A Broadband Spectro-polarimetric View of the NVSS Rotation Measure Catalogue I: Breaking the nπ-ambiguity
Yik Ki Ma, S. A. Mao, Jeroen Stil, Aritra Basu, Jennifer West, Carl Heiles, Alex S. Hill, S. K. Betti
MMNRAS , 1–27 (2019) Preprint 14 May 2019 Compiled using MNRAS L A TEX style file v3.0
A Broadband Spectro-polarimetric View of the NVSS RotationMeasure Catalogue I: Breaking the n π -ambiguity Yik Ki Ma (cid:63) † , S. A. Mao , Jeroen Stil , Aritra Basu , , Jennifer West , Carl Heiles ,Alex S. Hill , , , and S. K. Betti Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany Department of Physics and Astronomy, University of Calgary, 2500 University Drive NW, Calgary, AB T2N 1N4, Canada Fakultät für Physik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany Dunlap Institute for Astronomy and Astrophysics, The University of Toronto, 50 St. George Street, Toronto, ON M5S 3H4, Canada Department of Astronomy, University of California, Berkeley, CA 94720-3411, USA Department of Physics and Astronomy, The University of British Columbia, Vancouver, BC V6T 1Z1, Canada Space Science Institute, Boulder, CO, USA National Research Council Canada, Herzberg Program in Astronomy and Astrophysics, Dominion Radio Astrophysical Observatory, PO Box 248, Penticton,BC V2A 6J9, Canada Department of Astronomy, University of Massachusetts, 710 North Pleasant Street, Amherst, MA 01003-9305, USA
Accepted 2019 May 8. Received 2019 May 7; in original form 2018 September 7
ABSTRACT
The NRAO VLA Sky Survey (NVSS) Rotation Measure (RM) catalogue is invaluable forthe study of cosmic magnetism. However, the RM values reported in it can be affected by n π -ambiguity, resulting in deviations of the reported RM from the true values by multiples of ± . − . We therefore set off to observationally constrain the fraction of sources in theRM catalogue affected by this ambiguity. New broadband spectro-polarimetric observationswere performed with the Karl G. Jansky Very Large Array (VLA) at 1–2 GHz, with 23 n π -ambiguity candidates selected by their peculiarly high | RM | values. We identified nine sourceswith erroneous RM values due to n π -ambiguity and 11 with reliable RM values. In addition,we found two sources to be unpolarised and one source to be inconsistent with neither n π -ambiguity nor reliable RM cases. By comparing the statistical distributions of the above twomain classes, we devised a measure of how much a source’s RM deviates from that of itsneighbours: ∆ / σ , which we found to be a good diagnostic of n π -ambiguity. With this, weestimate that there are at least 50 sources affected by n π -ambiguity among the 37,543 sourcesin the catalogue. Finally, we explored the Faraday complexities of our sources revealed by ourbroadband observations. Key words: galaxies: active – galaxies: magnetic fields – ISM: magnetic fields – radiocontinuum: galaxies
Magnetic fields are ubiquitous in the Universe. For astrophysicalprocesses such as star formation, cosmic ray propagation, galacticoutflows, and galactic evolution, magnetic fields are critical andmust be considered (see review by Beck & Wielebinski 2013; Beck2016). Magnetic field structures of astrophysical objects can bedirectly measured through their polarised synchrotron diffuse emis-sion (e.g., Kothes et al. 2008; Heald et al. 2009; Ma et al. 2016;Basu et al. 2017). However, this technique is limited to probingvolumes filled with synchrotron-emitting cosmic ray electrons. Po- (cid:63)
Contact e-mail: [email protected] † Member of the International Max Planck Research School (IMPRS) forAstronomy and Astrophysics at the Universities of Bonn and Cologne larised emission from background sources can illuminate the fore-ground magneto-ionic media through the Faraday rotation effect,allowing the study of physical conditions in the intervening magne-tised plasma.Radio polarimetric observations of background extragalacticradio sources (EGSs) have been successful in revealing the mag-netic fields in foreground astrophysical objects, such as discrete H iiregions in the Milky Way (Harvey-Smith et al. 2011; Purcell et al.2015; Costa et al. 2016), Galactic high velocity clouds (McClure-Griffiths et al. 2010; Hill et al. 2013; Betti et al. 2019), the Galacticdisk (Van Eck et al. 2011), the Galactic halo (Mao et al. 2010, 2012;Terral & Ferrière 2017), the Magellanic system (Gaensler et al.2005; Mao et al. 2008; Kaczmarek et al. 2017), nearby galaxiessuch as M31 (Han et al. 1998; Gießübel et al. 2013), and cosmolog-ically distant galaxies (Mao et al. 2017). As the polarised radiation © 2019 The Authors a r X i v : . [ a s t r o - ph . GA ] M a y Y. K. Ma et al. traverses through the foreground media, its polarisation positionangle (PA; [rad]) will be rotated by ∆ PA = (cid:20) . ∫ (cid:96) n e ( s ) B (cid:107) ( s ) d s (cid:21) · λ ≡ φ · λ , (1)where (cid:96) [pc] is the (physical) distance of the emitting volume fromthe observer, n e [cm − ] is the electron density, B (cid:107) [ µ G] is thestrength of the magnetic field component along the line of sight ( s [pc]; increasing away from the observer), λ [m] is the wavelengthof the electromagnetic wave, and φ [rad m − ] is the Faraday depth(FD) of the emission region. This Faraday rotation effect encodesthe physical conditions of the foreground magneto-ionic media, inparticular n e and B (cid:107) , into FD. The traditional way to extract the FDvalues of polarised sources is by PA measurements at two or moredistinct frequency bands and perform a linear fit to PA against λ .In this case, FD is commonly referred to as Rotation Measure (RM)instead, which is the slope of the resulting fit. For situations wherePA measurements are only available at two frequencies, the resultingFD (or RM) values can be ambiguous because wrapping(s) of PAcan occur between the two bands. This is the so-called n π -ambiguityproblem, and can be best mitigated by additional PA measurementsat other frequency bands.Modern radio telescopes equipped with broadband backends,such as the Karl G. Jansky Very Large Array (VLA), have starteda new era in the study of cosmic magnetism. They opened upthe possibility of spectro-polarimetric observations with unprece-dented bandwidths (e.g. 1–2 GHz in L-band and 2–4 GHz in S-band for the VLA) and fine frequency resolutions (1–2 MHz in theabove-mentioned bands). This allows a simple eradication of n π -ambiguity in FD (or RM) measurements, since PAs at hundreds oreven thousands of closely spaced frequencies can be measured si-multaneously, ensuring no wrappings of PA between the channels.The even more important aspect of broadband spectro-polarimetricstudies is the possibility to apply analysis methods such as RM-Synthesis (Brentjens & de Bruyn 2005) and Stokes QU -fitting (e.g.Farnsworth et al. 2011; O’Sullivan et al. 2012). The former makesuse of the Fourier-like behaviour of polarisation signal, such thatinput complex polarisation ( P = Q + iU ) as a function of λ can betransformed into output Faraday spectrum ( F ; which is the complexpolarisation as a function of φ ): P ( λ ) = ∫ + ∞−∞ F ( φ ) e i φλ d φ, (2) F ( φ ) = ∫ + ∞−∞ P ( λ ) e − i φλ d λ . (3)The latter technique is to fit the observed Stokes Q and U values asa function of λ by using models of magnetised plasma along theline of sight. Both of the techniques allow exploration of Faradaycomplex sources (e.g. Burn 1966; Sokoloff et al. 1998), which emitat multiple FDs. These sources have varying polarisation fractionsas a function of λ , and sometimes deviate from the linear relation-ship between PA and λ . Given sufficient λ coverage, these sourceswould exhibit multiple peaks and/or extended component(s) in Fara-day spectra. In contrast, Faraday simple sources emit at a single FDonly, with constant polarisation fractions across λ , and have PAvalues varying linearly with λ . RM-Synthesis and QU -fitting arewidely used in broadband radio polarisation studies, with a grow-ing success in revealing the Faraday complexities of a significantnumber of the observed EGSs (e.g. Law et al. 2011; Anderson et al.2015, 2016; Kim et al. 2016; O’Sullivan et al. 2017; Kaczmareket al. 2018; Pasetto et al. 2018; Schnitzeler et al. 2019).The largest RM catalogue of polarised radio sources to date is the Taylor et al. (2009, hereafter TSS09) catalogue, which con-tains RM values of 37,543 radio sources north of δ = − ◦ at asource density of higher than one per square degree. This makes itinvaluable for the study of cosmic magnetism (e.g. Stil et al. 2011;Oppermann et al. 2012; Purcell et al. 2015; Terral & Ferrière 2017).TSS09 constructed the catalogue by re-analysing the NRAO VLASky Survey (NVSS; Condon et al. 1998) data, and thus it is alsocalled the NVSS RM catalogue. While in the original NVSS cata-logue the two intermediate frequencies (IFs; centred at 1364.9 and1435.1 MHz with bandwidths of 42 MHz each) were combined,TSS09 processed data from the two IFs independently, allowingdetermination of RM from these two frequency bands. However,these RM values could then be susceptible to n π -ambiguity as dis-cussed above. For each of the sources in their catalogue, the authorscompared the observed amount of depolarisation with that expectedfrom bandwidth depolarisation at the different allowed RM values,and also used the RM values of neighbouring sources within 3 ◦ , tominimise n π -ambiguity. However, it is not clear how effective thismethod really is at picking the correct RM values. Understandingthe limits of the NVSS RM catalogue is vital to the study of cosmicmagnetism. While upcoming polarisation surveys such as Polariza-tion Sky Survey of the Universe’s Magnetism (POSSUM; Gaensleret al. 2010) in 1130–1430 MHz and VLA Sky Survey (VLASS; My-ers et al. 2014) in 2–4 GHz are expected to bring vastly higher RMdensities compared to TSS09, the two surveys either do not have ex-act sky or frequency coverage as TSS09. This means the NVSS RMcatalogue will remain a unique dataset for studying the magnetisedUniverse, complementing the VLASS in the frequency domain andPOSSUM in the sky domain, in addition to both in the time domain.A prior deeper understanding in the systematics of TSS09 will fa-cilitate future robust comparisons among these surveys. The focusof our work here is to effectively test the reliability of the TSS09RM values by validating a small sample of TSS09 sources usingbroadband polarimetry, which provides us with n π -ambiguity-freeFD. In this paper, we report the results from new broadband ob-servations of 23 candidates from the NVSS RM catalogue whichcould suffer from n π -ambiguity. The observational setup and datareduction procedures are described in Section 2, and the resultsare presented in Section 3. In Section 4, we discuss the implica-tions of the results on the n π -ambiguity in the TSS09 catalogue,and also explore the Faraday complexities of the targets revealedby the new broadband observations. Finally, we conclude this workin Section 5. In the companion Paper II (Ma et al. 2019), we fur-ther compare this dataset with the TSS09 catalogue in matchingfrequency ranges to quantify the effects of the off-axis instrumen-tal polarisation on the TSS09 RM measurements. Throughout thepaper, we adopt a cosmology in accordance to the latest Planck results (i.e., H = . − Mpc − and Ω m = . MNRAS , 1–27 (2019) he n π -ambiguity in the NVSS RM Catalogue Table 1.
Summary of the VLA Observations on 2014 July 03Start Time End Time Flux and Bandpass Leakage Phase Target Source Target Source Angular(UTC) (UTC) Calibrator Calibrator Calibrator (NVSS) (Other Name) Resolution a (cid:48)(cid:48) × (cid:48)(cid:48) J1623 − − (cid:48)(cid:48) × (cid:48)(cid:48) J163927 − (cid:48)(cid:48) × (cid:48)(cid:48) J1733 − − (cid:48)(cid:48) × (cid:48)(cid:48) J1924+3329 J190255+315942 3C 395 52 (cid:48)(cid:48) × (cid:48)(cid:48) (cid:48)(cid:48) × (cid:48)(cid:48) J2202+4216 J220205+394913 — 56 (cid:48)(cid:48) × (cid:48)(cid:48) J220927+415834 — 57 (cid:48)(cid:48) × (cid:48)(cid:48) J224412+405715 — 53 (cid:48)(cid:48) × (cid:48)(cid:48) J224549+394122 3C 452 51 (cid:48)(cid:48) × (cid:48)(cid:48) J2340+1333 J234033+133300 4C +13.88 49 (cid:48)(cid:48) × (cid:48)(cid:48) J235728+230226 4C +22.65 46 (cid:48)(cid:48) × (cid:48)(cid:48) − − (cid:48)(cid:48) × (cid:48)(cid:48) J084600 − (cid:48)(cid:48) × (cid:48)(cid:48) J084701 − (cid:48)(cid:48) × (cid:48)(cid:48) J0921 − − (cid:48)(cid:48) × (cid:48)(cid:48) J091145 − (cid:48)(cid:48) × (cid:48)(cid:48) J092410 − (cid:48)(cid:48) × (cid:48)(cid:48) J093349 − (cid:48)(cid:48) × (cid:48)(cid:48) J1018 − − (cid:48)(cid:48) × (cid:48)(cid:48) J094750 − (cid:48)(cid:48) × (cid:48)(cid:48) J094808 − (cid:48)(cid:48) × (cid:48)(cid:48) J1120+1420 J111857+123442 4C +12.39 50 (cid:48)(cid:48) × (cid:48)(cid:48) a From channel maps at 1.5 GHz
We selected the 23 target sources from the TSS09 catalogue. Theyhave high | RM TSS09 | (cid:38)
300 rad m − and are situated away from theGalactic plane ( | b | > ◦ ) . In this region, the Galactic FD (or RM)contributions are less significant, with ≈
99 per cent of the TSS09sources with | RM TSS09 | <
150 rad m − . The peculiar populationwe selected, with high | RM TSS09 | , could be statistical outliers fromthe generally low | RM TSS09 | population, either because they havehigh intrinsic FD (or RM) values or they are positioned along spe-cial lines of sight with high foreground FD (or RM) contributions.On the other hand, our target sources could also be out-liars witherroneous RM TSS09 values, deviating from the true RM by multi-ples of ± . − due to n π -ambiguity (TSS09) and causingthem to stand out from the majority. However, we note that ourselection criteria does not allow us to study sources with high true | RM | having low reported | RM TSS09 | due to n π -ambiguity, andthus our study here only focuses on cases where sources with lowtrue | RM | are “boosted” to high | RM TSS09 | due to n π -ambiguity.We further selected only bright sources with NVSS total intensities Except for J234033 + TSS09 = + . ± . − .This source was also observed because it was thought to have a high emis-sion measure (EM; ∼
140 cm − pc) but low | RM | , which could be anothermanifestation of n π -ambiguity. However, upon close examination after theobservation was conducted, we found that the EM along this sightline isactually (cid:46)
10 cm − pc, thus disqualifying this source as an n π -ambiguitycandidate. This source will not be included in the statistical analysis on n π -ambiguity in this work. However, we later found that this source is un-polarised, which leads to implications on the residual off-axis polarisationleakage of TSS09 (see Paper II). larger than 100 mJy to ensure that sufficient signal-to-noise ratiocould be achieved.Our new broadband data were acquired using the VLA in L-band (1–2 GHz) in D array configuration. The observations werecarried out on 2014 July 03 in three observing blocks, and are sum-marised in Table 1 where the observing time, calibrators, targetsources, and angular resolutions are listed. For each of the targetsources, the integration time is about 3–4 minutes. We used theCommon Astronomy Software Applications (CASA) package (ver-sion 4.4.0; McMullin et al. 2007) for all of the data reductionprocedures.The three measurement sets were calibrated independently.Hanning smoothing is first applied to all the visibilities in frequencydomain to remove the Gibbs phenomenon, and the antenna positioncalibration is applied to the dataset. Then, we flagged out times whenthe antennas were not performing as intended or when prominentradio frequency interferences (RFI) were seen. Next, we determinedthe delay, bandpass, and gain solutions using the flux and/or phasecalibrators, with the absolute flux densities following the Perley& Butler (2013a) scales. The PA calibration was done by using thepreviously determined PAs of the flux calibrators 3C 286 and 3C 138(Perley & Butler 2013b), while the on-axis instrumental leakage wascorrected for by observing standard unpolarised leakage calibrators(see Table 1). Finally, we applied one round of phase self calibrationto all our target sources to further improve the gain solution. With the calibrated visibilities, we formed a series of Stokes I , Q ,and U images for each target source at different frequencies acrossL-band, combining 4 MHz of visibility data to form the images foreach step in the frequency axis. The Clark deconvolution algorithmin CASA task CLEAN was adopted, with Briggs visibilities weight-
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Table 2.
Positions of Individual Components of the Spatial DoublesSource Right Ascension Declination(NVSS) (J2000; h m s) (J2000; ◦ (cid:48) (cid:48)(cid:48) )J091145 − · · · a 09 11 42.47 ± −
30 13 19.26 ± · · · b 09 11 46.33 ± −
30 12 58.63 ± − · · · a 09 24 10.09 ± −
29 05 45.36 ± · · · b 09 24 11.44 ± −
29 06 26.66 ± − · · · a 09 35 43.98 ± −
32 28 48.51 ± · · · b 09 35 43.79 ± −
32 29 40.03 ± − · · · a 16 27 04.53 ± −
09 16 55.99 ± · · · b 16 27 06.78 ± −
09 17 06.50 ± − · · · a 16 39 27.09 ± −
12 41 26.41 ± · · · b 16 39 28.20 ± −
12 42 09.07 ± ing of robust = . − in Stokes I ,and 1 . − in Stokes Q and U .We measured the Stokes I , Q , and U values of our targetsources per frequency step. We used different methods dependingon whether the sources are spatially resolved with our observa-tional setup. For spatial singles and doubles, we used the CASAtask IMFIT to extract the flux densities and their uncertainties.The full-width at half-maximum (FWHM) of the Gaussian compo-nents are fixed as that of the synthesised beam at each frequencystep, and the fitted source locations in Stokes I are also used forStokes Q and U . The positions of the individual components ofthe five double sources in our sample are listed in Table 2. For ex-tended sources (J094750 − + nterms = I images using the entire L-band for each of the sources,from which 6 σ contours in Stokes I enclosing the target sourcesare defined. The CASA task IMSTAT is then used to integrate theStokes I , Q , and U flux densities within the contour for each channelmap. We note that using integrated flux densities discards all thespatial information we have of these two sources, and may increaseFaraday complexity and/or cause beam depolarisation. A detailedspatial analysis of them is included in Appendix A. The radio spec-tra of our targets are reported in Paper II, in which we address thepotential Stokes I and RM time variabilities of our sample. Using the extracted Stokes I , Q , and U values for every 4 MHzchannel map (Section 2.2), we performed RM-Synthesis (Brentjens& de Bruyn 2005) on all our target sources. For double sources, eachof the spatial components are analysed independently. We used the Python-based RM-Synthesis code, pyrmsynth , to perform thisanalysis, including RM-Clean algorithm (e.g. Heald et al. 2009) todeconvolve the Faraday spectra. The q = Q / I and u = U / I valuesare used as the inputs, and therefore the resulting complex Faradayspectra (sometimes referred to as Faraday dispersion functions inthe literature) are in units of polarisation fraction ( p ) per RotationMeasure Transfer Function (RMTF). With our observational setup,the resolution of Faraday spectrum, maximum detectable scale, andmaximum detectable FD are (equations 61–63 in Brentjens & deBruyn 2005) δφ ≈ √ ∆ λ ≈ − , (4)max-scale ≈ πλ ≈
143 rad m − , and (5) || φ max || ≈ √ δλ ≈ ( ) × rad m − , (6)respectively. The quoted range for δφ is due to the slightly different λ coverage for each source as the result of flagging (individualvalues listed in Table 3), and that for || φ max || is because of thedifference in widths of the 4 MHz channels in λ space across L-band. We adopted a normalised inverse noise variance weightingfunction of (e.g. Schnitzeler & Lee 2017) W ( λ ) ∝ σ q ( λ ) + σ u ( λ ) , (7)where σ q and σ u are the uncertainties in q and u respectively.The Faraday spectra were formed within − (cid:54) φ (rad m − ) (cid:54) + − . We first perform trial cleansto determine the rms noise (denoted as σ here) in the source-free FDranges of | φ | (cid:62) − from the q φ and u φ Faraday spectra.The final Faraday spectra are cleaned down to 6 σ only so as toavoid over-cleaning, which can introduce artefacts to the resultingspectra.The Faraday spectra amplitudes ( | F φ | = (cid:113) q φ + u φ ) are shownin Figure 1. For each amplitude spectrum, we counted the numberof peaks higher than 6 σ , and then we fitted the spectrum with thecorresponding number of Gaussian components plus a y -offset toextract the FD values and widths of the peaks. This 6 σ cutoff grantsus an insignificant false detection rate of (cid:46) . (cid:46) . δφ ) of a peakis within 10 per cent from the theoretical RMTF FWHM value (i.e. δφ ; obtained from pyrmsynth output), we re-fit the spectrum with δφ being fixed at δφ . The uncertainties in FD are obtained by (e.g.Mao et al. 2010; Iacobelli et al. 2013) δφ · ( S / N ) , (8)where S / N is the signal-to-noise ratio of the peak. The FD and δφ of the peaks are then fixed and used to fit the q φ and u φ Faradayspectra to extract the complex polarisation of the Faraday compo-nents that they correspond to. The obtained values, namely φ , δφ ,and complex polarisation, are then used to calculate p and intrinsicPA (PA ) of each Faraday component. The uncertainties are propa-gated by Monte Carlo simulations with 10 realisations per source,starting from assuming that q , u , φ , and δφ obtained from RM-Synthesis above follow Gaussian statistics. We evaluated the 68.3 Available on .MNRAS , 1–27 (2019) he n π -ambiguity in the NVSS RM Catalogue Table 3.
Results of RM-Synthesis on Broadband VLA DataSource p PA φ φ RM TSS09 | φ − RM TSS09 | δφ δφ (NVSS) (%) ( ◦ ) (rad m − ) (rad m − ) (rad m − ) (rad m − ) (rad m − ) (rad m − ) Outliers (Reliable RM
TSS09 ) J083930 − . + . − . + . + . − . + . ± . + . ± . + . ± . . . − . + . − . − . + . − . + . ± . + . ± . + . ± . . . − . + . − . + . + . − . + . ± . + . ± . + . ± . . . − (cid:63)(cid:63) — — — + . + . − . + . ± . · · · a 8 . + . − . + . + . − . + . ± . . · · · b 3 . + . − . + . + . − . + . ± . . − . + . − . − . + . − . + . ± . + . ± . + . ± . . . − (cid:63)(cid:63) — — — + . + . − . + . ± . · · · a 7 . + . − . − . + . − . + . ± . . · · · b 5 . + . − . + . + . − . + . ± . . − (cid:12)‡ . + . − . − . + . − . + . ± . + . ± . + . ± . . ± . . − (cid:63)(cid:63) — — — − . ± . − . ± . · · · a — — — — — — — — · · · b 11 . + . − . + . + . − . − . ± . . − (cid:63)(cid:63) — — — − . + . − . − . ± . · · · a 9 . + . − . + . + . − . − . ± . . · · · b 10 . + . − . + . + . − . − . ± . . + . + . − . + . + . − . − . ± . − . ± . − . ± . . . + . + . − . − . + . − . − . ± . − . ± . − . ± . . . Out- liars ( n π -ambiguity) J022915 + ! ‡ — — — + . ± . + . ± . . · · · FC 1 0 . + . − . + . + . − . − . ± . . ± . · · · FC 2 4 . + . − . − . + . − . + . ± . . ± . · · · FC 3 0 . + . − . + . + . − . + . ± . . ± . − (cid:63)(cid:63) — — — + . + . − . − . ± . · · · a 8 . + . − . − . + . − . + . ± . . · · · b 17 . + . − . − . + . − . + . ± . . − ‡ — — — + . + . − . − . ± . . · · · FC 1 4 . + . − . + . + . − . + . ± . . · · · FC 2 0 . + . − . − . + . − . + . ± . . + ‡ — — — + . + . − . − . ± . . · · · FC 1 0 . + . − . − . + . − . − . ± . . · · · FC 2 0 . + . − . + . + . − . + . ± . . − ‡ — — — + . + . − . − . ± . . · · · FC 1 0 . + . − . − . + . − . − . ± . . · · · FC 2 3 . + . − . − . + . − . + . ± . . + ‡ — — — + . + . − . − . ± . . · · · FC 1 1 . + . − . − . + . − . + . ± . . · · · FC 2 2 . + . − . + . + . − . + . ± . . + . + . − . + . + . − . − . ± . − . ± . + . ± . . . + (cid:12)‡ — — — − . + . − . + . ± . . · · · FC 1 0 . + . − . + . + . − . − . ± . . ± . · · · FC 2 5 . + . − . + . + . − . − . ± . . ± . + . + . − . + . + . − . + . ± . + . ± . − . ± . . . Others
J084600 − × — — — — + . ± . . + ? ‡ — — — − . + . − . − . ± . . · · · FC 1 0 . + . − . + . + . − . − . ± . . ± . · · · FC 2 0 . + . − . + . + . − . − . ± . . ± . · · · FC 3 0 . + . − . − . + . − . + . ± . . ± . + × — — — — + . ± . . × Unpolarised sources ? Special case compared to TSS09 catalogue (see Section 4.1.8) (cid:63)(cid:63)
Double point sources (cid:12)
Extended sources ! Faraday components (FCs) 1 and 3 may be artefacts corresponding to RMTF sidelobes (see text) ‡ Faraday complex from RM-SynthesisMNRAS000
Extended sources ! Faraday components (FCs) 1 and 3 may be artefacts corresponding to RMTF sidelobes (see text) ‡ Faraday complex from RM-SynthesisMNRAS000 , 1–27 (2019)
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Figure 1.
Faraday spectra of our target sources. Blue lines show the amplitude of the complex Faraday spectra after deconvolution, with the black barsrepresenting the clean components. The RM values from the NVSS RM catalogue (TSS09) are represented by the red vertical solid lines, while those RMvalues corresponding to ± π -ambiguity are indicated by red vertical dashed lines. We only show the spectra within the FD range of − + − ,as significant Faraday components are not found outside of this range. per cent confidence interval (corresponding to 1 σ under normaldistribution), which are listed as the asymmetric errors in Table 3.Such an error propagation method is needed, since strictly speakingthe uncertainties of both p and PA do not follow Gaussian distri-butions. A caveat to the results here is that the polarisation fraction p is the polarised intensity of the Faraday component divided bythe total intensity of the entire spatial component. The Ricean po-larisation bias is not corrected for because it is insignificant at oursignal-to-noise levels (see above).We also formed Faraday spectra for the leakage calibratorsJ0319+4130, J0713+4349, and J1407+2827, in order to constrainthe remaining instrumental polarisation leakage of our observations. Their spectra are also shown in Figure 1, with peak values of 0 . ± . . ± . . ± .
003 per cent, respectively.Note that these values could be due to random noise fluctuationsleading to polarisation bias (e.g. George et al. 2012) instead of due toresidual instrumental polarisation leakage, and are therefore upperlimits to the actual remaining leakage levels of our calibrated data.We conclude that the residual polarisation leakage in our data is at < .
02 per cent level.One point to note is that for one of our sources,J022915+085125, Faraday components (FCs) 1 and 3 are likelyartefacts corresponding to the sidelobes of the RMTF (see Table 3),most likely because the main (physical) peak is Faraday thick, lead-
MNRAS , 1–27 (2019) he n π -ambiguity in the NVSS RM Catalogue Figure 1. (Continued) Faraday spectra of our target sources. ing to sub-optimal deconvolution with RM-Clean. The two com-ponents are symmetric about the prominent Faraday component 2,having the same polarisation fraction of 0.27 per cent, PA and φ off-sets from component 2 by about 90 ◦ and 260 rad m − respectively,and δφ ≈
73 rad m − , less than the theoretical value of 124 rad m − .Upon inspection of the (complex) RMTF of this source, we find thatthe secondary maxima are offset from the primary by 161 rad m − ,phase offset by 180 ◦ (i.e. 90 ◦ in PA), and have FWHM of about108 rad m − . We have therefore ignored these two components inthe remainder of this paper.For the six sources that can be decomposed into multiple spatialor Faraday components, it is not trivial to directly compare themultiple FD values against the single RM TSS09 value of each source. Therefore, we define a polarisation-weighted FD as φ = (cid:213) i p i · S . , i · φ i (cid:205) j p j · S . , j , (9)where i and j are indices representing the spatial and/or Faradaycomponents, and S . is the flux density of the correspondingspatial component at 1.4 GHz (listed in Paper II). This formulationis a modified version of that from O’Sullivan et al. (2017), wherethe FDs were weighted by p instead. The uncertainties in φ are againpropagated by Monte Carlo simulations as above. We will comparethe φ values against the TSS09 RM values to determine whether thesource suffers from n π -ambiguity. The results are listed in Table 3.We find that two of our target sources (J084600 − + σ cutoff levelsat 0 .
07 and 0 .
06 per cent, respectively), and therefore are excluded
MNRAS000
MNRAS000 , 1–27 (2019)
Y. K. Ma et al.
Figure 1. (Continued) Faraday spectra of our target sources, as well as that of the leakage calibrators. The typical RMTF of our L-band observations is shownin the last panel. in the subsequent stages of our study in this paper . Furthermore,the spatial double J162706 − φ disagreeing with the TSS09 RM values by about ± . − , and 11 have the two sets of values agreeing within60 rad m − . The only remaining source J154936 + φ and RM TSS09 values deviating by 307 . − (see Section 4.1.8 for discussion on this source).We further performed a per-pixel RM-Synthesis analysis tothe extended sources J094750 − + QU -fitting results in Section 3.2. QU -fitting We complement our RM-Synthesis results in Section 3.1 withStokes QU -fitting analysis (e.g. Farnsworth et al. 2011; O’Sullivan We believe this discrepancy with TSS09 in polarisation level is due to theoff-axis polarisation leakage in the NVSS data, and we shall investigate thisin detail in the companion Paper II. et al. 2012). Tests with synthetic data have shown that QU -fittingcan perform better than RM-Synthesis for sources composed of twoFaraday thin components (Sun et al. 2015). The main differencebetween these two techniques is that the former is non-parametric,while the latter requires input astrophysical models. These modelsconsist of one or more Faraday components added together, whichcan correspond to discrete astrophysical sources or emitting vol-umes with different physical parameters within our telescope beamor flux integration region. For our study, we considered the followingFaraday components (Burn 1966; Sokoloff et al. 1998; O’Sullivanet al. 2012):(i) Thin:
A purely synchrotron-emitting volume, with Faradayrotation occurring in a foreground screen with a homogeneous mag-netic field and thermal electron density. The complex polarisationis given by p j ( λ ) = p , j e i ( PA , j + φ j λ ) . (10)(ii) Burn slab:
This depicts a volume that is simultaneouslysynchrotron-emitting and Faraday rotating, with no foregroundFaraday rotating screens. The magnetic fields, thermal electron den-sities, and cosmic rays densities in the slab are all uniform. The
MNRAS , 1–27 (2019) he n π -ambiguity in the NVSS RM Catalogue complex polarisation is given by p j ( λ ) = p , j sin ( φ j λ ) φ j λ e i ( PA , j + φ j λ ) . (11)(iii) Burn slab with foreground screen:
This is the same as a Burnslab component, except there is a homogeneous foreground rotatingscreen giving rise to an extra FD of φ fg . The complex polarisationis given by p j ( λ ) = p , j sin ( φ j λ ) φ j λ e i [ PA , j + ( φ j + φ fg ) λ ] . (12)(iv) External Faraday dispersion:
In addition to the homoge-neous Faraday screen for a thin component, an external turbulentFaraday screen lies in front of the synchrotron-emitting volume.This turbulent screen leads to a dispersion in FD ( σ φ ) through dif-ferent lines of sight to the emitting volume (within the telescopebeam or the flux integration region), causing depolarisation effects.The complex polarisation is given by p j ( λ ) = p , j e − σ φ, j λ e i ( PA , j + φ j λ ) . (13)(v) Internal Faraday dispersion:
This is similar to the Burn slababove, except that in the simultaneously emitting and Faraday rotat-ing volume there is also a turbulent magnetic field component. Thecomplex polarisation is given by p j ( λ ) = p , j e i PA , j (cid:32) − e i φ j λ − σ φ, j λ σ φ, j λ − i φ j λ (cid:33) . (14)(vi) Internal Faraday dispersion with foreground screen:
This isthe same as the internal Faraday dispersion component, but there isa homogeneous foreground rotating screen leading to an extra FDof φ fg . The complex polarisation is given by p j ( λ ) = p , j e i ( PA , j + φ fg λ ) (cid:32) − e i φ j λ − σ φ, j λ σ φ, j λ − i φ j λ (cid:33) . (15)A caveat of the QU -fitting technique here is that, similar toRM-Synthesis in Section 3.1, the intrinsic polarisation fraction p , j obtained from this analysis is the polarised intensity of the compo-nent j divided by the total intensity of the entire spatial component,since this analysis does not separate the total intensity into corre-sponding Faraday components.We deployed 10 different models to fit the observed q and u values of our target sources: single thin (1T), double thin (2T),triple thin (3T), single Burn slab (1B), double Burn slab (2B), sin-gle Burn slab with foreground screen (1B+fg), double Burn slabwith foreground screen (2B+fg), single external Faraday dispersion(1Ed), single internal Faraday dispersion (1Id), and single internalFaraday dispersion with foreground screen (1Id+fg). The complexpolarisation of the models are constructed by adding together that ofthe constituent Faraday components [ p ( λ ) = (cid:205) j p j ( λ ) ]. In otherwords, the Faraday components of each model are assumed to bespatially distributed perpendicular to the line of sight. For the dou-ble Burn slab with foreground screen model, both of the Burn slabcomponents are subjected to the same foreground FD, instead ofhaving individual φ fg values. The best-fit parameters and their un-certainties of each of the models for each target source are obtained,along with the reduced chi squared values ( χ ) and the Bayesianinformation criterion (BIC; e.g. O’Sullivan et al. 2012; Schnitzeleret al. 2019). We rejected models where the p , j and/or σ φ, j valuesare less than two times of the uncertainties. The remaining modelsfor each source are ranked according to the BIC values (with a lower value signifying a better model), and the best for each source is listedin Table 4 and plotted in Figure C1 in the Online SupplementaryMaterials (Appendix C). n π -ambiguity in the NVSS RM Catalogue In Section 3.1, we compared our φ values from RM-Synthesis per-formed on the new broadband data with narrowband RMs from theNVSS RM catalogue (TSS09). Nine out of 21 of our polarised tar-get sources (43 per cent) have φ values deviating by approximately ± . − from the corresponding RM TSS09 . The discrepancyis almost certainly due to n π -ambiguity in the TSS09 catalogue. Inan attempt to unveil the cause(s) and possible diagnostic(s) of this,we divided our sources into the two classes – out- liars and outliers– and compared select observed quantities. Specifically, we investi-gated the distributions of spectral index from our L-band observa-tions ( α L ; reported in Paper II), NVSS flux density ( S NVSS ), TSS09polarised intensity (PI
TSS09 ), TSS09 polarisation fraction ( p TSS09 ),RM
TSS09 , φ , | RM TSS09 − RM ◦ | , and | RM TSS09 − RM ◦ |/ σ ◦ , withRM ◦ and σ ◦ defined below. For each parameter, we performedtwo-sample Kolmogorov–Smirnov test (KS-test) with the null hy-pothesis being that the two samples are drawn from the same popu-lation. The above parameters are plotted in Figure 2, with their cor-responding KS-test p-values also reported. We adopted the standardp-value cutoff of 0.05 (a larger p-value favours the null hypothesis),and concluded that the two populations have different distributionsin α L , p TSS09 , | RM TSS09 − RM ◦ | , and | RM TSS09 − RM ◦ |/ σ ◦ ,which we will discuss in detail below. On the other hand, our KS-test results suggest that the two classes of sources likely originatefrom the same population in S NVSS , PI
TSS09 , RM
TSS09 , and φ ,with p-values of 0.168, 0.471, 0.058, and 0.085, respectively. How-ever, note that given this small sample size (nine and 11 in the twoclasses), we cannot rule out the possibility that our statistical analy-sis here could be biased by random statistical anomalies. Below, wewill also explore the effects of FD ranges and Faraday complexities(Table 5) on n π -ambiguity in TSS09 catalogue, and investigate thespecial case J154936 + φ and RM TSS09 consistent with neither the outlier nor the out- liar cases. n π -ambiguity Rejection Algorithm Before looking into the dependence of n π -ambiguity on variousparameters, we review the algorithm devised by TSS09 to minimise n π -ambiguity in their catalogue. This algorithm picks the mostprobable RM value for each source based on the following threeconstraints. First, they assumed that at most only a single PA wrapcan occur between the two NVSS IFs. This imposes a limit of | RM TSS09 | (cid:54) − for all sources. Second, they introducedthe parameter R = PI + PI c , (16)where PI , PI , and PI c are the polarised intensities in NVSS IF1,IF2, and combined band, respectively. Since the measured PI is afunction of the source | RM | due to bandwidth depolarisation inthe NVSS observational setup, the parameter R in turn is alsoa function of | RM | (Figure 3). TSS09 compared the observed R of each source with the predicted R values at the few possibleRM values, with a likelihood assigned to each possible RM. This MNRAS000
TSS09 , and φ ,with p-values of 0.168, 0.471, 0.058, and 0.085, respectively. How-ever, note that given this small sample size (nine and 11 in the twoclasses), we cannot rule out the possibility that our statistical analy-sis here could be biased by random statistical anomalies. Below, wewill also explore the effects of FD ranges and Faraday complexities(Table 5) on n π -ambiguity in TSS09 catalogue, and investigate thespecial case J154936 + φ and RM TSS09 consistent with neither the outlier nor the out- liar cases. n π -ambiguity Rejection Algorithm Before looking into the dependence of n π -ambiguity on variousparameters, we review the algorithm devised by TSS09 to minimise n π -ambiguity in their catalogue. This algorithm picks the mostprobable RM value for each source based on the following threeconstraints. First, they assumed that at most only a single PA wrapcan occur between the two NVSS IFs. This imposes a limit of | RM TSS09 | (cid:54) − for all sources. Second, they introducedthe parameter R = PI + PI c , (16)where PI , PI , and PI c are the polarised intensities in NVSS IF1,IF2, and combined band, respectively. Since the measured PI is afunction of the source | RM | due to bandwidth depolarisation inthe NVSS observational setup, the parameter R in turn is alsoa function of | RM | (Figure 3). TSS09 compared the observed R of each source with the predicted R values at the few possibleRM values, with a likelihood assigned to each possible RM. This MNRAS000 , 1–27 (2019) Y. K. Ma et al.
Out-liars(nπ-ambiguity) Outliers(Correct RM) −1.2−1.0−0.8−0.6−0.4−0.20.0+0.2 α L (a) p-value = 0.045 Out-liars(nπ-ambiguity) Outliers(Correct RM)10 S NV SS ( m Jy ) (b) p-value = 0.168 P I T SS ( m Jy ) (c) p-value = 0.471 p T SS ( % ) (d) p-value = 0.034 −600−400−2000+200+400+600 R M T SS (r ad m − ) (e) p-value = 0.058 −600−400−2000+200+400+600 φ (r ad m − ) (f) p-value = 0.085 Out-liars(nπ-ambiguity) Outliers(Correct RM)0100200300400500600 | R M T SS − R M ◦ | (r ad m − ) (g) p-value = 1.53E-04 Out-liars(nπ-ambiguity) Outliers(Reliable RM)051015202530 ∆ / σ (h) p-value = 2.10E-05 Figure 2.
Select parameters of our 20 target sources separated into out- liars (left; in blue) and outliers (right; in red). The p-value from two-sample KS-testis reported in each panel. In relevant cases, the medians of the two populations are plotted as blue dashed (for out- liars ) and red dotted (for outliers) lines. Inpanel (f), the areas highlighted in green corresponds to the | RM | ranges of < > ≈ . − , within which the R parameter has limitedreliability (TSS09). The grey solid line in panel (h) indicates the cutoff level at 2.85 (see text). MNRAS , 1–27 (2019) he n π -ambiguity in the NVSS RM Catalogue Table 4.
Results of QU -fitting on the New Broadband VLA DataSource Faraday φ p PA φ fg σ φ χ ∆ PI / PI a (NVSS) Model (rad m − ) (%) ( ◦ ) (rad m − ) (rad m − ) (%) Outliers (Reliable RM
TSS09 ) J083930 − + . ± . . ± . + . ± . . + . − + . ± . . ± . + . ± . . + . − + . ± . . ± . + . ± . . + . − + . ± . . ± . + . ± . . + . + . ± . . ± . + . ± . − + . ± . . ± . + . ± . . ± . . + . − + . ± . . ± . − . ± . + . ± . . + . − + . ± . . ± . − . ± . . + . − + . ± . . ± . + . ± . . + . − (cid:12) + . ± . . ± . − . ± . + . ± . . − . − − . ± . . ± . + . ± . . + . − . ± . . ± . − . ± . − − . ± . . ± . + . ± . . + . − + . ± . . ± . + . ± . − . ± . . + . + − . ± . . ± . + . ± . . + . + − . ± . . ± . − . ± . . + . Out- liars ( n π -ambiguity) J022915 + + . ± . . ± . + . ± . . − . − . ± . . ± . + . ± . − + . ± . . ± . − . ± . . + . − + . ± . . ± . − . ± . . + . − + . ± . . ± . + . ± . . − . + . ± . . ± . − . ± . + + . ± . . ± . + . ± . . − . + . ± . . ± . + . ± . − + . ± . . ± . − . ± . + . ± . . + . − . ± . . ± . + . ± . + . ± . + + . ± . . ± . + . ± . . + . + . ± . . ± . − . ± . + − . ± . . ± . + . ± . . ± . . − . + (cid:12) − . ± . . ± . + . ± . − . ± . . ± . . − . + + . ± . . ± . + . ± . . + . + . ± . . ± . − . ± . Others
J084600 − × — — — — — — — —J154936 + ? − . ± . . ± . + . ± . . − . + . ± . . ± . − . ± . − . ± . . ± . + . ± . + × — — — — — — — — NOTE — Key to the Faraday components: T: Thin; B: Burn slab; B+fg: Burn slab with foreground screen; Ed: External Faradaydispersion; Id: Internal Faraday dispersion; Id+fg: Internal Faraday dispersion with foreground screen a ∆ PI = PI − PI and PI = ( PI + PI )/
2, where PI and PI are the predicted polarised intensities at the two NVSS IFs according tothe best-fit model and fitted α L (reported in Paper II), without taking bandwidth depolarisation into account × Unpolarised sources ? Special case compared to TSS09 catalogue (see Section 4.1.8) (cid:12)
Extended sources means TSS09 assumed that the differences among PI , PI , andPI c are only due to bandwidth depolarisation, but not caused byother effects such as spectral indices and Faraday complexities.Lastly, they rejected candidate RM values that deviated significantlyfrom the RM values of surrounding sources. Specifically, for eachsource they computed the median RM TSS09 of neighbouring sourceswithin a radius of 3 ◦ , and only accepted candidate RM values within520 rad m − from the median RM. This implicitly assumes that theRM of individual sources cannot deviate significantly from that of their neighbours due to intrinsic RM or spatial fluctuations offoreground RM. The most likely candidate RM remaining is thenreported as the RM TSS09 . As seen in panel (a) of Figure 2, the out- liars and outliers appearto exhibit different distributions in α L . This is further supported bythe KS-test p-value of 0.045. The out- liars have α L spread evenly MNRAS000
Extended sources means TSS09 assumed that the differences among PI , PI , andPI c are only due to bandwidth depolarisation, but not caused byother effects such as spectral indices and Faraday complexities.Lastly, they rejected candidate RM values that deviated significantlyfrom the RM values of surrounding sources. Specifically, for eachsource they computed the median RM TSS09 of neighbouring sourceswithin a radius of 3 ◦ , and only accepted candidate RM values within520 rad m − from the median RM. This implicitly assumes that theRM of individual sources cannot deviate significantly from that of their neighbours due to intrinsic RM or spatial fluctuations offoreground RM. The most likely candidate RM remaining is thenreported as the RM TSS09 . As seen in panel (a) of Figure 2, the out- liars and outliers appearto exhibit different distributions in α L . This is further supported bythe KS-test p-value of 0.045. The out- liars have α L spread evenly MNRAS000 , 1–27 (2019) Y. K. Ma et al. |RM| (rad m −2 )0.00.20.40.60.81.0 D epo l a r i s a t i on NVSS IF1; α =0.0NVSS IF2; α =0.0NVSS Combined; α =0.0R (|RM|); α =−0.9R (|RM|); α =−0.6R (|RM|); α =−0.3R (|RM|); α =0.0 R Figure 3.
Bandwidth depolarisation and R = ( PI + PI )/ c of theNVSS observational setup, assuming a Faraday simple source with constant p ( λ ) . The bandwidth depolarisation of the two NVSS IFs, as well as thatcombining both bands, are shown as the black curves for the case withspectral index of α = . R with different α values are plotted as thecoloured curves, with the y -axis truncated at R =
150 because the peak of R for α = − . ≈
340 rad m − reaches about 900, which can haveobscured the other lower peaks. over a wide range from − .
99 to + .
02, with a median of − . α L of most (ten out of 11) of the outliers cluster between − .
14 and − .
69, with a median of − .
9. It would be natural todirectly link this discrepancy to the R parameter used in the TSS09algorithm. This is because the change in PI across λ caused byspectral index effects could be mistaken as bandwidth depolarisa-tion by the R algorithm, possibly leading to n π -ambiguity. To testthis hypothesis, we simulated R as a function of | RM | for severaldifferent α L values (0 . − . − .
6, and − . R values at any given | RM | are only weakly dependent on α L exceptnear the peak at | RM | ≈
340 rad m − where the predicted R di-verge. This means that R could be less effective in distinguishingdifferent RM values for sources with true | RM | ≈
340 rad m − .TSS09 also reached similar conclusion regarding this | RM | rangebut for different reasons (see Section 4.1.4). However, only two outof nine out- liars (J094808 − + | RM | range, and therefore spectral index dependence cannotexplain most of our n π -ambiguity sources. We show in panel (c) of Figure 2 the distribution of the two classesin PI
TSS09 , which is the average PI in the two IFs [i.e. ( PI + PI )/ TSS09 . It is worth noting, however,that while a previous study of 37 radio sources with high PI
TSS09 ( >
200 mJy) also at 1–2 GHz with the Allen Telescope Array (ATA)found no n π -ambiguity in the TSS09 catalogue, we have identifiedJ224549 + ≈ . ± . TSS09 = . ± . Q and U values for spatially resolved sources), as an out- liar . This showsthat not all sources with high PI have reliable TSS09 RM values. Table 5.
Number of Faraday Simple/Complex SourcesRM-Synthesis QU -fittingFaraday Faraday Faraday FaradaySimple Complex Simple ComplexOut- liars n π -ambiguity)Outliers 10 1 6 5(Reliable RM TSS09 ) The discrepancies between the two populations in the fractionalpolarisation reported in the TSS09 catalogue ( p TSS09 ) is more ap-parent, with KS-test p-value of 0.034. The p TSS09 values are plottedin panel (d) of Figure 2. Most (79 per cent) out- liars are concen-trated at 0.8–3.1 per cent, while the outliers spread more evenlybetween 2.6 and 8.7 per cent. The median p TSS09 of out- liars andoutliers are respectively 1.9 and 4.2 per cent. As we show in PaperII, sources with lower fractional polarisation are more susceptibleto instrumental effects, particularly off-axis polarisation leakage,which can diminish the effectiveness of the TSS09 algorithm. How-ever, this alone cannot explain all the out- liars we identified, sincetwo of them are highly polarised at 9 . ± . + . ± . − TSS09 pointed out that R (Equation 16) could be less effec-tive in selecting the correct RM values for sources with true | RM | falling within ranges of <
50 rad m − , >
520 rad m − , and ≈ . − (taken as 301–352 rad m − here). For the case of <
50 and >
520 rad m − , that is because for both cases R ≈ | RM | valueof ≈ . − is also believed to be a challenge for the TSS09algorithm, since (1) there is almost complete bandwidth depolarisa-tion for the combined band, and (2) R cannot distinguish betweenthe case of + . − . − as the predicted R are thesame.We plotted φ of the two samples in panel (f) of Figure 2,with the above ranges shaded in green. Out of our sample, tensources fall into these ranges, with only three being out- liars – J022915 + + . ± . − , J224412 + − . ± . − , and J235728 + + . ± . − . It is apparent that out- liars do not preferentially fallinto the above RM ranges, and our samples within those ranges aremore likely to have correct RM TSS09 than suffer from n π -ambiguity. Faraday complexity (formally defined in Section 4.3.1) could beone of the reasons for the presence of n π -ambiguity in the NVSSRM catalogue. As summarised in Table 5, six out of the nine (67per cent) out- liars are Faraday complex from our RM-Synthesisresults, while only one out of the 11 (9 per cent) outliers showFaraday complexities from the same analysis. We can draw similarconclusion from the QU -fitting results, with eight out of nine (89per cent) and five out of 11 (45 per cent) sources being Faradaycomplex, respectively. This may be because the R algorithm canbe affected by both non-linear PA and varying p across λ .We test the possibility of the latter by quantifying the amountof Faraday depolarisation due to Faraday complexities. For each MNRAS , 1–27 (2019) he n π -ambiguity in the NVSS RM Catalogue source, we adopted the best-fit model from QU -fitting, as well asspectral index α L from Paper II, to compute the PI at the twoNVSS IFs (PI and PI ) without taking bandwidth depolarisationinto account. A depolarisation parameter is defined as ∆ PIPI = PI − PI ( PI + PI )/ , (17)which is listed in Table 4 for each source. Note that the valuesfor even sources best characterised by the single thin model arenon-zero because of the effect of the spectral index, which leads toa positive ∆ PI / PI with negative α L . Apart from J022915 + | ∆ PI / PI | = . p ( λ ) at NVSS IF1approaches zero, we do not see clear signs of out- liars having larger | ∆ PI / PI | , as would be expected if the Faraday depolarisation affectsthe R algorithm leading to n π -ambiguity. For each of our target sources, we evaluated the medians (RM ◦ ) andstandard deviations ( σ ◦ ) of RM TSS09 values of the neighbouringsources within a radius of 3 ◦ . These values are listed in column 4 ofTable 6. On average, there are 28 neighbours to our target sourceswithin the 3 ◦ radius circles. Assuming that RM TSS09 values arecorrect for most of the neighbouring sources, RM ◦ and σ ◦ wouldrespectively represent the RM contribution by large-scale Galacticand/or intergalactic component(s), and the spatial fluctuations of theabove-mentioned foreground RM components superimposed on thestatistical spread of intrinsic RM of the neighbouring sources.The out- liars clearly deviate from the outliers in | RM TSS09 − RM ◦ | values, as we show in panel (g) of Figure 2. The out- liars gather within 416–522 rad m − , with a median of 502 rad m − ,while the outliers spread through 109 to 469 rad m − , with a me-dian of 199 rad m − . The large values for out- liars are clearly dueto n π -ambiguity, leading to discrepancies between the individualRM TSS09 and the respective RM ◦ . Large values are also found forthree outliers, which could stem from spatial variations of the fore-ground RM structures around the positions of those of our targets. Ifthis is the case, we would expect high σ ◦ values from those outliersas well.An even clearer diagnostic is therefore the deviation in RM inunits of σ ◦ . We plotted this ( | RM TSS09 − RM ◦ | / σ ◦ ; shortenedas ∆ / σ in text below) in panel (h) of Figure 2. Indeed, we foundthat all outliers converged to 0 . .
71 in ∆ / σ , meaning that thosewith high | RM TSS09 − RM ◦ | also have high σ ◦ , matching ourexpectation above. The out- liars , on the other hand, have ∆ / σ spreadover 3 .
02 to 27 .
24, since the large | RM TSS09 − RM ◦ | are due to n π -ambiguity and not necessarily accompanied by high σ ◦ dueto spatial variations of the foreground. There is an apparent cutoffbetween the two populations at about 2 .
85. Since this ∆ / σ parametercan be computed from the listed information from the NVSS RMcatalogue without any extra information, this could be useful foridentification of n π -ambiguity sources in the TSS09 catalogue (seeSection 4.1.7). n π -ambiguity? We apply our findings from Section 4.1.6 to estimate how manyTSS09 sources suffer from n π -ambiguity. The ∆ / σ values for all ofthe 37,543 TSS09 sources have been computed. There is an averageof 33 neighbouring sources for each TSS09 source. Sources with ∆ / σ larger than 2 .
70, 2 .
85, and 3 .
00 are identified, corresponding to loose, moderate, and strict cutoffs respectively according to Sec-tion 4.1.6. Although we found that 837, 701, and 603 sources satisfythe above lower limits in ∆ / σ respectively, we also noted that someof such sources have low | RM TSS09 − RM ◦ | . These sources may belocated at regions with smooth RM foreground leading to low σ ◦ and high ∆ / σ , but not suffering from n π -ambiguity. We thereforeimposed another constraint of | RM TSS09 − RM ◦ | (cid:62)
400 rad m − .This results in 56, 49, and 48 n π -ambiguity candidates in the entireTSS09 catalogue, depending on whether we adopt the loose, mod-erate, or strict cutoffs as defined above, respectively. Note that this isa lower limit estimated by the ∆ / σ parameter only, which may notexhaust the entire n π -ambiguity population of TSS09 (see below).On the other hand, EGSs with high intrinsic FD (or RM) magnitudesof (cid:38)
400 rad m − might also be included under the above selectioncriteria.We further compared our list of n π -ambiguity candidates withthe literature to verify the accuracy of our ∆ / σ criterion. Thewrongly classified sources (if any) can be separated into two cate-gories – false-positives ( n π -ambiguity candidates that actually havereliable RM TSS09 ) and false-negatives (sources that actually sufferfrom n π -ambiguity but not picked up by our algorithm above). Nofalse-positives have been identified after consulting catalogues ofpolarised sources verified to have reliable RM TSS09 (Mao et al.2010; Law et al. 2011; Van Eck et al. 2011; Mao et al. 2012; Raweset al. 2018; Betti et al. 2019), suggesting that our list of n π -ambiguitycandidates is accurate. We further compare our findings with theknown TSS09 n π -ambiguity sources listed in the literature to lookfor the false-negatives. Van Eck et al. (2011) reported RM valuesof 194 EGSs on the Galactic plane ( | b | (cid:54) ◦ ) with their obser-vations, of which 146 were cross-matched with TSS09. From thissample, 13 sources (9 %) were found to suffer from n π -ambiguityin TSS09. Most of these 13 sources are concentrated in the innerGalaxy (35 ◦ (cid:54) l (cid:54) ◦ ), with 11 out of the 15 cross-matches inthat region suffering from n π -ambiguity. This is likely linked tothe complex large-scale magnetic field structure of the Milky Waymanifested as large | RM | and rapid changes in RM in small spatialscales of a few degrees (e.g., Sun et al. 2008; Van Eck et al. 2011;Jansson & Farrar 2012), ultimately leading to the concentrationof n π -ambiguity sources there. However, using our ∆ / σ parame-ter defined above only one out of those 11 n π -ambiguity sourcesfound there is correctly classified as an n π -ambiguity candidate.This means that our n π -ambiguity candidates list from ∆ / σ is con-servative, i.e. there can be more than 50 n π -ambiguity sources inthe entire TSS09 catalogue. + Upon comparison between our broadband φ with narrowbandRM TSS09 (Section 3.1), we identified J154936 + . − . This source canneither be classified as an out- liar nor an outlier, as these twoclasses of sources should have deviating φ and RM TSS09 by about652 . − respectively. To rule out the possibility thatthis discrepancy of 307 . − is due to RM time variabil-ities, we compared the RM TSS09 of this source with RM
VLA from our Paper II. This RM
VLA is obtained from our new ob-servations within the NVSS frequency ranges only. We find thatthis source has RM
VLA = − . ± . − , similar to itsRM TSS09 = − . ± . − . In other words, the differenceof 307 . − above cannot be attributed to time variabilities.This peculiar difference in φ versus RM TSS09 is likely due toits Faraday complexity. Both RM-Synthesis and QU -fitting suggest MNRAS000
VLA = − . ± . − , similar to itsRM TSS09 = − . ± . − . In other words, the differenceof 307 . − above cannot be attributed to time variabilities.This peculiar difference in φ versus RM TSS09 is likely due toits Faraday complexity. Both RM-Synthesis and QU -fitting suggest MNRAS000 , 1–27 (2019) Y. K. Ma et al. that this source contains three Faraday components at FDs of about − −
20, and +
75 rad m − , with p of about 0 .
46, 0.27, and 0.33per cent respectively. Such a wide spread of Faraday componentsover FD, combined with their similar fractional polarisation, resultsin highly non-linear PA across λ in the NVSS bands, as well as inour broadband L-band. This leads to a poor agreement between φ and RM TSS09 .J154936 + ∼
100 rad m − ).For such case, narrowband RM values are clearly poor represen-tations of the foreground magneto-ionic media, while techniquesapplied to broadband data such as extraction of absolute maximain Faraday spectra (e.g. Mao et al. 2010; Betti et al. 2019) andusing polarisation-weighted FD (e.g. O’Sullivan et al. 2017, andthis work) also may not give satisfactory results. This highlightsthe power of broadband spectro-polarimetric observations, whichhave opened up the possibility to identify such sources for carefultreatments in RM grid experiments and/or further studies of theirintrinsic polarisation properties. In this Section, we showed the differences in the statistical distribu-tions for several parameters of out- liars ( n π -ambiguity sources inTSS09) versus outliers (sources with reliable RM TSS09 ). We sug-gest that low p TSS09 could cause n π -ambiguity in TSS09 values.Also, out- liars are found to have larger spread in α L and tends tobe Faraday complex, while outliers are concentrated at steeper α L and are more likely Faraday simple. However, there may not be adirect relationship between these and n π -ambiguity. Out- liars donot appear to preferentially fall within | FD | ranges of < > ≈ . − . We further compared, for each of our targetsources, their RM TSS09 with the median (RM ◦ ) and standard devi-ation ( σ ◦ ) of RM TSS09 of neighbouring sources within a radius of3 ◦ . All out- liars cluster at | RM TSS09 − RM ◦ | ≈
500 rad m − , whileoutliers span a range between 110 to 470 rad m − . Most interest-ingly, we found that ∆ / σ = | RM TSS09 − RM ◦ |/ σ ◦ is an excellentdiagnostic for n π -ambiguity in TSS09 catalogue. This parameter isan indicator of how much the RM TSS09 value of each source devi-ates from the RM caused by foreground structures, in units of howmuch such foreground structures fluctuate spatially. There is a cutoffat ∼ .
85 between the two classes of sources, with out- liars beingabove this cutoff and outliers below. Using this ∆ / σ parameter, com-bined with a further constraint of | RM TSS09 − RM ◦ | (cid:38)
400 rad m − to discard sources situated behind smooth RM foregrounds with low σ ◦ , we estimate that at least 50 out of the 37,543 TSS09 sourcescan be affected by the n π -ambiguity effect. Out of our 21 polarised target sources, we found (from RM-Synthesis) that 15 of them have | φ | >
200 rad m − . Such high | φ | values are peculiar for sources away from the Galactic plane, asis the case for our targets ( | b | > ◦ ). While the FD could originatefrom within the EGSs themselves or from their immediate ambi-ent media, it is challenging to directly confirm this scenario withthe available information. We therefore explore the possibility ofexplaining the FD values from Galactic contributions and/or from foreground galaxy clusters. For the former, as Galactic FD (or RM)structures are often associated with warm and/or cold phases of theinterstellar medium (e.g. Heiles & Haverkorn 2012), we looked intotheir respective tracers (H α and H i) as an attempt to unveil theorigin of the FDs. α Maps
Upon inspection of the Wisconsin H-Alpha Mapper Sky Survey(WHAMSS; Haffner et al. 2003, 2010) images, we found that nineof our high | φ | EGSs (J083930 − − − − − − − − − − − + + + α intensities ( I H α >
10 Rayleighs;see column 7 of Table 6). Thus, the high | φ | values of these 14sources could be attributed to foreground Galactic H ii structures.The only remaining EGS (J220927 + α intensity of only 5 .
75 R. The high FD ofthis source may have originated from a foreground galaxy cluster(Section 4.2.3).To assess the link between the H ii structures and the high | φ | , we estimate the regular magnetic field strengths ( B reg ) in thoseH ii clouds needed to produce the observed φ . We omit the GumNebula and Sh 2-27 here, as their magnetic field structures arealready studied in detail in Purcell et al. (2015) and Harvey-Smithet al. (2011) respectively using many of the above-mentioned EGSs.Following Harvey-Smith et al. (2011), emission measure (EM) and B reg are given byEM = . (cid:18) T e K (cid:19) . (cid:18) I H α R (cid:19) e τ cm − pc , (18) B reg ∼ √ B reg , (cid:107) = √ φ . √ EM √ f L µ G , (19)where T e is the electron temperature, τ is the optical depth due todust extinction, B reg , (cid:107) is the strength of the regular magnetic fieldcomponent parallel to the line of sight, f is the filling factor, and L is the integration path length through the H ii filament (in pc). Therelationship between B reg and B reg , (cid:107) stem from statistical argument,and it is implicitly assumed that n e is homogeneous. Furthermore, B reg , (cid:107) is assumed to be uniform in both strength and direction alongthe lines of sight. In particular, since EM is proportional to < n e >while FD is only proportional to < n e >, clumps of free electrons canresult in large EM values but only moderate FD values. We alsoassumed here that the observed φ of our EGSs come entirely fromthe H ii structures (i.e. zero intrinsic FD contributions). To get anupper limit of B reg , we assume a low T e of 7000 K (e.g. Peimbertet al. 2017), a typical f = . τ = . .
5) to obtain B reg (cid:46) . . (cid:18) φ rad m − (cid:19) (cid:18) L pc (cid:19) − (cid:18) I H α R (cid:19) − µ G . (20)Here, a lower optical depth would lead to a larger coefficient inthe above Equation. The only undetermined variable, L , can be ap-proximated by simple modelling of the geometry of the individualH ii filament. Sh 2-126 is located at a distance of about 370–600 pc MNRAS , 1–27 (2019) he n π -ambiguity in the NVSS RM Catalogue Table 6.
Foreground Diagnostics to Our Target SourcesSource φ RM TSS09 RM ◦ ± σ ◦ a RM O15 b N Hi c I H α d (NVSS) (rad m − ) (rad m − ) (rad m − ) (rad m − ) (10 cm − ) (Rayleighs)J022915 + † + . ± . + . ± . + . ± . + . ± . . E . − + . ± . + . ± . + . ± . + . ± . . G . − × — + . ± . + . ± . + . ± . . G . − + . ± . + . ± . − . ± . + . ± . . G . − + . ± . + . ± . + . ± . + . ± . . G . − (cid:63)(cid:63) † + . + . − . − . ± . + . ± . + . ± . . G . − (cid:63)(cid:63) + . + . − . + . ± . + . ± . + . ± . . G . − + . ± . + . ± . + . ± . + . ± . . G . − (cid:63)(cid:63) + . + . − . + . ± . + . ± . + . ± . . G . − (cid:12) + . ± . + . ± . − . ± . − . ± . . G . − † + . + . − . − . ± . + . ± . + . ± . . G . + † + . + . − . − . ± . + . ± . + . ± . . E . + ? − . + . − . − . ± . + . ± . + . ± . . E . − (cid:63)(cid:63) − . ± . − . ± . − . ± . − . ± . . G . − (cid:63)(cid:63) − . + . − . − . ± . − . ± . − . ± . . G . − † + . + . − . − . ± . + . ± . − . ± . . G . + † + . + . − . − . ± . + . ± . + . ± . . E . + − . ± . − . ± . − . ± . − . ± . . E . + − . ± . − . ± . − . ± . − . ± . . E . + † − . ± . + . ± . − . ± . − . ± . . E . + (cid:12)† − . + . − . + . ± . − . ± . − . ± . . E . + × — + . ± . − . ± . + . ± . . E . + † + . ± . − . ± . − . ± . − . ± . . E . a Median RM values of TSS09 sources within 3 ◦ radius, with the uncertainties being the standard deviations of thoseneighbouring sources b Galactic contribution to RM from Oppermann et al. (2015) c Neutral hydrogen column density from H i observations d Velocity-integrated H α intensity from the WHAMSS (Haffner et al. 2003, 2010) † Suffers n π -ambiguity in the TSS09 catalogue × Unpolarised sources ? Special case compared to TSS09 catalogue (see Section 4.1.8) (cid:63)(cid:63)
Double point sources (cid:12)
Extended sources E From the Effelsberg-Bonn H i Survey (EBHIS; Winkel et al. 2016) G From the Galactic All-Sky Survey (GASS; McClure-Griffiths et al. 2009; Kalberla & Haud 2015) from us (Chen & Lee 2008), and has an intriguing morphology con-sisting of filamentary/sheet-like structures with widths of about 50 (cid:48) ,translating to ∼ + + . + (cid:48) . It has not been studied in detail and thushas an unknown distance, but its spatial proximity and similarity inradial velocity ( v LSR ≈ − . − from WHAMSS) with nearbyH ii structures Sh 2-118 and Sh 2-123 suggest physical associationsamong these objects. The latter two clouds have kinematic distancesof 3 . B reg is only weaklysensitive to L , as an over-/under-estimation of the latter by 10 timeswould only result in the former being weaker/stronger by a factorof 3.2, which would not affect our order-of-magnitude estimationhere.Substituting in the adopted values of L for each H ii structure,as well as I H α from the WHAMSS (see Table 6) and φ as our φ values into Equation 20, we obtain B reg (cid:46) µ G for Sh 2-126and 52–86 µ G for the H ii filament in front of J220205 + ∼ µ G; e.g. Heiles et al. 1981;Gaensler et al. 2001; Harvey-Smith et al. 2011; Rodríguez et al.2012), though note that these H ii filaments might not be typicalH ii regions. Since our crude assumptions above would yield upperlimits in field strengths, and for these two clouds we only have veryrough estimates on the physical scales, we cannot draw a concreteconclusion on whether these two H ii structures can contribute tothe bulk of the observed | φ | of the three target sources. i Column Densities
We also looked into the Galactic H i column densities ( N Hi ) to-wards our target sources, using the result from the Effelsberg-BonnH i Survey (EBHIS; Winkel et al. 2016) for the northern sky andthe Galactic All-Sky Survey (GASS; McClure-Griffiths et al. 2009;Kalberla & Haud 2015) for the southern hemisphere. The fore-ground N Hi values for our target sources are listed in column 6 ofTable 6. However, we do not see any clear trends between | φ | and N Hi . MNRAS000
We also looked into the Galactic H i column densities ( N Hi ) to-wards our target sources, using the result from the Effelsberg-BonnH i Survey (EBHIS; Winkel et al. 2016) for the northern sky andthe Galactic All-Sky Survey (GASS; McClure-Griffiths et al. 2009;Kalberla & Haud 2015) for the southern hemisphere. The fore-ground N Hi values for our target sources are listed in column 6 ofTable 6. However, we do not see any clear trends between | φ | and N Hi . MNRAS000 , 1–27 (2019) Y. K. Ma et al.
We explore the possibility of high FD stemming from the hot magne-tised intracluster medium of foreground galaxy clusters (see Govoni& Feretti 2004), which can have | FD | contributions to embedded /background polarised sources of ∼
100 rad m − (e.g. Taylor et al.2001; Bonafede et al. 2009; Govoni et al. 2010). The NASA/IPACExtragalactic Database (NED) was consulted for galaxy clusterswithin 2 ◦ of our 15 high | φ | target sources, and we found matchesfor the six sources below.(i) J084701 − z = . ± . φ = + . ± . − is situated at 34 . (cid:48) . (cid:48) + z = . − z unknown) with φ = + . ± . − is accompanied by Abell 3421 at 92 . (cid:48) . (cid:48) z nor angular sizes.(iii) J094808 − z unknown) with φ =+ . + . − . rad m − is 81 . (cid:48) z = . + z unknown) with φ = − . ± . − is situated next to ZwCl 2200.7 + . (cid:48) (cid:48) (Zwicky et al. 1961), andtherefore cannot contribute to the FD of our target.(v) J220927 + z = . ± .
029 (photometric; Abol-fathi et al. 2018) with φ = − . ± . − is neighbouringUGCL 467 (also known as ZwCl 2207.8 + . (cid:48) z = . (cid:48) =
21 kpc) has a diameter of 164 (cid:48) (Baiesi-Pillastrini et al. 1984), and could be the prime contributor toFD of J220927 + α nor H i above.(vi) J224412 + z = .
171 (Ackermann et al. 2011) with φ = − . ± . − is accompanied by two galaxy clusters– 1RXS J223758.3 + . (cid:48) − . (cid:48) z nor cluster diameter.To summarise, J220927 + α or H i structures to) may have attainedits high FD from the foreground galaxy cluster UGCL 467. We can-not confidently attribute the high FD of the rest of our target sourcesto foreground clusters, given the ill-constrained parameters, partic-ularly redshifts, to the sources themselves and/or to the foregroundclusters. We find it necessary to formally define Faraday complex sourcesbefore proceeding further. The main reason is to facilitate com-parisons with the literature, as a growing number of broadbandspectro-polarimetric studies of EGSs choose to extract the flux den-sities of their samples by integrating within a source region (e.g.,Anderson et al. 2016; O’Sullivan et al. 2017). While this would besimilar to the strategies we adopted for our point sources and ex-tended sources, it is in contrast to our spatial doubles, for which we fitted two Gaussian functions to each image (per frequency channeland per Stokes parameter; Section 2.2) and analysed the two spatialcomponents independently. In other words, although we may be ableto identify small differences in FD between two spatial components,the same source may be classified as Faraday simple when the spa-tial information is discarded. We therefore carefully define Faradaycomplexity for our target sources here to match the expected out-come if our sources were not spatially resolved. Also, in additionto angular resolution (see Appendix A), we note that whether theFaraday complexity of a source can be correctly identified can alsodepend on the S / N ratio (e.g. Anderson et al. 2015; O’Sullivan et al.2017) and λ coverage (e.g. Anderson et al. 2016).We therefore define Faraday complex sources as follows. FromRM-Synthesis, an unresolved or extended source is considered asFaraday complex if it is decomposed into multiple Faraday com-ponents, or the only Faraday component is Faraday thick. Here, wedefine “Faraday thick component” as one with the fitted FWHM( δφ ) at least 10 per cent more than the theoretical FWHM of theRMTF ( δφ ), while “Faraday thin component” is one with δφ lessthan 1 . δφ . For a spatial double source, it is deemedFaraday complex if at least one of the spatial components is furtherdivided into multiple Faraday components, or one/each of the spa-tial components hosts a Faraday thick component, or each spatialcomponent contains one and only one Faraday thin component butthe FDs of these two components are separated by more than 37per cent of δφ (the choice of this factor is explained below). Onthe other hand, from QU -fitting a spatially unresolved or extendedsource is defined as Faraday complex if its best-fit model is notsingle thin (1T), while a double source is categorised as Faradaycomplex if either/both of the spatial components is/are best-fittedby models other than single thin, or both spatial components arebest characterised by the single thin model but the difference in FDsof the two Faraday simple components is larger than 37 per centof the δφ from RM-Synthesis (again, the choice of this factor isexplained below).As mentioned above, the most critical part of this formal defini-tion here is for spatial double sources, particularly for cases whereeach spatial component hosts a Faraday thin component. In suchcases, the two spatially resolved Faraday components could be in-distinguishable from a single Faraday thin component if we discardthe spatial information by combining them within a source integra-tion region. While previous works showed by simulations that Fara-day components with FDs separated by less than ≈ δφ cannot be confidently distinguished by RM-Synthesis and QU -fitting (e.g. Farnsworth et al. 2011; Sun et al. 2015; Schnitzeler2018; Miyashita et al. 2019), we chose to adopt a smaller cutoffvalue of 37 per cent here. This is because although the two Faradaythin components cannot be separated if they are situated too closetogether in Faraday space, the two combined could be identified asa single Faraday thick component. We calculated for the simplestcase of adding two Faraday thin components with equal amplitudestogether, and found that when they are separated by about 37 percent of δφ the combined function resembles a Gaussian functionwith δφ being 1.1 times of δφ , satisfying our definition of Faradaythick component above. Nonetheless, the choice of this cutoff valuewould not affect the results of our work here, as the most extremecase we have is J093544 − δφ only. MNRAS , 1–27 (2019) he n π -ambiguity in the NVSS RM Catalogue One of the major strengths of radio broadband spectro-polarimetricobservations is its ability to decompose spatially unresolved sources(e.g. EGSs) into multiple Faraday components. These componentscould be located anywhere in the volume traced by the telescopebeam, both parallel or perpendicular to the line of sight. This opensup the possibility of identification or even study of discrete physicalregions that are spatially unresolved by the observations, but thiswould require prior studies associating the Faraday componentswith spatial components for a sample of spatially resolved sources.There appears to be some correspondences between the number ofFaraday and spatial components of EGSs (e.g. Anderson et al. 2016;O’Sullivan et al. 2017, both with angular resolution of ∼ (cid:48)(cid:48) ). Thismotivates us to carry out similar investigations to our sample ofEGSs below.We first look at sources that are spatially resolved with our ∼ (cid:48)(cid:48) beam. The only such sources that are resolved into multipleFaraday components are J224549 + − − QU -fitting. All threeof them host two Faraday components each, with J224549 + − − − − − σ limit of 3 per cent). For the remain-ing three spatial doubles (J091145 − − − QU -fitting, we can still obtain the differencein FD between the two components, and compute what λ coveragesare required to resolve them into two Faraday components if thesesources were spatially unresolved. From RM-Synthesis and QU -fitting, our spatial doubles have differences in FD of 2 . . − and 2 . . − , respectively. Assuming that in both analyseswe can distinguish Faraday components separated by more than 50per cent of the theoretical δφ in RM-Synthesis (e.g. Schnitzeler2018), a λ coverage of more than 0 . .
72 m would be neededto resolve our spatial doubles into the multiple Faraday compo-nents. These translate to frequency coverages from 1 GHz downto 660 and 330 MHz, respectively. From this, we argue that forspectro-polarimetric studies of EGSs in GHz regime, we should not combine multiple spatial components together with flux integrationregions. This is because the spatially resolved Faraday componentswould then become a single unresolved Faraday component, leadingto loss of physical information of the sources. We draw similar con-clusions in Appendix A for our spatially extended sources, wherewe found that the QU -fitting results after spatial flux integrationsdiffers from our spatially resolved RM-Synthesis analysis.Furthermore, we searched for Faint Images of the Radio Skyat Twenty-Centimeters (FIRST; angular resolution ≈ (cid:48)(cid:48) ; Beckeret al. 1995), as well as Very Long Baseline Interferometry (VLBI;angular resolution ∼ mas; Fey & Charlot 1997, 2000) total intensityimages of all of our target sources. We found that four of them haveexisting higher angular resolution radio images. These sources arediscussed individually below.(i) J111857 + (cid:48)(cid:48) to the northwest ofthe main component. At VLBI resolution the source is extended at2 . . + QU -fitting. These Faraday components have vastlydifferent FD values ( − . ± . − . ± .
5, and + . ± . − from RM-Synthesis; similar to that from QU -fitting).This source is also spatially resolved into three components in theFIRST image – two bright blobs together resembling an FR II ra-dio galaxy with an angular scale of about 15 (cid:48)(cid:48) (corresponding to aprojected physical scale of about 130 kpc at z = . (cid:48)(cid:48) away to the southwest.(iii) J170934 − QU -fitting analysis. The source appearsin the VLBI image at 2 . . + . z = . . (cid:48)(cid:48) (FIRST) or mas (VLBI) resolutions. A caveat here is that because ofthe missing short uv -spacing, there could be missing flux from struc-tures on large angular scales, particularly in the VLBI images. Notethat this suggested association between the number of spatial andFaraday components is only speculative, and requires confirmationfrom high angular resolution spectro-polarimetric studies. Indeed,a more comprehensive study on the connection between Faradaycomponents and structures of EGSs at different angular and physi-cal scales, as well as for different source types, would be necessarybefore we can confidently interpret their Faraday complexities fromlow angular resolution observations alone.Finally, it has been suggested that lines of sight with GalacticH i column density of 1 . . × cm − may pass through mag-netised plasma in the Milky Way which could cause observed Fara-day complexities in background EGSs (Anderson et al. 2015). Thiswould imply that the turbulence scale of the magneto-ionic mediumcausing such complexities is less than ∼ . ∼ (cid:48)(cid:48) . All of our target sources have foreground H i columndensities higher than the above-mentioned range (Table 6), withJ111857 + . × cm − . It is rep-resented by double Faraday thin components in both RM-Synthesisand QU -fitting, with differences in FD of about 100 rad m − . Thissource is resolved into two spatial components separated by about10 (cid:48)(cid:48) in FIRST (see above). At such a small angular scale, the MilkyWay contribution to FD is not expected to vary by such a largeamount. We suggest that for this source, Faraday complexity is notcaused by the magneto-ionic medium in the Milky Way. MNRAS000
5, and + . ± . − from RM-Synthesis; similar to that from QU -fitting).This source is also spatially resolved into three components in theFIRST image – two bright blobs together resembling an FR II ra-dio galaxy with an angular scale of about 15 (cid:48)(cid:48) (corresponding to aprojected physical scale of about 130 kpc at z = . (cid:48)(cid:48) away to the southwest.(iii) J170934 − QU -fitting analysis. The source appearsin the VLBI image at 2 . . + . z = . . (cid:48)(cid:48) (FIRST) or mas (VLBI) resolutions. A caveat here is that because ofthe missing short uv -spacing, there could be missing flux from struc-tures on large angular scales, particularly in the VLBI images. Notethat this suggested association between the number of spatial andFaraday components is only speculative, and requires confirmationfrom high angular resolution spectro-polarimetric studies. Indeed,a more comprehensive study on the connection between Faradaycomponents and structures of EGSs at different angular and physi-cal scales, as well as for different source types, would be necessarybefore we can confidently interpret their Faraday complexities fromlow angular resolution observations alone.Finally, it has been suggested that lines of sight with GalacticH i column density of 1 . . × cm − may pass through mag-netised plasma in the Milky Way which could cause observed Fara-day complexities in background EGSs (Anderson et al. 2015). Thiswould imply that the turbulence scale of the magneto-ionic mediumcausing such complexities is less than ∼ . ∼ (cid:48)(cid:48) . All of our target sources have foreground H i columndensities higher than the above-mentioned range (Table 6), withJ111857 + . × cm − . It is rep-resented by double Faraday thin components in both RM-Synthesisand QU -fitting, with differences in FD of about 100 rad m − . Thissource is resolved into two spatial components separated by about10 (cid:48)(cid:48) in FIRST (see above). At such a small angular scale, the MilkyWay contribution to FD is not expected to vary by such a largeamount. We suggest that for this source, Faraday complexity is notcaused by the magneto-ionic medium in the Milky Way. MNRAS000 , 1–27 (2019) Y. K. Ma et al.
Our RM-Synthesis and QU -fitting results show respectively thateight (38 per cent) and 14 (67 per cent) out of the 21 polarised targetsources are Faraday complex. We briefly discuss the differencebetween these two algorithms in Appendix B. The RM-Synthesisfraction is similar to the 29 per cent (12 out of 42) obtained fromthe RM-Synthesis analysis on ATA data of bright radio sourcesin 1–2 GHz (angular resolution ∼ (cid:48)(cid:48) ; Law et al. 2011). Thissimilarity may be because of the similar λ coverages, as well asthe high signal-to-noise ratio in polarisation, in both studies. Incontrast, Anderson et al. (2015) reported with their 1 . . ∼ (cid:48) that only 12 per cent (19 outof 160) of their polarised sources appeared to be Faraday complexwith their observational setup. This can be attributed to the lowersignal-to-noise ratio in PI ( (cid:46)
10) of some of their target sources.As they suggested in their paper, sources that are genuinely Faradaycomplex might appear Faraday simple in the low S/N regime.There are spectro-polarimetric studies of EGSs at other wave-lengths that reported a much higher fraction of Faraday complexsources. Pasetto et al. (2018) found by QU -fitting analysis that, all oftheir 14 high RM sources are Faraday complex with their 4–12 GHzobservations (angular resolution (cid:46) (cid:48)(cid:48) ), though this could be biaseddue to their source selection criteria. They chose sources that areunpolarised in the NVSS at 1 . | RM | ranges ( ≈
350 or (cid:38) − ) and/or Faradaydepolarisation due to complexities. Nonetheless, O’Sullivan et al.(2017) reported that 90 per cent (90 out of 100) of their targetsare Faraday complex from their 1–3 GHz observations with angularresolution of ∼ (cid:48)(cid:48) , also with QU -fitting analysis. This is similar tothe findings of Anderson et al. (2016), who observed at 1 . ∼ (cid:48)(cid:48) ) a total of 36 EGSs selectedsuch that, based on archival narrowband 1 . λ range, with the remaining one consistentwith being unpolarised. By re-analysing their data at different λ coverages, they suggested that the detection of Faraday complexityof EGSs could be hindered by limited λ ranges. This suggests thatmany of our Faraday simple sources could become Faraday complexif they are observed at a wider λ range with sufficient S / N ratio. With new broadband spectro-polarimetric observations of 23 n π -ambiguity candidates with the VLA in L-band, we revealed nineout- liars (sources that suffer from n π -ambiguity in the NVSS RMcatalogue). By comparing the statistics of their observed parameterswith that of the 11 outliers (sources with reliable RM TSS09 ), wefind noticeable differences between the two classes in α L , p TSS09 , | RM TSS09 − RM ◦ | , ∆ / σ = | RM TSS09 − RM ◦ |/ σ ◦ , and Faradaycomplexities. In particular, we find ∆ / σ , which is a measure ofhow much a source’s RM deviates from the RMs of its surroundingsources, to be a good diagnostic for n π -ambiguity in the NVSSRM catalogue. There is an apparent cutoff at ∆ / σ ≈ .
85 betweenthe two populations, which we used to estimate that there are atleast 50 n π -ambiguity sources in the TSS09 catalogue out of thetotal of 37,543 sources. This is an important result for us to gaugethe reliability of the TSS09 catalogue, and merits further studies to verify these n π -ambiguity candidates. We further identified twosources that are polarised in TSS09 at 0.5–0.6 per cent levels, butare unpolarised (below the 6 σ cutoffs of ≈ .
07 per cent) in ournew broadband observations. These two sources have motivated adetailed study on the effects of the off-axis instrumental polarisationin the NVSS RM catalogue, presented in the companion Paper II.We found that 15 of our target sources have large | φ | >
200 rad m − despite being situated away from the Galactic plane( | b | > ◦ ). 14 of them are found to be lying behind Galac-tic H ii structures, which are likely the prime contributors to theobserved high | φ | of these sources. The only remaining source,J220927 + | φ | .Finally, we studied the Faraday complexities of our targetsources with our broadband 1–2 GHz observations. We found goodcorrespondence between the number of identified Faraday compo-nents from our analysis with the number of spatial components in to-tal intensities at ≈ (cid:48)(cid:48) and milli-arcsecond resolutions in FIRST andVLBI images, respectively. However, this speculated associationsbetween the Faraday and spatial components require confirmationfrom future polarisation studies at high angular resolution. In oursample of 21 polarised sources, eight (38 percent) and 14 (67 percent) are Faraday complex from our RM-Synthesis and QU -fittinganalysis respectively. The former value agrees with the 29 per centreported by Law et al. (2011) with their RM-Synthesis study ofEGSs at similar frequency range. We noted that if our target sourcesare re-observed with a wider λ coverage than that of our L-bandobservations here, many of our current Faraday simple sources willlikely become Faraday complex at sufficient signal-to-noise ratio inpolarisation. ACKNOWLEDGEMENTS
This is a pre-copyedited, author-produced PDF of an article acceptedfor publication in the Monthly Notices of the Royal AstronomicalSociety following peer review. The version of record is availableat: xxxxxxx. We thank the anonymous referee for the comments,especially for the suggestion to separate our original manuscriptinto two stand-alone publications, which have improved the clarityof the papers. We also thank Rainer Beck for his careful reading andvaluable suggestions and comments as the MPIfR internal referee,and Aristeidis Noutsos, Shane O’Sullivan, and Dominic Schnitzelerfor insightful discussions about this project. Y.K.M. was supportedfor this research by the International Max Planck Research School(IMPRS) for Astronomy and Astrophysics at the Universities ofBonn and Cologne. Y.K.M. acknowledges partial support throughthe Bonn-Cologne Graduate School of Physics and Astronomy. A.B.acknowledges financial support by the German Federal Ministry ofEducation and Research (BMBF) under grant 05A17PB1 (Verbund-projekt D-MeerKAT). A.S.H. and S.K.B. acknowledge support byNASA through grant number HST-AR-14297 to Haverford Col-lege from Space Telescope Science Institute, which is operated byAURA, Inc. under NASA contract NAS 5-26555. A.S.H. is par-tially supported by the Dunlap Institute, which is funded through anendowment established by the David Dunlap family and the Uni-versity of Toronto. The National Radio Astronomy Observatory is afacility of the National Science Foundation operated under cooper-ative agreement by Associated Universities, Inc. This research hasmade use of the NASA/IPAC Extragalactic Database (NED) whichis operated by the Jet Propulsion Laboratory, California Instituteof Technology, under contract with the National Aeronautics and
MNRAS , 1–27 (2019) he n π -ambiguity in the NVSS RM Catalogue Space Administration. The Wisconsin H-Alpha Mapper and its SkySurvey have been funded primarily through awards from the U.S.National Science Foundation.
This paper has been typeset from a TEX/L A TEX file prepared by the author.
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APPENDIX A: SPATIALLY RESOLVED SOURCES
In our new VLA D array observations, two of our target sources are spatiallyresolved, namely J094750 − + z = . z = . I , Q , and U images for detailed spatial analysis. Channelimages were again formed by binning 4 MHz of visibilities with identicalalgorithm and weighting scheme as the full band data (Section 2.2). The onlydifference is that the images formed here are re-smoothed to a common beamsize (210 (cid:48)(cid:48) × (cid:48)(cid:48) for J094750 − (cid:48)(cid:48) × (cid:48)(cid:48) for J224549 + − + I total intensity at 1 . S . ) and spectral index ( α L ) by fitting simple power law to each indi-vidual pixels in the maps: S ν = S . · (cid:16) ν . (cid:17) α L , (A1)where ν represents the observed frequency. Only pixels where the Stokes I values are larger than 6 σ in all channels are fitted. In addition, we per-formed RM-Synthesis for each pixel as per Section 3.1. The maps of numberof Faraday components (N Comp.), as well as that of p , FD, and FWHM ofthe strongest Faraday component, were created. All of the above-mentionedmaps, along with their uncertainties (if applicable) are presented in Fig-ures A1 (for J094750 − + − p ≈ . φ ≈ +
502 rad m − ,and PA ≈ − ◦ , while for J224549 + p ≈ . φ ≈ −
416 rad m − , and PA ≈ + ◦ .The polarisation maps we obtained here allow us to make interestingcomparisons with our results from the main text, where we used flux inte-gration regions for analysis, which discarded all spatial information of thesetwo sources. We note that J094750 − ≈ FWHM =
77 rad m − ). There isalso a significant FD gradient from southeast ( ≈ +
320 rad m − ) to northwest( ≈ +
350 rad m − ). This source was found to be Faraday thick in both ourRM-Synthesis and QU -fitting results. The analysis here shows that the Fara-day thickness is caused by spatial variations of FD across the sky plane (i.e.side-to-side variations in FD), but not due to Faraday rotation within thesynchrotron-emitting medium (i.e. back-to-front changes in FD). The po-larisation maps of J224549 + p ≈
17 per cent, which coin-cides spatially with a knot-like structure in the radio jet seen from Harwoodet al. (2017). Second, there is an FD gradient from east ( ≈ −
270 rad m − )to west ( ≈ −
300 rad m − ), with an FD filament with φ ≈ −
290 rad m − inthe western part of the source. On this filament we also find a lower p ≈ ≈
100 rad m − than the surroundings. Similarstructures in RM maps were also seen in other systems (e.g. 3C 270, 3C353, 4C 35.03, and M 84), with those “RM-bands” having differences inRM from the off-band regions of ≈ − (Guidetti et al. 2011). Table B1.
Comparison between RM-Synthesis and QU -fitting on the De-termination of Faraday ComplexitiesSource RM-Synthesis QU -fitting(NVSS) Results Results Sources with Differing Results
J022915 + − − − − − − + + + Sources with Agreeing Results
J083930 − − − − − − − − − + + − + + + APPENDIX B: COMPARISON BETWEENRM-SYNTHESIS AND QU -FITTING RESULTS We have presented in Sections 3.1 and 3.2 the results from RM-Synthesisand QU -fitting, respectively, and noted the discrepant results for several ofour target sources from the two analysis methods. This allows comparisonsof the two algorithms for the study of polarised EGSs in the frequency rangeof 1–2 GHz.We have divided our RM-Synthesis and QU -fitting results into fiveclasses – thin, double, triple, thick, and double thick. The former threecorresponds to one, two, and three unresolved Faraday components in RM-Synthesis, and 1T, 2T, and 3T models in QU -fitting, respectively, whilethe latter two (thick and double thick) maps to single and double resolvedFaraday components in RM-Synthesis, and 1B, 1B+fg, 1Ed, 1Id, or 1Id+fgfor thick, and 2B or 2B+fg for double thick in QU -fitting, respectively. Wethen compared if the two algorithms agreed on the source class, with theresults listed in Table B1. Out of the 25 polarised sources in our sample(spatial doubles are counted as two distinct sources), 15 have agreeingresults, while 10 are categorised into different classes. Further studies ofthese sources at a wider λ coverage are needed to determine whetherRM-Synthesis or QU -fitting are more reliable in uncovering the Faradaycomplexities of these sources correctly (Ma et al. in prep). Nonetheless, wenote that when RM-Synthesis identifies a source as Faraday complex, it isdefinitely so in QU -fitting, but the converse is not true. APPENDIX C: ONLINE SUPPLEMENTARY MATERIALS
We include here plots of the QU -fitting results in Figure C1. Plots of Stokes q = Q / I and u = U / I , along with polarisation fraction ( p ) and polarisationposition angle (PA) are shown. MNRAS , 1–27 (2019) he n π -ambiguity in the NVSS RM Catalogue Figure A1.
Images of J094750 − I values are greater than 6 σ in all of the individualchannels are shown. The white contours represent NVSS Stokes I map at [ . , . , . , . ] × . − . The beam sizes of the maps are shown in thelower left of each panel, with the open and filled ellipses representing that of our new observation and NVSS, respectively. We also plot the intrinsic polarisation B -orientations ( B = E + ◦ ; corrected for Faraday rotation) in the polarisation fraction panel as cyan lines. The green arrow in the “N Comp.” panel pointsto the region with a marginal detection of a secondary Faraday component as discussed in text.MNRAS000
Images of J094750 − I values are greater than 6 σ in all of the individualchannels are shown. The white contours represent NVSS Stokes I map at [ . , . , . , . ] × . − . The beam sizes of the maps are shown in thelower left of each panel, with the open and filled ellipses representing that of our new observation and NVSS, respectively. We also plot the intrinsic polarisation B -orientations ( B = E + ◦ ; corrected for Faraday rotation) in the polarisation fraction panel as cyan lines. The green arrow in the “N Comp.” panel pointsto the region with a marginal detection of a secondary Faraday component as discussed in text.MNRAS000 , 1–27 (2019) Y. K. Ma et al.
Figure A2.
Same as Figure A1, for J224549 + I map at [ . , . , . , . ] × . − , andthe green arrow in the “N Comp.” panel points to the finger-like patch with a marginal detection of a secondary Faraday component as discussed in text.MNRAS , 1–27 (2019) he n π -ambiguity in the NVSS RM Catalogue Figure C1. QU -fitting results of our polarised target sources, showing the best-fit model from our analysis. Each source spans one of the rows in the figure. TheStokes q = Q / I and u = U / I values are plotted in the left column in blue and red respectively, with polarisation fraction ( p ) shown in the middle column,and PA in the right column. Note that the error bars in PA are not shown here.MNRAS000
Same as Figure A1, for J224549 + I map at [ . , . , . , . ] × . − , andthe green arrow in the “N Comp.” panel points to the finger-like patch with a marginal detection of a secondary Faraday component as discussed in text.MNRAS , 1–27 (2019) he n π -ambiguity in the NVSS RM Catalogue Figure C1. QU -fitting results of our polarised target sources, showing the best-fit model from our analysis. Each source spans one of the rows in the figure. TheStokes q = Q / I and u = U / I values are plotted in the left column in blue and red respectively, with polarisation fraction ( p ) shown in the middle column,and PA in the right column. Note that the error bars in PA are not shown here.MNRAS000 , 1–27 (2019) Y. K. Ma et al.
Figure C1. (Continued) QU -fitting results of our polarised target sources. MNRAS , 1–27 (2019) he n π -ambiguity in the NVSS RM Catalogue Figure C1. (Continued) QU -fitting results of our polarised target sources.MNRAS000
Figure C1. (Continued) QU -fitting results of our polarised target sources. MNRAS , 1–27 (2019) he n π -ambiguity in the NVSS RM Catalogue Figure C1. (Continued) QU -fitting results of our polarised target sources.MNRAS000 , 1–27 (2019) Y. K. Ma et al.
Figure C1. (Continued) QU -fitting results of our polarised target sources. MNRAS , 1–27 (2019) he n π -ambiguity in the NVSS RM Catalogue Figure C1. (Continued) QU -fitting results of our polarised target sources.MNRAS000