Radiative Transfer for Simulations of Stellar Envelope Convection By Regner Trampedach 8/19/04

Hydro-dynamics Solve Euler equations Solve Euler equations Conservation of: Conservation of: – Mass: dρ / dt = - u ∙ ∇ ρ - ρ ∇ ∙ u – Momentum: ρ du / dt = - ρ u ∙ ∇ u + ∇ (T - P gas ) + ρ g – Energy: dE / dt = - ∇ ∙ uE + (T - P gas ) ∇ ∙u + ρ q rad Regular horizontal and optimized vertical grid Regular horizontal and optimized vertical grid

Vertical Temperature-cut of η -Boo

Applications of the Simulations Improving stellar structure models Improving stellar structure models – T- τ -relations – atmospheric boundary cond. –Calibration of the mixing-length parameter, α Abundance analysis Abundance analysis –Agreement between FeI, FeII and meteoritic –Lower C, N and O abundances – at odds with helioseismology Synthetic spectra/line-profiles Synthetic spectra/line-profiles –No free parameters, e.g., micro-/macro-turb.

Input Physics Equation of State (EOS) Equation of State (EOS) –Pressure for hydro-static support –Response to temperature-/density-changes Opacity: ff + bf + bb Opacity: ff + bf + bb –radiative transfer => –radiative heating: q rad,λ = 4π κ λ ( J λ - S λ )

FeI Opacity According to LAOL H ü bner et. Al (1977) H ü bner et. Al (1977) Semi-hydrogenic wave-functions Semi-hydrogenic wave-functions Hundreds of lines... Hundreds of lines...

FeI Opacity According to OP Seaton et. Al (1994) Seaton et. Al (1994) Intermediate S-L coupling Intermediate S-L coupling Hundreds of millions of lines! Hundreds of millions of lines!

bf-Opacity Before OP/OPAL From Peach (1962)

Confronting Experiment From Nahar, S. N., 2003, Phys. Rev. A (submitted)

Radiative Transfer Determines heating/cooling => structure Determines heating/cooling => structure Determines emergent flux/intensity => link to observations Determines emergent flux/intensity => link to observations Transfer Eq. d I λ /dτ λ = ( I λ – S λ ) solved for more than 10 5 wavelengths Transfer Eq. d I λ /dτ λ = ( I λ – S λ ) solved for more than 10 5 wavelengths Not possible in convection simulations Not possible in convection simulations Yet... Yet...

Statistical Methods Have used opacity binning (Nordlund 1982) a.k.a. the multi-group method Have used opacity binning (Nordlund 1982) a.k.a. the multi-group method Works well, and has correct asymptotic behaviour in optical thick/thin cases Works well, and has correct asymptotic behaviour in optical thick/thin cases Employs a number of somewhat arbitrary bridging functions and extrapolations Employs a number of somewhat arbitrary bridging functions and extrapolations Does not converge for N bin → ∞ Does not converge for N bin → ∞

S elective/ S parse O pacity S ampling

S O SS O SS O SS O S Carefully select N SOS wavelengths Carefully select N SOS wavelengths –covering the whole energy spectrum –that reproduce the full solution, e.g., heating; q rad, flux; F rad, and J and K. Perform radiative transfer on those λ Perform radiative transfer on those λ Paves the way for including velocity- effects Paves the way for including velocity- effects Spans the convective fluctuations better than the opacity binning method Spans the convective fluctuations better than the opacity binning method Converges for N SOS → ∞ Converges for N SOS → ∞

Multi-group vs. SOS SOS, N λ =50 SOS, N λ =50 Monochrome, ODF, N λ =2750 Monochrome, ODF, N λ =2750 Multi-group, N bin =4 Multi-group, N bin =4

Horizontal and temporal averages 50 bins same as 4 bins! 50 bins same as 4 bins! Too little cooling in conv/rad trans. Too little cooling in conv/rad trans. Too little heating in lower photosph. Too little heating in lower photosph. No action at or above T -min No action at or above T -min

- and their differences ___ straight average ___ straight average RMS average RMS average ● Systematic diffs for multi-group ● >4 times larger RMS differences

Summary Developed new radiative transfer scheme Developed new radiative transfer scheme Performs better than multi-group method Performs better than multi-group method –Much closer to monochromatic solution –More stable against convective fluctuations –Reproduce first three moments of I( μ ) – Convergent for N SOS → ∞

Prospects for the Future Calculate new and improved EOS-tables Calculate new and improved EOS-tables Use it as basis for new opacity calculation using the newest cross-section data Use it as basis for new opacity calculation using the newest cross-section data Implement the SOS radiative transfer scheme in the convection simulations Implement the SOS radiative transfer scheme in the convection simulations Build a grid of convection models, using the new EOS, opacities and SOS scheme Build a grid of convection models, using the new EOS, opacities and SOS scheme