THE LOW END OF THE INITIAL MASS FUNCTION IN YOUNG CLUSTERS. II.EVIDENCE FOR PRIMORDIAL MASS SEGREGATION IN NGC 330

IN THE SMALL MAGELLANIC CLOUD1

Marco Sirianni,2Antonella Nota,

3,4Guido DeMarchi,

3,4Claus Leitherer,

3and Mark Clampin

3

Received 2002 February 14; accepted 2002 July 9

ABSTRACT

As part of a larger program aimed at investigating the universality of the initial mass function (IMF) atlow masses in a number of young clusters in the LMC and SMC, we present a new study of the low end of thestellar IMF of NGC 330, the richest young star cluster in the SMC, from deep broadband V and I imagesobtained with HST/WFPC2. We detect stars down to a limiting magnitude of m555 ¼ 24:9, which corre-sponds to stellar masses of � 0:8 M� at the distance of the SMC. A comparison of the cluster color-magni-tude diagram with theoretical evolutionary tracks indicates an age of �30 Myr for NGC 330, in agreementwith previous published results. We derive the cluster luminosity function, which we correct for backgroundcontamination using an adjacent SMC field, and construct the mass function in the 1 7 M� mass range.Given the young cluster age, theMF can well approximate the IMF.We find that the IMF in the central clus-ter regions (within 3000) is well reproduced by a power law with a slope consistent with Salpeter’s. In addition,the richness of the cluster allows us to investigate the IMF as a function of radial distance from the center.We find that the IMF becomes steeper at increasing distances from the cluster center (between 3000 and 9000),with the number of massive stars (> 5 M�) decreasing from the core to the outskirts of the cluster 5 timesmore rapidly than the less-massive objects (’ 1 M�). We believe the observed mass segregation to be of aprimordial nature rather than dynamical since the age of NGC 330 is 10 times shorter than the expectedrelaxation time of the cluster.

Subject headings: galaxies: star clusters — Magellanic Clouds — stars: evolution —stars: luminosity function, mass function

1. INTRODUCTION

The stellar initial mass function (IMF) and its possibleuniversality has been the object of many studies and intensediscussion since stars could be individually counted andclusters could be resolved. With the advent of the HubbleSpace Telescope (HST), the commissioning of largeground-based telescopes (Keck, VLT), and the increasedsophistication of the available instrumentation, new energyhas infused the debate.

How universal is the IMF (Kroupa 2001)? In star clustersand associations, where the distance effects are removed andthe star formation history is simple, the answer should bestraightforward. At high masses (10 100 M�) there is evi-dence for an IMF having a slope � ¼ dðlog �Þ=dðlogMÞ ¼ �1:3� 0:5 (Scalo 1998). Any variations, ifpresent, would be masked by the observational uncertain-ties (Scalo 1998).

At small and intermediate masses (1� 10 M�), the situa-tion is very different. Even in the LMC itself, where the fieldpopulation shows a very steep IMF (� ’ �4; Massey et al.1995), deep photometry of clusters has produced IMFparameters ranging from the steep IMFs found by Mateo

(1988; � ’ �2) to the much shallower slopes (� ’ 0;�1)derived by Elson, Fall, & Freeman (1989) and Hunter et al.(1995, 1996). The low- and intermediate-mass end of theIMF in starburst galaxies may be even more peculiar. Themass-to-light ratio of the stellar population in the prototypi-cal starburst galaxy M82 can be understood if stars withmasses below�3 5 M� are absent (Rieke et al. 1993).

At the smallest masses, the spread of IMF slopes deter-mined is even larger: Are we observing true deviations fromthe Salpeter (1955) IMF, most likely triggered by local con-ditions of stellar density or star formation history? Or arewe simply dominated by the observational uncertaintiesrelated to the data analysis and interpretation such asthe choice of the evolutionary models or treatment ofcompleteness?

The conflicting claims of steep or shallow IMFs in LMC/SMC clusters (see references above) could also be attributedto the limitations of the data so far. In fact, ground-basedobservations were typically restricted to the clusters’ periph-ery; and, as such, they often ignored the mass contributionfrommassive stars in the core: the shape of the resultingMFcould then depend on the combination of the age of the clus-ter and the radial location at which the observations wereconducted, as we discuss in x 9, without necessarily reflect-ing the properties of the IMF. When investigations arerestricted to a limited portion of the cluster, the effects ofdynamical evolution on the shape of the local MF must betaken into account (G. De Marchi, F. Paresce, & S. Port-egies Zwart 2002, in preparation).

In order to put the observational results on an equal foot-ing, we have undertaken a systematic HST study of a num-ber of young clusters in the LMC and in the SMC atdifferent conditions of stellar density, age, and metallicity,

1 Based on observations with the NASA/ESA Hubble Space Telescopeobtained at the Space Telescope Science Institute, which is operated byAURA for NASA under contract NAS5-26555.

2 Johns Hopkins University, 3400 North Charles Street, Baltimore, MD21218; sirianni@pha.jhu.edu.

3 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore,MD 21218; nota@stsci.edu, demarchi@stsci.edu, leitherer@stsci.edu,clampin@stsci.edu.

4 Affiliated with the Research and Science Support Department of theEuropean Space Agency.

The Astrophysical Journal, 579:275–288, 2002 November 1

# 2002. The American Astronomical Society. All rights reserved. Printed in U.S.A.

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with the objective of studying the low end of the stellar IMFand understanding whether the IMF is constrained by localconditions at small masses. The advantage of such a study isto reduce the uncertainties associated with data analysis byestablishing a homogeneous data treatment procedure,including a unique choice of models. Moreover, all the clus-ters have the same known distance, and they are very popu-lous, providing good statistics to our sample.

This procedure has been already applied to the R136 clus-ter in the LMC (Sirianni et al. 2000, hereafter S00). Verydeep broadband V and I WFPC2 images of R136 from theHST archive were been used to sample the luminosity func-tion below the detection limit of 2:8 M� previously reached.In these new images, reaching a limiting magnitude ofmF555W ¼ 24:7, S00 found a population of red stars evenlydistributed in area surrounding the R136 cluster, most likelypre–main-sequence stars in the mass range 0:6 3 M�. Theyconstructed the IMF for the cluster in the 1:35 6:5 M�range and found that, after correcting for incompleteness,the IMF shows a flattening below’2 M�.

So far, this result is unique. Using the same methodology,HST images have been obtained for 10 additional clustersin the LMC (M. Sirianni et al. 2002, in preparation) and onein the SMC—NGC 330, for which we present the results inthis paper.

2. NGC 330

NGC 330 is one of the brightest young clusters in theSMC. It is young (age less than 50 Myr) and very populous.Initial indications of a very peculiar abundance (lower thana factor of 5 with respect to the SMC field young popula-tion; Richtler, Spite, & Spite 1990) have recently beenrevised by Gonzalez & Wallerstein (1999) and Hill (1999).Gonzalez & Wallerstein (1999) find a mean value of½Fe=H� ¼ �0:94� 0:02 dex for NGC 330, compared withthe SMC field of ½Fe=H� ¼ �0:69� 0:10 dex (Hill 1999).Hill (1999) finds a slightly different value ½Fe=H� ¼�0:82� 0:11 dex, which argues in favor of a formation ofthe cluster from gas that has experienced the same chemicalhistory of the SMC, and this strengthens the suitability ofNGC 330 for our comparative study.

NGC 330 is known for its intrinsic abundance of Be stars(Feast 1972; Grebel, Richtler, & de Boer 1992; Keller,Wood, & Bessell 1998; Keller & Bessell 1998). Keller,Wood, & Bessell (1998) found a much higher fraction of Bestars with respect to the total number of B stars for NGC330 than in the surrounding field. In Be stars, the emission isbelieved to arise from an optically thick rotating disk ofmaterial in the equatorial plane of the star (Struve 1931).Keller, Wood, & Bessell (1998) also found the Be popula-tion in NGC 330 to peak strongly at magnitudes corre-sponding to the main-sequence termination, whereas the Bestars in the field display a normal luminosity function. Theyconcluded that the unusual concentration of Be stars inNGC 330 can be explained by a combination of evolution-ary enhancement (due to an increase in the stellar rotationalvelocity during main-sequence evolution) and a distributionof intrinsically high rotational velocities within the cluster.Keller, Bessell, & Da Costa (2000) used the WFPC2 onHST to investigate the stellar population further. Theirimages reached a limiting magnitude of mF555W ’ 20. Theyobtained a color-magnitude diagram (CMD) that displays apeculiar feature: a clump of hot and luminous stars sepa-

rated from the main sequence (MS), with a bluer color(F160W F555W < �3:1). In lower-mass clusters, thisregion of the CMD typically contains blue stragglers, andKeller et al. (2000) concluded that there might be blue strag-glers even in the case of NGC 330. They also noted a finitewidth in the MS, larger than the photometric uncertainties,and they indicated as a possible explanation the combina-tion of differential reddening across the cluster with residualuncertainties in the position-dependent flat field.

Since our main interest lies in the characterization of thelow end of the IMF, we proposed to revisit NGC 330 withthe WFPC2 on HST and obtained much deeper imagesdesigned to sample the stellar IMF down to the lower limitof �1 M�. We present the results of this imaging programin the following sections of this paper.

3. OBSERVATIONS AND DATA REDUCTION

Multiple images of NGC 330 were obtained in 1999 Sep-tember using the WFPC2 on board the HST. Several expo-sures were taken through the filters F555W and F814W.These filters are described in detail by Biretta (1996) andclosely resemble the Johnson V and I filters in their photo-metric properties.

The images were dithered to allow a better removal of thedetector blemishes and to improve both point-spread func-tion (PSF) sampling and the photometric accuracy by aver-aging over the flat-field errors. For each filter, a ditheredpair of 350 s exposures shifted by 0>5 was obtained, in addi-tion to a single short (10 s) exposure to obtain photometryfor the brighter sources. The cluster was centered in theplanetary camera (PC), which has a field of view (FOV) of3500 � 3500 with an effective plate scale of 0>045 pixel�1. Theother three wide field (WF) chips observed the outer regionsof the cluster and flanking fields of the SMC with the samefilter configuration but a larger FOV of 7500 � 7500 chip�1

and a plate scale of 0>1 pixel�1. All the images were takenwith a gain of 7 e� ADU�1. The journal of all observationsis provided in Table 1.

The data set was processed adopting the standard STScIpipeline, which uses calibration observations and referencedata such as bias, flat-field, and dark frames that are con-stantly updated by the STScI WFPC2 team to track any

TABLE 1

Journal ofHST +WFPC2 Observations

Proposal Date Filter

Exposure Time

(s) ImageName

NGC 330 (�: 00h56m22 906, �:�72�28013>22; J2000.0)

8134 ......... 1999 Sep 4 F555W 350 U5AY1001R

8134 ......... 1999 Sep 4 F555W 350 U5AY1002R

8134 ......... 1999 Sep 4 F555W 10 U5AY1003R

8134 ......... 1999 Sep 4 F814W 350 U5AY1004R

8134 ......... 1999 Sep 4 F814W 350 U5AY1005R

8134 ......... 1999 Sep 4 F814W 10 U5AY1006R

NGC 330 (�: 00h56m22 910, �:�72�28013>51; J2000.0)

8134 ......... 1999 Sep 4 F814W 350 U5AY1007R

8134 ......... 1999 Sep 4 F814W 350 U5AY1008R

8134 ......... 1999 Sep 4 F555W 350 U5AY1009R

8134 ......... 1999 Sep 4 F555W 350 U5AY100AR

276 SIRIANNI ET AL. Vol. 579

changes in the performance of the camera and its detectors.The basic steps of the calibration include the correction forthe errors introduced by the analog to digital conversion,the removal of the bias level and bias pixel-to-pixelvariations, the dark subtraction, the flat-field application,and, for only the short 10 s exposure, a shutter shadingcorrection.

Pixels with elevated dark current were corrected ormarked as bad with the task WARMPIX in IRAF. Finally,in order to preserve the precision of our photometry acrossthe entire FOV, two additional corrections were applied tothe images after pipeline processing. First, a correction forthe 3% reduction in the quantum efficiency due to a manu-facturing pattern defect known as ‘‘ the 34th row error ’’(Anderson & King 1999); and, second, the correction forthe normalization of pixel area due to the geometric distor-tion removal caused by flat-fielding the images (Holtzmanet al. 1995).

The set of dithered observations was processed followingthe recipe of Gonzaga et al. (1998) using the STSDASDITHER-II package in IRAF. First, the individual imageswere cosmic-ray rejected and stacked into a single image per

filter per dithered position. Then, in order to measure theshifts and rotation between these images, they were geomet-rically corrected and resampled using the DRIZZLE task.This method applies a cross-correlation–based technique toeach of the three wide-field chips and averages the resultsfrom each chip to determine a better estimate of the overallimage offset with respect to the reference image. Once theshifts were measured, the images were aligned and com-bined using the DRIZZLE algorithm, which corrects forgeometric distortion and produces a final output imagesampled on a finer grid than the input images. The final pixelscale is 7/10 the original PC pixel, which translates into0>03185. Figure 1 shows the final combined image of NGC330 in the F555W filter. The entire field is shown (PC +WFchips) together with an expanded view of the PC frame, toshow at high resolution the inner regions of the cluster.

4. THE PHOTOMETRY

The strength of our proposed approach to the determina-tion of the IMF in the clusters of the sample rests on theconsistency of our data reduction and analysis. Therefore,

Fig. 1.—Final combinedWFPC2 image of theNGC 330 cluster in the F555Wfilter. Insert: Portion of the FOV of the Planetary Camera (34>4� 3400), wherethe cluster is centered. The orientation on the sky is indicated in the image.

No. 1, 2002 LOW END OF IMF IN YOUNG SMC CLUSTERS. II. 277

for NGC 330, we have followed the same methodology usedfor R136 (S00) and used the PSF fitting and aperturephotometry routines provided within DAOPHOT. Giventhe different characteristics of crowding of the PC and WFfields in the long and short exposures, we have opted to per-form PSF fitting photometry on the images generated by thelong exposures and aperture photometry in all shortexposures.

First, we performed a statistical study on a sample ofbona fide stars, carefully selected by eye in the F555Wimages, so as to define an appropriate range for the parame-ters that DAOFIND uses as selection criteria. The twoparameters that permit one to discriminate between truestars and spurious objects are the roundness, which allowsus to eliminate objects which are too elongated along rowsor columns, and the sharpness, which can be used to discardobjects whose profile largely differs from aGaussian.

From our sample of 215 stars in the PC frame, we foundmean values of 0:63� 0:08 and 0:14� 0:23 for the sharpnessand roundness, respectively. In the same way, from the sam-ple of 139 stars in the WF frames, we found mean values of0:75� 0:61 and 0:04� 0:30, respectively.

We then ran DAOFIND on our data and conservativelyset the detection threshold at 5 � above the local back-ground level. We excluded any object with sharpness androundness parameters exceeding�3 times the standard devi-ations of the average bona fide values. We carefully exam-ined the rejected objects and found that almost all of themwere noise peaks associated with hot columns, diffractionspikes, or highly saturated stars, with a small number beingisolated hot pixels and extended objects.

The list of stars detected in the F555W image was thenused to identify the stars in the F814W image: 3536 starswere found to be common to both filters in the PC and13,191 in the three WF chips. In the case of the short expo-sures, we combined the two single images in each filter toremove cosmic rays and ran the DAOFIND routine on theoutput image: 585 stars were found to be common toboth frames in the PC and 1129 in the three WF chips,respectively.

We then performed core aperture photometry on theresulting images. With this technique (De Marchi, Paresce,& Ferraro 1993), the stellar flux is measured in a very smallaperture 0>1 in size for the PC and 0>2 for the WF chips.The background is measured in an annulus located between5 and 10 pixels (0>16 0>32Þ from the PSF peak in the PCframe and between 3 and 5 pixels (0>3 0>5) in theWF chips.The background annulus is deliberately chosen in closeproximity to the star peak to ensure high-precision photo-metric measurements in the presence of crowding. Theannulus itself contains a nonnegligible amount of light fromthe PSF wings, which is subtracted together with the back-ground but needs to be estimated carefully and added backto the photometric measurement of the stellar source. Thisaperture correction is derived by assessing how the PSFencircled energy varies as a function of radial distance fromthe central peak. For each image, we have measured theencircled energy profile for a number of isolated stars andhave used these measurements to correct the fluxes derivedwith aperture photometry.

As previously mentioned, we used a PSF fitting techniqueon the images generated from the longest exposures. Weconstructed a sample PSF by combining seven moderatelybright and isolated stars uniformly distributed over the

image. We then performed PSF fitting photometry on theDRIZZLEd PC frame. As expected, the brightest stars weresaturated in the longest exposures and were discarded dur-ing the detection process. However, the magnitude of thesestars had been measured in the short integration frames andadded to the list of unsaturated objects. The final databasecontains the photometry obtained from the short exposures(stars with mF555W < 18:0) and the long exposures(mF555W � 18).

The absolute photometric calibration was performed byconverting the magnitudes of the individual stars to an aper-ture of radius 0>5 and by applying the transformation fromthe on-orbit system to the WFPC2 photometric system pro-vided by Holtzman et al. (1995). The zero points used were22.48 for the F555W filter and 21.60 for the F814W band.The original zero points were provided for a gain of 14 e�

ADU�1, to which we added the correction for the differentgain adopted in these observations (7 e� ADU�1; Holtzmanet al. 1995).

In Figures 2 and 3, we show the photometric errorsassigned by DAOPHOT to all our measurements for thelong and short exposures for the PC and for the WFs,respectively. For our study, we discarded all stars with anassociated error larger than 0.2 mags in both filters. Withthis limitation, our final photometric list contains 3052 starscommon to both filters in the PC frame, down to a limitingmagnitude mF555W ¼ 24:9 and 12,674 stars in the FOV ofthe three WF chips, down to mF555W ¼ 25:7. We finallyapplied corrections for the charge transfer efficiency and theshort versus long effect (see the Appendix). Although we donot include the full photometry database in this paper, thecomplete table is available in electronic form upon request.

5. THE COLOR-MAGNITUDE DIAGRAM

With the magnitudes obtained as described above, wehave constructed a CMD where mF555W is plotted as a func-tion of themF555W mF814W color for all the stars with photo-metric errors smaller than �0.2 mag in the combinedframes. Figures 4a and 4b show the CMD, referring to thePC andWF frames, respectively.

The brightest portion of the CMD of NGC 330 hasalready been extensively studied in previous papers. Thepresence of two well-detached clumps of red and blue super-giants made it problematic to understand the evolutionarystate of the most massive stars in the cluster (Caloi et al.1993) and to determine its age (Chiosi et al. 1995). Grebel,Roberts, & Brandner (1996), using spectroscopic and pho-tometric data, investigated the nature of the blue giants andsuggested that they represent a mixture of rapidly rotatingB/Be stars and blue straggler stars (BSSs) formed by theinteraction of binary systems. The interpretation of thesestars as a hot extension of the main sequence (MS) led to asystematic underestimate of the cluster age (Grebel et al.1966). The presence of BSS is also responsible of much ofthe previously inferred age spread (Chiosi et al. 1995), whichhas been largely overestimated. The rich population of Bestars, clearly visible as a strip at mF555W mF814W � 0:1between mF555W ¼ 18 and 15.5 parallel to the MS, hasrecently been investigated by Hummel et al. (1998) and Kel-ler et al. (1998, 2000). The BSSs are not, however, as easilydiscernible on our CMD as they are in the UV data of Kelleret al. (2000) because their colors saturate in the visible and

278 SIRIANNI ET AL. Vol. 579

near-IR bands (thence the uncertainties on the age men-tioned above).

The MS turnoff is located at mF555W ’ 16 mag. The simi-larity between the CMDs plotted in the two figures indicatesthat NGC 330 extends well beyond the PC FOV: the richpopulation of Be stars and the evolved red supergiants seenin the CMD of the PC are also present in the CMD of theWF. Moreover, the red giant branch visible in the PC dia-gram shows that the contamination by the surroundingLMC field population in the inner regions of the cluster can-not be ignored and must be taken into account when meas-uring the luminosity function (LF).

An interesting feature of the cluster MS is its nonnegli-gible broadening at all magnitude levels. Keller et al. (2000)have addressed this issue in detail and conclude that,although the broadening could be caused in part by flat-fielding errors in their data (WFPC2 images in the F160Wand F555W bands), it is most likely intrinsic to the clusterand/or due to differential reddening. They suggest both thepresence of binary systems containing a red supergiant andanMS star as well as patchy absorption as possible explana-tions. Comparing Figures 4a and 4b with one another, how-ever, reveals a marked increase of this effect with distancefrom the center: the spread in color around theMS ridge linein Figure 4a at 17 < mF555W < 20 is � ’ 0:06, whereas out-side of the core (Fig. 4b), it grows to � ’ 0:10 over the samemagnitude range. In the innermost regions, the extent of thespread is consistent with the error in the colors of the stars

of up to �0.25 mag stemming from the internal uncertaintyin our photometry alone (see Figs. 2 and 3). It is also consis-tent with the contamination due to SMC field stars, whosecolor spread can easily be judged from the shape of the redgiant branch shown in Figure 4b. In the WF frames, how-ever, the effect is even larger, but the photometric error issmaller than in the innermost regions at the same magnitudelevel due to the lower level of crowding. The area covered bytheWF fields is also larger (’4.65 arcmin2 vs.’0.32 arcmin2

of the PC field), thus making patchy absorption resultingfrom gas not yet dispersed the most likely explanation of theobservedMS broadening.

The amount of reddening toward NGC 330 is still anunsettled matter. Estimates range from EðB�VÞ ¼ 0:03(Carney, Janes, & Flower 1985) to EðB�VÞ ¼ 0:12 (Bessell1991). Reddening estimates from the most recent spectro-scopic observations seem to converge to an intermediatevalue (Caloi et al. 1993). In light of the possibility of patchyabsorption, however, all these numbers have to be takenwith care. For the purpose of measuring the MF, we adoptin the following a reddening correction of AV ¼ 0:16, whichis based on the EðB�VÞ ¼ 0:06 value of Gonzalez & Wal-lerstein (1999) and on the SMC extinction curve withR ¼ AV=EðB�VÞ ¼ 2:7 presented by Bouchet at al. (1985).Because we measure our LF in magnitude bins of DV ¼ 0:5mag, our determination of the MF is not significantlyaffected by the current uncertainties on the actual value ofAV .

Fig. 2.—Photometric errors assigned by DAOPHOT to all our measurements in the PC frame in long exposures (top) and short exposures (bottom). Wehave discarded all stars with an associated error larger than 0.2 mag in both filters.

No. 1, 2002 LOW END OF IMF IN YOUNG SMC CLUSTERS. II. 279

Fig. 3.—Same as in Fig. 3 but for theWFs frames

Fig. 4.—Observed color magnitude diagram of NGC 330 for all the stars with associated photometric error smaller than 0.2 in both filters. Objects 4252and 12708 are found in the image with the PC (left) andWFs (right), respectively. A reddening vector forAv ¼ 0:5 is also shown.

6. THE AGE OF NGC 330

An estimate of the age of NGC 330 can be obtained bycomparing the CMD of Figure 4a with the colors and mag-nitudes predicted by theoretical isochrones. Most models inthe recent literature provide isochrones in the Johnson/Cousin magnitude systems. In the following, we thereforemake the simplifying assumption that the HST F555W andF814W filters coincide in practice with the standard V and Ibands. In fact, as Holzman et al. (1995) show, the differencebetween F555W and F814W and their corresponding John-son/Cousin bands never exceeds �0.03 mag for stars in therange �0:5 < V � I < 1 spanned by our CMD. These dif-ferences are smaller than our photometric errors, and, there-fore, they will not significantly decrease the accuracy of ourage determination.

In order to derive the age of the cluster, we used (as inS00) isochrones constructed from the stellar evolutionarymodels provided by Siess, Fiorentini, & Dougados (1997).These tracks offer the choice of different metallicities:(Z ¼ 0:02, 0.04, 0.005). We adopted Z ¼ 0:005 for thisstudy, which is as close as possible to the value of Z ¼ 0:003estimated for NGC 330 (Hill 1999). However, the tracks ofSeiss et al. (1997) do not consider stars more massive than7 M�, yet the CMD clearly shows the presence of evolvedstars that can be more effectively used to estimate the age ofthe cluster with higher precision than simple isochrone fit-ting to the MS alone. To take advantage of red supergiantstars, and in order to assess the uncertainty affecting the agedetermination, we decided to also use the evolutionarytracks of Schaerer et al. (1993; published in the WFPC2photometric system by Lejeune & Schaerer 2001). In this

case, the choice for the metallicity is Z ¼ 0:004, slightly dif-ferent from theZ ¼ 0:005 of Siess et al. (1997).

Isochrone fitting for the sole purpose of age determina-tion has been restricted to the CMD of the innermost clus-ter regions (i.e., the PC field, or r < 2500), where the effectsof background/foreground contamination are less severe.To translate absolute magnitudes into the apparent ones,we assumed a distance modulus ðm�MÞV ¼ 18:85 andreddening AV ¼ 0:16, EðB�VÞ ¼ 0:06 as appropriate forNGC 330 (see x 5). Comparison with the isochrones sug-gests an age of 30 Myr, in excellent agreement with thelogðageÞ ¼ 7:5 determined by Keller et al. (2000). This isshown graphically in Figure 5, where the isochrones thatbest fitted the central cluster regions (Fig. 5a) are alsotraced on the CMD of the stars in the outskirts (Fig. 5b).The dotted line shows the model of Siess et al. (1997),extending up to the MS turnoff. The solid line shows themodel of Schaerer et al. (1993), which reaches the red giantbranch. Because of the errors in our photometry, the age ofNGC 330 estimated in this way can only be consideredaccurate to within �2 Myr. In Figure 5b, we also plot theisochrones of Schaerer et al. (1993) that seem to best fit thered giant branch of the older stellar population in theSMC. The age that we infer in this way is of the order of1–2 Gyr.

The isochrone that best fits the MS of the innermost clus-ter regions can also be used to derive the mass-luminosity(M-L) relation that we need to convert the LF into an MFas we explain in x 8 below. Since we are interested in the MFof MS stars, we decided to use the models of Siess et al.(1997) for compatibility with the determination of the MFin R136 (S00).

Fig. 5.—Observed color-magnitude diagram ofNGC 330 (left panel: PC, right panel: WFC) with the 30Myr isochrones from Schaerer et al. (1993; solid line)and Siess et al. (1997; dotted line). The 2Gyr isochrone in the right panel is from Schaerer et al. (1993; dashed line).Black dots: Original positions of the Be stars,which have been relocated to theMS after correction for the excess flux due to the circumstellar envelope (see x 8).

LOW END OF IMF IN YOUNG SMC CLUSTERS. II. 281

7. THE COMPLETENESS AND THEPHOTOMETRIC ERRORS

Before proceeding to the derivation of the IMF, it is alsonecessary to establish the completeness of our data. Forboth filters and for both the short and long combinedimages, the completeness has been assessed with the stan-dard procedure. Artificial stars were inserted into the imagefor each half-magnitude bin. The number of artificial starswas less than 10% of the total number of stars in order notto severely affect the crowding in the region considered. Theartificial stars have been retrieved using the same selectioncriteria of sharpness and roundness adopted in this work.Stars with a photometric error larger than 0.2 mag havebeen discarded. This test was repeated 10,000 times.

The results of the test are summarized in Figures 6a and6b, where we have plotted the completeness factor in eachfilter as a percentage of the stars successfully retrieved ver-sus the total number of stars artificially added in the PC(inner region) and the three WFs (outer regions). As can beseen in Figure 6a, the completeness is worse in the brightestmagnitude bin, where saturation effects prevent the detec-tion of other very bright objects, and toward the faint end,where signal-to-noise ratio (S/N) effects start to dominate.The drop in completeness at mF555W ¼ 18:2 for the innerregion of the cluster corresponds to the brightest magnitudebin from the long-exposure images, which is affected by thecontamination of saturated stars. The completeness is betterthan 50% down to mF555W ¼ 23:5, mF814W ¼ 22:8 for thephotometry in the center and down to mF555W ¼ 24:1,mF814W ¼ 23:2 for the outer regions.

8. THE LUMINOSITY AND MASS FUNCTIONS

The CMDs plotted in Figures 4a and 4b and the photo-metric completeness estimated as described above are thebasis for deriving the LF and, from it, the MF. While, in

principle, determining the LF would simply require thecounting of the number of stars along the cluster MS, inpractice, this task is complicated by a number of effects.

First and foremost, as Figure 4b demonstrates, the con-tamination due to stars in the field of the SMC is rathersevere. We have tackled this issue by adopting as back-ground the most external region of our FOV. The coreradius of NGC 330 is 9>5 (Vallenari, Ortolani, & Chiosi1994) and, as such, it is completely contained within the PCFOV (3400 � 3400). Its surface brightness profile extendsbeyond 8000, yet still within the three adjacent WFs FOV.We can, therefore, study the mass distribution along theentire extension of the cluster. We will assume that the LFmeasured in the most external region, between 9500 and 12500

from the center, represents an estimate of the LF of thebackground field.

Second, to obtain a reliable estimate of the LF for MSstars, we must make sure that we count only the MS stars,yet all of them. This requires that we properly place the Bestars on the MS at a luminosity that would correspond tostars of their mass. Because Be stars are all clumped togetherin the CMD (see Fig. 4), there is no doubt that they allbelong to the cluster. We thus need not apply the field starcorrection to them, and instead, we can directly place themonto the MS at the magnitude that corresponds to theirmass. The latter translation has already been derived byKeller et al. (2000) for most Be stars in our sample. Theseauthors have calculated a correction to apply to the effectivetemperature and luminosity of Be stars in order to removethe effect of the circumstellar reddening on the underlyingstars. We identified in our FOV the Be stars found by Kelleret al. (2000) and used their grid of colors and bolometriccorrection to assign to each Be star, with the corrected Teff

and L=L�, the appropriate photometry in the F555W andF814W bandpasses. In this way, we are able to place 99 Bestars on theMS position they would have without the excessflux due to their circumstellar envelope. Be stars that have

Fig. 6.—Completeness curves for F555W (left) and F814W (right). Horizontal lines show the 50% completeness level.

282 SIRIANNI ET AL. Vol. 579

been properly placed on the MS have been marked with ablack dot in Figures 5a and 5b. A handful of stars thatoccupy the Be star locus in the CMD (4 in the PC and 10 inthe WF frames) is not in the Keller et al. list due to a differ-ent FOV and, therefore, have not been corrected and werenot included in the LF. Their absence has negligible influ-ence on our conclusions.

We notice here that patchy absorption (that we suspect tobe at the origin of the observed MS broadening) has no sys-tematic effects on the slope of the LF: it merely increases thephotometric uncertainty and therefore smooths the LF as itallows stars of a given magnitude to spread into neighboringmagnitude bins. Such an effect is also not relevant for theconclusions of this work.

Due to the richness of our fields, we can study the varia-tion of the LF with radial distance. After conversion to anMF, we will be able to analyze in detail the radial distribu-tion of the stellar masses within the cluster to determinewhether the star forming process is uniform or changes sig-nificantly in the core.

We have chosen as cluster center the point of maximumstellar density in the frame, but our results are not signifi-cantly affected by a small uncertainty of its exact location.We have then traced five annuli between 000 and 2500, eachspanning 500, and 10 more annuli 1000 wide between 2500 and12500 from the center. We have extracted the stars whoseposition is defined by each annulus and constructed the LF.The artificial star experiments allow us to assess the com-pleteness within each annulus.

In Figure 7, we show the results of this analysis for severalrepresentative annuli. In the left column, we plot the CMDsderived from stars in our photometric catalog whose posi-tion is defined by each annulus. The LF after photometriccompleteness correction and relocation of the Be stars in theappropriate magnitude bin are shown as a solid line in Fig-ure 7 (second column). The vertical dashed lines in thoseplots indicate the magnitude level for 50% completeness,whereas the dotted lines show the LF before correction forcompleteness.

Converting the LF into an MF requires the knowledge ofthe ML relation to translate a given observed magnitudeinto a mass. To this end, we have used Siess et al.’s (1997) 30Myr isochrone (from 0.6 to 7 M�) as plotted in Figure 5aand discussed in x 6 above.

The derivative of the ML relationship is calculated byusing a spline interpolation. Any other fit to the isochrone(e.g., a cubic spline) would considerably increase the uncer-tainty of the derivative (Silvestri et al. 1998) and, as a result,of the MF as well. The MFs determined in this way in eachannulus are plotted in Figure 7 (middle column). Filled sym-bols mark the portion of the data with photometric com-pleteness greater than 50%, which we use for our analysis;whereas below the mass limit set by 50% completeness (ver-tical dashed lines), the data can no longer be reliably inter-preted. The dotted lines show theMFs derived from the LFsbefore correction for photometric incompleteness and provethat the latter is significant only at very lowmasses.

Comparing the slope of the LFs and MFs in all theannuli, we noticed sharp steepening with increasing radialdistance up to ’ 8000 from the center. At larger distance, theslopes of the MF appear constant, ��3.5, in very goodagreement with the value of �3:7� 0:5 for the SMC fieldstar population of Massey et al. (1995). This gives us someconfidence that the most external annuli are populated

mostly by background/foreground stars. Therefore, wehave constructed the LF in the region between 9500 and 12500

and subtracted the result from that measured in all otherannuli.

The panels in the fourth column in Figure 7 show the LFof each annulus corrected for completeness (dashed line),the background LF (dash-dotted line), and the final LF aftercorrection for field star contamination for photometriccompleteness and replacement of the Be stars in the appro-priate magnitude bin (solid line). The errors on the LFsreflect the Poisson statistics of the counting process (corre-sponding to the square root of the number of stars detectedin each bin, suitably corrected for completeness) as well asthe uncertainty stemming from the artificial star tests. Thetwo errors are combined quadratically and included in thefigure as�1 � error bars.

The progression of LFs in Figure 7 shows a noticeablesteepening with increasing radial distance. The total numberof objects in each annulus is, of course, not constant. If,however, all LFs were renormalized so that they have thesame number of stars atm555 ’ 22, one could notice that thesteepening of the LF with radial distance is due to the lackof more luminous (massive) stars in the periphery ratherthan to the lack of fainter stars in the core. This issue isaddressed in more detail in the discussion.

We finally calculated the MF from the corrected LF andplotted the results in the last column of Figure 7. The shapeof the MFs measured at different radial distances is consis-tent in that it features a monotonic increase with decreasingmass for stars more massive than�1 M�. The error bars onthe MFs come directly from those on the LF, as we haveassumed that the M-L relation is not affected by the statisti-cal uncertainty. A flattening could be indicated by the databelow that limit as is the case of R136 (S00). Yet, becausethe photometric completeness in that mass range is alwaysless than 50%, we cannot assess the reliability of this intrigu-ing conclusion, which would require deeper photometry.Consequently, we will concentrate on the mass range above�1 M�.

9. DISCUSSION AND CONCLUSIONS

The most remarkable feature of the MFs shown in Figure7 (right column) is the pronounced increase of the slope withincreasing distance from the cluster center at r > 300. Thistrend confirms the marked steepening already apparent inthe LFs. The MFs are well fitted by a power-law distribu-tion in the range 1 6 M� spanned by our observations asshown by the solid lines. The slopes were obtained with aweighted least-squares 1 degree polynomial fit. Only meas-urements with a finite error were considered for the fit (Fig.7, right column, solid circle). At an age of �30 Myr, NGC330 is young enough that its present-day MF should reflectthe cluster’s IMF, at least for stars that still are on their MS(i.e., for the objects less massive than � 8 M�). Figure 8shows the radial distribution of the index C of the IMF (Sal-peter’s 1955 IMF, � ¼ �1:35, is indicated in the upper rightcorner for reference). In the core (000–3000), with � ’ �1:5,the MF shows little variation, and it is very similar to Sal-peter’s IMF but becomes considerably steeper than the lat-ter in the periphery r > 3000.

A careful inspection of Figure 7 reveals that the steepen-ing of the MF with distance is due to the lack of massivestars in the periphery rather than to the excess of low-mass

No. 1, 2002 LOW END OF IMF IN YOUNG SMC CLUSTERS. II. 283

Fig.7.—

Inthiscompositediagram

(Figs.7aand7b),each

rowcorrespondsto

aspecificannulusaroundthecenteroftheNGC330cluster.F

romleftto

right:(a)Observed

CMD.(b)Observed

LF(dotted

line)

andcompletenesscorrectedLF(solidline).Verticallong-dashed

line:Magnitudelevelfor50%

completeness.(c)Mass

function.Filledsymbols:Portionofthedata

withphotometriccompletenessgreaterthan

50%

onwhichwebase

ouranalysis,whereasbelowthemasslimitsetby50%

completeness(verticaldashed

line),thedata

cannolonger

beconsidered

reliable.D

otted

lines:M

Fderived

from

theLFbefore

correc-

tionforphotometricincompleteness.(d

)Luminosity

functionbefore

(dashed

line)andafter

(solidline)thesubtractionofthefieldLF(dash-dotted

line).(e)MFderived

from

thecompletenesscorrectedLFbefore

(dashed

line)andafter

(filled

circles)thebackgroundsubtraction.F

illedcircles:MeasurementsusedforthefitoftheMFslope.Open

circles:Pointswithphotometriccompletenesslessthan50%

orwithanonfin-

iteerror.

284

Fig.7.—

Continued

285

objects there: the projected density (stars=D logm=pc2) ofthe latter decreases by a factor of �4 between the core andthe outskirts, whereas the former are depleted by a factor of20 (measured at�1.5 and�5:0 M�, respectively).

Concerning the actual slope of the MF, however, werecall that the presence of binaries has been ignored as isoften the case in studies of this kind. Obviously, we knowthat most stars in young clusters are binaries and when thelatter are included, the MF becomes steeper as a whole(Kroupa 2000). This does not modify any of the conclusionsthat follow, but the index of the MF (C) must be scaledaccordingly: it is thus well possible the MF is never as flat asindicated at the top of Figure 7.

The observed mass segregation (i.e., the preponderanceof massive stars in the core) can be understood the result oftwo mechanisms: (1) internal dynamical evolution and (2)local variations in star formation. In the following, we dis-cuss both effects.

Through repeated stellar encounters the stars in a clusterapproach energy equipartition, inducing more massive starsto give part of their kinetic energy to lighter objects and tosink deeper into the cluster’s potential well. Light stars, onthe other hand, can diffuse outward, passing onto larger,more energetic orbits (see, e.g., Spitzer 1987). The net resultof these effects is a flattening of theMF in the central regionsand a steepening in the periphery, a phenomenon calledmass segregation. In order for two- and three-body encoun-ters to allow enough objects to exchange energy and, thus,to impart a noticeable modification of the MF, a time of theorder of the half-mass relaxation time is required. For NGC330, the latter is about an order of magnitude longer thanthe cluster’s age, namely, �5� 108 yr (Kontizas 1984) ver-sus an estimated cluster age of �3� 107 yr. Although it ispossible that dynamical evolution has played some role inreshaping theMF in the innermost cluster regions where the

density is high and the relaxation time is short, it is neverthe-less quite unlikely that internal dynamical evolution mighthave had enough time to act on the stars throughout thewhole cluster and produce the effects that we see in Figure 7,particularly the shortage of massive stars in the outerregions.

In the scenario recently proposed by Bonnell et al.(2001b), however, mass segregation is a natural conse-quence of the mechanism of star formation itself, in such away that forming clusters are already segregated at birth. Asimilar hypothesis had previously been suggested by Larson(1982) and addressed observationally by Sagar et al. (1988).As a result of the competitive accretion that occurs in stellarclusters, stars forming near the center would have a higheraccretion rate, as the latter depends on the local gas density.The simulations of Bonnell et al. (2001a) show that,although low-mass stars form equally well throughout thecluster, more massive objects are almost exclusively rele-gated to the central regions (see their Fig. 4). As a conse-quence, they also predict that the MF should beconsiderably flatter in the core than it is in the periphery,much in the same way we see in Figure 8.

The prevalence of more massive stars in young clustercores has already been observed by Hillenbrand (1997) andHillenbrand & Hartmann (1998), who studied the OrionNebula Cluster (ONC) and found that the most massivestars are more concentrated than lower-mass objects. AsHillenbrand (1997) points out, however, the age distributionis also biased, with the youngest stars occupying the inner-most regions. Thus, because of the young age of the ONC(of order �1 Myr) and the considerable spread of the latter(a fewMyr; see Hillenbrand 1997 for a discussion), it is diffi-cult to assess whether the observed concentration of massivestars is due to mass segregation or to age segregation and tounderstand which of the two processes dominates. On theother hand, with an age of 30� 2 Myr and an age spreadnot larger than the uncertainty on the age (see x 6), NGC330 could offer a clear-cut opportunity for investigating thenature of the segregation of massive stars in a rather youngcluster.

Kontizas et al. (1998) present a good case for mass segre-gation in the intermediate-age LMC open clusters SL 666and, less convincingly, in NGC 2098. Although not asdetailed and robust as the ones that we present in Figure 8,their LFs of SL 666 show a clear steepening with increasingradial distance. These authors discuss at length the dynami-cal state of SL 666 and conclude that, in spite of an age ashigh as �1.2 Myr, this cluster is not likely to have under-gone extensive dynamical evolution and that the observedmass segregation must be primordial. A similar conclusionis reached by Fischer et al. (1998), who study NGC 2157 inthe LMC and detect a modest amount of mass segregation(although, within the quoted uncertainties, all their MFs areconsistent with the same � ’ �1:1� 0:45 index). Theyattribute this to the properties of the star formation processrather than to the dynamical evolution of the cluster, as thetwo-body relaxation timescale at the half-mass radius ofNGC 2157 is large (trh ’ 1� 109 yr) when compared withthe cluster age (�1� 108 yr). Another example of a young(3 4 Myr), segregated cluster is NGC 6231, studied byRaboud & Mermilliod (1998). These authors notice thatonly the most massive stars (> 25 M�) are centrally concen-trated, whereas the distribution of less-massive objects ismore uniform. They thus speculate that primordial mass

Fig. 8.—Slope of the IMFs constructed in equally spaced annuli are plot-ted as a function of distance from the cluster center. Dashed lines: Valuescorrected only for completeness, to be compared with the values correctedfor completeness and field star contamination (solid line).

286 SIRIANNI ET AL. Vol. 579

segregation might be completely washed out when the mostmassive stars evolve and disappear after�10Myr and causea violent relaxation, which could, in principle, also disruptthe cluster. None of these prescriptions seem to apply to thecase of NGC 330, where, as we show in Figure 8, mass segre-gation is still present at an age of �30 Myr, and the degreeof central concentration decreases proportionally with massat all levels, as one can see from the smooth steepening oftheMFwith increasing radial distance shown in Figure 7.

A body of observational evidence is mounting that sug-gests that the mass distribution at birth, namely the IMF, isa sensitive function of the location in which stars form. Dueto the uncertainties which have accompanied the various

claims of mass segregation detections in young clusters, lit-tle theoretical work has been done so far. Yet a robust andcomprehensive model of star formation should now incor-porate and treat primordial mass segregation as a con-straint, as suggested by Bonnell et al. (2001a).

We wish to thank Stefano Casertano for several helpfuldiscussions and for making his data available for our com-parison. Support for this work was provided by NASAgrant G0-8134-01-97 from the Space Telescope ScienceInstitute, which is operated by the Association of Univer-sities for Research in Astronomy, Inc., under NASA con-tract NAS5-26555.

APPENDIX

THE CTE CORRECTION

Shortly after the installation of WFPC2 on HST, it was discovered that the camera suffered from variable charge transferefficiency (CTE). In 1994March, a 10%–15% gradient in the photometric response of the CCDs along the column of each chipwas discovered in all four of the WFPC2 CCDs. This effect is due to the partial loss of signal when charge is transferred downthe chip during the readout, with the consequence that stars at higher row numbers appear fainter than they would if they wereat low row numbers (Holtzman et al. 1995). The main cause of CTE degradation is the continuous radiation damage thatoccurs for space-borne CCDs such as theWFPC2. A significant reduction to the CTE effect was achieved by cooling down thefour CCDs from �76�C to �88�C. The new temperature became operational on 1994 April 23, and the CTE stabilized at a�4% level.

The first detailed characterization of the CTE degradation for WFPC2 was published by Whitmore & Heyer (1997), whomade available an algorithm to calculate the CTE correction as function of position in the chip, background counts, and targetcounts. Later it was discovered that CTE increased with time, and an additional temporal dependence was included in the cor-rection formulae (Whitmore 1998). By the end of the fifth year of operation, the charge loss for the faintest sources is quite dra-matic: a star with 20–50 counts superposed on a low background was observed to be�40% fainter at Y ¼ 800 than at Y ¼ 1.The most updated CTE loss characterization with a longer time baseline (through 1999 February) is given in Whitmore,Heyer, & Casertano (1999), where a final set of formulae has been developed to correct for CTE loss when performing stellaraperture photometry with radii of 2 pixels.

In addition to these studies from the STScI WFPC2 group, there have been three independent determinations of the CTEloss. Stetson (1998) derived a calibration that produced results similar to those of Whitmore & Heyer (1997) but without find-ing a significant time dependence. Later, Saha, Labheardt, & Prosser (2000) found a possible correction for their data that dif-fers significantly from previous results and does not include anX position dependence, a time dependence, or a dependence oncounts in the Y position. Finally, very recently, Dolphin (2000b) analyzed all the corrections available in the literature andconcluded that the differences between the various methods might be related to the packages used to do photometry and, inparticular, in the different treatments of the background. Using their new photometry package created specifically forWFPC2data (Dolphin 2000a), they produced corrections similar to those ofWhitmore et al. (1999) with a more updated temporal cov-erage (2000March).

Recently, Riess, Biretta, & Casertano (1999) have applied a new method to monitor CTE with a finer time sampling bymeasuring the counts in cosmic ray tails in dark calibration frames. Their results confirmed a steady growth of both X and YCTE loss with time since installation of the instrument on board theHST. The CTE loss time dependence is very significant intheY direction, causing time-independent CTE solutions (Stetson 1998; Saha et al. 2000) to be valid only for small intervals oftime.

The data we have collected for our program span over more than a year. We performed our photometry with DAOPHOTand eventually decided to use the correction formulae provided byWhitmore et al. (1999).

In addition to the CTE effect, a position-independent charge loss was detected by P. Stetson (Kelson et al. 1993) and calledthe ‘‘ long versus short anomaly ’’ because it was first seen as difference in magnitude of a star in short and long exposures. Thiseffect has been confirmed in a number of studies and finally characterized by Casertano &Mutchler (1998, hereafter CM98) asa nonlinearity defect strictly dependent on total counts in a stellar image and independent on exposure time or position on thechip. CM98 provide a general formula to correct the observed counts in a variety of conditions. The presence of such effectsand the need for a correction has been recently discussed by Saha et al. (2000), who find such position-independent correctiononly in one filter, and by Stetson (1998), who finds no clear evidence of this anomaly. Dolphin (2000b) attempted to determinea position-independent loss in his CTE solution but found no evidence and concluded that all the detections of the long versusshort error are ultimately the result of sky overestimation.

We extensively examined our data in both filters to find evidence to rule out the presence of a short versus long effect. Of theentire sample of stars, 442 objects were found in both short and long exposures. After the CTE correction, we still have somediscrepancy between the resulting magnitudes in the short and long exposures. Such a discrepancy is not completely removedby applying the short versus long correction. In particular, we found that the CM98 formula overcorrects the faintest sources.

No. 1, 2002 LOW END OF IMF IN YOUNG SMC CLUSTERS. II. 287

In order to assess whether this effect was due to different aperture parameters used for the photometry or was, instead, intrinsicin our data, we extracted the NGC 2419 images used by CM98 from the archive. After performing aperture photometry on thenew data using the same parameters and selection rules as in our NGC 330 data, we confirmed the overcorrection for sourceswith signal lower than 120 counts. We therefore compared our photometry with the tables used by CM98 for their calibration,kindly provided by S. Casertano. We found perfect agreement between the two independent reductions; however, we alsofound that by selecting stars with photometric error less than 0.2 mag in the CM98 tables, we derive the overcorrection up to0.25 mags for stars with only 20 counts. The correction formula provided by CM98 has been obtained by fitting all the photo-metric measurements in the FOV regardless of the photometric error associated with the measurements and, therefore, mightnot be the best solution in our study, since we are interested only in stars with solid photometric measurement.

We decided to adopt the CM98 correction formula, which properly corrects the magnitude discrepancy for signal levelhigher than 120 counts, and to apply a further correction to remove the residuals. Given the similarities of the effect on ourNGC 330 and the NGC 2419 data sets, we decided to calibrate the short versus long correction on the latter due to the highernumber of sources (�1900) available for the calibration. The correction formula, to be applied to the source flux (DN) afterCTE and the CM98 corrections, has been obtained with a least-squares sixth-degree polynomial fit whose coefficients are listedin Table 2. We did not find any significant color dependence and, therefore, applied the same correction to both the F555Wand F814W data.

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TABLE 2

Coefficient of the Polynomial Fit

Degree Coefficient

0............................... �0.3537422

1............................... 0.0093738

2............................... �8.5775988e�5

3............................... 3.9170822e�7

4............................... �9.4029811e�10

5............................... 1.12550963e�12

6............................... �5.2748128e�16

288 SIRIANNI ET AL.