Steady Models of Optically Thin, Magnetically Supported Black Hole Accretion Disks

0.29 MB·English

by  H. Oda

Steady Models of Optically Thin, Magnetically Supported Black Hole
Accretion Disks
Hiroshi Oda
Graduate School of Science and Technology, Chiba University, 1–33 Yayoi-Cho, Inage-ku, Chiba 263–8522
[email protected]
Mami Machida
Division of Theoretical Astronomy, National Astronomical Observatory of Japan, 2–21–1 Osawa,
Mitaka, Tokyo 181–8588
[email protected]
7 Kenji E. Nakamura
0 Department of Sciences, Matsue National College of Technology, 14–4 Nishiikuma-Cho,
0 Matsue, Shimane 690–8518
2
[email protected]

n and
a Ryoji Matsumoto
J
Department of Physics, Faculty of Science, Chiba University, 1–33 Yayoi-Cho, Inage-ku, Chiba 263–8522

3 [email protected]
2

(Received hreception datei; accepted hacception datei)
1
v Abstract
8
5 Weobtainedsteadysolutionsofopticallythin,singletemperature,magnetizedblackholeaccretiondisks
6 assumingthermalbremsstrahlungcooling. Basedonthe resultsof3DMHDsimulationsofaccretiondisks,
1 weassumedthatthemagneticfieldsinsidethediskareturbulentanddominatedbyazimuthalcomponent.
0 We decomposed magnetic fields into an azimuthally averaged mean field and fluctuating fields. We also
7
assumed that the azimuthally averaged Maxwell stress is proportional to the total pressure. The radial
0
advection rate of the azimuthal magnetic flux Φ˙ is prescribed as being proportional to ̟−ζ, where ̟ is
/
h the radial coordinate and ζ is a parameter which parameterizes the radial variation of Φ˙. We found that
p when accretion rate M˙ exceeds the threshold for the onset of the thermal instability, a magnetic pressure
-
o dominated new branchappears. Thus the thermal equilibrium curve of optically thin disk has a ’Z’-shape
r in the plane of surface density and temperature. This indicates that as the mass accretion rate increases,
t
s a gas pressure dominated optically thin hot accretion disk undergoes a transition to a magnetic pressure
a dominated, optically thin cool disk. This disk corresponds to the X-ray hard, luminous disk in black hole
:
v candidates observed during the transition from a low/hard state to a high/soft state. We also obtained
i global steady transonic solutions containing such a transition layer.
X
Key words: accretion:accretiondisks — black holes — state transition – MHD
r
a
1. Introduction flows (RIAFs), were studied extensively by Narayan, Yi
(1994,1995)andAbramowiczetal.(1995). Abramowiczet
Optically thin, hot accretion disks have been stud- al.(1995)obtainedlocalthermalequilibriumcurvesofop-
ied to explain the X-ray hard state (low/hard state) of tically thin accretiondisks by solving Q =Q +Q ,
vis adv rad
black hole candidates. Thorne, Price (1975) proposed where Q is the radiative cooling rate. They showed
rad
that hard X-rays from Cyg X-1 are produced in the in- that when the accretion rate exceeds a critical accretion
ner optically thin hot disks. Shibazaki, Hoshi (1975) rate, optically thin solution disappears. Above this criti-
studied the structure and stability of optically thin hot cal accretion rate, the hard X-ray emitting optically thin
accretion disks. Eardley, Lightman, Shapiro (1975) and hot accretion disk undergoes a transition to an optically
Shapiro, Lightman, Eardley (1976) constructed a model thick cool disk which emits soft X-rays. The latter corre-
ofopticallythin,twotemperatureaccretiondisksinwhich sponds to the high/soft state (or thermal state) of black
ion temperature is higher than the electron tempera- hole candidates.
ture. Ichimaru (1977) pointed out the importance of Theimportanceofthe magneticfieldsontransitionbe-
advection of energy in hot accretion flows and obtained tween an optically thin hot disk and an optically thick
steady models of optically thin disks by equating the ra- cool disk was discussed by Ichimaru (1977). He sug-
dial heat advection Q and the viscous heating Q . gested that magnetic pressure p is limited below the
adv vis mag
Suchflows,whichnowarecalledadvectiondominatedac- gas pressure p because magnetic flux will escape from
gas
cretion flows (ADAFs) or radiatively inefficient accretion thediskbybuoyancy. However,thebuoyantescapeofthe
1
2 Astronomical Society of Japan
magnetic flux can be suppressed in strongly magnetized Recently, Machida, Nakamura, Matsumoto (2006) car-
disks. Shibata, Tajima, Matsumoto (1990) carried out ried out global three-dimensional MHD simulations of
magnetohydrodynamic (MHD) simulations of the buoy- black hole accretion disks including optically thin radia-
ant escape of the magnetic flux due to the Parker in- tive cooling. They started the simulation from a radia-
stability (Parker 1966) and showed that when the disk tively inefficient torus threaded by weak toroidal mag-
is dominated by magnetic pressure, it can stay in low-β netic fields. As MRI grows, an optically thin, hot accre-
state (β =p /p <1) because the growth rate of the tion disk is formed by transporting angular momentum
gas mag
Parkerinstabilityisreducedinlow-βdisksduetothemag- efficiently. They found by simulation including thermal
netic tension. Mineshige, Kusunose, Matsumoto (1995) bremsstrahlung cooling that when the density of the ac-
suggested that low-β disk emits hard X-rays. Pariev, cretion disk exceeds the critical density, cooling instabil-
Blackman, Boldyrev (2003) obtained steady solutions of ity takes place in the disk. The disk shrinks in the verti-
optically thick, magnetic pressure dominated disk but caldirectionwithalmostconservingthetoroidalmagnetic
theydidnotconsiderthe opticallythinsolutionincluding flux. They demonstrated that as the disk shrinks, mag-
advective cooling. Begelman, Pringle (2006) constructed netic pressure exceeds the gas pressure because the mag-
a model of optically thick low-β disk including radiation netic pressure increases due to flux conservation, mean-
pressure. while the gas pressure decreases due to cooling. When
Since magnetic fields contribute both to the angular the magnetic pressure becomes dominant, the disk stops
momentumtransportanddiskheating,weneedtounder- shrinking in the vertical direction because the magnetic
stand how magnetic fields are amplified and maintained pressure supports the disk. They found that the disk
inthedisk. Inconventionaltheoryofaccretiondisks(e.g., evolvestowardaquasi-equilibriumcoolstate. Duringthis
Shakura, Sunyaev 1973), phenomenological α-viscosity is transition from hot state to cool state, the disk remains
invoked. However, the physical mechanism which en- optically thin. They explained this quasi-steady state by
ables angular momentum transport efficient enough to equating the magnetic heating rate Q+ and the radia-
explain the activities of accretion powered sources such tive cooing rate Q assuming that the Maxwell stress is
rad
as dwarf nova was unknown until Balbus, Hawley (1991) proportionalto the total pressure and that the azimuthal
pointed out the importance of the magneto-rotationalin- magnetic flux B H is conserved during the transition,
ϕ
h i
stability (MRI) in accretion disks. MRI can excite and where B is the mean azimuthal magnetic field, and H
ϕ
h i
maintain magnetic turbulence. The Maxwell stress gen- is the half thickness of the disk.
erated by the turbulent magnetic fields efficiently trans- The purpose of this paper is to answer why hardX-ray
ports angular momentum and enables accretion of the emitting optically thin disks existabovethe criticallumi-
disk material. The growth and the saturation of MRI nosity. We show steady solutions of optically thin black
hasbeenstudiedbylocalthree-dimensionalMHDsimula- hole accretion disks by taking into account the magnetic
tions (e.g., Hawley et al. 1995, 1996; Matsumoto, Tajima fields,radiativecooling,andadvectionofenergyandmag-
1995;Brandenburgetal.1995;Sano,Inutuska2001;Sano netic fields. In section 2, we present basic equations. In
etal.2004)andbyglobalthree-dimensionalMHDsimula- section3,weobtainlocalthermalequilibriumcurves. We
tions(e.g.,Matsumoto1999;Hawley2000;Machidaetal. presentthe globalsteady solutions in section4. Section5
2000;Hawley,Krolik2001;Hawley,Krolik2002;Machida, is devoted for summary and discussion.
Matsumoto 2003). These global three-dimensional MHD
simulations of radiatively inefficient accretion disks indi- 2. Models and Assumptions
cate that the amplification of magnetic fields saturates
2.1. Basic Equations
when β 10 except in the plunging region of black hole
∼
accretion disk and in the disk corona, and that the disk We extended the basic equations for one-dimensional
approaches to a quasi-steady state. In this quasi-steady steady,opticallythinblackholeaccretionflows(e.g.,Kato
state, the average ratio of the Maxwell stress to gas et al. 1998) by incorporating the magnetic fields. We
pressure α , which corresponds to the α-parameter in adopt cylindrical coordinates (̟,ϕ,z). General relativis-
SS
the conventionalaccretiondisk theory(Shakura,Sunyaev tic effects are simulated by using the pseudo-Newtonian
1973), is 0.01 0.1 (e.g., Hawley 2000; Hawley, Krolik potential ψ = GM/(r r ) (Paczynsky, Wiita 1980),
s
∼ − − −
2001; Machida, Matsumoto 2003). where G is the gravitational constant, M is the mass
RXTEobservationsofstatetransitionsingalacticblack of the black hole (we assume M =10M⊙ in this paper),
hole candidates revealed that some black hole candidates r=(̟2+z2)1/2, and r =2GM/c2 is the Schwarzschild
s
(e.g., GX339-4, XTE J1550-564, and XTE J1859+226) radius.
stay in X-ray hard states even when their luminosities We start from the resistive MHD equations
exceed 20% of the Eddington luminosity (e.g., Done,
Gierlin´ski 2003; Gierlin´ski, Newton 2006). When α<0.1 ∂ρ
asindicatedbyglobalthree-dimensionalMHDsimula∼tions + (ρv)=0 , (1)
∂t ∇·
ofblackholeaccretionflowsandtheenergyconversionef-
ficiency η 0.1, their luminosities are higher than the
e ∼ ∂v j B
critical luminosity above which optically thin steady hot ρ +(v )v = ρ ψ p + × , (2)
gas
solutions disappear. (cid:20)∂t ·∇ (cid:21) − ∇ −∇ c
Magnetically Supported Accretion Disk 3
∂v ∂v ∂ψ 1∂p
z z tot
ρv +ρv = , (8)
̟ z
∂̟ ∂z −∂z −ρ ∂z
∂(ρǫ)
+ [(ρǫ+p )v] (v )p =q+ q− , (3)
gas gas
∂t ∇· − ·∇ −
∂ v ∂
̟
[(ρǫ+p )v ]+ (ρǫ+p )+ [(ρǫ+p )v ]
gas ̟ gas gas z
∂̟ ̟ ∂z
∂B 4π
= v B ηj , (4) ∂p ∂p
∂t ∇×(cid:18) × − c (cid:19) v gas v gas =q+ q− ,(9)
̟ z
− ∂̟ − ∂z −
whereρisthedensity,v isthevelocity,B isthemagnetic
where p =p +p is the total pressure and p =
field, j=c B/4π is the currentdensity, p = ρT/µ tot gas mag mag
∇× gas ℜ B2 /8π is the azimuthally averagedmagnetic pressure.
is the gas pressure, T is the temperature, ǫ=3 T/2µ is h ϕi
ℜ Global three-dimensional MHD simulations showed
the internal gas energy, is the gas constant, µ is the
mean molecular weight (wℜe assumed to be 0.617), q+ is that, in radiatively inefficient disks, azimuthally aver-
aged total stress inside the disk is dominated by the ̟ϕ-
the heating rate, q− is the radiativecooling rate,and η is
componentoftheMaxwellstress t B B /4π and
the resistivity. h ̟ϕi∼h ̟ ϕ i
that the average ratio of the Maxwell stress to the mag-
Global three-dimensional MHD simulations (e.g.,
netic pressure, α B B /4π / p , is nearly con-
Matsumoto 1999, Hawley 2000, Machida et al. 2000; m≡−h ̟ ϕ i h magi
stant( 0.2 0.5;e.g.,Hawley,Krolik2001,2002;Pessah
Hawley, Krolik 2001, 2002; Machida, Matsumoto 2003; ∼ −
et al. 2006). Since β 10 inside the disk except in the
Machida et al. 2004,2006) showedthat in radiatively in- ∼
plungingregion,α B B /4π /p 0.01 0.1(e.g.,
efficient accretion disks the amplified turbulent magnetic B≡−h ̟ ϕ i tot∼ −
Hawley, Krolik 2001). Machida, Nakamura, Matsumoto
fields saturates when β = p /p 10 except in the
plunging region and in the dgiassk cmoargon∼a. Inside the disk, (2006) showed that αB∼0.05−0.1 in cooling dominated
magnetic pressure supporteddisk. According to their nu-
the azimuthalcomponentofmagnetic fielddominatesthe
merical results, we assume that the azimuthally averaged
poloidal component.
̟ϕcomponentoftheMaxwellstressisproportionaltothe
Machida, Nakamura, Matsumoto (2006) carried out
total pressure ( B B /4π = α p ). Thus, equation
three-dimensionalMHDsimulationofblackholeaccretion h ̟ ϕ i − B tot
(7) becomes
disksincludingopticallythinradiativecoolingandshowed
that when the mass accretion rate exceeds the threshold
∂v ∂v ρv v
fortheonsetofthecoolinginstability,amagneticallysup- ρv ϕ +ρv ϕ + ̟ ϕ
̟ z
porteddisk isformed. The magneticpressureexceedsthe ∂̟ ∂z ̟
gas pressure because magnetic fields are amplified due to = 1 ∂ ̟2( α p ) + ∂ hBϕBzi . (10)
the vertical contraction of the disk, meanwhile the gas ̟2∂̟ − B tot ∂z(cid:18) 4π (cid:19)
(cid:2) (cid:3)
pressure decreases due to cooling.
We note that in the turbulent accretion disk, the av-
We have to take into account magnetic fields to study
erage of the radial component of magnetic fields, B ,
theevolutionofaccretiondisksinthisregime. Wedecom- h ̟i
is small because positive and negative B cancel out.
pose the magnetic fields into the axisymmetric toroidal ̟
On the other hand, the product of the radial and the
component (mean field) B¯ = B ˆe and the fluctuat-
ing fields δB=δB̟ˆe̟+δBϕˆehϕ+ϕiδBϕzˆez and decompose tnoorno-ildinaelacromstpatoene(nsetes,fiBg̟urBeϕ1,)d.oTeshunso,twcehacnagnensoigtnniengltehcet
the velocity into the mean velocity v¯=(v ,v ,v ) and
̟ ϕ z B B term.
fluctuating velocity δv. Here denotes the azimuthal h ̟ ϕi
h i
averageandB =δB , B = B +δB ,andB =δB . 2.2. Heating and Cooling Rates
̟ ̟ ϕ ϕ ϕ z z
h i
We assume that the fluctuating components vanish when
The other key factor is the heating rate expressed as
azimuthally averaged ( δv = δB =0). Thus, the az-
imuthally averagedmaghnetiic fiheld aind velocity are B = qv+is =t̟ϕ̟(dΩ/d̟) in conventional theory, where Ω is
h i the angularvelocity. Meanwhile, three-dimensionalMHD
B ˆe and v =v¯, respectively.
h ϕi ϕ h i simulationsindicatethatthetotaldissipativeheatingrate
Let us derive azimuthally averagedequations assuming
(a major part of dissipation is due to the thermalization
thatthediskisinasteadystate. Weassumethat B +
|h ϕi of magnetic energy via magnetic reconnection) is q+
δ(1B)ϕ-|≫(3)|δaBn̟d|i,g|nδoBrzin|.gBtyheazsiemcountdhaollrydearvetrearmginogfeδqBu̟atiaonnds thBhr̟ouBgϕh/o4uπti̟the(ddΩis/kd̟(e).ga.,nHdirtohsaetetthaisl.d2i0s0s6ip)a.tion occu∼rs
δB , we obtain
z We assume the magnetic heating as the heating mech-
∂ ∂ anism in the disk and set the heating term as follows:
(̟ρv )+ (ρv )=0 , (5)
̟ z
̟∂̟ ∂z
B B dΩ dΩ
∂v ∂v ρv2 ∂ψ ∂p B2 q+= h ̟ ϕi̟ = αBptot̟ . (11)
ρv ̟+ρv ̟ ϕ = ρ tot+h ϕi ,(6) 4π d̟ − d̟
̟ z
∂̟ ∂z − ̟ − ∂̟− ∂̟ 4π̟
As the cooling mechanism, we assume the radiative
cooling by the optically thin thermal bremsstrahlung
∂v ∂v ρv v
ϕ ϕ ̟ ϕ emission. Hence, the cooling rate is expressed as,
ρv +ρv +
̟ z
∂̟ ∂z ̟
q−=6.2 1020ρ2T1/2 ergs s−1 cm−3. (12)
= 1 ∂ ̟2hB̟Bϕi + ∂ hBϕBzi , (7) ×
̟2∂̟(cid:20) 4π (cid:21) ∂z(cid:18) 4π (cid:19)
4 Astronomical Society of Japan
2.3. Vertically Integrated Equations is the advective cooling rate, where
We assume that the temperature T, β(=p /p ), 3 dlnT dlnΣ dlnH
gas mag ξ= + , (21)
radial velocity v̟, and specific angular momentum ℓ(= −2dln̟ dln̟ − dln̟
̟vϕ) are independent of z. Machida et al. (2000) and and
Miller, Stone (2000) showed that the magnetic loops ∞ dΩ
emerge from the disk and form coronal magnetic loops. Q+= q+dz= α W̟ , (22)
B
In this paper, we focus on the disk and ignored the low Z−∞ − d̟
densityandlowβ corona. Weassumehydrostaticbalance ∞
Q−= q−dz=6.2 1020ρ2T1/2√πH (23)
intheverticaldirectionandthereforeignoretheleft-hand Z−∞ × 0
side of the z-component of momentum equation. Under
these assumptions, the surface density Σ, the vertically aretheheatingandcoolingratesintegratedinthevertical
integrated pressure W, and the half thickness of the disk direction.
H are given by 2.4. Prescription of the Advection Rate of the Toroidal
∞ Magnetic Flux
Σ= ρdz=√2πρ (̟)H , (13)
0
Z−∞ Thebasicequationsofthesteady,magnetizedaccretion
∞ T disks can be closed by specifying the radial distribution
W = ptotdz= ℜ (1+β−1)Σ , (14) of magnetic field. By azimuthally averagingequation (4),
Z−∞ µ
we obtain
1/2 1/2
1 W 1 T
H = = ℜ 1+β−1 , (15)
ΩK(cid:18)Σ (cid:19) ΩK(cid:20) µ (cid:0) (cid:1)(cid:21) ∂hBϕiˆeϕ = (v¯ B ˆe )
ϕ ϕ
∂t ∇× ×h i
where ρ (̟) is the equatorial density and Ω =
0 K
(GM/̟)1/2/(̟ r ) is the Keplerian angular speed. + δv δB + (η Bϕ ˆeϕ) ,(24)
s ∇×h × i ∇× ∇×h i
−
We now integrate other basic equations in the verti- where the second term on the right-hand side is the dy-
cal direction. Since the density vanishes at the disk sur- namo term and the last term is the magnetic diffusion
face,theverticallyintegratedequationofcontinuityisex- term. If we neglect these terms and assume that the disk
pressed as is in a steady state, we have
M˙ = 2π̟Σv̟ , (16) ∂ ∂
− 0= (v B ) (v B ) . (25)
z ϕ ̟ ϕ
where M˙ is the accretion rate. −∂z h i −∂̟ h i
The ̟-component of the vertically integrated mo- Assuming that the azimuthally averagedtoroidal mag-
mentum equation can be obtained by using B 2 = netic fields vanish at the disk surface, we can integrate
ϕ
8πp β−1=8πβ−1p / 1+β−1 , h i equation (25) in the vertical direction and obtain,
gas tot
(cid:0) (cid:1) ∞
v̟dv̟ + 1 dW = Φ˙ ≡Z−∞v̟hBϕidz=constant . (26)
d̟ Σ d̟
ℓ2 ℓ2 W dlnΩ 2 β−1 W The radial advection rate of the toroidal magnetic flux
− K K , (17) Φ˙ (hereafter we call it flux advection rate) can be evalu-
̟3 − Σ d̟ −̟1+β−1 Σ
ated by completing the vertical integration as
where ℓ = ̟2Ω is the Keplerian angular momentum
K K
andthesecondtermontheright-handsideisacorrection
Φ˙ = v B (̟)√4πH , (27)
resulting from the fact that the radial component of the − ̟ 0
gravitational force changes with height ( see Matsumoto where
et al. 1984).
B (̟)= B (̟;z=0)
Assuming that the last term of the ϕ-component of 0 h ϕi
momentum equation vanishes at the disk surface, the ϕ- =25/4π1/4( T/µ)1/2Σ1/2H−1/2β−1/2 (28)
ℜ
component of vertically integrated momentum equation
is the azimuthally averaged equatorial toroidal magnetic
is field. In accretion disks, Φ˙ is not always conserved be-
M˙ (ℓ ℓ )=2πα ̟2W , (18) cause Φ˙ can change with radius by the presence of the
in B
−
dynamo term and the magnetic diffusion term in equa-
where ℓ is the specific angular momentum swallowedby
in tion (24). According to Machida, Nakamura, Matsumoto
the black hole.
(2006), Φ˙ ̟−1 in the quasi steady state. Based on this
The vertically integrated energy equation is ∝
simulationresult,we parametrizethe dependence ofΦ˙ on
Q =Q+ Q− , (19)
adv ̟ by introducing a parameter ζ as
−
where
M˙ T ̟ −ζ
Qadv= 2π̟2ℜµ ξ (20) Φ˙(̟;ζ,M˙ )≡Φ˙out(M˙ )(cid:18)̟out(cid:19) , (29)
Magnetically Supported Accretion Disk 5
where Φ˙ is the flux advection rate at the outer bound- and equation (16) gives v M˙ /Σ, we obtain M˙ Σ
out ̟
∝ ∝
ary̟=̟ . Whenζ=0,themagneticfluxisconserved. and H constant. When the heating balances with the
out
Whenζ>0,themagneticfluxincreaseswithapproaching radiativ∼e cooling ( Q+ =Q−), since Q+ W and Q−
∝ ∝
to the black hole. Σ2T1/2/H, we find W M˙ Σ2T1/2/H. Using M˙ Σ
and H constant, we o∝btain∝T Σ−2. ∝
3. Local Model The ∼bottom panel of figure∝2 shows the effective
3.1. Local Approximation of Energy Equation 1o0p2t2icρal(̟d)eTp−th3.5cτmeff2g=−1√isκetshκeffopΣa/c2itywohfefrreee-κfrffee=ab6s.o4rp×-
0
Beforeobtainingglobaltransonicsolutions,wesolvethe tion. Note that, when ζ<0.3, the effective optical depth
energyequationQ+=Qadv+Q− locallyatsomespecified exceedsunitywhenM˙ >0.1M˙Edd,thus,opticallythinap-
radius to obtain thermal equilibrium curves of optically proximation is no longer valid. We will extend the low-β
thin black hole accretion flows. solution to optically thick regime in subsequent papers.
We approximated the energy equation in order to be One may wonder why T and β increase with ζ when
solved locally. We evaluated ξ by using results of global Σ is fixed in the low-β branch in figure 2. When Σ
three-dimensional MHD simulations of black hole accre- is fixed, since W (B /8π)H B 2W1/2 on the low-β
0 0
tion flows. According to the results by Machida et al. branch,Q+ W ∼B 4increases∝withζ. Thus,theequilib-
0
(2004), T ̟−1, Σ ̟1/2 , and H ̟, thus ξ = 1. riumtemper∝atur∝eincreasesaccordingtoW Q+=Q−
Furthermo∝re, we assum∝e Ω=ΩK. In ou∝r local model, we T1/2/H T1/2/W1/2 as T W3 Q+3. T∝he plasma∝β
adopt ℓin =ℓK(3rs)=1.8371, ̟ =5rs and αB =0.05 as on the lo∝w-β branch is given∝by β∝ ΣT/W T2/3 when
the fiducial value, hence, the free parameters are now M˙ Σ is fixed. Thus, β increases with ζ∼. ∝
and ζ. Figure3showsthethermalequilibriumcurvesforα =
B
The flux advection rate Φ˙ is determined from equa- 0.01. When α is smaller, the transition from the ADAF
B
tion (29) by specifying Φ˙out. We assume that at ̟ = branch to the low-β branch takes place at smaller Σ be-
̟out = 1000rs, ℓout = ℓK, Tout = Tvirial and βout = 10, cause heating rate is smaller. The bottom panel of figure
where Tvirial = (µc2/3 )(̟out/rs)−1 is the virial tem- 3 indicates that as the accretion rate increases, the disk
ℜ
perature at ̟ = ̟out. Since we fixed T, β, and ℓ at becomesopticallythickbefore themassaccretionrateat-
̟ = ̟out, equations (14) ,(15) and (18) indicate that tains M˙ 0.1M˙Edd when ζ<0.5.
W Σ , H constant and W M˙ , respec- In figu∼re 4, we show the radial dependence of the M˙
out out out out crit
tively.∝ We find Σ∼ M˙ , thus, equati∝on (16) indi- abovewhichthe ADAFbranchdisappearsforζ=0.4and
out
∝
cates that v constant. Therefore, equation (28) α =0.05 (solid), 0.03 (dashed), and 0.01 (dotted). The
̟out B
∼
gives B (̟ ) Σ 1/2 M˙ 1/2 and equation (27) gives critical accretion rate M˙ increases inward. This in-
0 out out crit
Φ˙ M˙ 1/2. ∝ ∝ dicates that as M˙ increases the transition from ADAF
out
∝
branch to low-β branch takes place first at large radius
3.2. Numerical Results
and propagates inward.
In figure 2, we show the thermal equilibrium curves at
̟=5r foropticallythinaccretiondisks. Theequilibrium 4. Global Model
s
curvesareplottedontheΣ M˙ ,Σ T,Σ β andΣ τ
eff
− − − −
plane for ζ =0, 0.4 and 0.8. In the top panel of figure 2, Wenumericallysolvedequations(16)-(19)inwardstart-
M˙ is normalizedby the EddingtonaccretionrateM˙Edd= ing from the outer boundary at ̟ =̟out by specifying
4πGM/(ηeκesc) = 1.64 1019(0.1/ηe)(M/10M⊙) g s−1, ℓout, Tout, βout, and four parameters αB, M˙ , ζ, and ℓin.
×
where the energy conversion efficiency ηe and the elec- Byadjustingtheparameterℓin,weobtainedaglobaltran-
tron scattering opacity κ are taken to be η =0.1 and sonicsolutionwhichsmoothlypassesthesonicpoint(e.g.,
es e
κes=0.34 cm2 g−1, respectively. The equilibrium curves Nakamura et al. 1997). The free parameters are now αB,
consist of three branches; (1) gas pressure supported, ad- M˙ , and ζ. We adopt α =0.05.
B
vectively cooled branch, which is thermally stable, i.e. We locate the outer boundary at ̟ =1000r , and
out s
ADAF branch, (2) gas pressure supported, radiatively imposed the boundary conditions ℓ = 0.45ℓ , T =
out K out
cooled branch, which is thermally unstable (SLE branch T and β =10. We adopt backward-Euler method
virial out
obtained by Shapiro, Lightman, Eardley 1976), and (3) as an integration method (see Matsumoto et al. 1984).
magnetic pressure supported, radiatively cooled branch, In figure 5, we show the radial structure of the disks
which is thermally stable. We call the last branch “low-β whenζ=1andM˙ /M˙ =0.0002(solid),0.2446(dashed),
Edd
branch”. We find that the low-β branchexists evenwhen 0.3181 (dashed-dotted), and 0.4094 (dotted). When M˙
M˙ >0.1M˙Edd, which corresponds to L>0.1LEdd. is small, since Σ is small in the whole disk, the radia-
I∼t maybe worthnoting that onthe lo∼w-β branch,M˙ tive cooling is inefficient. Thus the gas does not cool
Σ and T Σ−2. This dependence can be explained a∝s down. The heating is balancedmainly with the advective
∝
follows. When the magnetic pressure is dominant, W cooling. Furthermore, the flow is sub-Keplerian. When
B 2H (v B H)2/(v 2H) Φ˙2/(v 2H) M˙ /(v 2H∼). M˙ is large, the disk can be divided into four regions;
0 ̟ 0 ̟ ̟ ̟
We fin∝d v̟ H−1/2 because∝equation (18)∝gives W M˙ the outer boundary region ̟ >400rs, the outer transi-
. Since equ∝ation (15) gives H (W/Σ)1/2 (M˙ /Σ∝)1/2 tion region 400rs > ̟ > 150r∼s, the inner low-β region
∝ ∝ ∼ ∼
6 Astronomical Society of Japan
150r >̟ >50r and the inner ADAF region ̟ <50r ically dominated disks, since p p , the Maxwell
s s s tot mag
∼
when ∼M˙ /M∼˙ = 0.3181 . Near the outer bou∼ndary, stress is proportional to the magnetic pressure. In gas
Edd
the accretion flow is advection dominated; Q > Q−. pressure dominated disk, since β 10 inside the disk ex-
adv ∼
However,intheoutertransitionregion,theradiativecool- cept in the plunging region of black hole accretion disks,
ing overcomes the heating as Σ increases. Below this ra- pmag 0.1ptot. Thus our assumption is consistent with
∼
diusT decreasesinward,andβdecreases,thatis,themag- the results of global three-dimensional MHD simulations
netic pressure increases inward due to the magnetic flux byHawley,Krolik(2001,2002),inwhichtheyshowedthat
conservation. At ̟ 100r , T ceases to decrease inward Maxwell stress is proportional to magnetic pressure (see
s
because the heating∼balances with the radiative cooling. also Pessah et al. 2006).
In the inner ADAF region,the advectivecooling Qadv in- In this paper, we assumed that αB does not depend on
creasesasT increases,andbalanceswiththe heatingQ+. β and is uniform for simplicity. Let us discuss the depen-
Theradiativecoolingdecreasesintheinnermostregionas dence of αB on the strength of azimuthal magnetic fields.
Σ decreases. Pessah, Psaltis (2005) studied the stability of accretion
To check the validity of the optically thin approxima- diskswithsuperthermalazimuthalfieldsandshowedthat
tion, we show the radialdistribution ofτ inpanel (f) of MRI is suppressed when the Alfven speed inside the disk
eff
figure 5. In these models, τeff is less than unity. vA exceeds the geometrical mean of the sound speed cs
In the top panel of figure 6, we plotted the Σ M˙ re- and the Keplerian rotation speed vK (i.e. vA √csvK).
lation at ̟=5r obtained by global transonic so−lutions. Thus, the magnetic turbulence will be suppre≥ssed when
s
We also computed the luminosity L by integrating the this criterion is satisfied. In the three-dimensional MHD
radiative cooling all over the disk. In the bottom panel simulation reported by Machida, Nakamura, Matsumoto
of figure 6, we show the dependence of L, which is nor- (2006),fluctuatingmagneticfielddecreasesasβ decreases
malized by the Eddington luminosityL =η M˙ c2= to β 0.1 (see Fig. 5b in Machida et al. 2006) because
Edd e Edd ∼
1.47 1039(M/10M⊙) erg s−1, on the surface density at MRI is suppressed. Even in such disks, however, the an-
̟ =×5r . The optically thin low-β solution extends to gular momentum is transported by the Maxwell stress of
s
Lth∼ed0i.s1kLluEmddi.noNsoittyeitshoantlyw1h%enoΣft∼he1E0dgdicnmgt−o2nalutm̟in∼os5itrys (mMeaanchmidaag,nNetaikcafimeuldrsa,, aMndatαsuBmroetmoa2in0e0d6)α.BW∼e0.0n5ee−d0t.o1
eventhoughthemassaccretionrateattainsM˙ 0.1M˙ . carryoutsimulationsforlongerperiodtostudythedepen-
Edd
The disk is underluminous because the innerm∼ost region denceofαB onβ. WeshouldnotethatαB increasesinthe
of the disk is still advection dominated, that is, radia- innermostplungingregionofthedisk(e.g.,Hawley,Krolik
tively inefficient. The dotted curves in figure 6 (ζ =0.6) 2002; Machida et al. 2004). Thus, this dependence may
show that when 10 g cm−2 < Σ < 100 g cm−2, L in- affect on the global solutions passing through the sonic
creases although M˙ is nearly constant. It indicates that point in the plunging region. However, its effect will be
the disk luminosity increases as the area of the radiative small for local solutions outside the plunging region. We
cooling dominated region increases. The disk luminosity shouldalsonotethatweneglectedthelow-β coronaabove
can exceed 10% of the Eddington luminosity only when accretiondisks. Itwillbe ourfuture workto takeinto ac-
M˙ >M˙ . count the effects of such corona and outflows emerging
Edd
∼ from the disk.
5. Summary and Discussion The heating rate inside the disk originates from the
thermalizationofmagneticenergyviathemagneticrecon-
We obtained steady transonic solutions of optically nectionandisexpressedasq+= B̟Bϕ/4π ̟(dΩ/d̟)=
h i
thin black hole accretion flows by including magnetic αBptot̟(dΩ/d̟). Thisexpressionisbasedonthethree-
−
fields. We theoretically confirmed the simulation result dimensional MHD simulations both in gas pressure dom-
by Machida, Nakamura, Matsumoto (2006) that an opti- inant disks (Hirose et al. 2006) and in magnetic pres-
cally thin hot disk undergoes a transition to an optically sure dominant disks (Machida et al. 2006). If MRI is
thin magnetic pressure dominated disk when the accre- suppressed when vA > √csvK as Pessah, Psaltis (2005)
tion rate exceeds a threshold. In addition to the ADAF suggested, the dissipative heating rate Q+ may decrease.
branchandtheSLEbranchknownforopticallythindisks, Therefore,the disk temperaturewillbecomesmallerthan
we found a low-β branch in which the magnetic heating that shown in figure 3.
balances with the radiative cooling. As we showed in section 3.2, M˙ Σ and T Σ−2 on
∝ ∝
In section 3, we obtained localsolutions for single tem- the low-β branch. However, this dependence is differ-
perature disks dominated by the toroidalmagnetic fields. ent from that given by Machida, Nakamura, Matsumoto
As a consequence, we obtained three thermal equilibrium (2006) in which they derived M˙ Σ1/3 and T Σ−4.
∝ ∝
branches, ADAF, SLE and low-β branches. This difference comes from the fact that in our model
Based on the results of global three-dimensional MHD Φ˙ v̟BH M˙ 1/2 (that is, flux advection rate increases
∝ ∝
simulations by Machida, Nakamura, Matsumoto (2006), as mass accretion rate increases). Meanwhile Machida,
we assumed that the Maxwell stress is proportional to Nakamura, Matsumoto (2006) assumed BH constant.
∼
the total pressure, B B /4π = α p . Here α is a The criticalaccretionrate for the ADAF solutionM˙
̟ ϕ B tot B crit
h i −
parameter corresponding to the α-parameter in conven- increasesasdecreasingtheradiuswhen̟>10r ,andhas
s
tional theory of accretion disks. Note that in magnet- a maximum around ̟ 10r . This fact indicates that in
s
∼
Magnetically Supported Accretion Disk 7
the region ̟>10r , the low-β region propagates inward References
s
as M˙ increases. When M˙ exceeds the maximum of M˙ ,
crit
whole disk will become low β. Abramowicz,M.A.,Chen,X.,Kato,S.,Lasota,J.P.&Regev,
According to our local solutions, optically thin low- O. 1995, ApJ, 438, L37
β branches can exist above the critical accretion rate Balbus, S.A. & Hawley, J. F. 1991, ApJ, 376, 214
M˙ above which the ADAF branch disappears (M˙ Begelman, M. C. & Pringle, J. E. 2006, astro-ph/0612300
crit crit
0.1M˙ when α =0.05). The maximum luminosity o∼f Brandenburg,A.,Nordlund,A.,Stein,R.F.&Torkelsson, U.
Edd B 1995, ApJ,446, 741
the disk staying in the ADAF branch is only 10 % of the
Done, C. & Gierlin´ski, M. 2003, MNRAS,342, 1041
Eddingtonluminosity. ToexplaintheobservedX-rayhard
Eardley, D. M., Lightman, A. P. & Shapiro, S. L. 1975, ApJ,
statewithluminosityashighas20%ofthe Eddingtonlu- 199, L153
minosity, the disk should be kept in optically thin state. Gierlin´ski, M. & Newton, J. 2006, MNRAS,370, 837
Such disks can exist when ζ>0.5 in our models, where ζ Hawley,J.F.,Gammie, C.F.&Balbus,S.A.1995, ApJ,440,
is the parameter whichparam∼eterizes the radialvariation 742
of the advection rate of the toroidal magnetic flux. Hawley,J.F.,Gammie, C.F.&Balbus,S.A.1996, ApJ,464,
We also obtained steady global transonic solutions in- 690
cluding magnetic fields and obtained Σ M˙ relation. Hawley, J. F. 2000, ApJ, 528, 462
− Hawley, J. F. & Krolik, J. H. 2001, ApJ, 548, 348
When ζ 0, we could not obtain solutions above certain
∼ Hawley, J. F. & Krolik, J. H. 2002, ApJ, 566, 164
criticalaccretionrate. The reasonsimply comes fromthe
Hirose, S., Krolik, J. H. & Stone, J. M. 2006, ApJ, 640, 901
fact that we chose the ADAF-like outer boundary condi-
Ichimaru, S.1977, ApJ, 214, 840
tion. This means that there exists no solution connect- Kato, S., Fukue, J. & Mineshige, S. 1998, BLACK-HOLE
ing from such outer boundary to inner region with weak ACCRETION DISKS(Kyoto: KyotoUniversity Press)
magnetic field. When ζ 1, we can obtain solutions for Machida, M., Hayashi, M. R. & Matsumoto, R. 2000, ApJ,
larger M˙ . These solutio∼ns consist of four regions, the 532, L67
outer boundary region, the outer transition region, the Machida, M. & Matsumoto, R.2003, ApJ,585, 429
inner low-β region and the inner ADAF region. Machida,M.,Nakamura,K.E.&Matsumoto, R.2004, PASJ,
56, 671
In this paper, we assumed single temperature plasma
Machida,M.,Nakamura,K.E.&Matsumoto, R.2006, PASJ,
and included only the bremsstrahlung cooling. In mag-
58, 193
netized accretion disks, electrons will also be cooled by
Matsumoto, R., Kato, S., Fukue, J. & Okazaki, A. T. 1984,
synchrotronradiationandinverseComptoneffects. These
PASJ, 36, 71
processes will increase the luminosity of the disk and de-
Matsumoto, R. & Tajima, T. 1995, ApJ,445, 767
crease the critical accretion rate for the transition to the Matsumoto, R., 1999, NUMERICAL ASTROPHYSICS, ed.
low-β disk. In optically thin, hot accretion disks, since S. M. Miyama, T. Tomisaka & T. Hanawa (Amsterdam:
the gas temperature becomes so high while electrons are Kluwer Academic Publishers) 195
cooledbyradiation,theelectrontemperaturecanbecome Miller, K.A. & Stone, J. M. 2000, ApJ, 534, 398
lower than the ion temperature. Such two-temperature Mineshige,S.,Kusunose,M.&Matsumoto,R.1995,ApJ,445,
accretiondiskmodelswerestudiedbyShapiro,Lightman, L43
Nakamura, K. E., Kusunose, M. Matsumoto, R., & Kato, S.
Eardley (1976) and Ichimaru (1977). Nakamura et al.
1997, PASJ,49, 503
(1997) obtained transonic solutions of two temperature
Narayan, R.& Yi,I. 1994, ApJ,428, L13
accretion disks including synchrotron cooling and inverse
Narayan, R.& Yi,I. 1995, ApJ,452, 710
Compton effects. However, plasma β was assumed to be
Paczynsky,B. & Wiita, P. J. 1980, A&A,88, 23
uniform in their paper. In subsequent papers, we would Pariev, V. I., Blackman, E. G. & Boldyrev, S. A. 2003, A&A,
like to obtain solutions of magnetized two temperature 407, 403
disks subjecting to the cooling instability. Parker, E. N. 1966, ApJ, 145, 811
Pessah, M. E., & Psaltis, D.2005, ApJ, 628, 879
TheauthorsthankDrs. M.A.Abramowicz,C.Akizuki, Pessah, M.E.,Chan,C.&Psaltis, D.2006, astro-ph/0610565
C. Done, J. Fukue, S. Hirose, Y. Kato, S. Mineshige, K. Sano, T. & Inustuka,S. 2001. ApJ, 605, L179
Sano,T.,Inustuka,S.,Turner,N.J.&Stone,J.M.2004,ApJ,
Ohsuga, R.A. Remillard, K. Watarai, and S. Miyaji for
605, 321
their valuable comments and discussions. Part of this
Shakura,N. I. & Sunyaev,R. A.1973, A&A,24, 337
work was carried out when R.M. and M.M. attended the
Shapiro, S. L., Lightman, A. P. & Eardley, D. M. 1976, ApJ,
KITP program on ”Physics of Astrophysical Outflows
204, 187
and Accretion Disks” at UCSB. This work was sup-
Shibata,K.,Tajima, T.&Matsumoto,R.1990,ApJ,350,295
portedinpartbytheGrants-in-AidforScientificResearch Shibazaki, N. & Hoshi, R. 1975, Progress of Theoretical
of the Ministry of Education, Culture, Sports, Science Physics, 54, 706
and Technology (RM: 17030003), Japan Society for the Thorne, K. S., Price, R.H. 1975, ApJ, 195, L101
Promotion of Science (JSPS) Research Fellowships for
Young Scientists (MM: 17-1907, 18-1907), and by the
National Science Foundation under Grant No. PHY99-
07949.
8 Astronomical Society of Japan
Fig. 1. A schematic picture of a magnetic field line inside
theaccretiondisk. (a) Axisymmetricdiskthreaded bymean
azimuthal magneticfields. (b)Turbulentdiskwithmeanaz-
imuthalmagnetic fieldsandfluctuating magnetic fields. The
azimuthal average of the radial component of the magnetic
field,hB̟i,issmallbecausepositiveandnegativeB̟ cancel
out. Ontheotherhand,theazimuthalaverageoftheproduct
of the radial and toroidal component, hB̟Bϕi, has a large
negative value because B̟Bϕ does not change sign when
magnetic fields are deformed by nonlinear growth of MRI.
Fig. 2. Local thermal equilibrium curves of accretion disks
at ̟=5rs on the Σ−M˙, Σ−T, Σ−β, Σ−τeff plane for
opticallythinaccretiondiskswithM=10M⊙,αB=0.05and
ζ=0,0.4,0.8. SLEindicatesthecoolingdominatedbranchby
Shapiro,Lightman,Eardley(1976). M˙Edd=4πGM/(ηeκesc)
is the Eddington accretion rate where the energy
conversion efficiency ηe is taken to be ηe = 0.1.
Magnetically Supported Accretion Disk 9
Fig. 4. The radial dependence of M˙crit for ζ = 0.4 and
αB = 0.05 (solid), 0.03 (dashed) and 0.01 (dotted).
Fig. 3. Sameasfigure2butforαB=0.01.
10 Astronomical Society of Japan
Fig. 5. The radial structure of the disk when ζ = 1
and M˙/M˙Edd = 0.0002 (solid), 0.2446 (dashed),
0.3181(dashed-dotted), and 0.4094 (dotted) with
αB = 0.05. Thick solid curve in panel (e) shows
the Keplerian angular momentum distribution.