## 5. The reaction of the subphotospheric layers to irradiationFrom the stability analysis we have carried out in the previous
section it is clear that we need to know more about the functions
or
(cf. Eqs. 30 and 31) if we
wish to use the stability criteria in a quantitative way. So far we
know only the properties detailed in Eqs. (32)-(35). These,
however, are insufficient for our purposes. Therefore, in this section
we shall first use a simple model to derive
explicitly and thereafter discuss
results of numerical calculations. As in Sect. 4 we shall adopt
the weak irradiation assumption. In this approximation the relation
between the stellar flux ## 5.1. A one-zone model for the superadiabatic layerFor determining we need now a more detailed model of the stellar structure than the one we have assumed in Sect. 3. Whereas in Sect. 3 we have assumed with Kippenhahn & Weigert (1994) that the convective envelope remains adiabatic up to the photosphere, we shall now relax this assumption and take into account the existence of a thin superadiabatic convection zone below the photosphere where convection itself is ineffective as a means of energy transport and energy flows mainly via radiative diffusion. Sufficiently deep in the star, where convection is effective, i.e. adiabatic, the thermal and mechanical structure of the envelope remain spherically symmetric to a very good approximation even in the presence of anisotropic irradiation at the surface (cf. our discussion in Sect. 2). It is only the very thin superadiabatic layer (with a mass of typically ), where energy is mainly transported via radiation, which is strongly affected by irradiation. It is this property which allows us to make the weak irradiation assumption, i.e. to treat the effects of irradiation in a local approximation and with the following simple model. For this we adopt for the moment the temperature-pressure
stratification shown in Fig. 6. This means that we replace the
"true" structure by one in which we assume convection to be adiabatic
out to a point where the pressure is
and the temperature
(where the subscript
We can now to derive by making a simple one-zone model for the superadiabatic layer. For this we shall furthermore assume that the optical depth through this layer is large enough for the diffusion approximation to hold. Then the radiative flux is given by where is the density, the opacity and the other symbols have their usual meaning. Now we consider the transported flux in two different superadiabatic layers (using subscripts 1 and 2) on an arbitrary isobar at some level . The radiative fluxes are then Now we use for the opacity a power-law approximation of the form (10) and the ideal gas equation where is the gas constant and
Inserting now (10) and (65) into (63) we obtain Now we make the one-zone approximation by writing If we now identify layer 1 with the unirradiated one, i.e. set and , and layer 2 with an irradiated one, i.e. set and we obtain by inserting (67) into (66) where we have assumed for simplicity . Eq. (68) can be solved for , thus providing . Thus (68) together with (28) or special cases thereof (Eqs. 52 or 59) can be used as an outer boundary condition for numerical computations. Let us now briefly consider the consequences of our above
assumptions that and
. Because the dominant opacity
source is H We can also compute . Differentiation of (68) yields if . For consistency with Eq. (32) we require if . As can be seen from Eqs. (68) and (69) the functions
## 5.2. Stars with a massAs Fig. 7 shows, in stars
with a mass . This means on the one
hand that the total optical depth between the photosphere (at
) and the point where
is small, in fact too small for our
simple one-zone model, which assumes the diffusion approximation
(Eq. 61), to be applicable. On the other hand, this means also
that in these stars even in the photosphere a non-negligible fraction
of the flux is transported by convection. Therefore, we must ask how
irradiation changes the transported flux if the top of the (adiabatic)
convection zone is at low optical depth. Because the convective flux
is and the value of
is directly influenced by
irradiation, this situation cannot be described by a simple model.
Rather one ought to determine by
solving the full set of equations describing convective energy
transport, i.e. in the simplest case the equations of mixing length
theory. Considering the uncertainties inherent in current convection
theories it is not obvious whether it is possible to make general
statements about the functions ## 5.3. Results of numerical computationsNumerical computations of and have been carried out by HR97 for low-mass main sequence stars of Pop. I chemical composition using a full 1D stellar structure code (Hameury 1991), where the outer boundary condition was changed according to Eqs. (27) and (28) with The results relevant for this paper are shown in Fig. 8 and can be summarized as follows: -
a) With as shown in Fig. 7 and an appropriately chosen value for *b*, i.e. , typical for H^{-}opacity, the predictions of our simple one-zone model are in good agreement with numerical results as long as . In particular, the run of shown in Fig. 8 as a full line for the numerical computations and as a short-dashed line for the one-zone model show that within its range of validity the latter reproduces the numerical results quite well. -
b) for stars with a mass , as we have argued above (Sect. 5.2). -
c) Max 0.5 over the whole mass range of interest, i.e. . This is shown as a dash-dotted line in Fig. 8. -
d) In the point source model (cf Sect. 4.2.2) the integral in (57) has a maximum which is smaller than Max. Max is shown in Fig. 8 as a long dashed line. As can be seen, Max.
© European Southern Observatory (ESO) 2000 Online publication: August 23, 2000 |