Mon.Not.R.Astron.Soc.000,1–12(0000) Printed2February2008 (MNLATEXstylefilev1.4) A New approach for a Galactic Synchrotron Polarized Emission Template in the Microwave Range G. Bernardi,1,2 E. Carretti,1 R. Fabbri,3 C. Sbarra,1 S. Poppi,4 S. Cortiglioni1 1C.N.R./I.A.S.F. Bologna, ViaGobetti 101, I-40129 Bologna, Italy 2Dipartimento di Astronomia, Universitàdegli Studi di Bologna, ViaRanzani 1, I-40127 Bologna, Italy 3Dipartimento di Fisica, Universitàdi Firenze, Via Sansone 1, I-50019 SestoFiorentino (FI), Italy 4C.N.R./I.R.A. Bologna, Via Gobetti 101, I-40129 Bologna, Italy 3 0 0 2 24October 2002 n a J ABSTRACT We present a new approachin modelling the polarized Galactic synchrotronemission 7 in the microwave range (20-100 GHz), where this radiation is expected to play the 2 leading role in contaminating the Cosmic Microwave Background (CMB) data. Our 1 methodisbasedonrealsurveysandaimsatprovidingtherealspatialdistributionsof v both polarized intensity and polarization angles. Its main features are the modelling 1 of a polarization horizon to determine the polarized intensity and the use of starlight 4 optical data to model the polarization angle pattern. Our results are consistent with 5 several existing data, and our template is virtually free from Faraday rotation effects 1 as required at frequencies in the cosmologicalwindow. 0 3 Keywords: polarization,Galaxy,cosmicmicrowavebackground,Method:numerical. 0 / h p - o 1 INTRODUCTION detection are mainly related to the presence of foreground r noise from Galactic and extragalactic sources. Extragalac- st The polarized component of the diffuse background emis- tic foregrounds essentially consist of radio and infrared dis- sion inthemicrowaverangeisofgreat interestfor boththe a cretesources,whereasGalacticforegroundsaregeneratedby : GalacticstructureandtheCMB.Actually,itsmeasurement synchrotron, free-free, thermal dust and spinning/magnetic v leads to probing the structure of the interstellar medium i dust emissions. X (ISM)andtheGalacticmagneticfield.Moreover,thedetec- tion of CMB Polarization (CMBP) allows the investigation Synchrotron polarized emission should represent the r a of the early Universe. mostrelevantforegroundinthemicrowaverange:free-freeis CMB anisotropies and polarization are powerful tools fainter(<4µKat30GHzintotalintensity,seeReynolds& to determine cosmological parameters (Sazhin & Benitez Haffner2000)andalmostunpolarized,whereasthermaldust 1995, Zaldarriaga, Spergel & Seljak 1997, Kamionkowski & has a polarization degree much smaller than synchrotron Kosowsky 1998). However,although anisotropies havebeen (Prunetetal.1998,Tegmarketal.2000).Evidenceforspin- ⋆ ningormagneticdustemissionhasbeenfound(Kogutetal. already detected and space missions (MAP , PLANCK†) ◦ 1996,deOliveira–Costaetal.,2002)butitseemstoplayan are expected to make all-sky surveys down to 0 .1 angu- important role only up to 50 GHz. Moreover, it should lar resolution, CMBP still represents a challenge for as- ∼ havea low polarization degree (Lazarian & Prunet 2002). tronomers. The first detection has been just claimed by DASI (Kovac et al. 2002) and several experiments will ad- In spite of its importance, synchrotron emission is scarcely surveyed: existing data mainly cover the Galactic dressitsoon(SPOrt‡(seeCarrettietal.2002,Cortiglioniet Plane area at frequencies upto 2.7 GHz, far away from the al. 2002), MAP, PLANCK, B2K2 (Masi et al., 2002), BaR- cosmological window (Duncan et al. 1997, hereafter D97, SPOrt (Zannoni et al., 2002) and AMiBA (Kesteven et al., Duncan et al. 1999, hereafter D99, Uyaniker et al. 1999, 2002) among the others). Gaensler et al. 2001, Landecker et al. 2002). The Leiden Besides the CMBP low emission level (3-4 µK on sub- data (Brouw & Spoelstra 1976, hereafter BS76) cover high degree scales and < 1µK on large ones), difficulties in its Galactic latitudes, but are limited to < 1.4 GHz and are largelyundersampled.Thissituationmakeshavingareliable ⋆ http://map.gsfc.nasa.gov/ synchrotronpolarized emission templatein the20-100 GHz † http://astro.estec.esa.nl/SA-general/Projects/Planck/ range very important. This would allow, for instance, re- ‡ http://sport.bo.iasf.cnr.it liable numerical simulations to set-up and test destriping (cid:13)c 0000RAS 2 G. Bernardi et al. techniquesorforegroundseparationmethods(Revenuetal. have many advantages with respect to those in the radio 2000,Sbarraetal.2003,Tegmarketal.2000andreferences band: they are free from Faraday rotation and they cover therein). At present, only toy models exist, which do not almost all the sky. The catalogue lists polarization degree, accountfortherealspatialdistributionofbothpolarization position angle and distance of about 9000 stars from both intensity and polarization angles (Kogut & Hinshaw 2000, hemispheres. Further properties of this catalogue and con- Giardino et al. 2002). siderations confirming its validity as a polarization angle In this paper we present a new approach in modelling pattern for our model are described in section 2.4. the Galactic diffuse synchrotron polarized emission in the 20-100 GHz range. It is based on real surveys and fitted to the real spatial distribution of both polarized intensity and 2.2 Synchrotron Intensity Map polarization angles. Low frequency data are used to model The different frequency behaviour of free-free and syn- thepolarizedintensityandopticalstarlightisusedtomodel chrotron emissions allows the application of the Dodelson polarizationangles.ThisallowstheconstructionofQandU technique(Dodelson 1997) to Haslam (0.408 GHz) and Re- mapscoveringabouthalfoftheskywiththeSPOrtexperi- mentangularresolution(FWHM=7◦).Althoughourwork ich (1.4 GHz) maps. The original Dodelson formalism is centred on the CMB frequency dependence, so that here is finalized totheSPOrt experiment,themethod is general a slight modification is introduced to adjust the method to enoughtobesuitablealsoforsmallerangularscalesassoon as complete sets of data with sub-degreeangular resolution thesynchrotron–free-freecase.ForeachpixeliavectorTiis defined whose elements are the pixel antenna temperatures will be available. atthetwofrequencies.Itsexpressionintermsofsynchrotron Thegreatadvantageofthisnewapproachistoproduce and free-free components is given by: QandU mapsfreefromFaradayrotation,allowingadirect extrapolation to thecosmological window. Ti=Tsi +Tfif +Ni, (1) The outline of the paper is as follows: the synchrotron polarized emission model and the procedure to build the where Tsi and Tfif are the synchrotron and free-free contri- template are presented in Section 2, results and compar- butions, respectively,and Ni is thenoise. isons with existing data are described in Section 3 and 4, Assuming the noise from the two maps is completely respectively, whereas Section 5 contains the conclusions. uncorrelated(itcomes from differentexperiments),thecor- relation matrix Cab,i= Na,iNb,i (2) h i 2 THE MODEL isdiagonalwithpixelvariancesaselements(aandbindicate thefrequencies). 2.1 Ingredients Providedthefrequencybehaviours(shapes)FsandFff i i The aim of our work is to generate template maps of the are known, the estimators Θs and Θff of Ts and Tff can i i i i two linear Stokes parameters Q and U of the Galactic syn- beexpressed as: chrotron polarized radiation in the cosmological window Θs = θsFs (3) near 100 GHz with the SPOrt angular resolution (FWHM i i i =7◦). Wedivide theproblem in two parts: Θff = θffFff (4) i i i (i) constructing a polarized intensity (Ip) map: it can be whereθisandθiff aretheunknownamplitudesofsynchrotron obtained from existing total intensity (I) sky surveys as- and free-free, respectively. suming a model linking the polarized to the total intensity In therange 0.408 1.4 GHzfree-free and synchrotron − synchrotron emission; are known to follow power laws, so that their shapes are (ii) building a map of polarization angles not affected Fs = ν−βi (5) by the effects of Faraday rotation. At present, only optical i,a a starlight data fulfil thisrequirement. Fif,fa = νa−α (6) TheHaslam map(Haslam etal.1982) isthemost complete with the free-free spectral index α = 2.1 (RR88). The fre- skysurveyatradiowavelenghts,wheresynchrotronemission quency behaviour of synchrotron radiation depends on the is dominant. It is a full-sky map at 408 MHz with a reso- energy distribution of relativistic electrons and is spatially lution of 51 arcmin obtained combining observations taken varyingacross thesky. with different radiotelescopes. However,it is not perfect for Wedefinea scalar product between vectors as ouraimssincethefree-freeemissionisstillsignificant,expe- 2 cRiaRl8ly8)i.nCtohnesGeqaulaencttilcy,pidlaenneti(fiRcaeticiohn&anRdeiscuhbt1r9a8c8t,iohneroefatftheirs R·S=σθ(0)2 RaCa−b1Sb. (7) a,b=1 contribution is mandatory. As described in Section 2.2, we X perform this separation using the Dodelson (1997) formal- This expression is similar to that of Dodelson, apart from ism,whichrequiresasecondmapatdifferentfrequency.We thenormalization factor usetheReich(1982)mapat1.4GHzwithanangularresolu- 2 1 σ(0) = , (8) tionof35arcmin,theonlyotheravailablesurveywithabso- θ 2 FffFffC−1 lutecalibration coveringalarge partofthesky(δ 17◦). a,b=1 a b ab ≥− The polarization angles are taken from the Heiles which hePre is based on the free-free shape rather than on starlight polarization catalogue (Heiles 2000). These data that of CMB. Again following Dodelson, the best estimate (cid:13)c 0000RAS,MNRAS000,1–12 ANew approachforaGalactic Synchrotron PolarizedEmissionTemplateinthe Microwave Range 3 Figure1.Synchrotronspectralindexmapinthe0.408–1.4GHz range (see text and RR88 for details). The map is in Galactic coordinates withtheGalacticCentreinthemiddleandGalactic longitudesincreasingtowardleft. of the amplitudes is given by θis = K1−j1Fji ·Ti, (9) j X θiff = K0−j1Fji ·Ti with j =s,ff, (10) j X where thematrix K is defined as Kij =Fi Fj. (11) · Asaresult,thetwocomponentsareseparatedprovidingthe two maps of synchrotron (θs) and free-free (θff). i i Thespectralindexβihasbeenmodelledusingtheanal- ysis of RR88 who compared data at 408 MHz and 1.4 GHz obtaining the following results: ◦ β 2.85 toward the Galactic anticentre at b = 0 of • ∼ Galacticlatitude,withaflatteningwithincresingz(z isthe height above theGalactic plane); β 3.1 in the inner disk (Galactocentric distance < • ∼ 8 kpc and z < 2 kpc). In this region the spectral index is nearly cons|ta|nt, beginning to decrease from b 10◦ until ◦ | | ≃ reaching thevalue of β 2.65 at b 30 ; ∼ ◦ | |≃ β 2.65 for b > 30 independently of the Galactic • ∼ | | longitude l. The transition from the inner disk region to the rest of the Figure2.Spatialbehaviourofthesynchrotronspectralindexβ Galactic plane occurs between l = 45◦ and l = 55◦, where forthreespecialcases:l=0◦ (a), l=180◦ (b)andb=0◦ (c). a flattening from β = 3.1 to β = 2.85 is observed. From spectral index profiles (cfr. Figure 5 in RR88) we find the linear behaviour to be a good approximation for β in the 2.3 From Total to Polarized Intensity Map transition regions. From theseconsiderations, wemodelthe Acommonresultofradio-surveysinpolarizationistheiden- distribution of synchrotron spectral indeces as follows (see tification of two main components: a strong emission from Figure 1): discretesources (SupernovaRemnants-SNRs-andseveral 1) a region towards the Galactic centre (b < 10◦ and sources with no I-counterpart) and a weaker, diffuse emis- ◦ ◦ | | l<45 , l>315 ) with β =3.1; sion from a background component (D97, D99, Landecker 2)aregiontowardstheGalacticanticentreontheGalactic et al. 2002, Gaensler et al. 2001) that appears to be rather plane (b=0◦ and 55◦ 30 ) with β = larresolutionandfrequency.Thisisotropicbackgroundsug- | | 2.65. gests the presence of a polarization horizon, a local screen 4)thespectralindexfollowsalinearbehaviourinthetran- beyond which the polarized emission is cancelled out (D97, sitionregionsandcontinuityisimposedatborders.Figure2 Gaensler at al. 2001, Landecker et al. 2002). The horizon shows thespectral index behaviour in three special cases. can beimagined as a sort of bubblecentred in theobserver (cid:13)c 0000RAS,MNRAS000,1–12 4 G. Bernardi et al. whered=9kpcis theSundistancefrom GC(SeeFigure3 and AppendixA for details). The polarized brightness temperature Ts can be simi- p larly defined provided the emission is integrated out to the polarization horizon R and a polarization degree p is in- ph troduced: c2 Ts(ν,l,b)= pJs(ν,l,b)R . (14) p 2Kν2 ph Finally,equations(12)and(14)providetherelationbetween polarized and total intesity emissions: R Ts(ν,l,b)=p ph Ts(ν,l,b) (15) p L(l,b) ThequantitypR isunknownandrepresentsafreeparam- ph eter tobe calibrated with real data. 2.4 Polarization angle map Figure 3. Contributions to the Galactic polarized radiationare limited by a polarization horizon, a local bubble of radius Rph The propagation of an electromagnetic wave of wavelength centred on the Sun (see text for details). The thickness L(l,b) λ through a plasma in presence of a magnetic field B is ofthesynchrotronemittingregionofoursimplesphericalmodel affected by Faraday rotation. The net effect is a change in (the distance of the halo point P from the Sun position) is also thepolarization angle φ by shown. ∆φ = RMλ2 (16) pinotseigtiroante(dseaeloFnigguthree3li)n:etohfesnigehttpooulatrtiozetdhesihgnorailzoisno,nwlhyetrheaast RM = 812 ne(cm−3)B(µG)·dl(kpc)rad m2 Z signals beyond the horizon are depolarized by variations of polarization angles (changesandturbulencein theGalactic whereRMistherotationmeasure,neistheplasmaelectron densityanddlistheinfinitesimalpathalongthelineofsight. magnetic field). Estimates of RM from extragalactic radio sources give A further element suggesting the existence of such a typical values ranging from tens to hundreds rad/m2 at horizon comes from Landeckeret al. (2002), who show that mediumandhighGalacticlatitudes,andfromtenstothou- onlytheclosestSNRsarewellvisiblealsoinpolarizedemis- sands rad/m2 in the Galactic plane (Simard-Normandin & sion, whereas the most distant ones completely disappear. Kronberg 1980, Sofue & Fujimoto 1983, hereafter SF83, Thesizeofthehorizon isnotyetknown:it dependson Brown & Taylor 2001, hereafter BT01). In particular, the severaleffectsalongthelineofsight,likeGalactic magnetic behaviouralong theGalactic planeis well fitted by (BT01) fieldturbulenceandelectron densityvariations.However,it has been suggested that it can range from 2 kpc (Gaensler RM(l) = RM cos(l l ) (17) 0 0 − etal.2001)upto7kpc(D97,Landeckeretal.2002),sothat RM = 183 14 rad/m2 0 a few kpcappeares to be a quiteacceptable estimate. − ± ◦ ◦ l = 84 4 . Thepolarizationhorizonallowsustomodeltherelation 0 ± betweenpolarizedandtotalintensitysynchrotronemissions. Brown & Taylorsuggest thatthemodulation in RMoccurs Given the mean total synchrotron emissivity Js(ν,l,b) at because of a local constant magnetic field. Galactic coordinates (l,b) and the thickness L(l,b) of the Equation (17) gives, in the frequency range of the cos- synchrotron emitting region (see Figure 3) in the same di- mological window (20 100 GHz), negligible Faraday rota- rection, the brightness temperature Ts(ν,l,b) at frequency tion effects ( 2.5◦ at 2−0 GHz): all we need is a template of ν is: intrinsicpola±rizationangles.Whenusedat2.7GHz,whichis c2 thehighest frequencyof presentpolarization surveys,equa- Ts(ν,l,b)= Js(ν,l,b)L(l,b), (12) ◦ 2Kν2 tion (17) results in angular rotations up to 160 . This ± means that radio polarization data cannot be used to build where K is the Boltzmann constant. The thickness L(l,b) a reliable template of intrinsic polarization angles. dependson thegeometrical model describing thespace dis- To overcome this problem we use the Heiles catalogue tributionoftherelativistic–electron gasresponsibleforsyn- on starlight polarization, the optical frequency being unaf- chrotronemission.Asafirststepinmodellingthepolarized fected byFaraday rotation. synchrotron radiation we consider the simplest case where The polarization vector of starlight is parallel to the thegasisuniformlydistributedintheGalactichalo.Thisis Galactic magnetic field B because of selective absorption represented by a sphere of radius R = 15 kpc centred into byinterstellardustgrains,whoseminoraxisisaligned with theGalactic Centre(GC).Thus,in oursimplecase theline B(Fosalba et al., 2001). Sincethesynchrotronpolarization ofsightL(l,b)isthedistancebetweentheSunandtheedge vector is perpendicular to B, starlight polarization angles of this sphere: ◦ can be used as a template provided a 90 rotation is per- (R2/d2 1) formed. L(l,b)=d cos(b) cos(l) 1+ 1+ − , (13) cos2(b) cos2(l) Most of the Heiles catalogue stars ( 87%) are within (cid:20) r (cid:21) ≃ (cid:13)c 0000RAS,MNRAS000,1–12 ANew approachforaGalactic Synchrotron PolarizedEmissionTemplateinthe Microwave Range 5 Figure 5. Polarization angles of stars in the Heiles catalogue (degrees):eachstarisrepresentedbyapixelinaHEALPixformat mapwithNside=128(pixelsize≃0◦.5)).ThemapisinGalactic Figure 4.NumberofHeilesstarsversusSundistance. coordinates. 2kpc(seeFigure4),trackingthelocalmagneticfield.Their distance is in the order of thepolarization horizon size (see Section 2.3) confirming that the Heiles catalogue can be safely used asa templatefor thepolarization angles ofsyn- chrotron emission. The main problems to face with when using Heiles polarization angles are the irregular distribution of data and the variable sampling distance. However, if the uniformity scale of the angles is compatible with the sampling distance of the catalogue, the lack of data can be filled bylinear interpolation. An estimate of the uniformity scale can be obtained from Figures 9 and 10 of D97, showing that background emission regions have a polarization angle pattern varying ◦ ◦ slowly on scales of 5 10 .Onlyareas with strongsources − show a more complex structure, but their modellization is Figure6.Mapofsamplingdistances(degrees)oftheHeilescat- out of thepurposes of thispaper. alogue.ThemapisinGalacticcoordinates. ThesamplingdistanceoftheHeilescatalogue(Figure6) is compatible with the uniformity scale everywhere but in ◦ ◦ the region centred in (l=135 ,b=40 ) where it is greater than10◦.Weexcludethisregionfromtheinterpolationpro- (ii) the resulting maps are resampled in HEALPix§ for- matwithN =128,correspondingtoapixelsizeofabout cedureas well as from final template maps. side half a degree; Weperform theinterpolation bygenerating Q,U pairs (iii) Reichmap data aresmoothed totheHaslam resolu- corresponding tothe Heiles polarization angles θ: tion (FWHM=51 arcmin); Qθ = cos(2θ); (18) (iv) thetechniqueforcomponentseparation describedin U = sin(2θ). (19) Section 2.2 is applied by using the synchrotron spectral in- θ dex pattern previously described. This results in the I syn- Then, for each pixel of the template map under construc- chrotron emission map; tion we linearly interpolate the Qθ and Uθ values of the (v) the relation between I and Ip described by equa- threeclosest starsandcomputethecorrespondingpolariza- tion (15), which introduces the effects of the polarization tion angle. The interpolation methods uses parallel trans- horizon,isusedtoprovidetheshapeoftheIp map,thepa- port as described in Bruscoli et al. (2002). rameter pR being still free. Its calibration is performed ph a posteriori on BS76 data and a detailed discussion is presented in Section 3; 2.5 The Procedure (vi) from the Ip map and from the angle map obtained from starlight data, Q and U mapsare computed; Thepolarized synchrotronemission templateisbuiltasfol- (vii) the Q and U maps produced in this way are con- lows: volved with a FWHM = 7◦ Gaussian filter to match the (i) the CMB emission and the absolute calibration error SPOrtangularresolution.Thesmoothingprocedureapplies areremovedfromboththeReichandtheHaslammapsusing valuessuggested in RR88 (3.7 K and 2.8 K for Haslam and Reich maps, respectively); § http://www.eso.org/science/healpix/ (cid:13)c 0000RAS,MNRAS000,1–12 6 G. Bernardi et al. the parallel transport method described in Bruscoli et al. (2002) Table 1. Peak emission (Fan region) and Prms of our template atthefourSPOrtfrequencies.ThePrms iscomputed onthelow emissionareas(thefaintest50%pixels). ν (GHz) Ip peak(µK) Prms(µK) 3 THE POLARIZED SYNCHROTRON TEMPLATE 1.4 5×105 6.6×104 22 130 17 Our results are shown in Figure 7, where the Q & U tem- 32 43 5.6 plates at 1.4 GHzare presented.Figure 8shows a compari- 60 6.5 0.84 sonbetweenthepolarized intensityIp ofourmodelandthe 90 1.9 0.25 Ip map obtained from BS76 data. The comparison is performed only for Ip because polarization angles are strongly affected by Faraday rotation at 1.4 GHz (see Section 2.4). Areas not covered by our template are in grey: they corre- et al. 2001, Landecker et al. 2002). This is a further con- spond to the Southern sky not surveyed by Reich and the firmation provided by our model of the relation between regionaroundtheNorthCelestialPolewherestarlightangle synchrotron I and Ip includinga polarization horizon. data are too sparse. We extrapolate the Q and U templates at 1.4 GHz to The well known feature of real data, i.e. that Galactic the cosmological window, and in particular to the SPOrt plane and high Galactic latitudes having comparable emis- frequencies:22,32,60,90GHz.Weuseapowerlawwiththe sions, is well reproduced in ourtemplate. meansynchrotronspectralindexβ =3.0foundbyPlatania Our template is also able to reproduce the brightest et al. (1997) in the 1 19 GHz range. We do not show − structures in BS76 data, namely: the resulting maps being the same at 1.4 GHz apart from the normalization. Instead, we report the emission of the (i) the Fan region (the region situated in the Galactic most important structure (the Fan region) and the mean ◦ ◦ plane at 120 l 150 ); ≤ ≤ polarization level (ii) theNorth Galactic Spur; P = Q2 + U2 ) (22) though the latter is fainter than in real data. On the other rms h i h i hand, in our template a feature appears in the Galactic ofthelopwemissionareas(thefaintest50%pixels)inTable1 ◦ ◦ Plane towards l = 30 -40 , which is not present in BS76. for all theSPOrt frequencies. One reason might be the sparse sampling of BS76 in this area, where the sampling distance is 4◦ versus an angu- ◦ ∼ lar resolution of 0.6 , and a source might well have been missed. Another possibility could be due to Faraday depo- 4 COMPARISONS WITH EXISTING DATA larization effects. A qualitative analysis of the whole BS76 data set (0.408-1.4 GHz) suggests that at 1.4 GHzonly the 4.1 Power spectra FanregionandtheNorthGalacticSpurarefreefromdepo- As a first check we compute the Angular Power Spectra larization effects. This seems to be confirmed by Junkes et (APS) of our model and compare them with APS obtained al. (1990) who find the polarized intensity decreasing from from BS76 maps at 1.4 GHz (Bruscoli et al. 2002). ◦ l 50 towards the Galactic Centre with a relevant min- im∼um around l 30◦. They argue this behaviour might To account for the irregular sky coverage of our tem- ∼ ◦ plate we use the method described by Sbarra et al. (2003) be due to depolarization effects: in particular, at l 30 , ∼ and based on Q and U two-point correlation functions. thermal material in theforeground (Scutumarm) might be These are estimated directly on our maps as responsible for theobserved low polarization. The good agreement between the map obtained from C˜X(θ)=∆Xi ∆Xj X =Q,U,Ip (23) BS76 data and our model allows us to calibrate our tem- where ∆X is the pixel i content of map X, i and j identify plate (the parameter pRph is still free) by matching the Ip i pixelpairs at distance θ. emission of the two maps in a well defined area. We use ThepolarizedpowerspectraCE,BandCIp areobtained the Fan region, the most defined and morphologically simi- ℓ ℓ byintegration larareainboththetwomaps.Thecalibration isperformed withthe820MHzBS76data(seeFigure2inBruscolietal. π 2002) rather than with those at 1.4 GHz, because of their CℓE = Wℓ [C˜Q(θ)F1,ℓ2(θ)+C˜U(θ)F2,ℓ2(θ)]sin(θ)dθ bettersampling, providing Z0 π CB = W [C˜U(θ)F (θ)+C˜Q(θ)F (θ)]sin(θ)dθ pR =0.9 0.09 kpc, (20) ℓ ℓ 1,ℓ2 2,ℓ2 ph ± Z0 π wBSh7er6edtahteaecrarloibrriastidoonm.iAnsastuemdibnygtthhee∼po1la0r%izautniocnerptaoinnt7y◦oins CℓIp = 2πWℓ C˜Ip(θ)Pℓ(cosθ)dθ, (24) Z0 intherange 0.15–0.3 (Tegmarketal.2000) weobtain for ∼ where the functions F and F are described by Zal- thepolarization horizon: 1,ℓm 2,ℓm darriaga (1998), P are the Legendre polynomials, and the ℓ 3kpc

A New approach for a Galactic Synchrotron Polarized Emission Template in the Microwave Range

Similar A New approach for a Galactic Synchrotron Polarized Emission Template in the Microwave Range