Equilibrium Structure of Radiation-dominated Disk Segments

Transcription

1 Equilibrium Structure of Radiation-dominated Disk Segments Shigenobu Hirose The Earth Simulator Center, JAMSTEC, Japan collaborators Julian Krolik (JHU) Omer Blaes (UCSB)

2 Reconstruction of standard disk model with MRI stress by RMHD simulations Pgas << Prad Pgas >> Prad pressure dominance M/M = 6.62 C B A r electron scattering free-free opacity pressure dominance r r S Ω [/s] Σ [g/cm 2 ] references A Pgas >> Prad E+04 Hirose, Krolik, & Stone (2006) B Pgas ~ Prad E+04 Krolik, Hirose, & Blaes (2007) Blaes, Hirose, & Krolik (2007) C Pgas << Prad E+04 Hirose, Krolik, & Blaes in prep Omer s talk: implications to observation Julian s talk: toy model explaining the stability

3 Reconstruction of standard disk model with MRI stress by RMHD simulations Pgas << Prad Pgas >> Prad pressure dominance M/M = 6.62 C B A r electron scattering free-free opacity stable solutions here pressure dominance r r S Ω [/s] Σ [g/cm 2 ] references A Pgas >> Prad E+04 Hirose, Krolik, & Stone (2006) B Pgas ~ Prad E+04 Krolik, Hirose, & Blaes (2007) Blaes, Hirose, & Krolik (2007) C Pgas << Prad E+04 Hirose, Krolik, & Blaes in prep Omer s talk: implications to observation Julian s talk: toy model explaining the stability

4 Standard Accretion Disks optically thick and geometrically thin (nearly Keplerian) disks time scales T hydrostatic <T thermal T viscous vertical structure (z) radial structure (r) z vertical radial r central star accretion disk disk segment ( fixed surface density Σ )

5 alpha Model (Shakura & Sunyaev 1973) analogy with interior structure of stars momentum equation dp dz = ρω2 z energy equation ( ) d c dp = Q visc + dz κρ dz? alpha model momentum equation P H = ρω2 H energy equation cp κσ = 3 2 ΩT r φh α viscosity T r φ = αp steady state no magnetic field energy transport by radiation diffusion steady state no magnetic field energy transport by radiation diffusion one zone approximation alpha viscosity

6 alpha Model (Shakura & Sunyaev 1973) momentum equation P H = ρω2 H energy equation cp κσ = 3 2 ΩT r φh α viscosity T r φ = αp alpha model Σ = Σ(r ; M,Ṁ,α) H = H(r ; M,Ṁ,α) ρ = ρ(r ; M,Ṁ,α) P = P(r ; M,Ṁ,α) T r φ = T r φ (r ; M,Ṁ,α)... steady state no magnetic field energy transport by radiation diffusion one zone approximation alpha viscosity

7 Origin of Stress in Accretion Disks central star accretion disk MHD turbulence driven by MRI

8 Origin of Stress in Accretion Disks central star accretion disk MHD turbulence driven by MRI

9 Equations Governing C r.v/ D C r.vv/ D rp C j B C.N ff C es C r.ev/ D.4B ce/q ff C ce es 4k B.T T rad / rv m e c 2 P C r.ev/ D.4B ce/q ff ce es 4k B.T T rad /.r v/p C Q C m e c 2 MRI F D P D fe C r. v B/ D 0 c.n ff C es / re disk F 3D radiation MHD equations must be solved. ( Flux-limited diffusion approximation is adopted.)

10 Reconstruction of Standard Disk Model z M r H r Σ ρ t + (ρv) = 0 (ρv) + (ρvv) = p + j B+ χ Rρ t c F E + (Ev) = (4πκ B B cκ B E)ρ v : P F t e t + (ev) = (4πκ B B cκ B E)ρ ( v) p B t + ( v B) = 0 F = cλ χ R ρ E P = fe disk segment of column density Σ at radius r averaging in time and space Ṁ = H = ρ = P = T r φ =... α Ṁ(r ; M,Σ) H(r ; M,Σ) ρ(r ; M,Σ) P(r ; M,Σ) T r φ (r ; M,Σ) Tr φ dz Pdz

11 Basic Equations of Radiation MHD ( Flux-limited diffusion C r.v/ D 0 Rosseland C r.vv/ D rp C j BC.N ff C es Planck mean C r.ev/ D.4B ce/q ff C ce es 4k B.T T rad / rv m e c 2 P C r.ev/ D.4B ce/q ff ce es 4k B.T T rad /.r v/p C Q C m e c 2 MRI F D P D fe C r. v B/ D 0 c.n ff C es / re diffusion flux with limiter λ closure relation with Eddington tensor F κff : free-free opacity κes : electron scattering opacity explicit resistivity and viscosity are not included

12 C r.v/ D C r.vv/ D rp C j B C.N ff C es / c energy exchange between gas and radiation (Newton method) Explicitly solved (ZEUS + MOC-CT) Implicitly solved F radiative diffusion C r.ev/ D.4B ce/q ff C ce es 4k B.T T rad / rv W P r m e C r.ev/ D.4B ce/q ff ce es 4k B.T T rad /.r v/p C Q C m e c 2 MRI F D P D fe C r. v B/ D 0 c.n ff C es / re

13 Initial and Boundary Conditions A radial slice is simulated by a local stratified shearing box - rotating frame (Cartesian coordinate) vertical: z 8.4H 896 grids outflow boundary 2ρ v + 3ρ 2 x ˆx ρ 2 zẑ inertia force gravity - shearing periodic boundary condition in r g(z) = 2 z V y (x) = 3 2 x background shear flow M r azimuth: y 1.8H 96 grids periodic boundary radial: x 0.45H 48 grids shearing periodic boundary

14 Initial and Boundary Conditions A radial slice is simulated by a local stratified shearing box - rotating frame (Cartesian coordinate) vertical: z 8.4H 896 grids outflow boundary 2ρ v + 3ρ 2 x ˆx ρ 2 zẑ inertia force gravity - shearing periodic boundary condition in r g(z) = 2 z unbound V y (x) = 3 2 x background shear flow radiation M r bound gas magnetic field twisted azimuthal flux tube azimuth: y 1.8H dissipative periodic boundary 96 grids Parameters - angular velocity: Ω [/s] - surface density: Σ [g/cm 2 ] radial: x 0.45H 48 grids shearing periodic boundary

15 heating within R S of the midplane, where it is readily distinguished from expansion associated with magnetic buoyancy. Artificial viscous heating, mostly in shocks, makes up 12% of the overall dissipation. The dissipation is approximately balanced by cooling. The main cooling processes are diffusion and advection of radiation through the vertical boundaries, with diffusion carrying two-thirds of the energy flux. Mass lost through the boundaries is balanced by mass added to maintain densities above the floor, and total mass increases by less than 1% from 15 to 45 orbits. The energy flux is 73% of the energy input, and the remainder disappears partly through numerical losses of kinetic energy. A complete energy-conserving scheme may be useful for future calculations. Starting at 50 orbits, the disk heats, then cools. Total energy increases roughly linearly by a factor of 2.5 up to 90 orbits, decreases linearly until 145 orbits, then is again steady. The run ends at 170 orbits, or eight simulation thermal times after first saturation. No exponentially growing thermal instability is seen. However, during the hot period, the disk expands, and losses through the vertical boundaries reduce the mass by almost half. The absence of runaway heating may be due to the mass loss, rather than internal thermal stability. 5. CONCLUSIONS A patch of radiation-dominated accretion disk lying 100R S 8 from a 10 M, black hole is simulated including physical flux is zero. The structure the midplane, stresses grea and dissipation found thro space. The stresses result the surface layers are buo merical dissipation of the fi of the turbulence, while c vection of radiation throug ture may prove to vary location in the disk, surfa The time-averaged stru namically convectively sta vented by outflow through not tested, as there is no Photon bubble instability modes (Blaes & Socrates are found in the surface la the vertical grid spacing, with wavelengths greater grow more slowly than the are found. The likely direc reduce the cooling time. The methods were dev benefited also from discu The work was supported

16 heating within R S of the midplane, where it is readily distinguished from expansion associated with magnetic buoyancy. Artificial viscous heating, mostly in shocks, makes up 12% of the overall dissipation. The dissipation is approximately balanced by cooling. The main cooling processes are diffusion and advection of radiation through the vertical boundaries, with diffusion carrying two-thirds of the energy flux. Mass lost through the boundaries is balanced by mass added to maintain densities above the floor, and total mass increases by less than 1% from 15 to 45 orbits. The energy flux is 73% of the energy input, and the remainder disappears partly through numerical losses of kinetic energy. A complete energy-conserving scheme may be useful for future calculations. Starting at 50 orbits, the disk heats, then cools. Total energy increases roughly linearly by a factor of 2.5 up to 90 orbits, decreases linearly until 145 orbits, then is again steady. The run ends at 170 orbits, or eight simulation thermal times after first saturation. No exponentially growing thermal instability is seen. However, during the hot period, the disk expands, and losses through the vertical boundaries reduce the mass by almost half. The absence of runaway heating may be due to the mass loss, rather than internal thermal stability. 5. CONCLUSIONS A patch of radiation-dominated accretion disk lying 100R S 8 from a 10 M, black hole is simulated including physical flux is zero. The structure the midplane, stresses grea and dissipation found thro space. The stresses result the surface layers are buo merical dissipation of the fi of the turbulence, while c vection of radiation throug ture may prove to vary location in the disk, surfa The time-averaged stru namically convectively sta vented by outflow through not tested, as there is no Photon bubble instability modes (Blaes & Socrates are found in the surface la the vertical grid spacing, with wavelengths greater grow more slowly than the are found. The likely direc reduce the cooling time. The methods were dev benefited also from discu The work was supported

17 heating within R S of the midplane, where it is readily distinguished from expansion associated with magnetic buoyancy. Artificial viscous heating, mostly in shocks, makes up 12% of the overall dissipation. The dissipation is approximately balanced by cooling. The main cooling processes are diffusion and advection of radiation through the vertical boundaries, with diffusion carrying two-thirds of the energy flux. Mass lost through the boundaries is balanced by mass added to maintain densities above the floor, and total mass increases by less than 1% from 15 to 45 orbits. The energy flux is 73% of the energy input, and the remainder disappears partly through numerical losses of kinetic energy. A complete energy-conserving scheme may be useful for future calculations. Starting at 50 orbits, the disk heats, then cools. Total energy increases roughly linearly by a factor of 2.5 up to 90 orbits, decreases linearly until 145 orbits, then is again steady. The run ends at 170 orbits, or eight simulation thermal times after first saturation. No exponentially growing thermal instability is seen. However, during the hot period, the disk expands, and losses through the vertical boundaries reduce the mass by almost half. The absence of runaway heating may be due to the mass loss, rather than internal thermal stability. 5. CONCLUSIONS A patch of radiation-dominated accretion disk lying 100R S 8 from a 10 M, black hole is simulated including physical flux is zero. The structure the midplane, stresses grea and dissipation found thro space. The stresses result the surface layers are buo merical dissipation of the fi of the turbulence, while c vection of radiation throug ture may prove to vary location in the disk, surfa The time-averaged stru namically convectively sta vented by outflow through not tested, as there is no Photon bubble instability modes (Blaes & Socrates are found in the surface la the vertical grid spacing, with wavelengths greater grow more slowly than the are found. The likely direc reduce the cooling time. The methods were dev benefited also from discu The work was supported

18 Technical Consistency % time variation of surface density ΔΣ / Σ (%)!"2",-31 #!!#!%! "!! #!! $!! ()*+,-./0)(1 %!! &!! '!! % location of photosphere (f=0.5) 2,-31 #!!#!%! "!! #!! $!! ()*+,-./0)(1 %!! &!! '!! λmax / Δz,!, # % ' 4 "!,

19 MRI-driven Turbulence orbit: orbit: Eddington factor 1/3 < f < f = 1/3 1/3 < f < log10 (g/cm 3 ) log10 (erg/cm 3 ) density radiation energy

20 MRI-driven Turbulence orbit: orbit: Eddington factor 1/3 < f < f = 1/3 1/3 < f < log10 (erg/cm 3 ) (erg/s/cm 3 ) magnetic energy energy dissipation rate

21 Energetics in a disk segment 4B ce/q ff C ce es 4k B.T T rad / m e c 2 internal cooling rate ( χρ c F ) v radiation heating rate F radiation damping Q mag magnetic disk surface E B gravitational ±( p) v Q kin ± ( 2ρΩ v + 3ρΩ 2 xˆx ρω 2 zẑ ) v kinetic P : v ±(j B) v shearing boundary

22 Energies and Heating/Cooling Rates cooling rate heating rate thermal balance holds statistically ~ 40 thermal time radiation energy magnetic energy (x10) gas internal energy (x10) radiationdominated

23 Equilibrium of Time-averaged Quantities Dynamical Equilibrium 1 ρ(z) d P total (z) dz = Ω 2 z - pressure gradient = gravity Thermal Equilibrium Q + diss (z) = d F total (z) dz energy dissipation = energy transport

24 Dynamical Equilibrium 1 ρ(z) d P total (z) dz = Ω 2 z acceleration [cm/s 2 ] ! ! !4! z [H]

25 Dynamical Equilibrium acceleration [cm/s 2 ] ! ! !4! z [H] P total (z) P rad (z) + P mag (z) + P gas (z)

26 Dynamical Equilibrium acceleration [cm/s 2 ] ! magnetic! support radiation support magnetic support!4! z [H] P total (z) P rad (z) + P mag (z) + P gas (z)

27 Critical Value of Dissipation Rate (Shakura & Sunyaev 1976) dynamical equilibrium κ es ρ(z) c F(z) ρ(z)ω 2 z = 0 F(z) = cω2 κ es z thermal equilibrium df(z) dz + Q+ diss (z) = 0 cω2 Q+ diss (z) = κ es (const.)

28 Thermal Equilibrium Q + diss (z) = df total(z) dz dissipation rate [erg/cm 3 /s 1 ] ! !4! z [H] critical value c 2 es

29 Thermal Equilibrium dissipation rate [erg/cm 3 /s 1 ] ! !4! z [H] F total (z) D de rad dz (z) + E rad v(z) + critical value c 2 Poynting flux (E B) z E gas v(z) radiative diffusion radiation advection gas advection es

30 Thermal Equilibrium dissipation rate [erg/cm 3 /s 1 ] ! radiative diffusion radiative diffusion radiation advection!4! z [H] F total (z) D de rad dz (z) + radiative diffusion E rad v(z) + critical value c 2 Poynting flux (E B) z E gas v(z) radiative diffusion radiation advection gas advection es

31 Thermal Stability Time Trajectory for ~40 thermal time in log E - log Q space log Q + diss log Q rad d log Q d log E = 1 log E - dissipation rate is not proportional to energy - variance in heating rate is larger than that in cooling rate - change of heating precedes that of cooling

32 Thermal instability in the alpha model cooling by radiative diffusion cooling rate Q rad = cp κσ E Σ heating rate Q + diss = 3 2 ΩT r φ = 3 2 αωph E2 Σ alpha prescription T r φ = αp hydrostatic balance by radiation pressure

33 Thermal instability in the alpha model cooling by radiative diffusion cooling rate Q rad = cp κσ E Σ heating rate Q + diss = 3 2 ΩT r φ = 3 2 αωph E2 Σ alpha prescription T r φ = αp hydrostatic balance by radiation pressure

34 Thermal instability in the alpha model cooling by radiative diffusion cooling rate + radiative advection Q rad = cp κσ E Σ heating rate Q + diss = 3 2 ΩT r φ = 3 2 αωph E2 Σ alpha prescription T r φ = αp hydrostatic balance by radiation pressure

35 Summary Thermally (and dynamically) stable solutions are found in the radiation-dominated regime of the standard accretion disk by stratified shearing box simulations with radiation transport. Dynamical balance - The disk is supported by radiation pressure near the midplane while it is supported by magnetic pressure in the upper layers. - The magnetic field is generated near the midplane and supplied to the upper layers by magnetic buoyancy. Dissipation - The magnetic energy loss is ~ 70% and kinetic energy loss is ~ 30%. - The radiation damping is negligible as a dissipation mechanism. - The dissipation rate near the midplane is just above the critical dissipation rate, cω 2 /κ. Thermal balance - The main energy transport is radiation diffusion. The transport rate is the critical dissipation rate, cω 2 /κ. - The excess energy is exactly transported by radiation advection. It is associated with a vertical acoustic oscillation.