Distance and age of the massive stellar cluster Westerlund 1. I. Parallax method using Gaia-EDR3

ABSTRACT

Westerlund 1 (Wd 1) is one of the most massive young star clusters in the Milky Way. Although relevant for star formation and evolution, its fundamental parameters are not yet very well constrained. We aim to derive an accurate distance and provide constraints on the cluster age. We used the photometric and astrometric information available in the Gaia Early Data Release 3 (Gaia-EDR3) to infer its distance of 4.06|$^{+0.36}_{-0.34}$| kpc. Modelling of the eclipsing binary system W36, reported in Paper II, led to the distance of 4.03 ± 0.25 kpc, in agreement with the Gaia-EDR3 distance and, therefore, validating the parallax zero-point correction approach appropriate for red objects. The weighted average distance based on these two methods results in d_wd1  =  4.05 ± 0.20 kpc (m − M  =  13.04|$^{+0.11}_{-0.12}$| mag), which has an unprecedented accuracy of 5 per cent. Using the Binary Population and Spectral Synthesis (BPASS) models for the Red Supergiants with solar abundance, we derived an age of 10.7 ± 1 Myr, in excellent agreement with recent work by Beasor & Davies (10.4|$^{+1.3}_{-1.2}$| Myr) based on MIST evolutionary models. In Paper II, W36B was reported to be younger than 7.1 Myr, supporting recent claims of a temporal spread of several Myrs for the star-forming process within Wd 1 instead of a single monolithic starburst episode scenario.

1 INTRODUCTION

The formation and evolution of massive stars is poorly known because massive stellar clusters are extremely rare within distances that their stellar population can be resolved. In this context, Westerlund 1 (henceforth Wd 1) is relevant, since it is potentially the most massive (∼ 6 × 10⁴ M_⊙; Portegies Zwart, McMillan & Gieles 2010 and references therein) young stellar cluster (4–11 Myr; Brandner et al. 2008; Gennaro et al. 2011; Beasor et al. 2021) in our Galaxy. It is a local sibling of 30 Dor, although a little denser and less massive than this one. Wd 1 is located at ∼ 10 times closer distance (3–5 kpc; Kothes & Dougherty 2007; Andersen et al. 2017; Davies & Beasor 2019; Beasor et al. 2021), making it an ideal template for investigating the massive stellar formation and evolution process in the nearby Universe, but still at a considerable large distance and extinction (A_V ∼ 11.5; Damineli et al. 2016; Hosek et al. 2018) that precluded its full characterization until now. This massive cluster harbours a large population of confirmed massive stars evolved off the Main Sequence (MS): four Red Supergiants (RSG), 24 Wolf-Rayets (WR), more than 100 Blue Giant and Supergiants OB stars, one magnetar, one Luminous Blue Variable (LBV), one B[e], and six Yellow Hypergiants (YHG) – a short-lived phase (Clark et al. 2005; Crowther et al. 2006; Muno et al. 2006; Negueruela, Clark & Ritchie 2010). Given its large population of massive stars, Wd 1 is expected to generate more than a thousand supernovae explosions, producing a significant number of X-rays and γ-rays binaries, and the coalescence of some of them will likely generate gravitational waves (Abbott et al. 2020). Indeed, several X-ray binaries were already identified due to thermal X-rays produced by wind–wind collision in massive binaries (Skinner et al. 2006). The fraction of binaries is still not precisely known but is estimated to be a relatively large fraction of the Wd 1 stellar population (> 70 per cent; Crowther et al. 2006; Clark, Ritchie & Negueruela 2019; > 40 per cent, Ritchie et al. 2022). Binarity among massive stars is relevant since > 70 per cent of them will exchange mass with the companion, frequently leading to a binary merger (Sana et al. 2012) with potential for producing gravitational waves.

The question of whether Wd 1 serves as a ‘golden-standard’ laboratory for massive stellar evolution (Clark et al. 2014) is still under debate. There is a large age discrepancy between different methods for Wd 1. The pre-MS sequence population indicates an age of ∼ 5–7 Myr (Andersen et al. 2017), much younger than that of the less-luminous RSG (∼ 11 Myr, Beasor et al. 2021, leaving the door open for more than a unique episode of star formation over time. Since extinction and distance are not well constrained for Wd 1, absolute luminosities are uncertain, impacting the determination of individual stellar ages and, thus, the age of the cluster itself.

The presence of RSGs and WRs in a coeval cluster would indicate a narrow range of possible ages in a scenario of stars evolving in isolation (∼ 5 Myr; Clark et al. 2005). However, binary interactions shape the unique cohort of hypergiant stars, with mass-stripped primaries, rejuvenated secondaries, and merger products (Clark, Ritchie & Negueruela 2020). Beasor et al. (2021) proposed that stars in Wd 1 formed over a period of several Myr, similar to R136/NGC 2070, based on the evolved population (Schneider et al. 2018). However, this cluster shows many subclusters distributed throughout its confines which does not seem to be the case for Wd 1. The same authors point to the difference between the HRD of Wd 1 and RSGC1 (Davies et al. 2008). Although RSGC1 has a total mass and the age of RSGs similar to those in Wd 1, it does not show any WR and is prone to represent a single starburst.

The reddening law and extinction were accurately determined by Damineli et al. (2016, hereafter D16) using colour indices of bright cluster members calibrated from spectral types reported by Clark et al. (2005), Ritchie et al. (2009), Negueruela et al. (2010), Clark et al. (2020) and JHK_s photometry measurements combining telescopes of different diameters (4-m, 60-cm, and mask aperture of 2-cm). Hosek et al. (2018) used JHK_s photometry and modelled the colour excesses of MS members by adopting a 5 Myr isochrone. They found good agreement between the average extinction to the cluster (A_Ks  =  0.78 ± 0.16 mag) with that reported by D16 (A_Ks  =  0.74 ± 0.01 mag). Hosek et al. (2018) also derived a steep optical/NIR reddening law similar to that of D16 and showed that it is not compatible with previous ones, even with those tuned to the inner Galaxy (Galactic Bulge, outside of the Galactic Plane), such as reported by Nishiyama et al. (2009). To date, the reddening law and the mean A_Ks extinction are the best well-constrained parameters for the Wd 1 cluster.

Previous distance determinations, however, have not converged to a ∼ 10 per cent accuracy, which is an uncertainty level desired for this case. The first suggestion for the Wd 1 distance was made by Westerlund (1961) by assuming it was located on the Sagittarius arm, at d > 1.4 kpc. Kothes & Dougherty (2007) analysed the velocity of H i absorption features on a Galactic rotation curve model to derive the Wd 1 distance as d  =  3.9 ± 0.7 kpc. Other distances were derived through colour–magnitude diagram (CMD) analysis, using isochrone fitting procedures to evaluate the distance modulus for the cluster. Westerlund (1987) reported d  =  5.0|$^{+0.5}_{-1.0}$| kpc, while Piatti, Bica & Claria (1998) derived d  =  1.0 ± 0.4 kpc. Clark et al. (2005) suggested 2 < d < 5.5 kpc through calibrated luminosity of YHG and limiting luminosity for RSGs to be lower than the Humphreys–Davidson limit. The major issue in Clark et al. (2005) distance determinations is related to the assumption of a standard reddening law (R_V  =  3.1; Cardelli, Clayton & Mathis 1989) towards the cluster, in contradiction with their photometry which indicated that OB stars had a non-standard reddening. The minimum distance (2 kpc) was based on the non-detection of radio-emission from WR stars. Crowther et al. (2006) used JHK_s photometry of WR stars obtaining d ∼ 5.5 kpc. Their adoption of Indebetouw et al. (2005) reddening law is a weak point since it does not apply to inner regions of the Galactic plane (see D16). Negueruela et al. (2010) used OB supergiants and isochrone fitting to derive a distance of d ≈ 5 kpc. Despite those results being based on accurate spectral classification of the sources, they have uncertainties from using the Rieke & Lebofsky (1985) reddening law and the assumption those are single stars, while many OB supergiants are probably binaries with similar brightness companions (Crowther et al. 2006; Clark et al. 2019). Gennaro et al. (2011) reported d  =  4.0 ± 0.2 kpc based on fit to the isochrones of pre-MS stars. Hosek et al. (2018) obtained d  =  4.4 ± 0.3 kpc by fitting the MS assuming an age of 5 Myr. The weakness of the isochrone fitting method is the large degeneracy between extinction and distance, in addition to the strong dependence on the adopted reddening law.

Astrometric studies opened very promising horizons with the Gaia mission. Indeed, astrometric measurements (position, proper motion, and parallaxes) are independent of age, extinction, and reddening law. Unfortunately, the Gaia performance for objects at distances of ∼kpc scales, especially in dusty regions in the Galactic Centre direction, is still impacted because parallax zero-points are still uncertain for very red colour indices (Lindegren et al. 2021b). For Wd 1, the systematic effects are also affected by crowding, individual motions within the cluster, and motions due to binary orbits. To test the reliability of the second Data Release of the Gaia mission (Gaia -DR2) Aghakhanloo et al. (2020) used Bayesian inference to derive the distance to both the Wd 1 cluster members and the surrounding Galactic field stars, obtaining d  =  2.6|$^{+0.6}_{-0.4}$| kpc for Wd 1. Aghakhanloo et al. (2021) performed a similar study using the Gaia Early Data Release 3 (Gaia-EDR3), obtaining d  =  2.8|$^{+0.7}_{-0.6}$| kpc. Based on the parallax of OB-type stars in the Gaia- DR2, Davies & Beasor (2019) reported d  =  3.87|$^{+0.95}_{-0.64}$| kpc. Rate, Crowther & Parker (2020) used a sample of 18 WRs and OB stars present in Gaia-DR2 to derive the distance of d ∼ 3 kpc. Beasor et al. (2021) used the same method as Davies & Beasor (2019) using Gaia-EDR3 which shows an improved zero-point parallax determination (Lindegren et al. 2021b), reporting the distance of d  =  4.12|$^{+0.66}_{-0.33}$| kpc. In order to maximize cluster membership, they selected OB-type stars within 5 arcmin from the cluster centre and excluded stars with proper motions larger than 10 km s⁻¹, which is the approximate Virial velocity for a cluster with 10⁵ M_⊙ and size of 1 pc. More recently, Negueruela et al. (2022), reported a distance of d  =  4.23|$^{+0.23}_{-0.21}$| kpc based on the astrometry and photometry of the massive stellar population of Wd 1 from Clark et al. (2020).

In this work, we fine-tune procedures to improve the accuracy of the distance determination to the Westerlund 1 cluster using astrometry and photometry from the Gaia-EDR3 cluster members reported by Clark et al. (2020). We also adopt the distance of the W36 binary system from Rocha et al. (2022, hereafter Paper II), d_w36  =  4.03 ± 0.25 kpc, as an independent distance measurement to confirm the Gaia-EDR3 results.

This paper is organized as follows. In Section 2, we present the Gaia-EDR3 data used for deriving the distance to the Wd 1 cluster and the JHK_s photometry of the four RSGs. Section 3 presents the main results, including the analysis of the Gaia-EDR3 data and the distance to the Wd 1 cluster, followed by the age determination based on the Red Supergiant stars. We discuss the implications of our results in Section 4, and we summarize our results in Section 5.

2 DATA

2.1 Gaia-EDR3 archive

We downloaded the list containing all sources of the Gaia-EDR3 catalogue within a 15 arcmin radius around the central position of the Wd 1 cluster (RA = 16:47:04.00, Decl.  =  −45:51:04.9; Clark et al. 2005), and a total of 36 116 sources were found. We kept only sources associated with a Renormalised Unit Weight Error (RUWE) value ≤ 1.4 (for more details on the definition of RUWE, see Lindegren et al. 2018), and those detected above 3-σ in all three photometric bands (BP, G, and RP), leading to a final list of 25 501 objects.

The new astrometric solution for Gaia-EDR3 (Lindegren et al. 2021a, b) divides the sources into three main groups: (i) ‘2-p’: objects with two parameters solved (i.e. position); (ii) ‘5-p’: five parameters solved (i.e. position, parallax, proper motion); and (iii) ‘6-p’: six parameters solved, including the source colour (effective wavelength) defined as pseudocolour in the Gaia catalogue. The 6-p category encompasses the faintest sources, and/or objects with no reliable colour listed in Gaia-DR2 (Fabricius et al. 2021). Among the astrometric groups, the 5-p sources exhibit the most accurate astrometric solutions and, therefore, correspond to the best class of objects to investigate regions associated with high-extinction and/or at relatively large distances.

Among the selected 25 501 Gaia-EDR3 objects, 7736 and 17 765 are associated with 5-p and 6-p astrometric solutions, respectively.

2.1.1 Wd 1 members in the Gaia-EDR3 catalogue

The identification of cluster members in the Gaia-EDR3 catalogue was performed in two steps. First, we identified each member in one of the few catalogues which have been cross-matched with GAIA-DR3, such as the Guide Star Catalog II Lasker et al. (GSC-II 2008) in visible wavelengths, and the Two Micron All Sky Survey Skrutskie et al. (2MASS 2006) in the near-infrared. Then, we cross-matched the GSC-II IDs with the gsc23_best_neighbour table of the Gaia-EDR3 catalogue, and the 2MASS IDs with the tmass_best_neighbour table. The positions of 166 stellar members of Westerlund 1 reported by Clark et al. (2020) were cross-matched with the GSC-II and the 2MASS catalogues (assuming a maximum offset of 1 arcsec), resulting in a list of 85 sources with a 2MASS counterpart and 105 sources with a GSC counterpart.

Then, the lists were cross-matched with the Gaia-EDR3, leading to a final list with 106 Wd 1 members associated with a Gaia-EDR3 counterpart, from which 61 corresponds to 5-p sources (the first 10 rows are listed in Table 1, the full version is listed in Table A1), 29 sources are 6-p (Table  A2), and 16 have Gaia-EDR3 counterparts with no astrometric information. These last 16 objects were merged with 60 other Wd 1 members with no Gaia-EDR3 counterparts (Table A3).

2.1.2 Parallax zero-point correction: the analysis for Wd 1 members

We corrected Gaia-EDR3 parallaxes by using the methodology described in Lindegren et al. (2021b) and implemented in Python,¹ which estimates the parallax zero-point (π_ZP) separately for 5- and 6-parameter astrometric solutions as a function of the ecliptic latitude (ecl_lat), the G-band magnitude (phot_g_mean_mag), and the colour (effective wavenumber) for each source. The correction is well-determined for objects exhibiting G-band magnitudes between 6 and 21 mag, and effective wavenumber values between 1.1 and 1.9 for 5-p solutions (nu_eff_used_in_astrometry), and between 1.24 and 1.72 for 6-p solutions (pseudocolour).

Fig. 1 presents the distribution of the G-band magnitude against the effective wavenumber, for the known Wd 1 members classified as 5-p (61 sources, panel a) and as 6-p objects (27, panel b). In Fig. 1(a), the distribution of the 5-p sources shows that all points are located within the limits for the proper determination of their π_ZP values. On the other hand, Fig. 1(b) shows that all the 6-p sources exhibit pseudo-colour values outside the limits for obtaining a proper ZP correction, leading to dubious values.

Fig. 2 presents the G-band magnitude versus wavenumber plot exhibiting all the 25 501 sources selected in Section 2. From the 7736 sources with a five astrometric parameter solution (panel a), only two are located outside the ν_eff limit (1.1–1.9). However, ∼52 per cent of the 6-p sources (panel b) exhibit pseudo-colour values outside of the (1.24–1.72) limit, indicating that about half of the 17 765 sources does not have a reliable parallax zero-point estimate. Given that all the Wd 1 members associated with 6-p solutions do not have reliable parallax zero-point corrections (Fig. 1), we restricted the further analysis and results using only the 5-p sources from the Gaia-EDR3 catalogue, leading to a sample of 61 known Wd 1 members from a total of 7736 sources.

2.1.3 Selection of new Wd 1 member candidates based on the Gaia-EDR3 information

We used the Gaia-EDR3 astrometry and photometry to select additional Wd 1 member candidates based on the colour indices of the known members selected in Section 2.1.1. The main photometric and astrometric properties of the known members of Wd 1 are summarized in Table 2.

The sample exhibits BP − RP colour indices ranging between 3.8 and 6.9 mag with a median of 4.9|$_{-0.4}^{+0.6}$| mag, suggesting a relatively high reddening towards the cluster. For completeness, the BP − G colour index ranges from 2.3 to 5.3 mag with a median of 3.3|$_{-0.3}^{+0.6}$| mag, and the G − RP colour index ranges from 1.4 to 1.7 mag with a median of 1.54|$_{-0.05}^{+0.06}$| mag.

An additional astrometric criterion was defined by adopting a 5-σ threshold on the 2D proper motion space, filtering out objects with tangential velocities larger than ∼30 km s⁻¹ at a distance of ∼4.0 kpc from the median velocity of the cluster members (see Table  2).

The individual astrometric and photometric criteria led to samples of 1691 and 338 sources, respectively, showing that the photometric criteria are more effective for selecting the new candidates than the astrometric criterion alone. The combination of both selection criteria led to a sample of 166 new objects.

We computed the radial distribution of the density of sources and the median parallax in concentric rings from the centre of the Wd 1 cluster for the distinct samples. Fig. 3(a) shows the radial profile of the stellar density as a function of the distance from the centre of the Wd 1 cluster. The initial list of Gaia objects (in grey) exhibits a peak at the central position of FOV (r  =  0|${_{.}^{\prime}}$|5), decreasing towards larger radii. We used the sample containing the candidates and known Wd 1 members (in green) to estimate the effective radius of the cluster (⁠|$R_{50{{\ \rm per\ cent}}}$|⁠), defined as the radius containing about 50 per cent of the sample, leading to |$R_{50{{\ \rm per\ cent}}}$|  =  2|${_{.}^{\prime}}$|0 (shown as the vertical filled green line). The known Wd 1 members from Clark et al. (2020) (in red) are found within the inner 5 arcmin radius, which is about 3 × |$R_{50{{\ \rm per\ cent}}}$|  =  6|${_{.}^{\prime}}$|0 (shown as the vertical dashed green line) containing about ∼75 per cent of the cluster members (known members and candidates, shown by the green curve).

Fig. 3(b) exhibits the median parallax as a function of the distance to the centre of the cluster for the same samples presented in Fig. 3(a). The curve containing all Gaia-EDR3 sources (filled grey curve) shows an abrupt increase of the median parallax (from π ∼ 0.20 to ∼0.50 mas) in the inner 3 arcmin radius, reaching a roughly constant value for larger radii. The median parallax of the full Gaia-EDR3 sample is strongly biased by the relatively large fraction of Wd 1 members at the central region of the cluster (r ∼ 0 arcmin), while the bias diminishes towards larger distances. The r < 3|${_{.}^{\prime}}$|0 limit is roughly consistent with the effective radius of the cluster defined in Fig. 3(a), indicating that most of the stellar sources located within this radius are likely members of the cluster. As expected, the median parallax values for the known Wd 1 members (red curve) are significantly smaller, assuming values ranging from 0.17 to 0.3 mas within the inner 5 arcmin radius. The median parallax values of the cluster members (known members and candidates, in green) exhibit a smooth increase towards distances larger than the 3 |$R_{50{{\ \rm per\ cent}}}$| limit (vertical dashed green line), suggesting the existence of another population of stellar objects with similar astrometric and photometric properties as the Wd 1 cluster, but likely not bound to the cluster itself. For this reason, we excluded the sources located at distances larger than 6 arcmin, leading to a final sample of 172 sources, from which 111 are not present in the Clark et al. (2020) catalogue.

For completeness, the median parallax values of the field stars (dashed black curve) exhibit a relatively constant value around 0.5 mas within the entire range shown in the plot, indicating that the selection criteria adopted in this work were efficient to select the Wd 1 members even at the innermost region of the cluster.

2.2 Near-infrared photometry and far-red spectroscopy

Photometric JHK_s observations of the Wd 1 cluster were performed at the 60-cm Zeiss telescope of the Obsevatório Pico dos Dias (OPD/LNA, Brazil) on 2009 June 18. To avoid saturation with the minimum exposure time, the K_s-band filter was replaced by a narrow filter at λ 2.14 |$\mu$|m as a proxy for the K_s filter. In addition to this flux reduction setup, a black cardboard mask was placed in front of the telescope, on which four holes of 2 × 2 cm were cut. A range of exposure times from 1 to 100 s were used with and without the mask to sample the bright- and faint-end stars in the field. Additional images without mask were taken to reach a fainter magnitude limit (K_s ∼ 16).

Optical spectroscopy observations (λ8400–8900Å, R∼12 000) of the RGB stars were taken between 2012 and 2014 at OPD, and in 2017 at the SOAR telescope (R∼6000). We used results reported by Arévalo (2018) for the spectral types, but we did not assign temperatures with those spectral types as the lines defining them (e.g. TiO) are not formed in the stellar photosphere (Davies et al. 2013).

3 RESULTS

3.1 Gaia-EDR3

3.1.1 Astrometric and photometric properties of the cluster members

Fig. 4 presents three colour–magnitude diagrams (CMDs) showing the Gaia-EDR3 sources in the field (in grey), the known Wd 1 members (in red), the sample selected through the astrometric and effective radius criteria only (in blue) and the sample selected using the full selection criteria (in black). The known Wd 1 members are located in a reddened sequence in the CMDs (shown as red * symbols), justifying the usage of the photometric criteria (defined in Section 2.1.1 and indicated by the red dashed lines) to filter out a larger fraction of blue, foreground objects. For completeness, the sample obtained with the complementary selection criteria (i.e. astrometric and radial distance selection only) still exhibits a large fraction of foreground objects.

In addition, another group of relatively bright and intermediate-reddened objects (G ≳ 17, 1.5 ≤ [BP − RP] ≤ 3.5) is located between the foreground main-sequence line and the reddened sequence corresponding to the Wd 1 cluster. These objects are detected in all diagrams from Fig. 4 and they are consistent with a foreground red clump population, suggesting the existence of an evolved stellar population in the line of sight of the Wd 1 cluster that could introduce a bias on the analysis if not properly taken into account.

We also plotted the reddening vectors derived from the Damineli et al. (2016) extinction law in each of the CMDs from Fig.  4. The extinction coefficients were obtained for each Gaia filter assuming the following effective wavelengths of 532, 673, and 797 nm for the BP-, G-, and RP-bands, respectively (Jordi et al. 2010).

Fig. 5 exhibits the distribution of each sample in the proper motion space. The proper motion of the known Wd 1 members (in red) range −2.65 ≤ μ_α ≤ −1.57 mas yr⁻¹ (median of |$-2.26_{-0.23}^{+0.28}$| mas yr⁻¹) and −4.24 ≤ μ_δ ≤ −2.39 mas yr⁻¹ (median of |$-3.69_{-0.17}^{+0.24}$| mas yr⁻¹). The 1-σ values correspond to a velocity dispersion of about ± 5. km s⁻¹ at a distance of ∼ 4 kpc. The 5-σ confidence interval ellipsoid for the proper motions of the known Wd 1 members, shown as the red curve in Fig. 5, was used as the astrometric selection criterion defined in Section 2.1.1. The ellipsoid is based on the covariance matrix and the linear Pearson correlation coefficient of the 2D data (that is, μ_α versus μ_δ), and the size of the ellipsoid is defined as the number of standard deviations (σ) of the distribution.

The known Wd 1 members (in red) exhibit a compact distribution in terms of proper motions, as expected for a cluster of stars. The Wd 1 members also lie inside the 1-σ confidence interval of the initial list (indicated by the dashed grey curve). The centre of the distribution of the members is offset by (−0.49, −0.29) mas from the centre of the initial list of Gaia-EDR3 sources, corresponding to a linear projected velocity of about ∼ 10 km s⁻¹ at a distance of ∼ 4.0 kpc. About half of the sample selected through the photometric and angular distance criteria defined in Section 2.1.1 are located within the 5-σ confidence interval of the known members (dashed red curve), leading to the final list of 111 new Wd 1 member candidates.

Fig. 6 presents the distribution of the parallax values for the initial list of Gaia-EDR3 sources (in grey), the known Wd 1 members (in red), the samples selected through angular distance and effective radius (yellow), astrometric (green) and photometric criteria (blue), and the known cluster members + candidates (in black). As expected, the selection criteria applied to the Gaia-EDR3 sources was efficient to filter out objects associated with large parallax values (e.g. at closer distances), leading to a final sample exhibiting a relatively narrow distribution of parallaxes with a median value around 0.25 mas (indicated by the vertical black line). The resulting distribution is in good agreement with the value observed for the known Wd 1 members (0.236 mas, shown as the vertical red line).

Fig. 7 exhibits the spatial distribution of the sources in the FOV. As expected, the Wd 1 members and the new candidate Wd 1 members (shown as red ast and black circle symbols, respectively) exhibit a cusp distribution around the central position of Wd 1 in both axis. The sources selected through the photometric criteria alone (in blue) exhibits a tailed distribution extending to ∼12 arcmin to the West direction of Wd 1, while about 20 sources are located at r > 10 arcmin the SE direction of the cluster. These objects exhibit similar colours as the cluster members and are likely located at the same distance (i.e. the same spiral arm) as the cluster itself.

3.1.2 Distance inference from the Gaia-EDR3 parallaxes

As pointed out by those authors, this method does not take into account the physical depth of the cluster neither the position of the sources within the cluster, assuming that all members are located at the same distance. This approximation is true for distant clusters, from which the physical depth is relatively small when compared to the individual parallax uncertainties of its members. To verify if such approximation is valid for Wd 1, Fig. 8 shows the ratio between the parallax and their errors (π/σ_π) as a function of the G-band magnitude of the Gaia-EDR3 sources. The plot exhibits a large fraction of the selected Wd 1 sources (red * symbols) associated with relatively small parallax-to-error ratio values (0.01 ≲ π/σ_π ≲ 10) when compared to the values observed for the whole sample of Gaia-EDR3 sources in the field (grey points, with π/σ_π values up to ∼ 10²). These findings confirm the assumption of large uncertainties on the individual parallaxes of the cluster members and, therefore, its distance can be inferred using equation (4).

The result of equation (4) is a probability density function (PDF), peaking at the distance where the product P(π|d, σ_π) assumes its maximum value. The resulting PDF considering the 58 known Wd 1 members is shown as the red curve in Fig. 9. The PDF exhibits an asymmetric profile, with an elongated tail towards large distances. The PDF peaks at the distance of 4.14 kpc|$^{+0.60}_{-0.66}$| kpc, with the upper and lower uncertainties defined as the 68 per cent confidence interval, consistent with a 1-σ estimate from other methods.

The determination of the distance to the cluster and its associated errors was improved by considering the new Wd 1 member candidates, as shown in Fig. 9. Overlaid on the PDF of the known Wd 1 members (in red), the plot also shows the PDF considering only the new candidates (in blue), and the union of both samples (members + candidates, in black). The sample of 172 objects led to a more precise inference of the distance to the cluster, estimated as d_gedr3 = 4.06|$^{+0.36}_{-0.34}$| kpc.

For completeness, we computed the weighted-mean parallax of the field stars to infer their mean distance. The weighted mean parallax (〈π〉) was evaluated using equations (1)– (3) from Navarete, Galli & Damineli (2019), leading to 〈π〉  =  0.74 ± 0.20 mas. We estimated the distance of the field stars by simply inverting their mean parallax and propagating the errors, leading to a mean distance of 1.36|$^{+0.50}_{-0.29}$| kpc. Such distance is consistent with the location of the near-side of the Carinae arm at the line of sight of Wd 1 cluster (Reid et al. 2019).

A study of the W36 eclipsing binary system (Paper II) led to a independent distance estimate of d_w36  =  4.03 ± 0.25 kpc, in agreement with our Gaia-EDR3 results. From here on we adopt the weighted mean distance to the cluster as d_wd1  =  4.05 ± 0.20 kpc (further discussions are presented in Section 4.1).

3.2 Luminosity and age of the red supergiants

The improved accuracy of extinction and the distance obtained in this work has a direct impact on the determination of the luminosities of the cluster members. As a consequence, the parameters of the cluster as a whole can also be inferred with greater confidence.

The photometric results for the four RSGs and six YHGs are listed in Table  3. The photon statistics accuracy is ∼0.01 mag for the bright members (degrading for the fainter members), but due to the relatively small number of 2MASS calibrating sources in the FOV, the photometric accuracy was limited to ∼ 0.05 mag in the bright-end (K_s < 13). Using the references in the caption of Table  3 we derived the intrinsic colours and temperatures of our targets. We adopted the Damineli et al. (2016) reddening law to get the AK_s extinction: AK_s = 0.449|$E_{J-K_s}$| and AK_s = 1.300|$E_{H-K_s}$| (column 6 in their Table  3). The average extinction of the RSGs and YHGs A_Ks = 0.80 ± 0.14 is compatible with that derived from the OB stars (A_Ks = 0.74 ± 0.08, D16), indicating that these evolved cool stars do not have significant circumstellar extinction. The exception is W75 (in addition to W8a) which has a significantly larger than the average extinction, as suspected by Beasor et al. (2021).

The total luminosity of each RSG (column  7 of Table  3) was calculated by using the bolometric calibration suggested by Davies & Beasor (2018) – BC_K = 3.00 ± 0.18 – and the mean distance of d_wd1 = 4.05 kpc from Section 3.1.2. The range of RSGs luminosities is relatively narrow (Δ (log(L/|$\rm {L}_\odot$|⁠) = 0.24) indicating a narrow range of RSG progenitor masses. Column 8 of Table  3 shows luminosities from Beasor et al. (2021) (their table  4, column  6), scaled with our distance of d = 4.05 kpc. The results are in excellent agreement in luminosity indicating that the differences between the two works mostly arise from the AK_s extinction towards individual targets.

Table  3 also displays our photometric results for the six YHGs. The bolometric correction follows Flower (1996). Our results are in very good agreement with those reported by Beasor et al. (2021), which is based on independent photometry.

We adopted the Binary Population and Spectral Synthesis method (BPASS; Eldridge, Beasor & Britavskiy 2020) for estimating the age based on the mean of the luminosity of the four RSGs (interpolated into their Z = 0.020 models, column 6 of Table 1). The mean RSG luminosity, |$\log (L_{\rm RSG}/\rm {L}_\odot)$| = 5.10 ± 0.11 corresponds to a cluster age of 10.7 ± 1 Myr.

4 DISCUSSION

4.1 The distance to Wd1 confirmed by independent methods

We used the Gaia-EDR3 analysis to infer the distance to the Westerlund 1 cluster based on the sample of evolved, high-mass stars from Clark et al. (2020). We further included additional member candidates selected through astrometric and photometric criteria to obtain a final sample of 172 sources with Gaia-EDR3 counterparts, leading to the distance of d_gedr3  =  4.06|$^{+0.36}_{-0.34}$| kpc.

We also found that the sources in Clark et al. (2020) are relatively red objects, as expected for a cluster located at large distances in the Galactic Plane. The main impact of that fact is regarding the parallax zero-point derivation. For the range of brightness and colours of the cluster members in the Gaia-EDR3 catalogue, we found that the parallax zero-point correction is well-determined for Gaia objects with five astrometric parameters solution (5p), but not for objects with six astrometric parameters solution (6p, see Figs 1 and 2). Therefore, we excluded the 6p sources from our analysis. This criterion was fundamental for removing any systematic biases due to dubious parallax zero-points, a critical correction for objects at relatively large distances.

Our Gaia-EDR3 distance to Wd 1 is in excellent agreement with the distance reported by Beasor et al. (2021), 4.12|$^{+0.66}_{-0.33}$| kpc. The major difference between their results and ours is mainly related to the sample selection and the adoption of fine-tuning procedures in this work for excluding sources with unreliable zero-point parallax correction (see Section 2.1.2).

Negueruela et al. (2022) reported an independent confirmation of the distance to Wd 1 also based on the OB stellar population from Clark et al. (2020). They reported eight distance estimates (see their table 1), adopting case C as the distance to the cluster, 4.23|$^{+0.23}_{-0.21}$| kpc. Interestingly, cases F and H of their table 1 are very similar to our results (their source selection criteria for case H likely match the same as adopted by us). However, apparently they chose case C as an intermediate value of the eight cases instead, stating that their distance estimate is robust and independent in terms of their source selection methodology, the parallax zero-point correction, or choice of sources with 5- or 6-parameters astrometric solutions. Taking into account the independent distance of W36 in Paper II (d_w36  =  4.03 ± 0.25 kpc), the closest distances reported in their table 1 (cases F, H) are likely more reliable. They adopted a distinct parallax zero-point correction and a specific distance prior for the distribution of OB stars in the Galactic plane (see references therein), which led to a distance in agreement with our results and with those from Beasor et al. (2021).

These three Gaia-EDR3 distances indicate that the Wd 1 cluster is located between 4.0 and 4.3 kpc with unprecedented accuracy. In addition, the Gaia-EDR3 distances can be confirmed by independent measurements, such as the modelling of eclipsing binary systems. Indeed, Paper II presents the modelling of the eclipsing binary system W36, obtaining a distance of d_w36 = 4.03 ± 0.25 kpc. The distance to W36 is consistent within 1-σ with each of the three independent Gaia-EDR3 distances, being an important result once the parallax zero-point corrections are uncertain in the Gaia-EDR3 for most of objects associated with very red colour indices (Lindegren et al. 2021b), as likely observed for the bulk of the Wd 1 cluster members (see Section 2.1.2).

Table  4 summarizes the distances reported by Beasor et al. (2021), Negueruela et al. (2022), Paper I (this work), and also includes the distance to the massive binary system W36 from Paper II.

At the distance of 4.05 ± 0.20 kpc, the effective size of Wd 1, r  =  2|${_{.}^{\prime}}$|0 (Figs 3 and 7) corresponds to a linear radius of 2.33 ± 0.12 pc, and our sample is located within three times the effective radius, 7.0 ± 0.4 pc. It is worth mentioning that our sample is biased towards the high-mass stellar content of the cluster. In general, the low-mass counterparts are likely less gravitationally bound to the cluster, extending to larger distances from the centre of the cluster. Both the diameter of the cluster and the low-density halo concept are in agreement with the analysis presented by Negueruela et al. (2022).

Wd 1 is located at the galactic latitude of ℓ  =  339|${_{.}^{\circ}}$|55. Based on recent Galactic structure results (Reid et al. 2019), the cluster is located in the line of sight of three major galactic spiral arms: the Carina arm at a distance of 0.5 kpc, the Scutum-Crux near arm at 3.5 kpc, and the Norma near arm at 5.5 kpc. Our results suggest that the distance of Westerlund 1 is consistent with the position of the far side of the Scutum-Crux near arm, or even the near side of the Norma arm. Negueruela et al. (2022) provide strong arguments favouring that the cluster is located within the Norma arm, but further studies are still necessary to confirm the true association of the cluster with one or another spiral arm.

4.2 The Age of Westerlund 1

We further compare our results on the age of Wd 1 with those from Beasor et al. (2021). Those authors tested the robustness of different methods for evaluating the age of the cluster by using a wide range of stellar populations (RSGs, PMS, and the eclipsing binary W13). They reported the age of 7.2|$^{+1.1}_{-2.3}$| Myr for the pre-MS population, while the faintest RSG led to a older age between 9.2 and 11.7 Myr. Our results for the age of the RSGs based on BPASS models (11 ± 1 Myr, see Section 3.2) is in excellent agreement with those authors.

In Paper II, we derived the age of the eclipsing binary W36 (using Yusof et al. (2022) evolutionary models) as ∼6 Myr. Only the W36B component has a reliable age estimation, since the peeling of external layers of the W36A component shifts its effective temperature to the blue, mimicking an earlier age. Furthermore, the impact of mass accretion on to W36B is low since the mass lost by W36A is mainly through its stellar wind, so the W36B stellar parameters are little affected. The age of W36B is not far from the age limit of W13 (< 5 Myr) reported by Beasor et al. (2021) and also to ∼ 7 Myr suggested by Hosek et al. (2018). Compared to the ages of the RSGs, the much younger W36B and W13 sources are evidence of – at least – two episodes of star formation in the cluster.

5 CONCLUSIONS

Our astrometric and photometric analysis using the Gaia Early Data Release 3 (Gaia-EDR3) resulted in the distance of d_gedr3 = 4.06|$^{+0.36}_{-0.34}$| kpc to the massive stellar cluster Westerlund 1. The agreement between the Gaia-EDR3 distance and the distance of the W36 eclipsing binary system d_w36 = 4.03± 0.25 kpc (Paper II) validates our results and, in particular, the parallax zero-point determination for the reddest Wd 1 members classified as 5-p sources in the Gaia-EDR3 catalogue. The weighted mean distance of both independent estimates led to d_wd1  =  4.05 ± 0.20 kpc, with a unprecedented error of 5 per cent.

We reported the age of the RSG stars using BPASS models for the average luminosity as 9.7–11.7 Myr, in good agreement with that of W75 reported by Beasor et al. (2021, 9.2–11.7 Myr) based on a broader range of approaches and a different data set.

The age of the massive binary system W36 is between 3.5 and 7.1 Myr, in agreement with < 5 Myr reported by Beasor et al. (2021) for the binary system W13. These two works indicate a range of star formation in Wd1 of 5–12 Myr, in line with the synthetic cluster calculated by Yusof et al. (2022) combining rotating and non-rotating models with an age spread of 5–10 Myr. Yusof et al. (2022) showed that their result was able to reproduce qualitatively the observed population of RSGs, YHGs, and WRs in Wd 1.

ACKNOWLEDGEMENTS

The work of FN is supported by NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. FN and AD thanks to Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) for support through process number 2017/19181-9 (FN) and 2019/02029-2+2011/51680-6 (AD) and to Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) 301490/2019-8 (AD).

We thank B. Davies for careful reading of our manuscript, and for productive questions/remarks which has improved our manuscript. Based in part on observations carried out at Observatório do Pico dos Dias (OPD), which is operated by LNA/MCTI, Brazil. Based in part on observations obtained at the Southern Astrophysical Research (SOAR) telescope, which is a joint project of the Ministério da Ciência, Tecnologia e Inovações (MCTI/LNA) do Brasil, the US National Science Foundation’s NOIRLab, the University of North Carolina at Chapel Hill (UNC), and Michigan State University (MSU).

DATA AVAILABILITY

The data sets were derived from observations available in the public domain: https://www.cosmos.esa.int/web/gaia/earlydr3.

Tables 3, A1, A2, and A3 are also available in the CDS.

Footnotes

REFERENCES

APPENDIX A: TABLES