A New Generation of Cool White Dwarf Atmosphere Models. I

Accepted for publication in The Astrophysical Journal Preprint typeset using LATEX style emulateapj v. 08/22/09

A NEW GENERATION OF COOL WHITE DWARF ATMOSPHERE MODELS. I. THEORETICAL FRAMEWORK AND APPLICATIONS TO DZ STARS S. Blouin1, P. Dufour1, and N.F. Allard2,3 Accepted for publication in The Astrophysical Journal

ABSTRACT The photospheres of the coolest helium-atmosphere white dwarfs are characterized by fluid-like densities. Under those conditions, standard approximations used in model atmosphere codes are no longer appropriate. Unfortunately, the majority of cool He-rich white dwarfs show no spectral features, giving us no opportunities to put more elaborate models to the test. In the few cases where spectral features are observed (such as in cool DQ or DZ stars), current models completely fail to reproduce the spectroscopic data, signaling shortcomings in our theoretical framework. In order to fully trust parameters derived solely from the energy distribution, it is thus important to at least succeed in reproducing the spectra of the few coolest stars exhibiting spectral features, especially since such stars possess even less extreme physical conditions due to the presence of heavy elements. In this paper, we revise every building block of our model atmosphere code in order to eliminate low-density approximations. Our updated white dwarf atmosphere code incorporates state-of-the-art constitutive physics suitable for the conditions found in cool helium-rich stars (DC and DZ white dwarfs). This includes new high-density metal line profiles, nonideal continuum opacities, an accurate equation of state and a detailed description of the ionization equilibrium. In particular, we present new ab initio calculations to assess the ionization equilibrium of heavy elements (C, Ca, Fe, Mg and Na) in a dense helium medium and show how our improved models allow us to achieve better spectral fits for two cool DZ stars, Ross 640 and LP 658-2. Subject headings: equation of state — opacity — stars: atmospheres — stars: individual (LP 658-2, Ross 640) – white dwarfs

1. INTRODUCTION et al. 2005; Homeier et al. 2005, 2007, LP 658-2, Dufour Pure helium-rich white dwarfs do not show any spec- et al. 2007; Wolff et al. 2002). tral lines when T 10000 K. The same occurs for For all these stars, the discrepancies between mod- eff . els and observations can be related to nonideal high- Teff . 5000 K in the case of pure hydrogen-rich atmospheres. Together, these featureless white dwarfs are density effects arising at the photosphere since for cool known as DC stars. One is thus forced to rely solely (Teff < 6000 K) helium-rich white dwarfs, densities reach fluid-like values. At a Rosseland optical depth τR = 2/3, on the shape of the spectral energy distribution to de- −3 duce the chemical composition and effective temperature density can be as high as 1 g cm (Bergeron et al. 1995; of these white dwarfs (Bergeron et al. 1997, 2001). Al- Kowalski 2010b), which corresponds to a fluid where the though most cool white dwarfs have featureless spectra, separation between atoms is roughly equivalent to the some cool helium-rich white dwarfs do show significant dimension of atoms themselves. Clearly, under such con- spectral features that can be exploited to retrieve additions, interactions between species are no longer negli- ditional information on the physical conditions encoun- gible and the ideal gas approximation must be discarded. tered in their atmospheres. Some contain enough hy- The nonideal effects arising from this high density have remained mostly unnoticed for DC stars, since a feature- drogen to show strong H2-He collision-induced absorp- less spectrum provides little opportunity to test the action (CIA) features, some show C2 Swan bands (DQ and DQpec stars) and others show metal lines (DZ curacy of atmosphere models. In contrast, cool helium- stars). Interestingly, in all cases, models fail to repro- rich stars with spectral features (i.e., DQpec, DZ and arXiv:1807.06616v1 [astro-ph.SR] 17 Jul 2018 duce these spectra. For instance, the CIA is inadequately those with CIA features) provide a real challenge to at- modeled (e.g., LHS 3250, SDSS J123812.85+350249.1, mosphere models and an opportunity to test our under- SDSS J125106.11+440303.0, Gianninas et al. 2015), the standing of the chemistry and physics of warm dense helium. C2 bands are distorted (e.g., LHS 290, Kowalski 2010a) and the metal absorption lines often do not have the right In this series of papers, we present and apply our new strength or the right shape (e.g., WD 2356-209, Bergeron generation of atmosphere models for cool white dwarf stars. In the first paper of the series, we focus on im- 1 Département de Physique, Université de Montréal, Mon- proving our modeling of cool DZ stars. Note that ob- tréal, QC H3C 3J7, Canada; [email protected], taining better fits of these objects is far more than a [email protected]. mere aesthetic whim. Indeed, because they show spec- 2 GEPI, Observatoire de Paris, Université PSL, CNRS, UMR 8111, 61 avenue de l’Observatoire, 75014 Paris, France. tral lines, cool DZ stars represent a unique opportunity 3 Sorbonne Université, CNRS, UMR 7095, Institut to probe the physics and chemistry of cool helium-rich d’Astrophysique de Paris, 98bis boulevard Arago, 75014 Paris, atmospheres. In a way, they allow us to test the models France. used for DC stars. Once we will have proven that our 2 Blouin, Dufour & Allard new models are able to reproduce the rich and complex spectra of cool DZ stars, we will be confident that the TABLE 1 Metal line profiles computed using the constitutive physics is accurate and that the models can unified line shape theory described in reliably be used to measure the atmospheric parameters Allard et al. (1999). of all DC stars in general. This paper describes our new model atmosphere code Lines Source that includes all nonideal effects relevant for the mod- Ca I 4226 Å Allard, priv. comm. elling of the atmospheres of cool DZ and DC stars. This Ca II H & K Allard & Alekseev (2014) updated atmosphere code is based on the one described Mg I 2852 Å Allard et al. (in prep.) in Dufour et al. (2007). Building on other published Mg II 2795/2802 Å Allard et al. (2016a) works, as well as on our own new calculations, we have Mgb triplet Allard et al. (2016b) considerably improved the constitutive physics in our Na I D doublet Allard et al. (2014) code. Section 2 describes the additions made to correctly calculate radiative opacities and, in Section 3, we discuss broadening (Koester priv. comm.; Walkup et al. 1984). the improvements related to the equation of state and the Note that the exact treatment of these secondary transi- chemical equilibrium. Among the new physics added to tions has a limited impact on our atmospheric determi- the chemical equilibrium calculations, we used ab initio nations. techniques to implement a state-of-the-art description of We show in Figure 1 a comparison of line profiles the chemical equilibrium of heavy elements (C, Ca, Fe, calculated using the theory of Allard et al. (1999) to Mg and Na) in the dense atmosphere of cool DZ stars. those found assuming a Lorentzian profile, for temper- These calculations are detailed at length in Section 4. In ature and density conditions representative of the photo- Section 5, we present two applications that show how the sphere of cool DZ stars. Clearly, under such conditions, improvements included in our models translate in terms the Lorentzian function fails to provide a satisfactory de- of spectroscopic fits. Finally, in Section 6, we summa- scription of the line profiles. It underestimates the strong rize our results and outline the upcoming papers of this broadening observed in the more accurate line profiles series. and does not take into account the distortion and shift observed for many transitions. 2. RADIATIVE OPACITIES In this Section, we describe the additions brought to 2.2. Collision-induced absorption the code of Dufour et al. (2007) regarding the calculation The calculation of the H2-He CIA includes the high- of radiative opacities. This includes improved line pro- density distortion effects described in Blouin et al. files, high-density CIA distortion and continuum opaci- (2017). This pressure distortion effect alters the infrared ties corrected for collective interactions. energy distribution of cool DZ stars with hydrogen in their atmosphere and a photospheric density greater than −3 22 −3 2.1. Line profiles 0.1 g cm (nHe = 1.5 10 cm ). Moreover, we have ≈also included the He-He-He× CIA using the analytical fits In the atmosphere of cool DZ stars, the wings of heavy given in Kowalski (2014). element absorption lines are severely broadened by interactions with neutral helium. Hence, Lorentzian pro- 2.3. Rayleigh scattering files poorly reproduce observed spectral features. It is thus an absolute necessity to implement the unified line In a dense helium medium, collective interactions be- shape theory described in Allard et al. (1999) to treat tween atoms lead to a reduction of the Rayleigh scat- such line profiles. We implemented this formalism for tering cross section (Iglesias et al. 2002). For the wave- the strongest transitions found in cool DZ white dwarfs length domain relevant for white dwarf modeling (i.e., in (see Table 1). In particular, the line profiles described the low-frequency limit), the reduced cross section can in Allard & Alekseev (2014), Allard et al. (2014), Al- be expressed as (Rohrmann 2018; Kowalski 2006a) lard et al. (2016a), Allard et al. (2016b) and Allard et σ (ω) = S(0)σ0 (ω), (1) al. (in prep.) are used to compute the wings and a con- Rayleigh Rayleigh ventional Lorentzian profile is assumed for the core of where σ0 (ω) is the ideal gas result (e.g., Dalgarno spectral lines, where the density is low enough for this Rayleigh 1962) and S is the structure factor of the fluid at a approximation to hold. To connect the two profiles, we (0) wavenumber k . Therefore, to take into account the use a hyperbolic tangent function, which allows a smooth = 0 reduction of the Rayleigh scattering, we simply need to transition. It should also be noted that our Ca I 4226 Å know S , which is a function of the temperature and profile is still preliminary, as we do not yet have access (0) the density of the helium fluid. To compute S , we use to the high-quality ab initio potentials required for the (0) the analytical fit to the Monte Carlo results of Rohrmann computation of this particular line profile. To make up (2018). for this lack, we computed our own ab initio potentials through open-shell configuration-interaction singles cal- 2.4. He− free-free absorption culations with the ROCIS module of the ORCA quantum chemistry package4 (Neese 2012). Iglesias et al. (2002) also showed that the free-free ab- For transitions not listed in Table 1, our code assumes sorption cross section of the negative helium ion is re- a simple Lorentzian function or quasistatic van der Waals duced in a dense helium medium. Given that it is the dominant source of opacity in DZ stars, it is important 4 https://orcaforum.cec.mpg.de to take this reduction into account. The corrected cross A New Cool WD Atmosphere Code 3

Mg ii 2795/2802 A˚ Mg i 2852 A˚ Mgb triplet 2.5 0.25 1.5

2.0 0.20

1.0 1.5 0.15

1.0 0.10 0.5

0.5 0.05

) 0.0 0.0 0.00 2 2600 2700 2800 2900 3000 2600 2700 2800 2900 3000 4200 4600 5000 5400 5800 cm 15

− Ca i 4226 A˚ Ca ii H&K Na i D doublet

(10 1.5 1.2 1.2 σ

1.0 1.0

1.0 0.8 0.8

0.6 0.6

0.5 0.4 0.4

0.2 0.2

0.0 0.0 0.0 3900 4100 4300 4500 3700 3900 4100 4300 5000 5500 6000 6500 7000 λ (A)˚

Fig. 1.— Absorption cross section of metal spectral lines. The black lines correspond to the Lorenztian profiles and the red ones are the profiles obtained with the unified line shape theory of Allard et al. (1999). These line profiles were computed assuming T = 6000 K and 22 3 nHe = 10 cm− . Note that the improved line profile for Ca I 4226 Å relies on approximate potentials (see text). section for He− free-free absorption is given by (Iglesias by Equations 3.5 and 3.6 of Iglesias et al. (2002). From et al. 2002) these equations, it follows that two external inputs are 0 σ (ω) = δ (ω)σ (ω), (2) needed to compute δff (ω): (1) the structure factor S(k), ff ff ff p 0 and (2) the index of refraction of helium n(ω) = (ω). where σff (ω) is the ideal gas result (e.g., John 1994). The details regarding the calculation of the structure fac- δff (ω) can be computed as (Iglesias et al. 2002) tor are given below, while our evaluation of the index of R ∞ refraction is described in Section 2.5. 0 I(k)dk δff (ω) = R ∞ , (3) To compute S(k), we rely on the classical fluid the- 0 I0(k)dk ory and the Ornstein-Zernike (OZ) equation. To solve the OZ equation, we use the Percus-Yevick closure rela- where S(k) tion (Percus & Yevick 1958), since it is well-suited for I(k) = I0(k) (4) fluids dominated by short-range interactions (i.e., non- (ω) 2 | | coulombic interactions; Hansen & McDonald 2006). The and calculations are performed using a modified version of pyOZ5. Figure 2 compares our S(0) values to the S(0) " 2 2# 1 ~ k meω analytical fit given in Rohrmann (2018). The agreement I0(k) = exp −3 k −2mekBT 2 − k between both datasets is satisfactory under ρ = 1 g cm ~ 23 −3 (5) (nHe = 1.5 10 cm ), but worsens at higher densities. 2 2 × k [φe−He(r)] This disagreement reflects the limitations of the Percus- F . × 4πe2 Yevick closure relation at high densities, in a regime where the Monte Carlo calculations of Rohrmann (2018) In the last expressions, (ω) is the dielectric function, are more appropriate. Nevertheless, this small discrep- me and e are the electron mass and charge, kB is the ancy is of limited importance in the context of the mod- Boltzmann constant, ~ is the reduced Planck constant 5 and [φe−He(r)] is the Fourier transform of the electron- http://pyoz.vrbka.net heliumF potential, for which we use the simple form given 4 Blouin, Dufour & Allard

1.0 1.25 2000 K 4000 K 0.8 8000 K 1.20

0.6 1.15 n (0) S 0.4 1.10

1.05 0.2

1.00 0.0 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 3 ρ (g/cm3) ρ (g/cm )

Fig. 2.— Structure factor at k = 0 as a function of density and Fig. 3.— Index of refraction of helium as a function of density. for different temperatures. The solid lines show the analytical fits The line corresponds to the results of our calculations with Equa- obtained by Rohrmann (2018) from Monte Carlo calculations and tions 6, 7 and 8, and the circles are the laboratory measurements the circles show the results we found by solving the OZ equation. extracted from Dewaele et al. (2003). For both datasets, T = 300 K and λ = 6328 Å. eling of cool DZ stars, since the photospheric density of To validate our analytical model of the index of refrac- our models never exceeds 1 g cm−3. tion, we compared its predicted values with the high- ≈ pressure experimental measurements of Dewaele et al. 2.5. Index of refraction (2003). This comparison is shown in Figure 3 and reveals no significant deviation between our values and the The index of refraction, which is needed to compute the − laboratory measurements. Additionally, we checked that correction to the He free-free cross section (Equations our index of refraction values are virtually identical to 3 and 4), is obtained from the Lorentz-Lorenz equation, those obtained by Rohrmann (2018). 2 3 3 2 n 1 nHea0 nHea0 3 3. EQUATION OF STATE AND CHEMICAL 2 − = AR + BR + (nHe), (6) n + 2 NA NA O EQUILIBRIUM where AR and BR are the first and the second refractivity In this Section, we describe how the equation of state virial coefficients, nHe is the helium number density, ao and the chemical equilibrium calculations were modified is the Bohr radius and NA is the Avogadro constant. AR to take high-density nonideal effects into account. is proportional to the atomic polarizability α(ω) and is given by 3.1. Equation of state 4πNAα(ω) The total number density and the internal energy den- AR(ω) = . (7) 3 sity in each atmospheric layer are computed using the ab initio equations of state for hydrogen and helium pub- To compute AR, we use the helium polarizability val- lished by Becker et al. (2014). As in Blouin et al. (2017), ues reported in Masili & Starace (2003). For the second we resort to the additive volume rule for mixed H/He refractivity virial coefficient, we rely on the classical sta- compositions. The mass density ρ(P,T ) and the internal tistical mechanics expression (e.g., Fernández et al. 1999) energy density u(P,T ) are given by

2 2 Z ∞ 1 X Y 8NAπ φ(r) 2 = + , (10) BR(ω, T ) = ∆αave(ω, r) exp r dr, ρmix(P,T ) ρH(P,T ) ρHe(P,T ) 3 0 −kBT (8) umix(P,T ) = XuH(P,T ) + Y uHe(P,T ), (11) where ∆αave(ω, r) is the interaction-induced isotropic where X and Y are the mass fractions of hydrogen and polarizability and φ(r) is the helium-helium interatomic helium respectively. potential. To compute ∆αave(ω, r), we turn to the ex- For the densest cool DZ stars, the pressure at the pho- pansion tosphere exceeds 1011 dyn cm−2. Under such conditions, 2 4 using the ideal gas law can lead to an important over- ∆αave(ω, r) = ∆αave(0, r) + ω ∆S( 4, r) + (ω ), (9) − O estimation of the density. In fact, as shown in Figure where ∆αave(0, r) is given in Hättig et al. (1999) and 4, the ideal-gas density can be up to a factor 5 greater Maroulis (2000), and the Cauchy moment ∆S( 4, r) is than the value found when using the equation of state of given in Hättig et al. (1999). Finally, for the interaction− Becker et al. (2014). Such a difference can have a signif- potential φ(r) in Equation 8, we use the effective pair icant effect on the computed atmosphere structure and potential of Ross & Young (1986), which is calibrated to the synthetic spectrum, since most nonideal effects in- fit experimental data for high-density helium. cluded in the code (e.g., detailed line profiles, distorted A New Cool WD Atmosphere Code 5

101 (most importantly He− free-free), and hence the whole structure of the atmosphere. Unlike the rest of the nonideal eﬀects added to our atmosphere code, the equilibrium of heavy elements in the dense atmosphere of cool DZ stars has not yet been ex- 100 plored by other investigators using state-of-the-art meth-

) ods. Therefore, we had to perform our own calculations 3 before implementing this improved constitutive physics cm / in our code. In this Section, we ﬁrst give some theoreti- (g

ρ cal background and describe our strategy to compute the 1 10− ionization equilibrium (Section 4.1). Then, results from our ab initio calculations are presented in Section 4.2 and applied to white dwarf atmospheres in Section 4.3. 4000 K 8000 K 4.1. Theoretical framework 2 10− 4.1.1. The chemical picture 109 1010 1011 P (dyn/cm2) To tackle the problem of the ionization equilibrium of heavy elements in the dense atmosphere of cool white Fig. 4.— Density of a helium medium as a function of pressure dwarfs, we rely on the chemical picture. In this approach, and temperature. The solid lines show the results found when using the equation of state of Becker et al. (2014) and the dashed lines atoms, ions and electrons are considered as the basic par- correspond to the case where the ideal gas law is assumed. ticles and their interactions are modeled through interaction potentials. This is not as exact as the physical CIA profiles, high-density continuum opacities, nonideal picture, where nuclei and electrons are the basic parti- chemical equilibrium) are parametrized as functions of cles. However, using the chemical picture has several the density. For instance, using the ideal gas law would advantages. Since this approach is semi-analytical, the lead to an overestimation of the broadening of spectral results derived from it are more easily applicable in stel- lines due to an overestimation of the density of perturb- lar atmosphere codes (especially regarding opacity calcu- ing helium atoms. lations, where thousands of bound states must be taken into account to include the multitude of observed spectral 3.2. Chemical equilibrium lines). Moreover, it is easier to identify the contribution To compute the ionization equilibrium of helium, we of every physical effect and thus gain a better physical rely on the chemical model proposed by Kowalski et al. insight of the problem at hand (Winisdoerffer & Chabrier (2007). Since it does not rely on any free parameter, 2005). this ionization equilibrium model is a major improve- In the chemical picture, the ionization equilibrium ment over the occupation probability formalism (Hum- problem is reduced to the minimization of the Helmholtz mer & Mihalas 1988; Mihalas et al. 1988) used in most free energy F ( Ni ,V,T ) associated with a mixture { } white dwarf atmosphere codes. Compared to models made of species Ni in a volume V maintained at tem- where the ideal Saha equation is assumed, DZ models perature T (see for{ instance,} Fontaine et al. 1977; Magni that include the helium ionization equilibrium of Kowal- & Mazzitelli 1979; Hummer & Mihalas 1988; Saumon & ski et al. (2007) reach slightly lower densities in their Chabrier 1992). The total Helmholtz free energy of a deepest layers. This is the result of pressure ionization, mixture of atoms, ions and electrons can be expressed as id which increases the electronic density and, in turn, the the sum of the ideal free energy of the electron gas Fe , id opacity. However, this effect is not as important as in the ideal free energy of every ion from every species Fj,k, metal-free atmospheres since heavy elements provide the the contribution from the internal structure of bound majority of free electrons and therefore govern the atmo- species F int and the nonideal contribution related to the sphere structure. j,k interaction between species F nid, We have also included a detailed description of the X X X X ionization equilibrium of heavy elements, which is the F = F id + F id + F int + F nid, (12) subject of Section 4. e j,k j,k j k j k 4. IONIZATION EQUILIBRIUM OF HEAVY where k is an ionization state and j an atomic species. ELEMENTS Since F must be minimized, dF = 0 and the ionization Properly characterizing the ionization equilibrium of equilibrium of species J between ionization states K and heavy elements in the atmosphere of cool DZ stars is K + 1 imposes important from several perspectives. First, accurate ion- ! ization ratios are necessary to obtain the right spectral ∂F 0 = dNe line depths. For instance, in the case of a star that shows ∂N e Nj,k,V,T both Ca II H & K and Ca I 4226 Å in its spectrum, ob- ! taining the right Ca II/Ca I ratio is a prerequisite for ∂F reproducing simultaneously all spectral lines. Moreover, + dNK (13) ∂NK in cool DZ stars, heavy elements provide most of the elec- Ne,Nj,k6=K ,V,T ! trons. Therefore, a change in the ionization equilibrium ∂F of these trace species can influence other opacity sources + dNK+1, ∂N K+1 Ne,Nj,k6=K+1,V,T 6 Blouin, Dufour & Allard which, by definition of the chemical potential, is equiva- Pair Ab initio lent to the condition potential MD

µJ,K = µJ,K+1 + µe. (14) φij (r) Neglecting the interaction term F nid in Equation 12 φij (r) id id and taking Fe and Fj,k to be the free energy of an ideal non-relativistic non-degenerate gas (Landau & Lifchitz Ornstein- Zernike Classical 1980), Equation 14 leads to the well-known Saha equa- MD tion, equation 3/2 nK+1ne 2QK+1 2πmekBT −I/kB T Ri = 2 e , (15) { } nK QK h where h is the Planck constant, ni are number densities, Qi are partition functions and I is the ionization poten- DFT Ri tial. { } Now, if we keep the nonideal terms in the free energy equation, we find a result of the form of Equation 15, but Eexc with an effective ionization potential I + ∆I (Kowalski et al. 2007; Zaghloul 2009), Total µnid,ent 3/2 nonideal n n Q πm k T chemical K+1 e 2 K+1 2 e B −(I+∆I)/kB T = 2 e , potential nK QK h (16) Fig. 5.— Computational strategy used to retrieve the nonideal where chemical potential of ionic species. The dashed arrow indicates a validation step described in Section 4.2.2. ∆I = µnid + µnid µnid. (17) e K+1 − K Therefore, to compute the nonideal ionization equilibrium of heavy elements in dense helium-rich fluids, all dynamic equilibrium ionization potential since the con- that is needed is to compute the appropriate ∆I given figuration of atoms remains the same before and after by the above equation. the ionization takes place. However, plasmas encoun- In Equation 17, it is the difference in free energy of tered in white dwarf atmospheres have a finite coupling many-body systems in thermodynamic equilibrium with strength. When an atom is ionized, the medium responds different ionization states that is computed. This yields and additional energy is transferred between the atom an effective ionization potential applicable to thermody- and the surrounding particles (Crowley 2014). Therefore, namic ionization equilibrium calculations. As empha- the nonideal chemical potential of a species in ionization sized by Crowley (2014), this ionization potential is not state K can be expressed as the sum of two contributions, directly applicable to non-equilibrium processes (e.g., nid exc nid,ent photoionization). These are fast (adiabatic) processes µK = EK + µK , (18) that occur before the surrounding plasma has any time exc to respond. where EK is the excess of internal energy per particle nid,ent and µK is the entropic contribution to the nonideal 4.1.2. General strategy nid chemical potential. Note that this separation of µK into To compute ∆I, we have to evaluate the nonideal two distinct components directly follows from the defini- chemical potential of every species involved in the ionization of the Helmholtz free energy. As F = E + TS and nid nid nid µ = (∂F /∂NK ) , we can write tion process. The electronic term µe is already avail- K K Nk6=K ,V,T able in the literature. Kowalski et al. (2007) performed density functional theory (DFT) calculations to evaluate ∂ Enid + TSnid nid K K exc nid,ent the excess energy of an electron embedded in a dense µK = = EK + µK . ∂NK helium medium and found values that are in good agree- Nk6=K ,V,T ment with existing laboratory measurements (Broomall (19) et al. 1976). These calculations, published as polynomial Our general strategy is summarized in Figure 5. To expansions, were performed for a range of temperatures nid,ent compute the µK contribution, we follow the work of and densities suitable for our purpose. Kowalski (2006b) and Kowalski et al. (2007) and use the nid nid While µK+1 and µK were calculated by Kowalski et al. classical fluid theory and the OZ equation, as detailed in exc (2007) in the case of helium ionization, we are not aware Section 4.2.1. To retrieve EK , we turn to DFT to com- of any study where the nonideal chemical potentials were pute the excess energy of a metallic ion embedded in a computed for heavy elements surrounded by dense he- dense helium medium. This approach has the advantage lium. The central task of this Section is to compute of naturally taking into account many-body interaction exc these chemical potentials in order to obtain ∆I by virtue terms. Prior to using DFT to compute EK , we use of Equation 17. molecular dynamics (MD) simulations to obtain repre- In the limit of strongly-coupled systems, the role of sentative atomic configurations, as described in detail in entropy can be neglected for the calculation of thermo- Section 4.2.2. A New Cool WD Atmosphere Code 7

4.1.3. Comparison with previous studies 4.1.4. Approximations To take into account the nonideal ionization of heavy Before moving to the calculation of the nonideal chem- elements, white dwarf atmosphere models (Dufour et al. ical potentials and ∆I in Section 4.2, we take time 2007; Koester & Wolff 2000; Wolff et al. 2002) typically to justify three important approximations that we use rely on the Hummer-Mihalas occupation probability for- throughout Section 4. malism (Hummer & Mihalas 1988; Mihalas et al. 1988). Electrons and heavy elements as trace species — We are In this framework, an occupation probability wi is as- signed to every electronic level of every ion. If the level is interested in helium-rich plasmas, where heavy elements unperturbed, w = 1; if the level is completely destroyed and electrons can be considered as trace species. Hence, i we completely neglect the interaction of metallic ions by interparticle interactions, wi = 0. This occupation probability appears in the Boltzmann distribution and it with other metallic ions and with electrons. This ap- multiplies every term of the partition function, proximation is justified by the very low abundance of heavy elements in white dwarf atmospheres. Indeed, to our knowledge, the most metal-rich DZ star mentioned in X eiK QK = wiK giK exp , (20) the literature has an atmosphere with a number density −k T i B ratio of log Ca/He 6 (Ton 345, Wilson et al. 2015). As a consequence≈ of − this approximation, we completely where the sum is over all states i of species K, and g is a ignore the excess energy resulting from the interaction statistical weight. To compute wi in the particular case between charged species. Since electrostatic interactions of neutral interactions, Hummer & Mihalas (1988) use occur at long range, this approximation deserves some the second virial coefficient in the van der Waals equation additional justifications. To show that electrostatic in- of state to obtain teractions are negligible, we computed the contribution " # of electrostatic interactions to the Helmholtz free energy. 4π X 3 The latter can be broken down into three components wi = exp ni0 (ri + ri0 ) , (21) − 3 (Chabrier & Potekhin 1998), i0 F elec = F ee + F ii + F ie, (22) where ni is the number density of particles in state i and r is the radius of the particles in this state. The inter- i where Fee is the exchange-correlation contribution from pretation of Equation 21 is straightforward: when a state the electron fluid, F ii is the contribution from the one- occupies a volume of the same order as the mean volume component ion plasma and F ie is the electron screen- allowed per particle, it is gradually destroyed. Although ing contribution. To evaluate F elec, we used the equa- simple and easy to implement in atmosphere models, we tions reported in Ichimaru et al. (1987) for F ee and see three important drawbacks with this approach. those in Chabrier & Potekhin (1998) for F ii and F ie. If all electrons originate from singly-ionized species, then 1. This formalism is expected to break down above elec −3 F is a function of only the electronic density ne and 0.01 g cm (Hummer & Mihalas 1988), which is the plasma temperature T . Figure 6 shows ∆Ielec = insufficient≈ for many cool DZ white dwarfs. ∂ ∂ elec + F for different ne and T . The dashed ∂Ne ∂Nj,i+1 2. The excluded volume effect is only a caricature of line indicates the electronic density at the photosphere the real interaction potential between two neutral (τR = 2/3) of vMa2, a typical cool DZ star. At these particles. electronic densities and temperatures, the effect of electrostatic interactions on ∆I is of only a few meV and is 3. There is no theoretical prescription for the radii therefore negligible compared to the total ∆I reported later in this paper (which is of the order of a few eV). ri. For instance, for a ground state He I atom, should r be given by the hydrogenic approxima- The charged particles density is simply too low for elec- 2 trostatic interactions to have any significant effect. tion (r = n a0/Zeff = 0.39 Å) or should it be given by the van der Waals radius (1.40 Å, Bondi 1964)? Omission of the quantum behavior of ions — We do not To address this problem, it is always possible to take into account the quantum behavior of ions and calibrate the radii to fit the spectral lines observed atoms. To justify this approximation, we can compute in white dwarf stars. This was successfully done by the first quantum correction of the Helmholtz free energy Bergeron et al. (1991) for hydrogen, but it would (Wigner 1932), which can be seen as a correction for the be impracticable for DZ stars, where many ions overlapping wave functions of nearby particles. For an contribute to the total electronic density. m-component mixture, it can be expressed as (Saumon & Chabrier 1991) Our approach aims at answering these three concerns. 2 m Z First, by taking into account many-body interaction quant π~ X NaNb 2 2 terms, it is designed to remain physically exact up to F = φab(r)gab(r)r dr, 12kT V µab ∇ densities of the order of 1 g cm−3. Secondly, the inter- a,b action between species is modeled through ab initio cal- (23) culations that accurately describe the complex behavior where φab(r) and gab(r) are, respectively, the pair poten- of electrons under these high-density conditions. Finally, tial and the pair distribution function between species mamb a and b, and µab = is the reduced mass of par- since we rely only on first-principles physics, our method ma+mb does not require any free parameter. ticles a and b. The contribution of this term to ∆I is 8 Blouin, Dufour & Allard

101 Fundamental state contribution 2000 K NaII 0% Excited states contribution 100 20000 K NaI 7%

1 10− MgII 0%

vMa2 MgI 4% 2 10− (eV) FeII 42% elec I 3 FeI 27% ∆ 10− − CaII 15% 4 10− CaI 26%

5 10− CII 0%

CI 4% T = 5800 K 10 6 −1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 3 ne (cm− ) 0 10 20 30 40 50 Q Fig. 6.— Contribution of the electrostatic interaction to the effective ionization potential with respect to the electronic density Fig. 7.— Comparison of the contributions of the fundamen- and the temperature. The dashed line indicates the electronic den- tal state and the excited states to the partition function Q at sity at τR = 2/3 for vMa2, a typical cool DZ star. kB T = 0.5 eV for heavy ions found in cool DZ stars. The number at the end of each bar gives the fraction of Q resulting from excited states. This figure was made using the atomic data of the NIST Atomic Spectra Database (Kramida et al. 2015). computed as ∆Iquant = ∂ ∂ F quant. Using ∂Nj,k+1 − ∂Nj,k the pair distribution functions and the pair potentials in the worst case, the ionization fraction will be wrong described in Section 4.2.1, we find that ∆Iquant remains by a factor of 2. below 5 meV for all physical conditions relevant for the This maximum≈ error is not a cause of concern for the modeling of the atmosphere of cool DZ stars. As this modeling of the atmosphere of cool DZ stars. First, for is well below Eexc and µnid,ent, we can safely ignore the all other atomic species (C, Ca, Mg and Na) Q is far quantum behavior of ions. more dominated by the fundamental state contribution and the maximal error associated with this approxima- The ground-state approximation — To compute the ion- tion is thus much smaller than the value derived for Fe. ization equilibrium of heavy elements, we assume that Secondly, for the coolest DZ stars, the relative contribu- every atom is in its electronic ground state. This solely tion of the fundamental state to the partition function means that we consider all species to be in their ground is higher than for their warmer counterparts. Therefore, state when computing the ionization equilibrium. Once the ground-sate approximation becomes more accurate the ionization equilibrium is computed, the population of for the stars for which the departure for the ideal chem- every electronic state can be obtained through the Boltz- ical equilibrium is expected to be the most important. mann distribution. How good is this approximation? For Last but not least, for the conditions relevant for the helium atoms, this approximation is excellent. The first modeling of cool DZ stars, both this work and the for- excited state of He I lies at 19.8 eV, so almost all helium malism of Hummer & Mihalas (1988) predict deviations atoms are in their fundamental state for the temperature for the ideal gas equilibrium that are much more im- domain in which we are interested (kBT < 1 eV). portant than the aforementioned factor of 2 (see for For heavy elements, this approximation could be prob- instance Figure 15). ≈ lematic. It is well known that excited states are typically more affected by nonideal effects than the fundamental 4.2. Results state (e.g., Hummer & Mihalas 1988). Therefore, since the ∆I term in Equation 16 only takes into account the In this section, we detail the computations performed destruction of the fundamental state, an error could be to obtain ∆I for C, Ca, Fe, Mg and Na. In Sections 4.2.1 introduced in the ionization equilibrium if excited states and 4.2.2, we describe the computational setup and our are affected in a significantly different way and if they intermediate results, and our final results are given in account for a large portion of the partition function Q. Section 4.2.3. For the sake of clarity, we only refer to Ca To investigate the maximum error associated with this in the discussion of Sections 4.2.1 and 4.2.2, although all approximation, we computed the fundamental state con- the reported calculations were also performed for C, Fe, tribution to the partition function Q for C, Ca, Fe, Mg and Na. Mg and Na. The results are shown in Figure 7 for 4.2.1. Entropic contribution kBT = 0.5 eV. The worst possible error associated with this approximation will occur if all excited states are To compute the entropic contribution to the nonideal destroyed while the fundamental state remains unper- chemical potential, we first use the OZ equation (and turbed (see Equation 20). This scenario is highly un- the Percus-Yevick closure relation) to find the radial dis- likely, but provides an easy way of assessing the maxi- tribution function gHe−Ca(r) describing the spatial con- mum error. If it is the case, then, as shown in Figure 7, figuration of Ca relative to He atoms. Then, once the nid the maximum error on Q is 40% (see Fe II). Therefore, radial distribution function gHe−Ca(r) is obtained, µ ≈ Ca A New Cool WD Atmosphere Code 9

2.0 potential is pairwise additive, and an error may be in- CaI-He – This paper troduced if many-body terms are important. This is the CaI-He – Lovallo & Klobukowski 2004 main reason why we resort to the OZ equation only to CaI-He – Partridge et al. 2001 compute the entropic contribution and not to compute 1.5 CaII-He – This paper the excess energies. In fact, as described in Section 4.2.2, CaII-He – Czuchaj et al. 1996 we turn to DFT to compute excess energies, which guar- CaII-He – Allard & Alekseev 2014 antees that many-body interaction terms are properly taken into account. 1.0 (eV) φ 4.2.2. Excess energy contribution The excess energy of Ca embedded in a dense helium 0.5 medium made of N He atoms is given by exc E = ENHe+Ca ENHe ECa, (25) Ca−He − − 0.0 where ENCa+He is the total energy of the system, ENHe 1.5 2.0 2.5 3.0 3.5 4.0 4.5 is the energy of the N He atoms and ECa is the com- r (A)˚ puted energy of the isolated Ca atom. This calculation requires two steps. First, we need to find meaningful Fig. 8.— Comparison between the pair potentials for the Ca I– atomic configurations for the system (i.e., configurations He I and Ca II–He I interactions computed in this work and the values reported in Lovallo & Klobukowski (2004), Partridge et al. that are representative of the thermodynamic fluctua- (2001), Czuchaj et al. (1996) and Allard & Alekseev (2014) tions undergone by the real system). Then, we can use these configurations to compute the excess energy with Equation 25. can be obtained through Equations 9 and 12 of Kiselyov & Martynov (1990). From there, we simply substract the Molecular dynamics — To obtain representative atomic excess energy of Ca (as computed in the OZ framework) configurations of a system consisting of one Ca atom sur- nid,ent rounded by N He atoms at a given temperature and a to obtain µCa (Equation 18). To compute g r with the OZ equation, the pair given density, we turned to classical molecular dynamics He−Ca( ) 6 potentials φHe−He(r) and φHe−Ca(r) must be specified (in simulations. More precisely, we used LAMMPS (Plimp- accordance with the approximation detailed in Section ton 1995) and the pair potentials described in Section 4.1.4, φCa−Ca(r) = 0 since the metal-metal interactions 4.2.1. The simulations were performed in a cubic box are neglected). For the helium-helium pair potential, we with periodic boundary conditions. The box size and use the effective pair potential of Ross & Young (1986). the number of He atoms included in the simulations were As metal-helium pair potentials are not available in the chosen to attain the desired density (additional consid- literature for every metallic ion considered in this work, erations regarding finite-size effects are discussed in the we had to compute ab initio pair potentials between he- next paragraph) and the temperature was kept near the lium and metallic ions. To do so, we used the ORCA target value using a Nosé-Hoover thermostat (Nosé 1984; quantum chemistry package to obtain the potential en- Hoover 1985). The simulations were run for 5 ns using ergy φCa−He at various separations, 0.2 fs time steps. At regular time intervals, the atomic positions were saved and it is these configurations that φCa−He(r) = ECa−He(r) EHe ECa, (24) we use in the next Section to compute the excess energies. − − where ECa−He(r) is the total energy for a separation r DFT calculations — To compute the excess energy of and EHe and ECa are the computed energies of isolated Ca in the atomic configurations extracted from the He and Ca atoms. We rely on the CCSD(T) method molecular dynamics simulations, we used the Quantum (Raghavachari et al. 1989) as implemented in ORCA ESPRESSO7 DFT package (Giannozzi et al. 2009), with (Kollmar & Neese 2010; Neese et al. 2009) with the the PBE exchange-correlation functional (Perdew et al. aug-cc-pCVQZ basis sets (Dunning 1989; Kendall et al. 1996) and norm-conserving pseudopotentials. For all 1992; Woon & Dunning 1993). Using the counterpoise DFT calculations, we chose a kinetic energy cutoff of method (Boys & Bernardi 1970), we verified that the ba- 45 Ry (612 eV) and a charge density cutoff of 180 Ry. We sis set superposition error is small enough (< 2 meV) to checked that this cutoff is enough to achieve a < 0.05 eV be neglected for our purpose. convergence of the metal excess energy. To remove the In the particular case of Ca, a few interaction poten- electrostatic interaction associated with periodic bound- tials can be found in the literature for the Ca I–He I ary conditions, we used the Martyna-Tuckerman correc- (Lovallo & Klobukowski 2004; Partridge et al. 2001) and tion (Martyna & Tuckerman 1999) as implemented in the Ca II–He I interactions (Allard & Alekseev 2014; Quantum ESPRESSO, which allows to correct both Czuchaj et al. 1996). We used the values reported by the total energy and the SCF potential. these authors to validate our computational setup. This Furthermore, to make sure that the finite size of the comparison, which reveals no significant differences, is box does not result in undesired artifacts, we performed shown in Figure 8. simulations using different numbers of helium atoms per The main limitation of these pair potentials is that simulation box and different box sizes (up to N = 160 they were obtained in the infinite-dilution limit (i.e., Ca interacts with only one He atom). Therefore, when we 6 http://lammps.sandia.gov use these potentials, we implicitly assume that the total 7 http://quantum-espresso.org 10 Blouin, Dufour & Allard

10 1.0 ρ = 1.0 g/cm3 ρ = 1.0 g/cm3 ρ = 0.1 g/cm3 ρ = 0.1 g/cm3 8 0.8 exc E

0.6 6

(eV) 0.4

exc 4 E 0.2 2

Autocorrelation function of 0.0

0 0.2 0 20 40 60 80 100 120 140 160 180 − 0 5 10 15 20 25 30 35 40 Conﬁguration number Number of conﬁgurations

Excess energy of Ca at K for configurations Fig. 10.— Autocorrelation function of the excess energy time Fig. 9.— T = 4000 series shown in Figure 9. taken at 25 ps intervals from MD trajectories, for different helium densities. 0.25 helium atoms and up to a = 30 a.u.). We found that us- ρ = 1.0 g/cm3 ing at least N helium atoms and a simulation box 3 = 50 ρ = 0.1 g/cm of at least a = 15 a.u. (7.94 Å) allows a < 0.1 eV conver- 0.20 gence of the excess energy compared to results obtained at the same density with higher N and a values. This indicates that finite-size artifacts are negligible when these 0.15 two conditions are met. Hence, all DFT calculations re- (eV) ported in this work were performed with a a.u. and i 15 exc E N . ≥ h

50 σ ≥ 0.10 When computing the excess energy Eexc using configuration snapshots extracted from MD simulations, the results can fluctuate drastically from one configuration 0.05 to the other. This is shown in Figure 9, where the lines represent the evolution of Eexc from configuration to configuration. In Figure 10, we show the autocorrelation 0.00 0 20 40 60 80 100 120 140 160 180 function of the Eexc time series, Number of configurations PN−k Ei E Ei+k E i=1 exc exc exc exc Fig. 11.— Standard error of the mean of the Ca excess energy rk = − h i − h i . (26) PN i 2 at T = 4000 K with respect to the number of independent config- (E Eexc ) i=1 exc − h i urations used to compute the mean, for different helium densities. Since the autocorrelation function quickly decays to zero, we conclude that the time elapsed between each config- dilution limit, one could be worried about the exactitude uration snapshot is long enough for the Eexc time series of the atomic configurations obtained through molecu- values to be statistically independent. Therefore, we can lar dynamics using this potential. To check this point, safely apply the central-limit theorem to compute the we computed the excess energy of Ca using configura- standard error of the mean, tions extracted from ab initio molecular dynamics simulations. In this framework, no pair potential is as- σEexc σhEexci = . (27) sumed. The electronic density, energy and forces on √N ions are recomputed at every time step of the simula-

Figure 11 shows the evolution of σhEexci with respect to tion using DFT. This approach is expected to be more the number of configurations used to compute the mean. exact than the classical molecular dynamic approach, For both ρ = 0.1 and ρ = 1.0 g cm−3, we notice the 1/√N but its computational cost is larger by orders of mag- nitude. These calculations were performed using Born- decay of σhEexci. This implies that to improve the error by a factor of two, the number of configurations needs Oppenheimer molecular dynamics with the CPMD pack- to be quadrupled. From this analysis, we chose to use age8 (Hutter et al. 2008; Marx & Hutter 2000), with the 100 configurations for each (T, ρ) condition. This value PBE exchange-correlation functional and ultrasoft pseudopotentials (Vanderbilt 1990). We employed 0.5 fs time is enough to obtain σhEexci . 0.1 eV for most physical conditions considered in this work, which is an error that steps and an energy cut-off of 35 Ry. As before, we ex- we consider acceptable for our purpose. tracted atomic configurations from these simulations and used these configurations to compute the interaction en- Validation with ab initio molecular dynamics — Since 8 our φCa−He(r) potential was calculated in the infinite- http://cpmd.org A New Cool WD Atmosphere Code 11

7 3 DFT-MD (CPMD) 0.4 g/cm MD (LAMMPS) 15 0.8 g/cm3 6 + ∆ ion excess energy 1.5 g/cm3 5 T = 4000 K

4 10 (eV) exc 3 + electron excess energy E

2 6.11 5.66 Eﬀective ionization potential (eV) 5 + entropy contribution 4.61 1 4.04

Reference IP Effective IP 0 0.0 0.2 0.4 0.6 0.8 1.0 ρ (g/cm3) Fig. 13.— Contributions added to the reference ionization potential of Ca to obtain its effective ionization potential at various Excess energy of Ca at K for different Fig. 12.— T = 5000 densities (see legend). These results are for T = 4000 K. helium densities, obtained from configurations extracted from ab initio molecular dynamics (DFT-MD) and from classical molecular dynamics (MD) using the pair potentials described in Section 4.2.1. TABLE 2 Fitting parameters for ∆I(ρ, T ) ergy of Ca with the surrounding medium through DFT (Equation 28). calculations. Figure 12 compares the results obtained to those found Ion a1 b2 c3 with the classical molecular dynamics simulations. This comparison shows that there is only a negligible differ- C 1.91782 -3.24813 -1.19948 Ca -2.20703 -0.14431 0.57494 ence between the two approaches, at least below ρ = Fe -2.23142 0.48427 0.21301 1 g cm−3. We did not perform any comparison at higher Mg 0.45809 -0.85522 -1.01958 densities, because of the prohibitive calculation time of Na -0.52305 -0.62471 0.04833 −3 1 1 3 such calculations. In any case, densities above 1 g cm eV g− cm 2 4 1 1 3 are never encountered at the photosphere of cool DZ 10− eV g− K− cm 3 2 6 white dwarfs (Section 4.3). Therefore, we conclude that eV g− cm our infinite-dilution limit potential φCa−He(r) is sufficient to generate the atomic configurations used to compute reduction of the band gap of warm dense helium with the excess energy (and it is much faster than resorting increasing temperature. to ab initio molecular dynamics simulations). To easily implement these nonideal ionization potentials in atmosphere models, we have fitted our results 4.2.3. Ionization equilibrium with a simple function of ρ and T , Following the methodology described in the previous 2 nid,ent exc ∆I(ρ, T ) = min 0, (a + bT )ρ + cρ , (28) sections, we computed µK and EK for C I/C II, Ca I/Ca II, Fe I/Fe II, Mg I/Mg II and Na I/Na II. By where a, b and c are parameters found using a chi-squared adding these excess chemical potentials to the electron minimization algorithm, ρ is the helium density in g cm−3 excess energy, we computed by how much the ionization and T is the temperature in K. This expression allows a potential is altered at a given density and temperature satisfactory fit to the data and yields ∆I = 0 at ρ = 0. (Equation 17). Figure 13, which shows the three contri- The analytical fits are shown in Figure 14 and the fitting butions to ∆I (the free electron excess energy, the vari- parameters are reported in Table 2. Formally, in order ation of Eexc and the change in µnid,ent), illustrates this to stay within the limits of our calculations, the use of K K these analytical expressions should be limited to densities process in the case of Ca. −3 Figure 14 shows our final results. First, for every ion between 0 and 1.5 g cm and to temperatures between considered, we notice that ∆I 0 when ρ 0. This is 4000 and 8000 K. Nevertheless, we have verified that the expected behavior and it shows→ that our methodology→ Equation 28 can safely be extrapolated to lower (down to is consistent with the ideal regime when we push it to low 2000 K) or higher temperatures (at least up to 10000 K) densities. Secondly, we note that ∆I is always negative if needed. and that its absolute value increases with density. This result means that ionization becomes easier with increas- 4.2.4. Comparison with previous studies ing density, which also corresponds to the expected be- It is instructive to compare these results with the ion- havior. Finally, for all elements except Fe, we notice that ization equilibrium predicted by the Hummer-Mihalas higher temperatures are associated with slightly larger occupation probability formalism, which is widely used ionization potential depressions. This result is consistent in atmosphere codes. Since there is no theoretical pre- with the findings of Kowalski et al. (2007), who found a scription for the values of the hard sphere radii used 12 Blouin, Dufour & Allard

0.0 0.0 0 C Ca Fe

0.5 0.5 − − 1 − 1.0 − 1.0 − 2 − 1.5 − 1.5 − 3 2.0 − −

2.0 − 2.5 4 − −

0.00 0.25 0.50 0.75 1.00 1.25 1.50 0.00 0.25 0.50 0.75 1.00 1.25 1.50 0.00 0.25 0.50 0.75 1.00 1.25 1.50 (eV) I ∆ 0.0 Mg 0.0 Na

0.5 −

1.0 0.5 − −

1.5 −

2.0 1.0 − −

2.5 −

3.0 1.5 − − 0.00 0.25 0.50 0.75 1.00 1.25 1.50 0.00 0.25 0.50 0.75 1.00 1.25 1.50 ρ (g/cm3)

Fig. 14.— Depression of the ionization potential of C, Ca, Fe, Mg and Na embedded in a dense helium fluid. Circles show the results of our ab initio calculations and error bars indicate the statistical errors associated with the configuration sampling. The solid lines show the analytical fits found through a chi-squared minimization of Equation 28. Data in red are for T = 8000 K and data in yellow are for T = 4000 K. to compute the occupation probabilities (Equation 21), Figure 15 compares the multiplicative factors that need a somewhat arbitrary choice must be made to perform to be applied to the right-hand side of the Saha equa- this comparison. We chose to compute the hard sphere tion for the Ca I/Ca II ionization equilibrium to account radii with the hydrogenic approximation, as described by for nonideal effects (i.e., wCaII/wCaI in the case of the Beauchamp (1995). In this approximation, the radius of Hummer-Mihalas formalism and e−∆I/(kB T ) for our ion- a species in state i is given by ization model). The most obvious aspect of Figure 15 n2a is that we find a weaker pressure ionization than what i 0 is predicted using the Hummer-Mihalas formalism and ri = eff , (29) Zi hard sphere radii computed in the hydrogenic approximation. We checked that this result holds true for C, Fe, Mg where ni is the principal quantum number of the upper- and Na. This conclusion is consistent with the findings most electron, a0 is the Bohr radius and the effective nuclei charge Zeff is given by of Bergeron et al. (1991) for the ionization equilibrium of i hydrogen in cool DA stars. Using the Hummer-Mihalas r 2 eff Ii formalism and a hydrogen radius given by rn = n a0, Z = ni , (30) i 13.598 eV they found that the high Balmer lines are predicted to be too weak, indicating that pressure ionization in the where Ii is the energy needed to ionize an electron from Hummer-Mihalas formalism is too strong. They showed state i. In the Hummer-Mihalas formalism, every term that using a smaller radius in the computation of the oc- in the partition function is multiplied by the occupation 2 cupation probabilities, rn = 0.5n a0, allows better spec- probability (Equation 20). If we stick to the ground-state tral fits. approximation (Section 4.1.4), the occupation probabil- Unfortunately, we cannot compute the ionization po- ity is the same for every level and it can be factored out tential depression of H to directly confirm the conclu- of the partition function sum. Hence, the net effect of the sion of Bergeron et al. (1991). The problem is that the Hummer-Mihalas formalism is to multiply the right-hand H II–He potential (e.g., Pachucki 2012; Kołos & Peek side of the Saha equation (Equation 15) by the ratio of 1976) has a deep attractive well (since H+ and He can occupation probabilities, wZII/wZI. A New Cool WD Atmosphere Code 13

1010 Teﬀ = 4000 K 0 T = 4000 K 10 109

1 108 10−

7 2 10 10− 106 3 10− 105

) 4 Occupation probability 3 10 104 − cm / 3 5 10 10− 3) (g / Saha multiplicative factor 2 Teﬀ = 6000 K

10 = 2 This work 0 ν 10 τ

1 (

10 ρ 10 1 100 − 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 3 2 ρ (g/cm ) 10−

Fig. 15.— Multiplicative factor applied to the right-hand side 10 3 of the Ca I/Ca II Saha equation (Equation 15) to take nonideal − effects into account. The blue line is wCaII/wCaI, the result ob- 4 11 tained using the Hummer-Mihalas formalism, and the green curve 10− Ca/He = 10− ∆I/(kB T ) 10 is e− , the result obtained with our ionization model. Ca/He = 10− 9 5 Ca/He = 10− + 10− form the HeH molecule) that prevents proper conver- 2000 2500 3000 3500 4000 4500 5000 gence of the OZ equation solver. The same issue arises λ (A)˚ if we try to compute the ionization potential of H in a H-rich medium, since the H II–H I potential (e.g., Frost Fig. 16.— Density at an optical depth τν = 2/3 with respect to & Musulin 1954) also has an important attractive well λ. The top panel shows the results for Teff = 4000 K models and the bottom panel for . The Ca abundance is given in + + Teff = 6000 K (H and H can form the H2 molecule). the legend and a surface gravity log g = 8 is assumed. 4.3. Atmosphere models Using the analytical model described in the previous using the occupation probability formalism and the ideal Section, we implemented the improved ionization equi- Saha equilibrium (in each case, the atmosphere model librium of heavy elements in our atmosphere code to in- structure and the synthetic spectrum were computed us- vestigate how it affects the synthetic spectra of cool DZ ing the same ionization model). This figure focuses on stars. Before even examining any spectrum, we can get the region between 3500 and 4500 Å, since it contains an idea of the impact of the new nonideal ionization equi- several Ca, Fe and Mg absorption lines susceptible of be- librium by looking at the densities involved in the model ing affected by the choice of the ionization model. The atmospheres. Figure 16 shows the density at τν = 2/3 first thing to note is that for the high-density models as a function of λ for a few atmosphere models with dif- (i.e., those with a low metal abundance and a low ef- ferent effective temperatures and calcium abundances. 9 fective temperature) there are important differences be- This type of figure is useful to identify which densities are tween spectra obtained using the ideal Saha equilibrium probed at different wavelengths. In the previous Section, and our ionization model. These differences are mostly we saw that no important deviation from the ideal ioniza- due to a shift in the continuum associated with the in- tion equilibrium is expected below 0.1 g cm−3 (see Figure creased electronic density in models that take pressure 14). From Figure 16, it is clear that the probed densities ionization into account. Next, we notice that the spec- −10 tra computed using the Hummer-Mihalas formalism are are below this threshold for Ca/He & 10 and above −10 even further from the spectra obtained using the ideal this threshold for Ca/He . 10 . Therefore, it should become important to take into account the nonideal ion- Saha equilibrium than those computed with our ioniza- ization equilibrium for cool DZ atmosphere models with tion model. This is not surprising, since as seen in Fig- −10 ure 15, the Hummer-Mihalas formalism predicts a very Ca/He . 10 , but it is probably superfluous for mod- −10 strong pressure ionization. Finally, for the low-density els with Ca/He & 10 (note that nonideal effects on the opacities and the equation of state remain neverthe- models (i.e., those with a high metal abundance and/or less important in this regime). For intermediate densities a high effective temperature), all three sets of spectra are virtually identical, which is consistent with our analysis (Ca/He 10−10), using the nonideal ionization equilib- ≈ of Figure 16. rium should result in small changes in the spectral line The nonideal chemical equilibrium of heavy elements wings of the coolest models. also has a small impact on the model atmosphere struc- Figure 17 compares synthetic spectra computed with ture. The increased electronic density associated with our ionization equilibrium model to spectra computed pressure ionization leads to an increase of the Rosse- land mean opacity and therefore to a reduction of the 9 In this paper, the abundance of all metallic species, from C to Cu, is scaled to the abundance of Ca to match the abundance pressure at the photosphere. For instance, for Teff = ratios of chondrites reported in Lodders (2003). 4000 K, log g = 8 and Ca/He = 10−11, a model that 14 Blouin, Dufour & Allard

11 10 9 Ca/He = 10 Teff = 6000 K Ca/He = 10 Teff = 6000 K Ca/He = 10 Teff = 6000 K 8 − 8 − 8 − 6 6 6 4 4 4 2 2 2 ) 1

− 0 0 0 11 10 9 Hz Ca/He = 10− Teff = 5000 K Ca/He = 10− Teff = 5000 K Ca/He = 10− Teff = 5000 K 1 3 3 3 − s 2

− 2 2 2

1 1 1 erg cm 6

− 0 0 0

(10 11 10 9 Ca/He = 10− Teff = 4000 K Ca/He = 10− Teff = 4000 K Ca/He = 10− Teff = 4000 K ν

H 0.4 0.4 0.4 Occup. prob. Saha This paper 0.2 0.2 0.2

0.0 0.0 0.0 3600 3800 4000 4200 4400 3600 3800 4000 4200 4400 3600 3800 4000 4200 4400 λ (A)˚

Fig. 17.— Comparison between synthetic spectra computed using the Hummer & Mihalas (1988) formalism (in blue), the ionization equilibrium presented in this work (in red) and the ideal Saha equation (in black). All models were computed assuming log g = 8 and H/He = 0. The effective temperature and the metal abundance is indicated above each panel. assumes the ideal Saha equation has a photospheric density of 0.93 g cm−3, while an atmosphere structure based TABLE 3 on our ionization model has a photospheric density of Observational data. −3 0.89 g cm . Moreover, the occupation probability for- Ross 640 LP 658-2 malism predicts a density that is still lower (0.84 g cm−3). Given Figure 15, this result is not surprising: compared Parallax (mas) 62.915 0.022 155.250 0.029 B1 14.±02 15.±49 to our calculations, the Hummer-Mihalas formalism over- V 13.83 14.45 estimates the efficiency of pressure ionization. R 13.75 13.99 Our results constitute a physically-grounded answer to I 13.66 13.54 the question of the importance of pressure ionization in J 13.58 13.05 H 13.57 12.86 cool DZ stars, which will help to reduce the gap be- K 13.58 12.78 tween solutions found with different atmosphere codes. 1 There is a 3% uncertainty on all photometric mea- A good example to illustrate this point is vMa2 (WD surements. 0046+051). On one hand, using an ideal treatment of chemical equilibrium, Dufour et al. (2007) found a solution with Teff = (6220 240) K. On the other hand, At Teff 8000 K, Ross 640 is technically not a "cool" using the Hummer-Mihalas± occupation probability for- white dwarf.≈ Since the density at its photosphere is −3 21 −3 malism, Wolff et al. (2002) found Teff = (5700 200) K. 0.01 g cm (nHe = 1.5 10 cm ), nonideal effects In their analysis, Dufour et al. (2007) showed that± the dif- affecting≈ the equation of state× and the chemical equilib- ference between both solutions can largely be explained rium are minimal. However, this density is high enough by the different chemical equilibrium models used in both to induce important differences between Lorentzian pro- studies. This uncertainty can be removed by relying on files and the improved line profiles presented in Section the accurate description of the chemical equilibrium de- 2.1. This object is therefore the perfect candidate to test scribed in the current work. our line profiles separately, without the interference of other nonideal effects. 5. APPLICATIONS To fit this star, we follow the procedure described in To show how the improved constitutive physics pre- Dufour et al. (2007). In short, we first find Teff and sented in this work translates in terms of better spec- log g using the photometric technique described in Berg- troscopic fits, this Section presents the analysis of two eron et al. (2001). The photometric measurements are well-known DZ stars: Ross 640 (WD 1626+368) and LP first converted into fluxes using the constants reported 658-2 (WD 0552-041). Applications to other objects will in Holberg & Bergeron (2006). Then, these observed be presented in other papers of the series. fluxes fν are compared to the model fluxes Hν to obtain 2 Our new analysis of these two objects makes use of Teff and the solid angle π(R/D) , where R is the radius Gaia DR2 parallaxes (Prusti et al. 2016; Brown et al. of the star and D is its distance to the Earth. These 2018), BVRI and JHK photometry published in (Berg- parameters are found using a χ2 minimization technique eron et al. 2001, see Table 3), optical spectra published in relying on the Levenberg-Marquardt algorithm. Since D Giammichele et al. (2012) and UV spectra obtained with is known from the parallax measurement, the radius R HST and the Faint Object Spectrograph (FOS, Koester can be computed from the solid angle. The mass of the & Wolff 2000; Wolff et al. 2002) star and the corresponding surface gravity g = GM/R2 5.1. Ross 640 are then found using the evolutionary models of Fontaine et al. (2001). This log g value being generally different A New Cool WD Atmosphere Code 15

2.5 TABLE 4 ) Fitting parameters. 1 Fe ii − 2.0 Fe i/

˚ A Si i 1 − Ross 640 LP 658-2 s Fe ii 2 1.5 Ca ii T (K) 8070 140 4430 40 − eﬀ ± ± log g 7.923 0.008 7.967 0.022 1.0 log H/He 3.5 ± 0.2 < ±5 erg cm

− ± − 14

log Ca/He 9.12 0.05 11.38 0.05 − Si i Si i log Fe/He −8.44 ± 0.10 − -± 0.5 Si i Si i (10

log Mg/He −7.40 ± 0.10 8.66 0.20 λ Mg i f Fe ii log Si/He −7.90 ± 0.20 − -± Mg i − ± 0.0 Si i Mg ii 1800 2000 2200 2400 2600 2800 3000 3200 from our initial guess, we repeat the fitting procedure λ (A)˚ until all fitting parameters are converged. Once a consistent solution for Teff and log g is obtained 2.0 ) from the procedure described in the previous paragraph, 1 − we move to the determination of the abundances using ˚ A 1 1.5 − spectroscopic observations. We keep Teff and log g fixed s 2 to the values found using the photometric observations − Hβ and then fit the Ca/He and H/He ratios by minimizing 1.0 Mg i Mg i the χ2 between our synthetic spectra and the observed erg cm 14 spectrum. Since the abundances found with this tech- − 0.5 Hα (10

nique are generally different from those initially used for λ the photometric fit, the whole fitting procedure is re- f Ca ii 0.0 peated until internal consistency is reached. 4000 4500 5000 5500 6000 6500 Although the abundance ratio between the different λ (A)˚ heavy elements is kept constant during the χ2 minimization procedure, we manually adjust the abundance ratio 1.2 Teff = 8070 140 K of Mg, Fe and Si to fit the spectral lines labeled in Fig- ) ± 1 1.0 log g = 7.923 0.008 ure 18. All other heavy elements (from C to Cu) are − ± ˚ A 1 included in the models, but since we could not use any −

s 0.8 2

spectral line to ﬁt their abundances, we simply assume − the same abundance ratio with respect to Ca as in chon- 0.6

drites (Lodders 2003). erg cm

14 0.4

As shown in Figure 18, our solution is consistent with −

observations across all wavelengths. Our ﬁtting param- (10

λ 0.2 eters, given in Table 4, are roughly similar to those f 2 χν = 1.32 found by Dufour et al. (2007), Koester & Wolff (2000) 0.0 and Zeidler-KT et al. (1986) , although they all found a 0.5 1.0 1.5 2.0 higher effective temperature (8440 320 K, 8500 200 K λ (µm) ± ± and 8800 K, respectively). One major improvement com- Fig. 18.— Our best solution for Ross 640. The top panel shows pared to previous authors is our fit to the broad Mg II our fit to the UV spectrum, the middle panel is our fit to the visible 2795/2802 Å lines. To obtain a good fit, Koester & Wolff spectrum and the bottom panel shows our photometric fit to the (2000) arbitrarily multiplied the van der Waals broaden- BVRI and JHK bands. ing constant of these lines by 10. No arbitrary constants are needed using our new line profiles and a consistent TABLE 5 abundance is found from both the optical and ultraviolet Literature review of LP 658-2. magnesium lines.

Authors Teff (K) H/He 5.2. LP 658-2 Bergeron et al. (2001) 5060 60 He ± 4 LP 658-2 is a DZ star that exhibits a weak Ca II H& Wolff et al. (2002) 5060 60 H/He = 5 10− K doublet. During the last two decades, many authors Dufour et al. (2007) 4270 ± 70 He× Giammichele et al. (2012) 5180 ± 80 H have tried to fit this star, but none has reached a consis- ± tent solution across all wavelengths. Because they relied on different models and observations, the solutions they found are quite diverse (see Table 5). 658-2. The large absorption feature observed in the UV First, Bergeron et al. (2001) found that LP 658-2 has a was interpreted as strong broadening from the wing of helium-rich atmosphere with Teff = (5060 60) K. How- Lyα. Keeping the effective temperature fixed at the ± ever, their analysis was based on atmosphere models that Teff = 5060 K value found by Bergeron et al. (2001), did not include heavy elements, which strongly influence they used this UV absorption feature to fit the hydrogen UV opacities and the temperature profile. abundance and found that H/He = 5 10−4. However, Then, using HST data (FOS), Wolff et al. (2002) ex- contrarily to other stars in their sample× (e.g., LHS 1126 tended the analysis of Bergeron et al. (2001) with an and BPM 4729), they were not able to properly repro- investigation of the UV portion of the spectrum of LP duce the shape of this UV absorption feature. 16 Blouin, Dufour & Allard

Subsequently, using models that include heavy ele- 0.22

ments in the atmosphere structure, Dufour et al. (2007) ) 1 0.20 determined a much cooler temperature for LP 658-2 − ˚ A 1

(Teﬀ = 4270 70 K). At this temperature, the photomet- − s 0.18 Ca i

± 2 (absent) ric data can completely exclude the presence of traces of − hydrogen at the level found by Wolﬀ et al. (2002) since 0.16

H2-He CIA would cause a strong IR ﬂux depletion that erg cm

14 0.14 is not observed. However, the solution of Dufour et al. −

(2007) does not explain the UV absorption feature seen in (10

λ 0.12 Ca ii f the FOS data and their spectroscopic solution predicted Ca ii a large Ca I 4226 Å line, which is completely absent from 0.10 the observations. 3900 4000 4100 4200 4300 More recently, Giammichele et al. (2012) argued that λ (A)˚ the narrow H & K lines observed in the spectra of LP 658- 100 2 2 indicate that it is perhaps a hydrogen-rich star after χν = 0.39 Teﬀ = 4430 40 K ) ± 1 log g = 7.967 0.022 all. However, although an H-rich composition allowed − ± ˚ A a better ﬁt to the visible spectrum than Dufour et al. 1 − s

(2007), the photometric ﬁt was not as good (and it can 2

− 1 not explain the shape of the UV spectrum). 10− Using our improved models, we can now obtain a solu- erg cm tion that agrees perfectly with the observations across all 14 Observations − wavelengths assuming a helium-rich atmosphere (Figure Best ﬁt (10

λ 2 No Mg UV opacities

19). We can also constrain the amount of hydrogen to f 10− 5 H/He < 10−5, as a higher hydrogen abundance would No Mg UV opacities and H/He = 10− produce an IR flux depletion that is incompatible with 0.3 0.5 1.0 2.0 the observations. Given this limit, the shape of the UV λ (µm) continuum can no longer be explained by the wing of Lyα Fig. 19.— Our best solution for LP 658-2. The top panel shows (see the green dash-dot line in Figure 19). Instead, we our fit to the visible spectrum and the bottom panel displays our find that the absorption in the UV can naturally be ex- fit to the photometric observations and to the FOS data. The bot- plained by the presence of trace amounts of magnesium tom panel also shows two synthetic UV spectra computed without the Mg II 2795/2802 Å and the Mg I 2852 Å lines, one without (absorption from the Mg II 2795/2802 Å and the Mg I hydrogen (in blue) and one with H/He = 10 5 (in green). 2852 Å lines). While there is formally no lines detected, − the amount of magnesium needed to reproduce the UV continuum is small enough as to not produce features in While most of these nonideal effects were implemented the optical spectrum. using results previously published by various authors, we Finally, our new models do not predict the strong Ca I performed our own calculations to assess the chemical 4226 Å line that was predicted using the models of Du- equilibrium of heavy elements. four et al. (2007). This is mainly due to the use of our More precisely, we used the classical theory of fluid and improved line profiles (Section 2.1) as well as our new DFT calculations to characterize the ionization equilib- nonideal Ca ionization equilibrium calculation (Section rium of C, Ca, Fe, Mg and Na in a dense helium medium 4), the former effect being the most important. Our fit- and under the temperature and density conditions found ting parameters, given in Table 4, were found using the in the atmosphere of cool DZ stars. These calculations same fitting procedure as for Ross 640. show that the effective ionization potential begins to de- crease when the density exceeds 0.1 g cm−3, reaching a 6. CONCLUSION depression of 1 2 eV at ρ = 1 g cm−3. We provided ≈ − We have developed an updated atmosphere model code analytical fits to our data that can be implemented in at- that incorporates all the necessary constitutive physics mosphere model codes to obtain the effective ionization for an accurate description of cool DZ stars. This code potential for a given temperature and density. includes We computed atmosphere models using this improved description of the ionization of heavy elements and found The most important heavy element line profiles that under the right conditions (i.e., weakly polluted, • computed using the unified line shape theory of low-Teff objects) the synthetic spectrum can significantly Allard et al. (1999), differ from results obtained using the ideal Saha equation. Moreover, we found that the Hummer-Mihalas for- CIA profiles suitable for fluids where the density malism – when used in conjunction with hydrogenic hard • exceeds 0.1 g cm−3, sphere radii – leads to a much stronger pressure ionization than our model, which indicates an overestimation He Rayleigh scattering and He− free-free absorp- • of pressure ionization. This result is consistent with pre- tion corrected for collective interactions between vious findings based on comparisons between atmosphere atoms, models and observed spectra (Bergeron et al. 1991). Fi- An ab initio equation of state for H and He, nally, we showed how the improved constitutive physics • included in our code translates into better spectral fits A nonideal chemical equilibrium model for He, C, for Ross 640 and LP 658-2, two cool DZ stars that pre- • Ca, Fe, Mg and Na. sented a challenge to previous atmosphere model codes. A New Cool WD Atmosphere Code 17

In the next papers of this series, we will use our up- by the Gaia Data Processing and Analysis Consortium dated models to analyze in detail other cool white dwarfs, (DPAC, https://www.cosmos.esa.int/web/gaia/ in particular WD 2356-209 (a peculiar cool DZ star show- dpac/consortium). Funding for the DPAC has been ing an exceptionally strong Na D feature) and the ﬁrst provided by national institutions, in particular the cool DZ star to show CIA absorption. We will also an- institutions participating in the Gaia Multilateral alyze the bulk of the known cool white dwarfs taking Agreement. advantage of the Gaia data and revisit the spectral evo- This work has made use of the Montreal White Dwarf lution of these objects. Database (Dufour et al. 2017). This work used observations made with the NASA/ESA Hubble Space Telescope, and obtained We wish to thank Piotr M. Kowalski for useful discus- from the Hubble Legacy Archive, which is a collabo- sions regarding the DFT calculations presented in Sec- ration between the Space Telescope Science Institute tion 4. This work was supported in part by NSERC (STScI/NASA), the Space Telescope European Coor- (Canada). dinating Facility (ST-ECF/ESA) and the Canadian This work has made use of data from the Astronomy Data Centre (CADC/NRC/CSA). European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed

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