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

Light where  = wavelength  = frequency units: Hz or sec -1 c = 3.00 x cm sec -1 velocity of light in a vacuum Energy where h = x erg sec Planck’s constant

RegionWavelength Gamma Ray λ< 0.1 Å X-ray0.1 Å < λ< 100 Å Ultraviolet100 Å < λ< 4000Å Visible4000 Å < λ< 7000Å Infrared7000Å < λ<1 mm Microwave1 mm < λ< 10 cm Radio10 cm < λ

Units and other important facts Angstrom Å = cm = m Micron µm = cm = m Wave number = (2π)/ λ cm -1 (number of wavelengths / distance) 1eV = 1.6 x ergs Jansky = 1 Jy = W m -2 Hz -1 = ergs s -1 cm -2 Hz -1

Radio λ=few cm VLA, GBT Mm λ = 1,2,3mm ARO 12m, Interferometers – BIMA, OVRO ALMA Submm λ = 800 microns JCMT, CSO, SMT, South Pole 230, 345, 492 GHz Mid/far IR λ = microns Space only IRAS, ISO, Spitzer Near IR λ = 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

Shape arbitrary – see also UKIRT home page

Optical: λ = 3200 Å 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 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

Definitions: 1. Specific Intensity Consider photons flowing in direction Î, into solid angle dΩ, centered on Î The vector n is the normal to dA Î 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

n dΩdΩ 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

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

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

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

Comments: J ν has the same units as I ν ergs cm -2 sec -1 Hz -1 steradian -1 Even though you integrate over dΩ I ν to get J ν, you divide by 4π Some people define J ν without the 4 π, so the units are ergs cm -2 sec -1 Hz -1 J ν is what you need to know to compute photoionization rates

3. Net Flux F ν is the thing you observe: net energy crossing surface dA in normal direction n, from I ν integrated over all solid angle reduced by the effective area cosθ dA

If I ν is isotropic (not a function of angle), then  There is as much energy flowing in the +n direction as the –n direction Also, if I ν is isotropic, then

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

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.

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

Uniform, isotropic, homogeneous radiation field: F ν =0 J ν = I ν I ν is independent of distance from the source F ν obeys the inverse square law