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Stability of Accretion Disks

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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

- 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

- 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)

- Equilibrium: steady disk structure
- Perturbations to related quantities
- Perturbed equations
- Dispersion relation
- Solutions:
- perturbations growing: unstable
- perturbations damping: stable

- 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:

- Perturbations:
- Wavelength
- Ignore terms of order and comparing with terms of
- Perturbation form
Surface density

Half-thickness

- Perturbed eqs ( )

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

Radiation pressure dominated

Thermally unstable

Viscouslly unstable

- Define
- Dispersion relation

- 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)

- 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)

Piran (1978, ApJ)

- Disk Instability
- Diffusion eq:
- viscous instability:
- Thermal instability:
- limit cycle: A->B->D->C->A...
- Outbursts of Cataclysmic Variables

Smak (1984)

- Typical timescals
- Viscous timescale
- Thermal timescale

- Variation of soft component in BH X-ray binaries

Belloni et al. (1997)

GRS 1915+105

Viscous timescale

- 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

- Define
- Disk is stable to all modes when
- When , all modes are unstable if

- SLE: ion pressure dominates
- Ions lose energy to electrons
- Electrons lose energy for unsaturated Comptonization

--> Thermally unstable!

- 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

- Viscous heating:
- Radiative cooling:
- Advective cooling:
- Thermal stability:
- S-curve:

Slim disk branch

- 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)

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

- Time-dependent equations

- Perturbations
- Perturbed equations

- 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

- Perturbed equation and stability criteria for constant specific angular momentum tori

Dynamically unstable modes

- 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 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

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

- Self-similar solution

- 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)

- 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

- Equations for a 2-T ADAF

- Perturbed equations

- Dispersion relation

- Solutions
- 4 modes: thermal, viscous, 2 inertial-acoustic (O & I - modes)
- 2T ADAF is stable

- 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

- Disk model
- 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