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Analytical Relations for the Transfer Equation (Hubeny & Mihalas 11)

Analytical Relations for the Transfer Equation (Hubeny & Mihalas 11). Formal Solutions for I, J, H, K Moments of the TE w.r.t. Angle Diffusion Approximation. Schwarzschild – Milne Equations. Formal solution. Specific intensity Mean intensity Eddington flux Pressure term.

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Analytical Relations for the Transfer Equation (Hubeny & Mihalas 11)

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  1. Analytical Relations for the Transfer Equation (Hubeny & Mihalas 11) Formal Solutions for I, J, H, KMoments of the TE w.r.t. AngleDiffusion Approximation

  2. Schwarzschild – Milne Equations • Formal solution Specific intensity Mean intensity Eddington flux Pressure term

  3. Semi-infinite Atmosphere Case • Outgoing radiation, μ>0 • Incoming radiation, μ<0 z Τ=0

  4. Mean Field J (Schwarzschild eq.)

  5. F, K (Milne equations)

  6. Operator Short Forms J = Λ F = Φ K = ¼ Χ f(t) = S(t) Source function

  7. Properties of Exponential Integrals

  8. Linear Source FunctionS=a+bτ • J = • For large τ • At surface τ= 0

  9. Linear Source FunctionS=a+bτ • H = ¼Φ(a+bτ) • For large τ,H=b/3 (gradient of S) • At surface τ= 0

  10. Linear Source FunctionS=a+bτ • K = ¼ Χ(a+bτ) • 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

  11. Angular Moments of the Transfer Equation • Zeroth moment and one-D case:

  12. Radiative Equilibrium

  13. Next Angular Moment: Momentum Equation • First moment and one-D case:

  14. Next Angular Moment: Momentum Equation • Radiation force (per unit volume) = gradient of radiation pressure • Further moments don’t help …need closure to solve equations • Ahead will use variable Eddington factorf = K / J

  15. Diffusion Approximation (for solution deep in star)

  16. Diffusion Approximation Terms

  17. Only Need Leading Terms

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

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