Lecture 3 Radiative Transfer Lower Solar Atmosphere has

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 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 n. U AUL ( )/4 number of emissions per unit time, volume, frequency interval and solid angle. n. U : 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 n. U BULI ( )/4 emission n. U 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, 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 Si. I. 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 D 3 and He I 10830 line Ca. II 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/n. H) 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