1. CONNECTING RADIATION TO DYNAMICS THROUGH SIMULATIONS with Omer Blaes, Shigenobu Hirose, Jiming Shi and Jim Stone

2. Dynamics vs. Thermodynamics Angular momentum transport is the primary driver of accretion—Does thermodynamics matter? Yes— It determines the equation of state Radiation forces can determine the vertical profile of disk material Both issues can affect the magnetic saturation level

3. How to Include Thermodynamics Requirements: Good energy conservation Radiation transfer solution Include radiation in momentum, energy conservation

4. Specifics of Implementation (Hirose, K. & Stone 2006) Shearing box approximation Finite difference MHD (Zeus) Total and internal energy equations Internal energy acquires gridscale losses at every operator-split update Thermally-averaged opacity Time-implicit flux-limited diffusion approximation (multi-grid solver)

5. Program Overview: Disk Vertical Structure as Function of p r /p g What are the vertical profiles of density, stress, dissipation rate, pressure? What determines the magnetic saturation level? Is it numerically well-defined? What are the characteristics of the fluctuations, both in time and in position? Is the structure stable: dynamically, thermally, or in terms of inflow?

6. Simplest Example: A Gas-Dominated Disk

7. Time-Averaged Density and Dissipation Profiles Q = ½/ § ¡ 1 = 2

8. Time-Averaged Pressure Profiles

9. Saturated Parker Modes in the Upper Layers

10. Does Pressure Control Magnetic Stress? Or Does Magnetic Stress Control Pressure? Magnetic leads Pressure leads

11. Is the Stress Numerically Well-Defined?

12. standard resolution 2 x  2 x r 2 x z 2 x (r, ,z)  independent of resolution See also Davis, Stone & Pessah (2009) for the same conclusion Box = (2Hx8Hx8H) Standard = (32x64x256)

13. Buoyancy Creates Vertical Coherence black curve = standard resolution red curve = 2 x (r, ,z)

14. Thermodynamic Convergence Requires Better Resolution

15. When Radiation Forces Matter

16. Radiation-Dominance Is Generic (Shakura & Sunyaev 1973) r = r g < 170 ( L = L E ) 16 = 21 ( M = M ¯ ) 2 = 21 Radiation pressure exceeds gas pressure for That is, for the most interesting parts of all bright accretion disks around black holes

17.  – Model Predicts Thermal Instability When p r > p g $\int dz Q \propto p_r h$$ S h a k ura & S unyaev 1976 In the  model, Z d z Q » ­ Z d z T r Á » ®p r h When radiation pressure dominates, h / F = Z d z Q And p r » Q t coo l » Q ( h = c ) ¿ » ( ¿ = c ) Z d z Q T h erma l I ns t a b i l i t y

18. But Even When p r ~ 10p g, No Runaway! Hirose, K. & Blaes (2009) t cool = 15 orbits

19. Trends with p r /p g : Density Profile p r = p g » 10 p r = p g » 0 : 015

20. Trends with p r /p g : Pressure Profiles p r = p g » 0 : 015 p r = p g » 10

21. Trends with p r /p g : Dissipation Profile p r = p g » 0 : 015 p r = p g » 10

22. » ´ F ( 2 = 3 ) c ( a T 4 0 )= ¿

23. Trends with p r /p g : Stress vs. Pressure Magnetic leadsPressure leads

24. A New Dimensional Analysis Two quantities with dimensions stress: p (Shakura & Sunyaev 1973) c  (Shakura & Sunyaev 1976) ; Q = c  2 /  required for radiation-supported hydrostatic balance Fluctuation analysis reveals (weak) role of p; c  does not fluctuate Both enter through hydrostatic balance.

25. Averaging over T >> t cool Restores 

26. Is the Radiation-Dominated Branch Inflow Unstable? (Lightman & Eardley 1974) Mass conservation + angular momentum conservation give diffusion equation for surface density: @ § @ t + 1 R @ @ R ½ 1 ¼ ­ @ @ R h r 2 ­ _ M ( § ) i ¾ = 0 Linearizing gives a negative effective diffusion coefficient when the accretion rate varies inversely with the surface density.

27. Linear Analysis of Inflow in Shearing Boxes Incomplete Growth rate ~ k 2 ; what about horizontal heat flow? Shearing box can show bunching, but no net inflow There is a maximum  permitting thermal balance; what happens if  max ?

28. Summary Thermodynamics dynamics Hydrostatic balance/magnetic buoyancy determines vertical structure, influences magnetic saturation (two characteristic stresses) MHD dynamics/hydrostatic balance drives thermodynamics, stabilizing radiation-dominated disks MHD dynamics control surface density=optical depth, influence thermal state, which influences magnetic saturation: is radiation-dominated inflow stable?