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Mon. Not. R. Astron. Soc. 000, 1–12 (2007) Printed 11 November 2018 (MN LATEX style file v2.2)

The fraction of binary systems in the core of thirteen

low-density Galactic globular clusters⋆

A. Sollima1†, G. Beccari2,3,4, F. R. Ferraro1, F. Fusi Pecci2 and A. Sarajedini51Dipartimento di Astronomia, Universita di Bologna, via Ranzani 1, Bologna, 40127-I, Italy2INAF Osservatorio Astronomico di Bologna, via Ranzani 1, Bologna, 40127-I, Italy3INAF Osservatorio Astronomico di Collurania, via M. Maggioni, 64100-I Teramo, Italy4Dipartimento di Scienze della Comunicazione, Universita di Teramo, Teramo, 64100-I, Italy5Department of Astronomy, University of Florida, 32611 Gainesville, USA

Accepted 2007 June 15; Received 2007 March 19; in original form 2007 July ??

ABSTRACTWe used deep observations collected with ACS@HST to derive the fraction of binarysystems in a sample of thirteen low-density Galactic globular clusters. By analysing thecolor distribution of Main Sequence stars we derived the minimum fraction of binarysystems required to reproduce the observed color-magnitude diagram morphologies.We found that all the analysed globular clusters contain a minimum binary fractionlarger than 6% within the core radius. The estimated global fractions of binary systemsrange from 10% to 50% depending on the cluster. A dependence of the relative fractionof binary systems on the cluster age has been detected, suggesting that the binarydisruption process within the cluster core is active and can significantly reduce thebinary content in time.

Key words: stellar dynamics – methods: observational – techniques: photometric –binaries: general – stars: Population II – globular clusters: general

1 INTRODUCTION

Binary stars provide a unique tool to determine crucial in-formation about a variety of stellar characteristics includingmass, radius and luminosity. Fortunately, there are ampleopportunity to observe binary star systems: most stars arein fact in binary systems, at least in the solar neighbor-hood (Duquennoy & Mayor 1991). Since the first decades ofthe twentith century, the study of binary star systems pro-vided valuable information about the stellar structure andevolution, such as the mass-luminosity and mass-radius rela-tions (Kuiper 1938; Huang & Struve 1956). Binarity, underparticular conditions, induces the onset of nuclear reactionsleading to the formation of bright objects like novae anddetermines the fate of low-mass stars leading to SN Ia ex-plosions.

Binaries play also a key role in the dynamical evolutionof stellar systems and stellar populations studies. In colli-sional systems binaries provide the gravitational fuel thatcan delay and eventually stop and reverse the process ofcore collapse in globular clusters (see Hut et al. 1992 andreferences therein). Furthermore, the evolution of binaries

⋆ Based on ACS observations collected with the Hubble SpaceTelescope within the observing program GO 10755.† E-mail: antonio.sollima@bo.astro.it (AS)

in star clusters can produce peculiar stellar object of as-trophysic interest like blue stragglers, cataclysmic variables,low-mass X-ray binaries, millisecond pulsars, etc. (see Bai-lyn 1995 and reference therein). The binary fraction is a keyingredient in chemical and dynamical models to study theevolution of galaxies and stellar systems in general.

The main techniques used to derive the binary fractionin globular clusters are: i) radial velocity variability surveys(Latham 1996; Albrow et al. 2001) ii) searches for eclips-ing binaries (Mateo 1996) and iii) searches for secondarymain-sequences (MS) in color-magnitude diagrams (CMD,Rubenstein & Bailyn 1997). The first two methods rely onthe detection of individual binary systems in a given rangeof periods and mass-ratios. The studies carried out in thepast based on these methods argued for a deficiency of bi-nary stars in globular clusters compared to the field (Pryoret al. 1989; Hut 1992; Cote et al. 1996). However, the natureof these two methods leads to intrinsic observational biasesand a low detection efficiency. Conversely, the estimate ofthe binary fraction on the basis of the analysis of the num-ber of stars displaced in the secondary MS represents a moreefficient statistical approach and does not suffer of selectionbiases. In fact, any binary system in a globular cluster is seenas a single star with a flux equal to the sum of the fluxes ofthe two components. This effect locates any binary systemsistematically at brighter magnitudes with respect to single

c© 2007 RAS

2 Sollima et al.

MS stars, defining a secondary sequence in the CMD runningparallel to the cluster MS that allows to distinguish themfrom other single MS stars. Until now, the binary fractionhas been estimated following this approach only in few glob-ular clusters (Romani & Weinberg 1991; Bolte 1992; Ruben-stein & Bailyn 1997; Bellazzini et al. 2002; Clark, Sandquist& Bolte 2004; Zhao & Bailyn 2005).

In this paper we present an estimate of the binary frac-tion in thirteen low-density Galactic globular clusters. Weused the photometric survey carried out with the AdvancedCamera for Surveys (ACS) on board HST as a part of aTreasury program (Sarajedini et al. 2007).

In §2 we describe the observations, the data reduc-tion techniques and the photometric calibration. In §3 theadopted method to determine the fraction of binary systemsis presented. In §4 we derived the minimum binary fractionsin our target globular clusters. §5 is devoted to the esti-mate of the global binary fractions and to the comparisonof the measured relative fractions among the different glob-ular clusters of our sample. In §6 the radial distribution ofbinary systems is analysed. Finally, we summarize and dis-cuss our results in §7.

2 OBSERVATIONS AND DATA REDUCTION

The photometric data-set consists of a set of high-resolutionimages obtained with the ACS on board HST through theF606W (V606) and F814W (I814) filters. The target clusterswere selected on the basis of the following criteria:

• A high Galactic latitude (b > 15◦) in order to limit thefield contamination;

• A low reddening (E(B-V)<0.1) in order to avoid theoccurrence of differential reddening;

• A low apparent central density of stars1 (log ρ′0 <5 M⊙ arcmin−2) in order to limit the effects of crowdingand blending.

Thirteen cluster passed these criteria namely NGC288,NGC4590, NGC5053, NGC5466, NGC5897, NGC6101,NGC6362, NGC6723, NGC6981, M55, Arp 2, Terzan 7 andPalomar 12. In Table 1 the main physical parameters of theabove target clusters are listed. The central density ρ0, thecore radii rc and the half-mass relaxation times tr,rh arefrom Djorgovski (1993), the age t9 from Salaris & Weiss(2002) and the global metallicities [M/H ] from Ferraro etal. (1999)2. Note that the analysed sample spans a widerange in age and metallicity containing only low-density(log ρ0 < 2.75 M⊙pc

−3) globular clusters.

1 The apparent central density of stars has been calculated fromthe central surface density ρS,0 and the cluster distance d (fromMcLaughlin & Van der Marel 2005) according to the followingrelation

ρ′0 = ρS,0d2(

2π

21600)2

2 For the clusters NGC6101, NGC6362, NGC6723 and Palomar12 not included in the list of Ferraro et al. (1999) we trans-formed the metallicitiy [Fe/H] from Zinn & West (1984) into theglobal metallicity [M/H] following the prescriptions of Ferraro etal. (1999).

For each cluster the ACS field of view was centered onthe cluster center. We retrived all the available exposuresfrom the ESO/ST-ECF Science Archive. The exposure timesfor each cluster in each filter are listed in Table 2. All im-ages were passed through the standard ACS/WFC reduc-tion pipeline. Data reduction has been performed on theindividual pre-reduced images using the SExtractor photo-metric package (Bertin & Arnouts 1996). The choice of thedata-reduction software has been made after several trialsusing the most popular PSF-fitting softwares. However, theshape of the PSF quickly varies along the ACS chip ex-tension giving trouble to most PSF-fitting algorithms. Con-versely, given the small star density in these clusters, crowd-ing does not affect the aperture photometry, allowing toproperly estimate the magnitude of stars. This is evidentin Fig. 1 where a zoomed portion of the central region ofthe cluster NGC6723 (the most crowded GC of our sam-ple) is shown. Note that the surface density of stars in thisfield is 6 1.4 stars arcsec−2. For each star we measured theflux contained within a radius of 0.125” (corresponding to2.5 pixels ∼ FWHM) from the star center. The source de-tection and the photometric analysis have been performedindependently on each image. Only stars detected in threeout four frames have been included in the final catalog. Themost isolated and brightest stars in the field have been usedto link the aperture magnitudes at 0.5” to the instrumen-tal ones, after normalizing for exposure time. Instrumentalmagnitudes have been transformed into the VEGAMAG sys-tem by using the photometric zero-points by Sirianni et al.(2005). Finally, each ACS pointing has been corrected forgeometric distorsion using the prescriptions by Hack & Cox(2001).

Two globular clusters (NGC5053 and NGC5466) werealready analyzed by Sarajedini et al. (2007). Our photome-try has been compared with the photometric catalog alreadypublished by these authors. The mean magnitude differencesfound are ∆V606 = -0.004 ± 0.012 and ∆I814 = 0.004 ±0.012 for NGC5053 and ∆V606 = -0.031 ± 0.012 and ∆I814= -0.020 ± 0.012 for NGC5466, which are consistent with asmall systematic offset in both passbands.

Fig. 2 and 3 show the (I814, V606 − I814) CMDs of the13 globular clusters in our sample. The CMDs sample thecluster population from the sub-giant branch down to 5-6magnitudes below the MS turn-off. In all the target clustersthe binary sequence is well defined and distinguishable fromthe cluster’s MS. In the less dense clusters (e.g. Terzan 7,Pal 12) binary stars appears to populate preferentially aregion of the CMD ∼0.752 mag brighter than the clusterMS, approaching the equal-mass binary sequence (Eggleton,Mitton & Whelan 1978). In most clusters a number of bluestragglers stars populating the bright part of the CMD isalso evident.

3 METHOD

As quoted in §1, any binary system in a globular cluster isseen as a single star with a flux equal to the sum of thefluxes of the two components. This effect produces a sys-tematic overluminosity of these objects and a shift in colordepending on the magnitudes of the two components in eachpassband. In a simple stellar population the luminosity of a

c© 2007 RAS, MNRAS 000, 1–12

The binary fraction in thirteen globular clusters 3

Figure 2. I814, V606 − I814 CMDs of the target globular clusters NGC288, NGC4590, NGC5053, NGC5466, NGC5897 and NGC6101.

Figure 1. Zoomed image of the central region of the globularcluster NGC6723, the most crowded cluster of our sample.

Table 1. Main physical parameters of the target globular clusters

Name log ρ0 rc t9 log tr,rh [M/H]M⊙ pc−3 ” Gyr Gyr

NGC 288 1.80 85.20 11.3 8.99 -0.85NGC 4590 2.52 41.35 11.2 8.90 -1.81NGC 5053 0.51 134.40 10.8 9.59 -2.31NGC 5466 0.68 116.50 12.2 9.37 -1.94NGC 5897 1.32 118.70 12.3 9.31 -1.44NGC 6101 1.57 69.25 10.7 9.22 -1.40NGC 6362 2.23 79.15 11.0 8.83 -0.72NGC 6723 2.71 56.81 11.6 8.94 -0.73NGC 6981 2.26 32.09 9.5∗ 8.93 -1.10M55 2.12 170.8 12.3 8.89 -1.41Arp 2 -0.35 96.03 7-11.5 9.46 -1.44Terzan 7 1.97 36.51 7.4 9.03 -0.52Palomar 12 0.68 65.83 6.4 9.03 -0.76

∗ The age of NGC6981 has been taken from De Angeli et al.(2005)(see §5.3).

MS star is univocally connected with its mass. In particular,stars with smaller masses have fainter magnitudes followinga mass-luminosity relation. So, named M1 the mass of themost massive (primary) component in a given binary systemand M2 the mass of the less massive (secondary) one, themagnitude of the binary system can be written as:

msys = −2.5 log(FM1 + FM2) + c

c© 2007 RAS, MNRAS 000, 1–12

4 Sollima et al.

Figure 3. I814, V606 − I814 CMDs of the target globular clusters NGC6362, NGC723, NGC6981, Arp 2, M55, Terzan 7 and Palomar 12.

= mM1 − 2.5 log(1 +FM2

FM1

)

In this formulation the shift in magnitude of the binary sys-tem can be viewed as the effect of the secondary star that

perturbs the magnitude of the primary. The quantityFM2FM1

depends on the mass ratio of the two component (q = M2M1

).According to the definition of M1 and M2 given above, theparameter q is comprised in the range 0 < q < 1. Whenq=1 (equal mass binary) the binary system will appear−2.5 log(2) ∼ 0.752 mag brighter than the primary com-ponent. Conversely, when q approaches small values the ra-

tioFM2FM1

becomes close to zero, producing a negligible shift

in magnitude with respect to the primary star. Followingthese considerations, binary systems with small values of qbecomes indistinguishable from MS stars when photometricerrors are present. Hence, only binary systems with valuesof q larger than a minimum value (qmin) are unmistakablydistinguishable from single MS stars. For this reason, onlya lower limit to the binary fraction can be directly derivedwithout assuming a specific distribution of mass-ratios f(q).

In order to study the relative frequency of binary sys-tems in our target clusters we followed two different ap-proaches:

• We derived the minimum number of binary systems byconsidering only the fraction of binary systems with largemass-ratio values (q > qmin);

• We estimated the global binary fraction by assuminga given f(q) and comparing the simulated CMDs with theobserved ones.

A correct binary fraction estimation requires correctionsfor two important effects: i) blended sources contaminationand ii) field stars contamination. In the following sections wedescribe the adopted procedure to take into account theseeffects.

3.1 Blended sources

Chance superposition of two stars produces the same mag-nitude enhancement observed in a binary system. For thisreason it is impossible to discern whether a given objectis a physical binary or not. However, a statistical estimateof the distribution of blended sources expected to populatethe CMD as a function of magnitude and color is possibleby means of extensive artificial stars experiments (see Bel-lazzini et al. 2002).

For each individual cluster the adopted procedure forthe artificial star experiments has been performed as follows:

• The cluster mean ridge line has been calculated by av-eraging the colors of stars in the CMD over 0.2 mag boxesand applying a 2σ clipping algorithm;

• The magnitude of artificial stars has been randomlyextracted from a luminosity function (LF) modeled to re-

c© 2007 RAS, MNRAS 000, 1–12

The binary fraction in thirteen globular clusters 5

Table 2. Observing logs

Name # of exposures Filter Exposure time(s)

NGC 288 4 V606 1304 I814 150

NGC 4590 4 V606 1304 I814 150

NGC 5053 5 V606 3405 I814 350

NGC 5466 5 V606 3405 I814 350

NGC 5897 4 V606 3403 I814 350

NGC 6101 5 V606 3705 I814 380

NGC 6362 4 V606 1304 I814 150

NGC 6723 4 V606 1404 I814 150

NGC 6981 4 V606 1304 I814 150

M55 4 V606 704 I814 80

Arp 2 5 V606 3455 I814 345

Terzan 7 5 V606 3455 I814 345

Palomar 12 5 V606 3405 I814 340

produce the observed magnitude distribution of bright stars(F814W < 22) and to provide large numbers of faintstars down to below the detection limits of the observations(F814W > 26)3. The color of each star has been obtainedby deriving, for each extracted F814W magnitude, the corre-sponding F606W magnitude by interpolating on the clusterridge line. Thus, all the artificial stars lie on the cluster ridgeline in the CMD;

• We divided the frames into grids of cells of known width(30 pixels) and randomly positioned only one artificial starper cell for each run4;

• Artificial stars have been simulated using the Tiny Timmodel of the ACS PSF (Krist 19955) and added on theoriginal frames including Poisson photon noise. Each starhas been added to both F606W and F814W frames. Themeasurement process has been repeated adopting the same

3 Note that the assumption for the fainter stars is only for sta-tistical purposes, i.e. to simulate a large number of stars in therange of magnitude where significant losses due to incompletenessare expected.4 We constrain each artificial star to have a minimum distance (5pixels) from the edges of the cell. In this way we can control theminimum distance between adjacent artificial stars. At each runthe absolute position of the grid is randomly changed in a waythat, after a large number of experiments, the stars are uniformlydistributed in coordinates. Given the small stars density in theanalysed cluster areas, the radial dependence of the completenessfactor turns of to be neglegible.5 The Tiny Tim version 6.3 updated to model the ACS PSF isavailable at http://www.stsci.edu/software/tinytim/

Figure 4. Completeness factor c as a function of the F814W mag-nitude (upper panel) for the target cluster M55. In lower panels

the residuals between the input and output F606W and F814Wmagnitudes of artificial stars are shown.

procedure of the original measures and applying the sameselection criteria described in Sect. 2;

• The results of each single set of simulations have beenappended to a file until the desired total number of artificialstars has been reached. The final result for each subfield is alist containing the input and output values of positions andmagnitudes.

The residuals between the input and output V606 andI814 magnitudes and the completeness factor as a functionof the I814 magnitude are shown in Fig. 4 for the case of M55as an example. As expected, the distributions of the mag-nitude residuals are not symmetrical: a significant numberof stars have been recovered with a brigther output magni-tude than that assigned in input. This effect is due to thosestars that blended with nearby real stars with similar (orlarger) luminosity. More than 100,000 artificial stars havebeen produced for each cluster providing a robust estimateof the blending contamination together with the levels ofphotometric accuracy and completeness in all the regions ofthe CMD and throughout the cluster extension.

3.2 Field stars

Another potentially important contamination effect is dueto the presence of background and foreground field starsthat contaminate the binary region of the CMD. To ac-count for this effect, we used the Galaxy model of Robin etal. (2003). A catalog covering an area of 0.5 square degreearound each cluster center (from Djorgovski & Meylan 1993)has been retrived. A sub-sample of stars has been randomlyextracted from the entire catalog scaled to the ACS field ofview (202” × 202”). The V and I Johnson-Cousin magni-tudes were converted into the ACS photometric system by

c© 2007 RAS, MNRAS 000, 1–12

6 Sollima et al.

means of the transformations of Sirianni et al. (2005). Foreach synthetic field star, a star with similar input magnitude(∆I814 < 0.1) has been randomly extracted from the arti-ficial stars catalog. If the artificial star has been recoveredin the output catalog the V606 and I814 magnitude shiftswith respect to its input magnitudes have been added. Thisprocedure accounts for the effects of incompleteness, photo-metric errors and blending.

4 THE MINIMUM BINARY FRACTION

As pointed out in §3 there is a limited range of mass-ratiovalues (q > qmin) where it is possible to clearly distinguishbinary systems from single MS stars. The value of qmin de-pends on the photometric accuracy (i.e. the signal-to-noiseS/N ratio) of the data. The approach presented in this sec-tion allows to estimate the fraction of binaries with q > qmin

that represents a lower limit to the global cluster binary frac-tion.

In the following we will refer to the binary fraction ξas the ratio between the number of binary systems whoseprimary star has a mass comprised in a given mass range(Nb) and the number of cluster members in the same massrange (Ntot = NMS +Nb)

6. To derive an accurate estimateof this quantity we adopted the following procedure:

(i) We defined an I814 magnitude range that extends from1 to 4 magnitudes below the cluster turn-off. In this magni-tude range the completeness factor is always φ > 50%;

(ii) We converted the extremes of the adopted magnituderange (Iup and Idown) into masses (Mup and Mdown) usingthe mass-luminosity relation of Baraffe et al. (1997). To dothis, the V and I Johnson-Cousin magnitudes of the Baraffeet al. (1997) models were converted into the ACS photomet-ric system by means of the transformations by Sirianni etal. (2005). For our target clusters we assumed the metal-licities listed by Ferraro et al. (1999), the distance mod-uli and reddening coefficients listed by Harris (1996) andthe extinction coefficients AF814W = 2.809 E(B − V ) andAF814W = 1.825 E(B−V ) (Sirianni et al. 2005). Small shiftsin the distance moduli (∆(m−M)0 < 0.1) have been appliedin order to match the overall MS-TO shape;

(iii) We defined three regions of the CMD (see Fig. 5) asfollows:

• A region (A) containing all stars with Idown < I814 <Iup and a color difference from the MS mean ridge linesmaller then 4 times the photometric error correspondingto their magnitude (dark grey area in Fig. 5). This areacontains all the single MS stars in the above magnituderange and binary systems with q < qmin;

• We calculated the location in the CMD of a binarysystem formed by a primary star of mass Mup (andMdown

respectively) and different mass-ratios q ranging from 0to 1. These two tracks connect the MS mean ridge line

6 This quantity can be easily converted in the fraction ξ’ ofstars in binary systems (Nb,s) with respect to the cluster stars(Ntot,s) considering that Nb,s = 2 Nb according to the relation

ξ′ =2ξ

1 + ξ

with the equal mass binary sequence (which is 0.752 magbrighter than the MS ridge line) defining an area (B1)in the CMD. This area contains all the binary systemswith q < 1 and whose primary component has a massMdown < M1 < Mup;

• A region (B2) containing all stars with magnitudeIdown − 0.752 < I814 < Iup − 0.752 and whose color dif-ference from the equal mass binary sequence is comprisedbetween zero and 4 times the photometric error corre-sponding to their magnitude. This area is populated bybinary systems with q ∼ 1 that are shifted to the red sideof the equal-mass binary sequences because of photomet-ric errors;

(iv) We considered single MS stars all stars contained inA (MS sample), binary stars all stars contained in B1 andB2 but not in A (binary sample, grey area in Fig. 5);

(v) Since the selection boxes defined above cover two dif-ferent regions of the CMD with different completeness lev-els, we assigned to each star lying in the MS sample and inthe binary sample a completeness factor ci according to itsmagnitude (Bailyn et al. 1992). Then, the corrected num-ber of stars in each sample (NCMD

MS and NCMDbin ) has been

calculated as

N =∑

i

1

ci

(vi) We repeated steps (iv) and (v) for the samples ofartificial stars and field stars, obtaining the quantities Nart

MS

and Nartbin for the artificial stars sample and Nfield

MS and

Nfieldbin for the field stars sample;(vii) We calculated the normalization factor η for the

artificial stars sample by comparing the number of starsin the MS selection box

η =NCMD

MS

NartMS

(viii) The minimum binary fraction, corrected for fieldstars and blended sources, turns out to be

ξmin =NCMD

bin −Nfieldbin − η Nart

bin

(NCMDMS −Nfield

MS ) + (NCMDbin

−Nfield

bin− η Nart

bin)

Since the target clusters in our sample are located atdifferent distances, the ACS field of view covers differentfractions of the cluster’s extent. The procedure describedabove has been conducted considering only cluster stars (andartificial stars) located inside one core radius (rc, adoptedfrom Djorgovski 1993).

The obtained minimum binary fractions ξmin for theclusters in our sample are listed in Table 3. The typical er-ror (calculated by taking into account of the Poisson statis-tic and the uncertainties in the completeness corrections)is of the order of 1%. As can be noted, the minimum bi-nary fraction ξmin is larger than 6% in all the clustersof our sample. Therefore, this value seems to represent alower limit to the binary fraction at least in low-density(log ρ0 < 2.75M⊙pc−3, see Table 1) globular clusters.

5 THE GLOBAL BINARY FRACTION

The procedure described above allowed us to estimate theminimum binary fraction ξmin without any (arbitrary) as-

c© 2007 RAS, MNRAS 000, 1–12

The binary fraction in thirteen globular clusters 7

Figure 5. Selection boxes used to select the MSsample (darkgrey area) and the binary sample (grey area). The solid thickline marks the MS mean ridge line, the solid thin line marks theequal-mass binary sequence, dashed lines mark the 4 σ range usedto define the selection boxes A,B1 and B2 (see §4).

sumption on the distribution of mass-ratios f(q). However,caution must be used when comparing the derived binaryfraction among the different clusters of our sample. In fact,the definition of the MS sample and binary sample given in§4 depends on the photometric accuracy (e.g. the S/N ratio)that vary from cluster to cluster. An alternative approachconsists in the simulation of a binary population which fol-lows a given distribution f(q) and in the comparison be-tween the color distribution of simulated stars and the ob-served CMD. Until now there are neither theoretical argu-ments nor observational constraints to the shape of f(q) inglobular clusters. Studies on binary systems located in thelocal field suggest that the overall shape of f(q) can be re-produced by extracting randomly secondary stars from theobserved Initial Mass Function (IMF, Tout 1991). Fisher etal. (2005) estimated the mass-ratio distribution f(q) in thebinary population of the local field (at distances d < 100 pc).They found that most binary systems are formed by similarmass components (q ∼ 1).

In the following we calculate the binary fraction ξ inthe target clusters assuming two different shape of f(q): i) adistribution constructed by extracting random pairs of starsfrom the De Marchi et al. (2005) IMF (see Fig. 6 upperpanel) and ii) the distribution f(q) measured by Fisher etal. (2005, see Fig. 6 lower panel).

5.1 ξRA : f(q) from random associations

In the case of binary stars formed by random associationsbetween stars of different masses the general scheme adoptedfor an assumed binary fraction ξ has been the following:

(i) Artificial star I814 magnitudes have been converted

Figure 6. Distribution of mass-ratios of 100,000 binary stars sim-ulated in the magnitude range Idown < I814 < Iup from randomextractions from a De Marchi et al. (2005) IMF (upper panel).The distribution of mass-ratios adopted from Fisher et al. (2005)is shown in the bottom panel.

into masses by means of the mass-luminosity relation ofBaraffe et al. (1997). Then, a number of N (1 − ξ) arti-ficial stars were extracted from a De Marchi et al. (2005)IMF, where N is the number of stars in the observed cata-log. This sample of stars reproduces the MS population ofeach cluster taking into account also of blended sources;

(ii) The binary population has been simulated as follows:

a) A number of N ′(> ξ N) pairs of stars were extractedrandomly from a De Marchi et al. (2005) IMF;

b) The V606 and I814 magnitudes of the two compo-nents were derived adopting the mass-luminosity relationsof Baraffe et al. (1997) and the corresponding fluxes weresummed in order to obtain the V606 and I814 magnitudes ofthe unresolved binary system;

c) For each binary system, a star with similar input mag-nitude (∆I814 < 0.1) has been randomly extracted from theartificial stars catalog. If the artificial star has been recov-ered in the output catalog the V606 and I814 magnitude shiftswith respect to its input magnitudes have been added. Thisprocedure accounts for the effects of incompleteness, photo-metric errors and blending;

d) The final binary population has been simulated by ex-tracting a number of ξN objects from the entire catalog.

(iii) The field stars catalog (obtained as described in §3.2)was added to the simulated sample;

(iv) The ratio between the number of objects lying in

the selection boxes defined in §4 (rsim =Nsim

bin

NsimMS

) has been

calculated and compared to that measured in the observed

CMD (rCMD =NCMD

bin

NCMDMS

);

(v) Steps from (i) to (iv) have been repeated 100 times

c© 2007 RAS, MNRAS 000, 1–12

8 Sollima et al.

and a penalty function has been calculated as

χ2 =

100∑

i=1

(rsimi− rCMD)2

The whole procedure has been repeated for a wide grid ofbinary fractions ξ and a probability distribution as a func-tion of ξ has been produced. The value of ξ which minimizesthe penalty function χ2 has been adopted as the most prob-able. The error on the estimated binary fraction has beenestimated by estimating the interval where the χ2 accountfor the 68.2% probability (∼ 1σ) to recover the measuredquantity.

A typical iteration of the procedure described above isshowed in Fig. 7 where a simulated CMD of M55 is comparedwith the observed one. In Fig. 8 the distribution of χ2 andthe related probability as a function of the assumed value ofξ is shown.

The global binary fractions ξRA for the target clustersare listed in Table 3. As can be noted, most of the analysedclusters harbour a binary fractions 10% < ξ < 20% withthe exceptions of four cluster (NGC6981, Arp 2, Terzan 7and Palomar 12) which show a significantly larger binaryfraction (ξ > 35%). We want to stress that, although thismethod is independent on the S/N ratio and allows to de-rive the global binary fraction taking into account also of thehidden binary systems with q < qmin, it is subject to a num-ber of systematic uncertainties essentially due to unknowndistribution of binary mass-ratios. In fact, the binary frac-tions derived following the technique described above havea strong dependence on the low-mass end of the IMF whoseexact shape is still debated (see Kroupa 2002 and referencestherein). In particular, an increase of the fraction of low-mass stars significantly increases the probability to obtainbinaries with low-mass secondaries (i.e. with small mass ra-tios q). This effect would produce a significant overestimateof the binary fraction.

The mass-ratios distribution derived from the aboveprocedure, computed for a population of 100,000 binarieswith Idown < I814 < Iup, is shown in Fig. 6 (upper panel).This distribution significantly differs from that observed byHalbwachs et al. (2003) and Fisher et al. (2005) (but seealso Duquennoy & Mayor 1991). In particular, most binarystars present small values of q (q < 0.5) which produce alarge number of hidden binaries. Thus, the binary fractionsestimated in the observed clusters following this approachare probably sistematically overestimated. In the followingwe refer to this estimate as ξRA assuming it as a reasonableupper limit to the global binary fraction.

5.2 ξF : f(q) from Fisher et al. (2005)

As an alternative choice, we assumed a distribution of mass-ratios f(q) similar to that derived by Fisher et al. (2005)from observations of spectroscopic binaries in the solarneighborood (at distances d < 100 pc). The adopted mass-ratios distribution f(q) is shown in Fig. 6 (lower panel).Although this distribution is subject to significant observa-tional uncertainties and is derived for binary systems in adifferent environment, it represents one of the few observa-tional constraints to f(q) which can be found in literature.

The adopted procedure to derive the binary fraction ξ

Figure 8. Distribution of the calculated χ2 as a function of theassumed binary fraction for M55 (bottom panel). A parabolic fitto the data is showed. In the top panel the associated probabilityas a function of the assumed binary fraction is shown.

is the same described above but for the simulated binarypopulation (point (ii)a). In this case in fact, a number ofN ′(> ξ N) mass-ratios were extracted from the distributionf(q) shown in Fig. 6 (lower panel). Then, for each of theN ′ binary systems the mass of the primary component hasbeen extracted from a De Marchi et al. (2005) IMF and themass of the secondary component has been calculated. Allthe other steps of the procedure remain unchanged.

The calculated binary fractions ξF are listed in Table3. As expected, the values of ξF estimated following theassumption of a Fisher et al. (2005) f(q) are comprised be-tween the minimum binary fraction ξmin and the binaryfraction estimated by random associations ξRA. Note thatneither the ranking nor the relative proportions of the bi-nary fractions estimated among the different clusters of thesample appear to depend on the assumption of the shape off(q).

For some clusters of our sample the binary fraction werealready estimated in previous works. Bellazzini et al. (2002)and Bolte (1992) estimated a binary fraction comprised inthe range 10% < ξ < 20% for NGC288 by adopting a tech-nique similar to the one adopted here. These estimates arein good agreement with the result obtained in the presentanalysis (ξ ∼ 12%). Yan & Cohen (1996) measured a binaryfraction of 21% < ξ < 29% in NGC5053 on the basis of a ra-dial velocity survey. Our estimate suggests a slightly smallerbinary fraction in this cluster (ξ ∼ 11%). Note that the es-timate by Yan & Cohen (1996) is based on the detectionof 6 binary systems in a survey of 66 cluster members in alimited range of periods and mass-ratios. The uncertainty ofthis approach due to the small statistic is ∼ 10% and canaccount for the difference between their estimate and theone obtained in the present analysis.

c© 2007 RAS, MNRAS 000, 1–12

The binary fraction in thirteen globular clusters 9

Figure 7. Simulated (lower left panel) and observed (lower right panel) CMD of M55. In the upper panels the individual CMDs ofthe simulated single stars (upper left panel), binaries (upper central panel) and field stars (upper right panel) are shown.

Table 3. Binary fractions estimated for the target globular clus-ters

Name ξmin ξF ξRA σξ

% % % %

NGC 288 6 11.6 14.5 1.0NGC 4590 9 14.2 18.6 2.5NGC 5053 8 11.0 12.5 0.9NGC 5466 8 9.5 11.7 0.7NGC 5897 7 13.2 17.1 0.8NGC 6101 9 15.6 21.0 1.3NGC 6362 6 11.8 12.7 0.8NGC 6723 6 16.1 21.8 2.0NGC 6981 10 28.1 39.9 1.6M55 6 9.6 10.8 0.6Arp 2 8 32.9 52.1 3.6Terzan 7 21 50.9 64.9 2.9Palomar 12 18 40.8 50.6 6.6

In the following section we compare the obtained binaryfractions among the clusters of our sample as a function oftheir physical parameters.

5.3 Cluster to cluster comparison

Our sample contains thirteen low-density Galactic globularclusters spanning a large range of metallicity, age and struc-tural parameters (see Table 1). We used the results obtainedin the previous section to compare the core binary fractionξ among the clusters of our sample as a function of theirmain general and structural parameters in order to studythe efficiency of the different processes of formation and de-struction of binary systems.

We correlated the core binary fraction derived accord-ing to the different assumptions described in the previoussections (ξmin, ξF and ξRA) with the cluster’s ages (t9, fromSalaris & Weiss 2002), global metallicity ([M/H], from Fer-raro et al. 1999) central density (ρ0) and half-mass relax-ation time (tr,rh , from Djorgovski 1993), destruction rate (ν,from Gnedin & Ostriker 1997) and different structural pa-rameters (mass M , concentration c, binding energy Eb, half-mass radius rF , mass-luminosity ratio M/L, velocity disper-sion σv and escape velocity ve) adopted from McLaughlin& Van der Marel (2005). Of course, most of the quantitieslisted above are correlated.

The ages of two clusters, namely Arp2 and NGC6981,need a comment. According to Salaris & Weiss (2002), theage of Arp 2 is comparable to those of the oldest Galacticglobular clusters (t9 ∼ 11.3). The same conclusion has beenreached by Layden & Sarajedini (2000). Conversely, Buo-

c© 2007 RAS, MNRAS 000, 1–12

10 Sollima et al.

nanno et al. (1995) and Richer et al. (1996) classified it asa young globular cluster with an age comparable within 1Gyr to those of Terzan 7 and Palomar 12. Given the debatedquestion on the age of this cluster we excluded it from thefollowing analysis. The globular cluster NGC6981 is not in-cluded in the list of Salaris & Weiss (2002). An estimate ofthe age of this globular cluster has been presented by DeAngeli et al. (2005). We converted the ages measured by DeAngeli et al. (2005) into the Salaris & Weiss (2002) scale.Hence we adopted for this cluster an age of 9.5 Gyr.

In order to estimate the degree of dependence ofξ on the different clusters parameters we applied theBayesian Information Criterion test (Schwarz 1978) toour dataset. We assumed the binary fraction ξ as a linearcombination of a subsample of p parameters (λi) selectedamong those listed above.

ξf = αp+1 +

p∑

i=1

αiλi

Given a value of p, for any choice of the p parameters webest-fit our dataset with the above relation and calculatedthe quantity

BIC = ℓp − p

2log N

where ℓp is the logarithmic likelihood calculated as

ℓp = log Lp =

N∑

j=1

log Prj,p

=

N∑

j=1

log (e−

(ξj−ξf,j)2

2σ2ξ

σξ

√2π

)

Where N is the dimension of our sample (N=13) and σξ isthe residual of the fit. The p parameters that maximize thequantity BIC are the most probable correlators with ξ. Theabove analysis gives the maximum value of BIC for p=1and λp = t9. All the higher-order correlations appears asnon-significant.

The same result has been obtained considering all thethree estimates of ξ. A Spearman-rank correlation test givesprobabilities > 99% that the variables ξ and t9 are corre-lated, for all the considered estimates of ξ.

In Fig. 9 the core binary fractions ξmin, ξF and ξRA areplotted as a function of the clusters age. All the clusters ofour sample that present a large core binary fraction (ξF >25%) are sistematically younger than the other clusters.

Given the large systematic uncertainties involved in theestimate of the global binary fraction the above result can beconsidered only in a qualitative sense. However, the aboveanalysis indicates that the age seems to be the dominantparameter that determines the binary fraction in globularclusters belonging to this structural class.

6 BINARIES RADIAL DISTRIBUTION

Being bound systems, binary stars dynamically behave likea single star with a mass equal to the sum of the masses ofthe two components. After a time-scale comparable to the

Figure 9. Minimum (upper panel), estimated (middle panel)and maximum (lower panel) binary fractions as a function ofcluster age for the target clusters in our sample. For the clusterArp 2 the upper and lower limit are marked as open points.

cluster relaxation time, binary systems have smaller meanvelocities than single less massive stars, populating prefer-entially the most internal regions of the cluster. Since allthe globular clusters in our sample have a central relaxationtime shorter than their age, binary stars are expected to bemore centrally concentrated with respect to the other clus-ter stars. In order to test this hypothesis we calculated foreach target cluster the binary fraction ξ (following the pro-cedure described in §4) in three annuli of 500 pixels width lo-cated at three different distances from the cluster center. Wenoted that in seven (out of thirteen) globular clusters of oursample (namely NGC4590, NGC6101, NGC6362, NGC6723,NGC6981, Terzan 7 and Palomar 12) there is evidence of ra-dial segregation of binary systems toward the cluster center.In Fig. 10 the binary fractions (in unit of core fraction ξ)measured at different distances from the cluster’s centersin these seven clusters are shown. The binary fraction de-creases by a factor 2 at two core radii with respect to thecore binary fraction. A Kolmogorov-Smirnov test made onthe MS sample and binary sample (as defined in §4) yieldsfor these clusters probabilities smaller than 0.05% that thetwo samples are drawn from the same distribution. Notethat in most clusters the radial segregation of binary sys-tems is visible also within the core radius, indicating thatmass segregation is a very efficient process in these clusters.In the other six clusters the small number of stars and/orthe small radial coverage do not allow to detect a significantdifference in the radial distribution of binary stars.

c© 2007 RAS, MNRAS 000, 1–12

The binary fraction in thirteen globular clusters 11

Figure 10. Binary fraction (in unit of the core binary fraction) asa function of the distance from the cluster center (in unit of coreradii) for the target clusters NGC4590 (open circles), NGC6101(filled circles), NGC6362 (open triangles), NGC6723 (filled tri-angles), NGC6981 (open squares), Terzan 7 (filled squares) andPalomar 12 (asterisks).

7 DISCUSSION

In this paper we analysed the binary population of thirteenlow density Galactic globular clusters with the aim of study-ing their frequency and distribution.

In all the analysed globular clusters the minimum bi-nary fraction contained within one core radius is greaterthan 6%. This quantity seems to represent a lower limit tothe binary fraction in globular clusters of this structuralclass. This lower limit poses a firm constraint to the effi-ciency of the mechanism of binaries disruption. The existingestimates of the binary fraction in low-density globular clus-ters (Yan & Mateo 1994; Yan & Reid 1996; Yan & Cohen1996) agree with this lower limit. On the other hand, in high-density clusters the present day binary fraction appears tobe smaller (< 4−9% see Cool & Bolton 2002 and Romani &Weinberg 1991 for the case of NGC6397, M92 respectively)as expected because of the increasing efficiency of the dis-ruption through close encounters and of stellar evolution(Ivanova et al. 2005). According to the theoretical simula-tions of Ivanova et al. (2005) the present day binary fractionin a stellar system with a small central density (103M⊙pc

−3)should be < 30% of its initial fraction. Following these con-siderations the initial binary fraction in our target globularclusters could be > 20− 60%, comparable to that observedin the solar neighborhood (Abt & Levy 1976; Duquennoy &Mayor 1991; Reid & Gizis 1997).

The comparison between the estimated relative binaryfractions among the clusters of our sample suggests that theage is the dominant parameter that determines the fractionof surviving binary systems. This result can be interpretedas an indication that the disruption of soft binary systems

through close encounters with other single and/or binarystars is still efficient in low density globular clusters alsoin the last 5 Gyr of evolution. Unfortunately, there are noestimates of the binary fraction in globular clusters youngerthan 6 Gyr to test the efficiency of the process of binarydisruption in the early stages of evolution. Note however thatestimates of the binary fraction in open clusters (with ages< 3 Gyr) gives values as high as 30-50% (Bica & Bonatto2005).

The comparison between the radial distribution of bi-nary systems with respect to MS stars indicates that binarysystems are more concentrated toward the central region ofmost of the clusters of our sample. This evidence, alreadyfound in other past works (Yan & Reid 1996; Rubenstein &Baylin 1997; Albrow et al. 2001; Bellazzini et al. 2002; Zhao& Baylin 2005) is the result of the kinetic energy equiparti-tion that lead binary systems to settle in the deepest regionof the cluster potential well.

ACKNOWLEDGEMENTS

This research was supported by contract ASI-INAFI/023/05/0 and PRIN-INAF 2006. We warmly thankMichele Bellazzini and the anonymous referee for their help-ful comments and suggestions and Paolo Montegriffo for as-sistance during catalogs cross-correlation.

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