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Astron. Astrophys. 360, 969-990 (2000)

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5. The reaction of the subphotospheric layers to irradiation

From the stability analysis we have carried out in the previous section it is clear that we need to know more about the functions [FORMULA] or [FORMULA] (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 [FORMULA] 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 F and the irradiating flux [FORMULA] is a purely local property.

5.1. A one-zone model for the superadiabatic layer

For determining [FORMULA] 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 [FORMULA]), 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 [FORMULA] and the temperature [FORMULA] (where the subscript B stands for the base of the superadiabatic zone). The superadiabatic convection zone extends from the point [FORMULA] up to the photosphere where [FORMULA] and [FORMULA]. Because in this zone convection is ineffective, we make the simplifying assumption that energy transport is via radiation only, i.e. that [FORMULA], where [FORMULA] is the radiative temperature gradient. Because on an irradiated part [FORMULA], where [FORMULA] is the effective temperature in the absence of irradiation, but [FORMULA] is assumed to be the same irrespective of irradiation, we see that irradiation, by raising the effective temperature, reduces the temperature gradient [FORMULA] and thus the radiative energy loss through these layers. The superadiabatic layer works like a valve which is open if there is no external irradiation and which closes in progression with the irradiating flux. As sketched in Fig. 6 we assume for simplicity that [FORMULA] does not depend on [FORMULA], i.e. on [FORMULA].

[FIGURE] Fig. 6. Sketch of the temperature pressure stratification adopted in the framework of the one-zone model described in Sect. 5.1.

We can now to derive [FORMULA] 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 [FORMULA] is the density, [FORMULA] 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 [FORMULA]. 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 [FORMULA] is the gas constant and µ the mean molecular weight. Since in the regions of interest the dominant species (H and He) are neutral, we may assume [FORMULA]. Thus with [FORMULA] we have


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 [FORMULA] and [FORMULA], and layer 2 with an irradiated one, i.e. set [FORMULA] and [FORMULA] we obtain by inserting (67) into (66)


where we have assumed for simplicity [FORMULA]. Eq. (68) can be solved for [FORMULA], thus providing [FORMULA]. 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 [FORMULA] and [FORMULA]. Because the dominant opacity source is H- and therefore the opacity increases steeply with temperature, the photospheric pressure on an irradiated, hotter part of the star will be lower than on an unirradiated part (because [FORMULA], where g is the surface gravity). Furthermore, the hotter the surface, the further out the photospheric point will be, i.e. [FORMULA] in the above calculation. Because [FORMULA], the temperature gradient [FORMULA] below an irradiated part will be higher when setting [FORMULA] rather than using the proper value of [FORMULA]. Therefore by assuming [FORMULA] we overestimate the radiative flux on the irradiated part. We obtain the same result from using [FORMULA]: because in reality [FORMULA], the temperature gradient (67) on an irradiated part is lower than what our estimate with [FORMULA] yields. Therefore, our very simple one-zone model, i.e. Eq. (68) underestimates the blocking effect somewhat. Since, on the one hand, this deficit can easily be compensated for by slightly increasing the value of b, and since, on the other hand, the precise value of b which is appropriate is not exactly determined within our model (we shall later take an average value determined from published opacity tables), we consider (68) a fair approximation of the physical situation described. The really important aspect of our model is, however, that qualitatively it yields the correct behaviour of a stellar surface exposed to external irradiation, and that it is still simple enough to allow insight in the situation described.

We can also compute [FORMULA]. Differentiation of (68) yields


if [FORMULA]. For consistency with Eq. (32) we require [FORMULA] if [FORMULA].

As can be seen from Eqs. (68) and (69) the functions G and g depend only on two parameters characterizing the unirradiated star, namely on [FORMULA] and [FORMULA], and on the opacity law via b. While [FORMULA] is a well-defined quantity, [FORMULA] is not because in real stars the run of temperature T with pressure P is not as simple as the one assumed in our simple model (and sketched in Fig. 6). In particular, the transition from convective to radiative energy transport is smooth and does not occur at one particular point as we have assumed in our model. Since [FORMULA] stands for the temperature at which this transition occurs, we determine [FORMULA] by requiring that in a full stellar model the ratio [FORMULA] of convective flux [FORMULA] to radiative flux [FORMULA] reaches a prescribed value, say [FORMULA]. This is equivalent to the condition [FORMULA]. Of course the choice of k is somewhat arbitrary but a value [FORMULA] seems a natural choice. Our simple model is an acceptable description of the real situation only if for a given model [FORMULA] is sufficiently insensitive to k. In order to demonstrate that this is indeed the case we plot in Fig. 7 the run of [FORMULA] as a function of the stellar mass of zero-age main-sequence stars with Pop. I chemical composition (X = 0.70, Z = 0.02) for three different values of k, i.e. [FORMULA] (full line), [FORMULA] (dashed line) and [FORMULA] (dotted line). Fig. 7 shows two important properties of low-mass stars: The first one is that a significant superadiabatic convection zone exists only in stars with a mass [FORMULA]. Below [FORMULA] the stratification is essentially adiabatic up to the photosphere. This means that application of our simple one-zone model is restricted to stars in the mass range [FORMULA]. The second property shown in Fig. 7 is that the run of [FORMULA] for different k is qualitatively the same for all three values of k. This means that as long as a star has a significant superadiabatic convection zone the transition from convective to radiative energy transport occurs in a rather narrow temperature interval, thus justifying our simple approach.

[FIGURE] Fig. 7. Temperature [FORMULA] of the point in the superadiabatic convection zone of a main sequence star [FORMULA], or [FORMULA] for [FORMULA] (dashed line), [FORMULA] (full line), and [FORMULA] (dotted line) as a function of stellar mass. [FORMULA] is measured in units of [FORMULA], the effective temperature of the unirradiated star.

5.2. Stars with a mass [FORMULA]

As Fig. 7 shows, [FORMULA] in stars with a mass [FORMULA]. This means on the one hand that the total optical depth between the photosphere (at [FORMULA]) and the point where [FORMULA] 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 [FORMULA] and the value of [FORMULA] is directly influenced by irradiation, this situation cannot be described by a simple model. Rather one ought to determine [FORMULA] 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 g or G. After all it is at least conceivable that already a small irradiating flux could result in a significant reduction of [FORMULA] (which itself is a rather small number because convection is not far from adiabatic) and that therefore [FORMULA] could attain a large negative value. However, as the following argument shows, there is a limit to how fast F can drop in response to increasing [FORMULA]. If F drops too strongly this results in an effective temperature [FORMULA]. This in turn means that the temperature gradient must be steeper than in the unirradiated star and, therefore, that [FORMULA], in contradiction to the starting assumption [FORMULA]. In order to avoid this contradiction [FORMULA] must not decrease with increasing [FORMULA], i.e. [FORMULA], from which we immediately recover (33), i.e. [FORMULA]. Because [FORMULA] (cf. 34), g is maximal in the limit [FORMULA]. Therefore we need to determine [FORMULA] for the stars in question. For this we return to the one zone model (Sect. 5.1). From Eq. (69) we find that [FORMULA] increases as [FORMULA] decreases. In fact, in the limit [FORMULA], this model yields [FORMULA]. Thus the closer the adiabatic convection zone reaches to the surface the more sensitive the star is to irradiation, i.e. the larger g. On the other hand, we know that [FORMULA] in all cases. It is therefore plausible that for stars which are almost adiabatic up to the photosphere, i.e. [FORMULA], [FORMULA]. As we shall see below this is confirmed by numerical computations.

5.3. Results of numerical computations

Numerical computations of [FORMULA] and [FORMULA] 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 [FORMULA] The results relevant for this paper are shown in Fig. 8 and can be summarized as follows:

  • a) With [FORMULA] as shown in Fig. 7 and an appropriately chosen value for b, i.e. [FORMULA], typical for H- opacity, the predictions of our simple one-zone model are in good agreement with numerical results as long as [FORMULA]. In particular, the run of [FORMULA] 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) [FORMULA] for stars with a mass [FORMULA], as we have argued above (Sect. 5.2).

  • c) Max[FORMULA] [FORMULA] 0.5 over the whole mass range of interest, i.e. [FORMULA]. This is shown as a dash-dotted line in Fig. 8.

  • d) In the point source model (cf Sect. 4.2.2) the integral [FORMULA] in (57) has a maximum which is smaller than [FORMULA] Max[FORMULA]. Max[FORMULA] is shown in Fig. 8 as a long dashed line. As can be seen, Max[FORMULA].

[FIGURE] Fig. 8. The values of [FORMULA]=0) (see Eq. 31) for main sequence stars as a function of stellar mass. Results from full stellar models from HR97 are shown as a full line, results obtained from the one-zone model (Eq. 69) as short-dashed lines. For comparison we show also the run of [FORMULA] (dash-dotted line) and [FORMULA] (long dashed line) derived from the results of HR97).

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Online publication: August 23, 2000