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The Radiation Field

The Radiation Field. Description of the radiation field. Macroscopic description: Specific intensity as function of frequency, direction, location, and time; energy of radiation field (no polarization) in frequency interval ( ,+d ) in time interval ( t,t+dt ) in solid angle d around n

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The Radiation Field

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  1. The Radiation Field

  2. Description of the radiation field • Macroscopic description: • Specific intensity • as function of frequency, direction, location, and time; energy of radiation field (no polarization) • in frequency interval (,+d) • in time interval (t,t+dt) • in solid angle d around n • through area element d at location r  n

  3. The radiation field  d  dA d

  4. Relation • Energy in frequency interval (,+) • Energy in wavelength interval (,+) • i.e. • thus • with

  5. Invariance of specific intensity The specific intensity is distance independent if no sources or sinks are present. 

  6. Irradiance of two area elements  ´ d dA dA´

  7. Specific Intensity • Specific intensity can only be measured from extended objects, e.g. Sun, nebulae, planets • Detector measures energy per time and frequency interval

  8. Special symmetries • Time dependence unimportant for most problems • In most cases the stellar atmosphere can be described in plane-parallel geometry • Sun: z 

  9. For extended objects, e.g. giant stars (expanding atmospheres) spherical symmetry can be assumed r=R outer boundary z  pr 

  10. Integrals over angle, moments of intensity • The 0-th moment, mean intensity • In case of plane-parallel or spherical geometry

  11. J is related to the energy density u d dV dA l

  12. The 1st moment: radiation flux z y  x 

  13. Meaning of flux: • Radiation flux = netto energy going through area  z-axis • Decomposition into two half-spaces: • Special case: isotropic radiation field: • Other definitions:

  14. Idea behind definition of Eddington flux • In 1-dimensional geometry the n-th moments of intensity are

  15. Idea behind definition of astrophysical flux • Intensity averaged over stellar disk = astrophysical flux  p=Rsin R

  16. Idea behind definition of astrophysical flux • Intensity averaged over stellar disk = astrophysical flux dp p

  17. Flux at location of observer

  18. Total energy radiated away by the star, luminosity • Integral over frequency at outer boundary: • Multiplied by stellar surface area yields the luminosity 

  19. The photon gas pressure • Photon momentum: • Force: • Pressure: • Isotropic radiation field: 

  20. Special case: black body radiation (Hohlraumstrahlung) • Radiation field in Thermodynamic Equilibrium with matter of temperature T

  21. Hohlraumstrahlung

  22. Asymptotic behaviour • In the „red“ Rayleigh-Jeans domain • In the „blue“ Wien domain

  23. Wien´s law x:=hv/kT

  24. Integration over frequencies • Stefan-Boltzmann law • Energy density of blackbody radiation:

  25. Stars as black bodies – effective temperature • Surface as „open“ cavity (... physically nonsense) • Attention: definition dependent on stellar radius!  

  26. Stars as black bodies – effective temperature

  27. Examples and applications • Solar constant, effective temperature of the Sun • Sun‘s center 

  28. Examples and applications • Main sequence star, spectral type O • Interstellar dust • 3K background radiation

  29. Radiation temperature • ... is the temperature, at which the corresponding blackbody would have equal intensity • Comfortable quantity with Kelvin as unit • Often used in radio astronomy

  30. The Radiation Field- Summary -

  31. Summary: Definition of specific intensity  d  dA d

  32. Summary: Moments of radiation field • In 1-dim geometry (plane-parallel or spherically symmetric): Mean intensity Eddington flux K-integral

  33. Summary: Moments of radiation field

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