[1,2]G. Bono

[1]Department of Physics, Università di Roma Tor Vergata, via della Ricerca
Scientifica 1, Roma, 00133, Italy [2]INAF-Osservatorio Astronomico di Roma,
via Frascati 33, Monte Porzio Catone, 00078, Italy [3]INAF-Osservatorio
Astronomico d’Abruzzo, via M. Maggini s/n, Teramo, 64100, Italy

# Cepheids as distance indicators and stellar tracers

<EMAIL_ADDRESS>V. F. Braga<EMAIL_ADDRESS>A. Pietrinferni
<EMAIL_ADDRESS>*  *  *

###### Abstract

We review the phenomenology of classical Cepheids (CCs), Anomalous Cepheids
(ACs) and type II Cepheids (TIICs) in the Milky Way (MW) and in the Magellanic
Clouds (MCs). We also examine the Hertzsprung progression in different stellar
systems by using the shape of I-band light curves (Fourier parameters) and
observables based on the difference in magnitude and in phase between the bump
and the minimum in luminosity. The distribution of Cepheids in optical and in
optical-near infrared (NIR) color–magnitude diagrams is investigated to
constrain the topology of the instability strip. The use of Cepheids as
tracers of young (CCs), intermediate (ACs) and old (TIICs) stellar populations
are brought forward by the comparison between observations (MCs) and cluster
isochrones covering a broad range in stellar ages and in chemical
compositions. The different diagnostics adopted to estimate individual
distances (period–luminosity, period–Wesenheit, period–luminosity–color
relations) are reviewed together with pros and cons in the use of fundamental
and overtones, optical and NIR photometric bands, and reddening free pseudo
magnitudes (Wesenheit). We also discuss the use of CCs as stellar tracers and
the radial gradients among the different groups of elements (iron, $\alpha$,
neutron-capture) together with their age-dependence. Finally, we briefly
outline the role that near-future space and ground-based facilities will play
in the astrophysical and cosmological use of Cepheids.

###### keywords:

stars: variables: Cepheids — stars: stellar tracers — stars: stellar evolution
— stars: oscillations — stellar systems: Magellanic Clouds

Table 1: List of acronyms AC | Anomalous Cepheid
---|---
AGB | Asymptotic Giant Branch
AHB | Above Horizontal Branch
BC | Bolometric Correction
BHB | Blue Horizontal Branch
BLH | BL Herculis
CC | Classical Cepheid
CMD | Color-Magnitude Diagram
DORB | Classical Cepheids with a bump moving down along the rising branch
EHB | Extreme Horizontal Branch
ELT | Extremely Large Telescope
FO | First Overtone
FU | Fundamental
HB | Horizontal Branch
HRD | Hertzsprung–Russel Diagram
HP | Hertzsprung Progression
IDUP | First Dredge-up
LMC | Large Magellanic Cloud
LTE | Local Thermodynamic Equilibrium
MCs | Magellanic Clouds
MIR | Mid Infrared
ML | Mass-luminosity
MS | Main Sequence
MW | Milky Way
NIR | Near Infrared
OC | Open Cluster
PEAGB | Post-Early Asymptotic Giant Branch
P(HP) | Period at the center of the Hertzsprung Progression
PL | Period–Luminosity relation
PLC | Period–Luminosity-Color relation
PW | Period–Wesenheit relation
pWV | peculiar W Virginis
RGB | Red Giant Branch
RHB | Red Horizontal Branch
RRL | RR Lyrae
RVT | RV Tauri
SMC | Small Magellanic Cloud
SO | Second Overtone
TIIC | Type II Cepheid
TO | Third Overtone
TP | Thermal Pulse
TPAGB | Thermal-pulsing Asymptotic Giant Branch
TRGB | Tip of the Red Giant Branch
UPDB | Classical Cepheid with a bump moving up along the decreasing branch
WD | White Dwarf
WV | W Virginis
ZAHB | Zero-Age Horizontal Branch

###### Contents

  1. 1 Introduction
  2. 2 Phenomenology of Cepheids
    1. 2.1 Classical Cepheids
    2. 2.2 Anomalous Cepheids
    3. 2.3 Type II Cepheids
  3. 3 The Hertzsprung progression
  4. 4 Magellanic Cepheids and the color–magnitude diagram
    1. 4.1 Anomalous Cepheids
    2. 4.2 Type II Cepheids
  5. 5 Diagnostics to estimate individual Cepheid distances
    1. 5.1 Fundamental versus overtone PL relations
    2. 5.2 PL versus PW relations
      1. 5.2.1 Highlights concerning observed optical-NIR PL and PW relations
    3. 5.3 Global trend in the slopes of PL and PW relations
    4. 5.4 PL versus PLC relations
    5. 5.5 Metallicity dependence
  6. 6 Cepheids as stellar tracers
    1. 6.1 High-resolution spectroscopy
    2. 6.2 Atmospheric parameters
      1. 6.2.1 Effective temperature, surface gravity and micro-turbulent velocity
      2. 6.2.2 Effective temperature curve templates
    3. 6.3 Metallicity distribution in the Galactic thin disk
    4. 6.4 The iron distribution
    5. 6.5 Light metals and $\alpha$-element distribution
    6. 6.6 Neutron capture distribution
    7. 6.7 CNO distribution
    8. 6.8 A new spin on iron and $\alpha$-element abundance gradients
  7. 7 Omne ignoto pro magnifico
    1. 7.1 Open problems with possible near future solutions
    2. 7.2 Open problems with challenging near future solutions
  8. 8 Summary and future developments
  9. A Metallicity dependence of the Hertzsprung progression
  10. B Optical/near-infrared and near-infrared diagnostics
    1. B.1 Color magnitude diagrams
    2. B.2 Period-Luminosity and Period-Wesenheit relations
  11. C Evolutionary constraints for Cepheids
    1. C.1 Helium burning lifetimes
    2. C.2 On the mass luminosity relation of ACs
    3. C.3 First dredge up

##  1 Introduction

Cepheids111The first Classical Cepheid – $\eta$ Antinoi (the Antinous
constellation was later merged with the Aquila constellation and the currently
adopted name is $\eta$ Aql) was discovered by the British amateur astronomer
Edward Pigott on Septemebr 10, 1784, who observed it with his own telescope.
He provided a very accurate estimate of its pulsation period 7.18 days
compared with the current value of 7.176641 days. However, the official
discovery was attributed to another amateur astronomer, John Goodricke, who,
one month later, on October 20, discovered with his own telescope that the
star $\delta$ in the constellation of Cepheus was a variable star. Observing
the same object in subsequent nights, he estimated a period of 5.37 days (the
current estimate is 5.366249 days). Edward was much more experienced with
observations and he can be considered John’s mentor, since he was 17 years
old. John and Edward were friends and neighbours in York and they often
observed together. John also had some health problems (he was deaf and mute).
These are probably the reasons why the discovery was assigned to him and the
objects are now-days called Cepheids instead of Aquilaes (Hoskin, 1979;
Croswell, 1997). are radially-pulsating variables with periodic luminosity
variations used as standardizable candles and tracers of stellar populations.
This review focusses on the three different flavours of Cepheids: classical
Cepheids (CC, pulsation periods ranging from roughly one day to a few hundred
days, absolute visual magnitudes ranging from $M_{V}\sim-2.5$ mag to
$M_{V}\sim-7.5$ mag) tracers of stellar populations younger than 200–300 Myrs;
Anomalous Cepheids (ACs, pulsation periods ranging from half a day to a few
days, absolute visual magnitudes ranging from $M_{V}\sim-1$ mag to
$M_{V}\sim-2.5$ mag) tracers of intermediate-age (a few Gyrs) stellar
populations and type II Cepheids (TIICs, pulsation periods ranging from one to
a hubndred days, absolute visual magnitudes ranging from $M_{V}\sim-1$ mag to
$M_{V}\sim-4.5$ mag) tracers of stellar populations older than 10 Gyrs. They
have been the topic of many detailed theoretical and empirical investigations
and several reviews (Kraft, 1963; Sandage, 1972; Madore and Freedman, 1985,
1991; Gautschy and Saio, 1995, 1996; Sandage and Tammann, 2006; Tammann et al,
2008; Freedman and Madore, 2010; Bono et al, 1999a, b, 2010; Feast and
Whitelock, 2013; Subramanian et al, 2017; Bhardwaj, 2020)

Variable stars have played a key role in the development of modern,
quantitative Astrophysics. Baade’s discovery (Baade, 1956) that CCs and TIICs
trace young and old stellar populations and obey to two different
period–luminosity (PL) relations has had an unprecedented impact upon both
size and age of the Universe. Moreover, CCs one century ago were used to
estimate the distance to M31 and solve the Great Debate concerning the
extragalactic nature of the Nebulae (Hubble, 1925, 1929; Sandage, 2005) and to
trace, for the first time, the rotation of the Milky Way (MW) thin disc (Oort,
1927; Joy, 1939; de Grijs and Bono, 2017, and references therein). More
recently, they have been used as fundamental laboratories to constrain
evolutionary (Anderson et al, 2016; Bono et al, 2020b; De Somma et al, 2021)
and pulsation (Bono et al, 2000b; Marconi et al, 2005; Neilson and Lester,
2008; Neilson et al, 2011) properties of low and intermediate-mass stars.

The lively debate concerning the difference between evolutionary and pulsation
mass of classical Cepheids dates back to the 1970s/80s and required
significant improvements in micro-physics (radiative opacities, Rogers and
Iglesias, 1992; Seaton et al, 1994) and in macro-physics: time-dependent
convection, (Kippenhahn et al, 1980; Stellingwerf, 1982b, a, 1984a, 1984b;
Moskalik et al, 1992; Bono et al, 1994; Maeder and Meynet, 2000); extra-mixing
(Chiosi et al, 1992) and mass-loss (Neilson et al, 2011).

The quest for accurate empirical estimates of the current mass of CCs reached
a conclusion thanks to the discovery of CCs in double eclipsing binary systems
by large photometric surveys of variable stars in the Magellanic Clouds (MCs)
provided by microlensing experiments (MACHO, OGLE). The game changer in this
context was the early discovery by Udalski et al (1999) and by Welch and MACHO
Collaboration (1999), and in particular, the detailed and thorough
investigations by Soszyński et al (2008a) and by Udalski et al (2015).

The spectroscopic follow-up of these systems provided the opportunity to
measure with a geometrical method the mass of a classical Cepheid with a 1%
accuracy (Pietrzyński et al, 2010). This discovery was well complemented by
the discovery of a few systems in the Large Magellanic Cloud (LMC, Gieren et
al, 2015), in the Small Magellanic Cloud (SMC, Graczyk et al, 2012;
Pietrzyński et al, 2013) and with the measurement of the dynamical mass of
Polaris (Evans et al, 2008). This paramount observational effort provided the
opportunity to measure the SMC distance with 2% accuracy (Graczyk et al,
2020).

These measurements are crucial in order to fix the zero-point of the
mass–luminosity relation of core helium-burning stellar structures (Neilson et
al, 2011; Brott et al, 2011; Cassisi and Salaris, 2011; Prada Moroni et al,
2012), but the number is still too limited to constrain the slope over the
entire period range of both FO (Pilecki et al, 2015) and FU (Pilecki et al,
2021) CCs. These systems are also fundamental laboratories to improve the
physics of nonlinear pulsation models (Szabó et al, 2011; Marconi et al, 2013;
Paxton et al, 2019; Smolec et al, 2023).

An accurate dynamical mass-estimate has also been recently provided for a TIIC
by Pilecki et al (2018), confirming the evolutionary channel producing these
variables (see section 4.2). We still lack firm estimates of the dynamical
mass of an AC. In a recent very detailed investigation, Pilecki et al (2017)
suggested that OGLE-LMC-T2CEP-098 is an AC, but its period is too long and its
color is too red to be a canonical AC. Accurate measurements await for more
solid identifications and for detailed predictions concerning binary evolution
(mass-transfer phase) of pulsating stars.

The universality of the diagnostics adopted to estimate individual Cepheid
distances has also been a long-standing astrophysical problem. The
period–luminosity relation (PL) was discovered more than one century ago by
Henrietta Leavitt, (Leavitt, 1908; Leavitt and Pickering, 1912) in a titanic
effort to identify variable stars in the MCs. A few decades later, it became
clear - on the basis of observations and plain physical arguments - that
accurate CC individual distances would require the use of a
period–luminosity–color (PLC, Sandage, 1972; Sandage and Tammann, 2006)
relation. Individual distances based on the PL relation rely on the assumption
that the width in temperature of the Cepheid instability strip can be
neglected (Bono et al, 1999a). This assumption is less severe in the near-
infrared than in the optical regime. Nonlinear, convective pulsation models at
fixed chemical composition, stellar mass and luminosity, indicate that
Cepheids in the NIR regime become systematically brighter when moving from the
blue (hotter) to the red (cooler) edge of the instability strip, whereas their
magnitudes are almost constant in the optical regime. The difference is caused
by the stronger sensitivity to effective temperature of NIR bolometric
corrections when compared with optical bolometric corrections. As a
consequence, the spread in magnitude of the NIR PL relation, at fixed
pulsation periods, is on average a factor of two smaller than the optical PL
relations (see Fig. 6 in Bono et al, 1999b). This improvement was brought
forward long ago on an empirical basis by McGonegal et al (1982) and McAlary
et al (1983) and it is independent of the mild dependence of NIR bands on
extinction corrections.

However, the key question in using these diagnostics is: Are they Universal?
Can one calibrate them here and now and use them across the Local Universe? A
different way to rephrase the same question is: Are the PL and the PLC
relations dependent on the metallicity? This is far from being an academic
issue, since the current PL/PLC relations are calibrated by using Galactic
Cepheids to fix the zero-point of the relations and the MC Cepheids to fix the
slope of the same relations. However, the mean observed iron abundance of
Galactic Cepheids is around solar ([Fe/H]$\sim$0) while MC Cepheids are on
average a factor of two (LMC, [Fe/H]$\sim$–0.5) and a factor of four (SMC,
[Fe/H]$\sim$–0.7) more metal-poor.

##  2 Phenomenology of Cepheids

Cepheids are typically divided into three different sub–groups: classical
Cepheids, Anomalous Cepheids, and Type II Cepheids.

###  2.1 Classical Cepheids

Figure 1: Top – Distribution in Galactic coordinates of currently known CCs.
The current sample includes data from Pietrukowicz et al (2021, and references
therein). Galactic CCs are mainly distributed along the thin disk and the
regions across the bulge. The overdensities associated with the MCs are
labelled. Middle – Same as the top, but for ACs. The current sample includes
data available on the OGLE Download Site (http://ogle.astrouw.edu.pl/) and on
the Gaia catalog (Ripepi et al, 2022c). Bottom – Same as the middle, but for
TIICs.

CCs are central helium-burning, intermediate-mass stars with ages younger than
a few hundred Myrs; they are solid young stellar tracers, associated with HII
regions and young open clusters typical of Galactic thin disk and of stellar
systems experiencing recent star-formation episodes (spirals, dwarf
irregulars; see the top panel of Fig. 1). CCs pulsate in a variety of radial
modes, the most common are fundamental, first and second overtone. Together
with the single-mode oscillation, CCs are also characterized by several mixed
modes, i.e. radial oscillations in which at least two radial modes are
simultaneously excited. The occurrence of overtones among classical Cepheids
was still a matter of discussion till the late Eighties. In a seminal
investigation Bohm-Vitense (1988) suggested that a good fraction of short-
period Cepheids were overtone pulsators. In the General Catalog of Variable
Stars (GCVS) they were classified as “S Cepheids” where “S” stands for
Cepheids with sinusoidal light curves. The final empirical evidence was
provided by the microlensing experiments (MACHO, EROS, OGLE) showing
hundreds/thousands of Magellanic Cepheids pulsating in the first three radial
modes and in a variety of mixed modes. The name adopted for mixed mode CCs was
Beat Cepheids (Rodgers, 1970), so as to distinguish them from the so-called
bump Cepheids (Simon and Schmidt, 1976), i.e. the CCs showing the Hertzsprung
progression222The most appropriate way to introduce and summarize the
Hertzsprung progression is to use the sentences he wrote in his pioneering
paper: …At the shorter periods up to about 6 days the curves show the
characteristic regular $\delta$ Cephei form with quick rise and slow decrease,
without additional peculiarites. Above 6 days a secondary wave on the
descending branch of the light curve makes its appearance. This secondary wave
is a very characteristic feature of the following periods. For periods between
10 and 13 days, the secondary wave is, when present, situated on top of the
suppressed ordinary maximum, the form of the light curve being nearly
symmetrical. …The secondary wave has superseded the maximum shown at shorter
periods. At periods of about 14 or 15 days the new superposed maximum occurs
earlier, giving the light curve again an unsymmetrical shape with a hesitation
in the increase of the brightness about midway between minimum and maximum.
This hesitation is persistent at the periods mentioned and not found at any
other period materially different from them. …At periods longer than 16 days
several curves are found rather similar to that of $\delta$ Cephei, showing
quick rise and slow decrease apparently without complications. (Hertzsprung,
1926, see also section 3).  The identification of the different modes was
facilitated, since MC CCs are all located at the same distance. Therefore, the
typical diagnostics adopted to estimate individual distances (PL, PLC, Period-
Wesenheit [PW]) can also be adopted to identify the pulsation mode (see
section 5.2).

Figure 2:  Top: from left to right the light curves, based on the OGLE IV data
set, for SMC CCs pulsating in the fundamental (panel a), in the first overtone
(panel b), in the second overtone (panel c) and in a mixed-mode (panel d). The
ID and the period (days) are labelled. For the mixed mode are labelled both
primary and secondary period. Bottom: Same as the top, but for LMC CCs.

The top panels of Fig. 2 show from left to right the light curves of SMC CCs
pulsating in the fundamental (panel a), first overtone (panel b), second
overtone (panel c) and in a mixed mode (first overtone/fundamental, panel d),
while the bottom panels (e,f,g,h) display similar light curves, but for LMC
CCs. We are showing these typical light curves for MC CCs, because the
empirical scenario is quite rich and includes a variety of single and mixed
mode variables. The reader interested in a more detailed discussion concerning
their phenomenology is referred to Soszyński et al (2015b). The current MW
sample only includes a few first overtones, a single second overtone, and a
very limited number of mixed-modes. The lack of these objects is an
observational bias. The current Galactic photometric surveys are far from
being complete, since the current limiting magnitude in the optical regime is
$I\approx$19 mag. The observational bias becomes more severe in the
inner/outer disk and beyond the Galactic center, due to the high column
density of dust along the line of sight. The surface density of CCs (see Table
3) across the thin disk in the annulus in Galactocentric distance between 7
and 9 kpc is $\rho$=5.64 per kpc2, while in the annulus between 5 and 7 kpc it
is 20% smaller ($\rho$=4.51 per kpc2). However, the density gradient of stars
steadily increases when moving toward the inner disk and the surface density
of CCs should show the same trend. This is also the reason why the peak in the
iron distribution of CCs is at solar-iron abundance.

Pulsation models and observations indicate that the topology of the CC
instability strip depends on the metal content. Moreover, evolutionary models
also indicate that the width in temperature of the blue loop, and in turn, the
minimum mass crossing the CC instability strip depend on the metal content.
These circumstantial evidence takes account of the limited number of Galactic
second overtones and mixed-mode variables currently known. This bias is going
to be removed by the current ongoing long-term photometric surveys either of
the Galactic plane (OGLE, Skowron et al, 2019) or by an all-sky survey like
Gaia (Gaia DR3).

Figure 3:  Top: Panel a)– Period distribution for fundamental (red hatched
area), and first overtone (light blue hatched area) Galactic CCs. Note that
the Y-axis is logarithmic. Panel b)– $I$-band luminosity amplitude versus
logarithmic period (Bailey diagram) for FU (red), and FO (light blue) Galactic
CCs. Panel c)– Dereddened ($V-I$)0 color distribution for CCs. The sample
adopted for the color distribution is based on multi-band optical photometry
provided by van Leeuwen et al (2007, vL07) and on individual reddenings
provided by Tammann et al (2003). Middle: Same as the top, but for LMC CCs
(panels d,e,f). The green hatched area shows the period distribution of second
overtone CCs. Bottom: Same as the middle, but for SMC CCs (panels g,h,i).

Panel a) of Fig. 3 shows the period distribution of both fundamental (red
hatched area) and first overtone CCs (light blue hatched area). This sample
includes CCs collected by OGLE IV (Skowron et al, 2019; Pietrukowicz et al,
2021). We adopted this sample because accurate and homogeneous photometry and
pulsation parameters are available for both MC and MW variables. The mixed
mode Cepheids pulsating either in FO(primary)/FU(secondary) or in FU/FO were
included by using their dominant periods. The histograms plotted in this panel
display that FUs peak at $P\sim 4$ days ($\log P\sim 0.60$) and the
distribution is skewed towards longer periods, with a secondary maximum
(shoulder) located at $P\sim 12.5$ days ($\log P\sim 1.1$). As a whole, the
periods range from roughly one day to 50 days. The FOs peak at $P\sim 1.5$
days ($\log P\sim 0.20$), their distribution is skewed towards shorter periods
and their periods range from $\sim$0.18 days to $\sim$5.6 days.

Note that no single-mode second overtone CC is currently known in the MW. The
variable V473 Lyr is a second overtone CC, but shows strong amplitude and
phase modulations (Burki and Mayor, 1980; Breger, 1981; Molnár and Szabados,
2014; Molnár et al, 2017). Galactic double-mode and mixed-mode CCs, including
a second overtone mode, have also been identified (Pardo and Poretti, 1997;
Beltrame and Poretti, 2002; Sziládi et al, 2007; Poretti et al, 2014), but
they are few and were not included.

Panel b) of Fig. 3 shows the Bailey diagram (I-band amplitude versus
logarithmic period) for Galactic CCs. This panel only includes CCs with
accurate and homogeneous I-band light curves collected by OGLE IV along the
Galactic plane.

The FO amplitudes are on average a factor of two smaller than FU amplitudes.
The FU amplitudes display a broad distribution with amplitudes ranging from
$\Delta I\sim 0.2$ mag to $\Delta I\sim 0.8$ mag. There is a mild evidence for
a minimum in amplitudes for periods ranging from $P\sim 6.5$ to 10 days ($\log
P\sim 0.8$ to $\log P\sim 1$). The reason for this secondary minimum will
become clearer in the following.

Panel c) shows the dereddened color distribution of both FOs and FUs. This
panel only includes CCs for which are available accurate and homogeneous
estimates of $V$,$I$ photometry and individual reddenings. The main source for
the photometry is van Leeuwen et al (2007), while for the reddening is Tammann
et al (2003). The latter authors performed a critical analysis not only of the
reddening estimates available in the literature, but also on the absorption
coefficients and on their impact on the Period-Color relations (see also Laney
and Caldwell, 2007). As expected, the FOs are systematically bluer than FUs
and peak at ($V-I$)${}_{0}\sim 0.63$ mag, while the FU peak at
($V-I$)${}_{0}\sim 0.73$ mag (see Table 2).

The parameters adopted to discuss the pulsation properties of CCs are
independent of uncertainties affecting either distance and reddening estimates
(periods, luminosity amplitudes) or distance estimates (colors). The
dispersion in period and amplitude distribution is driven by differences in
evolutionary and/or in pulsation properties (Bono et al, 2020a).

The middle panels of the same figure show the same parameters, but for LMC
CCs. Data plotted in these panels display several interesting features worth
being discussed in detail.

i)–FO period distribution– The number of FOs (light blue diamonds) is
significantly larger than for Galactic Cepheids, indeed the population ratio
between FO/FU increases by $\sim$50% (0.49 versus 0.72). Moreover, their
periods range from 0.18 days to 6.3 days and their period distribution shows
two well-defined shoulders for periods of $\sim$0.3 days and $\sim$1 day
($\log P\sim-0.5$ and 0.0).

ii)–FU period distribution– The period distribution of FUs (red diamonds) is
broader, indeed their periods range from about a half day to more than one
hundred days. Moreover, the shoulder (secondary maximum) identified in
Galactic CCs is now located at longer periods, around 20 days ($\log P\sim
1.3$) but is less evident.

iii)–SO period distribution–The LMC include a sizable sample of SO CCs (green
diamonds) that are only minimally present among Galactic CCs.

Panel e) shows the Bailey diagram. A glance at the data plotted in this panel
shows that FUs display the typical “V”-shape with a well-defined maximum for
periods around 2.5 days ($\log P\sim$0.4) and a secondary minimum for periods
around ten days (Bono et al, 2000b). The main peak is connected with the peak
in the period distribution, while the secondary minimum is associated with the
Hertzsprung progression and to the so-called bump Cepheids (see section 3).
For periods around ten days, the phase of the bump along the pulsation cycle
approaches the maximum in surface brightness, and the luminosity amplitude
attains a minimum (associated with a minimum in radius) when compared with
shorter- and longer-period Cepheids. The amplitude distribution of FO Cepheids
is more complex, its maximum is located at periods around 1.26 days ($\log
P\sim 0.10$) and shows secondary maxima around 1 and 0.3 days ($\log P\sim
0.0$ and $-0.5$). The SOs are characterized by small amplitudes and also
cover, as expected, a narrow range in periods.

Panel f) shows the dereddened ($V-I$)0 color distribution. The difference in
color between Galactic and LMC CCs, the latter being systematically bluer, was
noted more than a half century ago by Gascoigne and Kron (1965) and more
recently by Laney and Stobie (1994). Data plotted in this figure show that FU,
FO and SO color distributions are quite symmetrical and the median ($V-I$)0
colors peak at ($V-I$)${}_{0}=0.651\pm 0.107$ mag (FU), ($V-I$)${}_{0}\sim
0.530\pm 0.091$ mag (FO) and ($V-I$)${}_{0}\sim 0.368\pm 0.129$ (SO) mag,
where the errors are the standard deviations (see Table 2).

Table 2: Median V-I colors (mag) of Magellanic Cloud and Milky Way Cepheids.

Mode |  V-I [mag]
---|---
| SMC | LMC | MW
—CCs—
FU | 0.613$\pm$ 0.102 | 0.651$\pm$ 0.107 | 0.726$\pm$ 0.094
FO | 0.509$\pm$ 0.095 | 0.530$\pm$ 0.091 | 0.629$\pm$ 0.059
SO | 0.373$\pm$ 0.056 | 0.368$\pm$ 0.129 | …
—ACs—
FU | 0.530$\pm$ 0.066 | 0.485$\pm$ 0.107 | 0.556$\pm$ 0.223
FO | 0.386$\pm$ 0.092 | 0.399$\pm$ 0.096 | 0.405$\pm$ 0.266

22footnotetext: The errors associated with the median colors are the standard
deviations.

The bottom panels show the same parameters of middle and top panels, but for
SMC CCs. The period distributions plotted in panel g) show that SOs are more
representative compared with the LMC, and indeed the population ratio SO/FO
increases by a factor of three (0.01 versus 0.03). They also cover a narrower
range in period ($0.40\leq P(\rm SO)\leq 0.92$ vs $0.58\leq P(\rm SO)\leq
1.46$ days) and a broader range in luminosity amplitudes (panel h) when
compared with LMC SOs.

The peak in the period distribution of SMC FUs and FOs shifts towards shorter
periods ($\sim$0.9 days (FO), $\sim$1.5 days (FU) vs $\sim$1.8 (FO) days,
$\sim$2.8 days (FU)). The range in periods shows the same trend: $0.25\leq
P(\rm FO)\leq 4.5$ vs $0.25\leq P(\rm FO)\leq 6.0$ days and $0.84\leq P(\rm
FU)\leq 129$333The SMC sample includes a long-period CC with $P\sim 208$ days.
vs $0.97\leq P(\rm FU)\leq 134$ days, respectively. Note that the shoulder
located in the long period tail of FU CCs is now placed at even longer periods
when compared with Galactic and LMC CCs, i.e. around 25 days ($\log P\sim
1.4$).

The $I$-band luminosity amplitudes plotted in panel h) display quite clearly
that the secondary minimum linked with the Hertzsprung progression moves SMC
bump Cepheids towards longer periods, $P\sim 9.8\pm 0.1$ days. Moreover, the
peak in the $I$-band luminosity amplitudes moves towards shorter periods,
namely 1.58 days (FU, $\log P\sim 0.2$), 0.79 days (FO, $\log P\sim-0.1$) and
0.63 days (SO, $\log P\sim-0.2$). Finally, the dereddened ($V-I$)0 colors of
SMC CCs plotted in panel f) and listed in Table 2 are either similar (SO) or
systematically bluer than LMC and Galactic CCs.

The circumstantial evidence emerging in the comparison between Galactic and
Magellanic CCs can be summarized as follows:

i)– The peak in the period distributions and the range in period covered by
FU, FO and SO CCs steadily move towards shorter periods when moving from
Galactic to LMC and SMC CCs. On the other hand, the shoulder located in the
long period tale of FU CCs, moves in the opposite direction: it is placed at
longer periods when moving from Galactic to LMC and SMC CCs.

ii)– The center of the Hertzsprung progression, i.e. the period in which bump
Cepheids attain a well-defined minimum in luminosity amplitude, steadily moves
towards longer periods when shifting from Galactic to LMC and SMC FU CCs.

iii)– The peak in the Bailey diagram for FU and FO CCs systematically shifts
towards shorter periods when moving from metal-rich to more metal-poor stellar
systems (Galaxy–LMC–SMC).

iv)– The color distribution of FU and FO CCs becomes systematically bluer when
moving from Galactic to LMC and to SMC (see Table 2). This evidence cannot be
extended to SO CCs, since they are vanishing in the MW and only two dozens are
currently known in the LMC. The evidence that the peak in color when moving
from FU to FO and to SO MC CCs becomes systematically bluer, fully supports
pulsation predictions concerning the topology of the instability strip.
Indeed, the regions in which they show a stable limit cycle become, at fixed
chemical composition, systematically hotter when moving from the fundamental
mode to the overtones.

Figure 4:  Panel a)– Scaled solar evolutionary tracks for young stellar
structures computed by neglecting convective overshooting during core
hydrogen-burning phases in the Hertzsprung–Russell diagram at fixed chemical
composition (metal mass fraction, Z=0.02) and stellar masses ranging from 2.5
to $12\,M/M_{\odot}$ (see labels). The dashed lines display hydrogen-burning
phases, while the magenta color marks central helium-burning phases, the cyan
color marks double-shell (H and He) burning phases (Asymptotic Giant Branch).
The almost vertical solid lines display the blue (hot, first overtone) and the
red (cool, fundamental) edge of the CC instability strip (Fiorentino et al,
2002; De Somma et al, 2021). Panel b)– Same as Panel a), but for a more metal-
poor chemical composition Z=0.01. Panel c)– Same as Panel b), but for a more
metal-poor chemical composition Z=0.001.

Theoretical and empirical evidence indicate that the difference in the
evolutionary and pulsation properties of CCs when moving from the MW to the
MCs is caused by the iron content. Indeed, current metallicity determinations
based on high-resolution spectra indicate that the mean observed iron
abundance of MW CCs is solar, whereas LMC CCs are on average a factor of two
more metal-poor and SMC CCs are a factor of four more metal-poor. Panels a),
b) and c) of Fig. 4 display the comparison between evolutionary and pulsation
predictions over a broad range of chemical compositions (for more details see
section C in Appendix).

In the following, we will discuss the population ratios of the different
pulsation modes. Before discussing them, we point out that, for the MW
samples, we will only derive the ratios based on the OGLE sample. In fact,
OGLE is the only survey allowing an extremely detailed and accurate analysis
of the mixed-mode classifications. We did not include Gaia classification,
because the sampling of the light curves is still based on a modest number of
phase points.

Data listed in Table 3 show that the population ratio between first overtone
and fundamental CCs is 0.65$\pm$0.02 in the SMC, 0.72$\pm$0.02 in the LMC and
0.49$\pm$0.01 in the MW. The observed population ratios do not show a clear
dependence on the observed iron abundance and probably hint to a possible
observational bias in the Galactic sample. The population ratio between second
overtone and fundamental CCs decreases, as expected, when moving from more
metal-poor to more metal-rich stellar systems (0.03 [SMC] vs 0.01 [LMC]) and
it is vanishing for the MW. On the other hand, the fraction of mixed mode
variables does not show a clear dependence on the metallicity (0.11$\pm$0.01,
[SMC]; 0.17$\pm$0.01, [LMC]; 0.14$\pm$0.01, [MW]).

The population ratio between first overtone and fundamental ACs, listed in
Table 3, is similar in the MCs (0.54$\pm$0.10, SMC; 0.44$\pm$0.08, LMC), while
in the MW attain a larger value (0.76$\pm$0.11). However, the Poissonian
uncertainties are large and do not allow us to reach firm conclusions. The
reader interested in a more detailed discussion concerning the statistics of
ACs in nearby dwarf galaxies is referred to Monelli and Fiorentino (2022).

The relative number of the different subgroups of TIICs changes when moving
from the MCs to the MW. The MW has the highest fraction of BLHer variables
(0.38$\pm$0.02) compared to the LMC (0.34$\pm$0.01) and to the SMC
(0.28$\pm$0.01). The fraction of WVir, including pWVir, follows an opposite
trend, decreasing from the SMC (0.51$\pm$0.01), to the LMC (0.46$\pm$0.01),
and the MW (0.43$\pm$0.02). Interestingly enough, the fraction of RVTau
variables is, within the errors, constant in the three different stellar
systems, namely 0.19$\pm$0.01 (MW), 0.19$\pm$0.01 (LMC) and, 0.21$\pm$0.01
(SMC). The population ratio between fundamental and first overtone is
meaningless for TIICs, due to the paucity of FOs which have been identified
sofar. Indeed, after the seminal discovery of two FO TIICs in the LMC by
Soszyński et al (2019) only three other variables were identified in the MW by
Ripepi et al (2022c, see also Table 3).

Table 3: Number of Magellanic Cloud and Milky Way Cepheids and their pulsation
mode.

Mode | SMC | LMC | MW (OGLE) | MW (PP) | MW (all)
---|---|---|---|---|---
—Classical Cepheids—
FU | 2754 | 2477 | 1216 | 2250 | …
FO | 1800 | 1778 | 591 | 1092 | …
SO | 91 | 26 | 0 | 1 | …
Mixed Mode
FU/FO | 69 | 96 | 59 | 100 | …
FO/SO | 239 | 322 | 161 | 201 | …
FO/TO | 0 | 1 | 0 | 0 | …
SO/TO | 0 | 1 | 3 | 3 | …
FU/FO/SO | 0 | 1 | 2 | 3 | …
FO/SO/TO | 1 | 7 | 8 | 8 | …
—Anomalous Cepheids—
FU | 79 | 102 | 38 | … | 118
FO | 43 | 45 | 81 | … | 90
MM | 0 | 0 | 1 | … | 1
—Type II Cepheids—
FU(tot) | 53 | 291 | 1666 | … | 2187
FU(BLHer) | 20 | 99 | 674 | … | 838
FU(WVir) | 15 | 108 | 609 | … | 869
FU(pWVir) | 7 | 27 | 67 | … | 67
FU(RVTau) | 11 | 55 | 313 | … | 410
FO | 0 | 2 | 3 | … | 3

33footnotetext: The source and the mode identification for Cepheids in the
“SMC”, “LMC” and “MW(OGLE)” column come from the OGLE IV data set
(http://ogle.astrouw.edu.pl/ and references therein). The source and the mode
identification for CCs listed in the column “MW (PP)” come from Pietrukowicz
et al (2021) and they are based on the cross-identification they performed
with the Gaia catalog (Ripepi et al, 2022c) and with catalogs available in the
literature. The source and the mode identification for ACs and TIICs listed in
the column “MW (all)” is based on the cross-identification (matching radius of
1 arcsec) that we performed between the catalogs available on the OGLE
Download Site (http://ogle.astrouw.edu.pl/) and the Gaia catalog (Ripepi et
al, 2022c).

###  2.2 Anomalous Cepheids

Anomalous Cepheids (ACs)444The definition of Anomalous Cepheids dates back to
Zinn and Searle (1976, see their Fig. 1) in their detailed analysis of the
mass-distribution of Draco dSph variable stars. The anomaly was driven by the
evidence that ACs, at fixed pulsation period, are brighter than RR Lyrae
(RRLs) and TIICs, and fainter than CCs. The increase of a factor of two in the
stellar mass for ACs, when compared with RRLs and TIICs, was originally
suggested by Christy (1970). The reader interested in a more detailed
discussion concerning the early discoveries of ACs in globulars and dwarf
galaxies is referred to Monelli and Fiorentino (2022). are low-mass stars with
stellar masses ranging from 0.8 to 1.8 (Fiorentino and Monelli, 2012) in which
central helium burning takes place in a partially electron-degenerate helium
core. They are associated with intermediate-age stellar tracers. The current
evidence indicates that they might be the aftermath of both single and binary
merging evolutionary channels. This is the reason why they are ubiquitous over
the entire Galactic spheroid and in nearby galaxies (see the middle panel of
Fig. 1). However, theory and observations suggest that only stellar
populations more metal-poor than [Fe/H]$\lesssim-1.5$ cross the Cepheid
instability strip. This is the reason why they mainly show up in metal-
poor/metal-intermediate stellar systems.

Figure 5:  Panel a): Light curve for an LMC AC pulsating in the fundamental
mode from the OGLE-IV data set. The ID and the period (days) are labelled.
Panel b): Same as the left, but for an SMC AC pulsating in the first overtone.
Panel c): Same as the left, but for the Galactic AC pulsating simultaneously
in the first three radial modes. Primary and secondary periods (days) are
labelled.

ACs display quite regular light curves for both fundamental and first overtone
variables. The majority of Magellanic ACs, the sample for which we have a more
detailed knowledge, mainly includes FU variables, indeed the population ratio
is $\sim$0.65–0.70. They were considered, until a few years ago, single mode
variables, but Soszyński et al (2020) found a Galactic triple mode AC. Figure
5 shows representative light curves for Galactic, LMC and SMC ACs in the three
different pulsation modes.

Figure 6:  Top: Panel a)– Period Distribution for fundamental (red-hatched
area) and first overtone (light-blue hatched area) Galactic ACs. Panel b)–
I-band luminosity amplitude versus period (Bailey diagram) for both FU and FO
ACs. Panel c)– dereddened ($V-I$)0 color distribution for the same ACs plotted
in the left and in the middle panel. Middle: Same as the top, but for LMC ACs.
Bottom: Same as the top, but for SMC ACs.

The AC pulsating in the fundamental-mode have pulsation periods ranging from
0.6 to 2.7 days, while the first-overtones oscillate with periods ranging from
0.4 to 1.2 days. The period distributions are quite similar when moving from
Galactic to Magellanic ACs (see left panels in Fig. 6). The same outcome
applies to the I-band luminosity amplitudes, the FO amplitudes are on average
a factor of two smaller than fundamental ones (see middle panels in Fig. 6).
The color ($V-I$) distribution covered by Magellanic ACs are quite similar,
thus suggesting a marginal dependence on the metallicity (see right panels in
Fig. 6).

Figure 7:  Panel a)– Evolutionary tracks for intermediate-age stellar
structures in the Hertzsprung–Russell diagram at fixed metallicity (Z=0.0008)
and stellar masses ranging from 1.1 to $2.3\,M/M_{\odot}$ (see labels). The
magenta color marks central helium-burning phases, whereas the light blue
color marks double-shell (H and He) burning phases (Asymptotic Giant Branch).
The almost vertical solid lines display the blue (hot) and the red (cool) edge
of the Anomalous Cepheid instability strip (Fiorentino and Monelli, 2012;
Monelli and Fiorentino, 2022). Panel b)– Same as panel a), but for Z=0.0004.
Panel c)– Same as panel a), but for Z=0.0002.

The comparison with Galactic ACs is hampered by the dependence of the
reddening correction in the Bulge and on the adopted reddening law in the
disk. In order to overcome possible systematics, the intrinsic (V-I)0 colors
of ACs in the Galactic Bulge were estimated using the reddening map provided
by Nataf et al (2013), while for the others we adopted the reddening maps
provided by Schlegel et al (1998) and by Schlafly and Finkbeiner (2011) and
the Cardelli et al (1989) reddening law. We found that their median intrinsic
colors are (V-I)0=0.556$\pm$0.223 mag (FU) and (V-I)0=0.405$\pm$0.266 mag (see
Table 2) and they are quite similar, within the errors, to the intrinsic
colors of MC ACs.

The predicted dependence of ACs evolutionary and pulsation properties on the
chemical composition are showed in Fig. 7). Panels a), b) and c) display
evolutionary tracks for metal-poor (see labelled values) intermediate-age
stellar structures (for more details see section C in Appendix). However,
spectroscopic abundances are only available for two Galactic ACs (V716 Oph, BF
Ser) that were originally classified as TIICs (Kovtyukh et al, 2018), but we
still lack a detailed abundance analysis for both Galactic and Magellanic ACs.

###  2.3 Type II Cepheids

TIICs555 They were originally called Globular Cluster Cepheids by (Baade,
1956) thanks to the discovery of stellar populations. The reader interested in
a more detailed discussion concerning early discoveries based on TIICs is
referred to Wallerstein and Cox (1984). are low-mass stars in a double shell,
helium and hydrogen, burning phase. According to the pulsation period, they
are either Asymptotic Giant Branch (AGB) or post-AGB radial variables. They
are solid old ($t\geq$10 Gyr) stellar tracers and have been identified in
stellar systems hosting old stellar populations (Halo, Bulge, globulars,
Magellanic Clouds, Andromeda group; see the bottom panel of Fig. 1). However,
they have not been identified in nearby dwarf spheroidal galaxies. It is not
clear yet whether this evidence is either an observational bias, or caused by
the lack of hot and extreme horizontal branch stars in these stellar systems,
as recently suggested by Bono et al (2020b).

The pulsation properties of TIICs are canonical, since they pulsate as single
mode variables in the fundamental and in the first overtone, but the latter
group only includes a few objects both in the Galaxy and in the Magellanic
Clouds (Soszyński et al, 2019). Interestingly enough, long-term photometric
surveys have also identified a few mixed-mode TIICs (Smolec et al, 2018;
Udalski et al, 2018). The top panels (a,b,c) of Fig. 8 shows representative
light curves for Galactic single and mixed-mode TIICs, while the bottom panels
display selected light curves for TIICs typical of the three sub-groups: a
Galactic Bulge BLH (panel d), an LMC WV (panel e) and an SMC RVT (panel f).

Figure 8:  Panel a): Light curve for a Galactic TIIC pulsating in the
fundamental mode from the OGLE IV dataset. The ID and the period are labelled.
Panel b): same as panel a), but for a Galactic first overtone TIIC. Panel c):
same as panel a), but for a Galactic mixed-mode TIIC. Primary and secondary
periods are labelled. Panel d): same as panel a), but for a Galactic Bulge BL
Herculis. Panel e): same as panel a), but for a LMC W Virginis. Panel f): same
as panel a), but for a SMC RV Tauri.

The separation between RRLs and TIICs is a long-standing problem. As a first
approximation, it is possible to adopt a period threshold, whose exact value
is still a matter of debate. A threshold of $\sim$0.8 days was set in the
review by Gautschy and Saio (1996) where type 1 variables (AHB1, above the
Horizontal Branch [HB]), as defined in Strom et al (1970) and Diethelm (1983,
1990), were considered as TIICs rather than evolved RRLs. This threshold is
obsolete, however, because a more extended and homogeneous investigation,
based on period distribution and on the Fourier parameters of the light curve
of RRLs in the Galactic bulge, has now set the threshold at one day (Soszyński
et al, 2008b, 2014). The RRLs in the bulge have a primordial (or minimally
enhanced) helium abundance (Marconi and Minniti, 2018), but there is
theoretical evidence that helium enhancement increases the periods of RRLs
(Marconi et al, 2018). This means that a one-day period threshold should be
considered a particular case of a more general chemical and evolution-
dependent threshold.

TIICs have periods ranging from 1 day to more than 100 days (see left panels
in Fig. 9). Moreover, they display two local minima in the period distribution
for $P\sim 5$ and $P\sim 20$ days. The former value was adopted for separating
BLHs from WVs, and the current data indeed suggest that it ranges from about
four days in the Bulge to about six days in the Galactic field. The latter
value ($P\sim 20$ days) was adopted for separating WVs from RVTs, and the
current data suggest that it shows up as a shoulder in the period distribution
of Bulge and field TIICs and as a local minimum in GCs and in Magellanic
TIICs.

The difference in the period distribution among BLHs, WVs, and RVTs is fully
supported by the Bailey diagrams, the $I$-band luminosity amplitudes versus
the logarithmic period, showed in the middle panels of Fig. 9. The Bailey
diagram shows that WVs attain a well-defined minimum at $P\sim 8$ days, with a
steady increase towards longer periods. The trend for RVTs is far from being
homogeneous, because the maximum around 20 days is broad. Moreover, RVTs in
the Bulge and in the LMC display a steady decrease towards longer periods and
a well-defined cutoff at periods longer than 60 days. On the other hand, RVTs
in the Galactic field approach 200 days and display at fixed periods a broad
range in luminosity amplitudes.

The current partition of TIICs into three sub-groups follows the
classification suggested by Soszyński et al (2008b, 2011). They also suggested
a new group of TIICs, the peculiar WVs (pWVs) which have peculiar light
curves. Moreover, the pWVs are, at fixed periods, brighter than typical TIICs.

The possible dependence on the metallicity requires a more detailed
discussion. We still lack spectroscopic measurements of Bulge TIICs, so we
assume that their metallicity distribution is either similar to that of Bulge
RRLs as measured by Walker and Terndrup (1991), suggesting a mean
[Fe/H]$=-1.0$ with a 0.16 dex standard deviation, or similar to Bulge red
giant stars, with average [Fe/H]$=-1.5$ and a standard deviation equal to 0.5
dex (Rich et al, 2012; Zoccali et al, 2017). The metallicity distribution of
TIICs in GCs and in the Galactic field ranges from [Fe/H]$\sim-2.4$ to
slightly super solar [Fe/H] (see Appendix in Bono et al, 2020b).

Figure 9:  Top: Panel a)– Period distribution for fundamental (red hatched
area) and first overtone (light blue hatched area) Galactic TIICs. Panel b)–
I-band luminosity amplitude versus period (Bailey diagram) for both FU and FO
TIICs. Panel c)– dereddened ($V-I$)0 color distribution for the same TIICs
plotted in the left and in the middle panel. Middle: Same as the top, but for
LMC TIICs. Bottom: Same as the top, but for SMC TIICs.

For LMC TIICs, we can follow two different paths. According to Gratton et al
(2004) the iron abundance of LMC RRLs based on low-resolution spectra range
from [Fe/H]$=-2.1$ to [Fe/H]$=-0.3$, but only a few stars are more metal-rich
than [Fe/H]$=-1.0$; the mean observed iron abundance for 98 RRLs is
[Fe/H]$=-1.48\pm 0.03\pm 0.06$. This metallicity range is also supported by
recent investigations of the mean metallicity of LMC globular clusters. Using
homogeneous Strömgren photometry, Piatti and Koch (2018) found, in agreement
with spectroscopic measurements, that the metallicity of the ten LMC globulars
ranges from $-2.1$ dex (NGC 1841) to $-1.1$ dex (ESO121-SC3). We still lack
direct measurements of the metallicity distribution of truly old SMC stellar
tracers. According to high-resolution spectroscopy (Dalessandro et al, 2016),
the metallicity of NGC 121, the only SMC globular cluster, is [Fe/H]$=-1.28$.
Metallicity estimates listed in Table A.1 of Bono et al (2020b) indicate that
TIICs cover roughly three dex in metal abundance. However, the population
ratios appear to be, within the errors, quite similar. This finding is also
supported by the similarity in the color distribution between Galactic and
Magellanic TIICs (see right panels in Fig. 9).

TIICs can be thought as the intersection of several theoretical and empirical
investigations; however, their evolutionary status is far from being well
established. A first analysis of the evolutionary properties of TIICs was
provided over 40 years ago by Gingold (1974, 1976, 1985). He recognized that a
significant fraction of hot (blue) HB stars evolve off the Zero-Age Horizontal
Branch (ZAHB) from the blue (hot) to the red (cool) region of the
color–magnitude diagram (CMD). In the approach to their AGB track, these stars
are in a double shell (hydrogen and helium) burning phase (Salaris and
Cassisi, 2005) and cross the instability strip at luminosities systematically
brighter than typical RRLs. The difference in luminosity and the lower mass
induce an increase in the pulsation period of TIICs when compared with RRLs.
Typically, the two classes are separated by a period threshold at one day.
This separation is supported by a well-defined minimum of the period
distribution when moving from RRLs to TIICs, but the physical mechanism(s)
causing this minimum are not yet clear, and the exact transition between RRLs
and TIICs has not been established (Braga et al, 2020).

The quoted calculations also suggest that blue HB stars after the first
crossing of the instability strip experience a “blue nose” (then dubbed
“Gingold’s nose”), a blue-loop in the CMD that causes two further crossings of
the instability strip before the tracks reach the AGB. These three consecutive
excursions were associated with the interplay between the helium and hydrogen
burning shells. After core-helium exhaustion, HB models with massive enough
envelopes evolve redward in the CMD. The subsequent shell-helium ignition
causes a further expansion of the envelope, and, in turn, a decrease in the
efficiency of the shell-hydrogen burning, which causes a temporary contraction
of the envelope, and a blueward evolution in the CMD. Once shell hydrogen
burning increases its energy production again, these models move back towards
the AGB track. Finally, these models would eventually experience a fourth
blueward crossing of the instability strip before approaching their white
dwarf (WD) cooling sequence (see Fig. 1 in Gingold 1985 and Fig. 2 in Maas et
al 2007). During their final crossing of the instability strip, these stellar
structures (in the post-AGB phase) are only supported by a vanishing shell
H-burning.

Basic arguments based on their evolutionary status and on the use of the
pulsation relation available at that time (van Albada and Baker, 1973) allowed
Gingold to associate the first three crossings (including Gingold’s nose) with
BLHs and the fourth one with the WVs variables. These early analyses, however,
lacked quantitative estimates of the time spent inside the instability strip
during the different crossings, and in particular the period distributions
associated with the different crossings. Moreover, HB evolutionary models
dating back to more than 25 years ago and based on updated input physics (Lee
et al, 1990; Castellani et al, 1991; Dorman and Rood, 1993; Brown et al, 2000;
Pietrinferni et al, 2006a; Dotter, 2008; VandenBerg et al, 2013) do not show
the Gingold’s nose.

The evolutionary properties of low-mass core helium burning models have been
discussed in several recent investigations (Cassisi and Salaris, 2011, and
references therein). Here we summarize the main relevant features in order to
explain the evolutionary channels producing TIICs.

The grey area displayed in the top panel of Fig. 10 outlines the region
between the ZAHB (faint envelope) and central helium exhaustion (bright
envelope) for a set of HB models with different masses and fixed chemical
composition (metal–Z=0.01–and helium–Y=0.259–mass fractions). We have assumed
an $\alpha$-enhanced chemical composition (Pietrinferni et al, 2006b) and a
progenitor mass according to a 13 Gyr isochrone (the mass at the main sequence
turn off, MSTO, is equal to $0.86\,M_{\odot}$).

Figure 10:  Top: HRD of HB evolutionary models covering a broad range of
stellar masses ($M/M_{\odot}$=0.48–0.90) and the same initial chemical
composition (Z=0.01, Y=0.259). The grey area outlines the region between ZAHB
(faint envelope) and central-helium exhaustion (bright envelope). The green
lines display HB models evolving as AGB-manqué, black lines the post early-AGB
models, and purple lines the thermal pulsing AGB models (see text for
details). The almost vertical blue and red solid lines indicate the hot and
cool edge of the TIIC instability strip. The minimum stellar mass (in solar
units) crossing the instability strip is labelled in black. The black dashed
lines show two iso-periodic lines for 5 and 20 days. Middle: Same as the top
panel, but for stellar masses ranging from $M/M_{\odot}$=0.4912 to 0.80 and
for a metal-intermediate chemical composition (Z=0.001, Y=0.246). Bottom: Same
as the top panel, but for stellar masses ranging from $M/M_{\odot}$=0.5035 to
0.70 and for a metal-poor chemical composition (Z=0.0001, Y=0.245). Image
reproduced with permission from Bono et al (2020b, Fig. 5), copyright by ESO.

The total mass along the ZAHB, as expected, decreases when moving from the red
HB (RHB) to the blue HB (BHB) and further on to the extreme HB (EHB). The
helium-core mass is constant along the ZAHB and is mainly fixed by the
chemical composition of the progenitor ($M_{c}^{\rm He}=0.4782\,M_{\odot}$)
and is roughly independent of age for ages above a few Gyr. The mass of the
envelope decreases from $0.4218\,M_{\odot}$ for RHB models to a few
thousandths solar masses for EHB models. In an actual old stellar population
with a fixed initial chemical composition, the mass lost along the RGB (more
efficient when approaching the tip of the RGB; Origlia et al, 2014) determines
the final mass distribution along the ZAHB.

The bright envelope of the grey area marks the central helium exhaustion,
corresponding formally to the beginning of the AGB phase. The small ripples
along the helium-exhaustion sequence ($\log L/L_{\odot}\sim 1.8$) show that
the lower the total mass of the HB model, the hotter the effective temperature
at which the helium exhaustion takes place. The luminosity of the ripples
ranges from $\log L/L_{\odot}\approx 2$ in the warm region to $\log
L/L_{\odot}\approx 1.6$ in the hot region of the HB. At this point, the He
burning moves smoothly to a shell around the carbon-oxygen core. The overlying
H-shell extinguishes, due to the expansion of the structure before reigniting
later with various efficiencies, depending on the mass thickness of the
envelope around the original He core.

Models with mass below $0.495\,M_{\odot}$ (corresponding to an envelope mass
lower than $\sim 0.017\,M_{\odot}$) never reach the AGB location; they do not
cross the instability strip and move to their WD cooling sequence, as a
carbon-oxygen (CO) WD (Castellani et al, 2006; Salaris et al, 2013). These
objects have been called AGB-manqué (Greggio and Renzini, 1990), and are shown
as green tracks in the top panel of Fig. 10).

More massive models cross the instability strip while moving towards their AGB
tracks. Models with $0.495\leq M/M_{\odot}<0.55$ reach the AGB, but move back
towards the WD sequence (hence they cross the instability strip again but at
higher luminosities) well before reaching the thermal pulse (TP) phase. They
are named post-early AGB (PEAGB), and are plotted as black lines in the top
panel of Fig. 10. These AGB models perform several gravo-nuclear loops in the
Hertzsprung–Russell Diagram (HRD), either during the AGB phase and/or in their
approach to the WD cooling sequence after leaving the AGB (during this post-
AGB transition models cross again the instability strip). Some of them may
take place inside the instability strip. The reader interested in a more
detailed discussion concerning their impact on the pulsation properties is
referred to Bono et al (1997). The evolutionary implications, and in
particular the impact of the loops concerning the AGB lifetime have recently
been discussed by Constantino et al (2016).

Models with $M\geq 0.54\,M_{\odot}$ (plotted as purple lines in Fig. 10)
evolve along the AGB and experience the TPs. The number of TPs, and in turn
the duration of their AGB phase, is once again dictated by the efficiency of
the mass loss and by their residual envelope mass (Weiss and Ferguson, 2009;
Cristallo et al, 2009). Calculations of TP evolution are quite demanding from
the computational point of view, hence we decided to use the fast and
simplified synthetic AGB technique originally developed by Iben and Truran
(1978) and more recently by Wagenhuber and Groenewegen (1998) to compute the
approach of these AGB models to the WD cooling sequence. In particular, the
synthetic AGB modelling started for thermal-pulsing AGB (TPAGB) models just
before the occurrence of the first TP, while for PEAGBs it was initiated at
the brightest and reddest point along the first crossing of the HRD, towards
the AGB.

The middle and the bottom panels of Fig. 10 show the same predictions, but for
two more metal-poor chemical compositions. The values of the stellar masses
plotted along the ZAHBs show the impact of the chemical composition. The mass
range of the tracks that cross the instability strip and produce TIICs
steadily decreases from 0.495–0.90 $M/M_{\odot}$ for Z=0.01, to 0.505–0.80
$M/M_{\odot}$ for Z=0.001, and to 0.515–0.70 $M/M_{\odot}$ for Z=0.0001. It is
worth mentioning that the range in luminosity covered by the different sets of
models is similar. The mild change in stellar mass and the similarity in
luminosities suggests a marginal dependence of the mass–luminosity (ML)
relation of TIICs on the chemical composition. In order to further define the
theoretical framework for RVTs stars (Wallerstein, 2002; Soszyński et al,
2011), we suggest that they are the progeny of both PEAGB and TPAGB. There are
two reasons supporting this hypothesis:

a) Period range – The theoretical periods for these models are systematically
longer than WVs and more typical of RVTs stars. The predicted mass for these
structures is uncertain because it depends on the efficiency of mass loss
during the TP phase. The theoretical framework is further complicated by the
fact that the number of TPs also depends on the initial mass of the progenitor
and on its initial chemical composition. This means that a contribution from
intermediate-mass stars cannot be excluded a priori. However, the lack of RVTs
in nearby dwarf spheroidal galaxies hosting a sizable fraction of
intermediate-mass stars with ages ranging from 1 Gyr to more than 6-8 Gyr,
such as Carina, Fornax and Sextans (Aparicio and Gallart, 2004; Beaton et al,
2018), suggests that this channel might not be very efficient. However, RVTs
have been identified in the MCs (Soszyński et al, 2008b; Ripepi et al, 2015).

b) Alternating cycle behaviour – There is evidence of an interaction between
the central star and the circumstellar envelope, possibly causing the
alternating-cycle behaviour (Feast et al, 2008; Rabidoux et al, 2010). The
final crossing of the instability strip before approaching the WD cooling
sequence either for PEAGB or for TPAGB models appears a very plausible
explanation.

The above circumstantial evidence suggests that the variable stars currently
classified as TIICs have a range of evolutionary properties. The BLHs and the
WVs appear to be either post-ZAHB (AGB, double shell burning) or post-AGB
(shell hydrogen burning) stars, whereas RVTs are mainly post-AGB objects.

##  3 The Hertzsprung progression

More than ninety years ago, Hertzsprung (1926) discovered that a sub-sample of
Galactic classical Cepheids presents a relationship between the bump along the
light curve and the pulsation period. The so-called “Hertzsprung progression”
(HP) was subsequently discovered among Andromeda and MC Cepheids by Shapley
and McKibben (1940), Kukarkin and Parenago (1949), and Payne-Gaposchkin
(1954). The HP observational scenario was enriched by Joy (1937) and by Ledoux
and Walraven (1958) who found a similar shift in the phase of the bump in
radial velocity curves.

The empirical fingerprint of the HP is the following: classical Cepheids in
the period range $6<P<16$ days show a bump along both the light and the
velocity curves. This secondary feature appears on the decreasing branch of
the light curve for Cepheids with periods up to 9 days, while it appears close
to maximum light for $9<P<11$ days and moves at earlier phases for longer
periods. On the basis of this observational evidence this group of variables
was christened “Bump Cepheids” for avoiding to be mixed-up with “Beat
Cepheids”. In fact, the latter group refer to mixed-mode variables i.e.
objects in which two or more modes are simultaneously excited, and therefore
both the shape of the light curves and the pulsation amplitudes change from
one cycle to the next, whereas bump Cepheids are single-mode variables and
their pulsation properties are characterized by a strong regularity over
consecutive cycles.

Figure 11:  Left: Phased $I$-band light curves of SMC bump Cepheids with a
bump moving up along the decreasing branch (UPDB). From top to bottom, the
different variables are plotted with increasing pulsation period and
artificially shifted in magnitude. The position of the bump is marked with a
pink circle, while the pulsation maximum is marked with a green circle. The
light curves are plotted twice, so as to emphasize the changes along the
pulsation cycle. The light curves are color-coded according to the luminosity
amplitude. They range from light blue for the largest amplitudes to dark blue
for the smallest ones. The labels on the left side show the name of the
variables, whereas those on the right side display the period (days) and the
$I$-band amplitude (mag). Right: Same as the left, but for SMC bump Cepheids
with a bump moving down along the rising branch (DORB).

A more quantitative approach concerning bump Cepheids was originally suggested
by Parenago and Kukarkin (1936); Payne-Gaposchkin (1947), and more recently by
Simon (1976) and by Simon and Lee (1981) who investigated the shape of the
light curves by means of Fourier analysis. The latter authors found out that
both the phase difference, $\phi_{21}$, and the amplitude ratio, $R_{21}$,
show a sharp minimum close to the center of the HP. Following this approach,
several investigations have been already devoted to Fourier parameters of
Galactic and Magellanic Cepheids. In particular, Moskalik et al (1992,
hereinafter MBM) suggested that the minimum in the Fourier parameters for
Galactic Cepheids takes place at P(HP) = 10.0 $\pm$ 0.5 days, while Moskalik
et al (2000) by investigating a sample of more than 100 radial velocity
curves, they found P(HP) = 9.95 $\pm$ 0.05 days (see also Hocdé et al, 2022,
and references therein). At the same time, Alcock et al (1999) by
investigating a large sample of LMC Cepheids estimated that the minimum in the
Fourier parameters is located at P(HP) = 11.2 $\pm$ 0.8 days. Thus supporting
the shift of the HP center toward longer periods originally suggested by
Payne-Gaposchkin (1951) and strengthened by Andreasen and Petersen (1987) and
by Andreasen (1988). More recently, Beaulieu (1998) suggested that the HP
center in LMC and in SMC Cepheids is located at P(HP) = 10.5 $\pm$ 0.5 days
and P(HP) = 11.0 $\pm$ 0.5 days, respectively. Since these three stellar
systems are characterized by different mean metallicities, namely Z=0.02 (MW),
Z=0.008 (LMC), and Z=0.004 (SMC), this empirical evidence suggests that a
decrease in metallicity moves the HP center toward longer periods.

Up to now, two distinct models have been proposed in the literature to explain
the appearance of the HP among bump Cepheids: the echo model and the resonance
model. The former was suggested by Whitney (1956) and discussed by Christy
(1968, 1975) on the basis of Cepheid nonlinear, radiative models. According to
Christy, during each cycle close to the phases of minimum radius and before
the phase of maximum expansion velocity a pressure excess is generated in the
first He ionization region. This pressure excess causes a rapid expansion
which, in turn, generates two pressure waves moving outward and inward. The
latter reaches the stellar core close to the phase of maximum radius, then
reflects and reaches the surface one cycle later causing the appearance of the
bump. The resonance model was suggested by Simon (1976) and is based on
linear, adiabatic periods. Within this theoretical framework, the bump would
be caused by a resonance between the second overtone and the fundamental mode,
and takes place when the period ratio between these two modes is close to 0.5.
In particular, they suggested that the instability of the fundamental mode
drives, due to a resonance, the second overtone instability. This explanation
lies on the evidence that the nonlinear, radiative models constructed by
Stobie (1969) show a bump along the radial velocity curves close to the
resonance line $P_{2}$/$P_{0}=0.5$.

Such an extensive observational and theoretical effort devoted to bump
Cepheids was not only aimed at understanding the HP, but also at providing
independent estimates of both the mass and the radius of these variables. In
fact, dating back to Christy (1968), Christy (1975), Stobie (1969) and Fricke
et al (1972) it was suggested that these two evolutionary parameters can be
constrained on the basis of period and phase of the bump. A different method
to estimate the mass, based on period ratios, was suggested by (Petersen,
1973). Mass determinations based on these two methods present a compelling
feature: they are based on observables, such as periods and phases of the
bump, which are not affected by systematic empirical uncertainties, since they
are only limited by photometric accuracy. However, pulsation masses based on
these methods are, with few exceptions (Carson and Stothers, 1988),
systematically smaller than the evolutionary masses. This longstanding puzzle
raised the so-called bump mass discrepancy (see also Cox et al, 1980) and at
the same time supported the use of a ML relation based on evolutionary models,
which include either a mild or a strong convective core overshooting (Simon,
1995; Wood, 1998).

Even though the new radiative opacities settled down this long-standing
problem (MBM; Kanbur and Simon 1994), recent linear (Buchler et al, 1996;
Simon and Young, 1997) and nonlinear (Wood et al, 1997) predictions for MC
Cepheids present a small discrepancy with the ML relations predicted by
current evolutionary models.

In order to provide a new quantitative spin on the HP, we decided to take
advantage of the homogeneous and accurate data set collected by OGLE-IV for
both Galactic and MC Cepheids to further constrain the metallicity dependence.

Figure 11 shows the $I$-band light curves of SMC bump Cepheids. The light
curves from top to bottom are plotted in order of increasing pulsation period
and their colors are correlated with the luminosity amplitude. Darker colors
mark low luminosity amplitude variables, whereas light blue colors mark large
amplitudes. To help the eye in the identification of secondary features, the
bump is marked with a pink circle, while the pulsation maximum is marked with
a green circle. To properly trace the transition of the bump from the
decreasing to the rising branch, the left panel only shows bump Cepheids with
the bump moving UP along the decreasing branch (UPDB), whereas the right panel
shows bump Cepheids with the bump moving DOwn along the rising branch (DORB).
The light curves plotted in this figure display several distinctive features
that are worth being discussed in detail.

a) The phase of the maximum among the UPDB Cepheids (left panel) is quite
constant and equal to $\phi\sim$0.1666The current light curves were phased by
using as reference epoch the phase along the rising branch in which the
magnitude of the light curve is equal to the mean magnitude of the object
(Inno et al, 2013)., while the phase of the bump steadily approaches the phase
of the maximum light when moving from shorter (top) to longer-period (bottom)
bump Cepheids.

b) The bump in UPDB Cepheids with periods shorter than $\sim$9 days is fainter
than the pulsation maximum, while for periods of the order of $\sim$9.3–9.5
days it attains magnitudes similar to the pulsation maximum and it becomes the
main maximum at longer periods. The variation in the shape of the light curve
is far from being smooth, indeed UPDB Cepheids with low luminosity amplitudes
have light curves that have been defined as “flat topped” , i.e. with a
luminosity maximum characterized by two secondary maxima.

c) The center of the HP is located at periods of P(HP)=9.8$\pm 0.2$ days, in
which the UPDB Cepheids attain a well-defined minimum in luminosity amplitude
(A($I$)$\sim$0.15 mag). The light curve of these variables is more sinusoidal,
almost featureless, with the bump showing up either as a small secondary
maximum along the rising branch, or a change in the slope along the rising
branch. To improve the identification of the bump in these variables we also
took advantage of the OGLE-IV V-band light curves.

d) The light curves of DORB Cepheids (right panel) show, as expected, an
opposite trend. Close to the minimum in luminosity amplitudes, the light
curves of UPDB and DORB Cepheids are similar. The phases of the maximum are
once again quite constant, and take place around phases $\phi$=0.1–0.2.
However, the phase of the bump steadily approaches the phase of the minimum in
luminosity when moving from shorter to longer period DORB Cepheids.

e) The variation of the light curve among LMC bump Cepheids (see Fig. 29 in
Appendix) is similar to SMC bump Cepheids (Fig. 11). The key difference is
that the center of the HP occurs at shorter periods, namely P(HP)$\sim
9.4\pm$0.2 days. In this period range, the luminosity amplitude is smaller
than 0.2 magnitudes, with a well-defined minimum for the Cepheid 2252 with
A($I$)$\sim$0.1 mag. There are a few exceptions concerning the shape of the
light curve, but the global trend is well defined among both UPDB and DORB
Cepheids.

f) The similarity in the variation of the light curves across the HP also
applies to Galactic bump Cepheids (see Fig. 29 in Appendix). The center of the
HP is located at even shorter periods P(HP)$\sim 9.0\pm 0.2$ days and $I$-band
luminosity amplitudes of the order of 0.25 magnitudes.

The reader interested in a more quantitative discussion concerning the
metallicity dependence of the HP is referred to section A in Appendix.

##  4 Magellanic Cepheids and the color–magnitude diagram

Optical, NIR and MIR photometric surveys of the MCs provide the unique
opportunity to investigate the Cepheid distribution across the CMD. The main
reason why the MCs play a crucial role in quantitative Astrophysics is mainly
that we can, as a first approximation, assume that the MC stellar populations
are all placed at the same distance, i.e., by neglecting the depth effect.
Data plotted in Fig. 12 display the distribution of SMC (left) and LMC (right)
Cepheids in the optical $I$,$V-I$ CMD. Different groups of variable stars are
plotted with different symbols, while variables pulsating in different modes
are plotted with different colors. Common stars are plotted as grey dots. Data
for both static and variables stars come from the OGLE IV data set. Helium-
burning variable stars are situated, as expected, between the main sequence
and the red giant branch in the so-called Hertzsprung gap. They typically
attain magnitudes either brighter (CCs, TIICs) or similar (ACs) to red clump
stars, the stellar over-density located in the SMC at $I\sim$18.55–18.65 mag
and $V-I\sim$0.90–0.95 mag and in the LMC at $I\sim$18.25–18.35 mag and
$V-I\sim$1.00–1.05 mag. The mean magnitude of the variable stars was estimated
as a mean in flux over the entire pulsation cycle, and then transformed into a
mean magnitude. Variables placed outside the well-defined, almost vertical,
variable sequences are either affected by differential reddening or by
blending (crowded regions). A glance at the data plotted in this figure
clearly show that optical CMDs can be adopted to validate variable
identification, but they are far from being an optimal diagnostic to identify
different groups of variable stars, since the different Cepheid groups overlap
both in magnitude and in color.

Figure 12: 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 $I$,$V-I$ CMD of the SMC. Field
SMC stars are plotted as grey dots. The optical CMD shows different triple
regions, i.e. regions in which the three different groups of variables
overlap. Data plotted in this panel come from the OGLE IV data set. The mean
magnitudes of variable stars were estimated with a fit of the light curves.
Right: Same as the left, but for LMC Cepheids.

Figure 13: Left: Comparison between theory and observations in the dereddened
$I_{0},(V-I)_{0}$ CMD for SMC CCs. CCs are marked with light (fundamental) and
dark (first overtone) red symbols, while static stars with grey dots. Variable
stars were dereddened by using the reddening map provided by Skowron et al
(2021), while static stars were dereddened through a mean reddening of
E($V-I$)=0.047 mag (Skowron et al, 2021). Solid lines display selected scaled-
solar stellar isochrones from the BASTI-IAC database at a fixed chemical
composition ([Fe/H]$=-0.60$, Y=0.252) and ages ranging from 30 to 300 Myrs
(see labelled values). Isochrones were plotted by assuming a true distance
modulus of $\mu=18.977\pm 0.028$ mag (Graczyk et al, 2020). Right: Same as the
left, but for LMC CCs and more metal-rich scaled-solar stellar isochrones
([Fe/H]$=-0.30$, Y=0.257). Isochrones were plotted by assuming a true distance
modulus of $\mu=18.477\pm 0.023$ mag (Pietrzyński et al, 2019). The mean
reddening adopted for static stars is E($V-I$)=0.100 mag (Skowron et al,
2021).

In order to find out more about the global properties of Magellanic CCs, the
right panel of Fig. 13 shows the comparison between theory and observations in
the optical dereddened $I_{0},(V-I)_{0}$ CMD for LMC CCs. The symbols are the
same as in Fig. 12, individual reddenings for CCs were estimated by using the
reddening map provided by Skowron et al (2021), while common stars were
dereddened through a mean reddening of E($V-I$)=0.100 mag (Skowron et al,
2021). The solid lines with different colors display stellar isochrones from
the BASTI-IAC database at fixed chemical composition ([Fe/H]$=-0.30$, iron
abundance; Y=0.252, helium mass fraction) and ages ranging from 40 to 300 Myrs
(see labelled values). Stellar isochrones are based on evolutionary models,
constructed by assuming a scaled-solar chemical mixture and by neglecting
convective core overshooting. They were plotted by assuming a true distance
modulus of $\mu=18.477\pm 0.023$ mag (Pietrzyński et al, 2019).

Detailed comparisons between theory (evolutionary and pulsation properties)
concerning MC variable stars have been widely discussed in the literature
(Cioni et al, 2014; Soszyński et al, 2017a; Ripepi et al, 2016). Here we are
mainly interested in providing a global picture of their properties. The
agreement between theory and observations is quite good over the entire
mass/age range. Indeed, the width in color of the blue loops (central helium
burning phases) takes globally into account the observed distribution of CCs
inside the instability strip. However, this pending issue requires more
detailed investigations, since the extension in temperature of the blue loop
depends on many different physical assumptions (Bono et al, 2000b) and we
still lack quantitative constraints on the impact that input physics
(opacities, equation of state), mass loss (Maeder and Meynet, 2000; Marengo et
al, 2010; Barmby et al, 2011), rotation (Maeder and Meynet, 2000; Costa et al,
2019) and extra-mixing have on these evolutionary phases. The current
theoretical framework also takes into consideration young MS stars (central
hydrogen burning phases), and RGB stars (shell hydrogen burning phases).

The left panel of Fig. 13 shows the same comparison, but for SMC CCs. The
solid lines display BASTI-IAC scaled-solar stellar isochrones at fixed
chemical composition ([He/H]$=0.60$, Y=0.257) and different ages (see labelled
values). Isochrones were plotted by assuming a true distance modulus of
$\mu=18.977\pm 0.028$ mag (Graczyk et al, 2020). The mean reddening adopted
for static stars is E($V-I$)=0.047 mag (Skowron et al, 2021). The agreement
between theory and observations is once again quite good over the entire
mass/age range.

The difference between stellar isochrones based on evolutionary models which
either neglect or take into account convective core overshooting during core
hydrogen-burning phases have been widely discussed in the literature. The main
difference is that the isochrones taking into account overshooting are
systematically brighter than canonical isochrones. This means that individual
Cepheid ages estimated by using the former set are systematically younger than
those based on the latter one. Moreover, the extent in temperature/color of
the blue loops associated with canonical and overshooting isochrones is
different: the former ones are on average larger. Once again, the morphology
of the blue loops depends on a variety of micro and macro physics phenomena
and we still lack firm theoretical predictions concerning who is doing what.
Note that stellar isochrones were downloaded from the new BASTI-IAC database
(http://basti-iac.oa-abruzzo.inaf.it).

However, in some specific cases, like the comparison between theory and
observations for the MC CCs, the resolution in mass of the BASTI-IAC
evolutionary tracks was improved, and in turn, we also re-computed stellar
isochrones.

###  4.1 Anomalous Cepheids

Figure 14: Left: Comparison between theory and observations in the dereddened
$I_{0},(V-I)_{0}$ CMD for SMC ACs. ACs are marked with light (fundamental) and
dark (first overtone) blue symbols, while static stars with grey dots. Color-
coded lines show stellar isochrones from the BASTI-IAC database at fixed
chemical composition ([Fe/H]$=-2.20$, Y=0.247) and stellar ages ranging from
0.5 Gyr to 2 Gyr (see labelled values). The green line shows the helium-
burning sequence (HeBS) for a more metal-rich ([Fe/H]$=-1.55$, Y=0.248)
chemical composition. Right: Same as the left, but for LMC ACs and more metal-
rich stellar isochrones ([Fe/H]$=-1.90$, Y=0.247).

The right panel of Fig. 14 shows the comparison between theory and
observations for LMC ACs. The LMC ACs and the static stars were dereddened
following the same approach adopted for LMC CCs. The solid lines display
stellar isochrones from the BASTI-IAC database at fixed chemical composition
([Fe/H]$=-1.90$, Y=0.247) and ages ranging from 0.8 to 2 Gyr (see labelled
values). Stellar isochrones are based on evolutionary models constructed
assuming a scaled-solar chemical mixture and by neglecting convective core
overshooting and they were plotted by assuming a true distance modulus of
$\mu=18.477\pm 0.023$ mag (Pietrzyński et al, 2019). A glance at the data
plotted in this panel brings forward two interesting features worth looking
into.

a) Metallicity distribution – The faint tail (m${}_{I}\lesssim 18.7$ mag) of
LMC ACs is quite metal-poor. The current comparison suggests a mean iron
abundance of the order of [Fe/H]=–1.9/–2.0, while the bright tail is less
metal-rich than [Fe/H]$=-1.55$, as suggested by the more metal-rich helium-
burning sequence (green line). The upper limit is quite well known and fixed
by the evidence that helium-burning loci for more metal-rich chemical
compositions attain effective temperatures (colors) that are systematically
hotter (bluer) than the Cepheid instability strip (Monelli and Fiorentino,
2022).

b) Luminosity function – There is evidence that both the I- and the V-band
luminosity function of LMC ACs shows a well-defined minimum, or a zone of
avoidance for m${}_{I}\approx 18.75$ mag. It is not clear whether this feature
might be associated with a difference in their origin (binary merging versus
single star evolution, Fiorentino and Monelli, 2012). The current evidence
indicates that stellar systems mainly dominated by old stellar populations
(Tucana, Cetus, Sculptor, LGS3) host ACs which are on average 1.0–1.5 mag
brighter than RRLs (Monelli and Fiorentino, 2022). Moreover, the bright tail
of the ACs shows up in stellar systems showing multiple star formation
episodes and a well sampled intermediate-age stellar population (MCs). This
working hypothesis requires more quantitative constraints on the possible
difference between faint and bright ACs.

The left panel of Fig. 14 shows the same comparison as the right panel, but
for SMC ACs. The stellar isochrones were computed at fixed chemical
composition ([Fe/H]$=-2.20$, Y=0.247) and the stellar ages are labelled. The
global properties of the SMC ACs appear to be quite similar to LMC ACs. They
only cover a narrower range in magnitudes and FO ACs also cover a narrower
range in period. The comparison between the LMC and the SMC luminosity
function for ACs indicates that the bright tail appears less extended. This
suggests a minor contribution in the younger (more massive) range.

###  4.2 Type II Cepheids

Figure 15: Left: Comparison between theory and observations in the dereddened
$I_{0},(V-I)_{0}$ CMD for SMC TIICs. TIICs are marked with light (fundamental)
green symbols, while static stars with grey dots. Solid lines display selected
HB evolutionary models from the BASTI-IAC database with a $\alpha$-enhanced
chemical composition ([Fe/H]$=-2.50$, Y=0.245) and stellar masses ranging from
$0.51\,M/M_{\odot}$ to $0.60\,M/M_{\odot}$ (see labelled values). The thick
green lines show the Zero Age Horizontal Branch (ZAHB) and the dashed line the
end of core helium burning. Right: Same as the left, but for LMC TIICs and
more metal-rich HB evolutionary models ([Fe/H]$=-1.55$, Y=0.246).

The right panel of Fig. 15 shows the comparison between theory and
observations for the LMC TIICs following the same approach adopted for CCs and
ACs. However, the solid lines display HB evolutionary models computed with an
$\alpha$-enhanced chemical mixture, at fixed chemical composition
([Fe/H]$=-1.55$, Y=0.246) and stellar masses ranging from $0.51\,M/M_{\odot}$
to $0.60\,M/M_{\odot}$ (see labelled values). Predictions plotted in this
panel display that the bulk of TIICs are AGB stars (hydrogen and helium shell
burning). The reader interested in a more detailed discussion concerning the
three different subgroups (BL Herculis, W Virginis, RV Tauri) is referred to
Bono et al (2020b) and to Braga et al (2020). The solid and the dashed green
lines plotted in the same panel show the ZAHB and the core helium exhaustion.
They enclose the region of the CMD, in which the bulk of the RRLs are expected
to be located. Clearly, this is a mere simplification, because LMC RRLs cover
a broad range in metallicity (Gratton et al, 2004; Skowron et al, 2016) and a
significant overlap in the CMD is expected between these two different groups
of variable stars.

The left panel of Fig. 15 shows the same comparison as the right panel, but
for SMC TIICs. The solid lines display similar HB evolutionary models, but
computed by assuming a more metal-poor chemical composition ([Fe/H]$=-2.50$,
Y=0.245). The global agreement is once again quite good over the entire
magnitude and color range.

##  5 Diagnostics to estimate individual Cepheid distances

More than one century ago, Henrietta Leavitt discovered that Cepheids obey to
well-defined Period Luminosity (PL) relations. The key advantage of these
relations is that they are linear over the entire period range. This outcome
applies to optical, NIR and MIR regime.777RRLs obey to PL relations only for
wavelengths longer than the $R$ bands (Bono et al, 2001; Braga et al, 2015) in
the visual band they obey to a mean absolute visual-magnitude metallicity
relation. There are a number of pros and cons in using similar diagnostics in
different wavelength regimes. The reader interested in a detailed empirical
discussion concerning the improvements in moving from the optical to the NIR
regime is referred to the seminal papers by McGonegal et al (1982, 1983) and
McAlary et al (1983). In the following, we will outline the most relevant
cons.

Figure 16: Panel a): dereddened $I$-band PL relations for SMC CCs. From
shorter to longer periods, the different symbols display FO (dark red) and
fundamental (light red) CCs. The solid black lines display the linear fits to
the PL relations (see Table 4). Panel b): Same as the left, but for FO (dark
blue) and FU (light blue) ACs and FU (green) TIICs. The black lines display
the linear fits to the PL relations (see Table 4). Panels c) and d): Same as
the top, but for LMC Cepheids. Note the increase in sample size of TIICs.

i)–Luminosity amplitude– The identification and characterization of regular
variables is significantly easier in the optical regime, because the
luminosity amplitude in the $B$-band is typically a factor of three-to-five
larger than in the NIR and MIR bands. This means that period determination and
mode identification are more straightforward.

ii)–Time series– Optical measurements, thanks to linearity, pixel scale and
the size of current CCDs are less demanding about telescope time than NIR
measurements.

iii)–Limiting magnitude– Current ground- and space-based observing facilities
allow us to identify and characterize CCs and TIICs in the optical regime not
only in Local Group (d$\lesssim$ 1 Mpc), but also in Local Volume (d$\sim$ 25
Mpc) galaxies (Freedman and Madore, 2010; Riess et al, 2021). The limiting
magnitudes in the NIR regime are systematically brighter, but JWST and ELTs
are going to open new paths (Fiorentino et al, 2020). There are also some
indisputable pros.

Figure 17: Top: Optical $I$,$V-I$ PW relations (see Table 5) for SMC CCs
(panel a) and for ACs plus TIICs (panel b). Symbols and color coding are the
same as in Fig. 16. Bottom: Same as the top, but for LMC Cepheids (CCs, panel
c; ACs plus TIICs, panel d).

i)–Reddening correction– Optical bands are more prone to systematics
concerning the reddening correction than NIR bands. The ratio between
selective absorption in the $K$ and in the $V$ band is, according to current
reddening law (Cardelli et al, 1989), of the order of 0.12. This means that
measurements in the $K$-band are roughly one order of magnitude less affected
by reddening uncertainties than the ones in the $V$-band.

ii)–Shape of the light curves– The shape of the light curves in the optical
regime is affected by both temperature and radius variation. This means that
light curves in the optical regime might display, according to the pulsation
period and the luminosity amplitude, sharp rising branches and cuspy
maxima/minima. The shape of the light curves in the NIR and in the MIR regime
are mainly dominated by radius variations and they are typically more
sinusoidal. This is the main reason why the light curve templates provide very
accurate estimates of the mean NIR magnitudes even with a single measurement,
once the period, the luminosity amplitude and the reference epoch are known
(Inno et al, 2015). In case three independent measurements are available, the
mean magnitude can be estimated once the period and the luminosity amplitude
are known (Inno et al, 2015; Braga et al, 2019).

iii)–Slope of the PL relation– The slope of the PL relations becomes
systematically steeper when moving from the optical to the NIR regime. This
means that NIR PL relations are, at fixed pulsation period, more accurate than
optical ones. In this context, it is worth mentioning that the luminosity
amplitudes becomes almost constant at wavelengths longer than the $K$-band,
and the same outcome applies to the slope of the PL relations. The approach to
a constant value is due to the fact that luminosity variations are mainly
dominated by radius variations.

iv)–Width in temperature of the Instability Strip– Radial variables obey to
PLC relation. The use of the PL relation for the estimate of individual
distances rely on the assumption that the width in temperature of the
instability strip can be neglected. This assumption is quite severe in the
optical regime, because optical bands are more affected by a variation in
intrinsic parameters and in metal content when compared with NIR bands. The
physical reason for this variation is mainly in the bolometric correction
(BC). The BC in the optical regime is almost constant when moving from the
blue to the red edge of the instability strip, while in the NIR regime it
increases. This means that regular variables in the NIR become intrinsically
brighter when moving from hotter (shorter periods) to cooler (longer periods)
variables. The consequence of this difference is that the cosmic variance
(standard deviation) of the $V$-band CC PL relation is, on average, a factor
of two to three larger than in the $K$-band (McGonegal et al, 1982; Inno et
al, 2016). This is the reason why individual distances based on $V$-band PL
relations need to be cautiously treated, since they are more affected by
possible systematics affecting the completeness of the sample over the entire
period range covered by the PL relation (Bono et al, 1999a).

###  5.1 Fundamental versus overtone PL relations

The use of overtone variables brings forward several key advantages when
compared with fundamental variables.

i)–Width in temperature of the Instability Strip– Overtones approach a stable
limit cycle in a region of the instability strip that is at least a factor of
two narrower than fundamental variables. This means that the PL relations
provide individual distances that are more accurate than distances based on
fundamental PL relations, because the dependence on the mean temperature is
milder (Bono et al, 1999a).

ii)–Comparison between theory and observations– Overtone variables are
systematically hotter than fundamental variables. This means that predicted
pulsation properties are less prone to uncertainties affecting the treatment
of convective transport.

However, overtones are also affected by two cons.

i)–Identification– Overtone variables are characterized by more sinusoidal
light curves due to the presence of a nodal line (Bono et al, 2000c). This
means that they have on average smaller amplitudes, and in turn, they need
very accurate time series to be identified. Moreover, the shape of their light
curves can be misidentified with eclipsing binaries.

ii)–Absolute calibration– Till a few years ago, the number of CCs pulsating in
the first overtone known in the MW was quite limited. An accurate geometrical
distance was only available for Polaris (Evans et al, 2002). Fortunately, the
current status concerning first overtone Cepheids rapidly improved (Ripepi et
al, 2019; Breuval et al, 2020). It is not clear yet whether the same outcome
applies to Galactic single-mode second overtones.

###  5.2 PL versus PW relations

Almost half a century ago, van den Bergh (1975) and Madore (1982) introduced
the so-called Wesenheit888Wesenheit is a German word used by philosophers and
its meaning is “essence”. magnitude, a pseudo magnitude that is reddening free
by construction. Given two generic photometric bands ($\xi$, $\chi$) it can be
defined as:

W($\chi$,$\xi$-$\chi$)= $\chi$ \- [Aχ / Aξ\- Aχ] $\times$($\xi$-$\chi$)

where the coefficient in square parentheses only depends on the adopted
reddening law. Therefore, if we assume that the reddening law is universal, we
can easily define a pseudo magnitude that is independent of reddening. The use
of the Wesenheit magnitudes has several indisputable advantages.

i)–Individual reddenings– CCs are young stars which trace the spiral arms of
the thin disk (Medina et al, 2021). Moreover, they are quite often still
embedded in the relics of the giant molecular cloud from which they originated
(Genovali et al, 2014). This means that they are either reddened or highly
reddened. This limitation applies not only to Galactic Cepheids, but also to
CCs in nearby galaxies. The use of the Wesenheit magnitudes overcome the
difficulty of individual reddening estimates/measurements.

Data plotted in Fig. 16 and in Fig. 38 (Appendix) display the dereddened $I$\-
and $K$-band PL relation. To overcome the overlap among different groups of
variable stars the top left panel shows the $I$-band PL relations for FO (dark
red) and FU (light red) SMC CCs, while the top right panel shows the PL
relations for SMC ACs (FO, dark blue; FU, light blue) and SMC TIICs (FU,
green). The bottom panels display the same data, but for LMC Cepheids. The
coefficients of the PL relations plotted in this figure are listed in Table 4
(Appendix). A few highlights concerning observed optical-NIR PL relations.

a)– Difference in standard deviation – Optical and NIR PL relations for LMC
Cepheids have, at fixed pulsation period, systematically smaller standard
deviations. The difference is due to the fact that the LMC is almost face-on,
while the SMC is elongated along the line of sight (Inno et al, 2015;
Jacyszyn-Dobrzeniecka et al, 2016, and references therein).

b)– AC number counts – The LMC is roughly one order of magnitude more massive
than the SMC ($\sim$2$\times 10^{11}$ Shipp et al 2021 vs $\sim$3$\times
10^{10}$ M⊙, Besla et al 2016). However, the number of ACs differ at the 20%
level (146 vs 122). The modest difference in the AC number counts is mainly
due to evolutionary effects: the SMC is, on average, more metal poor, and in
turn, the probability to produce ACs increases.

c)– TIICs number counts – The number of SMC TIICs is almost six times smaller
than LMC TIICs. It is worth mentioning that dwarf galaxies, typically, do not
host TIICs. The Sagittarius dSph (Soszyński et al, 2011) and the MCs are
exceptions, and their occurrence seems the consequence of the total mass
and/or of their dynamical evolution (Bono et al, 2020b; Braga et al, 2020;
Neeley et al, 2021). The difference between the number counts of SMC and LMC
TIICs seems to be the consequence of the difference in total mass, and in
particular, in the mass fraction of old stellar populations. The SMC hosts a
single GC (NGC 121), while the LMC hosts more than a dozen GC (van den Bergh,
2006).

ii)–Color information– The Wesenheit magnitude takes into account the color
information. Thus the PW relations are also pseudo PLC relations (Bono et al,
1999b). The main difference is that the color coefficient is fixed by the
adopted reddening law and not as a least squares solution of a three-
dimensional relation among periods, magnitudes and colors. The reason why the
intrinsic dispersion of the PW relations is smaller than the standard
deviations of the associated PL relations is also due to the use of color
information. Periods, magnitude and colors provide a more accurate location of
individual Cepheids inside the instability strip. Figures 17 and 18 show the
key advantage in using optical, NIR and optical–NIR Wesenheit magnitudes for
estimating individual Cepheid distances. The standard deviation decreases, on
average, by a factor of two to 20/30% in the NIR and in the optical-NIR
regime.

iii)–Triple band– Wesenheit magnitudes can be defined on the basis of two
magnitudes, but this means that magnitude and color are correlated. The use of
a Wesenheit pseudo magnitude based on three different bands overcome this
limitation. Moreover, the use of three bands brings forward a key advantage.
Optical/NIR/MIR magnitude and colors have opposite trends concerning the
metallicity dependence. Therefore, appropriate combinations of magnitude and
colors might be, within the errors, either independent of, or minimally
affected by the metal content. There are two key drawbacks in using the
Wesenheit pseudo magnitudes.

a)–Universality of the reddening law– The current empirical and theoretical
evidence indicates that the reddening law changes in stellar systems that
experienced different chemical enrichment histories (Calzetti et al, 2021).
However, the use of either optical-NIR or optical-MIR magnitudes has the key
advantage that the coefficients of the color term are tracing the derivative
of the reddening law rather than “local” value (Marconi et al, 2015; Braga et
al, 2015; Bono et al, 2019).

b)–Accuracy of mean colors– The observing strategy of long-term photometric
surveys is mainly based on a single band photometric survey. This band is used
to identify and characterize the variables. The number of measurements in the
companion band(s) is more limited, and in some cases they do not uniformly
cover the entire pulsation cycle. As a consequence, mean colors are affected
by photometric errors that are slightly larger than the repeatability errors.

####  5.2.1 Highlights concerning observed optical-NIR PL and PW relations

Figure 18: Top: Optical–NIR $K$,$V-K$ PW relations (see Table 5 (Appendix))
for SMC CCs (panel a) and for ACs plus TIICs (panel b). Symbols and color
coding are the same as in Fig. 16. Bottom: Same as the top, but for LMC
Cepheids (CCs, panel c; ACs plus TIICs, panel d).

The current empirical evidence indicates that PL and PW relations for CCs are
linear over the entire period range. However, there is a vast literature
concerning the possible occurrence of non-linearity in PL, Period-Color (PC)
and Amplitude-Color relations of CCs based on mean magnitudes (Ripepi et al,
2022b). The same outcome applies to predicted (Das et al, 2020) and observed
(MW, MC) diagnostics at maximum and minimum light (Kanbur et al, 2007). More
specifically, Ngeow and Kanbur (2005, 2006) by using optical (OGLE, MACHO) and
NIR (2MASS) mean magnitudes found that LMC CCs display either a break or a
non–linearity for periods around 10 days. They both adopted a chi-squared test
and an F-test and found that $V$,$R$,$J$,$H$ PL relations show evidence of
non-linearity for P$\approx$10 days, whereas the K-band PL relation appear to
be consistent with a single-line regression.

A similar investigation was also performed by Subramanian and Subramaniam
(2015) by using optical ($V$,$I$) band photometry provided by OGLE to
investigate the PL relations of SMC Cepheids. They found that optical PL
relations of both FU and FO CCs show a break for periods around 2.95 days and
around 1 day, respectively. These findings were supported and complemented by
Bhardwaj et al (2016b, a) using optical ($V$,$I$; OGLE) and NIR ($J$,$H$,$K$
Macri et al, 2015) photometry and several robust statistical tests. They found
that optical PL, PW and PC relations for LMC FU CCs are non-linear at periods
around 10 days, while the NIR PL and the optical-NIR, triple band PW relations
are non-linear around 18 days. Furthermore, they found that PL, PW and PC
relations for FO CCs display a significant change in the slope for periods
around 2.5 days only at optical wavelengths. More recently, Ripepi et al
(2022b) provided a variety of PL and PW relations for LMC CCs, by using data
collected by the VISTA NIR survey of the MCs, they found for the first time a
break for FO CCs at periods of 0.58 days. The reader interested in a more
detailed discussion addressing theoretical and empirical evidence concerning
breaks in PL relations and multi-phase relations is referred to Kurbah et al
(2023)

A few general considerations in dealing with this wealth of empirical
evidence. Photometric accuracy and completeness are two major issues which can
affect the slope of both PL/PW relations. Fortunately, long-term photometric
surveys (OGLE, Gaia) and space photometry (HST Riess et al, 2019) are
providing complete and accurate samples. However, we still lack accurate
estimates of the star formation episodes during the last 300 Myr in the MCs.
Moreover, spectroscopic surveys are still lagging, and we also lack accurate
estimates of the impact that chemical composition, and in particular iron
abundance, has on the slope of the PL/PW relations. Moreover, Soszyński et al
(2015a) suggested that the cleaning of the sample is a possible source of
systematics in the identification of break(s) in PL relations. He found that a
solid identification of ACs among SMC Cepheids significantly reduces the
evidence of a break for periods around 2.5 days among SMC FU CCs.

This is the case in which the use of the Occam’s razor can help in dealing
with this heuristic hypothesis. This precaution is supported by evolutionary
models: there is agreement on the evidence that the slope of the
mass–luminosity relation does not change, at fixed chemical composition, for
stellar masses typical of CCs (Bono et al, 2000b; Anderson, 2002; De Somma et
al, 2021).

The lively discussion concerning the linearity of PL/PW relations of CCs does
not apply to TIICs, indeed, the RV Tauri (long-period TIICs) appear to be in
the optical regime, at fixed pulsation period, brighter than expected on the
basis of the global PL relation. The difference is still present, but reduced
in the NIR regime and in the $I$,$V-I$ PW relation. This issue has been widely
discussed in the literature (Matsunaga et al, 2006; Soszyński et al, 2018;
Ripepi et al, 2014; Bhardwaj, 2022). In order to overcome these difficulties,
in the estimate of both PL and the PW relations, we only included TIICs with
periods shorter than 20 days.

Interestingly enough, data plotted in Fig. 16 and in Fig. 38 (Appendix)
display that SMC ACs pulsating in the FO appear to be for $\log P\sim-0.30$
(short period tail) systematically fainter than expected, according to the
slope of the PL relation. The same outcome applies to short period FU ACs, but
the difference is smaller. On the other hand, LMC ACs show a well-defined
slope over the entire period range.

The evidence that the change in SMC FO ACs occurs both in optical ($V$,$I$)
and in NIR ($J$,$K$) PL/PW relations further supports that it is intrinsic. In
order to detect the possible occurrence of a change in the slope, we computed
the PL and the PW relations, while neglecting the short period tail, and we
found that the slopes are significantly shallower (see Tables 4 5 (Appendix)
and Figs. 20, 21). The current data do not allow us to constrain whether the
difference is caused by a difference in chemical compositions and/or in the ML
relation.

The difference among optical, NIR and optical-NIR PW relations listed in Table
5 (Appendix) and plotted in Figs. 17 and 18 are mainly due to the adopted
color index, and to the coefficient of the color index. Optical-NIR colors
cover a broad range in wavelength, this means a larger sensitivity in
temperature. The range in $V-K$ color of LMC CCs is, on average, a factor of
2.5 larger than the range in $J-K$ color. The use of optical–NIR magnitudes
provides the unique opportunity to use color coefficients in the Wesenheit
magnitudes smaller than one, therefore limiting possible uncertainties in the
adopted mean colors.

Figure 19: Top: Optical–NIR, triple-band $K$,$V-I$ PW relations (see Table 5
(Appendix)) for SMC CCs (panel a) and for ACs plus TIICs (panel b). Symbols
and color coding are the same as in Fig. 16. Bottom: Same as the top, but for
LMC Cepheids (CCs, panel c; ACs plus TIICs, panel d).

###  5.3 Global trend in the slopes of PL and PW relations

In order to investigate on a more quantitative basis, the use of the triple-
bands in the definition of the Wesenheit pseudo magnitude, Fig. 19 shows the
$K$,$V-I$ PW relations for Cepheids in the Magellanic Clouds. The key
advantage in using this optical-NIR diagnostic is three-fold. a)– The mean
$K$-band magnitude is minimally affected by reddening. b)– The $V-I$ color has
a good sensitivity to the effective temperature. c)– The standard deviation is
slightly smaller when compared with similar optical-NIR PW relations (see
Table 5 (Appendix)).

Figure 20: Top-left: from left to right the slope of the PL relations for SMC
fundamental Cepheids: CCs (panel a), ACs (panel b), TIICs (panel c). The size
of the symbol is proportional to the error on the slope. The error bar shows
the standard deviation of the individual PL relation (see also Table 4
(Appendix)). Black diamonds display our own estimates of the PLs, while red
circles show the estimates by Soszyński et al (2015a), triangles (both blue
and cyan) the estimates by Ripepi et al (2014, 2015, 2017, 2022b). The blue
triangles display the slope of the PL relation for CC(FU) with logP$<$0.47;
the cyan triangles display the slope of the PL relation for CC(FU) with
logP$>$0.47. Top-right same as the top-left, but for SMC first-overtone CCs
(panel d) and ACs (panel e). Bottom-left: same as the top-left, but for the
LMC fundamental CCs (CCs, panel f; ACs, panel g; TIICs, panel h). Bottom-
right: same as the top-right, but for LMC first overtone CCs ((panel i) and
ACs (panel j). According to Ripepi et al (2022b) the blue triangles display
the slope of the PL relation for FO CCs with P$<$0.58 days while the cyan
triangles display slope of the PL relations for FO CCs with P$>$0.58 days.

The current optical (OGLE-IV) and NIR (priority: VISTA (Cioni et al, 2011),
secondary: IRSF (Ita et al, 2004), both converted to the 2MASS system by using
the transformations provided by Kato et al (2007); González-Fernández et al
(2018)) photometric data sets for Magellanic variables allow us to investigate
on a more quantitative basis several interesting features of the PL relations.

Data plotted in panel a) of Fig. 20 show the slopes of both optical ($V,I$)
and NIR ($J,H,K$) PL relations by using the same SMC fundamental variables
plotted in Fig. 16 and in Fig. 38 and listed in Table 4 (Appendix). The size
of the symbols is correlated with the standard deviation of the relations,
while the error bars display the uncertainties on the slopes. The slope, as
expected, steadily increases when moving from optical to NIR bands. The
increase is of the order of 20% in the transition from optical to NIR, and it
becomes negligible in the longer wavelength regime. Therefore the intrinsic
accuracy of K-band PL relation improves because the slope is steeper, and
because they are minimally affected by uncertainties on reddening (Bono et al,
2008).

The agreement with the slopes estimated by (Soszyński et al, 2015a, red
circles) is quite good, since it is based on the same data, but using
different selection criteria. The same outcome applies to the slopes (blue
triangles) of the NIR CC PL relations provided by Ripepi et al (2017, 2022b).
In this case we used the NIR mean magnitudes, based on the fit of the light
curves provided by Ripepi et al (2017, 2022b), together with the mean of the
NIR measurements provided by IRSF (Ita et al, 2004, 2018). The red circles and
the blue triangles have been slightly shifted along the X-axis to avoid the
overlap with symbols used for the current estimates (diamonds).

Panel b) of Fig. 20 shows the optical NIR slopes for SMC FU ACs. The symbols
are the same as in the left panel. The trend when moving from the optical to
the NIR bands is similar to the CCs and the slopes also attain similar values,
whereas the uncertainty on the slope and the standard deviations are
systematically larger (see Table 29 (Appendix)). The uncertainty in the
$H$-band is higher, because in this band fewer measurements have been
collected compared with the $J$ and the $K$-band.

Panel c) of Fig. 20 show the slopes for fundamental SMC TIICs. The trend is,
as expected, similar when moving from the optical to the NIR bands. However,
the slope of the PL relations are, on average, 30–40% shallower than the
slopes of ACs and CCs, while the standard deviations are 1.5–2 times larger.
The reasons for the difference have already been discussed in section 5.2.

Bottom left panels of Fig. 20 display the same slopes of the top panels, but
for LMC Cepheids. The trend is similar when moving from the optical to the NIR
bands as originally suggested by Matsunaga et al (2006, 2009). The blue and
the cyan triangles for CCs (panel f) display the comparison with Ripepi et al
(2017). The two estimates refer to the PL relations they derived for variables
with periods longer/shorter than 4.7 days and they agree, within the errors,
quite well. The agreement with similar estimates in the literature is, once
again, quite good. The similarity in the slope for LMC and SMC FU TIICs was
also suggested by Matsunaga et al (2011). The slopes for both ACs (panel g)
and TIICs (panel h) have larger errors due either to small statistics (ACs),
or to a large dispersion at a fixed pulsation period (TIICs). Moreover, the
slopes of both ACs and TIICs in the NIR bands appear to be either constant, or
display a mild decrease. They need to be cautiously treated, since this is
probably an observational bias caused by the limited accuracy of NIR mean
magnitudes in the short period range (fainter limit) of ACs.

The right panels of Fig. 20 display the same optical, NIR PL relations of the
left panels, but for first overtone CCs and ACs. The trends are similar, but
there are three distinctive features worth being discussed.

a)– The slopes are on average larger, when compared with fundamental variables
(panels d, i), since they range from $\sim$3 to $\sim$3.5. Moreover, the
current estimates are in remarkable agreement with literature estimates.

b)– The standard deviations of both optical and NIR PL relations are, on
average smaller, when compared with the FU CCs. This means that at a fixed
chemical composition and photometric band, the width in temperature of the
instability strip in which FOs are pulsationally stable is systematically
narrower than for FU CCs. These findings indicate that individual distances
based on PL relations for FO CCs are, at fixed chemical composition,
intrinsically more accurate (optical), or with an accuracy similar (NIR) to
individual distances for FU CCs. This evidence further supports predictions
provided by Bono et al (2001) by using pulsation models.

c)–Panel e) of Fig. 20 displays the slopes for SMC FO ACs. The empirical
scenario concerning the slopes of SMC FO ACs is more complex. Indeed, the
slopes based on the entire sample (crosses) display values similar to the
slopes of FO CCs. However, optical and NIR slopes become systematically
shallower once the short period tails are neglected (diamonds). The former
values agree, as expected, quite well with the estimates provided by Soszyński
et al (2015a, red circles). The slope of the $H$-band was not included,
because the number of measurements in this band is either missing (VVV) or
limited (IRSF).

Figure 21:  Top-left: from left to right the slope of optical, NIR and
optical-NIR PWrelations for SMC fundamental Cepheids: CCs (panel a), ACs
(panel b), TIICs (panel c). The symbols are the same as in Fig. 20. Top-right
same as the top-left, but for SMC first-overtone CCs (panel d) and ACs (panel
e). Bottom-left: same as the top-left, but for the LMC fundamental CCs (CCs,
panel f; ACs, panel g; TIICs, panel h). Bottom-right: same as the top-right,
but for LMC first overtone CCs ((panel i) and ACs (panel j).

Figure 21 shows the coefficients of the logarithmic period in both optical,
optical-NIR and NIR PW relations. The key feature of the data plotted in this
figure for both FU and FO Cepheids is that the slope is almost constant even
if the coefficient of the color term decreases by one order of magnitude when
moving from $I$,$V-I$ to $K$,$V-K$ (1.38 versus 0.13). Note once again that
the PW relations are pseudo PLC relations, in which the coefficient of the
color term is only fixed by the reddening law.

The comparison with similar estimates available in the literature is, within
the errors, quite good for the different variable groups and distance
diagnostics. The only difference is with slopes for FO CCs provided by Ripepi
et al (2017), but it is due to the split of the sample in short and long
period sub-sample. Indeed, the slopes they found using the entire sample
(green triangles) agree quite well.

The similarity between the slopes of the PW relations for FU and FO MC CCs is
quite remarkable and applies to optical, optical-NIR and NIR PW relations.
This similarity fully supports the simultaneous use of both FU and FO CCs
(fundamentalization of the periods) to estimate distances.

Finally, the PW relations for ACs including NIR bands need to be cautiously
treated for the same reasons we already mentioned for the NIR PL relations.

###  5.4 PL versus PLC relations

The pulsation relation dictates that the period of a variable depends on the
stellar mass, luminosity and effective temperature. Stellar mass and
luminosity are tightly correlated and provided by evolutionary predictions
through the so-called mass–luminosity relation. This means that pulsation
periods, if we neglect the dependence on the chemical composition, mainly
depend on luminosity and effective temperature. Therefore, an individual
correlation among the physical parameters governing the pulsation properties
of a radial variable requires the knowledge, together with the pulsation
period and the pulsation mode of both mean magnitude and color.

Intermediate-mass stars during core helium burning phases cross from one to
three times the instability strip. In multiple crossings CCs can have the same
effective temperature, and the same radius, and therefore the same luminosity,
but different masses. The consequence is a difference in the mean density, and
in turn in the pulsation period. In order to address this issue on a more
quantitative basis, we investigated the difference in luminosity among
different crossings for chemical compositions representative of MC and MW CCs
(see Fig. 22). Predictions plotted in this figure and listed in Table 6
(Appendix) are based on evolutionary models taking into account mild
convective core overshooting during central hydrogen burning phases and at the
bottom of the convective envelope along the red giant phases, show that the
difference in the mean luminosity across the instability strip between 2nd and
3rd crossing are at most of the order of a few hundredths of a dex. The
difference between 1st and 2nd/3rd is larger and of the order of 0.25–0.30
dex, slightly increasing from low to high metallicities. However, the 1st
crossing plays a marginal role, since the evolutionary time spent along this
crossing is one/two order of magnitude shorter than 2nd+3rd crossing.
According to the mass–luminosity relation provided by Bono et al (2000a) the
quoted difference ($\Delta\log(L/L_{\odot})\leq 0.05$) implies a difference in
stellar mass of $0.08\,M_{\odot}$ for $M=5\,M_{\odot}$. The pulsation relation
for fundamental CCs correlating stellar mass, luminosity and effective
temperature, the so-called van Albada & Baker relation (van Albada and Baker,
1971), provided by Bono et al (2000b, see their Table 6) gives, at fixed
effective temperature, a difference in logarithmic period of 0.06 dex. The
current analysis is suggesting that the spread either in mass or in luminosity
introduced by 2nd and 3rd crossing is negligible.

Figure 22:  Panel a) – Logarithmic stellar mass versus logarithmic luminosity
for the different crossings of the instability strip. Predictions plotted in
this panel have been computed for a scaled solar mixture and at fixed chemical
composition ([Fe/H]=-0.90) representative of CCs in metal–poor disk galaxies.
The three different crossings are plotted with different colors and symbols.
In the less massive regime only the 1st crossing is present, because in these
stellar structures the blue loop does not cross the instability strip. In the
more massive regime, only the 1st crossing is present because these stellar
structures already ignited core helium burning and they cross once the
instability strip. Panel b) – Same as Panel a), but for a chemical composition
representative of LMC CCs. Panel c) – Same as Panel a), but for a chemical
composition representative of SMC CCs. Panel d) – Same as Panel a), but for a
chemical composition representative of MW CCs.

The difficulties in using mean colors have already been discussed in section
2.1. Note that random-phase measurements in different bands introduce
systematic offsets in the mean color. Moreover, PLC relations are more
affected by reddening uncertainties, because they require dereddened
magnitudes and colors. These are the main reasons why individual Cepheid
distances are estimated by using either PL or PW relations.

In passing, it is worth mentioning that, on some occasions, the coefficient of
the color term in optical ($I$, $V$-$I$) and in optical-NIR ($K$,$V-K$; Ripepi
et al, 2017) PLC relations are, within the errors, quite similar to the ratio
between selective absorption coefficient and color excess. It is still not
clear whether this similarity is fortuitous, but the quoted PLC relations are
also minimally affected by uncertainties in reddening uncertainties.

###  5.5 Metallicity dependence

We have already discussed in section 2.1 that the pulsation properties of
Cepheids depend on the metal content. As a consequence, pulsation diagnostics
(PL/PW/PLC relation) adopted to estimate individual Cepheid distances are not
Universal. This working hypothesis implies that not only the slope but also
the zero-point of the adopted relations depend on the metallicity and they
must be properly calibrated.

There are two different paths that we can take in order to analyze the
dependence of Cepheid pulsation properties on the metallicity.

The first path relies on circumstantial empirical evidence. We can use either
Galactic or Magellanic Cepheids to investigate the impact that metallicity has
on individual distance determinations. This approach is still affected by
systematics, because quantitative constraints on the metallicity dependence
require accurate and homogeneous iron abundances and distances. Accurate iron
abundances based on high-resolution spectra are available for a significant
fraction of Galactic Cepheids. However, the current estimates are far from
being homogeneous. Different investigations rely on different line lists and
physical assumptions to estimate physical parameters (effective temperature,
surface gravity, micro-turbulent velocity) and elemental abundances (da Silva
et al, 2022). However, the strong limitation in the estimate of the
metallicity dependence for Galactic Cepheids is the need for accurate (1–2%
level) individual geometrical distances.

Accurate trigonometric parallaxes for a dozen Galactic CCs were originally
provided by HST, and more recently Gaia is significantly increasing the
accuracy and the sample size (Breuval et al, 2021). However, the role that the
current sample of Galactic Cepheids can play to assess the metallicity
dependence is still lively debated. In a recent investigation, Owens et al
(2022) derived new PL relations by using optical ($B$,$V$,$I$), NIR
($J$,$H$,$K$,) and MIR ([3.6], [4.5]) photometric bands for three dozen
Galactic CCs with periods ranging from 4 to 60 days and Gaia trigonometric
parallaxes. The distances they obtain by applying the new PL relations to the
MCs are discrepant when compared with geometrical distances based on detached
eclipsing binaries. Their main conclusion is that the coupling between
systematics affecting Gaia DR3 trigonometric parallaxes and uncertainties on
the metallicity dependence causes the error budget to be at the level of 3%.
The current findings indicate that there are still some limitations in using
bright Galactic CCs close to the saturation limit, but these thorny problems
will be solved in a few years.

In this context, MC Cepheids play a crucial role, because they are located, in
first approximation, at the same distance. This means that they are a perfect
laboratory to study the metallicity dependence. However, the number of
Cepheids for which accurate iron abundances are available, based on high-
resolution spectra, is only limited to less than one hundred LMC Cepheids
(Romaniello et al, 2022). The current estimates indicate a modest spread in
iron abundance. This finding is also supported by a large sample of
metallicity estimates based on the shape of both $V$\- and $I$-band light
curves (Fourier parameters) provided by Hocdé et al (2022). However, the
metallicity distribution of B-type stars (progenitors of Cepheids) indicate a
spread in N abundance of the order of one dex (Trundle et al, 2007; Hunter et
al, 2008, 2009)

Accurate spectroscopic abundances for SMC Cepheids are only available for a
few tens Cepheids and they suggest a modest chemical enrichment (Lemasle et
al, 2017). The same outcome applies to metallicity estimates of SMC CCs based
on photometric indices (Hocdé et al, 2022) and to the metallicity distribution
based on spectroscopy of A- and B-type stars (Venn and Przybilla, 2003;
Trundle and Lennon, 2005; Hunter et al, 2009)

The second path relies on theoretical predictions. This approach is based on
nonlinear, time-dependent convective models of CCs. The current hydro-
dynamical codes allow us to construct several series of pulsation models
covering a broad range in stellar masses, stellar luminosities and effective
temperatures typical of CCs. Moreover, the same calculations can be performed
for chemical compositions typical of both Galactic and Magellanic CCs
pulsating both in the fundamental and in the first overtone. The key advantage
of this approach concerning the metallicity dependence is that pulsation
predictions are used to estimate the difference in metallicity and not their
own metallicities.

The current predictions indicate that the visual magnitude of CCs, at fixed
pulsation period, becomes systematically fainter when moving from metal-poor
to more metal-rich stellar structures.

Empirical evidence based on individual heavy element abundances and on both
Galactic and Magellanic Cepheids covering a broad range in pulsation periods
also indicate that metal-rich CCs are, at a fixed pulsation period, fainter
than metal-poor ones. However, no general consensus has been reached on this
effect. This applies not only to the derivative, the so-called gamma
coefficient, but also to the sign of the coefficient (Storm et al, 2011;
Groenewegen, 2018; Ripepi et al, 2021; Breuval et al, 2022; Trentin et al,
2023). This is the typical long-standing open problem which requires a
significant improvement in the accuracy and in the sample size of the
spectroscopic measurements before we can reach a firm conclusion.

##  6 Cepheids as stellar tracers

###  6.1 High-resolution spectroscopy

Cepheids are typically bright stars, therefore we are dealing with very high
S/N, broad-wavelength range spectra999 In a short communication to the
National Academy of Sciences P.W. Merrill wrote: …Spectrograms …show several
lines of neutral technetium in the spectra of S-type stars especially of long-
period variables. …It is surprising to find an unstable element in the stars.
…S-type stars somehow produce technetium as they go along …(Merrill, 1952b,
see also Merrill (1952a)). These are probably the most dry sentences in the
history of Astronomy. The content of these sentences not only explained the
observations of Tc in AGB stars, but also paved the way to modern nuclear
Astrophysics, to stellar nucleo–synthesis and to the Astrophysical origin of
the Periodic Table.. The precision of the atmospheric parameters and of the
chemical abundances, however, depend not only on the quality of the spectra,
but also on the quality of the adopted line list. With this in mind, a large
portion of the investigation by da Silva et al (2022) was dedicated to finding
the best atomic transitions for iron and $\alpha$–elements that can be
detected in Cepheids. This means i) collecting the most precise transition
parameters, ii) removing absorption lines which are blended with other lines,
and iii) eliminating lines that for any reason deviate significantly from the
average value for their chemical species, or that display a dependence on
effective temperature, surface gravity and micro-turbulent velocity across the
pulsation cycle.

The first point is typically addressed by using a list of hundreds of atomic
transitions for iron and $\alpha$–elements commonly employed in the
literature. Their transition parameters were updated, whenever possible, with
laboratory measurements from Ruffoni et al (2014), Den Hartog et al (2014) and
Belmonte et al (2017) for Fe I, (Den Hartog et al, 2019) for Fe II, (Lawler et
al, 2013) for Ti I, and for Ti II (Wood et al, 2013). The astrophysical, but
homogeneous and precise compilation of Meléndez and Barbuy (2009) for Fe II
lines was also adopted. For the remaining lines, the transition parameters
collected and updated by the National Institute of Standards and Technology
(NIST) Atomic Spectra Database (Kramida et al, 2020) were used. If a line is
not available in any of these sources, it is eliminated from the preliminary
list and not used for the computation of atmospheric parameters, nor chemical
abundances.

The second point was addressed by removing blended lines, using as a reference
the Solar spectrum table by Moore et al (1966) and synthetic spectra. Then, da
Silva et al (2022) measured their equivalent width (EW) using the Automatic
Routine for line Equivalent widths in stellar Spectra (ARES, Sousa et al,
2007, 2015).

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

The reader interested in the final clean lists adopted for the estimate of the
atmospheric parameters and the chemical abundances (quality flag 1) and for
comparison purposes (quality flag 0) is referred to Tables 2 and 3, and to
Appendix A by da Silva et al (2022).

###  6.2 Atmospheric parameters

####  6.2.1 Effective temperature, surface gravity and micro-turbulent
velocity

The approach typically adopted to estimate the atmospheric parameters (i.e.,