## Spectral line polarization with angle-dependent partial frequency redistribution^{⋆}

### III. Single scattering approximation for the Hanle effect

Indian Institute of Astrophysics, Koramangala, Bangalore 560 034, India

e-mail: sampoorna@iiap.res.in

Received: 11 March 2011

Accepted: 6 June 2011

*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 coefficients , 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 term . Keeping only the contribution from 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.

Key words: line: formation / polarization / scattering / magnetic fields / methods: numerical / Sun: atmosphere

*© ESO, 2011*