Astron. Astrophys. 322, 896-910 (1997)

An operator perturbation method for polarized line transfer

I. Non-magnetic regime in 1D media

M. Faurobert-Scholl ¹, H. Frisch ¹ and K.N. Nagendra <sup>1, 2

1</sup> Laboratoire G.D. Cassini (CNRS, URA 1362), Observatoire de Nice, BP 229, F-06304 Nice Cedex 4, France
² Indian Institute of Astrophysics, Bangalore 560034, India

Received 22 May 1996 / Accepted 26 June 1996

Abstract

In this paper we generalize an Approximate Lambda Iteration (ALI) technique developed for scalar transfer problems to a vectorial transfer problem for polarized radiation. Scalar ALI techniques are based on a suitable decomposition of the Lambda operator governing the integral form of the transfer equation. Lambda operators for scalar transfer equations are diagonally dominant, offering thus the possibility to use iterative methods of the Jacobi type where the iteration process is based on the diagonal of the Lambda operator (Olson et al. 1986).

Here we consider resonance polarization, created by the scattering of an anisotropic radiation field, for spectral lines formed with complete frequency redistribution in a 1D axisymmetric medium. The problem can be formulated as an integral equation for a 2-component vector (Rees 1978) or, as shown by Ivanov (1995), as an integral equation for a () matrix source function which involves the same generalized Lambda operator as the vector integral equation. We find that this equation holds also in the presence of a weak turbulent magnetic field. The generalized Lambda operator is a () matrix operator. The element describes the propagation of the intensity and is identical to the Lambda operator of non-polarized problems. The element describes the propagation of the polarization. The off-diagonal terms weakly couple the intensity and the polarization. We propose a block Jacobi iterative method and show that its convergence properties are controlled by the propagator for the intensity. We also show that convergence can be accelerated by an Ng acceleration method applied to each element of the source matrix. We extend to polarized transfer a convergence criterion introduced by Auer et al. (1994) based on the grid truncation error of the converged solution.

Key words: stars: atmospheres radiative transfer polarization methods: numerical

Send offprint requests to: M. Faurobert-Scholl

Contents

• 1. Introduction
• 2. Vector and matrix polarized transfer equations
• 3. The generalized Lambda operator
  • 3.1. The propagating and mixing kernels
  • 3.2. The propagating and mixing operators
• 4. An approximate Lambda iteration method for polarized transfer
  • 4.1. Implementation of the iterative method
  • 4.2. Test problems
  • 4.3. Behavior of the correction terms
  • 4.4. Ng acceleration
  • 4.5. Some comments on the solution of the test problems
  • 4.6. Convergence criterion
  • 4.7. CPU requirements
• 5. Conclusions
• Acknowledgements
• Appendix
• Appendix A: the Jacobi iterative method
• Appendix B: A model for the convergence of the PALI iterative method
• References

© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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