Anisotropic Structure of Synchrotron Polarization

We first perform MHD turbulence simulation to obtain data for calculation of polarization. We use a code based on a third-order accurate hybrid essentially nonoscillatory (ENO) scheme in a periodic box of size 2π  with a resolution of 512³ grid points and solve the following equations:

$$\begin{eqnarray*}&&\begin{array}{l}\partial \rho /\partial t+\unicode{x025BF}\cdot (\rho {\boldsymbol{v}})=0,\\ \partial {\boldsymbol{v}}/\partial t+{\boldsymbol{v}}\cdot \unicode{x025BF}{\boldsymbol{v}}+{\rho }^{-1}\unicode{x025BF}({a}^{2}\rho )\\ \quad -\,(\unicode{x025BF}\times {\boldsymbol{B}})\times {\boldsymbol{B}}/4\pi \rho ={\boldsymbol{f}},\\ \partial {\boldsymbol{B}}/\partial t-\unicode{x025BF}\times ({\boldsymbol{v}}\times {\boldsymbol{B}})=0,\end{array}\end{eqnarray*}$$

with $\unicode{x025BF}\cdot {\boldsymbol{B}}=0$, where ρ is density, a is the sound speed, f is a random driving vector. The r.m.s. velocity (v_rms) is maintained to be approximately unity, so that v can be viewed as the velocity measured in units of the v_rms of the system. The magnetic field (B) consists of the mean field and a fluctuating field. The Alfvén speed (V_A) of the mean field is set to unity. We assume the direction of the mean magnetic field is parallel to the LOS in Section 4.1 and perpendicular to the LOS in other sections. Turbulence is driven solenoidally in Fourier space. The spectrum of magnetic field fluctuations follows a power law (Cho & Lazarian 2010). The resulting turbulence is characterized by the following parameters: the Alfvén mach number, ${M}_{{\rm{A}}}(={v}_{\mathrm{rms}}/{V}_{{\rm{A}}})\sim 0.7$, and the sonic Mach number, ${M}_{s}(={v}_{\mathrm{rms}}/a)\sim 0.7$. We assume that the thermal electron number density is proportional to ρ and use magnetic field directly from the simulation data for calculation of synchrotron polarization.

Using MHD turbulence data, we calculate synchrotron polarization described by the combination of the Stokes parameters Q and U:

$$\begin{eqnarray}&&P\equiv Q+{iU}\end{eqnarray} \tag{ 1 }$$

as we focus only on linear polarization in this paper.

We can write PI observed at a two-dimensional position X (=(x, y)) by using intrinsic PI density (P_j) arising from the source of size L along the LOS at wavelength λ as follows:

$$\begin{eqnarray}&&P\left({\boldsymbol{X}},{\lambda }^{2}\right)={\int }_{0}^{L}{{dzP}}_{j}({\boldsymbol{X}},z){e}^{2i{\lambda }^{2}{\rm{\Phi }}({\boldsymbol{X}},z)},\end{eqnarray} \tag{ 2 }$$

where the exponential factor describes Faraday rotation from the source at z along the LOS to the observer. The rotation of polarization angle due to Faraday rotation is proportional to ${\lambda }^{2}{\rm{\Phi }}({\boldsymbol{X}},z)$ and Φ(X, z) is Faraday rotation measure:

$$\begin{eqnarray}&&{\rm{\Phi }}({\boldsymbol{X}},z)\sim {\int }_{0}^{z}{n}_{e}(z){B}_{z}(z){dz},\end{eqnarray} \tag{ 3 }$$

where n_e is the number density of electrons, and B_z is the strength of the magnetic field parallel to the LOS.

Using synthetic data, Lee et al. (2016) found that the spectrum of PI changes as wavelength increases due to the effects of Faraday rotation, which is proportional to λ². In this paper, we use real MHD turbulence data and obtain the power spectrum of PI arising from fluctuations of synchrotron radiation and Faraday rotation at different wavelengths. Figure 2 shows the results for two different cases: the direction of mean magnetic field is either perpendicular (left panel of Figure 2) or parallel (right panel of Figure 2) to the LOS.

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In the left panel of Figure 2, we can see that, when the depolarization is insignificant, the spectrum goes up as the wavelength increases (λ = 0.02–1.4). This behavior of the power spectrum can be explained as follows. Since we have a strong mean magnetic field in the direction perpendicular to the LOS, the intrinsic synchrotron emission is more or less uniformly polarized. If we include Faraday rotation, it makes the polarization pattern deviate from the intrinsic state, which produces more fluctuations in polarization. Therefore, the spectrum goes up as wavelength increases. When the wavelength further increases, the depolarization effect affects large scales (i.e., at small-K's) more predominantly. As a result, the small-K part of the spectrum goes down, while the large-K part of the spectrum continues to move up, which is not affected by depolarization. Increase of the large-K part of spectrum implies that the Faraday rotation effect is more dominant than fluctuation of synchrotron polarization itself.

For the case of strong mean magnetic field along the LOS, the behavior of spectrum with different wavelengths is shown in the right panel of Figure 2. The spectrum at small-K goes down as wavelength increases unlike the spectrum for the mean field perpendicular to the LOS. Due to strong mean field along the LOS, which induces large Faraday rotation, polarization will be randomized at large scales (small-Ks) even at small wavelengths. However, we can still see the rise of spectrum at large-K (i.e., more fluctuations at small scales) for large wavelengths.

In the presence of a strong mean magnetic field, turbulence structures become elongated along the mean field direction. In this section, we focus on how to describe this anisotropic structure. The structure function can be used to study statistics of synchrotron polarization in addition to the spectrum. In particular, the structure function for PI is useful to study the degree of anisotropy from a map. To visualize anisotropic structure, we first utilize the two-dimensional second-order structure function of PI

$$\begin{eqnarray}&&{\mathrm{SF}}_{2}\left({r}_{\parallel },{r}_{\perp }\right)=\left\langle | \mathrm{PI}(x+{r}_{\parallel },y+{r}_{\perp })-\mathrm{PI}(x,y){| }^{2}\right\rangle ,\end{eqnarray} \tag{ 5 }$$

where r_∥ is distance along the mean magnetic field and r_⊥ is distance perpendicular to it. Since the mean magnetic field is along the x-direction, r_∥ and r_⊥ are increments in x- and y-directions, respectively.

We plot the structure functions for PI on the r_∥ and r_⊥ plane in Figure 3. The wavelengths for the panels are 0.02, 1.4, and 3.2 (left, middle, and right panels in Figure 3, respectively). In the left panel of Figure 3, we can clearly see that the structure is elongated along the mean field direction both on small scales and large scales. When Faraday depolarization effect begins to contribute, the structures on large scales start to lose their correlation (middle panel of Figure 3). As wavelength further increases, Faraday depolarization affects even smaller scales (right panel of Figure 3).

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We provide two types of complementary statistics to describe quantitatively the observed degree of anisotropy for PI. First, we plot the relation between ${\mathrm{SF}}_{2}({r}_{\parallel },0)$ and ${\mathrm{SF}}_{2}(0,{r}_{\perp })$ (left panel of Figure 4). The left panel of Figure 4 shows that all five curves representing five different wavelengths are below the red straight line, which denotes isotropic structures. The fact that all curves are below the red straight line implies that the structures are elongated along the mean field direction.⁵

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Second, we also calculate another statistic called QM, which is defined by

$$\begin{eqnarray}&&\mathrm{QM}(r)=\displaystyle \frac{2{\int }_{0}^{\pi }{e}^{-2i\phi }{\mathrm{SF}}_{2}(r,\phi )d\phi }{{\int }_{0}^{2\pi }{\mathrm{SF}}_{2}(r,\phi )d\phi },\end{eqnarray} \tag{ 6 }$$

where ϕ is the polar angle subtended by the horizontal axis and a line connecting the points on the plane of the sky. The QM is zero for isotropic structures and the absolute value of it gets larger as anisotropy increases. If anisotropic structure is elongated along the direction of the horizontal axis (here, x-axis), the moment has negative values. For the opposite case, it has positive values.

The right panel of Figure 4 is the resulting QM. From the panel, we find that the moment remains unchanged when the wavelength is so small that the depolarization effect is negligible. As wavelength increases, Faraday depolarization becomes effective on large scales, and the absolute value of the moment goes down and approaches zero on large scales first. The absolute value of moment continues to decrease and approaches zero on smaller scales at longer wavelengths. Even at very long wavelengths, we can see anisotropy at small scales.

In the left panel of Figure 6, we plot the second-order structure function of observed polarized synchrotron intensity. In the figure, we also plot those from the foreground region (middle panel of Figure 6) and the background region (right panel of Figure 6) only. The structure functions in Figure 6 are plotted for a short wavelength in order to ignore the depolarization effect. The structures in regions 1 and 2 are elongated with respect to their own mean field directions. The observed structures in the left panel are isotropic because the observed emission is a mixture of the background emission, whose structure is elongated along the x direction, and the foreground emission, whose structure is elongated along the y direction.

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We also visualize how the observed anisotropy of PI changes with wavelengths in Figure 7. At short wavelengths, we observe isotropic combined radiation emitted from both regions (left plot of Figure 7; the same as the left panel of Figure 6). However, when we observe it at longer wavelengths, we obtain anisotropic structures reflecting more dominantly those of the foreground region. This is because polarization arising from the background is negligible at longer wavelengths due to Faraday depolarization and, therefore, radiation from the region close to the observer contributes more to polarization structures. The middle and right panels of Figure 7 show structure functions at long wavelengths (λ = 1.4 and 3.2). As the wavelength increases from 1.4 to 3.2, the correlation on large scales gets gradually weaker and anisotropy is observed on progressively smaller scales. In both panels, the anisotropy on small scales is along the mean field direction of the foreground region. The very weak correlation on large scales at λ = 3.2 is due to enhanced Faraday rotation.

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As wavelength increases, observed anisotropy begins to reflect that of the foreground region. As a result, the behavior of the moment more or less resembles that of the single emitting region (see the middle panel of Figure 8). That is, the moment is a decreasing function of separation r and it is close to zero when r is larger than a certain value. As wavelength increases, the moment declines faster.

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From left to right in Figure 8, we plot the QM of PI from combined regions, the foreground region only, and the background region only. Note that, for the case of the background region only, the foreground region also contributes to Faraday rotation of the background emission. In the left panel of Figure 8, when the wavelength is very short (λ = 0.02; black solid curve), the intrinsic emissions are not affected by Faraday rotation and their structures are nearly isotropic due to the averaging effect at all scales. Therefore, the moment is almost zero on all scales. For λ = 0.56 (blue dotted line), emissions from both foreground and background regions are depolarized and become isotropic at the largest scales. However, only emission from the background region is depolarized in intermediate scales near R = 50 (see the middle and right panels of Figure 8). Therefore, the QM is positive due to anisotropic structure in the foreground region. At very small scales, emissions from both foreground and background regions are not strongly affected by Faraday depolarization and, hence, combined polarized emissions from foreground and background regions become isotropic due to the average effect. The resulting QM is nearly zero at very small scales (see the cyan curve in Figure 8). At a longer wavelength (e.g., λ ∼ 1.4), the emission from the background region is depolarized at all scales (see the right panel of Figure 8) and emission from the foreground region is depolarized at large scales (e.g., R > 50 for λ ∼ 1.4). As a result, the QM has large positive values at small scales and has nearly zero value at large scales (see the green curve in the left panel of Figure 8). In other words, we can see the anisotropy of polarized emission from the foreground region at small scales if we observe at a long wavelength.

A number of observations imply such a situation that synchrotron radiation mainly originates in the Galactic halo while the effect of Faraday rotation is stronger in the Galactic disk. The latter is supported by the fact that the r.m.s. fluctuation of Galactic rotation measure peaks strongly at the Galactic midplane (Taylor et al. 2009). Therefore, in this subsection, we consider three cases:

• 1.  
  Case 1. Both foreground and background produce similar amounts of Faraday rotation, but the background has stronger synchrotron emissivity.
• 2.  
  Case 2. Both regions have similar synchrotron emissivities, but the foreground produces stronger Faraday rotation.
• 3.  
  Case 3. The background has stronger synchrotron emissivity and the foreground produces stronger Faraday rotation.

Our main concern is Case 3. However, we also consider Case 1 and Case 2 for the sake of completeness.

Let us consider Case 3 first, in which we assume that the emission from the background region is five times stronger than that from the foreground region and the RM in the foreground region is five times larger than that in the background region. With these assumptions, we calculate structure function and its QM. Figure 9 is the structure function of the observed PI. At short wavelengths, Faraday rotation is negligible and anisotropic structure samples that of the stronger emission, which is the background emission in our case (left panel of Figure 9). As wavelength increases (λ ≳ 1.4), emission from the background goes through more Faraday depolarization than that from the foreground and therefore the observed structure is similar to that of the foreground. At very long wavelengths (λ ∼ 3.2), we can see that most structures are uncorrelated or weakly correlated and only small-scale structures maintain its anisotropy (Figure 9(c)). In fact, anisotropic structures are correlated when $l\lesssim \tfrac{2\pi }{{\lambda }^{2}\left\langle {n}_{e}\left|{B}_{| | }\right|\right\rangle }$, where l is the size of the foreground and ${B}_{| | }$ is magnetic field parallel to the LOS.⁶ We can obtain information on the magnetic field direction of foreground region from observed structure function below the scale. Note that the anisotropic structure on large scales $\left(l\gt \tfrac{2\pi }{{\lambda }^{2}\left\langle {n}_{e}\left|{B}_{| | }\right|\right\rangle }\right)$ in Figure 9(c) could be a numerical artifact. We can see artificial anisotropic structure on large scales caused by discrete Faraday sources along the LOS when we observe polarization structure at very long wavelengths (see A.3 of Lazarian & Yuen 2018b).

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To quantify anisotropic structures of polarized synchrotron emission from two regions, we calculate QM for three different cases mentioned above. We plot the QM for Case 1 in the left panel of Figure 10. At short wavelengths (λ < 1.4: black and blue lines), since Faraday rotation has no impact on the structure of polarized radiation, we see negative QM reflecting anisotropy of the background region where synchrotron emission is stronger. However, at a long wavelength (λ = 1.4: green dashed line), emission from the background is depolarized and the observed structure is indicative of QM in the foreground, which is positive. When wavelength is very long ((λ = 3.2: red dotted–dashed line), we see anisotropy (i.e., positive QM) only on very small scales. We also plot QM for Case 2 in the middle panel of Figure 10. At short wavelengths, QM is nearly zero. This is because, since both the foreground and background have similar synchrotron emissivities, the combined emission from them is virtually isotropic. As wavelength increases (λ ≳ 1.4), strong rotation measure of the foreground region attributes to fast depolarization of background emission and hence the observed QM reflects more predominantly that of the foreground. Therefore, compared with those for Case 1, the QMs for Case 2 have larger positive values. The right panel of Figure 10 shows the QM for Case 3. At short wavelengths, the QM mainly reveals that of the background region due to strong synchrotron emission from the background. At longer wavelengths (λ ≳ 1.4), QM has high positive values, representing elongated structure along the y-direction, within the scale, $l\sim \tfrac{2\pi }{{\lambda }^{2}\left\langle {n}_{e}\left|{B}_{| | }\right|\right\rangle }$. We also see negative values because of artificial anisotropic structure elongated along the x-direction at very long wavelengths (λ ∼ 3.2).

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As we mentioned earlier, the measurement of polarized synchrotron radiation is not a reliable way to probe the plane of the sky magnetic field structure from a volume that both emits synchrotron radiation and is subject to appreciable Faraday rotation. Using high frequencies at which Faraday rotation is negligible, one can obtain the 2D structure of magnetic field projected on the plane of the sky. The same information can also be obtained with Synchrotron Intensity Gradients (SIGs; Lazarian et al. 2017) for any frequency of measurements as the intensities are not subject to Faraday rotation.

Faraday tomography based on the approach proposed by Burn (1966) allows us to get insight into 3D distribution of the LOS component of magnetic field. The technique has its own problems for studying turbulent magnetic fields, however (see Lazarian & Yuen 2018b). Nevertheless, it can be considered as complimentary to what we suggested in this paper.

The closest in spirit technique is the SPG technique proposed in Lazarian & Yuen (2018b). The relation between our approach and that of the SPG is similar to the relation between the Correlation Function Anisotropy technique (Lazarian et al. 2002; Esquivel & Lazarian 2005, 2011) and the velocity gradients (see Yuen & Lazarian 2017; Lazarian & Yuen 2018a). The latter two techniques were compared in Yuen et al. (2018). The advantage of gradients is that they provide a more detailed structure of magnetic field. The advantage of studying anisotropy of correlation functions is that a more detailed analysis of the contribution of compressible and Alfvénic modes is possible. For the case of synchrotron emission this is possible due to the analytical description of the synchrotron statistics in Lazarian & Pogosyan (2012, 2016). We plan to perform the corresponding study elsewhere.

In this paper, we described observed anisotropic structure of polarized synchrotron intensity using second-order structure function and QM. In particular, second-order structure function visualizes anisotropic structure of PI and the QM quantifies the degree of its anisotropic structure. It is evident that the former can trace the direction of the mean magnetic field from the elongation of its contour map if there is no Faraday rotation.

The structure function can be used to identify the direction of the mean magnetic field even in the presence of Faraday rotation. Figure 1 shows how the polarization map changes as wavelength increases. At short wavelengths where the Faraday rotation effect is negligible, its structure is elongated along the horizontal direction, which is the direction of the mean magnetic field. The direction of polarization rotated by 90 degrees are also aligned with the direction of the mean magnetic field (left panel of Figure 1). At longer wavelengths where Faraday rotation is significant, depolarization makes polarization directions randomized, which results in large-scale structures more or less isotropic. Note, however, that the structures at small scales are still elongated along the mean field direction (middle and right panels of Figure 1). Therefore, we can reveal correct anisotropy, and the direction of the mean magnetic field, at small scales if we use the second-order structure function. In summary, Faraday rotation depending on observation wavelength affects large-scale structures first and we see uncorrelated large-scale structures with disordered direction of polarization at long wavelengths. Even though large-scale structures fail to present ordered shapes, the second-order structure function of the PI still indicates the direction of mean magnetic field at small scales when observation is performed at a long wavelength where Faraday rotation effect is too important to be ignored.

Comparison of QM with power spectrum enables us to help clarify the process of depolarization by Faraday rotation as wavelength increases. As we can see in the left panel of Figure 2, the spectrum goes up without changing its shape much as wavelength increases when wavelength is small (see black and blue curves). During this process, QM does not exhibit notable change (see black and blue curves in the right panel of Figure 4). However, as wavelength further increases, the spectrum and the quadrupole behave differently: the spectrum continuously goes up, but the QM deviates from those for small wavelengths (see green curves for λ ∼ 0.9). This result may imply that Faraday depolarization at its early stage does affect anisotropic structures, although it does not influence the magnitude of PI much. When λ ∼ 1.4, large-scale power becomes suppressed by Faraday depolarization and, as a result, the spectrum at small-K's stops increasing, while that at large-K's continues to go up (see the curve for λ ∼ 1.4). A detailed explanation for this spectral behavior can be found in Lee et al. (2016). By comparison, as depolarization begins, the QM decreases and goes to zero at large scales first (see the curve for λ ∼ 1.4 in the right panel of Figure 4), which means that depolarization destroys anisotropic structures.

The structure of polarization by synchrotron emission and Faraday rotation can be used to obtain properties of magnetic field in Galactic disk and halo. Observations of Faraday rotation measure have been carried out and found the dependence of rotation measure on the Galactic latitude: the RMs close to the Galactic plane are much larger than RMs at high Galactic latitudes (National Radio Astronomy Observatory VLS Sky Survey; Taylor et al. 2009). In fact, the amplitude of large-scale RM structures near the Galactic plane is ∼100 $\mathrm{rad}\,{{\rm{m}}}^{-2}$ and its amplitude for high latitudes is ∼20 $\mathrm{rad}\,{{\rm{m}}}^{-2}$ (Taylor et al. 2009; Schnitzeler 2010). If we adopt the free electron distribution model of Gaensler et al. (2008), the RM arising from the Galactic disk dominates that from the halo when Galactic latitude is roughly less than 30° (Taylor et al. 2009). However, when it comes to synchrotron emission, it is likely that emission from the Galactic halo is stronger. To model this, we have assumed the rotation measure in the Galactic disk is five times larger than that in the Galactic halo and synchrotron radiation and Faraday rotation occupy separate volumes in Section 5.2. Here, we assume that we are observing toward a low Galactic latitude (e.g., b ∼ 10°). In this case, the RM arising from the Galactic disk for the LOS is around 70 $\mathrm{rad}\,{{\rm{m}}}^{-2}$, if we assume n_e = 0.01 cm⁻³ (Gaensler et al. 2008 see also Cordes & Lazio 2004; Nota & Katgert 2010), ${b}_{| | }\sim 1.23\,\mu {\rm{G}}$, and L ∼ 7 kpc, where L is the path length. In the simulation, the RM of the foreground is ∼7.5 in code units. In our simulation, anisotropy of polarization map samples the direction of mean magnetic field in the background at λ ≲ 0.6, and it reflects the direction of mean magnetic field in the foreground at λ ≳ 1.4 in code units. If we convert λ = 1.4 in code units to real units, it corresponds to λ ∼ 45 cm. Therefore, we expect to study the direction of mean magnetic field in the background (i.e., Galactic halo) when λ ≲ 20 cm. It is likely that anisotropy reflects the direction of mean magnetic field in the foreground (i.e., Galactic disk) if λ ≳ 45 cm.

In this paper, we have assumed that a linearly polarized electromagnetic wave in a plasma can be decomposed into two independent eigenmodes, left and right circularly polarized waves. As they propagate independently (or, adiabatically) in a magnetized plasma, they produce Faraday rotation as given in Equation (3).

However, Broderick & Blandford (2010) showed that, at sufficiently low frequencies, "the character of Faraday rotation changes, entering what they term the super-adiabatic regime in which the rotation measure (RM) is proportional to the integrated absolute value of the LOS component of the field." That is, at very low frequencies, the left and the right circularly polarized waves become no longer independent and, hence, we cannot use Equation (3) anymore.

The critical frequency for the super-adiabatic regime is

$$\begin{eqnarray}&&{\nu }_{{SA}}=87{\left(\displaystyle \frac{n}{1{\mathrm{cm}}^{3}}\right)}^{1/4}{\left(\displaystyle \frac{B}{1{\rm{G}}}\right)}^{3/4}{\left(\displaystyle \frac{{l}_{B}}{{10}^{15}\mathrm{cm}}\right)}^{1/4}\mathrm{MHz},\end{eqnarray} \tag{ 7 }$$

where l_B is the local magnetic field reversal length scale and ν_SA typically ranges from 10 kHz to 10 GHz (Broderick & Blandford 2010). For Galactic halo, we expect that the critical frequency is ν_SA ∼ 15 kHz, which corresponds to λ_SA ∼ 2 × 10⁶ cm. Here, we use n ∼ 0.003 cm⁻³, B ∼ 1 μG, and l_B = 100 pc, which could be the outer scale of turbulence in the Galactic halo. This critical wavelength is much longer than the wavelengths discussed in the previous subsection. Therefore, it may not be necessary to consider the effects of nonadiabatic propagation for Faraday rotation in the Galactic halo.