Stability of Accretion Disks
WU Xue-Bing
(Peking University)
wuxb@pku.edu.cn
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
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
unstable
stable
• 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
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)
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)
Pr  Pg ,  es   ff
1. Inner part:
 16 / 21
R12  24 2 / 21M 13/ 21 M 16
f 4 / 21 (km), f  [1  Rin / R ]1/ 4
2. Middle part: Pg  Pr ,  es   ff
 2/ 3
R23  2.5 108 M11/ 3 M 16 f 8/ 3 (cm)
3. Outer part: P  P ,   
g
r
ff
es

M
 
[1  Rin / R ],
   cs H
3
1
Rin 1/ 2 
3 
V  
1 (
)
R
2R 

R



M  2 RVR 
VR  V ,
3GM M
[1 
8 R 3
V  (GM / R)1/ 2


Q   D( R ) 
(  2  H )
Rin / R ]
Ld
GM M

2 Rin
Shakura & Sunyaev (1976, MNRAS)
• Perturbations:
– Wavelength
– Ignore terms of order
comparing with terms of
– Perturbation form
and
Surface density
Half-thickness
– Perturbed eqs (
)
Shakura & Sunyaev (1976, MNRAS)
• Forms of u, h:
• For the real part of (R),
• Dispersion relation at <<R
Radiation
pressure
dominated
Thermally
unstable
Viscouslly
unstable
Piran (1978, ApJ)
• Define
• Dispersion relation
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)
Piran (1978, ApJ)
(b and c denote the signs of the 2nd and 3rd terms of the dispersion
relation)
• Can be used for studying the stability of accretion
disk models with different cooling mechanisms
Piran (1978, ApJ)
S-curve & Limit-cycle behavior
• Disk Instability
Diffusion eq:

3   1/ 2 
1/ 2 

R
(


R
)

t
R R 
R

viscous instability:
d () / d  0, exists in the inner disk
Thermal instability:
dQ  / dT  dQ  / dT , exists in the inner disk
limit cycle: A->B->D->C->A...
• Outbursts
of Cataclysmic Variables
Smak (1984)
•Typical timescals
Viscous timescale
Thermal timescale
tvisc ~ R 2 /  ~ R / Vr
tth ~ (cs2 / V2 ) R 2 /
•Variation of soft component in BH X-ray binaries
Belloni et al.
(1997)
GRS 1915+105
Viscous timescale
2. Stability studies on accretion
disk models
• Shapiro-Lightman-Eardley disk
– SLE (1976, ApJ, 207, 187): Hot, twotemperature (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!
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
Abramowicz et al. (1988, ApJ)
•
•
•
•
•
Viscous heating:
Radiative cooling:
Advective cooling:
Thermal stability:
S-curve:
Slim disk branch
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)
Papaloizou-Pringle Instability
• Dynamically (global) instability of thick
accretion disk (torus) to non-axisymmetric
perturbations (Papaloizou & Pringle 1984,
MNRAS, 208, 721)
• Equilibrium
Papaloizou-Pringle Instability
• Time-dependent equations
Papaloizou-Pringle Instability
• Perturbations
• Perturbed equations
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
Papaloizou-Pringle Instability
• Perturbed equation and stability criteria for
constant specific angular momentum tori
Dynamically
unstable modes
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
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+ - qq+~ q->> qadv,=> cooling dominated
(SS disk; SLE disk)
qadv~ q+>>q-,=> advection dominated
Advection dominated accretion flow
• Self-similar solution (Narayan & Yi, 1994, ApJ)
Advection dominated accretion flow
• Self-similar solution
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)
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
Wu (1997b, MNRAS, 292, 113)
• Equations for a 2-T ADAF
Wu (1997b, MNRAS, 292, 113)
• Perturbed equations
Wu (1997b, MNRAS, 292, 113)
• Dispersion relation
Wu (1997b, MNRAS, 292, 113)
• Solutions
– 4 modes: thermal,
viscous, 2 inertialacoustic (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
transition in XRBs (AGNs?)
– to help us to understand the source of
viscosity 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