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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)

• local energy balance:
• hydrostatic balance:
• “α” viscosity:

(a)inner region: (b)middle region: (c) outer region: Qvis

+ (r) = Qrad

• (r)

ΩK(r) H(r) = cs(r) ) ( ) ( r p r Tr α

φ

− = ⎪ ⎩ ⎪ ⎨ ⎧ Σ = Σ = L & & ) , , ; ( ) ( ) , , ; ( ) ( α α M M r r M M r H r H p ≈ pradiation, χ ≈ χThomson scattering p ≈ pgas, χ ≈ χThomson scattering p ≈ pgas, χ ≈ χfree free

Vertical Stratification of the Disks

MHD turbulence driven by MRI: most promising candidate for viscosity Dynamical equations of gas, radiation and magnetic field must be solved self-consistently. − dp(z) dz + χ(z)ρ(z) c F(z) − ρ(z)ΩK

2z = 0

− 4πB(z) − cE(z)

( )κ(z)ρ(z) + Qdiss

+ (z) = 0

− dP(z) dz − χ(z) c F(z) = 0 4πB(z) − cE(z)

( )κ(z)ρ(z) − dF(z)

dz = 0 ⎧ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪

momentum equation for gas energy equation for gas momentum equation for radiation energy equation for radiation

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

Local shearing box approximation is used to simulate a small patch of the disk.

r M

Azimuth y Height z Radius x

• H

H 12H

• 12H

2H

32 grids

8H

64 grids

24H

384 grids

shearing periodic boundary periodic boundary

• utflow boundary

α M &

Parameters in α model for the initial condition

300 0.03 0.1 6.62

M /M∗

Edd

/ M M & & α r/r

s

Basic Equations

Radiation MHD in the frequency-averaged flux limited diffusion (FLD) approximation

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

E e p E c F B v t B cE B p v ev t e F cE B v Ev t E z x v F c B j p vv t v v t f P ) 1 ( 4 4 P : 3 2

Rosseland Planck Planck Planck Planck 2 2 Rosseland

= − = ∇ − = = × × ∇ − ∂ ∂ − − ⋅ ∇ − = ⋅ ∇ + ∂ ∂ ⋅ ∇ − − + −∇ = ⋅ ∇ + ∂ ∂ Ω − Ω + × Ω − + × + −∇ = ⋅ ∇ + ∂ ∂ = ⋅ ∇ + ∂ ∂ γ ρ χ λ π ρ κ π ρ κ ρ ρ ρ ρ χ ρ ρ ρ ρ

• LTE: source function = Planck Function
• energy-mean opacity = Planck-mean opacity: κ

E = κ Planck

BPlanck

Basic Equations

3D equations of radiation MHD in the flux limited diffusion (FLD) approximation

• Eddington tensor:
• Eddington factor:
• flux limiter:
• opacity parameter:

f = 1 2 1− f

( )I + 1

2 3 f −1

( )nn, n ≡ ∇E

∇E R ≡ ∇E χRosselandρ I 3 1 P 3 1 lim , 3 1 ) ( lim 1 lim , 1 ) ( lim 3 6 2 ) (

2

E f R cE F f R R R R R R

R R R R

= ⇒ = = = ⇒ = = + + + =

→ → ∞ → ∞ →

λ λ λ

• ptically thin limit
• ptically thick limit

f = λ(R) + λ(R)2R2

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. Numerical dissipation rate is evaluated by solving adiabatic equation simultaneously. (For clarity, radiation and potential energies are not included in the above.)

∂ ∂t 1 2 ρv 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + ∇ ⋅ 1 2 ρv 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ v + pv ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = ∇ ⋅ v

( )p − ˜

Q

kin

∂ ∂t 1 2 B2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + ∇ ⋅ E × B

( )= − ˜

Q

mag

∂ ∂t 1 2 ρv 2 + e + 1 2 B2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + ∇ ⋅ 1 2 ρv 2 + e ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ v + pv + E × B ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = 0 ⇔ ∂e ∂t + ∇ ⋅ ev

( )= − ∇ ⋅ v ( )p + ˜

Q

kin + ˜

Q

mag

( ) ( )p

e t e v v ⋅ ∇ − = ⋅ ∇ + ∂ ∂ Q ~ : numerical

dissipation rate

cooling rate heating rate Qrad

− = F + Ev

Qdiss

+ = ˜

Q

mag + ˜

Q

kin

Thermodynamics

radiation internal kinetic magnetic

± j × B⋅ v ∇ ⋅ P

( )⋅ v

± ∇ ⋅ v

( )p

±κρ 4πB − cE

( )

B E ×

disk surface shearing boundary

χρ c F ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⋅ v F + Ev ˜ Q

kin +

˜ Q

mag +

Initial Condition

• ----- - gravity

− dp dz + χρ c F − ρΩK

2z = 0

− dF dz + Qdiss

+ (z) = 0

F = − c 3χρ dE dz E = aT 4 ⎧ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪

hydrostatic balance energy balance Eddington approximation Tgas = Trad

pressure density acceleration

magnetic radiation gas

Time Evolution of the System MHD Turbulence driven by MRI (T=25-30 orbits)

density magnetic energy dissipation rate radiation energy magnetic field lines

T=0 10 60 orbits

Initial transient statistically steady state ( ~ 5 cooling time )

Hydrostatic Balance

initial

• ----- - gravity
• |z| < 3H (disk body)
• β > 1
• gas pressure grad. ~ gravity
• constant magnetic pressure
• |z| > 3H (disk atmosphere)
• β < 1
• magnetic pressure grad. ~ gravity

density pressure acceleration

magnetic radiation gas

Local Dissipation and Energy Balance

+ diss

Q

stress

rad

F E × B Egasv

• Dissipation occurs mainly inside

the disk body.

• Dissipation rate is roughly uniform

inside the disk body.

• Dissipation distribution well agrees

with stress distribution.

• Dissipated energy is transferred to

the disk surface by radiation diffusion.

dissipation rate energy flux

Temperature Location of Photosphere

Eddington factor

f(y,z) f(x,z) f=1/3 photosphere (f=1/2) photosphere (f=1/2)

Gas and radiation well couples inside the disk body.

temperature

gas radiation initial

Alpha Value

Log10 ( β ) Log10 ( α ) β = Pgas / Pmag β > 1

inside disk

β < 1

• utside disk

t = 40

α = Trφ / Ptotal

• disk body ( β > 1 )

α ~ 0.3

• disk atmosphere ( β < 1 )

α ~ β−1 αmag = Trφ / Pmag αmag ~ 0.3 in the entire region

Summary

We calculate the vertical structure of disks where heating by dissipation of MRI-driven MHD turbulence is balanced by radiative cooling. photosphere (f = 1/2) photosphere (f = 1/2) f = 1/3

disk body ( β > 1) atmosphere ( β < 1 ) atmosphere ( β < 1 )

• optically thick
• gas-supported
• constant magnetic pressure
• thermal equilibrium
• roughly-constant dissipation
• α ~ β-1

z

7.5H 3H

• 3H
• 7.5H
• optically thick
• magnetically supported
• non thermal-equilibrium
• little dissipation
• α ~ 0.3