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High-Mass, OB Star Formation in M51: Hubble Space Telescope Hα and Paα Imaging

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THE ASTRONOMICAL JOURNAL, 122 :3017È3045, 2001 December( 2001. The American Astronomical Society. All rights reserved. Printed in U.S.A.

HIGH-MASS, OB STAR FORMATION IN M51: HUBBL E SPACE T EL ESCOPE Ha AND Paa IMAGING

N. Z. SCOVILLE

Department of Astronomy, California Institute of Technology, 1201 East California Boulevard, Pasadena, CA 91125

M. POLLETTA

Department of Astronomy, California Institute of Technology, 1201 East California Boulevard, Pasadena, CA 91125 ; andObservatoire de CH-1290 Sauverny, SwitzerlandGeneve,

S. EWALD AND S. R. STOLOVY

Department of Astronomy, California Institute of Technology, 1201 East California Boulevard, Pasadena, CA 91125

AND

R. THOMPSON AND M. RIEKE

Steward Observatory, University of Arizona, Tucson, AZ 85721Received 2001 February 8 ; accepted 2001 July 17

ABSTRACTWe have obtained Ha and Paa emission-line images covering the central 3@È4@ of M51 using the

WFPC2 and NICMOS instruments on the Hubble Space Telescope to study the high-mass stellar popu-lation. The pixels provide 4.6È9 pc resolution in M51, and the Ha/Paa line ratios are used to0A.1È0A.2obtain extinction estimates. A sample of 1373 Ha emission regions is cataloged using an automated anduniform measurement algorithm. Their sizes are typically 10È100 pc. The luminosity function for the Haemission regions is obtained over the range to 2 ] 1039 ergs s~1. The luminosity function isL Ha\ 1036Ðtted well by a power law with dN/d ln LP L~1.01. The power law is signiÐcantly truncated, and noregions were found with observed above 2 ] 1039 ergs s~1 (uncorrected for extinction). (TheL Hamaximum seen in ground-based studies is approximately a factor of 5 higher, very likely because of theblending of multiple regions.) The extinctions derived here increase the maximum intrinsic luminosity toabove 1040 ergs s~1. The logarithmically binned luminosity function is also somewhat steeper(a \ [1.01) than that found from ground-based imaging (a \ [0.5 to [0.8)Èprobably also a result ofour resolving regions that were blended in the ground-based images. The two-point correlation functionfor the H II regions exhibits strong clustering on scales ¹2A, or 96 pc.

To analyze the variations of H II region properties the galactic structure, the spiral arm areasvis-a-viswere deÐned independently from millimeter-CO and optical continuum imaging. Although the arms con-stitute only 25% of the disk surface area, the arms contain 45% of the cataloged H II regions. The lumi-nosity function is somewhat Ñatter in spiral arm regions than in the interarm areas ([0.72 to [0.95) ;however, this is very likely the result of increased blending of individual H II regions in the arms thathave higher surface density. No signiÐcant di†erence is seen in the sizes and electron densities of the H II

regions in spiral arm and interarm regions. For 209 regions that had º5 p detections in both Paa andHa, the observed line ratios indicate visual extinctions in the range mag. The mean extinctionA

V\ 0È6

was mag (weighting each region equally), 2.4 mag (weighting each by the observed HaAV

\ 3.1luminosity), and 3.0 mag (weighting by the extinction-corrected luminosity). On average, the observedHa luminosities should be increased by a factor of D10, implying comparable increases in global OBstar cluster luminosities and star formation rates. The full range of extinction-corrected Ha luminositiesis between 1037 and 2 ] 1040 ergs s~1.

The most luminous regions have sizes º100 pc, so it is very likely that they are blends of multipleregions. This is clear based on their sizes, which are much larger than the maximum diameter (¹50 pc)to which an H II region might conceivably expand within the D3 ] 106 yr lifetime of the OB stars. It isalso consistent with the observed correlation (L P D2) between the measured luminosities and sizes ofthe H II regions. We therefore generated a subsample of 1101 regions with sizes ¹50 pc, which is madeup of those regions that might conceivably be ionized by a single cluster. Their extinction-correctedluminosities range between 2] 1037 and 1039 ergs s~1, or between two-thirds of M42 (the OrionNebula) and W49 (the most luminous Galactic radio H II region). The upper limit for individual clustersis therefore conservatively ¹1039 ergs s~1, implying s~1 (with no corrections for dustQLyc,up^ 7 ] 1050absorption of the Lyman continuum or UV that escapes to the di†use medium). This corresponds tocluster masses ¹5000 (between 1 and 120M

_M

_).

The total star formation rate in M51 is estimated from the extinction-corrected Ha luminosities to beD4.2 yr~1 (assuming a Salpeter initial mass function between 1 and 120 and the cycling timeM

_M

_),

from the neutral interstellar medium into these stars is 1.2 ] 109 yr.We develop a simple model for the UV output from OB star clusters as a function of the cluster mass

and age in order to interpret constraints provided by the observed luminosity functions. The power-lawindex at the high-luminosity end of the luminosity function (a \ [1.01) implies N(Mcl)/dMclPMcl~2.01.This implies that high-mass star formation, cloud disruption due to OB stars, and UV production arecontributed to by a large range of cluster masses with equal e†ects per logarithmic interval of cluster

3017

3018 SCOVILLE ET AL.

mass. The high-mass clusters (D1000 have a mass such that the initial mass function is wellM_

)sampled up to D120 but this cluster mass is ¹1% of that available in a typical giant molecularM

_,

cloud. We suggest that OB star formation in a cloud core region is terminated at the point that radi-ation pressure on the surrounding dust exceeds the self-gravity of the core star cluster and that this iswhat limits the maximum mass of standard OB star clusters. This occurs at a stellar luminosity-to-massratio of D500È1000 which happens for clusters º750 We have modeled the core collapse(L /M)

_, M

_.

hydrodynamically and have found that a second wave of star formation may propagate outward in aradiatively compressed shell surrounding the core star clusterÈthis triggered, secondary star formationmay be the mechanism for formation of the superÈstar clusters seen in starburst galaxies.Key words : galaxies : ISM È galaxies : spiral È H II regions È stars : early-type

1. PREAMBLE

Were it not for the small number of youthful, luminousstars and their ongoing genesis, much of the beauty, vigor,and evolution that is our fascination in the universe wouldbe lost to the distant past. Energizing and enriching thedisks of galaxies are the most massive stars of each gener-ation. In youth, they illuminate the bright nebulae that soelegantly outline the spiral arms of distant galaxies ; indeath, their cataclysmic supernovae replenish the inter-stellar environment with gases, enriched in heavy elements.From these ashes, the future generations of stars will arise.The springs of rejuvenation are giant molecular cloudsencompassing millions of solar masses of cold gas. Insidethese ponderous cocoons, the metamorphosis of stars takesplace in dusty darkness.

2. INTRODUCTION

High-mass OB stars play a critical role in the energeticsand dynamics of the interstellar medium (ISM) and in thehighest luminosity phases of galactic structure and evolu-tion, speciÐcally the luminous spiral arm and starburstactivity. Nevertheless, the mechanisms for formation of OBassociations remain poorly understood, and indeed, it isuncertain whether high- and low-mass stars are formed bythe same or di†erent processes (see, e.g., Larson 1986). H II

regions have long been a primary probe of high-mass starformation and the properties of OB star clusters (Hodge1987 ; Kennicutt, Edgar, & Hodge 1989 ; Rand 1992 ;Thilker et al. 2001). The hydrogen recombination line Ñuxes(e.g., Ha), or radio free-free continuum, are proportional tothe volume-integrated emission measure of the H II regions.The latter is proportional to the total Lyman continuumemission rate of the associated high-mass stars under theassumption that all ionizing photons are locally absorbed.Thus, the Ha luminosity of an emission region is indicativeof the Lyman continuum emission and hence the mass ofhigh-mass stars (correcting for extinction and assuming aninitial mass function [IMF]). The luminosity function of theH II regions can then be used to study the distribution ofmasses and birthrates of OB associations. M51 is a Rosettastone for studies of OB star formation on account of itsproximity, 9.6 Mpc (Sandage & Tammann 1975) ; its grand-design spiral pattern ; its orientation, i\ 20¡ (Tully 1974) ;and its abundant, dense ISM (Scoville & Young 1983).

M51 has been the focus of numerous ground-based Hastudies (Kennicutt et al. 1989 ; Rand & Kulkarni 1990 ; vander Hulst et al. 1988 ; Rand 1992 ; Rozas, Beckman, &Knapen 1996 ; Petit et al. 1996 ; Thilker et al. 2000), andthese studies have contributed much of what is currentlyknown regarding OB associations in other spiral galaxies.The distribution of H II region Ha luminosities wasapproximately Ðtted by a truncated power law with

N(L )d ln L P L~0.55 to L ~0.75Èon the high-luminosityend. In several previous Ha studies (e.g., Rand 1992 ; Thilkeret al. 2000), the luminosity function appears steeper in theinterarm regions than in the arms (exponent [0.93 to[0.96 vs. [0.48 to [0.72). The bright end (L Ha º 1038.8ergs s~1) of the H II region luminosity function contributesthe bulk (º50%) of the discrete H II region luminosity(Rand 1992), but approximately 55% of the total Ha lumi-nosity originates from di†use ionized gas (DIG), i.e., not indiscrete regions (Rand 1992 ; Greenawalt et al. 1998). Basedon ground-based imaging, it remains uncertain whether theDIG is made of blended low-luminosity regions or trulydi†use gas.

We have recently completed a comprehensive study ofM51 made up of Hubble Space Telescope (HST ; WFPC2and NICMOS) imaging of the optical and near-infraredcontinuum (Polletta et al. 2001) and the Ha and Paa emis-sion lines (this paper). Related millimeter-CO interferome-try has been presented by Aalto et al. (1999). The formerprobes the stellar disk, the dust, and the OB star formation,while the latter probes the dense, molecular ISM that is thebirth site of OB star clusters.

The resolution available with HST imaging cor-0A.1È0A.2responds to 4.6È9.3 pc. These sizes correspond to those ofindividual resolved Galactic H II regions (e.g., the OrionNebula). For comparison, ground-based Ha imaging atresolutions º1AÈ2A corresponds to at least 50È100 pc, thesize of a large Galactic giant molecular cloud (GMC). Thelatter resolution will clearly blend multiple sites of OB starformation that occur within a single GMC (e.g., M42 andNGC 2024 in the Orion GMC). At the same time, theground-based resolution element will also contain enor-mous volumes of intervening neutral gas. The high angularresolution of the HST imaging is thus critical for the studyof extragalactic H II region properties.

To date, there have been surprisingly few studies of extra-galactic H II regions using HST . A recent study of M101 byPleuss, Heller, & Fricke (2000) clearly demonstrates theadvantages of HST . SpeciÐcally, the break in the luminosityfunction slope at seen in some ground-basedL Ha\ 1038.6studies (Beckman et al. 2000 ; attributed to the transitionfrom ionization-bounded to density-bounded regions) wasnot apparent. In addition, multiple H II regions were re-solved into individual regions of lower luminosity, resultingin a very di†erent distribution of H II region sizes andenabling an analysis of H II region clustering (Pleuss et al.2000).

2.1. Our StudyThis paper, which presents H II emission-line imaging,

addresses the following speciÐc issues :1. The global luminosity function of the H II regions

and their associated OB star clusters ;

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3020 SCOVILLE ET AL. Vol. 122

2. Comparison of the form of the luminosity functionwith theoretical expectations ;

3. Variations in the luminosity function from arm tointerarm regions and between the nuclear region and thegalactic disk ;

4. The H II region sizes and densities ; and5. Analysis of the reddening and extinction of the H II

regions based on the observed ratios of Ha/Paa lines.To address these issues in a meaningful way and fullyunderstand the observational limitations, we developautomated routines for the deÐnition of the H II regionboundaries and a model to simulate the blending of multi-ple regions (° 5.1). The temporal evolution of the Lymancontinuum emission from an OB star cluster is alsomodeled, and constraints on the cluster mass distributionare derived from the observed luminosity function of H II

regions. Lastly, we analyze the physical processes importantin determining the masses and sizes of OB star clustersforming within a molecular cloud core.

In the following sections, we present the observations andimages for Ha and Paa (°° 3È4) and discuss the deÐnitionand measurement of the H II regions (° 5). The clustering ofH II regions, in particular a two-point correlation function,is derived in ° 6. The observed Ha luminosity function andLyman continuum emission rates are presented in °° 7 and8. The size and density distributions for arm and interarmregions are discussed in ° 9. The Ha/Paa ratios are used toestimate extinctions of 209 H II regions in ° 10 and hencederive extinction-corrected luminosities. In ° 11, we presenta sample of H II regions with sizes ¹50 pc and thencompare these with Galactic H II regions in ° 12. In ° 13, wepresent a simple model invoking instantaneous OB starformation with a standard IMF to provide a context forinterpretation of the observational data. We also develop amodel for the formation of OB star clusters in which themass of the core cluster is limited by radiation pressure oncethe cluster has accumulated D1000 In ° 14, we use theM

_.

total Ñuxes in Ha to estimate the overall Lyman continuumproduction and star formation rate in M51.

3. OBSERVATIONS

HST imaging of M51 using both WFPC2 and NICMOSwas obtained in several observing programs, as discussed inPolletta et al. (2001) and summarized in Table 1.

3.1. W FPC2The WFPC2 continuum and Ha images were obtained in

1995 January (Ford, Tsvetanov, & Kriss 1996) and in 1999July by N. Z. S. and S. E. (Ðlling in the areas not covered inthe Ford et al. archive data). The raw images were Ñat-Ðelded using the automatic standard pipeline at the SpaceTelescope Science Institute (STScI), and cosmic rays wereremoved using our own procedure. The complete cali-bration procedures are described in Polletta et al. (2001).For subtraction of continuum from the F656N image,which includes Ha and continuum, the y-band (F547M)image was used in the earlier epoch and the I band (F814W)for the later epoch. The broadband continuum image wasÐrst scaled by a constant such that the signal strengths on asample of stars balanced those in the F656N image.

To test that the use of di†erent continuum Ðlters for thetwo Ðelds did not introduce photometric di†erences, wecompared the measured Ñuxes for regions in the overlaparea of the separate images. In the Ha]continuum image,

the average di†erence was 1.6%. The Ha Ñux (after contin-uum saturation) was di†erent by D2% for faint regions andby 6% for the brightest regions.

3.2. NICMOSNICMOS (187N and 190N) images were obtained as part

of the NICMOS GTO program with a mosaic of nineNICMOS camera 3 Ðelds covering most of the area of theWFPC2 images. NIC3 uses a 256] 256 HgCdTe arraywith plate scales of and pixel~1 in x and0A.203859 0A.203113y, providing a Ðeld of view (Thompson etD52A.19] 52A.00al. 1998). The F187N and F190N Ðlters, with e†ective wave-lengths of 1.87 and 1.90 km, were used to obtain on- ando†-band images. The FWHM resolution is at 1.87 km.0A.19Observations at each of the nine mosaic positions weredone using a square dither in each Ðlter setting, and at eachdither position, nondestructive reads (MULTIACCUM)were taken. The total integration times for each Ðlter arelisted in Table 1.

The data were reduced and calibrated using theCALNICA version 3.3 task (Bushouse & Stobie 1998) inIRAF/STSDAS1 and the reference Ðles (static data quality,detector read noise, detector nonlinearities Ðles) from STScINICMOS pipeline, with the exception of the Ñat-Ðelded anddark frame corrections that were provided by the NICMOSInstrument DeÐnition Team. The dithered images werethen shifted and mosaicked using the NICMOSAIC andNICSTIKUM IRAF tasks (D. Lytle 1998, privatecommunication). The plate scales of the Ðnal ““ drizzled ÏÏimages are and pixel~1 in x and y. The0A.0381 0A.0378images were mosaicked with relative o†sets determinedfrom common stars in the overlap regions and rotated withnorth up and east to the left using the data header orienta-tion angle (see Polletta et al. 2001).

3.3. W FPC2 and NICMOS Flux CalibrationFlux calibration of the WFPC2 images is done as

described in Polletta et al. (2001). For the NIC3 images, weemploy scale factors of 5.050] 10~5 and 5.033 ] 10~5 Jy(ADU s~1)~1 at 1.87 km (Paa) and 1.90 km (Rieke et al.2001). The rms noise in the Ðnal images is typically 62 and59 kJy arcsec~2 at 1.87 and 1.90 km. As a check on the Ñuxcalibration, three of the coauthors of this paper indepen-dently calibrated the Ha and Paa images with similarresults (within 10%).

At zero redshift, the Ha Ðlter on WFPC2 transmits the[N II] lines (6548.1 and 6583.4 in addition to Ha andÓ)stellar continuum. At the redshift of M51 (z\ 0.00154), thetransmissions are 98.7%, 0.0%, and 91.2% for the two red-shifted [N II] lines and Ha, respectively. Adopting a totalÑux in the [N II] lines of 0.4 Ha in M51 and a Ñux ratio of1 :3 for the two [N II] lines, we Ðnd that the detected lineÑux will be 1.012 times that of Ha. This fortuitouslyhappens because the redshift reduces the transmitted Ñux ofHa by nearly the same amount that the transmitted Ñux ofthe 6548.1 [N II] line is increased. We therefore neglectÓthis 1% correction.

ÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈ1 IRAF is distributed by the National Optical Astronomy Observatory,

which is operated by the Association of Universities for Research inAstronomy, Inc., under cooperative agreement with the National ScienceFoundation.

No. 6, 2001 M51 3021

TABLE 1

M51 Ha AND Paa OBSERVATIONS

j Exposures Orientation AngleaProgram ID Date Filter (km) (s) (deg)

WFPC2:5123 . . . . . . . . 1995 Jan 24 F656N (Ha) 0.6561 400] 1400 93.1998

F547M (y) 0.5361 260] 6007375 . . . . . . . . 1999 Jul 21 F656N (Ha) 0.6561 1300] 700

F814W (I) 0.8386 700] 300NICMOS:

7237 . . . . . . . . 1998 Jun 28 F187N 1.8740 576 [113.168F190N 1.9005 576 [113.168

a For WFPC2, this corresponds to P.A.[V3: angle between the axis V3 of the camera and thenorth. For NICMOS, it corresponds to the angle between the y-axis of the camera and the north.

4. IMAGES

The mosaic of continuum-subtracted Ha images is shownin Figure 1 for the central 281A ] 223A. The Ha is displayedin red, the continuum B band in blue, and the I or y band ingreen. Ha emission extends out 10A from nucleus atP.A.D [15¡, and bright, discrete Ha emission regionsoutline the spiral arms. The locations of bright Ha emissionare closely associated with the dark dust lanes, but relativeto the dust (and the millimeter-CO emission ; see Polletta etal. 2001), the Ha is often displaced to the outside or leadingedge of the arms. In the standard picture of spiral patternstreaming shown in Figure 2, this o†set implies that the H II

regions develop subsequent to the time of maximum con-centration of the dust and molecular clouds. Close inspec-tion of Figure 1 also reveals a number of smaller dust lanes

FIG. 2.ÈFlow stream lines for material at 3, 3.2, 6, and 6.2 kpc in aspiral potential like that of M51 based on the model of Roberts & Stewart(1987), with the spiral arms starting at R\ 2 kpc and P.A.\ 15¡. Since thepattern speed is approximately half the circular velocity, the gasapproaches the spiral arm from the underside, is deÑected along the armtoward smaller radius, and then leaves the arm on the front. Orbit crow-ding is due to the spiral arm streaming motions. The crosses correspond to250 Myr time intervals.

and H II regions in the interarm regions both to the east andthe west of the nucleus.

Two very luminous associations of stars are also seenÈone approximately 32A northeast of the nucleus with sur-rounding Ha emission, the other 99A west of the nucleus,just outside the spiral arm. The latter region is remarkablein being very large in extent (D7A, or 326 pc, in diameter)and in having very little Ha emission. The continuum colorsand the lack of Ha suggest that this is an aging association,in which the O stars have mostly evolved o† the mainsequence (º107 yr ; see Polletta et al. 2001). The nature ofthe larger association is not at all clear, since it is probablymuch more massive and more extended than any of theyounger clusters required to power the Ha emission regions.

The NICMOS Paa (red) image is shown in Figure 3 over-laid on the V (green) and B (blue) continuum. In Figure 4,the Paa (green) is combined with the Ha (red) in order tohighlight those regions with relatively high Paa/Ha ratios,indicating particularly high dust reddening. For a smallerarea west of the nucleus, Figure 5 shows the Ha and Paaemission with no optical continuum background. Althoughgenerally Paa shows the same emission regions as Ha, it isclear even from visual inspection of Figures 4 and 5 thatmany of the arm H II regions have considerable reddening.There are several regions appearing green in Figure 4,implying strong Paa but only very weak Ha. Many H II

regions also exhibit strong gradients in the Paa/Ha ratio.

5. H II REGION PARAMETERS

In measuring H II regions, the high angular resolution ofHST is critical to resolving regions associated with individ-ual OB star clusters. HST NICMOS also enables measure-ment of Paa (which cannot generally be done from theground) ; in combination with Ha, this line provides a probeof the dust extinction. These points are illustrated well inFigure 6, which shows a region approximately 60A north ofthe nucleus at the full-resolution Ha and the Ha smoothedto resolution (to simulate ground-based imaging). Com-1A.5parison of the two Ha images underscores the need for highangular resolution in order to resolve the multiple, distinctH II regions that often exist in a single complex.

In Figure 7, the Ha and Paa are shown for the sameregion, and large variations can be seen in their relativedistributions due to variable reddening. For example, thesouthern end of the complex is strongest in Paa, while thenorthern region is brightest in Ha.

5.1. H II Region DeÐnition and MeasurementMeasured H II region properties will be fundamentally


FIG. 3.ÈMosaic of NIC3 Paa (continuum-subtracted) images covering the central 186A ] 188A of M51. The Paa emission is shown in red, and continuumV and B bands are in green and blue, respectively. (In the red, we also added the I-band image in order to balance the colors on the stars.)

dependent on both the resolution of the images and theprocedures used to deÐne the boundaries of the emissionregions. For example, higher resolution images will resolvesome of the larger regions into multiple components (andhence reduce the number of the most luminous regions). Inaddition, the criteria adopted to resolve (or break up)blended regions can be critical. Algorithms to separate outthe regions from the background and break up blendedregions might invoke logic based on a priori understandingof the characteristics of Galactic H II regions. For example,virtually all Galactic H II regions have diameters much lessthan 50 pc, so it is reasonable to expect that extragalacticregions larger than this are likely to be blends, ionized by

multiple OB star clusters. Lastly, data with higher sensi-tivity might join multiple regions that would have appearedseparate at lesser sensitivity if the threshold for joiningregions was set at a sufficiently low level.

The H II region measurement process may be broken intothree steps : (1) the deÐnition of the boundaries of emissionregions at a speciÐed intensity threshold, (2) the resolutionof multiple strong peaks within a single boundary intoseparate, discrete emission regions, and (3) the measurementof H II region parameters, such as peak and integratedÑuxes, sizes, and positions. Clearly, the breaking up ofblended features is the most difficult (and subjective) stepÈfor this reason, we developed an automated procedure

No. 6, 2001 M51 3023

FIG. 4.ÈHa (red) and Paa (green) images for the area of overlap. The blue was made by the combination (V [Ha)] V in order to show the dust lanes andthe background disk stars. Areas with yellow color have strong Ha and Paa.

(rather than an interactive process). This has the advantagethat the results can be carefully compared for di†erent inputparameters, and the selection algorithm is uniform acrossthe entire image. Extensive trials of the selection parameterswere used to test their e†ects on the results.

Prior to measurements of the discrete H II regions, weremoved a very extended background in order to““ partially ÏÏ remove the DIG emission and to suppress anyresidual continuum that might result from color variationsof the continuum in di†erent regions of the galaxy. Thisbackground at each pixel was estimated from the 30th per-centile intensity for a histogram of pixels in an area of

64 ] 64 pixels centered on each pixel (that is, 70% of thepixels are brighter than the subtracted background). Thelocal histograms of pixel values in the Ðnal background-subtracted image have a large number of pixels close to zeroand a small fraction on the positive tail, representing realHa emission (assuming the H II regions do not Ðll a largefraction of the area). Sixty-four pixels correspond to anapproximately 300 pc linear scale, and this is much largerthan any discrete H II regions. The 30th percentile waschosen to avoid signiÐcant Malmquist bias and to avoidremoval of discrete H II region line emission in areas withlarge numbers of H II regions (none of which Ðll 70% of the


FIG. 5.ÈSame as Fig. 4, but for the area to the west of the nucleus with all continuum removed. Although virtually all reasonably bright Ha emissionregions are seen in Paa, extremely large variations in the line ratio are apparent from region to region, and the morphology of individual regions can be verydi†erent in Ha and Paa.

pixels in a surrounding 64 ] 64 pixel box). After removingthe background, the noise (p) was estimated in areas notincluding obvious emission. Typically, p ^ 4.8] 10~18ergs cm~2 s~1. The removed background varies between[0.9 p and 2.8 p, and the average values are [0.04 p(whole image), 0.08 p (arms), 1.8 p (nucleus), and [0.15 p(interarm) (as quoted above). Discrete but low surfacebrightness H II regions would still be cataloged, providedthey have at least 1 pixel exceeding 6 p (see below).

The logic adopted for deÐnition of the boundaries of theemission regions was to start from the brightest pixel, Ðndall pixels connected to this peak down to a level of 65% ofthe peak value, then repeat the procedure starting from thebrightest remaining pixel, and so on until there are no pixelsleft above the adopted threshold for a ““ signiÐcant ÏÏ peak(here taken to be 6 p). The only complication is that as oneis collecting the neighboring pixels around a peak, one mustalso check that those new pixels would not more appropri-ately associate with one of the peaks found earlier in theprocess. In such instances, where there were multipleregions neighboring on a pixel, the pixel was attached to theregion which had the highest average value for its neighbor-ing pixels.

In detail, our procedure involved the following steps :

1. Find the peak pixel remaining in the image.2. Let this pixel be the basis for starting a new ““ peak.ÏÏ3. Generate a list of all pixels above 65% of this peak

value.4. For each of the pixels found in step 3, associate it with

the new peak or whichever preexisting peak has the highestaverage in pixels that touch the pixel in question. As pixelsare assigned to H II regions, they are also removed from theoriginal image to avoid further consideration. Any of thepixels in the list from step 3 not touching on a previous H II

region are left in the image for later consideration. Thus,one is gradually working down in intensity through theimage.

5. If there are remaining pixels above 6 p that could formthe basis for a new peak, then go back to step 1.

6. Lastly, the algorithm examined all the H II regionsdeÐned in steps 1È5 to see if any should be collected into asingle region. Any region that did not have a valley of atleast a 4.5 p (down from its peak value) between it and theborder pixels of any neighboring H II region was added intothe neighboring region.

No. 6, 2001 M51 3025

FIG. 6.ÈHa emission structure apparent at using HST (left) compared with the same image smoothed with a FWHM Gaussian to simulate0A.1 1A.5typical ground-based resolution. The subimage area shown here is approximately 60A north of the nucleus. The length bar corresponds to 100 pc. GalacticGMCs have diameters typically 2È5 times smaller, and their associated H II regions have still smaller sizes. This comparison illustrates well that theground-based and HST imaging are clearly studying very di†erent physical structuresÈthe former, associations of OB star formation, the latter, perhapsindividual OB star formation regions.

For any regions with more than 3 pixels, a curve ofempirical growth was derived to correct for Ñux outlyingpixels beyond the lowest nonzero pixel value. The curve ofgrowth was computed individually for each H II region frommeasurements of the total integrated Ñux within each H II

region as a function of the pixel intensity, starting from thepeak value and working down to 2.5 p. For example,including all pixels in the region with pixel values above 2.5,4, 6, 8, 10, . . . , p, the separate Ñux sums were calculated.These Ñux integral measurements were then Ðtted by astraight line (as a function of the variable cuto† threshold),and the Ðt was extrapolated to the zero-intensity cuto†value. This correction resulted in a median increase of only15% in the Ñux. In this manner, we are not assuming a Ðxedmorphology for all H II regions, and the extrapolations areextremely modest, since we have collected all connectedpixels down to only 2.5 p.

Although we are using a criterion based on a percentageof peak for the initial association of the nearby pixels with agiven peak pixel, this technique is not the same as thatdeveloped by McCall, Straker, & Uomoto (1996). They col-lected pixels down to a Ðxed percentage of the local peakbut then extrapolated the Ñux within this region using aÐxed multiplicative factor to estimate the total Ñux of theH II region. Here, the 65% is simply used as the initialcriterion for associating neighboring pixels with a particularlocal maximum. In addition, as we continue working downin intensity, pixels below the 65% level (all the way down to2.5 p) are associated with the preexisting nearby peaks. Thetechnique employed by McCall et al. (1996) su†ers theobvious problem that it is essentially assuming a constant““ curve of growth ÏÏ as a function of the percentage of peakfor all H II regions, i.e., the H II regions must all havehomologous morphology.

Actually, our algorithm is somewhat similar in spirit tothat developed by Thilker et al. (2000). Both algorithms letall H II regions grow simultaneously as one works down tolower Ñux levels in an image (see step 3 above). This ensuresthat as much Ñux as possible is gathered into each H II

region without bias toward the order with which each H II

region was Ðrst found. In our case, new H II regions arealways started from the highest remaining pixel in the entireimage, and we require that every distinct H II region musthave at least a 4.5 p valley between its peak and any pixelsin a neighboring H II region (see step 6 above).

In the course of reÐning the measurement technique,several of the parameters were experimented with, and theresults were compared with what we would have selected by““ eye ÏÏ as distinct H II regions. For example, the percentageof the peak Ñux above which pixels can be added and thelevel below which the image was truncated (Ðnally adoptedat 2.5 p) were varied. The results were not extremely sensi-tive to the precise values of these parameters. The samplearea north of the nucleus is shown in contour form for Ha inthe top left panel of Figure 7, and the derived features areillustrated in the bottom right panel of Figure 7.

The algorithm was also tested on a simulation image inwhich spherical H II regions with a uniform distribution ofLyman continuum output (with a range of 102) were posi-tioned randomly within a three-dimensional spatial cube.The regions were postulated to have identical electron den-sities, so as the Lyman continuum increased, the H II regiondiameter would be larger. The cube was then projected ontotwo dimensions and processed with the H II region Ðndingalgorithm. The H II regions found agreed very closely withthe input H II region locations and sizes. The simulationand measurement was run for increasing numbers of inputH II regions ranging from 10 to 100 in order to test the


FIG. 7.ÈContours for Ha and Paa emissions (top) for the northwestern region shown in Fig. 6 (60A north of the nucleus). In the bottom right panel, the Ha(red) and Paa (blue) are shown together. It is clear that signiÐcant di†erences are seen in the Ha and Paa because of the higher extinction in Ha. The areas ofthe Ha emission regions, as deÐned by our automated algorithm, are shown (the peak positions are indicated by the black pixels). The length bar correspondsto 1A, or 46 pc.

e†ects of blending. (Given the input sizes of the H II regionsand the size of the three-dimensional cube, the case with 150regions corresponded to very severe blending.) In allinstances, the H II region Ðnding algorithm found thecorrect number and parameters for the input regions (towithin ¹20%). Test Ðts for the dependence of the total Ñuxon the measured diameter was typically L P D2.75, com-pared with the theoretical D3 dependenceÈprobablybecause of limited spatial resolution. The fact that theresults came so close to recovering the input parameterswas particularly reassuring in the case of the 100-regionsimulation, since the images were heavily blended in thiscase.

Once the boundaries of the Ha emission regions weredeÐned, the peak, the integrated Ñux, the emission centroidcoordinates, and the number of pixels within each regionwere cataloged. The size of each region was calculated as(area)1@2.

The Ðnal list has 1373 Ha emission regions : 1321 havingat least 3 pixels and 1273 having 5 or more pixels. Thelargest region associated with the nuclear AGN jet had 895pixels and was excluded from the following analysis. Thetotal Ñux of the 1373 discrete H II regions constitutes 31% ofthe total Ha Ñux from the area of the galaxy covered in theWFPC2 images ; the remaining Ñux constitutes the DIGemission, discrete H II regions with peak Ñux below our 6 ppeak requirement, and possibly some unremoved contin-uum. Given the low surface brightness of the residual Ñux(and its low signal-to-noise ratio), we do not feel that thedata enable an investigation of the source of the DIGemissionÈspeciÐcally whether it is truly di†use or simplydiscrete, but low surface brightness H II regions.

As a check of the H II region deÐnition and measurementroutines, we compared our results for selected individualregions with ground-based Ha measurements (Rand 1992 ;Petit et al. 1996 ; Thilker et al. 2000). Although Rand (1992)

No. 6, 2001 M51 3027

did not publish his individual H II region measurements, hedid provide us with his H II region list in tabular form.Figure 8 shows the locations of the Ha emission regionsfrom RandÏs and our samples. Generally, good agreementcan be seen in the areas of common coverage (i.e., exceptingthe nucleus, which was not measured by Rand, and theouter disk, which was not covered in our images). However,in most cases, the higher resolution HST data resolvesmultiple H II regions within individual regions identiÐed inthe ground-based imaging. This is illustrated in Figure 6,which shows the HST data in a small area north of thenucleus at the original HST resolution and smoothed to 1A.5resolution to simulate ground-based imaging. In the areacovered in our study, there were 17 regions cataloged byRand that were not cataloged by us. We therefore also com-pared with the H II region catalog of Petit et al. (1996), andin most cases, the regions cataloged by Rand but not seenby us were also not present in Petit et al. (1996) ; however,the Petit survey had lower sensitivity than RandÏs. As anadditional check, we also examined on a case-by-case basisthe 17 Rand regions that we did not Ðnd. For one, wesaw no evidence of a source in either the Ha or theHa]continuum image, one was on the edge of our Ðeld,

and the remaining regions had peaks between 3.4 p and5.6 p in our images. Thus, the latter sources did not meetour 6 p criterion for the peak Ñux. In fact, the excellentcorrespondence between our sample and RandÏs is quiteimpressive given the fact that our resolution is approx-imately 100 times smaller in area and therefore an extended,low surface brightness feature in RandÏs image could easilybe missed here.

Comparison of our Ñuxes with RandÏs was not straight-forward, since he did not remove the large-scale di†usebackground but rather a local background (immediatelyaround each region). Nevertheless, the results appear con-sistent : for 10 regions with well-deÐned, bright Ha emission,the integrated Ñuxes were found to agree within 25% in allcases. The peak Ñuxes and sizes are of course resolutiondependent, and perfect agreement should not be expected.Thilker et al. (2000) also compared their results with RandÏs(1992) measurements and found good agreement.

Thilker developed an iterative grouping algorithm fordelineation of H II region structures, and he kindly made hisroutines available. In the end, we developed our own algo-rithm for reasons of simplicity, familiarity with our routine,and speed of execution. Nevertheless, we did test the two

FIG. 8.ÈLocations of the Ha emission regions selected by our automated procedure shown together with those measured interactively by Rand (1992). Ingeneral, there is very close correspondence between the two samples, although in many instances, we have cataloged multiple regions where Rand had justone. The outer areas were not covered in our images, and therefore a number of RandÏs H II regions do not have a counterparts in our sample ; the converse istrue in the nuclear region, which was excluded from RandÏs analysis because of blending. There are 17 regions in RandÏs sample that do not appear in ours,although they are in the area analyzed by both studies. All of these regions have peak intensities just below our 6 p criterion for the peak intensity.


FIG. 9.ÈTwo-point angular correlation function for centroid positionsof the 1373 discrete H II regions (top) and pixels (bottom) with Ha emissionexceeding 3 p in the same Ha image with di†use background subtracted.The angular sampling of the images was normalized out by dividing thecorrelation functions with those computed for pixels sampled randomlywithin the observed image areas. This normalization does not removelarge-scale correlations, such as the concentration of H II regions towardsmall galactic radii and along spiral arms ; thus, neither correlation func-tion goes asymptotically to zero at large angles, and their integrals are notequal to zero.

procedures in the area of the northern spiral arm shown inFigure 6 ; the regions that were deÐned were consistent butnot identical.

Our algorithm was also run on the Paa image (using thesame parameters speciÐed in terms of the image noise level),yielding 232 Paa regions. All except three of these regionswere inside one of the previously found Ha regions ;however, in most cases, the boundaries were somewhat dif-

FIG. 10.ÈPercentage of H II regions with nearest neighbors within thespeciÐc angular o†set (h).

ferent. Usually, the Paa region was smaller in size, as aresult of the intrinsically lower Ñux and signal-to-noise ratioof Paa. In a few instances, the peak of the Paa was verysigniÐcantly shifted from the Ha peak, and it appears thatthese are regions with particularly high extinction at thelocation of maximum Paa emission.

6. CLUSTERING OF H II REGIONS

In order to quantify the e†ects of HST versus ground-based image resolution and to illustrate the clustering of theH II regions measured here, we have computed the two-point angular correlation function for the measured cen-troid positions of the 1373 H II regions (Fig. 9, top). In thelower panel of Figure 9, the correlation function is shownfor pixels with Ha emission exceeding 3 p (in the same Haimage with background subtraction that was used for deÐn-ing and measuring the discrete H II regions). In bothinstances, the angular sampling of the images was normal-ized by the correlation function for pixels sampled random-ly within the observed region. However, because of thelinear association of H II regions along the spiral arms andthe fact that their number is generally increasing towardsmall galactic radii, the purely random normalization doesnot remove large-scale correlationsÈthus, neither corre-lation function goes asymptotically to zero at large anglesnor does the integral equal zero.

Both correlation functions clearly show a strong clus-tering inside 2A, corresponding to 92 pc. This is inside thespatial scale sampled in ground-based Ha images, indicat-ing that a large number of the regions classiÐed fromground-based imaging are resolved into multiple regions atthe resolution used here. Pleuss et al. (2000) reached a0A.1similar conclusion based on a minimal spanning treeanalysis of HST images for M101.

The correlation function for the discrete H II regions (top)shows a decrease inside 1A, which is a result of the fact thatthe algorithm used to deÐne the H II regions requires atleast 1 pixel lower by 40% on the boundary between anytwo H II regions in order to subdivide an Ha emissionregion into two regions.

In Figure 10, the percentage of H II regions with nearestneighbors with separation less than h is shown as a functionof h. Forty-seven percent have their nearest neighbor within1A and 81% within 2A. Only 10% have their closest nearbyH II region more than 4A away. Both the correlation func-tion and Figure 10 clearly show that the H II regions arestrongly correlated on scales less than 2A.

7. LUMINOSITY FUNCTIONS

The observed (uncorrected for extinction) Ha luminosityfunction is shown in Figure 11, together with those derivedby Rand (1992) and Petit et al. (1996), using only H II

regions from their studies that are in the same area of thegalaxy covered by us. The two ground-based luminosityfunctions extend at least a factor of 5 higher in luminosity.This is due to the fact that all of the most luminous regionscataloged in the ground-based studies were resolved intomultiple regions in our sample.

To characterize the luminosity function, we express it as atruncated power law: in di†erential form,

dN(L Ha)d ln L Ha

\ Nup(L Ha/L up)a ; (1)

No. 6, 2001 M51 3029

FIG. 11.ÈObserved Ha luminosity function for all H II regions deÐned and measured in the WFPC2 images (not corrected for extinction) compared withthe luminosity functions derived from ground-based imaging by Rand (1992) and Petit et al. (1996). The fact that the HST luminosity function appears toshift to lower luminosity is due largely to the ability to separate individual H II regions that become blended in ground-based images (see text). Fits tologarithmically binned distributions are over the range ergs s~1. Typical extinctions of will shift theL Ha\ (40È800)] 1036 AHa\ 0.798A

V\ 0.798] 3.2

luminosity functions a factor of 5.9 higher.

and in integral form,

N(ºL Ha)\Nup[a

[(L Ha/L up)a [ 1] (2)

(McKee & Williams 1997). (Note that we adopt theopposite sign convention for a to that used by McKee &Williams.) As pointed out by McKee & Williams (1997), thisform has the advantage of having clear physical interpreta-tions for the parameters. SpeciÐcally, is the highestL upluminosity region, is approximately the number ofNupregions between and for a D [1, and is0.5L up L up Nup/[athe number of regions expected above if the distributionL upis not truncated (i.e., one can directly see whether the dis-tribution is ““ signiÐcantly ÏÏ truncated or terminated by lownumber statistics).

Fitting equation (2) to the apparent luminosity functionyields a \ [1.01^ 0.04, compared with [0.50 and [0.32derived for the Rand (1992) and Petit et al. (1996) samplesover the range (40È800) ] 1036 ergs s~1. [Since these Ðts arefor source counts binned logarithmically in the expo-L Ha,nents should be decreased by 1 for N(L )dL .] Our power-lawindex is substantially steeper than that derived from pre-vious studies (Kennicutt et al. 1989 ; Rand 1992 ; Petit et al.1996 ; Thilker et al. 2000) because of the fact that the higherangular resolution resolves the more luminous, apparentlyblended regions. The slope derived here is similar to that ofGalactic radio H II regions (a \ [1 to [1.3 ; see ° 12).

In addition to a steeper slope, we also Ðnd that theobserved luminosity function does not extend to as high aluminosity as the ground-based luminosity functions, as aresult of our resolving the larger regions. However, thelower maximum observed luminosity obtained here is com-pensated by high extinction corrections (see ° 10). The meanextinction derived in ° 10.2 for the discrete Ha emissionregions is mag. IfAHa \ 0.798SA

VT \ 0.798] 3.1\ 2.47

this is applied on average to all H II regions, then theobserved Ha luminosity functions are increased by a factorof D10 (° 10.2).

The observed luminosity functions di†erentiated betweenarm, interarm, and nuclear areas are shown in Figure 12.These three areas are deÐned and illustrated in Polletta etal. (2001). Over the luminosity range L Ha \ (12È500) ] 1036ergs s~1, logarithmic truncated power-law Ðts have expo-nents [0.72^ 0.03, [0.95^ 0.05, and [1.12^ 0.11 forthe arm, interarm, and nuclear (excluding the nuclear jet)regions, respectively (see Table 2). Thus, the luminosityfunction is signiÐcantly Ñatter in the spiral arms than in theinterarm regions. Rand (1992) and Thilker et al. (2000) alsofound steeper luminosity functions in the interarm regions(exponent [0.93 to [0.96 vs. [0.48 to [0.72) ; however,the actual values of their slopes in the interarm and armareas are somewhat di†erent from ours.

There are several possible explanations for the Ñatterluminosity functions in the arms and the nucleus comparedwith the interarm regions : (1) the surface density of H II


FIG. 12.ÈObserved Ha luminosity functions separated for arm, interarm, and nuclear regions. These areas are deÐned and shown in Polletta et al. (2001).Fits to logarithmically binned distributions are over the range ergs s~1. The power-law Ðts to the high-luminosity tail are also shown.L Ha \ (12È500)] 1036A signiÐcant Ñattening of the luminosity function is seen in the nucleus and spiral arms, compared with the interarm regions.

regions is higher in the arms, causing more blending and/orclustering, which in turn produces a large number of appar-ently high-luminosity regions ; (2) the OB star cluster massfunctions are signiÐcantly di†erent in the two areas ; and (3)the interarm H II regions and associated OB star clustersare, on average, older and therefore have evolved to lowerLyman continuum output levels. The only way there couldbe a systematically older population of clusters in the inter-arm regions than in the arms would be if the OB star clus-ters in the disk formed within the arms and then aged asthey moved into the interarm regions. However, the dura-tion of the Lyman continuum emission from a coevalcluster is far too short (¹3 ] 106 yr ; see ° 13.2) comparedwith the time (º3 ] 107 yr) needed to migrate into theinterarm regions for this explanation to be viable. Althoughthe second explanation cannot be ruled out at present, wethink that the level of blending and/or clustering expectedin the spiral arms is entirely consistent with the notion that

many of the most luminous regions are, in fact, blends. Onesupporting piece of evidence for this is the clear trend forthe most luminous regions to be the most extended and tohave the lowest densities (see ° 9).

The Ðtting results summarized in Table 2 clearly show asigniÐcant truncation of the luminosity functions at L up^4] 1038 ergs s~1. The last columns of Table 2 give the Ðtted

and As noted in the discussion followingNup Nobs(L [ L up).equation (2), the truncation is signiÐcant if Nup/[a ?This is indeed true for all four Ðts (based onNobs(L [ L up).comparison of the last two columns in Table 2), strongly

suggesting a physical or observational limitation to themaximum luminosity of the H II regions.

In none of the luminosity functions do we see evidence ofthe break in the slope at ergs s~1, which hasL HaD 1038.6been reported in some ground-based studies (Kennicutt etal. 1989 ; Rand 1992). (We did experiment with smoothingour Ha images by a factor of 4, then redeÐning the bound-

TABLE 2

M51 LUMINOSITY FUNCTION FITS

H II L upH II Region Sample Regions a (1038 ergs s~1) Nup Nobs(L [ L up)

All without jeta . . . . . . . . . . . . 1223 [1.01^ 0.04 4.0^ 0.5 12.5 ^ 3 5All without jetb . . . . . . . . . . . . 1223 [0.85^ 0.02 4.0^ 0.5 16.1 ^ 3 5Armb . . . . . . . . . . . . . . . . . . . . . . . 563 [0.72^ 0.03 3.3^ 0.5 13.1 ^ 3 4Interarmb . . . . . . . . . . . . . . . . . . 431 [0.95^ 0.05 2.0^ 0.9 8.1 ^ 2 1Nucleus without jetb . . . . . . 205 [1.12^ 0.11 1.1^ 0.4 4.3 ^ 2 0

NOTE.ÈObtained from Ðts to logarithmically binned distributions.a Fit range ergs s~1.L Ha \ (40È800) ] 1036b Fit range ergs s~1.L Ha\ (12È500) ] 1036

No. 6, 2001 M51 3031

aries and determining the luminosity function. The resultwas that the derived luminosity function became shallowerand developed a spectral break ; however, since the breakwas somewhat lower in luminosity than 1038.6 ergs s~1, wecannot say that it is a result of blending.)

8. LYMAN CONTINUUM EMISSION RATE

The observed H II regions range in luminosity fromto 2 ] 1039 ergs s~1. For case B recombi-L Ha \ 2 ] 1036

nation, the Ha luminosity is

L Ha\ 3.55] 10~25A T

e104 K

B~0.91nenpV ergs s~1 , (3)

where V is the volume of the H II region and and arene

npthe electron and proton volume densities (Osterbrock 1989).

Since absorptions by He do not signiÐcantly reduce thenumber of photons available to ionize H (Osterbrock 1989),the required Lyman continuum production rate, isQLyc,then given by

QLyc \ 7.32] 1011L HaA T

e104 K

B0.11s~1 . (4)

The observed Ha luminosities therefore translate to QLyc \2.2] 1048È1.5] 1051 s~1 for K. Extinction cor-T

e\ 104

rections are discussed in ° 10.

9. H II REGION SIZES AND DENSITIES

The H II region luminosities and their Lyman continuumemission rates constrain the implied masses of the OB star

clusters ; similarly, the sizes and electron densities of the H II

regions reÑect on the evolution of their spheresStro� mgrenand the surrounding ISM.

9.1. SizesThe distribution of H II region sizes [D\ 2(area/n)1@2] is

shown in Figure 13 for all regions with at least 2 pixels.They range from 10 to 250 pc in diameter ; those º120 pcwere not plotted, since they are clearly blends. The lowerlimit of 10 pc is due to the minimum 2 pixel criterion. Themean sizes are 30È34 pc in arm, interarm, and nuclearregions (Fig. 13). Both the shapes of the distributions andthe mean sizes are similar for all three areas. The majority ofthe H II regions have size ¹50 pc and thus could be associ-ated with a single or a few OB star clusters. Those regionswith larger sizes are very likely blended superpositions ofmultiple (but possibly related) OB associations (see ° 11).

9.2. Electron Densities (ne)

Since the Ha luminosities are proportional to thevolume-integrated emission measures, the mean electrondensity can be obtained using the measured sizes :

SneT \ 43

C(L Ha cor/1037 ergs s~1)(T /104 K)0.91(D/10 pc)3

D1@2cm~3 .

(5)

In equation (5), we normalized to typical values of L Ha cor \1037 ergs s~1 and a size of 10 pc. The derived for theSneT

H II regions are distributed mostly between 5 and 20 cm~3.

FIG. 13.ÈDistribution of H II region diameters for regions containing at least 2 pixels, separated between arm, interarm, and nuclear regions. NosigniÐcant di†erence is seen in the diameter distributions between the di†erent regions. The mean diameters are given for all regions smaller than 100 pc.Regions with diameter º120 pc were not plotted, since these are certainly blends of multiple regions.


FIG. 14.ÈApparent Ha luminosities (top) and H II region diameters (bottom) as functions of mean electron density, with the arm, interarm, and nuclearregions plotted in blue, red, and green, respectively.

In Figure 14, the H II region luminosities and sizes areplotted as functions of mean electron density. The sharpboundaries to the distributions on the lower side are dueto the surface brightness threshold and the minimum sizeof 1 pixel within the H II region. The average value is

cm~3 for sizes less than 40 pc. If the apparentSneT ^ 13

are corrected for the average extinction correctionne(eq. [8]), the mean densities are increased by a factor of 3 to

cm~3, which is similar to that of resolved Galac-SneT ^ 39

tic H II regions cm~3).(neD 100

In the above estimates, we have made no corrections forthe fact that many of the H II regions are only marginally

No. 6, 2001 M51 3033

resolved ; such corrections would of course increase themean densities further. Some of the larger regionsundoubtedly do include substantial empty or neutralvolumes. Moreover, there are substantial gradients in theHa surface brightness within individual regions, indicating

that the internal densities vary and commonly increase sev-eralfold at the peaks.

9.3. L uminosity versus Size : H II Region BlendingIn Figure 15, the H II region luminosities are plotted as a

FIG. 15.ÈObserved luminosities of the H II regions as a function of H II region diameter


function of their measured sizes. There is a clear trend forthe more luminous regions to be larger. Although such atrend is certainly expected, since more luminous OB associ-ations can ionize a larger volume of gas, the observeddependence is shallower than the D3 dependence expected ifthe electron densities are constant. SpeciÐcally, the moreluminous and larger regions appear to have lower average

The luminosity versus size data is Ðtted well by L P D2.ne.

(The lower limit of the data points in Fig. 15 is caused bythe observational threshold on the surface brightness.However, this threshold cannot account for the decrease indensity for larger regionsÈif the larger regions had thesame densities as the smaller regions, they would moreeasily make it into our sample.) The proper explanation forthe correlation is almost certainly that the larger regions areblended, and they include substantial empty or neutralvolumes or, equivalently, that the large H II regions mayhave expanded outside their progenitor GMCs.

In fact, it is easily shown that for superposed or blendedH II regions, one expects precisely the L P D2 correlationthat is observed. If the boundaries of ““ N ÏÏ multiple regionsof the same size just overlap as projected on the plane of thesky, their total projected ““ size ÏÏ computed as the(D

T),

square root of the sum of the total area, is given by DT2 \

Since the total luminosity of the blended region isNDi2.

then we expect This result isLT

\NLi, L

T\D2L

i/D

i2.

obviously not dependent on the H II regions being identical ;the observed correlation between luminosity and size is thusstrong evidence that the high-luminosity regions are blendsor superpositions of multiple lower luminosity regions. Thisis certainly not surprising based on observations of GalacticH II regions, which often showed multiple centers of ioniza-tion on scales of less than a few parsecs (see ° 12).

10. H II REGION EXTINCTIONS FROM Ha/Paa

For case B recombination in an ionization-bounded H II

region, the intrinsic Ñux ratio for Ha/Paa is 8.15, and thenonÈLyman series lines should be optically thin(Osterbrock 1989). (For a density-bounded H II region, thecase A recombination line ratio is Ha/Paa \ 6.14, but caseB is generally adopted as more appropriate for high surfacebrightness, discrete H II regions.) The observed Ñux ratioscan be less than 8.15 as a result of the higher extinction atthe line j \ 6563 (Ha) compared with 1.87 km (Paa). WeÓhave used the observed ratios to estimate the mean extinc-tions of each H II region detected in both lines, using therelation

AV

\ 3.75 logA8.15FPaa

FHa

Bmag . (6)

The constant in equation (10) was derived assuming thestandard Galactic extinction curve (Rieke & Lebofsky1985 ; Cardelli, Clayton, & Mathis 1989) and assuming thedust is distributed in a foreground screen, uniformly cover-ing each pixel.

After smoothing the Ha to the same resolution as Paa, wemeasured the Ha/Paa Ñux ratios and extinctions at allpixels within the H II region boundaries using only pixelsthat had both Ha and Paa detected at º5 p. Two hundrednine H II regions met these criteria. (With the Ha smoothedto the Paa resolution, the H II region Ðnding algorithm Ðnds702 regions. However, for the extinction analysis here, weadopted the original high-resolution H II region deÐnitionssimply because they yielded better boundary deÐnition.)

Flux ratios and extinctions were then estimated for each HII region by averaging the pixel ratios and extinctions,rather than from ratios of the H II region integrated Ñuxes.The distribution of these average Ñux ratios and for theA

VÏs

209 regions is shown in Figure 16. The mean extinctions aremag (weighting each region equally), 2.4 magA

V\ 3.1

(weighting each by the observed H II region luminosity), and3.0 mag (weighting by the extinction-corrected luminosity).The derived mean extinctions are in reasonable agreementwith the values mag) obtained by van der Hulst(A

V\ 0.8È4

et al. (1988) for a sample or 37 H II regions in M51, compar-ing Ha and radio free-free Ñuxes at 8A resolution. For 14 ofthe regions, the Balmer decrements yielded considerablylower values (in the range mag ; van der HulstA

V\ 0.4È2.4

et al. 1988), but these are optically biased toward lowerextinction regions. It should be pointed out that since the HII region boundaries were deÐned from the Ha images, theanalysis above excludes regions of extremely high extinctionin which the Paa emission is signiÐcantly o†set from Ha.The derived extinctions would have very likely been higherif the Paa had been used to deÐne the H II regions. We usedHa for delineating the H II region boundaries simplybecause of the higher signal-to-noise ratio and resolution.

In Figure 17, the derived extinctions of the 209 regionsare plotted against the observed (not extinction-corrected)Ha luminosities. In this Ðgure, the Ðlled circles are theaverage values in separate luminosity bins. No strong corre-

FIG. 16.ÈTop : Measured Ñux ratios (Ha/Paa) for 209 H II regions withboundaries deÐned from Ha (shown in Fig. 7), restricted to those pixelsdetected at º 5 p in both Paa and Ha. For case B recombination in anionization bounded H II region with no extinction, the intrinsic ratio is 8.15(dashed vertical line). The lower measured Ha/Paa ratios reÑect the higherextinction at the wavelength of Ha (6563 than that of Paa (1.87 km).Ó)Bottom : Distribution of average visual extinctions for the H II regionsinferred from the measured Ñux ratios, assuming an intrinsic ratio of 8.15and the standard ISM extinction curve (Rieke & Lebofsky 1985 ; Cardelliet al. 1989). The dashed vertical line is the average extinction A

V\ 3.2

mag. (The extinctions were determined at individual pixels and then aver-aged for each region.)

No. 6, 2001 M51 3035

FIG. 17.ÈDerived extinctions as a function of observed Ha luminosityfor 209 H II regions with boundaries deÐned from Ha (shown in Fig. 7),restricted to those pixels detected at º 5 p in Paa and Ha. The extinctionswere determined for individual pixels within each H II region and thenaveraged. The Ðlled circles are the averages for equal logarithmic bins inHa luminosity.

lation (or anticorrelation) is seen between the observed lumi-nosities and the derived extinctions as might be expected ifthe Ñux variations were largely due to extinction variations.The Ðlled-circle averages do show a weak correlation in thesense that the fainter regions have somewhat higher extinc-

FIG. 18.ÈTop : Measured Ñux ratios (Ha/Paa) for all individual pixelsdetected at º5 p in both Paa and Ha. For case B recombination in anionization-bounded H II region with no extinction, the intrinsic ratio is8.15 (dashed vertical line). Bottom : Derived visual extinctions for the samepixels obtained from the measured Ñux ratios assuming an intrinsic ratio of8.15 and the standard ISM extinction curve (Rieke & Lebofsky 1985 ;Cardelli et al. 1989).

tion, but the dispersion within bins is clearly much largerthan the trend.

We also measured the Ñux ratios and derived extinctionsfor all pixels in the Ha and Paa images that were bothdetected at º5 p, independent of whether or not they wereinside one of the discrete Ha regions. The results (shown inFig. 18) are entirely consistent with those found within thediscrete H II regions (Fig. 16). When the selection thresholdwas increased to 10 p, the results were also similar, implyingthat the larger extinction values are not simply the result ofpoor signal-to-noise ratio or a bias in either image.

10.1. Extinction Gradients across H II RegionsDetailed comparison of the Ha and Paa images reveals

that most individual H II regions also have very large varia-tions in the Ha/Paa ratio across their areas. (It is for thisreason that the ratios and extinctions for each region werecomputed in ° 8 as averages of the values from individualpixels rather than from ratios of the integrated Ñuxes.) InFigure 5, the area to the west of the nucleus is shown withHa in red and Paa in green (with no continuum added).Several of the emission regions show entirely di†erent mor-phology and peak locations in the optical and near-infraredlines. The gradients and peak o†sets vary from region toregion ; they therefore are not due to misalignment of theimages. In fact, there are several examples in this image ofemission regions in Paa that are hardly detected in Ha,indicating extinctions in excess of 4 mag. There are alsomany regions that are seen in Ha but not Paa, but this issimply due to the intrinsically high Ñux of Ha together withthe lower sensitivity of the Paa image.

10.2. Extinction-corrected H II Region L uminositiesThe extinction-corrected Ha luminosity is obtained from

L Ha cor \ 100.320AVL Ha uncor , (7)

using the standard ISM extinction curve (Rieke & Lebofsky1985) for which For the mean extinctionAHa \ 0.798A

V.

mag ; ° 10), the observed luminosities are(AV

\ 3.1increased by factors of D9.9, and we will adopt an averagecorrection factor of

S fHa extvcorT ^ 10 (8)

as an appropriate general extinction correction where indi-vidual extinctions are not available.

The extinction-corrected and observed luminosity dis-tributions are shown in Figure 19 for the 209 regionsdetected in both Ha and Paa. (The extinction-correctedluminosity of each region was calculated by correcting eachpixel for its extinction and then summing the extinction-corrected pixel luminosities.) The extinction-corrected dis-tribution exhibits a broad peak at approximately L Ha \1038 ergs s~1, and the most luminous region is at 3 ] 1039ergs s~1. The fallo† on the low-luminosity side is not real,since the sample has a detection threshold, and there are noregions from below the threshold that can populate the lowend after being corrected for extinction. The distributionsshown in Figure 19 should not be interpreted as luminosityfunctions, since they do not include all pixels in each region,only those pixels detected at º5 p in both lines. The criticalpoint we make from Figure 19 is that the net result ofextinction corrections, when they are derived on a case-by-case basis, is to shift the luminosity distribution up a factorof 10 in luminosity.


FIG. 19.ÈHa luminosities as observed for the 209 regions for which the extinction was derived and as corrected for extinction. The extinction-correctedluminosity of each region was calculated by correcting each pixel for its extinction and then summing the extinction-corrected pixel luminosities. Thesedistributions should not be interpreted as luminosity functions, since they include only those pixels in each region detected at º5 p in both lines.

11. H II REGION SAMPLE WITH DIAMETER ¹50 PARSECS

The most luminous emission regions in our sample arequite clearly blends or superpositions of H II regions withmultiple OB star clusters producing the ionization. Theirsizes are typically over 50 pc (see Fig. 15 and °° 9.1È9.2).Fifty parsecs is a very conservative upper limit to the size ofa region that might plausibly be ionized by a single compactOB cluster. This is because the main-sequence lifetime isonly 3] 106 yr for the OB stars that produce most of theionizing photons. Within this time, the ionized gas canexpand only to a radius of D30 pc, even if it is freelyexpanding at the sound speed (10 km s~1) of the 104 K gasand after the source of ionization is turned o†, the gasrecombines on a relatively short timescale (¹104 yr for

cm~3). Thirty parsecs is therefore a very conserva-neº 10

tive maximum radius for the H II region ionized by a single,coeval cluster of stars. It is a ““ conservative ÏÏ maximum,since the expansion is usually much slower than 10 km s~1,even if the surrounding, neutral gas has a density ¹10cm~3 (see, e.g., Osterbrock 1989).

Of course, the early growth of the H II region to its initialradius can be faster, but this initial phase takesStro� mgren

place in higher density regions, and the resulting Stro� mgrenradius is much smaller. The subsequent evolution is hydro-dynamic, and this is certainly the phase in which we see theobserved H II regions. In late phases, if the H II regionbecomes density bounded, the ionization front mightexpand at a higher speed, but this would correspond to theDIG H II regions.

In Figure 20, we show the distribution of observed Haluminosities for all 1101 H II regions in our sample withdiameters ¹50 pc. The range of observed is 2 ] 1036 toL Ha1 ] 1038 ergs s~1. Since the derived extinctions are approx-imately independent of apparent Ha luminosity (Fig. 17), weconclude that the maximum intrinsic luminosity ofunblended H II regions is a factor of 10 higher, or

max single cluster L Ha cor ^ 1039 ergs s~1 , (9)

and the maximum Lyman continuum production is

max single cluster QLyc^ 7 ] 1050 s~1 . (10)

Once again, this is a conservative upper limit, since some ofthe regions in the ¹50 pc sample are also likely to beblends.

12. COMPARISON WITH GALACTIC H II REGIONS

For comparison with Galactic H II regions, we make useof radio continuum observations of the free-free continuumto avoid Galactic line-of-sight dust obscuration. Radiostudies have concentrated on ““ compact ÏÏ H II regions,which are relatively young and have high surface bright-ness. Schraml & Mezger (1969) observed 18 free-free emis-sion complexes at j \ 2 cm with 2@ resolution (typicallycorresponding to 0.5È5 pc), and we make use of their samplefor our discussion. Their results are summarized in Table 3for M43, M42 (Orion Nebula), IC 1795 (W3), W51, andW49 (ordered with increasing luminosity).

Comparing the implied of the Galactic H II regionsL Hawith the extinction-corrected Ha luminosities (° 11) for the

No. 6, 2001 M51 3037

FIG. 20.ÈObserved Ha luminosities for the 1011 H II regions with diameters less than 50 pc. For the reasons discussed in the text, H II regions larger than50 pc are almost certainly ionized by multiple clusters ; the distribution shown here therefore shows the maximum luminosity likely to arise from a single OBstar cluster.

sample with D¹ 50 pc, it can be seen that the Ðrst value inthe M51 distribution function corresponds to a few timesM42, and the maximum single-cluster (eq. [9]) corre-L Ha corsponds to W49. Since W51 and W49 are the most luminousGalactic H II regions, we conclude that the M51 samplespans approximately the full range of Galactic compact H II

regions (perhaps not sampling to the lowest luminosities,but that depends on the order-of-magnitude extinction cor-rection adopted for M51). For reference, we also note thatthe 30 Dor cluster in the Large Magellanic Cloud (LMC) isa few times more luminous than W49. The diameter of the30 Dor stellar cluster is D20 pc, and the associated Ha

emission extends over D100 pc. In the full sample of M51regions (Fig. 11), the most luminous is at an observed L Ha \2 ] 1039 ergs s~1 ; if this region has a modest extinction, itsluminosity would be 20 times that of W49. This is anotherreason for believing that the most luminous Ha regions inM51 are blended.

Although the luminosity range of M51 H II regions withD¹ 50 pc is similar to that of the Galactic regions listed inTable 3, their sizes are typically several times larger, andtheir mean densities lower. These di†erences have severallikely explanations : the Galactic regions were selected forhigh surface brightness free-free emission (i.e., compact

TABLE 3

GALACTIC COMPACT H II REGIONS

Diameter ne

U QLyc L HaRegion (pc) (cm~3) (pc cm2@3)a (s~1) (ergs s~1)b

M43 . . . . . . . . . . . . . . . . 0.6 610 20 2.5 ] 1047 3.5] 1035M42 . . . . . . . . . . . . . . . . 0.7 2237 55 5.3 ] 1048 7.1] 1036IC 1795 (W3) . . . . . . 2.3 400 179 1.8 ] 1050 2.5] 1038W51 . . . . . . . . . . . . . . . . 4.9 372 208 2.9 ] 1050 3.9] 1038W49 . . . . . . . . . . . . . . . . 8.4 536 276 6.6 ] 1050 9.1] 1038

NOTE.ÈData obtained from 2 cm radio continuum observations of Schraml & Mezger1969.

a Excitation parameters from Schraml & Mezger 1969 have been summed(U \ Rsne2@3)

for all H II region components, except in the case of W51, for which we took only thestrongest component, since this direction is along a spiral arm tangent, and there is likely tobe confusion.

b Implied extinction-corrected (\1.37] 10~12Q).L Ha


radio H II regions) in order to avoid possible extended,missing Ñux ; the M51 regions are still only marginally re-solved and undoubtedly would include compact cores athigher resolution ; or some of the Galactic regions may be inan earlier stage of their evolution before they have fullyexpanded. Despite these di†erences, the ionizing lumi-nosities are similar.

The luminosity function of Galactic radio H II regionshas been Ðtted by Smith & Kennicutt (1989) and McKee &Williams (1997), who found power-law indices of [1.3 and[1.0, respectivelyÈconsistent with our value of [1.01 forM51. Although McKee & Williams (1997) derive an upperlimit for the Lyman continuum emission rate from GalacticH II regions of s~1 (including a correc-QLyc,up\ 4.9] 1051tion of 25% for absorption by internal dust), the actualmaximum observed (W49) is only 7 ] 1050 s~1. The lattervalue is consistent with our M51 extinction-corrected upperlimit s~1 (eq. [10], from the 50 pcQLyc,upD 7 ] 1050sample). Our estimate has no corrections for dust absorp-tion of the ionizing photons or UV escaping into the di†usemedium. The latter factor adopted in the McKee & Wil-liams (1997) study is very large (1/0.29 \ 3.45). Theyobtained this number by assuming the entire discrepancybetween the far-infrared COBE measurements of [N II] andthe sum of the Lyman continuum from discrete H II regionswas due to leakage of ionizing UV out of the regions mea-sured in radio H II regions. Alternatively, much of the dis-crepancy might be due to older or lower mass OBassociations (which are undetected in the Galactic radiosample), ionization by other sources, or even uncertaintiesin the analysis of the COBE data. We do not include thisfactor in view of its large uncertainty and because we wishto compare discrete H II regions in M51 with comparableobjects in the Galaxy.

13. OB STAR CLUSTERS AND H II REGION EVOLUTION

In order to understand the nature of the constraints pro-vided by the observed H II region Ha luminosities and theirdistribution, we develop here a simpliÐed model for OB starclusters and H II region evolution. The critical constraintsthat we wish to understand are

1. The slope of the luminosity function on the high-luminosity end eq. (1)]Èto what[dN(L )/d ln LP L Ha~1.01 ;extent is this due to the evolutionary decay of the Lymancontinuum emission from clusters as they age, as opposedto the mass spectra of the clusters and their high-massstars?Èand

2. The existence of an ““ approximate ÏÏ maximum lumi-nosity for individual OB star clusters at ergsL Ha D 1039s~1. (This maximum is surprising in view of the fact that itcorresponds to a cluster of only a few times 103 yetM

_,

GMCs contain 105È106 of molecular gas ; thus, there isM_sufficient mass available to form much more massive

clusters).

13.1. Model Stellar and Cluster ParametersFor each OB star cluster, we assume the star formation

occurs on a short timescale compared with stellar evolutiontimescales. We adopt a Salpeter IMF with N(m

*)dm

*P

over the range (here taken to be 1 and 120m*~2.35 m

lÈm

urespectively) rather than the local Galactic IMF (ScaloM_

,1987). The Salpeter IMF is consistent with the recent deter-minations for OB associations, which have power-law

indices in the range [2.1 to [2.45 (Massey et al. 1995) andthe 30 Dor cluster in the LMC (Selman et al. 1999).

The stellar lifetimes and Lyman continuum emissionrates are based on stellar models with constraints providedby observations of individual high-mass stars. Main-sequence lifetimes and bolometric luminosities are fromRenzini & Buzzoni (1986) and Maeder (1987) for ¹10 andº10 respectively. For reference, the main-sequenceM

_,

lifetimes are 8.6, 4.5, 3.8, and 3.3] 106 yr for 20, 40, 60, and80 The ionizing photon production rates areM

_. (QLyc)from Vacca, Garmany, & Shull (1996). These are shown in

Figure 21.In building up the stellar population of a star cluster, we

take advantage of the fact that the stellar IMF should beinterpreted as a probability distribution and that even thelow-mass clusters will have the same proportion of high-and low-mass stars, independent of the cluster mass (subjectto the condition that a very low mass cluster cannot have astar of mass greater than that of the cluster). Thus, if onesamples enough low-mass clusters, their average populationis the same as a single high-mass cluster with the entirerange of stellar masses sampled. An ensemble of lower massclusters will therefore have the same Lyman continuumoutput per unit cluster mass as a higher mass cluster with a““ saturated ÏÏ IMF (Oey & Clarke 1998). For this reason, ifone is interested in the behavior of the high-luminosity tailof the luminosity function (and not the dispersion of theluminosity function at lower ““ unsaturated ÏÏ luminosities),one can calculate the evolution of a single ““ saturated ÏÏcluster and scale the properties directly with cluster mass toderive the mean properties of lower mass clusters.

13.2. Evolution of Cluster L uminosity and QLycOnce assembled, we track the evolution of the cluster

Lyman continuum and luminosity, removing high-massstars after their main-sequence lifetimes. In Figure 22, thetemporal evolution of the Lyman continuum emission rateand total luminosity are shown scaled to a cluster mass of103 (between 1 and 120 For lower or higher massM

_M

_).

clusters, the mean and L can be scaled linearly withQLycmass (as long as one is analyzing a large population ofclusters). Assuming formation of all stars in the cluster in ashort time compared with the stellar evolution times, theLyman continuum emission rate will remain constant untilan age equal to the main-sequence lifetime of the most

FIG. 21.ÈAdopted Lyman continuum production, stellar luminosity,and main-sequence lifetimes as a function of stellar mass. The Lymanemission rate is from Vacca et al. (1996), and the main-sequence lifetimesand luminosities are from Renzini & Buzzoni (1986) and Maeder (1987)for ¹10 and º10 respectively.M

_,

No. 6, 2001 M51 3039

FIG. 22.ÈLyman continuum emission rate and luminosity for a““ saturated ÏÏ cluster as a function of time, scaled to a total cluster mass of103 with a Salpeter IMF from 1 to 120 Both the luminosity andM

_M

_.

can be scaled directly proportionally to cluster mass for lower andQLychigher mass clusters.

massive stars (D3 ] 106 yr ; see above and Fig. 21). Subse-quently, Q will decay exponentially in time with an e-foldingtime of D106 yr out to 107 yr, at which point the Lymancontinuum is relatively insigniÐcant (Fig. 22).

Integrating over the lifetime of the 103 cluster, theM_total number of Lyman continuum photons is 1.2 ] 1064.

This implies 1.2] 1061 Lyman continuum photons persolar mass of stars between 1 and 120 or equivalentlyM

_,

4.8] 1060 Lyman continuum photons per solar mass ofstars between 0.1 and 120 The latter estimate is 65%M

_.

greater than that derived by Kennicutt, Tamblyn, &Congdon (1994) for the same IMF and mass range. Theagreement is quite acceptable given the fact that their rela-tion was derived using Kurucz (1992) model atmospheresrather than the Vacca et al. (1996) observationally con-strained ionizing outputs, which are generally higher thanKuruczÏs.

Thus, for a steady state star formation rate, SFR, the OBclusters yield

QLyc/SFR\ 1.52] 1053 s~1 (M_

yr~1)~1 (11)

for a Salpeter IMF between 1 and 120 This result isM_

.useful for estimates of time- and global-averaged star for-mation rates (see ° 14).

13.3. Cluster MassesThe extinction-corrected Ha luminosities for the H II

region sample with diameters ¹50 pc (° 11) imply amaximum Lyman continuum emission rate of QLyc \ 7] 1050 s~1. This corresponds to cluster mass of 7 ] 103

(see Fig. 22), and a typical mean mass is D103M_

M_

.Above approximately 103 the cluster contains a rea-M

_,

sonable sampling of all stellar masses, and the stellar popu-lation is said to be ““ saturated ÏÏ (cf. Oey & Clarke 1998). TheLyman continuum production rate grows linearly forfurther increases in the cluster mass. If the Salpeter IMF isextended down to 0.1 the total cluster mass estimate isM

_,

increased by a factor of 2.5.We should stress that given the large number of H II

regions occupying the unsaturated (Ñat) part of the lumi-nosity function, estimates of the individual cluster masses forthese unsaturated H II regions is clearly very risky based onHa luminosities (or any Lyman continuum measure). Sincetheir stellar populations are poorly sampled, the ratio ofUV output to total stellar mass must Ñuctuate greatly from

one unsaturated region to another. And of course, sincethese regions are selected for having detectable Ha, they willbe biased toward high ratios of Lyc per solar mass of stars.

13.4. H II Region L uminosity FunctionsIn order to model the observed H II region luminosity

function, one must sample the modeled OB star clustersover a range of cluster masses and over the lifetime of theOB stars. Assuming approximately coeval star formation,the temporal evolution of the Lyman continuum luminosityfrom the clusters is determined by stellar evolution of thehighest mass stars. The luminosity function of the H II

regions can then be modeled by sampling uniformly in timethe Lyman output from each cluster mass, weighted by theinitial mass distribution of clusters.

In analogy with the treatment of stellar IMF, we assumethat the initial cluster masses can also be represented as apower law:

N(Mcl)dMclP Mcl! . (12)

In Figure 23, the luminosity functions are plotted for Ðvevalues of the mass power-law index ! between [1 and [3(eq. [12]). The slope of high-luminosity end of the lumi-nosity function is directly determined by the cluster massspectrum index ! (McKee & Williams 1997), since thehighest luminosities are contributed by ““ saturated ÏÏ clustersduring their initial constant-luminosity phase (\3 ] 106yr). The model luminosity functions shown in Figure 23clearly show that Ñat or nearly Ñat distributions of clustermass are ruled out by the observed luminosity functions. Anexcellent match to the observed luminosity function power-law index [1.01 is thus provided by N(Mcl)dMclPMcl~2.01with variations of *!\ ^0.15 between arm and interarmregions. The derived power-law index for the cluster massspectrum is close to, but not identical to, that of GalacticGMCs (!^[1.6 ; Scoville & Sanders 1987).

Thilker et al. (2001) did a similar analysis of their Hameasurements to obtain an initial cluster mass spectrum

(after exploring a range of [1.75 toN(Mcl)dMclPMcl~22.25). Both our and their results provide strong evidence ofa steeply falling cluster mass spectrum. These cluster massspectra are also in excellent agreement with the value!\ [2 derived by Larsen (2000) and Bik et al. (2001) from

FIG. 23.ÈExpected luminosity functions for Ðve values of the clustermass power-law index (!). The normalized luminosity functions perlogarithmic interval in Lyman continuum emission rate were computedassuming an equal probability of observing a given cluster at any timeduring its active lifetime.


UBV photometry of the clusters in M51 (as opposed to theionized gas).

The luminosity functions shown in Figure 23 were com-puted for a cluster mass range between 500 and 5000 M

_.

The adopted upper limit in the cluster mass produces thecuto† in the luminosity function at s~1.QLyc \ 4 ] 1050For cluster with a ““ saturated ÏÏ IMF, the location of thisbreak will scale linearly with the upper limit to TheMcl.upper limit to cluster mass is ¹7000 (between 1 andM

_120 for the D¹ 50 pc sample, assuming a factor of 10M_)

increase for extinction. Moreover, since some of the mostluminous regions in the sample are possibly blends, the realupper mass limit is probably less.

13.5. Upper ““Cuto† ÏÏ of Cluster MassHere, we brieÑy explore the possibility that there is a

physical mechanism limiting the buildup of clusters moremassive than a few times 103 In the following, we trackM

_.

the feedback of radiation pressure on buildup of a stellarcluster in a molecular cloud core. The important role ofradiation pressure in high-mass star formation has not beenamply recognized.

We posit a simple model for cluster buildup in which theprotocluster forms at the center of a molecular cloud core inwhich the density rises with decreasing radius. We assumethat initially the entire cloud core is in free-fall collapse. Thestars within the growing cluster have a Salpeter IMFÈthus,since lower mass stars form with a higher probability, theinitial low-mass cluster will have few high-mass stars.However, as the cluster grows and becomes more massivethe IMF is populated to higher mass. The surrounding gasfeels the gravitational attraction of the interior mass M

R(stars plus the interior core gas). Lastly, we assume that gasaccreted inside the adopted cluster radius is added to thecluster, and the total cluster mass is instantaneously redis-tributed with the Salpeter IMF.

This approach thus neglects the complication that low-or high-mass stars might preferentially form at di†erentepochs due to di†erences in the physics of their formationor their di†erent preÈmain-sequence evolutionary times.However, we should note that the assumptions made by usdo not really require that all stars form at the same time, butrather that the prestellar condensations that become starsof di†erent mass have formed and accreted into the cloudcore region. Since the low-mass stars have low luminosity,they are only important for their gravitational mass as faras our model is concerned. Thompson et al. (1998) Ðndevidence that the low- and high-mass star formation inNGC 2264 is coeval to within 103È104 yr.

Initially, the cluster will contain only a few low-massstars, and the gas dynamics will be entirely determined bythe self-gravity of the cloud core and cluster. Since weneglect rotational or magnetic stresses, this is clearly themost favorable situation for accretion and maximal growthof the cluster. On the other hand, as the cluster becomesmore massive and populates the upper main sequence, thehigher luminosity-to-mass ratio of high-mass stars willresult in increased radiation pressure on the surroundingdustÈeventually terminating further accretion to the cloudcore. The outward radiation pressure will dominate self-gravity at radius R when

L SiT4nR2cº

GMR

R2 , (13)

where SiT is the e†ective radiative absorption coefficientper unit mass.

Although the original stellar radiation is primarily UVand visible, the dust in the cloud core absorbs these photonsand reradiates the luminosity in the infrared. The““ e†ective ÏÏ absorption coefficient takes account of the factthat outside the radius where mag, the luminosity isA

VD 1

at longer wavelengths where the dust has a reduced absorp-tion efficiency. For the standard ISM dust-to-gas ratio(Bohlin 1975), mag corresponds to a columnA

V\ 1 NH \

2 ] 1021 cm~2. We therefore adopt

SiT \ 312jV

jeff(R)cm2 g~1 , (14)

with as the absorption coefficientÈweighted meanjeff(R)wavelength of the radiation Ðeld at radius R, and we adopta j~1 variation of the absorption efficiency with wave-length.

Combining equations (13) and (14), we Ðnd that the radi-ation pressure will exceed the gravity of the cluster starswhen

(L /M)clº 42jeff/jV(L /M)

_. (15)

If km, Thus, for clusters withjeff D 3 jeff/jV D 10.luminosity-to-mass ratios exceeding D500 radi-(L /M)

_,

ation pressure will halt further accretion. This luminosity-to-mass ratio is reached at about the point when the uppermain sequence is Ðrst fully populated, i.e., a cluster withapproximately 2000 distributed between 1 and 120M

_In the above discussion, we conservatively adopt lumi-M_

.nosities corresponding to the main sequence rather than themuch larger, short-term, preÈmain-sequence luminosities.This assumption is thus most conservative in the estimationof the radiation pressure e†ects. We have also assumed thedensity of dust and gas to be only a function of radius. If thematerial is instead contained in optically thick clumps(possible individual prestellar condensations) or a disk, thenthe strength of the radiation pressure compared withgravity is reduced by a factor that depends on the opacity ofthe clumps and their areal covering factor at each radius.Since neither of these factors are constrained by existingobservations, we have adopted the simplest case of uniformareal coverage and spherical symmetry. In reality, the gas islikely to be clumped and there will be multiple core regionswithin each GMC. The latter would certainly be unresolvedin ground-based Ha imaging and probably even at thehigher resolution used here. On the other hand, it may bethat even if there are multiple cores, not all will have theircluster formation synchronized to within 105 yr (the time-scale of the phenomena we have been discussing).

In the above, we assumed the most favorable conditionsfor cluster growthÈfree-fall collapse and no rotational ormagnetic impediments. Additional outward pressure can, ofcourse, arise from the hot ionized gas associated with theOB stars and their stellar winds ; these e†ects areundoubtedly important later in the evolution of the cluster,but they are also more dependent on the H II regiongeometry. By contrast, radiation pressure is inescapableduring the early phases of cluster formation and thereforemust play a central role in regulating the initial clustergrowth. The subsequent dynamic evolution of cloud corewill be a†ected by all three : radiation pressure, H II expan-sion, and stellar winds.

No. 6, 2001 M51 3041

Elmegreen (1983) has also pointed out the importance ofradiation pressure from a forming star cluster in possiblylimiting the maximum mass of star clusters. He used some-what di†erent numerical values for the dust opacity and didnot consider the reduced e†ective dust absorption crosssection due to the reprocessing into the infrared. Althoughhe used a higher e†ective dust cross section, this was com-pensated by his assumption that there will be substantialgas mass in the intracluster space, and he arrives at a similarlimiting cluster mass of D103 M

_.

13.6. OB Star Cluster Formation and ExpandingAssociations

To understand better the limit on the core stellar popu-lation of OB star clusters, we have numerically modeled thecollapse of a cloud core with a growing star cluster. Thecloud core, having an initial power-law radial densityproÐle,

n \ n0(R/Rcl)~2 , (16)

is assumed to start in free-fall collapse with VR

\Vff \In the subsequent dynamic evolution of the[(2GMR/R)1@2.

cloud core, envelope gas getting inside the adopted clusterradius (taken to be 0.01 pc) is added to the cluster withRclan efficiency of 50%. At each time step, the cluster mass isdistributed into stars with a Salpeter IMF and the clusterluminosity updated.

To compute the radiation pressure at each radius, weadopt an e†ective wavelength for the radiation from

jeff(R)^TV

jV

(1[ e~qV(R))TD(R)] Tcl e~qV(R) , (17)

where the dust temperature is given by

TD(R)\ 70

A L105 L

_

B1@5A2 ] 1017R

B2@5K . (18)

This scaling with L and R is based on infrared observationsof the OMC-1 cloud core (Werner et al. 1976). Equations(17) and (18) provide a reasonable approximation to thevariation of dust temperature in an optically thick cloudheated by a central luminosity source. Because of the highopacity of the cloud core, the dust is heated mainly byreradiated emission from interior shells (as opposed todirect stellar photons). For the e†ective temperature of thecluster luminosity, we adopt a constant value of Tcl\30,000 K, independent of the cluster mass.

The Lagrangian form of the hydrodynamic equation ofmotion was advanced in time with gravity, thermal, andradiation pressure terms. Mass shells were distributedlogarithmically with the less massive shells on the inside inorder that the transition from to was wellq

V\ 0 q

V\ 5

resolved. ArtiÐcial viscosity with a coefficient of 3 was intro-duced to stabilize the radiatively compressed shells (Christy1967). The gas was taken to be isothermal with an““ e†ective ÏÏ sound speed of 1 km s~1. The initial densityscale had cm~3 at 0.01 pc, and the cloud coren0\ 108extended out to a radius such that its total mass was2 ] 104 M

_.

The density and velocity proÐles at 50,000 yr intervalsafter the initial free-fall collapse are shown in Figure 24. Asexpected from the discussion above, the radiation pressurebecomes dominant over gravity for the inner shells of theinfalling gas when the cluster has reached D500 AtM

_.

this point, the infall of the inner shells is reversed, and theyrapidly accelerate outward and collide with the still infal-ling, exterior shells. The outer shells have high dust opa-cities (to the central cluster) and therefore are not subjectedto such high radiation pressures.

Where the outward and inward moving gases collide, adense, shock-compressed shell forms and acceleratesoutward. Typical outward radial velocities are D2È6 kms~1, and the density of the shocked layer is greatly enhancedover that of the initial density proÐle (Fig. 24). At this point,further accretion to the cluster core is e†ectively shut o†.For the parameters used in this calculation, the clusterended up with 881 after 500,000 yr. Without radiationM

_pressure, free-fall collapse with the same parameters wouldhave produced a cluster mass of 5800 in the sameM

_interval.It is interesting to note that the outward-moving shell

may propagate a second phase of stimulated star formationinto the cloud envelope. The compressed gas in this shellcan be unstable, collapse, and fragment into a second gener-ation of stars. The density of the compressed layer is typi-cally enhanced by a factor of 100È1000, becoming 105È106cm~3 at radii of a few parsecs. At T \ 50 K, these densitiesyield a Jeans length of pc and mass of 50 ThisD13 M

_.

second-generation star formation could become a signiÐ-cant enhancement to the initial cluster core mass providedthe density scale is high enough. Eventually, the densityn0in the outward-moving shell will drop because of sphericaldivergence and the fallo† of the outer envelope density. Theshell would then no longer be unstable, and stimulated starformation would stop. Stars formed within the expandingshell will, of course, have a net outward radial velocity of afew kilometers per second and be unbound from the corecluster. Evidence of radial expansion in OB associations isseen in the I Orion subgroup d (Trapezium cluster ; Blaauw1964) with a radial expansion of 2.5 km s~1 at the outeredge (RD 0.7 pc) of the cluster. It would be interesting toanalyze the Hipparcos data for nearby OB star clusters tolook for evidence of similar expansions. Dreher et al. (1984)discovered a ring (RD 0.4 pc) of ultracompact H II regionsin the W49 complex, each of which is ionized by an internalstar or star cluster ; this ring of high-mass star formationmight correspond to the second wave of triggered star for-mation discussed above.

Recent observations of luminous infrared galaxies haveshown considerably more luminous (presumably moremassive) superÈstar clusters (SSCs ; e.g., Whitmore et al.1999). In the context of the discussion above, it might bespeculated that such clusters could form as a result of aparticularly prodigious second wave of triggered star for-mation in unusually massive and dense molecular clouds.The galaxies hosting SSCs are largely interacting starburstsystems (e.g., M82 and the ““ Antennae ÏÏ galaxies) in whichthe molecular clouds are likely to be both more massive andof higher density. Clearly, if the density remains high furtherout from the star cluster core, the triggered wave of starformation can propagate further and generate a moremassive and more extended cluster.

A similar scenario for stimulated star formation in suc-cessive OB associations was suggested by Elmegreen &Lada (1970). In their model, the compression wave was dueto expansion of the hot, ionized sphere ratherStro� mgrenthan radiation pressure, and the motivation was to accountfor the temporal sequence of separate OB associations.

FIG. 24.ÈDensity proÐle for the cloud core. Cluster formation is shown at 0 (black), 50 (green), 100 (turquoise), 150 (blue), and 500 (red) ] 103 yr. Theinitial state was free-fall collapse in a R~2 density distribution normalized to 108 cm~3 at 0.01 pc. The Ðnal mass of the core cluster was 881 (comparedM

_with 5800 for free-fall collapse), and the luminosity was 106 The core accretion was fully halted by radiation pressure within 100,000 yr. After that,M_

L_

.the radiatively compressed shell moves outward at 2È6 km s~1. Since the compressed shell is likely to be Rayleigh-Taylor and Jeans unstable, star formationmay continue in the outward moving shell.

M51 3043

Their model is di†erent in both the physical mechanism forgas compression (H II region shock fronts) and ingeometryÈseparate OB associations formed in the com-pressed shell surrounding a large, expanding H II region.The model described above we discuss in more detail in aforthcoming paper.

A di†erent scenario for the formation of SSCs has beenmodeled by Tan & McKee (2001). They suggest that in aclumpy GMC with multiple cloud core regions, each of afew times 103 the SSC might form simply from havingM

_,

a simultaneously high efficiency of star formation in all ofthe cores. The SSC is then the composite of the multiple starclusters, and whether it is gravitationally bound or not thenjust depends on the overall efficiency of star formation forthe entire GMC. This is tantamount to assuming that alarge fraction of the original GMC mass is contained in thecore regions as opposed to the intraclump medium. (It isworth noting that the individual cores or clumps in theirmodel are posited to have masses of a few times 103 soM

_,

the clusters forming in each core still obey the radiationpressure limit discussed above.)

14. TOTAL LYMAN CONTINUUM AND STAR

FORMATION RATES

The total Ha luminosity can be used to estimate theoverall Lyman continuum output and formation rate of OBstars in M51. In doing this, we separate the di†use anddiscrete H II regions, and we must make corrections for theouter galaxy not covered in the WFPC2 images. In Table 4,we summarize the observed Ha luminosities, the adopted““ typical ÏÏ extinctions, extinction-corrected Ha luminosities,and implied Lyman continuum outputs for the separatecomponents and regions of the galaxy.

For the discrete H II regions, we adopt mag (seeAV

\ 3.1° 10) and therefore increase their luminosity by a factor of10 (eq. [8]). For the di†use gas, the extinction is probablyless, and we adopt mag and thus increase the di†useA

V^ 1

luminosity by a factor of 2.1 (eq. [7]). (The extinction of thedi†use gas could not be reliably estimated from our Paa,since it would require extremely accurate determinations ofthe backgrounds. The adopted 1 mag is simply based on thepresumption that since the di†use gas is more widespread,both between the dust clouds and at higher scale height, itsextinction is lower than for the discrete H II regions mea-sured here.) The Ha emission in the outer galaxy that was

not covered in the WFPC2 images was estimated by multi-plying the luminosity measured in the WFPC2 area by asimple scale factor (\1.92). This factor is equal to the ratioof the total observed Ha Ñux in M51 (Rand 1992) to thatmeasured by us in the WFPC2 area.

The total Lyman emission rates for the WFPC2 area andM51 are 3.0 and 7.3] 1053 s~1. The steady state star for-mation rates are 2.00 and 4.79 yr~1 for stars betweenM

_1 and 120 for the WFPC2 and total M51 areas (usingM_eq. [11]). The inferred rates are increased by a factor of 2.5 if

the IMF is extended down to 0.1 M_

.The total mass of star-forming gas can be estimated from

single-dish CO observations (e.g., Scoville & Young 1983).The mass within a 4 kpc radius is approximatelyH23 ] 109 and the total for the galaxy is 6 ] 109M

_, M

_(Scoville & Young 1983 after scaling for a conversion factorof cm~2/K km s~1 rather than 50%NH2

/ICO\ 2.2] 1022larger value used there). The implied cycling times for atypical H nucleus to pass into a new generation of stars istherefore 1.2] 109 yr for both the inner galaxy and totalM51. On the one hand, this time would be shortened if thestar formation associated with the H II regions extendsbelow 1 on the other hand, approximately 50% of theM

_;

mass absorbed in the stars actually gets recycled back intothe ISM through stellar evolution on a timescale of 109 yr(see Norman & Scoville 1988). If the same clusters formstars down to 0.1 the cycling time is reduced anotherM

_,

factor of 2.5 ; the shortness of this time suggests perhaps thatthe OB star clusters do not fully populate the lower massIMF.

15. CONCLUSIONS

High-resolution HST imaging of Ha and Paa has beenused to analyze the properties of H II regions and OB starformation in M51. The critical aspects of these data are thehigh resolution (4È9 pc, as compared with D100 pc inearlier ground-based imaging), which enables the clearseparation of individual star formation regions ; the abilityto determine and correct for dust extinction using theHa/Paa Ñux ratio ; and the high sensitivity, which enables usto probe H II regions with luminosity well below that ofM42 (the Orion Nebula). From the observational data, weÐnd the following :

1. A total of 1373 Ha emission regions were deÐned(using automated procedures). The total Ñux of these dis-

TABLE 4

M51 Ha AND LYMAN CONTINUUM

Obs. L Hab SAVTc Ext. Cor. L Hae QLycf

Region H II Typea (ergs s~1) (mag) fHa,extvcord (ergs s~1) (s~1)

WFPC area . . . . . . Discrete 2.81 ] 1040 3.1 10.0 2.8] 1041 2.0] 1053Di†use 6.56 ] 1040 1 2.1 1.4] 1041 1.0] 1053All 9.37] 1040 . . . . . . 5.2 ] 1041 3.0] 1053

M51 totalg . . . . . . . . All 1.80] 1041 . . . . . . 1.0 ] 1042 7.3] 1053

a ““ Discrete ÏÏ if contained in our H II region sample ; otherwise ““ di†use.ÏÏb Ha luminosity.c Adopted visual extinction.d Factor to multiply the observed Ha by ; see eq. (8).e Extinction-corrected Ha luminosity.f Required Lyman continuum emission rate.g The total Ha Ñux in M51 (Rand 1992). The total extinction-corrected luminosity and Q were computed by

multiplying the WFPC2 totals by a factor of 1.92, based on the ratio of the observed Ha in the WFPC2 andtotal areas.


crete regions constitutes about 31% of the total Ha emis-sion in the central 281A ] 223A of M51. The observed Haluminosities range from 1036 to 2 ] 1039 ergs s~1, and theirdiameters are mostly from 10 to 100 pc, with a mean valueof D30 pc. (For the higher luminosity and usually largerregions, the mean electron density is generally lower,strongly suggesting that some of these regions are likely tobe blends of multiple H II regions.)

2. The observed Ha luminosity function exhibits a broadpeak at ergs s~1, falling as on the high-L Ha D 1037 L Ha~1.01luminosity side. No evidence is seen for the break in theluminosity function at ergs s~1 reported inL Ha \ 1038.6ground-based studies. Compared with ground-based deter-minations, the luminosity function derived here is much lesspopulated at high luminosityÈvery likely most of theregions above 1039 ergs s~1 seen in ground-based studieswere blends of multiple lower luminosity regions that areseparated here.

3. SigniÐcant di†erences are seen in the luminosity func-tions of spiral arm, interarm, and nuclear H II regions, withthe spiral arm luminosity functions being Ñatter.

4. The observed correlation between H II region lumi-nosity and apparent size (L P D2) strongly implies that thehigh-luminosity regions are blends or superpositions ofmultiple lower luminosity regions.

5. For 209 regions that had º5 p detections in both Paaand Ha, the observed line ratios were used to derive theoverlying visual extinction assuming standard Galactic dustextinction curves and intrinsic line ratios given by case Brecombination. The implied extinctions range from A

V\ 0

to 6 mag, with an intensity-weighted mean value of SAVT \

3.1 mag. Thus, the observed Ha luminosities should beincreased by an average factor of 10, implying similarincreases in the OB star cluster luminosities and impliedstar formation rates. The high extinctions underscore theneed for extinction estimates in analyzing the properties ofH II regions and their associated OB star clusters.

6. The extinctions are also highly variable from region toregion and across individual regions. (In deriving the esti-mates quoted above, the line ratios were computed pixel bypixel and then averaged over each region.)

7. The most luminous regions have sizes º100 pc andare undoubtedly blends of multiple regions. This is clear,since their sizes are much larger than the maximum diam-eter (¹50 pc) to which an H II region might conceivablyexpand within the D3 ] 106 yr lifetime of the OB stars, andit is also consistent with observed correlation (L P D2)found between the measured luminosities and sizes of theH II regions.

8. A subsample of 1101 regions with sizes ¹50 pc there-fore constitutes those regions that might conceivably beionized by a single cluster. Their extinction-corrected lumi-nosities range between 2] 1037 and 1039 ergs s~1 (with nocorrections for dust absorption of the Lyman continuum orUV that escapes to the di†use medium). The range isroughly comparable to H II regions ranging between two-thirds of M42 (the Orion Nebula) and W49 (the most lumi-nous Galactic radio H II region). The upper limit forindividual cluster H II regions is therefore conservatively¹1039 ergs s~1.

In addition to the observational data, we have modeledthe Lyman continuum luminosity as a function of totalcluster mass with stars distributed with a Salpeter IMF

(1È120 and as a function of cluster age. This model isM_)

used to derive the cluster properties. We Ðnd the following :

1. The upper limit to the ¹50 pc sample luminosity func-tion corresponds to an ionizing photon production rate of

s~1, and the corresponding upper limitQLyc up^7 ] 1050for cluster masses is ¹5000 (The implied masses areM

_.

increased by a factor of 2.5, if the Salpeter IMF is extendeddown to 0.1 M

_.)

2. For a power-law distribution of cluster masses, we Ðndthat the observed peak in the luminosity function at 1037ergs s~1 clearly rules out Ñat or slowly falling cluster massspectra. The observed [1.01 power-law index on the high-luminosity tail of the luminosity distribution implies acluster mass spectrum of Thus, theN(Mcl)/dMclPMcl~2.01.e†ects of high-mass OB star cluster formation are contrib-uted to equally per logarithmic interval in cluster mass.

3. The highest mass clusters are approximately 1000 M_.

The parent molecular clouds are much more massive, andone must ask : why do the OB star clusters generally notbuild up to much greater mass? We suggest that the forma-tion of a massive cluster in a molecular cloud core is likelyto be terminated at the point when the luminosity-to-massratio of the cluster reaches D1000 at which radi-(L /M)

_ation pressure begins to dominate the self-gravity of thecluster. This is roughly the point at which the Salpeter IMFÐrst becomes fully populated.

4. A hydrodynamic model with an initial R~2 densitydistribution in free-fall collapse veriÐes that the core starcluster is likely to self-limit at D103 At this point,M

_.

radiation pressure e†ectively terminates further gas anddust accretion to the central core. However, we also Ðndthat a radiatively compressed shell will then propagateoutward at a few kilometers per second, possibly triggeringa second wave of star formation out to a few parsecsÏ radius.This may, in fact, be the mechanism for forming the moremassive star clusters (SSCs) seen in starburst galaxies, wherethe densities are likely to be higher out to larger radii in thecloud envelopes. The stars formed in the expanding shellwill have signiÐcant outward radial motion and will prob-ably be unbound.

Lastly, we have combined our modeling results with mea-surements of the total line emission to evaluate the totalLyman continuum output and global star formation rate.From the extinction-corrected Ha luminosity, we Ðnd

s~1 andQLyc,total\ 7 ] 1053 SFR(1È120M_) \ 4.17 M

_yr~1.

The NICMOS project has been supported by NASAgrant NAG 5-3042 to the NICMOS Instrument DeÐnitionTeam. This paper is based on observations with the NASA/ESA Hubble Space Telescope, obtained at the STScI, whichis operated by the Association of Universities for Researchin Astronomy, Inc., under NASA contract NAS 5-26555.We are very grateful to Rich Rand for several discussionsduring the course of our work and for permitting us accessto his Ha images. We also thank David Thilker for sharinghis H II region photometry software with us and forextremely constructive and thorough refereeing of thispaper. We also thank Jonathan Tan, Chris McKee, andCrystal Martin for very useful comments and suggestionson the paper. We thank Zara Turgel for careful proof-reading and comments on the manuscript.

No. 6, 2001 M51 3045

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