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Astron. Astrophys. 359, 1085-1106 (2000)

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2. Model calculations

2.1. Model atmospheres and programs

The calculations are performed using the standard assumptions of plane-parallel, homogeneous and stationary stellar atmospheres in hydrostatic and radiative equilibrium. Oxygen, although the most abundant metal, is supposed to be a trace element. The photoionization edge of the OI ground state coincides with the Lyman edge and the resonance lines fall into the same wavelength range as the Lyman series, thus contributing only a small fraction to the dominating hydrogen opacity; changes in the oxygen populations therefore have no appreciable effects on the atmospheric structure. Thus we obtain statistical equilibrium populations for OI while keeping the run of the atmospheric parameters fixed.

The non-LTE OI line profiles are computed on the basis of ATLAS9 (Kurucz 1979, 1991) LTE line-blanketed model atmospheres using LTE Opacity Distribution Functions (Kurucz 1992) to account for line blocking in the spectrum synthesis. For the calculation of model atmospheres for the most extreme supergiants close to the Eddington limit, modifications in the treatment of opacities and the radiative pressure in the outermost depth points within ATLAS9 have been made to achieve convergence. Comparisons at stellar parameters for which standard ATLAS9 converges have shown that the stratification deeper in the atmosphere is only marginally affected by these modifications.

With the assumptions cited above reliable analyses can be performed in the given temperature range from main sequence stars through to bright giants as indicated by Kudritzki (1988). Late A-type stars are an exception since an outer convection zone may be of importance as the observation of chromospheres in these stars proves, see Simon & Landsman (1997) for the latest discoveries. Convection is therefore taken into account in the standard way for ATLAS9 atmospheric models with [FORMULA] applying a value [FORMULA] of mixing length to scale height.

A thorough discussion of the possible deviations from the standard assumptions on the model atmosphere structure of supergiants is given by Venn (1995b). Additionally, non-LTE effects are often less significant on the model structure than line blanketing as demonstrated by Przybilla (1997).

The line formation calculations are performed using the programs DETAIL and SURFACE (Giddings 1981, Butler & Giddings 1985), with the former solving the radiative transfer and the statistical equilibrium equations and the latter computing the emergent flux. Recent improvement by inclusion of an ALI scheme (using the treatment of Rybicki & Hummer 1991) allows the utilisation of quite elaborate model atoms while the necessary computational resources remain at a low level. Optionally, effects of wind outflow velocity fields can be studied by adopting the ALI operator in the comoving frame as described by Puls (1991). Unified model atmospheres accounting for spherical extension and stellar wind but lacking line blanketing (Santolaya-Rey et al. 1997, SPH) replace the ATLAS9 atmospheres in the latter case.

2.2. The model atom

2.2.1. Energy levels

Neutral oxygen lines are most prominent in late A-/early F-type stars where the population of the excited energy levels reaches its maximum. At higher temperatures within our parameter range oxygen becomes rapidly ionized but only a negligible population is expected for excited OII energy levels connected to the ground state via radiatively permitted transitions as high excitation energies ([FORMULA]) are involved. It is therefore sufficient to take only the ground state of OII into account.

Since the OI lines in the visible originate from excited levels ([FORMULA] above the ground state), the OI model atom has to be fairly complete with respect to these levels. All energy levels below an excitation energy of 13.4 eV as listed by Moore (1976) are included explicitly in the model atom, together with additional P terms for principal quantum number [FORMULA], 8 and F terms for [FORMULA], adopted from OP computations (Butler & Zeippen, 2000). Sub-levels belonging to the same term are combined into a single level.

Additionally, level populations of OI up to [FORMULA] are computed in LTE relative to the ground state of OII with energies derived from their quantum defects. They are considered only in the number conservation equation.

2.2.2. Radiative transitions

All optically allowed bound-bound transitions between energy levels with non-LTE populations are taken into consideration. The required LS-coupling oscillator strengths are adopted from OP data (Butler & Zeippen, 1991, 2000). The reduction of individual lines of a multiplet into a single effective line introduces only small errors in the transition rates and occupation numbers as Baschek et al. (1977) have shown. In order to improve the computational efficiency we therefore ignore fine-structure splitting.

Grotrian diagrams of the triplet and quintet spin systems are shown in Fig. 1 and Fig. 2. Both spin systems together with the singlets ([FORMULA], [FORMULA], [FORMULA]) are treated simultaneously, the latter couple to the triplets via intercombination and forbidden transitions. The only direct coupling between the triplets and quintets is provided by the intersystem line [FORMULA] - [FORMULA]; oscillator strengths for the latter transitions are taken from the compilation of Wiese et al. (1996). A detailed comparison of the adopted oscillator strengths with measurements and theoretical work by other authors is performed by Butler & Zeippen (1991); the majority of the data is expected to be accurate to within 10% outdating most of the older data used in previous studies of non-LTE effects on OI .

[FIGURE] Fig. 1. Grotrian diagram for the OI triplet system.

[FIGURE] Fig. 2. Grotrian diagram for the OI quintet system.

Photoionizations from all energy levels with non-LTE populations are treated with cross sections fitted to the OP data (Butler & Zeippen, 1990, 2000). A carefully chosen frequency grid ensures a thorough representation of the numerous resonances present in the results of R-matrix calculations. Butler & Zeippen (1990) discuss the reliability of the cross sections for the first three states of OI in the context of the available experimental data and theoretical results from the literature; excellent agreement was found. The expected accuracy of the bulk of the OP photoionization cross sections amounts to approx. [FORMULA]10%.

A comparison of photoionization cross sections for the ground state and the quintet metastable state is presented in Fig. 3. Discrepancies up to an order of magnitude at threshold are present between the OP calculations and the data of Hofsäß (1970) used in the OI study of Baschek et al. (1977). Takeda (1992) uses cross sections provided by Henry (1970) and calculated according to Peach (1967). These are in better agreement with the OP data but for the cross sections of levels at higher energies he uses the hydrogenic approximation which provides reasonable results only for high quantum numbers [FORMULA]. In summary, the use of the OP data significantly improves the description of the photoionization processes compared to previous OI non-LTE studies.

[FIGURE] Fig. 3. Comparison of different photoionization cross sections (left panel) and excitation cross sections for electron collisions (right panel) on a logarithmic scale. Displayed are cross sections from OP computations (full line) and data from Hofsäß (1970, dotted), Henry (1970, dashed) and Peach (1967, dashed-dotted) for the photoionizations. In the case of the electron collisions our analytical fit (full line in upper diagram) to measured cross sections (Wang & McConkey, 1992, dots) is shown, as well as results from the approximations from Van Regemorter (1962, dotted) and from Allen (1973, dashed) and the theoretical data of Tayal & Henry (1989, full line in lower diagram) and Sawada & Ganas (1973, diamonds). Note that the Van Regemorter results are multiplied by a factor of 10000 in the lower right diagram.

In the first step of the computations, i.e. in DETAIL , the level populations are calculated using depth dependent Doppler profiles assuming LS coupling; microturbulence is explicitly accounted for by inclusion of an additional term in the Doppler width ([FORMULA]):

[EQUATION]

where [FORMULA] is the rest wavelength of the transition, c the speed of light, [FORMULA] the thermal velocity for the chemical species of interest and [FORMULA] the microturbulent velocity; see Sect. 5 for a discussion. Both continuous opacities and ATLAS9 line-distribution functions are accounted for in solving the radiation transfer.

These LS-coupling populations are then split according to the statistical weights of the individual sub-levels in order to calculate line profiles via the program SURFACE ; Voigt profile functions are adopted and the same microturbulent velocity as in DETAIL is applied. The damping parameters are calculated from radiative lifetimes given by Butler & Zeippen (2000) for the radiative widths and from the approximation of Cowley (1971) for collisional damping. A comparison with profiles computed with the more sophisticated data from Griem (1974) shows only negligible differences.

2.2.3. Collisional transitions

Special attention should be paid to an accurate description of electron collisions in view of their importance for the balance between the different spin system populations. Energy-resolved measurements for only a few transitions from the ground state are found in the literature; the results of Wang & McConkey (1992) are adopted. Theoretical cross sections for collisions between the ground state and the low excited singlet levels are taken from Tayal (1992). Results for the transitions to quintet terms were calculated by Tayal & Henry (1989) together with some additional cross sections in the triplet spin system. From a comparison with data presented in former work an accuracy significantly better than a factor of 2 is expected. For all remaining transitions up to the [FORMULA] energy levels the collision strengths from the distorted wave calculations of Bhatia & Kastner (1995) are used. The authors claim a generally good accuracy, even in the worst case being better than an order of magnitude, despite the fact that this method is not well suited for neutral species. For all other optically allowed transitions the Van Regemorter formula (Van Regemorter 1962) is applied with OP oscillator strengths. All the remaining bound-bound transitions are treated according to the semiempirical Allen formula (Allen 1973) with the collision strength [FORMULA] set equal to 1.0.

In Fig. 3 the collisional cross sections used in this work are compared to those derived from the approximation formula of Van Regemorter (1962) as adopted by Baschek et al. (1977). Large discrepancies up to several orders of magnitude are found. For the optically forbidden transition the cross section according to Allen (1973) and the results from the distorted-wave calculations of Sawada & Ganas (1973) as used by Takeda (1992) are also displayed. The former offers a better description than the Van Regemorter approximation; the latter are in good agreement with the data preferred by us. In summary, the preference of experimental and sophisticated theoretical data over results from approximation formula for many important transitions improves the reliability of our statistical equilibrium calculations.

Experimental cross sections from Thompson et al. (1995) are adopted for the collisional ionization of the ground state. Agreement better than a factor of 2 within the measured energy range is expected confirming previous results from other authors. Supplementary cross sections for ionization from [FORMULA] triplet levels are provided by Chung et al. (1993). Their comparison for different theoretical approaches suggests an accuracy better than 50%. The Seaton formula (Seaton 1962) is used for collisional ionization of the remaining non-LTE levels; threshold photoionization cross sections are taken from the OP data.

2.2.4. Charge exchange reactions

The nearly resonant charge exchange reaction [FORMULA] has been taken into account in the non-LTE calculations with rate coefficients determined according to the analytic fits of Arnaud & Rothenflug (1985). Non-LTE level populations [FORMULA] of hydrogen are calculated with DETAIL on the ATLAS9 model structure in advance. They show departure coefficients [FORMULA]=[FORMULA] differing significantly from unity only close to the Eddington limit (Kudritzki 1973). This process dominates the ionization balance of oxygen as the departures of the n(HI [FORMULA]/n(HII ) ratio are forced upon n(OI [FORMULA]/n(OII [FORMULA] - and the first two low-lying terms [FORMULA] and [FORMULA] which are in detailed balance with the OI ground state. Note that the OI/II ground states are already thermalized throughout most of the atmosphere when the charge exchange reaction is neglected (Fig. 4). As a consequence, the formation depths for the spectral lines discussed here are only marginally affected by charge exchange even for supergiants, resulting in negligible effects on the line strengths as tests have shown. Small effects on the [FORMULA] are expected for the resonance lines in the UV only.

[FIGURE] Fig. 4. Effects of the charge exchange reaction [FORMULA] on the departure coefficients and the ionization balance of oxygen as a function of the Rosseland optical depth [FORMULA] in a supergiant model for [FORMULA]=10000 K, [FORMULA]=1.5, [FORMULA]=8 [FORMULA]. Solid line: with, dotted line: without charge exchange. The formation depths of the line core ([FORMULA]) for several transitions are indicated. Term identifiers - 1: [FORMULA]; 4: [FORMULA]; 5: [FORMULA]; 6: [FORMULA]; 7: [FORMULA]; 52: [FORMULA] (OII )

For the case of spherically extended unified model atmospheres with strong winds (cf. Santolaya-Rey et al. 1997) significant non-LTE departures for hydrogen are present even in the line formation layers. But again, by omitting the charge exchange reaction negligible effects on the strengths of OI lines originating from the excited levels are found. Strong effects on the departure coefficients can only be expected beyond the sonic point.

2.3. Error estimates for the oxygen non-LTE calculations

To assess the importance of various parameters entering the non-LTE computations and to estimate systematic errors on the abundance analysis test calculations are performed for typical supergiant atmospheric parameters in the given temperature range: [FORMULA] of 8500 K/1.0, 10000 K/1.5 and 15000 K/2.0 with the microturbulence fixed at [FORMULA] assuming solar metallicity. Below [FORMULA][FORMULA]8500 K the hydrostatical model atmospheres for supergiants develop pressure inversion and should therefore - as well as for other reasons - be viewed with caution as a reliable description of the physical conditions. For a given synthetic spectrum of OI the abundance is adjusted in the model with modified parameters to reproduce the original line strengths. The results of the tests (mean values from the analysed lines) are summarised in Table 1.


[TABLE]

Table 1. Uncertainties in the non-LTE analysis of OI .
Notes:
a) mean values without the near-infrared transitions
b) mean values without the near-infrared transitions which can show deviations [FORMULA]0.5 dex at a factor 10


Uncertainties in the atmospheric parameters (at typical values of our analyses) have the largest effects on the non-LTE abundance determination for oxygen. Here one has to rely on the weak lines in the visible as the strong near-infrared lines react sensitively to even small changes in the microturbulent velocity. In any case, these lines are abundance indicators of only limited reliability for reasons discussed below.

Errors in the abundances derived from weak lines are directly proportional to inaccuracies in the gf values. The 10% overall error anticipated from the OP data manifests itself in an abundance uncertainty of [FORMULA]0.05 dex. Nevertheless, this might be an underestimate for individual transitions, cf. Table 4.

Generally, uncertainties in the line broadening due to radiative lifetimes are negligible. In the case of collision broadening the comparison of our values with Stark-broadening parameters from Griem (1974) for selected lines results in almost identical line profiles. But it should be noted that a typical error of a factor 2 in the collisional damping half-widths mainly affects the strong near-infrared lines.

No systematic error on the abundance analysis is expected from the variation of OP photoionization cross-sections within the given error bars. A factor of 5 in the absolute values of the cross-sections - a difference easily reached in comparison with former studies - on the other hand results in significant abundance corrections. The near-infrared lines are most strongly affected. But even this is an underestimation, larger differences are present even in some of the threshold values, amounting to factors of [FORMULA] in the - sometimes broad - resonances. The use of the OP data certainly results in a significant improvement compared to former OI studies.

Collisional excitation cross-sections also prove to be critical parameters in our statistical equilibrium computations. Unfortunately, for most of the transitions the data are based on an approximate formula, giving an accuracy within a factor 2 or 3 at best near threshold. Nevertheless, we do not expect larger overall systematic errors for our model than given in the corresponding entry of Table 1 as for the most important transitions quite accurate data are used. In-/decreasing the cross-sections by a factor 10 and therefore shifting farther to/from LTE conditions results in comparatively large abundance uncertainties and demonstrates again the need for accurate atomic data. The importance of detailed collisional data for the development of the strong non-LTE effect in the near-infrared triplet will be discussed later.

It is found that the accuracy of the collisional ionization cross sections is not a critical factor in the non-LTE computations as scaling them by a factor as large as 10 has virtually no effect on the calculated equivalent widths. Due to the small mean kinetic energy of the colliding electrons the low lying levels are not depopulated and thus the ionization balance is only marginally affected.

Inaccuracies in the rate coefficients of the charge exchange reaction have no effect on the abundance analysis in our approach as the changes occur only outside the line forming region.

Another source of systematic error is the continuum placement in the observed spectra. This strongly depends on the S/N ratio obtained. Our estimate in Table 1 should be applicable to high quality data with S/N[FORMULA]150 only. In general, equivalent widths studies are more susceptible to this systematic error than the spectrum synthesis technique which also accounts for the continuum regions.

The total uncertainties are computed from the sum of the squares of the appropriate uncertainties listed above. They should be viewed as the systematic errors applicable to our non-LTE calculations on OI within our methodology (cf. Sect. 2.1).

For our model of the supergiant HD 92207 (see Sect. 4) we performed the same parameter study as above to examine the reliability of our abundance analysis close to the Eddington limit. We find an estimated total uncertainty of [FORMULA]0.13 dex with errors due to the atmospheric/atomic parameters similar to those from Table 1. Note that the uncertainties in the atmospheric parameters of this star might be larger than adopted here thus increasing the systematic error. Finally, for main sequence stars we expect slightly smaller systematical errors than from Table 1 as most of the observed lines are formed under LTE conditions, the detailed atomic structure becoming irrelevant. The given uncertainties should therefore be viewed as upper limits for objects on the main sequence. In particular, for Vega (see Sect. 4) we derive a total uncertainty of [FORMULA]0.08 dex from the parameter study.

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Online publication: July 13, 2000
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