11institutetext:  Istituto ricerche solari Aldo e Cele Daccò (IRSOL), Faculty
of informatics, Università della Svizzera italiana, Locarno, Switzerland
22institutetext: Euler Institute, Università della Svizzera italiana, Lugano,
Switzerland 33institutetext: Institut für Sonnenphysik (KIS), Freiburg i. Br.,
Germany 44institutetext: Simula Research Laboratory, Oslo, Norway
55institutetext: Instituto de Astrofísica de Canarias, La Laguna, Tenerife,
Spain 66institutetext: Departamento de Astrofísica, Universidad de La Laguna,
La Laguna, Tenerife, Spain 77institutetext: Consejo Superior de
Investigaciones Científicas, Spain 88institutetext: Astronomical Institute
ASCR, Ondřejov, Czech Republic
88email<EMAIL_ADDRESS>
#  Accurate PRD modeling of the forward-scattering Hanle effect
in the chromospheric Ca i 4227 Å line

Luca Belluzzi 112233 Simone Riva 22 Gioele Janett 1122 Nuno Guerreiro 1122
Fabio Riva 1122 Pietro Benedusi 2244
Tanausú del Pino Alemán 5566 Ernest Alsina Ballester 5566 Javier Trujillo
Bueno 556677 Jiří Štěpán 88

###### Abstract

Context.  Measurable linear scattering polarization signals have been
predicted and detected at the solar disk center in the core of chromospheric
lines. These forward-scattering polarization signals, which are of high
interest for magnetic field diagnostics, have always been modeled either under
the assumption of complete frequency redistribution (CRD), or taking partial
frequency redistribution (PRD) effects into account under the angle-averaged
(AA) approximation.

Aims.  This work aims at assessing the suitability of the CRD and PRD–AA
approximations for modeling the forward-scattering polarization signals
produced by the presence of an inclined magnetic field, the so-called forward-
scattering Hanle effect, in the chromospheric Ca i 4227 Å line.

Methods.  Radiative transfer calculations for polarized radiation are
performed in semi-empirical 1D solar atmospheres, out of local thermodynamic
equilibrium (LTE). A two-step solution strategy is applied: the non-LTE RT
problem is first solved considering a multilevel atom and neglecting
polarization phenomena. The same problem is then solved including polarization
and magnetic fields, considering a two-level atom and keeping fixed the
population of the lower level calculated at the previous step. The problem of
step two is linear and it is solved with a preconditioned FGMRES iterative
method. The emergent fractional linear polarization signals calculated under
the CRD and PRD–AA approximations are analyzed and compared to those obtained
by modeling PRD effects in their general angle-dependent (AD) formulation.

Results.  With respect to the PRD–AD case, the CRD and PRD–AA calculations
significantly underestimate the amplitude of the line-center polarization
signals produced by the forward-scattering Hanle effect.

Conclusions.  The results of this work suggest that a PRD–AD modeling is
required in order to develop reliable diagnostic techniques exploiting the
forward-scattering polarization signals observed in the Ca i 4227 Å line.
These results need to be confirmed by full 3D calculations including non-
magnetic symmetry-breaking effects.

###### Key Words.:

Magnetic fields – Polarization – Radiative transfer – Scattering – Sun:
chromosphere

##  1 Introduction

When observed in quiet or moderately active regions of the solar disk, many
lines of the solar spectrum, from the near infrared to the far ultraviolet,
show measurable linear polarization signals, produced by the scattering of
anisotropic radiation (e.g., Stenflo & Keller, 1997; Gandorfer, 2000, 2002,
2005; Kano et al., 2017; Rachmeler et al., 2022). These scattering
polarization signals are receiving increasing attention by the scientific
community because they encode precious information on the thermodynamic and
magnetic properties of the solar atmosphere (e.g., Trujillo Bueno & del Pino
Alemán, 2022, and references therein).

When observed with low spatio-temporal resolution, scattering polarization
signals are generally strongest at the edge of the solar disk (limb) and
decrease towards the disk center, where they typically vanish. Consistently
with this observational evidence, it can be theoretically shown that in a
setting with cylindrical symmetry around the vertical, the amplitude of
scattering polarization scales as $(1-\mu^{2})$, with $\mu$ the cosine of the
inclination of the line-of-sight with respect to the vertical (e.g., Landi
Degl’Innocenti & Landolfi, 2004).111We note that $\mu$ also corresponds to the
cosine of the heliocentric angle of the observed region. On the other hand, if
the problem is not axially symmetric, non-negligible scattering polarization
signals can also be obtained at $\mu=1$, and thus also at the solar disk
center, in a forward-scattering geometry (Trujillo Bueno, 2001). The first
detection of forward-scattering polarization on the Sun was achieved by
Trujillo Bueno et al. (2002) while observing a quiescent filament in the He i
10830 Å triplet. Shortly afterward, while pointing an active region, Stenflo
(2003) detected a forward-scattering signal in the chromospheric Ca i line at
4227 Å. Other remarkable disk-center observations in this line were carried
out by Bianda et al. (2011) using the Zurich Imaging Polarimeter (ZIMPOL-III,
Ramelli et al., 2010).

A lack of axial symmetry in the solar atmosphere, capable of producing
forward-scattering polarization, can be due to (i) horizontal inhomogeneities
in the density and temperature of the solar plasma (Manso Sainz & Trujillo
Bueno, 2011), (ii) spatial gradients of the plasma bulk velocity (e.g., Štěpán
& Trujillo Bueno, 2016; del Pino Alemán et al., 2018), and (iii) a
deterministic magnetic field inclined with respect to the vertical (Trujillo
Bueno, 2001; Trujillo Bueno et al., 2002). In the first two scenarios, the
signals are produced by a lack of axial symmetry in the radiation field that
illuminates the atoms; in the third one, generally referred to as forward-
scattering Hanle effect, they arise directly from the magnetic field. In
general, these symmetry-breaking effects operate simultaneously, and the
individual contributions can hardly be isolated. Interestingly, radiative
transfer (RT) calculations carried out in state-of-the-art 3D models of the
solar atmosphere, in the limit of complete frequency redistribution (CRD),
show that the forward-scattering signals produced by the symmetry breaking
effects (i) and (ii) are reduced by the presence of magnetic fields (Štěpán &
Trujillo Bueno, 2016; del Pino Alemán et al., 2018; Jaume Bestard et al.,
2021). Moreover, it can be argued that the inherent averaging happening in
observations with low spatio-temporal resolution significantly conceals the
impact of mechanisms (i) and (ii), and indeed it is currently rather
challenging to detect forward-scattering polarization signals in very quiet
solar regions. As a matter of fact, the aforementioned observations in the Ca
i 4227 Å line were all performed in relatively strongly magnetized regions
(i.e., with noticeable $V/I$ signals), suggesting that the forward-scattering
Hanle effect might be the main mechanism responsible for their generation. To
routinely detect forward-scattering polarization signals in quiet solar
regions is one of the challenges for the new generation of large-aperture
solar telescopes, such as DKIST (Rimmele et al., 2020) and the future EST
(Quintero Noda et al., 2022).

In order to interpret the available observational data, various theoretical
works have already been carried out on the modeling of forward-scattering
polarization signals in the chromospheric Ca i 4227 Å line. Anusha et al.
(2011) modeled the observations of Bianda et al. (2011) in 1D semi-empirical
solar atmospheric models, taking partial frequency redistribution (PRD)
effects into account under the angle-average (AA) approximation (see Bommier,
1997b, Sect. 4.3). Their work allowed inferring information on the magnetic
fields of the low chromosphere, and showed that various observed features
cannot be reproduced if the CRD limit is considered. Subsequently, Carlin &
Bianda (2016, 2017) modeled the Ca i 4227 forward-scattering polarization
signals in 3D models of the solar atmosphere, in the limit of CRD. Their
calculations neglected both the effects of horizontal RT (i.e., they
considered the so-called 1.5D approximation) and the horizontal component of
the model’s bulk velocity. This work highlighted the key role played by the
dynamics and temporal evolution of the chromospheric plasma during the
integration time of the observations. In the investigations mentioned so far,
the symmetry-breaking mechanisms (i) and (ii) were not taken into account. By
contrast, Jaume Bestard et al. (2021) investigated the polarization of the Ca
i 4227 Å line performing full 3D calculations in realistic models of the solar
atmosphere with the radiative transfer code PORTA (Štěpán & Trujillo Bueno,
2013) under the assumption of CRD. Accounting for all symmetry-breaking
effects, their study showed that the spatial gradients in the horizontal
component of the plasma bulk velocity can produce conspicuous forward-
scattering polarization signals, comparatively larger than those produced by
horizontal inhomogeneities in the solar plasma, and that these signals are
reduced by the presence of a magnetic field.

So far, all the theoretical investigations on the forward-scattering
polarization of the Ca i 4227 Å line have been carried out either in the limit
of CRD or considering PRD effects under the AA simplifying approximation. This
is ultimately due to the formidable computational complexity of modeling
scattering polarization while accounting for PRD phenomena in their most
general angle-dependent (AD) formulation. Performing RT calculations in
realistic atmospheric models, taking AD PRD effects into account, is today
feasible (e.g., del Pino Alemán et al., 2020; Benedusi et al., 2023), and a
series of 1D applications to Ca i 4227 have been presented (e.g., Janett et
al., 2021a; Riva et al., 2023; Guerreiro et al., 2024). However, none of these
works specifically investigated the forward-scattering Hanle effect. The
present study focuses on this effect and aims at assessing the suitability of
the CRD and PRD–AA approximations for its modeling in the Ca i 4227 Å line.

The article is organized as follows: Section 2 exposes the considered RT
problem for polarized radiation and the adopted solution strategy. In Sect. 3,
we report and analyze the emergent fractional linear polarization of the Ca i
4227 line, comparing CRD, PRD–AA, and PRD–AD calculations. Section 4 discusses
the main results and their implications, while Sect. 5 provides remarks and
conclusions.

##  2 Problem formulation and solution strategy

The Ca i line at 4227 Å is produced by the transition between the ground level
of neutral calcium, $4s^{2}\,{}^{1}\rm{S}_{0}$, and the excited level
$4s4p\,^{1}\rm{P}^{\rm{o}}_{1}$. In quiet regions close to the solar limb,
this line shows a large scattering polarization signal, characterized by broad
lobes in the wings and a sharper peak in the core (e.g., Gandorfer, 2002).
Both the line-core peak and the wing lobes are sensitive to the magnetic
field, the former through the Hanle effect (which only operates in the line-
core region, e.g., Landi Degl’Innocenti & Landolfi, 2004) and the latter
through magneto-optical effects (Alsina Ballester et al., 2018). In low-
resolution observations, the forward-scattering polarization signals detected
close to the disk center consist instead in a single peak in the line-core
region (e.g., Bianda et al., 2011). A correct modeling of the scattering
polarization profiles of this line, and in particular of the wing lobes,
requires taking PRD effects into account (e.g., Faurobert, 1988). Nonetheless,
the limit of CRD can be used to approximately model the line-core peak, in
both limb and forward-scattering geometries (e.g., Sampoorna et al., 2010;
Jaume Bestard et al., 2021).

The scattering polarization signal of the Ca i 4227 Å line can be suitably
modeled considering a two-level atom. Indeed, the line is not part of a
multiplet (the upper and lower levels have spin zero) and, since the upper
level is not connected to lower-energy levels other than the ground level, its
population is mainly determined by this line itself (assuming that the
ionization fraction is known). In this work, we account for PRD effects using
the redistribution matrix for a two-level atom with unpolarized and
infinitely-sharp lower level as derived by Bommier (1997a, b). The long-lived
ground level of neutral calcium can indeed be treated as infinitely sharp and,
having total angular momentum $J=0$, it cannot carry atomic polarization. The
redistribution matrix is given by the sum of two terms: one that describes
scattering processes that are coherent in the atomic reference frame
($R^{\scriptscriptstyle\mathrm{II}}$), and the other describing scattering
processes that are totally incoherent in the same reference frame
($R^{\scriptscriptstyle\mathrm{III}}$). In the observer’s reference frame, we
consider the exact AD expression of $R^{\scriptscriptstyle\mathrm{II}}$, while
we make the assumption of totally incoherent scattering for
$R^{\scriptscriptstyle\mathrm{III}}$ (e.g., Bommier, 1997b; Sampoorna et al.,
2017; Riva et al., 2023). The CRD modeling is instead performed by applying
the theoretical approach described in Landi Degl’Innocenti & Landolfi (2004).

In this work, we only focus on the forward-scattering Hanle effect, neglecting
the two other mechanisms (i) and (ii). We thus consider the static 1D semi-
empirical atmospheric model C of Fontenla et al. (1993, hereafter FAL-C), and
we analyze the impact of inclined deterministic magnetic fields. The intensity
and polarization profiles of the emergent radiation are calculated by solving
the non-LTE RT problem for polarized radiation. The adopted solution strategy
requires two steps: first, we solve the non-LTE RT problem neglecting
polarization phenomena. This first step is carried out using the RH code
(Uitenbroek, 2001), and considering an atomic model for calcium composed of 25
levels (including five levels of Ca ii and the ground level of Ca iii).
Second, we solve the non-LTE RT problem including polarization, but keeping
fixed the population of the lower level calculated in step 1, so that the
problem is linear (e.g., Janett et al., 2021b). This is achieved with a
preconditioned FGMRES (Flexible Generalized Minimal RESidual) iterative
method, as described in Benedusi et al. (2021, 2022) and Janett et al. (2024).
All the physical and numerical parameters are the same as in Guerreiro et al.
(2024).

##  3 Results

This section presents the results of our calculations of the linear scattering
polarization profiles of the Ca i line at 4227 Å both in the absence and in
the presence of magnetic fields. The problem is formulated in a right-handed
Cartesian coordinate system with the $z$-axis directed along the vertical, and
the $x$-axis directed so that the line-of-sight (LOS) lies in the $x-z$ plane
($x>0$, $z>0$ quadrant). The LOS is thus fully specified by the cosine of its
inclination $\theta$ with respect to the vertical, that is
$\mu=\cos(\theta)\in[0,1]$. The reference direction for positive Stokes $Q$ is
taken parallel to the $y$-axis. The magnetic field vector is specified by the
intensity $B$, the inclination $\theta_{B}$ with respect to the vertical, and
the azimuth $\chi_{B}$ measured on the $x-y$ plane, counter-clockwise (for an
observer at $z>0$) from the $x$-axis. In Sect. 3.1, we analyze the case of
horizontal (i.e., $\theta_{B}=\pi/2$) magnetic fields, while the case of non-
horizontal inclined magnetic fields (i.e., $\theta_{B}=\pi/4$) is considered
in Sect. 3.2.

Figure 1: Center-to-limb variation of the Ca i 4227 line-center fractional
linear polarization $Q/I$ (_upper panel_) and $U/I$ (_lower panel_), obtained
from CRD, PRD–AA, and PRD–AD calculations in the FAL-C atmospheric model, both
in the absence (_dotted curves_) and in the presence (_solid curves_) of a
magnetic field. The dotted curves for $U/I$ are equal to zero. In the magnetic
case, a height-independent horizontal ($\theta_{B}=\pi/2$, $\chi_{B}=0$)
magnetic field of 20 G is considered. The reference direction for positive
Stokes $Q$ is parallel to the $y$-axis of the considered reference system,
that is perpendicular to the magnetic field.

Figure 2: Emergent $Q/I$ profiles of Ca i 4227 at $\mu=1$, obtained from CRD,
PRD–AA, and PRD–AD calculations in the FAL-C atmospheric model, in the
presence of a height-independent horizontal ($\theta_{B}=\pi/2$, $\chi_{B}=0$)
magnetic field of 20 G. The reference direction for positive Stokes $Q$ is
parallel to the $y$-axis of the considered reference system, that is,
perpendicular to the magnetic field. In this geometry, the $U/I$ profile
vanishes, and it is thus omitted. Figure 3: Polarization diagrams of the Ca i
4227 line-center emergent radiation at $\mu=1$, obtained from CRD, PRD–AA, and
PRD–AD calculations in the FAL-C atmospheric model, in the presence of a
height-independent horizontal ($\theta_{B}=\pi/2$) magnetic field of 20 G. The
circular markers indicate the effective calculations, carried out for the
azimuths $\chi_{B}=n\,\pi/8$ (with $n=0,...,15$). _Left panel_ : comparison
between CRD and PRD–AA calculations. _Right panel_ : comparison between CRD,
PRD–AA, and PRD–AD calculations (in the PRD–AA and CRD diagrams the circular
marker indicates a magnetic field direction with azimuth $\chi_{B}=0$, whereas
the arrow refers to $\chi_{B}=\pi/4$). The reference direction for positive
Stokes $Q$ is parallel to the $y$-axis of the considered reference system.

###  3.1 Horizontal magnetic fields

Figure 2 shows the center-to-limb variation of the amplitude of the line-
center scattering polarization $Q/I$ (upper panel) and $U/I$ (lower panel)
signals of the Ca i 4227 Å line, resulting from CRD, PRD–AA, and PRD–AD
calculations. As expected, in the absence of magnetic fields (dotted curves),
the amplitude of the $Q/I$ peak monotonically decreases while moving from the
limb towards the disk center, where it vanishes. The $U/I$ signal is always
zero, consistently with our choice of the reference direction for positive
Stokes $Q$. For any limb distance, the largest signals are obtained in the
PRD–AD setting.

The presence of a height-independent horizontal ($\theta_{B}=\pi/2$,
$\chi_{B}=0$) magnetic field of 20 G produces a significant Hanle rotation at
the limb. This leads to a depolarization of the line-center $Q/I$ signal while
giving rise to an appreciable $U/I$ signal (see solid curves). At the solar
disk center, the forward-scattering Hanle effect yields a non-zero $Q/I$
signal, while $U/I$ is zero, consistently with the geometry of the problem.
The CRD, PRD–AA, and PRD–AD calculations show significant differences for all
$\mu$, in both $Q/I$ and $U/I$, with the PRD–AD results always showing the
largest signal. A very interesting finding is that the PRD–AD results for
$Q/I$ drift away from the others while approaching the disk center. At
$\mu=1$, the PRD–AD signal is well above 1%, that is one order of magnitude
larger than in the PRD–AA and CRD cases. Moving from the limb to the disk
center, the $Q/I$ signal obtained from PRD–AA and PRD–AD calculations first
decreases, reaches a minimum, and then increases, always remaining positive.
On the contrary, in the CRD case, it shows a sign reversal at around
$\mu\approx 0.6$ and then remains negative until $\mu=1$, with amplitudes
always below 0.1% (see also Fig. 2). We note that the differences between
PRD–AA and PRD–AD calculations in Fig. 2 are magnified by the artificial
depolarization that is introduced by the AA approximation in the line-core of
$Q/I$ and $U/I$ for $\mu\neq 1$, both in the absence and presence of magnetic
fields (see Janett et al., 2021a). To better visualize the impact of the
different scattering modelings on the forward-scattering Hanle effect signal,
Fig. 2 shows the emergent $Q/I$ profiles at $\mu=1$. This clearly highlights
the significantly stronger polarization signals resulting from PRD–AD
calculations.

Figure 4: Line-center linear polarization degree as a function of the strength
of a horizontal ($\theta_{B}=\pi/2$, $\chi_{B}=0$) magnetic field, obtained
with PRD–AD (_red_), PRD–AA (_blue_) and CRD (_green_) calculations, for
different LOSs. The vertical dash-dotted line indicates the Hanle critical
field of Ca i 4227, that is $B_{H}=25$ G.

Figure 3 compares the polarization diagrams for the line-center radiation
emitted at $\mu=1$, obtained with PRD–AD, PRD–AA and CRD calculations,
considering height-independent horizontal (i.e., $\theta_{B}=\pi/2$) magnetic
fields of 20 G with different azimuths $\chi_{B}$. These diagrams further
highlight the large polarization amplitudes of the forward-scattering Hanle
effect signals obtained in the PRD–AD setting. We find differences of
approximately one order of magnitude (in both $Q/I$ and $U/I$) when comparing
PRD–AD with PRD–AA and CRD results. Much smaller, yet relevant, differences
are instead found between the CRD and PRD–AA calculations. Consistently with
the geometry of the problem, $Q/I$ and $U/I$ vanish for
$\chi_{B}=\pi/4+n\,\pi/2$ and $\chi_{B}=n\,\pi/2$ ($n=0,...,3$), respectively.
It can be noticed that the evolution of the amplitudes of $Q/I$ and $U/I$
shows a counter-clockwise rotation in all cases. Interestingly, the line-
center signal provided by CRD calculations always has the opposite sign with
respect to the one provided by PRD–AA and PRD–AD calculations.

Figure 4 shows the line-center linear polarization degree

$P_{L}=\frac{\sqrt{Q^{2}+U^{2}}}{I},$ (1)

for different LOSs and magnetic field strengths of a horizontal magnetic field
($\theta_{B}=\pi/2$, $\chi_{B}=0$), obtained with PRD–AD, PRD–AA, and CRD
calculations. At the solar disk center ($\mu=1$, solid lines), where the
polarization is fully produced by the forward-scattering Hanle effect, $P_{L}$
is already appreciable for magnetic fields of a few gauss only. Its value
quickly grows as the magnetic field strength increases further, and finally
stabilizes above the Hanle critical field. The increase of $P_{L}$ is much
steeper and significant in the PRD–AD case than in the PRD–AA and CRD ones.
Notably, the increase is not monotonic in the CRD case. As expected, for a
near-limb LOS ($\mu=0.3$, dotted lines), the Hanle effect produces a monotonic
decrease of $P_{L}$ with the magnetic field strength, until reaching a
saturation regime. For an intermediate LOS ($\mu=0.8$, dashed lines), the
value of $P_{L}$ first increases with the field strength, it reaches a
maximum, and it finally decreases till the saturation regime. In the PRD–AD
case, the initial increase is much steeper and significant, and the maximum is
reached for stronger fields (around the Hanle critical field) than in the
other cases. The PRD–AD calculations show significantly larger values of
$P_{L}$ for all LOS and for any magnetic field strength.

###  3.2 Inclined magnetic fields

The results presented in the previous section are obtained considering
horizontal magnetic fields. This geometry maximizes the breaking of the axial
symmetry for a given value of $B$, and thus the amplitude of the polarization
signals produced at the disk center via the forward-scattering Hanle effect.
On the other hand, the discrepancies between the emergent Stokes profiles
resulting from PRD–AD, PRD–AA, and CRD calculations change significantly if
magnetic fields with different inclinations with respect to the vertical are
considered. For this reason, we now analyze the case of a non-horizontal
inclined magnetic field. Figure 5 shows the center-to-limb variation of the
amplitude of the line-center scattering polarization $Q/I$ (upper panel) and
$U/I$ (lower panel) signals of the Ca i 4227 Å line, resulting from PRD–AD,
PRD–AA, and CRD calculations in the presence of a height-independent inclined
($\theta_{B}=\pi/4$, $\chi_{B}=0$) magnetic field of 20 G. This setting
confirms the presence of relevant differences between the different scattering
descriptions for all $\mu$, and in particular for the forward-scattering
geometry $\mu=1$. To better visualize the impact of the different scattering
modelings on the forward-scattering Hanle effect signal, Fig. 6 shows the
emergent $Q/I$ and $U/I$ profiles at $\mu=1$ for the same inclined magnetic
field. Unlike in the case of a horizontal magnetic field shown in Fig. 2, this
geometry produces an appreciable $U/I$ signal. The significantly stronger
linear polarization signals resulting from PRD–AD calculations are found both
in $Q/I$ and $U/I$.

Figure 7 compares the polarization diagrams for the line-center radiation
emitted at $\mu=1$ obtained with PRD–AD, PRD–AA and CRD calculations,
considering height-independent inclined (i.e., $\theta_{B}=\pi/4$) magnetic
fields of 20 G with different azimuths $\chi_{B}$. These diagrams reveal
smaller forward-scattering signals than those presented in Fig. 3, due to the
smaller horizontal component of the 20 G magnetic field. Moreover, both the
CRD and PRD–AA diagrams are rotated with respect to the PRD–AD one. This means
that both approximations could lead to a wrong sign of the $Q/I$ and $U/I$
line-center signals. Figure 8 shows a similar polarization diagram, but for
$\chi_{B}=0$ and different inclinations $\theta_{B}$ of the 20 G magnetic
field. As expected, the $Q/I$ and $U/I$ signals vanish in the presence of a
vertical magnetic field, that is $\theta_{B}=0,\pi$. As soon as the magnetic
field is sufficiently inclined, the CRD and PRD–AA calculations significantly
underestimate the amplitude of the line-center fractional linear polarization
signals with respect to the PRD–AD modeling. Interestingly, the CRD diagram
shows always negative line-center $Q/I$ signals.

Figure 5: Same as Fig. 2, but for a height-independent inclined
($\theta_{B}=\pi/4$, $\chi_{B}=0$) magnetic field of 20 G. Figure 6: Same as
Fig. 2, but for a height-independent inclined ($\theta_{B}=\pi/4$,
$\chi_{B}=0$) magnetic field of 20 G. Figure 7: Same as Fig. 3, but for a
height-independent inclined ($\theta_{B}=\pi/4$) magnetic field of 20 G.
Figure 8: Polarization diagrams obtained considering a height-independent
magnetic field of 20 G with a fixed azimuth $\chi_{B}=0$, and varying its
inclination $\theta_{B}\in[0,\pi]$. The circular markers indicate the
effective calculations, carried out for $\theta_{B}=n\,\pi/12$ with
$n=0,...,12$. The other parameters are the same as in Fig. 3.

##  4 Discussion

In general, CRD and PRD–AA calculations significantly underestimate the
amplitude of the line-center fractional linear polarization signals, with
respect to the PRD–AD modeling, and even the sign can be wrong. Providing a
simple and intuitive interpretation of these results is, unfortunately, not
straightforward. Here, we provide a few qualitative insights.

The polarization properties of scattered radiation strongly depend on the
detailed spectral and angular dependencies of the incident radiation field.
These dependencies can only be fully taken into account through a PRD–AD
description of scattering processes, while they are necessarily smoothed by
the averages inherent to the PRD–AA and CRD approximations. Indeed, it can be
shown that the CRD, PRD–AA, and PRD–AD calculations coincide in the limit of a
spectrally flat and isotropic radiation field. It appears reasonable that
scattering polarization signals, which are ultimately due to the geometry of
the problem and symmetry-breaking effects, are enhanced by the PRD–AD
approach, which exactly accounts for the complex coupling between the
frequencies and propagation directions of the incoming and scattered
radiation. Besides, the particular spectral structure and anisotropy of the
solar radiation field depend on the thermal and density stratification of the
atmospheric model through non-local RT effects. Inspecting and predicting
these effects in non-academic scenarios is notoriously difficult, and an even
more complex task is to predict how they are impacted by the considered
approximations in the modeling of scattering processes. In conclusion, it is
not possible to clearly identify a single specific reason that explains the
large differences observed between the CRD or PRD–AA calculations with respect
to the general PRD–AD modeling. Nonetheless, these results clearly manifest
the necessity of considering PRD effects in their general AD formulation. This
fact is even more crucial for full 3D RT calculations, which consider the
detailed geometrical structure of the solar plasma. At the dawn of the new
generation of the big solar spectropolarimetric facilities like the Daniel K.
Inouye Solar Telescope (DKIST, Rimmele et al., 2020) and the future European
Solar Telescope (EST, Quintero Noda et al., 2022), the call to fully include
the physical processes affecting the polarization of spectral lines is indeed
stronger than ever before.

A last remark concerns the significant discrepancies between the theoretical
calculations presented in Sect. 3.1, which predict polarization signals around
1 %, and the observations of Bianda et al. (2011), which show signals one
order of magnitude weaker (around 0.1 %). This difference is not surprising,
noticing that a horizontal magnetic field maximizes the breaking of the axial
symmetry and thus the forward-scattering Hanle effect. Moreover, the 3D
calculations in the CRD limit by Jaume Bestard et al. (2021) indicate that the
presence of an inclined magnetic field reduces (and does not enhance) the
amplitude of the scattering polarization signals produced by both horizontal
inhomogeneities of the solar plasma and spatial gradients of the plasma bulk
velocity (which are neglected in this work). The inclusion of mechanisms (i)
and (ii) and the use of state-of-the-art 3D MHD atmospheric models are needed
to obtain reliable information on the magnetic field strength. We also note
that the impact of instrumental effects on the observed line-core signals is
significant, especially in forward scattering observations (e.g., Zeuner et
al., 2020; del Pino Alemán & Trujillo Bueno, 2021, for the Sr i 4607 Å line).

##  5 Conclusions

The results of this work show that a reliable modeling of the fractional
linear polarization signals produced through the forward-scattering Hanle
effect in the Ca i line at 4227 Å requires taking PRD effects into account in
their general AD formulation. If the CRD or PRD–AA approximations are
considered, the amplitude of the line-center $Q/I$ and $U/I$ signals close to
$\mu=1$ could be significantly underestimated, and even the sign can be wrong.
This finding is of clear relevance, especially for the development of new
methods for solar magnetic field diagnostics based on forward-scattering
polarization signals in the Ca i 4227 Å line.

It can be expected that the results presented here for the Ca i 4227 Å line
generalize to other strong resonance lines, for which PRD effects are
relevant. For this reason, it is important to extend this work to other
spectral lines of interest for Hanle diagnostics, such as those recently
observed by the CLASP experiments (e.g., Kano et al., 2017; Rachmeler et al.,
2022), or those that DKIST (and EST) will allow observing with unprecedented
spatial and temporal resolutions.

Finally, it will be of high interest to consider realistic 3D atmospheric
models, which allow including horizontal inhomogeneities of the solar plasma
and spatial gradients of the bulk velocity. In this respect, the first 3D non-
LTE RT calculations, taking scattering polarization and AD PRD effects into
account, will soon be available thanks to recent software and algorithmic
developments (see Benedusi et al., 2023).

###### Acknowledgements.

This work was financed by the Swiss National Science Foundation (SNSF) through
grant CRSII5$\\_$180238\. T.P.A.’s participation in the publication is part of
the Project RYC2021-034006-I, funded by MICIN/AEI/10.13039/501100011033, and
the European Union “NextGenerationEU”/RTRP. T.P.A., J.T.B, and E.A.B.
acknowledge support from the Agencia Estatal de Investigación del Ministerio
de Ciencia, Innovación y Universidades (MCIU/AEI) under grant “Polarimetric
Inference of Magnetic Fields” and the European Regional Development Fund
(ERDF) with reference PID2022-136563NB-I00/10.13039/501100011033. J.Š.
acknowledges the financial support from project RVO:67985815 of the
Astronomical Institute of the Czech Academy of Sciences.

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