Reconciling the diffuse Galactic gamma-ray and the cosmic ray spectra

2.6 MB·

by  Lara Nava

MNRAS000,1–8(2015) Preprint12January2017 CompiledusingMNRASLATEXstylefilev3.0
Reconciling the diffuse Galactic γ-ray and the cosmic ray spectra
Lara Nava1 (cid:63), David Benyamin1, Tsvi Piran1, and Nir J. Shaviv1,2
1RacahInstituteofPhysics,TheHebrewUniversityofJerusalem,91904,Israel
2SchoolofNaturalSciences,InstituteforAdvancedStudy,Princeton08540,NJ,USA
AcceptedXXX.ReceivedYYY;inoriginalformZZZ
7
1 ABSTRACT
0 MostofthediffuseGalacticGeVγ-rayemissionisproducedviacollisionsofcosmicray(CR)
2 protonswithISMprotons.Assuchtheobservedspectraoftheγ-raysandtheCRsshouldbe
stronglylinked.RecentobservationsofFermi-LATexhibitahardeningoftheγ-rayspectrum
n
a ataroundahundredGeV,betweentheSagittariusandCarinatangents,andafurtherharden-
J ingatafewdegreesaboveandbelowtheGalacticplane.However,standardCRpropagation
0 modelsthatassumeatimeindependentsourcedistributionandalocationindependentdiffu-
1 sioncannotgiverisetoaspatiallydependentCR(andhenceγ-ray)spectralslopes.Herewe
consideradynamicspiralarmmodelinwhichthedistributionofCRsourcesisconcentrated

] inthe(dynamic)spiralarms,andwestudytheeffectsofthismodelontheπ0-decayproduced
E
γ-rayspectra.Withinthismodel,neartheGalacticarmstheobservedγ-rayspectralslopeis
H
nottriviallyrelatedtotheCRinjectionspectrumandenergydependenceofthediffusionco-
. efficient.WefinduniquesignaturesthatagreewiththeFermi-LATobservations.Thismodel
h
also provides a physical explanation for the difference between the local CR spectral slope
p
andtheCRslopeinferredfromtheaverageγ-rayspectrum.
-
o
r Keywords: diffusion–cosmicrays–ISM:general–Galaxy:general–gammarays:ISM
t
s
a
[

1 1 INTRODUCTION theGalacticarms)andassuchsimplemodificationsofthesource
v orthediffusioninsidethearmsandoutsideofthemareinsufficient
6 ObservationsoftheGeVγ-rayskyrevealsignificantinformation
toexplaintheobservations.
4 concerning Galactic cosmic rays (CR) at somewhat higher ener-
Withtheaboveinmind,weconsiderwithinthiscontextthe
7 gies. The π0 produced γ-rays can be used to determine the CR
“dynamic spiral arm" model that alleviates the assumption of an
2 interactionwiththeinterstellarmedium(ISM)atvariouslocations
0 in the Galaxy. The γ-ray sky can therefore be used as a tool to azimuthally symmetric source distribution and more realistically
1. studyCRphysics,andinparticular,CRaccelerationandpropaga- assumes that a large fraction of the CR sources are in the vicin-
ityofarms(Shaviv2002,2003;Gaggeroetal.2013;Werneretal.
0 tion.Fermiobservations(Seligetal.2015),whichimprovedearlier
2015).ThismodelissuccessfulinexplainingthevariableCRflux
7 COSB(Rogersetal.1988;Bloemen1989)andEGRETobserva-
asreconstructedfromIronMeteorites(Shaviv2002,2003),itnat-
1 tions(Fatoohietal.1996),revealedsmallbutnoticeabledifferences
: intheγ-rayspectralindexalongbothdifferentGalacticlongitudes urallyexplainsthesocalledPamelaanomalyinwhichthepositron
v toelectronratioincreasesabove10GeV(Shavivetal.2009;Gag-
i andlatitudes,particularlysowhenthelinesofsightaretangential
X tothespiralarms.Weexploreheretheimplicationsoftheseobser- gero et al. 2013), and more recently it also explains the apparent
differenceincolumndensityinferredfromthesub-IrontoIronand
r vationsonmodelsofCRpropagationintheGalaxy.
a BorontoCarbonmeasurements(Benyaminetal.2016).
Spatialvariationinthespectracannotbeexplainedawayby
One interesting aspect of the spiral arm model is that it re-
variationsintheintensityofCRsorvariationsofthedensityofthe
quires a smaller halo. This is because its path length distribution
target ISM protons. The observations of a different CR spectrum
ismissingshortpaths.Asaconsequence,thesamehalosizepro-
atdifferentlocationsimplyeitherthatdifferentsources,locatedat
duceslesssecondariessincetheCRstendtospendlesstimenear
differentregionsoftheGalaxy,havedifferentintrinsicspectraor
theplane,wherethedensityishigh.Inordertoreproducetheob-
thatCRpropagation(inparticularthediffusioncoefficient)hasa
served amount of secondaries the model then requires halo sizes
differentenergydependenceatdifferentregions.Thustheseobser-
that are typically a few hundred pc high (Benyamin et al. 2014,
vationsruleoutimmediatelythe“standard"axisymmetricGalactic
2016). The smaller halo and 10Be age constraints imply that the
CRmodel,oratleastimplythatitrequirestheincorporationofnew
diffusioncoefficientisalsomuchsmallerthanthecanonicalrange.
CRphysics.Asweshowlater,thenatureoftheobservedangular
Boththeseeffectsaresupportedwithrecentdirectmeasurements
dependenceisquitecomplicated(itisasymmetricwithrespectto
of the CR density in the vicinity of high velocity clouds located
outsidetheGalacticplane,revealingthatinsomeregionstheden-
(cid:63) E-mail:[email protected] sityfallsmuchfasterthanpredictedbystandardmodels,andthatit
(cid:13)c 2015TheAuthors
2 Navaetal.
haslargehorizontalvariations(Tibaldoetal.2015).Itwouldalso behaviourishardtoexplaininthestandardaxisymmetricCRdiffu-
explain the indirect inference of the CR density as a function of sionmodels.IftheCRinjectionspectrumanddiffusioncoefficient
distance from the Galactic plane using paleoclimate data (Shaviv isthesameeverywhere,thereshouldbenovariationsintheCRand
etal.2014). ensuingγ-rayspectraatall,exceptforthenormalization.However,
However,withoutanythingelse,astaticspiralarmmodeldoes alsostraightforwardextensionschangingthecharacteristicofei-
not introduce any additional time-scale such that the CR spectral thertheinjectionspectraorenergydependenceofthediffusionco-
form(butnotnormalization)shouldstillbelocationindependent. efficientinsidethespiralarmscannotexplainthisbehaviourasit
However, Benyamin et al. (2014) considered that the location of would give rise to harder spectra in the direction of the arm tan-
theCRsourcesisprogressivelymovingwiththespiralarms.They gents,notbetweenthem.
haveshownthatadvectionoftheISMrelativetothearmintroduces
a new effect (and time-scale) at low energies and it naturally ex-
plainstheriseinthesecondarytoprimaryratiobelow1GeV/nuc.
3 THEMODEL
WecompareherethepredictionsofthismodelwiththeFermiGeV
γ-rayspectralindex.Oneoftheinterestingpointsthatwillbeborne The CR distribution is inferred from the Monte Carlo simulation
outfromthiscomparisonisanexplanationtowhatappearsasan developed by Benyamin et al. (2014). While the details of the
inconsistencybetweenthelocalCRspectrumandthespectrumin- modelcanbefoundinBenyaminetal.(2014),Shaviv(2003)and
ferredfromtheaverageγ-rayspectralindex.Thiswillarisefrom Shavivetal.(2009),wesummarizeherethemainfeaturesofour
theeffectthatthearmdynamicshasontheCRspectrum,introduc- model,focusingonthoseingredientsthatarerelevantforthestudy
inganupstream/downstreamasymmetryaroundthearms. presentedinthiswork.CRsofenergiesuptothekneeareassumed
Webeginbybrieflydescribingtheall-skyFermi/LATspectral to be accelerated at SNR shocks. Both core collapse and type Ia
indexmap§2,comparingthemtostandardCRpropagationmodel SNe are considered, with the latter assumed to comprise 20% of
predictions.Wethendiscussin§3thedynamicspiralarmCRprop- the whole SN population. We assume here that Type Ia SNe and
agationmodelweuse.Wecontinuein§4withadescriptionofthe 10% of the core collapse SNe are distributed in a homogeneous
modelresultsandacomparisonwiththeobservations(§5).Wecon- disc, and take their radial and vertical distributions from Ferrière
cludewithadiscussionin§6. (2001). The rest of core collapse SNe are instead located in the
arms—since the progenitors of core collapse SNe live less than
30Myr,theyexplodenotfarfromtheirbirthsites,whicharepri-
marilythespiralarmsoftheGalaxy(e.g.,seeLacey&Duric2001).
2 OBSERVATIONS
Forthespiralstructurewefollowthegeometryofasuperpositionof
AlthoughsomerecentmeasurementsoftheinferredCRspectrain afour-armedsetandatwo-armedset,withtherationalanddetails
thedirectionsoppositetheGalacticcentreareconsistentwiththe describedinBenyaminetal.(2014).Thisisbasedoncumulative
standardmodel(Abdoetal.2010),thereareotherindicationstothe evidencebornefromdifferentempiricalresults(Amaral&Lepine
contrary. COS B measurements of the γ-ray spectra around 300- 1997;Dameetal.2001;Shaviv2003;Naoz&Shaviv2007).
700MeV(correspondingtoprotonenergiesoftypically7-20GeV) The CR spatial distribution from the axisymmetric compo-
revealedthattheCRspectralindexinthedirectionoftheGalactic nentsdoesnotdependontherotationoftheunderlyinginterstellar
armsisharderby0.4±0.2thanthatinotherdirections(Rogersetal. medium.Thespiralarmsbreaktheaxialsymmetry.Furthermore,
1988;Bloemen1989).ThiswascorroboratedbyEGRET(Fatoohi whenconsideringthepropagationfromthespiralarms,therotation
etal.1996).Similarly,theelectronsynchrotronspectrumat22-30 ofthearmsandtherotationoftheISMshouldbeconsidered.This
GHzproducedbyCRelectronsataround30GeVisharderinthe introducesanadditionalpropagationeffect,namely,“advection"of
directionofthearms(Bennett2012). the interstellar medium relative to the spiral arms. At energies of
Recently, Selig et al. (2015) analyzed the 6.5yr all-sky data uptoafewGeV/nuc.advectioncancompetewithdiffusionanda
from the Fermi/LAT in the range of energies between 0.6 and breakintheCRspectraisthereforeexpectedinthismodel.Italso
307.2GeV. They derived an all-sky spectral index map for the naturally explains the apparent rise at low energies in the Boron
diffuse anisotropic component, giving us the opportunity to test toCarbonratio(Benyaminetal.2014).Theinterplaybetweenthe
the prediction of the dynamic armed model against observations. advectionanddiffusionalsobreaksthesymmetryaroundthespiral
The map of the γ-ray spectral index δ, estimated in the range arms.Whilediffusionoperatesineitherdirectionbothaheadand
0.85 − 6.79GeV is shown in Fig. 8 (lower panel) for latitudes beyond the spiral arm, advection is only downstream. The break
|b| < 7◦.Thespectralindexδ isonaverageharderinthecentral in the CR spectrum will manifest itself as a break in the photon
region(−90◦ < l < 40◦),whereitalsodisplaysaclearvariation spectrumproducedininteractionsbetweenCRsandtheISM.The
asafunctionofthelatitude.Atsmalllatitudes|b| < 1◦,δ hasan sourcedistributionistheninhomogeneous,non-axisymmetricand
intermediatedvalueδ ∼ −2.5,whileatlargerlatitudesaharden- timedependent.
ingtovaluesδ (cid:38) −2.4isobserved.Largerlongitudesareinstead WecarryoutMonteCarlosimulationsfollowingthefullprop-
characterizedbyasofterspectrumwithoutapronouncedlatitudi- agationofalargenumberofCR“bundles".Theirinitiallocation
naldependence,withanindexδ(cid:46)−2.6.Thus,theoverallspectral in the Galaxy is randomly chosen according to the source distri-
slopemapappearstohaveahard“oval".Twoexamplesofγ-ray butiondescribedabove.Wefollowthebundlesfromtheirforma-
spectra (integrated over two different regions of the interest) are tionuntiltheyleavetheGalaxy,takingintoaccountenergylosses
showninFig.7(dashedlines,fromSeligetal.2015). (mainlyintheformofCoulombandionizationcooling).Thediffu-
Although the hard oval can be described as asymmetry be- sioncoefficientasafunctionofrigidityRisdescribedbyasingle
tweeninsideandoutsidetheGalaxy,amorenaturalinterpretation power-law,D = D (R/R )s,withs = 0.4,R = 3GeV/nuc,
0 0 0
given the longitudinal asymmetry is that it is associated with the D = 1.2×1027cm2/sandahalosizeZ = 250pc.Weassume
0 h
spiralarms.Thatis,thespectralslopeappearstobeharderbetween thattheinjectionprotonspectrumasafunctionofthemomentum
thearmtangents(Carinaat∼−76◦andSagittariusat∼50◦).This pisdescribedbyapower-lawdN /dp ∝ pα.Fortheindexα,
inj
MNRAS000,1–8(2015)
Galacticγ-rayandCRspectra 3
Figure1.AtopviewmapoftheMilkyWay(atZ = 0)showingthespectralindexoftheCRprotonspectrum,inferredfromsimulations,intherange
10−100GeV(left-handpanel)andintherange100−3000GeV(right-handpanel),asobtainedinthesmallhalomodel.TheSunislocatedatX=8.5kpc
andY =0(starsymbol).Notetheharder(softer)spectraaroundthespiralarmsthatarisefromadvectionoftheCRthere.
weconsidertwodifferentvalues.First,weinvestigatethestandard (8.5kpc,0,0),asindicatedbytheopenstarsymbol.Asexpected,
hypothesis α = −2.3, leading to an average CR spectral slope thespectrumisharderintheupstreamsideofthearms,andsofterin
(after diffusion) equal to α−s = −2.7. With these values, the thedownstreamside.ThisisbecauselowerenergiesCRsaremore
Boron to Carbon and Beryllium isotopes ratios (Benyamin et al. strongly affected by the advection of the ISM through the arms.
2014) and sub-Iron to Iron ratios (Benyamin et al. 2016) are re- Thevariationofthespectralindexacrosseacharmismoreevident
covered.Then,duetomotivationsthatwillbeclearlater,wewill inthelowerenergyrangeof10-100GeV(left-handpanel)andbe-
considerthecaseofaslightlyharderCRinjectionspectrum,with comeslessevidentathigherenergies(right-handpanel),wherethe
α=−2.25.Wealsoconsiderasecondmodel,withdynamicarms, relative effect of the advection, as compared to diffusion, is less
butmuchlargerdiffusioncoefficient(D =1028cm2/s)andahalo important.
0
sizeofZh = 2kpc,whichbetterresembles“standard"CRpropa- Fig.2(left-handpanel)showsthehistogramoftheCRspec-
gationmodels. tralindexvaluesintheplaneZ =0,estimatedintheenergyrange
The final CR distribution is recorded on a grid that spans a 10GeV-3TeV. As expected, the distribution is peaked at a value
volumeof30kpc×30kpc×2Zh.Itiscomposedof(0.1kpc)2× ∼ −2.7(openhistogram).Wenotehowever,thattheslopeatthe
(0.01kpc)sizedcells(0.1kpcintheX andY directionsand0.01 location of the Earth is somewhat steeper (∼ −2.75), due to the
kpcintheZ direction).Ineachcell,thespectrumat43different factthat,inthissimulation,theEarthliesclosertothedownstream
energybinsiscalculated,from0.5GeVupto500TeV.Toestimate sideofthearm.Namely,thelocalCRspectrumisnotrepresentative
thelocalISMdensity,weconsiderallthedifferentcomponentsof oftheaverageCRGalacticspectrumforthischoiceofparameters.
thegas(neutral,ionizedandmolecular)anddescribetheirdensity A slightly different location of the Earth (or, a slightly different
distribution in the Galaxy according to the parametrization given choiceoftheparametersdescribingthedynamicsand/ormorphol-
in Ferrière (2001). The local π0 production and the resulting γ- ogyoftheGalaxy)wouldallowustorecoveravalueequalto−2.7.
ray emission are then computed in each cell, from the local CR However,wenotethattheresultsforγ-rayspectrapresentasimilar
spectrumandISMdensity,followingKamaeetal.(2006).Oncethe feature—theyaretoosoftascomparedtoobservations.Wecom-
γ-rayspectrumiscomputedineachcellofthesimulationbox,we putetheγ-rayspectralindexintheenergyrange0.8−7GeVat
caninvestigate,inourdynamicspiralarmmodel,howthespectral different Galactic coordinates (l, b) and, for each position in the
slope ispredicted tochange across aspiral arm.Finally, in order sky,weestimatedthedifferencebetweenthespectralslopeofsim-
to compare simulations with observations, we integrate the γ-ray ulatedandobservedγ-rayspectra(twoexamplesofobservedand
emission along different lines of sight (l.o.s.) and derive the sky simulated γ-ray spectra can be found in Fig. 7, dashed and solid
mapofthespectralphotonindex. lines, respectively). For the observed spectra, the slope has been
computedinthesameenergyrange,fromSeligetal.(2015).The
resultsareshowninFig.2,right-handpanel.Theopenhistogram
showsthedistributionofthedifferencebetweenthesimulatedand
4 RESULTS
observed slopes. We limit this investigation to the region of the
Beforediscussingtheresultsofoursimulations,webeginwitha mapatsmalllatitudes,−5◦ <b<5◦.Theaveragevalueisaround
qualitativedescriptionoftheexpectedfeatures.Fig.1showstheCR −0.08,indicatingthatthesimulatedγ-rayspectraareonaverage
spectralindexspatialdistributionaspredictedbyoursimulations, softerthanwhatinferredfromobservations.Tosolvetheinconsis-
performed with α = −2.3. The Sun is located at (X,Y,Z) = tency between observations and both the simulated CR spectrum
MNRAS000,1–8(2015)
4 Navaetal.
atEarthandtheγ-rayspectra,weconsideramodelwithaharder opposite direction is considered (i.e., the one represented by the
injection spectrum α. The shaded histograms in Fig. 2 show that dot-dashedlineinFig.3).
inthiscasepredictionsoftheCRslopearoundthelocationofthe The spectra derived from integration along those l.o.s. that
Earth (left-hand panel) and of the map of γ-ray radiation (right- mostlyintersectthecentreofthearmareshownintheupperand
handpanel)areinbetteragreementwithobservationalconstraints. lower panels of Fig. 4 (green curves). The upper (lower) panel
Forthisreason,fromnowonwepresentanddiscussindetailonly refers to the l.o.s. represented by the dashed (dot-dashed) line in
theresultsobtainedwithα=−2.25.Fig.2(left-handpanel)also Fig. 3, and corresponds to a Galactic longitude l (cid:39) 50◦ (l (cid:39)
showsthatthepredictedCRspectrumatthelocationoftheEarth −104◦)andlatitudeb = 0◦.AlsodepictedinFig.4arethespec-
(thickverticalline)isnotrepresentativeoftheaverageCRspectral traonbothsidesofthearm(redandbluecurves).Thelongitudes
slope (peak of the distribution). While in these examples the dif- andthespectralslopes(intherange1-10GeV)arereportedineach
ferencebetweenthelocalandaveragespectralslopesisonly0.05, panel, while the latitude is always b = 0◦ in all these examples.
largerdifferencescanbeobtainedbyslightlychangingthelocation Spectraarenormalisedsothattheyoverlapatlowenergies.
ofthearmascomparedtotheEarth.Inthissense,themodelisnot FromthecomparisonbetweenFigs.3and4itisevidentthat
predictive,andthereisdegeneracyamongthesimulationparame- when integration along the l.o.s. is performed, the difference be-
ters.Althoughacompletestudyoftheparameterspaceisbeyond tween the spectral slopes at the two different sides of the arm is
thescopeofthispaper,weoutlinethattheproposedmodelgives stronglyreduced.Thisisduetothefactthatforb = 0◦,theemis-
anaturalexplanationfortheincreasingevidencethatthelocalCR sioncomingfrommoredistantregionsofthediscisalsoplayinga
spectrummightnotberepresentativeoftheaverageGalacticone roleinshapingthespectra.
(Ackermannetal.2012;Neronov&Malyshev2015). Wenowextendthestudyofthespectralindextoallthedif-
Thespatialdistributionofγ-rayspectraisexpectedtofollow ferentdirectionsofthesky,andbuildthespectralmapsindiffer-
ent energy ranges. The results are shown in Fig. 5 in the range
apatternsimilartotheonecharacterizingprotonspectra.Asacon-
sequence,alsotheγ-rayemissionintegratedalongdifferentl.o.s. 1-10GeV(upperpanel)and10-300GeV(bottompanel),forlati-
tudes|b|<7◦.Thegeneraltrendissimilarinbothenergyranges,
will show variations in its spectral shape, depending whether the
althoughthevariationinthespectralindexismuchsmalleratthe
l.o.s. intersects harder or softer regions. In order to quantify how
thespectrumoftheγ-rayradiationfromπ0decaychangesacrossa higherenergies(∆δ∼0.1).WiththehelpofFig.3,wecanunder-
spiralarm,weinferthespectralindexδbymodellingthesimulated standtheoriginofthelargevariationinthelowenergymap(upper
photonspectrawith apower-lawfunction:dN/dE ∝ Eδ.Since panel,Fig.5)asafunctionoflongitudeandlatitudeasfollows:
weexpectadifferentbehaviouratlowandhighenergies,thespec- • |b| < 2◦ corresponds to l.o.s. that cross the arm, but are
tralindexisestimatedintwodifferentenergyranges:1-10GeVand
dominated by the disc component. For this reason the spectral
10-300GeV.Webeginbyanalyzingthesmallhalomodel(witha
indexhasanintermediatevaluerangingbetween−2.5and−2.6.
halosizethatreproducestheBorontoCarbonratiointhedynamic
In particular, as can be understood from Fig. 3, at longitudes
spiralarmmodel). −100◦ <l <50◦,δ ∼−2.5,becausethehardestpartofthearm
Fig. 3 shows the map of the photon spectral index δ in the iscrossed,whileatl >50◦andl <−100◦thesoftestpartofthe
plane of the disc (Z = 0) in the low and high energy ranges armiscrossed,andδ∼−2.65;
(left and right-hand panel, respectively). The Sun is located at
(X,Y,Z) = (8.5kpc,0,0), as indicated by the open star sym- • |b| > 2◦ attheselatitudes,contrarytothepreviouscase,the
bol.Wefindthatthespectrumisharderintheupstreamsideofthe intersectionbetweenthel.o.s.andthedisccomponentisreduced,
arms,andsofterinthedownstreamside.Thisshouldbeexpected and the contribution from the arm is dominating the emission,
becauselowerenergiesaremorestronglyaffectedbytheadvection irrespectiveofthelongitude,anditdeterminesthespectralshape.
oftheISMthroughthearms.Asaconsequence,lowerenergyCRs The variation as a function of the longitude is determined by
are more abundant downstream, making the spectrum softer, and whichsideofthearmisthel.o.s.crossing.Thisexplainswhythe
lessabundantupstream,makingthespectrumthereharder.More- spectrumisharder at−100◦ < l < 50◦,with anindexcloseto
over, the variation of the spectral index across each arm is more δ ∼ −2.4, while for larger values of |l| the spectrum is softer
evident in the lower energy range of 1-10GeV (where we find (δ(cid:46)−2.65).
−3.0 < δ < −2.2)andbecomeslessevidentathigherenergies
(−2.9 < δ < −2.5),wheretherelativeeffectoftheadvection,as
Movingtowardsprogressivelyhigherlatitudes,thespectrum
comparedtodiffusion,islessimportant.
tendstowardssoftervalues,atalllongitudes.Theregionscharac-
Notethatmodelsincludingthespiralarmsbutneglectingtheir
terizedbythemostextreme(softandhard)spectralindicesarein
motion (Shaviv 2003; Shaviv et al. 2009; Gaggero et al. 2013)
factmissedbythel.o.s.andanintermediatespectralindexisrecov-
would instead predict the same spectrum across the Galaxy, with
ered.
noarmsignatures.
Summarizing, two main features are predicted by our
Thevariationofthespectralindexacrossthearmsaffectsthe small halo, dynamic spiral arm model. Spectra are hard at
γ-ray spectrum detected at Earth. The effect is more pronounced small/intermediateGalacticlongitudes(correspondingtothecen-
nearthedirectionsofthearmtangents.Considerforexamplethe tral part of the map), since these longitudes correspond to l.o.s.
closest arm and the arm tangents identified by the dashed line in crossingthesideofthearmoppositetothedirectionofthemotion
Fig.3(left-handpanel).Intherange1-10GeV,theγ-rayspectrum (redinFig.3).Onbothsidesofthiscentralhardregion,wherelat-
measuredatEarthisexpectedtohaveaspectralindexaroundδ ∼ itudesarehigher,spectraaresoft,sincetheselatitudescorrespond
−2.5, as can be guessed from Fig. 3. A slightly different l.o.s., tol.o.s.thatcrosstheothersideofthearm(blueinFig.3).Thesec-
pointingnowtoeithersideofthearmwouldcorrespondinsteadto ondpredictionisthatthistrendisparticularlyevidentatlatitudes
a softer or harder spectrum, depending on which side of the arm 2◦ < b < 4◦,wherethecontributionfromthearmisdominating
we are looking at. The same effect is expected when the almost theemission,andisreducedbothatverysmall(b < 2◦)latitudes
MNRAS000,1–8(2015)
Galacticγ-rayandCRspectra 5
Figure2.Left-handpanel:thedistributionofthesimulatedCRspectralindex,estimatedintherange10−3000GeV,intheplaneZ=0.Theopenhistogram
showstheresultsassumingainjectionspectralindexα = −2.3,whiletheshadedhistogramisobtainedforaharderinjectionindex,α = −2.25.Inboth
cases,thediffusioncoefficientisproportionaltoE0.4.TheverticalblueandgreylinesdepicttheobservedspectrumontheEarth,respectively.Right-hand
panel:thedistributionofthedifferencebetweentheγ-rayspectralindexintheplaneoftheskyobtainedfromsimulationsandfromobservations.Asmall
latituderegionhasbeenselected:−5◦<b<5◦.ThetwodifferenthistogramsrespectivelyrefertothetwodifferentchoicesforCRinjectionspectralindex.
Figure3.AtopviewmapoftheMilkyWay(atZ=0)showingthephotonindexoftheγ-rayspectrumfromπ0decay,asmeasuredintherange1−10GeV
(left-handpanel)andintherange10−300GeV(right-handpanel),asobtainedinthesmallhalomodel.TheSunislocatedatX=8.5kpcandY =0(star
symbol).Thedashedanddot-dashedlinesshowthetwolinesofsightthatinterceptthespiralarm.
(wheretheemissionistheresultofcontributionfromboththearm diffusioncoefficient(inparticular,tofitthe“age"derivedfromthe
andthedisc)andatlargelatitudes(b>4◦),wherethelocalemis- Berylliumisotoperatios).Asaconsequence,characteristicsofthe
siongovernsthespectralslope. CRs in the vicinity of the Galactic arms are smeared over large
Fig. 6 depicts the prediction of the large halo model, with regions,wipingawaydetails.
Z =2kpc,intheenergyrange1−10GeV.Whencomparedwith
h
thesmallhalomodel(Fig.5,upperpanel),thesamequalitativefea-
turesexist,butquantitativelytheyaredifferent.First,insteadofa
5 ACOMPARISONWITHOBSERVATIONS
welldefined“oval"withahigherspectralindex,therespectiveoval
issignificantlylessdefined.Second,insteadofhavingvariationsin Recently,aspectralmap(asmeasuredbyFermi)oftheanisotropic
theindexoforder∼0.2,therearevariationsofonly∼0.05. component (dominated, especially at low energies, by π0 decay)
The differences between the two different halo size models of the diffuse γ-ray emission has been presented by Selig et al.
arenotsurprisinggiventhatthelargehalomodelrequiresalarger (2015),allowingustocompareournumericalresultswithobser-
MNRAS000,1–8(2015)
6 Navaetal.
Figure6.Mapofthespectralslopeintheenergyrange1-10GeVoftheπ0-
producedγ-rayradiationobtainedfromthelargehalosimulation(Zh =
2kpc).Thecontourlevelsaredenotedbythecolourbaronthebottom,and
itisthesameasinFig.5.
Figure4. Simulatedγ-rayspectraatEarthfromπ0decayobtainedafter
integrationalongthel.o.s. inthesmallhalomodel.Ineachpanel,three
differentl.o.s.areshown:inthedirectionofthespiralarm(green),andon
bothsidesofthespiralarm(redandblue),atlatitudesb=0◦.Theupper
panelreferstodirectionsaroundl ∼ 50◦ (dashedlineinFig.3),while
thebottompanelreferstol∼−104◦(dot-dashedlineinFig3).Foreach
spectrum,thelongitudeandthespectralindex(intherange1−10GeV)
arereported.Unitsonthey-axisarearbitrary,andthespectrahavebeen
normalizedsothattheyoverlapatlowenergies.
Figure 7. Comparison between observed and simulated γ-ray spectra in
twodifferentregionsofinterest:|l| < 40◦and|b| < 1.5◦(upperpanel),
and −150◦ < l < −120◦ and |b| < 3◦ (lower panel). Observations
(takenfromSeligetal.2015)areshownwithadashedline.Theshaded
areasrepresenttheuncertainties,includingstatisticalandsystematicerrors.
Thespectrumderivedfromoursimulationinthesameregionoftheskyis
insteadshownwithasolidline.Normalizationsarearbitrary.
Figure5.Mapsofthespectralslopeintheenergyrange1-10GeV(top) vations.Theiranalysisisperformedinnineenergybins,from0.6
and10-300GeV(bottom)oftheπ0-producedγ-rayradiation,inthesmall to307.2GeV.First,wecomparetheirmeasuredγ-rayspectraesti-
halomodel.Thecontourlevelsaredenotedbythecolourbarontheright, matedintwodifferentregionsoftheskywithoursimulations.The
anditisthesameforbothmaps. resultsareshowninFig.7:datafromSeligetal.(2015)areshown
asadashedline.Uncertaintiesarerepresentedwithashadedarea.
The spectrum derived from our simulation is instead shown with
a solid line. The upper and bottom panels refer to two different
regionsofthesky.Normalizationsandunitsonthey-axisarear-
bitrary.Aninconsistencybetweenthesimulatedspectrumandthe
observedoneintheouterregionisevidentatenergies> 10GeV,
MNRAS000,1–8(2015)
Galacticγ-rayandCRspectra 7
wherethedatashowasteepeningofthespectralslope.Thissteep-
eningfoundbySeligetal.(2015)isinconsistentwithresultsfrom
Ackermannetal.(2012)andtheauthorsidentifytheoriginofthis
inconstancywiththelowsignaltonoiseratiointhehighestenergy
bins.
Weusethefluxesprovidedinthefirstfourenergybins1(cor-
responding to the energy range 0.85-6.79GeV) and estimate the
spectralindexbyfittingthedatawithasimplepower-lawfunction,
consistentlywiththemethodadoptedtoderivetheslopeofthesim-
ulatedspectra.
The Fermi spectral map at |b| < 7◦ in the range 0.85-
6.79GeV is depicted in the lower panel of Fig. 8. We focus on
thisrangeofenergiesandlatitudesbecauseitiswherethediffuse
emissionofGalacticoriginisexpectedtobedominatedbyπ0de-
cay,andthecomparisonbetweenoursimulationsandobservations
isthereforejustified.Inordertoperformapropercomparison,we
estimatethespectralindexofoursimulatedspectrainthesameen- Figure8.Mapsofthespectralslopeoftheπ0-producedγ-rayradiationin
ergyrange.Thesimulatedmapforthesmallhalomodelisshowed theenergyrange0.85-6.79GeV.Theupperpaneldepictstheresultsfrom
intheupperpanelofFig.8. thesmallhalosimulation,whilethebottompanelistheFermimapadapted
Firstofall,wenotethattherangeofδvaluesinthesimulated fromSeligetal.(2015).Thesamecolorbarisusedforboththesimulated
andobservedspectraissimilar:−2.65 < δ < −2.35.Moreover, andtheobservedmaps.
thetwomapsareverysimilar,i.e.,δvariesasafunctionsofband
laccordingtoasimilarpattern.Allthefeaturespresentinthesim-
6 DISCUSSIONANDSUMMARY
ulatedmapanddiscussedintheprevioussectionarevisiblealsoin
the map derived from Fermi data, albeit with some measurement A spiral arm source distribution was first incorporated into a CR
noise.Thisallowsusnotonlytoassessthevalidityofournumer- diffusion model to explain the apparent time variation in the CR
ical results, but also to provide an interpretation for the observed historyasreconstructedwithIronMeteorites(Shaviv2003).Asa
spectralmappresentedbySeligetal.(2015),atleastinthisrange “sideeffect"itwasfoundtoexplainthePamelaanomalyinwhich
oflatitudes. thepositronfractionappearstoincreaseaboveabout10GeV(Sha-
ByinspectingtheFermimapintheregion−90◦ <|l|<40◦, vivetal.2009;Gaggeroetal.2013).Thesemodels,however,do
wenotethatthespectraareonaverageharderthanthespectraon notintroduceanyadditionaltime-scalefortheprotonpopulation,
bothsidesofthecentralregion.Moreover,inthiscentralpartofthe becauseofwhichtheπ0 producedγ-raysshouldexhibitthesame
map,avariationofthespectralindexwiththelatitudeisvisible. power-law index in every direction, unless new physics is intro-
At|b| < 1◦ thespectralindexhasanintermediatevalue,and duced.
it becomes harder at around |b| ≈ 2◦ −3◦. We have interpreted SuchnewphysicswasintroducedbyBenyaminetal.(2014)
this variation of δ as due to the transition from l.o.s. where the whorealizedthatifoneconsidersthedynamicsofthespiralarms,
contributionfromthediscisrelevant,tol.o.s.wheretheclosestarm namely,thattheISMmovesrelativetothearms,theCRsundergo
isdominatingtheemission.Atevenhigherlatitudesthespectrum advectionrelativetotheCRsources(inadditiontodiffusion).This
becomessofter,consistentlywiththefactthatthecontributionfrom advection naturally predicted a rise in the Boron to Carbon ratio
thesideofthearmbecomesnegligibleagain. at energies below 1 GeV/nuc., as observed. However, several ad-
Alsotheregionatlargerlongitudes|l|>40◦exhibitsaverage ditionalconsequencesensuedfromthisanalysis.First,becauseof
propertiesverysimilartothosefoundinthesimulatedspectra.The thedifferentpathlengthdistribution,itwasfoundthatinorderto
spectrumissignificantlysofter,whichweinterpretedasduetothe correctlyrecovertheoverallamountofsecondariesproduced,one
fact that these l.o.s. mainly intersect the downstream side of the musttakeasmallerhaloandasmallerdiffusioncoefficient.Sec-
arm. ond,itwasfoundtoconsistentlyexplainthesub-IrontoIroniso-
Whilethesmallhalosimulationreproducesthemainfeatures topes,whichinstandardmodelsrequireaninconsistentlydifferent
intheFermiindexmaps,thisisnotthecasewiththelargehalosim- averagecolumndensity(Benyaminetal.2016).
ulation,theresultsofwhicharedepictedinFig.6.Specifically,the The next natural step was to simulate the γ-ray production
largehalomodelhassignificantlylessdefinedfeaturescompared through π0 decay, and compare with observations. This is more
withboththesmallhalosimulationandtheactualFermiobserva- robust than other γ-ray components that require the distribution
tions.Inotherwords,thelatterstronglypointtowardsasmallhalo. of electrons and the magnetic field as well, and therefore left for
Wehavealsocomparedtheobservationswiththespectralmap future analysis. In particular, given that standard models predict
derivedwithα = −2.3.Inthiscase,apatternsimilartotheone afixedpower-lawindex,bystudyingdirectionalvariationsofthe
showninFig.2isrecovered,but(asanticipatedin§4)thevalues power-lawindexwecanconcentrateondifferencesfromthestan-
are shifted towards softer spectral indices, and do not match the dardmodel.
observedrangeofvaluesobservedbyFermi. In the analysis, we have found that the dynamic spiral arm
modelexhibitsinterestingcharacteristics.First,thereisaveryclear
upstream/downstream asymmetry, such that the upstream side of
spiralarmsisharder.Inaddition,theGalacticplaneregioninwards
(towards the Galactic centre) of the two opposite arm tangents is
harder than the plane outside the tangents. Note that because the
1 https://www.mpa-garching.mpg.de/ift/fermi/ arm tangents are not symmetric around the Galactic centre, this
MNRAS000,1–8(2015)
8 Navaetal.
asymmetryisnotexpectedtobesymmetriceither.TheFermidata ACKNOWLEDGEMENTS
exhibit this signature–the spectral slope at the plane towards the
LNwaspartiallysupportedbyaMarieCurieIntra-EuropeanFel-
centreoftheGalaxyisharderthanintheotherdirections.More-
lowshipoftheEuropeanCommunity’s7thFrameworkProgramme
over, the data appear to exhibit an asymmetry about the Galactic
(PIEF-GA-2013-627715).TPwaspartiallysupportedbyAdvERC
centreexpectedfromtheasymmetricarmtangents,thoughnotun-
grantTREX.NJSgratefullyacknowledgesthesupportoftheIBM
equivocallygiventhedataquality.
EinsteinFellowshipoftheInstituteforAdvancedStudy,andIsrael
The second interesting signature, apparent in both the small Science Foundation (grant no. 1423/15). This research was also
halosimulationandtheFermiobservations,istheverticaldepen- supported by the I-CORE Program of the Planning and Budget-
dence.Whenobservingbetweenthearmtangentsthenthespectral ingCommitteeandtheIsraelScienceFoundation(center1829/12).
indexincreasesfromtheplaneuptoanaltitudeofabout1.5◦,and Theauthorsacknowledgetheuseoftheopenaccessmaterialpre-
thendecreasesfurtherawayfromtheplane.Thisseeminglyoddbe- sentedbySeligetal.(2015)ontheapplicationoftheD3POinfer-
haviourisduetothefactthatattheGalacticplane,thel.o.s.crosses encealgorithmtotheγ-rayskyseenbytheFermi-LAT.
theharderupstreamsideofthearmbutthesignificantcontribution
fromtheGalacticdiscfurtherinsidetheGalaxyflattensthespec-
trum.Ontheotherhand,atasomewhathigherangle,theGalactic
REFERENCES
discismostlyabsentsincethel.o.s.passesaboveit,andthecon-
tributionfromthehardarmdominatesthespectralshape.Ateven AbdoA.A.,etal.,2010,ApJ,710,133
higherlatitudes,thel.o.s.alreadypassesabovethearmandthelo- AckermannM.,etal.,2012,ApJ,750,3
AmaralL.H.,LepineJ.R.D.,1997,MNRAS,286,885
calsoftercontributionislefttodominate.Sincetheaforementioned
BennettC.L.e.a.,2012,preprint, (arXiv:1212.5225)
featuresaresignificantlyattenuatedinthelargehalomodelthatre-
BenyaminD.,NakarE.,PiranT.,ShavivN.J.,2014,ApJ,782,34
quiresalargerdiffusioncoefficient,wecanconcludethattheFermi
Benyamin D., Nakar E., Piran T., Shaviv N. J., 2016, preprint,
indexmapsaresurprisinglyconsistentwiththesmallhalosimula-
(arXiv:1601.03072)
tion,butnotthelargehalo. BloemenH.,1989,ARA&A,27,469
Itshouldbeemphasizedthatunlikeothermodelsthathaveto DameT.M.,HartmannD.,ThaddeusP.,2001,ApJ,547,792
FatoohiL.J.,GillerM.,WdowczykJ.,WolfendaleA.W.,ZhangL.,1996,
addadditionaladhocassumptionstoobtainspectralchanges(e.g.,
A&AS,120,407
onthediffusioncharacteristicsinsideandoutsidearms,oracontri-
FerrièreK.M.,2001,ReviewsofModernPhysics,73,1031
butionfromdarkmatter),thedynamicsofthearmsisanunavoid-
GaggeroD.,MaccioneL.,DiBernardoG.,EvoliC.,GrassoD.,2013,Phys-
ablecomponentthatshouldnecessarilybeincluded.Weknowthat
icalReviewLetters,111,021102
theCRsourcesareinthearmsandthattheISMisadvectedthrough KamaeT.,KarlssonN.,MizunoT.,AbeT.,KoiT.,2006,ApJ,647,692
them. LaceyC.K.,DuricN.,2001,ApJ,560,719
NaozS.,ShavivN.J.,2007,NewAstron.,12,410
Anotherinterestingpointisthatbecausethismodelpredicts
NeronovA.,MalyshevD.,2015,preprint, (arXiv:1505.07601)
thattheCRspectrumvariesacrosstheGalaxy,thelocalCRspec-
RogersM.J.,SadzinskaM.,SzabelskiJ.,vanderWaltD.J.,Wolfendale
trummeasuredattheEarthandtheaverageCRspectruminferred
A.W.,1988,JournalofPhysicsGNuclearPhysics,14,1147
fromtheaverageoftheγ-rayspectraarenotnecessarilythesame.
SeligM.,VaccaV.,OppermannN.,EnßlinT.A.,2015,A&A,581,A126
This is consistent with (and provides an explanation for) the ap- ShavivN.J.,2002,Phys.Rev.Lett.,89,051102
parentinconsistencybetweenthelocalandaveragespectralslopes. ShavivN.J.,2003,NewAstron.,8,39
The difference we need to introduce in our model is of 0.05, but ShavivN.J.,NakarE.,PiranT.,2009,PhysicalReviewLetters,103,111302
larger differences can also be explained by slightly changing the ShavivN.J.,ProkophA.,VeizerJ.,2014,ScientificReports,4,6150
locationofthearmwithrespecttotheEarthinthesimulation. SuM.,SlatyerT.R.,FinkbeinerD.P.,2010,ApJ,724,1044
TibaldoL.,etal.,2015,ApJ,807,161
Oneshouldalsomentionseveralcaveatsinthemodel.First,
Werner M., Kissmann R., Strong A. W., Reimer O., 2015, Astroparticle
theISMdensityisthesameinthespiralarmsandoutsideofthem. Physics,64,18
Thisisanaiveapproximation,theeffectofwhichisnotsupposed
to change the slopes in the XY map, however, it will emphasize
ThispaperhasbeentypesetfromaTEX/LATEXfilepreparedbytheauthor.
the γ-rays emission coming from the arms, and therefore should
somewhat increase the predicted spectral slope contrasts. On the
otherhand,themodeldoesnotconsiderpossiblevariationsofthe
diffusioncoefficient(e.g.,becauseofthemagneticfieldandISM
turbulencecharacteristics)thatcouldgivedifferentspectralslopes
inthearmsandoutside.
AnotherinterestingpointtoconsideristheeffectoftheFermi
bubbles (Su et al. 2010). It is clear that at a high enough Galac-
ticlatitudeitwilldominatethespectrum(Seligetal.2015).This
wasnotconsideredhere,butisshouldstartdominatingthespectral
indexatsomelatitudeabovetheplane.
Nevertheless, even with the caveats considered, the general
agreement between the theoretical predictions and observations
suggeststhattheCRssourcesareindeedconcentratedtowardsthe
arms(asexpectedfromSNR),andthatthehalosizeanddiffusion
coefficientaresmallerthanthecanonicalvaluesarisinginstandard
azimuthallysymmetricmodels.
MNRAS000,1–8(2015)