BROADBAND RADIO POLARIMETRY AND FARADAY ROTATION OF 563 EXTRAGALACTIC RADIO SOURCES

To obtain the frequency-dependent source flux densities required for our spectropolarimetric analysis, we generated a second set of mosaic images of the field in Stokes I, Q, U, and V at 8 MHz frequency intervals between 1.35 and 3.1 GHz. We refer to these as our "spectropolarimetric images." We chose an 8 MHz imaging subband to achieve acceptable imaging times while preventing bandwidth depolarization for Faraday depths of ϕ ≤ 4500 rad m². For each 8 MHz subband, we imaged each of the 342 mosaic pointings as follows.

We generated the Stokes I, Q, U, and V dirty maps out to three times the primary beam width (beyond which the primary beam response is negligible) with seven resolution elements across the synthesized beam. We employed a Briggs (1995) robust weighting scheme with robustness = −2. We discarded data for baselines incorporating antenna 6 (thereby reducing the resolution of the spectropolarimetric images by a factor of ∼4 from that of the source-finding image—see below for final resolution) to decrease the amplitude of side-lobe structure and to keep the image-processing and data-storage requirements achievable. The heavily RFI-affected visibilities in the uv distance range 0–1 kλ were also discarded.

We deconvolved the dirty maps using clean. For Stokes Q and U images, we imposed a flux density cutoff threshold of four times the rms noise of the Stokes V map. For the Stokes I images, our dynamic range limit of ∼100 (limited by uv coverage) meant that clean sometimes diverged when deconvolving the brightest sources in our field. To handle this, we set the clean cutoff threshold for Stokes I images to whichever was greater out of (1) eight times the Stokes V noise or (2) 1% of the flux density of the brightest source in the field. This resulted in side lobes being cleaned to a level of about twice the noise level measured in Stokes V in almost all pointings, below the typical measured noise of ∼3–5 mJy beam⁻¹ in the Stokes I image at each frequency. Compared to theoretical and source-finding image sensitivities quoted in Sections 2 and 3, respectively, the sensitivity of our spectropolarimetric images is reduced mainly through the effect of the robustness = −2 weighting scheme.

We created restored maps, then smoothed them to a common resolution of 90'' × 45''—slightly lower than the resolution of our images at 1.35 GHz produced under the weighting scheme described above.

After imaging each of the 342 mosaic pointings in this way, we primary beam-corrected and linearly combined the maps to form the final mosaic image of the field at each frequency.

We extracted the frequency-dependent Stokes I, Q, U, and V flux densities for each source cataloged in Table 1 (Section 3) directly from the spectropolarimetric images. We determined whether sources were resolved by assessing the goodness of fit of a 90'' × 45'' Gaussian (i.e., the restoring beam of the spectropolarimetric images) to the source. The result is recorded in column 6 of Table 1. For unresolved sources, flux densities were extracted by measuring the value of the image pixel centered on the source positions obtained in Section 3. For resolved sources, we generated an image mask for each source by smoothing the source-finding image to 90'' × 45'' and masking pixels for which the signal was <5σ. We then applied this mask to the Stokes I, Q, U, and V spectropolarimetric mosaics at each frequency, summing the spectral brightness in the unmasked regions to obtain the integrated flux density for each Stokes parameter in units of Jy.

For both resolved and unresolved sources, the uncertainties on the data were estimated via direct measurement of the rms noise adjacent to the source in the Stokes I, Q, and U clean residual images and the Stokes V dirty map. These values were appropriately adjusted for resolved sources to reflect the number of statistically independent spatial regions sampled in the unmasked region.

A robust protocol for calibrating off-axis polarization leakage is not currently available for ATCA data. Instead, we used a statistical analysis of the target data (described in Section A.2) to estimate upper limits on the leakage as a function of both frequency and position in the primary beam. On this basis, we excluded from subsequent analysis data at frequencies >2 GHz and sources lying >02 from the beam center (the latter condition affected sources at the mosaic edges only). While some frequency-dependent leakage persists in the remaining data, its impact on our work is diminished, due to incoherent summation under RM synthesis (Section 4.4). An analysis of each of the sources in the field (also presented in Section A.2) shows that the maximum polarization leakage that appears in RM synthesis spectra is ≲0.3% of Stokes I for sources ≲016 from the beam center (which applies to >80% of our sample). Because of the limited number of sufficiently bright sources ≳016 from the beam center, we can only derive a weak leakage limit of <1% of Stokes I for sources occupying these beam positions.

RM synthesis (Burn 1966; Brentjens & de Bruyn 2005) exploits the following Fourier relationship to directly calculate the intensity of polarized emission from a source over a vector of ϕ values:

$$\begin{eqnarray}&&{{\boldsymbol{F}}}_{\phi }=\displaystyle \sum _{j=1}^{n}{w}_{j}{{\boldsymbol{p}}}_{j}{e}^{-2i\phi ({\lambda }_{j}^{2}-{\lambda }_{0}^{2})}/\displaystyle \sum _{j=1}^{n}{w}_{j}\end{eqnarray} \tag{ 7 }$$

where F_ϕ is the complex value of the Faraday Dispersion Spectrum (FDS) at Faraday depth ϕ, n is the number of channels, ${{\boldsymbol{p}}}_{j}$ is the complex value of the fractional polarization vector in channel j possessing a mean wavelength λ_j , w_j is a weighting term acting on ${{\boldsymbol{p}}}_{j},$ and λ₀ is the weighted mean of λ² over all channels:

$$\begin{eqnarray}&&{\lambda }_{0}^{2}=\displaystyle \sum _{j=1}^{n}{w}_{j}{\lambda }_{j}^{2}/\displaystyle \sum _{j=1}^{n}{w}_{j}.\end{eqnarray} \tag{ 8 }$$

We use λ₀ as a convenient reference wavelength (recorded in column 7 of Table 1) at which we evaluate various models and quantities. We denote such quantities using a λ₀ subscript (e.g., ${I}_{{\lambda }_{0}}$, the interpolated value of Stokes I at λ₀).

The ability of RM synthesis to reconstruct polarized emission is characterized by three main quantities: the maximum detectable Faraday depth, the maximum detectable scale of emission structures in ϕ space, and the resolution in ϕ space (see Brentjens & de Bruyn 2005). For our data, the values are (respectively) ϕ_max ≈ 4100 rad m⁻², ϕ_max−scale ≈ 140 rad m⁻², and δ ϕ ≈ 120 rad m⁻².

We applied RM synthesis to each source as follows:

• 1.  
  Fit a polynomial of order 1 to $\mathrm{log}(I)/\mathrm{log}(\nu )$ to get log_model(λ), a power-law model of the total intensity spectrum.
• 2.  
  Divide out first-order spectral index effects by dividing Stokes Q and U by I_model to obtain Stokes q and u.
• 3.  
  Apply RM synthesis to the Stokes q and u spectra, using either constant weighting (i.e., w_j  = 1) or weighting by the inverse of its variance (our results vary depending on the weighting scheme adopted; see Section 6.1.2).
• 4.  
  Deconvolve the FDS output by RM synthesis using rmclean (Heald et al. 2009). We ran our analysis using three different rmclean cutoff levels (the FDS amplitude below which the deconvolution procedure ceases to be applied); the values adopted for these levels and the reasons for their use are discussed in the following section.

The FDS and associated rmclean component model can then be used to assess the Faraday complexity of a polarized radio source.

The question of how to best detect Faraday complexity is a subtle and complicated one, and an optimal approach has not yet been found. Previous studies have identified complexity using nonlinearity in χ versus λ² (e.g., Goldstein & Reed 1984), frequency-dependent change in p (e.g., Farnes et al. 2014a), nonsinusoidal variation in Stokes q and u (e.g., O'Sullivan et al. 2012), or detection of multiple components in the FDS (e.g., Law et al. 2011). The first two of these are unsuitable for our study because

• 1.  
  linearity of χ versus λ² does not imply Faraday simplicity, despite the converse being true; and
• 2.  
  p is significantly affected by Ricean bias for faint sources. The optimal debiasing scheme is signal-dependent, as is the residual bias on the estimate of the true value of p obtained using any given scheme (Simmons & Stewart 1985).

As part of a 1.1–1.4 GHz study into characterization of Faraday complexity, Sun et al. (2015) found that Faraday complexity arising from the interference of two Faraday-thin components could be detected more reliably by identifying nonsinusoidal variations in Stokes q and u than by using any other existing method. However, no such advantage was evident for the detection of complexity arising from Faraday-thick components, and the authors conclude that, over their limited bandwidth, all currently available methods are subject to type 2 (false negative) errors for detecting Faraday-thick emission components. As we will show, the majority of Faraday complexity that we observe is probably of this type.

We proceeded by noting that our primary requirement is to make a binary classification of sources into Faraday-complex or Faraday-simple categories and that we wish to employ conservative detection criteria in any case so as to minimize type 1 (false positive) errors. We therefore made a choice to detect complexity by searching for multiple components in the FDS resulting from RM synthesis and rmclean, which is both simple and intuitive, has proven effective for wavelength coverage and spectral sampling comparable to our own (see Law et al. 2011), and, at the same time, provides estimates of the peak and dispersion in Faraday depth of the emission from our sources.

Specifically, our method identifies complexity using the second moment of the rmclean component model, as described by Brown (2011). Denoting the rmclean model as F _M and the second moment of this quantity as σ_ϕ , it is calculated it as

$$\begin{eqnarray}&&{\sigma }_{\phi }={K}^{-1}\displaystyle \sum _{i=1}^{n}{({\phi }_{i}-{\mu }_{\phi })}^{2}| {{\boldsymbol{F}}}_{M}({\phi }_{i})| \end{eqnarray} \tag{ 9 }$$

where the normalization constant K is given by

$$\begin{eqnarray}&&K=\displaystyle \sum _{i=0}^{n}| {{\boldsymbol{F}}}_{M}({\phi }_{i})| \end{eqnarray} \tag{ 10 }$$

and μ_ϕ , the first moment of the distribution, is given by

$$\begin{eqnarray}&&{\mu }_{\phi }={K}^{-1}\displaystyle \sum _{i=0}^{n}{\phi }_{i}| {{\boldsymbol{F}}}_{M}({\phi }_{i})| \end{eqnarray} \tag{ 11 }$$

where n is the number of distinct ϕ values at which F _M is nonzero, while ϕ_i is the Faraday depth of the ith Faraday depth vector component (i.e., the vector of ϕ values over which Equation (7) was evaluated to calculate the FDS). Here, σ_ϕ is a measure of the dispersion of the rmclean model components; Faraday-simple sources will show zero or very little spread (i.e., they are dominated by polarized emission from a single Faraday depth), while Faraday-complex sources with multiple rmclean components should reflect this as a larger value of σ_ϕ . In Section 6.4.3, we show that this algorithm correctly selects the most heavily depolarizing/repolarizing sources as being Faraday-complex, which acts as a posteriori evidence that our method is effective. Further comparison of spectropolarimetric analysis techniques is beyond the scope of this work, but note that this is a topic of ongoing research (see Sun et al. 2015).

The practical implementation of our method requires choices to be made for two parameters. First, a threshold must be adopted for σ_ϕ above which a source is classified as Faraday-complex. We show that a value for this threshold is naturally suggested by our data in Section 6.1. The second parameter is the rmclean cutoff, which must be set low enough to detect faint emission components, while remaining high enough to render the false-detection probability negligible. Since no formalism currently exists that describes the statistical behavior of FDS containing multiple unresolved components, this is not a trivial problem. Instead, we make use of the analytic formalism of Hales et al. (2012), which fully characterizes the detection significance of polarized emission in FDS when multiple unresolved components are absent (see also Macquart et al. 2012). The equations derived therein relate the polarized S/N of an emission component in a Faraday-simple FDS to a Gaussian-equivalent significance (GES) level; that is, a 3σ GES detection for a non-Gaussian probability density function (PDF) is equivalent to a 3σ detection for a Gaussian PDF. If the rmclean cutoff is set to a sufficiently high GES (one that ensures a negligible false-detection probability), subsequent detection of multiple FDS components then implies complexity, even if the detection significance of the additional components cannot be formally calculated.

To provide a gauge of detection significance and degree of complexity, we calculated σ_ϕ at three different rmclean cutoffs—6σ GES, 8σ GES, and 10σ GES—then used the complexity classification assigned at each level as a proxy in this regard. For example, when we refer to a source as being "complex at the 8σ GES level," we mean it was detected as complex using an 8σ GES rmclean cutoff (and therefore also a 6σ GES cutoff), but not using a 10σ GES cutoff. We then attribute a higher degree of complexity or a more robust detection to this source than to a 6σ GES complex detection, but vice versa for a 10σ GES detection. We discuss detection reliability under these rmclean cutoffs in Appendix A.4.

Using the source-finding image (see Section 3), we classified the Stokes I morphology of our sources as "unresolved" if they were well fit by a Gaussian of the same dimensions as the restoring beam; "lobe/jet component" if they either (1) possessed obvious radio lobe or jet morphology or (2) possessed a counterpart radio source within 3' of similar brightness (Hammond et al. 2012 and refs therein); "core-jet" if the total flux was dominated by an unresolved component in the presence of additional resolved components that were either (1) substantially fainter than the core or (2) radiated away from the core with linear morphology; or "extended" for resolved sources not meeting the above criteria. Our morphological classifications are presented in column 22 of Table 1. Note that these morphological classifications (made using the 15'' resolution source-finding image) are distinct from the resolution status (resolved/unresolved in the 1' resolution spectropolarimetric images) that we assigned to the sources for the purpose of spectropolarimetric data extraction in Section 4.2.

We cross-matched sources with counterparts in infrared (WISE; Wright et al. 2010), optical (SuperCOSMOS; Hambly et al. 2001), and ultraviolet (GALEX) images and in the ROSAT All Sky Survey Bright Source Catalog (RASS-BSC; Voges et al. 1999) and Faint Source Catalog (RASS-FSC; Voges et al. 2000). In IR, optical, and UV, we manually identified counterparts by overlaying contours from the source-finding image onto the relevant survey images. We assigned a "match" when objects were present within 15'' of the radio contour centroid (i.e., within the beam FWHM of the source-finding image); a "match—off source" if either (1) the radio source was extended and a candidate counterpart source lay inside the 10% radio flux contour, (2) a second radio source was located within 3' and a candidate counterpart was located within 15'' of the position of the flux centroid of the two sources (e.g., Best et al. 2005), or (3) NASA Extragalactic Database (NED) queries revealed an existing association in the literature; or "no match" otherwise. The counterpart statuses in IR, optical, and UV are listed in columns 23, 24, and 25 of Table 1, respectively. Our X-ray cross matches were made in a binary manner based on the distance to the nearest cataloged sources and the positional errors for the RASS-BSC and RASS-FSC (Parejko et al. 2008). We assigned a "match" if the radio emission centroid fell within 20'' of a cataloged RASS-BSC source position or 40'' of a RASS-FSC source position, and "no match" otherwise. The X-ray counterpart classifications are listed for each source in column 26 of Table 1.

We furthermore cross-matched our sample with redshift data in the literature using NED and a matching radius of 1', approximately the FWHM of the spectropolarimetric image restoring beam. We accepted 61 redshift matches for our sample, 21 of which were polarized sources. These are recorded in column 27 of Table 1.

In this section we present the Faraday-complexity classifications obtained by applying the methods described in Sections 4.4 and 4.5. Initially we describe the results obtained using a 6σ GES rmclean cutoff and natural weighting in RM synthesis (defined as ${{w}}_{j}={\sigma }_{{qu},j}^{-2}$ in Equations (7) and (8)). We then explain how these classifications are affected when the rmclean cutoff or RM synthesis weighting scheme is changed in Sections 6.1.1 and 6.1.2.

We classified as unpolarized the 403 of 563 sources for which max($| \mathrm{FDS}|$) < rmclean cutoff (6σ GES). The remaining 160 were classified as polarized, yielding ∼5 polarized sources per square degree. This value is consistent with estimates of polarized source density in the literature (e.g., Stil et al. 2014), given the median 6σ GES rmclean cutoff we obtain is equivalent to P ≈ 0.6 mJy beam⁻¹ (rmsf beam)⁻¹.

We calculated σ_ϕ for each polarized source (recorded in column 15 of Table 1) and present a histogram of the resulting values in Figure 2. Three populations are evident. The first consists of sources with σ_ϕ  = 0: all rmclean components lie at precisely the same Faraday depth. The second at σ_ϕ ≈ 0.1 rad m⁻² occurs when all rmclean components are detected in two adjacent bins in ϕ space (separated by 0.1 rad m⁻²), with only a small minority of components in the second bin. We classified the 146 sources comprising both this and the σ_ϕ  = 0 population together as Faraday-simple.

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The third population contains 14 sources (12% of the polarized source sample), with σ_ϕ values in the range 10–300 rad m⁻². A clear division exists between the first two populations and the third, which suggests a σ_ϕ threshold to use for distinguishing between Faraday-simple and Faraday-complex sources. We classified this last population of sources as Faraday-complex, noting here that the Faraday classifications thereby obtained agree well with the depolarization characteristics of sample sources presented in Section 6.4.3.

6.1.1. Effect of rmclean Cutoff Threshold on Classifications

6.1.2. Effect of RM Synthesis Weighting on Classifications

The weighting scheme used in RM synthesis can affect sensitivity to Faraday complexity. In our observations, primary beam attenuation causes a small but systematic increase in the uncertainty of (q, u) data for off-axis sources toward higher frequencies. Under these circumstances, natural weighting will decrease sensitivity to depolarizing signals. We therefore repeated the analysis of the previous two sections using constant weighting (i.e., ${{w}}_{j}=1$ in Equations (7) and (8)). The resulting classifications are presented in Table 3, and we compare the classifications made under the natural- and constant-weighting schemes in Table 4. The majority of 10σ GES detections are unaffected by the weighting scheme employed. At 6σ GES, however, only a single source is classified as complex under both schemes, with the majority of complex detections occurring when constant weighting is used. Regardless of the cause of this effect, it is clear that choosing between the two weighting schemes may affect our ability to detect specific types of complexity (e.g., strongly depolarizing signals). Thus, we consider sources to be complex if they are detected as such under either natural or constant weighting. This results in a total of 19 sources being classified as complex using a 6σ GES rmclean cutoff.

Table 3. Number of Sources in Each Faraday Category as a Function of rmclean Cutoff GES Level Using Constant Weighting in RM Synthesis

Faraday classification	6σ GES	8σ GES	10σ GES
Unpolarized	400	438	461
Simple	145	114	93
Complex	18	11	9

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Table 4. Number of Elements in the Union, Intersection, and Differences of Sets Comprising the Faraday-complex Source Detections Using Constant (C) and Natural (N) RM Synthesis Weighting Schemes and That Are Detected at the Specified GES rmclean Level but Not Higher (e.g., Sources Detected as Complex Using a 10σ GES Cutoff Are Also Detected as Such Using a 6σ GES Cutoff but Are Not Counted in the 6σ GES Column)

Set	6σ GES	8σ GES	10σ GES
$| C\cup N|$	7	1	11
$| C\cap N|$	1	1	7
$| N\setminus C|$	1	0	2
$| C\setminus N|$	5	0	2

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We now present the spectropolarimetric data for all sources classified as complex, regardless of the weighting scheme and rmclean cutoff under which they were detected as such. These data are presented in Figures 3–21. We also include data for selected Faraday-simple sources (Figures 22–24) and unpolarized sources (Figures 25–27). The complexity classification of each source is noted in the figure captions. The data are presented in seven panels. The uppermost panel (panel (a)) shows the result of applying RM synthesis to the source data. It includes plots of the magnitude of the FDS, the rmclean component distribution, and the rmclean cutoff level, as described in the figure captions. To more clearly show the multiple rmclean components characteristic of complex sources, we include an inset panel zoomed in on $| \phi | \lt 750\;{\rm{rad}}$ m⁻² and $0\lt | \mathrm{FDS}| \lt 3\times$ the rmclean cutoff. Panel (b) shows Stokes I(λ²) with its fitted model. Panels (c) and (e) show the Stokes q(λ²) and u(λ²) and Stokes v(λ²) spectra, respectively. Panel (d) plots fractional polarization against λ², along with the best-fit depolarization models described in Section 6.4.3. Panel (f) plots polarization angle versus λ².

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In the following sections, we examine several nonpolarimetric properties of the sample with the aim of identifying characteristics that distinguish Faraday-complex sources.

6.3.1. Source Flux Density

6.3.2. Spectral Index

6.3.3. Radio Morphology

6.3.4. Multiwavelength Counterparts

6.3.5. Redshift

In Table 6 we present the median value of cross-matched redshifts as a function of Faraday category. The overlapping 66% confidence intervals for the median of each category demonstrate that any relationship between Faraday complexity and redshift must be substantially weaker than can be detected with our limited number of redshift cross matches.

Table 6. Number of Redshift Matches and Median Redshift for Cross-matched Sources in Our Sample, Split by Faraday Classification

	Complex	Simple	Unpolarized
# Matches	8	12	41
Median ${}_{66\%\;\mathrm{CI}\;\mathrm{lower}}^{66\%\;\mathrm{CI}\mathrm{upper}}$	${0.11}_{0.11}^{0.28}$	${0.18}_{0.14}^{0.21}$	${0.12}_{0.11}^{0.19}$

Note. Upper and lower bounds on the 66% confidence intervals for the median are also shown.

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6.3.6. Location in the Mosaic Field

We now examine several polarimetric properties of the sample sources, with the aim of determining the characteristics of complex sources and how these characteristics distinguish complex sources from simple ones.

6.4.1. Structure in Complex Faraday Dispersion Spectra

6.4.2. Faraday Depth

Here we determine whether the Faraday depth at which a source's emission peaks is correlated with its Faraday complexity classification. The existence of such a correlation would indicate that complexity is generated in regions possessing anisotropic ordered ${\boldsymbol{B}}$ field components—for example, jets in AGN.

We first calculate the residual peak Faraday depth of each source (${\phi }_{\mathrm{res},\mathrm{peak}}$) by subtracting the smooth Galactic contribution (${\phi }_{\mathrm{gal},\mathrm{peak}}$) to ${\phi }_{\mathrm{peak}},$ which is well fit by the following paraboloid:

$$\begin{eqnarray}&&{\phi }_{\mathrm{gal},\mathrm{peak}}({{\rm{R.A.}}}_{\mathrm{rel}},{{\rm{Decl.}}}_{\mathrm{rel}})=0.792{({{\rm{R.A.}}}_{\mathrm{rel}}-0.149)}^{2}+...\\ &&\quad 1.040{({{\rm{Decl.}}}_{\mathrm{rel}}+2.052)}^{2}+1.6331\mathrm{rad}\;{{\rm{m}}}^{-2},\end{eqnarray} \tag{ 12 }$$

where, with R.A. and decl. in J2000 coordinates

$$\begin{eqnarray*}&&{{\rm{R.A.}}}_{\mathrm{rel}}={\rm{R.A.}}-52\buildrel{\circ}\over{.} 385\;\;\mathrm{and}\;\;{{\rm{Decl.}}}_{\mathrm{rel}}={\rm{Decl.}}+35\buildrel{\circ}\over{.} 793.\end{eqnarray*}$$

The residual Faraday depth of each source is then calculated as ${\phi }_{\mathrm{res},\mathrm{peak}}={\phi }_{\mathrm{peak}}-{\phi }_{\mathrm{gal},\mathrm{peak}}.$ It is difficult to use ${\phi }_{\mathrm{res},\mathrm{peak}}$ directly to compare the dispersion of ${\phi }_{\mathrm{res},\mathrm{peak}}$ (not to be confused with σ_ϕ ) for simple and complex sources because the dominant contribution to this dispersion is signal-dependent. Instead we calculate the standardized residual (SR) of ϕ_res,peak for each source. This is defined as

$$\begin{eqnarray}&&\mathrm{SR}={\phi }_{\mathrm{res},\mathrm{peak}}/{\sigma }_{\mathrm{ex}+\mathrm{err}}\end{eqnarray} \tag{ 13 }$$

where

$$\begin{eqnarray}&&{\sigma }_{\mathrm{ex}+\mathrm{err}}^{2}={\sigma }_{\mathrm{err}}^{2}+{\sigma }_{\mathrm{ex}}^{2}\end{eqnarray} \tag{ 14 }$$

and ${\sigma }_{\mathrm{err}}^{2}$ and ${\sigma }_{\mathrm{ex}}^{2}$ are the contributions to ${\sigma }_{{ex}+{err}}$ from measurement error and the mean extragalactic contribution to the Faraday depth of a radio source, respectively. We use ${\sigma }_{\mathrm{ex}}^{2}=49$ rad² m⁻⁴ (Schnitzeler 2010). If the value adopted for ${\sigma }_{\mathrm{ex}}^{2}$ and the function describing the Galactic foreground are both accurate, SR should be Gaussian distributed with μ = 0 and σ = 1. We confirm that this is the case, finding μ = −0.06 ± 0.08 and σ = 1.09 ± 0.08 for the best-fit Gaussian to the histogram of SR data (both plotted in the bottom panel of Figure 33).

In the upper panel of Figure 33, we plot ${P}_{{\lambda }_{0}}$ versus SR so the variance in SR can be easily seen. Most sources, both complex and simple, are located within 1–2σ of zero and thus show no statistically significant enhancement in ${\phi }_{\mathrm{res},\mathrm{peak}}.$ However, several Faraday-complex sources do show statistically significant enhancements in ϕ_res,peak. While no Faraday-simple sources lie >5.5σ from the mean, four complex sources have ϕ_res,peak values between 6.5σ and 50σ. This is clearly statistically unlikely, even allowing for the slight nonnormality of the SR distribution (bottom panel). The names and $| {\phi }_{\mathrm{peak}}|$ values of these sources are 033843-352335 (Figure 5): 59 rad m⁻², 033829-352818 (Figure 11): 145 rad m⁻², 033754-351735 (Figure 13): 165 rad m⁻², and 032006-362044 (Figure 12): 300 rad m⁻². We note that the first three of these sources are among the prominent cluster of complex sources identified in Section 6.3.6.

6.4.3. Depolarization or Repolarization

Here we determine the extent to which Faraday complexity is associated with either depolarization or repolarization, which broadly constrains the possible mechanisms through which complexity is generated. For depolarizing sources, we then fit (existing) depolarization models to p(λ²) to obtain additional information about the depolarization behavior.

First we identify depolarizing/repolarizing sources by generating two sets of channels $L=\{i| {\lambda }_{i}\gt {\lambda }_{0}\}$ and $H=\{i| {\lambda }_{i}\lt {\lambda }_{0}\}$ and calculating the following depolarization ratio:

$$\begin{eqnarray}&&{\mathrm{DP}}_{L/H}=\displaystyle \frac{{\displaystyle \sum }_{i\in L}{p}_{i,\mathrm{corr}}/| L| }{{\displaystyle \sum }_{i\in H}{p}_{i,\mathrm{corr}}/| H| }\end{eqnarray} \tag{ 15 }$$

where ${p}_{i,\mathrm{corr}}=\sqrt{{p}_{i}^{2}-{\sigma }_{{qu},i}^{2}}$ is the bias-corrected polarization in channel i (see Simmons & Stewart 1985), $| L|$ and $| H|$ are the number of channels in the sets L and H, and the subscript L/H denotes that this is a ratio of the lower frequency band to the higher frequency band. We plot DP_L/H against ${P}_{{\lambda }_{0}}$ in Figure 34, indicating the complexity/polarization category of each source with the plot markers (see caption) and the median value of ${\mathrm{DP}}_{L/H}$ for Faraday-simple sources with a blue dashed line.

The simple sources are located in an envelope clustered around a median of 0.9, which broadens as the S/N drops with decreasing ${P}_{{\lambda }_{0}}.$ The offset from ${\mathrm{DP}}_{L/H}=1$ is statistically significant but caused by the mean residual bias from our polarization bias correction. The complex sources are mostly cleanly separated from the simple sources in the ${P}_{{\lambda }_{0}}$–${\mathrm{DP}}_{L/H}$ plane, being found primarily along the high ${P}_{{\lambda }_{0}},$ low ${\mathrm{DP}}_{L/H}$ (i.e., higher S/N, stronger depolarization) edge of the source distribution. The observed agreement between the presence or lack of depolarization or repolarization and the assigned complex or simple Faraday classification represents a posteriori evidence that our complexity classification algorithm is effective at distinguishing genuinely complex sources from simple ones.

Eighteen (of 19 total) complex sources have ${\mathrm{DP}}_{L/H}\lt 1$, and 17 have ${\mathrm{DP}}_{L/H}\lt 0.9$ (i.e., the simple source median). The complexity in our sample is therefore predominantly a result of source depolarization. The single exception is the repolarizing source 033653-361606 (Figure 14).

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Information about the physical nature of depolarizing Faraday screens can be obtained by modeling the behavior of p(λ²). We consider three depolarization models that, in combination, are capable of generating most of the p(λ²) behaviors that we observe. They are

(1) the Burn (1966) foreground screen, in which depolarization results from a large number of unresolved cells in the telescope beam:

$$\begin{eqnarray}&&p(\lambda )={p}_{[\lambda =0]}{e}^{-2{\sigma }_{\mathrm{RM}}^{2}{\lambda }^{4}};\end{eqnarray} \tag{ 16 }$$

(2) a Tribble (1991, 1992) screen, with N independent Faraday depth cells in the synthesized beam area (where $N\propto {t}^{2}/{s}_{0}^{2},$ s₀ is the cell scale length, and the synthesized beam FWHM = 2t $\sqrt{\mathrm{ln}2}):$

$$\begin{eqnarray}{p}^{2}(\lambda ) & = & {p}_{[\lambda =0]}^{2}\displaystyle \frac{1-\mathrm{exp}(1-{s}_{0}^{2}/2{t}^{2}-4{\sigma }_{\mathrm{RM}}^{2}{\lambda }^{4})}{1+8{\sigma }_{\mathrm{RM}}^{2}{\lambda }^{4}{t}^{2}/{s}_{0}^{2}}...\\ & & +{p}_{[\lambda =0]}^{2}\mathrm{exp}(1-{s}_{0}^{2}/2{t}^{2}-4{\sigma }_{\mathrm{RM}}^{2}{\lambda }^{4});\end{eqnarray} \tag{ 17 }$$

(3) interference of polarized components at two separate Faraday depths:

$$\begin{eqnarray}&&p(\lambda )=| {p}_{1}{e}^{2i{\chi }_{1}+{\phi }_{1}{\lambda }^{2}}+{p}_{2}{e}^{2i{\chi }_{2}+{\phi }_{2}{\lambda }^{2}}| .\end{eqnarray} \tag{ 18 }$$

In the equations above, p(λ) is the fractional polarization amplitude at wavelength λ, ${p}_{[\lambda =0]}$ is the fractional polarization at λ = 0, σ_RM is the dispersion in RM across a source due to a foreground Faraday screen, and p_i , ϕ_i , and χ_i are the fractional polarization amplitude, Faraday depth, and polarization angle of the ith polarized emission component.

We used the "Emcee" sampler package (Foreman-Mackey et al. 2013) to find maximum likelihood values and errors for parameters in these models when fit to p(λ²). For the Tribble model, single-resolution observations cannot break the degeneracy between σ_RM and s₀/t (Tribble 1991). We present maximum likelihood values for these parameters (without errors) regardless to show that the Tribble model provides a demonstrably superior reproduction of p(λ²) for some sources. We list best-fit parameter values and ${\bar{\chi }}^{2}$ in Table 7. We also record the value of the Akaike information criterion (AIC; Akaike 1974), where AIC $=\;2k-2\mathrm{ln}(L)$, k is the number of model parameters, and L is the maximum of the likelihood distribution returned by Emcee. Briefly, the AIC is a model selection criteria that is optimal in the sense that, for two models 1 and 2 in which model 1 has the lowest AIC value, model 2 is $\mathrm{exp}(({\mathrm{AIC}}_{1}-{\mathrm{AIC}}_{2})/2)$ times as likely as model 1 to minimize the information lost by either underfitting or overfitting the data. We consider model 1 significantly favored when ${\mathrm{AIC}}_{1}+5\lt {\mathrm{AIC}}_{2}.$ For a model where this is true against both other models, we list its AIC and ${\bar{\chi }}^{2}$ values in underlined bold text in Table 7. The best fit obtained with each model is overplotted on the p(λ²) plots presented in Section 6.2.

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Table 7. Reduced χ² and Values for Selected Best-fit Parameters for the Burn, Tribble, and Two Polarized Component Model Fits to p(λ²) for Each Complex Source as Described in the Text

	Burn				2 comp.					Tribble
Src. Name	${\bar{\chi }}^{2}$	AIC	p0	σRM	${\bar{\chi }}^{2}$	AIC	p1	p2	$| {\phi }_{1}-{\phi }_{2}|$	${\bar{\chi }}^{2}$	AIC	p0	σRM	s0/t
				(rad m−2)					(rad m−2)				(rad m−2)
032006-362044	1.1	−257.0	0.12 ± 0.02	9 ± 3	1.1	−238.7	0.12 ± 0.03	0.13 ± 0.02	41 ± 1	1.1	−232.9	0.16	28.0	2.1
032228-384841	10.6	−137.1	0.0470 ± 0.0005	8.2 ± 0.2	9.4	−229.9	0.023 ± 0.008	0.023 ± 0.008	16 ± 2	8.9	−228.8	0.047	8.7	0.56
033019-365308	1.8	−496.0	0.089 ± 0.002	10.2 ± 0.4	1.8	−492.9	0.021 ± 0.004	0.071 ± 0.004	27 ± 4	1.8	−496.0	0.099	51.0	4.7
033123-361041	0.9	−475.4	0.045 ± 0.005	18 ± 1	1.1	−469.8	0.019 ± 0.006	0.019 ± 0.006	29 ± 3	1.0	−473.4	0.065	26.0	0.6
033147-332912	3.5	−471.3	0.106 ± 0.002	19.3 ± 0.2	4.0	−449.9	0.0586 ± 0.0007	0.0370 ± 0.0006	33.5 ± 0.3	2.9	−503.0	0.13	24.0	0.51
033242-363645	2.5	−310.9	0.329 ± 0.009	10.4 ± 0.5	2.6	−306.5	0.06 ± 0.1	0.2 ± 0.1	31 ± 40	2.5	−277.6	0.3	13	1
033329-384204	2.7	−347.1	0.079 ± 0.005	14.1 ± 0.9	2.8	−342.2	0.048 ± 0.009	0.03 ± 0.01	27 ± 5	2.6	−350.3	1.2	55.0	0.21
033653-361606	8.6	−257.3	0.007 ± 0.02	0.009 ± 5	1.1	−644.5	0.0046 ± 0.0003	0.0073 ± 0.0003	64 ± 1	8.8	−255.0	0.0068	0.31	15.0
033725-375958	3.2	−298.9	0.239 ± 0.005	8.4 ± 0.4	3.3	−295.2	0.20 ± 0.01	0.05 ± 0.02	24 ± 6	3.2	−297.4	0.25	58.0	7.6
033726-380229	0.7	−312.8	0.20 ± 0.03	20 ± 2	0.9	−308.4	0.08 ± 0.02	0.08 ± 0.02	31 ± 3	0.7	−310.5	0.2	19.0	0.13
033754-351735	0.2	3.6	19 ± 30	60 ± 700	0.1	7.1	0.4245 ± 0.001	0.1812 ± 0.001	272 ± 1	0.1	5.3	1.7	35.0	0.68
033828-352659	5.7	−273.1	0.0156 ± 0.0001	0.03 ± 0.01	5.4	−327.8	0.019 ± 0.001	0.007 ± 0.002	49 ± 3	5.9	−298.7	0.016	1.9	8.0
033829-352818	1.8	−331.8	0.25 ± 0.02	24.2 ± 1	2.1	−319.2	0.076 ± 0.005	0.094 ± 0.004	33.9 ± 0.7	1.8	−330.9	0.28	26.0	0.19
033843-352335	5.3	−412.5	0.090 ± 0.001	16.9 ± 0.2	4.6	−452.5	0.0320 ± 0.0006	0.0597 ± 0.0006	34.2 ± 0.3	2.8	−546.9	0.15	30.0	0.73
033848-352215	6.1	−444.5	0.0843 ± 0.0004	13.39 ± 0.07	6.7	−422.2	0.057 ± 0.001	0.0259 ± 0.0006	27.2 ± 0.6	6.9	−410.6	0.088	16.0	0.75
034133-362252	4.1	−487.4	0.0358 ± 0.0009	15.1 ± 0.3	4.2	−485.4	0.024 ± 0.001	0.0117 ± 0.0005	30 ± 1	4.1	−484.7	0.036	16.0	0.47
034202-361520	1.0	−116.7	1.7 ± 0.2	22 ± 1	1.0	−115.5	0.58 ± 0.06	0.80 ± 0.05	34.6 ± 0.8	1.2	−107.3	1.6	23.0	0.51
034205-370322	5.5	−530.3	0.0656 ± 0.0002	7.63 ± 0.06	3.6	−629.3	0.0602 ± 0.0003	0.0096 ± 0.0002	29.9 ± 0.6	4.6	−580.1	0.068	56.0	8.4
034437-382640	3.4	−388.8	0.084 ± 0.002	9.4 ± 0.4	3.5	−389.4	0.074 ± 0.003	0.017 ± 0.002	31 ± 3	3.4	−391.0	0.091	30.0	3.1

Note. Fit parameters are bolded where specific models provide a clearly superior AIC value, as described in the main text.

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The results show that a variety of depolarizing behaviors are present. The sources 032228-384841 (Figure 7), 032006-362044 (Figure 12), and possibly 033829-352818 (Figure 11) show complicated oscillatory depolarization that the models we fit to the data are not capable of reproducing. We do not attempt to model this behavior here, but note that doing so would require >2 polarized components or even Faraday-thick components. The Tribble model is strongly favored for the sources 033147-332912 (Figure 6) and 033843-352335 (Figure 5), implying Faraday screens structured on angular scales not much smaller than the sources themselves. Finally, the core-dominated source 033653-361606 (Figure 14) cannot be fit with depolarization models; two Faraday-thin components are required to fit its observed repolarization.

We generally observe σ_ϕ in the range 0–25 rad m⁻² for sources best fit by the Burn model and $| {\phi }_{1}-{\phi }_{2}|$ in the range 20–50 rad m⁻² for those best fit by a double-component model. While the model parameters are poorly constrained for the Tribble model fits, we note that it requires σ_ϕ at least as large as the Burn model, but generally larger depending on the factor s₀/t.