## 3. Reaction of a low-mass star upon reducing its effective surfaceWe 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 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 where 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 the Eddington approximation yields the photospheric solution where is the photospheric pressure at an optical depth . The interior solution can be approximated by a polytrope with index and yields Taking (12) at the photospheric point
, i.e. equating (11) and (12),
together with (9) and (10) yields the luminosity 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 defines the thermal equilibrium values of the irradiated star. These can be expressed in terms of the thermal equilibrium values of the unirradiated star as follows: Since the dominant opacity source in the photosphere of cool stars
is bound-free absorption,
appropriate values for and 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 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
Since the radius in thermal equilibrium with irradiation is larger by the time scale for thermal relaxation becomes 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
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 |