Properties of the Magneto-ionic Medium in the Halo of M51 revealed by Wide-band Polarimetry
DDraft version July 31, 2018
Preprint typeset using L A TEX style emulateapj v. 05/12/14
PROPERTIES OF THE MAGNETO-IONIC MEDIUM IN THE HALO OF M 51 REVEALED BY WIDE-BANDPOLARIMETRY
S. A. Mao , E. Zweibel , A. Fletcher , J. Ott , F. Tabatabaei Draft version July 31, 2018
ABSTRACTWe present a study of the magneto-ionic medium in the Whirlpool galaxy (M51) using new wide-bandmulti-configuration polarization data at L band (1-2 GHz) obtained at the Karl G. Jansky Very LargeArray. By fitting the observed diffuse complex polarization Q + iU as a function of wavelength directlyto various depolarization models, we find that polarized emission from M51 at 1-2 GHz originates fromthe top of the synchrotron disk and then experiences Faraday rotation in the near-side thermal haloof the galaxy. Thus, the scale height of the thermal gas must exceed that of the synchrotron emittinggas at L band. The observed Faraday depth distribution at L band is consistent with a halo fieldthat comprises of a plane-parallel bisymmetric component and a vertical component which producesa Faraday rotation of ∼ − − . The derived rotation measure structure functions indicate acharacteristic scale of rotation measure fluctuations of less than 560 pc in the disk and approximately1 kpc in the halo. The outer scale of turbulence of 1 kpc found in the halo of M51 is consistent withsuperbubbles and the Parker instability being the main energy injection mechanisms in galactic halos. Keywords:
ISM: magnetic fields—polarization—Galaxy: halo INTRODUCTION
The origin and evolution of large-scale magnetic fieldsin galaxies remains an unresolved fundamental questionin astrophysics (for a review, see e.g., Kulsrud & Zweibel2008). Most of our knowledge of galactic-scale magneticfields come from studies of our own Milky Way. Differ-ent probes such as diffuse polarized synchrotron emis-sion (e.g., Carretti 2010; Wolleben et al. 2010), opticalstarlight polarization (e.g., Heiles 1996), Zeeman split-ting (e.g. Green et al. 2012) and Faraday rotation to-wards background extragalactic polarized sources andpulsars (e.g., van Eck et al. 2011; Mao et al. 2012a;Han et al. 2006) have been used to infer magnetic fieldstructures in the Galaxy. However, interpreting theseobservations is difficult due to complicated sight linesthrough the Galactic plane, our inability to distinguishsignatures of large-scale fields from local magnetic fea-tures, and pulsar distance uncertainties. Furthermore,extracting the magnetic field structure of the Milky Wayfrom Faraday rotation measures (RMs) of extragalacticsources and pulsars relies heavily on the assumed ther-mal electron density distribution. Therefore, we turn toexternal galaxies in order to obtain a clear picture oflarge-scale magnetic fields in spiral galaxies. National Radio Astronomy Observatory, P.O. Box O, Socorro,NM 87801, USA Department of Astronomy, The University of Wisconsin, Madi-son, WI 53706, USA Max-Planck-Institut f¨ur Radioastronomie, Auf dem H¨ugel 69,53121 Bonn, Germany; [email protected] Department of Physics, The University of Wisconsin, Madison,WI 53706, USA Center for Magnetic Self-Organization in Laboratory and As-trophysical Plasmas, The University of Wisconsin, Madison, WI53706, USA School of Mathematics and Statistics, University of Newcastle,Newcastle upon Tyne, NE1 7RU, UK Max-Planck-Institut f¨ur Astronomie, K¨onigstuhl 17, D-69117Heidelberg, Germany S. A. Mao is a Jansky Fellow of the National Radio AstronomyObservatory
M51, the Whirlpool galaxy, is a prime candidate forinvestigating the large-scale magnetic field structures.In fact, it was the first external galaxy detected in lin-ear polarization at radio wavelengths (Mathewson et al.1972). Shorter wavelength observations soon followed(Segalovitz 1976; Beck et al. 1987) and led to the tenta-tive suggestion of a bisymmetric magnetic field configu-ration (Tosa & Fujimoto 1978). A major step towardsunderstanding the polarized synchrotron emission fromM51 was made by Horellou et al. (1992), who derivedRM using Very Large Array data at 18 and 20 cm anddemonstrated that the resulting RM magnitude is toosmall compared to the equipartition estimation of theregular magnetic field strength. The authors concludedthat M51 is not transparent in polarization at 18 and 20cm due to severe Faraday depolarization effects.A general picture of M51’s large-scale magnetic fieldemerges from modeling works by Berkhuijsen et al.(1997) and Fletcher et al. (2011) using data at severalsparsely sampled wavelengths (3, 6, 18 and 20 cm). Bothfound that the galaxy hosts distinct large-scale magneticfields in its disk and halo. Then Fletcher et al. (2011)deduced that the disk field is dominated by an axisym-metric mode, whereas the halo field is dominated by abisymmetric mode. Based on polarization observationsof a sample of Spitzer Infrared Nearby Galaxies Survey(SINGS) galaxies at 18 and 22 cm, including M51, Braunet al. (2010) proposed that the observed characteristicpolarization and Faraday depth modulations as a func-tion of azimuth are produced by an axisymmetric spiralfield with a quadrupolar out-of-plane extension in thegalaxy’s near-side halo. M51 was also recently observedat the Giant Metrewave Radio Telescope (GMRT) at 610MHz (Farnes et al. 2013) and at the Low-Frequency Ar-ray (LOFAR) at 150 MHz (Mulcahy et al. 2014), andwas found to be completely depolarized within the sen-sitivity limit. At optical wavelengths, Scarrott et al.(1987) showed that linear polarization of M51 forms a a r X i v : . [ a s t r o - ph . GA ] D ec Mao et al. spiral pattern from the nucleus out to a radius of 4-5kpc. However, NIR observations at 1.6 µ m conducted us-ing the Mimir instrument on the Perkins telescope showthat the galaxy is unpolarized at a level of 0.05% (Pavel& Clemens 2012). Clearly, much work is needed to con-verge on a consistent picture of M51’s large-scale mag-netic field to explain its polarized emission across a widewavelength range.Structure functions of rotation measure can provideinformation on turbulence in the interstellar medium(ISM). Using RM structure functions of background ex-tragalactic sources behind the Galactic plane, Haverkornet al. (2006, 2008) derived an energy injection scale ofseveral parsecs (comparable to the typical size of HIIregions) inside spiral arms and a scale around 100 pc(comparable to the typical size of supernova remnants)in inter-arm regions. Relatively little is known aboutthe outer scale of turbulence in the diffused ionized gasin external galaxies – the structure function constructedusing RMs of extragalactic polarized sources behind theLarge Magellanic Cloud (LMC) led to an estimated scaleof 90 pc (Gaensler et al. 2005). In M51, Houde et al.(2013), using angular dispersion analysis, derived a tur-bulent correlation scale ∼
100 pc in the disk of M51,consistent with the Fletcher et al. (2011) result of 50 pcobtained based on the dispersion of RM.In this paper, we present new L band (1-2 GHz) KarlG. Jansky Very Large Array (VLA) polarization obser-vations of M51 which has the widest frequency coveragecompared to previous L band polarization observationsof the galaxy. With this broadband dataset in hand, wedirectly model the measured Stokes Q and U as a func-tion of wavelength to study both small and large-scalestructures in the magneto-ionic medium of the galaxy. InSection 2, we describe the Jansky VLA observations anddata reduction procedures. We present Faraday depthsof background extragalactic sources and compute theMilky Way foreground RM in the direction of M51 inSection 3. In Section 3.3, we present the Faraday depthcube obtained from the rotation measure synthesis tech-nique. In Section 3.4, we present the results of directlyfitting to Stokes Q and U to various depolarization mod-els. In Section 4, we interpret results of the QU fittingand discuss the nature of the Faraday rotating mediumat L band. The observed Faraday depth distribution to-wards M51 is compared to predictions from the Fletcheret al. (2011) and the Braun et al. (2010) models in Sec-tion 5. In Section 6, the RM structure functions arepresented and the outer scales of turbulence in the diskand halo of M51 are extracted. OBSERVATIONS AND DATA REDUCTION
M51 was observed as part of the science demonstra-tion of the Expanded Very Large Array (EVLA) project.Observations were carried out on 2010 September 6th,2010 December 5th and 2011 April 10th in D, C and Bconfigurations respectively. Data were taken at L band(971 MHz – 1995 MHz) in 8 spectral windows, each con-sisting 64 2-MHz channels. The total time on sourcewas approximately 7 hours. 3C286 was observed at thebeginning and the end of each run for absolute flux den-sity, bandpass and polarization angle calibration. Either3C295 (J1411+5212) or J1313+5458 was used as thecomplex gain and leakage calibrator and was observed every 30 mins during each run.Data calibration and reduction were carried out us-ing the Common Astronomy Software Applications (CASA). Visibilities affected by radio frequency inter-ference (RFI) were flagged using the automated flaggingalgorithm RFLAG in CASA. The measurement sets werecarefully inspected for further RFI excisions. After flag-ging, approximately 400 MHz of the total 1 GHz band-width at L band remain usable. Raw data were thenbinned into 8 MHz channels before calibration. Fluxdensities were scaled to the Perley & Butler (2013a)3C286 model. The measured polarization angles werecalibrated to 3C286, whose polarization angle is +33 ◦ across L band (Perley & Butler 2013b). Time dependentantennae gains were solved on a per-spectral-window ba-sis, while polarization leakage and absolute polarizationangle calibration were solved on a per-channel basis. Fol-lowing the leakage calibration, the residual instrumen-tal polarization level is below 0.25%. We have subse-quently masked all pixels with fractional polarization be-low 0.25%. We have verified, using Faraday rotation syn-thesis that 3C286 has a Faraday depth consistent withzero ( − ± − ). Imaging Total Intensity
After separately calibrating the measurement sets ofdifferent array configurations, we have constructed aStokes I natural-weighted image across L band. Wehave utilized the multi-scale, multi-frequency deconvo-lution as implemented in CASA (Rau & Cornwell 2011)which models the sky brightness as a linear combinationof both spectral and spatial basis functions. Two Tay-lor terms were used to model the spectral dependence.In order to model the extended emission in the field ofview, we used angular scales ranging from 0” (which cor-responds to point sources) to 5.3’ (comparable to theextent of the galaxy). We imaged a 30’ ×
30’ field aroundthe phase center.The resulting total intensity map, after primary beamcorrection, is shown in Figure 1, with a noise level of 23 µ Jy beam − at a resolution of 10.9” × map across L band is also produced and is shownin Figure 2, with a typical uncertainty of 0.2. The totalintegrated flux density of M51 is measured to be 1.4 Jyat 1.478 GHz, which is in good agreement with previoussingle-dish (Lequeux 1971) and interferometric observa-tions (Segalovitz 1977; Dumas et al. 2011). Therefore,we conclude that no large-scale flux is missing from ournew Jansky VLA multi-configuration data. Imaging Stokes Q and U To facilitate a Faraday rotation study, images of Stokes Q and U are made at each of the 45 8-MHz channels usingmulti-scale CLEAN, with angular scales spanning from“zero” to 2.7’. Natural weighting scheme is used in orderto maximize the sensitivity to weak and extended polar-ized emission. These channel maps are first made at theiroriginal resolutions and then corrected for the responseof the primary beam. Subsequently, these images haveall been smoothed to a common resolution of 10.9” × http://casa.nrao.edu We define spectral index α as I = I ν α . to match the resolution at the lowest frequency. The typ-ical noise in a Stokes Q or U channel map is about 56 µ Jy beam − , close to the theoretical noise level of about51 µ Jy beam − . Computing Faraday Depths
When linearly polarized light travels through a magne-tized medium, its plane of polarization rotates due to theFaraday rotation effect. In the case where backgroundpolarized emission experiences pure Faraday rotation ina foreground medium, the angle of rotation in radians isgiven by ∆ ψ = RM λ (1)where λ is the wavelength of the radiation measured inmeters and RM is the rotation measure. In the more gen-eral case of mixed emitting and rotating medium alongthe line of sight, or of multiple RM components withinthe telescope beam, Faraday depth φ is defined as: φ = 0 . (cid:90) observersource n e ( l ) B (cid:107) ( l ) dl rad m − (2)In the above equation, n e ( l ) (in cm − ) is the thermalelectron density, B (cid:107) ( l ) (in µ G) is the line of sight mag-netic field strength and d l (in pc) is a line element alongthe line of sight. RM is equivalent to the Faraday depth φ in the simple case of pure rotation caused by a singlecomponent. We note that the electron column densityis more fundamental than the volume density becausea path length assumption is needed in order to convertcolumn density into volume density.We have computed the Faraday depth of M51 at Lband using the RM synthesis technique (Brentjens & deBruyn 2005), followed by the deconvolution of RM spec-tra using the RMCLEAN algorithm (Heald et al. 2009).We stop cleaning when the peak of the spectrum falls be-low 4 times the noise level. Given the frequency setup ofour observations, the rotation measure spread function(RMSF) has a full width at half-maximum (FWHM) of90 rad m − , a great improvement over the restoring beamof 144 rad m − of the Westerbork (WSRT) SINGS ob-servation of M51 (Heald et al. 2009; Braun et al. 2010).Moreover, the much improved frequency coverage of ournew VLA observations have significantly suppressed theside-lobe level in the RMSF: the first side-lobe is at a levelof approximately 35% of the main peak (compared to78% for the WSRT work). The higher resolution and themuch lower side-lobe levels of our Faraday depth spectraenable us to better interpret structures in Faraday depth( φ ) space. The deconvolution algorithm will also performbetter because high side-lobes are less likely to be mis-taken as real signal during RMCLEAN. For illustration,we have plotted the RMSF of our VLA observations andthat of the WSRT observations (Heald et al. 2009) inFigure 3. The channel width (8 MHz) limits our sensi-tivity to Faraday depths < − . The highestobserving frequency (1.835 GHz) sets our sensitivity toextended structure in φ space: the sensitivity drops by50% for structures with φ extents greater than 118 radm − . RESULTS
Faraday Depths of Polarized ExtragalacticBackground Sources
To extract reliable polarization information from ex-tragalactic sources across the field and to avoid confu-sion with diffuse emission associated with M51, we havere-imaged the data in Stokes I , Q and U using only base-lines longer than 350 m. We imaged a field of size 40’ × I map has a noise levelof 25 µ Jy beam − , whereas the Stokes Q and U 8-MHzchannel maps have an rms noise of about 45 µ Jy beam − with a beam size of 13.2” × Q/I and
U/I cubes using thetotal intensity and spectral index map at the referencefrequency of 1.478 GHz.Although RM Synthesis is an excellent tool to vi-sualize polarized emission at various Faraday depths,there is considerable ambiguity in extracting proper-ties of the underlying Faraday structure using this ap-proach, especially when there is complexity in the spec-trum (Farnsworth et al. 2011; O’Sullivan et al. 2012).The latest RM benchmark test for different Faraday de-composition methods (Sun et al. 2014b) demonstratesthat directly fitting for Stokes Q , U as a function of λ ismost successful in recovering the correct components in φ space. In addition, modeling the depolarization trendcan reveal properties of the magneto-ionic medium thatare otherwise hard to obtain, such as the relative dis-tribution of synchrotron emitting and Faraday rotatingmaterial as well as turbulent properties in the medium.We first estimate the Faraday depth of each source (orcomponent) at the pixel with the highest signal-to-noisedetection in polarization using RM synthesis (Brentjens& de Bruyn 2005). The result is subsequently used asthe initial parameter guess for a maximum likelihood fitto Q / I , U / I as a function of λ to several models ofthe synchrotron emitting and Faraday rotating region.We consider the following models (i) a uniform externalFaraday rotating screen ; (ii) an inhomogeneous externalFaraday screen; (iii) an inhomogeneous Faraday screenwith a partial covering fraction; (iv) a Burn slab (con-sists of well-mixed thermal and synchrotron emitting gas)with a regular magnetic field; (v) a slab with both reg-ular and random fields and (vi) two spatially unresolvedFaraday depth components. Expressions correspondingto these models are listed in Appendix A. In addition tomaximum likelihood fits, least-square fits were performedas well. We confirm that the two methods converge tothe same solutions. We select a model that provides asatisfactory fit without introducing too many free pa-rameters. More complicated models are accepted onlywhen they significantly improve the reduced χ of thefit (at least at 5 σ level) based on the F-test. In addi-tion, we utilize the Bayesian information criterion (BIC) The term screen is used throughout this paper to represent aregion which consists only of thermal electrons with no relativisticelectrons.
Mao et al. defined in O’Sullivan et al. (2012) to select the best-fitmodel. The BIC penalizes models with a large numberof free parameters and at the same time it allows for aneasy comparison of non-nested models. We have chosena criterion for the BIC such that only when BIC model1 − BIC model2 >
30 do we favor the more complex model 2.We list the fitted parameters, their uncertainties, χ r and the BIC for each source (or component) and for eachmodel in Table 2. The best fit models are indicated usingboldface font in the Model column. We show the polar-ization data and the corresponding best fit models tosource J1330+4703b and J1329+4717 in Figure 4 and 5respectively. Since off-axis frequency-dependent instru-mental polarization response was not corrected for in ourdata set, we choose to only report Faraday depths ofsources within 20’ from the pointing center. In Figure 6,we plot our derived L-band Faraday depths (FD) againstthose derived by Heald et al. (2009) for extragalactic po-larized sources that overlap in both studies. Approxi-mately 60% of sources in both samples have consistentFDs within their measurement errors. This suggests agood agreement between the two L band datasets.We list the fractional polarizations and the Faradaydepths of sources that are present in both our VLAdata set and the Farnes et al. (2013) GMRT observa-tions in Table 3. For these sources, we predict the frac-tional polarization at 610 MHz using our best-fit modelsand compare them to the observed values . For 3 outof the 6 extragalactic sources, predictions of the frac-tional polarization at 610 MHz from our direct Q U fitsare consistent with the observed values. However, weoverpredict the fractional polarization for the remain-ing 3 sources. This could be due to an additional unre-solved and weakly-polarized steep-spectrum componentthat dominates at low frequencies, resulting in the muchlower total fractional polarization at 610 MHz. We sug-gest that a rigorous comparison should be performed onlyafter all off-axis instrumental polarization effects havebeen calibrated out in both our Jansky VLA and theGMRT Farnes et al. (2013) data sets. Using the 6 ex-tragalactic sources with both 610 MHz and L-band po-larization information, we find an average depolarizationratio of 0.4 between 610 MHz and 1.4 GHz. We note thatGieߨubel et al. (2013) found an average depolarization ofextragalactic sources around M31 between 325 MHz and1.4 GHz of ∼ > Milky Way Foreground RM in the Direction of M51
Previous estimates of the Milky Way foreground rota-tion measure in the direction of M51 have yielded verydifferent results. Horellou et al. (1992) estimated a fore-ground RM of − ±
12 rad m − by averaging RMs of9 extragalactic sources in the Simard-Normandin et al.(1981) catalog within 20 ◦ of M51. In the WSRT M51work, Heald et al. (2009) obtained a foreground RM We note that the value N in their Equation (14) should betwice the total number of frequency channels. The percent polarization listed in their Table 1 is an upperlimit because off-axis Stokes I , Q and U effects were not correctedfor. value of +12 ± − using the RMs of 5 extragalacticsources detected in the field of M51.To determine the Milky Way foreground RM, we onlyuse statistics of polarized extragalactic sources within 20’of M51 detected in our VLA observations to minimize ef-fects from uncorrected off-axis instrumental polarization.Moreover, this criterion limits the variance in RM intro-duced by fluctuations in thermal electron density andmagnetic fields in the Milky Way foreground. At theGalactic latitude of M51 ( b = +68 ◦ ), RM variance in theMilky Way remains strong, up to 9 rad m − (Schnitzeler2010). Since some extragalactic sources are located be-hind neutral hydrogen tidal features with substantial col-umn densities that could produce enhancements in Fara-day rotation with even a low ionization fraction, we fur-ther restrict our foreground sight lines to have HI columndensities < cm − in the Rots et al. (1990) HI map.The median Faraday depth of the 6 sources (denotedwith * in Table 1) meeting these criteria is +13 rad m − with a standard error of 1 rad m − , which we adopt asthe constant Milky Way foreground RM in the directionof M51. Our Milky Way foreground estimate is consis-tent with the Heald et al. (2009) estimation, but it isvery different from the Horellou et al. (1992) estimationdue to the large circle (radius 20 ◦ ) within which RMs ofextragalactic sources have been averaged in their study. RM Synthesis Results of the Diffuse PolarizedEmission from M51
We determine the Faraday depth of diffuse polarizedemission from M51 using the RM synthesis technique(Brentjens & de Bruyn 2005). Along each sight line, weextract the peak polarized intensity and the correspond-ing Faraday depth from the Faraday depth spectrum.The resulting polarized intensity and Faraday depth dis-tributions across the galaxy are displayed in Figures 7and 8 respectively. Only pixels with signal-to-noise de-tection in polarization greater than 7 are displayed . As-suming a single Faraday depth component along the lineof sight, we plot the magnetic field position angles (af-ter correcting for Faraday rotation) on the Hubble SpaceTelescope B band image (Mutchler et al. 2005) in Fig-ure 9. There appears to be polarized emission which coin-cides with the companion galaxy NGC 5195 and extendswestward. However, the relative distance of M51 andNGC 5195 is required to determine if this polarization istruly intrinsic to NGC 5195. Polarized synchrotron emit-ting arms and optical arms form a complicated patternacross the galaxy. The relative locations of the magneticand material arms were investigated in details by Patri-keev et al. (2006). In this paper, we concentrate on thestructure of the synchrotron emitting and Faraday ro-tating material along the line of sight. The integratedpolarized flux of M51 is measured to be 82 mJy whichis in excellent agreement with the value 81 ± This is equivalent to 6 σ in standard Gaussian statistics (Haleset al. 2012). Direct Fit to Q( λ ) and U( λ ) We have performed a direct fit to Stokes Q and U asa function of λ on a pixel-by-pixel basis across M51 us-ing the O’Sullivan et al. (2012) maximum likelihood ap-proach. We note that this is the first time this approachhas been applied to wide-band data of extended polar-ized emission from galaxies: previous works involve fit-ting Q ( λ ) and U ( λ ) of unresolved extragalactic sources.In order to decouple depolarization from simple spec-tral index effects, care has to be taken to estimate thesynchrotron fractional polarization. Unlike extragalacticradio sources whose emission at L band is mostly non-thermal so that one can simply divide out the observed Q and U by I to correct for spectral index effects, thesituation for radio emission from a galaxy is complicatedby the contamination of non-negligible free-free emissionat L band. Based on radio data spanning frequenciesfrom 0.4 to 22.8 GHz, Klein et al. (1984) found the totalthermal fraction of M51 at 1.4 GHz to be 5% with an in-tegrated synchrotron spectral index of − − . − − model1 − BIC model2 >
30. We note that this BIC crite-ria roughly corresponds to favoring model 2 over model1 at ≥ σ level according to the F-test. We allowedthe Milky Way foreground RM in these models to be afree parameter. Both Model (iii) and (iv) produce best-fit Milky Way foreground RM values in the range of ∼−
200 rad m − to ∼ +140 rad m − with large pixel-to-pixel variations. Since model (iii) and (iv) both yield Milky way foreground RM values inconsistent with ourestimation of +13 ± − , we do not consider thesetwo models any further.The Stokes Q and U versus λ trends can be wellfitted by Faraday rotation occurring in an externalscreen. More specifically, approximately 84% of thesight lines are consistent with a uniform screen, with-out wavelength-dependent depolarization across L band,while around 16% of the sight lines are consistent withexternal Faraday dispersion produced by an inhomoge-neous Faraday screen. Sight lines well described by ex-ternal Faraday dispersion have σ RM in the range 9 - 29rad m − , well exceeding the σ RM due to the Milky Wayforeground at angular separations < ◦ . Therefore, weattribute the depolarization to M51 instead of the MilkyWay. We show the spatial distribution of the derivedbest-fit model in Figure 10. We will discuss implicationsof this result in the next section. Since the best fit mod-els of the medium do not cause any changes to the valueof Faraday depth along each sight line, we continue touse the Faraday depth map derived using RM synthesis( § THE NATURE OF THE FARADAY ROTATING MEDIUMOF M51 AT 1-2 GHZ
Relative Scale Heights of Thermal and Cosmic RayElectrons
In Section 3.4, we showed that the observed trends ofStokes Q and U against λ can be well fitted by modelswhere Faraday rotation takes place outside of the syn-chrotron emitting region. This result on the geometry ofthe Faraday rotating and synchrotron emitting mediumimmediately implies that the scale height of the thermalgas (the warm ionized medium) is greater than that of thesynchrotron emitting gas at 1.4 GHz. Statistically speak-ing, scale heights of these two constituents of the ISM arefound to be similar: the typical synchrotron scale heightof edge-on galaxies is 1.8 kpc at 6 cm (Krause 2009),while the mean H α scale height ( ∝ n e ) is in the rangeof 1 − < ν GHz is given by τ syn ≈ B − / µ G ν − / yrs (3) Mao et al.
For a total plane-of-the-sky magnetic field strength of ∼ µ G at 1.5 GHz (Fletcher et al. 2011), the life timeof cosmic ray electron is 1.4 × yrs. If we assume thatthe cosmic ray propagation is diffusive and a diffusioncoefficient D of 3 × cm s − as in the Milky Way(e.g. Strong et al. 2007), then within the lifetime of thesecosmic ray electrons, they can diffuse up to a height of (cid:112) Dτ syn ∼ v A = 2 × B µ G √ n i cm s − (4)For an ion number density of n i ∼ − , the Alfv´enspeed is 95 km s − . Within the life time of these cosmicray electrons, they can travel up to a height of approx-imately 1.4 kpc from the mid-plane if the field lines arecompletely vertically directed. Since the thermal gas hasa larger extent than the cosmic ray electrons, the scaleheight of the thermal gas must exceed 1.2 kpc. We sug-gest that a systematic wide-band polarization survey andcareful modeling of Stokes Q and U against λ at L bandof nearby low inclination galaxies would be a novel ap-proach to extract relative scale heights of different phasesof the ISM, hence allowing for a better understandingof the source of ionization of the warm ionized medium(WIM) and cosmic ray confinement in galaxies. External Faraday Dispersion in the halo of M51
Most sight lines towards M51 do not exhibit any λ -dependent depolarization across L band, suggesting thatthe external Faraday dispersion effect is not severe acrossM51 in this wavelength range. The Faraday screen is ei-ther completely homogeneous, or that our observationshave a resolution much smaller than the characteristicturbulent scale in the screen, leading to little / no exter-nal Faraday dispersion effects. We show, in Section 6,using RM structure function analysis that the latter isindeed the case. We constructed the structure functionof the complex polarization ( SF CP ) and that of the po-larized intensity ( SF pi ) SF CP ( δθ ) = (cid:104)| P ( θ ) − P ( θ + δθ ) | (cid:105) (5) SF pi ( δθ ) = (cid:104) ( p ( θ ) − p ( θ + δθ )) (cid:105) , (6)where P = Q + iU and p = | P | . According to Sun et al.(2014a), the relative steepness of SF CP ( δθ ) and SF pi ( δθ )can be used to diagnose whether the observed polar-ized emission is intrinsic or heavily modified by a fore-ground turbulent Faraday screen which randomizes thedistribution of polarization angles, reduces the observedpolarized intensity and moves power to smaller scales.The authors showed mathematically that if the observedpolarized structure is mainly produced by a foregroundFaraday screen, then the slope of SF CP ( δθ ) would be sig-nificantly shallower than SF pi ( δθ ). Intuitively, this canbe understood as SF CP ( δθ ) being dependent on the dis-tributions of both the intensity and the angle, and hencethe slope of SF CP ( δθ ) would be flatter than SF pi ( δθ ) ifexternal Faraday dispersion is in action. In Figure 13, weplot the structure functions SF CP and SF pi computedusing our L band M51 data. It is clear that SF CP ( δθ ) does not have a shallower slope than SF pi ( δθ ). This sug-gests that the observed polarized structures are mostlyintrinsic with little modification/depolarization due tothe foreground Faraday screen in M51’s halo, which isconsistent with the result of our direct QU fit in § Small-scale RM Fluctuation in the Halo of M51
While most of the sight lines through M51 do not show λ -dependent depolarization within L band, 16% of thesight lines through the galaxy do exhibit λ -dependentdepolarization described by external Faraday dispersion( ∝ e − σ λ ). Several regions of M51, including the nu-cleus, the inner spiral arms and the region between M51and the companion galaxy appear to experience more se-vere λ -dependent depolarization: the dominant charac-teristic turbulent cell size in these regions must be smallercompared to the physical scale encompassed by the beam(370 pc). Depolarization in these regions can be well de-scribed by a median RM variance ( σ RM ) of 15 rad m − .By evoking a simple single-cell random magnetic fieldmodel (e.g. Gaensler et al. 2001), one can relate therandom magnetic field strength in the halo B r,halo to thevariance of RM ( σ RM ), electron density ( n e ), cell size ( l ),filling factor ( f ) and total path length D : B r,halo = σ RM . n e (cid:114) fDl . (7)Assuming n e = 0.05 cm − , l <
370 pc, D = 1 kpc and f = 1, we estimate a lower limit of the random fieldstrength in the halo B r,halo to be 1 µ G.We search for spatial correlations of these more tur-bulent regions in M51 with properties of the underlyingstar-forming disk. HII regions on pc scales are a viablesource of energy input into the magneto-ionic mediumand they have been shown to drive RM fluctuations inthe Milky Way spiral arms (e.g., Haverkorn et al. 2006).A large concentration of HII regions within the tele-scope beam can effectively depolarize background emis-sion. While we find some correspondence between lo-cations of HII regions (Lee et al. 2011) and pixels thatexhibit external Faraday dispersion within ∼ λ -dependentdepolarization within L band appear to have a slightlyhigher mean velocity dispersion (24.0 ± − ) thanthe rest of the galaxy (21.60 ± − ). We note thatHI velocity dispersion and depolarization of synchrotronemission are completely different approaches that probeturbulence in two different phases of the ISM. The factthat these two very different measurements converge sug-gests that the general ISM near M51’s nucleus and in the Magnetic field strength in each individual cell has the samemagnitude but different orientation. region between M51 and the companion NGC 5195 hasbeen stirred up. Modeling depolarization trends of syn-chrotron emission is thus an alternative way to identifyand characterize turbulent regions in a galaxy.
Reconciling the Derived Faraday Medium Structurewith Low Frequency Observations of M51
We check the consistency of our derived Faraday struc-ture in M51’s halo with low frequency polarization obser-vations of the galaxy, specifically the recent Giant Metre-wave Radio Telescope (GMRT) observations at 610 MHz(Farnes et al. 2013). M51 is found to be completely de-polarized down to a sensitivity limit of 44 µ Jy beam − at a resolution of 24” with a bandwidth of 16 MHz. Werepeated our direct Q U fit (Section 3.4) after smooth-ing the input Stokes Q and U cubes to a resolution of24”. Assuming a synchrotron spectral index of − > σ , the predicted median polarizedintensity of M51 at 610 MHz is 30 µ Jy beam − , belowthe sensitivity limit of the GMRT observations. Thus,the non-detection of M51 in polarization at 610 MHz isconsistent with our model at the 4 σ level. Since cosmicray electrons suffer less from energy losses and can prop-agate further away from their acceleration sites at lowfrequencies, the synchrotron scale height is likely largerat lower frequencies. For example, if cosmic ray diffusiondominates, then the synchrotron scale height should varyas ν − / . We suggest that at frequencies below 1 GHz,a part of the near-side halo may emit as well as Faradayrotate. Since this emission will suffer from differentialFaraday rotation and internal Faraday dispersion, it isunlikely to produce a significant amount of polarization.Future deep wide-band polarization observations of M51at low frequencies are much needed to further constrainthe properties of M51’s magnetized near-side halo. LARGE-SCALE MAGNETIC FIELD STRUCTURES INTHE HALO OF M51
The Fletcher Model
In the comprehensive modeling work of Fletcher et al.(2011), based on the drastically different Faraday depthsand degree of polarization measured at long (18cm and20cm) and short wavelengths (3cm and 6cm), the au-thors concluded that there must be two distinct layerswhere Faraday rotation occur in M51, designated as thedisk and the halo. In particular, polarized emission at18 and 20cm originates from the top of the synchrotronemitting disk, and only experiences Faraday rotation inthe thermal halo. Therefore, the Faraday structure of theFletcher et al. (2011) model agrees well with the resultof our direct QU fit (see Section 3.4): the thermal gashas a larger scale height than the synchrotron emittinggas at L band.Next, we compare the observed Faraday depths at Lband with predictions from the Fletcher et al. (2011)model. By fitting Fourier modes to the observed po-larization angles averaged in azimuth sectors at foursparsely sampled wavelengths, Fletcher et al. (2011)found that M51 hosts disk and halo fields of differenttopologies: the disk field is axisymmetric (with a weakm=2 component), while the halo field is bisymmetric. Inthe top panel of Figure 14, we show the observed Faraday depth of M51 after removing the Milky Way foregroundRM (+13 rad m − ). In the middle panel of the same fig-ure, we show the predicted Faraday depth distribution ofM51 at L band produced by the halo bisymmetric fieldin the Fletcher model . It is clear that the Fletcher et al.(2011) prediction is on average much too positive com-pared to the observed values.The RM due to the Milky Way foreground is a freeparameter in the Fletcher model that is found to be ∼ +4 rad m − across all radii. This value is differentfrom our estimated Milky Way RM based on the me-dian Faraday depths of background extragalactic sourcesin the field (+13 rad m − ). There exists a degeneracybetween the Milky Way RM and an RM produced bya vertical magnetic field in the halo of M51. Like theMilky way foreground, a vertical magnetic field in thehalo of M51 will produce additional Faraday rotation atboth short (3 and 6 cm) and long (18 and 20 cm) wave-lengths. In the bottom panel of Figure 14, we show thepredicted Faraday depths of M51 at L band from theFletcher et al. (2011) model with the addition of a ver-tical field that produces an RM of − − . Thismap clearly resembles the observed Faraday depth dis-tribution much better – the value of the summed resid-ual Σ i (RM model , i − RM observed , i ) decreases significantlyfrom 1.6 × to 1.3 × . We suggest that the discrep-ancy between the observed Faraday depths of M51 at Lband and the Fletcher model prediction can be reconciledif we include a coherent vertical magnetic field in M51’shalo in addition to the bisymmetric plane-parallel com-ponent . A similarly weak but coherent vertical magneticfield is also present in the halo of the nearby Large Mag-ellanic Cloud (Mao et al. 2012b) and M33 (Tabatabaeiet al. 2008).This coherent vertical field could be of either primor-dial or of dynamo origin. While azimuthal fields in thedisk of the galaxy can diffuse away easily, the verticalmagnetic flux is essentially conserved (e.g. Sofue & Fu-jimoto 1987). Thus, the measured vertical field in thehalo of M51 could reflect the field preserved since thecollapse of the protogalaxy. The expected magnitude ofthe vertical field in this scenario depends heavily on theinitial seed field strength. On the other hand, dynamotheory predicts vertical fields ∼
10% of the plane-parallelcomponent. In particular, the ratio of vertical field to ra-dial field follows from dynamo theory is given by (cid:112) h/R (Ruzmaikin et al. 1988) where h is the disk scale heightand R is the radius of the galaxy. Assuming h ∼ R ∼
10 kpc for M51 and the derived radial component ofthe magnetic field in the Fletcher et al. (2011) model, weobtain | RM B z | in the range of 7 −
21 rad m − in the halo.Our derived magnitude of ∼ − falls comfortablyin this range. The Braun Model We have extended the model prediction out to a galacto-centricradius of 9 kpc. This coherent vertical field has a small component projectedonto the sky plane but this component does not emit synchrotronradiation due to the lack of cosmic ray electrons in the halo. The vertical magnetic flux is conserved unless there is a radialflow that drags field lines out of the galaxy.
Mao et al.
Large-Scale Azimuthal Trends of Faraday Depth andPolarized Intensity
Using a sample of WSRT SINGS galaxies, Braun et al.(2010) proposed that galaxies have axisymmetric spiralfields with a quadrupole field geometry in the nearsidehalo. At L band, the polarized emitting zone is con-centrated 2-4 kpc above the mid-plane, this emission isthen Faraday rotated at a height of 4 −
10 kpc above themid-plane in the near-side halo. This picture of Faradayrotation taking place in a purely thermal halo is consis-tent with the result of our direct QU fit in Section 3.4,though the exact vertical extents of the emitting and ro-tating zones may not agree.The geometry of the magnetic field manifests itself inthe polarized intensity versus azimuth and Faraday depthversus azimuth trends. In particular, one expects asym-metry in polarized emission at L band with a globalminimum in polarized intensity near the receding ma-jor axis . Meanwhile, one expects a minimum Faradaydepth (a value closest to the Galactic foreground) nearthe approaching major axis (position angle of − ◦ ). InFigure 15 and 16, we plot the measured peak polarizedintensity as a function of azimuth (AZ = 0 ◦ at the reced-ing major axis) and the peak Faraday depth as a functionof azimuth averaged across all radii in 10 ◦ bins. Errorbars in the plot represent the standard deviations of po-larized intensity and Faraday depths within a bin. Wecan not readily identify any visible azimuthal trends inthese plots. Our derived polarized intensity and Faradaydepth modulations as a function of azimuth are visiblydifferent from those presented by Braun et al. (2010).This disparity could potentially be caused by a differentde-projection of the galaxy: we have adopted an inclina-tion angle of 20 ◦ (Tully 1974; Shetty et al. 2007), whereasBraun et al. (2010) assumed an inclination of 42 ◦ . Theuse of a different signal-to-noise cutoff in polarization de-tection when making the azimuth plots may also lead todifferent azimuthal modulations.In the Braun et al. (2010) work, instead of directly pre-dicting the azimuth modulations of the Faraday depthand polarized intensity, trends of the integrated line-of-sight ( (cid:82) B || dl ) and plane-of-the-sky ( (cid:82) B ⊥ dl ) magneticfields were presented. The observed Faraday depth isa convolution of the line-of-sight magnetic field and thethermal electron density distribution while the observedpolarization is a convolution of the plane-of-the-sky mag-netic field and the cosmic ray electron density distribu-tion. Unless the thermal electron and cosmic ray elec-tron distributions are completely uniform across the en-tire disk, trends of Faraday depth and polarized intensitymay not reflect those of (cid:82) B || dl and (cid:82) B ⊥ dl . Moreover, thepredicted modulations from their magnetic field modelwere plotted at a single fixed galacto-centric radius, butnot averaged across all radii as how the observed trendswere presented. It is unclear whether these character-istic modulations persist when Faraday depth and po-larization averaging has been performed across all radiiwithin an azimuth bin. In order to quantitatively com-pare the Braun et al. (2010) model and our observedFaraday depth and polarized intensity trends, a realisticdiffuse ionized gas and cosmic ray electron distribution For M51, the position angle of the receding major axis is 170 ◦ (Shetty et al. 2007). of the modeled galaxy is needed so that trends of Fara-day depth and polarized intensity as a function of az-imuth can be predicted directly and compared with theobserved values. This topic is outside of the scope of thecurrent paper and will be addressed in our forthcomingwork. A Search for Polarized Emission from the Far-sideHalo
Braun et al. (2010) proposed that polarized emissionof a sample of SINGS galaxies at cm wavelengths arisesfrom a zone 2-4 kpc above the mid-plane in the near-sidehalo. If the galaxy is symmetric, then there should exista similar synchrotron emitting zone 2-4 kpc below themid-plane. Braun et al. (2010) found evidence for thisfar-side halo in their WSRT Faraday cube of M51 at alevel of 0.6 mJy beam − at a resolution of 90”. Sincethis polarized emission from the far-side halo must firstpropagate through the galactic disk before reaching us,it will be dispersed to Faraday depths of ∼ ±
200 radm − due to the disk magnetic field and it will also be de-polarized (by external Faraday dispersion effects) due tostrong fluctuations of magnetic fields and electron den-sities in the turbulent disk. We search for the polarizedemission from the far-side halo in our new Jansky VLAL band data of M51. To increase our sensitivity to weakand extended emission, we smooth the input Stokes Q and U image cubes to a resolution of 90”. We then pro-ceed by remaking the Faraday depth cube at this coarserresolution. Noise in the resulting Faraday depth cubeis ∼ − . Hence, if there exists polarizedemission > − , it should be easily detectedat a signal-to-noise level above 5. In Figure 17, we showthe polarized emission of M51 at Faraday depths of − − and +200 rad m − .Contrary to the findings of Braun et al. (2010), we donot detect significant polarized emission at these Fara-day depths. There are several possible reasons for thisnon-detection. First, the polarized emission detected byBraun et al. (2010) may be associated with the first side-lobes of the WSRT RMSF, as the authors have also cau-tioned in their work. The first side-lobes of the WSRTRMSF are located at roughly the same Faraday depthsas the expected emission from the far-side halo ( ±
200 radm − ), and they have an amplitude ∼
78% of the mainpeak (see Figure 3). Our new Jansky VLA data covera more continuous frequency range and have suppressedthe first side lobes amplitude down to <
35% of the mainpeak. This makes it much more difficult to confuse side-lobes with real emission when deconvolving the dirtyFaraday depth spectra. Secondly, we have performedspatial smoothing in the input
Q U cubes (i.e. in λ do-main) while Braun et al. (2010) smoothed the complexpolarization in Faraday depth space. Since RMCLEANis a non-linear operation, artifacts can arise when spa-tial smoothing is done to the deconvolved Faraday depthcube.If the far-side halo indeed exists, our non-detection canbe used to estimate turbulent properties in the galac-tic mid-plane. We assume that the intrinsic brightnessof the far-side halo is the same as that of the near-sidehalo (median value ∼ − at 90” resolution).From our Faraday depth cube at 90” resolution, we finda 3 σ upper limit of the emission from the far-side haloto be 0.36 mJy beam − . We further assume that theturbulent mid-plane acts as an external inhomogeneousFaraday screen that depolarizes emission from the far-side halo according to P/P = e − σ λ . This leads to alower limit estimation for the RM variance σ RM ≥
20 radm − within the 90” beam. We note that this is consistentwith the RM variance inferred from the structure func-tion analysis using 3cm and 6cm RM data (Section 6.1). ROTATION MEASURE STRUCTURE FUNCTIONANALYSIS
Structure function of rotation measure contains uniqueinformation on the structure in the magneto-ionicmedium, including turbulence. The second order struc-ture function is defined as SF RM ( δθ ) = (cid:104) [RM( θ ) − RM( θ + δθ )] (cid:105) , (8)where θ is the projected angular separation between twosight lines and the angular brackets denote the averageof independent measurements having the same range ofangular separation δθ . Uncertainties in the RM measure-ments contribute to the observed RM structure functionin the form of a DC offset (cid:104) δ RM( θ ) + δ RM( θ + δθ ) (cid:105) ,which is subtracted from the measured SF RM ( δθ ).Obtaining robust uncertainties associated with SF RM ( δθ ) is essential to correctly interpret the struc-ture function. Since [RM( θ ) − RM( θ + δθ )] is apositive-only quantity, values within a δθ bin are notGaussianly distributed. Therefore, Gaussian statisti-cal indicators of the scatter (standard deviation andstandard error in the mean) may no longer be goodrepresentations of the true scatter within a δθ bin. Toobtain realistic error estimates, we utilize the bootstrapmethod. While the regular bootstrap approach willspatially de-correlate Faraday rotation between sightlines and erase all features in SF RM ( δθ ), the markedpoint bootstrap method is well suited for spatiallycorrelated data : instead of resampling the individualRM values, we resample [RM( θ ) − RM( θ + δθ )] in orderto preserve spatial correlations (Loh 2008). The originalRM sample is bootstrapped 500 times and the standarddeviation of the computed SF RM resampled ( δθ ) in each δθ bin provides a good estimate for the error in SF RM ( δθ ).Throughout this analysis, we assume that the MilkyWay foreground towards M51 is constant (+ 13 radm − ), and hence it does not contribute to the structurefunction. Any features seen in SF RM ( δθ ) must thereforebe intrinsic to M51.Since RM is the line-of-sight integral of the magneticfield weighted by the thermal electron density, features inthe RM structure function could reflect structures in elec-tron densities, magnetic fields and/or the path length. Inthe following analysis, we assume that the polarized emis-sion at a given wavelength (3/6 cm and L band) acrossM51 emerges from the same physical depth in the galaxy:for the 3/6cm data through the entire galaxy, and for theL band data from the top of the synchrotron disk. Hence,structure function features reflect only structures in elec-tron densities and magnetic fields but not the path lengthdifference. Since both the short wavelength (3,6 cm) and This variant of the traditional bootstrap method has beenused to estimate errors of the two-point correlation function ofgalaxy distribution. long wavelength (L band) RM maps exhibit sign rever-sals, we are confident that fluctuations in magnetic fieldsmust be present and must contribute to the observedRM structure functions. To disentangle which featuresin the RM structure function are due to the thermal elec-tron distribution or magnetic field structures requiresindependent information on the path-length integratedelectron density distribution. This is provided by emis-sion measure (EM) structure functions using extinction-corrected H α data or by pulsar dispersion measure (DM)structure functions (only possible for Milky Way works).However, a joint interpretation is often challenging sincethe actual path length probed by EM, DM and RM islikely to be very different. Turbulence in the Disk: RM Structure FunctionUsing 3 and 6 cm Data
We construct SF RM ( δθ ) using the rotation measuremap derived from 3 and 6 cm VLA data by Fletcheret al. (2011) which has a typical RM error of 10 radm − . The resulting RM structure function is shown inFigure 18. We do not display the structure function onscales smaller than the resolution of the RM map (log δθ< − δθ< δθ = − δθ =3’, or a physical scale 6.7 kpc). Below thisscale, SF RM ( δθ ) is flat with a slope of 0.10 ± ± σ = 10 . rad m − , which corresponds to σ RM = 56 rad m − (sincewe do not spatially resolve the turbulence, the true RMvariance is a factor of √ N larger, where N is the numberof turbulent cells within the beam). Since 3 and 6 cmpolarization data suffer much less Faraday depolariza-tion effects, the emission likely probes through the entiregalaxy. Hence, the large RM variance reflects the strongfluctuation in electron densities and magnetic fields inthe turbulent mid-plane of M51. The implied σ RM isin agreement with the non-detection of the far-side syn-chrotron emitting halo in Section 5.2.2, from which wederived a lower limit for σ RM ≥
20 rad m − . We notethat σ RM in M51’s disk is similar in magnitude to thatfound in the Large Magellanic Cloud ( ∼
80 rad m − )(Gaensler et al. 2005; Mao et al. 2012b) using struc-ture function of polarized extragalactic sources behindthe LMC.On scales smaller than 3’, SF RM ( δθ ) has a much shal-lower slope than the expected value of 5/3 (or 2/3) for3D (or 2D) Kolmogorov turbulence. If fluctuations inelectron densities and magnetic fields in M51’s disk in-deed follow the Kolmogorov spectrum, then the structurefunction must turnover at a scale below the resolution ofthe current 3 and 6 cm RM map – the outer scale ofturbulence in the mid-plane of the galaxy thus must besmaller than 15”, which corresponds to 555 pc at the dis-tance of M51. This is consistent with the recent result ofHoude et al. (2013), who found a turbulent correlation0 Mao et al. scale <
100 pc by applying angular dispersion analysisto VLA polarization angle data at 6 cm. Based on thedispersion of Faraday rotation derived from 3 and 6 cmVLA data, Fletcher et al. (2011) also independently es-timated a turbulent cell size of ∼
50 pc. We note thatthe lack of features in SF RM ( δθ ) on scales <
3’ indicatesthat the random magnetic field and electron density fluc-tuations far dominates over any features produced by thecoherent magnetic field in the disk and the halo of M51.To verify that the rise in the structure function onscales larger than 3’ is real and not merely due to thesampling of pixels in the field of view, we scramble theRM values across the image and recompute the structurefunction. Since the break at 3’ disappears from the struc-ture function and SF RM ( δθ ) becomes completely flat, weconclude that this feature is unlikely an artifact. A large-scale RM gradient produced by coherent fields in mag-netic arms of M51 could be responsible for producingthis steep feature in SF RM ( δθ ). Similar features havealso been detected in the RM structure function of NGC6946 (Beck 2007), for which the authors also suggesteda magnetic-arm origin. We verify that a large-scale az-imuthal magnetic field in a ring-like configuration ori-ented at an inclination angle of 20 ◦ could indeed producean RM structure function with a slope of ∼ Turbulence in the Halo: RM Structure Functionusing L band data
The RM structure function computed using our newJansky VLA Faraday depth map at L band is shownin Figure 19. We do not display the structure func-tion on scales smaller than the beam size (log δθ< − δθ< SF RM ( δθ ) rises at small scales and has a turnover atlog δθ = − δθ = 0.5’, or a physical scale of 1 kpc).At larger angular scales, the structure function remainsrather flat but with a clear bump at log δθ = − δθ =2.4’, or a physical scale of 5.3 kpc). SF RM ( δθ ) then steep-ens once again at log δθ = − δθ = 4.8’, or a physicalscale of 10 kpc) to a slope of ∼ σ = 200 rad m − , which corresponds to an RM variance of 10 radm − , is similar to the value computed from RMs of ex-tragalactic sources at high Galactic latitudes in our own Milky Way (Mao et al. 2010; Schnitzeler 2010). This σ RM is a factor of 5 smaller than that implied from the shortwavelength RM structure function in Section 6.1. Thisonce again reflects the difference in the polarization hori-zon when observing at different wavelengths at similarresolutions: unlike short wavelength polarized emissionthat probe through the entire galaxy, L band polarizedemission originates from a shallower depth in the galaxy(i.e. the near-side halo) and hence do not probe throughthe turbulent mid-plane where large fluctuations in elec-tron densities and magnetic fields produce a much largerRM variance.In order to interpret the RM structure function pro-duced in the less turbulent halo, where random fields nolonger dominate over the uniform component, we mustfirst remove any structure function features produced bythe large-scale halo magnetic field. In Figure 19, we plotthe structure function expected from the Fletcher et al.(2011) best fit large-scale halo field model as the dashedcurve (after subtracting the noise contribution due touncertainties in the modeled parameters). Although thelarge-scale halo field alone cannot account for the over-all amplitude of the observed RM structure function atL band and it cannot produce the break at δθ =0.5’, itis evident that the bump in SF RM ( δθ ) at δθ ∼ > δθ ) ∼− σ =10 rad m − ( σ RM ∼ − ) as the dottedline in Figure 19. The sum of these two structure func-tions is denoted as the solid curve in the same Figure.One immediately recognizes that several key features of SF RM ( δθ ) can be reproduced by the solid curve: (i) theslope of the structure function below < ∼ QU fit in Section 3.4, where we find lit-tle wavelength-dependent depolarization across L band.This is because any beam depolarization effects causedby an external inhomogeneous Faraday screen is domi-nated by the largest fluctuations on scales comparableto the outer scale of turbulence. In our case, the tur-bulence is spatially resolved (the beam of our L bandobservations ∼ (cid:28) The Source of Turbulence in the Halo of M51
The implied 1 kpc outer scale of turbulence in the haloof M51 is considerably larger than that inferred in thedisk of M51 ( <
100 pc). It also exceeds the scales in-ferred in the Milky Way based on structure functionsof extragalactic RMs: few pc in arm regions and ∼ −
638 pc) than in the disk (40 −
52 pc) in M51by fitting an analytical depolarization model to observa-tions at 3, 6 and 20 cm. However, the authors acknowl-edge that the derived halo cell size is likely very uncer-tain since the fitting algorithm has the assumption thatthe beam diameter (600pc) largely exceeds the turbulentcell size, which is likely not true in the halo. In galacticdisks, HII regions and supernova remnants are thought tobe the main drivers of turbulence, while other processesmay dominate the energy input in galactic halos (seee.g., Beck et al. 1996; Elmegreen & Scalo 2004). In thissection, we specifically consider two possible drivers ofturbulence in the halo of M51: superbubbles and Parkerinstability.The co-location of a hole in neutral hydrogen anda rotation measure gradient of diffuse polarized emis-sion at L band in the nearby face-on galaxy NGC 6946(Heald 2012) hints at a possible connection between en-ergetic events that form superbubbles and structures themagneto-ionic medium. We search for coincidence of RMgradients and HI holes in M51 by using the HI hole cata-logue from the THINGS survey (Bagetakos et al. 2011).More than 2/3 of the HI holes in the disk of M51 arelocated in regions devoid of L band polarized emissionbecause holes identified in HI maps are likely in thedisk where the density contrast is higher but where syn-chrotron emission is completely depolarized at L band.We do not find significant RM gradients across HI holesthat do fall within the polarized regions. The rarity ofsuch a detection is due to the fact that (1) the HI holehas to be at a certain azimuthal position in order to pro-duce an observable RM gradient, and (2) vertical shear-ing will destroy polarization signatures of the holes sothey have to be in a certain age range. Therefore, thelack of correlation between RM gradient and HI holesshould not be taken as an evidence against superbub-bles being the energy injection scale in the halo of M51.Although the median diameter ∼
700 pc of HI holes inM51 is seemingly too small compared to the outer scaleof 1 kpc implied from our structure function analysis,this can be reconciled by taking into account that hotgas parcels from supernova remnants and superbubbleswould expand as they rise above/below the mid-plane. Inthe recent numerical simulation of the magnetized ISMof Gent et al. (2013), the authors found an increase inthe correlation length scale by a factor of 1.5 − < (Parker 1966) which is theinstability experienced by a purely horizontal magneticfield under vertical perturbations. Cosmic ray pressureinflates magnetic field lines into the halo causing materialto slide down the field lines, which in turn further inflatesthe field due to buoyancy. In fact, Parker (1992) sug-gested that the entire surface of a galactic disk (on bothsides) is packed with Parker loops with a characteristicscale of 0.1 − ∼ . We note that it ischallenging to identify by eye positive and negative RMpatches separated by ∼ << also known as magnetic buoyancy instability. We suggest that it is much easier to search for signatures ofParker loops in RM maps produced by long wavelength data be-cause only the near-side halo of the galaxy is visible in polarization.At shorter wavelengths, polarized emission probes through the tur-bulent mid-plane, and thus relatively weak features produced byParker instabilities would be completely washed out by strong RMfluctuations in the mid-plane. In fact, Fletcher et al. (2011) werenot able to find any periodic RM pattern in the short wavelength(3 and 6 cm) RM map and took this as a lack of evidence for Parkerloops. Mao et al.
RM fluctuations perpendicular to the direction of thelarge-scale fields. While a magnetic field reversal alongan arc feature in the halo of M31 had been interpreted asa Parker instability loop anchored in a massive HI cloud(Beck et al. 1989), the evidence that we present in thiswork is the first to suggest that Parker loops could bepresent across the entire galactic halo, and not just in asingle loop associated with an individual gas cloud. Wenote that our argument presented here is solely basedon the characteristic scale of RM fluctuation. As futurework, comparing mock Faraday rotation maps and struc-ture functions generated from numerical simulations ofParker instability with wide-band polarization observa-tions is crucial to further constrain properties of turbu-lence in galactic halos.If the observed 1 kpc correlation length in the halo ofM51 indeed corresponds to Parker instability loops, ithas important implications on the large-scale magneticfield amplification mechanism in galaxies. Inflated mag-netized loops in the halo together with magnetic recon-nections can lead to the fast cosmic-ray-driven dynamoeffect (Parker 1992; Hanasz et al. 2004). This more effi-cient α -effect can lead to much shorter growth time thanthe classical α - ω dynamo, on times scales of few hundredMyrs. This process will allow coherent magnetic fieldsto be established in young galaxies, as well as galax-ies undergoing tidal interactions, such as the MagellanicClouds (Gaensler et al. 2005; Mao et al. 2008, 2012b)and M51 itself. Parker loops allow the transportation ofmagnetic flux into the halo and may be the key ingre-dient for the existence of large-scale magnetic fields ingalactic halos. CONCLUSIONS AND FUTURE WORK
In this paper, we have presented new wide-band, multi-configuration Jansky VLA polarization observations ofM51 at L band which have the best λ coverage in thisfrequency range to date. For the first time, we have fittedthe observed diffuse polarized emission from an externalgalaxy as a function of λ directly to various models ofthe Faraday rotating and synchrotron emitting medium.The majority of sight lines through M51 do not exhibit λ -dependent depolarization across L band and thereforetheir Q ( λ ) and U ( λ ) are well fitted by Faraday rotationoccurring in an external uniform screen. This is consis-tent with the picture of polarized emission at L bandbeing originated from the top layer of the synchrotronemitting disk and then Faraday rotated in the thermalhalo. Using the deduced relative extent of the syn- chrotron emitting gas and the thermal gas (warm ionizedmedium), the synchrotron cooling time scale, and a cos-mic ray diffusion coefficient similar to that in the MilkyWay, the warm ionized medium scale height in M51 isestimated to be at least 1.2 kpc. The predicted Fara-day depth distribution from the Fletcher et al. (2011)bisymmetric halo magnetic field model at L band and ourobserved RMs can be reconciled by introducing an addi-tional vertical coherent magnetic field component (RM= − − ) in the halo, which could be of primordialor dynamo origin. With the improved wavelength cov-erage of our observations and hence a narrower rotationmeasure spread function with lower side-lobe levels, wedo not detect any polarized emission from the far-sidehalo at Faraday depths of ∼ ±
200 rad m − as suggestedby Braun et al. (2010). This non-detection implies thatthe turbulent mid-plane must have an RM variance thatexceeds 20 rad m − , which is consistent with the shortwavelength (3 cm and 6 cm) RM variance. The RMstructure function derived using 3 and 6 cm data (whichprobe through the entire galaxy) shows no break on smallangular scales, implying that the energy injection scalein M51’s disk is smaller than the resolution of the obser-vation ( <
560 pc). On the other hand, the RM structurefunction of L band data (which only probe through thehalo) has a break at 1 kpc. Both superbubble-drivenand Parker instability-driven turbulence could producecorrelation lengths of approximately 1 kpc in the halo ofM51.Throughout this paper, we have demonstrated the im-portance of wide-band polarization data when extract-ing properties of magnetic fields and turbulence in theunderlying magneto-ionic medium. Future wide-bandpolarization observations between 6 cm and 18 cm (Sband) can probe deeper into the galaxy and may revealconnections between large-scale fields in the disk and inthe halo and possibly reveal a stronger correlation be-tween HI holes and the RM distribution. We can con-firm or rule out the existences of Parker loops in M51’shalo by obtaining even higher spatial resolution L bandpolarization data to search for short wavelength fluctua-tions in Faraday rotation perpendicular to the directionof the large-scale magnetic field. Finally, to understandglobal large-scale halo magnetic field configurations andto evaluate the universality of superbubble and Parkerinstability-driven turbulence in galactic halos, a system-atic polarization survey of mildly inclined galaxies at Land S bands would be of crucial importance.
APPENDIX
MODELS FOR THE FARADAY ROTATING MEDIUM USED FOR THE DIRECT
Q U
FITS
We summarize in the different models which are used to fit the observed Q / I and U / I of extragalactic sources andthe diffuse polarized emission.First, we consider the simplest model where intrinsic polarized background emission with a fractional polarization p and an intrinsic polarization angle φ propagates through a uniform foreground screen which has a rotation measureRM. In this case, the observed fractional complex polarization (cid:126)p is (cid:126)p = p e i ( φ +RM λ ) . (A1)In the case of an inhomogeneous foreground screen, under the assumption that the turbulent cell size is much smallerthan the beam size, polarized intensity is reduced due to the varying RM within the beam by the so-called external3Faraday dispersion effect (Burn 1966; Tribble 1991). The fractional complex polarization becomes (cid:126)p = p e − σ λ e i ( φ +RM λ ) , (A2)where the RM variance σ RM quantifies the fluctuations in electron densities and magnetic fields within the beam.If only part of the background polarized radiation passes through the foreground inhomogeneous screen, the complexpolarization depends on the fraction of the background source covered by the inhomogeneous screen f c , (cid:126)p = p [ f c e − σ λ + (1 − f c )] e i ( φ +RM λ ) . (A3)We then consider the case where there is differential depolarization intrinsic to the background source (the classicalBurn slab with a total Faraday depth R, Burn (1966)) and the radiation then pass through a uniform foregroundscreen with a Faraday rotation of RM (cid:126)p = p sin Rλ Rλ e i ( φ + Rλ +RM λ ) . (A4)If there is internal Faraday dispersion occurring within the background source characterized by σ RM , then theexpected complex fractional polarization is: (cid:126)p = p − e − S S e i ( φ +RM λ ) , where S = 2 λ σ − λ R (A5)Finally, there is the possibility that the observed polarized emission is the sum of two different polarization compo-nents, A and B: (cid:126)p = p ,A e i ( φ ,A +RM A λ ) + p ,B e i ( φ ,B +RM B λ ) . (A6) Facilities:
The Very Large Array The National Radio Astronomy Observatory is a facility of the National ScienceFoundation operated under cooperative agreement by Associated Universities, Inc.The authors thank Rainer Beck and Henrik Junklewitz for a close reading of the manuscript. The authors acknowl-edge fruitful discussions with Judith Irwin, Bob Lindner, Snezana Stanimirovi´c, Shane O’Sullivan, Jay Gallagher,Tony Wong, Cathy Horellou, Chris Hales, Amanda Kepley, Jamie Farnes, Blakesley Burkhart, Xiao Hui Sun, LarryRudnick, Peter Tribble, Steve Spangler, George Heald, Anvar Shukurov, Tim Robishaw, Rick Perley, Aritra Basu andWalter Max-Moerbeck.REFERENCES
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Coordinates of polarized extragalactic background sources in the field of viewSource Name RA(J2000)(hms) DEC(J2000)(dms) Distance from the center of FOV(’)J1330+4703a 13:30:45.085 +47:03:08.76 12.1J1330+4703b 13:30:45.286 +47:03:28.76 11.9J1329+4658a* 13:29:32.315 +46:58:45.38 13.2J1329+4658b* 13:29:28.796 +46:58:49.34 13.3J1329+4658c* 13:29:39.153 +46:59:09.45 12.5J1331+4713a* 13:31:22.554 +47:13:21.32 15.3J1331+4713b* 13:31:27.452 +47:13:01.08 16.1J1330+4710 13:30:15.978 +47:10:23.36 4.0J1329+4717* 13:29:41.628 +47:17:35.46 6.5J1330+4730 13:30:32.513 +47:30:55.08 20.6J1329+4706 13:29:31.089 +47:6:27.37 6.2 M a o e t a l . Table 2
Results of Stokes QU fit for all polarized extragalactic sources within 20’ ofthe field center.Source Name Model p ,A φ ,A RM A σ RM f c R p ,B φ ,B RM B χ r DOF BIC(fraction) (rad) (rad m − ) (rad m − ) (fraction) (rad m − ) (fraction) (rad) (rad m − )J1330+4703a i 0.067 ± ± ±
1. - - - - - - 2.727 87. -524. ii ± ± ± ± ± ± ± ±
3. 0.58 ± ± ± ±
1. - - +27. ±
2. - - - 1.611 86. -619.v 0.098 ± ± ±
4. 11. ±
3. - +14. ±
9. - - - 1.544 85. -622.vi 0.067 ± ± ±
1. - - - 0.007 ± ± ±
7. 2.344 84. -553.J1330+4703b i ± ± ±
2. - - - - - - 2.235 87. -459.ii 0.046 ± ± ±
2. 0. ±
4. - - - - - 2.261 86. -455.iii 0.046 ± ± ±
2. 0. ±
0. 1. ±
0. - - - - 2.287 85. -455.iv 0.046 ± ± ±
7. - - -0. ±
10 - - - 2.261 86. -455.v 0.046 ± ± ±
7. 0. ±
5. - +0. ±
10 - - - 2.287 85. -452.vi 0.005 ± ±
1. +112. ±
20 - - - 0.045 ± ± ±
3. 2.222 84. -456.J1329+4658a i 0.24 ± ± ±
2. - - - - - - 3.171 87. -117. ii ± ± ±
2. 12. ±
1. - - - - - 2.375 86. -185.iii 1. ±
0. 0.26 ± ±
2. 26. ±
2. 0.82 ± ± ± ±
2. - - +34. ±
3. - - - 2.449 86. -179.v 0.6 ± ± ±
4. 20. ±
5. - +22. ±
10 - - - 2.281 85. -192.vi 1. ±
0. -0.6 ± ±
3. - - - 0.83 ± ± ±
4. 2.280 84. -190.J1329+4658b i 0.16 ± ± ±
3. - - - - - - 2.889 87. -139. ii ± ± ±
3. 14. ±
1. - - - - - 2.005 86. -214.iii 0.7 ± ± ±
2. 21. ±
3. 0.86 ± ± ± ±
3. - - +38. ±
2. - - - 2.104 86. -206.v 0.7 ± ± ±
4. 23. ±
6. - -42. ±
10 - - - 1.882 85. -223.vi 0.5 ± ± ±
7. - - - 0.6 ± ± ±
6. 2.060 84. -206.J1329+4658c i 0.124 ± ± ±
1. - - - - - - 35.119 87. 2163.ii 0.202 ± ± ± ± iii ± ± ± ± ± ± ± ± ± ± ± ±
1. 16.0 ± ±
4. - - - 3.747 85. -567.vi 0.121 ± ± ±
1. - - - 0.011 ± ± ±
8. 30.130 84. 1650.J1331+4713a i 0.106 ± ± ± ii ± ± ± ± ± ± ± ±
4. 0.9 ± ± ± ± ± ± ± ±
3. 8. ±
2. - -14. ±
6. - - - 2.623 85. -655.vi 0.04 ± ± ±
5. - - - 0.11 ± ± ±
2. 2.587 84. -656.J1331+4713b i 0.117 ± ± ± ii ± ± ± ±
1. - - - - - 5.725 86. -340.iii 0.127 ± ± ± ±
1. 1. ±
0. - - - - 5.792 85. -338.iv 0.127 ± ± ±
2. - - -15. ±
3. - - - 5.718 86. -341.v 0.129 ± ± ±
4. -5. ±
3. - -10. ±
7. - - - 5.727 85. -342.vi 0.011 ± ± ±
20 - - - 0.111 ± ± ±
2. 5.347 84. -376.J1330+4710 i ± ± ± ± ± ± ± ± ± ± ± ±
0. - - - - 0.973 85. -756.iv 0.082 ± ± ± ±
2. - - - 0.961 86. -760.v 0.082 ± ± ±
3. 4. ±
2. - +8. ±
6. - - - 0.961 85. -757.vi 0.074 ± ± ± ± ± ±
9. 0.851 84. -764.J1329+4717 i 0.111 ± ± ±
1. - - - - - - 2.555 87. -394. ii ± ± ± ± ± ± ± ±
2. 0.70 ± ± ± ±
1. - - +31. ±
1. - - - 1.185 86. -510.v 0.21 ± ± ±
3. 17. ±
1. - -2. ±
10 - - - 1.085 85. -516.vi 0.8 ± ± ±
6. - - - 0.8 ± ± ±
6. 1.136 84. -509.J1329+4706 i 0.172 ± ±
6. +22. ±
2. - - - - - - 1.703 87. -265. ii ± ±
4. +21. ±
2. 12.0 ± ± ± ±
2. 20. ±
3. 0.79 ± ± ±
5. +3. ±
2. - - +35. ±
2. - - - 1.082 86. -316.v 0.43 ± ±
10 +22. ±
3. 20. ±
3. - -16. ±
10 - - - 0.998 85. -321.vi 3. ±
9. 176. ±
10 +11. ±
6. - - - 3. ±
9. 4.56 ± ±
3. 1.047 84. -314.
Table 2 — Continued
Source Name Model p ,A φ ,A RM A σ RM f c R p ,B φ ,B RM B χ r DOF BIC(fraction) (rad) (rad m − ) (rad m − ) (fraction) (rad m − ) (fraction) (rad) (rad m − ) Table 3
Properties of extragalactic polarized background sources that are in both thiswork and Farnes et al. (2013)Source Name FD Fractional Polarization FD Fractional Polarizationat L band at L band at 610 MHz at 610 MHz(rad m − ) (rad m − )J1330+4703a +16.6 ± ± ± ± ± ± ± ± ± ± − ± ± ± ± − ± ± ± ± ± ± ± ± ± ± Mao et al.
Figure 1.
Stokes I multi-frequency synthesis map of M51 at 1.478 GHz using the new wide-band multi-configuration VLA data. Figure 2.
Spectral index map of M51 at L band using the new wide-band multi-configuration VLA data. Pixels with signal-to-noise lessthan 3 have been masked. Mao et al. −2000 −1000 0 1000 2000Faraday Depth (rad m − )0.00.20.40.60.81.0 R M S F This paperWSRT Heald et al. (2009) work
Figure 3.
Comparison between the rotation measure spread function of our data (solid line) and that of Heald et al. (2009) (dashed line).The more continuous frequency coverage of our Jansky VLA data not only provides an improved resolution in Faraday depth space, it alsosignificantly suppresses the amplitude of the first side-lobe from ∼
78% down to < λ (m )0.000.050.10 Q /I and U /I −0.04−0.02 0.00 0.02 0.04 0.06 0.08Q/I0.000.050.10 U /I λ (m )−0.050.000.050.100.150.20 F r a c t i ona l P o l a r i z a t i on λ (m )−3−2−10123 P o l a r i z a t i on A ng l e (r ad ) Figure 4.
Polarized emission of source J1330+4703b and the corresponding best fit (solid line) to the uniform external Faraday screenmodel. We show the fractional Q (filled squares) and U (filled circles) versus λ in the top left plot. The top right plot shows the Q/I - U/I track. The bottom left plot shows the fractional polarization versus λ trend. The bottom right plot shows how polarization angles varyacross L band. Mao et al. λ (m )−0.050.000.050.100.15 Q /I and U /I −0.05 0.00 0.05 0.10 0.15 0.20Q/I0.000.020.040.060.080.100.12 U /I λ (m )0.000.050.100.150.200.25 F r a c t i ona l P o l a r i z a t i on λ (m )−0.50.00.51.01.5 P o l a r i z a t i on A ng l e (r ad ) Figure 5.
Polarized emission of source J1329+4717 and the corresponding best fit (solid line) to the inhomogeneous external Faradayscreen model. This figure has the same layout as in Figure 4. − )05101520253035 H ea l d e t a l. L b a nd F D ( r a d m − ) Figure 6.
Comparison of Faraday depths of extragalactic background sources reported in Heald et al. (2009) against the Faraday depthsderived from our Jansky VLA L band data. The solid line of slope 1 indicates where Heald et al. (2009) and our VLA Faraday depthsagree with each other. Mao et al.
Figure 7.
Faraday depth distribution of M51 at L band derived from RM synthesis. The color scale is in the unit of rad m − . Figure 8.
The de-biased polarized intensity of M51 at the peak of the Faraday depth spectrum. Mao et al.
Figure 9.
Faraday rotation-corrected intrinsic magnetic field orientations (orange line segments) and the polarized intensity contour (blue)at 65 µ Jy beam − overlaid on a B band Hubble Space Telescope image of M51 (Mutchler et al. 2005). Figure 10.
A map of M51 showing the spatial distribution of the best-fit model from directly fitting to Stokes Q and U . Sight lines with Q U versus λ behaviors best fitted by a uniform external Faraday screen are denoted by the white pixels. Sight lines with Q U versus λ behaviors best fitted by an inhomogeneous external Faraday screen are denoted by the black pixels. Pixels with insufficient signal-to-noisedetection in polarization have been blanked (grey). Mao et al. λ (m )−0.10.00.10.2 Q and U ( m Jy bea m − ) −0.1 0.0 0.1 0.2 0.3 0.4Q (mJy beam − )−0.2−0.10.00.10.2 U ( m Jy bea m − ) λ (m )−0.10.00.10.20.30.40.5 P o l a r i z ed I n t en s i t y ( m Jy bea m − ) λ (m )−10123 P o l a r i z a t i on A ng l e (r ad ) Figure 11.
Polarized emission along a sight line towards M51 and its corresponding best fit (solid line) to the uniform external Faradayscreen model. This figure has the same layout as in Figure 4. λ (m )−0.2−0.10.00.10.2 Q and U ( m Jy bea m − ) −0.3 −0.2 −0.1 0.0Q (mJy beam − )−0.2−0.10.00.10.20.30.4 U ( m Jy bea m − ) λ (m )−0.10.00.10.20.30.4 P o l a r i z ed I n t en s i t y ( m Jy bea m − ) λ (m )−4−3−2−10 P o l a r i z a t i on A ng l e (r ad ) Figure 12.
Polarized emission along a sight line towards M51 and its corresponding best fit (solid line) to the inhomogeneous externalFaraday screen model. This figure has the same layout as in Figure 4. Mao et al. −2.5 −2.0 −1.5 −1.0 −0.5log( δθ [deg] )−4.0−3.5−3.0−2.5−2.0−1.5 l og ( S F [ ( m Jy bea m − ) ] complex polarization polarized intensity Figure 13.
Structure function of complex polarization (dots) and polarized intensity (crosses) across M51 at L band. Figure 14.
Top: Faraday depth distribution of M51 at L band after removing the constant Milky Way contribution of +13 rad m − .Middle : The predicted Faraday depth distribution of M51 from the Fletcher et al. (2011) bisymmetric halo model. Bottom: The predictedFaraday depth distribution of M51 from the Fletcher et al. (2011) bisymmetric halo field with the addition of a vertical field that producesa Faraday depth of − − . Mao et al. ° )0.000.020.040.060.080.100.120.14 P eak P o l a r i z e d I n t e n s i t y ( m Jy b ea m − ) Figure 15.
Peak polarized intensity in our Faraday depth cube plotted as a function of azimuth. Only pixels with signal-to-noise detectionin polarization greater than 7 have been averaged in 10 ◦ bins across all radii. Error bars represent the standard deviation of polarizedintensity within a bin. ° )−20−10010203040 P eak F a r a d ay D e p t h ( r a d m − ) Figure 16.
Peak Faraday depth in our Faraday depth cube plotted as a function of azimuth. Only pixels with signal-to-noise detection inpolarization greater than 7 have been averaged in 10 ◦ bins across all radii. Error bars represent the standard deviation of Faraday depthwithin a bin. Mao et al.
Figure 17.
Polarized emission of M51 at a Faraday depth of −
180 rad m − (left) and +200 rad m − (right) at a resolution of 90”. Thecontour roughly outlines the location of M51: it is the polarized intensity contour at the peak of the Faraday depth cube at a level of 0.96mJy beam − . No significant polarized emission is detected. −2.5 −2.0 −1.5 −1.0 −0.5log( δθ [deg] )3.63.84.04.24.44.6 l og ( S F R M [ r ad m − ] ) Figure 18.
Rotation measure structure function of M51 constructed from the short wavelength (3 and 6 cm) RM map. The structurefunction has been binned in equal log interval of 0.1. The error bars denote the bootstrapped uncertainty of SF RM in each bin. The solidline is a broken power law fit to SF RM with a break at 3’. Mao et al. −2.5 −2.0 −1.5 −1.0 −0.5log( δθ [deg] )1.01.52.02.53.0 l og ( S F R M [ r ad m − ] ) Figure 19.
Rotation measure structure function of M51 constructed from our newly derived L band RMs. The structure function has beenbinned in equal log interval of 0.1. The error bars denote the bootstrapped uncertainty of SF RM in each bin. The dashed line representsthe RM structure function expected from the best fit large-scale halo magnetic field in Fletcher et al. (2011). The dotted line representsthe structure function expected from three-dimensional Kolmogorov turbulence that has an outer scale of 29” (approximately 1 kpc at thedistance of M51) and a variance of 7 rad m −2