1 / 20

Grey Atmosphere (Mihalas 3)

This study examines the grey atmosphere model and applies the Eddington approximation to find solutions that account for temperature stratification and limb darkening. By implementing Λ-iteration and Unsold iteration methods of discrete ordinates, we refine the grey solution towards greater accuracy. The importance of mean opacities, including Rosseland and Planck means, is highlighted in relation to radiative equilibrium. While initial grey atmosphere models outline general trends, it is crucial to incorporate frequency-dependent opacities and consider convection's role in temperature gradients for a comprehensive understanding of stellar atmospheres.

xander-david
Télécharger la présentation

Grey Atmosphere (Mihalas 3)

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Grey Atmosphere(Mihalas 3) Eddington Approximation SolutionTemperature StratificationLimb Darkening LawΛ-iteration, Unsőld iterationMethod of Discrete Ordinates

  2. 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

  3. Mean Opacities • Flux weighted mean(radiation pressure) • Rosseland mean(good at depth; low opacity weighted) • Planck mean(good near surface;near rad. equil.)

  4. Frequency Integrated Form of TE • TE • Radiative equilibrium • Recall moments of TE: • Apply Eddington approximation K/J = 1/3 H=F/4=conserved quantity q= Hopf functionin general

  5. Constant from Surface Flux

  6. Grey E.A. Limb Darkening Law

  7. 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 Λand Unsőld iteration methods

  8. Λ Iteration Further iterations possible, but convergence is slow since operator important only over photon free path.

  9. Unsőld Iteration * *

  10. Unsőld Iteration

  11. Unsőld Iteration • Initial estimate • Work out ΔH and ΔB • Next estimate • Converges at all depths

  12. Discrete Ordinates: Use S=J

  13. Trial Solution & Substitution

  14. Roots of Characteristic Function

  15. Roots of Characteristic Function

  16. Linear term & Full Solution

  17. Boundary Conditions • Lower limit on semi-infinite atmosphere • No incident radiation from space at top(n equations, n unknowns for Q, Lα) • Set b according to flux

  18. Final Solution • Good even with n small (better than 1% for n=3)

  19. Exact Solution

  20. 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

More Related