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Astron. Astrophys. 354, 193-215 (2000)

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7. Discussion

The first thing which is evident from the spectral analysis is the increasing difficulty in fitting the spectra of the earliest types with plane-parallel model atmospheres. In particular, for 50 000 K and above we were unable to determine the stellar parameters in the same way as we did for stars in Paper I and for the rest of the objects in the present study. This is true even for the relatively large gravity star Cyg OB2 [FORMULA]7, and was nearly the case for the relatively cool star HD 15 558. Plane-parallel analyses are still useful, however, because they apply up to temperatures of 50 000 K and because they can be used as constraints for the analysis in the larger parameter space demanded by more sophisticated models.

The temperature scale defined by plane-parallel, hydrostatic, non-blanketed models is probably too hot (see Vacca et al. 1996; Harries & Hilditch 1998). For example, Hubeny et al. (1998) have shown that the same quality fit can be achieved for 10 Lac with a line-blanketed model at 35 000 K and with the non-blanketed model used in Paper I at 37 500 K (all other parameters remaining the same). The effect can be even larger for the stars analysed here, since the line-blocking effects in plane-parallel, hydrostatic models move the stars towards higher temperatures.

In the spherical, non-hydrostatic models we have used here, line-blocking has been simulated, and we have found that it has the same effect as to keep the He II resonance lines in detailed balance (which were already in detailed balance in the plane-parallel models, so that they did not show the influence on the He II lines of the Pickering series we observe in these spherical models). This effect had important consequences in our analyses. It led us to change the temperature criterion and adopt He II [FORMULA]4200 as the main line for the fit, which resulted in lower temperatures than if we had adopted the He II [FORMULA]4541 in those cases in which we could not fit both lines simultaneously. The simulations indicate that the reduction of the strong upward rates of the He II resonance lines (whatever the real physical cause) will help to bring both lines into agreement (although at the moment we cannot say whether it will bring them completely into agreement). It also suggested to us that we should fit the red wing of [FORMULA] whenever the fit to the whole line was impossible. What could be important for the future is that we have shown that He II [FORMULA]4686 is also affected and its fit is highly improved (although still qualitatively) when the resonance lines of He II are kept in detailed balance, or the departure coefficient of the second level is kept below its detailed balance value through additional background opacity.

The plane-parallel spectroscopic masses are as usual lower than evolutionary masses (see Paper I or Vacca et al. 1996). The new temperature criterion does not strongly influence the mass discrepancy. In Paper I we showed that lowering the temperature (for whatever reason) will not bring the masses into agreement. In the present case, however, we find the mass discrepancy even for luminosity class V stars. This finding is contrary to the conclusion we obtained in Paper I (and is in agreement with other authors; see, for example, Vacca et al. 1996), namely that luminosity class V stars do not show a mass discrepancy. The reason for this apparent contradiction is that the gravities we derive here are also low for these luminosity class V stars. Thus the conclusion should be rather that `high-gravity stars do not show a mass discrepancy,' where the term `high' actually depends on the strength of the radiation field.

This indicates a problem of the hydrostatic models due to the intense radiation pressure, and in fact larger gravities are derived when using the spherical models with mass-loss, largely reducing the mass discrepancy, which is now about 50[FORMULA]. This result is even reinforced when we realize that most temperatures are now lower due to sphericity and a new temperature indicator. An additional contribution came in some cases from the fact that the wings of [FORMULA] can be strongly affected by wind contamination. In one case (HD 15 570) this contamination is so strong that actual information about the gravity from the wings of [FORMULA] is lost. In all other cases, however, the systematic effect that spectroscopic masses (without line-blanketing or blocking) are lower than evolutionary ones (without mixing mechanisms) is still present (note that formally the error bars overlap, and it is only because the effect is systematic that we can speak of a mass discrepancy).

The problem can be alternatively formulated using the escape velocities. (Table 6 gives the escape velocities and related values). Escape velocities have been derived using [FORMULA] values uncorrected for centrifugal forces, i.e., the measured values implicitely including the centrifugal force acceleration (cf. Sect. 4) which determine the effective escape velocities.


Table 6. [FORMULA] values, terminal velocities and escape velocities (in km s-1) obtained using the spectroscopic (sp) and evolutionary (ev) masses. In both cases we have taken into account the effect of the centrifugal force in reducing the escape velocity.

Fig. 28 shows the correlation between escape velocities and terminal wind velocities. We see that the diagram using evolutionary masses shows a good linear correlation between both quantities, whereas that with the spectroscopic masses shows a weaker correlation. However, the last diagram is in better agreement with the theory of radiatively driven winds. This theory predicts for the O-star domain (i.e., if the force-multiplier parameter [FORMULA] is small, cf. Friend & Abbott 1986; Kudritzki et al. 1989)


with [FORMULA] the escape velocity and [FORMULA] one of the line force multiplier parameters (the coefficient in the exponent of the line strength distribution function). With typical values for OB stars of [FORMULA]= 0.6-0.7 we obtain values of 3.3-5.2 for the ratio of terminal-to-escape velocity. This range of values is in agreement with those found here for the spectroscopic masses (see Table 6 ; note that the only point deviating from this range corresponds to HD 15 570, whose gravity and spectroscopic mass are very uncertain, and that HD 5 689, which is the leftmost point in the upper part of Fig. 28, would lie in the middle of the range if the absolute magnitude of a luminosity class III object were assumed).

[FIGURE] Fig. 28. The escape velocities obtained from spectroscopic (above) and evolutionary (below) masses, against the wind terminal velocities, all in km s-1. The lines have the slopes predicted by theory for [FORMULA] values of 0.6 and 0.7, e.g., 3.3 and 5.2. Plus signs mark the position of HD 5 689 if this star is assigned a magnitude of -5.78, typical of a O6 III star. Typical error bars have been plotted in the upper right corner. The abscissa error of Cyg OB2[FORMULA]7 in the upper plot is twice the corresponding error bar, due to its high [FORMULA] value

On the other hand, we derive very low ratios between terminal and escape velocities when using evolutionary masses (see Table 6), with an average ratio of 2.57, corresponding to [FORMULA]= 0.53 (which would be very low for the considered spectral range). Thus, Fig. 28 seems to indicate that the evolutionary masses are systematically too large (via the corresponding escape velocities), whereas the spectroscopic ratios are closer to the theoretically expected range. In Fig. 29 we can see this result, already contained in former papers (Groenewegen et al. 1989; Lamers & Leitherer 1993), from a slightly different point of view. Here we have plotted the ratio of evolutionary-to-spectroscopic mass versus the ratio of spectroscopic-to-evolutionary [FORMULA]. We see that, except for the odd case HD 5 689, there is a strong correlation, indicating the relation between mass discrepancy and our present knowledge of radiatively driven winds. (Note moreover that HD 5 689 would perfectly fit into the correlation when an absolute magnitude of -5.78 is assumed, as appropriate for an O6 III star, but note also that the terminal velocity for this star was derived from the spectral type-terminal velocity relation from Haser 1995, as was that of Cyg OB2[FORMULA]7). We should stress that we adopt a [FORMULA]-law for the wind velocity and determine then the [FORMULA]'s directly from the derived relation between [FORMULA] and escape velocity. Thus the agreement of the spectroscopic [FORMULA]'s with the predictions of the radiatively driven wind velocity results in mutual support.

[FIGURE] Fig. 29. The ratio [FORMULA](sp)/[FORMULA](ev) against the mass ratio [FORMULA]/[FORMULA]. Asterisks refer to the values quoted in Table 3 (in particular, the asterisk in the upper right corner corresponds to HD 5 689), whereas the cross marks the position of HD 5 689 if this star is assigned a magnitude of -5.78, typical of an O6 III star.

At present it is still unclear whether the inconsistency found is due to some physical effects that should be incorporated into the evolutionary or into the atmospheric models, although the former are perfectly able to explain the discrepancy, at least qualitatively, when introducing mixing effects (Heger 1998) that could also affect surface abundances.

Spherical models with mass-loss do not contribute to reduce the helium abundances found with plane-parallel models. In previous analyses (Herrero et al. 1995; Israelian et al. 1999) in which we used He I [FORMULA]4387, 4922 and He II [FORMULA]4541 as temperature indicators, the derived helium abundances were similar. As a result of the new indicator, the helium abundances had now to increase. However, we see that the stars of larger gravity (HD 15 558 and HD 15 629) do not seem to show He overabundances. Normal supergiants lie between the former and the fast rotators, that show the largest overabundances, in agreement with results from Paper I and the possibility that rotation plays a fundamental role in the chemical evolution of single stars. However, it is not clear whether a difference exists in the mass discrepancy between the rapid rotators and the other stars, or a correlation between the mass and helium discrepancies, that could indicate an overluminous evolution, as predicted by evolutionary rotating models (see Langer & Heger 1998, Maeder 1998 or Meynet 1998 and references therein). We should stress here that we do not consider HD 5 689 as really showing such a large mass discrepancy as it appears to do in Tables 2 and 3, but that it is a problem of the stellar classification (or a problem of assigning the star to the Cas OB7 association).

Derivation of the mass-loss rates has been hindered by our finding of the inconsistency between [FORMULA] and [FORMULA] in those cases in which the wind is particularly strong. Thus we have found that both lines can demand mass-loss rates that differ up to a factor of two. We prefer the values given by [FORMULA] because it is more sensitive to mass-loss, and attribute the problem to difficulties in describing the wind in the transition zone. IR observations and analyses giving information about this zone should help in the future to solve the problem.

From the three stars for which we had some information about mass-loss rates derived from radio emission, only one (for which in addition only an upper limit from radio fluxes is available) shows agreement between the [FORMULA] and radio values. For the other two, radio mass-loss rates are a factor of three to four lower than [FORMULA] mass-loss rates (and thus [FORMULA] values would be in between). However, several facts should be taken into account before claiming that there is a contradiction between both sets of values. First, the concerned stars are only two particular, rather special cases (one, HD 15 570, is a very extreme O star, and the other, HD 210 839, is a strong non-radial pulsator, rapid rotator with a probably non-spherically symmetric wind, cf. Sect. 5.2); secondly, what we actually obtain from the observations are Q values, proportional to ([FORMULA]) for [FORMULA], and to ([FORMULA]) for radio and thus the derived values depend on the particular set of chosen stellar parameters; and thirdly, also the proportionality constants in the case above depend on model details (for example, Lamers & Leitherer 1993, assume that He is doubly ionized in the wind of HD 210 839 between 10 and 100 stellar radii, whereas our calculations indicate that from 28 stellar radii upwards, He is single ionized, which would have an immediate effect on the derived radio mass-loss rates, increasing them). Thus, we cannot state that there is a general problem (or even a particular one), or decide which value we should prefer for each object.

Having derived all parameters we can obtain the modified wind momenta of all stars for which the radiation-driven wind predicts a tight correlation with luminosity. Our results are given in Table 3 and plotted in Fig. 30, together with the data given by Puls et al. (1996) for Galactic supergiants. We see that our points for supergiants agree well with theirs (we have discussed the differences in the individual studies). We have also plotted the regression lines derived from the supergiants from Puls et al. and from all the supergiants (the former and ours) to stress this point. We see that both regression lines are quite parallel, indicating that the new values do not significantly change the known WLR.

[FIGURE] Fig. 30. The wind momentum-luminosity relation for the stars in our sample. The abcissa is [FORMULA] and the ordinate is the logarithm of the modified wind momentum (MWM), log (M_[FORMULA]). Asterisks represent OB supergiants from Puls et al. (1996). Open diamonds, squares and triangles are respectively OB supergiants, giants and dwarfs from the present work. The positions of HD 14 947 and HD 210 839 in both studies are joined by a line, where the major difference results from the different effective temperatures and helium-abundances attributed to the objects.

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Online publication: January 31, 2000