Search for Classical Cepheids in Galactic Open Clusters and Calibration of the Period–Wesenheit–Metallicity Relation in the Gaia Bands

It is beneficial to calibrate the period–Wesenheit–metallicity relation (PWZR) of Delta Cephei stars (DCEPs), i.e., classical Cepheids, using accurate parallaxes of associated open clusters (OCs) from Gaia data release 3 (DR3). To this aim, we obtain a total of 43 OC–DCEPs (including 33 fundamental mode, 9 first overtone mode, and 1 multimode DCEPs) and calibrate the PWZR as WG=(−3.356±0.033)(logP−1)+(−5.947±0.025) + (−0.285 ± 0.064)[Fe/H]. The concurrently obtained residual parallax offset in OCs, zp = −4 ± 5 μas, demonstrates the adequacy of the parallax corrections within the magnitude range of OC member stars. By comparing the field DCEPs’ DR3 parallaxes with their photometric parallaxes derived by our PWZR, we estimated the residual parallax offset in field DCEPs as zp = −15 ± 3 μas. Using our PWZR, we estimate the distance modulus of the Large Magellanic Cloud to be 18.482 ± 0.040 mag, which aligns well with the most accurate published value obtained through geometric methods.

Delta Cephei stars (DCEPs), i.e., classical Cepheids, constitute Population I of Cepheids and are located in the instability strip above the main sequence in color-magnitude diagrams (CMDs;Turner et al. 2006).A DCEP is a youthful, periodic pulsating yellow giant or supergiant that has been around for tens to hundreds of millions of years, with a pulsation period of approximately 1 to 100 days.
The period-luminosity relation, also known as Leavitt's law (Leavitt & Pickering 1912), is a wellknown characteristic of DCEPs.Accurate distances derived from the period-luminosity relationship of DECPs are widely used.For example, DCEPs have been used to study the structure of the Milky Way (e.g., Chen et al. 2019;Skowron et al. 2019), measure the distances to other galaxies (e.g., Freedman et al. 2001;Sandage & Tammann 2006), and serve as a critical step in measuring the Hubble constant H 0 (e.g., Freedman et al. 2011;Riess et al. 2021).
Various reddening effects are produced by the different lines of sight and distances to DCEPs in the Milky Way.The period-Wesenheit relation (PWR) overcomes the limitations of reddening by transforming multiband magnitudes into Wesenheit magnitudes (Madore 1982;Majaess et al. 2008).
Traditionally, the PWR of DCEPs is calibrated using the parallaxes of individual stars (e.g., Ripepi et al. 2019;Poggio et al. 2021;Ripepi et al. 2022).However, this method suffers from uncertain residual parallax offset in the Gaia parallaxes of the individual stars (Lindegren et al. 2021).Besides, it is predicted that a variation in metal abundance affects the shape and width of the DCEP instability strip (e.g., Caputo et al. 2000), which consequently affects the coefficients of the PWR (Marconi et al. 2005(Marconi et al. , 2010;;De Somma et al. 2022, and references therein).Limitations in parallax accuracy led to a stagnation in studies of the period-Wesenheit-metallicity relation (PWZR) until the advent of the Gaia mission (Collaboration et al. 2016), which has provided accurate parallaxes for a total of 1.8 billion objects to date, resulting in a large number of works on the PWZR to spring up (e.g., Groenewegen 2018;Ripepi et al. 2019Ripepi et al. , 2020Ripepi et al. , 2021;;Riess et al. 2021;Breuval et al. 2022;Ripepi et al. 2022;Cruz Reyes & Anderson 2023;Trentin et al. 2024).
An alternative method to calibrating the PWR or PWZR is to use the parallaxes of open clusters (OCs) harboring DECPs.This newly developed method takes advantage of the stars in the OCs all having a similar distance, extinction, age, and metallicity, as well as the fact that the age distribution of OCs (ranging from several million years to several billion years; Kharchenko et al. 2013) partially overlaps with the age range of DECPs.After parallax corrections (Lindegren et al. 2021, hereafter L21), this method has been proven to eliminate residual parallax offset (Riess et al. 2022;Cruz Reyes & Anderson 2023), resulting in more accurate PWRs and PWZRs (e.g., Breuval et al. 2020;Zhou & Chen 2021;lin et al. 2022;Riess et al. 2022;Cruz Reyes & Anderson 2023).For example, Riess et al. (2022) used 17 OC-DCEPs with Hubble Space Telescope (HST) photometry to calibrate the PWZR in the HST photometric system and determine a precise Hubble constant of H 0 = 73.01 ± 0.99 km The first OC-DCEP was discovered by Doig (1925), and searches for OC-DECPs have been active in the past decade (Anderson et al. 2013;Chen et al. 2015;Clark et al. 2015;Chen et al. 2017;Lohr et al. 2018;Alonso-Santiago et al. 2020;Breuval et al. 2020;Negueruela et al. 2020;Medina et al. 2021;Zhou & Chen 2021;Hao et al. 2022;lin et al. 2022;Cruz Reyes & Anderson 2023).
Recently, there have been new searches for OCs (e.g., Hunt & Reffert 2023) based on Gaia data release 3 (DR3; Collaboration et al. 2023).In this current study, by cross-matching Gaia sources with the 3655 Galactic DCEPs compiled by Pietrukowicz et al. (2021), we assemble a larger sample of OC-DECPs, which allows us to derive more accurate PWZRs.
The structure of this paper is arranged as follows.In Section 2, we introduce our extended OC-DECP sample.In Section 3, we describe the calibration results for the PWZR derived with our samples.In Section 4, we test the reliability of our PWZR on Galactic field DCEPs and Large Magellanic Cloud (LMC) field DCEPs.Finally, in Section 5, we summarize this work.

Open Clusters
There are a total of 7167 clusters in the Hunt & Reffert (2024) catalog, which covers almost all previously published OCs.Among these 7167 clusters, we only utilize 3530 high-quality OCs, which all identify clear isochrones by network training methods and filter out moving groups by Jacobi radius.The OCs'coordinates, proper motions, and parallaxes were extracted from Hunt & Reffert Wang et al. (2024).It should be noted that OC parallaxes from Hunt & Reffert (2024) were derived through the maximum likelihood distances, where L21 corrections had been considered.We obtained the OC parallax error σ ϖ OC by the quadrature sum of the statistical uncertainty and the angular covariance 1 defined by Maíz Apellániz et al. (2021).

Classical Cepheids
Pietrukowicz et al. ( 2021) compiled a sample of 3,655 DCEPs in the Milky Way.They also supplied the DR3 source id of each DCEP by applying a matching radius of 0."5.We used their DR3 source id to match the gaiadr3.gaiasource and extracted the required parameters (e.g., α, δ, ϖ, µ α * , and µ δ ) for each DCEP.To obtain more reliable photometry of DCEPs, we extracted the intensity-averaged magnitudes (m G BP , m G RP , and m G ) from gaiadr3.vari cepheid 2 for 3,046 DCEPs.

Cross match
OC-DCEPs are identified if the following criteria are met: (1) The projected distance between DCEPs and OCs should be less than 25 pc, assuming that the parallaxes of DCEPs are equal to those of OCs.(2) The µ α * , µ δ , and ϖ of DCEPs should be within 3σ (σ is the standard deviation of OC) of those of OCs.Additionally, an expanded sample is taken into account, in which a few dimensions are slightly higher than 3σ but less than 3.5σ.(3) DCEPs should be located on the instability strip of their host OC's CMD (Turner et al. 2006).After filtering using the above criteria, we obtained 43 OC-DCEPs, whose astrometry and photometry are given in Table B. should be noted that among the 43 OC-DCEPs we obtained, there is an association of U Sgr with the OC IC 4725, but U Sgr is not in gaiadr3.varicepheid and is hence not used for PWZR calibration.

ANALYSIS
1 Due to the large number of member stars in an OC, the statistical uncertainty of the OC's parallax will benefit from the √ N improvement.However, as the angular covariance of the Gaia parallaxes is much larger (Lindegren et al. 2021;Maíz Apellániz et al. 2021;Vasiliev & Baumgardt 2021;Zinn 2021), we took into account the angular covariance. 2 https://gea.esac.esa.int/archive/Our 43 OC-DCEPs are composed of 33 DCEPs pulsating in the fundamental mode (F-mode), nine pulsating in the first overtone (1O-mode), and one multimode (F1O-mode) pulsator.To establish the PWZR of the DCEPs including the 1O-mode DCEPs, we used the equation P F = P 1O /(0.716 − 0.027 log P 1O ) (Feast & Catchpole 1997) to obtain their period in the F-mode, where P F and P 1O represent pulsations in the F-mode and 1O-mode, respectively.For that F1O-mode DCEP, we adopted its period in the F-mode.
To obtain the metal abundances of our OC-DCEPs, we matched our OC-DCEPs with Trentin et al.To calibrate the PWZR in the Gaia bands, we refer to the method in Riess et al. (2022) andRipepi et al. (2022).The photometric parallax (in milliarcseconds) is defined as: where w G is the apparent Wesenheit magnitude and can be defined as . We adopted the empirical result λ = 1.9 (Ripepi et al. 2019).W G is the absolute Wesenheit magnitude, which can be defined as: We used the optimize.minimizemethod from the Python Scipy library to minimize the following quantity: where zp is the residual parallax offset in OCs.For σ ϖ phot , we refer to the definition given in Ripepi et al. ( 2022): assuming a conservative error of 0.02 mag for the three Gaia bands (G BP , G RP , and G).We adopted a conservative dispersion of 0.1 mag for σ W G (De Somma et al. 2020).
To ensure the robustness of the fit, we performed 10,000 Monte Carlo simulations, where for each simulation we randomly varied ϖ OC and σ w G within their errors to obtain the distribution of each We present our PWZR and compare them with other works in Table 1.The zp of Case 3 and Case 4 in Table 1 are −4 ± 5 µas and 1 ± 5 µas, respectively, which proves the adequacy of L21 corrections within the magnitude range of OC member stars.As in most works, negative metallicity terms γ are obtained.Specifically, our γ of F-mode OC-DCEPs is −0.162 ± 0.070, but the γ of F+1O-mode OC-DCEPs is −0.285 ± 0.064.This is consistent with the conclusions of De Somma et al. ( 2022  the linear relation of OC-DCEPs is tighter than that of field DCEPs.Then, we applied our F+1Omode PWZR (i.e., Case 3 in Table 1) on the field DCEPs to derive their photometric parallaxes and parallax offsets, ∆ϖ, between the photometric parallaxes and DR3 parallaxes after L21 corrections (see the distribution of ∆ϖ in Figure 3).We convoluted this distribution using a Gaussian kernel density estimate (see the orange curve in Figure 3) with a bandwidth chosen according to Silverman (1986), and the ∆ϖ with the highest probability density (see the red dashed line in Figure 3) is the estimate of the zp in field DCEPs.To estimate the error of the zp in field DCEPs, we performed 10,000 Monte Carlo simulations, and for each simulation, we randomly varied the coefficients of PWZR (α, β, and γ) within the error to obtain 10,000 estimates of the zp in field DCEPs, and then calculated their standard deviation as the error.Finally, we obtained an estimate of the zp in field

Reliability Testing of PWZR on LMC Field DCEPs
The PWZR can be used to measure the distances to LMC field DCEPs and thus infer the distance modulus of LMC (µ LMC ).One of the most accurate published distance modulus measurements is µ LMC = 18.477 ± 0.004 (statistical error) ±0.026 (systematic error) mag, obtained from geometric measurements of eclipsing binaries (Pietrzyński et al. 2019).By comparing our PWZR-based distance modulus with the published µ LMC , it is possible to test the reliability of our PWZR on LMC field DCEPs.we randomly varied the coefficients (α, β, and γ) within their errors to obtain 10,000 medians, and then calculated their standard deviation, σ, as the error of µ LMC .
We list the derived µ LMC in Table 1  This PWZR estimates a µ LMC value of 18.482 ± 0.040 mag, which aligns well with the result derived by Pietrzyński et al. (2019) based on the geometric measurements of eclipsing binaries in the LMC.
1. Representative examples of OC-DCEPs and rejected associations are shown in panels (a) and (b) of Figure A.1, respectively.It

(
2024), who compiled 910 DCEPs with literature metal abundances from high-resolution spectroscopy or metal abundances from the Gaia Radial Velocity Spectrometer (see Section 2.2 in Trentin et al. 2024, for details).Finally, we obtained the metal abundances of 40 OC-DCEPs and compiled them in Table B.1.

Figure 1 .
Figure 1.PWR and PWZR fitting results from our OC-DCEPs.The color represents the value of metal abundance.The black dots in the two subfigures on the left are OC-DCEPs without literature metal abundances.
), who discovered the γ of ∼ −0.1 to ∼ −0.2 for the F-mode and ∼ −0.1 to ∼ −0.3 for the F+1O-mode based on stellar pulsation models.We also found that the absolute value of the γ obtained by the empirical relation of field DCEPs is larger than what we obtained.Our OC-DCEPs have a smaller range of metal abundances, which may explain the smaller absolute value of the γ we obtained.In the future, obtaining more OC-DCEPs with a wider range of metal abundances will help us better constrain the γ.CruzReyes & Anderson (2023) fixed the γ to −0.384 ± 0.051(Breuval et al. 2022) and then calibrated the PWZR in the Gaia bands using 26 F-mode OC-DCEPs and 225 field DCEPs as W G = (−3.242± 0.044) log(P − 1) + (−6.004 ± 0.019) + (−0.384 ± 0.051)[Fe/H].To compare with it, we adopted λ = 1.921 and fixed the slope α to −3.242 ± 0.044 and the γ to −0.384 ± 0.051.For our F+1O-mode and F-mode OC-DCEPs, the fitting results of intercept β are −6.008± 0.010 mag and −6.046 ± 0.010 mag, respectively.Both fitting results are consistent with Cruz Reyes & Anderson (2023) within 3σ, and the error is smaller.
Reliability Testing of PWZR on Galactic Field DCEPs We chose 758 F+1O-mode Galactic field DCEPs from Trentin et al. (2024) using the following criteria: (1) RUWE <1.4; and (2) ϖ/σ ϖ >5.The OC-DCEPs we obtained and the field DCEPs are plotted together in Figure 2. It can be seen that the linear relation between the two is consistent, and

Figure 2 .
Figure 2. Blue dots are the field DCEPs in Trentin et al. (2024).Red dots are our 42 F+1O-mode OC-DCEPs.The black line is the fitting result of our F+1O-mode PWR.

Figure 3 .
Figure 3. Normalized histogram of parallax offset (∆ϖ) estimated using our F+1O mode PWZR.The orange curve represents the Gaussian kernel density estimation for this distribution.The red dashed line represents the highest probability density of ∆ϖ.

Figure 4 .
Figure 4. Literature estimates of the zp in field DCEPs.
total of 43 OC-DCEPS, which is the largest sample of OC-DCEPs to date.Benefiting from OC's high-precision parallax, we calibrated the PWZR in the Gaia bands and estimated the zp in OCs simultaneously.We found that the zp in OCs is negligible, demonstrating the adequacy of L21 corrections within the magnitude range of OC member stars.For the metallicity term γ, we obtained that γ = −0.285± 0.064 for the F+1O-mode OC-DCEPs and γ = −0.162± 0.070 for the F-mode OC-DCEPs, which is consistent with the conclusions of DeSomma et al. (2022).Applying our F+1O model PWZR on field DCEPs and using a Gaussian kernel density estimate, we found that the zp = −15 ± 3 µas in field DCEPs, which is in good agreement withRiess et al. (2021).Our best PWZR is W G = (−3.356± 0.033) log(P − 1) + (−5.947 ± 0.025) + (−0.285 ± 0.064)[Fe/H].

APPENDIXA.
figure A.1.Because DCEPs have left the main sequence and entered the instability strip, they should be brighter than the main sequence member stars.

Figure
Figure C.1.The upper and lower corner plots represent the marginalised posterior distributions of the free parameters in PWR and PWZR, respectively.The vertical blue lines represent the median values, and the black dashed lines represent the 16th and 84th percentiles.

Table 1 .
Pietrzyński et al. (2019)nds PWZR Obtained in This Work with Other WorksNote-α, β, γ, and zp are the slope, intercept, metallicity term, and residual parallax offset, respectively.µLMC is the distance modulus of the LMC derived from the PWZR.∆µLMC is the difference between µ LMC measured by PWZR and µ LMC measured byPietrzyński et al. (2019).
means that this parameter does not join in the fitting as a free parameter.* Residual parallax offset in field DCEPs after L21 corrections.
Pietrzyński et al. (2019)r works' PWZR (i.e., Case 5 to Case 12) in Table1.All of their µ LMC are consistent with Pietrzyński et al. (2019) within 3σ, with the exception of Case 12, which deviates fromPietrzyński et al. (2019) Pietrzyński et al. (2019)ifference between µ LMC measured by PWZR and µ LMC measured byPietrzyński et al. (2019).It is evident from a comparison of Table1that the LMC's distance modulus determined by our PWZR is more accurate than found with our PWR, indicating that the latter is indeed affected by the metal abundances of the calibrating DECPs.The ∆µ LMC of Case 3 and Case 4 in Table1are 0.005 (0.1σ) and 0.043 (1.1σ), respectively, which are consistent withPietrzyński et al. (2019)within 3σ, confirming the reliability of our PWZR applied to LMC field DCEPs.We consider Case 3 as the optimal PWZR in this work because it best matches the result derived byPietrzyński et al. (2019).We also list the µ

Table B .
1 continued on next page Table B.1 (continued)