Equipos utilizados para los parámetros de calidad

Carbon is the 2nd most abundant metal in the universe. Similar to the case of nitrogen two stable isotopes are found,12_{C and}13_{C, with the former contributing almost 99% to the total carbon abundance. Carbon is produced} by the triple-αprocess during the helium-burning phase of stars via the reaction 4_{He(α, )}8_{Be(α, )}12_C∗_{( ,γγ)}12_C. Massive stars are the main source for the enrichment of the ISM with carbon, which is released in the final SN explosion.

Energy levels

Carbon is almost completely ionized throughout the atmosphere of BA-type stars, with a small fraction (on the order of several parts in a thousand) of CI at line-formation depths. Non-LTE effects are expected to be of importance for the interpretation of the CI/IIlines at visual and near-IR wavelengths, which are all emerging from highly-excited levels (7.5/14.5 eV above the ground states of CI/II).

Energy levels up to∼0.37/0.67 eV below the ionization threshold as listed by Moore (1993) are included explicitly in the CI/II model. Missing data for highly excited CIIlevels are adopted from Sigut (1996). This includes all energy levels withn≤6 and the 7s states in CIand all levels withn≤9 andA≤4 in CII. Only the ground state of CIIIis considered, as the ionization energy of 47.89 eV is large. Fine structure and intermediate- coupling terms are treated in analogy to nitrogen.

The populations of the remaining levels of CI(up ton=10) and of CII(up ton=14) are considered in the number conservation equation. They are computed in LTE relative to the ground state of the higher ionization stage, with level energies derived from their quantum defects.

Radiative transitions

Oscillator strengths for all optically allowed bound-bound transitions between energy levels with non-LTE pop- ulations are adopted from OP data (Luo & Pradhan 1989, for CI) and from Fernley et al. (TOPBASE, for CII). Again, the individual lines of a multiplet are reduced to a single effective ‘multiplet line’ in order to improve the computational efficiency.

Grotrian diagrams for the different spin systems of the present CI/IImodel atom are displayed in Fig. 3.8. All ionization stages under consideration are treated simultaneously in the non-LTE calculations. Additional radiative coupling between the different spin systems of CIis provided by all intercombination transitions listed by Wiese et al. (1996). On the other hand, intercombinations in CIIare neglected due to the very small transition probabilities. Luo & Pradhan (1989) and Yan et al. (1987) have compared the OP oscillator strengths for CI/II

with experimental and theoretical work by other authors; the majority of the data is expected to be accurate to within 10%, superceding most of the older data used in previous studies of non-LTE effects on CIor CII.

Opacity Project data are also adopted for the photoionization cross-sections (Luo & Pradhan (1989) for CI

and Fernley et al. for CII), as available through the TOPBASE database. The full resonance structure, as indicated by theR-matrix calculations, is accounted for in detail. These photoionization data are expected to be of similar accuracy as the oscillator strengths. Cross-sections for energy levels missing in the OP data (forA=4 in CII) are calculated in the hydrogenic approximation (Mihalas 1978, p. 99). A comparison of OP cross-sections with those of Henry (1970) – usually used in previous non-LTE studies on CI/II– is given in Fig. 3.9. The OP cross- section for the CII ground state agrees well with that of the less elaborate approach, except for several sharp

resonances. On the other hand the OP data for the three lowest CIstates are systematically larger on the average,

thus increasing the rate of photoionization. In addition, broad resonances corresponding to autoionizing states dominate the important near-threshold region, enlarging the cross-sections considerably. For the higher excited levels of CI/II, the use of detailed OP data also significantly improves the description of the photoionization processes compared to most of the previous non-LTE studies of these ions.

The further computations with DETAILand SURFACEwere made as in the case of nitrogen. Transition wave- lengths for the final spectrum synthesis are taken from Wiese et al. (1996) and Moore (1993) and transition probabilities from the sources indicated in Table A.1. The damping parameters are calculated from OP radiative lifetimes for the radiative widths and adopted from Griem (1964, 1974) for electron impact and ion broadening in CI/II. Missing collisional-damping data are computed from the approximation of Cowley (1971). Van der Waals damping is neglected in the parameter range considered here, as the atmospheric plasma is mostly ionized.

Figure 3.8:Grotrian diagrams for CI, singlet (upper left) and triplet spin system (upper right) and for CII, doublets (lower left) and quartets (lower right). Both ionic species and all spin systems (with an additional quintet level in CI) are treated simultaneously. Displayed are the energy levels and the radiative bound-bound transitions treated explicitly in non-LTE. The intercombination lines connecting the spin systems of CIare not shown in the diagram; in CIIintercombinations are neglected due to the small transition probabilities.

Figure 3.9: Comparison of photoionization cross-sections from OP computations (full line) and from Henry (1970, dotted line). On the left hand side data for the ground states of CIand CIIare displayed, on the right hand side data for the first and second excited levels of CI. The near-threshold regions of the latter two are dominated by broad resonances corresponding to autoionizing states.

Collisional transitions

Substantial interest in accurate cross-sections for collisional excitation in CI/IIfor fusion-plasma modelling (pol- lution of the plasma by carbon becomes a problem as the tokamak inner walls are made of graphite, which is released in traces during operation and therefore has to be monitored) and astrophysical applications has led to computational efforts on a large scale recently. For a significant fraction (over 300 transitions) of the data needed in the present work, detailed results are available from calculations carried out in the CCapproximation using theR-matrix method. Effective collision-strengths for CIare calculated from the collisional cross-section com- putations of Dunseath et al. (1997), following Burgess & Tully (1992). Additional cross-sections for collisional excitation between the four lowest-lying energy levels are adopted from Thomas & Nesbet (1975). For CII, effective collision-strengths are taken from the work of Blum & Pradhan (1992). Error estimates for these data are difficult to obtain, as no comparison with other experimental or theoretical results is available for almost all transitions. For test purposes (see below), a factor of two in accuracy is assumed. This is a rather conservative assumption, as the uncertainties in data obtained by the R-matrix method in theCCapproximation are known to be typically on the order of a few 10%. For the remaining bulk of the transitions, however, approximate for- mulae must be used, giving threshold values accurate to a factor 2–3 at best. Van Regemorter’s formula (Van Regemorter 1962) is applied for radiatively permitted transitions with OP oscillator strengths. For the optically forbidden transitions, the Allen formula (Allen 1973) is used with the collision strength set to 1.0.

Experimental cross-sections from Brook et al. (1978) and Yamada et al. (1989) are adopted for the collisional ionization of the ground states of CIand CII. The authors expect uncertainties<30% and<20%, respectively, for their data at energies near the ionization threshold, with significant improvements in the accuracy at higher energies (at a few percent uncertainty). All the remaining collisional ionization data are computed using the Seaton formula (Seaton 1962) with threshold photoionization cross-sections from the OP data, where available, or from the hydrogenic approximation.

Table 3.4: Uncertainties in the non-LTE analysis of carbon

changes inlogε(C)NLTE

9500/4.0 9500/2.0 9500/1.2 12000/1.8 CI CI CII CI CII CII

Atmospheric parameters:

Teff −150K σTeff −0.11 −0.09 +0.10 −0.06 +0.12 +0.07

logg+ 0.15dex σlogg −0.03 −0.04 +0.10 −0.10 +0.08 +0.08

ξ+1 km s−1 σξ −0.05 −0.02 −0.03 −0.01 −0.03 −0.06

y+ 0.15dex ±0.00 −0.03 −0.03 −0.04 −0.04 −0.06

[M/H]−0.2dex σ[M/H] +0.01 −0.03 ±0.00 ±0.00 ±0.00 −0.01

Line transitions:

Oscillator strengths+10% σloggf −0.05 −0.05 −0.05 −0.05 −0.04 −0.05 Damping constant∗2 σdamp −0.01 ±0.00 ±0.00 ±0.00 ±0.00 ±0.00

Photoionisations: Cross-sections+10% σrbf ±0.00 ±0.00 ±0.00 ±0.00 ±0.00 ±0.00 Cross-sections∗5 +0.08 +0.17 +0.02 +0.13 +0.03 +0.07 Collisional transitions: Cross-sections∗0.1 −0.16 −0.09 −0.06 −0.02 −0.06 −0.10 Cross-sections∗0.5 σcbb −0.06 −0.03 −0.02 −0.01 −0.02 −0.04 Cross-sections∗2 σcbb +0.06 +0.04 +0.02 +0.02 +0.03 +0.04 Cross-sections∗10 +0.18 +0.15 +0.08 +0.14 +0.15 +0.18 Collisional ionization: Cross-sections∗0.1 σcbf +0.03 ±0.00 ±0.00 ±0.00 ±0.00 ±0.00 Cross-sections∗10 σcbf −0.01 ±0.00 −0.01 ±0.00 ±0.00 ±0.00

Continuum placement σcont ±0.05 ±0.05 ±0.05 ±0.05 ±0.05 ±0.05

Estimated total uncertainty σsys ±0.16 ±0.13 ±0.16 ±0.14 ±0.16 ±0.15

Error estimates for the non-LTE calculations

To assess the effects of atmospheric parameters and crucial atomic data on the accuracy of carbon abundance determinations by the present non-LTE approach, test calculations were performed for typical atmospheric pa- rameters of BA-type stars as in the case of nitrogen. The results of the tests are summarised in Table 3.4. They correspond to the mean value obtained from a study of the linesλλ4228, 4771, 5052, 5380, 6013, 6828, 7111, 7115, 9078, 9088, 9111 (CI) andλλ3918, 4267, 6578, 6582 (CII).

The carbon non-LTE calculations turn out to react sensitively to inaccuracies in the stellar parameters and the atomic data. In general, the stronger lines are more susceptible, as they show stronger non-LTE effects. Changes in the effective temperature and surface gravity affect the delicate ionization balance of CI/II: within the assumed error bars for these parameters, the carbon abundance can be typically fixed only to approximately

±0.1 dex. Inaccuracies in the microturbulent velocity primarily alter the strengths of lines on the flat part of the curve of growth. Effects of an increased microturbulence are therefore small, as most of the lines remain weak in the parameter range of the present study. Only the near-IR lines of CIin the main sequence model and the CII lines in the hotter supergiant model are affected, resulting in abundance uncertainties of ∼0.15/0.06 dex. An increase of the atmospheric helium content systematically lowers the derived carbon abundances for both ionization stages in all the supergiant models, while the lines in the main-sequence model are insensitive to such a change. In the following estimation of the systematic error this effect is ignored as the helium abundance is explicitly determined and accounted for in the model atmosphere computations. Line-blocking effects introduced by metallicity variations on the order of 0.2 dex have a negligible impact on the CI/IIline-formation calculations. Inaccurategf-values enter the abundance determination linearly, as long as the carbon lines remain weak as is mostly the case in the present study. Modifications of the collisional-damping widths within a factor of 2 result in irrelevant changes of the line strengths, except for the near-IR CIlines in the main sequence model, where small abundance corrections of∼0.05 dex become necessary. No corrections have to be applied for variations in the photoionization cross-sections at the 10% level. On the other hand larger inaccuracies, by a factor of 5, strongly affect the lines of the minor ionic species of CIby way of a displaced ionization balance. The near-IR CIlines react most sensitively, resulting in abundance corrections of∼0.2 dex. In the hotter supergiant model a similar sensitivity is found for CIIλ4267. Accurate photoionization data are therefore an important ingredient in the non-LTE calculations for CI/II. The new model atom improves the situation in comparison with former studies.

In the present work, detailed collisional excitation data are used for the first time for a considerable number of transitions in CI/II. Collisions play an important rˆole in the abundance analyses, as is inferred from the test calculations. Even a systematic inaccuracy of a factor 2 in the collisional rates results in a ∼0.2 dex change in abundance derived from the CIλλ9061–9111 lines in the main sequence model. This uncertainty increases to

∼0.5 dex in abundance when the collisional rates are varied by a factor of 10. The weaker lines of CIand CII

react less sensitively – abundance uncertainties typically amount to less than 0.1 dex. Due to the lower particle densities, accurate collisional data become less important in supergiants. Nevertheless, CIIλ4267 is found to react sensitively to a reduction of the collisional rates in the supergiant models as well. Practically no influence on line strengths is found for a variation of the collisional ionization rates by an order of magnitude. A further source of systematic error is the continuum placement in the observed spectra, see the discussion of systematic error sources in nitrogen for further comments on this.

The total uncertainties are computed from the sum of the squares of the appropriate uncertainties in Table 3.4, according to Eq. (3.13). They should be viewed as the (conservative) systematic errors applicable to the non-LTE calculations on CI/IIwithin the limitations set by the method applied in the present work. The total uncertainty (systematic+statistic on the 1σ-level, cf. also Sect. 5.2) typically amounts to an unsatisfactory∼0.3 dex. The non-LTE effects in CI/II

The ionization balance of carbon in the photospheres of typical early A-type stars is displayed in Fig. 3.10. The dominant ionization stage in the line-formation region is CII, with small admixtures of CI(typically a few parts in a thousand) and CIII, the latter showing a strong increase from∼10−7_to10−3 _{in fraction with increasing} Rosseland optical depthτR. In the main sequence model, the ionization balance is shifted in favour of the lower ionization stage, due to the larger surface gravity and hence higher particle densities. The impact of an increase inTeff on the ionization balance is also shown in Fig. 3.10 for the late B-type supergiantβOri. The changes are in the expected direction, i.e. shifting the balance to the higher ionization stage. In general, the total CIIand CIII

populations are enhanced in non-LTE, while CIbecomes overionized.

In Fig. 3.11 departure coefficients bi are displayed as a function ofτR for the models of several objects discussed in Sect. 5.2. In addition, departure coefficients for a Vega-like model at solar elemental composition (Grevesse & Sauval 1998) are also given to study the influence of metallicity on the carbon non-LTE calculations. All the observed CI/IIlines in the visual/near-IR originate from fairly high-excited energy levels (7.5/14.5 eV). In Table 3.5, identifiers for the consecutively numbered energy terms of the CI/IImodel are given and the levels involved in the observed transitions are identified in Table 3.6.

The most striking feature in the main sequence models is the marked underpopulation of the ground state and the energetically close singlet and quintet states, which are coupled collisionally; these four levels are largely decoupled from the remaining CI levels. Photoionizations depopulate these lowest levels efficiently and the departures increase as the line blocking is reduced (Vega model atZ=−0.5 dex vs. theZmodel). On the other hand, neglecting the non-LTE radiation field by replacing the mean intensity by the Planck function,Jν=Bν, in the photoionization continua results in a complete relaxation of the situation and the low-energy levels of CIstay close to their detailed-equilibrium value.

The levels at higher excitation show departure coefficients slightly greater than unity in the line-formation region. But they contribute only little to the total CIpopulation, resulting in the net overionization of this ioniza- tion stage. As the excitation energy of CIlevels approaches the ionization threshold, the departures diminish due to the increasing collisional coupling with the ground state of CII, which is in LTE.

The CI lines are progressively weakened at higher luminosities due to the increasing overionization and become unobservable in the objects at highest luminosity class (for the temperature range under consideration). Departure coefficients for a number of CIlevels are therefore displayed for theηLeo model only, as the situation in HD 111613 is found to be similar. The ground state is even more depopulated than in the main sequence case and the collisional coupling of the singlet and quintet terms at low excitation energy is weakened. Again, the more highly excited CI levels show only marginal deviations from detailed equilibrium populations in the line-formation region.

CIIlines become observable in early A-type supergiants and strengthen rapidly with increasingTeff. The doublet ground state and the collisionally coupled lowest quartet term remain in detailed balance throughout the depth range covered by the models, as expected for the main ionization stage. The populations of the excited energy levels important for the formation of the observed lines (see Table 3.6) deviate only marginally from LTE, as in the CIcase. A marked overpopulation of the CIIIground state is developing at small optical depths.

Figure 3.10: Non-LTE and LTE ionization equilibrium of carbon for Vega (solid lines and filled circles, respectively),

ηLeo (dotted lines/open circles) and HD 92207 (dashed lines/ open diamonds; left panel) andβOri (above). Displayed are the ratios of the total level populations of the three ionization stagesn(CI/II/III) to the total carbon populationN(C) as a function of Rosseland optical depthτR.

Table 3.5: Term identifiers for carbon

Ion No. Term Ion No. Term Ion No. Term Ion No. Term CI 1 2p2 3P CI 27 4p1D CII 81 2p2Po CIII128 2s2 1S 2 2p2 1D 28 4p1S 82 2p2 4P 3 2p2 1S 29 4d1Do 86 3s2S 4 2p3 5So 30 5s3Po 87 3p2Po 5 3s3Po 31 4d3Fo 89 3d2D 6 3s1Po 33 5p1S 91 4s2S 8 3p1_{P 41 4d}1_Po _{93 3s}4_Po 9 3p3D 43 5p1P 94 4d2D 11 3p3_{P 49 5d}3_Fo _{96 4f}2_Fo 23 3p3P 50 6s3Po 108 6f2Fo 26 4p1P 60 5d1Po 112 3p4P

Table 3.6: Line identification for carbon

Ion λ( ˚A) Transition l – u Ion λ( ˚A) Transition l – u CI 4228.33 3s1_Po_{– 5p}1_{S 6 – 33 C} I 7115.17 3p3_{D – 4d}3_Fo _{9 – 31} 4371.37 3s1Po– 5p1P 6 – 43 7115.18 3p3D – 5s3Po 9 – 30 4771.74 3s3Po– 4p1P 5 – 26 7116.99 3p3D – 5s3Po 9 – 30 4775.90 3s3_Po_{– 4p}1_{P 5 – 26 7119.66 3p}3_{D – 5s}3_Po _{9 – 30} 4932.05 3s1Po– 4p1S 6 – 28 9078.29 3s3Po – 3p3P 5 – 11 5052.17 3s1_Po_{– 4p}1_{D 6 – 27 9088.51 3s}3_Po _{– 3p}3_{P 5 – 11} 5380.34 3s1Po– 3p3P 6 – 23 9111.81 3s3Po – 3p3P 5 – 11 5668.94 3p1_{P – 5d}1_Po _{8 – 60 C} II 3918.97 3p2_Po_{– 4s}2_{S 87 – 91} 6013.17 3p3D – 6s3Po 9 – 50 3920.68 3p2Po– 4s2S 87 – 91 6013.21 3p3D – 5d3Fo 9 – 49 4267.00 3d2D – 4f2Fo 89 – 96 6014.83 3p3D – 6s3Po 9 – 50 4267.26 3d2D – 4f2Fo 89 – 96 6587.61 3p1P – 4d1Po 8 – 41 5132.95 3s4Po– 3p4P 93 – 112 6828.12 3p1_{P – 4d}1_Do _{8 – 29 5133.28 3s}4_Po_{– 3p}4_{P 93 – 112} 7100.12 3p3D – 5s3Po 9 – 30 6151.27 4d2D – 6f2Fo 94 – 108 7108.93 3p3_{D – 5s}3_Po _{9 – 30 6151.53 4d}2_{D – 6f}2_Fo _{94 – 108} 7111.47 3p3D – 4d3Fo 9 – 31 6578.05 3s2S – 3p2Po 86 – 87 7113.18 3p3_{D – 4d}3_Fo _{9 – 31 6582.88 3s}2_{S – 3p}2_Po _{86 – 87}