3 A test case: 10 Lac

Before applying the modified code to the analysis of our Cyg OB2 sample, we have analyzed the O9V star 10 Lac as a test case. This star is well suited for calibration purposes because it is a luminosity class V star with a low projected rotational velocity, has already been used as standard by Herrero et al. (1992) and was later considered by Hubeny et al. (1998) to study the effects of line blanketing in plane-parallel, hydrostatic atmospheres.

Figure 1 gives the fit diagram for the H/He spectrum of 10 Lac. It is very similar to the corresponding fit diagram in Herrero et al. (1992), but now it is centered at a lower effective temperature (and a slightly lower gravity). A comparison of both diagrams also reveals that the dispersion around the final model is now smaller. Therefore, the present error box has a width of only 1000 K instead of 2000 K, as the one in Herrero et al. (1992). It is reassuring that all lines, including He II $\lambda$ 4200 that could not be fitted by Herrero et al. (1992), lie within the error box.

Figure 1: Fit diagram to the H/He spectrum of 10 Lac using FASTWIND plus approximated line blocking/blanketing. The filled circle marks the location of the chosen model, and the box around gives the size of the adopted error box.

The stellar parameters of 10 Lac we have derived here are $T_{\rm eff}=$ 35 500 $\pm$ 500 K, $\log g=$ 3.95 $\pm$ 0.10 and $\epsilon =$  $\frac{{N({\rm He})}}{{N({\rm H})+N({\rm He})} }=$ 0.09 $\pm$ 0.03, N(X) being the abundance of element X by number. The only significant difference compared to the results from Herrero et al. (1992) is the effective temperature, now lower by 2000 K. This is in complete agreement with the result obtained by Hubeny et al. (1998). These authors estimated a temperature of 35 000 K for 10 Lac using TLUSTY, a plane-parallel, hydrostatic, line-blanketed model.

Fit diagrams have the drawback that they only give the best possible fit for the chosen constraints. They rely on interpolations and sometimes (when using EWs) do not account for the profile shape. The actual final fit may still be poor. Figure 2 shows the line spectra for our final model of 10 Lac. Good agreement is found for all lines, although a few details are not perfectly reproduced. In particular, the core of He II $\lambda$ 4200 is too weak, which is also true for the forbidden component of He I $\lambda$ 4471. Besides this, however, the final fit is perfectly consistent in all other aspects.

Figure 2: The fit to the H/He spectrum of 10 Lac using FASTWIND with approximated line blocking/blanketing. The ordinate gives the relative flux values. Note that the scales are different for each line. See text for comments.

Our result also agrees with the study of Martins et al. (2002), who recently found that using pure H/He models (as Herrero et al. 1992 did) results in an effective temperature scale for O dwarfs hotter by 1500 to 4000 K compared to using line-blanketed models.

These authors have derived a new effective temperature scale for O dwarfs using CMFGEN (Hillier & Miller 1998), a spherical code including mass-loss and blanketing. In their $T_{\rm eff}$  scale, O9V stars are located at 33 000 K. However, their scale is calibrated using the equivalent width (EW) - spectral classification relations of Conti & Alschuler (1971) and Mathys (1998), and thus we have to compare our result with Martins et al. using the classification of 10 Lac in this system.

Conti & Alschuler (1971) have classified 10 Lac as O8 III and not as a luminosity class V star (which it has been considered by Martins et al. 2002), although the star lies just at the border between both luminosity classes. The luminosity class III is mainly due to the low EW quoted by Conti & Alschuler (1971) for the He I $\lambda$4143 line. Other EWs quoted by these authors are consistent with our observations which show a much larger value for this line, resulting in a luminosity class V within their classification scheme. Thus, we conclude that 10 Lac should be classified as O8 V in the system of Conti & Alschuler (1971) (whatever the reason for the low EW in Conti & Alschuler 1971 was).

The effective temperature in the Martins et al. scale for O8 dwarfs lies between 36 000 and 35 000 K, in perfect agreement with our result. Therefore we regard our result as fully consistent with recent findings using more elaborate but also more expensive fully blanketed NLTE model atmospheres.

Nevertheless we have analyzed 10 Lac with our version of CMFGEN that includes a consistent photospheric structure (Najarro 2002) and have fitted the line profiles instead of only using their EWs. Our results from CMFGEN completely agree with those from FASTWIND.

The reason for the lower temperatures derived is twofold (see also Martins et al. 2002). On the one hand, the radiation field which is backscattered due to the additional opacity produces a larger (E)UV radiation field. On the other hand, due to line-blanketing the electron temperature rises in photospheric regions. Both effects favour a higher ionization degree at lower effective temperatures, compared to unblanketed models. This effect can be clearly seen in Fig. 3.

The analysis of 10 Lac gives us an indication of what we can expect when introducing metal line opacity (namely lower effective temperatures), either in the form of traditional line-blanketing (as Hubeny et al. 1998; Martins et al. 2002 and references therein) or including the line-blanketing via adapted Hopf-parameters (as here). Although with our method we do not force flux conservations, in all models calculated here the flux is conserved to better than 3$\%$ at all depths, where the remaining deviations have no impact on the emergent fluxes and profiles.

Our analysis of 10 Lac also gives us an idea of the error bar we can expect for the stellar parameters. For a resolution of 8000, a S/N of 200 and a projected rotational velocity of 40 km s^-1, the estimated errors are $\pm$500 K in $T_{\rm eff}$, $\pm$0.1 dex in $\log g$ and $\pm$0.03 in $\epsilon$. For what follows we shall note here that this analysis does not give us information about the mass loss rate or the $\beta$ exponent in the wind velocity law, as the wind of 10 Lac is negligible. We obtain an upper limit of 10^-8 $M_{\odot}$ yr^-1, but the fit has the same quality for any value below that limit. (The fit presented here was performed with $\dot{M}=$ 10 $^{-10}~M_{\odot}$ yr^-1.)

Figure 3: The He ionization fractions in two models with $T_{\rm eff}=$ 35 500 K, $\log g=$ 3.95 and $\epsilon =$ 0.09, with approximated line-blocking/blanketing (solid line) and without (dashed line). At typical photospheric line formation depths the ionization degree of the model including metal opacities is larger. Thus, lower temperatures are required in these models to reproduce the He spectrum.