Radiative Transfer for Radiative Transfer for Simulations of StellarSimulations of StellarEnvelope ConvectionEnvelope Convection

ByBy

Regner TrampedachRegner Trampedach

8/19/048/19/04

Hydro-dynamicsHydro-dynamics

• Solve Euler equationsSolve Euler equations• Conservation of:Conservation of:

– Mass: Mass: ddρ ρ //ddtt = -= -uu ∙∙∇∇ ρρ --ρρ ∇∇ ∙∙uu– Momentum: Momentum: ρρ dduu //ddtt = -= -ρρ uu ∙∙∇∇ uu ++∇∇ ((TT -- PPgasgas))

++ρρ gg– Energy: Energy: ddEE //ddtt = - = -∇∇ ∙∙uuEE + + ((TT -- PPgasgas))∇∇ ∙∙uu

++ρ ρ qqradrad

• Regular horizontal and optimized vertical gridRegular horizontal and optimized vertical grid

Vertical Temperature-cut of Vertical Temperature-cut of η η --BooBoo

Applications of the Applications of the SimulationsSimulations• Improving stellar structure modelsImproving stellar structure models

– T-T-ττ -relations – atmospheric boundary cond.-relations – atmospheric boundary cond.– Calibration of the mixing-length parameter, Calibration of the mixing-length parameter,

αα

• Abundance analysisAbundance analysis– Agreement between FeI, FeII and meteoriticAgreement between FeI, FeII and meteoritic– Lower C, N and O abundances – at odds with Lower C, N and O abundances – at odds with

helioseismologyhelioseismology

• Synthetic spectra/line-profilesSynthetic spectra/line-profiles– No free parameters, e.g., micro-/macro-turb.No free parameters, e.g., micro-/macro-turb.

Input PhysicsInput Physics

• Equation of State (EOS)Equation of State (EOS)– Pressure for hydro-static supportPressure for hydro-static support– Response to temperature-/density-Response to temperature-/density-

changeschanges

• Opacity: ff + bf + bbOpacity: ff + bf + bb– radiative transfer =>radiative transfer =>

– radiative heating: radiative heating: qqrad,rad,λλ = 4= 4ππ κ κ λλ ((JJλλ--SSλ λ ))

FeI Opacity According to FeI Opacity According to LAOLLAOL

• HHüübner et. Al (1977)bner et. Al (1977)

• Semi-hydrogenic wave-functionsSemi-hydrogenic wave-functions

• Hundreds of lines...Hundreds of lines...

FeI Opacity According to OPFeI Opacity According to OP

• Seaton et. Al (1994)Seaton et. Al (1994)

• Intermediate Intermediate S-LS-L coupling coupling

• Hundreds of millions of lines!Hundreds of millions of lines!

bf-Opacity Before OP/OPALbf-Opacity Before OP/OPAL

From Peach (1962)From Peach (1962)

Confronting ExperimentConfronting Experiment

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

Radiative TransferRadiative Transfer

• Determines heating/cooling => Determines heating/cooling => structurestructure

• Determines Determines emergentemergent flux/intensity => flux/intensity => link to observationslink to observations

• Transfer Eq. Transfer Eq. dd IIλ λ /d/dτ τ λλ = ( = (IIλλ – – SSλ λ )) solved solved for more than 10for more than 1055 wavelengths wavelengths

• Not possible in convection simulationsNot possible in convection simulations Yet...Yet...

Statistical MethodsStatistical Methods

• Have used opacity binning (Nordlund Have used opacity binning (Nordlund 1982) a.k.a. the multi-group method1982) a.k.a. the multi-group method

• Works well, and has correct asymptotic Works well, and has correct asymptotic behaviour in optical thick/thin casesbehaviour in optical thick/thin cases

• Employs a number of somewhat arbitrary Employs a number of somewhat arbitrary bridging functions and extrapolationsbridging functions and extrapolations

• Does not converge for Does not converge for NN binbin→ ∞→ ∞

SSelective/elective/SSparse parse OOpacity pacity

SSamplingampling

SS OO SS• Carefully select Carefully select NNSOSSOS wavelengths wavelengths

– covering the whole energy spectrumcovering the whole energy spectrum– that reproduce the full solution, e.g., that reproduce the full solution, e.g.,

heating; heating; qqradrad, flux; , flux; FFradrad, and , and JJ and and KK..

• Perform radiative transfer on thosePerform radiative transfer on those λλ

• Paves the way for including velocity-Paves the way for including velocity-effectseffects• Spans the convective fluctuations Spans the convective fluctuations

better than the opacity binning better than the opacity binning methodmethod• Converges for Converges for NNSOSSOS→ ∞→ ∞

Multi-group vs. SOSMulti-group vs. SOS

• SOS, SOS, NNλ λ =50 =50

• MonochromeMonochrome, ODF, , ODF, NNλ λ

=2750 =2750

• Multi-group, Multi-group, NNbinbin=4=4

Horizontal and temporal Horizontal and temporal averagesaverages

• 50 bins same as 4 50 bins same as 4 bins!bins!

• Too little cooling Too little cooling in conv/rad trans.in conv/rad trans.

• Too little heating Too little heating in lower in lower photosph.photosph.

• No action at or No action at or above above TT-min-min

- and their differences- and their differences

______ straight straight

averageaverage

- - - RMS average- - - RMS average● Systematic diffs Systematic diffs

for multi-groupfor multi-group● >4 times larger >4 times larger

RMS differencesRMS differences

SummarySummary

• Developed new radiative transfer Developed new radiative transfer schemescheme

• Performs better than multi-group Performs better than multi-group methodmethod– Much closer to monochromatic solutionMuch closer to monochromatic solution– More stable against convective fluctuationsMore stable against convective fluctuations

– Reproduce first three moments of Reproduce first three moments of II((μ μ ))– Convergent forConvergent for NNSOSSOS→ ∞→ ∞

Prospects for the FutureProspects for the Future

• Calculate new and improved EOS-tablesCalculate new and improved EOS-tables

• Use it as basis for new opacity calculation Use it as basis for new opacity calculation using the newest cross-section datausing the newest cross-section data

• Implement the SOS radiative transfer Implement the SOS radiative transfer scheme in the convection simulationsscheme in the convection simulations

• Build a grid of convection models, using Build a grid of convection models, using the new EOS, opacities and SOS schemethe new EOS, opacities and SOS scheme