1. Stability of Accretion Disks
WU Xue-Bing
(Peking University)

2. Thanks to three professors who helped me a lot in studying accretion disks in last 20 years
Prof. LU Jufu
Prof. YANG Lantian
Prof. LI Qibin

3. 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

4. 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

5. 1. Why we need to study stability?
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)

6. How to study stability? Equilibrium: steady disk structure
Perturbations to related quantities
Perturbed equations
Dispersion relation
Solutions:
perturbations growing: unstable
perturbations damping: stable

7. 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)

8. Disk structure (Shakura & Sunyaev 1973)
1. Inner part:
2. Middle part:
3. Outer part:

9. Shakura & Sunyaev (1976, MNRAS)
Perturbations:
Wavelength
Ignore terms of order and comparing with terms of
Perturbation form
Surface density
Half-thickness
Perturbed eqs ( )

10. Shakura & Sunyaev (1976, MNRAS)
Forms of u, h:
For the real part of (R),
Dispersion relation at <<R

11. Radiation pressure dominated
Thermally unstable
Viscouslly unstable

12. Piran (1978, ApJ)
Define
Dispersion relation

13. 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 disk
Thermally stable
Viscously stable (LE mode)

14. 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)

15. Piran (1978, ApJ)

16. S-curve & Limit-cycle behavior
Disk Instability
Diffusion eq:
viscous instability:
Thermal instability:
limit cycle: A->B->D->C->A...
Outbursts of Cataclysmic Variables
Smak (1984)

17. Variation of soft component in BH X-ray binaries
Typical timescals
Viscous timescale
Thermal timescale
Variation of soft component in BH X-ray binaries
Belloni et al. (1997)
GRS
Viscous timescale

18. 2. Stability studies on accretion disk models
Shapiro-Lightman-Eardley disk
SLE (1976, ApJ, 207, 187): Hot, two-temperature (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

19. Pringle (1976) Define Disk is stable to all modes when
When , all modes are unstable if

20. Pringle (1976) SLE: ion pressure dominates
Ions lose energy to electrons
Electrons lose energy for unsaturated Comptonization
--> Thermally unstable!

21. 2. Stability studies on accretion disk models
Slim disk
Disk model: Abramowicz et al. (1988, ApJ, 332, 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

22. Abramowicz et al. (1988, ApJ) Viscous heating: Radiative cooling:
Advective cooling:
Thermal stability:
S-curve:
Slim disk branch

23. Papaloizou-Pringle Instability
Balbus & Hawley (1998, Rev. Mod. Phys.)
One of the most striking and unexpected results in accretion theory was the discovery of Papaloizou-Pringle instability
Movie (Produced by Joel E. Tohline, Louisiana State University's Astrophysics Theory Group)

24. Papaloizou-Pringle Instability
Dynamically (global) instability of thick accretion disk (torus) to non-axisymmetric perturbations (Papaloizou & Pringle 1984, MNRAS, 208, 721)
Equilibrium

25. Papaloizou-Pringle Instability
Time-dependent equations

26. Papaloizou-Pringle Instability
Perturbations
Perturbed equations

27. 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

28. Papaloizou-Pringle Instability
Perturbed equation and stability criteria for constant specific angular momentum tori
Dynamically unstable modes

29. 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

30. 2. Stability studies on accretion disk models
Advection dominated accretion flow
Narayan & Yi (1994, ApJ, 428, L13): Optically thin, geometrically thick, advection dominated
The bulk of liberated gravitational energy is carried 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

31. Advection dominated accretion flow
Self-similar solution (Narayan & Yi, 1994, ApJ)

32. Advection dominated accretion flow
Self-similar solution

33. Advection dominated accretion flow
Stability of ADAF
Analyzing the slope and comparing the heating & 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)

34. Advection dominated accretion flow
Stability of ADAF
Quantitative studies: Kato, Amramowicz & Chen (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

35. Wu (1997b, MNRAS, 292, 113)
Equations for a 2-T ADAF

36. Wu (1997b, MNRAS, 292, 113)
Perturbed equations

37. Wu (1997b, MNRAS, 292, 113)
Dispersion relation

38. Wu (1997b, MNRAS, 292, 113)
Solutions
4 modes: thermal, viscous, 2 inertial-acoustic (O & I - modes)
2T ADAF is stable

39. 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 transition in XRBs (AGNs?)
to help us to understand the source of viscosity and the mechanisms of angular momentum transfer in the AD

40. Discussions Disk model Stability analysis
May not be so simple as we thought
Disk + corona; inner ADAF + outer SSD; CDAF? disk + jet (or wind); shock?
Different stability properties for different disk structure
Stability analysis
Local or global
Effects of boundary condition
Numerical simulations