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Spectral Line Transfer Hubeny & Mihalas Chap. 8 Mihalas Chap. 10

Explore the definitions, equations, and solutions related to spectral line transfer in this informative chapter. Learn about absorption lines, scattering lines, line depth, and more.

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Spectral Line Transfer Hubeny & Mihalas Chap. 8 Mihalas Chap. 10

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  1. Spectral Line TransferHubeny & Mihalas Chap. 8Mihalas Chap. 10 Definitions Equation of Transfer No Scattering Solution Milne-Eddington Model Scattering Lines, Absorption Lines

  2. Definitions • Line depth • Equivalent width Aλ

  3. Equation of Transfer • Classical approach: absorption of photons by line has two parts(1-ε) of absorbed photons are scattered (e- returns to original state)ε of absorbed photons are destroyed (into thermal energy of gas) (for LTE: ε=1)

  4. Equation of Transfer • Χlϕν = line opacity × line profile -absorbed +thermal +scattered • +thermal line em. +scattered line emission (coherent) • Non-coherent scattering: redistribution function

  5. Milne-Eddington Eqtn.Solve at each frequencypoint across profile.

  6. Simple Case: No Scattering, Weak Line • Transfer equation (source function = Planck) • Recall relation with optical depth • Then from continuum and line flux estimates

  7. Simple Case: No Scattering, Weak Line • Consider weak lines: line << cont. opacity • At line center (maximum optical depth) • Find incremental change in cont. optical depth • Comparing above:

  8. No Scattering, Weak Line • Line depth expression • Line depth depends upon- ratio of line to continuum opacity- gradient of Planck function- line shape same as Φν- cont. opacity tends to increase with λ; T gradient smaller higher in atmosphere; lines weaker in red part of spectrum evaluated at τc = 2/3

  9. Formal Solution for Linear Source Function:Assume ρ, ε, λ constant with depth • Equation of Transfer • Moments Solution • Apply Eddington approximation • Linear source function (so zero second derivative)

  10. Formal Solution for Linear Source Function • Differential equation to solve: • General Solution • Apply boundary condition at depth

  11. Formal Solution for Linear Source Function • Apply boundary condition at surface: • From grey atmosphere solution, get J(τ=0): • Eddington approximation and first moment to get Hν

  12. Formal Solution for Linear Source Function • Set surface Jν equal: • Final solution: • Surface flux Hν

  13. Apply Milne Eddington for Lines • Ratio of line and continuum optical depths • Replace in source function • Apply to emergent flux expression

  14. Apply Milne Eddington for Lines • In continuum away from line: • Normalized flux profile:

  15. Scattering Lines • no scattering in continuum ρ=0 • pure scattering in line ε=0 • Normalized profile

  16. Scattering Lines • βν can be large for strong lines • Normalized profile can have black core

  17. Absorption Lines • no scattering in continuum ρ=0 • pure absorption in line ε=1 • Normalized profile

  18. Absorption Lines • Now for strong lines • Non-zero because we see Bν at upper level with non-zero temperature • For grey atmosphere, strongest lines:

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