Issue 
A&A
Volume 527, March 2011



Article Number  A89  
Number of page(s)  15  
Section  Stellar atmospheres  
DOI  https://doi.org/10.1051/00046361/201015813  
Published online  31 January 2011 
Spectral line polarization with angledependent partial frequency redistribution
II. Accelerated lambda iteration and scattering expansion methods for the Rayleigh scattering
^{1}
Indian Institute of Astrophysics,
Koramangala,
560 034
Bangalore,
India
email: sampoorna@iiap.res.in
^{2}
UNS, CNRS, Observatoire de la Côte d’Azur,
Lab. Cassiopée,
BP 4229, 06304
Nice Cedex 4,
France
Received:
24
September
2010
Accepted:
1
December
2010
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 angleaveraged PRD functions. With an increase in the polarimetric sensitivity and resolving power of the presentday 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 angledependent PRD, in planeparallel cylindrically symmetrical media. We try to bring out the essential differences between the polarized spectra formed under angleaveraged and the more realistic case of angledependent PRD functions.
Methods. We use a recently developed Stokes vector decomposition technique to formulate three different iterative methods tailored for angledependent PRD functions. Two of them are of the accelerated lambda iteration type, one is based on the corewing approach, and the other one on the frequency by frequency approach suitably generalized to handle angledependent 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 angledependent 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 angledependent PRD effects. For weak lines, we find no significant effects when the angledependence 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.
Key words: line: formation / polarization / scattering / magnetic fields / methods: numerical / Sun: atmosphere
© ESO, 2011
1. Introduction
This paper is concerned with a study of the linear polarization of spectral lines due to the resonance scattering of anisotropic radiation. We deal only with twolevel atoms with an unpolarized groundlevel. The theory of resonance scattering shows that in general correlations exist between the directions and frequencies of the incident and scattered photons, a situation referred to as “partial frequency redistribution” (PRD). In some cases, these correlations can be ignored and one has the socalled “complete frequency redistribution” (CRD). The line polarization is very sensitive to the nature of the frequency redistribution mechanism. Polarized radiative transfer problems with PRD are much harder to solve than those with CRD. The complexity is particularly significant because the scattering process is in general described by a (4 × 4) redistribution matrix. In the special case of the nonmagnetic resonance (Rayleigh) scattering in planar atmospheres with axisymmetric boundary conditions, it is sufficient to use (2 × 2) redistribution matrices for the Stokes vector (I,Q).
We are particularly interested in radiative transfer with an angledependent PRD. Dumont et al. (1977) were the first to consider line polarization with angledependent PRD, using the Hummer (1962)’s type I redistribution function. McKenna (1985) used Hummer’s types I, II, and III, and Faurobert (1987, 1988) the type II^{1}. Nagendra et al. (2002, 2003) considered a more general problem of angledependent PRD for the weakfield Hanle effect using the PRD matrices derived by Bommier (1997b). An even more general redistribution matrix for the HanleZeeman 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.
The difficulty in using angledependent PRD matrices comes mainly from the evaluation of the scattering terms because the redistribution matrices depend in an intricate way on the directions and frequencies of the incident and scattered beams. For this reason, most of the linear polarization investigations taking into account PRD effects have been carried out with an approximate angleaveraged PRD, as suggested by Rees & Saliba (1982). In this approximation, frequency redistribution is independent of the scattering angle and also of the polarization phase matrix. This socalled “hybrid” approximation has been employed in works by Rees & Saliba (1982), McKenna (1985), Faurobert (1987, 1988), Nagendra (1988, 1994), FaurobertScholl (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 angleaveraged PRD function is not unjustified as the quantitative differences between the Q/I profiles calculated with angledependent (AD) and angleaveraged (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 angledependent frequency redistribution.
Here we describe three new numerical methods to handle the Rayleigh scattering with an angledependent PRD. Two of them are of the accelerated lambda iteration (ALI) type and the other is based on a Neumann series expansion, which amounts to an expansion in the mean number of scattering events. They differ rather strongly from previous methods used for angledependent PRD. In Dumont et al. (1977), Faurobert (1987, 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 Feautriertype 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.
ALI methods introduced for scalar radiative transfer problems have been generalized to handle Rayleigh scattering and the weakfield Hanle effect (see the review by Nagendra 2003; and also Nagendra & Sampoorna 2009). These methods have the advantage of being much faster while remaining as accurate as the traditional exact or perturbative methods (see Nagendra et al. 1999). They have been so far applied to the angleaveraged PRD only. They make use of a decomposition of the Stokes parameters into a set of new fields, referred to as “reduced intensities” in FaurobertScholl (1991) and Nagendra et al. (1998) and as “spherical irreducible components” in Frisch (2007). These decompositions were introduced to study the Hanle effect. The decomposition method in Frisch (2007) is based on the decomposition of the polarization phase matrix in terms of the spherical irreducible tensors for polarimetry introduced by Landi Degl’Innocenti (1984, see also Bommier 1997a; Landi Degl’Innocenti & Landolfi 2004). In FaurobertScholl (1991, see also Nagendra et al. 1998), the decomposition relies on an azimuthal Fourier expansion method. It is a generalization of Chandrasekhar’s (1950) azimuthal Fourier expansion method for Rayleigh scattering, to the case of Hanle scattering. The based expansion technique turns out to be simpler than the azimuthal Fourier expansion method.
The irreducible spherical components satisfy transfer equations that are simpler than the equations for I and Q. For angleaveraged 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 (2009, 2010) that a similar decomposition of the Stokes parameters can be performed with angledependent PRD functions. The Hanle effect is considered in Frisch (2009) and the Rayleigh scattering in Frisch (2010, hereafter HF10). Here we show how the decomposition described in HF10 can be used to construct ALI and a Neumann series expansion methods.
The outline of the paper is as follows. In Sect. 2, we present transfer equations for the irreducible components of the radiation field and the main steps of the decomposition technique. In Sect. 3, we present two different ALI methods, one of which generalizes the frequencybyfrequency (FBF) method of Paletou & Auer (1995) into a frequencyangle by frequencyangle (FABFA) method and the other that generalizes the corewing separation method, also of Paletou & Auer (1995). In this section, we present a third iterative method, which relies on a Neumann series expansion of the irreducible components of the source terms contributing to the polarization. We refer to this approach here as the “scattering expansion method” because it is equivalent to an expansion in the mean number of scattering events. The three methods described in Sect. 3 make use of the azimuthal Fourier coefficients of order 0, 1, and 2 of the Hummer’s PRD functions of types II and III. In Sect. 4, we show how these Fourier coefficients can be calculated and discuss their main properties. Numerical validation and convergence properties of the iterative methods are presented in Sect. 5. Results are presented in Sect. 6, where we discuss in detail the angledependent PRD effects on the Q/I profiles. Some concluding remarks are presented in Sect. 7.
2. Governing equations
The atmosphere is assumed to be planeparallel, the primary source of photons to be of thermal origin, the incident radiation to be either zero or axisymmetric, and the magnetic field to be zero or microturbulent. These assumptions imply that the radiation field is cylindrically symmetric and can be described by the two Stokes parameters I and Q, if one chooses the reference direction for the measurement of Q such that positive Q is perpendicular to the surface of the atmosphere. Because of the cylindrical symmetry Stokes U = 0.
The polarized transfer equation for the Stokes parameters I and Q can be written in a component form as (1)where μ = cosθ, with θ being the colatitude with respect to the atmospheric normal, τ the line optical depth defined by dτ = −k_{l}dz, with k_{l} the frequencyaveraged line absorption coefficient, and ϕ(x) the normalized Voigt function. The frequency x is measured in units of the Doppler width, assumed to be constant, with x = 0 at line center. The ratio of continuum to line absorption coefficient is denoted by r. The total source vector is given by (2)where S_{c,i} are the components of the unpolarized continuum source vector. We assume that S_{c,0} = B, where B is the Planck function at line center, and S_{c,1} = 0. The line source vector can be written as (3)where dΩ′ = sinθ′ dθ′ dχ′. The outgoing and incoming ray directions Ω and Ω′ are defined, respectively, by their polar angles (θ,χ) and (θ′,χ′). For simplicity, we assume that the primary source is unpolarized, namely that only G_{0}(τ) is nonzero. It is proportional to the Planck function at line center. The term R_{ij}(x,Ω,x′,Ω′) denote the elements of the redistribution matrix for Rayleigh scattering (Domke & Hubeny 1988; Bommier 1997a).
According to the decomposition technique described in HF10, we can write I_{i} and S_{l,i} as (4)(5)We have four terms in the summation over K and Q corresponding to K = Q = 0, K = 2 with Q = 0,1,2. The irreducible tensors are defined in HF10. An explicit expression of Eq. (4) can be found in Eq. (42). The total source vector S_{i} has a decomposition similar to Eq. (5). The irreducible line source vector components may be written as (6)where (7)The coefficients are given in the Appendix of HF10. The functions in Eq. (6) take the form (8)(9)where W_{2}(J_{l},J_{u}) is an atomic depolarization factor depending on the angular momentum of the lower and upper levels of the transition. The coefficients α and β^{(K)} (Bommier 1997b) are branching ratios (10)and (11)where Γ_{R} is the radiative rate, Γ_{I} and Γ_{E} the inelastic and elastic collisional rates, and D^{(K)} the collisional depolarization rate such that D^{(0)} = 0. The coefficient μ_{2} takes the effects of a microturbulent magnetic field into account. It depends on the magnetic field probability density function (see e.g., Landi Degl’Innocenti & Landolfi 2004, p. 215) and is unity in the absence of magnetic fields. Since the Hanle effect acts only in the line core, the coefficient μ_{2} should be set to unity outside the line core. The (with X = II or III) are the azimuthal Fourier coefficients of order 0, 1, and 2 of Hummer (1962)’s PRD functions r_{II} and r_{III}. They are defined by (12)The components satisfy a transfer equation similar to Eq. (1). The source term is given by Eq. (2), where S_{l,i} and S_{c,i} are replaced by and . Introducing the fourcomponent vectors and , we can rewrite Eq. (6) in vector form as (13)The primary source vector is , where G_{0}(τ) = ϵB with ϵ = Γ_{I}/(Γ_{I} + Γ_{R}). The (4 × 4) matrix is diagonal, i.e., . Its elements are defined in Eqs. (8) and (9). The (4 × 4) matrix Γ is a full matrix with elements . Owing to its symmetry, it has only ten independent elements (see Eq. (7) and also HF10). When is independent of μ and μ′ (CRD or angleaveraged PRD), only the two components of the source vector corresponding to the index Q = 0 are nonzero. In this case, one recovers the usual Rayleigh scattering whereby the radiation field and source vector are fully described by twocomponent vectors.
3. Numerical methods of solution
We present three iterative methods to solve the problem of angledependent PRD. The first two are ALI type methods and the third is a scattering expansion method. We assume a planeparallel slab geometry with a given total optical thickness.
3.1. The polarized accelerated lambda iteration approaches
For notational simplicity, we neglect the explicit dependence of ℐ and on τ. The dependence on x and μ appear as subscripts. The formal solution of the transfer equation for the fourcomponent irreducible vector ℐ can be written as (14)where T_{xμ} is the directly transmitted part of the intensity vector and Λ_{xμ} is the frequency and angledependent (4 × 4) integral operator. We introduce the operator splitting written as (15)where is the diagonal approximate operator (see Olson et al. 1986). We now write the total source vector and the line source vector as (16)where n is the iteration index. Combining Eqs. (15) and (16) with Eq. (13), we derive an equation for the line source vector corrections that can be written as (17)where p_{x} = ϕ_{x}/(ϕ_{x} + r) and (18)For this investigation, we chose the Jacobi decomposition of the Λ operator. Superior iterative methods such as the GaussSeidel (GS) and the successive overrelaxation (SOR) technique were introduced into scalar radiative transfer theory by Trujillo Bueno & Fabiani Bendicho (1995). These methods provide faster convergence rates than the Jacobi method. Trujillo Bueno & Manso Sainz (1999) extended these methods to the case of resonance scattering. The generalization to the Hanle effect was developed by Manso Sainz & Trujillo Bueno (1999). In these references, CRD is assumed for the frequency redistribution. The GS and SOR methods were extended to the case of angleaveraged PRD in Sampoorna & Trujillo Bueno (2010). In this first attempt at developing ALI methods for angledependent PRD functions, we chose the simpler Jacobi decomposition. We now describe two different methods of solution for Eq. (17).
3.1.1. Frequencyangle by frequencyangle (FABFA) method
This FABFA method is a generalization of the FBF method introduced by Paletou & Auer (1995). In matrix form, Eq. (17) can be written as (19)where the residual vector r^{n} is given by the righthand side of Eq. (17). At each depth point, r^{n} and are vectors of length 4N_{x} 2N_{μ}, where N_{x} is the number of frequency points in the range [0,x_{max}] ^{2} and N_{μ} is the number of angle points in the range [0 < μ ≤ 1] . The matrix A thus has dimensions (4N_{x} 2N_{μ} × 4N_{x} 2N_{μ}). For a given x, x′, μ, and μ′, the matrix A can be decomposed into (N_{x} 2N_{μ} × N_{x} 2N_{μ}) blocks of 4 × 4 elements. In each block, denoted by , the elements may be written as (20)where m = 1,...,N_{x}, n = 1,...,N_{x′}, α = 1,...,2N_{μ}, and β = 1,...,2N_{μ′}, and E is the identity operator. The coefficients w_{β} denote the μ′ integration weights and g_{mα,nβ} are defined by (21)where are the frequency (x′) integration weights. The FABFA method requires the calculation of the matrix A^{1} before the iteration cycle.
3.1.2. The corewing separation method for angledependent PRD
Here we describe a generalization of the corewing method of Paletou & Auer (1995). According to the ALI methods, the righthand side of Eq. (17) must be calculated as accurately as possible but in the lefthand side there is some flexibility in the choice of the operator acting on the corrections . The choice of this operator affects the speed of convergence of the process but not the final solution. For Rayleigh scattering, we already know that the angledependence of the PRD functions has only a mild effect on the Q/I profiles. Hence, in the matrix in the lefthand side, we retain the two terms corresponding to the index Q = 0 and set to zero the two other terms corresponding to Q = 1 and Q = 2. This is equivalent to making the approximation (22)where ℰ and are (4 × 4) diagonal matrices defined by ℰ = diag [1,1,0,0] and , with (23)(24)and and are the typeII and typeIII angleaveraged (AA) redistribution functions. By comparing with the FABFA method, we find that this approximation leads to a correct converged solution, but at a much less computational cost (see Sect. 5).
Substituting Eq. (22) in Eq. (17), we obtain (25)To the above equation, we now apply the corewing method (Paletou & Auer 1995; Nagendra et al. 1999; Fluri et al. 2003), namely (26)(27)where x_{c} is the frequency that distinguishes the line core and the wing. A value of x_{c} = 3.5 Doppler widths for the separation frequency is a reasonable choice. Combining Eqs. (26) and (27) with Eq. (25), we obtain the expression for the line source vector corrections (28)where and . The frequency and angleindependent fourdimensional vector ΔT^{core} is given by (29)and the frequencydependent but angleindependent fourdimensional vector is given by (30)In Eq. (28), α_{x} is the corewing separation coefficient. In the core α_{x} = 0 and in the wings .
To evaluate ΔT^{core}, we consider Eq. (28) with α_{x} = 0 and apply the operator from left. After some algebra, we obtain (31)where (32)Similarly, to evaluate we apply the operator from the left to Eq. (28), to obtain (33)where (34)The equations for and ΔT^{core} are then incorporated into Eq. (28) to calculate the corrections to the source vector.
3.2. Scattering expansion approach
We present an iterative method of a different type that can also be used with an angledependent PRD. It is based on a Neumann series expansion of the components of the source vector contributing to the polarization. This Neumann series amounts to an expansion in the mean number of scattering events (see Frisch et al. 2009). Its first term yields the socalled single scattering solution. For Rayleigh scattering with an angledependent PRD, the single scattering solution is given in Eq. (25) of HF10. Here, following Frisch et al. (2009), we include higher order terms. The main steps of this method are given below.
We first neglect polarization in the calculation of Stokes I, i.e., we assume that Stokes I is given by the component . This component is the solution of a nonLTE unpolarized radiative transfer equation. We calculate it with a scalar version of the corewing ALI method described in Sect. 3.1.2, using as a PRD function. Knowing , we can calculate the single scattering approximation for each component (Q = 0,1,2). It may be written as (35)The superscript 1 stands for single scattering. The corresponding radiation field is calculated with a formal solver and it serves as a starting point for calculating the higher order terms. For order (n), (36)The iteration is continued until a convergence criterion defined in Sect. 5 is satisfied. The component is calculated only once. We emphasize that this method will be reliable only if the polarization rate remains small, say smaller than 20%.
Fig. 1 Surface plots of azimuth averaged redistribution functions of type II (left panels) and of type III (right panels) with Q = 0. The Xaxis represents the outgoing frequency x, and the Yaxis represents outgoing direction μ. The incoming direction is μ′ = 0.3. The damping parameter a = 0.001. The top two panels correspond to the incoming frequency x′ = 3, and the bottom panels to x′ = 4. 

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Fig. 2 Azimuth averaged redistribution functions of type II (left panels) and of type III (right panels), plotted as a function of the outgoing frequency x, for different choices of μ, and μ′. The damping parameter a = 0.001. Thin lines correspond to x′ = 1 and thick lines to x′ = 4. Solid, dotted and dashed lines correspond respectively to Q = 0,1, and 2. 

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4. The Fourier azimuthal averages and coefficients of the PRD functions
The numerical methods described in the preceding section involve the azimuthal Fourier coefficients and for Q = 0,1,2 defined in Eq. (12). The azimuthal averages with Q = 0 were considered in the classical work by Milkey et al. (1975). Subsequently Vardavas (1976), Faurobert (1987), and Wallace & Yelle (1989) devised methods to evaluate various azimuthal moments of the angledependent frequency redistribution functions. We stress that the moments of order Q = 0 are normalized to the absorption profile when integrated over all the incoming frequencies and angles, while the moments of orders Q = 1 and Q = 2 are normalized to zero. For reasons that we explain below, these normalizations should be satisfied to great accuracy.
Here, we use a 31point GaussLegendre quadrature to perform the integration over (χ − χ′). Since r_{X}(x,μ,x′,μ′,χ − χ′) (X = II and III) are even functions of (χ − χ′), the integration can be limited to the range 0 ≤ (χ − χ′) ≤ π.
In Fig. 1, we show surface plots of and (left panels and right panels, respectively) as a function of the scattered frequency x and the scattered direction μ for the incoming frequencies x′ = 3 and x′ = 4, and the incoming direction μ′ = 0.3. The damping parameter a of the Voigt profile ϕ_{x} is taken to be equal to 0.001. The behaviors at x′ = 3 and x′ = 4 are typical of line core and line wings, respectively. We can observe that the functions and have similar behaviors for x′ = 3 but quite different ones for x′ = 4. For the wing frequency x′ = 4, one recovers features that are typical of the angleaveraged PRD functions, namely as a function of x is peaked at the incident frequency x′, while shows a CRD–type behavior with a single peak at x = 0 for all values of μ. The results shown for μ′ = 0.3 hold for all incoming directions 0 < μ′ < 1.
Figure 2 shows and for all the three values Q = 0,1,2 as a function of the outgoing frequency x for different choices of (μ, μ′) and different incoming frequencies x′. Our choice of (μ,μ′) values for Fig. 2 are identical to those of Wallace & Yelle (1989) in their analysis of the azimuthaveraged type II redistribution function. The frequencies x′ = 1 and x′ = 4 are representative of the line center and wing behaviors, respectively. Figure 2 clearly shows that the sharp peaks appearing in Fig. 1 for Q = 0 also exist for Q = 1,2 for forward and backward scattering (see panels b, c, e, f). Actually, surface plots for Q = 1,2 are very similar to the surface plots shown in Fig. 1. However, because they are normalized to zero, and for Q = 1,2 can take negative values as can be observed in Fig. 2. In the upper panels of this figure, one can observe that the coefficients decrease in magnitude with increasing Q. This result agrees with the curves shown by Domke & Hubeny (1988) for μ = 0.8 and μ′ = 0.2 (see their Fig. 3). However, for the special cases of forward and backward scattering (panels b, c, e, f), the Fourier coefficients for Q = 1 and 2 can become larger than the Q = 0 coefficients.
A significant difference between angleaveraged and angledependent PRD functions is the presence of very sharp and narrow peaks for x ≃ x′ and μ ≃ μ′ for the angledependent PRD functions. This has important implications for the numerical evaluation of the scattering integrals (see Eq. (6)). It is well known in scalar nonLTE radiative transfer calculations that an accurate normalization is required for the profile function and the PRD functions, especially for lines with a large optical thickness and a very small thermalization parameter. Any error will indeed act as a spurious sink or source of photons. To achieve the normalization to a high accuracy, a very fine frequency grid is needed. However, the use of such fine frequency grids in the transfer calculations require large computing resources. Hence, strategies have been developed in the past (see for e.g., Wallace & Yelle 1989) to handle frequency quadratures in angledependent PRD problems. For unpolarized transfer problems, Adams et al. (1971) proposed a natural cubic spline representation for the radiation field. In this paper, we have developed our own strategy to maintain the computational cost of the radiative transfer calculations at a reasonable level while ensuring a high accuracy: is normalized to ϕ_{x} with an accuracy of 99% and the integrals of , Q = 1,2, over incoming frequencies and directions are around 10^{7}.
We first start with a frequency grid typical of PRD line transfer problems on which the transfer equation will be solved. The choice of the actual frequency grid is given below in Sect. 5. We then subdivide each frequency interval into a fine mesh of Simpson quadrature points (e.g., 41point Simpson formula) on which we calculate the coefficients . A sevenpoint Gaussian quadrature formula with μ in the range 0 < μ ≤ 1 is used for the angular grid.
To handle the peaks that occur in the forward and backward scattering situations (see e.g. Fig. 1) we proceed in the following way. We introduce a cutoff scattering angle Θ_{cut − off} = 10^{6} radians and assume that the PRD functions keep a constant value, given by the values at the cutoff, when Θ < Θ_{cut − off} or (π − Θ) < Θ_{cut − off}. This practical trick has implications for the normalization accuracy of the redistribution functions, but we have verified that Θ_{cut − off} = 10^{6} radians is a reasonable choice. To approach the two extreme values of Θ as close as possible, it is sufficient to employ a sevenpoint Gaussian quadrature in [0 < μ ≤ 1] .
5. Validation and convergence properties of the iterative methods
We consider isothermal, selfemitting planeparallel slab atmospheres with no incident radiation at the boundaries. These slab models are characterized by a set of input parameters (T,a,ϵ,r,Γ_{E}/Γ_{R}), where T is the optical thickness of the slab. The Planck function is set to unity. The depolarizing collisional rate D^{(2)} is assumed to be 0.5 × Γ_{E}. The thermalization parameter ϵ, which is actually a photon destruction probability, is defined by ϵ = Γ_{I}/(Γ_{R} + Γ_{I}) = 1 − β^{(0)}. The PRD function defined in Eq. (8) can also be written as (37)where γ_{coh} = (1 + Γ_{E}/(Γ_{R} + Γ_{I}))^{1}. For Γ_{E} = 0, one has pure r_{II} since γ_{coh} = 1. For small values of ϵ, γ_{coh} is about 1/(1 + Γ_{E}/Γ_{R}). In all our calculations, W_{2} = μ_{2} = 1.
For all the figures presented in this paper, we use a logarithmically spaced τgrid with 5 points per decade, with the first depth point at τ_{1} = 10^{2}. For the frequency grid, we use equally spaced points in the line core and logarithmically spaced ones in the wings. Furthermore, the maximum frequency x_{max} is chosen such that the condition ϕ(x_{max})T ≪ 1 is satisfied. We have typically 70 points in the interval [0,x_{max}] .
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 angledependent PRD functions. For the thin slab, the model parameters are (T,a,ϵ,r,Γ_{E}/Γ_{R}) = (10,10^{3},10^{4},0,1) and for the thick slab they are (T,a,ϵ,r,Γ_{E}/Γ_{R}) = (10^{3},10^{3},10^{4},0,0). We have found that our new iterative methods yield emergent solutions that are in very good agreement with the results of the perturbation method used in Nagendra et al. (2002, differences in the ratio Q/I are at most 6%).
Fig. 3 Maximum relative change of the Stokes I source vector component and of the surface polarization as a function of the iteration number. Slab model with parameters (T,a,ϵ,r,Γ_{E}/Γ_{R}) = (2 × 10^{4},10^{3},10^{4},0,0) are used. The panel a) corresponds to the FABFA method and the panel b) to the corewing method for angledependent PRD. The panel c) corresponds to the scattering expansion method. Different line types are: solid – and dotted – . 

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Figure 3 shows the variations in and versus the iteration number. The model parameters used are (T,a,ϵ,r,Γ_{E}/Γ_{R}) = (2 × 10^{4},10^{3},10^{4},0,0). The panel (a) corresponds to the FABFA method, and the panel (b) to the corewing method. The convergence behavior of these two new ALI methods is nearly similar to the standard Polarized ALI (PALI) methods (Nagendra et al. 1999). The approximation introduced for the corewing method (see Eq. (22)) does not seem to affect the speed of convergence significantly. We also find that the speeds of convergence are about the same for angleaveraged and angledependent PRD. The main parameters that can affect the speed of convergence are the optical thickness of the line (there being faster convergence for optically thin than optically thick lines), and the frequency grid, coarser grids leading to faster convergence (Olson et al. 1986). The panel (c) of Fig. 3 shows the variation in and for the scattering expansion method. The number of iterations in the second iteration stage is clearly small (~60) compared to the first iteration stage (~140).
To illustrate the convergence properties of the ALI methods, we show in Fig. 4 the convergence history of the four components of at x = 0 calculated with the corewing method. To reduce the number of lines, every fourth iteration solutions are plotted. We note that for x = 0 satisfies the law at the surface. As the slab is not optically very thick (T = 2 × 10^{4}), has not fully reached unity at the midslab. All the other components go to zero at the mid slab. In the wing frequencies, say x = 4, the rate of convergence for all the four components of is quite large (figure not shown here).
Fig. 4 Convergence history of the four components of the source vector for x = 0 and μ = 0.025 calculated with the corewing method. Same model as in Fig. 3. The dotted lines show the initial solutions (ϵB for , and zero for all other components). Since the slab is symmetric about the midplane, the results are shown only for a halfslab, τ ∈ [0,T/2] . 

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Figure 5 shows the convergence history of the components for x = 0 and μ = 0.025 calculated with the scattering expansion method, for the same atmospheric model as in Figs. 3 and 4. The dotted lines indicate the single scattered solution. For and , the single scattered solution is very close to the converged solution, while for a few iterations are needed to reach the converged solution. In the last panel of Fig. 5, we show the convergence history for the ratio Q/I. We have not performed a systematic investigation of the variation in the speed of convergence with the slab thickness as was done for CRD in Frisch et al. (2009), but the calculations that we have performed suggest that the behavior observed with CRD will also hold for angledependent and angleaveraged PRD, namely, a very good approximation to the emergent polarization is provided by the single scattered solution for optically thin slabs (T ≪ 10) and optically very thick ones (T ≥ 2 × 10^{6}), but is not reliable for intermediate optical thicknesses. As shown here, the problem can be cured by considering higher order terms in the scattering expansion. For example, for a very strong line such as Ca i 4227 Å the Q/I profile can be fitted fairly well with a single scattered solution (Anusha et al. 2010), although the line core is somewhat underestimated. The fit could probably be improved by considering higher order terms in the scattering expansion.
Fig. 5 The scattering expansion method. The three upper panels show the convergence history of the components for x = 0 and μ = 0.025. The last panel shows the convergence history of the ratio Q/I. In all the panels, the dotted lines represent the single scattered solution. The same slab model as in Figs. 3 and 4 is used. 

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In terms of computing resources, in particular computing time, the three iterative methods behave quite differently. In Table 1, we present the CPU time requirements for the three iterative methods in the case of angledependent PRD. The CPU times spent in specific parts of the computational process are listed, for each of the methods. The model used is the same as in Figs. 3–5. Since the computation of the azimuthal Fourier coefficients are common to all the three iterative methods, they are excluded when estimating the CPU time requirements. The computing time for can be huge (several hours) depending on the frequency and angle grids. For each method, the last but one column in Table 1 gives the number of iterations multiplied by the CPU time per iteration. This number of iterations is smaller than the results presented in Fig. 3 because we have applied the Ng acceleration to all the three iterative methods.
CPU time requirements for the iterative methods after the calculation of the azimuthal coefficients , Q = 0,1,2.
Fig. 6 A comparison of emergent Stokes profiles at μ = 0.11 for angledependent (solid lines) and angleaveraged (dashed lines) PRD functions. Left panel: slab model with parameters (T,a,ϵ,r,Γ_{E}/Γ_{R}) = (10,10^{3},10^{4},0,1). Right panel: slab model with parameters (T,a,ϵ,r,Γ_{E}/Γ_{R}) = (2 × 10^{4},10^{3},10^{4},0,0). 

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The scattering expansion method requires the construction of a scalar Λ^{∗} operator needed to calculate in the first stage of the method. The computation of Λ^{∗}, , and the single scattered solution requires roughly 3/20th of the total computing time. Thus, most of the time is spent in calculating the higher order terms in the scattering expansion. For the corewing method, almost the entire CPU time is spent in the iterative cycle itself. Each iterative cycle is about 20% shorter than the time per iteration of the scattering expansion method. In addition, a larger number of iterations are needed for convergence. It is the need to invert the large matrix A^{1} (see Eq. (19)) that considerably slows down the FABFA method. We also note that the time per iteration is about three times as long as with the corewing method. The CPU times given in Table 1 correspond to the pure R_{II} case (Γ_{E}/Γ_{R} = 0). The inclusion of collisions does not change the CPU times for different stages of computational process. The only difference occurs in the number of iterations required for convergence. The larger the value of Γ_{E}/Γ_{R}, the smaller the total number of iterations.
For the corewing and the scattering expansion methods, the computing time per iteration scales as N_{d}(N_{x}2N_{μ}), where N_{d} is the total number of depth points. For the FABFA method, the total computing time scales as N_{d}(N_{x}2N_{μ})^{2}. For the FABFA and corewing methods, the number of iterations can be reduced by using a GaussSeidel decomposition or SOR method instead of a Jacobi decomposition. In any case, the FABFA method will remain slow, by a large factor, compared to the corewing method. We recall that for the perturbation method used in Nagendra et al. (2002), the computing time per iteration scales as N_{d}(N_{x}2N_{μ}N_{ϕ}), with N_{ϕ} the number of azimuths needed to describe the azimuthal variation of the Stokes source vector. The number of iterations required for convergence is around 20.
6. Stokes profiles calculated with angledependent and angleaveraged PRD functions
We compare the Stokes parameters I, Q, and the ratio Q/I calculated with angleaveraged and angledependent PRD functions, for several atmospheric models. The Stokes parameters calculated with angledependent PRD functions are shown as solid lines and those calculated with angleaveraged PRD functions as dashed lines. The results are analyzed with the help of the decomposition, which may be written as (42)We recall that the irreducible components depend on τ, x, and μ. The contributions of the component go to zero at both the limb and the disk center, for both Stokes I and Stokes Q.
In Fig. 6, we show the emergent I, Q/I, and Q profiles at μ = 0.11 for angledependent and angleaveraged PRD functions. The angleaveraged solution is calculated with the PALI method of Fluri et al. (2003) and the angledependent solution with the corewing method introduced here. In the lefthand side panels, T = 10 and the PRD used is an equal mixture of r_{II} and r_{III}. In the righthand side panels, T = 2 × 10^{4} and the PRD used is pure r_{II}. Figure 6 clearly shows that the angleaveraged and angledependent solutions may differ significantly and that the angleaveraged emergent polarization may be smaller or larger than the angledependent one. We now try to explain the reasons for this, by considering the components .
In Fig. 7, we show the components corresponding to the two atmospheric models in Fig. 6. The components with Q ≠ 0 are zero for the angleaveraged PRD functions. The components are essentially equal for the AD and AA cases when T = 10, while they slightly differ around x = 3 when T = 2 × 10^{4}. Since Stokes I is dominated by , it is also nearly independent of the choice of the PRD function. Stokes Q consists of the three components , , and . We can observe that has very different values in the AD and AA cases, being smaller in absolute value in the AD case. This is a somewhat unexpected result. The reason for the difference between the AA and AD results is the use of in the AD case and the angleaveraged redistribution function in the AA case.
Fig. 7 A comparison of emergent at μ = 0.11 computed for the angleaveraged (dashed lines) and angledependent (solid lines) PRD. Left panel: slab model with parameters (T,a,ϵ,r,Γ_{E}/Γ_{R}) = (10,10^{3},10^{4},0,1). Right panel: slab model with parameters (T,a,ϵ,r,Γ_{E}/Γ_{R}) = (2 × 10^{4},10^{3},10^{4},0,0). 

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Fig. 8 Stokes I and Q/I profiles at the surface for different values of the heliocentric angle computed for the angleaveraged (dashed lines) and the angledependent (solid lines) PRD. The bottom three panels show the absolute difference in Q/I between the AA and AD profiles. The atmospheric model used is (T,a,ϵ,r,Γ_{E}/Γ_{R}) = (2 × 10^{4},10^{3},10^{4},0,0). 

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Fig. 9 The μ dependence of the irreducible components of ℐ (left panels) and (right panels) at the surface of a slab. Different line types are: thin solid line – μ = 0.025, dotted – μ = 0.129, dashed – μ = 0.297, dotdashed – μ = 0.5, dashtripledotted – μ = 0.7, longdashed – μ = 0.87, and thick solid – μ = 0.97. The atmospheric model is the same as in Fig. 8. 

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Equation (42) shows that for small values of μ, only and contribute to Stokes Q and that . For T = 10, the two terms have the same sign and when added together give a value of Stokes Q that is slightly smaller in the AD case than in the AA case. For spectral lines with a small optical thickness, the role of PRD is essentially negligible. These lines can be modeled reasonably well with CRD (Sampoorna et al. 2010). Thus, one may not expect large differences between angleaveraged and angledependent polarization profiles. For T = 2 × 10^{4}, and are mainly negative, with having a slightly larger absolute value than . When added together, they give a negative value, but because of the minus sign, one ends up with a positive Stokes Q, which is however smaller than the corresponding AA Stokes Q.
We now examine how the differences between the AD and AA polarizations vary with the position on the disk, the optical thickness T of the slab, the thermalization parameter ϵ, the value of the continuum absorption, and the ratio Γ_{E}/Γ_{R}.
6.1. Centertolimb variations
We know that the angledependent PRD functions become azimuthally symmetric at the disk center. Thus we can expect the AD and AA polarization profiles to become very close to each other when μ → 1. This is indeed what we observe in Fig. 8 where we show the emergent profiles of Stokes I and the ratio Q/I. In the bottom three panels, we show the absolute difference between the AA and AD Q/I profiles. As μ → 1, the absolute difference clearly goes to zero.
We show in Fig. 9 the dependence on μ of the components and at the surface. For , the dependence on μ comes from the AD redistribution functions (see Eq. (13)). We recall that the source vector components are independent of μ for AAPRD. For the components , the variations with μ are caused by the limbdarkening and the μvariation in the source vector components. In the right panels of Fig. 9, we can observe that is almost independent of μ and we can verify that , Q = 1,2, go to zero when μ is close to one. The variation in is rather monotonic, but that of is not. The component increases towards the disk center, as it is controlled by the magnitude of (see dotted lines in panel of Fig. 5). In the left panels of Fig. 9, one can verify that increases from the limb to the disk center, while the components , Q = 1,2, go to zero at the disk center.
Fig. 10 The emergent I, Q/I, and Q profiles at μ = 0.11 computed for the angleaveraged (dashed lines) and the angledependent (solid lines) PRD. Left panel shows the effect of optical thickness T when ϵ = 10^{4}. Different line types are: thin lines T = 2 × 10^{4}, medium thick lines T = 2 × 10^{6}, and thick lines T = 2 × 10^{8}. Inset in Q/I panel shows T = 2 × 10^{8} case for a larger frequency range. Right panel shows the effect of ϵ for optical thickness T = 2 × 10^{4}. Different line types are: thin lines ϵ = 10^{2}, medium thick lines ϵ = 10^{6}, and thick lines ϵ = 0. Remaining common parameters are (a,r,Γ_{E}/Γ_{R}) = (10^{3},0,0). 

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6.2. Effects of the optical thickness T and thermalization parameter ϵ
In Fig. 10, we compare the angleaveraged and the corresponding angledependent Stokes I, Q, and Q/I profiles for slabs with different values of the optical thickness T and the thermalization parameter ϵ. For the angledependent case, the calculations were performed with the corewing method. For all the examples shown in Fig. 10, the optical thickness is equal to or larger than 2 × 10^{4}. One can observe that the ratio Q/I is always smaller for the angledependent than for the angleaveraged PRD functions. This situation appears to be typical of optically thick lines in isothermal atmospheres (see also Nagendra et al. 2002).
In the left panels, ϵ = 10^{4}. The three values chosen for T correspond to effectively thin (ϵT = 2), effectively thick (ϵT = 200), and semiinfinite like conditions (ϵT = 2 × 10^{4}). The amplitudes of Q/I and Q in the near wing peaks decrease with increasing T. For Q/I, this is accompanied by a decrease in the differences between the angleaveraged and angledependent values. In the particular case of T = 2 × 10^{8}, typical features of semiinfinite atmospheres can be observed. For example, the appearance of double peaks – one narrow peak in the near wings, and a second broader one in the far wings – is a typical behavior of Q/I in strong resonance lines such as that of the Ca ii K line (see e.g., Stenflo 1980; Holzreuter et al. 2006).
Fig. 11 The emergent Q/I profiles at μ = 0.11 computed for the angleaveraged (dashed lines) and the angledependent (solid lines) PRD. The model parameters are (T,a,ϵ,Γ_{E}/Γ_{R}) = (2 × 10^{4},10^{3},10^{4},0). The parameter r takes the values 10^{8}, 10^{6}, and 10^{4}. 

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In the right panels of Fig. 10, the optical thickness is T = 2 × 10^{4}. As ϵ increases from 10^{6} (effectively thin, ϵT = 2 × 10^{2}) to 10^{2} (effectively thick, ϵT = 2 × 10^{2}), the magnitudes of both I and Q increase because the number of photons that are emitted increases, as does the strength of the coupling between the radiation field and the thermal pool. However, the ratio Q/I decreases because the number of polarized photons is reduced by the thermalization of the radiation field (see e.g., Saliba 1986). For ϵ = 10^{6}, the angleaveraged and angledependent profiles display significant differences in the Q/I profiles. This difference decreases when ϵ = 10^{2}.
We also considered the case ϵ = 0 (conservative scattering). In this case, there are no internal sources of photons (ϵB = 0), but there is an incident field at the lower boundary I(τ = T,x,μ) = B. For this model, the primary source of photons G_{0}(τ) decreases towards the surface at τ = 0. For the emergent Stokes I and Q profiles, the differences between the angleaveraged and angledependent cases are insignificant. Some differences appear however in the ratio Q/I around the line center (0 < x < 4).
Fig. 12 The emergent Stokes parameters I and Q and the ratio Q/I at μ = 0.11 computed for the angleaveraged (dashed lines) and the angledependent (solid lines) PRD. The model parameters are (T,a,ϵ,r) = (2 × 10^{4},10^{3},10^{4},0). The parameter Γ_{E}/Γ_{R} takes the values 1 and 10. 

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6.3. Effect of the continuum strength parameter r
All the results shown in the previous figures were obtained with a continuum absorption set to zero. In Fig. 11, we show the Q/I profile for values of the continuum strength parameter r set to 10^{8}, 10^{6}, and 10^{4}. The atmospheric model is the same as in Fig. 6b except for the value of r. When r increases, one can observe a significant decrease in amplitude of the near wing peak, an effect already investigated in Faurobert (1988), and a gradual disappearance of the differences between the angleaveraged and angledependent Q/I profiles. For r = 10^{2} (result not shown here), the differences become essentially zero.
6.4. Effect of the elastic collisions Γ_{E}
A change in the value of Γ_{E}, modifies the coherency factor γ_{coh} defined in Sect. 5, hence the relative contributions of r_{II} and r_{III}. It is well established that the wings of Stokes I and the linear polarization are quite sensitive to the PRD mechanism (Nagendra et al. 2002). We show in Fig. 12 the Stokes parameters I and Q, and the ratio Q/I calculated with the angleaveraged and angledependent PRD functions with the corewing method. We show two examples, Γ_{E}/Γ_{R} = 1, which corresponds to an even mixture of r_{II} and r_{III}, and Γ_{E}/Γ_{R} = 10, which corresponds to an uneven mixture of r_{II} and r_{III} with a dominant contribution from r_{III}. For D^{(2)}, we assume the relation D^{(2)} = 0.5Γ_{E}. We can observe that the polarization rate decreases as Γ_{E}/Γ_{R} increases, a phenomenon discussed in detail in Nagendra et al. (2002). The differences between the AA and AD Stokes I and Q profiles decrease as Γ_{E}/Γ_{R} increases. For Γ_{E}/Γ_{R} > 100, these differences become insignificant.
7. Conclusions
We have investigated the Rayleigh scattering with angledependent partial redistribution (PRD) functions to evaluate the reliability of the usual angleaveraged approximation. The analysis has been carried out for a twolevel atom with unpolarized ground level using the angledependent PRD functions established in Domke & Hubeny (1988); Bommier (1997a). The scattering medium has been assumed to be a planeparallel, cylindrically symmetrical, isothermal slab. For this model, the polarized radiation field is cylindrically symmetrical (i.e. depends only on the inclination θ with respect to the direction perpendicular to the slab surface) and can be represented by the two Stokes parameters I and Q. The source terms in the transfer equations for I and Q have a complicated dependence on the angle θ, because it appears in the polarization phase matrix and also in the angledependent PRD functions. It was shown in HF10 that the polarized radiation field can be decomposed into four components that are cylindrically symmetrical and satisfy standard radiative transfer equations. The source terms still depend on θ but in a much simpler way. Thanks to this decomposition method, we have been able to construct three different iterative methods of solution for the calculation of the Stokes parameters. Two are of the accelerated lambda iteration (ALI) type; they are generalizations of the frequencybyfrequency (FBF) and corewing methods originally formulated for scalar PRD transfer in Paletou & Auer (1995). The other is a scattering expansion (Neumann series) method developed for the Hanle effect with complete frequency redistribution in Frisch et al. (2009).
Crucial ingredients for the three methods are the azimuthal Fourier coefficients of order 0,1, and 2 of the angledependent Hummer’s (1962) PRD functions r_{II} and r_{III}. The calculations of these Fourier coefficients is quite time consuming, in particular for r_{III}, because very fine grids are needed to properly represent their sharp variations with the frequency and inclination of the incident and scattered beams. Once these coefficients have been calculated, and also an approximate solution of Stokes I (neglecting polarization) is obtained in the scattering expansion method, the corewing and scattering expansion approaches both provide fast and accurate solutions. The scattering expansion method appears particularly interesting in problems of large dimensionality (very large number of frequency, angle, and depth grid points).
We have found that the angleaveraged PRD functions overestimate the emergent polarization rate Q/I between 10% and 30% for slabs with a large optical thickness (between 2 × 10^{4} and 2 × 10^{8}), the largest differences occurring in the near wing peaks. For optically thin slabs, the differences are much smaller. We have not considered semiinfinite atmospheres. As the difference between the polarization rates obtained with angleaveraged and angledependent PRD functions remain fairly small, the sign of this difference may easily depend on the particular choice of the atomic and atmospheric model. It would certainly be interesting to consider realistic solar atmosphere models as has recently been done in Sampoorna et al. (2010). The numerical methods we have proposed here are able to do this. This type of analysis would certainly help us to understand the discrepancies between observed polarization rates and theoretical predictions (Anusha et al. 2010).
For the weakfield Hanle effect, the reliability of the angleaveraged approximation for PRD functions is clearly questionable, for Stokes Q and especially Stokes U (Nagendra et al. 2002). In the presence of a weak magnetic field, the polarized radiation field can be decomposed into a set of six irreducible components. For an angledependent PRD, these components have no cylindrical symmetry, but a Fourier azimuthal expansion allows one to construct an infinite set of integral equations similar to the equations given here (Frisch 2009). The corewing or a scattering expansion method applied to a truncated set of these equations offer some hope in calculating the polarization with a reasonable amount of numerical work.
Acknowledgments
The authors are grateful to Dr. V. Bommier for providing a FORTRAN PROGRAM to compute angledependent type III redistribution function and to L. S. Anusha for helpful discussions.
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All Tables
CPU time requirements for the iterative methods after the calculation of the azimuthal coefficients , Q = 0,1,2.
All Figures
Fig. 1 Surface plots of azimuth averaged redistribution functions of type II (left panels) and of type III (right panels) with Q = 0. The Xaxis represents the outgoing frequency x, and the Yaxis represents outgoing direction μ. The incoming direction is μ′ = 0.3. The damping parameter a = 0.001. The top two panels correspond to the incoming frequency x′ = 3, and the bottom panels to x′ = 4. 

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In the text 
Fig. 2 Azimuth averaged redistribution functions of type II (left panels) and of type III (right panels), plotted as a function of the outgoing frequency x, for different choices of μ, and μ′. The damping parameter a = 0.001. Thin lines correspond to x′ = 1 and thick lines to x′ = 4. Solid, dotted and dashed lines correspond respectively to Q = 0,1, and 2. 

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In the text 
Fig. 3 Maximum relative change of the Stokes I source vector component and of the surface polarization as a function of the iteration number. Slab model with parameters (T,a,ϵ,r,Γ_{E}/Γ_{R}) = (2 × 10^{4},10^{3},10^{4},0,0) are used. The panel a) corresponds to the FABFA method and the panel b) to the corewing method for angledependent PRD. The panel c) corresponds to the scattering expansion method. Different line types are: solid – and dotted – . 

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In the text 
Fig. 4 Convergence history of the four components of the source vector for x = 0 and μ = 0.025 calculated with the corewing method. Same model as in Fig. 3. The dotted lines show the initial solutions (ϵB for , and zero for all other components). Since the slab is symmetric about the midplane, the results are shown only for a halfslab, τ ∈ [0,T/2] . 

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In the text 
Fig. 5 The scattering expansion method. The three upper panels show the convergence history of the components for x = 0 and μ = 0.025. The last panel shows the convergence history of the ratio Q/I. In all the panels, the dotted lines represent the single scattered solution. The same slab model as in Figs. 3 and 4 is used. 

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In the text 
Fig. 6 A comparison of emergent Stokes profiles at μ = 0.11 for angledependent (solid lines) and angleaveraged (dashed lines) PRD functions. Left panel: slab model with parameters (T,a,ϵ,r,Γ_{E}/Γ_{R}) = (10,10^{3},10^{4},0,1). Right panel: slab model with parameters (T,a,ϵ,r,Γ_{E}/Γ_{R}) = (2 × 10^{4},10^{3},10^{4},0,0). 

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In the text 
Fig. 7 A comparison of emergent at μ = 0.11 computed for the angleaveraged (dashed lines) and angledependent (solid lines) PRD. Left panel: slab model with parameters (T,a,ϵ,r,Γ_{E}/Γ_{R}) = (10,10^{3},10^{4},0,1). Right panel: slab model with parameters (T,a,ϵ,r,Γ_{E}/Γ_{R}) = (2 × 10^{4},10^{3},10^{4},0,0). 

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In the text 
Fig. 8 Stokes I and Q/I profiles at the surface for different values of the heliocentric angle computed for the angleaveraged (dashed lines) and the angledependent (solid lines) PRD. The bottom three panels show the absolute difference in Q/I between the AA and AD profiles. The atmospheric model used is (T,a,ϵ,r,Γ_{E}/Γ_{R}) = (2 × 10^{4},10^{3},10^{4},0,0). 

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In the text 
Fig. 9 The μ dependence of the irreducible components of ℐ (left panels) and (right panels) at the surface of a slab. Different line types are: thin solid line – μ = 0.025, dotted – μ = 0.129, dashed – μ = 0.297, dotdashed – μ = 0.5, dashtripledotted – μ = 0.7, longdashed – μ = 0.87, and thick solid – μ = 0.97. The atmospheric model is the same as in Fig. 8. 

Open with DEXTER  
In the text 
Fig. 10 The emergent I, Q/I, and Q profiles at μ = 0.11 computed for the angleaveraged (dashed lines) and the angledependent (solid lines) PRD. Left panel shows the effect of optical thickness T when ϵ = 10^{4}. Different line types are: thin lines T = 2 × 10^{4}, medium thick lines T = 2 × 10^{6}, and thick lines T = 2 × 10^{8}. Inset in Q/I panel shows T = 2 × 10^{8} case for a larger frequency range. Right panel shows the effect of ϵ for optical thickness T = 2 × 10^{4}. Different line types are: thin lines ϵ = 10^{2}, medium thick lines ϵ = 10^{6}, and thick lines ϵ = 0. Remaining common parameters are (a,r,Γ_{E}/Γ_{R}) = (10^{3},0,0). 

Open with DEXTER  
In the text 
Fig. 11 The emergent Q/I profiles at μ = 0.11 computed for the angleaveraged (dashed lines) and the angledependent (solid lines) PRD. The model parameters are (T,a,ϵ,Γ_{E}/Γ_{R}) = (2 × 10^{4},10^{3},10^{4},0). The parameter r takes the values 10^{8}, 10^{6}, and 10^{4}. 

Open with DEXTER  
In the text 
Fig. 12 The emergent Stokes parameters I and Q and the ratio Q/I at μ = 0.11 computed for the angleaveraged (dashed lines) and the angledependent (solid lines) PRD. The model parameters are (T,a,ϵ,r) = (2 × 10^{4},10^{3},10^{4},0). The parameter Γ_{E}/Γ_{R} takes the values 1 and 10. 

Open with DEXTER  
In the text 
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