DRAFTVERSIONFEBRUARY2,2008 PreprinttypesetusingLATEXstyleemulateapj SPORADICALLYTORQUEDACCRETIONDISKSAROUNDBLACKHOLES DAVIDGAROFALO1ANDCHRISTOPHERS.REYNOLDS2 DraftversionFebruary2,2008 ABSTRACT Theassumptionthatblackholeaccretiondiskspossessanuntorquedinnerboundary,theso-calledzerotorque boundarycondition,hasbeenemployedbymodelsofblackholedisksformanyyears.However,recenttheoretical and observational work suggests that magnetic forces may appreciably torque the inner disk. This raises the 5 questionoftheeffectthatatime-changingmagnetictorquemayhaveontheevolutionofsuchadisk.Inparticular, 0 weexplorethesuggestionthatthe“DeepMinimumState”oftheSeyfertgalaxyMCG–6-30-15canbeidentified 0 asasporadicinnerdisktorquingevent.Thissuggestionismotivatedbydetailedanalysesofchangesintheprofile 2 ofthebroadfluorescenceironlineinXMM-Newtonspectra.Wefindthattheresponseofsuchadisktoatorquing eventhastwophases;aninitialdammingoftheaccretionflowtogetherwithapartialdrainingofthediskinterior n a tothetorquelocation,followedbyareplenishmentoftheinnerdiskasthesystemachievesanew(torqued)steady- J state. IftheDeepMinimumStateofMCG–6-30-15isindeedduetoasporadictorquingevent,weshowthatthe 1 fraction of the dissipated energygoing into X-rays must be smaller in the torqued state. We propose one such 2 scenarioinwhichComptoncoolingofthediskcoronaby“returningradiation”accompanyingacentral-torquing eventsuppressesthe0.5–10keVX-rayfluxcomingfromallbuttheinnermostregionsofthedisk. 2 Subjectheadings: accretion,accretiondisks–blackholephysics–magnetohydrodynamics–galaxies:active– v galaxies:individual(MCG–6-30-15)–X-rays:galaxies 5 5 1. INTRODUCTION 4 1 There is direct evidence that active galactic nuclei (AGN) are powered by disk accretion onto supermassive black holes. For 0 example, the centralregionsofthe giantellipticalgalaxyM87 werespatially resolvedbythe HubbleSpaceTelescope sufficiently 5 toactuallyseea∼ 100pc-scalediskofionizedgasinorbitaboutanunseenmassof3×109M ,presumedtobeasupermassive ⊙ 0 black hole (Fordetal, 1994; Harmsetal, 1994). The factthat this gasdisk is approximatelynormalto the famousrelativistic jet / h displayedby this objectis circumstantialevidencethatit is, indeed, the outer regionsof the accretiondisk thatpowersthis AGN. p In another important case, radio observations of megamasers in the spiral galaxy NGC 4258 (M106) reveal an almost perfectly - Keplerian∼0.5pcgasdiskorbitingacompactobjectwithmass3.5×107M (Miyoshietal. 1995;Greenhilletal. 1995). o ⊙ Evenbeforetheobservationalevidenceforblackholeaccretiondisksbecamecompelling,thebasictheoryofsuchdiskshadbeen r t extensivelydeveloped.Buildinguponthenon-relativistictheoryofShakura&Sunyaev(1973),Novikov&Thorne(1974)andPage s a &Thorne(1974)developedthe“standard”modelofageometrically-thin,radiatively-efficient,steady-state,viscousaccretiondisk : around an isolated Kerr black hole. In addition to the assumptions already listed, it is assumed that the viscous torque operating v i withinthediskbecomeszeroattheradiusofmarginalstability,r=rms. Physically,thiswasjustifiedbyassumingthattheaccretion X flowwouldpassthroughasonicpointclosetor =rmsandhenceflowballistically(i.e.,“plunge”)intotheblackhole. Even while setting up this boundary condition, Page & Thorne (1974) noted that magnetic fields may allow this zero-torque r a boundarycondition(ZTBC)tobeviolated.Giventhemodernviewpointofaccretiondisks,thatthevery“viscosity”drivingaccretion is dueto magnetohydrodynamic(MHD)turbulence,theideathatthe ZTBCcanbeviolatedhasbeenrevivedbyrecenttheoretical work,startingwithGammie(1999)andKrolik(1999a). Inindependenttreatments,theseauthorsshowthatsignificantenergyand angular momentum can be extracted from matter within the radius of marginal stability via magnetic connections with the main bodyoftheaccretiondisk. Agol&Krolik(2000)haveperformedtheformalextensionofthestandardmodeltoincludeatorqueat r =rmsandshowthattheextradissipationassociatedwiththistorqueproducesaverycentrallyconcentrateddissipationprofile.As shownbyGammie(1999),Agol&Krolik(2000),andLi(2002),thisprocesscanleadtoanextraction(andsubsequentdissipation) ofspinenergyandangularmomentumfromtherotatingblackholebytheaccretiondisk. Inthesecases,themagneticforcesmight be capableofplacingthe innermostpartof theflow onnegativeenergyorbits, allowinga Penroseprocesstobe realized(wenote thatWilliams[2003]hasalsoarguedfortheimportanceofanon-magnetic,particle-particleandparticle-photonscatteringmediated Penrose process). A second mechanism by which the central accretion disk can be torqued is via a direct magnetic connection betweentheinneraccretiondiskandthe(rotating)eventhorizonoftheblackhole. Inthiscase, aslongastheangularvelocityof theeventhorizonexceedsthatoftheinnerdisk,energyandangularmomentumofthespinningblackholecanbeextractedviathe Blandford-Znajekmechanism(Blandford&Znajek1977). We notethatfieldlinesthatdirectlyconnecttherotatingeventhorizon with the bodyoftheaccretiondiskthroughthe plungingregionare seenin recentGeneralRelativistic MHDsimulationsofblack holeaccretion(e.g.,Hiroseetal. 2004). InterestinthesetorquedrelativisticdiskshasreceivedaboostfromrecentX-rayobservations. TheX-rayspectraofmanyAGN (and,indeed,GalacticBlackHoleCandidates)revealthesignaturesof“X-rayreflection”fromoptically-thickmatter. Insomecases, examinationofthesefeaturesallowsustodetectstrongDopplerandgravitationalshiftsindicativeofcircularmotionclosetoablack 1Dept.ofPhysics,UniversityofMaryland,CollegePark,MD20742 2Dept.ofAstronomy,UniversityofMaryland,CollegePark,MD20742 1 2 hole (Fabian et al. 1989; Tanaka et al. 1995; Fabian et al. 2000; Reynolds & Nowak 2003). In these cases, the observed range ofDopplerandgravitationalshiftscanbeusedtomaptheX-rayemission/irradiationacrossthesurfaceoftheaccretiondisk. The XMM-Newtonsatelliteisparticularlywellsuitedtothestudyofthesefeaturesduetoitscombinationofgoodspectralresolutionand highthroughput.UsingXMM-Newton,Wilmsetal. (2001)andReynoldsetal. (2004)studiedtheSeyfert-1galaxyMCG–6-30-15in itspeculiar“DeepMinimumState”firstdiscoveredbyASCA(Iwasawaetal. 1996).ConfirmingtheprincipalresultofIwasawaetal. (1996),theX-rayreflectionfeatureswerefoundtobeextremelybroad.Thedegreeofgravitationalredshiftingrequiredthemajority of the X-ray emission to emerge within a radius of r ∼ 2GM/c2 from a near-extremalKerr black hole (i.e., a black hole with a spinparameterofa = 0.998). AsexplicitlyshowninReynoldsetal. (2004),itisveryproblematictoexplainthesedatawithinthe frameworkofthestandardaccretiondiskmodel. Fabian&Vaughan(2003)andMiniutti&Fabian(2004)suggestthatgravitational focusingoftheprimarycontinuumX-raysmightproducesuchacentrally-concentratedemissivityprofile.Alternatively,Reynoldset al. (2004)hasshownthatatorqueddiskcanreadilyexplaintheDeepMinimumspectrumprovidedthesourceisassumedtobeina torque-dominatedstate(or,intheterminologyofAgol&Krolik[2000],an“infinite-efficiency”state)wherebythepowerassociated withtheinnermosttorqueisinstantaneouslydominatingtheaccretionpower. Inotherwords,theX-raydatasuggestthatduringthis DeepMinimumstateofMCG–6-30-15thepowerderivedfromtheblackholespingreatlyexceedsthatderivedfromaccretion. Ofcourse,thisstateofaffairscannotlastforeverorelsethecentralblackholeinMCG–6-30-15wouldspindowntoapointwhere itcouldnolongerprovidethispower. Atsomepointinitshistory,thesystemmustbeinanaccretion-dominatedphaseinwhichthe blackholeisspunup. However,eveninitsspin-dominatedstate,thespin-downtimescaleofthecentralblackholeisoftheorderof 100millionyearsormore. Thuswecouldenvisageasituationinwhichthesystemshinesviaaquasi-steady-state,spin-dominated accretiondisk. Therearehints,though,thataccretiondisksmayswitchbetweenspin-dominatedandaccretion-dominatedonmuch shortertimescales. Initsnormalspectralstate, theX-rayreflectionfeaturesinMCG–6-30-15aremuchlesscentrallyconcentrated than in the Deep MinimumState, suggestingthat the normalstate mightbe accretion-dominated. Itis also importantto note that thissystemcanswitchbetweenitsnormalstateandtheDeepMinimumStateinaslittleas5–10ksec(Iwasawaetal. 1996),which corresponds to only a few dynamical timescales of the inner accretion disk. Thus it is of interest to consider the physics of an accretiondiskthatundergoesarapidtorquingevent.Thatistheprimemotivationforthispaper. In Section 2 we will begin our study of sporadicallytorquedaccretion disks by investigatingan analytic solution for a torqued Newtoniandisk.InSection3,wegeneralizetothefullyrelativisticequationsandobtainnumericalsolutions. InSection4,werelate our resultswith the observedpropertiesof the “Deep MinimumState” of MCG–6-30-15,and consequentlydiscussthe effectthat atorquingeventmayhaveonthephysicsoftheX-rayemittingdiskcorona. Inparticular,wesuggestthattheenhancedReturning Radiation associated with a torquing event might suppress 0.5–10keV coronal emission in all but the inner portion of the disk. Section5summarizesourmainconclusions. 2. ANANALYTIC“TOY”MODELOFATORQUEDDISK Webeginourinvestigationofsporadicallytorqueddisksviathestudyofasimplecasethatlendsitselftoastraightforwardanalytic solution. Weshallconstructamodelofaradiatively-efficientaccretiondiskfollowingtheusualapproachofPringle(1981).Weshallassume thattheaccretiondiskisaxisymmetric,geometrically-thinandinKeplerianmotionaboutapoint-massM. Usingacylindricalpolar coordinatesystem(r,z,φ)withtheaxispassingthroughthecentralmassnormaltothediskplane,weshalldenotethesurfacedensity ofthediskbyΣ(r,t),theangularvelocityofthediskaboutM asΩ(r)andtheradialvelocityofthediskmaterialasv (r,t). The r equationsthatdeterminethestructureofthethindiskassumingradiativeefficiencyaremassandangularmomentumconservation, ∂Σ ∂(rv Σ) r + r =0, (1) ∂t ∂r ∂(Σr2Ω) ∂(rΣv r2Ω) 1 ∂G r + r = , (2) ∂t ∂r 2π ∂r respectively,whereG(r,t)isthetorqueexertedbythediskoutsideofradiusronthediskinsideofthatradius. Instandarddiskmodels,thetorqueGistheintegratedvalueoftheonlystresstensorcomponent(S )thatsurvivesthecondition rφ ofaxisymmetryandgeometric-thinness.FromKrolik(1999b)wehave, ∂Ω 3 G = rS dz rdφ = 2πr νΣ , (3) rφ ∂r Z Z where we have introducedan ”effective kinematic viscosity”, ν. To generalize these modelsto the case of an externallyimposed torque,weset ∂Ω 3 G=2πr νΣ +G . (4) T ∂r Combiningeqns.1, 2and4, andspecializingtoaKeplerianrotationcurve,wegettheusualdiffusionequationforsurfacedensity modifiedfortheeffectsoftheexternaltorque, ∂Σ 3 ∂ ∂(νΣr1/2) 1 ∂ ∂G = r1/2 − r1/2 T . (5) ∂t r∂r ∂r rπ(GM)1/2 ∂r ∂r (cid:20) (cid:21) (cid:18) (cid:19) 3 Fortherestofthispaper,weshallworkinunitswhereGM = 1. Changingvariablestox = r1/2 andψ = νΣxandassumingthat ν hasnoexplicittimedependence,weget ∂ψ 3ν ∂2 G = ψ− T . (6) ∂t 4x2∂x2 3π (cid:18) (cid:19) Wenowconsideraparticulartorquingevent.Supposethatthedisksuffersnoexternaltorquesfortheperiodt<0.Then,att=0, weengageanexternaltorque(possiblyresultingfromamagneticconnectiontotheplungingregionorspinningeventhorizon)that depositsangularmomentuminto a narrowannulusatr = r0. If therate at whichangularmomentumis beingdepositedis β, we have ∂G T =βδ(r−r0)Θ(t), (7) ∂r giving GT(r,t)=βΘ(r−r0)Θ(t), (8) whereΘistheHeavysidestepfunction. Atthispoint,wespecializetoaparticularviscositylaw. Wesetν = kr formathematical convenience,althoughitwillnotmakequalitativedifference.Wecannowrewriteeqn.6as, ∂ξ 3k ∂2ξ β ∂t = 4 ∂x2 − 3πΘ(x−x0)δ(t), (9) (cid:20) (cid:21) where, β ξ = ψ− Θ(x−x0)Θ(t), (10) 3π andthedelta-functionintimeresultsfromthetime-derivativeoftheΘ(t)term. Supposethatthediskisintheuntorquedsteadystateatt<0. Fromeqn.9,onecaneasilyseethatsuchasteadystateisgivenby, ξ =ψ =A(x−x ) (t<0), (11) ss i whereAisanormalizationconstantandr ≡x2 istheinneredgeoftheuntorqueddiskdefinedasthelocationwherethe“viscous” i i torquesvanish. Examinationof eqn. 9 shows that the time-dependentbehaviorof the torqueddisk at times t > 0 is given by the simplediffusionequation, ∂ξ 3k ∂2ξ = , (12) ∂t 4 ∂x2 withaninitialconditionsetbyintegratingthroughthedelta-functionintime,ξ(x,t=0)=ξss−βΘ(x−x0)/3π. Theappropriate boundaryconditionisξ →ξ ast→∞. Standardmethods(i.e.,separationofvariables)givethefollowingsolution: ss β ∞ 1 ξ(x,t)=ξss(x)+ 3π2 0 λ[sinλx0 cosλx −(cosλx0 +1)sinλx]exp −3kλ2t/4 dλ. (13) Z (cid:2) (cid:3) Withthissolution,wecancomputethesurfacedensityofthediskatanygivenradiusandtime. Armedwiththesurfacedensity, wecanthencomputeallotherquantitiesofinterestincludingtheviscousdissipationrateperunitsurfaceareaofthedisk: νΣr2 D(r)= Ω′2, (14) 2 whereΩ′ =dΩ/dr. PlotsoftheradialdependenceofΣ(r)andD(r)forvarioustimesareshowninFigures1–2. Alsoshownisthe time dependenceofthetotalviscousdissipationobtainedbyintegratingD(r) acrossthewholedisk (i.e. luminosity). We cansee thattheresponseofthedisktotheonsetofanexternaltorquecanbeseparatedintotwophases. Inthefirstphase,theaccretionflow is“dammed”atr =r0duetotheinabilityoftheaccretionflowtotransporttheangularmomentumdepositedbytheexternaltorque. This leads to a build-upof mass (i.e., an increase in the surface density) in the regionr > r0. Concurrently,matter in the region r < r0 continues to accrete thereby partially draining away the surface density. The inevitable result is a growing discontinuity in the surface density at r = r0. The angular momentumtransportassociated with this discontinuitygrows until mass can, once again, flow inwards across this radius. One then enters the second phase of evolution, whereby the surface density in the region r <r0isreplenishedbacktoitsoriginallevelwhilethesurfacedensitydiscontinuityismaintainedatapproximatelyaconstantlevel. Eventually, one achievesthe torquedsteady-state solution (e.g., Agol& Krolik 2000). The two sets of figures are for an external torqueatr = 4andoneclosertotheinneredgeatr = 2. Notehow,foragiveninjectionrateofangularmomentum,theeffecton thediskstructureismuchmoredramaticforsmallerradius. Since our disks are assumed to be radiatively-efficient,the instantaneoustotal luminosity of the accretion disk can be formally decomposedintotwocomponents,oneduetothedecreaseingravitationalpotentialenergyoftheaccretinggas,andaseconddueto theworkdonebytheexternaltorque,i.e., 1 GMM˙ ∂G L=2 2πrD(r)dr = dr+ Ω Tdr. (15) 2 r2 ∂r Z Z Z 4 FIG.1.—Evolutionofour“toy”modeldiskwithatorqueactingatr=4.Panel(a)showstheinitialstateofthesurfacedensityprofileforthenontorqueddisk (solidline)andtheresultingtorquedsteady-state(dashedline).Panel(b)showsfourtimesintheearlyevolutionofthesurfacedensityprofile(solid-line:t=0the untorquedsteady-state,dashed-line:t =10−3,dot-dashed-line:t =10−2,dotted-line:t =10−1;weuseunitssuchthatk=1whichcorrespondstoscalingwith respecttotheviscoustimescaleoftheinnerdisk). Noticehowtheinitialevolutionissuchthatdensitydropsinwardofthetorquelocationandincreasesoutward ofitduetothe“damming”oftheaccretionflow. Panel(c)showsthesubsequentlateevolutiontowardsthetorquedsteady-state(solid-line:t = 0,oruntorqued steady-state,dashed-line:t = 1,dot-dashed-line:t =10,dotted-line:t = 100). Panels(d)and(e)showthedissipationprofilesD(r)intheearlyandlatestages, respectively,oftheevolutionwiththetypeoflineandtimecorrespondingtothoseoffigures(b)and(c).Inthiscase,theeffectonD(r)issubtle. Inpanel(f),we showtheluminosityprofileobtainedbyintegratingthedissipationprofileoverthedisksurface. Thefinalsteady-statetorquedluminosityprofileisenhancedwith respecttothenon-torquedsteady-stateprofileduetoworkdonebythetorque. 5 FIG.2.—Evolutionofour“toy”modeldiskwithatorqueactingatr=2.Panel(a)showstheinitialstateofthesurfacedensityprofileforthenontorqueddisk (solidline)andtheresultingtorquedsteady-state(dashedline). Panel(b)showsfourtimesintheearlyevolutionofthesurfacedensityprofile(solid-line:t = 0, dashed-line:t=10−3,dot-dashed-line:t=10−2,dotted-line:t=10−1).Panel(c)showsthesubsequentlateevolutiontowardsthetorquedsteady-state(dashed- line:t = 1,dot-dashed-line:t = 10,dotted-line:t = 100). Forreference,thelowersolid-lineshowstheuntorquedsteady-statedisk. Panels(d)and(e)showthe dissipationprofilesD(r)intheearlyandlatestages,respectively,oftheevolutionwiththesameline-type/timearrangementoffigure1. Inotherwords,theline typesoffigure(b)matchthoseoffigure(d)andthoseoffigure(c)matchthoseoffigure(e).Inpanel(f),weshowtheluminosityprofileobtainedbyintegratingthe dissipationprofileoverthedisksurfaceforbothtorquesatr=2(dashedline)andr=4(solidline). 6 As canbe seen fromFig. 1(f)andFig. 2(f), the luminositydipsbeforeclimbingupto a new elevatedlevelthatincludesthe work donebytheexternaltorqueaswellastheaccretionenergy. Thetemporarydipinluminosityisduetothedammingoftheaccretion flowintheearlyevolutionofthetorquingevent. Nowthatwehaveexploredatorquingeventviatheanalyticalsolutionofanextremelysimplifiedaccretiondiskmodel,wemove on to somewhat more realistic models. In the next section, we present a semi-analytic analysis of a geometrically-thin general relativisticaccretiondisk. 3. RELATIVISTICTORQUEDACCRETIONDISKS Relativity produces two complications to the analysis. Firstly, the equations governing the structure of the accretion disk are rathermorecomplexanddriveustousenumericalratherthananalytictechniques. Secondly,therelationshipbetweentheemitted andobservedfluxesbecomesnon-trivial,withgravitationallightbending,relativisticaberration/beaming,andDoppler/gravitational redshiftingallbecomingimportant.Weshalldealwiththeseissuesinturn. The time-dependent equations describing the structure of a geometrically-thin accretion disk in the θ = π/2 plane of a Kerr spacetimearegiveninBoyer-Lindquistcoordinates(t,R,θ,φ)byEardley&Lightman(1974).TakingΣ(R)tobethepropersurface densityofthedisk(i.e.,thesurfacedensitymeasuredbyalocalobservermovingwiththefluid),thediskevolutionisdescribedby ∂Σ C1/2 ∂ Γ ∂ = (WR2D) , (16) ∂t BR ∂R ∂L† ∂R " # ∂R where, aM1/2 B = (1+ ), (17) R3/2 3M 2aM1/2 C = (1− + ), (18) R R3/2 2M a2 D = (1− + ), (19) R R2 B Γ = , (20) C1/2 2aM3/2 a2M2 L† = M1/2R1/2(1− + ). (21) R3/2 R2 Here, a isthe dimensionlessspinparameterof theblackhole (denotedas a byEardley& Lightman1974)andL† is the specific ∗ angularmomentumofthefluidforprogradeorbits.Thelocalrφshearinthisflowisσ =−3ΩC−1D,whereΩ=(M/r3)1/2. Thus, guided by the non-relativisticprescription, we set the vertically-integratedrφ componentof the stress tensor in the absence of an externaltorquetobe 3 M1/2 D W =−νσΣ= νΣ , (22) 2 R3/2 C where ν is the same effective viscosity that appeared in the non-relativistic expressions. Noting that the total torque is given by G=2πWR2D,weseethattheappropriaterelativisticdiffusionequationdescribingasporadicallytorqueddisk(i.e.,thecounterpart toeqn.6)is ∂Σ C1/2 ∂ Γ ∂ 3D2 G = νΣM1/2R1/2− T . (23) ∂t 2πBR∂R ∂L† ∂R 2C 2π " ∂R (cid:18) (cid:19)# It isstraightforwardto verifythateqn.23 reducesto eqn.5 in the non-relativisticlimit(i.e., B,C,D → 1;L† → (MR)1/2). The complicationsintroducedby the relativistic factors renderthis equationintractable to elementarysolution methods. Thus, we use a simpleexplicitschemetonumericallysolvethisdiffusionequationfollowingthetreatmentofPressetal. (1992). Figures3and 4showthetemporalbehaviorofanaccretiondiskaroundSchwarzschild(a = 0)andnearmaximalKerr(a = 0.998)blackholes respectively. Tofacilitatecomparisonwiththenon-relativisticcase,wehavechosenthesameviscositylaw,ν = kR(i.e. ν scales withtheradialBoyer-Lindquistcoordinateandnottheproperdistance).AsintheNewtoniancase,thisprescriptiondoesnotchange theresultsqualitatively.Notethatthebehaviorissimilartothatfoundinthenon-relativisticmodel.Thetwophasesofevolution,the dammingphaseandthereplenishingphase,arereproduced.Thedifferencesdonotcomefromthedynamicsbutfromtheboundary conditionsthataredeterminedbytheequilibriumconditionsforcirculargeodesics.Thepresenceofaninnermoststablecircularorbit forgeneralrelativisticpotentialssuggestsplacingthediskinnerboundaryatR=6M forSchwarszchildspacetimeandR=1.23M forthenear-maximalKerrspacetime. Relating the observed flux to the fundamental disk structure is substantially more complex in the relativistic case due to the complexitiesofgeneralrelativisticphotonpropagation. Fora givenvalueofthestressW, energyconservationgivesthatthetotal radiativefluxfromonesideofthedisk,measuredinthelocallyorbitingframe,is(Novikov&Thorne1974), 3D F(R)= ΩW. (24) 4C 7 Supposethatthecorrespondingenergy-integrated(butangle-dependent)intensityisI (R,θ),whereθismeasuredfromthenormal e to the disk plane (in the locally orbiting frame of reference). Following Cunningham (1975), an observer at infinity will see an integratedluminosity, L0 = 2πIeΥg3(g∗−g∗2)−1/2dg∗d(πR2) (25) Z Z wherewehavefollowedthenotationofCunningham(1975)withtheexceptionofΥ. HereΥistherelativistictransferfunctionthat results from ray-tracingnull-geodesicsthroughthe Kerr metric fromthe disk to the observer. We have used the code of R.Speith (Speith, Riffert & Ruder 1995) to compute Υ and hence perform this integral in order to examine how the observed luminosity changesthroughthetorquingevent. These calculations uncover a fundamentaldifference between the Newtonian and relativistic cases. In the relativistic case, the observedchangesinluminosityareafunctionoftheinclinationangleoftheobserverduetotheeffectsoflightbendingandrelativistic beamingofthe disk emission. Figures3f and4fshow the temporalbehaviourofthe observedflux(normalizedtothe flux forthe untorqueddisk)forvariousobservinganglesforourSchwarzschildandnear-extremalKerrcases,respectively. Thefinalfractional increase in observed flux depends on the beaming pattern of the torque-energize region of the disk compared with that of the untorqueddisk. ForourSchwarzschildcase(Fig.3f),onecanseethatthefinalfractionalincreaseinobservedluminositydepends veryweaklyontheobservingangle,implyingthattheuntorqueddiskandthetorque-energerizedregionofthetorqueddiskhavevery similar beamingpatterns. Thereis, however,a muchmorepronouncedtemporarydecrease in observedflux athigherinclinations duetothetemporarydimmingofthe(morehighlybeamed)innerregionsofthedisk. ForourKerrcase,thefinalfractionalincrease inobservedluminosityincreasesbyalmostafactoroftwoasonemovesfromalmostface-ontoalmostedge-ondisks,implyingthat thetorque-energizedregionofthediskissignificantlymorebeamedthantheuntorqueddisk.Atemporarydecreaseisonlyobserved forthemostedge-oncases,againduetoatemporarydimmingoftheinnermostregionsofthedisk. These features are all symptomatic of the fact that our black hole torques our accretion disk and deposits energy and angular momentum in the disk. This extra source of energy that is dumped into the disk and that tends to affect the disk outward of the externaltorquelocationconstitutesthestartingpointfortheanalysisofSection4whereweattempttoexplainthe”deepminimum state”astheresultofjustsuchasporadictorquingevent. 4. CANWEINTERPRETTHE“DEEPMINIMUMSTATE”OFMCG–6-30-15ASASPORADICTORQUINGEVENT? Inadditiontoexploringthegeneralcharacteristicsofsporadically-torqueddisks,acentralmotivationforthisstudyaretherecent XMM-NewtonobservationsoftheSeyfertgalaxyMCG–6-30-15.Inparticular,wewouldliketoexplorewhethertheenigmatic“Deep Minimum State” of this AGN could correspond to a sporadic torquing event, possibly induced by the formation of a temporary magnetic connection between the inner accretion disk and either the plunging region of the disk or the rotating event horizon. TherearetwodefiningcharacteristicsoftheDeepMinimumStatethatmustbereproducedbyanysuccessfulmodel,theextremely broadenedX-rayreflectionfeatures(implyingaverycentrallyconcentratedX-rayirradiationpattern)andthefactor2–3dropinthe observedX-raycontinuumflux. AmajoruncertaintywhenrelatingdiskmodelstoX-rayobservationsisalwaystherelationbetweenthedissipationwithinthedisk (predictedbythemodels)andtheemissionoftheobservedX-rays. Ifwesupposethatalocaldisk-coronaradiatesafixedfraction oftheunderlyingdissipationintotheX-rayband,theresultsofthispaperquicklyleadtoacontradictionbetweenthesporadically- torqueddiskmodelandtheobservations. Whilethemodeldoespredictatemporarydipinobservedluminosityforsomeobserver inclinations(thatonemightbetemptedtoidentifywiththecontinuumdropintheDeepMinimum),thisdipisduetoadimmingof theinnermostregionsoftheaccretionflowasaresultofthedammingofthemassflux. Thisispreciselythepartoftheflowthatwe wishtobeenhancedinordertoexplainthesimultaneousbroadeningoftheX-rayreflectionfeatures. Withthe(standard)accretiondiskcoronaframwork,therelationbetweenthedissipationwithinthediskandtheemissionofthe observed X-rays depends on the reprocessing of disk photons by the corona. We suggest this relation changes when the system departsfromsteady-stateasthesporadictorqueengagesandwillusethis,inthenextsection,tomodeltheDeepMinimumspectrum. 4.1. QuenchingtheX-raycoronawithreturningradiation The assumption that the X-ray emission from the disk corona locally tracks the dissipation in the underlying accretion disk is clearly an oversimplication. For example, Krolik & Hawley (2001) have used high-resolution pseudo-Newtonian simulations to show that there is a rather extended transition (occurring near but slightly outside of the radius of marginal stability) from the pureMHD turbulentregioncharacterizingthe bulk of the disk to the morelaminarflow presentin the plungingregion. Since the heatingofthecoronaisalmostcertainlyduetoreconnectionandMHDwave heatingfromtheunderlyingdisk, thefractionofthe dissipated energytransportedto the coronawill certainly changewithin this transition region,leading to a violationof the simple assumptionemployedinourtoymodels.Atime-variablemagnetictorqueofthekindweenvisioninthispapermightaltertheMHD and thermodynamic properties of the gas and such a scenario might not be compatible with the one we describe in the thin-disk approximation. Inotherwords, we canimaginethatthe externaltorquechangesboththe radiativeefficiencyofthe gasaswellas localMHDpropertiestherebyinvalidatingthetreatmentofthemagnetorotationalinstabilityasalocalkinematicviscosity.Thisfact is most likely more important for thick disks where the degrees of freedom are greater. Global disk simulations focusing on the formationandpropertiesofthecoronaarerequiredtoaddressthisissueand,hence,arebeyondthescopeofthispaper. We do, however, note an importantand mostly neglected physicaleffect that could substantially change the structure of a disk coronainastronglytorqueddisk—Comptoncoolingbyfluxemittedelsewhereintheaccretiondiskand,inparticular,by“Returning Radiation”. Consider a geometrically-thinaccretion disk around a near-extremalKerr black hole, and suppose that it possesses a disk-huggingX-raycoronaenergizedfromthe underlyingdisk. Now supposethat the centralregionsof the disk are subjected to 8 FIG. 3.—EvolutionofdiskinSchwarzschildspacetimefortorqueatR/M=10. Panel(a)showsthesurfacedensityprofilejustafterthetorquingeventbegins aswellasthesteady-statetorquedprofileisapproached(dashed-line:t = 10000). Panel(b)showstheearlystagesintheevolutionofthesurfacedensityprofile withthesolidlinebeingtheuntorquedsteady-stateprofile(dashed-line:t = 0.8,dot-dashed-line:t = 2.53,dotted-line:t = 8.0). Panel(c)showstheuntorqued steady-stateprofile(solid-line)aswellasthelate-timeevolutionofthetorquedprofile(dashed-line:t =25,dot-dashed-line:t =80,dotted-line:t =253). Panel (d)showstheearlyevolutionofthedissipationfunctionwithlinesandtimescorrespondingtothoseofpanel(b). Thequalitativefeatureisagainadropinward ofthetorquelocationandanincreaseoutward. Panel(e)showsthelate-timeevolutionofdissipationfunctionwithlinesandtimesanalogoustothoseofpanel (c)whilepanel(f)showstheobservedluminositystartingatuntorquedsteady-statewitht=0. Theobservedluminosityisdeterminedforanglesof10(solid-line), 30(dashed-line),60(dot-dashedline)degrees,and80degrees(dottedline). Althoughthemagnitudeoftheobservedluminosityisnotthesameintheuntorqued steady-stateforallangles,wehavenormalizedtheminordertoseethechangewithrespecttotheuntorquedstate.Notethepresenceofadropintheluminosityas theangleofinclinationdecreases. 9 FIG. 4.—EvolutionofdiskinKerrspacetimefortorqueatR/M=2andspinparametera = 0.998. Panel(a)showsthesurfacedensityprofileinuntorqued steady-state(solid-line)aswellastheapproachtosteady-statetorquedprofile(dashed-line:t = 10000). Panel(b)showstheearlystagesintheevolutionofthe surfacedensityprofilewiththesolidlinebeingtheuntorquedsteady-stateprofile(solid-line:t = 0)andtheotherprofilesmatchingthetimesandlinestylesfor panel(b)oftheSchwarzschildfigure.Panel(c)showstheuntorquedsteady-stateprofile(solid-line)aswellasthelate-timeevolutionofthetorquedprofile(dashed- line:t =25,dot-dashed-line:t =80,dotted-line:t =253). Panel(d)showstheearlyevolutionofthedissipationprofileinadditiontotheuntorquedsteady-state (solid-line)forthesametimesandlinestylesoftheSchwarzschildpanel(d). Panel(e)showsthelatestageevolutionofthedissipationfunctionwithtimesand line-stylescompatiblewiththoseoffigure3e. Panel(f)showstheluminosityobservedatthesameanglesasintheSchwarzschildcase(10degrees,solid-line;30 degrees,dashed-line;60degrees,dotted-dashed-line;80degrees,dottedline). Notehowthesmallesttorquedsteady-stateriseoccursfortheintermediateangleof 55degrees.Thelackofadropintheobservedluminositycomesfromthepresenceoftheexternaltorquenearertotheinnerboundaryintheradialcoordinatethan intheSchwarzschildcase. 10 a significant torquing event. As shown above, the work done by the torque is rapidly radiated from the accretion disk in a very centrallyconcentratedmanner. Thetorque-inducedemissionwillbea combinationofboththermaloptical/UVradiationandhard X-rayemissionproducedbythecoronaassociatedwiththetorque-energizedregionsofthedisk. Now,somefractionofthetorque- inducedemissionwillstrikethediskatlargerradii—thiswillbeparticularlyprevalentifthediskisflaredorwarped,butwilloccur eveninflatdisksduetorelativisticlightbendingeffects(i.e.,ReturningRadiation;Cunningham1973). Thisextrairradiationwill enhancetheComptoncoolingofthecoronaattheselargerradii. Attheveryleast,theadditionalcoolingwillleadtoadecreasein theComptonamplificationfactorofthecoronaandasteepeningofthecoronalemission. Onecouldenvisageasituation,however, inwhichtheComptoncoolingbecomessoextremethatthecoronacompletelycollapsesandlocalEUV/X-rayemissionceases. Some essential aspects of this scenario can be captured in a simple model based on energy conservation, following Haardt & Maraschi(1991,1993). ConsidertheX-rayemittingcoronaaboveaunit-areapatchofthediskataradiusr. Ifafractionf ofthe energydissipatedintheunderlyingdiskgoesintoheatingthecorona,theheatingrateis H(r)=fD(r). (26) Bydefinition,the(Compton)coolingrateofthecoronais(A−1)Fs,whereAistheComptonamplificationfactorandFsisthesoft photonfluxpassingthroughthecoronawhichwillactasseedphotonsfortheinverseComptonscatteringprocessthatgeneratesthe X-rays. Equatingheatingandcoolinggives, fD(r)=(A−1)Fs. (27) WenowdetermineFs byexaminingenergyconservationofthecolderdiskunderlyingthecorona. Therearethreecontributionsto thatwemustconsider. Firstly,theportionoftheinternaldissipationwithinthediskthatdoesnotgettransportedintothediskwill becomethermalizedinthecolddiskandcontributeanamount(1−f)D(r)tothediskheating.Secondly,somefractionofthelocally generatedcoronalfluxξ1fD(r)willimpingeonthediskandbereprocessedintosoftflux. Theparameterξ1 encapsulatespossible anisotropiesin the coronalflux and the albedoof the disk, but will typically be of the order of ξ1 ∼ 0.2−0.5. Finally, as noted above,irradiationofourcoronalpatchfromotherradiiinthediskwillcontributetothe softfluxandhencetheComptoncooling. Thiswillcoolthecoronaduetoboththedirectactionoftheirradiatingsoftflux,andthereprocessing/thermalizationofthesoftand hardirradiatingflux.Supposethatthenon-localirradiatingfluxisR(r)timesthelocallyproducedflux.Thecorrespondingsoftflux contributingtotheComptoncoolingwillbeξ2RD(r),whereξ2 ∼<1parameterizesthefractionofthisnon-localemissionthatends upassoftflux. Hence,thetotalsoftfluxataparticularlocationinthediskwillbe Fs =ξ1fD(r)+ξ2R(r)D(r)+(1−f)D(r). (28) SolvingforA,weget f A=1+ . (29) ξ2R(r)+1−f(1−ξ1) Ofcourse,withinthissimplemodelthetotalenergydissipatedwithinthecoronaisafixedfractionoftheunderlyingdissipation irrespective of the (cooling) soft flux. However, the amplication factor is significantly reduced by Returning Radtion if R(r) ∼> ξ2−1[1−f(1−ξ1)] which, for canonicalvalues of f = 1 and ξ1 = ξ2 = 0.5, correspondsto R(r) ∼> 1. The resulting coronal spectrum from the affectedregionsof the disk would be expectedto steepen significantly, possibly placing a large fractionof the emissionintotheunobservableEUVband. Foraflatdiskatlargeradiisubjectedtoreturningradiation,wehaveR(r)=R0(a)+∆ηR∞(a),where∆ηistheenhancementin theefficiencyofthediskduetotheinnertorqueandR∞(a)andR0(a)aredimensionlessfunctionsoftheblackholespinparameter givenby the fitting formulaeof Agol& Krolik(2001). For a near-extremalKerrblack hole (a = 0.998),we have R0 ≈ 0.2and R ≈ 1. Thus,wecanseethatevenintheabsenceofdiskflaringorwarping,returningradiationalonecouldsignificantlydepress ∞ coronalX-rayactivityatlargeradiiif ∆η ∼> 1−f[1−ξ1] −0.2. (30) ξ2 Thus,althoughthereissomedependenceontheproperties(e.g.,isotropyandpatchiness)ofthecoronaandtheabilityofthediskto reprocessandthermalizeanyincomingflux,thecoronawillbedepressedifthediskisina“spin-dominated”state(∆η ∼>1),i.e.,a stateinwhichthediskisshiningviathereleaseofblackholespinenergyratherthangravitationalpotentialenergy. 4.2. AproposedscenariofortheMCG-6-30-15DeepMinimumState Let us now return to MCG–6-30-15and the sporadic external torque modelfor its Deep Minimum State. We suppose that the normal state of this system is that of a standard untorqued accretion disk that might well be described by the standard accretion modelsofNovikov&Thorne(1974)andPage&Thorne(1974).Wethensupposethatsomeshiftinmagneticconfigurationcaused theaccretiondisktobecomemagneticallytorquedbyeithertheplungingregionortherotatingblackholeitself. Wehypothesizethat thiseventsignalstheonsetofaDeepMinimumState. On timescales shorter than the viscous timescale of the inner disk (tvisc ∼ 1hour), we expect this torquing event to lead to a dammingoftheaccretionflowandatruedimmingofthediskinteriortothelocationwheretheconnectionhasoccurred. However, onlongertimescales,thediskwilltendtothenewtorquedsteady-state(providedthetorqueissufficientlylong-lived). If the magnetic torquing occurs in the very centralmost regions of the disk (which is likely in all of the scenarios that we are envisaging),thetorquedsteady-statewillpossessamuchmorecentrallyconcentrateddissipationpattern.Asdescribedabove,some