AGNFEEDBACKANDDELIVERYMETHODSFORSIMULATIONSOFCOOL-COREGALAXYCLUSTERSByGregoryRobertMeeceJr.ADISSERTATIONSubmittedtoMichiganStateUniversityinpartialfulllmentoftherequirementsforthedegreeofAstrophysicsandAstronomy-DoctorofPhilosophy2016ABSTRACTAGNFEEDBACKANDDELIVERYMETHODSFORSIMULATIONSOFCOOL-COREGALAXYCLUSTERSByGregoryRobertMeeceJr.Asthelargestgravitationallyboundstructuresintheuniverse,galaxyclustersstandatacrossroadbetweenastrophysicsandcosmology.Observationsofgalaxyclusterscanrevealinformationaboutthecompositionandevolutionoftheuniverse,butinordertointerprettheseobservations,astrophysicistsneedtounderstandtheprocessesshapingthegasintheintraclustermedium(ICM).Thisdissertationexplorestheroleofactivegalacticnuclei(AGN)inregulatingcoolingintheICMandfocusesonhowcondensationtriggeredbythermalinstabilitymayformafeedbackcyclethatpreventsclustersfromcooling.Roughlyhalfofgalaxyclustersareobservedtohavecoolingtimesmuchshorterthantheageofthecluster.However,theseclustersdonotseemtobecoolingdownanddonotappeartoformstarsatthepredictedrate.AGNfeedbackcansolvethisproblembyreheatingtheICM,butonlyiftheAGNpowerislinkedtotheICMcoolingrate.Thisdissertationexploresthe`Precipitation-regulatedfeedback'hypothesis,inwhichthecoolingICMbecomesthermallyunstable,leadingtotheformationofclumpsofcoldgas.ThiscoldgastriggersAGNactivity,whichreheatsthecluster.TheheatingstabilizestheICMagainstfurthercondensation,leadingtotheAGNshuttingoandallowingtheICMtocoolagain.Thus,thebalanceofradiativecoolingandAGNheatingservetoregulatethetemperatureoftheICMandkeepgalaxyclustersinaroughlystablethermalstate.Dedicatedtomyfriends,family,advisors,cats,andmywonderfulanceGabrielle,allofwhomhavesupportedmeinthisendeavor,havenoddedtheirheadsatappropriatetimes,andpretendedtounderstandwhatIamtalkingabout.Specialthankstomyfamily,whohavealwaysbelievedinmeandencouragedmeinacademicpursuits.MuchwritingcreditgoestomycatsPeterandFranciswhohavehelpedmekeepthetextsuccinctbymeowingandstandinginfrontofmymonitor.ThankstoGabrielleforputtingupwithmethroughallofthis.Finally,thankyoutothelargerastronomycommunity.KnowingthatothersndmyworkinterestingandusefulisthehighestrewardthatIcanimagine.iiiACKNOWLEDGMENTSThisworkwascompletedatMichiganStateUniversityfrom2011-2016,andIwouldliketothanktheUniversityandtheDepartmentofPhysicsandAstronomyfortheirsupportduringthattime.Iand/ormyresearchhasbeensupportedthroughNSFgrantOCI-1053575,NASAthroughgrantNNX12AC98GandNNX15AP39G,andHubbleTheoryGrantHST-AR-13261.01-AandHST-AR-14315.001-A,andbytheNationalScienceFoundationthroughgrantPHY-0941373.SimulationswereperformedonTeraGrid,XSEDE,andNASAresourcesthroughgrantsTG-AST090040,TG-AST100004,andSMD-15-6514.IhavealsobeensupportedasagraduateteachingassistantbythedepartmentandasaresearchassociatebytheLyman-BriggsCollegeandtheCollegeofNaturalSciencesthroughMarkVoit.AdditionalresearchfundingwasprovidedunderafellowshipfromtheMichiganInstituteforPlasmaScienceandEngineering(MIPSE).Iwouldalsoliketoacknowledgetheresearchersthathavecontributedtothiswork.Inadditiontomyadvisors(Drs.BrianO'SheaandMarkVoit)andmydissertationcommittee,Dr.MeganDonahue,Dr.BrittonSmith,Dr.DevinSilva,andBrianCrosbyatMichiganStatehavecontributedfrequentwisdomandadvice.IamalsoindebtedtoDrs.YuanLi(UniversityofMichigan),GregBryan(ColumbiaUniversity),PrateekSharma(IndianInstituteofScience),andMateuszRuszkowski(UniversityofMichigan)forusefuldiscussions.Thisworkisanextensionofworkdonebyearlierresearchersandwouldnothavebeenpossiblewithouttheireortsandcontributions.Meeceetal.(2015)extendsMikeMcCourt's(Harvard)dissertationwork,andMeeceetal.(2016)extendsYuanLi's.ThisdissertationisinspiredandbasedontheworkofMarkVoitandhiscollaborators.Finally,theEnzoandytcodesaredevelopedbyalargecollaborationofusers.Withouttheireorts,thisworkwouldnothavebeenpossible.ivTABLEOFCONTENTSLISTOFTABLES.............................................viiiLISTOFFIGURES............................................ix1Introduction...............................................11.1FormationofGalaxyClustersandtheCosmologicalContext..................11.1.1TheCDMModelofCosmology..............................11.1.2CosmologicalStructureFormation.............................31.2GalaxyClusters............................................51.2.1DarkMatterinClusters...................................61.2.2BaryonsinClusters.....................................71.2.3ObservingGalaxyClusters.................................81.2.3.1Radio........................................91.2.3.2Microwave.....................................91.2.3.3Infrared,Optical,andUV.............................91.2.3.4X-ray........................................101.2.3.5GammaRays....................................101.2.4MagneticFieldsinGalaxyClusters.............................101.2.4.1ObservationofMagneticFieldsinClusters...................111.2.4.2TheOriginofMagneticFieldsinClusters....................121.3ActiveGalacticNucleiandSupermassiveBlackHoles......................131.3.1AShortHistoryofAGN...................................131.3.2TheUniedmodelofAGN.................................141.3.3InteractionsBetweenAGNandtheirEnvironment....................161.3.4TheOriginofSupermassiveBlackHoles..........................171.4TheCoolingFlowProblemandPrecipitationRegulatedAGNFeedback............191.5PlanofThisDissertation......................................202LiteratureReview............................................212.1Introduction..............................................212.2TheCoolingFlowProblem.....................................212.3ObservationalEvidence(orlackthereof)forCoolingFlows...................232.3.1ProposedSolutionstotheCoolingFlowProblem.....................252.3.2ProposedHeatingSources..................................252.3.2.1Conduction.....................................262.3.2.2TurbulenceandMergers..............................262.3.2.3Supernovae.....................................272.3.2.4AGN........................................272.4AGNTriggering:Hotvs.ColdGas.................................282.4.1HotMode(Bondi-like)Accretion..............................282.4.2ColdModeAccretion....................................292.5ThermalInstability..........................................292.5.1Field1965...........................................302.5.2Defouw1970.........................................332.5.3Nulsen1986..........................................342.5.4Malagoli1987.........................................352.5.5Balbus1988..........................................352.5.6Balbus&Soker1989.....................................362.6ObservationsofMultiphaseGas...................................362.6.1Cavagnolo2008........................................372.6.2Werner2010.........................................38v2.7SimulationsofMultiphaseGasFormation.............................392.7.1Malagoli1990.........................................392.7.2McCourt2012........................................402.7.3Sharma2012.........................................402.7.4Joung2012..........................................412.7.5Scannapieco2012.......................................422.7.6Banerjee2014.........................................432.8AGNandEnergyTransfer......................................442.8.1BuoyantBubbles.......................................442.8.2CosmicRays.........................................452.8.3Turbulence..........................................452.8.4ShocksandSoundWaves..................................452.9Precipitation-RegulatedFeedback..................................483GrowthandEvolutionofThermalInstabilitiesinIdealizedGalaxyClusterCores...503.1Introduction..............................................503.2Method................................................533.2.1ProblemSetup........................................533.3Results:TheGrowthofThermalInstabilities...........................573.3.1ValidationofMethod....................................573.3.2InstabilityGrowthintheStrongCoolingRegime.....................583.3.3InstabilityGrowthintheConvectiveRegime.......................593.3.4TransitiontotheCondensedState.............................623.3.5CondensationRate......................................643.3.6EectofGeometry......................................643.4DiscussionandRelationshiptoRelatedWork...........................653.4.1ObservationsofMultiphaseGas..............................663.4.2SimulationsofMultiphaseGas...............................663.4.3CaveatsandLimitations...................................673.4.4Self-Regulation........................................683.5ConclusionsandFutureWork....................................703.6Acknowledgments...........................................704TriggeringandDeliveryAlgorithmsforAGNFeedback....................724.1Introduction..............................................724.2Method................................................764.2.1SimulationEnvironment...................................764.2.2ClusterSetup.........................................764.2.2.1TracerFluid....................................784.2.3FeedbackandJetModeling.................................784.2.3.1TriggeringMechanisms..............................794.2.3.2JetImplementation................................804.2.4HydroMethod........................................824.3Results.................................................824.3.1DeliveryofFeedback:Thermalvs.Kinetic........................824.3.1.1FeedbackPower..................................824.3.1.2ColdGasAccumulation..............................834.3.1.3RadialProles...................................844.3.1.4JetPrecession...................................884.3.2AGNTriggeringMechanisms................................894.3.3AccretionRadius.......................................934.4Discussion...............................................944.4.1PrecipitationandAGNFueling...............................944.4.2CaveatsandAdditionalPhysics...............................97vi4.4.2.1Conduction.....................................974.4.2.2MagneticFields..................................974.4.2.3StarFormation...................................984.4.3ComparisonWithSimilarStudies..............................994.4.3.1Li&Bryan2012-2015...............................994.4.3.2Gasparietal.2011................................1004.4.3.3Yangetal.2012..................................1004.4.3.4Duboisetal.2012.................................1014.5Conclusions..............................................1014.6Acknowledgments...........................................1025Conclusions................................................1045.1The`LastKpc'Problem.......................................1045.2Galaxy/AGNInteractions......................................1055.3AdditionalPhysicsandJet/ICMInteractions...........................1055.3.1Cooling............................................1055.3.2PlasmaPhysics........................................1065.4CosmologicalSimulations......................................107APPENDICES................................................109AppendixAEnzo:AnAdaptiveMeshHydrodynamicsCodeforAstrophysics...........110AppendixBTheIso-coolingSetup...................................116AppendixCIdealizedClusterSetup..................................120AppendixDFragmentationindustylow-metallicitystarforminghalos...............124AppendixEUseofCopyrightedMaterial...............................167BIBLIOGRAPHY..............................................171viiLISTOFTABLESTable4.1ParametersforAGNSimulations..............................80TableA.1SymbolsusedintheEulerequations.............................111TableD.1VaryingMetallicity......................................165TableD.2VaryingJeansRenement..................................165TableD.3VaryingSpin..........................................165TableD.4VaryingTurbulence......................................166TableD.5OtherRuns..........................................166viiiLISTOFFIGURESFigure2.1HowAGNFeedbackAectsClusterCores.........................46Figure2.2AGNFeedbackEnergyTransferIllustreatedwithPerturbedQuantities........47Figure3.1MultiphaseGasDensityandTemperatureCongurations................55Figure3.2ComparisonofMeece(2015)withMcCourt(2012)....................56Figure3.3CoolingtimeoverfreefalltimeforMultiphaseGasSimulations.............58Figure3.4GasDensityEvolutionOverTimeinMultiphaseSimulations..............59Figure3.5EvolutionoftheTimescaleRatioinMultiphaseSimulations...............60Figure3.6EvolutionoftheTimescaleforMoreStableInitialConditions..............61Figure3.7EvolutionoftheTimeScaleRatioforMoreStableICs..................61Figure3.8EvolutionoftheTimeScaleRationinFreefallTimes...................62Figure3.9PDFofGasDensityandTemperature...........................63Figure3.10TracerParticleEvolution..................................63Figure3.11ColdGasFractionintheMultiphaseSimulations.....................64Figure3.12EectofGeometryinMultiphaseGasSimulations....................65Figure3.13NecessaryAGNEciencytoBalanceCoolinginMultiphaseGasSimulations.....69Figure4.1AGNPowervs.CoolingRateinIdealizedAGnSimulations...............83Figure4.2EectofKineticFractiononJetPower..........................84Figure4.3ColdgasMassforDierentAGNTriggeringAlgorithms.................85Figure4.4EvolutionofGasQuantitiesforDierentKineticFractions...............86Figure4.5JetPowervs.CoolingRateforDierentTriggeringMethods..............87Figure4.6EectofTriggeringMethodonVelocityProles......................88Figure4.7EectofJetPrecession...................................89Figure4.8ComparisonofSimulationswithDierentTriggeringAlgorithmsandKineticFractions90Figure4.9EectofTriggeringMechanismonJetPower.......................91Figure4.10EectofTriggeringMechanismonGasQuantities....................92Figure4.11EectofAccretionRadiusonJetPower..........................93Figure4.12PDFofTimescaleRatioforThermalandKineticFeedback...............95FigureA.1CoolingRatesinEnzo....................................114FigureD.1InitialConditionsforLowMetallicityStarFormationSimulations...........127FigureD.2DensityEvolutionoftheFiducialRun...........................134ixFigureD.3GasPropertiesfortheFiducialRun............................135FigureD.4ChemicalStateoftheGasfortheFiducialRun......................136FigureD.5ProjectionsofDensityfortheFiducialRun........................139FigureD.6MaximumRenementvs.Density.............................141FigureD.7ImportanceofRenementCriterion............................142FigureD.8EectofRenementCriteriononFragmentation.....................144FigureD.9EectofRenementCriteriononGasDensity......................145FigureD.10GasProlesforSimulationsWithDierentMetallicities.................147FigureD.11ProjectionsofGasDensityforHighMassHalos......................148FigureD.12BoundClumpsforSimulationswithDierentMetallicity................149FigureD.13FragmentationinHighMetallicityHalos..........................150FigureD.14EectofSpinParameteronGasDensity..........................152FigureD.15EectofSpinParameteronAngularMomentumDistribution..............153FigureD.16EectofSpinParameteronFragmentation........................154FigureD.17EectofTurbulenceonGasDensity............................155FigureD.18EectofTurbulenceonFragmentation...........................155FigureD.19EectsofDustChemistryonGasProles.........................157FigureD.20EectofDeuteriumChemistryonTemperatureProles.................158FigureD.21BoundClumpsinSimulationswithHighDensityICs...................159FigureD.22GasProlesforSimulationsWithHighDensityICs...................161x1Introduction
Galaxyclustersstandatthecrossroadsofastrophysicsandcosmology.Throughobservations,galaxyclustershavebeenusedtoinvestigatethecontents,structure,andevolutionoftheuniverse.Inordertousegalaxyclustersinthismanner,itisnecessarythatastronomersunderstandthephysicalprocessesthatshapethelightemitting,baryonicmatter.Thisdissertationwillfocusonaparticularprocess{feedbackfromactivegalacticnuclei{anditsroleinregulatingthethermalstateoftheintraclustermedium.Galaxyclustersarelledwithahotplasma,knownastheintraclustermedium,orICM.ThisplasmaemitsX-raysandshouldcoolontimescalesmuchshorterthantheagesofmanyclusters.However,observationsshowthatmostoftheplasmadoesnotcooldown.Therefore,somephysicalmechanismmustbeheatingtheplasmaataratethatapproximatelybalancesradiativecooling.Thisisthecruxofthecooling-owproblem.InthisdissertationIexploreascenarioinwhichpowerfuljets,triggeredbytheaccretionofcoldgasontosupermassiveblackholesdeepintheclustercores,providethenecessaryheatingandkeepcool-coreclustersinaroughstateofthermalequilibrium.Thisintroductionbeginsbydiscussingaselectionofastronomicalconceptsthathavebearingontherestofthisdissertation,includinggalaxyclusterformation,galaxyclusterevolution,andactivegalacticnuclei(AGN).Thosewithastronggraspoftheseconceptscanskimthesesections.Section1.4outlinesthecoolingowproblemandtheprecipitationregulationtheoryoffeedback,whichformsthebasisforthiswork.Finally,Section1.5presentstheoutlineoftherestofthisdissertation.1.1FormationofGalaxyClustersandtheCosmologicalContext1.1.1TheCDMModelofCosmologyIntheCDMcosmologicalmodel,theuniverseconsistsofdarkenergy(theterm)and`cold'darkmatter(CDM),alongwithbaryons,electrons,neutrinos,andradiationallevolvinginageneralrelativisticframe-work.TheuniversebeganintheBigBangapproximately13.6billionyearsagoasahot,dense,expandingmedium.ThephysicsoftheuniverseimmediatelyfollowingtheBigBangisnotfullyunderstood,andwillrequirethedevelopmentofaquantumtheoryofgravity.TheevidencesuggeststhattheuniverseunderwentaperiodofexponentialexpansionsoonaftertheBigBang,termed`Ination'(rstdescribedinthepioneeringworksofGuth,1981;Linde,1982;Bardeenetal.,1983).Astheuniverseexpandedthetemperatureofthebackgroundradiationdropped,resultinginthecreationofbaryons,leptons,andatomicnuclei(Alpheretal.,1948).Althoughtheprimordialuniversewasnearlyhomogeneous,smallperturbationsinthedarkmatter1densitygrewthroughhierarchicalmergingtocreateawebofstructurethroughouttheuniverse.Forthepastseveralbillionyears,darkenergyhasbeendrivingtheexpansionofthecosmos.DarkenergyistheleastwellunderstoodcomponentoftheCDMmodelandalsothemostdynami-callyimportantatlargescales.Thenatureandoriginofdarkenergyarenotunderstood,butforastrophysicists,itismoreimportanttoknowitsdistributionandequationofstate.TheCDMmodelassumesthatdarkenergyisa`cosmologicalconstant'thathasauniform,constant,andlowdensitythroughoutspace.Alternativeandextendedmodelshavebeenproposed,buttheycannotcur-rentlybeobservationallydistinguishedfromthecosmologicalconstantmodel(e.g.,Andersonetal.,2012;PlanckCollaborationetal.,2015b).Darkenergyhastheeectofexertingnegativepressureonitssur-roundings,causingtheexpansionoftheuniversetoaccelerate.Duetoitsrelativelylowenergydensity,darkenergywasdynamicallyunimportantintheearlyuniversebutasspaceexpanded,theconstantdensityofdarkenergymeantthatitcametodominatetheexpansionoftheuniverse.Roughly10billionyearsaftertheBigBang,darkenergybegantodominatethelargescaleexpansionoftheuniverseandhasdonesoeversince.Ingravitationallyboundsystems(includinggalaxiesandgalaxyclusters)however,dynamicsaredominatedbygravity,andthedynamicaleectsofdarkenergycanbesafelyneglected.CDMassumesthatthebulkofthemassintheuniverseisintheformofcolddarkmatter,whichisbelievedtobesomesortofparticlethatonlyinteractsthroughgravityandpossiblytheweaknuclearforce(seeFeng,2010,forareview).Theparticlesare`cold'inthesensethattheirinitialthermalenergywasnegligiblysmallcomparedtotheirrestmassenergy,makingthemnon-relativistic.Aswithdarkenergy,thenatureoftheseparticlesandtheiroriginsarenotknown.Severaltheorieshavebeenproposed,includingsuper-symmetricpartnersofStandardModelparticles,sterileneutrinos,andaxions.CurrentexperimentssuchastheLargeHadronCollidermayconrmordisprovesomeofthesecandidatesinthenearfuture.Alternativestoparticleformulationsofdarkmatter,chieyModiedNewtonianDynamics(MOND),havebeenproposed(Milgrom,1983),buthavesofarfailedtogaintractionintheastrophysicscommunity.MONDtypetheoriesareoftenformulatedtoexplainoneparticularaspectofstructureformation,butsofarnotheoryhasbeenabletomatchobservationsonallscales.Onceagain,thecompositionofdarkmatterisnotasimportantforastrophysicsasitsdynamicproperties.WhiletheCDMmodelhasbeensuccessfulatexplainingstructureintheuniverseatlargescales,discrepanciesremainonthescaleofdwarfgalaxies.Thesediscrepanciesarenotdirectlyrelevantforthesystemsconsideredinthiswork,butanycomplicationstothecolddarkmatterhypothesisareworthnoting.Hotdarkmattermodels(e.g.Zel'dovich,1970;Doroshkevichetal.,1980),inwhichconstituentparticlesdomoverelativistically,areruledoutastheywouldresultinstructureformationoccurringatlatetimesandstartingatthelargestphysicalscales(`topdownstructureformation'),whichisnotconsistentwith2observations.'Warmdarkmatter'models,wherethethermalenergyoftheparticlesissmallbutnon-negligible,havebeenproposed(e.g.Bodeetal.,2001;Abazajianetal.,2001),butstrongconstraintshavebeenplacedonthecontributionofWDMtostructureformation(e.g.Vieletal.,2013),atleastonthescalesoflargegalaxiesandgalaxyclusters.Self-interactingdarkmattermodels,inwhichdarkmatterparticlesexperiencebinaryinteractionswitheachother,havebeenproposed(e.g.Spergel&Steinhardt,2000),butobservationsplacelimitsonsuchinteractionsthataresimilartothenon-interactingcase.Finally,itispossiblethatdarkmatterparticlescoulddecay,possiblyemittingelectromagneticradiation.Searchesforthisareongoingbuthaveturnedupnoconclusiveresults(Jeltema&Profumo,2016;Stormetal.,2013).EvenifdarkmatterdoesdierfromtheCDMmodel,observationsplacelimitsonhowlargethedeviationindynamicbehaviorcanbe.Thus,anydeparturefromtheCDMmodelwouldnotbeexpectedtohavealargeeectontheconclusionsofthiswork.
1.1.2CosmologicalStructureFormation
Galaxyclustersareimportantasprobesofbothastrophysicsandcosmology.ThelattercomesfromthefactthatgalaxyclustersarethelargestgravitationallyboundstructurestoformintheCDMuniverse,andtheirmassdistributioncontainsimportantinformationaboutthecontentsandhistoryoftheuniverse.Touseclustersinthismanner,itisnecessarytounderstandhowstructureslikegalaxyclustersformedfromthehomogeneousearlyconditionsofthepostBigBanguniverse.Thestudyofstructureformationisalsonecessarytounderstandthehistoryofbaryonsingalaxyclusters,givingacontextfortheuseofclustersastestsofastrophysicalprocesses.Theearlyuniverse(z>1100)consistedofanearlyhomogeneousmixtureofdarkmatter,baryons,leptons,andphotons.Inhomogeneities,possiblyrelicsofquantumuctuationsfrombeforetheinationaryepoch,tooktheformofanearlyGaussianrandomeld(PlanckCollaborationetal.,2014b)withverysmalldeviationsfromsmoothness.Althoughtheuniversewasexpanding,overdenseregionswouldhaveexpandedslightlymoreslowlythanunderdenseregions.Asthedominantcomponentofthemass-energydensityatthatepoch,darkmatterwouldhavedominatedthegravitationalinteractions.Baryonsandleptonswerepulledintothedarkmatterpotentialwellsandphotons,trappedintheopaquepre-recombinationplasma,followed.Radiationpressureprovidedarestoringforceasoverdensitiescollapse,leadingtooscillationofthematteroverdensities.Ataredshiftofz˘1100(PlanckCollaborationetal.,2015a),thetemperatureoftheuniversedecreasedtothepointthatelectronsandatomicnucleicouldcombine,lettingphotonsstreamfreelyoutofoverdensities.Thisradiation,todayredshiftedintothemicrowavepartofthespectrum,isobservedasthecosmicmicrowavebackground(CMB)(Smootetal.,1990;Bennettetal.,2003;PlanckCollaborationetal.,32015a;Ruhletal.,2004,andrelatedpublications).Followingthereleaseofphotons,overdensitiesproceededtocollapseuninhibited.Overdensitiesthathadcompletedanintegerorhalfintegernumberofoscillationsatthetimeofrecombinationformed,respectively,thepeaksandtroughsinthecosmicdensitydistribution.ThesepeaksandtroughswereimprintedontheCMBandformoneofthemostpowerfulprobesofcosmologyknowntoastronomers.Atthetimeofrecombination,theRMSdensityuctuationsintheuniversewereonly105(Smootetal.,1992),butwithouttherestoringpressureofthetrappedphotonsoverdenseregionswouldcollapseintothediversityofcosmologicalstructureobservedtoday.Overdensitiesintheearlyuniversewerenotperfectlyspherical,causingcollapsetoproceedatdierentratesalongdierentaxes.TheCDMmodelpredictsthatoverdensitieswouldhaverstcollapsedintosheets.Theintersectionofthesesheetsbecamethesitesoflamentsofgalaxies,andtheintersectionoflamentsbecamethesitesofgalacticsuperclusters.Onalargescale,theuniversebecamestructuredasacosmicweb,withoverdensesheetsandlamentssurroundinglarge,under-densevoids.Thecosmicwebhasbeenseeninobservations(Gottetal.,2005),lendingfurthercredencetotheCDMpicture.Overdenseregionsoftheuniversecollapsedhierarchically,withthesmallestregionscollapsingrstandlatercoalescingintolargerstructures.Baryonsfollowedthedarkmatterandsettledintothecoresofgravitationallyboundhalos.Oncethedensityofthebaryonshadincreasedandradiativecoolingbecameecient,therststarsbegantoformaroundz=20.Sinceprimordialgasisapoorcoolant(Galli&Palla,1998),therststarformingcloudswerenotassusceptibletofragmentationasarestarformingregionsinthepresentuniverse(Meeceetal.,2014;Abeletal.,2002;O'Shea&Norman,2007;Turketal.,2012,andAppendixD).Thus,itisexpectedthattherststarsweremassiveandshort-lived.Nevertheless,thesestarstransformedthebaryonicuniversebyreionizingtheinterstellargas(Wiseetal.,2014;Xuetal.,2014)andproducingtherstheavyelementsHeger&Woosley(2010).Therststarsarethoughttohaveendedtheirlivesinmassiveexplosions(Whalenetal.,2013;Ohkuboetal.,2009)thatwouldhaveexpelledgasfromtheirhosthalo(Smithetal.,2015).Hierarchicalcollapse,however,wouldeventuallycreatehalosmassiveenoughtohostongoingstarformation.Galaxieshavebeenobservedouttoaredshiftofz=11:1(Oeschetal.,2016).Moregalaxiesformedinthedensestregions,growingtherstproto-clusters(Kravtsov&Borgani,2012).Theseprotoclusterscontinuedtoaccretegasandgalaxiesalongcosmiclaments,mergedwithotherprotoclusters,andformedthemassivegalaxyclustersthatweobservetoday.41.2GalaxyClusters
Whataregalaxyclusters?Asthenamewouldimply,agalaxyclusterisaregioninwhichseveralgalaxiesarefoundincloseproximitytooneanotherrelativetothemeandistributionofgalaxiesintheuniverse.Agalaxyclusterismoreproperlydenedasalarge,virialized(dynamicallyrelaxed),gravitationallyboundstructurecontainingamixofbaryonicmatter1anddarkmatter.Infact,thegalaxiesthatdominatetheopticalemissionfromtheclusterencompassonlyasmallpercentageofthecluster'stotalmass.Darkmatteraccountsforthebulkofthegravitationalmassinaclusterwhilemostofthebaryoniccomponentresidesinadiuseplasmaofhotgasspreadthroughoutthecluster.Thisplasmaiscalledtheintraclustermedium,orICM.Whilethedenitionofagalaxyclusteras`large'issomewhatimprecise,therearevariousconstraintsandmetricsthatmaybeusedtolabelcollectionsofgalaxiesasclusters.AnuppermasslimitisprovidedbytheCDMhierarchicalcollapsemodelofstructureformationintheuniverse.InCDMthelargestgravitationallyboundstructureswereformedthemostrecently.Therefore,sincegalaxyclustersarethelargestgravitationallyboundobjectsintheuniverse,theuppermasslimitissetbytherateofstructureformation.AnalyticalmodelssuchastheworkofPress&Schechter(1974)providearoughestimateoftheclustermassfunction,withanuppermasslimitthatisstronglydependentonthechoiceofcosmology.SimulationssuchastheMillenniumRun(Springeletal.,2005b)andobservationssuchasthoseof`ElGordo'-themostmassivegalaxyclusterknown(Menanteauetal.,2012)-areconsistentwithanalyticalresultsandindependentcosmologicalprobes.Whilegalaxyclustersarethelargestgravitationallyboundstructuresintheuniverse,thisdoesnotmeanthattheyarethelargeststructuresthatexist.TheCDMmodelpredictsthatmatterintheuniverseisorganizedintoawebofover-denselamentsandwallssurroundingunder-densevoids.Additionally,galaxyclustersarenotscatteredrandomlyabouttheskybutinsteadarefoundinproximitytoeachotherinover-denseregionsknownassuperclusters,whichformwherelamentsintersect.Superclustershavenotfullycollapsedandvirializedandinmanycasesarenotgravitationallybound,insteadcontinuingtopartakeincosmicexpansion.TheformationofthecosmicwebhasbeenwellstudiedwithsimulationssuchastheMillenniumrun(Springeletal.,2005b).Galaxyclustersareoftencharacterizedbyrichness,denedasthenumberofgalaxiesaboveagivenmagnitudewithinagivendistanceofabrightcentralgalaxy.Thestandardforgalaxyclusterclassication1Technically,theterm`baryonicmatter'onlyreferstomattercomposedofbaryons,e.g.protonsandneutrons.Baryonicmatterthusdoesnotincludefreeelectronsorneutrinos.Althoughthesespeciesarepresentinclusters,theydonotcontributesignicantlytothemass.Thus,thetermbaryonicmattercanbereasonablyunderstoodinthisworktoincludeall`normalmatter'-thatis,everythingotherthandarkmatter-unlessotherwisespecied.5istheAbellcatalogue(Abell,1958;Abelletal.,1989),whichdenedrichnessclassesforclusterscontainingatleast30brightmembers.Inpractice,clusterswithlessthan50membersareknownas`galaxygroups'.Indeed,mostgalaxies(includingourown)arefoundingroups,butfewgalaxiesarefoundinrichclusters.Ifgravitationistheonlyforceshapingthepropertiesofgalaxyclusters,galaxygroupsshouldactlikescaleddownversionsofclusters.Whennon-gravitationaleectssuchasradiativecoolingandfeedbackareincluded,however,dierenceswillemergebetweenclustersofdierentmasses.Thus,dierencesinthebehaviorofgalaxygroupsandclustersisanactiveareaofresearch.Whendeningagalaxycluster,itisimportanttorememberthattheopticallighttracesonlythestarsingalaxies,whichmakeuponlyasmallpercentageofacluster'smass.Inordertoknowwhatagalaxyclusterisandtounderstandwhichphysicalprocessesareimportanttotheirbehavior,itisnecessarytohaveadescriptionofthedierentcomponentsofaclusterandofhowthosecomponentsaredistributed.Galaxyclustersaregravitationallybound,over-denseregionsoftheuniverseandarecom-posedofroughly85%darkmatterand15%baryonicmatter.Thecompositionofgalaxyclustersisrelativelywellknown{infact,clustersaretheonlystructuresintheuniversewherewecanobserveallofthecomponents.Thedarkmatterdistributioncanbestudiedthroughthemotionofclustergalaxiesorthroughgravitationallensingofbackgroundgalaxies.Thebaryoniccomponentisdividedbetweenstarsandgas.Thestellarcomponentiseasilystudiedthroughopticallight,whilethegasphaseisstudiedthroughX-rayemissionorthroughtheSunyaev-Zel'Dovicheect.
1.2.1DarkMatterinClusters
IntheCDMmodelofcosmology,thebulkofthematterinclustersiscomposedofsometypeofparticlethatonlyallowsweakandgravitationalinteractions.Duetothelackofstronginteractions,darkmatterparticlesdonotinteractwitheachother,andduetothelackofelectromagneticinteractions,donotemitradiation(hencetheterm`darkmatter')orinteractcloselywithbaryonicmatter.Darkmatterparticlesareassumedtobe`cold',meaningthattheyarecreatedwithnegligiblethermalenergy.Fromacosmologicalperspective,thecoldnessofdarkmatterandthepropertiesoftheallowedinteractionswillhavealargeeectonstructureformation.Fortheworkpresentedinthisdissertation,however,itisadequatetoassumethattheCDMpictureiscorrect,andthattheexactnatureofdarkmatterparticlesisnotimportant.Infact,formostastrophysicalapplicationsitissucienttotreatdarkmatterasacontinuousmediumandignoretheparticlenatureentirely.Ofgreaterinterestforclusterstudiesistheformofthedarkmatterdensityprole.Thedynamicsofdarkmatterinhalosaredominatedbygravity,whichisscalefree.Thus,darkmatterhalosofdierentmasses6wouldbeexpectedtohavesimilarforms.Asaself-gravitatingmedium,thedarkmatterdensitywillpeaknearthecenterofthecluster.Duetothecollisionlessnatureoftheparticles,itwouldnotbeexpectedtoformcomplexstructures.Simulationsshowthatthefunctionalformofdarkmatterprolesissimilarforawiderangeofgravitationallydominatedsystems.ThemostcommonlyusedformistheNavarro-Frenk-White(NFW)prole(Navarroetal.,1997)givenbyˆ(r)=ˆ0rRS1+rRS2(1.1)whereˆ0isequalto4timesthedensityatthescaleradiusRS.AsecondcommonlyusedformforthedarkmatterproleistheEinastoprole,rstdescribedbyJ.Einastoataconference(Einasto,1965)andlaterfoundtobeabettertfordarkmatterhaloes(e.g.Navarroetal.,2004,2010).TheEinastoproleisgivenbyˆ(r)=ˆ0exp2(1(r=RS))(1.2)whereˆ0isthedensityatthescaleradiusandisgenerallybetween0.2and0.3(Kravtsov&Borgani,2012,andreferencestherein)forclusters,withsomepowerlawdependenceonredshift(Gaoetal.,2008).Measurementsofvelocitydispersionsinclustersprovidedsomeoftherstevidencefortheexistenceofdarkmatter.Inparticular,FritzZwicky(Zwicky,1933)realizedthatinorderfortheComaclusterofgalaxiestosatisfytheVirialtheorem,mostofthecluster'smasswouldneedtoconsistofnon-luminousmatter.AccordingtothebaryoncensusconductedbyGonzalezetal.(2013),darkmatterisestimatedtomakeupeither86.4%(assumingWMAPcosmology)or85.6%(assumingPlanckcosmology)ofthetotalmatterinaclusterwithinR500,whichisclosetotheuniversalvalueof83.4%(WMAP)or84.5%(Planck).AsdiscussedinGonzalezetal.(2013),thepartitioningbetweengasanddarkmatterisfoundtobeonlyweaklydependentonclustermass,withthebaryonfractionrisingslightlywithincreasingclustermass.1.2.2BaryonsinClusters
Theremainderofagalaxycluster'smassiscomposedofbaryonicmattereitherintheformofstarsorgas.Baryonsemitenergythroughelectromagneticinteractions,makingthemsuitableforobservationfromEarth.Thus,itisprimarilythroughobservationsofthebaryonicmassthatscientistsareabletostudygalaxyclusterproperties,eventhoughbaryonsonlycomposeasmallfractionoftheclustermass.Thebaryonicmatterinclustersisdividedintoastellarcomponentandagaseouscomponent.Thestellarcomponentiscomposedofstarswithingalaxiesandapopulationofextra-galacticstars,termedthe7intraclusterlight.Uniquelyamongastrophysicalstructures,itispossibleformodernobservationstorevealallformsofbaryonicmatterwithinacluster,resultinginthepossibilityofacompletebaryoncensus.SuchabaryoncensuswasconductedbyGonzalezetal.(2013)whoconcludedthatthebaryonfraction2inclustersisfbaryon=0:1360:005usingWMAP7Cosmologyorfbaryon=0:1440:005usingPlanckcosmology.Therewasfoundtobeaweakbutstatisticallysignicantcorrelationbetweenclustermassandbaryonfraction,withfbaryonrisingslightlyinmoremassiveclusters.Thisrepresentsanupdatetothegroup'searlierbaryoncensus(Gonzalezetal.,2007),whichfoundresultsconsistentwithauniversalbaryonfraction.Thepartitionofbaryonsbetweenstarsandgasdoesvarysignicantlywithclustermass.Gonzalezetal.(2013)ndsastellarmassfractionofbetween1%and4%,withmoremassiveclustershavingalowerstellarmassfraction.Correspondingly,thegasfractionrisesfromaround7%inclusterswithamassof1014Mtoaround15%inthemostmassiveclusters.Thedecreaseinthefractionofmatterinstarsreectsareductionintheeciencyofstarformationinmoremassiveclusters.Itiscuriousthatgalaxyclustersaresoinecientatturninggasintostars,andthereasonisthoughttoinvolvestellarandAGNfeedback.Thus,thestellarmassfractioncanbeusedasaprobeoffeedbackprocessesingalaxyclusters.Thegaseousportionofgalaxyclusters(theintraclustermedium,orICM)isprimarilycomposedofHydrogenandHeliumwithasmallfractionofheavyelements,whicharetermed`metals'.Variousstudies(e.g.Leccardi&Molendi,2008;Matsushita,2011)ndanaveragemetallicityofaroundZ=Z=0:3(around1/3ofthesolarmetallicity).However,themetallicitydistributioninclustersisnotgenerallyat.Gasinthecentersofclustersisenrichedtoahigherlevel,uptoZ=Z=0:45withinR180inLeccardi&Molendi(2008)orZ=Z=0:88withinR180inMatsushita(2011).ThemetallicitydistributionisgenerallyfoundtobesharplypeakedinthecenterandtoattenoaroundthevirialradiustoavalueofZ=Z=0:2.ThisisinroughagreementwiththeresultsofhydrodynamicssimulationsinFabjanetal.(2008),althoughthemetallicitydistributioninthesimulationsishighernearthecentersoftheclusters.ItshouldbenotedthatFabjanetal.(2008)studiedrelaxedclusters,whichwouldbeexpectedtohavemoresharplypeakedcentralmetallicitydistributionsduetoalackofturbulenceandmixingfrommajormergers.1.2.3ObservingGalaxyClusters
Likemostastrophysicalobjects,galaxyclustersemitlightacrosstheelectromagneticspectrum.Onlyafractionofthetotalclustermasscanbedirectlyobserved,buttheradiationthatcanbedetectedcansayanenormousamountaboutthecontentsanddynamicsofclusters.Thissectionbrieydiscusseswhatfeaturesareobservedineachwavelength.2ThemassofBaryonsasafractionofthetotalmass(BaryonsandDarkmatter).81.2.3.1Radio
Radioemissionfromgalaxyclustersprimarilytracessynchrotronradiationfromacceleratingelectrons.Thesecanbeproducedbyseveralmechanisms(Ferrarietal.,2008;Ferettietal.,2012;Zandaneletal.,2014),in-cludingmergers,AGN,turbulence,shocks,andstellaractivity.Inaddition,atomicandmoleculartransitionscanalsoemitradiowaves.Processesobservedintheradioinclude(butarenotlimitedto)Radiorelicsfrommergers(e.g.,Ensslinetal.,1998;Skillmanetal.,2013)Radiohalos(e.g.,Ferrarietal.,2008;Ferettietal.,2012;Zandaneletal.,2014)RadioloudAGN(e.g.,Bestetal.,2007;Sambrunaetal.,1999;McNamara&Nulsen,2007)2.6mmCOemission(e.g.,Edge,2001;Russelletal.,2016)1.2.3.2Microwave
Themicrowaveskyisdominatedbythecosmicmicrowavebackground(CMB;Smootetal.,1990;Bennettetal.,2003;PlanckCollaborationetal.,2015a;Ruhletal.,2004,andrelatedpublications).Ingalaxyclusters,theCMBup-scatters(InverseComptonScattering)oofthehotICM,producingasmallupwardshiftintheobservedfrequency.Thiseect{theSunyaev-Zeldovich(SZ)eect(Sunyaev&Zeldovich,1970){canbeusedtoinferthemassesofgalaxyclusters(e.g.,Carlstrometal.,2011;PlanckCollaborationetal.,2014a).AstheSZeectdependsontheintegratedpressureoverthecluster,andsincehighredshiftclustersweredenserandhotterthanclusterstoday,theSZeectisessentiallyindependentofredshiftandallowsmassestimatesofdistantclusters.Recently,CMBsurveyshaveusedtheSZeecttondmassiveclustersthathavelaterbeendetectedintheoptical.TheSZeectcanalsobeusedtoinferclusterpropertiesviathekineticSZeect(e.g.Sieversetal.,2013).
1.2.3.3Infrared,Optical,andUV
Mostoftheinfraredlightemittedbyclusterscomesfromstars{specicallyredder,lowmassstars.Infraredlightisafrequentlyusedtodeterminethetotalstellarmassofaclusterandofclustergalaxiessincelowmassstarsarelonglived,meaningthattheirabundancetracesintegratedstarformationhistoryratherthanrecent.Ontheotherhand,protostellarcloudssurroundedbydustglowbrightlyintheIR,meaningIRcanbeusedtoinferongoingstarformationaswell(O'Deaetal.,2008).Thefar-IRcanalsobeusedtotracecoldgas(Werneretal.,2014).LikeIRemissions,opticallightinclustersisemittedbystars.Observationsintheopticalarethereforeimportantformeasuringthestellarmassofclusters,studyingstellarpopulations,andinferringthestar9formationhistoryofthecluster(e.g.,Fogartyetal.,2015;McDonaldetal.,2015;Loubseretal.,2016).Theamountofintraclusterlight(ICL)mightsaysomethingaboutthemergerhistoryofclustergalaxies.Opticalemissioncanalsobeaectedbygravitationallensing,whichcangiveinformationaboutthemassdistributioninthecluster(e.g.Postmanetal.,2012).Lensingprovidesanindependentconstraintonclustermasses.Stronglensingoccurswhenlightfromabackgroundsourceisbentbyinterveningmat-ter,leadingtotheformationofrings,arcs,andmultipleimages(e.g.,Broadhurstetal.,2005;Kellyetal.,2015).Weaklensingoccurswhengravitationallensingdistortstheshapesofbackgroundgalaxies(e.g.,Bartelmann&Schneider,2001;vonderLindenetal.,2014).TheUVlightfromgalaxyclusters,liketheUVlightfrommostgalaxies,isproducedbyyoung,massivestars.Therefore,UVobservationscanbeusedtodeterminethestarformationrateofclustergalaxies(e.g.O'Deaetal.,2010).
1.2.3.4X-ray
X-raysarepossiblythemostimportantpartofthespectrumforstudyingtheICMingalaxyclusters.HotgasintheICMemitsX-raysviaBremmstrahlung,whichoccurswhenchargedparticlesareacceleratedthroughbinaryinteractions(e.g.,Cavaliere&Fusco-Femiano,1976;Sarazin,1988).Thisradiationcanbeusedtoinferthetemperatureanddensityofthegasand,assuminghydrostaticequilibrium(HSE),toderiveamassprole(e.g.,Donahueetal.,2014).Thismassestimatedependsonestablishedscalingrelations.ThemajorobservatoriesforstudyinggalaxyclustersintheX-rayaretheChandraX-rayObservatory(Weisskopfetal.,2000)andtheXMM-NewtonObservatory(Jansenetal.,2001).X-rayscanalsobeusedtoinferthemetallicitydistribution(e.g.,Mitchelletal.,1976;Matsushita,2011).1.2.3.5GammaRays
GalaxyclustershavenotbeendenitivelydetectedinGammarays(Ackermannetal.,2014).Intheory,someclassesofdarkmattercandidatescouldproduceX-raysthroughself-annihilation,butthishasnotbeenconclusivelyobserved(Jeltema&Profumo,2016).
1.2.4MagneticFieldsinGalaxyClusters
Galaxyclustersareknowntocontainweak,tangledmagneticeldswhichextendthroughoutthecluster(seeCarilli&Taylor,2002,forareview).TheseeldsaretypicallyontheorderofafewmicroGauss(G),althoughtheymaybesomewhatstrongerinthecoreregion,especiallyincoolingowclusters.Theeldsaretooweaktobedynamicallyimportant,butarelikelytoplayaroleinenergytransportthroughanisotropic10conduction.Inaddition,theLarmorradiusofathermalelectroninaclustermagneticeldissignicantlyshorterthanthecollisionalmeanfreepath,meaningthatthedynamicsofelectronsonlargescaleswillbedominatedbyeectsfromthemagneticeld.Thus,magneticeldswouldbeexpectedtosuppressthermalconductioninatangledmagneticmedium,thoughtheextenttowhichthishappensinclustersisdebated(Smithetal.,2013;Ruszkowski&Oh,2011;Waghetal.,2014).Additionally,thedynamiceectsofmagneticeldsmightbeexpectedtoincreasetheviscosityofthegas,thoughwhetherthisisappreciableinclustersisunknown.Forageneraloverviewofmagneticeldsinclusterswithadditionalreferences,seeMcNamara&Nulsen(2007).
1.2.4.1ObservationofMagneticFieldsinClusters
Magneticeldsinclusterscanbeobservedandstudiedwithseveralmethods.Theobservationalprobesdiscussedincludesynchrotronradiation,polarizedradioemission,Faradayrotationofbackgroundsources,andinverseComptonscattering.AllofthesemethodsaredescribedinmoredetailinCarilli&Taylor(2002).Ifclustersdohavemagneticeldsandareroughlyinequilibrium,thereshouldbesomeequipartitionbetweentheenergyinmagneticeldsandthekineticenergiesofparticles.Theamountofsynchrotronradiationisanindicatoroftheparticleenergy,andfromthiswecaninferthemagneticeldstrength.ThismethodgivesvaluesontheorderofafewG,withcool-coreclustershavinghighereldsthannon-cool-coreclusters.Synchrotronradiationfromgalaxiesshouldbepolarized,sincethemagneticeldcreatesapreferreddirectionforelectronstomove.Whilesomedegreeofpolarizationiscausedbyourowngalaxy,radioemissionfromclustersappearstobemorepolarized.Thisisasecondindicatorthatclustershavemagneticelds.Thirdly,amagneticeldinaclusterwouldcauseFaradayrotationofemissionfromsourcesbehindthecluster.Thisisoneofthemostimportantprobesofeldstrength,andagaingivesestimatesontheorderofafewG.Thesimplestestimatesofthemagneticeldsfrompolarizationassumethattheeldisuniform,butthisisunlikelytobethecase.Amoredetailedmethodistoassumethattheeldistangledandtoapproximateitasbeingcomposedofcellsofsomecharacteristiclengthl,eachwitharandomorientation.Polarizationmeasurementsindicatethatthislengthscaleisontheorderof5-10kpc.Thepolarizationthatisobservedistakentobetheproductofarandomwalkthroughthecellsthatcomposetheeld.Finally,clustermagneticeldscanbemeasuredbyderivingarelationbetweenthesynchrotronradiationluminosityandComptonup-scattering.Intheorybotharecausedbythesamepopulationofrelativisticelectrons.Comptonup-scatteringiscausedbyscatteringofbackgroundphotons,andthusmeasuresthephotoneldenergy,whilesynchrotroniscausedbyscatteringoofvirtualphotons,andmeasuresthe11magneticeldenergy.Thus,theratioofsynchrotrontoComptonup-scatteringscalesliketheratioofphotonenergytomagneticenergy.Byitself,thismethodgivesestimatesontheorderof0.1Gformostclusters,anorderofmagnitudelowerthanfromothermethods.However,someadditionalfactorshavebeenidentiedthatcanbringthesenumbersintoagreementwithotherestimates.Forexample,collisionswouldkeepthepitchangleofelectronsisotropiceventhoughclassicaltheorysaystheywouldnotremainthatwayforlong.Secondly,thesynchrotronradiationmeasurementassumesanenergyspectrumfortheelectronsthatmightinfactbesteeper.FortheinverseComptoncase,itispossiblethatsomeoftheX-rayscouldcomefromthermalBremmstrahlung,althoughitishardtomakethisworkenergeticallywithoutevaporatingthecluster.Finally,substructureinthemagneticeldcouldleadtoerrors.Ifthecharacteristiclengthscaleoftheregionwhereelectronsarerelativisticislargerthanthescaleofthemagneticelds,theX-rayscouldbecomingfromfurtherout,wheretheeldisweaker,whilethepolarizationiscomingfromthestrongeldinthecore,leadingtoaninaccuratecomparison.
1.2.4.2TheOriginofMagneticFieldsinClusters
Severaltheorieshavebeenproposedtoexplainwherethemagneticeldsingalaxyclusterscomefromandhowtheyareampliedtotheircurrentstrength.Intheory,ifasmallmagneticeldexistedintheIGMafterrecombination,itwouldhavebeenampliedbygascompressionduringstructureformation.Earlystarscouldhavealsogeneratedmagneticeldsandexpelledtheminoutows.Finally,AGNgeneratemagnetizedjetsandcouldinprincipledepositsomeoftheirenergyinthemagneticeldofthecluster.Howevertheyaregenerated,magneticeldscanbeampliedbycompressionoftheeldlines,eitherthroughcompressionoftheICMorbyturbulencethroughadynamoeect.Itislikelythatbothofthesefactorsplayaroleingeneratingtheeldamplitudesthatareobserved.Cool-coreclustershaveeldsthatarehigherthannon-cool-coreclusters,whichisexpectedfromthehigherdensityofgasinthecore.Mergersandshockscouldgenerateturbulence,amplifyingtheeldfurther.AGNfeedbackhasbeensuggestedasamethodforamplifyingmagneticeldsincoolcoreclusters,andthereisevidence(Duboisetal.,2009)thatthiscouldinfactbehappening.ItisnotobviouswhateectAGNwouldhaveonexistingmagneticelds{theycouldeitherstrengthen(throughturbulence)orweakenthem(byreducingthegasdensity.)Duboisetal.(2009)ndsthatmagneticeldsinidealizedclustersareenhancedeitherwithorwithoutAGNfeedback,butfordierentreasons.Withoutfeedback,acoolingcatastropheoccursthatcompressesthegas,strengtheningtheeld.Withfeedback,thecoolingcatastropheispreventedbutthegasbecomesmoreturbulent.Itisworthnotingthatevenifconductionisimportant,itwouldnotamplifythemagneticeld,leavingtheobservationsunexplained.121.3ActiveGalacticNucleiandSupermassiveBlackHoles
Manygalaxiesareseentohavebrightemissionregionsintheircores.Basedontheinferredenergiesneededtopowerthisemission,activegalacticnuclei(AGN)areamongthemostpowerfulphenomenaintheuniverse.Inrecentdecades,aconsensushasemergedthatAGNarecomposedofanaccretiondisksurroundingasupermassiveblackhole(SMBH).Magneticeldsbecometwistedinthedierentiallyrotatingaccretiondisk,funnellingchargedparticlesintopowerfuljets.Thesejets,alongwithwindsfromthehotaccretiondiskitself,areinprincipleenergeticenoughtobalanceradiativecoolinglossesandcausealargescaleredistributionofgasingalaxies.ObservedscalingrelationsbetweenSMBHmassandgalacticpropertieshavestrengthenedtheideathatSMBHsandAGNplayacriticalroleingalaxyevolution.Inparticular,ithasbeenarguedthatAGNinbrightestclustergalaxies(BCGs)areresponsibleforbalancingcoolinglossesandhaltingacoolingcatastropheincool-coregalaxyclusters.ThissectiongivesashortintroductiontoAGNandhowtheycancoupletotheirenvironments.1.3.1AShortHistoryofAGN
ThoughAGNhavebeenobservedinonewayoranotherforoveracentury,thenatureoftheseobjectshasonlybecomeclearinrecentdecades,andtherearestillmanymysterieslefttounravel.ThissectioncontainsashorthistoryofobservationsofAGNactivity.AGNwererstobservedserendipitouslyandatrstwerenotrecognizedasextragalacticobjects.AGNwererstnoticedinspectralemissionbyFath(1909),whodescribedstrongemissionlinesinthenucleusofNGC1068.Curtis(1918)describestherstopticaldetectionofanAGN{acuriousbrightfeatureinthenucleusofM87withextendedemission.Seyfert(1943)madeasurveyofspiralnebulaewithbrightnuclearemissionandfoundgreatvariabilityintheirspectralfeatures,withcombinationsofbroadandnarrowemissionlines.Thebroadlines,wheninterpretedasstemmingfromaDopplershift,impliedgasmovingat1000sofkm/s{muchhigherthangalacticescapevelocity.Additionally,therapidvariabilityimpliedthattheemittingregionwasverysmall.Radiosurveysinthe1950sandearly1960s(Edgeetal.,1959;Bennett,1962)identiedapopulationofpointsourceradioemissionswithnobrightopticalcounterpart.Therstoftheseobjectstobematchedwithanopticalsourcewas3C273,whichappearedasafaintstarwithasmalljet.Opticalspectraof3C273(Schmidt,1963)wereconsistentwithasourceataredshiftof1.58,implyingaverybright,verydistantobjectemittingfromasmallregion.Schmidt(1963)hypothesizedthattheemissioncouldbecomingfromthenucleusofadistantgalaxy.13SpectralobservationsofAGNrevealedpointsourcesthatlookedlikestars(hencetheterm`quasar',shortforquasi-stellarobject)butwithlargeredshiftsandluminositiesfarabovetypicalstellarvalues.Forexample,Baade&Minkowski(1954)foundopticalandradioluminositiesofover1042erg/sfortheAGNinCygnusA{over10billiontimeshigherthanthesolarluminosity.Althoughtheauthorserroneouslyattributedtheemissiontoapairofmerginggalaxies,furtherobservationsbySchmidt(1963)wouldshowthattheAGNemissionregionmustbesmallerthan1kpc{toosmalltobeagalaxy.Theobservationsalsoshowedthatjetsneartheobjectwereontheorderof50kpcinlength,implyingthatthesourcehadbeenactiveforatleast100,000yearsandhademittedatleast1059ergsinthattime.Atthetimeoftheirdiscovery,therewasnophysicalprocessknownthatcouldexplainthehighpowerandsmallsizeofAGN.Salpeter(1964)andZel'dovich&Novikov(1965)independentlyhypothesizedthataccretionontoasuper-massiveblackholecouldintheorygeneratetherequiredamountsofenergy,presumingthatangularmomentumcouldbetransportedoutwardsallowingmaterialtofallin.TheSMBHpoweredAGNhypothesiswaseshedoutbyLynden-Bell(1969),whoalsoarguedthatmanygalaxiesshouldcontainquiescentAGNthatcouldinprinciplebeobserved.AlthoughtheSMBHatthecentersofAGNwerenot(andstillhavenotbeen)directlyobserved,theSMBHparadigmcontinuedtogaintractionasdetailswereclariedandalternativeexplanationsruledout.
1.3.2TheUniedmodelofAGN
DespitetheplausibilityoftheSMBHargumentforexplainingAGNobservations,itwasnotclearthatallAGNbehavedthesamewayorindeedwerepoweredbythesamephysicalprocess.Fromthebeginning,itwasnoticedthatAGNexhibitedadiversityofspectralfeatures.ManyAGNexhibitednarrowlineemissionspectra,butsomealsoshowedbroademissionlines.Further,someAGNwerebrightradiosources{oftenamongthebrightestinthesky{whileotherswereradioquiet.AsmallpercentageofAGNseemedtoshowacontinuousspectrawithnoemissionorabsorptionlinesandapparentlyexhibitedsuperluminalmotionintheirjets.TheseobservationsledtoAGNbeinggroupedintoseveralempiricallydenedcategories(seeLawrence,1987,forareview).DespitethediversityofAGNobservations,itwassuspectedthatmanyofthesedierencescouldbeexplainedintermsoftheorientationoftheAGNsystemanditsrecentactivity.Inunicationmodels(seeUrry&Padovani(1995)forareview),allAGNconsistofadiskofac-cretingmaterialsurroundingasupermassiveblackhole.WhencloudsofmaterialcomeneartheSMBH,theywillsettleintorotationwithinatorus.Closerin,thetorusattensintoadierentiallyrotatingaccretiondisk.Theaccretingmaterialgenerallyhassome(weak)magneticeld,theeldlinesofwhichtendtobedraggedalongwiththeow.Throughacombinationoffrictioninthediskandelectromagnetic14eects,angularmomentumistransferedoutwardsandgascanowtowardstheSMBH.Dierentialrotationcausesthedisktoheatup,causingittoemitUVandX-raylightaswellasblastingoahotwind.Withinaradius3ofRISCO=6GMSMBHc2(1.3)generalrelativitypredictsthatnostablecircularorbitexists|henceEquation1.3istheradiusoftheinnermoststablecircularorbit(ISCO).RISCOthereforeformstheinnerboundaryoftheaccretiondisk,withinwhichmaterialfallsintotheSMBH.Duringtheaccretionprocess,thegravitationalenergyoftheaccretingmaterialcanbereleasedinseveralways.Asmentionedabove,dierentialrotationheatstheaccretiondisk,drivingawindandcausingthehotmaterialtoemitlikeablackbody.Secondly,dierentialrotationwillcausemagneticeldlinestobecomestretchedandtwisted.Thisprocess,analogoustotwistingandstretchingrubberbands,storesmagneticenergyandchannelschargedparticlesintotheobservedrelativisticjets.Itisestimatedthatduringthein-spiralperiod,aparticlebeingaccretedontoanSMBHcanradiateawayupto10%ofitsrestmassenergy{evenmoreecientthanH-Hefusion,whichonlyreleases0.7%oftherestmassenergy.Asidefromtheseclassicalprocesses,generalrelativitypredictsthatenergycanbeextractedfromtheSMBHitself,furtherboostingtheenergyoftherelativisticjets.Energycannotbeextractedfromwithintheeventhorizonofablackhole,buttheKerrMetric(Kerr,1963)predicts4thatarotatingblackholewilldragspace-timeinaregionoutsidetheeventhorizon.Energycaninprinciplebeextractedfromthisregion,termedthe`Ergosphere',allowingforenergytobeextractedfromthespinoftheSMBH.IntheBlandford-Znajecprocess(Blandford&Znajek,1977),magneticeldlinespassthroughtheErgosphere,wheretheyarerotatedandtwisted,transferringenergyoutwardsattheexpenseoftheSMBH'sangularmomentum.InthePenroseprocess(Penrose&Floyd,1971),aclumpofmatterfallsintotheergosphereandsplitsintwo,withonepartfallingintotheSMBHandtheotherbeingejectedwithmoreenergythantheoriginalclump.Thejetproductionprocessisstillnotfullyunderstood(seeTchekhovskoyetal.,2011;S˘adowski&Narayan,2015;McKinneyetal.,2012,forsomerecentwork),butitisclearthatSMBH-diskinteractionsareeasilycapableofproducingtheenormousluminositiesobservedfromAGN.WiththismodelofanAGNinmind,thediversityofobservationsbecomesclearer.Therapidlyrotatinginneraccretiondiskwillproducebroadlineemission,whilethesloweroutertoruswillhavenarroweremissionlines.Thus,Type1AGN,whichexhibitbroadlineemissionfeatures,canbeunderstoodasAGNwherewe3Foranon-rotatingblackhole{theleadingfactordecreasesfrom6to1foramaximallyrotatingblackhole4Asahistoricalaside,thediscoverythatarotatingSMBHcouldpoweranAGNwasoneoftherstknownproblemsinwhichgeneralrelativitypredictedasignicantandobservabledeviationfromtheclassicalexpectation.15haveaviewoftheinneraccretiondisksincetheyareorientednearlyface-ontous.Type2AGN,withonlynarrowlinefeatures,areviewedatalargerangle(moreedge-on)suchthatthetorusblocksourviewoftheinnerregion,sothatonlynarrowlinesareseen.BLLacobjects(namedfortheirprototype5),whichshowneitherabsorptionnoremissionlines,areseennearlyalongtheaxisofthejet.Dierencesinradioemissioncanbedierencesinthespinoftheblackhole(Wilson&Colbert,1995),whichwouldaectthepropertiesoftherelativisticjets,inturnaectingtheamountofhot,synchrotronemittingplasmaproducedbytheAGN.
1.3.3InteractionsBetweenAGNandtheirEnvironment
Onthesurface,itwouldseemunlikelythatSMBHswouldhaveanoticeableeectontheirsurroundinggalaxies.Thephysicalsizeofablackholeisverysmall(ontheorderofAUfortheeventhorizonradius)comparedtothesizeofgalaxies(tensofkpc)orgalaxyclusters(oforderMpc).ThemassofaSMBHisalsoverysmallcomparedtotypicalmassesforgalaxiesandclusters,withgalaxiesgenerallyoutweighingtheircentralSMBHbyfactorsof103ormore.AlthoughthegravitationalaccelerationproducedbyanSMBHislarge,themagnitudefallsoasr2andshouldnotdominatedynamicsbeyondafewtensofpc.Asoutlinedintheprevioussection,however,AGNarecapableofreleasingenoughenergytosignicantlyaectthethermalstructureofthegasortodrivesignicantgasmotion(Voitetal.,2015c,a).Therefore,itiscorrecttothinkthatthroughAGN,SMBHsshouldhaveastronginuenceontheirenvironment.NeartheSMBH,theenergyreleasedfromtheAGNactsontheaccretingmaterialasatypeofnegativefeedback.Whentheaccretionrateincreases,sodoesthepowerreleasedbytheAGN,whichwillgenerateradiationpressureandlimitfurtherincreasesintheaccretionrate.ItiscommontomeasureAGNluminosityintermsoftheEddingtonluminosity(Eddington,1916)LEDD=4ˇGMBHmpc˙T(1.4)whichistheluminosityatwhichradiationpressurebalancesgravity.Whilethiswouldinprinciplegiveanupperlimittotheblackholeaccretionrate,theEddingtonluminositycalculationassumessphericalaccretion,whichisalmostcertainlyinvalid.Still,whilesuper-Eddingtonaccretionmayoccur,theEddingtonluminosityprovidesagoodreferencefortherealmwherenegativefeedbackbeginstostieaccretion,anditisdiculttoimagineanAGNexceedingtheEddingtonluminositybymorethananorderofmagnitude.Onlargerscales,themassesofSMBHscorrelatewithvariousgalacticproperties,includinggalacticluminosity(Magorrianetal.,1998),thecentralstellarvelocitydispersion(theM˙relation;5Ironically,BLLacishasbeenobservedtohaveweakemissionlines,andthusisnotaBLLacobject16Merritt,2000;Ferrarese&Merritt,2000;Gebhardtetal.,2000),andthegalacticvirialmass(Ferrareseetal.,2006).ThatthesequantitiesarecorrelatedimpliessomeconnectionbetweenthegrowthoftheSMBHandthehostgalaxy.Thisconnectioncouldmanifestinanumberofways:eithertheSMBHgrowthregulatestheformationofstarsthroughAGNfeedback,starformationdrivesthegrowthofAGNthroughwindsandstellarmassloss,oraccretionontothegalaxyfuelsbothstarformationandSMBHgrowthatproportionalrates.Althoughthetruecausesoftherelationshavenotbeenrmlyestablished,severaltheorieshavebeenproposed.Onewidelyacceptedtheory,thatofKing(2003),holdsthatoutowsfromAGNpushgasoutofthehostgalaxy,limitingtherateatwhichstarscanform.AGNfeedbackismostapparentinobservationsofmassivegalaxiesandgalaxyclusters.Incool-coregalaxyclusters,theroleofAGNinpreventingacoolingcatastrophehasbeenwellestablishedthroughthelackofobservedcoldgas(e.g.,Petersonetal.,2003;Peterson&Fabian,2006),theavailabilityofenergygeneratedbytheAGNtobalancecooling,andthelackofviablealternativeexplanations(Skoryetal.,2013).Studyingthecoolingowproblemformsthebasisforthisdissertation,anditissummarizedinSection1.4AmoredetailedjourneythroughtheliteratureonthecoolingowproblemispresentedinChapter2.Inadditiontothecooling-owproblem,evidenceforAGNfeedbackatclusterscalesisseeninICMshocks(Fabianetal.,2003),X-raybubbles(Fanaro&Riley,1974),andjet-drivenredistributionofmetals(Kirkpatrick&McNamara,2015).ForathoroughreviewoftheobservationalevidencerelatingtoAGNfeedback,seeFabian(2012).
1.3.4TheOriginofSupermassiveBlackHoles
Finally,withthemechanicsofAGNfeedbackinhand,thereremainsthequestionofwhere,when,andhowtheSMBHsthatpowerAGNform.Thisremainsanactivetopicofresearch(seeVolonteri(2010)forareview),andnoSMBHcreationtheoryhasgainedthefullacceptanceoftheastrophysicscommunity.ItisclearthatessentiallyallnearbylargegalaxiescontaincentralSMBHs(Ferrarese&Ford,2005),indicatingthattheirformationisreasonablycommon.Observations(Momjianetal.,2014;Willottetal.,2015)showthatpowerfulAGNwereinplaceinsomegalaxiesbyaredshiftofz=67,indicatingthatmassiveSMBHshadalreadyformedatthattime.Willottetal.(2003)ndsevidencefora3108MSMBHpoweringaquasaratz=6:41{lessthan1GyraftertheBigBang.ForsuchanSMBHtoexist,itmusthaveformedearlyintheuniverseandgrownattheEddingtonrateorfasterforitsentirelifetime.SeveraltheorieshavebeenpositedtoexplaintheformationofSMBHseeds.Inthesinglestellarprogenitormodel,massivePopIIIstars(M>260M)collapsedintotheseedsofSMBHs.Proponentsofthistheoryarguethattheinitialmassfunction(IMF)oftherststarswaslikelytopheavy,leadingtohighermass17stars(e.g.Brommetal.,2002),butlargeuncertaintiesinthePopIIIIMFstillexist(e.g.Glover&Abel,2008;Turketal.,2009),andthelowmassofPopIIIremnants(˘100M)wouldhavedicultygrowingtolargermassesunlesstheyformedveryearlyandwerenotejectedfromtheirhostgalaxies(Tanaka&Haiman,2009).Asecondtheory(Begelman&Rees,1978;Gurkanetal.,2006)isthatSMBHseedsformthroughmergersofmultiplestellarremnants.Thetheoryholdsthataprimordialproto-stellarcloudcanreachhighdensitiesbeforefragmenting,resultinginseveralmassivestarsformingclosetogether.Theremnantsofthesestarsthenmergehierarchically,creatingaSMBHseedofmass103104M.Theseseedsaremoremassivethanthoseinthesinglestellarprogenitormodel,andthuswouldhaveaneasiertimegrowingtohighmasses.Onceagain,however,themassfunctionoftherststarsisnotwellunderstoodenoughtopredictwhetherthisscenariocanexplainallSMBHs.Thirdly,ithasbeenposited(e.g,Begelmanetal.,2006;Bromm&Loeb,2003)thatSMBHseedscouldformthrougha`directcollapse'scenario,inwhichaprimordialcloudwithmass104105McollapsesdirectlyintoanSMBHwithoutformingstars.ThedicultyinthisscenarioisthatordinarilysuchacloudwouldformH2,whichwouldbeabletocoolthegas,makingitunstabletoJeansfragmentation.IfsomemechanismexistedtopreventtheformationofH2,however,acloudmightbestableenoughagainstfragmentationtocollapseintoasinglemassiveobject.Possiblesuppressionmechanismsincludeahalowithvirialtemperature>104KorastrongUVbackgroundcapableofdisassociatingH2(Dijkstraetal.,2008).Alternately,ithasbeenproposed(Shlosmanetal.,1989;Begelmanetal.,2006)thatgravitationalinstabilitieswithinlowangularmomentumgascouldconcentrateenoughmaterialinoneplacetomakethedirectcollapsescenarioplausible.Finally,theseedsofSMBHscouldhavebeenprimordialblackholesproducedbyavarietyofprocessesintheearlyuniverse(reviewedinCarr,2003).Suchprimordialblackholescouldhavemassesofupto105M(Khlopovetal.,2005),butthereisstillmuchuncertaintyinhow(orif)suchblackholesformedandwhattheirinitialmasseswouldhavebeen.Alargepopulationofprimordialblackholeswouldhaveobservableconsequence,causinggravitationallensingordisruptingstellarorbits,therebyhelpingtoconstrainthecontributionofprimordialblackholestoSMBHformation.181.4TheCoolingFlowProblemandPrecipitationRegulatedAGNFeedbackSincethedawnofX-rayobservations(e.g.Feltenetal.,1966;Bridle&Feldman,1972),ithasbeenapparentthatgalaxyclustersareemittingcopiousamountsofX-rays.Forthemajorityofclusters,thisradiationshouldbesucienttocooltheclustercoreonatimescalemuchshorterthantheageofthecluster(e.g.Leaetal.,1973;Mitchelletal.,1976;Edgeetal.,1992).Intheory,thisshouldleadtohundredsofsolarmassesofgascoolingperyear(Fabian,1994),whichwouldbeexpectedtoaccumulateinthecore,possiblyformingstars,andleadingtoaslowowofgastowardsthecenterofthecluster.Instead,galaxyclustersshowlittleevidenceofcoldgas(Peterson&Fabian,2006)andhavelowratesofstarformation(O'Deaetal.,2010).Itwouldthereforeappearthatthegasisradiatingstronglybutnotcooling.Thisisthecruxofthecoolingowproblem.Ifthegasisnotcooling,someadditionalheatsourcemustexistthatisabletomaintainthethermalequilibriumofthegasoverlongperiodsoftime.Severalmechanismshavebeenproposed,butonlyAGNfeedbackseemsenergeticenoughtocountercoolinglosses.AGNfeedbackisinprinciplepowerfulenoughtobalancecooling(seeMcNamara&Nulsen,2007,2012,forareview)buthowthefeedbackandthecoolingcouplearenotwellunderstood.Recently,evidencehasgrownfora`precipitation-regulated'modelofAGNfeedbackingalaxyclusters(Voitetal.,2015b).TheICMissubjecttobothheatingandcoolingprocessesandthereforemaybethermallyunstable(Field,1965,andsubsequentpapers),meaningthatcoolerregionsmaycoolfasterthantheyarebeingheated,causingcoldcloudsto`condense'outoftheICM.IfthesecloudsareaccretedbytheSMBHintheBCG,afeedbackloopmaybeestablished.ColdclumpswillformastheICMcools,triggeringAGNfeedbackthatwillreheatthecluster,prohibitingfurthercondensation.Thiscyclecouldintheorymaintainthermalbalanceintheclustercore,assumingthattheAGNfeedbackcancoupletotheICM.Simulationsofgalaxyclusterssupportthispictureofprecipitation-regulatedfeedback.AnalysisofthethermalstabilityoftheICM(McCourtetal.,2012)indicatethatcondensationisexpectedtooccurun-dercertainconditions,producingthecoldgasneededtopowertheAGN.SimulationsofAGNfeedback(Li&Bryan,2014a,b;Lietal.,2015)showthatAGNfeedbacktriggeredbycoldgasaccretioncanpreventacoolingowandproducesimulatedclusterswithpropertiesthatagreewithobservations.ThisdissertationwillfurtherexplorethesusceptibilityoftheICMtothermalinstabilityandthecouplingofAGNfeedbacktotheICM,withtheaimofapplyingtheseresultstothecoolingowproblem.191.5PlanofThisDissertation
Inthisdissertation,IexplorethephysicalprocessesthatregulatethestateoftheICM,withaparticularfocusontheroleofprecipitation-triggeredAGNfeedback.Chapter2providesadiscussionoftheliteraturerelatedtoAGNfeedbackingalaxyclustersandincludesreviewsofthecoolingowproblemfromanobservationalandtheoreticalstandpoint,simulationsofgalaxyclusters,theprecipitationtriggeringtheoryoffeedbackregulation,andsimulationsofAGNfeedback.Chapter3presentsoriginalresearchonthedevelopmentofthermalinstabilityandtheproductionofmultiphasegasingalaxyclusters.FurtheroriginalresearchonthetopicofmodelingAGNfeedbackingalaxyclustersispresentedinchapter4.Finally,Chapter5presentsthisworkwithinthebroadercontextofastrophysics,discussesunansweredquestionsintheeld,andoersavenuesforfutureresearch.ThesimulationsinthisworkwereperformedusingtheEnzohydrodynamicscode,whichisdescribedinAppendixA.AppendiciesBandCcontaindetailsabouttheimplementationandsetupofthesimulationsdiscussedinthiswork.AppendixDcontainsworkonPopIIIandlow-metallicitystarformation.ThisworkwascompletedwhileIwasagraduatestudentatMichiganStateUniversity,butwhichdoesnotrelatetothemainfocusofmydissertationwork.202LiteratureReview
2.1Introduction
Thissectionpresentsanoverviewofthehistoricalliteraturerelatingtotheprecipitation-regulatedtheoryofAGNfeedbackintheICM.Thissectionwillattempttopresentresultsinapedagogicalfashionandfocusesonhistoricalworks.Morerecentstudiesandmyownresearcharediscussedinsubsequentchapters.Section2.2presentsevidencefromearlyX-rayobservationsofgalaxyclustersthatindicatesthattheICMinmanyclustersshouldbecoolingrapidly.Suchcoolingshouldleadtoanaccumulationofcoldgasandotherobservableconsequences|however,highresolutionX-raydatapresentedinSection2.3dierfromthepredictionsofthecoolingowmodel.TheoriesthatexplaintheobservationsbyallowingforcoolinggastoremainundetectedarediscussedinSection2.3.1butareultimatelyunconvincing,indicatingthatsomeheatsourcemustbalanceradiativecooling.Section2.2presentsseveralproposedheatingmechanisms.ThemostplausiblecandidateisAGNheating,butthiscanonlyexplaintheobservationsiftheheatingis1.)stronglytiedtothecoolingrateoftheICMand2.)distributedthroughouttheclustercore.Therstconditioncanbesatisedwithtriggeringbytheaccretionofcoldgasproducedviathermalinstability,andthesecondthroughavarietyofcouplingmechanisms.Section2.5arguesthatthermalinstabilityleadingtoamultiphasemediumisplausibleintheICM,andSection2.8describeshowAGNfeedbackcanbedispersed.Alloftheseprocessescanbewrappedintoafulltheoryofprecipitation-regulatedAGNfeedback,whichispresentedandadvocatedinSection2.9.
2.2TheCoolingFlowProblem
AsdiscussedinSection1.2,galaxyclustersarelledwithahot(107108K,diuse(ne˘104102cm3)plasmacalledtheintraclustermedium(ICM;Fabian,1994).Roughlyhalfofgalaxyclustersareclassiedas`cool-core'clusters,inwhichtheICMisgenerallysphericalandundisturbed,thedensitydistributioncentrallypeaked,andthetemperaturecentrallydecreasing.Theorysuggeststhatalthoughtheagesoftheseclustersarelarge(severalGyr),thetimefortheICMintheircorestoradiateawayitsthermalenergyiscomparativelyshort(tensorhundredsofMyr).Thisshouldleadtoanaccumulationofcoldgasinthecore,fuellingstarformationandresultinginpeakedX-rayemission.Sincethiscoolingshouldleadtoaninwardowofgas,thisphenomenonisknownasa`coolingow'.Attemperaturesabove107K,theplasmaisfullyionized,meaningthatthedominantradiativeemission21mechanismisBremsstrahlung,orfree-free,emission.ForaplasmawithelectronandionnumberdensitiesneandniandtemperatureT,theemissivityperunitvolumeis L=2ˇkBT3m25ˇe63hmec3Z2nenigB(2.1)=(1:41027ergcm3s1K1=2)Z2nenigB(2.2)wheremistheaverageparticlemass,Zistheaveragenuclearnumber,andgBistheGauntfactor,whichisoforderunityandaccountsforquantumeects.Thecoolingtimescaleisdenedasthetimethatitwouldtaketheplasmatoradiateawayitsthermalenergyatitscurrentcoolingrate,andisgivenbytcool=3nkBT2L:(2.3)whereListhevolumetriccoolingrate.Whileobviouslyinexact(sincethecoolingrateistemperaturedependent),thecoolingtimecanbeusedtoestimatethetimescaleoverwhichcoolingisimportant,orhowlongitwouldtakeforcoolingtoalterthetemperatureoftheplasmabyasignicantamount.Fabian(1994)summarizestheexpectedevolutionofacoolingowinacool-coreclusterwithanidealized,sphericalprole.Thegasinsuchaclusterwouldbeexpectedtobeinhydrostaticequilibrium.OnecandeneacoolingradiusRcoolinwhichthecoolingtimeislessthantheageoftheuniverse,ortcool(Rcool)<H10(2.4)whereH0istheHubbleconstant.Althoughthegaswithinthisradiuswillbecooling,itmuststillsupporttheweightofthegasoutsideofRcool,implyingthatthepressuremustrise.Ifthegasiscooling,thismeansthatthedensitymustbeincreasing,whichcanonlybeachievedthroughacompressiveowofcoolinggastowardsthecenter.Evenifadiabaticheatingpreventstheinowinggasfromdecreasingintemperatureatrst,gasinthecenteroftheowwillradiateawayitsenergyandcoolcatastrophically.Assumingthatalloftheradiatedenergyescapesthecluster(validfortheopticallythinICMplasma),theclusterluminosityisrelatedtothemasscoolingrate(themassofgascoolinginagiventime)byLcool=52_MmkBT(2.5)whereLcoolisthetotalluminosityand_Misthemasscoolingrate.Thefactorof5=2isduetothethermal22energyofthegas(3/2kBT)andthedecreaseingasvolume(kBT).Fortypicalclusterpropertiesandluminosities(discussedinthenextsection),thevalueof_Mmaybegreaterthan100Myr1.Thisgascouldeitheraccumulateascoldclumpsinthecentralregionorfuelstarformation.Whateverthefateofthegas,thecoolingowtheorypredictsthatalargevolumeofcoldmassshouldaccumulateinthecentersofcool-coreclusters.Observations,however,donotndthistobethecase.2.3ObservationalEvidence(orlackthereof)forCoolingFlowsTherstX-rayobservations(forexample,theGeiger-counter-in-a-rocketobservationsofByrametal.,1966)revealedstrongextragalacticX-raysourcesthatwerelateridentied(Cavaliereetal.,1971)asgalaxyclusterswithX-rayluminositiesof10431045ergs1.MoredetailedobservationsbyX-rayobservationalsatelliteslikeUhuru(Giacconietal.,1972,1974;Formanetal.,1978)andHEAO-1(Formanetal.,1978)foundmoreX-rayclustersandprovidedfurtherconstraintsontheirluminosity.ForamoredetailedreviewofthehistoryofX-rayobservationsofclusters,seethereviewbySarazin(1988).Atrst,theemissionmechanismforclusterX-rayswasnotknown,thoughseveraltheorieswereproposed.Some(e.g.,Katz,1976;Fabianetal.,1976)favoredmodelsinwhichalargenumberofX-raypointsourcesproducedtheemission.Others(e.g.,Brecher&Burbidge,1972;Bridle&Feldman,1972)proposedmodelswhereincosmicrayswereresponsiblefortheX-rays.Finally,itwastheorizedthattheemissionmightbecomingfromBremsstrahlunginahotplasma(Feltenetal.,1966;Leaetal.,1973).Bremsstrahlungemissionbecamethemostconvincingexplanationfollowingthedetectionof7KeVX-raysfromheavilyionizedFe(Mitchelletal.,1976).TheFeemissioncouldonlyhavebeenproducedinahotgas(>107K),whichsetalowerlimitontheICMtemperatureandmadetheBremsstahlungemissionmodelmorecompelling.GiventhelargeX-rayluminositiesobserved,Equation2.5suggestsmasscoolingratesofseveralhundredSolarmassesperyear.Leaetal.(1973)andFabian(1994)foundthatthisisthecaseformanyoftheX-raybrightestclusters.Edgeetal.(1992)foundthatbetween70%and90%ofobservedclustershadcentralcoolingtimes<H10.Evenifthecoolingtimethresholdwasreducedtotcool<H10=2,themajorityofgalaxyclusterswerestillobservedtohostcoolingows.Thatthefractionwassohighimpliedthatthisstrongcoolingwasprobablynotarecentortransientphenomena,butwasinsteadapersistentandcommonfeature.ThelaunchofnewX-rayobservatoriesliketheChandraX-rayObservatory(Weisskopfetal.,2000)andtheXMM-NewtonObservatory(Jansenetal.,2001)allowedfordeeperspectroscopicinvestigation.Petersonetal.(2001)wastherststudytouseXMM-Newtonforstudyinggalaxyclusters(specicallyAbell1835),butfoundmuchlessemissionfromgasatcoldtemperaturesthanwasexpected.Atthevery23leastthishintedthatthecoolingowprocesswasmorecomplicatedthanoriginallythought,andoeredthepossibilitythatsomeunknownheatsourcewaspreventingthegasfromcoolingastheorized.Similarstudiesinthesameyear(Tamuraetal.,2001a,b;Kaastraetal.,2001)foundnoevidenceofgascolderthanaround1/3ofthemaximumtemperatureinthecoresofX-raybrightclusters.Petersonetal.(2003),usingasampleof14galaxyclustersobservedwithXMM-Newton,conrmedthatcoolingowclustersdonotseemtoaccumulatemuchcoldgas.BythetimeofthePeterson&Fabian(2006)review,itwasestablishedthatthepredictionsofthestandardcoolingowmodeldidnotmatchobservations.242.3.1ProposedSolutionstotheCoolingFlowProblem
Withthehighresolutionspectra,itbecameclearthatthebehaviorofgasincoolingowclusterswassignicantlydierentthanoriginallythought.SeveralmechanismswereproposedtoexplainthelackofsoftX-rayemissionfromgasatlessthan1/3ofthemaximumtemperature.Thesemechanismsincluded1.)absorptionofsoftX-raysbyinterveningmaterial,2.)non-radiativecoolingofgasbelowtheobservedcuto,3.)reductionofthecoldgasbystarformation,and4.)distributedheatingthatpreventedgasfromcoolingcompletely.Earlymodels(Johnstoneetal.,1992)attemptedtoexplaintheabsenceofsoftX-raysviaabsorptionbymaterialintheclustercores.Ifthiswerethecase,however,theabsorptionshouldhavebeenseeninotherlightsourcesfromthecentersofclusters.ObservationsofthejetinM87andthePerseuscluster(Bohringeretal.,2002;Churazovetal.,2003)didnotseemuchevidenceforabsorptionofsoftX-rays.AsecondpossibilityforthelackofX-rayswasthatgasbelowthecutowascoolingwithoutradiatinginX-rays.Inthisscenario,coolinggaswasmixedwithcoldgas,whichthenradiatedtheenergyawayintheopticalorUV.Alternatively,thecoolinggascouldhavetransferreditsenergytothecoldgasviaconduction.Fabianetal.(2002)examinedsuchascenarioandfounditplausible,butitdoesnotexplaintheultimatefateofthecoldgasandisstilldiculttoreconcilewiththehighresolutionX-rayspectra.Athirdpossibilityoeredwasthatthecoolinggaswasturningintostars.Thiswouldbetheexpectedfateofcoldgastrappedinapotentialwellsuchasaclustercore.However,thecoolinggasshouldstillemitsoftX-rays.Moreimportantly,thestarformationrateinclusterswasestimatedtobeatleastanorderofmagnitudebelowthemasscoolingrate(Crawfordetal.,1999;Donahueetal.,2000).Indeed,mostBCGshostlowratesofstarformation,implyingthatthepredictedcoolingowgaswasnotbeingturnedintostars.Modelsincludingmodiedstellarinitialmassfunctions(IMFs)werealsoproposed(Prestwichetal.,1997),butwouldhaverequirednostarstoformabove0.2M,inconsistentwiththeoryandobservation(e.g.Salpeter,1955).
2.3.2ProposedHeatingSources
Finally,itwasproposedthatthereasonforalackofobservationsofgascoolingbelow1/3ofthemaximumtemperaturemightbethatsomeheatingsourceexistedthatpreventedthegasfromcoolingbelowthattemperature.Whilephysicallyplausible,suchaheatsourcewouldneedtofulllanumberofimportantrequirements.First,theheatingneededtobedistributedovertheentireclustercore.Thisisdiculttodosincethecoolingrateisstronglydensity-dependentwhileheatingratesgenerallyarenot.Second,heatingwouldneedtobemoreorlessconstantoverseveralGyr,atleastwhenaveragedoverperiodsof108years25(theminimumobservedcoolingtime).Third,theheatingratewouldneedtobecloselycoupledtothecoolingrateinordertopreventtheclusterfromoverorunderheating.Variousstudieshaveproposedthermalconduction,turbulentdecay,mergers,supernova,andAGNasheatsourcesincool-coreclusters.Theevidenceforandagainsteachoftheseissummarizedbelow.2.3.2.1Conduction
BecauseasthecoolingcoresofgalaxyclustersaresurroundedbythehotgasoftheICM,thermalconductionhasbeeninvokedasapossiblesolutionoratleastacontributingfactortothecoolingowproblem(e.g.Zakamska&Narayan,2003;Voitetal.,2008).TheionizedplasmaoftheICMisexpectedtoconductheatviaSpitzerconduction(Spitzer,1962),givenbyFourier'sLaw~j=S~rT(2.6)where~jistheheatux,istheconductivitycoecient,and~rTisthetemperaturegradient.MakingtheassumptionsdescribedinSmithetal.(2013),theconductivitycoecientisgivenbyS=4:9107T5=2ergs1cm1K1(2.7)Observationsrevealthatwhileconductionmaywellhaveaneectoncoolingows(Voigtetal.,2002;Voit,2011),itisnotpowerfulenoughtopreventcoolinginallcases(Voigt&Fabian,2004).Moreimpor-tantly,conductivebalanceisanelytunedandunstableequilibrium(Bregman&David,1988;Voitetal.,2015b).Ifaclusterishotterthantheprolespeciedbyconductivebalance,conductionwillheatthecorefasterthanitcancool,drivingtheclustertoisothermality.Ifthecoreiscoolingfasterthanconductioncanstabilizeit,thecorewillcoolcatastrophically,leadingtotheclassiccoolingow.SimulationssuchasSmithetal.(2013)havefoundthatconductionmaybeimportantinhotterclustersbutisnotsucienttoaectthethermalstructureofcool-coreclusters.
2.3.2.2TurbulenceandMergers
Severalobservations(e.g.,Inogamov&Sunyaev,2003;Zhuravlevaetal.,2014,2015)havenotedsomelevelofturbulenceintheICM.Thisturbulentmotionwouldbeexpectedtodecayintothermalenergy,heatingthegasintheclustercore.AsanalyzedinZhuravlevaetal.(2014),turbulentheatingcouldbeofthesameorderofmagnitudeasheatlossesthroughouttheclustercoreandcouldthusformasolutiontothecoolingowproblem.26Currently,thestatisticsofturbulenceinthebroaderclusterpopulationisnotknownandthedrivingforceisnotwellunderstood.Mergerscouldinprincipledriveturbulence(Valdarnini,2006;Markevitch&Vikhlinin,2007;Burnsetal.,2008;ZuHoneetal.,2010),butmostcool-coreclustersdonotappeartohaveundergonemajormergersinthelastseveralGyr.Furthermore,itisdiculttoseehowmomentumfromamergercouldpenetratedownintotheclustercore.Othermethodsfordrivingturbulenceincludesupernova,AGN,andconvection.Inshort,whileturbulencemaywellbeimportantfortransferringenergytotheICM,someadditionalprocessisneededtoproducetheenergyintherstplace.2.3.2.3Supernovae
Supernovaeareanotherphenomenathatcouldintheorydepositheatintothecentersofcool-coreclusters(Voit&Bryan,2001),butobservationalconstraintslimittheircontributiontothecoolingowproblem.BCGstendtohavestarformationratesof10sofsolarmassesperyear,onlyasmallfractionofwhichgoesintomassivesupernovaprogenitors.SimulationssuchasDuboisetal.(2010)andSkoryetal.(2013)haveincludedormodelledsupernovaheatingbutndittobemorethananorderofmagnitudeweakerthanneededtopreventacoolingcatastrophe.Infact,Skoryetal.(2013)ndsthatevenwhenthesupernovaeciencyisturnedupto10xtheexpectedpower,itisstillnotenoughtopreventthegasfromcooling.Althoughsupernovafeedbackmaybemoreimportantonthegalaxyscale(Voitetal.,2015a),itcannotbyitselfpreventacoolingow.Additionally,supernovadepositmetals(whichincreasethecoolingrate)inthesameplacethattheydepositenergy,furtherdecreasingtheireectiveness).2.3.2.4AGN
Asmentionedearlier,AGNarecurrentlyconsideredtobethemostlikelysolutiontothecoolingowproblem.AGNareobservedtoexistinthevastmajorityofcool-coreclusters,asevidencedbyemissionfromjetsandlargeradiobubbles.Estimatesoftheenergyneededtoinatethebubbles(e.g.Churazovetal.,2003)indicatethatAGNareeasilypowerfulenoughtobalancecooling,providedtheirenergycanbeecientlytransferredtotheICMandthattherateofAGNfeedbackcanbecoupledtotheICMcooling.AfulldescriptionofAGNfeedbackwouldrequiregeneralrelativity,magnetohydrodynamics,andradiativetransfer.Duetothiscomplexity,thephysicsofAGNfeedbackremainuncertain.InorderforAGNfeedbacktoregulatethethermalstructureoftheICMandprevent(orgreatlyreduce)acoolingow,twoprocessesmustbeunderstood:1)howthecoolingoftheICMcantriggerAGNfeedbackand2)howenergyfromtheAGNisdistributedthroughouttheclustercore.Anumberoftriggeringmethodshavebeenproposed,includingaccretionofhotgasfromtheambientmedium(\BondiAccretion";Bondi&Hoyle,1944)andaccretionofcoldgas(Pizzolato&Soker,2005).TheAGNmaytransferenergytothesurrounding27mediumthroughanumberofprocessessuchasinatingcavities(Churazovetal.,2001),cosmicrays,drivingturbulence(Ruszkowski&Oh,2010),anddredgingcoldgasfromthecore(Meeceetal.,2016).ThenextsectionwilldiscusspossibleAGNtriggeringmethods,withanemphasisontheprecipitationtheory.ThefollowingsectionwilldiscusshowtheAGNcanreturnenergytoitssurroundings.2.4AGNTriggering:Hotvs.ColdGas
AsoutlinedinChapter1,AGNarepoweredbytheaccretionofmaterialontoanSMBH.IftheSMBHisaccretingmaterialatarate_M,thepoweroftheoutowisexpectedtoscalewiththerestmassenergyoftheaccretionas_E=_Mc2(2.8)whereisaneciencyfactorthatincludesthefractionofaccretingmaterialthatultimatelyreachestheSMBH(ratherthanbeingejectedbyoutows),theeciencyofconvertingrestmassintoenergy,andthefractionoffeedbackenergythatcouplestotheICM.TwoprincipalscenarioshavebeenproposedfortriggeringAGNfeedbackingalaxyclusters:1.)accretionofhotgasfromtheambientmediumor2.)accretionofcloudsofcold,densegas.EachscenarioresultsingasbeingchanneledtowardstheSMBHandproducingoutows,buttheydierinhowtheycoupletheaccretionratetothebulkpropertiesoftheICM,whichmayhaveimportantconsequencesfortheeectsofAGNheatingandcycling.UnderstandingwhatsetstheAGNaccretionrate(andthereforejetpower)iscriticaltodecidingwhetherAGNcansolvethecoolingowproblem.2.4.1HotMode(Bondi-like)Accretion
AstheSMBHismovingthrough(orjustsittingin)theICM,itwouldbeexpectedtoaccretematerialthatcomeswithinitsgravitationalradius.IntheBondi-Hoyle-Lyttletonaccretionscheme(Hoyle&Lyttleton,1939;Bondi&Hoyle,1944)1,theaccretingobjectistakentomovethroughaninnitegascloudandaccretematterasitgoes.InBondiaccretion,theaccretingobjectistakentobestationarywithinthecloud,leadingtoasteadyaccretionow.1Hoyle&Lyttleton(1939)isaninterestingreadfromahistoricalperspective.ThefocusofthearticleistosuggestthatchangesintheSolaraccretionratecouldintheoryleadtochangesintheSolarluminosityandcauseiceagesorwarmperiods.Basically,thearticlesuggeststhatiftheaccretionratescalesasˆ=v3,smallchangesinthedensityoftheISMorintherelativesvelocityofthesunwiththeISMcouldpotentiallyleadtolargechangesintheaccretionrate.Ifthekineticenergyoftheaccretedmaterialwereconvertedintoradiation,thechangesintheEarth'stemperaturecouldbeexplained.Thepaperdoesnotgosofartoarguethatthisisactuallythecase,butdoessuggestthatSolaraccretionshouldbetakenintoaccountinfutureclimatestudies.Althoughthistheorydoesnotseemtohavegainedanytraction(andIwouldassumehasbeeninvalidatedbythediscoveryoftheheliosphereandSolarwinds),thepaperlivedontoserveadierentpurposeandnowformsaspartofthefoundationofaccretiontheory.28Bondiaccretiongivesanaccretionrate_M=2ˇG2M2BHˆ1c3
s;1(2.9)whereˆ1andcs;1arethedensityandsoundspeedoftheambientmediumfarfromtheSMBH.TheassumptionofBondiaccretionisthattheaccretionowissteadyandhasacharacteristicradiusRB=GMBHc2
s;1(2.10)whichisaround50pcfortypicalSMBHsingalaxyclusters.TheBondiaccretionratecoupleslinearlytotheICMdensity(andthereforetemperatureforasubsonicow),meaningthatacoldergaswillhaveahigheraccretionrate.Bondiaccretionissimple,anditisunlikelythatmanyoftheassumptionsbackingthetheorywillbesatisedinpractice.Accretionisalmostcertainlynotasteady-stateow,andtheICMislikelytobemixedandturbulent,ratherthanhomogeneousandstatic.Further,outowsmeanthataccretionisunlikelytobesphericalwithintheBondiradius.Duetotheweakdependenceongastemperature,Bondiaccretionwouldlikelyleadtoamore-or-lesssteadyaccretionrate,ratherthanthehighlyvariableratesthatareobserved.2.4.2ColdModeAccretion
In`coldmode'accretiontheory(e.g.Pizzolato&Soker,2005),theSMBHprimarilyaccretesgasthroughthestochasticaccretionofcoldcloudsfromtheICM.ThistheorypresupposesthattheICMhasamultiphasetemperaturestructure,withcold,denseclumpsembeddedwithinahot,lowdensityplasma.StochasticaccretioncouldnaturallyexplaintherapidvariabilityofobservedAGNandmaybettercoupletheaccretionratetothetemperatureoftheICM.Intuitively,theamountofcoldgaswillbelinkedtotheaveragetemperatureoftheICM,withacolder,denserICMhostingmorecoldgas.ForadeeperunderstandingofhowconditionsintheICMcangenerateamultiphasemediumwithcoldgas,andtoestimatewhetherthecoldmodeaccretionrateprovidesaplausibleaccretionratetothecoolingowproblem,wemustdelveintotheexcitingrealmofthermalinstabilitytheory.
2.5ThermalInstability
ThedensityandtemperaturestructuresoftheICMareshapedbytheunderlyinggravitationalpotential,hydrostaticpressure,magneticelds,turbulence,andheattransportprocesses.Inparticular,thelastof29thesemaydrivechangesinthethermodynamicstructureoftheICMandcanstronglyinduceorinhibitthegrowthofsubstructure.Ifthebalancebetweenheatingandcoolingissuchthatdensityandtemperatureperturbationsareisobaricallyamplied,thegasissaidtobethermallyunstable,andmaydevelopamulti-phasestructureinwhichregionsofhotandcoldgascoexistatconstantpressure.ThequestionaddressedinthissectioniswhetherthermalinstabilityislikelytoaecttheICMand,ifso,inwhatmanner.ThissectiondiscussesthehistoryofthermalinstabilitytheoryasappliedtotheICM.ThermalinstabilityintheICMhasbeenstudiedusingseveraltechniques,eachwithitsownstrengthsandweaknesses.Theoreticalargumentsdeliveradeepunderstandingoftheprocessesunderlyingthermalinstabilityandmaybeapplicableoverawiderangeofenvironments,buttheresultsobtainedstronglydependontheunderlyingassumptionsandsimplicationsemployed.Simulationsareabletofollowtheevolutionofgasunderconditionsnotamenabletoanalyticstudy,butarestilllimitedbytheinitialconditionsandincludedphysics.Additionally,theresultsofasimulationaredependentontheinterpretationoftheresearcher,whomustinterpretthedatathroughthelensoftheirownknowledgeandprejudice.Observationscanintheoryprovidetheultimatetruthaboutthestateofclusters,butdonotalwaysrevealwhatphysicalprocessesareimportantorhowtheICMreachestheobservedstate.Further,currentobservationsareonlycapableofrevealingselectcomponentsofgalaxyclusters,meaningthatfurtherknowledgeisnecessarytocompletethepictureandllinthegaps.ThissectionsummarizesthehistoricandcurrenttheoreticalunderstandingofthermalinstabilityintheICM.Theoreticianstendtotreatthermalinstabilitybyrunningaperturbationanalysisonaninitiallystatic(oratleaststeady-state)setup.Thegrowthofperturbationsisfollowedinthelinearregime.WhilethemediastudiedbytheoryareveryapproximatetothetruestateoftheICM,theyrevealimportanttruthsaboutthecriteriaforthermalinstability.Additionally,theyshowhowthesecriteriamaybemodiedbyotherphysicalprocessessuchasthermalconduction,magneticelds,rotation,expansion,anddensitystratication.2.5.1Field1965
GeorgeField's1965treatiseonthermalinstability(Field,1965)isoftentakenasastartingpointinthestudyofastrophysicalmultiphasemedia.Whilepreviousauthors(FieldcitesParker,1953;Zanstra,1955a,b)hadstudiedthermalinstabilityinastrophysicalmedia,theseauthorshadincorrectlyderivedthecriterionforthermalinstability,leadingtoerroneousresults.ThecorrectcriterionwasderivedbyWeymann(1960),althoughthesignicancewasnotrealizedatthetime.Foramediuminthermalequilibriumwhereaperturbationisadded,suchthatsomethermodynamicvariableAiskeptconstant,Field(1965)givesthe30criterionas@@SA>0(2.11)whereSistheentropyandisthenetheatlossfunction(heatlostperunitmassperunittime)givenby(ˆ;T)=(ˆ;T)(ˆ;T)(2.12)whereisthecoolingrate(thermalenergyperunitmassperunittime)andistheheatingcontribution.Therefore,intheequilibriumstate(ˆ0;T0)=0(2.13)whereˆ0andT0representthedensityandtemperatureoftheunperturbedstates,respectively.Dependingonwhichthermodynamicvariableisheldconstant,theresultinginstabilitycriteriabecome@@Tˆ<0(Isochoric)@@TP=@@Tˆˆ0T0@@ˆT<0(Isobaric)@@TS=@@Tˆ+11ˆ0T0@@ˆT<0(Isentropic)Theisochoriccaseisnotveryinteresting,asconstant-densityperturbationsleadtopressuredierenceswhichwouldcausethestatetobeoutofequilibrium.Itistheisobariccasewhichismostinterestingwhenstudyingcondensationmodes.Isentropicperturbationsmayleadtoorresultfromsoundwaves.Assoundwavesmaybeampliedordampedinagaseousmedium,thestudyofisentropicperturbationsmayalsobeofinterest.Field(1965)goesontodiscussthegrowthrateofaperturbationofforma(~r;t)=a1exp(nt+i~k~r)(2.14)toamediuminthermalequilibriumwhereaisdensity,velocity,temperature,orsomeotherpropertyofthegas.Here,itisassumedthata1˝a0,theequilibriumvalueofa,meaningthattheperturbationmaybetreatedinthelinearregime.Thegrowthratenoftheperturbationiscontrolledbythethermalpropertiesofthemedium.BysubstitutingtheperturbationsintotheEulerequations,Field(1965)arrivesatacubicequationforn.Positiveroots(realorimaginaryrootswithpositiverealcomponents)willcorrespondtogrowingperturbations,whilenegativerootswillcorrespondtodampedperturbations.Thecubicequationallows31threeroots.Twooftherootscorrespondtosoundwaves,inwhichtemperatureanddensityvary.Ifnispositive,themediumisunstabletoisentropicdisturbances,andthesoundwaveswillbeamplied.Forthethirdroot,temperatureanddensityvaryoutofphasewithoneanother,leadingtonooverallchangeinpressureastheperturbationevolves.Thisisthecondensationmodewhichmayleadtotheformationofamultiphasemedium.Ifthermalconductionisimportantinthemedium,perturbationsbelowsomecriticalwavelengthwillbestabilizedbyconductionbeforetheycangrow.Forlowconductivity,thegrowthrateofthecondensa-tionmoderiseswithwave-numberandplateaustowardssmallerwavelengths.Asconductivityrises,thegrowthratewillfallforwave-numbersabovethecriticalvalue,meaningthatperturbationgrowthwillbeastrongerfunctionofwave-numberandwillpeaksomewherearoundthewavelengthofwavescorrespondingtoisothermalperturbations.Forhighconductivity,thegrowthratefunctionwillbemorestronglypeakedasafunctionofwave-number,sinceconductionwillbebetterabletostabilizethemediumatsmallscales.Addingamagneticeldtothemedium(Field,1965,specicallyconsidersauniformeld)hasanumberofeectsonthecondensationprocess.First,theeldintroducestwoadditionalwaverootsinthegrowthrateequation,correspondingtoAlfvenwaves.Oneofthenewrootscorrespondstoawavemovingper-pendiculartotheeld,whichisstable.Theotherrootcorrespondstoawavemovingalongtheeldandcangrow.Secondly,themagneticeldinhibitscondensationperpendiculartotheeld,ascompressionofthegasmustovercomethepressureofthemagneticeld.Condensationalongthedirectionoftheeldisunaected.Lastly,conductionperpendiculartothedirectionoftheeldwillbereduced.Thisanisotropicconductionmeansthatamagneticeldcanstabilizethemediumperpendiculartotheeldbutwillnotaectcondensationparalleltotheeld.Inthecaseofarotatingmediumwherethecentrifugalforceiscomparabletogravity,radialcondensationwillbeinhibitedforperturbationsaboveacriticalwavelength.Forsmallerperturbationsandforazimuthalcondensation,thegrowthrateisunaected.Thesituationsmentionedaboveareallworkedoutforthecaseofauniform,staticmedium.Foragravitationallystratiedmedium,thegrowthofperturbationsmightbeexpectedtovarywhenthescaleoftheperturbationsissimilartothatofthescaleheight,suchthattheeectsofdensityand/ortemperaturestraticationwouldbenon-negligible.Onemajorchangeisthatsmallscaleperturbationsmaygrowevenwhenthermalconductivityislarge.Intheuniformcase,temperaturevariationsarenecessarytobalancedensityperturbations(keepingpressureconstant).Ifconductionwaslarge,thetemperaturevariationswouldbewipedoutandtheperturbationswouldnotgrow.Inthestratiedcase,pressurevariationscanbebalancedbygravity,allowingperturbationstogroweveniftemperaturevariationsareerased.Thepaperalsoconsidersthecaseofcondensationinanexpandingmedium.Thisisrelevantfornovae32andotherexplosionsbutnotforthepresentwork2.Finally,thepaperconcludeswithapplicationsofthethermalinstabilitytheorytotheChromosphere,thesolarcorona,thegalactichalo,planetarynebula,andtheformationofgalaxyclusters.Whilemanyofthesetopicshavesincebeenrevisedbyothers,thispaperisinterestingasahistoricaltreatmentofthesetopics.
2.5.2Defouw1970
Defouw(1970)explorestheconnectionbetweenthermalinstabilityandconvectiveinstabilityandreachestheconclusionthatanythermallyunstableatmospherewillalsobeconvectivlyunstable.ThoughField(1965)studiesthecaseofagravitationallystratiedatmosphere,onlyverticalmotionofthegasisconsidered,leadingtooverallexpansionorcontractionoftheatmosphere.Inaddition,theaccelerationtermisnotconsideredinthemotionofthegas.Therefore,convectiondoesnotoccurinthatwork.However,Defouw(1970)arguesthatinarealisticmedium,thermalinstabilityismorelikelytoleadtoconvectionthantoexpansionorcontraction.Themanneroftheonsetofconvectioninathermallyunstablemediumdependsonthetemperaturegradientofthegas.Asanasidefromthispaper,considerthecasewithoutexternalheatingorradiativecooling.Thetemperaturegradientcanbeeithersuperadiabatic,adiabatic,orsubadiabatic.Consideramediumingravitationalequilibriumwheretemperatureanddensityincreasewithdepth.Now,imaginethatasmallparcelofuidisgivenaslightupwardnudge.Sincethepressuredecreasesastheparcelascends,theparcelexpands,cools,anddecreasesindensity.Adiabatic:Whentheparcelexpandstoequilibratewiththereducedpressure,thedensityofmaterialintheparcelisthesameasthedensityofthesurroundingmedium.Theparcelexperiencesnoforce.Themediumisstabletoconvection.Superadiabatic:Theparcelexpandstoequilibrate,butwhenthepressureisequaltothesurroundingpressure,thedensityisnowlowerthanthesurroundingdensity.Infact,thedensitycontrastisevengreaterthanitwasatthelowerheight,meaningthattheparcelcontinuestoriseupwards.Thegasisnowunstabletoconvection.Subadiabatic:Theparcelexpandsuntilthepressurematchesthenewsurroundingpressure,butthedensityendsupbeinggreaterthanthesurroundingpressure.Theparcelexperiencesarestoringforceandmustdescend.Itthenproceedstooscillatearounditsoriginalposition.Themediumisoverstabletoconvection.2Also,thissectionreferencesStephenHawkingwhenhewasstillagraduatestudentatCambridge.Hisnameisspelledwrong(Stephan)33Therefore,thesuperadiabaticcaseisunconditionallyunstable,whilethesubadiabaticcaseisoverlystable.Now,considerthecasewheretheheatlossfunctionisnon-zeroandthegasisthermallyunstable.Inthesubadiabaticcase,theparcelwillriseasdescribedabove.Asitrises,however,itwillremaindenserthanitssurroundings.Thismeansthattheparcelwillcoolfasterthanitssurroundings,sincethemediumisunstable.Therefore,thedensitycontrastwillincreaseastheparticlerisesandfalls,meaningthatwhenitreturnstoitsoriginalheight,theparcelwillbedenserthanitssurroundingsandwilldescendfurtherthanitascended.Theparcelwillthendescendtoalevelwhereitisunder-dense,atwhichpointthethermalinstabilitywillleadtoitrisingtoahigherheightthanonitspreviousascent.Thus,theoscillationabouttheoriginalpointwillgrowwithtime,leadingtoconvection.Ofcourse,theparcelapproachisaveryidealizedmethodwhichignoressomeimportantfactorssuchasviscosity,conduction,andmagneticelds.RepeatingtheanalysisusingtheBoussinesqframeworkandincludingviscosityandconductionconrmstheresultthatthereisalwayssomesortofthermal-convectiveinstability,unlesstheperturbationsaresmallenoughthatconductionorviscositycanstabilizethem.Thus,theresultisbasicallythesameasinField(1965).Addinginrotationandmagneticeldsallowsthepossibilityofmonotonicinstability345insteadofoscillatoryinstability.Theauthorpointsoutthatwhiletheoscillatorymotionmayinfactresultinsomeenergytransferbetweenuidlayers,dampeningtheoscillationsandweakeningthevalidityoftheresultsderivedinthiswork,monotonicinstabilitywouldbehardertodamp.
2.5.3Nulsen1986
Nulsen(1986)takestheideasofthermalinstabilitytheoryandappliesthemtothecontextofcoolingowsingalaxyclusters.ThispapercitestheworkofCowieetal.(1980),whichaspartofacasestudyofthePerseusclusterdescribeshowconvectioncouldpotentiallystabilizethegasinacoolingowagainstthermalinstability.Nulsen(1986)howeverarguesthatconvectiveblobswouldbedisrupted,makingthiseectunimportant.Theconclusionofthisworkisthatallbutthemostoverdenseperturbationswillbedisruptedlongbeforetheyreachtheirequilibriumposition.Astheblobisdisrupted,itsvelocityrelativetoitssurroundingswilldecrease,andtheowwillbecomeeectivelyco-moving.Thepapergoesontostudythedynamicsofthecoolingowallowingformassdropoutandconvection.Theconclusionseemstobethatcoldmasswouldbedepositedoveralargerangeofradii,notonlynear3Iassumethatthismeansexponentialgrowth,ratherthanacomplexgrowthrate.4Searchingforthistermonlineshowsthatitnearlyalwaysoccursinthecontextofuiddynamics,mostofthetimeinreferencetothispaper.5MarkVoitconrmsthatthismeansexponentialgrowth.34thecenterwheretheowissteady.Convectionwouldnotstabilizetheow,sinceparcelsofgaswouldbedisruptedbeforereachingequilibrium.However,thisanalysisdoesnotseemtoincludeacentralheatingtermandthuspredictsmorecondensationthanisobservedbymorerecentstudies.Balbus&Soker(1989)pointoutthatthispaperdealsmorewiththefateofnonlinearperturbationswithinacoolingow,ratherthanwiththethermalstabilityoftheowintherstplace.Nevertheless,thispaperisimportantforconsideringtheevolutionofperturbationsinadynamicratherthaninastaticmedium.
2.5.4Malagoli1987
Malagolietal.(1987)reconsiderstheanalysisofDefouw(1970),Cowieetal.(1980),andNulsen(1986)andappliestheiranalysistoanumericalmodelofM87.Theyconcludethatitisincorrecttotreatoverdensitiesasfree-oatingblobs,andinsteadagreewiththeresultsofDefouw(1970)inthatthebuoyancywillpushthemediumtoover-stability.Theynotethatconductionwillmakethegasstableonscalesofafewkpc,butatlargerscalesthegasshouldbeover-stable,leadingtocondensation.2.5.5Balbus1988
Balbus(1988)re-analysestheconditionsunderwhichgasincoolingowscanbethermallyunstableusingaLagrangianapproachratherthananEulerianapproach.Indoingthis,theyarriveatanumberofsurprisingconclusions:1.Radialthermalinstabilitydoesnotoccurinsphericalsystemsinhydrostaticequilibrium(contrastingwithmanyearlierworks.)2.Localisobaricinstability(e.g.condensation)canonlyoccurifperturbationsaregrowingonsomethingsimilartothethermaltimescale.3.BuildingoofDefouw(1970),gaswhichisstableagainstconvectionaccordingtotheSchwartzchildcriteriamayinfactbeconvectivelyunstableifthegasiscoolingradiatively.Themainresultsofthisworkarethatradialthermalinstabilitydoesnotoccurandthatthegrowthrateofnon-radialmodeswouldbeweak{thus,weshouldnotexpectthegasincoolingowstobeunstable.Inanalysingtheradialinstability,Balbus(1988)ndsthattheonlygrowthofradialmodesoccursonthesamescaleasthegrowthofthecoolingow,andisthusindistinguishable.Animportantcaveattothisstudyisthattheanalysisdoesnotincludetheeectsofthebackgroundmotionofthemedium.ThisistakenintoaccountinBalbus&Soker(1989),whichmodiestheseresults.352.5.6Balbus&Soker1989
Balbus&Soker(1989)presentsare-analysisofBalbus(1988)aswellasamoregeneralstudyofthermalinstabilityincoolingowsusingLagrangiandynamics.Intheintroduction,Balbus&Soker(1989)pointsoutanerrorintheanalysisofDefouw(1970).Whilethatstudytookintoaccountazimuthalstructure(thelackofwhichwascitedasadecitofField(1965),itdidnotallowforazimuthaldynamics.Movementintheazimuthaldirectioncouldinfactbeimportant,sincehavingsmallPsmalldoesnotnecessarilymeanthatr(P)issmall.Theseazimuthaldisplacementscouldcontributetothermalinstabilitymuchmorethantheoscillatingradialmodes.Thispapermakesapointofdierentiatingbetweenlocalandglobalinstabilities.PastanalysisusingEulerianplanewavesstudiedtheglobalinstabilityofthegastocondensationoroscillation.Thistypeofanalysisdoesnotnecessarilycapturealocalinstabilitythatmightexistevenwhenthemediumisgloballystable.UsingaLagrangianframeworkratherthananEulerianonecouldthereforegiveamoreaccuratepictureoftheconditionsforthermalinstability.InexaminingDefouw(1970),Balbus&Soker(1989)contendsthatinastaticmediumwheretheheat-lossratepergramisconstantthroughout,thermalandconvectiveinstabilitymustbelinked;thatis,ifamediumisthermallyunstableitwillnecessarilybeconvectivelyunstable,andifitisthermallystableitwillalsobestabletoconvection.Thus,over-stabilitywillnotbeafactorsincethetwoinstabilityconditionsdonotoccurindependently.If,ontheotherhand,thebackgroundisdynamic,thermalequilibriumisnotenforced,ortheheat-lossfunctionexplicitlydependsonposition(asinthecaseofanAGN),thetwocriterioncanoccurindependently,andover-stabilitymaybeimportant.
2.6ObservationsofMultiphaseGas
Asthepreceedingsectionmakesclear,thermalinstabilityisdiculttostudyfromapurelytheoreticalstandpoint,andsignicantuncertaintyremainsindeterminingthesusceptabilityoftheICMtotheformationofamultiphasemedium.Withtheadventofspace-basedobservatories,ithasbecomefeasibletoconductmulti-wavelengthsearchesformultiphasegasinclusters.Althoughtheamountsofcoldgasobservedaremuchlowerthanwhatispredictedbythecoolingowmodel,itisclearthatinsomeclustersatleast,coldgasisforming.IthasbeenhypothesizedthatthisgasmayoriginatefromthermalinstabilityandcouldbedrivingAGNfeedback.Thissectiondiscussessomeoftheobservationalevidenceformultiphasegasingalaxyclusters.362.6.1Cavagnolo2008
Cavagnoloetal.(2008)examinestheentropyproles,Hemission,andradioemissionfromalargenumberofgalaxyclusters.TheclusterdataistakenfromtheChandraarchives,andisnamedtheACCEPT(ArchiveofChandraClusterEntropyProleTables)sample.Foreachcluster,theauthortstheentropyproletotheformK(r)=K0+K100(r=100kpc)(2.15)WhereK(r)istheentropyasafunctionofradius,K0isthecentralentropy,andK100anddescribethepowerlawbehavioroftheentropyproleatlargeradii.Hisusedinthestudyasatracerofstarformationandcoldgas.WhiletheyassumethatstarformationistheprincipleproducerofH,theyallowthatthisisnotnecessarilythecase.AlthoughtheHvaluesaretakenfromdierentobservations,thestudydoesmakeaneorttomakethemconsistent.Moreimportantly,theirresultsonlydependonwhetherHwasdetected,ratherthantheexactamount.IntheACCEPTsample,64/110clustershaveonlyupperlimitsonHdetection.Theclustersthatdohavedetectionsalmostallhavecentralentropiesbelow30keVcm2,correspondingtoacoolingtime<1billionyears,whilethoseclusterswithupperlimitsliealmostentirelyabove30keVcm2.Thus,thepresenceofHisstronglycorrelatedwiththecentralentropyvalue,producingabimodalityofclustercoreproperties.Voitetal.(2015b)explainsthisasadividebetweenfeedback-regulatedcores(lowentropy,H)andconduction-regulatedcores(highentropy,noH).Thisstudydoesnotproposeanexplanationforthedichotomy.TheradioobservationswereusedtodeducewhetherornottheBCGhostedanAGN.OnthereasoningthatBCGsaremorelikelytoshowradioloudAGNthanothergalaxiesinacluster,radioemissionwastakenasevidenceofAGNactivity.Theradiodataalsoyieldsasplitinclusterproperties,thoughthedistinctionisnotasclearasintheHcase.Itisseenthatallclusterswithradioluminositiesabove1040erg/shavecentralentropiesbelow30keV,andthatallclusterswithcentralentropyabove30keVhaveradioluminositiesbelow1040erg/s.Theredoesnotappeartobeacorrelationbetweenradiopowerandentropy,however.SomeclusterswithHdonothaveradiodetections,andseveralwithradiodetectionsdonothaveH.ThiscouldsimplybeduetoepisodicorvariableAGNactivity.372.6.2Werner2010
Werneretal.(2010)describesobservationsofmultiphasegasinM876,oneofthebrightestcentralgalaxiesintheVirgoCluster.M87isagiantellipticalgalaxywithavisibleAGNjet.Theobservationsshowthatwhilesomeofthegasupliftedbythejetsisradiativelycooling,thebulkofthematerialremainshot,indicatingthatsomesortofheatingmechanismispreventingmostofthegasfromcooling.M87iswellstudied.Indierentobservations,itshowsmultiphasegasinsomesortoflamentarystructure,radioemission,andX-raycavities.ThispaperusesChandradatatoprobethestructureofthemultiphasegasathigherresolutions.TheX-rayobservationsshowthatthereiscoldgas(0.5keV)inM87inlamentsnearthejet.The1keVgasisspatiallycoincidentwithradioemissionandisthoughttollthecavities,whicharemoreorlessisothermal.2keVemissionisfairlysmoothandformsasphere.Thisgasissurroundedby3keVemissionoutsideoftheinnercore.The0.5keVgasisspatiallycoincidentwithHemissionnearthecore.Bothhavealamentarynature.Thereisalsoahorseshoe-shapedfeatureafewkpcfromthecenter.UsingtheChandradata,thegroupplotsapressureproleforthecore,whichshowsadiscontinuityoutsideofahighpressuresphere,indicatingsomeformofshock.AlloftheHisfoundinsideofthehighpressureregion.X-rayspectraindicatethatverylittleofthegasiscold,withamaximumcoolingrateof0.06Mperyear.TheX-rayarmsobservedinM87arerelativelysmoothandstraight,indicatingthatthegasisnotbeingdisruptedbyturbulentmotion.ThelamentsofHdonotshowpolarization,indicatingthatifmagneticeldsarepresent,theyaretangledonscalesbelow0.1kpc.WhileX-rayemittinggasisseenat0.5keV,noX-raygasisseenbelow0.5keV.Ifthegaswereinasteadystateandwerebeingheatedbyavolumeaveragedterm,heatingstrongenoughtokeepthe0.5keVgasfromcoolingwouldoverheatthesurroundinggas.Thus,theyauthorsconcludethattheheatingisnotduetoavolumeaveragedmechanism.Instead,theysuggestthatthecoolergasisheatedbymixingwithwarmergas.6TheintroductiontothispapercontainssomegoodinformationonM87aswellaspossibleAGNheatingmechanisms.382.7SimulationsofMultiphaseGasFormation
Analyticalstudiesofthermalinstabilityarelimitedinthattheycanonlydescribethelinearregimewhereperturbationsaresmall,andgenerallyonlyapplytocontrivedsetupsthatmaynotapproximatereality.Simulationsoeranalternativeapproach,allowingresearcherstofollowtheevolutionofgasbehavioroverlongperiodsoftimeintothenon-linearregime.Thissectiondiscussesseveralmilestonestudiesofmultiphasegasinsimulations.Avarietyoftypesofsimulationshavebeendevelopedtostudythermalinstability.2Dsimulationsarequicktorun,butdonotnecessarilycapturephenomenalikeconvectionormagneticelds,whichareinherently3Dprocesses.Further,thermalinstabilityrequiresaheatingandacoolingterm,theformsofwhichmustbeassumed.Thesimulationsdiscussedintheseworks(andinChapter3)generallyassumeheatingfunctionsthatareabletobalancecoolingintheunperturbedcase.MostofthesimulationsfollowtheoutlineofField(1965)andaddsmallperturbationstoanidealized,equilibriumsetup,thoughsomedealwithlargeroverdensities.
2.7.1Malagoli1990
Malagolietal.(1990)isanearlypaperthatuses2Dsimulationstostudythegrowthandevolutionofperturbationsinionizedgas.Thisworkisafollowuptoanearlierpaper(Malagolietal.,1987)inwhichtheauthorsanalyticallystudiedthegrowthofisobaricperturbationsinthelinearregime.Inthispaper,theauthorssimulatetheevolutionofasphericaloverdensityinordertodeterminethesetofconditionsforwhichtheblobcondensesandunderwhichitisshreddedbytheKelvin-Helmholtz(Thomson,1871)ortheRaleigh-Taylor(Rayleigh,1882)instabilities.Thisworkconsidersanisothermalsetupwithgasinhydrostaticequilibriumwiththestrengthofthecoolingrateasanindependentparameterinordertostudytheeectofcoolingrateonthegrowthofinstabilities.Aconstantheatingtermequaltothe(negative)coolingintheunperturbedstatewasimposed.ThesimulationswereperformedonaCartesianmeshwiththeperturbationonthezaxisandsymmetryimposedacrossthataxis.Theperturbationwasxedataninitialoverdensityofˆ=0:0067.Thermalconductionwasneglected,thoughtheymentionedthatitsinclusiondoesnotsignicantlyaecttheirresults.Forthisstudy,dierentvaluesforthecoolingratewereconsidered.Intherstcase,thebuoyancytimescaleismuchshorterthanthecoolingtime,whileinthesecondcasethetwoaresimilar.Forbothcases,theblobisshreddedbyinstabilitiesasitoscillates.Therunwithmorecoolingstaystogetherlonger,butbothareultimatelytornapart.392.7.2McCourt2012
McCourtetal.(2012)(whichformstheprecursorfortheworkinChapter3)studiedthepossibilityofformingamultiphasemediumina2DsimulationoftheICM.ThesesimulationsapproximatedtheICMasastratiedmedium,constructedinaplanarfashionwithgravitypullingtowardsthemid-plane.Inthissimulation,gaswasinitializedinhydrostaticequilibriumwithacoolingratethatscaledasn2T1=2.Theheatingratewasthenconstructedsuchthattheinstantaneousheatingrateatagivenheightwasequaltotheaveragecoolingrateatthatheight.Althoughsomewhatarticial,observationsshowthatheatinginrealclustersroughlybalancescoolingatallheights,meaningthatthisheatingmethodisnotaterribleapproximation.Thestudystartsoutwithananalyticstabilityanalysisfortheplasmaandconcludesthatperturbationsshoulddevelopwhenthecoolingtime(tcool)islessthanthefreefalltime(t)atagivenheight.Whentheratioislessthanone,overdenseclumpscancoolfasterthantheyfall,resultingincoldblobsoccurringinplace.Whentheratioisgreaterthanone,blobsofgasareabletofallasignicantdistancebeforecooling,leadingtoconvectionratherthancondensation.Thesimulationsinthispaperlargelyconrmtheresultsoftheanalyticstudy.Theyndthatthekeyparametercontrollingthedevelopmentofmultiphasegasisthetimescaleratiotcool=t.Thepaperalsoconsidersadditionalphysicslikemagneticeldsandconduction,thoughthesearenotfoundtohavesignicanteectsunlessthestrengthofconductionisveryhigh.Notably,thisstudydoesnottreatheatingorcoolinginthevicinityofthemid-plane,astheysaythattheirheatingroutineisnotanaccuratemeasureoffeedbacknearthecenterofclusters.ThisisaddressedinMeeceetal.(2015),whichincludesheatingandcoolingnearthemid-planeandndsthatitmakesanimportantdierenceintheresults.
2.7.3Sharma2012
Sharmaetal.(2012b)explorestheformationofmultiphasegasina3Dsphericalenvironmentwhereheatingbalancescoolingatallradii.Theythentestasecondheatingmethodinwhichheatingisproportionaltothemassuxthroughsomeinnerregion.Thispaperndsthatmultiphasegasisabletoformwhentheratioofcoolingtimetofree-falltimeislessthan10.ThesimulationsinSharmaetal.(2012b)arecarriedoutinsphericalcoordinates,withlogarithmicbinsinradiusandequallyspacedbinsintheazimuthal(˚)andpolar()directions.Inmostcases,theyuse1˚bin,buttheydoatestwith32bins.ThegasisinitializedtohaveapowerlawentropyproleandresideinanNFWhalo.Theconcentrationparameterofthehaloisxedat3.3,butthemassofthehaloisvariedbetweenruns.Gasisinitiallysmoothandinhydrostaticequilibriumwithdensityperturbations40uptoˆ=ˆ=0:3addedon.GascoolsaccordingtothecurvegiveninSutherland&Dopita(1993)andisheatedsuchthatheatingisotropicallybalancescoolingatallradii.Afewrunsexperimentwithvariationsintheheatingfunction.Theyalsotryaheatingfunctionwhichisproportionaltotheuxofgasthroughtheinnershellandheatsasapowerlawfunctionofradius.First,theauthorsstudytheeectsofheatingonthemassaccretionratebycarryingoutsimulationsofclustermasshaloswithandwithoutanidealizedheatingterm.Withoutheating,allsimulations(exceptforonewithveryhighinitialentropy)showstrong,steadymassaccretionratesofbetween100and1000M/yearasintheclassiccoolingowmodel.Withheating,theaccretionrateismuchlowerinallsimulations.Thetworunswithlowinitialentropydoshowstrongaccretionatrst,butitsoonsettlesdowntobetween1and10M/year.Therunwithmiddleentropyneverdevelopsastrongcoolingow.Lateron,theyshowthattheaccretionrateandtheportionoftimethataclusterspendswithagivenaccretionrateisnotsensitivetothetypeofheatingordetailsoftheimplementation.Theyalsocarryoutaseriesofrunswithdierenthalomassesbutsimilarentropyproles.Thelowermasshaloshavelowertemperatures(andthuslowertcool)butsimilarfreefalltimessincetheNFWproleisself-similar.Therefore,theyaremorelikelytoformmultiphasegas.Thesimulationsshowthathotmodeaccretionislowerforthelowermassruns,butthattherateofcoldmodeaccretionissimilar.Thisleadstosimilaramountsoffeedbackinlowermasshalos,whichmeansthatthegasgetsheatedmore.Thiscausesthetcool=tratiotoriseabove10.Haloswithdierentmassestendedtoendupwiththesamecoreentropy.Sincelowermasshaloshadlowercoretemperatures,thedensitywashigher,meaningthatthecoolcoreinlowermasshaloswaslarger.Verylittleemissionisseenfromgaswithtemperaturesbelowabout1=3ofthevirialtemperature.Forlargerhalos,feedbackneedstobefairlyecient,butatlowmassesitdoesnot,sincethecoldgasaccretionrateissimilarbutlessfeedbackisneededtoheatthegas.Thus,AGNaremorenecessaryforheatinglargegroupsandclustersthanforgalactic-scalehalos.2.7.4Joung2012
Joungetal.(2012)looksatthepossibilityofcondensationinthegalactichaloandwhethertheresultingcoldgascouldprovidefuelforstarformationingalaxies.Simulationsandgalaxyevolutionmodelsrevealthatmoststarforminggalaxiesseemtoberunningoutoffueltoformstars(seeforexampleSommer-Larsenetal.,2003).Asstarformationappearstobehappeningataroughlysteadyrate,itishypothesizedthatthesupplyofcoldgasneededforstarformationisbeingconstantlyreplenished.Onetheoryisthatcoldgascondensesoutofthehotgalacticcoronaandrainsdownonthegalaxy,providingthenecessaryfueltoformstars.Thispaperinvestigateswhethersuchacondensation/accretionprocessisfeasiblebysimulatingthegrowth41ofthermalinstabilitiesinthegalacticcorona.ThisworkusesEnzotostudytheevolutionofanisolatedoverdensityinastratied,isothermalcolumnofgas.Initially,thegasininhydrostaticequilibrium(HSE)withgravitypullingtowardsthecenter.GravityistreatedasastaticpotentialappropriateforaMilkyWaylikegalaxy.Asphericaloverdensityisaddedatagivenheightabovethecenter.Thesizeoftheoverdensityisvariedfromˆ=ˆ=1:01to100:0.Gasisallowedtocoolradiatively,withsomesimulationsusingametal-freecoolingfunctionandothersusingametallicityof0.3Solar.Thegasisradiativelyheatedwithusingaphotoelectricmodelwhichissaidtodependweaklyondensity.Withoutcooling,theblobessentiallybobsupanddown,ingoodagreementwiththeory.Whencoolingisturnedon,theyndthatcloudsareabletocoolifthetcool=taccelratioislessthan1.Otherwise,thecloudgetsshreddedbytheKelvin-Helmholtzinstabilitybeforeitcanradiateitsenergy.Thekeyresultisthatsmallperturbationsinthegalacticcoronashouldbeessentiallystable,butlargeoverdensitiescouldconceivablycoolandfall,providingfuelforstarformation.2.7.5Scannapieco2012
Scannapiecoetal.(2012)describestheformationofmultiphasegasinasimulationwithdriventurbulence.WhilethisworkfocusesontheISMinstarforminggalaxies,theresultsarerelevanttoclusters.Inanutshell,star-forminggalaxiesareobservedtohostlargeoutowsthatarenotaccuratelymodeledbysimulations.Onereasonforthiscouldbethatturbulentmotiononscalesbelowtheresolutionofmostsimulationsisimportantforseparatingthegasintoamultiphasemedium.Thisprocessisexploredinthiswork.ThesimulationsdescribedinthispaperwererunusingFLASH.Gaswasinitializedina128128512boxcovering4gravitationalscaleheightsinthezdirections.GravityisthesameasinMcCourtetal.(2012).Thegasinitiallyhasanexponentialdensityproleandisinhydrostaticequilibrium.TheyuseatabulatedcoolingrateforSolarmetallicitygasandincludeprimordialchemistry.Turbulentforcingisaddedtoexactlybalanceradiativecoolingateachtime-step.Intheducialrun,thegasremainsstableforapproximately3dynamicaltimesbeforeseparatingintoahotandacoldphase.Sincethehotgasisunabletocoolyetisbeingvolumetricallyheatedbydecayingturbulence,thereisarunawaypushtohighertemperatures,intheorydrivinganoutow.Inthiscase,theturbulentvelocityreachesroughly˙=45km/sbeforetheformationofamultiphasemedium.Anotherrunisconductedwithalowerinitialdensityandgravitationalscale.Here,theturbulentvelocityonlyreaches˙=20km/s,andthemediumremainsstableforover40dynamicaltimes.Whenthedensityisincreasedsothat˙=29km/s,thefractionofcoldgasincreases,butthemediumstillremainsstable.In42arunwithstrongerturbulencethanintheducialrun,amultiphasemediumagaindevelops.Thus,theyinferthatsomewherearound˙=35km/sthereisatransitionfromstabilitytoinstability.Finally,theyndthatincreasingtheresolutionleadstothesamebehaviorwithregardstoformingamultiphasemedium,butthattheoutowrateisincreasedabit.ThediscussionsectionofthisworkarguesthattheresultscouldwellbeapplicabletotheISM.Whilethesesimulationsdonotincluderotation,selfgravity,magneticelds,orconduction,itisarguedthatthesewouldbeunlikelytohavealargeimpactunderISMconditions.Starformationandfeedbackcouldprovideanadditionalheatsource,buttheauthorarguesthatobservationsofoutowscouldindicatethatthedrivingmechanismismorespreadout.Itisalsopossiblethatmassivestarsaredrivingturbulence,whichisinturncreatingtheoutowsobserved.
2.7.6Banerjee2014
Banerjee&Sharma(2014a)studieshowturbulentmixingcancoupleAGNfeedbacktotheintraclustermedium.TheynotethatthisworkissimilartoSharmaetal.(2010),exceptthatinsteadofaddingthejetenergytotheenergypartofthehydroequations,theyaddittothemomentumpartasturbulence.Thus,energyisinjectedasturbulencewhichlaterdecaysandheatsthegas.ThesimulationsareperformedwithMHDusingtheATHENAMHDcodeincludinganisotropicthermalconduction.Thestudyusestwoinitialsetups.Oneusesacubeofgaswithuniforminitialconditionsappropriateforaclustercore.Thesecondsetup(the`mixingsetup'astheyrefertoit)usestworegionswithdierentdensitiesinpressurebalance.Smalldensityperturbationsandturbulentforcingarethenaddedtothebox.CoolingiscarriedoutusingthesamecoolingfunctionfromSharmaetal.(2010).Ratherthanputtinginexplicitheating,theyaddforcedturbulencewhichiscalibratedtobalancelossesfromcooling.Asthegasevolves,itgoesthroughtwophases.First,thegasevolvesasteadyturbulentstate.Second,thegasdevelopsmultiphasestructureduetocooling.Otherresults:
Changingtheboxsizechangesthedrivingscaleoftheturbulence.Thisinturnaectshowmuchmultiphasegasisformed.Magneticeldsmakethecondensedgasmorelamentary.Inthemixingruns,theheatingmechanismismixingbetweenthehotandcoolgasratherthanturbulentdecay.432.8AGNandEnergyTransfer
AGNcanprovideasolutiontothecoolingowproblemiftheenergygenerationratecanbecoupledtotheICMcoolingrateandthefeedbackenergycanbecoupledtotheICMinamannerthatosetscooling.ThelastsectionshowedthatcoldgascondensingoutoftheICMduetothermalinstabilitycanprovideafuelsourcefortheAGN.ThenextquestionthereforeiswhethertheAGNfeedbackenergycanbereturnedtotheICM.Foraheatingsourcetobeeectiveinosettingcooling,itmustsatisfyanumberofcriterion.First,thefeedbackenergymustbeoftherightmagnitudetobalancecoolinginthegas.Second,theheatmustbedistributedthroughouttheICMratherthanbeingdepositedveryclosetotheAGN.Third,thefeedbackratemustbeabletoadjustontimescalesshorterthanthecoolingtime.Fourth,thefeedbackcannotgenerateverystrongshocks,astheseareruledoutbyobservations.AGNarepowerful,buthowtheytransferenergytotheICMisstillunclear(seeMcNamara&Nulsen,2007,2012;Fabian,2012,forreviews).Intheory,therearemanywaysinwhichAGNheatingcansatisfythesecriterion.Thesemethodsinclude1.)Inatingbuoyantbubbles,2.)Shocksorsoundwaves,3.)Turbulence,and4.)Cosmicrays.Eachofthesemethods,alongwiththeobservationalevidence,isdiscussedbelow.2.8.1BuoyantBubbles
ThejetsfromAGNareknowntoinatemassivebubbles(e.g.Churazovetal.,2001)ofhotgasthatproceedtorisebuoyantlythroughtheICM.ThesebubblescanheattheICMinanumberofways.First,expandingthebubblesdoesPdVworkonthesurroundinggas,heatingitthroughcompression.Theserisingbubblescanthereforebeusedascalorimeters(Churazovetal.,2002)oftheAGNpower,byassumingthatthecumulativeoutputfromtheAGNoverthebubbleinationtimeisequaltotheenergyneededtoinatethebubble.Thebubbleinationtimescalecanbemeasuredfromtheriseofthebubbleduetobuoyancy.Risingbubbleswilleventuallybedisruptedbyuidinstabilities,whichcompletestheprocessoftransfer-ringtheirenergyintotheICM.Thus,theabilityofbubblestoheatgasthroughouttheICMwillbeinpartdeterminedbytheprocessesthatdisruptorpreventdisruptionofbubbles.PurehydrodynamiccalculationssuggestthatKelvin-Helmholtz(KH)andRayleigh-Taylor(RT)instabilitiesshouldshredbubblesrapidly,butobservations(Fabianetal.,2011)ndthatmanybubblesarelongerlivedthanexpected,indicatingthatadditionalphysicalprocessesarepreventingbubbledisruption.Viscosity(Reynoldsetal.,2005)andmagneticdraping(wheremagneticeldswraparoundarisingbubble;seeRuszkowskietal.,2007)couldintheorystabilizethebubblesandallowthemtopropagatetolargerradii.44InadditiontoPdVworkanddissipation,buoyantbubblescouldalsoprovidedistributedheatingthroughcosmicrayinjectionandbystirringturbulence,bothofwhicharediscussedinmoredetailbelow.Risingbubblesmayalsostabilizethecoolingowbydredginguplowentropygasintheirwake(Churazovetal.,2001),whichisthenreplacedwithhigherentropygasfromfurtherout.2.8.2CosmicRays
AGNcanalsoheattheICMinadistributedmannerthroughdiusionofcosmicraysfromhotbubbles.Cosmicrayscaneasilybegeneratedintherelativisticplasmawithinthebubbles(Sijackietal.,2008)andcandiuseoutintothesurroundingmedium.StudieslikeSijackietal.(2008)indicatethatcosmicrayscanprovidesignicantheating.Unfortunately,thecreationanddiusionofcosmicraysrequirescomplexplasmaphysicsandisdiculttostudy.
2.8.3Turbulence
RisingbubblescanstirturbulenceintheirwakethatcanheattheICMeitherthroughdissipationorbymixingcoldgaswithhot.Simulationsndthatrisingbubblescanindeedcreateasignicantamountofturbulence(Walgetal.,2013).ObservationssuchasZhuravlevaetal.(2014)showsignicantdensityuctuationsintheICMofclusterswithAGNthat,ifduetoturbulence,holdenoughenergytolargelyosetcooling.ThetruedriverofthesedensityuctuationsremainunknownandwillrequiremoredetailedinvestigationusingnextgenerationX-raytelescopeslikeAthena(Nandraetal.,2013).2.8.4ShocksandSoundWaves
HotandfastAGNoutowsareexpectedtointeractwiththeICM,producingshocksthatcandistributeen-ergymoreisotropicallythanthejetitself.Whilemostclustersdonotexhibitstrongshocks(McNamara&Nulsen,2007),weakshockslinkedtotheAGNarecommon(McNamaraetal.,2005).Theseshocksareroughlyspher-ical,mayextendfor100sofkpc,haveMachnumbersbetween1.2and1.7,andcanhaveenergiesofupto1061ergs.Manyclusters(suchasthePerseuscluster{seeFabianetal.,2006)showseveralweakshocksemanatingfromthecoreregion.WeakeroutburstscouldalsoproducesoundwavesthatcouldtravelthroughtheICManddepositheatthroughdissipation.ThisprocesshasbeenexploredinRuszkowskietal.(2004),whichtentativelyndsthatsoundwavesmaycarryenoughenergytoosetradiativecooling.Inreality,itislikelythatsomecombinationofallofthesemethodsisnecessaryfordistributingAGNheatingthroughouttheICM.Figures2.1and2.2showhowtheseprocessescanbeobservedusingdatafrom45Figure2.1HowAGNFeedbackectsClusterCores:TheAGNfeedbackprocessasseeninsimu-lationsperformedbyMeeceetal.(2016).Eachframeshowsdierentphysicalquantitiesforthesamesliceofthesimulation.TheAGNitselfislocatedatthecenteroftheslice.Formoredetailsonthesimulations,seeChapter4.Dierentfeedbackprocessesandeectsarehighlighted.A:AGNoutowsinatehot,lowdensitycavitieswithintheICM.B:Cavityinationproducesweak,roughlysphericalshocks.C:InteractionsbetweentheoutowandtheICMproduceturbulencethatcandissipateortransporthotgasawayfromthejetaxis.D:Risingbubblesdredgeupmetalsandlowentropygasintheirwake.Meeceetal.(2016).Inparticular,theuseofdierenceimaging(Figure2.2;inspiredbyZhuravlevaetal.,2016)candisentangledierentheatingprocessesandcanbeusedtoquantifytheirimportance.46Figure2.2AGNFeedbackEnergyTransferIllustreatedwithPerturbedQuantities:PerturbationsindierentquantitiesfortheslicedepictedinFigure2.1areshown.Theaveragequantitieshqiarethevolumeaveragedvaluesofqateachradius.Thistypeofperturbationanalysisisusefulfordisentanglingtheeectsofdierentenergytransportprocesses.A:Directinjectionofhotgasismostclearlyvisibleintemperatureperturbations,astheoutowsarelargelyisothermalandmuchhotterthantheirsurroundings.B:Subsonicmixingisisobaric.Thechurninggasisprominentinthetemperaturemapbutinvisibleinthepressuremap.C:Weakshockarelargelyisentropic.Shocksaremostvisibleinthedensityandpressuremapsbutinvisibleintheperturbedentropy.472.9Precipitation-RegulatedFeedback
Astheprecedingsectionsmakeclear,AGNfeedbackisenergeticenoughtobalanceradiativecooling,canbecomestronglycoupledtothecoolingpropertiesoftheICMthroughthermalinstability,andcandistributeenergythroughouttheICM.Thesefactsmotivatetheprecipitation-regulationtheoryoffeedbackdescribedbyVoitetal.(2015b).AsdiscussedinSection2.5,acoolerICMismoresusceptibletothermalinstability,whichcanleadtothecondensationofcoldclumpsofgas.Intheprecipitation-regulationmodel,thesecoldclumpsthen`precipitate'downtowardstheSMBH,triggeringtheAGN.TheAGNthenproducesfeedbackthatheatstheICM,stabilizingitagainstfurtherprecipitationbyreducingtheamountofcoldgasavailableforfuel.ThepoweroftheAGNthendropsuntiltheICMcoolsdown,andthecyclerepeats.Thus,precipitationandAGNfeedbackcoupleandmaintaintheICMinastateofthermalequilibrium.TheACCEPTsampleofCavagnoloetal.(2009)ndsthatclustersseemtohavean`entropyoor'ofaround30KeVcm2intheircores.Whenconvertedtoatcool=tratio,asdoneinVoit&Donahue(2015),thiscorrespondstoaoorofaroundtcool=t=10.ThisisinagreementwiththeresultsofSharmaetal.(2012b)andMeeceetal.(2015),whichndthataratiooftcool=t=10isathresholdfortherapidformationofmultiphasegas.Theratioofcoolingtimeandfree-falltimeisseenasastrongtriggerfortheformationofmultiphasegasinobservationsofgalaxyclusters,asdiscussedinVoit&Donahue(2015)andVoitetal.(2015b).Clusterwithhighcentralentropyandlargeratiosoftcool=tdonothavedetectedHemission,indicatingthatlittle,ifany,multiphasegasispresent.Asthetimescaleratiodrops,however,astrongrampupintheamountofmultiphasegaspresentisseen.Fromthisevidence,Voitetal.(2015b)oerstheassertionthattcool=t=10isaoorbelowwhichclustersandgalaxiescannotfall.Whenevertheratiodropsbelow10,precipitationwilldrivestrongfeedbackthatheatsthecluster,increasingthecoolingtimeandrestoringbalance.ThenatureofthefeedbackistakentobeAGNfeedbackinclusters,butmayincludesupernovaandstarformationinlowermasssystems(Voitetal.,2015a,c).Precipitationdrivenfeedbackandconductionworktoconstrainclusterentropyandcoolingtimeprolestoanarrowband,asseeninFigures1and2ofVoitetal.(2015b).Thisconstraintnaturallyexplainstheobservedbimodalityofclustersintocool-coreandnon-cool-coreclusters.Conductiveequilibrium(Voitetal.,2008)isanunstableequilibrium.Clustersbelowtheconductionbalancelinewillcoolfasterthanconductioncanheatthem,pushingthemdowntotheprecipitationline,wheretheywillbecomecool-coreclusterswith48tcool=tslightlyabove10.Clustersabovetheconductionlinewillbeheatedbyconductionuntiltheyareisothermal,makingthemnon-cool-coreclusters.Clustersmaytransitionfromcool-coretonon-cool-coreifandonlyiftheycanbepushedabovetheconductivebalanceline,eitherbystrongAGNoutburstsormajormergersthatdisruptthecool-core.Therestofthisdissertationcontainsoriginalresearchexaminingdierentfacetsoftheprecipitation-regulationmodel.Chapter3presentsadetailedanalysisofmultiphasegasformationintheICM,andChapter4discussesthecreationofsub-gridmodelsofAGNfeedbackthataredrivenbyprecipitationandcanlargelysolvethecool-coreproblem.Chapter5concludesbydiscussingunansweredquestionsandcomplicationswiththemodel.493GrowthandEvolutionofThermalInstabil-
itiesinIdealizedGalaxyClusterCores
3.1Introduction
1X-rayobservationsofgalaxyclustershaverevealedthattheradiativecoolingtimeofgasinmanyclustercoresismuchshorterthantheHubbletime.Ifradiativecoolingwereuncompensatedbyheating,thegaswouldradiateawayitsthermalenergy,causingcoolinggastoowtowardthecenterofthecluster.Thiswouldbeaclassicalcoolingow,inwhichtheaccumulatingcoldgaswouldbeobservableandwouldleadtostarformationratesof&100Myr1(seeFabian(1994)forareview).Instead,X-rayobservationsreveallittlegascoolingbelowX-rayemittingtemperatures(e.g.Petersonetal.,2003;Peterson&Fabian,2006)andobservedstar-formationratesthatareoneortwoordersofmagnitudelowerthanpredictedbytheclassiccooling-owmodel(O'Deaetal.,2010;McDonaldetal.,2011).Thus,anadditionalpro-cessorprocessesmustbeheatingtheICMtomaintainapproximatethermalequilibrium.Severalmech-anismshavebeenproposedandtestedthroughsimulations,includingenergyinjectionfromsupernovae(Nagaietal.,2007;Burnsetal.,2008;Skoryetal.,2013),conductionofheatfromoutsideofthecore(Voigtetal.,2002;Zakamska&Narayan,2003;Smithetal.,2013),heatingthroughmergers(Valdarnini,2006;Markevitch&Vikhlinin,2007;ZuHoneetal.,2010),dynamicalfrictionfromgalaxyclustermotion(Ruszkowski&Oh,2011;Kimetal.,2005),turbulentdissipation(Zhuravlevaetal.,2014),andfeed-backfromAGNoutbursts(reviewedbyMcNamara&Nulsen,2007),whichisthemechanismweexploreinthiswork.AGNfeedbackisattractivebecauseasimpleestimateshowsthatanaccretingsupermassiveblackhole(SMBH)canprovideenoughenergytoosetcooling.Forexample,a109MSMBHaccretingoverthelifetimeoftheuniverseandradiatingwithamass-energyconversioneciencyofaround10%wouldreleaseatotalof˘1062ergs,correspondingtoanaveragepoweroutputofaround1044ergspersecond|easilyenoughtoosetradiativecoolingifalargefractionofthatpowerisinjectedintotheICM(seeChurazovetal.(2002)forfurtherdiscussion).Theoreticalandobservationalstudiessupporttheconclusionthatmanycool-coreclustershostAGNwithenoughpowertobalancecooling(e.g.,McNamara&Nulsen,2007;Dunn&Fabian,2006;B^rzanetal.,2004)ifasignicantfractionoftheAGNenergyistransferedtotheICM.Nevertheless,1ThischapterwasoriginallypublishedinTheAstrophysicalJournal(Meeceetal.,2015).Ithasbeenreformattedforinclusionhere.Forinformationaboutcopyrightandreuse,seeAppendixE.50thedetailsoftheAGNfuelingprocessandfeedbackmodearenotfullyunderstood.IfSMBHaccretionistoexplainthermalregulationofthecore,thentheaccretionratemustbelinkedtothethermalpropertiesoftheICM.AspointedoutbyMcNamara&Nulsen(2007),ifthetime-averagedheatingrateexceedsthecoolingrate,thecorewillheatbeyondwhatisobserved,andifitisloweritwillfailtopreventgasfromcooling.Moreimportantly,theshortcoolingtimesobservedinmanyclustercoresrequiretheheatingmechanismtorespondonshorttimescales.Itisthereforedesirablethatheatingbecoupledtothecoolingrate,toensurethatfeedbackisabletobalancecoolingbothonshorttimescalesandoverthelifetimeofthecluster.TwoqualitativelydierentaccretionmodeshavebeendescribedintheliteratureandimplementedinnumericalsimulationsofAGNfeedback.MostimplementationsbasetheblackholeaccretionrateonthepropertiesoftheambienthotgasusingmodicationsoftheclassicBondi(1952)analysisofsmooth,adiabaticaccretion,whileothersrelyoncondensationandinfallofcoldcloudstofueltheblackhole(e.g.,Pizzolato&Soker,2005;Gasparietal.,2012b,a).TheanalysisofVoitetal.(2014)stronglysuggeststhatthelatter\coldfeedback"modeismoreimportant,becauseofauniversaloorobservedintheradialcooling-timeprolesofgalaxyclustersthatcorrespondstothepredictedthresholdforcondensationofcoldclouds(Sharmaetal.,2012a).\Coldmode"accretioncouldbefueledbycoldgascondensingoutoftheICMinresponsetothermalinstability.ThetransitionoftheICMfromahomogeneoustoahetrogeneous,multiphasestructurehasalonghistoryofinvestigationusingtheoreticalargumentsandsimulations.Fromatheoreticalstandpoint,Field(1965)studiedtheevolutionofsmallperturbationsincoolingplasmasanddescribedanisobariccondensationmode,inwhichvariationsintemperatureanddensitymaybeamplied.Defouw(1970)extendedthisanalysis,ndingthatthermalandconvectivestabilityaretightlycoupled,aconnectionfurtherexploredinBalbus&Soker(1989).Theproblemofthermalinstabilityinthecontextofcoolingowsinclusterswassubsequentlyconsideredbynumerousauthors(e.g.Cowieetal.,1980;Nulsen,1986;Malagolietal.,1987;Loewenstein,1990)whoconcludedthatthecoolingICMshouldindeedbesubjecttothermalinstability.However,furtheranalysisbyBalbus(1988)andBalbus&Soker(1989)usingaLagrangianframework(incontrasttotheEulerianapproachoftheearlierworks)indicatedthattheICMmightbelesssusceptibletothermalinstabilitythanpreviouslythought,especiallywithouttheinclusionofaheatingterm.Thesestudiesgenerallytakeastheirstartingpointanequilibriumorsteady-statecongurationofgasthatmaynotaccuratelycapturethebehaviorofthedynamicICM.Further,theoreticalstudiesareoftenincapableofdealingwithspatiallydependentheatingterms,suchaswouldbeexpectedfromstarformationandAGNfeedback.Thereisgrowingevidencesuggestingthatthedominantparametercontrollingthetransitiontoamul-tiphasestateandtheamountofcoldgasthatcondensesistheratioofgascoolingtime,tcool,tofreefall51time,t.Bothnumericalsimulationsofthermalinstability(McCourtetal.,2012)andobservationsofgalaxy-clustercores(Cavagnoloetal.,2008;Raertyetal.,2008;Voitetal.,2014)supportthisconclusion.Withoutgravitytorestoreequilibrium,multiphasestructurecandevelopwithinafewcoolingtimes,be-causecollisionalcoolingprocessesscalewiththesquareofthegasdensity,allowingdenserregionstocoolfasterthantheirsurroundings.Ifthemediumisinoverallthermalbalance,gasclumpsthataredenserthanaveragecoolandcondense,whileunderdenseregionsheatandexpandfasterthantheycancool.Inagravitationalpotential,however,buoyancycomplicatesthedevelopmentofthermalinstabilityandcaninhibitcondensation(Malagolietal.,1987;Balbus&Soker,1989).Ifthefreefalltimeisshorterthantheradiativecoolingtime,anoverdenseclumpcansinktoadenserlayerbeforeitcansignicantlycool.WhiletheoreticalstudiesprovideinsightintothegeneralphysicsbehindcondensationintheICM,theyarenecessarilylimitedbymodelassumptionsandcansaylittleaboutthefateofinstabilitiesthatenterthenonlinearregime.Inrecentdecades,hydrodynamicsimulationssuchasthoseofMalagolietal.(1990),McCourtetal.(2012),andLi&Bryan(2014b)haveexploredthedevelopmentofthermalinstabilityinastrophysicalenvironments.TheseworksdemonstratedthatcondensationcanindeedbeexpectedtooccurinenvironmentscomparabletotheICM,atalevelexceedingthepredictionsofBalbus&Soker(1989).CondensationhasbeenexploredintheidealizedsimulationsofMcCourtetal.(2012),whichshowthatthegrowthofthermalinstabilitiesissignicantlyinhibitediftcool=t&1.However,furtherstud-iesbySharmaetal.(2012b)havefoundthatinasphericalgeometry,multiphasegascanstillcondensewhenevertcool=t.10duetogeometriccompression(seeSingh&Sharma(2015)forfurtherdiscussion.)Gasparietal.(2012b)alsondsthataratioofaround10isrequiredfortheformationofcoldclumps.Alternately,recentworkbyLi&Bryan(2014b)ndsthatcondensationoccurswhentcool=tisbetween3and10.There,condensationisstimulatedbyinteractionsbetweentheICMandanAGNjet.ThejetentrainscoldgasfromneartheSMBH,pushingittolessdenseregions.Theclump'spositiveradialvelocitypreventsitfromreturningtoanequilibriumposition,andthegasrapidlycools.Finally,observationsbyVoit&Donahue(2015)ndthattheminimumvalueoftcool=tinclusterswithmultiphasegasintheformofHnebulaegenerallyliesbetween5and30.Inthispaper,weuseidealized2Dand3Dhydrodynamicsimulationstostudyhowtheonsetofconden-sationdependsontheratioofcoolingtimetofreefalltimeandwhythereappearstobeachangeinclustercorepropertiesaroundaratioof10.SectionD.2presentssimulationsbasedonMcCourtetal.(2012)inwhichweexploreawiderrangeofinitialconditions.Section4.3analyzeshowthermalinstabilitiesgrowinthesesimulationsandinvestigatehowthatgrowthdependsontheinitialconditions.SectionD.6relatesthisworktoprevioustheoreticalworkanddiscussesthevalidityoftheseresultsinthecontextofrealgalaxyclusters.Section4.5concludesbydiscussingfutureworkalongtheselines.523.2Method
Inthisstudy,weconsidersimulationsofidealizedclustercoreswithplanar,cylindrical,andsphericalge-ometriesin2and3dimensions.ThesimulationswerecarriedoutusingtheAMRHydrodynamicscodeEnzo2(Bryanetal.,2014).Unlessotherwisenoted,2Drunswereconductedona300x300cellgridwithnoadaptivemesh,and3Drunsemployeda1283cellrootgridwith2layersofadaptivemesh,withrenementbasedonoverdensity,densitygradient,andcoolingtime.Wedonotincludemagneticelds,conduction,ortheselfgravityofthegas.Thesimulationswereanalyzedusingtheyt3analysistoolkit(Turketal.,2011).3.2.1ProblemSetup
Wesetupthegasinoursimulationssubjecttotheconstraintofhydrostaticequilibrium(HSE)andan`iso-cooling'initialcondition,underwhichthetcool=tratioisuniformthroughoutthevolume.Additionally,werunanumberofsimulationsusinganisothermalinitialconditioninsteadoftheiso-coolingone.Thesetupdescribedinthissectionappliestoallgeometries,aslongasthedenitionoftheheightcoordinatezchangesaccordingly.Inplanargeometries,zisthedistancefromthemidplane,incylindricalgeometriesitisthedistancefromtheaxisofsymmetry,andinsphericalgeometriesitisthedistancefromtheorigin.WechooseascaleheightofzS=100kpc(roughlycorrespondingtoalargecluster),aboxsizeofRS=2zS,ascaletemperatureofTS=108K,andagravitationalaccelerationscalegS=kBTSmpzS(3.1)sothatthegravitationalpotentialenergyandthermalenergyareofsimilarmagnitudeatthescaleheightzS.Thecoolingtimeisgivenbytcool(n;T)Ej_Ej=32nkBTnenH(T)(3.2)whereEisthethermalenergyperunitvolumeandtheformofthecoolingfunction(T)istakenfromSarazin&White(1987)forgasofhalf-solarmetallicity.Thestandardnormalizationof(T)isusedforgaswithaniso-coolinginitialconditionoftcool=t=1,andweobtaininitialconditionscorrespondingtootherinitialvaluesoftcool=tbyadjustingthenormalizationofwhilekeepingthegasdensityandtemperatureprolesxed.Twotimescalescharacterizingtheinitialconditionswillbeusefulinouranalysisoftheonsetofthermalinstability.Oneisthecoolingtimetcool;Satonescaleheight(z=zS)atthebeginningofthesimulation.Theotheristhefreefalltimeatonescale2http://enzo-project.org/3http://yt-project.org/53height,t;S,whichstaysconstantthroughoutthesimulation.Ingeneral,thefreefalltimeofthegasatheightzist(z)=s2zg(z)(3.3)wherethegravitationalaccelerationdeningthepotentialwellisg(z)=gStanh(z=zS)(3.4)forRSzRSandisdirectedtowardeitherthemidplane,thesymmetryaxis,ortheorigin,dependingonthegeometryofthepotential.Therelativeconstancyofg(z)awayfromtheoriginismeanttomimictheinnerregionofasphericalgravitationalpotentialinwhichthemassdensityisproportionalto1=z.Forjzj˝zS,thetanhfunctionensurescontinuityofthepotential,whiletheparameterallowsadjustmentofitscuspiness.Tomitigateboundaryeects,weaddasmallbuerregionof0:3zSaroundtheoutsideofoursimulationvolumeandhaveg(z)decreaserapidlyinthatregion.WerestrictouranalysistotheregionzRs.Thesimulationswepresenthereuse=1:0,whichresultsinarelativelysmoothpotentialwithagradualsofteningnearthemidplane.FollowingtheworkofMcCourtetal.(2012),weimplementaheatingratethatexactlybalancestheaveragecoolingrateateachheight.Todothis,wesumthetotalamountofcoolingineachbinofz,dividebythetotalvolumeofthebin,andchangethesigntogetthevolumetricheatingrateatheightz.Whileclearlyidealized,thisheatingprescriptionensuresthatthegasremainsinoverallthermalbalance,inagreementwiththeobservedthermalbehaviorofclusters.ThevalidityofthisprescriptionisdiscussedinSection3.4.3.Foriso-coolinginitialconditions,theinitialtemperatureatzSisTS.Equations3.2and3.3relatedensitytotemperatureviatcoolt=32nkBT(T)nenprg(z)2z;(3.5)andtheHSEconditionforanidealgasiskBmpT(z)dˆdz+ˆ(z)dTdz=ˆ(z)g(z)(3.6)Combiningthesetwoexpressionsgivesthetemperaturederivativeforiso-coolinginitialconditionsintheformofanODE:dlnTdlnz=mpg(z)zkBT(z)+12dlngdlnz1dlndlnT21(3.7)54Figure3.1MultiphaseGasDensityandTemperatureCongurations:Theinitialtemperature(blue)anddensity(red)prolesusedinthisworkisshownforplanargeometry.Insimulationswithcylindricalandsphericalgeometries,thegasisisothermalbeyondz=2.Theiso-coolingsetup,withaconstantvalueoftcool=t,isshownwithasolidline.Theisothermalsetupisshownwithadashedline.Bothsetupsareinhydrostaticequilibriumandhavearatiooftcool=t=1:0at1scaleheight.Theseprolesareusedforallsimulations;runswithdierentinitialvaluesoftcool=tareachievedbyscaling(T)afterinitialization.WeintegratethisequationtondT(z)anddeterminethedensityfromtheiso-coolingcondition.Forcylindricalandplanarsimulations,gasoutsideofRSistakentobeisothermalandinHSE,withT(z)=T(RS).TheresultingdensityandtemperatureprolesareshowninFigureD.1.WeimposeatemperatureooratToor=5:0106K,asweassumethatgasbelowthattemperatureinevitablycoolsrapidlytomuchlowertemperature.Thedetailsofthegasowbelowthattemperatureoccuratnerresolutionsthanareemployedinourmodels,anddonotaecttheoverallcondensationrate.Finally,weaddrandomlygeneratedisobaricperturbationstothegaswithanRMSoverdensityof0.01andaatspectrumwithwavenumbersbetween2and20(withk=1correspondingtotheboxsize).Thesamerealizationofperturbationsisusedacrossallsimulationstoensureconsistency.Asthegasquicklysettlesintoaconvectivestate,thedetailsoftheinitialperturbationsaresoonforgotten.FigureD.1alsoshowsacomparisonbetweentheiso-coolingandisothermalsetups.Inbothsetups,theinitialdensityandtemperatureprolesdonotvarybymorethanafactorof4throughoutthevolume.Thedensityismoresharplypeakedintheisothermalcase,leadingtoashortercoolingtimeinthecenter.Consequently,thegrowthofthermalinstabilitiesintheisothermalcaseismoredependentonheightandtheinitialconditionsthanintheiso-coolingsetup.55Figure3.2ComparisonofMeece(2015)withMcCourt(2012):Slicesofgasdensityareshownfor2Dplanarsimulationswithinitialvaluesoftcool=tatonescaleheightof0.1,0.5,1.0,and5.0afterthesimulationhasevolvedforatimet=20tcool;S.Inthetoprow,thegasisinitiallyisothermal.Inthebottomrow,theinitialtimescaleratioisidenticalthroughouttheentireregion.Bothmodelsproducequalitativelysimilarresults.Intheisothermalcase,gasnearthemidplanehasashortercoolingtimethangasaboveascaleheight,leadingtoearliercondensationnearthemidplaneandthecreationofhotbubblesthatriseupthroughlayersthathavenotyetbeguntocondense.Inbothcases,condensationoccursnearthemidplaneinsimulationswithananinitialvalueoftcool=t=5atonesaleheight.563.3Results:TheGrowthofThermalInstabilities
3.3.1ValidationofMethod
WebeginbyconductingsimulationswithinitialconditionssimilartothoseinMcCourtetal.(2012)tocheckifourmodelproducesqualitativelysimilarresults.Oursetupdiersfromtheirsinanumberofminordetails,includingtheshapeofthegravitationalprole(oursislesscuspynearthemidplane)andtheformofthecoolingfunction(T).Moreimportantly,theregionnearthemidplanedoesnotreceivespecialtreatmentinoursimulations,whereasMcCourtetal.(2012)shutoheatingandcoolinginthisregionandexcludethemidplaneregionfromanalysis.Inspiteofthesedierences,weobtainqualitativelysimilarresultsintheregimetcool=t.1:0.Weseecloseagreementfortheisothermalsetupforhigherratios,andsomewhatmorecondensationisseenfortheiso-coolingcasefortcool=t&1:0.Figure3.2showsslicesofdensityin2DCartesiansimulationswithsimilarinitialconditionstothoseexploredbyMcCourtetal.(2012).ThetoprowofFigure3.2usesisothermalinitialconditionsdeterminedbytheinitialvalueoftcool=tatonescaleheight.Incomparison,thesimulationsinthebottomrowusetheiso-coolinginitialconditionsdescribedinSectionD.2forwhichtcool=tisinitiallyconstantthroughoutthesimulationvolume.TheoverallbehaviorisqualitativelysimilarinbothcasesandresemblestheresultsobtainedbyMcCourtetal.(2012)outsideofthemidplaneregion.Whentheratiooftimescalesisbelowunity,thegascoolsinplaceandformsdropletsofcondensatethatraindowntowardsthemidplane.Inthesecases,convectiondoesnothinderthermalinstabilitybecausethegasisabletoadjustitsthermalstatefasterthanitisabletoconvect.Astheratiooftimescalesisincreased,thedynamicsofthegasbecomeincreasinglydominatedbyconvection,althoughgascontinuestocondensearoundthemidplane.WhileeachverticalpairofmodelsillustratedinFigure3.2behavessimilarly,anumberofminordierencescanbeobserved.Principally,condensationoccursmoreuniformlyforiso-coolinginitialconditionsthanforisothermalones.ThisresultarisesfromthedieringdensityprolesneededtosatisfytheHSEconstraint|isothermalinitialconditionshaveasteepergas-densitygradientandconsequentlyalargerrangeincoolingtimeacrossthesimulationdomain.Shortercoolingtimesnearthemidplaneleadtoa`crosstalk'eectthatismorepronouncedforisothermalinitialconditions.Condensationofgasnearthemidplanecauseshot,low-densitybubblestoformthereandtorisetogreateraltitudes,creatinginhomogeneityatthosealtitudesonafreefalltimescaleinsteadofacoolingtimescale.Thiscrosstalkbetweenlowerandupperlayerscomplicatesthetaskofinterpretinghowthermalinstabilityandcondensationdependonthechoiceofinitialtimescaleratioatonescaleheight.The`iso-cooling'condition,whilenotnecessarilymorephysicallyvalid,reducesthiscrosstalkandallowsforclearerinterpretationoftherelationshipbetweentheinitialtimescaleratioand57Figure3.3CoolingtimeoverfreefalltimeforMultiphaseGasSimulations:Theaverageratioofcoolingtimetofreefalltimeintheambientgasisshownasafunctionoftimeforsimulationswithlowinitialvaluesoftcool=t.Thexaxisisinunitsoftheinitialcoolingtimeatonescaleheight,tcool;S,ratherthanabsolutetime.Valuesareshownforthe2Dplanargeometrycase.Solidlinesindicatethevolume-averagedvalueoftcool=twithinazone0:8zS<z<1:2zS.Dashedlinesshowtheminimumvalueoftcool=twithintheentirebox.Atlowvaluesoftheinitialtimescaleratio,thegasisabletocoolinplacewithinafewcoolingtimes,drivingtherestofthegastotcool=t>10.Asthegascoolslargelyinplace,instabilitiesgrowpurelyonthecoolingtime,leadingtosimilarbehaviorforallruns.theonsetofcondensation.Incontrast,densemidplanegasinmodelswithisothermalinitialconditionsisabletocondensequicklyevenwhentheinitialratiooftimescalesatonescaleheightislarge.Thishappensbecausethegasnearthemidplanewillhavealowerratiooftcool=t,leadingtolocalizedcondensation.3.3.2InstabilityGrowthintheStrongCoolingRegime
Usingtheiso-coolingsimulationspresentedintheprevioussection,wehaveexaminedtheevolutionofperturbationsforthecaseinwhichrapidcoolingdominatesthedynamicsofthegas.Ifperturbationsareabletocoolandcollapsemorerapidlythantheycansink,condensationproceedsonatimescale˘tcool.Whenglobalthermalbalanceismaintained,theaveragetcool=toftheambientgasquicklyincreasesascondensationlowersthegasmassanddensityoftheambientmedium.Withinafewcoolingtimes,tcool=trisesto&10,asshowninFigure3.3.Inthisstrong-coolingregime,theonsetofcondensationisdeterminedbythegrowthoftheinitialperturbationsanddoesnotdependstronglyontheinitialratiooftcool=t.Condensationcontinuesunabateduntilthetimescaleratioisabove10,atwhichpointthecoolingisweakenoughthatthecondensationrateslows.58Figure3.4GasDensityEvolutionOverTimeinMultiphaseSimulationsEvolutionofgasdensityina2Dplanarsimulationwithaniso-coolinginitialconditionoftcool=t=5:0.Aftertwocoolingtimes,thegasisclearlyconvecting.Atfourcoolingtimesnocoldgashascondensed,buttheamplitudeoftheperturbationshasincreased.Theperturbationshavebeenfurtherampliedaftersixcoolingtimes,andtherstcondensatehasformed.Aftereightcoolingtimes,thedensestgasnearthecenterhasenteredintorunawaycooling,leadingtocontinuouscondensation.
3.3.3InstabilityGrowthintheConvectiveRegime
Whentheinitialratiooftcool=tislarge,incipientcondensingregionssinkintothegravitationalpotentialfasterthantheycancool,leadingtoaroiling,convectivestate.Theconvectionissubsonic,andalthoughthepressureremainsnearlyconstantatagivenheight,convectiondoesnotpreventthetemperatureanddensityperturbationsgeneratedbycoolingfromgrowing.Figure3.4illustratesthegrowthofperturbationsinamediumwithaninitialtimescaleratiooftcool=t=5:0.After2coolingtimes(1coolingtime=585Myr),thegasisconvecting.After4coolingtimes,thegascontinuestoconvect,butthedensityperturbationshaveincreased.After6coolingtimes,convectioncannolongersuppresscondensationofgasnearthemidplane,anditcoolscatastrophically.After8coolingtimes,asignicantamountofthedensegashascondensed.Itisthusclearthatthecondensationdoesnotsimplyswitchonwhentheaverageratioofcoolingtimetothedynamicaltimedropsbelowsomespecialvalue.Toquantifythetransitionofthegasfromarelativelysmooth,convectivestatetoamultiphasemedium,weplotinFigure3.5theprobabilitydistributionfunctionofthethermalstateofthegasastherunwithinitialtcool=t=5:0evolves.Afterseveralcoolingtimesthedistributionofgasinthez{(tcool=t)planehaswidenedconsiderably.After4coolingtimes,gasinthetailofthedistributionhasreachedaratioofaround3.Atthispoint,furtherperturbationgrowthisinevitableandcondensationbegins.59Figure3.5EvolutionoftheTimescaleRatioinMultiphaseSimulations:Evolutionofthemass-weightedprobabilitydistributionfortheratioofcoolingtimetofreefalltimefora2Dplanargeometrywithinitialtcool=t=5:0.Thedashedblacklineshowsthevolume-weightedaverageratioasafunctionofheight.Notethatwhengascondenses,mostofthevolumeisoccupiedbythehotgas,meaningthatthevolume-averagedratiowilltendtolieabovethemass-weightedmean.Therstpanelshowstheinitialstateofthegaswherethetimescaleratioisheldconstantthroughout(withsomespreadduetotheinitialperturbationspectrum).Att=2:0tcool;S,thegashasenteredintoaconvectivestateandalthoughcondensationhasnotyetcommenced,aspreadingaspropertiesisevident.Byt=6:0tcool;S,aportionofthegashasreachedastatewithtcool=tˇ23,andthecondensationprocesshasbegun.Althoughsomegasisenteringintothecoldphase,thevolumeaveragedratiooftcool=tremainsnearitsinitialvalueasthecoldgasoccupiesnegligiblevolume.Increasingtheinitialtimescaleratiototcool=t=20:0slowsthecondensationprocessandfurtherrestrictsittothemidplaneregion,asshowninFigure3.6.CondensationfollowsthesamegeneralpatternasinFigure3.5,exceptthatitisdelayedformorethan10timestheinitialcoolingtimeandismuchmorepronouncednearthemidplane.Theconcentrationtowardthemidplaneoccursbecausecoolinggasblobscansettleoveralargernumberoffreefalltimesandpreferentiallyaccumulateinthemidplanebeforecondensing.Inallofoursimulations,whichhaveiso-coolinginitialconditionsuptotcool=t=30,condensationeventuallyoccursaslongasitisgivenenoughtimetodevelop.Figure3.7showshowboththeaverageandminimumvaluesoftcool=tevolveduringeachrun.Condensationintherunswithlargevaluesoftcool=tmaybesurprisinginlightofrecenttheoreticalstudiespredictingthatthemediumshouldbecomemultiphaseonlyiftcool=t.10(Sharmaetal.,2012b;Gasparietal.,2012b;Singh&Sharma,2015),andwewilldiscusspossibleexplanationsforthisdierenceinSectionD.6.Figure3.8showsthesamesimulationsasFigure3.7plottedwithtimeinunitsofthefreefalltimeatonescaleheightratherthantheinitialcoolingtime.Asallsimulationsusethesamegravitationalpotential,thefreefalltimeisastandardclockandcorrespondstothesametimeintervalinphysicalunits(approximately117millionyears).Whentheinitialratioofcoolingtimetofreefalltimeexceeds˘10,condensationoccursafter˘100freefalltimes,correspondingtoatimescalecomparabletotheHubbletime.60Figure3.6EvolutionoftheTimescaleforMoreStableInitialConditions:SameasFigure3.5exceptforaninitialtimescaleratiooftcool=t=20:0.Byt=5:0tcool,perturbationshavestartedtogrowbuthavenotyetledtocondensation.Asthegasisabletoundergomorefreefalltimespercoolingtimethaninthecaseoftcool=t=5:0,coolergasisabletoeectivelysettletowardsthemidplane.Nevertheless,condensationisstillabletooccurnearthemidplaneeventhoughthevolumeaveragedvalueoftcool=tremainsnear10.Figure3.7EvolutionoftheTimeScaleRatioforMoreStableICs:SameasFigure3.3,exceptforrunswithlargerinitialvaluesoftcool=t.Ineachsimulation,theminimumvalueoftcool=tdecreasesonatimescaleroughlyproportionaltothecoolingtime.Whentheinitialratioishigher,ittakesseveralcoolingtimesforgastodevelopregionswithaminimumtimescalerationearunity;therefore,condensationisdelayedintheseruns.Notethatrunswithlargervaluesoftcool=thavealowoverallcoolingratewhich,combinedwiththetemperatureoorofToor=5106K,producestheoorinthetimescaleratio.61Figure3.8EvolutionoftheTimeScaleRationinFreefallTimes:SameasFigure3.7exceptwithtimeplottedinunitsofthefreefalltimeatonescaleheightratherthantheinitialcoolingtimeatonescaleheight.
3.3.4TransitiontotheCondensedState
Whilestudyingthegrowthofthermalinstabilitiesgivesinsightintotheconditionsunderwhichgaswillcondense,itdoesnotnecessarilyexplainhowthegasreachesacondensedstateorhowanindividualparcelofgasbehaves.Figure3.9depictsthegasdistributionintheˆTplaneintegratedover30coolingtimes.Tocomputethisdistribution,webingasmassinˆTspaceineachdataoutput(whichareevenlyspacedintime),sumoveralloftheoutputs,andnormalizesothattheintegraloverthedistributionisequalto1.Thisprobabilitydistributioncorrespondstotheprobabilityofaparcelofgasbeingfoundinagiventhermodynamicstateatsomepointduringthesimulation.Thegasisforthemostpartconstrainedtoalineofconstantpressurewithspreadduetogravitationalstratication.Thedistributionhastwopeaks;alowdensity,hightemperaturenodeinwhichthegasisconvecting,andacool,lowtemperaturenoderepresentingthecondensedstateofthegas.Theprobabilityofndinggasintheconnectingregionislow,indicatingthatcondensationfromthehotphaseintothecoldphaseproceedsrapidlyonceitbegins.Figure3.10illustratesthedynamicsofthegasduringtheconvectivestageandthecondensationprocessusingthemotionofaLagrangiantracerparticleinphasespace.Thegureshowsthepathofarepresentativeparticlewhichcondensesearlyinthesimulation.Forseveralcoolingtimes,thegassimplyconvectswithinanarrowportionofphasespace.Asthethermalperturbationsareamplied,thegasisdriventoacolder,denserstatewhichiswherecondensationoccurs.Whenthegasdoescondense,thecondensationprocessisveryrapid,andthegasstaysinthecondensedphaseafterwards.62˘˛˜ˆ˙˛˚˙˙ ˙ˇ˛˜Figure3.9PDFofGasDensityandTemperature:TheprobabilitydistributionfunctionofthegasintheˆTplane,averagedover30coolingtimesinthe2Dplanarsimulationwithaninitialtimescaleratiooftcool=t=5.Notethatalargefractionofthegasislocatedatthetemperatureoor,nearthex-axisatadensityslightlyabove1023gcm3.Astheconvectionandcondensationprocessesproceedsubsonically,theprocessislargelyisobaric,withamodestspreadduetogravitationalstratication.Linesofconstantcoolingtimeareshownasdashedlines,labeledwiththeratioofcoolingtimetofreefalltimeat1scaleheight.Notethatthegasspendsverylittletimebetweenthelinetcool=t=2andthetemperatureoor,indicatingthatoncethethresholdisreached,condensationproceedsrapidly.Figure3.10TracerParticleEvolution:ThedynamicsofuidduringthecondensationprocessareshowninthedynamicsofaLagrangiantracerparticlethroughphasespace.Theparticlesareinsertedduringinitializationinthe2Dplanarsimulationwithaninitialtimescaleratioof5.Theupperleftpanelshowstheparticle'spaththroughˆTspace,withthecolorofthelineshowingelapsedtimeincoolingtimes.TheupperrightpanelalsoshowsthepaththroughˆTspace,butiscoloredbytheratioofcoolingtofreefalltime.Thebottomleftpanelshowsparticleheightvs.time,andthebottomrightshowsthetimescaleratioasafunctionoftime.63Figure3.11ColdGasFractionintheMultiphaseSimulations:Thefractionofmassinthecondensedstateisshownasafunctionoftimefor2Dplanarrunswithlargeinitialvaluesoftcool=t.Thecondensedfractionismeasuredovertheentiredomain.
3.3.5CondensationRate
Inoursimulations,condensedgasremainsinthecondensedstateandsettlestowardsthecenter.Aftertheonsetofcondensationthegassegregatesintotwophases-thecoolcondensedmaterialinthecenterandthehot,convectivegasthatremainsuncondensed.Thisdeparturefromtheexpectationofself-regulationisaconsequenceofourfeedbackimplementationandisdiscussedfurtherinSection3.4.4.Still,itisinstructivetoexaminetherateofcondensationinoursimulations,asisshowninFigure3.11.Followingtheonsetofcondensation,instabilitiescontinuetogrowonthecoolingtimescale.Eachsimulationbehavessimilarlyonathermaltimescale,witharoughlylineargrowthinthetotalcondensedfraction.3.3.6EectofGeometry
SimulationswithdierentgeometriesareshowninFigure3.12.Allsimulationsusethesameinitialconditions(tcool=t=5:0).Inthesphericalcase,gravitypullstowardstheorigin,whileinthecylindricalsetupgravitypullstowardsthesymmetryaxis.Allsimulationsexhibitsimilarthermalbehavior.After10coolingtimes,thegashasenteredintoaconvectivestateandcondensationhasbegunnearthecenterofthepotential.Inthenon-planarruns,lessgascondensesastheregionnearthecenteroccupieslessvolume.Nevertheless,wedonotobserveasignicantchangeinthecondensationprocessamongsimulationswithdierentgeometries.64Figure3.12EectofGeometryinMultiphaseGasSimulations:Theevolutionofrunswithaninitialtcool=tof5.0areshownfordierentgeometriesatt=10tcool.Allrunsuseanidenticalsetupwithrespecttotheradialcoordinate,thoughthedenitionortheradialcoordinateischangedbasedonthegeometryofthesimulation.The2Drunsuseastaticgridof300300cells,whilethe3Drunsusearootgridof1283cellswith2layersofadaptivemesh.
3.4DiscussionandRelationshiptoRelatedWork
Oursimulationswouldseemtoindicatethatanymediumsubjecttoaheating/coolingbalanceaswehavedescribedinourmodelwilleventuallysuccumbtothermalinstabilityandproducecondensation.Neverthe-less,observationsseemtoindicatethatclusterswithtimescaleratiosaboveroughly10donotproducemuchmultiphasegas.Toexplainthisdiscrepancy,wenotethatataradiusofaround30kpc,atimescaleratioof10inalargegalaxyclustercorrespondstoacoolingtimeontheorderofaGyr.Physicalprocessessuchasmergers,starformation,andAGNfeedbackoccuronshortertimescales,renderingthecondensationprocesssub-dominantinthesecases.Therefore,inarealisticclusterenvironmentonlyclusterswithacoolingtimetofreefalltimeratioof.10arelikelytodevelopcondensation.Animportantcaveattothisobservationisthatwhilethegrowthofthermalinstabilitiesfrominitiallysmallperturbationsmaybeunimportantonclustertimescales,ifthegasisinhomogeneousduetootherphysicalprocesses(suchasanAGNjet)condensationmayoccurinthetailofthethermaldistributionsshowninFigure3.5and3.6.Thus,predictingtheonsetofcondensationisnotassimpleasmeasuringthevalueoftcool=t;thelevelofinhomogeneitymustalsobetakenintoaccount.Oursimulationsexaminetheformationofmultiphasegasinanidealizedsettingwhereinglobalbalancebetweenheatingandcoolingisstrictlyenforced.Whilethismodelgivesrisetoresultsthatarequalitativelyconsistentwithobservations,itclearlyneglectsthecomplexphysicsofAGNfeedbackandheattransportwhichoccurinrealclusters.Inthissection,wediscussourresultsinlightofcurrentobservationsofmultiphasegasandprevioussimulationsofcondensationandconsiderthecomplicationsthatinclusionof65additionalphysicalprocesseswouldcause.
3.4.1ObservationsofMultiphaseGas
Owingtothetimescalesinvolvedandthelimitsofcurrenttelescopes,astronomerscannotdirectlyobservethecondensationprocessintheICM.Nevertheless,thepastdecadehasdeepenedtheeld'sappreciationofafascinatingdichotomyinclusterpropertieswhenclustercoresareprobedforcoldgasandsignaturesofAGNfeedback.Whilecoolingisgenerallysuppressedincool-coregalaxyclusters(Petersonetal.,2003;Peterson&Fabian,2006),atleastsomecoldgasisobservedingalaxieswithlowcentraltemperatures,asseenintheworksofMcDonaldetal.(2010)andWerneretal.(2010).Cavagnoloetal.(2008)considerstheentropyproles,radioemissions,andpresenceofHintheACCEPTsampleof222galaxyclusters.AsHemissionrequiresthepresenceofcold(relativetotheICM)gas,thepresenceorabsenceofHinaclustermaybetakenasanindicatorofmultiphasegas.IntheclusterswithHobservations,Hisconclusivelydetectedinslightlyoverhalfofthesample.AstrongcorrelationisseenbetweenthepresenceofHandthecoreentropy;clusterswithHhavecentralentropiesbelow30keVcm2,whilethosewithoutHdetectionstendtolieabovethe30KeVline.Whentheentropyproleisusedtoinferacoolingtime,(asinVoit&Donahue,2015;Voitetal.,2014)acentralentropyof30KeVcorrespondstoacoolingtimeofaround1billionyears,consistentwithacoolingtimetofreefalltimeratioofaround10.InclustersintheACCEPTsample,thosewithdetectedHemissionconsistentlyhavetcool=tvaluesbelow'20,whilethosewithoutHdetectionslieentirelyabovethatvalue.3.4.2SimulationsofMultiphaseGas
McCourtetal.(2012),uponwhichthisstudyisbased,ndsthatprecipitationwilloccurrapidlyifthegasisabletocoolinplace,whichoccurswhentcool=t.1.Theauthorsalsoconcludethatthecondensa-tionprocessisrelativelyinsensitivetovariationsintheheatingrateandmechanism.Employingasimilarmethod,Sharmaetal.(2012b)ndsthatcondensationmayoccuringaswithatimescaleratioof.10inasphericalsimulation,anenhancementtheyattributetothecompressionofoverdenseblobsdescendinginasphericalgeometry.Additionally,Sharmaetal.(2012b)concludesthatcondensationdoesnotoccurwhenthetimescaleratiorisesabove10.Analyticworkhaslentfurthercredencetotheideathattcool=t.10representsacriticalthresholdforcondensation.Singh&Sharma(2015),extendingtheanalysisofPizzolato&Soker(2005),ndsthatsmallinstabilitiesmaygrowwhentcool=t.1forplanargeometriesand,whentheeectsofgeometriccompressionareincluded,maygrowfortcool=t.10forsphericalgeometries.WhiletheresultspresentedinSection4.3of66thispapersuggestamoderatelyhigherthresholdfortheplanarcase,webelievethattheseresultsarelargelyconsistentwithSingh&Sharma(2015)inthecontextofindividualoverdensitiescoolingandcondensinginplace.Inoursimulations,however,weseethatoverdensitiesinamediumabovethecriticalthresholdoscillate,leadingtoaroilingstatethatdevelopsfurtherperturbations.Thiscrosstalkeectbetweenlayersgeneratesnon-linearperturbationsandcausesthetemperaturedispersioninthemediumtogrowonthecoolingtimescale.Whenthecoldtailofthedistributionhasdroppedtotcool=t.23thecondensationprocessbegins.Thuswendthatthemechanismresponsibleforcondensationabovethecriticalthresholdisnotgeometriccompressionbutthecontinuedgrowthofperturbationsfollowingtheonsetofconvectioninthegas.Simulationsthatemploymorerealisticheatingmechanismsalsondthatcondensationoccursingalaxyclusters,albeitundersomewhatdierentcircumstancesthaninsimulationswithidealizedheating.Li&Bryan(2014b)employanAGNfeedbackalgorithminwhichheatingistriggeredbycoldgasaccretion.Thestudyndsthatcondensationoccurswhen3.tcool=t.10.Thiscondensationoccursalongtheaxisofthejet,wheredensegasisdraggedupandisabletocoolasitfalls.Thisisconsistentwithourndings,inwhichthethermalinstabilitycangrowwhentcool=t.10,butonlywhengasissucientlyhetrogeneous.Similarly,Gasparietal.(2012b)employsjetheatinginresponsetoaccretionandndsthatmultiphasegascanformwhentcool=t.10.3.4.3CaveatsandLimitations
Inthisstudy,wehaveusedanidealizedmodeltosimulatetheonsetofcondensationingalaxyclusters.Whilethesimplicitymakesthismodeleasytoanalyze,wehaveleftoutphysicsthatmayhavesignicantimpactonthedevelopmentofamultiphasemedium.Inparticular,conductionandthepresenceofmagneticeldsmayinhibitorshapethegrowthofcondensation.Conductionworkstosmoothouttemperatureperturbations,whilemagneticeldswillleadtoconductionbeinganisotropic.Magneticeldsinclustersarepoorlyunderstood.Whileweak,theyareknowntobepresentandmaybedynamicallyimportantinclustercores(Carilli&Taylor(2002)andreferencestherein).Moreimportantforthiswork,magneticeldsinaplasmawillleadtoanisotropicconduction,channelingheatalongthedirectionofmagneticeldlinesasexploredinRuszkowskietal.(2011).Similarly,Waghetal.(2014)studiesthegrowthofthermalinstabilitiesinasphericalsetupandincludesbothconductionandmagneticelds.Anisotropicconductionisnotfoundtoinhibitcondensation,butdoesleadtotheformationoflamentsratherthanglobulesofdensegas.Conductionisfoundtoinhibitcondensationiftheeciencyisabove0.3oftheSpitzervalue.67Inadditiontotheseomissions,ourassumedheatingfunctiondoesnotcapturethetruephysicalprocessresponsiblefortransferringenergyfromtheAGNtotheICM.WhilethedetailsofAGNheattransferarenotcurrentlyunderstood,severalmechanismshavebeenproposed,includingshocks(McNamara&Nulsen,2012;Ruszkowskietal.,2004),cosmicrays(Sharmaetal.,2010;Fujitaetal.,2013;Fujita&Ohira,2011,2012),turbulentmixing(Sharmaetal.,2009;Banerjee&Sharma,2014b),PdVworkfromtheinationofhotbubbles(McNamara&Nulsen,2007;B^rzanetal.,2004),andtheupliftingofcoolgasbyrisingbubbles(Millionetal.,2010).Theactualheatingfunctionisunlikelytomaintainperfectthermalbalance,andpresumablydoesnotactinastrictlyvolumetricsenseasassumedinthiswork.Still,thelackofcoldgasandstarformationincool-coreclustersimpliesthattheheatingfunctionmustbroadlymaintainthermalequilibrium,makingthemodelconsideredinthisworkphysicallyrelevant.3.4.4Self-Regulation
Thegasdoesnotreachasteadystate,asmightbeexpectedforanidealself-regulatingsystem.Instead,condensationcontinuesintheconvectivegasafterthecondensationprocesshasbegun,increasingthesepa-rationbetweenthehotandcoldphases.Inrealclusters,feedbackisexpectedtooperateinathermostat-likemanner,whichshouldproducearoughthermalequilibrium.Thelackofself-regulationinoursimulationsispurelyaneectoftheheatingmodelthatweemploy,anddoesnotaccuratelycapturetheresponseoffeedbacktocondensation.However,ifweimaginefeedbacktobepoweredbycondensation,wecanusethecalculatedheatingratetodeterminewhatfeedbackeciencywouldbenecessaryforthesystemtobalanceradiativelosses.Duringaccretion,AGNareexpectedtoconvertasignicantfractionoftheinfallingmassintoenergythatisthenreturnedtothesurroundingmedium.Thefeedbackratecanberelatedtothemassaccretionvia_Eˇ_Mc2(3.8)where_Eisthetotalenergyoutput,isaneciencyparameter,and_Misthemassaccretionrate.UndertheassumptionsthatallofthecondensinggasisusedtopowerfeedbackandthatallofthefeedbackenergyistransferredtotheICM,wehaveestimatedtheconversioneciencynecessarytomaintainthermalbalance.TheestimateisshowninFigure3.13forseveralinitialvaluesoftcool=t.Oncecondensationhasbegun,therequiredeciencyinallrunsreachesavalueofaround103,inlinewiththevaluesfoundinSharmaetal.(2012b).Astheaccumulationofcoldgasnearthemidplaneisanartifactofoursetupandwouldnotbeexpectedinarealcluster,wecalculatethecoolingrateovertheambientgas,whichisthatgasthatisabovethetemperatureoorof5:0106K.Althoughwedonotexplorethemechanismforreleasingmass-energy68Figure3.13NecessaryAGNEciencytoBalanceCoolinginMultiphaseGasSimulations:Thefeedbackeciencynecessarytomaintainthermalbalanceinthehotgasisshownfor2Dplanarrunsstartingwithdierentvaluesoftcool=t.Therequiredeciencyiscalculatedas=_E=_MColdc2,where_Eisthecoolingrateofallgasabovethetemperatureoor.Boththecoolingrateandthecondensationratehavebeensmoothedoveracoolingtime.
fromcondensedgasinthiswork,ifcondensationresultingfromthegrowthofthermalperturbationsisinprinciplecapableofbalancingradiativecoolingintheICM,thermalinstabilitymustbetakenseriouslyasafeatureofaself-regulatingenergycycleincool-coreclusters.693.5ConclusionsandFutureWork
Inthisstudy,wehaveinvestigatedtheonsetofconvectioninathermallyunstablemediumusinganidealizedmodel,includingaheatingschemethatstrictlyenforcesaglobalheating-coolingbalance.Althoughasimplication,thismodelgivesinsightintotheconditionsnecessaryfortheonsetofcondensationinagravitationallystratiedmediumsuchasthatinacool-coregalaxycluster.Thisstudyindicatesthatcondensationproceedsasfollows:Ifheatingisabletobalancecoolingatallradii,thermalinstabilitieswillgrowinamplitude,regardlessoftheinitialconditions.Iftheratioofthecoolingtothefreefalltimeis.2,(thestrongcoolingregime)thegaswillcondenseinplace,drivingthevolume-averagedtcool=tvalueabove10.Abovearatiooftcool=tˇ10,perturbationswillgrowonatimescaleproportionaltothecoolingtime.Oncetheperturbationdistributionhasbroadened,gaswithtcool=tˇ24willcondense,evenifthevolume-averagedratiooftcool=tisabove10.Ifthetimescaleratiois&10,thetimescaleforcondensationtooccuringaswithtcool˘1GyriscomparabletotheHubbleTimeandgreatlyexceedsotherrelevantclustertimescales.Afundamentallimitationofthisworkisthatthemodelassumesaheatingfunctionthatisidealizedanddoesnotmimicaspecicphysicalprocess.Inpreparationforfuturework,itwillbenecessarytoexamineagreatervarietyofheatingmodes,includingmodelsmoreanalogoustojetfeedbackandquasarwindsfromaccretingsupermassiveblackholes.Thephysicalprocessesunderlyingblackholeaccretion,feedback,andheattransfertotheICMarestillpoorlyunderstood,andelucidatingthemwillformthefocusoffuturestudies.
3.6Acknowledgments
TheauthorswouldliketothankGusEvrard,DevinSilvia,GregBryan,andYuanLiforhelpfuldiscussionsduringthepreparationofthispaper.ThisworkwassupportedbyNASAthroughgrantNNX12AC98GandHubbleTheoryGrantHST-AR-13261.01-A,andbytheNationalScienceFoundationthroughgrantPHY-0941373.ThesimulationspresentedinthispaperwereperformedandanalyzedontheTACCStampedesupercomputerunderXSEDEallocationsTG-AST090040andTG-AST100004.Thisworkwassupported70inpartbyMichiganStateUniversitythroughcomputationalresourcesprovidedbytheInstituteforCyber-EnabledResearch.BWOwassupportedinpartbythesabbaticalvisitorprogramattheMichiganInstituteforResearchinAstrophysics(MIRA)attheUniversityofMichiganinAnnArbor,andgratefullyacknowl-edgestheirhospitality.Enzoandytaredevelopedbyalargenumberofindependentresearchersfromnumerousinstitutionsaroundtheworld.Theircommitmenttoopensciencehashelpedmakethisworkpossible.714TriggeringandDeliveryAlgorithmsforAGN
Feedback
4.1Introduction
1Anactivegalacticnucleus(AGN)isthoughttobepresentinthecoreofnearlyeverymassivegalaxyandgalaxycluster.BasedonestimatesofjetpowerfromAGN-inatedcavities,ithasbecomeclearthatanAGNcanstronglyinuencecoolingandcondensationofgasinitshostgalaxy(e.g.,McNamara&Nulsen,2012),potentiallyexplainingtherelationshipsobservedbetweenthemassofagalaxy'scentralblackholeandthevelocitydispersionofitsstars(e.g.,Merritt,2000;Ferrarese&Merritt,2000),aswellasthestar-formationpropertiesofgalaxieswithAGNs(e.g.,Kaumannetal.,2003).Directlysimulatingtheco-evolutionofAGNstogetherwiththeirhostgalaxiesisnotcomputationallyfeasibleduetothelargedierencesinmassandsizebetweenanAGNanditshostgalaxy.Also,thediversesetofcomplexphysicalprocessesthatgovernAGNaccretionandoutowproductionremainpoorlyunderstood.Tocircumventthesedicultiesinstudiesofgalaxyevolution,simulatorshavedevelopedanumberof\subgrid"implementationsofAGNfeedbackthatareintendedtocapturetheinterplaybetweentheAGNanditsenvironmentwithoutrepresentingthedetailsofAGNaccretiononsmallerscales(see,forexampleOmmaetal.,2004;Springeletal.,2005a;Puchweinetal.,2008;Booth&Schaye,2009;Gasparietal.,2011b;Duboisetal.,2012;Li&Bryan,2014a;Steinbornetal.,2015).However,subgridimplementationsofAGNfeedbackvarywidely,andtherehasbeenlittlesystematiccomparison(butseeWurster&Thacker,2013;Yangetal.,2012).Fromtheexplorationsofparameterspacecarriedoutinthesestudies,ithasbecomeevidentthatvaryingcertainAGNfeedbackparameterscanleadtostrongdierencesinfeedbackpowerandenergypropogation.Inthispaper,wecompareseveralofthepopularmethodsforimplementingAGNfeedback.AnAGNconsistsofasupermassiveblackhole(SMBH)surroundedbyadiskofaccretingmaterial.Twistedmagneticeldsinthediskarethoughttochannelchargedparticlesintojets,whichcandrawadditionalpowerfromthespinoftheSMBHviatheBlandford-Znajekeect(Blandford&Znajek,1977;Blandford&Payne,1982).Theserelativisticjetsproducesynchrotronemission,whichisobservedintheradioband.Hence,thismodeofAGNenergyoutputistermed\radio-mode"feedback(e.g.,Churazovetal.,2001;Springeletal.,2005b).Additionally,dierentialrotationintheaccretiondiskwillheattheaccreting1ThischapterconsistsofmaterialsubmittedtoTheAstrophysicalJournal(Meeceetal.,2016).Ithasbeenreformattedforinclusionhere.Forinformationaboutcopyrightandreuse,seeAppendixE.72material,producingastrongUVux.IftheSMBHisaccretingneartheEddingtonLimit,radiationpressuremayalsodrivelargeoutows.This\quasar-mode"feedback(e.g.,Churazovetal.,2005)iscomposedofnon-relativisticmaterialandismoreisotropicthanradio-modefeedback.ThelengthandmassscalesthatareimportantforAGNaremuchsmallerthantherangecoveredbygalaxiesandgalaxyclusters.Forexample,ablackholewithamasssimilartotheonedwellingatthecenterofthePerseusCluster(˘3:4108M;seeWilmanetal.,2005)hasaSchwarzchildradiusofonlyafewAU,whereasthevirialradiusofagalaxyclusteris&1Mpc.Cosmologicalsimulationscapableofmodelingentiregalaxyclusterstypicallyhaveamaximumresolutionof˘1kpc,meaningthattheAGN'sbehaviormustbeapproximatedwithasubgridmodel.Furthermore,AGNareknowntobevariableontimescalesmuchshorterthanthedynamicaltimeofagalaxy.CosmologicalsimulationsthatmodelstructureformationoveraHubbleTimemustthereforerelyonanAGNmodelthatsmoothsoutthisshort-termvariabilitywhilepreservingthelarge-scalebehavioroftheresultingfeedback.Inorderforaself-regulatedfeedbacklooptoarise,asubgridAGNmodelmustcapturethecouplingbetweenanAGNanditsfuelsupply.AtriggeringalgorithmmustsomehowestimatethemassaccretionrateontotheSMBH,whichtranslatesintoaproportionalreleaseoffeedbackenergy.Thenadeliveryalgorithmmustprescribehowthatfeedbackenergyinteractswiththelocalenvironment.Incosmologicalsimulations,thesubgridAGNmodelmustalsoincludeprescriptionsforfollowingthecreation,advection,andmergerofSMBHs,althoughthesearenotdiscussedinthiswork(seeSijackietal.,2007;DiMatteoetal.,2008;Wurster&Thacker,2013;Vogelsbergeretal.,2013,formorediscussiononthesetopics).Instead,wearefocusingonjustthetriggeringanddeliveryalgorithms.ThereareanumberofreasonstobelievethatAGNfeedbackinmassivegalaxiesisself-regulated.First,asstatedearlier,strongrelationshipshavebeenfoundbetweenthemassesofSMBHsandtheproper-tiesoftheirhostgalaxies.Second,AGNsareconsideredthebestcandidatesforsolvingthe\CoolingFlow"problemingalaxyclustersandellipticalgalaxies(Binney&Tabor,1995;McNamara&Nulsen,2007;Nulsen&McNamara,2013).Manygalaxyclustershavecentralcoolingtimesthatarefarshorterthantheageoftheclusters,buttheobservedstar-formationratesareanorderofmagnitudeormorebelowwhatwouldbeexpectedfromuninhibitedcooling(O'Deaetal.,2008,2010;McDonaldetal.,2011).Also,theamountsofcoldgasthatseemtobeaccumulatingaremuchlessthanonewouldnaivelyexpect(Petersonetal.,2003;Peterson&Fabian,2006).Thistensionimpliestheexistenceofaheatsourcethatroughlybalancescoolinglosses.Non-AGNheatsources,suchassupernovae,mergers,conduction,andpreheatinghavebeenproposed,butareeithernotpowerfulenoughtobalancecooling(e.g.,Skoryetal.,2013)orareinconsistentwithobservations.AGNs,however,areknowntobepresentinthecentralgalaxiesofgalaxyclustersandproducefeedbackenergycomparabletothecoolingrate.Binney&Tabor(1995)showedthatwhencooling-73triggeredjetsareaddedtomodelsofcool-coreclusters,alternatingperiodscoolingandjetheatingcanleadtoaquasi-steadystatefortheICMintheclustercore.AsdiscussedinMcNamara&Nulsen(2012),thejetpowermustbecloselycoupledtothecoolingrateifAGNarebalancingcoolinginclusters.Otherwise,theAGNwouldeitheroverheatorunderheattheintraclustermedium(ICM).OnesimplealgorithmforestimatingtheAGNaccretionratebasesthescalingpropertiesontheBondi-Hoyleaccretionmodel,setoutinBondi(1952),whichimplicitlyassumesthattheSMBHisaccretinghotambientgasdirectlyfromtheICM.Strictlyspeaking,suchaBondiaccretionowshouldbesteady-state,sphericallysymmetric,andisentropic.AccretionthenbecomessupersonicwithintheBondiradiusgivenbyRBondiˇ2GMBH=c2
swhereMBHistheblackholemassandcsisthesoundspeedofthegasnearRBondi,andproceedsatarate_MBondithatdependsonMBHandcs.However,theBondiradiusisunresolvedinmanynumericalsimulationsofAGNfeedback,asisitsimpactongalaxyevolution.Springeletal.(2005a)thereforeproposedaparameterizedBondiaccretionratewithanarticialboostfactorsuchthat_MBH=_MBondi.TheboostfactoristypicallychosentobelargebecausetheactualgaspropertiesattheBondiradiusarelikelytopermitagreateraccretionratethanwouldariseinunder-resolvedsimulations.Springeletal.(2005a)andsubsequentstudiesfollowinguponthatworkuse=100,whileKhalatyanetal.(2008)use=300.Hopkins&Hernquist(2006),incontrast,useafactorthatisnearunity,albeitforstudiesofgalaxy-scalephenomena.Inreality,theassumptionsofBondiaccretion|steadyhomogeneousow,sphericalsymmetry,andadia-baticity|areunlikelytobevalidneartheBondiradiusaroundamassivegalaxy'sSMBH(see,forexample,thediscussioninMathews&Guo,2012).Furthermore,modelsrelyingonstandardBondiaccretionhaveproblemsgeneratingsucientlypowerfuloutowswithoutalargeboostingfactor(seeBS09fordiscussion)andwithreproducingthepropertiesofobservedcool-coreclustersabsentnetuning.Analternativemodelforself-regulatedaccretion,describedbyPizzolato&Soker(2005),positsthattheAGNisprimarilyfueledbyaccretionofcold,densegasthatrainsdowninastochasticmanner.AsidefromthepriorassumptionthattheAGNheatingrateislinkedtocoolingintheICM,thismodelissupportedbyobservationsofcoldgasandstarformationwhichindicatethatatleastsomegasisabletocool(e.g.,Edge,2001;Cavagnoloetal.,2008;O'Deaetal.,2008).Inthis\moderatecoolingow"or\coldfeedback"model,radialmixingresultingfromstrongAGNoutburstscreateslargeinhomogeneitiesthatcoolandcondenseatradiibetween5and30kpcfromtheAGN.ThecondensatesthenraindownontheSMBH,poweringsubsequentoutbursts.ThismodelcouplestheAGNtothecoolingpropertiesoftheentireclustercoreratherthanonlytotheregiondirectlysurroundingtheSMBH.Importantly,thecouplingalsooccursovertimescaleslongerthanthefreefallinthecore,leavingtimeforgastocoolandcondense.Coldmodefeedbackhasbeenimplementedinrecentsimulations,notablythoseofGasparietal.(2011a,b,2012b);Li&Bryan(2014a,b);Lietal.(2015),which74attainself-regulatedstatessimilartothoseobservedingalaxy-clustercores.Tomimictheeectsofbothcoldandhotaccretionmodes,Booth&Schaye(2009,hereafterreferredtoasBS09)proposedamodelthatinvokesBondiaccretionwithadensity-dependentboostfactor.Thisboostfactorequalsunityatlowdensities,givingtheclassicalBondiaccretionrate,butrampsupquicklyaboveapre-chosendensitythresholdinordertoaccount,rathercrudely,forcooling,condensation,andaccretionofcondensedgas.InthesimplestmodelsfordeliveryofAGNfeedback,allofthefeedbackenergyisassumedtothermalizeatscalesbelowtheresolutionofthegridandisdepositedasthermalenergyinasmallcentralregion.ThisapproachisusedinSpringeletal.(2005a)andsubsequentworks,includingrecentsimulationssuchastheIllustrissimulation(Vogelsbergeretal.,2014),Rhapsody-G(Hahnetal.,2015)andthesimulationsofRasiaetal.(2015).Inreality,AGNoutowsarelikelyasymmetriconscalesofseveralkpcwithasignicantproportionoftheirenergyinkineticform.SuchbipolaroutowsmaybeimportantfortransportingfeedbackenergytolargedistancesfromtheAGNandformixingmetalsouttodistancesof˘100kpcfromthecentralgalaxy(Kirkpatricketal.,2011;Kirkpatrick&McNamara,2015).Althoughmuchworkhasbeendonestudyinghighlycollimatedoutowsonsmallscalesovershorttimeperiods(Vernaleo&Reynolds,2006;Ommaetal.,2004),itisnotstraightforwardtoimplementtheminlarge-scalesimulationswithcoarserresolution.GiventheincreasingawarenessthatapropertreatmentofAGNfeedbackisessentialforaccuratemod-elingoftheevolutionoflargegalaxies,itisimportantthattheconsequencesofdierentAGNfeedbackimplementationsbeunderstood.TherestofthispapercomparesseveralcommonlyusedalgorithmsfortriggeringanddeliveryofAGNfeedbackinthecontextofanidealizedgalaxy-clustercore,inordertoex-plorehowtheydierinrepresentingthecouplingofanAGNtoitsenvironment,thetotalAGNfeedbackenergyproduced,andtheresultingthermodynamicprolesoftheambientmedium.SectionD.2discussesoursimulationsetupandoutlinesthetriggeringanddeliverymethodswestudy.Section4.3describestheresultsofchangingthetriggeringanddeliveryalgorithms.SectionD.6discussesourresultsinthecontextofthermal-instabilityanalysesofcoldgasaccumulationandselfregulation,alongwithadiscussionofhowphysicalprocessesthatwerenotincludedmighthaveaectedourresultsiftheyhadbeenincluded.Finally,Section4.5summarizesthekeyresultsandpointsoutavenuesandopportunitiesforfurtherstudy.754.2Method
Inthiswork,weconsidertheinterplaybetweenICMcoolingandAGNfeedbackusingasimpliedAGNmodelinanidealizedgalaxyclusterenvironment.ThesimulationsareperformedusingtheadaptivemeshhydrodynamicscodeEnzo2(Bryanetal.,2014)andanalyzedusingtheyt3analysistoolkit(Turketal.,2011).
4.2.1SimulationEnvironment
Oursimulationsincludehydrodynamics,gravity,radiativecooling,andAGNfeedback.WeuseastaticgravitationalpotentialrepresentingboththeclusteranditsBCGbutdonotaccountfortheself-gravityofthegas,whichweassumetobenegligible.WeuseatabulatedcoolingfunctiontakenfromSchureetal.(2009),assumingauniformmetallicityofhalftheSolarvalue.Thiscoolingfunctiondoesnotallowgastocoolbelow104K,whichdoesnotaectthequalitativebehaviorofoursimulations,sinceanyprocessesoccurringatlowertemperatureswouldtakeplacebelowourspatialresolutionlimit.Foranalysispurposes,wedeneanygasbelow3104Kas\cold."Section4.4.2discussesthepotentialeectsofincludingadditionalphysicalprocessessuchasmagneticelds,conduction,andstarformation,whichmayaectAGNfeedbackbutarenotincludedinoursimulations.Unlessotherwisenoted,thesimulationsetupencompassedaboxoflength3.2Mpcpersidewitha643cellrootgridand8levelsofAMRrenement,givingamaximumspatialresolutionof196pc.Asetof8nestedgrids,centeredontheclustercore,withtwicetheresolutionandhalfthewidthofthepreviouslevel,werecreatedduringinitializationandwereneverde-rened.Additionalrenementwasallowedtooccurbasedonstrongdensityorenergygradients,baryonoverdensity,andcooling.Allcellscontainingmaterialthatwasejectedfromthecentral10kpc,asindicatedbyapassivetracereldaddedtomaterialwithinthatregion,werecoveredbyatleast4levelsofrenement.Finally,thezonearoundtheAGNwhereaccretionwasmeasuredandfeedbackwasappliedwasalwaysrenedtothemaximumlevel.Wedonottakecosmologicalexpansionintoaccount.
4.2.2ClusterSetup
FollowingtheworkofLi&Bryan(2012),weinitializetheICMasahydrostaticsphereofgaswithinastaticsphericalgravitationalpotential.Thegravitationalpotentialcomprisestwocomponents:anNFWhaloandthestellarmassproleoftheBCG.ThevirialmassM200andconcentrationparametercoftheNFWhalo2http://enzo-project.org/3http://yt-project.org/76aredenedwithrespecttotheradiuswithinwhichthemeanmassdensityis200timesthecriticaldensity.FortheBCGweassumeamassproleoftheformM(r)=M4"2(r=4kpc)(1+r=4kpc)#;(4.1)whereM4isthestellarmasswithin4kpcandandareconstants.AsinLi&Bryan(2012)andMathewsetal.(2006),weusedthePerseusclusterasatemplate,choosingM200=8:51014M,c=6:81fortheNFWhalo,M4=7:51010M,=0:1,and=1:43fortheBCG.Withthesemassproles,theBCGisgravitationallydominant.10kpcfromthecenter,whileoutsideofthisradiustheNFWhalodominatesthepotential.Althoughwedonottakecosmologicalexpansionintoaccountinoursimulations,wedouseavanillaCDMmodelinordertospecifythevirialmassoftheNFWhaloandtosetitsgastemperature.Forinitialization,weassumeaclusteratredshiftz=0andacosmologywithM=0:3,=0:7,andH0=70km/s/Mpc.Wedonotexpectourresultstobesensitivetosmallchangesintheseparametervalues.ThehydrostaticgasinthehaloisinitializedwithanentropyproleoftheformK(r)=K0+K100(r=100kpc)K(4.2)whereweusethedenitionofspecicentropyusedintheACCEPTdatabase(Cavagnoloetal.,2009):KkBTn2=3e:(4.3)ForthePerseuscluster,ACCEPTgivesvaluesofK0=19:38keVcm2,K100=119:87keVcm2,andK=1:74,andweusethemforourinitialconguration.TheconditionforhydrostaticequilibriumisdPdr=ˆg:(4.4)Together,thethespeciedentropyproleandthehydrostaticcondition(equations4.2,4.3,and4.4)giveadierentialequationrelatingtemperatureandentropy.Itremainstospecifyaboundaryconditionsothatthisequationcanbeintegrated.FollowingVoit(2005),thetemperatureofahydrostaticICMcanbeapproximatedaskBT200=mp2[10GM200H(z)]2=3(4.5)WetakethisasacharacteristictemperaturefortheICMnearthevirialradiusandintegrateinwardsand77outwardstondthetemperatureanddensityprolesfortherestofthecluster.4.2.2.1TracerFluid
InordertotrackthegasdirectlyaectedbyAGNfeedback,wecontinuouslyinjecta(passive)traceruidintothecentral10kpc.ThispassivetraceralsoallowsustomeasuretheradialextentoffeedbackheatingandalsoindicatestheamountofmetaltransportfacilitatedbyAGNjetsandrisingcavities,whicharethoughttoplayanimportantroleinshapingthemetallicityprolesofclusters.TheamountoftracerinjectedperunitmassˆTisgivenbyˆT=SSFRY(4.6)whereweassumeaspecicstar-formationrateSSFR=1011yr1andayieldY=0:02.TheseassumptionsaremeanttobeacrudeapproximationformetalinjectionbytheoldstellarpopulationoftheBCG.Weemphasizethatallwearedoingisinjectingpassivetraceruid.Noactualstarformationtakesplace,andthetraceruiddoesnotaecttheradiativecoolingrate.Ourprimaryinterestisradialtransportanddistributionofthetraceruid.WedonotexpectitsconcentrationtomatchmetallicityvaluesintheICMofobservedclusters.
4.2.3FeedbackandJetModeling
AGNarecomplicatedsystemsgovernedbyphysicalprocessesthatarepoorlyconstrainedandspanmanyordersofmagnitudeinspaceandtime.OurgoalhereisnottounderstandallthedetailsofAGNphysicsbutrathertostudytheinterplaybetweenaccretion,jetoutows,andthethermalstateoftheICM.Tothisend,weimplementasimplied\AGNParticle"model,whereinaccretionontotheAGNlaunchesoutowsthatareinsensitivetothedetailsofgasaccretiononscales<200pc.Weimplementseveraltriggeringmechanisms,eachwithadierentalgorithmfordeterminingtheaccretionrate_Mintotheregionsurroundingthecentralsupermassiveblackhole,whichsetsthescaleoftheAGNfeedbackresponse.Ineachcase,theresultingoutputoffeedbackpoweristakentobe_E=_Mc2,whereisafeedbackeciencyfactorandcisthespeedoflight.Theaccretionrate_Misnotnecessarilytheactualaccretionrateontothecentralblackhole,andinouridealizedimplementationsnogasisremovedfromthesimulationvolume.Instead,itisassumedtobereheatedandexpelledfromthevicinityoftheblackholebyfeedback.Regardlessofthetriggeringmechanism,precessingjetsarelaunchedfromdisk-shapedregionsoneithersideoftheAGNasdescribedinthefollowingsubsections.PleaserefertoTable4.1forducialvaluesoftheAGNfeedbackparameters.784.2.3.1TriggeringMechanisms
Eachofthefollowingtriggeringmethodscalculates_MandremovesgasmassfromthegridwithinaspeciedradiusgivenbytheparameterRacc.Cold-GasTriggeredFeedback:MeanttoreplicatethetriggeringmechanismusedinLi&Bryan(2014a),feedbackistriggeredbythepresenceofgaswithinRaccandatorbelowathresholdtemperatureToor.Theaccretionratecorrespondingtoasinglecellis_Mcell=Mcelltacc(4.7)wheretaccisaconstanttimescale.FollowingLi&Bryan(2014a),wechoosetacc=5Myr,whichisclosetotheaveragefreefalltimeneartheaccretionradius.BoostedBondi-likeTriggering:TheaccretionrateissettotheBondiaccretionratederivedfromcondi-tionswithinRaccandmultipliedbyaconstantboostfactorsothat_M=2ˇG2M2BH^ˆ(^v2+^cs2)3=2(4.8)whereGisthegravitationalconstant,MBHisthemassoftheblackhole,and^ˆ,^v,and^csarethemass-averageddensity,velocitymagnitude,andsoundspeedwithinRacc.Inthisworkweadopt=100.MassisremovedfromeachcellwithinRaccinamass-averagedsense,suchthatMcell=McellM(<Racc)_Mt(4.9)BoothandSchayeAccretion:AsdescribedinBS09,theaccretionratefollowstheBondiformulabutwithaboostthatdependsonthegasdensityas=8
>
>
<
>
>
:1nn0(n=n0)n>n0(4.10)FollowingBooth&Schaye(2009),wetaken0=0:1cm3and=2.79ParameterValueDescription103JeteciencyRacc0.5kpcAccretionradiusTFloor3104KTemperatureoorMBH1:0108MSMBHmass˚Jet0.15radiansJetprecessionangle˝Jet10myrJetprecessionperiodRJ0.5kpcRadiusatwhichjetsarelaunchedRD0.5kpcInitialradialthicknessofjetsTable4.1ParametersforAGNSimulations:Theseparametervaluesareusedforallsimulationsunlessotherwisenotedinthetext.
4.2.3.2JetImplementation
Afterthetotalaccretionrate_Mduringatimesteptiscalculatedwithoneofthesetriggeringmethods,acorrespondingamountoffeedbackenergy_Mc2tisaddedtotheejectedgas.Weassumetheejectedmasstobeequalto_Mt,whichisanidealization.Inreality,themass-loadingfactorofthejetswilldependonsubgridphysicsthatisnotyetwellunderstood.However,Duboisetal.(2012)ndthatthechoiceofmass-loadingfactordoesnotstronglyaecttheirresults.Afractionfkofthefeedbackenergyisaddedtotheejectedmassaskineticenergy,whiletherestisaddedasthermalenergy.ThisnaturallyresultsinajetvelocityofvJet=cp2fkinetic(4.11)oraroundvˇ0:045cforourducialparameterchoices.KineticenergyandtheassociatedmassareputintothegridthroughtwodiskseachofradiusRDlocatedoneithersideoftheAGNatadistanceRJfromthecenter.Thejetsareorientedataxedangle˚Jetwithrespecttothezaxisandprecessarounditwithaperiod˝jet.Forsimulationswithpurethermalfeedback(fk=0),weagainfollowthemethodofBS09inordertopreventtheinjectedthermalenergyfromimmediatelybeingradiatedaway.FeedbackenergyisstoredupuntilenoughaccumulatestoheatthegasintheinjectionzonetoatleastTmin=107K.ExploratorysimulationswithTmin=108didnotshowanoticeabledierenceinbehavior,inagreementwithBS09.WeobservethatthisalgorithmresultsinaseriesofthermalpulsesasAGNfeedbackisrampingup,butcomesclosetosteadyinjectionwhentheAGNpowerishigh.Weperformedtestsusingthisinjectionthresholdwith80somekineticfeedback(fK>0:0)butdidnotobserveanoticeabledierencewhencomparedtosimulationswithcontinuousenergyinjection.UnlessotherwisenotedweusetheparametersgiveninTable4.1forallsimulations.814.2.4HydroMethod
Thesimulationsinthisworkusea3DversionoftheZEUShydrodynamicsmethod(Stone&Norman,1992)becauseofitsrobustnessandspeed.ZEUSisknowntobearelativelydiusivemethodandrequiresanarti-cialviscositytermthatmayaecttheaccuracyofourhydrodynamicscalculations.Wehaveexperimentedwithusingapiecewise-parabolicmethod(PPM)(Colella&Woodward,1984),butencounterednumericaldicultiesrelatingtothestrongdiscontinuitiesoccurringattheinjectionsite.4.3Results
ThemoststrikingdierenceswithinoursuiteofsimulationsarebetweenAGNfeedbackalgorithmsthatdeliverallofthefeedbackinthermalform(fk=0)andthosethatdeliveratleastsomekineticfeedback(fk>0).Changesinthetriggeringmethodproducesmallerdierencesinqualitativebehavior,probablybecauseallthreetriggeringmethodsimplementedhereendupstronglyboostingthefeedbackresponsewhensignicantamountsofcoldgasaccumulatenearthecentralblackhole.Wewillthereforepresentourresultsondeliverymechanismsrstandtriggeringmechanismssecond.4.3.1DeliveryofFeedback:Thermalvs.Kinetic
InjectionofAGNfeedbackenergyheatsthesurroundinggasthroughseveralprocesses.First,iffk<1:0,thentheAGNdirectlyinjectsthermalenergyintotheICM.Second,interactionsbetweentheAGNoutowandtheICMproduceshocksthatpropagateoutwardandheattheambientgasinaquasi-isotropicmanner.Third,outowsdriveturbulencethatcanheattheICMastheturbulencedecays.Finally,momentumfromtheAGNoutow|eitherdirectlyinjectedintheformofakineticjetordrivenbythermalexpansionofhotbubbles|candredgelow-entropygasoutofthecoreandmixitwithhigherentropygasatlargerradii.4.3.1.1FeedbackPower
Allofoursimulationswithfk>0:0followsimilarpatternsofevolution.Initially,theclustercoreissmooth,sphericallysymmetric,andcontainsnocoldgas.Thecoregasthencools,contracts,andgrowsdenserfor˘0:3Gyruntilcoldcloudsbegintocondenseatthecenterandstronglyboostthejetpower.Figure4.1showsboththejetpowerandcoolingluminositywithindierentradiiduringtherst2Gyrofacold-gastriggeredfeedbacksimulationthatdelivers50%ofthefeedbackpoweraskineticenergy.Noticethatthecoreachievesapproximatelong-termbalancewhenthejetpowerrisestomatchthecoolingluminosityfromwithinthecentral˘100kpc.Thisistypicalofoursimulationsthathaveasignicantfractionofthefeedback82˚˙˘ˇˇ˘˙˛ˆˆˆFigure4.1AGNPowervs.CoolingRateinIdealizedAGnSimulations:TotalAGNpower(thermal+kinetic)andcoolingluminosityforasimulationwithcoldgastriggeredfeedbackandfk=0:5.Thejaggedblacklineshowsinstantaneousjetpowersampledevery5Myr.Red,green,andbluelinesshowthetotalcoolingluminosityofgaswithin10,30,and100kpcrespectively,sampledatthesamecadence.powerinkineticform.However,thetotalfeedbackpowerbecomesmuchgreaterinsimulationswithpurelythermalfeedback.Figure4.2illustratesthevastdierenceinfeedbackpowerbetweenoursimulationswithfk=0andthosewithfk>0.Purethermalfeedbackeventuallysaturatesatapowerlevelmorethantwoordersofmagnitudegreaterthaninthesimulationswithsomekineticfeedback,evenwhencomparedtothecasewithfk=0:25.Furthermore,itcanbeseenthattheaveragefeedbackpowerintheself-regulatedsystemswithatleastsomekineticpowerisnotmonotonicallydependentonfk.Aslongassomeofthefeedbackpoweriskinetic,self-regulationhappensatapowerlevelof˘1045ergs1,whichissimilartothetime-averagedAGNpowerinferredfromobservationsofX-raycavitiesingalaxyclustercores(e.g.,McNamara&Nulsen,2012).
4.3.1.2ColdGasAccumulation
Feedbackpowerbecomesexcessivelylargeinthefk=0casebecausepurethermalfeedbackisineectiveatpreventinglargeamountsofcoldgasfromaccumulating.Figure4.3showsthat˘1012Mofcoldgasaccumulatesinlessthan1Gyrwhenfk=0,whereas.1010Maccumulatesduringthesametimeperiodinsimulationswithatleastsomekineticpower.Thelargecold-gasreservoirinthefk=0caseisnotsucientlydisruptedbythermalfeedbackandthereforeprovidesenoughcoldfuelfortheAGNtomaintainafeedback83ˆ˛#˜˜ˇ˘˙˘˙˘˙˘˙˘˙ˆ˛#˜˜ˇˆ˛#˜˚˝˚"˜˜ˇFigure4.2EectofKineticFractiononJetPower:Feedbackpower(_E)(thermal+kinetic)asafunctionoftimeforsimulationswithcoldgastriggeringandvaryingvaluesoffk,thefractionoffeedbackpowerinkineticform.PanelAshowstheinstantaneousvalueof_E.InPanelB,_Ehasbeensmoothedovera50Myruniformsmoothingkernel.PanelCshowsthecumulativeenergyreleasedbytheAGN.powerexceeding1047ergs1.Evenatthispowerlevel,theAGNfailstoejectoreliminatemuchofthecoldgasbecausethecoldgasisveryecientatradiatingawayfeedbackenergyowingtothen2dependenceofthecoolingrate.Theresultisthatthefeedbackenergythatdoesgointothecoldgasisalmostimmediatelyradiatedaway.Further,feedbackenergytendstopropagatemorereadilythroughthehotambientmediumalongthepathsofleastresistance,andendsupincreasingthethermalenergyofthediuse,volume-llinggaswithoutdiminishingthemassofcoldgasembeddedwithinit.Ifstarformationhadbeenallowedtoproceedinoursimulations,muchofthecoldgasthataccumulatesnearthecenterwouldeventuallyhaveformedstars.TheresultsshowninFigure4.3thereforeindicatethatthatpurethermalfeedbackwouldpermitatime-averagedstar-formationrate˘1023Myr1duringtherst˘1Gyr,whichismuchlargerthanobservedinallbutthemostactivelystar-forminggalaxyclustercores(O'Deaetal.,2008).Inordertounderstandwhykineticfeedbackissomuchmoresuccessfulthanthermalfeedbackinsuppressingcoldgasaccumulationandthestarformationthatwouldresult,weneedtolookathowthechoiceoffkaectstheradialdistributionofdensity,temperature,andentropyinthehotambientmedium.
4.3.1.3RadialProles
Figure4.4showshowtheaveragevaluesofdensity,temperature,entropy,andconcentrationoftraceruidchangeovertimeateachradiusinsimulationswithmixedkineticandthermalfeedback(fk=0:5,leftpanels)andpurethermalfeedback(fk=0,rightpanels).Gasoutsideof˘100kpcisnotshownbecauseitdoesnotevolveappreciablyover2Gyr,asthecoolingtimeatlargeradiiislongandlittleoftheAGN84˘˝ˆˇFigure4.3ColdgasMassforDierentAGNTriggeringAlgorithms:Totalmassofcoldgasasfunctionoftimeforsimulationswithcoldgastriggeringanddieringvaluesoffk.Theamountofcoldgasthataccumulatesinthesimulationwithfk=0istwoordersofmagnitudegreaterthaninanyofthesimulationswithsomeofthefeedbackenergyinkineticform.feedbackenergypropagatestothoseradii.Insideof100kpc,thefk=0:5simulationreachesanearlysteadystatein˘0:5Gyr,withdensity,temperature,andentropycontinuingtouctuatewithinnarrowrangesafterthattime.Theprolesofthetraceruidconcentrationdonotreachasteadystate,asthetraceriscontinuouslyinjectedovertimeanddistributedoutwardbythejet.However,thoseprolesdoshowthattraceruidisquicklymixedwiththeambientgasoutto&50kpcfromthecenter.Incontrast,thesimulationwithfk=0doesnotreachasteadystate.Inparticular,theazimuthallyaveragedspecicentropyofgasoutsideofthecentralfewkpcsteadilyriseswithtime,causingasteadydropinambientdensityandasteadyriseinambienttemperature.Initially,someoftheincreaseinmeanentropycomesfromtheremovaloflow-entropygasthroughcondensation(Voit&Bryan,2001;Voitetal.,2002;Nagaietal.,2007).However,themeanentropyat&10kpccontinuestoriseduringthesecondGyrofthesimulation,aftercondensationofcoldgashasleveledo.ThisriseisduetocontinualinputofthermalenergybyAGNfeedback,asmallfractionofwhichescapestheinnerfewkpcandpropagatesintotheICM,causingpressure-drivenexpansionoftheambientmedium.Theright-handpanelofFigure4.2showsthatAGNfeedbackinthefk=0simulationhasinjected˘1064ergafter2Gyr,whichiscomparabletothebindingenergyoftheentireintraclustermedium,althoughmostofthisisradiatedawaybythecoldgas.Thefractionthatdoesescapethecentralclumpofcoldgasslowsthecondensationprocessbyinatingtheclustercoreanddrivingthecoolingtimeoftheambientmediumat˘10kpcto˘5Gyrbutfailstoestablish85 #˛$'˝ˇ˜$'"$'"$'"$'"$'"$'"$'"$'"$'"$'"$'"˚˙&ˇ˜˘ˆ˛%˚!ˇ˘ˆ˛%˚!ˇFigure4.4EvolutionofGasQuantitiesforDierentKineticFractions:Prolesofvariousquantitiesasthesimulationevolvesforsimulationswithfk=0:5(left)andfk=0:0(right)withcoldgastriggering.Thickblacklinesdenotetheinitialconditions,whileotherlinecolorsindicatevaluesatlatertimes.Densityisweightedbyvolume,temperaturebymass,andmetallicitymymass.Entropyiscomputedusingthevolumeweightedtemperatureandthemassweighteddensity,asdiscussedinSection3.2.3ofSkoryetal.(2013).86ˇ˝˘# "ˇ˛˘˚˚˙ˇ˛˘ ˘ˆˆ˜ˆ˜ˆ˜"Figure4.5JetPowervs.CoolingRateforDierentTriggeringMethods:Theradiativecoolingrateofthegaswithindierentradiiiscomparedtothetotaljetpower(thermal+kinetic)forthesimulationwithfk=0:0.Attimesgreaterthan0.5Gyr,allofthecoolingisoccuringwithin10kpc,andthethreelinesoverlap.
aself-regulatedfeedbackloop.DespitethehighAGNpower,thissimulationisnotabletopreventabuildupofcoldgasfortworeasons:(1)feedbackenergydoesnotpropagatefarenoughfromthecenter,and(2)thermalfeedbackcannotdestroyalargecold-gasreservoir,onceitdevelops.AsillustratedinFigure4.5,almostallofthecoolingcomesfromthecentral10kpcwherethereisaconcentrationofcoldgaswithaveryshortcoolingtime.Althoughgasatthetemperatureoordoesnotcool,asmallriseintemperaturegreatlyincreasesitscoolingrateandpreventsthecoldgasfromheatingtotheambienttemperature.Thedevelopmentofalargecold-gasreservoiriscloselyrelatedtothefailureofthermalfeedbacktopropagatefeedbackenergybeyondthecentral˘30kpc.Figure4.6showsthermsgasvelocityasafunctionofradiusinsimulationswithdierentproportionsofkineticfeedback(fk=0.0,0.5,and1.0,respectively).Inthepurethermalcase,thereisasharpdropinrmsvelocitybeyond˘30kpcwhichisnotseeninthesimulationshavingsomekineticfeedback.Apparently,kineticfeedbackismoreeectiveattransportingfeedbackenergytolargeradii.Thisdiscrepancyarisesbecauseoutwardpropagationofcentrallyinjectedthermalfeedbackislimitedbytheamountofentropyitcangenerate.Itcreatescentralbubblesofhotgaswhichcanbuoyantlyriseonlyuntiltheyreachalayerofequivalententropy.Thenthebubblesblendwiththeirsurroundings.Inthissetofsimulations,centrallyinjectedhotbubblesstoprisingandblendwiththeambientmediumat˘30kpc,87Figure4.6EectofTriggeringMethodonVelocityProles:MassweightedproleofRMSvelocityinthehotambientmediumforsimulationswithcoldgastriggeringanddierentkineticfractions.Allprolesarecomputed1.75Gyrafterthebeginningofthesimulation.
asindicatedbythermsvelocitycurvesinFigure4.6,aswellasthepropagationoftraceruidinpanelHofFigure4.4.Wethereforeconcludethatourimplementationofpurethermalfeedbackdoesnotaddmuchheattogasinthe30{100kpcrangeofradiibutinsteadsteadilyraisestheentropyofambientgasat10{30kpc,whichattensitsentropygradient.Kineticfeedback,ontheotherhanddoespropagatebeyond30kpcandconsequentlyallowsgasintheentire10{100kpcrangetosettleintoaquasi-steady,self-regulatedstate.4.3.1.4JetPrecession
Inadditiontothebreakdownbetweenkineticandthermalfeedback,wehavealsoinvestigatedtheroleofjetprecession.JetsfromAGNcanreorientthemselvesontimescalesofafewtensofMyr(Dunnetal.,2006;Babuletal.,2013),butthedetailsofthisprocessarestilluncertain,andoursubgridmodelandidealizedsetuparenotcapableofself-consistentlymodelingjetprecession.Instead,wefollowLi&Bryan(2014a)andforcethejetstoprecessaroundaxedaxis.Previousstudieshavefoundthatsomeprecessionisnecessaryforself-regulationifthejetsarehighlycollimated.Otherwise,theydrilllong,narrowchannelsthroughtheICManddepositthebulkoftheirenergyfarfromthezoneinwhichself-regulationcanhappenVernaleo&Reynolds(2006).Figure4.7showsslicesoffoursimulationsperformedwithdierentjetprecessionangles.Whenthejetsdonotprecess(jet=0),theycarvechannelsthroughtheICMthatextendwellbeyond˘40kpc.However,duetoKelvin-HelmholzinstabilitiesandarticialviscosityintheZEUScode,theystillproducesomeheating88ˆ˙˙˘ˆ˙˙˘Figure4.7EectofJetPrecession:Densityslicesforsimulationswithdierentvaluesofjet.Inourmodel,theAGNjetprecessesaroundthezaxiswithaperiodof10Myrataconstantanglejetwiththezaxis.Allsimulationsusecoldgastriggeringandhavefk=0:5.closetotheAGNbutdonotdrivestrongshocks.Withasmallprecessionangle(jet=0:15;0:25),eachjetcontinuallyencounterscoldcloudsofcondensingmaterialthatblockitspath.Thesejet-cloudinteractionsrandomlydivertthejets,depositingtheirenergyinawiderrangeofdirections,whichcausesmoreoftheirkineticenergytothermalizeatsmallerradii.Precessionalsoproducesmoreturbulenceandcreatesshocksthatpropagateoutwardoveralargerangeofsolidangles.Astheprecessionangleincreases,thejetenergyspreadsoveralargerrangeofsolidanglesateversmallerradii,andjet-cloudcollisionsbecomemorefrequent.AsseeninthelastpanelofFigure4.7,thisleadstoamoredisturbedmorphologyat.20kpcandalargermassofaccumulatedcoldgas.Inthatrespect,kineticfeedbackwithaverylargeprecessionanglebecomesmorelikethermalfeedback,inthatfeedbackenergydoesnotpropagateasfarfromthecenterbeforeitbecomesthermalized.
4.3.2AGNTriggeringMechanisms
WedonotseestrongdierencesbetweenoursimulationswithdierentAGNtriggeringmechanisms,aslongasweareusingamaximumspatialresolutionof196pc(seeFigure4.8).ThisislikelyaconsequenceofbeingabletoresolvethemultiphasemediumintheregionsurroundingtheAGN.Sinceallofthetriggeringmechanismsconsideredaredependentongasdensity,acold,denseclumpofgasaccretingwilltriggeralargeoutburstregardlessofthedetailsofthetriggeringalgorithm.Theoutburstwillcontinueuntilthecoldgasisgone,ensuringthatroughlythesameamountofenergyisreleasedinallcases.Thecoldgaswouldnot89402002040z (Kpc)Cold, fk=01.0000 GyrCold, fk=0.5Cold, fk=1402002040z (Kpc)Bondi*100, fk=0Bondi*100, fk=0.5Bondi*100, f  = 1402002040x (Kpc)402002040z (Kpc)Booth & Schaye, fk=0402002040x (Kpc)Booth & Schaye, fk=0.5402002040x (Kpc)Booth & Schaye, fk=1k1e-261e-251e-241e-23Density [g cm-3]Figure4.8ComparisonofSimulationswithDierentTriggeringAlgorithmsandKineticFrac-tions:Slicesofdensitythrough9simulationswithdierenttriggeringmechanismsandkineticfeedbacklevels.Allsimulationsareshown1Gyrafterthebeginningoftherun.Simulationsinthetoprowaretriggeredbycoldgasaccretion,themiddlerowbyBondiaccretionwithaconstantboostfactor,andthebottomrowusingthemethodofBS09.fkgivesthefractionofthefeedbackthatisreturnedaskineticenergy.90˚ ˙(#˙#˝"˜ˆ""%˛ˇ˛˘(˙"!ˆ˚'˚ ˙(#˙#˝˚ ˙(#"%˘˜"&˙#˙#˝Figure4.9EectofTriggeringMechanismonJetPower:LikeFigure4.2,butforsimulationswithdierenttriggeringmechanisms.Allsimulationshavefk=0:5.AsinFigure4.2,PanelAshowstheinstantaneousvalueof_E,PanelBshows_Esmoothedover50Myr,andPanelCshowsthecumulativejetpower.
necessaryberesolvedbysimulationswithcoarserresolution,implyingthatthosesimulationsmightbemoresensitivetothechoiceoftriggeringalgorithm.Eachtriggeringmechanismdependsongasdensityeitherdirectly(BSandBoostedBondi-likeaccretion)orindirectly(coldgastriggering),andintheBSandBoostedBondi-likecasestheboostparametershavebeenchosentoprovidethe\right"amountoffeedback.Duetothedensity-dependentaccretionrate,acoldclumpfallingintotheaccretionzonethenalwaysproducesasurgeinjetpowerthatcontinuesuntiltheclumpiseithercompletelyheatedorcompletelyaccreted.Figure4.9showstotaljetpower(thermal+kinetic)versustimeforsimulationswithdierenttriggeringmechanisms.IntheBondi-likeandBSruns,feedbackisalwaysactive,butthepowerlevelisrelativelylowbeforecoldgasstartstocondenseanddrivesuptheAGNaccretionrate.Aftercondensationbegins,allthreetriggeringmechanismsleadtoself-regulatedjetpowerlevelsthatarenearlyidentical.Figure4.10showstheradialprolesofvariousquantitiesintheambienthotICMafter2Gyrforeachtriggeringmechanism.Therearesomedierencesintheinner10kpc,butthiszoneisstronglyaectedbythequicklyvaryingjet,producingprolesthatarevariablewithtime(seeFigure4.4).Between10and30kpc,therunwithcold-gastriggeringisslightlycolderandmoresusceptibletothermalinstabilitythantheotherruns,basedonthelowertcool=tratio.Beyond30kpc,therearenosignicantdierencesbetweenrunswithdierenttriggeringmechanisms.91ˆ$˙ ˚˛˜˝˜˜#ˇˇ%˜˚ˆˆ$˙ ˛ !#$!ˆ$˙ ˚#!˜ %˛ˆ$˙ #˜˜˝˘˘Figure4.10EectofTriggeringMechanismonGasQuantities:Prolesofdierentquantitiesforsimulationswithdierenttriggeringmechanismsafter2Gyr.Allsimulationsusefk=0:5.neisweightedbyvolume,whiletheotherprolesareweightedbymass.Allprolesexcisegasbelow3104K.92˚ ˙(#˙#˝"˜ˆ""%˛ˇ˛˘(˙"!ˆ˚'˚ ˙(#˙#˝˚ ˙(#"%˘˜"&˙#˙#˝Figure4.11EectofAccretionRadiusonJetPower:Jetpowervs.timeforsimulationswithracc=rdisk=2kpc.Allsimulationshavefk=0:5.LikeinFigure4.2,PanelAshowstheinstantaneousvalueof_E,PanelBshows_Esmoothedover50Myr,andPanelCshowsthecumulativejetpower.4.3.3AccretionRadius
Incosmologicalsimulationsofgalaxyclusterevolution,onewouldlikeasubgridmodelforAGNfeedbackthatgivesreliableresultsfortheSMBHaccretionrateevenwhentheBondiradius(letalonetheSchwarzschildradius)isnotresolved.Atspatialresolutionscoarserthan˘0:5kpc,thesizeofthe\accretionzone"thatdeterminesAGNfeedbackpowerwillnecessarilybelargerthanthatusedinourducialsimulations.WiththisincreaseinRacc,theresponsesofAGNtriggeringalgorithmswilldependonconditionsatlargerradii,whichcancoupleAGNfeedbacktoICMpropertiesatgreaterdistancesbutmayalsopermitlargeramountsofgastocondensebeforetheAGNfeedbackresponsebecomesstrongenoughtoopposecooling.TounderstandhowthesizeoftheaccretionzoneaectsAGNtriggering,wehavecarriedoutsimulationsinwhichRaccisincreasedto2kpc.Themaximumspatialresolutionremainsthesame,withasmallestcellwidthof196pc,meaningthattheaccretionradiusisalwaysresolvedbymultiplecells.IncarryingoutthesesimulationswesetthedistanceRJofthedisk-shapedjetinjectionregionequaltoRacc,sothatthejetemanatesfromtheedgeoftheaccretionsphere,notfromwithinit.Figure4.11showsjetpowerasafunctionoftimeforthesimulationswithalargeraccretionradius.Inallthreesimulations,uctuationsinjetpowerarenoticeablysmallerthanintheducialcase.WiththeexceptionoftheBoostedBondi-likerunthecumulativejetpoweriscomparabletotheearlierruns,andwedidnotobservequantitativedierencesintheICMproperties.However,theBoostedBondi-likesimulation,inwhichAGNfeedbackpowernowdependsonaveragegaspropertieswithinalargervolume,takeslongertorampup,resultinginalarge(>1012)massofcoldgasandahighercumulativejetpower.934.4Discussion
OursimulationshaveshownthatdierenttriggeringanddeliverymethodsforsubgridmodelsofAGNfeedbackcanhaveprofoundlydierenteectsontheresultingpropertiesoftheICM.WearenotattemptingtodeterminewhichmethodisthemostaccuratemodelofanAGNbutrathertoanalyzethereasonsforthosedierences.Ineachcase,trackingtheaccumulationofcold-gasfueliscritical,meaningthatwemustconsiderwhatallowsgastotransitionfromthehotambientmediumintothecold-gasfuelreservoir.Inthissection,wediscussthattransitionandconsiderthepotentialeectsofphysicalprocessesnotincludedinourmodels.
4.4.1PrecipitationandAGNFueling
Clearly,oursimulationswithpurethermalfeedbackbehavemarkedlydierentlythansimulationswithkineticfeedback,evenwhenfkissmall.ThepurethermalfeedbackrunsexperiencealargebuildupofcoldgasthatessentiallysmotherstheAGN,causingittoghtbackwithincreasinglypowerfulbursts.TheICMinthevicinityoftheAGNissubjecttobothradiativecoolingandheatingfrommixing,dissipation,andshocks.Thus,analyzingthethermalstabilityoftheICMmaygiveinsightintotheaccumulationofcoldgasandhelptoexplainthedierencesthatarisefromamongthesefeedbackalgorithms.Voitetal.(2015b)presentedevidencefora\precipitationtriggered"modelforcouplingtheAGNpowertothecoolingrateoftheICM.Intheprecipitationmodel,thecoolingICMbecomesthermallyunstable,leadingtothecondensationofcoldgas.ThiscoldgasisthenaccretedbytheAGN,triggeringfeedback.ThefeedbackheatstheICM,restoringthermalstabilityandreducingfurtheraccretion.Ascosmologicalsimulationstypicallylacktheresolutiontomodelthecondensationprocessitself,thethermalinstabilitycriterioncanbeusedtopredicttheamountofcoldgasavailableforaccretion.Foragravitationallystratiedmedium,onewouldexpectthatthermalstabilitywouldberelatedtotwonaturaltimescales|thecoolingtimescaletcoolandthedynamicaltimescalet.Simulations(McCourtetal.,2012;Sharmaetal.,2012b)ndthattheformationofcoldgasfromathermallyunstablemediumcanoccurwhenevertcool=t.10(ButseeMeeceetal.(2015),whichndsthatcondensationcanoccurforlargervaluesinsomecircumstances.)Similarly,theobservationsofVoit&Donahue(2015)andCavagnoloetal.(2008)showthatclusterswithtcool=t.10arelikelytoexhibitmultiphasegas,whileclustersabovethatratiodonot.Figure4.12showsthedistributionsoftcool=tandspecicentropy(K)forsimulationswithpurethermal(fk=0:0)andpartkinetic(fk=0:5)feedback.PanelAofFigure4.12showsthattheICMinthethermalfeedbacksimulationisdividedintotwophases.First,thereisahotphasewithtcool=t˛10thatoccupies94$$$!˙$$$#˙˚ˆˆ˜˙˚ˆˆ˚˝ˆˇˆˇˇ˜˙ˇ˚˝ˆ˙" ˘˝˚ˆˇˆˇˇ$$!˙$$$#˙˚ˆˆ˜˙˚ˆˆ˚˝ˆˇˆˇˇ˜˙ˇ˚˝ˆ˙" ˘˝˚ˆˇˆˇˇFigure4.12PDFofTimescaleRatioforThermalandKineticFeedback:Distributionofthetcool=tffratioforsimulationswithcoldtriggeringandeitherfk=0:0(leftcolumn)orfk=0:5(rightcolumn),shownat1.46Gyrafterthebeginningofthesimulation.PanelsAandBshowslicesofthelocaltcool=tforeachsimulation.Thecolor-breakinthescaleattcool=t=10indicatestheprecipitationthresholdidentiedbyearlierstudies.PanelsCandDshowthedistributionoftcool=tvaluesnormalizedbythetotalmassateachradius.Thecolorsshowthemassineachbindividedbythetotalmassinthatradialshell.Similarly,PanelsEandFshowtheEntropydistribution.Coldgas(<3104K)isexcludedfromtheanalysis.95thebulkofthevolumeoutsideof10kpc.Second,thereisalargeaccumulationofcoldgasthatnearlysmotherstheAGN.ThecoldgasmassbuildsupquicklyandthenstopsgrowingwhenthecoolingtimeinthehotICMrisestoseveralGyr.OutburstsofthermalfeedbacksporadicallypropelstreamersandblobsofcoolgasradiallyoutwardsfromtheAGN.Thesecoldstreamerstraveloutseveraltensofkpcbeforeturningaroundandrainingbackdownontothecore.PanelCshowsthatthereisalargespreadintcool=tandKinthe10{30kpcrange.Mostofthegasatintermediatevaluesoftcool=tdoesnotrepresentcondensationintheusualsense.Instead,itisgasintheboundarylayersofthestreamersthatiseithercoolingontothemorbeingheatedbyinteractionswiththehotICM.AsPanelsB,D,andFofFigure4.12illustrate,thegaspropertiesoftheICMforthesimulationwithfk=0:5areverydierent.TheICMhasmuchlowermeanvaluesoftcool=tandKateachradiusoutto30kpc.PanelBistypicalofthestateoftheclusterafterthejethasformed,withthevolumeinwhichtcool=t.10occupyingaroughlysphericalregionofradius˘20kpc,excludingahotchannelnearthejetaxis.Overallradiativelossesarenearlybalancedbygentleshockheatingoverseveralcoolingtimes.However,atradiiof˘10kpc,wheretcool=treachesaminimumvalue.10,weobserverelativelysmallamountsofcondensinggas.ConsistentwithLi&Bryan(2014b),thiscondensationoccursatthejet/ICMinterfacewherethejetgeneratesnon-linearentropyuctuationsbyupliftinglow-entropygasclosetotheAGNtogreaterheights,whereitthencondensesandfallsbacktowardthecenter.ThesecondensatesarethenaccretedbytheAGN,poweringthejetsandmaintainingthermalbalanceinthecluster.ThedramaticdierencesinthebehaviorofthecoldgasandthejetinthesesimulationshastodowithhowtheAGNdistributesenergytothesurroundinggas.Inthepurethermalfeedbackcase,thegasheatedbytheAGNatrsttendstofollowthepathofleastresistance,bypassingthedensergasnearthecore.Thisleadstoanaccumulationofcoldgaswithaveryshortcoolingtime,whichisabletoabsorbandreradiatetheAGNfeedbackatlatertimes.TheAGNinjectsenergyveryclosetothecenterofthecluster,whereitisimmediatelyradiatedawaybycoldgas.Inthe(fksimulation,theoutowcreatesahotcocoonarounditselfthatrapidlyrises.Thisoutowliftsthecentralgasoutward,whichhelpsdisruptthecoolingowandpullssomelow-entropygasupwardalongwiththejets.Thekineticoutowalsoallowsthefeedbackenergytopenetratetolargerradii(>10kpc)andheattheICMatgreaterradii.ThishelpstomaintainthebalanceofheatingandcoolinggloballyandpreventstheICMfromdividingintoahotandacoldphase.Thejetisabletoheatgasfurtherout,throughmixing,turbulentdecayandweakshocks,whichpreventsthecoolingtimeofalargefractionoftheICMfromgoingbelow10t.Thus,aclusterwithawarmer,lessdensecorewillrequirelessenergyinputtoregulatethanaclusterwithacold,densecore.Inadditiontodepositingfeedbackfurtherout,thekineticoutowsallowcoolinggastomixwiththehotgasinthejet.ThisincreasesthecoolingtimeoftheICMandstronglyinhibitstheformationofmorecold96gas.Thecoldorcoolinggasthatisnotaccretedissoonsweptupinthejet,whereitisdisruptedorheated.ThejetthuspreventsthecoldgasfromsmotheringtheAGN,allowingthefeedbacktoheattheICMratherthanquicklyradiatingaway.ThisexplainswhyaweakerAGNisabletoregulatetheICMinthekineticjetcasethaninthemorepowerfulpurethermalfeedbackcase.
4.4.2CaveatsandAdditionalPhysics
Inthisstudy,bothoursetupandourimplementationofAGNfeedbackhavebeensimpliedinordertofocusontheessentialfeaturesofcouplingbetweentheAGNandtheICM.Ofcourse,thesituationinrealclustersismorecomplicatedthanourmodel.Inadditiontothesesimplications,thereareanumberofpossiblyrelevantphysicalprocessesthatwehavenotincludedinourmodel,bothtosimplifytheproblemandtoreducethecomputationalresourcesrequired.Theseprocessesandtheirpotentialeectsarediscussedinthissection.
4.4.2.1Conduction
Oursimulationsdonotincludethermalconduction,eitherisotropicoralongmagneticelds.Fromatheo-reticalpointofview(Voitetal.,2015b,2008),whileconductionmaywellbeimportantforregulatingthethermalstateofwarm-coreclusters,cool-coreclustersliebelowthetcoolproleatwhichconductivetransportcanbalanceradiativelosses.Smithetal.(2013)hassimulatedcool-coreclusterswiththermalconductionbutwithoutAGNfeedback,andconcludesthatthermalconductionisnotabletopreventthecoolingcatas-tropheonitsownanddoesnothavealargeimpactonglobalclusterproperties.However,conductioncouldwellbeimportantfortheprecipitationtheory,asstrongconductioncouldsmoothouttheperturbationsthatevolveintonon-linearoverdensities.Waghetal.(2014)haveinvestigatedtheeectsofconductiononthermalstabilityandfoundthatconductionwouldneedtobequitestrongtopreventcondensation.4.4.2.2MagneticFields
Theintraclustermediumisknowntobeweaklymagnetized(Carilli&Taylor,2002).Overall,themagneticeldisbelievedtobetangledanddynamicallyunimportant.However,magneticeldsmayaectheattransportinthecorebymakingconductionanisotropic,astheelectronsthatmediateconductionwilltravelmoreeasilyalongeldlinesthanperpendiculartothem.Theimportanceofanisotropicconductionwilldependonthemagneticeldconguration,thedevelopmentofplasmainstabilities,andstirringoftheplasmabygalaxymotionsorAGNoutows.Atangledmagneticeldwouldbeexpectedtosupressconductiontoroughly1/3oftheSpitzervalue.However,aweaklymagnetized,conductingICMwithatemperature97gradientmightbesusceptabletoeitherthemagnetothermalinstability(MTI;Balbus,2000,2001;Quataert,2008)ortheheat-ux-drivenbuoyancyinstability(HBI;Quataert,2008;Parrishetal.,2009).Incool-coreclusters,theHBIwouldalignthemagneticeldperpendiculartoanoutwardtemperaturegradient,limitingtheinwardheatux.However,simulationssuchasRuszkowskietal.(2011)havefoundthatanisotropicthermalconductionisnotstrongenoughtoreorientthemagneticelds,andYang&Reynolds(2015)ndthatstirringbytheAGNwouldovercometheHBI,leadingtoconductionwithaneectivenessof>0:2timestheSpitzervalue.Whilenotdynamicallyimportantonlargescales,magneticeldsmayaecttheprecipitationandAGNfeedbackprocesses.Waghetal.(2014)foundthatanisotropicconductionwillnotpreventcondensationunlesstheeldisverystrong.MagneticeldsmaybestrongeranddynamicallyimportantclosetotheAGN,wherejetinducedturbulenceandeldinjectionfromthejetmayamplifythemagneticeld(Duboisetal.,2009;Sutteretal.,2012;Ruszkowskietal.,2011).AlongtheAGNjets,magneticdrapingisthoughttoplayanimportantroleinpreservingcavitiesandcoldfrontsagainstdisruptionfromKelvin-Helmholzinstabilities(Ruszkowskietal.,2007;Dursi&Pfrommer,2008).Thepreservationofcavitieswouldchangethemodeofheattransportinthecluster,becauseinatingcavitiesandrisingbubbleswouldbebetterabletostirturbulence,transporthotgastolargerradii,anddredgeupcoldgasintheirwake.4.4.2.3StarFormation
BCGsinmanycoolcoreclustersareobservedtobeformingstars(O'Deaetal.,2008,2010;Loubseretal.,2015;McDonaldetal.,2015),butstellarfeedbackalonecannotpreventthecoolingcatastropheincool-coreclusters(e.g.Skoryetal.,2013).Althoughwedonotincludestarformationinourmodel,Lietal.(2015)useasetupverysimilartoourducialmodeltoperformanextensiveinvestigationoftheroleofstarformationinregulatingAGNfeedback.Oneexpectsthestarformationrate(SFR)ofaBCGtoberelatedtotheamountofmultiphasegaspresent.Lietal.(2015)doseeacorrelationbetweenAGNfeedbackandtheSFR.Inthosesimulations,stellarfeedbackislesseectivethantheAGNatheatingtheICMbutmoreeectiveatconsumingcoldgas.IftheAGNisinalow-powerstate,acentralreservoirofcoldgasbuildsupandbooststheAGNpowerona˘100Myrtimescale.AGNfeedbackthenheatstheICMandslowstherateofgascondensation.However,theAGNremainspowerfuluntilstarformationconsumesthecoldgasinthecentralreservoirona˘2Gyrtimescale.Withoutcoldcloudstofuelit,theAGNfeedbackpowersubsides,andanothercyclesoonbeginsastheambientmediumonceagaincoolsandbecomesthermallyunstable.Thus,theprimaryeectofstarformationistoregulatethecyclingbehavioroftheAGNonGyrtimescales.984.4.3ComparisonWithSimilarStudies
AstheimportanceofAGNfeedbackhasgainedgreaterappreciationinrecentyears,severalstudieshavebeencarriedouttoinvestigatethebestwaytoimplementAGNfeedbackinsimulations.ItisdiculttodoacomprehensivecomparisonbetweenourresultsandthoseofpreviousstudiesasthoseworkshavegenerallysampledalimitedfractionoftheAGNfeedbackparameterspaceorassumevastlydierentinitialconditionsthanwedohere.ThechiefaimofthispaperistobetterunderstandwhichaspectsofAGNfeedbackimplementationsaremostdecisiveindeterminingthequalitativeconsequencesofsub-gridmodelsforAGNfeedback.Withthisinmind,wediscussthemajordierencesbetweenourimplementationandsomeAGNimplementationsusedinrelatedstudiesofAGNfeedback.Wherepossible,wecompareourresultstothoseobtainedusingtheseotheralgorithms.Notethatinadditiontothemajordierencesdiscussedhere,therearemanyothersmalldierencesinthedetailsofhowAGNfeedbackisimplementedandinthechoicesofphysicalmodelsconsidered.AsdemonstratedbySection4.3,theresultsofanAGNfeedbacksimulationmaybesensitivetoseeminglysmalldierencesinimplementation,andcautionshouldbetakenwhencomparingonesetofresultstoanother.
4.4.3.1Li&Bryan2012-2015
TheclusterandAGNmodelemployedinourpaperarelargelyanextensionoftheLi&BryansimulationsofAGNfeedback(Li&Bryan,2012,2014a,b;Lietal.,2015),withonlysmallchangestotheclusterandjetmodel(althoughweextendtherangeoftriggeringandfeedbackparameters).BothourstudyandtheirsuseEnzo.Giventhesimilaritiesofoursetups,itisnotsurprisingthatoursimulationsgivesimilarresults.Ourmaximumspatialresolutionisslightlycoarser(196pcvs.60pc),butweobtainsimilarbehaviorforsimilarchoicesoffeedbackparameters.OurndingsindicatethattheLi&Bryanresultsshouldberelativelyinsensitivetovariationsinthetriggeringmechanism,theamountofAGNprecession,andthedetailsoftheaccretionprocess.BothstudiesndthatthebehavioroftheAGNisrelativelyinsensitivetothekineticfractionoftheoutowaslongasthekineticfractionisnon-zero.OurstudydoesndthatthemassofcoldgasformeddependsstronglyontheAGNimplementation,butdoesnotaectthelongtermbehaviorofthesimulation.WegenerallyseeamassofcoldgasthatisanorderofmagnitudelessthanwhatLi&Bryanfoundintheirducialmodelbutobtainasimilarmasswhenweusethesamesetofparameters.ThisvariabilityinthecoldgasmassisconsistentwiththeparametervariationstudiesinLi&Bryan(2014a).994.4.3.2Gasparietal.2011
Gasparietal.(2011b)simulateAGNfeedbackusingtheFLASHcode(Fryxelletal.,2000).TheymodelanidealizedversionoftheclusterAbell1795withinastaticandsphericallysymmetricgravitationalpotentialusingasetofphysicalprocessessimilartothoseusedhere.Theminimumresolutionintheirstudyis2.7kpc.AGNfeedbackismodeledasapurelymechanicaljetwitheithercoldorhot(Bondi-like)triggeringanddierentjeteciencies.Theyalsoconsiderbothsteadyandintermittentjets.ForBondi-liketriggering,theaccretionrateiscalculatedfromthepropertiesofgaswithin5or10kpc.Gasparietal.(2011a)usesasimilarAGNmodelbutagravitationalpotentialappropriateforagalaxygroup.Gasparietal.(2011b)ndsthatbothacoldgastriggeredandaBonditriggeredAGNimplementationareabletobalanceradiativecoolingandpreserveacool-corestate.Themostsuccessfulcoldgasmodel(modelA3inthatpaper)issignicantlymoreburstythanoursimulations,withadutycycleofonly6%,resultinginonlyaround50outburstseachwithpowerontheorderof1048erg/s.Thetotalinjectedenergyafter2Gyrisontheorderof1061erg,consistentwithourresults.Weattributetheobserveddierenceinoutburstpoweranddutycycletothechoiceofaccretionradius,whereweuse0.5kpcandtheyuse10kpc.AsseeninFigure4.11inourpaper,increasingthesizeoftheaccretionradiusresultsinalargervariationinAGNpower.Thisfollowsfrommorecoldgasbeingabletotinsidethelargeraccretionzoneandfromthedicultyofexpellingcoldgasfromalargergravitationalwell.Inagreementwithourresults,Gasparietal.(2011b)ndsthatBondifeedbackwithalargeaveragingzone(10kpcintheirsimulations)isnotabletohaltthecoolingcatastrophe.Theirmodelwithanaveragingzoneof5kpcisabletobalancecoolingoveralongperiodoftime.Unlikethecoldgastriggeredcase,theBondiimplementationresultsinalowpower(order1044erg/s)jetwithlittlevariationinintensity.Inoursimulations,theBondiandcold-triggeredimplementationsactsimilarlywhenusinganaccretionradius/averagingzoneof0.5kpc.Weascribethistothehigherresolutionofoursimulations,whichareabletoresolvethecoldgasdirectly.
4.4.3.3Yangetal.2012
Yangetal.(2012)examinestheeectofdierentAGNsubgridmodelsonobservablepropertiesofsimulatedgalaxyclusters.Theymodelanidealizedclusterwithvirialmass1:51014Mandapolytropicequationofstate,alsousingFLASH.Theminimumresolutionofthesesimulationsis1.0kpc.Thephysicalprocessesconsideredareagainsimilartoours,whiletheAGNfeedbackmodelissomewhatdierent,consistingofeitherlarge(tensofkpc)thermalbubblesosetfromthecoreorjetswithwidthsofafewkpc.TheaccretionratewasdeterminedfromtheBondirate,withaconstantboostfactorrangingfrom1to100indierentruns.100AlthoughYangetal.(2012)doconsiderjetswithpurethermalfeedback(aswellasthermalbubblesoriginatingneartheAGN),theydonotseethesamesmotheringbehaviorthatwedo.Infact,thegasintheirsimulationsdoesnotbecomeverydense,rarelyexceedingdensitiesofne=101cm3.Weattributethesedierencestothenerresolutionofoursimulations,whichallowustoresolvethecondensationprocessandtheformationofcoldgasneartheAGN.
4.4.3.4Duboisetal.2012
Duboisetal.(2012)comparethermalandmechanicalfeedbackincosmologicalsimulationsusingthecodeRAMSES.Thesimulationsgenerallyhaveaminimumresolutionof1.52kpc,butsomerunshavehigherresolution.TheAGNpowerisdeterminedusingtheBS09method.ThermalenergyisreleasedinasphereofaafewcellsneartheAGN,whilekineticfeedbackisreleasedinajet.SimilartoYangetal.(2012),theydonotobserveAGNsmothering,butagainemployacoarserresolutionthanweuseinoursimulations.4.5Conclusions
Wehavecarriedoutacontrolledcomparisonofseveralcommonlyusedsub-gridimplementationsofAGNfeedback.OurmodeltreatstheAGNasaparticlesittinginthecoreofanidealizedcool-corecluster.TheAGNistriggeredbasedonlocalconditions(eithertheamountofcoldgasortheBondirate,witheitheraxedoradensitydependentboost)andreturnsenergytotheICMaseithercentralizedthermalblasts,akineticjet,oramixofthermalandkineticenergy.Ourmainconclusionsare:1.Purelythermalfeedbackproducesverydierentresultsthanfeedbackwithevenasmallkineticcom-ponent.Inthepurethermalcase,theAGNisinitiallyunabletoinhibitcoolingimmediatelyoutsideofthecore,leadingtoabuildupofcoldgas.ThisgassmotherstheAGNandimmediatelyradiatesawaythefeedbackenergy,evenifthefeedbackzoneitselfisheatedtoahightemperature.ThisalsoresultsinheatingoftheICMoutsideofthecorethroughacombinationofshockheatingandpreferentialcondensationoflowentropygas.AddingakineticcomponentallowstheAGNtopropagateenergyoutsideofthecoreandpreventssmotheringoftheAGN.2.Whensomefractionofthefeedbackisreturnedasakineticjet,theAGNisabletopreventthelargeaccumulationofcoldgasthatresultsfromacoolingcatastrophe.Instead,AGNfeedbackself-regulatestheICMinaquasi-steadystatewithtcool=t˘10at.20kpc.Theclustercoreiscooleroverallthanthecasewithpurethermalfeedback,butcontainsmuchlesscoldgasaroundtheAGN.1013.Wedoobservelargedierencesbetweencold-gastriggeredfeedback,boostedBondi-liketriggeringorBoothandSchayeaccretion,aslongasthe\accretionzone"usedtodeterminetheAGNfuelingrateissucientlysmall(˘200pc).Thisisprobablybecauseallthreemethods,bydesign,enduptriggeringstrongAGNfeedbackwhencoldcloudsbegintoaccumulateintheaccretionzone.4.Increasingthesizeoftheaccretionzone(to2kpc)reducesshort-termvariationinjetpowerbutdoesnotsignicantlyalterthetotalamountofAGNfeedbackortheglobalICMpropertiesinthecold-gastriggeredorBoothandSchayecases.However,theboostedBondi-likesimulationdoesnotachieveself-regulation,becauseAGNfeedbackdoesnotrampupfastenoughtopreventacoolingcatastrophe,resultinginalargecentralaccumulationofcoldgas.5.VerylargejetprecessionanglesdistributetheAGNfeedbackenergy,makingsimulationswithsignicantkineticoutputbehavemorelikesimulationshavingpurethermalfeedback.ThishappensbecausethekineticenergydoesnotescapetolargeradiiandthermalizesclosertotheAGNwhenitisspreadovertoolargeasolidangle.Furtherimprovementstosub-gridfeedbackmodelsandabetterunderstandingoftheAGNfeedbackprocessarenecessaryforthenextgenerationofgalaxyandgalaxyclustersimulations.Onatheoreticallevel,muchworkiscurrentlybeingdoneonthelinkbetweenthermalinstability,coldgasformation,anditsroleintriggeringfeedback.OngoingobservationswithChandraandXMM-NewtonaswellasfutureobservationswiththeSmart-XandAthenamissionswillgiveabetterunderstandinghowtheAGNfeedbackprocessoperatesinrealclusters.Finally,newimplementationsmustbedevelopedforcapturingtheconnectionbetweenAGNandtheirenvironments.Whilethesimulationsinthisworkhaveamaximumresolutionof˘200pc,cosmologicalsimulationsandsimulationswithmorecomplicatedphysicsgenerallyhave&kpcresolutionduetocomputationalresourcelimits.Infuturework,wewillaimattranslatingtheresultsfromthisprojectintoasub-gridimplementationthatcanbeusedatthesecoarserresolutions.4.6Acknowledgments
TheauthorswouldliketothankGusEvrard,MateuszRuszkowski,GregBryan,andYuanLiforhelpfuldiscussionsduringthepreparationofthispaper.ThisworkwassupportedinpartbyMichiganStateUni-versitythroughcomputationalresourcesprovidedbytheInstituteforCyber-EnabledResearch.BWOwassupportedinpartbythesabbaticalvisitorprogramattheMichiganInstituteforResearchinAstrophysics(MIRA)attheUniversityofMichiganinAnnArbor,andgratefullyacknowledgestheirhospitality.ThisworkwassupportedinpartbyNASAthroughgrantsNNX12AC98G,NNX15AP39G,andHubbleTheory102GrantsHST-AR-13261.01-AandHST-AR-14315.001-A.ThesimulationswererunontheNASAPleiadessupercomputerunderallocationSMD-15-6514.Theactiveparticleframework,uponwhichourAGNimple-mentationisbased,wasdevelopedbyNathanGoldbaumatNCSA,andwethankhimforhissupport.Enzoandytaredevelopedbyalargenumberofindependentresearchersfromnumerousinstitutionsaroundtheworld.Theircommitmenttoopensciencehashelpedmakethisworkpossible.1035Conclusions
TheworkdiscussedinthisdissertationhasfocusedontheroleofAGNinregulatingthethermalstateofcool-coregalaxyclusters.Chapters1and2laidoutthephysicalbackgroundofthecoolingowproblem,theevidenceforsomeformoffeedbackgainedfromobservations,andmotivatedtheprecipitation-regulatedfeedbackscenarioinwhichcoldgascondensesoutofathermallyunstablemedium,accretesontotheSMBH,andpowersanAGNthatrestoresthermalbalancetothecluster.Chapter3demonstratedhowthermalinstabilitycanleadtothecondensationofcoldgasclumps,andledtotheconclusionthatcoldgascanrapidlycondenseoutoftheICMwhentcool=t<10.Chapter4discussedtheimplementationandrobustnessofsubgridmodelsofAGNfeedback,ndingthatmodelswithsomedegreeofkineticfeedbackcandoagoodjobofsolvingthecoolingowproblem.Here,IconcludewithadiscussionofopenquestionsrelatingtoAGNfeedbackinclustersandpresentsomeavenuesforfuturework.5.1The`LastKpc'Problem
ThesubgridmodelsdiscussedinMeeceetal.(2016)andreferencesthereinbasetheaccretionrateonthegaswithinsomexedradius,andthusmaketheimplicitassumptionthatgaswithinthatradiusmakesitswaytotheSMBH,whereitisaccretedandpowerstheAGN.Thisassumptionisnecessary,asthesizeoftheSMBHistypicallywellbelowtheresolutionofthesimulations|theSMBHeventhorizonisontheorderofAUandtheBondiradiusandgravitationalinuenceradiusontheorderof10sofpc,whilesimulationsgenerallyhaveresolutionontheorderof100sofpcormore.However,itisnotatallobvioushowandwhetheracoldgascloudatseveral100sofpcwouldreachtheSMBH.ClumpsofgascondensingoutoftheturbulentICMwouldbeexpectedtohavesomeamountofangularmomentumrelativetotheSMBH,butneedtosomehowshedalmostallofitinordertoreachtheSMBH.Otherwise,thecloudswouldmoveballisticallyandwouldhavelittlechanceofhittingtherelativelysmallSMBH.Itispossiblethatsomecloudsarecreatedwithvirtuallynoangularmomentum,butthentherestofthegaswouldremain.Thatgaswouldlikelysettleintoalarge,rotatingdiskaroundtheSMBH,whichisnotobserved(Li&Bryan,2014a).Analternativeexplanationisthatclumpsofgasmovinginoppositedirectionswouldcollideandfallin1.AfullexplanationofhowgasreachestheSMBHwillrequiremodellingadditionalphysicsinhighres-olutionsimulations.Magneticeldsandradiativetransferarelikelytobedynamicallyimportantcloseto1Termedthe`ThreeStooges'modelbyBrianO'Shea.104theAGN,eveniftheyarenotfurtherout.Thephysicsofaccretiondisksarenotwellunderstood(seeTchekhovskoyetal.,2011),anditislikelythatdiskinteractions/instabilitiesareimportantforfunnellinggastowardstheSMBH.
5.2Galaxy/AGNInteractions
Thisdissertationhasalmostentirelyignoredthemostobviouscomponentofgalaxyclusters,namelythegalaxies.Whilethestellarmassofgalaxiesisonlyasmallfractionofthetotalclustermass,feedbackfromstellarprocesseshaveanimportantimpactongasdynamics,chemicalevolution,andobservablepropertiesofclusters.Afullunderstandingofgalaxyclusterswillneedtotakegalaxiesintoaccount.Starformationisgenerallymodeledinsimulationsusingasub-gridmodel,similartowhatIhavedonewithAGN.Unfortunately,thecurrentgenerationofsubgridmodelsistunedforsimulationsofgalaxiesanddonotproducerealisticresultswhenappliedtothescaleofclusters(seeArielietal.,2008).Specically,classicstarformationalgorithmsappliedtoclustersfailtoreproducetheobservednumber,distribution,andcolorsofgalaxies.Thisinturnleadstoproblemsinthechemicalevolutionofgalaxyclusters.NewalgorithmsforgeneratinggalaxiesinsimulationsofclustershavebeendevelopedbyArielietal.(2008),Arielietal.(2010),andCrosby(2016,inprep.).Thesemodelsusetheunderlyingdarkmatterdistributiontogenerategalaxiesandmodeltheinteractionofgalacticwinds,starformation,andrampressurestrippingfromtheICM.Initialresultsshowthatthesemodelsprovideabettermatchtoobservationsthanpreviousstudies.FuturesimulationswillneedtocombinegalaxyandAGNsubgridmodelsinordertoprovideafulldescriptionoftheevolutionofbaryonsinclusters.5.3AdditionalPhysicsandJet/ICMInteractions
Boththeoryandsimulationsmustnecessarilyadoptarangeofassumptionsandalimitedsetofphysicsduetothecomplexityoftheunderlyingsystem.Thereremainanumberofphysicalprocessesthatmightbeimportanttothedynamicsandthermalevolutionofgalaxyclustersbutthatarenotwidelyincorporatedinsimulations(includingthosepresentedinthisdissertation).HereIdiscusssomeoftheseprocessesandconsidertheirlikelyeectontheworkinthisdissertation.5.3.1Cooling
Inprinciple,thecoolingratedependsontheabundanceanddistributionofeachionizationstateofeachatomicspeciesaswellasthefreeelectrondensityandthebackgroundradiationeld.Thisisobviously105impracticaltocomputeatruntimewhilefollowingthenon-equilibriumabundancesofeachspecies,aseachtypeofatomwillhaveseveralionizationstates,eachwithdierentatomiclevels.Inaddition,manyreactioncoecientsarepoorlyconstrained.Finally,chemicalabundancesinclustersareonlyknownwithaccuracyforafeweasilyobservedelements(andtheseforonlyselectionizationstates),andthesearestilluncertainduetoobservationalandmodelinglimits.Therefore,simulationsmustmakeapproximationswhenitcomestocooling.Thecoolingroutinesinthisworkuseatabulatedcoolingrate(seeAppendixA)basedonaconstantanduniformchemicalcompositionandassumingequilibriumabundancesofionizationstates.Inreality,coolingismetallicity-dependent.Theassumptionofionizationequilibriumisusually,butnotalways,valid.Skoryetal.(2013)variedthecoolingratemodelusedforstudyinggalaxyclustersandfoundthatusingmetallicitydependentcoolinghadanoticeableeect,butdidnotchangetheoutcomeofthecoolingow.Similarly,IdonotexpectuncertaintiesinthecoolingratetochangetheultimateconclusionsofMeeceetal.(2015)orMeeceetal.(2016),thoughtheresultsaresomewhatdependentonthecoolingmodelthatisassumed.Fortheformer,variationsinthecoolingratemightchangethelocationofthetcool=tthresholdforprecipitation,butwouldnoteliminateit.Therefore,precipitationwouldstillbelikelytooccurinthelatterstudy,andtheAGNwouldbeexpectedtorespondaccordingly.Coolingroutinesmoreadvancedthanthoseusedinthisdissertationexist,andusingthemwouldprovideafullerunderstandingofthebehaviorofgasingalaxyclusters.Grackle(createdbyBrittonSmith,describedinKimetal.,2014)implementsmetallicity-dependentcoolingandcancomputenon-equilibriumabundancesforseveralprimordialspecies(H,D,Heandassociatedmolecules)atlowtemperature.Dengo(Silvia,2013),createdbyDevinSilvia,isabletotracknon-equilibriumabundancesforalargerrangeofelementsandmolecules.
5.3.2PlasmaPhysics
MostsimulationstreattheICMasapurelyhydrodynamicalproblemandneglecttheroleofmagneticeldsorplasmaeects.Magneticeldsinclustersaregenerallyweak(Carilli&Taylor,2002),sothisassumptionisgenerallyappropriate.Inaddition,plasmaphysicsiscomplicated(Schekochihinetal.,2009)anddiculttoimplement.ThestateoftheplasmaintheICMisalsouncertain(Eganetal.,2016).Magneticeldsandconductionmaybeimportantfordeterminingthermalstability,butonlyiftheseeectsarestrong.Field(1965)treatsmagneticeldsinhisstudyofthermalinstability,butndsthatmagneticeectsareonlyabletosuppressinstabilityiftheeldisstrong.Similarly,McCourtetal.(2012)considersmagneticeldsintheirsimulationsbutndsthattheydonotpreventtheinstability.Conduction106canstabilizethemedium,butonlyifconductionisverystrong.PlasmaeectsaremostlikelytoaectthedetailsofheattransportviaconductionandinteractionsbetweentheAGNjetandtheICM.Conductioninmagnetically-threadedplasmaisanisotropicsinceelectronscanmovealongmagneticeldlinesmoreeasilythanperpendiculartothem.Thiscanleadtovariousplasmaeectssuchastheheat-uxdrivenbuoyancyinstabilityormagneto-thermalinstabilities(Yang&Reynolds,2015).MagneticeldsmaybecomedynamicallyimportantclosertotheSMBH,andarealmostcertainlyimportantinthedynamicsofthediskandthejet.Finally,plasmaeectswillaltertheinteractionoftheAGNjetwiththeICM.AsdiscussedinChapter2,purelyhydrodynamiccavitieswouldbeshreddedbyinstabilitiesastheyrise,butmagneticeldsandviscositycouldacttokeepthemtogether.Thiscouldinturnaecttheabilityofjetstodredgeupgasandmetalsfromthecoreoftheclusteranddeliverthemfurtherout.5.4CosmologicalSimulations
ThesimulationsdiscussedinChapters3and4useidealizedinitialconditionsspeciedintermsofanalyticalformulae.Idealizedsetupsareusefulforperformingcontrolledexperiments,butdonotcapturetheeectsofinhomogeneityorcosmologicalstructureformation.Foratruetestoftheprecipitation-regulatedfeedbackhypothesis,thealgorithmsandadditionalphysicsdiscussedinthisdissertationmustbetestedinsimulationsthatusecosmologicalinitialconditions.Cosmologicalsimulationsaremorediculttorunthanidealizedsimulationsduetotheirincreasedcom-putationalcomplexity.Thedarkmattermustbefollowedusingn-bodydynamicsratherthanapproximatedwithastaticformula.Theeectsofcosmologicalexpansionmustalsobetakenintoaccount.Finally,anadaptivemeshcodelikeEnzomayendupreningmorecellsinacosmologicalsimulationthaninanidealizedone,asthe`interesting'cellsmightbespreadoveralargervolume.Thisincreasedcomplexityalsolimitstheresolutionofcosmologicalsimulations.TheidealizedsetupofMeeceetal.(2016)attainsaspatialresolutionof200pcinthesmallestcell,butcosmologicalsimulationsofgalaxyclustersgenerallyhaveresolutionsontheorderofafewkpc.AstheoutowsfromAGNarealsooforderkpcinwidth,thealgorithmsdiscussedherewillneedtobeadaptedforuseincosmologicalsimulations.Similarly,thecoarserresolutionwilllimittheabilitytoresolvecoldgasdirectly,meaningthatalternatecriteria(perhapsthetcool=tratio)willneedtobeusedtoapproximatetheaccretionrate.Galaxyclustersarethemostmassivestructuresintheuniverseandhostsomeofthemostenergeticpro-cessestoeveroccur.Thedynamicsandevolutionofthebaryoniccomponentsinvolvescalesrangingfromthemicrophysicsoftheatomstostructureformationinthecosmicweb.Theobservablecomponentsofclusters107encompassonlyatinyfractionoftheirtotalmassandenergy,andmuchofwhatisnotobservableisnotwellunderstood.Nevertheless,moderntheoriesandalgorithmscoupledwithpowerfulsupercomputershaveachievedresultsthat,consideringthebreadthandcomplexityoftheproblemtobesolved,areastonishing.Byincorporatingbettermodelsoffeedbackandphysicsinsimulationsofgalaxyclusters,itispromisingthatwewillattainabetterunderstandingofgalaxyclustersandthehistoryandevolutionoftheuniverse.108APPENDICES109AppendixAEnzo:AnAdaptiveMeshHy-
drodynamicsCodeforAstrophysics
Inthepasttwodecades,simulationhasjoinedtheoryandobservationasthethirdpillarofastrophysics.Hydrodynamicsimulations,whichattempttofollowthedynamicsofgas,areoftenthemostinformativetypeofastrophysicssimulation.ThesimulationsinthispapermakeuseoftheastrophysicshydrodynamicssimulationcodeEnzo.AfulldescriptionofEnzoisgivenintheEnzoMethodPaper(Bryanetal.,2014).Inthischapter,IgiveabriefoverviewofhydrodynamicssimulationsingeneralandtheEnzocodeinparticular.A.1EulerianHydrodynamics
EnzoisahydrodynamicscodethatusesanadaptivelyrenedmeshtosolvetheEulerequations.Foranidealuidwithaknownequationofstateandnegligibleviscosity,thedynamicsoftheuidcanbedescribedwithasetofconservationlaws:110SymbolDescriptiontTimeˆDensityvVelocityvectoreSpecicenergyPPressure˚GravitationalPotential LVolumetricCoolingrateVolumetricHeatingrateTableA.1SymbolsusedintheEulerequations(A.1,A.2,andA.3)@ˆ@t+r(ˆv)=0(A.1)@ˆv@t+r(ˆvv+IP)=ˆr˚(A.2)@e@t+r[(e+P)v]=ˆvr˚L+(A.3)whichdescribetheconservationofmass,momentum,andenergyrespectively.ThemeaningofthesymbolsinEquationsA.1,A.2,andA.3aregiveninTableA.1.TheEulerequationsyieldasetof5hyperbolicpartialdierentialequations(mass,energy,andthreemomentumcomponents)thatdescribethemotionoftheuid.Hyperbolicdierentialequationsdonotingeneralyieldanalyticalsolutionsandmustbesolvednumerically.Inpractice,thisinvolvesdiscretizingtheequationsineithermass(theLagrangianapproach)orspace(theEulerianapproach.)EnzoisanEuleriancodethatdiscretizesspacebysolvingthehydrodynamicsequationsonaCartesianmesh.Typicalproblemsinastrophysics,includingthesituationsdiscussedinthiswork,ofteninvolvespatialscalesthatcanvarybyseveralordersofmagnitude.However,onlyasmallvolumeofasimulationmaybe`interesting'enoughtowarranthighresolution,andtheextracomputationalresourcesneededtomodelthe`non-interesting'volumeofthesimulationmaynotbejustied.Enzocircumventsthisproblembyusinganadaptivemeshingalgorithmtooverlaygridswithhigherresolutionoverlowresolutiongrids.Withasmartchoiceofrenementcriterion,Enzoisthusabletoachieveahigheectivespatialresolutionwithoutsignicantcomputationaloverhead.CommonrenementstrategiesinEnzoarereningregionsbasedonover-density,densityortemperatureslope,highcoolingrates,ortheJeanscriterion.AlthoughadaptiverenementallowsEnzotoecientlysimulatelargevolumeswithhighresolutionwhereneeded,issuesrelatingtothetime-stepandloadbalancingmaylimittheadvantagesofrenementincertainsituations.Thetime-stepinEnzo(andallexplicithydrodynamicscodes)islimitedbytheCourant-111Freidrichs-Levy(CFL)conditionthydrominxics+jviji(A.4)foreachdimensioni.Conceptually,EquationA.4meansthatthetime-stepmustbeshorterthanthetimethatittakesinformation(intheformoftranslationalmotionorsoundwaves)tocrossacell.TheCFLconditionmeansthatthemosthighlyrenedregionsgenerallyrequirethesmallesttime-step.Forsimulationswithawiderangeofrenementlevels,thiscanresultinloadbalancingissues,wheremostoftheprocessingworkinasimulationisdedicatedtoupdatingasmall,highlyrenedregionwhiletherestofthesimulationwaits.Inaparallelcomputingenvironment,thismeansthatonlytheprocessorsupdatingthenestgridcellsareinuse,limitingopportunitiesforparallelization.EnzoimplementstwomainnumericalmethodsforsolvingtheEulerequations.TheZEUSmethod(Stone&Norman,1992)usesanitedierencealgorithmtocomputeuxes.Themethodissecondor-derinspaceandrstorderintime.ZEUSismorediusivethanothermethodsbutisrobust.Thesecondmethod,thePiecewiseParabolicMethod(PPM;Colella&Woodward,1984),usesparabolicinterpolationtocalculateuxes.ThePPMmethodisformallythirdorderaccurateinspaceandsecondorderintime,althoughtheuseofadaptivemeshingcanimpairthissomewhat.AlthoughPPMisformallymoreaccuratethanZEUS,PPMcanfailinregionsofstronggradientsorinthepresenceofstrongcooling.ThislackofrobustnesscanmakeZEUSanattractivechoiceforsimulationswithstrongfeedback,suchasthosedis-cussedinchapter4.EnzoalsocontainssolversforMagneto-Hydrodynamics(MHD)anduidscosmologicalcomovingcoordinates,butthesearenotusedinthecurrentwork.A.2DarkMatterandGravitation
Thegridbasedapproachusedforcomputinguiddynamicscannotbeappliedtothedarkmatteringalaxiesandclusters.Unlikethecollisionalparticlesintheplasma,darkmatterisassumedtobecollisionless(seeSection1.1.1).Darkmatterparticlesthatareclosetooneanotherinspacewillnotgenerallyhavethesamevelocity,meaningthattheparticleswilloccupya6Dphasespaceofpositionandvelocity.Thiswouldquicklybecomeintractableforevenacoarselyresolvedgrid.Enzomodelsthepotentialduetodarkmatterbyusingtracer`DarkMatterParticles'thatareassumedtorepresentrandomsamplingsofphasespace.TheparticlesevolveaccordingtoN-bodydynamics,computedusinganadaptiveparticle-meshalgorithm.Thesameparticle-meshalgorithmisusedtocomputetheselfgravityofthegasbyassigninggastonearbygridpoints.Inidealizedsimulationssuchasthosediscussedinthiswork,directlymodelingthedarkmattercan112introduceanunnecessarydegreeofcomplexityintothesimulation.Forthisreason,idealizedsimulationsoftenuseananalyticformforthegravitationalpotential.Furthermore,theselfgravityofthegasisoftensub-dominantonthelengthscalesofinterest(forexampleintheICMawayfromdensestellarstructuressuchasgalaxies),andthistoomaybeneglectedintheinterestofcomputationaleciency.A.3RadiativeCooling
Thegasandplasmathatllsspacebetweenthestarscoolsviaradiationoriginatingfromatomicandmoleculartransitionsorfree-freeemission.ThesecoolingprocessesarecapturedbytheLterminEquationA.3.ThesimplestformofcoolingcurrentlyimplementedinEnzousesatabulatedformoftheanalyticcoolingfunction(T)fromSarazin&White(1987).Agraphof(T)isshowninFigureA.1.ThecoolingrateisdenedsuchthatthecoolingrateperunitvolumeL(T;ne;np)isgivenbyL(T;ne;np)=(T)nenp:(A.5)Theenergyofthephotonsreleasedbyeachtransitiondependsonthestructureoftheatomsandmolecules,whiletheratesofeachtransitiondependonthedensitiesofeachatomicormolecularspecies,thetemperature,andmicrophysics.Ideally,thesimulationwouldtracktheabundancesofeachspeciesandcalculatedirectly,butthiswouldbeimpracticaldotothelargenumberofpossiblestatesandtransitionsaswellasuncertaintiesinmanyoftheratecoecients.However,simulatorscanoftenmakeassumptionsaboutequilibriumorthenatureofthedominanttransitionstoapproximateinatractableamountoftime.Below104Kthegasispartiallymolecular.Coolingisdominatedbymoleculartransitions,theratesofwhichdependondensity,abundance,andtemperature.Theratecalculationiscomplicated,andequilibriumcannotbeassumed.Fortunately,thegasinthisworkdoesnotcoolbelow104K,andsotheseprocessesarenotdiscussedhere(seeSmithetal.,2008,formoreinformation).Between104and105K,coolingisdominatedbyrecombinationlinesfromHandHe.CandOrecombinationdominatesnear105K,withNeandFerecombinationdominatingnear106K.Theimportanceofheavyelementsathighertemperaturesisanaturalconsequenceoftheincreasedbindingenergythatleadstohigherionizationtemperatures.Above˘106Ktheplasmaisfullyionized,andemissionisdominatedbyBremsstrahlungradiation,whichgoesasT1=2.SeeingashighZelementsdominatethecoolingrate(despitetheirlowabundancerelativetoHandHe)outsideofthefree-freeregime,isclearlymetallicitydependent,withmoremetalrichgashavingahighercoolingrate(seeSutherland&Dopita,1993,formoreinformation).Forsimplicity,thesimulations113ˇ˙˝˚˜˝˝ˇˆ˘˝˘ˆ˘˝FigureA.1CoolingRatesinEnzo:ThecoolingratesgivenbytheanalyticexpressioninSarazin&White(1987)andstoredintheEnzolecoolrates.in.Thesolidlinegivesthecoolingrateforgasofhalfsolarmetallicitywhilethedashedlineshowstherateforasolarabundance.TheformerisgenerallyusedinEnzosimulations.114inthiswork(andmanyothers)assumeaconstantmetallicityofZ=0:5Z.Methodsformetaldependentcoolingandnon-equilibriumcooling(Abeletal.,1997;Silviaetal.,2015)areavailableforsimulationswheredetailedcoolingisimportant.Inourwork,however,weareinterestedintheinterplayoffeedbackandtheICM,andwedonotexpectourresultstodependstronglyonthedetailsofthecoolingcurve.A.4StructureandDevelopmentofEnzoTheEnzocode1wasoriginallydevelopedbyGregBryanandMikeNormanattheUniversityofIllinoisinUrban-Champagneinthemid1990sandfurtherdevelopedattheUniversityofCaliforniainSanDiego(seeGregBryan'sthesis,Bryan,1996)withtheexplicitgoalofbeingusefultoresearchersbeyondtheinitialauthors.EnzowasutilizedandexpandedbysubsequentmembersoftheNormangroupatUCSD,andwaseventuallymadeavailabletoawideraudiencethroughversioncontrolsystemssuchasSubversionandMercurial2.Today,Enzoisusedbyhundredsofresearchersworldwideandisactivelydevelopedbydozensofusers.Enzoisopensource,andusersareencouragedtocontributetheirworktothecodebase.EnzoisprimarilywritteninC++,withFortranusedforcertainnumericalroutines.ThecodeusesMPIparallelizationwithadaptiveloadbalancingandhasdemonstratedtheabilitytoscaletoseveralthousandCPUs.DataiswrittenoutintheHDF5format.
A.5TheytAnalysisCodeTheytanalysiscode3wascreatedbyMattTurk(Turketal.,2011)atColumbiaUniversityasatoolforanalysingEnzosimulationdata.ytiscapableofautomaticallyparsingtheoutputofEnzosimulations,abstractingthedetailsofthesimulationstructure,andreturningphysicallyrelevantquantities.ytcanalsobeusedtogenerateproles,slices,projections,orvolumerenderingsofdierentquantities.AlthoughinitiallydevelopedforusewithEnzodata,ythasbeenextendedforusewithavarietyofastrophysicalcodes(Kimetal.,2014).ytisprimarilycodedinPython.LikeEnzo,thecodebaseisopensource4andisdevelopedbyawidecommunityofresearchers.1http://enzo-project.org2https://bitbucket.org/enzo/enzo-dev3http://yt-project.org4https://bitbucket.org/yt_analysis/yt115AppendixBTheIso-coolingSetup
ThissectiongivesthederivationfortheinitialconditionsusedinChapter3andMeeceetal.(2015).Theendresultisincludedinthepaper,butthefullderivationispresentedhere.Theiso-coolingsetupisgovernedbythreeconditions:
ThegasisinhydrostaticequilibriumTheratioofcoolingtimetofreefalltimeisthesameatallheights.Pressure,density,andtemperaturearerelatedbytheidealgaslaw.B.1Denitions
Thegasdensityisgivenbyˆ,whichismassperunitvolume(g/ccorappropriatecodeunits.)Thenumberdensity-thenumberofparticlesperunitvolume,isgivenbynˆm(B.1)whereisthemeanmolecularweightandmistheatomicmassunit.TheHydrogennumberdensityisthenumberdensityofprotons.nHfHˆmH(B.2)wherefHisthemassfractionofHydrogen(0.76forprimordialgas).Thecoolingrateintermsofchangeinenergyperunitvolumeperunittimeisgivenby_E(T)nenH:(B.3)FollowingthederivationusedinEnzo,thisbecomes_E(T)nenH(B.4)=0:52(fH+1)fHn2:(B.5)116Thecoolingtimeisdenedasthethermalenergyovertheenergylossratetcool32nkBT(T)nenH(B.6)=32fH(fH+1)kBTn:(B.7)Thefreefalltimeisdenedastr2zg(B.8)wherethegravitationalacelerationisg(z)gstanh(ˇz=zs):(B.9)Here,gsandzsarescalefactorschosentomatchthedesiredpropertiesatthescaleheights.Forthissetup,tcool=tisassumedtobeconstantwithinacertainrangeofheights.Forbrevity,thisratiowillbewrittentcool=t:(B.10)B.2Derivation
Theequationforhydrostaticequilibriuminvolvesbothtemperatureanddensity.Thetimescaleratiogivesarelationbetweentemperatureanddensity,whichcanbeusedtoturnthisintoanordinarydierentialequation.CombiningequationsB.7andB.8givestherelationbetweendensityandtemperature.=32fH(fH+1)kBTnrg2z:(B.11)Rearrangingthisgivesthedensityintermsofthetemperatureandheight,n=32fH(fH+1)kBTrg2z(B.12)Groupingtheconstantsgivesn=3kBp22fH(fH+1)Tg1=21z1=2(B.13)orˆ=3mkBp2fH(fH+1)Tg1=21z1=2:(B.14)117Next,theconditionforHSEisdPdz=ˆg:(B.15)Foranidealgas,P=ˆkBTm:(B.16)SubstitutingthisintotheHSEequationgiveskBmTdˆdz+ˆdTdz=ˆg:(B.17)RearranginganddividingthroughbyˆgivesdTdz=mgkBTˆdˆdz:(B.18)Calculatingthedensityderivativeisnotascomplicatedasitmightappear.First,writethedensityasˆ=Tg1=21z1=2(B.19)whereisthegroupofconstantsabove.Thederivativeisthendˆdz=dTdzg1=21z1=2+12Tdgdzg1=21z1=2Tg1=2ddz2z1=212Tg1=21z3=2(B.20)whichisadmittedlylong,butpullingoutTg1=21z1=2givesdˆdz=Tg1=21z1=2dTdzT1+12dgdzg1ddz112z1(B.21)whichisjustdˆdz=ˆdTdzT1+12dgdzg1ddz112z1:(B.22)Notethatyoucouldalsodothiswithlogsandgetthesameresult.Anyway,pluggingthisintotheHSEequationgivesdTdz=mgkBTdTdzT1+12dgdzg1ddz112z1:(B.23)OneimportantnoteisthatthecoolingrateisafunctionofT,notz.Thecoolingratederivativereallyonly118makessenseatthebeginningofthesimulation,whenTisafunctionofz.Therefore,dTdz=mgkBTdTdzT1+12dgdzg1ddTdTdz112z1:(B.24)RearrangingagaingivesdTdz+dTdzddTdTdzT=mgkBT12dgdzg112z1(B.25)ordTdz2ddTT=mgkBT12dgdzg112z1(B.26)andnally,dTdz=mgkB+T12dgdzg112z1ddTT2:(B.27)119AppendixCIdealizedClusterSetup
ThissectiondiscussestheproblemsetupusedinMeeceetal.(2016).C.1Motivation
Galaxyclusterscanroughlybedescribedasspheresofgasanddarkmatter.Thegascomponent(theICM)ismadeupofahotplasmathatisroughlyinhydrostaticequilibrium(HSE).Theory(Voit,2005)andobservationCavagnoloetal.(2008,2009)predictsthattheentropyprolesofmostcool-coreclustersaredescribedbyacommonprolethatdecreasestowardsthecenteroftheclusterandthenlevelsotowardsanentropyoor.
C.2Denitions
Theentropy(seeVoit,2005,forathoroughdiscussion)isdenedusingthedenitionofCavagnoloetal.(2008)asKkBTn1e(C.1)whereKistheentropy,kBistheBoltzmannconstant,TthetemperatureinKelvin,neistheelectrondensity,and=5=3isthegasconstant.Theelectrondensityisrelatedtotheparticledensitybyne=(fH+(1fH)=2)(C.2)=n(C.3)whereisthemeanmolecularweight(around0.6forionizedplasma)andfH=0:76istheHydrogenmassfraction.Thesymbolhasbeenintroducedhereforbrevity.Foranidealgas,thepressurePisgivenbyP=nkBT(C.4)which,usingthedenitionofentropy,becomesP=1Kn:(C.5)120Assumethatallquantitiesareincgsunits.NotethatentropyisusuallygiveninkeVcm2whichwillneedtobeconverted.
C.3TheGravitationalProle
ThegravitationalaccelerationofthesimulationisdominatedbythedarkmatterhaloandbytheBCGatsmallradii.Thus,theaccelerationprolecanbewritteng(r)=gNFW(r)+gBCG(r):(C.6)SincetheBCGandtheclusterarebothcenteredontheorigin,thisbecomesg(r)=G(MNFW(r)+MBCG(r)r2(C.7)TheNFWhalohasthedensityproleˆ=ˆS(r=RS)(1+r=RS)2(C.8)whereˆSandRSarethecharacteristicdensityandradiusrespectively.However,darkmatterhalosareusuallydescribedintermsofthevirialmass(M200,themassenclosedwithinR200,withingwhichthedensityis200timesthecriticaldensity)andtheconcentrationparameterR200=cRS(C.9)FromthedenitionsofcandM200,43ˇc3R3S(200ˆc)=M200(C.10)whichcanberearrangedtogetthescaleradiusRS=3M2004ˇc3(200ˆc)1=3:(C.11)ThemassenclosedwithinagivenradiusrinanNFWhaloisM(r)=4ˇˆSR3Slnr+RSRSrr+RS(C.12)121takingthemassatthevirialradiusgivesthescaledensityˆS=M2004ˇR3Sln(1+c)c1+c1:(C.13)WeassumeamassprolefortheBCGoftheformM(r)=M4"2(r=4kpc)(1+r=4kpc)#;(C.14)whereM4isthestellarmasswithin4kpcandandareconstants.ThisformgivesagoodmatchtotheempiricallyderivedformusedinLi&Bryan(2012)whenusingtheconstantsgiveninMeeceetal.(2016)andcaneasilybeadaptedforothergalaxies.
C.4DensityProle
WeassumethattheclusterisinHSEandhasastaticgravitationalpotentialg(r)(basedontheNFWhaloandBCG).TheentropyproleisgivenbyK(r)=K0+KS(r=RS)(C.15)whereK0,KS,RSandareconstants.TheACCEPTsample(Cavagnoloetal.,2009)usesascaleradiusRS=100kpc.TheconditionforHSEisdPdr=mg:(C.16)SubstitutingtheexpressionforP,takingthederivatives,andsolvinggivesthenalequationforthedensityproledndr=mng1ndKdrKn11:(C.17)C.5TemperatureProle
TheentropydenitionandprolerelatethedensityandtemperatureatagivenradiusT=K(n)2=3kB(C.18)Thisisusedtondthetemperatureproleoncethedensityprolehasbeenintegrated.122C.6IntegratingtheDensityProle
Thedensityproleisanon-lineardierentialequationandcanbeintegratednumericallyusinganRK4integrator.First,weneedaboundarycondition.FollowingVoit(2005),thetemperatureofahydrostaticICMcanbeapproximatedaskBT=mp2[10GM200H(z)]2=3(C.19)wherempistheprotonmass,Gisthegravitationalconstant,M200isthemasswithinR200,andH(z)istheHubbleconstantatredshiftz.IusethisasthetemperatureatR200,usethattondnatR200,andintegrateinwards(towardsthecenter)andoutwardstondtheentiredensityprole.Finally,IuseEquationC.18andtheentropyproletondthetemperature,andthesetupiscomplete.123AppendixDFragmentationindusty
low-metallicitystarforminghalosAbstractTherststarsintheuniverse,termedPopulationIII,arethoughttohavebeenverymassivecomparedtothestarsthatforminthepresentepoch.Asfeedbackfromtherstgenerationofstarsalteredthecontentsoftheinterstellarmedium,theuniverseswitchedtoalow-massmodeofstarformation,whichcontinuesinthehighmetallicitystarsformedinthepresentera.Severalstudieshaveinvestigatedthetransitionbetweenmetal-freeandmetal-enrichedstarformation,withtentativeevidencebeingfoundforametallicitythresholdnear103:5Zduetoatomicandmoleculartransitionsandanotherthresholdnear105:5Zduetodust.Inthiswork,wesimulatethefragmentationofcoolinggasinidealized,low-metallicityhalosusingtheAMRcodeEnzo.Weconductseveralsimulationsof106Mand107Mhalosatz=20inwhichthemetalcontent,initialrotation,anddegreeofturbulencearevariedinordertostudytheeectofthesepropertiesongasfragmentationoverarangeofdensities.Wendtentativesupportfortheideaofacriticalmetallicity,buttheeectofvaryingmetallicityonthegasweobserveisnotasdramaticaswhathasbeenreportedinearlierstudies.ItistheorizedthatatlowerredshiftswithalowerCMBtemperature,variationsinmetallicitymighthavealargereectoncoolingandfragmentation.Wendnoclearrelationbetweentheinitialspinortheinitiallevelofturbulenceinthehaloandthenalpropertiesofthegascontainedtherein.Additionally,wendthatthedegreetowhichtheJeanslengthisrened,theinitialdensityproleofthegas,andtheinclusionofdeuteriumchemistryeachhaveasignicanteectontheevolutionandfragmentationofthegasinthehalo{inparticular,wendthatatleast64gridcellsareneededtocovertheJeanslengthinordertoproperlyresolvethefragmentation.D.1Introduction
1Itiswellestablishedthattheveryearlyuniversecontainedonlytraceamountslithiumandessentiallynootherelementsheavierthanhydrogenandhelium(Steigman,2007;Wagoner,1973).AfterBigBangnucleosynthesis,virtuallyallheavyelementsaresynthesizedinstars.Itfollowsthattherststars,termedPopulationIIIstars,musthavebeenfreeofheavyelements.However,observationshaveyettoidentifyanyofthesemetalfreestars(Ryanetal.,1996;Beers&Christlieb,2005;Caauetal.,2012;Yongetal.,2013)ob-1ThischapterwasoriginallypublishedinTheAstrophysicalJournal(Meeceetal.,2014).Ithasbeenreformattedforinclusionhere.Forinformationaboutcopyrightandreuse,seeAppendixE.124servationsofLyman-systemsrevealthatevenlowdensitygasathighredshiftsiscontaminatedbyheavyel-ements,indicatingsignicantenrichmentbyearliergenerationsofstars(Cowie&Songaila,1998).Thisleadstotheconclusionthattherststarsweremassiveandshort-lived(Barkana&Loeb,2001;Ripamonti&Abel,2004;Bromm&Larson,2004;Glover,2005;Norman,2010).Toproduceastellarinitialmassfunction(IMF)consistingofmostlyhighmassstars,theprocessofstarformationintheprimordialuniversemusthavedieredsubstantiallyfrommoderndaystarformation.Starsformwhenover-densecloudsofgasradiateenergyandcollapseduetoselfgravity.DensityperturbationslargerthantheJeanslengthwilltendtocollapsefasterthanthesurroundinggas.Ifthegasisabletoecientlyradiateenergyasitcollapses,suchthattheJeansMassdecreaseswithincreasingtemperature,thegaswillcontinuouslyfragment.Thus,thenalmassoftheprotostellarcloudwillbesetbytheJeansmassatthepointwherethegascannolongercooleciently.Theinitialstellarmasswillbesetbythesizeoftheprotostellarcloudandaccretion,althoughthedetailsofthisprocessarequitecomplicated,andthePopulationIIIIMFishighlyuncertainasaresult(e.g.,Tan&McKee,2004;McKee&Tan,2008;Norman,2010;Clarketal.,2011;Greifetal.,2011).Inparticular,Clarketal.(2011)andGreifetal.(2011)ndthatfragmentationintheprotostellardiskcanresultinaclusteroflowmassstars,ratherthantheisolatedmassivestardescribedbyAbeletal.(2002).Radiativefeedbackfromtheprotostarorprotostarswilleventuallyhaltaccretion,settingthenalmassesofthestars(Hosokawaetal.,2011;Stacyetal.,2012).WhilemanyresultsforPopulationIIIstellarmasseshavebeengiven,afullunderstandingoftheprimordialIMFwillnotbepossiblewithouttheuseofdetailedsimulationsusingafullradiativetransfermodel.Nevertheless,itisalsonecessarytounderstandthegrowthandlargescalestructureofthepre-stellarhalo.Theabilityofthegascloudtocoolwillbesetbythemicro-physicsofthegas.Inthelocaluniverse,rotationallinesinCOandlinecoolingfromCIandOIareprimarilyresponsibleforcooling(Omukai,2000),andareabletolowerthetemperaturesofstarformingcloudstoaround10K.Intheearlyuniverse,however,theonlysignicantsourcesofcoolingwereH2andHDmolecules.DuetothelackofapermanentdipoleinH2,therotationalenergylevelsarerelativelywidelyspaced,androtationaltransitionsarenotabletocoolthegasbelowatemperatureofaround200K(Galli&Palla,1998).WhileHDisamoreeectivecoolantowingtoapermanentdipolemoment,thelowinitialfractionofdeuteriumpreventsahighHDfractionfromforming,typicallypreventingHDfromcontributingtothetotalcoolingasmuchasH2(Galli&Palla,2002;Ripamonti,2007).Ifheavyelementsarepresent,thegaswillbeabletocoolfasterandtolowertemperaturesthanispossibleinprimordialgas.Metalsintheformofdustwillbeabletocoolthegasthroughthermalradiation(Omukaietal.,2005;Schneider&Omukai,2010).DustcanalsoserveasacatalystforH2formation,providinganadditionalsourceofcooling.Manyauthorshavestudiedthetransitionfrommetal-freetometal-enrichedstarformationusingidealized125models.Whiletheinitialconditionsofthesemodelsarenecessarilylessaccuratethanthoseofcosmologicalsimulations,theirfullyspeciednatureallowsoneparametertobevariedatatime,facilitatingourabilitytoisolateandunderstandtheeectsofindividualphysicalprocesses.Overthepastdecade,idealizedsimulationshaveexploredmoreoftherelevantphysicalandchemicalprocessesunderlyingstarformationintheearlyuniverse,resultingintunablemodelsthatmoreaccuratelycapturetheconditionsofprimordialandlowmetallicitystarformation.Brommetal.(2001)modeleda2:0106Mtop-hatoverdensitycollapsingatz=30andfoundtherstevidenceofa`criticalmetallicity'ofapproximately5104Z.Omukaietal.(2005)hasstudiedthethermodynamicsofcollapsingprimordialandlowmetallicitygasusingone-zonemodels.Morerecently,aseriesofworks(Glover&Jappsen,2007,?;Jappsenetal.,2009a,b)modelledthecollapseofahot,ionizedgaswhichhadbeenallowedtorelaxtohydrostaticequilibriumwithinanNFWpotential(Navarroetal.,1997)beforecoolingwasturnedon.Thisgroupconcludedthatthereisnoclearcriticalmetallicity,andthatfragmentationismoredependentonthechoiceofinitialconditions.Severalworks(Omukai,2000;Omukaietal.,2005;Schneideretal.,2006;Schneider&Omukai,2010;Schneideretal.,2012)havefocusedontheeectsofdustcoolingonfragmentationinlowmetallicityclouds.DustcoolingistypicallyeectiveatdensitiesabovenH=1010cm3,wherethegasanddusttemperaturesarecoupled.Thesestudieshavefoundevidenceofalowermetallicitythresholdaround105:5Zduetodustcoolingwhendustisincludedinsimulations.Inaddition,thestarSDSSJ1029151+172927(Caauetal.,2011,2012)hasbeenfoundtohave[X=H]<104forallelementsmeasured,indicatingthatsomecoolingprocessotherthanmolecularcoolingisoperating.Klessenetal.(2012)hasattributedtheformationofSDSSJ1029151+172927todustcoolinginducedfragmentation,indicatingthatdustcanproducelow-massstarsbelowthemetalcoolingthreshold.Inthiswork,weextendthestudyofthetransitionfrommetal-freetometal-enrichedstarformationbyusinganidealizedmodelbasedontheresultsofcosmologicalsimulations.Ourmodelusesseveralparameterstosetthemetallicity,chemistry,andtheshapeoftheinitialdensity,temperature,androtationalproles,aswellasallowingfordierentlevelsofturbulenceanddierenthalomasses.InSectionD.2,wediscussoursimulationcodeandtheinitialsetupofourstar-forminghalosindetail.InSectionD.3,weprovideanoverviewoftheevolutionofourducialmodel.SectionD.4discussestheeectsofvaryingourrenementcriteriaandestablishesthecriterianecessarytoadequatelyresolvethecollapse.InSectionD.5,wediscusstheevolutionofourmodelfordierentpointsintheparameterspaceofmetallicity,rotation,turbulence,anddust.InSectionD.6,wediscusstheassumptionsinoursimulationsthatmayinuenceourresults,includingtheeectsofdeuteriumchemistry,theshapeoftheinitialdensityprole,andthevalidityofourchemicalmodel.WesummarizeandconcludeinSectionD.7.126ˇ˝&"˘ $˝%'˙˘˜ˇ˝&"˘ˆ˜"ˆ#%&#ˆˇ˝&"˘ ˙ˆ %˝˚ˆ˚!˘˝%'˛˜FigureD.1InitialConditionsforLowMetallicityStarFormationSimulations:Theinitialcondi-tionsofourmodelareshownforourhighandmassducialmodels.Solidlinesrepresentthetheoreticalvalues,whilethedashedlinesarethevaluesrealizedinoursimulation.PanelAshowsdensityasafunctionofradius.Theinitialtemperatureproleisderivedbyassuminghydrostaticequilibriuminthecoreandapowerlawfallointheenvelope,andisshowninPanelB.Therotationalvelocity,showninPanelC,isderivedbyassumingthataverageangularmomentumfollowsapowerlawrelationshipasafunctionofmassenclosed.
D.2Method
D.2.1TheSimulationCodeandIncludedPhysics
WemodelthecollapseofthehalousingtheEulerianadaptivemeshrenementcodeEnzo(O'Sheaetal.,2004;Normanetal.,2007;Bryanetal.,2014).ThehydrodynamicsarecalculatedusingthepiecewiseparabolicmethodofColella&Woodward(1984).Inordertoensureconservationofmasswithinoursim-ulation,weemployperiodicboundaryconditionsforthegas.Tocalculatethegravitationalpotential,weassumeisolatedboundaryconditions.Inadditiontotheselfgravityofthebaryons,wecalculatethegrav-itationalpotentialofastaticNFWhalo.Eachsimulationisinitializedwithatoplevelgridresolutionof1283cellsandisrenedduringsetup.Althoughwedonotusecomovingcoordinatesinthiswork,weassumearedshiftofz=20wherenecessaryduringinitialization,andalldistancesinthispaperinareinproperparsecsatthatredshift.Eachhaloisplacedinthecenterofaboxwithapropersizeof2000pcperside.Duringinitialization,werequirethattheinner100pcbecoveredbyfourlevelsofgridrenement,givingamaximumspatialresolutionof0.977pcatthebeginningofthesimulation.Asthevirialradiusofthedarkmatterhalo(takentobetheedgeofthesphere)isanorderofmagnitudesmallerthantheboxsize,theeectsofboundaryconditionsontheevolutionofthehaloshouldbenegligible.127D.2.1.1RenementConditions
Weemployfourcriteriafordeterminingwhentorenegridcells.Inallsimulations,renementiscarriedoutbysubdividingagridcellbyafactorof2alongeachdimension,thusinto8equal-sizedcells.Lagrangianrenementwouldthereforerequirethatwereneagridcellwhenevertheenclosedmassexceedstheaveragemassinonetoplevelgridcellbyafactorofeight.Tobetterunderstandtheevolutionofthedensestregions,weimposesuper-LagrangianrenementbyreningwhenevercellmassexceedsMcell>Mtop80:3l(D.1)Wherelisthecurrentrenementlevel.Oursecondrenementconditionsplitsacellwheneverthelocalcoolingtime,tcool,isshorterthanthesoundcrossingtimeofthecell,x=cs.Thisrequirementisnecessarytojustifyourassumptionthatthegasisthermodynamicallystableatscalessmallerthanthegridresolution.Thirdly,werenewhenthesizeofacellislargerthansomefractionofthelocalJeanslength,whenJ<NJx,whereJistheJeanslength(calculatedinthatcell)andNJisthenumberofcellsoverwhichtheJeanslengthmustberened.Unlessotherwisenoted,werequirethattheJeanslengthberesolvedbyatleast64cellsatalltimes,e.g.NJ=64.InSectionD.4,wediscussaseriesofteststodeterminetheminimumvalueofNJnecessaryinordertoaccuratelymodelthefragmentationofthecollapsinghalo.Finally,werequirethattheregionwithinacubewithsidelength100pccenteredonthecenterofthesphereisalwayscoveredbyaspatialresolutionoflessthan1pc.Thiscriterionensuresthattheconditionsintheinnerregionofthehalo,asdenedbyourinitialsetup,areaccuratelycapturedbythemesh.Weallowthesimulationtodynamicallyreneusingupto25levelsofgrids,foramaximumspatialresolutionof0.0962AU.D.2.1.2ChemistryModel
Ourchemistrymodelfollowsthenon-equilibriumreactionsfor12primordialchemicalspecies(H,H+,He,He+,He++,e,H2,H+
2,H,D,D+,andHD)(Anninosetal.,1997;Abeletal.,1997)andincludesH2chemistrywiththreebodyH2formation(Abeletal.,2002)andH2formationheating(Turketal.,2009).Inaddition,weusethecoolingmodelofSmithetal.(2008)totrackcoolingfrommetalsinsimulationswheremetalsarepresent.Unlikethechemicalmodelusedforprimordialspecies,themetalcoolingmodeldoesnotexplicitlytracktheabundanceofindividualmetalions.Instead,weusedatageneratedbythephotoionizationcodeCLOUDY(seeFerlandetal.(1998))tocalculatemetalcoolingratesforawiderangeofdensities.Throughoutthispaper,weassumeascaledsolarabundancepattern.ThevalidityofthischoiceisdiscussedinSectionD.6.5.InSectionD.6.1,wediscusstheeectsofusingareducedchemicalmodelthat128doesnotincludedeuteriumchemistry.
D.2.1.3DustModel
ThepresenceofdustgrainsaltersthethermalstateofthegasbyprovidingaveryecientchannelforH2formationandthroughheattransferviaelasticcollisionswiththegas.Dustgrainscoolviacontinuumthermalemissionandareheatedbyincidentradiation.Currently,weonlyconsiderincidentradiationfromtheCMB,butinprincipleanadditionalheatingtermcanbeeasilyadded.TheratesofH2formationongrainsurfacesandheatexchangewiththegasaredependentonthegraintemperature,Tgr,whichweassumetobeininstantaneousequilibrium.TheimplementationemployedherecloselyfollowsthatofOmukai(2000)andOmukaietal.(2005).Wecalculatethegraintemperaturebysolvingtheheatbalanceequationgivenby4˙T4grgr=gas=grain+4˙T4radgr;(D.2)where˙istheStefan-BoltzmannconstantandTradistheradiationtemperature,specicallytheCMBtemperaturehere.Therateofheatexchangebetweenthegasanddustperunitdustmass,gas=grain,isgivenbygas=grain=1:21031n2
HˆgrT1000K1=2(10:8e75=T)(TTgr)ergs1g1(D.3)(Hollenbach&McKee,1989),wherenHistheHnumberdensityandˆgristhedustmassdensity.Weassumethatasmetallicityincreases,dustremainsaconstantfractionofthemetallicity.WeadoptthepiecewisepolynomialapproximationofthegrainopacityofDopckeetal.(2011),givenby(Tgr)/8
>
>
>
>
<
>
>
>
>
:T2gr,Tgr<200K,constant,200K<Tgr<1500K,T12gr,Tgr>1500K;(D.4)withanormalizationofgr(Tgr=200K)=16cm2g1(Pollacketal.,1994;Omukai,2000).ThesteeppowerlawindexforT>1500Kmimicstheeectofgrainsmelting.WetaketheexactformoftherateforH2129formationongrainsgiveninOmukai(2000),whichisderivedfromtheworkofTielens&Hollenbach(1985).Weincludetheheating/coolingfromH2formation/destructionfollowingOmukai(2000)andHollenbach&McKee(1979).
D.2.2InitialConditions
WemodelthestarformingregionsasasphericallysymmetricbaryonichalowithaturbulentvelocityeldwithinastaticNFWpotential.Ourmodelsareempiricallymotivatedbytheresultsofcosmologicalsimu-lations(O'Shea&Norman,2007;Smithetal.,2009)andinformedbytheone-zonemodelsofOmukaietal.(2005).Althoughoursimulationsarenon-cosmological,weassumethatthecalculationproceedsataxedredshiftofz=20forthepurposesofcalculatingheatingandcoolingratesduetothecosmicmicrowavebackground.WeassumeanCDMuniversewith=0:7,M=0:3,andH0=70kms1Mpc1whererelevantduringtheinitialization.Theseparametersareusedwhencalculatingthevirialradiusofthehalo,andsmallvariationsincosmologicalparameterswouldnothavealargeeectonoursimulations.WeassumethatthehaloisdecoupledfromtheHubbleow,anddonottakecosmologicalexpansionintoaccountduringthesimulation(whichisreasonable,asthehaloisoverdenseenoughtobedecoupledfromtheexpansionoftheuniverse.)Thus,alldistancesquotedinthisworkareinphysical(i.e.,proper)units.D.2.2.1DarkMatterHalo
ThedarkmattercomponentofthehaloisassumedtoresideinanNFWhalo(Navarroetal.,1997),ˆDM(r)=ˆc(r=RS)(1+r=RS)2(D.5)whereˆcisequaltofourtimesthedensityatthevirialradiusandRSisthescaleradius.Theconcentrationparameterofthehalo,denedasc=R178RS(D.6)issettoc=2foroursimulations.Here,R178isthevirialradius,calculatedastheradiusatwhichtheaverageencloseddensityis178timesthecriticaldensityoftheuniverse(seeBryan&Norman(1998)formorediscussion).Ourvalueofciswithintheexpectedrangeforthehaloswearestudying,aspredictedbyDavis&Natarajan(2010).Inthiswork,themassofthehaloistakentomeanthemassofdarkmatterwithintheviralradiusofthehalo.Duetothelowinitialbaryondensity,thetotalmassofthehaloisnotsubstantiallyhigher.Westudymodelswithdarkmattermassesof106Mand107M,withcorresponding130virialradiiof153and329pc.Thesehalosarehereafterreferredtoasthe\lowmass"and\highmass"halos,respectively.
D.2.2.2BaryonDensityandTemperature
Thebaryoniccomponentofthehaloismodeledbyacoreinroughlyhydrostaticequilibriumwithadiuseenvelope.TheenvelopedropsorapidlyuntilitreachesthebackgrounddensityofnH=102cm3.ThedensityproleisdescribedbyˆB(r)=ˆB(r=Rcore)(1+r=Rcore)(D.7)andisshowninthePanelAofFigureD.1.Foroursimulations,weuse=0:1and=2:5.ThesevalueswerechosenbyttingtheresultsofthecosmologicalsimulationsperformedinO'Shea&Norman(2007)andadditionalunpublishedsimulationsperformedbyourgroupforthisstudy.Fortheinitialdensityprole,weuseacentralbaryonnumberdensityofnH=1cm3forboththehighandlowmassducialcases.Forthelowmasshalo,wechooseacoreradiusofRcore=8pcandforthehighmasscaseRcore=16pc.Ourchoiceofalowinitialcentraldensity(comparedtothedarkmatterdensity)ismotivatedbythedesirethatthesimulationhavetimeto`forget'thedetailsoftheinitialconditionsandreachastablecongurationbeforecollapsesetsin.InSectionD.6.3,wediscusstheresultsofstartingasimulationwithahigherinitialcentraldensity.Thetemperatureproleiscalculatedbyassumingthatthegasisinhydrostaticequilibriumwithinthecoreandisbeingadiabaticallyheatedintheenvelope.TheinitialtemperatureprolesisshowninPanelBofFigureD.1.
D.2.2.3ChemistryandMetallicity
Weinitializethegasinourmodeltoacompositionconsistentwithconditionsinthez=20universeforgasthathasnotbeenaectedbyrecentstarformation.Atinitialization,allsimulationshaveauniformelectronfractionof˜e=1:69104,basedoncalculationsperformedwiththecodeRECFAST(Seageretal.,1999,2000),andacorrespondingHIfractionoffHI=0:999831.TheHfractionisfH=1010.TheinitialmolecularhydrogenfractionisfH2I=104.InitialvaluesforDandHDarescaledtotheHandH2valuesusingaD/Hmassratioof6:8105.Inmodelswheremetalsarepresent,weassumeascaledsolarabundanceofheavyelements.Metallicityiskeptuniformthroughoutthesimulation.Insimulationswheredustispresent,itisassumedthatthemassfractionofheavyelementsindustis9:23103.Theeectsofourchoiceofinitialchemistryisdiscussed131inSectionD.6.
D.2.2.4VelocityProle
Thehaloisgivenaninitialangularmomentumdistributioncharacterizedbythedimensionlessbaryonicspinparameter,denedinPeebles(1971)as=JjEj1=2GM5=2(D.8)whereJisthetotalangularmomentumofthebaryons,Eisthebindingenergyofthebaryons,Gisthegravitationalconstant,andMisthemassofthebaryons.BasedontheresultsofO'Shea&Norman(2007),theangularmomentumisdistributedsothatthespecicangularmomentumasafunctionofmassenclosedisgivenbyjlj(M<r)/M(<r)MTotal0:9(D.9)whichisscaledbythespinparameter.
Gasoutsideofthevirialradiushasnoinitialvelocity,andnoneofthegashasaninitialradialvelocitybeforeturbulenceisadded.Thedarkmattercomponentofthehaloistreatedasastaticpotential,andthushasnovelocity.TheinitialrotationproleforourducialmodelsisshowninPanelCofFigureD.1.D.2.2.5Turbulence
TherotationalvelocityismodiedbyaddingaturbulentvelocityeldwithapowerspectrumP(k)/k4,suitableforcompressiblegas(Clarketal.,2011).TheturbulenteldisgeneratedusingthemethoddescribedinRogallo(1981).Theturbulenteldisappliedonlywithinthevirialradiusofthehalo,andisnormalizedsuchthattheRMSvelocityisaspeciedfractionofthesoundspeedofthehalo,asdenedinBarkana&Loeb(2001).Forconsistency,thesameturbulenceeldwasusedforallsimulations.D.2.3VaryingtheInitialConditions
Inordertostudytheimportanceofdierentmodelparameterstotheevolutionandfragmentationofthegas,weconductseveralrunswhereinoneparameterissystematicallyvaried.Foreachparameter,weconductsimulationsforboththehighandlowmasshalos,asdescribedabove.ThefulllistofsimulationsperformedinthisworkisgiveninTables1-5.Werunoursimulationsuntilthecentraldensityhasreachedahydrogennumberdensityofatleast132nH=1010cm3,whichisnearthelimitsofthevalidityofourcoolingmodelandisapproachingthepointwhereourassumptionthatthegasisopticallythinbeginstofail.Thus,wedonotmodeltheactualformationofstars,whichoccursathigherdensitiesandrequiresafullradiativetransfermodel.Thenalmassofthestarwillbesetbyadditionalphysics,includingadditionalcoolingprocesses(suchascollisionalionization),radiativefeedback,anddiskformation.Additionally,whiledustcoolinghadbeentheorizedtobeimportantathighdensities(seeSectionD.6.2fordiscussion),wedonotfullyexplorethisregime.D.2.4ClumpFinding
Toquantifythedegreeoffragmentation,werunaclumpnderonthecentral20pcofthenaloutputfromeachsimulation.Theclumpndingalgorithm,describedindetailinSmithetal.(2009)andimplementedintheyt2simulationanalysistoolkit(Turketal.,2011),worksbyidentifyingtopologicallydisconnectedstructuresindensityspace.FollowingSmithetal.(2009),wedeneaclumporfragmentas\themasscon-tainedbetweenalocaldensitymaximumandthelowestisodensitysurfacesurroundingonlythatmaximum."Smithetal.(2009)onlyconsiderclumpsthatarestrictlygravitationallybound,buthereweuseamodiedcriteriontoincludeclumpsthataremarginallyunboundbutrapidlycooling,sincetheseobjectswilllikelybecomeboundinthefuture.Clumpsareconsideredvalidiftheysatisfythefollowingrequirement:KE+TEXi(itdyn;i)<PE;(D.10)whereKE,TE,andPEarethetotalkinetic,thermal,andpotentialenergyoftheclumpandiandtdyn;iarethecoolingrateanddynamicaltimeforreachgridcellthatisamemberoftheclump.Singlecellclumpsarenotconsideredvalidunderanycircumstance.Inordertoquantifythedegreeoffragmentationasdensityincreases,wecountthenumberofclumpsmeetingtheabovecriteriawhichforminagivendensityinterval.Tobegin,wesearchforclumpsinhalfdexdensityintervalsrunningfromˆmintoˆmax(inthiswork,nH=1andnH=1010respectively).Foreachclump,werecordthedensityofthesurroundingisodensitycontour.Secondly,ineachdensityintervalwecountthenumberofclumpsforwhichthesurroundingcontourfallswithintheinterval.Intheory,densityregimesinwhichthegasundergoesfragmentationshouldbemarkedbyanincreaseinthenumberofclumpsformingatthosedensities.Thedegreeoffragmentationindierentsimulationsmaybecomparedvisually,asinthestyleofFigureD.8.Additionaltestshaveshownthatthetrendproducedbyourfragmentationquanticationprocedurearemoderatelyinsensitivetothemannerinwhichwedeneaclump.Specically,countingclumpsinwhich2http://yt-project.org/133FigureD.2DensityEvolutionoftheFiducialRun:Thecentraldensityofthegasisshownasafunctionoftime.Thexaxisshowstimeinmillionsofyearsbeforethelastdataoutput.Thetimefromthebeginningofthesimulationtothelastdataoutputis55.36millionyears.Thedottedlineshowsthefreefalltimeofasphereofgaswithagivendensity.Onthelefthandsideoftheplot,thecentraldensityisincreasingfasterthanthefreefalltimescale(asindicatedbytheslopesofthelines),indicatingthatthedarkmatterratherthanselfgravityiscontrollingthedynamicsofthegas.Duringthelastmillionyearsofthesimulation,theselfgravityofthegasdominatesinthecenter.Thegasevolvesinfreefalluntilroughly10,000yearsbeforetheendofthesimulation,inwhichpressuresupportdelaysfurthercollapse.KE<0:1PEandcountingallclumpscontaining>40cellsreproducesthetrendsseeninsectionD.5.1.D.3EvolutionoftheFiducialModel
Forourducialmodel,wechoosehighandlowmasshaloswithametallicityofZ=103Z,aspinparameterof=0:05,turbulencenormalizedto0.4timesthehalosoundspeed,andwithdustpresent.Thismodelischosentosimulateatypicalstarforminghaloatz=20,whichhasnothostedrecentstarformation(e.g.,O'Shea&Norman(2007);Smithetal.(2009)).Wechooseametallicitywhichisinthemiddleofourrangeofvalues,andisnearthetheoretical\criticalmetallicity."WemandatethattheJeanslengthbecoveredbyatleast64cellsatalltimesbysettingNJ=64.Thehighmassducialhalocollapses55.36millionyearsafterthebeginningofoursimulation.TheevolutionofthecentraldensityasafunctionoftimeisshowninFigureD.2.Thecollapsebeginsslowlyandacceleratesasdensityincreases.Whilethedarkmatterdominatesduringtheearlystagesofthecollapse,thebaryonscometodominatethepotentialduringthelastmillionyears,makingourresultsrelativelyinsensitivetothehaloproleatdensitiesabovenH=105cm3,roughlycorrespondingtotheinner1pc.134˙ "$.˚*!˜(˚!'˜˙ "$.˚*!˜ ˆ (!&),#"˚ˆ˜(˚!'˜˝#'*#+ -.+#˚ˇ˜ ˆ (!&),#"˚ˆ˜&%',˘($-$ &'/+'/+'/+'/+'/+ ˆ (!&),#"˚ˆ˜˛+˚%',˜ ˆ (!&),#"˚ˆ˜0˛0˚%',˜FigureD.3GasPropertiesfortheFiducialRun:Thephysicalstateofthegasinourhighmassducialmodelisshownforaseriesofoutputs.Thegreen,yellow,blue,red,andblacklinesshowtherstoutputsinwhichthecentraldensityreaches102,104,106,108,and1010cm3,respectively.Foreachoutput,thelegendshowsthetimeremaininguntiltheendofthesimulation.PanelAshowssphericallyaveragedgasdensityasafunctionofradius,centeredonthedensestpointinthesimulation.PanelBshowsthetotalgasmassenclosedasafunctionofradius.PanelCshowsthemassweightedsphericallyaveragedtemperatureofthegasasafunctionofdensity.PanelDshowsthemassaveragedangularmomentumasafunctionofenclosedmass.Forasphericallysymmetriccollapsewithnotangularmomentumtransport,theangularmomentumprolewouldnotchangewithtime.Thefactthatitdoesindicatesthatangularmomentumisbeingtransportedoutofthecorebyturbulence.PanelsEandFshowthemassweightedsphericallyaveragedradialvelocityandvelocitymagnitudeofthegasasafunctionofmassenclosed.IneachpanelexceptPanelE,theinitialconditionsarerepresentedbyadottedblackline.Thegashasnoinitialnetradialvelocity.135FigureD.4ChemicalStateoftheGasfortheFiducialRun:Thechemicalstateofthegasforthehighmassducialmodelisshown.PanelsAandBshowthemassfractionsofH2andHDrespectively.PanelCshowstheionizationfraction.PanelDshowstheratiooftheHDtoH2massfractions.ThedottedlinesinPanelsAandBshowthemassfractionswhenHandDrespectivelyarefullymolecular,andthedottedlineinPanelDshowsthemassfractionratiowhenbothspeciesarefullymolecular.136Whenthesimulationisinitialized,thereisaperiodlastingaround10millionyearsduringwhichthevelocityproleevolvesintoasteadystate.Duringthistime,thedetailsoftheinitialconditionsarewipedout.Atthispoint,thegasintheenvelopeiscollapsinginfreefallandisbeingheatedthroughadiabaticcompression,whilethegasinthecoreispressuresupported.Anaccretionshockformsattheedgeofthesphere,nearthevirialradius.Asthegasisdecelerated,itisheatedtothevirialtemperatureofthehalo.AsshowninPanelAofFigureD.3,thecollapseevolvesselfsimilarly,withthesizeofthecoreshrinkingasthegascollapsestohighercentraldensities.Thevelocityproleofthegasasafunctionofenclosedmass,showninPanelsEandFofFigureD.3,remainsroughlyconstant,withthegasintheenvelopeinfreefallandthegasinthecorecollapsingslowly.Thegasinthecoreevolvesquasi-staticallyuntilneartheendofthesimulation,atwhichpointthegasisabletoecientlycoolandcollapsesonafreefalltimescale.PanelDofFigureD.3showsthemassaveragedangularmomentumofthegasasafunctionofenclosedgasmass,denedasl(M)=J(M)M(D.11)whereMisthemassofgasenclosedwithinasphericalshellandJ(M)isthetotalangularmomentumofthegaswithintheshell.Withnoangularmomentumtransferandnoexternaltorque,l(M)wouldstayconstantthroughoutthecollapse.Inoursimulations,l(M)decreases,indicatingthatangularmomentumisbeingtransportedoutward(relativetotheLagrangianmasscoordinate)inthecentralregions.Thephysicalevolutionofthemodelmaybeunderstoodbylookingatthethermodynamicevolutionofthegas,showninPanelCofFigureD.3,andthechemicalevolution,showninFigureD.4.Inlowdensityregions,thegasisinfreefall,andisheatedbyadiabaticcompression.Therelevantreactionratesaretooslowtochangetheinitialmolecularchemistry,andradiativecoolingisnegligiblecomparedtocompressiveheating.Attheaccretionshock,thegasisrapidlyheatedtothevirialtemperatureofthehalo.Inthehighmasshalo,thevirialtemperatureishighenoughthatthegasenterstheregimeinwhichasmallfractionoftheH2andHDmoleculesaredissociatedandsomeofthegasisionized.Asthegasbecomesdenser,thegascoolsandthemolecularfractionbeginstoincrease.ThemaincoolantsinthegasatareH2,HD,andmetals.Attemperaturesbelow10,000K,atomichydrogenlinecoolingbecomesnegligible.H2isthemostabundantspeciesthatiscapableofradiativecooling,butisinecientowingtothelackofapermanentdipole.Instead,theH2moleculemustrelyonrarequadrupoletransitionsbetweenwidelyspacedenergylevels,andbyitselfisunabletocoolthegasbelowatemperatureofaround200K(Galli&Palla,1998).HD,thoughrarer,hasapermanentdipolemoment137andthusisabletocoolmoreeciently.Together,rotationaltransitionsinHDandnestructuretransitionsinmetalscaneectivelycoolthegastotheCMBtemperatureoor.TheratioofHDandH2issetbytheequilibriumrateofthereactionsH2+D+)HD+H+(D.12)HD+H+)H2+D+(D.13)asdescribedinOmukaietal.(2005).BecauseofthedierencesintheenergylevelsofH2andHD,EquationD.12ispreferredoverEquationD.13,resultinginanHD/H2fractionthatishigherthantheoverallD/Hfractionbyroughly2ordersofmagnitude(Galli&Palla,1998).ThisfractionationisobservedinPanelDofFigureD.4,whichshowstheHD/H2ratio.Asthegascools,theequilibriumabundancerapidlybeginstofavorHDproduction,whichfurtherincreasescoolingandinturnleadstomoreHDformation.FordensitieshigherthannH˘105cm3,thedeuteriumisfullymolecular.ThegascontinuestocollapseuntileitherthetemperatureistoolowtopopulateexcitedstatesinthecoolantsorthegasreachestheCMBtemperature.Forhalosatz=20,weimposeaCMBwithtemperatureTCMB=2:725(1+z)=57:225K(D.14)whichentersintotheheatingequationforthegasanddust.ThegasremainsattheCMBtemperatureooruntiladensityofnH˘107cm3isreached,atwhichpointrapidformationofH2ondustgrainsbrieyreheatsthegas.Atthehighestdensities,coolingviadustemissionisabletoecientlylowerthetemperatureofthegas,resultingincoolinginthehigher-metallicitysimulations.Theformationofstructureinthehaloisgovernedbythethermodynamicsofthegasduringcollapse.IfthecollapsinggasisabletocoolwithincreasingdensityorifthetemperatureincreaseswithdensityataslowerratethanT/ˆ1=2,thelocalJeansmasswilldecrease.AsthelocalJeansmasssetsthescaleforfragmentation,thegaswillbeexpectedtofragmentwhenevertheJeanslengthisdecreasing.FigureD.5showsprojectionsofdensitythroughthegasasthecentraldensityincreases.Atlowdensities,themassofgasinthecenterregionisbelowthelocalJeansmass.AsindicatedinPanelCofFigureD.3,thegasisabletocoolwithincreasingdensityfordensitiesbetweennH˘101cm3andnH˘105cm3.Asthegascoolsanddensityincreases,thelocalJeansmassisloweredbelowthecentralgasmass,causingperturbationstogrowintheregimewherethedensityisabovenH˘102cm3.138FigureD.5ProjectionsofDensityfortheFiducialRun:Projectionsofaveragedensitythroughthedensestpointareshownasthecentraldensityincreases.Eachprojectionhasascaleof10pc.ThegasisunstabletofragmentationwhenevertheJeansmassdecreaseswithincreasingdensity,whichoccursfordensitiesbetweennH˘101andnH˘104cm3,butstructurewillonlyformwhenthecentralgasmassexceedsthelocalJeansmass,whichonlyoccursoncethecentraldensityhasincreasedabovenH˘103cm3.139D.4RenementCriteria
Toachievethelargedynamicrangestudiedinoursimulations,weselectivelyrenegridcellsbasedondensity,coolingtime,andJeanslength,asdescribedinSectionD.2.1.1.Aspartofthiswork,wehavecarriedoutanumberofsimulationswhereinwevarythenumberofcellsrequiredtocovertheJeanslength,NJ,inordertodeterminetheminimumsetofcriterianeededtoresolvethecollapse.AsdescribedinTrueloveetal.(1998),under-resolvingtheJeanslengthingridbasedcodescanleadtoarticialsuper-Jeansperturbationsthatmayleadtospuriousfragmentation.IntestsofthecollapseofacloudwithaGaussiandensityprole,Trueloveetal.(1998)concludesthattheJeanslengthshouldbecoveredbyatleast4cellsatalltimes.However,thisdoesnotnecessarilyimplythatthesimulationisresolvedenoughtorevealpertinentdetailsoffragmentationinthecollapsinggas.Indeed,severalstudies(Federrathetal.(2011);Turketal.(2012);Latifetal.(2013)andreferencestherein)havefoundthatatleast32-64cellsperJeanslengtharenecessaryforresolvingvorticitywhenmodelingmagneticeldsinPopulationIIIstarformation.TounderstandtheeectsofvaryingthestrictnessoftheJeanscriteriononthephysicalphenomenonweareinterestedin,itisimportanttounderstandwhichrenementcriteriadominateatdierentdensities.InFigureD.6,weshowtheminimumleveltowhichacellmustberesolvedasafunctionofdensityforeachrenementcriterioninourducialmodel.FromEquationD.1,itiseasytocalculatetheminimumgridlevelforwhichthemassrenementcriteriaissatisedforagivendensity.TocalculatetherenementlevelnecessarytosatisfytheJeansandcoolingcriteria,whichrelyonthetemperatureandthecoolingtimeinadditiontothedensity,weusetheaveragevaluesofthesequantitiesateachdensityfromourducialmodel.Sinceacellwillbereneduntilallrenementcriteriaaremet,thecriterionwiththelargestminimumvaluewillbethedominantcriterionatagivendensity.Infact,iftherequiredJeanslengthcoverageissettoNJ=64orhigher,theonlyplacewheretheJeanslengthwillnotbethedominantcriterionisatthelowestdensities,wherefragmentationhasnotyetbegun.FromFigureD.6,itcanbeseenthattheJeansrenementcriterionisthedominantcriterionatalmostalldensities.Inaddition,itisseenthatforarangeofdensities,theJeanscriterionwillbethedominantcriterionevenwhentheminimumnumberofcellscoveringtheJeanslengthislowered.Thus,increasingthemandatedJeanslengthcoveragewillchangetheresolutionoverawiderangeofdensitiesandingeneralwillincreasetheresolutionofthesimulationoveralargerangeofspatialandmassscalesascomparedtothestandarddensity-basedcriteria.ThelinesshowninFigureD.6arecalculatedusingthemass-weightedaverageofthetemperatureandcoolingtimeatagivendensity.Whilethisapproachisusefulforndingtheregimeswheneachcriterionis140ˆˇ˚˘ˆ˝˛˜ˆ˝˘˘˝˙˙˘ˆFigureD.6MaximumRenementvs.Density:Theminimumrenementlevelforeachrenementcriterionisshownasafunctionofdensityforourhighmassducialhalo.Inotherwords,eachlinerepresentstheleveltowhichthesimulationwouldreneifonlythatcriteriawereapplied.Acellwillbereneduntilitisatthehighestnecessaryrenementlevel,meaningthattheactuallevelofrenementatagivendensityisindicatedbythehighestlevelintheplotabove.TheJeanscriterionandcoolingtimecriterionareevaluatedusingthemassweightedaveragetemperatureandcoolingtimeforeachdensity.ThesolidblacklineshowstheJeansrenementlevelwith64cellscoveringtheJeanslength.Fromtoptobottom,thedottedblacklinesshowthelevelwith32,16,8,and4cellscoveringtheJeanslength.141!ˇ"˜˜˘˚ˇˆ˝˜ˇ˚ˇ˜$˜˘˚˜ˇˆ˝˜ˇ˚ˇ˜$˜˘˚  ˛˝˜˙˝˚ˇˇˆ˝˜ˇ˚ˇ˜$FigureD.7ImportanceofRenementCriterion:Theimportanceofdierentrenementcriteriaareshownforour107Mducialmodel.Foreachcriteria,acellisrenedifonequantity(e.g.cellmass)isgreaterthanasecondquantity(e.gaminimummassforrenement).Theratioofthetwoquantitiesaredenotedby˘,where˘Massistheratioformassrenement,˘JeansistheratioforJeansrenement,and˘Coolingistheratioforcoolingbasedrenement.Acellshouldbeaggedforrenementif˘foranycriteriaisgreaterthan1.0.Atmostdensities,theJeanscriterionisthemostdominantrenementcriterion,withcoolingtimebeingimportantatlowdensities.Densitybasedrenementisneverimportant.dominant,itdoesnottakeintoaccountvariationsinthetemperatureorcoolingtimeofthegasatagivendensity,whichmaycausetheminimumrenementleveltovary.Inparticular,anaveragecoolingtimeforgasneartheCMBoorisnotrepresentative.Gasinthatdensityregimewithatemperatureabovetheoorwillcool,whilegaswithatemperaturebelowtheoorwillheat,givinganaveragecoolingtimethatisverylongbutignoringthattheactualcoolingorheatingtimeofthegasmaybesignicantlyshorter.Inordertoassesstheimportanceofthedierentcriteriaonacell-by-cellbasis,welookathowcloseeachcellinthesimulationistobeingrened.Todothis,weevaluatetheratios˘Mass=McellMtop80:3l(D.15)˘Jeans=NJxJ(D.16)˘Cooling=tcooltsound(D.17)whereMcellisthemassofthecell,Mtopisthemassofatoplevelgridcell,NJisthenumberofcellsthatmustcoverthelocalJeanslength,xisthecellwidth,JisthelocalJeanslength,tcoolisthecooling142time,andtsoundisthesoundcrossingtimeofacell,tsound=x=cs.Ifanyoftheseratiosaregreaterthan1,acellwillberened.Forourducialmodel,thedistributionsof˘Mass,˘Jeans,and˘CoolingareshowninFigureD.7.Asexpected,thedensityrenementcriterionisnotimportantforcellswithdensitiesgreaterthannH˘101cm3.Atlowdensities,boththeJeansandcoolingtimecriteriaareclosetobeingmetinalargenumberofcells.Athigherdensities,onlytheJeanscriterionisclosetobeingmet,indicatingthatitisindeedtheonlyimportantrenementcriterion.PanelCofFigureD.7,however,doesindicatethatthecoolingtimerenementislikelytobedominantatlowdensitiesforsomecells.FurthertestsofourducialmodelinwhichonlytheJeanscriterionisusedshowsimilaroverallevolutiontotherunswithallthreerenementcriteria,butforthehighmassrunthereareasmallnumberofcellsthatarenotrened,butordinarilywouldmeetthecoolingcriterionforrenement.Thus,thecoolingtimecriterionisnecessaryinsomecircumstancestofullyresolvethecollapseofthegas.Forourlowmassmodel,wendthattheJeanscriteriaisalwaysdominantbecausetemperaturesarelowerandthusthecoolingtimeislongerthaninthehighmasscase.HavingestablishedthattheJeanscriteriaisnearlyalwaysthedominantfactorinsettinggridresolution,thequestionbecomeshowstrictourrenementcriterianeedstobeinordertoproperlyresolvethecollapseandfragmentationofthecloud.Tounderstandtheeectsofresolutioncriteria,wehavecarriedoutaseriesofruns(describedinTableD.2)wherewevarythenumberofcellsthatmustcovertheJeanslengthfromtheTruelovecriterionofNJ=4toamaximumofNJ=64,thelimitofwhatiscomputationallyfeasibleforourstudy.Thenalstateofthissuiteofsimulationsisshownatascaleof10pcinFigureD.9forourhighmasshalo.143˘ˇ˜˜˛˜ˆ "ˇ˜ˇ˚˙˝˛˙˙˜˚!˜˜˛˜ˆ "ˇ˜ˇ˚˙˝˛˙˙˜FigureD.8EectofRenementCriteriononFragmentation:ThenumberofgravitationallyboundornearlyboundclumpsisshownforrunswithdierentlevelsofJeansrenement.They-axisshowsthenumberofcellsthatmustcoverthelocalJeanslengthatalltimes.Weidentifyclumpsusingacontouringalgorithm,andkeeponlythoseclumpsthatareclosetobeinggravitationallyboundandwhichwillbecomeboundifcoolingcontinues.Thenumberofclumpsineachhalfdexcontourintervalisshownabove.Clumpndingisperformedwheneachrunreachesacentraldensityof1010cm3.Forafulldescriptionoftheclumpndingroutine,seeSectionD.2.4.Fromtheseprojections,itisclearthattheevolutionofthegasisaectedbythelevelofresolution,evenwhentheTruelovecriterionissubstantiallyexceeded.TherunswithNJ=32andNJ=64showfragmentationonsmallscalesthatisnotpresentintheotherruns.Thedierencesstemfromincreasedresolutionofthegasatthedensitieswherefragmentationoccurs,whichleadstoanincreaseinthestrengthofperturbationswithlargewavenumber.Fromtheprojections,itisclearthata\phasetransition"ofsortsoccursbetweenNJ=16andNJ=32cells,buttherearealsohintsofadditionalfragmentationatNJ=64cells.WenotethatusingNJ=128whensimulatingthehighmasshalocausedasharpincreaseinthenumberofgridcellsinthesimulation,andtherunwasterminatedwhenitwasdeterminedtobecomputationallyinfeasible.Fortherestoftherunsinthisstudy,wechoosetouse64cellstocovertheJeanslengthinordertoresolvesmallscaleperturbationswhilemaintainingcomputationalfeasibility,butcautionthereaderthatfurtherincreasingtheresolutionmayhavenon-negligibleeects.InordertoquantifytheeectofJeansresolutiononfragmentation,weusetheclumpndingalgorithmdescribedinSectionD.2.4tondthenumberofpotentiallyboundclumpsineachsimulation,whichisshowninFigureD.8.Weobserveatrendofincreasingfragmentation(inferredfromtheincreaseinthenumberofidentiedclumps)withhigherresolutionoftheJeanslength.Theincreaseinfragmentationisseenatalldensities,andisparticularlyevidentathigherdensities(nH>103cm3).Thelowmasshaloshowslessfragmentationoverall,butthetrendofincreasingfragmentationwithincreasingstrictnessoftheJeansresolutioncriteriaholds.144FigureD.9EectofRenementCriteriononGasDensity:ProjectionsofaveragedensitythroughthedensestpointinthesimulationforrunswithdierentJeansrenementcriteriaforour107Mducialhalo.Eachprojectionhasawidthof10pc,andistakenwhenthecentraldensityhasreachednH=1010cm3.145D.5EectsofPhysicalParameterVariationonGasFragmenta-tionD.5.1Metallicity
Severalstudieshavefoundthattheintroductionofmetalshasastrongeectonthecoolingpropertiesofstarforminggas.Asthefractionofmetalsincreases,thegasisabletocoolmoreeciently.Thisinturnmayleadtoincreasedfragmentation,andisthereasonthatincreasingmetallicityhasbeenproposedasthedrivingfactorbehindthepurportedtransitionbetweenahighcharacteristicmassPopulationIIIIMFandalowercharacteristicmassmetal-enrichedIMF.Tounderstandtheeectsofmetallicityinourmodel,weperformaseriesofruns(seeTableD.1)wherethemetalcontentofthegasisvariedfromauniformmetallicityofZ=0(metalfree,primordialgas)toZ=102Z.Inthesesimulations,weassumethataxedfraction(9:23103bymass)ofthemetalsareintheformofdust.TheeectofvaryingmetallicityonthephysicalandthermodynamicevolutionofthegasisshowninFigureD.10forthehighmasshalo.Theevolutionofthelowmasshaloisvisuallysimilar,withtheexceptionofthelowervirialtemperature.Increasingtheamountofmetalsaltersthecoolingrateinthreeways.First,metalsdirectlycoolthegas,allowingthetemperaturetoreachtheCMBoormorequicklyandatlowerdensities.Second,increasingthemetallicityincreasestheamountofdustpresent.Asdust-mediatedreactionsbecomethedominantmolecularformationchannel,H2andHDareabletoformecientlyatdensitiesbelownH=108cm3,when3-bodyreactionsbecomeeective.Thisleadstomorecoolingatlowerdensities.Thirdly,dustitselfbecomesaneectivecoolantatdensitiesabovenH=109cm3.Theeectsofmetalsonthephysicalstateofthegasmaybeunderstoodbylookingatprojectionsofgasdensityinthecoreofthehalo.InFigureD.11,weshowthenalstateofthehighmasshaloatascaleof3pc.Projectionsthroughthesameregionsofthelowmasshaloshowsimilarbehavior.Athighmetallicities,thegasisabletocoolrapidly,meaningthatthedensestregionwillbeabletocollapsebeforealargemassofgashasbuiltupinthecore.Forlowmetallicityandmetal-freegas,however,thegasmustrelyonH2andHDtocool.Thecollapseisdelayed,whichgivesthecoremoretimetogrow,leadingtoalargermassofdensegasinthecentralregion.ThisisclearlyseeninFigureD.11(particularlyinPanelA),wherethedensestregionsinthelowmetallicityrunsaresurroundedbymoregasthaninthehighmetallicityruns.Theaccretionrateislowerinthelowmasshalo,andthetrendisnotasclear.FromthecollapsetimesgiveninTableD.1,itisevidentthatthereisatrendoffastercollapsewithincreasingmetallicity.146ˆ˝˚&˘"˙ˇ ˘˙˜ˇ ˘˙˜ˇ˛˜"˛#ˆ%&#˛˘ˇ ˘˙˜ˇ#ˆ˙%˚!  ˘˙˜ˇ#ˆ˙%˚! FigureD.10GasProlesforSimulationsWithDierentMetallicities:Comparisonofthephysical,thermal,andchemicalstateofourhighmasshaloasmetallicityisvaried,atthepointwhenthesimulationreachesacentraldensityofnH=1010cm3.PanelAshowssphericallyaveraged,massweighteddensityasafunctionofradius.Runswithhighermetallicitycollapsefaster,leadingtolowerdensitiesintheregionssurroundingthedensestpoint.PanelBshowssphericallyaveraged,massweightedtemperatureasafunctionofdensity.Asmetallicityisincreased,thegasisabletocooltolowertemperatures.TheCMBtemperatureisindicatedbyadashedline.PanelCshowsthemolecularhydrogenmassfractionasafunctionofdensity.Inthemetalfreecase,molecularhydrogenisformedprimarilythroughthethreebodyprocess,whichdoesnotbecomeeectiveuntildensitiesofnH˘108cm3.Asmetallicityisincreased,dustcatalyzedreactionsbecomethedominantmodeofH2formation.PanelDshowstheratiooftheHDtoH2massfractions,whichisenhancedovertheatomicvaluethroughchemicalfractionation.147FigureD.11ProjectionsofGasDensityforHighMassHalos:Projectionsofaveragedensitythroughthedensestpointinthesimulationforrunswithdierentmetallicityforour107Mhalo.Eachprojectionhasawidthof3pc,andistakenwhenthecentraldensityhasreachednH=1010cm3.148˘˚˚˝˚ˇ˜!˘˚˘˛ˆ˙˜ˆˆˇˇ˜!˛ ˚˚˝˚ˇ˜!˘˚˘˛ˆ˙˜ˆˆˇˇ˜!FigureD.12BoundClumpsforSimulationswithDierentMetallicity:Thenumberofboundorpotentiallyboundclumpsidentiedbyourclumpndingalgorithmaspartiallyboundforrunsinwhichmetallicityisvaried.Weexpectthatasthecoolingrateincreases,theamountoffragmentationwillincrease.InFigureD.12,weshowthedistributionofclumpsasafunctionofnHforthehighandlowmasshalos.Inthedensityrange101<nH<105cm3,allrunsshowevidenceoffragmentation,withslightlymorefragmentationathighermetallicities.Here,thegasisabletofragmentbecausethetemperatureisdecreasingwithincreasingdensity.AfterthegasreachestheCMBooritisnolongerabletocoolasdensityincreases,inhibitingfurtherfragmentation.AsH2isformed,thermalenergyisinjectedintothegas.Thehaloswithhighermetallicityareabletoeectivelyradiateawaythisenergy,whichallowsformorefragmentationathigherdensities.Inourhighmasshalo,onlythesimulationswithmetallicitiesabout103ZshowevidenceoffragmentationatnH&107cm3.Wenotethatcoolingfromdustmayleadtofurtherfragmentationinthelowermetallicityrunsathigherdensities.AsnotedbySchneideretal.(2006)andSchneider&Omukai(2010),dustcoolingmayleadtofragmentationinhaloeswithmetallicitiesabove106ZatdensitiesabovenH=1012,wherethegasanddusttemperaturescouple.Forthosesimulationswheretheclumpnderindicatesfragmentationathigherdensities,itisinterestingtoseewhateectcoolingishavingonthegasstructureattherelevantscales.FigureD.13showsprojectionsthroughthecoreatawidthof0.05pc,whichencompassesthedensityrangeabovenH˘106cm3,atthedensityregimetheclumpnderindicatesthatmetallicityaectsfragmentation.Theprojectionsindicatethatthegasinthecoredoesindeedformawhispysubstructureforhighmetallicitygas.Althoughthedensitycontrastissmall,theclumpnderindicatesthatsomeofthesestructuresaremarginallygravitationallybound,givingrisetothepossibilitythatsomeofthesestructurescouldbecomeprotostars.149FigureD.13FragmentationinHighMetallicityHalos:Projectionsofaveragedensitythroughthedensestpointinthesimulationforrunswithdierentmetallicityforour107Mhaloatthescaleswheretheclumpnderndsevidenceoffragmentationinhighmasshalos.Eachprojectionhasawidthof0.05pc,andistakenwhenthecentraldensityhasreached1010cm3.Substructureisevidentinthehighermetallicityruns(showninthebottomrow),andtheclumpnderconrmsthatsomeofthesestructuresmaybecomegravitationallyboundifcoolingpersists.150D.5.2Spin
Wehaveperformedaseriesofrunswherewevarytheinitialmagnitudeofangularmomentuminourhalos,asdescribedbythespinparameter(EquationD.8inSectionD.2.2.4).Forourhighandlowmasshalos,wevarythespinparameterfrom=0:0to=0:1,spanningthelikelyrangeofspinparametersforcosmologicalhalos(e.g.Yoshidaetal.(2003)).Theturbulentcomponentofthevelocityalsoimpartssomeangularmomentumtothegas,andiskeptconstantthroughoutthesesimulations.Astheturbulenteldisisotropicandthelargestlengthscaleissubstantiallysmallerthanthevirialradiusofthehalo,theturbulenteldshouldprovidenonetrotation,butwilldominatethemotionofthegasonlocalscales.Thefulllistofspinparameter-relatedsimulationscanbefoundinTableD.3.ProjectionsofthecoresofsimulationswithdierentspinparametersinthehighmasshaloareshowninFigureD.14atascaleof3pc.Whiletherearesubstantialdierencesbetweenthedierentruns,nocleartrendrelatingfragmentationorgasdensityproletoinitialspinemerges.ThisresultiseasytounderstandinthecontextofFiguresD.15,inwhichweplotaverageangularmomentumasafunctionofmassenclosedforthehighandlowmasshalosrespectively.Theangularmomentumdistributionissimilarinthelowmasshalo.Turbulenceisabletoecientlytransportangularmomentumawayfromtheinnerregionsofthehaloandnormalizetheangularmomentumdistributionforallrunstoroughlythesamelevel,suchthatthedynamicsofthecorearedominatedbytheturbulentmotionratherthanbyinitialrotation.Duringthecollapse,angularmomentumistransportedtolargerradiibyturbulence,thoughthetotalangularmomentumofthesphereisconserved.Thedierencesinfragmentationthatweobservearemorelikelyduetostochasticeectsresultingfromtheperturbationoftheinitialconditions.Inordertoconrmthattheinitialspindoesnotcorrelatewiththeamountoffragmentationinthehalo,weidentiedclumpsineachsimulationusingthemechanismasdescribedintheprevioussections.Thenumberofclumpsineachhalf-dexbin,showninFigureD.16,shownoidentiablerelationshipbetweenfragmentationandspinparameter.
D.5.3Turbulence
Tostudytheeectsofthelevelofturbulentmotionontheevolutionofourmodel,wesimulatedourhighandlowmasshaloswithvaryinglevelsofturbulence.Inoursimulations,theRMSvelocityoftheinitialturbulenteldisnormalizedtosomefractionfcsofthevirialsoundspeedofthehalo.Here,weshowresultsforhaloswithvaluesoffcsfrom0.0(noturbulence)to0.8(trans-sonicturbulence).ThefulllistofturbulencerelatedsimulationscanbefoundinTableD.4.AprojectionthroughthecoreofthehighmasshaloforthedierentrunsisshowninFigureD.17.151FigureD.14EectofSpinParameteronGasDensity:Projectionsthroughthecoreofoursimulationsinwhichthespinparameterisvaried.Althoughdierencesinstructureareobserved,thereisnosystematictrendinthefragmentationwithincreasingspin.Thewidthofeachimageis3.0pc.152˙ˇ˝˛˘ˇˆ˛FigureD.15EectofSpinParameteronAngularMomentumDistributionAverageangularmo-mentumvs.massenclosedforrunswithdierentinitialspinparametersforourhighmasshaloswhenthecentraldensityofeachsimulationreachesnH=1010cm3.Bythispoint,turbulencehasrandomizedthedistributionofangularmomentum.153ˇˆ""˚"˙#%ˆ˘"ˆ˜˝˛ ˙˚!˛˘#˘!˜$""˚"˙#%ˆ˘"ˆ˜˝˛ ˙˚!˛˘#˘!FigureD.16EectofSpinParameteronFragmentationThenumberofboundornearlyboundclumpsisshownforrunswithwithdierentinitialspinparameterasidentiedbyourclumpnder.Thenumberofclumpsineachhalfdexcontourintervalisshownabove.ClumpndingisperformedwheneachrunreachesacentraldensityofnH=1010cm3.Thefragmentationproleconrmsthatspinhasnocleareectonfragmentationbeyondperturbingtheinitialconditions.Itisclearthatevenasmallamountofturbulencehasadramaticeectontheevolutionofthehalo.Intherunswithnoturbulence,thehalosimplycollapsesradiallyanddoesnotfragment.Whenevenasmallamountofturbulenceisadded,thecorebecomesasymmetricandformsconsiderablymoresubstructure.Onceagain,weattempttoquantifyfragmentationbylookingforpotentiallyboundcollapsingclumpswithinoursimulation.InFigureD.18,weshowthedistributionofclumpsasafunctionofdensityfordierentlevelsofturbulence.Whiletheleveloffragmentationdierswiththeinitiallevelofturbulence,thereisnotaclearrelationshipbetweenthetwo.Infact,themostfragmentationseemstooccurforintermediatelevelsofturbulence.Wespeculatethathigherlevelsofturbulenceresultinsubstantialshockheatingofthegasastheturbulencedecaysaway,whichsuppressestheformationofgravitationallyboundclumpsandalsodelaysthecollapseofgasinthehalo.ThisissupportedbythecollapsetimesshowninTableD.4,whenbothlowandhighmasshaloswiththehighestlevelofturbulencetakelongertocollapsethantheirintermediate-turbulencecounterparts.
D.5.4Dust
Tobetterunderstandtheeectsofdustonhaloevolution,weranourducialmodelwithoutdustchemistry{thatis,assumingthatallmetalsareinthegaseousphase.ComparisonsofthephysicalandthermalevolutionoftherunswithandwithoutdustareshowninFigureD.19.Withoutdust,H2isnotabletoforminsignicantquantitiesuntiltheonsetof3-bodyreactions,whichoccursaroundadensityofnH˘108cm3.Thisinhibitstheabilityofthegastocoolatlowdensities.Additionally,thegasdoesnotundergoa154FigureD.17EectofTurbulenceonGasDensity:Projectionsthroughthecoreofoursimulationsinwhichtheamountofinitialturbulenceisvaried.Thescaleofeachimageis3.0pc.˘˜˜˝˜ˇ #˘˜˘˛ˆ˙!˚!ˆ˝˚ ˇ˛˝˛"˜˜˝˜ˇ #˘˜˘˛ˆ˙!˚!ˆ˝˚ ˇ˛˝FigureD.18EectofTurbulenceonFragmentation:Thenumberofboundorpartiallyboundclumpsisshownforrunswithwithdierentlevelsofturbulence.TheRMSvelocityoftheinitialturbulenteldisnormalizedtothefractionofthesoundspeedthatisplottedonthey-axis.Thenumberofclumpsineachhalf-dexcontourintervalisshownabove.ClumpndingisperformedwheneachrunreachesacentraldensityofnH=1010cm3.155secondperiodofdust-drivencoolingathighdensities.
D.6Discussion
Withtheresultsofourmodelinhand,wereturntothequestionswesetouttoinvestigate:whichphysicalpropertiesofthestar-forminghaloaectfragmentationinlowmetallicitygas,andisthereatrue`criticalmetallicity'thatgovernsthetransitionfromahigh-massPopulationIIIstellarIMFtoonethatismorelikethegalacticIMF?WendthatmetallicitydoeshaveaneectongasfragmentationatdensitiesabovenH˘106cm3,correspondingtophysicalscalessmallerthan0.1pc.OurresultslendtentativesupporttotheideaofacriticalmetallicityfoundbyBrommetal.(2001)andSmithetal.(2009),amongothers.However,ondensityscalesbelownH˘106cm3,correspondingtoaphysicalscaleofgreaterthan0.1pc,wedonotndmetallicitytohaveastrongimpactonfragmentation.Wealsondthattheeectofvaryingmetallicityonthethermodynamicpropertiesofthegas,asshowninPanelBofFiguresD.10,isnotasdramaticasisseeninotherstudiessuchasOmukaietal.(2005)andSmithetal.(2009).Inparticular,thegasinallofoursimulationsisabletocoolbelowthe200Koorsetbymolecularhydrogencooling.Weexplainthedierencesbetweenourresultsandpreviousworkasreectingourchoiceofinitialconditionsandthephysics,inparticulartheinclusionofdustanddeuteriumchemistry,inourmodel.D.6.1TheRoleofHDCooling
AsdiscussedinSectionD.3,HDisapowerfulcoolantthatcanlowerthetemperatureofthegassubstantiallybelowthelimitsetbyH2cooling.BecausetheformationofHDfromH2isenergeticallyfavored(seeEquationsD.12andD.13andthediscussionthatfollows),enoughHDcanformtohaveasignicantimpactonthethermodynamicevolutionofthegas.Using1Dsimulations,InordertoconrmthatHDisresponsibleforcoolingthegasbelow200Kintheabsenceofmetals,wehavererunourducialmodelwithareducedchemicalmodelthatdoesnotincludedeuteriumchemistry.Theresults,showninFigureD.20,conrmthattheinclusionofdeuteriumchemistryisabletolowerthetemperatureofthegaswellbelowthetemperaturesreachedbyH2coolingalone,evenintheprimordialruns.Inthecaseofthe103Zrun,thegasisabletocoolallthewaytotheCMBoor.AsthecombinedH2andHDcoolingratesdominatethemetalcoolingrateformetallicitiesbelow103Z,thereislittlevariationinthecoolingpropertiesofthegasatlowdensities,leadingtolittlechangeinfragmentationatthosedensitiesasmetallicityisvaried(seeFiguresD.10andD.12).Forhighermetallicities,themetalcoolingratedominates,leadingtoincreasingfragmentationwithincreasingmetallicity.TheimportanceofHDcoolinginoursimulationsissomewhatsurprising,giventhatotherworkshave156˝˚"*&˛$ˆ˛#˙$ˆ˛#˙˘˜#&˜'˝)*'˜ˆ˙$ˆ˛#˙'˝˛)"%$$ˆ˛#˙˝)"%" !˝((ˇ")!*()" !˝((%*()%+˝((ˇ")!*()%+˝((%*()FigureD.19EectsofDustChemistryonGasProles:Theeectsofdustonthephysicalandthermalevolutionofthehaloareshownforthehighandlowmasshalo.PanelAshowsthephysicalevolution,whichisnotgreatlyaected,asdustcoolingisonlyimportantathighdensities,asshowninPanelB.TheprimaryeectofdustistoserveasacatalystfortheformationofH2andHDatlowdensities.TheH2massfractionisshowninPanelC.Withoutdust,moleculesareonlyformedoncethreebodyreactionsbecomeimportantatnH˘108cm3.Theadditionofdustallowsthegastocoolatlowerdensitiesthroughmoleculartransitions,andathighdensitieswherethedustitselfisabletoradiateenergyfromthegas.InPanelD,theHD/H2ratioisdepressedbydustasmoreH2isformedatlowdensities,decreasingthedenominator.157ˆˇˇ˝˛˜ ˛˘˚˚˙ˇ˘˚˚ˇ˙!˚˚˙ˇ˙!˚˚ˇFigureD.20EectofDeuteriumChemistryonTemperatureProles:Themass-weightedaveragetemperatureisshownasafunctionofdensityforsimulationsthatincludeandneglectdeuteriumchemistry.Theconditionsarethesameasinourhighmassandlowmassducialruns.foundHDcoolingtobenegligible(Omukaietal.,2005;Brommetal.,2002)ingaswithlowinitialionizationbutisinagreementwiththeworksofRipamonti(2007)andGreifetal.(2011).Ripamonti(2007)concludesthatHDcoolingcanaecttheintermediatestagesofthefragmentationiftheHDfractionisgiventimetoincreaseasthegascoolsandcollapses.Inthatwork,HDcoolingisfoundtobemostimportantinlowmasshalos,wherethegascollapsesoveralongertimespan.AfterinvestigatingthefactorsaectingtheHDcoolingrate,weconcludethattheamountofHDthatformsisaectedbytheinitialconditionsofoursimulation.Startingfromalowcentraldensity,thehaloisabletofullybuildupanaccretionshockbeforecollapsing.Inourmodel,thehighandlowmasshalosbothformenoughHDforthetemperaturetodropwellbelowthetemperatureoorsetbyH2cooling,evenintherunswithprimordialgas.Thecollapsereacheshighertemperatures,possiblyionizingthegas,andtakeslonger.Incontrast,theonezonemodelsusedbyOmukaietal.(2005)donotdeveloptheaccretionshockandassumeacollapsewhichoccursonadynamicaltimescale.Brommetal.(2002)startsfromahigherinitialdensityandusesatophatdensityprole,whichagaindoesnotdeveloptheaccretionshockandallowsforafastercollapse.158˘˚˚˝˚ˇ˜!˘˚˘˛ˆ˙˛ ˚˚˝˚ˇ˜!˘˚˘˛ˆ˙FigureD.21BoundClumpsinSimulationswithHighDensityICs:Thenumberofgravitationallyboundornearlyboundclumpsisshownforrunswithdierentinitialdensityproles.Thelefthandpanelshowsresultsforthehighmasssetupwithmetallicitiesof103andprimordialgas.Therighthandplotshowsthesamerunsforthelowmasshalo.RunsmarkedwithHDstartwithaninitialcentralbaryondensity100xhigherthanintheducialmodel.
D.6.2TheRoleofDustCooling
Recentstudies(Schneideretal.,2006;Clarketal.,2008;Dopckeetal.,2011;Schneideretal.,2012,e.g.,)havefoundthatradiationfromdustgrainscandominatethecoolingofthegasathigh(nH>1012cm3)densitiesforgaswithmetallicityabove106Z.Thesesimulationsndsupportforasecondmetallic-itythresholdaround105Z,abovewhichdustcoolingleadstoasuddendropintemperature,spurringadditionalfragmentation.ThefragmentswhichareformedpredictastellarIMFpeakingaround1M,consistentwithmodern-daystarformation.AsoursimulationsdonotfollowtheevolutionofthegastodensitiesabovenH=1010cm3,wearenotabletoobservethisfragmentation.However,inthesimulationswithmetallicitieshighenoughfordustcoolingtooccuratdensitiesbelownH˘1010cm3,weobservethatthisdustcoolingphasecanleadtosignicantfragmentation.Therefore,weexpectthatwewouldhaveobservedfragmentationinourlowermetallicitiesrunshadwecarriedthemuptohigherdensities.D.6.3InitialDensityProle
Whenwestartourmodelfromarelativelyhighinitialdensity(nH=102cm3),thehalobeginstocollapseandfragmentbeforethegashasbeenfullyheatedbytheaccretionshock.ThisspeedsupthecollapseanddoesnotgivethegasenoughtimetobuildupasignicantamountofHD.Thisinturninhibitsthegasfromcoolingatlowdensities.ThisisshowninFigureD.22,whichcomparesthephysical,thermal,andchemicalpropertiesofourhighandlowmassducialmodels,aswellasourhighandlowmassprimordialmodels,withrunsstartedfromahighercentraldensity.Divergenceintheevolutionofthegasismostevidentin159PanelbatdensitiesbelownH˘102cm3.Thegascollapsesbeforetheaccretionshockhasfullydeveloped,andthetemperaturereachesamaximumvaluebelow1,000K,comparedtoover10,000Kfortheducialhighmasshalo.Ourclumpndingmethodconrmsthattheshapeoftheinitialdensityproleleadstoamarkedchangeinthefragmentationpropertiesofthehalo.Thenumberofclumps,showninFigureD.21,ismuchhigherinrunswheretheinitialbaryondensityisincreased.Inthreeofourfourruns,wenoteseveraladditionalclumpsbeingformedwhentheinitialdensityishigher,andthegascollapsesbeforeithasevolvedtoanequilibriumstate.
D.6.4ChoiceofRedshift
AsdescribedinSectionD.1,weassumearedshiftofz=20inallofoursimulations,whichaectsthepropertiesoftheNFWhaloandthetemperatureoftheCMB.Ourassumptionofz=20isbasedonseveralworks,includingTrenti&Stiavelli(2009),Norman(2010)andCrosbyetal.(2013),whichallowfortheformationofbothprimordialandlowmetallicitystarsinlargenumbersataredshiftof20.Asthecollapseisdominatedbytheselfgravityofthebaryonsatlatetimes(seeFigureD.2),variationsintheNFWdensityproleduetosmallvariationsinredshiftwillnothaveastrongimpactonourresults.AsthegasdoescooltotheCMBtemperatureinourhighmetallicityruns(seeFiguresD.10,itislikelythatchangestoTCMBduetoredshiftcouldaectthefragmentation.Smithetal.(2009)haslookedattheeectsofastrongCMBonprimordialandlowmetallicitystarformationandhasfoundthatathighredshifts,aCMBtemperatureoorcaninhibitfragmentationinhighmetallicityhalos.Thus,weexpectthatifwecarriedoutoursimulationsatalowerredshift,wewouldseemorefragmentationintherunswithmetallicitiesof102and103Z,whichcooltotheCMBoor,butthatwewouldseelittlechangeintheotherruns,whichdonotreachtheCMBtemperature.Thiswouldstrengthentheobserveddichotomyinfragmentationpropertiesandwouldlendfurthersupporttotheideaofacriticalmetallicity.
D.6.5LimitationsofThisWork
Whileoursimulationsattempttoaccuratelymodelthecollapseandevolutionofstarforminghalosofprimordialcompositionandatlowmetallicities,ourmodelisnecessarilylimitedbyouridealizedinitialsetupandfromthechoiceofphysicsincludedinoursimulation.Bymodelingthecollapseassphericallysymmetric,weignoretheeectsofgasaccretingalonglaments,whichmightmodifytheaccretionshockweobserve.Similarly,wemodelanisolatedhalo,whicheliminateshalogrowthandheatingduetomergers.Thishasbeenshowntoaectthethermodynamicbehaviorofthegas(e.g.,O'Shea&Norman,2007),but160˙˛!*&˝$ˇ˝#ˆ$ˇ˝#ˆ˘˚#&˚'˙)*'˚ˇˆ$ˇ˝#ˆ'˙˝)!%$$ˇ˝#ˆ˙)!%!˜ ˙((!˛*˝!˙"!˜ ˙((!˛*˝!˙"!˜ ˚$(!),%+˙((!˛*˝!˙"%+˙((!˛*˝!˙"!˜ ˚$(!),!˜ ˙(('!#%'˛!˙"!˜ ˙(('!#%'˛!˙"!˜ ˚$(!),%+˙(('!#%'˛!˙"%+˙(('!#%'˛!˙"!˜ ˚$(!),FigureD.22GasProlesforSimulationsWithHighDensityICs:Theeectsonthephysical,thermal,andchemicalevolutionofthegaswhenthesimulationisstartedfromahigherinitialgasdensity.Conditionsarethesameasinourducialrun,exceptthatthecentralgasdensityisˆc=100cm3insteadofˆc=1inourducialmodel.Whenthecentraldensityishigher,thegascollapsesbeforeithastimetoerasetheimprintoftheinitialconditions.Theaccretionshockisnotfullyformed,resultinginnoionizationandtheformationoflessHDatlowdensities.ThelowerHDfractionpreventsthegasfromcoolingtothelevelseenintheducialmodel.161insomewhatunpredictableways.Ourmodelmakesseveralassumptionsaboutthechemistryofthegasthatmayeecttheevolutionofthehalo.Whileourassumptionofuniformmetallictyislikelynotcorrect,violentrelaxationinpost-mergerhaloscouldecientlymixthegas,makingourapproximationofuniformityappropriate.Throughoutthiswork,wehaveassumedthatthedistributionofmetalsfollowsascaledsolarabundance.Whileitisquitepossiblethattheheavyelementsproducedbytherstgenerationofstarsmightnothavehadasolarabundancepattern(ashasbeenimpliedbyobservationsofmetalpoorstars{seeBeers&Christlieb(2005)),whatmattersinoursimulationistheoverallcoolingrateofthegas,notthedetailsofthecomposition.Ifthecompositionofthegaswerevariedcomparedtosolar,itwouldpotentiallychangethemetallicitywherechangestothefragmentationbecomeevident,butnotthequalitativebehaviorshowninthiswork.Throughoutthiswork,wehaveassumedthatthegasinthehaloofinterestismostlyneutralandhasnotbeenionizedbypreviousstarformation.Itisimportanttonotethatmanylowmetallicitystarsmayforminhalosthathavehostedpreviousgenerationsofstarformation,meaningthatourassumptionofnopreviousionizationmaynotbevalidinallcases.AsdiscussedinSmithetal.(2009)andGlover&Abel(2008)amongothers,previousionizationandsubsequentrecombinationcouldaectthemolecularfraction,asfreeelectronsserveascatalystsduringthemoleculeformationprocess.Anothersourceofuncertaintycomesintheassumedpropertiesofdustinourdustmodel.Throughoutthiswork,wehaveassumeddustgrainswithasizedistributionandcompositionsimilartograinsinthesolarneighborhood.Ifthepropertiesofdustgrainsinearlyuniverseorlowmetallicityenvironmentsweredrasticallydierentfromthoseinourmodel,ourassumptionsoftheH2formationrateondustgrainsandtherateofcoolingfromdustwouldnotnecessarilybevalid.Untildustpropertiescanbefurtherconstrained,theeectsofdustontheformationoftherststarscannotbefullyunderstood,andourassumptionofsolarneighborhoodlikedustpropertiesisreasonable(seeOmukaietal.(2005);Schneideretal.(2006);Schneider&Omukai(2010)formorediscussion).Asdiscussedintheprecedingsection,theinitialconditionsofthesimulationareimportantfordeterminingtheoutcomeofthefragmentationprocess.Perhapsthelargestuncertaintyinthisstudycomesinthechoiceofinitialconditionsforourlowmetallicityhalos.Althoughwehaveattemptedtobaseoursimulationsontheresultsofcosmologicalsimulations,thepropertiesof`typical'earlyuniversestarforminghalosarestillintheprocessofbeingconstrained(e.g.,seeCrosbyetal.,2013).Futuregenerationsofsemi-analyticandcosmologicalhydrodynamicsimulationswillbeabletobetterconstraintheconditionsunderwhichthePopulationIIItometalenrichedstarformationtransitionoccurred.162D.7SummaryandConclusions
Inthiswork,wehaveexploredtheparameterspaceoffragmentationinlowmetallicitystarforminghaloswiththegoalofbetterunderstandingthetransitionbetweenmetal-freeandmetal-enrichedstarformation.WehavedonesousingtheadaptivemeshhydrodynamicscodeEnzo,usinganidealizedsphericalsetupatz=20withinitialconditionsmodeledontheresultsofsimulationsthatstartfromcosmologicalinitialconditions.Oursimulationsutilizeachemicalmodelthatincludesdeuteriumchemistry.WehavealsoincludedadustmodelthattrackstheformationofH2ondustgrainsaswellasheatingandcoolingbydustgrains.Inourstudy,wehavesystematicallyvariedthemetallicity,theinitialspinrate,andthelevelofturbulenceofhaloswithinitialdarkmattermassesof106Mand107M,withtheaimofdeterminingtheeectsofeachparameterongasevolutionandfragmentation.Additionally,wehaveconductedsimulationswherewevarythephysicsthatareincludedinourmodelandtheformofourinitialconditionsinordertoinvestigatehowthesepropertiesaectourresults.WehavecarriedoutanumberofsimulationswhereNJ,thenumberofcellsrequiredtocoverthelocalJeanslength,isvaried.WendthatachangeinthequalitativepropertiesofthefragmentationoccursafterbetweenNJ=32andNJ=64.WeuseNJ=64inourstudies,butcautionthatincreasingNJfurthermighthavenon-negligibleeectsonthefragmentation.Weconcludethatvaryingthemetallicityofthecloudhasthelargestimpactonfragmentation,althoughitsinuenceinourmodelsislessimportantthaninpreviousworks.Asmetallicityisincreased,thegasisabletocoolandcollapsefaster,whichincreasesfragmentation.Aboveametallicityof104Z,thegasisabletofragmentathigherdensities,leadingtotheformationofsubstructureonsub-parsecscalesandamultitudeofpossiblestarformationsites.Wewouldlikelyseedustinducedfragmentationatlowermetallicitiesifwecarriedoursimulationsuptohigherdensities.Wendtentativesupportfortheideaofacriticalmetallicity,butdonotseeasmuchofavariationinevolutionashasbeenreportedinpreviousworks.However,giventhatourassumedredshiftofz=20resultsinarelativelyhighTCMB,andgiventhatourtwosimulationsabovethecriticalmetallicitycooltotheCMBoor,wetheorizethatatlowerredshiftsvaryingmetallicitywouldresultsinagreatervariationincoolingandfragmentationthanwhatweobserve.Wendthattheinitialspinhasnegligibleeectonfragmentation.Thelevelofturbulenceintheinitialvelocityeldhasbeenshowntoalterthefragmentationofthecloud,butdoesnotdosoinasystematicway,withintermediatelevelsofturbulencetypicallyresultinginmorefragmentationthaneitherhighorlowlevels.Ournalresultswerefoundtobeinuencedbytheinitialconditionsofoursimulationaswellasthe163physicsincludedinourcode,andareingenerallygoodagreementwithpreviousworks.WefoundthattheinclusionofdeuteriumchemistryaltersthethermalevolutionofthegasatallmetallicitiesbyallowingthegastocoolbelowthelowerlimitofH2atdensitieslowerthantheregimeinwhichmetalcoolingdominates.TheamountofHDthatisformedandthedensitieswhereitformsisheavilydependentontheinitialdensityproleandthesubsequentevolutionofthecloud.Inthisstudywehavepurposelystartedfromalowinitialdensitysothatthesimulationwillhavetimetoevolve,andthuserasethedetailsoftheinitialconditions.Whenwehavestartedthesimulationfromahighercentraldensity,thehalocollapsesbeforethegashashadtimetofullyformanaccretionshock.TheresultingcollapsedoesnotformasignicantamountofHD,resultinginhighertemperaturesduringthecollapse.Theinitialmassfunctionoftherststarsandthenatureofthetransitionfrommetalfreetolowmetallicitystarformationremainopenquestions.Ascurrentobservationscannotdirectlydetecttherstgenerationofstars,simulationhasemergedasthemainmethodforstudyingtheevolutionofbaryonsintheearlyuniverse.Semi-idealizedsimulationsareapowerfultoolforexploringtheformationandevolutionoftherststars,buttheirresultscanonlybeconsideredvalidifthesimulationsincludetherelevantphysicsandinitialconditions,whichmustbeinferredfromsimulationsbasedoncosmologicalinitialconditions.Further,thesecalculationsmustberesolvednumerically;inadequatespatialresolutionsuppressesfragmentation,thusfundamentallyaectingresults.Thesesimulationsinturncanbenetfromsemi-idealizedmodelsinordertodeterminewhatregionsaremostlikelytohostthesitesoflowmetallicitystarformation.Itisourhopethatwithfutureincreasesincomputingpowerandabetterunderstandingoftheconditionsintheearlyuniverse,thetransitionfromPopulationIIItometal-enrichedstarformationandthehistoryoftherststarsintheuniversecanbefullyunderstood.
D.8Acknowledgments
ThisworkusedtheExtremeScienceandEngineeringDiscoveryEnvironment(XSEDE),whichissupportedbyNationalScienceFoundationgrantnumberOCI-1053575.AllsimulationswerefundedbyXSEDEawardTG-AST090040andperformedontheTACCRangerandStampederesources.ThisworkwasfundedbytheNASAATFPprogram(NNX09AD80GandNNX12AC98G),theNSFASTprogram(AST-0908819),theLANLInstituteforGeophysicsandPlanetaryPhysics,andMSU'sInstituteforCyber-EnabledResearch.TheauthorswouldliketothankMikeNorman,MatthewTurk,andJeOishiforusefuldiscussions.WewouldalsoliketothankMarkVoitofMSUforhissupportofthisproject.Enzoandytaredevelopedbyalargenumberofindependentresearchfromnumerousinstitutionsaroundtheworld.Theircommittmenttoopensciencehashelpedmakethisworkpossible.164TableD.1VaryingMetallicityRunHaloMass(M)Metallicity(Z)CollapseTimelmzmi1060.095.156lmzm610610695.070lmzm510610594.677lmzm410610493.430lmzm310610391.196lmzm210610285.805hmzmi1070.057.229hmzm610710657.506hmzm510710556.532hmzm410710456.472hmzm310710355.360hmzm210710251.314Note.|VaryingMetallicity:Thesearetherunsperformedinthisworktotesttheectsofvaryingmetallicity.Asidefromthemetallicity,allrunshavethesameparametersastheducialmodels.ThelastcolumngivesthetimeinmillionsofyearsforthesimulationtoreachamaximumdensityofnH=1010cm3.TableD.2VaryingJeansRenementRunHaloMass(M)JeansCellsCollapseTime(Myr)lmj4106491.114lmj8106891.150lmj161061691.912lmj321063291.531lmj641066491.606hmj4107454.381hmj8107854.841hmj161071652.673hmj321073254.407hmj641076454.257Note.|VaryingJeansRenement:ThesearetherunsperformedinthisworktotesttheectsofvaryingtheJeansrnementcriteria.OtherthanthenumberofcellsrequiredtocovertheJeanslength,allrunshavethesameparametersastheducialmodels.ThelastcolumngivesthetimeinmillionsofyearsforthesimulationtoreachamaximumdensityofnH=1010cm3.TableD.3VaryingSpinRunHaloMass(M)SpinParameterCollapseTime(Myr)lmsp001060.0091.614lmsp011060.0191.572lmsp031060.0391.519lmsp051060.0591.603lmsp071060.0791.582lmsp091060.0991.533hmsp001070.0053.731hmsp011070.0154.054hmsp031070.0354.427hmsp051070.0554.051hmsp071070.0753.023hmsp091070.0954.302Note.|VaryingSpin:Thesearetherunsperformedinthisworktotesttheectsofvaryingtherotation,ascharacterizedbythedimensionlessspinparameter.Otherthanthespinparameter,allrunshavethesameparametersastheducialmodels.ThelastcolumngivesthetimeinmillionsofyearsforthesimulationtoreachamaximumdensityofnH=1010cm3.165TableD.4VaryingTurbulenceRunHaloMass(M)TurbulenceFactorCollapseTime(Myr)lmt001060.070.278lmt021060.286.686lmt041060.491.599lmt061060.698.232lmt081060.8108.922hmt001070.019.760hmt021070.255.234hmt041070.454.087hmt061070.661.464hmt081070.862.704Note.|VaryingTurbulence:Thesearetherunsperformedinthisworktotesttheectsofvaryingthedegreeofturbulence.TheRMSoftheinitialturbulentvelocityeldisnormalizedtosomefractionofthehalosoundspeed.Thisisshowninthethirdcolumn.Otherthanthedegreeofturbulence,allrunshavethesameparametersastheducialmodels.ThelastcolumngivesthetimeinmillionsofyearsforthesimulationtoreachamaximumdensityofnH=1010cm3.TableD.5OtherRunsRunHaloMass(M)CollapseTime(Myr)Descriptionlmnd10692.711NoDusthmnd10755.270NoDustlmm210693.013ReducedChemicalNetworkhmm210755.571ReducedChemicalNetworkhmhd1074.651HigherInitialDensityhmhdp1075.677HigherInitialDensity(Primordial)lmhd10612.315HigherInitialDensitylmhdp10616.300HigherInitialDensity(Primordial)Note.|OtherRuns:Thistableshowsadditionalrunsperformedinthispaper.Inallcases,theparametersarethesameasthoseofourducialmodelunlessotherwisenoted.Intherstsetofruns,wedonotincludedustchemistry.Inthesecondsetofruns,weuseareducedchemicalmodelwhichdoesnotincludeDeuteriumchemistry.Inthethirdsetofruns,westartwithaninitialbaryondensity100timeshigherthaninourducialmodel.166AppendixEUseofCopyrightedMaterial
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IamcompletingadoctoraldissertationMichiganStateUniversityentitled"AGNFeedbackandDeliveryMethodsforSimulationsofCool-CoreGalaxyClusters".Iwouldlikeyourpermissiontoreprintinmydissertation/thesisthefollowing:Meece,G.R.,OShea,B.W.,&Voit,G.M.GrowthandEvolutionofThermalInstabilitiesinIdealizedGalaxyClusterCores,2015ApJ,808,43(http://iopscience.iop.org/article/10.1088/0004-637X/808/1/43/meta)Meece,G.R.,Smith,B.D.,&OShea,B.W.FragmentationinDustyLow-metallicityStar-formingHalos.2014,ApJ,783,75
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Forfurtherinformation:http://iopscience.iop.org/page/copyright169Pleasenote:Wedonotprovidesignedpermissionformsasaseparateattachment.Pleaseprintthisemailandprovideittoyourpublisherasproofofpermission.170BIBLIOGRAPHY171BIBLIOGRAPHYAbazajian,K.,Fuller,G.M.,&Patel,M.2001,Phys.Rev.D,64,023501Abel,T.,Anninos,P.,Zhang,Y.,&Norman,M.L.1997,NewA,2,181Abel,T.,Bryan,G.L.,&Norman,M.L.2002,Science,295,93
Abell,G.O.1958,ApJS,3,211
Abell,G.O.,Corwin,Jr.,H.G.,&Olowin,R.P.1989,ApJS,70,1Ackermann,M.,etal.2014,ApJ,787,18
Alpher,R.A.,Bethe,H.,&Gamow,G.1948,PhysicalReview,73,803Anderson,L.,etal.2012,MNRAS,427,3435
Anninos,P.,Zhang,Y.,Abel,T.,&Norman,M.L.1997,NewA,2,209Arieli,Y.,Rephaeli,Y.,&Norman,M.L.2008,ApJ,683,L111
Arieli,Y.,Rephaeli,Y.,&Norman,M.L.2010,ApJ,716,918
Baade,W.,&Minkowski,R.1954,ApJ,119,206
Babul,A.,Sharma,P.,&Reynolds,C.S.2013,ApJ,768,11
Balbus,S.A.1988,ApJ,328,395
Balbus,S.A.2000,ApJ,534,420
Balbus,S.A.2001,ApJ,562,909
Balbus,S.A.,&Soker,N.1989,ApJ,341,611
Banerjee,N.,&Sharma,P.2014a,ArXive-prints
Banerjee,N.,&Sharma,P.2014b,MNRAS,443,687
Bardeen,J.M.,Steinhardt,P.J.,&Turner,M.S.1983,Phys.Rev.D,28,679Barkana,R.,&Loeb,A.2001,Phys.Rep.,349,125
Bartelmann,M.,&Schneider,P.2001,Phys.Rep.,340,291
Beers,T.C.,&Christlieb,N.2005,ARA&A,43,531
Begelman,M.C.,&Rees,M.J.1978,MNRAS,185,847
Begelman,M.C.,Volonteri,M.,&Rees,M.J.2006,MNRAS,370,289Bennett,A.S.1962,MmRAS,68,163
Bennett,C.L.,etal.2003,ApJ,583,1
Best,P.N.,vonderLinden,A.,Kaumann,G.,Heckman,T.M.,&Kaiser,C.R.2007,MNRAS,379,894Binney,J.,&Tabor,G.1995,MNRAS,276,663
B^rzan,L.,Raerty,D.A.,McNamara,B.R.,Wise,M.W.,&Nulsen,P.E.J.2004,ApJ,607,800Blandford,R.D.,&Payne,D.G.1982,MNRAS,199,883172Blandford,R.D.,&Znajek,R.L.1977,MNRAS,179,433
Bode,P.,Ostriker,J.P.,&Turok,N.2001,ApJ,556,93
Bohringer,H.,Matsushita,K.,Churazov,E.,Ikebe,Y.,&Chen,Y.2002,A&A,382,804Bondi,H.1952,MNRAS,112,195
Bondi,H.,&Hoyle,F.1944,MNRAS,104,273
Booth,C.M.,&Schaye,J.2009,MNRAS,398,53
Brecher,K.,&Burbidge,G.R.1972,ApJ,174,253
Bregman,J.N.,&David,L.P.1988,ApJ,326,639
Bridle,A.H.,&Feldman,P.A.1972,NaturePhysicalScience,235,168Broadhurst,T.,etal.2005,ApJ,621,53
Bromm,V.,Coppi,P.S.,&Larson,R.B.2002,ApJ,564,23
Bromm,V.,Ferrara,A.,Coppi,P.S.,&Larson,R.B.2001,MNRAS,328,969Bromm,V.,&Larson,R.B.2004,ARA&A,42,79
Bromm,V.,&Loeb,A.2003,ApJ,596,34
Bryan,G.1996,PhDthesis,,Univ.Illinois,(1996)
Bryan,G.L.,&Norman,M.L.1998,ApJ,495,80
Bryan,G.L.,etal.2014,ApJS,211,19
Burns,J.O.,Hallman,E.J.,Gantner,B.,Motl,P.M.,&Norman,M.L.2008,ApJ,675,1125Byram,E.T.,Chubb,T.A.,&Friedman,H.1966,Science,152,66Caau,E.,etal.2011,Nature,477,67
Caau,E.,etal.2012,A&A,542,A51
Carilli,C.L.,&Taylor,G.B.2002,ARA&A,40,319
Carlstrom,J.E.,etal.2011,PASP,123,568
Carr,B.J.2003,inLectureNotesinPhysics,BerlinSpringerVerlag,Vol.631,QuantumGravity:FromTheorytoExperimentalSearch,ed.D.Giulini,C.Kiefer,&C.Laemmerzahl,301{321Cavagnolo,K.W.,Donahue,M.,Voit,G.M.,&Sun,M.2008,ApJ,683,L107Cavagnolo,K.W.,Donahue,M.,Voit,G.M.,&Sun,M.2009,ApJS,182,12Cavaliere,A.,&Fusco-Femiano,R.1976,A&A,49,137
Cavaliere,A.G.,Gursky,H.,&Tucker,W.H.1971,Nature,231,437Churazov,E.,Bruggen,M.,Kaiser,C.R.,Bohringer,H.,&Forman,W.2001,ApJ,554,261Churazov,E.,Forman,W.,Jones,C.,&Bohringer,H.2003,ApJ,590,225Churazov,E.,Sazonov,S.,Sunyaev,R.,Forman,W.,Jones,C.,&Bohringer,H.2005,MNRAS,363,L91Churazov,E.,Sunyaev,R.,Forman,W.,&Bohringer,H.2002,MNRAS,332,729173Clark,P.C.,Glover,S.C.O.,&Klessen,R.S.2008,ApJ,672,757Clark,P.C.,Glover,S.C.O.,Klessen,R.S.,&Bromm,V.2011,ApJ,727,110Colella,P.,&Woodward,P.R.1984,JournalofComputationalPhysics,54,174Cowie,L.L.,Fabian,A.C.,&Nulsen,P.E.J.1980,MNRAS,191,399Cowie,L.L.,&Songaila,A.1998,Nature,394,44
Crawford,C.S.,Allen,S.W.,Ebeling,H.,Edge,A.C.,&Fabian,A.C.1999,MNRAS,306,857Crosby,B.D.,O'Shea,B.W.,Smith,B.D.,Turk,M.J.,&Hahn,O.2013,ApJ,773,108Curtis,H.D.1918,PublicationsofLickObservatory,13,9
Davis,A.J.,&Natarajan,P.2010,MNRAS,407,691
Defouw,R.J.1970,ApJ,160,659
DiMatteo,T.,Colberg,J.,Springel,V.,Hernquist,L.,&Sijacki,D.2008,ApJ,676,33Dijkstra,M.,Haiman,Z.,Mesinger,A.,&Wyithe,J.S.B.2008,MNRAS,391,1961Donahue,M.,Mack,J.,Voit,G.M.,Sparks,W.,Elston,R.,&Maloney,P.R.2000,ApJ,545,670Donahue,M.,etal.2014,ApJ,794,136
Dopcke,G.,Glover,S.C.O.,Clark,P.C.,&Klessen,R.S.2011,ApJ,729,L3Doroshkevich,A.G.,Zeldovich,Y.B.,Syunyaev,R.A.,&Khlopov,M.Y.1980,PismavAstronomicheskiiZhurnal,6,457Dubois,Y.,Devriendt,J.,Slyz,A.,&Silk,J.2009,MNRAS,399,L49Dubois,Y.,Devriendt,J.,Slyz,A.,&Teyssier,R.2010,MNRAS,409,985Dubois,Y.,Devriendt,J.,Slyz,A.,&Teyssier,R.2012,MNRAS,420,2662Dunn,R.J.H.,&Fabian,A.C.2006,MNRAS,373,959
Dunn,R.J.H.,Fabian,A.C.,&Sanders,J.S.2006,MNRAS,366,758Dursi,L.J.,&Pfrommer,C.2008,ApJ,677,993
Eddington,A.S.1916,MNRAS,77,16
Edge,A.C.2001,MNRAS,328,762
Edge,A.C.,Stewart,G.C.,&Fabian,A.C.1992,MNRAS,258,177Edge,D.O.,Shakeshaft,J.R.,McAdam,W.B.,Baldwin,J.E.,&Archer,S.1959,MmRAS,68,37Egan,H.,O'Shea,B.W.,Hallman,E.,Burns,J.,Xu,H.,Collins,D.,Li,H.,&Norman,M.L.2016,ArXive-printsEinasto,J.1965,TrudyAstrozicheskogoInstitutaAlma-Ata,5,87Ensslin,T.A.,Biermann,P.L.,Klein,U.,&Kohle,S.1998,A&A,332,395Fabian,A.C.1994,AnualReviewsofAstronomyandAstrophysics,32,277Fabian,A.C.2012,ARA&A,50,455174Fabian,A.C.,Allen,S.W.,Crawford,C.S.,Johnstone,R.M.,Morris,R.G.,Sanders,J.S.,&Schmidt,R.W.2002,MNRAS,332,L50Fabian,A.C.,Pringle,J.E.,&Rees,M.J.1976,Nature,263,301Fabian,A.C.,Sanders,J.S.,Allen,S.W.,Crawford,C.S.,Iwasawa,K.,Johnstone,R.M.,Schmidt,R.W.,&Taylor,G.B.2003,MNRAS,344,L43Fabian,A.C.,Sanders,J.S.,Taylor,G.B.,Allen,S.W.,Crawford,C.S.,Johnstone,R.M.,&Iwasawa,K.2006,MNRAS,366,417Fabian,A.C.,etal.2011,MNRAS,418,2154
Fabjan,D.,Tornatore,L.,Borgani,S.,Saro,A.,&Dolag,K.2008,MNRAS,386,1265Fanaro,B.L.,&Riley,J.M.1974,MNRAS,167,31P
Fath,E.A.1909,LickObservatoryBulletin,5,71
Federrath,C.,Sur,S.,Schleicher,D.R.G.,Banerjee,R.,&Klessen,R.S.2011,ApJ,731,62Felten,J.E.,Gould,R.J.,Stein,W.A.,&Woolf,N.J.1966,ApJ,146,955Feng,J.L.2010,ARA&A,48,495
Feretti,L.,Giovannini,G.,Govoni,F.,&Murgia,M.2012,A&ARev.,20,54Ferland,G.J.,Korista,K.T.,Verner,D.A.,Ferguson,J.W.,Kingdon,J.B.,&Verner,E.M.1998,PASP,110,761Ferrarese,L.,&Ford,H.2005,SpaceSci.Rev.,116,523
Ferrarese,L.,&Merritt,D.2000,ApJ,539,L9
Ferrarese,L.,etal.2006,ApJ,644,L21
Ferrari,C.,Govoni,F.,Schindler,S.,Bykov,A.M.,&Rephaeli,Y.2008,SpaceSci.Rev.,134,93Field,G.B.1965,ApJ,142,531
Fogarty,K.,Postman,M.,Connor,T.,Donahue,M.,&Moustakas,J.2015,ApJ,813,117Forman,W.,Jones,C.,Cominsky,L.,Julien,P.,Murray,S.,Peters,G.,Tananbaum,H.,&Giacconi,R.1978,ApJS,38,357Fryxell,B.,etal.2000,ApJS,131,273
Fujita,Y.,Kimura,S.,&Ohira,Y.2013,MNRAS,432,1434
Fujita,Y.,&Ohira,Y.2011,ApJ,738,182
Fujita,Y.,&Ohira,Y.2012,ApJ,746,53
Galli,D.,&Palla,F.1998,A&A,335,403
Galli,D.,&Palla,F.2002,Planet.SpaceSci.,50,1197
Gao,L.,Navarro,J.F.,Cole,S.,Frenk,C.S.,White,S.D.M.,Springel,V.,Jenkins,A.,&Neto,A.F.2008,MNRAS,387,536Gaspari,M.,Brighenti,F.,D'Ercole,A.,&Melioli,C.2011a,MNRAS,415,1549Gaspari,M.,Brighenti,F.,&Temi,P.2012a,MNRAS,424,190175Gaspari,M.,Melioli,C.,Brighenti,F.,&D'Ercole,A.2011b,MNRAS,411,349Gaspari,M.,Ruszkowski,M.,&Sharma,P.2012b,ApJ,746,94
Gebhardt,K.,etal.2000,ApJ,539,L13
Giacconi,R.,Murray,S.,Gursky,H.,Kellogg,E.,Schreier,E.,Matilsky,T.,Koch,D.,&Tananbaum,H.1974,ApJS,27,37Giacconi,R.,Murray,S.,Gursky,H.,Kellogg,E.,Schreier,E.,&Tananbaum,H.1972,ApJ,178,281Glover,S.2005,SpaceSci.Rev.,117,445
Glover,S.C.O.,&Abel,T.2008,MNRAS,388,1627
Glover,S.C.O.,&Jappsen,A.-K.2007,ApJ,666,1
Gonzalez,A.H.,Sivanandam,S.,Zabludo,A.I.,&Zaritsky,D.2013,ApJ,778,14Gonzalez,A.H.,Zaritsky,D.,&Zabludo,A.I.2007,ApJ,666,147Gott,III,J.R.,Juric,M.,Schlegel,D.,Hoyle,F.,Vogeley,M.,Tegmark,M.,Bahcall,N.,&Brinkmann,J.2005,ApJ,624,463Greif,T.H.,Springel,V.,White,S.D.M.,Glover,S.C.O.,Clark,P.C.,Smith,R.J.,Klessen,R.S.,&Bromm,V.2011,ApJ,737,75Gurkan,M.A.,Fregeau,J.M.,&Rasio,F.A.2006,ApJ,640,L39Guth,A.H.1981,Phys.Rev.D,23,347
Hahn,O.,Martizzi,D.,Wu,H.-Y.,Evrard,A.E.,Teyssier,R.,&Wechsler,R.H.2015,ArXive-printsHeger,A.,&Woosley,S.E.2010,ApJ,724,341
Hollenbach,D.,&McKee,C.F.1979,ApJS,41,555
Hollenbach,D.,&McKee,C.F.1989,ApJ,342,306
Hopkins,P.F.,&Hernquist,L.2006,ApJS,166,1
Hosokawa,T.,Omukai,K.,Yoshida,N.,&Yorke,H.W.2011,Science,334,1250Hoyle,F.,&Lyttleton,R.A.1939,ProceedingsoftheCambridgePhilosophicalSociety,35,405Inogamov,N.A.,&Sunyaev,R.A.2003,AstronomyLetters,29,791Jansen,F.,etal.2001,A&A,365,L1
Jappsen,A.-K.,Klessen,R.S.,Glover,S.C.O.,&MacLow,M.-M.2009a,ApJ,696,1065Jappsen,A.-K.,MacLow,M.-M.,Glover,S.C.O.,Klessen,R.S.,&Kitsionas,S.2009b,ApJ,694,1161Jeltema,T.,&Profumo,S.2016,MNRAS
Johnstone,R.M.,Fabian,A.C.,Edge,A.C.,&Thomas,P.A.1992,MNRAS,255,431Joung,M.R.,Bryan,G.L.,&Putman,M.E.2012,ApJ,745,148
Kaastra,J.S.,Ferrigno,C.,Tamura,T.,Paerels,F.B.S.,Peterson,J.R.,&Mittaz,J.P.D.2001,A&A,365,L99Katz,J.I.1976,ApJ,207,25
Kaumann,G.,etal.2003,MNRAS,346,1055176Kelly,P.L.,etal.2015,Science,347,1123
Kerr,R.P.1963,PhysicalReviewLetters,11,237
Khalatyan,A.,Cattaneo,A.,Schramm,M.,Gottlober,S.,Steinmetz,M.,&Wisotzki,L.2008,MNRAS,387,13Khlopov,M.Y.,Rubin,S.G.,&Sakharov,A.S.2005,AstroparticlePhysics,23,265Kim,J.-h.,etal.2014,ApJS,210,14
Kim,W.-T.,El-Zant,A.A.,&Kamionkowski,M.2005,ApJ,632,157King,A.2003,ApJ,596,L27
Kirkpatrick,C.C.,&McNamara,B.R.2015,MNRAS,452,4361
Kirkpatrick,C.C.,McNamara,B.R.,&Cavagnolo,K.W.2011,ApJ,731,L23Klessen,R.S.,Glover,S.C.O.,&Clark,P.C.2012,MNRAS,421,3217Kravtsov,A.V.,&Borgani,S.2012,ARA&A,50,353
Latif,M.A.,Schleicher,D.R.G.,Schmidt,W.,&Niemeyer,J.2013,MNRAS,432,668Lawrence,A.1987,PASP,99,309
Lea,S.M.,Silk,J.,Kellogg,E.,&Murray,S.1973,ApJ,184,L105Leccardi,A.,&Molendi,S.2008,A&A,487,461
Li,Y.,&Bryan,G.L.2012,ApJ,747,26
Li,Y.,&Bryan,G.L.2014a,ApJ,789,54
Li,Y.,&Bryan,G.L.2014b,ApJ,789,153
Li,Y.,Bryan,G.L.,Ruszkowski,M.,Voit,G.M.,O'Shea,B.W.,&Donahue,M.2015,ApJ,811,73Linde,A.D.1982,PhysicsLettersB,108,389
Loewenstein,M.1990,ApJ,349,471
Loubser,S.I.,Babul,A.,Hoekstra,H.,Mahdavi,A.,Donahue,M.,Bildfell,C.,&Voit,G.M.2015,ArXive-printsLoubser,S.I.,Babul,A.,Hoekstra,H.,Mahdavi,A.,Donahue,M.,Bildfell,C.,&Voit,G.M.2016,MNRAS,456,1565Lynden-Bell,D.1969,Nature,223,690
Magorrian,J.,etal.1998,AJ,115,2285
Malagoli,A.,Rosner,R.,&Bodo,G.1987,ApJ,319,632
Malagoli,A.,Rosner,R.,&Fryxell,B.1990,MNRAS,247,367
Markevitch,M.,&Vikhlinin,A.2007,Phys.Rep.,443,1
Mathews,W.G.,Faltenbacher,A.,&Brighenti,F.2006,ApJ,638,659Mathews,W.G.,&Guo,F.2012,ApJ,754,154
Matsushita,K.2011,A&A,527,A134177McCourt,M.,Sharma,P.,Quataert,E.,&Parrish,I.J.2012,MNRAS,419,3319McDonald,M.,Veilleux,S.,Rupke,D.S.N.,&Mushotzky,R.2010,ApJ,721,1262McDonald,M.,Veilleux,S.,Rupke,D.S.N.,Mushotzky,R.,&Reynolds,C.2011,ApJ,734,95McDonald,M.,etal.2015,ArXive-prints
McKee,C.F.,&Tan,J.C.2008,ApJ,681,771
McKinney,J.C.,Tchekhovskoy,A.,&Blandford,R.D.2012,MNRAS,423,3083McNamara,B.R.,&Nulsen,P.E.J.2007,ARA&A,45,117
McNamara,B.R.,&Nulsen,P.E.J.2012,NewJournalofPhysics,14,055023McNamara,B.R.,Nulsen,P.E.J.,Wise,M.W.,Raerty,D.A.,Carilli,C.,Sarazin,C.L.,&Blanton,E.L.2005,Nature,433,45Meece,G.R.,O'Shea,B.W.,&Voit,G.M.2015,ApJ,808,43
Meece,G.R.,Smith,B.D.,&O'Shea,B.W.2014,ApJ,783,75
Meece,G.R.,Voit,G.M.,&O'Shea,B.W.2016,ArXive-prints
Menanteau,F.,etal.2012,ApJ,748,7
Merritt,D.2000,inAstronomicalSocietyofthePacicConferenceSeries,Vol.197,DynamicsofGalaxies:fromtheEarlyUniversetothePresent,ed.F.Combes,G.A.Mamon,&V.Charmandaris,221Milgrom,M.1983,ApJ,270,365
Million,E.T.,Werner,N.,Simionescu,A.,Allen,S.W.,Nulsen,P.E.J.,Fabian,A.C.,Bohringer,H.,&Sanders,J.S.2010,MNRAS,407,2046Mitchell,R.J.,Culhane,J.L.,Davison,P.J.N.,&Ives,J.C.1976,MNRAS,175,29PMomjian,E.,Carilli,C.L.,Walter,F.,&Venemans,B.2014,AJ,147,6Nagai,D.,Kravtsov,A.V.,&Vikhlinin,A.2007,ApJ,668,1
Nandra,K.,etal.2013,ArXive-prints
Navarro,J.F.,Frenk,C.S.,&White,S.D.M.1997,ApJ,490,493Navarro,J.F.,etal.2004,MNRAS,349,1039
Navarro,J.F.,etal.2010,MNRAS,402,21
Norman,M.L.2010,inAmericanInstituteofPhysicsConferenceSeries,Vol.1294,AmericanInstituteofPhysicsConferenceSeries,ed.D.J.Whalen,V.Bromm,&N.Yoshida,17{27Norman,M.L.,Bryan,G.L.,Harkness,R.,Bordner,J.,Reynolds,D.,O'Shea,B.,&Wagner,R.2007,ArXive-printsNulsen,P.E.J.1986,MNRAS,221,377
Nulsen,P.E.J.,&McNamara,B.R.2013,AstronomischeNachrichten,334,386O'Dea,C.P.,etal.2008,ApJ,681,1035
O'Dea,K.P.,etal.2010,ApJ,719,1619
Oesch,P.A.,etal.2016,ApJ,819,129178Ohkubo,T.,Nomoto,K.,Umeda,H.,Yoshida,N.,&Tsuruta,S.2009,ApJ,706,1184Omma,H.,Binney,J.,Bryan,G.,&Slyz,A.2004,MNRAS,348,1105Omukai,K.2000,ApJ,534,809
Omukai,K.,Tsuribe,T.,Schneider,R.,&Ferrara,A.2005,ApJ,626,627O'Shea,B.W.,Bryan,G.,Bordner,J.,Norman,M.L.,Abel,T.,Harkness,R.,&Kritsuk,A.2004,ArXivAstrophysicse-printsO'Shea,B.W.,&Norman,M.L.2007,ApJ,654,66
Parker,E.N.1953,ApJ,117,431
Parrish,I.J.,Quataert,E.,&Sharma,P.2009,ApJ,703,96
Peebles,P.J.E.1971,A&A,11,377
Penrose,R.,&Floyd,R.M.1971,NaturePhysicalScience,229,177Peterson,J.R.,&Fabian,A.C.2006,Phys.Rep.,427,1
Peterson,J.R.,Kahn,S.M.,Paerels,F.B.S.,Kaastra,J.S.,Tamura,T.,Bleeker,J.A.M.,Ferrigno,C.,&Jernigan,J.G.2003,ApJ,590,207Peterson,J.R.,etal.2001,A&A,365,L104
Pizzolato,F.,&Soker,N.2005,ApJ,632,821
PlanckCollaborationetal.2014a,A&A,571,A20
PlanckCollaborationetal.2014b,A&A,571,A24
PlanckCollaborationetal.2015a,ArXive-prints
PlanckCollaborationetal.2015b,ArXive-prints
Pollack,J.B.,Hollenbach,D.,Beckwith,S.,Simonelli,D.P.,Roush,T.,&Fong,W.1994,ApJ,421,615Postman,M.,etal.2012,ApJS,199,25
Press,W.H.,&Schechter,P.1974,ApJ,187,425
Prestwich,A.H.,Joy,M.,Luginbuhl,C.B.,Sulkanen,M.,&Newberry,M.1997,ApJ,477,144Puchwein,E.,Sijacki,D.,&Springel,V.2008,ApJ,687,L53
Quataert,E.2008,ApJ,673,758
Raerty,D.A.,McNamara,B.R.,&Nulsen,P.E.J.2008,ApJ,687,899Rasia,E.,etal.2015,ApJ,813,L17
Rayleigh.1882,ProceedingsoftheLondonMathematicalSociety,s1-14,170Reynolds,C.S.,McKernan,B.,Fabian,A.C.,Stone,J.M.,&Vernaleo,J.C.2005,MNRAS,357,242Ripamonti,E.2007,MNRAS,376,709
Ripamonti,E.,&Abel,T.2004,MNRAS,348,1019
Rogallo,R.S.1981,NASASTI/ReconTechnicalReportN,81179Ruhl,J.,etal.2004,inSocietyofPhoto-OpticalInstrumentationEngineers(SPIE)ConferenceSeries,Vol.5498,Z-Spec:abroadbandmillimeter-wavegratingspectrometer:design,construction,andrstcryogenicmeasurements,ed.C.M.Bradford,P.A.R.Ade,J.E.Aguirre,J.J.Bock,M.Dragovan,L.Duband,L.Earle,J.Glenn,H.Matsuhara,B.J.Naylor,H.T.Nguyen,M.Yun,&J.Zmuidzinas,11{29Russell,H.R.,etal.2016,MNRAS,458,3134
Ruszkowski,M.,Bruggen,M.,&Begelman,M.C.2004,ApJ,611,158Ruszkowski,M.,Enˇlin,T.A.,Bruggen,M.,Heinz,S.,&Pfrommer,C.2007,MNRAS,378,662Ruszkowski,M.,Lee,D.,Bruggen,M.,Parrish,I.,&Oh,S.P.2011,ApJ,740,81Ruszkowski,M.,&Oh,S.P.2010,ApJ,713,1332
Ruszkowski,M.,&Oh,S.P.2011,MNRAS,414,1493
Ryan,S.G.,Norris,J.E.,&Beers,T.C.1996,ApJ,471,254
Salpeter,E.E.1955,ApJ,121,161
Salpeter,E.E.1964,ApJ,140,796
Sambruna,R.M.,Eracleous,M.,&Mushotzky,R.F.1999,ApJ,526,60Sarazin,C.L.1988,X-rayemissionfromclustersofgalaxiesSarazin,C.L.,&White,III,R.E.1987,ApJ,320,32
S˘adowski,A.,&Narayan,R.2015,MNRAS,453,3213
Scannapieco,E.,Gray,W.J.,&Pan,L.2012,ApJ,746,57
Schekochihin,A.A.,Cowley,S.C.,Dorland,W.,Hammett,G.W.,Howes,G.G.,Quataert,E.,&Tatsuno,T.2009,ApJS,182,310Schmidt,M.1963,Nature,197,1040
Schneider,R.,&Omukai,K.2010,MNRAS,402,429
Schneider,R.,Omukai,K.,Bianchi,S.,&Valiante,R.2012,MNRAS,419,1566Schneider,R.,Omukai,K.,Inoue,A.K.,&Ferrara,A.2006,MNRAS,369,1437Schure,K.M.,Kosenko,D.,Kaastra,J.S.,Keppens,R.,&Vink,J.2009,A&A,508,751Seager,S.,Sasselov,D.D.,&Scott,D.1999,ApJ,523,L1
Seager,S.,Sasselov,D.D.,&Scott,D.2000,ApJS,128,407
Seyfert,C.K.1943,ApJ,97,28
Sharma,P.,Chandran,B.D.G.,Quataert,E.,&Parrish,I.J.2009,ApJ,699,348Sharma,P.,McCourt,M.,Parrish,I.J.,&Quataert,E.2012a,MNRAS,427,1219Sharma,P.,McCourt,M.,Quataert,E.,&Parrish,I.J.2012b,MNRAS,420,3174Sharma,P.,Parrish,I.J.,&Quataert,E.2010,ApJ,720,652
Shlosman,I.,Frank,J.,&Begelman,M.C.1989,Nature,338,45Sievers,J.L.,etal.2013,J.CosmologyAstropart.Phys.,10,060Sijacki,D.,Pfrommer,C.,Springel,V.,&Enˇlin,T.A.2008,MNRAS,387,1403180Sijacki,D.,Springel,V.,DiMatteo,T.,&Hernquist,L.2007,MNRAS,380,877Silvia,D.W.2013,PhDthesis,UniversityofColoradoatBoulderSilvia,D.W.,O'Shea,B.W.,Smith,B.D.,Shull,J.M.,Turk,M.,&Reynolds,D.2015,inAmericanAstronomicalSocietyMeetingAbstracts,Vol.225,AmericanAstronomicalSocietyMeetingAbstracts,253.07Singh,A.,&Sharma,P.2015,MNRAS,446,1895
Skillman,S.W.,Xu,H.,Hallman,E.J.,O'Shea,B.W.,Burns,J.O.,Li,H.,Collins,D.C.,&Norman,M.L.2013,ApJ,765,21Skory,S.,Hallman,E.,Burns,J.O.,Skillman,S.W.,O'Shea,B.W.,&Smith,B.D.2013,ApJ,763,38Smith,B.,O'Shea,B.W.,Voit,G.M.,Ventimiglia,D.,&Skillman,S.W.2013,ApJ,778,152Smith,B.,Sigurdsson,S.,&Abel,T.2008,MNRAS,385,1443
Smith,B.D.,Turk,M.J.,Sigurdsson,S.,O'Shea,B.W.,&Norman,M.L.2009,ApJ,691,441Smith,B.D.,Wise,J.H.,O'Shea,B.W.,Norman,M.L.,&Khochfar,S.2015,MNRAS,452,2822Smoot,G.,etal.1990,ApJ,360,685
Smoot,G.F.,etal.1992,ApJ,396,L1
Sommer-Larsen,J.,Gotz,M.,&Portinari,L.2003,ApJ,596,47Spergel,D.N.,&Steinhardt,P.J.2000,PhysicalReviewLetters,84,3760Spitzer,L.1962,PhysicsofFullyIonizedGases
Springel,V.,DiMatteo,T.,&Hernquist,L.2005a,MNRAS,361,776Springel,V.,etal.2005b,Nature,435,629
Stacy,A.,Greif,T.H.,&Bromm,V.2012,MNRAS,422,290
Steigman,G.2007,AnnualReviewofNuclearandParticleScience,57,463Steinborn,L.K.,Dolag,K.,Hirschmann,M.,Prieto,M.A.,&Remus,R.-S.2015,MNRAS,448,1504Stone,J.M.,&Norman,M.L.1992,ApJS,80,753
Storm,E.,Jeltema,T.E.,Profumo,S.,&Rudnick,L.2013,ApJ,768,106Sunyaev,R.A.,&Zeldovich,Y.B.1970,Ap&SS,7,3
Sutherland,R.S.,&Dopita,M.A.1993,ApJS,88,253
Sutter,P.M.,Yang,H.-Y.K.,Ricker,P.M.,Foreman,G.,&Pugmire,D.2012,MNRAS,419,2293Tamura,T.,Bleeker,J.A.M.,Kaastra,J.S.,Ferrigno,C.,&Molendi,S.2001a,A&A,379,107Tamura,T.,etal.2001b,A&A,365,L87
Tan,J.C.,&McKee,C.F.2004,ApJ,603,383
Tanaka,T.,&Haiman,Z.2009,ApJ,696,1798
Tchekhovskoy,A.,Narayan,R.,&McKinney,J.C.2011,MNRAS,418,L79Thomson,W.1871,TheLondon,Edinburgh,andDublinPhilosophicalMagazineandJournalofScience,42,362181Tielens,A.G.G.M.,&Hollenbach,D.1985,ApJ,291,722
Trenti,M.,&Stiavelli,M.2009,ApJ,694,879
Truelove,J.K.,Klein,R.I.,McKee,C.F.,Holliman,II,J.H.,Howell,L.H.,Greenough,J.A.,&Woods,D.T.1998,ApJ,495,821Turk,M.J.,Abel,T.,&O'Shea,B.2009,Science,325,601
Turk,M.J.,Oishi,J.S.,Abel,T.,&Bryan,G.L.2012,ApJ,745,154Turk,M.J.,Smith,B.D.,Oishi,J.S.,Skory,S.,Skillman,S.W.,Abel,T.,&Norman,M.L.2011,ApJS,192,9Urry,C.M.,&Padovani,P.1995,PASP,107,803
Valdarnini,R.2006,NewA,12,71
Vernaleo,J.C.,&Reynolds,C.S.2006,ApJ,645,83
Viel,M.,Becker,G.D.,Bolton,J.S.,&Haehnelt,M.G.2013,Phys.Rev.D,88,043502Vogelsberger,M.,Genel,S.,Sijacki,D.,Torrey,P.,Springel,V.,&Hernquist,L.2013,MNRAS,436,3031Vogelsberger,M.,etal.2014,MNRAS,444,1518
Voigt,L.M.,&Fabian,A.C.2004,MNRAS,347,1130
Voigt,L.M.,Schmidt,R.W.,Fabian,A.C.,Allen,S.W.,&Johnstone,R.M.2002,MNRAS,335,L7Voit,G.M.2005,ReviewsofModernPhysics,77,207
Voit,G.M.2011,ApJ,740,28
Voit,G.M.,&Bryan,G.L.2001,Nature,414,425
Voit,G.M.,Bryan,G.L.,Balogh,M.L.,&Bower,R.G.2002,ApJ,576,601Voit,G.M.,Bryan,G.L.,O'Shea,B.W.,&Donahue,M.2015a,ApJ,808,L30Voit,G.M.,Cavagnolo,K.W.,Donahue,M.,Raerty,D.A.,McNamara,B.R.,&Nulsen,P.E.J.2008,ApJ,681,L5Voit,G.M.,&Donahue,M.2015,ApJ,799,L1
Voit,G.M.,Donahue,M.,Bryan,G.L.,&McDonald,M.2014,ArXive-printsVoit,G.M.,Donahue,M.,Bryan,G.L.,&McDonald,M.2015b,Nature,519,203Voit,G.M.,Donahue,M.,O'Shea,B.W.,Bryan,G.L.,Sun,M.,&Werner,N.2015c,ApJ,803,L21Volonteri,M.2010,A&ARev.,18,279
vonderLinden,A.,etal.2014,MNRAS,439,2
Wagh,B.,Sharma,P.,&McCourt,M.2014,MNRAS,439,2822
Wagoner,R.V.1973,ApJ,179,343
Walg,S.,Achterberg,A.,Marko,S.,Keppens,R.,&Meliani,Z.2013,MNRAS,433,1453Weisskopf,M.C.,Tananbaum,H.D.,VanSpeybroeck,L.P.,&O'Dell,S.L.2000,inProc.SPIE,Vol.4012,X-RayOptics,Instruments,andMissionsIII,ed.J.E.Truemper&B.Aschenbach,2{16Werner,N.,etal.2010,MNRAS,407,2063182Werner,N.,etal.2014,MNRAS,439,2291
Weymann,R.1960,ApJ,132,452
Whalen,D.J.,etal.2013,ApJ,777,110
Willott,C.J.,Bergeron,J.,&Omont,A.2015,ApJ,801,123
Willott,C.J.,McLure,R.J.,&Jarvis,M.J.2003,ApJ,587,L15Wilman,R.J.,Edge,A.C.,&Johnstone,R.M.2005,MNRAS,359,755Wilson,A.S.,&Colbert,E.J.M.1995,ApJ,438,62
Wise,J.H.,Demchenko,V.G.,Halicek,M.T.,Norman,M.L.,Turk,M.J.,Abel,T.,&Smith,B.D.2014,MNRAS,442,2560Wurster,J.,&Thacker,R.J.2013,MNRAS,431,2513
Xu,H.,Ahn,K.,Wise,J.H.,Norman,M.L.,&O'Shea,B.W.2014,ApJ,791,110Yang,H.-Y.K.,&Reynolds,C.S.2015,ArXive-prints
Yang,H.-Y.K.,Sutter,P.M.,&Ricker,P.M.2012,MNRAS,427,1614Yong,D.,etal.2013,ApJ,762,27
Yoshida,N.,Abel,T.,Hernquist,L.,&Sugiyama,N.2003,ApJ,592,645Zakamska,N.L.,&Narayan,R.2003,ApJ,582,162
Zandanel,F.,Pfrommer,C.,&Prada,F.2014,MNRAS,438,124
Zanstra,H.1955a,VistasinAstronomy,1,256
Zanstra,H.1955b,inIAUSymposium,Vol.2,GasDynamicsofCosmicClouds,70Zel'dovich,Y.B.1970,A&A,5,84
Zel'dovich,Y.B.,&Novikov,I.D.1965,SovietPhysicsDoklady,9,834Zhuravleva,I.,etal.2014,Nature,515,85
Zhuravleva,I.,etal.2015,MNRAS,450,4184
Zhuravleva,I.,etal.2016,MNRAS
ZuHone,J.A.,Markevitch,M.,&Johnson,R.E.2010,ApJ,717,908Zwicky,F.1933,HelveticaPhysicaActa,6,110183