Presenter:	Kanakatte Nanjundarao Nagendra
Affiliation:	Indian Institute of Astrophysics
Title:	Numerical methods in polarized line formation theory
Authors:	K.N. Nagendra
Form:	invited
Abstract:	In this article we present the governing equations of the scattering theory, the numerical methods of solution, sample results, and benchmark solutions of the polarized spectral line formation theory. While it is possible to develop methods of solving the polarized transfer equation for axisymmetric radiation fields, they become more involved for non-axisymmetric radiation fields -- that too in the presence of external magnetic fields. Basically it becomes necessary to expand away in a Fourier series, the azimuthal dependence of the polarized scattering redistribution matrices. The transfer equation in this so called `reduced form' remains axisymmetric in the Fourier domain. The relevant transfer equation is solved in the Fourier space itself, and then easily transformed into the real space. In the non-magnetic and weak field limits, such exact Fourier expansions have been achieved. However for moderately strong fields, the algebra of the Fourier expansion becomes formidable, but once achieved the computational gains are enormous. Because of the advantage of axisymmetry, the problem lends itself to be solved by appropriately organized PALI (Polarized Approximate Lambda Iteration) methods. The best example of this challenge is the solution of Hanle-Zeeman scattering line formation theory, including partial frequency redistribution and collisional redistribution effects. We compare few methods developed over the years, to solve a variety of polarized line formation problems of various levels of complexity and generality. We dwell upon the PALI method that we developed in the past decade and employed successfully to solve several problems. The PALI technology involves operator perturbation and construction of a suitable method to evaluate an iterated source function correction. In the PRD and polarized problems the frequency and angular coupling together make it a non-trivial task to set up such an operator perturbation algorithm. An example of such a generalization is what we named as Generalized PALI which involved domain based evaluation of PRD, instead of the simple core-wing approach. Another important component of PALI is the `Formal Solver'. A suitable adaptation of it (Feautrier, short characteristic, DELOPAR etc.) is not a difficult task. The PALI methods are extremely fast on a computer and require very small memory. The main difficulty is that they are not easy to generalize for physically complex problems (for example Hanle-Zeeman effect with angle dependent PRD). Therefore, the perturbation method has come to stay in polarized line transfer. It is a very useful tool when complicated and entirely new types of problems have to be approached. We outline how such an iterative + perturbative method performs in the difficult problem of Hanle-Zeeman radiative transfer. We present few benchmark results computed using different methods as simple illustrations.
Session:	4. Polarized radiative transfer, theory and modeling
Presentation date:	Tuesday 18th September
Presentation time:	14:30:00