Local Dissipation and Radiative Transfer in Accretion Disks - PowerPoint PPT Presentation

Local Dissipation and Radiative Transfer in Accretion Disks Shigenobu Hirose (Earth Simulator Center) Julian H. Krolik (Johns Hopkins University) James M. Stone (Princeton University)

Radial Structure of Geometrically Thin and Optically Thick Disks One zone model by Shakura and Sunyaev (1973) + ( r ) = Q rad - ( r ) Q vis • local energy balance: Ω K ( r ) H ( r ) = c s ( r ) • hydrostatic balance: “ α ” viscosity: = − α • T r ( r ) p ( r ) φ ⎧ & = α H ( r ) H ( r ; M , M , ) ⎪ & Σ = Σ α ⎨ ( r ) ( r ; M , M , ) ⎪ L ⎩ p ≈ p radiation , χ ≈ χ Thomson scattering (a)inner region: p ≈ p gas , χ ≈ χ Thomson scattering (b)middle region: p ≈ p gas , χ ≈ χ free free (c) outer region:

Vertical Stratification of the Disks ⎧ + χ ( z ) ρ ( z ) − dp ( z ) F ( z ) − ρ ( z ) Ω K 2 z = 0 ⎪ momentum equation for gas dz c ⎪ ( ) κ ( z ) ρ ( z ) + Q diss + ( z ) = 0 − 4 π B ( z ) − cE ( z ) ⎪ energy equation for gas ⎪ ⎨ − χ ( z ) − dP ( z ) F ( z ) = 0 ⎪ momentum equation for radiation dz c ⎪ ⎪ ) κ ( z ) ρ ( z ) − dF ( z ) ( energy equation for radiation 4 π B ( z ) − cE ( z ) = 0 ⎪ ⎩ dz MHD turbulence driven by MRI: most promising candidate for viscosity Dynamical equations of gas, radiation and magnetic field must be solved self-consistently.

Purpose of This Work is to Obtain … Vertical stratification of MRI-driven (gas-dominated) disks using 3D Radiation MHD simulation with FLD approximation, where dissipation process and radiative transfer are explicitly solved. ( Hirose, Krolik and Stone 2006 ) Related works • Miller & Stone (2000): gas-dominated disk (iso-thermal) • Turner (2004): radiation-dominated disk (FLD)

Simulation Domain Height z outflow boundary Local shearing box approximation is used to simulate a small patch of the disk. 12H α 24H 384 grids & M H r M -H Azimuth y periodic boundary 8H Parameters in α model for the initial condition 64 grids Radius x -12H & & α M / M ∗ r / r M / M 2H shearing periodic boundary s Edd 32 grids 6.62 0.1 0.03 300

Basic Equations Radiation MHD in the frequency-averaged flux limited diffusion (FLD) approximation ∂ ρ ( ) + ∇ ⋅ ρ = v 0 ∂ t ( ) ∂ ρ χ ρ ( ) v + ∇ ⋅ ρ = −∇ + × + − ρ Ω × + ρ Ω − ρ Ω 2 2 Rosseland vv p j B F 2 v 3 x z ∂ t c ∂ ( ) ( ) E + ∇ ⋅ = −∇ + κ ρ π − − ∇ ⋅ Ev v : P 4 B cE F ∂ Planck Planck t ∂ ( ) ( ) ( ) e + ∇ ⋅ = − ∇ ⋅ − κ ρ π − ev v p 4 B cE ∂ Planck Planck t ∂ B ( ) − ∇ × × = v B 0 ∂ t λ c = − ∇ F E χ ρ Rosseland = γ − p ( 1 ) e = P f E B Planck - LTE: source function = Planck Function - energy-mean opacity = Planck-mean opacity: κ E = κ Planck

Basic Equations 3D equations of radiation MHD in the flux limited diffusion (FLD) approximation ) nn , n ≡ ∇ E f = 1 ) I + 1 ( ( 2 1 − f 2 3 f − 1 - Eddington tensor: ∇ E f = λ ( R ) + λ ( R ) 2 R 2 - Eddington factor: + 2 R λ = ( R ) - flux limiter: + + 2 6 3 R R 1 λ = = ⇒ = lim ( R ) , lim f 1 F cE optically thin limit → ∞ → ∞ R R R 1 1 1 λ = = ⇒ = optically thick limit lim ( R ) , lim f P E I → → 3 3 3 R 0 R 0 ∇ E R ≡ - opacity parameter: χ Rosseland ρ ZEUS code with FLD module (Turner & Stone 2001) is modified and used. - energy conservation - implicit scheme for diffusion equation: Gauss-Seidel method accelerated by FMG

Energy Dissipation Explicit viscosity and resistivity are not included in the basic equations. Kinetic and magnetic energies that are numerically lost are captured as internal energy. ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ∂ 1 1 ( ) p − ˜ 2 ρ v 2 ⎟ + ∇ ⋅ 2 ρ v 2 v + pv ⎟ = ∇ ⋅ v ⎜ ⎜ ⎟ ⎜ Q ∂ t ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ kin ~ : numerical Q ⎛ ⎞ ∂ dissipation rate 1 ( ) = − ˜ ⎟ + ∇ ⋅ E × B ⎜ 2 B 2 Q ⎝ ⎠ ∂ t mag ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ∂ 2 ρ v 2 + e + 1 2 ρ v 2 + e 1 1 ⎟ + ∇ ⋅ v + pv + E × B ⎟ = 0 ⎜ ⎜ ⎟ 2 B 2 ⎜ ⎝ ⎠ ⎝ ⎠ ∂ t ⎝ ⎠ ⇔ ∂ e ( ) = − ∇ ⋅ v ( ) p + ˜ ∂ t + ∇ ⋅ ev kin + ˜ Q Q mag Numerical dissipation rate is evaluated by solving adiabatic equation simultaneously. ∂ ( ) ( ) p e + ∇ ⋅ = − ∇ ⋅ e v v ∂ t (For clarity, radiation and potential energies are not included in the above.)

Thermodynamics F + Ev disk surface radiation ( ) ± κρ 4 π B − cE ⎛ ⎞ χρ ⎟ ⋅ v ⎜ c F ⎝ ⎠ E × B ˜ + Q internal magnetic mag ( ) ⋅ v shearing boundary ∇ ⋅ P ˜ + Q kin ± j × B ⋅ v ( ) p ± ∇ ⋅ v kinetic − = F + Ev + = ˜ mag + ˜ cooling rate heating rate Q rad Q diss Q Q kin

Initial Condition ⎧ dz + χρ density − dp c F − ρΩ K 2 z = 0 ⎪ hydrostatic balance ⎪ ⎪ − dF + ( z ) = 0 dz + Q diss ⎪ energy balance ⎨ ⎪ F = − c dE Eddington approximation ⎪ 3 χρ dz ⎪ ⎪ E = aT 4 ⎩ T gas = T rad magnetic radiation gas pressure acceleration ------ - gravity

Time Evolution of the System T=0 10 60 orbits Initial statistically steady state transient ( ~ 5 cooling time ) MHD Turbulence driven by MRI (T=25-30 orbits) density magnetic energy dissipation rate radiation energy magnetic field lines

Hydrostatic Balance density • |z| < 3H (disk body) • β > 1 • gas pressure grad. ~ gravity • constant magnetic pressure initial • |z| > 3H (disk atmosphere) • β < 1 • magnetic pressure grad. ~ gravity magnetic radiation gas pressure acceleration ------ - gravity

Local Dissipation and Energy Balance dissipation rate • Dissipation occurs mainly inside the disk body. • Dissipation rate is roughly uniform + inside the disk body. Q diss • Dissipation distribution well agrees with stress distribution. stress energy flux • Dissipated energy is transferred to F rad the disk surface by radiation diffusion. E gas v E × B

Temperature Location of Photosphere Eddington factor temperature f (y,z) f (x,z) gas photosphere radiation ( f =1/2) initial f =1/3 Gas and radiation well couples inside the disk body. photosphere ( f =1/2)

Alpha Value β < 1 β > 1 α = T r φ / P total outside disk inside disk • disk body ( β > 1 ) α ~ 0.3 Log 10 ( α ) • disk atmosphere ( β < 1 ) α ~ β −1 α mag = T r φ / P mag t = 40 α mag ~ 0.3 in the entire region Log 10 ( β ) β = P gas / P mag

Summary We calculate the vertical structure of disks where heating by dissipation of MRI-driven MHD turbulence is balanced by radiative cooling. z photosphere 7.5H ( f = 1/2) atmosphere ( β < 1 ) 3H • optically thick • gas-supported • constant magnetic pressure disk body 0 f = 1/3 • thermal equilibrium ( β > 1) • roughly-constant dissipation • α ~ β -1 -3H • optically thick atmosphere • magnetically supported ( β < 1 ) • non thermal-equilibrium • little dissipation • α ~ 0.3 photosphere -7.5H ( f = 1/2)