The Role of the IGIMF in the chemical evolution of the solar neighbourhood

0.41 MB·English

by  Francesco Calura

UP:HaveObservationsRevealedaVariableUpperEndoftheInitialMassFunction?
ASPConferenceSeries,Vol.**VolumeNumber**
**Author**
(cid:13)c2010AstronomicalSocietyofthePacific
The Roleofthe IGIMFinthe chemical evolutionofthe solar
neighbourhood
1
1
0 FrancescoCalura1,SimoneRecchi2,FrancescaMatteucci3,PavelKroupa4
2
1JeremiahHorrocksInstitute forAstrophysics andSupercomputing, University
n
a ofCentralLancashire, PrestonPR12HE,UnitedKingdom
J
2Institute ofAstronomy,ViennaUniversity, Tu¨rkenschanzstrasse 17,A-1180,
5
2 Vienna,Austria

3Dipartimento diFisica,Sezione diAstronomia, Universita` diTrieste,viaG.B.
]
A Tiepolo11,34131, Trieste,Italy
G
4Argelander Institute forAstronomy, BonnUniversity, AufdemHu¨gel71,
.
h 53121Bonn,Germany
p
-
o Abstract. The integratedgalacticinitial massfunction(IGIMF)is computedfrom
r the combination of the stellar initial mass function (IMF) and the embedded cluster
t
s massfunction,describedbyapowerlawwithindexβ. Theresultofthecombinationis
a atime-varyingIMFwhichdependsonthestar formationrate. WeappliedtheIGIMF
[
formalism to a chemical evolution model for the solar neighbourhoodand compared

1 the results obtained by assuming three possible values for β with the ones obtained
v by meansof a standard, well-tested, constantIMF. In general, a lower absolutevalue
0 of β implies a flatter IGIMF, hence a larger number of massive stars, higher Type Ia
0 and II supernovarates, highermassejection ratesand higher[α/Fe] valuesata given
9 metallicity.Oursuggestedfiducialvalueforβis2,sincewiththisvaluewecanaccount
4 formostofthelocalobservables. Wediscussourresultsinabroaderperspective,with
. someimplicationsregardingthepossibleuniversalityoftheIMFandtheimportanceof
1
0 thestarformationthreshold.
1
1
:
v
i
X
1. Introduction
r
a
The initial stellar mass function (IMF) is of primary importance in galactic chemical
evolution models. TheIMFregulates therelative fractions ofstars ofdifferent masses,
hence their relative contribution tothe chemical enrichment oftheinterstellar medium
(ISM)istightlyrelatedtothisquantity. Forthisreason,theanalysisofabundanceratios
in galaxies may allow one to put robust constraints on both the normalization and the
slopeoftheIMF(Recchietal. 2009;Caluraetal. 2010).
TheSolarNeighbourhood(S.N.hereinafter)canbeconsideredthemostvaluableenvi-
ronment toachieve constraints onthe mainparameters regulationg chemical evolution
models, since it is definitely the best studied Galactic environment and many obser-
vational investigations devoted to its study provide us with a large set of observables
againstwhichmodelscanbetested. Theseobservables includediagramsofabundance
ratios versusmetallicity, whichareparticularly useful whentheyinvolve twoelements
1
2 Calura,Recchi,MatteucciandKroupa
synthesised by stars on different timescales. An example is the [α/Fe] vs [Fe/H] dia-
gram,sinceαelementsareproduced mostlybymassivestars(m > 8M )onveryshort
⊙
(< 0.03 Gyr) timescales, while type Ia supernovae (SNe) produce most of the Fe on
timescales ranging from 0.03GyruptooneHubbletime(Matteucci 2001). Thisdiag-
nostic is astrong function of the IMF,but depends also on the assumed star formation
(SF) history (Matteucci 2001; Calura et al. 2009). Another fundamental constraint is
the stellar metallicity distribution (SMD), which depends mainly on the IMF and on
the infall history (hence on the star formation history) of the studied system. Another
exampleofausefuldiagnostictestfortheIMFisthepresent-daymassfunction,i.e. the
mass function of living stars observed now in the Solar Vicinity (Elmegreen & Scalo
2006).
Theintegrated galactic initial mass function (IGIMF)originates from thecombination
of the stellar IMF within each star cluster and of the embedded cluster mass function
(CMF).Itreliesontheobservational evidencethatsmallclustersaremorenumerousin
galaxies andthat themostmassivestars tendtoformpreferentially inmassiveclusters
(Weidner & Kroupa 2006). The IGIMF is star-formation dependent, hence it is time-
dependent and its evolution with time is sensitive to the star formation history of the
environment.
Inthispaper, weuseallthelocalobservables tostudy theIGIMFanditseffectsonthe
chemical evolution of the solar neighbourhood. The results obtained with the IGIMF
arecomparedtothoseobtainedwithanon-star-formation dependent(henceconstantin
time),fiducialIMF.Theaimistoderivesomecontraints onthemainunknown param-
eteroftheIGIMF,i.e. theindexβofthepowerlawexpressing theembeddedCMF.
Thispaperisorganizedasfollows. InSection2wepresentadescriptionofthetheoret-
icalscenariobehindtheIGIMF.InSection3wedescribethechemicalevolutionmodel
for the Solar Neighbourhood. In Sect. 4 we present our results and finally in Sect. 5
someopenproblemsregardingtheIMFarediscussedandsomeconclusionsaredrawn.
Table 1. Solar neighbourhood observables, parameters to which they are most
sensititveandreferences.
Observable Parameter Reference
SFRSurfacedensity SFefficency Rana(1991)
typeIaSNR Integrated SFhistory, IMF Cappellaro (1996)
typeIISNR SFefficiency, IMF Cappellaro (1996)
Gassurfacedensity SFefficiency, IMF Kulkarni&Heiles(1987)
Olling&Merrifield(2001)
Stellarsurfacedensity SFhistory, IMF Weber&deBoer(2009)
Stellarabundance ratios SFhistory, IMF various authors
StellarMetallicity distribution SFhistory, IMF Jorgensen (2000)
Present-day massfunction SFhistory, IMF Miller&Scalo(1979)
TheIGIMFinthesolarneighbourhood 3
Figure1. Leftpanel: IGIMFsfordifferentclustermassfunctions. Upperpanel:
β=1;centralpanel:β=2;lowerpanel:β=2.35.Ineachpanelwehaveconsidered
7 possible values of SFRs, ranging from 10−4 M yr−1 (lowermost solid lines) to
⊙
100 M yr−1 (uppermostsolidlines),equallyspacedinlogarithm. Rightpanel: the
⊙
opencirclesarethe“fitness”asafunctionofβforvariousmodelswithdifferentSF
efficiencies (indicated by the numbers beside each open circle in units of Gyr−1.).
Thehorizontallineindicatesthefitnessvaluecomputedforthestandardmodel.
2. The(integrated galactic) initialmassfunction
The main equation used to calculate the IGIMF is (see the contributions by Pflamm-
Altenburg etal. andWeidneretal.):
Mecl,max(ψ(t))
ξ (m;ψ(t)) = ξ(m ≤ m )ξ (M )dM , (1)
IGIMF max ecl ecl ecl
Z
Mecl,min
where ψ is the star formation rate (SFR). The canonical stellar IMF is ξ(m) = km−α,
with α = 1.3 for 0.1 M ≤ m < 0.5 M and α = 2.35 for 0.5 M ≤ m < m . The
⊙ ⊙ ⊙ max
uppermassm isafunction ofthemassoftheembedded cluster M : thisislogical
max ecl
if one considers that small clusters do not have enough mass to produce very massive
stars. Star clusters are also apparently distributed according to a single-slope power
law, ξ ∝ M−β (Lada&Lada2003). Inthis workwehaveassumed 3possible values
ecl ecl
ofβ: 1,2and2.35.
M and M (ψ(t))aretheminimumandmaximumpossiblemassesoftheclus-
ecl,min ecl,max
ters in a population of clusters, respectively, and m = m (M ). For M we
max max ecl ecl,min
take5M (themassofaTaurus-Aurigaaggregate, whichisarguably thesmalleststar-
⊙
forming ”cluster” known). The upper mass of the cluster population depends instead
ontheSFRandthatmakesthewholeIGIMFdependent onψ.
Thestandard IMFisatwo-slopepowerlaw,definedinnumberas:
0.19 ·m−2.35 if m < 2M
ξ (m) = ⊙ (2)
std ( 0.24 ·m−2.70 if m > 2M⊙,
Thisequation represents asimplifiedtwo-slopeapproximation oftheactualScalo
(1986)IMF.TheIMFandalltheIGIMFsarenormalisedinmasstounityasthestandard
4 Calura,Recchi,MatteucciandKroupa
Figure 2. Upper panels: SFR, type Ia and II SN rates, stellar and gas density
versustime(left),abundanceratios(center)andstellarmetallicitydistribution(right)
calculatedwithβ= 1. Middlepanels: sameasabovebutwithβ= 2. Lowerpanels:
sameasabovebutwithβ = 2.35(intheleftpanelsonlythecasewithSFefficiency
1.5Gyr−1isshown).FordetailsseeCaluraetal. (2010).
IMF:
mξ(m)dm = 1. (3)
Z
In Fig. 1, we show the IGIMF as a function of the SFR for the three values of
β considered in this work, compared to our standard IMF. In general, a lower value
of β implies a flatter IGIMF, and a hence higher relative fraction of stars with masses
m > 1M .
⊙
3. Thechemicalevolution modelfortheSolarNeighbourhood
The main chemical evolution model is described in detail in Calura et al. (2010). The
modeliscalibrated inordertoreproduce alargesetofobservational constraints forthe
MilkyWaygalaxy(Chiappinietal. 2001). TheGalacticdiscisapproximatedbyseveral
TheIGIMFinthesolarneighbourhood 5
independent rings, 2 kpc wide, without exchange of matter between them. The Milky
Wayisassumedtoformasaresultoftwomaininfallepisodes. Duringthefirstepisode,
the halo and the thick disc are formed. During the second episode, a slower infall of
externalgasformsthethindiscwiththegasaccumulatingfasterintheinnerthaninthe
outer region (”inside-out” scenario, Matteucci & Franc¸ois 1989). The process of disc
formation is much longer than the halo and bulge formation, with time scales varying
from ∼ 2 Gyr in the inner disc to ∼ 7 Gyr in the solar region and up to 20 Gyr in the
outermost disc. Inthis paper, weare interested in the effects of atime-variyng IMFin
theSolarNeighbourhood. Forthispurpose,wefocusonaringlocatedat8kpcfromthe
Galacticcentre, 2kpcwide. Themodelincludes thecontributions oftypeIaSNe,type
II SNeand low and intermediate mass stars to the chemical enrichment of the ISM.A
SFthresholdisadopted(σ ∼ 7M pc−2)toreproducevariouslocalfeatures,including
th ⊙
abundancegradients(Colavittietal. 2009). TheIGIMFisallowedtovaryasafunction
of the SFR, which in turn is a function of cosmic time. The IGIMF is calculated as a
function of the SFRaccording to Eq. 1. In Table 1 we show the solar neighbourhood
observables used in this paper, with the main parameters on which they depend and
references.
In order to quantitatively compare our results with the observables considered in this
paper, wedefinethe fitnessquantity as:
1 w(i)[obs(i)−theo(i)]2
fitness = ; δ= Σ (4)
1+δ imax [obs(i)]2,[theo(i)]2
where, for the i-th value of each considered p(cid:8)arameter, obs (i) an(cid:9)d theo (i) are the
observedvaluesandthepredictions ofthemodel,respectively. Theweightw(i)isused
togiveeachsetofobservables thesamestatistical weight.
The closer fitness is to 1, the better the model is in reproducing the observations. In
the right panel of Fig. 1, we show the “fitness” as a function of β for all the models
considered inthispaper(seeSect. 4).
4. Results
Some results of our study are summarized in Fig. 2. Here we show the results of
the comparison between observables and model predictions computed with different
assumptions fortheclustermassfunctionindexβ. Ingeneral, thelocalobservedquan-
tities are plotted with their error bars. For a given value of β, the curves of different
types are the model results computed with different assumptions for the SF effiency,
reported in the legends. From the time evolution of the SFR,type Ia and II supernova
rate (SNR), gas and stellar mass density (left panels) computed with various assump-
tionsfortheparameter β,wecanseethatlowervalues ofβimplyhigher SNRs,higher
gas mass densities and in general lower mass locked up in living stars and remnants.
This is visible from the comparison of the results computed with the same values for
the SF efficiencies and different values for β. Another important issue regards the de-
pendence on the star formation threshold: some values of β (β = 1, β = 2) show SF
histories indendent from the SF threshold. This is basically due to the large mass re-
turnedbydyingstars,whichmaintainsthegasdensityalwaysabovethethresholdlevel
and which produces SF histories substantially different from those obtained with the
standard IMF, for which the effect of the threshold is remarkable, in particualr in the
last4Gyrofevolution. Ontheotherhand,thecasewithβ = 2.35showsadependence
6 Calura,Recchi,MatteucciandKroupa
ontheSFthresholdevenlargerthanthestandardcase,andthisisduetothelowermass
ejection ratesfromdyingstarswhichstemfromasteeper IGIMF(seeFig. 1).
Inthemiddlepanel,weshowtheabundance patternpredicted forthethreevaluesofβ,
compared to those predicted in the standard case. The elements studied here are O,Si
and Fe, since the theoretical understanding of their production is quite clear, and their
study allows us to neglect uncertainties related to their nucleosynthesis. In general,
lower values of β produce higher [Fe/H] values and higher [α/Fe] values (i.e. more
α-enhancedelementalabundances)atagivenmetallicity. Thisisrelatedtothefactthat
the fraction of massive stars increases with decreasing β and considering that massive
starsarethemainproducers ofαelements. Similarconclusions canbedrawnbylook-
ingattheplotsofthestellarmetallicitydistributions: thelargerthevalueofβ,thelarger
the relative fraction of stars producing Fe, i.e. mostly type Ia SNe, i.e. stars in binary
systemswithinitialmassrangingfrom0.8M to8M ,hencelarger theFeabundances
⊙ ⊙
atanygivenepoch. Thistranslates inSMDspeaking athigher [Fe/H]values forlower
valuesofβ,assumingthesameSFefficiency.
AscanbeseenfromtherightpanelofFig.1,themodelcalculatedwiththeIGIMFpro-
vidingthebestresultsistheonewithβ= 2andSFefficiency0.3-0.5Gyr−1. Theresults
obtained with this choice of β are quite similar to the ones achieved with the standard
IMF. This should not be a surprise since, as shown in Fig. 1, in the intermediate case
with β = 2 the IGIMF is very similar to the standard IMF. The assumption of β = 2
allows us to satisfactorily reproduce the set of observational constraints considered in
this work. Thisis animportant result, given thefact that the IGIMFiscomputed from
first principles. On the basis of the results described in this section, it may be difficut
todiscriminatebetweenthescenariowiththestandardIMFandtheIGIMFwithβ = 2.
In the next section, the use of a diagnostic possibly useful to disentangle between the
standard IMFandtheIGIMFwillbediscussed.
5. Discussion
Inthis paper, Wehave modelled the physical properties ofthe S.N.within theIGIMF
theory. In this scenario, the IGIMF can be calculated by combining the cluster mass
function with the stellar IMF, which represents the mass function of stars born within
clustersandwhichcanbedescribedbyadouble-slopepowerlaw. Animportantfeature
of the IGIMF is that it depends on the star formation rate, which in turn evolves with
time.
The parameter β regulating the cluster mass function may have an important im-
pact on the predicted properties of the Solar Neighbourhood. In general alower value
for β corresponds to a flatter IGIMF. In terms of chemical evolution, a flatter IGIMF
translatesintohighermassejectionratesfromdyingstars,hencegloballyalowermass
fraction incorporated intostellar remnants and higher gasmassdensities. Thisimplies
that the evolution of all the models computed assuming β = 1 and β = 2 are not sen-
sitivetothestar formation threshold andthestar formation histories donot exhibit the
“gasping”featurestypicalofthestandardmodel,whichinturnisdominatedbythresh-
oldeffects atevolutionary timesgreater than 10Gyr. Moreover, thelowerthevalue of
β, the higher the SN rate, the higher the metallicity and the larger the α-enhancement
visible intheabundance pattern. Thestatistical testusedtocomparemodelresults and
the obervables indicates that the model which best reproduces the local observables is
carachterized byβ = 2astheindexoftheCMF.Theresults ofthebestmodelarevery
TheIGIMFinthesolarneighbourhood 7
similar to those obtained with the standard case. A possible diagnostic which could
helpusdisentangling betweenthetwoisrepresented bythepresent-day massfunction
(PDMF).ThePDMFrepresents themassfunctionoflivingstarsasobserved intheso-
larneighbourhood. Thisquantity isanimportant diagnostic since itprovides pieces of
information complementary totheonesfromthepreviously discussed observables.
IntheleftpanelofFig.3,weshowthePDMFobservedintheS.N.andpredicted
bymeans ofour models. ThePDMFcomputed withthestandard IMFagrees withthe
observations intherange0.4 M -2 M . Atverylowstellar masses,thestandard IMF
⊙ ⊙
seems too steep, whereas the distribution of stars with masses > 30M is underesti-
⊙
mated. Once again, this is due to the SFthreshold, which has strong effects on the SF
historyofthesolarneighbourhood atlatetimes,inhibitingrecentSFandhencecausing
theunderabundanceorabsenceofverymassivestars. Incontrast,themodelscalculated
withtheIGIMFprovide allsimilarlyaverygoodfittotheobservedPDMF.
TheanalysisofFig.3seemstosuggestthattheSFthresholdshouldnotplayadominant
role in the late evolution of the S. N. Within the IGIMF theory, the existence of a SF
thresholdmaybeanobservationalselectioneffect,naturallyexplainedinthiscontextas
shown by Pflamm-Altenburg et al. (these proceedings). However, as shown by chem-
ical evolution results, the SF threshold is fundamental in reproducing the metallicity
gradientsobservedintheMWandinlocalgalaxies,unlessavariablestarformationef-
ficiencythroughthediscisassumed(Colavittietal. 2009). Thestudyoftheabundance
gradients withintheIGIMFtheorymaybeofcrucialhelpinsheding lightonthisissue
andwillbeconsidered infuturework.
Another important issue concerns the time evolution of the IGIMF. In the right panel
ofFig. 3, weshow how the IGIMFvaries asafunction of timeinthe case ofthethree
best models computed with different values of β. The best model (β = 2) shows very
small variation of the IGIMF with cosmic time. Strong variations are predicted by as-
suming β = 2.35, since in this case the star formation history is very much influenced
bytheeffectsoftheSFthreshold. Itmaybeinteresting totesttheeffectsoftheIGIMF
in spiral, Milky Way-like galaxies in a cosmological context. Cosmological semiana-
lytical models predic strong variations in thestar formation histories ofspiral galaxies
(Calura&Menci2009), whichpresentalargenumberofspikesduetomergingevents
andwhichshouldmanifestintostrongvariations oftheIGIMFwithredshift.
The universality of the IGIMF is another issue that deserves particular attention in the
future. AchemicalevolutionstudyofellipticalgalaxieswithintheIGIMFtheoryshows
that the best value for β is 2.35, allowing to reproduce best the integrated α/Fe ratios
observed in the local early-type galaxies. This value is in contrast with the best value
suggested by the analysis of the S. N. features. A further study of the IGIMF in local
dwarf irregular galaxies and dwarf spheroidals could certainly be of some help in this
regard.
Currently, the slope ofthe IGIMF in extreme SFconditions isanother largely debated
topic. Various indirect indications inexternal galaxies (Dabringhausen, these proceed-
ings)andinourGalaxy(Stolte,theseproceedings)seemtosuggestthatinstronglystar
forming systems, the slope of the stellar IMF should be flatter than the Salpeter one.
Moreover, the assumption of a slightly top-heavy IMF in starbursts helps alleviating
the discrepancy between cosmological models and observations regarding theα/Fe-σ
relation observed in local ellipticals (Calura & Menci 2009). Addressing this subject
withintheIGIMFtheorywillbeofprimaryimportance inthenearestfuture.
8 Calura,Recchi,MatteucciandKroupa
Figure 3. Left panel: present-day mass function as predicted by means of our
modelscomparedto localobservations. Rightpanel: time evolutionofthe IGIMF
assumingvariousvaluesfortheparameterβ.
Acknowledgments. FCwould like to thank the S.O.C.for the kind invitation and
forthefinancialsupport,whereastheL.O.C.isacknowledgedforbeingabletoestablish
apleasant andcomfortable environment todiscussexcitingscientifictopics.
References
Calura,F.,Menci,N.,2009,MNRAS,400,1347
Calura,F.,Pipino,A.,Chiappini,C.,Matteucci,F.,Maiolino,R.,2009,A&A,504,373
Calura,F.,Recchi,S.,Matteucci,F.,Kroupa,P.,2010,MNRAS,406,1985
Chiappini,C.,Matteucci,F.,Romano,D.,2001,ApJ,554,1044
Cappellaro,E.,1996,in, eds,Proc.IAUSymp.171,NewLightonGalaxyEvolution.Kluwer
Academic,Dordrecht,p.81
Colavitti,E.,Cescutti,G.,Matteucci,F.,Murante,G.2009,A&A,496,429
Elmegreen,B.G.,Scalo,J.,2006,ApJ,636,149
Kulkarni, S. R., Heiles, C. 1987, in Interstellar Processes, ed. D. Hollenbach, H. Thronson
(Dordrecht:Kluwer),87
Lada,C.J.,Lada,E.A.2003,ARA&A,41,57
Matteucci,F.,2001,ASSLVol.253,TheChemicalEvolutionoftheGalaxy.Kluwer,Dordrecht,
293
Matteucci,F.,Franc¸oisP.,1989,MNRAS,239,885
Miller,G.E.,Scalo,J.M.,1979,ApJS,41,513
Olling,R.P.,Merrifield,M.R.,2001,MNRAS,326,164
Rana,N.1991,ARA&A,29,129
Recchi,S.,Calura,F.,Kroupa,P.,2009,A&A,499,711
Scalo,J.M.,1986,FCPh,11,1
Weber,M.,deBoer,W.,2010,A&A,509,25
Weidner,C.,Kroupa,P.2006,MNRAS,365,1333