Stochastic star formation and a (nearly) uniform stellar initial mass function
aa r X i v : . [ a s t r o - ph . C O ] O c t Draft version November 14, 2018
Preprint typeset using L A TEX style emulateapj v. 11/10/09
STOCHASTIC STAR FORMATION AND A (NEARLY) UNIFORM STELLAR INITIAL MASS FUNCTION
Michele Fumagalli , Robert L. da Silva , and Mark R. Krumholz Draft version November 14, 2018
ABSTRACTRecent observations indicate a lower H α to FUV ratio in dwarf galaxies than in brighter systems,a trend that could be explained by a truncated and/or steeper IMF in small galaxies. However, atlow star formation rates (SFRs), the H α to FUV ratio can vary due to stochastic sampling even for auniversal IMF, a hypothesis that has, prior to this work, received limited investigation. Using slug ,a fully stochastic code for synthetic photometry in star clusters and galaxies, we compare the H α andFUV luminosity in a sample of ∼
450 nearby galaxies with models drawn from a universal Kroupa IMFand a modified IMF, the integrated galactic initial mass function (IGIMF). Once random sampling andtime evolution are included, a Kroupa IMF convolved with the cluster mass function reproduces theobserved H α distribution at all FUV luminosities, while a truncated IMF as implemented in currentIGIMF models underpredicts the H α luminosity by more than an order of magnitude at the lowestSFRs. We conclude that the observed luminosity is the result of the joint probability distributionfunction of the SFR, cluster mass function, and a universal IMF, consistent with parts of the IGIMFtheory, but that a truncation in the IMF in clusters is inconsistent with the observations. Future workwill examine stochastic star formation and its time dependence in detail to study whether randomsampling can explain other observations that suggest a varying IMF. Subject headings: galaxies: dwarf — galaxies: statistics — galaxies: star formation — galaxies: stellarcontent — ultraviolet: galaxies — stars: statistics INTRODUCTION
The stellar initial mass function (IMF), an essentialingredient for numerous astrophysical problems, is com-monly assumed to be invariant with time and with galac-tic properties. This ansatz follows from a lack of evi-dence for its variation, despite searches covering a widerange of environments (e.g. Kroupa 2001; Chabrier 2003;Bastian et al. 2010), more than from theoretical under-standing of the processes that regulate the mass distribu-tion of stars. However, several recent studies have ques-tioned the idea of a universal IMF in both nearby anddistant galaxies.Considering a few examples in the local universe,a truncated and/or steeper IMF has been invoked toexplain an apparent correlation between the massesof clusters and the masses of their largest members(Weidner & Kroupa 2006, though see Lamb et al. 2010for an opposing view), or between galaxy colors andH α equivalent widths (Hoversten & Glazebrook 2008;Gunawardhana et al. 2011), as well as between the ra-tio of H α to FUV luminosity ( L H α /L FUV ) and the sur-face brightness (Meurer et al. 2009) or the star forma-tion rates (SFRs; Lee et al. 2009; Weidner et al. 2011)in galaxies. However, these claims remain controver-sial, and alternative interpretations have been offered,without resorting to IMF variations (Boselli et al. 2009;Corbelli et al. 2009; Calzetti et al. 2010; Lamb et al.2010).In this letter we focus on observations of system-atic variations in L H α /L FUV (e.g. Meurer et al. 2009;Lee et al. 2009; Boselli et al. 2009) that have been [email protected] Department of Astronomy and Astrophysics, University ofCalifornia, 1156 High Street, Santa Cruz, CA 95064. NSF Graduate Research Fellow used to argue strongly for a non-universal IMF (e.g.Meurer et al. 2009; Weidner et al. 2011). Recombina-tion lines such as H α ultimately come from the ionizingphotons produced primarily by very massive stars, whileFUV luminosity is driven by less massive stars ( ∼
50 M ⊙ for H α , ∼
10 M ⊙ for FUV, using a Salpeter 1955 IMF).As a result, the ratio L H α /L FUV can be used as a probefor the massive end of the IMF, although the interpreta-tion of observations is complicated by additional factors(e.g. dust). Recently, both Meurer et al. (2009, hereafterM09) and Lee et al. (2009, hereafter L09) reported that L H α /L FUV varies systematically with galaxy properties,which may imply a variation of the IMF. Boselli et al.(2009, hereafter B09) have cautioned that dust correc-tion or a bursty star formation history (SFH) can pro-duce the observed trends even if the underlying IMF isuniversal, but this idea has not been explored in detailwith models. While they cannot rule out variations inthe IMF, L09 also argue that the underlying cause forthe observed trend is not clear and stochastic effects atlow SFR need to be explored further.Following these suggestions (see alsoWilliams & McKee 1997; Cervi˜no & Valls-Gabaud2003; Haas & Anders 2010; Eldridge 2011), in this workwe ask whether the observed L H α /L FUV is consistentwith a random but incomplete sampling of a canonicalor modified IMF. To test these hypotheses, we combineobservations from B09, L09 and M09 with models from slug (Fumagalli et al. 2011; da Silva et al. 2011), anovel fully-stochastic code for synthetic photometry ofstellar clusters and galaxies. THE OBSERVED GALAXY SAMPLE http://sites.google.com/site/runslug/ Fumagalli et al.
24 25 26 27 28 2911121314 l og ( L H α / L F UV ) ( H z ) Kroupa f c = 1
24 25 26 27 28 2911121314 l og ( L H α / L F UV ) ( H z )
24 25 26 27 28 2911121314 l og ( L H α / L F UV ) ( H z ) Kroupa f c = 0
24 25 26 27 28 2911121314 l og ( L H α / L F UV ) ( H z )
24 25 26 27 28 29log L
FUV (erg s −1 Hz −1 )11121314 l og ( L H α / L F UV ) ( H z ) Meurer+2009Lee+2009Boselli+2009IGIMFKroupa
IGIMF
24 25 26 27 28 29log L
FUV (erg s −1 Hz −1 )11121314 l og ( L H α / L F UV ) ( H z ) Figure 1.
Observed H α and FUV luminosity in a sample ofnearby galaxies (L09 red circles; B09 gray squares; M09 blue tri-angles). We also show (in green), ∼ slug models for a KroupaIMF with and without clusters (top and middle panel, respectively)and for the IGIMF (bottom panel). Analytic predictions for theKroupa IMF and IGIMF are superimposed (purple dashed and or-ange triple-dot dashed lines). White crosses mark the mean of thesimulated distributions, while the cyan crosses (top panel) are fora f c = 1 model with M cl , min = 500 M ⊙ . We compile a sample of 457 galaxies from B09, L09 andM09 with integrated H α and FUV luminosities, togetherwith corrections for dust extinction and [N II ] contami-nation. Due to differential absorption, particular care isrequired when performing dust corrections. Here we fol-low the procedure adopted by each author in their stud-ies, a choice motivated by the fact that the different dustextinctions are consistent across the three samples. Werefer to the original works for additional discussion.Similarly, a careful analysis of the selection biasesshould be a prerequisite for comparison with models. Un-fortunately, these data-sets have been assembled adopt-ing different selection criteria (a nearly complete volumelimited sample for L09, a subset of galaxies selected fromthe H I mass function for M09 and a sample of galaxieswith multi-wavelength observations for B09) that are dif-ficult to characterize. We will therefore emphasize onlythose results that are believed to be less sensitive to se-lection biases. STOCHASTIC EFFECTS AND
SLUG
MODELS
The IMF describes the relative probability with whichstars at any mass are formed. At high SFRs, the largenumber of stars guarantees a nearly complete sampling ofthe IMF, and the maximum stellar mass m max that canbe found in a galaxy approaches the theoretical upperend m max , ∗ of the IMF. At low SFRs instead, due to thesmaller number of stars that are formed, the probabil-ity to find massive stars decreases. The IMF is not fullysampled, and m max ≪ m max , ∗ . In this regime, stochas-tic effects become important to describe the observed L H α /L FUV distribution, regardless of the IMF functionalform. In addition to finite sampling in mass, L H α /L FUV is also affected by finite sampling in time since massivestars experience evolutionary phases of short duration(e.g. WN or WC phases) during which their ionizing lu-minosities can vary significantly.Since the majority of stars is believed to form in em-bedded star clusters (e.g. Lada & Lada 2003), two addi-tional effects should be considered. First, the maximumstellar mass in a cluster cannot exceed the cluster mass M ecl . Second, at low SFRs, a time-averaged continuousSFR results in a series of small “bursts” separated intime and associated with the formation of a new clus-ter. Provided that stars form with a modest spread inages, gaps in the ages of clusters increase the probabilityto observe a galaxy when the most massive stars havealready left the main sequence.Because it treats these types of stochastic samplingcorrectly, slug is the ideal tool to test whether incom-plete sampling of a canonical IMF can reproduce the de-ficiency of H α observed in low-luminosity galaxies with-out resorting to a modified IMF. For any given SFH,a fraction f c of the total stellar population is assumedto form in clusters. In this case, slug randomly drawsa set of clusters from a cluster mass function (CMF) ψ ( M ecl ) = M − β ecl and populates them with stars randomlyselected from an IMF φ ( m ) = m − γ over some specifiedmass interval. Currently, all the stars within a clusterare assumed to be coeval. The remaining fraction 1 − f c of the total stellar population are formed in the field,simulated by randomly selecting stars from an IMF. Theensemble of clusters and stars is then evolved with timeand at each time step the luminosity of individual starsis combined to quantify the number of ionizing photonsand L FUV for the simulated galaxy.In this work, we consider two classes of models. The“Kroupa model” is based on a universal Kroupa (2001)IMF in the mass interval 0 . −
120 M ⊙ and a CMF with β = 2 between 20 − M ⊙ . To test the effects of clus-tering we run two sets of simulations, one with f c = 0(all stars in the field, similar to the Monte Carlo in L09)and one with f c = 1 (all stars in clusters). These ex-treme cases bracket all possible solutions, although theobservations of Lada & Lada (2003) suggest that realityis likely closer to f c = 1 than f c = 0. The “IGIMFmodel” is instead based on the integrated galactic ini-tial mass function (IGIMF; Kroupa & Weidner 2003), amodified IMF in which m max is a function of M ecl that,in turn, depends on the galaxy SFR. The IGIMF imple-mentation follows the minimal-1 and minimal-2 formu-lations (Pflamm-Altenburg et al. 2007), with a Kroupa(2001) IMF and a CMF with β = 2, minimum mass of20 M ⊙ and maximum mass derived from the SFR av- (nearly) uniform stellar initial mass function 3eraged over 10 year. In both the Kroupa f c = 1 andthe IGIMF model, the IMF integrated over a galaxy ismodulated by the galaxy star formation and the CMF.While in the f c = 1 model the relations between thesequantities emerge from a pure stochastic realization ofthe IMF and CMF, the IGIMF relies on prescriptions forthese correlations that imply a truncation in the CMFand the IMF.For both models, we adopt stellar libraries with thePadova AGB tracks at solar metallicity (Bressan et al.1993), the Smith et al. (2002) implementation ofHillier & Miller (1998) and Lejeune et al. (1997) atmo-spheres, and Maeder & Meynet (1988) winds. We run10 simulated galaxies randomly drawn from a Schechterfunction, similar to the observed UV luminosity function(Wyder et al. 2005), with SFR 4 . × − − ⊙ yr − .We use 10 time steps of 80 Myr per galaxy to improvethe statistics. Each of them is independent since the80 Myr interval ensures that all massive stars that formin one time step are gone by the next one. To furtherimprove the statistics at higher luminosity (observationsare biased towards higher L FUV ), we add 2000 galaxiesuniformly distributed in logarithmic bins of SFR. Dueto escape or dust absorption before ionization, only afraction f H α of the Lyman continuum photons producesionizations that ultimately yield H α photons. To accountfor this effect, we correct the H α luminosity in the mod-els by a factor f H α = 0 .
95 (B09). Although the exactvalue for f H α is uncertain (L09), any choice in the range f H α = 1 − . Slug simulations and theoretical predictions are dis-played in Figure 1. Stochastic effects are evident in thedistributions, with simulated galaxies scattered in prox-imity to the expected luminosities for a fully sampledIMF. Since previous studies (e.g. Haas & Anders 2010)have highlighted the importance of the minimum massin the CMF ( M cl , min ) for this type of calculations, wedisplay in the top panel of Figure 1 the mean valuesfor additional simulations that are similar to the f c = 1model, but with M cl , min = 500 M ⊙ . DISCUSSION
The main result of this analysis is summarized in Fig-ure 1. At L FUV > erg s − Hz − , observed galaxieslie close to the value of L H α /L FUV expected for botha fully-sampled Kroupa IMF and the IGIMF, but atfainter luminosity and lower SFR the data deviates fromthe fully-sampled Kroupa IMF curve towards the fully-sampled IGIMF curve. This trend has been taken asevidence in support of the IGIMF, but both IMFs aresubject to stochastic effects. When we properly includethese using slug , we see that realizations drawn fromthe IGIMF are completely inconsistent with a signifi-cant fraction of the observed galaxies, particularly at L FUV ∼ − erg s − Hz − . Conversely, realiza-tions drawn from a Kroupa IMF span a larger range ofluminosity and overlap with most of the observed sam-ple, with clustering responsible for a further increase inthe luminosity spread (compare the f c = 1 and f c = 0models). N / N t o t FUV < 25.0
Obs.Krou. f c =1Krou. f c =0IGIMF N / N t o t FUV < 26.0 −3 −2 −1 0 1log L H α / < log L H α > 0.000.050.100.150.200.250.30 N / N t o t FUV < 27.0−3 −2 −1 0 1log L H α / < log L H α > 0.000.050.100.150.200.250.30 N / N t o t −3 −2 −1 0 1log L H α / < log L H α > 27.0 < log L FUV < 28.0−3 −2 −1 0 1log L H α / < log L H α > Figure 2.
Distributions of H α luminosity in intervals of L FUV forthe observed galaxies (solid black histogram) and for slug modelsbased on the Kroupa IMF with and without clusters (blue dash-triple-dotted and green dash-dotted histograms) and for the IGIMF(red dashed histogram). Distributions are centered to h L H α i , thelogarithmic mean of the observed H α luminosity in each bin. Thetwo downward triangles indicate the mean L H α for a fully sampledKroupa IMF (blue) and IGIMF (red). The width in the simulated distributions follows fromthe treatment of m max and clustering. For a universalIMF, m max can assume any value up to m max , ∗ , regard-less of the SFR. At low SFRs, realizations that lack mas-sive stars are frequent and skew the distribution to low L H α and low L FUV . At the same time, realizations withmassive stars are still possible and some models are dis-tributed near or even above the theoretical expectationfor a fully sampled IMF. The narrower scatter found forthe f c = 0 model emphasizes that stochastic samplingof the IMF alone cannot reproduce the entire range ofobserved luminosity (cf. L09). In the f c = 1 model,simulated galaxies that lie at the lowest L H α for anygiven L FUV have an excess of older and massive clus-ters. This is because at low SFR, a massive cluster rep-resents a significant event in the galaxy SFH that, onaverage, increases the time interval between the forma-tions of clusters and produces an intrinsic level of bursti-ness that results in a wider luminosity distribution (seealso da Silva et al. 2011). This effect is obviously am-plified in models with M cl , min = 500 M ⊙ , as evidentfrom the lower H α/ FUV luminosity in these simulations.Note that irregular SFHs are typical of dwarf galaxies(Weisz et al. 2008) and models with bursty star forma-tion may reproduce the observed luminosity distributionequally well. Conversely, in the IGIMF theory, at lowerSFRs, m max ≪ m max , ∗ . Further, only clusters with lowmass can be drawn. The narrower mass range that isaccessible translates in a narrower spread in luminosity,and none of the models can significantly exceed the ex-pected luminosities for a fully sampled IGIMF (see L09).Due to incompleteness, statistical comparisons be-tween data and simulations are not straightforward. Thisis complicated by the fact that, at any SFR, both H α andFUV luminosities are subject to scatter and a truly inde-pendent variable is lacking. Nevertheless, in Figure 2 andTable 1, we attempt to quantify the agreement between Fumagalli et al. Table 1
Summary of the H α statistics in four bins of FUV luminosity from Figure 2.Type a Number b Mean log L H α c Fully-sampled log L H α d Dispersion e Probability f N( σ ) g (erg s − ) (erg s − )24 . < log L FUV < . · · · · · · · · · Kroupa f c = 1 241817 37.0 37.7 0.86 0.5995 0.5Kroupa f c = 0 362593 37.3 37.7 0.55 0.5515 0.6IGIMF 64253 35.6 36.4 0.72 0.0034 2.925 . < log L FUV < . · · · · · · · · · Kroupa f c = 1 25571 38.3 38.7 0.55 0.1073 1.6Kroupa f c = 0 24705 38.5 38.7 0.30 0.1205 1.6IGIMF 9363 37.7 38.2 0.45 0.0000 6.926 . < log L FUV < . · · · · · · · · · Kroupa f c = 1 4248 39.5 39.7 0.40 0.3524 0.9Kroupa f c = 0 4238 39.6 39.7 0.29 0.0002 3.7IGIMF 3576 39.2 39.6 0.37 0.0003 3.627 . < log L FUV < . · · · · · · · · · Kroupa f c = 1 3494 40.6 40.7 0.34 0.4042 0.8Kroupa f c = 0 3562 40.6 40.7 0.29 0.0090 2.6IGIMF 3296 40.4 40.7 0.33 0.0365 2.1 a Type of distribution for which the statistics are listed. b Number of observed and simulated galaxies included in each log L FUV bin. c Mean of the log L H α distributions. d log L H α for a fully sampled IMF. e Standard deviation of the log L H α distributions. f Kolmogorov-Smirnov probability P associated to the hypothesis that the observed and simulated distributionsare drawn from the same parent population. Although the absolute values for the listed probabilities are difficultto interpret given the poorly characterized selection biases, the relative differences between the f c = 1 and theIGIMF models support quantitatively our conclusion. g Equivalent number of standard deviations N = √ − (P). models and observations by comparing the statistics ofthe H α distributions in intervals of L FUV .Considering the center of the H α distributions, theIGIMF and f c = 1 realizations diverge, moving towardslower FUV luminosity. At the lowest L FUV , the two dis-tributions are separated by ∼ L H α than observed. This is particularlyevident for the interval 25 < log L FUV <
26 that is char-acterized by large enough statistics ( ∼
100 galaxies) andlikely not affected by severe incompleteness. To quan-tify this claim, we perform a Kolmogorov-Smirnov testcomparing the observed distribution with 10000 randomsub-samples of the models, extracted to have a size com-parable to the data. We report the results in Table 1.We see that the observed data and the Kroupa f c = 1models are generally consistent with being drawn fromthe same parent distribution, while the hypothesis thatthe data and the IGIMF models originate from the sameparent distribution can be ruled out. The f c = 0 modelalso appears to be inconsistent with the data, though notby as much as the IGIMF model.Finally, in Figure 3 we compare for each individ-ual data point the observed L H α, obs with the simu-lated L H α, mod , averaged over an interval of FUV that is the larger of 0.1 dex and the error on the observed L FUV . This difference is then normalized to σ tot = p σ + σ where σ obs is the error on L H α, obs and σ mod is the standard deviation of the models. This quan-tity is defined such that a distribution centered on zerowith dispersion of unity indicates perfect agreement be-tween models and data (see right panels).In Figure 3, the center of the f c = 1 model (toppanel) is consistent with the observed galaxies within2 σ tot for L FUV < erg s − Hz − and within 3 σ tot for brighter luminosity, where σ obs becomes comparableto σ mod . Conversely, due to the narrower scatter, the f c = 0 model (central panel) is only partially consis-tent with observations, as previously suggested by theKS test. Also, simulated galaxies have higher L H α com-pared to the observed galaxies. Based on results of twoindependent tests, we favor the f c = 1 model over the f c = 0 simulations. The IGIMF model (bottom panel)reproduces instead the observed L H α only at high L FUV and there is a clear systematic offset between the dataand the models at L FUV < erg s − Hz − . Althoughthe usual notion of probability associated to σ tot doesnot apply since the H α distributions are not Gaussian,we conclude that observations are better described bymodels based on the Kroupa IMF than on the IGIMF,particularly at fainter luminosity and when clusters are (nearly) uniform stellar initial mass function 5
24 25 26 27 28log L
FUV (erg s −1 Hz −1 )−505 ( L H α , m o d − L H α , o b s ) σ t o t − IGIMF −505 ( L H α , m o d − L H α , o b s ) σ t o t − Krou. f c = 0 −505 ( L H α , m o d − L H α , o b s ) σ t o t − Krou. f c = 1 Meurer+2009Lee+2009Boselli+2009 .1 .2 .3N/N tot M e a n − . S t d . . M e a n . S t d . . M e a n . S t d . . Figure 3.
Difference between L H α in individual observations andthe mean of the simulated galaxies at comparable L FUV , normal-ized to the model standard deviation and observational uncertainty.Histograms and Gaussian statistics are shown in the right panels. included.Data are very sparse below L FUV ∼ erg s − Hz − ,where the separation between a universal IMF and asteeper/truncated IMF is most evident, precluding usfrom concluding that no variation in the IMF occurs atvery low luminosity. Moreover, we have not explored cor-relations with the galaxy surface brightness, suggested byM09 to be the physical quantity related to the IMF varia-tion. Also, selection biases are not well characterized anda putative population of galaxies with L H α < ergs − and L FUV ∼ − erg s − Hz − would call intoquestion a universal IMF. Further we assume that allstars born in an individual cluster are coeval, and fu-ture work will be required to investigate the effects ofrelaxing this assumption. Conversely, the observed ex-istence of galaxies with L H α ∼ − erg s − and L FUV ∼ − erg s − Hz − poses a direct challengeto the truncation in the IMF and CMF as currently im-plemented in IGIMF theory. As illustrated in Figure 1, ifthe IGIMF model is correct, it should be impossible forgalaxies to occupy this region of luminosity space. Theconclusion is strengthened by the fact that we have im-plemented only a minimal IGIMF. The standard IGIMFwould predict even less L H α , exacerbating the discrep-ancy with the observations. Uncertainties in the mea-surements, particularly in the dust corrections, togetherwith the fact that our models are highly idealized (sim-ple SFH, single metallicity, lack of any feedback), mayexplain the existence of galaxies non-overlapping withIGIMF models, but the discrepancy we found is mostlikely larger than the errors at faint L H α . SUMMARY AND CONCLUSION
Using slug , a fully stochastic code for synthetic pho-tometry in star clusters and galaxies, we have comparedthe H α to FUV luminosities in a sample of ∼
450 nearbygalaxies with models from a universal Kroupa IMF and amodified IMF, the IGIMF. Our principal findings are: i)simulated galaxies based on a Kroupa IMF and stochas-tic sampling of stellar masses are consistent with the ob-served L H α distribution; ii) only models where stars areformed in clusters account for the full scatter in the ob-served luminosity; iii) realizations based on a truncatedIMF as currently implemented in the IGIMF underesti-mate the mean H α luminosity.Based on this result, and since other factors not in-cluded in our simulations (e.g. dust, escape fraction ofionizing radiation, bursty SFHs) can mimic some of thefeatures of the observed luminosity distribution (B09,L09), we conclude that present observations of the inte-grated luminosity in nearby galaxies are consistent witha universal IMF and do not demand a truncation at itsupper end.While we show that the current IGIMF implementa-tion provides a poor description of available observations,our work is consistent with the fundamental idea behindthe IGIMF, i.e. that the SFR, the CMF, and the IMFjointly produce the observed luminosity distribution ofgalaxies. However, our analysis emphasizes that the cor-relations between these quantities emerge naturally fromstochastic sampling and that a further modification tothe IMF in clusters as proposed in the IGIMF model isnot needed to account for the integrated luminosities ingalaxies. Further, our calculation highlights how clustersintroduce the level of burstiness required to fully accountfor the observed luminosities, owing to the combined ef-fects of the cluster age distribution and the short life-timeof massive stars. Time dependence might be a key ele-ment currently missing in models of the integrated IMFin galaxies.We are most grateful to A.Boselli and G.Meurer forsharing their data and for valuable comments. Weacknowledge useful discussions with M.Cervi˜no, J.Lee,P.Kroupa, J.Scalo, X.Prochaska, J.Werk, G.Gavazzi andK.Schlaufman. We thank F.Bigiel for motivating us towrite slug . RLdS is supported under a NSF Gradu-ate Research Fellowship. MRK acknowledges supportfrom: Alfred P. Sloan Fellowship; NSF grants AST-0807739 and CAREER-0955300; NASA AstrophysicsTheory and Fundamental Physics grant NNX09AK31G; Spitzer Space Telescope
Theoretical Research Programgrant.
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