5 Discussion

The results obtained in the preceding section allow us to address four important aspects of the properties of massive OB supergiants: the effective temperature scale, the ionizing fluxes, the mass discrepancy and the wind momentum-luminosity relationship.

5.1 The effective temperature scale for massive O supergiants

With the inclusion of mass-loss, sphericity and line-blocking/blanketing we derived lower temperatures than those quoted in Herrero et al. (2001), where we have used plane-parallel, hydrostatic models without line-blocking for all stars except Cyg OB2 #7, for which a spherical model with mass-loss, but without line-blocking was used. As we have covered various spectral types from O3 to B1 in our analysis, we can obtain a temperature scale for O supergiants. We note however, that Cyg OB2 #4 is actually a luminosity class III star, while #2 is probably of class II.

For our earlier supergiants (O3-O7) we find temperatures that are 4000 to 8000 K cooler than in Herrero et al. (2001), while for the O9.5 I supergiant we obtain 2000 K less and for the B1 star the temperature is now hotter, but with a lower He abundance. These findings are in qualitative agreement with theoretical expectations (e.g., Schaerer & Schmutz 1994) and with the recent temperature scale for O dwarfs presented by Martins et al. (2002). These authors obtain lower temperatures for O dwarfs by 1500-4000 K when including line-blanketing and sphericity.

Our temperature scale is shown in Fig. 12, together with the one from Vacca et al. (1996) for O supergiants. The effective temperature scale from these authors has been mainly derived from analyses using pure H-He, plane-parallel, hydrostatic models, like those performed by Herrero et al. (1992). It is thus not surprising that our new scale is cooler.

Figure 12: Our temperature scale for O supergiants (circles) compared with the scale by Vacca et al. (1996) (triangles). New temperatures are much lower, except for the relatively cool Cyg OB2 #2 (B1I). Note that the entry at O7 is actually a luminosity class III star.

In spite of the still low number statistics, we already appreciate some interesting features in Fig. 12. First, from O3 to O9 we see a smooth temperature decline into which Cyg OB2 #4 fits perfectly, despite its luminosity class III. From this smoothness is excluded Cyg OB2 #11, much cooler than the other O5 supergiant, Cyg OB2 #8C, and even cooler than the O5.5 supergiant, Cyg OB2 #8A. The main difference in their properties is the extreme Of character of Cyg OB2 #11, which thus appears to be related to cooler temperatures. This is indicating to us that all spectral signatures have a significance in terms of stellar parameters, and thus a temperature scale using only spectral subtypes of O supergiants, with the various nuances in their classification scheme, will neccesarily be of limited accuracy.

We also see that both temperature scales converge towards later spectral types, until the B1 star, Cyg OB2 #2. This object has a temperature that does not seem to fit into the general behaviour, although data are still too scarce to know whether this has any significance.

The new temperature scale and the lower luminosities will have an influence on other aspects, e.g., on the emergent ionizing fluxes that are now much lower than in the older models, as we have seen in the discussion of Cyg OB2 #7.

5.2 Ionizing fluxes from O supergiants

Our treatment of UV metal line opacity is approximate (in the sense that we use an approximate NLTE approach and suitably averaged lines opacities), and we do not pretend to give a detailed description of the UV radiation field. However, the emergent fluxes, should be correct in an average sense (i.e., neglecting distinct spectral features, see again Fig. 5), in particular concerning frequency integrated quantities. (Note that the differences bluewards from He II 228 Å are mostly due to different temperature structures in the outer wind.) Thus, we should obtain a rather correct description of quantities extending over broad spectral regions, like the number of photons capable of ionizing hydrogen.

This number, of course, is of extreme relevance to studies of H II regions surrounding the stars. Vacca et al. (1992) have calculated ionizing fluxes from plane-parallel, hydrostatic, LTE, line-blanketed Kurucz-models (Kurucz 1992) and concluded, from a comparison with more elaborate models available at that time (Schaerer & Schmutz 1994), that their ionizing fluxes should be reliable. That would indicate that line-blocking/blanketing is the major effect when calculating the UV continuum ionizing flux at a given effective temperature.

Our calculations seem to support this conclusion, but also indicate that for an application to H II regions further effects have to be considered.

Figure 13: a) left: ionizing fluxes (photons cm-2 s-1) at the stellar surface versus effective temperature. b) right: ionizing luminosities (photons s-1) versus spectral type. Dots correspond to the stellar analyses presented here and lines represent the calibrations by Vacca et al. (1996) for O supergiants.

In Fig. 13a we have plotted our calculated H ionizing fluxes versus the effective temperature of the corresponding model. The dashed line gives the calibration by Vacca et al. (1996) for O supergiants (their Table 7). We see that the agreement is good. Even more, the entry departing slightly from the relation corresponds to Cyg OB2 #4, that has a luminosity class III. Note that the ionizing fluxes are basically independent of stellar radius. Thus, at given temperature the effects of metal line opacity remain the major ingredient which detemine the emergent UV flux, and the differences between NLTE and LTE models seem to be small regarding the integrated hydrogen Lyman fluxes. (This behaviour should become significantly different when, considering, e.g., He II ionizing fluxes because of the extreme sensitivity of this ion to NLTE effects, in particular as function of mass-loss.)

In Fig. 13b we have plotted the H ionizing luminosities as function of spectral type. Here, the differences with the calibration by Vacca et al. (1996) are apparent. They clearly originate from our new relation between spectral type and effective temperature (and our lower radii for most stars, compared to the calibrations by Vacca et al. 1996), i.e., the stellar luminosities at given subtype are now smaller. (This discrepancy between both plots, of course, results from the former inconsistency of calibrating spectral types via unblanketed models, however calculating the number of ionizing photons from blanketed ones). Furthermore, for one case (Cyg OB2 #8A) we find that our ionizing luminosities are larger as a consequence of the larger radius in this particular case. Since the ionizing luminosity is the quantity which really matters for the ionization of H II regions, our results indicate that statistically there are fewer photons available compared to earlier findings, but also that individual cases have to be studied in detail, because they can depart from the general trend.

5.3 The mass discrepancy

The mass discrepancy is a term that was used by Herrero et al. (1992) to refer to the systematic difference between spectroscopically determined stellar masses of OB supergiants and their evolutionary counterparts. The latter were always systematically larger, well beyond the error bars.

In Fig. 14 we have plotted the evolutionary mass (derived from the "classical'' models by Schaller et al. (1992) without rotation) and the spectroscopic masses obtained in our analysis for the Cyg OB2 supergiants. We see that the situation is by no means satisfactory: despite the (very) large errors adopted for the spectroscopic masses, three of the seven stars still do not cross the one-to-one relation and for two other we find only a marginal agreement. In fact, only two out of the seven stars have masses that agree reasonably well. However, compared to previous diagrams the situation seems to have improved: there is no clear systematic trend any longer. Roughly the same number of data points lie on each side of the one-to-one relation and the apparent scatter might be related to problems in the individual analyses.

Nevertheless it is somewhat too early to conclude that the atmosphere models with sphericity, mass-loss and metal opacity agree with the evolutionary models without rotation, and thus give the same answer in a statistical sense. First, the large scatter in Fig. 14 still poses a question for the masses of both set of models; and second, the stars in Cyg OB2 have moderate projected rotational velocities and are very young, with only the earliest type exhibiting an enhanced He abundance. Thus, older or faster rotating stars may have masses that disagree even more when derived using different methods.

This result indicates that we badly need both a calibration of present mass scales based on early type binaries (or any other reliable method), and CNO abundances for isolated massive stars.

Figure 14: A comparison of the spectroscopic and evolutionary masses for the Cyg OB2 supergiants. Although there are still serious differences, no obvious systematic trend is present. See text for details and discussion.

5.4 The Wind momentum-Luminosity Relationship in Cyg OB2

The original purpose of our work was to obtain a better constrained WLR for Galactic O stars by using objects belonging to the same association, and thus minimizing the scatter introduced by uncertainties in the relative distances. The distance to Cyg OB2 and thus the derived luminosities may be in error, but all determinations will be affected in a similar way.

Figure 15 displays the WLR obtained for our Cyg OB2 sample. Errors for $v_{\rm\infty}$  are taken from Herrero et al. (1991). The stars #8A, #8C, #4 and #10 follow a nice sequence with low scatter. Cyg OB2 #7 and #11 seem to lie above this sequence. This is interesting because these stars display the most extreme Of signatures in their spectra, which might be related to an ionization change in the wind that could result in a different line-driving force or to clumping effects that would produce an overestimation of the mass-loss rate. Cyg OB2 #2 seems to lie below that relation, which is consistent with the results by Kudritzki et al. (1991), who found a different WLR (with a lower offset) for the winds of early B supergiants, compared to the O-star case. Our observed relation also agrees well with the theoretical WLR derived by Vink et al. (2000) (the dashed line in Fig. 15).

We have investigated whether a reduction in the outer minimum electron temperature (that the model is allowed to reach, see Santolaya-Rey et al. 1997, Sect. 3.1) might result in lower mass-loss rates for Cyg OB2 #7 and #11. While we usually assume that $T_{\rm min}=$ 0.75 $T_{\rm eff}$ (a typical value for OB stars), calculations by Pauldrach (private communication) indicate that this value can reach values as low as 0.40  $T_{\rm eff}$ in extreme cases. This effect has usually no influence on an analyis as performed here, where almost all considered lines are formed in a region with temperatures beyond this minimum. "Only'' H $_{\rm\alpha}$ (and He II $\lambda$4686) might be affected in cases of extreme mass-loss, which is the reason that we have investigated here this question. Nevertheless, in all considered cases the resulting reduction of the derived mass-loss rate was less than 20$\%$. Thus, even accounting for this uncertainty we cannot bring the position of these two stars into agreement with the WLR defined by the other four O stars.

However, taking into account the error bars, our data are still compatible with a unique relation including all seven stars. This is also shown in Fig. 15, where we display two different regressions, one including only Cyg OB2 #8A, #8C, #4 and #10, and one including all seven stars, respectively. Interestingly, the relation obtained by including only the O-supergiants, not shown for clarity, is still marginally compatible with the position of the B-supergiant.

Figure 15: The wind momentum-luminosity relationship obtained for the Cyg OB2 supergiants. The dotted line gives the regression obtained by including all stars, while the solid line gives the one obtained by including only Cyg OB2 #8A, #8C, #4 and #10. The dashed line very close to the solid one is the theoretical relation by Vink et al. (2000). The entry with the highest luminosity is Cyg OB2 #8A and those close together are Cyg OB2 #7 and #11, the stars with extreme Of character.

In Fig. 16 we show a comparison of our data points with those obtained by Puls et al. (1996), updated by Herrero et al. (2000) for some entries. We see that both sets compare well, the most important difference being the fact that the scatter in our data points is lower. However, at present it is not possible to conclude whether this is a real improvement (as we have expected) or an artefact of the low number statistics. We note that the first possibility would imply that stars with extreme Of characteristics follow a slightly different relation than normal O stars, although we could not determine the reason for the relatively high position of Cyg OB2 #7 and #11.

Figure 16: A comparison of our results for Cyg OB2 OB-supergiants (filled dots) with the results by Puls et al. (1996) and Herrero et al. (2000).

In Table 4 we compare the coefficients we have obtained for the WLR (by weighted least-squares fits) using different samples (see above) as well as those quoted by Kudritzki & Puls (2000) in their Table 2, and the coefficients of the theoretical relation provided by Vink et al. (2000). A correct error treatment would require to take into account errors in both axes and their correlation. This treatment is not simple and will be presented elsewhere (Markova & Puls 2002). For the scope of the present paper we have adopted as error an average of the errors obtained when considering only those in the ordinate values (an underestimation) and when considering the errors in abscissa and ordinate as uncorrelated (an overestimation). Using both estimates for the errors, we have calculated the corresponding regression, where the resulting values for slope and offset turned out to be only marginally different. The final values quoted in Table 4 have been obtained from a straight average of these results.

 

 

Table 4: Coefficients for the wind momentum-luminosity relationship (see equation in p. 950) obtained in this work and taken from Kudritzki & Puls (2000) and Vink et al. (2000).

$\log D_0$	x	$\alpha^\prime_{\rm eff}$	Comments
16.81 $\pm$ 1.16	2.18 $\pm$ 0.21	0.46 $\pm$ 0.04	Using all Cyg OB2 stars
18.30 $\pm$ 1.82	1.92 $\pm$ 0.31	0.52 +0.10-0.07	Cyg OB2 #2 not included
19.27 $\pm$ 1.82	1.74 $\pm$ 0.32	0.58 +0.12-0.09	Including only Cyg OB2 #8A, #8C, #4, #10
20.69 $\pm$ 1.04	1.51 $\pm$ 0.18	0.66 $\pm$ 0.06	From Kudritzki & Puls 2000 (Table 2)
18.68 $\pm$ 0.26	1.826 $\pm$ 0.044	0.548 $\pm$ 0.013	From Vink et al. 2000 (Eq. (15))

We see that in spite of the visual agreement found in Fig. 16, our results differ significantly from those given by Kudritzki & Puls (2000), in particular concerning the slope of the relation. Kudritzki & Puls (2000) found a slope of 1.51 $\pm$ 0.18, while, when excluding the B supergiant, we obtain values from 1.74 $\pm$ 0.24 to 1.92 $\pm$ 0.22. In addition, our values for the vertical offset vary between 19.27 and 18.30, below the value of 20.69 of Kudritzki & Puls (2000). Although the error bars allow for marginal agreement, the conclusion is that our relation is steeper and thus $\alpha^\prime_{\rm eff}$ (=1/x) is smaller in our case. This indicates a different slope of the line-strength distribution function (see Puls et al. 2000 and Kudritzki & Puls 2000, Sect. 4.1), i.e., a larger number of weak lines.

In contrast, we obtain very good agreement with the theoretical relation by Vink et al. (2000). Their relation, both in slope and offset, lies right between our relation when considering all supergiants or only the moderate O, Of stars, respectively. Our observations are thus compatible with a unique WLR for all O supergiants, but favour a separation of the extreme O, Of stars.