1. THE ASTROPHYSICAL JOURNAL, 474 : 701È709, 1997 January 10
( 1997. The American Astronomical Society. All rights reserved. Printed in U.S.A.
HUBBL E SPACE T EL ESCOPE OBSERVATIONS OF THE POSTÈCORE-COLLAPSE
GLOBULAR CLUSTER NGC 6752. II. A LARGE MAIN-SEQUENCE
BINARY POPULATION
ERIC P. RUBENSTEIN AND CHARLES D. BAILYN
Yale University, Department of Astronomy, P.O. Box 208101, New Haven, CT, 06520-8101 ; ericr=astro.yale.edu
Received 1996 May 22 ; accepted 1996 August 6
ABSTRACT
We 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 be
expected if there were large numbers of binary stars. We use artiÐcial star experiments to characterize
the broadening of the main sequence that is expected, due to both photometric errors and the e†ect of
chance superposition of stars. The observed broadening is signiÐcantly greater than can be explained by
these 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 fraction
is 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 able stars. The current paucity of evidence for large
numbers of cluster binaries is generally taken to reÑect these
One of the main impediments to a more complete under-
observational difficulties.
standing of globular cluster (GC) dynamics and evolution is
Here we report the discovery of a population of binaries
the present uncertainty in binary frequency. Recent studies
constituting greater than 15% of the observable stars in the
(see review by Hut et al. 1992) have shown that binary
core of NGC 6752 using data from the Wide Field Planet-
systems are required to explain the high degree of mass
ary Camera 2 (WFPC2) instrument on the HST . This par-
segregation and the Ñat central surface density proÐle
ticular cluster was chosen because it is a nearby
observed in GCs such as M71 (Richer & Fahlman 1989).
postÈcore-collapse globular cluster (Djorgovski 1993) that
Since even a small initial binary population (as little as 3%
lies in the middle of the HST Ïs continuous viewing zone, so
according to Heggie & Aarseth 1992) can have a signiÐcant
we were able to monitor it continually for 20 hr. Paper I in
inÑuence on the dynamical evolution of a cluster, it is
this series reports the discovery of two candidate cataclys-
crucial to constrain the present binary frequency. Further-
mic variables in the core of NGC 6752 (Bailyn et al. 1996) ;
more, the dynamical state of GCs can alter the underlying
subsequent papers will discuss other variable stars and iso-
stellar population, particularly in the dense core of postÈ
chrone Ðtting to the color-magnitude diagram (CMD).
core-collapse globular clusters (see review by Bailyn 1995).
Here we report the existence in the CMD of a binary
Therefore, constraints on the binary population in the cores
sequence in the inner regions of this GC. We discuss the
of globular clusters are essential for both dynamical and
observations and reductions in ° 2. In ° 3, the artiÐcial star
stellar population studies.
tests and consequent analysis are presented. Sections 4 and
While a variety of individual binary systems have been
5 are discussions and conclusions, respectively.
found in GCs (see the recent conference proceedings edited
by Milone & Mermilliod 1996), there has not yet been an 2. OBSERVATIONS AND REDUCTIONS
unambiguous detection of a large population of main-
sequence binaries. Main-sequence binaries can be observed 2.1. Observations
either through variability (Mateo 1993) or as a ““ binary Our HST observations of the postÈcore-collapse GC
sequence ÏÏ of stars displaced to the red of the main sequence NGC 6752 were made on 1994 August 18. These obser-
proper. Variability studies have been quite successful of late vations were made with the cluster in the continuous
(e.g., Yan & Mateo 1994 ; Edmonds et al. 1996 ; Kaluzny & viewing zone (Gilliland et al. 1995), so that an uninterrupted
Krzeminski 1993 ; Rubenstein & Bailyn 1996), but they are
 time series over a 20 hr baseline could be collected. Three
strongly biased in favor of short-period binaries. Typically, hundred and six WFPC2 observations with the F555W and
[0.1% of main-sequence stars are found to be binaries in F814W Ðlters (hereafter referred to as ““ V ÏÏ and ““ I,ÏÏ
this way. Binary sequences have been observed in open respectively) were made of the clusterÏs core, while another
clusters (Kaluzny, Krzeminski, & Mazur 1996) and in E3
 16 observations were made with o†sets of 1 of the Ðeld of
(McClure et al. 1985), but in general the e†ects of photo- view. Long, medium, and short exposures3 were made to
metric errors and chance superpositions make such maximize the dynamic range of the data. The images were
sequences difficult to detect unambiguously (Romani & split among nine pointings (in a 3 ] 3 grid) o†set from each
Weinberg 1991). A survey for binaries in NGC 6752 with other by 0A5 (11 pixels) to reduce Ñat-Ðelding errors in the
.
the prerepair Hubble Space T elescope (HST ) (Shara et al. Ðnal photometry. The 16 o†set images were made to cali-
1995) found neither a population of binaries nor individual brate the charge transfer e†ect (CTE) problems discussed in
variable stars. However, the extreme crowding in the center Holtzman et al. (1995a, 1995b). The observing log is shown
of such clusters and the small amplitude of variability for in Table 1.
many variables conspire to obscure binaries and other vari- Due to HST operational constraints, it was not possible
701

2. 702 RUBENSTEIN & BAILYN Vol. 474
TABLE 1 Therefore, only the Planetary Camera (PC) data were
OBSERVING LOG retained after the detector was read out (see gray scale in
Fig. 1).
V FRAMES (s) I FRAMES (s) The raw data were calibrated at STScI via the pipeline
DITHER
POSITION 2 26 80 3 50 160 (Burrows 1994). The only unusual problem with this data
set was that six images were truncated such that the upper
1 ........ 1 13 3 1 12 3 one-third of the images were missing. Although the lower
2 ........ 0 13 3 1 12 3
3 ........ 1 13 3 1 11 3
portion of these images appear to be uncorrupted, we chose
4 ........ 1 13 3 1 12 3 not to use them in the subsequent analysis.
5 ........ 1 13 2 1 12 3
6 ........ 1 12 3 1 12 3 2.2. Reductions
7 ........ 1 13 3 1 12 3
8 ........ 0 13 3 1 12 3 Due to the undersampling of stellar proÐles, we used a
9 ........ 1 14 3 1 12 3 hybrid data reduction scheme. The locations of stars were
O†sets
determined by DAOPHOT2 and ALLSTAR2 (Stetson,
Davis, & Crabtree 1991). Then we used the stellar photo-
1È9 . . . . . . 0 1 0 0 1 0 metry software (SPS V1.5) package (Janes & Heasley 1993)
to determine the magnitude of the stars on the PC images
without permitting SPS to recentroid the stars. This
to transmit all of the WFPC2 data to the ground receiving package allowed us to perform aperture photometry
stations without interrupting the observing. Since we sequentially on each star after removing the nearby stars
wanted unbroken time series data to maximize the likeli- with a scaled point-spread function (PSF) Ðt in a manner
hood of observing short-period eclipsing variables, we similar to that described in Yanny et al. (1994). SPS has a
decided instead to sacriÐce the Wide Field (WF) data. 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 the
inner core radius.

3. 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 partially 12
corrupted proÐle only out to the radius of the defect. A
correction was then applied to this partial aperture photo-
metry ; this (usually) small o†set was determined from the
average proÐle calculated from many stars that have pris-
tine proÐles. The result of this secondary aperture correc- 16
tion is that stars with cosmic rays in the wing, or very faint
stars that rise above the sky only in the central pixels, are
V Mag
still measured e†ectively.
After all of the frames were reduced, StetsonÏs (1992)
DAOMASTER routine was used to match stars in di†erent
frames. We only retained stars that appeared in at least 100 20
frames in each Ðlter. We produced both time series light
curves and average magnitudes for each star ; the results of
the time series study will appear separately. Since the PC
data is undersampled, unlike most ground-based images,
the errors in the Ñat-Ðeld corrections become an important
24
source of scatter. In the Appendix, we demonstrate that
averaging photometric results from each frame reduces the
0.0 1.0 2.0 3.0
errors in the magnitudes by averaging over the residual
Ñat-Ðelding corrections. V-I Mag
FIG. 2.ÈCMD of NGC 6752 produced from 107 I and 116 V PC2
2.3. Charge T ransfer Calibrations images. Note the precise ridgeline in the turno† region and the clear evi-
Holtzman et al. (1995a, 1995b) report that, for images dence for main-sequence binaries. The stars brighter than V 16.2 are
with a low background count level, the WFPC2 CCDs have from the short exposures (see Table 1).
many small charge traps. The result of these traps is that the
stars near the top of the CCD are measured as having fewer
counts than equally bright stars near the bottom of the to the red of the main sequence. There are two main mecha-
CCD. Our o†set images permit us to determine the correc- nisms for producing such a spread : chance superposition of
tions for our observations. stars and a true binary population. Due to the exceptional
We used the SPS package to determine the magnitudes of resolution of the HST PC2 images, we are able to separate
stars on each of the 50 s I and 26 s V o†set exposures, and the contributions from these two components using artiÐ-
on an image from position 1 with the same exposure dura- cial star tests (e.g., Bolte 1994 and references therein).
tion. The same PSF stars, or the subset of those PSF stars Note that photometric error and foreground objects are
that fell on the o†set frame, were used to deÐne a PSF for not possible mechanisms. Errors in the photometry will
neighbor subtraction. The typical photometric zero-point appear as a nearly symmetric ““ spread ÏÏ in the main
o†sets are D0.01 mag. These o†sets were removed when the sequence to the blue and the red. Foreground objects are of
photometry was assembled into a single star list by negligible concern since there are very few in a 36A square
StetsonÏs (1992) DAOMATCH/DAOMASTER routines. area. We estimate from Ratnatunga & BahcallÏs (1985)
We checked for the systematic variation in a starÏs magni- models that a total of D1.4 Ðeld stars brighter than V 20
tude as a function of its Y -coordinate reported by Holtz- mag might be present in our Ðeld, while perhaps D4.7 Ðeld
man et al. (1995a, 1995b). A least-squares Ðt to the stars brighter than V 24 mag might be present. Of these,
magnitudes of the individual stars as a function of the Y - only D0.6 and D1.3, respectively, would lie within ^0.5
location yields a 2% ^ 1% variation in stellar brightness in mag of the main-sequence ridgeline (MSRL) in B[V . The
the V Ðlter, and a 0.5% ^ 1% e†ect in I. There is no signiÐ- small group of stars blueward of the MSRL below 19 mag
cant correlation between the X-location and the measured in Figure 2 are probably a combination of these foreground
magnitude. We conclude that, for our medium and long and background halo stars, and possibly cataclysmic vari-
exposures of NGC 6752, there is no signiÐcant CTE ables or faint galaxies. In any event, the few objects brighter
residual to correct, presumably because there is so much than V 19 mag blueward of the MSRL in Figure 2 indi-
charge throughout the chip. cate that there are also probably very few noncluster
The short exposures show a CTE e†ect with a 0.05 ^ 0.01 members near the MSRL between V 16.5 and V 19.0,
mag amplitude over the full range of the Y -position. This the region of the CMD relevant to the subsequent analysis.
reinforces the hypothesis that the total amount of charge on In ° 3.1, we discuss our artiÐcial star tests. In ° 3.2, we
the CCD determines which exposures will su†er from present the evidence that the main-sequence broadening is
charge transfer degradation. We also conÐrm Casertano & due to a large population of binary stars. To quantify the
StiavelliÏs (1995) report of a zero-point o†set between expo- fraction of stars that must be binaries, we perform Monte
sures of a few seconds duration and those that are longer Carlo tests in which we compare the redward spread of real
than a few tens of seconds. and artiÐcial stars from the MSRL. This Monte Carlo pro-
cedure and its results are discussed in ° 3.3 ; in that section,
3. BINARY FREQUENCY IN CORE OF NGC 6752 we also report a radial dependence of the binary fraction
The CMD (Fig. 2) derived from the photometric results found by comparing data from the inner core radius of the
obtained above shows evidence of a broadening above and cluster with that from more distant regions.

4. 704 RUBENSTEIN & BAILYN Vol. 474
3.1. ArtiÐcial Star T ests distribution of the real stars could be drawn from the same
We performed artiÐcial star tests to empirically measure underlying binary-free population as that of the artiÐcial
the accuracy of our photometry and to ascertain whether stars.
there is evidence for a true binary sequence. These artiÐcial
3.2.1. Ridgeline Color Dispersion Method : T esting for the Presence
stars were digitally added using the PSFs calculated from of Binaries in the CMD
the data but with Gaussian noise added. SPS V1.5 construc-
ts PSFs as the sum of a Gaussian analytic model and an We begin by deÐning the main-sequence ridgeline for
empirical look-up table (Janes & Heasley 1993). both the real and the artiÐcial data sets, and then calcu-
We added a total of 43,373 artiÐcial stars in 145 separate lating the deviation in color for each real and artiÐcial star
runs with the ““ Fake Star ÏÏ routines of SPS. The same artiÐ- from their respective MSRLs. Although the initial magni-
cial stars were added to all V and I images with V and I tudes of the artiÐcial stars placed them on the observed
magnitudes that initially placed them on the main sequence MSRL, their magnitudes after reduction were slightly o†set
(Bolte 1994). For each of the 145 artiÐcial star test runs, the to the red ; at a V mag of 17.1, this di†erence was only 0.01
same D300 stars were added to all 223 medium-exposure V in V [I, while at V 19.6, this o†set is 0.04 mag. This
and I images. These stars had randomly selected V magni- di†erence probably arises from an imperfect sky determi-
tudes that ranged from the saturation limit, 15.8, to well nation. A small error in calculating the sky background
below the faintest recovered real stars, 28.4. To ensure that level would a†ect the faint stars more than the bright stars ;
the addition of these artiÐcial stars did not alter the crowd- this is in agreement with the observed trend. This small
ing of the regions into which they were placed, only one star e†ect would not tend to disperse stars in the CMD, but
was added to each 40 ] 40 pixel box. The resultant frames merely move all stars of a given magnitude slightly. Because
were reduced in a manner identical to that described in ° 2. of this small di†erence, we deÐne an MSRL for the real stars
The same matching criteria used above were used to deter- and another for the reduced artiÐcial stars. In both cases, we
mine which real and artiÐcial stars were successfully recov- bin the stars according to V magnitude, with the Ðrst bin
ered. Although D12,000 of the artiÐcial stars were below starting at 17.1 and each bin including 0.25 mag up to a
the detection limit in the V frames, a total of 16,238 artiÐcial maximum magnitude of 19.6. We then make a color histo-
stars were recovered in at least 100 V and I images. gram of the stars in the V -magnitude range. The color bins
To conÐrm the similarity of the photometric errors of the are 0.008 mag in size. In each V -magnitude bin the MSRL is
real and artiÐcial stars, we compared the distribution of deÐned as the mode of this histogram.
both sets of stars blueward of the MSRL. For the purpose In the absence of a binary population, the real and artiÐ-
of checking the relative photometric accuracy of real and cial stars would exhibit similar distributions in deviation
artiÐcial stars, we looked at the blue side of the MSRL since from the observed ridgeline. For each star we determine the
these stars will be una†ected by binaries and chance super- di†erence in color, *C, between that star and the MSRL.
positions. We binned the stars according to V magnitude The stars are divided into those with *C [ 0 and those with
with a bin size of 0.25 mag. The real and artiÐcial star *C 0, that is, according to whether they are redder or
distributions were very similar, with neither being consis- bluer than the empirical MSRL. In this manner, a value of
tently broader than the other. For example, in the *C is determined for each real and artiÐcial star. For each
V 16.75 mag bin, the half-widths at half-maximum real star we compile a list of all artiÐcial stars that are
(HWHMs) di†er by less than 0.001 mag with the artiÐcial within ^0.15 mag of the real star and whose radial distance
stars having the broader HWHM, while in the V 17.75 from the cluster center is within 100 pixels of the real starÏs
mag bin, the HWHMs di†er by less than 0.001 mag with the radial distance from the cluster center. This cohort therefore
real stars having the broader HWHM. This is a strong consists of artiÐcial stars of nearly the same magnitude and
indication that the photometric errors of the real and artiÐ- in essentially similar levels of crowding as the real star, and
cial stars are similar in size and distribution. therefore the photometric errors and probability of chance
superposition should be similar. We then construct a cumu-
3.2. Existence of a Binary Sequence lative histogram of *C values from this cohort of artiÐcial
Romani & Weinberg (1991) and Hut et al. (1992) have stars selected in order to have photometric and crowding
discussed maximum likelihood estimates of the binary frac- properties similar to the real star. For each real star, we
tion in GCs in which observations are compared with theo- calculate the fraction of artiÐcial stars that have a *C
retical models of the CMD. However, it is difficult to *C , which we call Y . As a check on our procedure, we
real star
separate chance superposition from true binary stars in this varied the selection criteria for the cohort of artiÐcial stars
way. Therefore, we use a purely empirical technique. The that is compared with each real star. We found that chang-
artiÐcial star tests described above allow us to separate, in a ing the size of the allowed magnitude range from 0.15 to
statistical sense, the contributions from the chance super- 0.08 mag did not alter the results. Furthermore, a di†erent
position of two stars from that arising from a putative spatial test for selecting the cohort was tried and also did
underlying binary population. not change our conclusions.
BrieÑy, we determine the color distribution in the CMD With this list of Y -values we can test the hypothesis that
of the real main-sequence stars and how they are spread out the real stars have a di†erent *C distribution than the artiÐ-
redward of their ridgeline. We compare this color distribu- cial stars. The artiÐcial stars have the same photometric
tion with the color spread on the CMD of the artiÐcial stars errors that the real stars do, and they have the same likeli-
whose true magnitudes lie on the MSRL. The magnitudes hood of chance superposition with other stars on the sky as
obtained from reducing these artiÐcial stars include the real stars do. If the individual real stars were drawn from
e†ects of photometric errors and chance superposition but the same population as the artiÐcial stars, we would expect
not of a true binary population. A Kolmogrov-Smirnov the Y -values to be distributed randomly from 0 to 1 (see
(K-S) test is used to calculate the probability that the color Fig. 3). However, if there is a concentration of Y -values in

5. No. 2, 1997 BINARIES IN THE CORE OF NGC 6752 705
than 250 pixels, which is approximately equal to one core
1.0
radius (Djorgovski 1993) ; a circle with this radius, centered
on the cluster center, is plotted in Figure 1. The stars more
distant from the center were included in the second group.
Cumulative Histogram of Y
0.8 We then perform the ridgeline color dispersion test
described above on the set of stars in the inner and outer
groups (see Fig. 4). For the inner region, we found that the
0.6 formal probability that the real stars have the same under-
lying color distribution as the artiÐcial stars is 10~16. This
analysis was repeated for the outer group of stars. In this
case, the formal result of the one-sided K-S test, 0.21, is
0.4 inconclusive and suggests at most a small binary popu-
lation. The large disparity in the ridgeline color dispersion
results between the inner and the outer groups indicates
0.2 that their binary fractions are very di†erent. Note that this
di†erence is also visible when the CMD of each region is
plotted separately (see Fig. 5).
0.2 0.4 0.6 0.8 1.0 3.3. Binary Star Mass Segregation in the Center of
Fraction of Artificial Stars, Y, with ∆C<∆Creal star NGC 6752
We performed Monte Carlo tests to quantify the fraction
FIG. 3.ÈA cumulative histogram (see ° 3.2) that shows that the real star of stars in the core of NGC 6752 that are binaries. These
population ( jagged line) deviates signiÐcantly in color distribution from a
population devoid of binaries (bold, straight line).
tests were designed to determine what fraction of the artiÐ-
cial stars discussed above would have to be altered in order
to lie on a binary sequence and to make the color distribu-
some range of values, then the real stars and the artiÐcial tion of the artiÐcial stars similar to that of the real stars.
stars must have di†erent distributions in *C. If the Y -values Successful matches are deÐned by the *C distributions
of the real stars are biased toward unity relative to the being statistically similar to those of the real data set.
artiÐcial stars, this implies that the real stars are spread The two free parameters in these tests are the fraction of
toward the red from the main sequence beyond what is detected stars that are binaries and the fraction of light
created by chance superpositions. coming from each component of the binary system. We
In Figure 3, the cumulative histogram of Y -values is perform a series of Monte Carlo calculations that vary both
plotted versus the line segment from 0, 0 to 1, 1. This line of these parameters. The binary frequency ranges from 0%
segment corresponds to the null hypothesis, which states to 100%. For the second variable, we choose a power-law
that the two populations are drawn from the same parent relation that governs how close the ratio of the luminosity
populations. The data plotted fall systematically below this of each stellar component is to unity. In this param-
line segment. A one-sided K-S test indicates that the formal
chance that the artiÐcial stars have the same underlying *C
distribution as the real stars is 10~13. Therefore, it appears 1.0
that a binary population is required to explain the degree of
redward dispersion observed from the MSRL in NGC 6752.
3.2.2. Radial Di†erences
Cumulative Histogram of Y
0.8
Mass segregation is likely to result in the binary popu-
lation being centrally condensed in a dynamically evolved
cluster like NGC 6752. We searched for the e†ects of mass 0.6
segregation by examining the inner core radius and the rest
of the regions surveyed separately. The Ðrst step is to derive
a cluster center from our data. We use the iterative cen-
troiding method, described by Picard & Johnston (1994), to 0.4
Ðnd the cluster center at (229, 499) pixel coordinates (see
Sams 1995 concerning the intrinsic limitations of any such
technique). We also derive the uncertainty in centroid loca- 0.2
tion, 2.5 pixels 0A1. It is somewhat larger than the error
.
limit due to Ðnite sampling that Sams (1995) calculates, 1
pixel 0A04 (corresponding to 0A1 at 10 kpc). However, the
. .
total size of the centroidÏs uncertainty is small compared 0.2 0.4 0.6 0.8 1.0
with the radial bin we use and therefore is not a signiÐcant Fraction of Artificial Stars, Y, with ∆C<∆Creal star
concern.
Having determined the cluster center, we split the stars FIG. 4.ÈTwo cumulative histograms (see ° 3.2.2) that show that the real
into di†erent radial groups. Since there were only 2421 stars star population in the inner core radius (solid, jagged line) deviates signiÐ-
cantly in color distribution from a population devoid of binaries (bold,
in the Ðnal CMD, we could only construct two radial bins straight line), whereas the real star population outside this region (dashed
without seriously reducing the statistical conÐdence of our line) is not conclusively di†erent from a stellar population that has no
Ðndings. One group is composed of stars closer to the center binaries.

6. 706 RUBENSTEIN & BAILYN Vol. 474
0
10
ξ=2.0 –1
16.0 10
–2
10
0
18.0 10
–1
ξ=1.0
log(P)
10
–2
10
20.0
0
10
ξ=0.5 –1
V mag
10
22.0 –2
10
0
10
–1
ξ=0.25
log(P)
16.0 10
–2
10
0
10
18.0
ξ=0.125 –1
10
–2
10
20.0 0
10
–1
ξ=0.0
log(P)
10
22.0 –2
10
0.0 0.5 1.0 1.5 2.0 0 10 20 30 40 50
V-I mag Binary Fraction (%)
FIG. 5.ÈTwo CMDs (see ° 3.2.2) that show that the MSRL in the inner FIG. 6.ÈResults of Monte Carlo experiments performed on stars in the
core radius (top panel) is markedly skewed toward the red compared with inner core radius as described in ° 3.3. Each point shows the result of an
the stars farther from the center (bottom panel). individual Monte Carlo experiment. There are two free parameters in these
experiments, the binary fraction and m (a quantity appearing in the equa-
tion in ° 3.3) ; m 0 indicates equal luminosities for the primary and sec-
eterization, the secondary starÏs V magnitude, V , is ran- ondary, while larger values of m indicate a typically fainter distribution in
2 the secondaryÏs magnitude. The six panels show the e†ect of varying m
domly chosen with a distribution :
between 0.0 and 2.0. In each panel, the y-axis shows the log probability for
V each Monte Carlo experiment in which the artiÐcial star data (see ° 3.1) are
V 1, statistically similar to the observed data. The line plotted through the data
2 Rm is the median of the points at the indicated binary fraction. Note that the
value of m does not alter the required binary fraction by more than 10% at
where R is a random number between 0 and 1, and m is a the upper limits, and hardly at all at the lower limits.
free parameter. For the case m 0, each component of the
binaries contributes equally to the luminosity of the system,
i.e., that the luminosity ratio is always unity. As m increases, dependent on m ranging from D28% to D38%. It is encour-
the luminosity distribution of the secondaries is more aging that the required binary fraction goes up as m goes up
skewed toward lower luminosities, but a lower bound of since large m implies more of the binaries have a secondary
V 25.0 mag was also imposed. We chose six values of m : that contributes little light to the system. In such systems,
2
0.0, 0.125, 0.25, 0.5, 1.0, and 2.0. the binary lies very close to the single-star MSRL. The K-S
For each combination of binary fraction and m, we made test is not intended as a test to determine a ““ best Ðt ÏÏ the
1000 Monte Carlo tests. In each of these tests, we randomly way s2 tests do. Therefore, the most appropriate interpreta-
select an artiÐcial star from each real starÏs cohort. Each of tion of these results is as a preferred range in binary frac-
these artiÐcial stars has a chance equal to the binary frac- tion, e.g., 15%È38%.
tion of being designated a ““ pseudobinary ÏÏ and having a The tests in ° 3.2.2 are insufficient to determine whether
secondary star randomly selected as described above. The mass segregation has moved all of the clusterÏs binaries into
resulting set of stars is compared to the real stars with the the central core radius. Even though we cannot deÐnitively
ridgeline color dispersion technique described in ° 3.2 demonstrate whether or not a binary population exists in
above. The result of each individual Monte Carlo test is a this outer region of the clusterÏs core, we can place upper
probability that the artiÐcial star distribution, enhanced by limits on the binary frequency. To do this, we carried out
pseudobinaries, has the same *C distribution that the real tests identical to those described above, except using the
stars have. stars in the annulus surrounding the inner core region.
The results of these tests (the points in Fig. 6) indicate The results in this outer region (see Fig. 7) were less
that regardless of the choice of m, the lower limit (read from dependent on the value of m than in the core. Over the full
the graph at log (Probability) [2.5 99.7% conÐdence range of m, the 3 p upper limit on the binary frequency,
level) on the binary fraction in the inner core radius is 15%. consistent with the observations, is 16%. However, a binary
In discussing our results, we will refer to the line plotted frequency of zero cannot be ruled out. It is clear, however,
through the points, which is the median of the values at a that the binary frequency is di†erent in the inner and outer
given binary fraction. The upper limits are somewhat more regions studied.

7. No. 2, 1997 BINARIES IN THE CORE OF NGC 6752 707
10
0
sions may be dramatically altered.1
ξ=2.0 –1 The large number of binary systems also serve as a
10
–2
reservoir of heavy objects in the core of this cluster. The
10 total mass of many of the binaries will be signiÐcantly
10
0
greater than the turno† mass, but the luminosity of most of
–1
ξ=1.0 the binaries is less than that of a turno† mass star. There-
log(P)
10
–2
fore, the binaries constitute a signiÐcant, centrally concen-
10 trated population of objects with higher M/L ratios than
10
0
the main-sequence turno† (MSTO) stars that contribute
ξ=0.5 –1 most of the cluster light. Studies of clusters such as M15
10
–2
have demonstrated the existence of large numbers of dim
10 massive objects in their core (Phinney 1993). These objects
10
0
are usually interpreted as neutron stars or massive white
–1
ξ=0.25 dwarfs, but our results suggest that doubleÈmain-sequence
log(P)
10
binaries may contribute strongly to this population.
–2
10 Finally, a large population of binaries may alter the
10
0 main-sequence ridgeline and luminosity function of the
ξ=0.125 –1
cluster. If our photometric accuracy were somewhat less
10
than it is, we might have included the many binary systems
–2
10 in our determination of the main-sequence ridgeline, which
10
0 would then be displaced to the red from its true location.
–1
ξ=0.0 This would be a particular problem well below the MSTO,
log(P)
10
where the number of single stars is depleted by mass segre-
10
–2
gation. Similarly, the main-sequence mass function will be
distorted by the presence of binaries. Without the binaries,
0 10 20 30 40 50 it is likely that the inferred mass functions in the cores of
Binary Fraction (%) dense GCs will be even more depleted than is suggested by
FIG. 7.ÈResults of Monte Carlo experiments performed on the stars
the work of DeMarchi & Paresce (1995a, 1995b). A quanti-
outside the inner core radius out to the edge of the Ðeld, D3.3 core radii as tative study, which is under way, of the luminosity and mass
described in ° 3.3. The meaning of the six panels are the same as in Fig. 6. functions from this data will require careful completeness
Note that the binary frequency is nearly independent of m, and not conclu- corrections.
sively di†erent from 0%.
5. CONCLUSION
Photometry of the core of NGC 6752 with the HST
shows an asymmetric spread along the main-sequence
ridgeline. ArtiÐcial star tests demonstrate that the distribu-
4. DISCUSSION tion of stars away from the ridgeline requires a large binary
We have shown that a signiÐcant fraction of the main- population in the core, and a smaller but possibly still sig-
sequence stars in the center of NGC 6752 are likely to have niÐcant binary frequency in the adjacent few core radii. The
binary companions. This result has broad implications for lower and upper 99.7% conÐdence limits on the binary fre-
the stellar populations and dynamics of globular clusters. quency in the inner core radius is 15% and 28%È38%,
First, the large fraction of binaries implies that binary- depending on the distribution of luminosity ratios in the
binary interactions may be the dominant dynamical heating binaries. The binary frequency in the outer annulus is
process. It has long been known that a small population of ¹16%.
binary stars can contribute signiÐcant energy to the cluster The authors would like to thank Peter Stetson, Ken
as a whole through binaryÈsingle-star scattering (Heggie Janes, and Jim Heasley for making newer versions available
1975 ; Hut & Bahcall 1983). However, since binary stars of DAOFIND and SPS. C. D. B. is grateful for a National
have much larger scattering cross sections than single stars Young Investigator award from the NSF. We thank Mary
(Leonard 1989 ; Leonard & Fahlman 1991), binary-binary Katherine McGovern, Jerry Orosz, Richard Larson, Alison
scattering events will be even more important in popu- Sills, and Ken Sills for comments, suggestions, and help
lations with a signiÐcant binary fraction. These interactions with the data analysis. We would also like to thank the
have not been as well studied as binaryÈsingle-star inter- referee, Mario Mateo, for several suggestions that helped
actions (see review by Hut et al. 1992), but it seems likely clarify the presentation of this material. This research has
that in NGC 6752, at least, binary-binary interactions will made use of the SIMBAD database, operated at CDS,
dominate the dynamical evolution of the cluster. Strasbourg, France. This work has been supported by
Similarly, the collisions and close encounters responsible NASA through LTSA grant NAGW-2469 and grant
for a variety of anomalous objects, such as blue stragglers number HST-GO-5318 from the Space Telescope Science
and X-ray sources (see Bailyn 1995), are likely to be trig- Institute, which is operated by the Association of Uni-
gered by binary-binary encounters. The actual merger versities for Research in Astronomy, Inc., under NASA con-
process for two colliding stars in the binary-binary encoun- tract NAS 5-26555.
ter is likely to be similar to that in a single-star collision,
since the inÑuence of the other stars in the system will be 1 It is worth noting that the 17 blue stragglers seen in Fig. 2 are cen-
relatively small during the encounter. However, the colli- trally concentrated with respect to the other stars in the cluster, as is
sion rate and the distribution of the input stars to the colli- commonly the case for blue straggler systems.

8. 16.0
17.0
16.0
17.0
16.0
V mag
17.0
16.0
17.0
16.0
17.0
0.1 0.4 0.7
V-I mag
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 the
same 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 nominal
telescope position. The frames were analyzed separately using the same coordinates for the stars in successive frames ; recentroiding was turned o†. The next
panel (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 between
successive frames. The bottom panel was made by averaging magnitudes obtained from nine V and nine I images taken at di†erent locations. Note the clear
increase in precision from top to bottom, although the same total exposure time and reduction software were used.

9. 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 a
dramatically di†erent quality of results. Figure 8 shows a comparison of results near the MSTO of NGC 6752. In all cases,
nine V and nine I exposures were used, along with the same data reduction procedure described above.
The top panel of Figure 8 shows the results when the nine images were shifted by the small subpixel o†sets before being
averaged. The combined frames were subsequently reduced. The data in the next panel were handled the same way, except
that the individual frames were not shifted prior to averaging. It is clear from the comparison between these two sets of results
that noninteger pixel shifts should be avoided when dealing with WFPC2 images, and undersampled images generally. Such
noninteger shifts require interpolations in the undersampled cores of the stellar images. We believe that this is the cause of the
large errors in the top panel.
The middle panel shows the results when nine images at the same nominal position were reduced separately, and the
resulting magnitudes for each star were averaged afterward. In this case, the same star list was used for each of the nine V
frames (and one list for the nine I frames), without shifting the coordinates to account for the subpixel motions of the
telescope. Recentroiding was turned o† here, as in all of our analysis. The results of this panel are nearly an exact duplicate of
the 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 contrast
to 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 the
stellar proÐle, which is much less undersampled than the core. In this case, the interpolation errors are outweighed by the
improvement gained from using the most accurate stellar positions, which vary slightly from frame to frame even at the same
nominal pointing.
Finally, the bottom panel presents the results for a procedure similar to that of the previous panel, except that one image
from each of the nine di†erent dither positions was used. In this case, we not only gain the beneÐt from the previous
procedure, but we also average out Ñat-Ðelding errors. The improvement in quality from top to bottom is particularly
impressive 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 noted
the advantages of ““ dithering ÏÏ (see Biretta et al. 1996 for a discussion of commonly implemented ““ dithering strategies ÏÏ with
the HST ). Further improvements are also obtained by using individual input star lists with o†sets that account for subpixel
shifts between frames. For very faint or low surface brightness objects, summed frames may be crucial ; actually shifting the
data by subinteger values, however, should be avoided. For high-precision stellar photometry, it appears that separate
reductions for large numbers of frames with slightly di†erent pointings should be the recommended procedure.
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