Figure 8.3: **CAPTION
movement of the energy curve through the sonic point means that the gradient of M(r) and w(r) have square-root singularities there - a consequence of the violation of the transonic rule. From a physical standpoint, what is important is not so much the singularity as the fact that solutions cease to exist altogether once r is moved past the sonic point.
An examination of Eq. 8.32 shows that the temperature decreases as the sonic point is approached; this increases the Mach number by decreasing the speed of sound. In fact, where the Mach number is small enough that kinetic energy is negligible, the energy equation reduces to conservation of dry static energy cpT + gz, and tells us that the atmosphere ascends along the dry adiabat. This means it can't ascend far, because the temperature falls to zero at a finite height for the dry adiabat. Incorporation of the kinetic energy term causes the atmosphere to run out of energy before the temperature falls to zero. This is the situation we are fighting if we try to make an atmosphere hydrodynamically escape while starting from a realistic base temperature: without some additional supply of energy, the atmosphere runs out of energy before it gets very far. In this case, energetics requires the atmosphere to settle into a state of rest without a mean outflow. When starting from a low temperature on a planet with Earthlike gravity, energy must be deposited in the upper atmosphere in order to allow it to escape. Generally, this energy is supplied in the form of extreme ultraviolet.
Next we'll examine the geometry of the energy curves in a case where the transonic rule is satisfied and adiabatic hydrodynamic escape is possible. The situation is depicted in Fig. 8.4. To pass from a subsonic to a supersonic state, the energy curve first comes up to where the subsonic and supersonic solutions coalesce at the sonic point, but then moves down again as r is further increased, allowing the solution to continue on the supersonic branch. The transonic rule is equivalent to the requirement that when expressed as a function of r, the minimum of the energy curve E(M = 1, r/ro) has a maximum at r = rc, so that the curve first goes up to the sonic point then turns around and heads back downwards again.
Exercise 8.7.8 Verify this property by using the expression for E(M = 1 ,r/ro) and locating its minimum by taking the derivative with respect to r. To keep things simple, you can restrict attention to the adiabatic case 0 = const..
Figure 8.4: Sequence of energy curves for the situation in which the transonic rule is satisfied for r/rs = 10. Left panel: The energy curves move upward as r/rs is increased from 2 to 10. Right panel: The energy curves move downward as r/rs is increased further from 10 to 100.
It is now time to bring heating into the picture. Heating alters the preceding construction in two ways: Firstly, instead of the expression E being independent of r, it will vary on account of the deposition of energy from solar radiation or other sources. This moves the energy curve up or down with r, without changing its shape. Secondly, heating will cause 0 to vary with r, which changes the shape of the energy curve. It remains true, however, that the energy curve for any r has a unique minimum where the Mach number is unity, and that in order to effect the transition from subsonic to supersonic flow, the line defining the available amount of energy at any given r must start above the minimum, be brought to tangency at the minimum (either by moving the curve upward or moving the line downward or some combination of the two), and then moving the line defining available energy to some distance above the minimum as r is increased further.
In order to incorporate heating into the energy equation, we rewrite the diabatic term on the right hand side of Eq. 8.28 as follows:
dr pSQ/dt pdr/dt
where q is the heating rate per unit volume. Now let F(r) be the flux of energy due to means other than the bulk fluid flow, with the convention that inward fluxes are taken as positive. We will mostly deal with radiative flux, but F could equally well represent flux due to molecular diffusion of heat. In terms of the flux, the heating rate is given by q constant the heating term in the energy equation becomes
Since this is the gradient of a flux, Eq. 8.28 tells us that it can be combined with the previous expression for E to yield the revised conservation law
where Eo is a constant. Note that F/$ is an energy flux divided by a mass flux, and therefore has dimensions of velocity squared. For a given variation in energy flux, reducing the mass flux $ leads to a greater radial variation in the energy density E (the first pair of terms in the equation), because fluid has more time to accumulate energy when it is moving slowly.
The treatment of the radiative part of the heating is a bit tricky. As in our earlier calculations of planetary temperature, we are doing a globally averaged energy budget assuming uniform atmospheric conditions over the globe. That requires figuring the amount of EUV flux intercepted by the planet's atmosphere, and distributing it uniformly over the sphere. The assumption of effective redistribution of solar heating over the outer atmosphere is highly questionable, but it is quite customary in hydrodynamic escape calculations and is in any event the only approximation which allows us to make headway without very extensive and complex numerical simulations. That notwithstanding, there is an additional issue that did not arise in our earlier treatments of globally averaged energy budgets. As usual, the planet intercepts a disk of light of some radius, and the intercepted flux is spread uniformly over the surface of sphere of the corresponding radius. The difference in the present case is that the portion of the atmosphere that is dense enough to absorb significant amounts of EUV radiation can extend many planetary radii out from the surface. Therefore, the radius of the disk of intercepted radiation depends on the optical thickness (hence density) of the atmosphere. Let's suppose that the atmosphere is transparent to EUV for radii farther out than some radius rabs, but that the closer-in atmosphere absorbs strongly. Then, if Fq is the incoming flux of EUV from the sun (analogous to the solar constant), the intercepted power is nr2bsFQ. If we want this to be the total power entering the system, then the corresponding uniform radial flux 4nr2F(r) through a shell of radius r must be held constant at the value nr2absFQ for r > rabs. Therefore, we model radiative absorption by stipulating that r2F(r) = 1 r2absFQ for r > rabs while allowing the flux to decay to zero as radiation is absorbed deeper in the atmosphere.
Eq. 8.36 imposes a powerful constraint on the mass flux that can be sustained by a given level of heating. We'll suppose as usual that w is small at the base of the escaping atmosphere, but this time we'll also assume that the base is cool, so that cpT is negligible compared to the gravitational term at the base. This is the typical situation, in which the atmosphere is strongly gravitationally bound and the escape parameter Xc is small. Unlike the adiabatic escape case, we do not endow the base of the atmosphere with enough thermal energy to sustain the outflow, but rather deposit it gradually through absorption of stellar flux, generally in the extreme ultraviolet spectrum. Equating energy at the base (where radiative flux falls to zero) and at infinity yields the relation
Since w2 > 0, this imposes an upper bound on the mass flux $ for any given amount of radiative absorption. Note that this energetic constraint survives the addition of heat diffusion to the flux term, since diffusion only redistributes energy in the vertical and does not add new energy to the system. If (rabs/rs)2 « 1 and rs/rb « 1 as is typically the case, than the constraint is simply $ < 1 F(.^,/gsrs. It is important to recognize that this bound on the escape flux applies only in the low temperature limit, in which cpT at the base is negligible compared to the gravitational potential. For any finite temperature, the escape flux can exceed the limiting flux, by an amount that increases with temperature. As the base temperature approaches the temperature at which adiabatic escape becomes possible, the escape flux can become arbitrarily large, limited only by the density at the base.
The physical content of the constraint is simple: The escaping atmosphere carries kinetic and potential energy with it, and this outward energy flux must be matched by the supply of radiant energy absorbed within the escaping atmosphere. The amount of energy flux escaping due to mass flow is negligible from a standpoint of planetary energy balance; that energy loss is still by far dominated by infrared emission. However, from the standpoint of the energy budget of the outer atmosphere alone, the energy loss due to mass outflow can be the dominant term - at least for gases like H2 which are poor infrared emitters. Infrared emission, to the extent that it occurs at all, can be thought of as stealing energy from the supply of EUV heating available to sustain escape.
In order to complete the solution, we need to know how the potential temperature varies with radius. Once the potential temperature is known, we have enough thermodynamic information to compute the profiles of pressure and density as well. The radial variation of the potential temperature is obtained from the entropy equation:
The heating term (r/rs)2Q can be written as the radial gradient of a flux as before, but because of the factor 1/T appearing in the entropy equation, this equation cannot be integrated to yield a pointwise relation between entropy and flux the way we did for energy. The entropy change between a point r^ and rB depends on the shape of the heating curve between those points, and not just the amount of heat added; heat added at low temperature has a greater effect on entropy than heat added at high temperature.
We'll now exhibit some numerical solutions for the case in which the heating Q is a known function of r. Radiative heating depends on the density and temperature so strictly speaking it must be solved for together with the atmospheric structure; the extension to this case is straightforward once one understands how to solve the simpler problem. Given the heating, the numerical solution proceeds as follows. One starts by choosing the critical point position rc/rs, from which one can compute the critical point sound speed and temperature. Then, one integrates the differential equation 8.38 in a direction toward the planet. Since the escape flux $ appears as a parameter in this equation, one must guess a value of $ to carry out the integration. The chosen value of $ also fixes the density at the critical point since the velocity at the critical point is the local speed of sound. At each step of the integration of Eq. 8.38, one obtains value of ln 0, but to proceed one also needs to update the value of T. This is done by solving the energy equation, Eq. 8.36 (rewritten in terms of Mach number using Eq. 8.33), for the Mach number, following the subsonic branch. The Mach number, in turn, determines the new temperature and allows the integration to proceed further. As the integration proceeds, a point may be encountered in which solutions to the energy equation no longer exist, in which case the chosen value of $ is not realizable. If this situation doesnt arise, the integration is continued until the base rb is reached. One now knows the value of 0 there, which for positive heating will be less (usually much less) than the value at the critical point. The integration has already completely determined the temperature structure of the atmosphere, and he ratio of potential temperature at rb to the value at rc determines the proportionality constant between the density at rb and the (known) density at rc. This procedure yields a family of solutions, with rc and $ as parameters. When $ is large, Eq. 8.38 says that the potential tempreature becomes constant, in which case we recover the adiabatic escape solutions which typically have very high temperatures at the base. As $ is made smaller, heating causes the potential temperature at the base to be much smaller than the potential temperature at the critical point, which results in cooler temperatures at the base. When $ is made too small, however, the temperatures are driven to excessively low values and one encounters at some point a supersonic transition at a radius where the transonic rule is not satisfied, whereupon the solution ceases to exist. Carrying out this procedure requires only the integration of a first order differential equation and the solution of the energy equation at each step using Newton's method. It is quite straightforward to implement.
Before showing a specific family of solutions obtained using the above procedure, it is useful to identify a few relevant nondimensional parameters. The energy-limited escape flux identified earlier provides a convenient scale for nondimensionalization of $. Let's call the limiting flux $*. It can be written 2Fq/w"^sc, where wesc is the escape velocity from the surface, namely ^J2gsrs. We can define a characteristic temperature T* such that cpT* = w\sc. The heating can be written Q = (d/dr)(r2/r2sF), so it can be nondimensionalized by multiplying both sides of the entropy equation by rs (amounting to taking the planetary radius as the unit of length) and then writing F = Fq ■ (F/Fq) If the temperature is nondimensionalized against T*, then the entropy equation depends on the EUV flux, the escape velocity and the escape flux only through the dimensionless combination $/$*. Specifically, the nondimensional entropy equation becomes d ln 0 =1 ^ (8.39)
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