Lecture 3 Radiative Transfer

Lower Solar Atmosphere has two layers: Photosphere - 100 km thick; opaque Chromosphere - 10,000 km thick; optically thin; cooler

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

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

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

Absorption lines in LTE

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

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

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.)

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.

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

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

Source Function Radiative Transfer:

Equations of Statistical Equilibrium

Limb Darkening (Fig 4.2, Fig 4.3)

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()

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. 30. 1981, Ap.J Supp. 45, 635

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 10830 line CaII H and K lines A Simple Atmosphere Model: S is constant =B

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

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 = 0.248 +/- 0.002 Lithium depletion: due to burning of lithium at T=2.510 6 K, t = 5  107 yrs.

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