Analytic and numerical realisations of a disk galaxy

ABSTRACT Recentfocusontheimportanceofcold,unshockedgasaccretioningalaxyformation–notexplicitlyincludedinsemi-analyticstudies–motivatesthefollowingdetailedcom-parisonbetweentwoinherentlydiﬀerentmodellingtechniques:directhydrodynamicalsimulationandsemi-analyticmodelling.Byanalysingthephysicalassumptionsbuiltintothegasoline simulation,formulaefortheemergentbehaviourarederivedwhichallowimmediateandaccuratetranslationoftheseassumptionstothegalform semi-analyticmodel.Thesimulatedhalomergerhistoryisthenextractedandevolvedusingtheseequivalentequations,predictingastrikinglysimilargalacticsystem.Thisexer-cisedemonstratesthatitistheinitialconditionsandphysicalassumptionswhichareresponsibleforthepredictedevolution,notthechoiceofmodellingtechnique. Onthislevelplayingﬁeld,apreviously published galform modelisapplied(includingaddi-tionalphysicssuchaschemicalenrichmentandfeedbackfromactivegalacticnuclei)whichleadstostarklydiﬀerentpredictions. 1 INTRODUCTIONTwo diﬀerent approaches are widely used to test theoriesof galaxy formation. Both make use of developing compu-tational resources to integrate a set of coupled diﬀerentialequationsforwardintime,whereeachequationappliesphys-ical constraints, or empirical laws, to some of the system’sproperties.The semi-analytic method (White & Frenk 1991) canbecharacterisedbytheintenttoencapsulateasmuchofthephysicalbehaviouraspossibleintheequationsthemselves,chosentodescribesystematicproperties(virialisation,con-servationofangularmomentum,etc.)andthusleavingtheminimumtonumericalcalculation.Inherently diﬀerent to this is the approach taken bysimulations,wheretheethosistoapplyequationswhichareknown to govern the particular properties (e.g. the gravi-tational force between two particles) and demonstrate, byskilfully programmed numerical calculation, that these re-producemacroscopicproperties.Inpractice,adjustmentwillalwaysneedtobemadetoaccountforthefactthateachsimulatedparticlerepresentsmany,manyrealparticles.Intraditionalsimulationswhichconsideredonlygravity,thisadjustmentispurelymathemat-ical. When gravity is the dominant interaction, results aretherefore very robust and simulations are mostly acceptedasbeingafaithfulrealisationoftheUniverseatlargescales,wherethisisthecase.Anotablemeasureofthisacceptanceis that semi-analytic models have taken to using samplesof halos taken directly from simulations (Kauﬀmann et al.1999;Hellyetal.2003;Hattonetal.2003),ratherthansam-ples created statistically (Bower et al. 1993; Lacey & Cole1993) using methods based on the analytic arguments ofPress&Schechter(1974). Ingeneral,though,theremustcomeascalebelowwhichthegoverningequationsarenolongerdescribingparticlein-teractionsbutemergentphenomena.Atthesesmallerscales,the two largely contrasting approaches begin to look moresimilar. As an immediate example, simulations which en-compassanentiregalaxyhavenotattemptedtoresolvein-dividualstarsandmustthereforeapplyanequationwhichapproximatestheconversionrateoflargebodiesofgasintostars,basedonlocallyaverageddensity.Anequivalentandsometimesidenticalequationappearsinsemi-analyticmod-els(§3.7). Both approaches have had some success in accountingfortheobservedpropertiesofgalaxies.Predictivepower,ontheotherhand,reliesonsuretyintheunderlyingequations,oraclearunderstandingofhowdiﬀeringunderlyingassump-tionsmapontoﬁnalobservablequantities. To bolster this crucial understanding, it is extremelyhelpfultostudy,indetail,thepredictedevolutionofasingle,largegalaxyanditssatellites,asfollowedbythesetwodiﬀer-enttechniques.Iftheanalyticmodelbeginsfromtheprecisehalomergerhistoryfoundinthesimulation,andappliesthesame assumptions about the physics at smaller scales, willitpredictthesameobservablepropertiesand,ifnot,whereandwhydothetwocalculationsdiverge? 1.1 Previous researchExistingquantitativecomparisonsbetweensimulationsandsemi-analytic models have focussed on samplesof galaxies.