9 Discussion

The optically thick radiation driven wind models for WR-stars derived in this paper differ from the "standard'' models of WR-winds (Hamann & Koesterke 1998a; Koesterke & Hamann 1995; Dessart et al. 2000) in several ways: the velocity-law parameter $\beta \approx$ 4 to 6 of our models is considerably higher than the adopted $\beta=1$of the "standard'' models and the sonic point radius of the optically thick models is smaller than the inner radius of the "standard'' models for early type WR-stars. The low value of $\beta$is approximately correct for the winds of O-stars (e.g. Haser et al. 1995; Puls et al. 1996; Herrero et al. 2000), but for the winds of WR-stars $\beta$ might be significantly higher. This is because in O-star winds the radiative acceleration is mainly due to spectral lines, and hence the radiation force is sensitive to the Doppler shifts produced by the velocity gradient. This results in a fast acceleration of the wind and a small value of $\beta \simeq 0.7$ to 1 (e.g. Lamers & Cassinelli 1999, p. 240). However, the optically thick winds of WR-stars are largely driven by opacity sources which are less sensitive to Doppler shifts and hence we can expect a slower acceleration and higher values of $\beta$. This is supported by the analysis of the spectroscopic data of WR-stars by Lépine & Moffat (1999) (see also Moffat & Lépine 2000). They studied the variations of subpeaks in the line profiles of WR-stars and, assuming that these subpeaks are due to propagating wind inhomogeneities, they find that $\beta \simeq 5$ to 10. (A similar analysis of the line profile variations of the star $\zeta$ Pup (O4If) by Eversberg et al. (1998) gives $\beta \approx 1 - 1.2$, in very good agreement with $\beta \simeq 1.15$ derived from modeling of the $H_{\alpha}$-profile (Puls et al. 1996). This supports the assumption that the study of the kinematics of the subpeaks provides a reasonably good estimate of the value of $\beta$. Further support for the high value of $\beta$ for WR-stars comes from the study of the hydrodynamical modeling of the wind of WN4b star HD50896 by Schmutz (1997), who derived $\beta \simeq 8$. Hillier & Miller (1999) concluded that the atmospheric models with values of $\beta$ higher than 1 can not be excluded, but that it is very difficult to constrain the velocity law in the standard spectroscopic modeling studies, particularly for WC-stars with severe blending of lines.

We are aware of the fact that optically thick (in continuum) wind models are more sensitive to the velocity law in the inner part of the wind than in the outer wind and therefore the actual value of $\beta$ may be smaller in the part of the wind where the observed spectral lines are effectively formed. We assumed in our paper that $\beta$ is the same in the whole wind. However this was not a crucial assumption. It only allowed us to give a rough estimate of the "mean'' value of $\beta$ that is needed for a sufficiently large optical depth at the sonic point.

Support for our conclusion that the sonic radius $R_{\bf s}$ is smaller than the inner radius ("core'' radius) of the "standard'' models comes from the analysis of eclipsing and spectroscopic binaries. Cherepashchuk (1991, 2000) found from the modeling of light curves of eclipsing binary system V444 Cyg (HD 193576, WR139) that the optical photospheric radius where $\tau_{\rm V} \approx 1$ is at $R_{\rm ph} \simeq 3 ~R_{\odot}$ for the WN5-component. Hamann & Schwarz (1992) derived by the standard atmospheric modeling that $R_{\tau20} \approx 6~ R_{\odot}$ for this WN 5 star, which is twice as large. For the another eclipsing binary system CQ Cep (HD 214419, WR155) it is estimated from the analysis of the eclipses and the orbital motion that $R_{\rm ph} \leq 10~ R_{\odot}$ for the WN6-component (Cherepashchuk 1991; Marchenko et al. 1995; Moffat & Marchenko 1996). On the other hand Hamann & Koesterke (1998a) derived from the standard atmospheric modeling study that $R_{\tau20} \approx 25~ R_{\odot}$ for this WN6-component. The radius $R_{\tau 20}$ of the inner boundary of the atmospheric models at $\tau \simeq 20$ should be smaller than the optical radius. However, we see that for both binary systems the derived values of $R_{\tau 20}$ are about twice larger than the empirically derived photospheric radii. We can conclude that the "stardard'' atmospheric models of WR-stars overestimate the core radii of early type WR-stars by about a factor 2 to 3. Therefore, the fact that our models require a smaller sonic radius than $R_{\tau 20}$ of the "standard'' models and a higher value of $\beta$ is very reasonable.