7 Optically thick wind models for typical WR-stars with fixed ${\vec R}_{\rm s}$

We have applied the method for calculating optically thick radiatively driven wind models to the set of five WN-stars and three WC-stars of different spectral types. The parameters of these stars were discussed in Sect. 6 and are listed in Table 4. We have calculated optically thick wind models with input parameters: the mass and luminosity of the star, the chemical composition, mass-loss rate and the terminal velocity of the wind. The sonic radius was adopted according to the dependence between $R_{\rm s}/\mbox{$R_{\rm evol}$ }$ and spectral subclass (Table 5). The results of these models (variants A0 for constant L(r) and B0 for $\chi \sim \rho / T^{3.5}$) are presented in Table 6.

   

Table 5: Core radii $R_{\tau 20}$ and the adopted sonic radii $R_{\rm s}$.

Subclass	$R_{\tau 20}$	Nr	$R_{\tau 20}$	Nr	$R_{\rm s}/\mbox{$R_{\rm evol}$ }$
	( $R_{\rm evol}$)	Stars	( $R_{\rm evol}$)	Stars	adopted
WN 2-4	$2.6\pm0.3$	14 H			2
WN 5	$6.9\pm1.2$	12 H			2
WN 6	$10.4\pm1.6$	10 H			5
WN 7	$14.3\pm1.0$	12 H			10-20
WN 8-9	$19.0\pm0.4$	14 H			20
WC 4	$4.3\pm1.9$	2 K			2
WC 5	$3.4\pm0.4$	7 K	2.0	1 D	2
WC 6	$2.1\pm0.2$	7 K	6.0	1 D	2
WC 7	$5.2\pm0.9$	7 K	3.6	1 D	3
WC 8	$2.7\pm0.3$	2 K	$4.4\pm0.3$	2 D	4
WC 9					10-20

H = Hamann & Koesterke (1998a): WN-stars.
K = Koesterke & Hamann (1995): WC-stars.
D = Dessart et al. (2000): WC-stars.

We see that the sonic points are at optical depths between about 3 and 33. The temperatures at the sonic point fall in two intervals, $40~000 < T_{\rm s} < 80~000$ K and $140~000 < T_{\rm s} < 190~000$ K. We will show below that this is due to the dependence of the opacity in WR-stars on density and temperature: optically thick winds of WR-stars can only exist if the temperature of the sonic point is near about 160 000 K or near 50 000 K. The value of the velocity parameter $\beta$ is between about 3 and 6 for the WN-stars and between 4 and 7 for the WC-stars.

   
7.1 Comparison with OPAL opacities

The opacities in the vicinity of the sonic point for the models A0 and B0 can be compared with the OPAL opacities, which are the Rosseland mean opacities for non-expanding media (Iglesias & Rogers 1993, 1996). In the wind models we need the flux-averaged opacities for the expanding media. In the case of optically thick winds the sonic points are located at large optical depths ( $\tau^{\prime} \approx 20$) with the velocities being around 30 km s^-1 and at such conditions the Rosseland mean opacities are expected not to differ very much from the flux-averaged opacities. The models presented above showed that near the sonic point of WR-winds the temperatures are around 160 000 K or 50 000 K. Therefore we only concentrate on the OPAL-opacities in the range of $4.5 < \log(T) < 5.5$. In deriving the OPAL data we used the standard OPAL tables and the OPAL supportive codes for the interpolation from the tabulated data (hhtp://www-phys.llnl.gov/Research/OPAL). We used the subroutine packet OPACGN93(Z, X, T6, R) for WN-stars with metallicity Z=0.02 and with a hydrogen mass fraction X, with a temperature T₆ in millions of Kelvin, and with the OPAL parameter, $R=\rho/T_{6}^{3}$. For WC-stars we used the subroutine package OPAC(Z, X, $X_{\rm C}$, $X_{\rm O}$, T6, R) with Z=0.02, X=0.0 and the enhanced carbon mass fraction accounted for by the parameter $X_{\rm C}$ (the total mass fraction of C is the sum of the initial amount included in the metal mass fraction, Z, and $X_{\rm C}$), and the enhanced oxygen mass fraction accounted by the parameter $X_{\rm O}$ (the total mass fraction of O is the sum of the initial amount included in metal mass fraction, Z, and $X_{\rm O}$). The optically thick wind models of the WR-stars show that the parameter R is in the range of $-7.0 < \log (R)<-6.5$. The OPAL opacities for different values of X and Y with Z=0.02 are shown in the upper panels of Fig. 2 and those for C-enhanced and H-free gas in the lower panels. Notice the strong bump in opacity in all models around $\log~T \simeq 5.2$, which is mainly due to many bound-bound transitions of Fe, and a very small bump around $\log~T \simeq 4.6$ for some models, especially the C-rich and H-free models.

 

 

Table 6: Optically thick wind models for standard WR-stars: Variants A0 and B0.

Star	Sp.	$\log~\mbox{$L_*$ }$	$\log~\mbox{$\dot{M}$ }$	$R_{\rm s}$	$\taup_{\rm s}$	$\taup_{\rm s}$	$T_{\rm s}$	$T_{\rm s}$	$\beta$	$\beta$	$\left(\frac{{\rm d}\chi}{{\rm d}r}\right)_{\rm s}$	$\left(\frac{{\rm d}\chi}{{\rm d}r}\right)_{\rm s}$	$\chi_{\rm s}$	$\chi_{\rm s}$
					A0	B0	A0	B0	A0	B0	A0	B0	A0	B0
WR2	WN2	5.27	-5.40	1.73	9.52	8.06	151 600	145 900	5.86	4.89	0.96	5.91	0.70	0.70
WR139	WN5	5.21	-5.04	1.65	22.6	19.2	183 700	176 800	5.92	5.03	1.45	10.4	0.76	0.76
WR136	WN6	5.73	-4.20	4.94	33.1	28.3	156 900	150 900	6.02	5.09	0.41	3.01	0.47	0.47
WR22	WN7	6.08	-4.38	16.22	9.57	8.10	79 000	76 000	5.26	4.34	0.13	0.76	0.60	0.60
WR22	WN7	6.08	-4.38	32.44	5.76	4.84	49 700	47 900	4.75	3.83	0.08	0.44	0.60	0.60
WR105	WN9	5.81	-4.55	26.3	3.67	3.05	42 900	41 300	3.77	2.93	0.12	0.62	0.44	0.44
WR111	WC5	5.31	-5.00	1.78	24.6	21.0	191 200	184 000	6.65	5.73	0.81	6.52	0.69	0.68
WR111	WC5	5.31	-5.00	2.67	18.5	15.7	145 900	140 400	6.53	5.56	0.61	4.14	0.68	0.68
WR42	WC7	5.23	-4.89	2.51	25.7	22.0	155 500	149 700	6.24	5.33	0.84	6.06	0.74	0.74
WR103	WC9	5.20	-4.62	8.18	20.6	17.5	80 300	77 300	5.89	4.89	0.48	2.74	0.76	0.76
WR103	WC9	5.20	-4.62	16.36	12.6	10.7	50 600	48 700	5.56	4.45	0.30	1.58	0.75	0.75

(1) Variants A0 and B0 are for constant L(r) and for $\chi \sim \rho / T^{3.5}$ respectively.
(2) $R_{\rm s}$ is in units of $\mbox{$R_\odot$ }$ and $\mbox{$\dot{M}$ }$ is in $\mbox{$M_\odot$ }~{\rm yr}^{-1}$.
(3) $({\rm d} \chi /{\rm d}r)_{\rm s}$ is in units of 10^-14 cm g^-1 and $\chi_{\rm s}$is in units of cm² g^-1.

Figure 2: The OPAL opacity for different abundances as a function of temperature and $R \equiv \rho /(T_6^3)$. The plotted data are for $\log~R=-8.0$, -7.5, -7.0, -6.5 and -6.0. The higher the value of R the higher the values of $\chi$ at $\log~T=4.5$ and 5.5. The top figure is for models with Z=0.02 but different H and He abundances. The lower figure is for H-free gas with Z=0.02 and different abundances of He and C. Notice the strong bump near $\log (T) \approx 5.2$ and the small bump near $\log (T) \approx 4.6$. The sonic points of optically thick winds can only occur in the temperature regions where ${\rm d} \chi /{\rm d}T > 0$.

We have compared the values of $\chi_{\rm s}$ and $({\rm d} \chi /{\rm d}r)_{\rm s}$ of the optically thick wind models A0 and B0 of the WR-stars (Table 6) with the OPAL values. We find that the values of  $\chi_{\rm s}$ of the models are similar to the OPAL values. However, for all models there is a significant difference between the values of $({\rm d} \chi /{\rm d}r)_{\rm s}$ of the models and those of the OPAL opacities. In all models this gradient is positive (this is a requirement of optically thick radiation driven winds, see Sect. 4.2), whereas the gradient of the OPAL opacities, for the temperatures and densities at the sonic point of the models, is in some cases even negative. We stress, however, that by changing somewhat the sonic radius $R_{\rm s}$ for models A0 and B0 it is possible to achieve exact agreement with the OPAL opacity $\chi_{\rm s}$ and the gradient $({\rm d} \chi /{\rm d}r)_{\rm s}$ at the sonic point. This is because the OPAL opacity has a strong peak around $\log{T} \approx 5.2$and a small peak near $\log{T} \approx 4.6$ (for $R \approx {\rm const.}$). Our models show that R is about constant near the sonic point and so the plots of $\chi$ as function of T in Fig. 2 for constant R show approximately at which temperatures $({\rm d} \chi /{\rm d}r)_{\rm s}$ can be positive.