NLTE stellar parameters of UMP stars

Ultra-Metal poor stars: Spectroscopic determination of stellar atmospheric parameters using iron non-LTE line abundances


We present new ultra-metal-poor (UMP) stars parameters with [Fe/H] 4.0 based on line-by-line non-local thermodynamic equilibrium (NLTE) abundances using an up-to-date iron model atom with a new recipe for non-elastic hydrogen collision rates. We study the departures from LTE in their atmospheric parameter and show that they can grow up to  dex in [Fe/H],  K in and  dex in log  toward the lowest metallicities. Accurate NLTE atmospheric stellar parameters, in particular [Fe/H] being significantly higher, are the first step to eventually providing full NLTE abundance patterns that can be compared with Population III supernova nucleosynthesis yields to derive properties of the first stars. Overall, this maximizes the potential of these likely second-generation stars to investigate the early universe and how the chemical elements were formed.

line: formation — stars: abundances — stars: fundamental parameters — stars: Population II

1 Introduction

Ancient ultra-metal-poor (UMP) stars (with , e.g. Beers & Christlieb 2005) are rare relics of the early Universe. They provide unique insights into the first nucleosynthesis events and the first (Population III; Pop III hereafter) stars (Klessen et al., 2012; Bromm, 2013), the earliest phases of chemical enrichment (Frebel & Norris, 2015), as well as the formation of the first low-mass stars (Frebel et al., 2007; Chiaki et al., 2012; Ji et al., 2015). For example, detailed comparisons of supernova nucleosynthesis yields with stellar abundances have shown that Pop III stars were likely massive (20-60 M; Keller et al. 2014; Tominaga et al. 2014; Placco et al. 2015), in agreement with theoretical expectations (Bromm et al., 2002; Abel et al., 2001). As more stars and more detailed yield calculations become available, the nature and shape of the initial mass function of Pop III stars can ultimately be reconstructed that way.

Key ingredients are as accurate and precise as possible stellar abundances of many elements. For example, Placco et al. (2015) investigated how abundance availability and precision affected the results of fitting the abundance patterns with nucleosynthetic yields to derive Pop III stellar masses. They found that the exclusion of nitrogen from the abundance pattern had a significant impact on the final derived Pop III progenitor mass. To obtain high quality chemical abundances, a necessary prerequisite is accurate and precise stellar atmospheric parameters, i.e., effective temperature , surface gravity , iron abundance [Fe/H], and microturbulence  that characterize the star. Stellar parameters can be determined in different ways, but in addition to the metallicity, in most cases as least the microturbulence and/or the surface gravity are determined spectroscopically by demanding no abundance trend with reduced equivalent width and invoking ionization equilibrium of neutral and singly ionized iron lines, Fe i and Fe ii, respectively. Effective temperature can also be determined spectroscopically, by invoking the “excitation balance”, i.e. no abundance trend with excitation potential of Fe i lines.

For most abundance analyses, one-dimensional stellar model atmospheres are used, together with radiative transfer codes assuming local thermodynamic equilibrium (LTE). This method, however, is affected by unaccounted departures from LTE that can introduce significant systematic uncertainties since line formation and populations of non-dominant species (in this case Fe i) can potentially deviate from Saha-Boltzmann equilibrium assumed in LTE (e.g. Rutten 2003).

Deviations from LTE have been shown to increase toward lower metallicities and for extended atmospheres (giants and super-giants) (Mashonkina et al., 2011; Bergemann et al., 2012; Lind et al., 2012; Bergemann et al., 2015; Mashonkina et al., 2016). The decreasing number of electrons donated by metals leads to decreased collision rates. In cool, late-type stars (4000 K  K), lines arising from minority species are most affected by these deviations. To account for such departures, especially in cool UMP stars, it is necessary to investigate the formation of iron lines in non-local thermodynamic equilibrium (non-LTE, hereafter NLTE). For such an investigation, atomic data and other inputs are required for each element. Previous iron NLTE studies have reported abundance changes ( 0.1 dex) simply due to uncertainties in the input atomic data used in their model atoms (Bergemann et al., 2012; Mashonkina et al., 2011; Collet et al., 2005). Particularly problematic in this context are uncertainties arising from estimating the rate of inelastic collisions with neutral hydrogen atoms. They are usually obtained from the classical Drawin approximation (Drawin, 1968, 1969a, 1969b; Lambert, 1993) because full quantum calculations are lacking. However, the approximation is known to overestimate the collision rates by several orders of magnitude (Barklem et al., 2010; Lind et al., 2011; Osorio et al., 2015). Several attempts to calibrate these rates were made by applying a global multiplicative fudge factor (denoted ) to all the rates, calibrated against different benchmark stars. However, unlike what is expected for an intrinsic atomic property, different were obtained by different studies which were found to be star and model atom dependent (e.g. Thévenin & Idiart 1999; Korn et al. 2003; Mashonkina et al. 2016; Bergemann et al. 2012; Ezzeddine et al. 2016a).

To obtain more accurate collision rates, a new semi-empirical recipe was proposed by Ezzeddine et al. (2016b) who developed a quantum fitting method (QFM) to estimate the hydrogen collision rates involved in iron line formation and other elements in the lack of available published quantum rates. Transition energies dependent recipes for charge transfer and excitation rates were introduced to determine more reliable Fe abundances. The QFM has already been successfully applied to line formation calculations in solar-type FGK giants and metal-poor stars, such as the Sun, HD140283 and Arcturus. In this paper, we apply the QFM and our Fe model atom to 20 known UMP stars with . We use high-resolution spectra available in the literature to spectroscopically determine their stellar parameters using NLTE calculations of available Fe lines. This is the first step toward eventually determining their full NLTE abundance patterns.

The paper is structured as follows: In Section 2, we describe the input data used in the NLTE calculations, the iron model atom, and input atomic linelists and model atmospheres. In Section 3, we introduce the method used. Both our LTE and NLTE iron abundances are presented in Section 4, as well as the light element enhancement effects on the final results. In Section 5, we report the NLTE corrections for the LTE iron line abundances, as well as implications for the stellar parameters of the UMP stars. Finally, conclusions are presented in Section 6.

2 Input data for NLTE line formation calculations

2.1 Iron model atom

The iron model atom used in this work was built from all available energy levels for Fe i and Fe ii (846 Fe i and 1027 Fe ii levels and the Fe iii continuum) from the NIST6 database which were then collapsed into superlevels, disregarding fine-structure splitting for all levels except those of Fe i and Fe ii ground levels. The model also includes the predicted high lying Fe i levels from Peterson & Kurucz (2015) that correspond to UV and IR transitions, and establish important collisional couplings with the ground Fe ii level. All levels in the atom are coupled via an extensive linelist of radiative bound-bound (extracted from VALD37 database) and photoionization transitions (extracted from the NORAD8 database). A detailed description of the iron model atom can be found in Ezzeddine et al. (2016b).

Additionally, all levels in the atom are coupled via electron and hydrogen collisions. These collisions have been shown to have important effects on the final NLTE abundances (an extensive study showing this for Mg is presented in Osorio et al., 2015). While quantum atomic data for the hydrogen collision cross-sections have been computed for some light atoms (Li, Be, Na, Mg, Al, Si and Ca; Belyaev & Barklem, 2003, Yakovleva et al., 2016, Barklem et al., 2010, Belyaev et al., 2012, Belyaev, 2013, Belyaev et al., 2014 and Belyaev et al., 2016 respectively), data for larger atoms remain scarce and more difficult to compute.

In ultra metal-poor stellar atmospheres, the hydrogen to electron density ratios can reach up to due to the scarcity of free electron donors (i.e., metals). This enhances the role that inelastic hydrogen collision rates can play for a NLTE abundance determination. Recently, it has been shown that the charge-transfer (i.e., ion-pair production9) processes can dominate over excitation processes (Osorio et al., 2015; Osorio & Barklem, 2016; Ezzeddine et al., 2016b). Cross-section calculations for Li+H (Barklem et al., 2003) and Na+H (Barklem et al., 2010), for example, have shown that the largest cross sections for excitation are small compared with those for ion-pair production from certain energy states. Ion-pair production cross-sections calculations for Fe have not been published yet and were thus excluded in most previous NLTE iron studies for UMP stars.

A new semi-empirical quantum fitting method (QFM) to estimate the hydrogen collision rates, including the ion-pair production process, was developed by Ezzeddine et al. (2016b). This method is based on a general fitting recipe deduced from the quantum collision rates of several elements (Be, Na, Mg, Al, Si and Ca) and then applied to Fe. Tested on 24 Gaia benchmark stars (Jofré et al., 2014; Heiter et al., 2015a) with different stellar parameters, it was shown to improve the Fe i and Fe ii ionization balance and decrease the obtained abundance scatter, especially for the more metal-poor stars in the sample. This motivates the present work with UMP stars which are expected to experience significant NLTE abundance effects.

2.2 Linelists and equivalent widths

For our Fe abundance determination, we use absorption lines and equivalent widths measurements for each UMP star from relevant references in the literature. Details are presented in Section 3.1 below. values for Fe i and Fe ii lines from Gaia-ESO “golden” linelist v4 (Heiter et al., 2015b) were used. The linelist used for each star can be found in Table 3.

2.3 Model atmospheres

We employ 1D, plane-parallel MARCS atmospheric models (Gustafsson et al., 1975, 2008; Plez, 2008) which were interpolated10 to the corresponding input stellar parameters for each star listed in Table 1. We use models with a metallicity of dex for all stars with . Standard -element enhancement of was adopted for all UMP stars. Blanketing effects were taken into account by including background line opacity tables (excluding Fe) as a function of [Fe/H] and  (B. Plez priv. communication). Throughout, we adopt a reference solar iron abundance of (Fe) = 7.50 from Asplund et al. (2009).

3 Method

We use the input data introduced in Section 2 (including the Fe model atom, atomic linelists and 1D MARCS atmospheric models) to determine Fe i and Fe ii line-by-line NLTE abundances. We also report the LTE Fe abundance for each UMP star.

The NLTE radiative transfer code MULTI2.3 (Carlsson, 1986, 1992) was used to compute NLTE line profiles using the Accelerated Lambda Iteration (ALI) approximation method (Scharmer, 1981). It computes the level populations by solving the statistical equilibrium and radiative transfer equations simultaneously with no feedback from the element in question back into the atmosphere (i.e. considered as a trace element). For each line, the NLTE equivalent width was computed using a Voigt profile function with a maximum of 80 frequency points. We also compute the LTE equivalent width from the departure coefficient of each line such as (Wijbenga & Zwaan, 1972). A curve-of-growth (COG) method is then used to determine the line abundances that correspond to the observed . All lines used in the abundance analysis lie on the linear part of the COG ().

3.1 Stellar atmospheric parameters

We determined NLTE spectroscopic atmospheric parameters for the UMP stars, using NLTE calculations of the abundances of individual Fe i and Fe ii lines and upper limits following the method outlined in Section 3. To guide our calculations, we make use of the fact that stellar parameters for all stars have previously been determined under the assumption of LTE, either fully spectroscopically (, ,  and [Fe/H]) or partially (either  or ,  and [Fe/H]). We first computed a small grid of NLTE at stellar parameters centered around the LTE (or photometric) stellar parameters from the literature (see Section 4). We then compared the corresponding grid of computed NLTE equivalent widths, , with the measured observed ones . This was done for all stars for which of at least 5 iron lines could be measured. A first approximation of the initial stellar parameters for each star (in terms of , , [Fe/H]) was obtained using non-linear fitting of the computed to the observed in a --[Fe/H] parameter space. This procedure (using a Levenberg-Marquardt algorithm) takes into account that all stellar parameters depend on each other.

In a second step, the excitation balance of Fe i lines abundances as a function of excitation potential of the lower level, , of each line, as well as the ionization equilibrium of Fe i and Fe ii abundances (when available) were inspected. In the case of an abundance trend with or a mismatch between Fe i and Fe ii abundances, stellar parameters were adjusted accordingly. In the process, we derived the microturbulent velocity by removing any Fe i line abundance trend with reduced equivalent widths ().

Star [Fe/H]
[K] [K] [K] [cgs] [cgs] [cgs] [] [] [] [K] [cgs] [] [dex]
SDSS J22090028 644011 4.012 1.3 0.32
HE 21395432 5270 100 43 3.2 0.30 0.15 1.0 0.2 0.2 0.48
CS 30336049 4685 80 35 1.4 0.30 0.22 2.1 0.2 0.1 0.20 0.31
HE 14240241 5140 60 46 2.8 0.40 0.37 2.2 0.3 0.2 0.30 0.46
HE 00575959 5200 110 68 2.8 0.40 0.40 1.9 0.2 0.3 0.4 0.45
SDSS J0140+2344 5600 100 77 4.6 0.40 0.40 1.0 0.5 0.2 0.26
HE 22395019 6000 80 49 3.5 0.40 0.40 1.8 0.2 0.1 0.42
HE 13100536 5000 70 43 1.9 0.40 0.40 2.2 2.0 1.0 0.48
CD38 245 4700 60 38 2.0 0.40 0.21 2.1 0.2 0.1 0.50 0.25
SDSS J1204+1201 5350 100 45 3.3 0.40 0.40 1.5 0.2 0.2 0.48
CS 22949037 4800 90 67 1.9 0.30 0.20 1.9 0.2 0.2 0.40 0.51
HE 02330343 6020 80 52 3.4 0.40 0.40 1.8 0.3 0.2 0.45
HE 05574840 4800 80 67 2.4 0.30 0.49 1.8 0.4 0.3 0.20 0.38
SDSS J1742+2531 634513 4.014 1.5 0.48
SDSS J1029+1729 581115 4.016 1.5 0.40
SDSS J13130019 5100 80 67 2.7 0.40 0.40 1.8 0.2 0.2 0.10 0.61
SDSS J10350641 626017 4.0/4.418 1.5 0.54
HE 01075240 5050 60 43 2.3 0.40 0.40 2.2 0.3 0.3 0.10 0.75
HE 13272326 6130 100 32 3.7 0.40 0.40 1.7 0.4 0.3 0.4 0.66
SMSS J03136708 512519 2.320 2.0
Table 1: Derived stellar parameters of ultra metal-poor stars with their slope (, for  and ), variation (, for ) and fitting (, for ,  and ) uncertainties on their values (see Section 4.6 for detailed description of each uncertainty). For stars with not enough Fe lines to derive spectroscopic atmospheric parameters, corresponding literature values were adopted (indicated by the table notes below). Columns 11-13 present the NLTE corrections obtained from the differences between our spectroscopic NLTE stellar parameters and previously derived LTE or photometric values. Column 14 shows the NLTE Fe abundance corrections derived in this work.

4 Results

To proceed with our analysis, we divided our sample of 20 UMP stars into three metallicity subgroups following the classification in Placco et al. (2015) as shown in Table 1: Hyper-metal-poor (HMP) stars with , stars with and stars with . We exclude the carbon rich star G 7761 from our UMP sample due to the complexity of its spectrum showing very strong CH, CN, and C2 bands around all iron line regions. Its analysis would require the inclusion of CN and C2 lines in the continuum and line background opacities in the NLTE analysis code which is beyond the scope of the present work. Below we present a brief description of all stars in each subgroup and our stellar parameters in NLTE. We also discuss differences to LTE stellar parameters. All respective parameters are listed in Table 1. Our final NLTE Fe abundances and associated NLTE corrections are further analyzed in Sections 4 and 5 respectively.

4.1 HMP stars with [Fe/H]

This group includes five stars, SMSS J03136708, HE 13272326, HE 01075240, SDSS J10350641 and SDSS J13130019, with the lowest iron abundance of , as determined from their LTE analysis.

SMSS J03136708 is a warm red giant with the lowest known iron abundance (Keller et al., 2014). Only an upper limit of the Fe abundance could be derived because no Fe lines were detected in the spectrum. The authors determined  K and from spectro-photometry, consistent with results from stellar hydrogen line profiles and the derived lithium abundance. Using an equivalent width upper limit of the strongest Fe I lines (at 3859.91 Å), Keller et al. (2014) determined an upper iron abundance limit of . Also employing a , NLTE correction for this line from Lind et al. (2012), led to . Bessell et al. (2015) redetermined the upper limit in NLTE to using a model atmosphere and a spectrum with higher signal-to-noise. A microturbulent velocity of 2.0  was adopted for the star in both studies. More recently, Nordlander et al. (2017) performed a full 3D, NLTE analysis of this star using up to date atomic and hydrogen collisional data independent of classical approximations and free parameters. This led to higher iron abundances than Bessell et al. (2015), of and by fitting a stacked spectra in the vicinity of unblended Fe i lines at 3440.6 Å, 3581.2 Å, 3719.9 Å, 3737.1 Å, 3820.4 Å and 3859.9 Å respectively. This discrepancy with the Bessell et al. (2015) value was explained being due to differences in atomic data and the use of full 3D model as compared to an averaged model.

In this work, we adopt an upper limit for  mÅ for the strongest Fe i lines at 3608.859 Å and 3859.911 Å respectively. We find using only the resonance line at 3859.911 Å. We also compute from the non-resonance line at 3608.859 Å. Our  Å result agrees with that of Bessell et al. (2015), , who used the same line. Our 3608.859 Å LTE value agrees with that of Nordlander et al. (2017) within 0.1 dex (they report ). In NLTE, we determine an upper limit of from the 3608.859 Å line, and from the resonance 3859.911 Å. The abundance obtained from the resonance line is 0.2 dex lower than the 3608.859 Å line. We thus adopt the upper limit of from the resonant line at 3859.911 Å as our final result. This leads to an iron abundance that is 0.80 dex higher than the 3D NLTE result determined by Bessell et al. (2015) (who reported ). This result is in perfect agreement with the 1D, NLTE result of Nordlander et al. (2017) (who reported ). Our 1D, NLTE result is, however, 0.19 dex lower than their 3D, NLTE value. Overall, Nordlander et al. (2017) report agreement between their 1D and 3D NLTE results which adds confidence to our result and the use of our NLTE method. As no Fe lines could be detected, we did not compute any NLTE stellar parameters but instead adopt the temperature, gravity and microturbulent velocity from Bessell et al. (2015).

HE 13272326 is a relatively unevolved star located on either the main-sequence or the subgiant branch (Frebel et al., 2005; Aoki et al., 2006; Frebel et al., 2008). Frebel et al. (2005) used color-effective temperature relations from Alonso et al. (1996), to determine  K from broad-band photometry. They used the proper motion to set limits on the distance, and from a 12 Gyr isochrone with , two solutions, and 4.5, were obtained. Korn et al. (2008) favored a subgiant scenario after carrying out a NLTE Ca i/Ca ii ionization equilibrium analysis. The iron abundance of HE 13272326 was determined using 10 Fe i lines from Frebel et al. (2008) as no Fe ii lines could be detected. For the subgiant case, Frebel et al. (2008) derived and . A nominal NLTE correction of 0.2 dex (without any tailored calculation) was adopted in Frebel et al. (2005) following Asplund (2005). A microturbulent velocity of was adopted throughout.

We use 10 Fe i lines from Frebel et al. (2008) for our analysis of HE 13272326, and additionally a strong Fe ii line at 5018.45 Å for which we use an upper limit of  mÅ. Applying our stellar parameters fitting method described in Section 3.1, we find a best fit at =3.7, =6130 K but adopt  as in Frebel et al. (2008), given the paucity of lines. These values satisfy both the excitation and ionization equilibrium (to the extent the upper limit allows). We thus also favor the subgiant scenario, in agreement with Korn et al. (2008). Our  result agrees well with that of Frebel et al. (2005), however, detecting and measuring any Fe ii lines in this star would provide a better constraint on . Using our derived stellar parameters, we determine two sets of abundances for each  scenario. As such, we derive iron abundances of and for the subgiant case, and and for the dwarf case. Our subgiant LTE Fe abundance agrees with Frebel et al. (2008) within acceptable 0.11 dex.

HE 01075240 is a red giant star (Christlieb et al., 2002). Christlieb et al. (2004) derived  K following - relations by Alonso et al. (1999, 2001). They used different methods including relative strengths of Balmer line wings and evolutionary tracks to constraint the surface gravity.  dex was eventually adopted. of 25 Fe i lines were measured and one upper limit of  mÅ for the Fe ii line at 5018.440 Å. was determined by forcing the abundances of Fe i lines to have no trend with line strengths. This led to iron abundances of . Adopting a nominal NLTE correction of 0.11 dex, they report (without carrying any detailed NLTE calculation).

Using our fitting method, we determine atmospheric parameters of  K, and , in good agreement with those presented in Christlieb et al. (2004). Our LTE abundance of is in very good agreement with that of Christlieb et al. (2004). We then determine and from the same lines as in Christlieb et al. (2004). We note that the upper limit for the Fe ii line (5018.44 Å) is already at the level of the Fe i abundance. Should the true Fe ii abundance be significantly lower, the surface gravity of the star would need to be significantly increased.

SDSS J10350641 is a warm dwarf star (Bonifacio et al., 2015). No metal lines were found in its spectrum except for the Ca ii K line and the G-band. Bonifacio et al. (2015) derived  K from a calibration, found to be consistent with what was determined from the H line wings. Using a 12 Gyr isochrone, two possible values for  of 4.0 and 4.4 were found. An upper iron abundance limit of was set from synthesizing the wavelength region of 3820-3860 Å where the three strongest Fe i lines are found. Following Caffau et al. (2013), they assumed   for the microturbulent velocity due to the lack of any Fe lines.

As no iron lines were detected in this star, we could not derive NLTE stellar parameters with our spectroscopic fitting method. Using the   and both  values, and the upper limit on the equivalent width for the Fe i line at 3820.425 Å ( mÅ) from Bonifacio et al. (2015), we determine identical Fe i upper limit abundances in LTE and NLTE for both cases of of and . The LTE upper limit agrees well with that of Bonifacio et al. (2015) within 0.13 dex.

SDSS J13130019 is a star at the base of the red giant branch (Frebel et al., 2015). Its effective temperature  K was determined spectroscopically (LTE) using excitation balance and applying a temperature correction following Frebel et al. (2013). As no Fe ii lines could be detected in the spectrum, a surface gravity of was obtained using a 12 Gyr isochrone at . Iteratively, a microturbulent velocity of and iron abundance of were determined.

We determine atmospheric parameters of  K and . As no Fe ii lines were detected, we could not derive  via ionization equilibrium, however with our fitting method we obtain a best fit at . This value could be further investigated with the detection and measurement of Fe ii lines. We then used 36 Fe i lines from Frebel et al. (2015) to determine iron abundances of and .

4.2 UMP stars with [Fe/H]

These include SDSS J1029+1729, SDSS J1742+2531, HE 05574840 and HE 02330343.

SDSS J1029+1729 is a Turn-Off (TO) star first analyzed by Caffau et al. (2011, 2012). They derived  K from color relations from Ludwig et al. (2008) and from Ca i/Ca ii ionization equilibrium (due to the lack of any Fe ii lines). They obtained following the relation of Edvardsson et al. (1993). Iron abundances were derived as in LTE and , as obtained from adding 3D corrections to a NLTE analysis.

Only 3 Fe i lines could be measured by Caffau et al. (2012). Due to the scarcity of lines, we did not derive any NLTE stellar parameters for this star. Using stellar parameters and the three Fe i  measurements from Caffau et al. (2012), we derive and . Our LTE value is in agreement with that from Caffau et al. (2012) within error bars.

SDSS J1742+2531 is a warm TO star for which Bonifacio et al. (2015) derived  K, from colors, and in agreement with that derived from H line wings by the authors. Two values for  were determined using a 12 Gyr, isochrone: 4.0 for an evolved past turn-off (TO) star case and 4.3 for an un-evolved case. was eventually adopted, arguing accordance with a metal-poor TO star at this temperature. Using three measured Fe i lines, they derive .

We use the same lines to determine and . Due to scarcity of Fe lines, we refrain from determining other NLTE stellar parameters for this star, and adopt the temperature, gravity and from Bonifacio et al. (2015).

HE 05574840 is an evolved red giant star with  K, as determined from fitting Balmer lines (Norris et al., 2007). was determined from the Fe i/Fe ii and Ca i/Ca ii ionization equilibrium, and subsequently was obtained. From these stellar parameters they derived .

Using 59 Fe i and 1 Fe ii lines from Norris et al. (2007), we determine  K, and . Subsequently, we derive and , where our LTE value agrees with that of Norris et al. (2013).

HE 02330343 is a warm subgiant, and one of three UMP stars first analyzed by Hansen et al. (2014).  K was derived by fitting spectro-photometric observations with 1D, LTE MARCS model atmospheric fluxes and by employing -element enhanced, 10 Gyr isochrone at . No Fe ii lines were detected in the spectrum. and were derived from 11 Fe i lines.

Using measurements of Fe i lines provided by T. Hansen (priv. communication), we determine in NLTE  K, and using the fitting method, and derive and .

4.3 UMP stars with [Fe/H]

These include CS 22949037, SDSS J1204+1201, CD38 245, HE 13100536, HE 22395019, SDSS J0140+2344, HE 00575959, HE 14240241, CS 30336049, HE 21395432 and SDSS J22090028.

CS 22949037 is a well studied red giant star (Beers et al., 1992; Depagne et al., 2002; Roederer et al., 2014). Beers et al. (1992) determined from medium resolution spectrum, using the strength of the Ca ii K line. This value was then re-determined by many authors, e.g. Depagne et al. (2002) who derived , and Roederer et al. (2014) who reported . Depagne et al. (2002) estimated  K from giant stars color-calibrations relations from Alonso et al. (1999). was derived using Fe i/Fe ii and Ti i/Ti ii ionization equilibrium. Finally, was deduced by minimizing the Fe abundance trend as a function of .

We determine  K, =1.9 and , and subsequently derive and from 65 Fe i and 5 Fe ii lines from Depagne et al. (2002). Our LTE result agrees very well with that of Depagne et al. (2002).

SDSS J1204+1201 is an evolved subgiant star (Placco et al., 2015). The authors determined  K from excitation balance of Fe i lines, by employing a 12 Gyr isochrone at and from Fe i lines balance with reduced . From the above parameters, they derive .

We employ our fitting technique to determine  K, and . and were then derived using 20 Fe i lines from Placco et al. (2015). No Fe ii lines were detected.

CD38 245 is a red giant star first studied by Bessell & Norris (1984). Cayrel et al. (2004) determined  K following Alonso et al.’s (1999) color-indices calibration, from Fe i/Fe ii and Ti i/Ti ii ionization equilibrium and by minimizing Fe i abundance- trend. Using these stellar parameters, they deduced .

We determine and using 102 Fe i and 7 Fe ii lines from Cayrel et al. (2004). Our NLTE stellar parameters are  K, log  and  km s. While our  agrees with that of Cayrel et al. (2004) within error bars, it is worth noting that our  value is 0.5 dex higher than theirs.

HE 13100536 & HE 22395019 are two stars also analyzed by Hansen et al. (2014). Their stellar parameters were derived in the same way as that of HE 02330343. The authors derived  K, , and for HE 13100536, and  K, , and for HE 22395019 respectively.

We determine and from 17 Fe i lines for HE 13100536 and and from 15 Fe i lines for HE 22395019. Our best fit NLTE stellar parameters agree with those derived by Hansen et al. (2014). The Fe i NLTE line abundances are found to have no trend with . As no Fe ii lines could be detected, we could not test whether the Fe i/Fe ii agreement was satisfied. However, future detection and measurements of Fe ii lines can better validate our  results.

SDSS J0140+2344, CS 30336049 & HE 21395432 are three stars analyzed by Norris et al. (2013). The authors used spectro-photometry, Balmer lines fitting and H line indices to determine  K,  K and  K for SDSS J0140+2344, CS 30336049 and HE 21395432, respectively. For SDSS J0140+2344, no Fe ii lines were measured, and the authors employed a 12 Gyr isochrone to determine two  values: 4.7 for a dwarf case and 3.4 for a subgiant. Two corresponding values of and for  and and for [Fe/H](LTE) were determined, respectively. For CS 30336049, 3 Fe ii lines were detected, and 1 Fe ii line for HE 21395432. Using Fe i/Fe ii ionization equilibrium, they derived and for CS 30336049 and HE 21395432, respectively. From these parameters, and for CS 30336049 and and for HE 21395432 were determined.

We employed our fitting parameter technique to determine NLTE stellar parameters for these stars. For SDSS J0140+2344, we find  K, and . Based upon these results, we favor the dwarf scenario over the subgiant case, since our determined resulted from a much better fit of the computed to the than in the subgiant case. Using 35 Fe i lines from Norris et al. (2013), we subsequently determine and for SDSS J0140+2344. For CS 30336049 and HE 21395432, we use 74 Fe i and 3 Fe ii lines and 32 Fe i and 1 Fe ii lines from Norris et al. (2013) respectively to determine  K, and and  K, and , respectively. It follows that we obtain and for CS 30336049 and and for HE 21395432.

HE 00575959 & HE 14240241 are two evolved red giant stars studied by Cohen et al. (2004) and Cohen et al. (2008), respectively. Both studies used color indices to determine  K and  K for HE 00575959 and HE 14240241, and 12 Gyr isochrones to derive and , respectively. Removing abundances vs. line strength trends, they obtained and . Using these parameters, they derived and for HE 00575959 and HE 14240241, respectively.

For HE 00575959, we determine and from 53 Fe i lines from Cohen et al. (2004) (no Fe ii lines were detected). With our fitting method, we determine NLTE  K, and . For HE 14240241, we use 39 Fe i and 5 Fe ii lines to determine and . The LTE results are in good agreement with Cohen et al. (2004) and Cohen et al. (2008). We obtain  K, and for this star.

SDSS J22090028 is a warm dwarf star (Spite et al., 2013). The authors determined  K using color calibrations from Ludwig et al. (2008). They assumed which they found to satisfy the Fe i/Fe ii ionization equilibrium. They adopted and determined .

Using 5 Fe i lines from Spite et al. (2013), we determine and . Due to scarcity of Fe detected lines, we do not determine other NLTE stellar parameters, but instead adopt those from Spite et al. (2013).

Star [Fe i/H] [Fe ii/H] [Fe i/H] [Fe ii/H] [Fe/H] [Fe/H] N Fe i N Fe ii
SDSS J22090028 5 0
HE 21395432 32 1
CS 30336049 74 3
HE 14240241 39 5
HE 00575959 53 0
SDSS J0140+2344 35 0
HE 22395019 15 0
HE 13100536 17 0
CD38 245 102 7
SDSS J1204+1201 20 0
CS 22949037 65 5
HE 02330343 11 0
HE 05574840 59 1
SDSS J1742+2531 3 0
SDSS J1029+1729 3 0
SDSS J13130019 36 0
SDSS J10350641 1 0
HE 01075240 25 1
HE 13272326 10 1
SMSS J03136708 1 0
Table 2: Our results for the average Fe i, Fe ii and total Fe abundances with their standard deviation errors obtained for the UMP stars both in LTE and NLTE respectively. The number of Fe i and Fe ii lines used in the analysis are shown in the last two columns. Values are reported relative to the solar iron abundance of from Asplund et al. (2009).

4.4 Final Fe i and Fe ii abundances

We present our final NLTE Fe i and Fe ii abundances (whenever possible) and their standard deviations () in Table 2, computed with the spectroscopically determined NLTE stellar parameters given in Table 1. For comparison, we also present our corresponding LTE values. Despite the scarcity of Fe i and even more of Fe ii lines, we find that our 4-dimensional spectroscopic fitting method gives consistent Fe i and Fe ii abundances to within  dex without having to force this agreement. This adds confidence in our iron atomic model and method used, in addition to the NLTE derived stellar parameter. Additionally, we find slightly smaller standard deviations in the NLTE Fe abundances compared to LTE for most stars.

Line abundance dispersion can be due to a number of factors including uncertainties in oscillator strengths and other atomic data, measurements, model hypotheses (1D/3D, LTE/NLTE, ). We try in this work to address these possible causes by using the best available gf-values, and include a new approximation of hydrogen inelastic collisions in our NLTE modeling. The scatter is indeed reduced for most stars (See Table 2). However, a full 3D, NLTE analysis would likely decrease the scatter even more, but it is still challenging and computationally expensive to employ. In this context, it is encouraging that for SMSS J03136708, the only UMP star for which a full 3D, NLTE analysis has been performed, Nordlander et al. (2017) report fairly similar 1D NLTE and 3D NLTE results (within 0.2 dex), whereas much larger differences were obtained between LTE and NLTE models. This highlights that accurate abundances can presently be most efficiently obtained with 1D, NLTE models, such as the present study, whenever reliable atomic data are included. A few other full 3D NLTE calculations should however be performed to confirm this conclusion.

4.5 Light element enhancement effects on final Fe abundances

While Fe is usually considered a good proxy of the overall metal content of most stars, UMP stars can have large abundance enhancements in light and elements such C, N, O, Na, Mg, Si, Ca and Ti relative to iron. These elements can be important electron donors and can thus potentially affect the final Fe derived abundances. The feedback contribution from these elements are customarily treated by using -enhanced input model atmospheres of for all stars of .

Some of the most metal-poor stars have [C/Fe] values of 3 dex or more, with similar O and N abundances, see Table 6 of Placco et al. 2015 for C and N abundances for this sample of UMP stars. Zhao et al. (2016) showed that -elements such as Mg, Ti and Ca maintained constant values of relative to Fe for , but with [Ca/Fe] having tendencies to increase below , up to 0.6 dex at . Other studies of UMP stars, have shown that this ratio can potentially be higher than the canonical . Often it also varies from one element to another.

A detailed systematic elemental abundance study of UMP stars is therefore needed to quantify how these light element enhancements might affect the iron abundances as well as the stellar parameters of our sample stars. We therefore tested the effects of potential light element enhancements. We thus arbitrarily increased the input metal abundances of the stellar model atmosphere of each UMP star by +0.50 dex, in addition to the standard enhancement.

We then recomputed the final [Fe/H] NLTE abundances, and along the way recorded any potential changes in the spectroscopically determined stellar parameters. The changes are found to be  independent but  dependent. Hence, the abundances from Fe i lines are affected, while those from Fe ii are hardly changing. Cooler stars display stronger effects upon increasing the metal enhancement than warmer stars. For one of the coolest star in our UMP sample, CD38 245 (=4700 K), a change of  dex in the input model metal abundance results in a  dex decrease in the average Fe i abundances and a slight 0.02 dex increase in the Fe ii abundances. This corresponds to a compensated  dex change in . For a hotter star, HE 21395432 (=5270 K), smaller changes of 0.06 dex were obtained for Fe i and 0.02 dex for Fe ii, while negligible differences were noted for the other stellar parameters, including  change of . For the even hotter star, J1204+1201 (=5350 K), smaller differences of 0.01 dex for Fe i were obtained.

On average, for most UMP stars within  K, this additional metal abundance enhancement of +0.50 dex thus causes a slight decrease in the final Fe abundance by at most  dex. This value is much smaller than the typical error bars and can be thus considered negligible. For cooler stars below 5000 K, however, the enhancement results in larger decreases in the final Fe i abundances of typically 0.2-0.3 dex. If the metal abundance enhancement was pushed to +1.00 dex, the Fe i abundance would decrease by 0.3-0.4 dex. This decrease in the Fe i abundances is due to electron pumping in the atmospheric model upon enhancing the metal model metallicity, thus increasing the electron collisional rates by decreasing the photon mean free path, and hereby pushing the abundances lower toward LTE. Fe ii abundances, always being the dominant species, are much less affected by these changes. We note that the second most iron-poor star HE 13272326 that is also extremely enhanced in C, N and O relative to Fe, is a warm main-sequence star near the turnoff and thus likely not affected by such model metallicity enhancement changes. The warm giants J13030019 and HE 01075240 also with large C, N, and O enhancements, on the other hand, might be affected by this at the 0.3-0.5 dex level.

4.6 Uncertainties in stellar parameters

We report random and systematic uncertainties on , , [Fe/H] and  of our stellar parameters in Table 1. Our procedure for obtaining stellar parameters results in multiple uncertainties which we discuss in the following:

Our initial stellar parameters are obtained with a non-linear fitting method whose uncertainties arise from the covariance matrix in a --[Fe/H]- parameter space which reflect how constrained the parameters are by the data (both measured and computed ). Uncertainties depend on the error estimates of the measured which typically vary from 1-5 mÅ. Unfortunately, the used in this work have been adopted from different reference studies that do not always report measurements uncertainties. In those cases, we adopt a nominal value of 2 mÅ for all the lines. This value is typical for low S/N spectra of UMP stars (e.g., Bonifacio et al., 2015; Caffau et al., 2012). The resulting typical fitting uncertainties for all our stars are 50 K in , 0.2 dex in  and 0.2  in . We note that employing a higher value for nominal measurement uncertainty of 5 mÅ would increase the uncertainties up to 100 K in , 0.4 dex in  and 0.3  in . It is important to note that the level fitting uncertainties obtained for  are underestimated for stars lacking any Fe ii line detections. The  values obtained from the fitting method for HE 13272326, HE 01075240, J13130019, HE 02332343, J12041201, HE 13100536, HE 22395019, J01402344, HE 00575959 are driven solely by Fe i lines and thus their uncertainties from the method do not properly reflect the surface gravity dependence on Fe ii lines. We therefore decided to use a constant value of 0.40 dex as a  fitting uncertainty for these stars, as an average of the values reported by previous studies for  of UMP stars with no Fe ii lines (e.g. Frebel et al. 2008; Caffau et al. 2012; Frebel & Norris 2015; Bonifacio et al. 2015). Other random uncertainties arise from the uncertainty of the slopes of Fe line abundances versus and when obtaining the final effective temperature and   respectively. Varying the slope within its uncertainty (as determined by the data points) induces a change in the slopes which results in typical uncertainties of 100 K in  and 0.2  in . Individual results () for each star are listed in Table 1. Random uncertainties of  are determined from varying the ionization equilibrium of Fe i and Fe ii within their uncertainties. We adopt the corresponding change as final  uncertainty (), with typical values of 0.3 dex. For stars with no Fe ii detection, the same constant value as for the fitting uncertainties of 0.40 dex was used.

We now report uncertainties in our final Fe i and Fe ii abundances. First, the dispersion in individual line measurements, quantified by the standard deviation (). Typical values are 0.12 dex in NLTE and 0.20 dex in LTE for Fe i. There are not enough Fe ii lines to meaningfully quantify this for Fe ii so we adopt the Fe i results instead. We take the standard deviation as our Fe abundance uncertainties because the standard errors of Fe i would be unrealistically small (e.g., 0.02 and less). () are reported in Table 2. Second, systematic uncertainties arising from varying the stellar parameters ,  and  by about their uncertainty of  K,  cgs and  respectively. The resulting changes in the average Fe abundances typically are  dex in Fe i and  dex for Fe ii for changes in changes in ,  dex for Fe i and  dex for Fe ii for changes in  and finally  dex for Fe i and  dex for Fe ii for changes in . Total Fe abundance uncertainties are obtained by summing individual uncertainties ( and ) in quadrature. This leads to a typical total average value of 0.13 dex.

Similarly, the total uncertainties in the other stellar parameters are obtained by summing individual uncertainties (, , and ) in quadrature. This leads to typical total uncertainties of 112 K in , 0.45 dex in  for stars with Fe ii detection and 0.55 dex for stars without, and 0.4 in . The individual uncertainties for each star are listed in Table 1. These uncertainties well reflect the challenge of having available only a limited number of Fe lines in these most iron-poor stars.

5 NLTE corrections

We now discuss the differences between our NLTE and LTE iron abundances [Fe/H] for the UMP stars. We also report the differences between previously determined stellar parameters (,  and ) from the literature (where either full LTE or partial LTE and photometric methods were used). These NLTE corrections for [Fe/H] are shown in Table 2, while those for ,  and  are listed in Table 1.

5.1 [Fe/H] abundance corrections

We define the NLTE Fe line abundance correction for a specific spectral line as the difference between the NLTE and LTE Fe abundance for a given measured equivalent width. We calculate [Fe/H] [Fe/H] - [Fe/H], based on the average abundance differences across all individual Fe lines. The results as well as the number of Fe i and Fe ii lines used for each UMP star are listed in Table 2. The corrections are found to increase with decreasing [Fe/H] which can be understood due to the increasing magnitude of the over-ionization ( excess) in the UV. This over-ionization shifts the ionization-recombination balance towards more efficient ionization, thus de-populating the lower levels relative to LTE. This effect grows larger at lower metallicities as radiative rates become more efficient due to the decrease in electron number densities in the optically transparent atmospheric layers (Mashonkina et al., 2011; Lind et al., 2012; Mashonkina et al., 2016). The deviation from LTE in the line formation within the depth of the stellar atmosphere can be seen in Figure 2, where the relative populations (NLTE to LTE) of the ground Fe i level for the UMP stars with are displayed along their atmospheric depths at 5000 Å (). While the departures from LTE increase with decreasing Fe abundances, other factors such as lower gravities and higher effective temperatures can also play a role in the population deviations from LTE throughout the stellar atmospheres (Lind et al., 2012; Mashonkina et al., 2016).

The NLTE corrections as a function of [Fe/H](LTE) for the UMP stars are shown in Figure 1. The data are easily fit with a linear relation:

Figure 1: Differences between LTE and NLTE Fe abundances as a function of [Fe/H](LTE). UMP stars are shown as star symbols, and an additional three metal-poor reference stars as open circles.  [K] for all stars is color-coded. Black stars, the square and the triangle represent different possible NLTE corrections obtained for SMSS J03136708 as follows: (i) Filled black star is based on using the resonance line at 3859.911 Å. (ii) Open black star is the correction based on the line at 3608.859 Å. (iii) Open square is based on 1D NLTE corrections from Nordlander et al. (2017). (iv) Open triangle is based on the difference between 3D, NLTE to the 1D, LTE results by Nordlander et al. (2017) (see text for discussion). The red dashed line represents the fit for the UMP stars excluding SMSS J03136708. Open circles correspond to NLTE corrections obtained using Equation 1 for the benchmark metal-poor stars HD 140283, HD 84937 and G 6412. Red crosses correspond to the NLTE corrections obtained by Amarsi et al. (2016). The reference stars agree perfectly with the suggested UMP trend of [Fe/H]. All stars lie within the fitting slope error bars region of (gray shaded region).

The upper limit correction of for SMSS J03136708 was excluded from the fit as no iron lines detection were made in this star. The fitting fitting slope and -intercept standard errors are respectively (shown by the gray shaded area in Figure 1) and . All the stars, including SMSS J03136708, lie within this error bar (gray shaded) region.

This tight relation allows extending the NLTE corrections to other stars, and potentially also towards higher metallicities (). We test this on the benchmark metal-poor stars HD 84937 (), HD 140283 () and G 6412 () (Amarsi et al., 2016). Using Equation 1, we calculate NLTE corrections of 0.14 dex, 0.22 dex and 0.29 dex for HD 84937, HD 140283 and G 6412, respectively. Amarsi et al. (2016) studied these three stars using a full 3D and 1D NLTE analyses, using for the first time quantum mechanical atomic data for hydrogen collisions, and reliable non-spectroscopic atmospheric parameters. The authors report 0.14 dex and 0.21 dex and 0.24 dex as 1D NLTE corrections for HD 84937, HD 140283 and G 6412, respectively. These values are in excellent agreement with our values. Our fit can thus be used to predict NLTE corrections of metal-poor stars though the whole range of metallicities [Fe/H] from at least -8.00 to -2.00 dex, which further asserts that our relation can be used and applied to LTE Fe abundances of a variety of metal-poor stars.

5.2 Consequences for spectroscopic determination of stellar parameters  and

We present in Table 2 the difference in stellar parameters ,  and  between our NLTE and previously derived LTE spectroscopic or photometric values, whenever possible. This illustrates the changes by going to a full NLTE Fe line analysis. We obtain positive of 0.1 - 0.5 dex for all UMP stars whenever a NLTE  derivation was possible. An important consequence is that surface gravities derived by LTE analyses tend to be lower than what is expected in NLTE. LTE values should thus be corrected before any further elemental abundance determination. Our positive NLTE  corrections are in agreement with previous studies, e.g., Thévenin & Idiart (1999) who have found positive for a large number of metal-poor stars. Their values were found to be in agreement with spectroscopic independent  determinations, e.g. those derived from HIPPARCOS parallaxes.

Our NLTE  agree within error bars with the those photometrically derived values using color- calibration relations (e.g. Alonso et al. 1999, 2001). However, comparing values spectroscopically determined through LTE Fe i excitation equilibrium with our results shows deviations of up to  K, where NLTE  values are lower (). These deviations can be expected as the LTE Boltzmann equilibrium of atoms cannot pertain at lower metallicities, especially for the non-dominant neutral Fe species, which will affect the excitation balance of Fe i lines.

We treat the microturbulent velocity  as a free parameter but do not further consider the obtained results for . Nevertheless, we report the obtained values in Table. 2.

Given that there is a strong correlation of as a function of [Fe/H], we also attempted to determine as a function of  and . This requires inspecting lines of similar strengths (), and lines with the same lower levels excitation potential , as abundances derived from similar lines depend on the thermal stratification of the atmosphere. For example, low excitation lines are mostly prone to 3D effects which can lower Fe i abundances of metal-poor stars by  dex (Amarsi et al., 2016). However, due to the small number of UMP stars, and scarcity of Fe lines, there is not enough data available to map out temperature and surface gravity dependent . It is thus rather difficult to quantify the dependence of NLTE effects on other parameters than [Fe/H] in our sample of stars. If more stars are found in ongoing and future surveys, this question should be revisited. Nevertheless, spectroscopically derived stellar parameters using the LTE formalism have to be corrected for NLTE effects.

Figure 2: Departure coefficients for eight illustrative UMP stars with . Departures are shown for the ground level populations of both Fe i and Fe ii as a function of optical atmospheric depth at 5000 Å. Fe i populations deviate strongly from LTE down to at the lowest [Fe/H] for SMSS J03136708, while Fe ii population deviations can be considered negligible compared to Fe i.

6 Conclusions

We have presented 1D, NLTE Fe line-by-line formation computations for 20 UMP stars with . We use NLTE Fe i and Fe ii lines abundances, when available, to also determine spectroscopic stellar parameters ,  and   in addition to [Fe/H]. Our results show that:

  • Our NLTE Fe abundance corrections for the UMP stars are larger than any previous determinations, up to  dex at the lowest iron abundances. These results set a new scale of NLTE corrections to be applied to LTE abundances of other metal-poor stars. The larger corrections are mainly due to performing a full NLTE analysis, using new estimates of hydrogen collision rates and the inclusion of charge transfer rates for the first time in the NLTE analysis of UMP stars.

  • The line-by-line abundance scatter in NLTE is decreased for most stars down to  dex as compared to LTE.

  • The NLTE corrections we calculated over the range can be extrapolated up to at least , to predict NLTE corrections, in perfect agreement with independent (1D and 3D, NLTE) determinations.

Even though the number of known UMP stars has greatly increased over the last few years to a sample of 20 stars, the relatively small number remains a shortcoming to a full stellar population analysis. Future surveys are expected to deliver additional UMP stars, hopefully extending to . However, now is the time to revisit existing data and to analyze the known stars as precisely and uniformly as possible, as is presented for Fe abundances in this work.

Our results provide a fist step towards a full NLTE chemical species analysis of UMP and EMP stars. A full NLTE abundance pattern will enable us to put constraints on the Initial Mass Function (IMF) and other properties of Pop III stars, by comparing accurately computed NLTE abundances of a full set of elements to model supernova yields (e.g., as has been done in LTE by Placco et al. 2015).

We thank Piercarlo Bonifacio for the useful conversation and advice. We are thankful to Terese Hansen for providing us with equivalent width measurements for some of our studied stars. R.E. acknowledges support from a JINA-CEE fellowship (Joint Institute for Nuclear Astrophysics - Center for the Evolution of the Elements), funded in part by the National Science Foundation under Grant No. PHY-1430152 (JINA-CEE). A.F. is supported by NSF-CAREER grant AST-1255160. AF acknowledges support from the Silverman (1968) Family Career Development Professorship. This work was supported in part by the French-Lebanese Program PHC Cedre. This work has made use of the VALD database, operated at Uppsala University, the Institute of Astronomy RAS in Moscow, and the University of Vienna.
Star name Ion log  (X) (X)
[Å] [eV] [mÅ] [dex] [dex]
SDSS J22090028 Fe i 4045.810 1.48 0.28 36.4 3.50 3.82
Fe i 4063.590 1.56 0.07 31.7 3.67 3.95
Fe i 4071.740 1.61 0.02 28.9 3.72 3.99
Fe i 4383.549 1.48 0.20 28.8 3.36 3.72
Fe i 4404.750 1.56 0.14 17.4 3.46 3.78

Note. – This table is published in its entirety in the machine-readable format. A portion is shown here for guidance regarding its form and content.

Table 3: Atomic linelists and obtained Fe i and Fe ii abundances of the UMP stars.


  1. affiliation: Joint Institute for Nuclear Astrophysics, Center for the Evolution of the Elements, East Lansing, MI 48824, USA
  2. affiliation: Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
  3. affiliation: Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
  4. affiliation: Joint Institute for Nuclear Astrophysics, Center for the Evolution of the Elements, East Lansing, MI 48824, USA
  5. affiliation: Laboratoire Univers et Particules de Montpellier, Université de Montpellier, CNRS, UMR 5299, Montpellier, France
  9. During an atomic collision with hydrogen, i.e., A+H, the valence electron associated with the atom A has a certain probability to tunnel into the H atom, resulting in a predominantly ionic charge distribution or an ion-pair production A+H.
  10. The interpolation routine interpol_modeles.f from Thomas Masseron available on was used.
  11. From photometry; Spite et al. (2013)
  12. Fixed adopted value; Spite et al. (2013)
  13. From photometry and H- wings fitting; Bonifacio et al. (2015)
  14. From 12 Gyr isochrone; Bonifacio et al. (2015)
  15. From photometry; Caffau et al. (2012)
  16. From Ca i/Ca ii ionization equilibrium; Caffau et al. (2012)
  17. From photometry and H- wings fitting; Bonifacio et al. (2015)
  18. From 12 Gyr isochrone; Bonifacio et al. (2015)
  19. From spectrophotometry and H line profiles fitting; Bessell et al. (2015)
  20. From spectrophotometry and H line profiles fitting; Bessell et al. (2015)
  21. footnotetext: No line detection, upper limit only.


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