Stability of Accretion Disks
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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. YANG Lantian. Prof. LI Qibin. Content. Why we need to study disk stability

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Stability of Accretion Disks

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Stability of accretion disks

Stability of Accretion Disks

WU Xue-Bing

(Peking University)

[email protected]


Stability of accretion disks

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

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

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


1 why we need to study stability1

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

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

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)


Stability of accretion disks

  • Disk structure (Shakura & Sunyaev 1973)

  • 1. Inner part:

  • 2. Middle part:

  • 3. Outer part:


Shakura sunyaev 1976 mnras

Shakura & Sunyaev (1976, MNRAS)

  • Perturbations:

    • Wavelength

    • Ignore terms of order and comparing with terms of

    • Perturbation form

      Surface density

      Half-thickness

    • Perturbed eqs ( )


Shakura sunyaev 1976 mnras1

Shakura & Sunyaev (1976, MNRAS)

  • Forms of u, h:

  • For the real part of (R),

  • Dispersion relation at <<R


Stability of accretion disks

Radiation pressure dominated

Thermally unstable

Viscouslly unstable


Piran 1978 apj

Piran (1978, ApJ)

  • Define

  • Dispersion relation


Piran 1978 apj1

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 apj2

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)


Stability of accretion disks

Piran (1978, ApJ)


S curve limit cycle behavior

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)


Stability of accretion disks

  • Typical timescals

    • Viscous timescale

    • Thermal timescale

  • Variation of soft component in BH X-ray binaries

Belloni et al. (1997)

GRS 1915+105

Viscous timescale


2 stability studies on accretion disk models1

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


Pringle 1976

Pringle (1976)

  • Define

  • Disk is stable to all modes when

  • When , all modes are unstable if


Pringle 19761

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 models2

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

Abramowicz et al. (1988, ApJ)

  • Viscous heating:

  • Radiative cooling:

  • Advective cooling:

  • Thermal stability:

  • S-curve:

Slim disk branch


Papaloizou pringle instability

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 instability1

Papaloizou-Pringle Instability

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

  • Equilibrium


Papaloizou pringle instability2

Papaloizou-Pringle Instability

  • Time-dependent equations


Papaloizou pringle instability3

Papaloizou-Pringle Instability

  • Perturbations

  • Perturbed equations


Papaloizou pringle instability4

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 instability5

Papaloizou-Pringle Instability

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

Dynamically unstable modes


Papaloizou pringle instability6

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


Stability of accretion disks

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


Advection dominated accretion flow

Advection dominated accretion flow

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


Advection dominated accretion flow1

Advection dominated accretion flow

  • Self-similar solution


Advection dominated accretion flow2

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 flow3

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

Wu (1997b, MNRAS, 292, 113)

  • Equations for a 2-T ADAF


Wu 1997b mnras 292 1131

Wu (1997b, MNRAS, 292, 113)

  • Perturbed equations


Wu 1997b mnras 292 1132

Wu (1997b, MNRAS, 292, 113)

  • Dispersion relation


Wu 1997b mnras 292 1133

Wu (1997b, MNRAS, 292, 113)

  • Solutions

    • 4 modes: thermal, viscous, 2 inertial-acoustic (O & I - modes)

    • 2T ADAF is stable


Discussions

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


Discussions1

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


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