The Vertical Structure of Radiation Dominated Accretion Disks Omer Blaes with Shigenobu Hirose and Julian Krolik
Huge Theoretical Uncertainties Have Plagued Us for Years - Even In the Standard Geometrically Thin, Optically Thick Disk Model • How is (turbulent!) dissipation distributed vertically? • What role (if any) do convection and Poynting flux play • in the vertical transport of energy? • Are thermal (and “viscous”) instabilities in the radiation • dominated regime real, or are they merely artifacts of • a bad choice of stress prescription? • How big are fluctuations about equilibrium? • Do magnetic forces play any role in hydrostatic support?
Expectations in Radiation-Dominated Regime Hydrostatic equilibrium: A radially constant disk half-thickness:
Expectations in Radiation-Dominated Regime Radiative equilibrium: A vertically constant dissipation rate per unit volume: After vertical and time-averaging, this must be given by turbulent stress times rate of strain: (Shakura & Sunyaev 1976)
IF we choose the dissipation per unit MASS to be spatially constant as well, then the density must be constant. Ad Hoc Prescription I: The Density Profile The vertical density profile is completely unconstrained! Convective instability! (Bisnovatyi-Kogan & Blinnikov 1977) But perhaps the dissipation per unit mass is not constant???
Ad Hoc Prescription II: The Stress/Pressure Relation The radial midplane temperature and surface density profiles are completely unconstrained unless we adopt, e.g., an alpha prescription for the vertically averaged stress. If we choose , then the disk is thermally and “viscously” unstable if . or But other choices are possible, e.g. (Sakimoto & Coroniti 1981, …, Merloni & Fabian 2002, …) Or perhaps much of the accretion power is dissipated in a corona above the disk? (Svensson & Zdziarski 1994)
No Observational Evidence for Radiation Pressure Thermal/“Viscous” Instabilities - Except Perhaps in GRS 1915+105 -Belloni et al. (1997)
MRI Turbulence Can be Highly Compressible in Radiation Dominated Regime -Turner et al. (2003) Silk damping of turbulence may be important. (Agol & Krolik 1998)
“Photon Bubble Instability” F g -Turner et al. (2005)
z (vertical) y (azimuthal) x (radial) Stratified Shearing Box Simulations of MRI Turbulence Cartesian box corotating with fluid at center of box. Boundary conditions are shearing periodic in x, periodic in y, outflow in z.
Altitude Time Miller & Stone (2000) 25% of magnetic energy generated in turbulence buoyantly rises and dissipates in outer layers. Does this produce a hot corona? Uncertain, as simulation was isothermal.
Thermodynamically consistent, radiation MHD simulations in vertically stratified shearing boxes:
z (vertical) y (azimuthal) x (radial) The Simulation Domain Lx=0.45H, 48 zones Ly=1.8H, 96 zones Lz=8.4H, 896 zones (Apologies to Arthur C. Clarke and Stanley Kubrick)
Energy Balance - NO Thermal Instability! Heating vs. Cooling Radiation, Gas Internal, Magnetic, and Turbulent Kinetic Energies
Heating vs. Cooling Radiation, Gas Internal, Magnetic, and Turbulent Kinetic Energies
Time-Averaged Vertical Dissipation Profile Most of the dissipation is concentrated near midplane.
Turbulence near Midplane is Incompressible -----Silk Damping is Negligible
Density is far from constant with height. Density profile at 200 orbits. Time-averaged density profile.
r/Pgas r/(PtotPgas)1/2 r/Ptot
Vertical Hydrostatic Balance t = 200 orbits
Parker is Clearly Present t = 200 orbits
Large Density Fluctuations at Effective and Scattering Photospheres -upper effective photosphere at t=200 orbits
Photospheric Density Fluctuations Strong density fluctuations, at both scattering and effective photospheres. Strong fluctuations also seen at effective Photosphere in previous simulations with Prad>>Pgas and Prad~Pgas.
Overall Vertical Structure for all Prad/Pgas Regimes Photospheres Pmag>Prad, Pgas Parker Unstable Regions Prad, Pgas>Pmag MRI - the source of accretion power Parker Unstable Regions Pmag>Prad, Pgas Photospheres
Conclusions • No evidence for radiation pressure driven thermal instability, • despite fact that turbulent stresses may be tracking total • pressure (causal direction is OTHER way around, though!). • Dissipation is concentrated near disk midplane, with no • energetically significant corona. • Upper layers are always supported by magnetic fields, even • well beneath the photospheres. (Reflection modelers beware!) • Parker instability dominates, and drives strong density • fluctuations in all radiation/gas pressure regimes. Photon • bubble instability is unresolved in this simulation. • Spectra and color correction factors: magnetic field support • should harden spectra, density fluctuations should soften • spectra. Which dominates?
Caveats and Uncertainties • Simulations are expensive, and much more work needs to be • done to address the following issues: • Numerical convergence with increased resolution and box • size in all three directions (particularly radial and azimuthal). • How does initial magnetic field topology affect things? (We • start with a twisted azimuthal flux tube with net azimuthal • flux, but no net poloidal flux.) • Most of our dissipation is numerical and is captured at the • grid scale. Viscous and resistive scales are therefore • identical (i.e. Prandtl number is unity). Simulations in • non-stratified shearing boxes show that this might matter.
More Details on the (Time-Averaged) Energy Balance Stress Dissipation Rate Divergences of Poynting flux, Gas energy advection, Radiation energy advection, and Radiative diffusion, Total of last three matches dissipation rate.
Flux from top Flux from bottom