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1 Introduction↩︎

Classical Cepheids (CCs) are pulsating variable stars that have shaped our understanding of cosmic distance measurement. Their unique period-luminosity relation, first established by [1], provides a fundamental tool for studying stellar populations across different scales. Within the Milky Way, they enable precise mapping of Galactic structure and kinematics [2]–[4]. In cosmology, the Hubble constant is calibrated using CCs [5], providing constraints on the universe’s expansion history that are independent of early-universe CMB inferences within the \(\Lambda\mathrm{CDM}\) framework. The reliability of CCs as standard candles depends on thorough characterization of their properties in various stellar environments. This requirement drives ongoing efforts to improve calibration accuracy through multi-wavelength observations and theoretical modeling [6], [7]. Such work bridges Galactic and extragalactic studies, making CCs indispensable across astrophysical disciplines.

Open clusters (OCs) are young, metal-rich stellar systems concentrated in the Galactic disk, whose members formed nearly simultaneously and share common astrometric and chemical properties. These uniform characteristics allow the clusters’ ages, distances, and extinctions to be determined with high precision. For CCs belonging to these OCs, such uniformity allows their physical parameters, particularly ages to be constrained more robustly. Since the first identification of an OC Cepheid, subsequent studies [8]–[11] have steadily expanded this field. Large-scale surveys such as the VISTA Variables in the Via Lactea survey (VVV; [12]), the Gaia astrometric missions [13]–[15], the All-Sky Automated Survey for Supernovae (ASAS-SN; [16]), the Asteroid Terrestrial-impact Last Alert System (ATLAS; [17]), the Wide-field Infrared Survey Explorer (WISE; [18], [19]), the Optical Gravitational Lensing Experiment (OGLE; [20], [21]), and the Zwicky Transient Facility (ZTF; [22]) have delivered extensive catalogs of Galactic CCs. Despite these advances, accurately determining CCs’ properties remains challenging, which makes OC associations particularly valuable. In particular, Gaia’s high-precision astrometry has played a pivotal role in enabling the discovery of both new OCs, substantially increasing the number of confirmed OC Cepheids in recent years.

CCs are FGK-type giants or supergiants, and longer-period variables exhibit higher luminosities and greater masses, which implies younger ages. The clear correlation between period and age is referred to as the period-age relation (PAR) [23]–[25]. [23] developed theoretical PAR models using nonlinear convective pulsation and updated evolutionary tracks. These models were subsequently refined by [24] to incorporate the effects of stellar rotation. Since the ages of OCs can be directly determined through isochrone fitting, they offer the most robust empirical constraints on the PAR.

Observational calibrations of the PAR using OCs have a long history [26]. More recently, [27] calibrated the PAR using 33 OC Cepheids with 27 fitted OC ages, finding a small offset relative to the rotating models of Anderson16 at short periods. Subsequent studies have identified additional OC Cepheids [11], [28], [29], while reliable cluster age determinations remain limited, leaving the empirical calibration of the PAR open for further refinement.

Recent studies have refined the period-Wesenheit relation (PWR) for OC Cepheids [11], [27], [29], [30]. These works have taken into account not only the effects of parallax and extinction on the luminosities of CCs, but also the influence of metallicity, resulting in more precise calibrations. Nevertheless, the derived PWR fits show variations across different studies. Such discrepancies likely arise from differences in sample selection and membership determination, also reflecting the current limitations of the available OC Cepheids. Although a large number of OC Cepheids and candidates have been reported, the membership between many CCs and their host clusters still needs to be confirmed.

In this paper, we compiled a homogeneous sample of OC Cepheids based on Gaia Data Release 3 (DR3) [15], using astrometric and photometric membership analyses. The host cluster ages, as well as extinction, and distance parameters, were then refined through isochrone fitting. The PAR and PWR were recalibrated using refined OC ages and updated OC parallaxes, respectively, which resulted in more reliable calibrations and improved precision in Cepheid-based distance measurements.

We organize this paper as follows. In Section 2 and Section  3, we describe the datasets and methodology employed to identify OC Cepheids, followed by the data filtering process and the resulting sample in Section 4. Section 5 presents our analysis of the PAR and PWR, and Section 6 summarizes the study.

2 Data↩︎

2.1 Classical Cepheid↩︎

The CC sample used in this study is compiled from six catalogs published since 2019, including those by [2], [3], [22], [31]–[33]. [3] and [2] employed CCs as tracers to map the structure of the Milky Way. [31] reported 3,659 CCs from a wide-field survey that combined data from multiple projects. [32] combined Gaia astrometry with ASAS-SN photometry and identified nearly 1,900 CCs, over 90% of which are considered reliable. [22] performed an extensive search for variable stars using ZTF data and identified about 700 CCs. [33] obtained metallicity measurements for 499 CCs to refine the period-Wesenheit-metallicity (PWZ) relation.

2.2 Open Cluster↩︎

The OC data used in this study, collected from the Gaia astrometric mission [13]–[15], provide stellar positions and proper motions (\(l\), \(b\), \(\varpi\), \(\mu_{\alpha^{*}}\), and \(\mu_{\delta}\)), along with photometry in the Gaia bands (\(G\), \(G_{BP}\), and \(G_{RP}\)). With the release of Gaia DR3, the number of sources increased from 1.3 billion in DR2 to 1.8 billion. In particular, the precision of parallax measurements for sources with \(G\) band fainter than 18 mag improved by 30% in DR3, enhancing the capability to search for distant OCs in the Milky Way.

Our OC sample is compiled from the catalogs of [36], [37], and [38]. He22 employed the unsupervised Density-Based Spatial Clustering of Applications with Noise (DBSCAN) algorithm to identify 1,656 nearby OCs in the Milky Way, limited to Galactic latitudes \(|b| < 20^{\circ}\) and distances beyond 1.2 kpc. Extending this work, He23 restricted the latitude range to \(|b| < 10^{\circ}\), which yielded nearly three times more distant OCs (\(d > 5\) kpc), including systems with extinction above 5 mag and ages older than 100 Myr. Consequently, the catalog expanded to 2,085 OCs, including candidates. An updated catalog of Milky Way OCs based on Gaia DR3 was presented by HR24. Using the Hierarchical Density-Based Spatial Clustering of Applications with Noise (HDBSCAN) algorithm, they identified 7,167 OCs and provided detailed information including positions, parallaxes, proper motions, photometry, ages, and extinction values. Among these, 3,530 OCs are considered high-quality, characterized by well-defined isochrones derived from neural-network fitting and by the exclusion of kinematic outliers through Jacobi radius filtering.

We also supplemented our sample with additional OCs from [34] and [11], incorporating 3 OCs from each catalog.

2.3 Previous OC Cepheids↩︎

OC Cepheid samples have expanded with the advent of updated and higher-precision data, enabling the identification of new objects and the refinement of both the PAR and PWR. ZC21 conducted a comprehensive survey of CCs in Galactic OCs and identified 33 OC Cepheids, including 13 new discoveries. Using Gaia EDR3 [14] parallaxes and proper motions, they confirmed cluster membership for the CCs and re-estimated the cluster ages, extinctions, and distances via isochrone fitting, thereby enabling refined calibrations of the PAR and PWR.

[28] further expanded the selection of OC Cepheids by cross-matching the CC catalog of [31] with several OC catalogs [34], [39]. This effort resulted in the identification of 50 CCs associated with 45 OCs, of which 39 were confirmed as reliable OC Cepheids. [30] focused on calibrating the PWR for 51 OC Cepheids using different distance estimates, including individual CCs’ parallaxes, cluster-averaged parallaxes, and distance modulus, based on Gaia photometry.

The impact of metallicity on the PWR has been highlighted in recent studies. CRA23 performed a systematic search for OC Cepheids using Gaia DR3 [15], selecting candidates based on spatial position, parallax, and proper motion, further constrained by the color-magnitude diagram (CMD). They identified 34 CCs in 28 OCs as their primary sample, along with 4 candidates. By incorporating cluster-averaged parallaxes, they recalibrated the Leavitt law and further refined it with metallicity constraints. [29] employed the OC catalog of [38] with the CC catalog from [31], which yielded 43 OC Cepheids used to calibrate the PWZ relation.

By compiling Galactic OC Cepheids from these five studies (ZC21, Hao22, Lin22, CRA23 and Wang24), we obtained a total of 66 CCs associated with 59 OCs. Additionally, we took into account the OC Cepheid candidates reported by [40].

3 Method↩︎

3.1 Cross-Matching and Identifying OC Members with CCs↩︎

The selection of member stars in OCs based on spatial and kinematic properties is a well-established approach, as stars within the same OC generally share similar positions and motions [11], [27]–[29]. Accordingly, we performed an initial selection of CCs and OC members using five spatial parameters (\(l\), \(b\), \(\varpi\), \(\mu_{\alpha^{*}}\), and \(\mu_{\delta}\)), with a cross-matching radius of one arcsecond applied in TOPCAT [41].

To further ensure accurate membership, we defined the dispersions \(\delta_\varpi\) and \(\delta_{\rm pm}\) to quantify the deviation of a CC from its host OC in parallax (\(\varpi\)) and proper motion (\(pm\)), with \(R\) denoting its angular distance from the cluster center, respectively: \[\begin{align} \label{eq95delta} \delta_\varpi &= \frac{|\varpi - \bar{\varpi}|}{\sqrt{\sigma_\varpi^2 + \sigma_{\bar{\varpi}}^2}} \\[2mm] \delta_{pm} &= \frac{\sqrt{(\mu_{\alpha*} - \bar{\mu}_{\alpha*})^2 + (\mu_\delta - \bar{\mu}_\delta)^2}}{\sqrt{\sigma_{\mu_{\alpha*}}^2 + \sigma_{\bar{\mu}_{\alpha*}}^2 + \sigma_{\mu_\delta}^2 + \sigma_{\bar{\mu}_\delta}^2 + (\bar{\varpi}/4.74)^2}} \\[2mm] R &= \sqrt{ \big[(l - \bar{l}) \cos b \big]^2 + (b - \bar{b})^2 } \end{align}\tag{1}\] where (\(l\), \(b\)), \(\varpi\), and (\(\mu_{\alpha^{*}}\), \(\mu_{\delta}\)) denote the Galactic coordinates, parallax, and proper motions in right ascension and declination of the CCs, respectively. The parameters \(\bar{l}\) and \(\bar{b}\) are defined as the average Galactic coordinates of the member stars in each OC, and their uncertainties, \(\sigma_l\) and \(\sigma_b\), are given by the corresponding standard deviations. The values of \(\bar{\mu}_{\alpha^{*}}\), and \(\bar{\mu}_\delta\) represent the weighted means of the host OC member stars and (\(\bar{\varpi}/4.74\)) corresponds to the typical intrinsic velocity dispersion of OCs (\(\sim\)​1 km s\(^{-1}\), [42]). The individual member star parallaxes were first corrected for the zero-point (zpt) offset [43] and then used to calculate the weighted mean parallax (\(\bar{\varpi}\)) of each host OC. Although the weighted mean of a large number of member stars would nominally benefit from the \(\sqrt{N}\) reduction in statistical uncertainty, the angular covariance of Gaia parallaxes and proper motions exhibits many times larger than the weighted mean [44]–[48]. To account for this effect, we incorporated the angular covariance between member stars, with values of \(\sim 700~\mu\rm as^2\) for parallaxes and \(\sim 550~\mu\rm as^2~yr^{-2}\) for proper motions from [44], in the calculation the uncertainties of \(\sigma_{{\varpi}}\), \(\sigma_{\bar{\varpi}}\), \(\sigma_{\mu_{\alpha^{*}}}\), \(\sigma_{\mu_\delta}\), \(\sigma_{\bar\mu_{\alpha^{*}}}\), and \(\sigma_{\bar\mu_\delta}\).

Larger dispersions indicate stronger deviations of CCs from their host OCs, thereby reflecting lower astrometric membership likelihood. Accordingly, we adopted a \(3\sigma\) threshold in \(\delta_\varpi\) and \(\delta_{\rm pm}\), and further required that CCs lie within the spatial extent of the cluster core, to identify reliable cluster members.

3.2 Isochrone Fitting↩︎

After filtering OC Cepheids based on astrometric dispersions, we further applied isochrone fitting to validate their membership. The distinct position of CCs in the CMD of their host OCs, corresponding to the blue loop evolutionary stage [49], enables reliable separation from other member stars. Using theoretical data from the PARSEC models [50] and Gaia EDR3 photometry, we then performed isochrone fitting for each OC to re-estimate age, extinction, and distance parameters, thereby establishing the basis for the subsequent PAR and PWR analyses. We note that the precise location of the instability strip boundaries for CCs is sensitive to the adopted input physics in stellar models, including the treatment of convection, turbulent pressure, metallicity, and stellar rotation [51], [52]. To account for these uncertainties, we visually inspected the CMDs and adopted an empirical \(\sim\)​0.5 mag range when selecting OC Cepheids.

During the isochrone fitting, we systematically considered the effects of distance, metallicity, extinction, and age. The cluster distances were derived from the weighted mean parallaxes of its member stars. To better constrain the fitting, we incorporated metallicity as well. Given that member stars of an OC typically exhibit homogeneous metallicities, the CC metallicity was taken as representative of the host cluster. For CCs without spectroscopic metallicity, we used the spectroscopic metallicities of their host OCs from APOGEE [53], [54], LAMOST [55], and Gaia-ESO [56]. When such data were unavailable for both OCs and CCs, metallicities were estimated from the Galactic metallicity gradient of [33].

We initially constructed CMDs using age and extinction parameters from published OC catalogs. In some clusters, however, the fitted isochrones appeared shifted toward the red edge of the main sequence, likely due to systematic overestimation of BP-band photometry for faint stars [57]. To address this, we adjusted the extinction values downward, moving the isochrones toward the blue edge and thereby establishing the final extinction estimates. The cluster ages were then derived by locating the main-sequence turn-off of member stars in the CMD. To improve the precision of the isochrone fits, we restricted the age search range to [6.8, 10.1] dex with a step size of 0.01 dex.

As described in our previous studies (e.g., He22; He23), the extinction coefficient \(c\) for each OC was calculated using a polynomial function: \[\label{extinction95coefficient} \begin{align} c = c_1 + c_2 \ast bp\_rp_0 + c_3 \ast bp\_rp_0^2 + c_4 \ast bp\_rp_0^3 \\+ c_5 \ast A_0 + c_6 \ast A_0^2 + c_7 \ast A_0^3 + c_8 \ast bp\_rp_0 \ast A_0 \\+ c_9 \ast A_0 \ast bp\_rp_0^2 + c_{10} \ast bp\_rp_0 \ast A_0^2 \end{align}\tag{2}\] where the coefficients \(c_1\)-\(c_{10}\) were adopted from the auxiliary data provided by ESA/Gaia/DPAC Coordination Unit 5 (CU5; Photometric Processing) and compiled by Carine Babusiaux. The \(bp\_rp_0\) denotes the intrinsic Gaia color index corrected for extinction, and \(A_0\) is derived from our isochrone fitting and serves as the standard measure of interstellar extinction in the Gaia photometric system.

Based on the OC parameters reported in He22, He23, and HR24, we performed isochrone fitting for 102 OCs to refine their age and extinction parameters. The results were then compared with those from literature catalogs [34]–[38]. Figure 1 presents the distribution of ages from our isochrone fitting and from the literature. All four references exhibit an overall upward trend within the range \(\log Age =\) [6.0, 7.5] dex. Both [35] and HR24 display similar upward discrepancies at \(\log Age < 7.5\), followed by downward deviations at older ages. In contrast, CG20 shows both upward and downward deviations at younger ages. Results from He22 and He23 exhibit mainly upward discrepancies at younger ages and downward discrepancies at older ages.

Figure 1 also shows the extinction distributions. CG20 predominantly exhibit upward discrepancies, while HR24, He22 and He23 generally display downward deviations, with a few upward cases between 0 and 3 dex. Dias21, by comparison, display downward discrepancies in the range of [0, 3] dex and upward deviations from 3 to 5 dex. These differences likely arise from the diverse fitting methodologies employed in the literature. In contrast, our isochrone fitting is performed by visual inspections, yielding a homogeneous set of OC ages and extinctions and providing a consistent framework for calibrating OC Cepheids.

4 Results↩︎

4.1 Open Cluster housing Cepheids↩︎

Using the OC and CC catalogs introduced in Section 2, we applied the method described in Section 3.1 to perform an initial cross-identification of OC members and CCs. This process established preliminary associations between CCs and OCs, resulting in 210 CCs linked to 202 OCs. Since the OCs were drawn from multiple catalogs, occasional discrepancies in naming conventions appeared for clusters with consistent parameters. After resolving such duplications, the sample was refined to 193 CCs associated with 181 OCs. We then evaluated the astrometric membership of CCs in OCs using the dispersions \(\delta_\varpi\) and \(\delta_{pm}\) relative to the cluster center, as defined in Equation 1 , adopting a 3\(\sigma\) threshold as the membership criterion. We also took into account the angular distances of CCs relative to those of the member stars within each OC to ensure they are located near the cluster core. This step led to the exclusion of 55 CCs associated with 51 OCs.

After updating the cluster parameters through isochrone fitting, we further removed CCs that exhibited deviations of more than 0.5 mag from their expected position near the blue loop stage in the CMD of their host OCs, indicating that they are unlikely genuine OC Cepheids. This resulted in the exclusion of 43 CCs associated with 39 OCs. We note that our sample does not fully match those of ZC21, Hao22, Lin22, CRA23, and Wang24, with 8, 3, 19, 11, and 11 CCs excluded, respectively. This discrepancy primarily arises from differences in the OC member stars and CCs considered. By merging these datasets, we identified 27 additional CCs associated with 25 OCs. The final sample thus consists of 110 CCs associated with 102 OCs.

4.2 Classification of OC Cepheids↩︎

Among the identified OC-Cepheid associations, some CCs exhibit parallaxes, proper motions, and angular distances from the cluster center that deviate significantly from those of the OC members, suggesting they may not be physically associated with the clusters. In addition, the positions of certain CCs in the CMD are inconsistent with the expected location of CCs near the blue loop of their host OCs. These discrepancies indicate that CCs may not be associated with their host OCs. Consequently, we classified the 112 OC-Cepheid pairs, comprising 112 CCs associated with 102 OCs, into three categories: reliable OC Cepheids, OC Cepheid candidates, and rejected samples.

For the reliable OC Cepheids, the astrometric dispersions in parallax and proper motion both fell within 3\(\sigma\) of the OC members and they are situated near the cluster core. Simultaneously, their positions in the CMD are consistent with the expected location near the blue loop of their host clusters. These criteria indicate that the CCs are reliably associated with their host OCs. Based on this selection, we identified 41 CCs linked to 37 OCs, including 4 newly discovered OC Cepheids. For comparison, ZC21, Hao22, CRA23, and Wang24 reported 33, 39, 34, and 43 reliable OC Cepheids, respectively. In particular, RS Ori, previously suggested to be associated with FSR 0951, does not fully satisfy our membership criteria. By taking into account the distribution of red giant branch stars during isochrone fitting, we obtained an older age of 8.85 dex for FSR 0951. As a result, RS Ori’s position in the CMD deviates from the expected blue loop evolutionary stage by more than 0.5 mag. Previous studies have suggested the presence of a possible hot companion (e.g., [58], [59]), which could make the system appear brighter and bluer, thereby explaining its apparent deviation from the blue loop. Given its cluster membership and these peculiarities, RS Ori is classified as an OC Cepheid.

The OC Cepheid candidates showed three types of anomalies relative to their host OCs. The first group exhibited \(\delta_\varpi\) and \(\delta_{pm}\) were within 3\(\sigma\), and was located near the cluster core, but the CCs deviated from the blue loop in the CMD by more than 0.5 mag with atypical luminosities. The second group had \(\delta_\varpi\) and \(\delta_{pm}\) between 3\(\sigma\) and 5\(\sigma\), but their CMD positions remained close to the blue loop. Another group exhibited astrometric dispersions consistent with cluster membership and CMD positions near the blue loop, but with the largest angular distances among OC members, placing them near the outskirts of the cluster (e.g., V0379 Cas associated with NGC 129). Based on these criteria, we identified 19 CCs associated with 19 OCs as OC Cepheid candidates.

Rejected samples included CCs whose \(\delta_\varpi\) and \(\delta_{pm}\) differed from their host OCs by more than 5\(\sigma\) or with the largest angular distances within OC members, indicating unreliable membership. We also rejected CCs with \(\delta_\varpi\) and \(\delta_{pm}\) between 3\(\sigma\) and 5\(\sigma\), and situated near the cluster core if they showed significant deviations by more than 0.5 mag from the blue loop in the CMD. In addition, OCs with fewer than 20 identified members were considered unreliable. As a result, we identified 50 rejected samples, comprising 50 CCs associated with 49 OCs.

The 41 reliable OC Cepheids and their associated OC and stellar parameters are listed in Tables 1, 2, and 3. The 19 OC Cepheid candidates and 50 rejected samples are summarized in Appendix 7. The CMDs of the 4 newly identified OC Cepheids are presented in Figures 2, while those of the remaining reliable, candidate, and rejected samples are presented in Appendix 8.

4.3 Newly found OC Cepheids↩︎

4.3.1 Bochum 14 and OGLE-BLG-CEP-098 (5.80 days)↩︎

OGLE-BLG-CEP-098 is associated with Bochum 14 from HR24. Bochum 14 is located at a distance of 2696 \(\pm\) 189 pc. This CC is located \(\sim\)​0.84 arcmin from the OC center, well within the cluster core. OGLE-BLG-CEP-098 meets the astrometric membership criteria with \(\delta_{\varpi} = 1.11\,\sigma\) and \(\delta_{{pm}} = 0.47\,\sigma\). Isochrone fitting refines the cluster parameters to \(A_V = 4.12\), DM = 12.41, and \(\log Age = 8.00\). Notably, Bochum 14 exhibits a high extinction and significant differential extinction across the cluster field. Together with the case of HSC 413, this underscores the importance of near-infrared observations for more reliable determinations of cluster parameters.

4.3.2 Theia 358 and R Mus (7.51 days)↩︎

R Mus is associated with Theia 358 (HR24) and is located at approximately 15.33 arcmin from the cluster center, well inside the cluster boundaries. Its host OC is at a distance of \(939 \pm 16\) pc. The astrometric membership criteria for R Mus yield \(\delta_{\varpi} = 0.07\sigma\) and \(\delta_{pm} = 0.64\,\sigma\). In the CMD, R Mus is positioned near the blue loop. Its agreement with both the PAR and PWR further supports the reliability of our fitting and underscores its significance as a representative sample for studying CC properties in nearby OCs.

4.3.3 FSR 0158 and NSVS 8389949 (20.76 days)↩︎

OC Cepheid NSVS 8389949 is offset by approximately 10.83 arcmin from the center of FSR 0158 (HR24), yet still lies within the cluster’s core region. Its host OC is at a distance of \(4220 \pm 445\) pc. The astrometric membership criteria for NSVS 8389949 yield \(\delta_{\varpi} = 1.00\,\sigma\) and \(\delta_{pm} = 1.79\,\sigma\). This CC is long-period with \(\log P = 1.32\), helping to address the scarcity of long-period samples, which are essential for calibrating distances to extragalactic galaxies [60]. Additionally, FSR 0158 hosts one other OC Cepheid and one candidate, further enhancing the cluster’s contribution to calibrating the PAR and PWR relations.

4.3.4 CWNU 3123 and BM Per (22.96 days)↩︎

BM Per, with a period of 22.96 days, is classified in this study as a long-period OC Cepheid (\(\log P > 1.1\)). Its host OC is at a distance of \(3472 \pm 253\) pc. BM Per is located at about 2.70 arcmin from the center of CWNU 3123, well within the cluster core. The CC satisfies the astrometric membership criteria, yielding \(\delta_{\varpi} = 0.58\,\sigma\) and \(\delta_{pm} = 0.55\,\sigma\). In both the PAR and PWR diagrams, BM Per occupies an underpopulated region of long-period OC Cepheids, highlighting its significance within our sample.

5 Analayses↩︎

5.1 Fitting of Period-Age Relations for OC Cepheids↩︎

Over the past two decades, numerous studies have explored the theoretical PAR models for CCs [23]–[25]. However, observational calibrations of the PAR remain limited due to the small number of reliable OC Cepheids, especially those with long periods. In addition, deriving accurate age estimates for CCs is inherently challenging. Recently, ZC21 used Gaia EDR3 data to identify 33 OC Cepheids and calibrated the PAR based on cluster ages determined from isochrone fitting.

The PAR derived by ZC21 exhibits differences from the theoretical models of Anderson16 in the short-period range, while showing good agreement in the long-pefriod regime. Compared to Bono05, deviations are present across the full period range. ZC21 includes 3 long-period CCs (\(\log P > 1.1\)) and 30 short-period CCs (\(\log P < 1.1\)), which underscores the relative scarcity of long-period CCs in their sample.

In this study, we performed the search for OC Cepheids using an expanded and updated observational dataset of OCs and CCs. We identified 41 OC Cepheids, including 30 fundamental (F) mode pulsators, 10 first overtone (1O) mode pulsators and 1 multimode pulsators. The number of long-period CCs (\(\log P > 1.1\)) increased to 5, while the short-period CCs (\(\log P < 1.1\)) grew to 35. In PAR fitting, the periods of 1O mode pulsators were converted to their F mode equivalents using the relation from [61]: \[\label{PF1O} \begin{align} P_{\text{F}} / P_{\text{1O}} = 1.371 + 0.106 \log P_{\text{1O}} \end{align}\tag{3}\] allowing for a combined PAR fit of OC Cepheids across different pulsation modes. The final relation, shown in Figure 3 (red curve), is \[\begin{align} \log \textit{Age} = (-0.595 \pm 0.044) \log P + (8.430 \pm 0.042), \;\sigma_{fit} = 0.057 \end{align}\] The 1\(\sigma_{fit}\) scatter of 0.057 dex in \(\log\textit{Age}\) corresponds to an age uncertainty of approximately 14%, indicating a high correlation suitable for precise age determination.

Figure 3 compares our PAR with those of Bono05, Anderson16, and ZC21. Similar to ZC21, our results show some discrepancies with Bono05 across the entire period range. Across the whole period range, our fit exhibits a shallower slope and smaller intercept compared to ZC21. Interestingly, our fitted slope is nearly identical to Anderson16, although the intercept shows some deviation. Compared with ZC21, we identify three newly discovered long-period CCs (\(\log P > 1.1\)), all of which closely follow our fitting relation with deviations smaller than 3\(\sigma\). Overall, our fitted PAR lies systematically below previous calibrations of Anderson16 and ZC21 over the entire period range, suggesting slightly younger ages by \(\sim\)​0.06 dex and \(\sim\)​0.15 dex relative to Anderson16 and ZC21, respectively.

Notably, we excluded one long-period sample from ZC21, 2MASS J17184740-3817292 associated with BH 222 (gray open circle), because it was classified as a rejected sample due to a spatial dispersion exceeding 3\(\sigma\). In Figure 3, this object is indicated by gray open circles. ZC21 samples considered as candidates are marked by gray filled circles in the same figure and were not included in our PAR fitting. Additionally, one outlier VW Cru linked to CWNU 175 (red open circle) was excluded from the fitting. The outlier may be influenced by binary stars in OCs, which can lead to overestimated extinction values during isochrone fitting and thus biased age estimates. Another potential factor is the effect of stellar rotation on CCs [62], [63]. As noted by Anderson16, rotation modifies the distribution of metallicity both internally and on the stellar surface, significantly affecting stellar evolution.

We further compared the period-age distributions of the OC Cepheid candidates and rejected samples with the PAR fitting results for both F and 1O mode pulsators. As shown in Figure 4, most OC Cepheid candidates exhibit only minor deviations from the fitted curve (less than 0.2 dex). Remarkably, KQ Sco, associated with UBC 1558, shows a discrepancy of just 0.002 dex. Four candidates, however, display larger deviations. In contrast, most rejected samples display substantial discrepancies from the fitting results, although 13 samples show relatively small deviations (less than 0.2 dex). These results indicate that some OC Cepheid candidates may indeed be reliable, while the rejected samples are likely not genuine OC Cepheids or may reflect inaccuracies in the age estimates of their host OCs. With improved astrometric parameters for CCs and OCs from Gaia Data Release 4 (DR4), future work will facilitate a more robust confirmation of the membership for these OC Cepheid candidates.

Finally, we evaluated the performance of our PAR fitting by comparing the age weighted average residuals of OCs from several studies [34]–[38] with the results from this work, ZC21, and Anderson16, as shown in Figure 5. The comparison indicates that our PAR produces smaller weighted average residuals across both short-period and long-period samples. This improvement highlights the advantage of calibrating OC ages through manual isochrone fitting. Nevertheless, discrepancies in literature ages still remain, suggesting that further refinement is warranted. We expect that more precise age estimates in future studies will enhance the PAR, thereby providing stronger constraints on the formation and evolution of the Galactic disk and its spiral structure.

5.2 Fitting of Period-Wesenheit Relations for Cepheids↩︎

The PWR of CCs is a crucial tool for measuring cosmic distances. In recent years, substantial progress has been made toward refining the PWR. Many studies directly employed individual parameters of CCs for PWR calibration [33], [64], [65], but these results were limited by Gaia’s systematic uncertainty in parallax [43] and the underestimation of the formal uncertainties [66]. Subsequent efforts incorporated systematic error estimates and adopted parallax zpt corrections to mitigate these effects [48]. For OC Cepheids, the distances derived from their host clusters effectively reduce both random and systematic errors in the individual parallax measurements of the CCs (Lin22).

In this study, we calibrate the PWR using 41 OC Cepheids, including 30 F mode pulsators, 10 1O mode pulsators and 1 multimode pulsator. Consistent with the PAR, the periods of the 1O mode pulsators are converted to their F mode equivalents following Equation 3. The PWR fitting is then performed on the combined sample, adopting a linear relation: \[\begin{align} W_G &= a \log P + b \\ W_G &= W(G, \textit{G}_{\textit{BP}}, \textit{G}_{\textit{RP}}) = G - \lambda \left(\textit{G}_{\textit{BP}} - \textit{G}_{\textit{RP}}\right) \end{align}\] where \(W_G\) denotes the Wesenheit absolute magnitude [67], [68], with \(G\), \(G_{BP}\), and \(G_{RP}\) denoting Gaia band magnitudes. To ensure more reliable photometry for the CCs, we used the intensity-averaged magnitudes from gaiadr3.vari_cepheid [64] for fitting the PWR. We note that two OC Cepheids do not appear in gaiadr3.vari_cepheid. For the extinction coefficient, we adopt \(\lambda = 1.9\) as recommended by [65], which is appropriate for Gaia photometric bands. We adopt cluster-averaged parallaxes with zpt corrections instead of individual CCs parallaxes to reduce systematic uncertainties. As shown in Figure 6, the PWR fit for 41 OC Cepheids yields the following result (red curve): \[\begin{align} W_\textit{G} = (-3.615 \pm 0.083) \log P + (-2.379 \pm 0.096), \sigma_{fit} = 0.153 \end{align}\] which appears steeper than those from previous studies [11], [27], [30], [64]. This difference is likely due to the different samples used for the fitting. Simultaneously, we adopt cluster-averaged parallaxes obtained from the weighted average of the parallaxes of member stars, which effectively reduces both random and systematic errors. In contrast, when using the parallaxes of individual CCs for the fitting, the relation becomes steeper. Using cluster-averaged parallaxes, on the other hand, reduces the scatter.

Further, we analyze the effect of applying the parallax zpt correction in PWR fitting. Using the same cluster-averaged parallaxes and \(\lambda\), the zpt correction yields a PWR with both a smaller slope and intercept, leading to a shallower relation than the uncorrected fit. Consistent with previous studies (ZC21; Lin22), we also performed fits by adopting a fixed slope (\(a = -3.32\)) and \(\lambda\) = 1.9. Our analysis reveals that applying the zpt correction results in a shift of the fitted PWR intercept by approximately 0.1 mag compared to the case without the correction. This demonstrates that applying the zpt correction helps reduce systematic errors in Gaia parallaxes.

Additionally, in Figure 6, one outlier J194806.54+260526.1 associated with FSR 0158 was marked by red open circle. Its RUWE of 0.934 indicates a reliable astrometric solution, while its ipd_frac_multi_peak¹ signals multiple peaks in the point-spread function, suggesting a potentially blended or complex source. Future Gaia DR4, with its expanded sample and higher precision, is expected to yield more reliable parallaxes for OC Cepheids and enable a more accurate calibration of the PWR.

6 Summary↩︎

In this study, we utilized Gaia DR3 data to search for OC Cepheids, focusing on identifying distant and long-period samples and updating OCs parameters to calibrate the PAR and PWR of CCs. By integrating multiple up-to-date OC and CC catalogs, we built a more comprehensive database. We compiled three OC catalogs and five CC catalogs including their spatial positions, kinematic parameters and photometry (\(l\), \(b\), \(\varpi\), \(\mu_{\alpha^{*}}\), \(\mu_{\delta}\), \(G\), \(G_{BP}\), and \(G_{RP}\)). Initial associations were established through positional cross-matching within 1 arcsecond using TOPCAT [41]. Membership was then confirmed by applying astrometric dispersions (\(\delta_\varpi\), \(\delta_\mu\)) and angular distance relative to OCs and by examining CCs’ distinct positions near the blue loop in the CMD. Isochrone fitting was further used to refine key OC parameters such as age and extinction, providing the basis for subsequent PAR and PWR calibration. As a result, we identified 110 CCs associated with 102 OCs, of which 41 CCs linked to 37 OCs were classified as OC Cepheids. Among these, 4 are newly discovered. Notably, BM Per linked to CWNU 3123 is the newly longest-period confirmed OC Cepheid with \(\log P = 1.36\). Additionally, two newly discovered OC Cepheid candidates have distances exceeding 6 kpc.

Next, we performed fitting for the PAR and PWR using 41 OC Cepheids. Based on the ages derived from isochrone fitting, we calibrated the PAR as \(\log \textit{Age}\) = (-0.595 \(\pm\) 0.044) \(\log P\) + (8.430 \(\pm\) 0.042), \(\sigma_{fit} = 0.057\). The scatter corresponds to an age uncertainty of about 14%, demonstrating a precise and reliable relation for cluster age determination. Our results lie systematically below those of previous studies, indicating that the derived OC ages are slightly smaller. This discrepancy can be attributed to our use of more detailed isochrone fitting, including consideration of the red giant branch distribution and age determination, which may differ from the approaches adopted in previous studies. Moreover, when comparing our PAR calibration with other determinations of OC ages, additional differences are evident, suggesting that further optimization is required. For the PWR, we systematically examined the effects of parallax selection and zero-point correction. Using cluster-averaged parallaxes effectively mitigates Gaia’s systematics, while applying zpt corrections further improves the fit, yielding \(W_G\) = (-3.615 \(\pm\) 0.083) \(\log P\) + (-2.379 \(\pm\) 0.096), \(\sigma_{fit} = 0.153\).

With the improved accuracy of observational data, we have expanded the sample of OC Cepheids. Nevertheless, even in the Gaia DR3 era, the census of OC Cepheids remains relatively small and incomplete, particularly for long-period and distant cases. Future releases, especially Gaia DR4, will enlarge the sample and enable more refined PAR and PWR calibrations, with broad applications to studies of the Milky Way’s spiral structure and stellar evolution.

Acknowledgments↩︎

We sincerely thank all those who provided valuable feedback and assistance for this work, especially the anonymous referee for the helpful comments and suggestions that improved the manuscript. This work was supported by National Natural Science Foundation of China (NSFC) through grants 12573023,12303024, the Natural Science Foundation of Sichuan Province (2024NSFSC0453), and the "Young Data Scientists" project of the National Astronomical Data Center (NADC2023YDS-07). A.R. is supported by the China Manned Space Program with grant no.CMS-CSST-2025-A13. Y.L. is supported by the NSFC12573035, and the CSST project: CMS-CSST-2021-A10. K.W. is supported by NSFC12373035,12173028. And this work has made use of data from the European Space Agency mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC,https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.

7 Data for OC Cepheid candidates and rejected samples↩︎

These tables list the OC Cepheid candidates and rejected samples identified in this work, continuing from Tables 1, 2, and 3. The complete table is available in machine-readable format in the online journal.

8 CMDs of remaining samples↩︎

Analogous to Figure 2, this figure presents the CMDs of the remaining 106 OCs, including 37 OC Cepheids, 19 OC Cepheid candidates, and 50 rejected samples.