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Stability of Accretion Disks

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

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Stability of Accretion Disks

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  1. Stability of Accretion Disks WU Xue-Bing (Peking University) wuxb@pku.edu.cn

  2. 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

  3. 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

  4. 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

  5. 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)

  6. How to study stability? • Equilibrium: steady disk structure • Perturbations to related quantities • Perturbed equations • Dispersion relation • Solutions: • perturbations growing: unstable • perturbations damping: stable

  7. 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)

  8. Disk structure (Shakura & Sunyaev 1973) • 1. Inner part: • 2. Middle part: • 3. Outer part:

  9. Shakura & Sunyaev (1976, MNRAS) • Perturbations: • Wavelength • Ignore terms of order and comparing with terms of • Perturbation form Surface density Half-thickness • Perturbed eqs ( )

  10. Shakura & Sunyaev (1976, MNRAS) • Forms of u, h: • For the real part of (R), • Dispersion relation at <<R

  11. Radiation pressure dominated Thermally unstable Viscouslly unstable

  12. Piran (1978, ApJ) • Define • Dispersion relation

  13. 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)

  14. 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)

  15. Piran (1978, ApJ)

  16. 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)

  17. Typical timescals • Viscous timescale • Thermal timescale • Variation of soft component in BH X-ray binaries Belloni et al. (1997) GRS 1915+105 Viscous timescale

  18. 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

  19. Pringle (1976) • Define • Disk is stable to all modes when • When , all modes are unstable if

  20. Pringle (1976) • SLE: ion pressure dominates • Ions lose energy to electrons • Electrons lose energy for unsaturated Comptonization --> Thermally unstable!

  21. 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

  22. Abramowicz et al. (1988, ApJ) • Viscous heating: • Radiative cooling: • Advective cooling: • Thermal stability: • S-curve: Slim disk branch

  23. 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)

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

  25. Papaloizou-Pringle Instability • Time-dependent equations

  26. Papaloizou-Pringle Instability • Perturbations • Perturbed equations

  27. 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

  28. Papaloizou-Pringle Instability • Perturbed equation and stability criteria for constant specific angular momentum tori Dynamically unstable modes

  29. 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

  30. 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

  31. Advection dominated accretion flow • Self-similar solution (Narayan & Yi, 1994, ApJ)

  32. Advection dominated accretion flow • Self-similar solution

  33. 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)

  34. 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

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

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

  37. Wu (1997b, MNRAS, 292, 113) • Dispersion relation

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

  39. 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

  40. 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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