km deep in Titan's lower atmosphere, there are

1.3 ■ 1030 hydrogen atoms per square meter of Titan's surface, stored in the form of CH4. The calculated Jeans escape rate would be sufficient to remove this entire inventory in under a million years. The precise rate of hydrogen loss depends on the rate of decomposition of methane and the rate of delivery of hydrogen to the exobase, but it seems quite likely that Jeans escape can get rid of the hydrogen resulting from methane decomposition.

Before taking on more complicated and effective means of escape, there is one more basic concept we need to take on: diffusion and gravitational segregation of atmospheric species. Let's track the position of a molecule of species A moving with typical speed v, and colliding with background molecules from time to time. The typical distance the molecule moves between collisions is the mean free path i computed earlier. If we idealize the collisions as causing a randomization of the particle's direction, then the particle will undergo a random walk. For particles undergoing random motions of this sort, the flux is proportional do the gradient of particle concentration; the process is called diffusion, and the proportionality constant is the diffusion coefficient, which we shall call D. It is closely related to the heat diffusion coefficient we have introduced in previous chapters. In addition, molecules or atoms in a gravitational field will accelerate downward under the action of gravity until the drag force due to collisions with the rest of the gas equals the gravitational force. This equilibration happens quickly, so that the particles attain a terminal fall velocity w f. The terminal velocity is proportional to the local acceleration of gravity, and leads to a downward particle flux which is the product of the fall speed with the particle density.

The diffusion coefficient has units of length squared over time, and by dimensional analysis must be proportional to the product of mean free path i with the typical thermal velocity ^JkT/m where m is the particle mass of the species we ar tracking. Since i is inversely proportional to the total particle density n, the diffusion coefficient increases in inverse proportion to n. For this reason, it is often expressed in terms of a binary diffusion parameter b, via the expression b = Dn. For any given pair of species, b is a function of temperature alone. For ideal hard-sphere collisions between a particles with masses mi and m2 and radii ri and r2, the binary collision parameter is given by the expression

64 x where x is the collision cross section area based on radius (ri + r2)/2 and v = \JkT/m is the thermal velocity based on the harmonic mean of the masses, m = mim2/(mi + m2). For an ideal hard-sphere gas the binary parameter, and hence the diffusion coefficient, increases with the square root of temperature. For actual gases, however the effective collision diameter goes down somewhat with temperature, leading to other empirical temperature scaling laws, generally with temperature exponents in the range of .7 to 1. As with collision radius, atomic hydrogen is an exception, having a temperature scaling exponents between 1.6 and 1.7 for collisions with most species. The magnitude of the binary parameter for atomic hydrogen is also greater than one would expect from hard-sphere theory, since the effective collision radius is that of atomic hydrogen even when it is colliding with a substantially larger particle. Thus, atomic hydrogen has anomalously large diffusion, which increases anomalously strongly with temperature. The diffusion coefficient of atomic hydrogen is even larger than one would expect solely on the basis of it's low mass and hence large thermal velocity.

Next, we remark that the fall speed scales with the velocity acquired by the particle through gravitational acceleration in the time between collisions. Thus Wf scales with gi/^JkT/m. The ratio D/wf thus is proportional to (kT/m)/g = RAT/g, where Ra is the gas constant for the species A. This is just the isothermal scale height Ha that the species would have in isolation. In fact a more detailed calculation shows that for an isothermal ideal gas, the ratio D/wf is not just proportional to Ha, but is actually exactly equal to it. (As a short cut to this result, we may argue that it is implied by the requirement that the scale height be equal to the usual hydrostatic result when the species A dominates the atmospheric composition). In the following, we'll keep things simple by restricting attention to the isothermal case, which will be sufficient for our purposes.

Let nA be the particle density of species A, which may be one of many constituents of the gas. Putting together the flux due to diffusion and the gravitational settling, the net flux of the species (upward positive) is

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