A Broadband Spectro-polarimetric View of the NVSS Rotation Measure Catalogue II: Effects of Off-axis Instrumental Polarisation
Yik Ki Ma, S. A. Mao, Jeroen Stil, Aritra Basu, Jennifer West, Carl Heiles, Alex S. Hill, S. K. Betti
aa r X i v : . [ a s t r o - ph . GA ] M a y MNRAS , 1–16 (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 II: Effects of Off-axis Instrumental Polarisation
Yik Ki Ma ⋆ † , 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 10. Received 2019 May 7; in original form 2019 March 7
ABSTRACT
The NRAO VLA Sky Survey (NVSS) Rotation Measure (RM) catalogue has enabled numerousstudies in cosmic magnetism, and will continue being a unique dataset complementing futurepolarisation surveys. Robust comparisons with these new surveys will however require furtherunderstandings in the systematic effects present in the NVSS RM catalogue. In this paper,we make careful comparisons between our new on-axis broadband observations with the KarlG. Jansky Very Large Array and the NVSS RM results for 23 sources. We found that twounpolarised sources were reported as polarised at about 0.5 % level in the RM catalogue, andnoted significant differences between our newly derived RM values and the catalogue valuesfor the remaining 21 sources. These discrepancies are attributed to off-axis instrumentalpolarisation in the NVSS RM catalogue. By adopting the 0.5 % above as the typical off-axis instrumental polarisation amplitude, we quantified its effect on the reported RMs witha simulation, and found that on average the RM uncertainties in the catalogue have to beincreased by ≈
10 % to account for the off-axis instrumental polarisation effect. This effect ismore substantial for sources with lower fractional polarisation, and is a function of the source’strue RM. Moreover, the distribution of the resulting RM uncertainty is highly non-Gaussian.With the extra RM uncertainty incorporated, we found that the RM values from the twoobservations for most (18 out of 21) of our polarised targets can be reconciled. The remainingthree are interpreted as showing hints of time variabilities in RM.
Key words: galaxies: active – galaxies: magnetic fields – ISM: magnetic fields – radiocontinuum: galaxies
Magnetic fields are known to be crucial for astrophysical processessuch as star formation, cosmic ray propagation, galactic outflows,and galactic evolution (see, e.g., Beck & Wielebinski 2013; Beck2016). While the magnetic field strength and structure of astrophys-ical objects can be probed by measurements of their polarised syn-chrotron emission (e.g., Fletcher et al. 2011; Gießübel et al. 2013; ⋆ Contact e-mail: [email protected] † Member of the International Max Planck Research School (IMPRS) forAstronomy and Astrophysics at the Universities of Bonn and Cologne
Mao et al. 2015; Kierdorf et al. 2017), this method is only sensitiveto the magnetic field component in the plane of the sky in volumespopulated with cosmic ray electrons. A complementary method isto use background polarised sources as probes to the foregroundsubjects of interest – polarised emission experiences the Faradayrotation effect as it traverses through the foreground interveningmagnetised plasma, leading to a change in the polarisation positionangle (PA; [rad]) given by ∆ PA = (cid:20) . ∫ ℓ n e ( s ) B k ( s ) d s (cid:21) · λ ≡ RM · λ , (1) © 2019 The Authors Y. K. Ma et al. where ℓ [pc] is the (physical) distance to the source from the ob-server, n e [cm − ] is the thermal electron density, B k [ µ G] is thestrength of the magnetic field component along the line of sight ( s [pc]), λ [m] is the wavelength of the emission, and RM [rad m − ]is the rotation measure of the source . The integrated value of themagnetic field strength along the line of sight, weighted by n e , istherefore encrypted in the RM values. The RM of any given sight-line can be obtained by PA measurements at two or more frequencybands, followed by a linear fit to PA against λ . For example, theresulting RM from observations at two frequencies only is given byRM = PA − PA + n πλ − λ , (2)where the subscripts denote the two frequency bands, and n is aninteger corresponding to n π -ambiguity resulting from the possiblewrapping(s) of PA between the two bands (see Paper I, Ma et al.2019, for more details).Extragalactic radio sources (EGSs) have commonly been usedto uncover the magnetic fields in foreground astrophysical objects.Such RM-grid experiments can be broadly divided into two cate-gories: blind surveys and pointed observations. The NRAO VLASky Survey (NVSS) RM catalogue (Taylor et al. 2009, hereafterTSS09) is the largest RM catalogue to date, and is a notable exam-ple of blind surveys. This RM catalogue was built by re-analysingthe original NVSS data (Condon et al. 1998), which were taken byscanning through a regularly spaced hexagonal grid in the north-ern sky ( δ > − ◦ ). The wide sky coverage of the TSS09 cata-logue has enabled numerous studies of cosmic magnetism (e.g.,McClure-Griffiths et al. 2010; Harvey-Smith et al. 2011; Stil et al.2011; Oppermann et al. 2012; Hill et al. 2013; Oppermann et al.2015; Purcell et al. 2015; Terral & Ferrière 2017). Similar blind sur-vey strategies were also adopted by other works for specific parts ofthe sky (e.g., Gaensler et al. 2005; Mao et al. 2008; Gießübel et al.2013). With such surveying strategies, the target EGSs are ingeneral not on the pointing axis of the telescopes, which meansthe resulting data can be affected by off-axis instrumental effectsthat need to be accounted for (see below). On the other hand,the strategy of pointed observations is also commonly used (e.g.,Mao et al. 2010; Van Eck et al. 2011; Mao et al. 2012; Costa et al.2016; Kaczmarek et al. 2017; Mao et al. 2017; Betti et al. 2019),where target EGSs are selected from existing catalogues of po-larised radio sources. They are then observed with the EGSs placedon the pointing axis of the telescopes. Compared to blind surveys,the resulting data of the targets from these pointed observations arefree of off-axis instrumental artefacts.An ideal radio telescope with dual polarised feeds should haveindependent polarisation channels, each being insensitive to its or-thogonal counterpart. In reality, however, imperfections of the tele-scopes allow these polarisation channels to “see” the orthogonallypolarised components. This is known as the instrumental polarisa-tion (also known as the polarisation leakage) of radio telescopes,which can alter the measured polarisation signals. The polarisationleakage can be seen as comprised of two distinct elements – the on-axis and the off-axis components. The former is routinely calibratedout in polarisation studies (see, e.g., Hales 2017), usually by either(1) observing a known unpolarised calibrator and attributing themeasured polarisation signals as the instrumental response of the In this work, we investigate the narrowband results presented inTaylor et al. (2009), and therefore follow the traditional notion of RM insteadof the more generalised notion of Faraday depth. telescope, or (2) observing a calibrator over a range of parallactic an-gles to simultaneously determine the astrophysical and instrumentalpolarisation, given that the telescope is driven by altitude-azimuthal(alt-az) mounts. Both these strategies will remove the polarisationleakage at the pointing centre where the calibrator has been placed at(down to, e.g., . .
02 per cent in our new VLA observations; see Pa-per I), but residual polarisation leakage remains for positions withinthe primary beam away from the pointing axis (thus “off-axis”).This off-axis instrumental polarisation can be determined by holog-raphy scans (e.g. in the NVSS; Cotton 1994; Condon et al. 1998)and subsequently be calibrated out. Alternatively, the A-projectionalgorithm (Bhatnagar et al. 2008, 2013) can be further developed tocharacterise and correct for the off-axis polarisation leakage (see,e.g., Jagannathan et al. 2017, 2018). This full Mueller A-projectionrequires an adequate knowledge of the antenna optics, and can beapplied during the imaging step of data reduction.The off-axis polarisation leakage present in the NVSS data, ifcompletely uncorrected, can be up to 2.5 per cent (Cotton 1994).However, as calibrations for this off-axis leakage have been ap-plied in the image domain, the residual leakage remaining in thedata products of the original NVSS (namely, images and the sourcecatalogue) is ≈ . not applied, thoughthe mosaicking done to form their images could have smoothed outthe off-axis leakage pattern with respect to the NVSS pointing cen-tres. It is therefore likely that the reported RM values in the TSS09catalogue have been affected by off-axis polarisation leakage, withits effect still remain unaccounted for.It is crucial to fully understand the limits of this NVSS RMcatalogue given its relevance. Although ongoing polarisation sur-veys such as Polarization Sky Survey of the Universe’s Magnetism(POSSUM; Gaensler et al. 2010) in 1130–1430 MHz and VLA SkySurvey (VLASS; Myers et al. 2014) in 2–4 GHz will provide uswith drastically higher RM densities than TSS09, these two sur-veys either do not have exact sky or frequency coverage, and bothcover different time domains, compared to TSS09. This means theTSS09 catalogue will continue being a unique dataset depicting themagnetised Universe.In our Paper I (Ma et al. 2019), we have explored the n π -ambiguity problem in TSS09 and concluded that there are likelymore than 50 n π -ambiguity sources (with erroneous RM by ± . − ) out of the total 37,543 in the NVSS RM cata-logue. In addition, we found two sources that were reported as ≈ . MNRAS000
02 per cent in our new VLA observations; see Pa-per I), but residual polarisation leakage remains for positions withinthe primary beam away from the pointing axis (thus “off-axis”).This off-axis instrumental polarisation can be determined by holog-raphy scans (e.g. in the NVSS; Cotton 1994; Condon et al. 1998)and subsequently be calibrated out. Alternatively, the A-projectionalgorithm (Bhatnagar et al. 2008, 2013) can be further developed tocharacterise and correct for the off-axis polarisation leakage (see,e.g., Jagannathan et al. 2017, 2018). This full Mueller A-projectionrequires an adequate knowledge of the antenna optics, and can beapplied during the imaging step of data reduction.The off-axis polarisation leakage present in the NVSS data, ifcompletely uncorrected, can be up to 2.5 per cent (Cotton 1994).However, as calibrations for this off-axis leakage have been ap-plied in the image domain, the residual leakage remaining in thedata products of the original NVSS (namely, images and the sourcecatalogue) is ≈ . not applied, thoughthe mosaicking done to form their images could have smoothed outthe off-axis leakage pattern with respect to the NVSS pointing cen-tres. It is therefore likely that the reported RM values in the TSS09catalogue have been affected by off-axis polarisation leakage, withits effect still remain unaccounted for.It is crucial to fully understand the limits of this NVSS RMcatalogue given its relevance. Although ongoing polarisation sur-veys such as Polarization Sky Survey of the Universe’s Magnetism(POSSUM; Gaensler et al. 2010) in 1130–1430 MHz and VLA SkySurvey (VLASS; Myers et al. 2014) in 2–4 GHz will provide uswith drastically higher RM densities than TSS09, these two sur-veys either do not have exact sky or frequency coverage, and bothcover different time domains, compared to TSS09. This means theTSS09 catalogue will continue being a unique dataset depicting themagnetised Universe.In our Paper I (Ma et al. 2019), we have explored the n π -ambiguity problem in TSS09 and concluded that there are likelymore than 50 n π -ambiguity sources (with erroneous RM by ± . − ) out of the total 37,543 in the NVSS RM cata-logue. In addition, we found two sources that were reported as ≈ . MNRAS000 , 1–16 (2019) ff-axis Instrumental Polarisation of the NVSS RM Catalogue Table 1.
Comparison between new VLA and TSS09 resultsSource RM
VLA a RM Tcut ∆ RM b | ∆ RM |/ σ b ∆ S / S . c α L (NVSS) (rad m − ) (rad m − ) (rad m − ) (%)J111857 + † + . ± . + . ± . − . ± . . − . d − . ± . − + . ± . + . ± . − . ± . . + . − . ± . − † + . ± . + . ± . − . ± . . + . − . ± . + †⊙ − . ± . − . ± . + . ± . . + . − . ± . − † + . ± . + . ± . + . ± . . − . − . ± . − + . ± . + . ± . + . ± . . + . − . ± . + † + . ± . + . ± . + . ± . . − . − . ± . − ⊙ + . ± . + . ± . + . ± . . − . − . ± . + − . ± . − . ± . + . ± . . − . − . ± . + † − . ± . − . ± . − . ± . . + . − . ± . − ⋆⋆ + . ± . + . ± . + . ± . . − . − . ± . + ? − . ± . − . ± . − . ± . . − . − . ± . − † ⋆⋆ + . ± . + . ± . + . ± . . − . − . ± . − + . ± . + . ± . + . ± . . − . − . ± . + − . ± . − . ± . + . ± . . − . − . ± . − ⋆⋆ + . ± . + . ± . + . ± . . − . − . ± . − ⋆⋆ − . ± . − . ± . − . ± . . + . − . ± . + † + . ± . + . ± . − . ± . . − . − . ± . − + . ± . + . ± . − . ± . . − . − . ± . − ⋆⋆ − . ± . − . ± . − . ± . . + . − . ± . + † − . ± . − . ± . − . ± . . + . + . ± . − × — — — — − . − . ± . + × — — — — + . − . ± . NOTE —Sorted by | ∆ RM |/ σ in descending order a Using our new data in the NVSS bands only b ∆ RM = RM VLA − RM Tcut c ∆ S = S . − S Ncut d S Ncut from the original NVSS has been replaced by S Tcut from TSS09 instead (see Section 4.1) † Out- liars ( n π -ambiguity sources; see Paper I) × Unpolarised sources (less than the 6 σ cutoff level) ? Special case compared to TSS09 catalogue (see Paper I) ⋆⋆ Double point sources ⊙ Extended sources
A total of 23 target sources were selected from the TSS09 catalogue,with the original primary goal of identifying n π -ambiguity sources(addressed in Paper I). These sources were selected based on theirhigh | RM TSS09 | &
300 rad m − , despite being situated away fromthe Galactic plane ( | b | > ◦ ) . Furthermore, all of our targets arebright with NVSS total intensities larger than 100 mJy.Our new broadband observations were performed using theVLA in L-band (1–2 GHz) in D array configuration on 2014 July03. The same array configuration as used by the NVSS means thatour uv -coverages are similar to that of TSS09, and the observationsfrom both works are sensitive to emission at the same ranges ofangular scales. The typical integration time per source was about3–4 minutes. Standard calibration procedures were followed usingthe Common Astronomy Software Applications (CASA) package(version 4.4.0; McMullin et al. 2007), and are described in detail inPaper I. Except for J234033 + TSS09 = + . ± . − .This included source turned out to be unpolarised in our new observations,and is pivotal to our study of off-axis instrumental polarisation in this paper. −600 −400 −200 0 +200 +400 +600RM Tcut (rad m −2 )−600−400−2000+200+400+600 R M V L A ( r a d m − ) Figure 1.
Narrowband RM from our observations against that from TSS09cutout images. The n π -ambiguities have been corrected by using our broad-band φ values from RM-Synthesis. The solid line shows where the RMvalues of the two measurements are identical.MNRAS , 1–16 (2019) Y. K. Ma et al.
A careful comparison in polarisation properties between our newdata and the NVSS RM catalogue requires that the two datasets havenear identical frequency and uv -coverages, with the source proper-ties extracted following the same method. We have therefore formedtwo sets of radio images using our calibrated broadband VLA data inthe two NVSS intermediate frequency (IF) bands only. Although theoriginal NVSS bands had frequency ranges of 1343 . . . . . . . . I , Q , and U imagesfor each source in the NVSS IF1 and IF2 respectively. All theimages were made with Briggs visibilities weighting of robust = ′′ × ′′ matching that of the TSS09 images (see below). On the other hand,we obtained cutout images of our target sources from TSS09. TheTSS09 images were formed using the calibrated NVSS visibilitydata, independently for NVSS IF1 and IF2. A mild uv -taper wasapplied and led to their resulting beam of 60 ′′ × ′′ . We decided todetermine the RM values from the TSS09 images instead of directlyadopting the listed RM TSS09 values in their catalogue to ensure thatthe most direct comparison is performed. The Stokes I , Q , and U values of our sources were extracted from the two observations inidentical ways, with the flux densities of the unresolved sourcesdetermined from the CASA task IMFIT , and that of double andextended sources by integrating within 6 σ contours in total intensity. We perform a rigorous comparison between the polarisation proper-ties of our new VLA results and that in TSS09. By using the Stokes Q and U values obtained from our VLA data in NVSS bands andTSS09 cutout images, we computed the PA values byPA j =
12 tan − (cid:18) U j Q j (cid:19) , (3)where the subscripts denote the IFs. Equation 2 is then used to com-pute the RM values. We used λ = . λ = . λ = . λ = . n were chosen such that the RM values wouldmost closely match the broadband polarisation-weighted Faradaydepths ( φ ) reported in Paper I, which is free of n π -ambiguity. Theonly exception is J154936 + φ and the narrowband RM TSS09 for this source , and therefore we chose n for this source such We determined in our Paper I that the disagreement between φ andRM TSS09 for this source is because of its highly non-linear PA across λ resulting from its significant Faraday complexities. that the resulting RM values are the closest to its TSS09 value of − . − . As a sanity check, we further compared our RMvalues from TSS09 cutout images with the officially listed RM TSS09 (corrected for n π -ambiguity as above) to ensure that these two setsof RM values agree with each other (within ≈ σ ), as would beexpected since they were computed from the same dataset.The RM values obtained in the two datasets are shown inFigure 1, and listed in Table 1. The RM values from our new ob-servations are denoted as RM VLA , and that from TSS09 cutoutimages as RM
Tcut . We also listed the difference in RM ( ∆ RM = RM VLA − RM Tcut ), as well as its magnitude divided by RM uncer-tainties (i.e., RM differences in units of σ ; | ∆ RM |/ σ ) in the sameTable 1. It is found that nine (43 per cent) and five (24 per cent)out of our 21 polarised sources have deviating RM values by morethan 2 and 3 σ respectively. Assuming that the RM uncertaintiesare Gaussian (which is approximately true given the high signal-to-noise ratio in polarisation of &
15 for these sources), | ∆ RM |/ σ should follow a folded normal distribution with µ = σ = σ respectively. It is evident that, with the entire sampleof 21 sources considered as a whole, the RM values from TSS09and our observations do not agree within their uncertainties. We report here the full-band (1–2 GHz) radio spectra of our targets.This has been deferred to this paper from Paper I because of therelevance between Stokes I and RM time variabilities. The fluxdensities were extracted from 4 MHz channel images using the entireL-band (see Paper I), and were fitted for each source with a simplepower law: S ν = S . · (cid:16) ν . (cid:17) α L , (4)where S ν is the flux density at frequency ν , and α L is the spectralindex in L-band. The values of S . and α L are listed in Table 2,with the radio spectra shown in Figure A1. For double sources,apart from fitting the radio spectra of each of the spatial compo-nents (listed in Table 3), we also added the flux densities together(with uncertainties added in quadrature) to obtain a joint fit, whichfacilitates comparison with the integrated flux densities reported inthe NVSS ( S NVSS ; also in Table 2), as well as other lower resolutionradio studies. We do not see clear evidence of deviations of the radiospectra from the simple power law for any of the sources. We notethe significant discrepancies between S . and S NVSS for thedouble / extended sources J093544 − − + S Ncut ), also listed in Table 2. Indeed, we see much better agree-ment between S . and S Ncut for the above three sources, butfor four of our other targets (J094808 − + − + MNRAS000
15 for these sources), | ∆ RM |/ σ should follow a folded normal distribution with µ = σ = σ respectively. It is evident that, with the entire sampleof 21 sources considered as a whole, the RM values from TSS09and our observations do not agree within their uncertainties. We report here the full-band (1–2 GHz) radio spectra of our targets.This has been deferred to this paper from Paper I because of therelevance between Stokes I and RM time variabilities. The fluxdensities were extracted from 4 MHz channel images using the entireL-band (see Paper I), and were fitted for each source with a simplepower law: S ν = S . · (cid:16) ν . (cid:17) α L , (4)where S ν is the flux density at frequency ν , and α L is the spectralindex in L-band. The values of S . and α L are listed in Table 2,with the radio spectra shown in Figure A1. For double sources,apart from fitting the radio spectra of each of the spatial compo-nents (listed in Table 3), we also added the flux densities together(with uncertainties added in quadrature) to obtain a joint fit, whichfacilitates comparison with the integrated flux densities reported inthe NVSS ( S NVSS ; also in Table 2), as well as other lower resolutionradio studies. We do not see clear evidence of deviations of the radiospectra from the simple power law for any of the sources. We notethe significant discrepancies between S . and S NVSS for thedouble / extended sources J093544 − − + S Ncut ), also listed in Table 2. Indeed, we see much better agree-ment between S . and S Ncut for the above three sources, butfor four of our other targets (J094808 − + − + MNRAS000 , 1–16 (2019) ff-axis Instrumental Polarisation of the NVSS RM Catalogue Table 2.
Total flux densities and redshifts of the targetsSource α L S . S Ncut a S NVSS b z Reference(NVSS) ( S ν ∝ ν α L ) (mJy) (mJy) (mJy) (z)J022915 + − . ± .
006 649 . ± . . ± . . ± . − − . ± .
010 261 . ± . . ± . . ± . − − . ± .
007 1787 . ± . . ± . . ± . − − . ± .
012 162 . ± . . ± . . ± . . ± . s Huchra et al. (2012)J090015 − − . ± .
006 530 . ± . . ± . . ± . . s Perlman et al. (1998)J091145 − ⋆⋆ − . ± .
011 238 . ± . . ± . . ± . · · · a − . ± .
022 81 . ± . · · · b − . ± .
010 157 . ± . − ⋆⋆ − . ± .
006 590 . ± . . ± . . ± . · · · a − . ± .
010 291 . ± . · · · b − . ± .
009 299 . ± . − − . ± .
007 253 . ± . . ± . . ± . − ⋆⋆ − . ± .
006 499 . ± . . ± . . ± . · · · a − . ± .
010 227 . ± . · · · b − . ± .
008 271 . ± . − ⊙ − . ± .
013 602 . ± . . ± . . ± . . ± . s Jones et al. (2009)J094808 − − . ± .
009 245 . ± . . ± . . ± . + − . ± .
003 2041 . ± . . ± . c . ± . c . ± . s Abolfathi et al. (2018)J154936 + − . ± .
003 583 . ± . . ± . . ± . . s Hewitt & Burbidge (1987)J162706 − ⋆⋆ − . ± .
013 130 . ± . . ± . . ± . · · · a − . ± .
042 30 . ± . · · · b − . ± .
015 100 . ± . − ⋆⋆ − . ± .
005 394 . ± . . ± . . ± . · · · a − . ± .
006 247 . ± . · · · b − . ± .
010 146 . ± . − − . ± .
007 644 . ± . . ± . . ± . + − . ± .
003 2923 . ± . . ± . . ± . . s Hewitt & Burbidge (1987)J220205 + − . ± .
009 143 . ± . . ± . . ± . + − . ± .
006 283 . ± . . ± . . ± . . ± . p Abolfathi et al. (2018)J224412 + + . ± .
008 260 . ± . . ± . . ± . . s Ackermann et al. (2011)J224549 + ⊙ − . ± .
008 11181 . ± . . ± . . ± . s Hewitt & Burbidge (1991)J234033 + − . ± .
005 1868 . ± . . ± . . ± . + − . ± .
007 634 . ± . . ± . . ± . . ± . p Abolfathi et al. (2018) a Our integrated flux densities from NVSS cutout images b Listed integrated flux densities from the NVSS catalogue (Condon et al. 1998) c Likely erroneous due to missing pointing in NVSS (see Section 4.1) ⋆⋆ Double point sources ⊙ Extended sources p Photometric redshifts s Spectroscopic redshifts
Before comparing the flux densities between our 2014 observationsand that of the NVSS in the 1990s, we note the different absoluteflux density scales applied. The NVSS used 3C 295 as the fluxcalibrator adopting the Baars et al. (1977) scale, while we used3C 286 and 3C 138 following the Perley & Butler (2013) scale. Ithas been suggested that the systematic differences between thesetwo scales at 1 . . ≈
10 yr at 1 . S . with S Ncut obtained from NVSS cutout images by introducing a parameter (see Table 1) ∆ SS . = S . − S Ncut S . . (5)Four of our sources (namely, J094808 − + − + I timevariabilities between NVSS and our observations (over roughly 20years). As all these four sources are listed as compact in the originalNVSS catalogue (angular size < ′′ ) , this is consistent with thegeneral picture that the variable radio emission originates fromthe core of AGNs. We gathered flux density measurements of thesesources in the literature up to ∼
10 GHz and plotted them in Figure 2to facilitate comparisons. The angular size of J111857 + ′′ × ′′ in the FIRSTsurvey (Becker et al. 1995).MNRAS , 1–16 (2019) Y. K. Ma et al.
200 1000 5000 200001005001000 NVSS J094808-344010 α L =−0.373 ±0.009
50 100 1000 100001000500010000 NVSS J111857+123442 α L =−0.232 ±0.003 Gower et al. (1967)Adgie et al. (1972)Shimmins & Wall (1973)Wills (1975)Large et al. (1981)Kuehr et al. (1981)Becker et al. (1991)Gregory & Condon (1991)Griffith et al. (1994)Douglas et al. (1996)Wright et al. (1996)Condon et al. (1998)Mauch et al. (2003)Cohen et al. (2007)Massardi et al. (2008)Murphy et al. (2010)Richards et al. (2011)This work
200 1000 5000 20000300500800 NVSS J170934-172853 α L =−0.077 ±0.007
50 100 1000 100001005001000 NVSS J224412+405715 α L = +0.015 ±0.008 Frequency (MHz) F l u x D en s i t y ( m Jy ) Figure 2.
Radio spectra of the four sources with signs of Stokes I variabilities. Flux densities obtained from our new broadband measurements are shown asthe black points, with the black solid line representing the best-fit power law spectra. The coloured data points represent flux densities from various studies. -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 +0.2050100150200250300 | R M V L A − R M T c u t | ( r a d m − ) L | R M V L A − R M T c u t | / σ −S Ncut |/S (%)012345678
Figure 3.
Plots of measures of RM time variabilities against that of Stokes I time variabilities of our 21 polarised sources. The cyan points representJ111857 + I value instead of that from NVSS cutout image (see Section 4.1). J111857 + I disparityof about 45 per cent when compared with its NVSS value. However,as seen in Figure 2 the NVSS flux density is inconsistent with oth-ers’ as well as ours. We looked into the NVSS image containing thissource (C1112P12), and noticed that at the centre of NVSS pointing11195 + + ′ where the intensity is exactly zero.Note that (4 ◦ × ◦ ) NVSS images are weighted averages of theirconstituent snapshot images (one from each pointing; truncated atradius of 24 ′ ), with the weights defined to be proportional to thesquare of the primary beam attenuation (Condon et al. 1998). Sinceindividual NVSS pointings are separated by 26 ′ , the snapshot im- age of a missing pointing would result in a nearly circular areawith 2 ′ radius of zero pixel values, just as we found above. Thissuggests that the NVSS pointing 11195 + + + + + + . ′
9, 14 . ′
6, and 19 . ′ . . .
338 respectively (equation 5 ofCondon et al. 1998). If we replace the source flux density in the
MNRAS000
MNRAS000 , 1–16 (2019) ff-axis Instrumental Polarisation of the NVSS RM Catalogue Table 3.
Positions of individual components of the spatial doublesSource Right Ascension Declination(NVSS) (J2000; h m s) (J2000; ◦ ′ ′′ )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 ± NOTE – Identical to Table 2 of Paper I first pointing by zero, we obtain a weighted average flux density of47.9 per cent of the true value, exactly matching the 47.9 per centby comparing the NVSS value with that from TSS09 image. Eventhough the TSS09 images were also formed using the publishedNVSS visibility data, they were created by mosaicking with the pri-mary beam response divided out, and any missing pointings wouldlead to an increase in root-mean-square (rms) noise instead of erro-neous flux densities. Indeed, the flux density of 2041 . ± . . ± . . ± . I variability. In below, we compare our broadband S . with the flux density we obtained from TSS09 cutout forJ111857 + S Ncut from NVSS cutout images.We first look into α L of the four sources with significant Stokes I variabilities of more than 10 per cent. As expected, they all exhibitflat radio spectra ( − . α L + . I variabilities (Figure 3). Itappears that flat spectrum sources are more likely to have signif-icant RM discrepancies, though such differences in RM are notnecessarily accompanied by Stokes I variabilities. Note that the RMdifferences between our new observations and TSS09 could be at-tributed to off-axis polarisation leakage instead of true RM timevariabilities (see Section 4.2). By comparing the polarisation properties of our sample in TSS09with that from our new VLA observations (Paper I), we found thattwo of the sources are unpolarised. Furthermore, the RM values ofour 21 polarised target sources from our new VLA observations donot agree with that from TSS09 within measurement uncertainties.As we will show below, both are likely linked to off-axis instrumentalpolarisation in the NVSS observations.
From the RM-Synthesis analysis on our new broadband (1–2 GHz)data in Paper I, we found that two of our targets (J084600 − + σ detection limits of 0.07 and 0.06 per cent, respectively. These polar-isation levels are much lower than the respective values of 0 . ± . . ± .
02 per cent reported in the TSS09 catalogue, as wellas the 0 . ± .
03 and 0 . ± .
02 per cent in the original NVSScatalogue. Our on-axis broadband results are free of off-axis in-strumental effects of the VLA, and are resilient against bandwidthdepolarisation for sources with Faraday depths . rad m − (seePaper I). On the other hand, sources are in general placed signifi-cantly away from the pointing axis for surveys such as the NVSS, andtherefore the off-axis polarisation leakage has to be taken care of forboth the original NVSS catalogue (Condon et al. 1998) and TSS09.Corrections determined from holography scans were applied in theimage plane in the original NVSS but not in TSS09, although theoff-axis leakage pattern has been smoothed out by the mosaickingdone to produce the TSS09 images. Therefore, the residual leakagelevel in the original NVSS ( ≈ . ≈ . − + . ′ . ′ − + ∼ By comparing our results from new VLA data in NVSS bandswith TSS09, we found that the RM values derived from the twoobservations do not agree within their measurement uncertainties(see Section 3.1). We first rule out the possibility that the observedRM discrepancies are due to PA calibration errors, as we do not seesystematic trends in ∆ RM by grouping the target sources by the asso-ciated PA calibrators used in our observations. Another explanationto the RM disparity is genuine RM time variabilities, which canoccur when a jet component near the AGN core is traversing alongthe jet and illuminating different parts of the foreground magnetisedplasma near the jet at different epochs. Changes in RM of ≈ − have been noted for observations at &
10 GHz withinas short as a few months (e.g. Zavala & Taylor 2001; Hovatta et al.2012), and much lower values of ≈
10 rad m − at 1.4 GHz havebeen reported over ≈
20 yr (e.g. Anderson et al. 2016). This ap-parent variability for our targets will be further investigated withfollowup broadband polarisation observations (Ma et al. in prep).Finally, the discrepancies in measured RMs could also be attributed
MNRAS , 1–16 (2019)
Y. K. Ma et al. to some unaccounted systematic uncertainties in either or both ofthe observations, leading to underestimated uncertainties in RM.Here we propose that the disagreement in RM is mostly caused byresidual off-axis polarisation leakage in the NVSS RM catalogue,as it can be thought of as adding a leakage vector to the sourcepolarisation vector, modifying the measured PA values and thus theresulting RM. In the following, we quantify the effect of off-axisleakage on RM TSS09 by simulations.
We took the reported radio properties of the 37,543 sources inthe NVSS RM catalogue to construct the source true polarisationvectors at the two NVSS bands by P j , src = Q j , src + iU j , src = PI src · e i ( PA + RM src · λ j ) , (6)where the subscript j denotes the two NVSS IFs, and λ j is the centre(in frequency space) wavelength of the two IFs with λ = . λ = . src and the RM from TSS09 as RM src .Although we have set up the simulation by taking the listed TSS09values as the source true values while in reality they should betaken as the observed values with polarisation leakage added in,we argue that our simulation results would still be representative ofthe general statistics of the TSS09 catalogue given its large samplesize. A full treatment taking the listed TSS09 values as the observedvalues from our simulations will be presented in the future Paper III.It is assumed that all the sources have flat spectral indices ( α L = were not reported in the TSS09catalogue, we constructed 1,000 realisations for each source, eachwith a randomly picked PA value within [− π / + π / ] from auniform distribution. This results in a total of 37,543,000 inputrealisations.Then, we added leakage vectors ( P leak ) to the true polarisationvectors to obtain the observed polarisation vectors: P j , obs = P j , src + P leak , (7) P leak = ( S NVSS × . ) · e i PA leak . (8)The leakage vector has an amplitude fixed at 0.5 per cent of theNVSS Stokes I values, and PA leak again randomly picked within [− π / + π / ] from uniform distribution for each realisation, iden-tical at IF1 and IF2. In other words, it is assumed that the leakageamplitude does not depend on the source position within the tele-scope primary beam with respect to the pointing centre, which isjustifiable as the mosaicking done to produce the TSS09 images isexpected to smooth out the leakage amplitude within the primarybeam compared to the original leakage pattern of Cotton (1994).Furthermore, it is assumed that the instrumental polarisation (bothamplitude and PA) is not frequency dependent, although the VLAoff-axis leakage pattern in the NVSS configurations are in realityweakly dependent of both direction and frequency (Condon et al.1998; Cotton 1994). The 0.5 per cent leakage level we adopted Complex polarisations behave as vectors in the QU -plane, and can besimply added together. However, note that polarisation planes in the physicalspace do not add up as vectors (e.g. orthogonal polarisation planes cancelout each other). is motivated by the TSS09 reported polarisation fractions of thetwo unpolarised sources that we identified (J084600 − + ′ , 10 ′ , 15 ′ , and 20 ′ away from the pointing centrerespectively (see table 1 of Cotton 1994), and TSS09 sources havean average offset from the closest pointing centre by about 9 . ′
5. Wetherefore argue that an input leakage level of 0.5 per cent is a rea-sonable value to adopt, though note again that the residual leakagepattern in TSS09 is expected to be smoothed out from mosaicking.With the resulting polarisation vectors after adding in polar-isation leakages, we computed the observed RM values (RM obs )for each of the 37,543,000 realisations by Equations 3 and 2. The n π -ambiguity is resolved by choosing the closest possible RM valueto RM src . To quantify the effect of off-axis polarisation leakage on the mea-sured RM, we compared | RM src − RM obs | against the true polari-sation fraction ( p src = PI src / S NVSS ) for the 37,543,000 simulationrealisations. The RM differences reflect how much the injectedleakage vectors alter the source true RM values. To clearly see theunderlying statistics, instead of plotting each of the realisations, weperformed boxcar binning of these data points with a binning widthof 0.1 per cent in p src , with the results at the 50.0th, 68.3th, 95.5th,and 100.0th percentiles plotted as the colour solid lines in Figure 4.We also tested different binning widths (0.05, 0.2, and 0.5 per cent)to ensure that they all show consistent results. As expected, sourceswith lower fractional polarisation are more susceptible to changesin RM due to the injected polarisation leakage. At true polarisationlevel of 1 per cent, the RM difference at 50th percentile (i.e. me-dian) is 7 . − , while that at 68.3th percentile (correspondingto 1 σ significance) is 13 . − . These numbers are compara-ble to the median RM uncertainties of 10 . − reported in theTSS09 catalogue. This means the leakage effect introduces a sig-nificant extra RM uncertainty which has not been accounted for inthe TSS09 RM catalogue, and therefore we suggest that care has tobe taken when using the reported RM values of individual sourceswith p . p src . | RM src − RM obs | in Figure 5, from which wefound that 2, 6, 11, 15, and 29 per cent of the NVSS RM sourceshave | RM src − RM obs | >
10 rad m − at injected leakage levels of0.25, 0.50, 0.75, 1.00, and 2.00 per cent respectively.We further estimate how much TSS09 have underestimatedtheir RM uncertainties for not having the off-axis leakage effecttaken into account. To achieve this, we determined the rms valuein | RM src − RM obs | from the 1,000 simulation realisation for eachof the 37,543 sources individually (denoted as σ RM , leak ). This isthen added in quadrature to the listed TSS09 RM uncertainties ofthe corresponding source ( σ RM , TSS09 ) to yield the new RM un-certainties including the effect of off-axis leakage for each of the
MNRAS000
MNRAS000 , 1–16 (2019) ff-axis Instrumental Polarisation of the NVSS RM Catalogue src (%)050100150200250300350 | R M s r c − R M o b s | ( r a d m − ) Figure 4.
Simulation results of the off-axis polarisation leakage effects on the measured RM values reported in the TSS09 NVSS RM catalogue. The y -axisshows the difference between true and observed RM due to the added leakage, and x -axis is the true polarisation percentage. With injected leakage level of 0.5per cent of the Stokes I flux densities, boxcar percentiles (with binning width of 0.1 per cent) from the 37,543,000 realisations were computed and shown as thecolour solid lines. The 100.0th percentile (i.e. maximum) lines with leakage levels of 1.0 and 2.0 per cent are also shown for comparison. We also over-plot our21 polarised target sources as red points, and the 282 Smith Cloud sources (Betti et al. 2019) as black points. For these sources, the y -values are the differencein RM between TSS09 and new VLA results. src −RM obs | (rad m −2 )0.000.050.100.150.200.250.30 F r a c t i on o f S ou r c e s Figure 5.
The inverse cumulative distribution function (1 − CDF) of thedifference in RM due to off-axis polarisation leakage from our simulation.Injected leakage levels of 0.25, 0.50, 0.75, 1.00, and 2.00 per cent are adoptedand shown here. σ RM , new = q σ , TSS09 + σ , leak ). We calcu-lated σ RM , new / σ RM , TSS09 at our default injected leakage level of0.50 per cent, and obtained a median value of 1.09 out of the entiresample of 37,543 TSS09 sources. In other words, the TSS09 RMuncertainties should be increased by an average of nine per centto incorporate the effect of off-axis polarisation leakage. Choos-ing different leakage levels of 0.25, 0.75, 1.00, and 2.00 per centwould yield median σ RM , new / σ RM , TSS09 of 1.03, 1.14, 1.19, and1.39 instead. Finally, we stress here that one should be cautious when ap-plying the above results to individual TSS09 sources, since we findthat the RM uncertainty from this off-axis polarisation leakage isalso dependent on the actual RM of the source (Section 5.3). Thestatistics reported above are the average values out of the entireRM catalogue. Another point to note is that the distribution of ( RM src − RM obs ) is found to be asymmetric, and is highly non-Gaussian (see Section 5.3). src To investigate the relationship between the RM uncertainties dueto off-axis leakage and the source true RM values, we repeatedthe simulation outlined in Section 5.1 but with manually selectedRM src values instead. In particular, we adopted the source prop-erties of one of our targets, J091145 − S NVSS = . = . except for theRM value. Instead, we have manually put in RM src values of 0to +
800 rad m − at 1 rad m − interval, resulting in 801 artificialsources. We chose J091145 − p = . simulation realisations foreach artificial source (each with randomised PA and PA leak as be-fore; see Section 5.1), and plotted the distribution of the resulting ( RM src − RM obs ) individually for each artificial source as a 2D-histogram in the left panel of Figure 6. We further repeated theabove by using the TSS09 source properties of two of our other tar-gets – J093349 − S NVSS = . = . p = . + S NVSS = . = . p = . MNRAS , 1–16 (2019) Y. K. Ma et al. src (rad m −2 )−15−10−5051015 R M s r c − R M o b s (r ad m − ) src (rad m −2 )−40−30−20−10010203040 R M s r c − R M o b s (r ad m − ) src (rad m −2 )−200−150−100−50050100150200 R M s r c − R M o b s (r ad m − ) Figure 6.
Simulation results showing the relationship between ( RM src − RM obs ) and RM src as 2D-histograms (see Section 5.3). The left, middle, and rightpanels show the cases where the artificial target sources are strongly ( p = . p = . p = . y -axis at a particular RM src represents the artificial source with that corresponding RM src , chosen at a 1 rad m − interval from 0 to +
800 rad m − . The same binning width of 0 . − along the y -axis has been used for all three panels, and we have normalised thehistogram along each y -cut. Note that the y -axis and colour bar scales are different among the panels. −20 −15 −10 −5 0 5 10 15 20RM src −RM obs (rad m −2 )10 -2 -1 N u m be r o f R ea li s a t i on s −2 +150 rad m −2 +300 rad m −2 +450 rad m −2 +600 rad m −2 −40 −30 −20 −10 0 10 20 30 40RM src −RM obs (rad m −2 )10 -3 -2 -1 N u m be r o f R ea li s a t i on s −200−150−100 −50 0 50 100 150 200RM src −RM obs (rad m −2 )10 -4 -3 -2 -1 N u m be r o f R ea li s a t i on s Figure 7.
Simulation results showing the relationship between ( RM src − RM obs ) and RM src (see Section 5.3), plotting cuts along the y -axis from Figure 6 atRM src of 0, + + + +
600 rad m − as the blue, green, red, cyan, and magneta lines, respectively. Note that the x - and y -axis scales are differentamong the panels, and the y -axis is in logarithmic scale. are also shown in Figure 6 as the middle and right panels, respec-tively.From these simulations, we noted several interesting propertiesof the distribution of ( RM src − RM obs ) due to the injected off-axispolarisation leakage. Firstly, the distribution is a strong function ofRM src , with the spread in ( RM src − RM obs ) being the widest atRM src ∼
300 rad m − , while it is identical to zero for the case ofRM src = ± . − (i.e., the injected leakage wouldnot alter the RM value at all). These reported numbers, however,are only valid under the assumption that PA leak are identical inthe two IFs. Otherwise (i.e., if the off-axis leakage has a non-zeroRM), the distributions shown in Figure 6 will be shifted horizontallyby the leakage RM. Secondly, the distribution at any given RM src (except for the identical zero cases mentioned above) is highly non-Gaussian, shaped as a double horn. This can be seen more clearly inFigure 7 where we show cuts of the 2D-histograms at RM src of 0, + + + +
600 rad m − . An interesting implicationof this is that, the listed RM TSS09 values are in general not the mostlikely true RM of the sources, even for the highly polarised examplewith p = . ( RM src − RM obs ) distribution at any given RM src is almost identicalto 0 rad m − for all cases. Finally, it can be seen that the distribu-tions are asymmetric, with this property being more pronounced forsources with lower fractional polarisation. We will briefly explorethe underlying causes of these properties in Appendix B. We return to comparing the RM from our new observations(RM
VLA ) with that from TSS09 cutout images (RM
Tcut ) after tak-ing this leakage effect into account. For each of our polarised targetsources, we computed the RM uncertainty due to this off-axis leak-age ( σ RM , leak ) by repeating the simulation (Section 5.1) but onlyconsidering our targets one at a time, and with a much higher num-ber of simulation realisation of 10 per source. The rms of theresulting | RM src − RM obs | values is taken as the σ RM , leak of thatsource. We reiterate here that the distribution of ( RM src − RM obs ) is highly non-Gaussian (see Section 5.3).The RM discrepancies between the two observations areplotted in the form of histograms in Figure 8, in units of q σ , Tcut + σ , VLA (i.e. without taking leakage into account), aswell as q σ , Tcut + σ , VLA + σ , leak (i.e. with leakage takeninto account), with σ RM , Tcut and σ RM , VLA being the RM uncer-tainties in TSS09 cutout and our VLA observations, respectively.If the difference in RM is only due to random Gaussian noise, thehistogram should follow a folded normal distribution with µ = σ =
1, which is shown as the black curves in both plots. Indeed,the histogram follows the expected distribution much more closelyafter taking into consideration the effect of the off-axis leakage.This means the RM discrepancies between our new VLA obser-vations with that from TSS09 can be largely explained by the off-axis instrumental polarisation leakage at about 0.5 per cent level.The three sources still with significant RM discrepancies even afterconsidering the off-axis leakage are J084701 − . σ ), MNRAS , 1–16 (2019) ff-axis Instrumental Polarisation of the NVSS RM Catalogue | RM VLA −RM
Tcut | / (cid:0) σ +σ N o r m a li s ed C oun t | RM VLA −RM
Tcut | / (cid:0) σ +σ +σ N o r m a li s ed C oun t Figure 8.
Histograms of RM differences between our new VLA obser-vations and that from TSS09 cutout images, in units of RM uncertain-ties. The upper plot is in units of q σ , Tcut + σ , VLA (i.e. withouttaking off-axis leakage into account), and the lower plot is in units of q σ , Tcut + σ , leak + σ , VLA (i.e. with leakage taken into account).The black curves in both plots show a folded normal distribution with µ = σ = J111857 + . σ ), and J170934 − . σ ). Thiscould be due to genuine RM time variabilities, and will be investi-gated in a forthcoming paper (Ma et al. in prep). To further confirm that the RM discrepancies are due to off-axispolarisation leakage, we supplemented our 21 sources with the 282Smith Cloud sources from Betti et al. (2019) that are cross-matchedwith the TSS09 catalogue. Their sources were also placed on-axisin their broadband VLA observations in L-band and are thus un-affected by off-axis polarisation leakage. However, there are sig-nificant differences between their data and that of TSS09, namely(1) they used broadband 1–2 GHz data while TSS09 used narrow-band observations, (2) their observations were conducted in A arrayconfiguration instead of D array as did NVSS, and (3) their RMvalues were taken as the peak from RM-Synthesis analysis, whileTSS09 performed two-point λ fit to PA. Such differences in λ coverage, uv -coverage, and analysis method can lead to disagree-ments between their RM values and the corresponding TSS09 RM values, which could be mistaken as caused by the off-axis polari-sation leakage of TSS09 and/or RM time variabilities. A detailed,quantitative comparison between Betti et al. (2019) and TSS09 istherefore outside the scope of this work.The combined 303 sources are plotted in Figure 4. We com-puted the boxcar median line for these 303 observed sources, asshown in Figure 9 (denoted as “original”). A boxcar binning widthof 0.4 per cent is used here instead of the 0.1 per cent we used inFigure 4 because of the lack of observed sources in some p bins.We do not quantitatively compare this observed median line withthe simulation results (Figure 4) directly, because the selection bi-ases of the two observations preclude meaningful conclusions tobe drawn. In particular, our target sources were selected to have | RM TSS09 | &
300 rad m − , while the Betti et al. sources are inclose proximity to the Smith Cloud and therefore have a differentRM distribution than that of the entire sample of TSS09. As thedistribution of RM uncertainties due to off-axis leakage is a func-tion of the source RM (Section 5.3), it would not be surprising ifthe median line here does not closely match the simulation resultspresented above in Section 5.2.We focus here on the qualitative trend of the observed medianline, from which we find a peak at the lower end of p ( . | RM VLA − RM TSS09 | might just have low p coincidentally. In suchcase, | RM VLA − RM TSS09 | would actually have no correlation with p . We test this hypothesis with the bootstrapping method. With the303 observed sources, we shuffle the | RM VLA − RM TSS09 | valueswith respect to p , and then construct a new boxcar median line (alsoat binning width of 0.4 per cent). This shuffling process is repeatedfor 10 times, yielding 10 median lines. As the final step, for each p value we evaluated the 99.73th, 99.95th, and 99.99th percentilesof | RM VLA − RM TSS09 | of the 10 median lines.These percentile lines are plotted in Figure 9 together with theoriginal observed median line. The percentile lines peak at several p values – at 0.5 per cent and between 8.0 and 10.0 per cent. Thesepeaks are due to the scarcity of data points there, as can be seen inFigure 4. The original line is mostly featureless and lie well belowthe 99.73th percentile line for p & p . | RM VLA − RM TSS09 | at lower p is statistically robust (at about 99.95 per cent confidence level),and that residual off-axis polarisation leakage is indeed introducingextra RM uncertainties to TSS09. When Stil et al. (2011) compared the RM structure function of theGalactic poles they derived from TSS09 results with that fromMao et al. (2010), they suggested that the RM uncertainties inTSS09 catalogue could be underestimated by a factor 1.22. Thiswould explain the discrepancies between the two studies, thoughthe authors did not explore the cause of such underestimated un-certainties. As we have shown above, off-axis polarisation leakagecan introduce extra RM uncertainties to TSS09 results, and canpotentially explain the factor 1.22 that they suggested.To test this, we compiled the list of sources taken as the NorthGalactic Pole (NGP; 1,019 sources) and South Galactic Pole (SGP;752 sources) samples by Stil et al. (2011), and investigated the σ RM , new / σ RM , TSS09 ratio of these sources. This ratio quantifies
MNRAS , 1–16 (2019) Y. K. Ma et al.
TSS09 (%)020406080100120140160180 | R M V L A − R M T SS | ( r a d m − ) Original99.73th Percentile99.95th Percentile99.99th Percentile
Figure 9.
Difference in RM between TSS09 and new VLA observations. The boxcar median of the 303 sources (21 from this work, plus 282 from Betti et al.2019) is shown by the blue solid line. The colour dashed lines are the 99.73th, 99.95th, and 99.99th percentile lines respectively from 10 shuffles of the y -with respective to the x -values (see Section 5.4). A boxcar binning width of 0.4 per cent is used here. by what factor we should increase the TSS09 RM uncertainties inorder to incorporate the effects of off-axis polarisation leakage, andwere obtained in Section 5.2. Using our default leakage level of 0.5per cent, we find that the median values of σ RM , new / σ RM , TSS09 forthe NGP and SGP sources are both 1.02, much lower than the factorof 1.22 suggested by Stil et al. (2011). This suggests that there canbe other sources of TSS09 RM uncertainties that have not beenaccounted for yet.
From our new broadband spectro-polarimetric observations of 23NVSS RM sources with the VLA, we identified two unpolarisedsources that are listed as ≈ . . − for TSS09 sources with p . . − reported inthe TSS09 catalogue. For a typical TSS09 source, the RM uncer-tainties should be increased by nine per cent in order to incorporatethe effects of off-axis leakage. We further demonstrated that theprobability distribution of this extra RM uncertainty is asymmetric,highly non-Gaussian, and is a function of the source RM value.These properties must be carefully taken into account if one wishesto incorporate the off-axis leakage effects to individual sources,which will be the goal of our forthcoming Paper III.The RM discrepancies of our sources can be largely explained by taking the extra RM uncertainties due to leakage into account,though three sources still show hints of RM time variabilities. Fur-thermore, by supplementing our dataset with the 282 Smith Cloudsources from Betti et al. (2019), we confirmed (at confidence levelof about 99.95 per cent) that sources with lower fractional polar-isation have larger RM discrepancies between new on-axis VLAobservations and that from TSS09. This is almost certainly due tothe residual off-axis polarisation leakage in TSS09 catalogue. 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-
MNRAS , 1–16 (2019) ff-axis Instrumental Polarisation of the NVSS RM Catalogue 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 andSpace 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: RADIO SPECTRA OF OUR TARGETS
We show the radio spectra of our target sources in Figure A1. A simplepower law is fitted to our broadband VLA data (from Paper I). For sourcesidentified as spatial doubles, we fitted each component individually, as wellas their sum (with uncertainties added in quadrature). The best-fit parametersare listed in Table 2.
APPENDIX B: THE ROLE OF PA
LEAK
We further investigate here the effects of off-axis polarisation leakage inSection 5.3. Specifically, we aim to understand the cause of the asymmetricdouble-horn ( RM src − RM obs ) distributions seen in Figures 6 and 7. Toachieve this, we repeated our simulation with the artificial sources, again withthe TSS09 source properties of J091145 − S NVSS = . = . − S NVSS = . = . + S NVSS = . = . , 1–16 (2019) Y. K. Ma et al. L =−0.581 ±0.006 1000 1500 2000100200400 NVSS J083930-240723α L =−0.904 ±0.010 1000 1500 2000100020003000 NVSS J084600-261054α L =−0.437 ±0.0071000 1500 2000100150200 NVSS J084701-233701α L =−0.233 ±0.012 1000 1500 2000300500800 NVSS J090015-281758α L =−0.693 ±0.006 1000 1500 200010100400 NVSS J091145-301305α L =−0.923 ±0.011α L =−1.054 ±0.022α L =−0.858 ±0.0101000 1500 2000502001000 NVSS J092410-290606α L =−0.955 ±0.006α L =−1.011 ±0.010α L =−0.901 ±0.009 1000 1500 2000100200400 NVSS J093349-302700α L =−0.972 ±0.007 1000 1500 200050200800 NVSS J093544-322845α L =−0.857 ±0.006α L =−0.761 ±0.010α L =−0.941 ±0.0081000 1500 2000300500900 NVSS J094750-371528α L =−0.755 ±0.013 1000 1500 2000100200500 NVSS J094808-344010α L =−0.373 ±0.009 1000 1500 2000100020003000 NVSS J111857+123442α L =−0.232 ±0.003 Frequency (MHz) F l u x D en s i t y ( m Jy ) Figure A1.
Radio spectra across L-band of our target sources from the new VLA observations. The total flux densities are represented by black data points. Forsources that are resolved into two spatial components, the flux densities of the individual components are plotted in red (component a) and blue (componentb). The best-fit power law spectra ( S ν ∝ ν α L ) are shown as solid lines with corresponding colours to the data points. The flux densities we determined fromNVSS cutout images ( S cutout ) are plotted as the cyan circles, while that listed in the NVSS catalogue ( S NVSS ; Condon et al. 1998) are plotted as the magentadiamonds.polarised case). For each case, we manually selected RM src values of 0, + + + +
600 rad m − . However, instead of randomisingPA and PA leak as we did in Section 5.3, here we chose PA for eachartificial source such that the source PA in the NVSS IF1 is 0 ◦ , and PA leak is uniformly sampled within [− π / , + π / ] to see its effect on ( RM src − RM obs ) . The results of this are presented in Figures B1 and B2, with theformer showing the trend of ( RM src − RM obs ) and the latter showing thatof ( PI src − PI obs )/ S NVSS . By consulting these Figures, we can pinpointthe situations (in terms of the relative PA between the source and leakagevectors) that resulted in the double horn.From Figure B1, we can see that ( RM src − RM obs ) shows a nearlysinusoidal variation across PA leak , with the peaks and troughs corresponding to the two horns in Figures 6 and 7. For the strongly and intermediatelypolarised cases, the widths of the peaks and troughs are very similar, leadingto the symmetric ( RM src − RM obs ) distributions. On the other hand, for theweakly polarised case the differences in the widths of the peaks comparedto that of the troughs are much more apparent. This in turn leads to theextreme asymmetry in the double horns – the wider (or narrower) part in ( RM src − RM obs ) against PA leak (Figure B1) corresponds to the taller (orshorter) of the double horn of the histogram in Figures 6 and 7.MNRAS000
600 rad m − . However, instead of randomisingPA and PA leak as we did in Section 5.3, here we chose PA for eachartificial source such that the source PA in the NVSS IF1 is 0 ◦ , and PA leak is uniformly sampled within [− π / , + π / ] to see its effect on ( RM src − RM obs ) . The results of this are presented in Figures B1 and B2, with theformer showing the trend of ( RM src − RM obs ) and the latter showing thatof ( PI src − PI obs )/ S NVSS . By consulting these Figures, we can pinpointthe situations (in terms of the relative PA between the source and leakagevectors) that resulted in the double horn.From Figure B1, we can see that ( RM src − RM obs ) shows a nearlysinusoidal variation across PA leak , with the peaks and troughs corresponding to the two horns in Figures 6 and 7. For the strongly and intermediatelypolarised cases, the widths of the peaks and troughs are very similar, leadingto the symmetric ( RM src − RM obs ) distributions. On the other hand, for theweakly polarised case the differences in the widths of the peaks comparedto that of the troughs are much more apparent. This in turn leads to theextreme asymmetry in the double horns – the wider (or narrower) part in ( RM src − RM obs ) against PA leak (Figure B1) corresponds to the taller (orshorter) of the double horn of the histogram in Figures 6 and 7.MNRAS000 , 1–16 (2019) ff-axis Instrumental Polarisation of the NVSS RM Catalogue L =−0.797 ±0.003 1000 1500 20005100300 NVSS J162706-091705α L =−1.061 ±0.013α L =−0.730 ±0.042α L =−1.171 ±0.015 1000 1500 200050200600 NVSS J163927-124139α L =−0.822 ±0.005α L =−0.810 ±0.006α L =−0.843 ±0.0101000 1500 2000300500700 NVSS J170934-172853α L =−0.077 ±0.007 1000 1500 2000200030004000 NVSS J190255+315942α L =−0.320 ±0.003 1000 1500 200040200400 NVSS J220205+394913α L =−1.140 ±0.0091000 1500 2000100300600 NVSS J220927+415834α L =−0.964 ±0.006 1000 1500 2000100200400 NVSS J224412+405715α L = +0.015 ±0.008 1000 1500 20004000900015000 NVSS J224549+394122α L =−0.988 ±0.0081000 1500 2000100020004000 NVSS J234033+133300α L =−1.252 ±0.005 1000 1500 20004006001000 NVSS J235728+230226α L =−0.863 ±0.007 Frequency (MHz) F l u x D en s i t y ( m Jy ) Figure A1. (Continued) Radio spectra of our target sources from new VLA observations.MNRAS , 1–16 (2019) Y. K. Ma et al. −90 0 90PA leak ( ◦ )−20−15−10−505101520 R M s r c − R M o b s (r ad m − ) −90 0 90PA leak ( ◦ )−40−30−20−10010203040 R M s r c − R M o b s (r ad m − ) −90 0 90PA leak ( ◦ )−200−150−100−50050100150200 R M s r c − R M o b s (r ad m − ) Figure B1.
Simulation results showing the relationship between ( RM src − RM obs ) and PA leak . The left, middle, and right panels show the cases where theartificial target sources are strongly ( p = . p = . p = . in of 0, + + + +
600 rad m − are shown as the blue, green, red, cyan, and magenta lines, respectively. The input PA has beenchosen such that the true PA at the NVSS IF1 is 0 ◦ . Note that the y -axis scales are different among the panels. −90 0 90PA leak ( ◦ )−0.6−0.4−0.20.00.20.40.6 ( P I s r c − P I o b s ) / S NV SS ( % ) −90 0 90PA leak ( ◦ )−0.6−0.4−0.20.00.20.40.6 ( P I s r c − P I o b s ) / S NV SS ( % ) −90 0 90PA leak ( ◦ )−0.6−0.4−0.20.00.20.40.6 ( P I s r c − P I o b s ) / S NV SS ( % ) Figure B2.
Simulation results similar to those in Figure B1, but showing ( PI src − PI obs )/ S NVSS instead as the y -axis.MNRAS000