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Astron. Astrophys. 342, 201-211 (1999)

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4. The formal solution in a particular atmosphere

In the previous two sections, we have seen how to transform matrix [FORMULA] 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 [FORMULA] of the magnetic field and in the angles [FORMULA] and [FORMULA], variations in [FORMULA] and [FORMULA], treat analytically a general function [FORMULA] in the final matrix [FORMULA], and integrate for any emission vector [FORMULA].

In this context we assume an ideal atmosphere satisfying condition (27). We write a formal solution (3) for the diagonalized transfer equation (57):

[EQUATION]

Except for the last singular case, we can apply a diagonal matrix [FORMULA], and write its elements as

[EQUATION]

where the [FORMULA]'s coincide with the eigenvalues of the off-diagonal matrix [FORMULA]. We can now write a scalar disentangled solution for each one of the four components of the generalized Stokes vector as

[EQUATION]

where we can define four generalized optical depths

[EQUATION]

complex in general, with which the solutions are

[EQUATION]

We can relate now the generalized Stokes vector [FORMULA] with the physical Stokes vector [FORMULA] by writting in order all the transformations that have been made, i.e.

[EQUATION]

where [FORMULA] is the complete transformation, product of all others in the proper order. Evidently [FORMULA] is invertible, so that we can make back way from the generalized Stokes vector to the physical one:

[EQUATION]

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© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998
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