Stability of Accretion Disks WU Xue-Bing (Peking University) [email protected].

Stability of Accretion Disks

WU Xue-Bing

(Peking University)

[email protected]

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

Thermally unstable

Viscouslly unstable

Radiation pressure dominated

Piran (1978, ApJ)

• Define


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)

Piran (1978, ApJ)

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


• 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)

• Define

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

Pringle (1976)

• 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, 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


• 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


• 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

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


• Self-similar solution


• 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)


• 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)

• Equations for a 2-T ADAF

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

Wu (1997b, MNRAS, 292, 113)


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

Discussions• 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

Stability of Accretion Disks WU Xue-Bing (Peking University) [email protected].

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