Opacity and Optical Depth Absorption dominated intensity change can be written as di λ = κ λ ρ I λ ds with κ λ the absorption coefficient, or opacity The initial intensity I λ 0 of a light beam will be reduced by I λ = I λ 0 e κ λ ρ s Over characteristic length l = κλ ρ the intensity declines by /e
Opacity and Optical Depth In the solar photosphere the density is about ρ = 2. 0-4 kg m -3 and κ 500 = 0.03 m 2 kg - (at λ = 500 nm) This yields for the characteristic distance l = κ 500 ρ = 60 km which is in fact the mean free path for scattered photons defined earlier l = n σ λ
Opacity and Optical Depth Define the optical depth τ λ as dτ λ = κ λ ρ ds yields I λ = I λ 0 e τ λ The intensity of a ray starting at τ λ = will decline by a factor /e before escaping from the star The optical depth is a measure of the distance from the stars surface in units of mean free paths As a result, we see usually no deeper than one optical depth Gas is classified as Optically thin for τ λ << Optically thick for τ λ >>
Opacity The opacity of stellar material depends on the interactions between photons and particles Consider five primary sources Excitation and de-excitation of atoms and ions κ λ bb Ionization κ λ bf Electron scattering off photons and Bremsstrahlung κ λ ff Thomson scattering of photons off free electrons κ es Photoionization of H ions (typical for stars in F0 stage and later) κ H The total opacity is the sum of all five contributions Introduce a mean opacity averaged over all wavelength κ = 0 0 B ν (T) dν T B ν (T) dν T κ ν Rosseland Mean Opacity
Opacity Approximation formulae have been developed Ionization g bf ρ κ λ bf = 4.34 0 2 t Z ( + X ) T 3.5 m 2 kg - Electron scattering off photons and Bremsstrahlung κ λ ff = 3.68 0 8 g ff ( Z ) ( + X ) ρ T 3.5 m2 kg - Mass fractions of hydrogen X, helium Y and metals Z Gaunt factors g bf and g ff about Guillotine factor t between and 00
Opacity Cross section for electron scattering is independent of wavelength κ es = 0.02 ( + X ) m 2 kg - Photoionization of loosely bound (0.75 ev) second electron in hydrogen is main source of continuum opacity in the stars atmosphere Photons with less than 640 nm wavelength can remove second electron κ H = 7.9 0-34 ( Z / 0.02 ) ρ T 9 m 2 kg -
Radiative Transfer Emission is the process that scatters photons into the beam or directly produces them into the beam direction through atomic de-excitation Consider random walk of a photon with N randomly directed steps of fixed length l leading to a distance d traveled d = l N With the optical depth being roughly the number of mean free paths to the surface we obtain for the distance traveled d = τ λ l = l N This gives the average number of steps needed for a photon to travel the distance d before leaving the surface N = τ λ 2 for τ λ»
Limb Darkening The edge, or limb, of the Solar disk looks darker than its center Near the limb, one cannot see to the same radial depth as compared to the center
Radiation Transfer Equation Adding an emission term with emission coefficient j λ to the equation describing the change in intensity di λ = κ λ ρ I λ ds + j λ ρ ds di λ = I λ S λ κ λ ρ ds S λ = j λ κ λ Source Function In case the intensity does not change, the source function is equal to the intensity This is also true for the special case of blackbody radiation
Radiation Transfer Equation Making two assumptions, which are true for stars with large radii The atmosphere is plane-parallel (no curvature) Defining a vertical optical depth τ λ v = τ λ cosθ cosθ di λ dτ λ v = I λ S λ Integration over all wavelengths and assuming the opacity to be wavelength independent yields (grey atmosphere) cosθ di dτ v = I S
Radiation Transfer Equation Integration over all solid angles yields (S depends only on local conditions) d I cosθ dω = I dω S dω dτ v With F rad = I cosθ dω I = 4 π I dω We obtain df rad dτ v = 4 π ( I S )
Radiation Transfer Equation Multiplying by cosθ before integrating over all solid angles yields d dτ v I cos 2 θ dω = I cosθ dω S cosθ dω With P rad = c I cos 2 θ dω cosθ dω = 0 We obtain dp rad dτ v = F c rad
Radiation Transfer Equation In a spherical coordinate system with the star center as the origin dp rad dτ v = F c rad dp rad dr κ ρ = F rad c The net radiative flux is driven by differences in the radiation pressure In an equilibrium stellar atmosphere, absorption and emission are balanced, hence, the radiative flux must be constant F rad = F surface = σ T e 4 T e = effective surface temperature I = S Integration yields P rad = c F rad τ v + C
The Eddington Approximation I = 2 ( I out + I in ) Sir Arthur Eddington F rad = π ( I out I in ) (882-944) 2 π P rad = ( I out + I in ) 3 c With P rad we obtain for the transfer equation 4 π 3 c I = F rad τ v + C c
The Eddington Approximation At the top of the atmosphere where τ v = 0 and I in = 0 we obtain I (τ v = 0) = 2 π F rad which results in C = 2 3 c F rad This gives for the transfer equation 4 π 3 I = F rad τ v + 2 3 3 σ 2 4 π 3 I = T e 4 τ v +
The Eddington Approximation For a local thermodynamic equilibrium we had obtained σ T 4 I = S = B = π which finally yields 3 2 4 3 T 4 = T 4 e τ v + Observation For τ v = 2 / 3 we have T = T e which defines the surface of the star