Disk Instabilities and Cooling Fronts
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Accretion disk outbursts, and their subsequent decline, offer a unique opportunity to constrain the physics of angular momentum transport in hot accretion disks. Recent work has centered on the claim by Cannizzo et al. that the exponential decay of luminos
DiskInstabilitiesandCoolingFronts
EthanT.Vishniac
Dept.ofAstronomy,UniversityofTexas,Austin,TX78712
arXiv:astro-ph/9802232v1 18 Feb 1998Abstract.Accretiondiskoutbursts,andtheirsubsequentdecline,o erauniqueopportunitytoconstrainthephysicsofangularmomentumtransportinhotaccretiondisks.RecentworkhascenteredontheclaimbyCannizzoetal.thattheexponentialdecayofluminosityfollowinganoutburstinblackholeaccretiondisksystemsisonlyconsistentwithaparticularformforthedimensionlessviscosity,α=35(cs/r )3/2.Thisresultcanbeunderstoodintermsofasimplemodeloftheevolutionofcoolingfrontsinaccretiondisks.Inparticular,thecoolingfrontspeedduringdeclineis~αFcs,F(cs,F/r )n,whereFdenotesthepositionofthecoolingfront,andtheexactvalueofndependsonthehotstateopacity,(althoughgenerallyn≈1/2).Settingthisspeedproportionaltorconstrainsthefunctionalformofαinthehotphaseofthedisk,whichsetsitapartfrompreviousargumentsbasedontherelativedurationsofoutburstandquiescence.However,itremainsuncertainhowwellweknowtheexponentn.Inaddition,moreworkisneededtoclarifytheroleofirradiationinthesesystemsanditse ectonthecoolingfrontevolution.INTRODUCTIONThemostpopularmodelforsoftX-raytransientsanddwarfnovaeisthatbothareduetothermalinstabilitiesintheaccretiondisksaroundcollapsedobjects.Thespeci cmechanismthoughttodrivethethermalinstabilityisthechangeinopacityassociatedwiththeionizationofhydrogen(forarecentreviewseeCannizzo1993[1],orOsaki1996[2]).Inthesesystemstheaccretiondiskscanbemodeled
asgeometricallythin,H r,andwithverticalgravitysuppliedbythecentralobject.Asaresultthesoundspeedtoorbitalvelocityratio,cs/(r )scaleswiththethicknesstoradiusratio,H/r,andtheorbitalfrequency (r)isproportionaltor 3/2.
Theoutburstcycleinthesesystemsconsistsofalternatingperiodsofquiescenceandoutburst.Duringquiescencethediskluminosityislowandgasisaccretedfromthecompanionwithoutbeingtransferredthroughthedisk.Inthisstatethediskismostlyneutralandcold.Eventuallytheaccumulationofmaterialatlargeradiileadstotherunawayionizationofhydrogenandthediskmaterialjumpstoahotstate.Themasstransferrateincreasesdramaticallyandaheatingwavepropagatesinwardatthethermalspeed,~αcs,whereαisthedimensionlessdiskviscosityandcsisthemidplanesoundspeed.Inashorttimetheentirediskisionizedandhot,
Accretion disk outbursts, and their subsequent decline, offer a unique opportunity to constrain the physics of angular momentum transport in hot accretion disks. Recent work has centered on the claim by Cannizzo et al. that the exponential decay of luminos
andremainsinthatstateuntilthesteadydepletionofmaterialmakesitimpossibletosustaintheoutburst.Atthispointthediskstartstofallbackintothecoldstate,usuallystartingatsomelargeradius,andacoolingwavesweepsinwardwhiletheluminosityofthediskdecreases.Thedurationoftheoutburstandquiescentphasesisconsistentwithanoutburstvalueofαcloseto0.1andacoldstatevalueseveraltimessmaller.
Oneofthemorestrikingfeaturesofthesesystemsistheextenttowhichtheluminositydeclinecanbemodeledbyanexponentialdecaylaw,inextremecasesstretchingofthreeordersofmagnitudeinX-rayluminosity(Chenetal.1997[3])Mineshigeetal.(1993)[4]notedthatthisimpliesanexponentialdecayofthediskhotphase,andthatthiscanbeachievedinthecontextofthediskinstabilitymodelbyrequiringthecoolingwavevelocitytoscalewiththeradiusofthehotphase.MorerecentlyCannizzo,Chen&Livio(1995)[5]haveexploredthisargumentthroughaseriesofhighresolutionsimulations,andfoundthatthisscalingcanbeobtainedonlyforarestrictedclassofviscositylaws,andthatmodelswithaconstantαcanbestronglyexcluded.Herewewillreviewrecentprogressontheconnectionbetweencoolingwavesandmodelsforangularmomentumtransportinaccretiondisks.
THECOOLINGFRONTSPEED
Ifthehot,innerpartsoftheaccretiondiskareevolvingexponentially,thenthesimplestwaytodrivethisevolutionistohavetheboundaryofthehotphaseofthediskevolveexponentially,thatis
VF
r .(2)
Inotherwords,thatthecoolingfrontspeedisthelocalcoolingrate,α ,timesthelocaldiskheightmodi edbyauniversalspreadingfactor,thoughttobe~0.1.Thisresultre ectstheviewthatthecoolingfrontshouldbethoughtofasapurelylocalphenomena.
However,recentlyCCL[5]showedinsteadthatthecoolingfrontspeedcanbeapproximatedas
Accretion disk outbursts, and their subsequent decline, offer a unique opportunity to constrain the physics of angular momentum transport in hot accretion disks. Recent work has centered on the claim by Cannizzo et al. that the exponential decay of luminos
VF~αFcs,F cs,F
r 3/2,(4)
followingtheformproposedbyMeyerandMeyer-Hofmeister(1983)[8].Thetimescalefortheluminositydecayisthen
τF≈0.4GMBH
7M⊙.(6)
TheyshowedthatthisprescriptionleadstoreasonableagreementwithobservationsofX-raynovae,andwereunableto ndanyothersimpleprescriptionthatdidaswell.Anaddedadvantageofthismodelisapparentfromequation(5).Oneimmediatelyobtainsdramaticallyshortertimescalesfordwarfnovae,inagreementwithobservations.Infact,thesameprescriptioncanbeusedtomodeltheoutburstcycleofSSCygni(Cannizzo1996[9]).
Theseresultsraiseseveralimportantquestions.First,whyisthereafactorof(cs/r )1/2(or(H/r)1/2)inequation(3)?Thereareonlytwoobviousvelocitiesintheproblem,theaccretionvelocity,αc2
s/(r ),andthethermalspeed,αcs.Isthis
somesortofgeometricaverageorisittotallyunrelated?Second,isequation(4)anotherwayofdescribingthedropinalphafromthehotstatetothecoldstate,orisitameasurementofαinthehotstatealone?Third,ifthisisthecorrectformforα,whyistheexponent3/2?Finally,thecoe cientα0ispresumablyaconstantoforderunity.However,35isapeculiarlylargevalueforthenumberone.Wheredoesthisnumbercomefrom?
Toanticipatetherestofthispresentation,wenowhaveananalytictheoryforcoolingwavesinalocallyheateddisk(VishniacandWheeler1996[10],Vishniac1997[11])andthe rsttwoquestionscanbeanswered.Thethirdandfourthquestionsremainconfusing.
Theanalytictheoryofcoolingwavesstartsfromthethreeequationswhichde-scribeaverticallyaverageddiskmodel:
tΣ= 1
Accretion disk outbursts, and their subsequent decline, offer a unique opportunity to constrain the physics of angular momentum transport in hot accretion disks. Recent work has centered on the claim by Cannizzo et al. that the exponential decay of luminos
Vr=2
3c,(9)
followingCCL,wherea=3/7,b=1/7,andc=3/7.Weneedtoaddtheconditionthatbelowsomecriticaltemperature,whichisaveryweakfunctionofradius,coolingbecomesrapid.Atthisradiustheradialthermalgradientbecomesverylarge.Thisinturnimpliesthatalargefractionoftheangularmomentum uxfromsmallerradiiisdeposited,followingequation(8)withinaradialscalelength,rT.Theresultinggasvelocityis
Vr~αFc2
s,F
αF F.(11)
WeconcludethatrTissomeconstanttimesthediskheightandVrisdirectedoutward.
Thisrapidoutwardmotionatthecoolingfrontimpliesthatthecoolingfrontisprecededbyararefactionwave.Thissuggeststhatwecanunderstandtheprogressofthecoolingwavebyconsideringthedynamicsoftherarefactionwave.Ifwelabelapointjustaheadoftherarefactionwavewitha‘p’,thenconservationofmassimpliesthat
ΣpVF≈ΣFαFcs,F,
or
VF≈αFcs,FΣF(12)
r ~αFcs,FΣF
Accretion disk outbursts, and their subsequent decline, offer a unique opportunity to constrain the physics of angular momentum transport in hot accretion disks. Recent work has centered on the claim by Cannizzo et al. that the exponential decay of luminos
coolingwaveplowsthroughit.Thisdecouplesthecoolingwavebehaviorfromthe˙,butitalsoimpliesthatΣpclimbssharplyasthecoolingwavemovesevolutionofM
in.IfwerememberthatTFisnearlyconstant,thenequation(9)impliesthatΣFactuallydropsasatsmallerradii.ThisimpliesasteadydecreaseinΣF/Σp,andaconsequentdropinVF(followingequation(13).Ontheotherhand,ifVFismuchsmallerthantheaccretionvelocityaheadoftherarefactionwave,thenthediskevolvesfasterthanthecoolingwavecanmove.Thisisparadoxicalsincethedrainingoftheinnerdiskautomaticallydropsthediskintothecoldstate.Wecanuseequations(9)and(14)tosolveforVFforanyassumedformforα.If
α=α0 cs
r
where
q 1 q,(16)=1+1+ n
1 (n/2)b.(17)
ForaKramersopacitylaw,theusualchoiceforthediskhotstate,andn=3/2thisgivesq=25/46.Ingeneral,itisverydi culttogetavalueofqwhichisverydi erentfrom1/2foranyoftheusualopacitylaws.
Whathavewelearnedfromthisscalinglawargument?First,thisisanobserva-tionaltestforthefunctionalformofαinthehotstate,inspiteofthefactthatacoolingtransitionisinvolved.Thedynamicsofthewavearecontrolledbytherar-efactionwaveinthehotmaterial,anddonotdependonthestructureoftherapidcoolingregion.Thereisapossibleloopholehere,whichwewillreturntolater.Thecoldmaterialwillpileupafterithas nishedcooling,withacolumndensitynotmuchbelowΣp.Wehaveassumedthatthisisnotmorethanthemaximumcolumndensityonthelowerbranchofthe‘S’curve.Ifthisconditionisnotsatis ed,thengasisnotactuallyejectedataconstantfractionofthethermalspeedacrossthecoolingfrontandwehave,instead,
VF~αFc2s,F
Σcold,maximum 1 1/q.(18)
Asthe‘S’curvecentralbranchbecomesverticalthisgivesacoolingfrontvelocitywhichisclosetotheaccretionspeed.
Second,asatestofthemodelwecanaskhowitbehavesiftheupperbranchofthe‘S’curveisalmosthorizontal,i.e.ifthetemperatureisinsensitivetothecolumndensity.Inthislimit,a=0andconsequentlyq=1.Asexpected,thecoolingfrontmovesattheaccretionvelocity.Intheoppositelimit,whenthehot
Accretion disk outbursts, and their subsequent decline, offer a unique opportunity to constrain the physics of angular momentum transport in hot accretion disks. Recent work has centered on the claim by Cannizzo et al. that the exponential decay of luminos
stateisjustbarelythermallystable,a→∞andq→0.Thatis,thecoolingfrontmovesatclosetothethermalspeed.WecanunderstandtheintermediatebehaviorfoundbyCCLasare ectionofthefactthatrealisticdiskshaveahotstatewhichliesbetweenthesetwoextremecases.
Third,wenotethatonceweallowforthevariationoftheminimumhotstatetemperaturewithradius,aperfectexponentialdeclineactuallyfollowsforn=1.63.Thishasbeencon rmedbysubsequentnumericaltests(Cannizzo1997[12]).Thisdoesnotimplythatthisisthecorrectvaluefortheexponentn,sincetheobserva-tionsaren’tnecessarilyperfectlyexponential.Instead,thisshouldberegardedastestoftheabilityofthescalinglawtopredicttheresultsofnumericalsimulations.Wecanimproveonthisscalinglawbyconstructingasimilaritysolutionforthediskgasusingtheequationsofcontinuity,torque,andthermalstructure(Vishniac1997[11]).Theresultisonlyanapproximate ttotheactualsolution,sincethe˙atproblemitselfhasabuilt-inscale.TheappropriateboundaryconditionforMthefrontis
˙F=2πrFΣF(αFcs,F ),M(19)
where isaconstantrelatingtheout owspeedatthefronttothethermalspeedatthefront.Theanalytictheorydoesnotconstrainthevalueof soweneedtotakeitfromthesimulationsofCannizzoetal..(Thisgives ≈1/6).Thesecondboundaryconditionlooksabsurd.Itis
ΣF=0.(20)
Inpracticethismeansonlythatthedensitycontrastthroughtherarefactionwaveislarge.
Fromthiswederiveasimilaritysolutionwith
˙inner tM
r q(6 )1/2.(22)
Bothofwhichareconsistentwiththesimulationresults.Atanyonetimemorethanhalfthehotphaseofthediskisactuallymovingoutward,http://www.77cn.com.cnparingthesimilaritysolutiontothenumericalworkwe ndthatthetwodi erbyabout5-10%whichgivesuscon dencethatweunderstandbothofthem.
Accretion disk outbursts, and their subsequent decline, offer a unique opportunity to constrain the physics of angular momentum transport in hot accretion disks. Recent work has centered on the claim by Cannizzo et al. that the exponential decay of luminos
IMPLICATIONSFORATHEORYOFANGULAR
MOMENTUMTRANSPORT
Themodelpresentedherefortheevolutionofthecoolingwaveisbasedonassumingthatthediskisheatedthroughthelocaldissipationoforbitalenergyandthatαcanbedescribedintermsoflocalvariables.Assumingforthemomentthatthisisbasicallycorrect,whataretheimplicationsforafundamentaltheoryofα?Knowingthataparticularfunctionalformofα tstheavailableobservationalevidencefallswellshortofafundamentaltheoryforα.Ontheotherhand,wecanuseourresultstoidentifyparticularlypromisingmodels,ifanyexist.Sinceweareconcernedherewithangularmomentumtransportinthehot,ionizedphasewecanrestrictourattentiontomodelswhicharemoste cientinthisphase.
AllcurrentlypopularmodelsforthisregimemakeuseoftheVelikhov-Chandrasekharinstability(Velikhov1959[13],Chandrasekhar1961[14]), rstiden-ti edbyBalbusandHawley(1991)[15]asthemost,andperhapsonly,promisingmechanismforangularmomentumtransportinconductingaccretiondisks.Thisisaninstabilityofamagnetic eldembeddedinastronglyshearing ow.Thelinearinstabilityhasatypicalgrowthrate~ andatypicallengthscale~VA/ witha22resultantα∝VA/cs.Numericalsimulations(cf.thereviewbyGammie1997[16]andreferencestherein)con rmtheexistenceoftheinstabilityanditssaturationinaturbulentstate.Interestinglyenough,theyalsoindicatethatverticalstructureislargelyirrelevantforthedynamoprocess,whichallowsustoexcludemagneticbuoyancyasadrivingforceforthedynamo.Di erencesamongthemodelsbasedontheseresultsaredueentirelytodi erentproposalsfortheunderlyingdynamomechanism.
First,theturbulencemightsupportaninversecascade,inwhichalargescale eldisgeneratedspontaneouslyfromtheunderlyingturbulencewithoutanyappealtothekindofsymmetrybreakingusedinmean- elddynamotheory(BalbusandHawley1991[15],seealsoGammie1997[16]andreferencestherein).Sincethisisapurelylocalprocess,itproducesanαwhichissomeconstantoforderunity,witharelaxationtimewhichissomeconstanttimestheeddyturnovertime(~ 1).Theavailablesimulationsseemconsistentwiththismodeloveralimitedrangeofboxsizes,buttheresultantαisafunctionofgridresolution(e.g.Brandenburgetal.1996[17]).Periodicshearingboxsizeshavealengthwhichvariesbetween1and4times2πtimestheboxheight,whichisprobablyameasureofthesensitivitytodiskgeometryovertherange1<(r/H)<4inathindisk.However,thismodelisinconsistentwiththecoolingwaveresults.Ifthelatterarecorrect,thenthemodelneedstobemodi edtoallowsomenon-locale ects.
Second,thereistheinternalwave-drivendynamo(Vishniac&Diamond1992
[18],andreferencestherein)inwhichtidallygeneratedwavesdriveadynamowithagrowthrate~(H/r)n,wherenisabitlessthan1.5inastationarydisk.Balancingdissipationwithgrowthgivesα~Γ/ .Thisistherightscaling,butthisdynamomodelisinherentlynonlocalinabadsense.Thedependenceoninternalwaves
Accretion disk outbursts, and their subsequent decline, offer a unique opportunity to constrain the physics of angular momentum transport in hot accretion disks. Recent work has centered on the claim by Cannizzo et al. that the exponential decay of luminos
propagatingthroughthecoolregionshouldmakethisdynamoveryweakjustinsidethecoolingfront,wipingoutthe‘S’curvee ectandcompletelychangingtheevolutionofthecoolingfront.Althoughnodetailedcalculationofthesee ectsexist,itseemslikelythatthiswouldleadtoabroadcoolingfront,propagatingattheviscousspeed,withadependenceon(H/r)whichistoosteep.
Finally,therehasbeenanattempttoconstructamean elddynamotheorywithnosymmetrybreaking,theincoherentmean elddynamo(VishniacandBranden-burg1997[19]).Inthismodelthehelicitynecessaryforgeneratinglargescaleradialmagnetic eldemergesfromtherandom uctuationsprovidedbythesmallscaleinstabilitiesofthemagnetic eld.Itcanbeshownthatthisprocesswilldrivetheexponentialgrowthoflargescale,axisymmetricmagnetic elddomainsinthepresenceofstrongshearing.However,thisprocessbecomeslesse cientasthenumberofeddiesperannulusincreases,i.e.thisdynamorunsmoreslowlyasH/rdecreases(or,insimulations,astheheighttolengthratiooftheperiodicshearingboxdecreases).Inasense,thisisanattempttoimplementageometrydependentinversecascadeinthecontextofthe rstmodel.Unfortunately,whilethismodelseemsqualitativelypromising,itpredicts
α~ VA
r 2,(23)
whichistoosteep.Moreover,thenumericalsimulationsdonotshowtheexpectedsensitivitytogeometry.Thismaybebecausethelargevalueofα0restrictstheasymptoticregime,wherethescalinglawgiveninequation(23)isvalid,toboxesconsiderablylongerandthinnerthantherangeexploredtodate.Inanycase,inordertoreconcilethismodeltothecoolingwavecalculations,werequiredwarfnovaeandX-raynovaediskstobesu cientlythickthatsmallcorrectionstothedynamogrowthsoftenthesaturatedvalueofα.Moredetailedmodelingofthisdynamoprocesswillbenecessarytojudgewhetherornotthisisarealistichope.Weconcludethatnocurrentmodelisfullyconsistentwiththecoolingwaveresults,althoughthereissomechancethatamorerealisticversionoftheincoherentdynamomodelwillproveadequate.
FUTUREPROSPECTS
Whatcanweexpectbywayofprogressinthisarea?Ourabilitytosetlimitsontheallowedformsofαislimitedbytherangeofluminositieswecanobserveandthepresenceofnon-exponentialfeaturesinaccretiondisklightcurves,e.g.re aresandsecondarymaxima.Aslongasthesefeaturesremainpoorlyunderstooditwillbedi culttopindowntherequiredformofα.However,evenwithinthelimitsimposedbyourcurrentstateofunderstandingthecomputationalmodelscouldbemadetoyieldsigni cantlymoreinformation.Forexample,wehavenopreciseideaoftheappropriateerrorbarsareontheparametersoftheproposedfunctionalformofα.CCLoriginallyestimatedthatthevalueofnallowedbyobservational
Accretion disk outbursts, and their subsequent decline, offer a unique opportunity to constrain the physics of angular momentum transport in hot accretion disks. Recent work has centered on the claim by Cannizzo et al. that the exponential decay of luminos
constraintswasprobablywithin15%ofthe‘exponential’valueof1.5,butthevalueofnwhichactuallyyieldsaperfectexponentialdecayis10%higher,andtheavailablegridofmodelshasnotyetnaileddowntheallowedrangeofn.
Amoreseriousdangeristhepossibilityisthatthecoolingwavemodelusedhereneglectssomeimportantphysicale ects.Therelaxationofthemagnetic eldasthecoolingwaveapproachesisdi culttocalculate,sincetheactualdynamomechanismis,asnotedabove,acontroversialtopic.However,weexpectthetypicalrateforthisrelaxationtobecomparabletothedynamogrowthrate,whichshouldbe~α .Thisisalsothethermalrelaxationrateinanopticallythickdisk.Thediskstructurewillevolveatslowerratesuntiltheonsetofrapidcooling,whichistoolatetoa ectthecoolingwavearguments.Moreover,evenifweassumethatthemagnetic eldisfrozenintotheplasmafromtheverybeginningoftherarefactionwavewecanshowthatithasnoperceptiblee ectonourresults.
Another,andpotentiallymoreseriousproblemisthatthistreatmentassumesthatthedisktemperatureisdeterminedbylocaldissipation,thatis,itignoresirradiation.AttemptstoaccountfortheopticallightfromBHsystemssuggestthatirradiationmaybeassigni cantaslocalenergydissipation(Cannizzo1997
[12]),althoughitprobablydoesnotdominatebyalargefactor.Ifwerestrictourattentiontodwarfnovaesystems,whereirradiationislessimportant,thenwe ndsimilar,butlessprecise,constraintsonα(Canizzo1996[9]).Inanycase,thee ectofmoderateirradiationonthecoolingwavestructureneedstobeexplored.
Arethereacceptablealternativestothecoolingwavemodelforthethesesystems,perhapsonesthatnotinvolveavaryingα?Atpresenttheanswerisno,althoughathoroughsearchhasnotyetbeendone.Ifwewishtoconsideralternativeex-planationsfortheexponentialdecayofX-rayemissionfromblackholeaccretiondisksystems,weneedmodelsthatmaintainaconstantaccretiontimescaleintherelevantmassreservoir,thatis
1τacc=αc2
s
Accretion disk outbursts, and their subsequent decline, offer a unique opportunity to constrain the physics of angular momentum transport in hot accretion disks. Recent work has centered on the claim by Cannizzo et al. that the exponential decay of luminos
anddensityincreasessmoothlyasLxdecreases.Itremainstobeseenwhetherornotaphysicallyreasonablemodelcanbeconstructedalongtheselines.ThisworkhasbeensupportedinpartbyNASAgrantNAG5-2773.
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