THE BINARY FRACTION IN THE GLOBULAR CLUSTER M10 (NGC 6254): COMPARING CORE AND OUTER REGIONS^*

The binary fraction is an essential component in the formation and evolution of dynamically active systems, like globular clusters (GCs). In such dense environments, where stellar gravitational interactions are very frequent, binaries can exert a significant influence on both the dynamical evolution of the system and the properties of its stellar populations.

Being, on average, more massive than the other stars, binaries tend to sink into the highly crowded cluster centers, because of equipartition. The characteristic timescale of this process (the relaxation time) depends on the cluster structure and can be even longer than a Hubble time in the outskirts. Hence, in the outer regions of GCs we essentially expect to observe primordial binary systems, i.e., binaries created as part of the star formation process and evolving undisturbed. In the cluster core, on the other hand, a variety of dynamical processes (exchange interactions, three-body encounters, tidal captures, etc.) can take place, with competing effects on the binary population: binaries can be destroyed, created or just modified (e.g., Hut et al. 1992), with relative efficiencies that still are a matter of debate in the literature (e.g., Ivanova et al. 2005; Hurley et al. 2007; Sollima 2008; Fregeau et al. 2009). In general, however, the gravitational encounters occurring in the core tend to make the binaries harder (more tightly bound), thus providing a central energy source able to slow down the cluster core collapse (Goodman & Hut 1989). Following the N-body simulations of Gill et al. (2008), this energy source could also suppress the mass-segregation process, with a detectable effect on the radial behavior of the mass function of main-sequence (MS) stars. Since the same effect could alternatively be due to a central intermediate-mass black hole (IMBH; see also Pasquato et al. 2009; Beccari et al. 2010, hereafter B10), if we can measure the fraction of binaries, then we can say whether or not we need an IMBH to explain the low level of mass segregation that has been observed. Hence, the empirical estimate of the binary fraction in a sample of GCs representative of different environments is a prime ingredient for dynamical models, which helps us to understand the internal cluster dynamics.

Knowledge of the binary fraction is also crucial for understanding the properties of puzzling objects such as blue stragglers, millisecond pulsars, and cataclysmic variables, which are all thought to be the by-products of binary evolution (e.g., McCrea 1964; Romani et al. 1987; Ferraro et al. 2001; Leigh et al. 2011, and references therein). In particular, the analysis of the bimodal radial distribution of blue stragglers observed in a number of GCs (e.g., Ferraro et al. 1997, 2004; Dalessandro et al. 2008) suggests that a non-negligible fraction of these stars is generated by primordial binaries, which still orbit in isolation in the cluster outskirts and produce the observed rising branch of the distribution (Mapelli et al. 2004, 2006; Lanzoni et al. 2007a, 2007b). The interpretation of the double blue straggler sequence recently discovered in the core of M30 also requires a significant fraction of primordial binaries (Ferraro et al. 2009).

Despite its implications, however, the binary fraction in GCs still remains badly constrained, because of the challenging observational requirements. The main techniques commonly used for its estimate are: radial velocity variability surveys (e.g., Pryor et al. 1989; Latham 1996; Albrow et al. 2001), searches for eclipsing binaries (e.g., Mateo 1996; Cote et al. 1996), and the study of the distribution of stars along the cluster MS in color–magnitude diagrams (CMDs; e.g., Romani et al. 1991; Bolte 1992; Rubenstein & Bailyn 1997; Bellazzini et al. 2002; Clark et al. 2004; Zhao & Bailyn 2005; Sollima et al. 2007; Milone et al. 2008). The first two methods rely on the detection of individual binary systems in a given range of periods and mass ratios. Hence, the nature of these methods leads to intrinsic observational biases and a low detection efficiency. The latter approach relies on the simple fact that, since the flux of unresolved binaries is equal to the sum of the fluxes of the two components, the binaries composed by MS companions are shifted toward brighter magnitudes with respect to the single-star MS. This technique has the advantage of being more efficient and detecting binary systems regardless of their orbital periods and inclinations.

For the present paper, we used this latter technique to estimate the binary fraction in the core and the outskirts of M10 (NGC 6254). This is an "ordinary," dynamically relaxed GC, with absolute visual magnitude M_V = −7.48 (Harris 1996, 2010 edition), central mass density log ρ₀ = 3.8 (ρ₀ being in units of M_☉ pc⁻³; Pryor & Meylan 1993), and half-mass relaxation time t_h ∼ 0.8 Gyr (Harris 1996; see also McLaughlin & van der Marel 2005). The deep and high-quality photometry that B10 obtained for both the center and beyond the half-mass radius, allowed them to study the cluster mass function at different radial distances. The resulting mass-segregation profile is moderately flattened and can be explained by the presence of either an IMBH of ∼10³ M_☉, or a population of binaries with an initial fraction of 3%–5% (B10). Hence, within the framework proposed by Gill et al. (2008), any empirical constraint on the binary content in this system would allow us to assess the possible presence of a central IMBH. In addition, it will provide precious clues and constraints that would be useful for a robust interpretation of the properties the blue straggler population in this cluster (E. Dalessandro et al. 2011, in preparation).

The paper is organized as follows. The used data sets are presented in Section 2. The method adopted to estimate the binary fraction is outlined in Section 3. The results and the discussion are presented in Sections 4 and 5.

The data set used in the present work (the same as in B10) consists of a sample of 4 × 90 s images acquired in the F606W (V) and 4 × 90 s images in F814W filters obtained with the Advanced Camera for Surveys (ACS; GO-10775, PI: Sarajedini), complemented with 2 × 1100 s and 2 × 1200 s images in F606W and 2 × 1100 s and 2 × 1200 s images obtained with the Wide Field Planetary Camera 2 (WFPC2; GO-6113, PI: Paresce) on board the Hubble Space Telescope. The ACS data set samples the cluster central regions, while the WFPC2 set covers an area located between one and two half-mass radii (see Figure 1). The detailed description of the data reduction, photometric calibration,⁴ and astrometric solution procedures is given in B10 and the sample used in the present analysis contains the same "bona-fide" stars selected on the basis of the quality of the point-spread-function fitting (as measured by the DAOPHOTII sharpness parameter; Stetson 1987). The CMDs of the two data sets are shown in Figure 2. Stars brighter than I = 16 and I = 19.5 are saturated in the ACS and WFPC2 samples, respectively. B10 also performed a detailed photometric completeness study, based on artificial star experiments, where artificial stars were added to the original FLT frames and the whole data reduction procedure was repeated. From the resulting catalog, listing the input and output positions, and magnitudes for more than 500,000 artificial stars, B10 estimated that the photometric completeness drops below 50% at I ∼ 22.5 in the innermost region of the cluster and at I ∼ 25 for the WFPC2 data set.

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In order to estimate the binary fraction of M10, we followed the method extensively described in Bellazzini et al. (2002) and Sollima et al. (2007, hereafter S07). The basic idea is that the magnitude of a binary system corresponds to the luminosity of the primary (more massive) star, increased by the contribution of the companion of an amount that depends on the mass ratio of the two components (q = M₂/M₁). In fact, since the stars along the MS follow a mass–luminosity relation, the luminosity of a binary can be written in terms of the mass ratio of the two components. By definition, 0 < q ⩽ 1 and for q = 1 (equal-mass binary) the system appears ∼0.75 mag brighter than the single component, while the luminosity enhancement decreases for decreasing q. The spanning of all the possible values of q, at different magnitudes of the primary component, produces a broadening of the single-star MS, to its bright- and red-hand side. In principle, the ratio between the number of stars lying on the red side of the single-star MS and the total number of stars observed along the "broadened MS" provides the cluster binary fraction. In practice, depending on the photometric error of the data, a minimum value of the mass ratio (q_min) exists below which it is impossible to observationally distinguish a binary system from a single MS star. Moreover, it is necessary to take into account a number of effects, such as stellar blends and the contamination by foreground/background field stars, which can add spurious sources in the CMD.

Indeed, chance superpositions of two stars (blends) can produce a luminosity enhancement that mimics the magnitude shift characteristic of a genuine binary system. In order to correct for this effect, we analyzed the distribution of the residuals between the input and the output magnitudes of the artificial-star catalog built by B10 for the completeness study (see previous section). From the asymmetry of the distribution (which is skewed toward brighter output magnitudes because of the blending between artificial and real stars) we estimated that the percentage of blended sources that would mimic binary systems with q > q_min, varies from ∼6% in the core, to less than 0.2% in the external regions.

B10 also estimated the Galactic field contamination in the direction of M10, finding that it is very low: even in the worst case (the WFPC2 data set), where the number of cluster sources is small, the field stars are just ∼3% of the total sample. Despite such a low value, for a proper measurement of the binary fraction we performed a detailed study of the field contamination as a function of the magnitude. From the Galaxy model of Robin et al. (2003)⁵ we retrieved a catalog covering an area of $0.5\deg ^2$ in the direction of M10, and we randomly extracted two sub-samples of synthetic stars, scaled to the fields of view of the ACS and WFPC2 data sets. Their magnitudes were converted from the Johnson to the VEGAMAG photometric system adopting the prescriptions of Sirianni et al. (2005). Finally, by exploiting the artificial-star catalog used for the completeness study⁶ (Section 2), we obtained a catalog of synthetic field stars that includes the observational biases (incompleteness and blending), for both the ACS and WFPC2 data sets.

Once all the contaminant effects are taken into account, the binary fraction was estimated as the number of stars in the "binary population" divided by the total number of stars, i.e., binaries plus genuine, single, MS stars (hereafter the "MS population"). The "MS population" is defined as the set of stars having a color difference from the MS ridge line (MSRL) smaller than three times the typical photometric error at that magnitude level (see Figure 3). The operational definition of the "binary population" is given in Sections 4.1 and 4.2.

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The high photometric quality and the spatial coverage of the data sets previously described allowed us to study the binary fraction at different distances from the cluster center. In particular, here we have defined three concentric annuli bounded by the core radius and the half-mass radius. The adopted center of gravity and structural parameters have been recently determined from resolved star counts (Dalessandro et al. 2011): the coordinates of the center are $\alpha _{\rm J2000} = 16^{\rm h} 57^{\rm m} 8\mbox{$.\!\!^{\mathrm s}$}92$, δ_J2000 = −4° 5' 5807; the core, half-mass, and tidal radii are r_c = 48'', r_h = 147'', and r_t = 193, respectively. This center is located at ∼35 northwest from the one quoted by Goldsbury et al. (2010), a difference that has no impact on the following analysis and the obtained results. Hence, the first two radial bins (r < r_c and r_c < r < r_h) are sampled by the ACS data set, while the third one (r > r_h) is covered by the WFPC2 data (see Figure 1). Since the two data sets have different saturation and completeness levels (see Section 2), we performed the analysis in two different magnitude ranges: the adopted cuts along the MSRL are 18.8 < I < 21.5 for the ACS sample and 20.3 < I < 23 for the WFPC2 sample (see Figures 2 and 3). These intervals define what we call the "full magnitude range" of the two data sets. Then, with the aim of having an interval of magnitudes in common between the two samples where to directly compare the computed binary fractions, we considered three magnitude sub-ranges defined as follows: a "bright range" corresponding to 18.8 < I < 20.3, an "intermediate range" at 20.3 < I < 21.5, and a "faint range" at 21.5 < I < 23 (all the quoted magnitude values are measured along the MSRL). As is apparent from Figures 2 and 3, the bright range is probed only by the ACS data set, the faint range is found only in the WFPC2 sample, while the intermediate range is in common between the two.

4.1. The Minimum Binary Fraction

4.2. The Global Binary Fraction

In order to estimate the overall binary content of M10, independently of the value of q, we also computed the global binary fraction (ξ_TOT). This requires us to perform simulations of single and binary star populations assuming different input values of the global binary fraction (ξ_in) and then determining ξ_TOT from the comparison between the artificial and the observed CMDs: the value of ξ_in that provides the best match between the two CMDs is adopted as the global binary fraction ξ_TOT (see Bellazzini et al. 2002 and S07 for a detailed description of the procedure).

Once we assumed an input value of the binary fraction (ξ_in), for each of the considered radial and magnitude bins we have built a sample of N_MS and N_bin stars, with N_bin = Nξ_in, N being the number of observed objects (after having taken into account the number of contaminating field stars, discussed in Section 3) in that bin, and N_MS being N(1 − ξ_min). The MS stars have been simulated by randomly extracting N_MS values of the mass from the present-day cluster mass function derived by B10,⁷ and transforming the masses into luminosities by using the Baraffe et al. (1997) isochrones. Then, from the artificial-star catalog previously described, we have randomly selected an object with similar (ΔI < 0.1) magnitude and, if recovered, we assigned its output I and V magnitudes to the considered MS star. In order to simulate the binary systems we randomly extracted N_bin values of the mass of the primary component from the Kroupa (2002) initial mass function, and N_bin values of the binary mass ratio from the f(q) distribution observed by Fisher et al. (2005) in the solar neighborhood, thus also obtaining the mass of the secondary. After transforming masses into luminosities and summing up the fluxes of the two components, an object with similar magnitude was randomly extracted from the artificial-star catalog and, if recovered by the photometric analysis, the shifts between its input and output magnitudes were assigned to the considered binary system. Finally, the field stars were added to the sample. The result of this procedure is a list of synthetic stars with the same characteristics of real stars and containing a given fraction of binaries (ξ_in). To be precise, the MSs of the resulting artificial CMDs are narrower than the observed ones, because the formal photometric errors of the artificial-star catalog systematically underestimate the true observational uncertainties. This is apparent in Figure 4, where, for the magnitude range 19 < I < 19.5, the histogram corresponds to the distribution of the observed color differences Δ(V − I) with respect to the MSRL, the solid line is the best-fitting Gaussian of the blue side of this distribution (gray histogram; the red side has been ignored because it also includes the contribution of binaries and blends), and the dashed line is a Gaussian with a dispersion obtained by adopting the formal photometric error of the artificial-star catalog. In order to correct for this bias and adopt realistic values of the photometric uncertainty, we increased the formal errors σ_I and σ_V thus to reproduce the observed error distribution as a function of magnitude. As a check, we verified that the width of the resulting color distribution with respect to the MSRL well matches the observed one. An example of the synthetic CMD thus obtained, compared to the observed one is shown in Figure 5. From the simulated catalog we then computed the ratio r_sim = N^sim_bin/N^sim_MS between the number of synthetic stars belonging to the "binary population" defined in Section 4.1, and that of the synthetic "MS population." The same was done for the observed data sets, thus obtaining r_obs = N^obs_bin/N^obs_MS.

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Table 3. Global Binary Fraction (ξ_TOT) of M10 for the Considered Radial and Magnitude Intervals

Radial	Full	Bright	Intermediate	Faint
Bin	Range	(18.8 < I < 20.3)	(20.3 < I < 21.5)	(21.5 < I < 23)
r < rc	(13.8 ± 1.4)%	(15.1 ± 1.9)%	(10.0 ± 1.6)%	⋅⋅⋅
rc < r < rh	(7.4 ± 0.6)%	(7.6 ± 0.8)%	(6.3 ± 0.6)%	⋅⋅⋅
r > rh	(1.5 ± 0.6)%	⋅⋅⋅	(1.5 ± 1.0)%	(1.5 ± 0.7)%

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For every value of ξ_in, from 0.5% to 25% with steps of 0.5%, the entire procedure was repeated 100 times. Then, the penalty function χ²(ξ_in) was computed as the summation of (r_{sim, i} − r_obs)² for i = 1, 100, and the associated probability P(ξ_in) was derived. To illustrate this, Figure 6 shows the distribution of P as a function of the adopted values of ξ_in, ranging from 3% to 10%. The mean of the best-fitting Gaussian gives the global binary fraction (ξ_TOT) and its dispersion has been adopted as the error. The values of ξ_TOT obtained in the various radial and magnitude ranges are reported in Table 3. The global binary fraction shows the same radial behavior observed for ξ_min, varying from ∼14% or ∼10% in the cluster core (for the full and the intermediate magnitude ranges, respectively), down to ∼1.5% in the outskirts (for both). As before we find a dependence of the binary fraction on the magnitude. This could be an effect of mass segregation, since the average binary mass in the bright, intermediate, and faint ranges is M ∼ 1.1, 0.8, 0.5 M_☉, respectively. However, it could also depend on the assumed mass–ratio distribution and the estimate of blended sources, and future studies will be required to resolve this.

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We have presented a homogeneous analysis of the binary fraction in M10 as a function of the radial distance from the cluster center, from the core region, out to ∼2r_h. Within the errors, the derived core binary fraction is consistent with that measured in other GCs, which have typical values of ξ_TOT spanning from ∼10% to ∼25% (S07; Davis et al. 2008) but it is significantly smaller than that estimated for the faintest clusters in the sample of S07, which reach also binary fractions ξ_TOT ∼ 50%. This is in agreement with the quoted anti-correlation between binary fraction and total luminosity (Milone et al. 2008; Sollima 2008; Sollima et al. 2010). Also the binary fraction beyond the half-mass radius (∼1%) is consistent with previous estimates in GCs (Davis et al. 2008, see Table 1).

The minimum binary fraction decreases from ∼6% within r_c, to ∼1% beyond the half-mass radius. An analogous trend was found for the fraction of binaries with q ⩾ 0.6 and for the global binary fraction (Figure 7), the latter varying from ∼14% to ∼1.5% from the core to beyond the half-mass radius. Such a radial behavior is in agreement with what has been previously found in the few other GCs where this kind of investigation has been performed (Rubenstein & Bailyn 1997; Bellazzini et al. 2002; Zhao & Bailyn 2005; Sommariva et al. 2009, and references in Table 1 of Davis et al. 2008). It is also in agreement with the expectations of dynamical models, where the effect is essentially due to the mass-segregation process, which leads to an increase in the number of binaries in the cluster cores (e.g., Hurley et al. 2007; Sollima 2008; Fregeau et al. 2009; Ivanova 2011). Indeed, the half-mass relaxation time of M10 (∼0.8 Gyr, Harris 1996; see also Gnedin et al. 1999; McLaughlin & van der Marel 2005) is just a small fraction (∼4%) of the cluster age (t ∼ 13 Gyr; Dotter et al. 2010), so it seems safe to conclude that the system has already had time to achieve equipartition.

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By comparing the radial variation of the MS stellar mass function derived from the observations, with that obtained in N-body simulations, B10 suggested that either an IMBH or a population of binaries should be present and act as a central energy source in M10, suppressing the mass-segregation profile. In particular, the shallow mass-segregation profile could be modeled without an IMBH only when the simulations started with a primordial binary fraction of about 3%–5%. Within this framework, in Figure 7 we compare our derived values of ξ_TOT, with those obtained from the dynamical evolution of the 5% primordial binary population in the 32 K particle simulation of B10. For a proper comparison we considered a simulation snapshot at ∼7 relaxation times, and only those binaries made of two MS stars and with the primary component in the mass range 0.44–0.56 M_☉, corresponding to the lower and upper cuts of the intermediate magnitude range along the MSRL. The resulting binary fractions for the three considered radial bins are ξ_{N − body} = (0.070 ± 0.02), (0.032 ± 0.007), (0.026 ± 0.006), from the center to the outskirts. It is apparent from Figure 7 that the observed binary fraction is larger than the simulated one, especially in the core. This indicates that the binary content of M10 is indeed sufficient to account for the observed mass-segregation suppression, with no need to invoke an IMBH as additional energy source.

We thank the anonymous referee for the careful reading and useful comments that improved the presentation of this work. This research is part of the project COSMIC-LAB funded by the European Research Council (under contract ERC-2010-AdG-267675). The financial contributions of Istituto Nazionale di Astrofisica (INAF, under contract PRIN-INAF 2008) and the Agenzia Spaziale Italiana (under contract ASI/INAF I/009/10/0) are also acknowledged.