1. Lecture 3 Radiative Transfer

2. Lower Solar Atmosphere has two layers:
Photosphere km thick; opaque
Chromosphere - 10,000 km thick; optically thin; cooler

3. (I) Local Thermodynamic Equilibrium
Equation of Transfer:
Optical Depth

5. Thermodynamic Equilibrium
A single value of temperature T is sufficient to describe the thermodynamic state everywhere
state of excitation is governed by Boltzman equation
state of ionization is governed by Saha equation
radiation field is homogeneous and isotropic black body

6. Local Thermodynamic Equilibrium (LTE)
Above conditions are satisfied in a local area.
Such conditions are usually satisfied in the continuum of visible and near infrared, and wings of most spectral lines

7. Absorption lines in LTE

9. The key is to find two quantum mechanics quantities , in the expression of a, damping constant and f, the oscillator strength in addition Doppler width is modified in the existence of turbulence velocity

10. Non-LTE: Statistical Equilibrium
Temperature is now defined as electron temperature, Te , velocity distribution is still Maxwellian, because of frequent collision.

11. Einstein Coefficients
Considering a line radiation between two energy levels EL (lower) and EU (upper), h= EU- EL
Spontaneous emission from upper to lower energy nU AUL()/4 number of emissions per unit time, volume, frequency interval and solid angle. nU : number of atoms in upper level/volume; (): frequency distribution of emitted photon; AUL: Einstein coefficient for spontaneous emission (It’s dimension is 1/time; 108 s-1 is typical.)

12. Induced Emission and Absorption. nU BULI()/4 emission
Induced Emission and Absorption nU BULI()/4 emission nU BLUI()/4 absorption I : radiation intensity; (): line profile; BUL and BLU : Einstein coefficient of induced emission and absorption. B, I have the same unit as A.

13. Continuum Radiation
Photoionization Number of photoionizations from level j, unit time, volume, frequency interval and solid angle:
Radiative Recombination

14. Collision collision transition rate: Cij, which has no direct influence on radiation field, could be for line transition or continuum transition.

15. Source Function Radiative Transfer:

18. Equations of Statistical Equilibrium

19. Limb Darkening (Fig 4.2, Fig 4.3)

22. Model Calculation of LTE (Fig 4.4)
Usually, we select =5000 Angstrom as a reference wavelength as it is free of absorption line. For LTE, we have S=B , from B , we derive T()

25. Non-LTE At temperature minimum T=4200 K,
Non-LTE At temperature minimum T=4200 K, 5000=10 -4, LTE model is no longer applicable. Reason is that photo-ionization dominates over radiative recombination, i.e., neutral population is lower. Figs 4.7 & 4.8 give an example for SiI. Two famous models:
HSRA: Harrard-Smithsonian Reference atmosphere Gingerich et al. (1971) Solar Physics, 18, 347
VAL model: Vernazza, Avrett and Loeser 1976, Ap.J Supp , Ap.J Supp. 45, 635

27. Fig 4.9, Fig 4.10 The models here are called semi-empirical as T is adapted in order to reproduce observed intensity I. Table 4.1
Special Lines:
H line
D3 and He I line
CaII H and K lines
A Simple Atmosphere Model: S is constant =B

28. Forbidden lines
Violates selection rule, so normally the radiative probability is much smaller than the collisional de-excitation, but in the corona, the opposite is true.
Resonant Lines (Strong lines, such as H and K)
Equivalent Width
Integrated Line Intensity
Curve of Growth
Equivalent Width (W) as a function of number of absorbing atoms (N). It is used to determine abundance and temperature.
For weaker lines, W is proportional to N
For strong lines, W is proportional to N 1/2

37. Chemical Composition Chemical composition can be derived from spectrum analyses - Spectrum Synthesis
Standard Symbol: log A = 12 + log(ni/nH) ni=1012 particle/unit volume Table 4.2
Helium: It was discovered in 1868 by Lockyer. Most accurate determination of Y is from inversion of helium seismology. Y = /
Lithium depletion: due to burning of lithium at T=2.510 6 K, t = 5  107 yrs.

39. HW Set # 3
Problems 4.1, 4.2, 4.4, 4.10