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Issue
A&A
Volume 624, April 2019
Article Number A60
Number of page(s) 51
Section Atomic, molecular, and nuclear data
DOI https://doi.org/10.1051/0004-6361/201833553
Published online 10 April 2019

© ESO 2019

1. Introduction

Modern quantitative stellar spectroscopy routinely measures spectral line properties, such as elemental abundances, to a precision of 0.01 dex. However, systematic errors often dominate derived spectral line properties resulting in actual errors that are typically an order of magnitude larger (Barklem 2016). The work of Lindegren & Feltzing (2013) shows that the uncertainties in elemental abundances are a limiting factor in discriminating stellar populations, and that small improvements in uncertainty can significantly reduce the number of stars required for galactic archaeology. Such improvements are important for current and upcoming large-scale galactic surveys such as Gaia (Gaia Collaboration 2016), GALAH (De Silva et al. 2015), WEAVE (Dalton et al. 2012), and 4MOST (de Jong et al. 2012).

One of the main uncertainties in elemental abundances is the quality of the adopted atomic data used in spectral synthesis calculations (Bigot & Thevenin 2008). The Belgian repository of fundamental atomic data and stellar spectra (BRASS) aims to provide astronomers with quality information for the large amount of atomic data available for high-resolution optical spectroscopy, in an attempt to help reduce systematic input errors in quantitative spectroscopy from atomic data and line selection (Lobel et al. 2017). Previously, we retrieved and cross-matched a large quantity of atomic data from several major atomic databases such as the Vienna Atomic Line Database (VALD3; Ryabchikova et al. 2015), the National Institute of Standards and Technology Atomic Spectra Database (NIST ASD; Kramida et al. 2018), and providers within the Virtual Atomic and Molecular Data Centre (VAMDC; Dubernet et al. 2016), in preparation for quality assessment work (Laverick et al. 2018; hereafter Paper 1). In this work the atomic data of seemingly “unblended” spectral lines are quality assessed against several benchmark dwarf stars, including the Sun, spanning late F-type to early K-type stars, for the spectral range 4200–6800 Å1. Unblended spectral lines for stars of ∼G2V spectral type are identified in a homogeneous manner using both the observed benchmark spectra, and the theoretical input line list of Paper I. Astrophysical oscillator strengths2 are derived for these unblended lines, using two commonly utilised methods, to gauge the reliability of the spectral line for quantitative spectroscopy and to produce “benchmark” log(g f) values for quality assessment of atomic data. The literature log(g f) values are then compared against these benchmark log(g f) values to determine which literature values reliably reproduce the stellar spectra of cool dwarf stars, and thus whether the values can be recommended for spectroscopic modelling. This paper presents three main sets of results:

  • a set of extremely high-quality stellar spectra, with signal-to-noise ratios of S/N ∼ 800−1200 and a resolution R ∼ 85 000, for six well-studied stars between 5000–6000 K;

  • an identification of unblended and reliable atomic lines in these stellar spectra, determined using theoretical input line lists and the behaviour of the observed spectra;

  • a set of benchmark log(g f) values for these unblended lines, including a quality assessment of the available atomic data log(g f) values per spectral line.

Section 2 outlines the selection of the 5000–6000 K benchmark stars, the data reduction of the observed spectra, the details of the spectral synthesis calculations, and the stellar parameter determination of the benchmark stars. Section 3 covers the selection of theoretically unblended spectral lines and the measurement of their observed counterparts. Section 4 details the determination of astrophysical oscillator strengths for the selected lines, as well as their uncertainties. Finally, Sect. 5 explains how the derived oscillator strengths and uncertainties are used as benchmark values to perform a quality assessment of the available literature atomic data for the selected spectral lines. The findings of our quality assessment for over 800 spectral lines are also discussed.

2. Benchmark stars between 5000 and 6000 K

2.1. Selection and observations of benchmark stars

For the benchmark star selection the following criteria were imposed: The stars must

  • fall within the temperature range 5000–6000 K;

  • not belong to any known stellar variability classes;

  • not be chemically peculiar or metal-poor, i.e. have roughly solar-like abundances;

  • have low v sin i-values so that spectral lines are narrow;

  • ideally be dwarf stars to avoid non-LTE effects due to extended atmospheres;

  • be brighter than seventh magnitude in V so that an extremely high-S/N spectrum of the object can be obtained.

Using these criteria the following stars were chosen: ϵ Eri, 70 Oph A, 70 Vir, 51 Peg, 10 Tau, β Com, and the Sun. The benchmark spectra were obtained using the HERMES echelle spectrograph mounted on the Mercator 1.2 m telescope at the Roque de los Muchachos Observatory, La Palma, Spain, which is able to observe the complete wavelength range 3800–9000 Å in a single exposure at a resolution of R ∼ 85 000 (Raskin et al. 2011). Each object was observed 10–50 times, depending on the V mag, in succession throughout a single night. Extra flat-fields were also taken during the night to avoid introducing systematic noise in the reduction process. The resultant exposures were reduced using the dedicated HERMES pipeline (release V6.0) and co-added to produce a single stacked spectrum per object with a S/N of ∼800–1200. The solar spectrum was obtained from Neckel & Labs (1984), taken using the NSO/KPNO Fourier Transform Spectrograph (FTS) with a spectral resolution of R ∼ 350 000 and a S/N of ∼2500. The wavelength range in this work is limited to 4200–6800 Å to avoid heavily blended features in shorter wavelengths, and to avoid telluric contamination in the longer wavelengths.

2.2. Theoretical spectral modelling

The synthesis work is performed using the atomic line list detailed in Paper I, and 1D hydrostatic, plane-parallel atmospheric models computed using the publicly available ATLAS9 code (Kurucz 1992). The models adopt the updated opacity distribution functions (ODFs) calculated by Castelli & Kurucz (2003), use the “mixing length” approximation detailed by Castelli et al. (1997) in conjunction with a mixing length of L/Hρ = 1.25, and with the ATLAS9 “approximate treatment of overshooting” turned off as recommended by Bonifacio et al. (2012). The model grid of Castelli et al. (1997), calculated with the updated ODFs, was used as a starting point to calculate a much finer grid of stellar models in order to derive stellar parameters for the benchmark spectra. The grid stellar parameters cover 4000 <  Teff <  15 000 K (step size of 50 K), 0.0 <  log g <  5.0 dex (step size of 0.1 dex), −5.0 <  [M/H] <  1.0 dex (step size of 0.2 dex), 0 <  ζμ <  20 km s−1 (step size of 0.5 km s−1), and 0 <  v sin i <  300 km s−1 (step size of 1 km s−1).

The calculated models used the same mixing length and overshooting treatment as Castelli, but adopt the solar abundances of Grevesse et al. (2007) to be consistent with the transfer calculations. All synthetic spectra are computed in local thermal equilibrium (LTE), using the publicly available radiative transfer code TurboSpectrum V12.1.1 (Alvarez & Plez 1998; Plez 2012). The synthetic spectra are computed using the solar abundances of Grevesse et al. (2007). We assume all benchmark stars to have the same elemental abundance distribution as the Sun, but scaled according to the metallicity of the given benchmark star. The spectra are rotationally broadened to the v sin i of the given star, and instrumentally broadened to the spectral resolution of HERMES (or the FTS for the solar spectrum). Finally, the SpectRes Python resampling tool was used when comparing synthetic and observed spectra (Carnall 2017). The choice of synthesis code and stellar models was governed by the aim of the project, namely to constrain the atomic data of many lines, as homogeneously as possible, for stars of B-, A-, F-, G-, and K-spectral types. Section 4.3.2. provides a brief discussion of some of the uncertainties in the adopted modelling procedure.

2.3. Stellar parameter determination

For consistency the spectral parameters of the six benchmark stars observed by HERMES, initially outlined in Lobel et al. (2017), were determined as follows, taking account of the spectral modelling details discussed in Sect. 2.2. The stellar parameters are initially estimated using a limited number of diagnostic Balmer, Fe, and Mg absorption lines. The method then iterates over Teff, log g, [M/H], and ζμ until the best fit is found to the detailed shapes of a more extensive set of ∼30 diagnostic Fe I and Fe II photospheric lines. The fit method iterates until the best fit to the continuum-normalised Fe-line profiles has been accomplished using χ2 minimisation. The iterations over the ζμ proceed until the iron abundance value determined from the Fe I lines is in agreement with the iron abundance value determined from the Fe II lines. The iteration procedure assumes the ζμ stays constant with depth in the line formation regions of these medium-strong Fe lines. During each iteration, the spectrum is continuum-flux normalised using a semi-automatic template normalisation routine described in Appendix A.

The v sin i value, that is, the convolution of the projected rotational velocity and macro-broadening, is also iterated in steps of 1 km s−1 while obtaining the best fit to the detailed Fe line profiles. The final stellar parameter errors are: Teff ± 50 K, log g ± 0.20 dex, [M/H] ± 0.01 dex, ζμ ± 0.10 km s−1, and v sin i ± 1.00 km s−1. The standard solar parameters of Teff = 5777 K and log g = 4.44 dex were adopted, with the remaining parameters determined in this work. The FTS solar spectrum normalisation is described by Neckel & Labs (1984). The stellar parameters of the benchmark stars are listed in Table 1, and are found to be in good agreement with other literature surveys deriving astrophysical parameters for these stellar objects (da Silva et al. 2015; Brewer et al. 2016).

Table 1.

Stellar parameters of the FGK benchmark stars with effective temperatures in the range 5000 ≤ Teff ≤ ∼6000 K.

3. Selection of unblended spectral lines

3.1. Quantification of theoretical line blending

We define the theoretical core blending, Ωcore, of a given line as follows:

Ω core = 1 − ∫ λ 0 − x λ 0 + x ( 1 − F ( λ ) norm line ) d λ ∫ λ 0 − x λ 0 + x ( 1 − F ( λ ) norm tot ) d λ , $$ \begin{aligned} \Omega _{\rm core} = 1 - \frac{\int ^{\lambda _0+x}_{\lambda _0-x}\,\left( 1 - F(\lambda )_{\rm norm}^\mathrm{line}\right)\,\mathrm{d} \lambda }{ \int ^{\lambda _0+x}_{\lambda _0-x}\,\left( 1 - F(\lambda )_{\rm norm}^\mathrm{tot} \right)\,\mathrm{d} \lambda },\end{aligned} $$(1)

where

x = 1 2 W λ line d , $$ \begin{aligned} x = \frac{1}{2} \frac{W_{\lambda }^\mathrm{line}}{d} ,\end{aligned} $$(2)

where W λ line $ W_{\lambda}^{\mathrm{line}} $ is the equivalent width of the given line, d is the central depth of the line relative to the normalised continuum, λ0 is the rest wavelength of the line, F ( λ ) norm line $ F(\lambda)_{\mathrm{norm}}^{\mathrm{line}} $ is the normalised flux of the line, and F ( λ ) norm tot $ F(\lambda)_{\mathrm{norm}}^{\mathrm{tot}} $ is the normalised flux of the total spectrum, including all other spectral features. A line with d = 0 does not appear in the spectrum, whereas a line with d = 1 absorbs all available flux at λ0. A line with Ωcore = 0 (or 0%) is considered completely unblended in its core wavelength region λ0 − x to λ0 + x, whereas a line with Ωcore ≈ 1 (or ∼100%) is completely blended with other spectral features.

The core blending, Ωcore, is calculated for all 82337 lines in the input atomic line list, for the spectra of the Sun and 51 Peg at the resolutions of R ∼ 350 000 and R ∼ 85 000 respectively. Figure 1 shows the Ωcore as a percentage, plotted against the normalised central line depth, d, for the Sun and 51 Peg. Both the Sun and 51 Peg exhibit the same distributions: a substantial number of shallow, heavily-blended background lines, in addition to a number of relatively unblended lines. To reduce the impact of blending on the quality assessment (discussed later in Sect. 4.3.7) a cut-off of Ωcore  ≤ 10% is imposed, marked by the blue vertical lines in Fig. 1. This cut-off was selected as a balance between Ωcore and the number of investigated lines, and therefore it encompasses the full peak in unblended lines for 51 Peg. An additional cut-off on central line depth d ≥ 0.02 is imposed to help ensure that observed line profiles are actually measurable and less affected by any noise in the observed spectra.

thumbnail Fig. 1.

Spectral line depth, d, plotted against theoretical core-blending, Ωcore, (defined in Eqs. (1) and (2)) for all 82337 lines of the input atomic line list for the Sun (G2V) and 51 Peg (G2V). Solid blue lines show the selected cut-offs for the 5000–6000 K benchmark stars of d ≥ 0.02 and Ωcore ≤ 10%. Analysis of hotter benchmark stars will likely need different cut-offs, or potentially a different approach to treating blended spectral lines.

Using these constraints, 1515 atomic lines are retained for 51 Peg, and 1954 lines in the Sun. All 1515 lines are present in both 51 Peg and the Sun, the difference in quantity being attributed to the differing resolutions of the two spectra. Given that the majority of the benchmark spectra are taken with HERMES, the 1515 spectral lines were selected for measurements, and are henceforth described as “unblended” lines. No limits have been placed on the investigated species, other than mandating that they must fall in the investigated wavelength range, and must be deep and relatively unblended according to the theoretical line list.

3.2. Measurement of observed line profiles

The 1515 unblended lines were automatically measured in each of the seven observed benchmark spectra using a single Gaussian profile fit to determine if the feature exists and to determine its equivalent width, Wλ.

The Gaussian fit is optimised using Gauss–Newton non-linear regression, or a Nelder–Mead minimisation in the case of slow convergence, as described by Lobel et al. (2018). The best fit to the observed line flux is limited to the wavelength interval between the two local flux maxima either side of λ0 that exceed 2% of the normalised continuum flux level. A goodness-of-fit value of χ2 ≥ 0.95, in addition to manual inspection, was used to filter out poorly fitted, non-existent, heavily blended, or telluric-contaminated features. After measurements, 1091 spectral lines were found to be suitable for astrophysical log(g f) determination. A small number of these spectral lines cannot be assessed in all seven benchmark stars due to poor fitting in only some of the profiles, however the line can still be investigated at the cost of increased statistical uncertainty, discussed in Sect. 4.3.8. Additionally, the adoption of a Gaussian fit in place of a Voigt profile fit is discussed in Sect. 4.3.5.

4. Determination of astrophysical oscillator strengths for selected lines

There are two commonly employed methods to determine astrophysical log(g f) values: measuring a line equivalent width, Wλ, and converting it into a log(g f) value via the curve of growth (e.g. Sousa et al. 2007; Önehag et al. 2012; Andreasen et al. 2016), or by using detailed transfer calculations to vary a log(g f) value and determine a best-fitting value (e.g. Boeche & Grebel 2016; Tsantaki et al. 2018). Both these methods are subject to a number of assumptions, especially in their treatment of blended lines, which can lead to significant systematic differences in derived log(g f) values if not properly accounted for. As such, both the curve of growth and iterative modelling methods have been explored, with the following section detailing their implementation and associated uncertainties.

4.1. Determination of oscillator strengths using theoretical curves of growth

The measured Wλ value of a spectral line can be converted into a log(g f) value, on a star-by-star basis, using the following curve of growth relationship:

log ( W λ λ 0 ) = log ( g f λ 0 ) + log ( A el ) − 5040 T e E low + log ( C ) , $$ \begin{aligned} \log \left(\frac{W_{\lambda }}{\lambda _0} \right) = \log ({g}f\lambda _0) + \log (A_{\rm el}) - \frac{5040}{T_{\rm e}}E_{\rm low} +\log (C) ,\end{aligned} $$(3)

where Wλ is the equivalent width, λ0 is the transition wavelength, Ael is the elemental abundance of the species, Te is the excitation temperature, Elow is the transition lower energy level, and C contains the remaining terms such as the Saha population factor and the continuum opacity. Theoretical curves of growth are built by varying the log(g f) value of each of the 1091 selected lines over the range −6.0 <  log(g f) < 4.0 dex in steps of 0.1 dex, and then calculating the equivalent widths of the resultant line profiles. The curves are synthesised using the rest-wavelength and energy levels of the selected line, for each of the seven benchmark stars, resulting in a total of 7658 individual curves of growth. A high-order polynomial is fitted through each of the curves, with maximum fit deviation of Δlog(g f) ≪ 0.005 dex. The measured Wλ values are converted into log(g f) values using their respective curve of growth per line per star, as shown in Fig. 2. These individual log(g f)cog values per star are averaged over the benchmark stars to produce a final mean log ( g f ) ¯ cog $ \overline{\log({g}f)}_{\mathrm{cog}} $ value and associated uncertainty, as discussed in Sect. 4.3.8.

thumbnail Fig. 2.

Conversion of observed Cr II line equivalent width values into log(g f) values using theoretical curves of growth for a given star. Blue lines represent theoretical curves of growth for different Cr II lines. Black circles are measured equivalent width values plotted using the log(g f)input values, while black crosses are measured equivalent width values plotted using derived log(g f) values, i.e. the intersect of the equivalent width value with its respective curve of growth.

4.2. Determination of wavelengths and oscillator strengths via iterative detailed modelling

Unlike the curve of growth line profiles, the grid spectra are computed using the line of interest and all other nearby lines present in the input atomic and molecular line lists. A grid of synthetic spectra is calculated for each selected line per benchmark star, in a wavelength window of ±1 Å centred on the rest wavelength of the line. The grid method uses the rest-wavelength and the previously determined log ( g f ) ¯ cog $ \overline{\log({g}f)}_{\mathrm{cog}} $ values of the selected line as a central grid point, and covers the parameter ranges of −0.05 <  λinput <  +0.05 Å (in steps of 0.01 Å) and −0.5 <  log(g f) <  + 0.5 dex (in steps of 0.05 dex). An example of the calculated grid line profiles is shown in Fig. 3 for the central and extrema wavelength grid values.

thumbnail Fig. 3.

An example of the iterative detailed modelling of the λ6750.2 Fe I line in 51 Peg. The observed line profile is shown in black with solid markers, and the line profiles synthesised with varying input atomic parameters are shown in coloured lines (corresponding to different log(g f) values). The panels show three of the synthesised wavelength values: λinput −0.05 Å (left panel), λinput (middle panel), and λinput + 0.05 Å (right panel). Each panel shows the full −0.5 dex <  log(g f) <  + 0.5 dex grid (in steps of 0.05 dex).

The observed benchmark spectra and calculated line profiles are used to calculate a corresponding grid of χ red 2 $ \chi_{\rm red}^2 $ values, which are interpolated in steps of Δλ = 0.005 Å and Δ log(g f) = 0.01 dex using a bivariate cubic spline fit. The smallest χ red 2 $ \chi_{\rm red}^2 $ value of the interpolated grid provides the final set of wavelength and log(g f)grid values for the given line, for a given star, and confidence intervals at a 68.3% confidence limit are calculated for the pair of parameters to produce upper and lower error estimates on the parameters. These individual wavelengths, log(g f)grid values, and errors per star can be combined to produce a final mean wavelength, mean log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ value, and errors for the line (see Sect. 4.3.8). Figure 4 shows the χ red 2 $ \chi_{\rm red}^2 $ grids calculated for the λ5682.2 Ni I line in the several benchmark stars, in addition to the λgrid and log(g f)grid values corresponding to the χ2 minimum per star, 68.3% confidence limit error bars, and the log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ value of the line.

thumbnail Fig. 4.

log(g f)grid values and χ red 2 $ \chi_{\rm red}^2 $ distributions for the λ5682.2 Ni I line. Top left panel: individual log(g f)grid values per star, including 68.3% confidence-limit error bars, plotted against benchmark Teff. The solid black line denotes the weighted log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ and the hatched region denotes the standard deviation estimation, discussed in Sect. 4.3.8. Remaining panels: χ red 2 $ \chi_{\rm red}^2 $ distributions of the synthesised grid of λ and log(g f) values for the seven benchmark stars. Calculated grid points are denoted by black dots, the individual log(g f)grid values are denoted by black crosses, and the 68.3% confidence limits are shown with black contours. Each plot has a normalised χ red 2 $ \chi_{\rm red}^2 $ colour-scale, where yellow represents the χ red 2 $ \chi_{\rm red}^2 $ minimum, and dark blue represents the χ red 2 $ \chi_{\rm red}^2 $ maximum.

4.3. Uncertainties in the astrophysical oscillator strengths

4.3.1. Uncertainties due to S/N

Cayrel (1988) provides a formula for estimating the statistical uncertainty in measuring the Wλ of an observed profile using a Gaussian fit. Using Eq. (7) of their work, the HERMES FWHM ∼ 0.065 Å, a pixel size of δx ∼ 0.026 Å, and a S/N ≈ 800, a statistical uncertainty of ∼0.08 mÅ is determined for the equivalent width measurements. This statistical uncertainty only yields differences in log(g f) values on the order of the fourth decimal place, and so is negligible compared to the other sources of uncertainty.

4.3.2. Uncertainties due to modelling assumptions

We do not attempt an exhaustive investigation into all aspects of spectral modelling, such as adopting different model atmospheres and transfer codes, however we have adopted stellar objects for which the modelling uncertainties should be minimal. The plane-parallel atmospheric models of ATLAS9 were adopted, rather than the public MARCS models, as the MARCS model opacities do not contain enough ionised species to accurately model early-F stars and hotter (Plez 2011), which is required for future plans to extend the analysis to hotter stars and ionised lines. The benchmark star selection mandated dwarf-like objects so that the plane-parallel assumption of ATLAS9 is valid. In the context of cool stars, Gustafsson et al. (2008) show that the differences in temperature structures between ATLAS9 and MARCS are negligible for G-type dwarf stars of solar metallicity. Differences between ATLAS9, which uses ODFs, and ATLAS12, using opacity sampling, are shown to be extremely small when computed with the same chemical composition and line data (Plez 2011).

As shown in Lind et al. (2012), the magnitude of non-LTE effects on unsaturated, high-excitation Fe I lines for stars with stellar parameters of 5000 ≤ Teff ≤ 6000 K, 4.0 ≤ log g ≤ 4.5, and [M/H] ≈ [M/H]⊙ are between 0.00 and 0.02 dex at most, and therefore the majority of investigated lines should be free of non-LTE effects. We do however expect the adopted stellar model atmospheres to begin to suffer from shortcomings for the hottest A-type benchmark stars, and the resultant non-LTE effects will lead to systematic errors in astrophysical log(g f) values. It is hard to predict the exact magnitudes of non-LTE effects for individual spectral lines, but lines of lower excitation energies belonging to easily ionised species are more likely to exhibit non-LTE effects. As discussed in Sect. 5.2, we do not find any obvious evidence of non-LTE effects for the majority of our astrophysical log ( g f ) ¯ $ \overline{\log({g}f)} $ values, except for transitions belonging to the lowest energy levels of Elow ≤ 0.25 eV.

4.3.3. Uncertainties due to stellar parameters

Differences between the physical stellar parameters and the adopted stellar parameters will impact the derived log(g f) values, as any difference in number densities will be attributed to the derived log(g f) values. We also assume that the benchmark stars have the same elemental abundance distribution as the Sun, but scaled with metallicity. This means that any changes to the relative abundance ratios of an individual element, for a given benchmark star, will manifest as Δ log(g f) values for all lines belonging to that element. Due to the highly degenerate relationships between the stellar parameters, individual elemental abundances, and their dependence on the adopted atomic data, we do not propagate the stellar parameter errors into our log(g f) derivations or attempt to determine individual abundances. An accurate determination of these uncertainties warrants its own extensive investigation. Instead we adopt the position that the stellar parameters and metallicity scaled solar abundance ratios are correct, and that using multiple benchmark stars in the log ( g f ) ¯ $ \overline{\log({g}f)} $ derivations makes the final log ( g f ) ¯ $ \overline{\log({g}f)} $ values less susceptible to an erroneous set of stellar parameters or changes in abundance ratios. The standard deviation associated with the log ( g f ) ¯ $ \overline{\log({g}f)} $ value of a line will reflect the uncertainties of any of the sets of stellar parameters or abundance ratios.

4.3.4. Uncertainties due to normalisation

The benchmark spectra presented in this work are normalised using an automatic template normalisation in order to prevent the introduction of “human factors” into the individual stellar spectra. The result should be a more consistent normalisation across the spectra, leading to a decrease in scatter, or random error, in the derived log(g f) values of a given line. To account for any deviations from unity in local continuum placement, the Gaussian Wλ measurements are performed with a flat continuum offset term. In practice, this offset term allows the single Gaussian fit to mitigate the influence of nearby broad-winged lines. The iterative fitting does not attempt to measure any deviations in local continuum placement.

4.3.5. Impact of Gaussian profile for Wλ measurements

Spectral lines are often approximated as Gaussian profiles rather than Voigt profiles, however this assumption breaks down as lines become saturated and the profile wings begin to grow. Fitting a line with a Gaussian profile will lead to an underestimation of the line Wλ as a function of increasing Wλ. This issue is discussed by Luck (2017), in addition to the difficulties of performing physically accurate Voigt profile fits automatically for many lines.

In order to theoretically quantify this underestimation, Gaussian fits are performed on all synthetic line profiles of the curves of growth calculated in Sect. 4.1. Figure 5 shows the difference in Wλ between the two profile shapes as a function of Wλ, computed for the 1091 lines in the Solar spectrum. The Wλ difference was calculated on a line-by-line, star-by-star basis and applied to each of the corresponding Wλ measurements discussed in Sect. 3.2. Figure 6 shows the difference between the log ( g f ) ¯ cog $ \overline{\log({g}f)}_{\mathrm{cog}} $ values, measured using a Gaussian profile fit, and log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ values, calculated using the full radiative-transfer broadening, before and after the correction to the Wλ values. After correction there is a distinct improvement in the observed negative trend between Δ log ( g f ) ¯ cog-grid $ \Delta\overline{\log({g}f)}_{\text{ cog-grid}} $ and Wλ, with the remaining positive scatter caused by differences in the treatment of theoretical blending between the two methods, as discussed in Sect. 4.3.7.

thumbnail Fig. 5.

Predicted ΔWλ between Gaussian fits and transfer-calculation Voigt profiles plotted as a function of calculated Wλ shown for the 1091 investigated lines in the Sun (black lines). Blue crosses denote the proposed ΔWλ corrections for the 1091 spectral lines.

thumbnail Fig. 6.

Left panel: Δ log ( g f ) ¯ cog-grid $ \Delta\overline{\log({g}f)}_{\text{ cog-grid}} $ plotted against the calculated W λ ¯ $ \overline{W_{\lambda}} $ prior to the corrections to the Gaussian profiles. Right panel: as in Left panel but after corrections are applied on a line-per-line, star-by-star basis. The colour scale corresponds to the mean Ωcore (in percentage) of the line in question, recalculated using the new log ( g f ) ¯ cog $ \overline{\log({g}f)}_{\mathrm{cog}} $ values. The stratification of the Ωcore is explained in Sect. 4.3.7.

4.3.6. Uncertainties due to line blending in observed spectra

Observable blends, that is, those that exhibit clear asymmetries or multiple peaks, are manually flagged and excluded on a line-by-line, star-by-star basis, to mitigate their impact on the log(g f) derivations. Hidden blends, that is, those that do not cause measurable asymmetries or additional peaks in the line profile of interest, will cause an overestimation of the derived log(g f) value unless there is prior theoretical knowledge of this blend (see Sect. 4.3.7). If the two blending lines behave differently with temperature then it is expected that the benchmark stars will produce systematic differences in the derived log(g f) value. As such, the standard deviation of the log(g f) values for a given line can be used as an indicator as to whether or not a line may be affected by hidden blends, and thus whether or not it is reliable to use as a single unblended line in our analysis and quality assessment for this paper.

4.3.7. Uncertainties due to theoretical line blending

The curve of growth approach assumes that a measured spectral feature is completely unblended, attributing the measured Wλ to a single spectral line, whereas the iterative modelling approach may contain some knowledge of potential blending components in the input line list. This difference will lead the curve of growth approach to overestimate the log(g f)cog of the line of interest relative to the iterative modelling approach, assuming that the blending component is actually present in the observed spectrum.

Figure 7 shows log ( g f ) ¯ cog − log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{cog}} - \overline{\log({g}f)}_{\mathrm{grid}} $ plotted against the theoretical Wλ of the Sun, for intervals of Ωcore that have been recalculated using the new log ( g f ) ¯ $ \overline{\log({g}f)} $ values. Figure 8 shows the mean Δ log ( g f ) ¯ cog-grid $ \Delta\overline{\log({g}f)}_{\text{ cog-grid}} $ plotted as a function of Ωcore, as well as the standard deviation of the log ( g f ) ¯ cog-grid $ \overline{\log({g}f)}_{\text{ cog-grid}} $ values and σ ¯ grid $ \overline{\sigma}_{\mathrm{grid}} $ values plotted as functions of Ωcore. The mean Δ log ( g f ) ¯ $ \Delta\overline{\log({g}f)} $ is 0.00 dex when Ωcore ≪ 1%, and increases steadily with Ωcore, confirming that the systematic difference between the methods is due to theoretical blending. The standard deviation similarly increases with Ωcore, but levels out at roughly Ωcore ≤ 5% at a value of 1σ = ±0.04 dex, which therefore seems to represent the intrinsic scatter between the two methods. Properly correcting for this difference in methods is difficult as one must know how the background line presence would affect the single Gaussian fit, and rely on the assumption that the background line atomic data are accurate. Given that properly rectifying this difference would require one to simultaneously quality assess two or more blended lines, we instead recommend using the intrinsic scatter value of 0.04 dex as a constraint on line selection, in order to reduce the impact of this systematic difference on the quality assessment results (see Sect. 5.1).

thumbnail Fig. 7.

Δ log ( g f ) ¯ cog-grid $ \Delta\overline{\log({g}f)}_{\text{ cog-grid}} $ plotted against theoretical Wλ values for the Sun, shown for different steps of Ωcore (left to right, top to bottom). The mean difference of the Δ log ( g f ) ¯ cog-grid $ \Delta\overline{\log({g}f)}_{\text{ cog-grid}} $ values is shown by the dashed blue lines, and the standard deviation is denoted by the hatched area.

thumbnail Fig. 8.

Top panel: mean Δ log ( g f ) ¯ cog-grid $ \Delta\overline{\log({g}f)}_{\text{ cog-grid}} $ values plotted against Ωcore. Black error bars show the standard deviation σ of the Δ log ( g f ) ¯ cog-grid $ \Delta\overline{\log({g}f)}_{\text{ cog-grid}} $ values, and green error bars show mean of the σgrid values. Lower panel: σ and σ ¯ grid $ \overline{\sigma}_{\mathrm{grid}} $ values plotted as a function of Ωcore. The mean of the Δ log ( g f ) ¯ cog-grid $ \Delta\overline{\log({g}f)}_{\text{ cog-grid}} $ values is shown to increase as Ωcore increases. The σ also increases with Ωcore, however the σ ¯ grid $ \overline{\sigma}_{\mathrm{grid}} $ exhibits a flatter trend. At Ωcore ≤ 5%, σ is almost constant suggesting that the intrinsic scatter between the two methods is σ = 0.04 dex.

4.3.8. Final log ( g f ) ¯ $ \overline{\log({g}f)} $ values and uncertainties

The log ( g f ) ¯ cog $ \overline{\log({g}f)}_{\mathrm{cog}} $ values are calculated by taking the mean of the n individually derived stellar log(g f)cog values, after individual corrections for the Gaussian approximation discussed in Sect. 4.3.5. The standard deviation of the log(g f)cog values, σcog, are calculated as shown in Eq. (4):

σ cog = 1 n − 1 ∑ i = 1 n ( log ( g f ) i − log ( g f ) ¯ cog ) 2 . $$ \begin{aligned} {\sigma }_{\rm cog} = \sqrt{ \dfrac{1}{n - 1} \mathop {\sum }\limits _{i=1}^{n} \left( \log ({g}f)_{i} - \overline{\log ({g}f)}_{\rm cog} \right) ^2 } .\end{aligned} $$(4)

The log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ values are calculated by using a weighted mean of the n individually derived stellar log(g f)grid values, shown by Eq. (5), and the corresponding standard deviation estimations are shown by Eqs. (6) and (7), where the weights per star are equal to the inverse square of the log(g f)grid uncertainty estimates:

log ( g f ) ¯ grid = ∑ i = 1 n ( log ( g f ) i / σ i 2 ) ∑ i = 1 n 1 / σ i 2 , $$ \begin{aligned} \overline{\log ({g}f)}_{\rm grid} = \dfrac{\mathop {\sum }\limits _{i=1}^{n} \left( \log ({g}f)_{i} / \sigma _i^2 \right) }{\mathop {\sum }\limits _{i=1}^{n} 1/\sigma _i^2 } ,\end{aligned} $$(5)

and

σ grid = ∑ i = 1 n ( log ( g f ) i − log ( g f ) ¯ grid ) 2 / σ i 2 V 1 − ( V 2 / V 1 ) , $$ \begin{aligned} {\sigma }_{\rm grid} = \sqrt{ \dfrac{\mathop {\sum }\limits _{i=1}^{n} \left( \log ({g}f)_i - \overline{\log ({g}f)}_{\rm grid} \right)^2 / \sigma _i^2 }{ V_1 - ( V_2/ V_1 ) } } ,\end{aligned} $$(6)

where V1 and V2 are defined as follows:

V 1 = ∑ i = 1 n 1 / σ i 2 V 2 = ∑ i = 1 n ( 1 / σ i 2 ) 2 . $$ \begin{aligned} V_1 = \mathop {\sum }\limits _{i=1}^{n} 1/\sigma _i^2 \qquad V_2 = \mathop {\sum }\limits _{i=1}^{n} \left( 1/\sigma _i^2 \right)^2 .\end{aligned} $$(7)

In the case that all the weights are equal, Eqs. (6) and (7) simplify to take the same form as the standard deviation shown in Eq. (4).

5. Quality assessment and discussion

5.1. Selection of quality-assessable lines

A line is considered quality-assessable if both the curve of growth and iterative methods derive log ( g f ) ¯ $ \overline{\log({g}f)} $ values with overlapping errors, otherwise the line is deemed too difficult to accurately work with, be it due to measuring, modelling, or blending issues. An additional constraint, requiring the Δ log ( g f ) ¯ $ \Delta\overline{\log({g}f)} $ of the two methods to be ≤0.04 dex, can be imposed to increase the likelihood that the selected lines are free of systematic differences in analysis methods due to theoretical blending. Lines that fulfil this criterion are referred to as analysis-independent lines. As an example of the quality-assessable line selection, Figs. B.1–B.3 show the synthesised line profiles belonging to the log ( g f ) ¯ cog $ \overline{\log({g}f)}_{\mathrm{cog}} $ and log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ values, plotted against the observed line profile, for a non-quality-assessable line (Fig. B.1), a quality-assessable line (Fig. B.2), and an analysis-independent spectral line (Fig. B.3), for the several benchmark stars. The quality of the observed data is sufficiently high that even lines as weak as the central line depth d ≈ 0.02 can be reliably quality-assessed.

5.2. Distributions of derived λ and log(gf) values

Figure 9 shows the Δ λ ¯ grid-input $ \Delta\overline{\lambda}_{\text{ grid-input}} $ plotted against λinput for the 1091 investigated lines. For most lines there is excellent agreement between the literature wavelengths and those derived via the iterative modelling. A small fraction of transitions have significant offsets of up to |Δλ| ≤ 0.06 Å, which are caused by inaccuracies in the atomic data. There is a slight negative offset in Δ λ ¯ grid-input $ \Delta\overline{\lambda}_{\text{ grid-input}} $, however this is well within the uncertainties of the HERMES spectrograph wavelength calibration.

thumbnail Fig. 9.

Δ λ ¯ grid-input $ \Delta\overline{\lambda}_{\text{ grid-input}} $ vs. λinput for the 1091 investigated lines. 1σ error bars are shown in blue. A slight negative offset is present, however this is within the uncertainty in the HERMES spectrograph wavelength calibration. A small fraction of transitions have wavelength offsets that are attributed to inaccuracies in the atomic data.

Figure 10 shows the Δ log ( g f ) ¯ grid-input $ \Delta\overline{\log({g}f)}_{\text{ grid-input}} $ and Δ log ( g f ) ¯ cog-input $ \Delta\overline{\log({g}f)}_{\text{ cog-input}} $ plotted against log(g f)input for the 408 analysis-independent lines. We find that for a number of transitions both methods show significant revisions of the log(g f)input values as large as ±0.50 dex or more. We do not find any obvious trends between the log(g f) and log(g f)input values resulting from the two methods, nor is there a correlation between the uncertainties and either (a) the Δ log(g f)method-input values or (b) the log(g f)input values. Figure 11 shows the Δ log ( g f ) ¯ grid-input $ \Delta\overline{\log({g}f)}_{\text{ grid-input}} $ plotted against log(g f)input for the 1091 investigated lines, in addition to the log ( g f ) ¯ cog-grid $ \overline{\log({g}f)}_{\text{ cog-grid}} $ plotted against log(g f)input. Both the 845 quality-assessable lines and the 1091 investigated lines contain a number of transitions for which we find a significant downward log(g f) revision of up to 1.50 dex.

thumbnail Fig. 10.

Δ log ( g f ) ¯ grid-input $ \Delta\overline{\log({g}f)}_{\text{ grid-input}} $ (in blue) and Δ log ( g f ) ¯ cog-input $ \Delta\overline{\log({g}f)}_{\text{ cog-input}} $ (in black) plotted against log(g f)input for the 408 analysis-independent lines discussed in Sect. 5.1. There are no clear trends in either the Δ log ( g f ) ¯ grid-input $ \Delta\overline{\log({g}f)}_{\text{ grid-input}} $ values or Δ log ( g f ) ¯ cog-input $ \Delta\overline{\log({g}f)}_{\text{ cog-input}} $ values with respect to log(g f)input, however there are a significant number of transitions for which both methods, which agree within 0.04 dex of each other, find sizeable offsets compared to the log(g f)input. Some transitions even show differences between log(g f)input and our log ( g f ) ¯ $ \overline{\log({g}f)} $ values of ±0.50 dex or more. The zoomed subplot shows transitions that have significantly smaller differences between the derived and input log(g f) values, however a number of them still do not agree within the derived uncertainties.

thumbnail Fig. 11.

Δ log ( g f ) ¯ grid-input $ \Delta\overline{\log({g}f)}_{\text{ grid-input}} $ vs. log(g f)input for the 1091 investigated lines. Bottom panel: as above but for Δ log ( g f ) ¯ cog-grid $ \Delta\overline{\log({g}f)}_{\text{ cog-grid}} $. The 408 analysis-independent lines are shown in light green, the remainder of the 845 quality-assessable lines are shown in dark green, and the remainder of the 1091 investigated lines are shown in black. There are no obvious trends in either Δ log ( g f ) ¯ grid-input $ \Delta\overline{\log({g}f)}_{\text{ grid-input}} $ or Δ log ( g f ) ¯ cog-grid $ \Delta\overline{\log({g}f)}_{\text{ cog-grid}} $ with respect to log(g f)input, except for a number of transitions where both methods, agreeing with each other within errors, find significant upward revisions of the log(g f)input values as large as Δ log(g f) = 1.5 dex.

Figure 12 shows the Δ log ( g f ) ¯ grid-input $ \Delta\overline{\log({g}f)}_{\text{ grid-input}} $ and Δ log ( g f ) ¯ cog-input $ \Delta\overline{\log({g}f)}_{\text{ cog-input}} $ plotted against log(g f)input for the 408 analysis-independent lines. We find an increase in scatter of the derived log(g f) values for higher excitation transitions. This increase in scatter is uncorrelated with the uncertainties of the derived log(g f) values, and is emphasised even more by the quality-assessable lines shown in Fig. 13. We also find a positive offset for transitions belonging to the lowest excitation energies. This is possibly due to non-LTE effects in a small number of lines belonging to either the lowest terms of easily ionised species, or resonant lines of other species. Figure 13 shows the Δ log ( g f ) ¯ grid-input $ \Delta\overline{\log({g}f)}_{\text{ grid-input}} $ plotted against Elow for the 1091 investigated lines. The increase in scatter of Δ log ( g f ) ¯ grid-input $ \Delta\overline{\log({g}f)}_{\text{ grid-input}} $ with increasing Elow is present for the species Fe I and Si I, with the vast majority of the log(g f)input values originating from the theoretical Kurucz 2007 line lists (Kurucz 1999-2014). The likely cause for this apparent relationship is transitions with a high degree of atomic configuration mixing, for which log(g f) values are difficult to calculate.

thumbnail Fig. 12.

As in Fig. 10, but plotted as a function of Elow. There is a slight upward inflection of log ( g f ) ¯ $ \overline{\log({g}f)} $ values at excitation energies of Elow ≤ 0.25 eV which is possibly a symptom of non-LTE effects in these low-level transitions. Between 3.5 ≤ Elow ≤ 5.0 eV we find an increase in scatter of log ( g f ) ¯ $ \overline{\log({g}f)} $ values, often belonging to higher excitation Fe I transitions. The zoomed subplot shows that a significant number of analysis-independent transitions have multiplets in common, and that the log(g f) values of these multiplets can show significant scatter.

thumbnail Fig. 13.

As in Fig. 11, but plotted as a function of Elow. The large downward revisions pertain to Fe I transitions originating from higher excitation energies. The upward trend towards Elow = 0 eV is still visible, however to a lesser degree than the distribution of solely the analysis-independent lines.

Figure 14 shows the median of the 408 analysis-independent Δ log ( g f ) ¯ method-input $ \Delta\overline{\log({g}f)}_{\text{ method-input}} $ values, for both the curve of growth and iterative modelling methods, plotted against the benchmark stellar parameters. There is a small offset between the curve of growth and iterative modelling methods which is caused by the different treatments of line blending. There is no clear trend between the median Δ log ( g f ) ¯ method-input $ \Delta\overline{\log({g}f)}_{\text{ method-input}} $ values and any of the stellar parameters, with the median Δ log ( g f ) ¯ grid-input $ \Delta\overline{\log({g}f)}_{\text{ grid-input}} $ values centred around ∼0.00 dex, suggesting that the analysis is not subject to significant systematic modelling issues with respect to the stellar parameters. The exception to this is V I, which shows a negative trend with respect to increasing effective temperature. The cause of this is not clear, and could be linked to a number of possibilities: hyperfine splitting, which can easily be seen in the profile shapes of the λ4379.2 and λ4594.1 V I lines (see Table F.1 hyperlinks for interactive line plots); non-LTE effects, as V I is an easily ionised species and several of the V I lines originate from low-lying energy levels; or even differences in the [V/Fe] ratio between the different benchmark stars. Given that the same trend appears in multiple V I lines we believe the trend is less likely to be caused by line blending issues.

thumbnail Fig. 14.

Median of the 408 analysis-independent Δ log ( g f ) ¯ method-input $ \Delta\overline{\log({g}f)}_{\text{ method-input}} $ values, for both the curve of growth (black circles) and iterative modelling (blue circles) methods, are plotted against the stellar parameters of the seven benchmark stars. The median of the σgrid values are denoted by the blue error bars. The overestimation of the log ( g f ) cog ¯ $ \overline{\log({g}f)_{\mathrm{cog}}} $ relative to the log ( g f ) grid ¯ $ \overline{\log({g}f)_{\mathrm{grid}}} $ values can be seen. Apart from V I, we do not find any clear trends with respect to the stellar parameters for any of the 14 remaining species in the analysis-independent subset of lines.

5.3. Discussion of quality assessment results

Of the 1091 atomic lines investigated, 845 are found to be quality-assessable and 408 of them are found to be analysis-independent lines. The ionic distribution of these lines is listed in Table 2. Approximately half of the investigated and quality-assessable lines belong to Fe I, and another ∼10% of the lines belong to singly ionised species. The literature atomic data belonging to a quality-assessable line are considered in agreement with our work, and therefore we can be recommend the value for use, if it agrees within the errors of the derived log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ and σgrid values. We do not make use of any literature errors as these are only available in a handful of the retrieved databases. The log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ value is adopted as the benchmark value as in almost all cases it produces better χ red 2 $ \chi_{\rm red}^2 $ values than the log ( g f ) ¯ cog $ \overline{\log({g}f)}_{\mathrm{cog}} $ value. In addition, the log ( g f ) ¯ cog $ \overline{\log({g}f)}_{\mathrm{cog}} $ value relies upon theoretical corrections to the adopted line profile shape when measuring Wλ values. Of the 845 quality-assessable lines, 451 lines are found to have literature atomic data in agreement with the Δ log ( g f ) ¯ grid $ \Delta\overline{\log({g}f)}_{\mathrm{grid}} $ values, corresponding to ∼53% of the quality-assessable lines. This fraction is also representative of the analysis-independent lines, where 210 lines have sufficiently accurate atomic data. The majority of Fe-group species (Sc II, Ti I-II, V I, Cr I, Mn I, Fe II, Co I, Ni I) have a reasonable number of lines with accurate atomic data, typically 70–75% of lines, however the Fe I lines only have ∼38% of lines with accurate atomic data.

Table 2.

Elemental and ionic distribution of the investigated, quality-assessable, and analysis-independent lines. In brackets are the number of lines with atomic data within σgrid of the log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ values.

Table F.1 provides the detailed break down of the findings for the 1091 investigated lines. In an effort to help ascertain the quality of the log(g f) value of a given spectral line, interested readers can find interactive online plots of all 1091 investigated spectral lines, for all seven benchmark spectra, as hyperlinks in Table F.1. The hyperlinks display all cross-matched atomic data for the line in question, including additional information such as level configurations and terms, in addition to interactive plots of the observed line profiles and synthetic line profiles using all log(g f) values derived here and found in the literature, and their respective equivalent widths per log(g f) value, per star.

To better visualise the quality assessment process, Figs. C.1–C.3 show the observed lines profiles of three transitions in the seven benchmark spectra, including synthetic line profiles calculated using the log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ and available literature values. The three transitions show examples of the three quality-assessment outcomes: an example for which all available literature is within the errors of the log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ value (Fig. C.1); an example for which only some of the available literature values are within the errors of the log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ value (Fig. C.2); and an example for which none of the available literature values are within the errors of the derived log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ value (Fig. C.3). It is clear that the log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ values reproduce the several line profiles extremely well, and that in some cases the available literature data are insufficient to reproduce the observed line profiles.

It is worth mentioning important limitations of this work. The astrophysical log(g f) values derived in this work were determined using the solar abundances of Grevesse et al. (2007), and any updates to these elemental abundances would require a correction to the log(g f) values of lines belonging to updated species. Fortunately, the majority of the lines investigated here would only require directly inverse corrections, that is, if an abundance is revised by Δ[X/H] = 0.01 dex, then all lines of element X will require a simple update of Δ log(g f) = −0.01 dex. This means that the astrophysical log(g f) values should remain useful, especially in the context of differential abundance analysis, for years to come.

Blended lines, even accurately known ones, have been avoided here due to the complexity of simultaneous quality assessment of multiple components. Hyperfine and isotopically split lines have not received any special treatment throughout this analysis. In the case of isotopic lines we find a clear under-representation of s-process elements in the investigated lines. This is because the multiple components are often represented as individual entries in a line list, and so the Sect. 3.1 selection method treats the components as individual lines, leading to their rejection from analysis due to their high Ωcore values. The hyperfine splitting of lines has not been addressed here, as while hyperfine splitting information is becoming more abundant in the literature, it was not included in the investigated atomic databases at the time of retrieval3. Unlike the s-process elements, lines that require hyperfine splitting may be present in the quality assessment results, though inaccuracies in their derived log(g f) values depend heavily on the observed profile shape. Isotopic and hyperfine lines may benefit from an additional dedicated investigation, where the methods described in this paper are expanded to account for the multiple components of such transitions.

6. Summary

In this work we present a homogeneous quality assessment of the literature atomic data available for unblended spectral lines, in the wavelength range 4200–6800 Å, present in seven benchmark stars that span the effective temperature range 5000–6000 K. The work presented here was performed for multiple benchmark stars to help make the analysis more robust against any erroneous stellar parameter derivations, and also to investigate whether spectral lines are subject to different degrees of blending that would otherwise be unnoticed when analysing a single star. Apart from Teff, the stars are chosen to have similar stellar parameters to minimise any systematic modelling differences between objects. The objects were also chosen to minimise the impact of potential non-LTE effects which could systematically affect the derived log(g f) values. No clear correlations were found between the derived log(g f) values and the stellar parameters. The high-quality benchmark spectra were obtained using the Mercator-HERMES spectrograph, each with a resolution of R ∼ 85 000 and S/N ≈ 1000. The spectra are available online4 in both normalised and unnormalised format.

Of the 82337 atomic transitions found in the literature for this wavelength range, 1091 were found to be theoretically deep, unblended, and to have observable line profiles that can be fitted accurately with a single Gaussian profile. Given the high S/N and high resolution of our benchmark spectra, not all of our investigated lines can be used in other spectroscopic works, however the quality assessment results and log(g f) values themselves are valid at all spectral resolutions and S/Ns. In addition, this work provides quality information for many lines previously unused in spectroscopic analysis.

Astrophysical log(g f) values were derived for each of the 1091 transitions using two commonly employed methods: equivalent width and spectral synthesis fitting methods. Care was taken to address, minimise, and remove as many potential systematic errors as possible from the derived log(g f) values, and to produce representative error bars. Agreement between the two derived log(g f) values was used as a criterion to select a subset of 845 well-behaved lines which could be reliably quality-assessed. An additional constraint, mandating that the log(g f) values of the two methods are within 0.04 dex of each other, was imposed to produce a further subset of 408 spectral lines which are free of systematic differences caused by the adopted analysis methods.

A key conclusion of this work is that the treatment of blending lines is vital when aiming to reduce systematic errors and uncertainty from quantitative spectroscopy. Careful comparison between the curve of growth and detailed modelling methods revealed that even small background lines, contributing as little as 5% of the total equivalent width of a spectral feature, can lead to systematic offsets of ∼0.10 dex or more between the two methods.

The available atomic data for these 845 spectral lines were compared against the log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ values to determine whether the literature values were in agreement with the presented work, as discussed in Sect. 5.3. It was found that ∼53% of the quality-assessed lines have at least one literature log(g f) value in agreement with our work. For Fe I this value is ∼38% and for the remaining Fe-group elements the value is around 70–75% on average.

We provide recommendations, where possible, as to which literature atomic data will produce sufficiently accurate spectroscopic results. Where the literature is insufficient, such as in some cases where given values are over 0.5 dex from our values, we suggest the use of astrophysical log(g f) values derived in this work. While the quality assessment presented here is limited to the atomic data compiled in Paper 1, the astrophysical log(g f) values and errors can easily be used to perform quality assessments of other literature log(g f) values, allowing astronomers and atomic physicists to evaluate the quality of newly produced atomic data. All results of the presented investigation are available in the appendix, as well as in digital format via the CDS and at brass.sdf.org.

1

Throughout this paper we use the term “unblended” to refer to spectral lines that produce the majority of the flux absorption of their respective spectral feature, in the context of theoretical spectral calculations, as explained in Sect. 3.1.

2

Often written in terms of log weighted-oscillator strengths, or log(g f).

3

We note that in the time since our atomic data retrievals, VALD3 has expanded their database so that they can now provide HFS data for a large number of atomic species (Pakhomov et al. 2017).

Acknowledgments

We thank the atomic data producers and providers for their invaluable work towards improving the accuracy of stellar spectroscopy and the ease at which vast quantities of data can be retrieved. We also thank the anonymous referee for helping to substantially improve the paper. The research for the present results has been subsidised by the Belgian Federal Science policy Office under contract No. BR/143/A2/BRASS. T.M and M.V.d.S are supported by a grant from the Fondation ULB. 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. This work is based on observations made with the Mercator Telescope, operated on the island of La Palma by the Flemish Community, at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias. This work is also based on observations obtained with the HERMES spectrograph, which is supported by the Research Foundation – Flanders (FWO), Belgium, the Research Council of KU Leuven, Belgium, the Fonds National de la Recherche Scientifique (F.R.S.-FNRS), Belgium, the Royal Observatory of Belgium, the Observatoire de Genàve, Switzerland and the Thüringer Landessternwarte Tautenburg, Germany.

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Appendix A: Automatic normalisation of the observed benchmark spectra

The observed benchmark spectra are continuum flux normalised using a semi-automatic spectral-template normalisation procedure. This procedure searches for a series of wavelength points over sufficiently continuous flux regions close to the stellar continuum level in the theoretical spectra between 4000 Å and 6800 Å. The selected wavelength regions are then used as continuum anchor points for a polynomial fit to normalise the observed spectrum. We opt to use an automatic template normalisation procedure to remove the “human-factor” from the spectrum normalisation, leading to repeatable and consistent global continuum flux normalisations. The final results are scrutinised to ensure they behave as expected. There are no obvious issues in the continuum placement of the automatically normalised spectra (see Sect. 4.3.4 for discussion on the log(g f) uncertainties due to normalisation).

Appendix B: Line profile comparisons with the curve of growth and iterative modelling methods

Figures B.1–B.3 show the synthesised line profiles of the log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ and log ( g f ) ¯ cog $ \overline{\log({g}f)}_{\mathrm{cog}} $ values, including line profile error bars, alongside the observed line profiles in the seven observed benchmark stars. Figure B.1 shows an example where the log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ and log ( g f ) ¯ cog $ \overline{\log({g}f)}_{\mathrm{cog}} $ values do not agree within error of each other, and thus the spectral line is not suitable for quality assessment. Figure B.2 shows an example where the log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ and log ( g f ) ¯ cog $ \overline{\log({g}f)}_{\mathrm{cog}} $ values do agree within error of each other, making the line suitable for quality assessment. Figure B.2 shows a quality assessable line where the log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ and log ( g f ) ¯ cog $ \overline{\log({g}f)}_{\mathrm{cog}} $ values agree within 0.04 dex of each other, indicating the line is free of systematic issues due to the treatment of line blending by the two methods.

thumbnail Fig. B.1.

log(g f) determinations for the λ5708.1 Fe I line, which is deemed unsuitable for quality assessment due to the disagreement between the log ( g f ) ¯ cog $ \overline{\log({g}f)}_{\mathrm{cog}} $ and log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ values. Observed line profiles are shown in black dots, line profiles calculated using the log(g f)input values are shown in dashed grey, line profiles calculated using the log ( g f ) ¯ cog $ \overline{\log({g}f)}_{\mathrm{cog}} $ values are shown in green, line profiles calculated using the log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ values are shown in blue, and errors are shown as green and blue shaded areas, respectively. Vertical blue and dashed grey lines denote the λ ¯ grid $ \overline{\lambda}_{\mathrm{grid}} $ and λinput values respectively. The disagreement in this case is likely caused by the presence of a nearby spectral line affecting the single Gaussian fit of the curve of growth method.

thumbnail Fig. B.2.

log(g f) determinations for the λ5435.8 Ni I line which is deemed to be quality-assessable. Observed line profiles are shown in black dots, line profiles calculated using the log(g f)input values are shown in dashed grey, line profiles calculated using the log ( g f ) ¯ cog $ \overline{\log({g}f)}_{\mathrm{cog}} $ values are shown in green, line profiles calculated using the log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ values are shown in blue, and errors are shown as green and blue shaded areas, respectively. Vertical blue and dashed grey lines denote the λ ¯ grid $ \overline{\lambda}_{\mathrm{grid}} $ and λinput values, respectively.

thumbnail Fig. B.3.

log(g f) determinations for the λ5292.6 Fe I line deemed suitable for quality assessment and deemed to be robust against adopted analysis method. Observed line profiles are shown in black dots, line profiles calculated using the log(g f)input values are shown in dashed grey, line profiles calculated using the log ( g f ) ¯ cog $ \overline{\log({g}f)}_{\mathrm{cog}} $ values are shown in green, line profiles calculated using the log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ values are shown in blue, and errors are shown as green and blue shaded areas, respectively. Vertical blue and dashed grey lines denote the λ ¯ grid $ \overline{\lambda}_{\mathrm{grid}} $ and λinput values, respectively. This line also shows a measurable Δλ correction of ∼0.02 Å to the input wavelength.

Appendix C: Line profile comparisons with the cross-matched literature data

Figures C.1–C.3 show the synthesised line profiles of the log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ values, including line profile error bars, alongside the observed line profiles in the seven observed benchmark stars. In addition the figures show synthetic line profiles corresponding to the available literature log(g f) values of the given transition. All three examples show a analysis-independent line. Figure C.1 shows an example where all literature log(g f) values agree within error of the log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ value. Figure C.2 shows an example where only one of the literature log(g f) values agrees within error of the log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ value. Finally, Fig. C.3 shows an example where none of the literature log(g f) values agree within error of the log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ value.

thumbnail Fig. C.1.

Quality assessment for the λ4555.5 Ti I line. The style follows that of Figs. B.1–B.3, minus the log ( g f ) ¯ cog $ \overline{\log({g}f)}_{\mathrm{cog}} $ line profiles, and with several additional line profiles corresponding to the available literature log(g f) values of the transition. In this case all literature log(g f) values are within error of the log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ value and can be recommended for use.

thumbnail Fig. C.2.

Quality assessment for the λ6456.4 Fe II line. The style follows that of Figs. B.1–B.3, minus the log ( g f ) ¯ cog $ \overline{\log({g}f)}_{\mathrm{cog}} $ line profiles, and with several additional line profiles corresponding to the available literature log(g f) values of the transition. In this case only one literature log(g f) value is within error of the log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ value, and thus only this value is recommended for use.

thumbnail Fig. C.3.

Quality assessment for the λ6786.8 Fe I line. The style follows that of Figs. B.1–B.3, minus the log ( g f ) ¯ cog $ \overline{\log({g}f)}_{\mathrm{cog}} $ line profiles, and with several additional line profiles corresponding to the available literature log(g f) values of the transition. In this case none of the literature log(g f) values can accurately reproduce the observed profile, with Δ log(g f) values ranging between 0.1–0.25 dex, thus we would propose our log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ as a more accurate alternative.

Appendix D: Tables of individual log(gf)cog and log(gf)grid values per benchmark star for the 1091 investigated lines

The individual log(g f)cog, log(g f)grid, and σgrid values per benchmark star are available for each of the 1091 investigated lines in digital format via the CDS and at brass.sdf.org. Table D.1 shows an example of the available individual log(g f)cog values per benchmark star. Table D.2 shows the same but for the individual log(g f)grid values, which have the additional σgrid values corresponding to the 68.3% confidence intervals discussed in Sect. 4.2.

Table D.1.

An excerpt of the individual log(g f)cog values derived per benchmark star for each of the 1091 investigated lines.

Table D.2.

An excerpt of the individual log(g f)grid values, and corresponding σgrid values, derived per benchmark star for each of the 1091 investigated lines.

Appendix E: Tables of Wλ values per benchmark star for the 1091 investigated lines

The measured Wλ values, ΔWλ corrections (as discussed in Sect. 4.3.5), and calculated Wλ values for the log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ values, log ( g f ) ¯ cog $ \overline{\log({g}f)}_{\mathrm{cog}} $ values, and available literature log(g f) values are available, for each of the benchmark stars, in digital format via the CDS and at brass.sdf.org. Table E.1 shows an example of the Wλ values associated with the benchmark star ϵ Eri for the 1091 investigated spectral lines. Similar tables are available for each of the benchmark stars.

Table E.1.

An excerpt of the individual various Wλ values available for ϵ Eri for each of the 1091 investigated lines.

Appendix F: Table of quality assessment information for the 1091 investigated lines

Table F.1 presents both the line selection and quality assessment results for the full 1091 spectral lines investigated in this work. The table is available in digital format via the CDS and at brass.sdf.org.

Table F.1.

Results for the 1091 investigated lines in FGK stars, belonging to 28 different ionic species.

All Tables

Table 1.

Stellar parameters of the FGK benchmark stars with effective temperatures in the range 5000 ≤ Teff ≤ ∼6000 K.

In the text
Table 2.

Elemental and ionic distribution of the investigated, quality-assessable, and analysis-independent lines. In brackets are the number of lines with atomic data within σgrid of the log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ values.

In the text
Table D.1.

An excerpt of the individual log(g f)cog values derived per benchmark star for each of the 1091 investigated lines.

In the text
Table D.2.

An excerpt of the individual log(g f)grid values, and corresponding σgrid values, derived per benchmark star for each of the 1091 investigated lines.

In the text
Table E.1.

An excerpt of the individual various Wλ values available for ϵ Eri for each of the 1091 investigated lines.

In the text
Table F.1.

Results for the 1091 investigated lines in FGK stars, belonging to 28 different ionic species.

In the text

All Figures

thumbnail Fig. 1.

Spectral line depth, d, plotted against theoretical core-blending, Ωcore, (defined in Eqs. (1) and (2)) for all 82337 lines of the input atomic line list for the Sun (G2V) and 51 Peg (G2V). Solid blue lines show the selected cut-offs for the 5000–6000 K benchmark stars of d ≥ 0.02 and Ωcore ≤ 10%. Analysis of hotter benchmark stars will likely need different cut-offs, or potentially a different approach to treating blended spectral lines.
In the text
thumbnail Fig. 2.

Conversion of observed Cr II line equivalent width values into log(g f) values using theoretical curves of growth for a given star. Blue lines represent theoretical curves of growth for different Cr II lines. Black circles are measured equivalent width values plotted using the log(g f)input values, while black crosses are measured equivalent width values plotted using derived log(g f) values, i.e. the intersect of the equivalent width value with its respective curve of growth.
In the text
thumbnail Fig. 3.

An example of the iterative detailed modelling of the λ6750.2 Fe I line in 51 Peg. The observed line profile is shown in black with solid markers, and the line profiles synthesised with varying input atomic parameters are shown in coloured lines (corresponding to different log(g f) values). The panels show three of the synthesised wavelength values: λinput −0.05 Å (left panel), λinput (middle panel), and λinput + 0.05 Å (right panel). Each panel shows the full −0.5 dex <  log(g f) <  + 0.5 dex grid (in steps of 0.05 dex).
In the text
thumbnail Fig. 4.

log(g f)grid values and χ red 2 $ \chi_{\rm red}^2 $ distributions for the λ5682.2 Ni I line. Top left panel: individual log(g f)grid values per star, including 68.3% confidence-limit error bars, plotted against benchmark Teff. The solid black line denotes the weighted log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ and the hatched region denotes the standard deviation estimation, discussed in Sect. 4.3.8. Remaining panels: χ red 2 $ \chi_{\rm red}^2 $ distributions of the synthesised grid of λ and log(g f) values for the seven benchmark stars. Calculated grid points are denoted by black dots, the individual log(g f)grid values are denoted by black crosses, and the 68.3% confidence limits are shown with black contours. Each plot has a normalised χ red 2 $ \chi_{\rm red}^2 $ colour-scale, where yellow represents the χ red 2 $ \chi_{\rm red}^2 $ minimum, and dark blue represents the χ red 2 $ \chi_{\rm red}^2 $ maximum.
In the text
thumbnail Fig. 5.

Predicted ΔWλ between Gaussian fits and transfer-calculation Voigt profiles plotted as a function of calculated Wλ shown for the 1091 investigated lines in the Sun (black lines). Blue crosses denote the proposed ΔWλ corrections for the 1091 spectral lines.
In the text
thumbnail Fig. 6.

Left panel: Δ log ( g f ) ¯ cog-grid $ \Delta\overline{\log({g}f)}_{\text{ cog-grid}} $ plotted against the calculated W λ ¯ $ \overline{W_{\lambda}} $ prior to the corrections to the Gaussian profiles. Right panel: as in Left panel but after corrections are applied on a line-per-line, star-by-star basis. The colour scale corresponds to the mean Ωcore (in percentage) of the line in question, recalculated using the new log ( g f ) ¯ cog $ \overline{\log({g}f)}_{\mathrm{cog}} $ values. The stratification of the Ωcore is explained in Sect. 4.3.7.
In the text
thumbnail Fig. 7.

Δ log ( g f ) ¯ cog-grid $ \Delta\overline{\log({g}f)}_{\text{ cog-grid}} $ plotted against theoretical Wλ values for the Sun, shown for different steps of Ωcore (left to right, top to bottom). The mean difference of the Δ log ( g f ) ¯ cog-grid $ \Delta\overline{\log({g}f)}_{\text{ cog-grid}} $ values is shown by the dashed blue lines, and the standard deviation is denoted by the hatched area.
In the text
thumbnail Fig. 8.

Top panel: mean Δ log ( g f ) ¯ cog-grid $ \Delta\overline{\log({g}f)}_{\text{ cog-grid}} $ values plotted against Ωcore. Black error bars show the standard deviation σ of the Δ log ( g f ) ¯ cog-grid $ \Delta\overline{\log({g}f)}_{\text{ cog-grid}} $ values, and green error bars show mean of the σgrid values. Lower panel: σ and σ ¯ grid $ \overline{\sigma}_{\mathrm{grid}} $ values plotted as a function of Ωcore. The mean of the Δ log ( g f ) ¯ cog-grid $ \Delta\overline{\log({g}f)}_{\text{ cog-grid}} $ values is shown to increase as Ωcore increases. The σ also increases with Ωcore, however the σ ¯ grid $ \overline{\sigma}_{\mathrm{grid}} $ exhibits a flatter trend. At Ωcore ≤ 5%, σ is almost constant suggesting that the intrinsic scatter between the two methods is σ = 0.04 dex.
In the text
thumbnail Fig. 9.

Δ λ ¯ grid-input $ \Delta\overline{\lambda}_{\text{ grid-input}} $ vs. λinput for the 1091 investigated lines. 1σ error bars are shown in blue. A slight negative offset is present, however this is within the uncertainty in the HERMES spectrograph wavelength calibration. A small fraction of transitions have wavelength offsets that are attributed to inaccuracies in the atomic data.
In the text
thumbnail Fig. 10.

Δ log ( g f ) ¯ grid-input $ \Delta\overline{\log({g}f)}_{\text{ grid-input}} $ (in blue) and Δ log ( g f ) ¯ cog-input $ \Delta\overline{\log({g}f)}_{\text{ cog-input}} $ (in black) plotted against log(g f)input for the 408 analysis-independent lines discussed in Sect. 5.1. There are no clear trends in either the Δ log ( g f ) ¯ grid-input $ \Delta\overline{\log({g}f)}_{\text{ grid-input}} $ values or Δ log ( g f ) ¯ cog-input $ \Delta\overline{\log({g}f)}_{\text{ cog-input}} $ values with respect to log(g f)input, however there are a significant number of transitions for which both methods, which agree within 0.04 dex of each other, find sizeable offsets compared to the log(g f)input. Some transitions even show differences between log(g f)input and our log ( g f ) ¯ $ \overline{\log({g}f)} $ values of ±0.50 dex or more. The zoomed subplot shows transitions that have significantly smaller differences between the derived and input log(g f) values, however a number of them still do not agree within the derived uncertainties.
In the text
thumbnail Fig. 11.

Δ log ( g f ) ¯ grid-input $ \Delta\overline{\log({g}f)}_{\text{ grid-input}} $ vs. log(g f)input for the 1091 investigated lines. Bottom panel: as above but for Δ log ( g f ) ¯ cog-grid $ \Delta\overline{\log({g}f)}_{\text{ cog-grid}} $. The 408 analysis-independent lines are shown in light green, the remainder of the 845 quality-assessable lines are shown in dark green, and the remainder of the 1091 investigated lines are shown in black. There are no obvious trends in either Δ log ( g f ) ¯ grid-input $ \Delta\overline{\log({g}f)}_{\text{ grid-input}} $ or Δ log ( g f ) ¯ cog-grid $ \Delta\overline{\log({g}f)}_{\text{ cog-grid}} $ with respect to log(g f)input, except for a number of transitions where both methods, agreeing with each other within errors, find significant upward revisions of the log(g f)input values as large as Δ log(g f) = 1.5 dex.
In the text
thumbnail Fig. 12.

As in Fig. 10, but plotted as a function of Elow. There is a slight upward inflection of log ( g f ) ¯ $ \overline{\log({g}f)} $ values at excitation energies of Elow ≤ 0.25 eV which is possibly a symptom of non-LTE effects in these low-level transitions. Between 3.5 ≤ Elow ≤ 5.0 eV we find an increase in scatter of log ( g f ) ¯ $ \overline{\log({g}f)} $ values, often belonging to higher excitation Fe I transitions. The zoomed subplot shows that a significant number of analysis-independent transitions have multiplets in common, and that the log(g f) values of these multiplets can show significant scatter.
In the text
thumbnail Fig. 13.

As in Fig. 11, but plotted as a function of Elow. The large downward revisions pertain to Fe I transitions originating from higher excitation energies. The upward trend towards Elow = 0 eV is still visible, however to a lesser degree than the distribution of solely the analysis-independent lines.
In the text
thumbnail Fig. 14.

Median of the 408 analysis-independent Δ log ( g f ) ¯ method-input $ \Delta\overline{\log({g}f)}_{\text{ method-input}} $ values, for both the curve of growth (black circles) and iterative modelling (blue circles) methods, are plotted against the stellar parameters of the seven benchmark stars. The median of the σgrid values are denoted by the blue error bars. The overestimation of the log ( g f ) cog ¯ $ \overline{\log({g}f)_{\mathrm{cog}}} $ relative to the log ( g f ) grid ¯ $ \overline{\log({g}f)_{\mathrm{grid}}} $ values can be seen. Apart from V I, we do not find any clear trends with respect to the stellar parameters for any of the 14 remaining species in the analysis-independent subset of lines.
In the text
thumbnail Fig. B.1.

log(g f) determinations for the λ5708.1 Fe I line, which is deemed unsuitable for quality assessment due to the disagreement between the log ( g f ) ¯ cog $ \overline{\log({g}f)}_{\mathrm{cog}} $ and log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ values. Observed line profiles are shown in black dots, line profiles calculated using the log(g f)input values are shown in dashed grey, line profiles calculated using the log ( g f ) ¯ cog $ \overline{\log({g}f)}_{\mathrm{cog}} $ values are shown in green, line profiles calculated using the log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ values are shown in blue, and errors are shown as green and blue shaded areas, respectively. Vertical blue and dashed grey lines denote the λ ¯ grid $ \overline{\lambda}_{\mathrm{grid}} $ and λinput values respectively. The disagreement in this case is likely caused by the presence of a nearby spectral line affecting the single Gaussian fit of the curve of growth method.
In the text
thumbnail Fig. B.2.

log(g f) determinations for the λ5435.8 Ni I line which is deemed to be quality-assessable. Observed line profiles are shown in black dots, line profiles calculated using the log(g f)input values are shown in dashed grey, line profiles calculated using the log ( g f ) ¯ cog $ \overline{\log({g}f)}_{\mathrm{cog}} $ values are shown in green, line profiles calculated using the log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ values are shown in blue, and errors are shown as green and blue shaded areas, respectively. Vertical blue and dashed grey lines denote the λ ¯ grid $ \overline{\lambda}_{\mathrm{grid}} $ and λinput values, respectively.
In the text
thumbnail Fig. B.3.

log(g f) determinations for the λ5292.6 Fe I line deemed suitable for quality assessment and deemed to be robust against adopted analysis method. Observed line profiles are shown in black dots, line profiles calculated using the log(g f)input values are shown in dashed grey, line profiles calculated using the log ( g f ) ¯ cog $ \overline{\log({g}f)}_{\mathrm{cog}} $ values are shown in green, line profiles calculated using the log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ values are shown in blue, and errors are shown as green and blue shaded areas, respectively. Vertical blue and dashed grey lines denote the λ ¯ grid $ \overline{\lambda}_{\mathrm{grid}} $ and λinput values, respectively. This line also shows a measurable Δλ correction of ∼0.02 Å to the input wavelength.
In the text
thumbnail Fig. C.1.

Quality assessment for the λ4555.5 Ti I line. The style follows that of Figs. B.1–B.3, minus the log ( g f ) ¯ cog $ \overline{\log({g}f)}_{\mathrm{cog}} $ line profiles, and with several additional line profiles corresponding to the available literature log(g f) values of the transition. In this case all literature log(g f) values are within error of the log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ value and can be recommended for use.
In the text
thumbnail Fig. C.2.

Quality assessment for the λ6456.4 Fe II line. The style follows that of Figs. B.1–B.3, minus the log ( g f ) ¯ cog $ \overline{\log({g}f)}_{\mathrm{cog}} $ line profiles, and with several additional line profiles corresponding to the available literature log(g f) values of the transition. In this case only one literature log(g f) value is within error of the log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ value, and thus only this value is recommended for use.
In the text
thumbnail Fig. C.3.

Quality assessment for the λ6786.8 Fe I line. The style follows that of Figs. B.1–B.3, minus the log ( g f ) ¯ cog $ \overline{\log({g}f)}_{\mathrm{cog}} $ line profiles, and with several additional line profiles corresponding to the available literature log(g f) values of the transition. In this case none of the literature log(g f) values can accurately reproduce the observed profile, with Δ log(g f) values ranging between 0.1–0.25 dex, thus we would propose our log ( g f ) ¯ grid $ \overline{\log({g}f)}_{\mathrm{grid}} $ as a more accurate alternative.
In the text

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