ANALYSIS OF THE FORWARD-SCATTERING HANLE EFFECT IN THE Ca i 4227 Å LINE

2.1.1. Stokes Parameter Formulation

We use the standard notation of line formation theory (Mihalas 1978; Stenflo 1994). The Stokes vector RT equation for a two-level model atom with unpolarized ground level in a one-dimensional planar medium in the presence of weak-field Hanle effect may be written as

$$\begin{eqnarray} \mu \frac{\partial \bm {I}(\lambda, \bm {\Omega }, z)}{\partial z}&=& -[\kappa _l(z) \phi (\lambda, z) + \kappa _c(\lambda, z)+\sigma _c(\lambda, z) ] \nonumber \\ && \times\left[\bm {I}(\lambda, \bm {\Omega }, z)- \bm {S}(\lambda, \bm {\Omega }, z)\right],\qquad \end{eqnarray} \tag{ 1 }$$

where $\bm {I}$ = (I, Q, U)^T is the Stokes vector and $\bm {S}=(S_I, S_Q, S_U)^T$ the source vector. Here, $\bm {\Omega }=(\theta, \varphi)$ defines the ray direction with θ and φ being the inclination and azimuth of the scattered ray (see Figure 1). The Voigt profile function is denoted by ϕ. The dependence of ϕ on z comes from the damping parameter a = Γ_tot/4πΔν_D. Here

$$\begin{equation} \Gamma _{\rm tot}=\Gamma _R+\Gamma _E+\Gamma _I, \end{equation} \tag{ 2 }$$

with Γ_R being the radiative de-excitation rate. For the Ca i 4227 Å line Γ_R = 2.18 × 10⁸ s⁻¹. Γ_E and Γ_I are the elastic and inelastic collision rates, respectively. Γ_E is computed taking into account the van der Waals broadening (arising due to elastic collisions with neutral hydrogen) and Stark broadening (arising due to interactions with free electrons). Γ_I includes the inelastic collision processes like collisional de-excitation by electrons and protons, collisional ionization by electrons, and charge exchange processes (see Uitenbroek 2001, and the multi-level atom computer program provided by him). The Doppler width $\Delta \nu _D= \sqrt{\vphantom{A^A}\smash{\hbox{${{2 k_B T}/{M_a}+v^2_{{\rm turb}}}$}}}/\lambda _0$, where k_B is the Boltzmann constant, T is the temperature, M_a is the mass of the atom, v_turb is the micro-turbulent velocity (taken as 1 km s⁻¹), and λ₀ is the line center wavelength. κ_l is the wavelength-integrated line absorption coefficient, while σ_c and κ_c are the continuum scattering and continuum absorption coefficients, respectively. Other atomic parameters are given in Section 4.1 and Appendix A.

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The total opacity coefficient is κ_tot(λ, z) = κ_l(z)ϕ(λ, z) + σ_c(λ, z) + κ_c(λ, z). In a two-level model atom with unpolarized ground level, the total source vector $\bm {S}$ is defined as

$$\begin{eqnarray} \bm {S}(\lambda, \bm {\Omega }, z)&=&\frac{\kappa _l(z)\phi (\lambda, z) \bm {S}_{l}(\lambda, \bm {\Omega }, z)}{\kappa _{\rm tot}(\lambda, z)} \nonumber \\ &&+\frac{\sigma _c(\lambda, z) \bm {S}_{c}(\lambda, \bm {\Omega }, z) + \kappa _c(\lambda, z) {B}_{\lambda }(z){\bm U}}{\kappa _{\rm tot}(\lambda, z)}. \nonumber \\ \end{eqnarray} \tag{ 3 }$$

Here, $\bm {U}=(1,0,0)^T$ and B_λ is the Planck function at the line center. The line source vector is

$$\begin{eqnarray} {\bm {S}}_{l}(\lambda, \bm {\Omega }, z)&=& \epsilon {B}_{\lambda }(z)\bm {U}+\int _{-\infty }^{+\infty }\oint \frac{\hat{R}(\lambda, \lambda ^{\prime }, \bm {\Omega }, \bm {\Omega }^{\prime }, z, \bm {B})}{\phi(\lambda, z)} \nonumber \\ && \times \, {\bm {I}}(\lambda ^{\prime }, \bm {\Omega }^{\prime }, z) \,\frac{d\bm {\Omega }^{\prime }}{4\pi }\,d\lambda ^{\prime }. \end{eqnarray} \tag{ 4 }$$

Here, $\hat{R}$ is the Hanle redistribution matrix (Bommier 1997b). $\bm {B}$ is the vector magnetic field taken to be a free parameter. The continuum scattering source vector is

$$\begin{equation} {\bm {S}}_{c}(\lambda, \bm {\Omega }, z)= \oint \hat{P}(\bm {\Omega }, \bm {\Omega }^{\prime }) {\bm {I}}(\lambda, \bm {\Omega }^{\prime }, z)\,\frac{d\bm {\Omega }^{\prime }}{4\pi }, \end{equation} \tag{ 5 }$$

where $\hat{P}$ is the Rayleigh scattering phase matrix (Chandrasekhar 1960). For simplicity, frequency coherent scattering is assumed for the continuum. The thermalization parameter is defined by = Γ_I/(Γ_R + Γ_I). In Equation (4) (λ', $\bm {\Omega }^{\prime }$) and (λ, $\bm {\Omega }$) refer to the wavelength and direction of the incoming and the outgoing rays, respectively.

2.1.2. Spherical Irreducible Tensor Decomposition

In general the Stokes source vector $\bm {S}$ and the Stokes vector $\bm {I}$ in Equation (1) depend on $\bm {\Omega }=(\theta,\varphi)$. As shown in Frisch (2007), the vectors $\bm {S}$ and $\bm {I}$ can be decomposed into six cylindrically symmetric components, ${\mathcal I}^K_Q$ and ${\mathcal S}^K_Q$, with K = 0, 2 and Q ∈ [ − K, +K], if one represents them in terms of the spherical irreducible tensors for polarimetry defined in, e.g., Landi Degl'Innocenti & Landolfi (2004). With these components one can construct an irreducible source vector $\bm {\mathcal S}$ and an irreducible Stokes vector $\bm {\mathcal I}$. This decomposition is useful because $\bm {\mathcal S}$ becomes independent of the ray direction and $\bm {\mathcal I}$ becomes independent of the azimuthal angle φ. The transfer equation for $\bm {\mathcal I}$ may be written as

$$\begin{eqnarray} &&\mu \frac{{\partial {\bm {\mathcal I}}(\lambda, \mu, z)} }{\partial z}=-\kappa _{\rm tot}(\lambda, z) \left[{\bm {\mathcal I}}(\lambda, \mu, z) - {\bm {\mathcal S}}(\lambda, z)\right]. \end{eqnarray} \tag{ 6 }$$

In a two-level model atom with unpolarized ground level, the total irreducible source vector $\bm {\mathcal S}$ is defined as

$$\begin{eqnarray} \bm {\mathcal S}(\lambda, z)&=&\frac{\kappa _l(z)\phi (\lambda, z) \bm {\mathcal S}_{l}(\lambda, z)}{\kappa _{\rm tot}(\lambda, z)} \nonumber \\ &&+\frac{\sigma _c(\lambda, z) \bm {\mathcal S}_{c}(\lambda, z) + \kappa _c(\lambda, z) {B}_{\lambda }(z)\bm {\mathcal U}}{\kappa _{\rm tot}(\lambda, z)}. \end{eqnarray} \tag{ 7 }$$

Here, $\bm {\mathcal U}=(1,0,0,0,0,0)^T$. The irreducible line source vector is

$$\begin{eqnarray} {\bm {\mathcal S}}_{l}(\lambda, z)&=& \epsilon {B}_{\lambda }(z)\bm {\mathcal U}+\int _{-\infty }^{+\infty }\frac{1}{2}\int _{-1}^{+1} \frac{\hat{\mathcal R}(\lambda, \lambda ^{\prime }, z, \bm {B})}{\phi(\lambda,z)} \hat{\Psi }(\mu ^{\prime }) \nonumber \\ &&\times \,{\bm {\mathcal I}}(\lambda ^{\prime }, \mu ^{\prime }, z)\,d\mu ^{\prime }\,d\lambda ^{\prime }. \end{eqnarray} \tag{ 8 }$$

Here, μ' represents the incoming ray direction. The irreducible polarized continuum scattering source vector is

$$\begin{equation} {\bm {\mathcal S}}_{c}(\lambda, z)= \frac{1}{2}\int _{-1}^{+1} \hat{\Psi }(\mu ^{\prime }) {\bm {\mathcal I}}(\lambda, \mu ^{\prime }, z)\,d\mu ^{\prime }. \end{equation} \tag{ 9 }$$

$\hat{\mathcal R}$ is the angle-averaged PRD matrix in the irreducible basis for the Hanle effect (see Bommier 1997a, 1997b) given in Equation (A1) of Appendix A. The matrix $\hat{\Psi }$ is the Rayleigh scattering phase matrix in the irreducible basis. Its elements are given in Appendix A of Frisch (2007; see also Faurobert-Scholl 1991). We define the total optical depth scale through dτ_λ = −κ_tot(λ, z)dz.

The formal solution of Equation (6) can be written as

$$\begin{eqnarray} \bm {\mathcal I}(\lambda, \mu, \tau _{\lambda }) &=& \bm {\mathcal I}_0(\lambda, \mu, T_{\lambda }) \exp {\left[-\left(\frac{T_{\lambda }-\tau _{\lambda }}{\mu }\right)\right]} \nonumber \\ &&+ \int _{\tau _{\lambda }}^{T_{\lambda }} \exp {\left[-\left(\frac{\tau ^{\prime }_{\lambda }- \tau _{\lambda }}{\mu }\right)\right]} \bm {\mathcal S}(\lambda, \tau ^{\prime }_{\lambda }) \frac{d\tau ^{\prime }_{\lambda }}{\mu }\qquad \end{eqnarray} \tag{ 10 }$$

for μ > 0, and

$$\begin{eqnarray} \bm {\mathcal I}(\lambda, \mu, \tau _{\lambda }) &=& \bm {\mathcal I}_0(\lambda, \mu, 0) \exp {\left(-\frac{\tau _{\lambda }}{\mu } \right)} \nonumber \\ &&- \int _0^{\tau _{\lambda }} \exp {\left[-\left(\frac{\tau ^{\prime }_{\lambda }- \tau _{\lambda }}{\mu }\right)\right]} \bm {\mathcal S}(\lambda, \tau ^{\prime }_{\lambda }) \frac{d \tau ^{\prime }_{\lambda }}{\mu } \qquad \end{eqnarray} \tag{ 11 }$$

for μ < 0. $\bm {\mathcal I}_0$ represents a radiation field that is incident on the medium. We assume that no radiation is incident on the upper free boundary (τ_λ = 0), while at the lower boundary T_λ, $\bm {\mathcal I}_0(\lambda, \mu, T_{\lambda })= (B_\lambda (T_{\lambda }), 0, 0, 0, 0, 0)^T$.

The redistribution matrices used here are defined under the approximation III of Bommier (1997b; see our Appendix A). We take into account the depth dependence of the branching ratios that appear in the redistribution matrix $\hat{\mathcal R}$, in contrast to Nagendra et al. (2002) and Fluri et al. (2003), where these coefficients were kept independent of depth, because only isothermal atmospheres were considered. The polarized Hanle RT equation is solved for the irreducible Stokes vector $\bm {\mathcal I}$. The Stokes vector (I, Q, U)^T can then be deduced from $\bm {\mathcal I}$ (see, e.g., Frisch 2007; Anusha & Nagendra 2011). The components of $\bm {\mathcal I}$ are (I⁰₀, I²₀, I^2x₁, I^2y₁, I^2x₂, I^2y₂). If the magnetic field is zero, or micro-turbulent, only the two components I⁰₀ and I²₀ are non-zero. It is easy to see from the expression of Stokes Q that Q = 0 at the disk-center (μ = 1) and maximum at the limb (μ = 0) and that U = 0. The same argument applies to S⁰₀ and S²₀. When the magnetic field is non-zero and has some fixed orientation, then all six components are non-zero. Therefore, Stokes Q will be non-zero at the disk center and U will be non-zero with contributions coming from the four components I^2x₁, I^2y₁, I^2x₂, I^2y₂. This is what is referred to as the forward-scattering Hanle effect.

The non-LTE Zeeman-effect line transfer equation for the Stokes vector (I, Q, U, V)^T is given by

$$\begin{eqnarray} \mu \frac{\partial \bm {I}(\lambda, \bm {\Omega }, z)}{\partial z}&=& -\hat{K}(\lambda, z, \bm {B})\nonumber \\ &&\times \left[\bm {I}(\lambda, \bm {\Omega }, z)- \bm {S}(\lambda, z)\right], \end{eqnarray} \tag{ 12 }$$

where $\hat{K}$ is the absorption matrix. In Appendix B we give the relevant trigonometric functions that are required to write the Zeeman absorption matrix in the atmospheric reference frame. The general form of $\bm {S}(\lambda, z)$ for a multi-level model atom can be found in Landi Degl'Innocenti & Landolfi (2004).

We recall that Equation (1) is the Hanle RT equation written for the Stokes parameters (I, Q, U)^T (Stokes V is not included because it is not generated by the weak-field Hanle effect). Stokes I is not very sensitive to the magnetic field in the weak field limit (see Equation (13) below). Equation (12) is the Zeeman RT equation written for the Stokes vector (I, Q, U, V)^T. The Stokes parameters Q and U appearing in Equations (1) and (12) are the same quantities. In Equation (1) the Stokes Q is generated by resonance scattering but modified by a weak magnetic field ${\bm B}$, while Stokes U is generated entirely by the Hanle effect. In Equation (12) Stokes Q and U are generated by the transverse Zeeman effect. However, for the range of fields indicated by the observed V/I amplitudes (see below), Stokes Q and U generated from the transverse Zeeman effect are negligible, and therefore Stokes V decouples from Stokes Q and U in Equation (12). Further, we assume that the amount of atomic polarization in the upper level of the line is weak (due to the small value of the anisotropy of the solar radiation field). Thus, under this approximation combined with the weak field limit the RT equation for the linear polarization (Equation (1)) decouples from the RT equation for the circular polarization (Equation (12)). With the assumption of height independent magnetic fields the weak field approximation (see, e.g., Stenflo 1994; Landi Degl'Innocenti & Landolfi 2004) is defined as

$$\begin{equation} \frac{\Delta \lambda _H}{\Delta \lambda _D} \ll 1, \end{equation} \tag{ 13 }$$

where

$$\begin{equation} \Delta \lambda _H {\hbox{(m\AA)}}=4.67 \times 10^{-10} g_{{\rm eff}} \lambda _0^2 B, \end{equation} \tag{ 14 }$$

with g_eff being the effective Landé factor (g_eff = 1 for the Ca i 4227 Å line). The line center wavelength λ₀ is given in Å and B in Gauss. Δλ_D is the Doppler width expressed in wavelength units.

For the Ca i atom Δλ_D = 31 mÅ at the top of the FALC model atmosphere (where T = 9257 K). For the Ca i 4227 Å line we get Δλ_H = 8.33487 × 10⁻³B. The Zeeman splitting equals the Doppler width for a field strength of about B = 3720 G . In Table 1 we list the ratio Δλ_H/Δλ_D for various B values. For B ⩽ 40 G, we have Δλ_H/Δλ_D ⩽ 1%, so for B ⩾ 50 G the ratio Δλ_H/Δλ_D starts to become significant.

Under the weak field approximation (Equation (13)) V/I is simply given by

$$\begin{equation} \frac{V(\lambda,\mu)}{I(\lambda,\mu)}=\frac{1}{I(\lambda,\mu)} \left[-\Delta \lambda _H \cos \gamma \frac{\partial I(\lambda,\mu)}{\partial \lambda }\right], \end{equation} \tag{ 15 }$$

where I(λ, μ) is the emergent Stokes I profile, and

$$\begin{equation} \cos \gamma = \cos \theta _B \cos \theta + \sin \theta _B \sin \theta \cos (\varphi -\chi _B). \end{equation} \tag{ 16 }$$

Here, B, θ_B, and χ_B represent the strength, inclination, and azimuth of the magnetic field vector (see Figure 1). In all our calculations we set φ = 0° and cos θ = μ = 0.9. Use of Equation (15) allows us to bypass the numerically expensive task of solving the Zeeman RT equation for a range of B, θ_B, and χ_B values.

We use the FALC model atmosphere (Fontenla et al. 1993), which we find to be better for reproducing the observed linear polarization profiles from the Hanle effect. For comparison we also show theoretical profiles computed with the FALX model atmosphere (Avrett 1995; see our Figure 11).

In the multi-level RH-code, the atomic model of Ca i consists of 20 levels, with 17 line transitions and 19 continuum transitions. The main line is treated in PRD, using the angle-averaged PRD functions of Hummer (1962). All other lines of the multiplet are treated in complete frequency redistribution (CRD). However, for computing the polarization we restrict ourselves to a two-level atom model for the main line transition. All the blend lines are assumed to be depolarizing. They are treated in LTE in the RH-code. Therefore, the blend line absorption coefficient is implicitly included in the continuum absorption coefficient κ_c.

In this step we first derive the value of Bcos γ (longitudinal component) that is uniquely determined by the observed V/I profile when compared with the derivative of the observed Stokes I profile (see Equation (15)). We then compute ensembles of B, θ_B, and χ_B values that all reproduce the uniquely determined Bcos γ. These ensembles are model independent because they are directly based on the observed V/I and I profiles. For more details see Section 5.1.

While Step 1 fixes the value of the LOS component of $\bm {B}$ by using Stokes V, the components in the transversal plane are entirely unconstrained. To determine all three components of the field vector we need two additional observables, namely, the observed Q/I and U/I. We generate theoretical polarization diagrams for Q/I and U/I at the line center computed using the RT equation with the Hanle effect (see Equation (1)) for the sets of $\bm {B}$ values already extracted in Step 1. The observed line center values of Q/I and U/I are marked on these diagrams. With the polarization diagrams we can remove the ambiguities in the B, θ_B, and χ_B values. For more details see Section 5.2.

In this step we compare the observed profiles with the theoretical profiles calculated with the B, θ_B, and χ_B values extracted from Steps 1 and 2. Although in Step 2 we use the Q/I and U/I values only at line center, it turns out that these B, θ_B, and χ_B values are also able to reproduce the wavelength dependence of the observed Q/I and U/I profiles in the line core. For more details see Section 5.3.

Here we discuss in more detail how Step 1 is performed (Section 4.2). For the computation of the V/I profiles in Step 1 using Equation (15) we consider a grid of magnetic field parameters B, θ_B, and χ_B, which are in the range B ∈ [0 G, 50 G] in steps of 5 G, θ_B ∈ [0°, 180°] in steps of 5°, and χ_B ∈ [0°, 360°] in steps of 1125. For the computation of the derivative of the intensity with respect to the wavelength in Equation (15) we use the observed Stokes I itself. For differentiation we use an in-built subroutine (in the IDL programming language) that performs a numerical differentiation using a three-point Lagrangian interpolation. For a given location and a given field strength, we get several sets of (θ_B, χ_B) values (a minimum of 0 to a maximum of 120 sets) consistent with the observed V/I profiles.⁶ The field strengths derived here always correspond to weak values of B (10 G–50 G), consistent with the fact that the observed V/I signals are in the range 0.1%–0.9%. We have investigated the behavior of the V/I profiles in the full range of B ∈ [0 G, 50 G], θ_B ∈ [0°, 180°]. For each observation point the V/I model fitting restricts the values of B and θ_B to be, respectively, in the smaller sub-intervals of [0 G, 50 G] and [0°, 180°]. No such restriction is imposed on the χ_B values.

The values of B, θ_B, and χ_B deduced from Stokes V should also fit the observed Q/I and U/I profiles. For field strengths below a few hundred G the transverse Zeeman effect does not produce any significant Q/I and U/I. In contrast, at these field strengths, the Hanle effect plays an important role: it modifies the Q/I generated by the resonance scattering and creates a non-zero U/I. Hence, Q/I and U/I provide constraints on the magnetic field vector which are independent of those stemming from V/I. We compute the theoretical Stokes profiles (I, Q/I, and U/I) considering only the Hanle effect (see Equation (1)) with B, θ_B, and χ_B in the range restricted by the V/I model fitting. Since in the weak field approximation the transfer equation for Stokes V decouples from the transfer equation for (I, Q, U)^T, Hanle scattering does not produce any V/I signal unless there exists an initial source of the circular polarization.

We recall that the polarization diagrams are plots of Q/I versus U/I for given values of field strength B, wavelength λ, and LOS defined by μ, φ (see, e.g., Bommier 1977; Landi Degl'Innocenti 1982; Bommier et al. 1991; Faurobert-Scholl 1992; Stenflo 1994; Nagendra et al. 1998). The values of θ_B and χ_B are varied in finite steps. A closed curve (hereafter loop) is produced in the (Q/I, U/I) plane as χ_B is varied from 0° to 360°. Each value of θ_B corresponds to a different loop. The size and tilt of the polarization pattern in a loop with respect to the vertical (Q/I = 0 line) depend on the value of B and θ_B. Examples of polarization diagrams are shown in Figures 4 and 5 for B = 10 G, 15 G, 30 G, and 50 G. They are constructed with the theoretical Q/I and U/I values at line center calculated with μ = 0.9 and φ = 0°. Different loops correspond to different values of θ_B ∈ [100°, 175°], incremented in steps of 5°. In each of the loops the small square symbol corresponds to the first grid point χ_B = 0°, and as we proceed in the counterclockwise direction, χ_B takes values between 0 and 360 in steps of 11.25 deg.

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The observed Q/I and U/I at the line center, averaged over three pixels in the wavelength domain (in order to reduce noise) are marked in the diagrams as big squares-i, where i = 1,... , 8 stand for the eight locations on the slit. We recall that we refer to these eight locations as observation points 1–8.

We find that among the different solution sets (B, θ_B, and χ_B) given by the weak field approximation of the Zeeman effect, there is only one choice which can simultaneously fit the observed line center values of Q/I and U/I.

Our results strongly suggest that observed Q/I and U/I are signatures of the Hanle effect and V/I of the Zeeman effect. Whether resolved inhomogeneities in the atmosphere and velocity fields also contribute to the linear and circular polarizations is a question which goes far beyond the scope of this paper.

In Figure 7 we show the Q/I, U/I, and V/I fits for the observation points 1 and 2. The best fit values of the vector magnetic field for these points are (B, θ_B, χ_B) = (15 G, 125°, 56°) and (B, θ_B, χ_B) = (15 G, 110°, 191°), respectively. Figures 8–10 are similar to Figure 7, but for the observation points 3–8.

It can be observed on the CCD image shown in Figure 3 and on the observed spectra shown in Figures 7–10 that the shapes and signs of the Q/I and U/I profiles in the line core region vary significantly along the slit. These variations can be ascribed predominantly to changes in the magnetic field direction. In Figure 6 we show how sensitive indeed are the cores of Q/I and U/I profiles to the values of θ_B and χ_B. We also find that the values of B, θ_B, and χ_B, which give the best fits to the line center values of the observed Q/I and U/I, also give the best fits to the observed Q/I and U/I at other wavelengths in the line core. The most likely reason is that the frequency dependence of the line core polarization is essentially controlled by the anisotropy of the radiation field. In the wings, the values of U/I tend to zero and those of Q/I are controlled by Rayleigh scattering and PRD effects. A comparison of I, Q/I, and U/I profiles formed under CRD and PRD mechanisms are presented in Figure 7. See Appendix A for the definition of CRD. In the left panel, we compare the theoretical profiles of Q/I calculated with PRD (thick solid lines) and CRD (thin solid lines) for the observation point 1. This figure clearly shows that the wing shapes and peak values can be nicely fitted with PRD but not with CRD. Also note that the line center values of Q/I and U/I computed with CRD are smaller than those computed with PRD (by about 15% in terms of relative difference in Q/I). Moreover, we find that with CRD we are unable to reproduce the line center values of Q/I and U/I, for many of the observation points on the slit. Indeed, most of the observed values of Q/I and U/I lie outside the polarization diagrams constructed with CRD. Our results clearly show that PRD effects need to be taken into account in the determination of the magnetic field from the Ca i 4227 Å line polarization. The analysis of the second solar spectrum by Belluzzi & Landi Degl'Innocenti (2009) suggests that this conclusion could also hold for many other lines.

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We note here that the irreducible components ${\mathcal I}_Q^2$, Q ≠ 0, only contribute to the line cores of Q/I and U/I, because they are created by the Hanle effect. The component ${\mathcal I}^2_0$ on the other hand contributes to both the line core and the wings, but for Q/I alone.

In the left panels of Figure 10 we see positive peaks in the observed profiles of Q/I at 4226.65 Å and in both Q/I and U/I at 4226.8 Å (region of core minima). We have explored the possibility that these peaks could be due to the transverse Zeeman effect. Although these peaks can well be explained in terms of the σ components of the transverse Zeeman effect (but with fields much stronger than the range indicated by the observed Stokes V), it is not possible to fit the core peaks in terms of the π component of the transverse Zeeman effect. Only scattering and the Hanle effect can generate core signatures of sufficient amplitude in the linear polarization. This fact can be seen in Figure 11(a), where we compare the theoretical Stokes profiles computed using the Zeeman effect (using the RH-code) and the Hanle effect with the observational point 7. The $\bm {B}$ values used for the profiles computed using the Hanle effect are (30 G, 115°, 293°) and those computed using the Zeeman effect are (100 G, 90°, 203°). The curves showing the fit from the Hanle effect are the same as in the left panel of Figure 10.

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Another explanation of the mentioned peaks could be in terms of a different temperature structure of the atmosphere. To verify this we have tested a cooler model atmosphere (FALX). While the FALX model can generate these peaks, it fails to reproduce the observed line center Q/I and U/I amplitudes. The FALC model on the other hand can reproduce the line center peaks but not the peaks near the region of core minima. These results are shown in Figure 11(b), where we compare the theoretical Stokes profiles computed with the FALC and FALX model atmospheres, which give the best fit to the observation point 7. The FALC fit is the same as the one in the left panel of Figure 10. The $\bm {B}$ values derived from the FALC and FALX model atmospheres are, respectively, (30 G, 115°, 293°) and (30 G, 115°, 315°). We note that the two model atmospheres give two different azimuths χ_B. However, the FALC model gives an overall better fit than the FALX model. For all other observation points we find that FALX cannot reproduce the line center peaks, and that the overall fit is better in terms of the FALC model. Therefore, we have chosen the FALC model for all our calculations in the present paper.

We note that the positive red and blue peaks in the left panels of Figure 10 are asymmetric in nature. These differences between the red and the blue peaks in the observed Q/I and U/I profiles cannot be interpreted in terms of the Hanle effect, the transverse Zeeman effect, or a different temperature structure. This is because our theoretical profiles always produce symmetric red and blue peaks. In reality we observe such asymmetries often in Stokes spectra, and it is well known that they are due to spatially unresolved, correlated gradients of the magnetic and velocity fields in the solar atmosphere (see Stenflo et al. 1984; Stenflo 2010). However we have not taken up such studies in the present paper.

In Anusha et al. (2010) we found that the FALX model atmosphere gave a better fit to the observed profiles when modeling the non-magnetic limb observations of the Ca i 4227 Å line, while our present disk-center work favors the FALC model. In Figure 12(a) we compare the I/I_c, Q/I profiles computed using FALC and FALX model atmospheres for μ = 0.1 with the observations. We need three free parameters to model the non-magnetic observations. They are (1) an enhancement parameter c, associated with the elastic collision rate Γ_{E, vW} (of the van der Waals type), (2) a global scaling parameter s, and (3) a micro-turbulent magnetic field B_turb (see Anusha et al. 2010, for more details). One can notice that with a proper choice of free parameters FALX gives a better fit than the FALC model atmosphere to the non-magnetic limb observations of the Q/I profiles. This indicates that the actual temperature structure in the solar atmosphere might be better represented by a combination of model atmospheres (two-component models), but such studies are outside the scope of the present paper.

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The depolarizing effect of a magnetic field is often mimicked by a similar effect due to collisions and, in many cases, it is difficult to disentangle the two effects. In this paper we assume that the main contribution to the elastic collisions come from the van der Waals type collisions (Γ_{E, vW}). In Anusha et al. (2010) we found that in fitting the limb observations, particularly in reproducing asymmetric Q/I wing shapes, an enhancement of Γ_{E, vW} by a factor of c = 1.5 became necessary. A similar enhancement was also applied by Faurobert-Scholl (1992) to fit the observed Q/I wing polarization. The justification for this can be found in Derouich et al. (2003) and Barklem & O'Mara (1997), who respectively show that the old theories for the depolarizing elastic collision rate D⁽²⁾ and line broadening elastic collision rate Γ_E actually underestimate Γ_{E, vW}.

However, in the present paper, in fitting the near disk center observations we are able to reproduce the wing shapes of Q/I without any enhancement of Γ_{E, vW}. We examine the effect of such an enhancement of Γ_{E, vW} by taking c = 1.5 in Figure 12(b). It is clear that the differences arising due to this effect are small and the effect is mainly in the wings of the Q/I profiles, which are insensitive to the magnetic fields. The line core is unaffected by this modification of Γ_{E, vW}, because the line core is formed higher in the atmosphere where elastic collisions do not play any significant role. Thus our magnetic field determinations are unaffected by the inconsistencies in the theories for the elastic collision rates.

When interpreting Zeeman-effect observations of photospheric magnetic fields it is generally necessary to introduce a filling factor as a free parameter, since photospheric fields are extremely intermittent, with most of the flux in the form of strong (kG) elements occupying typically only 1% of the quiet photosphere (Stenflo 1973). The introduction of a filling factor as an additional free model parameter opens up a family of new solutions, which is destructive of the uniqueness of the solution unless some appropriate additional observational constraint is brought in. In the case of the photospheric Zeeman effect, the crucial additional constraint has been the introduction of the Stokes V line ratio, using simultaneous observations in two spectral lines that are formed in the same way but differ in their Landé factors.

Adding a magnetic filling factor as an additional degree of freedom in our interpretations of the Hanle forward-scattering observations might therefore seem to open up a Pandora's box of new possibilities, for which we, with our single-line observations, have no useful constraint. The actual situation is however not at all as unconstrained as it may seem earlier, since there are good reasons to believe that the filling factor is close to unity in our case and therefore does not play the role that it does in photospheric Zeeman-effect observations.

The argument for this is as follows: many of our observed polarization amplitudes in the core of the 4227 Å line are so large that theoretical modeling with the FALX atmosphere is unable to produce sufficient polarization, although the implicitly assumed filling factor is the maximum of 100%. Only with a hot atmosphere like FALC does a fit become possible. With any other filling factor the theoretically predicted polarization amplitudes will be smaller, in proportion to the assumed filling factor. This simply brings the observed amplitudes out of reach for a fit. Only with a filling factor close to unity is a fit at all possible in many of the observed cases.

In the case of the photospheric Zeeman-effect observations the situation is very different. The observed polarization amplitudes are mostly very small (on the quiet Sun). Since there is proportionality between circular polarization and field strength over a very wide range, from zero to typically about 500 G, where saturation gradually begins to set in, a tiny filling factor can easily be compensated for by a large field strength to fit the observed polarizations. With the line ratio (determining the differential Zeeman-effect nonlinearity between the two lines) the value of the field strength can be fixed. In contrast to the Zeeman-effect case, however, the Hanle polarization effects do not scale in proportion to the field strength. A small filling factor therefore cannot be compensated for by a large field strength or by any other field parameter, both because the field dependence is very nonlinear, and saturation takes place already for rather weak fields.

The question may then arise why we here conclude that the filling factor must be close to unity, while with Zeeman-effect observations we are used to filling factors that are a couple of orders of magnitude smaller. There are two reasons for this.

The first reason is that our 4227 Å observations refer to the chromosphere, and it is well known that the behavior of the Hanle effect in photospheric lines (like Sr i 4607 Å) and chromospheric lines (like Ca i 4227 Å) is entirely different (Stenflo 2003b). While the photospheric lines do not show much of any spatial structure and rarely any trace of a Stokes U signal, the chromospheric lines are full of spatial structures along the slit in both Q and U. The absence of U signals in the photospheric lines can be understood in terms of tangled fields on scales much smaller than the spatial resolution element, causing cancellation of the positive and negative contributions to Stokes U over each resolution element. The circumstance that we see substantial polarization signals in Stokes U with no evidence for cancellation effects in the chromospheric lines is evidence for resolved fields (filling factor near unity).

The second reason is that the 4227 Å observations that we are trying to fit here do not represent entirely quiet regions on the Sun, but were recorded in the outskirts of active regions, where we found an abundance of clear forward-scattering polarization signatures. In contrast, much less polarization is found far from active regions. The filling-factor concept may be more applicable in such truly quiet regions, but there the signals are so small that they are hard to measure with any acceptable signal-to-noise ratio. Application of the forward-scattering Hanle effect to quiet regions is therefore presently out of reach for two reasons: insufficient signal-to-noise ratio of the observations, and insufficient observational constraints (with single-line observations) when the fields are not spatially resolved.