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Stability of Accretion Disks

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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

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