Issue 
A&A
Volume 535, November 2011



Article Number  A32  
Number of page(s)  25  
Section  Interstellar and circumstellar matter  
DOI  https://doi.org/10.1051/00046361/201016003  
Published online  27 October 2011 
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 datasets and tools. As outlined in Sect. 2 (cf. Table 1), most previous investigations of our targets have been analyzed by means of fastwind^{14}, 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 windparameters 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 massloss 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 invariants^{15}(A.1)which describe the actual measurement quantities related to (i) ρ^{2}dependent processes (Q), if f_{v} is the average clumping factor in the corresponding formation region and the clumps are optically thin, and (ii) ρdependent 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. T_{eff}; all panels have the same scale, corresponding to an extent of 5000 K in T_{eff} 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 ρ^{2}diagnostics, all Q values have been normalized to f_{v} = 1 (see Eq. (A.1)). The asterisks provide the Q_{res}values which have to be compared with the corresponding values from ref# 1 (Fullerton et al. 2006, ρdiagnostics, but including the product with the ionization fraction of Pv). Note that all Q_{res} values have been scaled by a factor of 10 to fit into the individual figures. 

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Comparison of stellar and wind parameters as derived in the present analysis with results from previous investigations.
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 massloss 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 massloss 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 Xray/EUV radiation field), or (iii) the line formation calculations of UVresonance 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. 1^{16}.
In summary, we conclude that at least the analysis of Q seems to be wellconstrained, and that different investigations give rather similar results. The remaining problem is the determination of actual massloss rates, which involves the “measurement” of (absolute) values of clumping factors. As we have shown in this investigation, Lband spectroscopy turns out to be a promising tool for this objective. Let us note that only a measurement of actual massloss 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 stateoftheart 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 spectraltypeT_{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 Otype 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 form^{17}, again neglecting collisions, and assuming that ionization is only possible to the groundstate of the next higher ion (as it is the case for hydrogen): (B.1)where A_{ij} are the Einsteincoefficients for spontaneous decay, R_{ik} and R_{ki} the ratecoefficients 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_{k}n_{e}Φ_{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 netcontribution 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 netcontribution 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 boundbound and boundfree processes, (B.3)For all our simulations 16 we have now calculated those two terms which determine b_{4} and b_{5}. 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 (A_{41}Z_{41} + A_{42}Z_{42} + A_{43}Z_{43}) is much larger than the corresponding quantity from level 5 to lower levels (A_{51}Z_{51} + A_{52}Z_{52} + ...). This behavior, finally, can be traced down to the run of the oscillatorstrengths in hydrogen: On the one side, e.g., A_{41} is larger than A_{51}, 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, b_{4} ≈ 0.75 and b_{5} ≈ 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 b_{4} ≈ 0.48 and b_{5} ≈ 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 nebulalike 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 Planckfunction (i.e., the departure coefficients of n_{4} and n_{5} 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 massloss 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 groundstate has a major impact. We stress again that we are dealing here with (very) weak winds, i.e., the continuumedges 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 groundstate depopulation of Heii in dense hot star winds by Gabler et al. (1989).
Within our assumptions of ionizations to the groundstate of the next higher ion only and neglecting collisional processes, we obtain an alternative formulation of the rate equation for the groundstate, by summing up the rate equations for all levels i (Eq. (B.1)), (B.5)since the line contributions cancel out. Solving for the groundstate 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 groundstate (=Lyman) continuum, ν ≥ ν_{0}, 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 lineblocked 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 radiationfield on both sides of the Lyman edge.
For nonblocked models (see Fig. 14), the windtemperature is not too different from T_{rad}, and b_{1} becomes strongly overpopulated ∝ 1/W = r^{2}. Moreover, all net radiative brackets coupled to the groundstate, (B.8)become strongly negative, since (i) b_{1} is severely overpopulated and (ii) the Doppler effect in the wind allows for an illumination by the continuum bluewards from the resonanceline restframe frequencies ν_{j1}, i.e., (optically thin (Sobolev)approximation), with T_{rad,j1} ≫ T_{rad} due to missing lineblocking.
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 b_{2} > b_{3} > b_{4}... > 1 (cf. Fig. 14).
For lineblocked 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 Otype 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 groundstate even becomes underpopulated (b_{1} → 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 n_{5} > n_{4}, which is obvious also from the final increase of the linesource function for Br_{α} in Fig. 12 for all massloss 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 groundstate, and effective pumping due to the onset of the Dopplershift), though at a rather similar rate, with b_{4} ≳ b_{5}. Consequently, the sourcefunction approaches the LTE level, which explains its abrupt decrease in the transition region.
© ESO, 2011
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