We compute a new set of models, called A1 and B1, using the OPAL opacities, with the abundances of the individual stars from Nugis & Lamers (2000). In these models the sonic radius is varied (starting from the value of Table 5) until $({\rm d} \chi /{\rm d}r)_{\rm s}$ becomes equal to that from the OPAL opacity tables. We were not trying to get exact agreement of the model values of $\chi_{\rm s}$ with the OPAL opacities, because the opacities in our models are the flux-mean opacities whereas the OPAL-opacities are the Rosseland-mean values. The flux-mean opacities are expected to be (slightly) higher than the Rosseland-mean values. In Variant A1 we assume that L(r) is constant through the sonic point (similar to Variant A0) and in Variant B1 we assume that the opacity behaves like a power-law of the type $\chi=a\rho / T^{3.5}$ at the sonic point (similar to Variant B0).

The results are listed in Tables 7 and 8 for WN- and WC-stars. For WR105 (WN9) and WR103 (WC9) we present several models with different values of $R_{\rm s}$ . These will be discussed below. The tables show the adopted input values for the stellar parameters L_* and $\mbox{$\dot{M}$ }$ . The output values are: the radius, optical depth, temperature, opacity gradient and opacity, all at the sonic point. The OPAL-opacity gradient at the sonic point is per definition equal to that of the models. The opacity at the sonic point, that follows from the models, is compared with the OPAL-opacity at the same temperature and density. We see that the OPAL-opacities at the sonic point are typically a few tens of percent smaller than $\chi_{\rm s}$ (except for WR103). This could partly be due to the difference between the Rosseland-mean and the flux-mean opacities.

For the late type WR-stars WR105 and WR103 we present more than one model. This is because the sonic point radius $R_{\rm s}$ is not well known for the WN9 and WC9 stars. Starting with different values, we get different answers. If the resulting value of $R_{\rm s}$ , that follows from the model calculations, differs more than a factor two from what we think is a reasonable value, the result is considered doubtful. The doubtful models in Tables 7 and 8 are given in brackets.

The sonic point temperatures of models A1 and B1 for each star are very similar. We see that the temperatures of the models at the sonic point fall into two regions, $156~000 < T_{\rm s} < 162~000$ K and $37~000 <T_{\rm s} < 71~000$ K. It is a consequence of the fact that the opacity gradient ${\rm d}\chi/{\rm d}r$ has to be positive at the sonic point, to allow a transonic solution. The gradient is larger for models with a sonic point in the high temperature range, than for models with $T_{\rm s}$ in the low temperature range. However, the values of the opacity themselves are very similar in all cases. The models for stars with subtypes WN2 - WN6 and WC5 - WC7 all have high sonic point temperatures of $T_{\rm s} \approx 160~000$ K. The star WR22 (WN7) has a solution both in the high and in the low temperature range. The star WR105 (WN9) has a sonic point in the low temperature range.

There is a problem with the models of WC9 star WR103. The low $T_{\rm s}$ models of this star require an opacity at the sonic point of $\chi_{\rm s} \approx 0.75$ cm² g^-1, but the OPAL opacity at the sonic point is less than 0.30 cm² g^-1. On the other hand, the high temperature models for this star have about the right sonic point opacity, but the sonic point radius is much smaller than reasonable for this type. This discrepancy points either to a higher mass and luminosity than adopted for this star (this leads to a lower value of $\chi_{\rm s}$ ) or to a significant clumping at the sonic radius already, which would increase the Rosseland mean opacity.

Table 7: Optically thick wind models for standard WR-stars: Variant A1 (constant L(r)).
Star Sp. $\log~\mbox{$L_*$ }$ $\log~\mbox{$\dot{M}$ }$ $R_{\rm s}$ $\taup_{\rm s}$ $T_{\rm s}$ $\left(\frac{{\rm d}\chi}{{\rm d}r}\right)_{\rm s}$ $\chi_{\rm s}$ $\chi_{\rm OPAL}$ $\beta$

$R_\odot$ K 10^-14 cm g^-1 cm² g^-1 cm² g^-1

WR2 WN2 5.27 -5.40 1.64 9.87 156 800 0.99 0.700 0.601 5.89

WR139 WN5 5.21 -5.04 2.06 19.3 158 500 1.24 0.753 0.632 5.87

WR136 WN6 5.73 -4.20 4.82 33.7 159 500 0.42 0.473 0.694 6.02

WR22 WN7 6.08 -4.38 22.7 7.49 63 100 0.11 0.597 0.369 5.04

WR105 WN9 5.81 -4.55 14.5 5.81 63 810 0.18 0.437 0.361 4.29

WR105 WN9 5.81 -4.55 32.7 3.09 37 120 0.10 0.435 0.371 3.54

WR111 WC5 5.31 -5.00 2.39 20.0 157 300 0.66 0.681 0.539 6.57

WR42 WC7 5.23 -4.89 2.44 26.2 158 300 0.86 0.737 0.568 6.24

WR103 WC9 5.20 -4.62 9.90 18.0 70 750 0.42 0.755 0.261 5.82

(WR103 WC9 5.20 -4.62 2.93 42.3 159 300 0.98 0.774 0.602 6.03)

(WR103 WC9 5.20 -4.62 23.4 9.79 39 900 0.23 0.750 0.296 5.28)

**Table 7:** Optically thick wind models for standard WR-stars: Variant A1 (constant L(r)).
Star	Sp.	$\log~\mbox{$L_*$ }$	$\log~\mbox{$\dot{M}$ }$	$R_{\rm s}$	$\taup_{\rm s}$	$T_{\rm s}$	$\left(\frac{{\rm d}\chi}{{\rm d}r}\right)_{\rm s}$	$\chi_{\rm s}$	$\chi_{\rm OPAL}$	$\beta$
				$R_\odot$		K	10^-14 cm g^-1	cm² g^-1	cm² g^-1
WR2	WN2	5.27	-5.40	1.64	9.87	156 800	0.99	0.700	0.601	5.89
WR139	WN5	5.21	-5.04	2.06	19.3	158 500	1.24	0.753	0.632	5.87
WR136	WN6	5.73	-4.20	4.82	33.7	159 500	0.42	0.473	0.694	6.02
WR22	WN7	6.08	-4.38	22.7	7.49	63 100	0.11	0.597	0.369	5.04
WR105	WN9	5.81	-4.55	14.5	5.81	63 810	0.18	0.437	0.361	4.29
WR105	WN9	5.81	-4.55	32.7	3.09	37 120	0.10	0.435	0.371	3.54
WR111	WC5	5.31	-5.00	2.39	20.0	157 300	0.66	0.681	0.539	6.57
WR42	WC7	5.23	-4.89	2.44	26.2	158 300	0.86	0.737	0.568	6.24
WR103	WC9	5.20	-4.62	9.90	18.0	70 750	0.42	0.755	0.261	5.82
(WR103	WC9	5.20	-4.62	2.93	42.3	159 300	0.98	0.774	0.602	6.03)
(WR103	WC9	5.20	-4.62	23.4	9.79	39 900	0.23	0.750	0.296	5.28)

Table 8: Optically thick wind models for standard WR-stars: Variant B1 ( $\chi \sim \rho / T^{3.5}$ ).
Star Sp. $\log~\mbox{$L_*$ }$ $\log~\mbox{$\dot{M}$ }$ $R_{\rm s}$ $\taup_{\rm s}$ $T_{\rm s}$ $\left(\frac{{\rm d}\chi}{{\rm d}r}\right)_{\rm s}$ $\chi_{\rm s}$ $\chi_{\rm OPAL}$ $\beta$

$R_\odot$ K 10^-14 cm g^-1 cm² g^-1 cm² g^-1

WR2 WN2 5.27 -5.40 1.53 8.82 158 500 6.64 0.700 0.609 4.97

WR139 WN5 5.21 -5.04 1.91 17.3 160 100 8.91 0.753 0.650 4.98

WR136 WN6 5.73 -4.20 4.47 30.4 161 300 3.37 0.473 0.705 5.12

WR22 WN7 6.08 -4.38 5.40 17.9 158 200 2.31 0.608 0.756 4.83)

WR105 WN9 5.81 -4.55 28.9 2.82 38 730 0.58 0.435 0.374 2.84

WR111 WC5 5.31 -5.00 2.22 18.0 158 900 5.06 0.681 0.548 5.65

WR42 WC7 5.23 -4.89 2.28 23.5 159 700 6.76 0.737 0.578 5.36

WR103 WC9 5.20 -4.62 9.82 15.4 68 450 2.36 0.755 0.266 4.79

(WR103 WC9 5.20 -4.62 2.72 38.1 161 300 8.38 0.773 0.613 5.17)

(WR103 WC9 5.20 -4.62 20.8 9.03 41 510 1.32 0.750 0.300 4.26)

**Table 8:** Optically thick wind models for standard WR-stars: Variant B1 ( $\chi \sim \rho / T^{3.5}$ ).
Star	Sp.	$\log~\mbox{$L_*$ }$	$\log~\mbox{$\dot{M}$ }$	$R_{\rm s}$	$\taup_{\rm s}$	$T_{\rm s}$	$\left(\frac{{\rm d}\chi}{{\rm d}r}\right)_{\rm s}$	$\chi_{\rm s}$	$\chi_{\rm OPAL}$	$\beta$
				$R_\odot$		K	10^-14 cm g^-1	cm² g^-1	cm² g^-1
WR2	WN2	5.27	-5.40	1.53	8.82	158 500	6.64	0.700	0.609	4.97
WR139	WN5	5.21	-5.04	1.91	17.3	160 100	8.91	0.753	0.650	4.98
WR136	WN6	5.73	-4.20	4.47	30.4	161 300	3.37	0.473	0.705	5.12
WR22	WN7	6.08	-4.38	5.40	17.9	158 200	2.31	0.608	0.756	4.83)
WR105	WN9	5.81	-4.55	28.9	2.82	38 730	0.58	0.435	0.374	2.84
WR111	WC5	5.31	-5.00	2.22	18.0	158 900	5.06	0.681	0.548	5.65
WR42	WC7	5.23	-4.89	2.28	23.5	159 700	6.76	0.737	0.578	5.36
WR103	WC9	5.20	-4.62	9.82	15.4	68 450	2.36	0.755	0.266	4.79
(WR103	WC9	5.20	-4.62	2.72	38.1	161 300	8.38	0.773	0.613	5.17)
(WR103	WC9	5.20	-4.62	20.8	9.03	41 510	1.32	0.750	0.300	4.26)

8.1 A bifurcation in the optically thick winds

The optically thick wind models for WR-stars presented above clearly indicate the presence of two separate branches of solutions (bifurcation). These branches correspond to the intervals of sonic point temperature where it is possible to achieve positive opacity gradients ( $({\rm d}\chi/{\rm d}r)_{\rm s} > 0$ ): the high-temperature regime with $T \simeq 156~000$ K and the low-temperature regime with $37~000 \leq T \leq 71~000$ K. The high-temperature range is connected with the well-known iron opacity peak around $\log{T} \approx 5.2$ and the low-temperature range is connected with the weak opacity enhancement due to lower ions of iron and other metals in the range $37~000 \leq T \leq 71~000$ K. As can be seen from the OPAL-opacity tables, these ranges are clearly separated because ${\rm d}\chi/{\rm d}r < 0$ for $71~000 \leq T \leq 156~000$ K. This means that in the case of optically thick wind models with negligible contribution from the line driving force due to expansion, it is not possible to have a smooth evolution from the regime of mass loss with sonic point temperatures in the low-temperature range to the regime with sonic point temperatures with $T \geq 156~000$ K.

8 Optically thick wind models with OPAL opacities

8.1 A bifurcation in the optically thick winds