The Astrophysical Journal, 844:97 (10pp), 2017 August 1 https://doi.org/10.3847/1538-4357/aa7a15
© 2017. The American Astronomical Society. All rights reserved.
Polarized Line Formation in Arbitrary Strength Magnetic Fields Angle-averaged and
Angle-dependent Partial Frequency Redistribution
M. Sampoorna1, K. N. Nagendra1, and J. O. Sten 2,3flo
1 Indian Institute of Astrophysics, Koramangala, Bengaluru 560 034, India; sampoorna@iiap.res.in, knn@iiap.res.in
2 Institute of Astronomy, ETH Zurich, CH-8093 Zurich, Switzerland; stenflo@astro.phys.ethz.ch
3 Istituto Ricerche Solari Locarno, via Patocchi, CH-6605 Locarno-Monti, Switzerland
Received 2017 May 4; revised 2017 June 16; accepted 2017 June 16; published 2017 July 26
Abstract
Magnetic fields in the solar atmosphere leave their fingerprints in the polarized spectrum of the Sun via the Hanle
and Zeeman effects. While the Hanle and Zeeman effects dominate, respectively, in the weak and strong field
regimes, both these effects jointly operate in the intermediate field strength regime. Therefore, it is necessary to
solve the polarized line transfer equation, including the combined influence of Hanle and Zeeman effects.
Furthermore, it is required to take into account the effects of partial frequency redistribution (PRD) in scattering
when dealing with strong chromospheric lines with broad damping wings. In this paper, we present a numerical
method to solve the problem of polarized PRD line formation in magnetic fields of arbitrary strength and
orientation. This numerical method is based on the concept of operator perturbation. For our studies, we consider a
two-level atom model without hyperfine structure and lower-level polarization. We compare the PRD idealization
of angle-averaged Hanle–Zeeman redistribution matrices with the full treatment of angle-dependent PRD, to
indicate when the idealized treatment is inadequate and what kind of polarization effects are specific to angle-
dependent PRD. Because the angle-dependent treatment is presently computationally prohibitive when applied to
realistic model atmospheres, we present the computed emergent Stokes profiles for a range of magnetic fields, with
the assumption of an isothermal one-dimensional medium.
Key words: line: profiles – polarization – radiative transfer – scattering – Sun: atmosphere
1. Introduction computational speed. A comparison of the direct methods and
The regime of strong (kilogauss) magnetic fields where the the iterative methods are given in Nagendra et al. (1999).
In the last few decades the iterative methods based on the
Zeeman effect dominates is a well explored topic over a long concept of operator perturbation (Cannon 1973) have been
period of time in solar magnetic field diagnostics (see the applied to solve a variety of polarized transfer problems (see
review by Stenflo 2013). In recent years, the regime of weak the reviews by Trujillo Bueno 2003; Nagendra 2017). In the
(milligauss to few tens of Gauss) magnetic fields where the present paper, we apply this operator perturbation technique to
Hanle effect dominates is also quite well explored (Stenflo solve the problem of polarized PRD line formation in arbitrary
1994; Landi Degl’Innocenti & Landolfi 2004; Stenflo 2013). strength magnetic fields. In order to make the treatment
However, the regime of intermediate fields (hecto-Gauss fields computationally tractable, here we consider an isothermal one-
like 50–500 Gauss for optical lines), is relatively less explored. dimensional planar atmosphere, and a two-level atom model
This is because in this regime both the Hanle and Zeeman with zero nuclear spin and infinitely sharp lower level. We
effects contribute significantly and hence have to be treated neglect lower-level polarization. The Hanle–Zeeman redistri-
simultaneously in the formulation and solution of the bution matrix corresponding to scattering on such a two-level
concerned polarized radiative transfer equation. Such a atom is derived in Bommier (1997a, 1997b) using a rigorous
problem of polarized line formation in the intermediate field QED approach, in Bommier & Stenflo (1999) using a classical
regime becomes numerically complex when partial frequency oscillator theory, and in Sampoorna (2011) using the Kramers–
redistribution (PRD) in scattering is taken into account. Heisenberg scattering approach of Stenflo (1994). The
A first numerical solution for this problem was presented in equivalence between these theoretical approaches is shown in
Sampoorna et al. (2008), where a perturbation method (similar Sampoorna et al. (2007a, 2007b) for a J = 0  1  0
to that of Nagendra et al. 2002) was proposed to solve the scattering transition and in Sampoorna (2011) for a
concerned polarized transfer equation. However, the perturba- Jl  Ju  Jl scattering transition (where Jʼs denote the total
tion method, wherein polarization is treated as a perturbation to angular momentum quantum number).
intensity, works well only for smaller degrees of polarization. More recently, this problem has also been addressed by
For larger degrees of polarization, direct numerical methods Alsina Ballester et al. (2017) for the approximation of an angle-
( averaged Hanle–Zeeman PRD matrix. An important result bysuch as the Feautrier method see, e.g., Dumont et al. 1977;
these authors (see also Alsina Ballester et al. 2016) is that, in
Faurobert 1987) and discrete space method (see, e.g., strong resonance lines for which the effects of PRD are
Nagendra 1986, 1988) would be required. However, these significant, the magneto-optical terms of the Stokes-vector
classical methods are computationally expensive for complex transfer equation produce a clear magnetic sensitivity in the
polarized transfer problems, like the ones described above. wings of the Q/I profile, as well as sizable U/I wing signals
Therefore, it is more advantageous to work with iterative that are also sensitive to the presence of magnetic fields with
techniques that provide accurate solutions at a much higher strengths similar to or larger than those needed for the onset of
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The Astrophysical Journal, 844:97 (10pp), 2017 August 1 Sampoorna, Nagendra, & Stenflo
the Hanle effect in the spectral line under consideration. dttot = dt. Equation (1) can then be rewritten as
Recently, we have pointed out that in the presence of weak
magnetic fields the angle-dependent PRD effects cannot be ¶ I (t, x, n) = I (t, x, n) - Seff (t, x, n). (5)
neglected (see Nagendra et al. 2002; Nagendra & Sampoorna ¶t
2011; Supriya et al. 2013; Sampoorna 2014). Therefore, in the Here the effective source vector is
present paper, we consider both the angle-averaged and angle-
dependent Hanle–Zeeman redistribution matrices and study the Seff (t, x, n) = Stot (t, x, n) - K¢I (t, x, n), (6)
validity of the angle-averaged approximation for not only the
Hanle regime but also the intermediate regime of eld where we have redefined the total absorption matrix offi
strengths. Equation (2) as
The paper is organized as follows. In Section 2, we briefly K
present the basic equations governing the problem at hand. The K¢ = - E. (7)(j + r)
numerical method of the solution is described in Section 3. The I
emergent Stokes profiles for different field strengths, and angle- The total source vector is defined as
averaged and angle-dependent cases are presented in Section 4.
Conclusions are presented in Section 5. In Appendices A and B, ( ) S(t, x, n)Stot t, x, n = . (8)
we present the Hanle–Zeeman redistribution matrix and Zeeman (jI + r)
line absorption matrix in the atmospheric reference frame.
Appendix C presents the normalization of the Hanle–Zeeman The formal solution of Equation (5) is obtained by using the
redistribution matrix. DELOPAR method of Trujillo Bueno (2003, see also
Sampoorna et al. 2008).
2. The Basic Equations
3. Numerical Method of the Solution
In the presence of a magnetic field, the polarized radiative
transfer equation for the Stokes vector I(t, x, n) = Here we present an iterative method based on the concept of
(I , Q, U, V )T may be written as operator perturbation to solve Equation (5). Hereafter, we omit
the τ dependence of the quantities, while their angle-frequency
¶
m I (t, x, n) = KI (t, x, n) - S(t, x, n), (1) dependences appear as subscripts. The formal solution of
¶t Equation (5) can be written in terms of the lambda operator as
where τ is the line-center optical depth, x = (n0 - n) Dn is Ixn = Lxn[Seff,xn]. (9)D
the frequency separation from line center n0 in Doppler width Following Olson et al. (1986), we define a local, “angle-
(DnD) units, the vector n(J, j) is the propagation direction of frequency dependent” approximate lambda operator L*xn
the ray (where ϑ is the co-latitude and j the azimuth), and through
m = cosJ. The absorption matrix K is given by
Lxn = L*xn + (Lxn - L*xn). (10)
K = F + rE, (2)
We can now set up an iterative scheme to compute the effective
where F is the 4 × 4 Zeeman line absorption matrix, E is the source vector as
4 × 4 unity matrix, and r is the ratio of continuous to line-
n+1 n n
center opacity. The source vector is given by Seff,xn = Seff,xn + dSeff,xn, (11)
S(t, x, n) = (rE + F)Bn0U + Sscat (t, x, n), (3) where the superscript n refers to the nth iteration step. From
Equation (6), we find that
where dSneff,xn = dS
n n
tot,xn - K¢dIxn. (12)
Sscat (t, x, n) For numerical simplicity, we neglect the dInxn term, and obtain
∮ dn¢
+¥
= ò dx¢R(x, n; x¢, n¢; B)I (t, x¢, n¢). (4) from Equations (8) and (3)4p -¥ n
n dSscat,xn
In Equation (3), ò denotes the photon destruction probability per dSeff,xn = . (13)jI,xn + r
scattering, B is the Planck function, andU = (1, 0, 0, 0)Tn0 . The
redistribution matrix R(x, n; x¢, n¢; B) accounts for the correla- Substituting Equations (6), (8), and (3) in Equations (11) and
tions in frequency, angle, and polarization between the incident (13), it is easy to find that
radiation field at frequency x¢ and direction n¢ and the reemitted
Sn+1 = Sn + dSn . (14)
radiation at frequency x and direction n in the presence of a vector scat,xn scat,xn scat,xn
magnetic field B. The quantity dn¢ is an element of solid angle Combining Equations (9)–(14) with Equation (4), we derive an
around n¢. The explicit form of the Hanle–Zeeman redistribution expression for dS nscat,xn as
matrix R and the Zeeman line absorption matrix F are given in
Appendices A and B respectively. dn¢ +¥ RdSn xn,x ¢n¢,B nLet us define a total optical depth dttot = dt (jI + r) m, scat,xn - ∮ ò dx¢ L*p j x ¢n¢[dSscat,x ¢n¢]4 -¥
wherej is the diagonal element of the Zeeman line absorption I,x ¢n¢ + rI n n
matrix F. For notational simplification, here we will call = [Sscat,xn]FS - Sscat,xn, (15)
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The Astrophysical Journal, 844:97 (10pp), 2017 August 1 Sampoorna, Nagendra, & Stenflo
Figure 1. Comparison of emergent Stokes profiles computed using angle-dependent type-II and type-III redistribution functions (dotted lines) and those using angle-
dependent type-II and CRD functions (solid lines). The line of sight is at m = 0.11 and j = 0. An isothermal self-emitting slab is considered with the following
model parameters: (T , a,  , r, GE GR) = (200, 10-3, 10-4, 10-9, 1); and magnetic field orientations: (JB, jB) = (30, 0). Panel (a) corresponds to GB = 1 and panel
(b) to GB = 100.
where Other polarization components of the source vector are
+¥ computed by the classical lambda iteration method, but with
[ n dn¢S nscat,xn]FS = ∮ ò dx¢Rxn,x ¢n¢,BLx ¢n¢[Seff,x ¢n¢], (16) the important difference that the Stokes I parameter on which4p -¥ these components depend is being improved in the previous
is obtained from a formal solution of the effective source vector iteration by applying the approximate lambda iteration scheme
Seff,xn of the previous iterate. The system of Equations (15) can to SI. In this case, Equation (15) simplifies to
be rewritten in the form dn¢ +¥ RdSn 00,xn,x ¢n¢,B n
AdSn = rn, (17) I,scat,xn
- ∮ dx¢ L*x ¢n¢[dSI,scat,x ¢n¢]
scat 4p ò-¥ jI,x ¢n¢ + r
n n
where residual vector rn is given by the right-hand side of = [SI,scat,xn]FS - SI,scat,xn, (18)
Equation (15). At each depth point, dS n nscat and r are vectors of
length 4Nx 2Nm Nj, where Nx is the number of frequency points for the intensity component of the source vector. It is important to
in the range of [0, x ], N is the number of angle points in the note that [S
n
I,scat,xn]FS is the intensity component of [S nscat,xn] ,max m FS
range [0 < m  1], and N is the number of azimuth points in which is obtained from the formal solution of the effective sourcej
the range [0  j  2p]. The matrix A thus has dimensions vector Seff,xn of the previous iterate. Thus the coupling of other
(4N 2N N ´ 4N 2N N ) at each depth point. It is compu- Stokes parameters to Stokes I is retained in the computation ofx m j x m j n
tationally formidable to compute this huge matrix and then [SI,scat,xn]FS. However, this coupling is neglected in the computa-
invert it. Thus it is necessary to nd a work around for this tion of the second term in the left-hand side of Equation (18), tofi
problem. reduce the computational cost. Again Equation (18) can be written
Following Trujillo Bueno & Manso Sainz (1999, see also in a form similar to Equation (17), but now the length of the
Alsina Ballester et al. 2017), we apply the approximate lambda residual vector and source vector correction is Nx 2Nm Nj and
iteration technique only to the intensity component of the correspondingly the dimension of the A matrix is
source vector, i.e., SI, because it drives the convergence rate. (Nx 2Nm Nj ´ Nx 2Nm Nj) at each depth point. With this
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The Astrophysical Journal, 844:97 (10pp), 2017 August 1 Sampoorna, Nagendra, & Stenflo
logarithmically spaced ones in the wings. Furthermore, the
maximum frequency xmax is chosen such that the monochromatic
optical thickness at xmax is much smaller than unity. We typically
have 70 points in the interval [0, xmax]. We use a seven-point
Gaussian quadrature in [0 < m  1] and an eight-point trapezoi-
dal grid for 0  j  2p. We consider a two-level atom model
with Jl= 0 and Ju= 1. To obtain accurate solution, particularly
for the case of angle-dependent redistribution functions, it is
necessary to correctly normalize the Hanle–Zeeman redistribution
matrix. This aspect is discussed in more detail in Appendix C.
4.1. Validity of Approximating the Type-III Redistribution
Function by CRD
From Equations (31)–(34), it is clear that the numerical
evaluation of type-III redistribution functions is computation-
ally expensive because it involves evaluating an integral.
Therefore, in the literature, the type-III redistribution function
is often approximated by the complete frequency redistribution
(CRD) function. The validity of this approximation for the
computation of intensity is discussed in detail in Mihalas
(1978). In this section, we present the validity of this
approximation for polarization.
For the problem at hand, the above-mentioned approx-
imation implies approximating the magnetic redistribution
functions of type-III by
R III,HHM M ,M ¢M ¢ (x, x¢, Q)= H (a, xM¢ M )H (a, x ),u l u l u l Mu¢Ml¢
R III,HFM M (x, x¢, Q)= H (a, x ¢ )F (a, xM ),u l,M Mu¢Ml¢ uMl u¢Ml¢
R III,FHM M ,M ¢M ¢ (x, x¢, Q)= F (a, xM¢ M )H (a, xM ¢M ¢),u l u l u l u l
R III,FF ¢ ¢ (x, x¢, Q)= F (a, xM¢M M ,M M )F (a, xuMl Mu¢Ml¢). (19)u l u l
Figures 1 and 2 show a comparison of emergent Stokes profiles
Figure 2. Comparison of emergent Stokes profiles computed using angle- computed using the angle-dependent type-II and type-III redis-
dependent type-II and type-III redistribution functions (dotted lines) and tribution functions (dotted lines) and those using angle-dependent
those using angle-dependent type-II and CRD functions (solid lines). The type-II and CRD functions (solid lines). For optically thin self-
line of sight is at m = 0.11 and j = 0. An isothermal semi-infinite
atmosphere is considered with the following model parameters: emitting slabs, small differences are noticed in Q/I and U/I for
(T , a,  , r, G G ) = (109, 10-3, 10-4, 10-7E R , 1); and magnetic field para- weak fields (see Figure 1(a)). However, these differences reduce
meters: (GB, JB, jB) = (1, 90, 45). and nearly vanish as the field strength increases (see Figure 1(b)).
This may be due to the increasing dominance of the Zeeman
effect with an increase in field strength. Intensity and circular
simplification, the problem becomes computationally feasible. We polarization are nearly identical for both the cases. For semi-
find that the numerical method presented in this section converges infinite atmospheres, the approximation of replacing the type-III
well for eld strengths up to 300 G. redistribution function by CRD does not seem to produce anyfi
noticeable effect on the emergent Stokes profiles already for weak
fields (see Figure 2). Therefore, replacing the type-III redistribu-
4. Results and Discussions tion function by CRD (see Equations (19)) is a good
We consider isothermal, plane-parallel atmospheres with either approximation not only for the computation of intensity but also
no incident radiation at the boundaries (self-emitting slabs) for the computation of polarized profiles in arbitrary field
or semi-in nite atmospheres with Planck function radiation strengths. From here on we use this approximation for all thefi
eld incident on the lower boundary (typical of optically thick illustrations presented in the rest of the paper.fi
stellar atmospheres). Such slab models are characterized by
(T , a,  , r, GE GR), where T is the optical thickness of the slab. 4.2. Effect of Anomalous Dispersion Coefficients on Emergent
The Planck function at the line center Bn0 is taken as unity. The
Stokes Profiles
depolarizing elastic collision parameter is assumed to be In Zeeman effect theory, the anomalous dispersion coefficients
D(1) = D(2) = 0.5GE , while D(0) = 0. The magnetic field ci (see Equation (42)) are known to play a significant role in the
parameters are (GB, JB, jB), where GB = gJ wL GR and angles emergent Stokes profiles only for strong fields (see, e.g.,u
(JB, jB) define the field orientation with respect to the Stenflo 1994; Landi Degl’Innocenti & Landolfi 2004). However,
atmospheric normal. We use a logarithmic depth grid with five in the case of optically thick scattering lines, these coefficients are
points per decade. The first depth point is at t1 = 10-4. For the shown to play a significant role already for fields that are in the
frequency grid, we use equally spaced points in the line core and Hanle regime (see Alsina Ballester et al. 2017). In this section, we
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The Astrophysical Journal, 844:97 (10pp), 2017 August 1 Sampoorna, Nagendra, & Stenflo
Figure 3. Effect of anomalous dispersion coefficients on emergent Stokes profiles computed using angle-dependent Hanle–Zeeman redistribution matrix. The line of
sight is at m = 0.11 and j = 0. An isothermal semi-infinite atmosphere is considered with the following model parameters: (T , a,  , r, GE GR) =
(109, 10-3, 10-4, 10-7, 1); and magnetic field orientations: (JB, jB) = (90, 45). Panel (a) corresponds to GB = 0.1, panel (b) to GB = 1, and panel (c) to
GB = 30. The solid line is computed including the anomalous dispersion coefficients in the Zeeman absorption matrix, while the dashed line is computed by neglecting
these coefficients.
re-confirm their finding, but now for the case of angle-dependent angle-averaged counterparts, which are defined as
Hanle–Zeeman redistribution matrix.
Figure 3 shows a comparison of emergent Stokes profiles p¯ II, X 1 II, X
computed including (solid lines) and neglecting (dashed lines) RM ¢M M (x, x¢)= ò RM ¢M M (x, x¢, Q)sinQdQ, (20)l u l 2 0 l u l
the anomalous dispersion coefficients in the Zeeman absorption p
R¯ III, XY ( 1x, x¢) = ò R III, XYmatrix. Even for very weak fields (GB = 0.1), we see significant MuMl,Mu¢Ml¢ 2 MuMl,Mu¢M (x, x¢, Q)0 l¢
differences between the two cases, particularly in the wings of ´ sinQdQ, (21)
the U/I profiles. Indeed, all of the wing signatures in U/I are
entirely due to the anomalous dispersion coefficients (particu- where X and Y stand for the symbols H and/or F.
larly due to the cV coefficient that couples the Stokes Q and We have studied the validity of the angle-averaged
the U). As the field strength increases, the effect of these approximation as a function of field strength. For this, we
coefficients are also seen in the wings of Q/I. We remark that considered the case of an isothermal semi-infinite atmosphere
these wing signatures are not directly caused by the PRD and varied the field strength parameter GB from 0.1 to 100. We
functions or the Hanle effect. Indeed, the PRD is responsible recall that, the damping parameter a is given by
for the generation of Q/I signals in the line wings, which are G + G + G
modified subsequently by the cV coefficient to give rise to the a =
R I E . (22)
full magnetic sensitivity noticed in the wings of Q/I and U/I 4pDnD
profiles. The intensity and V/I on the other hand do not show For a spectral line at 5000 Å, a typical Doppler width of 30 mÅ,
much sensitivity to these coefficients for the magnetic field a damping parameter a = 10-3, and GE GR = 1, we obtain
parameters considered in Figure 3. from Equation (22) G 7 −1R = 2.26 ´ 10 s (where we have
assumed GI GR  1). In terms of the field strength B in Gauss,
4.3. Stokes Profiles Computed with the Angle-dependent and G 7B = 0.88 ´ 10 gJ (B GR). For a J = 0  1  0 scatteringu
Angle-averaged Hanle–Zeeman PRD Matrix transition considered for the illustrations presented in this
The solution of the polarized transfer equation, including paper, the Landé factor of the upper level gJ = 1. Clearly, au
angle-dependent PRD functions is known to be computation- variation of GB between 0.1 and 100, then corresponds to a
ally very expensive. To avoid such computationally expensive variation in the range between 0.25 and 256 G in field strength.
problems, it is often a common practice to replace the angle- Figures 4 and 5 show a comparison of the emergent Stokes
dependent PRD functions by their angle-averaged versions profiles computed with the angle-dependent (solid lines) and
(see, e.g., Rees & Saliba 1982; Faurobert 1987; Nagendra et al. angle-averaged (dashed lines) Hanle–Zeeman redistribution
2002). For the problem at hand, the angle-averaged approx- matrix for different values of the field strength parameter GB.
imation implies replacing the angle-dependent magnetic For optically thick lines, the differences between the two cases
redistribution functions defined in Equations (29)–(34) by their are seen mainly in the line core region (which is in agreement
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The Astrophysical Journal, 844:97 (10pp), 2017 August 1 Sampoorna, Nagendra, & Stenflo
Figure 4. Comparison of emergent Stokes profiles computed using an angle-dependent (solid lines) and those using an angle-averaged (dashed lines) Hanle–Zeeman
redistribution matrix. The line of sight is at m = 0.11 and j = 0. An isothermal semi-infinite atmosphere is considered with the following model parameters:
(T , a,  , r, GE GR) = (109, 10-3, 10-4, 10-7, 1); and magnetic field orientations: (JB, jB) = (90, 45). Panel (a) corresponds to GB = 1, and panel (b) to GB = 3.
with the results presented in Figure 6(b) of Nagendra & equation. Unlike the perturbation method presented in
Sampoorna 2011). The intensity and the V/I profiles are Sampoorna et al. (2008), the present numerical method is
insensitive to the choice of angle-dependent or angle-averaged robust and is able to handle atmospheres with any total optical
functions, while the linear polarization profiles are considerably thickness. However, like the perturbation method, the present
sensitive. From Figures 4 and 5, we see that the differences method converges well for field strengths up to 300 G. This is
between the U/I profiles computed with angle-dependent and due to the fact that the operator perturbation technique has been
angle-averaged PRD functions are particularly large for the applied only for the computation of source vector corresp-
Hanle saturation regime field strength represented by GB = 3. onding to the intensity component of the Stokes vector.
The differences between the two cases continue to persist for Although the operator perturbation technique can be applied to
fields in the intermediate Hanle–Zeeman regime and for GB as all four Stokes parameters, it becomes computationally
large as 100. This shows that angle-dependent effects are impractical to implement the same in reality.
important for the computation of linear polarization profiles, We have performed numerical studies to analyze the
particularly the U/I profiles. importance of the angle-dependent Hanle–Zeeman PRD matrix
in the computation of the emergent Stokes profiles of optically
5. Conclusions thick lines, for a range of field strength parameters GB between
In the present paper, we consider the problem of polarized 0.1 and 100. In agreement with the previous studies for the case
line formation including the effects of PRD and magnetic fields of the weak field (Nagendra & Sampoorna 2011), we find that
of arbitrary strength and orientation. We consider scattering on the angle-dependent effects are significant mainly in the line
a two-level atom with zero nuclear spin and an infinitely sharp core and particularly for the computation of linear polarization
unpolarized lower level. The Hanle–Zeeman redistribution profiles. Significant differences between the U/I profiles
matrix corresponding to this case has been derived in Bommier computed with angle-dependent and angle-averaged PRD
(1997a, 1997b, see also Bommier & Stenflo 1999; Sampoorna functions are noticed for fields in the saturation regime of the
et al. 2007a, 2007b; Sampoorna 2011). We have developed an Hanle effect. The differences in both Q/I and U/I persist
iterative technique based on the concept of operator perturba- for fields as large as GB = 100. Therefore, we conclude that
tion to solve the concerned polarized radiative transfer angle-dependent effects are important and should be taken into
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The Astrophysical Journal, 844:97 (10pp), 2017 August 1 Sampoorna, Nagendra, & Stenflo
Figure 5. Same as Figure 4, but for GB = 30 in panel (a) and GB = 100 in panel (b).
account for an accurate determination of the magnetic field. atmospheric reference frame: ARF). Transformation between the
Furthermore, we have shown that the approximation of two frames is described in Appendix D of Sampoorna et al.
replacing the angle-dependent type-III PRD functions by (2007b), where this transformation is performed numerically. On
CRD is a good approximation for optically thick lines. Finally, the other hand, the same transformation can be expressed in a
we re-confirm the importance of anomalous dispersion more compact analytic form by using the irreducible spherical
coefficients of the Zeeman absorption matrix on producing tensors for polarimetry introduced by Landi Degl’Innocenti
interesting signatures in the wings of linear polarization profiles (1984). The latter is presented in this Appendix.
even for very weak fields (as originally shown in Alsina For a Jl  Ju  Jl scattering transition with infinitely sharp
Ballester et al. 2017). lower level Jl, the Hanle–Zeeman redistribution matrix is given
by (see Equation (48) of Bommier 1997b)
We thank an anonymous referee for constructive comments
that helped improve the paper. Computations are performed on a II
20 node HYDRA cluster (dual Xeon X5675 with 6 cores per R(x, n; x¢, n¢; B)= R (x, n; x¢, n¢; B)
processor and 3.06 GHz clock speed), FORNAX (dual opteron + RIII (x, n; x¢, n¢; B). (23)
6220 with 8 cores and 3.0 GHz clock speed), and KASPAR (dual
Xeon X5675 with 6 cores per processor and 3.06GHz clock In the MRF, the elements of the type-II and type-III
speed) computing facilities at the Indian Institute of Astrophysics. redistribution matrices are given by (see Equations (51) and
(49) of Bommier 1997b, see also Equations (53) and (55) of
Appendix A Sampoorna 2011)
The Hanle–Zeeman Redistribution Matrix
The Hanle–Zeeman redistribution matrix derived in Bommier R II (x, n, x¢, n¢, B)
(1997b, see also Sampoorna et al. 2007a, 2007b; Sampoorna ij
2011) refers to a frame where the magnetic field is along the Z- = å (-1)Q K K ¢Q (i, n) -Q( j, n¢)K,K¢Q,II (x, x¢, Q, B),
axis (namely, the magnetic reference frame: MRF). However, for K¢KQ
transfer computations, we need this matrix defined in a frame (24)
where the Z-axis is along the atmospheric normal (namely, the
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The Astrophysical Journal, 844:97 (10pp), 2017 August 1 Sampoorna, Nagendra, & Stenflo
R III (x, n, x¢, n¢, B) Landé factor of the upper level. Furthermore, GR denotes theij
radiative de-excitation rate of the upper level, GI the inelastic
= å (-1)Q K(i, n) K ¢ K,K,K¢Q -Q( j, n¢)Q,III (x, x¢, Q, B). de-excitation rate, G the elastic collisional rate, and D(K )E the
KK¢KQ depolarizing collisional rate. The auxiliary quantities corresp-
(25) onding to type-II redistribution are given by
In the above equations, i and j denote the Stokes parameter
II 1
indices, which take values 0, 1, 2, and 3;  KQ(i, n) are the hM M ¢ (Ml¢M )= [R
II,H II,H
u u l 2 Ml¢M
+ R
uMl Ml¢M
];
u¢Ml
irreducible spherical tensors, where K takes values 0, 1, and 2,
1
while Q varies in the range of -K  Q  +K and n(q, f) f II (M¢M )= [R II,F - R II,FM M ¢ l l M ¢M ¢M M ¢M M ], (28)u u l u l l u l
refers to the ray direction with respect to the magnetic 2field; and
Θ denotes the scattering angle between the incident and where the magnetic redistribution functions of type-II are given
scattered rays. The laboratory frame redistribution functions by
K,K¢Q,II (x, x¢, Q, B) and 
K,K,K¢
Q,III (x, x¢, Q, B) are given by ⎧ ⎡ x - x¢ + x ⎤21 ⎪ ⎫MM ¢ ⎪
K,K¢
G II,H l l
Q,II (x, x ¢, Q, B) =
R 3(2J + 1) RM ¢M M (x, x¢, Q) = exp⎨-⎢ ⎥ ⎬u ⎪
G + G + G + iw g Q l u lR I E L p sinQ ⎩ ⎣ 2 sin(Q 2) ⎦J ⎭⎪u
´ (2K¢ + 1)(2K + 1) å (-1)Ju-Ml-1+Q ⎛⎜ a xMuMl + x ⎞´ H , M¢ + xuMl MlMl¢MuMu¢MlMl¢pp¢pp¢¢¢ ⎟
⎛ ⎝ cos(Q 2) 2 cos(Q 2) ⎠
,
( )J -M ¢-1+Q⎜ J 1 J ⎟⎞⎛ ⎞´ -1 u l l u⎝ ⎠⎝⎜
Jl 1 Ju
¢⎟-Ml -p Mu -Ml - (29)p¢ Mu⎠
⎛ Jl 1 Ju ⎞⎛ ⎞´ ⎜ ⎟⎜ Jl 1 Ju ⎟ ⎧⎪ ⎡II,F 1 ⎨ ⎢ x - x¢ + x ⎤
2 ⎫
¢ ⎪
⎝-M ¢ -p Mu⎠⎝-M ¢ -p¢¢¢ M ¢⎠ RM ¢M M (x, x¢, Q) = -
MlMexp l ⎬
l l u ⎥l u l
⎛ ⎞⎛ ⎞ p sinQ ⎩
⎪ ⎣ 2 sin(Q 2) ⎦ ⎪⎭
´ ⎜ 1 1 K¢⎟⎜ 1 1 K⎝ ⎟-p p¢ Q ⎠⎝-p p¢¢¢ Q ⎠ ´ F ⎜
⎛ a xMuMl + xM¢ M + xu l MlMl¢ ⎞, ⎟.
´ [h II (M ¢M ) + if II (M ¢M )], (26) ⎝ cos(Q 2) 2 cos(Q 2) ⎠MuMu¢ l l MuM lu¢ l
(30)
K,K,K¢( ) G RQ,III x, x¢, Q, B = G + G + D(K) + iw g Q In the above equations, H (a, x) and F (a, x) are the VoigtR I L Ju
[G - D(K)
and Faraday–Voigt functions, x
] M M
= x + (g
u l J Mu - gu J Ml)l
´ E 3(2Ju + 1)(2K + 1) (nL DnD) is the magnetically shifted non-dimensional fre-
GR + GI + GE + iwLgJ Qu quency (with a similar expression for the primed quantity),
´ (2K¢ + 1)(2K + 1) xM M¢ = nM Dn with n being the energy differencel l lMl¢ D MlMl¢
å (-1)Ju-Ml-1+Q(-1)Ju-Ml¢-1+Q between the magnetic substates Ml and Ml¢, and a =
M M ¢M M ¢¢¢MM ¢pp¢pp¢¢¢ (GR + GI + GE) (4pDnD) is the total damping width of theu u u u l l
⎜⎛ Jl 1 Ju ⎟⎞⎛⎜ Jl 1 J ⎞
line. The magnetic redistribution functions of type-III appear-
´ u⎝ ⎠⎝ ⎟⎠ ing in Equation (27) are given by-Ml -p Mu -Ml -p¢ Mu¢
⎛⎜ Jl 1 Ju ⎞⎟⎜⎛
+¥
J 1 J ⎞ 1 2
´ l u ⎟ R
III,HH
M M ,M ¢M ¢ (x, x¢, Q) = ò du e-u
⎝ u l u l 2-Ml¢ -p Mu⎠⎝-Ml¢ -p¢¢¢ Mu¢¢¢⎠ p sinQ -¥
⎛ ⎞⎛ ⎞⎛ ⎞ ⎢
⎡ a ⎥⎤ ⎛⎜ a xM ¢M ⎞J J K ´ H , u l¢
´ 1 1 K¢ 1 1 K u u
- u cotQ⎟,
⎝⎜ ⎟⎜ ⎟ ⎢ 2 2 ⎝ ⎠-p p¢ Q ⎠⎝-p p¢¢¢ Q ⎠⎜⎝ ¢ ⎟ a + (xM -M Q⎠ ⎣ M¢ M - u) ⎥ sinQ sinQu l ⎦u u
⎛ (31)⎜ Ju Ju K´ ⎟⎞ 1 [R III,HH III,HH⎝M ¢¢¢ -M Q⎠ 4 Mu¢Ml,Mu¢¢¢Ml¢ + RMu¢Ml,MuMl¢ +¥u u R III,HF 1 -u2
III,HH III,HH III,FH MuMl,Mu¢M ¢
(x, x¢, Q) = ò du e
+ 2RM M ,M ¢¢¢M ¢ + RM M ,M M ¢ + i(R
l p sinQ -¥
u l u l u l u l Mu¢Ml,Mu¢¢¢Ml¢ ⎡ ⎤
+ R III,FH - R III,FH - R III,FH a ⎛ a xM ⎞u¢Ml¢M ¢  ¢ ⎜M ,M M M M ,M ¢¢¢M ¢ M M ,M M ¢) ´ ⎢⎢ ⎥⎣ F ⎝ , - u cotQ
⎟,
u l u l u l u l u l u l a2 + (x 2 ⎠
III,HF III,HF III,HF M
¢ M - u) ⎥⎦ sinQ sinQ
+ ( u li RMu¢Ml,Mu¢¢¢M - Rl¢ Mu¢Ml,MuM + Rl¢ MuMl,Mu¢¢¢Ml¢ (32)
- R III,HF ) - R III,FF III,FFMuMl,MuMl¢ M ¢M + Ru l,Mu¢¢¢Ml¢ Mu¢Ml,MuMl¢
+ R III,FF III,FF +¥M M ,M ¢¢¢M ¢ - RM M ,M M ¢]. (27) R III,FH ( ) 1Q = -u2u l u l u l u l M M x, x¢, du eu l,Mu¢Ml¢ p2 sinQ ò-¥
In the above equations,Mu, Mu¢, Mu, Mu¢¢¢ andMl, Ml¢ denote the ⎡ (x ¢ - u) ⎤ ⎛ a xM ¢M ¢ ⎞
magnetic substates of the upper level Ju and the lower level J , ´ ⎢ MuMll ⎥H ⎜ , u l - u cotQ⎟, (33)2 2 ⎝ ⎠
respectively; wL = 2pnL is the Larmor frequency, and gJu is the ⎣⎢ a + (xM¢ uM - u) ⎥⎦ sinQ sinQl
8
The Astrophysical Journal, 844:97 (10pp), 2017 August 1 Sampoorna, Nagendra, & Stenflo
and Landolfi 2004)
+¥ K 0K
III,FF 1 -u2 ji = å  0 (i, n)F0 (Jl, J , x), (40)R uMuMl,M (x, x¢, Q) = du eu¢Ml¢ p2 sinQ ò-¥ K
⎡ (x ¢ - u) ⎤ ⎛ a x ¢ ¢ ⎞ where n(q, f) is the angle between the line of sight and the
´ ⎢ MuMl⎢ ⎥ ⎜
M
F , u
Ml
⎣ ⎥ ⎝ - u cotQ⎠
⎟.
2 ( ¢ )2 magnetic field, and the generalized profile function is given bya + xM - u sinQ sinQuMl ⎦ (see Landi Degl’Innocenti et al. 1991)
(34)
F0K q-10 (Jl, Ju, x) = å (-1) 3(2K + 1)
Following Section 5.12 of Landi Degl’Innocenti & Landolfi MuMlq
(2004, see also Frisch 2007), we transform Equations (24) and ⎛ ⎞2⎛ ⎞
(25) to the ARF. In the ARF, the magnetic field makes an angle J´ ⎜ u Jl 1⎟ ⎜ 1 1 K⎟H (a, xMuMl). (41)
JB with the atmospheric normal and has an azimuth jB. The ⎝-Mu Ml q⎠ ⎝-q q 0 ⎠
elements of the type-II and type-III redistribution matrices in The anomalous dispersion coefficients, c with i = 1, 2, 3
ARF are given by i
(corresponding to Stokes parameters Q, U, and V ) may be
R II (x, n, x¢, n¢, B) = å  K(i, n) written as (see Equation (A13.14) of Landi Degl’Innocenti &ij Q
KQ Landolfi 2004)
´ å NK,K¢QQ¢,II (x, x¢, Q, B)(-1)Q¢ K ¢-Q¢( j, n¢), (35) ci = å  K0 (i, n)Y0K0 (Jl, Ju, x), (42)
K¢Q¢ K
III K where the generalized dispersion profile function is given byRij (x, n, x¢, n¢, B) = å  Q (i, n) (see Landi Degl’Innocenti et al. 1991)
KQ
K,K,K¢ Q¢ K ¢ Y0K(J , J , x) = å (-1)q-1´ å N (x, x¢, Q, B)(-1)  ( j, n¢), (36) 0 l u 3(2K + 1)QQ¢,III -Q¢
KK¢Q¢ MuMlq
⎛ J J 1⎞2⎛ ⎞
where n(J, j) and n¢(J¢, j¢) 1 1 Know refer to the ray directions ´ ⎜ u l ⎟ ⎝⎜- ⎠⎟F (a, x-M M q q q 0 MuMl). (43)
with respect to the atmospheric normal. The type-II and type-III ⎝ u l ⎠
magnetic kernels have the form Equations (40) and (42) refer to the absorption and dispersion
coefficients in the MRF. They can be transformed to the ARF
NK,K¢ (x, x¢, Q, B) = ei(Q¢-Q)jQQ¢,II B using the rotation law obeyed by the irreducible spherical tensors.
´å d K (J )d K ¢ K,K¢QQ B QQ¢(-JB)Q,II (x, x¢, Q, B), (37)
From Equation (2.78) of Landi Degl’Innocenti & Landolfi
 (2004), we can writeQ
K,K,K¢ ( ) [
K(i, n)]MRF = å [ K (i, n)] KARFD (0, -JB, -j )*,
NQQ¢,III (x, x¢, Q, B) = ei Q¢-Q j
Q Q¢ QQ¢ B
B
Q¢
´ å d K (J )d K ¢ (-J )K,K,K¢(x, x¢, Q, B). (38) (44)QQ B QQ¢ B Q,III
Q where D KQQ¢ are the Wigner rotation matrices. Using this
Explicit forms of the reduced rotation matrices d K (JB) are transformation law, we obtain the absorption and dispersionQQ¢
given in Table 2.1 of Landi Degl’Innocenti & Landolfi (2004). coefficients in the ARF as
j =  K(i, n)e-iQjB d K (J )F0KB (Jl, Ju, x), (45)
Appendix B i å Q Q0 0KQ
The Zeeman Line Absorption Matrix
ci = å  KQ(i, n) e-iQjB d KQ0(JB)Y0K0 (Jl, Ju, x), (46)
For a two-level atom with an unpolarized lower level, the KQ
explicit form of the Zeeman line absorption matrix is given in
Stenflo (1994) and Landi Degl’Innocenti & Landolfi (2004). where n(J, j) now refers to the angle made by the line of sight
However, this matrix is usually given in a frame where the with respect to the ARF.
magnetic field is along the Z-axis. For the problem at hand, it is
necessary to transform this matrix to the ARF. For clarity, we Appendix C
present such a transformation in this appendix using the Normalization of the Hanle–Zeeman Redistribution Matrix
irreducible spherical tensors for polarimetry. As shown in Bommier (2017), the Hanle–Zeeman redis-
The Zeeman line absorption matrix has the form tribution matrix is normalized to
⎡j⎢ I
jQ jU jV ⎤⎥ dn dn¢j j c -c dx dx¢ R (x, n; x¢, n¢; B)
F ⎢ Q I V U= ⎢j -c j c ⎥⎥
ò ò ∮ ∮ ij
. (39) 4p 4p
U V I Q
⎢ GR⎣j c -c = di0dj0 . (47)V U Q jI ⎦⎥ GR + GI
The absorption coefficients, ji with i = 0, 1, 2, 3 (corresp- For accurate evaluation of the scattering integral, namely Sscat, it
onding to Stokes parameters I, Q, U, and V ) may be written is absolutely essential to correctly normalize the Hanle–
as (see Equation (A13.9) of Landi Degl’Innocenti & Zeeman redistribution matrix, particularly when angle-dependent
9
The Astrophysical Journal, 844:97 (10pp), 2017 August 1 Sampoorna, Nagendra, & Stenflo
redistribution functions are used. To achieve this numerically, we x, we obtain GR (GR + GI + GE) for the type-II redistribution and
first analytically derive the normalization of the type-II and type- [GR (GR + GI)] ´ [GE (GR + GI + GE)] for the type-III redis-
III redistribution matrices by integrating only over the incoming tribution, which when added gives the right-hand side of
angles and frequencies. The resulting analytic expressions (given Equation (47).
below) are used to re-normalize the redistribution matrices that
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10