Open Access
Issue
A&A
Volume 684, April 2024
Article Number A108
Number of page(s) 9
Section Galactic structure, stellar clusters and populations
DOI https://doi.org/10.1051/0004-6361/202347619
Published online 08 April 2024

© The Authors 2024

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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1. Introduction

The initial mass function (IMF) gives information about the initial mass distribution of stars in a stellar system (e.g. Bastian et al. 2010; Kroupa et al. 2013; Hopkins 2018). Together with previously formulated relations, it describes the mass distribution of a population of stars at birth of the embedded cluster. These previously formulated relations are the mmax − Mecl relation (Yan et al. 2023), with the maximum stellar mass mmax and the stellar mass of the embedded cluster Mecl, and the variation of the IMF with metallicity and density of a molecular cloud core in which the population forms as an embedded star cluster (Marks et al. 2012; Jeřábková et al. 2018; Yan et al. 2021). Another relation is the rh − Mecl relation by Marks & Kroupa (2012), in which rh is the half-mass radius of the embedded cluster at birth, just prior to gas expulsion. The concepts and relations are explained in more detail in Sect. 2.

With this contribution, we test whether the previously determined constraints on the variation of the IMF with metallicity and density of a star-forming gas cloud are consistent with the recently published observations of two young star clusters by Yasui et al. (2023). These clusters lie in the low-metallicity star-forming region Sh 2-209, which is located in the outer region of the Milky Way. Furthermore, we compare the observed clusters to the relation between rh and Mecl, obtained by Marks & Kroupa (2012).

The IMFs of solar metallicity regions have been well-studied. One example is the Taurus region (e.g. Luhman 2000; Briceño et al. 2002; Thies & Kroupa 2008) with a solar metallicity [Fe/H] = −0.01 ± 0.05 (D’Orazi et al. 2011) or NGC 2264 by Sung & Bessell (2010) with [Fe/H] ≈ −0.13 (Netopil et al. 2016). Another solar metallicity region is the Trapezium cluster in the Orion nebular cluster (ONC) with [Fe/H] = −0.01 ± 0.04 (D’Orazi et al. 2009), for which Muench et al. (2002) derived the mass function from B stars to the deuterium-burning limit. On the other hand, the IMF of an entire low-metallicity region is usually only known over a limited stellar-mass range. For example, Harayama et al. (2008) determined the IMF of the NGC 3603 star cluster within the NGC 3603 HII region in the Milky Way (Goss & Radhakrishnan 1969) within a stellar mass range of 0.4 − 20 M. Kuncarayakti et al. (2016) found it to have a metallicity between Z = 0.004 and 0.008. Because the two-body relaxation time for high-mass stars is similar to the cluster age, Harayama et al. (2008) suggested that a composition of primordial and dynamical effects leads to the observed top-heaviness of the IMF. Another example is NGC 346. This star-forming region in the Small Magellanic Cloud (SMC) has a metallicity of [Fe/H] ≈ −0.72 (Rochau et al. 2007): the IMF was determined within a stellar mass range of 0.8 − 60 M by Sabbi et al. (2008) over a region spanning ≈40 pc. The authors reported that they did not find any environmental effects on the IMF. This holds for the global IMF derived by Sabbi et al. (2008), who noted its flattening at the centre and attributed it to mass segregation, which must be primordial because NGC 346 is younger than its mass-segregation timescale (see Table 1 in Sabbi et al. 2008). This means that more massive stars formed in the centre, as underlined by Fig. 5 in Sabbi et al. (2008), which supports the idea of variations of the IMF. We discuss NGC 346 in Sect. 6.2.

S209 is the first low-metallicity ([O/H] = −0.5 dex) star-forming region whose IMF was determined over the wide mass range of 0.1 − 20 M (Yasui et al. 2023). Our approach is to assume the previously constrained variation of the IMF (see Sect. 2.3) and apply it to the two clusters, the main cluster (MC) and the sub-cluster (SC), given the available information on their metallicities. With this, we calculate the radii of the two clusters and check them for consistency with the variable IMF and the independently derived relation of rh and Mecl by Marks & Kroupa (2012, see Sect. 2.4). This places constraints on the distances of both clusters. We also check for consistency in the relation of mmax versus the Mecl (see Sect. 2.2) published by Yan et al. (2023). Section 2 documents the previously constrained birth relations, Sect. 3 introduces the observational data, Sect. 4 explains the methods, and Sect. 5 reports the results. The discussion and conclusion are provided in Sects. 6 and 7, respectively.

2. Previous results on the birth relations

In this section, we give an overview of the results from previous research on the properties of newly formed embedded clusters that we applied to the data of S209.

2.1. The canonical initial mass function

In a stellar system, the initial mass distribution of the stars is of special interest to resolve the formation process and the stellar system’s properties. The IMF gives the number of stars with a certain mass. While Salpeter introduced a one-power-law IMF (Salpeter 1955), Kroupa (2001) found that the mass distribution among stars in the solar neighbourhood follows a two-part power-law formulation, the canonical IMF. It is defined as the number of stars in the mass interval m to m + dm, dN = ξ(m) dm, where ξ(m) = k ⋅ kimαi, k and ki are constants, and the exponents are α1 = 1.3 for 0.08 ≤ m/M < 0.5 and α2 = 2.3 for 0.5 ≤ m/M < mmax, m is the stellar mass, and mmax is the most massive star in the embedded cluster (Sect. 2.2).

2.2. The mmax − Mecl relation

Based on the IMF being a probability density distribution function (PDF), it might be thought that the IMF of one massive star cluster is similar to that of multiple small star clusters. This contradicts the fact that in a massive embedded star cluster, more massive stars can form, which is not possible in low-mass clusters because the mass of a very massive star would exceed the mass of the small cluster. This concept is documented by the mmax − Mecl relation (Weidner & Kroupa 2006). A comparison of the results of the two sampling methods, namely the random and optimal sampling introduced by Kroupa et al. (2013), shows that the mmax − Mecl relation has a physical origin and is not a statistical effect (Yan et al. 2023). With this relation, an estimate of Mecl is obtained by only having mmax.

Yan et al. (2023) quantified the relation between mmax and Mecl (see Figs. 5 and 6 in Yan et al. 2023).

2.3. The varying initial mass function

While many Milky Way and Large Magellanic Cloud clusters follow the canoncial IMF (Kroupa 2002), subsequent data on the mass-to-light ratios (Dabringhausen et al. 2009) and the occurrence of low-mass X-ray bright binaries (Dabringhausen et al. 2012) and of the stellar content of low-concentration globular clusters led to the suggestion that the IMF becomes top-heavy at low metallicity and high gas density (Marks et al. 2012). Star counts in nearby low-metallicity star-forming regions support this result (Schneider et al. 2018; Kalari et al. 2018 at the 2σ confidence level). Assuming this variation of the IMF, Marks et al. (2012), Jeřábková et al. (2018), and Yan et al. (2021) introduced a metallicity-dependent relation between the initial gas density ρgas and the exponent α3.

Thus, the exponent α3 is related to the initial average cloud core density ρgas and the initial gas metallicity of the cluster [Z/H] (Jeřábková et al. 2018; Yan et al. 2021),

(1)

(2)

with the metal abundance by mass, [Z/H] = [Fe/H] + [α/Fe], [Fe/H], and [α/Fe] being the iron and alpha-element abundances, respectively. The typical value of [α/Fe] in the Milky Way is 0.3 (Forbes et al. 2011), which is also assumed in this work. Rearranging Eqs. (1) and (2), we obtain

(3)

where ρgas is the initial average density of the cloud core in units of 106M pc−3, and ρecl is the stellar mass density of the embedded cluster. These two are connected with the relation ρecl = ϵ ⋅ ρgas, where the canonical value of ϵ = 0.3 is adopted for the star formation efficiency (Lada & Lada 2003; André et al. 2014; Megeath et al. 2016; Banerjee & Kroupa 2018; Wirth et al. 2022). Furthermore, ϵ is assumed to be constant throughout the cluster, such that the gas exactly tracks the stars. We note that ρecl constitutes the mathematically idealised moment of the highest effective density of the embedded cluster at the idealised time when all stars are born instantly, such that the binary population subsequently evolves to the observed populations (Marks & Kroupa 2012). The real physical embedded cluster has a more complex formation history in which the stars form over one to a few crossing times of the contracting molecular cloud core before it expands due to the expulsion of residual gas (e.g. Kroupa et al. 2001; Della Croce et al. 2023).

For completeness, based on the observation by Kroupa (2002) that α1 and α2 appear to also depend on metallicity, Marks et al. (2012) and Yan et al. (2021) provided an updated formulation (Eq. (1)) that includes α1([Z/H]) and α2([Z/H]).

2.4. The rh − Mecl relation

Based on a study of binary populations, Marks & Kroupa (2012) introduced the canonical relation of rh and Mecl

(4)

This is the half-mass radius of the embedded cluster in the most compact configuration, that is, when it is deeply embedded, and prior to any gas expulsion. It is required in order for the emanating open star cluster to have the observed population of binary stars, given that star formation provides a binary fraction of almost 100% (Kroupa 1995a,b; Marks & Kroupa 2011).

3. Sh 2-209

S209 is an HII region located at a Galactic longitude and Galactic latitude with right ascension RA = 4 11 06.7 and declination Dec = +51 09 44 (Wenger et al. 2000). With data from Gaia Collaboration (2021), Yasui et al. (2023) determined an astrometric distance of ≈2.5 kpc. Assuming the solar galactocentric distance to be R = 8.0 kpc, S209 has a galactocentric distance of about 10.5 kpc (Yasui et al. 2023). On the other hand, Foster & Brunt (2015) determined a heliocentric distance of ≈10.58 kpc from radial velocity measurements of the Canadian Galactic Plane Survey in the radio regime. This result is consistent with the value determined by Chini & Wink (1984), using photometric and spectroscopic data from exciting stars. The galactocentric distance for this case, again assuming R = 8.0 kpc, is therefore approximately 18 kpc.

S209 is a star-forming environment consisting of a MC for which a sample of 1500 objects is identified and a SC with 350 members (Yasui et al. 2023). The observed mass range is 0.1 − 20 M. Depending on the age and distance, the observed mass range differs. For instance, for younger ages, masses down to 0.02 M are considered. When the first break mass is lower than the minimum mass of 0.1 M, the first break mass is considered as the minimum mass. Following Yasui et al. (2023), we adopt the term “break mass” as the mass at which the slope of the IMF changes. The observed mass range was determined from the colour-magnitude diagram. However, some stars that are below this limit but have a large K excess were considered in the derivation of the K-band luminosity functions that were used to obtain the IMFs (Yasui et al. 2023). To identify the objects in the clusters, near-infrared image data were used, which also show that the cluster is surrounded by residual gas. This means that gas expulsion has already occurred. Using the metallicity map from Eilers et al. (2022), we estimated the metallicity of the environment to be [Fe/H] ≈ −0.25 dex for the smaller distance and [Fe/H] ≈ −0.5 dex for the outer position (see Fig. 1). Based on previous studies, the result by Yasui et al. (2023) has an oxygen abundance of [O/H] = −0.5 dex. For a more detailed discussion of the metallicities, we refer to Sect. 6.

thumbnail Fig. 1.

Metallicity map from Eilers et al. (2022), where the Galactic centre at (0, 0) is indicated by the cross, the Sun is located at (−8.0 kpc, 0) and is indicated by the circled dot, and the two possible positions of S209 are indicated by an asterisk. The arrows show the rotation of the stars and are coloured according to their mean self-calibrated metallicity.

4. Methods

In order to test for the relations we introduced in Sect. 2, Mecl needs to be calculated. Ensuring that the IMF is smooth, we determined the constant k using the number of objects N in the star cluster and the equation . After constraining the IMF, the mass of the embedded cluster can be computed,

(5)

with Γi = 1 − αi.

Here, m1 (=0.5 M for the canonical IMF) is the first break mass, and m2 (=1 M in Eq. (1)) is the second break mass, while mmin = 0.08 M is approximately the hydrogen-burning limit.

In order to determine rh, ρecl needs to be computed. For this, Eqs. (2) and (3) were applied with the known metallicity. Using the relation ρecl = ϵ ⋅ ρgas with ϵ = 0.3 as explained in Sect. 2.3, ρecl was obtained.

After determining Mecl and ρecl, we calculated the pre-gas-expulsion rh, assuming spherical symmetry,

(6)

The results of these calculations are listed in Sect. 5.

The uncertainties were calculated by Gaussian error propagation (, where Δf is the uncertainty of the quantity f, xi is a variable, and Δxi is its corresponding error). Because the constant k was calculated via integration and was included in the determination of Mecl, where a second integration was applied, a large error can be obtained for Mecl. Furthermore, the errors for the exponents Γi contribute strongly. This can cause large uncertainties for the Mecl values, which lead to large uncertainties for some other results. Additional uncertainties such as extinction due to the interstellar medium or the intracluster medium were not accounted for here because a correction for them was already applied in the analysis by Yasui et al. (2023).

After computing Mecl and rh with Eqs. (5) and (6) based on the ρgas obtained with Eqs. (2) and (3), we compared the results to the rh − Mecl relation described in Sect. 2.4. This is displayed in Figs. 3 and 4 in Sect. 6.

5. Results

In Yasui et al. (2023), the parameters given for the IMF varied for the different galactocentric distances assumed (10.5 kpc or 18 kpc), for the ages from 0.5 to 10 Myr, and for the considered cluster (MC or SC). In order to give an overview, different cases were considered in this paper. In the main part, the parameters of the best-fit IMF are evaluated, and in the Appendix A the results for ages of 1, 5, and 10 Myr are listed.

Using the best-fit results that Yasui et al. (2023) obtained, displayed in Table 1 for the MC and in Table 2 for the SC, we plot the IMFs in Fig. 2. The uncertainties of the parameters were adopted from Table 9 and Sect. 7.1 in Yasui et al. (2023).

thumbnail Fig. 2.

Reconstructed IMF with values from Yasui et al. (2023). The dashed lines represent the canonical IMF (see Sect. 2.1). The upper line shows the MC, and the lower line shows the SC.

Table 1.

Best-fit values for the MC (Yasui et al. 2023).

Table 2.

Best-fit values for the SC (Yasui et al. 2023).

We proceeded as follows in order to test whether the observational constraints on α3 by Yasui et al. (2023) are consistent with the formulation of the variation of the IMF given by Eqs. (1) and (2) (Sect. 2.3). By adopting the reported values of α3, Eq. (3) was used to calculate ρgas and Eq. (5) to calculate Mecl. The application of Eq. (6) yields the value of rh that the embedded cluster should have. The masses of the cluster over these calculated rh are displayed in Figs. 3 and 4 in Sect. 6, where they are compared to the canonical half-mass radius relation (Eq. (4) in Sect. 2.4).

thumbnail Fig. 3.

Canonical relation of Mecl and rh (Eq. (4)) shown by the blue line. The data points are the calculated values for the MC in S209. The caps indicate the 1σ and 2σ uncertainty boundaries.

The resulting densities, radii, and masses are displayed in Table 3 for the MC and in Table 4 for the SC. The provided results are formal solutions with their 1σ uncertainties. The 1σ upper limit would be physically relevant.

Table 3.

Results for the MC.

Table 4.

Results for the SC.

Ideally, all stellar masses would be summed from star counts as a constraint on the stellar mass of the cluster, but these data were not available to us.

6. Discussion

6.1. Sh 2-209

Considering the mmax − Mecl relation that we explained in Sect. 2.2 and the results for the MC, mmax = 20 M (Yasui et al. 2023), we obtain Mecl ≈ 800 M, which is compatible with the derived Mecl for the best-fit parameters of MC at a galactocentric distance of 18 kpc (see Table 3). For older ages, it is more consistent with the MC being at a galactocentric distance of 10.5 kpc (see Table A.11). Based on radio emission data, Richards et al. (2012) found the ionised gas mass to be greater than 103M. This is more consistent with the values obtained for a galactocentric distance of 10.5 kpc (the authors even suggest a distance of 4.9 kpc).

With Fig. 1, we estimated the iron abundance [Fe/H] at the two possible positions of S209. Eilers et al. (2022) also provided a map for [O/Fe] (see their Fig. 6), from which we estimated [O/Fe] ≈ 0.03 dex for the inner and [O/Fe] ≈ 0.07 dex for the outer position. With the relation [O/H] = [O/Fe] + [Fe/H], we obtained an oxygen abundance [O/H] of ≈ − 0.22 dex for the smaller and ≈ − 0.43 dex for the larger distance. Because the maps by Eilers et al. (2022) do not account for small-scale variations in the metallicity, the individually observed metallicities may differ from those implied by the Eilers et al. (2022) map. Yasui et al. (2023) discussed different values for [O/H], namely ≈ − 0.43 dex (Vilchez & Esteban 1996), which is consistent with the value derived here. Rudolph et al. (2006) used the data of Vilchez & Esteban (1996) and derived [O/H] ≈ −0.29 dex, which is 30% lower than the value for the inner position and about 50% higher than the value for the larger distance. Using the work of Caplan et al. (2000), Rudolph et al. (2006) derived an oxygen abundance of ≈ − 0.58 dex, which is inconsistent with the metallicity determined using the map by Eilers et al. (2022) for the inner position and is about 35% lower than the value for the outer position. Finally, Yasui et al. (2023) suggested [O/H] ≈ −0.5 dex, which is consistent with Eilers et al. (2022) for the outer position of S209. This maintains the preference that S209 has a galactocentric distance of 18 kpc, as derived by Foster & Brunt (2015) and Chini & Wink (1984) using spectroscopic and photometric data.

In the determination above, no binary stars are assumed. In the binary-star theorem by Kroupa (2008), the majority of star formation results in binary systems. Therefore, a binary fraction of 95% is adopted in the following, which corresponds to the updated pre-main-sequence eigenevolution model by Belloni et al. (2017). In order to give a maximum correction, we assumed the binaries to be composed of equal-mass stars, but in reality, the mass of the secondaries is somewhat lower than that of the primaries (Moe & Di Stefano 2013; Sana et al. 2012). The value of α3 is not affected by a realistic binary, triple, and quadrupole population (Weidner et al. 2009; Kroupa & Jerabkova 2018). Hence, the mass Mecl is nearly doubled. Because the density is determined by the metallicity and α3 (see Eqs. (1) and (3)), which means that it is independent of Mecl, the half-mass radius in Eq. (6) increases accordingly.

The canonical relation between rh and Mecl (Eq. (4)) and the corresponding results are plotted in Fig. 3 for the MC and in Fig. 4 for the SC. The plots for the other ages are displayed in Appendix A. In Fig. 3, the value for a galactocentric distance of 18 kpc, which respects the binary-star theorem, lies in the error band and is therefore consistent within 1σ confidence, while the data points of the galactocentric distance of 10.5 kpc deviate more from the canonical Eq. (4), but still lie in the 1σ regime. The deviation of the result for a galactocentric distance of 10.5 kpc decreases with the binary assumption. In the plots in which the other ages are younger than 10 Myr, which are displayed in Appendix A, the data points for the MC at a galactocentric distance of 18 kpc fit the relation better, while the points for a galactocentric distance of 10.5 kpc do not deviate by more than 2σ. This changes for an older age (see Fig. A.8), where a galactocentric distance of 10.5 kpc corresponds better to the relation by Marks & Kroupa (2012).

thumbnail Fig. 4.

As Fig. 3, but for the SC.

All the half-mass radii are smaller than expected from the canonical Eq. (4), except for the age of 10 Myr (see Fig. A.8). This underlines the possibility that a low-metallicity star-forming region might be denser because the mass is included in a smaller volume than is determined for other very young star clusters of higher metallicity. This would be reminiscent of the smaller radii of low-metallicity stars compared to metal-rich stars of the same mass, suggesting that self-regulation may play an important role for the constitution of an embedded cluster (see also Yan et al. 2023).

In Fig. 4, the values for a galactocentric distance of 18 kpc of the SC again agree with the relation by Marks & Kroupa (2012) within the 1σ confidence region. The results for the 10.5 kpc distance are only consistent with Eq. (4) within the 3σ confidence. Hence, the prediction by Marks & Kroupa (2012), that is, Eq. (4), agrees with the determined values for the MC and SC for both distances, while the results for a galactocentric distance of 18 kpc represent the observational data better. However, given the large uncertainties, no highly significant conclusions can be drawn.

6.2. NGC 346

NGC 346 is the largest active star-forming region in the Small Magellanic Cloud (Rickard et al. 2022), for which Sabbi et al. (2008) argued that the IMF is the same as in the Galaxy based on their survey, which spanned ≈40 pc around the central embedded cluster. Figure 5 in Sabbi et al. (2008) showed, however, that the centremost 4 pc have a top-heavy IMF with α3 = (2.03 ± 0.14), which is consistent with a star cluster density of ρecl ≈ (0.25 ± 0.20)×106M pc−3, according to Eq. (3), assuming a metallicity [Fe/H] ≈ −0.72 (Rochau et al. 2007), which we adopted for [Z/H] assuming an error of 0.3 to account for [α/H]. With this, the total cluster mass is Mtot ≈ 3.9 × 105M, and with Eq. (6), we can give an upper limit on the half-mass radius of the central region of rh < (0.7 ± 0.2) pc, which is consistent with the foregoing calculation when the uncertainties caused by gas expulsion that lead to expansion are included. At larger distances between 4 and 9 pc, Sabbi et al. (2008) reported a canonical IMF (α3 ≈ 2.36 ± 0.09), which is likely a mixture of on-site lower-mass embedded clusters and ejected massive stars from the starburst cluster (Oh & Kroupa 2016; Oh et al. 2015). At larger distances, α3 ≈ (2.43 ± 0.09) and for distances larger than 14 pc, α3 ≈ (3.08 ± 0.14), which reflects a population of low-mass embedded clusters lacking massive stars (Weidner & Kroupa 2006; Yan et al. 2023). Although the ejected massive stars induce a steeper slope of the MF in the inner part, mass segregation is more present in this case, which leads to the flatter slope in the central region. The centre of this whole star-forming system on a scale of ≈40 pc across is reminiscent of the Orion-South star-forming cloud where only the ONC formed massive stars, while much of the southern part of the cloud formed many low-mass embedded clusters lacking massive stars (Hsu et al. 2012). In contrast to the assertion by Sabbi et al. (2008), NGC 346 thus appears to be consistent with the variation of the IMF given by Eqs. (1) and (2), but more work is needed to quantify details. For example, it will be necessary to quantify whether many low-mass embedded clusters formed with IMFs lacking massive stars compared to the centred starburst cluster at larger distances from the centre of NGC 346.

7. Conclusions

We analysed two clusters of the star-forming region S209 with respect to their initial conditions. In particular, the low metallicity was taken into account, which allowed us to determine ρecl (Eq. (3)), by using the determination of the IMF power-law index by Yasui et al. (2023). We constrained rh and compared it to the independently developed canonical relation of rh and Mecl (Eq. (4)).

Thus, the IMF data from Yasui et al. (2023) for the low-metallicity region S209 result in a top-heavy IMF. The determined value of Mecl for ages of about 0.5 Myr for the MC at a galactocentric distance of 18 kpc is most consistent with the value Mecl ≈ 800 M deduced with the relation of mmax and Mecl derived by Yan et al. (2023). This changes for older ages, where the resulting mass, Mecl, for the smaller galactocentric distance of 10.5 kpc is more consistent with the mmax − Mecl relation.

The comparison of the determined metallicity with the map by Eilers et al. (2022) for the outer position and the independently derived metallicities by Yasui et al. (2023) and Vilchez & Esteban (1996) indicate a preferred galactocentric distance of 18 kpc. The value for the inner position (10.5 kpc) is not consistent with the metallicity of S209.

The calculated value for the rh of the MC at a galactocentric distance of 18 kpc is consistent with the canonical rh − Mecl relation for young ages, namely being within the 1σ confidence region. The binary theorem introduced by Kroupa (2008) tightens these results. The values of rh for the MC and SC at a galactocentric distance of 10.5 kpc we found are only consistent within 2σ or 3σ confidence for young ages. On the other hand, they replicate the mmax − Mecl relation (Yan et al. 2021) better for an age of 10 Myr. This confirms the previously mentioned age-distance degeneracy.

In addition, we showed that the variation of the IMF (Eq. (2)) is consistent with NGC 346, which shows a top-heavy IMF in the innermost region and a lack of massive stars with increasing distance to the centre. This is reminiscent of the ONC.

In conclusion, S209 is consistent with a low-metallicity region resulting in a denser star-forming region with a top-heavy IMF, as quantified by Marks et al. (2012) and Yan et al. (2021). The findings for the MC with a galactocentric distance of 18 kpc are consistent with the relation of mmax and Mecl by Yan et al. (2023) for young ages, while a galactocentric distance of 10.5 kpc is more consistent for an older age. When the binary-star theorem by Kroupa (2008) is included, the values determined for Mecl and rh in this paper are consistent with the rh − Mecl relation derived by Marks & Kroupa (2012). To quantify this result further, more low-metallicity environments have to be examined in the future, and the heliocentric distance measurement of S209 needs to be improved.

Acknowledgments

We acknowledge support through the DAAD-Eastern-Europe Exchange grant at Bonn University and corresponding support from Charles University.

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Appendix A: Data and results for different ages

A.1. Age of 1 Myr

Table A.1.

Best-fit values for the MC with an age of 1 Myr (Yasui et al. 2023).

Table A.2.

Best-fit values for the SC with an age of 1 Myr (Yasui et al. 2023).

thumbnail Fig. A.1.

As Fig. 2, but for an age of 1 Myr.

Table A.3.

Results for the MC with an age of 1 Myr.

Table A.4.

Results for the SC with an age of 1 Myr.

thumbnail Fig. A.2.

As Fig. 3, but for the MC with an age of 1 Myr.

thumbnail Fig. A.3.

As Fig. 4, but for the SC with an age of 1 Myr.

A.2. Age of 5 Myr

Table A.5.

Best-fit values for the MC with an age of 5 Myr (Yasui et al. 2023).

Table A.6.

Best-fit values for the SC with an age of 5 Myr (Yasui et al. 2023).

thumbnail Fig. A.4.

As Fig. 2, but for an age of 5 Myr.

thumbnail Fig. A.5.

As Fig. 4, but for the MC with an age of 5 Myr.

Table A.7.

Results for the MC with an age of 5 Myr.

thumbnail Fig. A.6.

As Fig. 4, but for the SC with an age of 5 Myr.

Table A.8.

Results for the SC with an age of 5 Myr.

A.3. Age of 10 Myr

Table A.9.

Best-fit values for the MC with an age of 10 Myr (Yasui et al. 2023).

Table A.10.

Best-fit values for the SC with an age of 10 Myr (Yasui et al. 2023).

thumbnail Fig. A.7.

As Fig. 2, but for an age of 10 Myr.

Table A.11.

Results for the MC with an age of 10 Myr.

Table A.12.

Results for the SC with an age of 10 Myr.

thumbnail Fig. A.8.

As Fig. 3, but for the MC with an age of 10 Myr.

thumbnail Fig. A.9.

As Fig. 4, but for the SC with an age of 10 Myr.

All Tables

Table 1.

Best-fit values for the MC (Yasui et al. 2023).

Table 2.

Best-fit values for the SC (Yasui et al. 2023).

Table 3.

Results for the MC.

Table 4.

Results for the SC.

Table A.1.

Best-fit values for the MC with an age of 1 Myr (Yasui et al. 2023).

Table A.2.

Best-fit values for the SC with an age of 1 Myr (Yasui et al. 2023).

Table A.3.

Results for the MC with an age of 1 Myr.

Table A.4.

Results for the SC with an age of 1 Myr.

Table A.5.

Best-fit values for the MC with an age of 5 Myr (Yasui et al. 2023).

Table A.6.

Best-fit values for the SC with an age of 5 Myr (Yasui et al. 2023).

Table A.7.

Results for the MC with an age of 5 Myr.

Table A.8.

Results for the SC with an age of 5 Myr.

Table A.9.

Best-fit values for the MC with an age of 10 Myr (Yasui et al. 2023).

Table A.10.

Best-fit values for the SC with an age of 10 Myr (Yasui et al. 2023).

Table A.11.

Results for the MC with an age of 10 Myr.

Table A.12.

Results for the SC with an age of 10 Myr.

All Figures

thumbnail Fig. 1.

Metallicity map from Eilers et al. (2022), where the Galactic centre at (0, 0) is indicated by the cross, the Sun is located at (−8.0 kpc, 0) and is indicated by the circled dot, and the two possible positions of S209 are indicated by an asterisk. The arrows show the rotation of the stars and are coloured according to their mean self-calibrated metallicity.

In the text
thumbnail Fig. 2.

Reconstructed IMF with values from Yasui et al. (2023). The dashed lines represent the canonical IMF (see Sect. 2.1). The upper line shows the MC, and the lower line shows the SC.

In the text
thumbnail Fig. 3.

Canonical relation of Mecl and rh (Eq. (4)) shown by the blue line. The data points are the calculated values for the MC in S209. The caps indicate the 1σ and 2σ uncertainty boundaries.

In the text
thumbnail Fig. 4.

As Fig. 3, but for the SC.

In the text
thumbnail Fig. A.1.

As Fig. 2, but for an age of 1 Myr.

In the text
thumbnail Fig. A.2.

As Fig. 3, but for the MC with an age of 1 Myr.

In the text
thumbnail Fig. A.3.

As Fig. 4, but for the SC with an age of 1 Myr.

In the text
thumbnail Fig. A.4.

As Fig. 2, but for an age of 5 Myr.

In the text
thumbnail Fig. A.5.

As Fig. 4, but for the MC with an age of 5 Myr.

In the text
thumbnail Fig. A.6.

As Fig. 4, but for the SC with an age of 5 Myr.

In the text
thumbnail Fig. A.7.

As Fig. 2, but for an age of 10 Myr.

In the text
thumbnail Fig. A.8.

As Fig. 3, but for the MC with an age of 10 Myr.

In the text
thumbnail Fig. A.9.

As Fig. 4, but for the SC with an age of 10 Myr.

In the text

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