11institutetext:  INAF-Osservatorio di Astrofisica e Scienza dello Spazio di
Bologna, Via Gobetti 93/3, I-40129 Bologna, Italy

# Chemistry of multiple stellar populations in the mono-metallic, in situ,
bulge globular cluster NGC 6388††thanks: Based on observations collected at
ESO telescopes under programmes 073.D-0211 and 073.D-0760, 381.D-0329,
095.D-0834, and 099.D-0047. ††thanks: Full Tables A.1, A.2, A.3, and A.5 are
only available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsarc. u-strasbg.fr/viz-bin/cat/J/A+A/??/??.

Eugenio Carretta 11 Angela Bragaglia 11

We present the homogeneous abundance analysis for a combined sample of 185
giants in the bulge globular cluster (GC) NGC 6388. Our results are used to
describe the multiple stellar populations and differences or analogies with
bulge field stars. Proton-capture elements indicate that a single class of
first-generation polluters is sufficient to reproduce both the extreme and
intermediate parts of the anti-correlations among light elements O, Na, Mg,
and Al, which is at odds with our previous results based on a much smaller
sample. The abundance pattern of other species in NGC 6388 closely tracks the
trends observed in bulge field stars. In particular, the $\alpha$-elements,
including Si, rule out an accreted origin for NGC 6388, confirming our
previous results based on iron-peak elements, chemo-dynamical analysis, and
the age-metallicity relation. The neutron-capture elements are generally
uniform, although the [Zr/Fe] ratio shows an intrinsic scatter, correlated to
Na and Al abundances. Instead, we do not find enhancement in neutron-capture
elements for stars whose photometric properties would classify NGC 6388 as a
type II GC. Together with the homogeneity in [Fe/H] we found in a previous
paper, this indicates we need to better understand the criteria to separate
classes of GCs, coupling photometry, and spectroscopy. These results are based
on abundances of 22 species (O, Na, Mg, Al, Si, Ca, Ti, Sc, V, Cr, Mn, Fe, Co,
Ni, Zn, Y, Zr, Ba, La, Ce, Nd, and Eu) from UVES spectra sampling proton-,
$\alpha-$, neutron-capture elements, and Fe-peak elements. For 12 species, we
also obtain abundances in a large number of giants (up to 150) from GIRAFFE
spectra.

###### Key Words.:

Stars: abundances – Stars: atmospheres – Stars: Population II – Galaxy:
globular clusters: general – Galaxy: globular clusters: individual (NGC 6388)

††offprints: E. Carretta<EMAIL_ADDRESS>
##  1 Introduction

The high-mass, high-metallicity globular cluster (GC) NGC 6388, located in the
bulge of the Milky Way (MW) had been poorly studied using high resolution
spectroscopy, despite its relevance. The small apparent angular diameter and
the large contamination by field stars made it difficult to observe a large
number of cluster members (see, e.g. Wallerstein et al. 2007). This has been
amended with the use of multi-object spectrographs (such as FLAMES at the ESO
VLT; see Carretta et al. 2007a, Lanzoni et al. 2013) and large surveys such as
APOGEE (Mészáros et al. 2020: M20). Additionally, a more efficient choice of
candidate members can be provided by the $Gaia$ mission (see, i.e. the
astrometric analysis in Vasiliev and Baumgardt 2021).

This paper presents all the data from our series on NGC 6388 ‘reloaded’, aimed
at providing a detailed description of the chemical properties of multiple
stellar population in this GC. We started by using the existing data in the
ESO archive and added new observations purposely designed to complete the set
of setups needed to characterise the chemical composition. In particular, we
sought to include all the light elements involved in the (anti-)correlations
typical of multiple stellar populations detected in GCs (Carretta et al.
2009a,b, 2010a, Gratton et al. 2012a, 2019, Bastian and Lardo 2018).

In Carretta and Bragaglia (2018), we analysed UVES spectra of 24 giants
(adding 17 new stars to our original set of seven in Carretta et al. 2007a).
The resulting pattern of proton-capture elements seemed to imply the necessity
of two classes of polluters (the stars that produced the characterising set of
light elements in GCs) to explain the anti-correlations between O and Na, Mg,
and Al.

The next step (Carretta and Bragaglia 2019) was to discuss the abundance of
Mg, Ca, and Sc for the full set of 185 stars observed with both UVES and
GIRAFFE. By comparing the abundance ratios in NGC 6388 to those of field stars
of similar metallicity, we detected no significant differences for Ca and Sc
(at variance with NGC 2808 and a few other massive GCs: see Carretta and
Bragaglia 2021), whereas lighter proton-capture species, such as Si, showed
clear variations correlated to Mg depletion and Al enhancement. Together with
prediction from stellar nucleosynthesis, these observations allowed us to
pinpoint the range of internal temperature for the first generation (FG)
polluters in NGC 6388 ($\sim 100-150$ MK), with some difference between the
cases of the massive stars and asymptotic giant branch (AGB) stars.

We then concentrated on the metal abundance of this cluster and presented
(Carretta and Bragaglia 2022a) the atmospheric parameters for our new data
set. We also compared our results to literature results and addressed the
issue of a metallicity spread. The latter is predicted as a possible property
of a separate class of GCs (type II, see below), defined mainly on photometric
ground. However, we did not find any significant spread in [Fe/H], not even
considering stars belonging to the so-called anomalous red giants in the
chromosome map, a diagnostic based on Hubble Space Telescope UV photometry
used to define the hypothesised class of type II GCs (see Milone et al. 2017).
Either an intrinsic spread in iron (and neutron-capture elements, see below)
is not a necessary condition to belong to the so-called type II GCs, or NGC
6388 is not a type II GC, as claimed.

Finally, in Carretta and Bragaglia (2022b), we addressed the recent claim that
NGC 6388 may have been accreted to the MW, based on the proposed under-
abundance of Sc, V, and Zn in four stars (Minelli et al. 2021a). We instead
demonstrated how our derived abundances, for a much larger and significant
sample of cluster stars, are fully compatible with those of the MW bulge stars
and clusters of similar metallicity. Together with chemo-dynamical
considerations and the placement of NGC 6388 firmly on the in situ branch of
the age-metallicity diagram, our observations conclusively indicate an in situ
origin for this GC.

Figure 1:  Checks on stars with dubious membership. Upper panel: Radial
velocity profile of NGC 6388 based on our sample. The RV refers to the cluster
systemic velocity of 82.5 km s-1. The open circles indicate stars defined as
non-members in APOGEE DR17. The vertical lines indicate the half-mass and
tidal radii from Harris (2010). The dashed lines show the $\pm 3\sigma$
velocity dispersion profile of the best-fit King model as in Lanzoni et al.
(2013). All 185 stars fall well inside its limits. Lower panel: For stars in
common with APOGEE DR17 and considered non-members there, we show the offsets
from the cluster mean elemental ratios ([el/Fe]). For each element, the red
and blue lines indicate the $1\sigma$ and $3\sigma$ limits, respectively. Only
the Ba value for star l63p075 is off by more than $3\sigma,$ and we
conservatively excluded this abundance from the computed average in Table 1.

In the present paper, we complete and show the analysis of the dataset,
presenting the abundances of 22 species in NGC 6388 from high resolution
optical spectroscopy, which permitted a full and homogeneous chemical
characterisation of this GC. We discuss our results regarding the multiple
stellar population phenomenon, and similarities and differences with the
underlying bulge stellar populations.

The paper is organised as follows. A brief summary of data selection and
observations is provided in Sect. 2, whereas the abundance analysis and error
budget are described in Sect. 3. Results for proton capture and the heavier
elements ($\alpha$-capture, Fe-peak, and neutron-capture elements) are
discussed in Sect. 4 and Sect. 5, respectively. A summary of the properties of
NGC 6388 is provided in Sect. 6.

##  2 Summary of data selection and observations

Details on the full procedure we followed to select member stars to be
observed in NGC 6388, heavily contaminated by bulge and disc stars, are
discussed at length in Carretta and Bragaglia (2022a). We exploited previously
made spectroscopic observations (mainly for kinematics) in NGC 6388 to select
radial velocity (RV) members. The large value of the systemic RV (80 km s-1,
Harris 2010) allowed us to individuate bona fide cluster members. Culling our
targets from the programmes 073.D-0760 (PI Catelan), 381.D-0329 (PI Lanzoni),
and 095.D-0834 (PI Henault-Brunet), we analysed good quality archive data
(taken with GIRAFFE high resolution setups HR13 and HR21 or with UVES) or
acquired new observations using the same setups (programme 099.D-0047).

Our observing strategy was to obtain GIRAFFE HR13 spectra (to derive
atmospheric parameters and abundances of the light elements O, Na, Mg, Si) and
HR21 spectra (to derive Al abundances) for as many stars as possible in NGC
6388. Moreover, new UVES/FLAMES spectra were acquired for 12 giants. The new
observations were performed from April to August 2017. The wavelength coverage
is 6120-6405 Å for HR13, 8484-9001 Å for HR21, and 4800-6800 Å for
UVES/FLAMES. The median S/N values are 93, 116, and 50, respectively.

Coordinates, magnitudes, original samples (or new observations), and RVs are
listed in Carretta and Bragaglia (2022a) for the 12 and 150 stars with UVES
and GIRAFFE spectra, respectively. Similar data for 24 giants with previously
analysed UVES/FLAMES spectra can be found in Carretta and Bragaglia (2018).

We consider all stars in our sample to be good candidates as members of NGC
6388 based on the combination of RV and metallicity (presented in previous
papers), taking advantage of our derived very small spread in [Fe/H] (0.04 dex
from 185 stars as compared to 0.074 dex from 9 stars in M20). As already
discussed in CB22a, for a few stars membership is more dubious because of
their proper motions (measured by Gaia) or because they fall outside the tidal
radius (if we take the value in Harris 2010, but not if we consider the much
larger value in Baumgardt’s database, which would be outside the upper panel
in Fig. 1). Additionally, the referee noted that six of our targets are not
considered cluster members in APOGEE DR17. However, as shown in Fig. 1 (upper
panel) all stars in our sample fall well within the $3\sigma$ velocity
dispersion profile of the best-fit King model (for details, see Lanzoni et al.
2013, their Fig. 10 and Table 3). Additionally, if we consider the six
candidate non-members, we see that the abundances of elements not involved in
the multiple populations phenomenon are the same as the bulk of the cluster
sample (Fig. 1, lower panel). Furthermore, we find that four of them belong to
the second generation, based on their Na and O abundances (see Sect. 4), which
decreases the chance of them being field interlopers. In the following, we
treat all stars as members, bearing in mind that further studies (e.g. better
astrometric data in future Gaia data releases) will help settle the matter.

Figure 2: [Na/Fe] abundance ratios as a function of [O/Fe] ratios in NGC 6388.
Stars are colour-coded according their chemical composition following the
definitions in Carretta et al. (2009b): red for stars with primordial
composition (P), blue for those with intermediate composition (I), and green
for stars with heavily altered composition (E). Internal error bars are also
reported.

##  3 Abundance analysis and error budget

The same procedure adopted for GCs previously analysed in our FLAMES survey
(see Carretta et al. 2006, 2009a,b) was used here to derive atmospheric
parameters and abundances. The metallicity of NGC 6388 is reported and
discussed in Carretta and Bragaglia (2022a). As usual, our final effective
temperatures Teff were derived in two steps. First estimates were obtained
from the calibrations of Alonso et al. (1999, 2001). Afterwards, these
estimates were refined using a relation between the values from the first step
and $K$ magnitudes of the stars. Surface gravities were then obtained by using
the above temperatures, adopting distance modulus $(m-M)_{V}=16.14$ and
reddening $E(B-V)=0.37$ from Harris (2010), bolometric corrections from Alonso
et al. (1999), and masses of 0.90 $M_{\odot}$ and $M_{\rm bol,\odot}=4.75$.

For the abundance analysis we used equivalent widths (EW) measured with the
ROSA package (Gratton 1988), after correcting those measured on GIRAFFE
spectra to the system given by EWs from UVES spectra. The microturbulent
velocities $v_{t}$ were derived by minimising the slope of the relation
between Fe i abundances and expected line strength (Magain 1984). Models with
appropriate atmospheric parameters whose abundances matched those derived from
Fe i lines were then interpolated within the Kurucz (1993) grid of model
atmospheres. Adopted atmospheric parameters and derived abundances of Fe are
listed in Table 2 of Carretta and Bragaglia (2022a). The absence of trends as
a function of Teff and the good agreement between Fe i and Fe ii and between
Ti i and Ti ii are a good test of the reliability of the adopted scale of
atmospheric parameters.

Table 1: Mean abundances in NGC 6388 for the present and previous samples.

Element | stars | avg | rms | ref | stars | avg | rms | ref | stars | avg | rms | ref | Sun | ref
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$[$Fe/H$]$i | 12 |  $-$0.509 | 0.039 | (1) | 150 |  $-$0.488 | 0.040 | (2) | 35 |  $-$0.480 | 0.045 | (3) | 7.54 | 1
$[$Fe/H$]$ii | 12 |  $-$0.491 | 0.039 | (1) | 149 |  $-$0.478 | 0.053 | (2) | 35 |  $-$0.415 | 0.081 | (3) | 7.49 | 1
$[$O/Fe$]$i | 12 |  $-$0.210 | 0.187 | (1) | 148 |  $-$0.040 | 0.227 | (2) | 35 |  $-$0.201 | 0.190 | (3) | 8.79 | 1
$[$Na/Fe$]$i | 12 | +0.544 | 0.334 | (1) | 150 | +0.342 | 0.283 | (2) | 35 | +0.514 | 0.230 | (3) | 6.21 | 1
$[$Mg/Fe$]$i | 12 | +0.206 | 0.043 | (1) | 149 | +0.221 | 0.052 | (2) | 35 | +0.212 | 0.049 | (3) | 7.43 | 1
$[$Al/Fe$]$i | 12 | +0.385 | 0.391 | (1) | 125 | +0.464 | 0.303 | (2) | 35 | +0.423 | 0.350 | (3) | 6.23 | 3
$[$Si/Fe$]$i | 11 | +0.342 | 0.029 | (1) | 150 | +0.309 | 0.061 | (2) | 34 | +0.350 | 0.063 | (3) | 7.53 | 1
$[$Ca/Fe$]$ i | 12 | +0.082 | 0.047 | (1) | 150 | +0.069 | 0.046 | (2) | 35 | +0.062 | 0.049 | (3) | 6.27 | 1
$[$Sc/Fe$]$ii | 12 |  $-$0.016 | 0.059 | (1) | 150 |  $-$0.018 | 0.067 | (2) | 35 |  $-$0.027 | 0.084 | (3) | 3.13 | 1
$[$Ti/Fe$]$i | 12 | +0.292 | 0.050 | (1) | 150 | +0.270 | 0.098 | (2) | 35 | +0.295 | 0.086 | (3) | 5.00 | 1
$[$Ti/Fe$]$ii | 12 | +0.282 | 0.039 | (1) |  |  |  |  | 35 | +0.244 | 0.086 | (3) | 5.07 | 1
$[$V/Fe$]$i | 12 | +0.228 | 0.103 | (1) |  |  |  |  | 35 | +0.259 | 0.143 | (3) | 3.97 | 1
$[$Cr/Fe$]$i | 12 |  $-$0.138 | 0.039 | (1) |  |  |  |  | 35 |  $-$0.104 | 0.081 | (3) | 5.67 | 1
$[$Mn/Fe$]$i | 12 |  $-$0.220 | 0.061 | (1) |  |  |  |  | 35 |  $-$0.218 | 0.052 | (3) | 5.34 | 1
$[$Co/Fe$]$i | 12 | +0.043 | 0.069 | (1) |  |  |  |  | 35 | +0.041 | 0.078 | (3) | 4.92 | 2
$[$Ni/Fe$]$i | 12 | +0.023 | 0.022 | (1) | 150 | +0.034 | 0.030 | (2) | 35 | +0.032 | 0.029 | (3) | 6.28 | 1
$[$Zn/Fe$]$i | 9 | +0.128 | 0.125 | (1) |  |  |  |  | 31 | +0.102 | 0.239 | (3) | 4.59 | 1
$[$Y/Fe$]$i | 11 |  $-$0.323 | 0.183 | (1) |  |  |  |  | 32 |  $-$0.259 | 0.253 | (3) | 2.24 | 2
$[$Zr/Fe$]$i | 10 |  $-$0.101 | 0.096 | (1) | 138 |  $-$0.226 | 0.156 | (2) | 30 |  $-$0.113 | 0.156 | (3) | 2.60 | 2
$[$Zr/Fe$]$ii | 10 |  $-$0.186 | 0.203 | (1) |  |  |  |  | 29 |  $-$0.106 | 0.181 | (3) |  |
$[$Ba/Fe$]$ii | 11 |  $-$0.084 | 0.154 | (1) | 149 |  $-$0.016 | 0.135 | (2) | 33 |  $-$0.115 | 0.140 | (3) | 2.22 | 3
$[$La/Fe$]$ii | 12 | +0.155 | 0.065 | (1) |  |  |  |  | 35 | +0.191 | 0.141 | (3) | 1.22 | 2
$[$Ce/Fe$]$ii | 11 |  $-$0.295 | 0.059 | (1) |  |  |  |  | 34 |  $-$0.299 | 0.142 | (3) | 1.63 | 3
$[$Nd/Fe$]$ii | 11 | +0.032 | 0.082 | (1) |  |  |  |  | 34 | +0.032 | 0.091 | (3) | 1.50 | 2
$[$Eu/Fe$]$ii | 11 | +0.253 | 0.037 | (1) |  |  |  |  | 32 | +0.248 | 0.063 | (3) | 0.55 | 3

  * (1)

This work; new UVES spectra.

  * (2)

This work; GIRAFFE spectra.

  * (3)

Total sample with UVES spectra (this work + Carretta et al. 2007 + Carretta
and Bragaglia 2018).

  * References for solar: 1=Gratton et al. (2003); 2=Anders and Grevesse (1989); 3=R.Gratton (priv.comm.).]

  * The average [Ba/Fe] abundance from GIRAFFE is computed by excluding the star l63p075 (see text).

For the other elements, we proceeded as in our FLAMES survey. Oxygen
abundances were obtained from the forbidden [O i] line at 6300 Å (more rarely
also from the 6363 Å line) after cleaning the telluric lines as described in
Carretta et al. (2007a). To correct the Na abundances for non-LTE effects, we
adopted the prescriptions from Gratton et al. (1999). Finally, references for
the hyperfine structure corrections applied to Sc, V, Mn, and Co can be found
in Gratton et al. (2003).

Average abundances are given in Table 1 for the 12 stars observed with UVES
and the 150 stars with GIRAFFE spectra. We also list the combination of the
abundances of the present sample from UVES with those previously analysed in
Carretta and Bragaglia (2018) to show how good the consistency is between the
two studies that we merged together. In this table, the neutral species are
referred to as Fe i abudances, whereas abundance ratios of ionised species are
computed using singly ionised iron. Solar reference abundances are reported in
the penultimate column of Table 1, with their sources listed in the last
column.

Our procedure to estimate star to star errors due to uncertainties in the
adopted atmospheric parameters and in $EW$ measurements is described in detail
in Carretta et al. (2009a) for UVES and Carretta et al. (2009b) for GIRAFFE.
For the present work, the outcomes are summarised in Table 2 and Table 3 for
abundances obtained from UVES and GIRAFFE spectra, respectively. For the sake
of completeness, we also report the values relative to iron in these tables
(see Carretta and Bragaglia 2022a, Tables 4 and 5).

Table 2: Sensitivities of abundance ratios to variations in the atmospheric parameters and to errors in the equivalent widths and errors in abundances for stars of NGC 6388 observed with UVES. Element | Average | Teff | $\log g$ | [A/H] | $v_{t}$ | EWs | Total | Total
---|---|---|---|---|---|---|---|---
| n. lines | (K) | (dex) | (dex) | kms-1 | (dex) | Internal | Systematic
Variation |  | 50 | 0.20 | 0.10 | 0.10 |  |  |
Internal |  | 6 | 0.04 | 0.04 | 0.12 | 0.02 |  |
Systematic |  | 17 | 0.06 | 0.02 | 0.04 |  |  |
$[$Fe/H$]$i | 100 |  $-$0.027 | +0.041 | +0.023 |  $-$0.044 | 0.019 | 0.058 | 0.024
$[$Fe/H$]$ii | 13 |  $-$0.119 | +0.122 | +0.040 |  $-$0.030 | 0.052 | 0.072 | 0.055
$[$O/Fe$]$i | 2 | +0.041 | +0.038 | +0.017 | +0.039 | 0.134 | 0.142 | 0.058
$[$Na/Fe$]$i | 4 | +0.062 |  $-$0.088 | +0.017 | +0.055 | 0.095 | 0.117 | 0.103
$[$Mg/Fe$]$i | 4 | +0.004 |  $-$0.027 |  $-$0.000 | +0.018 | 0.095 | 0.097 | 0.016
$[$Al/Fe$]$i | 2 | +0.059 |  $-$0.037 |  $-$0.018 | +0.017 | 0.134 | 0.136 | 0.115
$[$Si/Fe$]$i | 8 |  $-$0.043 | +0.018 |  $-$0.002 | +0.026 | 0.067 | 0.074 | 0.020
$[$Ca/Fe$]$i | 16 | +0.081 |  $-$0.073 |  $-$0.009 |  $-$0.023 | 0.047 | 0.058 | 0.038
$[$Sc/Fe$]$ii | 8 | +0.094 |  $-$0.034 |  $-$0.006 |  $-$0.024 | 0.067 | 0.074 | 0.038
$[$Ti/Fe$]$i | 26 | +0.104 |  $-$0.045 |  $-$0.005 |  $-$0.035 | 0.037 | 0.058 | 0.042
$[$Ti/Fe$]$ii | 11 | +0.079 |  $-$0.043 |  $-$0.008 |  $-$0.027 | 0.057 | 0.067 | 0.033
$[$V/Fe$]$i | 12 | +0.109 |  $-$0.041 | +0.003 |  $-$0.039 | 0.055 | 0.074 | 0.051
$[$Cr/Fe$]$i | 20 | +0.067 |  $-$0.043 |  $-$0.005 | +0.005 | 0.042 | 0.044 | 0.033
$[$Mn/Fe$]$i | 7 | +0.045 |  $-$0.030 | +0.009 |  $-$0.011 | 0.071 | 0.073 | 0.025
$[$Co/Fe$]$i | 5 |  $-$0.001 | +0.006 |  $-$0.002 |  $-$0.009 | 0.085 | 0.085 | 0.020
$[$Ni/Fe$]$i | 36 |  $-$0.005 | +0.015 | +0.003 | +0.008 | 0.032 | 0.033 | 0.008
$[$Zn/Fe$]$i | 1 |  $-$0.035 | +0.006 |  $-$0.005 |  $-$0.013 | 0.189 | 0.190 | 0.044
$[$Y/Fe$]$i | 2 | +0.122 |  $-$0.023 |  $-$0.005 |  $-$0.012 | 0.134 | 0.135 | 0.069
$[$Zr/Fe$]$i | 5 | +0.105 |  $-$0.012 |  $-$0.001 |  $-$0.006 | 0.085 | 0.086 | 0.046
$[$Zr/Fe$]$ii | 1 | +0.089 |  $-$0.034 |  $-$0.003 | +0.015 | 0.189 | 0.190 | 0.060
$[$Ba/Fe$]$ii | 3 | +0.126 |  $-$0.062 | +0.006 |  $-$0.056 | 0.109 | 0.130 | 0.054
$[$La/Fe$]$ii | 2 | +0.128 |  $-$0.043 |  $-$0.002 |  $-$0.029 | 0.134 | 0.139 | 0.050
$[$Ce/Fe$]$ii | 1 | +0.122 |  $-$0.049 |  $-$0.005 |  $-$0.001 | 0.189 | 0.190 | 0.047
$[$Nd/Fe$]$ii | 5 | +0.122 |  $-$0.045 |  $-$0.004 |  $-$0.026 | 0.085 | 0.092 | 0.049
$[$Eu/Fe$]$ii | 2 | +0.109 |  $-$0.044 |  $-$0.004 | +0.013 | 0.134 | 0.135 | 0.041
Table 3: Sensitivities of abundance ratios to variations in the atmospheric parameters and to errors in the equivalent widths and errors in abundances for stars of NGC 6388 observed with GIRAFFE. Element | Average | Teff | $\log g$ | [A/H] | $v_{t}$ | EWs | Total | Total
---|---|---|---|---|---|---|---|---
| n. lines | (K) | (dex) | (dex) | kms-1 | (dex) | Internal | Systematic
Variation |  | 50 | 0.20 | 0.10 | 0.10 |  |  |
Internal |  | 6 | 0.04 | 0.04 | 0.22 | 0.03 |  |
Systematic |  | 57 | 0.06 | 0.02 | 0.01 |  |  |
$[$Fe/H$]$i | 20 | +0.014 | +0.032 | +0.022 |  $-$0.060 | 0.027 | 0.067 | 0.020
$[$Fe/H$]$ii | 3 |  $-$0.079 | +0.118 | +0.040 |  $-$0.018 | 0.070 | 0.079 | 0.097
$[$O/Fe$]$i | 2 |  $-$0.001 | +0.049 | +0.017 | +0.057 | 0.086 | 0.104 | 0.025
$[$Na/Fe$]$i | 2 | +0.025 |  $-$0.054 |  $-$0.009 | +0.031 | 0.086 | 0.092 | 0.040
$[$Mg/Fe$]$i | 2 |  $-$0.015 |  $-$0.009 |  $-$0.012 | +0.050 | 0.086 | 0.100 | 0.018
$[$Al/Fe$]$i | 2 | +0.016 |  $-$0.054 |  $-$0.020 | +0.033 | 0.086 | 0.093 | 0.037
$[$Si/Fe$]$i | 3 |  $-$0.056 | +0.014 | +0.001 | +0.047 | 0.070 | 0.085 | 0.064
$[$Ca/Fe$]$i | 5 | +0.048 |  $-$0.064 |  $-$0.016 | +0.007 | 0.055 | 0.057 | 0.058
$[$Sc/Fe$]$ii | 2 | +0.068 |  $-$0.033 |  $-$0.005 |  $-$0.015 | 0.086 | 0.088 | 0.078
$[$Ti/Fe$]$i | 3 | +0.069 |  $-$0.035 |  $-$0.019 |  $-$0.019 | 0.070 | 0.074 | 0.080
$[$Ni/Fe$]$i | 11 |  $-$0.022 | +0.019 | +0.001 | +0.035 | 0.037 | 0.051 | 0.026
$[$Zr/Fe$]$i | 4 | +0.084 |  $-$0.020 |  $-$0.019 | +0.037 | 0.061 | 0.073 | 0.097
$[$Ba/Fe$]$ii | 1 | +0.090 |  $-$0.067 | +0.005 |  $-$0.060 | 0.122 | 0.137 | 0.105

We varied one parameter at a time by the amount listed in the first row and
repeated the abundance analysis for all stars. The averages provide the
sensitivities of abundance ratios to changes in the atmospheric parameters
(i.e. $\Delta{\rm[el/Fe]}/\Delta{\rm(par)}$) and are listed in the main body
of the tables. Star-to-star (internal) errors and systematic errors in each
parameter are in the second and third rows. The typical internal errors in
abundances due to the measurements of EWs (0.02 dex and 0.03 dex for UVES and
GIRAFFE, respectively) are estimated as the average rms scatter in Fe
abundance divided by the square root of the typical number of measured Fe
lines (100 and 20 lines from UVES and GIRAFFE spectra, respectively). Finally,
the total internal and systematic errors in the derived abundances are
obtained by summing in quadrature the contributions of individual error
sources, weighted according to the errors relative to each parameter.

Abundances for individual stars are listed in the Appendix A. For tables
containing elements from the whole UVES+GIRAFFE sample, only an excerpt is
provided as a guidance of the content. The complete tables can be found at
CDS, Strasbourg.

##  4 Proton-capture elements

Sample and observations in NGC 6388 were purposely tailored to measure the
largest set of elements involved in proton-capture reactions resulting in the
network of correlations and anti-correlations observed in GC stars. Starting
from the lightest species (O, Na, Table A.1) up to the heavier Ca (Table A.2)
and Sc (Table A.3), almost all the interested elements are sampled (potassium
is still missing). We also show how, in NGC 6388, these elements define the
classical set of anti-correlations and correlations typical of massive GCs,
with a couple of notable exceptions that we discuss below.

###  4.1 The Na-O anti-correlation

The main chemical signature of multiple populations in any GC is evident in
Figure 2, where we show the classical anti-correlation between Na and O
abundances in 183 stars of NGC 6388. Stars are grouped (and colour-coded)
according to their chemical compositions following the PIE scheme as defined
in Carretta et al. (2009b). The stars with primordial composition (P, in red
in Figure 2) are those with the lowest Na abundances, comprised between
[Na/Fe]min111Conservatively, we excluded from this estimate the three stars
with the lowest Na values, which seem to be outliers separated from the bulk
of stars along the anti-correlation. ($=-$0.1 dex) and [Na/Fe]${}_{\rm
min}+0.3$ dex. This choice guarantees the interception of almost the totality
of stars with pristine composition, since 0.3 dex is typically a 3$\sigma$
star-to-star error in spectroscopic abundance analysis. The remainder of
stars, with chemical composition altered by FG polluters (whatever they were)
are dubbed second-generation (SG) stars and divided into two subgroups: those
with intermediate composition I (if the ratio [O/Na]$>-0.9$ dex, in blue in
Figure 2) and those with extreme E composition ([O/Na]$<-0.9$ dex, green
points).

Although more sophisticated methods of statistical cluster analysis (Valle et
al. 2021) only retrieve the main blocks of FG and SG stars, our finer
separation into I and E stars is not arbitrary, but motivated by the existence
of long tails in [O/Na] distributions observed in a number of GCs, such as NGC
2808 and NGC 5904 (M 5) (see Carretta et al. 2009b). In turn, the fractions of
stars within each group are correlated to physical properties of GCs,
including the total mass (see Table 4 and Figure 14 in Carretta et al. 2010a).
We also double checked the PIE attribution using a k-means algorithm. We
retrieved three groups almost identical to the ones adopted here, with minor
differences at the group edges, corroborating the adoption of the same
procedure of all papers of our series on Na-O anti-correlations for the sake
of homogeneity.

Figure 3: Anti-correlation of Na and O abundances in stars of NGC 6388 in
different evolutionary phases. In all three panels on the left, open circles
are our total sample. In the top left panel, we plot RGB stars as filled
circles. In the middle and lower panels, we superimpose values for RHB and AGB
stars, in red and blue filled circles, respectively. In the right column, we
show the observed distributions of the [O/Na] abundance ratios for RGB, RHB,
and AGB stars, from top to bottom, respectively.

In the present work, we confirm the fractions we found in Carretta and
Bragaglia (2018) for NGC 6388, reducing the associated Poisson errors by half
thanks to the much enlarged present sample. A third of stars show a
composition with O and Na levels identical to those of field Galactic stars of
similar metallicity (P=$29\pm 4\%$). The remaining SG stars are then split
into two components, the bulk consisting of stars with intermediate
composition (I=$54\pm 5\%$). Finally, we confirm once again that in NGC 6388
there is a noticeable fraction of stars showing an extremely modified
composition (E=$16\pm 3\%$). These computations are made over 184 stars,
because although two stars are lacking O abundances, one of them (l63p307) has
[Na/Fe]$=+0.102$ dex, a value that firmly places it in the P population.

The fraction E also includes the two stars with the highest Na abundances
([Na/Fe]$>1.2$ dex). They are members of NGC 6388 according to both the RV and
metallicity measured on our spectra and $Gaia$ astrometry (Vasiliev and
Baumgardt 2021). They lie nicely on the locus of the Na-O anti-correlation and
are also the stars with the lowest O abundances in our sample ([O/Fe]$<-0.6$
dex, see Table 6). We note that even if all the six stars flagged as possible
non-members in APOGEE DR17 were excluded, our results on multiple stellar
populations and their respective composition in NGC 6388 would not change.
Their exclusion would only change the fraction of P, I, and E stars within
their associated Poissonian errors.

Formally, we observe a monotonic increase of the metal abundance along the
sequence PIE. The average [Fe/H] of the three groups raises from $-0.493$ dex
for 55 P stars to $-0.485$ dex and $-0.481$ dex for I (100 objects) and E
stars (30 objects), respectively. Although statistically not significant, this
progression is what is expected following an increasing helium abundance in SG
stars of the I and E groups. The effect is to increase the strength of
metallic lines (see Böhm-Vitense 1979), and that of neutral lines more than
that of ionised lines. This is clearly confirmed by the iron difference
between P and E stars, which we find to be 0.006 dex for Fe ii, half of that
derived from Fe i lines. Although the combination of tiny differences in
abundance between the groups and internal errors associated with iron prevents
a stronger statement, these results qualitatively conform to expectations.

###  4.2 Horizontal and asymptotic giant branch stars

Our sample in NGC 6388 includes a small number of stars that are not on their
first ascent on the red giant branch (RGB), since our selection criterion for
membership was exclusively based on RVs (see Sect.2). This provides the
opportunity to homogeneously compare the behaviour of multiple stellar
populations in different evolutionary phases. The main spectroscopic feature,
the anti-correlation of Na-O abundances, is examined in Fig. 3. In the panels
of the left column, we highlight the abundances of ten red HB stars stars
(RHB, red filled circles) and 17 asymptotic giant branch stars (AGB, blue
filled circles) on the overall Na-O anti-correlation obtained for the whole
sample. While there are no differences in the metal abundances (the average
[Fe/H] values agree within 0.003 dex), as expected, the different patterns for
proton-capture species are evident.

Also this second finding is hardly surprising. At odds with the PIE
classification of RGB stars (fractions 25, 56, and 19%, respectively), when we
only consider RHB stars we find that the vast majority (70%) are in the P
component, with only 30% falling in the I group of SG stars (just the reverse
proportion with respect to RGB stars). Moreover, no RHB star is found with the
highly modified composition typical of the E component.

Our findings agree with the well-known scenario where stars with different He
contents are located on distinct portions of the HB. Spectroscopic detections
of He in HB stars are hard to obtain, but since He variations are correlated
to star to star abundance variations in light elements (see Gratton et al.
2004, 2012a, 2019; Bastian and Lardo 2018) it is much easier to prove this
scenario by analysing the pattern of other elements, including Na and O (see
Gratton et al. 2011, 2012b, 2013, 2014, 2015, Villanova et al. 2009, 2012,
Marino et al. 2011a). The almost complete segregation of Na-poor/O-rich stars
on the red extreme of the HB is a natural consequence of the strong
correlation linking the chemical variations in proton-capture species in GC
stars and the highest temperature that may be reached on the ZAHB in each GC,
as found by Carretta et al. (2007b). In turn, these observations agree with
the prediction by D’Antona et al. (2002) that variations in helium (or their
proxies, variations in light elements) are a key ingredient to explain the HB
morphology. In NGC 6388, all the analysed HB stars belong to the RHB, so they
behave almost like a simple stellar population, with the [O/Na] ratio peaking
at a high value (see Fig. 3, right column, middle panel).

The lower panels in Fig. 3 show the situation for the 17 AGB stars in our
sample. They are 11% of the RGB stars in our sample, so we are confident that
we did not miss a significant fraction of objects in this evolutionary phase,
since AGB stars are rarer (see discussion in Gratton et al. 2010 and their
Table 1). From the lower left panel in Fig. 3 and the number counts, we see
that the fraction of SG stars with intermediate composition (the I component)
is similar in RGB and AGB stars (56% and 53%, respectively). The largest
differences are for the primordial P component (25% for RGB and 41% for AGB
stars) and, in particular, for the extreme E component with only one AGB star
classified as such (18% for RGB and 6% for AGB stars). This agrees with
previous findings. Gratton et al. (2010) discussed the well-established lack
of CN-strong objects among AGB stars in GCs (e.g. Norris et al. 1981, Campbell
et al. 2010) and correlated it with the HB properties, in particular with the
expectation that the less massive HB stars do not even begin their AGB phase
(AGB manqué). That means that the most He-rich (Na-rich and O-poor) stars in
GCs, those we would classify as E, do not reach the AGB. More recent results,
sometimes contradictory, and further discussions on the different fractions of
FG and SG in RGB and AGB stars can be found, for instance, in Marino et al.
(2017), Wang et al. (2017), and MacLean et al. (2018).

Although the distributions of both RHB and AGB stars peak at higher values
than RGB stars (right column in Fig. 3), the average [O/Na] for AGB stars is
shifted to a lower value than for RHB stars. A look at the panels in the left
column shows that the effect is not due to different Na abundances (mean
[Na/Fe]$=+0.141$ rms=0.205 dex in RHB and +0.275 rms=0.206 dex in AGB), but
rather to an actual difference in the average O content (mean [O/Fe]$=+0.199$
rms=0.140 dex in RHB and $-0.023$ rms=0.173 dex in AGB). We used Student’s and
Welch’s tests with the null hypothesis that the groups of RHB and AGB stars
are extracted from a distribution with the same means. This hypothesis cannot
be rejected in the case of Na (two-tail probability $p=0.114$, 25 d.o.f.),
whereas it can be safely rejected regarding the [O/Fe] average content
($p=1.2\times 10^{-3}$), confirming that we are probably observing a real
difference. The above discussion also confirms well-known behaviours in NGC
6388 due to the global multiple population phenomenon, without the need to
invoke systematic or model-dependent effects.

Figure 4: Cumulative distributions of radial distances for stars in NGC 6388
in different evolutionary phases: RGB (black solid line), RHB (red short-long
dashed line), and AGB (blue dashed line). The probabilities of the Kolmogorov-
Smirnov test are labelled in the panel.

A further difference concerns the radial distribution of stars in different
evolutionary phases. In Fig. 4, we plot the cumulative distributions of radial
distances for stars in our sample, together with the results of a Kolmogorov-
Smirnov test to ascertain statistical differences between different groups. We
find that RHB stars are more externally distributed in the GC than both RGB
and AGB stars. Due to the limited size of the RHB and AGB samples, we prefer
not to expand further on this. On the other hand, we do not find any relevant
difference in the concentration of stars in the P, I, and E groups.

Figure 5: Extension of Na-O anti-correlation in 25 GCs of our FLAMES survey as
measured by the inter-quartile range of the [O/Na] ratio, as a function of the
total mass (using the total cluster absolute magnitude as proxy). NGC 6388 is
indicated with an open circle.

Figure 6: Mutual correlations and anti-correlations among proton-capture
elements in stars of NGC 6388. Internal, star to star errors can be found in
Table 2 and Table 3.

###  4.3 The peculiar Na-O anti-correlation of NGC 6388

The inter-quartile range of the [O/Na] ratio was proposed by Carretta (2006)
as a robust estimate of the extension of the Na-O anti-correlation because it
is more insensitive to outliers than other traditional indicators (e.g. the
intrinsic spread as measured by the rms scatter). From the 183 stars in NGC
6388 with measured O and Na abundances, we obtained IQR[O/Na]=0.638, which
agrees with the previous value (0.644) derived from a sample of only 49 stars
in Carretta and Bragaglia (2018). We then confirm that NGC 6388 belongs to the
group of massive GCs whose extent of the Na-O anti-correlation is too short
with respect to what expected on the basis of their total mass. In Fig. 5, we
show the IQR[O/Na]-mass relation from our FLAMES survey (see Carretta et al.
2010a), where we adopt the cluster total absolute magnitude $M_{V}$ as a
model-independent proxy of the present-day GC total mass. When we plot the 25
GCs of our FLAMES survey homogeneously analysed, we note that NGC 6388 falls
in a group of four GCs standing out of the main trend, the other ones being
NGC 6441, NGC 104 (47 Tuc), and NGC 7078 (M 15).

This behaviour is not dependent on our abundance analysis, since it is
confirmed by literature data such as 104 giants for NGC 104 (Cordero et al.
2014) or 18 giants in M 15 (Sneden et al. 1997). At present, we have no
explanation for these four GCs having an extension shorter than expected for
the Na-O anti-correlation, given their rather large total mass.

###  4.4 The other proton-capture elements

We determined the abundances of other elements (Mg, Al, Si, Ca, and Sc)
involved in the network of proton-capture reactions acting to modify the
primordial chemical composition of GC stars. The mutual relations among these
species in NGC 6388 are illustrated in Fig. 6. Internal errors can be found in
Table 2 and Table 3.

Figure 7: Run of different element abundances among PIE groups in NGC 6388,
indicated by the average differences with respect to the P group, whose
composition is representative of the primordial abundances in the proto-GC.

The trends visible in this figure point out once again that the whole pattern
we see among light elements in GC stars is clearly due to some form of
nucleosynthesis, because the variations in the chemical composition are not
randomly distributed, but follow well-known evolutions. There could be some
residual uncertainties related to the statistical grouping of stars, but the
three stellar populations are consistently ordered according to all
abundances. Enhancement in Al is accompanied by a depletion in Mg, Na is lower
when O is higher, and so on.

These trends are well summarised in Fig. 7, where we represent the average
abundances of different elements in the three components P, I, and E in NGC
6388. In this figure, we plot the average differences with respect to the P
group, so that the trends are a direct representation of the changes in the
chemical composition due to the formation of multiple populations with respect
to the floor of primordial abundances in the proto-GC.

Not surprisingly, we retrieve the large enhancements in Na and Al contents,
paralleled by a noticeable O depletion, and a smaller decrease in Mg
abundance. Both the Si-Al correlation and the Mg-Si anti-correlation act as a
precision thermometer that probes the inner temperature reached by the FG
polluters. The small but steady increase in the [Si/Fe] ratio when progressing
from P to I and to E component (and the simultaneous decrease in [Mg/Fe]) has
been explained (e.g. Karakas and Lattanzio 2003; Yong et al. 2005) by the
leakage from the Mg-Al cycle on 28Si. This occurs when the two reactions
27Al(p,$\gamma$)28Si and 27Al(p,$\alpha$)24Mg switch as relevance at a well-
defined temperature of $\sim 65$ MK (see Arnould et al. 1999, their Figure 8).
Concerning NGC 6388, this means that whatever the polluters were, in the early
evolution of the primordial population of the cluster they were able to reach
such a temperature.

In Carretta and Bragaglia (2019), we investigated the high extreme of the
temperature range possibly reached by the FG polluters in order to reproduce
the pattern of heavier proton-capture species, such as Sc and Ca. The idea was
to ascertain in NGC 6388 the presence (or lack) of the anti-correlations found
between the abundances of both Sc and Ca with those of Mg in the massive
cluster NGC 2808 (Carretta 2015). Their existence in NGC 2808 was interpreted
in the same framework of proton-capture reactions at very high temperature
used by Ventura et al. (2012) to explain the unique pattern of the K-Mg anti-
correlation in NGC 2419 (see Cohen and Kirby 2012, Mucciarelli et al. 2012).
The variations of Sc and Ca in NGC 2808 and the anti-correlation K-Mg, later
found by Mucciarelli et al. (2015), showed that this extreme regime of
H-burning can be traced even in more normal GCs than NGC 2419. However, in
Carretta and Bragaglia (2019) we were able to show that the abundances of Sc
and Ca in NGC 6388 cannot be distinguished from those of field stars of
similar metallicity. The latter represent fairly well the unpolluted,
primordial population of stars where only the effects of supernova
nucleosynthesis can be tracked. In turn, these observations mean that the FG
polluters in NGC 6388 were not able to reach the temperature of about 150 MK,
above which seed species including Ar and K start to be affected, allowing the
production of elements such as Sc and Ca (see Prantzos et al. 2017). The
constraints from the Si Al variations and the evidence of Sc and Ca being
largely unaffected would pinpoint a narrower range of 100-120 MK for the
temperature reached in the candidate polluters if these can be identified with
massive AGB stars (see D’Antona et al. 2016). In Fig. 7, we note a slight
enhancement in Sc and Ca in the extreme E fraction of stars in NGC 6388, but
it is not significant.

Table 4: Tests on (anti-)correlations. rel. | $r_{s}$ | $r_{p}$ | Nr. | t | prob.
---|---|---|---|---|---
| Spearm. | Pearson | stars | Stud. | two-tails
Al-O |  $-$0.602 |  $-$0.638 | 160 | 9.477 | $<1.0\times 10^{-6}$
Na-O |  $-$0.676 |  $-$0.693 | 183 | 12.342 | $<1.0\times 10^{-6}$
Na-Mg |  $-$0.242 |  $-$0.267 | 184 | 3.738 | $2.5\times 10^{-4}$
Na-Al | +0.691 | +0.704 | 160 | 12.460 | $<1.0\times 10^{-6}$
Na-Si | +0.443 | +0.443 | 184 | 6.666 | $<1.0\times 10^{-6}$
Al-Mg |  $-$0.255 |  $-$0.264 | 160 | 3.440 | $7.4\times 10^{-4}$
Si-Mg |  $-$0.286 |  $-$0.305 | 183 | 4.309 | $2.7\times 10^{-5}$
Si-Al | +0.275 | +0.266 | 159 | 3.458 | $7.0\times 10^{-4}$
O-Si |  $-$0.479 |  $-$0.468 | 182 | 7.105 | $<1.0\times 10^{-6}$
O-Mg | +0.397 | +0.402 | 182 | 5.890 | $<1.0\times 10^{-6}$
Ca-Mg | +0.032 | +0.041 | 184 | 0.554 | 0.581
Sc-Mg |  $-$0.067 | +0.024 | 184 | 0.324 | 0.746

The scenario and the conclusions discussed above strongly support statistics
as shown in Table 4, where we list the parameters of linear regressions
through the different relations among light elements shown in Fig. 2 and Fig.
6. We tested the level of significance for each regression, reporting the
results in the last two columns of Table 4. The two-tail probabilities listed
show that all the relations involving O, Na, Mg, Al, and Si in NGC 6388 are
real with a high level of significance. On the contrary, the two anti-
correlations between Mg and Sc, and Mg and Ca are not significant. We can then
confirm, based on robust evidence, that in NGC 6388 the FG polluters, whatever
they were, likely reached a maximum inner temperature restricted to a narrow
range between 100 and 120-150 MK, as found in Carretta and Bragaglia (2019).

Figure 8: Observed Al-O, Na-O, Al-Mg, and Si-Mg anti-correlations in NGC 6388
(empty red circles). Black filled points are the average values of each
element in the P, I, E groups. Solid lines are a simple dilution model
anchored to the primordial P and extreme E components.

###  4.5 Light elements, polluters, and dilution

In Carretta and Bragaglia (2018), we tried to ascertain how many classes or
kinds of polluters may have been contributing to the chemical budget of NGC
6388. We performed this exercise using the same approach adopted in Carretta
et al. (2012) to study the discrete components in NGC 6752.

We started with the simple dilution model (illustrated, e.g. in Carretta et
al. 2009b), where we reproduce the chemical pattern of the various stellar
populations by mixing the composition of the E group with different fractions
of primordial gas, which is simply represented by the composition of the P
component. This means that the I population would be obtained by mixing a
fraction $dil$ of matter with E-like composition together with a fraction
whose composition is P-like. It follows that if only a class of polluters was
in action, then the value of $dil$ should be the same for all elements:

$dil={{[A({\rm X})_{I}-A({\rm X})_{P}]}\over{[A({\rm X})_{E}-A({\rm
X})_{P}]}}={{[A({\rm Y})_{I}-A({\rm Y})_{P}]}\over{[A({\rm Y})_{E}-A({\rm
Y})_{P}]}},$ (1)

where $A(X)$ and $A(Y)$ are the abundances in number of atoms of elements X
and Y.

The results reached in Carretta and Bragaglia (2018) were not conclusive. The
main obstacle was the availability of Al abundances only for the limited
sample of 24 stars with UVES spectra. In turn, this provided two different
possible divisions in the P, I, and E groups, depending on whether the groups
were based on the Al-O or the Na-O plane. As a consequence, we cannot not
entirely exclude the possibility that more than a single class of polluters
could be necessary to produce the observed composition of the I component.

With the present large sample of stars for which homogeneous abundances of
several proton-capture species are obtained in NGC 6388, this problem is
solved. For instance, our sample of stars with derived Al abundances is
increased by a factor 6.7, with respect to Carretta and Bragaglia (2018).
Thus, we repeated the exercise.

We found that in the chemical planes Al-O, Na-O, Al-Mg, and Si-Mg a simple
dilution model, anchored to the mean values for P and E stars, now nicely
passes through the average value of the intermediate I population, as shown in
Fig. 8 and at variance with what was found in Carretta and Bragaglia (2018).
We thus conclude that the hints of multiple classes of polluters seen in
Carretta and Bragaglia (2018) were likely due to the limited size of the
available samples (especially Al abundances). Our present results are
compatible with the existence of a single class of FG polluters able to
reproduce the observed characteristics of multiple stellar populations in NGC
6388, when their ejecta were mixed with variable amount of pristine gas.

Figure 9: Summary of the dilution factors for different elements (indicated by
the atomic number. In the three panels we indicate the average values (solid
lines) and the associated $\pm 1\sigma$ (dashed line) resulting: i) from all
the five considered species, ii) neglecting Mg (empty symbol with large error
bar), and iii) considering only O, Na, and Al, (from top to bottom,
respectively).

A summary of the above exercise is given in Fig. 9, where we plot the
resulting dilution factors $dil$ for each considered species (indicated by the
atomic number). The value corresponding to Mg is indicated with a different
symbol (empty circle) to stress the large associated error. The same behaviour
was already noted in Carretta and Bragaglia (2018) and tentatively attributed
to the small variations in the abundance of Mg with respect to the primordial
value. However, we note that star-to-star variations in the Si content are
also small, yet its derived dilution factor is compatible with the others
species. At present, we do not have a satisfactory explanation for the
behaviour of the dilution of Mg.

The values we found for $dil$ are $0.41\pm 0.09$, $0.30\pm 0.06$, $0.14\pm
0.38$, $0.30\pm 0.06$, and $0.48\pm 0.16$ for O, Na, Mg, Al, and Si,
respectively. Apart from Mg, all these values lie within $\pm 1\sigma$ from
the average values labelled in Fig. 9, a strong indication for a unique class
of polluters acting in NGC 6388.

##  5 The chemical composition of NGC 6388 in context

###  5.1 $\alpha$-capture elements

Thanks to the spectral range of UVES spectra and to the large size of our
overall dataset, we were able to analyse the full set of $\alpha$-elements in
NGC 6388 for a large number of stars. Both products of hydrostatic burning (O,
Mg) and explosive nucleosynthesis (Si, Ca, Ti) were analysed. Their run as a
function of the effective temperature is summarised in Fig. 10 (red and blue
points for stars with GIRAFFE and UVES spectra, respectively). Internal error
bars for this figure are listed in Table 2 and Table 3 for abundances derived
from UVES and GIRAFFE spectra, respectively. No dependence of abundance on
Teff is present for any of these elements, over a range of luminosity of more
than four magnitudes.

Figure 10: Abundance ratios [Mg/Fe], [Si/Fe], [Ca/Fe], [Ti/Fe]i, and [Ti/Fe]ii
for stars in NGC 6388 as a function of the effective temperature. For the
present work, red filled circles indicate stars with GIRAFFE spectra, whereas
stars with UVES spectra are represented with blue filled squares. Large black
circles indicate the four stars analysed by Minelli et al. (2021a), and green
filled triangles are stars in APOGEE DR17 flagged as members of NGC 6388.

Since the key work by Tinsley (1979), abundance ratios were used as probe of
the birth properties of stars. The interplay between star formation and
stellar lifetimes (driven by the mass) means that elements from different
nucleosynthetic sites can be combined into abundance ratios useful to uncover
the primordial environment where the observed stars were born. In particular,
the $\alpha$-Fe plane is a privileged indicator, since a similarity in this
plane for two stellar populations indicates a similar chemical evolution of
the systems. However, the metallicity at which the so-called knee due to the
onset of Type Ia SNe occurs depends on the system’s stellar mass and on the
efficiency of star formation in the progenitor galaxy (see, e.g. the review by
Tolstoy et al. 2009). Owing to the impact of these abundance patterns on the
clues for the origin of NGC 6388, our results on $\alpha$-elements deserve a
detailed discussion.

The average amount of $\alpha$-elements in NGC 6388 is overabundant with
respect to the solar level, which is compatible with a chemical composition
dominated by the contribution of type II SNe acting before a significant
number of SN Ia started to add increasing amounts of iron and lower the ratio
[$\alpha$/Fe]. This evidence, together with the time delay expected for the
bulk explosions of SN Ia, is also consistent with GCs in the bulge being old
and nearly coeval to halo GCs, with similar levels of $\alpha$-element
overabundance.

The average [Si/Fe] ratio reflects the typical overabundance with respect to
the solar values found also in the majority of GCs and for the other
$\alpha$-elements in Fig. 10. We found a mean [Si/Fe] of 0.31 dex from 150
stars with GIRAFFE spectra and 0.35 dex from 34 stars observed with UVES,
after combining the results for the 12 new stars of the present study with the
sample analysed in Carretta and Bragaglia (2018). We thus confirm the high
value we found in our first analysis for NGC 6388 (Carretta et al. 2007a),
which is at odds with the low values derived by the APOGEE DR16
($<$[Si/Fe]$>=-0.03\pm 0.1$ dex, as reported in Horta et al. 2020) and DR17
($<$[Si/Fe]$>=+0.045\pm 0.061$ dex), from 53 stars flagged as members of NGC
6388 (Abdurro’uf et al. 2022).

Conversely, in NGC 6388 we found a value ($\sim 0.07$ dex, see Table 1) for
the average [Ca/Fe] ratio lower than for other $\alpha$-elements. However,
this is fully consistent with our previous analyses (Carretta et al. 2007a,
Carretta and Bragaglia 2018). Gratton et al. (2006) observed a similarly low
value in the massive bulge cluster NGC 6441, and they advanced the hypothesis
that this deficiency in Ca could be an artefact of the analysis, due to using
strong lines in bright and cool giants. However, this explanation hardly
applies to our sample, which includes stars spanning a range of about 1700 K
in Teff.

A comparison of the abundance of $\alpha$-elements in NGC 6388 from both
optical and infrared studies is provided in Fig. 10. To our derived abundances
we superimposed the four stars reanalysed by Minelli et al. (2021a) (large
black circles) and the about 50 stars flagged as members of NGC 6388 in the
APOGEE DR17 (filled green triangles). Looking at this Figure, clear offsets
are present between the abundances from infrared and optical spectra, with
both optical analyses giving consistent results (at least for the
$\alpha$-elements, see Carretta and Bragaglia 2022b and the next section). In
particular, Mg and Si abundances from APOGEE are lower than we find here,
whereas a good agreement is found for Ca abundances. For Ti, we are instead
seeing a larger spread in APOGEE abundances.

Part of the offsets may be explained by differences in the adopted solar
reference abundances. They are not given in the DR17 paper, but assuming that
they are not changed from DR16, we used the solar values provided by Smith et
al. (2021) for DR16 to compute the corrections needed to bring infrared data
on our solar scale (respectively +0.19, +0.11. +0.13, and -0.01 dex for Mg,
Si, Ca, and Ti). The overall agreement would improve for Mg and Si, but it
would worsen for Ca.

Table 5: Averages and rms scatters for $\alpha$-element ratios in NGC 6388 and
NGC 6441.

abund. | this work | this work | M21 | WKA07 | APOGEE | NGC 6441 | NGC 6441 | M20 | M20
---|---|---|---|---|---|---|---|---|---
ratio | UVES | GIRAFFE |  |  | DR17 | G06 | G07 | all | S/N$>$70
$[$Mg/Fe$]$ | +0.212 0.049 | +0.221 0.052 |  | +0.259 0.175 | +0.059 0.056 | +0.34 0.09 | +0.38 0.14 | +0.069 0.167 | +0.152 0.140
$[$Si/Fe$]$ | +0.350 0.063 | +0.309 0.061 | +0.240 0.100 | +0.368 0.158 | +0.045 0.061 | +0.33 0.11 | +0.41 0.19 | +0.118 0.190 | +0.223 0.219
$[$Ca/Fe$]$ | +0.062 0.049 | +0.069 0.046 |  $-$0.018 0.071 |  $-$0.185 0.223 | +0.032 0.074 | +0.03 0.04 | +0.21 0.19 | +0.070 0.193 | +0.130 0.243
$[$Ti/Fe$]$i | +0.295 0.086 | +0.270 0.098 | +0.115 0.087 |  $-$0.106 0.120 | +0.132 0.177 | +0.29 0.10 | +0.33 0.20 |  |
$[$Ti/Fe$]$ii | +0.244 0.086 |  |  | +0.294 0.128 | +0.304 0.221 | +0.33 0.14 |  |  |

  * M21:

Minelli et al. (2021a).

  * WKA07:

Wallerstein, Kovtyukh, Andrievsky (2007).

  * G06,G07:

Gratton et al. (2006: UVES); Gratton et al. (2007: GIRAFFE).

  * M20:

Mészáros et al. (2020).

For a more quantitative comparison, in Table 5 we report the averages and rms
scatters for $\alpha$-elements from the present and other studies: Minelli et
al. (2021a); Wallerstein et al. (2007), who studied eight cool giants in NGC
6388 (we report the values from their analysis with photometric gravities);
and the mean values from APOGEE DR17. As a further comparison, we added the
average values derived from UVES and GIRAFFE spectra by Gratton et al. (2006,
2007, respectively) in NGC 6441, often considered a twin GC of NGC 6388, with
similar origin and characteristics. Finally, the last two columns provide the
values for NGC 6388 derived by M20, both from all available stars and by
selecting stars with high S/Ns according to their criteria.

The values of [Mg/Fe] and [Si/Fe] from APOGEE DR17 are lower than those of the
analyses based on optical spectra. They are also lower than the values given
by M20, in particular when only high-S/N stars are used in their reanalysis
with BACCHUS of an earlier SDSS/APOGEE release. On the contrary, the Ca level
seems to be actually low in these GCs, with the notable exception of NGC 6441
from GIRAFFE spectra (see, however, the warning in Carretta and Bragaglia
2021, 2022b about the time on targets for NGC 6441 and the relative low S/N in
that cluster).

Figure 11: Comparison of abundances of $\alpha$-elements with literature data.
Left panels: Mean abundance ratios [Mg/Fe], [Si/Fe], [Ca/Fe], and [Ti/Fe]i
(from top to bottom) as a function of metallicity from the homogeneous
analysis of GCs in our FLAMES survey (filled blue points). Black filled
circles are the average ratios in four metal-rich GCs from Muñoz and
collaborators. The larger red point indicates NGC 6388 (present work). Grey
and cyan open circles are field stars in the disc and bulge of the Milky Way,
respectively, from different studies. References for all the data used in this
figure are reported in Appendix B, Tables B.1 and B.2. Green filled squares
are field stars in the dSph galaxies Fornax (Letarte et al. 2010, Lemasle et
al. 2014) and Sagittarius (Minelli et al. 2021b). Right panels: Enlargement
around the metallicity of NGC 6388.

In Fig. 11, we also compare the average abundances of Mg, Si, Ca, and Ti from
the present work with the abundances in the GCs homogeneously analysed in our
FLAMES survey (see Carretta et al. 2006 and following studies, referenced in
Table B.1), represented by blue filled circles with error bars. These data
have been complemented by four metal-rich GCs (three bulge clusters and one
disc cluster) from Muñoz and collaborators. As reference, in the same figure
we also plot field stars of the Galaxy, both in the halo and disc components
(open grey circles) and in the bulge (cyan open circles), from several
literature studies.

For the sake of clarity, all references to the data used are provided in
Appendix B (Tables B.1 and B.2), where we also give the identification of each
GC and its [Fe/H] value. Finally, we also plot abundances for the two more
massive dwarf spheroidal galaxies associated with the MW, one still distinct
(Fornax: Letarte et al. 2010, Lemasle et al. 2014) and the other already
accreted and disrupting in the MW (Sagittarius, Minelli et al. 2021b). These
stars are used as a robust benchmark for the pattern of $\alpha$-elements in
external systems of lower total mass than the Galaxy. The combination of lower
star formation and chemical evolution shows up in a lower metallicity knee and
resulting underabundance of these elements, once such systems were possibly
accreted into the main Galaxy, as is currently happening to Sgr.

The average abundance of GCs nicely follows the pattern of $\alpha$-elements
of field MW stars, a plateau at low metallicity followed by a decrease when
the metallicity increases after the ‘knee’, signalling the major onset of SN
Ia. As other metal-rich GCs ([Fe/H]$\;\lower
2.58334pt\hbox{$\sim$}\kern-7.75pt\raise 2.79857pt\hbox{$>$}\;-0.7$ dex), NGC
6388 participates in this trend well. In particular, the average [Si/Fe]
ratios in NGC 6388 and in NGC 6441 are in very good agreement with the mean
level shown by halo and disc GCs in the MW and with the abundance of bulge
field stars, both giants (as in GCs) and dwarfs, as shown in detail in Fig.
12, where we display some of the studies used in Fig. 11, in a restricted
range around the mean metallicity of NGC 6388.

Figure 12: Comparison of the average [Si/Fe] ratio in NGC 6388 (red point with
error bar) to six different studies of bulge field stars in the metallicity
range centred on the cluster: Johnson et al. (2014), Alves-Brito et al.
(2010), Bensby et al. (2017), Lucey et al. (2019, 2022), and Duong et al.
(2019). The evolutionary stage of each sample is reported in the panels.

Calcium abundances in NGC 6388 seem to lie at the lower envelope of the field
bulge star distribution (Fig. 11). Overall, NGC 6388 and NGC 6441, together
with the other bulge GCs (with the exception of NGC 6440, Muñoz et al. 2017)
seem to be more compatible with the disc stars. However, the mean ratio
[Ca/Fe] in NGC 6388 is still consistent with several studies focusing on bulge
stars (Fig. 13), again taking into account small offsets related to the
abundance analysis. Furthermore, we remind the reader that in NGC 6388, Ca is
not involved into the network of proton-capture reactions (Carretta and
Bragaglia 2021). This result is quantified in Fig. 7 and Table 4; among all
relations concerning proton-capture elements, the only ones resulting as
statistically not significant are those involving Ca and Sc.

Figure 13: Comparison of average [Ca/Fe] ratio in NGC 6388 (red point with
error bar) to seven different studies of bulge field stars in the metallicity
range centred on the cluster. To the studies listed in the previous figure, we
added the work by Jönsson et al. (2017).

Among the species examined in Fig. 11, only Ti does not show the typical
decline at high metallicity. We remark that this element is not a classical
$\alpha$-element and is located at the boundary between $\alpha$ and Fe-peak
species. Moreover, Ti abundances are only available for less than a half of
the GC sample, our FLAMES survey being focused on species (such as Mg, Si, and
sometimes Ca) involved in the multiple population phenomenon. For the stars
observed with UVES, both neutral and singly ionised lines were available. We
found, on average, a difference of [Ti/Fe]ii \- [Ti/Fe]i = $-$0.051 dex, with
rms=0.096 dex, which is not significant.

From this section, our results and the similarity between NGC 6388 and both
the bulge and disc field of the MW suggest a high-mass environment for the
progenitor of this cluster. Coupled with the chemo-dynamical evidence
presented in a previous paper (CB22b), this is indicative of a formation
within a massive component of the Galaxy, likely the bulge itself. The high
metallicity of NGC 6388 clearly excludes this GC from being associated with a
metal-poor structure recently reported from APOGEE data by Horta et al. (2021)
and considered possibly accreted in the early MW before the presently observed
main Galactic bulge (including its GC population and NGC 6388) was formed.

###  5.2 Fe-peak elements

Together with iron, we derived the abundances of seven species of the iron-
peak group in NGC 6388: [Sc/Fe]ii, [V/Fe], [Cr/Fe]i, [Mn/Fe], [Ni/Fe],
[Co/Fe], and [Zn/Fe]. Most abundances are derived from UVES spectra of the 12
stars newly observed in this work, integrated by the sample of Carretta and
Bragaglia (2018). However, abundances of Sc and Ni are also measured for the
larger sample of stars with GIRAFFE HR13 spectra, using on average 2 and 11
lines for Sc and Ni, respectively.

Figure 14: Abundance ratios [Sc/Fe]ii, [V/Fe], [Cr/Fe]i, [Mn/Fe], [Ni/Fe], and
[Zn/Fe] (from top to bottom) as a function of the effective temperature.
Symbols are as in Fig. 10.

The derived abundances present no dependence on the temperature of stars over
a range of about 1700 K (see Fig. 14), particularly evident in the cases of Sc
and Ni, and are roughly clustered within $\pm 0.10$ dex from the solar value,
apart from V, which shows a slight overabundance, and Mn, which shows a clear
deficiency, also reflected by the pattern of field stars at similar
metallicity (see below).

In Carretta and Bragaglia (2022b) the average abundances of Sc, V, and Zn from
the present work were used to check whether the level of these iron-peak
elements could be used for chemically tagging NGC 6388 (accreted or formed in
situ). Comparing the abundance of stars in NGC 6388 to a large ensemble of
field stars (both disc and bulge stars) at similar metallicity, we were able
to exclude a significant difference between cluster and field stars for all
the three species under scrutiny, thus rejecting the accretion origin.

The in situ nature of NGC 6388 is supported and strengthened by the abundance
of the other elements of the iron group derived here. In Fig. 15, we compare
the mean abundances of Cr, Mn, Co, and Ni obtained for NGC 6388 to several
samples of field stars in the Milky Way, together with the average abundances
for a number of GCs from our FLAMES survey and from Muñoz and collaborators.
No significant difference is found between cluster and field MW stars.

Figure 15: Comparison of average abundance ratios [Cr/Fe], [Mn/Fe], [Co/Fe],
and [Ni/Fe] as a function of metallicity for NGC 6388, the GCs in our FLAMES
survey, some metal-rich GCs from Muñoz and collaborators, and field stars in
the disc and bulge of the Milky Way from several studies. Symbols are as in
Fig. 11, and references are given in Appendix B.

The same pattern is followed well by other classical, less massive bulge GCs,
with the possible exception of Co, whose level seems to exceed the locus
defined by field bulge and disc stars. This is particularly evident for NGC
6528 (Muñoz et al. 2018). However, when NGC 6441 (Gratton et al. 2006) is also
considered among the bulge populations, we may stress again that in general
the iron-peak elements perform poorly, in picking up objects presumably of
accreted origin.

###  5.3 Neutron-capture elements: Zr and Ba

We obtained the abundances of the neutron-capture elements Y, Zr, Ba, La, Ce,
and Nd, sampling the first and second peak of species produced mainly by the
$s$-process. We also measured Eu, typically produced by the $r$-process in the
presence of higher neutron densities. Most abundances were obtained from UVES
spectra, due to their larger spectral coverage. However, we were also able to
derive Ba and Zr abundances from GIRAFFE HR13 spectra by measuring the Ba ii
6141 Å line in all of the 150 stars and up to a maximum of five Zr i lines in
138 stars. We first concentrate on these two elements and defer the analysis
of all others to the next sub-section.

Sources of atomic parameters for these Ba and Zr lines can be found in Table 8
of Gratton et al. (2007), with the exception of the $\log gf$ value for the Ba
ii line, taken instead from Sneden et al. (2003). Due to the strength of this
line and the consequent dependence on microturbulent velocity, abundances of
Ba were derived using the relation as a function of surface gravity for
$v_{t}$ (Worley et al. 2013) and a constant metallicity value ($-0.48$ dex)
for all stars. This expedient allows us to avoid spurious trend of abundances
as a function of the microturbulent velocity (see e.g. Carretta et al. 2015).
The absence of trends with effective temperature for all derived elements is
shown in Fig. 16.

Figure 16: Abundance ratios of neutron-capture elements in NGC 6388 as a
function of the effective temperature. Red filled circles indicate stars with
GIRAFFE spectra, whereas stars with UVES spectra are represented by blue
filled squares.

Large samples of stars with abundances of $s$-process elements are necessary
to explore another defining property of the type II GCs. Together with an
enhancement in metallicity [Fe/H], the metal-rich component is also proposed
to be enriched in neutron-capture elements. This characteristic may manifest
in a range of variations, going from the small amounts observed in NGC 1851
(Carretta et al. 2011) up to the large excesses detected in $\omega$ Cen
(Johnson and Pilachowski 2010; Marino et al. 2011b).

In Carretta and Bragaglia (2022a), we demonstrated that NGC 6388 is a typical
mono-metallic GC, whose intrinsic spread in [Fe/H] is fully compatible with
uncertainties derived from abundance analysis. Hence, finding here that
neither Ba (measured in 185 stars) nor Zr (in 168 stars) show evidence of
enhancement in part of the stars of this GC does not come as a surprise.

As a first test, we split our dataset at [Fe/H]$=-0.488$ dex (the mean
metallicity of the GIRAFFE sample). Then, we compared the cumulative
distribution of Ba (and Zr) abundances for all stars more metal rich and more
metal poor than this value using a Kolmogorov-Smirnov test. For both elements
we obtain a K-S probability $p\sim 0.30$. This means that it is statistically
not possible to safely reject the null hypothesis that the two distributions
are extracted from the same parent population. A second test involves the
pseudo-colour maps constructed using HST photometry (Milone et al. 2017).
Figure 17 shows the chromosome map derived by us using the public data
available from the HST archive (Nardiello et al. 2018; see Carretta and
Bragaglia 2022a for details). In the figure, the stars in our sample falling
in the small central region covered by the HST are indicated by larger
symbols, coloured according to the derived Ba (upper panel) and Zr i (lower
panel) abundances derived in the present work. There is no significant
difference between stars scattered to the red in the pseudo-colour map and the
other stars in this photometric plane. The direct implication of these tests
is that NGC 6388 does not qualify as a type II GC either for some enhancement
in metallicity or in the amount of $s$-process elements. In turn, the red RGB
stars scattered in the pseudo-colour maps obtained from HST photometry are
simply not yet explained by any observable change or alterations in their
chemical composition.

Figure 17: Pseudo-colour maps of NGC 6388 from Carretta and Bragaglia (2022a).
Larger symbols indicate stars of the present work cross-identified with the
HST photometry in Nardiello et al. (2018). The colour-coding in the upper and
bottom panels is done according to the abundances of Zr i and Ba,
respectively, obtained in the present work. Figure 18: Abundance ratio [Zr/Fe]
i as a function of [Na/Fe] in 168 stars of NGC 6388. Figure 19: As in Fig. 18,
but for 151 stars with both Zr and Al abundances.

On the other hand, we found statistically significant correlations between the
Zr abundance and a few light elements that are enhanced in the proton-capture
reactions typical of the FG polluters. In Fig. 18 and Fig. 19, we show the
ratio [Zr/Fe]i as a function of [Na/Fe] and [Al/Fe]. We tested the level of
significance for a linear regression between Zr and both Na and Al: the two-
tail probabilities ($p=2.4\times 10^{-5}$ and $p=0.033$ for Na and Al,
respectively) allow us to conclude that the observed relations are real, with
a high level of significance.

This is not the first time a similar correlation has been found. To our
knowledge, the first paper to discuss correlations between elements enhanced
in SG stars (such as Na) and others based on a large enough number of stars
was Yong et al. (2013) on NGC 6752. They used differential analysis to derive
very precise abundances (at 0.01 dex level) and found positive correlations,
generally statistically significant, between many elements and Na. This could
indicate that the same stars that enhanced Na over primordial values were also
responsible for the increase in the other elements, among them some neutron-
capture ones (unfortunately, they did not measure Zr). Their conclusion was
that the abundance trends are real and discussed three potential mechanisms to
explain them (besides the possibility of systematic errors in stellar
parameters, which were regarded very unlikely): a star-to-star variation in
CNO abundance, or in He (which is strictly connected to Na in multiple
populations), or inhomogeneous early chemical evolution (i.e. metallicity
variations). They favoured a combination of the last two, but encouraged
similar studies on other clusters to try and clarify this issue.

At odds with those findings, Schiappacasse-Ulloa & Lucatello (2023) did not
find correlation between Na and neutron-capture elements in the same cluster.
They analysed about 160 stars in NGC 6752, from the main-sequence turn-off up
to the RGB bump, deriving abundances of elements from different
nucleosynthetic chains, among which we find Na, Y, and Ba. While they saw a
mild Na-Y correlation, this is not statistically significant. Also, the Ba
abundance does not correlate with Na abundance, and they concluded that the
stars that enhanced the Na level did not contribute Y and Ba.

Finally, Kolomiecas et al. (2022) derived the Na and Zr abundances of about
240 RGB stars in 47 Tuc, finding a statistically significant positive
correlation between them. Their conclusion was that some amount of Zr should
have been produced by the same primordial stars which were enriched in Na the
SG stars. Unfortunately, this cannot be attributed unequivocally to a single
class of polluters, either AGBs or massive stars, or a combination of the two.
We follow Yong et al. (2013) and Kolomiecas et al. (2022) in encouraging the
extention of this kind of analysis to more elements and more clusters.

Figure 20: As in Fig. 11, but for average abundance ratios [Y/Fe], [Zr/Fe],
and [Ba/Fe]. References for the samples are in Appendix B. Figure 21: As in
Fig. 11, but for average abundance ratios [La/Fe], [Ce/Fe], [Nd/Fe], and
[Eu/Fe]. References for the samples are in Appendix B.

###  5.4 Neutron-capture elements: Overal pattern in NGC 6388

The average abundances in NGC 6388 are compared to those of field disc and
bulge stars, as well as to the mean abundances of previous analyses of GCs, in
Fig. 20 and Fig. 21 (symbols are as in Fig. 11). The neutron-capture elements
in NGC 6388 seem to be consistent with those of bulge field stars of similar
metallicity. Some offsets seem to exist with respect to the disc component
around [Fe/H]$\sim-0.5$ dex, but not enough to be very significant, maybe with
the exception of Zr. However, the deficiency in [Zr/Fe] (middle panel of Fig.
20) is also shared by NGC 6441 and two other more metal-rich GCs. Together
with the mean abundances of the four metal-poor GCs, the overall pattern could
be explained by the decreasing abundances of Zr (produced in the main
$s$-process component in AGB stars) as the metallicity increases due to the
continuous injection of fresh iron from SN Ia (e.g. Tinsley 1979).

Another possibility is the one highlighted by Kobayashi et al. (2020); the
contribution from nucleosynthetic sources with an intrinsically long time
delay is more affected by the star formation timescale. Inefficient star
formation (as in the halo) may give a higher level of neutron-capture elements
relatively to iron. On the other hand, rapid star formation implies a smaller
contribution of ejecta from longer lived, lower mass AGB stars, which are
producers of elements from the main $s$-process component. However, this
second alternative would apply to both the light and heavy $s$-process
elements (Zr, La, Ce) produced in the main $s$-process, whereas there is some
evidence that NGC 6388 and other bulge GCs show the same level of neutron-
capture elements as seen in halo metal-poor GCs, at least for La, Ba, and Nd.

Concerning the $r$-process, in the bottom panel of Fig. 21 we trace the run of
[Eu/Fe] as a function of metallicity. The pattern of constant values in the
halo phase, followed by a gradual decrease as [Fe/H] increases, is explained
well by the scenario pioneered by Tinsley (1979), pointing to the origin of Eu
in the same sites (massive stars) where the $\alpha$-elements are produced.
After the knee in [Fe/H] is reached in the main progenitor galaxy, the Eu
production would remain essentially flat, but the [Eu/Fe] ratio is lowered by
the increasing contribution of SN Ia. When our sample of clusters, mainly
constituted by metal-poor GCs, is complemented by metal-rich GCs, it is easy
to appreciate that the above scenario is satisfied by both field and GC stars.
The interplay between enrichment from core-collapse and thermonuclear SNe was
essentially the same, regardless of the star formation occurring in GCs or in
the general field.

Finally, in Fig. 22 we use the relative strengths of the $r$-process and
$s$-process to probe the relative contribution from high-mass stars (mainly
responsible for yields of the weak component of the $s$-process and the
$r$-process) and from low- or intermediate-mass AGB stars (main $s$-process).
In this figure, the pure $r$-process element Eu is compared to Y, chosen as
reference element for the $s$-process. Again, by plotting the ratio [Eu/Y] as
a function of the metallicity (liberally used as a chemical ‘chronometer’) we
are exploiting the different mass ranges, and therefore different evolutionary
timescales, of the involved stars to give a general picture of the enrichment
process in field and GC stars.

Figure 22: As in Fig. 11, but for the average abundance ratio [Eu/Y] of the Eu
and Y species, which are reference elements for the $r$ and $s$-processes,
respectively. Field bulge stars (cyan) are from Lucey et al. (2019) and
Gratton et al. (2006). Halo and disc stars (grey circles) are from James et
al. (2004) and the compilation by Venn et al. (2004).

At low metallicities, both GC and field stars show high [Eu/Y] ratios,
approaching the scaled Solar System pure $r$-process level, with scarce or no
contribution from the $s$-process in AGB stars. As lower mass stars with
longer evolutionary timescales appear on the enrichment scene, an increase in
the production of $s$-process elements lowers the [Eu/Y] ratio. The decrease
seems to happen in lockstep both in field and GC stars until about
[Fe/H]$\sim-0.5$. At this metallicity, our present analysis does confirm the
earlier results presented in Carretta et al. (2007a) for NGC 6388 and in
Gratton et al. (2006) for NGC 6441. In particular, NGC 6388 seems to have a
[Eu/Y] higher than observed in field stars of similar metallicities. Although
the larger errors associated with NGC 6441 make its ratio still compatible
with the field stars, the ratio in NGC 6388 is in better agreement with the
ratio in metal-poor GCs, where the contribution of AGB stars to the
$s$-process was not yet relevant.

In Carretta et al. (2007a), we put forward the hypothesis that this excess in
[Eu/Y] could be explained by an enhanced contribution of massive stars to the
enrichment in the bulge, since light $s$-process elements such as Y can be
also produced in the weak $s$-process component within the He-burning core of
massive stars (Couch et al. 1974, Travaglio et al. 2004). This idea would also
explain the high [Eu/Y] ratios measured in metal-rich bulge field stars
(Gratton et al. 2006, cyan circles in Fig. 22). However, we note that the disc
GC NGC 5927 (Mura-Guzmàn et al. 2018) also shares this excess ratio;
therefore, a larger sample of GCs at high metallicity is required to answer
the question.

Figure 23: Average abundances of Y, Zr, Ba, La, Ce, Nd, and Eu in NGC 6388
from UVES spectra. The respective rms scatters are also represented as
associated error bars. Our best-fit estimate using solar-scaled pure
$r$-process and $s$-process abundances is shown as a red line (see text for
the adopted procedure).

A more quantitative approach is summarised in Fig. 23, where we compare the
average abundances of neutron-capture elements derived from UVES spectra in
NGC 6388 to the ratios of $r-$ and $s-$ elements estimated by Simmerer et al.
(2004) in the Solar System. For this comparison we followed the approach
suggested by Raffaele Gratton and employed in Carretta et al. (2015) for NGC
6093. To reproduce the pattern of neutron-capture elements in NGC 6388, our
best fit must consider the sum of two contributions: a solar-scaled
$r$-process contribution, with a scaling by -0.27 dex, and a solar-scaled
$s$-process contribution, with a scaling by -0.51 dex. Taking into account the
derived metallicity for NGC 6388 ([Fe/H]$=-0.48$ dex), these scaling factors
imply abundance ratios of [$r$/Fe]$=+0.21$ dex and [$s$/Fe]$=-0.03$ dex.

The excess of elements produced by the $r$-process is very similar to the one
we obtain from the $\alpha$-elements. From GIRAFFE and UVES spectra, we derive
mean values of [$\alpha$/Fe]$=+0.22$ dex and +0.23 dex (regardless of
inclusion or exclusion of Mg in the average). The excess in $r$-process
elements with respect to the solar value may be interpreted as an iron
deficiency due to the fact that we see almost exclusively the contribution of
massive stars, whereas the one from SN Ia is missing. On the other side, our
data show that it is necessary to also consider a significant part of the
$s$-process, which scale almost exactly as Fe, to explain the observations in
NGC 6388 well.

##  6 Summary and conclusions

We present the homogeneous spectroscopic analysis of a large sample of stars
in NGC 6388. Concerning the proton-capture elements, we find that all stars
observed in NGC 6388 nicely trace the typical correlations and anti-
correlations that are the unique trademark of GCs. The exceptions are the
heaviest species (Ca, Sc) involved in the network of proton-capture reactions
in H-burning at high temperature. No statistically significant variation is
found for these two species between FG and SG stars, confirming qualitative
results shown in Carretta and Bragaglia (2019) for NGC 6388.

Star-to-star variations in Si, correlated to abundance changes in Al and anti-
correlated to Mg depletions, support leakage from the Mg-Al cycle on Si. In
turn, this requires temperatures as high as about 65 MK in the FG polluters.

A simple dilution model is compatible with a single class of polluters
injecting processed matter in the intra-cluster medium at early time in the
proro-GC. Mixing this polluted material (typical of the composition of the SG
group E) with different amounts of pristine gas (whose composition is
represented by the P group of stars), we can obtain a good agreement for the
composition of the intermediate SG group for all the involved species.

The extent of the Na-O anti-correlation, the privileged, unambiguous signature
of multiple stellar populations in GCs, seems to be too short in NGC 6388 with
respect to its large total mass. Together with a few other massive GCs (47
Tuc, M 15, NGC 6441), NGC 6388 lies slightly below the main trend in the
IQR[O/Na] versus $M_{V}$ relation describing the dependence of the extent of
Na-O anti-correlation as a function of cluster total mass (Carretta 2006;
Carretta et al. 2010a).

The inventory of $\alpha$-element abundances in NGC 6388 is very compatible
with the chemical pattern of field stars in the Milky Way, both in the disc
and bulge. The abundance ratios in NGC 6388 participate in the classical trend
defined by the interplay between star formation and lifetimes of the main
stellar nucleosynthesis sites, namely type II and type Ia SNe. The resulting
plateau at low metallicity, followed by a knee and the decrease of
$\alpha$/Fe] ratio at increasing [Fe/H] is followed by both field and GC
stars, including NGC 6388.

We found no evidence of a low level of Si, as derived by infrared APOGEE data.
Therefore, we cannot support an extragalactic origin for NGC 6388, as
suggested by Horta et al. (2020). We note that all the studies using optical
spectroscopy converge on finding normal, high values of [Si/Fe] for NGC 6388.
Its seems that there could be some offsets between optical and infrared
analyses due to still poorly understood effects concerning Si. A low value of
Ca is instead derived for NGC 6388, as well as for bulge stars in a similar
metallicity regime, regardless of whether optical or infrared spectra are
used.

The average abundances of elements of the iron group in NGC 6388 closely
follow the pattern of chemical enrichment typical of field stars in the Milky
Way. We then confirm and strengthen the results by Carretta and Bragaglia
(2022b): NGC 6388 is clearly not of extragalactic origin, but likely formed in
situ in the Galactic bulge, and the iron-peak species can only be used to
trace the GCs of the Sagittarius dwarf, whose content in such elements is
typically lower than in the autochthonous stars of our Galaxy. Consistently,
both these elements and the high Si level, normal for old GCs, point toward
the in situ origin of NGC 6388.

We do not detect any signature of enhancement in neutron-capture elements in a
fraction of stars of NCG 6388. In particular, there is no significant
difference in the abundance of Ba and Zr between FGs, SG stars, and the stars
scattered to the red of the RGB. This evidence corroborates the fact that NGC
6388 is not a GC of a distinct (type II) class and also leaves the red RGB
stars unexplained, at least from a chemical perspective.

Statistically significant correlations are found between Zr abundance and both
Na and Al abundances. Similar results are also found in a few other cases and
warrant being extended to a larger sample of GCs.

The excess of the $r$-process element Eu in NGC 6388 is consistent with the
values of more metal-poor old GCs and similar to the ratio of
$\alpha$-elements, showing the contribution of massive stars coupled to the
small injection of Fe from thermonuclear SNe at the epoch of cluster
formation. The overall pattern of neutron-capture elements from high-
resolution UVES spectra shows, however, that it is necessary to also consider
a contribution from the $s$-process in this GC.

###### Acknowledgements.

This research made use of the products of the Cosmic-Lab project funded by the
European Research Council. We thank E. Dalessandro for helpful discussions.
This research has made use of the SIMBAD database (in particular Vizier),
operated at CDS, Strasbourg, France, of the NASA’s Astrophysical Data System,
and of TOPCAT (http://www.starlink.ac.uk/topcat/). This paper makes use of the
data collected by the HST Treasury Program GO 13297. We acknowledge funding
from PRIN INAF 2019 “Building up the halo: chemo-dynamical tagging in the age
of large surveys”, PI Lucatello.

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##  Appendix A Abundances of individual stars

Abundances for individual stars in NGC 6388 are listed in Table 6 for proton-
capture elements, Table 7 for $\alpha$-elements, Table 8 for elements of the
Fe-group, and Table 9 for species from neutron-capture processes measured on
UVES spectra. Finally, in Table 10 we list the abundances of the neutron-
capture elements Zr i and Ba ii that could be derived also from GIRAFFE
spectra. For this table, as well as for tables relative to light elements,
$\alpha$-elements and Fe-peak elements (Tables A.1, A.2, A.3, and A.5), only
an excerpt is provided here as a guidance of the content. Complete tables can
be found at CDS Strasbourg.

Table 6: Light element abundances. Star | nr | [O/Fe]i | rms | lim | nr | [Na/Fe]i | rms | nr | [Mg/Fe]i | rms | nr | [Al/Fe]i | rms | phase
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
n63a | 2 |  $-$0.099 | 0.110 | 1 | 4 | +0.215 | 0.066 | 4 | +0.198 | 0.133 | 2 | +0.003 | 0.031 | RGB
n63b | 2 |  $-$0.149 | 0.137 | 1 | 3 | +0.111 | 0.041 | 3 | +0.265 | 0.067 | 2 | +0.265 | 0.027 | RGB
n63c | 2 |  $-$0.372 | 0.157 | 1 | 4 | +0.884 | 0.090 | 4 | +0.143 | 0.097 | 2 | +0.891 | 0.151 | RGB
n63d | 2 |  $-$0.242 | 0.075 | 1 | 4 | +0.497 | 0.060 | 4 | +0.234 | 0.099 | 2 | +0.091 | 0.083 | RGB
n63e | 2 |  $-$0.159 | 0.053 | 1 | 4 | +0.551 | 0.050 | 4 | +0.230 | 0.173 | 2 | +0.501 | 0.071 | RGB
n63f | 2 |  $-$0.112 | 0.002 | 1 | 4 | +0.639 | 0.085 | 4 | +0.223 | 0.116 | 2 | +0.199 | 0.126 | RGB
n63g | 1 |  $-$0.633 |  | 1 | 3 | +1.267 | 0.120 | 4 | +0.191 | 0.063 | 2 | +1.218 | 0.088 | RGB
n63h | 2 |  $-$0.061 | 0.013 | 1 | 4 | +0.196 | 0.044 | 4 | +0.202 | 0.104 | 2 | +0.081 | 0.071 | RGB
n63i | 2 |  $-$0.047 | 0.008 | 1 | 4 | +0.586 | 0.179 | 4 | +0.251 | 0.078 | 2 | +0.083 | 0.018 | RGB
n63l | 2 | +0.001 | 0.023 | 1 | 4 | +0.207 | 0.126 | 4 | +0.192 | 0.063 | 2 | +0.020 | 0.108 | RGB
n63m | 2 |  $-$0.200 | 0.083 | 1 | 4 | +0.735 | 0.112 | 4 | +0.222 | 0.146 | 2 | +0.633 | 0.159 | RGB
n63n | 2 |  $-$0.445 | 0.043 | 1 | 4 | +0.648 | 0.085 | 4 | +0.115 | 0.148 | 2 | +0.630 | 0.163 | RGB
l63p001 | 2 | +0.237 | 0.001 | 1 | 2 | +0.383 | 0.082 | 2 | +0.259 | 0.031 | 2 | +0.208 | 0.018 | RGB
l63p002 | 2 | +0.301 | 0.105 | 1 | 2 | +0.229 | 0.001 | 2 | +0.380 | 0.030 | 2 | +0.646 | 0.218 | RGB
l63p003 | 1 | +0.021 |  | 0 | 2 | +0.348 | 0.075 | 2 | +0.323 | 0.035 | 1 | +0.025 |  | RGB
l63p004 | 2 | +0.245 | 0.113 | 1 | 2 |  $-$0.315 | 0.142 | 2 | +0.215 | 0.083 | 2 | +0.002 | 0.080 | RGB
l63p006 | 1 | +0.120 |  | 1 | 2 | +0.426 | 0.063 | 2 | +0.246 | 0.104 | 0 |  |  | RGB
Table 7: $\alpha$-element abundances. Star | nr | [Si/Fe]i | rms | nr | [Ca/Fe]i | rms | nr | [Ti/Fe]i | rms | nr | [Ti/Fe]ii | rms
---|---|---|---|---|---|---|---|---|---|---|---|---
n63a | 9 | +0.358 | 0.166 | 16 | +0.034 | 0.140 | 25 | +0.171 | 0.211 | 11 | +0.264 | 0.254
n63b | 0 |  |  | 11 | +0.040 | 0.225 | 25 | +0.325 | 0.189 | 8 | +0.357 | 0.201
n63c | 8 | +0.364 | 0.167 | 18 | +0.098 | 0.222 | 29 | +0.288 | 0.271 | 11 | +0.274 | 0.143
n63d | 6 | +0.319 | 0.148 | 16 | +0.087 | 0.206 | 28 | +0.322 | 0.170 | 12 | +0.246 | 0.163
n63e | 8 | +0.327 | 0.180 | 19 | +0.072 | 0.177 | 26 | +0.272 | 0.151 | 12 | +0.307 | 0.194
n63f | 8 | +0.310 | 0.172 | 19 | +0.152 | 0.198 | 27 | +0.320 | 0.182 | 12 | +0.303 | 0.174
n63g | 8 | +0.288 | 0.278 | 9 | +0.081 | 0.217 | 21 | +0.378 | 0.220 | 10 | +0.324 | 0.230
n63h | 9 | +0.368 | 0.194 | 15 | +0.026 | 0.114 | 27 | +0.277 | 0.164 | 10 | +0.205 | 0.236
n63i | 8 | +0.353 | 0.151 | 19 | +0.052 | 0.164 | 27 | +0.309 | 0.153 | 12 | +0.263 | 0.149
n63l | 8 | +0.339 | 0.158 | 17 | +0.051 | 0.149 | 28 | +0.263 | 0.179 | 11 | +0.292 | 0.176
n63m | 7 | +0.359 | 0.198 | 17 | +0.121 | 0.174 | 27 | +0.314 | 0.261 | 10 | +0.265 | 0.176
n63n | 9 | +0.382 | 0.259 | 13 | +0.172 | 0.202 | 27 | +0.267 | 0.225 | 11 | +0.288 | 0.289
l63p001 | 3 | +0.262 | 0.060 | 6 | +0.139 | 0.088 | 3 | +0.298 | 0.074 | 0 |  |
l63p002 | 3 | +0.243 | 0.031 | 6 | +0.028 | 0.112 | 3 | +0.294 | 0.084 | 0 |  |
l63p003 | 3 | +0.166 | 0.006 | 6 | +0.159 | 0.116 | 2 | +0.288 | 0.098 | 0 |  |
l63p004 | 3 | +0.219 | 0.084 | 5 | +0.101 | 0.054 | 3 | +0.139 | 0.130 | 0 |  |
l63p006 | 3 | +0.270 | 0.062 | 6 | +0.061 | 0.160 | 4 | +0.256 | 0.172 | 0 |  |
Table 8: Iron-peak abundances. Star | nr |  [Sc/Fe]ii | rms | nr |  [V/Fe]i | rms | nr |  [Cr/Fe]i | rms | nr |  [Mn/Fe]i | rms | nr |  [Co/Fe]i | rms | nr |  [Ni/Fe]i | rms | nr |  [Zn/Fe]i | rms
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
n63a | 8 |  $-$0.035 | 0.134 | 13 | +0.069 | 0.261 | 17 |  $-$0.155 | 0.230 | 8 |  $-$0.247 | 0.244 | 5 |  $-$0.027 | 0.283 | 41 | +0.023 | 0.209 | 1 | +0.137 |
n63b | 3 | +0.072 | 0.239 | 8 | +0.100 | 0.266 | 15 |  $-$0.066 | 0.211 | 8 |  $-$0.231 | 0.295 | 5 | +0.083 | 0.334 | 26 | +0.018 | 0.358 | 0 |  |
n63c | 8 | +0.026 | 0.155 | 13 | +0.321 | 0.236 | 21 |  $-$0.107 | 0.284 | 7 |  $-$0.153 | 0.095 | 5 | +0.047 | 0.175 | 40 | +0.045 | 0.280 | 1 | +0.066 |
n63d | 7 | +0.004 | 0.220 | 13 | +0.167 | 0.246 | 17 |  $-$0.194 | 0.194 | 8 |  $-$0.204 | 0.182 | 5 | +0.111 | 0.178 | 31 | +0.054 | 0.171 | 0 |  |
n63e | 8 |  $-$0.086 | 0.125 | 11 | +0.205 | 0.205 | 25 |  $-$0.170 | 0.253 | 6 |  $-$0.202 | 0.204 | 5 | +0.077 | 0.143 | 43 |  $-$0.001 | 0.185 | 1 |  $-$0.067 |
n63f | 8 |  $-$0.016 | 0.151 | 13 | +0.328 | 0.204 | 25 |  $-$0.114 | 0.238 | 6 |  $-$0.126 | 0.196 | 5 | +0.134 | 0.175 | 41 | +0.029 | 0.169 | 1 | +0.363 |
n63g | 8 | +0.040 | 0.242 | 13 | +0.371 | 0.312 | 13 |  $-$0.138 | 0.259 | 6 |  $-$0.308 | 0.297 | 4 | +0.073 | 0.291 | 21 | +0.012 | 0.286 | 0 |  |
n63h | 8 |  $-$0.023 | 0.191 | 12 | +0.145 | 0.261 | 22 |  $-$0.156 | 0.251 | 7 |  $-$0.203 | 0.260 | 4 |  $-$0.026 | 0.157 | 37 | +0.004 | 0.177 | 1 | +0.218 |
n63i | 8 |  $-$0.109 | 0.186 | 13 | +0.191 | 0.175 | 25 |  $-$0.112 | 0.256 | 6 |  $-$0.317 | 0.119 | 5 |  $-$0.031 | 0.174 | 41 |  $-$0.015 | 0.160 | 1 | +0.020 |
n63l | 8 |  $-$0.112 | 0.178 | 12 | +0.184 | 0.194 | 22 |  $-$0.159 | 0.235 | 7 |  $-$0.169 | 0.194 | 5 | +0.137 | 0.106 | 39 | +0.013 | 0.155 | 1 | +0.109 |
n63m | 8 | +0.004 | 0.179 | 13 | +0.357 | 0.168 | 13 |  $-$0.098 | 0.172 | 6 |  $-$0.187 | 0.182 | 4 |  $-$0.044 | 0.174 | 40 | +0.049 | 0.231 | 1 | +0.208 |
n63n | 8 |  $-$0.001 | 0.096 | 13 | +0.297 | 0.201 | 21 |  $-$0.191 | 0.232 | 7 |  $-$0.288 | 0.183 | 5 |  $-$0.019 | 0.163 | 37 | +0.047 | 0.219 | 1 | +0.096 |
l63p001 | 2 | +0.003 | 0.012 |  |  |  | 1 |  $-$0.037 |  | 0 |  |  | 0 |  |  | 12 | +0.066 | 0.104 | 0 |  |
l63p002 | 2 |  $-$0.017 | 0.017 |  |  |  | 1 |  $-$0.126 |  | 0 |  |  | 0 |  |  | 12 | +0.051 | 0.057 | 0 |  |
l63p003 | 2 |  $-$0.122 | 0.071 |  |  |  | 1 | +0.256 |  | 0 |  |  | 0 |  |  | 8 | +0.019 | 0.100 | 0 |  |
l63p004 | 2 |  $-$0.066 | 0.007 |  |  |  | 1 |  $-$0.487 |  | 0 |  |  | 0 |  |  | 10 |  $-$0.068 | 0.093 | 0 |  |
l63p006 | 2 |  $-$0.044 | 0.014 |  |  |  | 1 | +0.080 |  | 0 |  |  | 0 |  |  | 10 | +0.028 | 0.152 | 0 |  |
Table 9: Neutron-capture abundances. Star | nr |  [Y/Fe]i | rms | nr |  [Zr/Fe]ii | rms | nr |  [La/Fe]ii | rms | nr |  [Ce/Fe]ii | rms | nr |  [Nd/Fe]ii | rms | nr |  [Eu/Fe]ii | rms
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
n63a | 2 |  $-$0.680 | 0.134 | 0 |  |  | 2 | +0.047 | 0.040 | 1 |  $-$0.365 |  | 3 | +0.070 | 0.082 | 2 | +0.238 | 0.064
n63b | 1 |  $-$0.300 |  | 1 |  |  | 2 | +0.214 | 0.022 | 0 |  |  | 3 | +0.003 | 0.066 | 0 |  |
n63c | 2 |  $-$0.424 | 0.018 | 1 |  $-$0.280 |  | 2 | +0.173 | 0.044 | 1 |  $-$0.285 |  | 3 | +0.012 | 0.157 | 2 | +0.283 | 0.020
n63d | 2 |  $-$0.340 | 0.221 | 1 |  $-$0.205 |  | 2 | +0.174 | 0.018 | 1 |  $-$0.323 |  | 3 | +0.029 | 0.203 | 2 | +0.288 | 0.168
n63e | 2 |  $-$0.287 | 0.202 | 1 | +0.133 |  | 2 | +0.094 | 0.175 | 1 |  $-$0.169 |  | 3 | +0.111 | 0.013 | 2 | +0.260 | 0.035
n63f | 2 |  $-$0.237 | 0.129 | 1 | +0.091 |  | 2 | +0.147 | 0.066 | 1 |  $-$0.233 |  | 3 | +0.183 | 0.059 | 2 | +0.265 | 0.000
n63g | 0 |  |  | 1 |  $-$0.371 |  | 2 | +0.238 | 0.279 | 1 |  $-$0.260 |  | 0 |  |  | 2 | +0.267 | 0.161
n63h | 2 |  $-$0.166 | 0.035 | 1 |  $-$0.066 |  | 2 | +0.149 | 0.027 | 1 |  $-$0.341 |  | 3 |  $-$0.123 | 0.099 | 2 | +0.183 | 0.013
n63i | 2 |  $-$0.173 | 0.216 | 1 |  $-$0.228 |  | 2 | +0.122 | 0.155 | 1 |  $-$0.290 |  | 3 |  $-$0.024 | 0.210 | 2 | +0.202 | 0.071
n63l | 2 |  $-$0.594 | 0.071 | 1 |  $-$0.071 |  | 2 | +0.145 | 0.053 | 1 |  $-$0.289 |  | 3 | +0.000 | 0.200 | 2 | +0.233 | 0.108
n63m | 2 |  $-$0.070 | 0.115 | 1 |  $-$0.395 |  | 2 | +0.273 | 0.115 | 1 |  $-$0.325 |  | 3 | +0.102 | 0.179 | 2 | +0.305 | 0.025
n63n | 2 |  $-$0.281 | 0.257 | 1 |  $-$0.464 |  | 2 | +0.088 | 0.009 | 1 |  $-$0.364 |  | 3 |  $-$0.010 | 0.133 | 2 | +0.257 | 0.084
Table 10: [Zr/Fe]i and [Ba/Fe]ii abundances. Star | nr |  [Zr/Fe]i | rms | nr |  [Ba/Fe]ii | rms
---|---|---|---|---|---|---
n63a | 4 |  $-$0.337 | 0.113 | 3 |  $-$0.478 | 0.025
n63b | 0 |  |  | 0 |  |
n63c | 4 | +0.008 | 0.252 | 3 | +0.016 | 0.149
n63d | 6 |  $-$0.091 | 0.275 | 3 |  $-$0.214 | 0.071
n63e | 4 |  $-$0.054 | 0.222 | 3 |  $-$0.083 | 0.069
n63f | 3 |  $-$0.008 | 0.029 | 3 |  $-$0.036 | 0.102
n63g | 0 |  |  | 3 | +0.049 | 0.221
n63h | 4 |  $-$0.094 | 0.228 | 3 |  $-$0.031 | 0.098
n63i | 3 |  $-$0.094 | 0.063 | 3 | +0.049 | 0.059
n63l | 3 |  $-$0.149 | 0.038 | 3 |  $-$0.148 | 0.045
n63m | 3 |  $-$0.122 | 0.280 | 3 | +0.011 | 0.082
n63n | 4 |  $-$0.068 | 0.162 | 3 |  $-$0.061 | 0.322
l63p001 | 4 |  $-$0.013 | 0.127 | 1 | +0.129 |
l63p002 | 4 |  $-$0.141 | 0.170 | 1 |  $-$0.091 |
l63p003 | 0 |  |  | 1 |  $-$0.077 |
l63p004 | 3 |  $-$0.116 | 0.197 | 1 |  $-$0.035 |
l63p006 | 1 |  $-$0.066 |  | 1 | +0.136 |

##  Appendix B References for studies listed in Figures 11, 15, 20, and 21

In Figures 11, 15, 20, and 21 we compare the average abundance ratios derived
in NGC 6388 for some species to a sample of GCs and field stars, both in the
disc and in the bulge, from a number of studies. Not all the elements were
available for GC stars, whereas all species were sampled in the abundance
analyses of comparison field stars in the Milky Way.

The GCs in our FLAMES survey, ordered by increasing metallicity, are listed in
Table LABEL:t:appB_noi (we give the [Fe/H] value from UVES spectra used in the
plots, from Carretta et al. 2009c or from individual papers, in parentheses).
We note that some elements were analysed in the same study for many GCs,
whereas other elements were obtained in the individual papers.

To GCs homogeneously analysed in our FLAMES survey (or with similar
procedures), we added four metal-rich GCs from the same group of investigators
(see Table 12). For field Milky Way stars (in halo, disc, or bulge
components), we used many studies, as detailed in the same table.

Table 11: Globular cluster in our FLAMES survey: Elements and references. GC and [Fe/H] | Ref for [Fe/H] | Ref for Mg, Si | Ref for Ca | Ref for other elem. | Other elements
---|---|---|---|---|---
NGC 7099 (M 30, $-2.344$) | Carretta et al. 2009c | Carretta et al. 2009a | Carretta et al. 2010b |  |
NGC 7078 (M 15, $-2.320$) | Carretta et al. 2009c | Carretta et al. 2009a | Carretta et al. 2010b |  |
Terzan 8 ($-2.270$) | Carretta et al. 2014a | Carretta et al. 2014a | Carretta et al. 2014a | Carretta et al. 2014a | Ti i, Cr, Mn, Ni, Y ii, Ba, Nd
NGC 4590 (M 68, $-2.265$) | Carretta et al. 2009c | Carretta et al. 2009a | Carretta et al. 2010b |  |
NGC 4833 ($-2.020$) | Carretta et al. 2014b | Carretta et al. 2014b | Carretta et al. 2014b | Carretta et al. 2014b | Ti i, Cr, Mn, Ni, Y ii, Y ii,Ba, Nd
NGC 6397 ($-1.988$) | Carretta et al. 2009c | Carretta et al. 2009a | Carretta et al. 2010b | James et al. 2004a | Y ii, Ba, Eu
NGC 6535 ($-1.952$) | Bragaglia et al. 2017 | Bragaglia et al. 2017 | Bragaglia et al. 2017 | Bragaglia et al. 2017 | Ti i, Cr, Mn, Ni
NGC 6809 (M 55, $-1.934$) | Carretta et al. 2009c | Carretta et al. 2009a | Carretta et al. 2010b |  |
NGC 5634 ($-1.867$) | Carretta et al. 2017 | Carretta et al. 2017 | Carretta et al. 2017 | Carretta et al. 2017 | Ti i, Cr, Mn, Co, Ni, Y ii, Ba, La, Nd, Eu
NGC 6093 (M 80, $-1.790$) | Carretta et al. 2015 | Carretta et al. 2015 | Carretta et al. 2015 | Carretta et al. 2015 | Ti i, Cr, Mn, Co, Ni, Y ii, Ba, La, Ce, Nd, Eu
NGC 1904 (M 79, $-1.579$) | Carretta et al. 2009c | Carretta et al. 2009a | Carretta et al. 2010b |  |
NGC 6254 (M 10, $-1.575$) | Carretta et al. 2009c | Carretta et al. 2009a | Carretta et al. 2010b |  |
NGC 6752 ($-1.555$) | Carretta et al. 2009c | Carretta et al. 2009a | Carretta et al. 2010b | James et al. 2004b | Y ii, Ba, Eu
NGC 3201 ($-1.512$) | Carretta et al. 2009c | Carretta et al. 2009a | Carretta et al. 2010b |  |
NGC 6715 (M 54, $-1.510$) | Carretta et al. 2010c | Carretta et al. 2010c | Carretta et al. 2010c | Carretta et al. 2010c | Ti i, Cr, Mn, Co, Ni
NGC 5904 (M 5, $-1.340$) | Carretta et al. 2009c | Carretta et al. 2009a | Carretta et al. 2010b |  |
NGC 6218 (M 12, $-1.330$) | Carretta et al. 2009c | Carretta et al. 2009a | Carretta et al. 2010b |  |
NGC 288 ($-1.305$) | Carretta et al. 2009c | Carretta et al. 2009a | Carretta et al. 2010b |  |
NGC 1851 ($-1.180$) | Carretta et al. 2011 | Carretta et al. 2011 | Carretta et al. 2011 | Carretta et al. 2011 | Ti i, Cr, Mn, Co, Ni, Y ii, Zr ii, Ba, La, Nd, Eu
NGC 362 ($-1.170$) | Carretta et al. 2013 | Carretta et al. 2013 | Carretta et al. 2013 | Carretta et al. 2013 | Ti i, Cr, Mn, Co, Ni, Y ii, Zr ii, Ba, La, Nd, Eu
NGC 6121 (M 4, ($-1.168$) | Carretta et al. 2009c | Carretta et al. 2009a | Carretta et al. 2010b |  |
NGC 2808 ($-1.130$) | Carretta 2015 | Carretta 2015 | Carretta 2015 | Carretta 2015 | Ti i, Cr, Mn, Ni
NGC 6171 (M 107, $-1.033$) | Carretta et al. 2009c | Carretta et al. 2009a | Carretta et al. 2010b |  |
NGC 6838 (M 71, $-0.832$) | Carretta et al. 2009c | Carretta et al. 2009a | Carretta et al. 2010b |  |
NGC 104 (47 Tuc, $-0.768$) | Carretta et al. 2009c | Carretta et al. 2009a | Carretta et al. 2010b | Carretta et al. 2004 | Cr, Mn, Ni; Y ii, Ba, Eu
|  |  |  | James et al. 2004a |
NGC 6388 ($-0.480$) | Carretta&Bragaglia 2022a | this work | this work | this work | Ti i, Cr i, Mn, Co, Ni, Y i, Zr sc i, Ba, La, Ce, Nd, Eu
NGC 6441 ($-0.430$) | Carretta et al. 2009c | Gratton et al. 2006 | Gratton et al. 2006 | Gratton et al. 2006 | Ti i, Cr, Mn, Ni, Y i, Zr i, Ba, Eu
Table 12: Literature data: Elements and references. GC or system | Reference | elements
---|---|---
NGC 6440 ([Fe/H]$=-0.50$) | Muñoz et al. 2017 | Mg, Si, Ca, Sc, Ti, Mn, Ni, Co, Ba, Eu
NGC 5927 ([Fe/H]$=-0.47$) | Mura-Guzmán et al. 2018 | Mg, Si, Ca, Sc, Ti, V, Cr, Mn, Co, Ni, Y, Zr, Ce, Ba, Eu
NGC 6528 ([Fe/H]$=-0.20$) | Muñoz et al. 2018 | Mg, Si, Ca, Sc, Ti, V, Cr, Mn, Co, Ni, Zr, Ba, Eu
NGC 6553 [Fe/H]$=-0.14$) | Muñoz et al. 2020 | Mg, Si, Ca, Sc, Ti, V, Cr, Mn, Ni, Y, Zr, Ce, Ba, Eu
halo/disc | Adibekyan et al. 2012 | Ca
| Battistini and Bensby 2016 | Zr, La, Ce, Nd
| Beirão et al. 2005 | Mg
| Bensby et al. 2014 | Mg, Ca, Cr, Ni, Y, Ba
| Brewer et al. 2016 | Mg, Mn, Ni, Y
| Delgado Mena et al. 2017 | Si
| Gratton et al. 2003 | Mg, Si, Ca, Ti, Cr, Mn, Ni
| Ishigaki et al. 2012, 2013 | Mg, Si, Ca, Ti, Cr, Mn, Co, Ni, Y, Zr, Ba, La, Nd, Eu
| Lai et al. 2008 | Si
| Neves et al. 2009 | Mg, Si. Ti, Cr, Mn, Co, Ni
| Reddy et al. 2003 | Si
| Reddy et al. 2006 | Ca
| Reggiani et al. 2017 | Mg, Cr, Mn, Co, Ni, Y, Zr, Ba
| Roederer et al. 2014. | Si, Ca, Ti, Cr, Mn, Co, Ni, Y, Zr, Ba, La, Ce, Nd, Eu
bulge | Alves-Brito et al. 2010 | Mg, Ca, Ti
| Barbuy et al. 2013 | Mn
| Bensby et al. 2017 | Mg, Si, Ca, Ti, Cr, Ni, Y, Ba
| da Silveira et al. 2018 | Mg, Si, Ca
| Duong et al. 2019a | Ti
| Duong et al. 2019b | Cr, Mn, Co, Ni, Zr, La, Nd, Eu
| Forsberg et al. 2019 | Zr, La, Ce, Eu
| Johnson et al. 2014 | Mg, Si, Ca, Cr, Co, Ni
| Jönsson et al. 2017 | Mg, Ca, Ti
| Lomaeva et al. 2019 | Cr, Mn, Co, Ni
| Lucey et al. 2019 | Mg, Si, Ca, Ti, Cr, Mn, Co, Ni, Y, Zr, Ba, La, Nd, Eu
| Lucey et al. 2022 | Ti