Large-scale magnetic field structure of NGC 3627 based on magnetic vector map

The data used in this study were published by Soida et al. (2001) and obtained from observations in the C-band (4.85 GHz) and X-band (8.46 GHz). They consist of a combination of single dish and interferometry data, obtained with the Effelsberg radio telescope and very large array (VLA), respectively. Observational information, such as the frequency, size of the synthesized beam, and root mean square (rms) noise, are summarized in Table 2.1.

Data analysis was performed using the software AIPS (Astronomical Image Processing System). We calculated the polarization intensity and polarization angle using the AIPS task COMB. Using the Stokes parameter, the polarization intensity and angle were calculated to be $P_{\rm POL}=\sqrt{Q^{2}+U^{2}}$ and $\chi=\frac{1}{2}{\rm arctan}(U/Q)$, respectively. The polarization intensity was corrected by subtracting the positive bias using the task POLCO. The primary beam was also corrected using the task PBCORL. The noise level $\sigma$ of each Stokes parameter was the variance of the emission-free region and measured using TVBOX and IMSTAT, both of which are the verbs of AIPS. The measured noise levels are summarized in table 2.1. The detection limit was defined as thrice the noise level, i.e., $3\sigma$.

We derived the total and polarized intensities in the C- and X-bands. In figure 1, we depict the contours of the total intensities and orientations of B-field ($90^{\circ}$ rotation of the E-field) overlaid on a black-and-white image from DSS2 B (McLean et al., 2000). The observed B-field is plotted only in pixels which the polarized intensities were detected with higher signal-to-noise (S/N) ratio than the detection limit in Stokes $Q$ and $U$ ($>3(\sigma_{\rm Q}+\sigma_{\rm U})/2$).

The total intensity is the brightest in the boundary region between the southern bar and spiral arm, and its values are 12.99 and 9.19 mJy/beam at the C- and X-bands, respectively. This region is also bright in CO and HI lines (Kuno et al., 2007; Walter et al., 2008). The polarized intensity is brightest in the galactic central region in the X-band and in the south-western spiral arm in the C-band.

To calculate the rotation measure ($RM$), Stokes $I$, $Q$ and $U$ maps of the X-band with higher angular resolution were convolved into a 13.5” $\times$ 13.5” beam using the AIPS task CONVL. We calculated the $RM$ at the pixels, which the polarized intensity was detected with better S/N ratio than the detection limit in both bands, based on the following equation:

The histogram of the calculated $RM$ is depicted in figure 2. We fitted the histogram of $RM$ with a Gaussian function by using the nonlinear least squares method, which is based on the Levenberg–Marquardt method. The peak value of the fitted Gaussian was 27.9 at $+56.5\ {\rm rad\ m^{-2}}$, and the dispersion was $48.3\ {\rm rad\ m^{-2}}$, which is consistent with the value of galaxies (Beck, 2016). We considered that $+56.5\ {\rm rad\ m^{-2}}$ was attributed to the foreground.

In figure 3, we depict an $RM$ map of NGC 3627, which the foreground $RM$ of $+56.5\ {\rm rad\ m^{-2}}$ was subtracted. The color range indicates the $RM$ value. The contour levels are $-$150.0, $-$131.25, $-$112.5, $-$93.75, $-$75.0, $-$56.25, $-$37.5, $-$18.75, 0.0, 18.75, 37.5, 56.25, 75.0, 93.75, 112.5, 131.25, and 150.0 ${\rm rad\ m^{-2}}$. This distribution is consistent with that reported by Soida et al. (2001).

We derived the strength of the total and ordered components of the magnetic field by using the total intensity and its polarization degree, respectively. The total magnetic field strength ($B_{\rm tot}$) is calculated using equation (2) based on an assumption of energy equipartition between cosmic rays and its magnetic field described in Beck & Krause (2005),

The strength of the ordered component can be obtained by using the equation $B_{\rm ord}=B_{\rm tot}/(1+q^{2})$, where $q$ is the ratio of the turbulent to ordered components. The ratio $q$ is given by $q\simeq\sqrt{2.1\ p_{0}/p}$, where $p$ is observed polarization degree and $p_{0}$ is intrinsic polarization degree. The polarization degree $p$ was calculated using Stokes $I$ and $P_{\rm POL,corrected}$ values ($p=P_{\rm POL,corrected}/I$). $P_{\rm POL,corrected}$ is corrected polarization intensity of positive bias. The intrinsic polarization degree $p_{0}$ is determined using the spectral index as $p_{0}=(3-3\alpha)/(5-3\alpha)$ , as described in Beck & Krause (2005). We adopted $\alpha=0.9$, $K_{0}=100$, $L=6.2\times 10^{21}\ {\rm cm}$, $E_{\rm p}=1.5\times 10^{-3}\ {\rm erg}$, $c_{2}=3.9\times 10^{-24}\ {\rm erg~{}G^{-1}sterad^{-1}}$, and $c_{4}=0.62^{2}$, to estimate the magnetic field strength, by following former works (Soida et al., 2001; Beck & Krause, 2005).

The radial distribution of the magnetic field strength and its equivalent energy density is depicted in figure 5. The averaged magnetic field strength over the galaxy is $B_{\rm tot}=18.8\pm 4.2\ \mu$G for the total field, and $B_{\rm ord}=6.1\pm 2.0\ \mu$G for the ordered field. Although the total field is the highest at the galactic center and decreases with radius, it is roughly constant in the range of r = 1.5–5.5 kpc. The ordered field is almost constant for the all the radii. Error bars are the standard deviation of each data within the corresponding radius range.

Soida et al. (2001) observed that $B_{\rm tot}=11\pm 2\ \mu$G for the total field, and $B_{\rm ord}=4\pm 1\ \mu$G for the ordered field, both of which are different than our results. However, we also observed that $B_{\rm tot,class}=13.2\pm 3.3\ \mu$G and $B_{\rm ord,class}=7.3\pm 1.2\ \mu$G using the classical method described by Beck & Krause (2005),

In Soida et al. (2001), the cutoff value of the lower electron energy was used to be 300 MeV. Therefore, we adapted $\nu_{\rm min}=500$ MHz, which was converted from 300 MeV (e.g., equation 2 of Reynolds & Keohane 1999), and $\nu_{\rm max}=10$ GHz. Furthermore, when we calculated $B_{\rm ord,class}$ using the classical method, the polarization intensity was used directly as $I_{\nu}$ instead of dividing it by $q$. From the above-mentioned results, it is considered that the difference between our results and that of Soida et al. (2001) is attributed to the difference between the methods used.

We derived a magnetic field vector map using the magnetic vector reconstruction method, as explained in Section 3. The left panel of figure 6 shows a magnetic field vector map of NGC 3627. The magnetic field vectors were plotted only at the points that satisfied the following four criteria: (1) Stokes $I$ and $P_{\rm POL}$ were higher than $3\sigma_{I}$ and $3(\sigma_{\rm Q}+\sigma_{\rm U})/2$ in the C- and X-bands, respectively; (2) the $RM$ value was higher than the detection limit ($\sigma_{\rm RM}$), which was calculated on the basis of error propagation using the expression $\sigma_{\rm RM}=\sqrt{(\delta\chi(\lambda_{1})^{2}-\delta\chi(\lambda_{2})^{2})/({\lambda_{1}^{2}}-{\lambda_{2}^{2}})^{2}}$; (3) the absolute value of RM is within 200 ${\rm rad\ m^{-2}}$, and (4) the difference in polarization angle in each band is larger than the sum of polarization angle errors. We used the polarization angle in the X-band as the orientation of the magnetic vector. We neglected the Faraday rotation due to the intervening media between the source and our location, as the typical $RM$ dispersion was $48.3{\rm rad\ m^{-2}}$, which corresponded to a few degrees of rotation in the X-band.

In the right panel of figure 6, we show a face-on view of the magnetic field vector map. It was made by rotating the left panel counter-clockwise by the position angle of $176^{\circ}$ to align the major axis with the vertical axis and horizontally enlarging it by a factor of 1/cos $i$. We observed that NGC 3627 has both inward (red) and outward (blue) magnetic field vectors, it suggests that the structure of the magnetic vector was not axis-symmetric spiral (ASS). In the left panel of figure 6, we show that the vectors in the northeastern and southwestern regions are inward and outward, respectively. Therefore, the mode of bi-symmetric spiral (BSS) seems dominant in this galaxy, as discussed in the following section.

As shown in the upper-right panel of figure 7, the angle of the magnetic vector is defined as the angle between the tangent of the circle and the magnetic vector. For instance, the pitch angle of $0^{\circ}$ corresponds to the vector angle of $0^{\circ}$ or $-180^{\circ}$. Azimuth is defined counter-clockwise from the major axis of the galaxy as shown in the upper-left of figure 7. The bottom-left and bottom-right panels show the vector map within a ring of 3.0 – 4.0 kpc, and a corresponding azimuthal profile of the magnetic vector angle, respectively.

We studied the modes of the magnetic field by focusing on the reversals of the magnetic vector. It can be seen from figure 6 that NGC 3627 has both inward and outward magnetic vectors. Panels of the first and third columns of figure 6 show magnetic vector maps within individual rings at radial intervals $18"$ ($=1.0$ kpc), and panels of the second and fourth columns show whether magnetic vectors directed inward or outward at azimuthal resolution $13.5"$ ($=$ beam size). Inward and outward vectors in figure 8 were defined as $-1$ and $+1$, respectively. We considered the most frequent value of three successive data points, which was obtained by the $most\_frequent\_value(s_{j-1},s_{j},s_{j+1})$ when the sign of the $j$-th data point was defined as $s_{j}$. The red and blue dots of the azimuthal profiles indicate the inward and outward vectors, respectively.

If the magnetic field has mode $m_{\rm B}$, $2m_{\rm B}$ reversals can be found azimuthally. Therefore, we studied the mode of the magnetic field by counting the number of magnetic field reversals. We found $m_{\rm B}=2$ in panel (d) of figure 8 and $m_{\rm B}=1$ in the others. Therefore, we found that the magnetic mode of $m_{\rm B}=1$ was dominant in NGC 3627. The radius with $m_{\rm B}=2$ corresponds to the bar-end region at a radius in the range of 4.0 to 5.0 kpc (Watanabe et al. (2011); Zhang & Buta (2015)). This is consistent with the result of Beck et al. (2005), in which regular magnetic fields change the sign in the bar-end region. For rings of radius in the range of 0.0 to 1.0 kpc and $>$ 8.0 kpc, the magnetic mode was not included because there were few data points ($<6$ points). These results for $m_{B}$ are consistent with those reported by Soida et al. (2001), who found mixed magnetic field modes.

The mode of $m_{\rm B}=1$ in the outer region (r$>5.0$ kpc) corresponds to the BSS structure. NGC 3627 has two spiral arms, which means that the mode of the spiral arms is $m_{\rm D}=2$ (Martínez-García & Puerari, 2014). Therefore, we observed that the relationship between the modes of the spiral arms and the magnetic field was $m_{\rm B}=m_{\rm D}/2$ in NGC 3627. This is in good agreement with the results of analytical galactic dynamo models reported by Chiba & Tosa (1990). They indicated that the two-armed spiral galaxies, that is, $m_{\rm D}=2$, can have the BSS structure due to the swing excitation. The growth of the magnetic field in this dynamo is described as a parametric resonance driven by the velocity profile around the density wave. It is not clear whether our results are universal or not in spiral galaxies. However, magnetic vector maps are useful for exploring reversal field structures such as BSS.

We understand that determination of the mode number simply by counting the number of field reversals is not a common method and that the Fourier transformation is often used. However, we noticed that the separations of spiral arms were not azimuthally equal and that the Fourier mode could be different from the actual apparent number of spiral arms. Therefore, we used this method instead of Fourier analysis to determine the relationship between the number of stellar spiral arms and the mode of the magnetic field.

We should also mention that the $RM$ foreground of $+56.5\ {\rm rad\ m^{-2}}$ estimated in our study is slightly different from Soida et al. (2001), who adopted $+50\ {\rm rad\ m^{-2}}$ as the $RM$ foreground. This seems attributed to difference in observational band as Soida et al. (2001) estimated the value using L-band data taken from Simard-Normandin & Kronberg (1980) while we used C- and X-band data. Since L-band data are strongly affected by depolarization compared to C- and X-bands data, we conclude that our estimation of the $RM$ foreground is more suitable to our study which is based on C- and X-bands.

Finally, we should mention how halo component of NGC 3627 affects our results. A study by Stepanov et al. (2008) showed that if the galactic inclination is more than $15^{\circ}$, the contribution of the vertical component of the halo to the observed $RM$ is negligible. Since the inclination of NGC 3627 is $52^{\circ}$, the obtained $RM$ map is not heavily affected by the halo component. In addition, Han (2017) mention the observed Faraday screen may be seen as different depending on the frequency, and the Faraday screens at the C- and X-bands represented a galactic disk. In this study, because we used C- and X-bands data, we considered the contribution of the halo component to be negligible.