Disentangling Magnetic Fields in NGC 6946 with Wide-band Polarimetry

We present λ13 cm polarization observations of the nearby spiral galaxy NGC 6946 with the Westerbork Synthesis Radio Telescope (WSRT) to examine the nearside halo magnetic fields. Despite λ13 cm exhibiting similar two-dimensional morphology as observed at longer (λ18–22 cm) or shorter (λ3 and λ6 cm) wavelengths, more complete frequency coverage will be required to explain the gap in polarization in the southwest quadrant of the galaxy. We fit models of the turbulent and coherent line-of-sight magnetic fields to the fractional degree of linearly polarized emission at λ3, λ6, λ13, λ18, and λ22 cm from observations taken with the WSRT, Karl G. Jansky Very Large Array, and Effelsberg telescopes. The results favor a multilayer turbulent magneto-ionized medium consistent with current observations of edge-on galaxies. We constrain the physical properties of the synchrotron-emitting thin and thick disks (scale heights of 300 pc and 1.4 kpc, respectively) along with the thermal thick disk and halo (scale heights of 1 and 5 kpc, respectively). Our preferred model indicates a clumpy and highly turbulent medium within 1 kpc of the midplane, and a diffuse extraplanar layer with a substantially lower degree of Faraday depolarization. In the halo, we estimate a regular magnetic field strength of 0.4–2.2 μG and that turbulence and a total magnetic field strength of ∼6 μG result in a Faraday dispersion of σ RM = 4–48 rad m−2. This work is an example of how the advanced capabilities of modern radio telescopes are opening a new frontier for the study of cosmic magnetism.


Introduction
Magnetic fields are observed in galactic systems of every size and shape-from dwarf galaxies (e.g., Gaensler et al. 2005;Chyży et al. 2011) to spiral galaxies (e.g., Beck 2007;Mao et al. 2015;Beck et al. 2019) and jets and lobes of active galactic nuclei (AGNs; e.g., Martí-Vidal et al. 2015) to cluster relics (e.g., Enßlin et al. 1998;Kierdorf et al. 2017).Observations such as these, along with computational studies, suggest that magnetic fields play a role in galaxy dynamics and evolution (e.g., Elstner et al. 2014;Chan & Del Popolo 2022;Khademi et al. 2023).In the interstellar medium (ISM) magnetic fields affect the energy balance by accelerating and confining cosmic rays (CRs; e.g., Beck 2007;Basu & Roy 2013;Becker Tjus & Merten 2020) and heating gas via magnetic reconnection events (Weżgowiec et al. 2016;Weżgowiec et al. 2020).They transport angular momentum and provide pressure support against gravitational collapse during star formation (McKee & Ostriker 2007;Krumholz & Federrath 2019;Stacy et al. 2022).The combination of magnetic and CR pressure generated by supernovae drives outflows of hot gas that alter the composition of the ISM and galactic halos (Everett et al. 2008;Ruszkowski et al. 2017), while magnetic fields within infalling clouds may prevent complete fragmentation (Hill et al. 2013).Strong magnetic fields are also key to understanding AGN where they are observed at the base of jets (Martí-Vidal et al. 2015) and are expected to be the primary source of jet collimation (e.g., Blandford & Payne 1982).Despite their prevalence, the origin and evolution of galactic magnetic fields are still not well understood.
One way to study the properties of magnetic fields and their role in galactic evolution is with radio continuum observations of nearby systems.Total synchrotron intensity can be translated to the total magnetic field strength in the plane of the sky under the assumption of energy equipartition between the cosmic rays (CRs) and magnetic fields (e.g., Beck & Krause 2005).The total magnetic field strength is comprised of both large-and small-scale magnetic fields.Linearly polarized intensity traces the larger-scale, ordered5 magnetic fields in the plane of the sky down to the scale of the observation resolution.Meanwhile, randomly oriented small-scale magnetic fields will cause the degree of polarization (DP = P/I, the ratio of linearly polarized intensity to total intensity) to decrease at longer wavelengths through depolarization (see, e.g., Sokoloff et al. 1998 for a review).Any magnetic fields directed along the line of sight will induce Faraday rotation, causing the polarization angle of the synchrotron emission to rotate as a function of wavelength squared.Different configurations of regions of synchrotron emission relative to the line-of-sight magnetic field can also cause wavelength-dependent depolarization.By observing polarized synchrotron emission across a broad range of wavelengths, both the depolarization and Faraday rotation can be studied.These observables allow the full, threedimensional magnetic field structure to be pieced together.
NGC 6946 is a prime candidate for such a comprehensive study.It is a nearly face-on (i = 38°; Boomsma et al. 2008) SABcd-type galaxy at a distance of 6.8 Mpc (15″ ∼ 500 pc) (Karachentsev et al. 2000), making it one of the largest galaxies on the sky with an optical diameter of D 11. 5 25 = ¢ (de Vaucouleurs et al. 1991) and an ideal target for well-resolved studies.It has multiple bright spiral arms, and an integrated star formation rate of 2.8 M e yr −1 (Calzetti et al. 2010).This star formation likely contributes to the numerous H I holes preferentially located in the spiral arms.There are also signs of extraplanar gas that may be part of an H I halo (Boomsma et al. 2008).Taken together, these properties could be signs that the star formation in NGC 6946 supports a galactic fountain where the gas and dust cycle in and out of the disk.This type of bulk motion, along with turbulence and differential disk rotation, is believed to contribute to the dynamo processes that amplify and sustain the magnetic fields in the galaxy (Beck et al. 2019).Its proximity and diversity of galactic processes make NGC 6946 an excellent target for studying magnetic fields.
Early narrowband radio studies of NGC 6946 at 610, 1415, and 5000 MHz (λ49, λ21, and λ6 cm, respectively) revealed a bright nonthermal disk generated by synchrotron emission (van der Kruit et al. 1977).Soon after, polarization was detected at λ2.8 cm by Klein et al. (1982) who found the orientation of the magnetic fields to be similar to the galaxy's spiral arms.Higher resolution observations at λ6.2 cm revealed the large-scale magnetic fields to be located in between the optical spiral arms (Beck & Hoernes 1996).These magnetic arms have widths of 500-1000 pc, which makes them narrower than the gaps between the optical arms.They also exhibit very high degrees of polarization.Beck & Hoernes (1996) found the northern arm to be 65% polarized at λ3 and λ6 cm.This is nearly the maximum possible degree of linear polarization suggesting that the magnetic fields are aligned.Through wavelet analysis, Frick et al. (2000) established that the magnetic features are structural rather than artifacts of depolarization.Further evidence for this interpretation was developed from λ18 and λ22 cm data by Beck (2007).Kurahara & Nakanishi (2019) presented evidence of magnetic field reversals associated with the optical and magnetic arms.
At longer wavelengths (λ > 18 cm), the observed DP decreases drastically, from 20% and 45% at λ3 and λ6 cm to 6%-10% at λ20.5 cm.The probable source of the depolarization is internal Faraday dispersion (IFD) caused by turbulence in the magnetoionic medium along the line of sight.Here, the Faraday rotating thermal electrons and synchrotron-emitting CRs are well mixed, and the magnetic fields are randomly oriented and small scale (smaller than the resolution of the observations).According to Beck (2007), the magnitude of the observed depolarization in NGC 6946 requires a characteristic Faraday dispersion of σ RM ; 38 rad m −2 , which is physically possible with standard ISM values.Despite this, the longer wavelength polarized emission in NGC 6946 traces extended structure and additional magnetic arms located at larger radii than observed at shorter wavelengths (Beck 2007;Heald et al. 2009).Studies of the spectral index found that an older CR population dominates in the magnetic arms and toward larger radii of NGC 6946, indicating aging as they propagate away from the place of acceleration in the star-forming arms (Tabatabaei et al. 2013;Heesen et al. 2014).The detection of these additional magnetic arms is likely due to a combination of two effects both seen at longer wavelengths: increased telescope sensitivity, and a decrease in emission from higher energy CR.
Curiously, the magnetic arms detected at λ3.6 and λ6.2 cm are only partially recovered at longer wavelengths, although Chupin et al. (2018) showed that they can be recognized by separating emission on different spatial scales through a wavelet transform.Portions of the magnetic arms located in the southwest (SW) quadrant are not visible at λ18-22 cm.This region is coincident with the receding major axis, and the Westerbork Synthesis Radio Telescope (WSRT)-SINGS survey (Braun et al. 2007) revealed a common pattern of strongly reduced polarized emission along the receding major axis in nearby, face-on spiral galaxies (Heald et al. 2009;Braun et al. 2010).The exact mechanism for the depolarization is still under debate, though its prevalence suggests a common largescale magnetic feature.Beck (1991) first proposed the presence of vertical fields along the line of sight as an explanation for this commonly observed phenomenon.Regular fields extending out of an inclined disk may be directed along the line of sight.This can cause differential Faraday rotation (DFR) and wavelength-dependent depolarization, which preferentially decreases the polarization at longer wavelengths.Using the larger sample of galaxies, Braun et al. (2007) were able to replicate the azimuthal variation in both linear polarization and Faraday depth by modeling an extended thick disk of synchrotron emission above and below the midplane.In the model, this region is threaded with an axisymmetric spiral magnetic field that is parallel to the disk and a quadrupole field extending out of the disk.When inclined, this topology aligns azimuthal magnetic fields along the line of sight in the quadrant of the receding major axis.This can have two effects when trying to observe polarized emission from this region: (1) any polarized synchrotron emission produced from the interaction of CRs with these magnetic fields will be aligned perpendicular to the direction of the magnetic fields, and thus out of the line of sight of the observer, and (2) the combination of emission and Faraday rotation in this region may lead to DFR that preferentially depolarizes longer wavelengths.Furthermore, an increase in Faraday dispersion at the midplane will depolarize longer wavelength polarized emission originating in the farside of the disk.In this quadrant, only shorter wavelength emission (e.g., λ3.6, λ6.2 cm) from the midplane is able to make it out of the galaxy.See Braun et al. (2010) for more details.
Despite years of observations, there is still much debate surrounding the exact global structure of these magnetic fields and the mechanisms required to form and maintain them.This is particularly true in the halo and at the disk-halo interface.In the disk, galactic dynamos produce large-scale fields through the shearing and twisting of small-scale fields generated by turbulent motion (e.g., Ferrière 1996;Kulsrud & Zweibel 2008;Gressel et al. 2013).Simulations are getting closer to reproducing the largescale magnetic features observed in disks with the addition of starformation-driven outflows and spiral perturbations to nonaxisymmetric gas flow (e.g., Otmianowska-Mazur et al. 2002;Chamandy et al. 2015).In the halo, observations suggest the presence of a separate dynamo.Fletcher et al. (2011) found indications that the halo field of nearby spiral galaxy M51 has a different pattern than the disk field.Mao et al. (2015) recently observed the correlation length of the Faraday depth structure function at 1-2 GHz to be ∼1 kpc in M51.This is the expected scale of Parker instability loops and super-bubbles.These processes, as well as CR-driven dynamos fed by circumnuclear outflows, may all contribute to the production of coherent structures outside of the disk (e.g., Elstner et al. 1992;Parker 1992;Hanasz et al. 2004;Chamandy et al. 2015).Detailed observations of both the small-and largescale three-dimensional magnetic field structure in and around galactic disks will help differentiate between these models, and in turn, increase our understanding of the flow of material between the disk and the halo.
In this paper, we present new λ13 cm observations of NGC 6946 from the WSRT.This wavelength regime is less affected by depolarization than longer wavelengths (λ > 18 cm), and allows investigation of the magnetic fields at the interface between the disk and halo.We combine these new data with previously published observations at λ3-6 cm and λ18-22 cm, to test different depolarization mechanisms and large-scale magnetic field structures by directly fitting models to the Stokes Q and U. We describe our λ13 cm WSRT observations in Section 2, and detail the analysis performed on the combined broadband data to produce maps of polarization, polarization position angle (PA), and rotation measure (RM).The results of the λ13 cm observations and their combination with λ3, λ6, λ18, and λ22 cm observations are presented in Section 3. We discuss the implications of the broadband model fitting and the value added by including the λ13 cm data in Section 4. Finally, we summarize our results in Section 5.

Data
To achieve broadband coverage of the radio polarization emission from NGC 6946, we combine new λ13 cm observations from the WSRT with previously published data at λ3.6, λ6.2, λ18, and λ22 cm.In the following section, we describe the λ13 cm observations (Section 2.1) and summarize the λ18-22 cm (Section 2.2) and λ3-6 cm (Section 2.4) observations.

13 cm WSRT Observations
Continuum observations of NGC 6946 at λ13 cm were obtained with the WSRT in the maxi-short array configuration.The 13 cm receiver covered the frequency range of 2200-2639 MHz, and the correlator was configured to produce eight overlapping 20 MHz sub-bands each with 64 × 312.5 kHz channels.Observations were conducted during four 12 hr observing sessions on 2008 August 19 (proposal code R08B/ 017) and 2013 July 8-10 (proposal code R13B/008).Observations of NGC 6946 were bracketed by two pairs of calibrators.Each pair consisted of a source with a known polarization angle and an unpolarized source.In the 2008 observation, the primary calibrators were 3C 286 (polarized) and 3C 147 (unpolarized), and the secondary calibrators were 3C 138 (polarized) and CTD93 (unpolarized).In 2013, the primary calibrators were 3C 286 (polarized) and 3C 48 (unpolarized), and the secondary calibrators were 3C 138 (polarized) and CTD93 (unpolarized).All four correlation products were observed.
Since the λ13 cm WSRT data are presented in a circular basis (RR, LL, RL, LR),6 the bulk of the data calibration was done using CASA 4.0.1 (McMullin et al. 2007).At the time, we were unable to use CASA for the system temperature correction so this was done in AIPS (Greisen 2003).Next, the data were imported into CASA, and split up by sub-band.Standard calibration routines, including bandpass, phase, and polarization calibrations, were done individually for each sub-band. 7or both observation epochs, 3C 286 was used for the bandpass and absolute flux calibrations as well as the linear polarization flux and polarization angle.At the time, CASA was not equipped with a model of the polarization properties of 3C 286 between 2.2 and 2.6 GHz data, so known results at 1.95 and 2.45 GHz were used to interpolate the percent of polarization and polarization angle.An unpolarized source was used to solve for the leakage.For the 2008 observation, calibrator 3C 147 was used for this instrumental polarization correction, and 3C 48 was used in the 2013 observations.Radio frequency interference was flagged using RFIGUI (Offringa et al. 2010) after an initial calibration, and then again after a second round of calibration.
Self-calibration and imaging were done with MIRIAD (Sault et al. 1995) following the procedures used to image the 18-22 cm WSRT-SINGS observations (Braun et al. 2010).For each set of observations, two iterations of phase-only selfcalibration with solution intervals of 2 minutes were executed for each individual sub-band.All four observations were combined for imaging.Two images were made for each subband resulting in 16 images spaced at ∼10 MHz intervals.Each image was generated with a restoring beam size of 15″ and CLEANed in MIRIAD using a mask derived from a smoothed total Stokes I map before primary beam correction with the task LINMOS.The noise for each Stokes Q and U image can be found in Table 1.
After combining these data with the previous observations detailed below, we found a systematic rotation in the λ13 cm polarization angle.A discussion of this issue, its possible origins, and the empirical correction we applied to the data are included in Appendix A.

18 and 22 cm WSRT Observations
Continuum observations at 18 and 22 cm were obtained for NGC 6946 during two observing sessions on 2003 March 23 and 2003 November 22 as part of the WSRT-SINGS Survey (Braun et al. 2007).Similar to the 13 cm data, each session consisted of a 12 hr integration in the maxi-short array configuration.Every 5 minutes the observing frequency was switched between two different frequency settings for an effective integration time of 12 hr at each setting.The two frequency settings were 1300-1432 MHz (22 cm) and 1631-1763 MHz (18 cm), and each consisted of eight sub-bands of 20 MHz nominal width, but spaced by 16 MHz to provide contiguous, non-attenuated coverage.All four correlation products (XX, YY, XY, YX) were observed.For details on the calibration procedures, see Braun et al. (2010).
As with the 13 cm observations, two maps were created for each sub-band in the 18 and 22 cm frequency range, resulting in 16 images per frequency setting, which were separated by approximately 10 MHz.Again, images were CLEANed using a mask derived from a smoothed total intensity map, and each image has a restoring beam of 15″.The noise in each 10 MHz Q and U image is listed in Table 9 in Appendix B.

RM Synthesis
We used RM synthesis (Burn 1966;Brentjens & de Bruyn 2005) to determine the total linear polarization across each individual broadband and the associated Faraday depth.Faraday depth (Φ) is a measure of the Faraday rotation that occurs as linearly polarized light travels through an ionized medium with a magnetic field directed along the path of propagation.The amount of rotation, or Φ, is determined by the path length (l), thermal electron density (n e ), and magnitude and direction of the magnetic field along the line of sight (B || ), such that Φ in rad m −2 is given by In the simplest case, where the source of polarized emission lies behind and separate from a region with constant electron density and magnetic field, the change in the polarization angle is where RM is in rad m −2 and λ is the wavelength of the radiation in meters.Positive Φ or RM are taken for magnetic fields pointing toward the observer (see also Ferrière et al. 2021).In reality, the structure and content of astrophysical magneto-ionized media tend to be more complex (e.g., the emitting medium is mixed with the rotating medium or there are multiple RM components within a single telescope beam).
Traditionally, Φ is determined by observing ψ at a few wavelengths and fitting a linear relationship as a function of λ 2 .This method suffers from a number of problems including nπ ambiguities and being constrained to high signal-to-noise (S/N) polarization observations (Ma et al. 2017).This is especially true when Φ is large as Q and U must be measured in narrow channels.With advances in telescope backends, an alternative approach called RM synthesis is now available (e.g., Burn 1966;Brentjens & de Bruyn 2005;Heald et al. 2009).
With 48 pairs of Stokes Q and U maps at λ13, λ18, and λ22 cm we employed the RM synthesis technique (Brentjens & de Bruyn 2005) followed by the deconvolution of Φ spectra using the RMCLEAN algorithm (Heald et al. 2009;Heald 2017) for different subsets of these long wavelength observations.As described in Sections 2.1 and 2.2, the frequency sampling of the images within each broadband (λ13, λ18, λ22 cm) was ≈10 MHz, which enables recovery of polarized emission at typical Faraday depth values at least as high as |Φ| ∼ 3600 rad m −2 .The polarized intensity maps at λ13 and λ22 cm were obtained by performing RM synthesis on each set of broadband images, separately, and extracting the peak polarized intensity from the output RMCLEANed Faraday depth cubes.Two Faraday depth maps were determined from RM synthesis: the first by combining the λ18 and λ22 cm broadband observations, and the second by combining the λ13 data with the λ18 and λ22 cm observations.The Faraday depth resolution, max scale, and errors of the resulting Faraday depth maps presented in Section 3.2 are given in Table 2.All of the final Faraday depth maps and results were derived using empirically corrected λ13 cm maps as discussed in Appendix A.

3 and 6 cm Very Large Array-Effelsberg Observations
Maps of Stokes I, Q, and U at λ3.6 and λ6.2 cm were obtained courtesy of Rainer Beck.The λ3.6 cm maps were made by combining single-band interferometric observations from the Karl G. Jansky Very Large Array (VLA; Beck 2007) with single-dish observations from the Effelsberg Telescope (Ehle & Beck 1993).This was also done at λ6.2 cm, again combining single-band observations with VLA (Beck & Hoernes 1996) and Effelsberg telescope (Beck 2007).Images at both wavelengths have 15″ resolution, equivalent to the λ13, λ18, and λ22 cm WSRT images.Calibration and image processing of the λ3.6 and λ6.2 cm data was completed before we received the data, and details of these steps can be found in the following papers: Ehle & Beck (1993), Beck &Hoernes (1996), andBeck (2007).We used MIRIAD to convert the coordinates from B1950 to J2000 and regrid the images to match the λ13, λ18, and λ22 cm data.The noise in each Q and U image is listed with the λ18 and λ22 cm data in Table 9 in Appendix B. RM synthesis could not be performed on these data since they are single band.Instead, the traditional method of computing RM 1 2 must be used to determine the Faraday rotation between λ3.6 and λ6.2 cm, where ψ is the observed linear polarization angle at each wavelength.

Maps of Fractional Linear Polarization
The spectral index of the synchrotron emission can also decrease as a function of λ, especially in older CR populations.That effect can mimic depolarization, and it is important to take this into account when modeling the impact of magnetic field structure on the observed polarization.We mitigated this effect by dividing the Stokes Q and U maps by maps of the nonthermal total intensity.We define q ≡ Q/I, u ≡ U/I, and p Q U I 2 2 º + .To calculate the fractional polarization p, we needed to isolate the nonthermal emission by subtracting the thermal component from the observed total intensity.We estimated the free-free radio continuum emission by using the standard conversion from a Hα map (Deeg et al. 1997).This was done using the relation between the thermal radio flux S th at a Following previous work on NGC 6946 by Heesen et al. (2014), we assumed the electron temperature of the ionized gas, T e , is 10 4 K, and used Hα emission observed by Ferguson et al. (1998).The Hα map was corrected for foreground Milky Way extinction (E(B − V ) = 0.342) and does not contain emission from the [N II] line.The map was not corrected for internal extinction, but as discussed by Heesen et al. (2014), this method of estimating the free-free thermal emission produces qualitatively similar maps as other methods (e.g., Beck 2007;Tabatabaei et al. 2013).When calculated at λ6.2 cm and compared to the observed total intensity map, the average thermal faction across the galaxy is 2.6%, and upward of 5%, peaking at 12%, in star-forming regions and the nucleus.The lack of short spacings for the λ13, λ18, and λ22 cm WSRT observations meant that these total intensity maps are missing flux on the largest angular scales.The shortest (unprojected) baseline is 36 m, although in practice visibilities are measured on projected baselines as short as 27 m.This means that angular scales 16. 6 28. 0> ¢ ¢ -(13-22 cm) are not captured.While this is larger than the optical diameter of NGC 6946 (D 11. 5;25 = ¢ de Vaucouleurs et al. 1991), negative troughs are still present in the images with a typical depth of around −5σ.Despite these artifacts, we found that the integrated flux densities are collectively consistent with a global spectral index of α = −0.9 and consistent with integrated flux densities from Effelsberg (Tabatabaei et al. 2017) to within 1.5σ.Given that the polarized intensity is produced on smaller scales due to RM fluctuations, we are not concerned with missing polarized flux.On the other hand, this systematic issue with the total intensity maps does prevent a sufficiently accurate estimate of the total nonthermal emission, and thus, we do not use these maps to determine the fractional linear polarization (q, u) at these wavelengths.
Instead, we estimated the thermal component at λ6.2 cm, subtracted it from the combined VLA and Effelsberg image at this wavelength, and then applied a single spectral index, α, to the image to extrapolate the nonthermal emission to the other wavelengths.We ran three iterations of the model fitting routine described in Section 3.5 using a different spectral index each time, α = −0.6,−0.85, and −1.1.This range of spectral indices was determined following the results of Tabatabaei et al. (2013) and Heesen et al. (2014).The effects of these different assumed spectral indices are discussed in Section 3.5 along with the details of the model fitting.
Given that the average thermal fraction at λ6.2 cm is less than 5%, as mentioned earlier in this section, we assume the subtraction of the estimated thermal component contributes little to the overall uncertainty of the resulting nonthermal map.Even if the thermal map was off by 50%, the resulting thermal fraction would only change by a few percent.We, therefore, assumed the uncertainty of the nonthermal map to be equal to that of the rms noise of the total intensity map, σ λ6.2,I = 20 μJy beam −1 , and used this value during the modeling procedure.

Polarized Intensity
Maps of λ6.2, λ13, and λ22 cm linear polarized intensity for NGC 6946 are shown in Figure 1 (left column).The λ6.2 cm polarization map was determined by adding in quadrature the combined Effelsberg and VLA Stokes Q and U maps.The corresponding λ13 and λ22 cm maps were determined by finding the peak polarization in the Faraday spectrum at each pixel.The resulting polarized intensity maps are positive definite and have Rician statistics instead of Gaussian (Wardle & Kronberg 1974).To correct for this bias we calculate the polarization maps using P et al. 2012).See Table 1 for the σ qu of the new λ13 cm images.
We also include maps of Hα emission (Ferguson et al. 1998) overplotted with line segments indicating the magnetic field PA in Figure 1 (right column).The top map illustrates the observed PA at λ3.6 cm for pixels with an S/N 4.0 following Beck (2007).Faraday rotation has a weak effect on shorter wavelengths and no correction was computed for these data.The middle and bottom maps show the PA determined from RM synthesis of λ13-22 cm and λ18-22 cm data, respectively.The PAs in both of these maps are corrected for Faraday rotation (PA = PA observed −Φλ 2 ).We used a stricter S/N threshold of >8 following George et al. (2012).The error for PA is determined by Brentjens &de Bruyn 2005 andMontier et al. 2015 for more details).The maximum error for the λ3.6 cm PA is then 7°. 2, and 11°.6 at λ13-22 cm and 21°.13 at λ18-22 cm.The uncertainty of the longer wavelength data corrected PAs is dominated by the uncertainty in Φ (see Section 3.2 for errors).
The PAs in Figure 1 show that ordered magnetic fields comprise the magnetic spiral arms detected by the polarized emission, and that these magnetic spiral arms are offset from the star-forming spiral arms.This property is present in all three wavelength regimes.Unfortunately, given the rotation correction on the λ13 cm data PA using the λ18-22 cm data (see Appendix B for details), any apparent variation between the λ13 cm PA and what was previously observed at λ3.6 and λ22 cm will be unreliable.
The magnetic arms first identified at λ6.2 cm by Beck & Hoernes (1996) are marked in regions B1, B2, and E in the polarized intensity maps.They are located between the material  arms as shown in the PA maps where the magnetic field line segments are located in between the Hα star-forming regions.While these features are visible at all wavelengths, there is a drop-off in their detection in the SW quadrant at longer wavelengths.This phenomenon has previously been observed in other nearby galaxies, and suggests a common origin for the large-scale structure (e.g., Heald et al. 2009;Braun et al. 2010).
Several different models have been proposed to explain this phenomenon, including a helical configuration extending out of the disk (e.g., Urbanik et al. 1997;Beck 2007) and a combination of an axisymmetric spiral in the disk and quadrupolar topology in a thick disk and halo (Braun et al. 2010).In both scenarios, the magnetic field configuration and orientation along the line of sight will cause the polarized emission and Faraday depolarization to vary across the disk.Regardless, longer wavelength emission experiences greater depolarization, and the emission originating in the thin disk and farside of the galaxy is more strongly depolarized at λ > 18 cm.Only shorter wavelength emission from these regions (λ < 18 cm) propagates beyond the galaxy.The λ13 cm emission clearly recovers part of the southern arm in region E and the inner arm in region B1 that is not detected at λ18-22 cm.This suggests the depolarization at the midplane has a weaker effect on the λ13 cm emission than λ > 18 cm, and confirms that the depolarization in the SW quadrant is indeed a wavelength-dependent phenomenon.As will be discussed in Section 3.2, the shorter wavelength data exhibit a larger Faraday rotation, which is likely due to the increased path length of emission originating at or beyond the midplane.The depolarization restricts our observations at longer wavelengths to emission originating at the nearside of the disk.With more reliable PA data at wavelengths between λ3 and 22 cm, Faraday rotation studies will be useful in determining the path length traversed by synchrotron emission.Full wavelength coverage from ∼λ3-22 cm will be useful for differentiating the exact structure of these large-scale features.
Additional structure detected at λ13 and λ22 cm but not at λ6.2 cm is marked in regions A, D, and F. Beck (2007) pointed out that there may be at least five magnetic arms extending to at least 10 kpc radius at λ18 and λ20.5 cm-two additional arms north and northwest of the northern arm visible at shorter wavelengths (region A), and possibly multiple arms in the broad feature southeast of the southern arm visible at shorter wavelengths (region F).These additional magnetic arms may be generated through shearing or higher-mode (m > 2) nonlinear dynamo action (e.g., Beck 2007;Beck et al. 2019).Numerical simulations of the mean dynamo by Chamandy et al. (2014) produced multiple and bifurcated magnetic arms by forcing a gaseous disk to have multiple spiral arms.The model was unable to reproduce magnetic arms with pitch angles comparable to the gaseous arms, a feature that is commonly observed.Other theoretical efforts have shown that pitch angle agreement may depend strongly on how the forcing spiral evolves (Chamandy et al. 2013(Chamandy et al. , 2015)), and that as our understanding of magnetic arm formation improves, we may be able to infer dynamical properties in the gaseous arms from future magnetic field observations (Beck et al. 2019).
The fact that none of these features located at extended radii are detected at the shortest wavelength is likely due to a combination of CR aging and telescope sensitivity.Previous studies found that the spectral index is steeper in the magnetic arms and toward larger radii suggesting that the emission from these regions is dominated by an older CRe population (Tabatabaei et al. 2013;Heesen et al. 2014).Older CR populations will emit far less radiation at higher frequencies, and the amount of observed polarization will likewise decrease.In addition to this, the sensitivity of the telescope falls off more quickly as a function of distance from the pointing center at shorter frequencies (although this has been mitigated to some extent with mosaicing and single-dish data).Therefore, the observing time required to detect extended polarized structures tends to be greatly increased.Additional observations of these multi-arm features at shorter wavelengths (λ3-6 cm) will ultimately help distinguish between these models.
Region D is most visible at λ22 cm.This extended feature may be an example of anomalous depolarization or repolarization, a term that indicates the observational recognition of an increasing DP at longer wavelengths, which is an atypical situation that provides insight into the underlying magnetic field structure.This phenomenon is likely produced by the specific geometric configuration of the large-scale regular fields in NGC 6946.It could be caused by DFR, which occurs when a synchrotron-emitting region is mixed with thermal electrons and a large-scale magnetic field directed along the line of sight.This causes the polarization versus λ 2 to vary like a sinc function.For this to be true, the effect of small-scale magnetic fields via IFD would need to be negligible.This is highly unlikely given the large extent of the nonthermal emission across the entire disk of the galaxy.It is more likely to be caused by twisted large-scale or helical fields, or a reversal between the disk and the halo (Sokoloff et al. 1998) as discussed by Beck (2007) and Urbanik et al. (1997).Such a reversal is suggested to exist between the disk and the halo of spiral galaxy M51 (Fletcher et al. 2011).
Lastly, region C marks the location of a known bright background source (R.A., decl.(2000) = 20 h 34 m 26 1, +60°10′ 32″).Previously, this source was observed to be 12% polarized at λ3 and λ6 cm (Beck 2007).In Figure 1, the source is visible at all wavelengths, and experiences a steep drop in polarization toward longer wavelengths.At λ13 cm, the polarized intensity drops to 53% of what is observed at λ6 cm, and at λ22 cm the polarized intensity drops to 8% of λ6 cm.While it is possible that there is depolarization intrinsic to the source causing it to be less polarized at longer wavelengths, the linear polarization of the source is subject to twice the path length of magnetoionized medium as the linear polarization produced at the midplane of NGC 6946 and intrinsic to the galaxy.This will increase the depolarization and the Faraday rotation experienced by the linear polarization.This was previously thought to be the case as discussed by Beck (2007).

Faraday Depth
While the polarized emission traces the ordered magnetic fields in the plane of the sky, Faraday depth, Φ, probes the coherent magnetic fields along the line of sight.This includes contributions from the target galaxy itself, intervening extragalactic material, and our own Milky Way.In order to study the Faraday rotation due to the magnetic fields within NGC 6946, the Faraday rotation induced in the foreground must first be removed.We do this by taking the average Φ across NGC 6946 and subtracting it from each pixel.This assumes that the variation in the foreground Φ occurs on scales larger than the observed physical extent of NGC 6946, ∼11′.A more detailed discussion of the validity of this assumption is given in Section 3.4 where we discuss the small-scale structure in Φ.The average Φ and maximum error, err Φ , for each map are listed in Table 2.The uncertainty associated with Φ in a given pixel depends on the S/N of the polarization at that location, and so we list the maximum err Φ for each map.The uncertainties associated with the mean Φ assume a normal distribution, which we know to be untrue given the Rician nature of polarization data.While these errors may thus be slightly inaccurate, they do allow for a better comparison of mean Φ values for each map.These values are in good agreement with each other and previous attempts to determine the contribution from the Milky Way foreground (e.g., Beck 2007;Braun et al. 2010).
The resulting maps of Faraday depth are shown in Figure 2. On the left is the RM between λ3 and 6 cm determined assuming simple Faraday rotation where the change in linear polarization angle, ψ, is a linear function of wavelength squared, λ 2 .Following Beck (2007), the Φ between λ3 and 6 cm is only shown for pixels with polarized intensity greater than or equal to 4× the rms noise at both wavelengths, restricting the maximum error in Φ to 69 rad m −2 .The middle and right maps were determined using RM synthesis of the λ13-22 cm and λ18-22 cm data, respectively.Here, pixels are masked with an S/N < 8 in the polarized intensity maps produced via RM synthesis, restricting the maximum error in the two maps to 5.9 and 9.0 rad m −2 in the λ13-22 cm and λ18-22 cm maps respectively.The stricter cutoff level for the RM synthesis maps was chosen to ensure that the error distribution of the selected pixels is well behaved (George et al. 2012).In all three cases, the Φ error depends on the S/N of the polarized intensity and is thus pixel dependent.The maximum error in each map is listed in Table 2.
For a galaxy that is directly face-on (i = 0°), locally coherent magnetic fields perpendicular to the disk will induce observed Faraday rotation.Alternatively, the Faraday rotation observed in an edge-on galaxy (i = 90°) will trace the magnetic fields parallel to the disk.In the case of NGC 6946, which has an inclination of i = 38° (Boomsma et al. 2008), the Faraday depth is produced by both the vertical fields extending out of the disk and the azimuthal fields parallel to the disk.
Previous observations at λ3-6 cm and λ18-22 cm revealed a gradient in Φ that stretches across the disk of NGC 6946.The Φ values tend to be negative toward the south and positive toward the north.This large-scale north-south gradient can be seen in all three maps in Figure 2, as well as in Figure 3 where the average Faraday depth for 10°azimuthal bins is plotted versus azimuth.The range in Φ probed by the λ3-6 cm observations is −200 to +200 rad m −2 and is much larger than the −30 to +30 rad m −2 difference across both the λ13-22 cm and λ18-22 cm Φ maps.Ehle & Beck (1993) found this large-scale variation in the λ3-6 cm Φ to be sinusoidal when examined as a function of azimuthal angle.This is predicted for an azimuthally symmetric magnetic field structure.In the case of NGC 6946, the variation in Φ is interpreted as a superposition of m = 0 and m = 2 dynamo modes (Rohde et al. 1999), where m = 0 is a regular spiral field with a strength and pitch angle that are constant along azimuthal angle and vary with radius.At λ18-22 cm, the azimuthal variation also appears to be sinusoidal, suggesting that these wavelengths may also be probing an axisymmetric structure on the nearside of the galaxy, possibly in the halo.As discussed, Braun et al. (2010) were able to reproduce the λ18-22 cm Φ pattern with an axisymmetric spiral magnetic field with out-of-disk quadrupolar topology.
This difference in Faraday depth between the two wavelength regimes can be explained by wavelength-dependent depolarization of small-scale magnetic fields.In general, there are two forms of wavelength-dependent depolarization.The first is an IFD, as described in Section 1.The second scenario is called an inhomogeneous Faraday screen (IFS).In this case, the Faraday rotating thermal electrons and magnetic fields reside in a region separate from the synchrotron-emitting CRs.Both depolarization mechanisms will cause longer wavelength polarized emission to experience a greater amount of depolarization than shorter wavelength emission.Any polarized emission originating at the midplane or on the farside of the disk must pass through a greater amount of Faraday rotating material, and thus, will be subject to more depolarization, regardless of the mechanism.Since the shorter wavelengths are less affected by this depolarization, we are able to observe polarized emission from the midplane and farside of the disk.Meanwhile, longer wavelength emission from these regions will experience heavy depolarization as it propagates through the disk, and the observed emission is more likely to originate from a nearside layer above the midplane.Thus, the increase in Φ observed at shorter wavelengths is due to the increased ISM column traversed by this emission.
We attempted to separate the disk and nearside halo components by fitting an axisymmetric magnetic field to the Figure 2. Faraday rotation maps detected using different wavelengths.On the left is the Faraday rotation determined by comparing the PAs of the electric field vector at 3 and 6 cm where the S/N of the polarized intensity exceeds 4 following Beck (2007).The middle map is from the RM synthesis of the 13-22 cm WSRT data, and the map on the right is from the RM synthesis of just the 18-22 cm WSRT data.The RMs in the latter two maps were determined at pixels where the polarized intensity was detected at an S/N > 4. The foreground Φ = 41.3 rad m −2 was determined by averaging across all wavelengths, and has been subtracted from all of the maps.The ellipses show 1, 4, 8, and 12 kpc radii.
foreground-subtracted λ13-22 cm Faraday depth map.Since we know there are multiple magnetic spiral arms, likely composed of m = 0 and m = 2 modes, causing the RM to vary in both azimuth and radius, we fit the axisymmetric structure in three different radial bins: 1-4, 4-8, and 8-12 kpc.Ellipses marking 1, 4, 8, and 12 kpc are shown in Figure 2. The weighted average of the data was determined in azimuthal bins of 10°and then fit by the following function: where Φ 0 is the Faraday depth corresponding to the maximum projection of the constant magnetic field, i is the inclination assumed to be 38°, f is the azimuth, p B is the PA of the magnetic field, and C is any residual foreground Faraday depth not subtracted.The results of fitting this function to the data are listed in Table 3 and shown in Figure 3.In Table 3, we can see that all three parameters vary with radius.There is good agreement between the results at λ13-22 cm and λ18-22 cm, which is what we expect if the λ13 cm polarized emission originates on the nearside of the disk but slightly closer to the midplane than the λ18-22 cm emission.At these wavelengths, we see a shift from positive to negative C as the radius increases, as well as a decrease in the amplitude in the middle radial bin.These results may hint at the presence of an additional vertical component to the magnetic field that points out of the disk in the inner region and back into the disk in the outer region.This could be evidence of the quadrupolar field described by Braun et al. (2010).A more in-depth analysis of this possibility would benefit from full radiative transfer modeling to break degeneracies in depth and opening angle, and is beyond the scope of this work.Future work with more complete wavelength coverage will enable a more robust examination.
The fit to the λ13-22 cm map is interpreted as the contribution from the halo.Residuals are shown as images in Figure 4, and as binned values in Figure 3.After subtracting the halo component there is little large-scale structure left in the λ13-22 cm and λ18-22 cm maps.This indicates that the simple model expressed in Equation ( 4) is a decent representation of the large-scale field structure in this region.The results of the fit suggest an axisymmetric halo component with Φ 0 ∼ 18 rad m −2 .Assuming a path length of 2-4 kpc and an electron density of ∼0.005-0.014cm −3 , the strength of the regular magnetic field parallel to the disk plane of NGC 6946 is ∼0.4-2.2 μG.This falls within the range of estimates for halo magnetic fields found in other comparable galaxies from 0.3 μG in the Milky Way (e.g., Mao et al. 2010) to 3-4 μG in M51 (Kierdorf et al. 2020).The remaining fluctuations in the λ13-22 cm and λ18-22 cm representations in Figures 4 and 3 are likely due to small-scale variations in the thick disk and halo as will be discussed in Section 3.4.
On the other hand, the residual map at λ3-6 cm (leftmost map in Figure 4) still displays a strong southeast-to-northwest gradient.This is also evident in the λ3-6 cm residuals in the bottom left plot of Figure 4, which shows the large-scale structure observed in the top left plot is still present in the shortest wavelength data after subtracting the halo component.As mentioned above, the emission at these higher frequencies will experience less depolarization, and thus, is expected to originate closer to the midplane.Since this emission probes a longer path through the Faraday rotating medium, it will undergo a greater amount of Faraday rotation than longer wavelength emission.The residuals shown in this map are plotted in the bottom left of Figure 3, and show a sinusoidal pattern in the azimuthally binned f.The remaining variation is best fit with a higher frequency (e.g., cos 2f) sinusoidal function.Assuming an electron density of ∼0.03 cm −3 and a path length through the synchrotron thin disk of ∼300 pc (Krause 2014), the average strength of the axisymmetric magnetic field at the midplane is ∼8 μG.This is in good agreement with estimates of the ordered field derived from the degree of linear polarization by Beck (2007).Furthermore, the similarity in PAs, p B , determined for 1-4 kpc across all wavelengths suggests that the field pattern is similar between the thin disk and nearside halo, at least in the inner region.Unlike M51, we do not see obvious signs of large-scale magnetic field reversal between these regions (Fletcher et al. 2011).In NGC 6946, this indicates that the large-scale magnetic field in the thin disk extends into the thick disk and possibly the halo.We therefore expect the origin of the largescale magnetic field in the thick disk and halo to be closely tied, if not directly dependent, on the large-scale dynamo generating the magnetic fields in the thin disk.
As might be expected given the λ13 cm PA correction (see Appendix A), there is little difference between the λ13-22 cm and λ18-22 cm data when comparing the mean Φ, the foreground-subtracted Φ maps, or the 2D large-scale halo fits.
From the linear polarization maps, we saw that the λ13 cm data begins to connect the morphological gap between the λ3-6 and the λ18-22 cm emission suggesting it experiences less depolarization than the longer wavelength observations.If this is indeed the case, we would expect the λ13 cm emission to pass through more magneto-ionized medium than the λ18-22 cm emission, and the observed Φ across the galaxy to be greater than at longer wavelengths, but smaller than at shorter wavelengths.Because we used the Φ from λ18-22 cm to perform the empirical correction of the 13 cm data, we forced the λ13 cm data to have the same Φ as the λ18-22 cm, and are therefore unable to look for such a difference.More complete wavelength coverage between λ3 and 22 cm will allow future observations to further test this idea.

Depolarization
As previously discussed in Sections 3.1 and 3.2, the varying polarization observed at different wavelengths in Figure 1 may be due to a combination of factors.The polarized emission is generated by CRs interacting with the ordered magnetic fields, which are embedded within a constantly evolving ISM.Stellar processes, such as protostellar outflows, stellar winds, and supernovae, will inject energy into the ISM.This energy will drive turbulence and the small-scale dynamo, which in turn, induce small-scale fluctuations in the electron density, gas velocity, and magnetic field strength.Observed polarized emission must then originate in and traverse regions containing both large-and small-scale magnetic fields.As mentioned briefly in Section 3.1, random fluctuations in the magnetic field strength and orientation will more strongly depolarize longer wavelength emission.If the magnetic field fluctuations reside in a region that contains both synchrotron emission and Faraday rotation, the depolarization will be caused by IFD.If the smallscale fluctuations reside in a region with thermal electrons, but separate from synchrotron emission, the depolarization will be caused by an IFS (also known as an external depolarization).
In this section, we examine the validity of the IFD as the main source of depolarization with respect to the addition of λ13 cm observations.In Section 3.5, we take a closer look at the IFS.
In its simplest form, the IFD only considers random magnetic fields in the emitting region, and is described by where S 2 RM 2 4 , and σ RM is the dispersion in intrinsic RM.This dispersion is a function of the random component of the magnetic field directed along the line of sight (B r,|| , μG), electron density (n e cm −3 ) within a region with dimensions of the turbulent scale (d pc), path length through the thermal gas (L pc), and filling factor ( f ).With these units, it can be written as σ RM = 0.81 n e B r,|| d(Lf/d) 0.5 .This expression assumes d is constant and that the fluctuations in B r follow a threedimensional Gaussian distribution such that B B 3 r r , =

||
. In reality, d is likely to vary as a power spectrum, and B r will be anisotropic.
We can begin to quantify this small-scale structure by looking at the ratio of the DP at different wavelengths, DP . To avoid introducing uncertainties associated with subtracting the thermal emission from the total intensity map, DP is typically computed by assuming a synchrotron spectral index, α, across the galaxy.Therefore, ) .We use α = −1.0.While this is a drastically simplified description of the synchrotron-emitting medium across the galaxy, it is a good approximation for the magnetic spiral arms, which is where the polarization is primarily detected and where the CR population tends to be older (see Tabatabaei et al. 2013 andHeesen et al. 2014 for detailed discussions of how α varies spatially in NGC 6946).
Maps of DP between λ22/λ6 (left) and λ13/λ6 (right) are shown in the top row of Figure 5.As may be expected from Equation (5), DP(λ22/λ6) tends to be smaller than DP(λ13/λ6), following the relation that longer wavelengths are more strongly depolarized than shorter wavelengths for a given Faraday dispersion, σ RM .We then use Equation (5) to solve for σ RM .The resulting maps of σ RM are shown in the bottom row of Figure 5.We calculate the mean and dispersion of DP and σ RM in three different areas: the inner 3 kpc (inner region), the SW quadrant, and everything outside of these regions (outer region).These results are listed in Table 4.
Although the amount of depolarization is different at λ13 and λ22, Figure 5 shows that they have similar spatial distributions.Both wavelengths experience greater depolarization in the central 3 kpc and SW quadrant than in either the northern arm or the southeast (SE) quadrant.The larger depolarization in the inner region compared to the outer region can be explained by an increase in the total magnetic field strength toward the center of the galaxy.The magnetic fields in this inner region were previously determined to be in equipartition with the turbulent medium, and there are also signs of amplification via compression (Beck 2007).As discussed in Section 3.1, the depolarization mechanism in the SW quadrant is still unknown, though it is likely due to largescale structure rather than IFD.If the increased depolarization was caused by IFD, we would expect to see a source of energy enhancing the small-scale magnetic field or turbulence, such as star formation.There are no such signs in the Hα or thermal radio continuum emission.Furthermore, there does not appear to be increased attenuation of the Hα emission in this region (Kessler et al. 2020).Therefore, do not expect there to be an undetected or hidden enhancement of star formation in this region, and rule out IFD as a dominant mechanism for the observed depolarization in the SW quadrant.
The σ RM values determined at these two wavelengths also agree within their respective levels of scatter for each region.The spatial correlation and matching σ RM suggest that the λ13 and λ22 cm polarized emission experience the same depolarization mechanism, and originate in and travel through the same medium.Since the λ13 cm emission experiences less depolarization (larger DP), more emission from closer to the midplane should be able to escape the galaxy compared to polarized emission at λ18-22 cm.This would mean that the path length traversed by the λ13 cm would be somewhat larger than that of the λ18-22 cm emission.As long as there are no line-of-sight field reversals closer to the midplane, the Faraday depth measured at λ13 cm would be larger than λ18-22 cm and smaller than λ3-6 cm.Future observations with improved measurements of the polarization angle at λ13 cm will enable better detection of the Faraday depth at this wavelength regime, and allow us to test this prediction.Previous estimates of σ RM found comparable values and determined that the observed depolarization can occur with typical ISM values: n e = 0.03 cm −3 , B r = 17 μG, L = 1000 pc, d = 50 pc, and f = 0.5 (e.g., Beck 2007).This picture does not seem to change with the addition of the λ13 cm data.
While B r agrees with observational estimations, we can check the validity of the assumption of equipartition in the interarm region.If magnetic fields in this region are being amplified by a small-scale dynamo driven by turbulence, we would expect the magnetic energy density of the random fields, U B , to be comparable to the kinetic energy density of the turbulent ionized gas, U KE (e.g., Kulsrud & Zweibel 2008).With U B B 8 r 2 = p , we assume the random fields to be isotropic and estimate the magnetic energy density to be U B = 1.1 × 10 −11 erg cm −3 .In a simple description of isotropic turbulence, we estimate the kinetic energy as U v . In this scenario, the small-scale dynamo is driven by the total gas motion, and thus the generation of the random fields would not solely rely on the kinetic energy of the thermal electrons with density n e , but the velocity dispersion, δv, of the total gas density, ρ.The velocity dispersion of the gas is estimated to be δv ∼ 12 km s −1 in the interarm region from Hα velocity dispersion maps courtesy of Fathi et al. (2009).From observations of H I (Boomsma et al. 2008;Walter et al. 2008), we assume a characteristic column density of N H I = 1.0 × 10 21 cm −2 where 80% of the gas is in a thin disk with a scale height of 200 pc and the remaining 20% is in a thick disk with a scale height of 1 kpc.This gives a typical gas density of ρ ∼ 1.2 cm −3 in the thin disk, resulting in U KE ∼ 4.7 × 10 −12 erg cm −3 , only about a factor of 2 lower than U B .In this analysis, we ignore molecular gas, and acknowledge that this likely contributes to an underestimation in U KE .While Leroy et al. (2009) found the total mass of molecular hydrogen to be ∼86% of the neutral hydrogen mass in NGC 6946, we expect this gas to be largely confined to the star-forming arms and not where the magnetic arms are located.If this gas mass were to be included and the magnetic field strength decreased by 15%, well within the uncertainties, the kinetic and magnetic energy densities would match.Therefore, we conclude that equipartition can hold in this region.

RM Fluctuations
The observed pixel-to-pixel fluctuations in the Faraday depth may be caused by several factors: noise in the Stokes Q and U observations, variation in the large-scale magnetic field, and variations in the small-scale magnetic field and electron density.To account for the large-scale structure, we use the residuals from subtracting the λ13-22 cm halo component, see Section 3.2.As previously discussed, there is little large-scale structure left in the residual maps at λ13-22 cm and λ18-22 cm.We next estimate the Faraday dispersion implied by the residual RM fluctuations.We correct the dispersion for  observational noise by subtracting in quadrature the median S/N-dependent Faraday depth errors from the standard deviation.The observational noise values from the λ13-22 cm and λ18-22 cm Faraday depth maps are 3.7 and 6.4 rad m −2 , respectively.The resulting estimates of the intrinsic Faraday dispersions are σ Φ (λ13-22) = 6.2 rad m −2 and σ Φ (λ18-22) = 5.6 rad m −2 .These are more than an order of magnitude smaller than the dispersion in Faraday depth observed at λ3-6 cm (σ Φ ∼ 96 rad m −2 ), another sign that the various wavelength regimes probe different depths into the galaxy.The Faraday dispersion at λ3-6 cm is much higher than at longer wavelengths since we are observing both the nearsides and farsides of the disk, while the λ13-22 cm emission only traces a relatively short depth into the nearside of the galaxy.Star formation in the midplane will create regions of higher thermal electron densities and pump energy into the small-scale dynamo that amplifies magnetic fields and alters the scale length of the turbulent cells and filling factors.This will lead to larger Faraday depths and steeper spatial changes observed at shorter wavelengths (e.g., λ3-6 cm).
The Faraday dispersions determined at λ13-22 cm and λ18-22 cm are comparable to what was determined in the halos of other galaxies, including σ Φ ∼ 9 rad m −2 in our own Milky Way (Mao et al. 2010;Schnitzeler 2010) and σ Φ ∼ 10 rad m −2 in M51 (Mao et al. 2015).The dispersion in the halo can be explained as turbulent isotropic fields produced by outflows from super-bubbles or the Parker instability.Previously, Heald (2012) reported a sharp RM gradient in Faraday depth maps determined from λ18-22 cm coincident with a H I bubble.This is evidence for a possible Parker loop and for magnetic transport between the disk and halo via the chimney process in NGC 6946.Only one such feature was found out of 121 H I holes cataloged by Boomsma et al. (2008).Higher frequency observations suffering from less depolarization and with greater sensitivity are expected to reveal this potentially ubiquitous phenomenon.
Recent observations at the S band with the VLA detected another such feature in the spiral galaxy NGC 628 (Mulcahy et al. 2017).Though this study was at a higher frequency, it did not recover more Parker loops.This could be due to the limit in Faraday depth resolution dictated by the 2.6-3.6 GHz observations, or signify that the timescale for these structures is too short for them to be observed in larger numbers at higher frequencies.The low reliability of our λ13 cm polarization angle prevents us from testing this idea with the current data, but future observations that more completely cover λ8-20 cm (L and S bands, 1-4 GHz) will provide better Faraday resolution and line-of-sight depth for such a study.
It is worth acknowledging the limitations of the initial assumption that the mean RM is a good proxy for the foreground contribution.In Galactic coordinates, NGC 6946 is located at l = 95°.719,b = 11°.673.This is relatively close to the Milky Way disk where the electron density, magnetic field strength, and turbulence are all at the highest values found within the Galaxy.Previous polarization studies of RM structure in the midplane of the Milky Way estimate the outer scale of turbulence to be ∼1 pc in star-forming arms and ∼100 pc between these regions (Haverkorn et al. 2008).Beyond the midplane, Schnitzeler (2010) determined the RM fluctuations across the sky to be dominated by the Galactic foreground at angular scales between 1°and 10°, which is much larger than the size of NGC 6946 on the sky (D 11. 5 25 = ¢ de Vaucouleurs et al. 1991).NGC 6946 is located beyond the second quadrant of the Milky Way where the known spiral features are the Perseus arm and the outer arm (Reid et al. 2016).Therefore, there is an increased likelihood that the RM induced by the Milky Way will fluctuate both along the line of sight and across the field.At distances of ∼4.0 and 7.6 kpc from the Sun, the line of sight to NGC 6946 passes ∼800 pc above the midplane over the Perseus arm and ∼1600 pc above the outer arm.This puts the line of sight well outside the regions of molecular gas and cold neutral medium, but it may still intersect regions of both warm ionized medium (WIM) and hot ionized medium, which have exponential scale heights around ∼1 kpc (e.g., Haffner et al. 1999).Assuming that the outer scale of turbulence found in the midplane of the material arms holds up to 1 kpc above the midplane, RM fluctuations above the Perseus arm would appear on the order of 51″ or ∼3 synthesized beams (15″) and those above the outer arm would be 27″ or comparable to two synthesized beams in our observations.While a structure function analysis of the observed RM fluctuation across NGC 6946 is beyond the scope of the paper, we note that it is possible that the Milky Way contributes to the small-scale fluctuations discussed in this section.This is particularly true if the outer scale at these heights above the spiral arms is more akin to the interarm regions than the star-forming arms themselves.

q, u modeling
Up to this point, we have used relatively simple descriptions and analysis to constrain the strength and structure of the magnetic fields in NGC 6946.The polarization and large-scale structure in Faraday depth suggested that an axisymmetric magnetic field in the thin disk also extends into the thick disk and possibly the halo.While changes in polarization between the different wavelength regimes can be described by IFD with realistic parameters, it cannot explain the variation in depolarization across the disk.Lastly, we examined how the small-scale fluctuations in the Faraday depth observed at longer wavelengths may be tied to the turbulent properties in the halo of NGC 6946.These piecemeal analyses are valuable indicative tools, but in the current era of broadband polarimetry, we can implement new methods to simultaneously use all of the available data to infer the magnetic field properties of extragalactic systems.
One such method is q, u-model fitting.In this section, we fit seven different models of the line-of-sight magnetic field properties to the available Stokes Q and U data from λ3-22 cm in order to simultaneously determine the large-and small-scale properties from the midplane to the halo of NGC 6946.O' Sullivan et al. (2012) applied this technique to distant radioloud quasars, and found that they could spectroscopically resolve multiple polarization components in objects that were unresolved spatially.More recently, Mao et al. (2015) applied this technique on a pixel-by-pixel basis to resolved L-band observations of M51.They found that the majority of the sightlines experienced little λ 2 -dependent depolarization across nearly 1 GHz of bandwidth, and that an external uniform screen fit well to these points.This is consistent with the idea that the L-band polarized emission originates from a top layer of the synchrotron-emitting disk and then experiences Faraday rotation in the thermal halo.Approximately 1/8 of the spatial area of M51 was found to exhibit λ 2 -dependent depolarization described by external Faraday dispersion.This tended to occur in the nucleus, inner spiral arms, and where the mean H I velocity dispersion was slightly higher suggesting that synchrotron depolarization may serve as an alternate method of identifying and characterizing turbulent regions in galaxies.These early efforts to fit models to the observed q and u spectra have provided a new probe into threedimensional magnetic field structure.
We apply the same maximum likelihood estimation technique as Mao et al. (2015) for fitting models on a pixelby-pixel basis to NGC 6946.This approach determines the parameters (p) for each model (q model , u model ) that maximize the probability (P) that the model represents the observed data (d).We calculated the likelihood using where N is the number of channels and σ q,u is the rms noise in each Q and U image.
We determined the best-fit parameters by using the PYTHON scipy.optimize.minimizeroutine to minimize |) ) .We distinguish between the goodness of fit for models with different degrees of freedom (k) by comparing the Bayesian information criterion (BIC; e.g., Trotta 2008), Despite using the most complete polarization spectrum available for NGC 6946, the model fitting suffers from large gaps in wavelength coverage.This makes it difficult to select a single best-fit model with a high degree of certainty.Instead, we reject the worst fits based on the following criterion: BIC model2 − BIC model1 > 30.
Here model 1 has the minimum BIC (note: a good BIC will be negative), and is favored over model 2 at the 6σ level according to the F-test.Models that satisfy BIC model2 − BIC model1 < 30 are retained.We further reject models whose best-fit parameters do not make physical sense with respect to the observed properties of the galaxy.For example, if the best-fit values for the foreground cannot be reconciled with what we expect from our previous estimates, then we can reject this model.The models that remain are used to get a sense of the line-of-sight magnetic field configuration.
When fitting models to the Stokes Q and U data it is important to take into consideration spectral index effects that can masquerade as depolarization.We are able to correct for this effect by dividing Stokes Q and U by the nonthermal total intensity to produce q and u.As noted in Section 2.5, we were unable to directly use the observed Stokes I data at longer wavelengths due to missing short spacings.Instead, we extrapolated the nonthermal map determined at λ6 cm to longer wavelengths by assuming a nonthermal spectral index, α.We completed the fitting three times using three different spectral indices, α = −1.1,−0.85, and −0.6, and assessed the differences in the following discussion.
We fit seven different functions that model the configuration of the magnetic fields, CRs, and electron density along the line of sight: Analytic functions and structural diagrams for these models are provided in Appendix C, Figures 11-13.Before we go into the details, it is worth mentioning the effects of the assumed spectral index on the fitting results.The spectral index is related to the energy distribution of the CR population, p 1 2 a = -where p is the slope for the power-law energy distribution.We set the upper and lower spectral index values based on previous studies that found the nonthermal spectral index to vary across the disk of NGC 6946.In regions dominated by star formation α > −0.6, while the interarm regions have α < −1.1 (e.g., Tabatabaei et al. 2013;Heesen et al. 2014).These spatial correlations agree well with the expectation that the CRs are accelerated via first-order Fermi acceleration in supernovae and then lose their energy by synchrotron emission and inverse-Compton scattering.Because we are extrapolating from λ6 cm, a steeper assumed spectral index (α = −1.1)would cause the input percent polarization to appear lower at longer wavelengths.This has the effect of producing a polarization spectrum that appears to be more depolarized than when a flatter spectral index (α = −0.6) is applied to the data.This causes the best-fit values for the depolarizing parameter in each model to be about 10% greater when adopting α = −1.1 compared to α = −0.6.While α is a key parameter, we find that the spectral index does not have a significant effect on which models are rejected based on the BIC.With both extreme assumptions for spectral index, DFR and the two spatially unresolved screens (EFS and TWO) are rejected in nearly every pixel, while internal Faraday depolarization and the IFSs are retained.Since the emission of interest tends to be in the interarm region and Tabatabaei et al. (2013) found the mean spectral index in this region to be α = −1.0,we will restrict the following discussion on the model fitting to the results produced with α = −1.1.
The initial best-fit results for each pixel are shown in Figure 6(a).The magnetic spiral arms, as traced by λ3-6 cm polarization, are best fit by mod.PIFS and IFD.This coincides with regions where there is the strongest signal from most wavelengths.Unsurprisingly, this is also where we are able to reject more models, as shown in Figure 6(b).Since mod.PIFS is one of the more complex models (k = 6), and thus most penalized by the BIC, the confluence of these results reinforces the importance of sensitive observations with broad wavelength coverage as we develop more complex descriptions.In the regions outside of the narrow magnetic arms detected at short wavelengths, PIFS and IFS provide the best fits.Here, the number of models satisfying the BIC limit increases.Future attempts at modeling will be greatly improved with better sensitivity and wavelength coverage.
We checked the best-fit values of parameters for the models that satisfy the BIC limit to see if any additional models should be rejected based on unphysical fits.We started by comparing the intrinsic percent polarization, p 0 (this is the amount of polarization produced prior to any depolarizing effects), and the alignment of the magnetic fields of the model fits to the observed properties at 3 and 6 cm.The best-fit values for polarization angle and polarized intensity are shown in Figure 7 overplotted on the observed Hα emission (Ferguson et al.  1998).Figure 7 shows that the mixed models overestimate the amount of polarization in the outer regions (e.g., SE corner).In this region, the mixed models also show more scatter in the magnetic field orientation, leading us to reject DFR and IFD.On the other hand, the IFS (Figure 7 bottom left) is not as good at recovering the amount of polarized intensity observed at longer wavelengths (λ > 13 cm) in this same region, and so we also reject this model.We also examine the best-fit Faraday depth that is determined for these models, as is shown in Figure 8.In each case, the mean Φ (reported in the title of each map) has been subtracted so that the maps are similar to those shown in Figure 2, and the extreme value for DFR reinforces our decision to reject that model.
We also reject the PIFS model on the basis of physicality.As discussed by Farnes et al. (2014), any model that describes depolarization through partial coverage must also account for depolarization in the uncovered region.This is considered in the mod.PIFS with the σ norm parameter, which estimates a layer of external Faraday dispersion proximal to the emission.
The model that is most consistently retained across the entire galaxy, and therefore, the best fit to the current data, is the mod.PIFS.This describes the line-of-sight magnetic structure of NGC 6946 as being composed of a region of synchrotron emission and two different regions containing IFS, one that completely covers the line of sight, and one that is partially covering.
It is worth noting that when the unresolved TWO model was not rejected, it was good at fitting only the long wavelength data, 13-22 cm, suggesting that there may be multiple emitting and rotating regions along the line of sight.Like all of these models, the unresolved TWO is an oversimplification of the line-of-sight magnetic field configuration.While it does not account for Faraday dispersion along the line of sight, it does consider large-scale line-of-sight magnetic fields that are unresolved within the beam.With limited λ 2 coverage, we can only broadly accept or reject these simple models.More complete λ 2 would allow a more detailed comparison fitting to more sophisticated models.One possible way to do this would be with the analytical models proposed by Shneider et al. (2014aShneider et al. ( , 2014b).This will be discussed further in Section 4.

Discussion
To examine the plausibility of the model fitting results, we created a sketch of the vertical profile of the magneto-ionized medium in NGC 6946.In Section 4.1, we draw from previous multiwavelength observations of nearby edge-on galaxies to estimate the vertical distribution of thermal electrons, synchrotron emission, and turbulent scale lengths.We use these values to predict the Faraday dispersion and the depolarization properties of the magnetic fields in different layers above the midplane.In Section 4.2, we compare these values to the bestfit model parameters.

A Vertical Profile Sketch of NGC 6946
A good proxy for estimating the vertical properties of NGC 6946 is the nearby edge-on galaxy NGC 891.Like NGC 6946, NGC 891 is relatively isolated, as confirmed by deep optical and near-infrared photometry, which show a smooth stellar disk that has not been perturbed by any recent gravitational disturbances (Morrison et al. 1997;Schechtman-Rook & Bershady 2013).The optical diameter of NGC 891 is greater than NGC 6946, ∼37 kpc (D 13. 5 25 = ¢ , distance 9.5 Mpc; de Vaucouleurs et al. 1991) compared to ∼23 kpc, but is forming stars at a rate of 3.8 M e yr −1 (Popescu et al. 2004) comparable to that of NGC 6946 (2.8M e yr −1 ).Early Hα observations of this galaxy showed the vertical distribution of the thermal electrons is well fit by the sum of two exponential disks, a thick disk component of H z,disk ∼ 1.0 kpc and halo component of H z,halo ∼ a few kiloparsecs (Rand 1997) , where 〈n e,0 〉 is the mean electron number density (cm −3 ) in the disk and H z is the electron scale height (pc).More recently, Boettcher et al. (2016) observed the extraplanar diffuse ionized gas (eDIG) in NGC 891 with integral field unit spectroscopy.They tested models of dynamical equilibrium in the eDIG, and found that their observations of NGC 891 support a two-component description with mean scale heights and electron densities for a thick disk and a halo of H z,disk = 1.0 kpc, f n 1.25 10 V e,0,disk ´cm −6 , H z,halo = 5.5 kpc, and f n 6.5 10 V e,0,halo = ´cm −6 .Here, f V is the volume filling factor that accounts for the clumpiness of the gas, and 〈n e 〉 = f V n e , where n e is the electron density within the volume.These values are in agreement with previous observations, and we use them to define different vertical layers of the WIM in NGC 6946.To estimate the filling factor for the WIM, we use observational results of the Milky Way that suggest f V ∼ 0.1 at the midplane and increasing to f V > 0.2-0.4 at |z| = 1 kpc (Haffner et al. 2009).
We define the heights of the synchrotron-emitting layers based on C-and L-band observations by the VLA CHANG-ES survey (Irwin et al. 2012).Averaging over a sample of 13 edgeon galaxies, Krause et al. (2018) found the radio scale heights of the halo to be H syn = 1.1 ± 0.3 kpc in the C band and H syn = 1.4 ± 0.7 kpc at the L band.The expected scale height for the synchrotron thin disk is H thin ∼ 300 pc.Assuming energy equipartition with the CRs, we estimate the vertical profile of the magnetic field strength to be exponential with, H B (3 − α)H syn , with α = −1.0.The observed radio scale heights are used to estimate the synchrotron thick disk and magnetic field scale height, H B ∼ 4-5 kpc.This estimate of H B agrees with a previous determination for the magnetic field scale height of NGC 891, which was measured to be ∼4 kpc by Hummel et al. (1991).We estimate the magnetic field strength at z = 0 to be equal to the average total magnetic field strength in the disk outside of the central region, ∼15 μG as determined by Beck (2007).
The last piece of information that is needed is the cell size of the turbulence, d.As discussed in Section 3.4, the outer scale of turbulence in the Milky Way varies with Galactic latitude.Closer to the midplane, the outer scale ranges from 1 < l < 100 pc, on the order of the typical sizes of supernovae and H II regions.Above the disk and into the halo, where the ambient pressure drops off, the outer scale of turbulence is l 100 pc.In our sketch, we follow this description and assume a smaller cell size closer to the midplane, d ∼ 50 pc, as was previously inferred for NGC 6946.We use larger cell sizes for the thick disks (d ∼ 50-100 pc) and halo (d ∼ 100-1000 pc).
The vertical description of NGC 6946 then takes the following form.At the midplane, a thin disk with scale height H thin ∼ 300 pc is dominated by synchrotron emission.This is embedded within a thick disk of thermal electrons with a scale height of H th ∼ 1000 pc.These layers are surrounded by an extended region of synchrotron emission, coined the synchrotron thick disk and estimated to have a scale height of H syn ∼ 1400 pc.Beyond these layers is the thermal halo, which has a scale height of H z,halo ∼ 5000 pc.We estimate the electron density by assuming a double exponential profile and using the values determined by Boettcher et al. (2016).With this framework in hand, we can estimate the amount of Faraday dispersion that can occur within each layer using n Ldf 0.812 e B 3 r s = F as described in Section 3.3.We also use this sketch to estimate the depth into the disk probed by the different observations.Berkhuijsen et al. (1997) proposed the following estimate for the minimum visible depth, Δz, into the thermal disk of a galaxy: where H λ2 is the height of the synchrotron-emitting layer, DP λ2/λ1 is the DP observed between wavelengths λ2 and λ1, and H th is the height of the thermal disk.It is likely that some depolarization will occur along Δz, making this a minimum estimate.With these minimum depths, we determine ranges for the maximum height above the disk where synchrotron emission will occur for the different wavelengths.This allows us to estimate the Faraday rotation experienced by each wavelength given the path length traveled through the galaxy.We use the average DP, at each wavelength as shown and discussed in Section 3.3.The sketch of the vertical profile is shown in Figure 9, and includes estimates of where the observed emission at different wavelengths originates.A summary of the assumed and predicted model values for Faraday depolarization is presented in Table 5, and the values of the emission region heights and subsequent RMs are presented in Table 6.With these values in hand, we can now compare them to the model fitting results.

Comparing the Sketch Predictions with q, u modeling
The q, u-model fitting preferred the mod.PIFS, though other models like the PIFS and IFD were not rejected outright.Figure 10 contains maps of the Faraday dispersion, σ RM , that were found to produce the best fits to the data for each model.
Several things are evident when these maps are compared to the predicted values from the vertical sketch.The first is that it is possible to fit the mod.PIFS values within the framework of Figure 9.If the λ13-22 cm polarized emission generated within ∼1 kpc of the disk is mostly depolarized, then the polarized emission that is observed primarily originates from the synchrotron thick disk.As this emission propagates out of the galaxy, it only needs to pass through the remainder of the synchrotron thick disk and the thermal halo.Using the observations of edge-on galaxies, we estimate the synchrotron thick disk of NGC 6946 to have turbulent magnetic fields with a Faraday dispersion of 26 < σ RM < 37 rad m −2 , while the thermal halo Faraday dispersion is 21 < σ RM < 69 rad m −2 .Either of these layers could account for the complete coverage (5 < σ RM < 30 rad m −2 ) fit by mod.PIFS.This suggests that the turbulent cells of these layers are ubiquitous across the disk, and that all observations of λ > 10 cm are affected by this source of depolarization.This fits well within the sketch since these upper layers have larger filling factors for the Faraday rotating thermal electrons, greater turbulent cell sizes, and longer path lengths.All of these factors increase the chance of intersection along the line of sight.
The mod.PIFS also predicts a partially covering inhomogeneous screen with a Faraday dispersion of 70 < σ RM < 100 rad m −2 (top right of Figure 10).These screens must be clumpier and more dispersed than in the synchrotron thick disk because they are only predicted to cover 80%-90% of all lines of sight toward the emitting source.There are two regions in Figure 9 that produce comparable σ RM , the synchrotron thin disk and thermal thick disk.Both of these regions have smaller filling factors and higher densities associated with the turbulent cells than the upper layers (see Table 5).It is then possible that the partially covering inhomogeneous screens reside within 1 kpc of the disk.A physical interpretation of this clumpy medium could be linked to the occurrence of super-bubbles.Boomsma et al. (2008) reported the average diameter of the H I holes in NGC 6946 to be 1.2 kpc.They are assumed to be at an average vertical height of 200 pc where the gas density is 10% of the midplane density, and it is likely that all of the detected holes have broken out of the thin disk into the halo (Boomsma et al. 2008).This is comparable to the 1 kpc correlation length in RM determined by Mao et al. (2015) for the halo of M51.Within a radius of 10 kpc, the covering fraction of H I holes falls between 0% and 50% as a function of radius, and are preferentially coincident with the material and star-forming arms.Since the magnetic field detections are located in between the material arms, it is possible that the low-density superbubbles only cover up ∼10% of the polarized disk.
The sketch also gives insight into the possibility of IFD.The Faraday dispersion values from the DP (Section 3.3) fall within the range of possible σ RM predicted for both the synchrotron  thin disk and the thermal thick disk.This is also true for the Faraday dispersion values determined from fitting the IFD model to the Stokes Q and U data from λ3-22 cm (bottom left of Figure 10).However, this model does not produce physical values when applied to pixels within the central ∼3 kpc and SW quadrant.But as was shown in Section 3.5, the IFD model is more frequently rejected than the multi-layer screens.The rejection of the simple interpretation of IFD as the only threedimensional description for the turbulent fields is further supported by the inference that the observed λ18-22 cm emission originates above z ∼ 1 kpc, beyond the regions of strongest Faraday dispersion.More sophisticated models that treat both IFD and external IFS are needed in combination with more complete wavelength coverage between λ3 and 22 cm to better differentiate between these sources of depolarization and where they occur.The addition of Faraday depth measurements for ∼λ10 cm will also help to constrain the depths of emission.Unfortunately, our current 13 cm data are not adequate for such an analysis, but our estimates of its physical depth of origin suggest that the λ13 cm polarized emission should undergo more Faraday rotation than the λ18-22 cm emission (Table 6).If this is the case, the difference between the Faraday depth at λ13 and λ18-22 cm will give some indication of the magnetic fields between their emitting regions.This information will help build the three-dimensional description of the large-scale magnetic field structure in NGC 6946.

Conclusions
Two main goals motivated our new λ13 cm observations of NGC 6946: (1) to characterize and quantify the depolarization, primarily in the SW quadrant, and (2) to link the magnetic field structure in the midplane, extended thick disk, and halo.The driving idea is that the disk is Faraday thick to longer wavelengths, while CR energy losses prevent the study of the halo and thick disk at shorter wavelengths.At longer wavelength (λ > 18 cm), polarized emission is prevented from escaping from the midplane, and must originate from layers of synchrotron emission on the nearside of the midplane so that the associated Faraday rotation is a unique probe of the magnetized medium in the thick disk and halo.Since the disk is Faraday thin for shorter wavelength emission (λ < 6 cm), observations at these wavelengths are able to detect the polarized emission arising from both sides of the disk.This emission experiences Faraday rotation from the entire midplane region, thick disk, and halo.By observing emission at an energy range between these two wavelength regimes we sought to examine an intermediate depth of the magnetized ISM and add new information to our understanding of the threedimensional magnetic field structure in NGC 6946.
The addition of the λ13 cm linearly polarized emission has revealed several insights into the magnetized medium in NGC 6946.First, the λ13 cm linearly polarized emission has intermediate morphological characteristics similar to both the λ3-6 cm and λ18-22 cm, suggesting that as anticipated the observed λ13 cm emission originates at depths somewhere between the λ3-6 cm and λ18-22 cm emitting regions.This is evident from the extended emission toward the SE and NE, which is similar to λ > 18 cm, and the detection of more structure along the SW magnetic arm, as seen at λ < 6 cm.Second, though it recovers more polarized emission along the SW magnetic arm, the λ13 cm polarized emission still exhibits a large gap in the SW quadrant and is not probing down to the midplane of NGC 6946.While DFR is still a possible explanation for both the field configuration and depolarization as suggested by previous modeling attempts (e.g., Braun et al. 2010), both the lack of emission and issues with the polarization angle calibration at λ13 cm prevent a more thorough investigation at this time.Additional observations that completely fill in the missing wavelength gaps between λ3 and 22 cm are required, and are possible with the VLA for galaxies in the northern sky including NGC 6946, and the Australia Telescope Compact Array for galaxies in the southern sky.Third, the amount of depolarization that occurs between λ13-6 cm and λ22-6 cm can be explained by the same Faraday dispersion.This is under the assumption that IFD is the primary source of depolarization.In this case, the λ3-22 cm emission all originates at the same physical depth, but because the λ13 cm emission is less affected by depolarization than λ > 18 cm, λ13 cm emission probes further into the thermal thick disk, while the λ18-22 cm polarized emission only reaches into the synchrotron thick disk.
By combining the λ13 cm WSRT data with previous observations at λ3, λ6, λ18, and λ22 cm, we were able to test different line-of-sight configurations of both large-and smallscale magnetic fields across the disk of NGC 6946.We fit seven different models to the q and u spectra, and found the modified partially covering inhomogeneous Faraday screen (mod.PIFS) to be the best description across the galaxy.This suggests the vertical structure of the magneto-ionized medium in NGC 6946 can be broken into a layer of clumpy, highly turbulent, strongly Faraday depolarizing (σ RM ∼ 70-100 rad m −2 ) medium along the midplane and a more diffuse layer that completely covers the disk, but has a low Faraday dispersion (σ RM ∼ 10 rad m −2 ).The first layer likely resides within 1 kpc of the midplane, making up the synchrotron thin disk and possibly the thermal thick disk, while the second fills the synchrotron thick disk and thermal halo.We show that the best-fit parameters of this model fit within our current understanding of thin and thick disks in edge-on galaxies.
We have demonstrated the importance of obtaining wideband polarization observations in the study of the threedimensional magnetic fields in galaxies, both on large and small scales.Polarization observations at λ13 cm reach closer to the disk-halo interface than is possible with either λ3-6 cm or λ18-22 cm observations.Future polarization observations that fill the wavelength range between λ6 and 18 cm will greatly improve our ability to untangle the three-dimensional magnetic field structure of NGC 6946 and other nearby galaxies.Additional data will allow for testing of more sophisticated models of the multilayered magnetized medium.Future projects like this will also benefit from more complete UV coverage, including single-dish observations to obtain flux at the shortest spacings.This will enable more accurate determinations of the thermal and nonthermal components, and avoid assumptions about the spectral index.These types of studies will continue to further our understanding of the diskhalo interface and how material and energy are transported between them.Modeling q and u spectra will be an important tool for extracting three-dimensional magnetic field structure from full wavelength coverage between λ3 and 22 cm.field vectors, which were consistently 20°lower than the 18-22 cm values and 10°-20°lower than the observed 3 cm values.
Close examination of the calibration steps and corrected polarization data of calibrator 3C 286 did not reveal the source of the systematic rotation.The observed properties can be found in Table 7.However, when the calibrations were applied to the secondary polarization calibrator 3C 138 an excess RM of 10 rad m −2 was, again, detected.Work by Perley & Butler (2013) shows the polarization angles of both 3C 138 and 3C 286 to be stable at the L band and C band over the past ∼30 yr.While we are probing a wavelength regime that is not as extensively studied, we do not expect the polarization properties between these two well-studied bands to vary on shorter timescales.Further analysis of any intrinsic variability of the calibrator sources is beyond the scope of this work.
To address this issue, we adopted and applied an empirical correction to the λ13 cm data.Assuming simple Faraday rotation, ψ = ψ 0 + RMλ 2 , we used the RM synthesis results of the λ18-22 cm data ( 6.87 rad 18 22 s = -F ¯m−2 ) to predict the intrinsic PA for each set of Stokes Q and U images within the λ13 cm band.Plotting the observed versus the predicted values on a pixel-by-pixel basis showed a linear trend with a slope close to 1, but offset by −0.3 radians.We determined the offset for each λ13 cm map by performing a least squares linear fit to the observed versus predicted values, and used this value to derotate the Stokes Q and U maps.In Table 8, the noise for each image is listed in columns (2) and (3), and the results of the linear fit are shown in columns (5) and (6).Following normal error propagation, we determined the new noise level in the rotated Stokes Q and U images which are listed in columns (2) and (3) of Table 1.There is a ∼10% increase in the uncertainty of the individual images, but the total polarization observed at λ13 cm is unchanged by the derotation.Furthermore, we performed q, u fitting (see Section 3.5) with both the original and de-rotated λ13 cm data, and the results were unchanged.While it is encouraging that the correction does not have large effects on our results, it does suggest that the incomplete wavelength coverage does not allow us more precision than Φ ± 10 rad m −2 .This correction prevents us from extracting reliable conclusions about the Faraday depth probed by the λ13 cm data.noise in the Stokes Q (column (2)), Stokes U (column (3)), and total linear polarization (column (4)) images.

Appendix C Models Fit to q, u
We present the functions, diagrams, and plots of the best fit in a given pixel for the seven different models used for q, u fitting of NGC 6946.In each diagram, a pink disk with a spiral pattern marks the midplane of the galaxy.We use straight arrows pointing toward the radio dish to represent the line-ofsight component of large-scale magnetic fields and curved lines to represent random components.Gray regions mark volumes containing thermal electrons and where Faraday rotation would occur.Different colored magnetic fields are used to represent magnetic fields located in separate regions that may not be distinguished by the observations.
In the plots of q, u, and p, we use the best-fit parameters for each model from the same pixel, and include the BIC value for the fit.We chose a typical pixel where mod.PIFS was the best fit, but where we were unable to reject all but one model based solely on the BIC.
We start with the mixed models.At the top of Figure 11, we show DFR where there is differential depolarization due to the large-scale line-of-sight magnetic field with total Faraday depth R in the region of synchrotron emission.The complex polarization p  is then a function of the intrinsic polarization p 0 , initial polarization angle f 0 , total Faraday depth R in the emitting region, and Faraday depth of other magnetic fields along the line of sight Φ . The bottom of Figure 11 shows IFD, where there are both large-scale magnetic fields with total Faraday depth R and small-scale magnetic fields characterized by dispersion σ Φ within the emitting region

Figure 1 .
Figure1.Left column: polarized intensity maps at 6.2 cm (top), 13 cm (middle), and 22 cm (bottom).All maps are constructed with a 15″ beam, which is shown in the bottom left corner of each map.Pixels with surface brightness less than 4 × σ QU are masked.This is 0.06, 0.102, and 0.039 mJy beam −1 for the 6, 13, and 22 cm maps, respectively.The 6 cm map is made by combining observations from the VLA and Effelsberg telescope.The two longer wavelength maps were observed with the WSRT determined using RM synthesis.Regions of interest discussed in Section 3.1 are demarcated and labeled in red.Right column: Hα (grayscale) maps(Ferguson et al. 1998) overplotted with red line segments indicating the magnetic field PA at λ3.6 (top), λ13-22 cm (middle), and λ18-22 cm (bottom).The length of the line segment indicates the relative strength of the polarized intensity where a length of 15″ is equivalent to 100 μJy beam −1 .

Figure 3 .
Figure3.Faraday depth vs. azimuth angle for the three different Faraday depth maps in Figure2.Each Faraday depth map has been divided into radial bins spanning 1-4 kpc (green), 4-8 kpc (orange), and 8-12 kpc (purple), as marked by ellipses in Figure2.The Faraday depth is averaged for azimuthal bins of 10°.The weighted average and standard error for these bins are marked by horizontal and vertical lines, respectively.The best fit for a constant axisymmetric magnetic field with Faraday depth Φ 0 and PA p B inclined by angle i = 38°to the line of sight (see Equation (4)) is overplotted.The top row shows the fit to the data.The bottom row shows the residuals after the best fit has been subtracted.

Figure 4 .
Figure 4. Residual Φ after subtracting the halo component determined by fits (Figure 3) to the λ13-22 cm from the Φ maps shown in Figure 2. Black ellipses are at radii of 4, 8, and 12 kpc.

Figure 5 .
Figure5.Top: ratio between polarized intensities at 22.0 and 6.2 cm (left) and 13 and 6.2 cm (right) corrected for spectral index (α = −1.0)at 15″ resolution.Ratios were computed for points with an S/N > 4 at both wavelengths.Bottom: estimated Faraday dispersion, σ RM , at 22 cm (left) and 13 cm (right) based on the observed depolarization (maps in the top row).
1. a slab with mixed emitting (CR electrons) and rotating (thermal electrons) plasma threaded by a regular magnetic field known as DFR (degrees of freedom k = 4) (Burn 1966); 2. a modified version of IFD with a slab of mixed emitting and rotating plasma with both regular and turbulent magnetic field components (hereafter IFD will refer to this version, k = 5) (Sokoloff et al. 1998); 3. an external Faraday screen (EFS) consisting of a region of thermal electrons threaded by a regular field located between the observer and region of polarized emission (EFS, k = 3) (O'Sullivan et al. 2012); 4. an IFS consisting of a region of thermal electrons and randomly oriented magnetic fields separate from the region of polarized emission (IFS, k = 4) (Sokoloff et al. 1998); 5. an IFS consisting of a region of thermal electrons and randomly oriented magnetic fields separate from the region of polarized emission, but that only partially covers the emitting region (PIFS, k = 5) (O'Sullivan et al. 2012); 6. an IFS consisting of a region of thermal electrons and randomly oriented magnetic fields separate from the region of polarized emission experiencing IFD, but that only partially covers the emitting region (mod.PIFS; k = 6) (Farnes et al. 2014); and 7. two spatially unresolved polarized components with independent Faraday rotation (TWO, k = 6) (O'Sullivan et al. 2012).

6.
Results of fitting models to q and u data with an assumed spectral index of α = −1.1 (a) Best-fit model in each pixel.(b) Comparison of the number of models that remain after assessing the BIC.

Figure 7 .
Figure 7. Intrinsic magnetic field orientation determined from model fitting with α = −1.1.Every model that we fit to the data included a parameter for the intrinsic magnetic field orientation, and here we show the best-fit value for five of the models as described by the title over each panel.Each panel shows the Hα emission of NGC 6946 with overplotted line segments depicting the orientation of the magnetic field.The length of each segment shows the polarized intensity where a length of 15″ is equivalent to 0.50 mJy beam −1 .The top row shows mixed models: DFR (left) and internal Faraday dispersion with a large-scale component (middle).The bottom row shows screen models: IFS (left), partial IFS (middle), and modified partial IFS (right).

Figure 8 .
Figure8.The foreground-subtracted Faraday depth determined toward each pixel from five models using α = −1.1.As with the observed data, the estimated foreground contribution was determined by taking the average Φ across the entire galaxy.This value is listed for each map, and has been subtracted from each pixel to show the best-fit Faraday depth due to the line-of-sight magnetic fields in NGC 6946.Similar to Figure7, mixed models are on the top, and the three screen models are on the bottom.

Figure
Figure Diagram of vertical ISM layers in NGC 6946 informed by observations of nearby edge-on galaxies.Transparent gray arrows are superimposed on these layers, marking the paths traveled toward the observer by different wavelength polarized emission: λ22, λ13, λ3-6 cm.The diagonally striped region represents the estimated heights where the polarized emission of a given wavelength could originate-that is, where emission could be emitted and still make it out of the galaxy without being fully depolarized.Regions marked in black represent regions where the Faraday depth is completely opaque and total depolarization occurs.

Figure 10 .
Figure 10.Maps of the best-fit values of Faraday dispersion for three different models: IFD, PIFS, and the two layers of mod.PIFS.The top row shows model fitting results for screens that are separate from regions of polarization emission and that only partially cover the line of sight (PIFS on the left and one layer of mod.PIFS on the right).The bottom row shows the Faraday dispersion in the emitting region (IFD) or in a layer separate from the emitting region that fully covers the line of sight (mod.PIFS).
Figure 12, we show the simple EFS at the top.This model consists of a Faraday rotating region with total Faraday depth Φ separate from the emitting region

Figure 11 .
Figure 11.Illustrations and examples of model fitting.Each panel depicts the line of sight for one of the models fit to the data.Examples of the best fit at specific pixels (solid line) are overplotted on the observational data (points with error bars).Stokes Q/I is shown in blue, Stokes U/I is shown in green, and total polarization, P/I, in black.This figure contains the DFR in the top panel (a) and internal Faraday dispersion with a large-scale line-of-sight magnetic field component (IFD) in the bottom panel (b).

Figure 12 .
Figure 12.Illustrations and examples of model fitting, continued.This figure contains the EFS in the top panel (a) and IFS in the bottom panel (b).

Figure 13 .
Figure 13.Illustrations and examples of model fitting, continued.This figure contains the partial coverage inhomogeneous screen (PIFS) in the top panel (a) and the mod.PIFS in the bottom panel (b).

Table 1
Properties of the Images and Rotation Results

Table 2
Average Φ and Errors

Table 4
NGC 6946 Depolarization and Faraday Dispersion