1. Radiative Transfer in 3D Numerical Simulations Robert Stein Department of Physics and Astronomy Michigan State University Åke Nordlund Niels Bohr Institute for Astronomy, Physics, and Geophysics University of Copenhagen

2. Radiative Transfer Produces low entropy plasma whose buoyancy work drives convection Determines (with convection and waves) mean atmospheric structure Provides diagnostics of velocity, temperature and magnetic field Reverses p-mode intensity vs. velocity asymmetry

3. Energy Conservation Radiative Heating/Cooling

4. Radiation Transfer LTE Non-gray Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface.

5. Solve Feautrier equations along rays through each grid point at the surface

6. 5 Rays Through Each Surface Grid Point Interpolate source function to rays at each height

7. Actually solve for q = P - B

8. Finite Difference Equation Problem: at small optical depth the 1 is lost re 1/  2  in B Solution: store the value - 1, (the sum of the elements in a row) and calculate B = - (1+A+B)

9. Interpolate q=P-B from slanted grid back to Cartesian grid

10. Simplifications Only 5 rays 4 Multi-group opacity bins Assume  L  C

11. Opacity is binned, according to its magnitude, into 4 bins.

12. Line opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bin

13. Advantage Wavelengths with same  (z) are grouped together, so integral over  and sum over commute

14. Radiative Heating/Cooling

15. Fluid Parcels that reach the surface Radiate away their Energy and Entropy Z S E  Q 

16. Granulation

17. Spectrum of granulation agrees with observations

18. Line Profiles

19. Convection produces line shifts, changes in line widths. No microturbulence, damping.

20. Never See Hot Gas

21. 3D atmosphere is hotter than 1D atmosphere emitting the same flux, 3D atmosphere is more extended

22. Extended atmosphere gives better agreement with p-mode frequencies

23. Stokes Profiles - Micropore Synthetic Observation - Perfect Telescope & Seeing

24. Stokes Image - Quiet Sun Synthetic Observation - La Palma Telescope MTF + Moderate Seeing Surface IntensityStokes V 6 Mm

25. Stokes Image - Quiet Sun Synthetic Observation - La Palma Telescope MTF + Excellent Seeing Surface IntensityStokes V 6 Mm

26. Stokes Image - Quiet Sun Synthetic Observation - Perfect Telescope & Seeing Surface IntensityStokes V 6 Mm

27. P-Mode Intensity - Velocity Phase

28. At fixed height V &  T have same profile. Radiation reduces  T - more at low

29. Height of  = 1 varies more at low frequencies where  T larger - Reduces  T more

30. P-Mode Excitation

32. The End