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Astron. Astrophys. 326, 249-256 (1997)

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3. The modelling

For each star, we computed a different semiempirical model. For "semiempirical" we mean that, given a T vs. z distribution, we self-consistently computed non-LTE populations for H, He, Fe, Si, Al, Ca, Na and Mg, solving simultaneously the equations of hydrostatic equilibrium, radiative transfer, and statistical equilibrium.

The modelling was done using the program Pandora developed, and kindly provided to us, by Dr. E. H. Avrett. For a detailed description of the code, we refer the reader to Vernazza et al. (1973, 1981). Avrett et al. (1986) explain in detail how the continuum opacities are treated. We assume that the absorption coefficient for each line has a Voigt profile


where [FORMULA] is the Doppler width, and a is the Voigt parameter


Here [FORMULA] and [FORMULA] are the atomic hydrogen and electron densities, T is the electron temperature, and the Doppler width [FORMULA] is in Å units. In the Voigt parameter we include radiative ([FORMULA]), and Stark ([FORMULA]) broadening, and Van der Waals ([FORMULA]) broadening due to hydrogen. For the hydrogen lines, we also include natural broadening. For details on how these mechanisms are considered, see Mauas et al. (1988). For hydrogen, the Stark broadening parameters were computed following Sutton (1978), and those for Van der Waals and natural broadening were computed as in Vernazza et al. (1981). [FORMULA] for the Na D lines and the Ca II K and H lines are from Konjevic et al (1984), and the [FORMULA] are from Lewis et al. (1972), for the Na D lines and from Monteiro et al. (1988), for Ca II K and H. More details on the modelling procedure will be given elsewhere (Mauas & Falchi 1997; Paper IV).

Once the calculations were completed for a given temperature distribution, we computed the continuum intensity and the emerging profiles for different lines, and compared them with the observations. We then modified the assumed T vs z distribution, until a satisfactory match between observations and calculations was obtained. The features we used for the comparison were the continuum between 3500 and 9000 Å, the four highest Balmer lines, the Ca II K line, and the Na D lines. Including many different lines removes, or at least reduces greatly, the underdetermination of the models. Moreover, we also checked our predictions against the observed fluxes in the U, B, V, R and I filters and in the infrared.

As can be seen in Table 1, the difference between the gravity values obtained for the three stars is smaller than their uncertainties, and therefore we used [FORMULA] for our models, as was done for AD Leo in Paper I. This is very similar to the value of [FORMULA] adopted by the Armagh group for their calculations (e.g. Houdebine and Doyle 1994a).

To our knowledge, there are no estimations for [FORMULA] for Gl 588 and Gl 628. However, for these inactive stars one should expect values smaller than 2 km s-1 (Marcy and Cheng 1992). In any case, given the resolution of our spectra, the observed line profiles are dominated by the instrumental broadening, and we did not include in our calculations the effect of rotational broadening.

A very important aspect when modelling cool stars is the inclusion in the opacity calculations of the line blanketing due to the numerous weak lines, both atomic and molecular, of the various species present in the atmosphere, in particular TiO and CaOH (Mould 1976). In this work we included the [FORMULA] atomic and molecular lines computed by Kurucz (1991).

Houdebine & Panagi (1990) studied the importance of hydrogen atomic parameters on the results of the modelling. In this paper, we used the collisional rates of Johnson (1972), for all transitions except [FORMULA], for which we used the results by Scholtz et al. (1990), and [FORMULA] and [FORMULA] for which we used the results by Giovanardi et al. (1987) and Giovanardi & Palla (1989). The Einstein coefficients were also taken from Johnson (1972) and the photoionization cross-sections from Mathisen (1984). The remaining parameters are as in Vernazza et al (1981). We point out that this atomic parameters differ from the ones we used in Papers I and II, were we adopted those in Vernazza et al (1981). For this reason, we have slightly modified the model for AD Leo, to be consistent with the other stars.

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© European Southern Observatory (ESO) 1997

Online publication: April 20, 1998