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

Definitions Line depth Equivalent width 2 AλAλ

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) 3

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

5 Milne-Eddington Eqtn. Solve at each frequency point across profile.

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

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: 7

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 Φ ν - cont. opacity tends to increase with λ; T gradient smaller higher in atmosphere; lines weaker in red part of spectrum 8 evaluated at τ c = 2/3

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) 9

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

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 H ν 11

Formal Solution for Linear Source Function Set surface J ν equal: Final solution: Surface flux H ν 12

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

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

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

Scattering Lines β ν can be large for strong lines Normalized profile can have black core 16

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

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