Extragalactic Magnetism with SOFIA (Legacy Program). I. The Magnetic Field in the Multiphase Interstellar Medium of M51^*

Publicly available SOFIA/HAWC+ observations of M51 obtained under proposals with IDs 70_0509 (Guaranteed Time Observations by the HAWC+ Team), 76_0003 (Discretionary Director Time), and 08_0260 (PI: D. Dowell) from 2017 to 2020 (see Figure 1) were used. Table 1 summarizes the observations combined in this work. Polarimetric observations with HAWC+ simultaneously measure two orthogonal components of linear polarization in two arrays of 32 × 40 pixels each. Observations were performed using Band D with a characteristic central wavelength of 154 μm, bandwidth of 34 μm, pixel scale of 690, and beam size (FWHM) of 136 (Harper et al. 2018). For M51, FWHM_HAWC+ = 0.565 kpc. Observations were performed in a four-position dither square pattern with a distance of several detector pixels in the equatorial sky coordinates system (ERF) as shown in Table 1 (column 8). The ERF for these observations was used, so a positive increase of angles is in the counterclockwise direction. In each dither position, four half-wave plate (HWP) position angles (PAs) were taken in the standard sequence 5°, 275, 50°, and 725. These dither sequences of four HWP PAs will be referred to as "sets" hereafter. A chop frequency of 10.2 Hz was used, with the chop angle, chop throw, and nod time as listed in Table 1. The chop angle is defined as the angle in the east-of-north direction along which the telescope chops with a given chop throw.

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The total observation time (on-source time + overheads) is 7.21 hr, of which 2.78 hr is the time on-source. Low-quality exposures due to bad tracking, vignetting by the observatory's door in flight F547, or other technical issues at the time of observations are listed in parentheses in the sets column. The observations require time on the off position due to the chop-nod technique, as well as time to take internal calibrators right before and after each set of four HWP PAs, which translates to an overhead of approximately × 2.6. Note that the previously published results of M51 by Jones et al. (2020) used only a subset of the data presented here. Specifically, Jones et al. (2020) used observations from 70_0509 and 76_0003, with a total time of 4.6 hr, where our observations encompass a total observing time of 7.21 hr. We present here observations with a larger integration time and better sensitivity, which allow us to perform a quantitative analysis of the inner and outer arms of M51. In addition, our data reduction pipeline, supported by the SOFIA Science Center, is the most updated version (v2.3.2), in comparison with that used by Jones et al. (2020), v1.3.0beta3. The new pipeline version corrects for background subtraction and propagation of errors from the time streams, so no inflated errors using a χ² analysis are required, and smoothing techniques have been implemented to account for correlated pixels. A direct comparison between both data sets is beyond the scope of this manuscript, and we refer the reader to the update of the pipeline by the SOFIA Science Center for further details.

The observations were reduced using the hawc_drp_pipeline v2.3.2. The pipeline procedure described by Harper et al. (2018) was used to background-subtract and flux-calibrate the data and compute Stokes parameters and their uncertainties. The final degree and PA of polarization are corrected for instrumental polarization, bias, and polarization efficiency. Typical standard deviations of the degree of polarization after subtraction of ∼0.8% are estimated. We generated final reduced images with a pixel scale equal to the half-beam size, which corresponds to 68. Further analysis and high-level displays were performed with custom Python routines, described in Section 3.1. We discard all of those measurements with an S/N lower than 2 in polarized intensity (${p}_{\mathrm{lim}}=0.05$, probability higher than 95% of having a signal higher than the noise level) in order to avoid regions dominated by noise. We also discard those pixels with an S/N in total intensity lower than $\sqrt{2}/{p}_{\mathrm{lim}}\sim 28.28$. We refer the reader to Section 4 of Gordon et al. (2018) for more details on SOFIA/HAWC+ quality cuts. The inferred B-field orientation at 154 μm is shown as streamlines using the line integral convolution (LIC; Cabral & Leedom 1993) technique in Figure 1 (left panel), where only polarization measurements with P/σ_P ≥ 3 were used, with σ_P as the uncertainty in the polarization fraction. A resample scale of 5 and a contrast of 2 were used to compute the LIC image. The total intensity and polarization map is shown in Figure 2. Inferred B-field orientations and the SOFIA/HAWC+ footprint at 154 μm are shown in Figure 4. In all figures, the observed PAs of polarization have been rotated by 90°. These observations are used to trace the magnetic fields in the cold and dense ISM regions of M51.

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We make use of the 3 and 6 cm radio polarimetric maps at a resolution of 8'' from Fletcher et al. (2011). These data sets were obtained using a combination of observations from the Karl G. Jansky Very Large Array (VLA) and the Effelsberg 100 m single-dish radio telescopes. We refer to the original paper for a complete description of the observations and data reductions of the data sets used in our work. Longer-wavelength (18 and 20 cm) observations from Fletcher et al. (2011) can be strongly affected by Faraday rotation (Beck & Wielebinski 2013) and are thus not considered in this work. For our analysis, Stokes IQU were convolved with a Gaussian kernel to match an FWHM_HAWC+ = 136 and reprojected to the HAWC+ observations. Then, the degree and PA of polarization and polarized flux were computed, accounting for the level of polarization bias as a function of the S/N (Wardle & Kronberg 1974). We show the magnetic field streamlines of the 6 cm data set in Figure 1 (right panel), compared to those of the 154 μm/HAWC+ observations (left panel). A resample scale of 5 and a contrast of 2 were used to compute the LIC image. Final inferred B-field orientations at 3 and 6 cm are shown in Figure 3 (left and right panels, respectively), where the observed PAs of polarization have the same length. To avoid biased results due to the number of measurements across the galaxy, we only use radio polarization measurements that are spatially coincident with the HAWC+ observations. The radio polarization maps are used to spatially correlate the polarization arising from synchrotron emission with that arising from thermal emission by means of magnetically aligned dust grains observed with HAWC+, as detailed in Section 4.

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The ¹²CO(1–0) observations were obtained from the Plateau de Bure interferometer (PdBI) and Arcsecond Whirlpool Survey (PAWS ²⁰ ), which uses the PdBI and IRAM 30 m data to image the emission from the molecular gas disk in M51 at high angular resolution. The data are described in Pety et al. (2013) and Colombo et al. (2014). Specifically, we used moments 0 (integrated emission line) and 2 (intensity-weighted dispersion, velocity dispersion) of the ¹²CO(1–0) emission line at angular resolutions of 6''. For our analysis, moments 0 and 2 were convolved using a Gaussian kernel to match the HAWC+ beam size of 136 and then reprojected to the grid of the HAWC+ observations. The H i data were obtained from The H i Nearby Galaxy Survey (THINGS ²¹ ), described in Walter et al. (2008). Moments 0 and 2 were used to trace the neutral gas in the disk of M51. For our analysis, these observations were processed using the same method as the ¹²CO(1–0) observations.

Both the ¹²CO(1–0) and H i data sets are used to trace the velocity dispersion as a proxy of the turbulence in the molecular and neutral gas. We note that the ¹²CO(1–0) integrated emission-line images of IRAM 30 m at a resolution of 23'' cover the full field of view (FOV) of the HAWC+ observations. However, due to the low angular resolution of the IRAM 30 m observations, any comparison between structures of the galaxy (arms, interarms) and polarization observations are not physically meaningful, as the structures are hardly distinguished. In addition, we used THINGS H i 21 cm data sets to generate the morphological mask that separates the arm and interarm regions (see Section 3.3).

The column density map, ${N}_{{\rm{H}}\,{\rm\small{I}}+2{{\rm{H}}}_{2}}$, was estimated using the integrated emission-line (moment 0) neutral, H i, and molecular ¹²CO(1–0) gas of M51. The IRAM 30 m ¹²CO(1–0) integrated emission-line observations with a resolution of 23'' were used for this analysis. These observations cover the full FOV of the HAWC+ observations, while the 6'' observations used in Section 2.3 only cover the central $\sim 3^{\prime}$ of M51. Specifically, we used the following H i and ¹²CO(1–0) conversions to N_{H i} and ${N}_{2{{\rm{H}}}_{2}}$:

$$\begin{eqnarray}&&{N}_{{\rm{H}}\,{\rm\small{I}}}=1.105\times {10}^{21}\displaystyle \frac{{I}_{{\rm{H}}\,{\rm\small{I}}}}{{\mathrm{FWHM}}_{\mathrm{HAWC}+}^{2}}\,({\mathrm{cm}}^{-2})\end{eqnarray} \tag{ 1 }$$

by Hunter et al. (2012), where I_{H i} is the integrated emission line (moment 0) of H i in units of Jy beam⁻¹ m s⁻¹, and FWHM_HAWC+ is the beam size of HAWC+ at 154 μm in units of arcseconds; and

$$\begin{eqnarray}&&{N}_{2{{\rm{H}}}_{2}}={X}_{\mathrm{CO}}{I}_{\mathrm{CO}}\,({\mathrm{cm}}^{-2})\end{eqnarray} \tag{ 2 }$$

by Bolatto et al. (2013), where I_CO is the integrated emission line (moment 0) of ¹²CO(1–0) in units of K km s⁻¹, and X_CO is the conversion factor of value 2 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹.

The final column density is estimated as ${N}_{{\rm{H}}\,{\rm\small{I}}+2{{\rm{H}}}_{2}}={N}_{{\rm{H}}\,{\rm\small{I}}}\,+{N}_{2{{\rm{H}}}_{2}}$. Column density values are in the range ${\mathrm{log}}_{10}({N}_{{\rm{H}}\,{\rm\small{I}}+2{{\rm{H}}}_{2}}[$ cm ⁻²]) = [20.4–22.11], in agreement with Mentuch Cooper et al. (2012). The computed column density is used for the analysis of the multiphase ISM, as well as the estimation of the SFR.

In this section, we describe the methodology used to estimate the magnetic and morphological pitch angles of M51. The algorithm described here is used to analyze the FIR, radio polarimetric observations, and velocity fields that result from the wavelet analysis of their total intensity maps (see Section 3.2).

The magnetic pitch angle profile is estimated as follows.

• 1.  
  The debiased polarization level and its associated uncertainty are computed using the Stokes IQU parameters and their uncertainties δ I, δ Q, δ U:

  $$\begin{eqnarray}&&{P}_{\mathrm{debias}}=\sqrt{{P}^{2}-\delta {P}^{2}},\end{eqnarray} \tag{ 3 }$$

  where

  $$\begin{eqnarray}&&P=\sqrt{{\left(\displaystyle \frac{Q}{I}\right)}^{2}+{\left(\displaystyle \frac{U}{I}\right)}^{2}}\end{eqnarray} \tag{ 4 }$$

  and

  $$\begin{eqnarray}&&\delta P=\displaystyle \frac{1}{I}\sqrt{\displaystyle \frac{{\left(Q\cdot \delta Q\right)}^{2}+{\left(U\cdot \delta U\right)}^{2}}{{Q}^{2}+{U}^{2}}+\delta {I}^{2}\displaystyle \frac{{Q}^{2}+{U}^{2}}{{I}^{2}}}.\end{eqnarray} \tag{ 5 }$$
• 2.  
  To reproject the observations, our method requires the coordinates of the galactic center (α, δ); the galactic disk inclination, i; and tilt angle, θ. Morphological parameters were adopted from Colombo et al. (2014) and have the following values: α = 2024699, δ = + 471952, i = 22° ± 5°, and θ = −70 ± 30, where α and δ are the equatorial coordinates of the center of M51, i is the apparent inclination of the disk with the LOS (where face-on corresponds to i = 0°), and θ is the apparent tilt angle of the major axis with positive values in the east-of-north direction (where north corresponds to θ = 0°).
• 3.  
  We compute the radius (R) and azimuthal position (ϕ) of every pixel in galactocentric coordinates, where all pixels are assumed to be located in the galactic plane (z = 0). Radial and angular masks are generated with the same tilt angle and inclination as M51.
• 4.  
  An azimuthal angular mask is created. This is generated such that the deprojected vector at each pixel location is perpendicular to the radial direction. We will refer to this idealized field as the "zero pitch angle" field, or Ξ. The observed debiased polarization measurements (P_debias) are deprojected to the galactic plane frame (${P}^{{\prime} }$) using two chained rotation matrices, one to account for the inclination of the galaxy, R_x [i], and a second for the tilt angle, R_z [θ]. The projected matrix is estimated to be ${P}^{{\prime} }={R}_{x}[i]{R}_{z}[\theta ]P$.
• 5.  
  A method was devised to account for the 180° degeneracy in the direction of HAWC+'s PAs. An effective averaging of the directions of several pixels requires resolution of the degeneracies. The Ξ zero pitch angle frame from the previous step is used to correct the PAs, arbitrarily setting them to a common outward-pointing direction. This is performed by measuring the relative angle difference with Ξ and adding or subtracting 180° as required. Note that the result is independent of the reference angle of choice, and it is only used for averaging purposes. As a consequence of this correction, the magnetic pitch angle profile also suffers a 180° degeneracy.
• 6.  
  We project the measured B-field orientations to a new reference frame in which the galaxy is observed face-on. We used the morphological parameters of the inclination and tilt angles (i, θ) and the measured PAs of the B-field orientation corrected for 180° degeneracy from the previous step.
• 7.  
  The pitch angle Ψ(x, y) is calculated as the difference between the measured PAs of the B-field orientation and the Ξ vector field.
• 8.  
  Then, Ψ(x, y) is averaged at each radius from the core. The radial bins are linearly spaced, and the number of them is optimized as a compromise between S/N and spatial resolution. The angular average is performed as follows:

  $$\begin{eqnarray}&&\overline{{\rm{\Psi }}}(R)=\mathrm{atan}2\left(\displaystyle \frac{\lt \cos \,{\rm{\Psi }}(x,y)\gt }{\lt \sin {\rm{\Psi }}(x,y)\gt }\right),\end{eqnarray} \tag{ 6 }$$

  where the < > operator indicates a robust median value (based on Monte Carlo simulations) and $\overline{{\rm{\Psi }}}$(R) is the averaged magnetic pitch angle value for a certain radial bin. For each map, the process detailed below is repeated 10,000 times, using Monte Carlo simulations to include the uncertainties of the tilt angle, inclination, and Stokes parameters. An independent Gaussian probability distribution for each parameter is assumed, with a standard deviation σ equal to their uncertainties. Each of these Monte Carlo simulations produces a magnetic pitch angle array. The results of the Monte Carlo simulations are stored in a data cube, which are later used to calculate the pitch angle profiles ($\overline{{\rm{\Psi }}}$(R)). Finally, for each radial bin, the median $\overline{{\rm{\Psi }}}$(r_i ) value and the 68% and 95% (equivalent to the 1σ and 2σ) uncertainty intervals are computed. For all analyses, we will consider a critical level of at least p = 0.05 (95%) to declare statistical significance.

>This method was implemented in Python and is available on the project website ²² (Borlaff 2021). In Appendix A, we test this method over a set of eight mock HAWC+ polarization observations using different tilt angles, inclinations, S/Ns, and magnetic pitch angles. Our method allows us to estimate the magnetic pitch angle profile without strong dominating systematic errors at an uncertainty level of p > 0.05. Using mock polarization observations with P/σ_p ≥ 2, an accuracy of ≤5° is expected in the $\overline{{\rm{\Psi }}}$(R).

Our magnetic pitch angle estimation method entails processing the data on a pixel-by-pixel basis, allowing the user to separate different regions of the galaxy by using masks. Section 4 describes how this masking technique was leveraged to produce measures of the magnetic pitch angle orientation for different regions in M51: (a) full disk, (b) arm versus interarm, and (c) arm 1 versus arm 2.

To compare the magnetic spiral structure with the morphology of the total intensity using several tracers, a measure of the pitch angle of the spiral arms is required. To identify the orientation of the spiral arms in the 154 μm, 3 cm, and 6 cm observations, we take advantage of the technique applied in Patrikeev et al. (2006) and Frick et al. (2016)—the two-dimensional anisotropic wavelet transform—for the identification of elongated structures. Wavelet transforms allow recovery of the PA of the maximum-amplitude wavelet at each pixel where the signal is significant, returning a map of wavelet orientations representing the local pitch angle of the image. The wavelet scale used is 138, twice the size of the pixel scale. We refer to the original articles (Patrikeev et al. 2006; Frick et al. 2016, and references therein) for a complete explanation of the method and its mathematical description.

In Section 4.4, we present the wavelet transform maps for the 154 μm FIR, 3 and 6 cm radio intensity images, ¹²CO(1–0), and 21 cm H i observations. The lines inside the spiral arms closely follow the local structure of the spiral arms for each tracer. Conveniently, the orientation of the wavelet transform can be decomposed into its corresponding Stokes Q and U, allowing analysis of their structure using the same pitch angle method and software described in Section 3.1.

The THINGS 21 cm observations of the H i gas disk (Section 2.3) and the morphological wavelet analysis from Section 3.2 are used to separate the different morphological regions of M51 (spiral arms, interarms, and core). The resulting masks are shown in Figure 4, and the polarization fields separated by the morphological masks for the different wavelengths are shown in Figure 5. We choose the H i gas to define the arm–interarm mask based on two factors: (1) we have high-resolution, deep observations of M51, and more importantly, (2) it allow us to trace the spiral arms closer to the inner core of the galaxy, something that is not possible with lower-resolution data such as those of our FIR observations.

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As a first step, the core region is defined by studying the surface brightness profile of the H i disk (see Figure 6). The inner region of the profile (R < 100'', <4.16 kpc) shows a nearly constant surface brightness, with a notable decrease of the 21 cm emission at R < 22'' (<0.9 kpc), corresponding to the core region. We fit the location of the break in the surface brightness profile using the software Elbow ²³ (Borlaff et al. 2017), obtaining a break radius of ${R}_{\mathrm{break}}={21.2}_{-1.6}^{{+1.8}^{\prime\prime}}$, ${0.88}_{-0.07}^{+0.08}$ kpc), statistically significant at a level of p < 10⁻⁵. We define this region as the radial limit for the core region in the morphological mask.

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In a second step, the intensity image of the H i observations is analyzed using the wavelet transformation method (see Section 3.2). The amplitude of the wavelet-transformed image provides us with a probability map of the spatial distribution of elongated structures, like spiral arms. We define as statistically significant (and thus part of a spiral arm) every pixel whose associated wavelet amplitude is higher than twice the standard deviation (2σ) of the background noise in the wavelet-transformed image. By doing this, we only select regions that have at least an ∼95% probability of being part of an elongated H i structure. Finally, we separate the two spiral arms using a visually defined polygon over the resulting mask, taking into account the morphology of the galaxy in the FIR, ¹²CO(1–0), 3 and 6 cm, and H i data sets (see Figure 4).

The properties of the magnetic pitch angle across the M51 galactic disk are first analyzed across the full disk mask, with no partition into arm and interarm regions (see Figures 2 and 3). The top panel of Figure 7 shows the radial profiles of the magnetic pitch angles for the full disk after applying the methodology presented in Section 3.1. For the radio polarization observations, we find that the magnetic pitch angle profile is mostly flat up to a radius of 220'' (9.15 kpc). Similarly, for our FIR observations, the magnetic pitch angle is mostly flat up to a radius of 160'' (6.66 kpc); for galactocentric radii larger than R > 160'' (>6.66 kpc), we find signs of a drop in the magnetic pitch angle profile. The central beam of the observations is shown as a black vertical dashed line in each panel. The pitch angle increases at the center due to resolution effects produced by the small number of polarization measurements available at the core.

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For the full disk (Figures 2, 3, and 7), we estimate an average magnetic pitch angle of ${\overline{{\rm{\Psi }}}}_{\mathrm{FIR}}^{\mathrm{FD}}=+{23.9}_{-1.2}^{+1.2\circ }$ for the 154 μm/HAWC+ data set. For the 3 and 6 cm observations, we obtain ${\overline{{\rm{\Psi }}}}_{3\,\mathrm{cm}}^{\mathrm{FD}}=+{26.0}_{-0.8}^{+0.9\circ }$ and ${\overline{{\rm{\Psi }}}}_{6\,\mathrm{cm}}^{\mathrm{FD}}=+{28.0}_{-0.6}^{+0.8\circ }$, which are compatible at some of the bins with the results from Fletcher et al. (2011; see their Table A1). The 3 and 6 cm magnetic pitch angle profiles presented in this work are slightly higher, on average, than those presented in Fletcher et al. (2011) but compatible on the low end in some regions. Specifically, comparing Figure 7 with line 3 of Table A1 in Fletcher et al. (2011), there is reasonable agreement within the error bars in the first three radial ranges. Only in the outer range (6.0–7.2 kpc) does the absolute value of the pitch angle from Fletcher et al. decrease, while it increases in Figure 7. Nevertheless, there is a substantial difference between the two analyses that we must consider. First, their profiles combine polarization observations from 3, 6, 18, and 20 cm data sets, while we are analyzing the 3 and 6 cm wavelengths independently. Second, their pitch angle (p₀) represents the average pitch angle for the dominant of two different large-scale modes of the regular magnetic field, while our profiles represent a nonparametric measurement of the magnetic pitch angle, including variations on smaller scales. For these reasons, we should consider a direct comparison between both profiles with care.

Given the angular resolution of the FIR and radio observations, the interarm and arm regions can be separated and analyzed independently. Using the mask described in Section 3.3, we generate three radial profiles of the magnetic pitch angle: arm 1, arm 2, and both spiral arms combined ("Arms region" in Figure 7). We adopt the same notation for the spiral arms of M51 as in Patrikeev et al. (2006; see their Figure 3). From the outskirts of the galaxy, arm 2 is the most northern arm close to M51b, while arm 1 is the most southern arm (Figure 4). We show the polarization measurements used for each region in Figure 5. The results of the magnetic pitch angle profile for both arms combined are shown in the middle panel of Figure 7, labeled as "Arms region." Figure 8 shows the magnetic pitch angle profiles for arms 1 and 2 separately at 154 μm, 3 cm, and 6 cm.

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For the arms region, we estimate an average magnetic pitch angle of ${\overline{{\rm{\Psi }}}}_{\mathrm{FIR}}^{\mathrm{arms}}$ $=\,+{16.9}_{-1.7}^{+1.8\circ }$ for the 154 μm observations and ${\overline{{\rm{\Psi }}}}_{3\,\mathrm{cm}}^{\mathrm{arms}}$ $=\,+{23.1}_{-1.0}^{+1.1\circ }$ and ${\overline{{\rm{\Psi }}}}_{6\,\mathrm{cm}}^{\mathrm{arms}}$ $=\,+{25.1}_{-0.8}^{+0.8\circ }$ for the 3 and 6 cm observations, respectively. The magnetic pitch angle profiles of the spiral arms reveal an interesting scenario. The radio polarization maps at 3 and 6 cm trace a relatively flat pitch angle up to a radius of 220'' (9.15 kpc), showing some steady increase with radius. The FIR magnetic pitch angle suffers a strong break at a radius of ∼150'' (∼6.24 kpc), decreasing suddenly toward negative values. Statistical analysis of the probability distributions obtained with the Monte Carlo simulations of each bin beyond the ∼150'' break reveals that the difference is significant (p < 0.05) and consistent up to the limiting radius of observation on M51.

The observed break in the magnetic pitch angle profile of the arms region has a significant impact on the average value. In Figure 9, we compare the global differences in the magnetic pitch angle between FIR and radio wavelengths. We measure the difference in average magnetic pitch angle for each pair of data sets (154 μm, 3 cm, and 6 cm) and arms regions of M51. The vertical histograms in Figure 9 represent the probability distribution for the difference in the median pitch angle as a function of the wavelengths and regions compared. These probability distributions are generated based on the 10,000 Monte Carlo simulations obtained for the magnetic pitch angle analysis. The distributions take into account the uncertainties in PA, inclination, and Stokes IQU from the different sets of polarization maps. Using these simulations, we are able to reconstruct the realistic probability density distribution of the average difference between the magnetic pitch angle profiles. We find a statistically significant difference in the magnetic pitch angle between FIR and radio wavelengths in the arms. Averaged across the complete extension of both arms, the FIR magnetic pitch angle is $-{6.2}_{-2.0}^{+2.1\circ }$ and $-{8.3}_{-1.9}^{+2.0\circ }$ lower than that measured in 3 and 6 cm, a result significant with p-values of 0.002 and <10⁻⁴, respectively.

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We now analyze the two arms separately in Figure 8. Results show that the two arms have different radial profiles of the magnetic pitch angles across the galactocentric radius. At small radii (R < 75'', >3.12 kpc), arm 1 shows a lower magnetic pitch angle than arm 2, ${\overline{{\rm{\Psi }}}}^{A1}\lt {\overline{{\rm{\Psi }}}}^{A2}$. The magnetic pitch angle profile is inverted at R > 75'' (>3.12 kpc), where ${\overline{{\rm{\Psi }}}}^{A1}\gt {\overline{{\rm{\Psi }}}}^{A2}$. This inversion is observed at all wavelengths up to R ∼ 160'' (6.66 kpc). At R > 160'' (>6.66 kpc), the magnetic pitch angle of both arms shows a sharp decrease toward zero and negative values in FIR but not in the 3 and 6 cm radio polarization observations. For the 3 and 6 cm radial profiles, the magnetic pitch angle of arm 1 is mostly flat beyond R > 75'' (>3.12 kpc), while arm 2 presents an upturn at R > 150'' (>6.24 kpc). A high pitch angle dispersion region is found in arm 1 at R ∼ 150'' (∼6.24 kpc) on the 154 μm/HAWC+ magnetic pitch angle profile.

We further explore the pitch angle difference for FIR and radio polarization observations in the northern section of arm 2, one of the closest—but not physically connected—spiral arm regions to M51b. We study the distribution of magnetic pitch angles in a rectangular aperture of 3.45 × 2.07 arcmin² (8.6 ×5.2 kpc²) centered at α = 20247, δ = 4723. Figure 10 shows the B-field orientations for the 154 μm, 3 cm, and 6 cm observations. Visual inspection of the three B-fields shows that, on average, the magnetic field at 154 μm shows a different orientation with a smaller pitch angle than those from radio polarimetric observations. In the left panel, we show the probability distributions for the average value of the pitch angle in that aperture. The results show a systematic difference (p < 10⁻⁴) between the FIR and the two radio observations. The magnetic pitch angles of 3 and 6 cm are compatible with each other. The average magnetic pitch angles in this region are $\overline{{\rm{\Psi }}}$ ${}_{\mathrm{FIR}}=-{8.5}_{-2.7}^{+2.8\circ }$, $\overline{{\rm{\Psi }}}$ ${}_{3\,\mathrm{cm}}=+{7.8}_{-1.8}^{+1.8\circ }$, and $\overline{{\rm{\Psi }}}$ ${}_{6\,\mathrm{cm}}=+{7.2}_{-1.3}^{+1.4\circ }$.

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We repeat the analysis on an equivalent aperture located in the southern region of arm 1, symmetrically separated from the core (α = 20246, δ = 4716, also with an area of 8.6 × 5.2 kpc²). The results show that the average magnetic pitch angle in this region is ${\overline{{\rm{\Psi }}}}_{\mathrm{FIR}}=+{5.8}_{-5.3}^{+5.2\circ }$, which is significantly (p <10⁻⁴) lower than those measured in 3 (${\overline{{\rm{\Psi }}}}_{3\,\mathrm{cm}}=+{29.1}_{-2.6}^{+2.8\circ }$) and 6 (${\overline{{\rm{\Psi }}}}_{6\,\mathrm{cm}}=+{28.3}_{-1.5}^{+1.2\circ }$) cm. These results—including the magnetic pitch angle profiles—confirm that the magnetic field in the outskirts of M51 traced by radio and FIR polarization observations is different.

Our results show that the structure of the magnetic field is not isotropic or homogeneous across the galactic disk. Interestingly, the independent trends of the two spiral arms in the inner region of the disk (R < 150'', <6.24 kpc) are detected in the three wavelengths independently, ensuring that the quality of the observations and the analysis are high enough to confirm that the radial changes in magnetic pitch angle are not caused by statistical uncertainty. In addition, we find that this feature is systematically present in both spiral arms at FIR wavelengths, confirming that the change in magnetic pitch angle is a detectable feature of the magnetic spiral structure of M51.

We analyze the interarm region in Figure 7, whose polarization measurements and models are shown in Figure 5. At all wavelengths, the interarm magnetic pitch angle shows a fairly constant structure up to 220'' (9.15 kpc). We estimate the average magnetic pitch angles to be ${\overline{{\rm{\Psi }}}}_{\mathrm{FIR}}^{\mathrm{IA}}$ $=\,+{28.6}_{-1.3}^{+1.3\circ }$, ${\overline{{\rm{\Psi }}}}_{3\,\mathrm{cm}}^{\mathrm{IA}}$ $=\,+{29.1}_{-1.0}^{+1.0\circ }$, and ${\overline{{\rm{\Psi }}}}_{6\,\mathrm{cm}}^{\mathrm{IA}}$ $=\,+{30.6}_{-0.8}^{+1.0\circ }$ for the 154 μm, 3 cm, and 6 cm observations, respectively. The magnetic pitch profiles and their average values show that the interarm magnetic field structure of M51 is the same at FIR and radio wavelengths. However, we find that the interarm magnetic pitch angles are higher than the corresponding values for the arm regions (Section 4.2). This is significant at a p-value <10⁻⁴ for 154 μm, 3 cm, and 6 cm.

The most striking result from the comparison of the interarm magnetic pitch angle profiles is that the 154 μm observations do not show signs of the same distortions or radial variations as those detected in the spiral arms (Section 4.2 and Figure 5). The interarm radial profile appears to be relatively smooth and constant across the galaxy disk up to the observed outer radius of 220'' (9.15 kpc). In Figure 9 (bottom row), we compare the global differences in the magnetic pitch angle between FIR and radio wavelengths, this time for the interarm region. We do not find any significant difference between the average magnetic pitch angle value of the FIR and radio polarization data sets in the interarm region, confirming the results from the previous profiles.

A summary of the average magnetic pitch angles within the radial range of 212–220'' (0.88–9.15 kpc) is shown in Table 2. Based on the results from previous sections, we conclude the following.

• 1.  
  The outer (R > 6.24 kpc) magnetic spiral structure of the spiral arms in M51 is wrapped tighter when measured in FIR than in radio wavelengths.
• 2.  
  The FIR interarm magnetic pitch angle structure is similar to that traced with the radio polarization observations in the diffuse ISM.

Table 2. Magnetic Field Pitch Angles in the Radial Range of 212–220'' (0.88–9.15 kpc) from Figure 7

Wavelength	Full Disk	Arm Region	Interarm Region
	(ΨFD, deg)	(ΨArms, deg)	(ΨIA, deg)
154 μm	${23.9}_{-1.2}^{+1.2}$	${16.9}_{-1.7}^{+1.8}$	${28.6}_{-1.3}^{+1.3}$
3 cm	${26.0}_{-0.8}^{+0.9}$	${23.1}_{-1.0}^{+1.1}$	${29.1}_{-1.0}^{+1.0}$
6 cm	${28.0}_{-0.6}^{+0.8}$	${25.1}_{-0.8}^{+0.8}$	${30.6}_{-0.8}^{+1.0}$

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These results suggest that the outer field decoupling of the FIR and radio magnetic fields is only associated with the spiral arms. This result is further confirmed with the observations of the magnetic pitch angle profiles and custom apertures studied in Section 4.2. We note that this difference is significant, despite the fact that the radial binning and the combination in azimuthal coordinates may be smoothing the differences found in the histograms from this section. We discuss the implications of these results in Section 6.

Figures 11 and 12 show the morphological pitch angle maps of the 154 μm, 3 cm, 6 cm, ¹²CO(1–0), and 21 cm H i observations. These maps have been constructed from the total intensity images and wavelet transform method described in Sections 3.1–3.2. To avoid selection effects due to the different resolution of the images, we convolved every data set to the 154 μm HAWC+ beam size, as we did in the previous section for the VLA/Effelsberg 100 m observations. In Figure 13, we present the morphological pitch angle profiles of the spiral arms for the five different data sets considered, plus the comparison of the magnetic and morphological pitch angle for 154 μm, 3 cm, and 6 cm.

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The morphological pitch angles of 154 μm, 3 cm, and 6 cm have a similar radial profile, i.e., ${\overline{{\rm{\Psi }}}}_{\mathrm{FIR}}^{\mathrm{morph}}\sim {\overline{{\rm{\Psi }}}}_{3\,\mathrm{cm}}^{\mathrm{morph}}\sim {\overline{{\rm{\Psi }}}}_{6\,\mathrm{cm}}^{\mathrm{morph}}$. At low radii (<120'', <5.0 kpc), the morphological pitch angle is relatively high, starting at ∼60°–70°. At higher radii (>120'', <5.0 kpc), the morphological pitch angle decreases to 0°–10° with a relatively slow increase showing some scatter in the outskirts (>200'', 8.32 kpc), especially for 154 μm. We also compare the distribution of the morphological pitch angle with the magnetic pitch angle profiles obtained in Sections 4.1–4.3 (see the third, fourth, and fifth panels of Figure 13). The analysis shows that for the three bands analyzed, the magnetic pitch angle is lower than the morphological equivalent up to a radius of ∼100'' (∼4.16 kpc). At larger radii (>100'', >4.16 kpc), the magnetic pitch angle is larger than the morphological pitch angle. The exception is in the outermost region (>175'', >7.28 kpc) of the 154 μm/HAWC+ data due to the magnetic pitch angle break reported in Section 4.2.

For ¹²CO(1–0), we find a relatively constant, albeit with large scatter, pitch angle profile of ${\overline{{\rm{\Psi }}}}_{\mathrm{CO}}^{\mathrm{Morph}}\sim 40^\circ$–60° up to the limit of the PAWS observations (R = 120'', ∼5 kpc), with an average of ${\overline{{\rm{\Psi }}}}_{\mathrm{CO}}^{\mathrm{Morph}}={30.7}_{-0.4}^{+0.5\circ }$. For the 21 cm H i observations, we find a relatively constant morphological pitch angle of ${\overline{{\rm{\Psi }}}}_{21\,\mathrm{cm}}^{\mathrm{Morph}}={9.9}_{-0.5}^{+0.3\circ }$ across the whole observable disk. We find that the morphological pitch angle of H i is smaller than at FIR, radio, and ¹²CO(1–0) within the central 120'' (5 kpc). But it is approximately similar to that of the outer region (R > 120'', >5 kpc) when compared with the 154 μm (${\overline{{\rm{\Psi }}}}_{\mathrm{FIR}}^{\mathrm{Morph}}$ $=\,{8.4}_{-0.5}^{+0.5\circ }$), 3 cm (${\overline{{\rm{\Psi }}}}_{3\,\mathrm{cm}}^{\mathrm{Morph}}$ $=\,{10.6}_{-0.9}^{+0.7\circ }$), and 6 cm (${\overline{{\rm{\Psi }}}}_{6\,\mathrm{cm}}^{\mathrm{Morph}}$ $=\,{13.0}_{-0.6}^{+0.7\circ }$). For reference, the average magnetic pitch angles in the outer region of the spiral arms are ${\overline{{\rm{\Psi }}}}_{\mathrm{FIR}}^{\mathrm{Arms}}={15.2}_{-4.2}^{+4.0\circ }$, ${\overline{{\rm{\Psi }}}}_{3\,\mathrm{cm}}^{\mathrm{Arms}}={25.9}_{-1.5}^{+1.5\circ }$, and ${\overline{{\rm{\Psi }}}}_{6\,\mathrm{cm}}^{\mathrm{Arms}}={27.5}_{-1.1}^{+1.1\circ }$.

We find that the magnetic field pitch angles are higher than the morphological pitch angles of the H i in the outskirts of the spiral arms of M51. The p-value for this difference in average values is lower than 10⁻⁴ for the 3 and 6 cm observations (highly significant) and p = 0.044 for the 154 μm observations. The higher values for the outermost bins of the FIR morphological pitch angle profile are possibly an artifact caused by the boundaries of the HAWC+ footprint with the wavelet algorithm; thus, we consider them negligible. In addition, the lower significance at 154 μm is caused by the observed magnetic pitch angle break in the outskirts, which, combined with the outer distortions on the morphological profile, reduces the difference between the morphological and magnetic values.

Close inspection of the total intensity distribution in Figures 11 and 12 reveals that the 154 μm, 3 cm, 6 cm, and ¹²CO(1–0) data sets show bright emission in the core of M51. Contrarily, 21 cm H i observations show no detectable emission at small radii, as previously mentioned in Section 3.3. These different distributions can be responsible for the difference in the morphological pitch angle distributions at inner radii (<120'', <5 kpc). The main reason is that the direction of the wavelet field is affected by the presence of a large, bright, radial central gradient from the core. Nevertheless, the facts that (1) we observe this relatively higher pitch angle value up to R ∼ 100'' (4.16 kpc), far away from the main component of the total intensity of the core, and (2) the ¹²CO(1–0) data set also shows a higher morphological pitch angle than the H i observations suggest that the morphological differences of the pitch angles for the spiral arms are not caused entirely by systematic effects from the central zone.

In summary, we find the following for the spiral arms.

• 1.  
  The morphological pitch angles change as a function of the multiphase ISM, such as ${\overline{{\rm{\Psi }}}}_{{\rm{H}}\,{\rm\small{I}}}^{\mathrm{Morph}}\lt {\overline{{\rm{\Psi }}}}_{\mathrm{CO}}^{\mathrm{Morph}}\,\lt {\overline{{\rm{\Psi }}}}_{\mathrm{FIR}}^{\mathrm{Morph}}\sim {\overline{{\rm{\Psi }}}}_{3\,\mathrm{cm}}^{\mathrm{Morph}}\sim {\overline{{\rm{\Psi }}}}_{6\,\mathrm{cm}}^{\mathrm{Morph}}$.
• 2.  
  The morphological pitch angles at FIR and radio wavelengths are similar across the full disk of M51.
• 3.  
  At FIR and radio and within the inner 100'' (4.16 kpc), the magnetic pitch angles are wrapped tighter than the morphological pitch angles.
• 4.  
  At FIR and radio and at a radius >100'' (>4.16 kpc), the magnetic pitch angles of the spiral arms are larger than those from the morphological structure. The exception is the FIR, whose magnetic pitch angle becomes tighter than the morphological pitch angle at a radius >200'' (>8.32 kpc).

To analyze how the different physical regimes of the multiphase ISM affect the B-fields in M51, we use the velocity dispersion of the neutral and molecular gas as a proxy for the kinetic energy of the turbulence in the ISM. We also use the column density of the galactic disk to study the effect of extinction as a function of the FIR and radio polarization.

In Figure 14, we analyze the variation of the total intensity (I), polarized intensity (PI), and polarization fraction (P) at 154 μm and radio wavelengths as functions of the column density (${N}_{{\rm{H}}\,{\rm\small{I}}+2{{\rm{H}}}_{2}}$). All ρ correlation coefficients in the figures are based on the Spearman nonparametric test. The distributions of the interarm, arm 1, and arm 2 regions are shown in the diagrams. We selected FIR polarization measurements with PI/σ_PI ≥ 3, σ_P ≤ 15%, and P ≤ 30%. For the selected measurements, the minimum S/N in polarization fraction equals 3. Note that we selected the cut in polarized flux such that it reduced any effects due to the positive bias of the polarization fraction. The medians of the physical parameters of the arm 1, arm 2, and interarm zones studied in this section are shown in Table 3. For simplicity, here we only show the diagrams in 3 cm, but the same results are obtained in 6 cm data sets (see Table 3 and the 6 cm radio polarization diagrams in Appendix C).

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Table 3. Medians of the Physical Parameters of the Arm 1, Arm 2, and Interarm Zones

	Parameter	Wavelength	Arm 1	Arm 2	Interarm
(1)	I (Jy arcsec−2)	154 μm	${6.58}_{-0.36}^{+0.54}\times {10}^{-3}$	${5.10}_{-0.11}^{+0.56}\times {10}^{-3}$	${4.08}_{-0.26}^{+0.48}\times {10}^{-3}$
(2)		3 cm	${3.48}_{-0.27}^{+0.26}\times {10}^{-6}$	${2.60}_{-0.16}^{+0.16}\times {10}^{-6}$	${1.88}_{-0.15}^{+0.17}\times {10}^{-6}$
(3)		6 cm	${6.58}_{-0.37}^{+0.30}\times {10}^{-6}$	${4.68}_{-0.31}^{+0.35}\times {10}^{-6}$	${3.30}_{-0.19}^{+0.26}\times {10}^{-6}$
(4)	PI (Jy arcsec−2)	154 μm	${2.04}_{-0.11}^{+0.15}\times {10}^{-4}$	${1.79}_{-0.08}^{+0.08}\times {10}^{-4}$	${1.86}_{-0.08}^{+0.08}\times {10}^{-4}$
(5)		3 cm	${5.41}_{-0.50}^{+0.62}\times {10}^{-7}$	${3.29}_{-0.26}^{+0.26}\times {10}^{-7}$	${3.43}_{-0.39}^{+0.18}\times {10}^{-7}$
(6)		6 cm	${9.80}_{-0.67}^{+0.80}\times {10}^{-7}$	${6.73}_{-0.34}^{+0.35}\times {10}^{-7}$	${7.58}_{-0.47}^{+0.63}\times {10}^{-7}$
(7)	P (%)	154 μm	${3.0}_{-0.3}^{+0.3}$	${3.5}_{-0.3}^{+0.4}$	${4.2}_{-0.3}^{+0.3}$
(8)		3 cm	${15.9}_{-1.5}^{+1.5}$	${13.5}_{-1.1}^{+1.0}$	${17.9}_{-1.1}^{+1.2}$
(9)		6 cm	${16.2}_{-1.3}^{+1.4}$	${14.7}_{-1.0}^{+1.0}$	${22.9}_{-1.3}^{+1.2}$
(10)	${\mathrm{log}}_{10}({N}_{{\rm{H}}\,{\rm\small{I}}+2{{\rm{H}}}_{2}})[{\mathrm{cm}}^{-2}])$		${21.66}_{-0.02}^{+0.01}$	${21.49}_{-0.02}^{+0.04}$	${21.40}_{-0.03}^{+0.02}$
(11)	${\sigma }_{{}^{12}\mathrm{CO}(1-0)}$ (km s−1)		${5.77}_{-0.08}^{+0.13}$	${5.61}_{-0.22}^{+0.15}$	${4.29}_{-0.07}^{+0.09}$
(12)	σH I (km s−1)		${18.22}_{-0.20}^{+0.35}$	${21.34}_{-0.30}^{+0.32}$	${23.40}_{-0.21}^{+0.34}$

Note. Rows from top to bottom: (1)–(3) total intensity for 154 μm, 3 cm, and 6 cm; (4)–(6) polarized intensity for 154 μm, 3 cm, and 6 cm; (7)–(9) polarization fraction for 154 μm, 3 cm, and 6 cm; (10) H i column density; (11) ¹²CO(1–0) velocity dispersion; (12) H i velocity dispersion.

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At 154 μm, we find a strong positive linear correlation between the total intensity and the column density ${N}_{{\rm{H}}\,{\rm\small{I}}+2{{\rm{H}}}_{2}}$. The polarization fraction decreases with increasing column density, while the polarized intensity remains fairly constant across the full range of column densities, i.e., ${\mathrm{log}}_{10}({N}_{{\rm{H}}\,{\rm\small{I}}+2{{\rm{H}}}_{2}}[{\mathrm{cm}}^{-2}])=[21.0$–22.1]. The FIR polarization fraction is found to change in slope at ${\mathrm{log}}_{10}({N}_{{\rm{H}}\,{\rm\small{I}}+2{{\rm{H}}}_{2}}[{\mathrm{cm}}^{-2}])\,={21.49}_{-0.02}^{+0.03}$ (we follow the same method used in Section 3.3 to measure the H i break). Using the relation between the optical extinction, A_V , and hydrogen column density, N_H, relation, N_H/A_V = (2.21 ± 0.09) × 10²¹ cm⁻² mag⁻¹ (Güver & Özel 2009), the change in slope corresponds to an extinction of ${A}_{V}={1.40}_{-0.12}^{+0.18}$ mag.

At radio wavelengths, the total intensity increases with the column density, with a slope of the ${\mathrm{log}}_{10}(I)$ versus ${\mathrm{log}}_{10}({N}_{{\rm{H}}\,{\rm\small{I}}+2{{\rm{H}}}_{2}}[$ cm ⁻²]) relation of 1.16 ± 0.03. The radio polarization fraction is fairly constant within the full range of column densities, while the polarized intensity increases with increasing column density (ρ > 0.5, p < 0.05 in all components). For both FIR and radio, we find no strong systematic differences in the trends and distributions of the arm 1, arm 2, and interarm zones in any case (see ρ correlation coefficients in Figure 14).

In Figure 15, we show the analysis as a function of the ¹²CO(1–0) velocity dispersion (${\sigma }_{v{,}^{12}\mathrm{CO}(1-0)}$). In the FIR, the total intensity increases with increasing the velocity dispersion of the molecular gas, the polarization fraction decreases with increasing the velocity dispersion of the molecular gas (p < 0.05 in all components), and the polarized intensity remains fairly constant (ρ < 0.3, not significant in arm 2). The interarm region has a lower dispersion velocity (p = 4.8 ×10⁻²², using the nonparametric two-sample comparison Anderson–Darling test; Scholz & Stephens 1987) than arms 1 and 2, dominating at ${\sigma }_{v{,}^{12}\mathrm{CO}(1-0)}\sim 3$–5 km s⁻¹. Arm 2 presents a more extended ${\sigma }_{v{,}^{12}\mathrm{CO}(1-0)}$ distribution than arm 1, reaching values as high as 10 km s⁻¹. Both arms present a 1.0% probability of having the same ¹²CO(1–0) velocity dispersion. At radio wavelengths, the total intensity increases with increasing the velocity dispersion of the molecular gas, the polarization fraction is fairly constant across the full range of the velocity dispersion of the molecular gas with ρ ≳ −0.3, and even this low trend is not statistically significant in arm 1. We find an upward trend in the polarized intensity (p < 0.05 in all components). As in the FIR, the radio polarization fraction is higher in the interarm with a probability of p = 4.7 × 10⁻³ in FIR and p = 1.2 × 10⁻⁴ in 3 cm. This result is consistent with the fact that the ¹²CO(1–0) velocity dispersion is lower in the interarm than in the arms. Fletcher et al. (2011) found an average polarization fraction of up to 40% in the interarm regions against a clearly reduced polarization fraction of up to 25% in the spiral arms.

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In Figure 16, we present diagrams for the H i velocity dispersion (σ_{v,H I}). In general, the results show weaker correlations with H i than with ¹²CO(1–0) velocity dispersion in FIR and radio. The relation between the total intensity of FIR and 3 cm with σ_{v,H I} presents a much lower correlation coefficient, which is only relatively mildly correlated in arm 2 (ρ ∼ 0.5) but not well correlated in the rest of the components. The results are similar for the polarization fraction and polarized intensity. The FIR polarization intensity does not show any correlation with σ_{v,H I} and is very low in the case of 3 cm. For the polarization fraction, we do not find any significant relation in FIR or 3 cm with the velocity dispersion of H i.

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In this section, we study the relation between star formation in the M51 disk and the magnetic fields. As described in Section 1, one of the hypotheses that could explain the potential differences between FIR and radio polarization maps is the effect of gas turbulence in star-forming regions. As supernova explosions and winds inject the ISM with some level of turbulence, these mechanisms will generate a relationship between turbulence-driven B-fields and SFR. In addition, due to the effects of gravitational collapse, winds, and star formation, the magnetic field in the molecular gas clouds can present systematically different directions when compared to that of the diffuse ISM (i.e., Pillai et al. 2020).

Therefore, SFR-induced turbulence is expected to be a dominant effect. To test this hypothesis, we study the relation between polarization fraction and polarized intensity with the SFR. We obtained the SFR map from Leroy et al. (2019), which combined UV, NIR, and mid-IR photometry based on GALEX(Martin et al. 2005), the Wide-field Infrared Survey Explorer (WISE; Wright et al. 2010), and stellar population synthesis models to calibrate integrated SFR estimators. The SFR scales with the ISM density, and this generally decreases with the galactocentric radius. Therefore, to compare different galactocentric radii in an equivalent way, we also normalize the SFR by the surface gas mass density to obtain the SFR efficiency per year. The gas mass density map was calculated by multiplying the column density maps used in Section 5 by the mean molecular weight μ and the hydrogen atomic mass (m_H; see Section 2).

In Figure 17, we show the SFR efficiency analysis for M51. The top panels show the SFR and SFR efficiency maps for the area of M51 (reprojected to the HAWC+ resolution) where we have available FIR and radio observations. As a reference, we display two dashed ellipses at galactocentric radii of 166'' and 183'' (6.9 and 7.6 kpc) as an approximate limiting radius where the magnetic pitch angle profiles of radio and FIR observations are compatible (Section 4.4). On the one hand, the SFR map shows a smooth distribution very similar to the total intensity in FIR, with two well-defined spiral arms and a bright inner region. On the other hand, the SFR efficiency map shows a clumpy structure, with knots of high efficiency in the outskirts of the spiral arms (R ∼ 150'', 6.2 kpc) and lower values in the interarms. The bottom panel presents the average SFR efficiency radial profile for the spiral arms of M51. Interestingly, both spiral arms do not show similar trends in SFR efficiency. Arm 2 shows a lower value closer to the galactic center than arm 1. Both arms present noncoincident peaks from the core to the outskirts. We find a decreasing trend in the SFR efficiency of both arms beyond ${6.47}_{-0.55}^{+0.41}$ kpc (${R}_{{\rm{break}}}\,={155.7}_{-13.2}^{+9.8}$ arcsec). This change in slope is significant at a p < 10⁻⁵ level. This might suggest that star formation processes might be playing a role in the same mechanism that produces the systematic differences between the FIR and radio magnetic pitch angle profiles found in Section 4.

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In Figure 18, we explore the overall effect of the SFR over the polarization fraction for the HAWC+, 3 cm, and 6 cm data sets. Interestingly, we found that there is a significant anticorrelation between the polarization fraction and the SFR in M51. This correlation is steeper and more correlated in FIR (ρ = −0.906) than in radio (ρ = −0.335 for 3 cm and −0.540 for 6 cm). For the three wavelengths, the correlation coefficients are significant at a level of p < 10⁻⁵. Linear modeling of the log-scaled SFR and polarization fraction diagrams (${\mathrm{log}}_{10}(P)=a{\mathrm{log}}_{10}\mathrm{SFR}+b$) for the different wavelengths shows that the 3 and 6 cm show a variation of P with the SFR shallower than that detected in the FIR data (see Table 4).

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Table 4. Linear Fits to the Relations between Total Intensity (rows 1—3), Polarized Intensity (4–6), and Polarized Fraction (7–9), for μm 154, 3 cm, and 6 cm as a Function of the SFR

ID	Equation	Slope	Intercept
1	${\mathrm{log}}_{10}{I}_{154\,\mu m}-{\mathrm{log}}_{10}\mathrm{SFR}$	${1.237}_{-0.016}^{+0.016}$	$-{0.358}_{-0.023}^{+0.023}$
2	${\mathrm{log}}_{10}{I}_{3\,\mathrm{cm}}-{\mathrm{log}}_{10}\mathrm{SFR}$	${1.061}_{-0.020}^{+0.019}$	$-{3.917}_{-0.030}^{+0.029}$
3	${\mathrm{log}}_{10}{I}_{6\,\mathrm{cm}}-{\mathrm{log}}_{10}\mathrm{SFR}$	${0.981}_{-0.022}^{+0.023}$	$-{3.781}_{-0.033}^{+0.035}$
4	${\mathrm{log}}_{10}{{PI}}_{154\,\mu m}-{\mathrm{log}}_{10}\mathrm{SFR}$	${0.024}_{-0.022}^{+0.021}$	$-{3.68}_{-0.034}^{+0.033}$
5	${\mathrm{log}}_{10}{{PI}}_{3\,\mathrm{cm}}-{\mathrm{log}}_{10}\mathrm{SFR}$	${0.603}_{-0.038}^{+0.039}$	$-{5.49}_{-0.061}^{+0.063}$
6	${\mathrm{log}}_{10}{{PI}}_{6\,\mathrm{cm}}-{\mathrm{log}}_{10}\mathrm{SFR}$	${0.360}_{-0.037}^{+0.038}$	$-{5.59}_{-0.063}^{+0.061}$
7	${\mathrm{log}}_{10}{P}_{154\,\mu m}-{\mathrm{log}}_{10}\mathrm{SFR}$	$-{1.212}_{-0.029}^{+0.028}$	$-{1.32}_{-0.044}^{+0.044}$
8	${\mathrm{log}}_{10}{P}_{3\,\mathrm{cm}}-{\mathrm{log}}_{10}\mathrm{SFR}$	$-{0.455}_{-0.033}^{+0.036}$	${0.420}_{-0.054}^{+0.058}$
9	${\mathrm{log}}_{10}{P}_{6\,\mathrm{cm}}-{\mathrm{log}}_{10}\mathrm{SFR}$	$-{0.620}_{-0.036}^{+0.037}$	${0.185}_{-0.061}^{+0.061}$
10 a	${\mathrm{log}}_{10}{P}_{154\,\mu m}-{\mathrm{log}}_{10}{I}_{154\,\mu m}$	$-{0.979}_{-0.018}^{+0.016}$	$-{1.670}_{-0.041}^{+0.036}$

^a Results for the ${P}_{154\mu m}$ vs. ${I}_{154\mu m}$ model.

We test the SFR correlation against the total and polarized intensity for the radio and FIR in Figure 19. We find that the FIR polarized intensity does not correlate with the SFR (p = 0.218), but we find a positive correlation in 3 and 6 cm (ρ ∼ 0.57 −0.43, p < 10⁻⁴). We find a positive correlation between the FIR and radio total intensity with the SFR (Table 4). This result is expected due to the FIR–radio correlation (de Jong et al. 1985) and the fact that the SFR is a function of total IR intensity, among other factors (Leroy et al. 2019). For radio, the total intensity increases faster than the polarized intensity with an increase of the SFR across the galaxy.

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In conclusion, we have found that there is a significant anticorrelation of the FIR polarization fraction with the SFR in M51, which does not translate into a correlation of the polarized intensity. In contrast, the radio polarized intensity does increase systematically at higher levels of SFR. The linear regression fit for the observed relation between the polarization fraction and SFR is compatible for the 3 and 6 cm observations but not the 154 μm FIR data set of M51. The observations of HAWC+ reveal that the polarization fraction in FIR is highly anticorrelated with the SFR, showing even lower values for the polarization fraction at similar levels of SFR when compared to that predicted by radio observations. For the polarized intensity, we also find a different behavior in the FIR and radio; 3 and 6 cm present a positive correlation between polarized intensity and SFR, while no correlation is observed in 154 μm. We discuss the relevance of these results in Section 6.3.

In this work, we find that the magnetic pitch angles at radio (3 and 6 cm) and FIR (154 μm) are well aligned in the inner R < 160'' (<6.7 kpc) radius of M51, i.e., R < 160'': Ψ_FIR ∼ Ψ_{3 cm} ∼ Ψ_{3 cm}. This result does not change when considering each of the spiral arms independently or combined, using only the interarm region, or analyzing the complete disk of M51 at once. Only for the interarm region are the FIR and radio magnetic pitch angles similar up to the largest radius (220'', 9.15 kpc) of our observations, i.e., R ≤ 220'': ${\overline{{\rm{\Psi }}}}_{\mathrm{FIR}}^{\mathrm{IA}}\sim {\overline{{\rm{\Psi }}}}_{3\,\mathrm{cm}}^{\mathrm{IA}}\sim {\overline{{\rm{\Psi }}}}_{6\,\mathrm{cm}}^{\mathrm{IA}}$. We find a significant difference between the magnetic pitch angles of the arms at radio and FIR in the outer region (R > 160''; > 6.7 kpc) of M51, i.e., R > 160'': ${\overline{{\rm{\Psi }}}}_{\mathrm{FIR}}^{\mathrm{Arms}}\lt {\overline{{\rm{\Psi }}}}_{3\,\mathrm{cm}}^{\mathrm{Arms}}\sim {\overline{{\rm{\Psi }}}}_{6\,\mathrm{cm}}^{\mathrm{Arms}}$. In the outskirts of M51, the FIR magnetic spiral arms are wrapped tighter than the radio ones. The radio magnetic pitch angle seems to be more open at increasing radius from the core. Our study provides the first observational evidence of a morphological difference between the kiloparsec-scale magnetic field structure of radio and FIR in external galaxies.

We find that the morphological and magnetic pitch angles vary as a function of the ISM component such as ${\overline{{\rm{\Psi }}}}_{{\rm{H}}\,{\rm\small{I}}}^{\mathrm{Morph}}\,\lt {\overline{{\rm{\Psi }}}}_{\mathrm{CO}}^{\mathrm{Morph}}\lt {\overline{{\rm{\Psi }}}}_{\mathrm{FIR}}^{\mathrm{Morph}}\sim {\overline{{\rm{\Psi }}}}_{3\,\mathrm{cm}}^{\mathrm{Morph}}\sim {\overline{{\rm{\Psi }}}}_{6\,\mathrm{cm}}^{\mathrm{Morph}}$ (see Section 4.4). The spiral arms traced by the neutral gas (H i) are wrapped tighter than those traced by the molecular gas observed in ¹²CO(1–0). Interestingly, the morphological pitch angles at radio and FIR are the same across the full extent (220'', 9.15 kpc) of the galaxy, i.e., ${\overline{{\rm{\Psi }}}}_{\mathrm{FIR}}^{\mathrm{Morph}}\sim {\overline{{\rm{\Psi }}}}_{3\,\mathrm{cm}}^{\mathrm{Morph}}\sim {\overline{{\rm{\Psi }}}}_{6\,\mathrm{cm}}^{\mathrm{Morph}}$. However, the magnetic and morphological angles show different behavior across the galaxy disk. At low radii (R < 120'', R < 5.0 kpc), ${{\rm{\Psi }}}_{\mathrm{FIR},3\,\mathrm{cm},6\,\mathrm{cm}}^{\mathrm{Morph}}\gt {{\rm{\Psi }}}_{\mathrm{FIR},3\,\mathrm{cm},6\,\mathrm{cm}}$, while at larger radii, ${{\rm{\Psi }}}_{\mathrm{FIR},3\,\mathrm{cm},6\,\mathrm{cm}}^{\mathrm{Morph}}\lt {{\rm{\Psi }}}_{\mathrm{FIR},3\,\mathrm{cm},6\,\mathrm{cm}}$. The exception is at FIR at radii R > 190'' (>7.9 kpc), where ${{\rm{\Psi }}}_{\mathrm{FIR}}^{\mathrm{Morph}}\gt {{\rm{\Psi }}}_{\mathrm{FIR}}$. Although radio and FIR may be tracing the same morphological regions of the galaxy disk, we found that the magnetic pitch angle of the FIR differs at the outskirts of the galaxy. The FIR may be affected by a different physical mechanism in the outer regions of M51 (see Section 6.2).

The statistical difference found between the morphological and magnetic pitch angles in the disk of M51 at the three wavelengths analyzed may be a direct hint of the independence of the α–Ω dynamo from the spiral density waves (Beck 2015b). Differences between the magnetic and morphological pitch angles have been repeatedly found by previous authors; the average magnetic pitch angle of M83 is about 20° larger than that of the morphological spiral arms (Frick et al. 2016). In M101, the ordered magnetic pitch angle is found to be ∼8° larger than those from the morphological pitch angle of the H i structures (Berkhuijsen et al. 2016). Van Eck et al. (2015) found that, on average, the magnetic pitch angle is ∼5°−10° more open than the morphological pitch angles using a sample of 20 nearby galaxies, a conclusion also found by Mulcahy et al. (2017) in M74.

In theory, spiral magnetic fields can be compressed by density waves, modifying the magnetic pitch angle. This mechanism would create a difference in the arm–interarm region across the galaxy disk. The regular magnetic field in the spiral arm should be more similar to that of the morphological pitch angle than the interarm magnetic field. The magnetic pitch angle may be first compressed and ordered in the interface between the arm and interarm regions. There may be a temporary and spatial disconnect between the morphological and magnetic spiral arms due to the relative action of the large- and small-scale dynamos. Detailed modeling of the M51 galactic system based on these observations would be required to test the interaction of the spiral density waves with the α–Ω dynamo.

In Section 5, we find that the radio and FIR total intensity emission are both tightly correlated with the column density ${N}_{{\rm{H}}\,{\rm\small{I}}+2{{\rm{H}}}_{2}}$ and the ¹²CO(1–0) velocity dispersion. This result and the implicit radio–FIR correlation were explained by Niklas & Beck (1997). In addition, we find that the FIR polarization decreases with increasing the velocity dispersion of the molecular gas and increasing column density. The interarm shows lower velocity dispersion and a higher degree of polarization than the arms. As the velocity dispersion is used as a proxy for the turbulent kinetic energy in the disk, a possible interpretation is that the small-scale turbulent magnetic field may be relatively more significant at a higher velocity dispersion of the molecular gas and column densities than the large-scale ordered field.

In addition, our results show that the FIR and radio total intensity, polarized intensity, and polarization fraction do not correlate with turbulence in H i. Using magnetohydrodynamic simulations, Dobbs & Price (2008) suggested that the small-scale turbulent component is produced by the velocity dispersion of the dust and cold gas. This turbulent component would be generated by the passage through a spiral shock. The authors found that without the cold gas component, the B-field remains well ordered apart from being compressed in the spiral shocks. Our results suggest that the small-scale turbulent field is then coupled to the molecular gas motions but not to the neutral gas of M51. The molecular gas motions are more concentrated in the densest regions of the spiral arm and spatially coincident with the star-forming regions along the arms.

These results suggest that the regions with higher column densities and levels of turbulence of the molecular gas ¹²CO(1–0) reduce the measured FIR polarization fraction inside each beam. The polarized intensity is not affected by these quantities. The polarization fraction at radio wavelengths seems to be insensitive to the column density and the level of turbulence of the molecular gas; instead, the polarized radio emission is affected by these quantities. We find that both FIR and radio are insensitive to the turbulence in the neutral gas (H i) across the galaxy disk. Interestingly, Beck et al. (2019) found no evidence of a spiral modulation of the rms turbulent speed when comparing the velocity dispersion of H i with the radio polarization of several spiral galaxies (M51 included).

In Section 5.2, we find a systematic anticorrelation between the polarization fraction and the SFR. Similar results were obtained earlier by Frick et al. (2001; using H_α emission and 6.2 cm radio polarization) and Tabatabaei et al. (2013) in NGC 6946. The results of our work indicate that both the FIR and radio polarization fraction are anticorrelated with the SFR. Interestingly, the polarized intensity at 154 μm shows a negligible correlation with the SFR, ${N}_{{\rm{H}}\,{\rm\small{I}}+2{{\rm{H}}}_{2}}$, and ¹²CO(1–0) velocity dispersion, whereas the polarized intensity increases at 3 and 6 cm. In the diffuse ISM, the polarization fraction will decrease due to (1) an increase of the relative contribution of unpolarized thermal emission from SFR, (2) Faraday depolarization, and (3) variations of the B-field orientation within the beam and along the LOS. Processes related to star formation (small-scale dynamo) would induce the formation of an anisotropic B-field component from the isotropic turbulent field, hence increasing the polarized intensity in 3 and 6 cm. The polarized intensity may increase if the relative contribution of anisotropic turbulent fields increases within the beam. However, the polarized intensity distributions in FIR show no correlation with SFR, ${N}_{{\rm{H}}\,{\rm\small{I}}+2{{\rm{H}}}_{2}}$, or turbulence. Two different scenarios may explain this result.

• 1.  
  Different magnetic field directions in the same LOS or within the same beam decrease the polarization intensity in FIR (Fissel et al. 2016).
• 2.  
  Dust grain alignment efficiency depends on the total intensity, especially toward regions of high column density (Hoang et al. 2021).

In the first scenario, the turbulence and morphological complexity of the B-field in and around the molecular clouds may cause beam depolarization at FIR wavelengths. Considering this hypothesis, the relative physical size of the HAWC+ beam at 154 μm is 136, approximately 565 pc at a distance of 8.58 Mpc. If we compare this with the size distribution of the giant molecular clouds of M51, which range from 9 to 190 pc in radius, with an average of ∼50 pc (Hughes et al. 2013), we find that the vast majority of these clouds and their structure are unresolved with our spatial resolution. Thus, the complex B-field in the POS within our beam and/or tangled B-field along the LOS toward the cores of these structures causes a drop of polarization in our observations (i.e., depolarization).

The second proposed mechanism is based on a loss of dust grain alignment efficiency toward regions of high column density and gas turbulence. According to the RATs (Dolginov & Mitrofanov 1976; Lazarian & Hoang 2007), dust grain alignment efficiency decreases for grains smaller than a certain size (a_crit) with column density due to the collision dumping effect (Hoang et al. 2021). Specifically, higher gas density causes a stronger loss of alignment by gas collision (which affects smaller grains more efficiently). This effect changes the population of aligned grain sizes to larger dust grains, i.e., the grain-size distribution of aligned grain is narrower, which makes polarization fraction decrease with increasing intensity and ${N}_{{\rm{H}}\,{\rm\small{I}}+2{{\rm{H}}}_{2}}$. In addition, polarization fraction decreases with increasing gas turbulence (velocity dispersion of gas) because the gas turbulence randomizes and/or changes the PA of polarization along the LOS. This effect results in a decrease of polarization fraction as ${\sigma }_{v{,}^{12}\mathrm{CO}(1-0)}$ increases, consistent with the first proposed scenario considering RATs.

To quantify this effect, the polarization fraction has been found to depend on a certain power of the total intensity (P ∝ I^ξ ; Hoang et al. 2021). The power depends on the dust grain alignment efficiency, where ξ = 0 corresponds to full alignment (perfectly polarized dust grain population), ξ = −1 to pure random alignment, and ξ = −0.5 to alignment dominated by gas turbulence. As PI = P · I, polarized intensity becomes constant as ξ decreases. In the case of M51, we measure $\xi =-{0.979}_{-0.018}^{+0.016}$ (see Table 4), which implies a pure random alignment regime. In this regime, polarized intensity is constant with total intensity, ${N}_{{\rm{H}}\,{\rm\small{I}}+2{{\rm{H}}}_{2}}$, and ${\sigma }_{v{,}^{12}\mathrm{CO}(1-0)}$.

These hypotheses for the variation of the polarization fraction of intensity, as well as the lower anticorrelation found in radio observations when compared to FIR, require further investigation, which is beyond the scope of this manuscript.

We cannot directly connect the variation of the polarization fraction with the inner structure of the magnetic field in radio. In order to do that, we would need to take into account the added factor of Faraday depolarization (Sokoloff et al. 1998) and the increase in unpolarized thermal emission, which can be significant at 3 and 6 cm. Regions with higher SFR present higher molecular gas densities, cold gas velocity dispersion, and higher neutral gas column densities. This can be associated with a decrease in the polarization fraction in FIR but also an increase of the total FIR intensity.

The SFR efficiency profile does show a significant decrease at $R={155.7}_{-13.2}^{+9.8}$'' (${6.47}_{-0.55}^{+0.41}$ kpc) of the galactic disk. The different SFR efficiency profiles between both arms suggest an asymmetric structure, possibly triggered by the interaction of the galactic disk with the companion galaxy, M51b. In addition, we do not observe the misalignment of the FIR magnetic field outside the spiral arms. This distortion is found in the outermost radius of M51, close to the radii where arm 2 is closer to M51b. While there is an agreement of the general structure between the magnetic field of the spiral arms in the molecular gas and the diffuse ISM for the inner region, the magnetic pitch angle break is only found in the FIR and not the radio polarization observations. These results may suggest that the molecular disk might be more affected by the interaction with M51b than the diffuse gas. This result is expected, since the molecular gas is a kinematically colder component of the galactic disk than the diffuse, more dispersion-supported gas. Iono et al. (2005) found significant differences (Δv > 50 km s⁻¹) between the diffuse gas and molecular disk kinematics in the rotation curves of a sample of galaxy interacting pairs observed in H i and CO. This suggests that the distortion of the magnetic pitch angle profile found in the outskirts could be produced by the interaction of M51b with the cold dense molecular disk, visible on both sides of the galaxy due to the effect of gravitational tidal forces (Duc & Renaud 2013). Galaxy interactions could affect the diffuse gas differently from the molecular gas, which is kinematically colder, with a highly rotation-supported distribution (Drzazga et al. 2011). In that case, the location of the molecular clouds preferentially in the spiral arms of M51 could explain the finding of a misalignment between the two components of the magnetic field. Indeed, a large angular dispersion in the measured magnetic field due to the interaction of galaxies has recently been found using 89 μm polarization data of Centaurus A by Lopez-Rodriguez (2021). This author found that the small-scale turbulent fields have a larger contribution than large-scale ordered fields in the molecular gas of the remnant warped disk. The fact that we find the distortion on both spiral arms will require a detailed magnetohydrodynamic study of the effects of tidal forces on galactic disks and the previous history of the M51 interaction. The most drastic feature is the down-bending break of the magnetic pitch angle profile in FIR.

Van Eck et al. (2015) and Chyży et al. (2017) found a tight relationship between the specific SFR and the total magnetic field strength, which implies that the process of amplifying magnetic fields in galaxies is mainly driven by small-scale dynamo mechanisms from the local SFR (Gressel et al. 2008a; Schleicher & Beck 2013). The results from Chyży et al. (2017) show that the total magnetic field is correlated with the density of the cold molecular gas (H₂) but not with the warm diffuse H i ISM, a result that is compatible with our findings in Section 5. This shows that the amplification of the B-fields may be taking place in the star-forming regions of M51. This amplification may be driven by small-scale turbulent dynamos, where small-scale refers to scales smaller than our beam size and spatially correlated with the star-forming regions along the spiral arms.