Ingredients for Accurate Simulations of Stellar Envelope Convection

1. Ingredients for Accurate Simulations of StellarEnvelope Convection by Regner Trampedach 03.12.03

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

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

5. Vertical Temperature-cut of η-Boo

6. Input Physics • Equation of State (EOS) • Pressure for hydro-static support • Response to temperature-/density-changes • Opacity: ff + bf + bb • radiative transfer => • radiative heating: qrad,λ = ρκλ(Jλ-Sλ)

7. Equation of StateTwo main purposes • Thermodynamic properties of plasma • Pressure, internal energy • Adiabatic exponent • Foundation of opacity calculations • ionization and dissociation balances • population of electronic- and roto-vibrational-states

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

9. MHD Equation of State • 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:

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

11. Micro-Field Effects in the Sun ___ OPAL ___ MHD2000 - - old Q(Fcd)

12. Quantum effects • Quantum diffractionfrom Heisenberg’suncertainty relation • Exchange interactionfrom Pauli’sexclusion principle

13. Exchange Interactions in the Sun ___OPAL ___ MHD2000 - - no Exch

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

15. Effective charges in the Sun ___OPAL ___ MHD2000 - - const. Z

16. Coulomb Interactions • Including the first-order (Debye-Hückel) term, had the largest effect on MHD/OPAL • OPAL includes terms up to n5/2 • Include results from Monte-Carlo sims.

17. Coulomb Interactions in the Sun ___OPAL ___ MHD2000 - - Debye-H

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

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

20. Opacity According to OP

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

22. Radiative Transfer • Determines heating/cooling => structure • Determines emergent flux/intensity => link to observations • Transfer Eq. dIλ/dτλ = (Iλ – Sλ) solved for more than 105 wavelengths • Not possible in convection simulations yet...

23. Statistical Methods • 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 • Employs a number of somewhat arbitrary bridging functions and extrapolations • Does not converge for Nbin→∞

24. Selective/Sparse Opacity Sampling • Carefully select NSOS wavelengths • covering the whole energy spectrum • that reproduce the full solution, e.g., heating; qrad, flux; Frad, and J and K.

25. SOS • Carefully select NSOS wavelengths • covering the whole energy spectrum • that reproduce the full solution, e.g., heating; qrad, flux; Frad, and J and K. • Perform radiative transfer on those λ • Paves the way for including velocity-effects • Spans the convective fluctuations better than the opacity binning method • Converges for NSOS→∞

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

27. Applications of the simulations • Improving stellar structure models • T-τ-relations – atmospheric boundary cond. • Calibration of the mixing-length parameter, α • Synthetic spectra/line-profiles • No free parameters, e.g., micro-/macro-turb. • Abundance analysis • Agreement between FeI, FeII and meteoritic • Lower C, N, O – helioseismology doesn’t agree!

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

29. Calibration of α • Use T(τ) from the simulations • Same atomic physics • Match ρ andT at commonP–point • Find significant variation of α over the Teff/gsurf-plane

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

31. Prospects for the Future • Calculate tables of MHD2000 • Use it as basis for new opacity calculation using the newest cross-section data • Implement the SOS radiative transfer scheme in the convection simulations • Build a grid of convection models, using the new EOS, opacities and SOS scheme