Stochasticity, a variable stellar upper-mass limit, binaries and star-formation rate indicators

0.45 MB·English

by  John J. Eldridge

Mon.Not.R.Astron.Soc.000,1–11(2005) Printed2February2012 (MNLATEXstylefilev2.2)
Stochasticity, a variable stellar upper-mass limit, binaries
and star-formation rate indicators
John J. Eldridge1,2 ⋆
2 1Department of Physics, University of Auckland, Private Bag 92019, Auckland, New Zealand
1 2Institute of Astronomy, The Observatories, Universityof Cambridge, Madingley Road, Cambridge, CB3 0HA
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2

n
a 2February2012
J

1
3 ABSTRACT
Using our Binary Population And Spectral Synthesis (BPASS) code we explore the
]
R effects on star-formation rate indicators of stochastically sampling the stellar initial
massfunction,addingaclustermassdependentstellarupper-masslimitandincluding
S
binary stars. We create synthetic spectra of young clusters and star-forming galaxies
.
h andcomparethese to observationsofHα emissionfromisolatedclustersandthe rela-
p tion between Hα and FUV emission from nearby galaxies. We find that observations
- of clusters tend to favour a purely stochastic sampling of the initial mass function
o
r for clustersless than 100M⊙,ratherthan the maximum stellarmass being dependant
t on the total cluster mass. It is more difficult to determine whether the same is true
s
a for more massive clusters. We also find that binary stars blur some of the observa-
[ tional differences that occur when a cluster-mass dependent stellar upper-mass limit

is imposed when filling the IMF. The effect is greatest when modelling the observed
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Hα andFUV star-formationrate ratiosin galaxies.This is because masstransferand
v
1 merging of stars owing to binary evolution creates more massive stars and stars that
1 have greater mass than the initial maximum imposed on the stellar population.
3
Key words: binaries:general–galaxies:starclusters–HIIregions–galaxies:stellar
4
. content
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1
: 1 INTRODUCTION theirownmassandage.Themassesoftheseclustersarealso
v
described by their own cluster initial mass function. Fur-
i A key problem when modelling stellar populations is how
X thermorefluctuationsintheIMFoftheseclusters,especially
r tpoopdueltaetrimonin.eTthheecdoinstvreinbtuiotinoanlomfeitnhitoidalisstteolladremfinaessaens iinnittihael iftheirmassislessthan104M⊙,canprovidelargevariation
a in the ionising fluxes from the cluster (Cervin˜o et al 2003;
massfunction(IMF)accordingtowhichthenumberofstars
Villaverde, Cervin˜o & Luridiana 2010a,b). The implication
ofagivenmassiscalculatedasafunctionofthemass.Typi-
of stars forming in clusters is that the stellar IMF (SIMF)
callyapower-lawofthemassisused.Thefirstwassuggested
by Salpeter (1955) where dN(M) ∝ M−2.35dM for 0.3 < does not apply across an entire galaxy. Instead to get the
galaxy-wide distribution of stellar masses we must model a
M/M⊙ < 10. Despite being 57 years old this IMF is still number of stellar clusters with different masses according
widely used and appears to be universal. This slope holds
to a cluster initial mass function (CIMF) and within each
over a wide range of stellar masses, only flattening in gra-
clusters apply a SIMF to produce an integrated galaxial
dient below stellar masses of around 1M⊙ (Miller & Scalo initial mass function (IGIMF) (e.g. Weidner& Kroupa
1979;Kroupa2001;Chabrier2003;Bastian, Covey & Meyer
2006; Pflamm-Altenburg,Weidner & Kroupa 2007).
2010).
This is the case even if not all the clusters are
The IMF provides the distribution of stellar initial
dense enough to remain bound over their life-
masses in a stellar population such as that found in stellar
times (Portegies Zwart, McMillan & Gieles 2010;
clusters. However when tryingtosimulate thestellar popu-
Bressert et al. 2010; Gieles & Portegies Zwart 2011).
lation inagalaxy it isimportant torecognise that agalaxy
Bastian, Covey & Meyer (2010) and Haas & Anders(2010)
is not made up of one unique stellar population. A galaxy
reviewed and investigated the importance of combining a
isactually madeup ofnumbersof stellar clusterseach with
CIMF and a SIMF to make a galaxy-wide SIMF. They
foundthattheresultantgalaxy-wideSIMFismostsensitive
to the minimum mass for a cluster, the slope of the CIMF
⋆ E-mail:[email protected]
(cid:13)c 2005RAS
2 John J. Eldridge
and whether the mass of the most massive star is limited Thenovelfeatureofthisworkisthatweareabletodemon-
by thecluster mass. strate how binary stars alter our population synthesis
The most extreme articulation of the last factor is predictions. Recent observations (Pinsonneault & Stanek
whether it is physically possible for a 100M⊙ cluster to be 2006; Kobulnicky & Fryer 2007; Kiminkiet al. 2009) in-
composed of asingle 100M⊙ star orthemore probablesce- dicate that the binary fraction in young massive stellar
nario of acluster composed of many lower-mass stars.. The populations is close to one. It is therefore vital to include
alternativeisthatsuchlow-massclusterscanonlyputsay10 binary stars especially those that interact. In our popu-
percentorlessoftheirtotalmassintothemostmassivestar. lations approximately two thirds of binaries interact. We
If theformer is possible it would suggest that some O stars first outline our stellar evolution models and the method
might form in isolation (here taken to be in isolation with of our spectral synthesis. We then describe our two ways
noother O or B stars in the same cluster so theotherclus- to determine the distribution of initial masses in synthetic
termembersareofamuchlowermass)andthiswouldhave clusters and galaxies. Next we discuss the observational
important implications for the star-formation process. I.e. implications of varying the IMF-filling method on the Hα
is star formation a pure-stochastic process or a bottom-up andFUV star-formation rate indicators. Finally we present
processwithlow-massstarsformedfirstandhigh-massstars ourconclusions.
only formed if there is enough material left. After account-
ingforrunawayOstarscontaminatingtheapparentnumber
ofisolated(withnocompanionsatall)OstarsdeWit et al. 2 NUMERICAL METHOD
(2005)suggestedthatatmost4±2percentofOstarsform
2.1 Binary population and spectral synthesis
outside a cluster environment. Parker & Goodwin (2007)
considered that an isolated O star only meant there was We have developed a novel and unique code to pro-
nootherOBstarin thesamecluster.Thisisthecase when duce synthetic stellar populations that include binary stars
a100M⊙ clusteriscomposedofonestarthatcontainsmost (Eldridge, Izzard & Tout 2008; Eldridge & Stanway 2009).
ofthemassoftheclusterwithafewverylow-masscompan- While similar codes exist our Binary Population and Spec-
ions. They modelled the populations of star clusters using tral Synthesis (BPASS) code has three important features
a standard CIMF with a slope of −2 and predicted that 5 each of which set it apart from other codes and enable it
per cent of O stars formed in clusters that had no other O to study stochastic effects on the IMF. First, and most im-
or B stars and would be observed as isolated when in fact portant,istheinclusionofbinaryevolutionwhenmodelling
theyarejustmassivestarsthathavebeenabletoform ina the stellar populations. The general effect of binaries is to
low-mass cluster. cause a population of stars to look bluerat older ages than
The argument against the conclusion that O stars predictedbysingle-starmodels.Secondly,alargenumberof
can form in low-mass clusters has been presented detailedstellar evolutionmodelsareusedtocreatethesyn-
by Vanbeveren (1982), Weidner& Kroupa (2006) and thetic populations rather than an approximate rapid popu-
Weidner,Kroupa & Bonnell(2010)whohavesuggestedthat lationsynthesismethod.Thirdly,weuseasmanytheoretical
observations indicate that, for a specific cluster mass, there inputsinoursynthesiswithasfewempiricalinputsaspossi-
is a maximum possible stellar mass well below the to- bletocreate acompletely syntheticmodeltocomparewith
tal cluster mass. If this were the case then, from their observations.
model, a 100M⊙ cluster could not form stars with masses BPASSusesapproximately15,000detailedstellarmod-
above 10M⊙. Therefore when a synthetic galaxy is cre- els calculated by the Cambridge STARS code as described
ated from the convolution of cluster and stellar IMFs byEldridge, Izzard & Tout(2008).Theseincludesinglestar
there would be a dearth of the most massive stars com- and binary models with initial masses between 0.5 and
pared with when there are no restrictions on the maxi- 120M⊙ and 5 and 120M⊙ respectively. We take 120M⊙
mum stellar mass. Pflamm-Altenburg,Weidner& Kroupa to be our most massive star possible because of our lim-
(2007, 2009) have suggested that there are differences in ited grid of binary evolution models. Above this mass the
star-formation rate indicators when stellar populations are mass-loss rates at solar metallicity on the main-sequence
modelled by the two different IMF filling methods. How- are high (Vink et al. 2011) and the evolutionary timescales
ever the observations used consider the differences that oc- of the stars vary little as the initial mass is increased fur-
cur at low star-formation rates and thus uncertainties and ther.Wenotethat, owing tostars merging ourbinary pop-
low-number fluctuations between observed systems make it ulations include some single stars that have effective ini-
difficult to determine whether the maximum stellar mass tial masses of 200M⊙ and above. Theminimum binary pri-
depends on the cluster mass. However the complementary mary mass of 5M⊙ is selected because initially our binary
studiesbyElmegreen(2006),Parker & Goodwin(2007)and models were specifically created to studytheprogenitors of
Maschberger & Clarke (2008) show that similar observa- core-collapse supernovae. The main-sequence lifetime of a
tions indicate that there is no evidence for restrictions on 5M⊙ star is 100Myrs, which is the period we use for the
themaximum stellar mass in clusters. duration of the star-burst in our constant star-formation
In this paper we examine recent observations models so there is no effect from low-mass binaries in
(Lee et al. 2009; Lamb et al. 2010; Calzetti et al. thesemodels. Furthermoreobservationsindicatethebinary
2010) that may provide a firmer constraint on fraction decreases at the low masses (Duquennoy& Mayor
how nature selects the masses of stars in clusters 1991; Leinert et al. 1997; Bouy et al. 2003). However the
and galaxies (Pflamm-Altenburg,Weidner& Kroupa binary fraction is more complicated than we assume here
2007, 2009; Villaverde, Cervin˜o & Luridiana 2010b; and is determined by a star cluster’s dynamics, environ-
Fumagalli, da Silva & Krumholz 2011; Weisz et al. 2012). ment and age. It is thought that stars of all masses can
(cid:13)c 2005RAS,MNRAS000,1–11
Stochasticity, a variable upper-mass limit, binaries and SFRIs 3
form in binary or multiple systems but these can be bro- ilar to those from the sorted-sampling method outlined by
ken up by dynamical interactions in young clusters (e.g. Weidner& Kroupa (2006).
Goodwin & Kroupa 2005; Fregeau, Ivanova& Rasio 2009). Wenotethatoursyntheticpopulationshavesomelim-
We note that Han, Podsiadlowski & Lynas-Gray (2007) itations. InFigures 1 and2 thereare diagonal and horizon-
foundlow-mass binarystarscan explaintheexcessUVflux tal linear features in the distribution of model populations.
observed in Elliptical galaxies. Such systems do not con- These arise at low cluster-masses and star-formation rates
tributestronglyuntil1Gyrafterformationatwhichtimeour owing to the limited resolution of the stellar model initial
estimated UV fluxes should increase only slightly. Creating masses and time bins used in our synthesis. This becomes
anew grid of low-mass detailed binary models for inclusion most noticeable when there is only one massive star in the
in BPASS is unnecessary for this work to demonstrate the stellarpopulation.Onesolutiontothiswouldbetointerpo-
importance of binary stars. late between stellar models but given that stellar evolution
Here we use models at solar metallicity with a metal- is non-linear and binary evolution is even less predictable
licity mass fraction of Z = 0.02. We include con- weavoid spuriousresults from interpolations and select the
vective overshooting and a mass-loss prescription that closest model available.
combines the mass-loss rates of Vink,deKoter & Lamers Alsoourbinarypopulationmodelsarenotcompleteand
(2001), de Jager, Nieuwenhuijzen & vander Hucht (1988) hereweareonlydemonstratingtheimportanceofincluding
and Nugis & Lamers (2000). The binary evolution ac- binary stars. For example, we do not include binaries with
counts for Roche-lobe overflow, common-envelope evo- initialprimarymassesbelow5M⊙ andasyetwedonotcon-
lution, mass transfer and neutron-star kicks which sider the emission from X-ray binaries. This would provide
affect the survival of binary stars after a super- anothersourceofionisingfluxthatwouldalsoeffecttheHα
nova. These models are combined with the stellar at- and UV flux ratio. The effect would be more important at
mosphere spectra of Smith, Norris & Crowther (2002), low cluster masses and low star-formation rates where one
Hamann, Gr¨afener & Liermann (2006) and Westera et al. X-ray binary would dominate the entire ionising flux from
(2002) to predict thespectra of thestellar populations. thestellar population (Mirabel et al. 2011).
A significant change we make here, compared with our
previouswork,istobreakfromourpreviousassumptionthat
the SIMF can be described by a simple Salpeter law over
2.2 Creating synthetic clusters
the entire mass range of stars. Our method requires us to
considerthatallstarsareborninclusters.Themassofthese
WeusethePSSandCMDMSMmethodstocreatesynthetic
clusters is described by a CIMF and the mass distribution
stellar populations in two regimes. In the first we consider
of stars within each cluster is described by the SIMF. This
individualstellarclusterswithallthestarscoeval.Wecreate
is achieved by first picking a cluster mass and then filling
models of stellar clusters with both PSS and CMDMSM
the cluster with stars from the SIMF. We model multiple
and investigate how they affect the Hα line flux per M⊙ in
clusters together to create synthetic galaxies with different
the cluster. Our process for creating a synthetic cluster to
star-formation histories but with the same mean constant
compare to the observations of Calzetti et al. (2010) is as
star-formation rate over a long period of time.
follows.
We use two methods of populating the SIMF for our
syntheticclusters.Theydifferbywhetherwelimitthemax- (i) We randomly generate a cluster mass between 10
imumstellarmassornot.Ourfirstmethodistoassumethat and 106M⊙ from the CIMF which has a slope of −2
any star can occur in any cluster such that, Mmax 6 Mcl. (deGrijs et al. 2003; Lada & Lada 2003).
I.e. the star cannot be more massive than the cluster it in- (ii) Wefilltheclusterwithstars,themassesofwhichare
habits. This we refer to as pure stochastic sampling (PSS) pickedatrandomfromtheSIMFwiththemaximumstellar
of the SIMF. In this SIMF we assume a Salpeter slope of mass given byPSS or CMDMSM.
-2.35 between 0.5 and 120 M⊙ and a slope of -1.3 between (iii) We add stars to the cluster until the total mass
0.1 and 0.5M⊙. It is similar to the constrained sampling is greater than our target cluster mass. We then consider
method outlined by Weidner& Kroupa(2006) andused by whetherthefinalclustermass iscloser tothetarget cluster
Villaverde, Cervin˜o & Luridiana (2010b). mass with or without the last star added to the cluster. If
Our second case has the maximum mass of the mass is closer without the last star we remove the last
a star in a cluster dependent on the total mass starfrom thecluster. Thisis similar tothesorted sampling
of the cluster. We use the relation calculated by of Weidner & Kroupa (2006) and makes it less likely that
Pflamm-Altenburg,Weidner& Kroupa (2007) which is a star can be added that is more massive than the target
given by, cluster mass as in the soft sampling of Elmegreen (2006).
(iv) We randomly generate the cluster age between 1
log10(Mmax/M⊙)=2.56log10(Mcl/M⊙) and 8Myr. This is to match the observed age range of
−1
×(cid:0)3.829.17+(log10(Mcl/M⊙))9.17(cid:1)9.17 −0.38, Calzetti et al. (2010).
(v) WecalculatetheHαfluxfortheresultantstellarpop-
whereMmax isthemaximumstellarmasspossibleinaclus- ulation. This is done with theoretical stellar atmospheres
ter of mass Mcl. We therefore use Mmax from this equa- and stellar models to predict the resultant total spectrum
tion as the maximum mass in our initial mass function up as described by Eldridge & Stanway (2009). We calculate
to a limit of 120M⊙ in our synthetic clusters. We refer to thenumberofionisingphotonsfromwavelengthsshortward
thismethodastheclustermassdependentmaximumstellar of 912˚A and convert this to the flux of Hα by assuming
mass (CMDMSM) method. The resulting clusters are sim- 1011.87 ionising photonsgive rise to 1ergs−1 of Hα flux.
(cid:13)c 2005RAS,MNRAS000,1–11
4 John J. Eldridge
This process is repeated for many different cluster CIMF suggest that there is no such dependence (Gieles
masses so that we can build up a picture of how Hα 2009; Larsen 2009). In this work we wish to concentrate on
flux varies with cluster mass for clusters aged between 1 whether the maximum stellar mass depends on the cluster
and8Myr.Calzetti et al.(2010)performedanobservational mass. We have calculated IGIMF models to see the effect
study of such clusters and provide the observed mean Hα of including such a limit and find our models are in agree-
flux per M⊙ for two different masses of clusters. We com- ment with those of Pflamm-Altenburg,Weidner& Kroupa
pare our models to these observed populations in Section (2007) and Pflamm-Altenburg,Weidner& Kroupa (2009).
3.1. In the binary population case we include a companion Also like Fumagalli, daSilva & Krumholz (2011) we find
foreverystarthathasaninitialmassgreaterthan5M⊙.We that IGIMF synthetic galaxies cannot reproduce the ob-
assign binaryparametersat randomfromaflatinitialmass served spread of star-formation rate ratios. This is because
ratioandflatdistributionofthelogarithmoftheinitialsep- restrictingthemaximumclustermassdecreasesthenumber
aration using the model closest to the parameters from our ofmassivestarsevenmoredramaticallythantheyareinour
gridofmodelscalculatedbyEldridge, Izzard & Tout(2008). CMDMSM models.
We include the mass of the companion in the total cluster In Section 3.2 we compare the synthetic populations
mass. to the observed galaxies of Lee et al. (2009). The novel
feature of our approach is in not forcing clusters to form
at the same time but allowing each to have a different
2.3 Synthetic galaxies
age. This leads to much more scatter in the predicted ob-
Oursecondsetofpopulationmodelsareforsyntheticgalax- servables of our synthetic galaxies. This was also found
ies with an assumed constant star-formation rate. Rather byFumagalli, da Silva& Krumholz(2011)and Weisz et al.
thanfillupthepopulationofagalaxyaccordingtoagalaxy- (2012). Wehavealso varied theage range used for thesyn-
wide IMF we create the galaxy from a set of clusters that thetic galaxies and find that increasing the age has little
eachhavetheirownindividualageandstellarpopulation.To effect on our results. Using a younger upper age limit in-
create a galaxy we first pick a star-formation rate between creases theamount ofHαfluxrelative totheUVflux.This
10−5 to10M⊙yr−1.Wethencreatethesyntheticgalaxy as is because the stars that cause Hα emission are more mas-
follows. siveandtypicallyhavelifetimesof10Myrorless,whilestars
thatcontributetotheFUVcontinuumspanamuchgreater
(i) WepickaclustermassatrandomfromaCIMFwhich
lifetime range of up to100Myr.
has a slope of −2 between 50 and 106M⊙.
(ii) We fill the cluster with a stellar population as de-
scribed in Section 2.2 and aged to between 0 and 100 Myr,
chosen at random from a uniform distribution.
3 RESULTS
(iii) Wecontinuethisprocessuntilthetotalmasscreated
inthegalaxyover100Myrgivestherequiredstar-formation 3.1 The Hα from individual clusters
rate.
Calzetti et al. (2010)suggested anoveltestfordetermining
(iv) Withthisstellarpopulationwecalculatethenumber
howtheIMFdefinesthepopulationofstellarclusters.They
ofionisingphotonsfromwavelengthsshortwardof912˚A and
convert this to the flux of Hα by assuming 1011.87 ionising studiedtheproductionofionisingphotonsbyyoungclusters
tphheotUoVnsflguivxedreinsesittyoa1teragws−av1eolefnHgtαhfloufx1.5W00e˚Aa.lso calculate itnheNnGoCne5119045.MIf⊙ancluIsMteFr sishopuolpduhlaatveedtphueresalymsetoscthelalastriccaolnly-
(v) FromtheseHαandUVfluxeswecalculateanappar- tentasahundred1000M⊙ clusters.Thereforebothsamples
ent star-formation rate from both and find their ratio. We would have the same Hα flux per M⊙ of stars. However if
floafsusxluomogfel(oaFg1s(0t1(a5Fr0-0(foH˚Arα)m/)ae/rtegirosgnss−sr−1a1tH)ez=−of14)11M=.1⊙a2n7y.dr8−5a1aUpsVroindfluuKcxeensdneanicsHuittαyt ttcholueastIleMirnsFkwbofoetuawld1e0ehn00aMMvem⊙laexcslsaunsHtdeαrMiflsCuldxtehvpeoneidrthoMef⊙mhuatnshsdairvneedas1ta01r00s05MMdu⊙⊙e
10 cluster.
(1998).
For our population of syntheticclusters we havecalcu-
Weperformthesesimulationsforsingleandbinarypop- latedthemeanHαfluxperM⊙ asfortheobservedclusters
ulations and for the PSS and CMDMSM methods of filling of Calzetti et al. (2010). Figure 1 shows our synthetic clus-
the IMF so that the differences can be compared. Here we ters as points along with the mean Hα flux.We see that at
use a different range of cluster masses based on the sugges- cluster masses below 104M⊙ the results diverge. With PSS
tion of Lada & Lada (2003) that thereis a turn-overin the itispossibletohaveonemassivestarmakingupmostofthe
massfunctionofmolecularcloudsataround50M⊙.Wealso massofaclusterwhilewith CMDMSMthisisnot possible.
onlyconsideraperiodof100Myrbecausethisisoftheorder Therefore for PSS and CMDMSM the observed mean Hα
of a typical star-formation burst duration (McQuinn et al. fluxdropsfrom themean valueofaround 1034.1ergs−1M−1
⊙
2009).Wealsofindthatincreasingtheagebeyond100Myr ataround 102 or104M⊙ respectively.Therefore bymeasur-
haslittleeffectonourresultsbecauseitisthetypicallifetime ingtheHαfluxfor clusters inbetween thesekeymasses we
of stars that contribute to the FUV. We note that, when should be able to determine how naturefills theIMF.
usedtocreateasyntheticgalaxy,ourCMDMSMmethodis Calzetti et al. (2010) provide two observed values for
based on the IGIMF method of Weidner & Kroupa (2006). their two mass bins. The first and higher value does not
However we do not limit the maximum cluster mass in a include clusters that are undetected in Hα. The second in-
synthetic galaxy by the total star-formation rate as they cludes these non-detections. The observations at a cluster
do in their IGIMF method. Recent investigations of the mass of 104.5M⊙ agree with the predicted mean Hα flux.
(cid:13)c 2005RAS,MNRAS000,1–11
Stochasticity, a variable upper-mass limit, binaries and SFRIs 5
Figure 1. Hα flux per M⊙ for our synthetic clusters. The shaded contours represent the density of individual realisations of different
clusters for two methods of fillingthe IMF: PSS forupper panels and CMDMSMfor the lower panels. Thelines show the meanfluxes
fromthesetwomethods,thesolidblacklineforPSSanddashedblacklineforCMDMSM.Thelinesareshowninbothsinglestarpanels
and both binary star panels. The black boxes represent the observations of Calzetti etal. (2010), the lower points include clusters not
detected inHαandthehighfluxvaluedoes notincludethem.Theblackopenboxrepresents therangeofpossiblevalues derivedfrom
the Velorum cluster estimated from Jefferiesetal. (2009) and DeMarcoetal. (2000). The diamonds and crosses are based on data of
Lambetal.(2010).Thediamondsrepresentclusterswithupperlimitstotheirmassandthecrosseswithmeasuredclustermasses.The
grey triangle indicates the position of the massiveHII region NGC604 with values from Eldridge&Relan˜o (2011). The left panels are
forsingle-starpopulationsandandtherightpanelsareforbinarypopulations.Thelinearfeaturesinthecontoursinthepanelsaredue
tothelimitedresolutionofthestellarmodelsinitialmassesandtimebinsusedinthesynthesis.
However the observed points at 103M⊙ are less conclusive. tially35and30M⊙ inaclusterwithatotalmassofbetween
The point without the non-detections lies on the the PSS 250and350M⊙ (DeMarco et al.2000;Jefferies et al.2009;
line,whilethepoint includingthenon-detectionslies inbe- Eldridge 2009). Wehaveindicated thelocation of thisclus-
tween thePSS and CMDMSM lines. ThusPSS givesa bet- ter in Figure 1. We see that the PSS clusters overlap with
terfitbuta refinedCMDMSM schemethat allows ahigher the parameters of this cluster. The CMDMSM models for
maximum mass for a certain cluster mass may match the a single star population do not reach this region. The bi-
Calzetti et al. (2010) data. nary CMDMSM models do reach the parameter space for
γ-Velorum.Thesmallnumberofsuchmodelsindicatessuch
AnalternativemethodtodiscriminatebetweenPSSand
clusterswouldberare.ThissuggeststhatPSSismorelikely
CMDMSMistosearchforindividualmassivestarsthatare
to be in action in nature although a more relaxed form of
in low-mass clusters. One example is the Wolf-Rayet star
CMDMSM would also fit the observed data.
γ-Velorum, the nearest Wolf-Rayet star to the Sun in the
Galaxy.Itisabinarysystemcontainingstarsthatwereini- Othermoreextremeexamplesoflow-massclusterswith
(cid:13)c 2005RAS,MNRAS000,1–11
6 John J. Eldridge
a single massive star were observed by Lamb et al. (2010). sivestars inthetotal stellar population.WeseeCMDMSM
They observed apparently isolated O stars and found low reproduces the lowest ratios at the highest star-formation
mass clusters associated with these stars. Using the stellar rates. PSS produces much higher ratios. However in this
and cluster masses derived by Lamb et al. (2010) and esti- modelwehaveassumed noleakage of anyionising photons.
matingtheionisingfluxforthemassivestarwehaveplotted Thiswould reducethecontributionfrom theHαfluxbyup
their clusters in Figure 1. They are only reproduced byour to around 50 per cent (see Zurita et al. 2002, for example).
PSS method. Thiswouldleadtolowerratiovaluesathighstar-formation
This agrees with previousstudiesbyTesti et al.(1997) rates for both PSS and CMDMSM.
Testi, Palla & Natta (1998, 1999) and Parker & Goodwin
To account for the leakage or loss of ionising photons
(2007) who use similar arguments. Maschberger & Clarke
from a galaxy or absorption by dust grains we have made
(2008)alsomadeadetailedstudyofallavailableinformation
a simple adjustment to our models. In Figure 3 we have
and also favour PSS. However Weidner, Kroupa& Bonnell
modified our syntheticratio distributions by assuming that
(2010) performed a similar analysis and found that for low
galaxieslosebetween0and50percentoftheirionisingpho-
massclusters,below100M⊙ PSSisfavouredbutmoremas- tons. We take our synthetic populations and smear them
siveclustersappearstohaveaCMDMSMrelation. Theob-
by this range of possible leakage fraction so that the mean
servations of Calzetti et al. (2010) do not currently favour
leakage is 25 per cent. Even for this modest loss of ionis-
eitherPSS orCMDMSM. Herewe can only agree that PSS
ing photons the ratio distribution changes the PSS model
occursinlow-mass,upto100M⊙,clusters.Formoremassive to match the range of observed galaxies. At the same time
clusters,ofaround1000M⊙,itisdifficulttodifferentiatebe- the CMDMSM method has a slightly worse agreement. To
tween PSS and CMDMSM. Finally for cluster masses more test the significance of these differences we have used a χ2
than about 104M⊙ thedifferences are less important. test to compare the observed distributions to the synthetic
Finallywenotethatourresultsareinlinewiththoseof
populations. We find that without leakage only CMDMSM
Villaverde, Cervin˜o & Luridiana(2010b).Theysuggestthat
withsinglestarsisaprobablematch.Howeverwithleakage
for cluster masses below 104M⊙ thereis a highly asymmet- onlytheCMDMSMsinglestarsyntheticpopulationisruled
ric scatter of the ionising flux around the mean integrated
out.
values from standard synthesis models because the single
mostmassivestardominatestheionisingfluxofthecluster. Fumagalli, daSilva & Krumholz (2011) used a leakage
This manifests itself in our results by the increased spread fraction of 5 percent and stated that their resultswere not
inHαfluxperM⊙ atlowclustermasses.Wenotethatthey dependent on the amount of leakage. This is because they
suggest that PSS is more favoured than CMDMSM. comparedtheamountofHαtomeanvaluesofHαfluxwhich
areless sensitive toleakage thantheHα/UVfluxratio (see
their figure 2). They found that for leakage fractions up to
3.2 The Hα and FUV in 11HUGS galaxies 40 percent their results were unaffected.This is within the
mean leakage of 25 per cent that we apply toour models.
Lee et al. (2009), Meurer et al. (2009) and Boselli et al.
(2009)haveattemptedtogaininsightintotheIMFbylook- Forthesinglestarpopulationthegreatestdifferencebe-
ingat emission from entiregalaxies. They broughttogether tween PSS and CMDMSM is seen in the different paths of
Hαobservations with farUVcontinuumobservations. Here the mean ratio values versus the star-formation rate deter-
we concentrate on the results of Lee et al. (2009) because minedfrom theUVflux.CMDMSMdecreasesmuchsooner
their set of galaxies are a volume limited sample of 315 than PSS at around 0.1M⊙yr−1. However there is a large
within 11Mpc. The emission of these two spectral star- possible range around these mean values in both cases and
formation rate indicators are determined by the number of the difference is only approximately 1σ. When we consider
stars with masses greater than 20 and 3M⊙ respectively. thebinarypopulationweseethatthedifferencebetweenthe
Thereforemeasuringtheratioofthetwofluxes,ortherela- twoIMFfillingmethodsissubstantiallyreduced.Thisisbe-
tivestar-formationratesmeasuredforthegalaxies,givesan cause, while the IMF initially leads to fewer massive stars
indication of the numberof stars in different mass regimes. in the CMDMSM case, binary interactions, such as merg-
Lee et al. (2009) found that as the Hα flux decreases the ingandmass-transfer, increase thenumberofmassivestars
Hα/UV ratio decreases so there is more UV flux than ex- relative to a single-star population. If we are to distinguish
pected.Pflamm-Altenburg,Weidner& Kroupa(2009) have between PSS and CMDMSM by means of the downturn in
suggested thatthis turndown isevidencefor IGIMFdeter- this ratio we must repeat the analysis that led to Figure 3
mining the galaxy-wide IMF of these galaxies. Their study for lower star-formation rates. We show the result for star-
wasbased onsinglestarmodelsalone. Herewerepeattheir formation rates between 10−2 and 10−4M⊙/yr−1 in Figure
analysis with binary as well as single star models and also 4. By eye PSS provides a better fit to the observed popu-
our stochastic approach to the star-formation history, with lation than CMDMSM. This is because CMDMSM has an
stellar clusters forming independentlyfrom one another. extendedtailofgalaxiestowardslowerratios.PSSdoesnot
WeplotoursyntheticgalaxiesinFigure2.Weseethat, have this tail. However including binary stars in the syn-
at star-formation rates above 10−2M⊙yr−1, the spread of thetic galaxies reduces it further in both CMDMSM and
models is similar but CMDMSM gives a slightly greater PSS.Alsothetailmightnotbepresentintheobserveddata
scatter towards lower values of the Hα/UV ratio. This owing to selection biases. A χ2 test reveals that both PSS
can be more easily seen in Figure 3 where we bin the populations are a probable match to the observed distribu-
synthetic and observed galaxies with star-formation rates tion.Thesingle-starCMDMSMdistributiondoesnotmatch
above 10−2M⊙yr−1 by their Hα/UV ratio. CMDMSM has the observed distribution. However our binary CMDMSM
agreaterrangeofratiosbecauseoftherelativelackofmas- population produces an equally likely fit to the observed
(cid:13)c 2005RAS,MNRAS000,1–11
Stochasticity, a variable upper-mass limit, binaries and SFRIs 7
Figure2.TheratioofSFRmeasuredbyHαandUVfluxesversestheSFRfromUVflux.TheasterisksaretheobservationsofLeeetal.
(2009) while the shaded region show the density of our individual realisations of synthetic galaxies. The thick solid lines indicate the
meanratiosforthesyntheticgalaxiesandtheir1σ limits.ThedashedlinesshowthemeanratiosfortheotherIMFfillingmethodwith
thesamestellarpopulation.TheupperpanelsareforPSSandthelowerpanelsareforCMDMSM.Whiletheleftpanelsareforasingle
star population and the right panels are for binary populations. Here we assume that a star formation rate of 1M⊙yr−1 is equivalent
to a log10(F(Hα)/ergss−1) = 41.1 and a UV flux density of log10(F(1500˚A)/ergss−1˚A) = 27.85. Linear features are due to limited
resolutionininitialmass,separationandmassratioparameterspaceofourbinarymodels.
data.Ourresultsalsoshowthat someionisingphotonleak- differentiatebetweenPSSandCMDMSMfromtheseobser-
ageisrequiredifourPSSmodelsaretomatchobservations. vations.Furthermoretheinclusionofbinariesinstellarpop-
ulation models means that any difference between PSS and
Lee et al. (2009) noted that their results indicate a
CMDMSM is only apparent at star-formation rates below
downturnintheHαtoUVratioatlowstar-formationrates.
thoseintheobservedsampleofLee et al.(2009).Therefore,
This could be explained by the IGIMF model put forward
fromtheobserveddistributionofHαtoFUVratio,itisnot
by Pflamm-Altenburg,Weidner& Kroupa (2009). At first
possibletodiscriminatebetweenPSSandCMDMSMowing
comparison of the synthetic and observed galaxies in Fig-
to the uncertainties in the importance of binary evolution
ures2and3temptsustoagreewiththisdeduction.Thisis
and ionising photon leakage.
mainlybecausethespreadoftheobservedgalaxiesathigher
star-formationratesisbetterreproducedbytheCMDMSM, Our conclusions are broadly in line with those of
single star models. The most significant difference between Fumagalli, da Silva& Krumholz(2011).Howevertheycom-
PSS and CMDMSM is in the region where the star forma- paredPSSmodelstoIGIMFmodels.TheIGIMFmodelsre-
tionratesdropbelow10−2M⊙yr−1.Allourmodelsareable strict the number of massive stars in the synthetic galaxies
to reproduce the observed galaxies with the lowest ratios further because they impose a maximum cluster mass that
at low star-formation rates. Therefore it is not possible to dependson thetotal star-formation rate. Wehaveonly im-
(cid:13)c 2005RAS,MNRAS000,1–11
8 John J. Eldridge
Figure3.ThedistributionofHαtoUVratioforobservedandsyntheticgalaxieswithstar-formationratesbetween10−2and1M⊙y−1.
TheredlinerepresentstheobservedsampleofLeeetal.(2009)whilethesolidlinerepresentstherelevantsyntheticgalaxiesfromFigure
2. The dashed line represents the synthetic observations smeared by a flat leakage of ionising photons distributed between a leakage
fraction of 0 and 50 per cent. The left panels are for PSS and the right panels are for CMDMSM.The first and third panels are for a
singlestarpopulationandthesecondandfourthpanelsareforbinarypopulations.
posedaclustermassthatdependentmaximumstellarmass each cluster has its own age independent of the other clus-
and have shown that a CMDMSM alone cannot be ruled ters. If there are enough clusters in a galaxy this leads to
out. an average stellar population. However if there are only a
Animportantconclusiontodrawfromourmodels(and few clusters the appearance of the galaxy-wide stellar pop-
those of Fumagalli, da Silva& Krumholz 2011; Weisz et al. ulation can be very different from what might be expected
2012)isthatthescatterandvariationoftheHα/UVratiois forasimplestellarpopulationwithasmoothstar-formation
notduetotheIMFfillingmethodbutitdependsmoreonthe history.
star-formation history of each individual galaxy. A general
trendwe findisthat thosesystemswith less star-formation
3.3 The importance of binaries
inthelast10Myrhavelowerratios,whilethosewithmostof
thestarformationinthelast10Myrhavehigherratioseven From our results it is possible to qualitatively demonstrate
at low mean star-formation rates. This is because the stars the need to use binary star models. For individual clusters
responsible for Hα emission typically have ages of 10 Myr binariesseemtohavelittleaffect.Thisisbecauseoftheshort
orless. Thisindicates thatanysimulation thatpredictsthe periodof8Myrwehaveusedtomatchtheobservedclusters
propertiesofasampleofgalaxiesmusttakeintoaccountthe inthiscase.Inoursyntheticgalaxies, with100Myrofstar-
stochastic nature of star-formation and recognise not only formation,weseethatthescatterofthesyntheticgalaxiesis
that each cluster has its own stellar content but also that reduced slightly if binaries are included and the mean SFR
(cid:13)c 2005RAS,MNRAS000,1–11
Stochasticity, a variable upper-mass limit, binaries and SFRIs 9
Figure 4. The distribution of Hα to UV ratio for observed and synthetic galaxies with star-formation rates between 10−4 and 10−2
M⊙yr−1.Theredlinerepresents the observedsampleofLeeetal.(2009)whilethesolidlinerepresents therelevant synthetic galaxies
fromFigure2.Thedashedlinerepresentsthesyntheticobservations smearedbyaflatleakageofionisingphotons distributionbetween
a leakage fraction of 0 and 50 per cent. The left panels are for PSS and the right panels are for CMDMSM. While the first and third
panelsareforasinglestarpopulationandthesecondandfourthpanelsareforbinarypopulations.
ratio starts to decrease at lower SFRs. This is more clearly narystarswecanalsoexpecttheapparentIMFtobeflatter.
shown in Figure 5 in which we compare populations with Furthermore the most massive star in a cluster might not
different IMFs and single-star to binary star ratios. We see havebeen themost massive star when it formed. Therefore
thatforasingle-starpopulationtheratiobeginstodropbe- interactingbinarystarshaveastrongeffectandmustbein-
tween 10−2 and 10−3M⊙yr−1 while for binary populations cludedwhenattemptsaremadetodeterminetheIMFfrom
this drop begins between 10−3 and 10−4M⊙yr−1. The bi- observations of stellar systems.
naryeffectmakesitdifficulttodistinguishbetweenthePSS
and CMDMSM IMF filling methods at any star-formation
rate.
4 CONCLUSIONS
Binary evolution affects the observed SFRs because
through mass transfer between and merging of stars it in- We have investigated two uncertainties in population syn-
creasesthenumberofmassivestarsattheexpenseoflower- thesis. These are how the IMF is filled and the effects
mass stars. We demonstrate this in Figure 5. We see here of interacting binary star evolution. The Hα flux per M⊙
that binary populations typically produce similar Hα/UV observed in samples of clusters is consistent with PSS of
flux ratios to a single star population when the IMF slope the SIMF for clusters around 100M⊙ because individual
isshallower. That isuntillow star-formation ratesatwhich low-mass clusters with one or two massive OB and WR
point,forasingleclusterwithasignificantpopulationofbi- stars, such as the Velorum cluster or those presented by
(cid:13)c 2005RAS,MNRAS000,1–11
10 John J. Eldridge
5 ACKNOWLEDGEMENTS
JJE would like to thank the anonymous referee for his
veryconstructive comments which havelead to a muchim-
proved paper. JJE is supported by the Institute of Astron-
omy’s STFC Theory Rolling grant. JJE would also like to
thank Joe Walmswell, Monica Relano, Ben Johnson, Dan
Weisz,JaniceLee,DaniellaCalzetti,SallyOey,MarkGieles,
Mieguel Cervin˜o, Michele Fumagalli, Robert da Silva and
ChristopherToutforveryhelpfuldiscussionsandcomments
on this paper.
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