Analytical Relations for the Transfer Equation (Mihalas 2)

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Analytical Relations for the Transfer Equation
(Mihalas 2)

• Formal Solutions for I, J, H, KMoments of the TE
  w.r.t. AngleDiffusion Approximation

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Schwarzschild Milne Equations

• Formal solution

Specific intensity Mean intensity Eddington
flux Pressure term
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Semi-infinite Atmosphere Case

• Outgoing radiation, µgt0
• Incoming radiation, µlt0

z
?0
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Mean Field J (Schwarzschild eq.)
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(No Transcript)
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F, K (Milne equations)
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Operator Short Forms
J ?
F F
K ¼ ?
f(t) S(t) Source function
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Properties of Exponential Integrals
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Linear Source FunctionSabt

• J
• For large t
• At surface t 0

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Linear Source FunctionSabt

• H ¼F(abt)
• For large t, Hb/3 (gradient of S)
• At surface t 0

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Linear Source FunctionSabt

• K ¼ ?(abt)
• Formal solutions are artificial because we
  imagine S is known
• If scattering is important then S will depend on
  the field for example
• Coupled integral equations

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Angular Moments of the Transfer Equation

• Zeroth moment and one-D case

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Radiative Equilibrium
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Next Angular Moment Momentum Equation

• First moment and one-D case

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Next Angular Moment Momentum Equation

• Radiation force (per unit volume) gradient of
  radiation pressure
• Further moments dont help need closure to
  solve equations
• Ahead will use variable Eddington factorf K /
  J

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Diffusion Approximation (for solution deep in
star)
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Diffusion Approximation Terms
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Only Need Leading Terms
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Results

• K / J 1/3
• Flux diffusion coefficient x T gradient
• Anisotropic term small at depth