GALA: AN AUTOMATIC TOOL FOR THE ABUNDANCE ANALYSIS OF STELLAR SPECTRA^*

The main parameters that define the model atmosphere, namely the effective temperature (T_eff), the surface gravity (log g), the microturbulent velocity (v_t), and the overall metallicity ([M/H]⁴), are constrained by the following.

• 1.  
  Temperature. The best value of T_eff is derived by imposing the so-called excitation equilibrium, requiring that there is no correlation between the abundance and the excitation potential χ of the neutral iron lines. The number of electrons populating each energy level is basically a function of T_eff according to the Boltzmann equation. If we assume a wrong T_eff in the analysis of a given stellar spectrum, we need different abundances for matching the observed profile of transitions with different values of χ. For instance, the use of a value of T_eff that is too large will lead to under-populating the lower energy levels, thus the predicted line profile for low χ transitions will be too shallow and a higher abundance will be needed to match the line profile. On the other hand, a wrong (too low) T_eff will lead to a deeper line profile for the low χ transitions. For this reason, a wrong, too large value of T_eff will introduce an anticorrelation between abundances and χ, and in the same way, a positive correlation is expected in the case of the adoption of a value of T_eff that is too small.
• 2.  
  Surface gravity. The best value of log g is derived with the so-called ionization equilibrium method, requiring that for a given species, the same abundance (within the uncertainties) has been obtained from lines of two ionization states (typically, neutral and singly ionized lines). Because gravity is a direct measure of the pressure of the photosphere, variations of log g lead to variations in the ionized lines (which are very sensitive to electronic pressure), while the neutral lines are basically insensitive to this parameter. This method implicitly assumes that the energy levels of a given species are populated according to the Boltzmann and Saha equations (thus, under local thermodynamical equilibrium (LTE) conditions). Possible departures from this assumption (which is especially critical for metal-poor and/or low-gravity stars) could alter the derived gravity when it is derived from the ionization balance because non-LTE effects affect mainly the neutral lines (even if the precise magnitude of the departures from LTE for the iron lines is still a matter of debate). As a sanity check, following the suggestion by Edvardsson (1988, p. 1), surface gravities determined from the ionization equilibria have to be "checked—when possible—with gravities determined from the wings of pressure-broadened metal lines".
• 3.  
  Microturbulent velocity. v_t is computed by requiring that there is no correlation between the iron abundance and the line strength (see Mucciarelli 2011 for a discussion about different approaches). The microturbulent velocity affects mainly the moderate/strong lines located along the flat regime of the curve of growth, while the lines along the linear part of the curve of growth are mainly sensitive to the abundance instead of the velocity fields. The necessity to introduce the microturbulent velocity as an additional broadening (added in quadrature to the Doppler broadening) arises from the fact that the non-thermal motions (basically due to the onset of the convection in the photosphere) are generally not well described by one-dimensional, static model atmospheres. Citing Kurucz (2005, p. 16), "microturbulent velocity is a parameter that is generally not considered physically except in the Sun" because in the Sun, velocity fields can be derived as a function of optical depth through analysis of the intensity spectrum (as performed by Fontenla et al. 1993). For other stars, v_t represents only a corrective factor that minimizes the line-to-line scatter for a given species and compensates (at least partially) for the incomplete description of the convection as implemented in one-dimensional model atmospheres.
• 4.  
  Metallicity. [M/H] is chosen according to the average iron content of a star, assuming [Fe/H] as a proxy for the overall metallicity. Generally, [Fe/H] is adopted as a good proxy of the metallicity because of its large number of available lines, but it does not necessarily indicate the overall metallicity of the studied star. In fact, iron is generally not the most abundant element in stars; elements such as C, N, and O would be the best tracers of stellar metallicity but they are difficult to measure.

Because of its statistical nature, the spectroscopic optimization of all the parameters can simultaneously be performed only if we have a sufficient number of Fe lines, distributed in a large range of EW and χ and in two levels of ionization. Alternatively, T_eff and log g can be inferred from the photometry (for instance with the isochrone-fitting technique or employing empirical or theoretical T_eff–color relations) or by fitting the wings of damped lines (such as hydrogen Balmer lines or the Mg b triplet) sensitive to the parameters, and only v_t needs to be tuned spectroscopically (following the approach described above). Note that some authors consider the method of deriving the parameters based on these constraints only as sanity checks performed a posteriori on the photometric parameters while other authors rely on these constraints to infer the best parameters.

In light of the method described above, it is worth bearing in mind that the atmospheric parameters are correlated with each other. In fact, the strongest lines are typically those with low χ: Figure 1 plots all the transitions in the range λ = 4000–8000 Å and with χ < 10 eV in the Kurucz/Castelli database⁵ in the plane χ versus log (gf) − θχ.⁶ As mentioned in Section 2.1, T_eff and χ are strictly linked and there is also a connection between v_t and the line strength. Hence, the statistical correlation between χ and the line strength leads to a correlation between T_eff and v_t. Thus, a variation of T_eff implies a variation of v_t. Also, variations of T_eff and v_t will change the abundances derived from different levels of ionization (hence, the gravity) in different ways.

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Let us consider an ATLAS9 model atmosphere computed with T_eff = 4500 K, log g = 1.5, v_t = 2 km s⁻¹, and [M/H] = −1.0 dex (Castelli & Kurucz 2004) and a set of neutral and singly ionized iron lines (predicted to be unblended through the inspection of a spectrum calculated with the same parameters at a spectral resolution of 45,000). The EWs of these transitions are computed by integrating the theoretical line profile through the WID subroutine implemented in the WIDTH9 code (Castelli 2005b). This means that each of these EWs will provide exactly [El/H] = −1.0 dex when they are analyzed using the model atmosphere described above.

The analysis of these lines (always adopting the same set of EWs) is repeated investigating a regular grid of the atmospheric parameters, namely T_eff = 3600–5400 K, log g = 0.5–2.5, v_t = 1.0–3.0 km s⁻¹, steps of δT_eff = 200 K, δlog g = 0.2, δv_t = 0.5 km s⁻¹, and assuming for all the models [M/H] = −1.0 dex. Figures 2, 3, and 4 display the quite complex interplay occurring among the atmospheric parameters.

• 1.  
  Figure 2 shows the behavior of the slope S_χ of the A(Fe)–χ relation for the above sample of lines as a function of T_eff, keeping gravity fixed, but varying the microturbulent velocity. The thick gray line connects points calculated with the original v_t of the model. The global trend is basically linear, at least if we consider a range of ±1000 K around the original T_eff. The inset panel shows the behavior of T_eff for which S_χ is zero (thus, the best T_eff) as a function of v_t and considering different gravities. The derived best temperature increases with increasing v_t (at fixed log g); gravity only has a second-order effect and it does not change the general behavior of the best T_eff as a function of v_t.
• 2.  
  Figure 3 summarizes the behavior of the slope S_EWR of the A(Fe)–EWR relation as a function of v_t (where EWR indicates the reduced EW, defined as EWR = log (EW/λ)), keeping gravity fixed at the original value but varying T_eff. For a given temperature, the slope decreases with increasing v_t, with a behavior that becomes less steep at v_t larger than the original value. The effect of the T_eff is appreciable for T_eff larger than the original value, while for lower T_eff all the curves are very similar to each other. The inset shows the change of the best value of v_t (for which the slope of the A(Fe)–EWR relation is zero) as a function of T_eff and for different values of log g. The observed trend is quite complex; basically, we note that the best value of v_t is very sensitive to T_eff when the latter is overestimated with respect to the true temperature, but with a negligible dependence on gravity, while the behavior is the opposite when T_eff is underestimated, with a degeneracy between T_eff and the best v_t but a consistent dependence on gravity.
• 3.  
  Figure 4 shows the behavior of the difference between A(Fe i) and A(Fe ii) as a function of log g, assuming v_t = 2 km s⁻¹ and for different values of T_eff. The general behavior is linear and the iron difference increases considerably with increasing temperature. The best value of gravity (see the inset in Figure 4) is highly sensitive to changes in T_eff, with δlog g/δT_eff ≃ 1 dex/300 K in the investigated case, but for a fixed T_eff turns out to be marginally sensitive to v_t (this is due to the fact that the Fe ii lines are basically distributed in strength in a similar way to the Fe i lines).

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It is important to bear in mind that these considerations are appropriate for the investigated case of a late-type star but the dependencies among the parameters can be different for different regimes of atmospheric parameters and/or metallicity. However, the example presented above demonstrates that an analytic approach to deriving the best model atmosphere is discouraged because it requires precise topography of the parameter space and inspection of a large number of model atmospheres.

GALA has been designed to perform classical chemical analysis based on the EWs in an automated way. The user can choose to perform a full spectroscopic optimization of the parameters or to optimize only some parameters, keeping the other parameters fixed to the specified input values.

The algorithm optimizes one parameter at a time, checking continuously if the new value of a given parameter changes the validity of the previously optimized ones.

For each atmospheric parameter X (corresponding to T_eff, log g, v_t, and [M/H]) we adopt a specific optimization parameter C(X), defined such that it turns out to be zero when the best value of the X parameter has been found. Hence, GALA varies X until a positive/negative pair of the C(X) is found, thus bracketing the zero value corresponding to the best value. Thus, the condition $C(\tilde{X}) = 0$ identifies $X = \tilde{X}$ as the best value of the given parameter. Finally, the best solution converges to a set of parameters that simultaneously verifies the constraints described in Section 2.1. When the atmospheric parameters have been found, the abundances of all the elements for which EWs have been provided are derived.

According to the literature, the adopted values of C(X) have been defined to parameterize the conditions listed in Section 2.1, namely, the following.

• 1.  
  The angular coefficient of the A(Fe)–χ relation (S_χ) for constraining T_eff. The lower panel of Figure 5 shows the change of this slope for a set of theoretical EWs computed with the correct T_eff (gray points) and with temperatures varied by ±500 K (empty points). The variation of T_eff produces a change in the slope (but also a change in the y-intercept).
• 2.  
  The angular coefficient of the A(Fe)–EWR relation (S_EWR) for constraining v_t. The upper panel of Figure 5 shows the same set of theoretical EWs analyzed with different values of v_t.
• 3.  
  The difference of the mean abundances obtained from Fe i and Fe ii lines for constraining the gravity.
• 4.  
  The average Fe abundance for constraining the metallicity of the model.

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If the errors in the EW measurement (σ_EW) are provided as input, the slopes are computed by taking into account the abundance uncertainties of the individual lines; the uncertainty on the iron abundance of a given line is estimated from the difference of the iron abundance computed for the input EW and for EW+σ_EW.⁷ In the A(Fe)–χ plane the uncertainties of χ are reasonably assumed to be negligible because the uncertainties of χ are typically less than 0.01 eV, while in the A(Fe)–EWR plane the least-squares fit takes into account the uncertainties in both the axes (following the prescriptions by Press et al. 1992).

The flexibility of GALA permits us to simultaneously deal with stars of different spectral types. This is done by considering that a large number of Fe i lines are generally available for F–G–K spectral type stars, whereas they are less numerous (or lacking) in O–B–A stars, for which a large number of Fe ii lines are typically available. Also, in some spectral regions there is a large number of lines for other iron-peak elements (mainly Ni, Cr, and Ti). For this reason, GALA is designed to optimize the parameters using lines other than Fe i by appropriately configuring the code. In the following, we will refer to the optimization made by using Fe i for T_eff and v_t, but our considerations are also valid for other elements with a sufficient number of lines.

GALA is structured in three main working blocks.

• 1.  
  The "guess" working block is aimed at finding the presumed, or guessed, atmospheric parameters in a fast way (this is especially useful in cases of large uncertainties or when one lacks a first-guess value for the parameters).
• 2.  
  The "analysis" working block finds the best model atmosphere through a local minimization starting with the guessed parameters provided by the user or obtained through the previous block.
• 3.  
  The "refinement" working block refines the solution, starting with the atmospheric parameters obtained in the previous block.

GALA can be flexibly configured to use different combinations of the three main working blocks. We defer to Section 8.4 the discussion of the effects of the working blocks. In the following, we describe the algorithm of each block and the cases in which each is recommended.

3.2.1. Guess Working Block

3.2.2. Analysis Working Block

This working block performs a complete optimization starting from the input parameters provided by the users or from those obtained with the guess working block. This block is developed to find a robust solution under the assumption that the input values are reasonably close to the real solution. When good priors are available (for instance, in the case of stellar cluster stars) this block is sufficient for finding the solution without the use of the guess block. Otherwise, when the guess model is uncertain (for instance, in the case of field stars for which reddening, distance, and evaluative mass could be highly uncertain, or, generally, in the case of inaccurate photometry), the analysis block is recommended to be used after the guess block. Figure 6 shows a flow chart the main steps of this working block. The following iterative procedure is performed.

• 1.  
  The procedure starts by computing the abundances using the guessed parameters and performing a new line rejection (independent of that of the previous block). At variance with the previous working block, now the parameters are varied by small steps configured by the user.
• 2.  
  The model metallicity is refined to match the average iron abundance.
• 3.  
  A new model with different values of T_eff is computed, according to the sign of S_χ of the previous model (i.e., a negative slope indicates an overestimated T_eff and vice versa). New models, varying only T_eff, are computed until a pair of negative/positive S_χ is found. Thus, these two values of T_eff identify the range of T_eff where the slope is zero. T_eff corresponding to the minimum |S_χ| is adopted.
• 4.  
  The same procedure is performed for v_t. If the final value of v_t is different from that used in the previous loop, GALA goes back to (2), checking if the new value of v_t needs a change in [M/H] and T_eff. Otherwise, the procedure moves on to the next loop.
• 5.  
  The surface gravity is varied until a positive/negative pair of Δ(Fe) is found. If the output log g differs from the input value, GALA returns to (2) with the last obtained model atmosphere and the entire procedure is repeated.
• 6.  
  When a model that satisfies all four constraints is found, the procedure ends and the next star is analyzed.

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We stress that the method employed in this working block is very robust but it has the disadvantage of being slow if the guessed parameters are far from the local solution.

3.2.3. Refinement Working Block

The most difficult parameter to constrain with the classical spectroscopic method is gravity. This is because of the relatively small number of available Fe ii lines, which can vary in the visual range from a handful of transitions up to ∼20, depending on the spectral region and/or the metallicity (for instance, some high-resolution spectra with a small wavelength coverage, such as the GIRAFFE at VLT or the Hydra at the BlancoTelescope spectra, can totally lack Fe ii lines).

GALA is equipped with different options for optimizing log g.

• 1.  
  The normal optimization is computed using the difference between the average abundances from neutral and singly ionized iron lines (as described above).
• 2.  
  The gravity is computed from the Stefan–Boltzmann equation by providing for input the term = log (4GMπσ/L), where G is the gravitational constant, σ is the Boltzmann constant, and M and L are, respectively, the mass and the luminosity of the star. Thus, during the optimization process, gravity is recomputed (as log g = − 4log T_eff) in each iteration according to the new value of T_eff.
• 3.  
  Gravity is computed by assuming a quadratic relation $\log g = A+B\, T_{\rm eff}+C\, T_{\rm eff}^2$ and providing as input the coefficients A, B, and C. This option is useful when, for instance, the investigated stars belong to the same stellar cluster and log g and T_eff can be parameterized by a simple relation (i.e., such as that described by a theoretical isochrone for a given evolutionary stage).

The user can choose the way to treat log g (fixed or optimized following one of the methods described above); if optimization of log g from the iron lines is requested but no Fe ii lines are available, GALA will try to use the second option (lines of other elements in different stages of ionization), or eventually will fix log g to the input value.

The algorithm used in GALA is basically independent of the code adopted to derive the abundances and of the model atmospheres. GALA is designed to manage the two most used and publicly available model atmospheres, namely, ATLAS9 and MARCS.

• 1.  
  ATLAS9. The suite of Kurucz codes represents the only suite of open-source and free programs to include the different aspects of the chemical analysis (model atmospheres, abundance calculations, spectral synthesis), allowing any user to compute new models and upgrade parts of the codes. GALA includes a dynamic call for the ATLAS9 code.⁸ Any time GALA needs to investigate a given set of atmospheric parameters, ATLAS9 is called, a new model atmosphere is computed, and finally, it is stored in a directory. The directory is checked by GALA whenever a model atmosphere is requested and ATLAS9 is called only if the model is lacking. The convergence of the new model atmosphere is checked for each atmospheric layer. Following the prescriptions by Castelli (1988), we require errors less than 1% and 10% for the flux and the flux derivative, respectively. Additional information about the calculation of each model atmosphere is saved. In the current version, GALA is able to manage the grid of ATLAS9 models by Castelli & Kurucz (2004) and the new grid of models calculated by Mészáros et al. (2012) for the APOGEE survey.
• 2.  
  MARCS. At variance with ATLAS9, for the MARCS models, the code to compute new model atmospheres is not released to the community. However, the Uppsala group provides a large grid of the MARCS models on their Web site.⁹ When GALA works with these grids (including both plane-parallel and spherical symmetry), new models are computed by interpolating the Uppsala grid using the code developed by T. Masseron (Masseron 2006).¹⁰ This code has been modified in order to put the interpolated MARCS models in ATLAS9 format to use with WIDTH9.

Note that the automatization of the chemical analysis based on EWs needs a wide grid of model atmospheres (both to interpolate and compute new models) linked to the code in order to freely explore the parameter space. Thus, GALA is linked to the ATLAS9 grid both with solar-scaled and α-enhanced chemical composition and to the MARCS grid with standard composition. Also, the use of other models or model grids can be easily implemented in the code. Sometimes, peculiar analyses or tests need to use specific models, for instance for the Sun (see the set of solar model atmospheres available from F. Castelli's Web site) or with arbitrary chemical compositions, such as those computed with the ATLAS12 code (Castelli 2005a). When a specific, single model is called, all the optimization options are automatically switched off.

GALA is equipped with a number of exit flags in order to avoid infinite loops or unforeseen cases stopping the analysis of the entire input list of stars. Here we summarize the main exit options.

• 1.  
  The user can set among the input parameters the maximum number of iterations allowed for each star. When the code reaches this value, it stops the analysis, moving to the next star. Generally, this parameter depends on the adopted grid steps and whether or not the input atmospheric parameters are close to the real parameters. When the guess working block is used, the analysis block typically converges in 3–5 iterations.
• 2.  
  If the dispersion around the mean of the abundances of the lines used for the optimization (after the line rejection) exceeds a threshold value, GALA skips the star. In fact, very large dispersions can possibly suggest some problems in the EW measurements.
• 3.  
  If the number of lines used for the optimization (after the line rejection) is smaller than a threshold, the optimization is not performed and the atmospheric parameters are fixed to the input guessed values.
• 4.  
  The procedure is stopped if the requested atmospheric parameter is outside the adopted grid of model atmospheres.
• 5.  
  GALA skips the analysis of the star if the call to the model atmosphere fails (problems in the ATLAS9 model computation or in the MARCS model interpolation) and the required model is not created. Otherwise, if the ATLAS9 model is calculated but some atmospheric layers do not converge (according to the criteria discussed above), GALA continues the analysis but it advises the user of the number of unconverged layers.

The statistical uncertainty on the abundance of each element is computed by considering only the surviving lines after the rejection process (see Section 4). When the uncertainty on the EW is provided for each individual line (thus allowing us to compute the abundance error for each transition), the mean abundance is computed by weighing the abundance of each line on its error, otherwise simple average and dispersion are computed. For those elements for which only one line is available, the error in abundance is obtained by varying the EW of 1σ_EW (if the EW uncertainties are provided). Otherwise, the adopted value is zero. As customary, the final statistical error on the abundance ratios is defined as $\sigma /\sqrt{N_{{\rm lines}}}$.

GALA estimates the internal error for each stellar parameter that has been derived from the spectroscopic analysis. The uncertainties of T_eff, v_t, and log g are estimated propagating the errors of the corresponding optimization parameter:

$$\begin{eqnarray*} &&\sigma _{X_{\rm i}}=\frac{\sigma _{C(X_{\rm i})}}{ \left(\frac{\delta C(X_{\rm i})}{\delta X_{\rm i}}\right)}, \end{eqnarray*}$$

where X_i are T_eff, log g, and v_t; C(X_i) indicates the optimization parameters defined in Section 3.1; and $\sigma _{C(X_{\rm i})}$ are the corresponding uncertainties.

The terms δC(X_i)/δX_i (which parameterize how the slopes and the iron difference vary with the appropriate parameters) are calculated numerically by varying X_i locally around the final best value, assuming the step used in the optimization process and recomputing the corresponding C(X_i).

The terms C(X_i) are computed by applying a jackknife bootstrapping technique (see Lupton 1993). The quoted quantities are recomputed by leaving out one different spectral line from the sample each time (thus, given a sample of N lines, each C(X) is computed N times by considering a subsample of N − 1 lines). The uncertainty on the parameter X is $\sigma _{{\rm Jack}}=\sqrt{N-1}\sigma _{{\rm sub}}$, where σ_sub is the standard deviation of the C(X) distribution derived from N subsamples. The σ_Jack takes into account the uncertainty arising from the sample size and the line distribution, and this resampling method is especially useful for estimating the bias arising from the lines' statistics. Note that the computation of the slopes is performed taking into account the effect of the EW and abundance uncertainties of each individual line. Thus, the uncertainty in the atmospheric parameter X_i becomes

$$\begin{eqnarray*} &&\sigma _{X_{\rm i}}=\frac{\sigma ^{{\rm Jack}}_{C(X_{\rm i})}}{ \left(\frac{\delta C(X_{\rm i})}{\delta X_{\rm i}} \right)}. \end{eqnarray*}$$

It is worth noting that these uncertainties represent the internal error in the derived parameters and are strongly dependent on the number of used lines and on the distribution of the lines (weak and strong transitions for the estimate of v_t and low- and high-χ lines for T_eff). Other factors that can affect the determination of the parameters (for instance, the threshold adopted in the EWs and in σ_EW) are not included in the error budgets and they can be considered external errors. Finally, the error due to the adopted grid size could be considered a systematic uncertainty (being the same for all the analyzed stars) and eventually added in quadrature to the internal error estimated by GALA.

The evaluation of the uncertainties arising from the atmospheric parameters is a more complex task. Generally, these errors are referred to as "systematic" uncertainties but this nomenclature is rather imprecise. In fact, the variation of a given parameter changes the abundance derived from the lines of the same element in a similar way (for instance, an increase of T_eff increases the abundance of all the iron lines). However, this error will be different from star to star due to the different numbers of lines, strength and χ distributions, EW quality, and so on. Thus, the uncertainties from the atmospheric parameters should be considered random errors when we compare different stars (but they are systematic uncertainties from line to line).

Several recipes are proposed in the literature. The most common method is to recompute the abundances, each time changing only one parameter, and keeping the other ones fixed to their best estimates. Then, the corresponding variations in the abundances are added in quadrature. This approach is the most conservative because it neglects the covariance terms arising from the interplay among the parameters (see Section 2.2), providing only an upper limit for the total error budget.

GALA follows the approach described by Cayrel et al. (2004) to naturally take into account the covariance terms. When the optimization process is ended, the analysis is repeated by altering the final T_eff by +$\sigma _{T_{\rm eff}}$ and $-\sigma _{T_{\rm eff}}$ (these uncertainties are calculated as described in Section 6.2), and re-optimizing the other parameters. The net variation of each chemical abundance with respect to the original value is assumed as final uncertainty due to the atmospheric parameters and naturally including the covariance terms. Additionally, upon request, the abundance variations are also calculated following the classical approach of varying only one parameter each time (keeping the other parameters fixed), leaving the user free to use this information as preferred.

GALA also provides a check parameter, useful for judging the quality of the global solution and for rapidly identifying stars with unsatisfactory solutions. For each model used during the optimization process, a merit function F_merit is defined as

$$\begin{eqnarray*} &&F_{{\rm merit}}=\sqrt{{\left(\frac{S_{\chi }}{\sigma _{{\rm Jack}}^{S_{\chi }}} \right)}^{2} + {\left(\frac{S_{{\rm EWR}}}{\sigma _{{\rm Jack}}^{{\rm EWR}}} \right)}^{2}+ {\left(\frac{\Delta {\rm Fe}}{\sigma _{{\rm Jack}}^{\Delta {\rm Fe}}} \right)}^{2}}, \end{eqnarray*}$$

taking into account the values of the optimization parameters and the corresponding uncertainties. In an ideal case, F_merit is zero if the three optimization parameters are exactly zero. Generally, all the solutions with F_merit ≃1 are valid and equally acceptable, while values of F_merit ≫1 are suspect and point out that at least one of the parameters is not well constrained within the quoted uncertainty. Note that F_merit provides only an indication of whether or not the solution is acceptable, but it does not specify which parameter is not well defined.

To summarize, when the full optimization process is completed, GALA will provide for each analyzed element the (weighted) mean abundance, the dispersion and the number of lines used (which provide the statistical uncertainty), the net variation in abundance due to the new optimization with $T_{\rm eff}+\sigma _{T_{\rm eff}}$ and that with $T_{\rm eff}-\sigma _{T_{\rm eff}}$ (which provides the uncertainty owing to the choice of stellar parameters). Also, for each atmospheric parameter the quoted internal uncertainties are computed. Finally, the quality parameter F_merit is provided to evaluate the goodness of the solution as a whole.

With GALA we analyzed a grid of synthetic spectra computed to mimic the UVES at VLT high-resolution spectra with the 580 Red Arm setup. The grid of synthetic spectra includes S/Ns of 20, 30, 50, and 100 per pixel for two different sets of atmospheric parameters: T_eff = 4500 K, log g = 1.5, v_t = 2 km s⁻¹, [M/H] = −1.0 dex to simulate a giant star, and T_eff = 6000 K, log g = 4.5, v_t = 1 km s⁻¹, [M/H] = −1.0 dex to simulate a dwarf star. The spectra were computed with the following procedure.

• 1.  
  For a given model atmosphere, two synthetic spectra were calculated with the SYNTHE code over the wavelength range covered by the two CCDs of the 580 UVES Red Arm grating and then convolved with a Gaussian profile in order to mimic the formal UVES instrumental broadening.
• 2.  
  The spectra were rebinned to a constant pixel size (δλ = 0.0147 and 0.0174 pixel Å⁻¹ for the lower and upper chips, respectively).
• 3.  
  The synthetic spectra (normalized to unity) were multiplied with the efficiency curve computed by the FLAMES-UVES ESO Time Calculator in order to model the shape of the templates as realistically as possible.
• 4.  
  Poissonian noise was injected in the spectra to simulate different noise conditions. Basically, the S/N varies along the spectrum as a function of the efficiency (and thus of the wavelength). The noise was added in any spectrum according to the curve of S/N as a function of λ provided by the FLAMES-UVES ESO Time Calculator. For each S/N a sample of 200 synthetic spectra was generated.

Figure 8 summarizes the average values obtained for each Monte Carlo sample for each atmospheric parameter as a function of S/N; the error bars indicate the dispersion around the mean. Results of the simulations of the giant star model atmosphere are shown in the upper panels of each window, while the lower panels summarize the results for the dwarf star simulations. Basically, the original parameters of the synthetic spectra (marked in Figure 8 as dashed horizontal lines) are recovered with small dispersions and without any significant bias. The major departure from the original values is observed in the microturbulent velocities (both dwarf and giant) at S/N = 20 because of the loss of weak lines.

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The previous test provides an indication about the stability of the method against the S/N, starting with spectra parameters that are known a priori. However, the employed noise model is a simplification because it does not take into account some effects that can also heavily affect the measurement of the EWs, such as the correlation of the noise among adjacent pixels, flat-fielding residuals, failures in the echelle orders merging, and the presence of spectral impurities. Also, the atomic data of the analyzed lines are the same used in the computation of the synthetic spectra, thus excluding from the final line-to-line dispersion random error due to the uncertainty on the atomic data.

In order to provide an additional test of the performance of GALA in conditions with different noise, we performed a simple experiment on the spectra acquired with UVES at FLAMES of the giant star NGC 1786−1501 in the Large Magellanic Cloud (LMC) globular cluster NGC 1786 (see Mucciarelli et al. 2009, 2010). This is a sample of eight spectra with the same exposure time (∼45 minutes) obtained under the same seeing conditions, with a typical S/N per pixel of 20 for each exposure. We used this data set in order to obtain eight spectra with different S/Ns ranging from ∼20 to ∼60 depending on the number of exposures averaged: the spectrum with the lowest S/N is just one exposure and the spectrum with the largest S/N is the average of all eight acquired exposures. The derived parameters for each spectrum are shown in Figure 9 as a function of S/N. The error bars are derived by applying the jackknife bootstrapping technique for T_eff, log g, and v_t, while for [Fe/H] we used the dispersion by the mean as an estimate of the error.

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We note that the parameters are well constrained with small uncertainties for spectra with low S/N: this is not a numerical artifact of the code but it is due to the large number of transitions available in the UVES spectra, coupled with an accurate rejection of the outliers and the use of the uncertainties for each individual line in the slopes' computations. Also, we note that major departures from the final parameters are found again in the determination of v_t at S/N = 20: the derived low value of v_t is due to the fact that at low S/Ns, several weak lines (useful for constraining the microturbulent velocity) are not well measured (then discarded by GALA) or not identified in the noise envelope by DAOSPEC. Note that the derived trend of v_t as a function of S/N shows the same behavior found and discussed by Mucciarelli (2011).

For each star we defined a suitable line list of Fe i and Fe ii transitions, starting from the most updated version of the Kurucz/Castelli lines data set.¹³ We apply an iterative procedure to define the line list. Assuming that the parameters of the targets are not known a priori, we performed a first analysis by using a preliminary line list including only laboratory transitions with χ < 6 eV and log (gf) > − 5 dex. Such a line list is not checked against the spectral blendings arising from the adopted spectral resolution and the atmospheric parameters and it is used only to perform a preliminary analysis. With the new parameters derived we define a new line list for each star. The lines are selected by the inspection of synthetic spectra computed with the new parameters and convolved with a Gaussian profile in order to reproduce the observed spectral resolution. At this step, only iron transitions predicted to be unblended are taken into account and used for the new analysis.

EWs are measured by using the code DAOSPEC which adopts a saturated Gaussian function to fit the line profile and a unique value for the FWHM for all the lines. We start with the FWHM derived from the nominal spectral resolution of the spectra, leaving DAOSPEC free to readjust the value of FWHM according to the global residual of the fitting procedure. The measurement of EWs is repeated using the optimized FWHM value as a new input value until convergence is reached at a level of 0.1 pixel. The formal error of the fit provided by DAOSPEC is used as 1σ uncertainty on the EW measurement.

The program stars are analyzed by employing all three working blocks, requiring a spectroscopic optimization of T_eff, log g, v_t, and [M/H] and in all cases starting with the same set of guessed parameters (namely, T_eff = 5000, log g = 2.5, [M/H] = −1.0 dex, and v_t = 1.5 km s⁻¹). The optimization is performed by exploring the parameter space in small steps of δT_eff = 50 K, δlog g = 0.1, and δv_t = 0.1 km s⁻¹; the metallicity is investigated by adopting the step of the ATLAS9 grids (δ[M/H] = 0.5 dex).

In the first run, we analyzed the program stars assuming the same configuration for the input parameters of GALA; in particular, we included only lines with σ_EW < 10% and with EWR >−5.8 (corresponding to ∼10 mÅ at 6000 Å). After a first run of GALA, we refined the maximum EWR allowed which depends mainly on the onset of the saturation along the curve of growth (and thus is different for stars with different atmospheric parameters). We adopted a maximum allowed value of EWR = −4.65 for Arcturus and μLeonis and −4.95 for the Sun and HD 84937. These values were chosen on the basis of visual inspection of the curve of growth, in order to exclude too strong lines, for which the Gaussian approximation can fail. After the first run of GALA, the line list is refined by using the new parameters obtained by GALA as described above and the procedure is repeated. Table 1 summarizes the derived atmospheric parameters (with the corresponding jackknife uncertainties) and the [Fe/H] i and [Fe/H] ii abundance ratios, together with the two error bars due to the atmospheric parameters and the internal error computed as $\sigma /\sqrt{(}N_{{\rm lines}})$.

Table 1. Iron Abundance (from Neutral and Singly Ionized Lines) and Atmospheric Parameters Derived with GALA for Arcturus, μLeonis, the Sun, and HD 84937

Star	[Fe i/H]	[Fe ii/H]	Teff	log g	vt
	(dex)	(dex)	(K)		(km s−1)
Sun	−0.01$^{+0.02}_{-0.03}\pm 0.009$	+0.01$^{+0.03}_{-0.07}\pm {0.003}$	5800 ± 64	4.50 ± 0.18	1.20 ± 0.13
Arcturus	−0.51$^{+0.05}_{-0.04}\pm 0.007$	−0.52$^{+0.03}_{-0.05}\pm {0.015}$	4300 ± 60	1.60 ± 0.06	1.50 ± 0.06
HD 84937	−2.28$^{+0.06}_{-0.04}\pm 0.007$	−2.27$^{+0.05}_{-0.04}\pm {0.020}$	6150 ± 56	3.20 ± 0.13	0.70 ± 0.24
μLeonis	+0.37$^{+0.04}_{-0.06}\pm 0.011$	+0.38$^{+0.04}_{-0.08}\pm {0.033}$	4500 ± 81	2.40 ± 0.26	1.40 ± 0.07
Sun	0.00	0.00	5778	4.438	0.8–1.35
Arcturus	−0.52 ± 0.02	−0.40 ± 0.03	4286 ± 30	1.66 ± 0.05	1.74
HD 84937	−2.14 ± 0.17	⋅⋅⋅	6251 ± 94	3.97 ± 0.18	0.8–1.7
μLeonis	+0.28 ± 0.13	⋅⋅⋅	4504 ± 121	2.33 ± 0.27	1.2–2.2

Notes. For the abundances, the first two error bars are the uncertainties arising from the atmospheric parameters following the prescriptions by Cayrel et al. (2004), while the last one is the internal error calculated as $\sigma /\sqrt{(}N_{{\rm lines}})$. The uncertainties in the derived atmospheric parameters are computed with a jackknife bootstrapping technique. In the lower part of the table, the values available in the literature are listed for comparison.

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We compare our results with those available in the literature (and listed in Table 1 as references). For the Sun we derive T_eff = 5800 ± 64 K, log g = 4.50 ± 0.18, v_t = 1.20 ± 0.13 km s⁻¹, and [Fe/H] = −0.01 ± 0.03 dex (where the error bar is the sum in quadrature of the individual uncertainties listed in Table 1). Our results for T_eff and log g agree very well with those listed in the compilation on the NASA Web site.¹⁴ Concerning the microturbulent velocities, values available in the literature range from 0.8 km s⁻¹ (Biemont et al. 1981) to 1.35 km s⁻¹ (Steffen et al. 2009). A value of 1 km s⁻¹ is typically adopted as representative for the Sun in several chemical analyses (see Caffau et al. 2011).

Our analysis of Arcturus provides T_eff = 4300 ± 60 K, log g = 1.60 ± 0.06, and v_t = 1.50 ± 0.06 km s⁻¹, with an iron abundance [Fe/H] = −0.51 ± 0.05. These results well match the recent analysis of Arcturus by Ramirez & Allende Prieto (2011) which provides an accurate determination of the atmospheric parameters and the chemical composition; in particular, T_eff and log g are derived in an independent way with respect to our approach, finding T_eff = 4286 ± 30 K (by fitting the observed spectral energy distribution), log g = 1.66 ± 0.05 (through the trigonometric parallax), while v_t turns out to be 1.74 km s⁻¹ (by using the same approach used in GALA). The final iron abundance is [Fe/H] = −0.52 ± 0.04 dex. In both cases, the agreement with the literature values is good. Finally, Figure 10 shows an example of the graphical output of GALA for Arcturus.

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For HD 84937 we derive T_eff = 6150 ± 56 K, log g = 3.20 ± 0.13, v_t = 0.70 ± 0.24 km s⁻¹, and [Fe/H] = −2.28 ± 0.06 dex, while for μLeonis we obtain T_eff = 4500 ± 81 K, log g = 2.40 ± 0.26, v_t = 1.40 ± 0.07 km s⁻¹, and [Fe/H] = +0.37 ± 0.06 dex. For these two stars, several determinations are available in the literature and we decide to use the average of the values listed in the classical compilation by Cayrel de Strobel et al. (2001) as a reference, T_eff = 6251 ± 94 K, log g = 3.97 ± 0.18, and [Fe/H] = −2.14 ± 0.17 dex for HD 84937 and T_eff = 4504 ± 121 K, log g = 2.33 ± 0.27, and [Fe/H] = +0.28 ± 0.13 dex for μLeonis. Note that the value of the microturbulent velocities is omitted by Cayrel de Strobel et al. (2001) because the authors listed in their compilation use different definitions for this parameter (see Mucciarelli 2011 for a review of the different approaches) or assume a representative value. For HD 84937, the values of v_t range from 0.8 up to 1.7 km s⁻¹, while our value is slightly lower. Also, for μLeonis, the range of values for v_t is wide (from 1.2 up to 2.2 km s⁻¹) but our value agrees with this range. Basically, the agreement with the values listed by Cayrel de Strobel et al. (2001) is good, but for the gravity of HD 84937, for which we derive a lower value, this difference can be partially explained in light of the different values of v_t.

A relevant feature of an automatic procedure to infer parameters and abundances is its stability against the input atmospheric parameters. In order to assess the effect of different first-guess parameters, we analyze the spectrum of Arcturus by investigating a regular grid of input parameters with T_eff ranging from 3800 to 4800 K (in steps of 100 K) and log g from 1.0 to 2.2 (in steps of 0.1). Figure 11 shows the grid of the input parameters in the T_eff–log g plane (empty points) with the position of the derived parameters (black points); the upper panel summarizes the results when GALA is used without the refinement working block, while the lower panel shows the results obtained by also employing the refinement option. The recovered parameters cover a small range: in the first run the dispersion of the mean is 46 K for T_eff and 0.09 for log g, while these values drop to 25 K and 0.05, respectively, when the refinement working block is enabled.

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