10 Summary and conclusions

The existing problems of the optically thin radiation driven wind theory in explaining the high momentum and the high mass-loss rates of WR-stars, has prompted our investigation of the possibility that the high mass-loss rates of WR-stars may be due to optically thick radiation driven winds.

Adopting the stellar parameters of the WR-stars and their observed high mass-loss rates and terminal wind velocities, we investigated the conditions at the sonic point of the wind that are needed to explain the high mass-loss rates. We reached the following conclusions:

1.

The mass-loss rates of optically thick radiation driven winds is approximately
        $\mbox{$\dot{M}$ }\simeq c_1 a_1^{1/2} R_{\rm s}^3 T_{\rm s}^{4.5} / GM$
with $c_1=64 \pi \sigma/(3c)$ and $a_{1}=k (\gamma+1) / (\mu m_{\rm H})$, where $R_{\rm s}$ and $T_{\rm s}$ are the radius and temperature at the sonic point ( $v_{\rm s}=c_{\rm s}$), $\gamma$ is the mean number of free electrons per ion and $\mu$ is the mean atomic weight of the atoms.

2.

Optically thick radiation driven wind models require that the opacity gradient at the sonic point is positive, i.e. $({\rm d}\chi/{\rm d}r)_{\rm s} > 0$. The OPAL-opacities for WR-stars show that this occurs in two temperature regimes: a hot regime with $155~000 < T_{\rm s} < 165~000$ K, and a cool regime with $37~000 <T_{\rm s} < 71~000$ K. The sonic points of early-type WR-stars (WN2-WN6 and WC5-WC7) are in the hot regime and those of late-type WR-stars are in the cool regime.

3.

The high sonic point temperatures imply a rather large effective optical depth at the sonic point of $\tau^{\prime}_{\rm s} \simeq 3$ to 10 for the low temperature regime and about 10 to 30 for the high temperature regime. Such high effective optical depths for the sonic points of WR-stars can only be achieved if the velocity law in the supersonic region is rather "soft'', with $\beta \approx$ 4 to 6. There is indeed observational evidence that winds of WR-stars have such soft velocity laws.

4.

The values of the opacities at the sonic points of our models that are required to explain the observed mass-loss rates of the WR-stars are close to the OPAL-opacities for the abundances of these WR-stars.

5.

The sonic radii $R_{\rm s}$ of our models are smaller than the so-called "core-radii'' where $\tau \simeq 20$ of the "standard'' models for early type WR-stars by about a factor two. Studies of WR-stars in eclipsing binary systems support the smaller radii of our models.

6.

Taking all these facts together, we find that the high mass-loss rates of WR-stars can be explained by optically thick radiation driven wind models.

We point out that we did not solve the structure of the whole wind. In fact, we only considered the conditions at the sonic point that are needed to start the wind with a high mass-loss rate. In particular, we did not consider the acceleration of the wind in the supersonic region. Therefore, if our assumption that the high mass-loss rate of WR-stars is due to radiation pressure in the optically thick transonic region is correct, we still have solved only half of the problem. The continuous acceleration of the outflowing gas up to the observed high terminal velocities still remains to be explained. The smaller radii and the more slowly increasing velocity laws that we derived, compared to the usually assumed values, may help in this respect, because a smaller radius implies a larger radiative flux (for the same luminosity) and a softer velocity law requires a smaller acceleration of the wind.

In a subsequent paper we will discuss the consequences of the possible occurrence of three types of radiation driven wind models (optically thin line-driven winds, optically thick radiation driven winds with $T_{\rm s} \approx 160~000$ K, and optically thick radiation driven winds with $T_{\rm s} \approx 50~000$ K) during the evolution of massive stars.

Acknowledgements

This work was supported by the Estonian Science Foundation grants No. 3166 and 5003, by a visitor-grant from NOVA, The Netherlands Research School for Astronomy and travel grants from the Leidsch Kerkhoven-Bosscha Foundation. We thank the referee, W.-R. Hamann, for constructive comments on an earlier version of the paper.