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Radiation Transfer

0. Radiation Transfer. z. I l. d W. d q. d f. q. y. dA. x. 0. Radiation Transfer. Emission → j l. I l 0. Absorption → k l. 0. Special Cases. Radiation Transfer. I l ( t l ) = I l (0) e - t l + S l (1 – e - t l ). I l ( t l ) = I l (0) e - t l. 1) Absorption spectra.

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Radiation Transfer

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  1. 0 Radiation Transfer z Il dW dq df q y dA x

  2. 0 Radiation Transfer Emission → jl Il0 Absorption → kl

  3. 0 Special Cases Radiation Transfer 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)

  4. 0 Special Cases Radiation Transfer (III) 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)

  5. 0 Special Cases Radiation Transfer (IV) Il (tl) = Il(0) e-tl + Sl (1 – e-tl) Il (tl) ≈ Sltl ≈ jl r Ds 2) Emission spectra No significant background source (Il (0)≈ 0) II) Optically thin emission: (tl << 1)

  6. 0 Einstein Coefficients E2 = E1 + hn0 E1 1) Prompt emission→ A21

  7. 0 Einstein Coefficients E2 = E1 + hn0 E1 1) Prompt emission→ A21 2) Absorption→ B12

  8. 0 Einstein Coefficients E2 = E1 + hn0 E1 1) Prompt emission→ A21 2) Absorption→ B12 3) Stimulated Emission→ B21

  9. 0 1) Bound-Bound transitions (lines) Radiation Mechanisms (I) 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

  10. 0 3) Free-free transitions (bremsstrahlung) Radiation Mechanisms (II) jn ~ e-(hn/kT) In Opt. thin Opt. thick ~ n2 n

  11. 0 4) Cyclotron/synchrotron Radiation Mechanisms (III) 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

  12. 0 Synchrotron Radiation Radiation Mechanisms (III) Relativistic electrons: nsy ~ 3.4*106 (B/G) g2 Hz e-n/nsy In n1/3 n nsy

  13. 0 Synchrotron Radiation Radiation Mechanisms (III) 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

  14. 0 5) Electron scattering Radiation Mechanisms (IV) 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

  15. 0 Plane Parallel Approximation z tl = tl,vsecq tl,v s = zsecq q

  16. 0 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]

  17. 0 Limb Darkening

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