I have the second edition of Houghton's "The Physics of Atmospheres". In section 2.2 he says one can do simple radiative transfer calculations in a plane parallel atmosphere by assuming that there are two fluxes, one going vertically up and one vertically down. In reality the radiation is going in all directions, but the two streams represent the amount integrated over the two respective hemispheres.
Now in reality different rays will see a given layer of atmosphere as having different optical depths, depending on the angle a given ray makes with the normal. Houghton says that detailed calculations show that one can take this into account by replacing the true thickness of a given layer dz with 5/3 dz.
This is my question-- Where does that 5/3 factor come from? When I do the calculation for an optically thin absorbing layer (as one would expect for something of infinitesmal thickness dz) I find that an initially isotropic stream (or rather, that half which is going either down or up) will have twice as much energy absorbed as would happen if all the radiation were travelling along a line normal to the layer. So I would have guessed you'd replace dz with 2 dz rather than 5/3 dz, but Houghton is the expert.
Houghton uses this 5/3 factor several times in the book.