Standard Accretion Disks Driven by MRI Stress comparison with the - PowerPoint PPT Presentation

Standard Accretion Disks Driven by MRI Stress — comparison with the α -viscosity model — Shigenobu Hirose (Institute for Research on Earth Evolution, JAMSTEC) collaboration with Omer Blaes (UCSB) and Julian Krolik (JHU) 2009/06/02 Workshop on MRI in Protoplanetary Disks Center for Planetary Science, Kobe University

Standard Accretion Disks (Shakura & Sunyaev 1973) definition ◮ optically thick ◮ geometrically thin: H ≪ R (nearly Keplerian: v sound ≪ R Ω K ) ◮ vertical hydrostatic balance ◮ local thermal balance: Q + diss (r) = Q − rad (r) z vertical M radial H r R

Timescales in Standard Accretion Disks local structure ◮ dynamical time: t dynamical ≡ H/v sound ◮ thermal time: t thermal ≡ E thermal /Q ± global structure ◮ inflow time: t inflow ≡ R/v r sharp difference in the timescales t orbital ∼ t dynamical < t thermal ≪ t inflow

Basic Equations of the α Model (Shakura & Sunyaev 1973) local structure (one zone approximation) H = 2 P c hydrostatic balance ΣΩ 2 K 4 T rφ Ω K = 2 acT 4 − 3 c thermal balance 3 κ Σ P c = a c + Σ k B T c 3 T 4 equation of state 2 µH T rφ = − 2 HαP c α prescription Σ = constant t dynamical , t thermal ≪ t inflow

Basic Equations of the α Model (Shakura & Sunyaev 1973) (continued) local solution H = H( Σ , Ω K , α) P c = P c ( Σ , Ω K , α) T c = T c ( Σ , Ω K , α) T rφ = T rφ ( Σ , Ω K , α) global structure ∂ Σ ∂t + 1 ∂ ∂r (r Σ v r ) = 0 mass conservation r Σ v r Ω K r 2 = − 2 ∂ ( ) r 2 T rφ angular momentum conservation ∂r

Thermal Stability of the α Model (Shakura & Sunyaev 1976) equation for δT c ( ≡ T c − T c | Q + = Q − ) ( � � ) ∂ log Q + − ∂ log Q − ∂δT c � � � � ∝ δT c � � ∂t ∂ log T c ∂ log T c � � Σ Σ note: Σ is assumed to be constant since t thermal ( ≪ t inflow ) . log Q + radiation dominated log Q − unstable ∼ T 8 log Q c ∼ T 4 c ∼ T c gas dominated stable log T c

Inflow Stability of the α Model (Lightman & Eardley 1974) diffusion equation for δ Σ ( ≡ Σ − Σ steady state ) � ∂δ Σ ∝ ∂ log T rφ � ∂δ Σ � � ∂t ∂ log Σ ∂r 2 � Q + = Q − note: Q + = Q − is assumed since t inflow ( ≫ t thermal ) . ∼ Σ − 1 radiation dominated unstable log T r φ gas dominated ∼ Σ 5 / 3 stable log Σ

Outline of This Work modern view of stress in accretion disks ◮ MHD turbulence driven by magneto-rotational instability (MRI) modern model of standard accretion disks ◮ vertical structure with local dissipation of turbulence and radiative transport ◮ 3D radiation MHD simulations in a stratified local shearing box ◮ local equilibrium solution in an averaged sense H = H( Σ , Ω K ) P c = P c ( Σ , Ω K ) T c = T c ( Σ , Ω K ) T rφ = T rφ ( Σ , Ω K ) ⇐ thermal equilibrium curve

Related Studies (stratified local shearing box simulations) ◮ Brandenburg et al.(1995) ◮ Stone et al.(1996) ◮ Miller & Stone (2000) ◮ Turner (2004) ◮ Hirose et al. (2006) ◮ Krolik et al. (2007) ◮ Blaes et al. (2007) ◮ Johansen & Levin (2008) ◮ Suzuki & Inutsuka (2009) ◮ Hirose et al. (2009) ◮ ...

Basic Equations radiation MHD equations with FLD approximation ∂ρ ∂t + ∇ · (ρ v ) = 0 κ R 4 π ( ∇ × B ) × B + ( ¯ ∂(ρ v ) 1 ff + κ es )ρ + ∇ · (ρ vv ) = −∇ (p + q) + F + f shearing box ∂t c ∂e ff ρ − cEκ es ρ 4 k B (T − T rad ) κ P ∂t + ∇ · (e v ) = − ( ∇ · v )(p + q) − ( 4 πB − cE) ¯ m e c 2 ∂E ff ρ + cEκ es ρ 4 k B (T − T rad ) κ P ∂t + ∇ · (E v ) = −∇ v : P + ( 4 πB − cE) ¯ − ∇ · F m e c 2 ∂ B ∂t − ∇ × ( v × B ) = 0 no explicit resistivity and cλ viscosity F = − ff + κ es )ρ ∇ E κ R ( ¯ numerical method ◮ hydro part: ZEUS ◮ magnetic part: MOC+CT ◮ radiation diffusion part (implicit): multigrid SOR

Simulation Setup simulation box outflow (no inflow) vertical: z ◮ stratfied shearing box ◮ Ω K = 190 s − 1 ( M/M ⊙ = 6 . 62 , r/r g = 30 ) g(z) = − Ω 2 K z initial condition 8 . 4 H ◮ gas and radiation 896 grids ◮ hydrostatic in z without B ◮ magnetic field ◮ twisted flux tube in y of β ≃ 20 periodic azimuth: y parameters 1 . 8 H ◮ surface density Σ 96 grids ◮ initial guess of Q + (, or thermal radial: x shearing periodic energy content) � ⇒ 0 . 45 H 48 grids gas/radiation-dominated

Parameter Space 10 7 Ω K = 190 s − 1 (fixed) ∂ T r φ < 0 0519b 0211b ∂ Σ 0520a 1126b T eff (K) 1112a 0320a 090304a 10 6 10 4 10 5 10 6 Σ (g cm − 2 ) Fig. 2.— Time averaged effective temperature of the radiation leaving each vertical face of the box, as a function of surface mass density for each simulation. From right to left, the solid curves show the predictions of alpha disk models with α = 0 . 01, 0.02, and 0.03, respectively. (See the Appendix for the equations used to define these alpha parameters.)

Radiation-dominated Disk Solution ◮ parameters ◮ Σ = 1 . 1 × 10 5 gcm − 2 ◮ guessed Q + = 9 . 4 × 10 21 ergcm − 2 s − 1 5 F (10 22 ergs cm ! 2 s ! 1 ) dissipation 4 radiative cooling 3 2 1 0 10 22 radiation E (ergs cm ! 2 ) gas 10 21 magnetic 10 20 10 19 0 100 200 300 400 500 600 t (orbits) ../mov/emag.mov ◮ radiation-dominated: E rad ∼ 20 E gas ◮ stable for 600 t orbit ∼ 40 t thermal ◮ time variations (quasi-steady state) ◮ MHD turbulence driven by MRI ◮ magnetic buoyancy (Parker instability) ◮ vertical oscillation (epicyclic mode, breathing mode)

Local Structure: Hydrostatic Balance t = 200 t orbit 3 (magnetic Ω 2 K z photosphere scattering tention) 2 acceleration (10 11 cm s ! 2 ) total acceleration 1 gas pressure 0 Lorentz force radiation ! 1 pressure ! 2 (magnetic pressure) ! 3 ! 4 ! 2 0 2 4 z / H ◮ | z | < 2 H : radiation pressure ../mov/lined.mov ◮ | z | > 2 H : magnetic pressure (+ magnetic tention) ◮ magnetic field is supplied to the upper (subphotospheric) layers by magnetic buoyancy (Parker instability)

Local Structure: Thermal Balance 6 d dz < (E + e)v z + F z > c Ω 2 dF/dz (10 15 ergs cm ! 3 s ! 1 ) 4 < q + > − < P : ∇ v > K κ d < F z > dz 2 0 d < Ev z > dz ! 4 ! 2 0 2 4 z / H < q + > − < P : ∇ v > = d dz < (E + e)v z + F z > ../mov/diss.mov ◮ dissipation: extended with double peaks ◮ radiation diffusion: d < F z > /dz ◮ ≃ c Ω 2 K /κ where radiation pressure competes the gravity ( | z | < H ) ◮ radiation advection: d < Ev z > /dz ◮ transports the excess energy ◮ associated with vertical oscillation, not buoyancy

Thermal Stability of MRI Disks thermal instability in the α model  4 T rφ (t) Ω K − 2 acT 4 d E (t) = − 3 c (t)   √ d E (t) = α Ω c Ω  dt 3 κ Σ √ 4 E (t) − E (t) dt κ 3 Σ    T rφ (t) = − αP(t) T rφ synchronized with P ◮ E B – E relation in the simulation (in place of T rφ – P relation) E B ∼ E 0 . 71 E B (t) ∼ E (t + t thermal ) 1.0 cross correlation with E B ∼ t thermal 0.8 cross correlation Magnetic Energy E log E B 0.6 0.4 1 0.2 0.0 ! 40 ! 20 0 20 40 100 lag (orbits) Radiation Energy log E time lag (orbits)

Thermal Stability of MRI Disks (continued) a toy model that allows a time lag between E B and E  d E (t) = E B (t) E (t)  −  t cool ( E (t 0 )) ( E (t)/ E (t 0 )) s   dt t diss   instability criterion ( E (t) ( 1 − s) < n ) n  d E B (t) = R(t) E B (t 0 ) − E B (t)      dt t grow E (t 0 ) t diss ◮ thermally stable solution: ( 1 − s) = 1 , n = 0 E B ∼ E 1 − s E 1.00 Normalized Magnetic Energy log E B Energy 1 0.10 E B 0.01 0 10 20 30 40 1 Cooling Times Normalized Radiation Energy t thermal log E

Thermal Equilibrium Curve 10 7 Ω K = 190 s − 1 (fixed) ∂ T r φ < 0 0519b 0211b ∂ Σ 0520a 1126b T eff (K) 1112a 0320a 090304a 10 6 10 4 10 5 10 6 Σ (g cm − 2 ) Fig. 2.— Time averaged effective temperature of the radiation leaving each vertical face of the box, as a function of surface mass density for each simulation. From right to left, the solid curves show the predictions of alpha disk models with α = 0 . 01, 0.02, and 0.03, respectively. (See the Appendix for the equations used to define these alpha parameters.)

Summary Comparison between the α disks and MRI disks α disks MRI disks hydrostatic thermal thermal magnetic a) pressure energy radiation diffusion radiation diffusion radiation advection b) transport yes c) stress–pressure yes correlation rad: stable d) thermal rad: unstable stability gas: stable gas: stable a) important in the upper subphotospheric layers b) important in the radiation dominated regime c) on timescales longer than t thermal d) – time lag between stress and pressureis necessary – intrinsic fluctuation of turbulence is longer than t cool