Astro 300B: Jan. 19, 2011 Radiative Transfer

1. Astro 300B: Jan. 19, 2011Radiative Transfer Read: Chapter 1, Rybicki & Lightman

2. Light whereλ = wavelength ν= frequency units: Hz or sec-1 c= 3.00 x 1010 cm sec-1 velocity of light in a vacuum where h = 6.625 x10-27 erg sec Planck’s constant Energy

4. Units and other important facts • Angstrom Å = 10-8 cm = 10-10 m • Micron µm =10-4 cm = 10-6 m • Wave number = (2π)/ λ cm-1 (number of wavelengths / distance) • eV = 1.6 x 10-12 ergs • Jansky = 1 Jy • = 10-26 W m-2 Hz -1 • = 10-23 ergs s-1 cm-2 Hz-1

5. Radioλ=few cm VLA, GBT Mmλ = 1,2,3mm ARO 12m, Interferometers – BIMA, OVRO ALMA Submmλ = 800 microns JCMT, CSO, SMT, South Pole 230, 345, 492 GHz Mid/far IRλ = 20-350 microns Space only IRAS, ISO, Spitzer Near IRλ = 1-10 microns J: 1.25 microns H: 1.60 microns K: 2.22 microns L: 3.4 microns N: 10.6 microns Sky emission lines very bright for λ= 8000 Å – 2 µm Many atmospheric absorption features λ > 2.2 µm: Thermal emission from telescope dominates NICMOS, WFPC3 on HST JWST

6. Shape arbitrary – see also UKIRT home page

7. Optical: λ = 3200 Å -- 9000 Å Earth’s atmosphere opaque for λ < 3200 Å silicon (ccds) transparent for λ > 9000 Å UV: 911 Å -- Milky Way is opaque for λ < 911 Å 1215 Å Lyman alpha, n=2  n=1 for Hydrogen IUE, HST, GALEX: 1150 – 3200 Å HUT, FUSE 911 -1200 Å (MgFl cutoff) X-ray E = 0.2 – 10 keV Einstein, ROSAT, Asca Chandra, XMM, Astro-E The γ- rays E > 10 keV XTE, GRO, INTEGRAL

9. Definitions: 1. Specific Intensity Î Consider photons flowing in direction Î, into solid angle dΩ, centered on Î The vector n is the normal to dA Energy (ergs) Projection of area dA perpendicular to Î radiant energy flowing through dA, in time dt, in solid angle dΩ, in direction Î Is defined as the constant of proportionality

10. n dΩ θ Perhaps it’s easier to visualize photons falling from the sky from all directions on a flat area,dA, on the surface of the Earth

11. Recall polar coordinates Solid angle: dΩ = sin θ dθ dφ units=steradian

12. Specific Intensity • Comments: • units: ergs cm-2 s-1 Hz-1steradian-1 • depends on location in space • on direction • on frequency • in the absence of interactions with matter, Iνis constant • in “thermodynamic equilibrium”, Iνis the blackbody, or Planck function – a universal function of temperature T only

13. Now consider various moments of the specific intensity: i.e. multiply Iν by powers of cosθ and integrate over dΩ 2. Mean Intensity Zeroth moment where

14. Comments: • Jν has the same units as Iν • ergs cm-2 sec-1 Hz-1 steradian-1 • Even though you integrate over dΩIν to get Jν, you divide by 4π • Some people define Jνwithoutthe 4 π, • so the units are ergs cm-2 sec-1 Hz-1 • Jν is what you need to know to compute photoionization rates

15. 3. Net Flux • Fνis the thing you observe: • net energy crossing surface dA • in normal direction n, • from Iν integrated over all solid angle • reduced by the effective area cosθ dA

16. If Iνis isotropic (not a function of angle), then • There is as much energy flowing in the +n direction • as the –n direction • Also, if Iν is isotropic, then

17. 4. Radiation pressure Momentum of a photon = E/c Pressure = momentum per unit time, per unit area  Momentum flux in direction θ is  Component of momentum flux normal to dA is  Radiation pressure is therefore dynes cm-2 Hz-1

18. We have written everything as a function of frequency, ν However, you can also integrate these quantities over ν, either ν= 0  or ν over some passband.

19. Iν=0 in all directions except towards the point source Fν vs. Iν Specific Intensity Point Source Iν=0 Iν=0 Fν= 0normal to the line of sight to the point source because cosθ = 0 FLUX Fν=0 In every other direction

20. Uniform, isotropic, homogeneous radiation field: • Fν=0 • Jν = Iν • Iνis independent of distance from the source • Fνobeys the inverse square law