© The Authors 2024

1. Introduction

The dense environments of star clusters are home to a variety of intriguing objects. Among them, binaries hold a position that is both fundamental and significant. Both open and globular clusters (GCs) host a substantial number of binaries, many of them primordial in nature (Hut et al. 1992). Binaries have a major role to play in the ever-ensuing dynamics of the clusters and are pivotal in determining their various parameters such as age, luminosity function, radius, and mass.

Binaries have higher interaction cross-sections compared to single stars. Through frequent interactions with other objects in the cluster, they form an efficient mechanism for kinetic energy distribution. This binary-burning phase of the cluster is the longest phase of cluster evolution in which most of the observed clusters are expected to be (Heggie 1975). The elastic scattering interactions of binaries are effective in withstanding the impeding gravitational collapse in the center, thereby, defining the dense cluster core (Heggie & Hut 2003). The high rate of scattering transforms the dense cluster neighborhood into an ideal cauldron for the production of exotic objects. In recent decades, blue straggler stars (BSSs; Piotto et al. 2004), cataclysmic variables (CVs; Cool et al. 1995), low-mass X-ray binaries (LMXBs; Kim et al. 2006), and millisecond pulsars (MSPs; Edmonds et al. 2002) have been detected in close binary systems in star clusters. In addition, their evolution is better understood against the backdrop of binary fractions in the cluster.

The evolution of binaries in a cluster largely depends on two factors: i) the secular binary stellar evolution and ii) their interaction with other cluster members. Secular binary evolution can depend on the mass and mass ratio of the companion stars. The interaction rate in clusters can be significant enough to cause the destruction of soft binaries even in low-density stellar systems. Hard binaries, on the other hand, become even harder with interactions (Heggie 1975). These interactions can guide the binary stellar evolution by altering the orbital parameters, assisting the escape and exchange of the binary members. The binaries with kinetic energy comparable to that of a typical cluster member are the main variables in the binary fraction. Their amount and influence on cluster evolution can significantly depend on the environment. By exploring the binary fractions in star clusters belonging to diverse environments in terms of their different stellar and cluster parameters, we can peek at the dynamics underway.

Among the various techniques in literature to identify and characterise binary systems, the method based on the color-magnitude diagram (CMD) provides an efficient approach for constraining the fraction of unresolved main sequence (MS) binaries (Rubenstein & Bailyn 1997). In the CMD of a star cluster, the MS-MS binaries populate the region on the red side of the MS fiducial line (MSFL). The two other prominent detection methods for binaries are (1) radial velocity variability analysis, which is employed to analyse binaries comprising a massive companion (Latham 1996) and (2) the method based on the stellar photometric variability (Kaluzny et al. 1996). These methods are biased toward the brightest systems or the binaries with short and eccentric orbits. CMD analysis not only overcomes these limitations but is also statistically robust. A limited amount of observational time in two filters alone is sufficient to examine the multitude of cluster members belonging to a large parameter space (Hut et al. 1992).

Photometric errors, differential reddening, and spatial dependent variations of the photometric zero points are among the major challenges for an accurate determination of photometric binaries. Field stars that sneak into the cluster region of the field of view (FoV) pose another hurdle. Hence, this approach necessitates high-precision photometry, astrometry, and high-resolution images which are achievable in the era of the Hubble Space Telescope (HST) and advanced data reduction software.

Pioneering studies on photometric binaries can be seen in the work of Bolte (1992), Aparicio et al. (1991), and Romani & Weinberg (1991). The first investigation of a large sample of clusters was done by Sollima et al. (2007), who studied the binaries in 13 low-density Galactic GCs and derived the binary fractions in the cluster cores. An extensive survey of photometric binaries in 67 Galactic GCs was conducted by Milone et al. (2012, 2016), using the images collected with the Advanced Camera for Survey (ACS) aboard HST. They analysed the binary fraction as a function of various cluster and stellar parameters and found that binaries in the GCs are typically more centrally concentrated than single stars, while the fraction of binaries in the cluster core anti-correlates with the cluster mass.

The main limitation of these works is that the Galactic GCs share similar ages, with most of them being older than ∼10 Gyr and massive (Dotter et al. 2010). Hence, it is challenging to investigate the dependence of binary fractions on cluster age and study the binary systems where one component is more massive than ∼0.8 M_⊙. On the contrary, the investigation of binaries in Galactic open clusters is limited to low-mass clusters with masses smaller than ∼10⁴ solar masses (e.g. Cordoni et al. 2023).

The present work surpasses these limitations on mass and age by analysing fourteen star clusters in the Large and Small Magellanic Clouds (LMC and SMC) spanning an age interval between ∼0.6 and 2.1 Gyr and masses between 10⁴ and 10⁵ solar masses.

The paper is structured as follows. Section 2 is dedicated to the details of observations, data sources, and data reduction techniques. In Sect. 3, we derive the structural parameters of the star cluster, whereas Sect. 4 discusses the method for deriving the binary fractions. Section 5 presents the results and a summary of the paper is provided in Sect. 6.

2. Data source and data reduction

This work investigates fourteen Magellanic Cloud (MC) star clusters younger than ∼2 Gyr that span an interval of mass (10⁴ and 10⁵ solar masses), which has been poorly explored in the context of binaries. To obtain robust results, our sample does not include clusters younger than ∼0.6 Gyr. Indeed, these star clusters exhibit split MSs associated with stellar populations with different rotation rates (Milone & Marino 2022, and references therein), which can alter our binary fraction analysis.

The clusters are observed with the Wide Field Channel (WFC) of the ACS and the Ultraviolet and Visible channel (UVIS) of the Wide Field Camera 3 (WFC3) on board HST. The details of the observations are summarised in Table 1. We have corrected the effect of the poor Charge Transfer Efficiency (CTE) by using the method described in Anderson & Bedin (2010).

High-precision photometry and astrometry, essential for an accurate binary analysis, are derived using the state-of-the-art data reduction programs developed by Jay Anderson (e.g. Anderson & King 2000; Anderson et al. 2008), which are based on the effective point spread function (ePSF) fitting. We used the FORTRAN software package KS2, which is an upgraded version of the program kitchen_sync (Anderson et al. 2008) that entails three different methods to measure the positions and magnitudes of stars. Method I measures the stars in each image, independently, by using the best available ePSF model. The results are then averaged to derive the best estimates of magnitude and position. This method provides the best photometry for relatively bright stars that define distinct peaks in a 5 × 5 pixel raster. Method II measures each star using aperture photometry, after subtracting all the neighboring stars. It works well for faint stars that do not have enough photons to be aptly fitted by the ePSF. Method III is similar to method II in terms of analysis and it works well in very crowded regions. We refer to papers by Sabbi et al. (2016), Bellini et al. (2017), and Milone et al. (2023) for details on KS2.

Since we are interested in stars with high-precision photometry, we excluded all the sources that were poorly fitted by the PSF model and the stars with large root mean square values in positions. To select these stars, we used the computer programs and the methods provided by Milone et al. (2012). The stellar coordinates are corrected for geometric distortion by using the solutions by Anderson & King (2006) and Bellini et al. (2011) the photometry has been calibrated to the Vega magnitude system as in Milone et al. (2023) using the zero points provided by Space Telescope Science Institute (STScI)¹.

The photometry was corrected for the effects of differential reddening by using the methods by Milone et al. (2012) and Legnardi et al. (2023). As an example, Fig. 1 compares the original CMD of NGC 1751 (left) with the CMD corrected for differential reddening (right).

Fig. 1. ifferential reddening correction. The figure comparison of the CMD of the star cluster NGC 1751 before (left panel) and after the correction for differential reddening (right panel).

Finally, we derived the ages, metallicities, distances, and reddening of the studied clusters by comparing the CMDs with isochrones, as in Cordoni et al. (2023). We used the isochrone from the Dartmouth Stellar Evolution Database² for clusters older than 1 Gyr and the MESA Isochrones and Stellar Tracks³ for clusters that are younger than 1 Gyr. The reddening coefficients for the different filters are provided by Aaron Dotter (priv. comm.). Thus obtained CMDs for all the clusters are shown in Fig. 2.

Fig. 2. Collection of CMDs for the investigated star clusters. For each cluster, we plot the F814W magnitude against the X−F814W color, where the X filter is quoted in the corresponding panel.

We performed artificial-star (AS) tests to estimate the photometric errors, the level of completeness of our sample, and the fraction of blended sources that contaminate the CMD region populated by binaries. To do so, we used the method by Anderson et al. (2008) and Milone et al. (2009). For each cluster, we generated a catalog of 100 000 ASs with instrumental magnitudes between −4 and −13.7 in the F814W bands. The colors of each AS are derived from the empirical fiducial line of MS stars. For the ASs, we adopted the same radial distribution and luminosity function as the real stars and reduced them using the same PSF model and procedure adopted for real stars. Moreover, we used the same criteria as for real stars to select the sample of relatively isolated ASs that are well fitted by the PSF. Completeness is derived for each star as in Milone et al. (2009; see their Sect. 2.2), by accounting for its magnitude and radial distance from the cluster center.

3. Structural parameters of the star clusters

In this section, we discuss how we derived the density profile of the studied clusters and inferred the values of the core radius, density, and mass of each cluster. These quantities are crucial in properly characterising the populations of binaries in star clusters.

3.1. Density profiles of the clusters

To estimate the density profile of each star cluster, we used the procedure illustrated in Fig. 3 for the cluster NGC 1751. We first derived the number of stars with mass greater than 0.9 M_⊙ in different annuli in the HST FoV and then corrected those numbers for completeness. The obtained numbers are normalised by the areas of the annuli to get the number density profile of the cluster.

Fig. 3. Density profile. Left panel: grey points correspond to the observed density profile of the cluster NGC 1751. The contribution from field contamination is denoted by the grey dotted horizontal line. The profile in blue is obtained after subtracting the background contamination from the actual density estimate, and it is fitted with the EEF profile. Right panel: corresponding quantities in a logarithmic scale. The vertical lines indicate the core radius, the corresponding half-mass radius denoted by Rhm, and the distances indicated as Rf and R2f. The background level (bg) is marked by a horizontal line.

We derived the parameters of the EEF profile (Elson et al. 1987) that provide the best fit with the observed density profile. Specifically, we adopted the relation:

$$\begin{array}{c}\hfill \mu (r)={\mu }_0{(1+\frac{r^2}{a^2})}^{-\frac{\gamma }{2}}+bg,\end{array}$$(1)

where μ₀ is the central density, a is the scaling factor, γ is the power law factor, and bg is the constant that accounts for the contamination from field stars. The core radius, R_c, is derived by the relation,

$$\begin{array}{c}\hfill R_c=a{(2^{\frac{2}{\gamma }}-1)}^{\frac{1}{2}}.\end{array}$$(2)

The left panel of Fig. 3 shows the EEF density profile that provides the best fit with the observations of NGC 1751. The right panel shows the corresponding plot in the logarithmic scale, namely, the EFF Profile. The grey and blue points denote the number density distribution with and without the contamination from field stars. Blue points are obtained by subtracting bg from the grey points, where the black horizontal line denotes the level of bg. The corresponding radius is R_f. The radius at which the number density becomes two times bg is denoted as R_2f.

For comprehensiveness, we derived the King profile (King 1962) parameters that provide the best-fit with the observed density profile. The results are listed in Table 2.

3.2. Mass functions and masses of the clusters

To derive the mass functions of each cluster, we used the procedure by Cordoni et al. (2023). We derived the number of MS stars within the radial distance of R_2f, divided them into intervals of equal mass, and normalised this quantity to the mass bin. The star counts are corrected for completeness, and only stars with completeness values larger than 0.5 are taken into account.

We fitted the observed mass function with the relation from Trenti et al. (2010):

$$\begin{array}{c}\hfill {\xi }_m={\xi }_0{m}^{-\alpha }.\end{array}$$(3)

To estimate the total mass of the cluster, M_Cluster, we integrated the mass function over the entire mass interval provided by the best-fit isochrone. Moreover, we accounted for the cluster stars with a radial distance larger than R_2f by using the cluster EEF profile and assuming a homogeneous radial distribution for stars with different masses.

Finally, we estimated the half-mass relaxation time by using the following equation,

$$\begin{array}{c}\hfill t_{\mathrm{rh}}=\frac{0.138M_{\mathrm{Cluster}}^{1/2}{R_{\mathrm{hm}}}^{3/2}}{G\overline{M}ln(0.11M_{\mathrm{Cluster}}/\overline{M})},\end{array}$$(4)

where G is the universal gravitational constant, $\overline{M}$ is the average mass of a star, and R_hm is the half-mass radius (Spitzer 1987). The results are provided in Table 3.

4. The binary fraction

Star clusters are dense systems. Surviving in such environments requires the stars in a binary to be in close proximity. Even with the high-resolution power of HST, it is hard to resolve the stars in such a system. Hence, the binaries are perceived as single stars of enhanced magnitude, where the observed flux will be the combination of the fluxes of the companion stars.

Supposing we are observing a binary system where the magnitudes and fluxes of the companion stars are m₁ and m₂, and F₁ and F₂, respectively. Then, the magnitude of the binary system is:

$$\begin{array}{c}\hfill m_{\mathrm{bin}}=m_1-2.5log(1+\frac{F_2}{F_1}),\end{array}$$(5)

where subscripts 1 and 2 denote the primary and secondary stars, respectively, with the primary star being the more massive component of the binary system.

In a simple stellar population, the fluxes of MS stars depend on stellar mass and follow a given mass-luminosity relation. Hence, the luminosity of a binary system depends on the mass ratio between the companion stars:

$$\begin{array}{c}\hfill q=\frac{M_2}{M_1},\end{array}$$(6)

where, 0 < q ≤ 1.

As illustrated in Fig. 4, MS-MS binaries populate the CMD region on the red and bright side of the MSFL, with equal-mass binaries defining a fiducial line that is parallel to the MSFL but is shifted by 0.7526 mag in brightness. The fiducial lines are derived from the observed MS stars of the cluster NGC 1718, which is taken as a test case for demonstration. Specifically, we selected the well-measured candidate MS stars and divided this sample into intervals of 0.5 mag in the F475W band. The MSFL is derived by linearly interpolating the median colors and magnitudes of the stars in each bin. The standard deviation of the colors of stars in each bin, σ, is considered a proxy for the average color error of the stars in that bin. For a fixed mass of the primary star, the binaries with q < 1 distribute on a curved line between the MSFL and the equal-mass binary fiducial line. Due to observational errors, it is not possible to disentangle binaries with small mass ratios from single stars in the observed CMD. The colors and magnitudes of the binaries with different mass ratios are determined with the mass-luminosity relation provided by the best-fitting isochrones provided by Dotter et al. (2008), Dotter (2016). The latter is derived by comparing the CMD of each cluster with a grid of solar-scaled isochrones that account for different ages, metallicities ([Fe/H]), distance modulus ((m − M)₀), and reddening (E(B − V), see Milone et al. 2009, for details). The distance to the cluster is calculated from (m − M)₀. The values of age, [Fe/H], (m − M)₀, and E(B − V) that provide the best match with the observed data are listed in Table 4⁴.

Fig. 4. Loci of binaries. CMD of NGC 1718 zoomed around the upper MS. The colored continuous lines are the fiducial lines of binary systems with different mass ratios, as indicated in the inset. The dashed lines represent the locus of binaries with primary masses of 1.19, 1.01, and 0.86 solar masses and mass ratios between 0 and 1.

The method for deriving the fraction of binaries is illustrated in Fig. 5 for NGC 1718. We divided the CMD into two regions. Region A includes all the single stars in the studied luminosity interval and the binary systems with a primary star in the same magnitude range. It is limited to the left by the MSFL, blueshifted by three times the color observational error (σ). The right boundary corresponds to the fiducial line of equal-mass binaries redshifted by three times the color error. The upper and lower boundaries are the sequences of binaries where the mass ratio ranges from zero to one and the primary stars exhibit magnitudes of $m_{F475W}=m_{F475W}^{\mathrm{bright}}$ and $m_{F475W}^{\mathrm{faint}}$, respectively. Here, $m_{F475W}^{\mathrm{bright}}$ and $m_{F475W}^{\mathrm{faint}}$, corresponding to masses M_u and M_l, are the bright and faint limit of the studied luminosity interval. Then, $m_{F475W}^{\mathrm{bright}}$ is selected such that region A does not intervene with the turnoff region, and the selected $m_{F475W}^{\mathrm{faint}}$ assures the completeness of the magnitude range of analysis remains above 0.5. Region B is the portion of region A that is predominantly populated by binaries with q ≥ q_min. The value of q_min is chosen with the criterion that the fiducial line of binaries with q = q_min is redder than the MSFL shifted by σ to the red.

Fig. 5. Method of analysis. The figure summarises the method that was used to analyse MS-MS binaries in NGC 1718. Top-left panel: region A of the mF475W vs. mF475W − mF814W CMD, which is shaded with light-blue colour. Top-right panel: region B, a subsection of region A, is shaded with a pink. Bottom-left: CMDs of stars in the cluster region. Bottom-centre: field region. Bottom-right: simulated CMD. The MS fiducial line and the fiducial of equal-mass binaries are represented with red and cyan dot-dashed lines, respectively, in all panels. The blue dot-dashed and blue-dashed lines mark the left and right color boundaries of region A, displaced by three times the colour error from the fiducial and equal mass binary lines, respectively. Green curves represent the mass limit of analysis, i.e., binaries with certain primary mass and the mass ratio with secondary mass ranging between 0 and 1. Orange lines represent the unequal mass binary line, i.e., binaries with a mass ratio, q = 0.6. See text for details.

Moreover, to derive the fraction of binaries, we defined two regions in the HST FoV: 1) The cluster-region that extends from the dense cluster centre to R_2f. 2) Starting from R_2f, the field-region stretches to the outermost parts of the HST FoV. Given the small FoV of HST, we can assume that the distributions of field stars in the cluster-region and field-region are nearly the same. The field stars appearing in cluster-region may contribute to the stars present in regions A and B of the cluster-region CMD. Their amount is estimated by constructing a CMD using the stars appearing in the field-region. As an example, the bottom-left and bottom-middle panels of Fig. 5 show the CMDs of stars in the cluster-region and the field-region of NGC 1718, respectively.

The binary fraction is derived using the Eq. (1) of Milone et al. (2012):

$$\begin{array}{c}\hfill F_{\mathrm{bin}}^{q\ge q_{\min }}=\frac{N_{\mathrm{cluster}}^{\mathrm{B}}-N_{\mathrm{field}}^{\mathrm{B}}}{N_{\mathrm{cluster}}^{\mathrm{A}}-N_{\mathrm{field}}^{\mathrm{A}}}-\frac{N_{\mathrm{art}}^{\mathrm{B}}}{N_{\mathrm{art}}^{\mathrm{A}}}.\end{array}$$(7)

Here, $N_{\mathrm{cluster}}^{\mathrm{A}}$ and $N_{\mathrm{cluster}}^{\mathrm{B}}$ are the number of stars in the cluster-region, corrected for completeness, in the regions A and B of the CMD, respectively. $N_{\mathrm{field}}^{\mathrm{A}}$ and $N_{\mathrm{field}}^{\mathrm{B}}$ are the corresponding number of stars in the field-region normalised by the ratio between the respective areas of the cluster and field region. Then, $N_{\mathrm{art}}^{\mathrm{A}}$ and $N_{\mathrm{art}}^{\mathrm{B}}$ are the numbers of artificial stars that populate the regions A and B of the CMD. The artificial star binary fraction is denoted by the term $\frac{N_{\mathrm{art}}^{\mathrm{B}}}{N_{\mathrm{art}}^{\mathrm{A}}}$ of the Eq. (7). This term accounts for the fraction of single stars that populate region B of the CMD due to large photometric errors or chance superposition of their images with other single stars.

5. Results and discussion

For each cluster, we derived the fraction of binaries with a mass ratio, q ≥ 0.7 by following the procedure described in Sect. 4. Moreover, in the clusters where the binaries are better distinguishable from single stars, namely NGC 1718, NGC 1751, NGC 1783, NGC 1846, NGC 1868, NGC 2203, and NGC 2213. We also studied binaries with q ≥ 0.6.

The results are listed in Table 5, where we provide the fraction of binaries within the radial distance of R_hm and the analysed stellar-mass interval, given by the difference between the corresponding M_u and M_l values. The binary fraction ranges between ∼0.07 in NGC 1846, to ∼0.24, in NGC 2108. Table 5 also provides the fraction of binaries within the core of each cluster⁵.

As shown in Fig. 6, the binary fraction exhibits no significant correlations with cluster age and metallicity. This conclusion is supported by Spearman’s rank correlation coefficients of 0.2 and −0.2, respectively.

Fig. 6. Binary fraction and cluster parameters. Fraction of binaries with q ≥ 0.7 and within the Rhm of the cluster is plotted against the age (left panel) and iron abundance (right panel) of the host cluster.

5.1. Binary fraction and the mass of the primary star

To investigate the binary fraction in different intervals of primary-star mass, we divided the studied magnitude interval into five magnitude bins of equal width⁶. To do this, we used the mass-luminosity relations inferred from the best-fitting isochrones. We estimated the fraction of binaries with q ≥ 0.7 and r ≤ R_hm in each bin and plotted this quantity as a function of the mean mass of the primary stars in that bin, as illustrated in Fig. 7. A visual inspection of this figure suggests that the behavior of $F_{\mathrm{bin},R_{\mathrm{hm}}}^{q\ge 0.7}$ as a function of the stellar mass varies from cluster to cluster.

Fig. 7. Binary fraction and mass of the primary star. The fraction of binaries with q ≥ 0.7 is plotted against the mass of the primary star.

Furthermore, for each cluster, we derived the χ² value⁷ with respect to the average binary fraction, and the corresponding P-value, which indicates the deviation from a flat distribution. The latter is estimated using 1000 Monte Carlo simulations. In each simulation, we assumed a flat binary fraction corresponding to the observed average binary fraction and the same number of stars as in the observed CMD. The P-value is defined as the fraction of simulations with χ² values greater than the observed χ². The χ² and P-values are listed in Table 6.

We find that various clusters manifest a trend very close to flat distribution, as evident in NGC 1806 (P-value ∼ 1). Conversely, NGC 2108 (P-value of ∼0) shows a decrease in the binary fraction with increasing mass of the primary star.

To further compare the results from the different clusters, we calculated the average binary fraction (${\overline{F}}_{\mathrm{bin},R_{\mathrm{hm}}}^{q\ge 0.7}$) in the different mass bins and plotted the $F_{\mathrm{bin},R_{\mathrm{hm}}}^{q\ge 0.7}$/${\overline{F}}_{\mathrm{bin},R_{\mathrm{hm}}}^{q\ge 0.7}$ ratio as a function of the mean mass of the primary stars in that bin. The results from all the analysed clusters are plotted together in Fig. 8, where we observe that the general trend is that of a flat distribution, with a Spearman’s rank correlation coefficient of −0.1.

Fig. 8. Mass distribution of binaries. Binary fraction with mass ratios larger than 0.7 as a function of the mass of the primary star for all studied clusters. The binary fraction is normalised to the average binary fraction in a mass bin.

5.2. Relations between the binary fraction and the mass ratio

To constrain the relation between binary population and mass ratio, we calculated the binary fraction in different mass ratio bins and with r ≤ R_hm. The q-intervals are selected within region B of the CMD with a prerequisite that the strips formed by the different q-loci should occupy the same area in the CMD. Therefore, we have selected loci of q values 0.6, 0.685, 0.755, 0.833, and 1. Though the subsequent q values are not equal in increment, they form equal-area strips in the CMD.

We calculated the fraction of binaries in each mass ratio interval and derived the equivalent binary fraction,

$$\begin{array}{c}\hfill V_{\mathrm{bin}}=\frac{F_{\mathrm{bin}}}{\mathrm{\Delta }q},\end{array}$$(8)

where F_bin is the fraction of binaries in a given mass ratio interval, Δq.

The values of V_bin, Rhm are plotted against q, as shown in Fig. 9, whereas the χ² and P-values are provided in Table 6. We note that some clusters such as NGC 1718, NGC 2203, and NGC 2213 (all with P-value = 0) manifest an increase in the frequency of binaries with higher mass ratios, while others, such as NGC 1651 (P-value ≥ 0.5) and NGC 1846 (P-value ∼ 0.5), exhibit a nearly flat distribution.

Fig. 9. Binary fraction and mass ratio parameter. The frequency of binaries is plotted as a function of the mass ratio. The horizontal bars mark the mass-ratio intervals corresponding to each point.

By combining the results from all clusters, Fig. 10 investigates the overall trend between binary fraction and mass ratio. To properly compare the different clusters, we have normalised the fraction of binaries by its average value. The distribution is nearly flat with a Spearman’s rank correlation coefficient of +0.4.

Fig. 10. Mass ratio distribution of binaries. Frequency of binaries as a function of the mass ratio for all the studied clusters.

5.3. The radial distribution of binaries

The radial distribution of binaries is examined by dividing the cluster region within the radial distance of R_2f into five annuli, each with an equal number of MS stars. The binary fraction in each annulus is calculated using the procedure mentioned in Sect. 4 and is plotted as a function of the mean radial distance of the stars in that annulus from the cluster center, as in Fig. 11. The corresponding χ² and P-values are reported in Table 6.

Fig. 11. Radial distribution of binaries. The core radii of the studied clusters are denoted with a dotted pink line.

We find that in some clusters, such as ESO057SC075 and NGC 1868, which are the studied clusters with a higher dynamical age of 7 and 10 respectively, the binaries are centrally concentrated. However, the binaries of other clusters, including NGC 1751 and NGC 2108, show a flat distribution with P-values ∼ 1 whereas NGC 1872 shows hints of a secondary peak. The change in radial distribution from flat to double peak and then to a concentration in the center can be a function of dynamical age. Mass segregation is the main driver of this behaviour, as observed in the case of BSSs (Ferraro et al. 2012).

The results for all the clusters are plotted together in the left panel of Fig. 12, where we normalise the fraction of binaries to the fraction of binaries in the core and express the radial distance in the units of the core radius. We do not find evidence for a correlation between the fraction of binaries and the radial distance from the cluster center, as denoted by the correlation rank of −0.3. However, when we separate the clusters into two groups with ages either smaller or greater than five times their half-mass relaxation time, t_rh (middle and right panels of Fig. 12, respectively), a pattern emerges. With a correlation rank of −0.6, most dynamically old clusters exhibit some hints of binary segregation into the centre, while the dynamically younger ones exhibit a flat distribution with a correlation rank of 0.

Fig. 12. Difference types of radial distribution. Binary fraction, normalised to the binary fraction in the core, as a function of the radial distance, in the unit of core radius.

5.4. Candidate blue straggler stars

The CMDs of all analysed clusters show a sequence of stars in the blue and brighter side of the MSTO. This region of the CMD is populated by the BSSs, whose origins are traditionally associated with stellar mergers or the evolution of binary stars (e.g. Sun et al. 2018, and references therein). Recent works argue that the sequences of blue stars observed in some intermediate-age LMC star clusters correspond to young stellar populations that emerged from the merging events in the MC (Li et al. 2016; Hong et al. 2017). Understanding the origin of the stars, which we refer to as candidate BSSs hereafter, is beyond the purposes of our study. Here, we focus on determining some of their observational properties.

To derive the fraction of candidate BSSs in each cluster, we defined two regions in their CMD, namely A1 and B1, as illustrated in Fig. 13 for the cluster NGC 2213. The A1 region (pink-shaded area) mostly hosts MS stars. It is similar to the region A of the CMD introduced in Sect. 4, but spans an interval of one magnitude, in the filter F814W, along the fiducial line. The B1 region corresponds to the portion of the CMD that hosts the candidate BSSs and is limited by the azure solid and dotted lines. The continuous line corresponds to the MSFL shifted by four times σ to the blue side, where σ is the color uncertainty. The vertical line has the same color as the MSTO, whereas the horizontal line is 0.25 mag fainter than the brightest limit of region A1.

Fig. 13. Procedure to derive the BSS fraction. mF814W vs. mF475W − mF814W CMD of stars in the cluster region of NGC 2213 (left panel) and in the field region (right panel). The pink-shaded areas mark the CMD region A1, which is mostly populated by MS stars, while the stars in the CMD region B1 which hosts the candidate BSSs are marked with azure dots. The azure solid and dotted lines separate the B1 region from the remaining CMD area. See the text for details.

The fraction of candidate BSSs is calculated as:

$$\begin{array}{c}\hfill F_{\mathrm{BSS}}=\frac{N_{\mathrm{cluster}}^{\mathrm{B}1}-N_{\mathrm{field}}^{\mathrm{B}1}}{N_{\mathrm{cluster}}^{\mathrm{A}1}-N_{\mathrm{field}}^{\mathrm{A}1}},\end{array}$$(9)

where $N_{\mathrm{cluster}}^{\mathrm{A}1}$ and $N_{\mathrm{cluster}}^{\mathrm{B}1}$ are the numbers of stars, corrected for completeness, in the regions A1 and B1 of the cluster-region CMD, whereas $N_{\mathrm{field}}^{\mathrm{A}1}$ and $N_{\mathrm{field}}^{\mathrm{B}1}$ are the corresponding quantities for the field-region CMD that have been normalised by the ratio between the areas of cluster-region and the field-region.

The results are listed in Table 7 and reveal that the fraction of candidate BSSs within R_hm ranges from 0.1%, in NGC 2108, to about 2% in NGC 1806 and NGC 2213. Figure 14 reveals that there is no correlation between the fraction of candidate BSSs in the core and the cluster age or metallicity, as demonstrated by the Spearman’s rank correlation coefficients of 0.3 and −0.3, respectively. Nevertheless, we confirm that clusters younger than ∼1 Gyr are unlikely to host BSSs (Rain et al. 2021; Cordoni et al. 2023).

Fig. 14. BSS fraction and cluster parameters. Fraction of candidate BSSs in the core as a function of cluster age (left panel) and iron abundance (right panel).

To further investigate the BSSs of the analysed clusters, we derived the A⁺ parameter, which quantifies the difference between the cumulative distribution of the BSSs, ϕ_BSS, and that of a reference population of stars with smaller masses, ϕ_REF (e.g. Lanzoni et al. 2016).

The A⁺ parameter is defined as:

$$\begin{array}{c}\hfill A^{+}(x)={\displaystyle {\int }_{x_{min}}^x{\varphi }_{\mathrm{BSS}}(x^{\prime })\mathrm{d}{x}^{\prime }-{\int }_{x_{min}}^x}{\varphi }_{\mathrm{REF}}(x^{\prime })\mathrm{d}{x}^{\prime },\end{array}$$(10)

where x = log(R/R_hm) and x_min is the minimum value that we sampled. We limited the analysis to stars within the half-mass radius and adopted the MS stars in the region A1 of the CMD as the reference population, as illustrated in Fig. 13 for the cluster NGC 2213. To account for the effect of field stars on the determination of A⁺, we adopted a method that is based on the distributions of 1000 groups of BSSs and reference stars.

To define each group, we first randomly associated each field-region star that populates the B1 and A1 portions of the CMD with a random position within the half-mass radius of the cluster. Then, we excluded them from the sample of BSSs and reference-population stars and derived a value of A⁺ by using the remaining stars. The best determination of A⁺ and the corresponding uncertainty are provided by the mean and root mean square (rms) of the 1000 determinations, respectively.

The results are listed in Table 7 and reveal that A⁺ ranges from ∼ − 0.16 in NGC 1783 to 0.30 in NGC 2213 and that eleven out of fourteen clusters have positive A⁺ values. Moreover, we find a mild correlation between A⁺ and the cluster dynamical age that yields a Spearman’s correlation rank of 0.6. This finding would provide constraints on the origin of the candidate BSSs (e.g. Li et al. 2016).

The fraction of candidate BSSs are further plotted as a function of the binary fraction in Fig. 15. We did not find a significant correlation between these two quantities, similar to what has been observed in Galactic open clusters (Cordoni et al. 2023). Conversely, there is evidence that clusters with different densities appear to populate different sequences.

Fig. 15. Binary fraction and BSS fraction. Fraction of candidate BSSs as a function of the fraction of binaries with mass ratio, q ≥ 0.7 in the studied clusters. The clusters are colour-coded based on their density, as shown in the colour bar.

5.5. Comparison with Galactic clusters

To compare the binary fractions of MC star clusters and Galactic open and globular clusters, we combined the results of this work with results from the literature where binaries were analysed homogeneously. The results are illustrated in Fig. 16 which compares the core binary fraction with q ≥ 0.7 derived in this paper (purple dots) and those measured in Galactic GCs (GGCs, green dots; Milone et al. 2012, 2016) and Galactic open clusters (pink dots; Cordoni et al. 2023). The left panel of Fig. 16 shows the fraction of binary as a function of the cluster mass in the logarithmic scale. We used mass and t_rh values for Galactic GCs from Baumgardt & Hilker (2018), except for NGC 6637, NGC 6652, NGC 6981, and Palomar 1, for which the adopted values come from McLaughlin & van der Marel (2005). The MC star clusters studied in our paper span a mass interval between 10⁴ and 10⁵ solar masses, which are poorly populated by the clusters studied in the previous studies.

Fig. 16. Binary fraction in different environments. The plot compares the core binary fraction for clusters belonging to different environments. Galactic GCs, Galactic open clusters, and the Magellanic Clouds globular clusters studied in this paper are denoted in lime green, magenta, and violet, respectively. Binary fractions in these clusters are explored as a function of mass (left panel), age (middle panel), and dynamical age (right panel) on a logarithmic scale.

When considering all clusters together, we find an anti-correlation between the binary fraction in the core and the cluster mass. However, for a fixed cluster mass, the binary fraction spans a wide range of values. In particular, the MC clusters studied in this paper exhibit a larger fraction of binaries than the bulk of GGCs with similar masses. Conversely, as shown in the middle and right panel of Fig. 16, there is no evidence for a correlation between the fraction of binaries and the cluster age or the ratio between cluster age and t_rh.

6. Summary and conclusions

We used data collected with the UVIS/WFC3 and ACS/WFC cameras on board HST to investigate the population of binaries and candidate BSSs along the MS of 14 MC star clusters. Moreover, we determined the structure parameters of the star clusters, including the core radius and the central density, and estimated the mass function and the total mass of the cluster by fitting the density profile with an EEF profile. The main results on binaries can be summarised as follows,

• When we analysed the entire FoV, the fraction of binaries with mass ratio, q ≥ 0.7 and r ≤ R_hm ranges from ∼7%, in NGC 1846, to more than ∼20%, in NGC 2108. These values correspond to a total binary fraction between ∼20% and 70%, by assuming a flat mass-ratio distribution. We obtain similar results for the binaries in the cluster cores.

• There is no correlation between the fraction of binaries and the fraction of candidate BSSs, similar to what has been observed in the case of Galactic open clusters (Cordoni et al. 2023).

• When we combine the results from all clusters, we find no evidence for correlations between the fraction of binaries and either the mass of the primary star or the mass ratio. However, we notice various remarkable exceptions. As an example, the binary fraction decreases towards large stellar masses in NGC 2108, while NGC 1806 follows a flat distribution. Moreover, NGC 1718, NGC 2203, and NGC 2213 exhibit a predominance of binaries with large mass ratios, whereas the fractions in NGC 1651 and NGC 1846 are distributed homogeneously.

• There is no evidence for significant differences in the radial distribution of binaries in most studied clusters. However, in clusters that are significantly older in comparison to their t_rh, the binaries are more centrally concentrated than single stars.

We have combined results on MC clusters with those by a recent paper where we have performed a similar analysis on 78 Galactic open clusters (Cordoni et al. 2023) and with results on 67 GGCs (Milone et al. 2012, 2016). In total, binaries have been now homogeneously studied in 159 star clusters.

The fraction of binaries does not correlate with either the cluster age or with the dynamical age. Conversely, we find a significant anti-correlation between the fraction of binaries in the core and the mass of the host cluster. However, clusters with similar masses exhibit a range of binary fractions that is wider than the observational errors. As an example, the star clusters studied in this paper typically host higher binary fractions than the GGCs with similar masses. This fact indicates that at least another parameter, in addition to cluster mass, determines the fraction of binaries in the core.

Acknowledgments

This work has received support from the European Research Council (ERC), under the European Union’s Horizon 2020 research innovation program (Grant Agreement ERC-StG 2016, No: 716082, GALFOR, PI: Milone, http://progetti.dfa.unipd.it/GALFOR. A.F.M., G.C., and A.P.M. acknowledge the support from the INAF-GTO-GRANTS 2022 (“Understanding the formation of globular clusters with their multiple stellar generations”, PI. A.F. Marino). S.J. acknowledges support from the NRF of Korea (2022R1A2C3002992, 2022R1A6A1A03053472). T.Z. has received funding from the European Union’s Horizon 2020 research and innovation program under the MarieSklodowska-Curie Grant Agreement No. 101034319 and the European Union – NextGenerationEU, beneficiary: Ziliotto. This study has used HST archival data from the Space Telescope Science Institute (STScI). We thank the anonymous referees for their constructive suggestions.

References

All Tables

All Figures

	Fig. 1. ifferential reddening correction. The figure comparison of the CMD of the star cluster NGC 1751 before (left panel) and after the correction for differential reddening (right panel).
In the text

	Fig. 2. Collection of CMDs for the investigated star clusters. For each cluster, we plot the F814W magnitude against the X−F814W color, where the X filter is quoted in the corresponding panel.
In the text

Fig. 3. Density profile. Left panel: grey points correspond to the observed density profile of the cluster NGC 1751. The contribution from field contamination is denoted by the grey dotted horizontal line. The profile in blue is obtained after subtracting the background contamination from the actual density estimate, and it is fitted with the EEF profile. Right panel: corresponding quantities in a logarithmic scale. The vertical lines indicate the core radius, the corresponding half-mass radius denoted by Rhm, and the distances indicated as Rf and R2f. The background level (bg) is marked by a horizontal line.

In the text

	Fig. 4. Loci of binaries. CMD of NGC 1718 zoomed around the upper MS. The colored continuous lines are the fiducial lines of binary systems with different mass ratios, as indicated in the inset. The dashed lines represent the locus of binaries with primary masses of 1.19, 1.01, and 0.86 solar masses and mass ratios between 0 and 1.
In the text

Fig. 5. Method of analysis. The figure summarises the method that was used to analyse MS-MS binaries in NGC 1718. Top-left panel: region A of the mF475W vs. mF475W − mF814W CMD, which is shaded with light-blue colour. Top-right panel: region B, a subsection of region A, is shaded with a pink. Bottom-left: CMDs of stars in the cluster region. Bottom-centre: field region. Bottom-right: simulated CMD. The MS fiducial line and the fiducial of equal-mass binaries are represented with red and cyan dot-dashed lines, respectively, in all panels. The blue dot-dashed and blue-dashed lines mark the left and right color boundaries of region A, displaced by three times the colour error from the fiducial and equal mass binary lines, respectively. Green curves represent the mass limit of analysis, i.e., binaries with certain primary mass and the mass ratio with secondary mass ranging between 0 and 1. Orange lines represent the unequal mass binary line, i.e., binaries with a mass ratio, q = 0.6. See text for details.

In the text

	Fig. 6. Binary fraction and cluster parameters. Fraction of binaries with q ≥ 0.7 and within the Rhm of the cluster is plotted against the age (left panel) and iron abundance (right panel) of the host cluster.
In the text

	Fig. 7. Binary fraction and mass of the primary star. The fraction of binaries with q ≥ 0.7 is plotted against the mass of the primary star.
In the text

	Fig. 8. Mass distribution of binaries. Binary fraction with mass ratios larger than 0.7 as a function of the mass of the primary star for all studied clusters. The binary fraction is normalised to the average binary fraction in a mass bin.
In the text

	Fig. 9. Binary fraction and mass ratio parameter. The frequency of binaries is plotted as a function of the mass ratio. The horizontal bars mark the mass-ratio intervals corresponding to each point.
In the text

	Fig. 10. Mass ratio distribution of binaries. Frequency of binaries as a function of the mass ratio for all the studied clusters.
In the text

	Fig. 11. Radial distribution of binaries. The core radii of the studied clusters are denoted with a dotted pink line.
In the text

	Fig. 12. Difference types of radial distribution. Binary fraction, normalised to the binary fraction in the core, as a function of the radial distance, in the unit of core radius.
In the text

Fig. 13. Procedure to derive the BSS fraction. mF814W vs. mF475W − mF814W CMD of stars in the cluster region of NGC 2213 (left panel) and in the field region (right panel). The pink-shaded areas mark the CMD region A1, which is mostly populated by MS stars, while the stars in the CMD region B1 which hosts the candidate BSSs are marked with azure dots. The azure solid and dotted lines separate the B1 region from the remaining CMD area. See the text for details.

In the text

	Fig. 14. BSS fraction and cluster parameters. Fraction of candidate BSSs in the core as a function of cluster age (left panel) and iron abundance (right panel).
In the text

	Fig. 15. Binary fraction and BSS fraction. Fraction of candidate BSSs as a function of the fraction of binaries with mass ratio, q ≥ 0.7 in the studied clusters. The clusters are colour-coded based on their density, as shown in the colour bar.
In the text

Fig. 16. Binary fraction in different environments. The plot compares the core binary fraction for clusters belonging to different environments. Galactic GCs, Galactic open clusters, and the Magellanic Clouds globular clusters studied in this paper are denoted in lime green, magenta, and violet, respectively. Binary fractions in these clusters are explored as a function of mass (left panel), age (middle panel), and dynamical age (right panel) on a logarithmic scale.

In the text