The early evolution of the star cluster mass function
aa r X i v : . [ a s t r o - ph . GA ] J a n Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 7 June 2018 (MN LaTEX style file v2.2)
The early evolution of the star cluster mass function
M. Gieles ⋆ European Southern Observatory, Casilla 19001, Santiago 19, Chile
Accepted 2009 January 6. Received 2009 January 6; in original form 2008 December 4
ABSTRACT
Several recent studies have shown that the star cluster initial mass function (CIMF) can bewell approximated by a power law, with indications for a steepening or truncation at highmasses. This contribution considers the evolution of such a mass function due to cluster dis-ruption, with emphasis on the part of the mass function that is observable in the first ∼ − at low massesand an exponential truncation at M ∗ . Cluster disruption due to the tidal field of the host galaxyand encounters with giant molecular clouds flattens the low-mass end of the mass function,but there is always a part of the ‘evolved Schechter function’ that can be approximated by apower law with index − . The mass range for which this holds depends on age, τ , and shifts tohigher masses roughly as τ . . Mean cluster masses derived from luminosity limited samplesincrease with age very similarly due to the evolutionary fading of clusters. Empirical massfunctions are, therefore, approximately power laws with index − , or slightly steeper, at allages. The results are illustrated by an application to the star cluster population of the interact-ing galaxy M51, which can be well described by a model with M ∗ = (1 . ± . × M ⊙ and a short (mass-dependent) disruption time destroying M ∗ clusters in roughly a Gyr. Key words: galaxies: star clusters – open clusters and assocations: general – globular clus-ters: general
The survival chances of young ( . M , andenvironment has been the topic of quite some debate re-cently. Two models for the early evolution of clusters have re-cently become available, with a very different role of M inthe disruption description: a mass-dependent disruption model(Boutloukos & Lamers 2003, hereafter BL03; Lamers et al. 2005a;Lamers, Gieles & Portegies Zwart 2005b) and a mass-independentdisruption model (Fall et al. 2005; Whitmore et al. 2007). Bothdisruption models assume a continuous power law cluster ini-tial mass function (CIMF) with index − , based on empiricalderivations of the mass function of young ( ∼
10 Myr) clusters withmasses between ∼ M ⊙ and ∼ M ⊙ in a variety of galac-tic environments (see for example Zhang & Fall (1999) for thecase of the clusters in the Antennae galaxies; Bik et al. (2003) forclusters in M51; McCrady & Graham (2007) for cluster in M82; ⋆ E-mail: [email protected] de Grijs & Goodwin (2008) for the SMC and de Grijs et al. (2003a)for a compilation of clusters in different galaxies) .Based on a study of the age and mass distributions of starclusters in four different galaxies, BL03 conclude that the lifetimeof star clusters, or disruption time-scale, t dis , depends on M as t dis ∝ M γ , with γ ≃ . ± . and with the proportionalityconstant dependent on galactic environment. This mass-dependentdisruption (MDD) model is supported by our theoretical under-standing of the disruption of star clusters, since the same scalingof t dis with M , i.e. the same value of γ , was found from N -bodysimulations of clusters dissolving under the combined effect of in-ternal relaxation and external tides (Baumgardt & Makino 2003;Gieles & Baumgardt 2008), and also for the disruption of clustersdue to external perturbations, or ‘shocks’. In fact, t dis due to shocksdepends on both M and the half-mass radius, r h , since t dis scaleswith the cluster density within r h (Spitzer 1958; Ostriker et al.1972), which combined with the shallow dependence of r h on M for (young) clusters ( r h ∝ M . , e.g. Larsen 2004), leads to a sim-ilar value of γ as for tidal disruption. The observationally derived CIMF from young ( .
10 Myr) clusters willbe referred to as the CIMF emp . This is because the number of massive clus-ters younger than ∼
10 Myr is not necessarily sufficient to reveal the shapeof the CIMF at high masses. See Section 2.1.1 for more details.c (cid:13)
M. Gieles
MDD flattens the low-mass end of the cluster mass function,d N/ d M , at old ages. Fall & Zhang (2001) show that for a con-stant mass-loss rate ( γ = 1 ), the mass function of a single-agecluster population evolves towards a flat distribution at the low-mass end, independent of the initial shape of the mass function.Such a mass function is in good agreement with that of the Galacticglobular cluster system. However, the globular cluster mass func-tion (GCMF) is usually derived from the luminosity function us-ing a constant mass-to-light ratio, M/L . Mandushev et al. (1991)find from dynamical mass estimates of Galactic globular clustersthat their
M/L is actually an increasing function of M , whichcould be the result of the preferential depletion of low-mass stars(Kruijssen & Lamers 2008). Such a mass-dependent M/L altersthe shape of the GCMF and could allow for smaller values of γ (Kruijssen & Portegies Zwart, in prep). A general expression for thelogarithmic slope at low masses due to MDD is γ − (Lamers et al.2005a).An MDD model for the early evolution of star clusters wasintroduced by BL03. In the four galaxies studied by BL03, onlymasses for clusters in M33 and M51 were available. The mass func-tions of clusters in these galaxies were presented for all ages, sono evidence for a flattening of the mass function for a selected setof old clusters was provided, although these integrated mass func-tions are slightly flatter at low masses than a power law with index − . de Grijs et al. (2003b) report a turnover in the mass function ofclusters in region B of M82 and they claim that the population isapproximately coeval ( ∼ steepens with age, from a power law function withan index of − . at ∼
10 Myr to one with an index of − . at ∼ − .Zhang & Fall (1999) determined the mass function of clus-ter in the Antennae aged between 25 and 160 Myr and show thatit has essentially the same shape as that for the clusters between2.5 and 6 Myr, namely a power law with index − . Fall et al.(2005) show that the age distribution, d N/ d τ , of mass limited sub-samples of clusters in the Antennae galaxies declines as τ − . Suchan age distribution can result from a constant formation historycombined with a 90% disruption fraction each age dex. Togetherwith the similarity of the mass functions at different ages the au-thors concluded that the disruption of star clusters in the first Gyris independent of their mass. Several claims have been made thatthis mass-independent disruption (MID) model also describes theage distribution of clusters in other galaxies, such as the SMC (Chandar et al. 2006) and M33 (Sarajedini & Mancone 2007), butthese results have later been ascribed to detection incompleteness,since luminosity limited cluster samples, not affected by disrup-tion, also have an age distribution that scales approximately as τ − (Gieles, Lamers & Portegies Zwart 2007).The consequence of the 90% MID model is that the number ofclusters in logarithmic age bins is constant, which for a continuouspower law CIMF results in a constant maximum cluster mass insuch bins. Gieles & Bastian (2008) showed that this is indeed thecase for the Antennae, but through a comparison to cluster popu-lations in six other galaxies, they also show that the cluster popu-lation of the Antennae galaxies is unique in that sense. The MIDmodel fails to reproduce the first few 100 Myr of the (mass limited)d N/ d τ in other galaxies (see the discussion in Gieles et al. 2007and Gieles & Bastian 2008), so it seems not to be a ‘universal’ sce-nario for cluster evolution, as claimed by Whitmore et al. (2007).In summary, the MDD model can not reproduce the correctmass functions at old ages and the MID model can not reproducethe correct age distribution at young ages in galaxies other than theAntennae.This study searches for the explanation of this problem byabandoning an assumption that both models make, namely a con-tinuous power law CIMF. Alternatively, a truncated Schechter(1976) function (equation 1) is considered for the parent distri-bution from which cluster masses are drawn and this function isevolved with mass-dependent disruption. In Section 2 argumentsfor the choice of this truncated distribution function are given andits basic properties are presented. In Section 3 an analytical modelfor evolved Schechter mass functions is presented and a compar-ison to the cluster population of M51 is provided in Section 4. Adiscussion and conclusions are given in Section 5. Assume that initial cluster masses, M i , are drawn from a Schechter(1976) parent distribution function of the formd N d M i = A M − i exp( − M i /M ∗ ) , (1)where M ∗ is the mass where the exponential drop occurs and A is a constant that scales with the cluster formation rate (CFR). Theconstant A can also be taken as a function of time, such that equa-tion (1) describes the cluster formation history and thus representsthe probability of a cluster with an initial mass between M i and M i + d M i forming at a time between t and t + d t . Since A willbe assumed to be constant in most cases throughout this study andsince the focus will be on the evolution of the mass function the no-tation d N/ d M i is preferred, though d N/ ( d M i d t ) would be moreprecise.As mentioned in footnote 1, this distribution function for thecluster initial mass function (CIMF) is not necessarily reflectedfrom the empirically derived CIMF (CIMF emp ). This is becauseusually only clusters younger than ∼
10 Myr are used to determinethe CIMF emp . This time interval is short compared to the range ofcluster ages in a typical cluster population (few Gyrs). When sam-pling only a low number of clusters from the distribution function c (cid:13) , 000–000 he star cluster mass function in equation (1), the most massive cluster actually formed, M max ,can be less massive than M ∗ , and then the truncation is not recog-nised from the CIMF emp . The analytical form of equation (1) wasproposed for the luminosity function galaxies, where L ∗ (instead of M ∗ ) is the characteristic galaxy luminosity (Schechter 1976). Thisfunctional form follows from a stochastic self-similar model for theorigin of galaxies from self-gravitating gas (e.g. Press & Schechter1974). On stellar scales a power law (self-similar) mass spectrum atlow masses with an exponential cut-off at high masses also followsfrom theoretical models of fragmentation in a collapsing molecularcloud (Silk & Takahashi 1979). For a constant cluster formation ef-ficiency the CIMF reflects the shape of the giant molecular clouds(GMCs) and cores from which the clusters form. Collisions be-tween cores within the GMCs and destruction of the most massivecores by the star formation process also leads to a Schechter type(equilibrium) mass function for clusters (McLaughlin & Pudritz1996).The choice for a truncation is also motivated by several indica-tions that the high-mass end of the cluster mass function is steeperwhen a larger age range, and thus a larger total number of clusters,is considered. The most important arguments are discussed below. Hunter et al. (2003) studied the evolution of the most massivecluster mass, M max , in equally spaced logarithmic age bins, M max (log τ ) , in the SMC and the LMC. They show that for acontinuous power law mass function M max (log τ ) increases withage. This is because for a power law mass function with index − α , M max scales with the number of clusters, N , as M max ∝ N / ( α − . Assuming a constant CFR, the number of clustersper logarithmic unit of age, d N/ d log τ , scales linearly with τ ,since d N/ d log τ ∝ τ d N/ d τ , with d N/ d τ constant. In this case M max (log τ ) ∝ τ / ( α − . For α = 2 this results in a linear scalingof M max (log τ ) with τ . Kumai, Basu & Fujimoto (1993) find sucha relation in the log( M ) vs. log( τ ) plane of clusters in the SMC,LMC, M31 and M33, which they attribute to time-dependent con-ditions for cluster formation. Elmegreen & Efremov (1997) werethe first to suggest that such an increase is purely a statistical effectthat arises from sampling cluster masses from a power law func-tion with index − . This power law form was suggested for themass function of cloud cores as the result of the fractal and turbu-lent nature of the interstellar gas (Elmegreen & Falgarone 1996).This scale-invariant structure of the gas combined with a near con-stant star formation efficiency then leads to the same simple powerlaw form for the mass function of clusters (Elmegreen & Efremov1997). For a constant star formation rate (SFR), the number of clus-ters is set by the time interval that is considered, which then deter-mines the maximum cluster mass that can (on average) be expected.What this means is that the probability of finding the physical con-ditions, such as density and pressure, to form a massive cluster isdetermined by the structure of the gas, which can be described bya simple functional form.Hunter et al. (2003) find M max (log τ ) ∝ τ . in the SMCand LMC, from which they conclude that α ≃ . . However,from direct fits to the CIMF emp de Grijs & Anders (2006) andde Grijs & Goodwin (2008) find indices of α = 1 . ± . and α = 2 . ± . for the LMC and the SMC, respectively. Thedifference between the results of Hunter et al. and de Grijs and co-workers can be explained when a Schechter distribution func-tion (equation 1) is assumed: Hunter et al. (2003) only use themost massive clusters, i.e. close to M ∗ where the mass functionis steeper, whereas the studies of de Grijs et al. use the full massrange of only the youngest clusters, for which M max . M ∗ . Fromthis it can be concluded that in these galaxies it takes more than10 Myr to sample enough clusters from the CIMF to reach M ∗ . Thisscenario is confirmed by the results of de Grijs & Anders (2006)who show that the mass function of LMC clusters in differentage bins gets steeper with age, from − . ± . at ∼
10 Myr to − . ± . at ∼ M ∗ = 2 × M ⊙ provides a good description of the high-mass end of the mass function of LMC clusters when the full agerange is considered (but excluding the globular clusters).Gieles & Bastian (2008) also looked at M max (log τ ) for clus-ters in the SMC and LMC (using the data from Hunter et al.) andadded cluster populations from five other galaxies. A linear in-crease of M max (log τ ) with τ holds for the Milky Way open clus-ters and for the clusters in the SMC, LMC, M33 and M83 up toages of ∼
100 Myr, which implies that α = 2 for the mass range(sampled in that age range) in those galaxies. It also means thatthere has been no disruption of massive clusters in these galaxies inthis age range. In the ‘universal’ MID model of Whitmore et al.(2007), 90% of all clusters gets destroyed each age dex, whichpredicts a constant number of clusters in logarithmic age bins (ifd N/ d τ ∝ τ − , then d N/ d log τ =constant), and therefore a con-stant M max in such bins (Gieles & Bastian 2008). Since this is notthe case in most of the galaxies, the linear scaling of M max (log τ ) with τ is an important argument against the ‘universal’ MID modelof Whitmore et al. (2007).When the full age range of the cluster population is consid-ered, the increase of M max (log τ ) with τ is slower than linear,roughly M max (log τ ) ∝ τ . (Gieles & Bastian 2008), in agree-ment with what Hunter et al. had found. For clusters in M51 theincrease of M max (log τ ) with τ is even slower and for the Anten-nae galaxies M max (log τ ) is essentially independent of τ , suggest-ing that due to the high SFR of these galaxies M max ≃ M ∗ alreadyat short intervals of τ , such that M max (log τ ) is approximately con-stant. However, a high MID fraction and a truncation in the CIMFhave more or less the same effect on the evolution of M max (log τ ) with τ (Gieles & Bastian 2008). In addition, a non-constant forma-tion history of clusters, one which was lower in the past, can alsoproduce a constant M max (log τ ) . Additional age and mass distribu-tions are needed to tell these effects apart.In Section 4 it is shown that the truncated CIMF scenario com-bined with mass-dependent disruption nicely explains the age dis-tribution and mass function of M51 clusters and the MID modelis rejected based on the age distribution of massive clusters. How-ever, a truncation in the mass function does not explain the τ − agedistribution of the Antennae clusters. Indirect evidence for a steepening of the cluster mass function at thehigh-mass end comes from the luminosity function (LF) of clusters. c (cid:13) , 000–000 M. Gieles
The LFs of clusters in different galaxies can be well approximatedby a power law, but with an index smaller than the index of theCIMF emp (between − . and − , e.g. Dolphin & Kennicutt 2002;Elmegreen et al. 2002; Larsen 2002) with the LF being steeperat higher luminosities (Whitmore et al. 1999; Zepf et al. 1999;Benedict et al. 2002; Larsen 2002; Mengel et al. 2005; Gieles et al.2006a,b; Hwang & Lee 2008). The LF consists of clusters with dif-ferent ages. Due to the age dependent light-to-mass ratio of clus-ters, the LF does not necessarily have the same shape as the massfunction. However, if the CIMF is a continuous power law with thesame index at all ages, i.e. with the physical maximum much higherthan M max , the LF will be a power law with the exact same index.Age dependent extinction or bursts in the formation rate would notcause a difference between the CIMF and the LF. An addition ofidentical power laws always results in the same power law. So thefact that the LF is slightly steeper than the CIMF emp is already astrong indication that the CIMF is not a continuous power law func-tion. When an abrupt truncation of the CIMF at some mass M up is assumed, it is possible to roughly estimate the index of thebright-end of the LF. Assume that the CIMF is fully populated,i.e. the mass of the most massive cluster actually formed, M max , isequal to M up . Then assume a constant formation history of clus-ters, so a constant number of the most massive clusters per unitof time: d N up / d τ = constant. The luminosities of these clusters, L up , however, depend strongly on age. The light-to-mass ratio,or the flux of a cluster of constant mass, scales roughly with τ as τ − ζ , with . . ζ . depending on the filter, such that ∂L up /∂τ ∝ τ − ζ − ∝ L /ζ up . Then the LF of such clusters isd N up d L up ∝ d N up d τ ˛˛˛˛ ∂ τ∂L up ˛˛˛˛ (2) ∝ L − − /ζ up , (3) ∝ L − . up , (4)where in the last step ζ = 0 . is used (BL03, Gieles et al. 2007),such that the index of − . holds for the V -band LF. The samearguments hold for the luminosities of the 2 nd , 3 d , etc. most massiveclusters, such that the bright-end of the LF of the entire populationis an addition of power law with index − . , resulting in a powerlaw with index − . .The faint-end of the LF should still be a power law with index − or flatter if MDD is important. This double power law shapefor the LF was found by Gieles et al. (2006b) and Whitmore et al.(1999) for the LF of clusters in M51 and the Antennae, respec-tively. When a Schechter function for the CIMF is considered, thelogarithmic slope of the LF shows a smooth decline from − toroughly − between M V ≃ − and M V ≃ − (Larsen 2008).MDD affects the faint-end ( M V & − ) of the LF and makes itslightly shallower (index > − ). M ∗ Is the value of M ∗ universal? Or does it somehow depend ongalactic conditions, such as the SFR? The two proxies for theshape of the high-mass end of the CIMF, namely the evolutionof M max (log τ ) with τ (Section 2.1.1) and the power law indexof the LF (Section 2.1.2), both show more convincing signaturesof a truncation in the mass function in galaxies with a high SFR. This suggests that in such environments there are numerous clus-ters with masses above M ∗ , as opposed to galaxies with a lowSFR. This excludes a linear dependence of M ∗ on the SFR. If M ∗ would scale linearly with the SFR, then it would take the sameamount of time in each galaxy for M max to reach M ∗ , since inthe power law regime of the Schechter function M max (log τ ) ∝ τ (Section 2.1.1). This was suggested by Weidner, Kroupa & Larsen(2004) and Maschberger & Kroupa (2007), who claim a formationepoch of Myr for an entire CIMF. If the CIMF is populatedup to the highest mass each 10 Myr, than there should be no in-crease of M max (log τ ) with age (assuming all age bins have a widthlarger than 10 Myr). But this is not what Hunter et al. (2003) andGieles & Bastian (2008) find. Weidner et al. (2004) base their re-sult on the relation between the luminosity of the brightest clusterin a galaxy, M brightest V , with the SFR of that galaxy using data fromLarsen (2002) and the assumption that the brightest cluster is alsothe most massive cluster. They then use a constant light-to-massratio to convert luminosities to masses. This assumption was fur-ther investigated by Larsen (2008), who determined the ages of all M brightest V clusters from Larsen (2002) and showed that the brightestclusters ( M brightest V . − ) are young ( .
30 Myr) while the clusterswith M brightest V & − have a large age spread (between ∼
10 Myrand ∼ M brightest V is M V dependent, then so isthe light-to-mass ratio. Assuming that it is constant can thus leadto a misinterpretation of the data. Larsen (2008) showed that the M brightest V vs. SFR relation for (quiescent) spiral galaxies is consis-tent with a constant M ∗ of M ∗ ≃ × M ⊙ .This value of M ∗ is probably not universal, since there areextremely massive clusters known ( − M ⊙ , Maraston et al.2004; Bastian et al. 2006), which are unlikely to have formed froma Schechter function with M ∗ = 2 × M ⊙ . Jord´an et al. (2007,hereafter J07) determine M ∗ values of globular cluster popula-tions of early-type galaxies in the Virgo Cluster by comparingthe cluster luminosity functions to ‘evolved Schechter functions’.They find a trend of M ∗ with host galaxy luminosity, with M ∗ being higher in brighter galaxies, and values for M ∗ between afew times M ⊙ and a few times M ⊙ . Several other studieshave also shown that a Schechter function fits the high-mass end ofthe globular cluster mass function better than a continuous powerlaw with index − (McLaughlin & Pudritz 1996; Burkert & Smith2000; Fall & Zhang 2001; Waters et al. 2006; Harris et al. 2008).Billett et al. (2002) suggest there should exist a maximumcluster mass based on the Kennicutt (1998) relation between theSFR and gas density and the assumption of a pressure equilibriumbetween the ambient interstellar medium and the cloud-cores fromwhich star clusters form. For a constant cluster density this rela-tion is M up ∝ SFR and for a constant cluster radius, it would be M up ∝ SFR / . With the latter increase of M up with the SFR, i.e.slower than linear, it is possible to explain the observational trendsdiscussed in Sections 2.1.1–2.1.2. The key point is that M ∗ is prob-ably environment dependent and for the masses of a cluster popula-tion in a given galaxy to be affected by the truncation, the product ofthe SFR times the age range has to be high. For M ∗ ∝ SFR / , lowSFR galaxies need longer time intervals for M max to reach M ∗ thanhigh SFR galaxies, which seems to be what is found from observa-tions. The relation between M max and the SFR will be quantified inmore detail in Section 2.2. c (cid:13) , 000–000 he star cluster mass function Figure 1.
Example of the relation between the mass of the most mas-sive cluster, M max , as a function of the total mass formed in clusters, Γ · SFR · ∆ t , when the CIMF is a Schechter function (equation 1) with M ∗ = 2 × M ⊙ (Larsen 2008). The dashed line shows a predictionfor M max in the case of an nontruncated mass function. The dotted linesindicate three masses and the corresponding logarithmic slope, − β , of theSchechter function. The top x -axis shows the time range that is needed toform the amount of mass that is represented on the bottom x -axis whenSFR = 1 M ⊙ yr − and Γ = 0 . . M max and the star formation rate The mass of the most massive cluster in a cluster population, M max ,depends on how many cluster are formed. This can be related tothe CFR which scales linearly with the constant A in equation (1),sinceCFR = Z ∞ M min M i d N d M i d M i , (5) = A E „ M min M ∗ « , (6) ≃ A, (7)where E n ( x ) with n = 1 is a generalised expression of the expo-nential integral . In the last step M min /M ∗ ≃ − is used. For aratio of M min /M ∗ = 10 − (10 − ) a relation of CFR ≃ A (6 A ) is found, showing that the relation between the CFR and A is rel-atively insensitive to the ratio M min /M ∗ . The variable A can be afunction of time, A ( t ) , to include variations in the CFR. This doesnot influence the shape of the mass function, but it will affect the The exponential integral is defined as E n = R ∞ t − n exp( − xt ) d t , andis related to the (upper) incomplete Gamma function, Γ( a, x ) , as E n ( x ) = x n − Γ(1 − n, x ) . See Section 6.3 in Numerical Recipes (Press et al.1992) for details on the exponential integral and how to implement thisfunction in a code. The exponential integral is predefined in
IDL as thefunction expint(n,x) . shape of the age distribution, which can be derived once the CIMFis evolved with disruption (Section 3).The relation between M max and A can be found from Z ∞ M max d N d M i d M i , (8) = A E ( M max /M ∗ ) M max . (9)For M max << M ∗ the term E ( M max /M ∗ ) ≃ , and A ≃ M max ,which is the result for a continuous power law with index − . Anexpression for M max as a function of the CFR can be found by sub-stituting A as a function of the CFR from equation (6) in equa-tion (9). Since this is then in units of M ⊙ yr − , a multiplication bysome time interval, ∆ t , is necessary to get an expression for M max in M ⊙ . Assuming that a constant fraction Γ of the SFR ends upin star clusters that survive the embedded phase (Bastian 2008), i.e.CFR = Γ · SFR, then a relation between M max and the SFR is found M max E ( M max /M ∗ ) = Γ · SFR · ∆ t E ( M min /M ∗ ) . (10)This relation between M max and Γ · SFR · ∆ t is illustrated inFig. 1. Since the exponential integral is not easily inverted the prod-uct Γ · SFR · ∆ t is calculated for a series of M max values. A valueof M ∗ = 2 × M ⊙ is used, which was found for spiral galax-ies by Larsen (2008), and E ( M min /M ∗ ) = 10 , corresponding to M min ≃ . The dotted lines indicate three masses where the loga-rithmic slope of the mass function, − β , has the values − . , − . and − . The logarithmic slope at M of a mass function d N/ d M isdefined as − β ( M ) ≡ d ln( d N/ d M ) d ln M , (11)which for the CIMF gives − β ( M i ) = − − M i /M ∗ . (12)Note that β is defined such that it is positive for a decliningd N/ d M .From Fig. 1 we see that M max ≃ . · SFR · ∆ t for M max . . M ∗ . This corresponds to the mass range of the CIMF where β ≃ . For higher values of Γ · SFR · ∆ t the relation between M max and Γ · SFR · ∆ t flattens. The dashed line shows the predicted M max for a nontruncated mass function, or a much higher M ∗ . Eventhough the two predictions diverge to a difference up to two dex in M max , a large increase of Γ · SFR · ∆ t (three dex) is needed to reachthis difference. When M max in the CIMF emp is sampled close to or abit above M ∗ , the difference between a continuous power law and aSchechter function will be hard to tell. Mainly because the numberof clusters in this high-mass tail is low.The top x -axis of Fig. 1 is labelled with ∆ t values corre-sponding to Γ · SFR · ∆ t values with a fixed SFR and Γ ofSFR = 1 M ⊙ yr − and Γ = 0 . . These ∆ t values can be in-terpreted as age ranges. In this illustrative example of a galaxywith a moderate SFR, the CIMF emp ( ∆ t . Myr) will con-tain clusters with masses up to ∼ × M ⊙ , while clusters withmasses up to ∼ M ⊙ have formed when an age range of ∼
10 Gyr is considered. However, without the exponential trun-cation clusters with masses up to M ⊙ are predicted to haveformed. The example in Fig. 1 can be applied to the MilkyWay. Its most massive young star cluster known to date is West- c (cid:13) , 000–000 M. Gieles
Figure 2.
Different stages of the evolved Schechter mass function for a constant mass loss rate (left panel, equation [17]) and a mass-dependent mass loss rate(right panel, equation [26]). The corresponding values of the ratio ∆ /M ∗ (left) and ∆ γ /M γ ∗ (right) are indicated. The dashed line in both panels shows theSchechter function at t = 0 (equation 1). erlund 1 which has a mass around M ⊙ (Clark et al. 2005;Mengel & Tacconi-Garman 2007). There are a handful of othermassive clusters in the Milky Way known, such as the Arches andQuintuplet cluster towards the Galactic centre, NGC 3603 and thetwo recently discovered red super giant clusters (Figer et al. 2006;Davies et al. 2007). They are all around ∼ M ⊙ and span an agerange of ∼ Myr. From Fig. 1 it can be seen that for this age rangevalues of − × M ⊙ are expected for M max . If the last Gyrof cluster formation is considered, the Galaxy has probably formedseveral clusters more massive than Westerlund 1, but not as many,nor as massive compared to the scenario in which the CIMF is acontinuous power law.With the properties of the CIMF introduced, an analytical ex-pression for the ‘evolved Schechter function’, i.e. the cluster massfunction after mass-dependent disruption is applied, can now bederived. In this section the ‘evolved Schechter function’ is presented, basedon the CIMF of equation (1) which is evolved with mass-dependentcluster disruption due to e.g. the tidal field of the host galaxy and/orencounters with GMCs. Cluster mass loss due to stellar evolutionis not considered at this stage. Also, the power law part of the massfunction is always assumed to have an index of − for the moment.It is a simple mathematical exercise to derive all expressionsin this section for a variable index and to include the effect of massloss by stellar evolution, but for reasons of simplicity this is notdone in the derivations of the formulae in the coming two sections.In Section 3.3 these two effects are added to the formalism. The evolved Schechter function is first introduced in the form de-scribed by J07. The mass loss rate of clusters, ˙ M , is assumed to be constant. The disruption time for a cluster of mass M , i.e. the timeneeded to reach M = 0 , is then given by t dis ≡ M ˙ M , (13) = t M, (14)where in the second step t ≡ / ˙ M is introduced. This linear scal-ing of t dis with M follows from the assumption that t dis is a con-stant times the half-mass relaxation time, t rh , and the assumption ofa constant cluster half-mass density and a constant Coulomb loga-rithm in t rh (e.g. Gnedin & Ostriker 1997; Fall & Zhang 2001). Amore general expression for the scaling of t dis with M of equa-tion (14) would be t dis = t M γ (equation 22), with γ = 1 .The mass evolution of a cluster with time is then M = M i − ˙ Mt, = M i − ∆ , (15)where the variable ∆ ≡ t/t (= ˙ M t ) is introduced, which is aproxy of time, but has the dimension of mass since it is the amountof mass with which M i has reduced at time t due to disruption.Though the assumptions that have to be made to arrive to thisconstant mass loss are arguable, the result is a mathematically ap-pealing description of the evolution of cluster masses, enabling usto make simple analytical predictions. In Section 3.2 a more realis-tic ˙ M , based on results of N -body simulations, is explored.The mass function as a function of time, d N/ d M , followsfrom the conservation of numberd N d M = d N d M i ˛˛˛˛ ∂M i ∂M ˛˛˛˛ . (16) To keep the formulae synoptic, the variables M and d N/ d M , i.e. withoutthe subscript i , are used for to the mass and the mass function as a functionof time, i.e. M ( t ) and d N/ d M ( t ) , respectively. The variables M and M i are in units of M ⊙ and t and t are in units of time (Myr).c (cid:13) , 000–000 he star cluster mass function Since ∂M i /∂M = 1 and M i = M + ∆ (equation 15) it thenfollows thatd N d M = A [ M + ∆] exp „ − M + ∆ M ∗ « . (17)The behaviour of this evolved Schechter function is shown in theleft panel of Fig. 2. The dashed line shows the CIMF (equation [1]or equation [17] with ∆ = 0 ) and the full lines show the result ofequation (17) at different times (i.e. for various ratios of ∆ /M ∗ ).For M . . and ∆ . M ∗ the d N/ d M is flat ( β = 0 ). This isthe direct consequence of the constant ˙ M (Fall & Zhang 2001). Thereader is referred to J07 for an equivalent functional form describ-ing the luminosity functions, i.e. the number of clusters in constantmagnitude interval, which is peaked and rises at low luminosities.From equation (17) an expression for the logarithmic slope(equation 11) can be derived − β = − M + ∆ − MM ∗ . (18)The three terms on the right-hand side are, respectively, the initialindex of the power law part of the Schechter function, a disruptionterm that makes the mass function shallower and a truncation termthat makes the mass function steeper. Solving for M gives the massat which the mass function has the logarithmic slope − βM ( β ) = − ∆+( β − M ∗ + p (∆+( β − M ∗ ) +8∆ M ∗ . (19)From this it can be seen that M ( β ) assumes its simplest form when β = 2 . This is where the truncation term in equation (18) equals thedisruption term and this could, therefore, be interpreted as a pointin the mass function where there is a balance between a steepeningdue to the truncation and a flattening by disruption. Using equa-tion (19) with β = 2 an expression for M ( β = 2) arises M ( β = 2) = − ∆ + √ ∆ + 8∆ M ∗ . (20)From equation (20) it can be seen that in the limit of ∆ << M ∗ ,i.e. at young ages where (massive) clusters are not yet affected bydisruption, the scaling of M ( β = 2) with ∆ converges to M ( β =2) = √ M ∗ . Since ∆ is a proxy for age, it means that the partof the mass function that can be approximated by a power law withindex − is found at higher masses at older ages. This can alsobe seen from the expression for β as a function of M and ∆ inequation (18). For the mass function to have a logarithmic slope of β = 2 , the two right-hand terms (i.e. the disruption and truncationterm) in equation (18) need to have a sum of 0, so ∆ is higher athigher masses.From equation (19) it can be seen that in the same regime of ∆ << M ∗ the behaviour of M ( β = 2) is different. For β < , M ( β ) also increases with ∆ , but linearly, so faster than M ( β = 2) .For β > , M ( β ) is constant. This is because at young ages thesteep part of the mass function is not affected by disruption at all.Equation (19) also allows us to find the mass where the massfunction, when presented as the number of clusters per logarithmicunit of mass, d N/ d log M , peaks or ‘turns over’. This is where β =1 and this M is defined as the turnover mass, M TO . Its dependenceon aforementioned variables is (J07) M TO = − ( M ∗ + ∆) + p (∆ + M ∗ ) + 4∆ M ∗ . (21) For ∆ << M ∗ the turnover mass scales as M TO ∝ ∆ , like M ( β < , meaning that the turnover shifts to higher masses as time pro-gresses. Notice that due to the different scaling of M ( β = 2) and M TO with ∆ , they approach each other for ∆ /M ∗ & . In fact,for ∆ >> M ∗ the shape of the mass function remains unchanged(J07) and approaches an equilibrium form with M TO = M ∗ and M ( β = 2) = 2 M ∗ . The number of clusters reduces as time pro-gresses, but the shape of the evolved Schechter function remains thesame. This means that clusters with initial masses much higher than M ∗ are replacing clusters with lower masses that have already beendisrupted. Since the number of clusters above M ∗ is very low, inpractise it means that the total number of remaining clusters quicklydrops to 0 after ∆ = M ∗ .The behaviour of the relations described above is illustrated inthe left panel of Fig. 3. It shows M vs. ∆ , with both quantities rel-ative to M ∗ , such that the x and y scales are in dimensionless unitsof mass and time, respectively. The results for β = 1 / . / / . (equations [19], [20] & [21]) are shown.The grey shaded region in Fig. 3 shows that there is a largeregion in the age vs. mass plane, where the logarithmic slope ofthe mass function equals − ± . . This region more or less coin-cides with the strip that observed cluster populations occupy in theage-mass plane, implying that the observable part of the evolvedSchechter function can be approximated by a power law with index − . This will be verified in Section 4 using empirically derivedages and masses of star clusters in M51. But first the case of amass-dependent mass loss rate is discussed in Section 3.2. In the previous section a (simplified) linear scaling between t dis and t rh was adopted. However, there are theoretical arguments fora de-coupling of t dis from t rh for clusters dissolving in a tidalfield (Fukushige & Heggie 2000). Baumgardt (2001) predicted that t dis ∝ t / rh for clusters that are initially Roche-lobe filling. Thisleads to a simple scaling of t dis with M of the form (Lamers et al.2005b) t dis = t M γ , (22)with t an environment dependent constant and γ ≃ . .This result for Roche-lobe filling clusters was indeed foundfrom N -body simulations of clusters dissolving in a tidalfield (Vesperini & Heggie 1997; Baumgardt & Makino 2003)and it was also found for Roche-lobe underfilling clusters(Gieles & Baumgardt 2008). The simple disruption law of equa-tion (22) was also used to describe age and mass distributions ofclusters, from which a mean value of γ = 0 . ± . was found(BL03).The mass loss rate, ˙ M , can be related to t dis as ˙ M = Mt dis , = M − γ t . (23)For γ = 1 this results in a mass loss rate that is no longer con-stant, but instead becomes a function of M . Note that then t isnot simply / ˙ M anymore and that t dis is no longer the time of totaldisruption as in Section 3.1. The total disruption time, i.e. the time c (cid:13) , 000–000 M. Gieles
Figure 3.
The evolution with age of the mass where the evolved Schechter function has the logarithmic slope, − β , for different values of β . In the left panela constant mass loss rate is considered (equation 17, Section 3.1) and in the right panel the mass loss rate depends on mass (equation 26, Section 3.2). The x -axes represent time or age since ∆ ≡ t/t and ∆ γ ≡ γt/t . The mass for which β = 1 , i.e. the mass at which the mass function, presented as the numberof clusters in logarithmic mass bins, turns over, is indicated as M TO . The grey shaded region indicates the part of the mass function where the logarithmicslope is − . ± . . it takes for all stars to become unbound, will be referred to as t totdis .The relation between t totdis and t dis will be derived a bit further down.With t dis defined as in equation (22) it is still possible toget an analytical expression for the mass evolution with time(Lamers et al. 2005a) M = [ M γi − ∆ γ ] /γ , (24)where ∆ γ is defined as ∆ γ ≡ γt/t to get a similar looking ex-pression as in the constant mass loss case (equation 15). The effectof stellar evolution can be taken into account by replacing M γi inequation (24) by ( µ ev M i ) γ , where µ ev is a time-dependent vari-able that represents the fraction of the initial stellar mass that hasnot been lost by stellar winds or super-nova explosions (typicallybetween 0.7 and 1). For reasons of simplicity µ ev is not consid-ered in the derivations that follow in this section. In Section 3.3 theresults including µ ev are given. The symbol ∆ γ is used when refer-ring to γ < and ∆ is used for the case of γ = 1 . The variable ∆ γ is a proxy of time, like ∆ , but note that ∆ γ does not represent theamount of mass that is lost from M i , like ∆ , because of the powersof γ and /γ in equation (24). The instantaneous disruption timefor a cluster with mass M is when ∆ γ /M γ = 1 (equation 24),such that t totdis ( M ) = ( t /γ ) M γ , or t dis = γt totdis .The partial derivative that is needed (equation 16) to get anexpression for the evolved Schechter function can be found fromequation (24) ∂M i ∂M = » γ M γ – /γ − , (25)which combined with equation (1) & (16) gives an expression forthe evolved Schechter function for a variable γ d N d M = A M γ − [ M γ +∆ γ ] ( γ +1) /γ exp − [ M γ + ∆ γ ] /γ M ∗ ! . (26) For M << M ∗ the exponential term in equation (26) is about 1and the result for an nontruncated mass function of Lamers et al.(2005a) is found, which at low masses and old ages ( M γ << ∆ γ )goes to d N/ d M ∝ M γ − . This is the consequence of the assumedrelation between ˙ M and t dis (equation 23). If ˙ M was assumed to beconstant in time, even when t totdis ( M i ) ∝ M γi , with γ < , the low-mass end of the mass function would be flat (index of 0). The shapeof the low-mass end of the evolved mass function is thus indepen-dent of the shape of the CIMF at those masses and depends onlyon the way clusters lose mass (Fall & Zhang 2001; Lamers et al.2005a).Using the definition of the logarithmic slope, − β , from equa-tion (11) an expression for β is found − β = − γ + 1)∆ γ M γ + ∆ γ − M γ ( M γ + ∆ γ ) /γ − M ∗ . (27)Unfortunately equation (27) can only be solved for M analyti-cally for a few specific values of γ and β and the solutions are rathercomplicated. Therefore, M ( β ) is solved numerically and the rightpanel of Fig. 3 shows the result for the same values of β as in theconstant mass loss case (left panel of Fig. 3) using γ = 0 . . For M << M ∗ , M TO scales as ∆ /γγ , which was also found under theassumption of a continuous power law in the instantaneous disrup-tion model of BL03. However, the increase of M TO with age slowsdown at high ages, due to the truncation. In contrast to the constantmass loss case, the evolved Schechter function with γ < does notapproach an equilibrium shape. For γ = 0 . , the value of M TO reaches its highest value ( M TO ≃ . M ∗ ) around ∆ γ ≃ M γ ∗ and then decreases again for older ages. This subtle differencemight be important. If globular clusters formed from a Schechtertype CIMF, with M ∗ ≃ M ⊙ , then together with the fact that M TO ≃ × M ⊙ , the constant mass loss model would say thatthe value of M TO can still increase by roughly a factor of five (seeleft panel of Fig. 3). However, the γ = 0 . model shows that M TO c (cid:13) , 000–000 he star cluster mass function already has its highest possible value when M TO /M ∗ ≃ . (rightpanel of Fig. 3). A possible explanation for the near universality ofthe value of M TO could thus be that most globular cluster systemshave reached already their maximum value of M TO /M ∗ due to dis-ruption and for a range of roughly an order of magnitude in ∆ γ , thevalue of M TO will be (within a factor of two) about . M ∗ . Sinceglobular clusters are roughly coeval, a spread in ∆ γ can be inter-preted as a spread in disruption time-scales, thereby allowing for arange of t dis values (i.e. different galactocentric distances) resultingin similar values of M TO . Perhaps this adds a piece to the puzzle ofthe problem of the universality of the globular cluster mass function(see also Ostriker & Gnedin 1997; Vesperini 2000; Whitmore et al.2002; Baumgardt et al. 2008).The mass for which the mass function has an index of − scales approximately as ∆ ηγ , with η ≃ . , i.e. slightly differentthan the √ ∆ scaling found for the γ = 1 case described in Sec-tion 3.1 . However, the ∆ . γ scaling is a bit closer to the increasewith age of the limiting mass due to the detection limit, M lim ( τ ) , ofempirically derived cluster masses. For a V -band detection limit, M lim ( τ ) scales as τ . . This implies that the observable part of theevolved Schechter function, i.e. above the detection limit, has a log-arithmic slope of approximately − at all ages. This is an importantresult, because it means that there is no significant difference to beexpected between the CIMF emp and the shape of the cluster massfunction at old ages. So the argument that disruption needs to bemass-independent because old clusters have a similar mass func-tion as young clusters does not hold when a Schechter function isassumed for the CIMF.All results presented for the constant mass loss case in Sec-tion 3.1 are simply a subset of the more general solutions presentedin this section. That is, when using γ = 1 , ∆ γ = ∆ and equa-tions (22), (24), (26) & (27) from this section are the same as equa-tions (14), (15), (17) & (18), respectively, from Section 3.1. Forcompleteness a general expression for the evolved Schechter func-tion, including mass loss by stellar evolution and a variable initialpower law index, is presented in Section 3.3. As mentioned in the beginning of this section all derivations aredone for a fixed power law index of − (equation 1) and without theinclusion of mass loss due to stellar evolution. The expressions forthe evolved Schechter function and the logarithmic slope − β arehere given for a variable initial power law index, − α , and an addi-tional term µ ev , which is the fraction of the original mass that is notlost due to stellar evolution. Therefore, µ ev is a function of time andcan be taken from an SSP model. See also Lamers et al. (2005a) foranalytical approximations of µ ev ( t ) . Following the same steps as inSection 3.2 the expressions for d N/ d M and β ared N d M = A µ α − ev M γ − [ M γ +∆ γ ] ( γ + α − /γ exp − [ M γ + ∆ γ ] /γ µ ev M ∗ ! , (28)and In fact, by numerically solving equation (27) for β = 2 and < γ the relation between η and γ is η ≃ − . γ + 0 . γ . − β = − α + ( γ + α − γ M γ + ∆ γ − M γ ( M γ + ∆ γ ) /γ − µ ev M ∗ . (29)From equations (28) & (29) it can be seen that µ ev does notaffect the shape of the mass function at low masses, it only affectsthat vertical offset by a bit. The exponential truncation occurs atslightly lower masses, at µ ev M ∗ instead of M ∗ . To illustrate the cluster population model of the previous section,some of its elements are compared to the cluster population ofthe interacting galaxy M51 (NGC 5194). The cluster ages andmasses were determined from HST/WFPC2 multi-band photom-etry by Bastian et al. (2005, hereafter B05). The masses are cor-rected for mass loss by stellar evolution, i.e. they represent the sumof the initial masses of the stars that are still bound in the clus-ters. The present day masses of the oldest clusters are, therefore, ∼
25% lower. The M51 cluster population is a good benchmark totest the evolved Schechter function for several reasons: firstly, it is arich cluster population hosting several massive star clusters (Larsen2000; B05; Lee et al. 2005; Hwang & Lee 2008). M51 hosts suffi-cient clusters above ∼ M ⊙ , which is the approximate minimummass for the optical fluxes not to be affected by stochastic effectsdue to sampling of the stellar IMF, allowing reliable age datingthrough multi-band photometry (e.g. Cervi˜no & Luridiana 2004).Secondly, for this cluster population it has been suggested that themass function is truncated (Gieles et al. 2006b; Haas et al. 2008).Lastly, the disruption time of clusters in M51 is thought to be veryshort (BL03, Gieles et al. 2005), due to the strong tidal field andthe high density of GMCs in the inner region of the galaxy, wheremost of the clusters of the B05 sample reside. These environmen-tal properties suggest a noticeable evolution of the mass function.Before a comparison between the model and the data is given, theparameters for the CIMF ( M ∗ ) and the disruption law (equation 22)are determined in Section 4.1. If a constant cluster formation history is assumed the evolvedSchechter function (equation 26) essentially represents the prob-ability of finding a cluster with mass between M and M + d M andage between τ and τ + d τ . It can thus be used as a two-dimensionaldistribution function to determine all the parameters of the evolvedSchechter function from empirically derived ages and masses us-ing a maximum likelihood estimate. The only thing which needsto be added is the detection limit. For this the result of the in-completeness analysis of B05 is used, who show that their clustersample is limited by a detection in the F435W (roughly B ) band,for which the 90% completeness fraction is a 22.6 mag. The samesimple stellar population (SSP) model was used as in B05 to de-rive the ages and mass, namely the GALEV models for Salpeterstellar IMF between . M ⊙ and M ⊙ (Schulz et al. 2002;Anders & Fritze-v. Alvensleben 2003). The photometric evolutionof a single mass cluster in the F435W band from the SSP model,combined with the 90% completeness limit and the distance toM51 (8.4 Mpc, Feldmeier et al. 1997) then gives the limiting clus-ter mass as a function of age, M lim ( τ ) (BL03; Gieles et al. 2007). c (cid:13) , 000–000 M. Gieles
This limiting mass, together with the cluster ages and masses, isshown in Fig. 4 and in panel (a) of Fig. 5.Using M lim ( τ ) and equation (26) artificial models for varying γ , t and M ∗ are built and a (simultaneous) maximisation of thelikelihood of these parameters gives • M ∗ = (1 . ± . × M ⊙ ; • t = 0 . ± . Myr; • γ = 0 . ± . .The (statistical) uncertainties on each variable are determined us-ing a bootstrap method. For this the original data-set is randomlyre-sampled 1000 times, allowing for multiple entries of the sameage-mass pair and omission of values, such that the total number ofage-mass pairs is the same. The likelihood of the three parametersis determined for each of these 1000 samples and the standard de-viation of the results for each parameter is used as the uncertainty.The disruption parameters ( t and γ ) imply a total dissolutiontime for a cluster with an initial mass M ∗ of t tot ∗ = ( t /γ ) M γ ∗ ≃ Myr, implying that there should not be many clusters that sur-vive longer than a Gyr. Indeed B05 found only a handful of clus-ters older than 1 Gyr, although this is probably not only causedby disruption, but also by detection incompleteness. A cluster withan initial mass of M ⊙ gets completely destroyed in t tot ≃ Myr. This latter value is in perfect agreement with the re-sults of Gieles et al. (2005), who found . t tot . , de-pending on what is assumed for the cluster formation history. Thisvery short lifetime of clusters in M51, as compared to the lifetimeof clusters in the solar neighbourhood or the Magellanic Clouds,was attributed to the high molecular cloud density in this galaxy(Gieles et al. 2006c).The value of γ = 0 . ± . agrees perfectly with the meanresult of BL03 who found γ = 0 . ± . from fits to the ageand mass distributions of clusters in several galaxies. This valuealso follows from theory and N -body simulations of clusters dis-solving in a tidal field, since for these clusters the disruption time-scale can be expressed as t dis ∝ ( N/ log Λ) . (Baumgardt 2001;Baumgardt & Makino 2003; Gieles & Baumgardt 2008). In thisequation, log Λ ≃ log(0 . N ) is the Coulomb logarithm and it fol-lows that this expression for t dis can be well approximated by t dis ∝ N . when a relevant range of N is considered (Lamers et al.2005b). The disruption time due to external perturbations scales (onaverage) in a similar way with mass. It actually scales linearly withthe density of clusters, but since the radius of clusters depends onlyweakly on their mass, t dis due to external perturbations (GMCs,spiral arms, etc.) also scales as M γ with γ slightly smaller than1 (Gieles et al. 2006c; Lamers & Gieles 2006). Gieles et al. (2005)also looked at the mass dependence of cluster disruption in M51and found that the value of γ depends on the mass range consid-ered. When excluding the most massive clusters ( M & M ⊙ ) avalue of γ = 0 . was found. Here it is shown that, when consider-ing a Schechter function for the CIMF, the full cluster populationcan be fit with a disruption law in which t dis depends on mass.The value of M ∗ is in excellent agreement with the result ofLarsen (2008) who found M ∗ = (2 . ± . × M ⊙ for a sam-ple of spiral galaxies. Gieles et al. (2006b) found a slightly lowervalue of M ∗ = 10 M ⊙ through a model comparison to the LF ofM51 clusters. Since the LF is usually constructed for apparent lu-minosities, i.e. not corrected for local extinction, the derived valueof M ∗ from the LF can be underestimating the true value of M ∗ . Gieles et al. (2006b) applied a correction of A V = 0 . mag to allclusters to roughly account for extinction, but B05 showed that theyoungest clusters, which are the brightest ones, are slightly moreextincted than this. Extinction of the brightest clusters has a similareffect on the LF as lowering M ∗ , so this could be why the value for M ∗ found by Gieles et al. (2006b) is slightly lower than that foundhere. Fig. 4 shows the M51 ages and masses in dimensionless units ontop of the predictions for the evolution of several values of β . Theresults of the maximum likelihood estimation from Section 4.1 areused for the calculation of M ( β ) and the normalisation of age andmass.The increase of the minimum observable cluster mass, M lim ( τ ) , is shown as a dot-dashed line marking the lower enve-lope of the data points. Most of the clusters fall in the grey region,for which the predicted logarithmic slope of the mass function is − ± . , with the M ( β = 2) line showing a very similar increasewith age as the data points. At old ages M lim ( τ ) approaches the M ( β = 2) line, implying that the mass function at these ages issomewhat steeper than a − power law. Although the disruptiontime of clusters in M51 is very short, M TO does not get closer thanroughly one mass dex below M lim ( τ ) and is, therefore, not observ-able at any age. The dashed line in Fig. 4 shows M ( β = 1 . forthe case of a continuous power law CIMF (BL03). This line showsthat disruption would clearly make the mass function of the old-est half of the data flatter than a − power law if the CIMF was acontinuous power law with that index, whereas the M ( β = 1 . line for the evolved Schechter function bends down at old ages andstays (almost) below the detection limit.A direct comparison between the evolved Schechter function(Section 3.2) and the M51 cluster mass functions at different agesis given in Section 4.3. The presentation of cluster ages and masses in dimensionlessunits in the previous section is not very common. Therefore, theage-mass diagram is presented again, but now in physical units(panel [a] of Fig. 5). Three age bins are created, with bound-aries , , Myr and 600 Myr. The upper boundaries are indi-cated by τ , τ and τ , respectively. Due to the increasing M lim ( τ ) with τ , three corresponding mass limits are defined as M j = M lim ( τ j ) , with < j < . The values of M j are roughly × M ⊙ , M ⊙ and × M ⊙ .The empirical mass functions in different age bins are shownin panels (b)-(d) of Fig. 5. In each panel an arrow denotes the lowermass limit of each sample, M j . The first mass bin starts at thislower limit and the number of clusters in each mass bin is countedand then divided by the width of the bin, such that d N/ d M is ob-tained. The d N/ d M is also divided by the age range of the sample.In this way the histogram points represent the number of clustersper unit of mass and time/age, or d N/ ( d M d τ ) , which can be com-pared to the evolved Schechter function. The variables for γ , t and M ∗ as determined in Section 4.1 are used for the model predictions.The dashed lines show the CIMFs (equation 1) and the evolvedSchechter functions (full lines, equation 26) are based on the mean c (cid:13) , 000–000 he star cluster mass function Figure 4.
The time evolution of the mass where the evolved Schechter func-tion has the logarithmic slope, − β , for β = 1 , . , and 2.3 (same as in theright panel of Fig. 3). The dots are ages and masses of M51 clusters fromB05, with the dot-dashed line defining the limiting mass due to a detectionlimit of 22.6 mag in the F435W filter. The dashed line shows M ( β = 1 . when a continuous power law CIMF is assumed. The values of γ, t and M ∗ , used to normalise the ages and masses to dimensionless units, followfrom a maximum likelihood estimation (Section 4.1). age of the M51 clusters in each age bin, being approximately Myr,50 Myr and
Myr for the panels (b), (c) and (d), respectively.The vertical offset is determined by A , and the best agreement isfound for A = 0 . M ⊙ yr − . The coincidence of the empiricalmass functions with the CIMF (dashed line) in the first two agebins, shows that the number of clusters per linear unit of time is ap-proximately constant in the first Myr. The implications of thiswill be discussed in more detail in Section 4.4. Only the d N/ d M in the oldest age bin falls visibly below the CIMF. If a continuouspower law distribution function would have been assumed for theCIMF, many more high-mass clusters would have to have been de-stroyed to make the model prediction agree with the data (alreadynoted by Gieles et al. [2005] who do not include the most massiveclusters when determining the disruption time of M51 clusters).The evolved Schechter functions describe the empirically derivedmass functions very well and as could be seen already in Fig. 4, theturnover of the evolved Schechter functions remains well below thedetection limit.The empirical mass functions are also approximated by powerlaws and their indices, − β fit , are indicated in each panel. The em-pirical mass functions in the first two age bins can be approximatedby a power law with index of roughly − , but the oldest mass func-tion is steeper. From a comparison to the (evolved) Schechter func-tion, we see this is because the limiting mass approaches M ∗ at thisage, where β = 3 . On the low-mass end of the evolved Schechterfunction the effect of mass-dependent disruption is seen at old ages(panel [d]), where the mass function flattens to a logarithmic slopeof γ − − . and the turnover is at M TO ≃ M ⊙ .Though this is only an application to one data-set, it showsthat a Schechter function evolved with mass-dependent disruptionnicely describes empirically derived age and mass distributions. The lack of a flattening of the empirical mass function at old ages,as compared to the CIMF emp , is due the exponential truncation athigh masses and the detection limit that shifts the mean observedmasses closer to M ∗ , where the CIMF is steeper. Assuming a constant formation history of clusters, the cluster agedistribution, d N/ d τ , can be found from a numerical integration ofthe evolved Schechter function from M j to ∞ at each τ .In panel (e) three (mass limited) age distributions are shown,for the three maximum ages, τ j , and corresponding lower masslimit, M j . The clusters are binned, such that the first bin starts at4 Myr, the age of the youngest cluster, and the last bin ends at τ j .The number of bins is chosen such that the bin widths are approx-imately constant for the three mass cuts. The number of clusters ineach bin is counted and divided by the bin width, such that d N/ d τ follows, i.e. the number of clusters per Myr.Over-plotted as full lines are the age distribution resultingfrom the numerical integrations of the evolved Schechter functionsagain using the best-fitting parameters from the maximum likeli-hood estimate of Section 4.1. A single value of A = 0 . M ⊙ yr − ,i.e. the same value as in Section 4.3, was used to describe all threeage distributions. The different vertical offsets are caused by thedifferent values of M j . The good agreement between the modeland the three empirical age distribution shows that both the disrup-tion parameters and the CIMF shape, with the parameters from themodel fit of Section 4.1, provide a good description of the clusterpopulation of M51. With the value of A and the global SFR of M51it is possible to derive Γ , i.e. the fraction of star formation occurringin star clusters that survive the embedded phase (CFR = Γ · SFR,Bastian 2008). The global SFR of M51 is around M ⊙ yr − (e.g.Scoville et al. 2001). Since Γ · SFR = A E ( M min /M ∗ ) (equa-tion 6), with E ( M min /M ∗ ) roughly between 6 and 10, depend-ing on M min , the value of Γ resulting from this comparison is then Γ ≃ . − . . So roughly 10% of the star formation in M51occurs in star clusters that are detectable in the optical. This isin good agreement with the general finding of Bastian (2008) that Γ = 0 . ± . in different galactic environments.The theoretical age distributions all show an initially flat partand then a rapid decline. For the lowest mass cut ( M ) the datado not allow us to verify this, due to the young age at which the M lim ( τ ) line cuts the sample ( τ = 10 Myr). The d N/ d τ of thesample with the highest mass cut nicely shows this flat part and thedecline. The solutions for the mass limited age distributions can beapproximated within ∼
10% accuracy byd N d τ ( M > M j ) ≃ AM ∗ „ t tot ∗ τ + t tot j « /γ E „ M j M ∗ « exp ` − τ /t tot ∗ ´ (30)where t tot ∗ is the total disruption time of a cluster with an initialmass M ∗ and t tot j is the same for a cluster with an initial mass M j .This approximation only holds for an initial power law index of − at low masses. The approximation in equation (30) has the τ − /γ scaling predicted by BL03 for a continuous power law mass func-tion, but it falls off faster (exponentially) at old ages, due to thetruncation in the CIMF.A flat part in d N/ d τ at young ages is a typical result of theMDD model (BL03), although BL03 present the age distributionsfor luminosity limited cluster samples, which decline at young ages c (cid:13)000
10% accuracy byd N d τ ( M > M j ) ≃ AM ∗ „ t tot ∗ τ + t tot j « /γ E „ M j M ∗ « exp ` − τ /t tot ∗ ´ (30)where t tot ∗ is the total disruption time of a cluster with an initialmass M ∗ and t tot j is the same for a cluster with an initial mass M j .This approximation only holds for an initial power law index of − at low masses. The approximation in equation (30) has the τ − /γ scaling predicted by BL03 for a continuous power law mass func-tion, but it falls off faster (exponentially) at old ages, due to thetruncation in the CIMF.A flat part in d N/ d τ at young ages is a typical result of theMDD model (BL03), although BL03 present the age distributionsfor luminosity limited cluster samples, which decline at young ages c (cid:13)000 , 000–000 M. Gieles
Figure 5.
Ages and masses of M51 clusters (panel [a]) and the resulting mass functions in different age bins (panels [b]-[d]) with the evolved Schechterfunctions (equation 26) over-plotted. The parameters were determined by a maximum likelihood fit of the model to the data (Section 4.1). The empirical massfunctions are approximated by power laws and the resulting index, − β fit , is indicated in each panel. The mass limited age distributions for different mass cutsare shown in panel (e). The vertical positioning of the age distributions with respect to one another is due the fact that each sample has a different lower masslimit ( M j ). All mass functions and age distributions are described by a single value of A = 0 . M ⊙ yr − , corresponding to Γ ≃ . , i.e. 10% of the starformation in M51 occurs in star clusters. as τ − ζ due to evolutionary fading of clusters. In the predictionfor d N/ d τ in this study the evolutionary fading does not enter be-cause for each age distribution a lower mass limit is adopted that isabove the fading line (Fig. 5). A flat d N/ d τ was also found for amass limited sub-sample of clusters in the SMC up to almost a Gyr(Gieles et al. 2007). Indirect evidence for a flat d N/ d τ for clustersin different galaxies follows from the linear scaling of M max (log τ ) with τ in the first 100 Myr (Section 2.1.1 and Gieles & Bastian2008).Note that there is no evidence for ‘infant mortality’ from thed N/ d τ in the first ∼
10 Myr, as was reported by B05 from the samedata set. B05 used slightly smaller age bins and found evidence fora drop of a factor of four between the first age bin ( . Myr)and the second age bin. They also report an enhancement of clus- c (cid:13) , 000–000 he star cluster mass function ters around 60 Myr, which coincides with the moment of the lastencounter between M51 and the companion galaxy (NGC 5195).However, there is an artificial age gap between 10 Myr and 20 Myrin the d N/ d τ , which is a common feature when broad-band pho-tometry is used to derive ages (Gieles et al. 2005; Lee et al. 2005;Whitmore et al. 2007). These three effects average out when usingslightly larger age bins and the details in the d N/ d τ reported byB05 are not separable anymore. The difference between the firstand the second bin for the sample with the highest mass cut iswithin σ still consistent with a decrease of a factor of two orthree. However, panel (e) of Fig. 5 shows that a flat d N/ d τ up to ∼ Myr describes the data very well.Since this is a quite different interpretation of the data thanwhat was concluded by B05, caused by a difference in how thedata are binned, it is interesting to have a closer look at the d N/ d τ of massive clusters with a method that does not rely on binning.One way of doing that is by creating a cumulative distributionfunction (CDF), which can then be compared to different mod-els using a Kolmogorov-Smirnoff (K-S) test. In Fig. 6 the CDFof the first 100 Myr of the d N/ d τ of clusters more massive than M ≃ × M ⊙ (Fig.5) is shown (dots). The best-fitting modelfrom Section 4.1 is shown as a full line. The null hypothesis thatthe data have been drawn from this model cannot be rejected. TheK-S probability is 10%, i.e. the model is within 2 σ consistent withthese data.Whitmore et al. (2007) claim that cluster evolution is ‘univer-sal’ in the first 100 Myr and that the fraction of disrupted clusters inthis period is independent of environment and mass. In their model,90% of the clusters gets destroyed each age dex, resulting in a τ − scaling of the d N/ d τ for mass limited samples. The CDF of thismodel is shown as a dashed line in Fig. 6. The K-S probability forthe Whitmore et al. model is × − and the hypothesis that thesedata are drawn from a τ − age distribution can, therefore, be safelyreject . Even if the disruption fraction is lowered to 80%(70%), re-sulting in d N/ d τ ∝ τ − . ( τ − . ) (Whitmore et al. 2007), the K-Sprobability is × − (10 − ) .The fact that the full optically selected cluster population ofM51 can be described by a CFR that is roughly 10% of the SFRindicates that if all stars form in embedded clusters, 90% of themare destroyed when the residual gas of the star formation process isremoved and that this process lasts only a few Myrs. And it seemsthat ‘infant mortality’ does not affect the optically detected clus-ters and that information on the number of embedded clusters isneeded to estimate the ‘infant mortality rate’, as was also done forthe clusters in the solar neighbourhood (Lada & Lada 2003). This study considers the early evolution (first ∼ Gyr) of the starcluster mass function, with particular focus on the mass rangethat is available through observations. A Schechter type function,i.e. a power law with an exponential truncation (equation 1), isused for the cluster initial mass function (CIMF). The use of thisfunction is motivated by observational indications that the high-mass end of the CIMF of young extra-galactic clusters is steeperthan the ‘canonical’ − power law (Gieles et al. 2006a; Larsen2008). The exponential truncation provides a good description Figure 6.
Cumulative distribution function (CDF) of ages ( Myr) forthe most massive clusters ( M > M ≃ × M ⊙ ). The dashed and fulllines show the CDF for the 90% MID model of Whitmore et al. (2007) andthe one presented in this study, respectively. of the high-mass tail of the globular cluster mass function (e.g.McLaughlin & Pudritz 1996).Empirical cluster masses are generally found to be higher atolder (logarithmic) ages. This is an observational bias because oftwo effects: the rapid fading of star clusters with age, making itincreasingly more difficult to see low-mass clusters with increas-ing age; also, masses increase due to a size-of-sample effect, sincelonger time intervals are considered at higher logarithmic ages. Ifthe CIMF has indeed an exponential truncation, this means that atolder (logarithmic) ages the steepening is more noticeable than atyoung ages.The shape of the cluster mass function at different ages de-pends on the functional form of the CIMF and the way clusters losemass due to disruptive effects. Two competing models exist for theevolution of clusters, a mass-dependent disruption model (MDD,e.g. Boutloukos & Lamers 2003) and a mass-independent disrup-tion model (MID, e.g. Whitmore et al. 2007) and both assume thatstar cluster masses are drawn from a continuous power law distri-bution with index − .In this contribution a Schechter function is used for the CIMF,with a power law index of − at low masses and an exponen-tial truncation at M ∗ (equation 1), which is evolved with mass-dependent cluster disruption. In summary, it is found that • the exponential truncation of the Schechter function is not nec-essarily detectable from a small cluster sample, the kind that is usedto create the mass function of clusters younger than ∼
10 Myr (herereferred to as CIMF emp ), because in this short time interval thereare generally not enough clusters sampled above M ∗ ; • the age distribution of mass limited sub-samples is flat duringthe first ∼
100 Myr (depending on the mass cut and the disruptiontime-scale) and drops exponentially at older ages. Through a com-parison to ages and masses of clusters in M51, it is shown that thisholds for a sub-sample of clusters more massive than × M ⊙ .The τ − age distribution, proposed by Whitmore et al. (2007) as c (cid:13) , 000–000 M. Gieles the result of a ‘universal’ cluster disruption model, is ruled out athigh significance for these data; • the mass for which the logarithmic slope of the evolvedSchechter function is − increases with age as τ . . This scaling issimilar to the increase of the limiting cluster mass, M lim ( τ ) , due tothe evolutionary fading of clusters. This means that the mass func-tion of clusters above the detection limit is approximately a powerlaw with index − at all ages.The MDD model, based on a continuous (nontruncated) powerlaw CIMF, predicts that the mass function at old ages shouldhave a logarithmic slope of γ − ≃ − . (Lamers et al.2005a), while empirically derived mass functions at old ages arepower laws with indices of approximately − , or even steeper(e.g. de Grijs & Anders 2006). This weakness of the MDD modelwas at the same time a supporting argument for the MID model.However, in this study it shown that a power law mass functionwith index − at old ages results naturally when a Schechter typeCIMF is assumed. This is illustrated in Section 4, where it is shownthat a Schechter function evolved with mass-dependent disruptionprovides an excellent description of the (mass limited) age distribu-tion and the mass function at different ages for star clusters in M51.The power law mass function of M51 clusters gets steeper with age,from an index of − at the youngest ages to an index of − . at ∼
250 Myr. However, simultaneous fitting of models with differentCIMF and disruption parameters to the ages and masses shows thatthe disruption time depends on mass as t dis ∝ M . ± . . Thismeans that the logarithmic slope of the CIMF that is observable at ∼
250 Myr is smaller than − . (i.e. steeper), since it has alreadybeen affected by disruption.It would be interesting to apply the evolved Schechter functionto more cluster populations for which age and mass information isavailable. As demonstrated in Section 4.1, the fundamental param-eters defining a cluster population, namely M ∗ of the CIMF and t and γ of the disruption law (equation 22) can easily be deter-mined from luminosity limited cluster samples using a maximumlikelihood estimate. Parmentier & de Grijs (2008) apply the MDDmodel to the age and mass distributions of LMC clusters and con-clude that it is hard to constrain the disruption time-scale. They findthat models with a long disruption times ( & Gyr) are needed todescribe the mass function, while short disruption times ( . Gyr)are preferred for the age distribution. Larsen (2008) showed that anexponential truncation in the CIMF around × M ⊙ providesa good description of the mass function of LMC clusters. Perhapsthe disruption time-scale of LMC clusters can be better constrainedwhen the evolved Schechter function from Section 3.2 is used.Most probably the model presented in this paper will not beable to explain the age distribution of clusters in the Antennaegalaxies (Fall et al. 2005). The τ − scaling of the age distributionand the mass-independent nature of the disruption model invokedto explain this is very peculiar and this has not been found in othergalaxies thus far. Since the galactic environment in the Antennaegalaxies is quite different than that of a quiescent spiral galaxy,it could be that clusters suffer from different (additional) mecha-nisms that disrupt clusters of different masses equally fast. Some-thing along these lines was recently suggested by Renaud et al.(2008) who studied Antennae like mergers with N -body simula-tions. They show that due to the interaction of the two spiral discsthere are strong compressive tides that probably induce star andcluster formation. Their models show that if clusters or associa- tions indeed form in such compressive tides, they can be held to-gether for 10 Myr or longer, thereby postponing the dissolution ofclusters due to gas removal, which in other galaxies seems to de-stroy the majority ( ∼ M ∗ ≃ × M ⊙ . It shouldbe possible to describe most of the age and mass distributions in(quiescent) galaxies with the model presented in this study.To validate the correctness of the model presented in thisstudy, it would be convincing to ‘detect’ the turnover in the massfunction at ages between ∼
500 Myr and ∼ M TO , above the detection limit.The improved sensitivity of the new HST/WFC3 camera will al-low us to trace the cluster mass function at old ages to the requireddepth to derive a mass function at intermediate ages down to theturnover ( M TO ≃ M ⊙ ) for cluster populations at distances of − Mpc.
ACKNOWLEDGEMENT
MG enjoyed stimulating discussions with Andr´es Jord´an andthanks Nate Bastian, Iraklis Konstantopoulos, Henny Lamers andSøren Larsen for discussions and critical reading of earlier versionsof this manuscript.
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