Numerical Model Atmospheres (Hubeny

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Numerical Model Atmospheres (Hubeny Mihalas
16, 17)

• EquationsHydrostatic EquilibriumTemperature
  Correction Schemes

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Summary Basic Equations
Equation CorrespondingState Parameter
Radiative transfer Mean intensities, J?
Radiative equilibrium Temperature, T
Hydrostatic equilibrium Total particle density, N
Statistical equilibrium Populations, ni
Charge conservation Electron density, ne
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Physical State

• Recall rate equations that link the populations
  in each ionization/excitation state
• Based primarily upon temperature and electron
  density
• Given abundances, ne, T we can find N, Pg, and ?
• With these state variables, we can calculate the
  gas opacity as a function of frequency

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Hydrostatic Equilibrium

• Gravitational force inward is balanced by the
  pressure gradient outwards,
• Pressure may have several components gas,
  radiation, turbulence, magnetic
• µ atomic mass units / free particle in gas

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Column Density

• Rewrite H.E. using column mass inwards (measured
  in g/cm2), RHOX in ATLAS
• Solution for constant T, µ (scale height)

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Gas Pressure Gradient

• Ignoring turbulence and magnetic fields
• Radiation pressure acts against gravity
  (important in O-stars, supergiants)

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Temperature Relations

• If we knew T(m) and P(m) then we could get ?(m)
  (gas law) and then find ?? and ??
• Then solve the transfer equation for the
  radiative field (S? ?? / ?? )
• But normally we start with T(t) not T(m)
• Since dm -? dz dt? / ?? we can transform
  results to an optical depth scale by considering
  the opacity

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ATLAS Approach (Kurucz)

• H.E.
• Start at top and estimate opacity ? from adopted
  gas pressure and temperature
• At next optical depth step down,
• Recalculate ? for mean between optical depth
  steps, then iterate to convergence
• Move down to next depth point and repeat

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Temperature Distributions

• If we have a good T(t) relation, then model is
  complete T(t) ? P(t) ? ?(t) ? radiation field
• However, usually first guess for T(t) will not
  satisfy flux conservation at every depth point
• Use temperature correction schemes based upon
  radiative equilibrium

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Solar Temperature Relation

• From Eddington-Barbier (limb darkening)

t0 t(5000 Å)
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Rescaling for Other Stars
Reasonable starting approximation
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Temperature Relations for Supergiants

• Differences smalldespite very different length
  scales

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Other Effects on T(t)
Including line opacity or line blanketing
Convection
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Temperature Correction Schemes

• The temperature correction need not be very
  accurate, because successive iterations of the
  model remove small errors. It should be
  emphasized that the criterion for judging the
  effectiveness of a temperature correction scheme
  is the total amount of computer time needed to
  calculate a model. Mathematical rigor is
  irrelevant. Any empirically derived tricks for
  speeding convergence are completely
  justified.(R. L. Kurucz)

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Some T Correction Methods

• ? iteration scheme
• Not too good at depth (cf. gray case)

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Some T Correction Methods

• Unsöld-Lucy methodsimilar to gray case find
  corrections to the source function Planck
  function that keep flux conserved (good for LTE,
  not non-LTE)
• Avrett and Krook method (ATLAS)develop
  perturbation equations for both T and t at
  discrete points (important for upper and lower
  depths, respectively) interpolate back to
  standard t grid at end (useful even when
  convection carries a significant fraction of flux)

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Some T Correction Methods

• Auer Mihalas (1969, ApJ, 158, 641)
  linearization method build in ?T correction in
  Feautrier method
• Matrices more complicated
• Solve for intensities then update ?T