11institutetext:  Université de Strasbourg, CNRS, Observatoire astronomique de
Strasbourg, UMR7550, F-67000, Strasbourg, France 22institutetext: Centre de
Recherche Astrophysique de Lyon UMR5574, Univ Lyon, Univ Lyon1, ENS de Lyon,
CNRS, F-69230 Saint-Genis-Laval, France 33institutetext: NAT- Universidade
Cidade de São Paulo/Universidade Cruzeiro do Sul, Rua Galvão Bueno, 868,
01506-000 Sao Paulo, Brazil

# NGC 6397: The metallicity trend along the isochrone revisited

Rashi Jain 11 Philippe Prugniel 22 Lucimara Martins 3322 Ariane Lançon 11

(Received xxxx; accepted xxxx)

###### Abstract

Context.  Recent work has used spectra of $\sim$5000 stars in NGC 6397 that
were extracted from a MUSE mosaic to determine the atmospheric parameters for
these stars by fitting the spectra against the Göttingen Spectral Library. A
significant change in metallicity between the turn off and the red giant
branch was found and was discussed as a possible manifestation of predicted
effects of atomic diffusion. However, the small amplitude of the effect and
inconsistency with earlier measurements call for more attention before this
result is interpreted. Systematic effects due to the interpolation or to the
synthetic spectra cannot be ruled out at this level of precision.

Aims. We reanalyze the data with : the ELODIE and MILES reference libraries in
order to assess the robustness of the result. These empirical libraries have a
finer metallicity coverage down to approximately the cluster metalicity turn-
off.

Methods.  We used the ULySS full-spectrum fitting package, together with the
library interpolators to remeasure the three atmospheric parameters effective
temperature, surface gravity, and [Fe/H] metallicity.

Results.  We find a very low [Fe/H] dispersion along the isochrone (0.07 dex),
consistent with our error estimate (0.05 dex). However, the [Fe/H] trend is
not reproducible. This shows that the data have the potential to reveal
patterns of the magnitude of the expected physical effects, but the analysis
methods need to be refined to cull systematic effects that currently dominate
the patterns.

###### Key Words.:

Methods: data analysis, Techniques: spectroscopic, Stars: fundamental
parameters.

##  1 Introduction

The introduction of the wide-field integral field spectrograph MUSE (Bacon et
al. 2010, 2014) mounted at UT4 (Yepun) of the ESO Very Large Telescope (VLT)
recently ushered in a new area for the exploration of Galactic globular
clusters. Husser et al. (2016, hereafter H16) carried out and analyzed
observations of NGC 6397. With a field of one arcminute and a spatial
resolution limited by the seeing, MUSE allows mapping a full globular cluster
in a reasonable number of pointings. The instrument allows obtaining many more
spectra in one shot than a multi-object spectrograph, and the continuous
mapping of the area makes it possible to use deblending methods similar to
those that are commonly applied for crowded-field photometry (e.g., DAOPHOT,
Stetson 1987). Although these spectra do not have the high spectral resolution
required for abundance analysis (Alves-Brito et al. 2012; Ernandes et al.
2018), they are suitable for determining the atmospheric parameters, effective
temperature ($T_{\mathrm{eff}}$), surface gravity (log g), and metallicity
([Fe/H]) using methods that are applicable at low resolution. Full-spectrum
fitting is one of these methods, and its reliability and efficiency has been
demonstrated in several instances. For example, Koleva & Vazdekis (2012) have
shown that the parameters can be retrieved from spectra with a resolution
$R=\lambda/\Delta\lambda\approx 1000$, and the authors found that the
precision was not degraded compared to high-resolution spectra or the
appearance of significant biases. Full spectrum fitting offers the advantage
of optimally using the entire signal, which is particularly valuable in the
present case, where the observations cover a range of apparent magnitudes and
hence of signal-to-noise ratios (S/N).

H16 analyzed 18 932 spectra of 12 307 stars extracted from a mosaic of 23 MUSE
pointings with full-spectrum fitting. The observations were compared to a grid
of models (Husser et al. 2013, the Göttingen Spectral Library, GSL) that were
computed with the PHOENIX synthesis code (Allard & Hauschildt 1995) which was
interpolated to minimize the residuals. One of the remarkable results of this
work is the finding of a metallicity trend along the isochrone, with an
amplitude of 0.2 to 0.3 dex. The metallicity is minimum at the turn-off (TO)
and rises both along the sub-giant branch (sGB) and down the main sequence
(MS). This is reminiscent of the expectations from atomic diffusion, which
reduces the surface abundances of metals in regimes where mixing processes
such as turbulence are relatively inefficient (Richard et al. 2002).

The mean observed effect in H16 is slightly stronger than found in the
detailed abundance analysis of individual cluster stars of Nordlander et al.
(2012), which was $\sim 0.1$ dex on average, but was based on a small number
of stars in each of the evolutionary phases of interest. The authors of this
study highlight the sensitivity of the metallicity trend to model aspects such
as the efficiency of the turbulent mixing, temperature scale, nonlocal thermal
equilibrium (NLTE) and 3D effects. The model grid used in H16 was computed
with 1D models.

In addition, the magnitude of the detected metallicity variations are only a
fraction of the mesh of the grid of available reference synthetic spectra. The
GSL was computed at regular spacing in the $T_{\mathrm{eff}}$, log g, [Fe/H],
and [$\alpha$/Fe] parameter space, and the steps in metallicity are 1.0 dex
for $\rm{[Fe/H]}<-2.0$ and 0.5 dex for $\rm{[Fe/H]}>-2.0$. Figure 1 (left
panel) shows the distribution of the GSL spectra in the log g versus
$T_{\mathrm{eff}}$ and [Fe/H] versus $T_{\mathrm{eff}}$ diagrams, plotted over
the distribution of the individual stars of the cluster. The mean metallicity
of the cluster is close to the $\rm{[Fe/H]}=-2.0$ dex slice of the GSL, and
the whole metallicity range is well within the two surrounding slices, at
$-1.5$ and $-3.0$ dex. We connot rule out that the measured trend is affected
by the spline interpolation in the grid.

The potential significance of the result for our understanding of mixing
processes in stars, may affect quantities of general interest in the study of
star clusters, such as stellar lifetimes or the location of the TO. We
therefore here revisit the result using a different approach. Studies of the
integrated light of globular clusters have shown that the residuals of full
spectrum fits are significantly lower when the artificial cluster spectra are
based on empirical spectral libraries than when they are based on theoretical
libraries (Martins et al. 2019).

Figure 1: Distribution of the library spectra in the region of the parameter
space occupied by the cluster stars. The gray dots represent the H16
measurements of the cluster members. The top graphics are the projections in
the log g vs. $T_{\mathrm{eff}}$ plane and the bottom graphics show the [Fe/H]
vs. $T_{\mathrm{eff}}$ projections. The overplotted colored points are the
location of the reference spectra in the three libraries used in this paper.
The left panel shows the Göttingen Spectral Library used by H16, the central
panel shows the ELODIE library, and the right panel the MILES library. These
last two libraries were used in tour reanalysis. Only the library stars
corresponding to the region mapped by the our sample are shown:
$T_{\mathrm{eff}}<7000$ K, and $-3.04<\rm{[Fe/H]}<-0.88$ dex (77 stars for
ELODIE and 137 for MILES, including 40 and 46 stars, respectively with
$\rm{[Fe/H]}<-1.7$ dex). The color scales for the ELODIE and MILES libraries
are indicated on the right.

We use two of these empirical spectral libraries to reanalyze the stellar
spectra of the MUSE observations of NGC 6397. By using other reference spectra
than H16, we introduce a different set of biases. Differences with the results
of H16 will provide an assessment of systematic uncertainties, and the basis
for future improvements.

Section 2 presents the data and their analysis, Sect. 3 our new analysis,
Sect. 4 discusses the results, and Sect. 5 draws the conclusions.

##  2 Data and earlier analysis

The observations of a 5 x 5 mosaic111Two of the planned fields, located at the
periphery, were not observed; 23 fields are available. reaching out to a
distance of $\sim$ 3.5 arcmin from the cluster center were made during the
MUSE commissioning, between 2014 July 26 to August 3 (ESO program identifier
60.A-9100(C)). They consisted of 127 pointings of exposure time of not more
than 60 s to avoid saturation of the bright giant stars, for a total
integration time of 95 min. Each pointing covers a field of 1 x 1 arcmin with
a spatial step of 0.2 arcmin, and the seeing was in the range 0.6” to 1”.

The data were reduced by the MUSE consortium using the official pipeline, and
the extraction of the individual stellar spectra was described in H16, with a
deblending approach presented in Kamann et al. (2013, implemented in the
PampelMuse software, publicly available). This method relies on a photometric
and astrometric input catalog, which in this particular case was the Anderson
et al. (2008) table derived from images obtained with the Advanced Cmaera for
Surveys (ACS) on board the Hubble Space Telescope (HST). An initial point
spread function (PSF) model, set to be the seeing, is used to create a mock
image that is cross-correlated with each wavelength layer of the MUSE cube to
find the coordinates transformations. The most isolated and bright spectra are
used to iteratively refine the PSF by subtracting all the other spectra. The
final wavelength-dependent PSF is derived by smoothing the PSF obtained for
each layer, and it is used to extract the spectra. In total, 18 932 spectra of
12 307 stars were extracted. The stars in the overlapping regions between the
different fields were observed multiple times. The line spread function (LSF)
varies with the wavelength from a full-width at half-maximum (FWHM) of about
2.82 Å at $\lambda=4750$Å (equivalent to an instrumental velocity dispersion,
$\sigma_{ins}=80$ $\rm{km}\,s^{-1}$) to about 2.54 Å at $\lambda=7000$Å
($\sigma_{ins}=46$ $\rm{km}\,s^{-1}$), and stays approximately constant
further in the red. The actual LSF also varies across the 24 spectrographs
forming MUSE by roughly 0.1 Å.

Out of the 18 932 extracted spectra, 14 271, with an SNR greater than 5 are
distributed on the MUSE website222http://muse-vlt.eu/science/globular-cluster-
ngc-6397 (the raw data are available in the ESO
archive333http://archive.eso.org/).

To determine the atmospheric parameters, H16 proceeded in three steps. First
they obtained $T_{\mathrm{eff}}$ and log g by fitting the HST photometry to an
isochrone. They then used these parameters to generate interpolated models
that they cross-correlated with the observations to derive the radial velocity
($v_{\mathrm{rad}}$). In the third step, they performed an optimization to
produce the final $T_{\mathrm{eff}}$, [Fe/H], $v_{\mathrm{rad}}$, line
broadening, and telluric absorption spectrum. They used their own full-
spectrum fitting procedure, using GSL interpolated by cubic spline as
reference. The interpolation cannot be a simple linear interpolation because
the optimization method requires its derivative to be continuous. The value of
log g was not optimized; the adopted value is the photometric one. The authors
justify this choice as a precautionary approach to prevent a possible
degeneracy between log g and the broadening.

This analysis provided parameters for 5 882 spectra of 4 132 stars with an S/N
$>20$, above which the formal error on $T_{\mathrm{eff}}$ is lower than 100 K,
and on [Fe/H] lower than 0.16 dex. The tables with results were kindly
provided to us by Tim-Oliver Husser. For 367 of these spectra log g was not
determined, and consequently, neither $T_{\mathrm{eff}}$ nor [Fe/H] were
measured. To clean the sample of the nonmember stars, we selected the spectra
within an ellipse centered on $v_{\mathrm{rad}}$ = 17.8 $\rm{km}\,s^{-1}$ and
[Fe/H] = -1.96 dex (mean parameters of the cluster), and semiaxes of 29
$\rm{km}\,s^{-1}$ and 1.08 dex (corresponding approximately to the selection
area measured in Figure 7 of H16; the actual values are not reported in the
text of this paper). The sample is reduced to 5 510 spectra of 4 089 distinct
cluster members. Furthermore, we are interested here in the TO and red giant
branch (RGB), and because the method of determining the stellar parameters by
full-spectrum fitting has never been validated for the hot stars, we followed
H16 and limited the sample to $T_{\mathrm{eff}}<7000$ K. This criterion
excludes the hot horizontal branch stars. Finally, the sample contains 5 400
spectra of 4 053 distinct cluster members.

##  3 Reanalysis

###  3.1 Method

The aim of this paper is to repeat H16’s analysis of the metallicity trend
along the cluster sequence with other reference libraries in order to
determine the reliability of this trend. Alternatives to GSL are the ELODIE
and MILES empirical libraries, among others. Synthetic and empirical libraries
are seen as complementary; each has their advantages and drawbacks. On the one
hand, synthetic spectra can be computed at any point in the parameter space
and are therefore not limited to the region populated with stars that can be
observed (i.e., essentially stars from the solar neighbourhood). They are also
free of noise and observational signatures, and their spectral resolution can
be as high as necessary. However, synthetic spectra still fail to accurately
match observed spectra, because the physical models and lists of atomic and
molecular transitions are limited (Martins et al. 2014). On the other hand,
empirical libraries can accurately reproduce observed spectra, and this is why
they are extensively used to model integrated spectra of stellar populations
(Martins et al. 2019).

ELODIE is an empirical library, initially presented in Prugniel & Soubiran
(2001) and later upgraded to increase the coverage in the parameter space. We
used the latest version (version 3.2), described in Wu et al. (2011),
containing 1962 spectra at a constant FWHM resolution of 0.55 Å over the
wavelength range 3900 – 6800 Å (R $\approx$ 10 000). The second library, MILES
(Sánchez-Blázquez et al. 2006), contains spectra of 985 stars at a resolution
about 2.55 Å (R $\approx$ 2000) in the wavelength range 3536-7410 Å. The
distribution of the stars populating the region of the cluster in the
parameter space are represented in the central and right panels of Fig. 1 for
ELODIE and MILES, respectively. Both libraries sample the region of the RGB
reasonably well down to almost the TO. Along the MS, most of the stars are on
the high-metallicity side compared to the cluster, and only a few have the
cluster metallicity. It is also noticeable that the library MSs are shifted
toward lower gravities by about 0.1 dex compared to the cluster star gravities
determined by isochrone fitting. We found indications that this may be at
least partly due to an underestimate of the gravities in the libraries.

The fitting procedure to be used for estimating stellar parameters requires
interpolating through the libraries to compute a spectrum for any set of
$T_{\mathrm{eff}}$, log g, and [Fe/H]. For this purpose, we used polynomial
interpolators computed using stellar parameters of the library stars compiled
from the literature, generally obtained through detailed abundance studies
using high-resolution spectroscopy. The ELODIE interpolator is described in Wu
et al. (2011), and for MILES, we used the interpolator described in Sharma et
al. (2016). The quality of the interpolated spectra naturally depends on the
distribution of the stars in the parameter space: the denser the distribution,
the better the interpolation. The quality also depends on the accuracy of the
input stellar parameters that were used when the interpolators were computed,
and on the intrinsic accuracy of the interpolators. At the margin of the
parameter space, the interpolators use synthetic spectra to extend the
validity range, but with a lower quality. The MS of the cluster is at the edge
of the validity region of the interpolators. However, using the procedure
described below, we successfully fit synthetic spectra from Coelho et al.
(2005) and from the GSL that were chosen to map this regime. This gave us some
confidence in our ability to analyze MS spectra. However, similar tests with
the cluster’s spectra failed to restore the correct gravity. The solutions
were found at a correct temperature, but located on the giant branch. The
reasons for this difficulty are not completely clear, but are probably related
to the drop in S/N from $\sim$60 near the TO to $\sim$20 on the lower MS, and
to stronger contamination by other cluster stars for the fainter objects. We
here prefer to restrict the spectral analysis to log g ¡4.2 and high S/N. and
are subject to a strong light contamination by the other cluster stars.
Although we here follow H16 and adopt the isochrone gravities, we prefer to
restrict our analysis to the RGB, $\rm{logg}<4.2$. This sample contains 1 587
spectra of 1 063 stars.

The atmospheric parameters were determined with the ULySS package (Koleva et
al. 2009) by minimizing the squared residuals between the MUSE observation and
an interpolated spectrum,

$S(\lambda)=P_{n}(\lambda)\times
G(v_{\mathrm{rad}},\sigma_{\mathrm{rel}})\otimes\mathrm{TGM}(T_{\mathrm{eff}},\rm{logg},\rm{[Fe/H]},\lambda),$
(1)

where $P_{n}(\lambda)$ is a series of Legendre polynomial up to degree $n$,
meant to absorb the instrumental spectral response and line-of-sight
extinction; we used $n=20$. $G(v_{\mathrm{rad}},\sigma_{\mathrm{rel}})$ is a
Gaussian function centered at $v_{\mathrm{rad}}$ and with standard deviation
$\sigma_{\mathrm{rel}}$. The spectra were binned in logarithmic wavelength, so
that the Doppler shift can be expressed by a convolution. TGM is the spectral
interpolator. The free parameters are $T_{\mathrm{eff}}$, [Fe/H],
$v_{\mathrm{rad}}$, $\sigma_{\mathrm{rel}}$ and the coefficients of
$P_{n}(\lambda)$. See Arentsen et al. (2019) for other details on the
procedure.

$\sigma_{\mathrm{rel}}$ was let free in order to account for the variation of
instrumental broadening from spectrum to spectrum (as described in Sect. 2,
the MUSE LSF varies between the individual spectrographs). The relative LSF
between MUSE and the libraries changes significantly over the wavelength
range, and we therefore ingested this variation pattern in the interpolated
spectra, before the minimization. This was made with a stepwise convolution of
the library (see the uly_lsf_convol function in the ULySS package). The LSF
injection improves the quality of the fits and slightly reduces the errors of
the estimated parameters, but it does not affect the general trends.

In the case of the ELODIE library, the injected LSF was chosen so that the
interpolated spectrum has a higher resolution than the MUSE spectrum. The
observation was then fit by letting $\sigma_{\mathrm{rel}}$ free. For the
MILES library, the resolution after the LSF injection is very close to that of
MUSE, but because of the spectrum-by-spectrum variations by 0.1 Å, it is
sometimes lower. In such cases, detected because the first fit fails to
determine $\sigma_{\mathrm{rel}}$, we convolved the observation with a
Gaussian with a dispersion of 30 $\rm{km}\,s^{-1}$, and then let the fitting
procedure determine $\sigma_{\mathrm{rel}}$.

A difference between the present procedure and that of H16 is that the latter
adjusted the telluric absorption together with the stellar parameters and
broadening. Telluric absorption features are prominent in the red part of the
spectra, which are outside the currently analyzed region, and we more simply
masked (i. e., ignored) the two strongest features. The first feature is the
water absorption near the NaD line that affects both the ELODIE and MILES, and
the second is the B band, near 6887 Å (Fraunhofer 1817, 1821), which affects
MILES.

No error spectra are associated with the extracted observations, but a mean
S/N is provided. We used it to determine a noise level that we assumed to be
identical throughout the spectrum, meaning that all wavelength bins have the
same weight. This assumed error sets the scale of the fitting errors on the
parameters that we tune in Sect. 3.2 into internal errors, by matching the
repeated observations of stars in the overlaps of the different pointings.

In order to keep the reanalysis as closely as possible to that of H16, we
carried it out by fixing log g to their photometric value (but see Sect. 4.2,
where we also determine log g spectroscopically). As first guesses, we used
for $T_{\mathrm{eff}}$, the photometric value, and for [Fe/H] we used -2.0
dex, which is an approximation of the cluster metallicity.

###  3.2 Assessment of the internal errors

The full catalog contains 5 510 measurements for 4 089 stars (including the
stars from the MS to the RGB). For the 1 200 stars with two or more repeated
observations, the spectra differ by the noise and by the propagation of
systematics introduced by the spectrograph and data reduction. We assessed the
precision by comparing the atmospheric parameters measured on each pair of
observations as in Arentsen et al. (2019). We computed for each pair of
observations $i$:

$\Delta\mathrm{P}_{i}=\frac{\mathrm{P}_{1,i}-\mathrm{P}_{2,i}}{\sqrt{\epsilon_{1,i}^{2}+\epsilon_{2,i}^{2}}},$
(2)

where P can be $T_{\mathrm{eff}}$ or [Fe/H], $\epsilon$ is the formal ULySS
error on the respective parameters, and $1$ and $2$ indicate the observation
of each spectrum of the pair. This distribution is expected to be nearly
Gaussian, and if the errors are properly scaled, their standard deviation is
expected to be one. For both parameters we find $\Delta\mathrm{P}=$ 2.2 and
1.8 for ELODIE and MILES, respectively. Because we assumed a constant noise
scaled to produce the average S/N determined by the data reduction and
spectrum extraction pipeline, we indeed expected deviations from unity. We
also assumed that all the pixels are independent, which cannot be the case
because the spectra were rebinned into evenly sampled wavelength (this process
necessarily introduces a correlation in the noise), and this leads to an
underestimation of the errors.

We verified that these distributions were similar throughout the entire
parameter space, and we found only a marginal indication that they may depend
on S/N, in a way consistent with an additional source of noise independent of
S/N. The effect is about 15 K and 0.01 dex to be quadratically added on the
errors on $T_{\mathrm{eff}}$ and [Fe/H],respectively. This is very small.

The fact that the distributions are similar for the two parameters is a
satisfactory sanity check of our analysis, and we did not investigate the
deviations from unity further because they are not relevant for our present
discussion. We rescaled the errors by the factors described above when we
computed the errors in the final table (see Sect. 3.4).

###  3.3 Assessment of the external errors (accuracy)

For an estimate of the external errors, we compared our measurements with
previous works. First of all, the comparison of 1587 measurements with H16
indicates $T_{\mathrm{eff}}({\rm\scriptstyle
H16})-T_{\mathrm{eff}}({\rm\scriptstyle ELODIE/MILES})=92$ K with a dispersion
of $40$ K. For the metallicity, the bias is different for ELODIE or MILES:
$\rm{[Fe/H]}({\rm\scriptstyle H16})-\rm{[Fe/H]}({\rm\scriptstyle
ELODIE})=-0.07$ dex, and $\rm{[Fe/H]}({\rm\scriptstyle
H16})-\rm{[Fe/H]}({\rm\scriptstyle MILES})=0.03$, with a dispersion of 0.15
dex. Because the three series used the same reduced spectra, they can be
similarly affected by systematics due to the data. These figures were
therefore be regarded as lower limits to the errors, and comparisons with
independent measurements will give a more complete assessment of the external
errors.

Five previous studies determined the parameters of a significant number of
stars in this cluster. Carretta et al. (2009a) measured 13 stars at high
spectral resolution, with the UVES spectrograph attached to the VLT, and
Carretta et al. (2009b) measured 144 stars at intermediate resolution, with
GIRAFFE on the same telescope, all giants with $T_{\mathrm{eff}}<5400$ K.
Lovisi et al. (2012) measured 146 stars, but those in common with the MUSE
sample are mostly blue stragglers, which are not discussed here. Finally, Korn
et al. (2007), and Lind et al. (2008) observed stars with the UVES (18 stars)
and GIRAFFE (116 stars) spectrographs, respectively, but in more peripheral
fields that lie outside the MUSE pointings.

The comparisons with the two first datasets are very similar, so that we can
merge them and report only the total comparison. We find
$T_{\mathrm{eff}}({\rm\scriptstyle lit})-T_{\mathrm{eff}}({\rm\scriptstyle
ELODIE})=15$ K with a dispersion of $30$ K, and
$T_{\mathrm{eff}}({\rm\scriptstyle lit})-T_{\mathrm{eff}}({\rm\scriptstyle
MILES})=48$ K with a dispersion of $25$ K. For the metallicity, we find no
differences with ELODIE, and $\rm{[Fe/H]}({\rm\scriptstyle
lit})-\rm{[Fe/H]}({\rm\scriptstyle MILES})=0.07$ dex, in both cases with a
dispersion of $0.05$ dex.

The comparison between the H16 measurements and the literature displays the
following differences: $T_{\mathrm{eff}}({\rm\scriptstyle
lit})-T_{\mathrm{eff}}({\rm\scriptstyle H16})=-100$ K and
$\rm{[Fe/H]}({\rm\scriptstyle lit})-\rm{[Fe/H]}({\rm\scriptstyle H16})=-0.08$
dex, with dispersions of $48$ K and $0.06$ dex on the two parameters. In all
cases the log g values are very consistent with the literature, which is
expected because they were determined photometrically in all cases with very
similar methods.

The picture that emerges from these comparisons is that (i) $T_{\mathrm{eff}}$
measured by H16 is on average hotter than the literature and our own
measurements. (ii) [Fe/H] is reasonably consistent between the different
series; our measurements are marginally more consistent with the literature
than those of H16. (iii) The consistency is similar for our two analyses, with
a marginally better performance of ELODIE.

For the observations compared with the literature, the mean estimated internal
errors after the rescaling described in Sect. 3.2 are $18$ and $25$ K on
$T_{\mathrm{eff}}$, and $0.04$ and $0.05$ dex on [Fe/H] for ELODIE and MILES,
respectively, which is about $1.25$ times lower than the external dispersions
estimated here. Because these dispersions include both the errors on our and
on the literature measurements, we conclude that our estimates of the internal
errors properly reflect the external errors, or accuracy of the measurements.

###  3.4 Results

Figure 2: Distribution of the parameter measurements for the 5 400 spectra of
cluster members with good-quality spectra in the log g vs. $T_{\mathrm{eff}}$
plane. The left panel shows the H16 catalogue, the central and right represent
our reanalysis with the ELODIE and MILES interpolators, respectively. The
color scale, shows the metallicity, in the right margin. The region of the
diagram that we do not discuss here ($\rm{logg}<4.2$) is shown with paler
colors.

Our measurements of the atmospheric parameters for the 1 587 spectra are
available in electronic form in Vizier.

Figure 2 presents the distribution of the measurements in the log g versus
$T_{\mathrm{eff}}$ plane. The vertical axis is the photometric gravity, and
the horizontal axis is the $T_{\mathrm{eff}}$ from H16, ELODIE, and MILES, for
the left, central, and right panel, respectively. The data points are color-
coded according to the metallicity. This can be compared with the Figure 8 of
H16. The RGB harbors an unphysical break in the H16 data near
$T_{\mathrm{eff}}=5000$ K. We cannot provide a final explanation for this
break, but it may correspond to a discontinuity of the opacities used for GSL,
at a threshold temperature where some molecular bands are considered. This RGB
is generally hotter than in our analysis, reflecting the systematic difference
quantified in Sect. 3.3 that was also found when H16 was compared to earlier
measurements. The small $T_{\mathrm{eff}}$ difference between ELODIE and MILES
is consistent with the uncertainties on the temperature calibration of the
RGB.

Figure 3: [Fe/H] vs. $T_{\mathrm{eff}}$ for the giants ($\rm{logg}<4.2$ dex).
The top panel shows the H16 catalog, and the middle and bottom panel represent
our reanalysis with the ELODIE and MILES interpolators,respectively (blue
dots). The solid red line connects the mean values and standard deviations of
the metallicity in 250 K wide bins. The black dots show the location of the
ELODIE and MILES library stars, in the middle and bottom panel, respectively.
The mean [Fe/H] dispersion in [Fe/H] of stars taken in bins of 250 K for H16
is 0.076, for ELODIE is 0.075 and that for MILES is 0.079 calculated for stars
with $T_{\mathrm{eff}}$ $<6\,500$K.

Figure 3 shows the distribution of the sample in the [Fe/H] versus
$T_{\mathrm{eff}}$ plane. The top panel shows the result from the H16
analysis, and the middle and lower panels are the results with the ELODIE, and
MILES interpolators respectively. In the three cases, the low dispersion
around the observed trend reveals a high internal consistency. The mean [Fe/H]
dispersion around the cluster isochrone, estimated as the average rms computed
in the $T_{\mathrm{eff}}$ bins shown in Fig. 3, is about $0.07$ dex. Compared
to the $0.05$ dex external error reported in Sect. 3.3, we see no clear
indication of a cosmic dispersion.

In addition to this internal consistency, the three analyses display different
patterns of metallicity variation. The results with ELODIE would be consistent
with a constant metallicity. The H16 metallicity steadily increases along the
sGB, while the MILES trend is wavy, with a pronounced [Fe/H] decline
immediately above the TO.

##  4 Discussion

###  4.1 Changes in surface metallicity above the turn-off

Considering that chemical tagging (the ability to associate individual stars
with their birth environment) has been a science driver for many of the large
surveys now underway, the ability to differentiate surface from initial
abundances is a crucial necessity of stellar evolution models. Many processes
can alter the surface abundance of a star during its lifetime. Atomic
diffusion is one of the most frequent discussed processes. It encompasses
different processes that operate in the radiative regions of the stars and
cause the redistribution of elements, mainly by the migration of heavier
elements toward its center, which consequently lowers the surface metallicity
of the stars. Although atomic diffusion in stars has been theoretically
predicted more than a century ago by Chapman (1917), its real effect on
stellar evolution remains debated (VandenBerg et al. 2002). Mixing processes
(e.g., convection and turbulent mixing), compete with gravitational settling
to weaken the actual depletion in heavy elements. The efficiency of these
processes, at variance with the pure atomic diffusion, has no well-established
physical description, and can only be determined through comparison with
observations. For instance, Dotter et al. (2017) investigated models with and
without these effects and found that the differences in ages obtained through
isochrones might reach up to 20%. Star clusters, with a coeval stellar
population that in principle is initially chemically homogeneous, are the
perfect laboratory for testing the balance between the different phenomena.

It is expected that because of atomic diffusion, the surface abundance of
heavy element is minimum at the TO, increasing both down the MS and toward the
RGB. Metallicity trends near the TO of globular clusters have previously been
discussed in NGC 6752 by Gruyters et al. (2013), and in M 30 Gruyters et al.
(2016), and in the M 67 open cluster by Bertelli Motta et al. (2018), Souto et
al. (2018), Gao et al. (2018), and Liu et al. (2019). In particular Gratton et
al. (2001) and more recently Lovisi et al. (2012) found no difference between
the abundances of TO stars and base-RGB stars for NGC 6397.In contrast, Korn
et al. (2007), Lind et al. (2008) and Nordlander et al. (2012) found a clear
trend. The latter three works found similar trends for the iron metallicity
from the TO to the RGB, obtaining maximum metallicity differences of about
0.15 $-$ 0.20 dex. They argued that their results indicate the need of atomic
diffusion with weak efficiency of turbulent mixing, but also that, based on
the uncertainties from modeling techniques, it cannot be ruled out that this
might be just an artifact. For example, the Figure 7 of Lind et al. (2008)
shows that although they and Korn et al. (2007) found a very similar trend in
[Fe/H] along the Hertzsprung-Ruseel diagram, the systematic differences
between their values is about 0.15 dex, which they explained as possible
differences between the NLTE corrections. In other clusters, the measured
amplitude of variations is smaller, or no effect was detected. All these works
attempted to determine chemical abundances with dedicated techniques,
addressing and trying to take into account model weaknesses such as the NLTE
and 3D effects.

We here found different [Fe/H] trends above the TO using three different
stellar libraries. This casts some doubt on the reality the reality of the
detection of atomic diffusion effects.

Regarding the H16 results, we recall that the magnitude of the trend is
significantly smaller than the GSL grid mesh and that the synthetic spectra
themselves can introduce systematics due to calculation details. The
nonuniform distribution of the cluster stars along the sequence in the the top
panel of Fig. 3 is an indication that some finite-grid effects are indeed
present along the temperature axis in this analysis, but the ramification for
other parameters remain unclear. In contrast, the ELODIE and MILES
interpolators do not suffer from this wide-mesh drawback, but they are limited
by the small number of stars in this region of the parameter space (about 40
stars for both libraries in the $\pm 0.3$ dex slice around the cluster
metallicity, see Fig. 3). These observed library spectra are also subject to
noise and to the effect of peculiarities of individual stars. In well-
populated regions of the parameter space (typically close to solar
metallicity), these effects are averaged out because a fair number of stars
have similar parameters, but in this low-metallicity regime, every single star
might have a strong influence on the interpolator.

The distribution of the ELODIE stars (the black dots in Fig. 3) is uniform
except in the region $T_{\mathrm{eff}}>6100$ K, which includes only one star
with $\rm{[Fe/H]}<-1.7$ dex. In the case of MILES, we note that in the range
$5500<T_{\mathrm{eff}}<5850$ K one library star has $\rm{[Fe/H]}<-1.7$ dex, HD
140283, with $\rm{[Fe/H]}\approx-2.57$ dex. The MILES pattern in this regime
may therefore be an artifact of the interpolation. The interpolator we used
here is an improvement on the interpolator that was initially published by
Prugniel et al. (2011). The improvement concerned the cool stars
($T_{\mathrm{eff}}<4800K$), and consisted of refining the stellar parameters
by critically scrutinizing the literature and tuning the interpolator. The
number of terms in the polynomial development was also increased, essentially
to reflect the variation between the giants and the dwarfs better, and this
can affect its behavior at warmer temperature. We repeated the analysis with
this first version interpolator, in order to verify the reliability of the
metallicity trend with respect to the change of the interpolator, and we did
not notice any significant difference. During the preparation of this improved
MILES interpolator, a series of interpolators were constructed, where for each
a single star was excluded. They are called X-interpolators in Sharma et al.
(2016), and they were used to verify the stability of the results. We repeated
the analysis here with all the X-interpolators of the library stars in the
region of the parameters we analyzed to determine whether the trend was
modified for some of these stars. We found the trend to be robust, and
therefore rule out any strong bias due to an individual star. We also compared
the parameters that were used to compute the MILES interpolator with an
updated compilation of the literature, and did not find any discrepancy.
Finally, we redetermined the parameters of the MILES spectra by fitting in the
same way as the MUSE observation (self-inversion of the interpolator), and we
found no systematic effect there either.

The difference between the trends found for the three analyses remains to be
understood. Each analysis approach has its merits and drawbacks, and we found
no reason prefer one more that the others.

###  4.2 Spectroscopic surface gravity

Figure 4: Comparison between the spectroscopic and photometric log g. The
dotted line is the 1:1 correspondence, the color of the symbol reflects S/R as
indicated in the figure. The top panel shows the MILES analysis and the bottom
panel that of ELODIE. Figure 5: [Fe/H] vs. $T_{\mathrm{eff}}$ for the giants
($\rm{logg}<4.2$ dex), with log g determined spectroscopically. The top panel
shows results obtained with the ELODIE interpolator, and the bottom panel
shows those obtained with the MILES interpolator. The solid red line connects
the mean values and standard deviations of the metallicity in 250 K wide bins.
The difference with Fig. 3 is that log g was this time determined by the full-
spectrum fitting, not by photometry.

H16 did not fit log g from the spectra because of concerns of degeneracy
between the instrumental broadening and the surface gravity. Because the MUSE
LSF cannot be precisely assessed for the spectra extracted from the final
cubes, the authors preferred to adopt log g from their photometric fit to the
isochrone. Although the line width is the main indicator for the gravity in
individual spectral lines that are observed at high spectral resolution, the
degeneracy between the broadening and the gravity may not be important at
medium resolution and for full-spectrum fitting, where all the lines are used,
notwithstanding their sensitivity to gravity.

To determine the effect of the gravity on the results for the metallicity, we
performed a separate analysis with log g as an additional free parameter.
Figure 4 compares the two values. For the high S/N spectra, (S/N$>50$), the
differences in log g do not exceed 0.5 dex between the two methods. When we
compare the new values with literature values (see Sect. 3.3), we find
$\rm{logg}({\rm\scriptstyle lit})-\rm{logg}({\rm\scriptstyle spectro})=0.15$
dex, with a dispersion of 0.25 dex. $T_{\mathrm{eff}}$ obtained with free log
g is lower by $\sim 40$ K (ELODIE) and $85$ K (MILES) compared to the values
when log g was fixed.

The [Fe/H] trends with this analysis, presented in Fig. 5, are not
significantly modified in this new analysis (compare with Fig. 3).

##  5 Conclusions

We repeated the analysis of 1 587 MUSE spectra of 1 063 stars located above
the TO of NGC 6397 that was originally performed by H16. While their analysis
used a grid of synthetic stellar spectra, we have used two empirical stellar
libraries, ELODIE and MILES, to determine $T_{\mathrm{eff}}$ and [Fe/H] of
these stars. These spectra have $S/N\gtrsim 50$, and comparisons with H16 and
earlier studies suggest that the errors in our determinations are $\sim 25$ K
and $0.05$ dex on the two parameters respectively. The metallicity dispersions
of the measurements, $\sim 0.07$ dex, agree with these figures.

Despite this fair consistency, the metallicity trends along the sGB and RGB
are different in the three analyses. No trend was found with ELODIE, and the
trend found with MILES does not match the theoretical expectations. Previous
observational studies of this and other clusters also lead to contradictory
results about the reality of this variation, and the detection of the effect
of atomic diffusion.

These empirical libraries contain only a limited number of stars in this low-
metallicity regime (they do not have enough stars on the MS, which makes it
impossible to explore the trend below the TO). Despite this limitation, the
empirical libraries are essential for validating the synthetic libraries, and
they are still required ingredients for models of integrated spectra of
stellar populations as are used in extragalactic astronomy. Therefore, the
motivation to continue improving the empirical libraries remains strong. We
here clearly identified the regime of [Fe/H] $\sim-2$ as an area where the
libraries deserve to be improved.

The metallicity trend found by H16 qualitatively reproduces the expected
effect of atomic diffusion. The reference grid of synthetic spectra has a wide
metallicity mesh, however, which casts doubts on the capability of resolving
the small physical effect. When the grid mesh are compared to the generally
accepted accuracy in the determinated atmospheric parameters, most of the
grids that are currently used for population analysis are too coarse in their
metallicity coverage. For $T_{\mathrm{eff}}\sim 6\,000$ K, the mesh size
typically is 200 K, which is roughly four times the admitted precision on
measurements. On log g the size is 0.5 dex, which about five times the
precision, and on [Fe/H], the size is 0.5 dex, which is about ten times the
precision. For example, for the GSL the meshes sizes are 100 K, 0.5, and 0.5
dex, and in Coelho (2014) they were 200 K, 0.5, and 0.5 dex on the three
parameters. Only recently have libraries with a finer metallicity sampling
become available. For example, INTRIGOS (Franchini et al. 2018) and Allende
Prieto et al. (2018) have mesh sizes of 250 K, 0.5, and 0.25 dex. Adopting
finer sampling is naturally challenging in terms of both the computing time
required to produce the large number of spectra, and the data volume which
needs to be manipulated, which is still in the 100s GB in Allende Prieto et
al. (2018). To properly resolve the metallicity trends due to atomic
diffusion, or to determine the other surface abundances alterations due to
other processes, a mesh size of 0.1 dex is required.

From point of view of high resolution abundance analysis, there is also large
space for improvement. Limitations such as the treatment of NLTE and 3D
effects are still a large source of uncertainty.

We illustrated the limits of the current libraries at low metallicity. While
observations of Galactic globular clusters are classical benchmark for
population models, the empirical libraries are at best limited near the TO of
these clusters, and the interpolated spectra used for the models are crudely
approximate. The most recent library is the X-Shooter Spectral Library, XSL
Arentsen et al. (2019). It has a more uniform coverage of the metallicity
range than earlier libraries, but also suffers from a relative shortage of TO
and MS stars in the low metallicity regime. For the stellar population models
and for the validation of synthetic spectra, it is desirable to extend the
coverage of these libraries.

###### Acknowledgements.

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 web site. We acknowlege the LABEX Lyon Institute of Origins
(ANR-10-LABX-0066) of the Université de Lyon for its financial support within
the program ”Investissements d’Avenir” (ANR-11-IDEX-0007) of the French
government operated by the National Research Agency (ANR), and IDEXLYON for
their support fo the Indo-French Astronomy school (IFAS4), where this project
was started. This work was supported by the Programme National Cosmology et
Galaxies (PNCG) of CNRS$/$INSU with INP and IN2P3, co-funded by CEA and CNES.
We thank Yasna Ordenes, Yue Wu, Jean Damascène Mbarubucyeye, Sina Chen, Sonali
Borah, Raghu Prasad, and Saumya Gupta, who contributed to early investigations
on this project. We thank the referee for the comments, which helped improve
the quality of the paper. L.P.M. thanks FAPESP (grant 2018/26381-4) and CNPQ
(grant 306359/2018-9). L.P.M. also thanks L.Tresse, director of CRAL, for the
hosting during the development of this work.

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