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

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6. Additional considerations

6.1. The [FORMULA] problem

As we have seen in the foregoing analysis, for three of our objects with dense winds the wind emission seen in [FORMULA] is inconsistent with the mass-loss rate derived from [FORMULA]. It may be questioned why this obvious problem has not already been discovered by Puls et al. (1996) who found no such discrepancy and derived- for the object common to both investigations (HD 14 947 3) -a lower value for [FORMULA] than follows from our analysis.

This difference has the following origin. For the problematic objects, the Doppler core of [FORMULA] becomes optically thick in the wind. Thus, it depends only on the wind conditions (mainly in the transonic region) and is independent of any underlying photospheric radiation. Consequently, it can be correctly fitted by the approach described in Puls et al. and would compare to our results.

On the other hand, the wings of [FORMULA] depend on the illuminating radiation from below the transonic region. In the concept applied by Puls et al., this radiation was taken from photospheric profiles calculated on the basis of hydrostatic models, and the wind contamination was correctly accounted for. The fitted [FORMULA] value was finally corrected for the difference between hydrodynamic and hydrostatic atmospheres, accounting for the different formation depths in an approximate and global way, where the correction turned out to be only moderate in the parameter space considered.

This procedure, however, is only justified if the major part of the wings is actually formed in some pseudo-hydrostatic environment, which is the case if the wind densities are not too high. For increasing wind density, however, the difference between hydrostatic and hydrodynamic stratification (e.g., Puls et al., Fig. 16) becomes increasingly larger, and the point where the transition between both regimes occurs shifts to correspondingly larger (mean) optical depths. Thus, for higher wind densities only the far wings are formed in a purely hydrostatic environment, whereas the inner wings are severely affected by the wind conditions, which display a lower density at given optical depth. Consequently, the use of hydrostatic profiles and the global correction applied by Puls et al. will inevitably fail under those conditions. Moreover, the derived values for [FORMULA] will be too small compared to "reality", since, for a given [FORMULA], the hydrostatic densities are always larger than those in the wind regime. As a result, the deviations between the profiles calculated by Puls et al. and ours should become largest just outside the Doppler core, and vanish at the extreme wings. Thus, it is not too surprising that the results of our consistent description deviate from the results given by Puls et al. with respect to gravity (derived from the wings), whereas the [FORMULA] Doppler cores would be consistent with the mass-loss rate derived from [FORMULA], however are unfortunately not visible due to rotational broadening.

We like to point out that for a (small) number of objects with denser winds the determination of the stellar mass becomes completely impossible, since in those cases even the continuum is formed in the wind and the reaction of any profile on pressure scale height becomes impossible.

In order to check under which conditions this problem will arise, we have calculated the (minimum) continuum optical depth given by electron scattering as function of wind parameters:


with [FORMULA] the electron scattering mass absorption coefficient, Y the helium abundance (by number) with respect to hydrogen and [FORMULA] the number of free electrons provided per He atom. By means of the equation of continuity and for the [FORMULA] velocity law which is used also in our model simulations,


(where [FORMULA] is the velocity at the transition point between pseudo-hydrostatic and wind region, cf Santolaya-Rey et al., 1997, their Sect. 2.2), we have


where [FORMULA]. Inserting typical parameters and denoting by A the optical depth-like quantity


the electron scattering optical depth is finally given by


(approximating b by unity). In Fig. 23 we have plotted this quantity, evaluated at [FORMULA] (which compares roughly with the velocity at the transition point for hot stars) as a function of A and [FORMULA]. The dotted line gives the optical depth of 2/3, and for all objects with [FORMULA] the continuum is definitely formed in the wind. We have indicated the position of our sample stars by asterisks, where the number refers to Table 1. Obviously, object no. 2 (= HD 15 570) lies just at the border line, and thus a gravity determination is almost impossible (cf. Fig. 19 for the influence of gravity on [FORMULA]). For the objects nos. 1 (Cyg OB2 [FORMULA]7), 5 (HD 14 947) and 6 (HD 210 839) at least the inner wings of [FORMULA] are severely affected by the wind, and thus result in a larger gravity than found by Puls et al. 1996 (if all other parameters remain the same). For the remaining three stars, the continuum is formed solely in the hydrostatic part, and the derived numbers should coincide with the approximate method, as is actually the case.

[FIGURE] Fig. 23. Electron scattering optical depth at transition velocity [FORMULA] as function of A (cf. Eq. (4)) and [FORMULA] = 1.3 (dashed-dotted), 1.0 (fully drawn) and 0.7 (dashed). The positions of our programme stars (according to the derived wind parameters) have been indicated by asterisks.

6.2. Influence of He II resonance lines

Those readers in particular who are not well familiarized with spectral analyses of hot stars may question why we did not take the most prominent line of He II in the optical part of the spectrum, namely He II  4686 ([FORMULA]), into consideration so far (although we have plotted it for all models and the discrepancy is obvious). This question is completely legitimate, since the upper level of this transition is just the lower one of our strategic lines He II [FORMULA] 4200, 4541, and should be reproduced with a similar degree of precision if our models were reliable.

However, it is well known that this line (if formed in the wind) is extremely difficult to fit, and, to our knowledge, has never been used in any kind of NLTE-analysis of luminous O-stars. Usually, if one compares the predicted profiles to observations, the synthetic line turns out to be too strong in emission, even if all other lines including [FORMULA] do perfectly fit.

This rather unsatisfactory behaviour, which is normally by-passed by simply excluding He II 4686 from the line list, relates to the extreme sensitivity of the participating levels on the treatment of the He II resonance lines and their sensitivity to line-blocking (for a discussion concerning this problem of the formation of He I -lines, cf. Santolaya-Rey et al. 1997; see also the related discussion concerning the ionization structure of WRs by Schmutz 1997). In standard simulations for wind conditions as described here, where line-blocking effects are excluded, the dominating background "opacity" below 303 Å (referring to He II [FORMULA]) is Thomson scattering , leading to extremely enhanced radiation temperatures at the resonance-line' frequencies. Compared, for example, to a detailed balance situation (see below), the ground-state becomes depopulated, which in consequence (and in connection with the increased escape-probabilities due to the velocity field) prohibits He III from recombining and gives rise to much weaker absorption edges at 229 Å, compared to plane-parallel simulations (cf. Gabler et al. 1989, especially Fig. A2).

Moreover, since the radiation temperature is increasing towards higher frequencies (due to the decreasing bf-opacity at lowest photospheric levels), the NLTE departure coefficients are larger for higher levels than for lower ones. Thus, in addition to the wind emission by geometrical effects, the lines between excited levels (predominately He II 4686) are contaminated by a strong source function [FORMULA], which leads to a much stronger total emission than would be the case if the resonance lines were of less importance. Actually, a pilot investigation by Sellmaier (1996) for the case of [FORMULA] Pup has shown that the emission of He II 4686 could be significantly reduced if line-blocking was accounted for correctly.

In order to investigate in how far the above effects are of influence for our analysis (especially for the strengths of He II 4200 and 4541, respectively), we have run a number of simulations with different treatments of the He II resonance lines, for the example of our final model for HD 14947, where the majority of lines is formed in the wind. To check our hypothesis that the dominating effect leading to erroneous results follows from the increased pumping by resonance lines, two principally different approaches were considered, which should give similar results if the hypothesis were correct.

On the one hand, we set all He II resonance lines into detailed balance. Alternatively, we simulated an additional [FORMULA]-square dependent background opacity in the decisive frequency range 227 Å [FORMULA] 400 Å, defined by


with different values of [FORMULA] between [FORMULA] and [FORMULA]. The frequential dependence [FORMULA] was assumed to be either increasing or decreasing,


and the appropriate emission component has been set to Planck. The value of [FORMULA] corresponds to the inverse of the electron density at that point where the additional opacity reaches the same value as the electron scattering opacity. From the numbers given above, it is obvious that our choice is rather low compared to what might be expected in reality. For our final discussion, we have selected four models with parameters given in Table 5.


Table 5. Different approximations for treatment of He II resonance lines: The value of [FORMULA] corresponds to the definition in Eq. (6), and the line styles are the same as in Figs. 24 and 26.

Fig. 24 verifies the expected behaviour for the He II ground-state. Both for the models with detailed balance as well as with simulated background opacities, He III begins to recombine in the outer atmosphere as long as the strong upward rates present in model 1 (dotted) are no longer active, so that a significantly enhanced ionization edge develops. Only for the model with the lowest value of [FORMULA] (not displayed), the influence of the background opacity becomes so weak that the model remains ionized throughout the wind.

[FIGURE] Fig. 24. Radiation temperatures of emergent fluxes for the model of HD 14947, with different treatment of He II resonance lines. Line styles as defined in Table 5.

Besides the reaction of the ground-state, also the expected behaviour of the excited states (reversal of population) takes place as long as the radiation temperature does not significantly increase towards higher frequencies (model 3) or the resonance lines are not active (model 2). A comparison between the corresponding He II 4686 profiles and the observations (Fig. 25) shows that our simulations are in almost perfect agreement, whereas our standard model displays much too much emission. It is interesting to note here a remark given by our anonymous referee. Henrichs (1991) reports that the equivalent width of He II [FORMULA]4686 in [FORMULA]Cep varies in concert with the high-velocity edge of the C IV [FORMULA]1550 line, suggesting that both variations share a common origin that could rely on the behaviour of the background opacity or the resonance lines of He II , as studied here.

[FIGURE] Fig. 25. Comparison of the He II 4686 profile observed in HD 14 947 with those obtained in our simulation of line-blocking (full drawn), of detailed balance for He II resonance transitions (dashed) and in our standard model (dotted).

A final check on how far our models compare to reality allows the comparison with the UV-line He II 1640 ([FORMULA]), since the lower state of this transition behaves differently from the other excited levels. In those cases where the resonance lines are no longer active (detailed balance) or are of negligible importance (background opacities) and the He II Lyman edge becomes optically thick throughout the atmosphere, this state becomes the effective ground state of the ion 4. Thus, its population is predominantly controlled by the photoionization balance at 911 Å coinciding with the hydrogen Lyman edge. Since this edge is optically thin, the 2nd level becomes strongly overpopulated because of the diluted radiation field, i.e., the absorption should be larger than for the standard model. A comparison with the observed IUE profile 5 shows that our models are on the right track, in contrast to the standard model which predicts too little absorption. Only at higher velocities, i.e., in the outermost wind, are they too strong compared to observations, whereas for the inner wind, which is the decisive part concerning our analysis, they are in perfect agreement. A comparison with model 4 shows that some fine-tuning might improve even the situation at larger velocities. (We note that in order to fit the position of the emission peak, we had to apply an artificial velocity dispersion of roughly 100 km/s, consistent with the values found from the analysis of UV resonance lines.)

[FIGURE] Fig. 26. IUE spectrum of HD 149475 , He II [FORMULA]1640, compared to profiles generated from models with different treatment of He II resonance lines. Observations shifted by 90 km/s to the blue, velocity dispersion [FORMULA] = 100 km/s applied to account for the red-shifted emission peak (see text). Line styles as defined in Table 5.

In conclusion, we found that suppressing the large upward rates from resonance lines present in our standard model gives rise to a different population of ground and excited levels. The exact mechanism for this suppression, however, seems to be irrelevant to the results, and the synthesized profiles of previously problematic lines compare well with the observations.

We are now able to check the consequences of the manipulations outlined with respect to the strategic lines analysed so far and to derive constraints on which lines are more robust concerning our present ignorance of the real situation.

Fig. 27 shows the profiles of He II [FORMULA] 4541, 4200 produced by the different simulations and the "standard" model. We see that the wings of the He II lines become stronger in the simulations as a consequence of the now reduced departure coefficients of the upper levels. However, the effect is smaller in the He II [FORMULA]4200 line (transition 4 [FORMULA] 11) than in the He II [FORMULA]4541 (transition 4 [FORMULA] 9) because transitions involving higher levels are weaker and form closer to the photosphere, so that the increased source function is not so visible. Since He II 4200 shows up to be more stable, this is the preferred line in case of any discrepancy. This is a change of criterion with respect to former analyses in our group, but we prefer always to follow a single criterion that allows us to understand physically changes in the derived parameters. In addition, we should mention that also [FORMULA] becomes weaker, especially in the blue wing, as a consequence of these effects in the overlapping He II line. That is the reason why we prefer to fit the red wing in case of difficulties like those in [FORMULA] Cep or HD 15 570 (and in concert with the findings by Puls et al. (1996), who had also to manipulate the He II departures predicted by unified models if the wind was strong). Finally, we point out that He I is not so strongly affected if these lines are formed purely in the photosphere, but, as has been shown by Santolaya-Rey et al. (1997) they are also influenced by any effects that modify the population of the He II ground level (see Sect. 3.4.1 in Santolaya-Rey et al., where this problem has already been discussed).

[FIGURE] Fig. 27. Line profiles of He II 4541 (left) and He II 4200 (right) for the simulation of line-blocking (fully drawn), of detailed-balance (dashed), and for the "standard" model (dotted). Calculations were made for the final model of HD 14 947.

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Online publication: January 31, 2000