Dispersion in modeled abundances

Chemical abundance data for the Galaxy show a wide dispersion. We try to check if this dispersion may also be found with chemical evolution models as an effect of variations of an Initial Mass Function (IMF) which follows a Poisson's distribution

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aDISPERSIONINMODELEDABUNDANCESJuanJos´eBaz´an,MercedesMoll´aDpto.deF´ sicaTe´orica.UniversidadAut´onomadeMadrid,28049Cantoblanco,Madrid.MiguelCervi noIAA/CSIC,Apartado3004,18080,Granada.1.IntroductionChemicalabundancedatafortheGalaxyshowsawidedispersion.Wetrytocheckifthisdispersionmayalsobefoundwithchemicalevolutionmodelsasane ectofvariationsofanInitialMassFunction(IMF)whichfollowsaPoisson’sdistribution.Wewillusetwodi erentmethods:a)ananalysisofthemodelinternalerrorandb)astatisticstudyonthee ectsoftheIMFdiscretization,basedonmontecarlomodels.BothmethodshavebeenusedinachemicalevolutionmodelappliedtotheSolarNeighborhood.PreliminaryresultsrelativetoOandNareshown.2.TheInitialMassFunctionWewilltrytojustifythedatadispersionbythepossiblevariationsinthemodelresultswhichcouldbeproducedbythebehavioroftheIMF,whosepropertiesaredescribedbyaprobabilitydistributionfunction:IfweassumethatNtotstarsareobservedinamassrange[m0,mmax],

themassmiofthei-thstarisarandomvariablewhoseprobabilitydistributionfunctionisgivenbythestellarinitialmassfunction:Φ(mi),whoseintegrationalongthewholemassrangeisnormalizedto1.ThenumberNiofstarsofmassmiisanotherrandomvariablewhosedistributionfunctionfollowsaPoissonianbehavior,withthevalueoftheIMFatthatmassastheonlyparameterofthedistribution:

=Φ(mi)dmi,sincethetotalnumberofstarsis:Ntot= dni=dNi/NtotdNi.AsimpletesttocheckthePoissoniannatureofthedistributionofniissimplytheratioofitsvariancetoitsaveragevalueasafunctionofmass.Thisratioshouldbecloseto1.Cervi noetal.(2001)illustratesintheir gure3the3resultsofatestwith1000Montecarlosimulationsofclusterswith10and104starswithaSalpeterIMFslope.Despite

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Chemical abundance data for the Galaxy show a wide dispersion. We try to check if this dispersion may also be found with chemical evolution models as an effect of variations of an Initial Mass Function (IMF) which follows a Poisson's distribution

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niistheratioofaPoissonvariablewithaconstant,itisPoissoniandistributedwithina

10%.

Figure1.DispersionofO–panela)–andN–panelb)–abundancedataoftheGalaxytakenfromdi erentsources

OnceweknowthatnicanbeapproximatedbyaPoissonvariable,itispossibletoapplyaproperstatisticalformalismintothecodeofourchemicalevolutionmodelinordertostudythe uctuationswhichwemayobtainfromthisstoscasticbehaviorfortheIMF.

3.Thetheoreticalmultiphasechemical

evolutionmodel

Ourmodeledphysicalsystemisacylindricalsub-galacticzonecen-teredontheSolarNeighborhood,extendingabout1kpcintheGalacticplane.ThemodelusedinthisworkisthecodefromFerrinietal.(1992),improvedwithrecentyields(Gavil´anetal.2002).Itcomputesapro-ductionmatrixQij(m),whereeach(i,j)elementrepresentsthefractionofthestarmassinitiallyintheformofchemicalspeciesj,transformedandejectedaschemicalspeciesi,for15chemicalelements.AmatrixiscomputedforeachstellarmassweigthedbytheIMF.Thenucleosynthe-sisproductionfromSNI,SNII,andnormalstarsistakenintoaccount.Weletthemodelevolvesfor13,2Gyrs,witha0.01timestep.TheIMFusedistakenfromFerrinietal.(1991):

Φ(m)=22.0865m 0.52x10 [2.07(logm)+1.92logm+0.73]1

Chemical abundance data for the Galaxy show a wide dispersion. We try to check if this dispersion may also be found with chemical evolution models as an effect of variations of an Initial Mass Function (IMF) which follows a Poisson's distribution

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3.1Error

analysis

Figure2.VariabilityoftheIMFasafunctionofmass.

ConsideringthePoissoniannaturefortheIMF,weassumeaninitialvarianceofσ2(Φ)=Φ.Then,wetrailtheerror,followingerrorpropa-gationrules,withinthemodelineveryequationinvolvedandrelatedtoourIMF.Asaresultwecanobtainameasureofthevariabilityofourmodelalongthewholemassintervalconsidered.

WemayobserveinFig.2thatthevariabilityincreasesstronglyinthemassintervalfrom4to8M⊙.ThisisexactlytheintervalofmassproducingFeandN.Therefore,inagreementwiththisresult,ourmodelmustshowalargerdispersionintheseelementsincomparisonwithotherelementsasO.

3.2MonteCarloSimulations

Weuseanuniformlydistributedrandomnumbersgeneratortogen-eratefromit,andusingtheaccumulatedprobabilitydistribution,asuc-cessionofnumberswithourIMFasprobabilityfunction.OuraimisthecomputationofacompletesetofQi,jMonteCarlosimulations,wheretheIMFisrandomlydi erentforeachofthem,withmassesfrom0.8upto100M⊙,dividedin800intervalswithlinearinterpolationwithinthem.Wealsowouldliketostudythee ectsofthediscretizationcausedbyroundinginthecode.Ouraimistocalculate～500 1000simu-lations,with～105 106stars.Wewillrepeatoursimulationsfortwodi erentrounddownvalues:10 3and10 6.ThewellcalibratedmodelfromGavil´anetal.(2002)willbeourbaseofmeanvalues.

Chemical abundance data for the Galaxy show a wide dispersion. We try to check if this dispersion may also be found with chemical evolution models as an effect of variations of an Initial Mass Function (IMF) which follows a Poisson's distribution

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4.Results

Thepreliminaryresultshavebeenobtainedbytherealizationof400montecarlosimulationsforthegalaxydiskzone,withouthavingintoac-counttheroundinginthecodeandwithnoncalibratedmodels.Despiteofthesefacts,adispersionof～10 2alreadyappearsinourOandNabundances.aswemayseeinFig.3.Thisfactseemstoindicatethatagoodpercentageoftheobserveddispersioncouldbeachievedwiththecompletesetof

simulations.

Figure3.PreliminaryresultsofdispersioninOandNabundances.

5.Conclusions

Withthiskindofcalculationswemightestimatethevariabilityandtheerrorsthatcanbeobtainedbythemodels.Thus,wecouldincludeintothecodethedispersionappearingasaconsequenceoftheuncer-taintiesintheIMF,andthiswaywecouldcheckifispossibletogetthesamedispersionasobserved.Itseems,fromourpreliminaryresults,thattakingintoaccountthesampling uctuationsfromtheIMFintothemodelscanreproducetheobserveddispersion.

References

Cervi no,M.,Moll´a,M.2002,A&A,394,525

Cervi no,M.etal.2001,A&A,376,422

Ferrini,F.etal.1992,ApJ,387,138

Gavil´an,M.,Buell&Moll´a,M.2002,inpreparation

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