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

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5. Analysis with spherical models with mass-loss

The difficulties in the analysis of some of the spectra discussed in the previous section result from the neglect of sphericity and mass-loss effects. Thus, we decided to use the program described by Santolaya-Rey et al. (1997) to account for these effects. Briefly, this program constructs a unified model of the stellar photosphere and wind regions, with a number of standard (stationarity, homogeneity) and non-standard assumptions and approximations (the most important of the latter concerns the way the temperature structure is obtained, which results in well approximated, but not forced, flux conservation throughout the atmosphere and a constant temperature in the wind). It solves the line-formation problem in an expanding atmosphere with spherical geometry, and Stark broadening is included in the final formal solution.

The atomic models used are the H and He models used also in our plane-parallel analysis, with minor changes to adapt them to the new program. Line-blocking has not been included so far. Its influence, however, will be considered in the next section (at least on a qualitative basis), and has affected the results decribed here.

We have adopted the following procedure. We begin with the parameters given by the plane-parallel models (see Table 2). The values of the terminal velocities are taken from Puls et al. (1996), except for HD 15 570 (from Lamers & Leitherer 1993) and for Cyg OB 2 [FORMULA]7 and HD 5689, where we have adopted the [FORMULA] - spectral classification relation provided by Haser (1995). The last object has been assigned a [FORMULA] corresponding to an O6 III star, which is a compromise between the luminosity class V given by Garmany & Stencel (1992) and the low gravity obtained from our analysis (see the discussion in the preceding section). Note, that this compromise is reasonable, since the difference between the mean [FORMULA] of O6V and O6I stars is only 450 km s-1.

Also needed are the values of the [FORMULA] exponent in the [FORMULA]-velocity law. For HD 15 558, HD 15 629 and HD 210 839 we adopted the values given by Puls et al. (1996). For HD 14 947, also analysed by Puls et al., we adopted [FORMULA] = 1.15 instead of 1.0 as quoted by Puls et al., for reasons explained later in the discussion of HD 14 947. For Cyg OB2 [FORMULA]7, HD 15 570 and HD 5 689 which were not analysed by Puls et al we used [FORMULA]= 0.8, 1.0 and 1.0, respectively.

For each star we have constructed a small model grid by varying gravity, temperature and mass-loss rate. We then tried to fit [FORMULA], [FORMULA] and the ratio He II [FORMULA]4200 to He I [FORMULA]4471, which is our new preferred temperature indicator, following the results described in the next section, where we will show that this He II line is less sensitive to model details than He II [FORMULA]4541. Of course, the use of a new temperature indicator introduces also an additional difference to the results from our plane-parallel considerations.

All other parameters remain fixed at the beginning at their values determined from the plane-parallel analysis. When necessary, the helium abundance has been changed later. The radius merits additional comments. Its value is derived from the emergent flux, which has been obtained for a given set of parameters (those from the plane-parallel analysis). Changing the parameters will change the derived radii, and thus an iteration process might be required. We have recalculated the radii of our objects with the new parameters, and have obtained an average change of only 2[FORMULA] (the maximum change being of only 4.5[FORMULA], from 22 to 23 [FORMULA]). Thus we used the plane-parallel values without further iteration. The finally adopted and derived parameters are summarized in Table 3. For those stars which have been analysed by Puls et al. (1996), we give their results in Table 4 for comparison. The new position of the stars in the HR diagram is shown in Fig. 13.

[FIGURE] Fig. 13. The programme stars on the Hertzsprung-Russell diagram after our analysis with spherical models with mass-loss. Compare the new diagram with Fig. 7. Details are as in Fig. 7.


Table 3. Parameters determined for the programme stars starting with the parameters from Table 2 and using the spherical models with mass-loss. Temperatures are in thousands of Kelvin, velocities in km s-1 and mass-losses in solar masses per year. MWM stands for modified wind momentum, and given is log (M_[FORMULA]), with [FORMULA] in solar masses per year, [FORMULA] in km s-1 and R in solar radii. [FORMULA], [FORMULA] and [FORMULA] are, respectively, the present spectroscopic and evolutionary masses, and the initial evolutionary mass, in solar units. The last column indicates whether we formally have a mass discrepancy considering an error of 0.22 in log([FORMULA]).


Table 4. Parameters as determined by Puls et al. (1996) for some of the stars in Table 3. Units are as in Table 3.

5.1. HD 5 689, O6 V

We can see the line fit to this star in Fig. 14, for a model with [FORMULA] = 37 000 K, [FORMULA] = 3.45 (which increases up to 3.57 due to the centrifugal correction) and log M _= -6.80. The temperature is lower than in the plane-parallel case, in part due to the change of our temperature criterion. The gravity, however, remains the same (which actually implies an increase with respect to plane-parallel models, since a lower temperature usually demands a lower gravity to fit a given line). Thus, the large mass discrepancy found above (see Table 2) cannot be reduced by the spherical models, since it originates from the low value of the radius, rather than from the low gravity. Neither can we decide whether the star has a larger radius in parallel with a higher mass-loss rate, since the [FORMULA] wind-emission depends only on the ratio (M_/[FORMULA]), whereas its photospheric component is independent of both. Assuming an absolute magnitude of a typical O6 III star ([FORMULA] = -5.78 mag, see Table 6 of Vacca et al. 1996) results in a radius of 15.9 [FORMULA], but does not change the fit quality (it affects only slightly the fit of He I [FORMULA]4471). Of course, mass, luminosity and mass-loss rate would be affected and we would obtain a spectroscopic mass of 30.0 [FORMULA], an evolutionary mass of 41.9 [FORMULA] (thus reducing the mass discrepancy to 30[FORMULA], a typical value), a luminosity of 5.6, closer to the other objects (see also the discussion in Sect. 7) and a logarithmic mass-loss rate of -6.17. However, this possibility would also imply that the wind in HD 5689 is much less efficient than in HD 210 839, as both stars would have roughly similar parameters, except for [FORMULA]. Unfortunately, without knowing the distance to HD 5689 more accurately, we cannot derive stronger conclusions. We also see that He II [FORMULA]4541 does not fit completely, and in particular the fit of He II [FORMULA]4686 is poor. This line always shows a poor fit, with an observed absorption that is much stronger than predicted, especially in the blue wing. As explained in the following section, this seems to be related to lack of line-blocking in our models, and He II [FORMULA]4541 seems also to be affected, although to a lesser extent.

[FIGURE] Fig. 14. Fit to the H and He lines in HD 5 689 using a spherical model with the parameters given in Table 3. From left to right and top to bottom we show [FORMULA], [FORMULA], He I 4471, He II 4200, He II 4541 and He II 4686. Wavelengths (along the abscissa) are given in Å. The ordinates give the relative fluxes.

5.2. HD 210 839, O6 I(n)fp

The line fit of this star, shown in Fig. 15, reveals two problems. First, the form of [FORMULA] suggest that the adopted rotational velocity is too large. A value of 200 km s-1 results in a much better fit 2

[FIGURE] Fig. 15. As Fig. 14, but for HD 210 839. The fit of [FORMULA] improves considerably if the adopted rotational velocity is lowered from 250 to 200 km s-1.

The second problem is that we were unable to fit the P Cygni form of the [FORMULA]-profile. As was pointed out previously, [FORMULA] Cep is a possible non-radial pulsator (Fullerton et al. 1996, de Jong et al. 1999), which might induce deviations from homogeneity (by exciting the line-driven wind instability already in the lower wind part, see Feldmeier et al. 1997), and the large rotation rate might have an additional influence on the wind structure and the resulting profile (cf. Owocki et al. 1998and references therein; Petrenz & Puls 1996). We decided to concentrate on the red wing of [FORMULA], since the theoretical simulations result in a blue wing affected by an extra He II absorption, which is inadequately described in our present models (see next section), and any wind variability becomes much more visible in the blue wing, compared to the red one. Again, the predicted emission in He II [FORMULA]4686 is much stronger than observed; also, the temperature is lower than in the plane-parallel case. However, the gravity is larger, and we do not find a mass discrepancy for this star.

[FORMULA] Cep has been also analysed by Puls et al. (1996), who adopted slightly different parameters. The main difference is the helium abundance ([FORMULA] = 0.09 instead of 0.25), which has a small influence on the gravity, however produces a larger wind momentum with a lower luminosity in our present results. (The value for [FORMULA] quoted by Puls et al. was not derived from a consistent spectral analysis, however taken from the literature). It is interesting to note that the high abundance favoured by our findings is in agreement with Blaauw's suggestion that all runaway stars are He enriched (Blaauw 1993). Note also that the possible runaway nature of HD 5 689 fits within this scenario.

Finally, we point out that our mass-loss rate coincides well with other determinations from [FORMULA] by Lamers & Leitherer (1993), but is larger than that derived from radio fluxes (Lamers & Leitherer 1993) by a factor of 3.

5.3. HD 14 947, O5 If+

The analysis of HD 14 947 has been hindered by an inconsistency in the radial velocities derived from the blue and the red spectrum (the spectra were taken in September 1991 and August 1992, see Table 1). While in the blue spectrum there are several metal lines to derive the radial velocity correction, in the red spectrum we only have three lines of He II and they give a result that is incompatible with that of the blue metal lines. Mason et al. (1998) list this star as having a constant radial velocity and thus we don't have a clear explanation for the radial velocity change. Therefore, the radial velocity correction for this star is particularly inaccurate. We have adopted the correction indicated by the blue metal lines, accounting for the different rest frames as a function of observation date.

The line fit of [FORMULA] is comparatively poor for this star (see Fig. 16). This reflects a problem that we have found for the first time, an inconsistency between the fit for [FORMULA] and [FORMULA]. The fit for [FORMULA] would need a mass-loss rate which is only half that of [FORMULA]. This inconsistency is only weakly dependent on the gravity.

[FIGURE] Fig. 16. The line fit to HD 14 947 using spherical models with mass-loss. We see the comparatively poor fit for [FORMULA] in the core, which can be largely improved by reducing the mass-loss rate by a factor of two.

The new temperature derived for HD 14 947 is much lower than that obtained from plane-parallel models. This is due to sphericity together with the new criterion of using He II [FORMULA] 4200. Again, in agreement with the large mass-loss rate, He II [FORMULA]4541, 4686 are poorly fitted, (see discussion in the next section).

The spectroscopic mass derived is clearly lower than the evolutionary one, although they agree within the errors. The analysis by Puls et al. (1996) gave slightly different stellar parameters, the most important difference being the now lower temperature implying a lower luminosity by 0.23 dex.

There is an upper limit of [FORMULA] [FORMULA]/yr for the mass-loss rate of HD 14 947 derived from radio fluxes (Lamers & Leitherer 1993). The value we find here is in agreement with this upper limit.

5.4. HD 15 558, O5 III(f)

The line fit to HD 15 558 obtained for the parameters given in Table 3 can be seen in Fig. 17. The mass-loss rate is rather large, in agreement with the high luminosity, especially when compared with HD 15 629, a star that could be considered similar at first inspection since both [FORMULA] profiles are in absorption. However, the value we obtain here for [FORMULA] is lower than that obtained by Puls et al. (1996) due to the change in the stellar parameters. The gravity is largely increased with respect to the value derived with plane-parallel, hydrostatic models (by 0.15 dex), and in fact the derived spectroscopic mass is nearly equal to the evolutionary one, thus making HD 15 558 one of the few cases for which we do not find any mass discrepancy. The generally good agreement in the line fit to HD 15 558 is again broken by He II [FORMULA]4686.

[FIGURE] Fig. 17. As Fig. 14, but now showing the line fit to HD 15 558.

5.5. HD 15 629, O5 V((f))

The line fit of HD 15 629 is obtained for the parameters given in Table 3 and is shown in Fig. 18. We see that the mass-loss rate is relatively low and agrees very well with the value given by Puls et al. (1996). The gravity we obtain is similar to that in the plane-parallel case, and thus the new mass is again lower than the evolutionary one, although there is formal agreement considering the error bars. The worst fit corresponds to He II [FORMULA]4686. A comparison with Puls et al. (1996) shows that both sets of values are compatible, although the general trends (lower temperatures, radii and luminosities in our case) remain.

[FIGURE] Fig. 18. As Fig. 14, for HD 15 629.

5.6. HD 15 570, O4 If+

This is an extreme object, and the adopted parameters are actually a compromise, reflected in very large uncertainties. For this star we derive [FORMULA] = 42 000 K, [FORMULA] = 3.80, [FORMULA] = 0.15, log M _= -4.75 (see Table 3 for the rest of the parameters). The very high mass-loss rate results in an extreme insensitivity of the wings of [FORMULA] to gravity variations, as can be seen in Fig. 19, where we have plotted the profiles for [FORMULA] = 4.05 and [FORMULA] = 3.55. These can be considered as limits beyond which the fits begin to become poorer than for the range [3.55,4.05]. Thus we have adopted [FORMULA] = 3.80[FORMULA]0.25. However, it should be borne in mind that even such a large error is optimistic, as the uncertainty of the radial velocity correction has not been included. Also the He profiles are largely insensitive to [FORMULA] variations, but they allow us to restrict [FORMULA] and [FORMULA].

[FIGURE] Fig. 19. The line fit to [FORMULA] in HD 15 570 with [FORMULA] = 4.05 (solid line) and [FORMULA] = 3.55 (dashed line). All other parameters are as in Table 3. The abscissa gives wavelength in Å and the ordinate relative fluxes.

The line fit to this star is shown in Fig. 20. A detailed fit of the strong [FORMULA] emission of HD 15 570 was not possible, since for the given parameters the simulations always showed a double peak due to the extra emission coming from the He II blend (see the discussion in the following section) not seen in the observations. Thus the adopted criterion was to fit the red wing of [FORMULA]. Again, we found the same problem encountered for HD 14 947. The best fit of [FORMULA] gives a mass-loss rate different from the best fit of [FORMULA], without any possibility to fit both simultaneously, with a discrepancy of roughly a factor of two. [FORMULA] indicates a value of about log M _= -4.75 and [FORMULA] one around log M _= -5.0, but this low value of log M&_nbsp;would also imply that the gravity is lower than in the case of higher mass-loss rate. This is shown in Fig. 21, where we see the fit for [FORMULA] at log M _= -4.75 and -5.0 at [FORMULA] = 3.80. Comparing with Fig. 19 we see that the effect of M _= lowering the mass-loss rate is larger than any change of gravity at log -4.75. At the lower log M&_nbsp;of -5.0 we would obtain a lower and more constrained gravity, but it is impossible to obtain even a moderate fit to [FORMULA]. In any case, the mass-loss rate is very large, although we should point out again that our mass-loss rate coincides well with other [FORMULA] determinations (Puls et al. 1996; Lamers & Leitherer 1993) but it is larger than that derived from radio fluxes (Lamers & Leitherer 1993) by a factor of 4 if we adopt the value from the fit of [FORMULA], and by a factor of 2 if we take the one more consistent with [FORMULA].

[FIGURE] Fig. 20. The line fit for HD 15 570. The fit shown corresponds to the model fitting the red wing of [FORMULA] (see text).

[FIGURE] Fig. 21. The line fit to [FORMULA] in HD 15 570 with log M _= -4.75 (solid line) and log M _= -5.00 (dashed line) at [FORMULA] = 3.80. All other parameters are as in Table 3. Comparing with Fig. 19 we see that for this strong wind, changing the mass-loss rate has a larger effect than changing the gravity. The abscissa gives wavelength in Å and the ordinate relative fluxes.

The temperature is now much lower than in the plane-parallel case, as for HD 14 947, again due to the combined effects of sphericity and the new criterion for the He ionization equilibrium. In agreement with this, the fit to He II [FORMULA]4541, 4686 is poor. The formal change in [FORMULA] is large with respect to the plane-parallel values (0.20 dex) and now HD 15 570 has also the largest spectroscopic mass, even formally exceeding 100 [FORMULA]. Little can be said about the actual mass and the mass discrepancy. In Table 3 we see that the spectroscopic mass is 30[FORMULA] larger than the evolutionary one, but we did not attemp to bring both into agreement (which could be possible adopting [FORMULA] = 3.65), since for HD 15 570 gravity and mass are only formal values in the centre of a large uncertainty area, and have to be regarded as rather inaccurate. It is clear however that HD 15 570 is one of the most extreme O stars in the Milky Way, and probably one of the most massive stars.

5.7. Cyg OB2 [FORMULA]7, O3 If*

The line fit for this star, displayed in Fig. 22, shows similar problems to those of HD 14 947 and HD 15 570 discussed before. The mass-loss rate is quite large, again exceeding 10-5 [FORMULA] /yr, and the parameters point to a very massive star. The large mass-loss rate results in a change of [FORMULA] with respect to the plane-parallel hydrostatic models, which is relatively large (0.15). However, the mass discrepancy is still present, since the spectroscopic mass is still a factor of 2 lower than the derived evolutionary mass. Another important change with respect to plane-parallel models is the new helium abundance, now [FORMULA]= 0.18. It is interesting to note that Herrero et al. (1999) could not find evidence of any helium discrepancy in their analyses of other, less extreme Cyg OB2 stars.

[FIGURE] Fig. 22. The line fit for Cyg OB2 [FORMULA] using a spherical model with the parameters given in Table 3.

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Online publication: January 31, 2000