Ingredients for Accurate Simulations of Stellar Envelope Convection by Regner Trampedach

Hydro-dynamics Solve Euler equationsSolve Euler equations Conservation of:Conservation of: –Mass: dρ/dt = -u ∙Δρ - ρΔ∙u –Momentum: ρdu /dt = -ρu ∙Δu + Δ(T-P gas )+ ρg –Energy: dE /dt = -Δ∙uE + (T-P gas )Δ∙u + Δ∙q rad Hyman 3 rd -order time-stepping (predictor/corr) Hyman 3 rd -order time-stepping (predictor/corr) Cubic-spline interpolation vertically and compact 6 th order interpolation horizontally Cubic-spline interpolation vertically and compact 6 th order interpolation horizontally Regular horizontal and optimized vertical grid Regular horizontal and optimized vertical grid

Numerical Stability Schemes using centered derivatives are unstable Schemes using centered derivatives are unstable Fixed with artificial diffusion Fixed with artificial diffusion

Vertical Temperature-cut of η-Boo

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 = ρκ λ (J λ -S λ )

Equation of State Two main purposes Thermodynamic properties of plasma Thermodynamic properties of plasma –Pressure, internal energy –Adiabatic exponent Foundation of opacity calculations Foundation of opacity calculations –ionization and dissociation balances –population of electronic- and roto-vibrational- states

The OP/MHD- and OPAL- projects Prompted by a plea by Simon (1982) Prompted by a plea by Simon (1982) Pulsations by κ-mechanism didn’t agree with observations Pulsations by κ-mechanism didn’t agree with observations Substantial disagreement with helioseismic structure of the Sun Substantial disagreement with helioseismic structure of the Sun MHD OPAL

MHD Equation of State Explicitly includes hundreds of energy- levels for each ion/atom/molecule Explicitly includes hundreds of energy- levels for each ion/atom/molecule Use occupation probabilities to account for destruction of states from “collisions” with other particles: Use occupation probabilities to account for destruction of states from “collisions” with other particles:

Micro-field Distributions Ionization by fluctuating fields from passing ions/electrons Ionization by fluctuating fields from passing ions/electrons With a state, i, being destroyed by a field of critical strength, F cr, the probability of it surviving is w i = Q (F cr ) = ∫ 0 Fcr P (F ) dF With a state, i, being destroyed by a field of critical strength, F cr, the probability of it surviving is w i = Q (F cr ) = ∫ 0 Fcr P (F ) dF

Micro-Field Effects in the Sun ___ OPAL ___ MHD old Q (F cd )

Quantum effects Quantum diffraction from Heisenberg’s uncertainty relation Quantum diffraction from Heisenberg’s uncertainty relation Exchange interaction from Pauli’s exclusion principle Exchange interaction from Pauli’s exclusion principle

Exchange Interactions in the Sun ___ OPAL ___ MHD no Exch

Interaction with Neutral Particles Original MHD used hard-sphere interacts. Original MHD used hard-sphere interacts. How do hard spheres interact? How do hard spheres interact? Through electric forces, of course... Through electric forces, of course... Assume Gaussian (s-orbital) e ̶̶̶̶̶̶ -distribution Assume Gaussian (s-orbital) e ̶̶̶̶̶̶ -distribution

Effective charges in the Sun ___ OPAL ___ MHD const. Z

Coulomb Interactions Including the first- order (Debye-Hückel) term, had the largest effect on MHD/OPAL Including the first- order (Debye-Hückel) term, had the largest effect on MHD/OPAL OPAL includes terms up to n 5/2 OPAL includes terms up to n 5/2 Include results from Monte-Carlo sims. Include results from Monte-Carlo sims.

Coulomb Interactions in the Sun ___ OPAL ___ MHD Debye-H

Additional Changes Relativistic effects – affects stellar centres Relativistic effects – affects stellar centres –The Sun has a relativistically degenerate core –well, - at least slightly... Molecules Molecules –315 di-atomic and 99 poly-atomic (+ ions) –Affects stellar atmospheres and the convection simulations

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

Opacity According to OP

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. dI λ /dτ λ = (I λ – S λ ) solved for more than 10 5 wavelengths Transfer Eq. dI 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 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.

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 → ∞

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 Synthetic spectra/line-profiles Synthetic spectra/line-profiles –No free parameters, e.g., micro-/macro-turb. Abundance analysis Abundance analysis –Agreement between Fe I, Fe II and meteoritic –Lower C, N, O

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 Synthetic spectra/line-profiles Synthetic spectra/line-profiles –No free parameters, e.g., micro-/macro-turb. Abundance analysis Abundance analysis –Agreement between Fe I, Fe II and meteoritic –Lower C, N, O – helioseismology doesn’t agree!

T -τ-relations Can indeed describe non-grey atmospheres Can indeed describe non-grey atmospheres Made fits to T (τ) for seven simulations Made fits to T (τ) for seven simulations Not necessarily in radiative equilibrium in radiative zone. Not necessarily in radiative equilibrium in radiative zone. Balance between radiative heating and adiabatic cooling by convective overshoot Balance between radiative heating and adiabatic cooling by convective overshoot

Calibration of α Use T (τ) from the simulations Use T (τ) from the simulations –Same atomic physics Match ρ and T at common P –point Match ρ and T at common P –point Find significant variation of α over the T eff /g surf -plane Find significant variation of α over the T eff /g surf -plane

Summary Developed new equation of state Developed new equation of state –With larger range of validity Developed new radiative transfer scheme Developed new radiative transfer scheme First published T (τ) to include convective effects First published T (τ) to include convective effects First calibration of α against 3D convection simulations First calibration of α against 3D convection simulations

Prospects for the Future Calculate tables of MHD2000 Calculate tables of MHD2000 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