1. Introduction
Faraday tomography aims to determine the distribution and properties of magnetoionic medium along the line of sight (LOS) toward, and within, astronomical objects through the Faraday effect. This technique requires wide-band low frequency (≲10 GHz) polarization data, and thus the Square Kilometre Array and its precursors/pathfinders make it possible to use the technique for studying cosmic magnetism. Cosmic magnetism studies using Faraday tomography require the interpretation of the Faraday spectrum,
, which gives the intensity of synchrotron polarization as a function of the Faraday depth,
.
is obtained from the observed polarization spectrum,
, through the equation
where the
is the wavelength (see e.g., [
1,
2]). The Faraday depth
is expressed as
where
is the thermal electron number density,
is the LOS component of the magnetic field,
is the integration variable in physical distance, and
r is an arbitrary physical distance. One of the difficulties in using the technique, as implied by Equation (
2), is that
can change its sign along the LOS due to turbulence and it causes the lack of a one-to-one relation between the Faraday depth and the physical depth.
There have been many challenges to apply Faraday tomography for studying objects with various scales including the external galaxies and the Milky Way. For studying the diffuse emission on the sky, Faraday depth cubes are often used, which consist of two-dimensional images as a function of the Faraday depth,
. Previous studies have found interesting emission features on large scales, which are associated with the Milky Way, from Faraday depth cubes (e.g., [
3,
4,
5]). However, the interpretation of the physical depth information from the
is necessary to fully extract the potential of Faraday tomography. For the purpose of localizing the emissions, previous studies have made various attempts: they simply assumed the uniform Faraday rotating medium as the Galactic halo (or thick-disk) (e.g., [
4]), used optical photometric data showing a morphological correspondence (e.g., [
5]), or used the information of H
line which is a good tracer of warm ionized gas (e.g., [
3,
6]).
Another complementary approach to interpret the properties of
can be to use simulation results. In simulations, we have the information of the magnetic field as a function of physical depth and also we can calculate the Faraday spectrum. Once we can know how the galactic properties in the physical depth are reflected in the Faraday depth, it can provide how to extract the information in the physical depth from the Faraday spectrum. The earlier works using simple models found that the magnetic field reversals along the LOS causes singularities in
(Faraday caustics) and that a turbulent magnetic field basically appears as many small-scale components on
(Faraday forests) [
7,
8,
9]. In Ideguchi et al. [
10], they studied
of a small portion of the external face-on galaxies using the realistic Galactic model constructed by Akahori et al. [
11]. They found that the realistic
of galaxies is complicated, but some global galactic properties can affect the shape of
. For instance, the presence of coherent, vertical magnetic field and the larger scale height of thermal electron density result in a wider width of
. As emphasized by Ideguchi et al. [
10], the study using the realistic Galactic model provides the realistic
of galaxies, but therefore, it is not straightforward to interpret the results due to its complexity. Following the studies by Ideguchi et al. [
10], Ideguchi et al. [
12] employ “simple, toy models” for the magnetic field in order for the deeper understanding of the behavior of
. They found that the shape of
will converge to a well defined average by observing multiple LOSs covering a region of ≳(10 coherent length of turbulence)
. They also showed that with these average
, the shape-characterizing parameters such as width, skewness, and kurtosis also converge to consistent values, and the information of the coherent, vertical magnetic field can be extracted.
In this paper, I demonstrate some additional analysis using the Galactic model by Akahori et al. [
11]. I calculate the
in the same manner as Ideguchi et al. [
10]. First, I focus on the effect of the Galactic disk and halo (or thick-disk) structures on
, which is motivated from the expectation in Ideguchi et al. [
10] that the near-side halo along the LOS constructs the peak at
. In addition, I study the presence of coherent, vertical magnetic field by separately looking at each contributions from the disk and halos on
. Then, I try to compare the realistic
with the study in Ideguchi et al. [
12]. Since Ideguchi et al. [
12] considered the only disk to make the situation as simple as possible, I calculate the realistic
by considering only the contribution from disk. Interpretation of
of external galaxies is useful not only to study magnetic fields associated with the galaxies themselves, but also could be useful to probe intergalactic magnetic field embedded between two galaxies as a gap between two components in
space (e.g., [
13,
14]). Though the previous works focused on one-dimensional
, I also construct Faraday depth cubes of the Milky Way using the Galactic model. I study how the presence of the vertical magnetic field and the halo affect the polarization structures in the Faraday depth cubes. Study of
of the Milky Way is important to understand our galaxy, and also the information of diffuse emissions from the Galactic interstellar medium is essential for studying the outside of our galaxy including the epoch of reionization. Below, I briefly explain the Galactic model in
Section 2. Then, I show the results in
Section 3 and the summary and discussion follows in
Section 4.
2. Galactic Model
In this paper, I employ a Galactic model constructed by Akahori et al. [
11] (see also [
10]). Note that any model can be used for this study as long as it contains information of 3-D magnetic field vector and thermal and cosmic-ray electron densities. This model consists of the global, regular component as well as the turbulent, random component. The regular component is modeled using the thermal electron density model of Cordes and Lazio [
15] and magnetic field models including an axisymmetric spiral field and a halo toroidal field [
16] as well as a dipole poloidal field that produces a coherent vertical field near the Earth [
17]. The random component is modeled by MHD turbulence simulations [
18]. They set the integral scale of turbulence as
pc. The computational region of the model is
kpc centered on the location of the Earth,
kpc, where the
plane coincides with the galactic midplane with
y pointing the anti-galactic center and
z penetrating the midplane. This is a model of the solar neighborhood and I study the Faraday spectrum of the Milky Way by using it, but I also use it as an external galaxy. I regard the
z direction as the line of sight (LOS), and this is equivalent to considering the Milky Way towards high galactic latitude or the external face-on galaxies. The region is divided by 32 pixels in
x and
y and by 1280 pixels in
z.
For the calculation of
, I consider two cases:
and
G, where
is the strength of coherent magnetic fields that vertically penetrates the disk. The coherent magnetic field is written as [
17]
where
. For
G case,
is set so that
G around
kpc, while it is absent for
case.
Figure 1 shows examples of the LOS distributions (through
z direction) of physical quantities for the case of
(a) and of
G (b). The top and the second top panels are the thermal electron density,
, in cm
and the parallel component of the magnetic field along the LOS,
, in
G, respectively. The third panels from the top show the Faraday depth,
, in rad m
, which are calculated from
and
using Equation (
2). Note that since these are the cases where the model is treated as an external galaxy,
by design. For the “
” case, since the
is random with the mean of 0, the
changes randomly with respect to the LOS. On the other hand, for the “
G” case,
is a monotonic function of the physical depth since the
is random but always positive along
z-axis (and thus it points away from us). The bottom panels show the distributions of polarization intensity,
, whose peaks are normalized to 1. One can see both the emissions from the disk at |z| ≲ 1 kpc and that from the halo at |z| ≳ 1 kpc. I consider the region of |z| ≲ 1 kpc as the disk and |z| ≳ 1 kpc as the halos for the following calculations.
4. Summary and Discussion
In this paper, following Ideguchi et al. [
10], I tried the further analysis of the realistic Faraday spectrum of galaxies using the realistic Galactic model constructed by Akahori et al. [
11]. First, I assumed observations of external face-on galaxies with two cases:
and
G, where
is the mean strength of coherent, vertical magnetic fields. I separated contributions of the disk and halos structures to the shape of
and looked at how a resultant, realistic
is constructed. It was shown that the near-halo causes a sharp structure at
for the both cases. In both cases, the disk and far-halo are broadly distributed in
space. In the
case this is caused by the turbulent
and
, while for
G case the non-zero
produces a monotonic relationship between physical depth and
. This relationship produces a distinction of halo-disk-halo structures even in
.
In Ideguchi et al. [
12], they showed that the shape of
becomes smooth by observing the region which covers ≥(10 coherent length of turbulence)
by using a simple toy model. Then, they mentioned that, in Ideguchi et al. [
10], the covering region of (500 pc)
with
pc that provides ∼44 LOSs is not enough for
to be converged to a well defined average. In this work, I suspect the structures of halos as a cause of complexity of the realistic
, since the simple model assumes only a disk while the realistic model contains both disk and halos. To check this, I calculated realistic
only from the disk structure (
) for 800 realizations of turbulence, and investigated them using moments of
. It became clear that distributions of the moments are barely changed between the case with and without the halos. Also, distributions of the moments of the realistic
seemed to be more scattered than the case using the simple model with 44 LOSs. These results imply that the complexity of realistic
also comes from the non-uniform distributions of physical quantities (e.g., exponential distribution for cosmic-ray electron density). Though the study using the simple toy model by Ideguchi et al. [
12] assume uniform distributions of physical quantities, for deeper understanding of the realistic
, it is necessary to study the simple model by adding realistic assumptions one by one.
Finally, I calculated the Faraday depth cubes of the Milky Way by using a half of the calculation region (i.e.,
) of the Galactic model. Specifically, I tried to study contributions of the halo to Faraday depth cubes. An interesting result is that for the case of “
G”, morphologies of the emission at large
are produced by the emission from the halo. Hence, if there is non-zero
in the Milky Way, it may be possible to separate emissions of the halo from that of the disk and to study them separately by means of Faraday tomography. The same goes for external face-on galaxies if there is non-zero
and we may be able to study the halo-disk-halo structures. Indeed, it is observationally found that there is a non-zero vertical magnetic field toward the south Galactic pole (e.g., [
19,
20]). This may enable us to make detailed studies of the magnetic field of the disk and halo in the Milky Way by Faraday tomography.
While this paper employed the realistic Galactic ISM/magnetism, several simple assumptions were still made. Let me briefly mention the effects of these simple assumptions. In this work, all LOSs are assumed to start from different locations in the midplane for simplicity, while in real observations, nearer parts of the LOS are closer together. This affects to less emissions in the nearer parts, and thus emissions from the disk is expected to get small. On the other hand, the result that a halo emission appears at large in the G case is barely affected by this assumption. In addition, this work adopted a model where the sign of is constant everywhere, which arises from the fact that coherent component of magnetic fields is stronger than random component. This may not be true in the case where is non-zero, but is still weaker than the random component. When strengths of coherent and random components of magnetic fields are comparable, it is expected that emissions at large mainly comes from halo but also partly from disk, which makes it more difficult to separate disk and halo contributions in . In a case where random component is much stronger than coherent component, the result is thought to be essentially no different from the case, and thus it is basically impossible to distinguish halo and disk structures in . Finally, I discuss an effect of inclination of galaxies. Although this paper only studied face-on galaxies, if a galactic disc is inclined, part of large scale magnetic fields which is parallel to the disc can be a LOS component. This could make complicated, but may also make it possible to address magnetic fields parallel to discs from the shape of . This study will be reported in a forthcoming paper (Eguchi et al. in preparation).
This study investigated the “intrinsic” Faraday spectrum of external galaxies and the Milky Way. However, it is impossible to fully reconstruct the intrinsic Faraday spectrum in practice. This is because of our limited frequency coverage, and the appearance of observational artifacts in data processing (e.g., instrumental polarization). Especially, the reconstruction of the Faraday spectrum highly depends on the observation frequency and its coverage. Some of important parameters introduced in Brentjens and de Bruyn [
2] explain that while frequency coverage at lower frequencies provides higher precision in
space, lower frequency observations also provide less sensitivity to broad structure in
space. For instance, the High Band Antenna of the Low Frequency Array [
21] (120–240 MHz) has ∼1 rad m
resolution and is sensitive for the scale up to ∼1 rad m
in
space, while the full band of the Australian Square Kilometre Array Pathfinder [
22] (700–1800 MHz) has the ∼22 rad m
resolution and is sensitive for the scale up to ∼113 rad m
. As shown in the results of this paper, the Faraday spectrum of galaxies tend to be diffuse in
space with the widths of larger than 10 rad m
as well as the small-scale components due to turbulence. For the reconstruction of such
, it is necessary to use wide-band polarization data from low frequency (∼100 MHz) to higher frequency (a few GHz). The necessary frequency coverage and appropriate Faraday tomography methods for reconstructing such complicated Faraday spectrum will be investigated in the forthcoming studies.