A&A 623, A71(2019)
https://doi.org/10.1051/0004-6361/201834777
©ESO 2019
Astronomy&AstrophysicsDiffuse polarized emission in the LOFAR Two-meter Sky Survey?
6
C.L.VanEck1,2,M.Haverkorn2,M.I.R.Alves2,R. Beck3,P. Best4,E. Carretti5,K.T.Chy˙zy,
T. Enßlin7,8,J.S.Farnes2,9,K.Ferrière10,G. Heald11,M. Iacobelli12,V. Jelic´13,W.Reich3,
H.J.A.Rttgering14, andD.H.F.M.Schnitzeler3
1 Dunlap InstituteforAstronomyandAstrophysics,UniversityofToronto,50St. GeorgeStreet,Toronto,ON M5S 3H4, Canada
e-mail: cameron.van.eck@dunlap.utoronto.ca
2 DepartmentofAstrophysics/IMAPP,RadboudUniversity,POBox9010, 6500GLNijmegen, TheNetherlands 3 Max-Planck-InstitutfRadioastronomie,AufdemHel69,53121Bonn,Germany 4 SUPA, InstituteforAstronomy,Royal Observatory, BlackfordHill, Edinburgh, EH9 3HJ,UK 5 INAF – OsservatorioAstronomicodiCagliari,Via della Scienza5, 09047Selargius(CA),Italy 6 Astronomical Observatory, JagiellonianUniversity, ul.Orla171, 30-244Krakw,Poland 7 Max PlanckInstituteforAstrophysics,Karl-Schwarzschild-Str.1,85748 Garching, Germany 8Ludwig-Maximilians-Universität Mchen, Geschwister-Scholl-Platz1, 80539 Mchen, Germany 9 Oxforde-ResearchCentre (OeRC),KebleRoad, OxfordOX1 3QG,UK
10 IRAP,UniversitédeToulouse, CNRS,9avenuedu ColonelRoche,BP44346,31028Toulouse, Cedex4,France 11 CSIROAstronomyand Space Science,POBox1130, Bentley,WA6102,Australia 12 ASTRON,theNetherlands InstituteforRadioAstronomy,Postbus2, 7990AA Dwingeloo, TheNetherlands 13 Ru¯c Institute,Bijeniˇ
der Boškovi´cka cesta54,10000Zagreb,Croatia 14 Leiden Observatory, LeidenUniversity,POBox9513, 2300RA Leiden, TheNetherlands
Received4December2018/Accepted25 January2019
ABSTRACT
Faradaytomography allows us to map diffuse polarized synchrotron emission from our Galaxy and use it to interpret the magnetic feldinthe interstellar medium (ISM).Wehave appliedFaraday tomography to60 observationsfromtheLOFARTwo-meterSky Survey(LOTSS)andproducedaFaradaydepthcube mosaiccovering568squaredegreesathigh Galactic latitudes,at4. 03angular resolution and1radm−2Faradaydepthresolution,witha typical noiselevelof50–100µJy per point spread function (PSF) perrotation measure spread function (RMSF; 40–80 mK RMSF−1). Whilepartsoftheimagesarestronglyaffectedbyinstrumentalpolarization, weobserveddiffusepolarized emissionthroughoutmostofthefeld,withtypicalbrightnessbetween1and6KRMSF−1, andFaraday depths between −7 and +25 rad m−2. We observed many new polarization features, some up to 15◦ in length. These include two regionswith veryuniformly structured, lineargradientsintheFaradaydepth;we measuredthesteepnessofthesegradientsas2.6and 13radm−2 deg−1.WealsoobservedarelationshipbetweenoneofthegradientsandanH IflamentinthelocalISM.OtherISMtracers werealsocheckedforcorrelationswithourpolarizationdataand nonewerefound,butverylittlesignalwas seeninmosttracersinthis region.We concludethattheLOTSSdataareverywell suitedforFaradaytomography,andthata full-scale surveywithalltheLOTSS datahasthepotentialtorevealmany new GalacticpolarizationfeaturesandmapoutdiffuseFaradaydepth structureacrosstheentire northern hemisphere.
Key words. ISM: magnetic felds – polarization
1. Introduction through magnetized plasma (which flls most of the volume of the ISM), which causes a frequency-dependent rotation of the
Magnetic feldsarepresentthroughout interstellar spaceandplay polarizationangle.Thechangeinthe polarizationangle(Δθ)is an important role in many aspects of the interstellar medium
givenby
(ISM), such as cloud collapse duringstarformation(van Loo
" !!#
Z observer
et al. 2012), energy transports and cascades in magnetohydro-ne B dl
Δθ = λ2 φ(d) = λ20.812 rad m−2 · , dynamic turbulence(Beresnyak&Lazarian2015), andpressure
source cm−3 µG pc
balancebetween differentgas phases(Boulares&Cox1990).
(1)
Interstellar magnetic felds can be measured using radio polarizationthrough two processes: synchrotron emission and where φ(d) is the Faraday depth which depends on the free Faradayrotation. Synchrotron emissionisproducedthroughout electron density(ne)and magnetic feld(B)along the line of interstellar spacebycosmic-rayelectrons astheyare accelerated sight integrated fromthe emission source at distance d to the byinterstellar magnetic felds, resulting in polarized radio emis-observer. sion. Faraday rotation occurs when polarized emission passes Polarized synchrotron emission is produced throughout the
Galaxy, and then undergoes Faraday rotation as it propagates
? The data shown in Figs. 2–7 are available at the CDS via through the ISM; therefore along any line of sight we see the anonymous ftp to cdsarc.u-strasbg.fr
(130.79.128.5) or via superposition of the emission at all distances and with corre-http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/623/A71
spondinglydifferentFaradaydepths.Therotation measure(RM)
Article publishedbyEDP Sciences A71,page1of 22
synthesis technique(Brentjens&deBruyn 2005)canbe usedto transform the observed wavelength-dependent polarization into the distributionof polarized emission asa functionofFaraday depth. When applied to 3D image-frequency data this is called Faradaytomography, whichproducesFaradaydepthcubesthat map outthe diffuse polarized emission asa functionof position ontheskyandFaradaydepth.Suchobservationscanbeusedto constrain magnetic felds in the ISM and study their properties (e.g., Schnitzeleret al. 2007;Jeli´
c et al. 2015;Lenc et al. 2016; Van Ecketal. 2017).
The resolution in Faraday depth of Faraday tomography depends on the range of wavelength-squared sampled by the observations, so low-frequency radio telescopes (whichcan pro-duce very large wavelength-squared coverage) are capable of achieving much fner Faraday depth resolution than higherfrequency instruments. The newest generation of very low-frequency radio telescopes such as the Low Frequency Array (LOFAR, van Haarlem et al. 2013)and the Murchison Wide-feld Array (MWA, Tingay et al. 2013), which operate in the 100–200MHzrange,arecapableofreachingFaradaydepthresolutions around1radm−2,approximatelytwoorders of magnitude smallerthantheresolutionthat canbeachievedat1.4GHzor higher frequencies.
In recent years, there have been many studies using Fara-daytomographytostudy diffuse polarization, eachwithdifferent observations balancing tradeoffs between feld-of-view, angular resolution,andFaradaydepthresolution:Jeli´cetal.(2014,2015), andVanEck et al. (2017)usedLOFAR observations,Bernardi etal. (2013)andLencetal. (2016)usedMWAobservations,Hill et al. (2017)used datafromthe Global Magneto-Ionic Medium Survey (GMIMS, Wolleben et al. 2008), while Iacobelli et al. (2013)used observationsfromtheWesterborkSynthesisRadio Telescope (WSRT). The observations withhigh angular resolu
tion tend to have smaller felds of view, and vice versa, making it diffcultto observethe full sizeof large(severaldegree) polar-izationfeatures (such asthe flamentin Jeli´cet al.2015 that extends outside the feld of view) while still minimizing the effectsof beam depolarization (whichreducesthe signalpresent in low-resolution observations). Inthis paper wereport onthe polarizationprocessingof 60LOFAR observationsfromtheLOFARTwo-meterSky Sur-vey(LOTSS, Shimwelletal.2017)andpresentaFaradaydepth mosaic covering an area of 568 square degrees. This is the frst work to combine multiple LOFAR observations into a single Faraday depth cube. This allows us to take advantage of the high angular and Faraday depth resolution of LOFAR while still looking at large-scalefeatures inthe diffuse polarization. In Sect. 2wepresent our datareduction andtheproductionof Faradaydepthcubesfor eachobservation.In Sect.3wepresent aFaradaydepthcube mosaicproducedfromthese observations, and highlight the diffuse polarization features seen. In Sect. 4 we compare our mosaic against other tracers for components of the ISM, and in Sect. 5 we discuss possible origins for the polarizedfeaturesweobserve.In Sect. 6weconsiderthe depolarizationeffectsofFaradayrotationgradients andtheimplications for our observations and others. In Sect. 7 we summarize our results.
2. Data processing
We analyzed 63 calibrated visibility datasets generated with observations from the LOFAR Two-meter Sky Survey; full details of the observational parameters and calibration methods canbefoundin Shimwellet al. (2017). The observations are from the LOTSS test region, right ascension from 10h30m to 15h30m and declination from 45◦ to 57◦, which covers the HETDEXSpring feld(Hilletal.2008),aregion nearthe Galac
tic north pole. Each observation hasa nominal durationof8h and coversthefrequencyrange120–168 MHz with488channels. Wereceivedthe observationsafter direction-independent ampli-tude and phase calibration with the LOTSS pipeline. Figure 1 shows the coverage of these observations in equatorial and Galactic coordinates. The polarization calibration and imaging closely followsthatof VanEck et al. (2017)andis described below.
Polarization calibrationintheformofa correctionfor ionosphericFaraday rotationwas performedpriorto imaging. This was done using the RMextract package1 written by Maaijke Mevius, combined with maps of the ionospheric total elec-tron content fromthe Centerfor Orbit Determination in Europe (CODE)2.Thispackageproducedpredictionsforthe ionospheric Faradayrotationthatwere usedbythe BlackBoardSelfcal soft-ware(BBS,Pandeyetal. 2009)toderotatethe polarizationofthe visibilities andtherebyremovethe ionosphericFaradayrotation. The estimated error in Faraday depth correction is approximately 0.1 rad m−2 (Van Eck et al. 2018). No correction for polarization leakagewasperformed,asitwas deemedtoo com-putationally expensive using the currently available methods. Theeffectsof polarization leakage ontheresults are discussed in Sect. 3.1.
Polarization imagingwas performed independentlyfor each channel usingAWImager(Tasseet al.2013).Abaseline length upper limit of 800λ was applied, as this made the synthesized beam (hereafter, point spread function or PSF) more uniform across the band, giving a FWHM of 4. 03. The short baselines betweenthetwohalvesof eachhigh-band antenna(HBA)station (i.e., substations CS(X)HBA0 and CS(X)HBA1, iterating (X) overallHBAstations; see van Haarlemetal.2013,fora descrip
tion of the substation layout and naming) were removed, as these were known to often suffer from signifcant mutual inter-ference. A robust weighting of 1.0 was used for imaging, and no CLEANing was performed, as the signal to noise ratio in individual channels was expected to be too low for CLEAN to be effective. By default AWimager produces images both withand without correctionfortheLOFARprimarybeam;the images without correctionwere usedforthequality controlstep described next, whilethe images with correctionwere usedfor producingFaradaydepthcubes.
Wefoundthata smallfractionofthe imageswerestrongly affectedby interference or calibration problems, which usually manifested as very strong patterns throughout the image. To identifythesechannelsin an automatedway,we calculatedthe standarddeviationoverallpixelsin eachofthe images without primarybeam correction (separatelyforStokes Q and U). The channelsaffectedbytheseproblemsstoodoutashaving abnormallylargestandarddeviations, butvariationsinthe background noise acrossthebandandbetweendifferent observationsmadeit diffcultto assigna singlethresholdvaluefor classifying images as bad.Wechoseto calculate,for each channel,the medianof thestandarddeviationsoftheimageswithin50channels above or belowinfrequency;ifthestandarddeviation(foreither Q or U separately)inachannelwas morethan1.5 timesthe medianstandarddeviationofthese neighbouringchannelsthenthatchannel wasfaggedasbad. ThesebadchannelswereremovedbeforeRM synthesis. Duringthisprocess,three observationswerefoundto
1 https://github.com/maaijke/RMextract/
2 http://aiuws.unibe.ch/ionosphere/
A71, page2of 22
C.L.VanEck etal.:LOTSS diffuse
Fig.
1.
LocationsoftheLOTSS HETDEX observations,inbothEquatorial(dotted lines)and Galactic (dashed lines) coordinates.Eachgraycircle coversthe area insidetheprimarybeam FWHM (3.4◦ diameter)for one observation,atthe highestfrequency(168MHz).Thisimage,andallthat follow,areinanorthographicprojection.Thegapsare causedbythethree feldsthatwereremovedforbeingverystronglyaffectedbyinstrumental leakage.
have muchhigherstandarddeviations acrossthe full bandwidth than the others, which was due to the presence of extremely bright radio sources; these observations were removed and not includedinthefollowingsteps.Afterthisstep,the typical noise level in a single channel was 2–5 mJy PSF−1.
RM synthesis was performed using pyrmsynth3. Channel weightswere calculated usingtheinversesquareofthestandard deviation of each channel that was calculated in the previous step. For most observations, the weights were fairly constant across the band, so this weighting choice was not expected to produce any signifcant changes from uniform weighting. Weightswereapplied independentlyperchannel; no adjustments for sampling density inλ2wereapplied.Fromthefrequencycoverage andweighting used,theresultingrotation measure spread function (RMSF) had a typical full-widthat half-max (FWHM) of1.2 rad m−2.From Eq.(61) of Brentjens&de Bruyn (2005), thetheoretical FWHM(for uniformweighting)is1.15 radm−2, sothesevalues are consistent and showthatthe naturalweightinghasnot signifcantlydegradedtheFaradaydepth resolution. From Eqs. (62) and (63) of Brentjens& de Bruyn (2005), our observations are not sensitivetoextendedstructuresinFaraday depthwiderthan about1.0radm−2,andwehavelimited sensitivitytoFaradaydepthslargerin absolutevaluethan170radm−2. Faraday depth measurements beyond this range may also be unreliable(Schnitzeler&Lee2015).Asis typicalforLOFAR observations, we are unable to resolve any features with Faraday thicknessgreaterthantheresolution,thereforewe are only abletopickup unresolvedfeatures(or sharpedgesofFaraday thickfeatures,as describedin VanEcketal.2017).No spectral index correctionwas includedintheRMsynthesisstep;thismay produce minor errors in the polarized intensity and small sec-ondarypeaksintheRM spectrum,butthesewerenotexpectedto affect ouranalysis.EachRMcubecoveredFaradaydepthsfrom −100to+100radm−2instepsof 0.25radm−2. The RM-CLEAN
3 https://github.com/mrbell/pyrmsynth
algorithm(Healdetal.2009)was appliedtoeachFaradaydepth cube,toadepthof2mJy PSF−1 RMSF−1.
In orderto combinetheFaradaydepthcubesfromthe different observations, we estimated the position dependent noise in the cubes. We did this on a per-pixel basis by taking the dis-tribution of polarized intensity values taken from the regions of the spectrum that were expected to be dominated by noise (|φ| > 20 rad m−2)and fttingaRayleigh distribution (whichis theexpected distributionfor polarized intensityinthe absence of signal, Macquart etal. 2012;Hales etal. 2012). The resulting Rayleighσ parameter (whichisequivalenttothe Gaussian σ of the underlyingStokesQ and U distributions)wastakento bethe noiseinthat pixel.
To mergethe separateFaradaydepthcubesweaveragedthe overlappingregions usinginverse-varianceweightingtoproduce a fnal mosaic coveringthe entire HETDEXregion. Alternative weighting schemes, where each pixel used only the observation with the lowest noise or the observation with the nearest pointing center,were alsotriedbutwerefoundtoproducesignifcant artifacts in the overlap regions of adjacent pointings. However, these were useful for assessing the quality of the data and the effectiveness of the mosaicing; we observed that manyfeaturesinthe cubes, such as depolarization canals, could be traced continuously across the boundaries between different observations.
3. The HETDEX mosaic
The combined Faraday depth cube resulting from the processing describedpreviously coversatotal areaofskyof 568square degrees,withatypical noiselevelbetween50and100 µJyPSF−1 RMSF−1, with higher noise at the edges and near very bright Stokes I sources. Figures 2–7 show selected slices from the cube. These images give the polarized intensity in mJy PSF−1 RMSF−1; the conversion to brightness temperature units is
0.8K(mJy PSF−1)−1.Figures 8–11show collapsedversionsof
A71, page3of 22
Fig.
2.
Selected slices from the Faraday depth cube mosaic, showing the polarized intensity between −7 and −4 rad m−2. Top panel: typical quiescent slice, withstrong artifacts around 3C295.Lower two panels:diffuse polarized emission appearinginthetopright.
sectionsofthecubewhere,perpixel,thehighestpolarizedinten-Whilethis mosaic covers alargeareaofskyand allows sitywas locatedandthepolarized intensityandFaradaydepthof largefeaturestobefollowed across several pointings,itis not thepeakwereusedtodeterminethebrightnessandcolor,respec-sensitivetovery large polarizedfeatures. The shortest base-tively.Figure 8showsthe full mosaic, whileFigs. 9–11focus on lines present in the observations are approximately 15λ in the specifcdiffusepolarizedfeaturesdescribedin Sects. 3.2.1– (projected)length, correspondingtoan angular scaleof about
3.2.3.Figure 12shows some typicalFaradaydepthspectrafrom 3.8◦; uniform polarization features larger than this scale will eachregion.Inthecubeweseemanydiffusepolarized emission be missedandthis shouldbe consideredwhen interpretingthe features, polarized point sources, and instrumental polarization mosaic. The distribution of baselines is shown inFig. 13. How
leakage, whichare discussed in detail below. ever, at such low frequencies this may not be as signifcant a
A71, page4of 22
C.L.VanEck etal.:LOTSS diffuse
Fig.
3.
Selected slicesfromtheFaradaydepthcube mosaic, showingthe polarized intensitybetween −3 and −1 rad m−2. Thefeatureinthetop righttravels south-westward,and anotherfeature appearsinthetop centerandspreads outwards.TheinstrumentalleakagefrombrightStokes I sources also appear stronglyat −2 and −1 rad m−2.
problemasat higherfrequencies,duetoFaraday rotationgra-rotationvariation necessary to cause this polarization change. dientsdriving emission to smaller angular scales(Schnitzeler For a typicalfrequencyof150 MHz,this corresponds toa etal. 2009).We can observe polarizationfeatures with achange Faraday depth change of 0.8 rad m−2 per 3.8◦ or 0.21 rad
−2
in polarization anglegreaterthan π radians over a 3.8◦ angular mdeg−1. Polarization features with greater variation in scale; if we assume that all of this change is due to Faraday Faraday depth than this value should be detected by our rotation fuctuations,we can calculatetheamplitudeofFaraday observations.
A71, page5of 22
Fig.
4.
Selected slicesfromtheFaradaydepthcube mosaic,showingthepolarized intensitybetween0and2radm−2. Thetop-rightfeaturecontinues totravel south-westward, becomingveryflamentary-looking, whilethetop-centerfeaturecontinuestoexpand outwards.Two morediffusefeatures appearinthelowerright corner andlower left-of-center.
3.1. Instrumental polarization leakage
The instrumental polarization leakageforLOFAR, whichcauses emission fromStokes I to incorrectlyappearintheotherStokes parameters, has the convenient property of being effectively independent of frequency. The result of this is that after RM synthesis, the leakage should appear only at Faraday depth 0 rad m−2, which would allow astrophysical emission at other Faraday depths to be identifed. However, the ionospheric Faraday rotation correction applies a frequency-dependent
A71, page6of 22
C.L.VanEck etal.:LOTSS diffuse
Fig.
5.
Selected slicesfromtheFaradaydepthcube mosaic,showingthepolarizedintensitybetween3and5 radm−2. Thetoprightfeature now appearsasaverylong flamentary structure spanningthe heightoftheimage,beforefadingaway.Thelowerrightfeaturealsofadesaway at higherFaradaydepths.Thetop centerandlowerleft-of-centerfeaturesnow appearasaseriesofpatchesrunning south-westwardjustleftof center. Afaint,verystraight flamentaryfeature appearsinthebottom mid-left,slowlytravelingupwards.
polarization anglerotationtothe data, which shiftsthe leak-correction varies with time,this also hastheresult of caus-agetothe oppositeFaradaydepth asthecorrection(e.g.,ifthe ingpartialdepolarizationoftheleakage,asthepost-correction ionospheric Faraday rotation is +2 rad m−2, the leakage will instrumental leakage will have different polarization angles at appear at −2radm−2 afterthe correction). Sincethe ionospheric different times.
A71, page7of 22
Fig.
6.
Selected slicesfromtheFaradaydepthcube mosaic,showingthepolarized intensitybetween7and12radm−2. The intensity scale has been adjustedasthe emissionatthesedepthsis muchfainter.Thetoprightandleft-of-centerfeatures continuetofadeout, whilethefaintvery straight flament continuesto travel upwards with new,thinfeatures appearing above it.
The neteffectofthe instrumental polarization leakageis conditions,the leakage sources appear at slightly different to cause all Stokes I sources to appear in the Faraday depth Faraday depths at different locations in the mosaic, and cubes between approximately −2 and0 radm−2 (as the iono-some regions also appear to have stronger or weaker leakspheric correction was typically between 0 and 2rad m−2). age depending on how muchdepolarizationthe time-variability Since the different observations had different ionospheric produced.
A71, page8of 22
C.L.VanEck etal.:LOTSS diffuse
Fig.
7.
Selected slices from the Faraday depth cube mosaic, showing the polarized intensity between 15 and 21 rad m−2. The faint, straight flamentaryfeatures slowlyconverge and disappear. Top panel:averybright polarized source appearstothe leftof 3C295. Some emission appears tobepresentatthebottomoftheimage,justleftof center,butitisnotclearifthisisrealor causedbyenhanced noiseattheedgeofthe mosaic.
In additionthere is signifcant off-source polarization leak-polarization causedbythe polarization leakage does not appear age,whichisthedominantcauseofthestrongartifactsappearing justatthe location ofthe source,butis convolvedwiththe widelydistributedinFaradaydepth aroundverybright sources PSF,just astheStokes I and real polarized emission is. Since such as 3C295(α = 14h11m20.59s, δ = +52◦1209.600, 84 Jy in the locations and intensities of the sidelobes of the PSF vary Stokes I at 178 MHz; Spinrad et al. 1985). The spurious with frequency, pixels near sources can pass in and out of
A71, page9of 22
Fig.
8.
Collapsedviewofthefullmosaic,wherethepolarized intensityandFaradaydepthofthebrightestfeatureintheFaradaydepthspectrum were usedtodeterminethebrightness and color, respectively. Instrumental leakagefrombright point sources, particularly3C295, dominate many areas withfeatures atFaradaydepths around −2rad m−2, but diffuse emission canbe seenthrough mostofthe mosaic.
sidelobes as a function of frequency. The result is a very com-plicated frequency-dependent spurious polarization signal that, when RM synthesis is applied, results in structure broadlydistributedinFaradaydepth. Thus, whilethe on-source polarization leakageis usuallyconfnedtoasingleFaradaydepth(determined bythe ionosphere),theoff-source leakageisbroadlydistributed inFaradaydepthandcoverslargerareasaroundbrighter sources.
In the HETDEX region mosaic, the off-source leakage of the two brightest sources occupies a large portion of the feld: 3C295 (84 Jy) is in the mid-east (left) part of the mosaic, and 3C280(α = 12h56m59s, δ = +47◦1804800,24Jy at178MHz; Spinrad et al. 1985)is in the bottom center of the mosaic. As aresulttheregion around 3C295is unusablefor polarization analysis withthese data.
3.2. Diffuse emission
Diffuse polarized emissionis seenthrough muchofthe mosaic, with polarized intensity levels ranging from 6–8 mJy PSF−1 RMSF−1forthebrighterfeaturesto1mJy PSF−1 RMSF−1forthe faintestidentifedfeatures. Thepresenceof polarization leakage has madequantitative analysisofthe diffuse emission diffcult, as it is not clear how to separate the diffuse emission from the off-source leakage. We have divided the diffuse emission into three regions, based on their morphologies and positions, and discuss eachin turn below.
3.2.1. Bright northwest gradient and southwest patchy emission
Thisregion, showninFig. 9, occupiesthe feldwest(right)of 12hright ascension,andis dominatedbytwobrightfeaturesthat appear connected.The moststrikingfeatureisagradientinFaraday depth that occurs in the northwest part of the feld, from approximately11hto12hinright ascensionand+53◦ to +57◦300 in declination4. This emissionfeature appears atFaradaydepths as low as −7 rad m−2 (Fig. 2, top panel) and can be traced as a single continuous sheet in the cube up to +7 rad m−2 (Fig. 6,
4 Whenreferencing coordinatesto locationsinthe images,itisimpor-tant to recall that this is not a rectilinear projection; the projection of equatorial coordinatesinto ourimagesisshowninFig. 1.
top panel), withtypical brightness of 6–8 mJy PSF−1 RMSF−1. Between −4 rad m−2 and +3 rad m−2(Fig. 2,bottom panel,to Fig.5,top panel),thisfeature hasthe appearanceofa “traveling flament”, causedbyaverylineargradientinFaradaydepth.We measured locations alongthe “flament” at −3.5 rad m−2 and at +3 rad m−2, andfoundthatthe separationbetween corresponding points was approximately2.5◦,resulting inaFaradaydepth gradient of 2.6 rad m−2 deg−1.
AtFaraday depths around −1 rad m−2, additional emission emergesfrom manylocationsinthewest-mostthirdofthe feld andexpands withincreasingFaradaydepth. Whilethis emission looks patchyand disconnected inindividual slices (particularly at0,+1, and+5radm−2) and in Fig. 9, careful inspection of the full cube shows that the emission is a continuous sheet “wrinkled”inFaradaydepth, with manylocal minima and max-imainFaradaydepth surroundedby a complexwebof emission at intermediateFaradaydepths.Thisisbest seenat+3radm−2 (Fig. 5,top panel) where many“loops” of emission can be seen (surrounding the locations of local Faraday depth maxima or minima); this slice also best shows the extent of the emission, whichextendsthe full heightofthe feld.
Near the region occupied by the linear gradient, a second emissionfeature is also present at higherFaradaydepths. It can be seenat+3radm−2 atthe samelocation occupiedbythe frst feature at −5 rad m−2 (as a result, it cannot be seen in Fig. 9, whichonlyshowsthebrightestFaraday-depthpeakfoundin each pixel).Itis seenovera muchsmaller areathanthe frstfeature, butthis maybe duetothe muchlowerbrightnessofthisfeature, 1–2 mJy PSF−1 RMSF−1.It canbe seenatFaradaydepthsupto +14 rad m−2, and while it has some morphological similarities tothe frstfeature(gradients and local maxima at similar loca-tions),italsohas some signifcantdifferences(anotable absence of emissionin certainregionsthat arebrightinthe frstfeature).
3.2.2. Central sheet
The region between 12h and 14h, shown in Fig. 10, is also dominatedby abrightfeature. Thisfeature appearstobea con-tinuous sheet of emission distributed inFaradaydepthbetween −3.5 and +12 rad m−2, with several local maxima and minima in Faraday depth at various locations and flamentary-looking
A71, page10of 22
C.L.VanEck etal.:LOTSS diffuse
Fig.
9.
Collapsedviewofthewest regionofthe mosaic,wherethepolarized intensityandFaradaydepthofthebrightestfeatureintheFaraday depthspectrumwereusedtodeterminethebrightnessand colorrespectively.Thefeaturesinthe north westand northcenter, markedbythegray box and describedin Sect. 3.2.1, canbe clearly seen as sheetsof polarized emission withFaraday rotationgradients, whilethefeatureinthe southwest shows much more patchy structurein polarized intensity. The instrumental leakage causesbrightStokes I sources to appear as small blue circularfeatures. Thethree crosses markthe locationsofthe spectra showninFig. 12.
emission. The typical brightness is 3–4 mJy PSF−1 RMSF−1. The brightest region occurs at the northern edge of the feld, where emission appears at −3.5 rad m−2 and spreads outwards in all directions withincreasingFaradaydepth, with a typicalbrightness of 6–8 mJy PSF−1 RMSF−1.A second bright spot occurs to the southeast, around (13h150,47◦), with Faraday depths of 5–7 rad m−2.
3.2.3. Southeast “flament”
Thethird region,showninFig. 11isbetween14h and15hright ascension, and between +45◦ and +50◦ declination, and contains another continuousfeature withaverylinearFaradaydepthgradient.Thefeature appearsasaverylongthin flamentinFaraday depthslicesfrom+1to+11radm−2,whichin some slices can be seentoextend over12◦ in length.As withtheotherfeatures,this can be interpreted as a continuous sheet of polarized emission.
Around +9 rad m−2, additional emission can be seen further north, and asFaraday depth increases these merge intothe fl-amentstoward several local maximaatFaradaydepthsbetween +19 and +25 rad m−2. The typicalbrightnessofthisfeatureis 1–2 mJy PSF−1 RMSF−1. Thechangein positionofthe flament was measuredbetween+3 and+11radm−2, andwasfoundtobe 0.6◦, indicatingaFaradaydepthgradientof13 radm−2 deg−1.
3.3. Point sources
In addition to diffuse Galactic polarized emission, polarized emission is also observed from some of the unresolved point sources. The identifcation, measurement, and analysis of these sources is done in a parallel paper(Van Eck et al. 2018), and will not be discussed here. The majority of sources arefound at Faradaydepths between +10and +25 rad m−2, higherthan most of the diffuse emission. This implies that the observed diffuse
A71, page11of 22
emission is from only a portion of the line of sight through the Galaxy, with the more distant parts contributing additional Faradayrotationtotheextragalactic sources.
4. Comparison with other tracers
Previously published studies of Faraday depth cubes have looked at relating the observed diffuse polarization to tracers ofother ISM components. Zaroubiet al. (2015)relatedFaraday depth structure observedin oneLOFAR feldto high-frequency (353 GHz) polarization maps, which trace dust emission, from the Planck mission, and found that the observed flaments in Faraday depth followed the magnetic feld orientation inferred from the Planck polarization. VanEck et al. (2017)suggested that neutral clouds could act as sources of Faraday-thin polar-ized emission (which would not be stronglydepolarized at low frequencies), and associated two observed polarizationfeatures withneutral cloudsinthe local ISM.
WeinvestigatedseveralISMtracersforfeatures correspondingtothosewe seein ourFaradaydepthcube.For each tracer, weregriddedthe data ontothe sameprojection as our feld,for easeof comparison. Thesetracersandtheir sourcesare:
–Hα (integratedovervelocity),from Finkbeiner(2003), with a resolution of1◦;
–
408 MHzradio continuum,from Remazeilles et al. (2015); Haslametal. (1982), with aresolutionof560;
–
Planck thermaldustemission,witharesolutionof600,atareferencefrequencyof545 GHz;all Planck mapstaken fromthe Commander component separationin PlanckCollaborationX
(2016);
–
Planck thermal dustpolarization, witharesolutionof100, ata reference frequency of 353 GHz;
–
Planck synchrotron emission, with a resolution of 600, at the reference frequency of 408 MHz;
–
Planck synchrotron polarization, with a resolution of 400, at the reference frequency of 30 GHz;
–
Planck CO(1–0), witha resolution of 600;
–
Planck free–free emission, witha resolution of 600;
–
integrated H I line emission from the Effelsberg-Bonn HI survey(EBHIS, Winkeletal.2016), with aresolutionof110;
A71, page12of 22
C.L.VanEck etal.:LOTSS diffuse
– dust extinction, calculated from Green et al. (2015) as described below, with a resolution of 420;
–1.4GHzpolarized intensity,fromWollebenetal.(2006),with
a resolution of 360 . The maps of eachofthese tracers are shown inFigs. 14and15. The Planck polarization maps were converted from Stokes Q and U to polarized intensity. The extinction maps were made byusingthe MWDUSTpackage(Bovyetal.2016)togetthe optical extinction as a function of distance, taking the numeri-cal derivative with respect to distance (whichshould serve as a proxyfor dustdensity). Thisproduceda3D data cubeofextinction per unit distance, which was then integrated over selected distance ranges to produce extinction maps for different dis-tances. The distancerangeswerechosentobe 0–75,75–250, and 250–1000pc.The boundary at75pcwaschosenasthe Green etal. (2015)extinctionmodelisoftenpoorlyconstrainedfordis
tances lessthanthis;the boundaryat1000pcwaschosenasno signifcantextinction contributionwas seenbeyondthisdistance; the boundary at 250 pc was chosen as the extinction showed different spatial structure on both sides of this boundary (i.e., the morphologicaldifferencesbetweenthetwowere maximized by placing the boundary at 250 pc). The extinction maps were observedtohavealarge scatterbetween adjacentpixels,sothey were smoothed using a Gaussian kernel with a FWHM of 420 (σ =180)toproducethe maps showninFig.15.
The synchrotron emission and polarization, 408 MHz radio continuum,Hα, free–free emission, andCOemission maps are alllargelyfeatureless,sono comparisonswiththelow-frequency radio polarizationwere possible. TheHI column density,ther-mal dust emission and polarization, andextinctionbetween75 and 250 pc all show similar structure, and the location of this structure does not appeartostrongly correlate withthe position ofthelow-frequency polarized emission.
However,thereisaveryinterestingrelationshipbetweenthe southeastFaradaydepthgradientfeature andtheH I flament in the southeastpartof our feld.Figure11showsthisregionofthe feld, comparing ourFaradaydepthcube withcontours madeby integratingtheH I dataoverthevelocityrangeofthe flament (−46 to −40 kms−1). TheHI flament hasa sharp boundary withminimal emission(at anyvelocity) southofthelowestcon-tour.The lineargradientinFaradaydepthoccurs southoftheHI flament,andatthe locationofthe flamenttheFaradaydepth structure is more complex, with several local maxima, and the polarized intensity is more patchy. This transition from linear gradient to more complicated structure is very closely aligned with the boundary of the flament, including the slight bend around α = 14h30m, δ = +47◦200, which is strong evidence that this is not coincidental positioning. We discuss interpretations ofthisrelationshipin Sect. 5.3.We notethat whilethe south
eastgradient showsthis clear alignment with anH I flament, thisis notthe caseforthe northwestgradient, which does also occurinaregionoflowH I emission but has noH Ifeatures in the area where the gradient transitions to a more complex morphology.
A71, page13of 22
The comparison of our 150 MHz polarization datatothe
1.4GHzpolarizationdatafromWollebenetal.(2006)is interest-ing,but complicatedby a numberoffactors.Firstisthatweare comparingaFaraday depth cube, which can separate out mul-tipleFaradaydepth components,toafrequency-averaged polar-izationmap,whichwillshowa combinationofallFaradaydepth components present. Second, and related to the frst, is that our low-frequency data will notbe sensitivetoFaraday-thick com-ponents while the higher frequency data may include emission from such components (albeit possibly partially depolarized). Third,the1.4GHzdataisata muchlowerresolution,360, so it will likelybe morestronglyaffectedbybeam depolarization.
With those caveats, there are some interesting relationships betweenthe1.4GHzpolarized intensityand someofthefeatures in ourFaradaydepthcube.Severalofthebrighterregionsinthe
1.4GHzdatacorrespondtobrighterregionsintheFaradaydepth cube:thebrighterregioninthe northwestregion,around(11h300 , +57◦),overlaps withthebright,low-Faradaydepth regioninthe northwestgradient(thegreen region inFig. 9);the bright region in the north-center, around (13h, +57◦), overlaps the brightest part of the central sheet (the bright green region at the top of Fig. 10); and the bright region slightly east and south of the center, around (13h300, +49◦), overlaps with the second bright regioninthe centralsheet(redregioninthelowerleftofFig.10). Itis particularly noteworthy that eachoftheseregions correspondstoa maximumora minimuminFaradaydepth,andthat the regions with intermediate, more spatially variable Faraday depthsshowweakerpolarized intensityat1.4GHz.Thisisnot
A71, page14of 22
C.L.VanEck etal.:LOTSS diffuse
aperfect correlation:thebright1.4GHzregioninthe southeast regionofthe feld,around(14h300,+47◦),overlaps withthesteep Faradaydepthgradient(the “flament”inFig.11).
One possible explanation is that this might be depolarization caused by the Faraday depth gradients. However, this is not supportedby theresultsof Appendix A, asthegradientof
2.5 rad m−2 deg−1wouldproduce only a4◦ change in polarization angle acrossthe1.4 GHz beam, so minimal depolarization by the gradient is expected. Based on this, we rule out this explanation.
Finding an explanation becomes more diffcult when the polarized intensities are compared.We estimatethe typicaltotalintensity spectral indexinthisregionis −2.6 (usingthe150 MHz data from Landecker & Wielebinski 1970 and the 1420 MHz data from Reich 1982), so under the simplest assumption of a constant polarizedfraction (perfectlyFaraday-thin emission)the expectedpolarized intensityat150MHzcanbeestimated.Inthe areasoffaintest1.4 GHz polarization,the polarizedbrightness temperatureat1.4GHzis typically20–40mK, which wouldbe equivalentto7–14Kat150 MHz.Correspondingly,thebrighter areas with polarized brightness temperatures of 100–150 mK wouldbeequivalentto 35–50K.Inthefainterregions, wherethe northwestgradientis,the typical polarized intensitywe observe at 150 MHz is about 3–5 K; correspondingly in the brightest regions of the central sheet we observe maximal intensities of about8K.
These values imply we are only detecting roughly 40% of the polarized intensity in the faint regions and about 20% in the brighter regions. The two most straightforward explanations for this are that the “missing” polarized intensity is either in Faraday-thick components and correspondingly depolarized at low-frequencies, or exists on the very large angular scales not observedbyLOFAR.However,it’s notclearwhyeitherofthese effects would cause a higher degree of missing emission in regions of higher polarized intensity.
Inretrospect,itmay notbe surprisingthat manyoftheISM tracers show very little signal or structure. The HETDEX feld was originally selected for high-redshift cosmology observationsintheoptical, andwaschosenfor minimal contamination from Galacticforegrounds.Asaresult, manyofthetracerswe investigated were barelydetected in this region. In this respect, the choice of initial region for LOTSS is somewhat unfortunate for Galactic foreground science, but it also suggests that future investigations using observations in other regions, particularlyat lower Galactic latitudes, will probablyhavemorevisible ISMstructureagainstwhichto comparetheFaradaytomography observations.
5. Interpretation of diffuse emission
The lackof correlationsbetweenthe polarized emission andthe ISM tracers provides constraints on the source of the polarized emission. Belowwe consider separatelythe sourceoftheFaraday thin polarized emission, and the Faraday rotation of that emission.
5.1. Origin of Faraday thin emission
Due to the frequency coverage of our observations, we are insensitiveto polarizedfeatures withFaraday thicknessgreater than about 1.0 rad m−2. This places strong constraints on the ISMstructuresthatwe can observe; VanEck etal. (2017)dis
cussed whichISM conditions could causeFaraday-thinfeatures. Herewe considerthese conditionsinrelationtothe polarization features were observed.
Alocalized enhancementinthe magnetic feldperpendicular tothe lineof sight, suchasina shock, couldproduceaFaradaythinfeature, butwould alsoproducea synchrotronexcess which should be seen intotal intensity. An enhancement inthe degree of orderinthe magnetic feld could alsoproduce suchfeatures. There are also no obvious sources of such enhancements, such as supernovaremnants,inthis feld,sobothofthese possibilities seem unlikely.
Strong localized enhancements or diminishments in the free electron density could alsoproduceFaraday thinfeatures. Enhancements would be associated with objects like shocks or HII regions, neither of which are observed in this feld. Typical locationsoflowfree electron densitywouldbe neutral clouds (associated with the warm neutral or cold neutral ISM phases)orregionsofthehot ionizedISMphase.Thetracersof neutral material(H I,thermaldust,extinction)donotshowany correlation to the observed polarized emission, and the local ISM modelsof Lallementet al. (2014)do not show anyclouds withinthe nearestfew100pcinthisdirection.The Local Bubble isavolumeofhotionized medium(HIM)surroundingtheSun, and may (depending on the path length in this direction) have internal Faraday rotation of less than 1 rad m−2 which would resultinaFaraday thinfeature(VanEcket al.2017).However, thisfeaturewould nothave anyforegroundtoproduceFaraday rotation,sothefeaturewould appearat0± 0.5radm−2and could notbethe sourceofthe emissionfeatures seenatotherFaraday depths. More distantregionsof HIM couldbethe sourceofthat emission,providedthey are not so large asto becomeFaraday thick.
Another possibility is a region where the component of the magnetic feld parallel to the line of sight is very small. If this occurs as a result of a reversal in the sign of the parallel magnetic feldthisis calledaFaraday caustic(Belletal.2011), but lines of sight where the parallel component is very small in some volume but has the same sign in front and behind this volume are also possible. Suchregions are possible in anyphase of the ISM. Bell et al. (2011) showed that Faraday caustics can form sheets of Faraday-thin emission, covering the area of sky with the sign-reversal. While Bell et al. (2011) show
A71, page15of 22
A71, page16of 22
C.L.VanEck etal.:LOTSS diffuse
A71, page17of 22
that Faraday caustics should have a distinctive one-sided tail in the Faraday depth spectrum, they also state that to resolve this structure requires that the ratio between the highest and lowest frequencies must be at least 1.5 (their Eq. (19)), which is not satisfed for our observations. As a result we cannot identify this characteristic feature of Faraday caustics, but future observations with larger bandwidth may be able to do so.
Finally,thesefeatures couldrepresent sharp “edges”inthe intrinsicFaradayspectra, causedbysharptransitionsintheFaradayspectrumamplitudeattheedgesofFaraday thickfeatures. Such transitions wouldhave similar causes astheFaraday-thin features described above: a sharptransition in the free electron density, the strengthof the parallel component of the magnetic feld, or the synchrotron emissivity. However, suchsharpedges tendtobestrongly depolarized(VanEck et al.2017), which in turn requires a very high intrinsic polarization to produce a detectablefeatureintheFaradaydepthcubes.
We are notabletodetermine whichofthese possible causes are the source of the Faraday-thin features we observe. The potential signatures in the other ISM tracers of these conditions are diffcult to measure due to limited sensitivity of the tracer observations,ortothefactthat mostofthetracerdataare line-of-sight integratedratherthantomographic measurements.
5.2. Gradients in Faraday rotation
TheFaradaydepth structure seeninthediffuse emissionshows awealthof complex morphologies, such asthe lineargradients and local minima/maximapreviouslydescribed.Adetailed analysisof allthese individualfeaturesisbeyondthe scopeofthis work, butwedo consider some possible causesoftheFaraday depthgradients.
A gradient in Faraday depth can be caused by a gradient in the three factors that determine the Faraday depth: the free electron density, the parallel magnetic feld, and the (physical) depth oftheFaraday-rotatingvolume.Agradient inthe depth oftheFaraday-rotatingregionwould naturally resultifthe emittingregionwasa sheetthatwasnotperpendiculartothe lineof sight and the rotation occurred directlyin front of the emitting region; if such an emitting sheet was reasonably fat the dis-tancetotheregion,andthe correspondingFaradaydepth, could increase linearlywithposition.Gradientsinthe electron density mightbeexpectedonlarger(kpc) scales,duetoeffectslikethe scale height of free electrons, and also onthe scales of individual clouds (e.g.,atransitionlayerbetweena neutral cloudandan ionized skin or exterior).
The lineargradientinthe northwestfeature (Sect. 3.2.1)con
tains both negative and positive Faraday depths. The simplest explanationforthisisagradientintheparallel componentof the magnetic feld that causes the sign to change from negative to positive. Alternative explanations would require two regions at different distances withoppositelydirected parallel magnetic feldsandan electron densityordistancegradientin oneofthe regions.However,thiswouldproduceaFaraday caustic associated with the reversal in parallel magnetic feld between these tworegions anda correspondingFaraday-thin emissionfeature (which wedo notobserve). Whilethere are manypossible mag-netic feld confgurationsthatwouldproduceagradientinthe parallel component of the magnetic feld, one possible confgu-ration that would be consistent both with the observations and theoretical models(as discussedin Beck etal.1996;Shukurov 2004)is a magnetic flament bent into a loop suchas shown in Fig.16.
Fig.
16.
A schematic diagram of a possible source for Faraday depth gradients, wherea magnetic loop (blacklines)would possessagradientinthe parallel componentofthe magnetic feld andin turnproduce agradient inFaradaydepth. An (unrelated)Faraday-thin emissionfeatureinthebackgroundwouldprovidethe observed sourceofpolarized emission(graylines).
5.3. Southeast Faraday depth gradient-HI correlation
Thetransitionfroma smooth lineargradientinFaradaydepth outsideoftheH I flament, showninFig. 11,toa more com-plicated morphologyoverlappingthe flamentisveryintriguing. Sinceweare ableto observe smooth transitionsintheFaraday depthandpolarized intensityinthe flamentregion,weconclude thattheHIflamentisnotassociatedwiththe sourceofthepolarized emission, and mustlieintheforeground withtherestofthe Faraday-rotatingstructure.Furthermore, sincethe orientationof thegradient so closely aligns with thatofthe flament bound-ary,we concludethatthegradientis most likely relatedto, and co-distant with,theH I flament.
One possible interpretationofthe sourceofthegradientisin terms of an envelope of ionized material surrounding the (predominantly neutral)H I flament.Ifthisisthe case, thenwe expectto seea similargradient ontheother sideofthe flament; unfortunatelythat correspondstotheregion where instrumental leakage from 3C295 overwhelms the real signal. Improvements inthe leakagecalibrationforLOFAR datashould allowfor future investigationofthis possibility.
6. Depolarization by Faraday depth gradients
In addition to constraining possible magnetic feld confgurations, the Faraday depth gradients seen in this feld are noteworthyinthatthey canbe diffcultto observe,requiring specifc observational parameters.At higherfrequencies,itisoftenthe casethattheFaradaydepth resolutionis solowthatgradients inFaradaydepth can onlybe measuredin cases with veryhigh signal-to-noiseratios.However,atlowerfrequenciesthe synthesized beamis typicallylargerthanat higherfrequencies, leading tostronger beam depolarization maskingthe polarized signal and makingthegradients undetectable. Appendix A describes this beam depolarization, and gives an approximate threshold for the conditions under whichthe depolarization becomes signifcant: polarization anglegradientsgreaterthan180◦ over the FWHM of the beam will be strongly depolarized, and gradients shallower than this may also be signifcantly depolarized depending ontheexact beam parameters.
A71, page18of 22
C.L.VanEck etal.:LOTSS diffuse
We can usethisthresholdto determinethe depolarization causedby theFaradaydepthgradientsin our observations.For our observations,withaFWHMof4. 03andwavelengthsbetween
1.9and2.5m,a polarizationanglegradientof180◦ FWHM−1 would be created by a Faraday depth gradient of 7.5–13 rad m−2 deg−1. In the HETDEX feld, we observed two verylinear gradients in Faraday depth, with values of 2.6 rad m−2 deg−1 (Sect. 3.2.1), which is well inside the predicted range of minimal depolarization, and 13 rad m−2 deg−1 (Sect. 3.2.3), at the upper limitofthisrange. Beam depolarizationby thegradient mayexplainthelower polarized fux densityofthesteepergradient comparedtotheother polarizedfeatures.Forthesegradients, the Gaussian modelpredictsthatthe shallowergradientwouldbe detectedat87%ofitsintrinsic intensityandthesteepergradient at 3%. Our synthesized beam likely falls somewhere between the two models, implying that the measured brightness of the shallower gradient is probably accurate within approximately 10% whilethesteepergradientisprobablysomewherebetween 1–30 timesbrighterthan our measurement.
We can also consider the detectability of Faraday depth gradients with other low-frequency radio observations. Lenc et al. (2016) observedfaint diffuse emission withthe MWA, at154 MHz(λ =1.95 m) with a resolution of 540 and similar sensitivity as our observations. With the simple beam model of Appendix A, the depolarization threshold corresponds to a Faradaydepthgradientof 0.9radm−2 deg−1.However,theyalso applieda Gaussian(u,v)tapertotheir data, whichprobablyhas theeffectof makingtheir beam more Gaussian-like and increasingthe depolarization present. These observations are probably stronglyaffectedby beam depolarizationduetoFaradaydepth gradients; the majority of the structure seen in the HETDEX feld could not be detected with these observational parameters. Observing such features may be possible with the MWA byincluding the longer baselines and minimizing the effects of the Gaussiantaper.
There are plansto usethe data fromthe CanadianH I Map-ping Experiment (CHIME) to measure polarized foregrounds. CHIME will operate at 400–800 MHz, with a resolution of approximately 300. The Faraday depth gradient threshold for these parametersvariesfrom11–45radm−2 deg−1 fromthebottomofthefrequency bandtothetop, indicatingthatthese observations should beless affectedbygradient beam depolarization than ourLOTSS observations.
The sensitivityof observationstosteepFaradaydepthgradients may be a consideration in planning future observations, as it may have strong consequences on the interpretation of suchobservations.Ifpolarizedfeatureswith steepgradientsare presentinan observedregionofsky,they may stronglydepolarizedandthusnotdetected. Thiswould leadtoan underestimate in the amount of polarized fux present, and could introduce a bias into anyinferred properties of the diffuse polarization and ISM magnetic feld. Also, polarizedfeaturesthat includeregions ofbothshallow andsteepgradientswouldhave missingregions, which could cause large connectedFaraday depth structuresto be interpreted as separatefeatures.
We have not used the full resolution available with the LOTSS data, so a reprocessing of these data at higher resolution could improve the sensitivity to steeper Faraday depth gradients.Atthe sameresolution asthe initialLOTSSStokesI data products, 2500,thethresholdfor depolarization increasesto 72–125 rad m−2 deg−1 at the bottom and top ends of the frequency band.However,the higherresolutionresultsina correspondingly lower sensitivity to diffuse emission, making it diffcultto selecta compromisebetween maximizing sensitivity to diffuse emission while preventing beam depolarization due to unresolved polarization anglegradients. Higherresolution also signifcantly increases the computational and data storage requirements of such reprocessing, so future surveys will need to balance computational expense against sensitivity to steeper Faradaydepthgradients.
7. Conclusions
We have used 60 8-h observations from the LOFAR TwometerSky SurveytoperformFaradaytomography coveringthe HETDEXSpring feld(right ascensionfrom10h30mto15h30m and declination from 45◦ to 57◦). We have produced a mosaic Faradaydepthcubeofthisregionat4. 03resolution, whichshows polarized emission as a function of Faraday depth. The low-frequency natureof ourdatagivesusaFaradaydepthresolution of1 rad m−2, allowing us to probe very small variations in Faraday depth. We achieve a typical sensitivity of 50–100 µJy PSF−1 RMSF−1.
In our Faraday depth cube we see diffuse polarized emission across most of the region, at Faraday depths between −7 and +25radm−2. This diffuse emission mostlytakestheformof “sheets”, where the emission appears to be flamentary at any single Faraday depth but can be seen as a continuous feature distributed smoothly over Faraday depth. We are able to map out several of these sheets in different positions in the region. Afewofthese showvery linearfeatures, where emissionover severalsquaredegreeshasa smooth lineargradientinFaraday depth.
We compared our Faraday depth cube with several other tracers of different ISM components, andfound two interesting relationships.OneofthelargelinearFaradaydepthgradientsfollowstheedgeofanH Iflament,withthegradient occurringjust outsidethe observededgeofthe flamentwiththeFaradaydepth decreasing with distancefromthe flament.We alsofoundthat the locationsof maximaand minimaintheFaradaydepthofthe featureswe observe tendtooverlap withregionsof high21-cm polarized intensity.
MotivatedbytheFaradaydepthgradients seenin our observations, we considered beam depolarization caused by the polarization angle gradients produced by such Faraday depth gradients. Many previous authors have considered the depolarization effects of a Gaussian PSF, which produces very strong depolarization. Wehave shownthat an idealized PSFfor an interferometer withuniform (u,v) coverage causes much weaker depolarization for some cases.Weexpect that mostrealistic observations willfall betweenthese two cases, and suggestthat Faradaydepthgradients maybe more likelyto be detectedthan previously expected.
We have shown that the LOTSS data are well suited for Faradaytomography. However,thequality oftheFaradaydepth cubes could be improved signifcantlybyperforming additional processing to remove the instrumental polarization leakage, forexamplethroughthe use of complex CLEAN(Pratley& Johnston-Hollitt2016)toremoveimage-plane sidelobes.Itmay also be worthwhile to explore higher (angular) resolution, as this could increase sensitivity to emission with stronger Faraday depth gradients. At the time of this work, LOTSS data withdirection-dependent calibrationwas not availablefor most observations. Early investigations into the effects of direction-dependent polarization onthe polarization properties suggest that there is a modest improvement due to fewer image-plane sidelobes of the instrumental leakage around bright sources
(S. O’Sullivan, priv. comm.), which is encouraging for future
A71, page19of 22
polarization work withLOTSS data.LOTSS will observethe entire sky north of declination zero, so a Faraday tomography survey, with excellent sensitivity and resolution (both image-planeandinFaradaydepth),will soonbe possible.
Acknowledgements. C.V.E.would liketothank Justin Bray and JeroenStilfor their helpful comments on the derivation in the appendix. This work is part oftheresearchprogram 639.042.915, whichis(partly) fnancedby theNetherlandsOrganisationfor ScientifcResearch(NWO).LOFAR,theLowFrequency Array designed and constructed by ASTRON, has facilities in several countries,that areownedby various parties (each with theirown funding sources), andthat are collectivelyoperatedby the InternationalLOFARTelescope(ILT) foundation under a joint scientifc policy. This research used ionospheric TEC maps produced by the Centre for Orbit Determination in Europe (CODE, http://aiuws.unibe.ch/ionosphere/). SomeoftheISMtracermapswere downloadedfromtheLegacyArchivefor Microwave Background Data Analysis (LAMBDA), part of the High Energy Astrophysics Science Archive Center (HEASARC). HEASARC/LAMBDA is a service of the Astrophysics Science DivisionattheNASAGoddardSpaceFlight Center5.Thisresearchmadeextensive useofAstropy,acommunity-developed corePythonpackageforAstronomy (AstropyCollaboration 2013); SciPy(Joneset al. 2001);NumPy(van derWalt et al. 2011); IPython(Pérez& Granger 2007); matplotlib(Hunter 2007);the CommonAstronomySoftware Applications(CASA, McMullinetal. 2007); and theKarma visualizationtools(Gooch1996).
References
AstropyCollaboration(Robitaille,T.P.,etal.)2013, A&A, 558, A33 Beck, R., Brandenburg, A., Moss, D., Shukurov, A., & Sokoloff, D. 1996,
ARA&A,34,155 Bell,M.R., Junklewitz,H.,&Enßlin,T.A.2011, A&A,535,A85 Beresnyak,A.,&Lazarian,A.2015, Astrophys. Space Sci. Lib.,407,163 Bernardi,G.,Greenhill,L.J., Mitchell,D.A.,etal.2013, ApJ,771,105 Boulares,A.,&Cox,D.P.1990, ApJ, 365,544 Bovy,J., Rix,H.-W.,Green,G.M.,Schlafy,E.F.,&Finkbeiner,D.P.2016,
ApJ,818,130 Brentjens,M.A.,&deBruynA.G. 2005, A&A,441,1217 Finkbeiner,D.P. 2003,ApJS,146,407 Gooch,R.1996,inAstronomical Data Analysis Software andSystemsV, eds.
G.H. Jacoby&J. Barnes, ASP Conf. Ser.,101,80 Green,G.M.,Schlafy,E.F.,Finkbeiner,D.P.,etal.2015, ApJ,810,25 Hales, C.A., Gaensler, B. M.,Norris, R.P.,&Middelberg, E. 2012, MNRAS,
424,2160 Haslam, C. G. T., Salter, C. J., Stoffel, H., & Wilson, W. E. 1982, A&AS,
47,1 Heald,G.,Braun,R.,&Edmonds,R.2009, A&A,503,409 Hill,G.J., Gebhardt,K.,Komatsu,E.,etal. 2008,inPanoramicViewsof Galaxy
FormationandEvolution, eds.T.Kodama,T.Yamada,&K.Aoki,ASP Conf. Ser., 399,115 Hill,A.S., Landecker,T.L.,Carretti,E.,etal.2017, MNRAS,467,4631 Hunter,J.D. 2007, Comput. Sci.&Eng.,9,90 Iacobelli,M.,Haverkorn,M.,&Katgert,P.2013, A&A,549, A56 Jeli´
c,V.,deBruyn,A.G.,Mevius,M.,etal.2014, A&A, 568,A101 Jeli´
c,V.,deBruyn,A.G.,Pandey,V.N.,etal.2015, A&A, 583,A137 Jones, E., Oliphant,T.,Peterson,P.,et al. 2001, SciPy: Open source scientifc
toolsfor Python,www.scipy.org
Lallement,R.,Vergely, J.-L.,Valette,B.,etal.2014, A&A,561,A91 Landecker,T.L.,&Wielebinski,R.1970, Aust.J.Phys.Astrophys. Suppl.,16,1 Lenc, E., Gaensler, B. M., Sun, X. H., etal. 2016, ApJ, 830, 38 Macquart, J.-P., Ekers, R. D.,Feain, I.,&Johnston-Hollitt, M. 2012, ApJ,750,
139
McMullin,J.P.,Waters,B.,Schiebel,D.,Young,W.,&Golap,K.2007,inAstronomicalData AnalysisSoftwareandSystemsXVI, eds.R.A.Shaw,F. Hill, &D.J. Bell,ASP Conf. Ser.,376,127
Pandey,V.N.,van Zwieten,J.E.,deBruyn,A.G.,&Nijboer,R. 2009,inThe Low-FrequencyRadioUniverse, eds. D. J. Saikia, D.A. Green,Y. Gupta,&
T.Venturi,ASP Conf. Ser.,407, 384 Pérez,F.,&Granger,B.E. 2007,Comput. Sci. Eng.,9,21 PlanckCollaboration X. 2016, A&A, 594, A10 Pratley,L.,&Johnston-Hollitt,M.2016, MNRAS,462,3483 Reich,W.1982,A&AS, 48,219 Remazeilles, M., Dickinson, C., Banday,A. J., Bigot-Sazy, M.-A.,&Ghosh,T.
2015, MNRAS, 451, 4311 Schnitzeler,D.H.F.M.,&Lee,K.J.2015, MNRAS,447,L26 Schnitzeler,D.H.F.M.,Katgert,P.,&deBruynA.G. 2007, A&A,471,L21 Schnitzeler, D. H. F. M., Katgert, P., & de Bruyn A. G. 2009, A&A, 494,
611 Shimwell, T. W., Rttgering, H. J. A., Best, P. N., et al. 2017, A&A, 598,
A104 Shukurov,A. 2004, ArXivAstrophysics e-prints [arXiv:astro-ph/0411739] Sokoloff,D.D.,Bykov,A.A.,Shukurov,A.,etal.1998, MNRAS,299,189 Spinrad,H., Marr,J.,Aguilar,L.,&Djorgovski,S.1985, PASP,97, 932 Tasse,C.,vanderTol,S.,van Zwieten,J.,van Diepen,G.,&Bhatnagar,S.2013,
A&A, 553, A105 Tingay,S.J., Goeke,R.,Bowman,J.D.,etal.2013,PASA,30,e007 Tribble,P.C.1991,MNRAS, 250,726 van derWalt, S., Colbert, S. C.,&Varoquaux, G. 2011, Comput. Sci. Eng.,13,
22 Van Eck, C. L., Haverkorn, M., Alves, M. I. R., etal. 2017,A&A, 597, A98 Van Eck, C. L., Haverkorn, M., Alves, M. I. R., et al. 2018, A&A, 613,
A58 van Haarlem,M.P.,Wise,M.W., Gunst,A.W.,etal.2013, A&A, 556,A2 van Loo, S., Hartquist,T.W.,&Falle, S.A. E. G. 2012, Astron. Geophys., 53,
5.31 Winkel,B.,Kerp,J., Flr,L.,etal.2016,A&A,585,A41 Wolleben,M., Landecker,T.L.,Reich,W.,&Wielebinski,R.2006,A&A,448,
411 Wolleben,M., Landecker,T.L.,Carretti,E.,etal.2008,Proc.Int.Astron.Union, 4, 89
Zaroubi, S., Jeli´
c,V.,deBruyn,A.G.,etal.2015, MNRAS,454,L46
5 https://lambda.gsfc.nasa.gov/
A71, page 20 of 22
C.L.VanEck etal.:LOTSS diffuse
Appendix A: Beam depolarization in a linear Faraday depth gradient
GradientsinFaradaydepthwith respectto positiononthesky resultin correspondinggradientsinthepolarizationangle(ifthe intrinsic emitted angles are uniform across the source), which in turn produce beam depolarization when observed with a fniteresolution.Hereweexplorethe depolarization causedby Faradaydepthgradientsby consideringthe caseofan infnite, uniform background emission source, witha(foreground)gradient in Faraday depth, corresponding to the sheets of emission and Faraday depth gradients we observe in our data. When a Gaussian beam is assumed, an analytical solution can befound, as describedin Sokoloff etal. (1998).However, morerealistic synthesized beams have not been considered in the literature, and below we show that this can make a signifcant difference intheexpected depolarization.
Consider a gradient in Faraday depth with respect to an angular position variable x,dφ . The resulting intrinsic (sky)
dx
polarization,inthe complex-polarization notation6,is
!
˜˜
Psky(x, y, λ) = P0 exp 2iλ2dφ x (A.1)
dx
where λ is the observing wavelength and P˜0 is the intrinsic emitted polarizationpriortoFaraday rotation7. The observed polarization isthe convolution ofthe intrinsic polarization with the telescope beam. In the case of a circular Gaussian beam
1 +y
( exp(− x22σ2 2 )), the convolution integral can be solved ana
2πσ2
lytically,resultinginthepreviouslyknownresult
!⎛ !2⎞ P˜obs(x, y, λ) = P˜0exp 2iλ2dφ x exp ⎜
⎝−2 λ2dφσ ⎟
. (A.2)
⎠
dx dx
The frst part of this equation is identical to the intrinsic polarization and contains allthe position dependence andthe only complex terms, which meansthatthe correct polarization angle,Faradaydepth, andFaradaydepthgradient arerecovered, butthe polarized intensityis modifedby the later partsofthe equation.The secondpartoftheequationgivesthe depolarization, and showsthatthe depolarizationisavery strong function ofthewavelength,gradient, and beam size.
However, a Gaussian is often not an accurate representationofthe synthesized beam, depending onthe(u,v)coverage of the instrument used. Previous calculations such as Tribble (1991), while not for the exact same model, have shown that Gaussian functions tendto cause much stronger depolarization thanother functions.Withthis motivation,wehavedevelopeda moregeneralformulationfor beam depolarizationofgradients.
Thegeneral case, andthe more naturalformulation for interferometric observations, canbefoundsimplyby exploiting properties of Fourier transforms. Since the observed polarizationisthe convolutionofthesky polarization withthe telescope beam,we can movetotheFourier domain wherethe convolution becomesa multiplication andthe beam becomesthe(u,v) coverageofthe observation:
F{ P˜obs(x,y)} = F{ P˜sky(x,y) ∗ Beam(x,y)}, (A.3)
= F{ P˜sky(x,y)}F{Beam(x,y)}, (A.4)
6 We use a tilde to denote complex quantities, specifcally phasors describing linear polarization. 7 For conveniencewehave alignedthegradient withthex-axis and set φ =0at x =0, butthese are arbitraryanddo notaffectthe fnalresult.
2λ2dφ !
= P˜0 δ u − δ(v) W(u,v), (A.5)
2π dx
where W(u,v)isthe samplingweight functioninthe (u,v) plane, and in the last step we have used the defnition of P˜sky(x) from Eq.(A.1)andrecalledthattheFouriertransformofa complex exponential is a suitablyshifted Dirac delta function. Inverting theFouriertransformtorecoverthe observedskydistribution
gives:
˜Pobs(x, y) = = ( ! )2λ2 dφ F −1 ˜P0 δ u − δ(v) W(u, v), 2π dx ! ! λ2 dφ ˜P0 exp 2iλ2 dφ x W , 0 . dx π dx (A.6) (A.7)
Tothispoint,theresultisgeneralandworksforanysampling function; using GaussianweightsgivestheresultfromEq.(A.2). The correctFaradaydepth,andbyextensionthegradientinFaradaydepth,isrecoveredatall positions,butthepolarized intensity is modifedbytheweighting functionatthe locationofthegradient inthe(u,v)plane. As a result, gradients at positions in the(u,v)plane that are not sampled bythe instrument or that havebeendown-weighted signifcantlywillhavezeroorstrongly reducedpolarized intensity and maynotbedetected;for suchan idealizedgradientthe beam depolarization depends only onthe weight function.
Thesteepestgradientthat canbe measuredforagiven observation can be determined from three parameters: the observing wavelength, λ,the largestsampled (u,v) coordinate, UVmax (nor
dφ
malizedbywavelength),andtheFaradaydepthgradient, dx .We foundthatwe couldreducethisto one parameterbyconverting our model into scale-free units: we replaced the UVmax param-eter with the synthesized beam full-width half max (FWHM), whichis defned as FWHM = UVmax −1;replaced the λ2dφ term
dx
in the intrinsic polarization with a single variable, the polar-ization angle gradient (with units of radians of rotation per beam FWHM); and expressed the position variable x in terms ofthe FWHM. Theresulting conditionfor being observableis
d(φλ2)
FWHM <π, or equivalently less than a 180◦ change in
dx
polarization angle acrossthe FWHMofthe synthesized beam.
ToderiveEq.(A.7),we assumedthatthe sourceofthe polar-ized emission has an infnite extent, so that it shows up in the (u,v)planeatasingle point.Real sources, whichhavea fnite extent, cover an area inthe uv-plane, not a single point.A more general argument can be constructed from Fourier properties, using the ideal case as a guide. Consider a polarized intensity feature of arbitrary morphology and angular size (and its corresponding Fourier transform to the (u,v) plane), and apply a linearFaraday depthgradient oftheform in Eq.(A.1)which has the effect of adding a position dependent phase modulation.FromtheFourier shift-modulationtheorem,the neteffect istoshifttheFouriertransformofthe polarizedfeatureinthe (u,v) plane, with the length and orientation of the shift determined by the steepness and orientation of the Faraday depth gradient. This shift can cause some or allofthe polarizedfeature to fall outside of the (u,v) coverage of the observation, causing depolarization.
Thisresult mirrors thatof Schnitzeler et al. (2009), who usedthe samegradient model andFourierpropertiesto derive the corresponding result for the shallowest gradient that can be detected, given the shortest baseline present in an observation. These canbe interpretedtogether:the longestbaseline sets thesteepestFaradaydepthgradientthat canbe observed, while
A71, page21of 22
the shortest baseline sets the shallowest gradient that can be observed. Single-dish observations, which measure around the centerofthe (u,v) plane, are sensitiveto large polarizedfeatures withsmallFaradaydepthgradients.
It should also be noted that this derivation describes beam depolarization in a single frequency channel. This depolarizationmayhaveaverystrongfrequency dependence;asfrequency decreases the location of the gradient in the (u,v) plane will move outwards to larger (u,v) distances, scaling as the square of the wavelength, while a baseline of a fxed length moves inwardsproportionallytothewavelength.Foragiven observation there will be a range of gradients that can be detected in a fraction of the observed bandwidth, producing a complex frequency-dependence which may introduce artifcial structure into the corresponding Faraday depth spectrum. Determining the specifceffectsofa“partialdetection”ofagradientisbeyond the scopeofthis derivation.
Overall,we fndthreeimportantresultsfromthis derivation. First,wehave identifedakeyfgureof meritforcharacterizing Faradaydepthgradientsforthepurposeofevaluatingtheir depo-larizationwithinagiven observation:thechangeinpolarization angle acrossthe FWHM ofthe synthesized beam, withgradients steeperthan180◦ per beam FWHM being beyondthe (u,v) coverageofthe observation
Second, beam depolarization may often not be as strong as may have been previously predicted using a Gaussian beam model. Justbelowthethresholdof180◦ per FWHMthe depolarizationof an idealgradientby the Gaussian beamwould leave only 2.9% of the original polarized fux, whereas a uniform-weighted observation withthe same beam FWHMwould notbe depolarized. Theeffects willbe lessextremefora morerealistic gradient, dependingontheweightingofthe(u,v) coverageofthe observation andthe angularextentofthegradient.
Third, and perhaps most importantly, this suggests that the choice of (u,v) weighting in the imaging process can have very strongeffectsonthe measuredpolarization.Afullstudyofthis effect is beyond the scope of this work, but these simple mod-els imply that the use of Gaussian tapers and similar weighting schemes that highly down-weight longer baselines may cause signifcant depolarization;the useof suchtapers shouldbe con-sidered cautiouslytoavoidremoving possible polarizedfeatures of interest.
A71, page 22 of 22