The B/C and sub-Iron/Iron Cosmic ray ratios - further evidence in favor of the spiral arm diffusion model

3.5 MB·

by  David Benyamin

PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
THEB/CANDSUB-IRON/IRONCOSMICRAYRATIOS-FURTHEREVIDENCEINFAVOROFTHESPIRALARM
DIFFUSIONMODEL
DAVIDBENYAMIN1,EHUDNAKAR2,TSVIPIRAN1&NIRJ.SHAVIV1,3
1.TheRacahInstituteofphysics,TheHebrewUniversityofJerusalem,Jerusalem91904,Israel
2.RaymondandBeverlySacklerSchoolofPhysics&Astronomy,TelAvivUniversity,TelAviv69978,Israeland
3.SchoolofNaturalSciences,InstituteforAdvancedStudy,EinsteinDrive,Princeton,NJ08540
ABSTRACT
TheBorontoCarbon(B/C)andsub-Fe/FeratiosprovidesanimportantclueonCosmicRay(CR)propagation
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within the Galaxy. These ratios estimate the grammage that the CR traverse as they propagate from their
1
sourcestoEarth. AttemptstoexplaintheseratioswithinthestandardCRpropagationmodelsrequireadhoc
0
modificationsandevenwiththosethesemodelsnecessitateinconsistentgrammagestoexplainbothratios. As
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analternative,physicallymotivatedmodel,wehaveproposedthatCRoriginatepreferablywithinthegalactic

r spiralarms. CRpropagationfromdynamicspiralarmshasimportantimprintsonvarioussecondarytoprimary
p
ratios,suchastheB/Cratioandthepositronfraction.Weuseourspiralarmdiffusionmodelwiththespallation
A
networkextendeduptoNickeltocalculatethesub-Fe/Feratio.Weshowthatwithoutanyadditionalparameters
thespiralarmmodelconsistentlyexplainsbothratioswiththesamegrammage,providingfurtherevidencein
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2 favorofthismodel.
Subjectheadings:cosmicrays—diffusion—Galaxy: kinematicsanddynamics

]
E
H 1. INTRODUCTION to find any combination of parameters that fit both ratios si-
multaneously, for the aforementioned reason. Namely, Iron
. Cosmic ray (CR) composition measurements have been
h nuclei do not produce enough secondaries given model pa-
collectedforalmostfourdecades. Theiranalysisprovidesin-
p rametersthatcorrectlyreproducetheB/C.
formationonthepropagationthroughtheinterstellarmedium
- Over the past decade, CRs propagation models evolved to
o (ISM),whichfurtherprovidesinformationontheCRsources
includespiralarmsasthesourceoftheCRparticles(Shaviv
r andvariouspropertiesoftheISM.Oneofthemostimportant
t aspects characterizing the propagation is the CR path length 2003;Shavivetal. 2009;Gaggeroetal. 2013a;Effenberger
s
et al. 2012; Werner et al. 2013). In these models, the CR
a distribution (PLD), which describes the probability distribu-
[ tion function of the path lengths traversed by CRs between sourcesprimarilyresideatafinitedistancefromthesolarsys-
tem,andthisnaturallysuppressestheshortpathlengths(e.g.,
theiroriginandtheirmeasurementnearEarth. ThisPLDde-
2 see figs. 4 and 6 in Benyamin et al. 2014, hereafter B14).
termines the amount of spallation and radioactive decay the
v Hence, spiral arm models could in principle fit both ratios
particles will undergo on the way to Earth. Thus, by mea-
2 with the same model parameters. We study this possibility
suring the ratio between secondary CR particles, formed en
7
here.
0 route, to primary particles accelerated at the source, one can
B14 considered the spiral arms to be dynamic as well,
3 constrainthePLD.
showingthatthismodelrecoversthelowenergybehaviorof
0 Until recently one of the stringent and often implicit as-
theB/Cratio. Morespecifically,secondariestoprimariesra-
. sumptions in CR propagation models was an azimuthally
1 tios such as the B/C, are expected to decrease with energy
symmetric source distribution (e.g. Strong et al. 2007; di
0 becausethetimerequiredforparticlestoreachthesolarsys-
Bernardo et al. 2010). The PLD resulting from such mod-
6 tem(i.e.,their“age”)decreaseswiththediffusivity,whichin-
elscloselyresemblesanexponentialPLD.
1 creaseswiththeenergy.However,below1GeV/nuc.,theB/C
Oneoftheproblemsinthesedisk-likemodelsistheamount
: ratio exhibits the opposite behavior. Although unexpected
v of grammage the models require to explain the B/C and the
i sub-Fe/Fe ratios given the same geometric parameters. Be- fromjustdiffusion,itcanbeexplainedbytakingintoaccount
X thedynamicsofthespiralarms.Thisisbecauseatsufficiently
cause of the higher spallation cross-sections, the sub-Fe/Fe
r ratioistypicallysensitivetotherelativefractionofshortpath low energies the CR age will be governed not by the diffu-
a sionfromthespiralarmsbutinsteadbythetimesincethelast
lengths in the PLD. The B/C ratio, with lower spallations
spiral arm passage, which would be shorter. Since CRs be-
crosssections,ismoresensitivetotherelativefractionoflong
low 1 GeV/nuc. are non-relativistic, an energy independent
path lengths. When trying to fit both measured ratios simul-
propagationtimethengiveslessspallationproductsatlower
taneously in a disk-like model, one finds inconsistency with
energies,thusreproducingtheB/Criseatlowenergies(B14),
thenearlyexponentialPLDofthedisk-likemodels—thesub-
withoutrequiringanyadditionalassumptionsonthepropaga-
Fe/Fe ratio requires a larger mean grammage than the B/C
tions(suchasagalacticwind,reaccalerationorbreaksinthe
ratio. Garcia-Munozetal. (1987)suggestedthatthisincon-
diffusioncoefficient).
sistency could be resolved with the ad hoc assumption that
One interesting ramification of the modified PLD of the
shortpathlengthsaresuppressed.
spiral arm model is that it requires a notably smaller halo
In a different analysis, Davis et al. (2000) investigated
size and diffusion coefficients to recover the observed B/C
whether the introduction of a second, low-energy CR source
and 10Be/Be ratio. Namely, by modifying the PLD one has
can reproduce the B/C and sub-Fe/Fe ratios, in particular, at
tochangethe“canonical”diffusionparameterscharacterizing
low energies. Although they found that one can recover the
thecosmicrays(B14). Thisisreasonableandevenexpected
B/Cratioandthesub-Fe/Feratioseparately,theywereunable
2 Benyaminetal.
giventhatthose“canonical”parameterswereobtainedwithin 1988) and SANRIKU (Hareyama 1999). We carry out the
the standard disk-like model whose propagation characteris- analysis twice, once for a disk-like model having no spiral
ticsarequitedifferentfromthoseofthespiralarmmodelthat arms,andonceforthespiralarmmodelofB14.
weconsiderhere. We begin in §2 by briefly describing the model we devel-
An interesting related phenomenon is the unexpected en- oped in B14 and the parameters we use. In §2.1 we detail
ergy dependence of the positron fraction, e+/(e+ + e−), the improvements carried out for the present work. The re-
of CRs. Since positrons in standard scenarios are sec- sults are then described §4, the main part of which includes
ondaryparticles,theratioisexpectedtodecreasewithenergy separate fits of δ, and D to the observed B/C ratio and to
0
(Moskalenko&Strong1998).However,thePAMELAsatellite thesub-Fe/Feratio. Theimplicationsoftheseresultsarethen
measurementsrevealedthattheratioincreaseswiththeenergy discussedin§5.
above10GeVuptoatleast100GeV(Adriani2009). More
2. THENUMERICALMODEL
recent measurements by AMS-02 show leveling at 300 GeV
(Accardoetal. 2014). InB14wedevelopedaCRadvection-diffusionmodelwith
Given that standard models could not explain this behav- which we recovered the B/C ratio. The nuclear network in
ior,severalsolutionsweresuggested. Theincreasedpositron the model included all stable and long lived radioactive iso-
fraction at higher energies can be explained in any scenario topes between Beryllium and Oxygen. The model followed
havinganadditionalprimarypopulationofpairswithahard the propagation of primary and secondary CRs from their
spectrum,suchthatitdominatesthepopulationofsecondary sources,primarilylocatedonspiralarms. Unlikepresentday
positronsabove10GeV.Pairscannaturallybecreatedbydark state of the art simulations (such as GALPROP, Strong et al.
matter annihilation (Bergstro¨m et al. 2008; Ibarra & Tran 2007 and DRAGON, di Bernardo et al. 2010) that solve the
2008),bypulsars(Harding&Ramaty1987,Chietal. 1996, partial differential equations of the describing advection and
Aharonian et al. 1995, Hooper et al. 2009, and Profumo diffusion,ourmodelusesaMonteCarlo(MC)algorithm.
2008) or in aged SNRs (Blasi 2009, Blasi & Serpico 2009a, Instead of simulating the small steps corresponding to the
Ahlersetal. 2009,andMertsch&Sarkar2009). physicalmeanfreepath(m.f.p.) oftheCRs,around1pc,our
AnothertypeofexplanationdirectlyrelatedtothePLDwas MCsimulationusesalargereffectivem.f.p.,whichlumpsto-
proposedbyShavivetal. (2009). IftheCRsourcesarecon- gethermanysmalleffectivephysicalsteps. Thiscorresponds
centrated at a finite distance from the solar system, as is ex- to a larger effective time step chosen to be τesc/100, where
pectedfromthespiralarmstructure,thepaucityofshortpath τ ≈ Z2/2D is the typical escape time of a CR from the
esc h
lengthsimpliesthatprimaryelectronswithahighenoughen- galaxy2. In the present simulations, we fix Z to be 250 pc,
h
ergy will cool before reaching the solar system. In contrast, likeournominalmodelinB14. Thiscorrespondstoaneffec-
thesecondarypositronsareformedbyprotonsthateffectively tivem.f.p.of25pc, Whichissmallerthanthetypicallength
donotcool,suchthattheycanbeformedinthesolarsystem’s scaleoverwhichthegasdensityvaries.
vicinity. For the local electron cooling rate determined by Each time step, we check whether the CR bundle had left
Synchrotronandinverse-Comptonscattering,andthetypical the galaxy. If it did not, we calculate the grammage that the
CR age determined by Beryllium isotope ratios, this behav- bundletraversedinthistimestep. Wethencalculatetheprob-
ior predicts that the Positron fraction should start increasing abilitythattheparticleshadundergoneanuclearreactionwith
above10GeV,thusexplainingthe“PamelaAnomaly”,effec- ISM. If the particles did undergo spallation into secondary
tivelywithoutanyfreeparameters. particles,wefollowthelatterwiththesamemethodology.We
In the present work we study the B/C and sub-Fe/Fe ra- nominally use the YIELDX subroutine of Silberberg & Tsao
tios using an extended version of the B14 model, as is de- (1990) for the nuclear cross-secitons. However, we repeat
scribedin§2.1. Oneofthemajoruncertaintiesarisingwhen the analysis with Webber et al. (2003) to obtain a handle
studyingthesub-IrontoIronratioanditsimplicationstothe on the uncertainty introduced by the nuclear cross-sections,
PLD,istheenergydependentspallationcross-sections. Since as described below. For long-lived radioactive isotopes, we
many of the partial cross-sections are poorly measured, the also check each time step for radioactive decay. The code
values used are often the results of fits and extrapolations, also considers Coulomb and ionization losses as elaborated
giving rise to large uncertainties (Moskalenko 2011; Web- inMannheim&Schlickeiser(1994)andappliedinGALPROP
ber et al. 2003). The problem is aggravated below a few (Strong&Moskalenko1998).
GeV/nuc.,wherethecross-sectionshavelargerenergydepen- Forthepresentsimulations,eachrunuses109 CRbundles
dences(Moskalenko&Mashnik2003;Schwalleretal. 1979), inthespiralarmssimulationsand108inthedisk-likesimula-
and for heavier elements (e.g., see appendix II in Garcia- tions.Foreachbundle,werandomlychooseitsinitialisotopic
Munozetal. 1987;Titarenkoetal. 2008;Sisterson&Vincent identity and initial location in the galaxy, and follow it until
2006).Welimittheanalysistoenergiesbetween10GeV/nuc. itescapesthegalaxy,coolbelowthe10GeV/nuc.,breakinto
to 1000 GeV/nuc. Although it is not clear by how much it isotopes lighter than the lightest element simulated or reach
decreases the uncertainties in the cross-sections, the weaker the solar vicinity. In the latter case we record all simulated
energydependencewilltranslateintoasmalleruncertaintyin isotopes (Beryllium to Silicon in the light elements simula-
theinferredspectralslopeofthediffusioncoefficient. Weuse tions and Scandium to Nickel in the heavy elements simula-
thenominalmodelparametersofB14,assummarizedin§2.3, tions),ina3Darray(Z,A,energy).Theenergiesarerecorded
andsearchtheoptimaldiffusioncoefficientspectralindexand in10binsperdecade, coveringtherangeof10GeV/nuc. to
normalization,namely,δ,andD inD =D (E/E )δ1,with 1000GeV/nuc..
0 0 0
E = 3 GeV/nuc., to fit both the B/C from AMS-02 (Oliva
0
etal. 2013)andthesub-Fe/FedatafromHEAO-3(Binnsetal. 2.1. Theextendedcode
1Throughoutthepaper,theterm“diffusioncoefficient”actuallyrefersto 2Althoughthetimestepδt(E)isenergydependent,theeffectivem.f.p.is
thenormalizationD0andnotD(E). not,asitdependsonthecombinationδt(E)×D(E).
3
Below we detail specific modifications and improvements
TABLE1
ofthecodeusedinthiswork.
NOMINALMODELPARAMETERS
2.1.1. Nuclearnetwork valuefor valuefor
parameter definition spiralarm disk-like
AsweareinterestedinsecondordercorrectionstotheB/C
model model
ratio,weextendedthecodetodescribethespallationnetwork Zh Halfhaloheight 250 pc 1kpc
allthewayuptoSilicon. τarm Lastspiralarmpassage 5 Myr
Inadditiontotherangeof“light”elementsnecessaryforthe i4 4-armsset’spitchangle 28◦
aforementionedcalculation,wearealsointerestedincalculat- i2 2-armsset’spitchangle 11◦
iwnogrkthoefshueba-Fviee/rFeelreamtieon.tsW, ferothmerSefcoarnedaiudmdetdotNheicnkuecllaesarwneeltl-, Ω4 Antghuela4r-avremloscsiteytof 15(km/s)kpc−1
usingthe YIELDX (Silberberg&Tsao1990),oralternatively Ω2 Antghuela2r-avremloscsiteytof 25(km/s)kpc−1
Webberetal. (2003)forthepartialspallationcross-sections
PercentageofSNin
describingtheinteractionwithHydrogen.Weusetheapprox- fSN,4 the4-armsset 48.4%
imate formulae of Karol (1975) for the total inelastic cross-
PercentageofSNin
sectionsofinteractionswithHelium. fSN,2 the2-armsset 24.2%
Percentageofcorecollapse
2.1.2. Fastercode fSN,CC SNeinthedisk 8.1% 80.7%
The code was accelerated by limiting the CR diffusion in Percentageof
the spiral-arms models to take place only within of a radius fSN,Ia SNTypeIa 19.3% 19.3%
5 times the halo size around the solar system. This captures
4. RESULTS
97% of the total CRs reaching the solar system, when com-
paring to a full calculation without the distance cutoff, but We have separately fitted the B/C data and the sub-Fe/Fe
increases the speed by about an order of magnitude to reach data to the two CR source models: the spiral arms sources,
similarstatistics. Forthedisk-likemodelsimulations, thisis andadisk-like,azimuthallysymmetric,one. Foreachmodel
unnecessary. Theazimuthalsymmetryallowsformuchbetter wethencheckeditsconsistency,asthedifferentdatasets(B/C
statisticsthanthespiral-armssimulationwithmuchfewerCR andsub-Fe/Fe)shouldbefittedwiththesamemodelsparame-
bundles. ters. Ifwefindconsistencytheoptimalparametersthenteach
Inordertoincreasetheaccuracy,wedecreasethetimestep usaboutCRdiffusionintheinterstellarmedium.
10-fold when the CR bundle is within a distance of 150 pc To obtain the best fit values, we minimize the χ2 for each
fromthesolarsystem,anadditional10-foldwhenthebundle simulation. Theuncertaintyusedinestimatingtheχ2 values
is within 120 pc, and we take a time step for which the ef- includeboththosearisingfromthe(MonteCarlo)simulation
fectivemeanfreepath(m.f.p.) isofordertheactualphysical andthequotedobservationalerrorswhicharetypicallyanor-
m.f.p. (oforder1pc), whentheCRbundleiswithin100pc derofmagnitudelarger.
fromthesolarsystem.
4.1. B/C
2.2. Initialcomposition Thefittotheoptimalspectralslopeδ anddiffusioncoeffi-
Theinitialcompositionforwhichweobtaintheoptimalfit cient D is carried by calculating the χ2 over a two dimen-
0
tothemeasuredsecondary/primaryisotoperatiosinthelight sionalarrayofvalues. Then,thesetofχ2 foreachvalueofδ
elementssimulationsis37%Carbon(bynumber),3%Nitro- intherangeof0.3to0.7isfittedwithapolynomial,andthe
gen,52%Oxygen,4%Neon,2%Magnesiumand2%Silicon. minimumfound.
Note that although the method for finding the initial compo- Theoptimaldiffusioncoefficientasafunctionofδisgiven
sition is the same as in B14, the composition is somewhat in table 2. By comparing the disk-like models to the spiral
differentbecausethecodenowfollowsmoreisotopes,asde- arm models, it is evident that the typical D required to fit
0
scribedin§2.1.1.Fortheheavyelementssimulations,thebest theobservationsin“standard”disk-likemodelsismuchlarger
fittotheobservedNi/Feratio,isobtainedforaninitialcom- thanthetypicalD requiredtofitthesamedatainthespiral
0
positionconsistingof95%Ironand5%Nickel. armmodel. Thisrecapturestheresultpreviouslydescribedin
B14,andarisesfromthesmallerhalointhespiralarmmodel.
2.3. Thenominalmodelparameters The modelfits tothe data, asa functionof δ, aredepicted
in fig. 1 for the disk-like model and fig. 2 for the spiral arm
Table. 1 summarizes the nominal model parameters ob-
model. Oneseesfromthesefitsthatonerequiresroughlythe
tained by or referenced within B14. These parameters are
sameδ ∼0.4(thickerpurpleline)tofitboththedisk-likeand
separately fixed for either the spiral-arms simulations or the
spiralarmmodels.
disk-likesimulations.
Figs.5&6depicttheχ2asafunctionofbothδandD .
0
3. OBSERVATIONALDATASETS Thelatterplotsalsodemonstratethatforagivenvalueofδ
thereisanarrowrangeofD forwhichthereisareasonable
We compare the model predictions for the B/C ratio with 0
fit. However, without prior knowledge of δ, there is a much
theAMS-02results,whoseadvantageoverpreviousdatasets
widerallowedrangeforoptimalD .
isthenotablybetterstatistics.Intotal,thereare10datapoints 0
between10GeV/nuc.and1000GeV/nuc.(Olivaetal. 2013).
4.2. Iron
Thesub-Fe/FedatasetsusedboththeHEAO-3(Binnsetal.
1988)andtheSANRIKUmeasurements(Hareyama1999).In IndependentlyoffittingtheB/Cdata,wefitthesub-Ironto
total,thereare21datapointsbetween10GeV/nuc.and1000 Irondatainasimilarway. ThemodelfitstotheIrondataas
GeV/nucleon. afunctionofδ, aredepictedinfig.3forthedisk-likemodel
4 Benyaminetal.
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(cid:230)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:230) (cid:230)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:230)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:230) (cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:230) (cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:230)(cid:230)(cid:230)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) AAMM(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:230)SS(cid:45)(cid:45)00(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)22 (cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:43)(cid:43)ScTiVFe0000....(cid:224)01125050(cid:224)(cid:72)(cid:76)(cid:144)(cid:230)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:224) (cid:230)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:224)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)∆∆∆∆∆∆∆∆∆(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:230)000000000.........(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)334455667(cid:224)5555(cid:230) (cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:230) (cid:224)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:230) (cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:230)(cid:224)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:230)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:230)(cid:224)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:230)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:224)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:230) (cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:224) (cid:230)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:224) (cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:230)(cid:230)(cid:224)(cid:224)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) SSHH(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)AAEENNAARROO(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)II(cid:224)KK(cid:45)(cid:45)33UU(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)
0.00 0.00
1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0
Log10EGeVnucleon Log10EGeVnucleon
FIG.1.—Theoptimalχ2fitoftheB/Cdataforseveralδvaluesintherange FIG.3.—Theoptimalχ2fitofthesub-Fe/Fedataforasetofδ’sbetween
(cid:64) (cid:144) (cid:68) (cid:64) (cid:144) (cid:68)
0.3to0.7,fordisk-likesimulations. DataistakenfromOlivaetal. (2013). 0.3to0.7,indisk-likesimulations. Thethickermustardcoloredlinecorre-
Thethickerpurplelinecorrespondstotheoverallbestfit,withδ=0.45. spondstotheoverallbestfit,withδ=0.5.DatatakenfromHEAO-3(Binns
etal. 1988)andSANRIKU(Hareyama1999).
(cid:230) 0.25 (cid:42)(cid:42)
BC(cid:230)00000(cid:230).....0011205050(cid:230)(cid:144)(cid:230)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:230) (cid:230)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:230) (cid:230)(cid:230)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)∆∆∆∆∆∆∆∆∆(cid:230)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)000000000.........(cid:230)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)334455667(cid:230)5555(cid:230)(cid:230)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:230)(cid:230)(cid:230)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:230)(cid:230) (cid:230)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:230)(cid:230)(cid:230)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:230)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:230)(cid:230)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:230)(cid:230)(cid:230)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:230)(cid:230)(cid:230)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:230) (cid:230)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:230) (cid:230)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:230)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:230) (cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:230) (cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:230)(cid:230)(cid:230)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) AAMM(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:230)SS(cid:45)(cid:45)00(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)22 (cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:43)(cid:43)ScTiVFe(cid:224) 0000....(cid:224)01125050(cid:224)(cid:72)(cid:76)(cid:144)(cid:230)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:224) (cid:230)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:224)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)∆∆∆∆∆∆∆∆(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:61)(cid:230)00000000........(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)33445566(cid:224)5555(cid:230) (cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:230) (cid:224)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:230) (cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:230)(cid:224)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:230)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:230)(cid:224)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:230)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:224)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:230) (cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:224) (cid:230)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:224) (cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:230)(cid:230)(cid:224)(cid:224)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) SSHH(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)AAEENNAARROO(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)II(cid:224)KK(cid:45)(cid:45)33UU(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)
1.0 1.5 2.0 2.5 3.0 0.00 ∆(cid:61)0.7
Log10EGeVnucleon 1.0 1.5 2.0 2.5 3.0
FIG.2.—Similartofig.1butforthespiralarmmodel.Theoveralloptimal Log10EGeVnucleon
fithereisalsoδ=0.45. (cid:64) (cid:144) (cid:68) FIG.4.—Similartofig.3butforthespiralarmmodel.Unlikethedisk-like
andfig.4forthespiralarmmodel. UnliketheB/Cfits,here model,theoveralloptimalfithereisδ(cid:64)=0(cid:144).35de(cid:68)pictedathickblueline.
the optimal fits require a different δ values for the two mod- D ∼1027cm2s−1to1.2×1027cm2s−1andδ ∼0.35−0.45.
0
els. For the disk-like model, the optimal is δ ∼ 0.5 while
Thiscanalsobeseenintable2. Whilethedisk-likemodelre-
it is δ ∼ 0.35 for the spiral arm model. The source of the quires a similar δ to fit both datasets, the iron data requires
difference points to an interesting physical effect associated
a diffusion coefficient that is of order 35% smaller from the
withthedifferentpathlengthdistribution. Becausethecross-
diffusioncoefficientrequiredtorecovertheB/Cdata,forany
sectionoftheheavierelementsislarger,theirmeanfreepaths given δ. The smaller diffusion coefficient corresponds to a
are much smaller relative to the distance traversed, thus giv-
largergrammage.ThisrecapturestheclaimsraisedbyGarcia-
ing rise to saturation (especially at lower energies), whereby
Munozetal. (1987),thatthesub-Fe/Fedatarequiresalarger
an increase in the grammage does not increase as much the
grammagethantheB/Cdatarequires.Herewelimitourselves
ratio between secondary and primary isotopes. This satura- tohighenergies(>10GeV/nuc.)whereweexpectsmallerun-
tioncausesadecreasesinthesecondary/primaryslope. How-
certaintiesintheenergydependenceofthecross-sections,and
ever,becauseofthedifferentpathlengthdistribution,theef-
reachthesameconclusion.
fect is notably larger for the disk like model with an expo- Thisinconsistencyismanifestedintheminimumχ2/d.o.f.
nentialPLD.Thus,inordertocompensateforthiseffect,one
(per degree of freedom) obtained in the combined analysis.
requires a steeper diffusion power law index to explain the For the disk-like model, we find χ2/d.o.f. = 3.66 with
small observed slope. For the spiral arm model, the satura- d.o.f. = 31−2 = 29. However, for the spiral arm model
tionislessimportantsuchthatthediffusionpowerlawindex it is χ2/d.o.f. = 0.95, for the same number of degrees of
shouldbeclosertotheobservedsub-Fe/Feslope.
freedom. Formally, this implies that the disk like model can
Figs.5&6showtheχ2asafunctionofbothδandD0. beruledoutatthe10−10level.
Toseehowsensitivethisconclusionistothenuclearcross-
4.3. B/Cvs. sub-Fe/Fecomparison
sections employed, we repeated the analysis with the cross-
TheaboveanalysiswascarriedoutseparatelywithB/Cand sections of Silberberg & Tsao (1990) replaced with those of
sub-Fe/Fe data sets. However, any consistent model should Webberetal. (2003). Theresultsaredepictedasthedashed
be able to fit both datasets with the same model parameters. contoursinfigs. 5and6. Evidently,thediscrepancyremains
This implies that the χ2 contours depicted in figs. 5 & 6 though it is more than twice smaller. In other words, the
should overlay each other. A quick inspection reveals that uncertainties in the cross-sections are sufficiently large that
only the spiral arm model gives consistent parameters, with one cannot unequivocally use the B/C and sub-Fe/Fe data to
5
rule out a disk-like model, though it is inconsistent with the
presentspallationcross-sections.
One should emphasize that the actual values of the diffu- 3
2
1
sion coefficients are less certain than from these fits. This is
because we set the nominal halo sizes for the two models. 1.4×1027
Changingthehalosizeswouldalsoscalethevaluesoftheop-
timal D obtained. While this introduces an uncertainty in
0
theactualvaluesofD thiswillnotresolvetheinconsistency
0
binettrwodeeuncethaentywioncdoantasissetetsnicnytfhoerdtihsek-slipkieraml aordmelsmaonddewl.ilTlnhoatt 2msec/]1.2×1027
iFse,/tFheessuccahlinthgawt tohueldcobnesitshteenscaym(einfotrhebostphirtahlearBm/Cmaonddels)ubo-r Dc[0
1
inconsistency(inthedisk-likemodel)betweenthedatasetfits
wouldremain. 1.0×1027 2
3
8.0×1026
1×1028
0.3 0.4 0.5 0.6 0.7
δ
FIG.6.—Acontourplotofχ2forthespiralarmmodel.Theredcontours
correspondtotheB/Cfit.Thebluecontourscorrespondtothesub-Fe/Fefit.
Thedashedcontoursaresimilartothoseinfig.5.
8×1027
ec]
2Dcms[/0 1 2 3 In previous wo5r.kD, IwSCeUdSeSvIOelNop&eSdUaMfMuAllRyY3D CR diffusion
model which not only considers that most CR acceleration
6×1027
takes place in the vicinity of spiral arms, but also that these
arms are dynamic (Benyamin et al. 2014). It was shown
thatbyaddingtheseobservationallybasedcomponentstothe
model, one recovers observed secondary to primary ratios,
4×1027 1 such as of Boron to Carbon. In particular, it was found that
2
the ratio increases with energy below 1 GeV/nuc. and de-
3
creasesabovethisenergy, insteadofthemonotonicdecrease
withenergyexpectedinthesimplestgalacticdiffusionmodel
0.3 0.4 0.5 0.6 0.7
δ (e.g.,Cesarsky1980,withouthavingtoassumeadditionalas-
FIG.5.—Acontourplotofχ2forthedisk-likemodel. Theredcontours sumptions, such as reacceleration or a galactic wind). This
correspondtotheB/Cfit.Thebluecontourscorrespondtothesub-Fe/Fefit. is because the age of the CRs at low energies is not deter-
Thedashedlinescorrespondtothesamecontoursasobtainedwhenreplac- minedbythediffusiontimefromthespiralarms,butinstead
ingthecross-sectionsofSilberberg&Tsao(1990)withthosecompiledby governedbythe(energyindependent)timesincethelastspi-
Webberetal. (2003). NotethatthediscrepancybetweentheB/Candthe
ral arm passage. Since below 1 GeV/nuc., the particles are
sub-Fe/Federivedmodelparametersdecreases,butstillremains.
non-relativistic,afixedagetranslatesintoagrammagewhich
isincreasingwithenergy, andcorrespondinglyanincreasing
secondarytoprimaryratioatlowenergies.
One very important aspect of this model is that the path
lengthdistributionisdifferentfromtheonefoundinstandard
TABLE2 diffusion models. In the latter, the PLD is typically close to
THEOPTIMALD0INUNITSOF1027CM2S−1ASOBTAINEDFORTHE being exponential. However, if most CRs arrive from a dis-
TWOMODELS,WHENSEPARATELYFITTINGTHEB/CANDTHE
SUB-FE/FEDATA. tance,thenthePLDwillshowapaucityofsmallpathlengths
(comparefig. 4tofig.6inB14).
D0fromspiralarmmodel D0fromdisk-likemodel ThedifferentPLDhasaninterestingramification. CRsar-
δ B/C sub-Fe/Fe B/C sub-Fe/Fe
riving after having passed a short path length necessarily re-
0.3 1.6 1.65 10.9 8.8
main close to the galactic plane. In contrast, CRs having a
0.35 1.37 1.4 9.5 7.4
longpathlengthcouldstrayfurtherfromtheplanebeforere-
0.4 1.17 1.2 8.2 6.2
0.45 1.03 1.01 7 5.3 turning back. Since the ISM density falls with the distance
0.5 0.89 0.9 6.2 4.5 from the plane, CRs with short paths therefore experience a
0.55 0.8 0.74 5.5 3.8 higheraverageISMdensitythanCRswithlongpaths. Thus,
0.6 0.65 0.66 4.7 3.2 a distribution which is missing the short path lengths, as is
0.65 0.58 0.56 4 2.6 the case in the spiral arm model, will have a lower average
0.7 0.5 0.5 3.6 2.3 interaction with the ISM for a given average path length. In
otherwords,theaveragegrammagewillbelowerforthesame
averagephysical length. Thisresultimplies thatifthe spiral
6 Benyaminetal.
arm model is to recover the observed secondary to primary Thebestfitvaluesareadiffusionspectralslopeofδ =0.35
ratio, the model has to keep the CRs closer to the galactic toδ =0.43,andadiffusioncoefficientnormalizationbetween
planewherethedensityishigher. Forthisreason,thetypical D =1×1027cm2s−1andD =1.5×1027cm2s−1(though
0 0
halo size required to fit the same secondary to primary ratio thediffusioncoefficientrangewouldscalewiththehalosize,
dataislower(typicallyafewhundredpc,comparedwiththe which was chosen here to be 250 pc). To summarize, the
1to4kpcinmorestandarddiffusionmodels). However,be- consistencyfoundhereisoneofseveralsuccessfulpredictions
causethetypicalageiscloselyrelatedtotheratiobetweenthe bornefromthespiralarmmodel:
radioactive and stable Beryllium isotopes, which is a model 1. Theincrease inthepositron fractionabove ∼ 10 GeV
independent observation, the smaller halo requires a smaller is a necessary outcome given that the primary elec-
diffusioncoefficient.At1GeV/nuc.itshouldbeoforder1027 trons which arrive from a finite distance cool through
andnot1028cm2s−1. inverse-Compton and synchrotron radiation while sec-
HereweextendedthemodeldevelopedinB14,asdetailed ondarypositronscanformlocally(Shavivetal. 2009;
in §2.1. In particular, we enlarged the spallation network to Gaggeroetal. 2013b).
describethespallationofIronandNickelaswellastheirspal-
lationproducts. Thus,wecouldpredictthesub-Fe/Feratioas 2. The increase in the B/C ratio for E/nucleon (cid:46)
a function of model parameters, in addition to the B/C ra- 1 GeV/nuc. is recovered since the CR age saturates
tio. We carried a parameter study whereby we fit both the over low energies at the time since the last spiral arm
spectral slope δ of the diffusion coefficient and its normal- passage(B14).
ization D (while keeping the other model parameters fixed
0 3. The concentration of CR sources around the galactic
at their nominal values). This was carried out twice, for a
spiral arms and lower diffusion coefficient give rise to
(“standard”) disk-like model and for a spiral arm model, in
a temporal variation in the CR flux, seen as 145 Myr
theenergyrangeof10GeV/nuc. -1000GeV/nuc.. Therea-
cyclesintheCRexposureagesbehaviorofironmete-
son for focusing on high energies is to avoid the notorious
orites(Shaviv2003).
uncertainties associated with the spallation cross-sections at
lowerenergies(e.g.,Moskalenko2011;Webberetal. 2003). 4. Thesmallerhaloofthespiralarmmodelexplainsthe32
These uncertainties become more acute as Z increases (see Myroscillationseeninthe550Myrlongpaleoclimatic
Titarenkoetal. 2008;Sisterson&Vincent2006andalsoAp- record(Shavivetal. 2014).
pendixIIinGarcia-Munozetal. 1987).
Inastandarddisk-likemodel,wefindaninconsistencybe- 5. The lower diffusion coefficient implies that the ex-
tweenthediffusioncoefficientrequiredtofittheB/Candthe pected anisotropy in the arrival direction of cosmic
value required to fit the sub-Fe/Fe ratio. Phrased differently, rays should be smaller by an order of magnitude from
it implies that the sub-Fe/Fe ratio requires more grammage predictions of disk-like models with a larger diffusion
than the B/C ratio. One way to resolve the inconsistency coefficient, and consistent with the measurements be-
present in the standard disk-like model would be to modify tween1to100TeV(Shavivetal. 2016).
the cross-sections at the higher energies over which the fit
to observational data was carried out. This is not inconceiv- 6. The arm dynamics are responsible for a CR “wake”
able given that by replacing the cross-sections of Silberberg behind the arms with softer CRs, which give rise to
&Tsao(1990)withthoseofWebberetal. (2003),wereduce a softer π0 produced spectrum. Together with the
thediscrepancybymorethanafactorof2. smallerhalo,thepredictedspectralslopemapinγ-rays
Another way to resolve the inconsistency within the stan- at around 1-10 GeV is consistent with the FERMI ob-
dard disk like model would be to truncate the short paths servations(Navaetal. 2016).
lengthsbylettingtheCRsinteractwithmaterialattheSNR.
7. Last, as we have shown here, the inconsistency be-
Thisexplanationwhichwasoriginallyproposedtoresolvethe
tween the required diffusion coefficient to fit the B/C
PAMELAanomaly(Blasi&Serpico2009b), wouldrequires
andthesub-Fe/Feratiosisresolvedgiventhedifferent
however additional free parameters. A priori it is not clear
PathLengthDistribution.
thattheparametersthatresolvethePAMELAanomalywould
alsoresolvethisinconsistency.
Thus, the present results introduces further circumstantial
A spiral arm source distribution consistently requires the
evidencetoalistthatstronglyimpliesthatCRssourceinho-
same diffusion coefficient (or grammage) when fitting the
mogeneityplaysanimportantroleinCRphysics.
sub-Fe/Fe and B/C datasets. This arises because the aver-
age mean free path for spallation of Iron is notably smaller ACKNOWLEDGEMENTS
than that of Carbon and Oxygen, such that the sub-Fe/Fe
ThisworkishasbeensupportedbyISFgrantno.1423/15
and the B/C are not sensitive to the same path lengths. As
(DB&NJS),aswellasbytheI-COREProgramofthePlan-
a consequence, modifying the PLD in the spiral arm model
ningandBudgetingCommitteeandTheIsraelScienceFoun-
doesn’t change the average grammage needed to explain the
dation(1829/12). NJSalsogratefullyacknowledgesthesup-
sub-Fe/FeandtheB/Cbythesamefactor.
port of the IBM Einstein Fellowship at the Institute for Ad-
vancedStudy.
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