The Astrophysical Journal, 828:84 (10pp), 2016 September 10 doi:10.3847/0004-637X/828/2/84
© 2016. The American Astronomical Society. All rights reserved.
POLARIZED LINE FORMATION WITH LOWER-LEVEL POLARIZATION AND PARTIAL FREQUENCY
REDISTRIBUTION
H. D. Supriya1, M. Sampoorna1, K. N. Nagendra1, J. O. Sten o2,3fl , and B. Ravindra1
1 Indian Institute of Astrophysics, Bangalore 560034, India
2 Institute of Astronomy, ETH Zurich, CH-8093 Zurich, Switzerland
3 Istituto Ricerche Solari Locarno, Via Patocchi, CH-6605 Locarno-Monti, Switzerland
Received 2016 January 29; revised 2016 June 10; accepted 2016 July 5; published 2016 September 7
ABSTRACT
In the well-established theories of polarized line formation with partial frequency redistribution (PRD) for a
two-level and two-term atom, it is generally assumed that the lower level of the scattering transition is
unpolarized. However, the existence of unexplained spectral features in some lines of the Second Solar
Spectrum points toward a need to relax this assumption. There exists a density matrix theory that accounts for
the polarization of all the atomic levels, but it is based on the flat-spectrum approximation (corresponding to
complete frequency redistribution). In the present paper we propose a numerical algorithm to solve the
problem of polarized line formation in magnetized media, which includes both the effects of PRD and the
lower level polarization (LLP) for a two-level atom. First we derive a collisionless redistribution matrix that
includes the combined effects of the PRD and the LLP. We then solve the relevant transfer equation using a
two-stage approach. For illustration purposes, we consider two case studies in the non-magnetic regime,
namely, the Ja = 1, Jb = 0 and Ja = Jb = 1, where Ja and Jb represent the total angular momentum quantum
numbers of the lower and upper states, respectively. Our studies show that the effects of LLP are significant
only in the line core. This leads us to propose a simplified numerical approach to solve the concerned radiative
transfer problem.
Key words: line: formation – methods: numerical – polarization – radiative transfer – scattering – Sun: atmosphere
1. INTRODUCTION upper level and the lower level is assumed to be unpolarized.
4
The linear polarization of the spectral lines is produced due Except for the case when the total angular momentum of the
lower level is J = 0 or 1/2, the assumption of zero atomic
to the absorption, emission, and scattering of radiation in the aalignment in the lower level is questionable, particularly when
solar atmosphere. The anisotropic illumination of the atom the lower level is different from the ground state. Trujillo
induces atomic alignment, which in turn gives rise to the Bueno & Landi Degl’Innocenti (1997) studied the influence of
polarization of the radiation (scattering polarization). There lower level atomic polarization on the scattering line polariza-
are two important theoretical approaches developed so far to tion for the case of a two-level atom with J =1 and J = 0.
study the physics of scattering polarization. The first one is a bThis is an example where the resulting polarization is
the self-consistent approach developed by Landi Degl’Inno- completely due to the population imbalances in the sublevels
centi (1983) using the density matrix formalism, starting of the lower atomic level. They used the density matrix
from the principles of quantum electrodynamics. One of the approach and solved simultaneously the statistical equilibrium
main advantages of this “density matrix” approach is that it equations (SEEs), neglecting stimulated emission and the
allows one to take into account the polarization of all levels transfer equation under complete frequency redistribution
of the atomic system under consideration. This naturally (CRD). This theory was later applied to explain many spectral
allows to take into account the lower level polarization features in the SSS. Landi Degl’Innocenti (1998) introduced
(LLP). The density matrix formalism is developed under the optical depopulation pumping of the lower levels as a possible
flat spectrum approximation and hence its main limitation is mechanism to explain the observed linear polarization in the
the difficulty to take into account the effects of partial Na I D
( ) 1
line. Trujillo Bueno (1999) showed the importance of
frequency redistribution PRD . The second theoretical LLP in the case of the Mg I b2 line. Also he pointed out the
approach is the semi-classical one, which provides the importance of the depolarizing elastic collisions and their role
advantage of including the effects of PRD by means of in decreasing the alignment of the atomic levels (see also
redistribution matrices (Stenflo 1994, hereafter S94). Using Casini et al. 2002). Trujillo Bueno et al. (2002) demonstrated
this “redistribution matrix” approach, our understanding of the operation of the ground-level Hanle effect and importance
the physics of resonance scattering has improved greatly and of the selective absorption from the ground level to the
the effects of PRD have been studied extensively. The generation of the polarization in the He I triplet system. Also
limitation of this theory is that using it we can deal with only the importance of atomic polarization of the metastable lower
two-level and two-term atoms with unpolarized, infinitely level of the Ca II infrared triplet was presented by Manso Sainz
sharp lower levels. & Trujillo Bueno (2003, 2010). However, in all the above-
The many anomalous spectral structures in the Second Solar mentioned studies, except in Landi Degl’Innocenti (1998), the
Spectrum (SSS, Stenflo & Keller 1997; Stenflo et al. 2000) cast effects of the PRD were neglected. In Nagendra (2003) the
doubt on the general assumption, made in the redistribution
matrix approach, that the anisotropic illumination of atoms in 4 Ja and Jb represent total angular momentum quantum numbers of the lower
the solar atmosphere induces population imbalances only in the and upper levels, respectively.
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The Astrophysical Journal, 828:84 (10pp), 2016 September 10 Supriya et al.
effects of PRD on linear polarization profiles have been Kramers–Heisenberg approach as given in Stenflo (1998) but
reviewed and the limitations of CRD approximation are pointed include the contribution from the lower level density matrix
out (see also Nagendra 2014, 2015). It is well known that CRD elements. In Stenflo (1998) the populations of the magnetic
approximation is sufficient in describing the line core sublevels were assumed to be the same for all the magnetic
polarization, whereas the PRD effects are important in the sublevels of the lower level. We relax this assumption here and
wings of strong resonance lines. thereby take into account the population imbalances among the
Formulation of a general self-consistent theory for lower magnetic sublevels. Following the procedure given in
radiative transfer problem including the effects of PRD and AppendicesA andB of Sampoorna et al. (2007), we express
LLP is a complex theoretical problem. Landi Degl’Innocenti the type II redistribution matrix with LLP in terms of
et al. (1997) have formulated a theory for coherent scattering irreducible spherical tensors of Landi Degl’Innocenti &
that takes into account the LLP. This theory is based on the Landolfi (2004, hereafter LL04). After an elaborate algebra
concept of “metalevels.” Based on this theory, Belluzzi et al. we obtain the expression for the type II redistribution matrix
(2015) have recently derived the collisionless redistribution
matrix for a two-term atom with hyper ne structure splitting with LLP in the atomic frame asfi
in the non-magnetic regime by including the polarizability of
the lower hyperfine levels ( F levels). They have applied this 2R II 2ij (x, n, x¢, n¢, B) = (2Ja + 1) å
theory to the problem of Na I D lines. In their studies they 3 mam m m¢f b bqq¢qq‴
have treated the LLP factor as a free parameter. Note that in
the present paper we do not treat the LLP factor as a free ´ å å (2K¢ + 1)(2K + 1)(2KL + 1)
parameter, but instead obtain it under the CRD limit for a K ¢KQKLQL
two-level atom. Casini et al. (2014) have also presented a G´ (-1)q-q¢ (-1)Ja-ma (-1)Q R
new quantum scattering theory with which they have derived GR + GI + iQgbwL
a generalized redistribution function for a polarized two-term ⎛ ⎞
atom with hyperfine structure splitting. As an alternative ⎛K ⎜ J J K´ r L a a L ⎟⎞⎜ Jb Ja 1 ⎟
attempt, in the present paper, we perform numerical QL ⎝ma -ma -QL⎠⎝-mb ma -q¢⎠
computations for a two-level atom by combining the ⎛⎜ J J 1 ⎟⎞⎜⎛b a Jb Ja 1 ⎞redistribution matrix approach and the density matrix ´ ⎟
approach. Using the redistribution matrix approach, we ⎝-mb mf -q⎠⎝-m¢b ma -q‴⎠
derive the collisionless PRD matrix (the so-called type II ⎛ J J 1 ⎞⎛ ⎞
redistribution matrix in the nomenclature of Hummer 1962), ⎜ b a´ ⎟⎜1 1 K⎟
including the effects of LLP. In the process, the density ⎝-m¢b mf -q⎠⎝q -q Q ⎠
matrix elements of the lower level are appropriately ⎛
incorporated in to the PRD matrix derived starting from the ´ ⎜ 1 1 K¢⎞⎟d (x - x¢ - naf )
Kramers–Heisenberg scattering formulation. We remark that ⎝q¢ -q‴ Q ⎠
only the population imbalances among the sublevels of the 1
lower level are taken into account, while the coherences ´ [f (n J m¢ ,J m - x¢) + f* ¢b a (n - x )]2 b a Jbmb,Jama
among them are ignored. This is consistent with the
K K ¢
assumption of an infinitely sharp lower level. The lower ´ (-1)QTQ (i, n)T-Q ( j, n¢). (1)
level density matrix elements are obtained by solving the
SEEs that are derived using the density matrix approach. The In the above equation μʼs denote the magnetic substates of a
type II redistribution matrix so derived is then included in the given J-state. The multipolar components of the lower level
radiative transfer equation. To this end we use the quantum density matrix are denoted by rKL . The multipolar index
field theory approach given by S94 to obtain the transfer QL
equation for the problem at hand. We further apply this 0  KL  2Ja with QL Î [-KL, +KL] is associated with the
K
theoretical formulation to the cases of 1  0  1 and lower level having total angular momentum Ja. r LQ can beL
1  1  1 transitions in the non-magnetic regime. obtained by solving the polarized SEEs as given in Equations
In Section 2 we present the collisionless redistribution matrix (10.1) and (10.2) of LL04. SEEs given in LL04 take into
for a two-level atom with PRD and LLP mechanisms properly account both population imbalances and coherences while the
taken into account. In Section 3 the radiative transfer equation redistribution matrix derived above takes into account only the
for solving the concerned problem is presented. Section 4 population imbalances. This is because the first 3j symbol
concerns a discussion on the influence of LLP on the polarized which arises due to the inclusion of LLP restricts the value of
line profiles formed under PRD. In Section 5 we propose a
QL to 0. All the different symbols appearing in the abovesimple alternative approach to solve the same problem. The
conclusions are presented in Section 6. The two-stage equation are consistent with Sampoorna (2011), therefore we
numerical procedure used to solve the transfer equation and do not elaborate on them. The profile function f (n Jbmb,J m - x)a a
SEEs is described in the Appendix. is defined in Equation(40) of Sampoorna (2011).
The above expression gives the RII redistribution matrix in
the atomic frame for a two-level atom without hyperfine
2. REDISTRIBUTION MATRIX WITH PRD AND LLP structure splitting. A more general expression for this matrix
As a first step we have derived the redistribution matrix for a multiplet (including also hyperfine structure) is given in
including the effects of PRD and LLP. We considered a general Landi Degl’Innocenti et al. (1997, see Equation(12) in their
case of Ja  Jb  Ja scattering transition. We follow the paper; see also Equation(1) in Landi Degl’Innocenti 1999).
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The Astrophysical Journal, 828:84 (10pp), 2016 September 10 Supriya et al.
The expression in Equation (1) is normalized to The components of the g and f matrices are given by
2 (2 + 1)2 Equations (8.94) and (8.95) of S94, respectively. We have toJa å å (2KL + 1) (-1)Ja+ma note that in S94 the total angular momentum quantum numbers
9 (2Jb + 1) mambKLQL of the lower and upper levels are denoted by Jμ and Jm,
⎛ J J K ⎞⎛ J J 1 ⎞2 respectively. However, to be consistent with the notations used
rK ⎜ a a L b a´ LQ ⎝m -m - ⎟Q ⎠⎜⎝ ¢⎠⎟ . (2)-m m - in the present paper, we denote them by Ja and Jb. Further, theL a a L b a q magnetic quantum numbers μ, μ′, m, and m′ of S94,
In order to transform the atomic frame RII matrix derived in respectively, are replaced by μa, m¢ ¢a, μb, and mb. Also the
Equation (1) to the laboratory reference system, we followed expressions in S94 are for a general transition in a multi-level
Section4 of Sampoorna (2011). This process simply involves system. In Section 2 we derived the redistribution matrix for a
the transformation two-level atomic system with LLP under the following
assumptions: (1) we neglect the off-diagonal terms of the
1 [f (n J m¢ ,J m - x¢) + f*(n J m ,J m - x¢)] lower level density matrix. This means that we consider only
2 b b a a b b a a the population imbalances in the lower level and neglect the
´ d (x - x¢ - naf )  h IIm m¢ (mf ma) + if
II
b m m¢
(mf ma)], (3) coherences between the magnetic substates. In other words,b b b
II II only the r terms contribute to the transfer equation; (2) wewhere h and f are the auxiliary functions which are defined mamaconsider the case of Rayleigh scattering, i.e., Ja = Jf. These
in Equations(22) and(23) of Sampoorna (2011). In the next assumptions are also taken into account while using the
section we include the type II redistribution matrix for a two- expressions of gaa¢ and faa¢ in the transfer equation.level atom with LLP, into the radiative transfer equation. The elements of the g matrix in the first two terms of the
right-hand side of Equation (4) represent radiative absorption.
3. RADIATIVE TRANSFER EQUATION FOR A TWO- The f matrix elements in the first two terms in the second line
LEVEL ATOM WITH LLP represent the stimulated emission. From Equations (8.113) and
We remark that in the density matrix approach of LL04 the (8.114) of S94 and the explanation that follows, we see that the
transfer equation is written in terms of emission and absorption terms inside the summation in Equation (5) can be written as a
coefficients. These emission and absorption coef -1ficients depend Mueller matrix M = TWT for the scattering of the Stokes
on the density matrix elements of the upper and lower levels vector Sk. In this manner the spontaneous emission term can be
respectively. On the other hand, in the redistribution matrix transformed from the coherency matrix formalism to the Stokes
approach, the transfer equation for a two-level atom without vector formalism. This transformation has to also be applied to
LLP is written in terms of a source vector that depends on the the absorption and stimulated emission terms in the radiative
scattering integral. The scattering integral basically contains the transfer Equation (4). We carried out these transformations and
redistribution matrix for the problem at hand. However, the found that the expressions we obtained are similar to those
transfer equation of LL04 (which can handle a two-level atom given in Section 6.7 of LL04. Thus the radiative transfer
with LLP) cannot be used for our purposes because the equation in Stokes vector basis can now be written as
emission vector is not written in terms of the scattering integral
3 3
involving the redistribution matrices. Therefore, we need to dSk =- åAkj Sj + åAS Sextend the transfer equation in the redistribution matrix kj jds j=0 j=0
approach to include the effects of LLP.
3
In order to derive the radiative transfer equation for a two- hn dW¢+ N B
level atom with LLP, we follow the quantum eld theory Ja JaJ
dx¢åMkjSj, (6)
fi 4p b ò 4p ò j=0
approach of S94 (see his Chapters 7 and 8). It is important to
note that the theory presented in S94 is in coherency matrix where Mkj are the elements of the Mueller scattering matrix, Akj
formalism. These equations are now converted to Stokes vector and AS are the elements of the radiative absorption and
formalism in the present paper. The notations used in the kjstimulated emission matrix, respectively. For the problem at
present section have the same meaning as in S94 unless
specified. From Equation (8.15) of S94 the radiative transfer hand, namely, a two-level atom with LLP, this matrix M is
equation can be written as simply the type II redistribution matrix described in Section 2.
dDaa¢ = - å[(g D + D g† ) 3.1. Contribution from Thermal Emissionab ba¢ abds ba¢b The transfer Equation (6) derived above does not take into
- ( f D + D †ba ab f )] + F , (4) account the contribution from the thermal emission. Thus, theab ¢ ba¢ c transfer equation obtained in Equation (6) represents only pure
where Fc represents the spontaneous emission term in the scattering. For practical applications, however, we need to take
coherency matrix formalism and is given by (see also Section into account the contribution from the thermal emission. For
8.10 of S94) this purpose we follow the procedure given in Section6.9 of
S94 to calculate the contribution from thermal emission.
hn 3 dW¢
F = ( f + f † ) ~ ò ò dx¢ Thermal emission is nothing but a limiting case in which thec c2 aa¢ aa¢ 4p scattering atom has completely lost its memory about how it
´år ååw w * D . (5) was excited (see Stenflo 1998). Hereafter we neglect them m ab bb ¢a a a¢b ¢
m m bb ¢ contribution from the stimulated emission (the second term ona f the right-hand side of Equation (6)). The radiative transfer
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The Astrophysical Journal, 828:84 (10pp), 2016 September 10 Supriya et al.
equation including the contribution from thermal emission of S94)
( j thermalk ) can be written as abs N (2Ja + 1)fq = årm m Sq (ma, mb)Hq. (12)
dS 3k hn dW¢ NJa pDn
a a
=- åA S + N B ò ò dx¢
D mamb
kj j J J J
ds j=0 4p
a a b 4p In Equation (12) the contribution from the population
3
II thermal imbalances in the lower level to the absorption processes is´ åRkj Sj + jk . (7) included via the density matrix element rm m . The transitionj=0 a a
strength Sq (ma, mb) is given by (see Equation (6.33) of S94)
Generally, the thermal emission is given by the absorption ⎛ ⎞2
matrix times the Planck function (B ). However, for the (m m ) = ⎜ Ja JS , 3 b 1n q ⎟o a b - m m q . (13)
problem at hand, it is necessary to distinguish the processes of ⎝ a b ⎠
absorption and thermal emission. While the LLP is relevant for In the case of thermal emission, the emission processes are
radiative absorption it is irrelevant for the thermal emission. independent of the absorption. Therefore, we define a separate
Therefore, to distinguish these two processes we define the profile matrix (Femi) to account for the emission processes. The
emission profile matrix asFemi and absorption profile matrix as matrix elements of theFemi matrix are now independent of the
Fabs. The absorption profile matrix is related to the absorption population imbalances in the lower level. The expression for
matrix hA derived starting from the quantum field theory of femiq is given by (see Equation (6.37) of S94)
S94 through
1
hA Fabs ( ) f
emi
q = åSq (m , m )H . (14)= kL , 8 qpDn a bD mamb
where kL is the line-averaged absorption coefficient (for the The form of the emission profile matrixFemi is the same as the
case when stimulated emission is neglected) defined as absorption profile matrix Fabs with the elements fabsI,Q,U,V ,D
hn
k = N B replaced by f
emi
I,Q,U,V ,D. For the problem at hand, the thermalL Ja J4p a
Jb. (9)
emission term is given by
The expression of the absorption profile matrix Fabs in the j thermal = kL (1 - a)Sba Femi1, (15)
atmospheric reference frame is the same as the expression
under the summation K, Q, K , Q of Equation (7.15a) of LL04. where 1 = (1, 0, 0, 0 )
T and α is the fraction of the scattering
l l
In the line of sight reference frame, this matrix is the same as process given by
Equation (6.59) of S94, which is given by NJa BJaJb òfa = x
Jxdx
⎛ , (16)fabs fabs abs abs⎜ ⎞ NJa BJaJb òfx Jxdx + NJa CJaJbI Q fU fV⎜ ⎟fabs abs abs abs⎜ f y - y ⎟⎟ with fx the area normalized profile function which is equal toFabs Q I V U= , (10) emi⎜ abs abs abs abs ⎟ fq when q = 0. The line source function Sba takes the⎜fU - yV f I yQ⎜ ⎟⎟ following simple form when stimulated emission is neglected:⎝fabs yabs abs absV U - yQ f I ⎠ N
S = Jb
AJbJa
ba . (17)
where NJa BJaJb
1 In the above equations, AJbJa and BJaJb are the Einstein
fabs abs 2 abs absI = fD sin g + (f2 +
+ f- ), coefficients for the spontaneous emission and absorption
abs abs 2 respectively, and CJaJb is the upward inelastic collisional rate.fQ = fD sin g cos 2c, By defining dt = -kLds we can rewrite the transfer
fabs = fabs sin2 g sin 2c, Equation (7) including the unpolarized continuum asU D
fabs
1
= (fabs - fabs)cos g, dI = (FabsV + - + rE)I - (rBn0 1 + S2 scatt
)
dt
abs 1 [ abs 1f = f - (fabs + fabs - (1 - a)S F
emi
)], (11) ba 1, (18)D 2 0 2 + -
where I = [S0, S T T1, S2, S3]  =[I , Q, U, V ] , r is the ratio of
with the corresponding expressions for yabsQ,U,V where fq is continuum to line-averaged opacity, and E is a 4×4 unit
replaced by ψq, the anomalous dispersion profile. In the above matrix. The scattering source vector Sscatt is given by
expressions γ and χ denote the inclination and azimuth of the
magnetic eld with respect to the line of sight. Since the dW
¢
fi Sscatt = ò ò dx¢RII (x, n, x¢, n¢, B)I (x¢, n¢). (19)
transfer equation is solved in the frame where the z-axis is 4p
along the atmospheric normal, we need to convert the angles γ The above equations take a simpler form in the absence of
and χ in the line-of-sight frame to the atmospheric reference magnetic fields. These equations are given in the next section.
frame. This can be done following Appendix B of Anusha et al. Furthermore, the equations presented in the present paper are
(2011). The expression for fabsq is given by (see Equation (6.52) for the case without elastic collisions. When elastic collisions
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The Astrophysical Journal, 828:84 (10pp), 2016 September 10 Supriya et al.
are included the contribution from the incoherent scattering
processes to the source vector should also be taken into
account. The details regarding this can be found in Sections
5.17 and 6.9 of S94.
3.2. The Non-magnetic Case
To numerically solve the problem of polarized radiative
transfer including the effects of LLP, we restrict our attention to
the non-magnetic case. For this particular case the Stokes V is
zero, and for the planar geometry Stokes U is zero. Therefore
the dimension of the problem reduces from 4×4 to 2×2.
The emission profile matrix Femi which contributes to the
thermal emission can be simplified further for this case. In the
absence of magnetic field, Equation (14) reduces to
emi Hfq = = fx. (20)pDnD
Therefore, femiD = 0 (cf. Equation (11)). Thus f
emi
I is the only
non-zero term in the emission profile matrixFemi, which takes
the form
⎛femi 0 ⎞
Femi = ⎜⎜ I⎝ 0 femi⎟
⎟. (21)
I ⎠
Figure 1. Emergent intensity and polarization for μ=0.11 computed using the
From Equations (15) and (21) we see that the thermal emission two-stage approach. The case of the 1  0  1 transition is considered with
the effects of LLP. Solid line represents the case of PRD and the dotted line that
contributes only to the Stokes I. Equation (18) can thus be of CRD. Other input parameters are A = 108 s-1; C 4 -1JbJa JbJa = 10 s . No
rewritten as background continuum opacity is used.
dI
= KI - [rBn0 1 + Sl], (22)dt 4. NUMERICAL RESULTS IN THE ABSENCE OF
MAGNETIC FIELDS
where K = Fabs + rE. The elements of the line source vector
S = (S ,S )T can be written as For our studies we consider two cases namely Ja = 1, Jb = 0l I,l Q,l and Ja = Jb = 1 to illustrate the effects of LLP on the polarized
1 II line formation. The governing equations and the concernedSI,l = ò dm¢ ò dx¢ [R00 (x, m, x¢, m¢)I (x¢, m¢)2 numerical method of solution are described for the Ja = Jb = 1
II case in Appendix. A similar procedure can be followed for+ R01(x, m, x¢, m¢)Q (x¢, m¢)] + (1 - a)S emiba fI , deriving corresponding expressions for the Ja = 1, Jb = 0 case
1
= ò m¢ ò (see also Trujillo Bueno & Landi Degl’Innocenti 1997). ForSQ,l d dx¢ [R II10 (x, m, x¢, m¢)I (x¢, m¢)2 our computations, we have considered a plane parallel
+ R II (x, m, x¢, m¢)Q (x¢, m¢)]. isothermal atmospheric slab with effective temperature of11 6000 K with no incident radiation at the boundaries. Back-
(23) ground continuum opacity is assumed to be zero. For all the
To solve the problem of polarized line formation including results presented in this paper we consider a thick slab of total
line center optical thickness T=1012. The effect of depolariz-
PRD and LLP, we follow a two-stage approach. In the first ing elastic collisions is neglected.
stage we solve the SEEs and the transfer equation simulta-
neously for a given Ja and Jb, taking into account the effects of 4.1. The Case of the 1  0  1 Transition
the polarization of the lower level but in the limit of CRD. The
density matrix elements obtained as output from the first stage We consider a two-level atom with Ja = 1 and Jb = 0. The
are used as input to compute the redistribution matrix that concerned SEEs and the transfer equations are given in Trujillo
enters the second stage (cf. Equation (23)). In the second stage Bueno & Landi Degl’Innocenti (1997). Because of the
( ( )) cylindrical symmetry of the problem, only three density matrixwe solve the radiative transfer equation see Equation 22 elements are needed to fully specify the excitation state of the
including the effects of PRD and LLP. Further details on the atoms, namely, r0 (a), r0 (b), and r2 (a). For this particular case
numerical method adopted are described in the Appendix. In 0 0 0
we consider the example of a hypothetical line at 5000Å
this two-stage approach, the density matrix elements are whose Einstein coefficient for spontaneous emission is
computed neglecting the effects of PRD, and they are kept Aba = 108 s-1 and whose downward inelastic collisional rate
fixed when computing the polarized PRD line profiles. Such a is Cba = 104 s-1.
two-stage approach is basically an approximation proposed in Figure 1 shows the emergent (I, Q/I) at μ=0.11 for the
this paper, which lacks a complete, self-consistent theory for 1  0  1 transition. Here we compare the results obtained
the problem at hand. under the limits of PRD and CRD when LLP is taken into
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The Astrophysical Journal, 828:84 (10pp), 2016 September 10 Supriya et al.
account. We see that the intensity profiles show the typical
signatures of the PRD and CRD mechanisms. In particular, in
the case of CRD we obtain an absorption line (see dotted line in
Figure 1), while in the case of PRD we obtain a self-absorbed
emission line (see solid line in Figure 1). The self-absorbed
emission type profiles in the intensity appear because of the
nature of the collisionless redistribution function exhibiting a
transition from the CRD-like behavior in the line core to the
coherent scattering-like behavior in the line wings (see e.g.,
Figure4(a) of Rees & Saliba 1982). The Q/I profiles are
identical for the PRD and CRD limits. This is because, for this
particular transition in the non-magnetic regime, only the
elements R II00 and R
II
01 of the redistribution matrix are non-zero
and all the other elements are zero. Hence the line source vector
corresponding to the polarization (SQ,l) is always zero (see
Equation (23)). This implies that the contribution to the emitted
polarization for this case does not come from the redistribution
processes but only from the dichroic absorption (see Trujillo
Bueno & Landi Degl’Innocenti 1997). In order to understand
the combined effects of PRD and LLP in a better way, we
consider another case study with Ja = Jb = 1.
4.2. The Case of the 1  1  1 Transition
For the case when Ja = Jb = 1, even in the absence of LLP,
since the upper level is polarized, a finite amount of emergent Figure 2. Emergent intensity and polarization for μ=0.11 computed using the
polarization is generated unlike the case of Ja = 1, Jb = 0. For two-stage approach for the 1  1  1 case. The different line types
all the computations of this particular transition ( J =J =1), correspond to: solid line—PRD + LLP, dotted line—CRD + LLP, dasheda b
we again consider the hypothetical case like that described for line—PRD + ULL, and the dotted–dashed line—CRD + ULL. Theabbreviation ULL stands for unpolarized lower level. Other input parameters
Ja = 1, Jb = 0. In this case, there are four density matrix are A 8 -1 4 -1JbJa = 10 s ; CJbJa = 10 s . No background continuum opacity is
elements to be determined, namely, r0 2 00 (a), r0 (a), r0 (b), and used. The inset in the Q/I panel shows the Q/I profiles for a shorter frequency
r2 (b), when polarizability of both levels are taken into account. bandwidth for the sake of clarity.0
Figure 2 shows the emergent (I, Q/I) at μ=0.11 for the
1  1  1 transition. The solid line in Figure 2 represents the two-stage approach that we have used. This possibility needs to
emergent profiles obtained when both the effects of PRD and be tested based on a more elaborate theory of PRD, like the
LLP are considered. In order to see the importance of both recent formulations by Casini et al. (2014) and Bom-
effects we have overplotted the (I, Q/I) profiles obtained when mier (2016).
only the effects of PRD are considered with an unpolarized
lower level (ULL—dashed line); when the effects of LLP is
considered in the limit of CRD (dotted line); and the case
( 5. AN ALTERNATIVE APPROACH TO INCLUDE THEwhere only the CRD effects are considered with ULL dotted–
) EFFECTS OF LLP IN POLARIZED LINES FORMEDdashed line . Figure 2 clearly shows that the LLP effects appear UNDER PRD
only in the emergent polarization and the intensity profiles
remain unchanged whether or not LLP is taken into account. The conclusion that LLP effects are only significant in the
We see that the LLP effects in the emergent Q/I are significant line core allows us to use an alternative approach to solve the
mainly in the core (up to ∼2 Doppler widths, see inset in the problem at hand. We refer to this approach as the correction
lower panel of Figure 2), and in the wings the effects of PRD method. In this method, we compute the line profiles taking
are dominant (compare solid and dashed lines). The enhance- into account PRD and neglecting LLP (in the standard two-
ment in the emergent polarization at the line center when LLP level approach) and later apply to it the corrections due to the
is included is around 5%. effects of LLP computed using the density matrix approach
In the first stage of the two-stage approach, the SEEs are with CRD. The actual procedure is described below.
solved under the approximation of CRD, and thus all the (i) We solve the SEEs and the transfer equation simulta-
transition rates that enter into the SEEs are frequency- neously for a given Ja and Jb, taking into account the effects of
integrated quantities. Therefore, all the redistribution effects a polarized lower level in the limit of CRD. For this purpose,
are integrated away. The contributions to the frequency- we use the relevant equations derived from the density matrix
integrated scattering probability come almost entirely from approach. We also neglect the stimulated emission. The Stokes
the Doppler core. In SEEs, we compare the individual Q parameter obtained through a simultaneous solution of SEEs
transition rates for a given radiation field. For each individual and the polarized transfer equation is denoted byQ LLPCRD. For the
transition, the contributions of the wing photons are insignif- numerical solution of this problem, we use the Rybicki and
icant compared with the core photons. Since it is only the core Hummer method (see Rybicki & Hummer 1991) appropriately
photons that are relevant to SEEs, the effects of LLP only show generalized to handle the polarized lower level. The governing
up in the core but are absent in the wings. We cannot exclude equations and the details of the numerical procedure followed
that the absence of LLP effects in the wings could be due to the are described in Appendix A.1.
6
The Astrophysical Journal, 828:84 (10pp), 2016 September 10 Supriya et al.
developed by Landi Degl’Innocenti (1983). The density matrix
elements thus obtained from the first stage are used as inputs to
the second stage to compute the collisionless redistribution
matrix elements. In the second stage, we use the DELOPAR
method to obtain the formal solution. Furthermore, we use the
frequency-angle-by-frequency-angle (FABFA) method to com-
pute the source vector corrections.
To demonstrate the effects of PRD and LLP, we consider
two examples, namely, the 1  0  1 transition and
1  1  1 transition. The case of the 1  0  1 transition
does not show any signatures of PRD in the emergent
polarization profile. This is because in this particular case,
the contribution to the emergent linear polarization does not
Figure 3. Emergent polarization for μ=0.11 computed using the correction come from the scattering processes but only from the dichroic
method (solid line) and the two-stage approach (dotted line). Other parameters
are the same as in Figure 2. absorption from the lower level. However, in the 1  1  1
transition, the PRD signatures in the emergent polarization
profile can be clearly seen in the line wings. Our studies
(ii)We solve the same problem as above, but now neglecting indicate that the LLP effects are confined mostly to the line
the effects of LLP. The resulting Stokes Q parameter is denoted core region. The reason behind this appears to be that the SEEs
by Q ULLCRD . are solved under the flat spectrum approximation, which makes
(iii) The difference between the solutions obtained with and the concerned transition rates frequency-integrated quantities.
without the effects of LLP is This leads us to a computationally simpler numerical approach
LLP ULL called the “correction method” to study the effects of PRD andDQCRD = QCRD - QCRD . (24) LLP on polarized line formation. We have verified that this
We refer to Δ Q computationally simpler correction method represents aCRD as the correction term.
(iv) We now solve the transfer equation for the atomic sufficiently good approximation and is therefore useful in
system under consideration using the standard two-level atom practical model calculations.
approach including PRD. For this purpose we use a polarized
approximate lambda iteration method (Nagendra et al. 1999).
The polarization thus obtained is referred to as QULL APPENDIXPRD .
(v) It is well known that the effects of PRD are noticeable NUMERICAL METHOD: TWO-STAGE APPROACH
mainly in the line wings. Our studies showed that the effects of In this section we describe the two-stage approach to solve
a polarized lower level of an atomic system are confined to the the problem of polarized line formation including LLP and
line core region (see Figure 2). To a good approximation a PRD for the non-magnetic case. In the first stage we solve the
solution including the effects of PRD and also the LLP can be polarized SEEs and transfer equation simultaneously in the
obtained from limit of CRD. For the numerical solution of this problem, we
LLP ULL use the Rybicki and Hummer method (see Rybicki &QPRD = QPRD + DQCRD. (25) Hummer 1991) appropriately generalized to handle polarized
In Figure 3 we compare the Q/I profiles obtained using the lower level (see also Trujillo Bueno 2003). In the second stage,
simple correction method described above and the elaborate we solve the polarized radiative transfer equation including the
two-stage approach proposed in this paper (see Appendix). We effects of LLP (through the density matrix elements derived in
see that the results from both methods match closely. Thus in the first stage) and PRD.
order to simplify the computational efforts one can use the
simple correction method instead of the two-stage approach.
A.1. Stage 1 of the Two-stage Approach
6. CONCLUSIONS The governing equations and concerned numerical method
In this paper we have presented a numerical algorithm to for the simultaneous solution of the SEEs and transfer equation
solve the polarized radiative transfer problem including the in the limit of CRD is illustrated here for the 1  1  1
effects of PRD and LLP in the general case of magnetic media. transition. For this particular transition, the SEEs and transfer
Following the Kramers–Heisenberg approach as given in equations are given in Trujillo Bueno (1999). There are four
Stenflo (1998), we have derived the general collisionless density matrix elements to be determined, namely, r
0
0 (a),
redistribution matrix including the effects of LLP for a two- r20 (a), r
0
0 (b), and r
2
0 (b) when polarizability of both the levels
level atomic system. This redistribution matrix now depends on are taken into account for this case. The SEEs to be solved
the density matrix elements of the lower atomic level. We then correspond to the K=0 and 2 components of the upper level,
include this redistribution matrix in the transfer equation. For the K=2 component of the lower level density matrix, and the
this purpose, we followed the quantum field theory approach number conservation equation. These equations can be derived
of S94. starting from the general Equations (10.4) given in LL04. The
A two-stage approach is proposed to solve the polarized upward and downward inelastic collisional rates are taken into
radiative transfer problem including the effects of PRD and account when solving the SEEs. J 00 and J
2
0 are two quantities
LLP in the non-magnetic regime. In the first stage we solve the related to the radiation field that enter the SEEs and the
SEEs and the transfer equation simultaneously under the flat expressions for these are given in Trujillo Bueno (1999). The
spectrum approximation using the density matrix formalism coupled transfer equations for the Stokes parameters I and Q
7
The Astrophysical Journal, 828:84 (10pp), 2016 September 10 Supriya et al.
are given by approximate operators L*+ and L*- are chosen to be the
=  - hA - hA , (26) diagonals of the respective actual lambda operators. ThedI ds I I I QQ radiation field tensors can now be written as
dQ ds =  - hAI - hAQ Q I Q. (27) 0
0 0,eff *0
r (b)
*0 0
Here s is the geometrical distance along the ray. In the J0 = J0 + (L 0 + L 2 ) , (35)r0 (a)
preceding equations, òI and òQ are the emission coefficients and
0
hAI and h
A
Q are the absorption coefficients. They depend on the r02 2,eff 2 2 0 (b)
atomic density matrix elements, which in turn depend on the J0 = J0 + (L*0 + L*2 ) , (36)r0 (a)
Stokes I and Q parameters. Thus the problem becomes both 0
nonlinear and nonlocal. This is referred to as the NLTE where J 0,eff and J2,eff0 0 are given by
problem of the second kind (see LL04). To solve this problem,
we use an iterative technique based on the approximate lambda 1J 0,eff
1 1
0 = dx fx dm
iteration (ALI) method. We note that Q ¹ 0 in the case of the 2 ò ò-1 2
1  1  1 transition, whereas it is equal to 0 in the case of the ´ [(L+ - L* † †+)[S+] + (L- - L*-)[S-]], (37)
1  0  1 transition as the upper level with Jb = 0 cannot
lead to the emission of polarized radiation. The coupled transfer 1
J2,eff = 1
1
dx f dm
Equations (26) and (27) can be decoupled by working with 0 4 2 ò x ò-1 2
I+ = I + Q and I- = I - Q (see Trujillo Bueno 2003). The ´ [((3m2 - 1) + 3(m2 - 1)) (L †+ - L*+)[S+]
decoupled transfer equations can be written as
+ ((3m2 - 1) - 3(m2 - 1)) (L †- - L*-)[S-]]. (38)
dI+ = + - h+I+, (28)ds The components of the Λ
* operator are given by
dI- = - - h-I-, (29) 1
1 1 2hn 3
ds L*
0
0 = ò dx fx ò dm L*+2 -1 2 c2
where + = I + Q; - = I - Q; h+ = hI + hQ and 1 - 1 [(3m2 - 1) + 3(m2 - 1)][r2 (b) r0 (b)]
h- = hI - hQ. Thus we can write the source functions S
4 2 0 0
+ ´ ,
1
and S− as 1 - [(3m2 - 1) + 3(m2 - 1)][r
2 0
4 2 0
(a) r0 (a)]

= + 1S *0 1 1 2hn
3
+
h L 2 = dx fx dm L*-+ 2 ò ò-1 2 c2
0 1 2 1 2 0
2hn 3 r
2 2 2 2
0 (b) - [(3m - 1) + 3(m - 1)]r0 (b) 1 - [(3m - 1) - 3(m - 1)][r0 (b) r0 (b)]
= 4 2 , ´ 4 2 ,
c2 r00 (a) -
1 [(3m2 - 1) + 3(m2 - 1)]r2 1 2 2 2 0
4 2 0
(a) 1 - [(3m - 1) - 3(m - 1)][r0 (a) r (a)]4 2 0
(30) 1
*2 1 1 2hn
3
 L 0 = ò dx fx ò dm4 2
S = - -1 2 c
2
-
h- ´ L*+[(3m2 - 1) + 3(m2 - 1)]
3 r0 (b) - 1 [(3m22hn 0 - 1) - 3(m
2 - 1)]r2 (b)
4 2 0 1 -
1 [(3m2 - 1) + 3(m2 - 1)][r2 (b) r0 (b)]
= . ´ 4 2
0 0
,
c2 r0 (a) - 1 [(3m2 - 1) - 3(m2 - 1)]r20 0 (a)4 2 1 -
1 [(3m2 - 1) + 3(m2 - 1)][r20 (a) r
0
0 (a)]4 2
(31) 1 3
L*2 1
1 2hn
2 = dx f dm
The formal solution of the transfer Equations (28) and (29) can 4 2 ò x ò-1 2 c2
be written as ´ L*-[(3m2 - 1) - 3(m2 - 1)]
I+ = L+[S+], (32) 1 - 1 [(3m2 - 1) - 3(m2 - 1)][r20 (b) r
0
0 (b)]4 2
I- = L-[S ´ .-], (33) 1 - 1 [(3m2 - 1) - 3(m2 - 1)][r20 (a) r
0
0 (a)]4 2
where Λ+ and Λ− are the operators that depend on the optical
distances between the grid points. We use the short (39)
characteristics method (Olson & Kunasz 1987) to find the In the computation of S+ and S− in the first iteration, we need
formal solution of the transfer Equations (28) and (29). the values of r0 (a), r0 (b), r2 (a), and r2 (b). These are obtained
Now, in order to linearize the SEEs, we introduce the 0 0 0 0by assuming the LTE populations. First we compute the
approximate operator based on the idea of operator splitting: number densities of the lower level (NJa), upper level (NJb), and
I+  L+[S+] + (L+ - L*+)[S†+], the total density (N = NJa + NJb). From this, we compute
N N
I- L-[S-] + (
0
L- - L*)[S†], (34) r0 (a) =
1 Ja and r0 (b) = 1 Jb . We assume
- - 3 N 0 3 N
2
where the “†” represents the quantities known from the r0 (a) = r
2
0 (b) = 0 in the first iteration. Preconditioning the
previous iteration. Following Olson et al. (1986), the quantity r00 (b) r
0
0 (a), we can linearize the SEEs to obtain the
8
The Astrophysical Journal, 828:84 (10pp), 2016 September 10 Supriya et al.
linearized equations A.2. Stage 2 of the Two-stage Approach
[B J 0,eff + C ]r0 (a) + [B (L* 0 + L* 0) The density matrix elements obtained from the first stageJaJb 0 JaJb 0 JaJb 0 2
⎡ described above are used to compute the elements of the
] 0 ( ) ⎢1- A - C r b - B J2,eff⎣ redistribution matrix R
II
J J J J J J 0 ij
(x, m, x¢, m¢) which is needed in the
b a b a 0 2 a b second stage. In this stage we solve the polarized transfer
⎛ 0 ( ) ⎞†⎤ equation given in Equation (22). By defining the total optical1 2 2 r b abs+ B (L* + L* )⎜⎜ 0 ⎟⎟ 2⎥⎥r (a) = 0, (40a) depth dttot = (fI + r)dt we can simplify Equation (22) asJ J 0 22 a b ⎝r00 (a) ⎠ ⎦ 0 dI
= I - Seff . (41)
⎡ B ⎤ ⎡B ⎤ dttot⎢ JaJb 2,eff 0 JaJb 2 2 0⎣- J ⎥r (a) - ⎢ (L*0 + L*2 )⎥r (b)2 0 ⎦ 0 ⎣ 2 ⎦ 0 Here, the effective source vector is
⎢⎡ (2) BJaJb 0,eff B ¢+ ⎣CJ J - J
JaJb 0
0 - (L*0 + L* 0)
Seff = Stot - K I, (42)
a b 22 2 where we have redefined the total absorption matrix as
⎛ 0⎜r (b) ⎟⎞
†
´ ⎜ 0 ⎟ B- J J Ka b J2,eff B- JaJb⎝ ⎠ 0 (L
* 2 + L* 20 2 ) K¢ = - E. (43)r00 (a) 2 2 f
abs
I + r
⎛ 0 ( ) ⎞†r b ⎥⎤⎜ ⎟ The total source vector is defined as´ ⎜ 0⎝ ⎟⎠ ⎥r
2
0 (a) + [CJbJa - AJbJa ]r
2
0 (b) = 0, (40b)r0 (a) ⎦ 10 Stot = [rBn0 1 + Sl]. (44)fabsI + r
[B J2,eff ]r0JaJb 0 0 (a) + [B
2 2
JaJb (L*0 + L*2 )]r
0
0 (b) With these expressions we can apply the same steps as in
⎡⎢ ⎛r0 (b) ⎞
† Equations(19)–(26) of Sampoorna et al. (2008) to obtain the
- ⎢2BJaJb J
0,eff + 2B (L* 0 00 J J 0 + L*2 )⎜a b ⎜ 0 ⎟⎝ 0 ⎠⎟ formal solution of the transfer equation using the DELOPAR⎣ r0 (a) method (see also Trujillo Bueno 2003). The transfer
B ⎛r0J J 2 2 ⎜ 0 (b) ⎞
† ⎤ Equation (41) is solved iteratively using an ALI method. To
+ a b (L* + L* )⎜ ⎟⎟ + 2C ⎥⎥
2
0 2 ⎝ ⎠ JaJb r0 (a)
compute the source vector corrections, we use the so-called
2 r00 (a) ⎦ FABFA method, similar to that given in Sampoorna et al.
+ [ (2) - ]r2 ( ) = ( ) (2011). Hereon the dependencies over x and μ appear as2CJ AbJa JbJa 0 b 0. 40c subscripts. The formal solution of the transfer Equation (41)
Here CJbJa is the downward inelastic collisional rate. In the can be written as
present case there is a contribution from the K=2 multipole Ixm = Lxm [Stot,xm]. (45)
component of the inelastic collision rates, namely, C (2)JaJ andb
(2) Lxm is the frequency- and angle-dependent integral operator,CJ J . The relation between the Kth multipole component andb a which can be split as
zeroth component of different collision rates is given in
Appendix 4 of LL04. Using that relation we get Lxm = L*xm + (Lxm - L*xm), (46)
C (2)J J = -CJ J 2 and C
(2)
a b a b JbJ = -CJ J 2 for the 1  1  1b a *~a where Lxm represents the diagonal approximate operator. Now
transition (we have taken K = 1). The above equations are then we can write the total source vector as
solved for the density matrix elements r00 (a), r
0
0 (b), r
2
0 (a), and S n+1 n ntot,xm = Stot,xm + dStot,xm. (47)
r20 (b). In the subsequent iterations the source functions and the
quantities J 00 and J
2
0 are updated. The iteration sequence is
Here n represents the iteration index. Using Equations (23),
continued until convergence is obtained over the density matrix (44), (46), and (47) we obtain
elements. In this way the SEEs and the transfer equations are RII
n 1 xm,x ¢m¢
solved simultaneously using the ALI method. The limit of ULL dS ntot,xm - pxm ò dm¢2 ò dx¢ L*f x ¢m¢[dStot,x ¢m¢]
is recovered by setting r20 (a) to zero in the above equations. It
x
= p J n + p (1 - a)1S + pc nhas to be noted that the time evolution equation of the lower xm xm xm ba xmBn0 1 - Stot,xm, (48)
level will then be given by rKQ (a) = dK0dQ0r
0
0 (a), and where p abs cxm = fx (fI + r) and pxm = r (f
abs + r). We have
therefore, the Equation (40c) will vanish. The rest of the Ito note that in the above equation, the thermal emission term
iteration procedure remains the same, which then involves
remains constant over iterations in the non-magnetic case. The
solving the SEEs for the three unknowns, namely, r2 (b), r00 0 (b), mean intensity is given by
and r00 (a). II
A similar procedure can also be followed for the case of J = 1 Ra n ¢ ¢ xm,x ¢m¢
1 and Jb =
J = dm dx I . (49)
0, which is simpler compared to the Ja=Jb=1 xm 2 ò ò x ¢m¢fx
case, with only three density matrix elements to be determined.
The SEEs and the transfer equation for this case ( J = 1 and J The standard steps of FABFA as given in Sampoorna et al.a b
= 0) in the non-magnetic regime are given in Trujillo Bueno & (2011) can now be applied to solve the system of linear
Landi Degl’Innocenti (1997). equation (48) to obtain the corrections to the total source vector
9
The Astrophysical Journal, 828:84 (10pp), 2016 September 10 Supriya et al.
(dStot,xm) in the iteration process. In this way, using the ALI Nagendra, K. N. 2014, in ASP Conf. Ser., 489, Solar Polarization 7, ed.
method, we solve the transfer equation, which now includes the K. N. Nagendra et al. (San Francisco, CA: ASP), 179
Nagendra, K. N. 2015, in Proc. IAU Symp. 305, Polarimetry: From the Sun to
effects of PRD and LLP. Stars and Stellar Environments, ed. K. N. Nagendra et al. (Cambridge:
Cambridge Univ. Press), 351
REFERENCES Nagendra, K. N., Paletou, F., Frisch, H., & Faurobert-Scholl, M. 1999, in Solar
Polarization, Vol. 243 ed. K. N. Nagendra & J. O. Stenflo (Boston, MA:
Anusha, L. S., Nagendra, K. N., Bianda, M., et al. 2011, ApJ, 737, 95 Kluwer), 127
Belluzzi, L., Trujillo Bueno, J., & Landi Degl’Innocenti, E. 2015, ApJ, Olson, G. L., Auer, L. H., & Buchler, J. R. 1986, JQSRT, 35, 431
814, 116 Olson, G. L., & Kunasz, P. 1987, JQSRT, 38, 325
Bommier, V. 2016, A&A, 591, 59 Rees, D. E., & Saliba, G. J. 1982, A&A, 115, 1
Casini, R., Landi Degl’Innocenti, E., Landolfi, M., & Trujillo Bueno, J. 2002, Rybicki, G. B., & Hummer, D. G. 1991, A&A, 245, 171
ApJ, 573, 864 Sampoorna, M. 2011, ApJ, 731, 114
Casini, R., Landi Degl’Innocenti, M., Manso Sainz, R., Sampoorna, M., Nagendra, K. N., & Frisch, H. 2011, A&A, 527, 89
Landi Degl’Innocenti, E., & Landolfi, M. 2014, ApJ, 791, 94 Sampoorna, M., Nagendra, K. N., & Stenflo, J. O. 2007, ApJ, 670, 1485
Hummer, D. G. 1962, MNRAS, 125, 21 Sampoorna, M., Nagendra, K. N., & Stenflo, J. O. 2008, ApJ, 679, 889
Landi Degl’Innocenti, E. 1983, SoPh, 85, 3 Stenflo, J. O. 1994, Solar Magnetic Fields: Polarized Radiation Diagnostics
Landi Degl’Innocenti, E. 1999, in Solar Polarization, Vol. 243 ed. (Dordrecht: Kluwer) (S94)
K. N. Nagendra & J. O. Stenflo (Boston, MA: Kluwer), 61 Stenflo, J. O. 1998, A&A, 338, 301
Landi Degl’Innocenti, E., Landi Degl’Innocenti, M., & Landolfi, M. 1997, in Stenflo, J. O., & Keller, C. U. 1997, A&A, 321, 927
Proc. Forum THEMIS, Science with THEMIS, ed. N. Mein & Stenflo, J. O., Keller, C. U., & Gandorfer, A. 2000, A&A, 355, 789
S. Sahal Bŕechot (Paris: Obs. Paris-Meudon), 59 Trujillo Bueno, J. 1999, in Solar Polarization, Vol. 243 ed. K. N. Nagendra &
Landi Degl’Innocenti, E. 1998, Natur, 392, 256 J. O. Stenflo (Boston, MA: Kluwer), 73
Landi Degl’Innocenti, E., & Landolfi, M. 2004, Polarization in Spectral Lines Trujillo Bueno, J. 2003, in ASP Conf. Ser. 288, Stellar Atmosphere
(Dordrecht: Kluwer) (LL04) Modeling, ed. I. Hubeny, D. Mihalas, & K. Werner (San Francisco:
Manso Sainz, R., & Trujillo Bueno, J. 2003, PhRvL, 91, 111102 ASP), 551
Manso Sainz, R., & Trujillo Bueno, J. 2010, ApJ, 722, 1416 Trujillo Bueno, J., & Landi Degl’Innocenti, E. 1997, ApJL, 482, L183
Nagendra, K. N. 2003, in ASP Conf. Proc. 288, Stellar Atmosphere Modeling, Trujillo Bueno, J., Landi Degl’Innocenti, E., Collados, M., Merenda, L., &
ed. I. Hubeny, D. Mihalas, & K. Werner (San Francisco, CA: ASP), 583 Manso Sainz, R. 2002, Natur, 415, 403
10