THE ASTROPHYSICAL JOURNAL, 474 :701È709, 1997 January 101997. The American Astronomical Society. All rights reserved. Printed in U.S.A.(

HUBBL E SPACE T EL ESCOPE OBSERVATIONS OF THE POSTÈCORE-COLLAPSEGLOBULAR CLUSTER NGC 6752. II. A LARGE MAIN-SEQUENCE

BINARY POPULATION

ERIC P. AND CHARLES D.RUBENSTEIN BAILYN

Yale University, Department of Astronomy, P.O. Box 208101, New Haven, CT, 06520-8101 ; ericr=astro.yale.eduReceived 1996 May 22; accepted 1996 August 6

ABSTRACTWe present a color-magnitude diagram (CMD) of NGC 6752 based on post-refurbishment Planetary

Camera 2 observations of its core. The main sequence is broadened and asymmetric, as would beexpected if there were large numbers of binary stars. We use artiÐcial star experiments to characterizethe broadening of the main sequence that is expected, due to both photometric errors and the e†ect ofchance superposition of stars. The observed broadening is signiÐcantly greater than can be explained bythese two e†ects alone, so a main-sequence binary population is required to explain the observations.We develop a Monte Carlo technique to calculate the binary frequency in the CMD. The binary fractionis probably in the range 15%È38% in the inner core radius (r \ 11A) but is probably less than 16%beyond that.Subject heading : binaries : eclipsing È globular clusters : individual (NGC 6752) È stars : statistics

1. INTRODUCTION

One of the main impediments to a more complete under-standing of globular cluster (GC) dynamics and evolution isthe present uncertainty in binary frequency. Recent studies(see review by et al. have shown that binaryHut 1992)systems are required to explain the high degree of masssegregation and the Ñat central surface density proÐleobserved in GCs such as M71 & Fahlman(Richer 1989).Since even a small initial binary population (as little as 3%according to & Aarseth can have a signiÐcantHeggie 1992)inÑuence on the dynamical evolution of a cluster, it iscrucial to constrain the present binary frequency. Further-more, the dynamical state of GCs can alter the underlyingstellar population, particularly in the dense core of postÈcore-collapse globular clusters (see review by Bailyn 1995).Therefore, constraints on the binary population in the coresof globular clusters are essential for both dynamical andstellar population studies.

While a variety of individual binary systems have beenfound in GCs (see the recent conference proceedings editedby & Mermilliod there has not yet been anMilone 1996),unambiguous detection of a large population of main-sequence binaries. Main-sequence binaries can be observedeither through variability or as a ““ binary(Mateo 1993)sequence ÏÏ of stars displaced to the red of the main sequenceproper. Variability studies have been quite successful of late(e.g., & Mateo et al. &Yan 1994 ; Edmonds 1996 ; KaluznyKrzemin� ski & Bailyn but they are1993 ; Rubenstein 1996),strongly biased in favor of short-period binaries. Typically,

of main-sequence stars are found to be binaries in[0.1%this way. Binary sequences have been observed in openclusters Krzemin� ski, & Mazur and in E3(Kaluzny, 1996)

et al. but in general the e†ects of photo-(McClure 1985),metric errors and chance superpositions make suchsequences difficult to detect unambiguously &(RomaniWeinberg A survey for binaries in NGC 6752 with1991).the prerepair Hubble Space T elescope (HST ) et al.(Shara

found neither a population of binaries nor individual1995)variable stars. However, the extreme crowding in the centerof such clusters and the small amplitude of variability formany variables conspire to obscure binaries and other vari-

able stars. The current paucity of evidence for largenumbers of cluster binaries is generally taken to reÑect theseobservational difficulties.

Here we report the discovery of a population of binariesconstituting greater than 15% of the observable stars in thecore of NGC 6752 using data from the Wide Field Planet-ary Camera 2 (WFPC2) instrument on the HST . This par-ticular cluster was chosen because it is a nearbypostÈcore-collapse globular cluster that(Djorgovski 1993)lies in the middle of the HST Ïs continuous viewing zone, sowe were able to monitor it continually for 20 hr. Paper I inthis series reports the discovery of two candidate cataclys-mic variables in the core of NGC 6752 (Bailyn et al. 1996) ;subsequent papers will discuss other variable stars and iso-chrone Ðtting to the color-magnitude diagram (CMD).

Here we report the existence in the CMD of a binarysequence in the inner regions of this GC. We discuss theobservations and reductions in In the artiÐcial star° 2. ° 3,tests and consequent analysis are presented. Sections and4

are discussions and conclusions, respectively.5

2. OBSERVATIONS AND REDUCTIONS

2.1. ObservationsOur HST observations of the postÈcore-collapse GC

NGC 6752 were made on 1994 August 18. These obser-vations were made with the cluster in the continuousviewing zone et al. so that an uninterrupted(Gilliland 1995),time series over a 20 hr baseline could be collected. Threehundred and six WFPC2 observations with the F555W andF814W Ðlters (hereafter referred to as ““V ÏÏ and ““ I,ÏÏrespectively) were made of the clusterÏs core, while another16 observations were made with o†sets of of the Ðeld of13view. Long, medium, and short exposures were made tomaximize the dynamic range of the data. The images weresplit among nine pointings (in a 3 ] 3 grid) o†set from eachother by (11 pixels) to reduce Ñat-Ðelding errors in the0A.5Ðnal photometry. The 16 o†set images were made to cali-brate the charge transfer e†ect (CTE) problems discussed inHoltzman et al. The observing log is shown(1995a, 1995b).in Table 1.

Due to HST operational constraints, it was not possible

701

702 RUBENSTEIN & BAILYN Vol. 474

TABLE 1

OBSERVING LOG

V FRAMES (s) I FRAMES (s)DITHER

POSITION 2 26 80 3 50 160

1 . . . . . . . . 1 13 3 1 12 32 . . . . . . . . 0 13 3 1 12 33 . . . . . . . . 1 13 3 1 11 34 . . . . . . . . 1 13 3 1 12 35 . . . . . . . . 1 13 2 1 12 36 . . . . . . . . 1 12 3 1 12 37 . . . . . . . . 1 13 3 1 12 38 . . . . . . . . 0 13 3 1 12 39 . . . . . . . . 1 14 3 1 12 3

O†sets

1È9 . . . . . . 0 1 0 0 1 0

to transmit all of the WFPC2 data to the ground receivingstations without interrupting the observing. Since wewanted unbroken time series data to maximize the likeli-hood of observing short-period eclipsing variables, wedecided instead to sacriÐce the Wide Field (WF) data.

Therefore, only the Planetary Camera (PC) data wereretained after the detector was read out (see gray scale inFig. 1).

The raw data were calibrated at STScI via the pipelineThe only unusual problem with this data(Burrows 1994).

set was that six images were truncated such that the upperone-third of the images were missing. Although the lowerportion of these images appear to be uncorrupted, we chosenot to use them in the subsequent analysis.

2.2. ReductionsDue to the undersampling of stellar proÐles, we used a

hybrid data reduction scheme. The locations of stars weredetermined by DAOPHOT2 and ALLSTAR2 (Stetson,Davis, & Crabtree Then we used the stellar photo-1991).metry software (SPS V1.5) package & Heasley(Janes 1993)to determine the magnitude of the stars on the PC imageswithout permitting SPS to recentroid the stars. Thispackage allowed us to perform aperture photometrysequentially on each star after removing the nearby starswith a scaled point-spread function (PSF) Ðt in a mannersimilar to that described in et al. SPS has aYanny (1994).high level of automation that allows for a very consistent

FIG. 1.ÈGray-scale image, 36A ] 36A, of NGC 6752 produced from a 26 s V PC2 image. The large circle is centered on the cluster center and encloses theinner core radius.

0.0 1.0 2.0 3.0V-I Mag

24

20

16

12

V M

ag

No. 2, 1997 BINARIES IN THE CORE OF NGC 6752 703

reduction procedure for each frame, minimizing frame-to-frame o†sets in the photometric zero point.

We measured the brightness of stars that have a partiallycorrupted proÐle only out to the radius of the defect. Acorrection was then applied to this partial aperture photo-metry ; this (usually) small o†set was determined from theaverage proÐle calculated from many stars that have pris-tine proÐles. The result of this secondary aperture correc-tion is that stars with cosmic rays in the wing, or very faintstars that rise above the sky only in the central pixels, arestill measured e†ectively.

After all of the frames were reduced, StetsonÏs (1992)DAOMASTER routine was used to match stars in di†erentframes. We only retained stars that appeared in at least 100frames in each Ðlter. We produced both time series lightcurves and average magnitudes for each star ; the results ofthe time series study will appear separately. Since the PCdata is undersampled, unlike most ground-based images,the errors in the Ñat-Ðeld corrections become an importantsource of scatter. In the Appendix, we demonstrate thataveraging photometric results from each frame reduces theerrors in the magnitudes by averaging over the residualÑat-Ðelding corrections.

2.3. Charge T ransfer CalibrationsHoltzman et al. report that, for images(1995a, 1995b)

with a low background count level, the WFPC2 CCDs havemany small charge traps. The result of these traps is that thestars near the top of the CCD are measured as having fewercounts than equally bright stars near the bottom of theCCD. Our o†set images permit us to determine the correc-tions for our observations.

We used the SPS package to determine the magnitudes ofstars on each of the 50 s I and 26 s V o†set exposures, andon an image from position 1 with the same exposure dura-tion. The same PSF stars, or the subset of those PSF starsthat fell on the o†set frame, were used to deÐne a PSF forneighbor subtraction. The typical photometric zero-pointo†sets are D0.01 mag. These o†sets were removed when thephotometry was assembled into a single star list by

DAOMATCH/DAOMASTER routines.StetsonÏs (1992)We checked for the systematic variation in a starÏs magni-tude as a function of its Y -coordinate reported by Holtz-man et al. A least-squares Ðt to the(1995a, 1995b).magnitudes of the individual stars as a function of the Y -location yields a 2%^ 1% variation in stellar brightness inthe V Ðlter, and a 0.5%^ 1% e†ect in I. There is no signiÐ-cant correlation between the X-location and the measuredmagnitude. We conclude that, for our medium and longexposures of NGC 6752, there is no signiÐcant CTEresidual to correct, presumably because there is so muchcharge throughout the chip.

The short exposures show a CTE e†ect with a 0.05^ 0.01mag amplitude over the full range of the Y -position. Thisreinforces the hypothesis that the total amount of charge onthe CCD determines which exposures will su†er fromcharge transfer degradation. We also conÐrm &CasertanoStiavelliÏs report of a zero-point o†set between expo-(1995)sures of a few seconds duration and those that are longerthan a few tens of seconds.

3. BINARY FREQUENCY IN CORE OF NGC 6752

The CMD derived from the photometric results(Fig. 2)obtained above shows evidence of a broadening above and

FIG. 2.ÈCMD of NGC 6752 produced from 107 I and 116 V PC2images. Note the precise ridgeline in the turno† region and the clear evi-dence for main-sequence binaries. The stars brighter than V \ 16.2 arefrom the short exposures (see Table 1).

to the red of the main sequence. There are two main mecha-nisms for producing such a spread : chance superposition ofstars and a true binary population. Due to the exceptionalresolution of the HST PC2 images, we are able to separatethe contributions from these two components using artiÐ-cial star tests (e.g., and references therein).Bolte 1994

Note that photometric error and foreground objects arenot possible mechanisms. Errors in the photometry willappear as a nearly symmetric ““ spread ÏÏ in the mainsequence to the blue and the red. Foreground objects are ofnegligible concern since there are very few in a 36A squarearea. We estimate from & BahcallÏsRatnatunga (1985)models that a total of D1.4 Ðeld stars brighter than V \ 20mag might be present in our Ðeld, while perhaps D4.7 Ðeldstars brighter than V \ 24 mag might be present. Of these,only D0.6 and D1.3, respectively, would lie within ^0.5mag of the main-sequence ridgeline (MSRL) in B[V . Thesmall group of stars blueward of the MSRL below 19 magin are probably a combination of these foregroundFigure 2and background halo stars, and possibly cataclysmic vari-ables or faint galaxies. In any event, the few objects brighterthan V \ 19 mag blueward of the MSRL in indi-Figure 2cate that there are also probably very few nonclustermembers near the MSRL between V \ 16.5 and V \ 19.0,the region of the CMD relevant to the subsequent analysis.

In we discuss our artiÐcial star tests. In we° 3.1, ° 3.2,present the evidence that the main-sequence broadening isdue to a large population of binary stars. To quantify thefraction of stars that must be binaries, we perform MonteCarlo tests in which we compare the redward spread of realand artiÐcial stars from the MSRL. This Monte Carlo pro-cedure and its results are discussed in in that section,° 3.3 ;we also report a radial dependence of the binary fractionfound by comparing data from the inner core radius of thecluster with that from more distant regions.

704 RUBENSTEIN & BAILYN Vol. 474

3.1. ArtiÐcial Star T estsWe performed artiÐcial star tests to empirically measure

the accuracy of our photometry and to ascertain whetherthere is evidence for a true binary sequence. These artiÐcialstars were digitally added using the PSFs calculated fromthe data but with Gaussian noise added. SPS V1.5 construc-ts PSFs as the sum of a Gaussian analytic model and anempirical look-up table & Heasley(Janes 1993).

We added a total of 43,373 artiÐcial stars in 145 separateruns with the ““ Fake Star ÏÏ routines of SPS. The same artiÐ-cial stars were added to all V and I images with V and Imagnitudes that initially placed them on the main sequence

For each of the 145 artiÐcial star test runs, the(Bolte 1994).same D300 stars were added to all 223 medium-exposure Vand I images. These stars had randomly selected V magni-tudes that ranged from the saturation limit, 15.8, to wellbelow the faintest recovered real stars, 28.4. To ensure thatthe addition of these artiÐcial stars did not alter the crowd-ing of the regions into which they were placed, only one starwas added to each 40] 40 pixel box. The resultant frameswere reduced in a manner identical to that described in ° 2.The same matching criteria used above were used to deter-mine which real and artiÐcial stars were successfully recov-ered. Although D12,000 of the artiÐcial stars were belowthe detection limit in the V frames, a total of 16,238 artiÐcialstars were recovered in at least 100 V and I images.

To conÐrm the similarity of the photometric errors of thereal and artiÐcial stars, we compared the distribution ofboth sets of stars blueward of the MSRL. For the purposeof checking the relative photometric accuracy of real andartiÐcial stars, we looked at the blue side of the MSRL sincethese stars will be una†ected by binaries and chance super-positions. We binned the stars according to V magnitudewith a bin size of 0.25 mag. The real and artiÐcial stardistributions were very similar, with neither being consis-tently broader than the other. For example, in theV \ 16.75 mag bin, the half-widths at half-maximum(HWHMs) di†er by less than 0.001 mag with the artiÐcialstars having the broader HWHM, while in the V \ 17.75mag bin, the HWHMs di†er by less than 0.001 mag with thereal stars having the broader HWHM. This is a strongindication that the photometric errors of the real and artiÐ-cial stars are similar in size and distribution.

3.2. Existence of a Binary Sequence& Weinberg and et al. haveRomani (1991) Hut (1992)

discussed maximum likelihood estimates of the binary frac-tion in GCs in which observations are compared with theo-retical models of the CMD. However, it is difficult toseparate chance superposition from true binary stars in thisway. Therefore, we use a purely empirical technique. TheartiÐcial star tests described above allow us to separate, in astatistical sense, the contributions from the chance super-position of two stars from that arising from a putativeunderlying binary population.

BrieÑy, we determine the color distribution in the CMDof the real main-sequence stars and how they are spread outredward of their ridgeline. We compare this color distribu-tion with the color spread on the CMD of the artiÐcial starswhose true magnitudes lie on the MSRL. The magnitudesobtained from reducing these artiÐcial stars include thee†ects of photometric errors and chance superposition butnot of a true binary population. A Kolmogrov-Smirnov(K-S) test is used to calculate the probability that the color

distribution of the real stars could be drawn from the sameunderlying binary-free population as that of the artiÐcialstars.

3.2.1. Ridgeline Color Dispersion Method : T esting for the Presenceof Binaries in the CMD

We begin by deÐning the main-sequence ridgeline forboth the real and the artiÐcial data sets, and then calcu-lating the deviation in color for each real and artiÐcial starfrom their respective MSRLs. Although the initial magni-tudes of the artiÐcial stars placed them on the observedMSRL, their magnitudes after reduction were slightly o†setto the red ; at a V mag of 17.1, this di†erence was only 0.01in V [I, while at V \ 19.6, this o†set is 0.04 mag. Thisdi†erence probably arises from an imperfect sky determi-nation. A small error in calculating the sky backgroundlevel would a†ect the faint stars more than the bright stars ;this is in agreement with the observed trend. This smalle†ect would not tend to disperse stars in the CMD, butmerely move all stars of a given magnitude slightly. Becauseof this small di†erence, we deÐne an MSRL for the real starsand another for the reduced artiÐcial stars. In both cases, webin the stars according to V magnitude, with the Ðrst binstarting at 17.1 and each bin including 0.25 mag up to amaximum magnitude of 19.6. We then make a color histo-gram of the stars in the V -magnitude range. The color binsare 0.008 mag in size. In each V -magnitude bin the MSRL isdeÐned as the mode of this histogram.

In the absence of a binary population, the real and artiÐ-cial stars would exhibit similar distributions in deviationfrom the observed ridgeline. For each star we determine thedi†erence in color, *C, between that star and the MSRL.The stars are divided into those with *C[ 0 and those with*C\ 0, that is, according to whether they are redder orbluer than the empirical MSRL. In this manner, a value of*C is determined for each real and artiÐcial star. For eachreal star we compile a list of all artiÐcial stars that arewithin ^0.15 mag of the real star and whose radial distancefrom the cluster center is within 100 pixels of the real starÏsradial distance from the cluster center. This cohort thereforeconsists of artiÐcial stars of nearly the same magnitude andin essentially similar levels of crowding as the real star, andtherefore the photometric errors and probability of chancesuperposition should be similar. We then construct a cumu-lative histogram of *C values from this cohort of artiÐcialstars selected in order to have photometric and crowdingproperties similar to the real star. For each real star, wecalculate the fraction of artiÐcial stars that have a *C\

which we call Y . As a check on our procedure, we*Creal star,varied the selection criteria for the cohort of artiÐcial starsthat is compared with each real star. We found that chang-ing the size of the allowed magnitude range from 0.15 to0.08 mag did not alter the results. Furthermore, a di†erentspatial test for selecting the cohort was tried and also didnot change our conclusions.

With this list of Y -values we can test the hypothesis thatthe real stars have a di†erent *C distribution than the artiÐ-cial stars. The artiÐcial stars have the same photometricerrors that the real stars do, and they have the same likeli-hood of chance superposition with other stars on the sky asreal stars do. If the individual real stars were drawn fromthe same population as the artiÐcial stars, we would expectthe Y -values to be distributed randomly from 0 to 1 (see

However, if there is a concentration of Y -values inFig. 3).

0.2 0.4 0.6 0.8 1.0Fraction of Artificial Stars, Y, with ∆C<∆Creal star

0.2

0.4

0.6

0.8

1.0C

umul

ativ

e H

isto

gram

of

Y

0.2 0.4 0.6 0.8 1.0Fraction of Artificial Stars, Y, with ∆C<∆Creal star

0.2

0.4

0.6

0.8

1.0

Cum

ulat

ive

His

togr

am o

f Y

No. 2, 1997 BINARIES IN THE CORE OF NGC 6752 705

FIG. 3.ÈA cumulative histogram (see that shows that the real star° 3.2)population ( jagged line) deviates signiÐcantly in color distribution from apopulation devoid of binaries (bold, straight line).

some range of values, then the real stars and the artiÐcialstars must have di†erent distributions in *C. If the Y -valuesof the real stars are biased toward unity relative to theartiÐcial stars, this implies that the real stars are spreadtoward the red from the main sequence beyond what iscreated by chance superpositions.

In the cumulative histogram of Y -values isFigure 3,plotted versus the line segment from 0, 0 to 1, 1. This linesegment corresponds to the null hypothesis, which statesthat the two populations are drawn from the same parentpopulations. The data plotted fall systematically below thisline segment. A one-sided K-S test indicates that the formalchance that the artiÐcial stars have the same underlying *Cdistribution as the real stars is 10~13. Therefore, it appearsthat a binary population is required to explain the degree ofredward dispersion observed from the MSRL in NGC 6752.

3.2.2. Radial Di†erences

Mass segregation is likely to result in the binary popu-lation being centrally condensed in a dynamically evolvedcluster like NGC 6752. We searched for the e†ects of masssegregation by examining the inner core radius and the restof the regions surveyed separately. The Ðrst step is to derivea cluster center from our data. We use the iterative cen-troiding method, described by & Johnston toPicard (1994),Ðnd the cluster center at (229, 499) pixel coordinates (see

concerning the intrinsic limitations of any suchSams 1995technique). We also derive the uncertainty in centroid loca-tion, 2.5 It is somewhat larger than the errorpixels\ 0A.1.limit due to Ðnite sampling that calculates, 1Sams (1995)

(corresponding to at 10 kpc). However, thepixel\ 0A.04 0A.1total size of the centroidÏs uncertainty is small comparedwith the radial bin we use and therefore is not a signiÐcantconcern.

Having determined the cluster center, we split the starsinto di†erent radial groups. Since there were only 2421 starsin the Ðnal CMD, we could only construct two radial binswithout seriously reducing the statistical conÐdence of ourÐndings. One group is composed of stars closer to the center

than 250 pixels, which is approximately equal to one coreradius a circle with this radius, centered(Djorgovski 1993) ;on the cluster center, is plotted in The stars moreFigure 1.distant from the center were included in the second group.

We then perform the ridgeline color dispersion testdescribed above on the set of stars in the inner and outergroups (see For the inner region, we found that theFig. 4).formal probability that the real stars have the same under-lying color distribution as the artiÐcial stars is 10~16. Thisanalysis was repeated for the outer group of stars. In thiscase, the formal result of the one-sided K-S test, 0.21, isinconclusive and suggests at most a small binary popu-lation. The large disparity in the ridgeline color dispersionresults between the inner and the outer groups indicatesthat their binary fractions are very di†erent. Note that thisdi†erence is also visible when the CMD of each region isplotted separately (see Fig. 5).

3.3. Binary Star Mass Segregation in the Center ofNGC 6752

We performed Monte Carlo tests to quantify the fractionof stars in the core of NGC 6752 that are binaries. Thesetests were designed to determine what fraction of the artiÐ-cial stars discussed above would have to be altered in orderto lie on a binary sequence and to make the color distribu-tion of the artiÐcial stars similar to that of the real stars.Successful matches are deÐned by the *C distributionsbeing statistically similar to those of the real data set.

The two free parameters in these tests are the fraction ofdetected stars that are binaries and the fraction of lightcoming from each component of the binary system. Weperform a series of Monte Carlo calculations that vary bothof these parameters. The binary frequency ranges from 0%to 100%. For the second variable, we choose a power-lawrelation that governs how close the ratio of the luminosityof each stellar component is to unity. In this param-

FIG. 4.ÈTwo cumulative histograms (see that show that the real° 3.2.2)star population in the inner core radius (solid, jagged line) deviates signiÐ-cantly in color distribution from a population devoid of binaries (bold,straight line), whereas the real star population outside this region (dashedline) is not conclusively di†erent from a stellar population that has nobinaries.

0.0 0.5 1.0 1.5 2.0V-I mag

22.0

20.0

18.0

16.0

22.0

20.0

18.0

16.0

V m

ag

0 10 20 30 40 50Binary Fraction (%)

10–2

10–1

100

log(

P)

10–2

10–1

100

10–2

10–1

100

log(

P)

10–2

10–1

100

10–2

10–1

100

log(

P)

10–2

10–1

100

ξ=0.0

ξ=0.125

ξ=0.25

ξ=0.5

ξ=1.0

ξ=2.0

706 RUBENSTEIN & BAILYN Vol. 474

FIG. 5.ÈTwo CMDs (see that show that the MSRL in the inner° 3.2.2)core radius (top panel) is markedly skewed toward the red compared withthe stars farther from the center (bottom panel).

eterization, the secondary starÏs V magnitude, is ran-V2,domly chosen with a distribution :

V2\ V1Rm

,

where R is a random number between 0 and 1, and m is afree parameter. For the case m \ 0, each component of thebinaries contributes equally to the luminosity of the system,i.e., that the luminosity ratio is always unity. As m increases,the luminosity distribution of the secondaries is moreskewed toward lower luminosities, but a lower bound of

mag was also imposed. We chose six values of m :V2\ 25.00.0, 0.125, 0.25, 0.5, 1.0, and 2.0.

For each combination of binary fraction and m, we made1000 Monte Carlo tests. In each of these tests, we randomlyselect an artiÐcial star from each real starÏs cohort. Each ofthese artiÐcial stars has a chance equal to the binary frac-tion of being designated a ““ pseudobinary ÏÏ and having asecondary star randomly selected as described above. Theresulting set of stars is compared to the real stars with theridgeline color dispersion technique described in ° 3.2above. The result of each individual Monte Carlo test is aprobability that the artiÐcial star distribution, enhanced bypseudobinaries, has the same *C distribution that the realstars have.

The results of these tests (the points in indicateFig. 6)that regardless of the choice of m, the lower limit (read fromthe graph at log (Probability) \ [2.5\ 99.7% conÐdencelevel) on the binary fraction in the inner core radius is 15%.In discussing our results, we will refer to the line plottedthrough the points, which is the median of the values at agiven binary fraction. The upper limits are somewhat more

FIG. 6.ÈResults of Monte Carlo experiments performed on stars in theinner core radius as described in Each point shows the result of an° 3.3.individual Monte Carlo experiment. There are two free parameters in theseexperiments, the binary fraction and m (a quantity appearing in the equa-tion in m \ 0 indicates equal luminosities for the primary and sec-° 3.3) ;ondary, while larger values of m indicate a typically fainter distribution inthe secondaryÏs magnitude. The six panels show the e†ect of varying mbetween 0.0 and 2.0. In each panel, the y-axis shows the log probability foreach Monte Carlo experiment in which the artiÐcial star data (see are° 3.1)statistically similar to the observed data. The line plotted through the datais the median of the points at the indicated binary fraction. Note that thevalue of m does not alter the required binary fraction by more than 10% atthe upper limits, and hardly at all at the lower limits.

dependent on m ranging from D28% to D38%. It is encour-aging that the required binary fraction goes up as m goes upsince large m implies more of the binaries have a secondarythat contributes little light to the system. In such systems,the binary lies very close to the single-star MSRL. The K-Stest is not intended as a test to determine a ““ best Ðt ÏÏ theway s2 tests do. Therefore, the most appropriate interpreta-tion of these results is as a preferred range in binary frac-tion, e.g., 15%È38%.

The tests in are insufficient to determine whether° 3.2.2mass segregation has moved all of the clusterÏs binaries intothe central core radius. Even though we cannot deÐnitivelydemonstrate whether or not a binary population exists inthis outer region of the clusterÏs core, we can place upperlimits on the binary frequency. To do this, we carried outtests identical to those described above, except using thestars in the annulus surrounding the inner core region.

The results in this outer region (see were lessFig. 7)dependent on the value of m than in the core. Over the fullrange of m, the 3 p upper limit on the binary frequency,consistent with the observations, is 16%. However, a binaryfrequency of zero cannot be ruled out. It is clear, however,that the binary frequency is di†erent in the inner and outerregions studied.

0 10 20 30 40 50Binary Fraction (%)

10–2

10–1

100

log(

P)

10–2

10–1

100

10–2

10–1

100

log(

P)

10–2

10–1

100

10–2

10–1

100

log(

P)

10–2

10–1

100

ξ=0.0

ξ=0.125

ξ=0.25

ξ=0.5

ξ=1.0

ξ=2.0

No. 2, 1997 BINARIES IN THE CORE OF NGC 6752 707

FIG. 7.ÈResults of Monte Carlo experiments performed on the starsoutside the inner core radius out to the edge of the Ðeld, D3.3 core radii asdescribed in The meaning of the six panels are the same as in° 3.3. Fig. 6.Note that the binary frequency is nearly independent of m, and not conclu-sively di†erent from 0%.

4. DISCUSSION

We have shown that a signiÐcant fraction of the main-sequence stars in the center of NGC 6752 are likely to havebinary companions. This result has broad implications forthe stellar populations and dynamics of globular clusters.

First, the large fraction of binaries implies that binary-binary interactions may be the dominant dynamical heatingprocess. It has long been known that a small population ofbinary stars can contribute signiÐcant energy to the clusteras a whole through binaryÈsingle-star scattering (Heggie

& Bahcall However, since binary stars1975 ; Hut 1983).have much larger scattering cross sections than single stars

& Fahlman binary-binary(Leonard 1989 ; Leonard 1991),scattering events will be even more important in popu-lations with a signiÐcant binary fraction. These interactionshave not been as well studied as binaryÈsingle-star inter-actions (see review by et al. but it seems likelyHut 1992),that in NGC 6752, at least, binary-binary interactions willdominate the dynamical evolution of the cluster.

Similarly, the collisions and close encounters responsiblefor a variety of anomalous objects, such as blue stragglersand X-ray sources (see are likely to be trig-Bailyn 1995),gered by binary-binary encounters. The actual mergerprocess for two colliding stars in the binary-binary encoun-ter is likely to be similar to that in a single-star collision,since the inÑuence of the other stars in the system will berelatively small during the encounter. However, the colli-sion rate and the distribution of the input stars to the colli-

sions may be dramatically altered.1The large number of binary systems also serve as a

reservoir of heavy objects in the core of this cluster. Thetotal mass of many of the binaries will be signiÐcantlygreater than the turno† mass, but the luminosity of most ofthe binaries is less than that of a turno† mass star. There-fore, the binaries constitute a signiÐcant, centrally concen-trated population of objects with higher M/L ratios thanthe main-sequence turno† (MSTO) stars that contributemost of the cluster light. Studies of clusters such as M15have demonstrated the existence of large numbers of dimmassive objects in their core These objects(Phinney 1993).are usually interpreted as neutron stars or massive whitedwarfs, but our results suggest that doubleÈmain-sequencebinaries may contribute strongly to this population.

Finally, a large population of binaries may alter themain-sequence ridgeline and luminosity function of thecluster. If our photometric accuracy were somewhat lessthan it is, we might have included the many binary systemsin our determination of the main-sequence ridgeline, whichwould then be displaced to the red from its true location.This would be a particular problem well below the MSTO,where the number of single stars is depleted by mass segre-gation. Similarly, the main-sequence mass function will bedistorted by the presence of binaries. Without the binaries,it is likely that the inferred mass functions in the cores ofdense GCs will be even more depleted than is suggested bythe work of DeMarchi & Paresce A quanti-(1995a, 1995b).tative study, which is under way, of the luminosity and massfunctions from this data will require careful completenesscorrections.

5. CONCLUSION

Photometry of the core of NGC 6752 with the HSTshows an asymmetric spread along the main-sequenceridgeline. ArtiÐcial star tests demonstrate that the distribu-tion of stars away from the ridgeline requires a large binarypopulation in the core, and a smaller but possibly still sig-niÐcant binary frequency in the adjacent few core radii. Thelower and upper 99.7% conÐdence limits on the binary fre-quency in the inner core radius is 15% and 28%È38%,depending on the distribution of luminosity ratios in thebinaries. The binary frequency in the outer annulus is¹16%.

The authors would like to thank Peter Stetson, KenJanes, and Jim Heasley for making newer versions availableof DAOFIND and SPS. C. D. B. is grateful for a NationalYoung Investigator award from the NSF. We thank MaryKatherine McGovern, Jerry Orosz, Richard Larson, AlisonSills, and Ken Sills for comments, suggestions, and helpwith the data analysis. We would also like to thank thereferee, Mario Mateo, for several suggestions that helpedclarify the presentation of this material. This research hasmade use of the SIMBAD database, operated at CDS,Strasbourg, France. This work has been supported byNASA through LTSA grant NAGW-2469 and grantnumber HST-GO-5318 from the Space Telescope ScienceInstitute, which is operated by the Association of Uni-versities for Research in Astronomy, Inc., under NASA con-tract NAS 5-26555.

1 It is worth noting that the 17 blue stragglers seen in are cen-Fig. 2trally concentrated with respect to the other stars in the cluster, as iscommonly the case for blue straggler systems.

0.1 0.4 0.7V-I mag

17.0

16.0

17.0

16.0

17.0

16.0

V m

ag17.0

16.0

17.0

16.0

FIG. 8.ÈCMDs of the turno† region of NGC 6752 constructed from subsets of the data collected. The top panel was made from nine images, all from thesame dither position, shifted by the subpixel o†sets and then averaged. The next panel was made by averaging nine images, all from the same dither position,without shifting them Ðrst. The middle panel was made by averaging magnitudes obtained from each of the nine V and nine I images, from the same nominaltelescope position. The frames were analyzed separately using the same coordinates for the stars in successive frames ; recentroiding was turned o†. The nextpanel (fourth from the top) was made the same way as the middle panel, except that the coordinates of the stars were shifted by the subpixel o†sets betweensuccessive frames. The bottom panel was made by averaging magnitudes obtained from nine V and nine I images taken at di†erent locations. Note the clearincrease in precision from top to bottom, although the same total exposure time and reduction software were used.

BINARIES IN THE CORE OF NGC 6752 709

APPENDIX

DATA REDUCTION STRATEGIES USING THE WFPC2

In the course of performing this investigation, we have found that di†erent ways of handling the data can result in adramatically di†erent quality of results. shows a comparison of results near the MSTO of NGC 6752. In all cases,Figure 8nine V and nine I exposures were used, along with the same data reduction procedure described above.

The top panel of shows the results when the nine images were shifted by the small subpixel o†sets before beingFigure 8averaged. The combined frames were subsequently reduced. The data in the next panel were handled the same way, exceptthat the individual frames were not shifted prior to averaging. It is clear from the comparison between these two sets of resultsthat noninteger pixel shifts should be avoided when dealing with WFPC2 images, and undersampled images generally. Suchnoninteger shifts require interpolations in the undersampled cores of the stellar images. We believe that this is the cause of thelarge errors in the top panel.

The middle panel shows the results when nine images at the same nominal position were reduced separately, and theresulting magnitudes for each star were averaged afterward. In this case, the same star list was used for each of the nine Vframes (and one list for the nine I frames), without shifting the coordinates to account for the subpixel motions of thetelescope. Recentroiding was turned o† here, as in all of our analysis. The results of this panel are nearly an exact duplicate ofthe previous panel, as would be expected, since the same data and star positions are used in both cases.

In the next panel (the fourth from the top), we again perform SPS photometry separately on each of the nine images.However, in this case, we shift the input star positions to account for the subpixel pointing shifts in the telescope. In contrastto the situation for the Ðrst panel, this procedure improves the quality of the photometry. This is because the only inter-polation required in this case is at the edge of the aperture used for the aperture photometry. This is located in the wings of thestellar proÐle, which is much less undersampled than the core. In this case, the interpolation errors are outweighed by theimprovement gained from using the most accurate stellar positions, which vary slightly from frame to frame even at the samenominal pointing.

Finally, the bottom panel presents the results for a procedure similar to that of the previous panel, except that one imagefrom each of the nine di†erent dither positions was used. In this case, we not only gain the beneÐt from the previousprocedure, but we also average out Ñat-Ðelding errors. The improvement in quality from top to bottom is particularlyimpressive given that the same total exposure time and data reduction software were used in all cases.

These results demonstrate the importance of compensating for residual Ñat-Ðelding errorsÈother authors have also notedthe advantages of ““ dithering ÏÏ (see et al. for a discussion of commonly implemented ““ dithering strategies ÏÏ withBiretta 1996the HST ). Further improvements are also obtained by using individual input star lists with o†sets that account for subpixelshifts between frames. For very faint or low surface brightness objects, summed frames may be crucial ; actually shifting thedata by subinteger values, however, should be avoided. For high-precision stellar photometry, it appears that separatereductions for large numbers of frames with slightly di†erent pointings should be the recommended procedure.

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