Lecture 8: Stellar Atmosphere

1. Lecture 8: Stellar Atmosphere 3. 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

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. 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:

5. 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

6. 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

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

8. 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

9. 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

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

11. 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

12. 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