Lecture 8 Radiative transfer

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

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:

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.

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.

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:

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

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

Break

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

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

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

Approximate solutions Approximation #3: An atmosphere in radiative equilibrium

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

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