Spectral line polarization with angle-dependent partial frequency redistribution

2011

A&A

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.

Section: A Decomposition Methods For Hanle Effect With Angle-dependentmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Spectral line polarization with angle-dependent partial frequency redistribution

2011

A&A

“…For angle-averaged PRD functions, the irreducible components of the source terms become independent of the ray direction, even in the presence of a magnetic field, and satisfy fairly standard integral equations that can be used to construct ALI numerical methods of solution. It has been shown in Frisch (2009Frisch ( , 2010 that a similar decomposition of the Stokes parameters can be performed with angle-dependent PRD functions. The Hanle effect is considered in Frisch (2009) and the Rayleigh scattering in Frisch (2010, hereafter HF10).…”

Section: Introductionmentioning

confidence: 99%

Spectral line polarization with angle-dependent partial frequency redistribution

Nagendra²,

Frisch³

2011

A&A

Self Cite

Context. The linear polarization of strong resonance lines observed in the solar spectrum is created by the scattering of the photospheric radiation field. This polarization is sensitive to the form of the partial frequency redistribution (PRD) function used in the line radiative transfer equation. Observations have been analyzed until now with angle-averaged PRD functions. With an increase in the polarimetric sensitivity and resolving power of the present-day telescopes, it will become possible to detect finer effects caused by the angle dependence of the PRD functions. Aims. We devise new efficient numerical methods to solve the polarized line transfer equation with angle-dependent PRD, in planeparallel cylindrically symmetrical media. We try to bring out the essential differences between the polarized spectra formed under angle-averaged and the more realistic case of angle-dependent PRD functions. Methods. We use a recently developed Stokes vector decomposition technique to formulate three different iterative methods tailored for angle-dependent PRD functions. Two of them are of the accelerated lambda iteration type, one is based on the core-wing approach, and the other one on the frequency by frequency approach suitably generalized to handle angle-dependent PRD. The third one is based on a series expansion in the mean number of scattering events (Neumann series expansion). Results. We show that all these methods work well on this difficult problem of polarized line formation with angle-dependent PRD. We present several benchmark solutions with isothermal atmospheres to show the performance of the three numerical methods and to analyze the role of the angle-dependent PRD effects. For weak lines, we find no significant effects when the angle-dependence of the PRD functions is taken into account. For strong lines, we find a significant decrease in the polarization, the largest effect occurring in the near wing maxima.

“…Its first term yields the so-called "single scattered solution". 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).…”

Section: Scattering Expansion Methods For Hanle Effect With Angle-depementioning

confidence: 99%

“…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).…”

Section: Introductionmentioning

confidence: 99%

Spectral line polarization with angle-dependent partial frequency redistribution

Nagendra

2011

A&A

Context. The partial frequency redistribution (PRD) effects in line scattering are necessary ingredients for interpreting the linear polarization observed in strong resonance lines. It is a common practice to use angle-averaged PRD functions for simplicity (obtained by averaging over all scattering angles). It has been established that the use of angle-dependent PRD functions instead of angleaveraged functions is essential for weak fields. Aims. Here we present an efficient iterative method to solve the polarized line radiative transfer equation in weak magnetic fields using angle-dependent PRD functions. Methods. Based on the theory of Stokes vector decomposition for the Hanle effect combined with the Fourier azimuthal expansion of the angle-dependent PRD function, we try to formulate an efficient numerical method of solving the concerned transfer problem in one-dimensional media. This iterative method (referred to as the scattering expansion method, SEM) is based on a series expansion of the polarized source vector in mean number of scatterings (Neumann series expansion). We apply the SEM approach to handle both the exact and various approximate forms of the Hanle scattering redistribution matrix. Results. The SEM is shown to be an efficient method to solve angle-dependent PRD problems involving the Hanle effect. We show that compared to the earlier methods such as the perturbation methods, the SEM is stable and faster. We find that angle-dependent PRD significantly affects the Stokes U parameter.