11institutetext: Leibniz-Institut für Astrophysik Potsdam (AIP), An der
Sternwarte 16, 14482 Potsdam, Germany
11email<EMAIL_ADDRESS>22institutetext: Indian Institute of Astrophysics,
Koramangala II Block, Bangalore-560034, India 33institutetext: Pondicherry
University, R.V. Nagar, Kalapet, 605014, Puducherry, India 44institutetext:
NAT - Universidade Cidade de São Paulo/Universidade Cruzeiro do Sul, Rua
Galvão Bueno 868, São Paulo-SP, 01506-000, Brazil 55institutetext: Centre de
Recherche Astrophysique de Lyon (CRAL, CNRS, UMR 5574), Université Lyon 1,
Ecole Nationale Supérieure de Lyon, Université de Lyon, France

# Prospects of measuring a metallicity trend and spread in globular clusters
from low-resolution spectroscopy ††thanks: The Table with the results for each
individual star is only available in electronic format the CDS via anonymous
ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/

M.Baratella Prospects of measuring a metallicity trend and spread in globular
clusters from low-resolution spectroscopy ††thanks: The Table with the results
for each individual star is only available in electronic format the CDS via
anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/Prospects of measuring a
metallicity trend and spread in globular clusters from low-resolution
spectroscopy ††thanks: The Table with the results for each individual star is
only available in electronic format the CDS via anonymous ftp to
cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-
bin/qcat?J/A+A/ Deepthi S. Prabhu Prospects of measuring a metallicity trend
and spread in globular clusters from low-resolution spectroscopy ††thanks: The
Table with the results for each individual star is only available in
electronic format the CDS via anonymous ftp to cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/Prospects
of measuring a metallicity trend and spread in globular clusters from low-
resolution spectroscopy ††thanks: The Table with the results for each
individual star is only available in electronic format the CDS via anonymous
ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/Prospects of measuring a
metallicity trend and spread in globular clusters from low-resolution
spectroscopy ††thanks: The Table with the results for each individual star is
only available in electronic format the CDS via anonymous ftp to
cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-
bin/qcat?J/A+A/Prospects of measuring a metallicity trend and spread in
globular clusters from low-resolution spectroscopy ††thanks: The Table with
the results for each individual star is only available in electronic format
the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/ L. Lima Prospects of measuring
a metallicity trend and spread in globular clusters from low-resolution
spectroscopy ††thanks: The Table with the results for each individual star is
only available in electronic format the CDS via anonymous ftp to
cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-
bin/qcat?J/A+A/Prospects of measuring a metallicity trend and spread in
globular clusters from low-resolution spectroscopy ††thanks: The Table with
the results for each individual star is only available in electronic format
the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/ P. Prugniel Prospects of
measuring a metallicity trend and spread in globular clusters from low-
resolution spectroscopy ††thanks: The Table with the results for each
individual star is only available in electronic format the CDS via anonymous
ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/Prospects of measuring a
metallicity trend and spread in globular clusters from low-resolution
spectroscopy ††thanks: The Table with the results for each individual star is
only available in electronic format the CDS via anonymous ftp to
cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-
bin/qcat?J/A+A/

(Received xxxx; accepted xxxx)

###### Abstract

Context.  The metallicity spread, or the metallicity trend along the
evolutionary sequence of a globular cluster, is a rich source of information
to help understand the cluster physics (e.g. multiple populations) and stellar
physics (e.g. atomic diffusion). Low-resolution integral-field-unit
spectroscopy in the optical with the MUSE spectrograph is an attractive
prospect if it can provide these diagnostics because it allows us to
efficiently extract spectra of a large fraction of the cluster stars with only
a few telescope pointings.

Aims. We investigate the possibilities of full-spectrum fitting to derive
stellar parameters and chemical abundances at low spectral resolution
(R$\sim$2 000).

Methods. We reanalysed 1584 MUSE spectra of 1061 stars above the turn-off of
NGC 6397 using FERRE and employing two different synthetic libraries.

Results. We derive the equivalent iron abundance $\rm{[Fe/H]_{e}}$ for fixed
values of [$\alpha$/Fe] (solar or enhanced). We find that (i) the
interpolation schema and grid mesh are not critical for the precision,
metallicity spread, and trend; (ii) with the two considered grids,
$\rm{[Fe/H]_{e}}$ increases by $\sim 0.2$ dex along the sub-giant branch,
starting from the turn-off of the main sequence; (iii) restricting the
wavelength range to the optical decreases the precision significantly; and
(iv) the precision obtained with the synthetic libraries is lower than the
precision obtained previously with empirical libraries.

Conclusions.  Full-spectrum fitting provides reproducible results that are
robust to the choice of the reference grid of synthetic spectra and to the
details of the analysis. The $\rm{[Fe/H]_{e}}$ increase along the sub-giant
branch is in stark contrast with the nearly constant iron abundance previously
found with empirical libraries. The precision of the measurements (0.05 dex on
$\rm{[Fe/H]_{e}}$) is currently not sufficient to assess the intrinsic
chemical abundance spreads, but this may change with deeper observations.
Improvements of the synthetic spectra are still needed to deliver the full
possibilities of full-spectrum fitting.

###### Key Words.:

Methods: data analysis, Techniques: spectroscopic, Stars: fundamental
parameters, Stars: abundances, (Galaxy:) globular clusters: individual: NGC
6397.

##  1 Introduction

Globular clusters (GCs) are studied for their own sakes, for instance, to
investigate the characteristics of their stellar population and infer their
formation and evolution processes. They are also benchmarks for probing
theories of stellar physics, such as the variations in the surface chemical
composition of stars with respect to their initial composition as expected
from atomic diffusion (Chapman 1917).

Significant abundance spreads were documented in Gratton et al. (2012) or
Mészáros et al. (2015), for example, with amplitudes larger by more than 0.1
dex for the $\alpha$-elements. With a few exceptions such as $\omega$ Cen and
Terzan 5, the iron ([Fe/H]) spreads are in contrast lower than the
observational uncertainties in general (about 0.05 dex; Carretta et al. 2009).
Some abundance trends (i.e. systematic change along the temperature sequence
of the cluster) were also suggested (Korn et al. 2007; Lind et al. 2008;
Nordlander et al. 2012, in NGC 6397). These studies are at the precision limit
of the measurements, and a number of observational or modelling effects can
mislead the interpretation.

In these investigations, the authors analysed observations with high signal-
to-noise ratio (S/N) and high spectral resolution of as many stars as
possible. However, these observations are expensive because the multiplexing
capability of high-resolution spectrographs is limited and exposure times are
long, and they are hampered by crowding in the central regions of the
clusters. The final samples are therefore often restricted to typically a
dozen stars in the outskirts of the cluster. Low-resolution integral-field-
unit (IFU) spectroscopy may be considered as an alternative. It benefits from
a large multiplexing and from the development of crowded-field spectroscopy
(Kamann et al. 2013).

The MUSE IFU spectrograph, attached to the ESO Very Large Telescope (Bacon et
al. 2010), allows observing a significant fraction of a Galactic GC in a
single telescope pointing. It has been used by Husser et al. (2016, hereafter
H16) to observe NGC 6397, where the parameters of about $1600$ stars above the
turn-off (TO) point of the main sequence were measured. Because of the low
spectral resolution (R$\sim$2000), individual spectral lines from different
chemical species are blended, and classical methods are not applicable.
Therefore, the authors used full-spectrum fitting (see e.g. Koleva et al.
2009), where all the spectral bins are compared at once to reference spectra
to derive the atmospheric parameters such as effective temperature
($T_{\mathrm{eff}}$), surface gravity ($\log g$), and [Fe/H]. This has
demonstrated that low-resolution IFU spectroscopy can be effective at
analysing large samples of stars in a GC with good internal precision. H16
found a [Fe/H] trend of $\sim 0.2$ dex along the effective temperature
sequence, but when the same dataset was reanalysed by Jain et al. (2020,
hereafter J20), the iron abundance was found to be nearly constant
(${\rm{[Fe/H]}}\sim-2.0$ dex). Both studies used the full-spectrum fitting
technique to derive the parameters, and the origins of the different results
lie in the details of the analyses. Specifically, while H16 was using
synthetic spectra from the Göttingen Spectral
Library111https://phoenix.astro.physik.uni-goettingen.de (hereafter GSL;
Husser et al. 2013) as reference, J20 used empirical libraries (ELODIE;
Prugniel & Soubiran 2001; Wu et al. 2011, and MILES; Sánchez-Blázquez et al.
2006). We also stress that with full-spectrum fitting, the derived iron
abundance depends on the abundance pattern inscribed in the set of reference
spectra. For this reason, we call it an equivalent iron abundance, hereafter
noted $\rm{[Fe/H]_{e}}$, whose relation to the true abundance has to be
considered with care.

To assess the external accuracy of these results, we need to understand the
systematics caused by the analysis methods, such as those due to the
interpolation performed within the spectral library and those due to the
adopted libraries. Comparing the results obtained with different libraries and
methods may give a lower limit of this second source of errors.

As a step in this direction, we reanalyse here the set of MUSE observations
previously used by H16 and J20. We use a different analysis tool and different
sets of reference models to explore the prospects of using full-spectrum
fitting methods to derive accurate parameters from the analysis of low-
resolution stellar spectra. The paper is organised as follows: in Sect. 2 we
present the observational material, and in Sect. 3 we describe the analysis
method based on FERRE222https://github.com/callendeprieto/ferre (Allende-
Prieto & Apogee Team 2015). Sect. 4 presents the analysis with the GSL grid
(the same grid as was used by H16), hence allowing us to compare the results
and investigate the effect of the interpolation scheme and of other details of
the setup. In Sect. 5 we carry out the analysis with the Allende Prieto et al.
(2018, hereafter AP18) grid of models to investigate the effect of the choice
of model and of the mesh size. We finally discuss the different aspects that
affect the ability of measuring metallicity trends and spreads in Sect. 6, and
we draw conclusions in Sect. 7.

##  2 Observational material

NGC 6397 was observed during the MUSE commissioning between July 26 and August
3, 2014 (ESO ID program 60.A-9100(C)). The observations consist of a 5x5
square mosaic (two tiles are missing) and reach about 3.5 arcmin from the
cluster centre on its diagonal. They are based on 127 pointings with exposure
times of about 60 s to avoid saturation of the brightest stars for a total
integration time of 95 min. Each pointing covers a field of 1x1 arcmin, and
the seeing is between 0.6 and 1 arcsec. All the data have already been reduced
by the MUSE consortium using the official pipeline. The MUSE spectra cover a
wide wavelength range, $\lambda\lambda=470-950$ nm, with a resolution R
$=\lambda/\Delta\lambda\sim 2\,000$, where $\lambda$ is the wavelength, and
$\Delta\lambda$ the full width at half maximum (FWHM) of an unresolved line.

The spectra were extracted by H16 (see that paper for further details) and
were downloaded from the website set for commissioning data333http://muse-
vlt.eu/science/globular-cluster-ngc-6397/. The typical S/N of stars at the TO
is $\sim 50$ on average, and the brightest giants reach S/N $\sim\,200$. The
table with photometry and measurements made in H16 were kindly provided by
Tim-Oliver Husser (priv. comm.). We corrected the spectra for the telluric
absorption by dividing each stellar spectrum by the corresponding telluric
spectrum determined by H16. We then reduced the spectra to the rest-frame
velocity by changing the world coordinate system (WCS), using the mean
velocity of the cluster. Because the sampling is not exactly the same in all
the spectra (starting wavelength at $4749.75$ or $4750$ Å), we rebinned all of
them to the same WCS to simplify the analysis with FERRE (see Sect. 3). The
normalisation of the spectra to their continuum was performed during the
analysis in order to employ the same algorithm and parameters for the
observations and the models. The weight of the individual spectral bins was
computed assuming a constant S/N throughout the spectrum. We experimented with
various weighting schemes and retained the scheme that provided the best
precision.

These commissioning observations are based on short observing time and
moderate image quality (they were obtained before the ground-layer adaptive
optics entered routine use to improve the image quality). Moreover, since that
early epoch, the data reduction software has been improved. Still, this public
collection of reduced spectra is unique in its the possibility of comparing
different approaches for measuring stellar atmospheric parameters.

###  2.1 Selected sample and surface gravities

H16 used this observational material to determine the atmospheric parameters
of 4132 stars from 5881 spectra (because the individual tiles of the mosaic
overlap, some stars were observed more than once) with S/N $>20$. With this
S/N cut-off, the formal errors on $T_{\mathrm{eff}}$ and $\rm{[Fe/H]_{e}}$ are
lower than 100 K and 0.16 dex for stars along the giant and sub-giant (SG)
branches, and near the TO, as discussed by H16. Later, J20 restricted this
sample further to the cluster members (based on the radial velocity), excluded
the hot stars (restricting to $T_{\mathrm{eff}}<7000$ K), and finally selected
only the stars with $\log g<4.2$. This resulted in a sample of 1587 spectra
for 1063 stars.

We adopt the values of $\log g$ computed by H16 using the photometry from
Anderson et al. (2008) and isochrones from Bressan et al. (2012). J20 has
shown that fitting $\log g$ at the same time as $T_{\mathrm{eff}}$ and
$\rm{[Fe/H]_{e}}$ produces results consistent with the photometric $\log g$
and does not affect the observed iron abundance trend. Adopting this
photometric $\log g$ simplifies the comparison with the earlier studies.

In the J20 sample, three of the 1587 stars have $\log g$ $<1$, but the two
model grids that we use (GSL and AP18) start at $\log g$ $=1$ because the
slice at $\log g$ $=0$ is incomplete (see Sect. 4 for further details). Thus,
we excluded them from our analysis, and the final sample consists of 1584
spectra for 1061 stars.

###  2.2 Injection of the line-spread function

The full-spectrum fitting method we used involves comparing each spectral bin
of an observed spectrum to the corresponding bin of model spectra and
searching for the physical parameters that provide the best match (in a
$\chi^{2}$ sense). Because models are computed at a high resolution, they must
be transformed to have the same line-spread function (LSF) as the
observations. This process is called LSF injection, and the function used in
the transformation is called relative LSF.

The mean LSF of the MUSE spectra varies with wavelength, and it varies by 0.1
Å across the 24 spectrographs that form MUSE. H16 has pointed out the
difficulty of accurately matching the LSF for each individual spectrum, and
following them and J20, we neglected the variation between the different
spectrograph. This simplification is justified a posteriori in Sect. 4.3 by
testing the effect of the LSF width and centring on the uncertainties.

To model its variation across the wavelength range, we defined the LSF in two
segments, and we used a piece-wise convolution similar to the implementation
in ULySS444http://ulyss.univ-lyon1.fr/ (Koleva et al. 2009). As the GSL and
AP18 models have the same spectral resolution (R $=10\,000$), we adopted the
same relative Gaussian LSF for both: at $\lambda=4750\,\AA$, the FWHM is
$2.75\,\AA$, at $\lambda=7000\,\AA$, it is $2.45\,\AA$, and at
$\lambda=9300\,\AA$, it is $2.50\,\AA$. Moreover, we noticed a small shift in
the NIR range, for which the spectra were corrected by adding a 0.20 Å shift
at $\lambda=9300\,\AA$. Each model spectrum was convolved with these three
LSFs, and the final model was obtained by interpolating linearly in wavelength
between the first and second spectra for wavelengths smaller than $7000\,\AA$,
and between the second and third spectra for larger wavelengths.

Table 1: Parameter space of the grids.  Variable | Range | Step |
---|---|---|---
GSL grids, [$\alpha$/Fe] = 0.4 & 0 dex
$T_{\mathrm{eff}}$ | 5100 : 7000 | 100 | K
$\log g$ | 1 : 6 | 0.5 | dex
[Fe/H] |  $-3$ : 0 | 0.5 | dex
AP18 fine grids, [$\alpha$/Fe] = 0.5 & 0 dex, $\xi=1$ & $2$ $\rm{km}\,s^{-1}$
$T_{\mathrm{eff}}$ | 4000 : 7000 | 250 | K
$\log g$ | 1 : 5 | 0.5 | dex
[Fe/H] |  $-3$ : 0 | 0.25 | dex
AP18 coarse grid, [$\alpha$/Fe] = 0.5 dex, $\xi=1$ $\rm{km}\,s^{-1}$
$T_{\mathrm{eff}}$ | 4000 : 7000 | 500 | K
$\log g$ | 1 : 5 | 1.0 | dex
[Fe/H] |  $-3$ : 0 | 0.50 | dex

##  3 Analysis code: FERRE

In H16 and J20, similar codes were used in the analysis, namely
spexxy555https://spexxy.readthedocs.io and ULySS, respectively. The main
difference is in the adopted models: while H16 used the GSL synthetic library,
J20, employed empirical spectra. We used FERRE, a public code for full-
spectrum fitting, notably used in the SEGUE stellar parameter pipeline (Lee et
al. 2008) and in the APOGEE pipeline (García Pérez et al. 2016, ASPCAP).
Briefly, it interpolates between the nodes of an evenly sampled grid of
synthetic stellar models and performs a $\chi^{2}$ minimisation to find the
parameters of the model that best reproduce the observed spectrum. We used two
different model grids: GSL and AP18 (described in Sect. 4 and 5).

FERRE offers a choice of optimisation algorithms (we used the default Nelder-
Mead minimisation algorithm666We checked that the other minimisation
algorithms accurately produce the same results for any interpolation scheme.
Therefore, we adopt this default algorithm, and will not discuss the other
algorithms.), of different interpolation schemes in the grid, and of different
normalisations of the spectra to their pseudo-continuum. Thus, it allows us to
investigate how the different algorithms of interpolation in the grids could
affect the measurements.

In spexxy or ULySS, the broadening function and systemic velocity are fitted
at the same time as the atmospheric parameters, as well as a multiplicative
polynomial that absorbs the differences in flux calibration between the
observation and the reference spectra. Fitting these parameters simultaneously
with the atmospheric parameters avoids biases that may occur when the
different parameters are partially degenerate, but it requires more
computations. In contrast, FERRE does not adjust the broadening and continuum:
the observations and the model grid should have the same LSF and be normalised
in the same way. These are the essential differences between the two
approaches.

###  3.1 Equivalent iron abundance

While classical abundance methods at high spectral resolution (R$\sim
40\,000$) measure individual resolved lines, full-spectrum fitting uses the
complete spectral information by combining all the spectral lines from any
chemical species present in the considered wavelength range. This is an
optimal use of the signal, and thus opens the method to the exploitation of
low S/N spectra (e.g. in SEGUE, [Fe/H] was determined for spectra with S/N ¿
10 by Lee et al. 2008), and of low- and medium-resolution spectra.

A rigorous application of this method would be to simultaneously determine the
abundances of all the elements that significantly contribute to the opacity in
the considered wavelength range. This would be very difficult for two reasons.
Firstly, because the larger the number of free parameters, the more unstable
the fit. Secondly, because this will require a grid with separate dimensions
sampling each chemical element. This is not practically feasible because of
the excessively large volume of data it would represent, and because of the
time needed to compute such a grid. The usual simplification is to fit only
one or two abundance parameters, typically [Fe/H] or [Fe/H] plus
[$\alpha$/Fe], and assume a fixed (solar) pattern for the other elements: for
this reason, we refer to the metallicity as the equivalent iron abundance
($\rm{[Fe/H]_{e}}$). A mismatch between the assumed and real pattern will
result in biases of the derived parameters. For example, if a spectrum
enhanced in $\alpha$-elements is fit with solar-scaled reference spectra, the
measured equivalent $\rm{[Fe/H]_{e}}$ would be larger than the true one: the
iron lines will be over-fitted, and the $\alpha$-elements will be under-
fitted. A strategy for measuring the abundance of individual $\alpha$-elements
has been tried at higher spectral resolution in the ASPCAP pipeline (García
Pérez et al. 2016). It consists of determining [Fe/H] and [$\alpha$/Fe] first,
and then using spectral masks to isolate the lines of each element to measure
its abundance. This approach is probably difficult to use at the MUSE
resolution, where the lines are blended.

In the following analysis, we use a fixed abundance pattern, either with
${\rm{[\alpha/Fe]}}\,=\,0$, or with enhanced $\alpha$-elements
(${\rm{[\alpha/Fe]}}\,=\,0.4$ or $0.5$ dex) corresponding to the pattern
expected in NGC 6397. The first equivalent $\rm{[Fe/H]_{e}}$ is expected to
over-estimate the real iron abundance, and the second is expected to be a more
accurate estimate.

###  3.2 Interpolation schemes

As already mentioned in the previous section, FERRE has different options for
interpolating the models at any arbitrary location in the parameter space,
namely, linear interpolation, and quadratic and cubic Beziér splines following
the description in Auer (2003). These methods are local interpolations,
computed on two (for linear) to four (for cubic interpolations) neighbours on
each axis.777FERRE also proposes a method called “cubic spline”, based on
Press et al. (1992), but while the scope of the cubic spline is to provide an
interpolating function with continuous first and second derivatives over the
whole range of the parameters, the implementation is “local” (solved in each
dimension in a four-point neighbourhood).

Although all the interpolation schemes implemented in FERRE are locally
infinitely derivable over their computation interval, the overall
interpolating function is continuous but not derivable at the nodes. On each
side of a node, a different interpolation neighbourhood is used, and the local
interpolating functions are therefore different. In contrast, a true spline
interpolation, like in Press et al. (1992), would be derivable at the nodes,
but it would be more demanding in computation time and memory storage.

Mészáros & Allende Prieto (2013) evaluated the accuracy of the different
interpolation schemes. They computed synthetic models at random locations in
the parameter space between the nodes of the grids and compared them to those
derived by interpolating the grid. They found that the cubic interpolation
schemes produce residuals that are about 40% smaller than the linear
interpolation, and are marginally better than the quadratic interpolation. We
therefore adopted the cubic Beziér spline interpolation as a default choice
(i.e. we use this method when not specified otherwise), and we used the other
methods to perform comparisons.

###  3.3 Normalisation to the pseudo-continuum

The observations and the models were normalised in the same way before the
minimisation was run. This is essential to avoid that a mismatch in the shape
of the continuum does not contribute to the $\chi^{2}$, which would
unavoidably happen even with the most carefully flux-calibrated spectra
(because of uncertainties in the flux calibration or in the correction for
Galactic extinction). This normalisation was made using a running average
implemented in FERRE. The code also has two other normalisation methods that
we did not find appropriate for the present work: polynomial, which is limited
to very low degrees, and segmented average, which does not remove the local
shape of the spectral energy distribution.

###  3.4 Default set-up

Unless otherwise indicated, we adopted the following setup. We run FERRE on
the complete MUSE wavelength range, $\lambda\lambda=4749.52-9300$ Å. The
effect of using different wavelength ranges is discussed in Sect. 6. We
adopted the cubic interpolation, because this option was preferred by Mészáros
& Allende Prieto (2013) and it indeed produces satisfactory results. We
evaluate the importance of this choice in Sect. 4.4. As normalisation scheme,
we chose the running mean with a default window of 60 pixels (i.e. 75 Å), as
with this value the residuals (i.e. the difference between the observed and
fitted spectra) appear flat. We evaluate the sensitivity of this parameter in
Sect. 4.5. As initial guesses for the stellar parameters, we set
$T_{\mathrm{eff}}=5500$ K and ${\rm{[Fe/H]}}=-1.9$ dex. We tested that other
choices, such as starting from the photometric $T_{\mathrm{eff}}$ or from
$T_{\mathrm{eff}}=4500$ K, do not affect the solutions. As in H16 and J20, we
kept $\log g$ fixed to the photometric value (see Sect. 2.1). We fixed
${\rm{[\alpha/Fe]}}\,=\,0.4$ dex (for GSL) or 0.5 dex (for AP), as it
corresponds to the abundance pattern expected in the cluster. We explore the
effect of using ${\rm{[\alpha/Fe]}}\,=\,0$ in Sect. 5.1.

Figure 1: Parameter space coverage of the grids. The grey dots represent the
analysed sample, and the black dots mark the nodes of the grids of synthetic
spectra. The top panels are the projections on the $\log g$ vs.
$T_{\mathrm{eff}}$ plane, and the bottom panels show the projections in [Fe/H]
vs. $T_{\mathrm{eff}}$. The left column is for the GSL (the
${\rm{[Fe/H]}}=-2.5$ dex slice, which is not originally a part of GSL, and is
produced by averaging the ${\rm{[Fe/H]}}=-3$ dex and ${\rm{[Fe/H]}}=-2$ dex
slices, see Sect. 4). The central column is the AP18 grid (see Sect. 5). The
right column is also from the AP18 grid, but with twice larger steps on each
axis to evaluate the effect of a coarser grid (see Sect. 5.2).

##  4 Analysis with the GSL grid

GSL is a grid of high-resolution synthetic spectra created with the PHOENIX
stellar atmosphere code (Hauschildt & Baron 1999). The spectra were computed
in spherical geometry (important in the low-gravity regime) and in local
thermal equilibrium (LTE). The structure of the models is similar to the
structure of MARCS models (Gustafsson et al. 2008), and they were computed
adopting the PHOENIX-ACES equation of state and opacities. The adopted line
list comes from Kurucz (2009). GSL covers a wide range of parameters:
$2\,300<\leavevmode\nobreak\ T_{\mathrm{eff}}<\leavevmode\nobreak\ 12\,000$
K888The original grid was limited to $T_{\mathrm{eff}}\leq 8\,000$ K. The
version available on the website extends the grid to hotter models.,
$0<\leavevmode\nobreak\ \log g<6$, $-4<{\rm{[Fe/H]}}<1$ dex, and
$-0.2<\leavevmode\nobreak\ {\rm{[\alpha/Fe]}}<\leavevmode\nobreak\ 1.2$ dex.

In GSL, as extensively described in Husser et al. (2013), the micro-turbulence
velocity parameter, $\xi$, is computed from the macro-turbulence velocity,
that is, from the convection, and it varies from one model to the other. Its
values are stored in the header of the fits files. Abundance ratios are
calculated with respect to the solar reference abundances given by Asplund et
al. (2009).

We used the version post-processed to a reduced spectral resolution, R
$=10\,000$, over the wavelength range from 3000 to 25000 Å. We selected grids
with fixed ${\rm{[\alpha/Fe]}}=0$ and $+0.4$ dex (the latter being a
reasonable assumption for low-metallicity GCs), and we restricted the range of
the other parameters to what is suitable for the cluster. At low gravity,
$\log g=0$ and $0.5$ dex, the temperature coverage of GSL is restricted. For
instance, at ${\rm{[Fe/H]}}=-2$ dex, for $\log g=0$ dex, the coverage is
$2300<\leavevmode\nobreak\ T_{\mathrm{eff}}\leavevmode\nobreak\ <6000$ K. The
absence of hotter models is an obstacle for making a grid, because FERRE
requires a strictly rectangular grid. We therefore chose to truncate the
sample to $\log g>1$.

(a)  (b)

(c)  (d)

(e)  (f)

Figure 2:  Typical fits of a spectrum (ID = id000010747jd2456866p4985f000).
The fit, performed over the complete wavelength range, 4750 – 9300 Å, returned
$T_{\mathrm{eff}}=5471$ K and $\rm{[Fe/H]_{e}}=-1.84$ dex for global rms
residuals of $0.01$ for both GSL and AP18. The left column shows the
${\rm{[\alpha/Fe]}}=0.4$ dex GSL grid, and the right column presents the
${\rm{[\alpha/Fe]}}=0.5$ dex fine AP18 grid. The top row shows a small
wavelength region around the Hβ line, the middle row shows a region around the
Mgb triplet, and the bottom one shows a region around the Ca ii triplet. For
each plot, the top panel presents the observation (black line) with the best-
fit model over-plotted in red. The bottom panel presents the residuals,
observation minus model.  Table 2: Mean iron abundance, errors, dispersion,
and trend obtained from different setups, compared with H16 and J20.

Lib. |  [$\alpha$/Fe] |  $\lambda\lambda$ (nm) | I | $\langle\rm{[Fe/H]}\rangle$ |  $\epsilon$($T_{\mathrm{eff}}$) |  $\epsilon$(${\rm{[Fe/H]}}$) |  $\sigma$(${\rm{[Fe/H]}}$) |  $\tau$(${\rm{[Fe/H]}}$)
---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
Husser et al. (2016)
GSL | 0 | $475-930$ | C | $-1.995$ | 46 | 0.080 | 0.103 | $0.21$
Jain et al. (2020)
MIL | SN | $475-740$ | P | $-2.086$ | 56 | 0.078 | 0.095 | $-0.05$
GSL | $0.4$ | $475-930$ | L | $-1.966$ | 41 | 0.060 | 0.081 | $0.15$
Q | $-1.942$ | 40 | 0.063 | 0.084 | $0.17$
C | $-1.945$ | 41 | 0.059 | 0.082 | $0.18$
GSL | $0.4$ | $475-740$ | L | $-2.186$ | 50 | 0.131 | 0.137 | $0.28$
Q | $-2.180$ | 50 | 0.134 | 0.142 | $0.28$
C | $-2.167$ | 51 | 0.133 | 0.139 | $0.25$
GSL | $0.4$ | $475-580$ | L | $-2.224$ | 56 | 0.146 | 0.159 | $0.33$
Q | $-2.218$ | 55 | 0.147 | 0.164 | $0.33$
C | $-2.203$ | 57 | 0.149 | 0.165 | $0.31$
GSL | $0$ | $475-930$ | L | $-1.727$ | 42 | 0.063 | 0.086 | $0.19$
Q | $-1.704$ | 42 | 0.061 | 0.082 | $0.16$
C | $-1.712$ | 42 | 0.061 | 0.082 | $0.16$
GSL | $0$ | $475-740$ | L | $-2.066$ | 48 | 0.130 | 0.135 | $0.22$
Q | $-2.055$ | 48 | 0.133 | 0.143 | $0.23$
C | $-2.047$ | 48 | 0.125 | 0.131 | $0.21$
GSL | $0$ | $475-580$ | L | $-2.129$ | 56 | 0.151 | 0.168 | $0.34$
Q | $-2.123$ | 56 | 0.154 | 0.174 | $0.34$
C | $-2.109$ | 56 | 0.154 | 0.171 | $0.33$
AP18 | $0.5$ | $475-930$ | L | $-2.031$ | 41 | 0.055 | 0.079 | $0.20$
Q | $-2.015$ | 40 | 0.052 | 0.075 | $0.21$
C | $-2.014$ | 41 | 0.052 | 0.075 | $0.21$
AP18 | $0.5$ | $475-740$ | L | $-1.945$ | 48 | 0.086 | 0.113 | $0.46$
Q | $-1.928$ | 47 | 0.086 | 0.112 | $0.46$
C | $-1.925$ | 47 | 0.084 | 0.111 | $0.46$
AP18 | $0.5$ | $475-580$ | L | $-2.092$ | 61 | 0.129 | 0.137 | $0.39$
Q | $-2.074$ | 62 | 0.128 | 0.134 | $0.40$
C | $-2.073$ | 62 | 0.127 | 0.134 | $0.41$
AP18 | $0$ | $475-930$ | L | $-1.650$ | 42 | 0.058 | 0.081 | $0.24$
Q | $-1.636$ | 41 | 0.056 | 0.079 | $0.25$
C | $-1.635$ | 42 | 0.056 | 0.079 | $0.25$
AP18 | $0$ | $475-740$ | L | $-1.758$ | 50 | 0.089 | 0.120 | $0.47$
Q | $-1.744$ | 50 | 0.091 | 0.122 | $0.47$
C | $-1.741$ | 50 | 0.091 | 0.121 | $0.47$
AP18 | $0$ | $475-580$ | L | $-1.925$ | 62 | 0.145 | 0.148 | $0.42$
Q | $-1.908$ | 63 | 0.143 | 0.147 | $0.44$
C | $-1.907$ | 63 | 0.143 | 0.146 | $0.45$
APC | $0.5$ | $475-930$ | L | $-2.068$ | 42 | 0.061 | 0.083 | $0.20$
Q | $-2.023$ | 41 | 0.055 | 0.077 | $0.20$
C | $-2.013$ | 40 | 0.051 | 0.074 | $0.22$

999(1) Grid or library; MIL stands for the MILES library (Sánchez-Blázquez et
al. 2006) used in J20, and APC for the AP18 coarse grid. (2) [$\alpha$/Fe]
value; SN stands for solar neighbourhood, as the abundance pattern of the
empirical library used by J20 is made of stars from this environment. (3)
Wavelength range in nm. (4) Interpolation schema; L: linear, Q: quadratic, C:
cubic, P: polynomial approximation; the cubic splines used in H16 are
different from the cubic Beziér interpolation of FERRE used in our analyses.
(5) $\langle\rm{[Fe/H]}\rangle$ is the average of the mean iron abundance
determined in five $T_{\mathrm{eff}}$ bins of 250 K, for
$5100<T_{\mathrm{eff}}<6350$ K. (6) $\epsilon(T_{\mathrm{eff}})$ is the mean
uncertainty on $T_{\mathrm{eff}}$ estimated from pairs of repeated
observations, averaged in the five $T_{\mathrm{eff}}$ bins. (7)
$\epsilon({\rm{[Fe/H]}}$) is the mean uncertainty on $\rm{[Fe/H]_{e}}$
estimated from pairs of repeated observations and computed in the same way as
$\epsilon(T_{\mathrm{eff}}$). (8) $\sigma({\rm{[Fe/H]}})$ is the square root
of the mean variance of $\rm{[Fe/H]_{e}}$ in the same five bins. (9)
$\tau({\rm{[Fe/H]}})$ is the equivalent iron abundance trend, computed as the
difference of mean $\rm{[Fe/H]_{e}}$ in the first and last bin.

###  4.1 Construction of the grids and analysis

In figure 8 of H16, we noticed an irregular distribution of the measurements
along the $T_{\mathrm{eff}}$ sequence. We investigate these issues in Appendix
A and find a non-physical discontinuity at $T_{\mathrm{eff}}=5000$ K. For this
reason, we limited the range of our grid to
$T_{\mathrm{eff}}\leavevmode\nobreak\ \geq\leavevmode\nobreak\ 5100$ K. The
characteristics of the resulting grid are presented in Table 1. FERRE requires
the grid to sample each axis evenly101010This aspect, not fundamental to the
FERRE principle, is due to the choice of describing the grid with a simple
coordinate system., but the GSL [Fe/H] sampling is not uniform. We show in
Fig. 1 (left panel) the parameter space coverage of the grids (the top and
bottom rows show the $\log g$ versus $T_{\mathrm{eff}}$ and [Fe/H] versus
$T_{\mathrm{eff}}$ planes, respectively), we added the missing
${\rm{[Fe/H]}}=-2.5$ dex slice to GSL by averaging the $-2.0$ and $-3.0$ dex
slices (red dots in Fig. 1). In Appendix A, we also found an outlier near the
TO, and we replaced it with the average between the spectra 100 K apart in the
grid. This outlier was found to affect the measurements, while the two others
detected in this region were not found to have an impact.

The procedure of building the grid in the FERRE format has three steps. i)
Conversion from vacuum into air wavelengths, ii) injection of the relative LSF
as described in Sect. 2, and iii) rebinning as the observations.

When the grid was ready, we ran FERRE. Figure 2 (left column) presents the
typical fit of a stellar spectrum. It shows three remarkable regions: around
Hβ (spectral window 4790-4940 Å), Mgb (spectral window 5000-5300 Å), and the
Ca ii triplet (spectral window 8400-8750 Å). The rms of the residuals for the
full wavelength range (not shown in the plot) is 0.01. The test spectrum has
an estimated S/N of 115, and the residuals indicate a normalised $\chi^{2}$ of
1.7. The figure shows that the LSF and the wavelength calibration are
adequately matched, but it also shows significant residuals on strong lines.
As already noted in literature, strong lines such as the H lines or the Ca ii
triplet are difficult to model for a number of reasons, for instance the LTE
approximation (see e.g. Martins & Coelho 2007).

(a)  (b)

Figure 3: Comparison between our measurements (y-axis) and H16 results
(x-axis) obtained with solar [$\alpha$/Fe]. The bottom panels show the
differences $\Delta$ (our $-$ H16). We report the comparison between the
$T_{\mathrm{eff}}$ on the left and the comparison between our
$\rm{[Fe/H]_{e}}$ and [Fe/H] of H16 on the right. The mean and rms differences
are indicated.

Table 9 reports the results obtained for the different setups we used. Our
measurements of $T_{\mathrm{eff}}$ and $\rm{[Fe/H]_{e}}$ for the individual
stars for each combination of wavelength, interpolation, and assumed
[$\alpha$/Fe] are available in electronic form at the CDS. In order to assess
the precision and to compare these solutions, we estimated (i) the measurement
errors and (ii) the iron abundance spread of the sequence. The measurement
errors (due to the noise in the observations, the data reduction, and the
analysis) can be reliably estimated using pairs of repeated observations. As
in J20, given two observations (1 and 2) of the same star i, we computed
$\Delta P_{i}=(P_{1,i}-P_{2,i})/\sqrt{2}$, where $P_{i}$ is any of the stellar
parameters ($\rm{[Fe/H]_{e}}$ or $T_{\mathrm{eff}}$), and the measurement
error $\epsilon(P_{i})$ is estimated as the rms of $\Delta P_{i}$. Because the
iron abundance may change along the sequence of the cluster, the
$\rm{[Fe/H]_{e}}$ dispersion measured over the entire sample would combine the
effects of the trend and of the spread. To isolate the spread, we divided the
$T_{\mathrm{eff}}$ range into bins of 250 K and computed the $\rm{[Fe/H]_{e}}$
dispersion, $\sigma_{j}({\rm{[Fe/H]}}),$ in each bin. For consistency, the
final errors and spread, $\epsilon(T_{\mathrm{eff}})$,
$\epsilon({\rm{[Fe/H]}})$ and $\sigma({\rm{[Fe/H]}})$, are the quadratic
average of the $\epsilon_{j}(P_{i})$ and $\sigma_{j}(P_{i})$ (columns 6, 7,
and 8), and the mean $\rm{[Fe/H]_{e}}$ (column 5) is the average of the mean
$\rm{[Fe/H]_{e}}$ in each bin. As expected, we note that
$\epsilon({\rm{[Fe/H]}})$ is always smaller than $\sigma({\rm{[Fe/H]}})$. This
reflects the fact that in addition to the effect of the noise and observation,
$\sigma({\rm{[Fe/H]}})$ includes other sources of errors (e.g. due to the
precision of the spectral models) and a possible cosmic spread. Finally, the
equivalent iron abundance trend $\tau({\rm{[Fe/H]}})$ (column 9) is defined as
the difference between the mean $\rm{[Fe/H]_{e}}$ in the first and last bins.

###  4.2 Comparison with H16

Figure 3 compares our measurements with those of H16, both obtained with
[$\alpha$/Fe] = 0. On the left, we compare our $T_{\mathrm{eff}}$ (y-axis) and
$T_{\mathrm{eff}}$ from H16 (x-axis) in the top panel, and at the bottom, we
report the difference between our $T_{\mathrm{eff}}$ and H16
($\Delta$$T_{\mathrm{eff}}$ (our-H16)) as a function of the $T_{\mathrm{eff}}$
from H16. On the right, we show the same set of plots for the
$\rm{[Fe/H]_{e}}$ measurements. Our average equivalent iron abundance is
higher by 0.30 dex than the one reported in H16. The rms $\rm{[Fe/H]_{e}}$
dispersion between the two series is 0.08 dex, which is consistent with the
measurement errors. In contrast, there is no overall bias in
$T_{\mathrm{eff}}$: the mean difference is $-29$ K with an rms of 41 K, but
there is a significant drift with a $\sim 100$ K amplitude; our values are
cooler than those of H16 at the TO.

The two analyses use the same data, the same reference synthetic spectra, and
the same technique (full-spectrum fitting). The source of the
$\rm{[Fe/H]_{e}}$ bias and of the temperature drift was searched for in the
several differences between the two approaches. The differences due to the
analysis programs are that (i) we derived only the atmospheric parameters,
while H16 also adjusted the broadening and shift of the spectra, (ii) we used
various schema implemented in FERRE to interpolate over the grid of models,
while H16 used a cubic spline interpolation, and (iii) we normalised the
observed and model spectra to their pseudo-continuum before fitting them,
while H16 fitted a multiplicative polynomial at the same time as all the other
parameters.

The significance of these three aspects is investigated in detail in the next
sections: apparently, they cannot explain the difference of $\rm{[Fe/H]_{e}}$
between the two analyses. After excluding differences in the programs as a
source for the bias, a possible explanation could be searched for in the
weighting schema of each individual wavelength bin. To support this
hypothesis, we note that the H16 solution is intermediate between our
solutions with the full and trimmed to $740$ nm wavelength ranges. By
decreasing the weight of the near-IR (NIR) region, we can indeed recover a
mean $\rm{[Fe/H]_{e}}$ consistent with H16, but with larger errors. When we
use a constant weight throughout the wavelength range, the $T_{\mathrm{eff}}$
drift with respect to H16 is cancelled (but the abundance difference remains).
We cannot attempt to precisely reproduce the H16 results because we lack
information about the weighting that was used. This sensitivity to the
weighting reflects inconsistencies in the modelling in different spectral
regions, reminiscent of those emphasised by Lançon et al. (2021).

###  4.3 Effect of the LSF matching

An accurate LSF injection is a priori more critical here than in H16 and J20,
where the velocity broadening adjusted to each individual spectrum was
absorbing part of its variation throughout the MUSE sample. We carried out two
tests to evaluate the effect of the LSF on the parameters. The first test
simulated the variation in resolution amongst the MUSE spectrographs, and the
second test simulated wavelength shifts that could result from errors in the
dispersion relation, or in the rest-frame reduction.

For the first test, we used a GSL grid with an LSF FWHM increased by 0.1 Å,
equivalent to the real uncertainty. We find that the $T_{\mathrm{eff}}$
increases by 3 K, and $\rm{[Fe/H]_{e}}$ by $0.01$ dex for the whole sample.
This is about 10 to 20% of the measurement errors, and therefore a minor
contribution to the error budget.

For the second test, we shifted the LSF by 0.5 Å (i.e. 0.4 times the
wavelength bin). This corresponds to about 30 $\rm{km}\,s^{-1}$, that is,
about five times higher than the velocity dispersion measured by H16. Here, we
find an increase in $T_{\mathrm{eff}}$ by 21 K and a decrease in
$\rm{[Fe/H]_{e}}$ by $0.03$ dex. Therefore, the wavelength calibration
uncertainty and internal velocity dispersion are also minor contributions to
the total error.

We note that this second test also mimics the effect of inaccurate line lists
that manifests as spectral lines modelled at incorrect wavelengths, or with
incorrect strengths. This well-known problem is probably responsible for a
significant fraction of the spectral mismatch between observation and models
(Franchini et al. 2018; Martins et al. 2019). By splitting the sample into two
temperature regimes, we find that $\rm{[Fe/H]_{e}}$ decreases by 0.04 dex for
the warm stars ($T_{\mathrm{eff}}>5800$ K), while it decreases by only $0.01$
dex for the cool stars. This suggests that inaccurate line lists may affect
the measured equivalent $\rm{[Fe/H]_{e}}$ trend.

Table 3: $\rm{[Fe/H]_{e}}$ dispersion: Effect of the normalisation window.

Norm. window | Interpolation | $\langle\rm{[Fe/H]}\rangle$ |  $\sigma$([Fe/H])
---|---|---|---
(1) | (2) | (3) | (4)
30 | L | -1.950 | 0.084
Q | -1.926 | 0.087
C | -1.930 | 0.085
60 | L | -1.966 | 0.081
Q | -1.942 | 0.084
C | -1.945 | 0.082
120 | L | -1.934 | 0.086
Q | -1.907 | 0.088
C | -1.911 | 0.089
111111Dispersion along the giant branch with different normalisation windows,
using GSL in the wavelength range 475 - 930 nm. (1) Running-mean normalisation
window given as the number of wavelength bins. (2) Interpolation schema; L:
linear, Q: quadratic, C: cubic (3) $\langle\rm{[Fe/H]}\rangle$ is the average
of the average $\rm{[Fe/H]_{e}}$ determined in $T_{\mathrm{eff}}$ bins of 250
K. (4) $\sigma({\rm{[Fe/H]}})$ is the square root of the mean variance of
$\rm{[Fe/H]_{e}}$ determined in $T_{\mathrm{eff}}$ bins of 250 K.

(a)  Figure 4: $\rm{[Fe/H]_{e}}$ vs. $T_{\mathrm{eff}}$ diagrams published in
H16 (top panel, with [$\alpha$/Fe] =0) and those obtained with FERRE using GSL
and the three different interpolation schemes (other panels, with
[$\alpha$/Fe] =+0.4). The measurements of individual spectra are shown as blue
dots, and the nodes of the grids are overlaid as thin crosses. The red dots
and line are the mean $\rm{[Fe/H]_{e}}$ in bins of 250 K, and the error bars
are the dispersion in these bins.

###  4.4 Effect of the interpolation schema

The possible importance of interpolation has been pointed out by J20, who
noted that the iron abundance trend along the SG branch is only a small
fraction of the mesh size. This trend is presented in Fig. 4, in which we
compare the H16 measurements with those obtained with FERRE using various
interpolation schemes, namely, linear (second panel), quadratic Beziér spline
(third panel), and cubic Beziér spline (fourth panel). It is evident that the
trend is similar with the different interpolation schemas. The measurement
errors ($\epsilon({\rm{[Fe/H]}})$ and $\epsilon(T_{\mathrm{eff}})$), the mean
and spread of equivalent iron abundance, $\langle\rm{[Fe/H]}\rangle$ and
$\sigma({\rm{[Fe/H]}})$ obtained in each case are comparable (see Table 9).
This implies that the choice of the interpolation scheme is not critical in
this analysis.

A peculiar pattern in Fig. 4 is to be mentioned: there is a concentration of
measurements aligned on $\rm{[Fe/H]_{e}}$ $=-2.0$ dex with the linear, and to
a lesser extent with the quadratic interpolation. This is an artefact of the
interpolation that is discussed in Appendix B.

###  4.5 Effect of the continuum normalisation

The observations and the models were both normalised by dividing them by a
running mean with a window of 60 wavelength bins, which corresponds to 75 Å.
With this width, the residuals of the fits (Fig. 2) do not show low
frequencies. We also tested different normalisation windows by fixing the
width to twice larger (120) and twice smaller (30). In Table 11 we report the
dispersion values corresponding to these tests. The trend is similar in the
three cases, and the smaller dispersions obtained with 60 bins justify our
choice.

(a)  Figure 5: Comparison between $T_{\mathrm{eff}}$ and $\rm{[Fe/H]_{e}}$
obtained with GSL and AP18 grids. The top panel shows the difference between
$T_{\mathrm{eff}}$ ($\Delta T_{\mathrm{eff}}$(AP18-GSL), computed by
subtracting the GSL results from the AP18 results) as a function of
$T_{\mathrm{eff}}$ values with GSL. The bottom panel shows the difference of
equivalent iron abundance ($\Delta\rm{[Fe/H]_{e}}$) as a function of the iron
abundance with GSL. The dashed blue line shows the expected bias due to the
different level of $\alpha$-element enhancement in the two grids (0.4 vs. 0.5
dex, see the text for details). In both panels we report the mean and rms of
the differences.  (a)  Figure 6: $\rm{[Fe/H]_{e}}$ vs. $T_{\mathrm{eff}}$
relation using the AP18 fine (top and second panels) and coarse (third panel)
grids. The top panel shows the linear interpolation, and the two other panels
show the cubic interpolation. The grey lines and symbols represent the
solution (with an offset of -0.076) obtained with the GSL grid (cubic
interpolation) and reported on Fig. 4. The other symbols and line styles are
as in Fig. 4.

##  5 Analysis with AP18 grid

Allende Prieto et al. (2018) provided a comprehensive set of synthetic
spectral grids obtained with ATLAS9 atmosphere models, computed by Mészáros et
al. (2012) using opacity distribution functions assuming plane-parallel and
LTE approximations. The spectra were synthesised with the radiative code
ASS$\epsilon$ET (Koesterke 2009). The line lists are from the Kurucz public
database (updated in 2007), plus some specific updates. The grids cover the
parameter space from $3500$ to $30\,000$ K, $0$ to $5$ dex, and $-5.0$ to
$0.5$ dex for $T_{\mathrm{eff}}$, $\log g$, and [Fe/H], respectively, and also
sample [$\alpha$/Fe] and $\xi$. The adopted solar reference abundance is from
Asplund et al. (2009), as for the GSL grid. Different grids are provided at
various spectral resolutions and with various coverage and sampling of this
wide domain of parameters.

We used their so-called fine grids ns1 and ns2, at the resolution R $=10\,000$
(see table 1 from Allende Prieto et al. 2018), with both solar and enhanced
[$\alpha$/Fe] (+0.50 dex), and with $\xi=1$ and 2 $\rm{km}\,s^{-1}$. From our
fine grid (AP18 in Table 9), we extracted a coarse grid (APC in Table 9) with
a twice larger mesh size. Details can be found in Table 1 and in Fig. 1.

To build our final AP18 and APC grids, we processed them as done for the GSL
grid, that is, with the same procedure of LSF injection and rebinning. The
only difference was that the AP18 grids were already in air wavelengths.

###  5.1 Comparison with our GSL results

We repeated the fits of the previous section with the AP18 fine-grained grid
with ${\rm{[\alpha/Fe]}}=0.5$ dex and $\xi=1$ $\rm{km}\,s^{-1}$ for the three
interpolation schemes. A typical fit is presented in Fig. 2 and can be
compared with the fit of the same spectrum with GSL made in Sect. 4. A
quantitative comparison of spectral fits with GSL and AP18 grids is made by
measuring the mean rms residuals over the whole sample in different spectral
regions. We find that these values in the Hβ, Mgb , and Ca ii triplet regions
are almost the same with both grids. In the full wavelength range (not shown
in the plot), as seen with the GSL grid, the rms residual is equal to 0.01.

A comparison of the solutions with AP18 and GSL grids and cubic interpolation
is presented in Fig. 5. A systematic difference of equivalent
$\rm{[Fe/H]_{e}}$ is expected because the two grids are computed at different
$\alpha$-elements enhancements, 0.5 dex for AP18, and 0.4 dex for GSL (see
Sect. 3.1). The solutions with solar-scaled and enhanced grids are reported in
Table 9. For solar-scaled models, when the full wavelength range is
considered, the equivalent $\rm{[Fe/H]_{e}}$ are increased by 0.233 and 0.379
dex for GSL and AP18, respectively. This important effect shows that the
$\alpha$-elements are a major contribution to the total opacity in the
considered spectral region. By extrapolating this effect linearly, we can
correct the solutions obtained with the two grids to a mid-point enhancement
of $0.45$ dex. For GSL, the correction is $-0.029$ dex, and for AP18, it is
$0.047$ dex. Hence, the mean difference is $\Delta\rm{[Fe/H]_{e}}({\rm
AP18-GSL})=-0.076$ dex. This is represented in the figure (dashed blue line),
and it closely matches the observed bias of $-0.073$ dex.

The mean biases (AP18 $-$ GSL, reported in the figure) are well within the
generally accepted external errors. The rms dispersions between the two series
are two to three times smaller than the uncertainties inferred from pair-wise
comparisons. This indicates that in our dataset, the random errors linked to
the choice of synthetic grid contribute marginally to the error budget.

The $\rm{[Fe/H]_{e}}$ distributions along the $T_{\mathrm{eff}}$ sequence
using either linear or cubic interpolation are represented in Fig. 6 (top and
central panels). The equivalent iron abundance trend is qualitatively similar
to the trend obtained before with GSL: this is also corroborated by the values
of $\tau$([Fe/H]) reported in Table 9. As with GSL, the choice of the
interpolation scheme does not affect the trend.

Table 9 also shows that the estimated errors ($\epsilon$) and the spreads
($\sigma$) are slightly smaller with AP18 than with GSL (by roughly 0.01–0.02
dex). This is consistent with Franchini et al. (2018), who also compared GSL
and other grids of synthetic spectra to high-resolution spectra (R $\sim$ 50
000) obtained in the Gaia-ESO survey (Gilmore et al. 2012) over a $500$ Å
spectral range around $5200$ Å. They found that GSL matches the spectra less
well than other libraries.

###  5.2 Effects of different mesh sizes

A valuable characteristic of the AP18 grid is its fine [Fe/H] sampling. To
test the importance of the mesh size, we used the coarse grid (see Table 1),
whose [Fe/H] sampling is the same as that of H16. The $\rm{[Fe/H]_{e}}$ mean
value and spread for the coarse grid with different interpolation schema are
given in Table 9, and the $\rm{[Fe/H]_{e}}$ versus $T_{\mathrm{eff}}$ diagram
with cubic interpolation is shown in Fig. 6 (bottom panel). The values of
$\epsilon$(${\rm{[Fe/H]}}$) with the coarse grid are not significantly higher
than those with the fine grid, and the mean iron abundance is not affected
when quadratic or cubic interpolation is used. With linear interpolation, the
dispersion is marginally increased and the mean iron abundance is marginally
decreased. We conclude that the mesh size is not critical, and that a $0.5$
dex iron abundance step does not affect the results dramatically even for
effects ten times smaller than the mesh. Together with the minor effect of
changing the interpolation schema seen in Sect. 4.4, this shows that (i) a
linear approximation within the considered grids of models is usually
satisfactory, and (ii) using a higher-order interpolation schema with a grid
mesh as large as $0.5$ dex in [Fe/H] allows us to recover the precision of a
twice finer mesh.

###  5.3 Effects of micro-turbulence velocity

Micro-turbulence velocity ($\xi$) is an ad hoc parameter introduced to account
for the difference between the measured and expected strength of spectral
lines. The $\xi$ parameter works as an additional broadening and can be
derived spectroscopically by imposing that weak and strong iron lines match
with the same abundance (see Mucciarelli 2011 for an extensive discussion). It
also depends on the other atmospheric parameters: $\xi$ increases with
$T_{\mathrm{eff}}$, and it has a tight correlation with $\log g$, being larger
in giants than in dwarf stars (Gray et al. 2001; Takeda et al. 2008).

In the series of models used by Lind et al. (2008) to analyse NGC 6397
spectra, $\xi$ decreases from $1.86\pm 0.13$ $\rm{km}\,s^{-1}$ at 6250 K (TO)
to $1.47\pm 0.09$ $\rm{km}\,s^{-1}$ at 5450 K (SG). In the APOGEE ASPCAP
analysis (García Pérez et al. 2016), $\xi$ was fixed to a value bound to $\log
g$ through the relation $\xi=2.478-0.325\log g$. Along the evolutionary
sequence of the cluster, this corresponds to 1.1 $\rm{km}\,s^{-1}$ at
$T_{\mathrm{eff}}=6250$ K and 1.5 $\rm{km}\,s^{-1}$ at $T_{\mathrm{eff}}=5250$
K. In GSL, $\xi$ was derived consistently from the macro-turbulence velocity
and was computed for each model. According to the indications in the header of
the individual files, it varies from $\xi=1.2$ $\rm{km}\,s^{-1}$ near the TO
(near 6250 K, at $\log g$ =4.0 dex) and $1.35$ $\rm{km}\,s^{-1}$ at the SG
(near 5450 K, at $\log g$ =3.0 dex), which is almost flat.

In AP18, $\xi$ is a free parameter. To assess its importance, we compared the
results obtained with the grids produced with either $\xi_{1}=1$ and
$\xi_{2}=2$ $\rm{km}\,s^{-1}$. The mean $T_{\mathrm{eff}}$ for the whole
sample is not affected by the value of $\xi$:
$T_{\mathrm{eff}}(\xi_{2})-T_{\mathrm{eff}}(\xi_{1})=6$ K (with a dispersion
of 6 K), but as expected, $\rm{[Fe/H]_{e}}$ depends on $\xi$:
$\langle\rm{[Fe/H]}\rangle(\xi_{2})-\langle\rm{[Fe/H]}\rangle(\xi_{1})=-0.062$
dex. Finally, the choice of the variation of $\xi$ along the sequence may
affect the trend by $\sim 0.03$ dex, that is, $\sim 15$% of the observed
trend. In other words, this is a noticeable effect, but it is unlikely to
account for the currently observed trend.

(a)  Figure 7: $\rm{[Fe/H]_{e}}$ vs. $T_{\mathrm{eff}}$ relation obtained with
GSL using different wavelength ranges of the observations. The top panel shows
the relation for the full range, and the bottom panel shows the relation for
the trimmed wavelength range (same as the MILES range in J20). The symbols and
decorations are the same as in Fig 4. (a)  Figure 8: $\rm{[Fe/H]_{e}}$ vs.
$T_{\mathrm{eff}}$ relation obtained with AP18 grids ([$\alpha$/Fe] =+0.5 dex)
using different wavelength ranges of the observations: the full range (top
panel), and the trimmed wavelength range (bottom panel). The symbols and
decorations are the same as in Fig 4.

##  6 Discussion

We analysed the set of NGC 6397 spectra using two synthetic libraries, and in
the previous sections, we assessed how several details of the analysis may
affect the precision and accuracy of the measurements. Table 4 summarises the
factors that we investigated. The first aspect discussed below is the
accuracy, as reflected by the mean metallicity computed over the whole sample.
We then focus on the trend of [Fe/H] with $T_{\mathrm{eff}}$ and on the [Fe/H]
spread around the cluster sequence.

###  6.1 Mean metallicity

NGC 6397 has ${\rm{[Fe/H]}}=-2.02$ dex in Harris (2010). Gratton et al. (2001)
obtained high-resolution (R $\sim 43\,000$) spectra of 5 TO stars and 3
subgiants, analysed them with LTE models, and measured individual abundances
in the range $-2.10<{\rm{[Fe/H]}}<-2.00$ dex, with a precision of about 0.04
dex on each measurement. Nordlander et al. (2012, revising the earlier
analysis by ) measured ${\rm{[Fe/H]}}=-2.23$ dex at the TO and
${\rm{[Fe/H]}}=-2.13$ dex for the SG, using 12 TO and SG stars observed with
FLAMES-UVES (R=47 000). In a companion paper by Lind et al. (2008) that used
$R=27\,000$ spectra of 116 stars, the same trend was obtained, but with an
abundance lower by $\sim 0.05$ dex. Lovisi et al. (2012) re-analysed the
latter data and obtained ${\rm{[Fe/H]}}=-2.12$ dex. Mészáros et al. (2020)
used 141 giants from the APOGEE survey at R$=22\,500$ and obtained
${\rm{[Fe/H]}}=-1.89$ dex, but their metallicity scale has been documented to
be $0.06-0.15$ dex larger than others. This is illustrative of the uncertainty
on the [Fe/H] scale due to a number of factors. The iron abundance at the TO
may be depleted because of atomic diffusion, and the value for the SG should
be more representative of the composition of the gas when the cluster formed:
we can therefore retain that the initial abundance reported in the literature
is ${\rm{[Fe/H]}}\sim-2.1$ dex.

This value compares well with the mean equivalent iron abundances measured
with the $\alpha$-enhanced models in the full wavelength range: $-1.945$ and
$-2.014$ dex, with GSL and AP18, respectively. Our values may be marginally
higher, but they are still in the range of values found in earlier studies. We
then compared our results with those of J20 by repeating the fits using the
wavelength ranges explored in their paper. The two empirical libraries used by
J20, ELODIE and MILES, are limited redwards to $\lambda=5800$ Å and $7400$ Å,
respectively. As these empirical libraries are made of stars from the solar
neighbourhood and therefore are enhanced in $\alpha$-elements in the low-
metallicity regime (similar to NGC 6397), we consistently compared the J20
results with those we obtained with the enhanced models. The results for the
MILES range are reported in Table 9: the mean equivalent iron abundances with
GSL and AP18 are $-2.17$ and $-1.93$ dex, respectively, compared with $-2.09$
dex in J20. Such $\sim 0.15$ dex differences are often observed between
different series of measurements. In our case, the analysed spectra are the
same and only some details in the analysis differ. We have shown in previous
sections that these details are unlikely to affect the results, and the origin
of the difference is certainly the choice of the set of reference spectra. For
instance, the mismatch of [$\alpha$/Fe] (${\rm{[\alpha/Fe]}}=0.4$ dex in GSL
and 0.5 dex in AP18) can account for a $0.07$ dex difference on the equivalent
iron abundance (Sect. 5.1). It should be noted that the mean solution obtained
with the empirical library falls between the solutions obtained with the two
synthetic libraries. We are therefore satisfied by the consistency between
these results. The precision of the metallicity measurements is lower with the
synthetic libraries than in J20 ($\epsilon({\rm{[Fe/H]}})$ is larger).

It should also be noted that with GSL, $\langle\rm{[Fe/H]}\rangle$ decreases
by $\sim 0.25$ dex when the wavelength is trimmed from the full range
$\lambda\lambda=4750-9300$ Å to the optical domain (Fig. 7). The systematic is
less clear with AP18 (Fig. 8): The bias on $\langle\rm{[Fe/H]}\rangle$ is
smaller, but it masks a decrease in abundance for the warm stars and an
increase for the cool stars. This difference between the different regions of
the spectrum reveals the limited quality of the synthetic spectra. This
includes the limitation on the atmosphere models, spectral synthesis, and
atomic and molecular data, and the possible unsuitability of the abundance
pattern (flat $\alpha$-enhancement). Access to the bluer region of the
spectrum, where $\alpha$-enhancements opacities become more important, would
certainly help to clarify this difference (this is unfortunately not possible
with MUSE).

We have discussed the degeneracy between $\rm{[Fe/H]_{e}}$ and [$\alpha$/Fe]
in the full wavelength range in Sect. 5.1. Comparing the solutions reported in
Table 9 for the solar-scaled and enhanced models, we note that this degeneracy
is a factor 2 smaller in the MILES range and 2.5 times smaller in the ELODIE
range. This certainly reflects the dominance of the iron lines in the optical
range.

Table 4: Summary of the different tests conducted in this work. Factor | Test | Result
---|---|---
LSF broadening accuracy (Sect. 4.3) |  Experiment with perturbed broadening within the LSF uncertainty |  Insignificant effect
Wavelength calibration accuracy (or line-list completeness) (Sect. 4.3) |  Experiment with shifted wavelengths |  Minor effect on $\tau({\rm{[Fe/H]}})$
Interpolation schema (Sect. 4.4) |  Compare analyses with linear, quadratic, and cubic interpolations |  Results with cubic interpolation are marginally more precise. Artefacts with linear interpolation.
Grid choice (Sect. 5.1) |  Compare analyses with the GSL and AP18 grids |  Results are consistent, but the precision with AP18 is marginally better
[$\alpha$/Fe] of the reference spectra (Sect. 5.1) |  Use reference grids with different [$\alpha$/Fe] |  Significant degeneracy between $\rm{[Fe/H]_{e}}$ and [$\alpha$/Fe]
Mesh size of the grid of reference spectra (Sect. 5.2) |  Compare solutions with full and half density grids extracted from AP18 |  Minor effects with linear interpolation, insignificant effects otherwise
Micro-turbulent velocity parameter, $\xi$ (Sect. 5.3) |  Compare solutions with AP18 grids with $\xi=1$ and $2$ $\rm{km}\,s^{-1}$ |  Minor effect on $\tau({\rm{[Fe/H]}})$
Wavelength range (Sect. 6.1) |  Compare analyses with the full MUSE range, and with the optical region only |  Significant degradation of the precision when the range is trimmed; bias on $\langle\rm{[Fe/H]}\rangle$; different behaviour with the two grids

###  6.2 Metallicity trend

A trend of the surface metallicity along the sequence of the cluster may be
due to atomic diffusion. This is a balance between the gravitational settling
and the radiative force that is counter-acted by the convective mixing. The
resulting depletion of metals is larger at the TO, where the stars have thin
convective layers. After the TO, these layers start to expand and to deepen,
so that convection acts against atomic diffusion, re-homogenizing the matter
(Salaris & Cassisi 2017). Consequently, the giant stars show the initial
chemical composition at their surface. Theoretical models (e.g. Dotter et al.
2017) predict that the Fe abundance will be $0.1-0.2$ dex lower at the TO.
These models predict depletions that vary for the different atomic species.

In Korn et al. (2007) and Nordlander et al. (2012), a 0.1 dex depletion of Fe
at the TO compared to the SGB of NGC 6397 was detected. They also measured
similar trends for other elements (Mg, Ca, Ti, and Cr). The follow-up paper by
Lind et al. (2008) confirmed the higher-resolution results. It should be noted
that while the former results were derived from a non-LTE analysis, the latter
are derived in LTE approximation: however, given the limited
$T_{\mathrm{eff}}$ range of the sample, the LTE correction is not expected to
induce a metallicity trend. These trends, compared to models of atomic
diffusion by Richard et al. (2005), allowed the authors to constrain the
mixing parameters, which in turn could be used to derive the initial or
primordial abundance of Li. Lovisi et al. (2012) independently re-analysed the
latter dataset, and their results did not show the above [Fe/H] trend. Studies
by other groups, albeit based on only a few stars, did show no or even
opposite trends (Gratton et al. 2001; Koch & McWilliam 2011). Metallicity
trends were subsequently detected in several other GCs by the same group. In
NGC 6752, using similar data, Gruyters et al. (2013, 2014) found a Fe
depletion of 0.08$\pm$0.06 dex that is significant at the 1-2 $\sigma$ level.
Finally, a similar depletion was found in M 30 (also known as NGC 7099)
(Gruyters et al. 2016; Gavel et al. 2021), a more distant cluster for which
measuring abundances of TO stars is challenging.

In other clusters with similar spectroscopic measurements extending to the TO,
no [Fe/H] trends were observed. This is the case in M 4 (also known as NGC
6121, with ${\rm{[Fe/H]}}=-1.1$ dex) (Mucciarelli et al. 2011).

In our analysis, the trend ($\tau({\rm{[Fe/H]}})$) is reported in Table 9.
With the two considered synthetic grids, $\rm{[Fe/H]_{e}}$ increases from the
TO along the SG. When the analysis is carried out over the full wavelength
range and models are used that are enhanced in $\alpha$-elements, we find
$\tau({\rm{[Fe/H]}})\sim 0.2$ dex, which agrees with earlier results by H16.
At first sight, it corroborates the earlier analyses of high-resolution
spectra reported above. However, while we tend to have more confidence in the
results based on the full spectral range because they have the best precision,
we do not ignore the solutions obtained on the shortened optical range, which
are notably different. They reach $\tau({\rm{[Fe/H]}})\sim 0.3-0.4$ dex (Fig.
7 and 8). As noted when we discussed the mean equivalent Fe abundance, there
appears to exist a difference between the optical and NIR regions, and it is
certainly due to imperfections in the synthetic spectra.

The discrepancy pointed out in J20 between the nearly constant Fe abundance
found with empirical libraries and the rising abundance obtained with
synthetic libraries is even starker when the comparison is made over the same
wavelength range. Our new analyses produced unrealistic values in some cases,
in particular for the coolest stars analysed with AP18
($\rm{[Fe/H]_{e}}\sim-1.6$ dex).

The libraries used in J20 are sets of observations of real stars, and they
provide a better spectral match with other observations than synthetic spectra
do. However, these libraries had to be calibrated in $T_{\mathrm{eff}}$, $\log
g$, and [Fe/H] using classical methods based on high spectral resolution
spectroscopy, and they therefore also rely on synthetic spectra. This means
that the advantage of a better spectral match may to some extend be dampened
by the uncertainties linked to this indirect calibration, and in particular
its inhomogeneity resulting from the many sources of measurements used to
build a library. A more arguable aspect of empirical libraries, however, is
their limited coverage of the parameter space. J20 questioned the small number
of low-metallicity stars in libraries as a possible source of the discrepancy.

On the other hand, synthetic libraries entail shortcomings in the lists of
atomic or molecular lines or in the implementation of the physics in the
models that are probably the origin of the poorer spectral match. They may
also cause systematics on the derived parameters that might also explain the
discrepancy.

The consistency between the solutions obtained with the two synthetic grids is
an interesting point, but it does not disqualify the results of J20. The two
types of reference libraries have their respective strengths and weaknesses,
and at this point, we cannot conclude about the accuracy of the empirical or
synthetic solutions. It would be premature to interpret our full-spectrum
fitting analyses of low-resolution spectra in physical terms and claim the
detection of signatures of atomic diffusion.

###  6.3 Cosmic metallicity spread

Some massive clusters, such as $\omega$ Cen, and Terzan 5, exhibit a large
[Fe/H] spread (see e.g. Origlia et al. 2003; Ferraro et al. 2009) and are
regarded as atypical. Most other clusters are homogeneous concerning iron:
Carretta et al. (2009) assessed the [Fe/H] spread in GCs using medium-
resolution (R$\leavevmode\nobreak\ =\leavevmode\nobreak\ 20\,000$) spectra of
2000 stars in 19 clusters (about 100 stars in each cluster). They found a
measured spread of $0.048$ dex, dominated by the uncertainties on the
measurements to a level at which it was impossible to measure reliable
intrinsic values.

Mészáros et al. (2020) more recently reached the same conclusion from a
homogeneous analysis of 30 clusters in the APOGEE survey. The compilation by
Bailin (2019) reported 55 clusters with a possibly detected [Fe/H] spread with
a median value of 0.045 dex. However, this is comparable to the errors on
individual measurements and therefore critically depends on a precise
knowledge of them (in this case, the errors are essentially derived from the
dispersion of the measurements obtained on individual spectral lines; by
ignoring the propagation of the random errors on the atmospheric parameters,
the errors may have been under-estimated and the spread over-estimated). In
some clusters, spreads of $\sim\leavevmode\nobreak\ 0.05$ dex were claimed,
but later disputed (see e.g. Mucciarelli et al. 2015, in NGC 6656), reflecting
the caveat of precision abundances measurements. For about 40 giants in NGC
6752, Yong et al. (2013) carried out a relatively detailed abundances analysis
using high-resolution spectra. They achieved an internal precision of about
0.01 dex and measured a 0.02 dex spread for [Fe/H]. These small spreads may be
more representative of the general population of Galactic cluster, but He
variation could also contribute to this apparent spread. In contrast, spreads
in lighter elements with amplitudes reaching 0.1 dex are widely observed (e.g.
Carretta et al. 2009), and are certainly associated with the presence of
multiple stellar populations.

In our analysis, the squared difference between the measured spread and the
dispersion due to the noise and data reduction,
$\Delta({\rm{[Fe/H]}})=\sqrt{\sigma({\rm{[Fe/H]}})^{2}-\epsilon({\rm{[Fe/H]}})^{2}}$
, reflects the other causes of uncertainty and the intrinsic equivalent
$\rm{[Fe/H]_{e}}$ spread. Considering the fits giving the smallest
$\epsilon({\rm{[Fe/H]}})$ in Table 9, those over the full wavelength range
with either GSL or AP18, the residual spread is $\Delta({\rm{[Fe/H]}})=0.057$
dex. If we consider the J20 solution, we obtain a similar estimate:
$\Delta({\rm{[Fe/H]}})=0.054$ dex (although it is based on a shorter
wavelength range, this solution has a precision comparable to that of the
synthetic libraries on the full wavelength range). This is an upper limit to
the intrinsic spread in equivalent $\rm{[Fe/H]_{e}}$, but the consistency
between the two estimates suggests that it is close to the real value.

$\Delta({\rm{[Fe/H]}})$ combines the spread on the iron abundance and the
effect of the spread on other elements. We discussed the degeneracy between
$\rm{[Fe/H]_{e}}$ and [$\alpha$/Fe] above. A change in
$\delta{\rm{[\alpha/Fe]}}$ results in a $\delta{\rm{[Fe/H]}}$ change in
$\delta{\rm{[Fe/H]}}=a\times\delta{\rm{[\alpha/Fe]}}$, where according to the
values in Table 9, $a=0.58$ for GSL and $a=0.75$ for AP18. An intrinsic
[$\alpha$/Fe] spread of $\sim 0.08-0.1$ dex would therefore be sufficient to
explain $\Delta({\rm{[Fe/H]}})$. This value is possible, and the observed
residual spread does not require a spread of iron abundance.

##  7 Conclusion

We re-analysed the MUSE commissioning observations of 1061 stars in NGC 6397
previously studied by H16 and J20 in order to evaluate whether such a large
set of low-resolution (R$\sim 2000$) spectra may allow us to probe subtle
effects in the stellar atmospheres. In particular, our aim was to determine
whether variations in the surface chemical composition along the sequence of
the cluster may be reliably studied or if they remain masked by effects due to
the analysis procedures or to the modelling of the spectra.

Specifically, we focused on the disagreement between H16 and J20 on the
$\rm{[Fe/H]_{e}}$ trend: H16 found that the equivalent iron abundance rises by
about 0.15 dex along the SG branch, while J20 did not find any trend. The two
studies used the same data and similar analysis methods, but the first used as
reference the GSL grid of synthetic spectra and the second used empirical
libraries (ELODIE and MILES). The open questions were (i) whether the details
of the analysis procedure, for instance the interpolation between spectra
within the grid, affect the measured metallicty trend and (ii) whether the
spectral models of stellar atmospheres have the required reliability to reveal
the fine physical details.

Our analysis was made with a different code, FERRE. This code allowed us to
explore the importance of the interpolation schema and different grids of
models, GSL and AP18, which enabled us to explore the effect of the mesh size.

We found that (i) the interpolation schema and grid sampling are not critical
with the considered grids, and (ii) GSL and AP18 are essentially consistent
for the measured iron abundance trend: In both cases, we find an abundance
increase along the SG branch, which also agrees with the results of H16; and
(iii) the precision of the measurements is mostly similar with both grids,
although it is marginally better with AP18. When it is restricted to the
optical wavelength range, the precision is strongly degraded and does not
rival the precision reached by J20 using empirical libraries.

Therefore, we give a robust and unambiguous answer to the first question: The
interpolation is not an issue, even for interpreting abundance trends that are
one-fivth or one-tenth of the mesh size. A linear approximation of the
variations in the grid is still satisfactory, although cubic interpolations
are better.

The second question is more difficult. The consistency between the results
obtained with the two grids of models is encouraging, but does not formally
imply that the results are reliable. It remains risky to simply reject the
results obtained by J20 with empirical libraries. Although these latter
libraries are used at the edge of their validity space and the results stand
on a limited number of reference stars with well-documented parameters, the
nearly constant iron abundance found with both MILES and ELODIE is striking.
Nothing in the process biases the result towards a constant abundance, and the
probability of obtaining this result by chance is certainly low. In addition,
the quality of the spectral fits is considerably better with the empirical
libraries, and the abundance uncertainties and spread are smaller.

We recall that the GSL and AP18 models share some physical assumptions and
ingredients. In particular, the atomic and molecular line lists have similar
limitations that may explain a large part of the observed spectral residuals.
The difference observed between the solutions obtained in the optical and in
the optical plus NIR (see Sec. 6.1) indicates that the models are not yet
accurate enough. Metallicity biases of 0.1 or 0.2 dex are possible, and the
divergence of the iron abundance trend when the wavelength range is shortened
(in particular with AP18) casts some doubts on the reality of this trend.

In contrast, we found the trend to be relatively robust toward the choice of
the micro-turbulence velocity parameter and of its variation along the cluster
sequence. The possible effect is only $\sim$0.03 dex on
$\tau$(${\rm{[Fe/H]}}$).

The difference between synthetic and empirical libraries is not explained, and
there are still some open questions that need to be further investigated in
the future. The use of the X-Shooter Spectral Library (Gonneau et al. 2020;
Verro et al. 2021) to fit the whole wavelength range may help to shed light on
the current difference observed between optical and NIR. Our dataset comes
from short exposures, and therefore the achieved S/N is not the best that
could be obtained with the instrument. The observed spread is definitely
dominated by the effect of the noise. With better observations, the main
source of error may be lowered, which would improve the determination of the
intrinsic equivalent iron abundance spread. It may also allow us to separate
the [$\alpha$/Fe] and [Fe/H] spreads. Improving the synthetic models is a more
difficult and critical issue. One route could be the empirical calibration of
the atomic and molecular line lists (Martins et al. 2014; Shetrone et al.
2015; Franchini et al. 2018), but this may mask some more fundamental
questions in the stellar models, such as the effects of non-LTE or 3D
treatment.

###### Acknowledgements.

We thank the anonymous referee for very helpful comments and suggestions. We
thank Tim-Oliver Husser who shared with us the tables with the results of the
H16 analysis, and the MUSE team for making public the extracted spectra on
their website. We are grateful to Lucimara Martins, Ariane Lançon, Anke
Arentsen, Paula Coelho, and Rashi Jain for fruitful discussions. We
acknowledge the LABEX Lyon Institute of Origins (ANR-10-LABX-0066) of the
Université de Lyon for its support within the program ”Investissements
d’Avenir” (ANR-11-IDEX-0007) of the French government operated by the National
Research Agency (ANR), IUCAA and CRAL for their support for the IFAS summer
school where this work was started. This work was supported in part by grant
ANR-19-CE31-0022 (POPSYCLE) of the French ANR.

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##  Appendix A Discontinuity and outliers in GSL

(a)  (b)

Figure 9: $\chi^{2}$ map with GSL in the region of the $T_{\mathrm{eff}}=5000$
K discontinuity. On the right, we show the spectrum ID =
id000009229jd2456865p5898f000 in the full wavelength range. On the left, we
show a stacked map for the 214 spectra with $4300<T_{\mathrm{eff}}<5300$ K,
selected from J20.

Natura non facit saltum (nature does not make jumps, Leibniz 1765) expresses a
“continuity principle” that has been called along the centuries to justify or
describe important theories in various fields of sciences, from the
infinitesimal calculus to the evolution of biological organisms. If not a
strong physical law, this is definitely an assumption behind interpolating in
a grid of models to determine physical parameters. In the considered region of
parameters, the stellar spectra are continuous with respect to
$T_{\mathrm{eff}}$, $\log g$, and chemical composition, but artificial
discontinuities, irregularities, or outlier spectra may arise from the
modelling. The irregular distribution of the measurements along the cluster
sequence (see H16 their figure 8 and J20 their figure 1) suggests such
discontinuities: there are gaps and concentrations in the $T_{\mathrm{eff}}$
coverage, in particular, near $T_{\mathrm{eff}}=5000$ K. Tim-Oliver Husser
reported suspicions of problems in GSL in this region for giant stars (private
communication).

Therefore, we explored the grid in search for outliers or discontinuities. For
each $\log g$, [Fe/H], and [$\alpha$/Fe], we computed the difference between a
spectrum on the grid and a spectrum that was linearly interpolated using the
preceding and following spectra along the temperature sequence. This allowed
us to clearly detect a sharp discontinuity separating the region with
$T_{\mathrm{eff}}\leq 5000$ K from the rest (this affects the entire range of
gravities, not only the giants). We also found isolated outliers at
$[T_{\mathrm{eff}},\log g,{\rm{[Fe/H]}}]=[5400,2.0,-1.5],[6900,2.0,-1.5]$, and
$[5600,2.0,-3.0]$, which are too far from the cluster sequence to affect our
analysis, and also at $[6400,4.,-2]$, $[6300,4.,-1.5]$, $[6400,4.,-3]$. These
latter outliers, and in particular the first, may affect the measurements near
the TO and may be due to undiagnosed failure in the convergence of the models.
Some convergence issues due to numerical instabilities where the convection
layer is thin are identified in the header of the GSL files, and in some
cases, the convection has been disabled and/or the $\xi$ parameter set to
zero, but this affects warm giants, not the outliers detected here.

To study the discontinuity at $T_{\mathrm{eff}}=5000$ K further, we selected
spectra from J20 with $4300<T_{\mathrm{eff}}<5300$ K and used FERRE to build
$\chi^{2}$ maps. For each point of a $T_{\mathrm{eff}}$ versus $\log g$ grid,
we linearly interpolated a spectrum in GSL at the $\log g$ obtained in J20,
and we computed the corresponding $\chi^{2}$ value. An example of the
resulting $\chi^{2}$ map is presented in the right panel of Fig. 9 for the
spectrum ID = id000009229jd2456865p5898f000. The map shows a step where the
region with $T_{\mathrm{eff}}\leq 5000$ K has a higher $\chi^{2}$, that is,
the quality of the grid is lower in the low-temperature regime, and the
temperature of the minimum $\chi^{2}$, $T_{\mathrm{eff}}=5035$ K has been
biased toward a higher value. The left panel presents the stacked $\chi^{2}$
map for the 214 spectra with $4300<T_{\mathrm{eff}}<5300$ K. The stacking
shows that the problem is not due to an individual spectrum, and is definitely
a sharp step in the $\chi^{2}$ space. The transition in the range
$5000<T_{\mathrm{eff}}<5100$ K is due to the linear interpolation used in the
visualisation process.

We are not certain about the origin of this discontinuity, but the GSL paper
(Husser et al. 2013) mentions that above 5000 K the reference wavelength used
to determine the mean optical depth changes from 12 000 Å to 5 000 Å. Other
discontinuities of the parameters are also mentioned at 4000 K and 8000 K, but
because they are not in the region of interest for the present work, we did
not explore them and we cannot report whether they are associated with
discontinuities in the spectra. The first is a threshold above which NLTE
corrections are used in the computations of some line profiles, and the second
marks the temperature over which the molecular line list is limited to a few
important species.

Lançon et al. (2021) carried out an extensive comparison between the empirical
X-Shooter Spectral Library (Gonneau et al. 2020, XSL) and GSL, and found
discrepancies arising below $T_{\mathrm{eff}}=5000$ K, particularly in the
blue region of the spectra. It is difficult to precisely relate this
degradation to the discontinuity we observe, however. There are indeed a
number of reasons why the model matches become more problematic at low
temperature. We limited the range of our grids to $T_{\mathrm{eff}}\geq 5100$
K in order to avoid problems with the interpolation at cooler temperatures.

##  Appendix B Interpolation artefacts

As mentioned in Sect. 4.4, there are obvious concentrations of points at
${\rm{[Fe/H]}}=-2.0$ dex (a slice of the grid) in Fig. 4 where the analysis
was made with GSL. The same pattern is seen in Fig. 6, with a second
concentration at ${\rm{[Fe/H]}}=-2.25$ dex, also a slice for the AP18 grid.
These patterns are particularly visible with the linear interpolation, but
they also present with the quadratic interpolation (as already noticed by
García Pérez et al. 2016, who analysed the APOGEE data with FERRE). These are
artefacts of the interpolation methods.

Whether it explicitly computes the derivatives or not, the minimisation
algorithm descends the slope, and may get contradictory indications on the two
sides of a slice where the derivatives are discontinuous. This may cause the
solution to diverge from the slice (which is not seen in the distribution of
the solutions), or to converge to the slice and produce concentration of
points aligned on slices. This difficulty is inherent to any local
interpolation, regardless of its order, as they are by nature non-derivable at
the nodes. However, with higher-order interpolation, the problem becomes
smaller, certainly because the discontinuities are reduced. It is barely
apparent with the quadratic interpolation and is invisible with the cubic one.
A global spline interpolation would avoid this artefact. These concentrations
will often not bias the solutions statistically: although they are easily
detected on graphs, they may concern only a small fraction of the solutions
and affect a very limited region of the parameter space in the close vicinity
of a slice of the grid.