Hubble Space Telescope observations of the globular cluster NGC 6752 reveal a broadened and asymmetric main sequence, indicative of a large binary population. Artificial star experiments show that photometric errors and chance star superpositions alone cannot account for the observed broadening. A Monte Carlo technique is used to estimate the binary fraction, which is likely 15-38% in the inner core but less than 16% beyond that radius. The discovery of a significant main-sequence binary population provides constraints on the dynamical evolution and stellar populations of globular clusters.

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. AND CHARLES D.RUBENSTEIN 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
One of the main impediments to a more complete under-
standing of globular cluster (GC) dynamics and evolution is
the 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 mass
segregation and the Ñat central surface density proÐle
observed 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 is
crucial to constrain the present binary frequency. Further-
more, the dynamical state of GCs can alter the underlying
stellar population, particularly in the dense core of postÈ
core-collapse globular clusters (see review by Bailyn 1995).
Therefore, constraints on the binary population in the cores
of globular clusters are essential for both dynamical and
stellar population studies.
While a variety of individual binary systems have been
found in GCs (see the recent conference proceedings edited
by & Mermilliod there has not yet been anMilone 1996),
unambiguous detection of a large population of main-
sequence binaries. Main-sequence binaries can be observed
either through variability or as a ““ binary(Mateo 1993)
sequence ÏÏ of stars displaced to the red of the main sequence
proper. Variability studies have been quite successful of late
(e.g., & Mateo et al. &Yan 1994; Edmonds 1996; Kaluzny
Krzemin 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 open
clusters 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 such
sequences difficult to detect unambiguously &(Romani
Weinberg 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 center
of such clusters and the small amplitude of variability for
many variables conspire to obscure binaries and other vari-
able stars. The current paucity of evidence for large
numbers of cluster binaries is generally taken to reÑect these
observational difficulties.
Here we report the discovery of a population of binaries
constituting greater than 15% of the observable stars in the
core 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 nearby
postÈcore-collapse globular cluster that(Djorgovski 1993)
lies in the middle of the HST Ïs continuous viewing zone, so
we were able to monitor it continually for 20 hr. Paper I in
this 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 binary
sequence in the inner regions of this GC. We discuss the
observations 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. Observations
Our 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 continuous
viewing zone et al. so that an uninterrupted(Gilliland 1995),
time series over a 20 hr baseline could be collected. Three
hundred and six WFPC2 observations with the F555W and
F814W Ðlters (hereafter referred to as ““ V ÏÏ and ““ I,ÏÏ
respectively) were made of the clusterÏs core, while another
16 observations were made with o†sets of of the Ðeld of1
3view. Long, medium, and short exposures were made to
maximize the dynamic range of the data. The images were
split among nine pointings (in a 3 ] 3 grid) o†set from each
other by (11 pixels) to reduce Ñat-Ðelding errors in the0A.5
Ðnal photometry. The 16 o†set images were made to cali-
brate the charge transfer e†ect (CTE) problems discussed in
Holtzman et al. The observing log is shown(1995a, 1995b).
in Table 1.
Due to HST operational constraints, it was not possible
701

2. 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 3
2 . . . . . . . . 0 13 3 1 12 3
3 . . . . . . . . 1 13 3 1 11 3
4 . . . . . . . . 1 13 3 1 12 3
5 . . . . . . . . 1 13 2 1 12 3
6 . . . . . . . . 1 12 3 1 12 3
7 . . . . . . . . 1 13 3 1 12 3
8 . . . . . . . . 0 13 3 1 12 3
9 . . . . . . . . 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 receiving
stations without interrupting the observing. Since we
wanted unbroken time series data to maximize the likeli-
hood of observing short-period eclipsing variables, we
decided instead to sacriÐce the Wide Field (WF) data.
Therefore, only the Planetary Camera (PC) data were
retained after the detector was read out (see gray scale in
Fig. 1).
The raw data were calibrated at STScI via the pipeline
The only unusual problem with this data(Burrows 1994).
set was that six images were truncated such that the upper
one-third of the images were missing. Although the lower
portion of these images appear to be uncorrupted, we chose
not to use them in the subsequent analysis.
2.2. Reductions
Due to the undersampling of stellar proÐles, we used a
hybrid data reduction scheme. The locations of stars were
determined 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 images
without permitting SPS to recentroid the stars. This
package allowed us to perform aperture photometry
sequentially on each star after removing the nearby stars
with a scaled point-spread function (PSF) Ðt in a manner
similar 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 the
inner core radius.

3. 0.0 1.0 2.0 3.0
V-I Mag
24
20
16
12
VMag
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
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-
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
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
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
source of scatter. In the Appendix, we demonstrate that
averaging photometric results from each frame reduces the
errors in the magnitudes by averaging over the residual
Ñat-Ðelding corrections.
2.3. Charge T ransfer Calibrations
Holtzman et al. report that, for images(1995a, 1995b)
with a low background count level, the WFPC2 CCDs have
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
CCD. Our o†set images permit us to determine the correc-
tions for our observations.
We used the SPS package to determine the magnitudes of
stars on each of the 50 s I and 26 s V o†set exposures, and
on an image from position 1 with the same exposure dura-
tion. The same PSF stars, or the subset of those PSF stars
that fell on the o†set frame, were used to deÐne a PSF for
neighbor subtraction. The typical photometric zero-point
o†sets are D0.01 mag. These o†sets were removed when the
photometry 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 in
the V Ðlter, and a 0.5% ^ 1% e†ect in I. There is no signiÐ-
cant correlation between the X-location and the measured
magnitude. We conclude that, for our medium and long
exposures of NGC 6752, there is no signiÐcant CTE
residual to correct, presumably because there is so much
charge throughout the chip.
The short exposures show a CTE e†ect with a 0.05 ^ 0.01
mag amplitude over the full range of the Y -position. This
reinforces the hypothesis that the total amount of charge on
the CCD determines which exposures will su†er from
charge transfer degradation. We also conÐrm &Casertano
StiavelliÏs report of a zero-point o†set between expo-(1995)
sures of a few seconds duration and those that are longer
than 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 PC2
images. Note the precise ridgeline in the turno† region and the clear evi-
dence for main-sequence binaries. The stars brighter than V 16.2 are
from 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 of
stars and a true binary population. Due to the exceptional
resolution of the HST PC2 images, we are able to separate
the contributions from these two components using artiÐ-
cial star tests (e.g., and references therein).Bolte 1994
Note that photometric error and foreground objects are
not possible mechanisms. Errors in the photometry will
appear as a nearly symmetric ““ spread ÏÏ in the main
sequence to the blue and the red. Foreground objects are of
negligible concern since there are very few in a 36A square
area. We estimate from & BahcallÏsRatnatunga (1985)
models that a total of D1.4 Ðeld stars brighter than V 20
mag might be present in our Ðeld, while perhaps D4.7 Ðeld
stars brighter than V 24 mag might be present. Of these,
only D0.6 and D1.3, respectively, would lie within ^0.5
mag of the main-sequence ridgeline (MSRL) in B[V . The
small group of stars blueward of the MSRL below 19 mag
in are probably a combination of these foregroundFigure 2
and background halo stars, and possibly cataclysmic vari-
ables or faint galaxies. In any event, the few objects brighter
than V 19 mag blueward of the MSRL in indi-Figure 2
cate that there are also probably very few noncluster
members 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 is
due to a large population of binary stars. To quantify the
fraction of stars that must be binaries, we perform Monte
Carlo tests in which we compare the redward spread of real
and 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 fraction
found by comparing data from the inner core radius of the
cluster with that from more distant regions.

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

5. 0.2 0.4 0.6 0.8 1.0
Fraction of Artificial Stars, Y, with ∆C<∆Creal star
0.2
0.4
0.6
0.8
1.0
CumulativeHistogramofY
0.2 0.4 0.6 0.8 1.0
Fraction of Artificial Stars, Y, with ∆C<∆Creal star
0.2
0.4
0.6
0.8
1.0
CumulativeHistogramofY
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 a
population devoid of binaries (bold, straight line).
some range of values, then the real stars and the artiÐcial
stars must have di†erent distributions in *C. If the Y -values
of the real stars are biased toward unity relative to the
artiÐcial stars, this implies that the real stars are spread
toward the red from the main sequence beyond what is
created by chance superpositions.
In the cumulative histogram of Y -values isFigure 3,
plotted versus the line segment from 0, 0 to 1, 1. This line
segment corresponds to the null hypothesis, which states
that the two populations are drawn from the same parent
populations. The data plotted fall systematically below this
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
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
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
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 & Johnston toPicard (1994),
Ðnd the cluster center at (229, 499) pixel coordinates (see
concerning the intrinsic limitations of any suchSams 1995
technique). 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.1
total size of the centroidÏs uncertainty is small compared
with the radial bin we use and therefore is not a signiÐcant
concern.
Having determined the cluster center, we split the stars
into di†erent radial groups. Since there were only 2421 stars
in the Ðnal CMD, we could only construct two radial bins
without seriously reducing the statistical conÐdence of our
Ðndings. One group is composed of stars closer to the center
than 250 pixels, which is approximately equal to one core
radius 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 test
described above on the set of stars in the inner and outer
groups (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. 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
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
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).
3.3. Binary Star Mass Segregation in the Center of
NGC 6752
We performed Monte Carlo tests to quantify the fraction
of stars in the core of NGC 6752 that are binaries. These
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-
tion of the artiÐcial stars similar to that of the real stars.
Successful matches are deÐned by the *C distributions
being statistically similar to those of the real data set.
The two free parameters in these tests are the fraction of
detected stars that are binaries and the fraction of light
coming from each component of the binary system. We
perform a series of Monte Carlo calculations that vary both
of these parameters. The binary frequency ranges from 0%
to 100%. For the second variable, we choose a power-law
relation that governs how close the ratio of the luminosity
of 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 (dashed
line) is not conclusively di†erent from a stellar population that has no
binaries.

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

7. 0 10 20 30 40 50
Binary Fraction (%)
10
–2
10
–1
10
0
log(P)
10
–2
10
–1
10
0
10
–2
10
–1
10
0
log(P)
10
–2
10
–1
10
0
10
–2
10
–1
10
0
log(P)
10
–2
10
–1
10
0
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 stars
outside the inner core radius out to the edge of the Ðeld, D3.3 core radii as
described 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 have
binary companions. This result has broad implications for
the stellar populations and dynamics of globular clusters.
First, the large fraction of binaries implies that binary-
binary interactions may be the dominant dynamical heating
process. It has long been known that a small population of
binary stars can contribute signiÐcant energy to the cluster
as 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 interactions
have 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 will
dominate the dynamical evolution of the cluster.
Similarly, the collisions and close encounters responsible
for a variety of anomalous objects, such as blue stragglers
and X-ray sources (see are likely to be trig-Bailyn 1995),
gered by binary-binary encounters. The actual merger
process 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 be
relatively small during the encounter. However, the colli-
sion rate and the distribution of the input stars to the colli-
sions may be dramatically altered.1
The large number of binary systems also serve as a
reservoir of heavy objects in the core of this cluster. The
total mass of many of the binaries will be signiÐcantly
greater than the turno† mass, but the luminosity of most of
the 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 than
the main-sequence turno† (MSTO) stars that contribute
most of the cluster light. Studies of clusters such as M15
have demonstrated the existence of large numbers of dim
massive objects in their core These objects(Phinney 1993).
are usually interpreted as neutron stars or massive white
dwarfs, but our results suggest that doubleÈmain-sequence
binaries may contribute strongly to this population.
Finally, a large population of binaries may alter the
main-sequence ridgeline and luminosity function of the
cluster. If our photometric accuracy were somewhat less
than it is, we might have included the many binary systems
in our determination of the main-sequence ridgeline, which
would 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 be
distorted by the presence of binaries. Without the binaries,
it is likely that the inferred mass functions in the cores of
dense GCs will be even more depleted than is suggested by
the work of DeMarchi & Paresce A quanti-(1995a, 1995b).
tative study, which is under way, of the luminosity and mass
functions from this data will require careful completeness
corrections.
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-
tion of stars away from the ridgeline requires a large binary
population in the core, and a smaller but possibly still sig-
niÐcant binary frequency in the adjacent few core radii. The
lower 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 the
binaries. The binary frequency in the outer annulus is
¹16%.
The authors would like to thank Peter Stetson, Ken
Janes, and Jim Heasley for making newer versions available
of DAOFIND and SPS. C. D. B. is grateful for a National
Young Investigator award from the NSF. We thank Mary
Katherine McGovern, Jerry Orosz, Richard Larson, Alison
Sills, and Ken Sills for comments, suggestions, and help
with the data analysis. We would also like to thank the
referee, Mario Mateo, for several suggestions that helped
clarify the presentation of this material. This research has
made use of the SIMBAD database, operated at CDS,
Strasbourg, France. This work has been supported by
NASA through LTSA grant NAGW-2469 and grant
number HST-GO-5318 from the Space Telescope Science
Institute, 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. 2
trally concentrated with respect to the other stars in the cluster, as is
commonly the case for blue straggler systems.

8. 0.1 0.4 0.7
V-I mag
17.0
16.0
17.0
16.0
17.0
16.0
Vmag
17.0
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 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. shows a comparison of results near the MSTO of NGC 6752. In all cases,Figure 8
nine 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 8
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 et al. for a discussion of commonly implemented ““ dithering strategies ÏÏ withBiretta 1996
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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