FARADAY DISPERSION FUNCTIONS OF GALAXIES

In this section, we explain Faraday RM synthesis as first proposed by Burn (1966) and extended by Brentjens & de Bruyn (2005). The complex polarized intensity P(λ²) as a function of wavelength λ is expressed as

$$\begin{eqnarray} P(\lambda ^2) &=& p(\lambda ^2)I(\lambda ^2) \nonumber \\ &=& Q(\lambda ^2)+iU(\lambda ^2) \nonumber \\ &=& \int _{-\infty }^{+\infty }F(\phi)e^{2i\phi \lambda ^2} d\phi, \end{eqnarray} \tag{ 1 }$$

where p is the complex fractional polarization, and I, Q, and U are the Stokes parameters. F(ϕ) is the FDF, which is a complex polarized intensity as a function of Faraday depth ϕ. The Faraday depth (or the RM when the FDF consists of a single delta function) is defined as

$$\begin{equation} \phi (x) = 810 \int _x^0 \left(\frac{n_{\rm e} (x^{\prime })}{1\; {\rm cm}^{-3}} \right) \left(\frac{B_\parallel (x^{\prime })}{1\; \mu {\rm G}} \right) \left(\frac{dx^{\prime }}{1\; {\rm kpc}} \right) {\rm rad\,m^{-2}}, \end{equation} \tag{ 2 }$$

where n_e is the electron density, B_∥ is the LOS component of magnetic fields, x' is the physical distance, and x is the distance to a source from an observer at x' = 0.

The FDF can be reconstructed by the inverted equation of Equation (1):

$$\begin{equation} F(\phi) =\int _{-\infty }^{+\infty }P(\lambda ^2)e^{-2i\phi \lambda ^2}d\lambda ^2. \end{equation} \tag{ 3 }$$

This transform is called Faraday RM synthesis, which gives the FDF from observed P(λ²).

It should be noted that there is, in general, no one-to-one correspondence between ϕ and x. The FDF thus does not straightforwardly give the distribution of polarized intensity in a physical space. Nevertheless, the distribution of magnetic fields and thermal and cosmic-ray electrons are reflected in the FDF and it is possible to extract information on distribution (Beck et al. 2012).

In order to study FDFs of galaxies, we employ a model of the Milky Way constructed by Akahori et al. (2013). Actually, this is a model of the solar neighbor, but we use it as a representative model because physical quantities have been better determined there. We expect that the model is reasonable also for external spiral galaxies. Below, we briefly summarize the model.

The model considers the domain toward high Galactic latitudes of the Milky Way. It 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 & Lazio (2002) and magnetic field models including an axisymmetric spiral field and a halo toroidal field (Sun et al. 2008) plus a dipole poloidal field that produces a coherent vertical field near the Earth (Giacinti et al. 2010). The random component is modeled by piling up computational boxes of MHD turbulence simulations (Kim et al. 1999) from the Galactic midplane up to ±10.0 kpc, where each box size is 500 pc with 512 grids on a side. They adopted a driving scale of 250 pc and a rms flow speed of 30 km s⁻¹ based on Hα observations (Hill et al. 2008). Then, Faraday depth is calculated with the above models.

Over the frequencies we are considering (∼100 MHz–10 GHz), polarized emissions of galaxies mostly come from synchrotron radiation of cosmic-ray electrons, and other radiation mechanisms can be safely neglected. This means that we need information of a cosmic-ray electron density in addition to magnetic fields derived from the above model. For this, we utilize an exponential model based on observations (Sun et al. 2008). With the cosmic-ray electron density and the amplitude and direction of magnetic fields, synchrotron emissivities and specific Stokes parameters are calculated following Equations (3)–(5) of Waelkens et al. (2009). In this paper, we calculate the Stokes parameters at 1 GHz.

We assume observations of a face-on spiral galaxy and use Cartesian coordinates (x, y, z), where the x−y plane coincides with the galactic midplane with y pointing the anti-galactic center and z penetrating the midplane. The LOS is set to be parallel to the z-axis, and the field-of-view center directs to (x, y) = (0, 8.5) in kiloparsecs (the location of the Earth in the Milky Way). We only consider the computational region of −0.25 < x < 0.25, 8.25 < y < 8.75, and −10 < z < 10 all in kiloparsecs. The simulated 500 pc × 500 pc region corresponds to ∼10'' × 10'' if we put the galaxy at a distance of 10 Mpc from the Earth (e.g., at Virgo cluster).

We divide the computational region by 32 grids each in x and y, and by 1280 grids in z. For each grid, we calculate the Faraday depth using Equation (2), and calculate the polarized intensity described in Section 2.2. We integrate the intensity along the LOS and then we obtain the FDF by sorting the intensity as a function of Faraday depth. We also obtain Faraday depth as a function of z. The Faraday depth for a grid cell is calculated at the middle of the cell, and the polarized intensity of a cell is an average within the cell. We set the resolution in Faraday depth for the calculation of FDFs to 0.1 rad m⁻², the resolution of which is sufficiently high to follow the variation of the FDF, so that the results do not change significantly with a higher resolution in Faraday depth.

In Akahori et al. (2013), the model parameters that determine global configurations of the galaxy such as the scale heights of thermal and cosmic-ray electron densities and the mean strength of coherent vertical magnetic fields were chosen based on observations of the Milky Way. In this work, we can vary the values of such global parameters because our purpose is not to reproduce observed galaxies, but to demonstrate what galaxies look like in terms of FDF. The above set of parameters are interesting for studying the intrinsic FDFs since they are all fundamental parameters of galaxies, and are major ones that mainly characterize the FDF for face-on galaxies. Even some extreme cases would be helpful for understanding the nature of the FDF. We take the "ADO model" in Akahori et al. (2013) as our fiducial model and consider three variants summarized in Table 1. Considerations for some other parameters will be commented on in Section 4.

The first parameter we vary is the mean strength of the vertical magnetic field. Here, the random vertical magnetic field always exists in the model and this component additionally incorporates the global, regular magnetic field along the z-axis. The component is modeled with the Galactic center dipole field, where the field is penetrating the galactic plane, and is written as (Giacinti et al. 2010)

$$\begin{equation} B_{{\rm p},z}(z) =\mu _{\rm p}(1-3(z/r)^2)/r^3, \end{equation} \tag{ 4 }$$

where r = (x² + y² + z²)^1/2. For Model 2, we set μ_p to a value such that the strength B_{p, z}(z) ∼ 1.0 μG around (x, y) = (0, 8.5) in kiloparsecs while it is absent for other models. The second parameter is the scale height of cosmic-ray electron density. The cosmic-ray electron density is written as (Sun et al. 2008)

$$\begin{equation} C(z) \propto C_0 \exp (-|z|/h_{\rm c}), \end{equation} \tag{ 5 }$$

where C₀ = 0.0001 cm⁻³ and the scale height h_c = 1 kpc for Models 1, 2, and 4 and 3 kpc for Model 3. The third parameter is the scale height of the thermal electron density of the thick disk component, which is given by (Cordes & Lazio 2002)

$$\begin{equation} n_1(z) \propto n_1 {\rm sech^2}(z/h_1), \end{equation} \tag{ 6 }$$

where n₁ = 0.02 cm⁻³ and the scale height h₁ = 1 kpc for Models 1–3 and 3 kpc for Model 4.

In this section, we show some examples of FDFs of the galaxies of Model 1. First, we calculate the FDFs of a single grid in the x − y plane (16 pc × 16 pc) which corresponds to a beam size of 03 × 03 at 10 Mpc.

Figure 1 shows the result for a relatively simple case where the Faraday depth is virtually a monotonic function of physical distance. The top left panel shows the distributions of thermal electron density and the LOS (z-axis) component of magnetic fields in the physical space along a LOS. As explained in the previous section, the density is concentrated near the galactic midplane within the scale height (|z| ≲ 1 kpc), while magnetic fields extend further to outer regions due to the halo magnetic fields. Both density and magnetic field distributions have violent fluctuations caused by turbulence.

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The resultant Faraday depth is shown as the blue dotted line in the bottom left panel of Figure 1. In this case, Faraday depth almost monotonically decreases with the physical distance because magnetic fields are mostly positive. The panel also shows the polarized intensity distribution, which shows that the intensity is peaked at the midplane with tails in the halo. We find that variations of the Faraday depth in the halo (|z| > 1 kpc) is small (<1 rad m⁻²), even though the sub-μG magnetic field extends to the scale of several kiloparsecs. This indicates the feature of Faraday caustics that polarized emissions in the halo tend to contribute to the FDF of the same value of a Faraday depth.

The right panels show the polarization angle (top) and the FDF (bottom). We can see that the absolute value of the FDF has a main peak at ϕ = −5 rad m⁻² and sub-peaks at ϕ = −13 and −1 rad m⁻² with different polarization angles to each other. The main peak and two sub-peaks can be attributed to the emissions in the midplane at z ∼ 0, the halo with z > 0, and the halo with z < 0, respectively, as implied by the relation between ϕ and z (bottom left panel). As expected, although polarized intensities are small in the halo, we can see them in the FDF as large peaks because they are accumulated to relatively large values at ϕ = −13 and −1 rad m⁻².

Since the sign of the LOS component of magnetic fields can change, in general, Faraday depth does not vary monotonically with physical distance. Figure 2 represents a less simple case. Again, the bottom left panel shows that the polarized intensity is peaked at the midplane with tails in the halo, but the behavior of the Faraday depth is no longer monotonic with respect to physical distance.

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As is shown in the bottom right panel of Figure 2, there are three peaks at ϕ = −1, 0, and 7 rad m⁻² in the FDF, similar to the previous case in Figure 1. However, the main peak at ϕ = 0 rad m⁻² is attributed to the emissions in the halo with z < 0 (see the bottom left panel), rather than the midplane at z ∼ 0. The emissions in the midplane and in the halo with z > 0 appear in the FDF at ϕ = −1 and 7 rad m⁻², respectively. Therefore, the order of the peak locations is inverted in the FDF compared with that in physical space, because of the non-monotonicity of the Faraday depth. Moreover, the peak amplitudes of the FDF do not straightforwardly suggest the origin of the emissions (midplane or halo) due to their degeneracies in a certain Faraday depth.

Figure 3 shows a case with a Faraday depth that is far from monotonic. It is seen that the FDF is limited to a region with small values of |ϕ| compared with the previous two cases. This is because the Faraday depth behaves like a random-walk and does not reach large values. The relation between ϕ and z is thus far from a one-to-one correspondence. The behavior of the Faraday depth is so complicated that it is difficult to find the correspondence between the peaks in the FDF and those in physical space. This is the generic situation and simple behaviors of the two cases in Figures 1 and 2 are rather exceptional. Thus, as we saw above, the FDF looks very different even for a fixed set of global parameters and is highly dependent on the different realizations of turbulent components that determine the behavior of Faraday depth.

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We also calculate FDFs of the entire region in the x − y plane (500 pc × 500 pc) which corresponds to a beam size of 10'' × 10''. Figure 4 shows the FDF integrated over this region as well as those of the four quadrants (250 pc × 250 pc). We see that the integrated FDF is much broader and smoother compared with those in Figures 1–3. Such broadening and smoothing is ascribed to the fact that each quadrant contributes to a different part of the total FDF, shown as colored lines. The result demonstrates that an observation with a larger beam size produces a broader and smoother FDF, in which various FDFs having peaks at different Faraday depths are mixed together.

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Note that the absolute value of the total FDF is not, in general, equal to the sum of the absolute values of FDFs for the quadrants because the polarization angles are different from each other. This is effect is the so-called beam depolarization, though it is not so significant at the scales shown because coherent magnetic fields dominate over the turbulent magnetic fields.

If the FDF of galaxies can be described with simple functions, we can quantify the FDF easily. However, as shown in the previous subsection, FDFs of galaxies are very complicated in general and it is very difficult to translate them into the distribution of physical quantities in physical space. Nevertheless, it would be possible to extract some global properties of galaxies, such as the average amplitude and the coherence length of magnetic fields and the scale heights of thermal and cosmic-ray electrons. This is because such properties could affect the FDF through the behavior of the Faraday depth as a function of physical distance. Below, we consider statistical properties of FDFs of the whole region for the four models in Table 1, mainly for the purpose of interpreting the global shape of the FDFs.

We carried out 800 runs with different realizations of turbulence for each model. To do this, we randomly shift and rotate computational boxes of MHD turbulence simulations when we pile them up. Figure 5 shows 19 examples of FDFs for Model 1. We find that the shape of FDF varies significantly for different configurations of turbulence. For example, some FDFs have a single thin peak and others have multiple peaks. Such variations can be also seen for Models 2–4 (Figures 6–8). Therefore, we see no universal shape for the FDF, even if we fix the parameters characterizing the global properties mentioned above.

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Apparently, Models 2 (Figure 6) and 4 (Figure 8) have broader FDFs and this feature may allow us to probe some global properties of galaxies. Since the reconstructed FDF tends to show a peaked profile with tails, to quantify the features of the global shape of FDFs, we calculate the standard deviation σ, skewness γ_s, and kurtosis γ_k of FDFs, defined as

$$\begin{eqnarray} \sigma ^2 &=& \frac{\sum _i |F(\phi _i)| (\phi _i-\mu)^2}{\sum _i |F(\phi _i)|} \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} \gamma _{\rm s} &=& \frac{\sum _i |F(\phi _i)| (\phi _i-\mu)^3}{\sigma ^3 \sum _i |F(\phi _i)|} \end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray} \gamma _{\rm k} &=& \frac{\sum _i |F(\phi _i)| (\phi _i-\mu)^4}{\sigma ^4 \sum _i |F(\phi _i)|}, \end{eqnarray} \tag{ 9 }$$

where ϕ_i is the Faraday depth of the i-th bin and

$$\begin{equation} \mu = \frac{\sum _i |F(\phi _i)| \phi _i}{\sum _i |F(\phi _i)|} \end{equation} \tag{ 10 }$$

is the average Faraday depth. We note that we take the definition for the kurtosis so that the Gaussian function equals 3. We calculate the probability distribution functions of these quantities, using 800 FDFs with different configuration of turbulence, for each model.

The probability distribution functions and the 68% confidence intervals for the standard deviation, skewness, and kurtosis are shown in Figures 9–11, respectively. The most obvious difference can be seen in the probability distribution of the standard deviation, where Models 2 and 4 have larger values. This can be ascribed to the fact that the Faraday depth tends to reach large absolute values, although the physical reason is different between Model 2 and Model 4. For Model 2, it is due to the presence of the coherent vertical magnetic field which pushes up the Faraday depth to large values. In contrast, in Model 4, the Faraday depth behaves as random walk but the extended thermal electrons increase the expected value of the variance of the Faraday depth.

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On the other hand, although 68% confidence intervalsmostly overlap with each other in the probability distribution of skewness and kurtosis, for Model 2, these quantities are limited to relatively narrow ranges. The skewness for Model 2 tends to be negative and this is due to the orientation of coherent vertical magnetic fields.