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ff Di use Gamma Rays in 3D Galactic Cosmic-ray Propagation Models R. Kissmann1,a), F. Niederwanger1, O. Reimer1and A. W. Strong2 7 1 0 1Institutfu¨rAstro-undTeilchenphysik,LeopoldFranzensUniversita¨tInnsbruckTechnikerstraße25/8, 2 6020Innsbruck,Austria n 2Max-Planck-Institutfu¨rextraterrestrischePhysik,Garching,Germany a J a)Correspondingauthor:[email protected] 5 2 Abstract.ThePicardcodeforthenumericalsolutionoftheGalacticcosmicraypropagationproblemallowsforhigh-resolution ] modelsthatacknowledgethe3DstructureofourGalaxy.Picardwasusedtodeterminediffusegamma-rayemissionoftheGalaxy E over the energy range from 100MeV to 100TeV. We discuss the impact of a cosmic-ray source distribution aligned with the H Galacticspiralarmsforarangeofsuchspiral-armmodels.Asexpected,theimpactonthegamma-rayemissionismostdistinct intheinverse-Comptonchannel,whereimprintsofthespiralarmsarevisibleandyieldpredictionsthatarenolongersymmetric . h totherotationalaxisoftheMilkyway.Wewillillustratethesedifferencesbyadirectcomparisontoresultsfrompreviousaxially p symmetric Galactic propagation models: we find differences in the gamma-ray flux both on global scales and on local scales - relatedtothespiralarmtangents.Wecomparegamma-rayfluxandspectraaton-armvs.off-armprojectionsandcharacterizethe o differencestoaxiallysymmetricmodels. r t s a [ INTRODUCTION 1 v ThefactthatourGalaxydoesnothaveanazimuthallysymmetricshapeislatelybeingacknowledgedinrecentGalactic 5 cosmic-raypropagationmodels[see,e.g.,1,2,3,4,5,6].ThesemodelsfocusontheobservationthatourGalaxyhas 8 a spiral structure [see 7] and implications for the transport of cosmic rays within the Galaxy. This respective spiral 2 structurewillmanifestitselfinthedistributionofISMgas[see8]andtheinterstellarmagneticfield[see,e.g.9,10] 7 0 butalsointhedistributionofcosmic-raysources[see,e.g.1,2,11]. . Theimpactofthesourcedistributiononthecosmic-rayfluxanddistributionwithintheGalaxyhasbeenstudied 1 0 byseveralgroups[see1,2,4,12,5,13,6].Itwasshownthatagoodagreementwiththeobservedcosmic-raydataat 7 Earth[see1,13]canbefoundbothinmodelswithanaxisymmetricsourcedistributionandinmodelsusingasource 1 distribution motivated by the observed three-dimensional structure of our Galaxy. Different spatial distributions of : the sources lead to different spatial distributions of the cosmic-ray flux, which should lead to distinct effects on the v i emissionofradiationbytheinteractionofcosmicrayswiththeinterstellarmediumresultinginthedifferentmodels. X Here, we focus on the imprint of non-axisymmetric source distribution on the gamma-ray emission from our r Galaxy.ForthisweusesimulationsperformedwiththePicardcode.Thiscodewasintroducedin[14]andisoptimised a forsuchspatiallythree-dimensionalnumericalmodelling.InthefollowingweintroducethePicardcodewithafocus on recent enhancements. Subsequently we discuss the ensuing gamma-ray emission for our models and show the relatedenergyspectra. THEPICARDCODE The Picard code was introduced in [14] with a focus on spatially three-dimensional numerical modelling of Galac- tic cosmic-ray transport problems. The code is optimised to find steady-state solutions to the cosmic-ray transport equation: ∂ψ ∂ ∂ 1 ∂ (cid:26) p (cid:27) 1 1 i =q((cid:126)r,p)+∇·D∇ψ + p2D ψ −∇·(cid:126)vψ − p˙ψ − (∇·(cid:126)v)ψ − ψ − ψ, (1) ∂t i ∂p pp∂p p2 i i ∂p i 3 i τ i τ i f r butitcanalsobeappliedtosolvetime-dependentproblems.Anumericalsolutiontothesteady-stateproblemisfound be discretising the differential operators in equation (1) in space and momentum and then finding an approximate solution to the resulting linear system of equations. In Picard the linear system of equations can be solved using differentstandardtechniques,includingarangeofmultigridmethods[see15]ortheBiCGStabmethodasdescribed in[16],withthepossibilitytoaddothersolversintothemodularframeworkofthecode. Picard features all relevant transport processes inherent in equation (1): spatial diffusion, spatial convection, momentumdiffusionandrelevantenergy-lossprocesses.Fluxesoflightspeciesdependonthoseofheavierspeciesdue tothepossiblefragmentationorradiative-decayprocesses.Thisisrealisedthroughthefragmentationandradiative- decay timescales τ and τ , respectively, together with a consideration of a related source term for other species. f r TheresultingnuclearnetworkisefficientlyimplementedinPicard,leadingtoanumericallyoptimisedandaccurate treatmentofcouplednuclearspecies[see13]. ProminentfeaturesofthePicardcodeincludethepossibilitytoconsideranisotropicspatialdiffusion[foradis- cussionsee,e.g.17,18]andarbitraryconvectionvelocities(cid:126)v.ThenumericalsolverisMPI-parallel,thusfacilitating the possibility to achieve up to deca-parsec scale resolution by making use of large-scale distributed-memory com- puters.ThenumericalsolutioncomputedusingPicardisacquiredwithauser-definedaccuracywithouttheneedto check the quality of the numerical solution a-posteriory. Picard is continuously expanded where, e.g., the currently ongoingimplementationofanimprovedinterstellarradiationfieldisdiscussedby[19]. Withthiscodewestudiedtheimplicationsofspiral-armcosmic-raysourcedistributionsonthespatialvariability of the cosmic-ray flux in [5]. Additionally, we showed in [13] that the transport parameters in equation (1) can be adaptedtosuchlocalisedsourcedistributions,thusleadingtoagoodagreementbetweenpredictedcosmic-rayfluxes at Earth and corresponding observations. For the secondary to primary ratios there appears a dependence on the distancetothenearestsourcesinmodulesusingsuchlocalisedsources.Inthefollowingwewilladdresstheimpact ofthesemodelsontheGalacticdiffusegamma-rayemission. GAMMA-RAYEMISSION In[13]wefoundthataGalacticcosmic-raypropagationmodelfeaturinglocalisedsourcescanreproducecosmic-ray measurementsatEarthaswellasamodelwithunderlyingaxisymmetry.Incontrasttosuchanaxisymmetricmodel,a spiral-armsourcedistributionleadstoalocalisationofthecosmic-rayflux,atleastfortheprimarycosmic-rayspecies. Thedistinctdifferenceinspatialdistributionbetweenthedifferentmodelscanbeexpectedtoleadtodifferencesinthe resultinggamma-rayfluxinthedifferentmodels.Inthefollowingwefocusonthecomparisonbetweenamodelusing anaxisymmetricsourcedistributionandoneusingaspiral-armsourcedistributionwithfourspiralarms.Detailscan befoundin[5]and[13],wherethisspecificmodelisreferredtoastheSteiman-model. Therelevantchannelsfordiffusegamma-rayemissionareπ0-decay,bremsstrahlung,andinverseCompton(IC) radiation.Theemissivityoftheformertwooftheseprocessesresultsfromtheproductofthegas-densitywiththere- spectivecosmic-raydensity–inthiscasenucleonsandelectrons,respectively.Therefore,itisnoteasytodistinguish whetheraspatialfeatureinthegamma-rayemissionresultsfromlocalstructuresinthegasdistributionorthecosmic raydistribution.SincecurrentmodelsoftheGalacticradiationfieldarespatiallymuchsmootherthanthegasdistribu- tion[see20,21,22],theICchannelisamuchclearertracerofthelocalisationofthesourceswithinourmodels.This is obvious when investigating the gamma-ray emission in Fig. 1. A comparison shows that the Galactic plane near theGalacticcentrebecomesslightlybrighterforthemodelwithaspiral-armsourcedistribution.Otherwiselocalised featuresaffectedbythedifferentsourcemodelsarenotreadilyapparent. InthefollowingwewillfocusontheregionneartheGalacticcentre,wherethelocalisedfeaturesofthecosmic- raydistributionintheSteiman-modelhavethemostobviousimpactonthegamma-rayemission.Thisiscausedbythe spiralarmsthataretightlywoundaroundtheGalacticcentreleadingtoseveralspiralarmtangentsneartheGalactic centre along which the projected source intensity is enhanced. This is shown in Fig. 2 where the projected source intensityneartheGalacticcentreregionisshown.Thespecificspiral-armtangentsareconsistentwithobservations [7]sincetheSteimanmodelisdrivenbyobservationsoftheGalacticspiralarms[23].Thespecificspiral-armtangents visible in Fig. 2 are tangents to the Sagittarius / Carina arm at l ∼50◦, l ∼-16◦, and l ∼-75◦, to the Scutum / Crux Centaurusarmatl∼30◦andl∼-50◦,theNormaarmatl∼19◦andatl∼-30◦,andthePerseusarmatl∼-22◦. 5e-21 1e-17 FIGURE1:All-skygamma-rayemissionat E = 10TeVforanaxisymmetric(top)andafour-arm(bottom)source γ distribution.Thegamma-rayemissionisshowninunitsofcm−2s−1sr−1MeV−1. InFig.2wealsoshowtheICemissionatanenergyof∼100GeV.Theincreaseingamma-rayfluxatthepositions ofthespiral-armtangentsisobviousatthisenergy.High-energyelectronsaretightlycoupledtotheirsourcesdueto theirveryhighenergylosses[5].Therefore,therelatedgamma-rayemissiontracesthesesourceregionsleadingtoan imprintofthespiral-armtangentsalsoonthegamma-rayemission.Inthetotalgamma-rayemissiontheseeffectsare not so obvious, because of the impact of the gas-distribution discussed above and also because the Galactic diffuse gamma-ray emission is in many directions dominated by π0-decay emission. In the following we investigate the quantitativeimpactofthespiral-armsourcedistributiononthegamma-rayemission. GAMMA-RAYSPECTRA Foraquantitativeanalysiswecomparegamma-rayspectraaton-armandoff-armpositions.InFig.3corresponding results are shown for spiral-arm tangents of the Norma and the Sagittarius arm. Obviously, the on-arm flux is sig- nificantlyhigher inbothcases andforall gamma-rayemissionchannels. Especially,forBremsstrahlung andICthe on-armspectraarealsoharder.Forthesechannelswefoundthatthepower-lawindexcanincreaseby0.1orevenmore ataspiral-armtangentascomparedtoanoff-armposition.ThisissomewhatmorepronouncedfortheSagittariusarm, becauseinthecaseoftheNormaarmtheoff-armregionisinaregionfeaturingseveralotherspiral-arms.Thus,the Normaoff-armregioniscontaminatedbythepresenceofotherclose-bycosmic-raysources. ThespectralhardeningoftheICfluxcanbeexplainedbytheenergydependenceoftheelectron’senergylosses. Duetotheirrapidincreasewithenergy,high-energyelectronssufferfromthemostsevereenergylosses.Therefore, high-energy electrons can only be found in the close vicinity of their sources. In [5] a significant hardening of the electronspectrumwasobservedforon-armpositions.Thisnaturallytranslatesintoahardeningoftheresultingphoton spectrumforasightlineinthedirectionofaspiral-armtangent. The π0 channel shows only mild change in spectral slope. This is also consistent with the observations in [5] wheretheprotonfluxwasfoundtomainlychangeinamplitudefromanon-armtoanoff-armposition.Thisreflects thefactthatthespatialdistributionofnuclearcosmicraysismainlydrivenbyspatialdiffusionduetotheirratherlow energy losses. Also in gamma rays this channel mainly features a change in amplitude when comparing an on-arm 40 −1] 30 10-13 sr 20 −2 s 10 −2 b b 0 m 10 10-14 −1c 20 V e 30 M 40 [ 50 0 50 l l FIGURE2:Left:Projectedsourceintensityforthefour-armsource-distributionmodel.Spiral-armtangentsarevisible as enhanced intensity in the Galactic plane. Right: Inverse Compton gamma-ray emission at 100 GeV for the same model. 100 100 total flux total flux ] 10-1 IC ] 10-1 IC −1 Bremsstrahlung −1 Bremsstrahlung sr pion decay sr pion decay −2 10-2 −2 10-2 s s −2 −2 m 10-3 m 10-3 c c V V Me 10-4 Me 10-4 [ [ γ γ J J 10-5 10-5 10-6 10-6 102 103 104 105 106 107 108 102 103 104 105 106 107 108 E[MeV] E[MeV] FIGURE 3: Spectra for the the different gamma-ray emission channels for two of the spiral-arm tangents. In each ploton-armandoff-armdataarecompared,wheretheon-armspectraaretheoneswiththehigherfluxes.Ontheleft resultsareshownforatangentoftheNormaarmwiththeon-armdatatakenat-30◦ longitudeandtheoff-armdata takenat-35◦longitude.OntherightcorrespondingdataforaSagittarius-armtangentareshownwiththeon-armdata takenat50◦longitudeandtheoff-armdatatakenat62◦longitude.Resultsareintegratedoveracirclewitharadiusof 2◦. andanoff-armregion. Since the total flux in both cases is dominated by π0 decay, the change in slope for the total flux is also rather small.OnlyintheenergyregimefromafewuptoafewhundredGeV,wherethecontributionofICemissiontothe totalgamma-rayfluxislargest,thespectrumatanon-armpositionbecomesharderwithachangeinpowerlawindex of up to 0.05. In the present case we used the same source-distribution for electrons and nuclei. In a model using differentsourcedistributionsthecorrespondingchangeswouldbecomelarger,forsuchlocalisedsourcedistributions aswerediscussedinthiscase.Infact,forhigh-energyelectronsitwillbeimportanttoconsidertheactualsources[see 24,25,26]whenaimingforanaccuratemodelforthediffusegamma-rayemissionintheGalacticplane.Ourmodel showsthatlocalisedsourceswillnotonlyhaveanimpactonthegamma-rayemissionintheGalacticplane,butcanalso affectthelocalspectralslope.Thiswillmakeatemplate-fittingapproachatleastmoreproblematicthanaphysically motivatedmodellingapproachbasedonthepropagationphysicsofGalacticcosmicrays.Oneshouldbeawarethat inthecurrentapproachonlythedistributionofthecosmic-raysourceswasadaptedtotheGalacticstructure.Withan adoptionofthegas-,magneticfield-,andpossibilitytheinterstellarradiationdistributionthedifferencesbetweenan axisymmetricapproachandoneacknowledgingtheactualstructurecanbecomeevenlarger,holdingthepossibilityto alsoexplainthesmall-scalestructureinthegamma-rayemissionintheGalacticplane. CONCLUSIONS InthisstudyweshowedthatPicardisreadytopredictgamma-rayfluxesforacomparisonwithdatainthecontextof Galacticcosmic-raytransportmodelstakingintoaccountthethree-dimensionalstructureofourGalaxy.Weshoweda firstquantitativediscussionoftheimpactoflocalisedsourcesontheGalacticdiffusegamma-rayemission.Wefound ahardeningofthegamma-rayemissionspectruminthedirectionofthecosmic-raysourcesfortheleptonicemission channels.Sinceinourmodeltheemissionisdominatedbyπ0 decaythechangeinslopeforthetotalgamma-rayflux is rather small and also limited to energies between 1 and a few hundred GeV, where the leptonic channels provide a comparatively large fraction of the total emission. The current analysis was focussed on the impact of the source distribution.Whilethiscanonlybeviewedasafirststep,withthefutureneedtoincludethefullGalacticstructurefor gas-, magnetic field-, and interstellar radiation-distribution, only a dissemination of the individual effects can show therespectiveimpactonchangesinthegamma-rayemission. ACKNOWLEDGMENTS The computational results presented have been achieved (in part) using the HPC infrastructure LEO and MACH of theUniversityofInnsbruck. FN acknowledges financial support from the Austrian Science Fund (FWF) project number I1345, in collaboration withtheFrenchScienceFund(ANR),projectIDANR-13-IS05-0001. REFERENCES [1] D. Benyamin, E. Nakar, T. Piran, et al. Recovering the Observed B/C Ratio in a Dynamic Spiral-armed CosmicRayModel. ApJ,782:34,February2014. [2] F.Effenberger,H.Fichtner,K.Scherer,etal. AnisotropicdiffusionofGalacticcosmicrayprotonsandtheir steady-stateazimuthaldistribution. A&A,547:A120,November2012. 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