Ingredients for Accurate Simulations of Stellar Envelope Convection

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Ingredients for Accurate Simulations of
StellarEnvelope Convection

• by
• Regner Trampedach
• 03.12.03

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Hydro-dynamics

• Solve Euler equations
• Conservation of
• Mass d?/dt -u ?? - ??u
• Momentum ?du /dt -?u ?u ?(T-Pgas) ?g
• Energy dE /dt -?uE (T-Pgas)?u
  ?qrad
• Hyman 3rd-order time-stepping (predictor/corr)
• Cubic-spline interpolation vertically and compact
  6th order interpolation horizontally
• Regular horizontal and optimized vertical grid

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Numerical Stability

• Schemes using centered derivatives are unstable
• Fixed with artificial diffusion

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Vertical Temperature-cut of ?-Boo
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Input Physics

• Equation of State (EOS)
• Pressure for hydro-static support
• Response to temperature-/density-changes
• Opacity ff bf bb
• radiative transfer gt
• radiative heating qrad,? ???(J?-S?)

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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-sta
  tes

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The OP/MHD- and OPAL-projects

• Prompted by a plea by Simon (1982)
• Pulsations by ?-mechanism didnt agree with
  observations
• Substantial disagreement with helioseismic
  structure of the Sun

OPAL
MHD
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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

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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 is wi Q (Fcr) ?0FcrP (F
  ) dF

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Micro-Field Effects in the Sun
___ OPAL ___ MHD2000 - - old Q (Fcd)
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Quantum effects

• Quantum diffractionfrom Heisenbergsuncertainty
  relation
• Exchange interactionfrom Paulisexclusion
  principle

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Exchange Interactions in the Sun
___ OPAL ___ MHD2000 - - no Exch
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Interaction with Neutral Particles

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

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Effective charges in the Sun
___ OPAL ___ MHD2000 - - const. Z
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Coulomb Interactions

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

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Coulomb Interactions in the Sun
___ OPAL ___ MHD2000 - - Debye-H
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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

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bf-Opacity Before OP/OPAL

• From Peach (1962)

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Opacity According to OP
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Confronting Experiment

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

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Radiative Transfer

• Determines heating/cooling gt structure
• Determines emergent flux/intensity gt link to
  observations
• Transfer Eq. dI ?/dt? (I ? S ?) solved
  for more than 105 wavelengths
• Not possible in convection simulations
• yet...

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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 N bin?8

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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.

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S O S

• 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 ?
• Paves the way for including velocity-effects
• Spans the convective fluctuations better than the
  opacity binning method
• Converges for NSOS?8

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Applications of the simulations

• Improving stellar structure models
• T-t-relations atmospheric boundary cond.
• Calibration of the mixing-length parameter, a
• Synthetic spectra/line-profiles
• No free parameters, e.g., micro-/macro-turb.
• Abundance analysis
• Agreement between Fe I, Fe II and meteoritic
• Lower C, N, O

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Applications of the simulations

• Improving stellar structure models
• T-t-relations atmospheric boundary cond.
• Calibration of the mixing-length parameter, a
• Synthetic spectra/line-profiles
• No free parameters, e.g., micro-/macro-turb.
• Abundance analysis
• Agreement between Fe I, Fe II and meteoritic
• Lower C, N, O helioseismology doesnt agree!

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T -t-relations

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

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Calibration of a

• Use T (t) from the simulations
• Same atomic physics
• Match ? and T at common P point
• Find significant variation of a over the
  Teff/gsurf-plane

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Summary

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

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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