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Accretion onto Black Hole : Advection Dominated Flow
K. Hayashida
Osaka University
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Free Fall & Escape Velocity
E=0 (at Infinite) E=1/2v2-GM/r=0 (at r )
v=sqrt(2GM/r) v=Free Fall Velocity=Escape Velocity
v=c … r=rg =2GM/c2 Schwartzshild radius 3km for 1Mo
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Kepler Motion
GM/r2 = v2/r = r2 v=sqrt(GM/r) ; =sqrt(GM/r3) l (angular momentum) = vr = sqrt(GMr) E=1/2 v2 –GM/r = –GM/2r = –(GM)2/2l2
To accrete from r1 to r2, particle must lose E=GM/2r2 – GM/2r1 … e.g. Radiation
Must lose l=sqrt(GMr1) - sqrt(GMr2) …Angular Momentum Transfer
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Viscosity
Viscosity force
dynamical viscotiy kinematic viscosity)
※Viscosity time scale >Hubble time unless turbulence or magnetic field exists.
r-r
r v(r)
v(r-r)
Angular MomemtumFlow
r
dt r
dr
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Effective Potential
Stable Circular Orbit r>=3rg
Binding Energy at r=3rg =0.0572c2
… Mass conversion
efficiency
2 2
2 2 2( ) 1
2eff
GM l lr
r c r r
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Accretion Flow (Disk) Models
Start from Kepler Motion Optically Thick Standard Disk Optically Thin Disk
Irradiation Effect, Relativistic Correction, Advection etc.
Slim Disk (Optically Thick ADAF) Optically Thin ADAF
Start from Free Fall Hydrodynamic Spherical Accretion Flow=Bondi
Accretion … transonic flow
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Standard Accretion Disk Model
Shakura and Sunyaev (1973) Optically Thick Geometrically Thin (r/H<<1) Rotation = Local Keplerian Steady, Axisymmetric Viscosity is proportional to Pressure
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Standard Disk Model-2
Mass Conservation Angular Velocity Angular Momentum Conservation
Hydrostatic Balance
2 rM rv
:
2
SurfaceDensity
H
3/K GM r
2 3( )2 in r
M dl l r T r
dr
22
( )
( )
z
z K
dp pg H
dz HGM H
g H Hr r
One zone approx.
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Standard Disk Model-3
Energy Balance
Equation of State Opacity Viscosity Prescription
2
4
9 3
4 2
2
vis r
rad eff
vis rad
Q T
Q T
Q Q
42
3B
gas rad cH
k aTp p p T
m
3.50es ff es T
r
dt r p
dr -disk model
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Standard Disk Thermal Equilibrium Curve
Double Valued Solutions for fixed
Correspondsto L~0.1LEdd
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Standard Disk Heating and Cooling
Low Temperature
High Temperature
4 8
2
gasvis
radff
pQ T
T TQ
8
4 4
2vis rad
rades
TQ p H
T TQ
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Disk Blackbody Spectra
2din
GMML
r
Total Disk
(see Mitsuda et al., 1984)
1/ 4
3
31 /
8eff in
GMMT r r
r
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Optically Thin Disk
Problem of Optically Thick Disk Fail to explain Hard X-ray, Gamma-ray
Emission Optically Thin Disk (Shapiro-Lightman-
Earley Disk) (1976) Radiation Temperature can reach Tvir
21 12 11
( ) 10 ( )2 10
p pvir
B B g g
GMm m c r rT K
k r k r r
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Optically Thin Disk-2
Energy Balance
Disk
( )3
2B i e
ie Ep
vis ie
ie rad
k T T
m
Q
Q
9
1/ 21/ 2
12
10
0.410
e
i
g
T K
TH r
r K r
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Stability (Secular, Thermal)
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Advection Terms
Energy Equation
Energy Balance
( ) ( )
( )
T v s v pdivv q q
pdiv v v p q q
2
2 2 2
2
2 1/ 23/ 20
1/ 2 1/ 2 1/ 20
2 4
3
4
8 8 8
3 (2 )
adv vis rad
Kadv
vis r k
rad kes es
Q Q Q
M MQ
r r
d dQ rT r M
dr dr
acT c c MQ
H H
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Optically Thick (& High dM/dt) ADAF
ADAF
18
turng
H rm
r r
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Optically Thin (& Low Density) ADAF
Depending on Number of Solutions Changes.
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Thermal Equilibrium ADAF (Optically Thin)
3 2 2 1/ 2max 2.0 10 ( / )gm r r
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Thermal Equilibrium ADAF
ADAF (thick or thin)… H/r ~1
Conical Flow
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ADAF (Opticallt Thick and Thin)
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Optically Thin, Two Temperature ADAF
i eT T
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Optically Thin, Two Temperature ADAF (Model fit to SgrA)
dM/dt is known from observation.
L is too low unless ADAF is considered.
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Presence of Event Horizon : BH vs NS
Luminosity at Quiescence Lmin
NS with Surface
BH without Surface
minL m
2minL m
Narayan et al., Theory of Black HoleAccretion Discs, 1998, p.177
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Slim Disk Model = Optically Thick ADAF Mineshige et al., 2000
NLS1
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Summary