A study of broadband Faraday rotation and polarization behaviour over 1.3–10 GHz in 36 discrete radio sources
We present a broadband polarization analysis of 36 discrete polarized radio sources over a very broad, densely-sampled frequency band. Our sample was selected on the basis of polarization behaviour apparent in narrowband archival data at 1.4 GHz: half the sample show complicated frequency-dependent polarization behaviour (i.e. Faraday complexity) at these frequencies, while half show comparatively simple behaviour (i.e. they appear Faraday simple). We re-observed the sample using the Australia Telescope Compact Array (ATCA) in full polarization, with 6 GHz of densely sampled frequency coverage spanning 1.3 to 10 GHz. We have devised a general polarization modelling technique that allows us to identify multiple polarized emission components in a source, and to characterize their properties. We detect Faraday complex behaviour in almost every source in our sample. Several sources exhibit particularly remarkable polarization behaviour. By comparing our new and archival data, we have identified temporal variability in the broadband integrated polarization spectra of some sources. In a number of cases, the characteristics of the polarized emission components, including the range of Faraday depths over which they emit, their temporal variability, spectral index, and the linear extent of the source, allow us to argue that the spectropolarimetric data encodes information about the magnetoionic environment of active galactic nuclei themselves. Furthermore, the data place direct constraints on the geometry and magnetoionic structure of this material. We discuss the consequences of restricted frequency bands on the detection and interpretation of polarization structures, and implications for upcoming spectropolarimetric surveys.
Subject headings:techniques: polarimetric – galaxies: magnetic field – radio continuum: galaxies
Radio-loud Active Galactic Nuclei (AGN) play a key role in driving the evolution and ecology of the Universe by feeding energy and material into their broader environment. However, we still lack a clear picture of the exact physical processes involved, and indeed, of the detailed physical structure of AGN, jets, and radio lobes more generally. Arguably, the main impediment to an improved understanding of these objects is the observational challenge of spatially resolving the structures of interest, which are small in physical extent and lie at great distance. Very Long Baseline Interferometry (VLBI) observations represent one powerful method of approaching this problem, but at the current time, still provides limited survey capability and limited instantaneous frequency coverage, and key structures of interest still remain near or below the resolution limit (e.g. Taylor & Zavala 2010).
Often, the unresolved issues surrounding AGN fuelling and feedback have, at their heart, a set of fundamental questions concerning the structure and dynamics of magnetized plasma associated with AGN. To address these questions, an alternate and complementary approach to VLBI-based studies is available: the frequency-dependant signal encoded in linearly polarized emission by Faraday rotation can be used to probe magnetoionized structure towards radio sources on spatial scales below the observing resolution (e.g. Gardner & Whiteoak 1966, Burn 1966, Gaensler et al. 2015, Heald et al. 2015).
The way in which this information is encoded can be described mathematically as follows: First, the linear polarization state of radio emission is described by a complex vector . It is related to the Stokes parameters & , the polarization angle , the fractional polarization and the total intensity as:
After being emitted at a distance , linearly polarized radiation will be Faraday rotated by magnetized plasma along the line of sight (LOS) to an observer by an amount equal to
where is the polarization angle, is the observing wavelength, and is the Faraday depth, given by
and, in turn, [cm] & [G] are the thermal electron density and magnetic field along the LOS respectively.
The net observable polarization is obtained by summing the polarized emission emerging from all possible Faraday depths within the synthesized beam of the telescope:
The function (the so-called Faraday Dispersion Function, henceforth FDF) specifies the distribution of polarized emission as a function of Faraday depth along the LOS, and possesses units of Jy m rad for a source which is extended both in the plane of the sky and in Faraday depth. Based on the form of this function, two broad types of behaviour can result. When is a -function (i.e. polarized emission emanates from a single Faraday depth), the polarization behaviour is characterized by constant , linear (modulo radians), and a quantity known as the rotation measure (RM) — the gradient of . The RM parameterizes the magnitude of the Faraday rotation effect, and can be related to magnetized structure along the LOS through a relation similar to Eqn. 3. Ultimately this Faraday simple behaviour is an idealization, but one which individual polarized emission components from a source can closely approximate (e.g. O’Sullivan et al. 2012). Conversely, when polarized emission occurs over a range of Faraday depths, or if multiple polarized emission components are emitted from different Faraday depths, we refer to a source as being Faraday complex. In this case, Stokes /, /, & can all show complicated behaviours which reveal magnetoionized material along the LOS, and can be modelled to reveal the structure and properties of this material on scales smaller than the resolution of the observing instrument.
A number of authors have exploited Faraday complex behaviour to study magnetoionized structures towards unresolved radio sources. Early work by Slysh (1965) and Goldstein & Reid (1984) established the existence of Faraday complex behaviour in discrete radio sources using narrowband polarization data. Observations over broad (but sparsely sampled) frequency bands ( GHz) appear to reveal the existence and properties of magnetized material in the vicinity of radio sources themselves (e.g. Conway et al. 1974, Farnes et al. 2014, Pasetto et al. 2015). Other studies have reached similar conclusions by studying narrower, GHz-wide bands with high sampling density (Law et al. 2011, O’Sullivan et al. 2012). In general though, it remains unclear exactly which structures in radio sources are being studied, or indeed, whether other structures along the LOS also make a substantial contribution to the observed polarization behaviour. For example, Anderson et al. (2015) recently analysed 160 polarized sources over 1.3–2 GHz and found evidence for a significant Galactic contribution to the Faraday complexity of background sources.
As a tool for studying the physics of unresolved magnetoionic structure in the radio source population, broadband spectropolarimetry will come of age in the approaching era of wide field, broadband surveys such as the Polarization Sky Survey of the Universe’s Magnetism (POSSUM; Gaensler et al. 2009) and the VLA Sky Survey (VLASS; VSSG 2015). However, even these surveys will have bandwidths which are narrow compared to what can be achieved with contemporary targeted observations of a (much) smaller sample. Currently, we lack a detailed survey of the near-continuously sampled, multi-GHz-bandwidth polarization properties of Faraday complex radio sources. It remains a largely open question what the broadband polarization spectra of AGN look like. The study we present here aims to address this issue by presenting a spectropolarimetric survey of confirmed Faraday complex sources over a very broad 1.3–10 GHz band. Specifically, we seek to address the following questions: To what extent and over what wavelength ranges do deviations from Faraday simple behaviour occur? What are the characteristic Faraday depths and dispersions of the emission components responsible for these deviations? Is it possible to model, in a simple way, polarization data over fractional bandwidths approaching 200%? To what extent can the properties of individual polarized emission components be distinguished and characterized? Where along the LOS are the emitting/rotating/Faraday dispersing medium(s) located? In particular, are the Faraday effects generated in the sources themselves (thus providing a method for their study) or in the foreground? What are the implications and opportunities for upcoming broadband spectropolarimetric surveys? Finally, we note what our study is not, which is an in-depth investigation of the physics of individual sources. This would require complementary multiwavelength data which is not presently available for most objects in our sample.
Our paper is set out as follows. We describe the construction of a sample to address our questions in Section 2, our observations and their calibration in Section 3, and our spectropolarimetric analysis in Section 4, including imaging, modelling and fitting the data. Ancillary non-spectropolarimetric analysis is described in Section 5. We present our results in Section 6 and our discussion in Section 7. A summary of our work and our conclusions are provided in Section 8.
2. Sample construction
The targets for our study were selected from amongst two archival polarization data sets (detailed in Table 1), which were observed and processed by several independent groups. Kim et al. (1998) conducted a 130 square degree survey of the Large Magellanic Cloud (LMC), during which polarization data was recorded commensally. Gaensler et al. (2005) reprocessed this data and imaged it in full polarization between 1.328 & 1.432 GHz. Feain et al. (2009) observed a 34 square degree region around the radio galaxy Centaurus A, imaging the data in full polarization between 1.296 & 1.480 GHz. For the combined total of 572 sources robustly detected in linear polarization in these data, Gaensler et al. (2005) and Feain et al. (2009) extracted Stokes , , & flux densities at 8 MHz intervals through the respective bands. We applied RM synthesis (Brentjens & DeBruyn 2005) & rmclean (Heald et al. 2009) to these data, identifying Faraday complex sources using the second moment () of the rmclean component distribution (Brown et al. 2011, Anderson et al. 2015). This provides a measure of the spread of rmclean components in space, and is given by:
where the normalization constant K is given by
and , the first moment of the distribution, is given by
where is the number of channels possessing non-zero values in the rmclean model and is the Faraday depth of channel . We identified sources for which rad m as candidates for our Faraday complex sample (e.g. see Anderson et al. 2015), and sources with rad m (i.e. Faraday simple sources) as a control sample. In addition, we only selected sources where the chance of foreground contamination from the LMC and Centaurus A radio lobes was minimal. In the former case, we selected only sources which fell outside the boundaries of the LMC, and in regions with an emission measure consistent with zero (i.e. with a low column density of free electrons — see the lefthand panel of Fig. 1). In the latter case, we generally selected sources which sat outside the Centaurus A radio lobes, though we chose to include several sources which do not meet this criteria (see the righthand panel of Fig. 1). We note that our choice of array configurations and weighting scheme (see Section 4.1) results in the diffuse emission from the lobes being completely resolved out. As such, the only way that the lobes could affect the level of Faraday complexity in our polarization data is if RM variations within the lobes act as a depolarising foreground screen. However, Feain et al. (2009) showed that the lobes do not appear to show RM structure on angular scales below , which all of our sources are considerably smaller than.
In total, 7 (13) complex and 7 (13) simple sources were selected from the LMC (Centaurus) datasets, with band-averaged, undebiased (see Simmons & Stewart 1985) polarized signal-to-noise (S/N) ratios of between 4 & 350. In this paper, we use source designations that begin with either ‘cena’ or ‘lmc’ to specify the data set that each source was selected from, followed by a ‘c’ or ‘s’ to specify whether the source was selected because it was Faraday complex or simple, and which end with a sequential numerical designation — e.g. lmc_c04. In addition, we have added our two phase calibrators to the sample (named 0515-674 and 1315-46 in this work; see Section 3). All of these sources are listed in column 1 of Table 2 along with their corresponding RA and Dec. (columns 2 & 3 respectively).
|[GHz]||[”]||[rad m]||[rad m]||[rad m]|
|LMC (Archival)||1.328–1.432||1994 Oct 26 – Nov 9||750D||6060||490||72||3200|
|”||”||1995 Feb 23 – Mar 11||750A||”||”||”||”|
|”||”||1995 Jun 2–7||750B||”||”||”||”|
|”||”||1995 Oct 15–31||750B||”||”||”||”|
|”||”||1996 Jan 27 – Feb 8||750C||”||”||”||”|
|Cen A (Archival)||1.296–1.480||2006 Dec 20 – 2007 Jan 7||750A||6633||280||77||3300|
|”||”||2007 Feb 23 – Mar 13||750D||”||”||”||”|
|”||”||2007 Dec 7 – 30||750C||”||”||”||”|
|”||”||2008 Feb 16 – Mar 13||750B||”||”||”||”|
|This paper||1.350–9.900||2012 Feb 10–12||6A||1515–11||70||3400||7900|
|”||”||2012 Jun 22–24||6D||”||”||”||”|
|”||”||2012 Aug 17–19||6A||”||”||”||”|
Note. – For the current observations, the quoted range in synthesized beam size (column 5) corresponds to the change between 1.3 & 9.9 GHz. (column 6), max-scale (column 7) and (column 8) are the resolution in -space, the maximum detectable emission scale in -space and the maximum detectable Faraday depth of emission respectively. Note that the current observations have , while the opposite is true for the archival observations.
|Source||RA||Decl.||Self calibrated||Maximum||AT20G HARC|
|bands||angular size||polynomial deg.|
|lmc_c16||04:35:41.95||-62:58:44.27||16, 6, 3||4||0.3707(8)||5||-0.849(7)||-|
|lmc_c15||04:38:17.75||-73:09:56.40||16, 6, 3||5||0.1224(7)||4||-0.915(6)||-|
|lmc_c07||04:45:13.38||-65:47:10.65||16, 6, 3||-||0.1208(3)||5||-0.324(5)||0.95, 0.95|
|lmc_s13||05:16:36.84||-72:37:10.86||16, 6, 3||-||0.2021(5)||7||+0.357(9)||0.99|
|0515-674||05:15:37.56||-67:21:27.78||16, 6, 3||-||1.5443(2)||4||-0.763(2)||0.98, 0.98|
|lmc_s11||05:41:50.69||-73:32:14.01||16, 6, 3||5||0.987(2)||7||-0.421(4)||0.87|
|lmc_c04||05:45:54.59||-64:53:28.20||16, 6, 3||-||0.445(1)||3||-0.552(3)||0.60, 0.72|
|lmc_c02||06:01:11.54||-70:36:07.20||16, 6, 3||-||0.2996(6)||7||+0.135(8)||-|
|lmc_c03||06:10:12.90||-74:32:05.95||16, 6, 3||35||0.733(2)||7||-0.833(6)||-|
|lmc_c01||06:20:02.55||-69:16:52.72||16, 6, 3||4||0.1014(5)||4||-0.685(5)||-|
|cen_c1093||13:16:06.06||-39:20:06.00||16, 6, 3||-||0.0200(3)||3||-0.664(9)||-|
|cen_c1972||13:17:13.00||-41:09:34.00||16, 6, 3||2||6.3||0.167(1)||3||-1.416(9)||-|
|0315-46||13:18:30.06||-46:20:34.90||16, 6, 3||-||2.197(4)||4||-1.004(3)||-|
|cen_c1573||13:19:31.30||-44:59:43.00||16, 6, 3||7||0.107(1)||5||-1.17(8)||-|
|cen_c1748||13:19:43.00||-44:59:04.00||16, 6, 3||-||0.0235(6)||3||-0.91(2)||-|
|cen_c1827||13:21:04.00||-41:14:51.00||16, 6, 3||38||0.285(2)||7||-1.33(2)||-|
|cen_c1466||13:23:04.00||-38:49:05.00||16, 6, 3||-||0.2925(8)||7||-0.16(1)||0.99|
|cen_s1568||13:27:57.30||-45:50:19.00||16, 6, 3||-||0.0096(4)||4||+0.31(1)||-|
|cen_s1349||13:30:11.00||-40:54:36.00||16, 6, 3||-||0.177(1)||4||-0.542(8)||-|
|cen_s1437||13:31:08.50||-44:29:54.00||16, 6, 3||-||0.0496(3)||4||-0.702(7)||-|
|cen_s1803||13:35:56.50||-39:30:06.00||16, 6, 3||13||0.0943(8)||4||-1.25(2)||-|
|cen_s1290||13:39:50.00||-45:50:59.00||16, 6, 3||-||0.0370(3)||4||-0.245(8)||-|
Note. – In column 5, the maximum angular size was determined from images at 1.4 GHz for extended sources, and from images at the highest frequency at which the source remained visible in cases where the source resolved into multiple separated sub-components towards higher frequencies. In column 6, a ‘’ symbol indicates that a source is unresolved over 1.3–10 GHz, by which we mean its peak-to-total flux density ratio is at least 0.95 over the full range. A ‘’ symbol indicates that a source is resolved at all frequencies between 1.3 & 10 GHz (i.e. a peak-to-total flux density ratio of ), while a number indicates the frequency in GHz above which the source becomes resolved. These designations apply to observations conducted with the ATCA array configurations listed in column 4 of Table 1 and imaged using robust weighting. (column 7) is the value of the Stokes I model at 1.4 GHz. (column 9) is the mean spectral index over the 1.3–10 GHz band. The uncertainties on the AT20G high angular resolution catalogue (HARC) visibility ratios listed in column 10 are all .
3. Observations and calibration
We observed the 40 sample sources using the Australia Telescope Compact Array (ATCA; Wilson et al. 2011) over 1.1–3.1 GHz, 4.5–6.5 GHz and 8.0–10.0 GHz in full polarization with nominal 1 MHz channel resolution. At each end of each of these bands, 100 channels were flagged due to diminished sensitivity in these regions, resulting in an effective frequency coverage of 1.2–3.0 GHz, 4.6–6.4 GHz and 8.1–9.9 GHz. Henceforth, we refer to these as the 16 cm, 6 cm and 3 cm bands respectively.
Each source was observed on axis over a full 12 hour synthesis, receiving a total integration time of 23 minutes split over 15 cuts for the 16 cm observations, and 50 minutes split over 33 cuts for the 6 & 3 cm band observations. We used 6 km array configurations to achieve high spatial resolution and to maximize overlap in coverage between bands (see Fig. 2) for sources that became resolved towards higher frequencies. We provide images of these sources in Appendix A. The minimum scale in the 16 cm, 6 cm & 3 cm bands is 1.4 k, 4.8 k & 8 k respectively, corresponding to a maximum angular scale sensitivity of approximately 130”, 40” & 22”. Important details of our observations are summarized in Table 1.
The data were reduced and calibrated using standard procedures for cm-band ATCA data. Radio-frequency interference (RFI) was flagged iteratively throughout the calibration process using the sumthreshold algorithm (Offringa et al. 2010). RFI was so severe below 1.3 GHz that all data below this frequency were discarded. Daily observations of either PKS B1934-638 or PKS B0823-500 were used to calibrate the bandpass response and primary flux scale. The time-dependent complex antenna gains and on-axis polarization leakage were calibrated using PKS B0515-674 and MRC B1315-460 for the LMC and Centaurus sources respectively, observed half-hourly for the 16 cm band observations and every 20 minutes for the 6 & 3 cm band observations. Independent calibration solutions were derived at 128 MHz intervals in each band then interpolated to account for the frequency dependence of the complex gain and leakage (Schnitzeler et al. 2011). We have included these two calibrators as sources in our sample; their properties are discussed alongside those of the other sample sources in the relevant sections of our paper.
We phase and amplitude self-calibrated the data for all sources in our sample for bands where adequate signal was present. To handle the large fractional bandwidths involved, we split the data for self calibration into sixteen 128 MHz sub bands for the 16 cm data, eight 256 MHz sub bands for the 6 cm data, and four 512 MHz sub bands for the 3 cm data, then derived separate self-calibration solutions in each sub band. The calibration solutions were interpolated and applied continuously across each of the 2 GHz bands. For sources and bands in which good phase and amplitude self-calibration solutions were found, we compared the data extracted from images made both before and after the procedure. We found that for the pre-self-calibration images, the source flux densities were systematically lower (by up to 20%), and the noise levels higher by a factor of up to 10, than for the post-self-calibrated images. This effect was particularly evident for the 3 cm band data. As such, we chose to discard data for sources in bands where both phase and amplitude self-calibration was unsuccessful. We list the frequency bands in which self calibration was successful for each source in column 4 of Table 2.
For the archival data extracted from mosaiced images, the estimated maximum polarization leakage in the mosaiced images is 0.1% of Stokes I (e.g. Feain et al. 2009). For the current on-axis observations, the frequency-dependent leakage is no greater than % of Stokes I (e.g. Schnitzeler et al. 2011), and as low as 0.002% of Stokes I when averaged over the full available bandwidth (see results for 0515-674 presented in Section 6.2.1).
4. Spectropolarimetric analysis
4.1. Imaging and polarized signal extraction
We imaged each source in Stokes , & at 20 MHz intervals through the 16 cm & 6 cm bands, and at a substantially coarser resolution of 200 MHz in the 3 cm band after verifying that no components with extreme Faraday depth were present in the calibrated data. We used robust weighting (Briggs 1995) to moderately down-weight data in sparsely populated regions of the plane. The higher spatial resolution of the new vs. archival data (Table 1) meant that several sources are resolved at all frequencies, some become resolved above some frequency , while the rest are fully unresolved. We convey this information by recording either ‘’, or ‘’ (respectively) in column 5 of Table 2. For the sources which became resolved above a frequency , we applied a taper to the data above this frequency to match the resolution of the observations at . For the fully resolved sources, we tapered the data to match the scale-size sensitivity at 1.3 GHz. Of the 40 sources originally selected for our new observations, six subsequently proved so faint and/or heavily resolved that we discarded them from the study. Including the two calibrators then, our final sample consists of the 36 sources listed in Table 2.
For our spectropolarimetric analysis, we extracted the integrated frequency dependent Stokes , & flux densities for each source. These quantities were measured at the location of the peak pixel in Stokes I for the fully unresolved and partially resolved sources (in the latter case, after convolving the images at to match the resolution at ). To extract the integrated polarization of the fully resolved sources, we applied a mask to the source at its 10% flux density radio contour and extracted the integrated flux densities from the unmasked region. The uncertainty in each Stokes parameter was estimated via direct measurement of the RMS noise adjacent to the source, modified appropriately for the number of independent beams sampled in the unmasked regions for ‘’ sources.
Our spectropolarimetric analysis was conducted on the fractional Stokes parameters & , which were obtained by fitting a polynomial to log() vs. log(), transforming this model into linear space then dividing it out of Stokes & . The 1.4 GHz Stokes I flux density predicted by the model, the degree of the polynomial fit required, and the spectral index of the model at 1.4 GHz are presented in columns 6, 7 & 8 of Table 2 respectively.
4.2. Modelling Stokes q() & u()
A key goal of our work is to identify and characterize Faraday thick components. This has traditionally been achieved by fitting with a depolarization model derived for a specific physical arrangement of magnetoionic material (e.g. O’Sullivan et al. 2012). However, this approach involves making assumptions about the structure of the magnetoionic material under investigation. Moreover, we found that existing models were incapable of reproducing the detailed polarization behaviour of several of our high S/N sources.
Instead, we modelled our data in a general way that allowed us to quantify the number of dominant emission components in each source, their width and shape in space, and their initial polarization angle. We did this by constructing model FDFs from elementary basis functions. For each model, polarized emission components (which could be Faraday thick or thin) are combined in space as:
then transformed into space (see Section 4.3) using
The goodness of fit of the model to the measured polarization data can then be assessed.
We modelled Faraday-thin FDF components as complex-valued functions of modulus with polarization angle at a Faraday depth of . For Faraday thick components, we used a parametric function capable of generating a variety of behaviours — a Super-Gaussian (e.g. Decker 1994) which takes on complex values, defined by:
where is an amplitude parameter, is the emitted polarization angle, is the mean Faraday depth of the emission component, is a shape parameter, and is a width parameter which is related to the traditional FWHM of the Gaussian parameter as:
With reasonably few parameters, the Super-Gaussian function provides a generic Faraday thick FDF component capable of modelling emission from an assortment of magnetoionic arrangements. For example, when , Eqn. 10 reduces to a normal Gaussian function and can be used to model depolarization from a large number of independent magnetoionized cells covering the source in a foreground Faraday rotating medium (i.e. the Burn foreground screen; Burn 1966). As increases, the Gaussian becomes progressively ‘squared’ (see Fig. 3) until at , becomes a reasonable approximation to a top hat function, which can be used to model a region of mixed emitting and Faraday rotating plasma (i.e. the “Burn slab”; Burn 1966), linear RM gradients or specific types of foreground depolarization screen (Schnitzeler et al. 2015). Other values of N provide the flexibility to approximate the polarization behaviours expected from non-uniform gradients in the RM of a foreground Faraday rotating medium or an illuminating background source, multiple layers of 2D turbulent foregrounds (Schnitzeler et al. 2015), among other possible structures. While a lone Super-Gaussian cannot describe magnetoionic structures that have a skewed distribution in space, different parametric curves could in principle be used for this purpose. In general, we found that this was not necessary to achieve reasonably good fits to our data (but see Section 6.2.1).
We point out that our method avoids the deconvolution ambiguities that afflict the so-called open-ended spectropolarimetric analysis algorithms (see Sun et al. 2015), while at the same time, it avoids making significant a-priori assumptions about the magnetoionic structure of the source under study. It therefore combines some of the benefits of open-ended methods with those of QU-fitting, though with the drawback that using Monte Carlo methods (see Section 4.3 below), the fitting process can be quite time-consuming.
In this work, we refer to FDF models which contain a specific combination of thick and/or thin components as a ‘model type’. We denote the model type with character strings in which each ‘S’ (‘T’) indicates the presence of a thin (thick) component (the characters S and T were chosen in reference to ‘simple’ and ‘thick’ respectively). For example, a model consisting of two thin components and a thick component is denoted by the string ‘SST’. We refer to a specific realization of a model type — i.e. when all parameters in the model have specified values — as a ‘polarization model’ or simply ‘a model’.
4.3. Fitting the models and selecting the best-fit model(s)
We fit our models using Monte Carlo methods. The posterior probability for a set of model parameters () given some data () and a model type (M) can be calculated using BayesÕ theorem:
The likelihood for polarization data (,) and a model (q,u) can be quantified as:
For — the prior belief that the model parameters should take on a set of values — we used a uniform PDF in ranges where these values were physically acceptable: [,) rad for (the initial polarization angle), [0,0.7] for (the fractional polarized amplitude), [-3000,3000] rad m for and [0,1500] rad m for . The PDF was set to zero outside these ranges. Note that the prior ranges for and do not probe the full range of parameter space allowed by our data (see Table 1). We imposed these limits to aid the computational tractability of our fitting procedure.
Finally, since we are only interested in relative probabilities for the purpose of model comparison, we set the normalization factor .
We used the affine-invariant MCMC sampler Emcee (Foreman-Mackay et al. 2013) to sample , deeming the parameter values which maximized to specify the best fit capable of being achieved with a given model type. The 1 uncertainties for each fitted parameter were measured directly from the posterior probability distribution (marginalized over all other model parameters). These values are only meaningful when a good fit to the data has been achieved.
Initially, we fit models of type S, T, SS, ST, TT & SSS to every source in our sample. Several sources required more complicated model types to achieve a good fit (lmc_c03, lmc_c04 & lmc_s11; described in Section 6.2.1). We constructed these models manually on a case-by-case basis, by adding additional S & T components to the models until a good fit was achieved. For the best fit obtained for each model type, we quantified the absolute goodness of fit using the reduced chi-squared () statistic.
We compared the relative merit of each fitted model type using the Akaike information criterion (AIC; Akaike 1974). Formally, for a model the AIC is calculated as
where = no. of fitted model parameters, and is the maximum of the likelihood distribution for model M. For two models M & M with AIC values of AIC & AIC respectively, M is then exp((AICAIC)/2) times as likely as M to minimize the information lost by over or under fitting the data. We consider a model to be substantially favoured by the AIC when AIC AIC (i.e. % confidence). We refer to the model with the lowest AIC value for a given source as the overall best fit model.
5. Ancillary analysis
5.1. RM synthesis & rmclean
RM synthesis (Brentjens & DeBruyn 2005) is a technique for directly calculating (also called the Faraday dispersion spectrum, or FDS) from observations of using a Fourier transform (see also Burn 1966). A procedure called rmclean (Heald et al. 2009) is usually then applied to the FDS to remove artefacts caused by imperfect sampling of . We applied RM synthesis & rmclean to each source in our sample so that the resulting FDS could be directly compared to our FDF models.
5.2. Morphological constraints
We crossmatched our sources with the AT20G high angular resolution catalogue (Chhetri et al. 2013) to provide morphological constraints on ” angular scales. This catalogue contains ratios of scalar-averaged visibilities measured on 6km and 4.5km baselines for sources observed by ATCA at 20 GHz. We record the visibility ratios for matched sources in column 9 of Table 2.
6.1. Spectropolarimetric data
We begin by presenting our spectropolarimetric data, and by pointing out several obvious and noteworthy features found therein. Our data (and our best-fit polarization models — discussed in the next section) are plotted on separate panels in Figs. 4–38. The sources are presented in order such that the value of the best fit S model decreases — i.e. roughly in order of decreasing Faraday complexity. In panels (a), (d) & (e) of these figures, we plot (,), & (respectively) against . The coverage is dominated the 16 cm band data which spans m, while the 6 cm & 3 cm data correspond to m and m respectively. The uncertainty in the plotted quantities is larger at frequencies where our sources are resolved (see column 5 of Table 2), reflecting the the loss of information brought about by the image convolution procedure described in Section 4.1. Panel (b) plots Stokes vs. , while panel (c) plots the magnitude of the FDS & rmclean components, along with the rmclean cutoff (see Section 5.1; also Heald et al. 2009). Panel (f) plots the overall best fit model FDF; the magnitude of each model component is plotted on the main axes, while the intrinsic polarization angle of each component is plotted on the inset polar axes. Panel (g) plots the total intensity spectrum and a polynomial model fit to these data (see Section 4.1, also col. 7 of Table 2). We also plot the instantaneous spectral index associated with the polynomial model, calculated as . Panels (a), (d), (e) & (g) additionally contain inset axes below the main axes, on which we plot the standardized residual (SR) between the data and the model fit, where SR , and , & are the values of the data, model and standard uncertainty (respectively) in channel . For panels (a) & (g), these data should be Gaussian distributed about SR with a standard deviation of 1.
The plots described above immediately reveal the existence of substantial variety in the spectropolarimetric behaviour of the sample. Particular features of note include:
The presence of Faraday thick structures: These are both obvious and resolved in the FDS for several sources in our sample (see panel (c) of relevant figures, noting that our observations have rad m but rad m — see Table 1). lmc_c04 (Fig. 5) is particularly striking in this regard. lmc_s11 (Fig. 4), lmc_c03 (Fig. 6), and 1315-46 (Fig. 10) provide further examples.
Varied ) behaviour: ) (panel (d) decreases monotonically to zero for some sources (e.g. cen_c1827; Fig. 11), apparently to finite limiting values for others (e.g. cen_c1466; Fig. 8), and oscillates for still others (e.g. lmc_c15; Fig. 16). lmc_s13 (Fig. 7) repolarizes (i.e. is positive) over the full band.
Linearity (or lack thereof) of : For some sources, near-linearity of (panel (e) is maintained over broad ranges despite strong concurrent depolarization — e.g. lmc_s11 (Fig. 4), lmc_c03 (Fig. 6), lmc_c02 (Fig. 9), cen_c1827 (Fig. 11), cen_c1972 (Fig. 12) & lmc_c01 (Fig. 14). For other sources, is strongly frequency dependent (and so therefore is the RM) — e.g. lmc_c15 (Fig. 16) and lmc_c04 (Fig. 5).
Structure in the total intensity spectra (panel (g)): vs. is rarely well described by a pure power law, but rather, often shows substantial structure. Concave (e.g cen_c1972; Fig. 12) and convex (e.g lmc_c04; Fig. 5) features are apparent — often in the same spectrum (e.g lmc_s11; Fig. 4). Several sources such as lmc_c02 (Fig. 9) and lmc_s13 (Fig. 7) possess particularly complicated Stokes I spectra.
We interpret these behaviours in light of our polarimetric models in Section 7.1.
6.2. 1.3–10 GHz spectropolarimetric modelling
We now report on the results of fitting the 1.3–10 GHz polarization data presented in Section 6.1, using the modelling techniques described in Sections 4.2 & 4.3. Table 3 contains a summary of the fitting results. The sources are ordered in the table by the value of the best fitting S model, from highest to lowest (i.e. most to least Faraday complex). We include results for the best fit S model for each source by default, but only include results for Faraday complex models if they have a better (i.e. lower) AIC value than the best fit S model, and are either a) the best fitting model overall, or b) indistinguishable from the overall best fit model on the basis of our AIC criteria (i.e. AIC AIC).
The results for each source occupy several rows in the table and are arranged as follows. The source names in column 1 indicate where the fitting data for each source begins and ends. In the first row entry for each source, the figure number displaying the source data, the polarized signal-to-noise ratio of the source, the overall best fit model type, and the corresponding & AIC values are listed in columns 2, 3, 4, 5 & 6 (respectively). For the first component in the best fit model, we list its type in column 7, contribution to fractional polarization as in column 8, and best fit parameter values – in columns 9–13. This is repeated for each of the remaining best fit model components in columns 9–f subsequent rows of the table. Further complex models warranting inclusion are presented in subsequent rows in the same way, followed by the best fit S model for each source. In all cases, the first row entry for a given model is indicated in column 4 by listing its model type. For all quantities, the uncertainties are indicated using standard parentheses notation. Note that for one source only in our sample (0515-674), a model with zero polarized emission provides an excellent fit to the data. We have entered ‘Unp’ as the model type in this isolated case.
|Source||Figure||Polarized||Model||AIC||Component||Best fit parameters for model component|
|[ ]aafootnotemark:||[ ]aafootnotemark:||[ ]aafootnotemark:||[ ]aafootnotemark:||[ ]aafootnotemark:|
|Source||Figure||Polarized||Model||AIC||Component||Best fit parameters for model component|
|[ ]aafootnotemark:||[ ]aafootnotemark:||[ ]aafootnotemark:||[ ]aafootnotemark:||[ ]aafootnotemark:|
|Source||Figure||Polarized||Model||AIC||Component||Best fit parameters for model component|
|[ ]aafootnotemark:||[ ]aafootnotemark:||[ ]aafootnotemark:||[ ]aafootnotemark:||[ ]aafootnotemark:|
|Source||Figure||Polarized||Model||AIC||Component||Best fit parameters for model component|
|[ ]aafootnotemark:||[ ]aafootnotemark:||[ ]aafootnotemark:||[ ]aafootnotemark:||[ ]aafootnotemark:|
Note. – The fitted model parameters P–P refer to different quantities possessing different units for S (Faraday simple/thin) and T (Faraday thick) model components. For T model components, P, P, P, P & P correspond to [non-dim], [rad m], [rad m], [non-dim] & [rad] in Eqn. 10 respectively. For S model components, P, P & P correspond to [non-dim], RM [rad m] & [rad] respectively. The column positions for S components are chosen such that similar quantities align between S & T components. P & P are therefore left blank. Note that a model with zero polarization provides an excellent fit to 0515-674. In this table for this isolated case, we designate this model type as ‘Unp’ (unpolarized).
6.2.1 Overall best fit models: Goodness of fit and model type
The overall best fit model for 30/36 sources has (column 5, Table 3), with an average value of 1.49. We used a Komogorov-Smirnov (K-S) test to compare the distribution of the standardized (,) residuals (defined in Section 6.1, see also inset axes in panel (a) of Figs. 4–38) to the standard normal distribution, and found no statistically significant difference for any of these sources. The remaining four sources — lmc_s11 (Fig. 4), lmc_c04 (Fig. 5), lmc_c03 (Fig. 6) & cen_c1466 (Fig. 8) — had higher values of 3.7, 4.4, 7.8 & 4.2 (respectively), but each have high polarized S/N and the fractional data-model residual values are low.
With the exception of a single source (0515-674), the overall best fit model improves on the AIC value of the best fitting S model by between 10 & 23000. Thus, we positively detect Faraday complex polarization structure in almost our entire sample, including those sources selected to be part of our Faraday simple control group (Section 2). The bright calibrator source 0515-674 represents the sole, remarkable exception (Fig. 39). The best-fit S model has a fractional polarization of less than 2, though an unpolarized model is also consistent with the data across the entire band.
The best fit model type is unambiguous for 23 sources, while the remaining thirteen sources — which are generally faint or heavily resolved — have multiple best fit candidates. The overall best fit model for each source contains between 1 & 4 emission components. Only in three cases was the polarization data well-described by a model with a single component (0515-674; Fig. 39, cen_c1764; Fig. 35 & cen_c1435; Fig. 36). 29 sources required two model components, while five sources required either 3 or 4 model components. Three of the sources in this latter group (lmc_s11, Fig. 4; lmc_c04, Fig. 5; lmc_c03, Fig. 6) also had larger values for the overall best fit models, suggesting that an even greater number of model components might be required to adequately describe their polarization behaviour. However, we were unsuccessful in identifying any such model. Alternately, it may indicate that a more sophisticated functional form for T model components is required to fit the subtle behaviour in data with high S/N.
Of the T, SS, ST, TT & SSS model types fit to the entire sample, each provided the best fit for at least one source. TT models were most commonly selected by the AIC (19 sources), followed by ST & SS models (6 & 4 sources respectively), SSS models (2 sources) and a T model (1 source). The custom models required for lmc_c04, lmc_s11 & lmc_c03 were of type STT, SSTT & SSTT respectively. In general then, both S & T components were required to model our data, with 14 (28) of the 36 sources requiring at least one S (T) component.
6.2.2 Spectropolarimetric structure of resolved sources
In Appendix A, we provide images of the resolved sources in our sample in both total and linearly polarized intensity. A range of morphological types are present. In both quantities, there are sources which resolve cleanly into multiple subcomponents which are themselves unresolved, sources which resolve into one or more extended components, and various other combinations of these possibilities.
It is interesting to consider whether the number and type of spectropolarimetric model components fit to these sources (remembering that our spectropolarimetric analysis is performed on the integrated flux for each source) shows any clear relationship with the morphological components present in images of the sources. In general, we do not find this to be the case. Sources with two FDF model components (either S or T-type) often show two spatial components in the appendix images (e.g. lmc_c16 — Figs. 17 & 46; lmc_s14 — Figs. 33 & 48; lmc_c06 — Figs. 15 & 49; cen_c1832 — Figs. 32 & 55; cen_c1573 — Figs. 13 & 56). However, there are also sources which require a greater number of components in the FDF model than there are visible morphological components (e.g. lmc_c15 — Figs. 16 & 47; lmc_s11 — Figs. 4 & 50; lmc_c03 — Figs. 6 & 51; cen_s1681 — Figs. 28 & 53; cen_c1972 — Figs. 12 & 54), and vice versa (e.g. cen_c1764 — Figs. 35 & 59; cen_c1827 — Figs. 11 & 58). Furthermore, we are unable to identify any clear relationship between the presence or absence of unresolved / extended subcomponents in the images, and the presence or absence of S / T components in the best fit FDF models. For example, spatially unresolved components in the source images give rise to both S components (e.g. lmc_c15 — Figs. 16 & 47) and T components (e.g. lmc_c16 — Figs. 17 & 46), while extended components in the source images can do likewise (e.g. cen_s1803 — Figs. 38 & 63 & lmc_c06 — Figs. 15 & 49, for S & T components respectively).
However, it is likely that this analysis is significantly hampered by a lack of spatial resolution and the range of source brightnesses present in our sample.
6.2.3 Requirement for broad components in the best-fit FDF models
A primary motivation for our study was to search for emission components with large peak Faraday depths and dispersions, since this can place strong constraints on the physical origin of the emission. Such components are required to fit the polarization data in the FDF models of 1315-46, lmc_s11, lmc_c02, lmc_c03 & lmc_c04. We now describe the characteristics of these components.
The phase calibrator source 1315-46 shows very strong depolarization in the 3 & 6 cm bands. Its FDF model (panel (f) of Fig. 10) possesses two broad emission components, centered on & rad m with & rad m respectively. The Super-Gaussian shape parameter is strongly constrained for the dominant T component with a value of . Thus, this T component depolarizes in a manner close to that expected for the Burn (1966) foreground screen until m, by which point their contribution to the integrated polarization is negligible.
The FDF model for lmc_s11 (panel (f) of Fig. 4) possesses two components with large Faraday depth/dispersion, centered on & rad m with & rad m respectively. This model achieves an excellent fit to the data with the exception of Stokes in the 3 cm band, which deviates from our model significantly (see inset axis in panel (a) of Fig. 4). While we could broadly reproduce this behaviour using models containing components with similar Faraday depths & dispersions, the goodness of fit at 6 cm was compromised and the result was strongly disfavoured by the AIC. The Super-Gaussian shape parameter is strongly constrained for both T components in the best fit model with & . Thus, each T component essentially depolarizes (where is some constant) until m, beyond which the polarized signal is effectively undetectable.
For lmc_c03 (Fig. 6), two T components successfully reproduce most aspects of the (,) behaviour at 6/3 cm, though minor discrepancies are also evident (see inset axis in panel (a) of Fig. 6). The components have & rad m — by far the the most highly Faraday dispersed emission components detected in our data — centered on & rad m respectively. Both components are mostly depolarized by m, but since for each, small oscillations persist in & out to m (see panel (b)). For m and m where we have no data, the model predicts pronounced structure which is likely an artefact of the unconstrained fit in these regions.
The model for lmc_c02 (Fig. 9) contains two T components. The component with the smaller amplitude has rad m, rad m, and . As with lmc_s11, this component depolarizes smoothly and monotonically for m.
lmc_c04 (Fig. 5) shows arguably the most spectacular behaviour, which is primarily caused by a dominant T component possessing , centred on rad m with (panel (f)). The and values combine to produce strong oscillatory depolarization in , with a deep null at m and several local maxima and minima evident (panel (d)). While this behaviour is reminiscent of that associated with the Burn (1966) slab, our model differs in its detailed behaviour (e.g. see Fig. 3, which shows the subtle effect which the parameter has on depolarization behaviour).
6.2.4 Detection & characteristics of bright emission components with low Faraday dispersion
The overall best fit models for the sources lmc_s13 (Fig. 7), cen_c1466 (Fig. 8), lmc_s11 (Fig. 4) & lmc_c15 (Fig. 16) contain at least one bright S component (with band-averaged S/N of between 20 & 160). Though components which are truly Faraday-thin will never be generated in Nature, the emission modelled by these S components must nevertheless approach this ideal, otherwise T components would have been selected over S components by the AIC criteria. At the same time, values of 10–120 rad m indicate that substantial Faraday rotation has occurred along the LOS. Together, this suggests that the Faraday rotating plasmas lying along the LOS possess comparatively uniform and structure. In this section, we derive upper limits on the Faraday dispersion of these components to more fully explore this possibility.
To derive these limits, we refit our data over ranges where the low-Faraday-dispersion components dominate (i.e. m for lmc_s13, cen_c1466 & lmc_s11, and the entire band for lmc_c15) using models consisting solely of T components. We present the results in Table 4. They show that emission components which contribute substantially to the fractional polarization of these sources (column 5) typically possess very low Faraday dispersions — the upper limits are generally a few rad m, while the lower limits are in some cases consistent with 0 rad m (column 7). We discuss these results further in Section 7.3.
|[rad m]||[rad m]|
Note. – This table is formatted similarly to Table 3 — see Section 6.2 for a detailed description of the layout. The superscript and subscript values quoted in column 7 are the standard errors on the fitted model parameters.
6.3. Spectropolarimetric variability over 1.3–1.5 GHz
Law et al. (2011) & O’Sullivan et al. (2012) detected multiple components in the broadband polarization spectra of a number of radio sources, and argued that these emission components likely arise in the inner parsec-scale regions of the associated AGN. Given that parsec-scale magnetoionic structure in AGN is observed to evolve over timescales of months to years (e.g. Hovatta et al. 2012, Zavala & Taylor 2001), these authors predict that the integrated polarization spectra of core-dominated radio sources may also vary with time. If this behaviour is commonplace, time domain broadband spectropolarimetry could become a powerful tool for studying magnetoionic structure in the immediate vicinity of AGN using non-VLBI radio telescopes.
We searched for the predicted variability by fitting polarization models to both the archival and 2012 data (for the non-calibrator sample sources only) over the 1.3–1.5 GHz band common to both epochs. We used models consisting of a single T component to account for the small amount of depolarization apparent in some sources over this narrow band. However, the & parameters (Eqn. 10) were poorly constrained by the narrowband data, so we ignore them in the subsequent analysis. We identified variability in the remaining parameters — , & (see Eqn. 10) — by calculating the standardized difference between the best fit model parameter values for each epoch, defined by:
where and represent the best fit parameters values for , & in the archival and 2012 epochs respectively, and and are the standard uncertainties on those parameter values. We flagged sources as potentially variable when (i.e. the best fit model parameter values differ between epochs with % confidence) for one or more of the best fit model parameters.
|[rad m ]||[rad m ]||[rad]||[rad]|
Note. – The line ruled under the first row of the table distinguishes sources in which we detect temporal variability (sources below the line) from cen_c1972 — an example of a source which we argue shows temporal change but not true polarization varability (see main text).
31 of the 34 sources either have for all model parameters and thus show no evidence of spectropolarimetric variability, or show significant changes which are nevertheless more readily attributed to other factors. An example of the latter case is cen_c1972. Its Stokes I, (,) & (,) data are plotted in row 1 of Fig. 40, while the fitted parameter values and uncertainties are listed in row 1 of Table 5. The polarization data and models are evidently similar between epochs. for both and , indicating no substantial change in either polarization property. At first, the 13% increase in appears significant with . However, the total intensity decreases by a similar amount over the same period, meaning the changes are probably unrelated to the polarization structure of the source. The Stokes (,) data show no statistically significant differences between epochs.
The remaining three sources — lmc_c03, lmc_s11 & lmc_s13 — vary in one or more of the fitted model parameters. lmc_s11 shows the biggest changes (4th row of Table 5 & Fig. 40), with , and . Between the archival and 2012 epochs, the fractional polarization (i.e. ) increases by 2.23(3)%, increases by rad m, and by rad. The total intensity changes during the same period, but the polarization changes are evident in both (,) and (,), and cannot be explained by total intensity variation. lmc_s13 (3rd row of Table 5 & Fig. 40) shows highly significant change in with , with moderately significant accompanying changes in () & (). From the archival to 2012 epochs, the fractional polarization has increased by 2.9(1)%, by 27(11) rad m, an by 1.3(9) rad. Finally, for lmc_c03 (2nd row of Table 5 & Fig. 40), Stokes I changes by % between epochs and the Stokes (,) data differ more between epochs than the (,) data. Nonetheless, moderately significant differences remain in the (,) data. The source shows small but significant changes of 6.4(2.5) rad m in () and 0.5(1) rad in (), with a tiny, marginal change of 0.16(7)% in ().
7.1. Interpreting complex behaviour using model FDFs
The FDF models derived using our method can often assist in interpreting behaviour which might otherwise remain cryptic. In this section we briefly illustrate this point by considering examples of the notable polarization behaviours described in Section 6.1.
In Section 6.1, we noted the existence of transitions in , or behaviour for the sources lmc_s11 (Fig. 4) at m, lmc_c02 (Fig. 9) at both m & m and lmc_c03 (Fig. 6) at m. This behaviour is invariably caused by a transition in the emission component(s) dominating the net polarization as a function of . Typically, one or more strongly depolarizing T components dominates at small values, transitioning to dominance by fainter component(s) with lower Faraday dispersion(s) at larger values. The resulting behaviour can be counter-intuitive: In the case of lmc_c02, over the range 0–0.01 m, two depolarizing T components (panel (f) of Fig. 9) interfere to produce a pronounced increase in the integrated fractional polarization (panel (d)), before the thicker of the two components depolarizes completely at m and net depolarization associated with the remaining component ensues.
The linearity of depends on the relative symmetry of the FDF (Burn 1966), after depolarization effects are accounted for. Sources with asymmetric FDFs exhibit non-linearity in (e.g. lmc_c15; Fig. 16) and vice versa (e.g. cen_s1349; Fig. 37). lmc_s11 (Fig. 4) provides an example of more complicated behaviour: Its FDF model is asymmetric at high frequencies due to the presence of two broad T components, but these depolarize by m. Two S components remain, providing comparative symmetry in the FDF and approximately linear behaviour.
The observed behaviour depends on the number, shape, and type of components in the model FDF. If the source contains T components, their shape in space strongly affects whether they depolarize monotonically to (e.g. cen_c1827; Fig. 11) or repolarize (e.g. lmc_c04; Fig. 5). The presence of an additional S component is associated with depolarization to a finite limiting value (e.g. cen_s1681; Fig. 28).
7.2. The origin of broad components in the FDF models
In Section 6.2.3, we described how broad FDF model components were required to model the polarization data for several of our sources. There are two possible sources for these polarization behaviours: the first is associated with Faraday rotation, while the second, which may apply to flat-spectrum core-dominated source, is associated with optical depth effects, whereby different parts of the radio core are observed at different frequencies. In the following, we assume that Faraday effects are responsible for the observed polarization properties, while acknowledging that latter effect may indeed be acting to some extent (most plausibly in the flat spectrum source lmc_c02). Disentangling the relative contribution of these effects in typical sources will require broadband polarimetric studies of sources at high spatial resolution, and is beyond the scope of this paper.
7.2.1 The location of the dispersive medium along the LOS
The Faraday thick emission components detected in 1315-46 (Fig. 10), lmc_s11 (Fig. 4), lmc_c02 (Fig. 9), lmc_c03 (Fig. 6), & lmc_c04 (Fig. 5) have in the range 65–1450 rad m and in the range 40–420 rad m. These values are lower limits — they must multiplied by a factor of to get the values in the rest frame of a Faraday rotating medium at redshift . In this section we discuss the origin of these components, the information they convey about magnetoionic structure along the LOS, and the nature of the associated radio sources.
Other than the Faraday rotating medium associated with the host galaxy, the most plausible origin of the large Faraday dispersions observed in 1315-46, lmc_s11, lmc_c02, lmc_c03, & lmc_c04 is variation in the RM across each source, caused by the the hot thermal plasma associated with the intracluster medium (ICM) of galaxy clusters. Typically, RM variations across radio sources embedded in ICMs are observed to be a few hundred rad m in the emitting frame (e.g. Murgia et al. 2004, Govoni et al. 2010, Bonafede et al. 2010), but can exceed thousands of rad m for the inner-most regions of cooling core clusters (Perley & Taylor 1991, Vogt & Enßlin 2006). However, lmc_s11, lmc_c02 & lmc_c04 are completely spatially unresolved over 1.3–10 GHz, while lmc_c03 is dominated by an unresolved component in the 6/3 cm bands (see Fig. 51). The corresponding upper limit on physical extent is a few kpc regardless of redshift. Moreover, the visibility ratios from the AT20G high angular resolution catalogue (col. 9, Table 2) indicate that lmc_s11 and lmc_c04 are compact on angular scales of 0.15” ( kpc maximum). The auto-correlation length for the magnetic field in cooling core clusters is typically estimated to be 5 kpc, with minimum fluctuation scales of 0.7–2 kpc (e.g. Bonafede 2010, Vacca et al. 2012). Thus, even if these sources were deeply embedded in such a cluster, their small physical extent means they are unlikely to intercept a sufficient number of depolarizing cells to account for their Faraday dispersion. Independent evidence against the the ICM-based origin is provided by the spectral index of the sources: Sources embedded in dense cluster environments typically have (e.g. Slee, Siegman & Wilson 1983, Klamer et al. 2006), whereas lmc_s11, lmc_c04 & lmc_c02 have , and lmc_c03 has (Table 2 , col. 8). The observed values of & are generally lower than seen at parsec-scales in AGN jets (with the exception of lmc_c03), but are broadly consistent with structure seen at kpc scales (e.g. Algaba et al. 2012). Thus, for the sources lmc_s11, lmc_c02, lmc_c03, & lmc_c04, we claim that the Faraday dispersion of the broad T components detected in our modelling is generated either inside, or in the immediate vicinity of, AGN jets on kpc scales or below.
The source 1315-46 is somewhat different to the other four discussed above. Though it is also unresolved over 1.3–10 GHz, it possesses a steep spectral index of , and two emission components with large Faraday dispersion (several hundred rad m