Lecture 10 - Radiative Transfer (2/16/99)

Stat Mech --- | --- Stellar Structure I

ASTR012 Reading:
Chapter 8-7, 10-2, 13-1 (ZG4)

Notes:
pages 38 - 42

This is an X-ray image of the Sun obtained on February 21, 1994. The brighter regions are sources of increased X-ray emissions. (Courtesy Calvin J. Hamilton, and Yohkoh)

	Key Question:	How does an atmosphere attenuate a beam of light?
	Key Principle:	Radiative Transfer
	Key Problem:	Construct a Model Photosphere

Investigations:

Radiative Transfer II - Opacity and Optical Depth
- How does the absorption depend upon the gas density?
- How does the absorption depend upon the incident intensity?
- How does the absorption depend upon the layer thickness?
- What is the opacity of a gas?
- What is the optical depth of an absorbing medium?
- How do you solve the differential equation for pure absorption?
- Why is the solution to the radiative transfer equation an exponential in the optical depth?
- What are the implications of this for the case of observed photons from the solar photosphere?
- How deep is the tau=1 layer in the solar photosphere?
The Plane-Parallel Atmosphere
- How does the radiative transfer equation scale with inclination angle?
- What is the airmass?
- In what limit can observations through the Earth's atmosphere be approximated as absorption in a plane-parallel medium?
- How can observations at different angles be used to solve for the zenith optical depth and the incident intensity?
- Why is this important for observational astronomy?
Radiative Transfer III - Emission + Absorption
- How does the emission depend upon the density and thickness?
- What is the source function?
- What is the radiative transfer equation?
- Where in the radiative transfer equation is all the physics contained?
- How does the source function "drag" the intensity toward its value?
- What is the solution to the radiative transfer equation for a uniform intensity medium?
- What is the source function for a medium in thermodynamic equilibrium?
- What is the solution for a constant source function?
- How fast is the incident intensity replaced by the thermal emission of the medium?
Opacity in the Solar Photosphere
- How do we know that the solar photosphere is opaque at optical wavelengths?
- How can the photosphere be opaque at the low densities and temperatures prevalent there?
- What is the primary opacity agent in the solar photosphere?
- What is the H- ion?
- What is the ionization potential of H- compared with that of H?
- What is meant by a bound-free transition?
- What is a bound-bound transition?
- What is free-free absorption and emission?
- What is electron scattering?
- Upon what things do the opacity and source function in a stellar atmosphere depend?
Interlude - Iterative Solution of Equations
- How do you find the maximum of the blackbody function (Wien's Law)?
- How do you solve the Saha equation for the temperature at a fixed ionization fraction?
- How can you iteratively solve an equation x=f(x)?
- How is this related to root-finding?
- What determines whether this procedure converges?

Iterative Solution of Numerical Problems

I mentioned in class that a wide range of numerical problems can be solved iteratively, often with just your calculator. These problems are of the type that can be put in the form

x = f(x)

and that starting with a guess x_1 you can iterate

x_i+1 = f( x_i )

to arrive at a stable solution. Note that there are often 2 or more ways to set up the equation, since by taking the inverse x = f^-1(x) should work also (that isnt an exponent, but a functional inverse).

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smyers@nrao.edu Steven T. Myers