Hydrodynamic escape is basically a more efficient means of deploying the energy available to the atmosphere in order to assist escape. The energy involved still comes from EUV absorption or the general thermal energy of the atmosphere, but instead of this accumulating in a more or less random set of motions, in some circumstances the energy can sustain a mean outward escaping flow which carries fluid to space without wasting energy on motions directed toward the planet or on a population of molecules with velocities too small to escape. In these circumstances, there is no longer an exobase from which particles escape directly to space. Instead, there is an outflow that acts like a collisional fluid out to distances so great that the atmosphere is no longer gravitationally bound. Hydrodynamic escape plays a very central role in hydrogen-rich outer atmospheres (including those that arise from dissociation of water vapor), and so we will accord it a great deal of attention. The phenomenon can also play a role in escape of heavier species in the case of small bodies or very hot planets. Hydrodynamic escape is a fascinating and important subject, and one which is very ripe for further research. It is also a subject where the student can attain a nearly complete level of understanding on the basis of some simple principles of thermodynamics and mechanics. Therefore, it is a subject which we will delve into at some considerable length.

The starting point for our discussion is Newton's Law - force equals mass time acceleration -written for the radial direction measured outward from the planet's center. Let r be the radial position, and suppose that the only nonvanishing velocity is the radial velocity w(r). We suppose further that the system is in a steady state, so that winds, temperature and so forth are time-independent when measured at any fixed position. This does not mean that acceleration vanishes, however, since the acceleration must be measured following the path of an outward-moving fluid particle, whose position can be written r(t). Since w = dr/dt, the acceleration following a fluid parcel is dw/dt = (dw/dr)(dr/dt) = w(dw/dr). Let rs be the radius of the planet's surface 4 , and gs = g(rs) be the surface gravity. Then Newton's law (expressed per unit volume) becomes dw dp r2 , pw— = - dp - pgs r2 (8.23)

dr dr r2

When w = 0 this reduces to the hydrostatic balance given in Eq. 8.12. Atmospheres are never exactly at rest, and so the hydrostatic approximation we have been using throughout this book amounts to an assumption that the accelerations on the left hand side of the equation are negligible compared to the individual terms on the right hand side. What we are up to now is the business of figuring out what happens when the radial acceleration becomes large enough to disrupt the hydrostatic balance. Basically, we only need to solve the radial momentum equation subject to suitable boundary conditions; the resulting solution determines the outward mass flux. However, there are a number of subtleties concerning the circumstances in which a steady solution can exist, and the nature of the boundary conditions that can be applied. Therefore, we proceed to the solution through a number of intermediate steps.

To obtain a solution, the momentum equation must be supplemented by mass conservation and thermodynamic relations. For steady flow, conservation of mass requires that the mass flux must be independent of r. Defining the area of the shell at radius r as A(r) = 4nr2, the mass flux p(r)w(r)A(r) is constant. It is convenient to define the mass flux per unit surface are of the planet,

which is of course also constant. The thermodynamic relations needed consist of the equation of state (p = pRT in the present discussion) and the corresponding equation for potential temperature (0) or entropy (cp ln 0). In the adiabatic case, the entropy is independent of r and is fixed by the boundary conditions. In the general case including heating, we need an equation for the radial variation of entropy, which we will bring in later.

The transition from flow speeds slower than that of sound (subsonic flow) to flow speeds greater than that of sound (supersonic flow) plays an important role throughout the following, so we will need to know the speed of sound. The sound speed will be denoted by the symbol c in this section, since there is little risk of confusion with the speed of light here. For an ideal gas with gas constant R and temperature T, the speed of sound is given by c2 = yRT, where 7 = cp/cv (see Problem ??). With a little manipulation, Eq. 8.23 can then be re-written to yield a powerful constraint on the circumstances in which a one-dimensional flow can smoothly make the transition from subsonic to supersonic. We start by dividing the equation by p and working with the resulting pressure gradient term p-1dp/dr. Using the definition of potential temperature, this term can be re-written

4 Any other convenient reference radius can be used in place of rs

Next, we need to use mass conservation to eliminate p. Specifically, we take the log of Phi = pwA(r)/A(rs), then take the derivative with respect to r, and solve for dln(p)/dr. Upon substitution of the result into the momentum equation and rearranging terms we find c2 dw 2 dln(A/9) rS,

w2 dr dr r2

The ratio w/c is the Mach number, for which we will use the symbol M. Eq. 8.26, called the transonic rule, implies that the right hand side must vanish at the point where M = 1, if we require that dw/dr be finite there. This relation is valid even in the presence of diabatic heating due to radiation, thermal diffusion or any other means; diabatic heating causes 0 to vary with r, and the entropy equation needs to be used to obtain this gradient in terms of the heating rate. The point where M = 1 is called the sonic point, or sometimes the critical point.

If gravity is set to zero, Eq. 8.26 also constrains the flow of fluid in a tube with cross section area A(r), with r being the distance along the axis of the tube (see Problem ??). In that context, it implies that if one wants to create an adiabatic supersonic jet by feeding subsonic flow into one end of a tube, then things must be arranged so that the sonic point occurs at a constriction of the tube where the area has a local minimum. In particular, one cannot make a supersonic nozzle in the intuitive shape of a cone with the point snipped off. This realization was the basis of the design of the de Laval nozzle 5.

If the heating vanishes in the vicinity of the transonic point, 0 is constant there and the transonic condition becomes

where wesc is the escape velocity at radius r. Thus, the transonic rule states that at the point of transition between subsonic and supersonic flow, the speed of sound must be half the escape velocity. Using the expression for c2 , this condition determines the temperature at the sonic point once the position is given. Except for small bodies with low surface gravity, the sonic point must be very far out from the planet if one is to avoid temperatures temperatures far higher than are likely to be sustainable given the supply of energy. For example, with H2 on Earth the sonic point temperature would be over 4800K if it were placed at 1.1 Earth radii from the planet's center. At 30 radii out, where the gravity is weaker, the sonic point temperature falls to 177K. For heavier molecules, the temperatures are much higher. For N2 the sonic point temperature would be 2500K even at 30 Earth radii. Numbers for Venus would be similar. Because it is difficult to sustain such high temperatures, hydrodynamic escape of gases much heavier than H2 from Earth- or Venus-sized bodies is not likely, except perhaps for planets in orbits where they receive much more radiation from the primary star than does Earth or Venus. For smaller bodies, escape of heavier gases starts to come more within the realm of possibility; for N2 on Mars, the sonic point temperature is 503K at 30 Mars radii, and on Titan it is 139K at 30 Titan radii. Before long, we will learn how to compute how much absorbed solar radiation is necessary to sustain such temperatures.

The next step is to derive an energy equation, which we do by rewriting the pressure gradient term in the momentum equation in a different form from that used in the preceding discussion. The first law of thermodynamics states that cpdT — p-1 dp = SQ, where SQ is the heat added per

5Gustaf Patrick de Laval (1845-1913) was a Swedish inventor who developed the de Laval nozzle as a way of making a more powerful steam turbine. Subsequent developments led to rotary separation of oil and water, which found even greater commercial applications in the dairy industry, to the problem of cream separation. Centrifugal cream separators were the mainstay of his company, Alfa Laval, which exists to this day. de Laval also invented the first commercially viable milking machine. The de Laval nozzle was first used for rocketry by Robert Goddard, and makes a cameo appearance in Homer Hickam's book, Rocket Boys (which became October Sky when it reached the silver screen).

unit mass, as defined in Chapter 2. If we divide by dr then the first law can be used to rewrite the pressure gradient term p-1dp/dr in the momentum equation. Assuming cp to be constant, the result can be put into the form

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