SOLAR OPACITY CALCULATIONS USING THE SUPER-TRANSITION-ARRAY METHOD

M. Krief, A. Feigel, and D. Gazit

The Racah Institute of Physics, The Hebrew University, 91904 Jerusalem, Israel; menahem.krief@mail.huji.ac.ilReceived 2016 January 8; accepted 2016 February 15; published 2016 April 7

ABSTRACT

A new opacity model has been developed based on the Super-Transition-Array (STA)method for the calculation ofmonochromatic opacities of plasmas in local thermodynamic equilibrium. The atomic code, named STAR(STA-Revised), is described and used to calculate spectral opacities for a solar model implementing the recentAGSS09 composition. Calculations are carried out throughout the solar radiative zone. The relative contributionsof different chemical elements and atomic processes to the total Rosseland mean opacity are analyzed in detail.Monochromatic opacities and charge-state distributions are compared with the widely used Opacity Project (OP)code, for several elements near the radiation–convection interface. STAR Rosseland opacities for the solar mixtureshow a very good agreement with OP and the OPAL opacity code throughout the radiation zone. Finally, anexplicit STA calculation was performed of the full AGSS09 photospheric mixture, including all heavy metals. Itwas shown that, due to their extremely low abundance, and despite being very good photon absorbers, the heavyelements do not affect the Rosseland opacity.

Key words: atomic data – atomic processes – dense matter – opacity – plasmas – Sun: interior

1. INTRODUCTION

Over the past decade, a new solar problem has emerged assolar photospheric abundances have been improved (Asplundet al. 2006, 2009; Caffau et al. 2011; Grevesse et al. 2015;Scott et al. 2015a, 2015b) and the indicated solar metallicity,which is mainly due to low-Z metallic elements, has beensignificantly revised downward from that previously assumed(Grevesse & Noels 1993; Grevesse & Sauval 1998). Standardsolar models (SSMs), which are the fundamental theoreticaltools to investigate the solar interior, do not reproducehelioseismic measurements (Basu & Antia 2004), such as theconvection zone radius, the surface helium abundance, and thesound speed profile (Serenelli et al. 2009), when using theserevised abundances. This gave rise to the solar compositionproblem, motivating a rapid increase in research effort inthe field.

Rosseland opacity is a key quantity describing the couplingbetween radiation and matter in the solar interior plasma.Metallic elements contribute significantly to the opacity,although they are present only as a few percent of the mixture,since bound–bound and bound–free absorption can become amajor source of opacity for the partially ionized metals. Thus,the amount of metals directly modifies the opacity profile of theSun, and the solar composition problem is strongly related tothe role of opacity in solar models.

Among a variety of proposed explanations, it has beenshown that in order to reproduce the helioseismic measure-ments, changes to the opacity are required to compensate thelower solar metallicity. Specifically, it was shown that a smoothincrease in the opacity from the range of 2%–5% in the centralregions to the range of 12%–15% at the radiation–convectioninterface is required (see Christensen-Dalsgaard et al. 2009;Serenelli et al. 2009; Villante 2010; Villante & Ricci 2010;Villante & Serenelli 2015 and references therein). It is believedthat the opacities of metals in the stellar mixture should berevised upward to compensate for the decreased abundances oflow-Zmetallic elements. In addition, the monochromaticopacity of iron, a major contributer to the solar opacity,wasrecently measured in very similar conditions to those

expected near the boundary of the solar convection zone(Bailey et al. 2015). The measured spectrum appears to belarger than calculations of various state-of-the-art and widelyused atomic models, by about a factor of two in several spectralregions near the L-shell photoabsorption lines. No satisfactoryexplanation for this discrepancy is known.We have recently developed (Krief & Feigel 2015a) an

atomic code, named STAR (STA-Revised), for the calculationof opacities of plasmas in local thermodynamic equilibrium bythe STA method. The STA method (Bar-Shalom et al. 1989) isused to group the large number of transition lines between anenormous number of configurations into large assemblies,called Super-Transition-Arrays (STAs). The moments of theSTAs are calculated analytically using the partition functionalgebra, and are split into smaller STAs until convergence isachieved. Unlike detailed-line-accounting (DLA) methods suchas the Opacity Project (OP, Seaton et al. 1994) and OPAL(Iglesias & Rogers 1996), the STA method is able to handlesituations where the number of relevant transitions is immense.On the other hand, unlike average-atom models (Rozsnyai1972; Shalitin et al. 1984), the STA method reveals theunresolved-transition-array (UTA) spectrum, otherwiseobtained by a very costly, sometimes intractable, detailed-configuration-accounting (DCA) calculation (see Section 3.4).In this work we use STAR to calculate and analyze solar

opacities. The atomic calculations are based on a fullyrelativistic quantum-mechanical theory via the Dirac equation.We analyze in detail the contribution of different chemicalelements in the solar mixture to the total Rosseland meanopacity and examine the role of different atomic processes. Theanalysis is carried out throughout the radiative zone (that is,from the solar core to the radiation–convection boundary).

2. THE SOLAR MODEL

The development of advanced stellar atmosphere modelsresulted in a downward revision of the solar photospheric metalabundances. In this work we use the recent set of chemicalabundances from Asplund et al. (2009, AGSS09) for volatileelements (C, N, O, and Ne) together with their recommended

The Astrophysical Journal, 821:45 (15pp), 2016 April 10 doi:10.3847/0004-637X/821/1/45© 2016. The American Astronomical Society. All rights reserved.

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meteoritic abundances for refractory elements (Mg, Si, S, andFe), where Si is the anchor of both scales. This compilation wasused as an input for evolutionary solar model calculationspresented in Villante et al. (2014), Villante (2010), Serenelliet al. (2009, 2011), and references therein, which we refer to as“the SSM.” The resulting solar profiles1 of various quantitiesare used throughout this work. The SSM temperature anddensity profiles are shown in Figure 1. The solar mixture of theSSM consists of the following 24 elements: H, He, C, N, O,Ne, Na, Mg, Al, Si, P, S, Cl, Ar, K, Ca, Sc, Ti, V, Cr, Mn, Fe,Co, and Ni. The elemental composition profiles are shown inFigure 2. These are determined from the initial abundances andfrom the SSM due to processes such as nuclear burning,diffusion, and gravitational settling.

The opacity used in the SSM was calculated by the OP(Seaton et al. 1994; Badnell et al. 2005). The effect of intrinsicchanges in opacity (that is, with a fixed composition) onobservable quantities, i.e., the convection zone radius, surfacehelium abundance, sound speed profile etc, can be studied bythe linear solar model (LSM) of Villante & Ricci (2010). Thisjustifies the calculation of different opacity models using thethermodynamic conditions obtained by the SSM.

3. THE OPACITY MODEL

In this section we briefly describe various parts of thecomputational model implemented by the STAR atomic code.A more comprehensive description of the STAR model will bereported elsewhere.

3.1. The Mixture Model

The initial step in the calculation of the opacity of a mixtureis the calculation of the effective densities of the differentcomponents (Nikiforov & Uvarov 1969). A mixture oftemperature T, density ρ, and mass fractions mi is describedas a collection of ion-spheres with different volumes, orequivalently, individual effective mass densities ri. The indivi-dual densities are found from the requirement that the chemicalpotentials mi of all species are equal:

m m= , 1i ( )

and from the condition that the volume of the mixture is equalto the sum of the volumes occupied by the individual

components:

å år r=m

m1. 2

ii

i

i

i

( )

The equality of chemical potentials ensures the equality ofelectronic pressures only, rather than the total pressure(including the ionic contribution, see Carson et al. 1968).However, the usual procedure adopted in the present work is toequalize the chemical potentials (Nikiforov & Uvarov1969;Rose1992; Klapisch & Busquet2013), which, if the freeelectrons are treated semiclassically, also ensures the equalityof the (free) electronic densities at the ion-sphere radius.The chemical potential can be calculated using a finite-

temperature Thomas–Fermi model (Feynman et al. 1949) forwhich bound and free electrons are treated semiclasically, oralternatively, with more sophisticated ion-sphere models suchas the relativistic Hartree–Fock–Slater average-atom model forwhich bound electrons are treated quantum-mechanically via

Figure 1. Temperature (red) and density (blue) profiles of the SSM.

Figure 2. Profile of mass fractions for the 24 elements in the SSM mixture.

1 Solar data by Serenelli and Villante can be found online: http://www.ice.csic.es/personal/aldos/Solar_Data.html.

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the Dirac equation (Rozsnyai 1972; Iacob et al. 2006), or theLiberman model for which bound and free electrons are treatedquantum-mechanically (Liberman 1979; Wilson et al. 2006;Pénicaud 2009; Murillo et al. 2013; Ovechkin et al. 2014). Inthis work we used the relativistic Hartree–Fock–Slater average-atom model for the calculation of effective mass densities forthe solar mixture. The solar mixture degeneracy parameterh m= k TB (where kB is the Boltzmann’s constant) is shown inFigure 3. It is evident that the degeneracy increases toward thecenter due to the large increase in matter density, and despitethe (lower) increase in temperature. Effective mass densities ofseveral elements are given in Figure 4.

The monochromatic opacity is evaluated as the sum of fourdifferent contributions involving photon scattering (SC) and thebound–bound (BB), bound–free (BF), and free–free (FF)photoabsorption processes. For each component of the mixture,the cross section for photon absorption (including stimulatedemission) is written as

s ss s s

=+ + + - -

E E

E E E e1 . 3i

E k T

sc

bb bf ff B

( ) ( )[ ( ) ( ) ( )]( ) ( )

The calculation of the cross section of each element is carriedout at the effective density as explained in the previousparagraph. The monochromatic opacity is given by theabsorption coefficient per unit mass:

k s=EN

AE , 4i

A

ii( ) ( ) ( )

where Ai is the atomic mass of component i and NA isAvogadro’s constant. The total monochromatic opacity of themixture is given by the sum of the individual opacities,weighted by the mass fraction:

åk k=E m E . 5i

i i( ) ( ) ( )

Total monochromatic opacities, together with the contributionof individual atomic processes, are shown in Figure 5, foroxygen, magnesium, and iron at the conditions of theradiation–convection boundary. The Rosseland mean opacityis given by

òk k=

¥ R u du

uk T

1, 6

R B0

( )( )

( )

with the Rosseland weight function

p=

-R u

u e

e

15

4 1, 7

u

u4

4

2( )

( )( )

where =u E k TB .We adopt the following simple way to measure fractional

contributions (i.e., the contribution of different elements ordifferent atomic processes) to the total Rosseland mean.Suppose an opacity spectrum is composed of N individualspectral contributions:

åk k==

E E . 8toti

N

i1

( ) ( ) ( )

We define the fractional Rosseland contributions by

dkk k

k=

- -

, 9iRi

Ri

RN

1

( )

where kRi is the Rosseland mean (6) of the cumulative spectra

kå = Eji

j1 ( ), while for the first contributer dk k k= R RN

11 .

3.2. Photon Scattering

In this work we calculate the photon scattering with severalcorrections as summarized in the book by Huebner & Barfield(2014). The photon scattering cross section is written as

s h h d s= ¢ ¢E G u T R T f Z, , , , 10f Tsc ( ) ( ) ( ) ( ) ( )

where Zf is the average number of free electrons and theThomson cross section is

spa= a

8

3, 11T

402 ( )

with α the fine-structure constant and a0 the Bohr radius.¢G u T,( ) corrects the Klein–Nishina cross section due to the

finite temperature and contains corrections for inelasticscattering. It was calculated by Sampson (1959) using arelativistic Maxwell–Boltzmann distribution for nondegenerateelectrons. For nonrelativistic temperatures it is given by the

Figure 3. Profile of the solar mixture degeneracy parameter m k TB .Figure 4. Effective densities for the 10 most abundant elements.

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expansion

⎜ ⎟⎛⎝

⎞⎠

¢ = + ¢ + ¢ + ¢

- + ¢ + ¢ ¢

+ + ¢ ¢ - ¢

G u T T T T

T T uT

T uT uT

, 1 2 515

41

516 103 408

21

2

609

5

2203

70, 12

2 3

2

2 3

( )

( )

( ) ( ) ( )

where ¢ =T k T m cB e2. The factor h ¢R T,( ), which was

calculated by Kilcrease & Magee (2001), contains correctionsdue to the Pauli blocking as a result of partial degeneracy andincludes a relativistic electron dispersion relation. For ¢T 1(which holds in the solar interior) it is given by

⎡⎣⎢

⎤⎦⎥

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥

h h hhh

hh

h

hh

hh

¢ = +¢

-

- -

´ - ¢

R T RT

RI

I

I

IR

I

I

I

IT

,15

81

345

1281

30

23

7

23, 13

NR NR 3 2

1 2

3 2

1 2

NR

3 2

1 2

5 2

3 2

2

( ) ( ) ( )( )( )

( )( )

[ ( )

( )( )

( )( )

( )

where RNR is the analogous nonrelativistic correction due toRose (1995):

hhh

= -RI

I2, 14NR 1 2

1 2( )

( )( )

( )

and the Fermi–Dirac integral is defined by

ò=+

¥

-I x

y dy

e 1. 15k

k

y x0( ) ( )

Finally, h df ,( ) is a correction due to plasma collective effects.We use the formula of Boercker (1987), which is based onelectron correlations at high densities and includes anexchange–correlation correction:

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

h dh

hll

hd

= +

- -

fI

RI

h

, 12

12

. 16D

e

1 2

3 2

NR2

1 2

3 2

( )( )

( )( )

( ) ( )

The ion and electron Debye lengths are given by

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜

⎞⎠⎟

l l= =

k T

Z n e

k T

n e; , 17i

B

f ie

B

e

02 2

1 20

2

1 2

( )

where 0 is the vacuum dielectric constant and e is the unitelectron charge; the total Debye length is

l l l= +

1 1 1, 18

D e i( )

and

⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

dd

d d ddd

d d

= + ++

+ + +

h3

82 2 ln

2

2 28

3, 19

3 2

2

( ) ( )

( )

with:

⎛⎝⎜

⎞⎠⎟

d

l=

c

E

1

2. 20

D

2

( )

We note that for long photon wavelengths where d 1, adirect numerical calculation of Equation (19) can be unstable.In practice, for d > 1000 it is preferable to use an expansion at

Figure 5. Total monochromatic opacity (solid line) and the contributions fromthe individual atomic processes (dashed lines): bound–bound (BB), bound–free(BF), free–free (FF), and scattering (SC), at =R 0.726 (for which=T 174.9 eV, r = -0.16 g cm 3), for oxygen (upper panel), magnesium

(middle panel) and iron (lower panel). The relative contributions of the atomicprocesses to the Rosseland mean are given in the legends, and the effectivemass densities are given above each panel.

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d = ¥:

dd d d

d d

» - + -

+ -

h 17

5

11

5

128

3544

7

232

21. 21

2 3

4 5

( )

( )

3.3. Free–Free

In this work, the FF photoabsorption is calculated via ascreened-hydrogenic approximation with a multiplicativedegeneracy correction. The FF absorption cross section isgiven by

s s f h=E E u g, , 22Kff ff( ) ( ) ( ) ¯ ( )

where the Kramers cross section (Kramers 1923) is

⎛⎝⎜

⎞⎠⎟

s

p p=E

e

m c

m

k T

Z n

E

16

3

2

3, 23K

e

e

B

f e2 2 6

2

1 2 2

3( ) ( )

with ne the number density of free electrons. To allow areasonably accurate and fast calculation of the (thermal)averaged FF Gaunt factor gff¯ , we use a screened-hydrogenicapproximation and a Maxwellian distribution via the tables ofvan Hoof et al. (2014). Degeneracy is incorporated using thecorrection factor (Green 1960; Rose 1993; Faussurier et al.2015)

⎛⎝⎜

⎞⎠⎟f h

ph

=-

++

h

h- -u

I e

e

e,

2 1ln

1

1. 24

u u1 2

( )( )( )

( )

We note that since f depends only on the temperature and thechemical potential, the degeneracy correction is the same for allcomponents in the mixture. The effect of degeneracy atdifferent values of the solar radius is presented in Figure 6. It isevident that the correction is vital in the vicinity of the corebecause the plasma is more degenerate (see also Figure 3).

3.4. Bound–Bound

In principle, the BB photoabsorption spectrum results fromall radiative transitions between all levels from all pairs ofelectronic configurations. Unfortunately, for a hot denseplasma, the number of lines between each pair of configura-tions may be enormous (Gilleron & Pain 2009; Scott & Hansen2010) and statistical methods must be used. The UTA method

(Moszkowski 1962; Bauche-Arnoult et al. 1979; Bauche et al.1988; Krief & Feigel 2015b) treats all levels between each pairof configurations statistically, using analytic expressions for theenergy moments of the transition array. In many cases, thenumber of configurations may also be extremely large and adetailed-configuration-accounting (DCA) calculation is intract-able as well.Let us estimate the number of populated configurations. A

(relativistic) configuration is defined by a set of occupationnumbers qs{ } on relativistic =s nlj( ) orbitals, which are fullsolutions of the Dirac equation. Neglecting electron correla-tions, the probability distribution of qs{ } is binomial (Iacobet al. 2006):

⎛⎝⎜

⎞⎠⎟= - -P q

g

qn n1 , 25s

s

s

ssq

sg qs s s({ }) ( ) ( )

where = +m- -n e1 1sk Ts B( )( ) is the Fermi–Dirac distribu-

tion, = +g j2 1s s is the orbital degeneracy, and s is the orbitalenergy. The variance of the population in each shell is given by

d = á - ñ = -q q q g n n1 , 26s s s s s s2( ) ( ) ( )

so that the occupation numbers of shells whose energies arenear the Fermi–Dirac step m- » k Ts B∣ ∣ fluctuate, while theother shells are either filled or empty. The number of populatedconfigurations C can be estimated from the “width” of themultivariate distribution (25):

d» +q6 1 . 27Cs

s( ) ( )

A calculation of C for various elements in the solar mixture isshown in Figure 7. The sharp jumps are a direct result of thesudden decrease in the number of bound orbitals as theydissolve into the continuum. This is the process of pressureionization, which is more dominant toward the solar core as thedensity increases. It is evident that the number of configura-tions grows exponentially with the atomic number. Near theconvection–radiation boundary, > 10C

8 for atomic numbers>Z 15 and a DCA calculation is very costly, while a full

detailed-line-accounting (DLA) calculation is probablyimpossible.The BB photoabsorption cross section is calculated by the

STA method (Bar-Shalom et al. 1989; Blenski et al. 1997;Hazak & Kurzweil 2012). We follow the notation of Bar-

Figure 6. Free–free degeneracy correction f h- u1 ,( ) at several values of thesolar radius.

Figure 7. Number of populated electronic configurations C for variouselements, across the radiation zone.

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Shalom et al. (1989), for which relativistic =s nlj( ) orbitals aregrouped into supershellss = sÎ ss ( ). Configurations arecollected into large groups called superconfigurations (SCs),defined by a set of occupation numbers sQ{ } over thesupershells and denoted by sX = s

sQ( ) . For each orbitaljump a b and SC, all transitions from levels in Ξ to levelsin X¢ (which is obtained from all configurations in Ξ via theelectron jump) are grouped into a single STA whose first threemoments—the intensity, average energy, and variance—arerespectively

å å=abX

ÎX¢ÎX¢

ÎÎ ¢

I I a, 28C

Ci Cj C

ij ( )

å å= -ababX

ÎX¢ÎX¢

ÎÎ ¢

X

EI

IE E b, 28

CC

i Cj C

ijj i( ) ( )

å åD = - -abab

abX

ÎX¢ÎX¢

ÎÎ ¢

XX

I

IE E E c, 28

CC

i Cj C

ijj i

2 2( ) ( ) ( )

where Ei is the energy of level i and the intensity of line i jis s=I Nij i ij with Ni the population of level i and sij the linecross section. The moments (28a) can be expressed as SCaverages of occupation number polynomials that can be writtenin terms of generalized partition functions and evaluatednumerically using recursion relations (Bar-Shalom et al. 1989;Wilson & Chen 1999; Gilleron & Pain 2004; Wilson et al.2007), known as the partition function algebra. The ionicdistribution

å å=

åX

=ÎX

ss

P P , 29Q

Q QC

Cwith:

( )

where Q is the number of bound electrons and PC is thepopulation of the configuration C, is also calculated via thepartition function algebra. Ion distributions for iron and oxygenare given in Figure 8 and the average ionization for variouschemical elements as a function of the solar radius is given inFigure 9. It is evident that the ionization increases toward thesolar core as a result of the higher temperature and pressureionization.

The BB spectrum contains the contribution of all lineprofiles from each orbital jump from all SCs. In this work weuse the Voigt profile, and the total BB spectrum is written as

åås s g= -a b

ab ab abab

XX X XE I V E E , , . 30bb ( ) ( ) ( )

The Voigt profile V is defined by the convolution of theGaussian and Lorentzian profiles:

òs gps

= ¢ s

-¥

¥- ¢V E dE e, ,

1

231E 2

2 2( ) ( )

pg

g´

- ¢ +E E

1 2

2. 32

2 2( ) ( )( )

The total Gaussian width

s = D +ab ababX X D 332 2( ) ( )

includes the Doppler width abD and the statistical width DabX

resulting from the fluctuations in the occupation numbers of thevarious configurations in Ξ and from the UTA widths (Bauche-Arnoult et al. 1985; Bar-Shalom et al. 1995; Krief & Feigel2015b). gab is the total Lorentz width due to the natural andStark broadening effects. We use the well known semi-empirical formulas for electron impact broadening in the one-perturber approximation by Dimitrijević & Konjević(1980, 1987).In the STA calculation, supershells are split recursively,

giving rise to a larger number of SCs and a larger number ofSTAs in (30). This procedure is repeated until the Rosselandmean opacity (or alternatively, any other criterion) hasconverged. For mid- or high-Z hot dense plasmas, theconvergence is very fast since the spectral lines are broadened

Figure 8. Ion charge distributions for iron (lower panel) and silicon (upperpanel) at several values of the solar radius.

Figure 9. Average ionization for various elements throughout the radia-tion zone.

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and significantly overlap (Bar-Shalom et al.1989). We notethat the supershells are split according to the fluctuations intheir occupation numbers d d= ås sÎQ qs s, which allows a veryrobust and fast convergence.

3.5. Bound–Free

The BB STA formalism is straightforwardly generalized forBF transitions (Bar-Shalom & Oreg 1996) and is presentedbriefly in this subsection. The generalization is obtained bytreating the continuum orbitals via the Fermi–Dirac statisticswhile not including them in the supershell structure. The totalBF cross section contains contributions due to photoionizationfrom all bound orbitals α:

ås = Ga

aE E , 34bf ( ) ( ) ( )

where Ga E( ) is given by the convolution

òG = -a a a¥

E d G E M , 350

( ) ( ) ( ) ( )

over the free orbital energy ò. aG is given by the sum

å- = -a a

XXG E G E , 36( ) ( ) ( )

of all individual STA profiles:

s g- = - -a a a aaX X X XG E A V E E , , , 37( ) ([ ] ) ( )

centered at the energy +aXE and includes a Gaussian width

s = + Daa

aX XD , 382 2( ) ( )

where aD , DaX, and ga are respectively the Doppler, STA, and

Lorentzian widths. The profile intensity is = á ñaaX X XA N q1( ) ,

where XN 1( ) is the total SC population and á ña Xq is the SCaverage of the number of electrons in orbital α. These momentsare easily calculated via the partition function algebra (Oreget al. 1997) analogously to the BB case. aM ( ) describes thetotal coupling of the orbital α to the continuum via photonabsorption:

å= + -aaM M j n2 1 1 , 39

ljlj,( ) ( )( ) ( )

where the summation is over the total and spatial angularmomentum quantum numbers of the free orbital lj( ), and thecoefficient + -j n2 1 1( )( ) is the average number of available“holes” in the free orbital. aM lj, is the total cross section for thetransition a lj( ) and is calculated via the relativisticexpressions of Grant (1970). The sum in (39) is restricted bythe selection rules of the transition.

4. SOLAR OPACITY CALCULATIONS

The opacity at the solar core calculated with a variety ofatomic models was compared by Rose (2001, 2004). Solaropacities calculated with the widely used OPAL (Rogers &Iglesias 1992, 1995; Iglesias & Rogers 1996; Iglesias 2015)and OP (Seaton et al. 1994; Badnell et al. 2005) codes werecompared in detail by Seaton & Badnell (2004). Additionalcalculations and comparisons with the LEDCOP and ATOMICcodes can be found in Neuforge-Verheecke et al. (2001) andColgan et al. (2013, 2016). Average atom solar opacitycalculations were presented by Rozsnyai (2001) and Wang

et al. (2004). A recent analysis of the solar opacity was carriedout by Blancard et al. (2012) and Mondet et al. (2015), usingthe OPAS DCA model.In this section we analyze the solar opacity throughout the

radiative zone, calculated by STAR. The resulting profiles ofRosseland opacity and mean free path rk=l 1R R are shown inFigure 10. We note that even though the solar mixture is moreopaque at larger radii, the mean free path is larger as well, dueto the (approximately) exponential decrease in density (seeFigures 1 and 4).

4.1. Relative Contribution of Different Elements Alongthe Radiation Zone

The total monochromatic opacity and the monochromaticcontributions of various chemical elements at three differentvalues of the solar radius are shown in Figure 11, together withthe Rosseland weight function (7). It is evident from the figurethat, as is well known, metals contribute significantly to theopacity despite their low abundance, because some of them arenot fully ionized and strong BB transition lines and BF edgesappear near the Rosseland peak, whose location varies as» k T3.83 B . In Figure 12 the contribution of various elements tothe Rosseland mean opacity along the radiation zone ispresented. These results are in agreement with Figure 10 ofVillante (2010) and Figure 2 of Blancard et al. (2012). We notethat the maxima and minima are caused by the metalliccontributions as a direct result of the variations (mainly due tothe varying temperature) in the positions of lines and edgesrelative to the position of the Rosseland peak. For example, atthe solar core or near the convection–radiation boundary, theiron K- and L-shell features are, respectively, near theRosseland peak, while at =R 0.3 they are both well outsidethe Rosseland distribution (as seen in Figure 11). This results ina peak near the core, a minimum near =R 0.3, and a peak nearthe radiation–convection interface, in the iron opacity con-tribution profile, as shown in Figure 12. A complete plot oforbital energies relative to k TB for the s3 , s2 , and s1 orbitalsand for various elements across the radiation zone is shown inFigure 13. The relevance of each element due to each of itsorbitals is inferred from the proximity to the Rosseland peak. Itis also evident that orbital energies become less negativetoward the core since the screening of the nucleus is increasingdue to the increase in density. As a result, some orbitals can bepressure-ionized into the continuum at a finite radius. Indeed,

Figure 10. Rosseland mean opacity (red curve) and mean free path (blue curve)calculated by the STAR atomic code, for the solar mixture across theradiation zone.

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Figure 11. Total spectral opacity k E( ) (green solid curve), the elemental spectral contributions km Ei i ( ), and the Rosseland weight (red solid curve) at three differentvalues of the solar radius. Only elements that contribute more than 1% to the Rosseland mean are shown (see Figure 12). The solar radius and temperature areindicated in each panel.

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as seen in Figure 13, the s1 orbital of hydrogen and helium, thes2 orbital of C, N, O, Ne, Mg, Al, and Si, and the s3 orbital ofall elements in the solar mixture pressure-ionize at a finiteradius.

4.2. Relative Contributions of Different Atomic ProcessesAlong the Radiation Zone

The contribution of the BB and BF processes to theRosseland opacity depends on the positions of photoabsorptionlines and edges relative to the position of the Rosseland peakand on the population of the various bound states. Relativecontributions of the four different atomic processes—SC, FF,BF, and BB—to the elemental Rosseland opacity of variouselements throughout the radiation zone are presented inFigure 14. The results are in agreement with Figure 2 ofBlancard et al. (2012). The behavior of the BB and BFcontributions in Figure 14 can be qualitatively understood fromFigures 13 and 11. For hydrogen and helium, all orbitals arepressure-ionized up to some finite radius ( »R 0.65 and

»R 0.38, respectively) and the BF contribution is zero up tothese threshold radii. In addition, orbitals with principalquantum number >n 1 are pressure-ionized throughout theradiation zone so that there is no BB contribution for hydrogenand helium. On the other hand, for all other elements theK-shell exists throughout the radiation zone and the BFcontribution is always non-zero. However, several elementshave a finite threshold radius for the existence of the L-shellorbitals and therefore a finite threshold radius for the BBcontribution (see Figures 14(c)–(k)). This threshold radius islower for heavier elements, for which the L-shell orbitals arecloser to the heavier nucleus and a larger density is required forpressure ionization. For the heaviest elements (Figures 14(l)–(o)), the L-shell exists throughout the radiation zone, andtherefore they always have a non-zero BB contribution. As wehave mentioned, as the temperature decreases outward, BBlines move with respect to the Rosseland peak (Figure 13).K-shell lines of certain light elements only approach theRosseland peak from below, explaining the monotonicbehavior of their BB contributions (Figures 14(c)–(f)), whilethe K-shell lines of heavier elements cross the Rosseland peakand have a local maximum in their BB contribution (Figures 14(g)–(j)). For heavier elements, after their K-shell lines pass theRosseland peak, at large enough radii, their L-shell lines alsobegin to approach it, giving rise to a local maximum followedby a local minimum in the BB contribution (Figures 14(k)–(m)). Finally, we note that the iron and nickel K-shell lines atR=0 are already past the Rosseland peak, while their L-shelllines cross it near the radiation–convection boundary, givingrise to a local minimum followed by a local maximum (Figures14(n) and (o)).The total contribution of the different atomic processes (due

to all elements combined) to the SSM mixture Rosselandopacity is shown in Figure 15.

4.3. Comparison with the OP

We compare STAR and OP monochromatic opacities andion charge-state distributions for O, Ne, Mg, and Fe, whichhave a major opacity contribution near the radiation–

Figure 12. Relative contributions of the 17 elements with the largestcontributions to the total solar Rosseland mean opacity, across theradiation zone.

Figure 13. Orbital energies divided by k TB for the s3 (upper panel), s2 (middlepanel), and s1 (lower panel) orbitals and for various elements, across theradiation zone. The Rosseland peak is located on the =y 3.83 dashed–dotted line.

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convection interface. Comparisons are made at =T 192.92 eVand =n 10e

23 cm–3available in the TOPbase database (Cuntoet al. 1994, p. 53), which is used to generate OP data. The samecomparisons were made by Blancard et al. (2012) for Fe and

Mg with OPAS and by Colgan et al. (2013) for Fe and O usingthe LEDCOP and ATOMIC opacity codes. The ion distribu-tions are given in Figures 16–19. The OPAS chargedistributions given in Blancard et al. (2012) for Mg and Fe

Figure 14. Relative contributions of scattering (red), free–free (blue), bound–free (magenta), and bound–bound (black) processes to the opacity of several elements (asspecified under each plot), across the radiation zone.

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are also shown in Figures 18 and 19. For O and Ne, meanionizations Z of OP and STAR are in excellent agreement,though the charge distributions differ slightly. We note thatlarger differences appear for Fe, and STAR populations aremuch larger for the lower charge states ( <Q 16). In theselower ionic states, ground and excited configurations must havea non-empty M-shell. This tendency is similar to that shown byOPAS, though the STAR populations are somewhat larger thanOPAS at the lower charge states.

A comparison between OP and STAR O, Ne, Mg, and Femonochromatic opacities is given in Figures 20–23, respec-tively. The Rosseland means are also given in the figures. It isevident that OP K-shell lines that are shown for O, Ne, and Mgare significantly broader than the STAR lines. The broad OPK-shell wings that lie near the Rosseland peak contributesignificantly to the Rosseland mean, as seen in Figures 20–22.Such differences were also observed by Blancard et al. (2012)for Mg and by Colgan et al. (2013) for O. Both STAR and OPlinewidth calculations include electron impact broadening inthe one-perturber approximation. However, OP uses collisionalrates obtained from comprehensive quantum calculations thatinclude unitarization (Seaton 1987; Burke 2011), while STAR

uses the widely used semi-empirical formulas of Dimitrijević &Konjević (1987). A comparison of the iron monochromaticopacity is shown in Figure 23. It is evident that the STARopacity is larger than OP near the M-shell BB and BF regions,which contribute significantly to the Rosseland mean. Similardifferences were also found by Blancard et al. (2012) andColgan et al. (2016). This larger M-shell contribution isexplained by the larger STAR population of the lower ionicstates (Figure 19), for which ground and excited configurationshave a non-empty M-shell. The STA method takes into accountthe huge number of excited configurations in these lower ionstates by the robust superconfiguration approach.A comparison of STAR and OPAS with OP Rosseland mean

opacities at =T 192.92 eV and = -n 10 cme23 3 for the 17

elements available in the TOPbase database is given inFigure 24. (The OPAS data were taken from Blancard et al.

Figure 15. Relative contributions of the different atomic processes to the totalSSM mixture opacity, across the radiation zone.

Figure 16. Ion charge distributions for oxygen at =T 192.92 eV and= -n 10 cme

23 3 calculated by OP (red solid line) and STAR (blue dashedline). The mean ionizations are given in the legend.

Figure 17. Ion charge distributions for neon at =T 192.92 eV and= -n 10 cme

23 3 calculated by OP (red solid line) and STAR (blue dashedline). The mean ionizations are given in the legend.

Figure 18. Ion charge distributions for magnesium at =T 192.92 eV and= -n 10 cme

23 3 calculated by OP (red solid line), STAR (blue dashed line),and OPAS (black dashed–dotted line). The mean ionizations are given in thelegend.

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Figure 19. Ion charge distributions for iron at =T 192.92 eV and= -n 10 cme

23 3 calculated by OP (red solid line), STAR (blue dashed line),and OPAS (black dashed–dotted line). The mean ionizations are given in thelegend.

Figure 20. Comparison of STAR (blue line) and OP (red line) monochromaticopacity calculations for oxygen at =T 192.92 eV and = -n 10 cme

23 3. TheRosseland mean opacities are given in the legend.

Figure 21. Same as Figure 20, but for neon at =T 192.92 eVand = -n 10 cme

23 3.

Figure 22. Same as Figure 20, but for magnesium at =T 192.92 eVand = -n 10 cme

23 3.

Figure 23. Same as Figure 20, but for iron at =T 192.92 eVand = -n 10 cme

23 3.

Figure 24. Relative difference k k - 1R ROP between OP and STAR (blue dots)

and between OP and OPAS (red dots, taken from Blancard et al. 2012), for allelements available in the TOPbase database at =T 192.92 eVand = -n 10 cme

23 3.

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2012.) For Ne, Na, Mg, Al, and Si, the OP opacity issignificantly higher than STAR due to the wider K-shell lines.On the other hand, for Cr, Mn, Fe, and Ni, STAR opacity issignificantly larger than OP due to higher population of non-empty M-shell configurations. It is also evident that the STARdifferences are in relatively good agreement with those ofOPAS except for carbon, for which the STAR agrees betterwith OP.

A comparison of the solar opacity profile throughout theradiation zone, between calculations via STAR and OPAL(given by Villante et al. 2014), with the opacity profilecalculated by OP (Seaton et al. 1994; Badnell et al. 2005) ispresented in Figure 25. It can be seen that the STAR and OPcalculations agree within 6% throughout the radiation zone.The opacity of the solar mixture is quite insensitive to theindividual element opacities, so that, as was shown by Blancardet al. (2012), despite the large differences between severalelemental opacities (Figure 24), the total STAR opacity of themixture is in good agreement with OP. This is possible sinceSTAR opacities are significantly lower than OP for Ne, Na,Mg, Al, and Si, while they are significantly higher for Cr, Mn,Fe, and Ni.

4.4. The Effect of Heavy Metals

Metals heavier than nickel have a very low abundance in thesolar mixture, and therefore are usually neglected in SSMs.However, in solar conditions, heavy metals may have a largenumber of bound electrons and therefore their spectra usuallycontain a huge number of strongly absorbing lines, which mayin principle affect the solar opacity. Due to the huge number oflines, a tractable calculation of the spectra of hot high-Zelements is only possible using the STA or average-atommethods. Solar opacities including heavier elements from thecomposition of Grevesse & Noels (1993) were calculated andanalyzed by Iglesias et al. (1995) using the STA program ofBar-Shalom et al. (1989). It was shown that even though theheavy elements are very strong photon absorbers, their verylow abundance leads to a negligible effect on the Rosselandmean opacity.

Here we consider the Rosseland mean opacity of the solarmixture including all elements in the recent AGSS09 (Asplund

Figure 25. Relative difference k k - 1R ROP between OP and OPAL (black

curve) and between OP and STAR (blue curve), across the radiation zone.

Table 1Specification of the Mass and Number Fractions of the AGSS09

Photospheric Admixture.

Z Element Mass Number OpacityFraction Fraction Fraction

1 H -7.36 01( ) -9.21 01( ) -4.02 02( )2 He -2.51 01( ) -7.84 02( ) -1.83 02( )3 Li -5.73 11( ) -1.03 11( ) -1.02 11( )4 Be -1.59 10( ) -2.21 11( ) -9.23 11( )5 B -3.99 09( ) -4.61 10( ) -8.07 09( )6 C -2.38 03( ) -2.48 04( ) -1.70 02( )7 N -6.97 04( ) -6.22 05( ) -1.59 02( )8 O -5.77 03( ) -4.51 04( ) -3.18 01( )9 F -5.08 07( ) -3.34 08( ) -5.33 05( )10 Ne -1.26 03( ) -7.84 05( ) -1.84 01( )11 Na -2.94 05( ) -1.60 06( ) -4.89 03( )12 Mg -7.12 04( ) -3.67 05( ) -7.31 02( )13 Al -5.60 05( ) -2.59 06( ) -4.26 03( )14 Si -6.69 04( ) -2.98 05( ) -2.38 02( )15 P -5.86 06( ) -2.37 07( ) -2.25 04( )16 S -3.11 04( ) -1.21 05( ) -1.22 02( )17 Cl -8.25 06( ) -2.91 07( ) -4.99 04( )18 Ar -7.39 05( ) -2.31 06( ) -5.93 03( )19 K -3.08 06( ) -9.87 08( ) -3.58 04( )20 Ca -6.45 05( ) -2.01 06( ) -8.44 03( )21 Sc -4.67 08( ) -1.30 09( ) -5.63 06( )22 Ti -3.14 06( ) -8.21 08( ) -4.17 04( )23 V -3.19 07( ) -7.84 09( ) -5.31 05( )24 Cr -1.67 05( ) -4.02 07( ) -3.55 03( )25 Mn -1.09 05( ) -2.48 07( ) -2.52 03( )26 Fe -1.30 03( ) -2.91 05( ) -2.49 01( )27 Co -4.24 06( ) -9.00 08( ) -7.17 04( )28 Ni -7.17 05( ) -1.53 06( ) -1.52 02( )29 Cu -7.24 07( ) -1.43 08( ) -1.67 04( )30 Zn -1.75 06( ) -3.34 08( ) -4.06 04( )31 Ga -5.63 08( ) -1.01 09( ) -1.22 05( )32 Ge -2.39 07( ) -4.11 09( ) -3.70 05( )36 Kr -1.10 07( ) -1.64 09( ) -2.10 05( )37 Rb -2.08 08( ) -3.05 10( ) -3.84 06( )38 Sr -4.78 08( ) -6.83 10( ) -1.01 05( )39 Y -1.06 08( ) -1.49 10( ) -2.24 06( )40 Zr -2.55 08( ) -3.50 10( ) -5.32 06( )41 Nb -1.97 09( ) -2.66 11( ) -4.14 07( )42 Mo -5.36 09( ) -6.98 11( ) -1.10 06( )44 Ru -4.18 09( ) -5.18 11( ) -8.36 07( )45 Rh -6.16 10( ) -7.48 12( ) -1.22 07( )46 Pd -2.91 09( ) -3.42 11( ) -5.70 07( )47 Ag -6.91 10( ) -8.02 12( ) -1.40 07( )49 In -5.33 10( ) -5.81 12( ) -1.17 07( )50 Sn -9.58 09( ) -1.01 10( ) -2.17 06( )54 Xe -1.68 08( ) -1.60 10( ) -3.43 06( )56 Ba -1.53 08( ) -1.39 10( ) -3.24 06( )57 La -1.29 09( ) -1.16 11( ) -2.99 07( )58 Ce -3.92 09( ) -3.50 11( ) -9.90 07( )59 Pr -5.44 10( ) -4.83 12( ) -1.40 07( )60 Nd -2.79 09( ) -2.42 11( ) -7.07 07( )62 Sm -1.01 09( ) -8.40 12( ) -2.63 07( )63 Eu -3.70 10( ) -3.05 12( ) -8.95 08( )64 Gd -1.36 09( ) -1.08 11( ) -2.75 07( )65 Tb -2.33 10( ) -1.84 12( ) -5.22 08( )66 Dy -1.51 09( ) -1.16 11( ) -3.69 07( )67 Ho -3.67 10( ) -2.78 12( ) -8.60 08( )68 Er -1.02 09( ) -7.66 12( ) -2.21 07( )69 Tm -1.57 10( ) -1.16 12( ) -3.38 08( )

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et al. 2009) photospheric composition. The calculation is doneat =T 176.3 eV and r = -0.16 g cm 3, which exist nearby theradiation–convection interface. In Table 1 the mass and numberfractions of all elements in the AGSS09 mixture are given,together with the elemental contributions to the Rosselandmean opacity of the mixture. The table is also visualized inFigure 26. It is evident that the opacity fractions of heavymetals are larger by more than four (two) orders of magnitudethan their number (mass) fractions, indicating that heavy metalsare indeed good absorbers. However, as was also shown byIglesias et al. (1995), it is seen that due to their extremely lowabundance, the effect of the heavy metals on the totalRosseland mean opacity is completely negligible. Thecontributions of the different atomic processes for the variouselements are given in Figure 27. We note that the local maximain the BB contribution represent the K-, L-, and M-shell linescrossing the Rosseland peak (here, the crossing occurs due tothe dependence of line energies on the atomic number). Theestimated number of populated configurations (27) of thedifferent elements is given in Figure 28. It is evident thataccording to the estimate (27), for mid- and high-Z metals, acomplete DCA (and, of course, DLA) calculation is absolutelyintractable, and STA or average-atom (Rozsnyai 1972; Shalitinet al. 1984) methods must be used.

5. CONCLUSION

A new opacity code, STAR, implementing the STA methodof Bar-Shalom et al. (1989), was presented. It was used tocalculate and analyze opacities throughout the radiation zonefor a solar model implementing the recent AGSS09 composi-tion. The relative contributions of various elements and atomicprocesses to the Rosseland mean opacity were compared indetail. STAR and OP monochromatic opacities and charge-state distributions were compared for several elements near theradiation–convection interface. For the heavier elements, suchas iron, STAR Rosseland opacity is significantly larger thanOP, which is explained by the larger population of lowerionization states. For the lighter elements, such as magnesium,OP Rosseland opacity is significantly larger than STAR, due toa much larger line broadening of the K-shell lines shown byOP. Very similar results were reported by Blancard et al.(2012) using the OPAS code. STAR Rosseland opacities forthe solar mixture were calculated throughout the radiation zoneand a good agreement with OP and OPAL was achieved.Finally, an explicit STAR calculation of the full AGSS09photospheric mixture, including all heavy metals, wasperformed. It was shown that due to their extremely lowabundance, and despite being very good photon absorbers, theheavy elements do not affect the Rosseland opacity, inagreement with Iglesias et al. (1995).

Table 1(Continued)

Z Element Mass Number OpacityFraction Fraction Fraction

70 Yb -8.81 10( ) -6.37 12( ) -1.93 07( )71 Lu -1.62 10( ) -1.16 12( ) -3.50 08( )72 Hf -9.30 10( ) -6.52 12( ) -1.86 07( )74 W -9.58 10( ) -6.52 12( ) -1.98 07( )76 Os -3.52 09( ) -2.31 11( ) -7.21 07( )77 Ir -3.39 09( ) -2.21 11( ) -6.90 07( )79 Au -1.21 09( ) -7.66 12( ) -2.46 07( )81 Tl -1.19 09( ) -7.31 12( ) -2.37 07( )82 Pb -8.58 09( ) -5.18 11( ) -1.69 06( )90 Th -1.79 10( ) -9.64 13( ) -3.54 08( )

Note. Also listed are the elemental contributions to the Rosseland opacity,calculated by STAR at T = 176.3 eV, ρ = 0.1623 g cm–3, that are found nearthe radiation–convection interface. a(–b) represents a×10–b.

Figure 26. Visualization of Table 1: mass fractions (red), number fractions(blue), and Rosseland opacity fractions (magenta) for the elements in the fullAGSS09 admixture. The mixture opacity was calculated near the radiation–convection boundary at r= = -T 176.3 eV, 0.1623 g cm 3.

Figure 27. Relative contributions to the opacity of the different atomicprocesses for each of the elements in the full AGSS09 admixture, calculatedat r= = -T 176.3 eV, 0.162 g cm 3.

Figure 28. Number of populated electronic configurations C for the elementsin the full AGSS09 admixture near the radiation–convection boundary,=T 176.3 eV and r = -0.1623 g cm 3.

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The authors thank Aldo Serenelli and Francesco Villante forproviding various solar profiles and for useful suggestions andcomments.

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