Spectral Line Transfer Hubeny Mihalas Chap 8 Mihalas

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 Aλ • Equivalent width 2

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 • Χl ϕν = line opacity × line profile -absorbed +thermal +scattered +thermal line em. +scattered line emission (coherent) • Non-coherent scattering: redistribution function 4

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

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 evaluated at τc = 2/3 • 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 8

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 Hν 11

Formal Solution for Linear Source Function • Set surface Jν equal: • Final solution: • Surface flux Hν 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 • βν 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 Bν at upper level with non-zero temperature • For grey atmosphere, strongest lines: 18