Spectral line polarization with angle-dependent partial frequency redistribution

Abstract

Context. The solar limb observations in spectral lines display evidence of linear polarization, caused by non-magnetic resonance scattering process. This polarization is modified by weak magnetic fields -the process of the Hanle effect. These two processes serve as diagnostic tools for weak solar magnetic field determination. In modeling the polarimetric observations the partial frequency redistribution (PRD) effects in line scattering have to be accounted for. For simplicity, it is common practice to use PRD functions averaged over all scattering angles. For weak fields, it has been established that the use of angle-dependent PRD functions instead of angle-averaged functions is essential. Aims. We introduce a single scattering approximation to the problem of polarized line radiative transfer in weak magnetic fields with an angle-dependent PRD. This helps us to rapidly compute an approximate solution to the difficult and numerically expensive problem of polarized line formation with angle-dependent PRD. Methods. We start from the recently developed Stokes vector decomposition technique combined with the Fourier azimuthal expansion for angle-dependent PRD with the Hanle effect. In this decomposition technique, the polarized radiation field (I, Q, U) is decomposed into an infinite set of cylindrically symmetric Fourier coefficientsĨ (k)K Q , where K = 0, 2, with −K ≤ Q ≤ +K, and k is the order of the Fourier coefficients (k takes values from −∞ to +∞). In the single scattering approximation, the effect of the magnetic field on the Stokes I is neglected, so that it can be computed using the standard non-local thermodynamic equilibrium (non-LTE) scalar line transfer equation. In the case of angle-dependent PRD, we further assume that the Stokes I is cylindrically symmetric and given by its dominant termĨ in the source terms for the K = 2 components (which give rise to Stokes Q and U), the value of k is limited to 0, ±1, ±2. As a result, the dimensionality of the problem is reduced from infinity to 25 for the K = 2 Fourier coefficients. Results. We show that the single scattered solution provides a reasonable approximation to the emergent polarization computed using the polarized line transfer equation including angle-dependent PRD and the Hanle effect. While the full problem is computationally expensive, the single scattering approximation provides a faster method of solution. The presence of elastic collisions particularly enhances the domain of applicability of this approximation.

The Transfer of Resonance Line Polarization with Partial Frequency Redistribution in the General Hanle–Zeeman Regime

“…Detailed information on the range of validity of this approximation can be found in Faurobert (1987Faurobert ( , 1988 in the absence of magnetic field. For a discussion of the validity of this approximation in the presence of a weak magnetic field see Sampoorna et al (2008) and Sampoorna (2011). Using such approximation, the frequencydependent part of the redistribution function becomes:…”

Spectral line polarization with angle-dependent partial frequency redistribution

“…Keeping only the contribution ofĨ (0)0 0 on the RHS of Eq. (11) to the K = 2 Fourier coefficients, we can show that values of k are limited to k = 0, ±1, and ±2 (see Sampoorna 2011b). Thus the single scattering approximation for each componentS (k)2 l,Q can be written as…”

“…(14) The difficulty in implementing the approximation II in the decomposition method given above was discussed in Sampoorna (2011b). Basically the domains of approximation II depend on both the frequencies (x, x ) and the scattering angle Θ, while we work in the Fourier basis which is inherently azimuth-angle independent.…”

“…For Rayleigh scattering with angle-dependent PRD the single scattering approximation to the polarized source vector components was presented by Frisch (2010). For the Hanle effect with angledependent PRD the corresponding approximation was presented recently by Sampoorna (2011b). The inclusion of higher-order terms in the Neumann series allows us to include multiple scattering effects in the expression for the polarized source vector components.…”

“…Equally important is the physics of line scatteringnamely the details of the frequency redistribution during scattering. It is well-known that for a finer analysis of the polarized spectrum, particularly of the strong resonance lines, the angledependent partial frequency redistribution (PRD) is necessary, as shown in the previous papers of this series (Frisch 2010;Sampoorna et al 2011;Sampoorna 2011b).…”