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**Stability of Accretion Disks**

WU Xue-Bing (Peking University)

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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. YANG Lantian Prof. LI Qibin

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**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

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**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

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**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)

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**How to study stability? Equilibrium: steady disk structure**

Perturbations to related quantities Perturbed equations Dispersion relation Solutions: perturbations growing: unstable perturbations damping: stable

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**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)

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**Disk structure (Shakura & Sunyaev 1973)**

1. Inner part: 2. Middle part: 3. Outer part:

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**Shakura & Sunyaev (1976, MNRAS)**

Perturbations: Wavelength Ignore terms of order and comparing with terms of Perturbation form Surface density Half-thickness Perturbed eqs ( )

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**Shakura & Sunyaev (1976, MNRAS)**

Forms of u, h: For the real part of (R), Dispersion relation at <<R

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**Radiation pressure dominated**

Thermally unstable Viscouslly unstable

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Piran (1978, ApJ) Define Dispersion relation

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**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)

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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)

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Piran (1978, ApJ)

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**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)

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**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

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**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

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**Pringle (1976) Define Disk is stable to all modes when**

When , all modes are unstable if

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**Pringle (1976) SLE: ion pressure dominates**

Ions lose energy to electrons Electrons lose energy for unsaturated Comptonization --> Thermally unstable!

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**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

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**Abramowicz et al. (1988, ApJ) Viscous heating: Radiative cooling:**

Advective cooling: Thermal stability: S-curve: Slim disk branch

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**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)

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**Papaloizou-Pringle Instability**

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

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**Papaloizou-Pringle Instability**

Time-dependent equations

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**Papaloizou-Pringle Instability**

Perturbations Perturbed equations

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**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

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**Papaloizou-Pringle Instability**

Perturbed equation and stability criteria for constant specific angular momentum tori Dynamically unstable modes

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**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

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**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

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**Advection dominated accretion flow**

Self-similar solution (Narayan & Yi, 1994, ApJ)

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**Advection dominated accretion flow**

Self-similar solution

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**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)

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**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

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

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

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

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

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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

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**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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