Accepted 23-Dec-2004totheAstrophysicalJournalLetters PreprinttypesetusingLATEXstyleemulateapjv.04/03/99 STATISTICAL CONFIRMATION OF A STELLAR UPPER MASS LIMIT M. S. Oey UniversityofMichigan,DepartmentofAstronomy,830DennisonBuilding,AnnArbor,MI 48109-1090,USA and C. J. Clarke InstituteofAstronomy,MadingleyRoad,CambridgeCB30HA,UK Accepted 23-Dec-2004 to the Astrophysical Journal Letters ABSTRACT 5 We derive the expectation value for the maximum stellar mass (m ) in an ensemble of N stars, as 0 max a function of the IMF upper-mass cutoff (m ) and N. We statistically demonstrate that the upper 0 up IMF of the local massive star census observedthus far in the Milky Way and Magellanic Clouds clearly 2 exhibits a universal upper mass cutoff around 120−200 M⊙ for a Salpeter IMF, although the result is n more ambiguous for a steeper IMF. a J Subject headings: stars: early-type — stars: fundamental parameters — stars: mass function — stars: statistics — open clusters and associations — galaxies: stellar content 7 1 1. INTRODUCTION 2. THEEXPECTATIONVALUEhmmaxi v The upper mass limit to the stellar initial mass func- Because of the decreasing power law form of the IMF, 5 3 tion(IMF) isa criticalparameterinunderstandingstellar the characteristic mass of the largest star formed in clus- 1 populations, star formation, and massive star feedback in tersofN starsdecreasesasN decreases. Figure 1demon- 1 galaxies. To date, the largest empirical mass estimate for strates this effect with a Monte Carlo simulation. N 0 an individual star is around 200−250 M⊙ for the Pis- is drawn for individual star clusters from the universal 5 tol Star (Figer et al. 1998) near the Galactic Center, and power-law distribution in N (e.g., Oey & Clarke 1998; 0 around 120−200 M⊙ for the most massive stars in the Elmegreen & Efremov 1997): / h Large Magellanic Cloud (e.g., Massey & Hunter 1998). In n(N) dN ∝N−2 dN , (1) p practice, most applications assume an upper mass limit o- to the IMF of mup ∼ 100 to 150 M⊙. However, there is andthe stellarmasses for eachcluster of N starsis drawn some confusion on whether the apparent observed upper r from the Salpeter (1955) IMF, within a mass range of 20 st limitsimplyrepresentsastatisticallimitowingtoalackof to 100 M⊙: sampledstarsinindividualclusters(Massey2003;Massey a : & Hunter 1998; Elmegreen 1997). φ(m) dm∝m−2.35 dm . (2) v BeforetheadventoftheHubbleSpaceTelescope(HST), i Figure1showsthedistributionofthemostmassivestarin X stars with extremely high masses ∼> 1000 M⊙ were sug- each cluster, m vs logN. For single stars, we confirm gestedtoexist. ThedensestellarknotR136ainthe30Do- max r that the bin of logN = 0 is described simply by the IMF a radusstar-formingregionoftheLMCwasthe best-known (equation 2). It is apparent that for large N, one can ex- candidate for harboringsuch a star (Cassinelli, Mathis, & pect that m ≃ m , but that for small N, the typical Savage 1981). The viable candidates for these supermas- max up most massive star is much lower in mass. For N =1, the sive stars were eventually resolved by HST and ground- basedimagingintosmallerstarswithintheconventionally typical mmax is the mean of the IMF, which is 37 M⊙ for observed mass range (e.g., Weigelt et al. 1991; Heydari- the distribution of 20≤m≤100 M⊙ used in Figure 1. Wecananalyticallyderivetheexpectationvaluehm i Malayeri, Remy, & Magain 1988). In recent years, how- max forthe mostmassivestarinanensembleofN starsasfol- ever, the possibility of supermassive stars is receiving re- lows. For N stars,the probability that all are in the mass newed attention as a possible mode of star formation in range 0 to M is, the early universe (e.g., Bond, Arnett, & Carr 1984; Lar- son 1998; Bromm, Kudritzki & Loeb 2001). M N It is therefore important to clarify expectations for the P(0,M)= φ(m) dm , (3) highest-mass stars compared to the existing observations. "Z0 # Elmegreen(2000)quantitativelydemonstratesthat,inthe whereφ(m)correspondstotheIMF,i.e.,aprobabilitydis- absence of an upper-mass cutoff, stellar masses should be observedupto40,000 M⊙ fortheentireMilkyWay,based tribution function whose integral is unity. It follows that the probability that all the stars are in the mass range 0 onestimatesforthecurrentstarformationrateandmolec- to M +dM is, ular gas mass. Here, we derive the behavior of the ex- pectation values for the most massive stars and demon- N N M M d strate that, for a universal IMF, current observations in- P(0,M+dM)≃ φ(m)dm + φ(m)dm dM deed show the existence of an upper-mass limit around dM "Z0 # "Z0 # mup ∼120−200 M⊙. (4) 1 2 Stellar Upper Mass Limit Fig. 1.— Monte Carlosimulationshowing the maximum stellar mass mmax per cluster vs the number of stars logN per cluster for5000 clustersinadistributionofN givenbyequation1. ASalpeterIMFisadoptedwithstellarmassesbetween20–100 M⊙ inthissimulation. by Taylor expansion. Thus we see that the probability value of m is considerably greater than the observed max that the most massive star is in the range M to M +dM maximumof∼120−200 M⊙ in R136a,unless mup is low is, (≪500 M⊙). Weidner&Kroupa(2004)reachedthesame conclusionfrom a similar analysis of R136a;Selman et al. N M (1999) also suggested a cutoff using a less rigorous analy- d P(M,M +dM)= φ(m) dm dM , (5) sis. We can furthermore assess the statistical significance dM "Z0 # of this result by calculating p(mmax|mup), the probability of obtaining an observed maximum stellar mass ≤ m max and the expectation value for the most massive star is, foragivenm . Table1listsp(m |m )calculatedfrom up max up equation 5 for R136a for a range of m . This demon- N up mup d M strates the negligible likelihood (< 10−5) that R136a is hm i= M φ(m) dm dM . (6) max dM drawn from a population which extends to 1000 M⊙. Z0 "Z0 # The results of Weidner and Kroupa (2004), and those presented here, do not support the suggestion by Massey Integrating by parts, this yields, (2003) and Massey & Hunter (1998) that the upper IMF m M N in the R136a is consistent with mup = ∞. Massey & up Hunter (1998) found that the penultimate mass bin in hm i=m − φ(m) dm dM . (7) max up the empirical mass function is fully consistent with the Z0 "Z0 # Salpeter slope. However, they omit from their mass func- ForlargeN,equation7confirmsthathm i→m ,cor- tion, and from their analysis, stars with inferred masses max up respondingtoanIMFthatiswell-sampleduptotheupper > 120 M⊙, because the lack of stellar models in the grid preclude reliable mass determinations. For the two effec- mass limit (termed “saturated” by Oey & Clarke 1998). tive temperature scales they adopted, there are 2 or 9 of We numericallyintegrateequation7using a lowermass limitmlo =10 M⊙ insteadof0,andassumingtheSalpeter these omitted, most-massive stars. Although we do not knowthe exactmasses,their numbers are sufficientto de- IMF. Figure 2 shows the expectation value for the most massive star hm i vs the upper mass limit m for termine whether an upper mass cutoff to the IMF power max up N = 100, 250, and 1000 stars (solid lines). The dotted lawexists. ForaSalpeterIMF,intheabsenceofanupper line shows the identical relation hm i = m for com- masscutoff,thereshouldbeatotalof1.7timesmorestars max up parison. ForlowerN,hm iissmalleratanygivenm , at m > 120 M⊙ than are found in the the mass bin 85 – max up as expected. 120M⊙. Massey & Hunter (1998) count (8, 11) stars in the latter mass bin, therefore implying that (14, 19) stars 3. RESULTS should be found at higher masses. This is significantly more than the (2, 9) stars found. Thus, R136a exhibits a 3.1. R136a cutoff around 120−200 M⊙, consistent with the finding We start by comparing Figure 2 to the R136a region by Weidner & Kroupa (2004). in 30 Doradus, which, at an age of 1–2 Myr (Massey & Hunter 1998), is sufficiently young that none of its stars 3.2. A sample of young OB associations have expired yet as supernovae. We consider stars hav- ing m > 10 M⊙, of which Hunter et al. (1997) found Although a truncated IMF in R136aseems conclusively N = 650 in this region. We however note that this value demonstrated,itispossiblethatthedense,richclusteren- represents a strong lower limit, since the star counts are vironmentofthis regionrepresentsa specialcase. Canwe significantlyincompletebetween10and15M⊙ (Massey& drawasimilarconclusionfromawidersampleofordinary Hunter1998). Figure2demonstratesthattheexpectation OBassociations? Toexaminethisfurther,weconsiderthe M. S. Oey & C. J. Clarke 3 Fig. 2.—The expectation value hmmaxivs upper mass limitmup, forN =100, 250, and 1000 starshaving massesabove mlo =10 M⊙, assumingaSalpeterIMF.Thedottedlineshowsmmax=mup forcomparison. upperIMFfromthesubstantialsampleofOBassociations probabilities that these values are uniformly distributed, thathavebeenuniformlystudiedbyMasseyandcollabora- ofP <0.002, <0.02, <0.12,and<0.47for,respectively, tors,whoestimatedstellarmassesfromspectroscopicclas- mup = 104, 200, 150, and 120 M⊙. We also compute P sifications. Massey,Johnson, & DeGioia-Eastwood(1995) foranadoptedobservedmmax =200M⊙,asmightbepos- tabulate the numbers of stars having m ≥ 10 M⊙ in the sibleforTr14/16andR136a(Table1). Figure4showsthe MilkyWayandLMCassociations. Tominimize thepossi- respective results in this case: P < 0.002, < 0.002, 0.47, bility that the most massive stars have already expired as and < 0.92 for mup = 104, 103, 200, and 150 M⊙. We supernovae, we count only stars in OB associations with therefore see that mup =∞, and even 103 M⊙, are effec- ages ≤ 3 Myr. Table 1 shows the observed N(≥ 10M⊙) tively ruled out. Hence the results from this wider total and m for these objects. sample of OB associations points to an upper-mass limit max We now compute p(mmax|mup) for all the objects (Ta- to the IMF around the observed values of 120−200 M⊙. ble 1). These show that, although none of these regions individually provide strong constraints onthe upper mass 4. CONCLUSION cutoff, they collectively point to a conclusion similar to that found for R136a. The total N =263 stars, for which We have analyzedthe upper IMF in a sample ofyoung, inspectionofFigure2againshowsthattheobservedmax- nearbyOB associationsthatbestrepresentsstellarcensus imum stellar masses imply that m should not exceed data in this regime. The clusters are young enough that up a few hundred M⊙. Elmegreen (2000) reached a similar their highest-mass members remain present, and the stel- conclusion based on the lack of supermassive stars in the larmassesarespectroscopicallydeterminedbyMasseyand entire population of the Milky Way. In considering the collaborators(Masseyetal. 1995;Massey&Hunter1998). total of 263 stars, or Milky Way population, we assume Ourresultsprovideclearevidenceforanuppertruncation that the IMF is a universal probability distribution func- intheIMF.Whilethisresulthasbeenpreviouslynotedby tion that is independent of specific conditions in individ- Weidner&KroupainthecaseofR136a,weshowherethat ualclustersandparentmolecularclouds. Indeed,theIMF italsoappliesto amuchwidersampleofOBassociations. is conventionallytreated as a universalfunction (see, e.g., Wehavefurthermorequantifiedthestatisticalsignificance Elmegreen2000). Wealsoemphasizethatourtotalcounts of such statements. For example, we find that the proba- of N are conservative lower limits, since additional young bilitythatthestellarpopulationofR136aisdrawnfroma massivestarscanbe countedfromassociationsstudiedby parent distribution having mup= 104 M⊙ is < 10−5, and other authors. We chose not to include these additional forotherassociationstheprobabilityisonlyafewpercent. stars in the interest of maintaining a uniform and well- It should be noted that our results are sensitive to the understood sample. slope of the IMF for stars more massive than 10 M⊙. In Furthermore, we can now evaluate the total probabili- this mass range, it is often reported that the IMF power- ties P that the values of p(m |m ) represent uniform law exponent is close to the Salpeter value ∼−2.35 (e.g., max up distributions between 0 and 1, as expected for any uni- Massey 2003; Schaerer 2003; Kroupa 2002). However, versal m . For example, we would expect 10% of the should the slope through some systematic observational up regions to fall into the category where p(m |m ) was bias be significantly steeper, then our demonstration of max up ≤ 0.1, 20% to have a p(m |m ) of ≤ 0.2, and so an upper-mass cutoff becomes less vivid. For example, max up on. Figure 3 shows, for each assumed m of the par- we find that adopting an IMF slope of −2.8 yields an up ent IMF, the distribution of p(m |m ) for the individ- aggregate probability that the clusters originate from an max up ualregions. Itisevidentthatforhighervaluesofmup,the IMF having mup= 104 M⊙ of P < 0.30, contrasted to valuesofp(m |m )areunacceptablyclusteredtowards P <0.002 for the Salpeter slope. Conversely,for a parent max up smallvalues. AK-Stestconfirmsthis conclusion,yielding IMFslope flatter thanthe Salpeter value, the existence of anupper-masscutoff is evenmorestronglydemonstrated. 4 Stellar Upper Mass Limit Table 1 Sample of OB Associations Name N(>10 M⊙) mmax p(104) p(103) p(200) p(150) p(120) R136aa 650 120 10−10 10−10 10−5 0.002 1.000 R136ab 650 200 10−5 10−5 1.000 ··· ··· Berkeley86 10 40 0.188 0.192 0.224 0.244 0.268 NGC7380 11 65 0.400 0.409 0.486 0.534 0.592 IC1805 24 100 0.335 0.350 0.510 0.626 0.784 NGC1893 19 65 0.206 0.213 0.288 0.338 0.404 NGC2244 12 70 0.407 0.416 0.502 0.556 0.623 Tr14/16a 82 120 0.055 0.064 0.231 0.464 1.000 Tr14/16b 82 200 0.236 0.276 1.000 ··· ··· LH10 65 90 0.032 0.037 0.102 0.176 0.324 LH117/118 40 100 0.161 0.174 0.326 0.458 0.666 aValuesobtainedbyadopting120 M⊙ forthemostmassiveobservedstars. aValuesobtainedbyadopting200 M⊙ forthemostmassiveobservedstars. Fig. 3.—Distributionofp(mmax|mup)forthesampleofOBassociations inTable1,adopting mmax =120 M⊙ forR136aandTr14/16. The aggregate probability that these distributions originate from a uniform distribution are P < 0.002, < 0.02, < 0.12, and < 0.47 for, respectively, mup =104, 200, 150,and120 M⊙. M. S. Oey & C. J. Clarke 5 Fig. 4.— Same as Figure 3, but adopting an observed mmax for R136a and Tr 14/16 of 200 M⊙. The aggregate probability that these distributions originate from a uniform distribution between 0 and 1 are P < 0.002, < 0.002, < 0.47, and < 0.92 for, respectively, mup =104, 103, 200,and150 M⊙. For R136a, p(104) = 6×10−4 for the steeper slope (cf. not alreadyexpired, and it therefore depends critically on Table 1),whichisstillanegligibleprobability. Weidner& the reliability of evolutionary models for the most mas- Kroupa (2004)examine the influence of the slope in more sivestars,andonthereliabilitywithwhichonecanassign detail. ages to OB associations. It also assumes that the stars Thus, given the standard Salpeter slope for massive in the OB associations are coeval. Should the star for- stars, it is hard to escape the conclusion that the IMF mation process indeed be suppressed at high masses, as is truncated near mup ∼ 120− 200 M⊙, based on this suggested by our results, a major goal for theorists will analysis. If these results are real, the only other possi- be to identify the physics, e.g., plausibly associated with bilities are that the IMF is not universal, or there is an stellar feedback, that introduces this mass scale into the extreme selection effect that prevents our observations of star formation process. the most massive stars. We note that m need not be up an absolute limit, but represents at least a dramatic drop We are pleased to acknowledge discussions with Phil from the power-law form of the IMF. Our conclusion de- Massey and Don Figer. We also thank the referee, Rolf pends onthe assumptionthat the highestmassstarshave Kudritzki, for useful comments. REFERENCES Bond,J.R.,Arnett,W.D.,&Carr,B.J.1984,ApJ,280,825 Massey,P.2003, ARA&A,41,15 Bromm,V.,Kudritzki,R.P.,&Loeb,A.2001,ApJ,552,464 Massey,P.&Hunter,D.A.1998,ApJ,493,180 Cassinelli, J. P., Mathis, J. S., & Savage, B. D. 1981, Science 212, Massey,P.,Johnson,K.E.,&DeGioia-Eastwood,K.1995,ApJ,454, 1497 151 Elmegreen,B.G.2000,ApJ,539,342 Oey,M.S.&Clarke,C.J.1998,AJ,115,1543 Figer,D.F.,Najarro,F.,Morris,M.,McLean,I.S.,Geballe,T.R., Salpeter,E.E.1955,ApJ,121,161 Ghez,A.M.,&Langer,N.1998,ApJ,506,384 Schaerer,D.2003, inAMassiveStarOdyssey, fromMainSequence Heydari-Malayeri,M.,Remy,M.,&Magain,P.1988,A&A,201,L41 to Supernova, IAU Symp. 212, K. A. van der Hucht, A. Herrero, Hunter,D.A.,Vacca,W.D.,Massey,P.,Lynds,R.,&O’Neil,E.J. &C.Esteban(eds.),642 1997,AJ,113,1691 Selman,F.,Melnick,J.,Bosch,G.,&Terlevich,R.1999,A&A,347, Kroupa,P.2002,Science, 295,82 352 Larson,R.B.1998,MNRAS,301,569 Weidner,C.&Kroupa,P.2004,MNRAS,348,187