Note that 2gsrs is the square of the escape velocity from the surface, and when multiplied by rs /r it becomes the square of the escape velocity from radius r. Let us defer for the moment the business of making sense of the term SQ/dr - which is a bit of a mathematical monstrosity - and explore the adiabatic case, for which SQ = 0. In this case, the expression
must be independent of r. E is the energy per unit mass of the fluid, the three terms representing kinetic energy, internal thermal energy, and gravitational potential energy. Note that when the kinetic energy is negligible and the gravitational term is expanded about rs by writing r = rs + Z with Z/rs ^ 1, E reduces to the dry static energy cpT + gZ derived in Chapter 2. The neglect of w is in fact why this quantity is called static energy.
Eq. 8.29 is much like the energy balance we used to determine the escape velocity, except this time the thermal energy cpT is also in play. The energy must be positive at infinity for escape to be possible, so the threshold for escape is defined by E = 0; evaluating this at r « rs and assuming w to be small there, we find that escape is possible when cpT > gsrs near rs. For atmospheres having temperature low enough that Jeans escape is small, the thermal term is negligible compared to the gravitational term, though, since it can be easily shown that cpT/gs rs is the same order of magnitude as the escape parameter Xc defined earlier.
Exercise 8.7.5 Write down the relation between cpT/gsrs and Ac.
The large r behavior is important, since it provides our boundary condition where the atmosphere meets the near-vacuum of outer space. The combination of mass conservation and energy conservation yields two possible kinds of large-r behavior in the adiabatic case. Since pwr2 is constant, then if w remains finite at large r, p ^ 0 there. If the far-field flow is adiabatic, then p is proportional to (p/po )1/gamma, and so vanishing density implies vanishing pressure, which turn (by the formula for potential temperature) implies that T ^ 0 at large r. Since w is finite but T gets small, the speed of sound approaches zero and the flow is supersonic at infinity. Thus, the branch that is supersonic at infinity has vanishing pressure, density and temperature at infinity, and patches smoothly to outer space. Moreover, since the flow is supersonic, information cannot travel upstream back toward the surface of the planet, so conditions at infinity do not affect conditions lower down in the escaping atmosphere. So far we haven't used the energy conservation equation directly, but only the adiabatic assumption; the supersonic conditions and vanishing of temperature and pressure at infinity can survive the addition of a moderate amount of heating in the exterior region. If there is no heating there, then using Eq. 8.29 energy conservation tells us that if w is nonzero at infinity, it in fact becomes constant there. At large r, both the temperature and gravitational potential energy vanish, meaning that all the energy of the lower atmosphere is converted to kinetic energy of the escaping flow.
On the other hand, w may vanish at large r. In this case, Eq. 8.29 immediately implies that T asymptotes to a constant at large r, whence density and pressure also asymptote to constants. Finite temperature with vanishing velocity implies that this solution is subsonic at infinity. This is not a viable steady state, because it requires that the interplanetary medium exert a back-pressure on the atmosphere to hold it in. The solution also has the unphysical property of filling interplanetary space with gas having a finite density. What happens if we try to set up a subsonically escaping atmosphere? Note that for subsonic flow, information can propagate upstream towards the planet's surface. Therefore, if there is no back-pressure to hold back the flow, it is likely that the escaping outer region of the atmosphere will accelerate to supersonic conditions, while sending a signal upstream that modifies the upstream boundary condition in such a fashion that the transonic rule is satisfied.
For an escaping atmosphere with nonzero outward mass flux, it is therefore generally believed that the solution must be supersonic at large r in order to be physically realizable. If the atmosphere starts with small w at the base of the escaping flow (as is generally required by the large density there), then the relatively high temperature at the base yields a large sound speed, implying that the lower atmosphere is subsonic. Therefore, an escaping atmosphere will have a sonic point, where the transonic rule will need to apply. This allows us to compute the temperature at the base of an adiabatic escaping atmosphere. There is no exobase for atmospheres undergoing hydrodynamic escape, but we will take the base to be at a position where a single gas, light enough to escape hydrodynamically, dominates the atmospheric composition. This position, which we'll denote by rb, could be at the planet's surface in some cases, but more typically would be somewhat above the homopause. In any event, it is generally not terribly far above the planet's surface, as compared to the planet's radius, so rb/rs is typically of order unity. We equate the energy at rb (assuming kinetic energy 2w2 to be negligible there) to the energy at the critical point rc as follows:
CpTb — 22gsrs — = 1 c(Tc)2 + CpTc — 12gsrs —
The second equality makes use of the transonic rule in order to eliminate the temperature and sound speed at the critical (i.e. sonic) point. This determines Tb in terms of the critical point position, the size and gravity of the planet, and the characteristics of the gas. It is a remarkable fact that for a gas made of spherical atoms with no internal degrees of freedom, the base temperature Tb becomes independent of the critical point position, since 7 = | for such gases. This is particularly important given that hydrodynamic escape of atomic hydrogen is of primary interest.
For more complex gases, Tb decreases gently as the sonic point is moved outward, approaching a limiting value at large distances. Assuming rs/rc ^ 1, and rs/rb « 1, then Tb « gsrs/cp. Note that this limiting basal temperature is precisely the same threshold temperature for escape which we computed earlier on the basis of simple energetic grounds. Eq. 8.30 becomes invalid if we move the critical point all the way to rb, since we assumed the kinetic energy to be negligible at low levels when deriving the equation. In the limit where the critical point approaches the base, we get the base temperature directly from the transonic rule, which tells us c2 = 7RTb = 2gsrs(rs/rb), or Tb = 2(7 — 1)-1gsrs/cp if rs/rb « 1. Hence, the maximum base temperature exceeds the minimum by a factor of 2(7 — 1)-1, which is 1.3 for H2, 1.25 for N2 and 1.7 for CO2. However, the high end of the temperature range where the critical point approaches the base is of little physical relevance, since it corresponds to the case where the lower atmosphere has somehow directly been given enough kinetic energy to escape.
Except for small bodies, the threshold temperature for escape is very high: over 3000K
for atomic hydrogen on Earth, or 4386K for H2 - rising to an outrageous 60, 000K for N2. The required temperatures are high simply because the potential temperature is constant, so that the atmosphere is expanding and cooling along the dry adiabat; in order to still be hot enough to match the temperature at the sonic point, it must thus start with an exceedingly high temperature. On Titan,where gravity is weaker adiabatic escape of atomic hydrogen requires a basal temperature of only 167K, while H2 requires 244K and N2 requires 3352K. For the Moon, escape of H2 would require a minimum temperature of 198K; this is somewhat cooler than the Titan case because the greater surface gravity of the Moon is more than made up for by its smaller radius. Adiabatic hydrodynamic escape of hydrogen from Titan or the Moon seems within the realm of possibility, but for larger,denser bodies it would be possible only in exotic high energy conditions. In fact, high-temperature adiabatic escape is much more relevant as a theory for the solar wind than it is for escape of most planetary atmospheres. Indeed, the fluid dynamics we have outlined above was first developed to explain the solar wind, and only later adapted for planetary purposes. For Earth or Mars sized bodies, it is conceivable that the required temperatures could be attained for H or maybe H2 in the aftermath of a giant impact, or perhaps while a magma ocean is still extant. In general, though, hydrodynamic escape of planetary atmospheres must be sustained by absorption of solar (or stellar) radiation at lower altitudes, which offsets the adiabatic cooling of the expanding atmosphere and allows the sonic condition to be met without unreasonably high temperatures at the base. The absorption acts like the boost phase of a rocket, launching the atmosphere toward escape velocity.
At this point we encounter a troubling difficulty regarding the lower boundary condition: for adiabatic hydrodynamic escape, we have little or no ability to control the temperature at the base of the escaping atmosphere, since the basal temperature is nearly (or completely) insensitive to the position of the sonic point. The density (or equivalently, the pressure) at the sonic point is a free parameter, but its value affects only the density at the base, not the temperature. We have already shown that when the atmosphere is too cold, then it runs out of energy if it tries to flow outward, so that the system must instead settle into the hydrostatically balanced solution with no outflow. But what happens if the atmosphere is too hot? What goes wrong when the lower atmosphere is too hot is that the entire atmosphere is too hot to have a sonic point, so it is either supersonic everywhere or subsonic everywhere. The first solution violates the condition that the velocity in the lower atmosphere start off small, while the second solution fails to meet the customary boundary condition at infinity. Yet, it is implausible that an atmosphere with a somewhat lower temperature that is happily escaping will suddenly lose its ability to escape if more energy is added to the system by increasing the temperature. Recognizing that in subsonic flow, the conditions at infinity can send a signal upstream modifying conditions near the planet's surface, it seems most likely that in the too-hot case, a supersonic flow will develop at infinity, which will reduce the air temperature near the planet's surface in such a way as to allow the transonic rule to be satisfied. This situation does not seem to have been explored by numerical simulation at the time of writing, however.
To illustrate how the calculation of escape flux works, let's consider adiabatic blowoff of a pure H2 atmosphere from the Moon, assuming the surface pressure to be 1 bar. In this case, we assume the atmosphere is heated from below by solar absorption at the lunar surface, idealizing the atmosphere as being transparent to solar radiation. The surface temperature is just the equilibrium no-atmosphere blackbody temperature, since H2 nearly transparent to infrared at these pressures. Under these conditions, we can take the base of the escaping atmosphere to be right at the surface. Now, Eq. 8.30 tightly constrains the allowable range of surface temperatures for an escaping atmosphere. Since w was assumed small at the base in the derivation of Eq. 8.30, the formula becomes invalid if we move the sonic point all the way to the surface. However, if we put it fairly close, at r/rs = 1.5, then Tb = 270.5K, while the transonic rule tells us that ths sonic point temperature is Tc = 163K. As Tb approaches the minimum temperature of 197.7K, the sonic point moves to infinity and Tc falls to zero. We'll assume the Moon to be in an orbit for which the surface temperature lies in this range; interestingly, the position of the actual Moon would satisfy this condition if the Moon had enough atmosphere to even out temperature fluctuations. Now, to determine the escape rate, we need the density at the sonic point. Since potential temperature is constant, the temperature ratio Tc/Tb determines the pressure at the sonic point in terms of the surface pressure, and from this and the temperature we get the required density, which is .026kg/m3 when Tb = 270.5K and the sonic point is at rc/rs = 1.5, or .00064kg/m3 when Tb = 208.7K and the sonic point is at rc/rs = 10. The velocity at the sonic point is just the sound speed there (known because we know the temperature), and the mass flux per unit planetary surface area is $ = pc(^RTc)1 (rc/rs)2. This is 56.8kg/m2s for Tb = 270.5K and 24.0kg/m2s for Tb = 208.6K. Note that for any given rc, the critical point density pc is proportional to the surface pressure, so changing the surface pressure just changes the escape flux proportionately. These are very large escape fluxes. The mass of the hypothetical lunar atmosphere is about 62,000kg/m2, and would be lost in just over a half hour in the hotter case and just over an hour in the colder case. We have thus learned that for a Moon-sized body in an orbit where the solar radiation is similar to that received at Earth's orbit today, H2 would be lost by adiabatic blowoff almost at once. As the temperature decreases toward the threshold temperature, then the sonic point moves out to infinity, and the escape flux very gradually reduces; in fact, it can be easily shown that the escape flux approaches zero as the surface temperature approaches the threshold temperature. Hydrodynamic escape is a process with dramatic thresholds: for a surface temperature of 208K the atmosphere is lost in a matter of hours, whereas the escape flow shuts off completely when the temperature drops below 198K (though other escape mechanisms may still cause loss of the atmosphere).
Exercise 8.7.6 Following the reasoning in the preceding paragraph, for an arbitrary body derive a formula for the escape flux $ in terms of the surface temperature and surface density. Show that the escape flux is proportional to surface density, and that the escape flux approaches zero as the surface temperature approaches the minimum surface temperature for adiabatic hydrodynamic escape.
Re-interpretation of the transonic rule in terms of energetics
We can gain a better appreciation of the meaning of the transonic rule, and of what happens when it is violated, by expressing the energy conservation relation in terms of the Mach number. This requires expressing c2 in terms of the Mach number, or equivalently writing T in terms of the Mach number, which can be done by making use of mass conservation, the ideal gas equation of state, and the expression for potential temperature. In particular, if we let (po,To, po, Mo) be the state of the atmosphere at some point ro, then the four relations
T = 0 • (^)R/cp,p = po^(^)1/Y, $ = pcM-^ = pocoMo4, - = (T)2 (8.31)
po 0 po s2 s2 co To can be solved for temperature, yielding
where 3 = 2(y — 1)/(y + 1). This result is valid whether or not the flow is adiabatic. In the adiabatic case, 0 is independent of and 0 = To if we define the potential temperature with regard to the pressure po at the reference point. With the relation 8.32 in hand, Eq. 8.29 can be written
The right hand side can be considered to be a function E(M, r/rs) with 2gsrs and conditions at ro as parameters. This expression is valid even in the presence of heating, in which case 0 is a function of r. For the sake of generality, we have written the expression in terms of an arbitrary reference point ro, but most commonly we will take ro to be a sonic point, at which Mo = 1 by definition and at which moreover the transonic rule is satisfied. The transonic rule gives To in terms of ro and the gravitational acceleration, so in this case the only free parameters governing the shape of the curve are ro (which we'll call rc in this case) and 0. When the flow is moreover adiabatic throughout the escaping atmosphere, 0 = To and the curve is governed by a single parameter, for any given planet.
Recall from Chapter 2 that 7 is determined by the number of excited degrees of freedom of the atoms or molecules making up the gas. For spherical particles with no internal degrees of freedom, 7 = 5/3, while 7 ^ 1 for complex molecules with very many excited degrees of freedom, though for most atmospheric gases 7 > 1.29. Hence, 0 < ft < 2, with the lower limit not being very closely approached in practice. Because ft < 2, it follows that E(M, r/rs) ^ to as M ^ to. In addition, since ft is positive, it follows that E(M,r/rs) ^ to as M ^ 0. For fixed r/rs, the function has a single minimum at M = 1, as can be verified by differentiation of the expression with respect to M. It is the shape of the energy curve, and the occurrence of the minimum at M =1, that underlies the transonic rule.
Exercise 8.7.7 Carry out the suggested differentiation, verify that the minimum of E occurs at M = 1, and write down the expression for Emi„(r/rs).
Some representative energy curves are sketched in Fig. 8.3. For any given energy Eo, there are three possible situations regarding the flows that satisfy E(M,r/rs) = Eo:
• There can be a pair of solutions, one of which is subsonic and the other supersonic.
• There can be a single solution, with M =1 There can be no solutions at all.
Now let's consider what happens if we start with a subsonic solution and track it's evolution as we increase r, which changes the energy curve. The gravitational potential term shifts the curve upward without changing its shape, while the r-dependent factor in the first term of Eq. 8.33 flattens the curve and moves it downward as r is increased. The situation is depicted in Figure 8.3, where we start with T = 300K and M = .01 at r/rs = 1.1. We already know from our previous results that this temperature is too low to allow the transonic rule to be satisfied for Earthlike gravity, so what happens as r is increased? Examining the intersection points between the initial energy and the energy curve, we see that the Mach number increases until it reaches the sonic point at r/rs = 1.167. When r is further increased, the energy curve continues to move upward, however, so there are no solutions compatible with the initial energy. This is the generic behavior when we start from too low a temperature and the transonic rule is not satisfied. The atmosphere can still approach M = 1 from either the subsonic or supersonic side, but the upward
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