Accretion onto Black Hole : Advection Dominated Flow

K. Hayashida

Osaka University

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

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

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

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

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

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

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.

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

Standard Disk Thermal Equilibrium Curve

Double Valued Solutions for fixed

Correspondsto L~0.1LEdd

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

Disk Blackbody Spectra

2din

GMML

r

Total Disk

(see Mitsuda et al., 1984)

1/ 4

3

31 /

8eff in

GMMT r r

r

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

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

Stability (Secular, Thermal)

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

Optically Thick (& High dM/dt) ADAF

ADAF

18

turng

H rm

r r

Optically Thin (& Low Density) ADAF

Depending on Number of Solutions Changes.

Thermal Equilibrium ADAF (Optically Thin)

3 2 2 1/ 2max 2.0 10 ( / )gm r r

Thermal Equilibrium ADAF

ADAF (thick or thin)… H/r ~1

Conical Flow

ADAF (Opticallt Thick and Thin)

Optically Thin, Two Temperature ADAF

i eT T

Optically Thin, Two Temperature ADAF (Model fit to SgrA)

dM/dt is known from observation.

L is too low unless ADAF is considered.

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

Slim 　 Disk 　 Model = Optically Thick ADAF Mineshige et al., 2000

NLS1

Summary