The stellar photosphere-hydrogen ionization front interaction in Classical Pulsators: a theoretical explanation for observed period-colour relations

RR Lyraes, classical Cepheids and Type II Cepheids (T2Cs) are well-known classical pulsators in different evolutionary stages. While RR Lyraes are horizontal-branch stars with helium-burning cores (Preston, 1964), classical Cepheids are located on the blue loops in the HR diagram (Anderson et al., 2016). T2Cs are old, low-mass (${\sim 0.5-0.6\;\text{M}_{\odot}}$), metal-poor pulsating variables with typical periods of about $1$ to $50$ days (Wallerstein, 2002; Sandage & Tammann, 2006). T2Cs are fainter than the classical Cepheids but brighter than the RR Lyrae stars. They have traditionally been divided into three subclasses: BL Her stars with periods between $1-4$ days, W Vir stars with periods ranging from $4-20$ days and RV Tau stars with periods longer than $20$ days (Soszyński et al., 2018). We note here that RV Tau variables are post AGB stars with typical masses of $\sim 0.7M_{\odot}$ (Tuchman et al., 1993; Fokin, 1994), higher than those for BL Her and W Vir stars. In addition, there is a fourth subclass, the peculiar W Vir (pW Vir) stars which are brighter and bluer than the W Vir stars (Soszyński et al., 2008). All these classical pulsating stars follow well-defined Period-Luminosity relations (PLRs) which make them excellent distance indicators (Muraveva et al., 2015; Bhardwaj et al., 2016a; Bhardwaj et al., 2017a; Ripepi et al., 2017; Subramanian et al., 2017; Beaton & Carnegie-Chicago Hubble Program Team, 2018).

The PCAC relations for classical Cepheids and RR Lyraes have been extensively studied to understand the hydrodynamics in their outer envelopes (Simon et al., 1993; Kanbur et al., 2004; Bhardwaj et al., 2014; Ngeow et al., 2017; Das et al., 2018). RR Lyrae stars are on the Horizontal Branch (HB) and evolve towards higher luminosities and cooler temperatures in the HR diagram. However, depending on the morphology of the evolutionary tracks, RR Lyraes can evolve bluewards from a relatively red zero-age HB before turning redwards towards the AGB. Their period-colour (PC) relations have a particular structure: shallow/steep sloped relations at minimum/maximum light respectively. Classical Cepheids lie in a region of higher luminosities and cooler temperatures on the HR diagram with respect to the RR Lyrae stars: their PC relations at minimum (steep slope) and maximum (shallow slope) light are the opposite of RR Lyraes. Kanbur et al. (2004); Bhardwaj et al. (2014); Ngeow et al. (2017) and references therein contend that the changes in the PCAC relations are caused by an interaction of the stellar photosphere and hydrogen ionization front (HIF) in the outer envelope. The photosphere is taken as being at optical depth ${\tau}=2/3$ and the HIF is defined where the majority of hydrogen becomes ionized. However, PCAC relations for T2Cs at the maximum and minimum light have not been studied previously.

The classical picture of the theory of stellar evolution suggests that the progenitors of T2Cs evolve redward from the blue edge of the HB and may move up the Asymptotic Giant Branch (AGB) going through the BL Her-type and W Vir-type pulsation stages (Gingold, 1985; Wallerstein, 2002). However, recent work by Gezer et al. (2015); Groenewegen & Jurkovic (2017a, b); Manick et al. (2017) among others questions this classical picture and the origin and evolution of T2Cs is not as well-understood as was once thought to be. Iwanek et al. (2018) confirmed BL Her stars to be as old as RR Lyraes while W Vir stars could be a mixture of old and intermediate-age stars. While Groenewegen & Jurkovic (2017a) found that BL Her stars may be explained by low-mass stars ($\sim 0.5-0.6M_{\odot}$) evolving off the zero-age HB, they state that the evolutionary status of W Vir stars is unclear. This is because for the same luminosity, W Vir stars have longer periods and thus, lower masses than their short period classical Cepheid counterparts. However, evolutionary tracks of $\sim 2.5-4M_{\odot}$ stars cross the W Vir region in the HR diagram. Groenewegen & Jurkovic (2017a) suggested that the origin of W Vir stars may have some relation to binarity.

From the theoretical perspective, several linear and nonlinear convective models for RR Lyraes, Cepheids and T2Cs have been computed to study their pulsation properties and PLRs (Fiorentino et al., 2007; Marconi & Di Criscienzo, 2007; Smolec et al., 2012; Marconi et al., 2013; Marconi et al., 2015; Bono et al., 2016; Smolec, 2016). In the case of Cepheids and RR Lyraes, their light curve structures have also been compared with observations (Bhardwaj et al., 2017b; Das et al., 2018). For T2Cs, Marconi & Di Criscienzo (2007); Smolec (2016) have employed non-linear models to investigate their pulsation modes. While they pulsate primarily in the fundamental mode, double-mode and overtone-mode T2Cs have also been found by Smolec et al. (2018) and Soszyński et al. (2019), respectively. Recently, Modules for Experiments in Stellar Astrophysics (MESA, Paxton et al., 2011; Paxton et al., 2013, 2015, 2018, 2019) incorporated radial stellar pulsation modules which can be used to generate light curves of these classical pulsators. Our aim is to compare the observed PCAC relations for RR Lyraes, classical Cepheids and T2Cs with a self-consistent set of pulsation models and relate the empirical variations to changes in the envelope structure and evolutionary states of these variables.

There are several reasons for comparing observed and theoretical PCAC relations. Firstly, PC relations at mean light are important in a wider astrophysical context because through the period-luminosity-colour (PLC) relations, they influence the PLRs which is vital for the distance scale and non-CMB estimates of Hubble’s constant (Beaton et al., 2016; Riess et al., 2016; Riess et al., 2019). Traditionally, PC relations are studied at mean light (Tammann et al., 2003) but these relations are the average of the corresponding relations at various phase points during a pulsation cycle. Therefore in order to fully understand the physics behind these relations at mean light, it is important to study such relations during the pulsation cycle. Previous work has found small nonlinearities in the Cepheid PLRs (Bhardwaj et al., 2016b) which may occur due to the significant variations in the PC relation at maximum and minimum light. Further, observed PCAC relations can be compared to theoretical PCAC relations obtained from extensive grids of theoretical pulsation models to constrain global stellar parameters (Bellinger et al., 2019) and hence constrain theories of stellar pulsation and evolution.

Here we focus on PC relations at maximum and minimum light and corresponding AC relations (Bhardwaj et al., 2014; Das et al., 2018, and references therein) because the behaviour at these phases has been shown to be strongly influenced by the interaction between the HIF and the stellar photosphere. In previous work, Simon et al. (1993); Kanbur (1995); Kanbur & Phillips (1996) investigated how such interactions can strongly influence the behavior of PC relations at maximum (classical Cepheids) and minimum light (RR Lyraes). Later Kanbur et al. (2004); Kanbur & Ngeow (2006); Kanbur et al. (2007) modelled this interaction using 1D radiation hydrodynamic models of Cepheids and RR Lyraes. These authors found that the stellar photosphere and the HIF are not always co-moving during a pulsation cycle. When the two are “engaged” (that is, the photosphere lies at the base of the HIF or the distance between them is small) at low temperatures, the temperature of the photosphere takes on the properties of the temperature at which hydrogen ionizes. From Saha ionization equilibrium theory, this temperature is independent of density at low temperatures, leading to a PC relation that is flatter or more independent of global stellar pulsation properties. This is the situation, for example, for Cepheids at maximum light. At minimum light for Cepheids, the photosphere and the HIF are not engaged and the temperature of the photosphere is dependent on global stellar properties - hence we have a sloped PC relation. In the case of RR Lyraes, the photosphere and the HIF are engaged through the pulsation cycle (Kanbur, 1995; Kanbur & Phillips, 1996). However the interaction is at low temperatures at minimum light and hence RR Lyraes exhibit a flat PC relation at minimum light. At maximum light, this interaction is at higher temperatures and the temperature at which hydrogen ionizes or the temperature of the stellar photosphere is strongly dependent on temperature and global stellar properties. Hence we have a definite slope for RRab PC relations at maximum light but a flat PC slope at minimum light.

One measure of “engagement” between the photosphere and HIF is the distance between them in mass coordinates (Kanbur et al., 2007, and references therein). In this work, we study how this distance varies across classical Cepheids, RR Lyraes and T2Cs and across pulsation phases. We emphasize here that for the purpose of this paper, the locations of different types of variable stars on the HR diagram and thus, in different evolutionary states determine the relative location of HIF and photosphere. A higher $L/M$ ratio and/or a cooler effective temperature indicates greater distance between the HIF and the stellar photosphere (Kanbur, 1995; Kanbur & Phillips, 1996). W Vir stars lie in the same region of the HR diagram as classical Cepheids. The HIF-stellar photosphere interaction theory (Kanbur, 1995; Kanbur & Phillips, 1996) implies that as a consequence of stellar evolution and because of similar locations of W Vir stars and classical Cepheids on the HR diagram, the PC relation obeyed by W Vir stars should be similar to that for classical Cepheids. We also note that previous work has found some statistically significant differences in PCAC relations between the Galaxy, LMC and SMC, though the general nature (that is, flat PC relation at maximum/minimum for Cepheids/RR Lyraes) is preserved. Hence is is important to understand the effect of metallicity with a view to using such observed PC relations at multiple phases to constrain models. It is in this context that we study the PCAC properties of T2Cs in the Bulge and the Magellanic Clouds for the very first time, to the best of our knowledge using the best available data and pulsation codes and compare with results for Cepheids and RR Lyraes. Hence this project aims to find direct observational evidence supporting the HIF-stellar photosphere interaction in the outer envelopes of variable stars and to probe the PC relation across a broad spectrum of classical pulsating stars in different evolutionary stages.

The structure of this paper is as follows: The theoretical motivation of this project is discussed in Section 2 while Section 3 describes the observational data used in this analysis and the Fourier decomposition technique. The extinction-corrected period-colour and amplitude-colour relations are briefly discussed in Section 4. We study the distance between the HIF and the stellar photosphere of RR Lyrae, BL Her and classical Cepheid models computed using MESA in Section 5. Finally, we summarise the results of this study in Section 6.

Kanbur (1995) found that the HIF was further out in the mass distribution in RR Lyrae stars as compared to classical Cepheids, i.e., the HIF is closer to the surface of the star with a higher effective temperature. In fact, Kanbur (1995) noted that the HIF lies further in the mass distribution as $L/M$ increases and as $T_{\text{eff}}$ decreases. While this result may be expected because Cepheids are cooler than RR Lyraes, it is nevertheless instructive and important to be able to demonstrate it both computationally and theoretically. To physically justify this statement, let $\tau$ be the optical depth in terms of the Rosseland mean opacity. Then the temperature $T$ in the stellar envelope can be approximated by

where $T_{\text{eff}}$ is the effective temperature and $q(\tau)$ is the Hopf function, a monotonically increasing function of $\tau$ (e.g., Hopf, 1930). We may consider $T_{\text{eff}}$ to be constant at $\sim 10000$K at the base of the HIF. Since $q(\tau)$ increases monotonically with $\tau$, the opacity of the HIF increases with decreasing effective temperature. Therefore, the HIF lies farther from the surface when the star is cooler.

To investigate and contextualize this within the stellar evolution theory, we used MESA to calculate the evolution of a star with an initial mass of $0.83\;\text{M}_{\odot}$ and initial composition of ${Y=0.25,\;Z=0.001}$ and study the relative location of the HIF and the stellar photosphere as it moves off the horizontal branch. We used the Grevesse & Sauval (1998, GS98) mixture for heavy-mass elements, nuclear reaction rates from the Nuclear Astrophysics Compilation of Reaction Rates (NACRE, Angulo et al., 1999), and the Opacity Project at Livermore (OPAL) equation of state and type-two opacities (Iglesias & Rogers, 1996; Rogers & Nayfonov, 2002). We treated convection using Böhm-Vitense (1958) mixing-length theory with a solar-calibrated mixing length (${\alpha_{\text{MLT}}\simeq 1.84}$). We calculated mass loss using Reimers’ prescription with a mass loss rate of ${\eta=0.5}$ (Reimers, 1975; Fadeyev, 2019). We tracked the evolution from the pre-main sequence through the main sequence (MS), up the red giant branch (RGB), past the helium flash, onto the horizontal branch (HB), and finally onto the asymptotic giant branch (AGB).

Fig. 1 displays the location of the HIF compared to the photosphere as a function of stellar age. We note that this relative location or distance varies considerably during the evolutionary track. During the HB phase of evolution, the HIF is located near to the stellar photosphere (Kanbur, 1995). After the exhaustion of helium in the core, when the star moves off the HB, it moves to brighter luminosities and cooler temperatures. That is, its $L/M$ ratio increases and its $T_{\text{eff}}$ decreases. Consequently, the HIF moves deeper into the mass distribution of the star. Figs. 1, 3 and 3 display these arguments. Fig. 1 presents the relative location of the photosphere and HIF at various stages during the life of a $0.632M_{\odot}$ star. The star has an initial mass of $0.83M_{\odot}$ and only reaches this low mass at the advanced stages of evolution. After the HB phase we see clearly that the HIF moves further away from the photosphere. Figs. 3 and 3 display this in greater detail. The left panel of Fig. 3 presents the location of this star on the HB whilst the right panel displays its outer envelope structure - specifically its temperature, opacity and ionized hydrogen structure. Fig. 3 is the same but for a star on the AGB. We clearly see that as the star moves to the AGB, its HIF moves further into the mass distribution.

In order to confirm these proposed HIF-photosphere interactions, we now obtain the PCAC relations from observations for stars at various stages of stellar evolution. The optical ($VI$) light curves of the classical Cepheids and T2Cs in the Galactic bulge are taken from the OGLE-IV catalog (Soszyński et al., 2017) while those in Large Magellanic Cloud (LMC) and Small Magellanic Cloud (SMC) are taken from Soszyński et al. (2015) for classical Cepheids and from Soszyński et al. (2018) for T2Cs. The classification for the different subclasses of T2Cs is as provided by OGLE-IV. For our analysis, we choose the stars with well-sampled light curves having more than 30 data points in both $V$ and $I$ bands. The number of stars available for classical Cepheids and T2Cs is summarized in Table 1.

The photometric light curve data are fitted with the Fourier sine series (see example, Deb & Singh, 2009; Bhardwaj et al., 2015; Das et al., 2018):

where the phase $x$ is calculated as:

where $t_{0}$ is the epoch of maximum brightness and $P$ is the pulsation period. The order of the fit, $N$ is obtained using the Bart’s criteria (Bart, 1982) by varying it from 4 to 8. From the Fourier-fitted light curves, we obtain colour at maximum, minimum and mean light as:

where $I_{\operatorname{phmax}}$ and $I_{\operatorname{phmin}}$ correspond to the $I$-magnitude at the same phase as that of $V_{\max}$ and $V_{\min}$, respectively. The amplitude in $V$ is simply defined as the difference between the maximum and minimum of the magnitude in the $V$-band.

To correct for extinction in the classical Cepheids and T2Cs in the LMC and SMC, we obtain their colour excess ${E(V-I)}$ values using the reddening maps of Haschke et al. (2011) and convert to their ${E(B-V)}$ values using $E(V-I)=1.38E(B-V)$(Tammann et al., 2003). The extinction values are found by adopting the reddening law of Cardelli et al. (1989): $A_{V}=3.32E(B-V);A_{I}=1.94E(B-V)$(Schlegel et al., 1998).

For the T2Cs in the Galactic bulge, we first use the extinction maps of Gonzalez et al. (2012) and adopt the standard reddening law from Cardelli et al. (1989) ($\frac{A_{K_{s}}}{A_{V}}=0.114;\frac{A_{I}}{A_{V}}=0.479$). Given the non-standard nature of the reddening law towards the central Galactic bulge (Popowski, 2000; Udalski, 2003; Nishiyama et al., 2009; Nataf et al., 2013; Matsunaga et al., 2013), we also use the extinction calculator provided by Nataf et al. (2013) to obtain the colour excess $E(V-I)$ and extinction $A_{I}$. This calculator primarily assumes the $E(J-K_{s})$ measurements of Gonzalez et al. (2012). The $A_{V}$ values are then estimated by using $E(V-I)=A_{V}-A_{I}$. We only use the first method of extinction correction for the classical Cepheids in the Galactic bulge.

These extinction corrections are applied to the magnitudes and colours at minimum, mean, and maximum light during the pulsation cycle. We fit linear-regression models to extinction corrected PC and AC relations for different classes of pulsating stars. Throughout this paper, we employ an iterative 3$\sigma$ outlier clipping during the regression analysis and points beyond this threshold are considered outliers.

Several aspects of our theoretical approach have been outlined in the introduction and earlier sections and in earlier papers (Simon et al., 1993; Kanbur, 1995; Kanbur & Phillips, 1996; Kanbur et al., 2004; Kanbur & Ngeow, 2006; Kanbur et al., 2007). In summary, the properties of mean light PC relations are determined by the properties of PC relations at different phases during a pulsation cycle and at maximum and minimum light, these are determined by the interaction of the stellar photosphere and the HIF.

In an effort to understand the interaction of the HIF and the stellar photosphere during the pulsation cycle, we theoretically study the HIF-stellar photosphere distance (see Kanbur et al., 2004) at minimum and maximum light using the radial stellar pulsation code in MESA. We use MESA r-$11701$ (Paxton et al., 2019) to compute RR Lyrae, BL Her and classical Cepheid models, the parameters of which are listed in Table 4. We chose this pulsation code because it is a state-of-the-art pulsation code that is open source. Further it allows some flexibility in treatments of turbulent convection and opens up the possibility of using future detailed studies to constrain theories of turbulent convection in classical Cepheids and RR Lyraes. Here we show that in terms of light curves, it matches observations well. In common with many other pulsation codes (Marconi et al., 2015) it uses the diffusion approximation to model radiative transfer, though future plans include more sophisticated treatments of radiative transfer.

The input parameters for these models ($Z$, $X$, $M/M_{\odot}$, $L/L_{\odot}$ and ${T_{\text{eff}}}$) have been taken from literature (see e.g. Marconi et al. 2015 for RR Lyraes, Marconi & Di Criscienzo 2007 for BL Her models and Table 1 of Kanbur et al. 2004 for classical Cepheids). We may choose to use any of the four sets of convection parameters as outlined in Table 4 of Paxton et al. (2019). Set A corresponds to the simplest convection model, set B adds radiative cooling, set C adds turbulent pressure and turbulent flux, and set D includes these effects simultaneously. In the present work, we choose to use the simplest (Set A) and the most complex (Set D) convection models for the same input parameters. A study of the effect of different sets of convection parameters on the light curve structures of the models is planned for the future. For our analysis, we proceed with only those models that have full-amplitude stable pulsations in the fundamental mode. Fourier analysis of these models shows considerably good match between the models and observations in the Galactic bulge, LMC and SMC for OGLE-IV RR Lyraes, BL Her stars and classical Cepheids with respect to their Fourier parameters and thus, their light curve structures.

We have analysed the largest available dataset of T2Cs in the Bulge, LMC and SMC from the OGLE-IV survey for the first time. We found that there exists a flat PC slope at maximum light for W Vir stars in Bulge, LMC and SMC within 3$\sigma$ uncertainties. This is similar to the flat PC${}_{\textrm{max}}$ slope observed in long period classical Cepheids from present work and Bhardwaj et al. (2014). Also, the PC${}_{\textrm{min}}$ slope of W Vir stars is much greater than the slope at maximum light. We observed a flat but slightly negative PC${}_{\textrm{min}}$ slope in BL Her stars from SMC ($-0.251\pm 0.339$) similar to that obtained for RR Lyrae stars from SMC at minimum light (Das et al., 2018). The slopes of the PC relations for RV Tau stars in Bulge, LMC and SMC are flat within 3$\sigma$ uncertainties at minimum and maximum light. However, there is high dispersion in all the cases. Using the statistical $F$-test, we find a break in the PC relations at a period of 10 days for classical Cepheids in LMC and SMC from OGLE-IV at both minimum and maximum light, similar to earlier results from Kanbur & Ngeow (2004) and Bhardwaj et al. (2014). There exists a flat PC${}_{\textrm{max}}$ within 3$\sigma$ uncertainties and a significantly sloped PC${}_{\textrm{min}}$ for the longer period classical Cepheids in both LMC and SMC. We have also carried out $t$-test to study the equivalence of PC slopes in LMC and SMC across a broad spectrum of variable star types and found the results to statistically support the above mentioned results.

We also computed a few RR Lyrae, BL Her and classical Cepheid models using the radial stellar pulsation code in MESA to theoretically study the distance between HIF and stellar photosphere at minimum and maximum light. We found that RRL models have a flat period-temperature relation at minimum and maximum light within 3$\sigma$ uncertainties using both sets of convection parameters A and D and a flat period-temperature relation at maximum light within 3$\sigma$ uncertainties exists for BL Her and Cepheid models using both sets of convection parameters. The period-temperature relation for the models has a higher dispersion at maximum light than at minimum light. We also found the models to have consistent results with observations with respect to both Fourier parameters as well as colour. The HIF and stellar photosphere are engaged and co-moving at maximum light for classical Cepheids, which results in a flat PC slope at maximum light. However, they are disengaged at minimum light and we observe a significant slope in PC${}_{\textrm{min}}$. For RR Lyraes, the HIF and stellar photosphere are engaged during both minimum and maximum light. However, at maximum light, the temperature at which hydrogen starts to ionize appreciably is in a range where Saha ionization equilibrium is much more sensitive to temperature than at the temperatures associated with minimum light, even though the range of densities are similar. This results in a flat PC${}_{\textrm{min}}$ slope but a significant slope in PC${}_{\textrm{max}}$. BL Her stars are cooler and brighter than RR Lyraes; thus we suggest that the HIF is further inside the mass distribution, thereby increasing the distance between the HIF and stellar photosphere. The models computed using MESA are broadly consistent with this picture. We have not computed W Vir stars using MESA; however, similar locations of W Vir stars and classical Cepheids on the HR diagram and similar behaviour of PC relations from observations for both suggest similar HIF and stellar photosphere interaction. Thus, in this work, we have incorporated the PC and AC relations for RR Lyraes, BL Her, W Vir and classical Cepheid stars in a single unifying theory that involves the interaction of the hydrogen ionization front and stellar photosphere and the theory of stellar evolution. Fig. 14 summarizes the main results in the form of a flowchart. An exception to these results are the flat PC relations for BL Her stars at both minimum and maximum light and for W Vir stars at minimum light in the SMC. However, we note here that the statistical sample of stars in the SMC is small ($\sim 20$ BL Her and $\sim 15$ W Vir stars).

Paxton et al. (2019) have outlined the physical and numerical assumptions incorporated in the MESA-RSP code and demonstrated that it produces stable, multi-wavelength light curve models for a broad spectrum of variable stars. To evaluate possible systematics in MESA-RSP models due to the input physics such as the equation of state, opacities or color-temperature transformations, it is important to compare these with different pulsation models in the literature. Since MESA-RSP has just been published (Paxton et al., 2019), detailed comparisons between pulsation results with this code and others in the literature have not yet been carried out. We note that comparisons between observations and data for a wide class of radially pulsating stars were reasonable (Paxton et al., 2019). MESA-RSP follows the work of Smolec & Moskalik (2008) in its treatment of stellar pulsation and uses a particular theory of turbulent convection (Kuhfuss, 1986). Other similar codes (Marconi et al., 2015) use the convection formulation outlined in Stellingwerf (1982a, b). MESA-RSP affords the possibility of varying the convection parameters used. Using turbulent convection parameter sets A and D (Paxton et al., 2019) suggest that our results are robust to this. In common with other pulsation codes in the literature, MESA-RSP uses the diffusion approximation to account for radiative transfer. Relaxing this assumption is certainly a project for the future but we do not anticipate that a more detailed radiative transfer will affect the general character of the results presented here. Previously, Simon et al. (1993) and Kanbur & Phillips (1996) used purely radiative models, whilst Kanbur et al. (2007) and Das et al. (2018) used convection theories from Kuhfuss (1986) and Stellingwerf (1982a), respectively and found similar qualitative results regarding the behaviour of PC relations of classical Cepheids and RR Lyraes. While a detailed comparison of input physics in different pulsation models is beyond the scope of this paper, we are computing a fine grid of Cepheid and RR Lyrae models covering the entire parameter space using MESA-RSP, which will be used to explore possible systematics and constrain pulsation models in the future.

The authors thank the referee for useful comments and suggestions that improved the quality of the manuscript. SD acknowledges the INSPIRE Senior Research Fellowship vide Sanction Order No. DST/INSPIRE Fellowship/2016/IF160068 under the INSPIRE Program from the Department of Science & Technology, Government of India. HPS and SMK thank the Indo-US Science and Technology Forum for funding the Indo-US virtual joint networked centre on “Theoretical analyses of variable star light curves in the era of large surveys”. SD acknowledges the travel support provided by SERB, Government of India vide file number ITS/2019/004781 to attend the RR Lyrae/Cepheid 2019 Conference “Frontiers of Classical Pulsators: Theory and Observations”, USA where the authors had useful discussions. Funding for the Stellar Astrophysics Centre is provided by The Danish National Research Foundation (Grant agreement no.: DNRF106). AB acknowledges research grant #11850410434 awarded by the National Natural Science Foundation of China through the Research Fund for International Young Scientists, China Postdoctoral General Grant (2018M640018), and Peking University Strategic Partnership Fund awarded to Peking-Tokyo Joint-collaboration on Research in Astronomy and Astrophysics. BM and NP acknowledge IUSSTF for supporting travel to University of Delhi and AC and RJ thank SUNY Oswego and the NSF STEP grant. The authors acknowledge the use of High Performance Computing facility Pegasus at IUCAA, Pune and the following software used in this project: MESA r$10108$ and MESA r$11701$ (Paxton et al., 2011; Paxton et al., 2013, 2015, 2018, 2019).