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Thanks to three professors who helped me a lot in studying
accretion disks in last 20 years
Prof. LU Jufu
Prof. LI Qibin
Prof. YANG Lantian
Content
• Why we need to study disk stability
• Stability studies on accretion disk models– Shakura-Sunyaev disk– Shapiro-Lightman-Eardley disk– Slim disk– Advection dominated accretion flow
• Discussions
1. Why we need to study stability?
• An unstable equilibrium can not exist for a long time in nature
• Some form of disk instabilities can be used to explain the observed variabilities (in CVs, XRBs, AGNs?)
• Disk instability can provide mechanisms for accretion mode transition
unstable
stable
• Some instabilities are needed to create efficient mechanisms for angular momentum transport within the disk (Magneto-rotational instability (MRI); Balbus & Hawley 1991, ApJ, 376, 214)
1. Why we need to study stability?
How to study stability?
• Equilibrium: steady disk structure
• Perturbations to related quantities
• Perturbed equations
• Dispersion relation
• Solutions:– perturbations growing: unstable– perturbations damping: stable
2. Stability studies on accretion disk models
• Shakura-Sunyaev disk– Disk model (Shakura & Sunyaev 1973, A&A,
24, 337): Geometrically thin, optically thick, three-zone (A,B,C) structure, multi-color blackbody spectrum
– Stability: unstable in A but stable in B & C • Pringle, Rees, Pacholczyk (1973)• Lightman & Eardley (1974), Lightman (1974)• Shakura & Sunyaev (1976, MNRAS, 175, 613)• Pringle (1976)• Piran (1978, ApJ, 221, 652)
• Disk structure (Shakura & Sunyaev 1973)
1. Inner part:
2. Middle part:
3. Outer part:
ffesgr PP ,
16/ 212/ 21 3/ 21 4/ 21 1/ 4
1612 124 (km), [1 / ]inR M M f f R R
ffesrg PP ,2/3
8 1/3 8/31623 12.5 10 (cm)R M M f
esffrg PP ,
s[1 / ], c3 in
MR R H
11/ 23
1 ( )2
inR
RV
R R
1/ 2, ( / )RV V V GM R
2 ( 2 )RM RV H
3
3( ) [1 / ]
8 in
GM MQ D R R R
R
2d
in
GM ML
R
Shakura & Sunyaev (1976, MNRAS)
• Perturbations:– Wavelength – Ignore terms of order and co
mparing with terms of – Perturbation form
Surface density
Half-thickness
– Perturbed eqs ( )
Shakura & Sunyaev (1976, MNRAS)
• Forms of u, h:
• For the real part of (R),
• Dispersion relation at <<R
Piran (1978, ApJ)
• Two solutions for the dispersion relation viscous (LE) mode; thermal mode
• An unstable mode has Re()>0
• A necessary condition for a stable diskThermally stable
Viscously stable (LE mode)
Piran (1978, ApJ)
• Can be used for studying the stability of accretion disk models with different cooling mechanisms
(b and c denote the signs of the 2nd and 3rd terms of the dispersion relation)
S-curve & Limit-cycle behavior• Disk Instability
Diffusion eq:
viscous instability:
Thermal instability:
limit cycle: A->B->D->C->A...
• Outbursts of Cataclysmic Variables
diskinner in the exists ,0/)( dd
diskinner in the exists ,// dTdQdTdQ
1/ 2 1/ 23( )R R
t R R R
Smak (1984)
•Variation of soft component in BH X-ray binaries
Viscous timescale
rvisc VRRt /~/~ 2 •Typical timescals
Viscous timescaleThermal timescale /)/(~ 222 RVct sth
Belloni et al. (1997)
GRS 1915+105
2. Stability studies on accretion disk models
• Shapiro-Lightman-Eardley disk– SLE (1976, ApJ, 207, 187): Hot, two-temperat
ure (Ti>>Te), optically thin, geometrically thick
– Pringle, Rees & Pacholczky (1973, A&A): a disk emitting optically-thin bremsstrahlung is thermally unstable
– Pringle (1976, MNRAS, 177, 65), Piran (1978): SLE is thermally unstable
Pringle (1976)
• SLE: ion pressure dominates
• Ions lose energy to electrons
• Electrons lose energy for unsaturated Comptonization
--> Thermally unstable!
2. Stability studies on accretion disk models
• Slim disk– Disk model: Abramowicz et al. (1988, ApJ, 3
32, 646); radial velocity, pressure and radial advection terms added
– Optically thick, geometrically slim, radiation pressure dominated, super-Eddington accretion rate
– Thermally stable if advection dominated
Abramowicz et al. (1988, ApJ)
• Viscous heating:
• Radiative cooling:
• Advective cooling:
• Thermal stability:
• S-curve: Slim disk branch
Papaloizou-Pringle Instability
• Movie (Produced by Joel E. Tohline, Louisiana State University's Astrophysics Theory Group)
• Balbus & Hawley (1998, Rev. Mod. Phys.)– One of the most striking and unexpected result
s in accretion theory was the discovery of Papaloizou-Pringle instability
Papaloizou-Pringle Instability• Dynamically (global) instability of thick acc
retion disk (torus) to non-axisymmetric perturbations (Papaloizou & Pringle 1984, MNRAS, 208, 721)
• Equilibrium
Papaloizou-Pringle Instability
• A single eigenvalue equation for which describes the stability of a polytropic torus with arbitrary angular velocity distribution
High wavenumber limit (local approximation), if
Rayleigh (1916) criterion for the stability of a differential rotating liquid
Papaloizou-Pringle Instability
• Perturbed equation and stability criteria for constant specific angular momentum tori
Dynamically unstable modes
Papaloizou-Pringle Instability
• Papaloizou-Pringle (1985, MNRAS): Case of a non-constant specific angular momentum torus
• Dynamical instabilities persist in this case
• Additional unrelated Kelvin-Helmholtz-like instabilities are introduced
• The general unstable mode is a mixture of these two
2. Stability studies on accretion disk models
• Advection dominated accretion flow– Narayan & Yi (1994, ApJ, 428, L13): Optically t
hin, geometrically thick, advection dominated– The bulk of liberated gravitational energy is carri
ed in by the accreting gas as entropy rather than being radiated
qadv=ρVTds/dt=q+ - q-
q+~ q->> qadv,=> cooling dominated (SS disk; SLE disk)
qadv~ q+>>q-,=> advection dominated
Advection dominated accretion flow
• Stability of ADAF– Analyzing the slope and comparing the hea
ting & cooling rate near the equilibrium, Chen et al. (1995, ApJ), Abramowicz et al. (1995. ApJ), Narayan & Yi (1995b, ApJ) suggested ADAF is both thermally and viscously stable (long wavelength limit)
Narayan & Yi (1995b)
Advection dominated accretion flow
• Stability of ADAF– Quantitative studies: Kato, Amramowicz & Ch
en (1996, PASJ); Wu & Li (1996, ApJ); Wu (1997a, ApJ); Wu (1997b, MNRAS)
– ADAF is thermally stable against short wavelength perturbations if optically thin but thermally unstable if optically thick
– A 2-T ADAF is both thermally and viscously stable
Wu (1997b, MNRAS, 292, 113)
• Solutions– 4 modes: thermal,
viscous, 2 inertial-acoustic (O & I - modes)
– 2T ADAF is stable
Discussions
• Stability study is an important part of accretion disk theory– to identify the real accretion disk equilibria– to explain variabilities of compact objects– to provide possible mechanisms for state tran
sition in XRBs (AGNs?)– to help us to understand the source of viscosi
ty and the mechanisms of angular momentum transfer in the AD