Open cluster age calibration from colour–magnitude morphological indices using Gaia DR3 data

ABSTRACT

Star clusters are crucial for understanding how stars evolve. Their colour–magnitude diagrams show the effects of stellar evolution of approximately coeval objects with the same chemical composition. Furthermore, the determination of their astrophysical parameters (age, distance, colour excess, and metallicity) together with their spatial distribution provides information about the structure and the evolution of the Galaxy itself. Using data from the Gaia DR3 and Two Micron All Sky Survey catalogues, we develop methodologies for characterizing open clusters. Precise membership lists, mean astrometric parameters, and radii are obtained. Using photometric data from both data sources, we carried out new age calibrations that rely on morphological indices based on colour (⁠|$\Delta BR$|⁠) and magnitude (⁠|$\Delta G$|⁠) differences between the red clump and the turn-off for a sample of 34 open clusters with ages covering the interval |$8.3 < \log [t({\rm yr})] < 9.9$|⁠. A set of age calibration functions based on Gaia morphological age indices are determined for the first time. We demonstrate their accuracy, obtaining a mean residual of 0.06 dex in |$\log [t({\rm {yr}})]$|⁠. Our results also show that stellar evolution models tend to predict the difference |$\Delta G$|⁠. However, they typically overestimate the difference |$\Delta BR$| for objects younger than |$\log [t({\rm yr})] = 8.8$|⁠.

1 INTRODUCTION

Open clusters (OCs) are important objects to study the history and structure of the Galactic disc. Young OCs reveal how stars form in embedded environments as well as the recent disc history (Lada & Lada 2003). On the other hand, the older OCs are the fingerprints of the chemical and dynamical evolution of the Galactic disc (e.g. Friel 1995; Carraro et al. 2007; Netopil et al. 2016 [hereafter N2016]; Cantat-Gaudin et al. 2020 [hereafter C2020]).

The Galactic OCs present a vast range of ages (⁠|$\log [t({\rm yr})] \lesssim 10.0$|⁠) and typically solar metallicities (⁠|$-0.5 \lesssim [\mathrm{ Fe/H}] \lesssim 0.5$|⁠) (Kharchenko et al. 2013; N2016; Dias et al. 2021 [hereafter D2021]). Their colour–magnitude diagrams (CMDs) contain the fiducial positions of evolving stars of different masses according to their evolutionary stages at a given age, allowing us to study in detail the stellar evolution.

The separation between fiducial OCs members from field stars, especially in high stellar density environments, like the Galactic disc, is a challenging task. However, due to the high precision of the astrometric and photometric data provided by the most recent releases of the Gaia catalogue (DR2: Evans et al. 2018; Lindegren et al. 2018; EDR3: Gaia Collaboration 2021, DR3: Gaia Collaboration 2023) it has been possible to disentangle both cluster and field populations, allowing a drastic increase of works providing precise member lists of OCs (Cantat-Gaudin et al. 2018; Angelo et al. 2019, 2021; D2021). Combined with efficient methodologies, a number of new objects has been discovered (Ferreira et al. 2019, 2020, 2021; Sim et al. 2019; Castro-Ginard et al. 2020; He et al. 2023; Hunt & Reffert 2023 [hereafter H2023]).

By taking advantage of the resulting decontaminated CMDs from such precise member star lists, the stellar populations from OCs can be better studied. An important feature, known as the red clump (RC), characteristic of intermediate age and older stellar populations whose more massive stars are passing through the evolutionary stage of Helium burning in the core, can be easily noticed (see figs 2, 3, and 10 from Gaia Collaboration 2018). The RC is a concentration of cold and luminous stars which occurs after the red giant branch (RGB) phase in the CMD, and is the longest subsequent evolutionary stage in the life of a star, after the main sequence (MS). The RC is not often observed in the CMD of very young clusters, because the He-burning phase is rapid in the more massive stars, thereby it has been reported for clusters within the age interval of |$8.3 \lesssim \log [t({\rm yr})] \lesssim 10.0$| (Grocholski & Sarajedini 2002; van Helshoecht & Groenewegen 2007; Onozato et al. 2019).

The RC is widely used as a standard candle, because although the colours of stars in this evolutionary phase depend strongly on age and metallicity, for low-mass stars (⁠|$M\lesssim 1.9 {\rm M}_{\odot }$|⁠), the absolute magnitude do not show large variations, especially in the infrared. In this way, by establishing a reliable RC mean magnitude value for old populations (⁠|$\log [t({\rm yr})] \gtrsim 9.0$|⁠), we are capable to determine distances to structures within our Galaxy and to neighbouring galaxies (Grocholski & Sarajedini 2002; Helshoecht & Groenewegen 2007; Bilir et al. 2013; Girardi 2016; Onozato et al. 2019).

For intermediate age and older clusters |$8.3 \lesssim \log [t({\rm yr})] \lesssim 10.0$|⁠, the average position of the RC stars in the CMD can also be used as an age indicator. Indices based on CMD morphology are common in the literature. For example, the difference in magnitude |$\Delta V$| between the MS turn-off (bluest point on the MS) and the mean magnitude of the RC tend to increase with age for star populations older than 1 Gyr. On the other hand, the difference in colour |$\Delta (B-V)$| of the same regions tend to decrease with age (Anthony-Twarog & Twarog 1985). A morphological age index called MAR is also defined by Anthony-Twarog & Twarog (1985), taking the ratio |$\Delta V$|/|$\Delta (B-V)$|⁠.

Phelps, Janes & Montgomery (1994) defined the morphological age indices |$\delta V$| and |$\delta 1$|⁠. The index |$\delta V$| is the difference between the inflection point of the MS or the base of the giant branch and the mean magnitude value of the RC, while |$\delta 1$| is the colour difference between the point one magnitude brighter than the turn-off and the base of the giant branch. The index |$\delta V$| established in Phelps et al. (1994) was later calibrated with the ages and metallicities of star clusters in Salaris, Weiss & Percival (2004).

Such indices based on the CMD morphology are common in the literature and are well-known age indicators for Galactic clusters (Anthony-Twarog & Twarog 1985; Phelps et al. 1994; Salaris et al. 2004; Beletsky, Carraro & Ivanov 2009; Piatti, Clariá & Ahumada 2010; Oralhan et al. 2015) and Magellanic Clouds clusters (Geisler et al. 1997; Parisi et al. 2014). Those age determination techniques are independent of distance and reddening, but it is necessary the consistent identification of those key evolutionary features in the CMDs.

In this work, we used data from the 2MASS (Two Micron All Sky Survey, Skrutskie et al. 2006) and Gaia DR3 catalogues to establish morphological age indices for a set of Galactic OCs, extending the morphological age indices for OCs as young as |$\log [t({\rm yr})]=8.3$|⁠. We adopted the magnitude difference between the turn-off point and the RC, an index widely employed as an age indicator, both in the visible (⁠|$\Delta V$|⁠, Carraro & Chiosi 1994) and in the infrared (⁠|$\Delta K$|⁠, Beletsky et al. 2009; Zasowski et al. 2013). We also present age calibrations with those indices, using the 2MASS index |$\Delta K$| as a benchmark.

This paper is structured as follows. In Section 2, the data are presented. In Section 3, the OC sample selection is described. In Section 4, the analysis procedures are discussed, including membership assessment and determination of mean astrometric parameters. The CMDs properties of our OC sample are explored in Section 5. The main results are presented in Section 6 and the concluding remarks are given in Section 7.

2 DATA

2.1 Gaia

The Gaia DR3 catalogue provides positions, proper motions in right ascension and declination, parallaxes, and photometry (G, |$G_{\mathrm{ BP}}$|⁠, and |$G_{\mathrm{ RP}}$| passbands) for nearly two billion sources. The Gaia@AIP (https://gaia.aip.de/) online services have been used to extract Gaia DR3 data for each OC in circular regions centred on the coordinates presented in D2021 catalogue. For a first guess, we adopted an extraction radius of 1, 2, and 3 deg for OCs of heliocentric distances |$d>2\, {\rm kpc}$|⁠, |$1\, {\rm kpc} < d < 2\, {\rm kpc}$|⁠, and |$d < 1\, {\rm kpc}$|⁠, respectively. Those sizes were large enough to restrict the studied OCs and an adjacent comparison star field. Confirmed OCs in D2021 present mean tidal radius (⁠|$r_{t}$|⁠) of |$9.85\, {\rm pc}$| with standard deviation of |$5.21\, {\rm pc}$| (Kharchenko et al. 2013). Assuming a typical superior limit in OC size of |$r_{t} \sim 15\, {\rm pc}$|⁠, an OC would appear to have apparent radii of |$\sim 26$| arcmin at |$2\, {\rm kpc}$|⁠, |$\sim 52$| arcmin at |$1\, {\rm kpc}$|⁠, and |$\sim 104$| arcmin at |$500\, {\rm pc}$|⁠. This means that those extracting radii are capable to encompass a typical OC at those ranges of distances.

To extract the data with the corrected version of the photometric flux excess factor (Riello et al. 2021), we adopted the query examples presented in Gaia EDR3 documentation (appendix B, Gaia Collaboration 2021). We also adopted a filter to keep only stars with |$G < 19$| to avoid less informative sources, which is also the nominal magnitude limit to ensure astrometric and photometric completeness (Lindegren et al. 2021).

In order to correct the original data for the parallax zero-point bias, we applied the available recipe presented in Lindegren et al. (2021), which is applied equally to Gaia EDR3 and to Gaia DR3 astrometry. This correction is provided separately for sources with available five- and six-parameter astrometric solutions and is given as a function of the source magnitude, colour, and celestial position. No further corrections in flux or G-band magnitudes have been applied, since they are already implemented in DR3.

We applied quality filters limiting our database to remove spurious astrometric and photometric solutions by keeping sources consistent with the two following equations:

where |$C^{*}$| is the corrected value of the BP and RP flux excess factor, |$\sigma _{C^{*}}$| is given by equation (18) of Riello et al. (2021), and RUWE is the renormalized unit weight error (Lindegren et al. 2021).

2.2 2MASS

The 2MASS provides positions and near-infrared photometry (J, H, and K passbands) for nearly 470 million sources, covering the entire sky. For each target, we have used the Vizier service to extract data from the 2MASS catalogue for stars inside the circular regions quoted above. To ensure quality to our sample, we only extracted stars with 2MASS JHKs photometric quality flag |$^{\prime }AAA^{\prime }$|⁠. We then cross-matched Gaia catalogue with 2MASS by selecting 2MASS sources within 1 arcsec from Gaia sources. The photometric depth of our catalogues is governed by Gaia photometry, in other words, if a star from Gaia do not have a counterpart in 2MASS survey, this star is not excluded.

3 THE OC SAMPLE

It is beyond the scope of this work to determine the astrophysical parameters of the OCs (age, distance, colour excess, and metallicity). Therefore, we decided to use the OC catalogue published by N2016, which contains a homogenized sample of 172 OCs with averaged ages and metallicities, based on [Fe/H] abundances from numerous individual studies, as well as their associated uncertainties. To select our OC sample, we restricted their catalogue to objects closer than 3 kpc from the Sun according to D2021 and for which metallicities were taken from high-resolution spectroscopy. For the remaining sample, we kept OCs for which we could visually identify in the CMD a concentration of stars around the RC position and for which the turn-off points are brighter than |$G = 19$|⁠, resulting in a final sample of 34 OCs, covering |$8.3< \mathrm{ log}t[\mathrm{ yr}]< 9.9$| and |$-0.44<[\mathrm{ Fe/H}]<0.37$|⁠.

The catalogue presented in N2016 is widely adopted, especially as a reference for metallicity (Bossini et al. 2019; Chen & Zhao 2020; Zhang, Chen & Zhao 2021; Im et al. 2023) and recent works still adopt their age values (Chen & Zhao 2020). To ensure the quality of their data as age references for the range of ages and distances explored in this work, we compared the selected sample of OCs with recent catalogues based on Gaia data (C2020; D2021; H2023). The comparison between the age values reported by these catalogues is presented in Table 1. The age values in N2016 do not show significant offsets from recent literature values, and their comparison with recent catalogues exhibits similar correlations and residuals, as observed when catalogues based on Gaia photometry are compared with each other. The only discrepant age value found was for the OC NGC2354, whose age value was not included in the comparisons. A discussion concerning this particular OC is presented in Appendix A1.

We have also compiled distances and colour excesses from D2021. Positions, ages, metallicities, distances, and colour excesses of our sample are presented in Table 2.

4 METHODOLOGY

To assess memberships and remove the field population from the clusters sample, we have employed proper motion and parallax selections of members. For this purpose, we have developed a methodology based on Ferreira et al. (2019) that includes:

• preliminary analysis of vector point diagrams (VPDs) to determine the mode of the cluster’s proper motion distribution for both components;

• construction of a proper motion mask around the determined proper motion mode, to create a subsample almost free of contamination from field stars;

• determination of the cluster centre and radius by building radial density profiles (RDPs);

• Gaussian fittings over the proper motion components and parallaxes distributions and filters restricting stars based on such distributions.

4.1 The clusters proper motion detection

To find the OCs signatures in the VPD, we used the same method adopted in Ferreira et al.(2019, 2020, 2021), where a colour filter is applied on the sample to discard very reddened field stars and maximize the contrast between cluster and field population. We started with a colour threshold value |$G_{\mathrm{ BP}}-G_{\mathrm{ RP}}< 2.5$|⁠. For the cases of more distant and highly reddened clusters, we increased this threshold value and, for cases of less reddened ones, this value is decreased, in order to make the initial detection of the cluster evident as an overdensity in the VPD. We then computed the mode of the proper motions in right ascension (⁠|$\mu _{\alpha }^{*}$|⁠) and declination (⁠|$\mu _{\delta }$|⁠).

In Fig. 1, we show how we identified the proper motion signature for the OC IC 4756. The top-left panel shows a VPD for all stars within 2 deg from its centre, containing 629 323 stars, showing that without any filter, we are not capable to find the cluster. In the top-right panel, a colour filter is represented by the red line |$G_{\mathrm{ BP}}-G_{\mathrm{ RP}}<1.20$|⁠, selecting the cyan sample, which corresponds to 8402 stars. In the bottom-left panel, the VPD with the sample filtered by colour (yellow and brown samples) is plotted over the entire sample (grey dots), exhibiting an overdensity of stars corresponding to the cluster, where the red lines mark its proper motion modes at |$\mu _{\alpha }^{*}=1.28$|  mas yr|$^{-1}$| and |$\mu _{\delta }=-4.98$|  mas yr|$^{-1}$|⁠. The same VPD with the colour filtered sample, including histograms where the proper motion modes are indicated, is shown in the bottom-right panel.

4.2 The proper motion filter

In order to estimate how the dispersion in proper motion components of real OCs behave (on average) as function of distance, we used the recent catalogue published by H2023, where the OCs members were obtained by using HDBSCAN algorithm over Gaia DR3 data. We built a sample by limiting the OCs to distances between 40 pc and 6 kpc and number of members larger than 200. The catalogue used provides, for each OC, its position, astrophysical parameters, and the mean astrometric parameters and dispersions presented by its probable members. We then separated the OCs into intervals of 50 pc and calculated the mean proper motion dispersion values for the entire group of OCs within each distance interval. In this procedure, we established a small catalogue of distances and the expected OC proper motion dispersions at those distances.

As shown in Fig. 2, we note that the nearest clusters tend to exhibit higher dispersion values, due to the apparent random motion of the stars being greater than the proper motion uncertainties. On the other hand, clusters located at distances farther than 2 kpc tend (on average) to exhibit an approximately fixed value, which shows that beyond this limit the physical dispersion of proper motions tends to be negligible in face of astrometric errors. In this way, we adopted an exponential fitting over these data to reproduce this behaviour for both proper motion components (Fig. 2).

This procedure was carried out to construct proper motion masks of sizes that fit the clusters proper motion spread for different distances. From the mean values |$\sigma _{\mathrm{ rep}}$| of proper motion dispersion, we adopted boxes with sides equal to 20 times this value, that is, we limited the samples of stars within 10 |$\sigma _{\mathrm{ rep}}$| around the mode of the proper motion distributions for both components (Fig. 3). The adopted sizes of the proper motion masks were:

where |$D_{{\rm {cluster}}}$| is the distance (in pc) according to D2021 and |$L_{{\rm {pmra}}}$| and |$L_{{\rm {pmde}}}$| represent the sizes of the mask in proper motion units in right ascension and declination, respectively. For subsequent analyses, we will abandon any colour filters, as they obviously exclude very cool low MS stars from nearby clusters and possible RGB stars for some older clusters. This procedure defines our database. We restricted the OCs proper motion space by employing the box-shaped filter centred on the modal values of |$\mu _{\alpha }^{*}$| and |$\mu _{\delta }$|⁠, as shown in the top panels of Fig. 3.

4.3 Centre and radial density profile

After establishing the size of the proper motion mask for our OCs sample, we applied it to the original database to reduce the contamination by field stars. We constructed frequency histograms with the equatorial coordinates of the stars and estimated the centre values of the clusters in both coordinates through Gaussian fittings (taking the average values of the fit). This average value is used as a first guess for the central coordinates to build the RDPs, however the final values are based on the best RDP constructed.

We built the RDPs by counting stars within concentric rings with same thickness as function of the distance to the cluster centre. We repeated this procedure for four ring thicknesses (75, 100, 125 and 150 arcsec, except for the closest clusters, for which larger bin sizes were used) in order to mitigate binning effects on the density distribution. We also computed the density values of the background far from the centre of the cluster. Then, we determined the value of the cluster’s limiting radius (⁠|$r_{\mathrm{ lim}}$|⁠) as the distance at which the density level reaches the mean value computed for the sky background (bottom panels, Fig. 3).

In this procedure, we applied small variations in the coordinates of the centre so that the density profile had a well-defined central maximum. The radius containing 50  per cent of the members (⁠|$r_{50}$|⁠) was also determined. The OCs Ruprecht 147, NGC 752, IC 4756, NGC 2527, and IC 4651 have their spatial distributions significantly affected by field stars due their projection towards dense star fields and/or due to the fact that they are poor stellar concentrations. Therefore, to build the RDPs for those objects, we adopted a subsample with parallaxes above 0.9 mas to increase the contrast with the field population. Taking into account the expected dispersion in parallax, this selection did not remove members from the respective OCs.

4.4 Two-dimensional proper motion filter and parallax filter

In order to obtain precise member star lists, we established a filter capable of better predicting the morphology of the distribution of stars in the VPD, since the box-shaped masks determined previously tend to encompass regions substantially larger than the dispersions expected for the clusters.

For this purpose, we constructed a 2D histogram of the VPD with the samples restricted by the limiting radius determined for the cluster. Then we also constructed 2D histograms for an adjacent concentric annular stellar field with the same area as the cluster: the internal radius of the control field is 1.3 times greater than the limiting radius. In order to remove the contribution of field stars in the proper motion space, we subtracted the histograms and performed 2D Gaussian fittings over the resulting ones. Subsequently, those stars with proper motion components outside 3|$\sigma$| of the 2D Gaussian mean values, considered as proper motion outliers, were removed from the sample.

To discard remaining probable field stars with discrepant parallaxes from the average OC parallax, we performed a 1D Gaussian fitting to the parallax distribution of the sample filtered by proper motion, limiting them to 3|$\sigma$| of the average value (Fig. 4).

4.5 Member lists

Our OCs member lists were obtained through the filters mentioned above. The panels in Fig. 4 show the members proper motions and parallaxes distributions and the cleaned CMD. We compared our OCs member lists with those of recent works (H2023; Alfonso, García-Varela & Vieira 2024 [hereafter Alf2024]) that used Gaia DR3 data to perform membership determination using similar techniques (HDBSCAN). Alf2024 provides member lists for OCs within 1 kpc. We compared our OCs with those present in both catalogues, resulting in 8 OCs that span a considerable range of sizes and numbers of members. Table 3 exhibits a comparison between the mean astrometric parameters derived in the studies for coincidental OCs. The mean astrometric parameters derived in this work are in good agreement with the literature, with no significant offsets in the compared values. We also compared the number of member stars in our study (⁠|$N_{\mathrm{ members}}$|⁠) with those reported in the literature, as shown in Fig. 5. Our number of members differs from that given by Alf2024, which did not apply a spatial restriction to their OCs, resulting in clusters that appear to encompass more members. In contrast, H2023 reported the total number of members within the tidal radius, which is in good agreement with our results (see Fig. 5).

The Galactic coordinates l and b, |$r_{\mathrm{ lim}}$|⁠, |$r_{50}$|⁠, proper motions in right ascension and declination (⁠|$\mu _{\alpha }^{*}$| and |$\mu _{\delta }$|⁠), parallaxes (⁠|$\varpi$|⁠), and their dispersions and the number of probable members N are presented in Table 4. Tables containing member lists and all parameters determined for our sample of OCs are available electronically through Vizier.¹

5 RC AND TURN-OFF POSITIONS

The position of the RC in the CMD is often obtained by applying a simple filter over the stars distribution to take its mean value in magnitude and/or colour (Grocholski & Sarajedini 2002; Helshoecht & Groenewegen 2007; Beletsky et al. 2009; Onozato et al. 2019). We determined the mean values and dispersion of colour (⁠|$G_{\mathrm{ BP}}-G_{\mathrm{ RP}}$| and |$J-K$|⁠) and magnitudes (G and K) of the RC by restricting the sample in a box with width of 1 mag for G, 0.4 mag for |$G_{\mathrm{ BP}}-G_{\mathrm{ RP}}$|⁠, 1.4 mag for K, and 0.4 mag for |$J-K$|⁠. Those sizes were chosen due to the visual dispersion of the stars in the RC of both visible and infrared CMDs, encompassing the region defined by RC stars. As uncertainties in the colour and magnitude of the RC were adopted the corresponding standard deviation from the mean.

We considered the bluest point of the MS as the turn-off position. However, some OCs of our sample present blue straggler stars and/or remaining field stars in the final member list that have colour indices bluer than the expected turn-off position. In order to remove those isolated stars, we identified the number of nearest neighbours of each star from the CMD within squared boxes of width 0.05 mag. We then removed stars with few close neighbours and, from the remaining sample, measured the colour and magnitude values of the bluest star left in the MS.

For CMDs with Gaia filters, the removal of stars with only one close neighbour worked for most of our sample, and for only four OCs we had to remove stars with more than three close neighbours. On the other hand, the 2MASS photometry presents larger errors and the removal of stars with one, two or three close neighbours worked for most of our sample, although in some cases we had to elevate this cut-off value (for five OCs we had to remove stars with more than six neighbours). To provide uncertainties, we adopted the standard deviation value of colour and magnitude of the five bluest remaining MS stars. For 2MASS data, we did not determine the turn-off position for the OCs NGC 6819, Trumpler 5, and Collinder 261 due the observational limit, which is brighter than the turn-off of such distant and old objects. The scheme used to identify the turnoff positions and RC average magnitudes is summarized in Fig. 6.

We also applied a similar procedure for these same evolutionary regions to the parsec isochrones (Bressan et al. 2012), with ages and metallicities representative of our OC sample. Initially, we selected stars where the isochrone table label identifier was equal to 1 (MS) and 4, 5, and 6 (He-burning stars  = RC). The turn-off position was taken as the bluest point of the MS, however we rejected the blue hook-like structure, that represents stars at the end of the MS, in order to keep consistency with the age range analysed.

The RC positions according to the parsec isochrones correspond to the point of maximum effective temperature (similar to those displayed in fig. 8 of Ruiz-Dern et al. 2018). Fig. 7 shows the procedure applied for different ages and the result for the entire range of ages and metallicities. The calculated RC colours and magnitudes, morphological age indices (see the next section) and their uncertainties are presented in Tables A1 and A2, respectively. Detailed tables with those parameters and all other useful information concerning our OC sample and also the calculations performed on the models are available electronically through Vizier.²

6 RESULTS AND DISCUSSION

6.1 The RC position

From the mean values of colour and magnitude for RC stars identified in Section 5, we obtained their absolute values. We took distances and colour excesses from D2021 to obtain the distance modulus and the interstellar extinction for each passband. To convert colour excesses values into interstellar extinction, we used values from the relationship |$A_{V}/A_{\lambda }$| based on an established extinction law (Cardelli, Clayton & Mathis 1989), with |$R_{V}=3.1$|⁠. Using these values, we obtained the absolute RC magnitudes |$M_{G}$| and |$M_{K}$| and intrinsic colour indices |$(G_{\mathrm{ BP}}-G_{\mathrm{ RP}})_0$| and |$(J-K)_{0}$|⁠.

We verified the variation of the RC absolute magnitude and intrinsic colour as a function of age, as can be seen in Fig. 8. We noted that for both visible (Gaia, upper panels) and infrared filters (2MASS, lower panels), RC stars tend to get significantly dimmer with age, within |$8.3<\log [t({\rm yr})]< 9.0$|⁠. Through this age interval, according to Gaia filters, the RC colour index seems not to change significantly, on the other hand, according to the infrared filters, the RC tends to get hotter. From |$\log [t({\rm yr})] \sim 9.0$| onwards, for both visible and infrared filters, the RC tend to become cooler and a slightly brighter until the age of |$\log [t({\rm yr})] \sim 9.2$|⁠. For older OCs, the values of |$M_{G}$| and |$M_{K}$| tend to remain approximately constant, but |$M_{K}$| is less affected by metallicity and exhibits a tighter distribution for older RC populations. On average the RC position tends to become cooler with age, in agreement with the parsec models and the analysis in Grocholski & Sarajedini (2002) for infrared bands.

The main structures formed in observational Hertzsprung-Russell (HR) diagrams of field stars have already been pointed out in the literature (see fig. 10 from Gaia Collaboration 2018). For example, the secondary red clump (SRC) is a structure more extended in its bluest part towards fainter magnitudes than the RC. It tends to appear around |$(G_{\mathrm{ BP}}-G_{\mathrm{ RP}})_{0} = 1.10$|⁠, |$M_{G} = 0.60$| in observational Gaia HR diagrams, which corresponds to younger more massive RC stars. Indeed, the RC magnitude and colour from our observed sample reflect such extended structure. In the left panels of Fig. 8, we see this structure in Gaia passbands around the expected position and in 2MASS passbands this structure is more remarkable and lies around |$(J-K)_{0} = 0.53$| and |$M_{K} = -1.30$|⁠. According to our data, those RC stars are comprised in the age range of |$8.80<\log [t({\rm yr})]< 9.20$|⁠.

A bluer and vertical structure that is called the vertical red clump is also present, in which core-helium-burning stars that are even more massive are more luminous than the RC and lie still on the blue part of it. This structure is present in the left panels of Fig. 8, for OCs younger than |$\log [t({\rm yr})]=8.80$| where the absolute magnitude decreases strongly with age.

In general, for both visible and infrared wavelengths, the values of |$M_{G}$| and |$M_{K}$| as a function of age are well represented by parsec isochrones (second column of panels in Fig. 8), especially for the older RC population (Ruiz-Dern et al. 2018; Onozato et al. 2019). We note that the infrared models tend to present values of |$M_{K}$| systematically brighter for objects younger than |$\log [t({\rm yr})]\sim 9.0$|⁠, as seen in figs 6 and 7 from Helshoecht & Groenewegen (2007).

When comparing the RC colour indices (third column of panels in Fig. 8), we noticed that the nearly solar metallicity models represent well the population older than |$\log [t({\rm yr})]\sim 9.0$|⁠. On the other hand, they are systematically redder than the observed younger OCs. The RC colours, according to parsec isochrones, also exhibit a minimum at |$\log [t({\rm yr})]\sim 9.0$| for metallicities |$[\mathrm{ Fe/H}] \gtrsim -0.2$|⁠, however this trend is better followed by infrared data compared with visible data (right panels of Fig. 8). This discrepancy with the models was already noted in Piatti, Clariá & Bica (1998) and can also be observed in recent works with objects of age |$\log [t({\rm yr})] \lesssim 8.8$| (Piatti et al. 2011; Bossini et al. 2019; Martinez et al. 2020; Holanda et al. 2022), in which the theoretical isochrones, when well fitted to the MS, do not pass through the RC stars, predicting an RC locus systematically redder. Such discrepancies may be caused by a combination of effects such as: unresolved binary stars, mass distribution of giant parent clump stars (including mass loss), differential reddening or the presence of SRC stellar populations simultaneously with the RC for some objects, making the observed average RC value bluer. Alternatively, this may indicate that the models still have deficiencies in bolometric corrections, colour transformations, and effective temperatures in this age range (Girardi 1999; An et al. 2019; Sandquist et al. 2020).

Our sample of clusters has approximately solar metallicity, with an average value of |$[\mathrm{ Fe/H}]=0.02$| and |$\sigma _{[\mathrm{ Fe/H}]}=0.12$| dex, with few clusters outside this range. We note that OCs with metallicities that are more discrepant from the average tend to present large variations in RC colour, but more moderate variations in magnitude, mainly for the infrared, in agreement with the expectation that the average RC magnitude is a good indicator of distance even with variations in metallicity.

We also determined the average RC magnitude for our sample by adopting an interval of ages less affected by population effects. For this purpose, we used objects older than |$\log [t({\rm yr})]= 9.2$|⁠, a similar value adopted in Grocholski & Sarajedini (2002) to estimate the mean RC value for |$M_{K}$|⁠. We found |$M_{G}=0.42 \pm 0.05$| and |$M_{K}=-1.66 \pm 0.04$|⁠, which is in good agreement with the literature (Table 5).

6.2 Morphological age indices

In this section, we investigated how turnoff-RC differences relate to age and metallicity. Using the quantities determined in Section 5, we determined the indices |$\Delta G$| and |$\Delta K$| as the difference in magnitude between the RC and the turnoff, as well as the colour difference |$\Delta BR$| and |$\Delta JK$| for both Gaia and 2MASS passbands, respectively.

The top panels of Fig. 9 show how the indices |$\Delta G$| and |$\Delta BR$| are related with age and metallicity and the bottom ones illustrate the same relations for |$\Delta K$| and |$\Delta JK$| indices. It is evident that there appear to be two approximate linear relationships of |$\Delta G$| with |$\log [t({\rm yr})]$|⁠: one for objects with ages younger than |$\log [t({\rm yr})] \sim 8.8$| and another for objects older than this limit. Our older OCs present a similar linear trend to that determined by Phelps et al. (1994) for the indices in the planes |$\delta _{1}(BV)$| versus |$\delta V$| and |$\delta _{1}(VI)$| versus |$\delta V$|⁠.

At least in the range spanned by OCs in our sample, metallicity does not appear to affect strongly the |$\Delta G$| values for |$\log [t({\rm yr})]>9.0$|⁠, which has already been verified by other authors (Carraro & Chiosi 1994; Phelps et al. 1994; Salaris et al. 2004; Beletsky et al. 2009). This effect is confirmed by the overplotted parsec models.

According to our data and the parsec models, the index |$\Delta BR$| also shows little dependence on metallicity for objects older than |$\log [t({\rm yr})] \sim 8.8$|⁠. The models indicate a more significant dependence on metallicity for younger objects (see Fig. 9, top-right panel).

In general, the models show good agreement with the data on the relation |$\Delta G$| versus |$\log [t({\rm yr})]$|⁠. Regarding the values of |$\Delta BR$|⁠, apparently for younger objects (⁠|$\log [t({\rm yr})]< 8.8$|⁠), the isochrones tend to overestimate this index, which is related to the prediction of redder RC in this age interval (see Section 6.1).

Regarding the infrared indices |$\Delta K$| and |$\Delta JK$| (bottom panels in Fig. 9), it is possible to note trends analogous to those for Gaia indices, but both indices present a wider distribution when compared with Gaia ones due the larger photometric errors. In general, parsec models show good agreement with the data for the relation |$\Delta K$| versus |$\log [t({\rm yr})]$|⁠. However, the index |$\Delta JK$| seems to be much more affected by the 2MASS colour indices errors, especially for older OCs (⁠|$\log [t({\rm yr})]\gtrsim 9.0$|⁠), which present a distribution within |$0.4 <\Delta JK< 0.5$|⁠. The difference |$\Delta JK$| of our OCs exhibits a range of |$\sim 0.35$| mag (considering the overall age interval), although the average error of the colour index |$(J-K)$| for our clusters members are about |$\sim 0.05$| mag. We did not see the same problem with Gaia photometry with the index |$\Delta BR$|⁠, because this index from our OCs star members spreads over a wider range (⁠|$\sim 1$| mag), where the mean colour index |$(G_{\mathrm{ BP}}-G_{\mathrm{ RP}})$| error is about |$\sim 0.008$| mag, in other words this index is not very affected by photometric errors.

We also calculated a similar age index to MAR, defined in Anthony-Twarog & Twarog (1985), taking the ratio |$\Delta G$|/|$\Delta BR$|⁠. We do not define the same index for 2MASS bands due to the scatter observed in index |$\Delta JK$|⁠. The same ratio was determined for parsec isochrones and a comparison with data is shown in Fig. 10. Despite the discrepancies of the index |$\Delta BR$| for younger OCs, the index MAR is very well reproduced by parsec isochrones, with the exception of OCs older than |$\log [t({\rm yr})]>9.5$|⁠. In Anthony-Twarog & Twarog (1985), a linear correlation with age is obtained from MAR (ratio |$\Delta V$|/|$\Delta (B-V)$|⁠) for objects older than 2 Gyr (⁠|$\log [t({\rm yr})]=9.3$|⁠), however, our work shows that this relation is clearly non-linear, especially for objects younger than 1 Gyr and older than 3 Gyr.

6.3 Age calibrations

As far as we are aware, age calibrations for Gaia data (⁠|$\Delta G$| and |$\Delta BR$| indices) are not yet available in the literature, so here we will provide them for both Gaia morphological indices for the first time. We also performed calibrations for the |$\Delta K$| index as a benchmark. In Fig. 9, it is possible to note that both indices |$\Delta G$| and |$\Delta K$| increase with age for OCs older than |$\log [t({\rm yr})] = 8.8$|⁠, for which we determined calibrations [equations (5) and (6)] similar to those present in Beletsky et al. (2009). In addition, we determined two more calibrations for |$\Delta G$|⁠: quadratic in |$\Delta G$| (equation 7) and a relation that takes into account the metallicity (equation 8). Fig. 11 summarizes all these calibrations. Mean residuals, correlation coefficients, and the age range of the calibration equations established in this work are present in Table 6.

Regarding the age calibration for the |$\Delta K$| index (equation 6), we see a very good agreement with Beletsky et al. (2009), showing that our method provides similar results in comparison with those obtained by visual inspection of the CMDs, which validates its application on Gaia data. As seen in Section 6.2 and already evidenced by other authors for visible and infrared morphologycal age indices |$\Delta V$| and |$\Delta K$| (Carraro & Chiosi 1994; Phelps et al. 1994; Salaris et al. 2004; Beletsky et al. 2009), the dependence of the index |$\Delta G$| with metallicity is small (see Fig. 9), so that in many cases the metallicity term can be neglected. As accurate metallicity determinations are scarce in the literature, equations (5) and (7) become useful tools for determining ages of Galactic OCs whenever their turnoffs and RC positions are measured with good accuracy on their CMDs.

6.4 General formula

As seen in Fig. 9, when we took into account the relation |$\log [t({\rm {yr}})]$| versus |$\Delta G$|⁠, we realize that objects with ages in the range |$8.3 < \log [t(\mathrm{ yr})] < 8.8$| may have the same |$\Delta G$| values as those of |$8.8 < \log [t(\mathrm{ yr})] < 9.5$|⁠. However, the parameter |$\Delta BR$| has an almost linear dependence on age, i.e. younger objects exhibit greater |$\Delta BR$| than the older ones. In this case, using the index |$\Delta BR$| breaks the degeneracy of the |$\Delta G$| values as an age indicator. Thus, by extending the age calibration to the interval |$\sim 8.3 < \log [t(\mathrm{ yr})] < 9.0$|⁠, we were able to establish a fitting function with quadratic and linear terms in both indices |$\Delta G$| and |$\Delta BR$| [equation (9) and Table 7]. A comparison of OCs ages from our sample with that determined using equation (9) is shown in Fig. 12, where we note small residuals. Therefore, equation (9) represents an important tool to determine OCs ages within a wide range using Gaia data.

7 CONCLUSIONS

In this work, we were able to obtain member star lists for a set of 34 star clusters from accurate astrometry and photometry from Gaia DR3 data using a uniform methodology. CMDs were constructed and morphological age indices were measured.

We presented an observational view of the RC population through the interval |$8.3< \log [t({\rm yr})]< 9.9$|⁠. In general, for both visible and infrared, the values of the absolute magnitudes of the RC, |$M_{G}$|⁠, and |$M_{K}$| as a function of age are well represented by parsec isochrones. We note that for the infrared the models tend to present values of |$M_{K}$| systematically brighter for younger objects.

When comparing the RC colour indices, we noticed that the models represent well the older population (⁠|$\log [t({\rm yr})]> 9.0$|⁠). However, they are systematically redder than observed younger OCs. The same conclusions were drawn by other authors for objects with age |$\log [t({\rm yr})] \lesssim 8.8$| (Piatti et al. 2011; Bossini et al. 2019; Martinez et al. 2020; Holanda et al. 2022). Such discrepancies may be caused by a combination of different effects. For example, the presence of unresolved binary stars, the mass distribution of the RC progenitors stars (including mass loss), presence of differential reddening, or the presence of SRC stellar populations simultaneously with the RC for some objects, making the observed average RC colour bluer than expected. Alternatively, this may indicate that the models still present limitations such as bolometric corrections, colour transformations, and effective temperatures for this age range. Uncertainties of transformations from the theoretical to the observational plane also play a role.

Comparisons of the established morphological indices showed that the models tend to satisfactorily predict the |$\Delta G$| index for the entire age range explored here. But for ages |$\log [t({\rm yr})]< 8.8$|⁠, the models tend to present the index |$\Delta BR$| systematically greater than observed (probably an effect caused by the inefficiency of predicting the RC colours for younger ages. According to our data, the models also face challenges in accurately predicting the index |$\Delta BR$| for older objects (⁠|$\log [t(\mathrm{ yr})] > 9.6$|⁠), although few OCs are that old to provide a useful comparison.

Finally, we provided a set of age calibration functions based on Gaia morphological indices for the first time, allowing an estimate of star cluster ages based on such indices. In particular, a non-linear fitting function was obtained by using both indices |$\Delta BR$| and |$\Delta G$|⁠, extending the age determination using such method to younger objects. We have demonstrated its accuracy for the range |$8.3<\log [t({\rm yr})]< 9.8$| by a direct comparison with ages from the literature, obtaining a mean residuals of 0.06 dex in |$\log [t({\rm {yr}})]$|⁠.

Although out the scope of the present paper, asteroseismology is an alternative approach to obtain precise ages of OCs, from which new colour calibrations can be derived. However, due to the lack of OCs with ages measured via asteroseismology (e.g. Brogaard et al. 2023; Ash et al. 2025), we still face difficulties incorporating a significant number of objects with isochrone-independent age measurements. In the future, as a sizeable sample of stars in several clusters is measured using asteroseismology, the results from this age-dating method may then be used to establish new colour calibrations.

ACKNOWLEDGEMENTS

We thank the referee for a critical, positive, and motivating report. The authors wish to thank the Brazilian financial agencies FAPEMIG, CNPq, and CAPES (finance code 001). WJBC acknowledges the support from CNPq–BRICS 440142/2022-9, FAPEMIG APQ 02493–22, and FNDCT/FINEP/REF 0180/22. FFSM acknowledges financial support from Conselho Nacional de Desenvolvimento Cientıf́ico e Tecnológico–CNPq (proc. 404482/2021-0) and from FAPERJ (proc. E-26/201.386/2022 and E-26/211.475/2021). This research has made use of the VizieR catalogue access tool, CDS, Strasbourg, France. This work has made use of data from the European Space Agency (ESA) 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. This research has made use of topcat (Taylor 2005).

DATA AVAILABILITY

The data underlying this article is publicly available (Gaia DR3 and 2MASS) or is available in the article.

Footnotes

REFERENCES

APPENDIX A: TABLES

This appendix contains information from OCs CMDs properties. In Table A1, the average RC apparent magnitudes (G and K) and colour indices (⁠|$(G_{\mathrm{ BP}}-G_{\mathrm{ RP})}$| and |$\sigma _{(J-K)}$|⁠) and their uncertainties are presented. The number of RC stars |$N_{G}$| and |$N_{K}$| filtered by the box-shaped filter for Gaia and 2MASS CMDs is also presented. In Table A2, the morphological age indices |$\Delta G$|⁠, |$\Delta BR$|⁠, |$\Delta K$|⁠, and |$\Delta JK$| and their uncertainties are presented. A review of the astrophysical parameters for the cluster NGC2354 taken from the literature is presented in Table A3.

A1 The NGC2354 age

NGC2354 is an OC located in the constellation of Canis Major which, according to our analysis from Gaia data, has an apparent diameter of 25 arcmin and 234 members. Our analysis proved to be efficient in obtaining reliable members of this cluster, as its CMD clearly presents an evolutionary sequence without many outliers (see Fig. A1) and the dispersion values of the members in astrometric space were compatible with those of the other clusters with similar distances (see Table 4).

However, we note that the age value used in this work (established in N2016) proved to be underestimated in relation to recent age values established in the literature, based on determinations with data from Gaia. We also noticed discrepancies in the properties presented by this cluster when compared to others. This object presented very discrepant values on the plane |$\Delta G$| versus |$\Delta BR$| and in the age relation of the MAR index, when compared to objects with the same age, as can be seen in Fig. A2. This reinforces the importance of such indices in charaterizing OCs morphologically.

We carried out a review of the astrophysical parameters of this object in the literature and noticed that in studies where the authors did not use precise proper motion data, low age values were assigned to this object (see Table A3). We note that prior to the availability of data from Gaia, the determinations of age, distance, and colour excess, on average, were: |$\overline{\mathrm{ log}[t]}=8.35$| and |$\sigma _{\log [t({\rm yr})]}=0.27$|⁠, |$\overline{d}=3027$| pc and |$\sigma _{d}=915$| pc, and |$\overline{E(B-V)}=0.31$| and |$\sigma _{E(B-V)}=0.17$|⁠. It is evident that the age value is compatible with the average established in N2016 of |$\overline{\log [t({\rm yr})]}= 8.30 \pm 0.23$|⁠. With the exception of Kharchenko et al. (2013), the other authors who characterized this cluster before the Gaia era, used UBV photometry and interpreted its CMD as a much younger and more distant object.

After the availability of Gaia, the same determinations led to the following average values: |$\overline{\log [t({\rm yr})]}=9.14$| and |$\sigma _{\log [t({\rm yr})]}=0.06$|⁠, |$\overline{d}=1259$| pc and |$\sigma _{d}=84$| pc, and |$\overline{E(B-V)}=0.16$| and |$\sigma _{E(B-V)}=0.06$|⁠. A smaller fluctuation of the values and a better convergence of the distance estimate (⁠|$d=1279^{+188}_{-145}$| pc) was obtained through the average parallax of the cluster members in Cantat-Gaudin et al. (2018). For this particular object, we replaced its age from N2016 by the averaged Gaia determinations and assumed the corresponding dispersion as its uncertainty: |$\overline{\log [t({\rm yr})]}=9.14 \pm 0.06$|⁠.