3. Reaction of a low-mass star upon reducing its effective surface
We start our examination of anisotropic irradiation by recalling one of the main results obtained in the studies of spherically symmetric irradiation (e.g. Podsiadlowski 1991; D'Antona & Ergma 1993), namely that the main effect of irradiating a low-mass main-sequence star is that the star cannot lose energy as effectively (or at all) through the irradiated parts of its surface. If irradiation is spherically symmetric, the star has no choice but to store the blocked energy in gravitational and internal energy with the well-known result that it swells. In the case of anisotropic irradiation, such as one-sided irradiation from an accreting companion, the situation is qualitatively different: in addition to storing the blocked luminosity in internal and gravitational energy, the star can also divert its energy flow to and lose energy from the unirradiated parts of its surface. This is easily possible in the adiabatic convection zone. Because in this zone the energy flow is almost fully decoupled from the mechanical and thermal structure (the flow being proportional to with rather than to (where is the actual temperature gradient and the adiabatic temperature gradient, and T and P are respectively the temperature and pressure) the mechanical and thermal structure of the star can still be considered to be spherically symmetric despite the fact that the energy flow might be highly anisotropic. The only thing which changes is the surface boundary condition which replaces the Stefan-Boltzmann law in the unirradiated case.
In the following we shall make a simple model for studying the situation described above. In this model we assume that over a fraction of the stellar surface the energy outflow is totally blocked (because of irradiation, or e.g. star spots, see Spruit & Ritter 1983) and that the remaining fraction of the surface radiates with an effective temperature . The surface luminosity of the star L can therefore be written as
where R is the stellar radius. It is clear that in a more realistic model itself must depend on the irradiating flux . In Sect. 4 we shall discuss results of numerical calculations and a model with which we can evaluate . For the moment we note only that as and that in the limit of high , approaches the surface fraction of the star which sees the irradiation source.
The purpose of this simple modelling is, on the one hand, to provide estimates for the magnitude of and the time scale associated with the thermal relaxation process enforced by anisotropic irradiation, and, on the other hand, to show that the effects of anisotropic irradiation are not only quantitatively but also qualitatively different from those obtained in the spherically symmetric case.
The internal structure of a low-mass star with a deep outer convection zone can be described by the simple analytical model by Kippenhahn & Weigert (1994) for stars on or near the Hayashi line. This model is particularly applicable in our case, since we are only interested in the differential behaviour. Assuming a power law approximation for the frequency independent, i.e. grey photospheric opacity on the unirradiated surface of the form
Here and are respectively the luminosity and the radius of the unirradiated star in thermal equilibrium. In general, the irradiated star will not be in thermal equilibrium, i.e. does not equal the nuclear luminosity . For the disturbed star, the latter can be estimated by using homology relations. If we write the rate of nuclear energy generation in the form appropriate for hydrogen burning via the pp-chain, i.e.
where is the density, we obtain (see e.g. Kippenhahn & Weigert 1994)
From (13) and (15) we obtain the gravitational luminosity
Since the dominant opacity source in the photosphere of cool stars is bound-free absorption, appropriate values for a and b in (10) are and . Nuclear energy generation in low-mass stars occurs mainly via the pp-I chain and the appropriate value of in (14) is . As a result, we find that
This means that the effective temperature on the unirradiated part of the surface hardly changes, reflecting a well-known property of stars on or near the Hayashi line. Furthermore, since , the response of the stellar radius to anisotropic irradiation is much weaker than if isotropic irradiation is assumed. Specifically, if , as is the case for one-sided irradiation by an accreting companion, the equilibrium radius with irradiation is larger than the one without irradiation by at most %. On the other hand, the total luminosity is significantly reduced: because of the rather large value of already a slight expansion of the star leads to a marked reduction of its nuclear luminosity.
Clearly, our model is not applicable if . This is because on the formal level if (see Eq. 9) and the values of R and in thermal equilibrium with and (Eqs. 17 and 19) diverge. There is also a physical reason why this model does not apply in this case: corresponds to strong, spherically symmetric irradiation where the star in thermal equilibrium is fully radiative (e.g. Podsiadlowski 1991), whereas our model applies only to the extent that the star in question retains a deep outer convective envelope, even when irradiated.
We can now estimate the maximum contribution to mass transfer arising from thermal relaxation due to irradiation and the associated time scale. Thermal relaxation is maximal right at the onset of irradiation. At that time the gravitational luminosity of the star is . Using now the bipolytrope model in the formulation of KR92, we can write for the thermal relaxation term due to irradiation
where is the Kelvin-Helmholtz time of the unirradiated star in thermal equilibrium, the thermal time scale of the convective envelope, and the quantity defines a dimensionless number which depends only on the relative size of and on the polytropic index in the radiative core. An explicit expression for is given in KFKR96). In particular, for a single polytrope , i.e. a fully convective star, where , one has . Furthermore, we show as a function of mass for zero age main sequence stars as a full line in Fig. 1. We note that scales roughly as the inverse of the relative mass of the convective envelope: the dashed line in Fig. 1 shows that within better than a factor of two. Thus for the purpose of an estimate we can rewrite (20) as
As can be seen from (21) the maximal rate of expansion is proportional to . Interestingly, the time over which the new thermal equilibrium is established is much shorter than if , unless is a small number and our model does not apply anyway.
Either (21) or (23) are to be inserted in (8) to get an estimate for the contribution of irradiation to mass transfer. Using (21) in (8) yields the peak contribution and (23) in (8) a time average.
In order to check the validity of our simple analytical model we have also performed numerical computations of full stellar models using the modified Stefan-Boltzmann law (9) as one of the outer boundary conditions and with as a free parameter. For our computations we have used a modified version of Mazzitelli's (1989) stellar evolution code which is described in more detail in KR92, and have assumed a standard Pop. I chemical composition with (in the usual notation) and .
One of the basic predictions of our simple model is that (see Eq. 17) scales linearly with and that the slope is small. This is nicely confirmed by the behaviour of full stellar models shown in Fig. 2. As can be seen, the prediction is valid, at least qualitatively, over more than two orders of magnitude of . Furthermore the slope in Fig. 2 is indeed small, confirming that is a small number. That the slope is different for stars of different mass is due to the fact that the effective values of the parameters a, b and change with stellar mass.
In Fig. 3 we show in a mass radius diagram the thermal equilibrium radius as a function of mass for three different values of , i.e. for the standard main sequence , and for and . As can be seen, the mass radius relations for and are shifted by a small amount to larger radii and run roughly "parallel"to the standard main sequence , as is predicted by our simple model.
Finally, in Fig. 4 we show as an example the thermal relaxation with time of an star with . If time is measured in units of , as is done in Fig. 4, it is seen that the relaxation process is characterized by this time scale and that new thermal equilibrium is reached after only 0.2-0.3 of these time units. Again this confirms our analytical result (cf. Eq. 23), according to which the relaxation process lasts a few in these units. It is also seen that the effective temperature on the unirradiated part of the star rises only very little, as predicted, but that the relative mass of the convective envelope is reduced significantly from at the beginning to in the new thermal equilibrium.
© European Southern Observatory (ESO) 2000
Online publication: August 23, 2000