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

## 6. Conclusions

The purpose of this paper was first to deepen our understanding of the integration of the RTE for polarised light and next to improve the basis for numerical codes. The main conclusions is: The fundamental key to solve the RTE of polarised light is the commutation of the absorption matrix and its integral.

When this commutation condition is satisfied:

1. A scalar-like solution can be proposed to the vector equation.

2. A constant absorption matrix satisfies the commutation requirement, however, it is only a sufficient condition, not necessary. After some elaboration one can show that only two constraints at all are necessary instead of the seven inherent in the fully constant absorption matrix.

3. In general, it will possible to diagonalize the absorption matrix and consequently also the RTE with its vector solution. This results in four scalar equations with four scalar solutions. The variables are no more the usual Stokes parameters, but generalized ones, corresponding to general states of polarisation.

4. The solution being analytical, it is valid for quite thick optical layers. The real numerical application is beyond the scope of this paper.

When the commutation condition does not hold, one can turn to Magnus' solution, described shortly in the appendix. Direct application of Magnus' solution to a numerical code seems immature at present. For a general atmosphere, the numerical strategy proposed is to integrate analytically what we can and approximate the rest, that is:

1. Divide the atmosphere into a reasonably number of layers, so that in each of them the commutation condition is only slightly violated.

2. Approximate the general absorption matrix in each layer by an average that satisfies the commutation condition.

3. Apply the solution developed in this paper using last matrix.

4. Applying an approximation for the residual matrix, eventually the one used in DELO (Rees et al., 1989).

As an objective, we intend to improve the efficiency of integration and inversion codes. This will be a must in treating the abundant data expected from multi-line spectropolarimetric observations to be provided by the French-Italian telescope THEMIS.    © European Southern Observatory (ESO) 1999

Online publication: December 22, 1998 