To help in understanding how the structure of the atmosphere
determines the polarization of the emergent radiation, we have
considered the simplest example which captures some of the behavior
seen in detailed calculations. This is a semi-infinite atmosphere
where the Planck function at the wavelength of interest is a linear
function: B_nu(\tau) = a + b*\tau. We also assume that the ratio 
of scattering to total opacity is small and constant with optical
depth. 
  Under this approximation, we are able to obtain an expression
for the behavior of a source term, p_nu(\tau), with depth.
Further, we integrate this expression over \tau to obtain the
Stokes parameters of the emergent radiation.
  The derivation of the analytic result is given in the pdf file
AP2.pdf. Two accompaning figures are Fig_1 and Fig_2.
  In the figures Sig0p02.pdf, Sig0p1.pdf and Sig0p3.pdf, we compare
the results of our analytic approximation with the exact solutions
for the same model obtained by the numerical "Matrix Method" which 
is presented in the parent folder. It is seen that the results of 
this approximation are quite close to the exact results for a  
scattering to opacity ratio of less than the order of 0.1.
 Finally, we show that we can obtain a more general result under
the assumption that the fraction of scattering to total opacity
varies with optical depth as exp(-s*tau), where s is a free 
parameter (and s=0 is the case of constant scattering ratio).

The file "Anal_Pol.ijs" is code in the "J" language to evaluate the
simple analytic expressions.