#### Transcript Grey Atmosphere (Mihalas 3)

Grey Atmosphere (Mihalas 3) Eddington Approximation Solution Temperature Stratification Limb Darkening Law Λ-iteration, Unsőld iteration Method of Discrete Ordinates 1 Grey or Constant Opacity Case • Simplifying assumption Χν = Χ independent of wavelength • OK in some cases (H- in Sun; Thomson scattering in hot stars) • Good starting point for iterative solutions • Use some kind of mean opacity 2 Mean Opacities • Flux weighted mean (radiation pressure) 0 0 F H d / H d R1 1 B d 3 T 4 T 0 • Rosseland mean (good at depth; low opacity weighted) • Planck mean P B d / B d 0 0 (good near surface; near rad. equil.) 3 Frequency Integrated Form of TE • TE I I S • Radiative equilibrium J S B T • Recall moments of TE: H=F/4=conserved quantity • Apply Eddington approximation K/J = 1/3 q= Hopf function in general dH JS 0 d dK F H K const . d 4 3 J S F const . 4 3 F q 4 4 Constant from Surface Flux 5 Grey E.A. Limb Darkening Law 6 Improvements by Iteration • All based on K/J=1/3 which is too small close to the surface • Flux is not rigorously conserved (close) • Two improvement schemes used to revise the grey solution and bring in closer to an exact solution: Lambda Λ and Unsőld iteration methods 7 Λ Iteration Further iterations possible, but convergence is slow since operator important only over photon free path. 8 Unsőld Iteration * * J 3 2 2 F 3H 2 H , 0 4 3 3 9 Unsőld Iteration 10 Unsőld Iteration • Initial estimate 1 1 3 H F F 4 4 4 • Work out ΔH and ΔB • Next estimate 1 H B0 B 4 2 3 • Converges at all depths 11 Discrete Ordinates: Use S=J 12 Trial Solution & Substitution 13 Roots of Characteristic Function T k 2 k2 1 1 1 12 22 32 14 Roots of Characteristic Function 15 Linear term & Full Solution 16 Boundary Conditions • Lower limit on semi-infinite atmosphere lim I e 0 L 0 • No incident radiation from space at top (n equations, n unknowns for Q, Lα) n 1 L 0 Q i 1 1 k i • Set b according to flux 3 b F 4 17 Final Solution • • Good even with n small (better than 1% for n=3) 18 Exact Solution 3 J F q 4 19 Next steps … • Grey atmosphere shows general trends • But need to account for real opacities that are frequency dependent • Need to check if temperature gradient is altered by convection, another way stars find to transport flux outwards 20