OPACITY CALCULATIONS FOR SOLAR MIXTURES

Radiation plays a major role in a wide variety of high energy density astrophysical plasmas. For example, it is well known that the internal temperature profiles of solar-like stars are controlled by the ability of stellar matter to absorb radiation, which can be described in terms of radiative opacity (Eddington 1926). Among the numerous elements making up the stellar chemical composition, metallic elements (elements with atomic number Z greater than two) significantly contribute to the opacity, although they are only present as a few percent of the mixture.

Based on recent refined solar photosphere spectral analysis (Asplund et al. 2009; Scott et al. 2015a, 2015b; Grevesse et al. 2015), the inferred amounts of low-Z metallic elements (mainly carbon, nitrogen, and oxygen) have been revised downward. Standard solar models using such an improved photospheric composition disagree with helioseismic and neutrino observations (Guzik 2008; Serenelli et al. 2009; Turck-Chièze & Couvidat 2011). Among the various proposed explanations, it has been estimated that a smooth increase of the opacity from the range of 2%–5% in the central regions to the range of 12%–15% at the base of the solar convection zone could reconcile models and observations (Serenelli et al. 2009). Such an assumption implies that the opacities of heavier elements in the stellar mixture should be revised upward to compensate for the decreased low-Z metallic element abundances. The focus is on heavier metallic elements because they are always partially ionized in solar-like star interiors. Due to high temperatures, numerous excited atomic states are noticeably populated and significantly contribute to the opacity. Calculations based on an incomplete sampling of such states could lead to opacity values that are too low.

Recently, measurements have been performed of the wavelength-resolved iron opacity at electron temperatures of ${T}_{{\rm{e}}}=1.9-2.3\times {10}^{6}\;{\rm{K}}$ and densities of ${N}_{{\rm{e}}}\;=0.7-4.0\times {10}^{22}\ {\mathrm{cm}}^{-3}$ (Bailey et al. 2015). These thermodynamic conditions are very similar to those expected near the solar radiation/convection zone boundary. The measured wavelength-resolved opacity appears to be 30%–400% higher than in calculations in a spectral range where L-shell photoexcitation and photoionization processes are prevalent.

Both helioseismic observations and laboratory measurements support the revision of opacity calculations in high energy density plasmas. A large part of the opacity tables commonly used by the astrophysicists community were calculated 20 years ago when computer resources were not what they are today. In order to reduce computational effort, various approximations were made to evaluate the total opacity of the considered mixtures (Iglesias & Rogers 1996; Badnell et al. 2005). Several groups are developing new opacity codes (Porcherot et al. 2011; Blancard et al. 2012; Colgan et al. 2013). Although they differ in their equation of state modeling, all of these codes allow for an extensive detailed line accounting (DLA) based on high-quality atomic structure data obtained in full intermediate coupling. The OPAS opacity model has recently been used to calculate the solar opacities along a specific thermodynamical path (Blancard et al. 2012). In the present work, we used this model to produce opacity tables that are useful for solar radiative zone modeling. According to the AGS'09 composition (Asplund et al. 2009), a mixture including the 22 most abundant chemical elements has been considered (abundance greater or almost equal to 5). Hence, cobalt has been added to the 21 element list considered in OPAL calculations (Iglesias & Rogers 1996). Possible sources of uncertainty in the calculations are examined and discussed. The Rosseland opacities that we obtained are compared to OP (http://cdsweb.u-strasbg.fr/topbase/topbase.html) and OPAL (http://opalopacity.llnl.gov) data.

The OPAS opacity code has been described previously (Blancard et al. 2012). We simply note that it has been developed in the detailed configuration accounting framework. The monochromatic opacity is evaluated as a sum of four contributions involving the diffusion process as free–free, bound–free, and bound–bound absorption processes. The bound–bound opacity is calculated by combining various approximations to take into account the level structure of configurations. In any case, statistical methods are used to describe the electric-dipole transition arrays connecting two non-relativistic configurations. When necessary, and computationally manageable, the statistical description of the transition arrays is replaced by an extensive DLA performed in the full intermediate coupling scheme from jj-coupling. For stellar opacity tabulations, such an approach is the best compromise between accuracy and computational time.

Since its first use to calculate solar opacities, the OPAS code has been improved by including a systematic ionic Stark broadening treatment. In hot dense plasmas, highly excited atomic states can be notably populated. Photoexcitation processes from these atomic states give rise to satellite lines that can significantly overlap and merge with resonance lines due to broadening mechanisms acting on the overall lines. The Stark effect can be an important source of line broadening. For hot dense plasmas, it results from the existence of fluctuating electric fields produced by both free electrons and slow-moving ions. Splitting levels into sublevels, these electric fields can significantly modify the atomic bound states, together with the wavelengths and strengths of the transitions connecting these states. The perturbing field induced by the ions is described here using the quasi-static approximation. The electric microfield distribution we considered depends on the considered mixture. It is evaluated as a sum over the chemical elements of the individual electric microfield distributions weighted by the element number fraction in the mixture. For hydrogen-like and helium-like ions, the initial line broadening treatment has been retained. We note that it is based on the standard theory including linear the Stark theory (Griem 1997). For a given electric microfield distribution, spectral line profiles are computed following methods developed by Lee (1988). For the other ionic stages, statistical ionic Stark broadening is applied using a model proposed by Rozsnyai (1977).

Opacity calculations dedicated to the solar interior have been performed for temperatures between $\mathrm{log}[T({\rm{K}})]=6$ and $\mathrm{log}[T({\rm{K}})]=7.2$. The electron density has been sampled between $\mathrm{log}[{N}_{{\rm{e}}}({\mathrm{cm}}^{-3})]=20$ and $\mathrm{log}[{N}_{{\rm{e}}}({\mathrm{cm}}^{-3})]=26$. An iso-thermal iso-N_e mixing model is used to evaluate the opacity of the mixture. For a given N_e value, the partial density of each element in the mixture is determined. The density of the mixture (ρ) is deduced from the additivity of the partial volumes. The opacity of the mixture is expressed as a weighted sum of individual opacities computed at the partial density of interest. The refined electron density sampling we used allows for an accurate interpolation of our Rosseland opacity values on a $\mathrm{log}[R]$-grid (where $R=\rho /{T}_{6}^{3}$, with ${T}_{6}={10}^{-6}\times T({\rm{K}})$ and ρ expressed in ${\rm{g}}\;{\mathrm{cm}}^{-3}$).

3.1. Equation of State

Because they are prevalent at solar interior temperatures, an exhaustive and accurate description of excited atomic states is crucial to correctly evaluate opacities. In stellar interiors, the number and energy of these atomic states strongly depend on the approach employed to describe the plasma environment effects.

In order to illustrate this point, we focused on iron opacity. Three thermodynamical conditions are considered, $\{\mathrm{log}[T({\rm{K}})],\mathrm{log}[{N}_{{\rm{e}}}({\mathrm{cm}}^{-3})])\}$ = $\{6.35,23\}$, $\{6.75,24.4\}$, and $\{7.20,25.8\}$, for which iron is a major metallic contributor to the mixture opacity (Figure 1). Concerning the Sun, these thermodynamical conditions are similar to those expected near the radiation/convection boundary, the middle of the radiative zone, and the center, respectively. For each case, the OPAS configuration probabilities are replaced by those obtained by solving Saha–Boltzmann equations and using the configuration energies of isolated ions corrected by plasma environment effects. These effects are deduced from either an ion sphere (IS) model (Massacrier & Dubau 1990) or the Stewart–Pyatt (SP) model (Stewart & Pyatt 1966). The isolated ion configuration energies are simply evaluated using a screened hydrogenic model including ℓ-splitting (Faussurier et al. 1997). The average ionization, the lowest bound subshell, the number of retained non-relativistic configurations, and the iron Rosseland opacity are given in Table 1, which also contains the OPAS values. IS and OPAS average ionizations are close to whatever thermodynamical conditions we considered. Except for the ($\mathrm{log}[T({\rm{K}})]=7.20$, $\mathrm{log}[{N}_{{\rm{e}}}({\mathrm{cm}}^{-3})]=25.8$) case, IS ionization standard deviations are lower than the OPAS values, leading to a lower iron Rosseland opacity. Compared to the OPAS values, the SP average ionizations are systematically lower because the plasma environment effect modeling induces smaller configuration energy corrections. Depending on the thermodynamical conditions, the ratios of IS and SP iron Rosseland opacities to the OPAS values equal $0.886,0.910,1.008$ and $1.167,1.065,0.886$, respectively. Mixture Rosseland opacities have been calculated by replacing the OPAS iron spectral opacity by the IS and SP calculations for $(X=0.35,Z=0.02).$ The ${\kappa }_{{\rm{R}}}^{\mathrm{IS}(\mathrm{SP})}/{\kappa }_{{\rm{R}}}^{\mathrm{OPAS}}$ ratio equals $0.990\;(1.018)$, $0.987\;(1.009)$, and $1.007\;(0.993)$ for the three thermodynamical conditions we considered.

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Table 1.  Equation of State of Iron and its Influence on the Rosseland Mean Opacities for Typical Solar Conditions

	Z*	${\sigma }_{{Z}^{*}}$	nℓmax	Conf	${\kappa }_{{\rm{R}}}(\mathrm{Fe})$	${\kappa }_{{\rm{R}}}(\mathrm{Mix})$
	16.29	1.255	6s	44746	1350.9	24.88
A	16.43	1.196	6h	23607	1196.4	24.64
	15.29	1.375	8k	60661	1576.6	25.34
	20.17	1.418	3d	1092	290.8	4.53
B	20.14	1.181	4f	890	264.5	4.47
	19.06	1.586	5g	6836	309.6	4.57
	22.08	1.173	2p	59	120.3	1.38
C	21.98	1.186	2p	51	121.3	1.39
	21.05	0.882	3d	369	106.9	1.37

Note. OPAS, IS (Ionic Sphere), and SP (Stewart Pyatt) values are given on the upper, mid, and lower lines, respectively. Opacity is expressed in ${\mathrm{cm}}^{2}\;{{\rm{g}}}^{-1}$.

Average Ionization (Z^*), Ionization Standard Deviation (${\sigma }_{{Z}^{*}}$), Lowest Bound Subshell (nℓ_max), Number of Selected Non-relativistic Configurations (Conf), Iron Rosseland Opacity (${\kappa }_{{\rm{R}}}(\mathrm{Fe})$), and 22 Chemical Element Mixture (AGS'09 Composition, $X=0.35,Z=0.02$) Rosseland Opacity (${\kappa }_{{\rm{R}}}(\mathrm{Mix})$) for $\{\mathrm{log}[T({\rm{K}})],\mathrm{log}[{N}_{{\rm{e}}}({\mathrm{cm}}^{-3})]\}$ = $\{6.35,23\}$ (A), $\{6.75,24.4\}$ (B), and $\{7.20,25.8\}$ (C)

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We also examined the impact of excited configuration sampling. Based on the IS model, Saha–Boltzmann equations have been solved considering, for each ionic state, multiple excitations from the ground configuration valence shell. The results in Table 1 have been obtained considering up to triple excitations. Downgraded calculations have also been performed where the excited configuration sampling has been restricted to single and double excitations. The resulting iron spectral opacities for $\mathrm{log}[T({\rm{K}})]=6.35$ and $\mathrm{log}[{N}_{{\rm{e}}}({\mathrm{cm}}^{-3})]=23$ are plotted in Figure 2. The contribution of M-shell bound–free absorption from excited configurations is clearly visible near u = 3 (u is the reduced photon energy defined as $u=h\nu /{k}_{{\rm{B}}}T$, where h, ν, and ${k}_{{\rm{B}}}$ are the Planck constant, the frequency, and the Boltzmann constant, respectively). The iron Rosseland opacities equal 906.0 and $1126.8\;{\mathrm{cm}}^{2}\;{{\rm{g}}}^{-1}$ for single and double excitations, respectively. Using these iron spectral opacities, mixture Rosseland opacities ($X=0.35,Z=0.02$) equal 24.603 and 24.641 for single and double excitations, which are very close to the nominal IS calculation (24.642).

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The evaluation of excited configuration energies can be another source of uncertainty. We recall that OPAS uses the notion of superconfiguration characterized by a set a supershells where each of them can be defined by one or several sub-shells up to several spectroscopic shells. For each superconfiguration, a set of one-electron radial wavefunctions is optimized while minimizing the statistical average of the energy levels that the superconfiguration contains. Forcing each spectroscopic shell to be a supershell, we increase the number of initially generated superconfigurations by approximately one order of magnitude, improving the energy calculation accuracy of the configurations contained in a given superconfiguration. Concerning the iron opacity for $\mathrm{log}[T({\rm{K}})]=6.35$ and $\mathrm{log}[{N}_{{\rm{e}}}({\mathrm{cm}}^{-3})]=23$, we found that ${\kappa }_{{\rm{R}}}$ has decreased by $2.2\%$. Concerning the stellar mixture for the same thermodynamical conditions, the Rosseland opacity decrease is less than $0.2\%$ for ($X=0.35,Z=0.02$).

3.2. Detailed Line Accounting

The OPAS bound–bound opacity calculation combines statistical (Unresolved Transition Array; UTA) and DLA descriptions of electric-dipole transition arrays. We note that opacities are calculated in the $[0.1,20]$ reduced photon energy range. For each transition array connecting two non-relativistic configurations, the DLA is applied for $u\in [0.2,13.5]$ only if the number of lines is smaller than 10⁶ and if the line overlap criteria, defined as the ratio of the average Doppler width to the average energy separation between two lines, is lower than 10. These values have been retained to ensure an exhaustive DLA treatment.

In order to examine the impact of a DLA treatment, the mixture opacity calculations have been performed while forcing a statistical treatment for the overall transition arrays. The results obtained using this downgraded OPAS version (dOPAS) have been compared to standard OPAS calculations. Figure 3 shows the relative variation ${\kappa }_{{\rm{R}}}^{\mathrm{dOPAS}}/{\kappa }_{{\rm{R}}}^{\mathrm{OPAS}}-1$ for ($X=0.35,Z=0.02$). If ratio values are always positive in the considered thermodynamical domain, then the impact of the DLA treatment remains moderate. The larger differences can be observed in the range of low $\mathrm{log}[T]$ and low $\mathrm{log}[R]$ values. The largest difference is $7.8\%$ for $\mathrm{log}[T({\rm{K}})]=6.0$, $\mathrm{log}[R]=-2.0$. At low temperatures, the Doppler width involved in the line overlap criteria is small enough to allow a DLA treatment both for the K-shell of low-Z elements and for the L-shell of the heavier elements for which a lot of UTA contain a large number of lines. At a fixed temperature, the spectral distribution of a given resolved transition array is smoothed when the density increases. Electron collision broadening is an efficient broadening process that acts on each line of the transition array. Such a process mainly contributes to fill the spectral gap between lines. The opacity of the stellar mixture results from the accumulation of such smoothed spectral distributions showing an absorption background not much different from that obtained using the systematic statistical treatment for the overall transition arrays. Remembering that the Rosseland opacity is a harmonic average of the monochromatic opacity, the ${\kappa }_{{\rm{R}}}$ values deduced from standard and downgraded OPAS calculations are quite similar when the density increases. Our results confirm those previously obtained for highly diluted and low-temperature astrophysical plasmas (Iglesias & Wilson 1994).

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3.3. Electric Microfield Distribution and Ionic Stark Broadening Modeling

In order to estimate the sensitivity of the opacities with the electric microfields, in Figure 4 we compare the nominal ${\kappa }_{{\rm{R}}}$ OPAS values for ($X=0.70,Z=0.02$) and the results deduced from downgraded computations where spectra are computed using the ($X=0.0,Z=0.02$) microfield distribution and mixed according to the ($X=0.70,Z=0.02$) composition. The difference between the two calculations appears to be quite small and always lower than or equal to zero. The largest discrepancy ($-1.13\%$) is found for ($\mathrm{log}[R]=-1.95$, $\mathrm{log}[T({\rm{K}})]=6.15)$. For these thermodynamical conditions, the average electric field values equal 0.112 and 0.100 atomic units for ($X=0.70,Z=0.02$) and ($X=0.00,Z=0.02$), respectively. These average values are used to evaluate the statistical ionic Stark broadening applied (except for H-like and He-like ions for which a refined treatment is used) to the overall transition arrays. If the mixture Rosseland opacities weakly depend on the electric microfield distribution modeling, then we find that the statistical ionic Stark broadening treatment recently implemented in the OPAS code noticeably increases their values for low-density and low-temperature plasmas. For such media, photoexcitation processes from partially ionized metallic elements significantly contribute to the total opacity (see upper panel on Figure 1). As an example, the ratio of the 22 chemical element mixture Rosseland opacity calculated with and without statistical ionic Stark broadening for ($\mathrm{log}[T({\rm{K}})]=6.35$, $\mathrm{log}[{N}_{{\rm{e}}}({\mathrm{cm}}^{-3})]=23$) and ($X=0.35,Z=0.02$) equals 1.015. The validity of our statistical ionic Stark broadening treatment remains a source of uncertainty that could be quantified by performing systematic detailed Stark broadening calculations. Unfortunately, such calculations are presently impracticable for complex N-electron atomic structures.

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3.4. Interpolation

OPAS Rosseland opacities have been tabulated for $\mathrm{log}[R]$ in the range $[-2,-1]$ (step = 0.05) and for $\mathrm{log}[T({\rm{K}})]$ in the range $[6,7.2]$ (step = 0.025). Hydrogen mass fractions equaling $0.$, 0.2, 0.35, 0.5, 0.7, and 0.8 have been considered. For each X value, metallic element mass fractions equal to 0.01, 0.013, 0.015, 0.017, and 0.02 have been retained. Table 2 illustrates the available data for $(X=0.35,Z=0.02)$. We note that for the overall considered thermodynamical conditions, the cobalt contribution to the mixture Rosseland opacity is negligible. Greater relative Co contributions are found for higher temperatures but are always lower than $0.2\%$.

OPAS Rosseland opacities are compared to OPAL data (http://opalopacity.llnl.gov). The xztrin21.f subroutine package has been used to interpolate OPAL data on the above-defined $(\mathrm{log}[T],\mathrm{log}[R])$ grid. Comparisons have been performed for Z = 0.01 and 0.02 showing the same tendency. Expressed as ${\kappa }_{{\rm{R}}}^{\mathrm{OPAS}}/{\kappa }_{{\rm{R}}}^{\mathrm{OPAL}}-1$, differences are shown in Figure 5 for $(X=0.35,Z=0.02)$. In a large part of the thermodynamical domain considered, the differences appear to be quite small (less than 5%), except in the low-temperature and low-density regime where the OPAS data are up to 11.4% greater than the OPAL values. Assuming that OPAS and OPAL calculations involved similar exhaustive DLA treatments, the use of statistical ionic Stark broadening in the OPAS code could explain such an opacity increase. Differences concerning the sampling and the description of excited atomic states could be also involved.

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We compared OPAS Rosseland opacities to OP data (http://cdsweb.u-strasbg.fr/topbase/topbase.html). Comparisons have been performed for integer values of $\mathrm{log}[{N}_{{\rm{e}}}({\mathrm{cm}}^{-3})]$ in the range $[22,26]$ and for $\mathrm{log}[T({\rm{K}})]$ between 6 and 7.2. Expressed as ${\kappa }_{{\rm{R}}}^{\mathrm{OPAS}}/{\kappa }_{{\rm{R}}}^{\mathrm{OP}}-1$, the differences are shown in Figure 6 for $(X=0.35,Z=0.02)$. The OPAS and OP calculations are in good agreement, except at the lower electron density where OPAS data are up to $12.7\%$. The largest difference is obtained for $\mathrm{log}[{N}_{{\rm{e}}}({\mathrm{cm}}^{-3})]=22$ and $\mathrm{log}[T({\rm{K}})]=6.2$. For these thermodynamical conditions, our calculations indicate that Cl, K, and Ti (three elements which are not included in the OP database) contribute together for 1% to the mixture Rosseland opacity. The analysis of the remaining $11.7\%$ difference is complicated because eight metallic elements (C, N, O, Si, S, Ar, Ca, and Fe) noticeably contribute. Oxygen and iron are the major metallic contributors (20% and 12% to the total opacity, respectively). Focusing on iron, the individual contribution we calculated is 40% higher than the value deduced from OP data. OPAS and OP iron average ionization are very close (16.10 and 16.06, respectively) and the charge state distributions are in good agreement. In the photon energy range where L-shell photoexcitation processes are prevalent, the OP spectrum exhibits deep gaps between the main resonant structures. The OPAS calculation involves an exhaustive sampling of excited atomic states. Bound-bound transitions from these states mainly contribute to reduce the gaps compared to the OP calculation. Furthermore, the use of statistical ionic Stark broadening in OPAS calculations enhances the gap filling efficiency near the L-shell photoionization threshold.

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According to the AGS'09 composition (Asplund et al. 2009), the OPAS opacity model has been used to calculate opacities of stellar plasmas in local thermodynamic equilibrium. Dedicated to solar interior modeling, calculations have been performed for temperatures between $\mathrm{log}[T({\rm{K}})]=6$ and $\mathrm{log}[T({\rm{K}})]=7.2$, and for electron densities between ${10}^{20}$ and ${10}^{26}\;{\mathrm{cm}}^{-3}$. Possible sources of uncertainty in the tabulations have been examined, and the associated precision has been estimated to be approximately 1%. Twenty-two chemical element mixture Rosseland opacities have been tabulated for various hydrogen and metallic element mass fractions. Comparisons with the OPAL and OP data have shown a good agreement, except in the low-temperature and low-density domain where the OPAS results are up to approximately $12.7\%$ larger.

This work has been performed under the auspice of the Agence Nationale de la Recherche (Contract No. ANR-12-BS05-0017 entitled OPACITY). The authors thank Sylvaine Turck-Chièze for having stimulated this work.