the effective temperature, the surface gravity, and the micro-turbulent
velocity, [$\xi$]) 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
micro-turbulent 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 at each step, and the adopted value is [Fe I/H], which is
the mean iron abundances provided by individual Fe I lines. The uncertainties
on $T_{\rm eff}$ are the standard deviations calculated using the LDR method,
they are typically of the order of 150 K while the uncertainties on the
individual estimates of $\log g$ and $\xi$ are assumed to be $\sim$0.3 dex and
0.5 km s-1 (see Genovali et al (2014) for a detailed discussion).

Thanks to these improvements, the errors on the derived amplitudes for the
effective temperature curves are a factor of two smaller than our previous
estimates, whereas for the surface gravity the differences are even more
significant, with typical errors almost three times smaller. Concerning the
micro-turbulent velocity variation, the errors on the current amplitude
estimates are, on average, more than twice smaller (Proxauf et al, 2018).

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. Together with the removal of blended
lines, and of lines that display a well-defined trend with effective
temperature or with equivalent width. 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 by da Silva et al (2022).
The reader interested in a more detailed discussion concerning the estimate of
the intrinsic parameters of classical Cepheids is referred to Vasilyev et al
(2018).

In this context, it is worth mentioning that the improvement in the accuracy
and in the sampling of the pulsation cycle improved the empirical framework
concerning the variation of atmospheric parameters along the pulsation cycle.
Indeed, the minimum in the effective temperature is often approached across
the same phases in which the minimum in surface gravity is also approached.
Moreover and even more importantly, these phases across the minimum anticipate
the increase in microturbulent velocity. This parameter, as expected, attains
its maximum just before the maximum in effective temperature (see Figs. B.2
and B.4 in da Silva et al 2022).

A new and innovative approach to constrain the temperature scale of classical
Cepheids was recently suggested by Lemasle et al (2020). The new approach
relies on a data-driven machine-learning technique applied to observed
spectra, in which the flux ratios were tied to temperatures derived using the
infrared surface-brightness method.

####  6.2.2 Effective temperature curve templates

By taking advantage of the substantial phase coverage of the sample of
calibrating Cepheids, the effective temperature measurements based on LDRs
were also adopted to compute new templates for different bins of pulsation
periods (da Silva et al, 2022). However, instead of using the $T_{\rm eff}$
curves directly, they preferred to provide templates for the $\theta$
parameter – defined as $\theta=5040/T_{\rm eff}$ – given its linear dependence
on the Johnson–Cousins $R-I$ color index (Taylor, 1994).

The approach is similar to the NIR light-curve templates provided by Inno et
al (2015): first, periods and reference epochs are adopted to fold the
$\theta$ curves ($\theta$Cs). Subsequently, the folded $\theta$Cs were
normalized by subtracting their average and dividing by their amplitudes. The
$\theta$Cs curves are well sampled, meaning that the phase points are large
enough 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, da Silva et al (2022) 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, they were able to provide
$\theta$C templates homogeneous with those adopted for the NIR light curve
templates. The reader interested in a more detailed and quantitative
discussion about the use of both 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 23: From top to bottom: Normalized $\theta=5040/T_{\rm eff}$ curves of
CCs in different period bins (3–5 days, panel a; 5–7 days, panel b; 7–9 days,
panel d; 9–11 days, panel e). Different colors are associated to different
CCs. The names of the CCs are labelled and the numbers in parentheses show the
pulsation period in days. The black line is the analytical fit to the
$\theta$-curve template. Adapted from da Silva et al (2022, Fig. 5). Image
credit: R. da Silva.

The normalized $\theta$ curves for the four period bins are plotted in Fig. 23
together with the analytical fits (black lines), based on Fourier series, to
the data (see Fig. 7 in da Silva et al, 2022). The separation of the Cepheids
into different period bins is a mandatory step, because not only the amplitude
of the $\theta$ curves, but also their shape changes with period. Note that
spectroscopic data for Cepheids with periods outside the selected period bins
are available, but they are either not as well-sampled as the data for the
other Cepheids, or the $\theta$ curves do not overlap very well because of the
limited number of measurements. More accurate and homogeneous data for FU and
FO CCs are required in order to overcome this limitation.

The use of the effective temperature curve templates to derive the mean
effective temperature requires the ephemerides, the luminosity amplitude
($\Delta$V), and a single spectroscopic measurement of the $T_{\rm eff}$
together with a linear relation between $\Delta$V and $\Delta\theta$:
$\Delta\theta=(0.184\pm 0.014)\Delta{\rm V}+(0.000\pm 0.011)$;
$(\sigma=0.013)$.

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

Figure 24: Effective temperature versus period for CCs with full coverage of
the pulsation cycle. The different colors indicate the different period bins
of the temperature curve templates. The dashed lines indicate the average
temperature for each bin in period. The orange line shows the linear fit to
the data.

However, the effective temperature curve templates are far from being an
academic issue, since once a period–mean temperature relation is known, they
can also provide the estimate of the effective temperature along the pulsation
cycle. This is a far-reaching opportunity, since effective temperature
estimates based on color-temperature relations are hampered by uncertainties
affecting reddening estimates. This is a trivial effort in the optical regime
with high-resolution and high signal-to-noise spectra, but it becomes more
difficult either in the NIR, or at low spectral resolution, or in the very
metal-poor regime due to the paucity of lines. Therefore, we performed a
linear fit between pulsation period and mean effective temperature (see Fig.
24) and we found:

$T_{\rm eff}[K]=6256.73(\pm 89.45)-71.23(\pm 13.29)\times P$ (1)

with a standard deviation of 286 K. Once the mean $T_{\rm eff}$ is available,
one can anchor the temperature template to the phase of the spectroscopic
measurement and provide an estimate of the $T_{\rm eff}$ at any phase.

Optical and NIR light curves of the calibrating Cepheids, together with
effective temperature and radial velocity curves, can be adopted 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.

###  6.3 Metallicity distribution in the Galactic thin disk

In discussing the metallicity distribution of Cepheids, we are following a
series of papers (Romaniello et al, 2008; Pedicelli et al, 2010; Lemasle et
al, 2013, 2017; Inno et al, 2019; Genovali et al, 2013, 2014; da Silva et al,
2016; Proxauf et al, 2018; da Silva et al, 2022) focused on the metallicity
distribution of CCs. We are following these papers because they rely on the
same strategy for the estimate of the atmospheric parameters and of the
abundances.

Spectroscopic abundances discussed in the following rely on 1D LTE analysis.
These investigations are based on different sets of 1D LTE atmosphere model
provided by Castelli and Kurucz (2004), and by Gustafsson et al (2008). Theory
and observations suggest that this approach is prone to systematics when
compared with 1D and/or 3D NLTE analysis. This effect becomes larger when
moving from dwarfs to giants and from metal-rich to more metal-poor stellar
populations (Thévenin and Idiart, 1999; Idiart and Thévenin, 2000; Bergemann
et al, 2012; Hansen et al, 2013; Fabrizio et al, 2021). Different lines of the
same element have different NLTE corrections (Collet et al, 2005; Merle et al,
2011) and neutral lines are more affected than ionized lines. Theory and
observations indicate that Fe ii lines are minimally affected by NLTE effects
(Lind et al, 2012; Hansen et al, 2013). Moreover, the forbidden [O i] line at
6300.3 Å is not subject to any NLTE effect in cool stars (Kiselman, 2001;
Tautvaišienė et al, 2015) and the same outcome applies to the S i triplet at
6757 Å as thoroughly discussed by Duffau et al (2017). Fortunately, CCs are
GK-type giants and they display in high resolution and high signal-to-noise
ratio spectra from several to a few dozen of FeII lines. Moreover, the S i
triplet is unblended and relatively strong and the quoted forbidden line can
be measured over a significant fraction of Galactic CCs (da Silva et al,
2023). The Galactic metallicity gradients based on other elements should be
cautiously treated until the 1D and the 3D NLTE analysis will be performed.

Figure 25: Top: Iron abundances as a function of the Galactocentric distance
for a sample of 20 calibrating Cepheids (blue circles) compared with results
from the literature (gray circles): Luck et al (2011, LII); Luck and 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 (2022a) (dashed
magenta line). The latter was artificially shifted in order 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: same as the top, but as a
function of the logarithmic pulsation period. The solid blue line plus the
equation show the linear regression fitted to the current sample. Image
reproduced with permission from da Silva et al (2022, Fig. 10), copyright by
ESO.

###  6.4 The iron distribution

During the last few years, several investigations have addressed the open
problem concerning the space and age dependence of the metallicity gradient in
the Galactic thin disk. This issue has been studied not only from the
empirical (see, e.g., Maciel et al, 2003; Nordström et al, 2004; Henry et al,
2010; Yong et al, 2012) but also from the theoretical point of view. In
particular, the role that different stellar tracers can play in constraining
the chemical tagging in spatial distribution and in age distribution (Freeman
and Bland-Hawthorn, 2002).

In order to investigate the iron distribution of the calibrating Cepheids
(blue circles) provided by da Silva et al (2022), the top panel of Fig. 25
shows the individual iron abundances as a function of Galactocentric distance
$R_{G}$. Note that da Silva et al (2022) defined calibrating CCs only
variables whose entire pulsation cycle is covered by high-resolution spectra.
The gray circles display similar Cepheid abundances, but only for CCs
available in the literature. A linear fit over the entire sample gives the
following relation:

${\rm[Fe/H]}=(-0.055\pm 0.003)\,R_{G}+(0.43\pm 0.03)$ (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 (2022a). To overcome
variations in the zero-point, mainly introduced by the sampling of innermost
and outermost Cepheids, the gradient from Ripepi et al (2022a) was
artificially shifted, so as to coincide with the current radial gradient at
$R_{G}=10$ kpc.

The bottom panel of Fig. 25 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 in iron abundance when moving from young (long-period)
to less young (short-period) classical Cepheids.

###  6.5 Light metals and $\alpha$-element distribution

Figure 26: From top to bottom: Abundances for CCs of Na, Al, and $\alpha$
elements (Mg, Si, Ca) as a function of Galactocentric distance. The abundances
by G15 (filled blue circles) are compared with those of Yong et al (2006, YON,
cyan asterisks), Luck et al (2011, LII, magenta crosses), Luck and Lambert
(2011, LIII, open green circles) and Lemasle et al (2013, LEM, red
triangles)). The solid and the dashed lines display the linear regressions to
only G15 data and to the entire sample. Image reproduced with permission from
Genovali et al (2015, Fig. 2), copyright by ESO.

This section deals with light metals and $\alpha$-element gradients traced by
CCs across the Galactic thin disk. Fig. 26 shows the abundances of Na, Al and
of three $\alpha$-elements (Mg, Si, Ca) as a function of the Galactocentric
distance. The five investigated elements display well-defined radial
gradients. The difference in standard deviations is mainly correlated with the
number of lines adopted to estimate the abundances. The dispersion of the Mg
gradient is almost a factor of two larger than the Si gradient, because the
former is based on a single line, while the latter on more than a dozen. The
radial gradients of the five investigated elements attain, within the errors,
quite similar slopes as originally found by Lemasle et al (2007, 2013) and by
LII+LIII. The above result becomes even more compelling if we consider the
similarity with the iron radial gradient ($-0.060\pm 0.002\,\mathrm{dex\
kpc^{-1}}$) found by G14 using the same CCs. The current radial gradients seem
to show a flattening for distances greater than $\sim$13 kpc. This supports a
similar trend, but based on open clusters (Carraro et al, 2007; Yong et al,
2012; Magrini et al, 2023). However, firm conclusions are hampered by the
increased spread in abundance and by the scarcity of outer-disk CCs.

One of the most relevant issue in dealing with the chemical enrichment history
of the thin disk is the age dependence. Indeed, chemo-dynamical models suggest
a significant flattening of the metallicity gradients for ages older than 1-3
Gyr (Minchev et al, 2014; Kubryk et al, 2015). This means a steady decrease in
the metallicity with increasing age. The same models also predict a strong
dependence on stellar migrations. To constrain the age dependence of the
metallicity gradients, Fig. 2 in Genovali et al (2015) shows the same
elemental abundances of Fig. 26, but as a function of the logarithmic period.
Data plotted in their figure show a well-defined positive gradient for
increasing pulsation period. The slopes of the $\alpha$-elements attain
similar values, while for Na and Al they are systematically larger (see
labelled values). This evidence further supports the hydrostatic nature
(Arnett, 1979; Limongi and Chieffi, 2018) of both Na and Al, owing to their
steady increase correlated with the pulsation period (stellar mass).

Data plotted in Fig. 5 of Genovali et al (2015) show that the ratio
[element/Fe] is, on average, quite flat across the entire thin disk. They
performed a bi-weight linear least squares fit over the entire sample, and
they found evidence of a positive slope for Ca. This finding brings forward a
few interesting consequences.

i)–Chemical enrichment history– Iron and the other five elements have had
quite similar chemical enrichment histories across the Galactic thin disk.
Moreover, the quoted ratios have small standard deviations suggesting that it
is dominated by measurement errors. The [$\alpha$/Fe] ratios are typically
considered as tracers of the star-formation activity and the lack of a clear
negative gradient between the high (inner) and the low (outer) density regions
of the thin disk suggests that either the iron and the quoted elements are
produced by the same polluters, or the [$\alpha$/Fe] radial gradients are
affected by other parameters, such as radial migration of stars or radial gas
flows.

ii)–Radial gradients– The occurrence of a mild positive gradient in [Ca/H] as
a function of period, and of a negative gradient of [Ca/H] as a function of
Galactocentric distance are not correlated, since CCs located in the outer
disk have a canonical period distribution (mostly between 2 and 20 days). The
negative trend shown by Ca when moving from older to younger CCs remains
unclear.

In order to further investigate the ratio between hydrostatic and explosive
elements, the left panel of Fig. 11 in Genovali et al (2015) shows [Mg/Ca] as
a function of the iron abundance. The CCs display typical solar abundances for
iron abundances that are more metal-poor than the Sun, but the spread
increases to 0.5–0.7 dex for iron abundances more metal-rich than the Sun. On
the other hand, the abundance ratio of thin- and thick-disk dwarfs (B05, M14)
display the typical decrease when moving from metal-poor to more metal-rich
objects. The metal-rich regime shows a similar trend, and indeed the metal-
rich dwarfs (green diamonds, B05) display a mild increase in the [Mg/Ca]
abundance ratio for [Fe/H]$\geq$0.15.

The reasons for the above difference are not clear, therefore G15 searched for
a possible radial dependence. Data plotted in the middle panel of their Fig.
11 display that a significant fraction of the spread in [Mg/Ca] abundance
($\sim$0.7 dex) showed by metal-rich CCs is evident among objects placed
either across or inside the solar circle ($\sim$8 kpc). However, the abundance
ratio is quite constant across the disk regions covered by CCs and dwarfs.
This trend is slightly counter intuitive, since the increase in iron, typical
of the innermost disk regions, should also be followed by a steady increase in
Ca, since they are explosive elements. Namely, there is a steady decrease in
the [Mg/Ca] abundance ratio when approaching the innermost disk regions.

The right panel of their Fig. 11 shows the [Mg/Ca] abundance ratio of CCs as a
function of the logarithmic period. Data plotted in this figure show no clear
trend. The spread in the [Mg/Ca] abundance ratio is almost constant over the
entire period range, thus suggesting that there is no solid evidence of a
dependence on stellar mass.

The scientific impact of the $\alpha$-element abundances on the chemical
enrichment of the Galactic thin disk, and the difference between explosive and
hydrostatic elements have been recently addressed by Trentin et al (2023).

###  6.6 Neutron capture distribution

Figure 27: From top to bottom: Abundances for CCs of neutron capture elements
(Y, La, Ce, Nd, Eu) as a function of the Galactocentric distance. Abundances
provided by da Silva et al (2016, filled blue circles) are compared with those
of LII (magenta crosses, Luck et al, 2011), LIII (open green circles, Luck and
Lambert, 2011), and LEM (red triangles, Lemasle et al, 2013). The blue solid
line shows the linear regression of our Cepheid sample, while the black dashed
line the linear regression of the entire Cepheid sample. The blue error bars
display the mean spectroscopic error of the current sample. The abundances
available in the literature have similar errors. Adapted from da Silva et al
(2016, Fig. 2). Image credit: R. da Silva.

In the following, we discuss the radial gradients of four s-process (Y, La,
Ce, Nd) elements and a single r-process element (Eu). The n-capture elements
can be split according to solar system abundances in pure s-process, pure
r-process, and mixed-percentage isotopes. Among them, Eu is a pure r-process
element, since the r-fraction abundance is 97% (Burris et al, 2000; Simmerer
et al, 2004), while for the s-process elements the s-fraction ranges from
roughly 50% (Nd, 58%, Sneden et al, 2008) to more than 70% (Y, 72%; La, 75%;
Ce, 81%). The quoted s- and r-fraction abundances should be treated cautiously
since Bisterzo et al (2011), by using a different approach, found similar
fractions for Eu (94%), Nd (52%), La (71%), and Ce (81%), but a significantly
larger s-fraction for Y (92%).

Figure 27 shows the abundance of the five selected neutron capture elements as
a function of the Galactocentric distance. Data plotted in this figure bring
out several interesting results.

i)–Neutron-capture radial gradients– The five investigated neutron-capture
elements display well-defined radial gradients. This evidence, combined with
similar results concerning iron peak and $\alpha$-elements (LII, LIII) further
indicates that young stellar tracers show radial gradients across the Galactic
thin disk. There is only one exception, Andrievsky et al (2014) and Martin et
al (2015) found that Ba does not display a radial gradient. However, Ba
abundances in classical Cepheids are affected by severe limits. In particular,
Luck (2014) noted that strong BaII lines are affected by line-formation
effects, while Andrievsky et al (2013) discussed isotopic shift effects. The
reason for the lack of a Ba gradient remains unclear.

ii)–Slopes of the radial gradients– The slopes are quite similar, on average
$-0.025\pm 0.004\,\mathrm{dex\ kpc^{-1}}$ for La, Ce, Nd, and Eu. The only
exception is Y, for which the slope is more than a factor of two steeper
($-0.053\pm 0.003\,\mathrm{dex\ kpc^{-1}}$). The current slopes agree quite
well, within the errors, with similar estimates available in the literature.

iii)–Spread of the radial gradients– The spread of the individual abundances
attains similar values across the thin disk. The outermost disk regions are an
exception, since the spread increases for $R_{G}$ larger than 13 kpc. The
neutron-capture elements display the same trend of iron and $\alpha$-element
abundances. Among the investigated elements, Y once again seems to be an
exception, since the spread is homogeneous across the disk regions covered by
the current CC sample.

iv)–Theory and observations– Radial gradients for La ($\sim 0.020\pm
0.003\,\mathrm{dex\ kpc^{-1}}$) and for Eu ($\sim 0.030\pm 0.004\,\mathrm{dex\
kpc^{-1}}$) agree quite well with theoretical predictions by Cescutti et al
(2007) for Galactocentric distances covering the entire thin disk (4$\leq
R_{G}\leq$22 kpc). They found a slope of $\sim 0.021\,\mathrm{dex\ kpc^{-1}}$
for La and of $\sim 0.030\,\mathrm{dex\ kpc^{-1}}$ for Eu. The predicted
slopes become steeper for Galactocentric distances shorter than 14 kpc, and
shallower for distances larger than 16 kpc (see their Table 5).

###  6.7 CNO distribution

CNO abundances are a solid diagnostic to constrain evolutionary predictions,
based on different physical assumptions, concerning the size of the helium
core and the efficiency of the first dredge-up. The key issue is that
evolutionary models accounting for core rotation display an enhancement in N
that is significantly larger when compared with models neglecting rotation and
only accounting for semi-convection and/or convective core overshooting during
central H-burning phases. This means that CNO abundances are a solid
observable to constrain the input physics of intermediate-mass models
(Anderson et al, 2014). Even though the abundances of these three elements can
play a key role to constrain the physics of intermediate-mass stars, we still
lack accurate and homogeneous measurements of short and long-period Galactic
and Magellanic CCs (Andrievsky et al, 2005; Kovtyukh et al, 2005; Luck and
Lambert, 2011; Romaniello et al, 2022; Trentin et al, 2023), to constrain the
impact that mixing due to the first dredge-up (see section C.3 in Appendix)
have on the surface abundances of CCs.

###  6.8 A new spin on iron and $\alpha$-element abundance gradients

In a recent investigation da Silva et al (2023) provided a new homogeneous and
accurate investigation of the radial gradients (iron, $\alpha$-elements [O,
S]) across the Galactic thin disk by using CCs. This paper offers several new
insights concerning the MW chemical enrichment history. The main novelties
when compared with similar investigations in the literature are the following.

i)–Sample selections– High-resolution, high SNR spectra for CCs were secured
with four different spectrographs, the sample includes more than 1100 spectra
for 356 variables. For 77 CCs accurate abundances are provided for the first
time. The heliocentric distances are mainly based on Gaia EDR3 distances
(Bailer-Jones et al, 2021) complemented with NIR and MIR diagnostics available
in the literature. Their Galactocentric distances range from the inner disk
($R_{G}$=5 kpc) to the outskirts of the disk ($R_{G}$=30 kpc) with a handful
(eight) of Cepheids located at distances larger than 18 kpc. Moreover, they
also investigated the orbital properties of their spectroscopic sample and
they found that the bulk of them have orbits typical of thin disk stars. The
sample of open clusters is from Randich et al (2022) and includes 62 stellar
systems observed by the Gaia-ESO Survey (GES) plus 18 OCs retrieved from the
ESO archive (Viscasillas Vázquez et al, 2022). The age of the OCs ranges from
a few Myrs to $\sim$7 Gyr, while their Galactocentric distances range from
$\sim$6 to 21 kpc.

ii)–Homogeneous abundances– Atmospheric parameters and elemental abundances
over the entire spectroscopic sample were based on individual spectra. The
abundances for each spectrum are the mean value computed from individual
lines. The final abundances for variables with multiple spectra are the mean
over the individual spectra.

iii)–Spectroscopic diagnostics– The O abundances are based on the forbidden [O
i] line at 6300.3 Å, in the LTE approximation, while S abundances are based on
the S i triplet at 6757 Å.

The radial gradient for iron, oxygen and sulphur are plotted as a function of
the Galactocentric distance in Fig. 28. Simple inspection to the data plotted
in this figure disclose several new findings.

i)–Linearity of the radial gradient– The iron radial gradient is linear for
Galactocentric distances smaller than $\sim$12 kpc, but it becomes flatter at
larger distances ($R_{G}\geq$15 kpc). They also found that a logarithmic fit
takes into account the linear gradient of the inner disk and for the
flattening of the outer disk. Their estimates of both linear and logarithmic
fits agrees quite well with similar estimates available in the literature
(Carraro et al, 2007; Yong et al, 2012; Donor et al, 2020; Magrini et al,
2023). However, the iron radial gradient provided by da Silva et al (2023)
shows a well defined departure from linearity for Galactocentric distances
larger than 12–14 kpc. A more quantitative analysis requires a larger sample
to assess whether the radial gradient shows either a change in the slope, or a
steady variation with distance, as suggested by the logarithmic fit.

ii)–Comparison between CCs and OCs– Data plotted in the top panel of Fig. 28.
show that CCs and OCs show similar iron radial gradients. Moreover, the ages
of the GES OC sample cover more than one order of magnitude, whereas their
radial distributions are similar. However, the flattening of the radial
gradient in the outer disk is only partially supported by OCs, since they
typically have Galactocentric distance smaller than $\sim$15 kpc with a single
cluster located at $\sim$20 kpc.

iii)–Oxygen radial gradient– A glance at the oxygen abundances plotted in the
middle panel of Fig. 28 shows that the oxygen radial gradient is 25% shallower
than the iron gradient. CCs and OCs display very similar radial trends and a
vanishing, if any, dependence on the cluster age. To overcome possible
systematics, da Silva et al (2023) used the same data provided by Magrini et
al (2023) and performed an independent fit following the same approach adopted
for CCs. They found that the two gradients, within the errors, agree quite
well with each other. The comparison with similar estimates available in the
literature based on CCs indicates that the current slope is shallower (Luck
and Lambert, 2011; Trentin et al, 2023). However, the comparison should be
used carefully, since the difference could be due to the different lines
adopted for measuring oxygen abundances, and to the spectroscopic analysis.

iv)–Slope of the sulfur radial gradient– Data plotted in the bottom panel of
Fig. 28 show that the linear fit of the sulfur radial gradient is twice
steeper than the slope of the iron gradient, while the difference in the
logarithmic fit is $\approx$0.7 dex ($\log{\rm kpc})^{-1}$. It is noteworthy
that the sulfur radial gradients for both CCs and OCs agree quite well.
Indeed, the zero-points and the slopes of the linear fits attain very similar
values (see labels). This circumstantial evidence brings forward an
$\alpha$-element with a radial gradient steeper than the iron gradient, and
points toward a more significant role played by massive stars in iron and
sulfur chemical enrichment.

Figure 28: Top: Iron abundances for CCs as a function of the Galactocentric
distance (kpc). CCs are marked with blue circles while OCs with colored
symbols. Iron abundances for OCs are from Magrini et al (2023). The solid
black line shows the linear regression for CCs with Galactocentric distances
ranging from 4 to 12.5 kpc, while the dashed black line is the extrapolation
up to 17 kpc. The green line displays the logarithmic fit over the entire
sample of CCs. The coefficients of the analytical relations are labelled. The
shaded area shows the running average, and its standard deviation, weighted
with a Gaussian function taking into account the errors. Middle: same as the
top, but for O abundances. Bottom: Same as the top, but for S abundances.
Adapted from da Silva et al (2023, Fig. 2). Image credit: R. da Silva.

The same authors also performed a detailed comparison between observations and
chemical evolution models of the Galactic thin disk. They adopted the best-fit
model recently provided by Palla et al (2020) and they found that the use of a
constant star formation efficiency for $R_{G}\geq 12$ kpc takes into account
the flattening observed in iron and in $\alpha$-element radial gradients.
Moreover, they found that inside the solar circle the current [S/H] and [S/Fe]
gradients are well reproduced by canonical yield prescriptions by Kobayashi et
al (2006) and by Kobayashi et al (2011). The same comparison indicates that a
decrease of a factor of four in the current S predictions is required to
explain the flattening in the outermost regions. This means a relevant
decrease in the S yields in the more metal-poor regime.

Finally, Matsunaga et al (2023) took advantage of high resolution, near
infrared spectra, covering the $Y$,$J$-bands, collected with WINERED at the
Magellan telescope to investigate the iron abundance of 16 CCs located in the
inner disk ($R_{G}$=3–5.6 kpc). They found that the iron radial gradient in
this region is either shallower or even flat (see their Fig. 2 and also
Kovtyukh et al 2022).

##  7 Omne ignoto pro magnifico

The next two subsections are focussed on open problems concerning Cepheids
with possible/challenging near-term solutions.

###  7.1 Open problems with possible near future solutions

The most interesting open problems at the crossroad between theory and
observations appear to be the following.

i)–Nonlinear, convective models for long-period TIIC Cepheids– The current
generation of nonlinear, convective hydrodynamical models of variable stars
shacked off their speculative nature and are providing solid predictions of
the current properties of radial variables. However, we are still missing
detailed pulsation predictions concerning long-period W Virginis and RV Tauri.

ii)–Metallicity dependence of both PL, PLC and PW relations– This is a sort of
message in the bottle for future spectroscopic surveys. Gaia and long-term
ground-based variability surveys (ASAS-SN101010https://asas-sn.osu.edu/, OGLE-
IV111111http://ogle.astrouw.edu.pl/, ZTF121212https://www.ztf.caltech.edu/,
PTF131313https://www.ptf.caltech.edu/page/ptf_about, KISOGP (Takase and
Miyauchi-Isobe, 1984) will provide solid identification of thousands of
Galactic Cepheids. In a few years, Gaia will also provide accurate geometrical
distances for a good fraction of them. Moreover, the use of high resolution
images collected with NIRCam on board James Webb Space
Telescope141414https://webb.nasa.gov is providing the unique opportunity to
test the crowding affecting extragalactic Cepheids and to validate earlier
results based on HST photometry (Riess et al, 2023, 2024) and in the near
future to improve the accuracy of the extragalactic Population I and
Population II distance scale (Madore and Freedman, 2023, and references
therein).

Moreover, the unique opportunity of using the panoramic imaging capabilities
of NIRCam on board James Webb Space Telescope151515https://webb.nasa.gov will
provide accurate NIR and MIR photometry to significantly improve the accuracy
of the extragalactic Population I and Population II distance scale (Madore and
Freedman, 2023, and references therein).

The near-future challenge for ongoing and forthcoming optical (SDSS
V161616https://www.sdss.org/, GALAH171717https://www.galah-survey.org/,
WEAVE181818https://ingconfluence.ing.iac.es/confluence/display/WEAV/The+WEAVE+Project,
4MOST191919https://www.4most.eu/cms/home/, MAVIS@VLT (Monty et al,
2021)202020https://mavis-ao.org/mavis/) and NIR
(MOONS@VLT212121https://vltmoons.org/,
APOGEE222222https://www.sdss.org/instruments/apogee-spectrographs/,
ERIS@VLT232323https://www.eso.org/public/teles-instr/paranal-
observatory/vlt/vlt-instr/eris/, WINERED@Magellan242424http://lihweb.kyoto-
su.ac.jp/WINERED/overview_winered.html) spectroscopic surveys is to provide
accurate and homogeneous abundances for iron peak, $\alpha$ and neutron
capture elements. All of this will bring forward reliable predictionss for the
impact of metallicity on pulsation properties and on the different diagnostics
to estimate individual distances. In his seminal review on the absolute
magnitude of CCs Bob Kraft suggested, on the basis of preliminary evidence,
that MC and MW open clusters hosting CCs should cover a range in chemical
compositions (Kraft, 1963). In particular, more than half century ago, he
argued

Would we then be justified in assuming universality of the period-luminosity
law even among so-called classical Cepheids?

This vexata questio has not been solved yet, but the solution, sixty years
later, seems just around the corner.

iii)–Hertzsprung et similia– Homogeneous and accurate photometric data
collected by microlensing (MACHO, EROS, OGLE) experiments paved the way for a
new golden age for the physics of stellar oscillations and for the use of
variable stars as stellar tracers. These surveys collected data mainly in a
single band, plus an ancillary secondary band. The same area is going to
experience a quantum jump in a couple of years thanks to the Vera C. Rubin
Observatory and the LSST survey. This survey is going to observe the entire
southern sky every three nights in six different photometric bands
($u,g,r,i,z,y$). In a time interval of ten years, the VRO will collect on
average more than eighty phase point per objects in each band down to limiting
magnitudes of 24–27. This will provide a complete census of helium-burning
variables across the Local Group, and of CCs and Miras across the Local
Volume. There will be the unique opportunity to study long-term secondary
modulations (mixed-mode, Blazhko), and in particular, pulsation phenomena
showing amplitude modulations over a broad range in wavelengths. In this
context, the spectroscopic follow-up will play a crucial role not only to
provide radial velocity measurements (low spectral resolution) and individual
elemental abundances (medium/high spectral resolution), but also to further
constrain the physics of variable stars. The most recent investigation
concerning the phase lag between light and radial velocity curves in CCs dates
back to Szabó et al (2007) and is based on a few dozen Galactic Cepheids. More
accurate data can shed new light on the role that the different ionization
regions play in shaping the light curves.

iv)–Homogeneous spectroscopic abundances – Radial gradients appear to show
quite similar trends when moving from the inner to the outer disk. Typically,
they agree quite well with predictions based on chemical and on chemo-
dynamical evolution models. There are a couple of exceptions like Ca among the
$\alpha$-elements and Y among the neutron capture elements. The current
empirical and theoretical evidence do not allow us to understand whether this
is a limit in the atomic data we are using for these elements, or whether they
are suggesting different paths in the chemical enrichment of the thin disk.
Homogeneous, high-resolution spectra based on a sizable number of lines can
shed new light on the nature or nurture of the quoted elements.

It is worth mentioning that the current abundances analyses are mainly based
on 1D stellar atmosphere models in LTE approximations, but the transition to
spectroscopic abundances based on 3D stellar atmosphere models in NLTE
approximations and with time-dependent convective energy transport is a
scientific goal, achievable in the next few years (Magic et al, 2015;
Chiavassa et al, 2018; Gerber et al, 2023). This transition is more difficult
in dealing with Cepheids, since they experience along the pulsation cycle
variations in effective temperature of several hundred degrees, in surface
gravity of half dex and almost a factor two in micro-turbulent velocity (see
section 6.2.1). Indeed, in a recent investigation Vasilyev et al (2019), by
using radiation hydrodynamics simulations of Cepheid atmospheres (2D NLTE line
formation), found a significant variation in the oxygen abundance (O i triplet
at 7774 Å) along the pulsation cycle.

Moreover, large and homogeneous open cluster parameters (distance, reddening,
proper motion, chemical composition) are required to trace the age dependence
of the radial gradients across the thin disk. This seems well within the reach
of Gaia and near future-optical/NIR spectroscopic survey.

###  7.2 Open problems with challenging near future solutions

Thinking about significant near-term achievements in the field of radial
variables, some of the most interesting open problems appear to be the
following.

i)–To loop or not to loop?– We are still missing the micro and macro physics
driving the morphology of the blue loops. The same drawback applies to the
efficiency of the mass loss during the red giant branch phase. Therefore, the
comparison between predicted and observed period distributions does depend on
the physical assumptions adopted to deal with these phenomena.

ii)–What is the physical mechanism driving the occurrence of bump Cepheids?–
In spite of the paramount theoretical and observational effort, we still lack
ab initio calculations to explain the occurrence of secondary features (bumps,
dips) along the pulsation cycle.

iii)–What is the physical mechanism driving the occurrence of mixed-mode
Cepheids? Is there any connection with Blazhko RR Lyrae?– The same outcome
applies to the physical mechanism(s) driving the occurrence of secondary
modulations. In particular, it is not clear whether the secondary modulations
present in RR Lyrae and in Cepheids have a common origin.

iv)–Reddening estimates– Accurate estimates of the extinction in low and high
reddened regions has been one the toughest problems in quantitative
astrophysics. Even in the era of Gaia, as well as of large spectroscopic
surveys, we are dealing with this long-standing problem. A significant and
quantitative improvement has been provided by diffuse interstellar bands
(Munari et al, 2008; Zhao et al, 2021; Saydjari et al, 2022), but this
approach requires at least medium resolution spectra. The photometric
diagnostics based either on variables stars, (Sturch, 1978) or on narrow-band
photometry (Strömgren) require large amount of telescope time. A diagnostic
that is solid and easy to use is still lacking.

##  8 Summary and future developments

This review deals with different aspects of Cepheid variable stars. the
content can be split into five different broad topics.

Phenomenology of variable stars– Ongoing space (Gaia, WISE) and ground-based
long-term variability surveys (OGLE IV) provided the opportunity to
investigate the phenomenology of variable stars. In particular, the occurrence
of single and mixed-mode variables in the MW and in the Magellanic Clouds.
This means a detailed investigation of the diagnostics adopted for the
identification (periods, pulsation amplitudes, shape of the light curves) of
the pulsation mode. Moreover, the opportunity to use homogeneous data for the
three different stellar systems, characterized by different chemical
enrichment histories, allowed us to investigate the dependence of the
pulsation parameters on the metal content. The circumstantial evidence
concerning the pulsation parameters are the following.

The period distribution of CCs steadily moves towards shorter periods when
moving from more metal-rich (MW) to more metal-poor (SMC) stellar systems.
Moreover, the mean colors of CCs become systematically bluer for a decrease in
the metal content.

The luminosity amplitude for FU and FO CCs display well-defined primary and
secondary peaks moving towards shorter periods for a decrease in the metal
content.

The use of different populations ratios among single-mode variables and/or
among mixed-mode variables provide an unique opportunity to analyze, on a
quantitative basis, pulsation (topology of the instability strip) and
evolutionary (the stars formation episodes) properties, together with the
chemical enrichment histories of the different stellar systems.

i)–The population ratio between second overtone and fundamental CCs decreases
for an increase in metallicity (0.03 [SMC] vs 0.01 [LMC]) and it is vanishing
for the MW. The fraction of mixed-mode variables, instead, does not seem to
depend on metallicity (0.11$\pm$0.01, [SMC]; 0.17$\pm$0.01, [LMC];
0.14$\pm$0.01, [MW]).

ii)–The population ratio between first overtone and fundamental ACs is clearly
higher in the MW (0.76$\pm$0.11) than in the SMC (0.54$\pm$0.10) and in the
LMC (0.44$\pm$0.08). However, the MC sample is limited and affected by large
Poissonian uncertainties.

iii)–The relative number of different subgroups of TIICs does display a
positive trend with metallicity for BLHer (0.28$\pm$0.01, [SMC];
0.34$\pm$0.01, [LMC]; 0.38$\pm$0.02, [MW]) and a negative one for WVir
(0.51$\pm$0.01, [SMC]; 0.46$\pm$0.01, [LMC]; 0.43$\pm$0.02, [MW]). On the
other hand, the fraction of RVTau remains, within the errors, constant:
0.21$\pm$0.01 (SMC), 0.19$\pm$0.01 (LMC) and 0.19$\pm$0.01 (MW). Note that
only a few FOs have been currently identified among MW and MC TIICs.

Hertzsprung progression– A detailed analysis of the pulsation amplitude and of
the difference between the bump and the minimum in luminosity, together with
their difference in phase, allowed us to investigate on a quantitative basis
the occurrence of the Hertzsprung progression in MW and Magellanic CCs. The
pulsation amplitudes were adopted because, in a specific period range, the
maximum of the light curve is given by the bump rather than by the pulsation
maximum. We found that the center of the Hertzsprung progression is anti-
correlated with the metal content. Indeed, the center moves from P(HP)=$9.0\pm
0.2$ days in the MW to P(HP)=$9.4\pm 0.2$ days in the LMC and to
P(HP)=$10.0\pm 0.2$ days in the SMC.

Evolutionary constraints for Cepheids– The evolutionary properties of Cepheids
were investigated by using optical, optical-NIR and NIR color–magnitude
diagrams. In order to overcome possible systematics, we adopted homogeneous
optical (OGLE IV) and NIR (IRSF) photometry for both variable stars and common
stars in the MW and in the MCs. The comparison between observations and theory
was performed by using homogeneous sets of stellar isochrones and evolutionary
models available in the BASTI-IAC data base. The key advantage of this
database is that we adopted cluster isochrones, either considering or
neglecting convective core overshooting during central hydrogen-burning
phases. Moreover, evolutionary predictions are available for scaled-solar and
$\alpha$-enhanced chemical mixtures. This is a crucial issue in dealing with
young (CCs), intermediate (ACs) and old (TIICs) stellar tracers. The
comparison provides ranges in age and in metallicity for the three different
groups of variable stars, which agree quite well with similar estimates
available in the literature.

We have studied in detail the transition between stellar structures that
ignite helium in a core partially affected by electron degeneracy, and those
that ignite helium quiescently. This difference marks the transition between
low- and intermediate-mass stars. It goes without saying that the two groups
obey to different mass-luminosity relations. The current circumstantial
evidence indicates that the bright portion of the ACs is located across this
boundary. This means that the brightest ACs are quite similar to CCs, whereas
the others are the truly Anomalous Cepheids.

It is worth mentioning that the central hydrogen-burning evolutionary time in
the quoted transition is only a factor of four longer than the central helium-
burning lifetime. The hydrogen-burning lifetime is typically two order of
magnitude longer in the low-mass regime, and roughly 6/8 times longer in the
intermediate-mass regime. This is why stellar structures in the transition
between low- and intermediate-mass stars are the main contributors of central
helium-burning stellar populations (red clump stars).

We have also examined the impact of the first dredge-up among different
evolutionary scenarios (classical, overshooting) to trace its efficiency. We
further support the use of CNO abundances as a solid diagnostic to
discriminate among the different physical assumptions driving the mixing
during central hydrogen-burning phases in intermediate-mass stars.

Cepheids as primary distance indicators– The use of the different diagnostics
have been investigated in detail. The current evidence based on MC Cepheids
fully support the use of FOs as distance indicators. Moreover, pros and cons
of different distance diagnostics PL, PW and PLC were addressed on a
quantitative basis. We also discussed advantages and drawbacks in using
optical, optical-NIR and NIR mean magnitudes.

Cepheids as stellar tracers– We studied the use of MW CCs as young stellar
tracers of the Galactic thin disk. The current evidence indicates well-defined
negative radial gradients in iron peak, $\alpha$\- and neutron capture
elements. The only exception is barium, and we still lack a quantitative
explanation. Plain physical arguments suggest that the chemical enrichment has
been more efficient in the high-density regions of the inner disk, and then
moved into the outermost disk regions (Spitoni et al, 2019; Matteucci, 2021,
and references therein). Preliminary evidence based on both CCs and open
clusters indicates a flattening of the gradient in the outskirts of the thin
disk. This global trend is only minimally affected by the flaring of the disk
(Lemasle et al, 2022).

Spectroscopic measurements based on high-resolution spectra indicate that the
slopes of several $\alpha$-elements are quite similar to the slope of iron.
Indeed, the trend of the [$\alpha$/Fe] abundance ratio is mildly positive
across the disk. This suggests a similar chemical enrichment history. This
evidence is further supported by the radial gradients of s- (La, Ce, Nd) and
r- (Eu) process elements. The slopes of these elements are steeper than those
of the $\alpha$-elements, and the [n-capture/Fe] abundance ratios are slightly
more positive across the entire disk. There is only one exception: Y shows a
slope quite similar to iron, and in turn, the [Y/Fe] abundance ratio is flat.

Interestingly enough, the same abundance ratios display a flat trend as a
function of the age (logarithmic period). The pulsation period is a solid
analog of the individual age of CCs, since the longer the period, the younger
is the Cepheid. Therefore, the chemical enrichment history of CCs has been
quite homogeneous during the last 200/300 Myrs. There is one exception: Ca,
showing a mild positive gradient as a function of the period. The positive
gradient in [Ca/H] versus period and the negative gradient of [Ca/H] vs
Galactocentric distance are not related, since outer disk CCs have a canonical
period distribution.

Acknowledgments We are grateful to two anonymous referees for their positive
words concerning the content and the cut of an earlier version of the present
paper, and for their very relevant suggestions, which improved its
readability. They also provided specific and appropriate hints on a couple of
sections which improved the content of the paper. Special thanks to G. Giobbi
for her valuable editing of the manuscript and V. D’Orazi, G. Fiorentino, M.
Marconi, N. Matsunaga and M. Salaris, for their detailed suggestions
concerning the content and the layout of an early version of this manuscript.
We are also happy to acknowledge informative interchanges and many interesting
conversations with D. Lennon, W. Freedman, B. Madore, A. Udalski and their
collaborators over more than 20 years.

It is a real pleasure to thank the many colleagues that supported me with
suggestions and advice along my research activity (V. Castellani, F. Caputo,
R. Buonanno, R.F. Stellingwerf, A.J. Cox, P.B. Stetson, H. Smith, A.R. Walker,
C. Sneden, H.-R. Kudritzki). Special thanks to my long-standing
collaborators—M. Marconi, M. Dall’Ora, M. Monelli, M. Fabrizio, R. da Silva,
B. Lemasle, L. Inno, V. D’Orazi, M. Marengo, J. Mullen, F. Thevenin, J. Storm,
B. Chaboyer, S. Cassisi, M. Salaris, G. Coppola, V. Ripepi, I. Musella, M. di
Criscienzo, G. Fiorentino, C. E. Martinez Vazquez, N. Matsunaga, W. Gieren, G.
Pietrzynski, B. Pilecki, V. Kovtyukh, M. Tantalo, E. Valenti, M. Zoccali—who
made and are still making the research in this field a fantastic adventure.

I am immensely grateful to my original family and to my current family, and in
particular to $F^{2}G^{2}$, they never pointed me out for the many absences
for astrophysical otium. This paper is dedicated to Francesco, who made our
recent life “really something special”. Special thanks to the A&A Review
editorial board, and in particular, to Francesca Matteucci, Bengt Gustafsson
and Frank Schulz for this unique opportunity, and for being so patient with us
over many years. This investigation was partially supported by Project PRIN
MUR 2022 (2022ARWP9C) “Early Formation and Evolution of Bulge and HalO
(EFEBHO)”, PI: M. Marconi, funded by European Union – Next Generation EU and
by the Large grant INAF 2022 MOVIE (PI: M. Marconi). This research was also
partially supported by the Munich Institute for Astro–Particle and BioPhysics
(MIAPbP) which is funded by the Deutsche
Forschungsgemeinschaft(DFG,GermanResearch Foundation) under Germany´s
Excellence Strategy – EXC-2094 – 390783311.

##  Appendix A Metallicity dependence of the Hertzsprung progression

In order to investigate on a more quantitative basis the variation of the bump
among UPDB and DORB Cepheids, Figs. 29 and 30 display the $I$-band light
curves for LMC and Galactic bump Cepheids.

Figure 29: Phased $I$-band light curves of LMC bump Cepheids with a bump
moving either up along the decreasing branch (UPDB) or down along the rising
branch (DORB).

Figure 30: Phased $I$-band light curves of LMC bump Cepheids with a bump
moving either up along the decreasing branch (UPDB) or down along the rising
branch (DORB).

Moreover, we also estimated the difference in phase between the pulsation
maximum and the minimum in luminosity ($\Phi$Puls) and the difference in phase
between the bump and the minimum in luminosity ($\Phi$bump). Data plotted in
the top panels of Fig. 31 show that UPDB Cepheids (squares) are characterized
by an almost constant phase difference between the pulsation maximum and the
minimum. The mean value are, within the errors, quite similar (0.26$\pm$0.02,
SMC; 0.29$\pm$0.03, LMC; 0.30$\pm$0.03, MW) when moving from the more metal-
poor to the more metal-rich stellar systems. In the MCs there are a few
outliers which attain larger values, but they are typically located beyond the
HP.

Conversely, the DORB Cepheids (diamonds) display a steady decrease from
$\approx$0.50 to $\approx$0.30 when moving from the short- to the long-period
range. There is a range in period between 9.5 and 11 days in which the two
subgroups overlap with each other. However, the separation between UPDB and
DORB Cepheids, with the exception of a few transitional objects, appears quite
clear.

Figure 31:  Top: from left to right difference between the phase of the
maximum and the phase of the minimum in $I$-band light curves as a function of
the period for SMC (panel a), LMC (panel b) and Galactic (panel c) bump
Cepheids. Squares and diamonds mark UPDB and DORB Cepheids. Bottom: Same as
the top, but the difference is between the phase of the bump and the phase of
the minimum along $I$-band light curves (SMC, panel d; LMC, panel e; Galactic,
panel f).

The bottom panels of the same figure display the difference between the phase
of the bump and the phase of the minimum in the $I$-band light curve. The
separation between UPDB and DORB Cepheids becomes even clearer. Indeed, the
UPDBs are clustering around a constant mean value ranging from 0.49$\pm$0.02
(SMC) to 0.50$\pm$0.03 (LMC) and to 0.51$\pm$0.03 (MW). The DORBs show a
steady decrease when moving from the short- to the long-period range. Once
again, the two subgroups display an overlap for periods ranging from 9.5 to 11
days, but they differ on average by a factor of two.

In order to further investigate on a quantitative basis the difference between
the UPDB and DORB Cepheids, the top panels of Fig. 32 display the difference
in magnitude between the pulsation maximum and the minimum in the $I$-band
light curve (pulsation amplitude, $\Delta$Puls). In this context, it is worth
mentioning that $\Delta$Puls is not always the luminosity amplitude, i.e. the
difference in magnitude between the absolute maximum and the minimum along the
light curve. Indeed, there is a range in period in which the bump is brighter
than the pulsation maximum (see Figs. 11, 29 and 30). Data plotted in these
panels display quite clearly that the pulsation amplitudes attain a minimum
that steadily decreases from 9.5–10 days for the SMC to 9.0–9.5 days for the
LMC and to $\approx$9.0 days for the Galactic UPDB Cepheids (squares). The
DORB Cepheids (diamonds) display a steady increase when moving from the short
to the long-period regime, and the global trend is steeper when moving from
metal-poor to more metal-rich stellar systems. They are characterized by a
large spread at fixed pulsation period, but the spread in magnitude is
intrinsic, because it is well beyond the typical photometric error along the
light curve.

The center of the HP in the MW and in the MCs were marked with vertical black
lines in Fig. 32. Data plotted in this figure display some outliers across the
minimum. This variation across the center of the HP is intrinsic and expected.
Indeed, MC and MW CCs are characterized by a well defined metallicity
distribution, with a standard deviation of the order of 0.5 dex (Romaniello et
al, 2008; Hocdé et al, 2022). In a recent investigation, Romaniello et al
(2022) by using high resolution spectra of 89 LMC CCs suggested that the
spread in iron abundance is quite modest. However, spectroscopic measurements
based on B-type stars (Hunter et al, 2009) and on red giants (Dobbie et al,
2014) indicate a spread in chemical composition larger than 0.5 dex.

Figure 32:  Top: from left to right difference between the pulsation maximum
and the minimum in the $I$-band light curves as a function of the period for
SMC (panel a), LMC (panel b) and Galactic (panel c) bump Cepheids. The black
vertical line shows the center of the HP. The symbols are the same as in Fig.
31. Bottom: Same as the top, but the difference is between the magnitude of
the bump and the minimum along $I$-band light curves (SMC, panel d; LMC, panel
e; Galactic, panel f).

The difference in magnitude between the bump and the minimum versus the
pulsation period (bottom panels of Fig. 32) display an opposite trend. The
UPDB Cepheids attain, as expected, the largest amplitudes, while DORB Cepheids
show once again a steady decrease when moving from the short- to the long-
period range. The bump Cepheids that trace the center of the HP, i.e. those
with the smallest pulsation amplitudes, attain also the smallest bump
amplitudes. However, the minimum is less clear, since UPDBs and DORBs
partially overlap across the HP.

Figure 33:  Top: $R_{21}$ (panel a) and $R_{31}$ (panel b) Fourier parameters
as a function of the pulsation period for $I$-band light curves of SMC bump
Cepheids provided by OGLE IV. The symbols are the same as in Fig. 31. Bottom:
Same as the top, but for $\phi_{21}$ (panel c) and $\phi_{31}$ (panel d)
Fourier parameters .

The current findings concerning the use of the Bailey (pulsation amplitude)
and pseudo-Bailey (difference in magnitude between the bump and the minimum)
diagrams together with the phase difference diagrams allow us to further
constrain the use of the Fourier parameters to trace the variation of the
shape of the light curves across the HP. In order to analyze this issue in
depth by using homogeneous and accurate estimates of the Fourier parameters,
we took advantage of the Fourier analysis provided by OGLE IV (Soszyński et
al, 2015a, b, 2017b, 2018). Data plotted in the top panels of Fig. 33, 34 and
35 display that the $R_{21}$ (left) and the $R_{31}$ (right) Fourier
parameters associated with the amplitudes of the first three harmonics display
either a mild quadratic trend ($R_{21}$), or an almost constant value
($R_{31}$) across the entire period range of the HP. The top right panels of
these figures show that the $R_{31}$ parameter shows minima similar to the
minima we identified by using the difference in magnitude between pulsation
maximum and minimum and the difference between bump and minimum. In order to
help the eye identify the minima, we plotted the same black vertical lines
plotted in Fig. 32. This suggests that the $R_{31}$ parameter is a solid
diagnostic to constrain the center of the HP.

The trend of the phase difference among the same harmonics, plotted in the
bottom panels of the same figure, shows several interesting features. The
distribution of the $\phi_{21}$ (left) has a wedge shape with two well-defined
sequences. The horizontal sequence, which attains the largest values and it is
almost constant over the entire period range, while the slant sequence moves
from the center of the HP and approaches the horizontal sequence in the long-
period range. The UPDB Cepheids partially overlap with DORB Cepheids. As a
whole, the $\phi_{21}$ parameter does not seem to be a solid diagnostic to
constrain the center of the HP. The separation between UPDBs and DORBs showed
by the $\phi_{31}$ Fourier parameter plotted in the bottom right panel is a
very solid diagnostic to separate bump Cepheids.

Figure 34:  Top: $R_{21}$ (panel a) and $R_{31}$ (panel b) Fourier parameters
as a function of the pulsation period for $I$-band light curves of LMC bump
Cepheids provided by OGLE IV ($R_{21}$, panel a; and $R_{31}$, panel b).
Symbols are the same as in Fig. 31. Bottom: Same as the top, but for
$\phi_{21}$ (panel c) and $\phi_{31}$ (panel d) Fourier parameters.

The key difference in the comparison between the Bailey and the pseudo-Bailey
diagram with the $R_{21}$ (left) and the $R_{31}$ Fourier parameters is that
the latter attain very similar values when moving from MC to Galactic
Cepheids, whereas the former ones display a clearer trend when moving from
more metal-poor to more metal-rich stellar systems. A similar outcome applies
to the comparison between the phase difference of pulsation maximum and of the
bump with the $\phi_{21}$ and $\phi_{31}$ Fourier parameters. The former
parameters display well separated linear trends for Cepheids, with a bump
located either along the decreasing or the rising branch of the light curve.

Figure 35:  Top: $R_{21}$ (panel a) and $R_{31}$ (panel b) Fourier parameters
as a function of the pulsation period for $I$-band light curves of Galactic
bump Cepheids provided by OGLE IV ($R_{21}$, panel a; R31, panel b). Symbols
are the same as in Fig. 31. Bottom: Same as the top, but for $\phi_{21}$
(panel c) and $\phi_{31}$ (panel d) Fourier parameters.

In closing this section it is worth mentioning that the HP can be considered
as the fingerprint of CCs. To our knowledge there is no other group of
variable stars showing a tight correlation of the phase of a secondary feature
along the light and the radial velocity curves with a specific range of
pulsation periods. This outcome applies to MW, MC and to extragalactic
CCs.252525This paragraph was triggered by a chat we have had with Adam Riess
concerning the evolutionary and pulsation properties of extragalactic
Cepheids.

Figure 36: Left: distribution of fundamental (light red) and first overtone
(dark red) CCs, fundamental (light blue) and first overtone (dark blue) ACs
and fundamental (green) TIICs in the NIR $K$,$J-K$ CMD of the SMC. The overlap
among the different groups of Cepheids is even more severe, due to the reduced
sensitivity of the $J-K$ color to the effective temperature. Data plotted in
this panel come from the IRSF survey of the Magellanic Clouds (Ita et al,
2004, 2018). The mean magnitudes of variable stars were estimated as the mean
of the measurements in magnitude. Right: Same as the left, but for LMC
Cepheids.

##  Appendix B Optical/near-infrared and near-infrared diagnostics

###  B.1 Color magnitude diagrams

This section deals with optical/NIR and NIR diagnostics, namely the $K$,$V-K$
and the $K$,$J-K$ CMDs of MC stellar populations discussed in Sect. 4.
Moreover, the $K$-band PL relations and the $K$,$J-K$ PW relations are further
investigated (see Sect. 6.2). Figure 36 shows the same variables discussed in
section˜4, but in the NIR $K$,$J-K$, CMD. Data for both static and variables
stars come from the NIR photometric survey of the MCs performed by IRSF (Ita
et al, 2004, 2018). The mean magnitude of variable stars is the mean of the
measurements. Data plotted in this figure display quite clearly that the
degeneracy among the different groups of variable stars and common stars is
even more severe, since they overlap over a significant fraction of the CMD.
The difference with optical CMDs is mainly due to the reduced sensitivity of
NIR colors to the effective temperature of F,G and early K-type stars.
Moreover, limiting magnitudes in NIR photometry are shallower than in the
optical regime, and evolutionary features such as the RC can hardly be
recognized.

Figure 37: Left: distribution of fundamental (light red) and first overtone
(dark red) CCs, fundamental (light blue) and first overtone (dark blue) ACs
and fundamental (green) TIICs in the Optical-NIR $K$,$V-K$ CMD of the SMC. The
different groups of Cepheids are more separated, due to the increased
sensitivity of the $V-K$ color to the effective temperature. Right: Same as
the left, but for LMC Cepheids.

To partially ease this difficulty, Fig. 37 shows the same variables, but
plotted in the optical-NIR $K$,$V-K$ CMD. There is still some overlap between
static and variable stars, but the increased sensitivity to the effective
temperature provides a better separation of the different groups of variable
stars. Indeed, they cover a range in color which is at least a factor of four
larger when compared with the optical CMD. Moreover, the distribution in
magnitude and in color is significantly more linear, since they become
intrinsically brighter when moving from hotter (shorter periods) to cooler
(longer periods) variables.

###  B.2 Period-Luminosity and Period-Wesenheit relations

This subsection deals with NIR and optical-NIR diagnostics to estimate
individual Cepheid distances. Data plotted in Fig. 38 clearly show the
advantages in using NIR mean magnitudes. The spread for both FO and FU
Cepheids is, at fixed pulsation period, systematically smaller and the slope
becomes systematically steeper (see Table 4). The same trend also applies to
optical, optical-NIR and NIR PW relations (see Table 5).

Figure 38: Top: Dereddened $K$-band PL relations (see Table 4) of SMC CCs
(left) and ACs plus TIICs (right). From shorter to longer periods, the dark
red and the light red symbols display FO and FU CCs, while the dark blue and
the light blue symbols display FO and FU ACs and the green symbols the FU
TIICs. The solid black lines display the linear fits to the PL relations.
Bottom: Same as the top, but for LMC Cepheids

Table 4: Optical and NIR PL relations ($M_{X}\,=\,a\,+\,b\times\log P$) for
Magellanic Cloud Cepheids.

band | mode |  a11footnotemark: 1 | b | $\epsilon_{a}$ | $\epsilon_{b}$ | $\sigma$
---|---|---|---|---|---|---
—SMC CCs—
$V$ | FO | 17.155 | –3.143 | 0.008 | 0.033 | 0.282
$I$ | FO | 16.723 | –3.299 | 0.006 | 0.026 | 0.231
$J$ | FO | 16.388 | –3.398 | 0.006 | 0.024 | 0.205
$H$ | FO | 16.162 | –3.537 | 0.006 | 0.023 | 0.179
$K$ | FO | 16.101 | –3.501 | 0.005 | 0.021 | 0.184
$V$ | FU | 17.783 | –2.907 | 0.009 | 0.018 | 0.267
$I$ | FU | 17.315 | –3.128 | 0.007 | 0.014 | 0.214
$J$ | FU | 16.935 | –3.211 | 0.006 | 0.013 | 0.202
$H$ | FU | 16.701 | –3.376 | 0.006 | 0.013 | 0.186
$K$ | FU | 16.633 | –3.365 | 0.005 | 0.011 | 0.172
—SMC ACs—
$V$ | FO | 17.601 | –2.370 | 0.048 | 0.344 | 0.171
$I$ | FO | 17.216 | –2.585 | 0.033 | 0.237 | 0.120
$J$ | FO | 16.825 | –2.859 | 0.027 | 0.214 | 0.055
$H$ | FO | 16.540 | –3.836 | 0.044 | 0.258 | 0.074
$K$ | FO | 16.638 | –2.820 | 0.044 | 0.272 | 0.124
$V$ | FU | 18.173 | –2.256 | 0.031 | 0.208 | 0.233
$I$ | FU | 17.674 | –2.233 | 0.018 | 0.113 | 0.105
$J$ | FU | 17.392 | –2.887 | 0.023 | 0.148 | 0.111
$H$ | FU | 17.095 | –2.649 | 0.069 | 0.393 | 0.230
$K$ | FU | 17.132 | –3.175 | 0.024 | 0.153 | 0.121
—SMC TIICs—
$V$ | FU | 18.885 | –1.677 | 0.092 | 0.127 | 0.325
$I$ | FU | 18.401 | –1.913 | 0.078 | 0.108 | 0.276
$J$ | FU | 17.973 | –2.027 | 0.080 | 0.112 | 0.286
$H$ | FU | 17.705 | –2.113 | 0.122 | 0.170 | 0.355
$K$ | FU | 17.694 | –2.211 | 0.070 | 0.097 | 0.248
—LMC CCs—
$V$ | FO | 16.618 | –3.241 | 0.007 | 0.018 | 0.187
$I$ | FO | 16.207 | –3.317 | 0.005 | 0.013 | 0.135
$J$ | FO | 15.829 | –3.364 | 0.004 | 0.011 | 0.115
$H$ | FO | 15.614 | –3.475 | 0.004 | 0.011 | 0.094
$K$ | FO | 15.549 | –3.441 | 0.003 | 0.009 | 0.089
$V$ | FU | 17.056 | –2.690 | 0.011 | 0.016 | 0.180
$I$ | FU | 16.660 | –2.924 | 0.008 | 0.012 | 0.123
$J$ | FU | 16.266 | –3.013 | 0.007 | 0.011 | 0.124
$H$ | FU | 16.094 | –3.232 | 0.007 | 0.011 | 0.117
$K$ | FU | 16.006 | –3.195 | 0.005 | 0.008 | 0.086
—LMC ACs—
$V$ | FO | 16.988 | –3.416 | 0.065 | 0.316 | 0.236
$I$ | FO | 16.679 | –3.362 | 0.046 | 0.214 | 0.168
$J$ | FO | 16.354 | –3.390 | 0.047 | 0.224 | 0.152
$H$ | FO | 16.219 | –3.504 | 0.064 | 0.264 | 0.158
$K$ | FO | 16.143 | –3.603 | 0.029 | 0.136 | 0.079
$V$ | FU | 17.719 | –2.582 | 0.028 | 0.186 | 0.252
$I$ | FU | 17.353 | –2.921 | 0.018 | 0.122 | 0.162
$J$ | FU | 16.980 | –3.136 | 0.020 | 0.139 | 0.162
$H$ | FU | 16.754 | –3.311 | 0.015 | 0.102 | 0.080
$K$ | FU | 16.715 | –3.284 | 0.015 | 0.112 | 0.122
—LMC TIICs—
$V$ | FU | 18.414 | –1.602 | 0.036 | 0.046 | 0.262
$I$ | FU | 18.041 | –1.954 | 0.022 | 0.028 | 0.170
$J$ | FU | 17.601 | –2.063 | 0.027 | 0.034 | 0.192
$H$ | FU | 17.497 | –2.326 | 0.035 | 0.043 | 0.206
$K$ | FU | 17.428 | –2.352 | 0.017 | 0.021 | 0.109

2525footnotetext: From left to right the columns list the photometric band,
the mode, the zero-point, the slope and their uncertainties and the standard
deviation (mag).11footnotemark: 1

To obtain the absolute zero points, one should subtract the geometrical
distance modulus of the LMC and SMC, based on eclipsing binaries
(18.477$\pm$0.026 mag and 18.977$\pm$0.028 mag for the LMC and SMC,
respectively, (Pietrzyński et al, 2019; Graczyk et al, 2020))

Table 5: Optical, NIR and optical–NIR PW relations ($M_{X}\,=\,a\,+\,b\times\log P\,+\,c\times CI$) for Magellanic Cloud Cepheids. band/CI | mode |  a11footnotemark: 1 | b | $\epsilon_{a}$ | $\epsilon_{b}$ | c | $\sigma$
---|---|---|---|---|---|---|---
—SMC CCs—
$I,V-I$ | FO | 15.966 | –3.570 | 0.005 | 0.020 | 1.550 | 0.173
$J,V-J$ | FO | 16.064 | –3.554 | 0.005 | 0.022 | 0.406 | 0.186
$H,V-H$ | FO | 15.940 | –3.642 | 0.005 | 0.021 | 0.217 | 0.165
$K,V-K$ | FO | 15.956 | –3.569 | 0.005 | 0.021 | 0.134 | 0.174
$K,I-K$ | FO | 15.949 | –3.570 | 0.005 | 0.020 | 0.250 | 0.175
$K,J-K$ | FO | 15.904 | –3.576 | 0.005 | 0.021 | 0.690 | 0.178
$K,V-I$ | FO | 16.047 | –3.528 | 0.005 | 0.021 | 0.288 | 0.179
$I,V-I$ | FU | 16.491 | –3.457 | 0.005 | 0.011 | 1.550 | 0.159
$J,V-J$ | FU | 16.580 | –3.359 | 0.006 | 0.013 | 0.406 | 0.187
$H,V-H$ | FU | 16.450 | –3.471 | 0.006 | 0.013 | 0.217 | 0.184
$K,V-K$ | FU | 16.473 | –3.429 | 0.005 | 0.011 | 0.134 | 0.163
$K,I-K$ | FU | 16.468 | –3.432 | 0.005 | 0.011 | 0.250 | 0.164
$K,J-K$ | FU | 16.422 | –3.466 | 0.005 | 0.011 | 0.690 | 0.172
$K,V-I$ | FU | 16.573 | –3.396 | 0.005 | 0.011 | 0.288 | 0.166
—SMC ACs—
$I,V-I$ | FO | 16.517 | –3.470 | 0.026 | 0.150 | 1.550 | 0.083
$J,V-J$ | FO | 16.519 | –3.217 | 0.062 | 0.381 | 0.406 | 0.143
$H,V-H$ | FO | 16.299 | –4.136 | 0.044 | 0.236 | 0.217 | 0.075
$K,V-K$ | FO | 16.502 | –2.763 | 0.045 | 0.267 | 0.134 | 0.119
$K,I-K$ | FO | 16.343 | –3.674 | 0.018 | 0.110 | 0.250 | 0.037
$K,J-K$ | FO | 16.424 | –2.776 | 0.048 | 0.286 | 0.690 | 0.128
$K,V-I$ | FO | 16.627 | –2.527 | 0.043 | 0.261 | 0.288 | 0.100
$I,V-I$ | FU | 16.936 | –2.918 | 0.017 | 0.114 | 1.550 | 0.121
$J,V-J$ | FU | 17.053 | –3.042 | 0.027 | 0.175 | 0.406 | 0.134
$H,V-H$ | FU | 16.842 | –2.680 | 0.078 | 0.442 | 0.217 | 0.259
$K,V-K$ | FU | 16.981 | –3.234 | 0.025 | 0.159 | 0.134 | 0.126
$K,I-K$ | FU | 16.976 | –3.252 | 0.024 | 0.159 | 0.250 | 0.126
$K,J-K$ | FU | 16.912 | –3.158 | 0.022 | 0.145 | 0.690 | 0.108
$K,V-I$ | FU | 17.073 | –3.195 | 0.024 | 0.156 | 0.288 | 0.124
—SMC TIICs—
$I,V-I$ | FU | 17.554 | –2.309 | 0.056 | 0.079 | 1.550 | 0.201
$J,V-J$ | FU | 17.604 | –2.193 | 0.079 | 0.111 | 0.406 | 0.283
$H,V-H$ | FU | 17.470 | –2.244 | 0.124 | 0.173 | 0.217 | 0.361
$K,V-K$ | FU | 17.536 | –2.290 | 0.067 | 0.094 | 0.134 | 0.239
$K,I-K$ | FU | 17.527 | –2.291 | 0.069 | 0.097 | 0.250 | 0.247
$K,J-K$ | FU | 17.502 | –2.338 | 0.072 | 0.101 | 0.690 | 0.257
$K,V-I$ | FU | 17.635 | –2.249 | 0.067 | 0.093 | 0.288 | 0.238
—LMC CCs—
$I,V-I$ | FO | 15.396 | –3.437 | 0.003 | 0.008 | 1.550 | 0.079
$J,V-J$ | FO | 15.479 | –3.448 | 0.004 | 0.010 | 0.406 | 0.099
$H,V-H$ | FO | 15.377 | –3.538 | 0.004 | 0.010 | 0.217 | 0.083
$K,V-K$ | FO | 15.398 | –3.490 | 0.003 | 0.009 | 0.134 | 0.079
$K,I-K$ | FO | 15.389 | –3.493 | 0.003 | 0.009 | 0.250 | 0.082
$K,J-K$ | FO | 15.355 | –3.502 | 0.003 | 0.009 | 0.690 | 0.082
$K,V-I$ | FO | 15.503 | –3.465 | 0.003 | 0.009 | 0.288 | 0.086
$I,V-I$ | FU | 15.889 | –3.315 | 0.005 | 0.008 | 1.550 | 0.077
$J,V-J$ | FU | 15.922 | –3.183 | 0.007 | 0.011 | 0.406 | 0.120
$H,V-H$ | FU | 15.858 | –3.352 | 0.008 | 0.012 | 0.217 | 0.120
$K,V-K$ | FU | 15.850 | –3.263 | 0.005 | 0.007 | 0.134 | 0.078
$K,I-K$ | FU | 15.844 | –3.269 | 0.005 | 0.008 | 0.250 | 0.079
$K,J-K$ | FU | 15.812 | –3.299 | 0.005 | 0.008 | 0.690 | 0.091
$K,V-I$ | FU | 15.961 | –3.233 | 0.005 | 0.008 | 0.288 | 0.082
continued on next page |  |  |  |
Table 5: continued.

band/CI | mode |  a11footnotemark: 1 | b | $\epsilon_{a}$ | $\epsilon_{b}$ | c | $\sigma$
---|---|---|---|---|---|---|---
—LMC ACs—
$I,V-I$ | FO | 16.012 | –3.475 | 0.038 | 0.183 | 1.550 | 0.127
$J,V-J$ | FO | 16.088 | –3.380 | 0.045 | 0.225 | 0.406 | 0.142
$H,V-H$ | FO | 16.003 | –3.690 | 0.076 | 0.326 | 0.217 | 0.181
$K,V-K$ | FO | 16.060 | –3.462 | 0.030 | 0.152 | 0.134 | 0.089
$K,I-K$ | FO | 16.022 | –3.535 | 0.028 | 0.133 | 0.250 | 0.074
$K,J-K$ | FO | 16.025 | –3.355 | 0.037 | 0.172 | 0.690 | 0.115
$K,V-I$ | FO | 16.112 | –3.610 | 0.030 | 0.148 | 0.288 | 0.079
$I,V-I$ | FU | 16.578 | –2.825 | 0.014 | 0.089 | 1.550 | 0.108
$K,J-K$ | FU | 16.514 | –3.327 | 0.016 | 0.120 | 0.406 | 0.131
$K,V-K$ | FU | 16.573 | –3.396 | 0.016 | 0.115 | 0.217 | 0.123
$K,V-I$ | FU | 16.669 | –3.332 | 0.016 | 0.113 | 0.134 | 0.123
$K,I-K$ | FU | 16.557 | –3.233 | 0.017 | 0.119 | 0.250 | 0.136
$J,V-J$ | FU | 16.642 | –3.160 | 0.020 | 0.137 | 0.690 | 0.156
$H,V-H$ | FU | 16.570 | –3.406 | 0.023 | 0.142 | 0.288 | 0.127
—LMC TIICs—
$I,V-I$ | FU | 17.326 | –2.496 | 0.014 | 0.018 | 1.550 | 0.101
$J,V-J$ | FU | 17.280 | –2.307 | 0.031 | 0.038 | 0.406 | 0.202
$H,V-H$ | FU | 17.260 | –2.470 | 0.042 | 0.052 | 0.217 | 0.237
$K,V-K$ | FU | 17.263 | –2.433 | 0.017 | 0.021 | 0.134 | 0.107
$K,I-K$ | FU | 17.264 | –2.448 | 0.017 | 0.021 | 0.250 | 0.114
$K,J-K$ | FU | 17.275 | –2.523 | 0.020 | 0.024 | 0.690 | 0.129
$K,V-I$ | FU | 17.368 | –2.375 | 0.016 | 0.020 | 0.288 | 0.101

2525footnotetext: From left to right the columns list the photometric band and
the color adopted to define the Wesenheit pseudo magnitude, the mode, the
zero-point, the slope of the PW relation, the uncertainties on zero-point and
slope, the color index and the standard deviation (mag).11footnotemark: 1

To obtain the absolute zero points, one should subtract the geometrical
distance modulus of the LMC and SMC, based on eclipsing binaries
(18.477$\pm$0.026 mag and 18.977$\pm$0.028 mag for the LMC and SMC,
respectively (Pietrzyński et al, 2019; Graczyk et al, 2020)).

##  Appendix C Evolutionary constraints for Cepheids

Cepheid pulsation properties are the convolution between evolutionary and
pulsation properties of low- and intermediate-mass stars. Panel a) of Fig. 4
shows the comparison in the HRD between BASTI-IAC canonical scaled solar
evolutionary models at solar chemical composition, covering a broad range in
stellar masses and the CC instability strip (Fiorentino et al, 2002). The
dashed lines display hydrogen-burning phases, while solid purple lines display
helium-burning phases, and the solid cyan lines the double shell burning
phases (AGB). The blue and the red almost vertical lines show the first
overtone blue edge and the fundamental red edge for the same chemical
composition. These boundaries bracket the region of the HRD, in which CCs are
expected to be pulsationally stable. Evolutionary and pulsation predictions
plotted in this panel indicate that CCs at solar iron abundance are expected
to cover a broad range in pulsation period. The minimum mass crossing the
Cepheid instability strip is of the order of $M=3.0$–$3.5\,M_{\odot}$, while
the most massive ones are still crossing the instability strip during helium-
burning phases.

Panels b) and c) of Fig. 4 display the same predictions, but for stellar
structures constructed by assuming chemical compositions that are half solar
Z=0.01 and Z=0.001, respectively. A glance at the predictions plotted in these
panels shows quite clearly that the minimum mass crossing the CC instability
strip steadily decreases when moving into the more metal-poor regimes. This
explains why the period distribution shifts toward shorter periods moving from
the MW to the SMC. Moreover, evolutionary tracks plotted in the bottom panel
show that a significant fraction of more massive stellar structures cross the
instability strip during the AGB (cyan color). This means a decrease in the
evolutionary time spent inside the instability strip, and in turn, it explains
the lack of long period Cepheids in metal-poor stellar systems like IC 1613
(Pietrzyński et al, 2006) and WLM (Pietrzyński et al, 2007).

In order to investigate this issue on a more quantitative basis, Table 6 gives
for four different chemical compositions the difference in luminosity and in
evolutionary lifetime for the three different crossings based on evolutionary
models taking into account convective core overshooting during central
hydrogen burning phases.

Table 6: Mean luminosity and evolutionary lifetime for the three different
crossings of intermediate-mass evolutionary models constructed by taking into
account convective core overshooting during central hydrogen burning phases.

$M/M_{\odot}$ | $\langle\log(L/L_{\odot})\rangle^{a}$ | t(Kyr) | $\langle\log(L/L_{\odot})\rangle^{a}$ | t(Kyr) | $\langle\log(L/L_{\odot})\rangle^{a}$ | t(Kyr)
---|---|---|---|---|---|---
| 1st crossing | 2nd crossing | 3rd crossing
[Fe/H]=+0.06
2.8 | 2.099 | 251.81 | … | … | … | …
3.0 | 2.205 | 170.61 | … | … | … | …
3.5 | 2.436 | 73.56 | … | … | … | …
4.0 | 2.636 | 39.95 | … | … | … | …
4.5 | 2.812 | 23.00 | … | … | … | …
5.0 | 2.970 | 14.50 | 3.163 |  2880.21b | … | …
6.0 | 3.242 | 7.00 | 3.459 | 138.81 | 3.501 | 191.63
7.0 | 3.472 | 4.00 | 3.704 | 53.40 | 3.739 | 59.52
8.0 | 3.649 | 2.50 | 3.922 | 16.96 | 3.933 | 27.52
9.0 | 3.818 | 1.56 | 4.109 | 9.45 | 4.098 | 15.28
10.0 | 3.937 | 1.20 | … | … | … | …
11.0 | 3.998 | 0.79 | 4.402 | 5.22 | 4.339 | 6.46
12.0 | 4.109 | 0.70 | 4.490 | 5.22 | 4.443 | 5.40
13.0 | 4.303 | 1.12 | … | … | … | …
14.0 | 4.622 | 9.09 | … | … | … | …
[Fe/H]=-0.30
2.8 | 2.210 | 159.59 | … | … | … | …
3.0 | 2.324 | 111.45 | … | … | … | …
3.5 | 2.539 | 52.57 | … | … | … | …
4.0 | 2.736 | 29.07 | 2.907 | 45.04 | 2.982 | 2225.45
4.5 | 2.910 | 17.79 | 3.113 | 364.28 | 3.179 | 365.68
5.0 | 3.068 | 11.90 | 3.289 | 307.13 | 3.348 | 192.83
6.0 | 3.340 | 6.09 | 3.570 | 140.94 | 3.612 | 36.92
7.0 | 3.565 | 3.70 | 3.807 | 41.16 | 3.821 | 16.35
8.0 | 3.749 | 2.58 | 4.009 | 17.75 | 3.998 | 10.73
9.0 | 3.896 | 1.90 | 4.157 | 12.57 | 4.103 | 5.28
10.0 | 3.991 | 1.29 | 4.314 | 12.29 | 4.273 | 9.08
11.0 | 4.099 | 1.16 | 4.406 | 9.48 | 4.291 | 2.25
12.0 | 4.239 | 1.31 | 4.498 | 10.30 | 4.380 | 2.24
13.0 | 4.421 | 2.23 | 4.610 | 15.56 | 4.601 | 11.20
[Fe/H]=-0.60
2.8 | 2.296 | 114.17 | … | … | … | …
3.0 | 2.397 | 82.87 | … | … | … | …
3.5 | 2.623 | 41.54 | 2.814 | 737.93 | 2.898 | 933.95
4.0 | 2.819 | 23.61 | 3.035 | 645.04 | 3.101 | 248.89
4.5 | 2.992 | 15.17 | 3.232 | 682.77 | 3.273 | 186.44
5.0 | 3.148 | 10.35 | 3.403 | 551.16 | 3.428 | 176.14
6.0 | 3.417 | 5.70 | 3.659 | 232.49 | 3.694 | 99.88
7.0 | 3.628 | 4.00 | 3.882 | 55.88 | 3.912 | 55.91
8.0 | 3.807 | 3.00 | 4.070 | 25.61 | 4.085 | 33.60
9.0 | 3.944 | 1.95 | 4.221 | 19.35 | 4.216 | 17.50
10.0 | 4.066 | 1.38 | 4.334 | 16.28 | 4.300 | 6.24
11.0 | 4.197 | 1.66 | 4.435 | 19.41 | 4.396 | 5.24
12.0 | 4.378 | 2.64 | 4.556 | 36.83 | 4.591 | 15.55
13.0 | 4.724 | 105.75 | … | … | … | …
[Fe/H]=-0.90
2.6 | 2.262 | 129.10 | … | … | … | …
2.8 | 2.371 | 91.97 | 2.615 | 3449.14 | 2.646 | 4483.62
3.0 | 2.472 | 66.21 | 2.737 |  6159.92b | … | …
3.5 | 2.698 | 35.14 | 2.964 |  5414.59b | … | …
4.0 | 2.893 | 20.54 | 3.132 | 1129.51 | 3.186 | 315.84
4.5 | 3.066 | 13.50 | 3.291 | 595.31 | 3,359 | 134.27
5.0 | 3.221 | 9.50 | 3.435 | 370.46 | 3.509 | 66.84
6.0 | 3.481 | 6.00 | 3.686 | 107.64 | 3.754 | 24.09
7.0 | 3.692 | 3.71 | 3.904 | 31.16 | 3.947 | 13.18
8.0 | 3.868 | 2.71 | 4.087 | 22.71 | 4.106 | 9.37
9.0 | 4.013 | 2.41 | 4.237 | 25.18 | 4.212 | 4.92
10.0 | 4.148 | 2.32 | 4.358 | 34.52 | 4.341 | 4.40
11.0 | 4.317 | 3.14 | 4.474 | 74.24 | 4.534 | 10.85
12.0 | 4.652 | 28.52 | … | … | … | …
13.0 | 4.738 | 18.84 | … | … | … | …

2525footnotetext: a The mean luminosity was estimated as the mean along the
crossing (see text for more details). b The blue loop is entirely located
inside the instability strip.

Figure 7 shows the comparison in the HRD between BASTI-IAC canonical
evolutionary models for intermediate-age stellar structures and the AC
instability strip (Monelli and Fiorentino, 2022). The blue tips (the hottest
point during helium burning phases) when moving from more metal-rich to more
metal-poor stellar structures become systematically hotter and cross the
Anomalous Cepheid instability strip (blue and red almost vertical lines). This
trend is even more evident for stellar structures taking into account
convective core overshooting.

Evolutionary and pulsation predictions for TIICs plotted in Fig. 10 cover two
dex in metal/hydrogen abundance ratio. However, TIICs evolutionary status is
minimally affected by the metal content, since they are either AGB or post AGB
stars during their lifetime. This property is independent of the pulsation
period, i.e. from BL Herculis to RV Tauri variables. Moreover, the current
evidence supports their presence both in metal-poor and in metal-rich stellar
systems. The key point is the presence of hot and extreme HB stars, i.e. the
low mass-tail of HB stars.

###  C.1 Helium burning lifetimes

The top left panels of Fig. 39 display three key parameters for investigating
central helium burning phases of both low- and intermediate-mass stellar
structures at solar metal content. The solid lines plotted in panel a) show
the helium-core mass (solar units) as a function of the stellar mass just
before the onset of central helium burning. The helium-core mass is, as
expected, almost constant ($M_{\rm CHe}/M_{\odot}\sim 0.5$) for stellar masses
ranging from 1 to $1.6\,M_{\odot}$, attains a well-defined minimum ($M_{\rm
CHe}/M_{\odot}\sim 0.32$) for $M\sim 1.8\,M_{\odot}$ and then steadily
increases. This minimum marks the transition between low-mass stellar
structures that ignite helium in electron-degenerate core through a core-
helium flash and intermediate-mass stars that ignite helium quiescently.
Hydrogen burning for stellar structures at solar iron abundance and more
massive than $1.6\,M_{\odot}$ is almost entirely driven by the CNO cycle. The
strong dependence of the CNO cycle on the temperature causes an increase in
the core temperature gradient, and in turn, the transition into the convective
regime. The onset of convective cores causes the steady increase in the helium
core mass soon after the minimum. This transition has been defined by Sweigart
et al (1989) as the RGB transition. Solid and dashed line display predictions
for evolutionary models constructed either taking account of or neglecting
convective core overshooting. Canonical models display, as expected, the same
transition, but at larger stellar masses ($M\sim 2.2\,M_{\odot}$).

Panel b) shows the logarithmic luminosity (solar units) at the onset of
central helium burning as a function of the stellar mass. Predictions plotted
in this panel display a well known result. The luminosity of evolutionary
models taking account for the convective core overshooting is, at fixed
stellar mass, systematically brighter than canonical models. Indeed, the
former models have, at fixed stellar mass, larger helium-core masses.

Figure 39:  Top left: Panel a)– Core helium mass (solar units) just before
central helium burning phases as a function of the stellar mass (solar units)
at solar metal content. The solid lines display predictions based on
evolutionary models taking account of convective core overshooting, while the
dashed lines on evolutionary models that neglect convective core overshooting.
Panel b)– Logarithmic luminosity (solar units) at the onset of central helium-
burning phases as a function of the stellar mass (solar units). Panel c)–
Logarithmic ratio between central hydrogen and central helium-burning
evolutionary times (black line). The blue line shows the central helium-
burning evolutionary time, the scale is showed on the right side of the panel.
Top right: Panels d,e,f) same as the top left, but for a more metal-poor
chemical composition [Fe/H]$\,=-0.90$. Bottom left: Panels g,h,i) same as the
top left, but for a more metal-poor chemical composition [Fe/H]$\,=-1.30$.
Bottom right: Panels j,k,l) same as the top left, but for a more metal-poor
chemical composition [Fe/H]$\,=-1.60$.

Panel c) shows the logarithm of the ratio between central hydrogen and helium
burning time. Predictions indicate that this ratio decreases by 1.5 dex when
moving from the low- to the intermediate-mass regime. This decrease is caused
by the fact that, across the transition, the luminosity of central helium-
burning structures decreases and the central helium-burning lifetime increases
by more than a factor of three. Indeed, the central helium-burning time (solid
blue line) increases from $\approx$100 Myr to almost $\approx$300 Myr. The
main consequence of this increase is that central hydrogen burning lifetime
across the transition is only a factor of four longer than central helium-
burning lifetime. The hydrogen burning lifetime is two order of magnitude
longer in the low-mass regime, and roughly one order of magnitude longer in
the intermediate-mass regime. The stellar structures at the transition between
low- and intermediate-mass stellar structures are strong contributors of
central helium-burning stellar populations (red clump, Girardi, 2016).
Predictions based on canonical evolutionary models display the same trend. The
main difference being that central hydrogen burning lifetime is, at the
transition, only a factor of two longer than central helium lifetime. Thus
further increases the fraction of red clump stars compared with main-sequence
stars.

Evolutionary predictions plotted in panels d), e), and f) and in the bottom
panels of Fig. 39 display the same predictions, but for more metal-poor
chemical compositions (see labelled values). The decrease in the metal content
causes, as expected, a mild decrease in the stellar mass at which the
transition between low- and intermediate-mass stars takes place. Indeed, it
decreases from $M\sim 2.3\,M_{\odot}$ at solar iron abundance to $M\sim
2.2\,M_{\odot}$ at [Fe/H]$=-1.90$. The decrease is caused by the fact that
more metal-poor stellar structure are, at fixed stellar mass, brighter and
hotter than more metal-rich stellar structures. Therefore, the former stellar
structures ignite quiescently helium at lower stellar masses (Castellani and
degl’Innocenti, 1995).

Evolutionary models constructed by taking into account convective core
overshooting during central hydrogen burning show a similar trend. The
difference is also discussed in section C.2; here we simply mention that the
transition decreases from $M\sim 2.0\,M_{\odot}$ to $M\sim 1.90\,M_{\odot}$
when moving from solar iron abundance to [Fe/H]$=-1.90$.

###  C.2 On the mass luminosity relation of ACs

Evolutionary predictions plotted in Fig. 40 display the mass–luminosity
relations in a log-log plane for stellar structures during central helium-
burning phases. Intermediate-mass stars perform the so-called blue loop,
during which they cross the Cepheid instability strip. The luminosity for
these stellar structures was estimated at the center of the instability strip
($\log T_{\rm eff}=3.85$), as the mean between the blue-ward and the red-ward
excursion. Stellar structures in which the helium ignition in the core takes
place in partially electron degenerate conditions (low-mass regime) do not
become hot enough to cross the Cepheid instability strip. The luminosity for
these stellar structures was defined as the mean luminosity between the
luminosity at the helium ignition and central helium exhaustion. The black
crosses show the transition ($M/M_{\odot}$=2.3–2.2) between stellar structures
experiencing a mild core-helium flash and stellar structures that ignite
helium quiescently. In order to provide a more quantitative estimate of the
transition between low and intermediate-mass stars, we also performed linear
fits of the two groups. We found that the ML relation for the intermediate-
mass stars is:

$\log(M/M_{\odot})=1.21\pm 0.03+3.06(\pm 0.05)\times\log(L/L_{\odot})$

where the symbols have their usual meaning, and the standard deviation is
$\sigma$= 0.03 dex, whereas for the low-mass stars it is

$\log(M/M_{\odot})=1.95\pm 0.02+0.98(\pm 0.08)\times\log(L/L_{\odot})$

with $\sigma$= 0.02 dex. The current findings are suggesting that the
coefficient of the luminosity term is three times larger when moving from the
low- to the intermediate-mass regime.

Figure 40: Left: Logarithmic stellar mass versus logarithmic luminosity.
Different symbols display canonical evolutionary models constructed by
assuming different scaled-solar chemical compositions (see labelled values).
The black crosses mark the transition between stellar structures, in which
core helium burning is partially affected by electron degeneracy (less
massive), and those with a quiescent ignition of core-helium burning (more
massive). The dashed and the solid black lines show the linear fit for the two
different groups. Right: Same as the left, but for evolutionary models
constructed by assuming a mild convective core overshooting during central
hydrogen burning phases.

In order to provide a comprehensive analysis of the transition between low-
and intermediate-mass stars we performed the same calculations, but for
evolutionary models constructed by assuming a mild convective core
overshooting during central hydrogen-burning phases. Predictions plotted in
the right panel of Fig. 40 clearly show that models taking account of core
overshooting are at a fixed stellar mass, as expected, systematically brighter
than canonical models. Moreover, the transition between low- and intermediate
mass stars takes place at lower stellar masses when compared with canonical
models, typically between $M/M_{\odot}$=2.0 and 1.9. The difference is due to
the fact that evolutionary models taking into account core overshooting have,
at a fixed stellar mass, larger core helium masses, and in turn hotter central
temperatures. Therefore, they are less affected by electron degeneracy. We
also performed the linear fits to the two different regimes and we found:

$\log(M/M_{\odot})=1.31\pm 0.01+3.16(\pm 0.01)\times\log(L/L_{\odot})$

with $\sigma$= 0.02 dex (intermediate-mass)

$\log(M/M_{\odot})=2.08\pm 0.07+0.25(\pm 0.33)\times\log(L/L_{\odot})$

with $\sigma$= 0.06 dex (low-mass). The difference between low- and
intermediate-mass regime is smaller than for canonical models, but the
variation is still at 50% level. This suggests, once again, a relevant change
between faint and bright ACs.

This direct evidence, together with the circumstantial evidence brought
forward in Sect. 5 discloses a more complex evolutionary framework for ACs.
Indeed, the current findings are suggesting, as originally found by Caputo et
al (2004) and Fiorentino et al (2014), that ACs are a mix of stellar
structures that are either partially affected by electron degeneracy or
stellar structures that quiescently ignite helium. The latter group is from
the evolutionary point of view identical to classical Cepheids. This means
that more massive ACs are indeed, nothing else but the more metal-poor tail of
classical Cepheids (Caputo et al, 2004). Therefore, the truly ACs appear to be
the less massive ACs. This plain evidence also explains why the PL relations
for ACs explicitly include a term for the stellar mass. The ACs identified in
dwarf galaxies are a mix of objects which are either minimally or partially
affected by electron degeneracy. This means that they are fundamental
laboratories to constrain the impact of electron degeneracy on the pulsation
properties of ACs.

###  C.3 First dredge up

The transition between low- and intermediate-mass stars brings forward several
interesting properties concerning the mixing and the variation of surface
abundances. Panel a) of Fig. 41 shows a set of evolutionary model at solar
iron abundance and stellar masses ranging from 2.3 to $13\,M_{\odot}$. The red
circles plotted along the individual tracks mark the central helium ignition.

Figure 41:  Panel a): Hertzsprung–Russell Diagram for young stellar structures
neglecting convective overshooting during core hydrogen-burning phase with a
scaled solar chemical mixture and solar iron abundance. The stellar masses
range from 2.8 to $13\,M_{\odot}$. The blue squares mark the position of the I
dredge up, whereas the red circles the ignition of the 3$\alpha$ reactions
(core He burning). Panel b): Same as the left, but for a more metal-poor
([Fe/H]$=-0.90$) chemical composition. Bottom: Same as the top, but for two
more metal-poor chemical compositions: [Fe/H]$=-1.30$ (panel c) and
[Fe/H]$=-1.90$ (panel d).

This key evolutionary phase moves from the tip of the red giant branch (TRGB)
in the low- and in the intermediate-mass regime to earlier and earlier phases
in the massive stellar regime. The helium ignition for $M\sim 10\,M_{\odot}$
takes place in the so-called Hertzsprung gap, well before the stellar
structure approached the TRGB, and soon after central hydrogen exhaustion for
$M\sim 13\,M_{\odot}$. The latter stellar structure does not perform the blue
loop, therefore, its contribution to classical Cepheids becomes either minimal
or vanishing.

The blue squares plotted along the same tracks mark the phases in which the I
dredge up (IDUP) takes place. This is a well-known mixing phenomenon, stellar
structures moving towards their Hayashi track become systematically cooler.
The increase in the envelope opacity causes the convective envelope to become
thicker and thicker. During subsequent evolutionary phases the hydrogen shell
moves outwards (the helium-core mass increases) and the bottom of the
convective envelope recedes. In this specific phase, the IDUP occurs, during
which nuclear processed material (central hydrogen burning phases) is dredged-
up into the surface regions. Predictions plotted in this panel show that
stellar structures in which central-helium ignition takes place before the
stellar structures approach their Hayashi track either do not experience or
experience a very mild IDUP.

Evolutionary tracks plotted in panel b) and in the panels c), d) display
similar predictions, but for more metal-poor stellar structures (see labelled
values). The main consequence of the decrease in metal content is that central
helium ignition in more metal-poor stellar structures takes place before the
approach to the Hayashi tracks at lower stellar masses, indeed it ranges from
$M=6.5\,M_{\odot}$ for [Fe/H]$=-0.90$ to $M=2.3\,M_{\odot}$ at [Fe/]$=-1.90$.
This means that only a minor fraction, if any, of intermediate mass stars in
the metal-poor regime experience the IDUP. This change has implications not
only for the surface chemical abundances, but also on the evolutionary
lifetime spent inside the instability strip.

The key evolutionary parameters at central He-burning ignition and at the IDUP
for canonical and convective core overshooting evolutionary models are listed
in Table 7 and in Table 8 available in this Appendix.

To further constrain the abundance patterns of different groups of variable
stars, we decided to analyze the impact of the I dredge up on their surface
abundance. Figure 42 shows the fractional variation in the abundances of the
elements affected by the I dredge up. The top panels display the relative
difference between the surface chemical composition at the I dredge up and the
surface chemical composition on the main sequence for He (panel a), C (panel
b), N (panel c) and O (panel d) normalized to the abundance on the main
sequence. These panels only includes stellar structures which undergo the I
dredge up and lines of different colors show predictions for different
chemical compositions. The bottom panels of Fig. 42 display the predicted
fractional variation in the abundance of He (panel e), C (panel f), N (panel
g) and O (panel h), but for more metal-poor chemical compositions.

Panel a) of Fig. 42 shows that the impact on the He abundance in the metal-
rich regime is, as expected, modest and typically smaller than 10% with a
local minimum for stellar masses of 1.5–2.0$\,M_{\odot}$. The variation in
helium abundance is vanishing for stellar masses larger than $\sim
2.5\,M_{\odot}$ over the entire metallicity range. The impact is also minimal
in the metal-poor regime. Indeed, canonical (dashed lines) and non-canonical
(solid lines) stellar structures display variations of the order of a few
percents.

Figure 42:  Top: Relative difference between the surface chemical composition
at the I dredge up and the surface chemical composition on the main sequence
for He (panel a), C (panel b), N (panel c) and O (panel d) normalized to the
abundance on the main sequence. The iron abundance of the different
evolutionary models is color-coded and changes from solar iron abundance
([Fe/H]=0.06, red) to half-solar ([Fe/H]$=-0.30$, blue) and to Z=0.001
([Fe/H]$=-0.90$, green). Dashed lines display predictions based on
evolutionary models neglecting convective core overshooting, while the solid
ones are based on models taking into account convective core overshooting
during central hydrogen-burning phases. Note that only evolutionary models
that underwent the I dredge up have been plotted. Bottom: Same as the top
panels, but for the three more metal-poor chemical compositions:
[Fe/H]$=-1.30$, red; [Fe/H]$=-1.55$, blue; [Fe/H]$=-1.90$, green.

The variation concerning the C abundance is more complex. Indeed, the surface
C abundance steadily decreases in more metal-rich stellar structures (left
panels) that undergo a core helium flash. The variation in C (panel b) ranges
from a few percent for old ($M/M_{\odot}=0.8$) stellar structures to 30%–40%
for stellar structures at the transition between low- and intermediate-mass
($M/M_{\odot}$=2.0–2.2) regime. The metallicity has a minimal impact, since
they remain under-abundant over the entire range in stellar mass. More metal-
poor stellar structures (panel f) display a steady increase soon after the
maximum, thus suggesting that these structures experience a very mild IDUP.
The difference is mainly caused by the fact that metal-poor stellar structures
are systematically brighter and hotter than metal-rich ones. This means that
along the RGB, convective transport is less efficient and the bottom of the
convective envelope is not deep enough to approach the regions in which the
CNO cycle has been efficient. The difference between canonical (dashed lines)
and non-canonical (solid) is minimal with the former ones experiencing the
IDUP in a very limited range in mass.

The variation of the surface N abundance is, as expected for the CN cycle, the
mirror image of the C abundance. The N abundance (panel c) steadily increases
by more than a factor of two when moving from old ($M/M_{\odot}=0.8$) to
intermediate-age ($M/M_{\odot}$=2.0–2.2) stellar structures. The current
predictions indicate that more metal-rich (solar, half solar) stellar
structures are overabundant in N over the entire mass range. On the other
hand, metal-intermediate and metal-poor stellar structures (panel g) are less
affected by the IDUP for stellar masses larger than $4\,M_{\odot}$. Canonical
evolutionary models indicate that only stellar structures less massive than
$4\,M_{\odot}$ experience the IDUP.

The variation of the surface O abundance for metal-rich (panel d) stellar
structures is similar to C, but there are some key differences. In the low-
mass regime ($M/M_{\odot}\leq 1.5$) O abundance is not affected by the IDUP.
This trend is expected because in these structures the NO cycle has a very low
efficiency. The variation becomes more relevant and approaches 5%–6% for
stellar structures with masses ranging from 2 to 4 $M_{\odot}$. The trend in
the metal-poor regime (panel h) is the mirror image of He abundance, indeed,
for stellar masses of 1.8–2.5 $M_{\odot}$ decreases by roughly 3%–4%, while it
is vanishing for more massive stellar structures.

The difference between metal-rich and metal-poor stellar structures is
twofold: the metal-poor ones are brighter and hotter; this means that with a
fixed stellar mass the central temperature is higher and the NO cycle becomes
more efficient. The convective envelopes are less extended, since convection
is less efficient along the RGB, but they deepen enough to dredge up material
processed by both CN and NO cycles.

The current predictions concerning the efficiency of the IDUP as a function of
stellar mass and chemical composition bring forward two key features worth
being discussed in more detail.

a) Stellar structures at the transition between low- and intermediate-mass
stars after the IDUP are predicted to show an enhancement in N abundance that
is two times the abundance on the main sequence. The decrease in C abundance
for the same stellar structures is 30%–40% when compared with the MS
abundance. Note that this metallicity range does not produce ACs, since their
helium-burning locus is systematically cooler (redder) than the instability
strip.

b) The current predictions indicate that Galactic CCs should show larger
abundance variations due to the IDUP when compared with LMC CCs, and in
particular, with SMC CCs. In this context, it is worth mentioning that
canonical evolutionary models for intermediate-mass stars in the metal-poor
regime do not experience any IDUP. Therefore, CCs in very metal-poor stellar
systems (IC 1613, SagDig, NGC 6822) should show N and C surface abundances
similar to their main sequence progenitors.

Table 7:  Evolutionary parameters at central He-burning ignition and at I dredge-up (IDUP) from BaSTI2 scaled solar (Caffau et al, 2011) canonical evolutionary models.  | He burning | IDUP
---|---|---
$M/M_{\odot}$ | $\log\,t$ | $\log\,L/L_{\odot}$ | $\log\,T_{\rm eff}$ | $\log\,t$ | $\log\,L/L_{\odot}$ | $\log\,T_{\rm eff}$
[Fe/H]=+0.06, Y=0.2695
2.5 | 8.726 | 2.353 | 3.653 | 2.268 | 3.659 | 8.724
2.6 | 8.676 | 2.342 | 3.656 | 2.347 | 3.656 | 8.676
2.8 | 8.587 | 2.365 | 3.659 | 2.400 | 3.657 | 8.587
3.0 | 8.505 | 2.409 | 3.661 | 2.436 | 3.659 | 8.506
3.5 | 8.328 | 2.562 | 3.659 | 2.534 | 3.661 | 8.330
4.0 | 8.177 | 2.716 | 3.655 | 2.621 | 3.663 | 8.177
4.5 | 8.048 | 2.863 | 3.651 | 2.691 | 3.666 | 8.047
5.0 | 7.931 | 3.000 | 3.646 | 3.064 | 3.641 | 7.931
5.5 | 7.833 | 3.129 | 3.642 | 3.194 | 3.636 | 7.833
6.0 | 7.745 | 3.249 | 3.637 | 3.320 | 3.631 | 7.745
6.5 | 7.667 | 3.364 | 3.633 | 3.437 | 3.626 | 7.668
7.0 | 7.598 | 3.473 | 3.628 | 3.549 | 3.621 | 7.598
7.5 | 7.536 | 3.568 | 3.625 | 3.655 | 3.617 | 7.536
8.0 | 7.479 | 3.601 | 3.627 | 3.755 | 3.612 | 7.480
8.5 | 7.428 | 3.493 | 3.642 | 3.848 | 3.608 | 7.429
9.0 | 7.381 | 3.449 | 3.665 | 3.937 | 3.604 | 7.382
9.5 | 7.338 | 3.702 | 3.767 | 4.011 | 3.601 | 7.339
10.0 | 7.299 | 3.900 | 3.913 | 4.057 | 3.600 | 7.300
11.0 | 7.229 | 4.139 | 4.091 | 4.070 | 3.605 | 7.230
12.0 | 7.189 | 4.324 | 4.210 | 4.328 | 3.596 | 7.257
13.0 | 7.147 | 4.447 | 4.238 | 4.537 | 3.577 | 7.216
14.0 | 7.140 | 4.558 | 4.279 | 4.715 | 3.564 | 7.197
[Fe/H]=–0.30, Y=0.2571
2.5 | 8.647 | 2.330 | 3.676 | 2.358 | 3.674 | 8.648
2.6 | 8.602 | 2.332 | 3.678 | 2.370 | 3.676 | 8.602
2.8 | 8.517 | 2.367 | 3.680 | 2.412 | 3.677 | 8.518
3.0 | 8.440 | 2.419 | 3.681 | 2.468 | 3.678 | 8.440
3.5 | 8.271 | 2.579 | 3.679 | 2.632 | 3.675 | 8.271
4.0 | 8.130 | 2.731 | 3.676 | 2.735 | 3.676 | 8.130
4.5 | 8.008 | 2.880 | 3.671 | 2.826 | 3.676 | 8.008
5.0 | 7.903 | 3.017 | 3.668 | 3.078 | 3.663 | 7.902
5.5 | 7.806 | 3.133 | 3.665 | 3.211 | 3.658 | 7.806
6.0 | 7.724 | 3.193 | 3.666 | 3.338 | 3.653 | 7.725
6.5 | 7.651 | 3.139 | 3.681 | 3.456 | 3.648 | 7.652
7.0 | 7.586 | 3.236 | 3.698 | 3.570 | 3.644 | 7.587
7.5 | 7.528 | 3.431 | 3.729 | 3.673 | 3.639 | 7.528
8.0 | 7.475 | 3.592 | 3.801 | 3.773 | 3.633 | 7.476
8.5 | 7.427 | 3.728 | 3.879 | 3.862 | 3.631 | 7.427
9.0 | 7.383 | 3.850 | 3.950 | 3.940 | 3.628 | 7.383
9.5 | 7.343 | 3.981 | 4.040 | 3.974 | 3.629 | 7.343
10.0 | 7.306 | 4.068 | 4.104 | 3.976 | 3.633 | 7.307
11.0 | 7.239 | 4.243 | 4.208 | 4.364 | 3.608 | 7.307
12.0 | 7.200 | 4.362 | 4.268 | 4.445 | 3.607 | 7.266
continued on next page
Table 7: continued. | He burning | IDUP
---|---|---
$M/M_{\odot}$ | $\log\,t$ | $\log\,L/L_{\odot}$ | $\log\,T_{\rm eff}$ | $\log\,t$ | $\log\,L/L_{\odot}$ | $\log\,T_{\rm eff}$
[Fe/H]=–0.60, Y=0.2521
2.5 | 8.598 | 2.296 | 3.692 | 2.341 | 3.690 | 8.599
2.6 | 8.554 | 2.309 | 3.694 | 2.330 | 3.693 | 8.554
2.8 | 8.472 | 2.355 | 3.695 | 2.404 | 3.692 | 8.473
3.0 | 8.398 | 2.418 | 3.695 | 2.466 | 3.692 | 8.398
3.5 | 8.236 | 2.581 | 3.693 | 2.633 | 3.689 | 8.236
4.0 | 8.099 | 2.737 | 3.690 | 2.795 | 3.686 | 8.100
4.5 | 7.982 | 2.878 | 3.687 | 2.944 | 3.682 | 7.983
5.0 | 7.880 | 2.917 | 3.692 | 3.057 | 3.681 | 7.885
5.5 | 7.788 | 2.960 | 3.714 | 3.225 | 3.673 | 7.789
6.0 | 7.709 | 3.196 | 3.756 | 3.351 | 3.668 | 7.710
6.5 | 7.640 | 3.381 | 3.837 | 3.468 | 3.664 | 7.640
7.0 | 7.577 | 3.530 | 3.894 | 3.582 | 3.660 | 7.578
7.5 | 7.521 | 3.656 | 3.939 | 3.688 | 3.656 | 7.522
8.0 | 7.470 | 3.764 | 3.986 | 3.782 | 3.652 | 7.470
8.5 | 7.423 | 3.866 | 4.031 | 3.866 | 3.648 | 7.424
9.0 | 7.381 | 3.964 | 4.088 | 3.913 | 3.648 | 7.382
9.5 | 7.342 | 4.056 | 4.149 | 4.183 | 3.634 | 7.409
10.0 | 7.306 | 4.130 | 4.186 | 4.256 | 3.631 | 7.373
[Fe/H]=–0.90, Y=0.2496
2.5 | 8.564 | 2.255 | 3.706 | 2.302 | 3.703 | 8.565
2.6 | 8.521 | 2.277 | 3.706 | 2.326 | 3.704 | 8.522
2.8 | 8.442 | 2.336 | 3.707 | 2.386 | 3.704 | 8.442
3.0 | 8.369 | 2.404 | 3.706 | 2.454 | 3.703 | 8.370
3.5 | 8.212 | 2.574 | 3.705 | 2.627 | 3.701 | 8.212
4.0 | 8.079 | 2.686 | 3.706 | 2.795 | 3.698 | 8.079
4.5 | 7.964 | 2.747 | 3.726 | 2.944 | 3.694 | 7.965
5.0 | 7.866 | 3.020 | 3.783 | 3.096 | 3.690 | 7.867
5.5 | 7.776 | 3.227 | 3.880 | 3.231 | 3.686 | 7.777
6.0 | 7.700 | 3.387 | 3.939 | 3.358 | 3.682 | 7.701
6.5 | 7.632 | 3.523 | 3.983 | 3.477 | 3.678 | 7.633
7.0 | 7.571 | 3.643 | 4.021 | 3.587 | 3.675 | 7.572
7.5 | 7.516 | 3.750 | 4.051 | 3.691 | 3.671 | 7.518
8.0 | 7.467 | 3.843 | 4.081 | 3.784 | 3.667 | 7.468
8.5 | 7.421 | 3.929 | 4.120 | 3.848 | 3.666 | 7.423
9.0 | 7.379 | 4.016 | 4.176 | 4.289 | 3.635 | 7.447
9.5 | 7.341 | 4.095 | 4.206 | 4.375 | 3.631 | 7.407
Table 8: Evolutionary parameters at central He-burning ignition and at I dredge-up (IDUP) from BaSTI2 scaled-solar (Caffau et al, 2011) evolutionary models taking into account of convective core overshooting during central hydrogen burning phases.  | He burning | IDUP
---|---|---
$M/M_{\odot}$ | $\log\,t$ | $\log\,L/L_{\odot}$ | $\log\,T_{\rm eff}$ | $\log\,t$ | $\log\,L/L_{\odot}$ | $\log\,T_{\rm eff}$
[Fe/H]=+0.06, Y=0.2695
2.5 | 8.796 | 2.501 | 3.645 | 2.538 | 3.642 | 8.797
2.6 | 8.749 | 2.523 | 3.645 | 2.567 | 3.642 | 8.749
2.8 | 8.660 | 2.590 | 3.645 | 2.635 | 3.641 | 8.660
3.0 | 8.578 | 2.661 | 3.644 | 2.712 | 3.640 | 8.579
3.5 | 8.400 | 2.840 | 3.639 | 2.894 | 3.635 | 8.400
4.0 | 8.249 | 3.013 | 3.634 | 3.071 | 3.629 | 8.250
4.5 | 8.120 | 3.173 | 3.628 | 3.231 | 3.622 | 8.121
5.0 | 8.004 | 3.312 | 3.622 | 3.380 | 3.616 | 8.004
5.5 | 7.905 | 3.449 | 3.617 | 3.523 | 3.610 | 7.906
6.0 | 7.818 | 3.576 | 3.612 | 3.649 | 3.604 | 7.818
6.5 | 7.740 | 3.692 | 3.607 | 3.765 | 3.599 | 7.741
7.0 | 7.670 | 3.798 | 3.602 | 3.872 | 3.595 | 7.671
7.5 | 7.607 | 3.896 | 3.598 | 3.972 | 3.590 | 7.608
8.0 | 7.550 | 3.987 | 3.594 | 4.064 | 3.586 | 7.551
8.5 | 7.498 | 4.069 | 3.591 | 4.150 | 3.583 | 7.499
9.0 | 7.451 | 4.153 | 3.587 | 4.229 | 3.579 | 7.451
9.5 | 7.407 | 4.236 | 3.583 | 4.303 | 3.576 | 7.408
10.0 | 7.367 | 4.312 | 3.580 | 4.372 | 3.574 | 7.367
11.0 | 7.295 | 3.893 | 3.640 | 4.496 | 3.569 | 7.295
12.0 | 7.234 | 4.329 | 3.942 | 4.583 | 3.568 | 7.234
13.0 | 7.181 | 4.513 | 4.081 | 4.556 | 3.578 | 7.181
14.0 | 7.135 | 4.653 | 4.173 | 4.787 | 3.561 | 7.177
[Fe/H]=–0.30, Y=0.2571
2.5 | 8.718 | 2.520 | 3.665 | 2.460 | 3.670 | 8.718
2.6 | 8.674 | 2.553 | 3.665 | 2.481 | 3.671 | 8.673
2.8 | 8.589 | 2.621 | 3.665 | 2.540 | 3.671 | 8.589
3.0 | 8.512 | 2.693 | 3.663 | 2.584 | 3.672 | 8.512
3.5 | 8.344 | 2.877 | 3.659 | 2.685 | 3.674 | 8.344
4.0 | 8.203 | 3.046 | 3.654 | 2.770 | 3.678 | 8.202
4.5 | 8.081 | 3.206 | 3.648 | 2.878 | 3.680 | 8.081
5.0 | 7.976 | 3.351 | 3.643 | 3.302 | 3.647 | 7.976
5.5 | 7.879 | 3.477 | 3.639 | 3.548 | 3.632 | 7.880
6.0 | 7.798 | 3.601 | 3.634 | 3.675 | 3.627 | 7.798
6.5 | 7.725 | 3.716 | 3.629 | 3.788 | 3.622 | 7.725
7.0 | 7.659 | 3.817 | 3.625 | 3.893 | 3.618 | 7.659
7.5 | 7.600 | 3.912 | 3.621 | 3.992 | 3.614 | 7.600
8.0 | 7.546 | 3.978 | 3.620 | 4.082 | 3.610 | 7.546
8.5 | 7.497 | 4.082 | 3.615 | 4.168 | 3.607 | 7.498
9.0 | 7.452 | 4.128 | 3.615 | 4.246 | 3.604 | 7.452
9.5 | 7.410 | 3.949 | 3.638 | 4.317 | 3.601 | 7.411
10.0 | 7.373 | 4.060 | 3.757 | 4.385 | 3.599 | 7.373
11.0 | 7.304 | 4.281 | 3.954 | 4.496 | 3.595 | 7.304
12.0 | 7.244 | 4.436 | 4.088 | 4.553 | 3.596 | 7.245
13.0 | 7.192 | 4.552 | 4.175 | 4.579 | 3.601 | 7.192
14.0 | 7.145 | 4.680 | 4.255 | 4.813 | 3.586 | 7.185
continued on next page
Table 8: continued. | He burning | IDUP
---|---|---
$M/M_{\odot}$ | $\log\,t$ | $\log\,L/L_{\odot}$ | $\log\,T_{\rm eff}$ | $\log\,t$ | $\log\,L/L_{\odot}$ | $\log\,T_{\rm eff}$
[Fe/H]=–0.60, Y=0.2521
2.5 | 8.670 | 2.527 | 3.680 | 2.573 | 3.677 | 8.670
2.6 | 8.626 | 2.562 | 3.679 | 2.610 | 3.676 | 8.627
2.8 | 8.545 | 2.636 | 3.679 | 2.685 | 3.675 | 8.546
3.0 | 8.471 | 2.713 | 3.677 | 2.718 | 3.677 | 8.471
3.5 | 8.309 | 2.895 | 3.673 | 2.824 | 3.678 | 8.308
4.0 | 8.173 | 3.066 | 3.668 | 2.931 | 3.680 | 8.172
4.5 | 8.056 | 3.223 | 3.663 | 3.244 | 3.662 | 8.056
5.0 | 7.954 | 3.369 | 3.658 | 3.426 | 3.654 | 7.955
5.5 | 7.862 | 3.493 | 3.655 | 3.565 | 3.649 | 7.862
6.0 | 7.783 | 3.614 | 3.650 | 3.687 | 3.644 | 7.784
6.5 | 7.713 | 3.720 | 3.647 | 3.801 | 3.640 | 7.714
7.0 | 7.650 | 3.759 | 3.649 | 3.905 | 3.635 | 7.650
7.5 | 7.593 | 3.670 | 3.666 | 4.002 | 3.632 | 7.593
8.0 | 7.541 | 3.733 | 3.681 | 4.091 | 3.628 | 7.541
8.5 | 7.493 | 3.896 | 3.722 | 4.174 | 3.625 | 7.493
9.0 | 7.450 | 4.019 | 3.797 | 4.252 | 3.622 | 7.450
9.5 | 7.409 | 4.122 | 3.878 | 4.323 | 3.620 | 7.410
10.0 | 7.372 | 4.209 | 3.954 | 4.386 | 3.618 | 7.373
[Fe/H]=–0.90, Y=0.2496
2.5 | 8.636 | 2.527 | 3.692 | 2.571 | 3.689 | 8.637
2.6 | 8.594 | 2.563 | 3.692 | 2.611 | 3.689 | 8.594
2.8 | 8.515 | 2.638 | 3.691 | 2.688 | 3.688 | 8.515
3.0 | 8.443 | 2.719 | 3.689 | 2.766 | 3.686 | 8.443
3.5 | 8.285 | 2.902 | 3.685 | 2.955 | 3.682 | 8.285
4.0 | 8.152 | 3.075 | 3.681 | 3.126 | 3.677 | 8.153
4.5 | 8.038 | 3.230 | 3.677 | 3.262 | 3.675 | 8.042
5.0 | 7.940 | 3.373 | 3.673 | 3.404 | 3.670 | 7.940
5.5 | 7.850 | 3.482 | 3.671 | 3.569 | 3.663 | 7.851
6.0 | 7.774 | 3.482 | 3.678 | 3.690 | 3.659 | 7.774
6.5 | 7.705 | 3.515 | 3.693 | 3.803 | 3.655 | 7.706
7.0 | 7.644 | 3.683 | 3.717 | 3.907 | 3.651 | 7.644
7.5 | 7.588 | 3.831 | 3.798 | 4.003 | 3.648 | 7.588
8.0 | 7.537 | 3.943 | 3.865 | 4.090 | 3.645 | 7.537
8.5 | 7.490 | 4.038 | 3.918 | 4.175 | 3.642 | 7.491
9.0 | 7.447 | 4.124 | 3.965 | 4.252 | 3.640 | 7.448
9.5 | 7.408 | 4.199 | 4.013 | 4.322 | 3.637 | 7.408

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