Online material

Appendix A: Detailed comparison of present results with other investigations

The present work comprises a detailed analysis of a small sample of hot stars, based on the combination of optical, NIR and UV spectra with one of the most sophisticated NLTE atmosphere codes presently available, cmfgen. Thus, a comparison of the derived results with those from more restricted investigations (with respect to wavelength range) based on alternative atmosphere codes provides an opportunity to address typical uncertainties inherent to the spectroscopic analysis of such objects, caused by different data-sets and tools. As outlined in Sect. 2 (cf. Table 1), most previous investigations of our targets have been analyzed by means of fastwind¹⁴, or by (quasi-) analytic methods designed for specific diagnostics such as H_α or the IR-/mm-/radio continuum (for an overview of these methods, see Kudritzki & Puls 2000; Puls et al. 2008, and references therein).

Brief comments on important differences between our and those results have been already given in Sects. 4 and 5, and the complete set of the various stellar and wind-parameters is presented in Table A.2. In the following, the various investigations are referred to following the enumeration provided at the end of this table (ref#). Note that a direct comparison of mass-loss rates is still not possible, due to the uncertainties in distances (and thus radii) for Galactic objects. Moreover, previous analyses were based on either unclumped models (ref# 2–7) or that the derived clumping factors had to be normalized to the clumping conditions in the outermost wind, which are still unclear (ref# 8). Thus, a meaningful comparison is possible only for the optical depth invariants¹⁵(A.1)which describe the actual measurement quantities related to (i) ρ²-dependent processes (Q), if f_v is the average clumping factor in the corresponding formation region and the clumps are optically thin, and (ii) ρ-dependent processes (Q_res), under the assumption that clumping plays only a minor role (but see Sect. 1).

Figure A.1 provides an impression of the differences in the most important parameters for the individual stars, by comparing the effective temperatures and gravities (upper 3  ×  3 panels), and the optical depth invariants and luminosities (lower 3  ×  3 panels). Note that all panels provide identical scales, to enable an easy visualization. From the figure, it is quite clear that typical differences in T_eff are of the order of 1000 to 2000 K, with corresponding differences of 0.1 to 0.2 dex in log g. The largest differences are found for the effective temperature of Cyg OB2 #8C (roughly 4000 K when comparing with ref# 3 and 6/8), which has been already discussed in Sect. 4, and most probably relates to an underestimation of vsini in these investigations. It is reassuring that in most cases the connecting lines between our (crosses) and the other results in the T_eff-log g plane have a positive slope, indicating that higher temperatures go in line with higher gravities and vice versa, which is consistent with the behavior of the gravity indicators (usually, the wings of the Balmer lines).

Fig. A.1 Comparison of the results obtained in this work (crosses) with results from other investigations (for data and reference identifiers, see Table A.2). Upper 3  ×  3 panels: log g vs. Teff; all panels have the same scale, corresponding to an extent of 5000 K in Teff and 0.4 dex in log g. Lower 3  ×  3 panels: log Q vs. log (L/L⊙), with axes extending over 0.4 dex in log (L/L⊙) and 3.5 dex in log Q. In order to facilitate the comparison with ρ2-diagnostics, all Q values have been normalized to fv = 1 (see Eq. (A.1)). The asterisks provide the Qres-values which have to be compared with the corresponding values from ref# 1 (Fullerton et al. 2006, ρ-diagnostics, but including the product with the ionization fraction of Pv). Note that all Qres values have been scaled by a factor of 10 to fit into the individual figures.

Open with DEXTER

The average differences with respect to effective temperature,  ⟨ΔT_eff⟩, are presented in Table A.1, discarding ref# 1 and 8 who adopted the stellar parameters, mostly from ref# 2–7. Major discrepancies seem to be present when comparing with ref# 3 and 6, who derived temperatures being on average 1700 K higher than our results. Note, however, that a large part of this discrepancy is caused by the results obtained for Cyg OB2 #8C (see above). The other three investigations (ref# 4, 5 and 7) deviate, at least on average, much less from the present one, by a few hundreds of Kelvin. The dispersion of the differences, however, is very similar in all cases, about 1700 K, except for ref# 5, with a dispersion of 600 K. Thus, overall, the dispersion of  ⟨ΔT_eff⟩  is somewhat larger than to be expected from the typically quoted individual uncertainties of 1000 K, which should give rise to a dispersion of 1400 K.

Regarding the optical depth invariants, the situation is satisfactory. Except for ref# 3, the mean differences are at or below 0.1 dex (25%), with a dispersion of typically 0.3 dex (factor of 2), which is consistent with the typical individual errors (see Markova et al. 2004, for a detailed analysis). This result is particularly obvious from Fig. A.1 (lower panels), where in most cases all investigations show rather similar Q values.

Regarding the Q_res values which are relevant when comparing with ref# 1 (the Pv investigation by Fullerton et al. 2006), the discrepancy is still large, particularly when accounting for the fact that our results indicate considerable clumping, thus reducing the absolute mass-loss rate significantly with respect to previous investigations. On average, we find  ⟨ Δlog Q_res ⟩  ≈  −1.4 dex, with a dispersion of 0.8 dex, Thus, either (i) the actual mass-loss rates are even smaller than derived here, or (ii) the ionization fraction of Pv (remember that the results from Fullerton et al. include the product with this quantity) is very low, of the order of 4% (which would require extreme conditions in the wind, e.g., a very strong X-ray/EUV radiation field), or (iii) the line formation calculations of UV-resonance lines (both in the investigation by Fullerton et al. and in our analysis) require some additional considerations, such as the presence of optically thick clumps and/or the inclusion of a porosity in velocity space, see Sect. 1¹⁶.

In summary, we conclude that at least the analysis of Q seems to be well-constrained, and that different investigations give rather similar results. The remaining problem is the determination of actual mass-loss rates, which involves the “measurement” of (absolute) values of clumping factors. As we have shown in this investigation, L-band spectroscopy turns out to be a promising tool for this objective. Let us note that only a measurement of actual mass-loss rates will enable a strict comparison with theoretical predictions (as performed in Sect. 6), to identify present shortcomings and to provide “hard numbers” for evolutionary calculations.

The precision of effective temperatures, on the other hand, is less satisfactory. Irrespective of the fact that we did not find a real trend in the average differences with respect to three from five investigations, the dispersion is quite large, and individual discrepancies amount to intolerable values. Because of our detailed analysis covering a large range of wavelength domains and using a state-of-the-art model atmosphere code based on an “exact” treatment of all processes, we are quite confident that the T_eff-errors in our work are of the order of 1000 K or less, which means that the corresponding errors in the previous investigations must be of the order of 1400 K or more. Additionally, two from five investigations gave a rather large average difference with respect to our results, which is alarming since all five investigations have been performed with the same NLTE atmosphere code. Insofar, recent attempts to provide reliable spectral-type-T_eff-calibrations have to be augmented by results from large samples to decrease the individual scatter in a statistical way.

Appendix B: Occupation numbers of the hydrogen n = 4 and n = 5 level in the outer atmospheres of late O-type stars with thin winds

Conditions in the outer photosphere.

As we have seen from Fig. 14, almost all of our simulations (and many more which have not been displayed) resulted in a stronger depopulation of level 4 compared to level 5 in the outer atmosphere, independent of the various processes considered. One might question how far this result can be explained (coincidence or not?). To obtain an impression on the relevant physics, we write the rate equations for level i > 1 in the following, condensed form¹⁷, again neglecting collisions, and assuming that ionization is only possible to the ground-state of the next higher ion (as it is the case for hydrogen): (B.1)where A_ij are the Einstein-coefficients for spontaneous decay, R_ik and R_ki the rate-coefficients for ionization/recombination, and Z_ij the net radiative brackets for the considered line transition, (the fraction denotes the ratio of mean line intensity and line source function). = n_kn_eΦ_ik(T_e) denotes the LTE population of level i, accounting for the actual electron and ion density, n_e and n_k (for further details see, e.g., Mihalas 1978), such that the departure coefficients are given by . Note that for purely spontaneous decays Z_ij = 1, for lines which are in detailed balance, Z_ij = 0, and for levels which are strongly pumped (e.g., by resonance lines with a significantly overpopulated lower level), Z_ij < 0. Solving for the departure coefficients, Eq. (B.1) results in (B.2)The sum in the nominator corresponds to the net-contribution of lines from “above” (i.e., with upper levels j > i), normalized to the LTE population of the considered level, whereas the sum in the denominator is the net-contribution of lines to lower levels (j < i). The complete fraction can be interpreted as the ratio of populating and depopulating rates, which can be split into the contributions from bound-bound and bound-free processes, (B.3)For all our simulations 1-6 we have now calculated those two terms which determine b₄ and b₅. At first, let us concentrate on the outer photosphere, as on the right of Fig. 14. In almost all cases (except for simulation 5), the 2nd term dominates the departure coefficient, and, moreover, the first term (the ratio!, not the individual components) remains rather similar, of order 0.2. Consequently, the stronger depopulation of level 4 compared to level 5 is due to the fact that the quantity (B.4)is usually larger for level 5 than for level 4, even though R_k5 < R_k4: the accumulated transition probability from level 4 to lower levels (A₄₁Z₄₁ + A₄₂Z₄₂ + A₄₃Z₄₃) is much larger than the corresponding quantity from level 5 to lower levels (A₅₁Z₅₁ + A₅₂Z₅₂ + ...). This behavior, finally, can be traced down to the run of the oscillator-strengths in hydrogen: On the one side, e.g., A₄₁ is larger than A₅₁, etc., whereas, on the other, the corresponding net radiative brackets (Z_4j vs. Z_5j etc.) do not differ too much.

Two examples shall illustrate our findings. For the complete solution, the first term in Eq. (B.3) is roughly 0.22, whereas the 2nd term amounts to 0.62 for level 5 and to 0.53 for level 4. Thus, b₄ ≈ 0.75 and b₅ ≈ 0.84. For simulation 2, with Z_ij = Z_ji = 1 and R_ik = 0 for i > 1, the first term  ≈ 0.18, and the 2nd one is 0.41 and 0.3, respectively, such that b₄ ≈ 0.48 and b₅ ≈ 0.6 (cf. Fig. 14).

In conclusion, the stronger depopulation of level 4 compared to level 5 in the outer photospheres of hot stars can indeed be regarded as the consequence of a typical nebula-like situation, namely as due to the competition between recombination and downwards transitions. Different approximations regarding the contributing lines do control the absolute size of the departures, but not the general trend.

Conditions in the wind.

Though the formation of the emission peak of Br_α for objects with thin winds is controlled by the processes in the upper part of the photosphere, it is also important to understand the conditions in the wind, since, as we have seen in Fig. 13, the onset of the wind prohibits a further growth of the corresponding source function: Immediately after the transition point between photosphere and wind, the source function drops to values corresponding to the local Planck-function (i.e., the departure coefficients of n₄ and n₅ become similar). Only in the outer wind the source function increases again (in contrast to the predictions of the pure H/He model, see below), which remains invisible in the profile, due to very low line optical depths. If this abrupt decrease would not happen, the monotonic behavior of the strength of the emission peak (Fig. 12) would no longer be warranted for mass-loss rates at the upper end of the scale considered here, and an important aspect of its diagnostic potential would be lost.

Let us first concentrate on the conditions in the outer wind, where the ground-state has a major impact. We stress again that we are dealing here with (very) weak winds, i.e., the continuum-edges are formed deep in the photosphere, whereas the wind and the transition region are already optically thin. Otherwise, we could no longer assume a “given” radiation temperature in the continuum, but would have to account for a simultaneous solution of radiation field and occupation numbers, as it was done, e.g., to explain the ground-state depopulation of Heii in dense hot star winds by Gabler et al. (1989).

Within our assumptions of ionizations to the ground-state of the next higher ion only and neglecting collisional processes, we obtain an alternative formulation of the rate equation for the ground-state, by summing up the rate equations for all levels i (Eq. (B.1)), (B.5)since the line contributions cancel out. Solving for the ground-state departure coefficient, we find (B.6)which can be approximated by the well known expression (B.7)where T_rad is the radiation temperature in the ground-state (=Lyman) continuum, ν ≥ ν₀, and W the dilution factor. A correction factor of order unity accounts for the ionization/recombination from the excited levels (for details, see, e.g., Puls et al. 2005).

From here on, we have to divide between line-blocked and unblocked (e.g., pure H/He-) models, as they behave different in the wind (though similar in the outer photosphere), due to a considerably different run of electron temperature and radiation-field on both sides of the Lyman edge.

For non-blocked models (see Fig. 14), the wind-temperature is not too different from T_rad, and b₁ becomes strongly overpopulated  ∝ 1/W = r². Moreover, all net radiative brackets coupled to the ground-state, (B.8)become strongly negative, since (i) b₁ is severely overpopulated and (ii) the Doppler effect in the wind allows for an illumination by the continuum bluewards from the resonance-line rest-frame frequencies ν_j1, i.e., (optically thin (Sobolev-)approximation), with T_rad,j1 ≫ T_rad due to missing line-blocking.

The consequence for the population of the excited levels (Eq. (B.2)) is twofold. Because of the strong pumping by the resonance lines, the (normalized) population of the higher levels () is much larger than in the photosphere, and the

line term becomes larger than the recombination term. Second, the denominator decreases significantly, due to the direct effect of Z_j1 and since the the ionization rates  ∝ W become negligible.

In total, now the first term dominates in Eq. (B.3), and the situation is just opposite to the conditions in the outer photosphere: The lower the considered level i, the larger is the nominator and the smaller the denominator, such that we obtain the sequence b₂ > b₃ > b₄... > 1 (cf. Fig. 14).

For line-blocked models, on the other hand, the cooling by the enormous number of metallic lines leads to a strong decrease of the electron temperature in the outer wind, and T_e becomes much smaller than the radiation temperature in the Lyman continuum (for our late O-type model, 10 000 K vs. 25 000 K). In this case, ionization, though diluted, outweighs recombination (the exponential term in Eq. (B.7)), and the ground-state even becomes underpopulated (b₁  →  0.5). Consequently, the resonance lines can no longer pump the excited levels (even more, since for blocked models the radiation temperatures close to the resonance lines, T_rad,j1, are much smaller than in the unblocked case). Thus, we find a situation similar to that in the outer photosphere, namely that the 2nd term in Eq. (B.3) is the decisive one, and n₅ > n₄, which is obvious also from the final increase of the line-source function for Br_α in Fig. 12 for all mass-loss rates considered.

Finally, in the region between the outer photosphere and the outer wind, the dilution of the radiation field is faster or similar to the decrease of T_e, both for the blocked and the unblocked models. Thus, the departure coefficients of level 4 and 5 increase in this region (due to an overpopulated ground-state, and effective pumping due to the onset of the Doppler-shift), though at a rather similar rate, with b₄ ≳ b₅. Consequently, the source-function approaches the LTE level, which explains its abrupt decrease in the transition region.

© ESO, 2011