1. Chapter 9 Stellar Atmospheres

2. Specific Intensity, I I ( or I ) is a vector (units: W m -2 Hz -1 sterad -1 )

3. Specific Intensity, I I is a vector (units: W m -2 Hz -1 sterad -1 ) Illustration of an Anisotopic Radiation Field

4. Specific Intensity, I Define: outward Flux, F, luminosity, L, mean intensity, J and radiation pressure, P. The main objective of all this math is to explain limb darkening!

5. The Radiative Transfer Equation dI   I ds +  j ds Where  is the opacity (note the minus sign) and j  is the emissivity.  is the density and ds is the path length along the beam.

6. Sources of Opacity Bound-Bound absorption, ,bb produces spectral absorption lines, consequently the opacity is small except at discrete frequencies corresponding to transitions in the atom. There is no simple expression for ,bb Bound-Free absorption, ,bf also known as photoionization, occurs when a photon has enough energy to ionize an atom. An approximate estimate for the photoionization cross- section of an H atom is given by σ bf = 1.31 x 10 -15 n -5 ( λ/5000 Å) 3 cm 2

7. Sources of Opacity (continued) Free-free absorption, ,ff is a scattering process, a free electron absorbs a photon in the vicinity of an ion, increasing the speed of the electron. This process can occur over a range of wavelengths and so contributes to the continuum opacity. Free-free absorption is also known as Bremstrahlung which is German for “breaking” as in “acceleration radiation. Electron (Thomson) scattering,  es a photon is scattered, but not absorbed. The cross-section for Thomson scattering is independent of frequency; σ T = 8 π (e 2 /m e c 2 ) 2 = 6.65 x 10 -25 cm 2 3

8. Sources of Opacity (continued) The total opacity,    bb +   bf +   ff +  es For the purposes of stellar atmosphere calculations, a frequency average opacity,  is employed, called the Rosseland mean where;  bf ~ ρT -3.5 m 2 kg -1 see eqn. 9.22  ff ~ ρT -3.5 m 2 kg -1 see eqn. 9.23  es ~ 0.02 (1 + X) m 2 kg -1 (for a gas of H mass fraction X) see eqn. 9.27

9. These opacity functions are plotted in Fig. 9.10

10. Consider a star made of Hydrogen and Helium, then the mass, m, of an average atom, m = (N H m H + N He m He ) / (N H + N He ) …..(1) Where N H and N He are the number of H and He atoms, and m H and m He are the masses of a single H and He atom, respectively. However, it is more useful to express the composition of a star in terms of mass fractions, where X = M H /M Tot and Y = M He /M tot So, invert equation 1, above, to get 1/m = (N H + N He )/ (N H m H + N He m He ) Mean molecular mass and mass fractions.

11. 1/m = (N H + N He )/ (N H m H + N He m He ) the denominator is the total mass of gas 1 = N H + N He m M Tot M Tot Multiply each term on the R.H.S by 1, by forming a ratio m H /m H for the first term and m He /m He for the second term; 1 = N H m H + N He m He m M Tot m H M Tot m He 1 = X + Y m m H m He By definition, m =  m H, so 1 = X + Y  m H m H m He

12. 1 = X + Y  m H m H m He Now cross multiply by m H 1 = X + m H Y  m He where m H /m He = ¼, is just the ratio of atomic weights, A. So, that 1 = X + Y  4 Additional terms may be added for additional elements, but usually one additional term, Z, which refers to the mass fraction of everything, other than H and He, collectively referred to as “metals”. An equivalent expression for the mean electron molecular weight is given by 1 = 1 X + 2 Y  e 4 Where the numerator is the mass fraction times the number of electrons for each element.

13.    is the ‘mean free path’, so that  is the number of mean free paths in distance s. Then the radiative transfer equation; dI    I ds +  j ds becomes dI  I  + j   ds   Usually written as - dI  I - S   ds  Where S = j   is the Source Function    The Source Function, S

14. Optical Depth,  d  =    ds    is the ‘mean free path’, so that  is the number of mean free paths in s. If z is the vertical displacement, then z = -s Cos  (see Fig 9.14)    ds =    dz Cos  Then the radiative transfer equation dI   I ds +  j ds becomes Cos  dI  I - j  d  ’   where  ’ =    dz, is the vertical optical depth.

15. Integrate R.T.E. for the Eddington Atmosphere the radiative transfer equation, integrated over d  is Cos  dI  I - S  d  ’ where  ’ is the vertical optical depth. The frequency dependence in   is taken care of by the Rossland Mean opacity. Take two moments of the R.T.E. which means integrate over solid angle, d , to get the first moment. Then multiply by Cos  and integrate again to get the second moment. After some algebra and using some previous results you can show that T 4 = ¾ T e 4 (  ’ + 2/3) where T is the temperature at a vertical optical depth,  ’ and T e is the effective temperature, characterizing the light we see from the star. The result shows that the light we see from the star emerges not from  ’ = 0 as you would expect, but rather from  ’ = 2/3. This explains limb darkening as at large , you see higher in the atmosphere where the atmosphere is cooler and hence emitting less light.

16. Spectral Line Formation Many of the same concepts are encountered in the formation of spectral lines. The line opacity is greatest at the center of the line. Consequently, the radiation at the line center emerges from higher in the photosphere. This would make a good research project……