5. Rosseland Mean Opacity of Free-Free Absorption
This exercise is intended to be a warm-up to an important problem in the next set, which deals with the Eddington standard model.
a. The frequency-dependent free-free absorption opacity is given by:
Here, gff is a quantum-mechanical correction factor to the classical result, known as the Gaunt factor.
a. Begin by first deriving an expression for the derivative of the Planck function B, with respect to temperature, B/T.
b. Next, introduce a dimensionless variable x = -hv/kT, and express your result in terms of x.
c. Derive an expression for B/Tok- and plot the resulting function as a function of the variable x. Use the plot to demonstrate that the Rosseland mean opacity is largely determined by K when the frequency is a few times kT/h.
d. Thus, show that the Rosseland mean opacity for free-free absorption obeys the Kramers relationship k pT-3.5. This will be important for the Eddington standard model.