## 4. The formal solution in a particular atmosphereIn the previous two sections, we have seen how to transform matrix in order to be able to write an exponential solution in a convenient form, and to be able to calculate that exponential as easily as the scalar case. We have shown that this procedure applies to absorption matrices whose depth dependence has a few degrees of freedom but still satisfying condition (5). Summing up the last two sections, we can manage gradients in the azimuth of the magnetic field and in the angles and , variations in and , treat analytically a general function in the final matrix , and integrate for any emission vector . In this context we assume an ideal atmosphere satisfying condition (27). We write a formal solution (3) for the diagonalized transfer equation (57): Except for the last singular case, we can apply a diagonal matrix , and write its elements as where the 's coincide with the eigenvalues of the off-diagonal matrix . We can now write a scalar disentangled solution for each one of the four components of the generalized Stokes vector as where we can define four generalized optical depths complex in general, with which the solutions are We can relate now the generalized Stokes vector with the physical Stokes vector by writting in order all the transformations that have been made, i.e. where is the complete transformation, product of all others in the proper order. Evidently is invertible, so that we can make back way from the generalized Stokes vector to the physical one: © European Southern Observatory (ESO) 1999 Online publication: December 22, 1998 |