Magnetic arms of NGC6946 traced in the Faraday cubes at low radio frequencies
Anton Chupin, Rainer Beck, Peter Frick, George Heald, Dmitry Sokoloff, Rodion Stepanov
aa r X i v : . [ a s t r o - ph . I M ] A ug Astronomische Nachrichten, 17 August 2018
Magnetic arms of NGC6946 traced in the Faraday cubes at low radiofrequencies
A. Chupin ⋆ , R. Beck , P. Frick , G. Heald , D. Sokolo ff , R. Stepanov Institute of Continuous Media Mechanics, Korolyov str. 1, 614061 Perm, Russia MPI f¨ur Radioastronomie, Auf dem H¨ugel 69, 53121 Bonn, Germany CSIRO Astronomy and Space Science, 26 Dick Perry Avenue, Kensington, WA 6151, Australia;ASTRON, the Netherlands Institute for Radio Astronomy, Postbus 2, NL-7990 AA Dwingeloo, the Netherlands;Kapteyn Astronomical Institute, University of Groningen, PO Box 800, NL-9700 AV Groningen, the Netherlands Department of Physics, Moscow University, 119992 Moscow, Russia Perm State National Research University, Perm, RussiaReceived XXXX, accepted XXXXPublished online XXXX
Key words methods: data analysis – techniques: polarimetric – galaxies: magnetic fields – turbulenceMagnetic fields in galaxies exist on various spatial scales. Large-scale magnetic fields are thought to be generated by the α − Ω dynamo. Small-scale galactic magnetic fields (1 kpc and below) can be generated by tangling the large-scale field orby the small-scale turbulent dynamo. The analysis of field structures with the help of polarized radio continuum emissionis hampered by the e ff ect of Faraday dispersion (due to fluctuations in magnetic field and / or thermal electron density)that shifts signals from large to small scales. At long observation wavelengths large-scale magnetic fields may becomeinvisible, as in the case of spectro-polarimetric data cube of the spiral galaxy NGC 6946 observed with the WesterborkRadio Synthesis Telescope in the wavelength range 17–23 cm. The application of RM Synthesis alone does not overcomethis problem. We propose to decompose the Faraday data cube into data cubes at di ff erent spatial scales by a wavelettransform. Signatures of the “magnetic arms” observed in NGC 6946 at shorter wavelengths become visible. Our methodallows us to search for large-scale field patterns in data cubes at long wavelengths, as provided by new-generation radiotelescopes. Copyright line will be provided by the publisher
Magnetic fields of nearby galaxies are quite well investi-gated. The observational results are compatible with a sce-nario of magnetic field excitation by a galactic α − Ω dynamo(for a review see e.g. Beck et al. 1996). The main bulk ofthese observations is obtained from analysis of galactic po-larized radio continuum radiation observed with the currentgeneration of radio telescopes, such as the E ff elsberg 100-mdish and the Very Large Array (VLA).Galactic magnetic fields are important for the evolutionof galaxies, the astrophysics of the interstellar medium, cos-mic ray propagation (e.g. Pshirkov et al. 2011), as well asof fundamental interest. In particular, observations demon-strate prominent magnetic structures in the form of mag-netic arms that are usually situated between material arms,as in NGC 6946 (Beck 2007). The relation between gas andmagnetic fields can also be more complicated, like e.g. inIC 342 (Beck 2015) and in M 83 (Frick et al. 2016). The ver-ification of various scenarios for the generation of such finemagnetic structures (e.g. Chamandy et al. 2013; Moss et al.2015) is limited by technical abilities (angular resolution ⋆ E-mail: [email protected] and sensitivity) of current radio telescopes. Further progresscan be expected with the new generation of radio telescopes,the European Low Frequency Array (LOFAR), the AustraliaSKA Pathfinder (ASKAP) and the South-African Karoo Ar-ray Telescope (MeerKAT), allowing high-resolution, multi-channel polarimetric observations.An important e ff ect for the quantification of mag-netic fields is Faraday rotation of polarized radio ra-diation that requires multi-wavelength observations (e.g.Ruzmaikin & Sokolo ff ff elsberg and VLA telescopes (3–20 cm)increases the accuracy of the measurements substantially(Beck et al. 2012). On the other hand, increasing the obser-vation wavelength leads to more severe Faraday depolariza-tion e ff ects, which complicates the extraction of physicallyvaluable information on magnetic fields from observationaldata (Burn 1966; Sokolo ff et al. 1998).Faraday depolarization, in particular Faraday dispersiondue to small-scale fluctuations in magnetic field and ther-mal electron density within the emitting medium or in the Copyright line will be provided by the publisher
A. Chupin et al.: Magnetic arms traced in the Faraday cube
Faraday-rotating medium in the foreground (Sokolo ff et al.1998), can already be strong at wavelengths of around20 cm. Maps of polarized radio emission obtained with theWesterbork Synthesis Radio Telescope (WSRT) for many(21) galaxies (Heald et al. 2009) do not reveal (at least not ina straightforward way) many structures that are well knownfrom observations at shorter wavelengths. In particular, themap of Faraday depths (a measure of the integral of theproduct of thermal electron density n e and magnetic fieldstrength B along the line of sight) of NGC 6946 obtained byHeald et al. (2009) in the wavelength range 17–23 cm (con-sisting of two bands of 17.0–18.4 cm and 20.9–23.1 cm)(Fig. 1, bottom) shows small-scale fluctuations superim-posed on a large-scale gradient and does not directly showthe well-defined “magnetic arms” known from a previousstudy at λ . B n e d λ (where B is the average strengthof the magnetic field along the line of sight, d is the coher-ence scale of the magnetic field, n e is electron density and λ the wavelength) are generally larger than π at λ ≃
20 cm.Fluctuations in B and n e lead to strong gradients in themaps of Stokes Q and U and hence to shifting the signals ofpolarized intensity from large to small angular scales. Thepower spectrum of polarized intensity becomes flatter (e.g.La Porta & Burigana 2006). The imprints of the large-scalefield remain recognizable at smaller spatial scales. By usingwavelets we can isolate small-scale magnetic fields obtainedas the result of decay of the large-scale ones and then rec-ognize the locations of large-scale fields.We will test our method on the observational data forNGC 6946 by Heald et al. (2009). The magnetic arm con-figuration in this galaxy is known from observations at shortwavelengths (Beck 2007), allowing us to verify the results. We base our analysis on observation of linearly polarizedradio continuum emission of NGC 6946 in the range 17–23 cm obtained as part of the Spitzer Infrared Nearby Galax-ies Survey (SINGS) by the Westerbork Synthesis RadioTelescope (WSRT) with an angular resolution (beam size)of 15 ′′ × ′′ (Heald et al. 2009) (Fig. 1). The data is avail-able in the form of cubes of Stokes parameters Q and U in803 frequency channels. The scale is 4 ′′ per pixel. Imagesat 11 cm from E ff elsberg and at 6 cm combined from data ¢ ² ¢ ² ¢ ² ¢ ² ¢ ² ¢ ² é ¢ é ¢ é ¢ é ¢ é ¢ Right ascension H - L D e c li na t i on
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Fig. 1
Observations of NGC 6946 in the wavelengthrange 17–23 cm and results of RM Synthesis: top - distri-bution of a (normalized) peak intensity | F max | of the Fara-day spectra with overlayed red contours at 15% of the max-imum, bottom - distribution of Faraday depths φ max (inrad m − ) at which the maximal values of | F max | are obtained,according to Heald et al. (2009).from the VLA and E ff elsberg radio telescopes (Beck 2007)are used for comparison.RM Synthesis has been introduced byBrentjens & de Bruyn (2005) for multichannel spectro-polarimetric data (data cube with two sky dimensions andwavelength) obtained by modern radio telescopes thatprovide observations of polarized intensity (via the Stokesparameters Q and U ) over a wide range of wavelengths λ .From the data cube RM Synthesis calculates the “Faradaydispersion function” F (also called “Faraday spectrum”)for each pixel on the sky plane, obtaining the Faraday cube F ( φ, l , b ) = π Z λ max λ min P ( λ , l , b ) e − i φλ d λ , (1)where φ is the Faraday depth, l and b are the sky coor-dinates of the pixel, and λ min and λ max define the rangeof available wavelengths. F is measured in units of Jan- Copyright line will be provided by the publisher sna header will be provided by the publisher 3 sky per half-width of the telescope beam and per half-width of the “Faraday beam” (Faraday Point Spread Func-tion, see Brentjens & de Bruyn (2005)). We use a range of − < φ < − which should be su ffi cient todetect all possible sources.The Faraday depth cube generated by the transform(1) requires additional e ff orts to recognize magnetic fieldstructures. A simple algorithm (exploited in particular byHeald et al. (2009)) is as follows. Along each line of sightthe value F max with a peak intensity at the Faraday depth φ max is found and stored at the corresponding pixel on thesky plane | F max ( l , b ) | = max φ | F ( φ, l , b ) | = | F ( φ max , l , b ) | , (2)where F max is measured in units of polarized intensity. Themaps of | F max | and φ max on the sky plane ( l , b ) of NGC 6946are shown in Fig. 1.A well-known tool for structure recognition and scale-by-scale data analysis is wavelet analysis, widely used inthe interpretation of galactic images (e.g. Frick et al. 2001;Patrikeev et al. 2006; Tabatabaei et al. 2013). The wavelettransform can be used for optimizing RM Synthesis (seeFrick et al. 2010, 2011). In this work, wavelets are used asa spatial-scale filtering tool.The wavelet transform of a 2D image f ( l , b ) produces a3D data cube (two coordinates l and b plus scale a ) w a ( l , b ) ≡ W { f ( l , b ) } == a Z ∞−∞ f ( l ′ , b ′ ) ψ ∗ l ′ − la , b ′ − ba ! dl ′ db ′ , (3)where ψ ∗ is the complex conjugated analysing wavelet andscale a is the characteristic radius of the wavelet func-tion. a is related to the full width at half power Θ as Θ = √ ln a ≃ . a .The wavelet decomposition of the Faraday spectrum forany fixed Faraday depth can be performed as w φ a ( l , b ) = W { F ( φ, l , b ) } . (4)As an example we show the wavelet decomposition ap-plied to the map of polarized intensity at a Faraday depth of φ =
40 rad m − and at three di ff erent scales a = , , ′′ in Fig. 2. The Faraday depth φ =
40 rad m − is the averagevalue of depth in Fig. 1, bottom. This is the Faraday rotationof the Galactic foreground. The patterns of ordered struc-tures (magnetic arms) are best visible at the scale a = ′′ ,which will be used and discussed below.In a second test, we apply the wavelet transform to themap of F max (Fig. 1, bottom) as˜ w a ( l , b ) = W { F max ( l , b ) } . (5)The normalized modulus of ˜ w a ( l , b ) for same three scales a = , , ′′ is shown in Fig. 3. All wavelet transformsare made using an isotropic analyzing wavelet, called the Mexican hat ψ ( x , y ) = (2 − x − y ) e − ( x + y ) / . We concludethat the magnetic arms are not clearly present in these plots.In general, the wavelet transform of the Faraday cuberesults in a 4D data array w a ( φ, l , b ), which characterizesthe intensity of structures of scale a at Faraday depth φ ata given pixel ( l , b ). The key point of our approach is thatwe first apply the wavelet transform (4) to map at each fre-quency F ( φ, l , b ), so we obtain one data cube per waveletscale, and then calculate the maximum intensity value of w a ( φ, l , b ) along each line of sight (similar to F max , but for agiven scale a ) | w max a ( l , b ) | = max φ | w φ a ( l , b ) | , (6)where the maximum is taken over the whole consideredrange of φ .The normalized modulus of w max a ( l , b ) for scales a = , , ′′ is shown in Fig. 4. At a = ′′ (corresponding toabout twice the beam size of the observations) the magneticarms have the higher contrast in comparison with ˜ w a ( l , b ) (inFig. 3). The structures enhanced in the w max16 ′′ ( l , b ) map arealso visible in the ˜ w ′′ ( l , b ) map. However, the ˜ w ′′ ( l , b )map has lot of additional structures of similar or even higherintensity that are spread over a much larger region.The di ff erence between the two approaches, i.e. between˜ w a ( l , b ) and w max a ( l , b ), is mathematically the inverse orderof operations taking the maximum over φ and the waveletdecomposition (schematically shown in Fig. 7). This di ff er-ence yields additional information on the structure of large-scale magnetic field, as discussed in the next Section. The application of RM Synthesis followed by the determi-nation of maxima of the Faraday spectra at each pixel in thesky plane at any Faraday depth does not clearly reveal anyelongated arm structures (Fig. 1, top). On the other hand, ifwe first apply a wavelet decomposition which isolates struc-tures at a given scale and then determine the maxima of theFaraday spectra at each pixel, the resulting image revealspronounced structures that are invisible when applying RMSynthesis only. We clearly see from Fig. 4, middle that theisolated structures are organized in the form of spiral arms.The elongated arms consist of a set of local maxima alongthe arms.Comparing the middle plots of Fig. 3 and Fig. 4, we findthat w max a ( l , b ) has a higher contrast between arm and inter-arm regions than ˜ w a ( l , b ). We measure the contrast c as fol-lows. We divide the image into the magnetic arm and inter-arm space as it comes from 6 cm data (boundaries are shownby black contours in Fig. 5, top) and exclude the very centralpart (distance from center up to 50 ′′ (equal to 1 .
67 kpc) andthe outer part (more than 340 ′′ or 11 . ffi cients in the arms and interarm areas and measure the Copyright line will be provided by the publisher
A. Chupin et al.: Magnetic arms traced in the Faraday cube ¢ ² ¢ ² ¢ ² é ¢ é ¢ é ¢ Right ascension H - L D e c li na t i on
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Fig. 2
Distribution of | w a ( l , b ) | for scales a = , , ′′ (from left to right) at Faraday depth φ =
40 rad m − . Thegrayscale shows the normalized modulus of w , the red contours depict 1 . × rms and 2 × rms levels of w calculated for allvalues in a map. ¢ ² ¢ ² ¢ ² é ¢ é ¢ é ¢ Right ascension H - L D e c li na t i on
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Fig. 3
Distribution of | ˜ w a ( l , b ) | for scales a = , , ′′ (from left to right). The grayscale shows the normalized modulusof w , the red contours depict 1 . × rms and 2 × rms levels of w calculated for all values in a map.contrast as the ratio between these intensities. The contrastsare given in Table 1. Our method enlarges this quantity byabout 10% for the northern arm and by about 15% for thesouthern arm. In spite of this improvement, the contrast isstill smaller than that at 6 cm.A straightforward recommendation would be to per-form observations at 6 cm and even 3 cm with higher sen-sitivity, e.g. with the SKA. As this remains unrealistic forthe near future, we have to restrict our demands to longerwavelengths, e.g. with the SKA precursors MeerKAT andASKAP. Then the suggested method is able to improve thecontrast up to 15%, which can be su ffi cient to isolate thearms in the images. We remind that we used NGC 6946as an illustrative example because the position of magneticarms is known from 6 cm data. Applications of the methodare recommended for galaxies where 6 cm data are absent.Structures of w max a ( l , b ) fit to the large-scale structuresthat are visible in the polarization map at 6 cm wavelength(see Fig. 5, top), located between the optical arms (seeFig. 5, bottom). We obtain enlargement of contrast for small a only. It means that the method is sensitive to small-scaledetails in the image. These details can correspond to real Table 1
Comparison of contrast for various distributionsbetween arm and interarm regions for wavelet coe ffi cients atscale a = ′′ ≈
535 pc, versus 6 cm. Errors were estimatedby standard deviation at 30% bootstrapping of the sets ˜ w a w max a . ± .
007 1 . ± .
007 2 . ± . . ± .
005 1 . ± .
005 2 . ± . . ± .
005 1 . ± .
004 2 . ± . small-scale structures of the magnetic field or be the resultof Faraday e ff ects on the polarized emission from the large-scale magnetic field, to be figured out with numerical testswhich is presented below. Note that the polarized intensitymap at 17-23 cm (see Fig. 1, top) revels some detail of mag-netic arms which are identified in Fig. 5. The point is how-ever that the details in Fig. 1, top are embedded in di ff usesurrounding and its relation with magnetic arms remains un-clear.To illustrate the idea of the method we constructed anartificial example, producing a data cube for the same setof channels as in data for NGC 6946 analysed before. We Copyright line will be provided by the publisher sna header will be provided by the publisher 5 ¢ ² ¢ ² ¢ ² é ¢ é ¢ é ¢ Right ascension H - L D e c li na t i on
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Fig. 4
Distribution of | w max a ( l , b ) | for scales a = , , ′′ (from left to right). The grayscale shows the normalizedmodulus of w , the red contours depict 1 . × rms and 2 × rms levels of w calculated for all values in a map.simulate the polarized emission (in the computational box25 × × B ϕ ( r , ϕ, z ) = B exp − rr ! − zz ! , (7)where the radial Gaussian scalelength is r =
10 kpc, thevertical Gaussian scale is z = B = µ G. The homogeneous tur-bulent field was simulated by a numerical model that al-lows to control the spectral law and the characteristic scaleof turbulence l t (providing the maximum of energy spec-trum) (Stepanov et al. 2014). We do not consider any pos-sible inhomogeneity of the turbulent field in the midplane.We choose a spectral slope of -5 / l t and + ff erent valuesof l t = , ,
400 pc in order to search for a possibledependence.The rms strength of the turbulent field is taken as 1 µ G.The number densities of relativistic and thermal electronsare assumed to be constant, namely n c = − and n e = . − . The large-scale disk magnetic field corre-sponds to a Faraday-thin source of the synchrotron emis-sion which does not cause significant Faraday depolariza-tion. The turbulent field acts as a Faraday screen that doesnot contribute much to the emission but depolarizes it anddisperses it to di ff erent Faraday depth. The output of themodel is two data cubes of Q and U with coordinates andfrequency.Next, we perform RM Synthesis on the artificial datacube using the same wavelength range as in the case ofNGC 6946 and calculate the wavelet coe ffi cients ˜ w a ( l , b )and w max a ( l , b ). The wavelet filter doesn’t influence on Fara-day spectra directly, however, it suppresses peaks in 2Dmap whose scales are di ff erent from scale a . The intensi-ties (spectral power) of ˜ w a ( l , b ) and w max a ( l , b ) versus scale a are shown in Fig. 6. The intensity of w a is on a low level on all scalesbecause the polarized intensity at Faraday depth φ =
40 rad m − does not contain spectral power at large scales.The intensity of ˜ w a ( l , b ) substantially increases with scalebecause only large scales are prominent, while the intensityof w max a ( l , b ) is practically constant, it detects power on allscales. If the large scales are well represented in the data(i.e. if there is no strong Faraday depolarization and mostspectral power is on large scales), the standard method ofstructure recognition ( ˜ w a ( l , b )) works best and reveals thestructure of the large-scale magnetic field. If, however, Fara-day depolarization is strong and there is no chance to rec-ognize the large scales directly, our new method opens anadditional possibility to find imprints of the large scales atsmall scales; the small scales are much better recognized by w max a ( l , b ) compared to ˜ w a ( l , b ). Late application of wavelettransform in w max a ( l , b ) suppresses the regions in the ex-tended disk but keeps the regions in the magnetic arms, be-cause the turbulent field is weaker there and hence there isless Faraday dispersion, so that the structures are less ran-domized. The scale of the crossing point in Fig. 6 dependsof the properties of turbulence and observation wavelengthand gives the upper limit in scale below which the suggestedtechnique is suitable.We checked the minimal requirement of recoveringthe Faraday spectrum from the observational frequencyband. The FWHM of the Faraday Point Spread Function(FPSF), or equivalently, the resolution in Faraday depthspace △ φ FPSF should be comparable at least to the rms dis-persion of Faraday rotation caused by the turbulent mag-netic field. Following Brentjens & de Bruyn (2005) the es-timate ∆ φ FPSF = √ λ − λ gives ∆ φ FPSF ≈
170 rad m − for the observations used here.This is larger than the dispersion of Faraday rotation (about40 rad m − ) in Fig. 1 (bottom). It explains why enlarge-ment of contrast obtained by our method (see Table 1) re-mains moderate however su ffi cient to isolate magnetic arms Copyright line will be provided by the publisher
A. Chupin et al.: Magnetic arms traced in the Faraday cube ¢ ² ¢ ² ¢ ² ¢ ² ¢ ² ¢ ² é ¢ é ¢ é ¢ é ¢ é ¢ é ¢ Right ascension H - L D e c li na t i on ´ Fig. 5
Top: Isolated magnetic arms (red contours at 1 . × rms ) obtained from the data at 17–23 cm with the wavelettransform for scale a = ′′ (Fig. 4, middle), and modelarms for methods comparison (black contours) overlayedon the image of polarized intensity at 6 cm wavelength(grayscale). The maximum intensity is 340 µ Jy / beam. Bot-tom: Isolated magnetic arms (shown in colour depicting theFaraday depths φ max ( rad m − ) of F max ), overlayed on anoptical image (grayscale).in Fig. 4. If ∆ φ FPSF is very large, then φ max is about the samefor all lines of sight, so that ˜ w a ≈ w max a and our method givethe same results as the traditional one.Our model is admittedly simplistic. Observations atlonger wavelengths generally probe magnetic structures thatare farther from the galaxy midplane and hence close to theobserver (e.g. Braun et al. 2010). Presuming that the mag-netic field morphology has some vertical structure, changesin the large-scale morphological features are expected whenobserving at progressively lower and lower frequencies. Ifthe large-scale fields in the magnetic arms are tied to thestar-forming ISM, we may expect to see them vanish atlower frequencies, where mainly the thick disk or halo is Fig. 6
The root-mean-square spectral power as a functionof scale a for the simulated data: thick line – w a , dashedline – ˜ w a , dotted line – w max a . The characteristic scale of tur-bulence is l t = ′′ , corresponding to 200 pc. The noiselevel of the synthetic signal is zero. RM Synthesis (cid:1849)(cid:1846) (cid:3109) ( (cid:942) ) max (cid:3109) ( (cid:942) ) max (cid:3109) ( (cid:942) ) (cid:1849)(cid:1846) ( (cid:942) ) compare large scale MF detection large scale MF traced by small scales at small a at large a ? max a w a w ~ aa ww ~ max (cid:33) aa ww ~ max (cid:31) aa ww ~ max (cid:124) Fig. 7
Scheme of a heuristic approach. Notation ( · ) meansan operator argument taken as a result from the previousstep.observed rather than emission from the disk that is depo-larized by star-formation induced turbulence. In the case ofNGC 6946, our result shows that the magnetic arms extendsu ffi ciently high into the thick disk or halo, so that their im-prints can still be detected at wavelengths around 20 cm.The overall scheme of our analysis is shown in Fig. 7.In summary, the distribution of small-scale magnetic fieldsas recognized by wavelet filtering of spatial scales tracesthe locations of the large-scale field (e.g. in the magneticarms), if the imprints of large-scale fields are random-ized by Faraday depolarization. This method is a pow-erful tool to analyze spectro-polarimetric data cubes ob-tained at long radio wavelengths. It should be applied to fur- Copyright line will be provided by the publisher sna header will be provided by the publisher 7 ther galaxies from the survey by Heald et al. (2009) and togalaxies observed with the VLA in L-band (1–2 GHz) andwith new-generation radio telescopes like LOFAR, ASKAP,MeerKAT and SKA.
Acknowledgements.
The authors thank local and anonimous ref-erees for substantial questions which are significantly improvesthe understandability and conclusiveness of the paper. Rainer Beckacknowledges support by DFG Research Unit FOR1254. Numeri-cal simulations were performed on the supercomputers URAN andTRITON of Russian Academy of Science, Ural Branch.
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