Spectral line polarization with angledependent partial frequency redistribution^{⋆}
III. Single scattering approximation for the Hanle effect
Indian Institute of Astrophysics, Koramangala, Bangalore 560 034, India
email: 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 nonmagnetic 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 angledependent PRD functions instead of angleaveraged functions is essential.
Aims. We introduce a single scattering approximation to the problem of polarized line radiative transfer in weak magnetic fields with an angledependent PRD. This helps us to rapidly compute an approximate solution to the difficult and numerically expensive problem of polarized line formation with angledependent PRD.
Methods. We start from the recently developed Stokes vector decomposition technique combined with the Fourier azimuthal expansion for angledependent 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 nonlocal thermodynamic equilibrium (nonLTE) scalar line transfer equation. In the case of angledependent 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 angledependent 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
Appendices A and B are available in electronic form at http://www.aanda.org
© ESO, 2011
1. Introduction
It is now well established that the Hanle effect serves as a diagnostic tool to determine the deterministic or turbulent weak magnetic fields on the Sun (see e.g., Stenflo 1982; Trujillo Bueno et al. 2004). The Hanle effect is a magnetic modification of the Rayleigh scattering spectral line polarization. The scattering polarization signatures in strong resonance lines are particularly sensitive to the frequency redistribution mechanism used in their computation. In the case of Rayleigh scattering, the differences in Q/I computed using angleaveraged and angledependent partial frequency redistribution (PRD) functions is between 10% and 30% (Sampoorna et al. 2011, see also Faurobert 1988). In the presence of weak magnetic fields, Nagendra et al. (2002) showed that the Stokes U is very sensitive to the angleaveraging of the redistribution function. However, these authors used a computationally expensive numerical method to solve the polarized line transfer equation including angledependent PRD function and the Hanle effect. Our aim here is to present an approximate method based on the concept of the single scattering approximation (see Frisch et al. 2009; Frisch 2010; Anusha et al. 2010; Sampoorna et al. 2011) to solve the abovementioned problem efficiently.
The problem of angledependent PRD is characterized by intricate coupling between angle and frequency variables. This makes the evaluation of scattering integrals, and the solution of polarized transfer equation a challenging problem. A decomposition technique to reduce the nonaxisymmetric Stokes transfer equation to cylindrically symmetric one was developed by Frisch (2007), for the case of an angleaveraged PRD with the Hanle effect. This technique was extended by Frisch (2009) to handle angledependent PRD with the Hanle effect. Basically the Stokes vector (I,Q,U) is first decomposed into a set of six irreducible components . In the particular case of angledependent PRD, these components are also nonaxisymmetric. An azimuthal Fourier expansion of the angledependent PRD
functions then allows us to further reduce into an infinite set of cylindrically symmetric Fourier coefficients . This method helps us to construct an infinite set of integral equations for the Fourier coefficients of the irreducible source vector components. As shown in Frisch (2010), the problem simplifies when the magnetic field is set to zero or assumed to be turbulent. In this case, the Fourier wavenumber k takes only the values k = 0,1,2, and four components are sufficient to represent the polarized radiation field. As shown in Sampoorna et al. (2011), the polarized radiation field can be calculated by solving a set of integral equations for the corresponding cylindrically symmetric irreducible source vector components. Several numerical methods are described in this reference : two accelerated lambda iteration (ALI) methods, associated with either a corewing or a frequencyangle by frequencyangle technique, and one scattering expansion method, based on a Neumann series expansion in terms of the mean number of scattering events.
For the Hanle effect with an angledependent PRD, the numerical solution is possible only if one truncates the infinite set of integral equations for the Fourier coefficients of the irreducible source vector components. For the type II PRD function (frequency coherent scattering in the atomic rest frame) of Hummer (1962), Domke & Hubeny (1988) showed that, at least five terms in the azimuthal Fourier expansion are needed to get a good agreement with the exact value of the function. This implies that the azimuthal order k takes values 0, ± 1, ± 2, ± 3, and ± 4, for each combination of K and Q of the irreducible source vector. For the Hanle problem, K = 0 and 2, with − K ≤ Q ≤ + K. Hence, we obtain a total of 54 coupled integral equations. Although they can be numerically solved, it is computationally demanding. To simplify this numerically difficult problem, we present an approximate method of solution based on the concept of a single scattering approximation.
In Sect. 2, we briefly recall the decomposition technique presented in Frisch (2009) for the Hanle effect with angledependent PRD. In Sect. 3, we derive approximate expressions for the Fourier coefficients of irreducible source vector components applying the single scattering approximation. A generic form for the Hanle scattering redistribution matrix is assumed in Sects. 2 and 3 (see Eq. (4) below). In Sect. 4, we generalize the results of Sects. 2 and 3 to the Hanle scattering redistribution matrix derived by Bommier (1997b, the socalled approximationII). Section 5 is devoted to the comparison of results computed using the single scattering approximation with those computed using the perturbation method of Nagendra et al. (2002). Conclusions are presented in Sect. 6. Some technical details are given in Appendices A and B.
2. A decomposition method for Hanle effect with angledependent PRD
For the sake of clarity, we recall some important equations from Frisch (2009). The polarized transfer equation for the Stokes vector can be written in component form as (1)where Ω (θ,χ) is the outgoing ray direction defined with respect to the atmospheric normal and μ = cosθ. The line optical depth τ is defined by dτ = −k_{l} dz, where k_{l} is the frequency–averaged line absorption coefficient, and ϕ(x) is the normalized Voigt function. The frequency x is measured in units of the Doppler width, with x = 0 at the 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_{ν0}, where B_{ν0} is the Planck function at the line center, and S_{c,1} = S_{c,2} = 0. The line source vector can be written as (3)where Ω′ (θ′,χ′) is the direction of the incoming ray defined with respect to the atmospheric normal, and dΩ′ = sinθ′ dθ′ dχ′. For simplicity, we assume that the primary source is unpolarized, namely that only G_{0}(τ) is nonzero, and it is proportional to B_{ν0}. In Sect. 3, we assume a generic form for the elements of the redistribution matrix in the presence of a weak magnetic field B given by (4)where r(x,x′,Θ) is any redistribution function with proper normalization conditions, such as the type I, II, or III PRD functions of Hummer (1962), and Θ is the angle between the incoming (Ω′) and outgoing (Ω) rays. In terms of the irreducible spherical tensors introduced by Landi Degl’Innocenti (1984), the Hanle phase matrix elements can be written as (see Chaps. 5 and 10 of Landi Degl’Innocenti & Landolfi 2004; or Frisch 2007; 2009) (5)For the coefficients we assume the simple form (6)where are coefficients that describe the effects of the magnetic field, ϵ the thermalization parameter, W_{K} the depolarizability factors depending on the Jquantum numbers of upper and lower levels, (θ_{B},χ_{B}) the polar angles defining the magnetic field orientation with respect to the atmospheric normal, and Γ = 2πν_{L}g_{u}/A_{ul} in standard notation is the Hanle parameter.
The Stokes vector and the source vector can each be decomposed into six irreducible components and that are nonaxisymmetric because of the angle dependence of the PRD function. To describe this angle dependence of the PRD function, we introduce an azimuthal Fourier expansion written as (see Eq. (15) of Frisch 2009) (7)where (8)It is easy to verify that . To describe the azimuthal dependence of the irreducible components , we introduce the Fourier decomposition (9)Assuming a similar expansion for the primary source term , it is easy to show that has an azimuthal expansion similar to Eq. (9).
The Fourier coefficients satisfy a nonlocal thermodynamic equilibrium (hereafter nonLTE) transfer equation (10)where the source term is given by Eq. (2) with and instead of S_{l,i} and S_{c,i}. Since the continuum is assumed to be unpolarized, . The Fourier coefficients of the line source vector are given by (11)where the primary source term , and k′ = k + Q′ − Q′′. For a given value of k, the values of k′ are clearly restricted to k − 4 ≤ k′ ≤ k + 4. The nonzero azimuthal Fourier coefficients are of the form (12)where K and K′ are both even or both odd. The are combinations of trigonometric functions that may be written as . It is easy to verify that satisfy the following conjugation property (13)We note that the range of k values extends a priori from − ∞ to + ∞, thereby giving rise to an infinite set of integral equations for . A solution is possible only if we truncate this infinite set. As already noted in Sect. 1, if we restrict k to k = 0, ± 1, ± 2, ± 3, ± 4, we obtain a set of 54 coupled integral equations. Before going into such a difficult numerical problem, we show in the following section how the problem can be simplified by using a single scattering approximation (see Frisch et al. 2009; Frisch 2010).
3. Approximate expressions for the Fourier coefficients
First we make the reasonable assumption that Stokes I is unaffected by the polarization. We can thus describe Stokes I with only, neglecting the contributions of the K ≠ 0 terms. For an angledependent PRD, is nonaxisymmetric and can be Fourier expanded as in Eq. (9). However, as Stokes I is almost independent of the magnetic field (in the weak field Hanle regime that we consider), we make a further assumption that is independent of χ. Hence, only the k = 0 term contributes to and we can write (14)With these approximations, the irreducible component takes a simple form, namely (15)where , and the thermalization parameter ϵ = Γ_{I}/(Γ_{R} + Γ_{I}), with Γ_{R} and Γ_{I} being the radiative and inelastic collisional deexcitation rates.
To obtain approximate expressions for the Fourier components of , we apply the single scattering approximation to Eq. (11), namely we keep in the righthand side of this equation only the contribution from . This amounts to assuming that k′ = K′ = Q′′ = 0. Since k′ = k + Q′ − Q′′, we obtain k = −Q′. Hence, only the components with k = 0, ± 1, ± 2 are nonzero. These 25 Fourier components are given by the following approximate expression (16)The explicit forms of the azimuthal Fourier coefficients are given in the Appendix A.
The irreducible components and their Fourier components for k ≠ 0 are complex numbers. We introduce here their real components. Following Frisch (2007, Eqs. (25) and (26)), we define the real components of to be (17)(18)For the Fourier components , we use the conjugation property given in Eq. (13) to define their real components as (19)(20)For k ≠ 0, using the conjugation property of (see Appendix A) and of (see Chapter 5 of Landi Degl’Innocenti & Landolfi 2004), we can deduce from Eq. (16) that (21)Combining Eqs. (17) and (18) with the azimuthal Fourier expansion of , we obtain (22)(23)(24)where (25)for k > 0 and Q > 0. There are 25 nonzero real components of . Explicit expressions are given in Appendix B.
4. The Hanle scattering redistribution matrix
In Sect. 2, the decomposition method is presented with an oversimplified redistribution matrix. Here we show how to extend it to a standard Hanle redistribution matrix that takes into account elastic collisions and that the Hanle effect only operates in the line core.
Using a QED approach, Bommier (1997a,b) derived the Hanle scattering redistribution matrix including the effects of elastic collisions for a twolevel atom with unpolarized lower level. For practical applications, she also presents the socalled approximationII and approximationIII, where the twodimensional (2D) frequency space (x,x′) is decomposed into several domains, in each domain the frequency redistribution being decoupled from the polarization. The approximationII uses the angledependent PRD functions and approximationIII the corresponding angleaveraged functions. The approximationsII and III of the Hanle redistribution matrix were considered in Nagendra et al. (2002, see also Nagendra et al. 2003), where a perturbation method is used to solve the polarized transfer equations. For approximationIII, a polarized ALI method based on the corewing approach was later developed by Fluri et al. (2003).
For approximationIII, the domains depend on (x, x′), but for approximationII they depend in addition on the scattering angle Θ between the incoming and outgoing rays (see Figs. 1 and 2 in Nagendra et al. 2002). Thus, one practical question that arises when applying the decomposition technique developed for angledependent PRD presented in Sect. 2 to the case of approximationII, is how to reduce or remove the (χ − χ′) dependence appearing in the expression of the redistribution matrix given in Eqs. (90) − (98) of Bommier (1997b). Here, to circumvent this problem, we use the angleaveraged domains of approximationIII, together with the angledependent redistribution functions of approximationII. With this simplifying assumption, which as shown in Sect. 5 yields a reasonable solution, it is straightforward to generalize all the equations given in Sects. 2 and 3 to handle the approximationII of Bommier (1997b). The only difference is that now depends on both x and x′. Therefore in Eqs. (11) and (16), should be inside the frequency integral, and should be replaced by (26)where the index m ( = 1,2,3,4,5) stands for different (x, x′) frequency domains. The domains relevant to the typeIII redistribution are represented by m = 1,2,3 and those relevant to the typeII redistribution by m = 4,5. Expressions for the can be found in Bommier (1997b, see also Nagendra et al. 2002; Frisch 2007; Anusha et al. 2011).
5. Results and discussions
Here we validate the single scattering approximation presented in Sect. 3, by comparing the solutions computed with this approximation, and those obtained by solving the full radiative transfer equation with an angledependent PRD and the Hanle effect. For the latter, we use a perturbation method developed by Nagendra et al. (2002). To obtain the single scattering solution, we first solve the scalar radiative transfer equation with the source term given by Eq. (15), using an ALI method based on the corewing approach or frequencyangle by frequencyangle approach (see Sampoorna et al. 2011). The scalar intensity obtained in this way is then used in Eq. (16) or Eqs. (B.1) and (B.2), to compute the single scattered source vector. A formal solution then gives the corresponding single scattered radiation field.
Fig. 1 Comparison of single scattered (dashed lines) and multiple scattered (solid lines) solutions computed with the angledependent PRD. A 1D cutoff assumption is used with x_{c} = 3. The model parameters are (a, ϵ, r, Γ_{E}/Γ_{R}) = (10^{3}, 10^{3}, 0, 0) and the magnetic field parameters are (Γ, θ_{B}, χ_{B}) = (1, 30°, 0°). The line of sight is represented by μ = 0.11 and χ = 0°. Panel a) corresponds to T = 1, panel b) to T = 100, and panel c) to T = 10^{4}. 

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We consider isothermal, selfemitting planeparallel atmospheres with no incident radiation at the boundaries. These slab models are characterized by (T, a, ϵ, r, Γ_{E}/Γ_{R}), where T is the optical thickness of the slab and Γ_{E} is the elastic collisional rate. The depolarizing collisional rate D^{(2)} is assumed to be 0.5 × Γ_{E}. For all the figures presented in this paper, a = 10^{3}, ϵ = 10^{3}, r = 0 (pure line case), and the line of sight is defined by μ = 0.11 and χ = 0°. The magnetic field parameters are taken as Γ = 1, θ_{B} = 30°, and χ_{B} = 0°.
In Fig. 1, we compare the single scattered solution (dashed lines) with the full radiative transfer solution (hereafter, multiple scattered solution; see solid lines) for T = 1 (panel (a)), 100 (panel (b)), and 10^{4} (panel (c)). For brevity, we only show the Q/I and U/I profiles. The comparison is carried out for the typeII angledependent redistribution function of Hummer (1962), i.e., with Γ_{E}/Γ_{R} = 0. We have used a onedimensional (1D) cutoff assumption, setting the Hanle effect to zero for x > x_{c} with x_{c} = 3. Owing to this abrupt cutoff, spikes or dents are observed around x = 3 (see Fig. 1). Smoother curves can be obtained by using the more realistic 2D domains of Bommier (1997b) (see Fig. 3).
Figure 1 shows that the single scattering approximation can capture the main features of the Q/I and U/I profiles. It is the first time that this approximation has been employed for Stokes U. We can see that it is able to follow the very nonmonotonic behavior of U/I. From a quantitative point of view, we observe at line center that the single scattering approximation underestimates the polarization for T = 1 and overestimates it for T = 100 and T = 10^{4}. The transition takes place around T = 100. As explained in Frisch et al. (2009) for the case of complete frequency redistribution (CRD), this change in behavior is related to the competition between a limbdarkened outgoing radiation and a limbbrightened incoming one (see also Trujillo Bueno 2001). In the near wings of Q/I, the single scattering approximation strongly underestimates the polarization. We recall that these wings are formed by Rayleigh scattering. For the angledependent case, Sampoorna et al. (2011) showed that the exact value of Q/I all along the profile can be obtained by an iterative method incorporating successive scattering orders (see Fig. 5 in this reference). It is likely that this method will also be able to improve the estimation of U/I.
In Fig. 2, we present the single scattered and multiple scattered solutions computed for T = 10^{4}, using the more realistic 2D domains of Bommier (1997b). Panel (a) corresponds to the collisionless case and panel (b) to an equal mix of typeII and typeIII scattering. We recall that single scattered solutions are calculated with the angleaveraged domains corresponding to approximationIII and the multiple scattered solutions with angledependent domains corresponding to approximationII. Figure 2 shows that this simplifying assumption gives solutions that compare reasonably with the corresponding multiple scattered solutions. The comparison between Figs. 2a and 2b shows that the single scattered solution can become a very good approximation to the multiple scattered solution in the presence of elastic collisions, particularly at the near wing maximum in Q/I profile. For angleaveraged PRD, we also have examples of single scattering solutions providing very accurate approximations (see Fig. 4 of Anusha et al. 2010).
Fig. 2 Comparison of single scattered (dashed lines) and multiple scattered (solid lines) solutions computed with approximationII of Bommier (1997b). The model parameters are (T, a, ϵ, r) = (10^{4}, 10^{3}, 10^{3}, 0) and the magnetic field parameters are (Γ, θ_{B}, χ_{B}) = (1, 30°, 0°). Panel a) corresponds to collisionless case (Γ_{E}/Γ_{R} = 0) and panel b) to an equal mix of typeII and III scattering (Γ_{E}/Γ_{R} = 1). 

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The comparison of Fig. 2a with Fig. 1c suggests that U/I is very sensitive to the type of cutoff approximation used. We now discuss this point in more detail. Figure 3 shows the Q/I and U/I profiles computed for the collisionless case, and with three types of frequency domains. The solid lines are computed using a 1D cutoff assumption (x_{c} = 3), and dashed lines with actual 2D domains of Bommier (1997b). The dotted lines are computed using a square domain (). Such square domains have been used by Fluri et al. (2003, see their Fig. 2b). Figure 3a shows the single scattered solution, while Fig. 3b shows the multiple scattered solution. Clearly Q/I is insensitive to the type of frequency domains used, while U/I is quite sensitive, particularly in the transition region (3 ≤ x < 5).
6. Conclusions
The solution of a polarized line transfer equation with an angledependent PRD and the Hanle effect is computationally very demanding. To reduce the complexity of the numerical problem, Frisch (2009) developed a decomposition technique for angledependent Hanle problem. In this technique, the nonaxisymmetric polarized radiation field is decomposed into an infinite set of cylindrically symmetric Fourier coefficients, by applying an azimuthal Fourier expansion of the angledependent PRD function. Such a decomposition method allows one to construct an infinite set of integral equations for the Fourier coefficients of the irreducible source vector components, the solution of which is possible only if we truncate this infinite set. Although the truncated integral equations can be numerically solved, this solution is still computationally demanding (at least 54 integral equations need to be solved simultaneously). Therefore, to simplify the problem, here we have presented an approximate method of solution, based on the single scattering approximation. An approximate solution can be obtained by keeping only the contribution of the zeroth order Fourier coefficient, namely, in the source terms for the K = 2 components (see Eq. (16)), which are responsible for the generation of Stokes Q and U. As a result, the number of Fourier coefficients reduces to 25 for K = 2 components. The Stokes I in this approximation is assumed to be cylindrically symmetric and decoupled from Stokes Q and U, so that it is computed through a solution of the scalar nonLTE transfer equation with an angledependent PRD function. An ALI method associated with a corewing technique or frequencyangle by frequencyangle technique is used to solve this scalar transfer problem (see Sampoorna et al. 2011).
We have presented the single scattering approximation for a Hanle scattering redistribution matrix with (i) 1D cutoff assumption (see Sect. 3) and (ii) the more realistic frequency domains (see Sect. 4) introduced by Bommier (1997b). We show that the single scattered solution provides a reasonable approximation to the emergent solutions computed by solving the full polarized line transfer equation with angledependent PRD and the Hanle effect. To compute the full solution, we use a perturbation method developed by Nagendra et al. (2002). While the perturbation method used by Nagendra et al. (2002) is computationally expensive, the single scattering approximation presented here provides a rapid method of solution. For example, for T = 10^{4} with 41 depth points, seven μ points (eight χ points required only in the case of perturbation method), and 39 frequency points, the perturbation method requires 24 min with 2.7 G Bytes of memory, while single scattered solution can be computed in 20 s with 22 M Bytes of memory, on a Sun Fire V20z Server, 2385 MHz, with a Singlecore AMD Opteron processor.
The differences between single scattered and multiple scattered solutions for angledependent, as well as angleaveraged PRD are somewhat similar to those for CRD in the line core region. In the case of an angledependent and an angleaveraged typeII PRD function, the differences are particularly large at the near wing maximum in Q/I (see Fig. 1c). This difference is reduced in the presence of elastic collisions (see Fig. 2b). Thus, elastic collisions seem to expand the domain of applicability of single scattering approximation. We have also presented the sensitivity of U/I profiles to the choice of different frequency domains (see Fig. 3).
Fig. 3 The effect of three types of frequency domains on Q/I and U/I profiles computed using angledependent PRD functions. Solid lines correspond to the 1D cutoff assumption (x_{c} = 3), dotted lines to 2D square domains (), and dashed lines to the approximationII of Bommier (1997b). The model parameters are the same as in Fig. 2a. Panel a) corresponds to single scattered solutions, and panel b) to the multiple scattered solutions. 

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The single scattered solution presented in this paper can be improved by including higher orders of scattering. Such a method was presented in Frisch et al. (2009) for the Hanle effect with CRD, and in Sampoorna et al. (2011)
for Rayleigh scattering with angledependent PRD, where it was referred to as “scattering expansion method”. It is also possible to construct such a scattering expansion method for the case of Hanle effect with an angledependent PRD discussed in this paper. The single scattered solution should be seen as an “approximate solution” to the exact solution, and can be used in many practical applications that do not require highly accurate solution of this rather difficult problem, involving angledependent PRD matrices.
Acknowledgments
I am very grateful to Drs. H. Frisch and K. N. Nagendra for stimulating discussions and critical comments on the manuscript. Thanks are also due to L. S. Anusha for useful comments.
References
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Online material
Appendix A: The azimuthal Fourier coefficients
The azimuthal Fourier coefficients are defined in Eq. (12). It is easy to verify that they satisfy the properties (A.1)(A.2)In this Appendix, we give only those Fourier coefficients that are of relevance to the problem at hand, namely those corresponding to K′ = Q′ = 0. Owing to the abovementioned symmetry relations, there are only four independent coefficients, which are given by (A.3)If we chose the reference angle γ = 0 (see Landi Degl’Innocenti & Landolfi 2004), then from Eq. (A.6) of Frisch (2010), it is clear that are real and hence that are also real. Using Eq. (A.6) of Frisch (2010) in our Eq. (A.3), we obtain the explicit form for these coefficients (A.4)
Appendix B: The explicit form of the real components of
In this Appendix, we list the explicit analytic forms of the 25 nonzero real components of , defined in Eqs. (19), (20), and (25) of the text. For K = 2, we define (B.1)We can express all the 25 nonzero real components of in terms of as follows : (B.2)The above equations are valid for x < x_{c} in the case of a 1D cutoff assumption. For x > x_{c}, we have to set . In the case of approximationII, the various appearing in the above equations should be taken inside the frequency integral appearing in Eq. (B.1), and Eqs. (106)–(113) of Bommier (1997b) have to be used.
The various appearing in the above set of equations are given by Eq. (6) in the case of the 1D cutoff assumption and Eqs. (A11)–(A18) of Anusha et al. (in press) in the case of the 2D frequency domains of Bommier (1997b). Below we give the explicit forms of appearing in those equations. We introduce the abbreviations, C_{B} = cosθ_{B}, S_{B} = sinθ_{B}, c_{1} = cosχ_{B}, s_{1} = sinχ_{B}, c_{2} = cos2χ_{B}, and s_{2} = sin2χ_{B}. In terms of the elements of the matrix given below (see also Appendix C in Frisch 2007), we have (B.3)where (B.4)
All Figures
Fig. 1 Comparison of single scattered (dashed lines) and multiple scattered (solid lines) solutions computed with the angledependent PRD. A 1D cutoff assumption is used with x_{c} = 3. The model parameters are (a, ϵ, r, Γ_{E}/Γ_{R}) = (10^{3}, 10^{3}, 0, 0) and the magnetic field parameters are (Γ, θ_{B}, χ_{B}) = (1, 30°, 0°). The line of sight is represented by μ = 0.11 and χ = 0°. Panel a) corresponds to T = 1, panel b) to T = 100, and panel c) to T = 10^{4}. 

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In the text 
Fig. 2 Comparison of single scattered (dashed lines) and multiple scattered (solid lines) solutions computed with approximationII of Bommier (1997b). The model parameters are (T, a, ϵ, r) = (10^{4}, 10^{3}, 10^{3}, 0) and the magnetic field parameters are (Γ, θ_{B}, χ_{B}) = (1, 30°, 0°). Panel a) corresponds to collisionless case (Γ_{E}/Γ_{R} = 0) and panel b) to an equal mix of typeII and III scattering (Γ_{E}/Γ_{R} = 1). 

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In the text 
Fig. 3 The effect of three types of frequency domains on Q/I and U/I profiles computed using angledependent PRD functions. Solid lines correspond to the 1D cutoff assumption (x_{c} = 3), dotted lines to 2D square domains (), and dashed lines to the approximationII of Bommier (1997b). The model parameters are the same as in Fig. 2a. Panel a) corresponds to single scattered solutions, and panel b) to the multiple scattered solutions. 

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In the text 