Radiation Transfer z Il dW dq df q y dA x

Radiation Transfer Emission → jl Il0 Absorption → kl

Bright background source behind a cold absorber Radiation Transfer Special Cases Il (tl) = Il(0) e-tl + Sl (1 – e-tl) Il (tl) = Il(0) e-tl 1) Absorption spectra Bright background source behind a cold absorber (Sl ≈ 0)

Radiation Transfer (III) Special Cases Il (tl) = Il(0) e-tl + Sl (1 – e-tl) Il (tl) = Sl (1 – e-tl) Il (tl) = Sl 2) Emission spectra No significant background source (Il (0) ≈ 0) I) Optically thick emission: (tl >> 1)

Radiation Transfer (IV) Special Cases Il (tl) = Il(0) e-tl + Sl (1 – e-tl) Il (tl) ≈ Sl tl ≈ jl r Ds 2) Emission spectra No significant background source (Il (0) ≈ 0) II) Optically thin emission: (tl << 1)

Einstein Coefficients E2 = E1 + hn0 E1 1) Prompt emission → A21

Einstein Coefficients E2 = E1 + hn0 E1 1) Prompt emission → A21 2) Absorption → B12

Einstein Coefficients E2 = E1 + hn0 E1 1) Prompt emission → A21 2) Absorption → B12 3) Stimulated Emission → B21

Radiation Mechanisms (I) 1) Bound-Bound transitions (lines) Get A21 = spontaneous transition probability per unit time, from quantum mechanics. 2) Bound-Free transitions (recombination / photoionization) Characteristic absorption edges: sabs ~ l3 ~ n-3 jn ~ (ehn/kT – 1) -1 hnthr = c In n

Radiation Mechanisms (II) 3) Free-free transitions (bremsstrahlung) jn ~ e-(hn/kT) In Opt. thin Opt. thick ~ n2 n

Radiation Mechanisms (III) 4) Cyclotron/synchrotron Cyclotron frequency: ncy = eB/(2pmec) ~ 2.8*106 (B/G) Hz Magnetic field B Nonrelativistic electrons Cyclotron radiation In Harmonics: In ~ (v/c)n ncy n

Radiation Mechanisms (III) Synchrotron Radiation Relativistic electrons: nsy ~ 3.4*106 (B/G) g2 Hz e-n/nsy In n1/3 n nsy

Radiation Mechanisms (III) Synchrotron Radiation Power-law distribution of relativistic electrons: Ne(g) ~ g-p jn ~ n-a a = (p-1)/2 kn ~ n-b b = (p+4)/2 Opt. thick In Opt. thin n5/2 n-(p-1)/2 n

Radiation Mechanisms (IV) 5) Electron scattering Most important in very hot (relativistic) plasmas Determined by Thomson cross section: sT = 6.65*10-25 cm2 Power-law distribution of relativistic electrons: Ne(g) ~ g-p jn ~ n-a a = (p-1)/2

Plane Parallel Approximation z tl = tl,v secq tl,v q s = z secq

Rosseland Mean Opacity Kramer’s Opacity Law aR ~ r T-7/2 log(aR [cm-1]) Gas fully ionized; opacity dominated by free-free absorption Gas gradually becoming ionized 104 105 106 107 Temperature [K]

Limb Darkening