The outflows accelerated by the magnetic fields and radiation force of accretion disks

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by  Xinwu Cao

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THEOUTFLOWSACCELERATEDBYTHEMAGNETICFIELDSANDRADIATIONFORCEOFACCRETIONDISKS
XINWUCAO
KeyLaboratoryforResearchinGalaxiesandCosmology,ShanghaiAstronomicalObservatory,ChineseAcademyofSciences,
80NandanRoad,Shanghai,200030,China;[email protected]
acceptedbyApJ
ABSTRACT
The inner region of a luminous accretion disk is radiation pressure dominated. We estimate the surface
4
1 temperatureof a radiationpressure dominatedaccretiondisk, Θ=c2s/r2Ω2K ≪(H/r)2, which is significantly
0 lowerthan thatof a gaspressuredominateddisk, Θ∼(H/r)2. Thismeansthatthe outflowcan be launched
2 magneticallyfromthephotosphereoftheradiationpressuredominatediskonlyiftheeffectivepotentialbarrier
along the magnetic field line is extremely shallow or no potential barrier is present. For the latter case, the
n
slowsonicpointintheoutflowmayprobablybeinthedisk,whichleadstoaslowcirculardenseflowabove
a
J the disk. This implies that hot gas (probably in the corona) is necessary for launching an outflow from the
radiation pressure dominated disk, which provides a natural explanation on the observational evidence that
8
therelativisticjetsarerelatedtohotplasmainsomeX-raybinariesandactivegalacticnuclei. Weinvestigate
1
the outflowsaccelerated from the hot coronaabove the disk by the magnetic field and radiation force of the
] accretiondisk.Wefindthat,withthehelpoftheradiationforce,themasslossrateintheoutflowishigh,which
E leadstoaslowoutflow. Thismaybethereasonwhythejetsinradio-loudnarrow-lineSeyfertgalaxiesarein
H generalmildrelativisticcomparedwiththoseinblazars.
. Subjectheadings:accretion,accretiondisks,galaxies:jets,magneticfields,galaxies:active
h
p
- 1. INTRODUCTION sub-Keplerian (Ogilvie&Livio 1998, 2001; Cao&Spruit
o
2002; Cao 2012). The radiation-magnetohydrodynamic
r The acceleration of the gas by the magnetic field lines
t simulations have been carried out on the global struc-
s threading the rotating disks is considered as a promising
ture of the accretion disks and outflows around black
a explanation for jets/outflows observed in different types of
[ the sources, i.e., young stellar objects, X-ray binaries, and holes (e.g., Moscibrodzkaetal. 2007; Mos´cibrodzkaetal.
active galactic nuclei (AGNs) (see reviews in Spruit 1996; 2009; Mos´cibrodzka&Proga 2009; Takeuchietal. 2010;
2 Ohsuga&Mineshige 2011; Vaidyaetal. 2011). It is found
Konigl&Pudritz 2000; Pudritzetal. 2007; Spruit2010). In
v
thatthemagneticforce,togetherwiththeradiationforceex-
0 this model, a fraction of gas is driven form the disk along
erted by accretion disks, can efficiently drive outflows from
2 thefieldlineco-rotatingwiththediskbythecentrifugalforce
luminousaccretiondisks.Cao(2012)investigatedthelaunch-
3 (Blandford&Payne1982). Thejets/outflowsaredominantly
ing condition for the cold outflows driven by the magnetic
2 poweredby the rotationalkinetic energyof the disk through
field and radiation force of an accretion disk, in which the
. theorderedlargescalefieldthreadingtheaccretiondisk.
1 diskissub-Keplerianduetotheforceexertedbythemagnetic
It has been also suggested that the outflow can be
0 field. Ithasbeenfoundthattheforceexertedbytheradiation
accelerated by the radiation pressure of the disk (e.g.,
4 fromthe disk does help to launch the outflow. The cold gas
Bisnovatyi-Kogan&Blinnikov 1977; Shlosmanetal. 1985;
1 can be driven from the disk surface even if the field line is
Murrayetal.1995). Thisscenarioisattractiveespeciallyfor
v: thejetsinthesourcesaccretingathighmassrates.Indeed,jets inclinedatanglegreaterthan60◦withrespecttothedisksur-
i havebeenobservedintheblackholesystemsaccretingathigh face,whichisobviouslydifferentfromthepuremagnetically
X drivencoldoutflow(Blandford&Payne1982).
rates, such as, some narrow-lineSeyfertI galaxies(NLS1s),
r young radio galaxies, and microquasars (e.g., Zhouetal. In this work, we explore the outflow accelerated from the
a hotcoronaabovethediskbythemagneticfieldandradiation
2003; Doietal. 2006; Yuanetal. 2008; Gu&Chen 2010;
forceofthedisk. Inourmodel,thediskstructure,especially
Czernyetal. 2009; Wu 2009; Fenderetal. 2004). The radi-
thecircularmotionvelocityofthegasesinthedisk,isaltered
ation pressure may also play an important role in the mag-
inthepresenceofastronglarge-scaleorderedmagneticfield
neticallydrivenoutflowsespeciallyfromluminousaccretion
threadingthedisk. WedescribethemodelandresultsinSec-
disks. Proga(2000)performednumericalsimulationsonthe
tions 2 and 3, and the discussion is in Sections 4. The final
radiation-driven outflows from a luminous Keplerian accre-
sectioncontainsasummary.
tion disk threaded by a strong large-scale ordered magnetic
field. It has been found that the radiation force is essential
in driving outflows from the disk if the thermal energy of
the gas is low or the field lines make an angle greater than 2. MODEL
60◦ withrespecttothediskmidplane. Strictlyspeaking,the
Weconsidertheoutflowacceleratedbythelargescalemag-
circularmotion of the gas in the accretiondisk always devi-
neticfieldandtheradiationforceofaradiationpressurepres-
ates from Keplerian motion in the presence of a large scale
suredominatedaccretiondisk. Therotationalvelocityofthe
magneticfield,whichusuallyexertsaradialforceagainstthe
gas in the disk is sub-Keplerian in the presence of a mag-
gravitationofthecentralobject.Therefore,theaccretiondisk
neticfield, dueto a radialmagneticforceagainstthegravity
threaded by ordered magnetic field lines is therefore always
of the central object. The outflow is fed by the gas at the
2 XinwuCao
disksurface. Thepropertiesofthetransitionregionbetween Thepressureatthemid-planeofanaccretiondiskis
thediskandoutflowarestillquiteuncertain,thoughsomeef-
4σ
forts have been devoted on this issue (e.g., Ogilvie&Livio p = T4, (3)
1998, 2001). In this work, we consider the outflows fed by d 3c c
the hotcoronaabovethe disk, and the temperatureand den- where T is the centraltemperatureof the disk. The flux ra-
c
sityofthecoronaaredescribedastheinputmodelparameters. diatedfromthe unitarea of the disk surface is relatedto the
Alargescalemagneticfieldthreadingthediskisakeyingre- temperatureanddensityofthediskby
dient in our model calculations, however, the origin of such
afieldisstillnotwellunderstood. Thefieldcanbeadvected 8σT4
f = c . (4)
inwardsbytheaccretionmatterfromtheinterstellarmedium rad 3Σ κ
d T
or a companion star (Bisnovatyi-Kogan&Ruzmaikin 1974,
1976). The inward advection of the field lines is balanced Substituting Equations (1), (3), and (4) into Equation (2),
by the outward movementof field lines caused by magnetic wefind
˜
diffusion(e.g.,vanBallegooijen1989;Lubowetal.1994). In 1- Ω˜2= 2βHd , (5)
principle,thefieldconfigurationcanbecalculatedifthestruc- κ (1+H˜2)3/2
0 d
tureofthediskisknown. Lubowetal.(1994)foundthatthe
advectionofthefieldinaconventionalviscouslydriventhin wherethedimensionlessquantitiesΩ˜,H˜ ,andβ,aredefined
d
accretiondiskisratherinefficient. Itwasarguedthattheac- as
cretion velocityof the gas in the region away from the mid-
plane of the disk can be larger than that at the midplane of Ω˜ = Ω ; Ω = GM 1/2; H˜ = H; β= B2z/p . (6)
the disk, which makes the field dragged in by the accretion Ω K r3 d r 8π d
K (cid:18) i (cid:19) i
disk more efficiently(Lovelaceetal. 2009; Guilet&Ogilvie
2012, 2013). Cao&Spruit (2013)’s calculations show that 2.2. Magneticfieldconfiguration
the external field can be advected inwards efficiently if the
Weuseamagneticfieldconfigurationwith
angularmomentumofthediskispredominatelyremovedby
tihsethoeutdflyonwasm.oApnroaclteesrsneastivinetfhoertdhieskosr,igainndotfhmeaoguntfletoicwfisecladns B (r, z=0)= 1+ r 2 - 1/2, (7)
be driven by the dynamo generated magnetic fields of the z r
" (cid:18) 0(cid:19) #
disks (vonRekowskietal. 2003; Campbell 2003). To avoid
the complexity of the detailed physics of the disk field for- at the mid-plane of the disk (Cao&Spruit 1994), which is
mation,weuseananalyticmagneticfieldconfigurationofan similar to the self-similar disk given in Blandford&Payne
accretiondiskgiveninCao&Spruit(1994)toinvestigatethe (1982) (see Spruit 1996, for the detailed discussion). The
dynamicsoftheoutflowdrivenbythemagneticfieldandra- streamfunctionofthepotentialfieldinthespaceabove/below
diationpressureofthedisk. thedisksatisfyingtheboundarycondition(7)isgivenby
2.1. Structureofaradiationpressuredominatedaccretion r 2 z 2 1/2 z
Φ= + 1+ - 1+ . (8)
disk
r r r
"(cid:18) 0(cid:19) (cid:18) (cid:12) 0(cid:12)(cid:19) # (cid:18) (cid:12) 0(cid:12)(cid:19)
The structure of a radiation pressure dominated accretion (cid:12) (cid:12) (cid:12) (cid:12)
diskwithalargescalemagneticfieldhasbeenstudiedindetail It satisfies the conditions ▽(cid:12)(cid:12)×B(cid:12)(cid:12) =0 and ▽·B(cid:12)(cid:12) =(cid:12)(cid:12)0, and it is
byCao(2012). Webrieflysummarizethemainresultsinthis usedinthisworktostudythedynamicsoftheoutflowabove
subsection. thedisk. Thecomponentsofthefieldinthespaceabovethe
Thehalf-thicknessH ofaradiationpressuredominatedac- diskaregivenby
d
cretiondiskisestimatedbyassumingtheverticalcomponent 1∂Φ
B (r, z)= , (9)
ofgravitytobebalancedwiththeradiationforceandtheverti- z r ∂r
calcomponentmagneticforceatthedisksurface. Thecurva-
and
tureofthefieldlineatthedisksurfaceisusuallyverysmall, 1∂Φ
and the vertical magnetic force can be neglected compared B (r, z)=- . (10)
r
r ∂z
withtheradiationforceatz=H . Thediskthicknesscanbe
d
calculatedwith In principle, the radial magnetic field will be sheared into
azimuthal field by the differential rotation of the gas in the
GMH f κ
d = rad T, (1) disk,whichleadstomagnetorotationalinstability(MRI)and
(r2+H2)3/2 c
i d the MRI-driven turbulence (Balbus&Hawley 1991, 1998).
This is the most promising mechanism for angular momen-
where f is the flux from the unit surface area of the disk,
rad tum transportationin accretion disks. In this work, we con-
andκ istheThompsonscatteringcross-section.
TheTangularvelocityΩofthediskiscalculatedwith siderthepropertiesofthediskastheboundaryconditionsfor
drivingtheoutflows,andthereforewedonotneedtoconsider
BSB B2 thedetailedphysicsoftheazimuthalfieldin thediskcaused
riΩ2K- riΩ2= 2πrΣzr = 2πΣzrκ , (2) bythedifferentialrotation.
d i d i 0
whereBSandB aretheradialandverticalcomponentsofthe 2.3. Radiationoftheaccretiondisk
r z
field at the disk surface, respectively, κ =B /BS, and Σ is The radiation force exerted by the disk is in z-direction at
0 z r d
thesurfacedensityofthedisk. Thefootpointofthefieldline the disk surface, which can be determined locally, F (z=
rad
atthedisksurfaceislocatedatr=r andz=H . H ) = f κ ρ/c , where ρ is the density of the gas at
i d d rad T i i
Theoutflowsacceleratedbythemagneticfieldsandradiationforce 3
the disk surface. In the space above the disk, the ra- 2.4. Outflowsdrivenbythemagneticfieldandradiation
diation force exserted by the disk has to be calculated pressure
by integrating over the contribution from the whole disk
Anisothermaloutflowdrivenbythemagneticfieldandra-
(Bisnovatyi-Kogan&Blinnikov1977).
diation force of the disk along the field line is described by
The r and z-components of the radiation force exerted on
thefollowingEquations(cf.Cao&Spruit1994):
thegaselementwithdensityρat(r, z)abovethediskare
F (r, z)= κTρ p=ρc2s,i, (17)
rad,r c f
v = B , (18)
p p
f (r′)µ′(1- µ′2)1/2(r- r′cosφ′) ρ
× rad r′dr′dφ′, (11)
πl(h2+l2) (vφ- Ωr)Bp=vpBφ, (19)
rZ′ φZ′
B B
and r vφ- p φ =ΩrA, (20)
F (r, z)= κTρ frad(r′)h2 r′dr′dφ′, (12) (cid:18) 4πρvp(cid:19)
rad,z c π(h2+l2)2 and
Zr′ φZ′ v2p+1(v - Ωr)2+c2 ln ρ +Ψ (r, z)=E, (21)
where 2 2 φ s,i ρA eff
h=z- H˜dr, wheretheeffectivepotentialalongthefieldlinethreadingthe
accretiondiskwithangularvelocityΩatthefieldfootpointr
l=[(r- r′cosφ′)2+r′2sin2φ′]1/2, is i
and GM 1
µ′= h . Ψeff(r, z)=- (r2+z2)1/2- 2Ω(ri)2r2+Ψrad(r, z). (22)
(h2+l2)1/2
ThelastterminEquation(22)iscontributedbytheradiation
We haveassumedthatthe relativedisk half-thicknessH˜ re- force from the disk (see Equation 13). Equation (18) is the
d
mains constant with radius for a thin disk, which is taken mass conservationof the outflow along the field line, where
as an input parameter in our model. The flux from the vp andBp arethe poloidalvelocityandfieldstrengthrespec-
unit surface of a standard thin disk is roughly f (r)∝r- 3 tively,and f describesthemasslossrateintheoutflow. The
rad
(Shakura&Sunyaev 1973). A pseudo potential Ψ con- conservationofangularmomentumisdescribedbyEquation
rad
tributed by the radiation force along the field line is defined (20), in which rA is the Alfvén radius of the outflow. The
fieldlineandthemotionoftheoutflowareparallelintheco-
as
c c rotating field line frame (Equation19). Equation (21) is the
Ψrad=- Frad,rdr- Frad,zdz, (13) BernoulliEquationoftheisothermaloutflow.
κ ρ κ ρ
T Z T Z Substituting Equations (17)-(20) into Equation (21), we
whichareintegratedalongthefieldlinefromthedisksurface. have
ThefieldlineshapeisdescribedbyEquation(8)withaspec-
ifiedvalueofΦ. SubstituteEquation(1)intoEquations(11), H(r, ρ, r , ρ )= B2p ρA 2+ Ω2(rA2 - r2)2
wecanre-writethesetwoequationsindimensionlessform, A A 8πρ ρ 2r2(1- ρ/ρ )2
A(cid:18) (cid:19) A
F˜rad,r(r, z)=Frad,r(r, z)GMcrκi2 ρ +c2s,ilnρρA +Ψeff=E, (23)
T
wherec isthesoundspeedofthegasatthedisksurface/the
= 1 µ′(1- µ′2)1/2(r- r′cosφ′)H˜d ri 3r′dr′dφ′, bottomos,iftheoutflow,andρA isthedensityoftheoutflowat
π l(h2+l2)(1+H˜2)3/2 r′ the Alfvén radius rA. It can be re-written in dimensionless
Zr′ φZ′ d (cid:16) (cid:17) formas
and (14) H˜(x, y, xi, yi)= ri H(r, ρ, rA, ρA)
GM
F˜rad,z(r, z)=Frad,z(r, z)GMcrκi2Tρ = βiyyi2Θi+2(x12-x2x(21)-2Ω˜y2)2+Θilny+Ψ˜eff=E˜, (24)
i
1 h2H˜ r 3
= d i r′dr′dφ′, (15) where
π (h2+l2)2(1+H˜2)3/2 r′ c2
Zr′ φZ′ d (cid:16) (cid:17) x=r/rA; y=ρ/ρA; Θi= r2Ωs,i2 ;
i K
where r is the footpointof the field line at the disk surface.
The dimiensionless pseudo potential along a magnetic field β = B2p,i/p = B2p,i/ρc2 , (25)
linecontributedbytheradiationforceis i 8π i 8π i s,i
Ψ˜rad(˜r, ˜z)=Ψrad(r, z)GrMi =- F˜rad,rdr˜- F˜rad,zd˜z, (16) athnedbtohtetosmubosfcrtihpetsou“it"florwef.er to the values of the quantities at
Z Z Substituting Equation(1) into Equation(22), the effective
wherer˜=r/r and˜z=z/r. potentialalongthefieldlinethreadingtheaccretiondiskwith
i i
4 XinwuCao
angularvelocityΩatradiusr canbere-writtenindimension- Substituting Equations (1) and (32) into Equation (33), we
i
lessform, have
Ψ˜eff(˜r, ˜z)=Ψeff(r, z)GrMi =- (˜r2+1˜z2)1/2- 21Ω˜2˜r2+Ψ˜rad(˜r(,2˜z6)), ri2cΩ2s,i2K =(cid:20)(1.895M⊙)26π02GκT3(1r+iΩH˜2Kd2H˜)3d/2(cid:21)1/4GMriµ
where ˜r=r/ri, ˜z=z/ri, and the last term is calculated with =4.946×10- 6µ- 1(r/r )1/2H˜1/4m- 1/4(1+H˜2)- 3/8, (34)
Equation(16). i S d d
Foragivenmagneticfieldconfiguration,theoutflowalong where
a field line passes through three critical points, i.e., the
2GM M
slow sonic, Alfvén, and fast sonic points (Sakurai 1985; r = , m= , (35)
S c2 M
Cao&Spruit 1994). At the two sonic points, the function ⊙
˜
H(x, y, xi, yi)satisfies and(¯hc/G)3/2m-p2=1.895M⊙ isused. Themolecularweight
˜ ˜ µ=0.5forpurehydrogenplasma.
∂H ∂H
= =0, (27) Thesurfacetemperatureofagaspressuredominatedaccre-
∂x ∂y tiondiskis
and ˜ ˜ ˜ c2s,i = 4 1/4 c2s,c = 4 1/4H˜2, (36)
H(xs, ys, xi, yi)=H(xf, yf, xi, yi)=E. (28) ri2Ω2K (cid:18)3τ(cid:19) ri2Ω2K (cid:18)3τ(cid:19) d
Anadditionalcondition, wherec isthesoundspeedofthegasatthemid-planeofthe
s,c
disk, andτ is theopticaldepthof thedisk in the verticaldi-
˜ ˜
H(x, y, xi, yi)=E, (29) rection. Theopticaldepthτ isintherangeof∼102- 105for
athinaccretiondiskaccretingatdifferentrates,whichmeans
is imposed at the disk surface, i.e., x =xi, y =yi. A set of thesurfacetemperatureofthedisk,c2 /r2Ω2 ∼0.06- 0.34H˜2
sevenEquations(27),(28),and(29),canbesolvedforseven s,i i K d
˜ (Cao&Spruit 2013). Compared with a gas pressure domi-
variables, x , y , x, y, x, y, and E, for an outflow along a
s s f f i i nated accretion disk, the surface temperature of a radiation
given magnetic field line, when the temperature and density
ofthegasatthebottomoftheoutflowarespecified. pressuredominatedaccretiondiskismuchlower,c2s,i/ri2Ω2K≪
Withthederivedoutflowsolution,themasslossrateinthe H˜2(seeEquation34).Theaccretiondiskisgaspressuredom-
d
outflowfromtheunitsurfaceareaisavailable, inatedintheouterregionor/anditsmassaccretionrateislow
(Laor&Netzer 1989; Shakura&Sunyaev 1973). This im-
B κ
m˙ =ρv z =ρv 0 , (30) plies gas pressure gradient in the outflow from the inner re-
w i p,iB i p,i(1+κ2)1/2 gionofaluminous(radiationpressuredominated)diskisal-
p,i 0
mostnegligible,andthereforethecoldgasapproximationisa
whichcanbere-writtenindimensionlessformas goodapproximation(Cao2012). Inthiscase,theoutflowwill
m˙ βH˜ (1+κ2)1/2 besuppressedif the field line is inclinedat anangle slightly
w = d 0 . (31) largerthanthecriticalone,becausetheinternalenergyofsuch
ΣdΩK κ0[2βiΘiyi(1+H˜d2)3]1/2 coldgasatthedisksurfaceistoolowtoovercometheeffec-
tivepotentialbarrieralongthefieldline. Iftheangleislower
Here,Equations(1),(3),(4)and(24)areused.Thereciprocal than the critical one, the slow sonic point may probably go
ofm˙w/ΣdΩK representsthenumberoforbitsinwhichallthe intothedisk,andthedensityofthegasinthediskisusually
gasinthediskischanneledintotheoutflow. high,whichusuallyleadstoaslowdensecircularflowabove
thedisk(seeCao&Spruit1994;Spruit1996,forthedetailed
2.5. Boundaryconditions discussion). It implies that hot gas is needed for feeding an
outflowfromaradiationpressuredominatedaccretiondisk.
Thesolutionofanoutflowalongagivenfieldlineisavail-
Theobservedultraviolet(UV)/opticalemissionofAGNis
ablebysolvinga setofsevennon-linearalgebraicequations
thought to be the thermal emission from the standard geo-
withsuitableboundaryconditionsatthedisksurface.Thedi-
mensionlesstemperatureΘ ofthegasisaninputparameter, metrically thin, optically thick accretion disks (e.g., Shields
i 1978;Malkan&Sargent1982;Sun&Malkan1989).Theob-
andthedensityofthegasisdescribedbyβ ifthevalueofβis
i servedpower-lawhardX-rayspectraofAGNaremostlikely
specified(seeEquation25).Fortheoutflowcanbeefficiently
due to the inverse Compton scattering of soft photons on a
drivenbythemagneticfield,thegaspressureshouldbelower
population of hot electrons in the coronas above the disk
thanthemagneticpressure,i.e.,β &1isrequired. Formost
i (Galeevetal. 1979; Haardt&Maraschi 1993; Haardtetal.
calculationsinthiswork,β =1isused.
i 1994). It has been found that the temperature of the hot
The temperature of the photosphere roughly equals to the
electrons in the corona is roughly around 109 K, which can
surfacetemperatureofthedisk,T,whichisgivenby
i
successfully reproduce a power-law hard X-ray spectrum as
f1/4 observed(e.g., Liuetal. 2003; Cao 2009). In this work, we
T = rad . (32) considerahotcoronaaboveathinaccretiondisk, whichisa
i σ1/4 reservoirto supplyhotgas to feed the outflowdrivenby the
Thesoundspeedofthegasinthephotosphereofthediskis magnetic field and radiation force of the disk. The detailed
propertiesofthecoronaarestillquiteunclear.Forsimplicity,
kT 1/2 we use two parameters, the temperature Θi and the density
i
c = . (33) of the corona, in our model calculations. As the magnetic
s,i µm
(cid:18) p(cid:19)
Theoutflowsacceleratedbythemagneticfieldsandradiationforce 5
field strengthat thedisk surfaceis describedbythe valueof 100
β specified,thedensityofthecoronaisgivenbytheratioof
magnetictogaspressureinthecorona(β)forgiventemper-
i
atureΘianddiskfieldstrengthβ (seeEquation25). 10−2
2)K
2Wc/( r s,ii10−4

10−6
3. RESULTS
Wecomparethetemperaturesofthephotospheresbetween 10−8
theradiationpressuredominateddisksandgaspressuredom- 0 0.1 0.2 0.3 0.4 0.5
H/ r
inated accretion disks in Figure 1. As discussed in Section d i
2.5, the temperature of the photosphere of a radiation pres- FIG.1.—Thetemperature ofthephotosphere ofthedisk. Thecoloured
suredominatedaccretiondiskisindeedmuchlowerthanthat lines represent theresults forradiation pressure dominated accretion disks
(seeSection 2.5forthe detailed discussion). Theredlines are theresults
ofagaspressuredominatedaccretiondisk. We notethatthe
calculatedwithablackholemassM=10M⊙,whilethegreenlinesarefora
surfacetemperatureoftheradiationpressuredominateddisk massiveblackholewithM=108M⊙.Thesolidlinesindicatethetemperature
decreaseswithincreasingblackholemass. atradiusr=10rS,andthedashedlinesareforr=100rS.Theblacksolidline
We use the magnetic field configuration above the disk representthetemperature ofthephotosphere ofanisothermalgaspressure
dominatedaccretion disk, whilethedashedanddottedlinesaretheresults
given in Section 2.2, of which the field line is given by set-
calculatedwithτ=100and105,respectively.
tingΦ=const.Inourcalculations,therelativediskthickness
˜ 20
H isassumedtobeindependentofradius,whichisaninput
d
parameter. Theangularvelocityofthediskcanbecalculated
withEquation(5),ifthestrengthofthemagneticfieldβatthe 18
˜
disksurface,thediskthicknessH ,andthefieldlineinclina-
d
tionκ arespecified. Thefieldlineinclinationκ atthedisk
0 0 16
surfaceisafunctionofr/r .
0
WiththespecifiedvaluesofΘ andβ ofthecorona,wecan
i i
calculatethedynamicsoftheoutflowalongagivenmagnetic 14
field line as described in Section 2. In Figures 2-4, we plot
thelocationofthecriticalpointsintheoutflowsolutionsde-
12
rivedwithdifferentvaluesofthemodelparameters.Weshow
howtheresultsvarywiththemagneticfieldstrengthβinFig-
0
ure 2. The rotational velocity Ω˜ of the disk decreases with z/r 10
increasingfieldstrengthβ (seeEquation5),andtheeffective
potentialbarrierincreaseswithdecreasingΩ˜. Thus,thecriti- 8
calpointsgofarawayfromthedisk. InFigure3,weplotthe
resultscalculatedwithdifferentvaluesofcoronatemperature
Θ. Inordertoshowtheroleoftheradiationforceontheac- 6
i
celerationoftheoutflows,wecomparetheresultswiththose
of the purely magnetically driven outflowsin Figure 4. The 4
locationofthecriticalpointsinthepuremagneticallydriven
outflows is farther away from the disk compared with those
2
oftheoutflowsdrivenbythemagneticfieldtogetherwiththe
radiation force. This is because the effective potential bar-
rierbecomesdeepforapuremagneticallydrivenoutflow(see 0
0 2 4 6 8 10
Figure5).
r/r
We can calculate the mass loss rate in the outflow with 0
Equation (31) when the outflow solution is derived. In Fig-
FIG.2.—Thelocationsofthecriticalpointsoftheoutflowsolutions(red:
ures 6, we plot the mass loss rates as functions of the field
slow sonic points; green: Alfvén points; blue: fast sonic points). The
line inclination κ0 at the disk surface. The velocities of the disk thickness H˜d =0.1, the dimensionless temperature of the hot corona
outflows along the field lines are given in Figures 7 and 8. Θi=0.01, andthe ratio ofmagnetic togaspressure inthecorona βi=1,
Themasslossratesdecreasewithκ , whilethevelocitiesof areadoptedinthecalculations. Differenttypesoflinesrepresenttheresults
0 fordifferentvaluesofβ=0.75(solid)and1(dashed),respectively.
the outflows increase with the field line inclination κ (see
0
Figures 6-8). We find that the dynamical properties of the
outflowsaredifferentnearthedisksurface,whilethepoloidal
velocitiesconvergeatlargedistancesalongthefieldlineswith
the same κ for low-κ case. It is found that the velocities
0 0
of the flows increase with decreasing density of the corona
(large-β), if the values of all the other parameters are fixed
i
(seeFigures7and8).
6 XinwuCao
−1.35
20
−1.4
−1M/ r)i
G
( eff
15 Y −1.45
0
r
z/
−1.5
1 1.1 1.2 1.3 1.4 1.5
10 r/ r
i
FIG.5.—Theeffectivepotentialalongthefieldlinethreadingadiskwith
˜
Hd=0.1andβ=0.75. Thesolidlinesrepresenttheresultsforthefieldline
inclinationκ0=1.5atthedisksurface,whilethedashedlinesareforκ0=3.
Forcomparison,weplottheresultscalculatedwithoutconsideringradiation
pressureeffectsasgreenlines.
5
100
10−1
0
0 2 4 6 8 10 K10−2
r/r Ω
0 Σd
/
FIG.3.—ThesameasFigure2. Inallcalculations, β=0.75isadopted. ˙mw10−3
DifferenttypesoflinesrepresenttheresultsfordifferentvaluesofΘi=0.005
(solid),0.01(dashed),and0.02(dotted),respectively.
10−4
10−5
1 1.5 2 2.5 3 3.5 4
20 k ( B/ BS)
0 z r
FIG.6.—Thedimensionlessmasslossratesintheoutflowsasfunctionsof
fieldlineinclinationκ0=Bz/BSr atthedisksurface. Inallthecalculations,
˜
β=0.75andHd=0.1areadopted.Thesolidlinesrepresenttheresultscalcu-
latedwithβi=1.0,whilethedashedlinesareforβi=3.Thedifferentcolors
15 arefordifferentcoronatemperature, Θi=0.01(red)and0.02(green). For
comparison,weplottheresultforpurelymagneticallydrivenoutflowswith
Θi=0.02asblacklines(solidline:βi=1,anddashedlines:βi=3).
0
r
z/

10
5
0
0 2 4 6 8 10
r/r
0
FIG.4.—Thelocationsofthecriticalpointsoftheoutflowsolutions(red:
slowsonicpoints; green: Alfvénpoints; blue: fastsonicpoints). Thedisk
thickness H˜d =0.1, the dimensionless temperature of the hotcorona Θi=
0.02, and the ratio of magnetic to gas pressure in the corona βi =1, and
β=0.75areadoptedinthecalculations. Forcomparison,theresultswithout
consideringtheeffectsofradiationpressurearealsoplotted(dashedlines).
Theoutflowsacceleratedbythemagneticfieldsandradiationforce 7
1 that for a gas pressure dominated accretion disk with the
10
˜
same relative disk thickness H (see Figure 1). The sur-
d
face temperatureof the radiation pressure disk is insensitive
0
10
to the disk thickness H /r, while it decreases with increas-
d
K ing black hole mass. The gas pressure gradient may help
Wr i 10−1 accelerating the outflow. The importance of the gas pres-
/ p sure on driving the outflow can be estimated by comparing
v 10−2 c2s with the effective potential barrier ∆Ψeff = Ψmeffax- Ψeff,i
alongthefieldline.Inthemagneticallydrivenoutflowmodel,
the mass loss rate in the outflow m˙ ∝exp(- ∆Ψ /c2) (see
−3 w eff s
10 Spruit1996;Ogilvie&Livio1998,forthedetails). Forsuch
1.5 2 4 6 8 alow-temperaturephotosphereatthesurfaceoftheradiation
pressuredominateddisk,anextremelyshallowpotentialbar-
rier(small-∆Ψ )isrequiredforlaunchinganoutflow,which
eff
meansthevaluesofβ andκ shouldbeinthenarrowranges
K 1 0
close to those of the cold gas launching condition given in
Wr i Cao (2012). Itimpliesstrict constraintsonthe outflowsthat
/
can be driven from the disk surface, especially for massive
vf
0.5 blackholecases,asthetemperaturesofthespheresareabove
twoordersofmagnitudelowerthanthoseofstellarblackhole
accretiondisks(see Figure1). The outflowcan be launched
0 along the field line if the effective potential Ψeff monotoni-
2 4 6 8 cally decreasesalongthe field line. In this case, the outflow
r/ r is alternativelyaccelerated within the disk, which is, for ex-
i
ample,similartothesolutionsoftheoutflowsalongthefield
FIG.7.—Thepoloidal andazimuthal velocities oftheoutflowalongthe linesinclinedatsmallangleswithrespecttothedisksurface
fieldline.Theratioofmagnetictogaspressureinthecoronaβi=1,andβ= giveninCao&Spruit(1994). Thegasovercomesapotential
0.75,areadoptedinallthecalculations. Thesolidlinesrepresenttheoutflow
alongthefieldlinewithκ0=1.5atthedisksurface,whilethedashedlines barrieralongthefieldlineinthedisk,andtheslowsonicpoint
areforκ0=3.Thecoloredlinesrepresentfortheresultswithdifferentvalues islocatedwithinthedisk. Thus,theslowdensecircularflows
ofmodelparameters(redlines: Θi=0.01,andgreenlines: Θi=0.02). The areusuallypresentabovethedisksinthiscase(Cao&Spruit
blackdashedlines aretheresults calculated withpuremagnetically driven
outflowmodel(κ0=3andΘi=0.02).Thedotsindicatethecriticalpointsin 1994).Suchcalculationscanbedonewithaknownfieldcon-
theoutflow,i.e.,slowsonic,Alfvén,andfastsonicpoints(fromlefttoright). figurationwithin the disk, which is beyondthe scope of this
work,andwewillnotinvestigatethiskindofsolutionsinthis
101 paper.
As discussed in Section 2.5, the outflow can be driven
from the hot gas in the corona above the disk. The
0
10 field configuration can be derived with the detailed physics
K of advection and diffusion of magnetic fields considered
W/ r pi 10−1 (2v0a0n9B; aGlluegiloeoti&jenOg1i9lv8i9e; 2L0u1b2o,w2e0t1a3l;. 1C9a9o4&; LSopvrueliatce20e1t3a)l.,
v whichisbeyondthescopeofthiswork.Weusetheconfigura-
10−2
tionofthefieldgiveninSection2.2,ofwhichthestrengthat
thedisk surfacedecreaseswith increasingradius. Aswe fo-
10−3 cusonhowthehotgasatthedisksurfaceisacceleratedalong
thefieldlinewiththehelpwiththeradiationforceofthedisk,
1.5 2 4 6 8
the above mentioned field configuration is sufficient for our
presentinvestigation.
The location of the critical pointsof the outflow solutions
K 1 withdifferentvaluesoftheaccretiondisk/coronaparameters
Wr i isplottedinFigures2-4. Itisfoundthattheslowsonicpoints
/ arealways close to the disk surface. In the centralregionof
vf
0.5 thedisk,thefieldlinesareinclinedatlargeangleswithrespect
tothedisksurface.Themasslossrateintheoutflowdecreases
with increasing inclinationκ (see Figure 6), because of the
0
effectivepotentialbarrierincreaseswithκ (seeFigure5)and
0 0
2 4 6 8 m˙w ∝exp(- ∆Ψeff/c2s). The Alfvén and fast sonic points go
r/ r farther away from the disk when κ is large (see Figures 2-
i 4), whichiscausedbya tenuouslo0w-m˙ outflow. Themass
w
FIG.8.—ThesameasFigure7,butβi=3isadopted. loss rate in the outflowincreases with the temperatureΘi of
the gas in the corona, if the values of all other parameters
4. DISCUSSION arefixed. Inordertoillustratetheroleoftheradiationforce
ontheaccelerationoftheoutflow,wecomparetheresultsof
Wefindthatthetemperatureofthephotosphereatthesur-
puremagneticallydrivenoutflowsinFigure6. Themassloss
face of a radiation pressure disk is significantly lower than
8 XinwuCao
ratesintheoutflowsdrivenbythemagneticfieldandradiation anoutflowfromtheradiationpressuredominateddisk,which
pressurearesignificantlyhigherthanthoseforpuremagnet- providesa naturalexplanationonthe observationalevidence
ically driven outflows, because the radiation pressure makes thatthe relativistic jets are related to hotplasma in some X-
theeffectivepotentialbarriershallow. raybinariesandactive galacticnuclei(Zdziarskietal. 2011;
Less mass is loaded in the outflow in high-κ cases, and Wuetal.2013).
0
the outflowcan be drivento a relativehigh velocity,while a Weinvestigatetheoutflowsacceleratedfromthehotcorona
slow dense outflow is present when κ is small. It is found abovethediskbythemagneticfieldandradiationforceofthe
0
thatthepoloidalvelocitiesoftheoutflowsalmostconvergeat accretion disk, and find that the outflow can be driven from
largedistancesalongthefieldlinewithagivenκ forlow-κ the corona with the help of radiation force even if the field
0 0
cases. Thisimpliesthatthegasinthedenseoutflowsisdomi- line is inclined at a large angle (>60◦) with respect to the
nantlyacceleratedbytheradiationpressureofthedisk,which disk surface. The potential barrier decreases due to the ra-
is almost independent of the values of the disk parameters. diation force of the disk, and therefore the mass loss rate in
However,the situationisdifferentfortheazimuthalvelocity the outflow increases. We find that slow outflows with high
of the outflows, which is independent of the radiation pres- masslossratesarepresentifthefieldlineinclinationκ .2.
0
sure. Wefindthattheoutflowswithlowmasslossrates,i.e., This may be the reason why the jets in radio-loud narrow-
low-Θ,or/andhigh-κ asdiscussedabove,canbeaccelerated lineSeyfertgalaxiesareingeneralmildrelativisticcompared
i 0
toalargevelocityintheazimuthaldirection. with those in blazars (Komossaetal. 2006; Doietal. 2006;
Gu&Chen2010;Doietal.2011).
5. SUMMARY
We estimate the temperatureof the photosphereabovethe
radiation dominated accretion disk, and find that Θ ≪H˜2. This work is supported by the NSFC (grants 11173043,
d
Therefore,itisdifficultfortheoutflowsdrivenfromthepho- 11121062, and 11233006), the CAS/SAFEA International
tospheresoftheradiationdominateddisks. Thisimpliesthat Partnership Program for Creative Research Teams (KJCX2-
hot gas (probably in the corona) is necessary for launching YW-T23),andShanghaiMunicipality.
REFERENCES
Balbus,S.A.,&Hawley,J.F.1991,ApJ,376,214 Moscibrodzka,M.,Proga,D.,Czerny,B.,&Siemiginowska,A.2007,A&A,
Balbus,S.A.,&Hawley,J.F.1998,ReviewsofModernPhysics,70,1 474,1
Bisnovatyi-Kogan,G.S.,&Blinnikov,S.I.1977,A&A,59,111 Murray,N.,Chiang,J.,Grossman,S.A.,&Voit,G.M.1995,ApJ,451,498
Bisnovatyi-Kogan,G.S.,&Ruzmaikin,A.A.1974,Ap&SS,28,45 Ogilvie,G.I.,&Livio,M.1998,ApJ,499,329
Bisnovatyi-Kogan,G.S.,&Ruzmaikin,A.A.1976,Ap&SS,42,401 Ogilvie,G.I.,&Livio,M.2001,ApJ,553,158
Blandford,R.D.,&Payne,D.G.1982,MNRAS,199,883 Ohsuga,K.,&Mineshige,S.2011,ApJ,736,2
Campbell,C.G.2003,MNRAS,345,123 Proga,D.2000,ApJ,538,684
Cao,X.2009,MNRAS,394,207 Proga,D.2003,ApJ,585,406
Cao,X.2012,MNRAS,426,2813 Proga,D.,&Begelman,M.C.2003,ApJ,592,767
Cao,X.,&Spruit,H.C.1994,A&A,287,80 Proga,D.,Stone,J.M.,&Kallman,T.R.2000,ApJ,543,686
Cao,X.,&Spruit,H.C.2002,A&A,385,289 Pudritz,R.E.,Ouyed,R.,Fendt,C.,&Brandenburg,A.2007,Protostars
Cao,X.,&Spruit,H.C.2013,ApJ,765,149 andPlanetsV,277
Czerny,B.,Siemiginowska,A.,Janiuk,A.,Nikiel-Wroczyn´ski,B.,& Sakurai,T.1985,A&A,152,121
Stawarz,Ł.2009,ApJ,698,840 Shakura,N.I.,&Sunyaev,R.A.1973,A&A,24,337
Doi,A.,Asada,K.,&Nagai,H.2011,ApJ,738,126 Shields,G.A.1978,Nature,272,706
Doi,A.,Fujisawa,K.,Habe,A.,etal.2006,PASJ,58,777 Shlosman,I.,Vitello,P.A.,&Shaviv,G.1985,ApJ,294,96
Fender,R.P.,Belloni,T.M.,&Gallo,E.2004,MNRAS,355,1105 Spruit,H.C.1996,NATOASICProc.477,EvolutionaryProcessesin
Galeev,A.A.,Rosner,R.,&Vaiana,G.S.1979,ApJ,229,318 BinaryStars,ed.R.A.M.J.Wijers,M.B.Davies,&C.A.Tout
Gu,M.,&Chen,Y.2010,AJ,139,2612 (Dordrecht:Kluwer),249
Guilet,J.,&Ogilvie,G.I.2012,MNRAS,424,2097 Spruit,H.C.2010,TheJetParadigm,ed.T.Belloni(LectureNotesin
Guilet,J.,&Ogilvie,G.I.2013,MNRAS,430,822 Physics,Vol.794;Berlin:Springer),233
Haardt,F.,&Maraschi,L.1993,ApJ,413,507 Sun,W.-H.,&Malkan,M.A.1989,ApJ,346,68
Haardt,F.,Maraschi,L.,&Ghisellini,G.1994,ApJ,432,L95 Takeuchi,S.,Ohsuga,K.,&Mineshige,S.2010,PASJ,62,L43
Koide,S.,Shibata,K.,&Kudoh,T.1999,ApJ,522,727 Vaidya,B.,Fendt,C.,Beuther,H.,&Porth,O.2011,ApJ,742,56
Komossa,S.,Voges,W.,Xu,D.,etal.2006,AJ,132,531 vanBallegooijen,A.A.1989,AccretionDisksandMagneticFieldsin
Konigl,A.,&Pudritz,R.E.2000,ProtostarsandPlanetsIV,759 Astrophysics,Vol.156,ed.G.Belvedere(Dordrecht:Kluwer),99
Laor,A.,&Netzer,H.1989,MNRAS,238,897 vonRekowski,B.,Brandenburg,A.,Dobler,W.,Dobler,W.,&Shukurov,A.
Liu,B.F.,Mineshige,S.,&Ohsuga,K.2003,ApJ,587,571 2003,A&A,398,825
Lovelace,R.V.E.,Rothstein,D.M.,&Bisnovatyi-Kogan,G.S.2009,ApJ, Wu,Q.2009,ApJ,701,L95
701,885 Wu,Q.,Cao,X.,Ho,L.C.,&Wang,D.-X.2013,ApJ,770,31
Lubow,S.H.,Papaloizou,J.C.B.,&Pringle,J.E.1994,MNRAS,267,235 Yuan,W.,Zhou,H.Y.,Komossa,S.,etal.2008,ApJ,685,801
Malkan,M.A.,&Sargent,W.L.W.1982,ApJ,254,22 Zdziarski,A.A.,Skinner,G.K.,Pooley,G.G.,&Lubin´ski,P.2011,
Mos´cibrodzka,M.,Gammie,C.F.,Dolence,J.C.,Shiokawa,H.,&Leung, MNRAS,416,1324
P.K.2009,ApJ,706,497 Zhou,H.-Y.,Wang,T.-G.,Dong,X.-B.,Zhou,Y.-Y.,&Li,C.2003,ApJ,
Mos´cibrodzka,M.,&Proga,D.2009,MNRAS,397,2087 584,147