Astron. Astrophys. 332, 610-628 (1998)

7. Concluding remarks

The Approximate Lambda Iteration method presented in this paper, has several advantages over the perturbation method introduced in FS91. Since it is an operator perturbation method, it does not explicitly make use of the smallness of the degree of polarization and is able to handle situations where the polarization is arbitrarily large. Also the speed of convergence is entirely independent of the vector magnetic field. By using a Fourier decomposition with respect to instead of , it offers the possibility to calculate the Hanle effect produced by a magnetic field with a depth-dependent co-latitude, strength and also azimuthal angle. The numerical accuracy of the PALI-H method is also somewhat superior to that of the FS91 perturbation method. It is accurate up to the 6th significant digit, when compared to a direct numerical solution obtained by a Feautrier method. The main advantage is the gain in CPU time, as the PALI-H method is 50 times faster than Feautrier method for most of the practical problems of interest in Solar Physics.

Introducing an irreducible radiation field, we are able to write a six-component vector radiative transfer equation with a vector source function depending only on the optical depth and we could thus give a very simple derivation, significantly simpler than in FS91, of the integral equation which is the starting point for the PALI-H method.

It should be stressed that the perturbation method developed in FS91 can handle both complete and partial frequency redistribution problems whereas here only complete frequency redistribution has been considered. A few preliminary tests have shown that the PALI-H method can be adapted to partial frequency redistribution. That such a generalization is feasible has already been shown for resonance polarization in zero magnetic field by Paletou & Faurobert-Scholl (1997).

It is needless to say that the PALI-H method can handle any radiative transfer problems involving a non-axisymmetric radiation field, as for example the scattering of a non-axisymmetric incident radiation field in a non-magnetic atmosphere. In this case reduces to the () identity matrix.

Here the PALI-H method has been presented and applied with the assumption that the absorption profile is independent of optical depth. It works equally well for real atmospheres where this assumption would not be correct. The depth dependence of the profile need only to be taken into account when calculating the formal solution of the transfer equation (50) and the mean irreducible intensity with the expression given in Eq. (49). Naturally, when is depth-dependent, the kernel of the integral equation (51) depends separately on both the arguments and , but the iterative method never makes explicitly use of this equation.

Considering the inherent uncertainties in the observed data, the simple perturbative approximation for the Hanle effect introduced in this paper should prove very useful for preliminary analyses. This approximation enables one to reduce the six-dimensional vector transfer problem for the irreducible radiation field into a two-dimensional modified resonance polarization problem which yields the azimuthal average of the irreducible radiation field. The azimuth dependent components can then be calculated by solving four scalar transfer problems with known source functions or simply by using an Eddington-Barbier approximation, when only the surface polarization is of interest.

© European Southern Observatory (ESO) 1998

Online publication: March 23, 1998
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