ACCEPTEDBYAPJ PreprinttypesetusingLATEXstyleemulateapjv.5/2/11 THELARGESCALEMAGNETICFIELDSOFTHINACCRETIONDISKS XINWUCAO1 ANDHENDRIKC.SPRUIT2 1KeyLaboratoryforResearchinGalaxiesandCosmology,ShanghaiAstronomicalObservatory,ChineseAcademyofSciences,80NandanRoad,Shanghai, 200030,China;[email protected] 2MaxPlanckInstituteforAstrophysics,Karl-Schwarzschild-Str. 1,85748,Garching,Germany;[email protected] acceptedbyApJ ABSTRACT 3 Largescale magnetic field threadingan accretiondisk is a key ingredientin the jet formationmodel. The 1 0 mostattractivescenariofortheoriginofsuchalargescale fieldistheadvectionofthefieldbythegasinthe 2 accretiondiskfromtheinterstellarmediumoracompanionstar. However,itisrealizedthatoutwarddiffusion oftheaccretedfieldisfastcomparedtotheinwardaccretionvelocityinageometricallythinaccretiondiskifthe n valueofthePrandtlnumberP isaroundunity. Inthiswork,werevisitthisproblemconsideringtheangular a m momentum of the disk is removed predominantly by the magnetically driven outflows. The radial velocity J of the disk is significantly increased due to the presence of the outflows. Using a simplified model for the 9 verticaldiskstructure,we findthatevenmoderatelyweakfieldscancausesufficientangularmomentumloss 1 viaamagneticwindtobalanceoutwarddiffusion. Therearetwoequilibriumpoints,oneatlowfieldstrengths correspondingto a plasma-beta at the midplane of orderseveralhundred,and one for strongaccreted fields, ] E β∼1. Wesurmisethatthefirstisrelevantfortheaccretionofweak,possiblyexternal,fieldsthroughtheouter partsofthedisk,whilethelatteronecouldexplainthetendency,observedinfull3Dnumericalsimulations,of H strongfluxbundlesatthecentersofdisktostayconfinedinspiteofstrongMRIturbulencesurroundingthem. . h Subjectheadings:accretion,accretiondisks,galaxies:jets,magneticfields p - o 1. INTRODUCTION advectiondominatedaccretionflow(Cao2011),whichishot r andgeometricallythick(Narayan&Yi1994,1995). t Jets/outflowsareobservedindifferenttypesofthesources, s Theadvectionof the field in a geometricallythin(H/r≪ a such as, active galactic nuclei (AGNs), X-ray binaries, and 1),turbulentaccretiondiskisinefficient,however,becausethe [ youngstellarobjects,whichareprobablydrivenfromtheac- radial componentof the magnetic field diffuses much faster cretion disk through the magnetic field lines threading the 1 disk (see reviews in Spruit 1996; Konigl&Pudritz 2000; acrossthedisk,onatimescale ∼H2/η. Asaresult,amag- v Pudritzetal. 2007; Spruit 2010). The large scale magnetic netic field with inclination Br/Bz ∼1 actually diffuses out- 3 fieldco-rotateswiththegasesinthedisk,andthejets/outflows wardonatimescaleoforderH/rshorterthanapurelyverti- 4 calfield(vanBallegooijen1989).Aninclinedfieldisaneces- arepoweredbythegravitationenergyreleasedbyaccretionof 5 saryconsequenceofeffectiveaccretionofthefield,however, thegasesthroughtheorderedfieldthreadingthedisk. Alarge 4 sincetheaccumulationoffieldlinesintheinnerdiskexertsa scale magnetic field, of uniformpolarity threadingthe inner . 1 partsofthediskisprobablyakeyingredientinthisjetforma- pressurethatcausesthefieldabovethedisktospreadoutward. 0 tionmodel. Whilesuchaninclinedconfigurationisfavorableforlaunch- 3 ing a flow (Blandford&Payne 1982; Cao&Spruit1994), it Theoriginofsuchafieldisnotwellunderstood,however, 1 since the net magnetic flux threading a disk cannot be pro- raisesthe problemhowitcan beaccretedeffectivelyagainst : theactionofmagneticdiffusion. v ducedorchangedby internalprocessesin thedisk (aconse- Severalalternativesweresuggestedtoresolvethedifficulty i quenceofthesolenoidalnatureofthemagneticfield,seee.g. X Spruit2010).Anetmagneticfluxintheinnerdiskmustthere- offieldadvectioninthinaccretiondisks. Spruit&Uzdensky r foreeitherbeinheritedfrominitialconditions,orsomehowbe (2005) suggested that a weak large-scale magnetic field a threadsthediskintheformoflocalizedpatchesinwhichthe accretedfroma largerdistance; ultimatelyforexamplefrom the interstellarmediumora companionstar(cf. Bisnovatyi- field is strong enough to cause efficient angular momentum throughamagneticwind. Generalrelativisticmagnetohydro- Kogan&Ruzmaikin1974,1976). dynamic(GRMHD)simulationsofanaccretiontorusembed- One could imagine a steady state in which the inward advection of the field lines is balanced by the outward dedinalarge-scalemagneticfieldshowedthatacentralmag- netic flux bundle, once formed from a suitable initial condi- movement of field lines due to magnetic diffusion. In tion, can survive in spite of MRI turbulence present in the conventional isotropic idealizations of a turbulent plasma, it is expected (e.g. Parker 1979) that ν ∼ η ∼ lv, in disk surroundingit (Beckwithetal. 2009). The calculations t by Guilet&Ogilvie (2012a) and Guilet & Ogilvie (2012b) which l is the largest eddy size, and v is turnover ve- t showthattheaccretionvelocityofthegasintheregionaway locity, i.e. P ∼ 1. Whether this is actually the case in m MRI turbulence has been investigated using numerical sim- from the midplane of the disk can be larger than that at the midplaneofthedisk, whichmaypartiallysolvetheproblem ulations (e.g., Yousefetal. 2003; Lesur&Longaretti 2009; oftooefficientdiffusionofthefieldinthinaccretiondisk. Fromang&Stone 2009; Guan&Gammie 2009). The re- sults all suggest that the effective magnetic Prandtl number Theoutwarddiffusionofthefieldcouldbebalancedbyac- cretionifa processcanbefoundthatincreasesthe accretion isaroundunity. Fromangetal.(2009),forexample,measure P ≈2. ForsuchPrandtlnumbersaverticalfield(perpendic- velocitybyafactor∼r/Hrelativetotherateduetotheturbu- m ulartothediskplane)canindeedbedraggedefficientlybyan lencealone. Weexploreheretheconditionsunderthiscanbe 2 achievedbyamagneticwindgeneratedbytheweakmagnetic Theangularmomentumequationforasteadyaccretiondisk fieldthatistobeaccreted. withoutflowsis d d dΩ 2. MODEL (2πrΣvrr2Ω)= (2πrνΣr2 )- 2πrTm, (4) dr dr dr 2.1. Modelassumptions whereΣ is thetotalsurfacemassdensityofthe disk(count- Apart from the accreted field, the disk model we use is a ingbothsides),T thetotaltorqueperunitofsurfaceareaof m standard α–disk model, i.e. with a viscosity ν parametrized thedisk(countingbothsides)duetothemagneticallydriven as ν =αcsH, where cs is the (isothermal) sound speed, H is outflows.IntegratingEq.(4),yields thescaleheightofthedisk,andα∼0.01- 0.1(therangeof dΩ valuesmeasuredinMRIsimulations).Thefieldtobeaccreted 2πrΣv r2Ω=2πrνΣr2 - 2πf (r)+C, (5) r m is assumed to be sufficiently weak that it does not suppress dr magnetororationalinstability.ThediskthuscontainsanMRI- wherethevalueoftheintegralconstantC canbedetermined generatedfield as well as a weaker field of uniformpolarity withaboundaryconditionontheaccretingobject,and threadingthedisk. df (r) Since the accretionvelocitythatis to beachievedexceeds rT = m . (6) m theviscousrate,wesimplifytheanalysisbyassumingtheac- dr cretionflow to bedominatedbythe angularmomentumloss Themagnetictorquein unitsurface disk areaexertedbythe fromtheaccretedfield,ignoringtheviscouscontributionfrom outflowsis the MRI turbulence. Theresultingaccretionrate then hasto T =2m˙ r2Ω, (7) exceedoutwarddiffusionduetotheMRIturbulence.Thetur- m w A bulenceisassumedtoproduceaneffectivemagneticdiffusiv- where m˙w is the mass loss rate in the outflow from a unit itycorrespondingtoamagneticPrandtlnumberPm∼1. surfacearea ofthe disk (single sided), rA is the (cylindrical) The model needs a prescription for the angular momen- Alfvén radius of the outflow, Ω(r) is the angularvelocity of tumlossproducedbytheaccretingweakmagneticfield. For thedisk, andr is theradiusofthe field line footpointatthe thisweusetheWeber-Davismodelforamagneticallydriven disksurface.Thedimensionlessmassloadparameterµofthe wind, in the ‘cold’ approximation (in which the gas pres- outflowis 4πρ v Ωr 4πΩr sure force is neglected). It leads to a simple description in µ= w w = m˙ , (8) terms of the field strength and mass loss rate (Mestel 2012, B2p BpBz w Spruit 1996). The Weber-Davis model strictly applies only where ρ v =m˙ B /B is the mass flux parallel to the field to the ‘split monopole’ configuration, in which the poloidal w w w p z line. InthecoldWeber-DavismodeltheAlfvénradiusis field is purely radial. Its properties are found to be a rather goodapproximationformoregeneralpoloidalfieldshapesas 3 1/2 well(Andersonetal. 2005),whichmakesitadequateforthe rA=r (1+µ- 2/3) . (9) 2 presentpurpose. (cid:20) (cid:21) The angular momentum loss in this model can be char- [FormoredetaileddiscussionofthismodelseeMestel(2012) acterized by a single dimensionless constant, a ‘mass load or Spruit (1996)]. MHD simulations of axisymmetric mag- parameter’µ(c.f.Michel1969,Mestel,2012): netically driven flows have shown that relations like (9) and (10) below are fair approximationsfor more general config- µ=χ4πΩr0/Bp, (1) urationsthantheWeber-Davis‘splitmonopole’(cf.Fig.7in Andersonetal.2005). SubstitutingEqs.(9)and(8)into(7), where B =(B2+B2)1/2 is the poloidalcomponentof the ac- p z r wefind cretedfield atradiusr onthe surfaceofthedisk, and Ω the 3 rotationrateat r0 (cyli0ndricalcoordinatesr,φ,z). χisacon- Tm= 4πrB2pµ(1+µ- 2/3). (10) stantalongtheflow,itisameasureofthe‘massfluxperfield Theradialvelocityofanaccretionflowinwhichtheangular line’: momentumisremovedpredominantlybytheoutflowscanbe χ=ρv /B , (2) p p estimatedfromEq.(4): where ρ, vp, Bp are the mass density, poloidal velocity and T d - 1 2T poloidalfield strength. Itis relatedto the mass flux perunit v ∼- m (r2Ω) ≃- m , (11) surfaceareafromthediskm˙w by r Σ (cid:20)dr (cid:21) ΣrΩ wherewehaveassumedthattherotationisapproximatelyKe- m˙w=Bzχ. (3) plerian, Ω≈Ω . [This is sufficient for the following esti- K mates,buthastobemademoreprecisewhenconsideringthe Theasymptoticvelocityofthewinddecreasesmonotonically windlaunchingconditions(section2.3).] Themassaccretion withincreasingµ,andforµ=1equalstherotationvelocityat rateoftheaccretiondiskis r . Theangularmomentumlossincreasesmonotonicallywith 0 µ. M˙ =- 2πrΣv ≃ 4πTm, (12) Itturnsoutthattheconditionsforeffectiveaccretionofthe r Ω field can be satisfied when µ&αP r/H. This is described m whereEq.(11)isused. Wecancomparethemasslossratein inthefollowingsections, usingaspecificmodelforthedisk the outflowswith the accretion rate throughthe disk. Using structure. (8),(10)and(12): 2.2. Themagneticfieldofthedisk dlnM˙ = 4πr2m˙w = 1(1+µ- 2/3)- 1. (13) ˙ dlnr M 3 Thelargescalemagneticfieldsofthinaccretiondisks 3 This shows that forlarge mass loading parameters, µ&1, a (moreaccurately,theslowmodecuspspeed). Themassflux, fraction∼1/3oftheaccretionratecanbelostinthewind,if parallelto the field, is thusapproximatelyρc , andperunit s,s thewindissustainedacrossthediskoveradistanceoforderr. ofsurfaceareaparalleltothedisksurface,themassfluxis Forµ.1themasslossinthewindisnotsignificantcompared B withtheaccretionrate. Substituting(10)into(11),weobtain m˙ ∼ zρc , (22) w B s,s p 6c g(µ) vr=- ΩsHβp cs, (14) whereBp=(B2z+B2r)1/2asbefore.Asmentioned,weconsider only the wind associated with the accreted field. The MRI whereg=µ(1+µ- 2/3),c isthesoundspeedofthegasinthe turbulencealsoproducesafield,butbeingalocalprocess,we s midplaneofthedisk,and assumethatitdoesnotproducealargescalefieldthatwould besignificantforwind-drivenangularmomentumloss. B2p The mass flux depends sensitively on the surface temper- β =p / (15) p c 8π ature Ts. Assume a radiative disk, i.e. the vertical energy transport through the disk is by radiation. In the diffusion isameasureofthepoloidalfieldstrengthatthesurfacerela- approximation,the surfacetemperatureis thenrelatedto the tive to p , the gaspressure at the midplane of the disk. The c temperatureT ofthediskatthemidplaneofthediskby magneticfieldisadvectedinwardsonatimescale c τadv∼ r = r ΩHβp . (16) 4σ3τTc4 =σTs4, (23) |v | c 6c g(µ) r s s where τ isthe opticaldepthof the diskin the verticaldirec- InthelimitH/r≪1,theradialcomponentofthefielddom- tion. Intermsofthesoundspeedsc ,c atthesurfaceand s,s s,c inates the outward diffusion timescale τdiff of the magnetic themidplane, field(vanBallegooijen1989,Lubowetal.1994).Let 4 1/8 c = c . (24) κ0=Bz/Br,s (17) s,s 3τ s,c (cid:18) (cid:19) be the inclination, with respect to the horizontal, of the ac- Alongafieldlinethatcorotateswithitsfootpointinthedisk cretedfieldatthedisksurface(s). Thediffusiontimescaleis theoutflowisgovernedbyaneffectivepotentialΨ , eff thenoforder τdif∼ rHηκ0, (18) Ψeff(r,z)= (r2+GzM2)1/2 - 12Ω2r2, (25) where η is the magneticdiffusivity. [Thisdependenceholds where Ω is the angular velocity of the footpoint at the disk as long as 1/κ > H/r; in the opposite case of a nearly surface. In this expression and in the following, the (small) 0 vertical field B /B < H/r, the diffusion time is of order differencebetweentherotationrateΩofthefieldlineandthe r 0 r2/η]. ThemagneticPrandtlnumberisdefinedasP =η/ν, Keplerian value ΩK has to be included consistently, since it m hasa strongeffectonthe launchingconditionsforthe wind, where ν is the turbulentviscosity. With the conventionalα- throughitseffectonΨ (OgilvieandLivio,2001). Thisdif- parametrization,ν=αc H wecanthenwriteEq.(18)as eff s ference,duetothemagneticstressexertedatthesurfacesfol- τ ∼ rκ0 1 . (19) lowsfromtheradialequationofmotion, dif c αP s m B B B2 rΩ2 - rΩ2= r,s z = z , (26) For a steady state, the advection of the field in the disk has K 2πΣ 2πΣκ 0 to balance the diffusion, i.e. τ =τ . With (16), (19) this yieldsaconditiononthemassfladuvxpadriafmeterµ: whereΣthetotal(two-sided)surfacedensityΣ≈2ρcH.This yields µ(1+µ- 2/3)= αΩHβpPm, (20) 2r c2 1/2 6csκ0 Ω=ΩK 1- β κ H r2Ωs2 , (27) (cid:20) z 0 K(cid:21) whichreducesto whichcanbeapproximatedas αβ P µ(1+µ- 2/3)= 6κp0m (21) Ω=Ω 1- H 2 1/2, (28) K in weak field approximation, cs =ΩH. If a significant mass (cid:20) r βzκ0(cid:21) flux,µ∼1canbelaunched,andassumingα=0.01,Pm=1, inweakfieldcase,whereβ =8πp /B2=β (1+κ2)/κ2isthe and a field inclination of 30◦ to the vertical, this shows that z c z p 0 0 plasma-betaoftheaccretedfieldevaluatedatthemidplane( ) a weak field with β ∼103 can still be accreted through the c ofthedisk. angularmomentumlossofthewindassociatedwithit. Inthe Next, we make the model more specific and simplify it a followingsectionsweinvestigatethiswithspecificmodelsfor bitbyapproximatingthesoundspeedinthelaunchingregion thestructureofthedisk. as constant with height. In addition we take the midplane temperatureasrepresentativefortheinteriorofthedisk. This 2.3. Themassfluxinthewind is sufficient for the evaluation of quantities like the rotation Themasslossrateintheoutflowisgovernedbythegasden- ratecorrectionin(28). sityatthepositionofthesonicpoint(inthefollowinglabeled Thelocationofthesonicpointisclosetothemaximumof with index ), where the flow speed equals the sound speed the effective potential. The mass flux is determined by the s 4 density at the sonic point; a fair approximationforthis is to Equation (32) should be a good approximation for the treat the subsonic regionas if it were in hydrostaticequilib- presentinvestigation,especiallyinthethinouterregionofthe rium.Thisyieldsthefollowingestimateforthemasslossrate diskwhichisprobablythemostcriticalregionfortheaccre- inanisothermaloutflow(perunitofdisksurfacearea), tionofanetmagneticflux.Substituting(29)into(8),themass flowparameteris κ m˙ ∼ 0 ρ c exp - (Ψ - Ψ )/c2 , (29) w (1+κ20)1/2 0 s,s eff,s eff,0 s,s µ= 4πΩrρ0cs,sexp - (Ψ - Ψ )/c2 . (37) (cid:2) (cid:3) B2 eff,s eff,0 s,s where ρ is the density of the gas in the base of the outflow p (still to0be specified), and Ψeff,s and Ψeff,0 are the effective Toestimate thedensityρ0(cid:2)atthe baseof theflo(cid:3)w, we note potentialat the sonic pointand the footpointat the disk sur- that Eq. (37) is applicable only at heights in the atmosphere face,respectively.Thefactorinvolvingκ0isequaltotheratio where the field is strong enough, relative to the plasma, to Bz/Br,sinEq.(17). enforcecorotationsotheeffectivepotentialΨisrelevantfor The vertical structure of an isothermal disk can be calcu- thelaunchingprocessofthewind.Insidethedisk,whereβ> latedwiththeverticalmomentumequation, 1,thisisnotthecase. Asbaseoftheflow,wherethepressure is p ,weassumetheheightwheretheplasma-betaisoforder dρ(z) B (z)dB (z) 0 c2 =- ρ(z)Ω2z- r r . (30) unity,i.e. s dz K 4π dz β ≡8πp /B2≈1, (38) s 0 p Inprinciple,thefieldlineshapeiscomputablebysolvingthe whichdetermines p if β isgiven. Formostofcalculations radialandverticalmomentumequationswithsuitablebound- 0 s reportedbelow, β =1 is used, some with a lowervalue0.1. ary conditions (for a detailed discussion see Cao & Spruit s Themassfluxisthenrelatedtothe poloidalfieldstrengthat 2002, hereafter CS02). For the isothermal case, an approx- thedisksurfaceby imate analytical expression is proposed for the shape of the fieldlinesintheflow: B2 p ρ c = . (39) r- r = H (1- η2+η2z2H- 2)1/2- H (1- η2)1/2, (31) 0 s,s 8πcs,s i κ η2 i i κ η2 i 0 i 0 i Using(24)Eq.(37)canthenbewrittenas where ri is the radius of the field line footpoint at the mid- Ωr plane of the disk, and η =tanh(1) (see CS02). This expres- µ= (3τ/4)1/8exp - (Ψ - Ψ )/c2 . (40) i 2c eff,s eff,0 s,s sionreproducesthebasicfeaturesoftheKippenhahn-Schlüter s,c modelforasheetofgassuspendedagainstgravitybyamag- Thefactorinfrontoftheexp(cid:2)onentialtendstobea(cid:3)largenum- neticfield(Kippenhahn&Schlüter1957),forweakaswellas ber, the exponential itself a small one. To evaluate the ef- strongfieldcases. AsdoneinCS02,weuseafittingformula fectivepotentialasafunctionofheight,fieldlineinclination, tocalculatethescaleheightofthediskintherestofthiswork, andthe(slightlynon-Keplerian)rotationrateΩ,themodelof CS02isused. Itisalsousedfortheopticaldepthconnecting 1/2 H 1 4c2 1 thesurfacetemperaturetothediskmidplanetemperature. = s,c +f2 - f, (32) r 2 r2Ω2 2 K ! 3. RESULTS where The disk is compressed in the vertical direction by the 1 B2 curved magnetic field line, which sets an upper limit on the f = 2(1- e- 1/2)κ 4πρrHzΩ2κ . (33) magnetic field strength. We plot the correspondingminimal 0 K 0 β as a function of the field inclination κ at the disk sur- z 0 SolvingforH: faceinFig.1. Thecurvedfieldlinealsoexertsaradialforce on the disk against the gravity of the central object, which c 1 1/2 providesanadditionalconstraintonthefieldstrength(Fig.1 H= Ωs,c 1- (1- e- 1/2)β (1+κ2) , (34) show the resultfordifferentvaluesof the disk temperature). K(cid:20) p 0 (cid:21) TheangularvelocityΩdeviatesfromtheKeplerianvaluedue whichrequires totheradialmagneticforce. Fig.2showsthedimensionless angular velocity Ω/Ω of the disk as a function of κ and K 0 βp>βp,min= (1- e- 1/12)(1+κ2), (35) ΘF=ocr2sg,ci/v(ern2Ωva2Kl)u.esofthediskparameters,i.e.,theviscosityα, 0 thetemperatureparameterΘinthedisk,andtheopticaldepth or 1 τ,thedependenceofmassloadingµonβz canbecalculated βz>βz,min= (1- e- 1/2)κ2. (36) withEqs.(20)and(40)respectively.Theserelationsareplot- 0 ted in Figs. 3-5 for disk-outflow systems with different val- The square bracket in (34) gives the magnetic correction to uesoftheparameters. Inallcalculations,Pm=1isadopted. the standard relation between disk thickness and the sound It is found that two branches of solutions usually exist for speedatthemidplane. Itreducestounityforaweakfield,or mostcases. Weplotthesolutionsofthedisk-outflowsystems whentheradialcomponentissmallcomparedwiththeverti- in Figs. 6 and 7. The solutions with different values of βs calcomponent.Foratypicalvalueofκ0∼1,wefindβpmust (βs =8πp0/B2p) at the base of the outflow forthe isothermal be&1.3.Thismeansthatthemagneticpressurecanbelarger diskarecomparedinFigs.8and9. Thetwobranchesofso- thanthegaspressureinthediskonlyiftheinclinationofthe lutionscorrespondtolowmass(µ≪1)loadedoutflowswith fieldlineatthedisksurfaceκ islargerthanunity. strong field strength (low-β), or high mass loaded outflows 0 Thelargescalemagneticfieldsofthinaccretiondisks 5 withrelativeweakfieldstrength.Theopticaldepthneededto atthedisksurface.WehaveusedthecoldWeber-Davismodel connectsurfaceandinternaltemperaturehasbeencomputed fordeterminingthemassloadparameterµoftheoutflowasa by includinga Rosseland mean opacityand electronscatter- functionofstrengthoftheaccretedfieldandparametersofthe ing;itisshowninFig.10. diskstructure: itsopticaldepth(aradiativediskisassumed), For comparison, we have repeated the calculations for a atemperature-parameterΘ,theα-viscosity,andthemagnetic uniformly isothermal disk, that is, the temperatures of disk Prandtl number P of the assumed MRI turbulence (cf. Eq. m andwindareassumedtobethesame(Figs.8and9). Though 20). Balancing the resulting accretionvelocitywith the out- thisisnotveryrealistic,itgivesanimpressionofthesensitiv- warddiffusionbymagneticturbulencedeterminesthecondi- ityoftheresultstothemodelassumptionsmade. tionsforexistenceofastationarydisk-outflowsystem. Figs. 3-5 illustrate the propertiesof the solutions. It is foundthat two solution branches exist for all cases. The lower branch correspondsto high field strength and low µ, i.e., low mass 101 lossrate,theupperonecorrespondstolowfieldstrengthand high µ, i.e. high mass loss rate (see Figs. 6 and 7). There is an upper limit on the field inclination κ at the disk sur- 0 face,whichincreaseswithdisktemperature.Overcomingthe 100 deepereffectivepotentialbarrierassociated with a largerin- clination requires a higher internal energy of the gas. The n z,mi mcaalxdiemputhmτinocflitnhaetidoinskκ,0atshtuhsedseucrrfeaacseestewmiptheriantcurreeadsiencgreoapsteis- b with increasing τ, (keeping other disk parameters fixed, see 10−1 Eq.24). We have also explored the sensitivity to model assump- tionssomewhatwith solutionsfora more drastically simpli- fiedcase,wherethetemperatureisassumeduniformthrough- 10−2 out,i.e. thetemperatureoftheoutflowisthesameasthedisk 0 2 4 6 8 10 k temperature.ThisisshowninFigs.8and9. Thesealsoshow 0 theeffectofassumingalowerdensityofthegasatthebaseof FIG.1.—Constraintsonthepoloidalmagneticfieldstrengthintermsofβz theoutflow(β =0.1). Theresultsarequalitativelysimilarto (theplasma-betaoftheaccretedfieldmeasuredatthemidplaneofthedisk, s seetext).Blackline:βz,minasconstrainedbytheverticalpressureexertedby thosewithβs=1. thecurvedfieldline(Eq.36). Colors: theminimalvaluesofβpconstrained bytheradialmagneticforce(seeEq.27)fordifferentdisktemperatures,Θ= 0.01(red),2.5 10- 4(green),and10- 4(blue). × 100 100 10−3 m 10−6 t =102 10−9 K W/ 10−12 W 100 100 101 102 10−3 10−1 m 10−6 t =103 10−1 100 101 102 10−9 b z FIG.2.—DeviationoftheangularvelocityofthediskΩrelativetoKep- 10−12 lerianduetomagneticstress,asafunctionofβz(Eq.27). Fieldlineincli- 100 100 101 102 nationκ0=1(red),√3(green),and2.5(blue). Disktemperatureparameter Θ=0.01(solid),2.5×10- 4(dashed),and10- 4(dash-dotted). 10−3 3.1. Discussion m 10−6 t =104 Themagneticfieldisdraggedinwardsbytheaccretiondisk, 10−9 anoutflowislaunchedbythisfield,andtheradialvelocityof the accretion disk is determined by the rate of angular mo- 10−12 mentum carried away by the outflows. This loss rate in the 10−1 100 101 102 103 outflows can be estimated by exploring the launching pro- b cess of the outflow, which depends sensitively on the field z strength/configurationandthedensity/temperatureofthe gas 6 thFeIwG.in3d.—lauMncahsisnglocaodnindgitiµonassEaqf.u(n4c0ti)o.nBorofkβezn.:Smolaisds:laosaddientgerrmeqinueidredfrofomr 100 effectiveaccretionofthefieldlines(20),fordiskviscosityα=0.1(dashed)) andα=1(dash-dotted).Theintersectionpointsarepossiblesolutionsforthe 10−3 stationarywinddrivenaccretionproblem.Fieldlineinclinationsareκ0=1.5 (mreadg)n,e√tic3P(rgarnedetnl)n,uamndbe2r(Pbmlu=e)1..Thedisktemperatureparameter Θ=0.01, m 10−6 t =102 10−9 10−12 100 100 100 101 102 10−3 10−3 m 10−6 t =102 m 10−6 t =103 10−9 10−9 10−12 10−12 100 100 101 102 100 100 101 102 10−3 10−3 m 10−6 t =103 m 10−6 t =104 10−9 10−9 10−12 10−12 100 100 101 102 10−1 100 1b01 102 103 z 10−3 FIG.5.—AsFig.3,forΘ=10- 4. m 10−6 t =104 10−9 10−12 102 10−1 100 101 102 103 b 100 z −2 FIG.4.—AsFig.3forΘ=2.5 10- 4. 10 × −4 m 10 −6 10 −8 10 1.5 2 2.5 3 3.5 3 10 2 10 z b 1 10 0 10 −1 10 1 1.5 2 2.5 3 3.5 4 k 0 FIG.6.—Resulting disk-outflow solutions fordifferent disktemperature parameters:Θ=0.01(red),2.5 10- 4(green),and10- 4(blue),anddifferent valuesofthediskopticaldepth:×τ=102(solid),103(dashed),and104(dash- dotted).Theviscosityparameterα=0.1. Thelargescalemagneticfieldsofthinaccretiondisks 7 2 3 10 10 100 100 −2 10 −3 10 −4 m 10 m −6 10 −6 10 −9 10 −8 10 −12 10 1.5 2 2.5 3 3.5 2 3 4 5 6 7 3 3 10 10 2 2 10 10 z z b 101 b 101 100 100 −1 −1 10 10 1 1.5 2 2.5 3 3.5 4 1 2 3 4 5 6 7 8 k k 0 0 FIG.7.—AsFig.6,forα=1. FIG.9.—ThesameasFig.8,forα=1. 103 105 0 10 104 −3 10 m 10−6 t 103 −9 10 102 −12 10 2 3 4 5 6 7 3 10 101 100 101 102 103 r/ r 102 S z FIG.10.— The optical depth of the standard thin accretion disks with- b 101 outmagnetic fieldasfunctions ofradius, where rS=2GM/c2. Theopac- ity κtot =κes+κR is adopted, where the Rosseland mean opacity κR = 100 5a×bla1c0k24hρoTlec- 7w/2ithcmM2=g-110.MT⊙h,ewrehdilelintheesbreluperelsiennetstahreerfeosrualtsblcaaclkcuhloalteedwfiothr M=108M⊙. Thedifferentlinetypescorrespondtothediskswithdifferent 10−1 parameters: solid(α=1andm˙ =0.1),dashed(α=0.1andm˙ =0.1),dash- 1 2 3 4 5 6 7 8 dotted(α=1andm˙ =0.01),anddottedlines(α=0.1andm˙ =0.01). k Thedependenceofthesolutionsonthevalueofαisshown 0 in Fig. 7. The magnetic diffusivity η is scaled with the tur- FIG.8.— The disk-outflow solutions for vertically isothermal accretion bulentviscosityν,andthereforethediffusionbecomesmore disks. Thecoloredlinesrepresentthesolutionsderivedwithdifferentdisk important for the cases with a higher value of the viscosity temperature,Θ=0.01(red),2.5 10- 4(green),and10- 4(blue),respectively. Thesolutionsderivedwithdiffe×rentratiosofgaspressuretomagneticpres- parameter α. In order to compete with the diffusion of the sureatthedisksurfaceareindicatedwithdifferentlinetypes,βs=1(solid), field,alargeradialvelocityofthediskisrequiredforhigh α and0.1(dashed),respectively.Viscosityparameterα=0.1. cases, which correspondsa high rate of angular momentum removalbytheoutflows. Wefindthatthevaluesofthemass loadparameterµoftheoutflowaresystematicallyhigherfor those derivedwith a larger α (comparethe resultsin Figs. 6 and7). 8 The accretion disk is vertically compressed by the curved thiswork. magneticfield,whichsetsanupperlimitonthefieldstrength Newisthehigh-fieldsolutionfound(leftmostintersection). (seeEqs.35and36). Thevalueofβ onlydependsonthe Bythesamelineofreasoningasabove,thispointisexpected z,min field inclination κ at the disk surface, and β decreases tobestable,sincetheslopesofthedashedandsolidcurvesare 0 z,min with increasing κ (see Fig. 1). There is a force exerted on reversedhere. Itis, however,somewhatoutside theassump- 0 thediskbythe curvedfield againstthegravityofthecentral tions made, since the MRI turbulence that was assumed for objectintheradialdirection,whichmakestherotationofthe the magnetic diffusion is probably suppressed at these field gas in the disk be sub-Keplerian. The rotational velocity of strengths.Instead,instabilityofthestrongfielditselfislikely the disk can be quite low if the field strength is sufficiently tocauseitsoutwarddiffusion(asinthesimulationsofStehle strong(seeEq.27andFig.2). Thediskisthenmagnetically &Spruit2001andIgumenschevetal.2003). Tothe (uncer- supported against gravity. Such configurations are likely to tain)extentthatthisprocesscanbeparametrizedintermsof beunstabletointerchangeinstabilities,however(Spruitetal. c2/Ω, the present analysis would still apply. We speculate s 1995), which effectively cause the magnetic field to spread thatthe stability ofthispointis actuallysignificant, andthat outward and limit the field strength (as observed in the nu- it is relevantfor the experimentallyobservedstability of the merical simulations of Stehle & Spruit 2001). The require- strongcentralfluxbundlesinnumericalsimulationsofaccre- ment that the magnetic force is less than the gravity in the tionontoblackholes. radialdirectionprovidesanadditionalconstraintonthefield strength.Wefindthattheconstraintsalmostoverlapwiththat 4. CONCLUSIONS constrainedbytheforceintheverticaldirectionwhenκ .3, 0 Wehaveconsideredthepossibilitythattheangularmomen- while maximal field strength becomes lower (a larger β ) min tum of an accretion disk is removed predominantly by out- foradiskwithrelativehightemperatureand κ &3(seeFig. 0 flowsdrivenbytheaccretedfield(Bisnovatyi-Kogan&Ruz- 1). maikin1974, Blandford1976). An obstacle to this proposal From Fig. 1 we see that the magnetic field can be very hasbeentherealization(vanBallegooijen1989,Lubowetal. strong, e.g., the magneticpressure can be more than one or- (1994)) that outward diffusions of the accreted field is fast derofmagnitudehigherthanthegaspressureinthediskifthe comparedtotheinwardaccretionvelocityinageometrically fieldinclinationκ issufficientlylarge.However,themagnet- 0 thin accretion disk if the value of the Prandtl number P is icallydrivenoutflowwillbesuppressedifκ istoolarge,be- m 0 aroundunity.Revisitingthisproblem,wefindthatevenmod- causethe effectivepotentialbarrierbecomesextremelydeep erately weak fields can in fact cause sufficient angular mo- inthiscase. Theresultsshowthatβ&0.5isalwayssatisfied mentum loss via a magnetic wind to balance outward diffu- inthedisk-outflowsolutions(seeFigs.6and7).Notethatthis sion. The estimate in Eq. (21) shows that, at P =1, a field doesnotmeanthemagneticfieldcannotbestrong,asthedisk withamagneticpressureaslowas∼10- 3ofthemgaspressure issignificantlycompressedintheverticaldirection,whichin- p atthediskmidplanehasachanceoffacilitatingitsownac- creases the density of the disk and then the gas pressure for c cretionbydrivingamoderatelystrongmagneticoutflow.This givendisktemperature. isduemoreorlesstocompoundingnumericalfactorsoforder unity. In particularwhen MRI turbulenceproducesthe rela- 3.2. Stability tivelyloweffectiveviscosityα∼0.01thatisseeninseveral Wehavecalculatedonlystationarysolutions,buttheirsta- numericalsimulations. bilitytotime-dependentperturbationscanalreadybeguessed Usingasimplifiedmodelfortheverticaldiskstructure,we at by inspection of the intersection points in Fig. 3. Near have studied the conditions for existence of such stationary thehigh-beta(lowfield)solutionthemassloadingparameter equilibriabetween wind-inducedadvectionand outwardtur- (solidline)isnearlyindependentofthefieldstrengthassumed bulentdiffusioninmorequantitativedetail. Twoequilibrium (thisisbecauseoftheassumedvalueoftheplasma-betaatthe pointsarefound,oneatlow field strengthscorrespondingto base of the flow). If the field strength were to decrease (to a plasma-betaat the midplaneof orderseveralhundred, and therightoftheintersectionpoint),themasslosswouldneed oneforstrongaccretedfields,β∼1.Wesurmisethatthefirst toincreaseinordertomaintainabalancebetweeninwardac- isrelevantfortheaccretionofweak,possiblyexternal,fields cretion and outward diffusion(dashed line). Since the mass throughtheouterpartsofthedisk,whilethelatteronecould loadingactuallydoesnotchangemuch,theangularmomen- explain the tendency, observed in full 3D numerical simu- tumlossisinsufficienttobalanceoutwarddiffusionforsucha lations, of strong flux bundles at the centers of disk to stay perturbation.Outwarddiffusionwillthentendtodecreasethe confinedinspiteofstrongMRIturbulencesurroundingthem field strength, providinga positive feedbackto the perturba- (e.g. Beckwith et al. 2009). These authorsalso identify the tion. Thisstationarysolutionisthusexpectedtobeunstable. mechanismresponsibleformaintenanceofthebundleagainst Thismechanismofinstabilityisthesameasthatidentified the outward diffusion that one might expect from the turbu- inthelinearstabilityanalysisofCS02. Thetimescaleofthe lence surroundingit. Unlike the present model, this mecha- instabilityiscomparablewiththedynamicaltimescaleofthe nismdoesnotdependonthepresenceofawind. disk if the magnetic torque is large, which becomes signif- icantly small when the magnetic torque is weak (see CS02 for the detailed results and discussion). At sufficiently low HS thanks the Shanghai Astronomical Observatory for fieldstrengthsor/andhigh-κ ,wherethemagnetictorquebe- theirgeneroushospitality duringthe work on the projectre- 0 comesweak,thisanalysispredictedaregimeofstabilitydue ported here. This work is supported by the National Ba- tomagneticdiffusion. Thisimpliesthatthegrowthtimescale sic Research Program of China (grant 2009CB824800), the of such instability considered in this work should be signif- NSFC (grants 11173043,11121062and 11233006),and the icantly lower than the dynamicaltimescale of the disk. The CAS/SAFEA InternationalPartnershipProgramforCreative detailed calculation of the instability is beyondthe scope of ResearchTeams(KJCX2-YW-T23). Thelargescalemagneticfieldsofthinaccretiondisks 9 REFERENCES Anderson,J.M.,Li,Z.-Y.,Krasnopolsky,R.,&Blandford,R.D.2005,ApJ, Mestel,L.2012,Stellarmagnetism,secondedition.Oxfordscience 630,945 publications(Internationalseriesofmonographsonphysics154) Beckwith,K.,Hawley,J.F.,&Krolik,J.H.2009,ApJ,707,428 Narayan,R.,&Yi,I.1994,ApJ,428,L13 Bisnovatyi-Kogan,G.S.,&Ruzmaikin,A.A.1974,Ap&SS,28,45 Narayan,R.,&Yi,I.1995,ApJ,452,710 Bisnovatyi-Kogan,G.S.,&Ruzmaikin,A.A.1976,Ap&SS,42,401 Ogilvie,G.I.,&Livio,M.2001,ApJ,553,158 Blandford,R.D.1976,MNRAS,176,465 Parker,E.N.1979,inChapter17,CosmicalMagneticFields Blandford,R.D.,&Payne,D.G.1982,MNRAS,199,883 (Oxford:ClarendonPress) Cao,X.2011,ApJ,737,94 Pudritz,R.E.,Ouyed,R.,Fendt,C.,&Brandenburg,A.2007,Protostars Cao,X.,&Spruit,H.C.1994,A&A,287,80 andPlanetsV,277 Cao,X.,&Spruit,H.C.2002,A&A,385,289(CS02) Spruit,H.C.1996,NATOASICProc.477:EvolutionaryProcessesin Fromang,S.,Papaloizou,J.,Lesur,G.,&Heinemann,T.2009,Numerical BinaryStars,249 ModelingofSpacePlasmaFlows:ASTRONUM-2008,406,9 Spruit,H.C.2010,LectureNotesinPhysics,BerlinSpringerVerlag,794, Fromang,S.,&Stone,J.M.2009,A&A,507,19 233 Guan,X.,&Gammie,C.F.2009,ApJ,697,1901 Spruit,H.C.,Stehle,R.,&Papaloizou,J.C.B.1995,MNRAS,275,1223 Guilet,J.,&Ogilvie,G.I.2012a,MNRAS,424,2097 Spruit,H.C.,&Uzdensky,D.A.2005,ApJ,629,960 Guilet,J.,&Ogilvie,G.I.2012b,arXiv:1212.0855 Stehle,R.,&Spruit,H.C.2001,MNRAS,323,587 Igumenshchev,I.V.,Narayan,R.,&Abramowicz,M.A.2003,ApJ,592, vanBallegooijen,A.A.1989,AccretionDisksandMagneticFieldsin 1042 Astrophysics,156,99 Kippenhahn,R.,&Schlüter,A.1957,ZAp,43,36 Yousef,T.A.,Brandenburg,A.,Rüdiger,G.2003,A&A,411,321 Konigl,A.,&Pudritz,R.E.2000,ProtostarsandPlanetsIV,759 Lesur,G.,&Longaretti,P.-Y.2009,A&A,504,309 Lubow,S.H.,Papaloizou,J.C.B.,&Pringle,J.E.1994,MNRAS,267,235 Lubow,S.H.,Papaloizou,J.C.B.,&Pringle,J.E.1994,MNRAS,268, 1010