The large scale magnetic fields of thin accretion disks

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by  Xinwu Cao

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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