11institutetext:  INAF - Osservatorio Astronomico di Roma, via Frascati 33,
00078 Monte Porzio Catone, Italy 22institutetext: Agenzia Spaziale Italiana,
Space Science Data Center, via del Politecnico snc, 00133 Rome, Italy
33institutetext: Dipartimento di Fisica, Università di Roma Tor Vergata, via
della Ricerca Scientifica 1, 00133 Roma, Italy 44institutetext: INAF -
Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, 35122 Padova,
Italy 55institutetext: Astronomisches Rechen-Institut, Zentrum für Astronomie
der Universität Heidelberg, Mönchhofstr. 12-14, 69120, Heidelberg, Germany
66institutetext: Max Planck Institute for Astronomy, D-69117 Heidelberg,
Germany; Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej
17, DK-2100 Copenhagen Ø, Denmark 77institutetext: INAF - Osservatorio
Astronomico di Capodimonte, Napoli, Italy 88institutetext: GEPI, Observatoire
de Paris, PSL Research University, CNRS, 61 Avenue de l’Observatoire, 75014,
Paris, France 99institutetext: UPJV, Université de Picardie Jules Verne, 33
rue St Leu, 80080, Amiens, France 1010institutetext: Koninklijke Sterrenwacht
van België, Ringlaan 3, 1180, Brussels, Belgium 1111institutetext: Science and
Technology Department, Parthenope University of Naples, Naples, Italy
1212institutetext: Astronomical Observatory, Odessa National University,
Shevchenko Park, 65014, Odessa, Ukraine 1313institutetext: LMU München,
Universitätssternwarte, Scheinerstr. 1, D-81679 München, Germany
1414institutetext: Institute for Astronomy, University of Hawaii at Manoa,
2680 Woodlawn Drive, Honolulu, HI 96822, USA 1515institutetext: Department of
Astronomy, School of Science, The University of Tokyo, 7-3-1, Hongo, Bunkyo-
ku, Tokyo 113-0033, Japan 1616institutetext: Laboratory of Infrared High-
Resolution spectroscopy (LiH), Koyama Astronomical Observatory, Kyoto Sangyo
University, Motoyama, Kamigamo, Kita-ku, Kyoto 603-8555, Japan
1717institutetext: Instituto de Astrofísica de Canarias (IAC), La Laguna,
E-38205, Spain 1818institutetext: Departamento de Astrofísica, Universidad de
La Laguna (ULL), E-38200, La Laguna, Spain 1919institutetext: INAF-
Osservatorio Astronomico d’Abruzzo, Via M. Maggini, s/n, I-64100, Teramo,
Italy 2020institutetext: Istituto Nazionale di Fisica Nucleare, Laboratori
Nazionali di Frascati (INFN-LNF), Frascati, Italy 2121institutetext: Leibniz-
Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482, Potsdam,
Germany 2222institutetext: Université Côte d’Azur, Observatoire de la Côte
d’Azur, CNRS, Lagrange UMR 7293, CS 34229, 06304, Nice Cedex 4, France

# A new and homogeneous metallicity scale
for Galactic classical Cepheids

II. The abundance of iron and $\alpha$ elements ††thanks:  Partly based on
observations made with ESO Telescopes at the La Silla/Paranal Observatories
under program IDs: 072.D-0419, 073.D-0136, and 190.D-0237 for HARPS spectra;
084.B-0029, 087.A-9013, 074.D-0008, 075.D-0676, and 60.A-9120 for FEROS
spectra; 081.D-0928, 082.D-0901, 089.D-0767, and 093.D-0816 for UVES spectra.,
††thanks:  Partly based on data obtained with the STELLA robotic telescopes in
Tenerife, a facility of The Leibniz Institute for Astrophysics Potsdam (AIP)
jointly operated by the AIP and by the Instituto de Astrofísica de Canarias
(IAC)., ††thanks:  Tables 2, 3, 4, 8, and 9 are only available in electronic
form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.

R. da Silva 1122 J. Crestani 3311 G. Bono 3311 V.F. Braga 1122 V. D’Orazi 44
B. Lemasle 55 M. Bergemann 66 M. Dall’Ora 77
G. Fiorentino 11 P. François 8899 M.A.T. Groenewegen 1010 L. Inno 1111 V.
Kovtyukh 1212 R.-P. Kudritzki 13131414
N. Matsunaga 15151616 M. Monelli 17171818 A. Pietrinferni 1919 L. Porcelli
2020 J. Storm 2121 M. Tantalo 3311 F. Thévénin 2222

(Received …; accepted …)

###### Abstract

Context. Classical Cepheids are the most popular distance indicators and
tracers of young stellar populations. The key advantage is that they are
bright and they can be easily identified in Local Group and Local Volume
galaxies. Their evolutionary and pulsation properties depend on their chemical
abundances.

Aims. The main aim of this investigation is to perform a new and accurate
abundance analysis of two tens of calibrating Galactic Cepheids using high
spectral resolution (R$\sim$40,000–115,000) and high S/N spectra ($\sim$400)
covering the entire pulsation cycle.

Methods. We focus our attention on possible systematics aﬀecting the estimate
of atmospheric parameters and elemental abundances along the pulsation cycle.
We cleaned the line list by using atomic transition parameters based on
laboratory measurements and by removing lines that are either blended or
display abundance variations along the pulsation cycle.

Results. The spectroscopic approach that we developed brings forward small
dispersions in the variation of the atmospheric parameters ($\sigma$($T_{\rm
eff}$)$\sim$50 K, $\sigma$($\log{g}$)$\sim$0.2 dex, and
$\sigma$($\xi$)$\sim$0.2 km s-1) and in the abundance of both iron ($\lesssim$
0.05 dex) and $\alpha$ elements ($\lesssim$ 0.10 dex) over the entire
pulsation cycle. We also provide new and accurate eﬀective temperature
templates by splitting the calibrating Cepheids into four diﬀerent period
bins, ranging from short to long periods. For each period bin, we performed an
analytical ﬁt with Fourier series providing $\theta=5040\leavevmode\nobreak\
/{T_{\rm eff}}$ as a function of the pulsation phase.

Conclusions. The current findings are a good viaticum to trace the chemical
enrichment of the Galactic thin disk by using classical Cepheids and a
fundamental stepping stone for further investigations into the more metal-poor
regime typical of Magellanic Cepheids.

###### Key Words.:

Galaxy: disk – stars: abundances – stars: fundamental parameters – stars:
variables: Cepheids

##  1 Introduction

The current empirical evidence indicates that the estimates of the Hubble
constant based on Cosmic Microwave Background (CMB) and on the standard
cosmological model ($\Lambda$ Cold Dark Matter, $\Lambda$CDM) are at odds with
the direct estimates based on primary (classical Cepheids) and secondary (type
Ia Supernovae) distance indicators (Planck Collaboration et al., 2020; Riess
et al., 2019). The current tension between early and late Universe
measurements is at a 5-$\sigma$ level. It is not clear whether it might be
explained by unknown systematics or whether this disagreement is opening the
path for new physics beyond the standard cosmological model (see, e.g.,
Niedermann & Sloth, 2020, and references therein). In this context, it is
worth mentioning that the use of the Tip of the Red Giant Branch (TRGB) as
primary distance indicators gives values of the Hubble constant that are
intermediate between those based on the CMB and those based on the Cepheid
distance scale (Freedman et al., 2020).

There are solid reasons to believe that the difference between the TRGB and
the Cepheid distance scale is caused by the dependence of the metal content on
the zero-point and/or the slope of the adopted diagnostics (Bono et al., 2010;
Pietrzyński et al., 2013). The dependence of optical and near-infrared
diagnostics on metallicity is still a controversial issue for both theoretical
(Marconi et al., 2005; Bono et al., 2010) and empirical (Groenewegen, 2018,
2020; Ripepi et al., 2019, 2021; Breuval et al., 2020, 2021) approaches.
However, the improved accuracy of geometrical distances from Gaia DR3, and the
homogeneity of spectroscopic abundances will provide solid constraints on this
longstanding problem (Ripepi et al., 2022, and references therein).

Table 1: Sample of 20 calibrating Cepheids with high-resolution spectra
covering either a substantial portion or the entire pulsation cycle.

Name | $\alpha_{\rm ICRS}$ | $\delta_{\rm ICRS}$ | Mode |  ${\rm[Fe/H]}_{\rm lit}$ $\pm$ $\sigma$ |  Ref.aa$a$ 1: Genovali et al. (2014); for T Vul, FF Aql, $δ$ Cep, and S Sge we assumed a typical error of 0.1 dex since they were not provided by the original authors; 2: Proxauf et al. (2018), from which the uncertainty adopted is the largest value between $\sigma$ and std.  | $N_{\rm F}$ | $N_{\rm H}$ | $N_{\rm U}$ | $N_{\rm S}$ | $N_{\rm tot}$
---|---|---|---|---|---|---|---|---|---|---
R TrA | 15:19:45.712 |  $-$66:29:45.742 | 0 |  $-$0.01 |  $\pm$ 0.03 | 2 | 1 | 14 | … | … | 15
T Vul | 20:51:28.238 | +28:15:01.817 | 0 | 0.07 |  $\pm$ 0.10 | 1 | … | … | … | 26 | 26
FF Aql | 18:58:14.748 | +17:21:39.296 | 1 | 0.10 |  $\pm$ 0.10 | 1 | … | … | … | 27 | 27
S Cru | 12:54:21.997 |  $-$58:25:50.215 | 0 | 0.09 |  $\pm$ 0.04 | 2 | … | … | … | 13 | 13
$δ$ Cep | 22:29:10.265 | +58:24:54.714 | 0 | 0.09 |  $\pm$ 0.10 | 1 | … | … | … | 18 | 18
Y Sgr | 18:21:22.986 |  $-$18:51:36.002 | 0 | 0.00 |  $\pm$ 0.06 | 2 | … | 20 | … | 4 | 24
XX Sgr | 18:24:44.500 |  $-$16:47:49.820 | 0 | 0.06 |  $\pm$ 0.04 | 2 | 4 | … | 5 | 3 | 12
$η$ Aql | 19:52:28.368 | +01:00:20.370 | 0B | 0.24 |  $\pm$ 0.09 | 2 | … | … | … | 11 | 11
S Sge | 19:56:01.261 | +16:38:05.236 | 0B | 0.14 |  $\pm$ 0.10 | 1 | … | … | … | 21 | 21
V500 Sco | 17:48:37.501 |  $-$30:28:33.461 | 0B |  $-$0.07 |  $\pm$ 0.08 | 1 | … | … | 4 | 3 | 7
$β$ Dor | 05:33:37.512 |  $-$62:29:23.323 | 0B |  $-$0.03 |  $\pm$ 0.05 | 2 | 1 | 46 | … | … | 47
$ζ$ Gem | 07:04:06.530 | +20:34:13.074 | 0B | 0.16 |  $\pm$ 0.05 | 2 | … | 47 | … | 81 | 128
VY Sgr | 18:12:04.566 |  $-$20:42:14.465 | 0B | 0.25 |  $\pm$ 0.08 | 2 | 29 | … | 4 | … | 33
UZ Sct | 18:31:22.367 |  $-$12:55:43.339 | 0B | 0.11 |  $\pm$ 0.09 | 2 | 28 | … | 6 | … | 34
Y Oph | 17:52:38.702 |  $-$06:08:36.875 | 0 | 0.08 |  $\pm$ 0.05 | 2 | … | 8 | … | … | 8
RY Sco | 17:50:52.345 |  $-$33:42:20.411 | 0 | 0.01 |  $\pm$ 0.06 | 1 | … | … | 5 | 3 | 8
RZ Vel | 08:37:01.303 |  $-$44:06:52.844 | 0 | 0.08 |  $\pm$ 0.06 | 2 | 1 | 11 | … | … | 12
V340 Ara | 16:45:19.112 |  $-$51:20:33.394 | 0 | 0.23 |  $\pm$ 0.07 | 2 | 26 | … | 6 | … | 32
WZ Sgr | 18:16:59.716 |  $-$19:04:32.989 | 0 | 0.28 |  $\pm$ 0.08 | 1 | 1 | … | 5 | 3 | 9
RS Pup | 08:13:04.216 |  $-$34:34:42.692 | 0 | 0.14 |  $\pm$ 0.07 | 2 | … | 15 | … | 3 | 18
111The first four columns give the star name, the right ascension and
declination, and the pulsation mode (0: fundamental; 1: first overtone; 0B:
fundamental with bump). The Cols. 5 and 6 give the iron abundance from
literature and the corresponding references. Columns from 7 to 10 show the
number of optical spectra used for each spectrograph: $N_{\rm F}$: FEROS;
$N_{\rm H}$: HARPS; $N_{\rm U}$: UVES; $N_{\rm S}$: STELLA. The last column
lists the total number of spectra per target.

Classical Cepheids are bright ($-2\leq\ M_{V}\leq\ -7$ mag) radially pulsating
stars with periods ranging from roughly one day to more than one hundred days
(see, e.g., Skowron et al., 2019, and references therein). Their iron
abundances range from solar values in the solar vicinity to $\sim$0.5 dex more
metal-rich in the inner Galactic disk and to $-$0.5 dex more metal-poor in the
outer disk (Inno et al., 2019). Cepheids in the Large Magellanic Cloud (LMC)
are on average a factor of two more metal-poor (Romaniello et al., 2021), and
those in the Small Magellanic Cloud (SMC) a factor of four more metal-poor
than solar (Lemasle et al., 2017). Cepheids are intermediate-mass stars
crossing the instability strip during central helium burning, the so-called
blue loops, and their progenitor mass ranges from $\sim$3 to $\sim$10
$M_{\odot}$ – the exact limit depends on the chemical composition (Bono et
al., 2000b; De Somma et al., 2021). Classical Cepheids are solid stellar
beacons to trace young stellar populations even in the far side of the
Galactic thin disk (Minniti et al., 2020, 2021) and in stellar systems
experiencing recent star formation events such as dwarf irregulars and star
forming galaxies (Neeley et al., 2021). Their period-luminosity relations
(Breuval et al., 2020; Ripepi et al., 2020) provide accurate (at the level of
$\sim$1-2%) measurements of individual distances. Individual distances
together with detailed information concerning their elemental abundances (iron
peak, $\alpha$, neutron capture), provide the unique opportunity to
investigate the radial gradients and, in turn, the recent (t $\leq$ 200-300
Myr) chemical enrichment history of the thin disk and nearby stellar systems
(Romaniello et al., 2008; Lemasle et al., 2013; Genovali et al., 2015; da
Silva et al., 2016).

Furthermore, classical Cepheids are fundamental laboratories for investigating
not only the advanced evolutionary phases of intermediate-mass stars (Bono et
al., 2000a; Neilson et al., 2011; Prada Moroni et al., 2012; De Somma et al.,
2021), but also for constraining the physical mechanisms driving their
variability (Bono et al., 1999; Marconi et al., 2013).

Although classical Cepheids are the cross-road of several recent and long-
standing astrophysical problems, we still lack accurate investigations
concerning the variation of their physical properties along the pulsation
cycle. The current studies are mainly based on both optical and near-infrared
(NIR) light curves, but we still lack detailed spectroscopic analysis
concerning the variation of effective temperature, surface gravity, and
microturbulent velocity (see, e.g., Lemasle et al., 2020; Kovtyukh, 2007;
Proxauf et al., 2018, and references therein).

Figure 1: Examples of high-resolution spectra for the stars $β$ Dor and $ζ$
Gem in different pulsation phases (values on the right side of each panel).
The vertical red lines indicate the position in the rest-frame wavelength of
atomic lines either only used for the abundance determination or also adopted
by the LDR method. For the sake of clarity the continuum of each spectrum was
shifted to arbitrary levels.

Pioneering investigations based on multiple (from a few to almost two dozen)
spectra along the pulsation cycle were provided more than ten years ago by
Luck & Andrievsky (2004), Kovtyukh et al. (2005), Andrievsky et al. (2005),
and Luck et al. (2008), in which they evaluated the variation of the
atmospheric parameters and the abundances in both short- and long-period
Cepheids. More recently, we performed a detailed investigation, based on a
large dataset of high-resolution spectra, for a sample of 14 calibrating
Cepheids (Proxauf et al., 2018). In that work, the number of spectra per
object ranges from five to more than 120, and the spectra covered a
significant portion of the pulsation cycle. In the same investigation we also
used a new and more homogeneous list of Line Depth Ratios (LDR) to estimate
the effective temperature directly from the spectra, an approach that is
independent of the photometric properties. The reader interested in a more
detailed discussion is referred to Kovtyukh (2007, and references therein).
However, all these investigations were hampered by three limitations:

  * a)

Line list – We compiled a list of iron lines using line lists available in the
literature. The sources of atomic transition parameters were inhomogeneous,
because they did not pass a detailed scrutiny.

  * b)

Phase coverage – In spite of the large number of spectra adopted in the
spectroscopic analysis, several objects had spectra collected at similar
phases and, consequently, they had a modest phase coverage.

  * c)

Intrinsic variation – The inhomogeneity of the atomic transition parameters
introduces uncertainties in the spectroscopic surface gravity and in the
microturbulent velocity. This caused an increase in the standard deviation at
fixed pulsation phase, and in turn, some limitation in the analysis of their
variation along the pulsation cycle.

To overcome these limitations, the current investigation is based on a new
approach including five steps: $i)$ We updated the atomic transition
parameters of our line lists with more recent and accurate laboratory
measurements available in the literature; $ii)$ We collected new high signal-
to-noise ratio (S/N) and high-resolution spectra for our sample of calibrating
Cepheids, and we also performed a detailed search for similar quality spectra
available in online science archives. This translates into an increase in the
number of calibrating Cepheids by almost 50% (from 14 to 20). Moreover, we
also added new spectra to Cepheids already included in the former sample. The
current sample includes 19 stars pulsating in the fundamental mode and, for
the first time used by us as a calibrating Cepheid, one star (FF Aql)
pulsating in the first overtone mode, and their metallicities range from
$-$0.08 ($β$ Dor) to 0.19 dex (VY Sgr); $iii)$ We performed a detailed visual
check of hundreds of lines along the pulsation cycle to create a new
homogeneous line list. Special attention was paid in the identification of
lines showing blends (or poor definition of the continuum), or having any
correlation with the effective temperature (due to possible NLTE effects) or
with the equivalent widths (due to saturated lines), or presenting any
systematic over- or under-abundance; $iv)$ We improved the algorithm that we
use for the estimate of the atmospheric parameters, and in turn, for the
estimate of the elemental abundances; $v)$ In addition to the atmospheric
parameters and iron abundances, the current study also includes the abundances
of five $\alpha$ elements: Mg, Si, S, Ca, and Ti.

We notice that the star AV Sgr, which was one of the calibrating Cepheids in
Proxauf et al. (2018), is not included in the current work. The reason is
that, after the revision of the Fe line list, we were not able to derive
accurate estimates of the atmospheric parameters for many of the spectra of
this variable. The number of spectra with accurate measurements was not enough
to provide a good coverage of the pulsation cycle, and therefore, it was
removed from the list of calibrating Cepheids. The parameters published in our
previous paper can still be used, but keeping in mind that they are based on a
different version of the line list.

The structure of the paper is the following: in Sect. 2 we discuss the
spectroscopic dataset and provide detailed information concerning the current
sample of calibrating Cepheids. In Sect. 3 we discuss the data reduction
process and introduce the atomic line lists used. Sect. 4 includes a
description of the radial-velocity and photometric data used. In Sect. 5 we
explain how we derived the atmospheric parameters and their variation along
the pulsation cycle. The iron and $\alpha$-element abundances and their
variation along the pulsation cycle are discussed in Sect. 6. A summary of the
results and some final remarks concerning this project are given in Sect. 7.

##  2 Spectroscopic datasets

The spectroscopic datasets analyzed in the current paper are spectra collected
using several different instruments. Three of them are mounted on telescopes
of the European Southern Observatory (ESO): the High Accuracy Radial velocity
Planet Searcher (HARPS; Mayor et al., 2003) at the 3.6 m, the Fiber-fed
Extended Range Optical Spectrograph (FEROS; Kaufer et al., 1999) at the 2.2 m
MPG/ESO, and the Ultraviolet and Visual Echelle Spectrograph (UVES; Dekker et
al., 2000) at the Very Large Telescope. The other instrument is the STELLA
Echelle Spectrograph (SES; Strassmeier et al., 2004, 2010) located at the
Izãna Observatory on Tenerife in the Canary islands.

The spectral resolution of the quoted spectrographs for the instrument
settings used are R$\sim$40 000 (UVES), R$\sim$115 000 (HARPS), R$\sim$48 000
(FEROS), and R$\sim$55 000 (STELLA). For details on the wavelength ranges of
our spectra we refer the reader to Sect. 2 of the papers Proxauf et al. (2018)
and Crestani et al. (2021).

The entire spectroscopic sample includes reduced spectra downloaded from the
ESO and the STELLA archives, for a total of 1383 spectra of 285 stars: 199
HARPS spectra of 9 stars, 419 FEROS spectra of 151 stars, 450 STELLA spectra
of 68 stars, and 315 UVES spectra of 123 stars. Their S/N ranges from a few
tens to more than 400 (about 70% of our spectra have S/N of at least 100).
From this sample we selected the spectra of 20 targets that cover either a
significant part (about half or more) or the entire pulsation cycle. From now
on we refer to this sub-sample as our sample of 20 calibrating Cepheids.
Details on the number of spectra are given in Table 1.

Figure 2: Iron abundances as a function of the pulsation phase, effective
temperature, and equivalent width. These are examples of good lines that were
kept in our initial line list for HARPS (green circles), FEROS (blue
triangle), and STELLA (red stars) spectra of $β$ Dor and $ζ$ Gem. The hatched
regions around the dashed lines indicate the 1-$\sigma$ uncertainty around the
mean.

##  3 Data reduction and analysis

As mentioned in Proxauf et al. (2018), the spectra from UVES and HARPS (Phase
3) were already pre-reduced (i.e., reduced up to the wavelength calibration
step) by their own pipeline. As previously done, the FEROS spectra were
reduced using a modified version of the Data Reduction System (DRS) pipeline.
A few tens of FEROS spectra collected before 2004, which initially could not
be reduced, were later on reduced and delivered by one of the Phase 3 Data
Releases (DR3 and DR3.1). However, they are all low-quality spectra that were
not included in our analysis.

After the pre-reduction steps, all the selected spectra were normalized to the
continuum using the Image Reduction and Analysis Facility (IRAF 222The IRAF
package is distributed by the National Optical Astronomy Observatories (NOAO),
USA.) by fitting cubic spline functions to a set of continuum windows. An
exception is the case of the STELLA spectra, which were reduced by a dedicated
pipeline (Weber et al., 2012) based on IRAF routines and that includes several
reduction steps up to the continuum normalization.

For more details about the quality of the spectra we refer the reader to Sect.
3 of Proxauf et al. (2018), which also includes a figure (their Fig. 1)
showing a few examples of HARPS, FEROS, UVES, and STELLA spectra of different
metallicities and S/N. In our Fig. 1, we plot an example of HARPS spectra for
$β$ Dor and of STELLA spectra for $ζ$ Gem in different pulsation phases,
showing how the profile of the absorption lines may change (in depth and in
shape) along the pulsation cycle. Studies of line asymmetry changes over the
pulsation cycle of Cepheid stars were also reported by Nardetto et al. (2006)
and Nardetto et al. (2008), in a series of papers on high-resolution
spectroscopy, for Fe i and H$\alpha$ lines in the optical and, more recently,
by Nardetto et al. (2018) for the Na i line at 2208.969 nm.

###  3.1 Line lists and equivalent widths

Similarly to what we did in our previous works, here we use three different
line lists:

  1. a)

one containing atomic lines of N, Si, S, Ca, Ti, V, Cr, Mn, Fe, Co, and Ni,
created from a combination of four line lists received from V. Kovtyukh (for a
total of 153 atomic lines). These lines are used to derive the effective
temperature ($T_{\rm eff}$) of our sample stars according to the procedure
described in Sect. 4.2 of Proxauf et al. (2018);

  2. b)

one containing 424 Fe i and 97 Fe ii lines, created from a combination of
lines from Genovali et al. (2013) with lines from the Gaia-ESO Survey (GES,
Gilmore et al., 2012; Randich et al., 2013). It is used as a reference line
list in the determination of both the stellar surface gravity ($\log{g}$), as
described in Sect. 5.1, and the stellar metallicity, as discussed in Sect. 6;

  3. c)

one containing lines of $\alpha$ elements (Mg: 9 lines; Si: 11 lines; S: 1
line; Ca: 36 lines; and Ti: 72 lines), based on line lists from For & Sneden
(2010), Venn et al. (2012), Lemasle et al. (2013), and McWilliam et al. (2013,
and references therein).

Table 2: Excerpt from the list of Fe i and Fe ii atomic lines.

$\lambda$ [Å] | Species |  $\xi$ [eV] | $\log{gf}$ |  Quality flag | Ref.
---|---|---|---|---|---
3763.789 | Fe i | 0.989 |  $-$0.220 | 1 | 1
3787.880 | Fe i | 1.010 |  $-$0.840 | 1 | 1
… | … | … | … | … |
7655.488 | Fe ii | 3.892 |  $-$3.560 | 1 | 2
7711.724 | Fe ii | 3.903 |  $-$2.500 | 1 | 2
333The quality flag column indicates if the line was used (1) or not used (0)
in the current study. The complete table is available at the CDS. 1: O’Brian
et al. (1991); 2: Meléndez & Barbuy (2009); 3: NIST: Kramida et al. (2020).

Table 3: The same as in Table 2 but showing an excerpt of atomic lines from the list of $\alpha$ elements. $\lambda$ [Å] | Species |  $\xi$ [eV] | $\log{gf}$ |  Quality flag | Ref.
---|---|---|---|---|---
3829.355 | Mg i | 2.709 |  $-$0.227 | 0 | 3
4571.096 | Mg i | 0.000 |  $-$5.620 | 1 | 3
… | … | … | … | … |
6606.956 | Ti ii | 2.061 |  $-$2.790 | 1 | 3
7214.729 | Ti ii | 2.590 |  $-$1.750 | 1 | 3

The Cepheids variables are bright stars and thus we have at our disposal very
high S/N, broad wavelength range spectra. The precision of the atmospheric
parameters and of the chemical abundances, however, depends not only on the
quality of the spectra, but also on the quality of the adopted line list. With
this in mind, a large portion of the present work was dedicated to finding the
best atomic transitions for iron and $\alpha$ elements that can be detected in
Cepheids. In practice, this means $i)$ collecting the most precise transition
parameters, $ii)$ removing absorption lines that are blended with other lines,
and $iii)$ eliminating lines that for any reason deviate significantly from
the average value for their chemical species, or that display a dependency on
any quantity that changes across the pulsation cycle, such as the effective
temperature.

In order to address the first point, we departed from a list of hundreds of
atomic transitions for iron and $\alpha$ elements commonly employed in the
literature. We updated their transition parameters whenever possible with the
updated laboratory measurements from Ruffoni et al. (2014), Den Hartog et al.
(2014), and Belmonte et al. (2017) for Fe i, Den Hartog et al. (2019) for Fe
ii, Lawler et al. (2013) for Ti i, and Wood et al. (2013) for Ti ii. We also
made ample use of the astrophysical, but homogeneous and precise compilation
of Meléndez & Barbuy (2009) for Fe ii. For the remaining lines, we adopted the
transition parameters collected and updated by the National Institute of
Standards and Technology (NIST) Atomic Spectra Database (Kramida et al.,
2020). If a line is not available in any of these sources, it is eliminated
from the preliminary list and not used for the computation of atmospheric
parameters nor chemical abundances.

With this preliminary list at hand, we addressed the second point mentioned
above by removing any blended lines, using as reference the Solar spectrum
table by Moore et al. (1966) and synthetic spectra. Then we measured their
equivalent width (EW) using the Automatic Routine for line Equivalent widths
in stellar Spectra (ARES, Sousa et al., 2007, 2015).

Figure 3: Top: RV curve of the fundamental pulsator S Cru. The red solid line
displays the PLOESS fit to the RV curve. The horizontal red dashed and dotted
lines display the mean RV ($\langle$RV$\rangle$) and its uncertainty. The
vertical red dotted line displays the phase where RV equals
$\langle$RV$\rangle$ on the rising branch. The name, $\langle$RV$\rangle$,
period and reference epoch are labelled. Bottom: same as in the top panel but
for the $V$-band light curve, its PLOESS fit and its mean value. We note that,
within the uncertainty, the light curve is co-phased with the RV curve.

Finally, we addressed the third point by removing deviant lines. In a first
iteration, we estimated initial values for the atmospheric parameters and the
chemical abundances. Lines are considered deviant if they result in abundances
diverging by at least 3 $\sigma$ from the mean for a given element, or if
their behavior across the pulsation cycle is irregular. This latter situation
is investigated by plotting all line-by-line measurements for each chemical
species and individual exposure of our calibrating stars versus phase,
effective temperature, and equivalent width, and removing lines that show
strong trends in any of those planes. The calibrating stars mostly used are
those with the largest number of individual spectra covering the whole
pulsation cycle, such as $β$ Dor, $ζ$ Gem, and FF Aql.

The final clean lists, shown in Tables 2 and 3 with quality flag 1, were used
to compute the final atmospheric parameters and chemical abundances for the
whole sample. The remaining lines, shown in the same tables with quality flag
0, were measured only for comparison purposes.

An example of a good Fe line is shown in Fig. 2 for $β$ Dor and $ζ$ Gem.
Examples and a description of the typical behavior of irregular (less
reliable) lines are found in the Appendix A, in the plots of Figs. 13-17.
These same irregular lines, however, might provide good results for other
stars, for instance, of different spectral types, or having spectra of
different quality or different resolution. This is the reason why we preferred
to exclude some lines only for this particular analysis, and keep the initial
line lists as they are to be used in future studies. It is worth mentioning
that we only excluded lines that were clearly having problems in most of the
stars in our sample.

##  4 Radial velocity and photometric data

The radial velocity (RV) measurements for our sample were done using the
following methods: $i)$ for HARPS and UVES spectra, we used IRAF packages to
cross-correlate the target spectrum with a solar template spectrum (Solar flux
atlas from 296 to 1300 nm, Kurucz et al., 1984) degraded to the UVES
resolution; $ii)$ for FEROS spectra we adopted the RVs derived by ARES, the
routine we used for the equivalent width measurements (see Sect. 3.1); $iii)$
the STELLA radial velocities are written in the header of the FITS files and
they are based on cross-correlations with a template spectrum, performed by a
dedicated reduction pipeline also using IRAF. In case of the Cepheids, a
G-type dwarf template is used.

Figure 4: Photometric amplitude as a function of the logarithmic period
(Bailey diagram). Our calibrating Cepheids are represented by colored symbols
whereas the Cepheids from literature (Luck et al., 2011; Luck & Lambert, 2011;
Lemasle et al., 2013; Yong et al., 2006) are shown in light gray circles.
Table 4: Excerpt from the list of atmospheric parameters, Fe abundances, and
radial velocities for each spectrum of the 20 calibrating Cepheids.

Name | Dataset |  MJD [d] |  $T_{\rm eff}$ $\pm$ $\sigma$ [K] | $\log{g}$ |  $\upsilon_{t}$ [km s-1] |  Fe i $\pm$ $\sigma$ | $N_{\rm{{Fe\textsc{i}}}}$ |  Fe ii $\pm$ $\sigma$ | $N_{\rm{{Fe\textsc{ii}}}}$ |  $RV$ $\pm$ $\sigma$ [km s-1]
---|---|---|---|---|---|---|---|---|---|---
R TrA | HARPS | 53150.1353086 | 5918 |  $\pm$ 154 | 2.3 | 3.5 |  $-$0.02 |  $\pm$ 0.09 | 86 |  $-$0.03 |  $\pm$ 0.03 | 7 | 2.57 |  $\pm$ 1.38
R TrA | HARPS | 53150.1453265 | 5892 |  $\pm$ 128 | 2.3 | 3.6 |  $-$0.04 |  $\pm$ 0.09 | 81 |  $-$0.05 |  $\pm$ 0.02 | 8 | 2.40 |  $\pm$ 1.37
… | … | … |  | … | … | … |  | … | … |  | … | … |  | …
RS Pup | STELLA | 57708.2167938 | 5508 |  $\pm$ 98 | 0.9 | 3.4 | 0.12 |  $\pm$ 0.11 | 107 | 0.12 |  $\pm$ 0.15 | 7 | 7.40 |  $\pm$ 0.20
RS Pup | STELLA | 57713.2628277 | 5298 |  $\pm$ 111 | … | … |  | … | … |  | … | … | 16.25 |  $\pm$ 0.20

444The first three columns give the target name, spectroscopic dataset, and
Modified Julian Date at which the spectrum was collected. Columns 4, 5, and 6
give, respectively, the effective temperature and its standard deviation,
surface gravity, and microturbulent velocity. The Cols. 7-8 and 9-10 list both
the Fe i and Fe ii abundances derived from individual lines together with the
standard deviations and the number of lines used (for the Fe ii abundances, a
minimal uncertainty of 0.11 dex was adopt when computing the mean abundances,
as described in the text). The last column gives the radial velocities with
their uncertainties. The complete table is available at the CDS.

Table 5: Photometric and radial-velocity parameters for our sample of 20
calibrating Cepheids.

Name |  V [mag] |  $\Delta$V [mag] |  $\gamma$ [km/s] |  $\Delta$RV [km/s] |  LC sourceaa$a$ 0: ASAS-SN (Shappee et al., 2014; Kochanek et al., 2017); 1: Groenewegen (2008, and references therein); 2: Pel (1976). |  Period [days] |  $T_{0}-2\,400\,000$ [days] |  Ephemerides sourcebb$b$ 0: Period and $T_{0}$ derived from our own radial velocity curves; 1: Period and $T_{0}$ derived from the radial velocities by Groenewegen (2008, and references therein); 2: $T_{0}$ derived from the light curve; 3: $T_{0}$ derived from the radial velocities by Groenewegen (2008, and references therein); 4: Period and $T_{0}$ derived from the light curve.
---|---|---|---|---|---|---|---|---
R TrA | 6.66 |  $\pm$ 0.02 | 0.56 |  $\pm$ 0.02 | –12.44 |  $\pm$ 2.54 | 28.1 |  $\pm$ 2.2 | 1 | 3 | .3892582 | 44 423.59377 | 1
T Vul | 5.75 |  $\pm$ 0.03 | 0.64 |  $\pm$ 0.04 | 0.85 |  $\pm$ 0.11 | 34.5 |  $\pm$ 3.5 | 1 | 4 | .435462 | 57 632.18950 | 0
|  |  |  |  |  |  |  |  |  |  |  | 50 735.20310 | 2
FF Aql | 5.19 |  $\pm$ 0.01 | 0.33 |  $\pm$ 0.03 | –11.36 |  $\pm$ 0.10 | 16.1 |  $\pm$ 1.5 | 2 | 4 | .470916 | 56 888.66207 | 0
S Cru | 6.60 |  $\pm$ 0.02 | 0.73 |  $\pm$ 0.03 | –4.70 |  $\pm$ 0.28 | 33.8 |  $\pm$ 4.3 | 1 | 4 | .68973 | 55 279.67331 | 0
$δ$ Cep | 3.94 |  $\pm$ 0.01 | 0.84 |  $\pm$ 0.04 | –16.32 |  $\pm$ 0.20 | 38.9 |  $\pm$ 4.9 | 1 | 5 | .3663 | 56 917.54900 | 0
Y Sgr | 5.74 |  $\pm$ 0.02 | 0.72 |  $\pm$ 0.03 | –2.84 |  $\pm$ 3.02 | 34.2 |  $\pm$ 3.0 | 1 | 5 | .7733866 | 54 946.78504 | 1
XX Sgr | 8.86 |  $\pm$ 0.02 | 0.87 |  $\pm$ 0.04 | 12.30 |  $\pm$ 0.46 | 41.6 |  $\pm$ 1.2 | 1 | 6 | .4243013 | 54 946.67631 | 1
$η$ Aql | 3.89 |  $\pm$ 0.02 | 0.78 |  $\pm$ 0.03 | –14.47 |  $\pm$ 0.63 | 42.1 |  $\pm$ 2.2 | 1 | 7 | .1768572 | 50 724.43545 | 1
S Sge | 5.61 |  $\pm$ 0.03 | 0.72 |  $\pm$ 0.03 | –16.40 |  $\pm$ 0.25 | 39.3 |  $\pm$ 4.7 | 1 | 8 | .3445707 | 57 636.28926 | 0
|  |  |  |  |  |  |  |  |  |  |  | 48 746.52000 | 3
|  |  |  |  |  |  |  |  |  | 8 | .3823514 | 50 339.17710 | 4
V500 Sco | 8.74 |  $\pm$ 0.01 | 0.71 |  $\pm$ 0.05 | –7.61 |  $\pm$ 3.37 | 32.2 |  $\pm$ 5.4 | 1 | 9 | .317 | 44 789.61621 | 1
$β$ Dor | 5.25 |  $\pm$ 0.01 | 0.63 |  $\pm$ 0.06 | 8.17 |  $\pm$ 0.11 | 32.9 |  $\pm$ 2.3 | 2 | 9 | .84308 | 53 284.79472 | 0
|  |  |  |  |  |  |  |  |  | 9 | .8527371 | 41 149.27150 | 4
$ζ$ Gem | 3.89 |  $\pm$ 0.03 | 0.49 |  $\pm$ 0.03 | 6.78 |  $\pm$ 0.42 | 26.5 |  $\pm$ 1.0 | 1 | 10 | .149816 | 54 805.78476 | 1
VY Sgr | 11.46 |  $\pm$ 0.02 | 1.14 |  $\pm$ 0.05 | 14.23 |  $\pm$ 1.58 | 50.4 |  $\pm$ 4.4 | 1 | 13 | .55845 | 54 171.32722 | 1
UZ Sct | 11.25 |  $\pm$ 0.03 | 0.75 |  $\pm$ 0.05 | 39.52 |  $\pm$ 0.18 | 44.7 |  $\pm$ 3.5 | 0 | 14 | .744162 | 56 171.62381 | 0
Y Oph | 6.15 |  $\pm$ 0.02 | 0.49 |  $\pm$ 0.03 | –7.56 |  $\pm$ 2.01 | 16.5 |  $\pm$ 2.0 | 1 | 17 | .12633896 | 53 209.96732 | 1
RY Sco | 8.02 |  $\pm$ 0.02 | 0.83 |  $\pm$ 0.03 | –17.73 |  $\pm$ 2.02 | 30.1 |  $\pm$ 3.0 | 1 | 20 | .321538 | 45 077.36644 | 1
RZ Vel | 7.09 |  $\pm$ 0.04 | 1.17 |  $\pm$ 0.05 | 24.11 |  $\pm$ 4.52 | 54.6 |  $\pm$ 5.4 | 1 | 20 | .399727 | 50 776.13608 | 1
V340 Ara | 10.20 |  $\pm$ 0.06 | 1.08 |  $\pm$ 0.07 | –76.25 |  $\pm$ 4.19 | 47.7 |  $\pm$ 9.3 | 1 | 20 | .811386 | 56 138.97597 | 0
WZ Sgr | 8.03 |  $\pm$ 0.03 | 1.08 |  $\pm$ 0.05 | –17.78 |  $\pm$ 1.17 | 54.2 |  $\pm$ 2.9 | 1 | 21 | .849708 | 50 691.28534 | 1
RS Pup | 6.99 |  $\pm$ 0.03 | 1.11 |  $\pm$ 0.04 | 25.80 |  $\pm$ 1.91 | 48.5 |  $\pm$ 2.1 | 1 | 41 | .488634 | 53 092.94483 | 1

555From left to right the columns give the star name, the mean magnitude in
visual bands, the magnitude amplitude, the mean radial velocity, the RV
amplitude, the light curve source, the pulsation period and zero-phase
reference epoch of mean magnitude (or mean radial velocity), and the
Ephemerides source. For the variables VY Sgr, RZ Vel, and WZ Sgr, our RV
measurements and those by Groenewegen (2008, and references therein) show a
shift in phase of $-$0.05, +0.03, and $-$0.03, respectively. To co-phase the
two datasets, the latter sample of radial velocities plotted in Fig. 18 were
shifted.

The uncertainties adopted for these RV estimates depend on the method used:
for HARPS, UVES, and STELLA spectra, the uncertainties are estimated by IRAF
during the cross-correlation; for FEROS spectra, they are estimated by ARES
using spectral windows free of lines. Typical values for our HARPS, UVES, and
FEROS spectra range from a few m/s in the best cases to a few km s-1,
depending on the spectral type. For the STELLA spectra, we adopted an
uncertainty of 0.2 km s-1. This is a conservative estimate considering that
typical formal errors for high S/N spectra of Cepheids stars is about 50 m s-1
and that the RV zero point for this spectrograph has RMS variations of 30 to
150 m s-1 over two years. The RV estimates and the corresponding uncertainties
for our sample of calibrating Cepheids are listed in Table 4.

In addition to our own RV measurements, we adopted literature RV curves and
light curves to provide accurate ephemerides, meaning reference epoch
($T_{0}$) plus pulsation period ($P$) for these variables. More specifically,
we collected RV curves and $V$-band light curves from Pel (1976) and
Groenewegen (2008, and references therein). For UZ Sct, we adopted the
$V$-band light curve from the All-Sky Automated Survey for Supernovae (ASAS-
SN, Shappee et al., 2014; Kochanek et al., 2017) because it is the only well-
sampled light curve that can be co-phased with our RV time series.

Table 6: Mean atmospheric parameters derived for the 20 calibrating Cepheids.

Name |  $\langle$$T_{\rm eff}$$\rangle$ $\pm$ $\sigma$ [K] |  $\langle\theta\rangle\pm\sigma$ |  $\Delta\theta\pm\sigma$ |  $N_{\rm spec}$ ($T_{\rm eff}$) |  $\langle$$\log{g}$$\rangle$ $\pm$ $\sigma$ |  $\langle$$\upsilon_{t}$$\rangle$ $\pm$ $\sigma$ [km s-1] |  [Fe i/H] $\pm$ $\sigma$ |  [Fe ii/H] $\pm$ $\sigma$ |  [Fe/H] $\pm$ $\sigma$ (std) | $N_{\rm spec}$
---|---|---|---|---|---|---|---|---|---|---
R TrA | 6035 |  $\pm$ 25 | 0.840 |  $\pm$ 0.010 | 0.098 |  $\pm$ 0.016 | 15 | 2.01 |  $\pm$ 0.08 | 3.23 |  $\pm$ 0.13 |  $-$0.03 |  $\pm$ 0.02 |  $-$0.03 |  $\pm$ 0.03 |  $-$0.03 |  $\pm$ 0.02 (0.02) | 15
T Vul | 5934 |  $\pm$ 20 | 0.855 |  $\pm$ 0.010 | 0.122 |  $\pm$ 0.024 | 26 | 1.43 |  $\pm$ 0.06 | 3.07 |  $\pm$ 0.10 |  $-$0.04 |  $\pm$ 0.02 |  $-$0.03 |  $\pm$ 0.02 |  $-$0.04 |  $\pm$ 0.02 (0.05) | 26
FF Aql | 6182 |  $\pm$ 18 | 0.820 |  $\pm$ 0.010 | 0.058 |  $\pm$ 0.012 | 27 | 1.35 |  $\pm$ 0.06 | 3.03 |  $\pm$ 0.10 | 0.05 |  $\pm$ 0.02 | 0.05 |  $\pm$ 0.03 | 0.05 |  $\pm$ 0.02 (0.05) | 27
S Cru | 6018 |  $\pm$ 20 | 0.862 |  $\pm$ 0.010 | 0.140 |  $\pm$ 0.028 | 13 | 1.57 |  $\pm$ 0.08 | 3.01 |  $\pm$ 0.14 |  $-$0.03 |  $\pm$ 0.02 |  $-$0.01 |  $\pm$ 0.03 |  $-$0.02 |  $\pm$ 0.02 (0.07) | 13
$δ$ Cep | 5905 |  $\pm$ 22 | 0.862 |  $\pm$ 0.010 | 0.154 |  $\pm$ 0.022 | 18 | 1.42 |  $\pm$ 0.07 | 3.08 |  $\pm$ 0.12 | 0.06 |  $\pm$ 0.02 | 0.05 |  $\pm$ 0.03 | 0.05 |  $\pm$ 0.02 (0.05) | 18
Y Sgr | 5914 |  $\pm$ 25 | 0.877 |  $\pm$ 0.020 | 0.128 |  $\pm$ 0.027 | 24 | 1.58 |  $\pm$ 0.06 | 3.48 |  $\pm$ 0.10 | 0.03 |  $\pm$ 0.02 | 0.02 |  $\pm$ 0.03 | 0.02 |  $\pm$ 0.02 (0.08) | 23
XX Sgr | 5884 |  $\pm$ 29 | 0.855 |  $\pm$ 0.010 | 0.151 |  $\pm$ 0.031 | 12 | 1.25 |  $\pm$ 0.09 | 2.92 |  $\pm$ 0.14 |  $-$0.01 |  $\pm$ 0.03 |  $-$0.01 |  $\pm$ 0.04 |  $-$0.01 |  $\pm$ 0.02 (0.05) | 12
$η$ Aql | 5480 |  $\pm$ 40 | 0.891 |  $\pm$ 0.030 | 0.113 |  $\pm$ 0.035 | 11 | 1.16 |  $\pm$ 0.09 | 3.43 |  $\pm$ 0.15 | 0.09 |  $\pm$ 0.04 | 0.08 |  $\pm$ 0.04 | 0.09 |  $\pm$ 0.03 (0.10) | 11
S Sge | 5777 |  $\pm$ 21 | 0.883 |  $\pm$ 0.012 | 0.134 |  $\pm$ 0.019 | 21 | 1.20 |  $\pm$ 0.07 | 3.13 |  $\pm$ 0.11 | 0.10 |  $\pm$ 0.02 | 0.09 |  $\pm$ 0.03 | 0.09 |  $\pm$ 0.02 (0.05) | 21
V500 Sco | 5797 |  $\pm$ 32 | 0.850 |  $\pm$ 0.053 |  |  | 7 | 1.21 |  $\pm$ 0.11 | 3.04 |  $\pm$ 0.19 |  $-$0.03 |  $\pm$ 0.04 |  $-$0.03 |  $\pm$ 0.04 |  $-$0.03 |  $\pm$ 0.03 (0.02) | 7
$β$ Dor | 5552 |  $\pm$ 13 | 0.903 |  $\pm$ 0.020 | 0.132 |  $\pm$ 0.023 | 47 | 1.15 |  $\pm$ 0.04 | 3.31 |  $\pm$ 0.07 |  $-$0.07 |  $\pm$ 0.01 |  $-$0.08 |  $\pm$ 0.02 |  $-$0.08 |  $\pm$ 0.01 (0.04) | 47
$ζ$ Gem | 5500 |  $\pm$ 8 | 0.914 |  $\pm$ 0.020 | 0.100 |  $\pm$ 0.011 | 128 | 1.14 |  $\pm$ 0.03 | 3.32 |  $\pm$ 0.04 | 0.02 |  $\pm$ 0.01 | 0.01 |  $\pm$ 0.01 | 0.01 |  $\pm$ 0.01 (0.06) | 128
VY Sgr | 5343 |  $\pm$ 27 | 0.954 |  $\pm$ 0.020 | 0.214 |  $\pm$ 0.048 | 24 | 1.09 |  $\pm$ 0.08 | 3.38 |  $\pm$ 0.13 | 0.19 |  $\pm$ 0.04 | 0.18 |  $\pm$ 0.04 | 0.19 |  $\pm$ 0.03 (0.11) | 14
UZ Sct | 5113 |  $\pm$ 25 | 0.979 |  $\pm$ 0.045 |  |  | 25 | 0.87 |  $\pm$ 0.09 | 3.35 |  $\pm$ 0.16 | 0.14 |  $\pm$ 0.04 | 0.10 |  $\pm$ 0.05 | 0.12 |  $\pm$ 0.03 (0.07) | 10
Y Oph | 5609 |  $\pm$ 33 | 0.900 |  $\pm$ 0.013 |  |  | 8 | 1.20 |  $\pm$ 0.11 | 3.79 |  $\pm$ 0.18 |  $-$0.05 |  $\pm$ 0.04 |  $-$0.05 |  $\pm$ 0.05 |  $-$0.05 |  $\pm$ 0.03 (0.02) | 8
RY Sco | 5743 |  $\pm$ 36 | 0.886 |  $\pm$ 0.071 |  |  | 8 | 0.99 |  $\pm$ 0.11 | 3.69 |  $\pm$ 0.18 | 0.05 |  $\pm$ 0.04 | 0.04 |  $\pm$ 0.04 | 0.04 |  $\pm$ 0.03 (0.07) | 8
RZ Vel | 5483 |  $\pm$ 29 | 0.919 |  $\pm$ 0.020 | 0.232 |  $\pm$ 0.037 | 12 | 1.02 |  $\pm$ 0.09 | 4.35 |  $\pm$ 0.15 | 0.04 |  $\pm$ 0.02 | 0.03 |  $\pm$ 0.05 | 0.03 |  $\pm$ 0.02 (0.11) | 11
V340 Ara | 5170 |  $\pm$ 26 | 0.933 |  $\pm$ 0.057 |  |  | 32 | 1.10 |  $\pm$ 0.09 | 4.60 |  $\pm$ 0.16 | 0.12 |  $\pm$ 0.04 | 0.08 |  $\pm$ 0.05 | 0.10 |  $\pm$ 0.03 (0.09) | 10
WZ Sgr | 5411 |  $\pm$ 33 | 0.924 |  $\pm$ 0.067 |  |  | 9 | 0.89 |  $\pm$ 0.11 | 3.77 |  $\pm$ 0.19 | 0.13 |  $\pm$ 0.04 | 0.11 |  $\pm$ 0.05 | 0.12 |  $\pm$ 0.03 (0.07) | 7
RS Pup | 5432 |  $\pm$ 25 | 0.901 |  $\pm$ 0.020 | 0.195 |  $\pm$ 0.032 | 18 | 0.81 |  $\pm$ 0.07 | 3.74 |  $\pm$ 0.12 | 0.07 |  $\pm$ 0.02 | 0.09 |  $\pm$ 0.03 | 0.08 |  $\pm$ 0.02 (0.06) | 16

666Columns from 1 to 5 give, respectively, the star name, the mean effective
temperature and mean $\theta$, the $\theta$ amplitude (see Section 5.2), and
the number of spectra used to compute these mean values. Columns from 6 to 9
lists the surface gravity, microturbulent velocity, and iron abundances from
neutral and ionized lines. These are the weighted mean values and their
standard errors computed from the individual measurements listed in Table 4.
Column 10 gives the weighted iron abundance and its standard error, calculated
from Cols. 8 and 9. The standard deviation calculated using all individual
measurements from both Fe i and Fe ii lines is also shown within parentheses.
The last column gives the number of spectra used to derive $\log{g}$,
$\upsilon_{t}$, and the iron abundances.

We estimated the pulsation periods of our targets by using our own interactive
method (Braga et al., 2016) based on the Lomb-Scargle periodogram (Scargle,
1982). Historically, the most commonly used $T_{0}$ is the time of maximum
light ($T_{\rm max}$), which matches the time of minimum velocity. Starting
from Inno et al. (2015), we have been promoting the use of another reference
for both Cepheids and RR Lyrae, that is, the epoch of the mean magnitude on
the rising branch of the light curve ($T_{\rm mean}^{\rm opt}$) or,
equivalently, the epoch of mean velocity on the decreasing branch of the RV
curve ($T_{\rm mean}^{\rm RV}$). See Fig. 3 for a visualization of this
reference epoch on both the light and RV curves. A detailed procedure to
derive $T_{\rm mean}^{\rm opt}$ is described in Braga et al. (2021). The
advantages of the $T_{\rm mean}^{\rm opt}$ reference epoch over $T_{\rm max}$
are well-explained in Inno et al. (2015) and quantitatively discussed in Braga
et al. (2021).

The light and radial-velocity curves for all the sample stars are shown in the
Appendix B, in Fig. 18. The estimated ephemerides of our targets are listed in
Table 5 together with the mean magnitudes and magnitude amplitudes in the
visual band, and the mean radial velocities and RV amplitudes. We noticed that
for the stars VY Sgr, RZ Vel, and WZ Sgr, for which the new period and $T_{0}$
values are based on the radial velocities derived by Groenewegen (2008, and
references therein), the new ephemerides caused a shift in phase between the
two datasets shown in Fig. 18. To overcome this problem, we applied a shift of
$-$0.05, +0.03, and $-$0.03, respectively, to the literature data plotted in
these panels.

Figure 4 shows the distribution of the current calibrating Cepheids (colored
symbols), together with Galactic Cepheids available in the literature (light
gray circles), in the so-called Bailey diagram (luminosity amplitude as a
function of the logarithmic period). The plotted data display the classical
”V” shape in which the local minimum around ten days is mainly caused by the
Hertzsprung progression (Bono et al., 2000b). By having a look also at Fig.
18, we see a subgroup of classical Cepheids showing a well defined bump on the
light curve. The phase of such bumps moves from the decreasing branch for
periods between $\sim$6 and $\sim$9 days, passing across the maximum for
periods between $\sim$9 and $\sim$12 days, and along the rising branch for
periods between $\sim$12 and $\sim$16 days. At the center of the Hertzsprung
progression (P$\sim$10 days) the light curve is flat topped and the luminosity
amplitude attains a well defined minimum (Bono et al., 2000b, 2002). These
variables are called Bump Cepheids for avoiding to be mixed-up with the Beat
Cepheids, that is, classical Cepheids pulsating simultaneously in two or more
modes. The calibrating Cepheids cover both the short- and the long-period
range, and in particular, the current sample includes seven Cepheids across
the Hertzsprung progression.

##  5 Atmospheric parameters and effective temperature curve templates

###  5.1 Effective temperature, surface gravity, and microturbulent velocity

The approach we adopted to estimate the atmospheric parameters (i.e., the
effective temperature, the surface gravity, and the microturbulent velocity)
was already discussed in detail by Proxauf et al. (2018). Here we only recap
the key points. The $T_{\rm eff}$ along the pulsation cycle was estimated by
using the LDR method, which relies on the correlation between the line depth
ratios of pairs of absorption lines in the spectra of different stars and the
effective temperature of the same stars. The surface gravity was derived
through the ionization equilibrium of Fe i and Fe ii lines, and the
microturbulent velocity was obtained by minimizing the dependence of the
abundances provided by single Fe i lines on their EWs. During this procedure,
the effective temperature was kept fixed, whereas $\log{g}$ and $\xi$ were
iteratively changed until convergence. The metallicity used as input by our
algorithm is updated in each step, and the adopted value is [Fe i/H], which is
the mean iron abundances provided by individual Fe i lines.

The estimates of $T_{\rm eff}$, $\log{g}$, and $\xi$ for the individual
spectrum of our sample are listed in Table 4. The uncertainties on $T_{\rm
eff}$ are the standard deviation calculated using the LDR method, that is,
from the effective temperatures provided by the individual pairs of lines. The
uncertainties on the individual estimates of $\log{g}$ and $\xi$ are assumed
to be of the order of 0.3 dex and 0.5 km s-1, respectively (see Genovali et
al., 2014, for a detailed discussion). The weighted mean values of the
atmospheric parameters are listed in Table 6 together with the corresponding
standard errors calculated for each calibrating Cepheid.

Figure 5: Normalized $\theta$ (5040/$T_{\rm eff}$) as a function of the
pulsation phase. The panels show the phase-folded curves separating the
Cepheids in different bins of pulsation period (values within parentheses):
from 3 to 5 days, from 5 to 7 days, from 7 to 9.5 days, and from 9.5 to 10.5
days. For the variable S Sge, the two outliers shown in Fig. 19 were not used
in the construction of the $T_{\rm eff}$ curve templates. The curves display
the fit of the Fourier series to the data (see Table 7). The standard
deviations of the fit are also shown.

Figure 6: Normalized $\theta$ (5040/$T_{\rm eff}$) as a function of the
pulsation phase showing the fits of the Fourier series to the data of Fig. 5.
The shaded areas display the 1-$\sigma$ standard deviation.

In the Appendix B, Figs. 19, 20, and 21 show the variation of the atmospheric
parameters along the pulsation cycle. To provide a quantitative estimate of
the differences between the current approach and our previous measurements, we
fitted Fourier series to the effective temperature, surface gravity, and
microturbulent velocity curves in order to evaluate their amplitudes,
similarly to what we did in Proxauf et al. (2018, their Figs. 3 and 4). We
found that, on average, the errors on the derived amplitudes for the effective
temperature curves are a factor 2 smaller than our previous estimates, whereas
for the surface gravity the differences are even more significant, with
typical errors almost 3 times smaller. For the microturbulent velocity
variation, the errors on the current amplitude estimates are more than twice
smaller (the dispersions are $\sim$50 K for $T_{\rm eff}$, $\sim$0.2 dex for
$\log{g}$, and $\sim$0.2 km s-1 for $\xi$). This means that the variation of
the atmospheric parameters along the pulsation cycle is now much better
constrained, thanks not only to the larger number of spectra but also to the
better determination of such parameters.

The improvement in the estimate of $\log{g}$ and $\xi$ is mainly due to the
improvement in the line list, which reduced the number of spurious abundance
values provided by unreliable iron lines. As described in the Appendix A, we
removed lines that are blended, lines that display a well defined trend with
effective temperature or with equivalent width, among others. The estimate of
$T_{\rm eff}$ is based on the same line list that we previously used,
therefore the improvement is mainly due to the larger number of spectra
analyzed in the current work. All in all, these changes increased the accuracy
in the atmospheric parameters. The reader interested in detailed discussion
concerning the estimate of the intrinsic parameters of classical Cepheids is
referred to Vasilyev et al. (2018) and Lemasle et al. (2020).

We note in passing that the improvement in the accuracy and in the sampling of
the pulsation cycle allow us to state that the minimum in the effective
temperature is often approached in the same phases in which the minimum in
surface gravity is also approached. Moreover, and even more importantly, these
minimum phases anticipate the increase in microturbulent velocity. This
parameter, as expected, attains it maximum just before the maximum in
effective temperature, which can be clearly observed by comparing several
panels of Figs. 19 and 21.

Table 7: Coefficients of the Fourier series fitted to the data shown in Fig. 5
for different bins of pulsation period.

|  $3<P<5$ [days] |  $5<P<7$ [days] |  $7<P<9.5$ [days] |  $9.5<P<10.5$ [days]
---|---|---|---|---
$c_{0}$ | 0.44 |  $\pm$ 0.21 | 0.44 |  $\pm$ 0.23 | 0.48 |  $\pm$ 0.26 | 0.47 |  $\pm$ 0.12
$c_{1}$ | 2.1 |  $\pm$ 0.5 | 2.1 |  $\pm$ 0.5 |  $-$4.4 |  $\pm$ 0.6 |  $-$4.4 |  $\pm$ 0.2
$c_{2}$ | 0.13 |  $\pm$ 0.22 | 0.15 |  $\pm$ 0.24 | 0.14 |  $\pm$ 0.28 | 0.04 |  $\pm$ 0.12
$c_{3}$ |  $-$4.5 |  $\pm$ 1.4 | 2.2 |  $\pm$ 1.4 | 2.6 |  $\pm$ 2.0 | 4.2 |  $\pm$ 2.9
$c_{4}$ | 0.06 |  $\pm$ 0.21 | 0.06 |  $\pm$ 0.23 | 0.05 |  $\pm$ 0.27 | 0.04 |  $\pm$ 0.11
$c_{5}$ | 1.4 |  $\pm$ 3.6 | 2.6 |  $\pm$ 3.7 | 2.2 |  $\pm$ 5.6 | 4.4 |  $\pm$ 2.7
$c_{6}$ | 0.03 |  $\pm$ 0.21 | 0.03 |  $\pm$ 0.22 | 0.03 |  $\pm$ 0.28 | 0.02 |  $\pm$ 0.11
$c_{7}$ | 1.3 |  $\pm$ 8.3 | 2.7 |  $\pm$ 7.6 |  $-$0.93 |  $\pm$ 9.56 | 4.2 |  $\pm$ 6.9
$c_{8}$ | 0.01 |  $\pm$ 0.19 |  | … |  | … |  | …
$c_{9}$ | 2.8 |  $\pm$ 16.7 |  | … |  | … |  | …
777The Fourier series are of the form $\sum c_{2i}\cos{\left[2(i+1)\pi
x+c_{2i+1}\right]}$ for $i=0,1,\dots,N-1$, where $N$ is the number of terms.

###  5.2 Effective temperature curve templates

By taking advantage of the substantial phase coverage of our sample of
calibrating Cepheids, we used the effective temperature measurements to
compute new templates for different bins of pulsation periods. However,
instead of using the $T_{\rm eff}$ curves directly, we preferred to provide
templates for the theta parameter — defined as $\theta=5040/{T_{\rm eff}}$ —
given its linear dependency on the Cousins R-I color index (Taylor, 1994).

The approach is similar to the NIR light-curve templates provided by Inno et
al. (2015): first, we adopted the ephemerides in Table 5 to fold the $\theta$
curves ($\theta$Cs). Subsequently, we normalized the folded $\theta$Cs by
subtracting their average and dividing by their amplitudes. Our $\theta$Cs are
well sampled, meaning that we have enough phase points to separate the
Cepheids into different period bins and to provide analytical relations for
the $\theta$ curve templates in each bin. We adopted the same period
thresholds introduced by Inno et al. (2015). Therefore, based on their Table
1, we generated four cumulative and normalized theta curves for the bins 2
(3-5 days, three Cepheids), 3 (5-7 days, three Cepheids), 4 (7-9.5 days, two
Cepheids), and 5 (9.5-10.5 days, two Cepheids). By adopting the same period
bins we are able to provide $\theta$C templates that are homogeneous with
those from the NIR light curves. The reader interested in a more detailed and
quantitative discussion concerning the use of cumulative and normalized curves
to derive the analytical fits, together with the adopted thresholds for the
different period bins, is referred to Inno et al. (2015).

Figure 7: Distribution of standard deviations computed from individual
abundances of Fe i and Fe ii lines. The panels show the histogram for spectra
of stars in common with Proxauf et al. (2018).

Figure 8: Atmospheric parameters differences for stars in common with Proxauf
et al. (2018). Each panel shows our previous determinations subtracted by the
current values.

The four normalized $\theta$ curves are separately shown in Fig. 5, and in
Fig. 6 we compare the Fourier series fitted to the data of each bin. The
corresponding coefficients of the fitted functions are listed in Table 7. The
separation of the Cepheids into different period bins is a mandatory step
because not only the amplitude of the theta curves (listed in Table 6), but
also their shape changes with period (as shown in Figs. 5 and 6). Although we
do have data for Cepheids with periods outside of the selected period bins,
they are either not as well sampled as the data for the other Cepheids, or the
theta curves do not overlap well because the number of measurements is too
limited. More accurate and homogeneous data are required to overcome this
limitation.

In order to allow the reader to use these curve templates to derive the mean
effective temperature by using the ephemerides, the luminosity amplitude
($\Delta V$), and a single spectroscopic measurement of the $T_{\rm eff}$, we
also provide a linear relation between $\Delta V$ and $\Delta\theta$:

$\Delta\theta=(0.184\pm 0.014)\Delta V+(0.000\pm 0.011);\,(\sigma=0.013)$ (1)

The estimate of the $\Delta\theta$ amplitude has to be applied to the
normalized template as a multiplicative factor. Only after this rescaling
operation the template can be anchored to the empirical data and used to
derive the mean $T_{\rm eff}$.

Table 8: Excerpt from the list of abundances of $\alpha$ elements for each
spectrum of the 20 calibrating Cepheids.

Name | Dataset |  MJD [d] | [Mg/H] $\pm$ $\sigma$ | $N_{\rm{Mg}}$ | … | Ti i $\pm$ $\sigma$ | $N_{\rm{{Ti\textsc{i}}}}$ | Ti ii $\pm$ $\sigma$ | $N_{\rm{{Ti\textsc{ii}}}}$ | [Ti/H] $\pm$ $\sigma$
---|---|---|---|---|---|---|---|---|---|---
R TrA | HARPS | 53150.1453265 | 0.20 |  $\pm$ 0.11 | 1 | … |  $-$0.21 |  $\pm$ 0.15 | 8 | 0.05 |  $\pm$ 0.10 | 2 |  $-$0.12 |  $\pm$ 0.19
R TrA | HARPS | 53150.1353086 | 0.17 |  $\pm$ 0.11 | 1 | … |  $-$0.17 |  $\pm$ 0.27 | 12 | 0.01 |  $\pm$ 0.10 | 3 |  $-$0.14 |  $\pm$ 0.25
… | … | … |  | … | … | … |  | … | … |  | … | … |  | …
RS Pup | STELLA | 57708.2167938 | 0.24 |  $\pm$ 0.11 | 2 | … |  $-$0.06 |  $\pm$ 0.28 | 15 | 0.22 |  $\pm$ 0.10 | 4 | 0.09 |  $\pm$ 0.19
RS Pup | STELLA | 57713.2628277 |  | … | … | … |  | … | … |  | … | … |  | …

888The first three columns give the target name, spectroscopic dataset, and
Modified Julian Date at which the spectrum was collected. The other columns
give the abundances from both neutral and ionized lines, their standard
deviations, and the number of lines used. For each element X, the weighted
mean of X i and X ii abundances (weighted by 1/$\sigma^{2}$) and its intrinsic
error is also shown. Magnesium and Sulfur abundances are from neutral lines
only. The complete table is available at the CDS.

Table 9: Excerpt from the list of mean abundances of $\alpha$ elements derived
for the 20 calibrating Cepheids.

Name |  [Mg/H] $\pm$ $\sigma$ (std) | $N_{\rm spec}$ | … |  [Ti i/H] $\pm$ $\sigma$ |  [Ti ii/H] $\pm$ $\sigma$ |  [Ti/H] $\pm$ $\sigma$ (std) | $N_{\rm spec}$
---|---|---|---|---|---|---|---
R TrA | 0.10 |  $\pm$ 0.03 (0.08) | 15 | … |  $-$0.19 |  $\pm$ 0.04 |  $-$0.09 |  $\pm$ 0.03 |  $-$0.18 |  $\pm$ 0.04 (0.10) | 15
T Vul | 0.07 |  $\pm$ 0.03 (0.15) | 26 | … |  $-$0.11 |  $\pm$ 0.03 |  $-$0.07 |  $\pm$ 0.02 |  $-$0.10 |  $\pm$ 0.03 (0.10) | 26
… |  | … | … | … |  | … |  | … |  | … | …
WZ Sgr | 0.12 |  $\pm$ 0.06 (0.03) | 4 | … | 0.08 |  $\pm$ 0.07 | 0.01 |  $\pm$ 0.05 | 0.06 |  $\pm$ 0.07 (0.12) | 7
RS Pup | 0.09 |  $\pm$ 0.03 (0.14) | 10 | … | 0.02 |  $\pm$ 0.04 | 0.16 |  $\pm$ 0.03 | 0.04 |  $\pm$ 0.04 (0.14) | 16

999From left to right the columns give the star name, the abundances from both
neutral and ionized lines, their uncertainties, and the number of spectra
used. These are the weighted mean values and their standard errors computed
from the individual measurements listed in Table 8. The standard deviation
calculated using all individual abundances from both neutral and ionized lines
is also shown. The complete table is available at the CDS.

Optical and NIR light curves of the calibrating Cepheids, together with
effective temperature and radial velocity curves, will be adopted by our group
to perform a detailed comparison with nonlinear, convective hydrodynamical
models of classical Cepheids. Dating back to Natale et al. (2008) it has been
found that the simultaneous fit of both luminosity and radial-velocity
variations provides solid constraints on the physical assumptions adopted to
build pulsation models (Marconi et al., 2013). Moreover, the use of the
effective temperature curves (shapes and amplitudes), covering a broad range
of pulsation periods, brings forward the opportunity to constrain, on a
quantitative basis, the efficiency of the convective transport over the entire
pulsation cycle.

Figure 9: Atmospheric parameters derived by keeping fixed the effective
temperature (previously obtained from the LDR method) in comparison with the
same parameters derived with $T_{\rm eff}$ as a free parameter. The panels
show the comparison for the $β$ Dor and $ζ$ Gem stars.

##  6 Iron and $\alpha$-element abundances

###  6.1 Iron abundance

The determination of the atmospheric parameters was done with the Python
wrapper 101010pyMOOGi: https://github.com/madamow/pymoogi of the MOOG code
(Sneden, 2002). This is a LTE radiative code that we applied to model
atmospheres interpolated on the grids of Castelli & Kurucz (2004). Once a
convergence is achieved in the determination of $T_{\rm eff}$, $\log{g}$,
$\xi$, and [Fe/H], as described in Sect. 5.1, MOOG also provides the final
iron abundance derived from individual Fe i and Fe ii lines. Such lines are
those included in the input line list and that passed the selection criteria
that we applied to choose the best atomic transitions.

Table 4 lists, for each spectrum in our sample, the mean Fe i and Fe ii
abundances derived from individual lines together with the standard deviations
and the number of lines used. The mean Fe i and Fe ii abundances derived for
each star from the individual spectra, weighted by the standard deviations
aforementioned, and the corresponding standard errors are shown in Table 6.
Column 8 gives our final determination for the stellar metallicity, which is
the weighted mean calculated from both the [Fe i/H] and the [Fe ii/H]
abundances. The adopted uncertainties are the largest values between the
standard error computed from the weighted mean ($\sigma$) and the standard
deviation (std).

Figure 10: Stellar metallicity measured for our sample of 20 calibrating
Cepheids compared with results from literature. Top panel: metallicity as a
function of the Galactocentric distance from the current work (blue circles)
and from literature (gray circles): Luck et al. (2011, LII); Luck & Lambert
(2011, LIII); Lemasle et al. (2013, LEM); Yong et al. (2006, YON). A linear
regression (solid black line plus equation) fitted to the entire sample is
compared with the radial gradient provided by Ripepi et al. (2022) (dashed
magenta line). The latter was artificially shifted to coincide with the
current radial gradient at ${\rm R}_{G}$ = 10 kpc. The ${\rm R}_{G}$ values
are from Genovali et al. (2014). Bottom panel: same as the top, but as a
function of the logarithmic pulsation period. A linear regression (solid blue
line plus equation) fitted to the current sample is also show.

Figure 22 shows the derived metallicities as a function of the pulsation
phase. Data plotted in this figure show that the iron abundances are quite
stable along the pulsation cycle. This outcome applies to short-/long-period
Cepheids and to Bump Cepheids. We note that the current approach improves the
results obtained by Proxauf et al. (2018), since in their iron abundances was
still present a mild variation along the rising branch of large amplitude
Cepheids (in our current results the dispersions are in most cases smaller
than 0.05 dex). The improvement is mainly due to the very careful selection of
the lines adopted to estimate the iron abundance. The difference is soundly
supported by the distribution of the standard deviations displayed in Fig. 7
for the Cepheids in common. The current standard deviations for both [Fe i/H]
(violet) and [Fe ii/H] (green) lines (bottom panel) are smaller by 30%-40%
when compared with our previous investigation. Moreover and even more
importantly, they also attain similar values.

To further constrain the difference between the current and our previous
investigation, Fig. 8 shows the comparison for both the atmospheric parameters
and the iron abundance. The agreement is remarkable over the entire
temperature range. The mean and the standard deviations for the atmospheric
parameters are well within the current uncertainties. The new iron abundances
are, on average, $\sim$0.1 dex more metal-poor. Owing to the similarity in the
atmospheric parameters, this difference seems an obvious consequence of the
new line list.

Several different approaches have been suggested in the literature to estimate
the atmospheric parameters of variable stars (Fukue et al., 2015; Jian et al.,
2020; Lemasle et al., 2020; Matsunaga et al., 2021; Taniguchi et al., 2021;
Romaniello et al., 2021). In the current investigation, the effective
temperature is firstly derived using the LDR method, then the other
atmospheric parameters are estimated at fixed effective temperature. The
advantages of this approach have already been discussed in several papers. To
provide a more quantitative estimate of the differences compared to another
method normally used, in which the minimization process also includes $T_{\rm
eff}$ as a free parameter, we performed a new and independent determination of
the three atmospheric parameters plus abundances simultaneously for two
variables in our sample ($β$ Dor and $ζ$ Gem). The differences between our
canonical approach and the literature approach are shown in Fig. 9. The
overall agreement is quite good over the entire temperature range. Indeed, the
mean and the standard deviations attain tiny values. The mean difference in
effective temperature is 55 K, and for the surface gravity it is 0.12 dex
only, while the microturbulent velocities attain almost identical values. The
quoted differences cause a small increase of 0.04 dex in the iron abundance
when $T_{\rm eff}$ is a free parameter. We note that we adopted the same line
list in the tests performed either with fixed or with free $T_{\rm eff}$.

Figure 11: [X/Fe] abundances as a function of metallicity (left panels) and of
the logarithmic pulsation period (right panels) comparing our 20 calibrating
Cepheids with stars from literature: LII: Luck et al. (2011); LIII: Luck &
Lambert (2011); LEM: Lemasle et al. (2013); YON: Yong et al. (2006). The error
bars indicate our typical errors.

In closing this subsection, we also investigate whether the current iron
abundances display any clear trend with the Galactocentric distance (${\rm
R}_{G}$). The top panel of Fig. 10 shows the iron abundance as a function of
${\rm R}_{G}$ for the calibrating Cepheids (blue circles) and for Cepheids
available in the literature (gray circles). We performed a linear fit over the
entire sample and we found the following relation:

${\rm[Fe/H]}=(-0.055\pm 0.003){\rm R}_{G}+(0.43\pm 0.03)$ (2)

The current slope agrees quite well with similar estimates available in the
literature and, in particular, with the recent estimate of the iron radial
gradient (dashed line) provided by Ripepi et al. (2022). We note that, to
overcome variations in the zero-point mainly introduced by innermost and
outermost Cepheids, the gradient from Ripepi et al. (2022) was artificially
shifted to coincide with the current radial gradient at ${\rm R}_{G}$ = 10
kpc.

The bottom panel of Fig. 10 shows the iron abundance as a function of the
logarithmic pulsation period. Data plotted in this panel agree quite well with
similar estimates available in the literature. Moreover, they do not show any
significant variation when moving from young (long-period) to less young
(short-period) classical Cepheids.

###  6.2 $\alpha$-element abundances

The abundances of the $\alpha$ elements were derived using pyMOOGi as well
(except sulfur, for which the abundances were estimated using the spectral
synthesis method - see Sect. 6.2.1). We used the $abfind$ driver, which
requires as input $i)$ the model atmospheres, calculated from the atmospheric
parameters obtained for each star, and $ii)$ a list of lines containing the
wavelength, the atomic number, the lower-level excitation potential, the
$\log{gf}$, and the measured equivalent widths. Table 8 lists the individual
$\alpha$-element abundances obtained for each spectrum together with the
standard deviations and the number of lines used for each element. The mean
abundances, weighted by the standard deviations, and the corresponding
standard errors are shown in Table 9. The same mean data are plotted in Figs.
11, 12, and 23 for our 20 calibrating Cepheids (blue circles) and for similar
data available in the literature (gray circles).

Figure 12: [X/H] and [X/Fe] abundance ratios as a function of the
Galactocentric distance ($R_{\rm G}$) for our 20 calibrating Cepheids compared
with stars from literature: LII: Luck et al. (2011); LIII: Luck & Lambert
(2011); LEM: Lemasle et al. (2013); YON: Yong et al. (2006). The error bars
indicate our typical errors. Linear regressions fitted to the entire sample
are also shown (solid lines)

For cases in which the abundance of a given element at a given phase is based
on only one or two lines, or if the estimated error is lower than a typical
value for each element/species, the uncertainty is set to that typical value,
namely: $\sigma$([Mg/H]) = 0.11, $\sigma$([Si i/H]) = 0.10, $\sigma$([Ca i/H])
= 0.18, $\sigma$([Ca ii/H]) = 0.18, $\sigma$([Ti i/H]) = 0.14, and
$\sigma$([Ti ii/H]) = 0.10 dex (for the sulfur abundances, the uncertainties
come from the spectral synthesis). The estimate of these typical errors was
done by calculating the median of the standard deviations listed in Table 8
for spectra having at least 2 lines of the same element/species.

The adopted standard solar abundances for both iron and $\alpha$ elements are
from Asplund et al. (2009), namely: $A({\rm Fe})_{\odot}$ = 7.50, $A({\rm
Mg})_{\odot}$ = 7.60, $A({\rm Si})_{\odot}$ = 7.51, $A({\rm S})_{\odot}$ =
7.12, $A({\rm Ca})_{\odot}$ = 6.34, and $A({\rm Ti})_{\odot}$ = 4.95, with $A$
representing the typical logarithmic notation where H is defined to be $A({\rm
H})_{\odot}$ = 12.00. These values are in good agreement with the very recent
determinations done by Asplund et al. (2021), which were obtained using the
most up-to-date atomic and molecular data.

For both iron and $\alpha$ abundances, we evaluated the systematic differences
for stars in common between the current investigation and measurements from
literature: Luck et al. (2011), Luck & Lambert (2011), Lemasle et al. (2013),
and Yong et al. (2006) for iron and $\alpha$ elements, and Proxauf et al.
(2018) for iron only. The differences for all the investigated elements are
listed in Table 10. In order to perform a direct comparison, we applied these
zero-point differences to the literature datasets by putting the iron and the
$\alpha$-element abundances in our metallicity scale. The spectroscopic
samples for which there are no variables in common between our investigation
and those available in the literature, the homogenization of the abundances
was performed in two steps. To be more specific, we have no variables in
common with Yong et al. (2006), therefore, we first scaled their abundances to
match those from Luck & Lambert (2011), and then we scaled them to match our
metallicity scale.

Data plotted in Fig. 23 display, once again, the remarkable agreement of the
$\alpha$-element abundances over the entire pulsation cycle (the dispersions
in most cases are smaller than 0.10 dex). This outcome applies to all the
investigated $\alpha$ elements (except sulfur, for which we have abundances at
a single phase only), and to both neutral and ionized species. We note in
passing that this finding seems more relevant when compared with the variation
of the iron abundances along the pulsation cycle. Indeed, the $\alpha$-element
abundances are only based on a modest number or, in some cases, just a few
lines. Therefore, they are more prone to possible systematics.

To further investigate the internal consistency of the current
$\alpha$-element abundances with similar estimates available in the
literature, we checked the behavior of the [X/Fe] abundance ratios as a
function of the stellar metallicity. Data plotted in the left panels of Fig.
11 show that the current results agree, within the errors, quite well with
similar determinations done in previous investigations (Luck & Lambert, 2011;
Luck et al., 2011; Lemasle et al., 2013; Yong et al., 2006). The key
difference seems to be the smaller dispersion at fixed iron abundance, thus
suggesting that a fraction of the spread currently observed might still be
caused by intrinsic errors. This finding is further supported by the evidence
that the current estimates do not show any clear trend when plotted against
the pulsation period (see the right panels of Fig. 11). We note that classical
Cepheids, when moving from short to long periods, become systematically cooler
and more expanded (lower surface gravity).

####  6.2.1 Sulfur abundances

To determine the sulfur abundances, we carried out spectral synthesis
calculations by exploiting the multiplet at 6757 Å, which is proven to be
reliable and not affected by NLTE effects (see, e.g., Takeda et al., 2016;
Duffau et al., 2017, and references therein). Synthetic spectra were computed
using the $synth$ driver of MOOG (2019 version), the same grid of model
atmospheres by Castelli & Kurucz (2004) used for the other elements, and
atomic parameters from Caffau et al. (2007). We synthesized a 30 Å wide region
around the S i multiplet in order to determine the broadening of the spectral
lines. Then we degraded the synthetic spectra to match the resolution of the
observational datasets. The best fit was found with $\chi$-square minimization
methods, as routinely done in the literature. Finally, we estimated the
internal errors, with a mean value of $\sigma$([S i/H]) = 0.10 dex. These
errors are due to continuum placement (related to the S/N, quality of the
spectra, best fitting procedure) and due to errors on the atmospheric
parameters (this was done in the standard way, which is by varying one
parameter at a time and inspecting the corresponding variations). We refer the
reader to D’Orazi et al. (2017, 2020) for details on error computation.

The reasons why we focused our attention on S i abundances for our sample of
calibrating Cepheids are manifold:

i)– Sulfur is a volatile element that is not locked in dust grains, which
means that no correction is required to compare the current sulfur abundances
with predictions based on chemical evolution models (Chiappini et al., 2001;
Cescutti et al., 2007). Therefore, sulfur is a more solid tracer of
$\alpha$-element abundances than Si and Ca. These three $\alpha$ elements are
produced during the oxygen burning and their abundances should show similar
chemical enrichment histories (Limongi & Chieffi, 2003).

ii)– Sulfur brings forward several key advantages when compared with other
$\alpha$ elements, since it shows several lines at both optical and NIR
wavelengths. The most popular sulfur lines are the multiplets at 6757 Å and at
8694 Å, but they are weak and they are typically measured on stars more metal-
rich than [Fe/H] $\sim$ $-$1.5 dex. More recently, the multiplet at 9237 Å has
also been used because it is very strong and it can be measured even in very
metal-poor stars (Nissen et al., 2004; Caffau et al., 2005, 2007). However,
these lines are also affected by NLTE effects, making them even stronger, and
the sulfur abundances need to be properly treated. Even more recently, high-
resolution NIR spectrographs also provided the opportunity to use the S i
multiplet at 1045 nm for both dwarf and giant stars (Caffau et al., 2016).

iii)– Possibility of comparison with the S abundances in external galaxies,
such as Blue Compact Galaxies, Damped Lyman Alpha systems (Dessauges-Zavadsky
et al., 2007), and Active Galactic Nuclei (Liu et al., 2015, Mizumoto et al.,
submitted to ApJ). Indeed, thanks to strong emission lines, the S abundances
can be measured even at large redshifts.

iv)– In our previous investigation focused on $\alpha$-element abundances in
classical Cepheids (Genovali et al., 2015) we neglected sulfur. In the current
study, we perform accurate measurements of the multiplet at 6757 Å. The
multiplet at 9237 Å is only available in a small subsets of spectra (UVES),
but the lines are too strong (EW $\geq$ 300 mÅ) and it was neglected. Data
plotted in Fig. 12 show the [X/H] (left panels) and the [X/Fe] (right panels)
$\alpha$-element abundances as a function of the Galactocentric distances,
estimated by Genovali et al. (2014) using NIR mean magnitudes and predicted
Period-Luminosity relations (Inno et al., 2013). The radial gradients plotted
in the left panels indicate that sulfur is the $\alpha$ element with the
steeper slope (see labeled values). Indeed, its slope is at least a factor of
two steeper than for Ca (0.076 vs. 0.031 ${\rm dex\,kpc}^{-1}$) and $\sim$50%
steeper than those for Mg and Si. In passing, we also note that the radial
gradient of the abundance ratios plotted in the right panels of the same
figure display, as expected (see Genovali et al., 2015), either a flat trend
(Si) or a marginal positive slope (Mg, Ti) across the thin disk. There are two
exceptions. Ca is steadily becoming overabundant (positive slope) when moving
from the innermost (more metal-rich) to the outermost (more metal-poor) disk
regions. This is a consequence of the fact that the slope of its radial
gradient is shallower than the iron gradient (0.31 vs. 0.55 ${\rm
dex\,kpc}^{-1}$). S shows a negative slope, because it is the only
$\alpha$-element with a slope in the radial gradient steeper than the iron
gradient (0.76 vs. 0.55 ${\rm dex\,kpc}^{-1}$).

##  7 Summary and final remarks

We performed a new spectroscopic analysis of almost two dozen of northern and
southern calibrating Cepheids. As a whole, we collected more than 500 high-
resolution (R$\sim$40 000–115 000), high S/N ($\sim$50–370) optical spectra
with four different spectrographs (STELLA, FEROS, HARPS, UVES) covering either
a substantial portion or the entire pulsation cycle. The main aim of this
investigation is twofold: $i)$ provide new and accurate constraints on
possible systematics affecting the estimate of atmospheric parameters and
elemental abundances along the pulsation cycle; $ii)$ develop new approaches
and/or diagnostics to provide accurate estimates of both atmospheric
parameters and chemical abundances.

Table 10: Mean abundance differences for stars in common between the current
sample of calibrating Cepheids and other datasets.

Abundance ratio | Datasets1 |  Zero-point difference | $N_{\rm Common}$
---|---|---|---
[Fe/H] | P18$-$TS | 0.07 |  $\pm$ 0.06 | 13
[Fe/H] | LII$-$TS | 0.05 |  $\pm$ 0.07 | 20
[Fe/H] | LIII$-$TS | 0.11 |  $\pm$ 0.08 | 6
[Fe/H] | LII$-$LEM | 0.07 |  $\pm$ 0.12 | 55
[Fe/H] | YON$-$LIII |  $-$0.33 |  $\pm$ 0.15 | 21
[Mg/H] | LII$-$TS |  $-$0.14 |  $\pm$ 0.11 | 16
[Mg/H] | LIII$-$TS | 0.11 |  $\pm$ 0.16 | 5
[Mg/H] | LII$-$LEM |  $-$0.26 |  $\pm$ 0.24 | 35
[Mg/H] | YON$-$LIII |  $-$0.09 |  $\pm$ 0.14 | 16
[Si/H] | LII$-$TS |  $-$0.13 |  $\pm$ 0.07 | 20
[Si/H] | LIII$-$TS |  $-$0.01 |  $\pm$ 0.09 | 6
[Si/H] | LII$-$LEM |  $-$0.06 |  $\pm$ 0.11 | 55
[Si/H] | YON$-$LIII |  $-$0.12 |  $\pm$ 0.07 | 20
[S/H] | LII$-$TS | 0.11 |  $\pm$ 0.13 | 17
[S/H] | LII$-$LEM |  $-$0.13 |  $\pm$ 0.19 | 45
[Ca/H] | LII$-$TS |  $-$0.07 |  $\pm$ 0.10 | 20
[Ca/H] | LIII$-$TS |  $-$0.01 |  $\pm$ 0.10 | 6
[Ca/H] | LII$-$LEM |  $-$0.06 |  $\pm$ 0.17 | 55
[Ca/H] | YON$-$LIII |  $-$0.16 |  $\pm$ 0.11 | 21
[Ti/H] | LII$-$TS | 0.12 |  $\pm$ 0.09 | 20
[Ti/H] | LIII$-$TS | 0.17 |  $\pm$ 0.12 | 6
[Ti/H] | YON$-$LIII | 0.33 |  $\pm$ 0.20 | 20
111111(1) TS: this study; P18: Proxauf et al. (2018); LII: Luck et al. (2011);
LIII: Luck & Lambert (2011); LEM: Lemasle et al. (2013); YON: Yong et al.
(2006).

To accomplish the former goal, we assembled three different line lists. By
using different line lists available in the literature, we compiled a very
detailed list of both Fe i and Fe ii lines. Moreover, we also collected a
sizable sample of lines for $\alpha$-elements: Mg, Si, S, Ca, and Ti. Special
care was also paid to the line list of both iron-peak and $\alpha$-elements
used for the estimate of the effective temperature using the LDR method.

In order to provide homogeneous and accurate estimates of both atmospheric
parameters and elemental abundances, we undertook a lengthy and detailed
analysis of the quoted line lists. Our approach moved along three different
paths: $i)$ we collected the three most precise atomic transition parameters
available in the literature; $ii)$ we removed absorption lines that are
blended with other lines; $iii)$ we removed lines that either show relevant
abundance variations from the average value of their chemical species or that
display a steady variation along the pulsation cycle with atmospheric
parameters. Concerning the transition parameters, we gave the priority to
recent laboratory measurements, then to homogeneous astrophysical estimates
and, for the remaining lines, to the transition parameters available on the
NIST Atomic Spectra Database. To identify blended lines, we adopted both the
solar spectrum by Moore et al. (1966) and a grid of synthetic spectra.
Finally, we applied an iterative approach to remove lines showing significant
variations. This latter step was done based on a preliminary estimate of the
atmospheric parameters and the abundances. We were then able to identify lines
that show either a variation in abundance at the 3-$\sigma$ level from the
mean value of the specific element and/or a steady variation along the
pulsation cycle. In more detail, to properly identify robust lines, we plotted
all the lines available in our initial line lists, for each chemical species,
individual exposure, and calibrating Cepheid, as a function of the pulsation
phase, eﬀective temperature, and equivalent width. Lastly, we performed a new
estimate of the atmospheric parameters and the individual elemental
abundances, using an improved version of our algorithm and using only the line
lists based on optimal lines.

The comparison between the current atmospheric parameters and similar
estimated provided by our group, using a similar approach but old line lists,
show smoother variation along the pulsation cycle and, in particular, smaller
standard deviations ($\sim$50 K for the effective temperature, $\sim$0.2 dex
for the surface gravity, and $\sim$0.2 km s-1 for the microturbulent
velocity). The estimate of both iron and $\alpha$-element abundances is also
significantly improved, since they display an almost constant value as a
function of the pulsation phase. The dispersions are $\lesssim$ 0.05 dex for
iron and $\lesssim$ 0.10 dex for the $\alpha$ elements. We note that this is
the first time that $\alpha$-element abundances are critically investigated
for possible variations along the pulsation cycle.

This finding is further supported by the comparison with similar abundance
estimates available in the literature. We found that the current calibrating
Cepheids display, at fixed either iron abundance or pulsation period, smaller
standard deviations. This result suggests that a good fraction of the spread
normally observed in elemental abundances is mainly caused by the adopted line
list and by the algorithm used to estimate the atmospheric parameters.

Moreover and even more importantly, we also derived for the first time new and
accurate effective temperature templates. Following Inno et al. (2015), the
calibrating Cepheids were split into four different period bins, ranging from
3 to 10.5 days in period. For each period bin, we performed an analytical fit
with Fourier series providing $\theta=5040\leavevmode\nobreak\ /{T_{\rm eff}}$
as a function of the pulsation phase. A similar approach was already applied
to fundamental RR Lyrae by For & Sneden (2010) and by Magurno et al. (2018),
but the variations along the pulsation cycle are, at fixed pulsation phase,
larger due to the larger luminosity amplitudes and probably to the possible
occurrence of nonlinear phenomena (shocks). The current findings appear quite
promising, because the new templates can provide accurate estimates of the
effective temperature once the pulsation period and the phase (and in turn the
reference epoch) of a Cepheid are known. We note that this information is
crucial in the abundance analysis of NIR spectra, because in the metal-poor
regimes only a modest number of ionized and neutral iron lines are present.

The current investigation was focused on Galactic Cepheids with iron
abundances close to the solar value. In a forthcoming paper we plan to apply
the same approach to more metal-poor Magellanic Cloud Cepheids.

###### Acknowledgements.

It is a real pleasure to thank the anonymous referee for her/his positive
appreciation of the current investigation and for her/his pertinent
suggestions that improved the content and the readability of the paper. It is
also a pleasure to thank F. Matteucci and E. Spitoni for several detailed
discussions concerning the role of $\alpha$ elements in constraining chemical
evolution models. R.P.K. gratefully acknowledges support by the Munich
Excellence Cluster Origins Funded by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) under Germany’s Excellence Strategy EXC-2094
39078331. B.L. acknowledges the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) – Project-ID 138713538 – SFB 881 (”The Milky Way System”,
sub-project A05).

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##  Appendix A Cleaning from irregular atomic lines

Many of the atomic lines present in our initial line lists of iron and
$\alpha$ elements turned out to be not reliable enough. The abundances
provided for our spectra by such irregular lines exhibit at least one of the
following problems:

  1. 1.

Dependence on EW: a clear trend, possibly caused by blended lines, is seen in
the [X/H] abundance ratios as a function of the equivalent widths. In these
cases, we have either eliminated the line or defined a new and acceptable EW
range. A few examples for Fe i and Fe ii, for the stars $β$ Dor and $ζ$ Gem,
are shown in Fig. 13.

  2. 2.

Abundance bump at some phases: there are cases in which the abundance derived
for a given element changes along the pulsation cycle, being under- or over-
estimated around phases between 0.8 and 0.9. These bumps are not caused by
saturation, so there is no clear correlation with EW. In such situations, we
only kept the lines where the bump was not very large in order to avoid a
wrong determination of the mean abundance. Bump examples are shown in Fig. 14.

  3. 3.

Chaotic behavior: the [X/H] abundance ratios have a very large dispersion and
no clear trend with any other parameter. This might be caused by very close
blends where the deblending with two or multiple Gaussian functions not always
worked. See examples in Fig. 15. The same figure also shows examples of lines
completely saturated.

  4. 4.

Dependence on $T_{\rm eff}$: Fig. 16 shows two examples for which the
abundances derived from Fe i lines have a clear trend with the effective
temperature, possibly caused by NLTE effects. An option to avoid this problem
could be to define, at least for these irregular lines, a range of $T_{\rm
eff}$ within which the abundances they provide can be accepted. For this work,
however, we preferred not to use such lines at all.

  5. 5.

Under- or over-abundances: Some of the atomic lines in our lists, though
exhibiting a coherent behavior in phase, $T_{\rm eff}$, and EW, provide
systematic under- or over-abundances. See examples in Fig. 17. These problems
are probably not caused by saturation but by a wrong value of $\log{gf}$, or
by a wrong continuum identification, which is not always an easy task in
regions with too many lines. For several cases we were able to adjust the
$\log{gf}$ values using a high-resolution spectrum of Arcturus as reference.
For other unsolved cases the lines were simply not used in our analysis.

Figure 13: Iron abundances as a function of the pulsation phase, effective
temperature, and equivalent width for lines excluded from the initial line
list. These examples, for HARPS (green circles), FEROS (blue triangle), and
STELLA (red stars) spectra of $β$ Dor and $ζ$ Gem, show a dependence of the
abundances on the equivalent widths possibly caused by blends. Lines too weak
or too strong (probably saturated) are represented by light gray symbols. The
hatched regions around the dashed lines indicate the 1-$\sigma$ uncertainty
around the mean.

Figure 14: Same as in Fig. 13 but showing an example of a line excluded
because of having a bump around pulsation phases 0.8-0.9, not caused by
saturation or by any correlation with the equivalent widths.

Figure 15: Same as in Fig. 13 but showing examples of lines excluded either
because of being completely saturated or because of having a chaotic
distribution, probably caused by blends.

Figure 16: Same as in Fig. 13 but showing an example of a line excluded
because of having some correlation with the effective temperature, possibly
caused by NLTE effects.

Figure 17: Same as in Fig. 13 but showing an example of a line that, even if
the dispersion in abundance is very small, was excluded from the initial line
list because of having an unexpected over-abundance in metallicity, possibly
caused by a wrong $\log{gf}$ value.

##  Appendix B Variations along the pulsation cycle

###  B.1 Light and radial-velocity curves

The panels of Fig. 18 show the phase-folded light and radial-velocity curves
for our sample of 20 calibrating Cepheids. The stellar magnitude in visual
bands were taken from literature, as explained in Sect. 4. The radial
velocities were either measured from our spectra or taken from literature.
Only FF Aql and $β$ Dor have RV values derived from our spectra only.

Our RV measurements agree quite well with the literature values. Significant
differences are seen only for two stars, T Vul and S Sge, which might be
related to instrumental drifts caused by their long baseline observations (for
both stars, the two datasets, ours and from literature, are separated by
almost 17 years) or, most likely, to the binarity of these objects (Evans
1992; Evans et al. 1993).

As detailed in Table 5, for T Vul we derived two different values of reference
epoch ($T_{0}$), one from our own radial velocities and another one from the
light curve. For S Sge we have independent measurements of $T_{0}$ and
pulsation period derived from both the RV datasets and from the light curve.
Another particular case is $β$ Dor, one of our bump Cepheids (see Table 1),
for which we have independent values of $T_{0}$ and period derived from our RV
curve and from the literature light curve.

The main criteria we adopted for selecting the light curves available in the
literature were: $i)$ well-sampled photometric data; $ii)$ minimization of the
difference in epoch between photometric and spectroscopic data. It goes
without saying that well-sampled light curves from Gaia DR3 will allow us to
provide accurate and homogeneous ephemerides for the majority of classical
Cepheids.

Figure 18: Light curves and radial velocities as a function of the pulsation
phase. Measurements from different spectrographs are indicated with different
colors and symbols. The error bars in some cases are smaller than the symbol
size.

Figure 18: continued.

###  B.2 Atmospheric parameters and metallicity

The variation of the atmospheric parameters ($T_{\rm eff}$, $\log{g}$, and
$\xi$) and the iron abundances along the pulsation cycle are shown in the
panels of figures from 19 to 22. It is worth mentioning that the number of
points in the $T_{\rm eff}$ panels are larger than in the other panels. This
is because the determination of the effective temperature and of the other
parameters are based on different and independent methods. For many spectra we
were only able to derive the effective temperature (using the LDR method),
whereas the derivation of the other parameters did not converge when we tried
to force the ionization equilibrium of Fe i and Fe ii lines and the
independence of the abundances on the equivalent widths (very often due to a
limited number of good lines).

The variable S Sge shows two outliers in the $T_{\rm eff}$ curve plotted in
the panel of Fig. 19. However, these values might be correct because their
shift with respect to the other spectra could be due to a limited accuracy in
their phase determination, caused either by uncertainties in the period
estimate or by any variation in period.

Figure 19: Effective temperature as a function of the pulsation phase.
Measurements from different spectrographs are indicated with different colors
and symbols. To help with the comparison, the panels are plotted with the same
y-axis range.

Figure 20: Same as in Fig. 19, but showing the surface gravity as a function
of the pulsation phase.

Figure 21: Same as in Fig. 19, but showing the microturbulent velocity as a
function of the pulsation phase.

Figure 22: Abundances from Fe i and Fe ii lines as a function of the pulsation
phase. The color coding of the various points is the same as in Fig. 19. To
help with the comparison, the panels are plotted with same y-axis range: 1.5
dex for both Fe i and Fe ii panels. The light gray shaded regions indicate the
$\pm$1-$\sigma$ uncertainty around the weighted mean (from Cols. 5 and 6 of
Table 6).

Figure 22: continued.

###  B.3 Abundances of $\alpha$ elements

Figures 23 shows the [$\alpha$/H] abundance ratios for our sample as a
function of the pulsation phase, in which we also compare the abundances from
neutral (colored symbols) and ionized (light gray symbols) elements, when
available. The agreement between the two species is normally within 2 sigma,
with only a few exceptions for the Ca abundances (the Ca ii abundances are
very often based on a few weak lines only).

Figure 23: [X/H] abundances as a function of the pulsation phase. Each panel
shows the abundances of $\alpha$ elements from neutral and ionized lines as
listed in Tables 8 and 9. As explained in the text, the sulfur abundances were
derived using the spectral synthesis method applied to one spectrum of each
star (an exception are the stars $η$Aql, VY Sgr, and V340 Ara, for which the
[S/H] abundances could not be derived due to the noisy profile of the sulfur
line used).

Figure 23: continued.

Figure 23: continued.

Figure 23: continued.

Figure 23: continued.