1. Lecture 8
Radiative transfer

2. Solar abundances from absorption lines
From an analysis of spectral lines, the following are the most abundant elements in the solar photosphere
Element
Atomic
Log Relative
Column Density
Number
Abundance
kg m-2
Hydrogen 1 11
Helium 2
-1.01 43
Oxygen 8
-3.07
0.15
Carbon 6
-3.4
0.053
Neon 10
-3.91
0.027
Nitrogen 7 -4
0.015
Iron 26
-4.33
0.029
Magnesium 12
-4.42
0.01
Silicon 14
-4.45
0.011
Sulfur 16
-4.79
0.0057

3. Emission coefficient
The emission coefficient is the opposite of the opacity: it quantifies processes which increase the intensity of radiation,
The emission coefficient has units of W/m/str/kg
Thus, accounting for both processes:

4. The source function
The intensity of radiation is therefore determined by the relative importance of the emission coefficient and the opacity
where we have defined the source function:
The source function has units of intensity, Wm-3sr-1
As the ratio of two inverse processes (emission and absorption), the source function is relatively insensitive to the detailed properties of the stellar material.

5. The source function
The intensity of radiation is therefore determined by the relative importance of the emission coefficient and the opacity
where we have defined the source function:
The source function has units of intensity, Wm-3sr-1
As the ratio of two inverse processes (emission and absorption), the source function is relatively insensitive to the detailed properties of the stellar material.

6. Radiative transfer
This is the time independent radiative transfer equation
For a system in thermodynamic equilibrium (e.g. a blackbody), every process of absorption is perfectly balanced by an inverse process of emission. Since the intensity is equal to the blackbody function and therefore constant throughout the box:

7. Radiative transfer: general solution
i.e. the final intensity is the initial intensity, reduced by absorption, plus the emission at every point along the path, also reduced by absorption

8. Example: homogeneous medium
Imagine a beam of light with Il,0 at s=0 entering a volume of gas of constant density, opacity and source function.
In the limit of high optical depth
In the limit of

9. Break

10. Approximate solutions
Approximation #1: Plane-parallel atmospheres
We can define a vertical optical depth such that
where
i.e.
and the transfer equation becomes

11. Approximate solutions
Approximation #2: Gray atmospheres
Integrating the intensity and source function over all wavelengths,
We get the following simplified transfer equation
Integrating over all solid angles,
where Frad is the radiative flux through unit area

12. The photon wind
In a spherically symmetric star with the origin at the centre
So the net radiative flux (i.e. movement of photons through the star) is driven by differences in the radiation pressure

13. Approximate solutions
Approximation #3: An atmosphere in radiative equilibrium

14. The Eddington approximation
To determine the temperature structure of the atmosphere, we need to establish the temperature dependence of the radiation pressure to solve:
Since
We need to assume something about the angular distribution of the intensity

15. The Eddington approximation
This is the Eddington-Barbier relation: the surface flux is determined by the value of the source function at a vertical optical depth of 2/3