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

2. Radiation Transfer
Emission → jl
Il0
Absorption → kl

3. 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)

4. 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)

5. 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)

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

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

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

9. 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

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

11. 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

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

13. 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

14. 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

15. Plane Parallel Approximationz
tl = tl,v secq
tl,v q
s = z secq

16. 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. Limb Darkening