1 Introduction

Wolf-Rayet stars (WR-stars) are believed to be the evolved hot massive stars which almost have reached the end of their nuclear burning phase. Mass loss is a dominant feature of massive star evolution deeply influencing all stellar properties. The cause of the high mass loss from WR-stars has remained unveiled up to the present time.

It is often assumed that the winds of WR-stars are due to radiation pressure, i.e. the transfer of momentum from the radiation to the gas by means of absorption or scattering of photons, in analogy to the radiation driven winds of O-stars. This suggestion is based on the fact that both the luminosities of the WR-stars and the terminal wind velocities are in the same range as the luminosities and wind velocities of the O-stars. However, the mass-loss rates of the WR-stars are about a factor ten higher than those of O-stars of the same luminosity. This implies that the transfer of momentum from the radiation to the gas must be about ten times more efficient than in the winds of O-stars. This high efficiency is needed both in the subsonic region deep in the wind, where the mass-loss rate is set, and in the supersonic region where the wind is accelerated to its high terminal velocity. The purpose of the present study is to find out whether optically thick wind models can provide sufficient radiative driving in the inner wind regions of WR-stars, i.e. whether the WR-winds are indeed radiatively driven.

Clumping-corrected mass-loss rates ($\dot{M}$) of WR-stars of different subtypes lie in the range of 0.2 to $10 \times 10^{-5}$  $\mbox{$M_\odot$ }~{\rm yr}^{-1}$ and terminal velocities ($v_\infty$) lie in the range 700 to 6000 km s^-1 (Nugis et al. 1998; Hamann et al. 2000; Nugis & Lamers 2000). The luminosities of WR-stars are nearly of the same magnitude as O-stars but their masses are on the average 3 times lower (Maeder & Meynet 1987; Nugis $\&$ Lamers 2000). The momentum transfer efficiency $\eta$ of WR-stars, i.e. the ratio between the wind momentum loss and radiative momentum loss ( $\eta=\mbox{$\dot{M}$ }\mbox{$v_\infty$ }/(L/c)$), lies in the range of $1 < \eta < 20$. This high value of $\eta$ requires a very efficient momentum transfer with multiple scatterings (Owocki & Gayley 1999). Lucy & Abbott (1993) showed that if the actual $\eta$ is around 10 then the multiscattering in ionization-stratified WR-winds can provide the necessary driving force. According to Schmutz (1997) and Gayley et al. (1995) it is indeed possible to have sufficient radiative force for models with $\eta \approx 10$ to support the acceleration of the outer wind, but the mechanism that provides the driving in the inner wind regions, where $v < \mbox{$v_{\rm esc}$ }$ is still missing.

Owocki & Gayley (1999) suggested that possibly a "two-stage'' driving process might be needed, by which strong stellar pulsations actually initiate the mass loss, with radiative forces taking over to drive the extended acceleration and high terminal speeds. Cassinelli (1991) has suggested earlier that the high mass-loss rates of WR-stars may require a fast magnetic rotator model. In this paper we investigate whether there is indeed a need for an alternative mechanism to initiate the winds of WR-stars, or whether radiation pressure in the optically thick transonic layers is sufficiently efficient to get the wind started.

Line-driven wind models are quite successful in explaining the observed properties of mass loss from OB-stars. Unfortunately, a straightforward application of these models to WR-stars is not justified. The main acceleration in the WR-winds takes place at large values of frequency-averaged optical depths, i.e. below the photosphere, where radiation forces due to true continuum absorption are important. In the line driven wind models of O-stars these can be ignored. The first attempts to apply "optically thick wind'' to WR-stars have been made by Kato & Iben (1992) and by Pistinner & Eichler (1995). An optically thick wind is usually regarded as a steady state wind in which the acceleration of matter is due to continuum absorption, occuring below the photosphere (Kato $\&$ Iben 1992). In the present paper we define an "optically thick wind'' as a steady state radiatively driven wind with the sonic point located at large optical depth ( $\tau \gg 1$). In our definition the opacity below the photosphere is not necessarily dominated by continuum absorption. All opacity sources must be taken into account: line absorption, continuum absorption and electron scattering. Recent recalculations of Rosseland mean opacities (Iglesias & Rogers 1993, 1996) have revealed increased opacities in the outer stellar layers as compared to older calculations and this is basically due to a more correct inclusion of metal bound-bound transitions. The full self-consistent solution of optically thick winds requires the computation of a model for the whole star which takes into account both the nuclear burning core and the radiatively expanding envelope. This is a complicated task that requires the knowledge of accurate opacities and physics in the dynamical, i.e. expanding envelope. At present, this knowledge is not available. Therefore we undertake a simpler task: we investigate whether optically thick wind models of WR-stars can explain the observed mass-loss rates and we derive the specific properties of WR-stars at low expansion velocities below the sonic point. To achieve this, we take the following steps:
(a) We adopt the relations between the luminosity and the mass of the WR-stars as predicted by the evolutionary calculations. This is justified because it is well known that the basic properties of stellar cores of WR-stars are not depending on the ways and processes how the mass is removed from outer layers (Maeder & Meynet 1987; Langer 1989; Maeder 1991).
(b) We derive the equations that describe the structure of the optically thick part of the wind. These are the coupled equations of mass continuity, radiative transfer, energy conservation and momentum conservation. The temperature structure follows from the energy equation and the velocity structure follows from the momentum equation. The opacities due to lines and continuum play a crucial role in determining both the temperature structure, via the heating and cooling terms and the optical depth, and the velocity structure because the radiative force depends on the flux-mean opacity.
(c) Since we do not know the opacities with sufficient accuracy to construct a full optically thick wind model, we will express the temperature and velocity structure as a function of the optical depth. The optical depth depends on the structure of the atmosphere in both the subsonic and the supersonic region. For the supersonic region we will use the empirically determined velocity and density structure for the calculation of the optical depth.
(d) The mass-loss rate of any stellar wind model is set by the condition that the solution of the momentum equation should smoothly pass through its critical point, which is usually the sonic point (e.g. Lamers & Cassinelli 1999). We investigate the conditions at the sonic point in the optically thick wind to see if they can explain the observed high mass-loss rates of the WR-stars.
(e) We will show that the observed mass-loss rates can be explained by radiation pressure in an optically thick wind, but only if the velocity law in the supersonic part of the wind is much softer (slower acceleration) than generally adopted. This requirement comes from the fact that the optical depth at the sonic point must already be very much larger than unity.
(f) We then compare the opacities that are needed to explain the high mass-loss rates of the WR-stars by means of optically thick radiation driven winds with new OPAL opacities.

The structure of the paper is as follows. In Sect. 2 we describe the physical processes and the mathematical equations of the optically thick radiation driven wind models. In Sect. 3 we describe the method that was used to calculate the optically thick wind models and in Sect. 4 we give some simple estimates based on general momentum and energy considerations of optically thick winds. In Sect. 5 we present the results of a few series of test models for the optically thick wind of the star WR139 (WN5). We will draw some important conclusions about the properties and the conditions of optically thick wind models. In Sects. 6 and 7 we apply the optically thick wind models to explain the observed mass-loss rates of a set of characteristic WN and WC stars. In Sect. 6 we first discuss the characteristic properties of these stars, such as the adopted radius of the sonic point. The models for these stars are presented in Sect. 7. In Sect. 8 we make new models with the opacity gradient at the sonic point derived from the OPAL-opacities for the abundances of WR-stars and we discuss the existence of a bifurcation in the wind models of WR-stars. The results are discussed in Sect. 9 and the conclusions are in Sect. 10.