galaxies: magnetic fields — polarization — radio continuum: galaxies

# Pseudo-observation of spiral galaxies in the radio band to verify
depolarization models

Yuta Tashima11affiliation: Department of Astronomical Science, School of
Physical Sciences, The Graduate University for Advanced Studies (SOKENDAI),
2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan 22affiliation: Department of
Physics, Faculty of Sciences, Kyushu University, 744 Motooka, Nishi-ku,
Fukuoka 819-0395, Japan<EMAIL_ADDRESS>Takumi Ohmura
33affiliation: Institute for Cosmic Ray Research, The University of Tokyo,
5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582 Japan<EMAIL_ADDRESS>Mami Machida44affiliation: Division of Science, National Astronomical
Observatory of Japan, 2-21-1, Osawa, Mitaka, 181-8588,
Japan,[0000-0001-6353-7639]

###### Abstract

Studies of the three-dimensional structures of galactic magnetic fields are
now entering a new era, with broadband, highly sensitive radio observations
and new analysis methods. To reveal the magnetic field configuration from the
observed intensities integrated along the line of sight, it is necessary to
derive an appropriate model involving various combinations of parameters that
can reproduce the same observational characteristics. We aim to clarify the
relationship between the radiation field and the spatial distribution of
physical quantities through pseudo-observations using global three-dimensional
magnetohydrodynamics (MHD) simulation results. In particular, we focus here on
using the depolarization effect, which is important in the meter-wave band, to
verify the polarization model and to identify the emission region. First, we
show that wavelength-independent depolarization, which takes into account
anisotropic turbulence, does not work efficiently because the polarized
emission is stronger in regions of ordered spiral fields than in regions
dominated by isotropic turbulent fields. Beam depolarization, specifically
internal depolarization, becomes more effective below 1$\mathrm{\>GHz}$.
Although in and close to the equatorial plane there will be strong
depolarization which increases with observing wavelength, this effect is less
in the halo, making halo magnetic fields detectable through their polarized
emission at meter-wavelength bands. Although polarized emission from the halo
is below the detection limit of current facilities, it will be detectable
within the SKA era. In addition, we found that the spiral polarization
projected on a screen is produced by overlapping magnetic flux tubes extending
to different heights from the equatorial plane. This suggests that the
traditional classification of global magnetic fields has difficulty
reproducing the global structure of the magnetic fields. Finally, we
demonstrate the method to separate magnetic flux tubes at different heights by
using peak frequencies which cause the decreasing of polarized flux.

##  1 Introduction

Spiral galaxies typically have mean magnetic fields of a few $\mu$G, with
additional turbulent magnetic fields that are stronger than the mean field
(Soida et al., 2011; Fletcher, 2010; Beck, 2015a). These magnetic fields have
a great influence on the time evolution of galaxies and on star-formation
activity (Krause, 2009; Birnboim et al., 2015). Studies of the structure,
origin, and evolution of galactic magnetic fields have therefore been
conducted for many years (e.g., Beck, 2015b; Kurahara et al., 2021).

The polarization of the radio continuum, which originates from synchrotron
radiation, is a useful tool for revealing the magnetic fields in galaxies and
the interstellar medium (ISM). The total energy of the magnetic field,
including both the turbulent field and the mean field, can be estimated from
the total intensity of the radiation. The polarized intensity provides
information about the ordered field, and the strength of the magnetic field
along the line of sight (LOS) can be estimated from multi-frequency
observations of the Faraday rotation Measure (RM), which is the coefficient of
proportionality between the polarization angle and the square of the observing
wavelength (e.g., Rand & Kulkarni, 1989; Stepanov et al., 2008; Mao et al.,
2012). We can calculate the RM theoretically as the integral over the product
of the electron number density $n_{\mathrm{e}}$ and the magnetic field
parallel to the LOS $B_{\parallel}$. Using the RM and the polarized intensity,
the strengths of both the mean and the ordered magnetic fields can be
determined. For example, Beck (2015a) summarized the results of radio
observations of the spiral galaxy IC 342 and reported that it has a total
field of 13 $\mu$G and an ordered field of 4 $\mu$G. The coherence length of
the turbulent magnetic field in a galactic disk is thought to be
10–100$\mathrm{\>pc}$ (Ohno & Shibata, 1993). The length scale of turbulent
magnetic fields are reported to be about 50 pc for M51 (Fletcher et al., 2011)
and about 70 pc for IC 342 (Beck, 2015a).

However, the RM sometimes cannot be interpolated linearly, especially in the
low frequency region (e.g., O’Sullivan & Gabuzda, 2009; O’Sullivan et al.,
2012). One of the reasons of this is that wavelength dependence of Faraday
rotation leads to cancelling out of polarized emission on the long wavelength
side. The existence of turbulent magnetic fields also depolarizes emission
since polarized angles are randomized by turbulent fields. The overlaying
emission region along the line of sight becomes the another reason for the
decreasing of the polarized emission. These phenomena are referred to as
Faraday depolarization (see detail in section 2). Burn (1966) modeled the
depolarization produced by isotropic turbulence with a Gaussian distribution.
Sokoloff et al. (1998) updated the depolarization model to treat the effect of
anisotropic turbulent fields. Arshakian & Beck (2011) investigated the effect
of anisotropic turbulence, and they studied the dependence of Faraday
depolarization on the wavelength and magnetic field strength. As an example of
depolarization in low frequency radio observations of spiral galaxies, Mulcahy
et al. (2014) reported that the 151 MHz polarized emission of M 51 is below
the detection limit of the Low Frequency Array (LOFAR).

Another important analysis method for magnetic fields is Faraday tomography,
which can separate the components of the integrated intensity along the LOS in
Faraday depth space (Brentjens & de Bruyn, 2005). Fletcher et al. (2011)
reported that M 51, a typical bi-symmetric spiral (BSS-type) galaxy, is
divided into an axisymmetric spiral in the disk and a BSS in the halo. In this
way, Faraday tomography is expected to enable the elucidation of the magnetic
field structure along the LOS. However, the interpretation of Faraday
tomography is not yet adequate because there is not a simple correspondence
between Faraday depth space and real space along the LOS.

Shneider et al. (2014a) developed a theoretical model of galactic disks and
halos as two-layer or three-layer magnetized plasmas. They reported that a
comparison between multi-wavelength polarization observations and two-layer or
three-layer depolarization models can characterize the strength of the mean
field and the anisotropy of the turbulent field in disks and halos. Using
three-band data for M51, Shneider et al. (2014b) showed that this galaxy can
be explained by a two-layer model, with the polarized emission originating
behind the disk being completely depolarized. They also indicated that the
turbulence characteristics of the halo are isotropic, while those of the disk
are anisotropic. Kierdorf et al. (2020) reported that even after adding S-band
results to the previous data (Shneider et al., 2014b), a two-layer model
remains suitable for M51. Importantly, they pointed out that the S-band
results are essential for distinguishing between the two-layer and three-layer
models. This model is unique in that it includes multiple types of mean fields
in the calculations. However, these mean fields models did not consider
whether the models satisfy physical constants or not. We therefore think that
it is important to investigate the depolarization characteristics using the
results of numerical experiments in order to elucidate the magnetic field
structure of the galaxy.

In the 2000s, several studies have begun to use magnetohydorodynamic (MHD)
simulations to clarify the time evolution and origin of magnetic fields and
their effects on galaxy structure formation (Nishikori et al., 2006; Machida
et al., 2009, 2013; Hanasz et al., 2009). Machida et al. (2013) performed a
three-dimensional MHD simulation of a galactic gas disk and reported that the
amplified magnetic field in the disk is constantly supplied to the halo.
Cosmological galaxy-formation simulations based on the $\Lambda$CDM model have
also been performed (Springel et al., 2008; Pakmor et al., 2014). These
numerical simulations of galactic disks show that galactic magnetic fields
have complex turbulent structures.

Since the physical quantities obtained in MHD simulations are different from
the observed quantities, we need to calculate the observables to understand
the correspondence between three-dimensional structures of the magnetic fields
in galaxies and integrated intensity map. Some groups are starting to study
pseudo-observations of radiation using the results of numerical simulations of
galactic disks (Machida et al., 2018, 2019; Reissl et al., 2019). Since the
turbulence scale of galaxies is shorter than the numerical grid size, Reissl
et al. (2019) performed pseudo-observations assuming Gaussian sub-grid
turbulence fields. They assumed the beam size of the pseudo-observations to be
larger than the assumed sub-grid turbulence scale. Therefore, the
depolarization effects due to the isotropic turbulent magnetic field in the
sub-grid are sufficient to affect the polarization image. This means that
depolarization strongly affects the given turbulence structure and thus that
both the appropriate coherence length scale of the turbulence and the
fluctuations of the magnetic field strength less than the grid size need to be
considered. Machida et al. (2018) performed pseudo-observations that take into
account the wavelength-independent depolarization produced by the isotropic
turbulent magnetic fields proposed by Sokoloff et al. (1998). Machida et al.
(2019) subsequently incorporated wavelength-dependent depolarization to
determine the polarization intensity at meter wavelengths. They reported that
the magnetic field in the disk is depolarized by Faraday rotation, but that a
dilute halo field may be observable at meter wavelengths.

In this paper, we present pseudo-observation results over the frequency range
0.15$\mathrm{\>GHz}$ to 8.5$\mathrm{\>GHz}$ using more accurate depolarization
models. Our purpose is to clarify the relationship between the actual
structure and the polarized radiation by comparing the three-dimensional
structure of the galactic magnetic field with the observed polarization. We
introduce the depolarization models in section 2 and discuss the numerical
methods used to obtain the pseudo-observations in section 3. We present
numerical results in section 4 and discuss our conclusions in section 5.
Throughout this paper, we choose the LOS direction to be the $z$-axis.

##  2 Depolarization

The polarization degree $p$ defines the fraction of the total intensity that
is polarized. It is defined in this paper by the following equation:

$p=p_{0}D,$ (1)

where $p_{0}$ is the intrinsic degree of polarization of synchrotron
radiation,

$p_{0}=\frac{3s+3}{3s+7}$ (2)

(Burn, 1966), where $s$ is the spectral index of cosmic-ray electrons, and $D$
is the depolarization coefficient. Since the depolarization effect depends on
the configuration of the magnetic field, several examples of magnetic field
structure and thermal electron number density have already been studied (Burn,
1966; Sokoloff et al., 1998). When depolarization models overlap, the effect
can be expressed as the product of the respective depolarization coefficients:

$D=D_{W}D_{\mathrm{F}}D_{A}D_{B},$ (3)

where $D_{W}$ represents the effect of wavelength-independent depolarization,
$D_{\mathrm{F}}$ represents differential Faraday rotation depolarization, and
$D_{A}$ and $D_{B}$ represent two types of beam depolarization. The following
sub-sections describe these depolarization models.

###  2.1 Wavelength-independent depolarization

Since synchrotron radiation is emitted with a direction of linear polarization
perpendicular to the magnetic field, the polarization angle in a turbulent
magnetic field is oriented in various directions. Therefore, the polarization
of the radiation cancels out even without considering Faraday rotation.
Wavelength-independent depolarization occurs in such a turbulent magnetic
field. In this section, we assume that the synchrotron emissivity scales as
$\varepsilon_{I}\propto B^{2}_{\perp}~{}~{}(s=3$, see equation (25)) below.
When there is a Gaussian turbulent magnetic field with dispersion
$\sigma_{\mathrm{i}}^{2}$ and average value $\overline{B}_{\mathrm{i}}$, the
depolarization coefficient can be represented as (Sokoloff et al., 1998)

$\begin{split}D_{W}=&\left(\overline{B^{2}}~{}\overline{B_{\perp}^{2}}\right)^{-1}\\\
&\times\left\\{\left[\overline{B}_{\sigma\mathrm{max}}^{4}-\overline{B}_{\sigma\mathrm{min}}^{4}+3\left(\sigma_{\mathrm{max}}^{4}-\sigma_{\mathrm{min}}^{4}\right)\right.\right.\\\
&\left.+6\left(\overline{B}_{\sigma\mathrm{max}}^{2}\sigma_{\mathrm{max}}^{2}-\overline{B}_{\sigma\mathrm{min}}^{2}\sigma_{\mathrm{min}}^{2}\right)\right.\\\
&\left.+\overline{B_{z}^{2}}\left(\overline{B^{2}}_{\sigma\mathrm{max}}-\overline{B^{2}}_{\sigma\mathrm{min}}\right)\right]^{2}\\\
&\left.+4\overline{B}_{\sigma\mathrm{max}}^{2}\overline{B}_{\sigma\mathrm{min}}^{2}\left[\overline{B^{2}}+2\left(\sigma_{\mathrm{max}}^{2}+\sigma_{\mathrm{min}}^{2}\right)\right]^{2}\right\\}^{1/2}\end{split}$
(4)

where $\overline{B_{i}^{2}}=\overline{B}_{i}^{2}+\sigma_{i}^{2}$, and
$\overline{B_{\perp}^{2}}=\overline{B}^{2}_{\perp}+\sigma_{\mathrm{max}}^{2}+\sigma_{\mathrm{min}}^{2}$.
Here, the subscript $i$ represents the direction of the magnetic field. The
quantity $\sigma_{\mathrm{max}}^{2}$ is the dispersion in the direction in
which the magnetic field is most dispersed in the plane perpendicular to the
LOS, and $\sigma_{\mathrm{min}}^{2}$ is the dispersion in the direction
perpendicular to direction of $\sigma_{\mathrm{max}}^{2}$. The quantities
$\overline{B}_{\sigma\mathrm{max}}$ and $\overline{B}_{\sigma\mathrm{min}}$
are the mean magnetic field strengths in each dispersion direction. The
quantities $\sigma_{\mathrm{max}}^{2}$ and $\sigma_{\mathrm{min}}^{2}$ are
obtained from the dispersion in the $x$ and $y$ directions and their
covariance. We define the covariance matrix as

$\left(\begin{array}[]{cc}\sigma_{x}^{2}&\sigma_{xy}\\\
\sigma_{xy}&\sigma_{y}^{2}\end{array}\right),$ (5)

where $\sigma_{x}^{2}$ and $\sigma_{y}^{2}$ are the dispersions in the $x$\-
and $y$-directions and $\sigma_{xy}$ is the covariance. The eigenvalues of
this matrix are $\sigma_{\mathrm{max}}^{2}$ and $\sigma_{\mathrm{min}}^{2}$,
and the corresponding eigenvectors are the directions of
$\sigma_{\mathrm{max}}^{2}$ and $\sigma_{\mathrm{min}}^{2}$.

In the special case where the turbulent magnetic field is isotropic
($\sigma_{x}=\sigma_{y}=\sigma_{z}\equiv\sigma$) and the mean field in the LOS
direction is zero, equation (4) can be rewritten as

$D_{W}=\frac{1+7\left(\sigma/\overline{B}_{\perp}\right)^{2}}{1+9\left(\sigma/\overline{B}_{\perp}\right)^{2}+10\left(\sigma/\overline{B}_{\perp}\right)^{4}}.$
(6)

Previous studies (Machida et al., 2018) have used equation (6) to calculate
the wavelength-independent depolarization.

###  2.2 Wavelength-dependent depolarization

Wavelength-dependent depolarization is caused by Faraday rotation. The amount
of rotation of the plane of polarization due to Faraday rotation is

$\Delta\chi_{\mathrm{F}}=\mathrm{RM}\lambda^{2},$ (7)

where $\lambda$ is the observing wavelength. We also introduce the Faraday
depth (FD) as an indicator of the amount of Faraday rotation that occurs in a
specific region; it is given by

$\mathrm{FD}=K\int^{z_{\mathrm{a}}}_{z_{\mathrm{b}}}n_{\mathrm{e}}B_{z}dz,$
(8) $K=\frac{e^{3}}{2\pi m_{\mathrm{e}}^{2}c^{4}}=\mathrm{constant},$ (9)

where $n_{e}$ is the thermal electron number density, and $B_{\parallel}$ is
the strength of the magnetic field parallel to the LOS. In addition, $e$ is
the elementary charge, $m_{\mathrm{e}}$ is the mass of the electron, and $c$
is the speed of light. The FD matches the RM if the range of integration in
equation (8) is extended to the entire LOS ($z_{a}\rightarrow
0,z_{b}\rightarrow\infty$).

####  2.2.1 Differential Faraday rotation depolarization

When plasma is composed of both thermal and cosmic-ray electrons, synchrotron
radiation and Faraday rotation occur simultaneously at the same position.
Therefore, the polarization angle inside that region becomes irregular, and
the polarized intensity averaged over that region decreases even if the
polarization angle emitted by the synchrotron radiation is uniform along the
LOS. This phenomenon is called differential Faraday rotation depolarization.
The depolarization coefficient can be written as

$D_{\mathrm{F}}=\frac{\sin(\mathrm{FD}\lambda^{2})}{\mathrm{FD}\lambda^{2}},$
(10)

in the region where $n_{\mathrm{e}}$ and magnetic fields
$\bm{B}=(B_{x},B_{y},B_{z})$ are constants.

####  2.2.2 Beam depolarization

When polarized radiation passes through a turbulent magnetic field, the
polarization angle becomes randomized due to Faraday rotation. These random
polarization angles all enter the beam and cancel each other out; this is
called “beam depolarization.”

Beam depolarization can be divided into two types: one is internal
depolarization that affects the radiating region and the other is external
depolarization caused by the foreground medium. We do not consider external
depolarization in this study.

When the emission of polarized radiation and Faraday rotation occur at the
same position, internal depolarization becomes important. Consider the case of
Gaussian turbulent magnetic field with the dispersion $\sigma_{z}^{2}$ and
constant polarized radiation between the path lengths $L$ and $L+\Delta z$.
The degree of polarization observed along this LOS is

$\begin{split}p=&p_{0}\exp\left\\{-2(Kn_{\mathrm{e}}\sigma_{z})^{2}L\lambda^{4}d\right\\}\\\
&\times\left(\frac{1-e^{-S^{\prime}}}{S^{\prime}}\right),\\\
S^{\prime}=&\left(Kn_{\mathrm{e}}\sigma_{z}\right)^{2}d\Delta
z\lambda^{4},\end{split}$ (11)

where $d$ is the coherence length scale of the turbulent field. Equation (11)
is the product of depolarization in the local radiation area (between $L$ and
$L+\Delta z$) and depolarization between the local radiation area and the
observer. We call the former ”internal depolarization A” and the latter
”internal depolarization B”. The respective depolarization coefficients are

$D_{A}=\left|\frac{1-e^{-S^{\prime}}}{S^{\prime}}\right|,$ (12)
$\begin{split}D_{B}&=\exp\left\\{-2(Kn_{\mathrm{e}}\sigma_{z})^{2}dL\lambda^{4}\right\\}\\\
&=\exp\left\\{-2\sigma^{2}_{\mathrm{FD}}\frac{L}{d}\lambda^{4}\right\\},\end{split}$
(13)

where $\sigma^{2}_{\mathrm{FD}}\equiv(Kn_{\mathrm{e}}\sigma_{z}d)^{2}$ is the
dispersion of the FD.

##  3 Pseudo-observations

###  3.1 Polarized radiative transfer equation

We first present the equations of radiative transfer that include
depolarization. Here, we use the Cartesian coordinate $S$-frame $(x,y,z)$ in
which the $z$-axis is the LOS direction of the radiation:

$\displaystyle\frac{dI}{dz}$ $\displaystyle=\varepsilon_{I}$ (14)
$\displaystyle\frac{dQ}{dz}$ $\displaystyle=D\varepsilon_{Q}\cos
2\chi_{B}-2U\frac{d\chi_{\mathrm{F}}}{dz}$ (15) $\displaystyle\frac{dU}{dz}$
$\displaystyle=D\varepsilon_{Q}\sin
2\chi_{B}+2Q\frac{d\chi_{\mathrm{F}}}{dz},$ (16)

where $(I,Q,U)$ are the Stokes parameters, and $\varepsilon_{I,Q}$ are the
radiation coefficients of each Stokes parameters. The quantity
$d\chi_{\mathrm{F}}/dz~{}~{}(\equiv Kn_{\mathrm{e}}B_{z}\lambda^{2})$ is the
Faraday rotation term, $D$ is the depolarization coefficient described in
section 2, and $\chi_{B}$ is the direction of $B_{\perp}$. The transfer
equations can be solved easily by taking the direction of the magnetic field
to be the direction of Stokes $U$. In this study, we confirm that synchrotron
self-absorption and free-free absorption are not effective, even at 0.1 GHz.

The polarization angle $\chi$ and the polarization intensity is calculated as
follows using the Stokes parameters:

$\chi=\left\\{\begin{array}[]{ll}\frac{1}{2}\tan^{-1}{\left(\frac{U}{Q}\right)}&\quad(Q>0),\\\
\frac{1}{2}\tan^{-1}{\left(\frac{U}{Q}\right)}+\frac{\pi}{2}&\quad(U>0,Q<0),\\\
\frac{1}{2}\tan^{-1}{\left(\frac{U}{Q}\right)}-\frac{\pi}{2}&\quad(U<0,Q<0),\\\
\frac{\pi}{4}&\quad(U>0,Q=0),\\\ -\frac{\pi}{4}&\quad(U<0,Q=0),\\\
0&\quad(U=0,Q=0),\\\ \end{array}\right.$ (17) $P=\sqrt{Q^{2}+U^{2}}.$ (18)

The number density of cosmic-ray electrons is required to calculate
synchrotron radiation. However, the cosmic-ray electron distribution cannot be
obtained from the MHD simulation. Consequently, we assume that the energy
density of cosmic-ray particles is equal to the magnetic energy density
(Akahori et al., 2018). Also, we assume a single power-law distribution for
the cosmic-ray electrons. Thus, the number density of cosmic-ray particles
between Lorentz factors $\gamma$ and $\gamma+d\gamma$ is given by

$n_{\mathrm{ce}}(\gamma)=N_{0}\gamma^{-s}d\gamma,$ (19)

where $s$ is the spectral index; in this study, $s=3$ (Sun et al., 2008).
Therefore, the cosmic-ray electron number density $n_{ce}$ is

$\begin{split}n_{\mathrm{ce}}&=N_{0}\int^{\gamma_{\mathrm{max}}}_{\gamma_{\mathrm{min}}}\gamma^{-s}d\gamma\\\
&=N_{0}N_{1}(s,\gamma_{\mathrm{min}},\gamma_{\mathrm{max}}),\end{split}$ (20)

where
$N_{1}=[\gamma^{1-s}]^{\gamma_{\mathrm{max}}}_{\gamma_{\mathrm{min}}}/(1-s)$.
Similarly, the energy density is

$\begin{split}\epsilon_{\mathrm{ce}}&=N_{0}m_{\mathrm{e}}c^{2}\int^{\gamma_{\mathrm{max}}}_{\gamma_{\mathrm{min}}}\gamma^{1-s}d\gamma\\\
&=N_{0}m_{\mathrm{e}}c^{2}N_{2}(s,\gamma_{\mathrm{min}},\gamma_{\mathrm{max}}),\end{split}$
(21)

where
$N_{2}=[\gamma^{2-s}]^{\gamma_{\mathrm{max}}}_{\gamma_{\mathrm{min}}}/(2-s)$,
and $\gamma_{\mathrm{min}}$ and $\gamma_{\mathrm{max}}$ are free parameters.
In this calculation, we used $\gamma_{\mathrm{min}}=200$,
$\gamma_{\mathrm{max}}=3000$. The quantities $n_{\mathrm{ce}}$ and
$\epsilon_{\mathrm{ce}}$ are almost independent of the value of $s$ when
$s=3$. Additionally, the selection of $\gamma_{\mathrm{max}}$—anywhere from
3000 to infinity—has little effect on radio emission in the GHz band. On the
low-energy side ($\gamma<200$), the cosmic-ray electrons lose energy by
Coulomb interactions with the thermal plasma (Sarazin, 1999).

We assume an equipartition between the total energy in relativistic particles,
which is twice the energy in relativistic electrons, and the magnetic energy:

$2\epsilon_{\mathrm{ce}}=\frac{B^{2}}{8\pi}.$ (22)

Therefore,

$N_{0}=\frac{B^{2}}{16\pi m_{\mathrm{e}}c^{2}N_{2}},$ (23)

and the cosmic-ray electron number density is

$n_{\mathrm{ce}}=\frac{N_{1}B^{2}}{16\pi m_{\mathrm{e}}c^{2}N_{2}}.$ (24)

Using $N_{0}$ obtained from Equation (23), we obtain the radiation
coefficients

$\varepsilon_{I}=c_{5}(s)N_{0}B_{\perp}^{\frac{s+1}{2}}\left(\frac{\nu}{2c_{1}}\right)^{\frac{1-s}{2}},\quad\varepsilon_{Q}=\frac{s+1}{s+7/3}\varepsilon_{I},$
(25)

where

$\begin{split}c_{1}&=\frac{3e}{4\pi m_{\mathrm{e}}^{3}c^{5}},\\\
c_{5}(s)&=\frac{\sqrt{3}}{16\pi}\frac{e^{3}}{m_{\mathrm{e}}c^{2}}\left(\frac{s+7/3}{s+1}\right)\Gamma\left(\frac{3s-1}{12}\right)\Gamma\left(\frac{3s+17}{12}\right),\\\
\end{split}$ (26)

The quantities $c_{1}$ and $c_{5}$ are dimensionless functions of $s$, which
have been determined by Pacholczyk (1970).

###  3.2 The galaxy model

####  3.2.1 3D MHD simulation

Table 1: Numerical units of the physical quantities from the MHD simulation used for the pseudo-observations Physical Quantity | Numerical Unit
---|---
Density |  $1.6\times 10^{-24}$ $\mathrm{g~{}cm^{-3}}$
Magnetic Field | 26 $\mathrm{\mu G}$
Temperature |  $5.2\times 10^{6}$ K
Cosmic-ray electron number density |  $3.5\times 10^{-3}$ $\mathrm{cm^{-3}}$
Absorption coefficient (0.1$\mathrm{\>GHz}$) | $6.7\times 10^{-16}$

In this paper, we calculate the observables using the results of MHD
simulations (Machida et al., 2013) that were performed in the cylindrical
coordinate system $(r,\phi,z^{\prime})$. The size of the simulation box is $r$
(kpc) $<56$ in the radial direction, $0\leq\phi\leq 2\pi$ in the azimuthal
direction, and $-10<z^{\prime}$ (kpc) $<10$ in the vertical direction. The
numbers of grid points are (250,256,640) in $(r,\phi,z^{\prime})$ direction,
respectively. For $0<r$ (kpc) $<6$ and $-2<z^{\prime}$ (kpc) $<2$—near the
center of the galaxy model—the spatial resolution of the simulation is $\Delta
r=50\mathrm{\>pc}$ and $\Delta z^{\prime}=10\mathrm{\>pc}$. The azimuthal
resolution is 25$\mathrm{\>pc}$ at $r=1\mathrm{\>kpc}$. The time resolution of
the simulation at this spatial resolution is $10^{3}$ yr. The growth timescale
of the magnetorotational instability (MRI) is comparable to the rotation time.
The rotation time at 1 kpc is about $10^{4}$ yr, while our numerical
simulation extends over $5.8\times 10^{9}$ yr; in other words, this time is
long enough to resolve the MRI growth time.

The initial condition of the simulation was an equilibrium torus threaded by a
weak azimuthal magnetic field embedded in a non-rotating, high-temperature
corona under Miyamoto and Nagai potentials (Miyamoto & Nagai, 1975). See
Machida et al. (2013) for more details of the simulation data. Since Machida
et al. (2013) is an ideal MHD simulation, it is a scale-free simulation. For
this study, we therefore chose parameters close to those appropriate for a
spiral galaxy, as shown in Table 1. The cosmic-ray electron number density and
the radiation coefficients required for the radiative transfer equation are
determined automatically from equations (23) and (25) using the physical
quantities obtained from the MHD simulation (see also Table 1)

####  3.2.2 Coordinates

Figure 1:  (a) The $z$-axis is defined to be the LOS direction, which is
normal to the sky projection plane $(x,y)$. The primed axes indicate the
original coordinates of the galaxy in the numerical simulation. The angle
theta is the inclination angle between the rotational axis of the galaxy (the
$z^{\prime}$-axis) and the LOS. (b) To determine the depolarization
coefficient, we need the dispersion for each magnetic field direction at point
$(x,y,z)$ in the S-frame, which is denoted by the orange dot in this figure.
In this work, the dispersions are computed from equation (27) by using the
magnetic fields at the nearest eight grids in the $S^{\prime}$-frame, which
are denoted by the blue dots.

We calculated the observational quantities—such as the Stokes parameters and
the RM—from the numerical simulation results for the galactic gaseous disk. We
set the projection plane for the pseudo-observables and its normal to be the
$S$-frame $(x,y,z)$. The $S^{\prime}$-frame indicates the coordinates for the
model galaxy obtained from the three-dimensional global simulation. Note that
the center of the $S$-frame is at the same location as that of the
$S^{\prime}$-frame and that the $x$-axis overlays the $x^{\prime}$-axis (see
figure 1a). The MHD variables in the $S$-frame are calculated by linear
interpolation of the variables in the S′-frame. The inclination angle between
the rotation axis of the galaxy (the $z^{\prime}$-axis) and the LOS (the
$z$-axis) is defined as $\theta$. Face-on and edge-on correspond to
$\theta=0^{\circ}$ and $90^{\circ}$, respectively. In this study, we report
the results for the face-on view ($\theta=0^{\circ}$). The spatial range of
the pseudo-observation coordinates is set to be $(x,y,z)\in\pm
10\mathrm{\>kpc}$ centered on the center of the galaxy, and the numbers of
grid points are $(n_{x},n_{y},n_{z})=(200,200,200)$. An observational map can
be obtained by numerical integration from $z=-10\mathrm{\>kpc}$ to
$z=10\mathrm{\>kpc}$ at all 2D coordinate points $(x,y)$. Therefore, a pseudo-
observation result can be obtained as an image with the numbers of pixels
$(N_{z},N_{y})=(200,200)$ centered on the galactic center.

####  3.2.3 Determination of the magnetic field dispersion

It is necessary to take into account the depolarization induced by turbulent
fields with length scales below the grid size. In this study, we implement the
depolarization effect of the turbulent field by using the model introduced in
section 2.

To use this depolarization model, we need to obtain the mean field and the
sub-grid-scale turbulence field. In this study, we estimate the property of
the magnetic field inside the grid from the surrounding magnetic fields,
following Machida et al. (2019). The validity of this method is described in
section 4.1 of Machida et al. (2019). First, we calculate the mean magnetic
field by averaging the magnetic field inside the $\pm 5$ grid points in the
$S^{\prime}$-frame. The mean magnetic field in the S-frame is calculated by
linear interpolation using the mean field of the S-frame. To calculate the
dispersion for each magnetic field direction at the grid point $(x,y,z)$ in
the S-frame, we use the eight grid points in the $S^{\prime}$-frame (h), that
are nearest to the given grid point, as shown in figure 1b. The dispersion of
the magnetic field in the $x$-direction is thus given by:

$\sigma^{2}_{x}(x,y,z)=\frac{1}{8}\sum_{(l,m,n)\in
h}\left[B_{x}(l,m,n)-\overline{B_{x}}(l,m,n)\right)^{2}.$ (27)

Dispersions in the $y$ and $z$ directions are obtained in the same way. The FD
dispersion $\sigma^{2}_{\mathrm{FD}}$ is also needed in order to use the
internal depolarization model [see equation (13)). The dispersion of FD in
grid point $(x,y,z)$ is

$\begin{split}\sigma^{2}_{\mathrm{FD}}(x,y,z)=\frac{1}{9}\sum^{x+1}_{x^{\prime\prime}=x-1}\sum^{y+1}_{y^{\prime\prime}=y-1}&\left\\{\mathrm{FD}_{\mathrm{turb}}(x^{\prime\prime},y^{\prime\prime},z)\right.\\\
&\left.-\overline{\mathrm{FD}}_{\mathrm{turb}}(x,y,z)\right\\}^{2},\end{split}$
(28)
$\begin{split}\mathrm{FD}_{\mathrm{turb}}(x,y,z)=&0.81\sum_{z^{\prime}=1}^{z}n_{\mathrm{e}}(x,y,z^{\prime})\\\
&\times\left(B_{z}(x,y,z^{\prime})-\overline{B_{z}}(x,y,z^{\prime})\right)\Delta
z,\end{split}$ (29)
$\overline{\mathrm{FD}}_{\mathrm{turb}}(x,y,z)=\frac{1}{9}\sum^{x+1}_{x^{\prime\prime}=x-1}\sum^{y+1}_{y^{\prime\prime}=y-1}\mathrm{FD}_{\mathrm{turb}}(x^{\prime\prime},y^{\prime\prime},z),$
(30)

where $\mathrm{FD}_{\rm turb}$ is the value of FD due to the sub-grid-scale
turbulence and $\overline{\mathrm{FD}}_{\rm turb}$ is the average of
$\mathrm{FD}_{\rm turb}$. In this study, we assumed the length scale of the
magnetic field [see equations (12) and (13)] to be $d=100\mathrm{\>pc}$, which
is the same as the length scale in a previous study (Machida et al., 2019).

##  4 Results

The pseudo-observations are performed using a dataset similar to that of
Machida et al. (2018). We first present the RM maps and then show the results
of the depolarization behavior in the radio band.

###  4.1 Comparison of two methods for calculating the Faraday RM

Figure 2:  (left) Face-on view color map of the FD of the spiral galaxy. The
FD is derived using equation (8), and it has units of
[$\mathrm{rad\>m^{-2}}$]. The vertical and horizontal axes have units of
[kpc]. (right) The RM color map calculated from equation (31), obtained using
8$\mathrm{\>GHz}$ and 10$\mathrm{\>GHz}$.

As mentioned in Section 1, the RM can be calculated using either the
theoretical relationship (equation (8)) and the conventional relationship
(equation (7)). In this paper, we call the former results FD and the latter
${\rm RM_{obs}}$.

####  4.1.1 Faraday depth

The left panel of figure 2 shows the spatial distribution of FD, which is the
same result as in a previous study [figure 2a of (Machida et al., 2018)]. The
length scale of the galactic magnetic field in the azimuthal direction is
longer than that of the vertical field, so the FD distribution shows a mottled
structure extending in the azimuthal direction. Moreover, since the gas
density at the outer edge of the disk ($>5\mathrm{\>kpc}$) is low, the
absolute value of FD is less than half that at the inner edge.

####  4.1.2 Faraday RM

The right panel of Figure 2 shows the distribution of ${\rm RM_{obs}}$
obtained using the observing frequencies 8$\mathrm{\>GHz}$ and
10$\mathrm{\>GHz}$.

${\rm RM_{obs}}=\frac{\chi_{\rm 10GHz}-\chi_{\rm 8GHz}}{\lambda^{2}_{\rm
10GHz}-\lambda^{2}_{\rm 8GHz}}.$ (31)

In this frequency range, the amount of Faraday rotation is small. Therefore,
the n-$\pi$ ambiguity do not appear on any pixel on the observation map.

Compared with FD, the absolute value of ${\rm RM_{obs}}$ is reduced overall.
This is consistent with the theory that the observed RM is halved when the
magnetic field is symmetric along the LOS (Sokoloff et al., 1998). On average,
the absolute value of ${\rm RM_{obs}}$ is about half the value of FD, although
there are some areas where it is reduced to 1/4. In addition, the overall
structure is almost the same, although if we examine the details, we can see
that the sign is reversed and that the absolute value is not simply smaller.

###  4.2 Effect of depolarization

Figure 3:  Total intensity map of a spiral galaxy for the viewing angle
$\theta=0^{\circ}$. This is the result at the pseudo-observation frequency of
8.5$\mathrm{\>GHz}$. The vertical and horizontal axes have units of [kpc], and
the total intensity has units of [$\mathrm{mJy\>str^{-1}}$]

In this section, we focus on the effect of depolarization. First, we show the
total intensity map at 8.5 GHz for comparison (Figure 3). This map reflects
the magnetic spiral arm structure of the galactic disk and is consistent with
our previous study (Machida et al., 2018).

####  4.2.1 Wavelength-independent depolarization

Figure 4:  These maps show the degree of polarization of a face-on spiral
galaxy at the observing frequency 8.5$\mathrm{\>GHz}$. (a) Map of the degree
of polarization without including depolarization models. (b) Wavelength-
independent depolarization model with isotropic turbulent magnetic fields
(equation 6). (c) Wavelength-independent depolarization model with anisotropic
turbulent magnetic fields (equation 4).  Figure 5:  (left) The ratio of
$\sigma_{\mathrm{max}}^{2}$ to $\sigma_{\mathrm{min}}^{2}$. This panel shows
the value in the equatorial plane ($z=0$). (right) The ratio of the values
integrated along the LOS by weighting each dispersion
($\sigma_{\mathrm{max}}^{2}$ and $\sigma_{\mathrm{min}}^{2}$) with the
emissivity.

Figure 4 shows the effects of wavelength-independent depolarization. Figures
4a, 4b, and 4c show the case with no depolarization, with isotropic
turbulence, and with anisotropic turbulence, respectively. The degree of
polarization reaches 0.7 for the anisotropic model, which is consistent with
almost fully polarized radiation. When we adopt an isotropic turbulent field
in the calculation, the depolarization works effectively, and the results are
similar to previous studies.

The azimuthal component of the galactic magnetic field is the most substantial
because of galactic rotation. Since the energy of the turbulent magnetic field
is calculated assuming a Gaussian distribution, the azimuthal turbulent
magnetic field is also the dominant component. In figure 4c, the wavelength
depolarization is calculated taking into account anisotropic turbulence. In
this case, the degree of polarization exceeds 0.6 over the entire region due
to the weakened effect of depolarization.

Next, we consider how and where anisotropic turbulence affects inefficient
depolarization. Figure 5 shows the ratio of the dispersion in the maximum and
minimum directions. The local values on the equatorial plane are shown in the
left panel, and the ratio of the emissivity-weighted dispersion integrated
along the LOS is plotted in the right panel.

When the turbulence is completely isotropic, the dispersion ratio $\sigma_{\rm
min}^{2}/\sigma_{\rm max}^{2}$ is 1. Since the mean fields have a
predominantly azimuthal component and form filamentary structures, the
dispersion ratios in such filaments show anisotropic features. Other regions
where isotropic turbulence develops have $\sigma_{\rm min}^{2}/\sigma_{\rm
max}^{2}\sim 1$. As the volume of the filamentary region is sufficiently
small, the structure of the local dispersion ratio becomes sub-isotropic
(figure 5, left). In the presence of strongly polarized radiation from a
filament, only the dispersion around the radiating region needs to be
considered, since depolarization due to the turbulent field only acts on such
a filamentary radiating region. The right side of figure 5 shows the ratio of
the integrals of the minimum and maximum dispersions weighted by the
emissivity. As a result, the effect of wavelength-independent depolarization
is suppressed due to the anisotropy of the dispersion and of the radiating
regions. In addition, when magnetic turbulence is superimposed on the mean
field, the effect of depolarization is overestimated by assuming isotropic
turbulence.

####  4.2.2 Differential Faraday rotation depolarization

Figure 6:  (left) Map of polarized intensity at the observing frequency
0.15$\mathrm{\>GHz}$. without including depolarization models. The polarized
intensity has units of [$\mathrm{mJy\>str^{-1}}$]. (right) Same as (left), but
including the model for differential Faraday rotation depolarization given by
equation (10).

Figure 6 shows the polarization intensity at 0.15$\mathrm{\>GHz}$ for the no
depolarization model (left) and including differential Faraday rotation
depolarization (right), respectively. The distribution is almost identical to
that in the left panel of figure 6 and in figure 3 because depolarization is
ignored, although the polarization map does highlight the spiral arms. When
differential polarization due to Faraday rotation is included in the
calculation, the polarized intensity is reduced by more than two orders of
magnitude compared to the unpolarized case (see the right panel of figure 6).
Although the magnetic spiral arms emit strongly polarized radiation, they are
effectively depolarized due to Faraday rotation. Therefore, the spiral arm
region and the inter-arm region emit similar amounts of polarized radiation.
As a result, the polarization map that includes depolarization has a smaller
range of intensities than that without depolarization. In particular, the
filamentary structure in the range $-8<x<-3$, $y=7.5$ is completely
depolarized and invisible. This occurs because differential Faraday rotation
depolarization is more effective when the RM is larger.

####  4.2.3 Beam depolarization

Figure 7:  (a) Maps of the polarized intensity for a face-on spiral galaxy at
the observing frequency 0.15$\mathrm{\>GHz}$. The polarized intensity has
units of [$\mathrm{mJy\>{str}^{-1}}$]. This map includes only internal
depolarization A derived from equation (12). (b) Same as (a), but the
depolarization model includes only internal depolarization B derived from
equation (13). (c) Same as (a), but including both internal depolarization A
and B.

Figure 7 shows the effect of internal depolarization that is caused by the
dispersion of the magnetic field along the LOS. As shown in section 2.2.2, the
internal depolarization is divided into two parts when the emitting source
itself becomes the source of depolarization. One is internal depolarization A,
which originates in the local emitting region, and the other is internal
depolarization B, that occurs when the polarized radiation passes through the
region causing Faraday rotation in the polarization source.

Figure 7a shows a map of the polarized intensity that includes only internal
depolarization A caused by the turbulent magnetic field. The remaining
polarized intensity due to internal depolarization A has a distribution
similar to that of the total intensity map (see figure 3). This means that the
strong-field region is hardly affected by internal depolarization A. The cause
of internal depolarization A is the turbulent magnetic field in the emitting
area. Therefore, this depolarization effect is weakened in a region with
ordered magnetic flux. Since the polarized intensity averaged along the
magnetic spiral arms is about $10^{7}\>\mathrm{mJy\>str^{-1}}$, internal
depolarization A is more efficient than differential Faraday rotation
depolarization (see Figure 6, right).

Next, we consider internal depolarization B. This corresponds to
depolarization along the LOS between the emitting region and the surface of
the polarized-light source. The dispersion that causes this depolarization is
therefore evaluated by integrating along the LOS where the radio waves pass.
Since the turbulent magnetic field is distributed over the galactic disk and
halo, internal depolarization B works very efficiently indeed (Figure 7b). The
region with the remaining polarized emission is not very consistent with the
total intensity map (figure 3). Figure 7c shows the polarized intensity map
including both internal depolarizations A and B. The overall structure is
similar to the result for internal polarization B.

####  4.2.4 All depolarization models

Figure 8:  (a) Map of the polarized intensity at the observing frequency
0.15$\mathrm{\>GHz}$. All the depolarization models introduced in section 2
are considered. The polarized intensity has units of
[$\mathrm{mJy\>str^{-1}}$]. The observed magnetic field directions,
$\chi+90^{\circ}$, at the same frequency are drawn on the map. (b) Same as
(a), but the observing frequency is 8.5$\mathrm{\>GHz}$. (c) Map of the degree
of polarization at the observing frequency 0.15$\mathrm{\>GHz}$. (d) Same as
(c), but at the observing frequency 8.5$\mathrm{\>GHz}$. The crosses labeled A
and B in panels (a) and (b) are the two lines of sight discussed in Section
5.2.

Figure 8 shows the polarized intensity and the degree of polarization at the
observing frequencies 0.15$\mathrm{\>GHz}$ and 8.5$\mathrm{\>GHz}$. At
8.5$\mathrm{\>GHz}$, the polarized intensity map shows a distribution similar
to the total intensity map (figure 3), and the degree of polarization shows
high values throughout the map. In addition, the polarization angle
$\chi+90^{\circ}$ indicates the direction of the magnetic field aligned with
the spiral structure. This represents the direction of the disk magnetic
field. The pseudo-observation result at 0.15$\mathrm{\>GHz}$ shows that the
polarized intensity is significantly reduced by the depolarization models; the
polarization degree is less than 1% almost everywhere in the map. The
polarized intensity map does not simply decrease uniformly but appears to
reflect radiation from different structures. The polarization angle completely
loses information about the original magnetic field direction because the very
large Faraday rotation effect makes it random.

##  5 Discussion

###  5.1 The relation between FD and RM

Figure 9:  The same as figure 2, but with the FD derived by integrating from
the position of the strongest magnetic field along the LOS.

In section 4.1, we compared the quantity FD calculated using equation (8) with
the quantity $\rm RM_{obs}$ obtained from the proportionality relationship
between the polarization angle and the square of the wavelength [equation
(7)]. We found that FD is about twice as large as $\rm RM_{obs}$. This may be
due to differences in the polarization sources. FD is the amount of rotation
that polarized waves from a background source experience as they pass through
the galaxy. In contrast, $\rm RM_{obs}$ corresponds to the rotation of
polarized waves emitted from the galaxy itself. Naively, we may guess that
polarization originating from behind the position where the maximum polarized
intensity is produced, i.e., the position where the magnetic fields are
strongest, does not affect the amount obtained by integration along the LOS.
In figure 9, we therefore show the FD integrated from the position of maximum
polarization toward the observer. This result shows that the overall absolute
value of this integral is about half the value of the FD and is close to the
value of $\rm RM_{obs}$. Also, the spatial distribution is similar to that of
$\rm RM_{obs}$. The RM obtained from an actual observation can thus be
considered to represent the FD from the position where the polarized radiation
at the local position is strongest along the LOS. Since it is difficult to
determine from RM observations alone where a polarized radiation source is
located along the LOS, one must consider methods to identify the three-
dimensional distribution of the magnetic field.

###  5.2 Depolarization

Figure 10:  The spectra of polarized intensity. The polarization intensities
along LOS A (blue), which has a large depolarization effect at low
frequencies, and along LOS B (orange), which has a small effect, are plotted.
The green line shows the slope of the frequency dependence of internal
depolarization B on the low frequency side, $p\propto\nu^{-3}$. LOS A:
$(x,y)=(2\mathrm{kpc},-1\mathrm{kpc})$, LOS B:
$(x,y)=(4\mathrm{kpc},-5\mathrm{kpc})$ Figure 11:  (a) Changes in the integral
of polarized intensity along LOS A (blue) and LOS B (orange). (b) Distribution
of normalized magnetic field strengths perpendicular to the LOS on LOS A and
LOS B. (c) Changes in the dispersion of the FD on LOS A and LOS B. The
dispersion of the FD is larger far from the observer because of the greater
length of the Faraday rotation integral.

Figure 10 shows the frequency dependence of the two lines of sight. LOS
A(2$\mathrm{\>kpc}$, $-1\mathrm{\>kpc}$) is the LOS that shows the highest
polarized intensity in the pseudo-observation at 0.15$\mathrm{\>GHz}$. In
contrast, LOS B (4$\mathrm{\>kpc}$, $-5\mathrm{\>kpc}$) shows a strong
polarized intensity at 8.5$\mathrm{\>GHz}$, but the polarized intensity is
significantly reduced at 0.15$\mathrm{\>GHz}$.

At LOS A, the polarized intensity begins to decrease from around
2$\mathrm{\>GHz}$ due to the depolarization effect, but the polarized
intensity increases again below about 0.5$\mathrm{\>GHz}$. This is thought to
occur because the position that produces the dominant radiation on the LOS has
changed. At LOS B, the polarized intensity begins to decrease below about
1$\mathrm{\>GHz}$ and continues to decrease significantly down to
0.15$\mathrm{\>GHz}$. The green straight line is the frequency dependence of
polarized intensity when the internal depolarization is strong. The internal
depolarization is known to follow a power law ($\propto\nu^{4-\alpha}$) when
the Faraday dispersion is large (Arshakian & Beck, 2011). Here, $\alpha$ is
the synchrotron spectral index, which can be written as $\alpha=(s-1)/2$.
Because the slope of the spectrum of LOS B approximately matches the green
line, this shows that the polarization is eliminated by internal
depolarization.

Figure 11a shows the relation between the length of the LOS and the polarized
intensity at 0.15$\mathrm{\>GHz}$. As in figure 10, the blue and orange curves
show the LOS positions where the polarized intensity is at its maximum and
minimum, respectively, at this frequency. The magnetic field strength
perpendicular to each LOS is shown in figure 11b. LOS A and LOS B have similar
magnetic field distributions within $|z|<2\mathrm{\>kpc}$, but there is a
difference for $z>2.5\mathrm{\>kpc}$, where a strong magnetic field is seen
only along LOS A. Figure 11c shows the FD dispersion. Both LOS A and B have a
strong perpendicular magnetic field at $z<2\mathrm{\>kpc}$, but the FD
dispersion is also large, so the radiation is completely depolarized. The
dispersion along LOS A decreases suddenly for $z>2.5\mathrm{\>kpc}$, even
though the perpendicular field remains high. This implies that LOS A has an
ordered magnetic field that is sufficiently large to cause polarized emission.
On the other hand, along LOS B, although the amplitude of the perpendicular
magnetic field is reduced, turbulent fields have developed, and dispersion
plays a role in depolarization. Therefore, the polarized emission along LOS A
originates in the ordered magnetic field in the halo region.

###  5.3 Comparison with actual observations

Figure 12:  Same as figure 8, but with the size of the observing beam changed.
The Gaussian widths of the beams in these panels are 800 pc. The circle at the
lower left in each panel shows the beam size. The polarized intensity has the
units [$\mathrm{mJy\>beam^{-1}}$].  Figure 13:  (left) Polarized intensity map
for a face-on spiral galaxy at the observing frequency 4.8$\mathrm{\>GHz}$.
Assuming this to be an observation of IC 342, the distance is
3.3$\mathrm{\>kpc}$, and the beam size is 25 arcsec. The units of the color
bar are [$\rm mJy\>beam^{-1}$]. (right) Three-dimensional distribution of the
polarized radiation reaching the observer. The orange lines associate arm
structures with the originating locations of the polarized radiation.  Figure
14:  The same as figure 13, but at the observing frequency
$0.15\mathrm{\>GHz}$. The units of the color bar are [$\rm\mu Jy\>beam^{-1}$].

In this section, we compare the results of our pseudo-observations with actual
observations. Because of the superposition of polarized waves in the beam, the
depolarization depends strongly on the beam size. Figure 12 shows the
polarized intensity and polarization degree at 0.15$\mathrm{\>GHz}$ and
8.5$\mathrm{\>GHz}$ with a Gaussian beamwidth of 800$\mathrm{\>pc}$. The beam-
averaged polarized intensity at 8.5$\mathrm{\>GHz}$ displayed in figure 12d
looks smoother than that in figure 8, but there is no significant change in
either the polarization or the intensity distribution. On the other hand, at
0.15 GHz, the degree of polarization $p$ is significantly reduced because the
polarization angles are random. As a result, $p<$ 0.1% for most pixels. To
validate our pseudo-observations, we rescaled the beam size to that used for
observations of the spiral galaxy IC 342 (figure 13). We set the distance from
the observer to be 3.3 Mpc and calculated the spatial resolution assuming
$25"$ (400$\mathrm{\>pc}$). The left panel of figure 13 shows a projection map
of the polarized intensity at 4.8$\mathrm{\>GHz}$. The polarization
intensities obtained in our pseudo-observations reproduce well the features of
the actual observations of IC 342 (Beck, 2015a). The right panel of figure 13
shows the three-dimensional distribution of the polarized intensity, which
enables us to understand where the projected polarized radiation originates in
the galactic disk. The green surfaces are isosurfaces of the polarized
intensity at $0.08\mathrm{\>Jy\>beam^{-1}}$; they show that the emission
region extends over $\pm 2\mathrm{\>kpc}$ from the equatorial plane. The
orange lines show the correspondence between the obtained arm structure and
the radiation position, which show that each arm is located at a different
height. For example, we can see that two or more spiral fields overlap in arm
1 in figure 13. Even though the spiral arms in the projection appear to be a
single structure, they are not located on the same slice in the height
direction. In the pseudo-observations, a region with strong polarization
exists at $|x|<2.5\mathrm{\>kpc}$, $|y|<2.5\mathrm{\>kpc}$, which can be seen
even at 1$\mathrm{\>GHz}$. However, in the actual observations, this feature
disappears due to depolarization at about 1.4$\mathrm{\>GHz}$ (Beck, 2015a).
This occurs because the model of Machida et al. (2013) neglects the cold HI
clouds that produce turbulent magnetic fields with short length scales in the
disk. If we were to treat this cold part of the disk, the depolarization due
to Faraday rotation would be more effective, and the polarized emission would
be reduced (Borlaff et al., 2021).

Next, we consider depolarization effects using pseudo-observations at 0.15 GHz
(figure 14). We assumed that these are observations of IC 342 at a resolution
of 25 arcseconds. The maximum polarization intensity at 0.15 GHz is weaker
than at 4.8 GHz, reaching only about 30$\mathrm{\>\mu Jy\>beam^{-1}}$. This is
because the polarized intensity produced by the disk is depolarized, and only
the dilute halo component remains in the integrated intensity. In addition,
the correlation length of magnetic turbulence in the present model galaxy
tends to be long because only the high-temperature plasma in the galactic disk
is included. The current model assumes that we are dealing with warm and hot
plasma. However, real galactic gas disks form high-density, cold gas disks,
and these regions have strong magnetic fields. In fact, the central region in
Figure 14 where the polarization intensity shows a maximum value should be
composed of cold gas. Such a high-density region would be expected to have
stronger depolarization than the current model. Therefore, in order to
calculate the depolarization effect correctly, a multi-temperature disk must
be taken into account. Compared with the three-dimensional structure of
polarized radiation shown in Figure 13, the right panel of figure 14 shows
that the magnetic field structure of the halo is being observed, as shown in
the previous section. If LOFAR were used to observe with the NL-remote, even
3$\sigma$ detection of such weak halo magnetic fields requires an observation
time of $10^{5}$ hours or more (van Haarlem et al., 2013). SKA1-Low has a
resolution of about 5 arcsec at 150 MHz. Halo radiation at this resolution is
about 2 $\mu$Jy/beam. To detect 3$\sigma$ of this radiation with 100 MHz
bandwidth observation requires about 80 hours of observation time. (SKA
Baseline Design document version2) 111SKA Baseline Design document version 2,
2015 $\langle$https://www.skatelescope.org/key-documents/$\rangle$.. The
sensitivity of SKA2 is about 10 times better than that of SKA1. Therefore,
using SKA2, even with 10 MHz bandwidth observation, 5$\sigma$ detection is
possible in a few tens hours of observation time.

Figure 15:  (a) Map of the observing frequency that maximizes the intensity of
polarized radiation. (b) Map showing the positions from which the strongest
polarized radiation is emitted along the LOS at 8.5 GHz.  Figure 16:  (a) Map
of FD dispersion obtained from the frequency dependence of the polarized
intensity using the internal depolarization model. (b) Map of FD dispersion
between the observer and the position where the polarized radiation is the
strongest, which is derived directly from the MHD simulation.

We next consider whether it is possible to retrieve information about the
spatial distribution along the LOS from the polarized intensity. The polarized
intensity shows a maximum at the frequencies where beam depolarization begins
to operate (e.g., figure 10, Arshakian & Beck (2011)). This feature has the
potential to be detectable at locations along the LOS. A region with a peak at
a low frequency means that there is no turbulent and dense plasma in front of
it, and it is expected to be distributed in front of the galactic disk, where
it is more susceptible to depolarization. In figure 15a, we therefore plot the
frequency at which the polarized intensity becomes the strongest along the
LOS, and we show the corresponding position in figure 15b. The color bar in
figure 15b represents the $z$ coordinate, with positive being the observer’s
side. The spiral structure extends over the range $y=0-5$ kpc, with an
emitting region at $z=2$ kpc on the observer’s side. A similar spiral can be
seen near $y=-2$ kpc and $x=0-4$ kpc. However, although the arm structure near
$y=-5$ kpc has a part with a low frequency peak, the actual radiating position
is near the equatorial plane. The difference between these correlations can be
determined from the length scale of the peak frequency map. When the frequency
peak is mottled, it is influenced by the random magnetic field in the disk. On
the other hand, a part where the length scale of the peak frequency structure
is long reflects the structure of the magnetic flux tube above the disk.

Since the peak frequency depends on the dispersion of the FD, we obtained that
quantity using the dispersion model and utilized the numerical results to
check the verification. Figures 16a and 16b are FD dispersion maps computed
from the internal depolarization model (see Equation 12) and from the
numerical simulation (Equation 28), respectively. The FD dispersion of the
front spiral arm ($y=1$–5) shown in figure 16b shows that the values are more
than one order of magnitude smaller than the values of
$\sigma^{2}_{\mathrm{FD}}$ around the spiral arm for both derivation methods.
Although the density of a galactic halo is 10–100 times lower than the density
of a galactic disk, magnetic flux tubes with lengths greater than the
turbulence length scale may be the origin of the polarized emission. Comparing
the absolute values of the FD dispersion, we find that the results obtained
using the depolarization model (figure 16a) are higher than those obtained
from the direct calculation (figure 16b). This means that the turbulent
magnetic field may be overestimated from the observations.

###  5.4 Validity of the pseudo-observation assumptions

Figure 17:  Maps of the degree of polarization at the observing frequency
0.15$\mathrm{\>GHz}$. Both maps have the same beam size. (left) Calculated by
applying sub-grid turbulence. (right) Calculated by our method using
depolarization models.

The total intensities obtained from pseudo-observations using cosmological
galaxy simulations have been reported to be lower than the actual observations
when cosmic-ray electron energies are assumed to be equivalent to the magnetic
energy (Reissl et al., 2019). Since our simulation also assumes equipartition
of the energy, the trend of the pseudo-observation is similar to the previous
results. On the other hand, the polarized intensity in the simulation agrees
with the actual intensity. This suggests that, although the mean field that
originates the polarized emission is reproduced, the current numerical
simulations do not provide enough turbulent field to explain the total
intensity. Observational evidence shows that the energy in cosmic-ray
electrons is about 50 times smaller than that in cosmic-ray ions (Cummings et
al., 2016). Since the energy in cosmic-ray electrons is balanced by the mean
magnetic field, the energy in invisible cosmic-ray ions may be higher. Also,
the total energy of the magnetic field may be underestimated by the amount of
invisible cosmic-ray ions. Since cosmic-ray ions enhance magnetic turbulence
(Kuwabara et al., 2004; Kuwabara & Ko, 2015), a magnetic instability may
easily be able to amplify the turbulent field to the required energy. It will
be necessary to simulate the dynamics of cosmic rays with MHD flows to obtain
more accurate observables. The galactic disk contains a mixture of multi-
temperature structures such as molecular clouds, HI regions, and high-
temperature plasma, each of which generates turbulence with its own length
scale (Inutsuka et al., 2015). In each component, the turbulent magnetic
energy is comparable to the gas pressure, and low-temperature components are
embedded in the high-temperature components. These various components are also
important as a cause of magnetic turbulence below the resolution limit.

To verify this method of using depolarization models, we compared our pseudo-
observation code with the method used by Reissl et al. (2019) to apply a
turbulent sub-grid magnetic field and perform beam averaging. Figure 17 shows
the polarization degree at 146 MHz obtained using the two methods. The left
panel of figure 17 shows the result obtained by applying sub-grid turbulence,
and the right panel shows the result obtained in this study using
depolarization models. The sub-grid width is 1/3 of the original grid width in
each direction, so the numerical cost of the sub-grid method is 9 times larger
than that of our method. There was almost no difference in the degree of
polarization, which shows that the calculations using depolarization models
give reasonable results. We have thus shown that Burn’s depolarization model
successfully reproduces internal depolarization when the turbulence scale is
smaller than the MHD grid size.

The right panel of figure 17 corresponds to the case where the length scale of
the turbulent magnetic field in figure 8c is reduced by the factor 1/3.
Although the turbulent length scale is shorter than in figure 8c, the spatial
distribution shows no significant difference. In other words, a reasonable
value for the length scale of the turbulent magnetic field is about 100 pc.

##  6 Summary

We have performed pseudo-observations using physical quantities obtained from
MHD simulations in order to clarify the relationship between the projected
intensity and the emitting region in real space. We have also focused on the
importance of depolarization at meter wavelengths. Here, we summarize our
findings:

  * •

Assuming a linear relationship between polarization angle and wavelength
squared, we found that the value of the RM is the amount of Faraday rotation
obtained from the brightest polarized emission region along the LOS to the
observer. This value may underestimate the strength of the magnetic field.

  * •

Depolarization depends strongly on the properties of the turbulence, so it is
necessary to handle the turbulent magnetic field carefully. When wavelength-
independent polarization is calculated assuming isotropic turbulence, the
effect of depolarization is overestimated because most of the polarized
intensity is emitted from the surroundings of the spiral arm, and the
turbulent magnetic field along the magnetic spiral arms has strong
inhomogeneity (Machida et al., 2019).

  * •

Internal depolarization can be divided into two parts: depolarization that
occurs within the emitting region and depolarization that occurs from the
emitting region to the observer. We found that the latter produces the most
efficient depolarization of all the depolarization models.

  * •

Polarized emission in the 0.15$\mathrm{\>GHz}$ band originates from the
magnetic field of the halo, since emission from the equatorial plane is
reduced by strong depolarization. This halo emission is difficult to detect
with current facilities. But it may be detectable if sufficient observation
time is obtained using SKA1. Also, SKA2 is expected to be sufficient to detect
these structures.

  * •

Since the projected intensity originates from several emitting regions located
at different heights in real space, it is necessary to pay attention to the
correlation between the spatial distribution of the emitting filaments and the
projected intensity map. By investigating the frequency at which beam
depolarization begins to operate, we found that it is possible to separate
emitting regions with different heights.

The present model does not take into account cooling, which is important for
the formation of molecular clouds and HI clouds. When cooling is neglected,
the coherence length of the turbulence increases because the thermal energy is
greater. This means that depolarization will be weaker than is the actual case
for both wavelength-independent and wavelength-dependent depolarization.
Recent high-resolution, far-infrared polarimetric observations of M 51 have
identified a magnetic field with a coherence length along the spiral arms that
is shorter than that of the magnetic field derived from radio observations
(Borlaff et al., 2021). Several groups, including ours, have already stared to
conduct 3D MHD simulations for the galactic gas disk, including radiative
cooling (Reissl et al., 2019; Kudoh et al., 2020). In the future, we will
consider pseudo-observations that take into account results for cold galactic
disks.

We are grateful to Drs. H. Sakemi, M. Haverkorn, S. Totorica, and Mr. R. Omae
for useful discussion and brushing up on our writing. We also thank the
anonymous referee for his/her useful comments and constructive suggestions.
This work was supported in part by JSPS KAKENHI (MM:19K03916, 20H01941,
TO:22K14032). Our numerical computations were carried out on SX-9 and on
analysis servers at the Center for Computational Astrophysics of the National
Astronomical Observatory of Japan. The computation was carried out using the
computer resources offered by the Research Institute for Information
Technology, Kyushu University.

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