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Astron. Astrophys. 318, 819-834 (1997)

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6. A model of the stationary wind

In order to describe the wind-profile variations by density variations, we first have to derive a stationary wind model.

Our wind models are based on the [FORMULA] code. It is a modification of the more general SEI code by Lamers et al. (1987). It solves the radiative transfer equation for lines in the two-level approximation, with pre-specified non-LTE departure coefficients for both levels. We ascribe the average Balmer profiles to an undisturbed wind for which we derived the parameters by modeling the mean profile.

The temperature in the wind is given by a power law


For [FORMULA] we use the temperature of the stellar surface given by the grey approximation:


Using radio observations, Leitherer & Robert (1991) determined the radius of a shell with an electron temperature of 7000 K to 13 [FORMULA] for [FORMULA]  Sco. We use this as outer wind temperature for all stars. The models are not very sensitive to the exponent of the temperature laws if it is not too steep. Therefore a value of [FORMULA] was taken for all stars.

The velocity law [FORMULA] in our models is given by


The values for [FORMULA] are found in Table 3. For the initial velocity at the lower boundary of the wind [FORMULA] we adopt the speed of sound in the stellar atmosphere. We expect [FORMULA] to be a good approximation for all stars. This value is also adopted for the turbulent velocity in the wind (cf. Lamers et al. 1987). The value for [FORMULA] is determined by the best fit.

The electron and proton densities in the wind are:


Here [FORMULA] is the local density, calculated from the equation of continuity, and [FORMULA] is Avogadro's constant. The average helium ionization in the wind, [FORMULA], and the helium abundance, [FORMULA], were taken from Leitherer & Robert (1991).

We calculate NLTE-departure coefficients with the Sobolev code described by Bastian (1982) and Stahl et al. (1983) for a wind which is not fully ionized. The original NLTE-departure coefficients for [FORMULA] calculated with this code show a local maximum inside a few stellar radii and a steep rise outside this radius. This rise cannot be taken at its face value, otherwise the profiles would be box-shaped. So we introduced a scale height [FORMULA] to suppress this rise:


The resulting NLTE-departure coefficients are displayed in Fig. 12.

[FIGURE] Fig. 12a-c. NLTE-departure coefficients applied to model the Balmer profiles

The underlying photospheric profiles of the Balmer lines and the neighboring lines are calculated using the [FORMULA] -code introduced in Sect.  3. The abundances were adopted from Kaufer et al. (1994) for galactic B-stars. The spectra were broadened with the rotational velocity given in Table 3. The velocities have determined from SiIII lines. Even if the core-halo separation in the model is a quite crude simplification, the models are able to fit the observed profiles. Not only the stronger emission lines but also the absorption components (Figs. 13-15) are modeled quite well. On the other hand, the broad emission wings cannot be explained by our models. If we introduce electron scattering, using the method described by Scuderi et al. (1994), the required electron column densities would be higher by several orders of magnitude than predicted by our models. The fact that the broad wings have the same strength in [FORMULA] and [FORMULA] also does not support the electron-scattering model. If caused by electron scattering, the strength of the wings should scale with the emission-line equivalent width. The predicted additional absorption on the red side of [FORMULA] could be an effect of the artificial separation of photosphere and wind mentioned before. It is caused by the Stark broadening of the underlying photospheric profile due to a hydrostatic LTE model atmosphere.

[FIGURE] Fig. 13. The calculated and the observed average Balmer-line profiles of [FORMULA]  Sco . The calculated profiles are overplotted as circles. The residuals are plotted under the profiles
[FIGURE] Fig. 14. Same as Fig. 13, but for HD 169454
[FIGURE] Fig. 15. Same as Fig. 13, but for HD 190603

The parameters adpted to model the Balmer profiles of the program stars are summarized in Table 6. For comparison, we have included radio mass-loss rates. These were rescaled with our adopted values of the distance d and the terminal velocity [FORMULA]. The radio mass-loss rates were derived by Abbott et al. (1980) for HD 190603 and Bieging et al. (1989) for the other two stars. Note the excellent agreement between the mass-loss rates derived from the Balmer lines and from the radio continuum.


Table 6. Model parameters for the observed stars

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998