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

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7. Modeling the wind variations

To model the time-dependent variations in the wind we have to choose a wind geometry. The two general possibilities are:

1. Spherical symmetry in the wind and in the variations ("shells").

2. Spherical steady wind with randomly distributed inhomogeneities ("blobs").

As Lamers (1994) showed, the first possibility would result not only in a varying absorption component, but also in emission variability all over the profile. By calculating models with spherical shells, the variations in the wings of the modeled emission are found to be much stronger than observed. This is due to additional emission caused by the shell which adds to the steady flux and therefore changes the shape of the profile.

We observe only variations in the center of the emission, but not in the wings. The profile strengthens and weakens simultaneously with the same factor over its full width. This excludes spherically symmetric shells, since such a shell produces a constant absolute flux contribution (box-shaped) to the profile which is broadening with time.

Thus we adopt a model with outwards moving blobs. These blobs have a large effect on the profile if they are in front of the stellar disk. They increase the optical depth of the line at the wavelength corresponding to their radial velocity. This produces discrete absorption components superimposed on the stationary line profiles. The effect of blobs is small when they are not in front of the stellar disk. In that case they only contribute to some extra radiation emitted in all directions (Lamers 1994). We neglect the effect of blobs that are not in front of the stellar disk on the emission component. This is justified later when we conclude that the density perturbations are only of the order of 20%.

To model the line profiles with density perturbations in the line-of-sight to the stellar disk, we keep the wind density unaffected for impact parameters [FORMULA]. Variations are allowed only for the column in front of the stellar disk, [FORMULA], which produces the absorption part of the profile. So the modeled density variations look more like "slices" rather than "blobs". A sketch of the geometry is given in Fig. 16.

[FIGURE] Fig. 16. Geometry applied to model the profile variations

To model the time-dependent variations in the wind we have to integrate [FORMULA] to get [FORMULA]. This specifies the propagation of variations through the wind in time:


At a given time [FORMULA] we calculate the date of ejection [FORMULA] from the star of every radius in the wind. A factor [FORMULA] is the applied to the density [FORMULA] in the above mentioned column in front of the star:


where [FORMULA] is the density of the ambient wind at distance R. R is given in units of stellar radius [FORMULA]. The function [FORMULA] gives the density perturbations at the base of the wind as a function of time.

For our example we took [FORMULA]  Sco in the year 1993. The applied density perturbations were derived by measuring the minimum flux of the absorption component in [FORMULA] in each spectrum and splining these points to get a smooth function [FORMULA] in time. The function was normalized to a mean value of unity and the variance of the density was chosen to model the flux variations as accurately as possible. Because of the normalization, the function [FORMULA] also reaches values below unity. This causes the disappearance of the absorption, which is impossible if we only allow for denser regions.

The function [FORMULA] was shifted backwards in time by 19 days, with respect to the model date of the variations in [FORMULA] (Fig. 17). This is due to the fact that, using a [FORMULA] velocity law, the variations at the base of the wind take some time to propagate to the [FORMULA] -forming region.

[FIGURE] Fig. 17. The graph shows the time dependence of the absorption depth of [FORMULA] in [FORMULA]  Sco in 1993 and the therefrom derived density variations [FORMULA]. The four strongest peaks seen here can be identified as corresponding events to the four features traced in Fig. 8

We derived a velocity law with [FORMULA] =1.5 to 1.7 and 2 to 3 from the modeling of the stationary wind and the velocity of the perturbations (Sect.  4.2), respectively.

In modeling the perturbations of the line profiles we assumed the same velocity law for the ambient wind and the blobs. This was done for consistency because the program is not able to describe a wind with two velocity fields in a consistent way.

The values of [FORMULA] are the same as in the stationary case shown in Table 6. So our models are not meant to fit the actual profiles very accurately but only to show that the suggested basic mechanisms can explain the observed absorption variability of this type of stars.

After the integration of the envelope flux, the apparent continuum radiation was normalized to unity. We used dynamical spectra as in Figs. 18 and 19 to compare the model with the observations. For a more detailed comparison, each Balmer line in each spectrum has been fitted with Gaussians both in absorption and in emission to derive the central velocity and the flux in the line center. Due to the P Cygni-type profiles of [FORMULA] and [FORMULA], this is not as accurate as for the photospheric lines in absolute terms, but allows a comparison of the variation patterns of the models and the star. Because of the larger wavelength steps of the models the error bars of the fits to the models are larger.

[FIGURE] Fig. 18a and b. Dynamical spectrum and model of [FORMULA] in [FORMULA]  Sco in 1993
[FIGURE] Fig. 19a and b. Dynamical spectrum and model of [FORMULA] for [FORMULA]  Sco in 1993

The model qualitatively reproduces the outward acceleration of the observed [FORMULA] and [FORMULA] variations (Fig. 20). The velocity of the absorption minimum for the steady model is slightly lower than observed while the variance is similar. Note that the modeled variability pattern is the same as the observed one, apart from offsets. These offsets can be explained by deviations already visible in the stationary models (cf. Fig. 13). Fig. 21 shows that the model also works for HD 169454.

[FIGURE] Fig. 20. Radial velocities of the absorption components of [FORMULA] and [FORMULA] for [FORMULA]  Sco in 1993 and radial velocities measured in the modeled profiles of [FORMULA] and [FORMULA]. The density variations applied to the model are shown in Fig. 17
[FIGURE] Fig. 21a and b. Observed (upper) and modeled (lower) dynamical spectra of [FORMULA] for HD 169454 in 1994

The same method could not be applied to HD 190603 because of the lower sampling rate.

Apart from single extraordinary strong events which occurred in the absorption of [FORMULA]  Sco 1993 (discussed above) and in the emission of HD 190603 in 1991, we found no significant differences in different years. For HD 190603 the emission was extremely low in [FORMULA] and vanished in [FORMULA] in two spectra in 1991, following a deep absorption in the photospheric lines. The typical density variations are of the order of about 13% for [FORMULA]  Sco, typical peak values being less than 20%. Only the above mentioned extreme event required a density different by 30% from the mean value to be modeled properly. If the density perturbation occur in random directions in the wind with a spherical distribution, the variation in the mass-loss rate will be of the same order of magnitude, i.e. about 10%.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998