###### Abstract

Globular clusters are both primary fossils of galactic evolution and formation
and are ideal laboratories for constraining the evolution of low-mass and
metal-poor stars. RR Lyrae and type II Cepheid variables are low-mass,
radially pulsating stars that trace old-age stellar populations. These stellar
standard candles in globular clusters are crucial for measuring their precise
distances and, in turn, absolute ages, and for the calibration of the
extragalactic distance scale. Herein, the evolutionary stages of RR Lyrae and
type II Cepheids are discussed, and their pulsation properties, including the
light curves, color–magnitude and period–amplitude diagrams, and
period–luminosity relations in globular clusters at optical and infrared
wavelengths are presented. The RR Lyrae visual magnitude–metallicity relation
and the multiband period–luminosity–metallicity relations in globular clusters
covering a wide metallicity range are also discussed in detail for their
application to the RR Lyrae-based distance scale.

###### keywords:

RR Lyrae variable stars; type II Cepheids; globular clusters; stellar
evolution; stellar pulsations; distance indicators

8 2 122 Academic Editor: Mikhail E. Sachkov 11 January 2022 10 February 2022
13 February 2022 https://doi.org/10.3390/ universe8020122 RR Lyrae and Type II
Cepheid Variables in Globular Clusters: Optical and Infrared Properties
Universe special issue on “Recent Advances in Pulsating Stars”, M. Marconi and
V. Ripepi, eds. Anupam Bhardwaj 1,2,†,‡,$ Bhardwaj, A.; EACOA Fellow. IAU
Gruber Foundation Fellow. Marie Skłodowska-Curie Fellow.

##  1 Introduction

Galactic globular clusters (GCs) have traditionally been used as isolated
laboratories for our understanding of many aspects of stellar structure,
evolution, and dynamics, and for constraining models of stellar populations.
These remarkable astronomical objects are compact, gravitationally bound
systems of up to a million stars and host some of the oldest stars. As the
oldest known stellar systems, the Milky Way GCs provide important clues to its
early formation and continuing evolution. In general, GCs are associated with
galaxies of all types and masses and their properties exhibit a strong
correlation with that of their host galaxies (Harris, 1991; Brodie and
Strader, 2006). With ubiquitously old ages comparable to the age of the
Universe, the cluster RR Lyrae (RRL) variables also trace the evolution
history of their host galaxies and can provide important clues to their
formation and evolution.

Variable stars in GCs play an important role in determining their distances
and, thus, constraining their intrinsic properties such as the absolute ages,
which also place a limit on the minimum age of the Universe (Marín-Franch et
al., 2009). The earliest studies of RRL variables were primarily based on GCs.
Following the discovery of variable stars in GCs (Pickering, 1889), Solon
Bailey discovered hundreds of “cluster variables” from the Harvard College
Observatory, and later separated those as subtypes of RRL variables (Bailey,
1902). Among the prototypes of different subclasses of population II Cepheids,
W Virginis was discovered first (Schönfeld, 1866), followed by RV Tauri
(Ceraski, 1905) and BL Herculis (Hoffmeister, 1929). The radial velocity
determinations of Cepheids by Joy (1937) hinted that W Virginis was quite
different from the classical or type I Cepheids of similar periods
(Wallerstein and Cox, 1984). Later, Baade (1944) showed in his pioneering work
that the stellar populations of the galaxies fall into two distinct groups of
solar-neighbourhood-like population I and GC-like population II stars. BL
Herculis, W Virginis, and RV Tauri fall into the latter group, and are now
representative of different subtypes111We exclusively use “BL Her”, “ W Vir”,
and “RV Tau” to refer to the subclasses of type II Cepheids based on their
prototype star—BL Herculis, W Virginis, and RV Tauri, respectively. of type II
Cepheid (T2C) variables.

The modern time-domain surveys have provided a plethora of data for RRL and
T2C variables at optical wavelengths. At the same time, infrared observations
have also increased extensively over the past two decades. High-precision
photometric and spectroscopic data have also revealed that GCs are not simple
stellar populations, and exhibit star-to-star abundance variations in specific
elements (see reviews (Gratton et al., 2012; Bastian and Lardo, 2018)).
However, the horizontal branch (HB) morphology in GCs is mainly determined by
metallicity, and the differences in HB morphologies of similar metallicity GCs
have been attributed to the second parameter problem  (Sandage and
Wallerstein, 1960). While several possible solutions have been proposed for
the second parameter (see (Catelan, 2009) for details), internal helium
variation is one of the main (second) parameters governing the HB morphology
(Milone et al., 2018). Despite the recent progress in observational and
theoretical frameworks, the impact of chemical composition on pulsation
properties and the period–luminosity relations (PLRs) of HB RRL and evolved
T2Cs is not well-understood at all wavelengths.

This review summarizes recent progress in studies of RRL and T2Cs in GCs based
on multiband photometric data. It is beyond the scope of this article to
provide a comprehensive overview of these variables in GCs, and therefore, it
will focus on a comparative study of optical and infrared pulsation properties
of population II variables. We refer readers to several excellent reviews on
RRL and T2C variables available in the literature: (Smith, 1995; Wallerstein,
2002; Catelan et al., 2004; Sandage and Tammann, 2006; Catelan, 2009;
McWilliam, 2011; Beaton et al., 2018; Matsunaga et al., 2018; Bhardwaj, 2020).
This article is organized as follows: the evolutionary and pulsational
scenario for RRL and T2Cs is discussed in Section 2. The optical and near-
infrared (NIR) pulsation properties of RRL and T2Cs in GCs are discussed in
Sections 3 and 4, and a brief summary is provided in Section 5.

##  2 Evolution of RR Lyrae and Type II Cepheids

RRL and T2Cs populate the well-known classical Cepheid instability strip
(IS)—a narrow, nearly vertical strip in temperature in the Hertzsprung–Russell
(HR) diagram. RRL variables are old ($\geq 10$ Gyr), low-mass
($\sim$0.5–0.8$~{}M_{\odot}$), metal-poor stars that are located at the cross-
section of the HB and the IS. RRL stars pulsate primarily in the fundamental
mode (RRab) and first-overtone mode (RRc), with periods in the range between
about 0.2 d and 0.8 d. A few RRL variables are classified as doule mode
pulsators (RRd) because they pulsate simultaneously in the fundamental and
overtone modes (for example, (Sandage et al., 1981; Jurcsik et al., 2015;
Soszyński et al., 2017)). T2Cs also belong to old, low-mass, metal-poor
stellar populations. A preliminary classification of T2Cs is performed based
on their pulsation periods: BL Her ($1\lesssim\\!P<5$ d), W Vir
($5\lesssim\\!P\\!\lesssim 20$ d), and RV Tau ($P>20$ d). Soszyński et al.
(2008) suggested another subtype, peculiar W Vir (pW Vir,
$5\lesssim\\!P\\!\lesssim 10$ d), which are mostly brighter and bluer than W
Vir stars. T2Cs primarily pulsate in the fundamental mode, but BL Her,
pulsating in the first-overtone mode, have also been discovered (Soszyński et
al., 2019). The location of RRL and T2Cs on a typical color–magnitude diagram
of a GC is shown in Figure 1.

RRL are in the core helium-burning evolutionary phase of low-mass stars and
occupy a region in the HR diagram which is at the intersection between the
Cepheid IS and the HB. A low-mass ($\sim$1 M⊙) star spends most of its
lifetime in the core hydrogen-burning main-sequence phase. After the
exhaustion of hydrogen in the core, the inert helium core contracts and the
hydrogen burning occurs in a thin shell surrounding the core. The heating of
the contracting helium core drives the fusion in the hydrogen shell faster;
the star expands as it becomes brighter and redder, moving to the giant
branch. At the red giant phase, the inert helium core continues contracting
and heating, and the fusion of helium starts at the tip of the red giant in a
flash. Once the helium burning in the core becomes the primary source of
energy, the star leaves the giant branch as it becomes fainter and bluer. The
stellar envelope contracts, attaining the thermal and hydrostatic equilibrium
as it reaches the HB phase. The evolution of low-mass stars in the central
helium-burning evolutionary phase is discussed in detail in the books of
Kippenhahn and Weigert (1991), Salaris and Cassisi (2005), and Catelan and
Smith (2015). The zero-age HB (ZAHB) star is characterized by the helium-
burning core and the hydrogen-burning shell surrounding the core. The location
of ZAHB stars on the HB depends on the total mass (or the envelope mass) for a
given helium core mass and envelope composition. For the complex morphology of
the HB, the reader is referred to the review by Catelan (2009, and references
therein). The HB stars have a wide range of effective temperatures, but only
stars with specific initial masses ($\lesssim$0.8 $M_{\odot}$) attain
temperatures that place them within the IS. The blue and the red edge of the
IS of RRL varies between $\sim$7100 K and $\sim$5200 K, respectively, for a
wider range of metal abundances (Marconi et al., 2015). The RRL IS shifts to
cooler temperatures as the metal abundance increases and becomes
systematically fainter for the metal-rich pulsators (Bono et al., 1997;
Marconi et al., 2015).

Figure 1: Color–magnitude diagram for a GC with the approximate positions of
RRL, BL Her, W Vir, and RV Tau indicated. The blue and red HB on either side
of the RRL “gap” are also marked.

T2Cs are in the post-HB evolutionary phase of low-mass stars. After the
exhaustion of core helium, the inert carbon-oxygen core contracts and heats up
in low-mass stars. The helium and hydrogen burning occurs in thin shells
outside the core. The star expands rapidly and becomes cooler and brighter
reaching the asymptotic giant branch (AGB). It suffers shell flashes at the
boundary between the core and the helium region. During these shell flashes,
the radius decreases temporarily, and thus, the star moves to the hotter
temperatures and inside the IS (Gingold, 1976; Wallerstein, 2002). T2Cs evolve
from the blue HB and reach the IS at higher luminosities than those of RRL.
The short-period T2Cs, BL Her, evolve from the HB, bluer than the RRL gap,
towards AGB, while depleting core helium. During this process, they cross the
IS at luminosities corresponding to a pulsation period of 1 to 5 days. The
intermediate-period W Vir evolve to higher luminosities after reaching the AGB
and begin to suffer helium shell flashes. This leads to their temporary
excursions within the IS at luminosities corresponding to stars with periods
between 12 and about 20 days (Wallerstein and Cox, 1984). Once the star
becomes more luminous, mass loss reduces the size of the hydrogen shell and it
passes through IS once again with periods of more than 20 days, representing
the post-AGB evolution of RV Tau variables (Bono et al., 1997; Wallerstein,
2002). However, Groenewegen and Jurkovic (2017a) suggested that the binarity
might be at the origin of the W Vir variables. Similarly, RV Tau stars may
also evolve from the more massive and younger objects, or represent binary
evolution (see (Groenewegen and Jurkovic, 2017b; Manick et al., 2018) for
details).

##  3 RR Lyrae Variables

Optical photometric surveys of GCs have been carried out over the past century
to investigate their variable star population, including RRL and T2Cs (Clement
et al., 2001). While optical time-series data is plentiful, homogeneous NIR
time-series for a statistically significant sample of RRL in a given GC is
limited to few studies, which utilized the increasing infrared observations in
the last decade. For a comparative investigation of the optical and NIR
properties of RRL, the time-series data available in the literature was
complied for GCs, covering a wide range of mean-metallicities. Table 1
summarizes the data along with their references.

Table 1: Source of multiband time-series data for RRL in GCs.

GC | [Fe/H] | _D_ | $\boldsymbol{E_{BV}}$ | $\boldsymbol{E_{JK_{s}}}$ | Optical | $\boldsymbol{N_{\rm{RRL}}}$ | NIR | $\boldsymbol{N_{\rm{RRL}}}$
---|---|---|---|---|---|---|---|---
|  | kpc | mag | mag |  |  |  |
M15 | $-2.33$ | 10.71 | 0.10 | — |  $BVI$ (Corwin et al., 2008) | 89 |  $JK_{s}$ (Bhardwaj et al., 2021) | 129
M53 | $-2.06$ | 18.50 | 0.02 | — |  $VI$ (Arellano Ferro et al., 2012) | 64 |  $JHK_{s}$ (Bhardwaj et al., 2021) | 63
M22 | $-1.70$ | 3.30 | — | 0.18 |  $VI$ (Soszyński et al., 2019) | 28 |  $K_{s}$ (Alonso-García et al., 2021) | 62
$\omega$ Cen | $-1.64$ | 5.43 | 0.11 | — |  $BVI$ (Braga et al., 2016) | 179 |  $JHK_{s}$ (Braga et al., 2018) | 182
M3 | $-1.50$ | 10.18 | 0.01 | — |  $BVI$ (Jurcsik et al., 2015, 2018) | 148 |  $JHK_{s}$ (Bhardwaj et al., 2020) | 228
M5 | $-1.33$ | 7.48 | 0.03 | — |  $VI$ (Arellano Ferro et al., 2016) | 28 |  $JK_{s}$ (Coppola et al., 2011) | 76
M4 | $-1.18$ | 1.85 | 0.35 | — |  $BVI$ (Stetson et al., 2014) | 44 |  $JHK_{s}$ (Stetson et al., 2014) | 44
N6569 | $-0.75$ | 10.53 | — | 0.25 |  $VI$ (Soszyński et al., 2019) | 21 |  $K_{s}$ (Alonso-García et al., 2021) | 11
N6441 | $-0.45$ | 12.73 | — | 0.18 |  $VI$ (Soszyński et al., 2019) | 37 |  $K_{s}$ (Alonso-García et al., 2021) | 23

Notes: The mean-metallicities ([Fe/H]) and the distance ($D$) to GCs are taken
from Carretta et al. (2009) and Baumgardt and Vasiliev (2021), respectively.
The color-excess values ($E_{BV}=E(B-V)$ and $E_{JK_{s}}=E(J-K_{s})$) are
adopted from the references provided with optical and NIR data.

###  3.1 Multiband Light Curves

The light curve structure of variable stars is not only crucial for their
identification and classification but also for gaining valuable information
about the underlying physical processes producing the changes in their
brightness. Figure 2 shows the multiband light curves of RRL in M3. A clear
decrease in the amplitudes of the light curves can be seen moving from optical
to NIR wavelengths. The skewness and acuteness of RRL light curves attain a
value close to unity at infrared wavelengths, implying symmetric sinusoidal
brightness variations. The shape of RRab light curves is asymmetric, with a
steep rise from minima to maxima and a slow decrease of brightness from maxima
to minima at optical wavelengths. Most RRab light curves have a saw-toothed
shape with a bump in the vicinity of the minimum brightness and are easily
identified. For RRab stars, the phase of maximum brightness shifts
significantly to later phases at longer wavelengths. This phase lag in the
maximum brightness between optical and infrared bands is similar to
fundamental-mode classical Cepheids (Madore and Freedman, 1991; Macri et al.,
2015; Bhardwaj et al., 2015), whereas the phase lag is close to zero for
overtone-mode variables. The overtone-mode RRc stars have more symmetric light
curves and often exhibit near-sinusoidal variations, even at optical
wavelengths. In some cases, RRc stars also exhibit a small bump along the
ascending branch. The smaller amplitudes, fainter brightness, and more
sinusoidal light curves make the identification and classifications of RRc
stars relatively more difficult than for RRab stars.

Figure 2: Multiband light curves of RRab (left and middle) and RRc (right)
stars in M3. A monotonic drop in amplitude is seen as a function of increasing
wavelength. The shape of RRab light curves also changes from a saw-tooth shape
at optical wavelengths to more sinusoidal shape in infrared bands. In case of
RRab, a clear variation in the phase of maximum light is also noted, which
shifts toward later phases in infrared bands. The variable star ID and the
pulsation periods from the catalog of Clement et al. (2001) are listed at the
top of each panel.

The stellar pulsation models can successfully predict multiband light and
radial velocity variations, and the light curve morphology depends on the
chemical composition (Bono et al., 2000; Marconi et al., 2015). The
quantification of the light curve structure based on Fourier decomposition
(Simon and Lee, 1981) or principal component analysis (Deb and Singh, 2009)
has been used for a rigorous comparison between observations and theory and to
provide constraints for the stellar pulsation models (Simon and Teays, 1982;
Bono et al., 1996; Marconi et al., 2005; Das et al., 2018). The light curve
parameters of RRL stars have been used to obtain photometric metallicity
measurements based on an empirical relation correlating period, Fourier phase
parameter ($\phi_{31}$), and [Fe/H] (Jurcsik and Kovacs, 1996; Nemec et al.,
2013). The application of machine-learning methods on theoretical and
observational data have also shown that the light curve parameters play an
important role in predicting the physical parameters of RRL variables
(Bellinger et al., 2020). The light curves shown in Figure 2 display stable
and periodic variations, but a significant fraction of RRL also exhibit the
Blazhko effect (Blažko, 1907). The Blazhko effect results in (multi-)periodic
modulations of the light curves on a timescale that is much longer than the
primary pulsation period. The Blazhko effect is not well understood, despite a
number of investigations that are even based on unprecedentedly high-precision
photometry from Kepler and TESS (Jurcsik et al., 2009; Kolenberg et al., 2010;
Szabó et al., 2010; Skarka et al., 2020; Molnár et al., 2022). The space-based
photometry has also revealed several low-amplitude additional modes and
dynamical effects in the pulsation, including period doubling of the
fundamental mode (Szabó et al., 2010; Molnár et al., 2022). While a
significant fraction of RRL stars in GCs show modulations (Arellano Ferro et
al., 2012; Jurcsik, 2019), we will refrain from a detailed discussion of the
Blazhko effect in this article.

###  3.2 Color–Magnitude Diagrams

Figures 3 and 4 show color–magnitude diagrams for M4 ([Fe/H]$\sim$$-$1.18),
M3([Fe/H]$\sim$$-$1.50), $\omega$ Cen ([Fe/H]$\sim$$-$1.64), and M53
([Fe/H]$\sim$$-$2.06) GCs in optical and NIR bands, respectively. The variable
RRL population is also shown in colored symbols. These color–magnitude
diagrams are not corrected for extinction; note that there is significant and
differential reddening towards M4 and $\omega$ Cen (Hendricks et al., 2012;
Calamida et al., 2005).

Figure 3: Optical color–magnitude diagrams for GCs. RRL variables are shown in
red symbols.

The horizontal branch morphology is quite interesting and complex, and has
been discussed extensively in several excellent reviews in the literature
((Smith, 1995; Bono et al., 2003; Catelan, 2009) and references therein). The
main components of the HB include blue HB, variable HB (RRL gap), red HB, and
the red clump (Catelan, 2009). The red HB stars are commonly present in metal-
rich and relatively young GCs. These occupy the horizontal part of the HB,
unlike the more massive and significantly brighter red clump stars. Figure 3
shows red HB stars in metal-rich M4 and M3, with the latter GC being more
populated. A smaller fraction of red HB stars can also be found in the more
metal-poor GCs, for example, as a result of the evolution of blue HB or RRL
variables to the right of the color–magnitude diagrams. The variable HB that
includes RRL variables is located in the region of the HB that crosses the
Cepheid IS. In typical color–magnitude diagrams based on single-epoch data,
this region appears as a “gap” because long-time monitoring is required to
obtain the mean magnitudes of variable stars and to properly place those on
the color–magnitude diagrams. However, the color–magnitude diagrams shown in
Figures 3 and in 4 are based on long-term photometric data, and therefore, the
variable HB is distinctly evident.

Figure 4: NIR color–magnitude diagrams for GCs. RRL variables are shown in red
symbols.

The blue HB consists of stars hotter than the RRL IS but is far from being
horizontal at the blue end. The blue HB includes a canonical, “horizontal”
part and a nearly “vertical” component which ends in a “blue HB tail”. Figure
3 and 4 show a distinctly clear blue HB for metal-poor GCs. The hottest stars
on the HB have temperatures above 20,000 K, and are referred to as extreme or
extended HB stars. These hotter HB stars are less luminous than the cooler
blue HB stars and may also end up with a “blue hook” of extremely hot stars.
Some studies have also suggested that the canonical blue HB and the blue HB
tail are separated by a “gap” (see (Catelan, 2009) for details).

Figure 5 shows a zoom-in of the RRL color–magnitude diagrams in four different
GCs. The magnitudes and colors are corrected for extinction using the
classical reddening law of Cardelli et al. (1989). Theoretically predicted
boundaries of the fundamental red edge and the first-overtone blue edge of the
IS are based on the empirical relations provided by Marconi et al. (2015).
Theoretically predicted magnitudes were converted to apparent magnitudes using
the distance to the GC in the respective studies (Braga et al., 2015, 2018;
Bhardwaj, 2020; Bhardwaj et al., 2021). As expected, RRc are located towards
the blue (hot) edge of the IS, while the RRab are towards the red (cool) edge
of the IS. The red edge of the IS is consistent in both optical and NIR
color–magnitude diagrams. The predicted blue edge appears to be redder in the
case of M4 and $\omega$ Cen in NIR data, but is consistent in optical
color–magnitude diagrams. While same cluster reddening is adopted for all RRL,
there is evidence of differential reddening in both these clusters. The
agreement between predicted boundaries and empirical IS is reasonably good if
we take into account the basic assumptions in pulsation models. The predicted
boundaries of the IS were estimated for a fixed composition and the pulsation
models also have a resolution of 50K in effective temperature. The differences
can be partly attributed to the observed metallicity spread in $\omega$ Cen
(Magurno et al., 2019). Note that the predicted boundaries of the IS are
metal-dependent in optical, but are derived to be independent of metallicity
in NIR.

Figure 5: Close-up of optical and NIR color–magnitude diagrams for RRL in
different GCs. Red circles and blue asterisk show RRab and RRc, respectively.
The theoretically predicted red and blue edges of the IS are also shown from
Marconi et al. (2015).

###  3.3 The Bailey Diagrams and the Oosterhoff Dichotomy

The pioneering work of Bailey on variable stars in GCs initiated the
investigations on period–amplitude diagrams for their RRL stars (Bailey,
1902). The Bailey diagrams, or the period–amplitude diagrams for RRL differ
from cluster to cluster, and these differences can be associated with the
Oosterhoff type (Oosterhoff, 1939) of the GC. The Oosterhoff dichotomy is one
of the long-standing problems in modern astronomy which shows that the
galactic GCs can be separated into two distinct groups according to the mean
period of their RRab variables. The RRab in Oosterhoff I (OoI) clusters have a
shorter mean period of 0.55 days, and are relatively metal-rich ([Fe/H]
$\gtrsim$$-1.5$ dex), while RRab in Oosterhoff II (OoII) have a longer mean
period of 0.65 days and [Fe/H] $\lesssim$$-1.5$ dex (Oosterhoff, 1939; Smith,
1995; Catelan, 2009; Catelan and Smith, 2015).

The left panel of Figure 6 displays the Oosterhoff dichotomy in the Galactic
GCs. These clusters are separated in two distinct groups of OoI and OoII, and
a gap is seen in the mean period around 0.6 days. However, a few metal-rich
clusters (NGC 6441, NGC 6388, [Fe/H] $>$$-$0.7 dex) have a longer value for
the mean-period of RRab (0.7 days) and are often called OoIII clusters.
Initially, the dichotomy was attributed to a dichotomy in the “transition
period” between RRab and RRc stars, which reflects a difference in the
effective temperature at the transition point (van Albada and Baker, 1973). It
was suggested that the temperature difference arising from the “hysteresis
effect” (van Albada and Baker, 1973) is responsible for the Oosterhoff
dichotomy. Sandage (1958), using a period–mean density equation, estimated
that the $V$-band HB magnitudes differ by 0.2 mag between OoII and OoI
clusters, such that the OoII clusters are brighter. Sandage et al. (1981)
quantified a period-shift at a fixed temperature and suggested that the
Oosterhoff dichotomy can be explained as the difference in the intrinsic
luminosity for the RRL in two clusters. Renzini (1983) suggested that, in
addition to Sandage period-shifts, the non-monotonic behavior of HB morphology
with decreasing metallicity can also contribute to the Oosterhoff dichotomy.
Despite several investigations on Oosterhoff dichotomy, a general consensus
has not yet been reached (see (Catelan, 2009)). The GCs and dwarf galaxies in
the Milky Way satellite systems do not show such a dichotomy (see (Catelan,
2009) for more details). This suggests that the Oosterhoff dichotomy is
specific to galactic GCs and can provide useful insights into the Milky Way’s
evolution history.

Figure 6: Left: Oosterhoff dichotomy in GCs as shown by plotting the mean
period of RRab stars versus [Fe/H] for Galactic GCs. The shaded region
represents the Oosterhoff gap separating the two Oosterhoff types. Two metal-
rich GCs (NGC 6441 and NGC 6388) are also shown as Oosterhoff III clusters.
Right: Period–amplitude diagram for RRab in Oosterhoff I (M3) and II ($\omega$
Cen) clusters. For a fixed composition, RRL models from Marconi et al. (2015);
Marconi and Minniti (2018) with different helium abundances, masses, and
luminosities are also overplotted.

The right panel of Figure 6 shows a period–amplitude diagram of RRab variables
in M3 and $\omega$ Cen. RRL in OoII clusters have longer periods than those in
OoI clusters at a fixed amplitude. Most observational period–amplitude
diagrams of RRab exhibit scatter in both OoI and OoII clusters. If the light
curves are well-sampled, the photometric uncertainties on amplitudes are
typically small (<0.05 mag). The Blazhko effect in RRL, which can change mean
amplitudes by more than $90\%$ (see Figure 4 of (Jurkovic, 2018)), is the
dominant source of scatter in the observational period–amplitude diagrams. In
the case of $\omega$ Cen, the metallicity spread of RRL can also contribute to
this scatter. RRL in OoI M3 also show a small fraction of OoII RRL (Cacciari
et al., 2005), presumably because the mean metallicity of the GC is nearly at
the boundary between OoI and OoII. The RRL models computed at fixed metal
contents (Z = 0.004) and primordial helium contents of Y=0.25 and Y=0.30 are
also shown in the left panel of Figure 6 (Marconi et al., 2015; Marconi and
Minniti, 2018; Das et al., 2018). The helium-enhancement results in a
systematic shift in periods, which primarily occurs due to increased
luminosity levels for similar masses (see (Rood, 1973; Sweigart and Catelan,
1998; Marconi et al., 2018) and references therein). Therefore, Bailey
diagrams can be used to constrain the helium content of stars in a GC. Marconi
and Minniti (2018) estimated the helium-abundance (Y = 0.245) of the RRL
population in the Milky Way bulge by comparing their minimum periods derived
from stellar pulsation models.

Figure 7: Optical and NIR period–amplitude diagrams for RRL in GCs of
different mean-metallicities. The solid and dashed lines represent loci of OoI
and OoII RRab, respectively.

Figure 7 shows optical and NIR period–amplitude diagrams for RRL stars in GCs
of different mean-metallicities. The amplitudes of RRab stars decrease as a
function of pulsation period, while the amplitudes for RRc seem to rise first
and then fall with increasing period. The solid and dashed lines in the $B$\-
and $J$-band represent the approximate loci of the OoI and OoII RRab in M3
from Cacciari et al. (2005) and Bhardwaj et al. (2020), respectively. The loci
in the $I$\- and $K_{s}$-bands are overplotted by scaling the $B$\- and
$J$-band loci with corresponding amplitude ratios (Braga et al., 2018;
Bhardwaj et al., 2020). Most RRab in M4 fall on the locus of RRab stars while
the majority of RRab in metal-poor M15 are located on the OoII locus. RRab in
M3 seem to have both OoI and OoII sequences, with the former being clearly
more populated. Infrared Bailey diagrams seem to exhibit larger scatter due to
smaller amplitudes, which exhibit larger uncertainties because of the sparsely
sampled light curves. In the case of $\omega$ Cen, there is more scatter,
presumably resulting from intrinsic metallicity spread in RRL stars (Sollima
et al., 2006; Magurno et al., 2019). While the impact of metallicity on Bailey
diagrams is debated (Bono et al., 2007), period–amplitude–[Fe/H] relations
have been used to determine metallicities for RRab stars (see, (Kinemuchi et
al., 2006; Kunder and Chaboyer, 2009)). The period–amplitude–[Fe/H] relations
suggest a continuous correlation between period and both the luminosity and
metallicity for RRL in GCs.

In metal-rich OoIII NGC 6441, RRab do not fall clearly into either of the OoI
or OoII groups in Figure 7, but are closer to the OoII locus, similar to
$\omega$ Cen. Even though NGC 6441 is metal-rich, previous studies of its RRab
have shown that they have period–amplitude diagrams similar to those in OoII
GCs (Corwin et al., 2006). Braga et al. (2020) showed that the long-period
RRab in $\omega$ Cen follows the locus of OoIII RRab, similar to NGC 6441
variables. Rich et al. (1997) first discovered that metal-rich NGC 6441 and
NGC 6388 not only have expected red HB but also blue HB components that extend
through the IS. These GCs are also suggested to host more than a single
stellar population, with the possibly helium-enhanced smaller population
responsible for the blue HB component (McWilliam and Zoccali, 2010; Bellini et
al., 2013). In metal-poor clusters such as M15, there is considerable scatter
in Bailey diagrams where a significant fraction of RRab fall between the two
Oosterhoff-type sequences (e.g., (Corwin et al., 2008)). While there is some
correlation between the location of Bailey diagrams and [Fe/H], several
authors have suggested that the amplitude at a fixed period is a function of
the Oosterhoff type, more than that of the metal abundance (e.g., (Bono et
al., 2007; Kunder and Chaboyer, 2009)). The importance of Oosterhoff type
becomes evident, for example, in OoI (M3) and OoII (M2) clusters, where the
difference of mean-metallicities is small but M2 RRab are significantly
shifted to longer periods than those in M3 (see Figure 4, p. 22 in (McWilliam
and Zoccali, 2010)).

Optical and NIR amplitude ratios have also been used as diagnostic tools for
Oosterhoff dichotomy in RRab in GCs. Braga et al. (2018) found that the
optical–NIR and NIR amplitude ratios for long-period ($\log(P)>0.7$ days) RRab
in $\omega$ Cen are systematically larger than the short-period RRab
variables. Bhardwaj et al. (2020) studied optical and NIR amplitude ratios in
M3 and found similar trends. The authors found a break period at
($\log(P)>0.6$ days), such that the amplitude ratios for the longer-period
RRab are significantly larger. Median values of amplitude ratios are
consistent between the two OoI and OoII clusters. The increase in amplitude
ratios occur at a smaller period for RRab in M3 than those in $\omega$ Cen,
and the difference in the break periods between the two clusters is similar to
the difference in mean periods of their RRab stars. Therefore, amplitude
ratios may also be indicators of the Oosterhoff type of the cluster.
Furthermore, amplitude ratios are also important for template light curves of
RRL, to correct for amplitudes when those are known only at a particular
wavelength (Braga et al., 2019).

###  3.4 The Visual Magnitude–Metallicity Relation

The absolute visual magnitudes of RRL stars have important astrophysical
implications, in particular to determine distance using the
magnitude–metallicity relation. Sandage et al. (1981); Sandage (1982)
correlated the location of RRL in the period–amplitude diagrams with both the
absolute magnitude and [Fe/H], and determined the empirical relations between
the period-shift and metallicity. From the observed period-shifts between
clusters, Sandage (1982) derived an empirical relation between the luminosity
and metallicity: $\Delta M_{bol}=0.35$[Fe/H]. The luminosity–metallicity
relation between RRL $V$-band absolute magnitude ($M_{V}$) and the metallicity
([Fe/H]) is expressed as $M_{V}=\alpha+\beta\mathrm{[Fe/H]}$, where the slope
($\beta$) and the zero-point ($\alpha$) have been determined through several
empirical and theoretical investigations in the literature ((Lee et al., 1990;
Fernley et al., 1998; Caputo et al., 2000; Clementini et al., 2003; Bono et
al., 2003; Muraveva et al., 2018), and references therein). The calibration of
the magnitude–metallicity relation for RRL in the LMC (Clementini et al.,
2003) and the Milky Way (Muraveva et al., 2018) are provided below:

$\displaystyle V_{0}$ $\displaystyle=19.064(\pm 0.017)+0.214(\pm
0.047)\mathrm{[Fe/H]},$ $\displaystyle M_{V}$ $\displaystyle=1.17(\pm
0.04)+0.34(\pm 0.03)\mathrm{[Fe/H]}.$

The zero-point in the first relation above includes the distance to LMC in the
case of apparent magnitude ($V_{0}$). The absolute calibration ($M_{V}$) has
been derived by Muraveva et al. (2018) using Gaia parallaxes for field RRL.
While the magnitude–metallicity relation is simple and straightforward in the
sense that it requires only two observables—the apparent visual magnitude and
the metallicity—its calibration is affected by deceptive systematic errors,
which leads to a range of metallicity coefficients (slopes) from 0.37 to 0.13
mag dex-1 (Muraveva et al., 2018). Several studies have also provided
theoretical and empirical evidences that the $M_{V}-$[Fe/H] relation is not
linear over the entire metallicity range covered by the GCs, and proposed a
quadratic relation (Caputo et al., 2000; Bono et al., 2003; Catelan et al.,
2004; Bono et al., 2007). The shallower slope for metal-poor systems becomes
significantly steeper for metal-rich ([Fe/H] $>$$-$1.5) RRL stars. However,
this departure from linearity is not observed for LMC RRL stars over a broad
metallicity range (Clementini et al., 2003), which indicates the difficulty in
disentangling the metallicity and evolutionary effects in the observational
magnitude–metallicity relation. With Gaia parallaxes, Muraveva et al. (2018)
found that the coefficient of the quadratic metallicity term is not
significant, and that the zero-points are consistent between both linear and
quadratic versions.

The magnitude–metallicity relation relies on the assumption that RRL stars are
on the ZAHB, but both the field and cluster RRL exhibit a spread in
luminosity. Therefore, the theoretical calibrations based on ZAHB models do
not account for evolutionary effects (Catelan et al., 2004; Bono et al.,
2007), while the empirical calibrations use both ZAHB RRL and evolved RRL
where the latter have higher luminosities (Beaton et al., 2018). The intrinsic
luminosity width of the ZAHB becomes larger when moving from metal-poor to
metal-rich GCs. Therefore, RRL with different metal abundances are affected by
the evolutionary effects of both off-ZAHB evolution and the HB morphology
(Caputo, 1998). One of the sources of systematics on empirical
magnitude–metallicity relations is the reddening uncertainties which are
significant, i.e., a small uncertainty of 0.01 mag in reddening values implies
an error of 0.03 mag in the visual magnitude. Furthermore, the variation in
the initial helium content, which may depend on environment, can also affect
the visual magnitudes of RRL variables (Beaton et al., 2018).

###  3.5 Multiband Period–Luminosity Relations

The pulsation theory predicts a period–luminosity–color relation for classical
pulsating stars such as RRL and Cepheid variables. While Cepheids follow well-
defined PLRs at both visual and infrared wavelengths, the dependence of RRL
luminosity on pulsation period is negligible at visual wavelengths. Unlike
Cepheids, which cover a wide range of luminosity within the IS, RRL have much
smaller variation in the luminosity within the HB phase of low-mass stars.
Given that the luminosity variations for RRL are much more modest, the
temperature range within the IS also plays an important role in determining
the range of RRL periods. Therefore, RRL stars exhibit a PLR only if the
bolometric corrections are significant and are such that the absolute
magnitudes cover a large range, as is the case at longer wavelengths. Indeed,
the bolometric correction’s sensitivity to effective temperature becomes
significant only at wavelengths longer than the $R$-band (Catelan et al.,
2004; Marconi et al., 2015), resulting in a well-defined PLR.

Figure 8: $I$-band PLRs for RRL in GCs of different mean-metallicities. Solid
lines display a universal slope of PLR relation to all GCs while dashed lines
represent $\pm 3\sigma$ scatter in the underlying relation. Mean-metallicities
for each cluster are also indicated in each panel. Red circles and blue
squares represent RRab and (fundamentalized) RRc stars, respectively. Grey
symbols represent $\pm 3\sigma$ outliers.

The pioneering work of Longmore et al. (1986) showed that RRL stars follow a
very tight PLR in the $K$-band. Using single-epoch $K$-band photometry in
three GCs (M3, M5, and M107), Longmore et al. (1986) found that the PLRs
exhibit remarkably small scatter as compared to optical bands. The infrared
PLRs come naturally from the pulsation equation because bolometric corrections
increase with effective temperature, such that redder RRL are brighter in the
$K$-band. NIR observations of RRL in the earliest studies on PLRs were limited
to a few epochs or sparsely sampled light curves (e.g., (Longmore et al.,
1990; Nemec et al., 1994; Butler, 2003)). With the advent of modern infrared
detectors, several empirical investigations have been carried out on infrared
PLRs for RRL stars in GCs ((Coppola et al., 2011; Braga et al., 2015; Neeley
et al., 2017; Braga et al., 2018; Bhardwaj, 2020; Bhardwaj et al., 2021), and
references therein).

We compared PLRs for RRL in GCs with different mean-metallicities for which
time-series optical and NIR data is available in the literature. The time-
series photometry is critical to obtain precise and accurate mean-magnitudes
for well-constrained PLRs. Figure 8 shows $I$-band PLR for RRL in GCs covering
a wide metallicity range of $\Delta$[Fe/H]$\sim$2.0 dex. Under the assumption
of the universality of RRL PLRs in GCs, we solved for a common slope and
variable zero-point in different GCs. Note that the periods of overtone-mode
RRc variables were fundamentalized using $\log(P_{RRab})=\log(P_{RRc})+0.127$
(Petersen, 1991; Coppola et al., 2015) to obtain a global (RRab+RRc) sample of
RRL. Outliers beyond $\pm 3\sigma$ were removed recursively until convergence.
Optical mean-magnitudes are based on long-term photometric light curves and
are, therefore, very accurate and precise. However, there are possible
systematics in the data adopted from different literature studies.
Nevertheless, a common slope of the PLR fits the data for all GCs very well,
regardless of their metallicity, thus validating the assumption of
universality of RRL PLRs.

Figure 9: $K_{s}$-band PLRs for RRL in GCs of different mean-metallicities.
Solid lines display a universal slope of PLR relation to all GCs, while dashed
lines represent $\pm 3\sigma$ scatter in the underlying relation. Mean-
metallicities for each cluster are also indicated in each panel. Red circles
and blue squares represent RRab and (fundamentalized) RRc stars, respectively.
Grey symbols represent $\pm 3\sigma$ outliers.

NIR photometric light curves are much more sparsely sampled than the optical
light curves (see Figure 2). Therefore, NIR templates for RRL become very
useful for determining accurate mean-magnitudes and deriving precise PLRs.
Given that the NIR light curves are much more symmetric and near-sinusoidal in
shape than the optical light curves, a simple sine-series fit is the simplest
template that can be adopted to estimate mean-magnitudes and peak-to-peak
amplitudes. While sine series is a good approximation for RRc light curves,
the RRab light curves exhibit clear evidence of asymmetry for different
periods (see Figure 2). Hajdu et al. (2018) used principal component analysis
to generate $J$ and $H$-band templates from well-sampled $K_{s}$-band light
curves of RRL in the VISTA VVV survey (Minniti et al., 2010). Similarly, Braga
et al. (2019) provided new $JHK_{s}$ template light curves of RRL variables
and showed that the mean magnitudes can be estimated with an accuracy of
$2\%$, even from single-epoch NIR observations. Figure 9 displays $K_{s}$-band
PLRs for RRL in different GCs. The mean-magnitudes for RRL in these GCs are
based on template-fits to $K_{s}$-band light curves. The significantly smaller
scatter, as compared to the optical bands, is distinctly evident in the PLRs.
Even in the $\omega$ Cen, which exhibits a spread in metallicity, the
$K_{s}$-band PLR does not exhibit large intrinsic dispersion (Navarrete et
al., 2015; Braga et al., 2018).

Similar to NIR bands, mid-infrared (MIR) observations also have indisputable
advantages with respect to shorter wavelengths because the extinction is more
than an order of magnitude smaller ($A_{V}$$\sim$$15A_{3.6\upmu\rm m}$) at the
3.6-$\upmu$m band. The luminosity variations are also small and mostly
insensitive to effective temperatures. A few studies utilized increasing MIR
observations, in particular, from the Wide Field Infrared Survey Explorer
(WISE) and the Spitzer Space Telescope to investigate MIR PLRs for field RRL
(Klein et al., 2011; Madore et al., 2013; Klein et al., 2014) and cluster RRL
(Dambis et al., 2014; Neeley et al., 2017; Muraveva et al., 2018). The
calibration of MIR PLRs of RRL in 15 GCs was first provided by Dambis et al.
(2014) using WISE data. Depending on the calibration based on either
statistical or trigonometric parallaxes, Dambis et al. (2014) found two
significantly different estimates of the absolute zero-point of PLRs. Together
with a theoretical framework, Neeley et al. (2017) employed Spitzer
observations of RRL in M4 to derive precise PLRs with a dispersion of $\sim
0.05$ mag in 3.6-$\upmu$m and 4.5-$\upmu$m bands as part of the Carnegie RR
Lyrae program. The authors showed that the metallicity spread introduces a
dispersion of 0.13 mag in the PLR if the sample includes a range of iron
abundances. However, the scatter in MIR PLRs is reduced to 0.02 mag when the
metallicity term is included, implying a percent-level precise distance
determination using RRL stars.

###  3.6 Period–Luminosity–Metallicity Relations

Theoretical studies on RRL predicted a strong dependence of PLRs on metal
abundance at all wavelengths. Bono et al. (2001) derived a theoretical
$K$-band PLZ relation and showed that the dependence of RRL luminosity on
metallicity ($\sim$0.17 mag/dex) is relatively smaller than that in the
optical bands (>0.2 mag/dex). Additionally, the uncertainties on the mass and
luminosity also do not effect the PLRs significantly at infrared wavelengths.
However, Sollima et al. (2006) derived the empirical
period–luminosity–metallicity (PLZ) relation in the $K_{s}$-band for RRL in
GCs and found a metallicity coefficient ($\sim$0.08 mag/dex) nearly two-times
smaller than the theoretical predictions of Bono et al. (2001). Catelan et al.
(2004) also derived metal-dependent PLRs for RRL based on synthetic HB models
and found a significant metallicity term (0.21–0.17 mag/dex) in $IJHK_{s}$
bands. Using stellar pulsation models, Marconi et al. (2015) generated RRL
models for a wide range of metal abundances (Z = 0.02 to 0.0001) at
$UBVRIJHK_{s}$ wavelengths. They also found a strong metallicity dependence
that varies from $\sim$0.14–0.19 mag/dex depending on the subclass of RRL and
the wavelength under consideration. Recently, Bhardwaj et al. (2021)
investigated the $K_{s}$-band PLZ relation for RRL in GCs and found a
metallicity coefficient consistent with theoretical predictions. The PLZ
relations for RRL are written in the following form:

$M=\alpha+\beta\log(P)+\gamma[Fe/H],\\\ $ (1)

where $M$ is the absolute magnitude, and $\alpha$, $\beta$, and $\gamma$ are
the slope, zero-point, and the metallicity coefficient, respectively.

Figure 10 displays the $I$-band PLZ relation for RRL in GCs. We used the mean-
magnitudes derived from the time-series data to obtain an accurate and precise
relative quantification of the metallicity coefficient. The mean-metallicity
for these GCs is taken from Carretta et al. (2009) for consistency with the
theoretical models of Marconi et al. (2015). Note that one main source of
systematics is the independent distances to GCs. As shown in Figure 8, we can
solve for a universal slope and separate the zero-point for each GC
simultaneously. For the absolute quantification of metallicity effects and the
zero-point, we adopted theoretical absolute magnitudes and the absolute
magnitudes based on parallaxes of field RRL from Gaia EDR3 (Lindegren et al.,
2021), as discussed in detail in Bhardwaj et al. (2021). The empirical PLZ
relation in the $I$-band is shown in Figure 10. The top panel displays clear
trend in RRL magnitudes as a function of metallicity, such that the metal-poor
RRL are brighter. When the metallicity term is taken into account, the
residuals of the PLZ relation, as shown in the bottom panel of Figure 10, do
not exhibit any trend with the metallicity.

Figure 10: Top: Combined $I$-band PLRs for RRL in different GCs. The absolute
magnitudes for RRL in GCs were determined based on the zero-point calibration
using Gaia EDR3 parallaxes. The color code in the legend corresponds to the
mean metallicity of GC. Bottom: residuals of the PLZ relation in the
$K_{s}$-band for RRL stars. Note the strong metallicity trend in the PLRs in
the top panel, which disappears in the residuals of the PLZ relation.

The empirical PLZ relation in the $I$-band derived in this work and the
theoretical PLZ relation from Marconi et al. (2015) are provided below:

$\displaystyle\textrm{M}_{I_{GC}}$
$\displaystyle=0.05-1.38\log(P)+0.21\textrm{[Fe/H]}~{}~{}~{}(\sigma=0.07),$
$\displaystyle\textrm{M}_{I_{TH}}$
$\displaystyle=-0.07-1.53\log(P)+0.17\textrm{[Fe/H]}~{}~{}~{}(\sigma=0.09).$
(2)

Here, the uncertainties on the coefficients are small, and therefore, the
difference in the slope of the empirical and predicted PLZ relation is
statistically significant. The slope of the empirical PLZ relation is
shallower than the theoretical relations, and the metallicity coefficients are
relatively larger than the theoretical predictions. This difference in the
slope of the empirical PLZ relation could possibly be due to the bulge GCs,
which seem to show shallower PLR than the more metal-poor clusters. Note that
the extinction in the bulge clusters is large at optical wavelengths, and the
reddening law also deviates from the standard Cardelli et al. (1989) law.
Moreover, the optical data is taken from several different studies in the
literature, and there could be possible systematic uncertainties due to the
inhomogeneous sample of optical photometry in GCs.

Figure 11 displays the $K_{s}$-band PLZ relation for RRL in GCs. The empirical
relation, together with the theoretical $K$-band relation from Marconi et al.
(2015), is:

$\displaystyle\textrm{M}_{K_{GC}}$
$\displaystyle=-0.86-2.35\log(P)+0.17\textrm{[Fe/H]}~{}~{}~{}(\sigma=0.04),$
$\displaystyle\textrm{M}_{K_{TH}}$
$\displaystyle=-0.82-2.25\log(P)+0.18\textrm{[Fe/H]}~{}~{}~{}(\sigma=0.04).$
(3)

Figure 11: Same as in Figure 10, but in the $K_{s}$-band.

The uncertainties in the coefficients of the $K_{s}$-band PLZ relations are
marginal, given the tight relation ($\sigma\sim$0.04 mag) for RRL in the GCs.
Note that all $K_{s}$-band photometry is homogenized in the 2MASS photometric
system. The empirical and theoretical calibrations of PLZ relation are in
excellent agreement in the $K_{s}$-band. The absolute calibration of the RRL
PLZ relation using Gaia EDR3 parallaxes also results in a zero-point
consistent with theoretical relations. While we did not discuss the
application of NIR PLZ relations for RRL to individual stellar systems,
several investigations have been carried out in the literature to obtain
precise and accurate distances to GCs, e.g., M92 (Del Principe et al., 2005),
M5 (Coppola et al., 2011), $\omega$ cen (Navarrete et al., 2015; Braga et al.,
2018), M4 (Braga et al., 2015), M3 (Bhardwaj et al., 2020), and M53 (Bhardwaj
et al., 2021).

In addition to PLZs for RRL, Wesenheit magnitudes (Madore, 1982; Bhardwaj et
al., 2016) have also been used to derive period–Wesenheit–metallicity (PWZ)
relations. Wesenheit magnitudes are, by construction, independent of
reddening, and therefore, two-color photometry allows for accurate distance
determinations using PWZ relations by minimizing extinction uncertainties.
Marconi et al. (2015) presented theoretical optical and NIR PWZ relations
using stellar pulsation models. The authors found that the optical Wesenheit,
using a combination of $B$\- and $V$-bands, is almost independent of
metallicity, but other Wesenheit relations exhibit relatively strong
metallicity dependence. The application of PWZ relations to obtain a precise
distance to $\omega$ Cen has been discussed in detail in Braga et al. (2016).

The absolute calibration of RRL PLZ and PWZ relations is critical for
providing classical Cepheid-independent calibrations of the extragalactic
distance scale for the determination of the Hubble constant (Beaton et al.,
2018). The Hubble constant measurements in the late and early Universe are
currently in tension, implying that the Universe is expanding faster than the
predictions of the standard cosmological model (see (Verde et al., 2019; Riess
et al., 2021; Freedman, 2021) for details). This Hubble tension, if confirmed,
could have significant consequences for both fundamental physics and modern
cosmology. Therefore, independent extragalactic distance scales based on
population I and population II distance indicators are crucial for exploring
residual systematics in the local Hubble constant measurements. RRL and the
Tip of the Red Giant Branch (TRGB) stars are population II distance indicators
that can be applied to galaxies of any Hubble type and of any inclination, and
they are found in less-extincted and less-crowded environments, unlike young
population I Cepheids (Freedman et al., 2020). Given the recent progress in
observational and theoretical frameworks, the stage is set for fully
exploiting population II distance indicators. RRL, despite being fainter than
Cepheids and TRGB, can be used to complement and test TRGB distances within
and beyond the local group. The GC RRL variables can be used to determine
their accurate distances and calibrate independent secondary distance
indicators such as the GC luminosity function. The improvement of RRL as
distance indicators will not only enable morphological studies of the Milky
Way and nearby galaxies, but also boost their application as stellar
population tracers of age, metallicity, and the extinction of their host
galaxies. The RRL distance scale will reach its full potential with upcoming
major observational facilities, which will predominantly operate in the
infrared regime where these standard candles become more precise and powerful.

##  4 Type II Cepheids

The discovery of T2Cs was a crucial step in the modernization of the
extragalactic distance scale in the 1950s, when it was suggested that the PLRs
of T2Cs are different than those of classical Cepheids (Baade, 1958). Since
then, T2Cs have received far less attention than classical Cepheids because
the former are fainter and typically less abundant. In the catalogue of
Clement et al. (2001), there are about hundred known candidate T2Cs in the
GCs. More than 300 T2Cs are known in the Magellanic Clouds, but the number is
nearly thirty-times smaller than that of known classical Cepheids (Soszyński
et al., 2018, 2015). However, T2Cs are very useful tracers in old stellar
systems such as the Galactic bulge (Soszyński et al., 2017), where population
I classical Cepheids are not present. Soszyński et al. (2017) found more than
900 T2Cs in the bulge. T2Cs are known to have evolved from the blue HB in GCs,
and therefore, they are likely to be found in stellar systems where blue HB is
well populated.

Most GCs host, at most, a couple of T2Cs; thus, homogeneous optical and NIR
photometry is not available for a statistically significant number of cluster
variables in the literature. Only three clusters (NGC 6388, NGC 6441, and
$\omega$ Cen) host seven or more T2Cs, with NGC 6388 known to host 10 T2C
candidates (Clement et al., 2001). Nemec et al. (1994) provided a compilation
of optical photometry of 40 GC T2Cs and derived their PLRs. Matsunaga et al.
(2006) used homogeneous NIR time-series for 46 T2Cs in 26 GCs to derive
$JHK_{s}$ PLRs. While Pritzl et al. (2003) added T2Cs in NGC 6388 and NGC 6441
to an optical sample, Braga et al. (2020) provided an updated sample of T2Cs
with NIR photometry by adding variables in $\omega$ Cen. We combined and
updated the optical and NIR dataset of T2Cs from these studies to obtain a
sample of 60 variables in GCs for a comparative study of their pulsation
properties. The distances, mean-metallicities, and the reddening values in
each GC were also compiled from the aforementioned studies. The mean-
magnitudes and colors of T2Cs were also corrected for extinction using the
reddening law of Cardelli et al. (1989).

###  4.1 Multiband Light Curves

T2Cs are known to exhibit complex light curve variations depending on their
subtypes, periods, and stellar environment. Figure 12 displays example light
curves of a BL Her, W Vir, and RV Tau variable in $\omega$ Cen. A clear
decrease in amplitude and a small phase lag from optical to NIR is evident in
these light curves. The optical light curves of BL Her show a sharp rise from
minima to maxima and a shallower decreasing branch similar to RRab stars (see
Figure 2). This saw-tooth RRab-like characteristic is associated with the
above HB variables originally suggested by (Diethelm, 1983) for short-period
(<3 days) T2Cs. NIR light curves are also more symmetric in $HK_{s}$-bands.
T2Cs in the bulge display secondary bumps on the descending branch for the
shortest periods, which moves to earlier phases, and to the ascending branch
for longer periods (Sandage et al., 1994; Soszyński et al., 2018). Note that
this is similar to the so-called “Hertzsprung Progression” in classical
Cepheids (Hertzsprung, 1926; Bhardwaj et al., 2015, 2017). This bump feature
disappears for longer-period ($P>3$ days) BL Her, which show a plateau or flat
maxima (e.g., V48 in Figure B.1 of (Braga et al., 2020)).

Figure 12: Multiband light curves of T2Cs in $\omega$ Cen from Braga et al.
(2020). Each panel displays example light curves of different subclasses of
T2Cs—Left: BL Her; Middle: W Vir; Right: RV Tau. The variable star ID and the
pulsation periods are listed on the top of each panel.

The multiband light curves of a W Vir T2C, shown in Figure 12, are near-
sinusoidal, in particular for wavelengths longer than $V$. The variations in
the light curves occur systematically, which can change amplitudes in
different pulsation cycles. For shorter-period ($5<P<8$ days) W Vir, the rise
time from minima to maxima is faster, but the light curves are more symmetric
for longer-period variables. For longer-period W Vir in the bulge and
Magellanic Clouds, a bump prior to the maxima is commonly seen, as first
noticed by Diethelm (1983). Another class of peculiar W Vir was also proposed
by Soszynski et al. (2008) in the Magellanic Clouds, which are brighter and
bluer than W Vir. The light curves of peculiar W Vir exhibit significant
scatter due to variations of periods and additional modulations, suggesting
their binary evolution.

The light curves of RV Tau are known to exhibit alternating deep and shallow
minima. However, the light curves of V1, shown in the right panel of Figure
12, do not display this feature, much like the other RV Tau variables in GCs
(Zsoldos, 1998). The alternative minima is typical of field RV Tauri, and is
also common in bulge and Magellanic Cloud variables. The occurrence of deep
and shallow minima also interchanges in some RV Tau variables. Furthermore,
the maxima can also exhibit alternations such that higher maxima follow deeper
minima. The light curves of RV Tau exhibit significant scatter due to their
irregular brightness variations from cycle to cycle, and this scatter
increases for longer-period RV Tau stars.

###  4.2 Color–Magnitude and Period–Amplitude Diagrams

Figure 13 displays optical and NIR color–magnitude diagrams of T2Cs after
correcting for distance and extinction. The short-period BL Her are slightly
bluer than the longer period T2Cs, as these evolve from the blue HB towards
the AGB and cross the IS while depleting helium in the core. The brighter W
Vir are considered to make temporary excursions through IS from cooler
temperatures, when they suffer shell flashes after reaching AGB (Wallerstein,
2002). However, these excursions are not expected to occur commonly, as
suggested by the theoretical models (Smolec, 2016; Bono et al., 2016). Note
that there are fewer T2Cs in optical color–magnitude diagrams because our
sample lacks quality data for long-period RV Tau. Long-term photometric
monitoring is essential to properly estimate the mean pulsation properties for
these variables.

Figure 13: Optical and NIR color–magnitude diagrams of different subtypes of
T2Cs variables. BLH—BL Her, WVI—W Vir, and RVT—RV Tau. The reference
color–magnitude diagram (small/black data points) is of M3, after correcting
for distance and extinction.

Figure 14 shows period–amplitude diagrams for T2Cs in GCs at optical and NIR
wavelengths. For a relative comparison, we have also shown amplitudes of T2Cs
in the LMC and the Galactic bulge. The optical amplitudes of BL Her show a
marginal increase with period for up to 3 days, albeit with large scatter. The
amplitudes of BL Her display an evidence of a decrease between 3 and 7 days.
This suggests that these above HB variables display a wide range of light
curve morphologies, even within a narrow period range. The amplitudes of W Vir
increase sharply as a function of period between 5 and 20 days, while the
amplitudes of RV Tau ($P>20$ days) decrease with increasing period. In the NIR
bands, a significantly smaller scatter is seen in amplitudes for BL Her
variables with periods less than 3 days. In this period range, the amplitudes
seem to increase almost linearly as a function of period in the $K_{s}$-band.
Similar trends to the optical bands are seen for W Vir and RV Tau at longer
wavelengths. Braga et al. (2020) investigated amplitude ratios for T2Cs and
found a small but significant dependence on pulsation period, unlike RRL
stars, which seem to have constant amplitude ratios for RRc and RRab stars
(Braga et al., 2018; Bhardwaj et al., 2020).

Figure 14: Optical and NIR period–amplitude diagrams for T2Cs in GCs. The grey
symbols are amplitudes of T2Cs in the LMC (left panels) and the Galactic bulge
(right panels).

###  4.3 Period–Luminosity Relations

From the earlier studies of T2Cs in GCs based on photographic plates, it was
known that these variables follow a PLR in optical bands (Harris, 1985).
Optical PLRs were also derived separately for short- and long-period T2Cs with
a break at 10 days (McNamara, 1995), and two PLRs were even fitted to T2Cs in
GCs, assuming their pulsations both in fundamental and overtone modes (Nemec
et al., 1994). However, modern data from time-domain surveys, in particular
those in the Magellanic Clouds, revealed that T2Cs follow solid, optical band
PLRs (Alcock et al., 1998; Majaess et al., 2009; Soszyński et al., 2008).
Using high-precision HST photometric data for the two metal-rich clusters NGC
6388 and NGC 6441, which host most T2Cs, Pritzl et al. (2003) derived well-
defined and tight PLRs for T2Cs in $BVI$-bands.

There is empirical evidence that T2Cs follow tight PLRs in GCs at NIR
wavelengths (Matsunaga et al., 2006) similar to RRL stars. Matsunaga et al.
(2006) used NIR light curves of 46 T2Cs in 21 GCs to derive a linear PLR for
BL Her, W Vir, and RV Tau variables. They used distances to individual GCs
based on HB–metallicity relations, which also showed a consistency between the
RRL and T2C distance scale. However, RV Tau variables in the Magellanic Clouds
appear to be brighter than the PLR followed by the BL Her and W Vir stars
(Ripepi et al., 2015). Bhardwaj et al. (2017a) also found an evidence of a
linear relation for all T2Cs in the LMC. NIR observations of BL Her and W Vir
in the bulge have also been used to investigate their PLRs and determine a
distance to the galactic center (Bhardwaj et al., 2017b; Braga et al., 2018).

Figure 15 displays optical and NIR PLRs for T2Cs in GCs. A single slope is
fitted to all three subclasses of T2Cs. The best-fitted PLRs are provided
below:

$\displaystyle\textrm{M}_{B}$
$\displaystyle=~{}~{}~{}0.23~{}(0.13)-1.64~{}(0.13)\log(P)~{}~{}~{}(\sigma=0.38,N=39)$
$\displaystyle\textrm{M}_{V}$
$\displaystyle=-0.02~{}(0.10)-1.90~{}(0.09)\log(P)~{}~{}~{}(\sigma=0.28,N=36)$
$\displaystyle\textrm{M}_{I}$
$\displaystyle=-0.54~{}(0.07)-2.14~{}(0.07)\log(P)~{}~{}~{}(\sigma=0.22,N=42)$
$\displaystyle\textrm{M}_{J}$
$\displaystyle=-0.88~{}(0.06)-2.23~{}(0.05)\log(P)~{}~{}~{}(\sigma=0.18,N=50)$
$\displaystyle\textrm{M}_{H}$
$\displaystyle=-1.11~{}(0.06)-2.35~{}(0.05)\log(P)~{}~{}~{}(\sigma=0.17,N=48)$
$\displaystyle\textrm{M}_{K_{S}}$
$\displaystyle=-1.11~{}(0.05)-2.41~{}(0.04)\log(P)~{}~{}~{}(\sigma=0.15,N=50)$

Figure 15: Optical and NIR PLRs of T2Cs in GCs. Dashed lines represent best-
fitting linear regression. Dotted lines display $\pm 2\sigma$ scatter in the
best-fitting relation. Grey plus symbols represent outlier variables.

It is evident that the PLRs exhibit consistently smaller scatter at longer
wavelengths. The slope and zero-point of optical $BVI$ PLRs in GCs are
statistically consistent with those derived in Nemec et al. (1994) and Pritzl
et al. (2003). The NIR PLRs are similar to those in Matsunaga et al. (2006)
because the T2C sample is the same with the addition of four more T2Cs (three
in $\omega$ Cen and one in M15). In Figure 15, RV Tau seem to fall
systematically on the brighter end of the best-fitting relation at optical
wavelengths. However, there is no such evidence in the NIR data.

There is theoretical and empirical evidence that the PLRs are minimally
affected by the metal-abundances, in contrast to RRL stars. Theoretical
predictions of Di Criscienzo et al. (2007) and Marconi and Di Criscienzo
(2007) suggested that the metallicity effects are of the order of $0.05$
mag/dex on T2C PLRs. Das et al. (2021) also computed BL Her models for a wide
range of metal-abundances and derived multiband PLRs, which were found to be
independent of metallicity. Matsunaga et al. (2006) also found a negligible
metallicity dependence on T2C PLRs at NIR wavelengths. We did not find any
correlation of the residuals of PLRs with metallicity for T2Cs, implying a
negligible dependence on metallicity. Braga et al. (2020) used T2C PLRs in
$\omega$ Cen to obtain a distance to the cluster that is consistent with
independent methods, including those based on RRL stars. Since T2Cs and RRL
follow the same PLRs in NIR bands, their combined PLRs can be used as an
alternative to classical Cepheids for providing the calibration of the first
rung of the extragalactic distance scale.

##  5 Summary

The article provides a review of recent advancements in the studies of the
optical and NIR pulsation properties of RRL and T2Cs in GCs. While the modern
time-domain surveys provide a vast amount of data at optical wavelengths, the
$JHK_{s}$ time-series is still limited to only a few GCs, covering a
statistically significant sample of RRL. Moreover, the crowded nature of GCs
makes it difficult to obtain homogeneous photometry, even from time-domain
surveys. We summarized light curve variations, color–magnitude and
period–amplitude diagrams, and provided an empirical calibration of RRL and
T2C PLRs at multiple wavelengths. The evolutionary scenario for RRL is well-
understood, but the evolution of T2Cs, in particular, W Vir and RV Tau, is
still debated in the literature. The pulsation properties of both RRL and T2Cs
have now been explored in great detail at both the optical and NIR bands,
which further strengthens their case for being potential primary distance
indicators for the extragalactic distance scale. However, there still remain
open questions related to the Oosterhoff dichotomy and the second parameter
for the HB morphology in GCs. While some progress has been made in quantifying
the metallicity effects on RRL and T2C PLRs, their absolute calibration is
also being addressed with parallaxes from the Gaia mission. Our understanding
of the impact of composition, metallicity, and helium effects in particular,
on RRL and T2C pulsation properties will benefit from ongoing and upcoming
large spectroscopic surveys. These cluster variables have played significant
roles in our understanding of the evolution of stars and our Galaxy, and will
continue enabling new insights into the mysteries of the Universe.

This research received no external funding.

Not applicable

Not applicable

Not applicable

The authors declare no conflict of interest.

[custom] References

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