1. Stellar Atmospheres: The Radiation Field 1 The Radiation Field

2. Stellar Atmospheres: 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. Stellar Atmospheres: The Radiation Field 3 The radiation field dA dd  dd

4. Stellar Atmospheres: The Radiation Field 4 Relation Energy in frequency interval (, +  ) Energy in wavelength interval (, +  ) i.e. thus with

5. Stellar Atmospheres: The Radiation Field 5 Invariance of specific intensity The specific intensity is distance independent if no sources or sinks are present. 

6. Stellar Atmospheres: The Radiation Field 6 Irradiance of two area elements dA dA´ d ´

7. Stellar Atmospheres: The Radiation Field 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. Stellar Atmospheres: The Radiation Field 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. Stellar Atmospheres: The Radiation Field 9 For extended objects, e.g. giant stars (expanding atmospheres) spherical symmetry can be assumed r=R outer boundary p r z

10. Stellar Atmospheres: The Radiation Field 10 Integrals over angle, moments of intensity The 0-th moment, mean intensity In case of plane-parallel or spherical geometry

11. Stellar Atmospheres: The Radiation Field 11 J is related to the energy density u l dV dA dd

12. Stellar Atmospheres: The Radiation Field 12 The 1st moment: radiation flux  x y z

13. Stellar Atmospheres: The Radiation Field 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. Stellar Atmospheres: The Radiation Field 14 Idea behind definition of Eddington flux In 1-dimensional geometry the n-th moments of intensity are

15. Stellar Atmospheres: The Radiation Field 15 Idea behind definition of astrophysical flux Intensity averaged over stellar disk = astrophysical flux R p=Rsin

16. Stellar Atmospheres: The Radiation Field 16 Idea behind definition of astrophysical flux Intensity averaged over stellar disk = astrophysical flux dp p

17. Stellar Atmospheres: The Radiation Field 17 Flux at location of observer 

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

19. Stellar Atmospheres: The Radiation Field 19 The photon gas pressure Photon momentum: Force: Pressure: Isotropic radiation field: 

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

21. Stellar Atmospheres: The Radiation Field 21 Hohlraumstrahlung

22. Stellar Atmospheres: The Radiation Field 22 Asymptotic behaviour In the „red“ Rayleigh- Jeans domain In the „blue“ Wien domain

23. Stellar Atmospheres: The Radiation Field 23 Wien´s law x:=hv/kT

24. Stellar Atmospheres: The Radiation Field 24 Integration over frequencies Stefan-Boltzmann law Energy density of blackbody radiation:

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

26. Stellar Atmospheres: The Radiation Field 26 Stars as black bodies – effective temperature

27. Stellar Atmospheres: The Radiation Field 27 Examples and applications Solar constant, effective temperature of the Sun Sun‘s center 

28. Stellar Atmospheres: The Radiation Field 28 Examples and applications Main sequence star, spectral type O Interstellar dust 3K background radiation

29. Stellar Atmospheres: The Radiation Field 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. Stellar Atmospheres: The Radiation Field 30 The Radiation Field - Summary -

31. Stellar Atmospheres: The Radiation Field 31 Summary: Definition of specific intensity dA dd  dd

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

33. Stellar Atmospheres: The Radiation Field 33 Summary: Moments of radiation field