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5. Radiation transport Reading: Shu, Vol.I, Ch.1 and Ch.3 5.1 The radiation transport equation Being equipped with techniques to describe non-relativistic matter we now turn our attention to radiation. We know that in many cirumstances we can use classical optics, i.e. describe radiation as freely propagating rays and waves or photons when interacting with matter. We also know that when interactions with individual photons are concerned, such as the emission or absorption of photons by atoms, a quantummechanical treatment is needed. Let us Frst deFne a few quantities. Assume a area element dA perpendicular to incoming radiation. All rays through dA , whose direction is within the solid angle element d Ω, transport the energy dE through dA in the time interval dt and frequency interval . We deFne I ν = dE dA dt dν d Ω speciFc intensity (5 . 1) Averaging over solid angle yields J ν = 1 4 π I I ν d Ω mean intensity (5 . 2) The energy density spectrum per solid angle element then is u ν (Ω) = dE dV dν d Ω = dE c dt dA dν d Ω = I ν c (5 . 3) And the total energy density spectrum u ν = I u ν (Ω) d Ω = I I ν c d Ω = 4 π c J ν (5 . 4) How do these quantities compare with the notion of a photon distribution function for spin state i ? Using p = hν/c we obtain X i f i ( ~x,~p, t ) = X i dN i d 3 x d 3 p = u ν (Ω) E p 2 dp = I ν c c 3 ( ) 2 h I ν = h 4 ν 3 c 2 X i f i ( ~x,~p, t ) (5 . 5) In thermodynamic equilibrium with matter the radiation Feld should be a black-body or Planck spectrum. Planck I ν = 2 h ν 3 c 2 1 exp ± kT ² - 1 1

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or because the blackbody emission is unpolarized f i ( ~x,~p, t ) = 1 h 3 1 exp ± kT ² - 1 = 1 h 3 n i ( ~x,~p, t ) (5 . 6) where n i is called the photon occupation number. If radiation passes through matter, its speciFc intensity may change. Energy may be added by emission or taken from the radiation Feld by absorption processes. Using dV = dA ds let us deFne the spontaneous emission coe±cient as j ν = dE dV dt dν d Ω = dE ds dA dt dν d Ω = dI ν ds (5 . 7) ²or the absorption let us visualize an ensemble of particles with density n , each of which blocks the radiation over a area σ . The total absorbing area in a test volume then is dA a = n σ dA ds . The absorbed energy is dE = I ν dA a d Ω dtdν = - dI ν dA d Ω dtdν dI ν = - nσ I ν ds (5 . 8) We now deFne the absorption coe±cient α ν and the optical depth τ as α ν = τ = Z s s 0 α ν ds = α ν ds (5 . 9) In total we have derived the radiation transport equation without scattering dI ν ds = j ν - α ν I ν or dI ν = S ν - I ν mit S ν = j ν α ν (5 . 10) The quantity S ν is called the source function. The radiation transport equation has the formal solution I ν ( τ ) = I ν (0) exp(
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