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Appendix A: Details about Kurucz’s and K-OPAL opacity data

	Fig. A.1 Rosseland-mean opacities lgκR [ cm2 g-1 ] calculated by Castelli & Kurucz for different chemical composition of elements and ξ = 0. The data plotted for the gas pressure, lgPg = 2,4,6, and 8 (in cgs units) were taken from the Castelli website on 2012. Black lines show opacity data Nos. 1–9 listed in Table A.1, while blue lines display opacity data Nos. 10–22.
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	Fig. A.2 Same as Fig. A.1, but the Rosseland-mean opacities lgκR [ cm2 g-1 ] calculated by Castelli & Kurucz for enhanced abundances of O, Ne, Mg, Si, S, Ar, Ca, and Ti elements by +0.4 dex, viz., opacity data Nos. 23–35 listed in Table A.1 were plotted.
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	Fig. A.3 Examples of the Rosseland-mean opacities lgκR [ cm2 g-1 ] calculated by Meszaros et al. for different chemical composition of elements and ξ = 2km s-1 (Table A.2). The data plotted for the gas pressure, lgPg = 2,4,6, and 8 (in cgs units) were taken from the ATLAS-APOGEE website on 2013.
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	Fig. A.4 Derivatives of the Rosseland-mean opacities over density vs. lgT for lgR = lgρ − 3lgT + 18 = −6.0, −5.0, −4.5, and −4.0. The OPAL (blue lines) and K-OPAL/A (red lines) data were plotted for Z = 0.0266,0.169,0.00542, viz., opacity data No. 1, 3, 7, 11, 13, and 16 listed in Table 1 are shown. The K-OPAL/A data with α-enhanced elements (m05a in Table A.1) were also plotted as green lines.
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	Fig. A.5 Same as Fig. A.4, but for vs. lgT.
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Fig. A.6 OP monochromatic cross-sections lgσ (in atomic units) vs. for lgNe = 16 and lgT = 4.8,5.0,5.2, and 5.3. The yellow lines show the total opacity produced by all elements considered in the OP project, the black lines display opacity caused by H, He, Fe, and σe, while the violet lines correspond to Fe. The Rosseland-mean weighting function is also plotted as smooth lines. The brown line in the two lowest panels correspond to H, He, and σe.

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	Fig. A.7 Derivatives of the Rosseland-mean opacities over temperature vs. lgT for lgR = −6.0, −5.0, −4.5, and −4.0. The K-OPAL/B data (Nos. 18–20 in Table 1) were compared with the OPAL (No. 1) and K-OPAL/A (No. 11) data.
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The Rosseland-mean opacities calculated by Castelli & Kurucz are available in tabular form for different metallicities and microturbulent velocities ξ = 0,1,2,4, and 8 km s^-1. Following Kurucz, the data (Table A.1) were named using the scheme, which describes abundances of metals relative to the solar abundances listed in Grevesse & Sauval (1998). In particular, p15 corresponds to enhanced abundances of metals by [m/H] = lg(m/H) − lg(m/H)_⊙ = +1.5 dex, the solar chemical composition ([m/H] = +0.0) is denoted as p00, and [m/H] = −4.0 as m40. The X, Y, and Z abundance parameters and the mass abundances of oxygen (Z_O) and iron (Z_Fe) were also shown in Table A.1. The opacity data Nos. 1–22 have the same relative abundances of metals taken from Grevesse & Sauval (1998), while data Nos. 23–35 have enhanced abundances of O, Ne, Mg, Si, S, Ar, Ca, and Ti by +0.4 dex. These elements are produced by stellar nucleosyntheses in the α-process. The last two rows in Table A.1 show the abundance parameters listed in Asplund et al. (2005) and Asplund et al. (2009).

The CK Rosseland-mean opacities, , were tabulated for 3.3 ≤ lgT ≤ 5.3 and −4.0 ≤ lgP_g ≤ 8.0, where P_g is the gas pressure in the cgs units. Figure A.1 shows as a function of lgT for lgP_g = 2,4,6, and 8, and ξ = 0. In each panel, the top line corresponds to X = 0.4786 and Z = 0.3596, while the bottom line to X = 0.7474 and Z = 1.49 × 10^-7 (Table A.1). Clearly, the opacity bump at lgT ≈ 5.06 is caused by metals and can be explained by much more detailed atom models of the new Kurucz data. The opacities with enhanced abundances of the α-rich elements (O, Ne, Mg, Si, S, Ar, Ca, and Ti) by +0.4 dex. were plotted in Fig. A.2. These elements participate in the opacity bump at lgT ≈ 5.06 in addition to other heavy elements (cf. also Fig. 2, where the data of m05 and m05a were plotted).

Recently, an extensive grid of atmosphere models, as well as the Rosseland-mean opacities, , were recalculated by Meszaros et al. (2012) for scaled solar abundances of metals given by Asplund et al. (2005). Examples are shown in Table A.2 and Fig. A.3. The data were named using the Kurucz scheme for the scaled metal (m) abundances composed with a standard/nonstandard content of C (c) and O (o).

We would like to note that Castelli & Kurucz’s mixture of elements has higher values of Z and Z_O than those of Asplund et al. (2009), but the same abundance of Fe. The Meszaros et al. data (M) have lower Z- and Z_Fe-values. In addition, the helium abundance of the CK and M standard (solar) mixtures is not far from the helioseismological value Y = 0.2485 given by Basu & Antia (2004). This helium abundance was adopted by Grevesse & Sauval (1998), Asplund et al. (2005) and Asplund et al. (2009). The CK and M elemental mixtures have Y = 0.1635−0.2526 (Tables A.1 and A.2).

The CK and M atmosphere models are available from Castelli and ATLAS-APOGEE websites (Sect. 3), respectively. The older models, available from Kurucz website, were calculated for older opacity distribution functions (ODF) with about 58 million lines of atoms and molecules in total. Despite this, according to Kurucz (2011), the line opacity was underestimated because not enough lines were included in calculations.

According to Kurucz (2012, priv. comm.), the CK opacities are less reliable for lgT ≥ 5.2. The K-OPAL models were therefore calculated using the CK (or M) opacities for lgT < 5.2 and the OPAL (or OP) data for higher temperatures. We adjusted these two kinds of Rosseland-mean opacities at lgT = 5.2, assuming . The opacity data are usually included as tables of κ_R(lgT,lgR) (the OPAL format) in computing codes for modeling stellar structure, where lgR = lgρ − 3lgT + 18. We attempted to achieve good accuracy in the transformation of κ_R from (lgT,lgP_g) to (lgT,lgR) variables. First, we solved the EOS with a Kurucz’s computing code to find out gas densities, ρ, for each pair of (lgT, lgP_g). Next, we interpolated the opacity data to (lgT, lgR)-points. In addition to κ_R, the derivatives of the Rosseland-mean opacities over temperature and density were calculated numerically with two methods. In the first method, having already calculated tables of κ_R(lgT,lgρ), we used Kurucz’s (1970) differentiation procedure to obtain and . Next, we used the thermodynamic identity for (A.1)Again, the CK (or M) and OPAL (or OP) derivatives were adjusted at lgT = 5.2. In the second method, we started from κ_R(lgT,lgR) and ran Seaton’s (1993) bicubic interpolation procedure, which involves 16 neighborhood points as well as the derivatives of κ_R over temperature and density. Both methods yield similar results, although the second method reveals a little bit more regular behavior of the derivatives because of the higher number of data points involved in the procedure. In the rest of this paper we used data obtained from the second method. No additional smoothing procedure was ran.

The K-OPAL opacity data discussed in Sect. 3 show two opacity bumps at lgT = 5.06 and 5.3. Examples of the hybrid opacities, κ_R were plotted in Fig. 2 and their derivatives over density, (κ_ρ)_T, and temperature (κ_T)_ρ, in Figs. A.4 and A.5, respectively. The local minimum of the opacity at lgT = 5.2 plays a crucial role in the K-OPAL seismic models of B stars as discussed in Sect. 5. A deeper insight into the problem follows from an analysis of the OP monochromatic opacities. Figure A.6 shows contributions of different elements to the total opacity (yellow lines) for lgT = 4.8,5.0, and 5.3, and lgN_e = 16. The black lines display opacity caused by H, He, Fe, and electron scattering (σ_e), while cross-sections of Fe are shown as violet lines. In Fig. A.6, the monochromatic cross-sections, σ(u) are given in atomic units, where . The Rosseland-mean opacity per unit mass is then (A.2)where and μ is the mean atomic weight. The smooth red lines in Fig. A.6, corresponding to lgF(u) + C, display that a spectral region near u = 3.83 contributes mostly to the Rosseland-mean opacity. The constant C was introduced for better visibility of the Rosseland-mean weighting function (corrected for stimulated emission), F(u). In the cases of lgT = 4.8 and 5.0, the Fe-curtain (violet lines in Fig. A.6) is located well below opacity caused by H, He, and σ_e at spectral region near u ≈ 3.83. Note that the contribution of metals other than Fe is significant at lgT = 5.0. This is the reason why the new Kurucz data were able to change the Rosseland-mean opacity mostly near lgT = 5.06. The Fe-curtain dominates at lgT = 5.3 (Fig. A.6). Now, the OP (and OPAL) opacity is caused mostly by very many lines due to Fe in electron configurations of the type 3s^x3p^y3d^z and 3s^x3p^y3d^znl, (Iglesias et al. 1990; Lynas-Gray et al. 1995; Seaton et al. 1994). As one can see, the synthetic Fe cross-section is located above the total cross-sections caused by H, He, and σ_e (a brown line) near u ≈ 3.83 (and u ≥ 7). This changes quickly depending on the temperature. At lgT = 5.2, the Fe-curtain is also seen, but its maximum is shifted to u ≈ 5. Similarly, the maximum is located at u ≈ 2.3 for lgT = 5.5. As a result, the OP (and OPAL) opacity bump at lgT = 5.3 is narrow and the K-OP (and K-OPAL) data have a two-peak structure of κ_R with the additional opacity bump at lgT = 5.06.

As mentioned in Sect. 3.2, the K-OPAL/B models were designed to study how the relative strength of the opacity bumps at lgT = 5.06 and 5.3 influence seismic models of early-type stars. They are basically the OPAL models for Z = 0.0264 and Z = 0.0168, but was selected near the opacity bump at lgT ≈ 5.06. The CK opacity data of m075,m05,m04,m02, and p00 were considered. Figures A.7 show κ_R and κ_T for models of Nos. 18–20 (Table 1) in comparison with the OPAL (No. 1) and K-OPAL/A data (No. 11).

Relatively little is known about the sensitivity of the OP and OPAL data to the pressure-broadening parameters for spectral lines. There is little sensitivity at high densities, where the lines become very broad and give a quasicontinuum, or at very low densities, where the profiles are mainly determined by Doppler broadening and radiation damping. At intermediate densities, like lgN_e = 15.5 and lgT = 5.2 (lgR = −5.8) discussed by Seaton et al. (1994), an increase of the pressure broadening parameters of Fe lines by a factor of 2 results in an increase in the Rosseland-mean opacity by 33 per cent for the solar mix of elements. The opacity is much lower when only doppler broadening of lines is assumed (Rogers & Iglesias 1992).

In addition, including more electron configurations and many more line transitions, as in Kurucz (2011), would probably also increase opacity at lgT> 5.2. Another potentially important reason why the OP (and OPAL) opacities should be upgraded is related to autoionizing transitions. In the OP project, these transitions were treated as resonances in photoionization cross-sections (Bautista 1997). To make the problem tractable, each LS-term was represented by only one level with its average energy. The splitting of spectroscopic LS terms

into individual levels may be necessary for final calculations of photoionization cross-sections with resonance structures. This is suggested by Kurucz’s treatment of autoionizing transitions (Castelli & Kurucz 2004) and by recent results published by Nahar et al. (2011), which include the fine structure explicitly for Fe XVII. Similarly, the new effective cross-sections with photoexcitation-of-core (PEC) resonances for O III and Fe XXI are much higher, up to two orders of magnitude, than in the OP data (Fig. 1 in Pradhan & Nahar 2009). The missing opacity in the OP data may lie in an energy range from a few eV to few hundreds of eV, depending on the element and ionization stage. Therefore, we also searched K-OPAL/C (and K-OP/C) models (with opacity data Nos. 23–25) with enhanced opacities at lgT> 5.2. In the case of No. 23, EOS and nuclear reaction rates were calculated for the initial (at ZAMS) chemical composition of X = 0.7394, Y = 0.2499, and Z = 0.0107. The CK opacity data corresponding to Z = 0.0107 were assumed at lgT < 5.2, but the OPAL opacities corresponding to Z = 0.0264 at lgT> 5.2. Similarly, models of Nos. 24 and 25, were calculated for Z = 0.00537 and 0.00303 with enhanced opacities at lgT> 5.2 (OPAL: Z = 0.0168 and 0.0134), respectively.

Examples of κ_R and κ_T are shown in Figs. 11 and 12 (Sect. 5), where differential work integrals were plotted for the K-OPAL models.

© ESO, 2014