Hanle effect with angle-dependent partial redistribution

Abstract

Abstract. The polarized line transfer equation for the Hanle effect is solved in the framework of an exact partial frequency redistribution (PRD) theory developed by Bommier (1997a,b). In that theory the effect of collisions on the Hanle effect is considered self-consistently. We follow that approach in the line transfer computations presented here. The theory formulated by Bommier clearly recognizes two levels of approximations for exact PRD, in order to facilitate the solution of the line transfer equation. The second level employs angle-dependent redistribution functions, and numerically represents a more difficult problem compared to the third level, which involves only the use of angle-averaged frequency redistribution functions. We present a method which can solve the problem in both the levels of approximation. The method is based on a perturbative approach to line polarization. Although computationally expensive, it offers the only practical means of solving the angle-dependent Hanle PRD problem. We discuss the numerical aspects of assembling the so called "frequency domain dependent redistribution matrices", and also an efficient way of computing the scattering integral. Some examples are presented to illustrate the interesting aspects of the Hanle-PRD problem with angle-dependent frequency redistribution. A comparison of the emergent profiles computed under angle-averaged and angle-dependent redistribution is carried out, and the effect of collisions is investigated. We show that it is necessary to incorporate an angle-dependent redistribution mechanism especially in the computation of the Stokes U parameter. We demonstrate that the use of simple frequency domains is good enough in practical applications of the Hanle PRD theory.

Spectral line polarization with angle-dependent partial frequency redistribution

“…To test the correctness of our three iterative methods, we compared the emergent solutions with those of the perturbative type method developed in Nagendra et al (2002), for an optically thin (T = 10) slab and a relatively thicker (T = 10 3 ) one. In Nagendra et al (2002), the radiation field is represented by the two Stokes parameters I and Q and there is no azimuthal Fourier decomposition of the angle-dependent PRD functions.…”

“…In Nagendra et al (2002), the radiation field is represented by the two Stokes parameters I and Q and there is no azimuthal Fourier decomposition of the angle-dependent PRD functions. For the thin slab, the model parameters are (T, a, , r, Γ E /Γ R ) = (10, 10 −3 , 10 −4 , 0, 1) and for the thick slab they are (T, a, , r, Γ E /Γ R ) = (10 3 , 10 −3 , 10 −4 , 0, 0).…”

“…McKenna (1985) used Hummer's types I, II, andIII, andFaurobert (1987, 1988) the type II 1 . Nagendra et al (2002Nagendra et al ( , 2003 considered a more general problem of angle-dependent PRD for the weakfield Hanle effect using the PRD matrices derived by Bommier (1997b). An even more general redistribution matrix for the Hanle-Zeeman scattering was proposed in Sampoorna et al (2007a,b) and used in Sampoorna et al (2008a) to calculate linear and circular polarizations of spectral lines in magnetic fields of arbitrary strength.…”

“…This so-called "hybrid" approximation has been employed in works by Rees & Saliba (1982), McKenna (1985), Faurobert (1987Faurobert ( , 1988, Nagendra (1988Nagendra ( , 1994, Faurobert-Scholl (1991), Nagendra et al (1999), Fluri et al (2003), Sampoorna et al (2008b), Sampoorna & Trujillo Bueno (2010), among others. For the Rayleigh scattering, the use of an angle-averaged PRD function is not unjustified as the quantitative differences between the Q/I profiles calculated with angle-dependent (AD) and angle-averaged (AA) frequency redistribution remain around 15% or less as shown by Faurobert (1987) and Nagendra et al (2002). Improvements in observational techniques is a strong incentive to develop more elaborate diagnostic tools by taking into account the angle-dependent frequency redistribution.…”

“…They differ rather strongly from previous methods used for angle-dependent PRD. In Dumont et al (1977), Faurobert (1987Faurobert ( , 1988, Nagendra et al (2002), Sampoorna et al (2008a), the radiation field is described by the two Stokes parameters I and Q and the transfer equations for I and Q are solved by Feautrier-type methods, sometimes associated with a perturbation method, or fully perturbative methods, which is based on the linear polarization created by resonance scattering being a few percent only. In McKenna (1985), the radiation field is also described by I and Q, but the radiative transfer equations are solved by a moment equation method.…”