30TH INTERNATIONAL COSMIC RAY CONFERENCE

The end of the Galactic spectrum

C. DE DONATO1 , G. A. MEDINA-TANCO2

1Dipartimento di Fisica dell’Universita degli Studi di Milano and INFN, Milano, Italy,2Dep. Altas Energias, Inst. de Ciencias Nucleares, Universidad Nacional Autonoma de Mexico, MexicoDF, CP 04510cinzia.dedonato@mi.infn.it

Abstract: We use a diffusion galactic model to analyze the end of the Galactic cosmic ray spectrumand its mixing with the extragalactic cosmic ray flux. We analyze the transition between Galactic andextragalactic components using two different extragalactic models. We compare the sum of the diffusivegalactic spectrum and extragalactic spectrum with the available experimental data.

Introduction

The cosmic ray energy spectrum extends for manyorders of magnitude with a power law index≈ 2.7.Along this range of energies, the three spectral fea-tures are known: the first knee at E ≈ 3 PeV ,the second knee at E ≈ 0.5 EeV and the ankle,a dip extending from the second knee to beyond10 EeV . The nature of the second knee and of theankle are still uncertain [1]; a possible interpreta-tion of the two features is the transition betweenthe galactic and extragalactic components. At en-ergies between 1017 − 1018 eV the galactic super-nova remnants (SNRs) are expected to become in-efficient as accelerators. This fact, combined withmagnetic deconfinement should mark the end ofthe Galactic component of cosmic rays, althoughthe picture could be confused by the existenceof additional Galactic accelerators at higher en-ergy. On the other hand, at energies above the sec-ond knee, extragalactic particles are able to travelfrom the nearest extragalactic sources in less thana Hubble time. Consequently, the spectrum maypresent above 1017.5 eV a growing extragalacticcomponent that becomes dominant above 1019 eV .The region between the second knee and the anklecould be the transition region between the galacticand extragalactic components.In this work we analyze the transition region com-paring the diffusive galactic spectrum from SNRswith two different models of extragalactic spec-

trum, one in which only protons [2] are injectedat the sources and another in which a mixed com-position containing heavy nuclei [3] is injected.

Diffusion Galactic model

We used the numerical diffusive propagation codeGALPROP [4, 5] to reproduce the galactic spec-trum from SuperNova Remnants (SNRs). The dif-fusive model is axisymmetric. The propagationregion is, in cylindrical coordinates, bounded byR = Rh = 30 kpc and z = zh = 4 kpc, be-yond which free escape is assumed. The propaga-tion equation is:

∂ψ

∂t= q(~r, p) + ~∇ · (Dxx

~∇ψ) +

− ∂

∂p(pψ)− 1

τfψ − 1

τrψ (1)

where ψ(~r, p, t) is the density per unit of total par-ticle momentum, q(~r, p) is the source term, Dxx

is the spatial diffusion coefficient, p = dp/dt isthe momentum loss rate and τf and τr are the timescale of fragmentation and the time scale of ra-dioactive decay respectively. The diffusion coeffi-cient is taken as βD0(ρ/ρD)δ , where ρ is the par-ticle rigidity,D0 is the diffusion coefficient at a ref-erence rigidity ρD and δ = 0.6. The distribution ofcosmic rays sources used is that of Galactic SNRs

Proceedings of the 30th International Cosmic Ray ConferenceRogelio Caballero, Juan Carlos D’Olivo, Gustavo Medina-Tanco,Lukas Nellen, Federico A. Sánchez, José F. Valdés-Galicia (eds.)Universidad Nacional Autónoma de México,Mexico City, Mexico, 2008

Vol. 2 (OG part 1), pages 219–222

ID 1249

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THE END OF THE GALACTIC SPECTRUM

Figure 1: Diffusive Galactic spectrum, ΦG,with renormalization of the CNO group (Z:6-12),ΦCNOG and of the heavy component (Z : 19− 26),ΦHG . Agreement with experimental data is obtainedfor k′ ≈ 1 and k ≈ 0.8− 1.2.

(in agreement with EGRET gamma-ray data) [4].The injection spectrum is a a power law function inrigidity with a break at rigidity ρ0, beyond whichit falls exponentially with a rigidity scale ρc:

I(ρ) =

(ρρ0

)−αρ ≤ ρ0

exp[−( ρρ0 − 1)/ ρc

ρ0)]

ρ > ρ0

where α = 2.05, ρ0 = 1.8 PV and ρc = 1.26 PV .Stable nuclei with Z < 26 are injected, withenergy independent isotopic abundances derivedfrom low energy CR measurements[5].Detailed and realistic interstellar molecular (H2),atomic (H) and ionized (HI) hydrogen distributionsare used [6].

Diffusive Galactic spectrum

The diffusive galactic spectrum has been normal-ized to match KASCADE data at ∼ 3 × 106

GeV. While with this renormalization our spec-trum agrees with JACEE and Sokol data at lowerenergies, beyond the knee the diffusive spectrumpresents a strong deficit of flux. Since at E >107 GeV the composition is dominated by inter-mediate (Z : 6 − 12) and heavier (Z : 19 − 26)nuclei, we renormalize these components by a fac-tor of 2, which produces a good agreement with theexperimental data (see Fig. 1).The renormalized diffusive galactic spectrum re-produces well the data up to E ≈ 108 GeV , be-

yond which the spectrum falls steeply because ofend of the SNR acceleration efficiency.

Extragalactic spectrum

In order to study how the transition between thegalactic and extragalactic components takes place,we compare the galactic spectrum originated inSNR with two different possible scenarios for theextragalactic component.In the first model [2], a pure proton extragalac-tic spectrum, accelerated by a homogeneous dis-tribution of cosmic sources, is considered. Localoverdensities/deficits of UHECR sources affect theshape of the GZK modulation, but do not affectthe low energy region where the matching with theGalactic spectrum occurs. Different cases of lo-cal overdensity/deficity of sources are consideredwithin this model.In the second model [3], the extragalactic spec-trum is calculated for a mixed composition at injec-tion typical of low energy cosmic rays for differentsource evolution models in red shift.In both cases, the various parameters of the modelsare tuned to fit the available CR data at UHE andare, in that highest energy regime, experimentallyindistinguishable at present.

Combined spectrum: matching Galac-tic and extragalactic components

In order to study how the transition between theGalactic and extragalactic components takes place,we subtract the combined theoretical (Galactic plusextragalactic) spectrum from the available data.Two different approaches are used.First, we try to match the experimental data byvarying the normalization of the heavy Galacticcomponent, while keeping constant the previousrenormalization of the CNO group. The best re-produced spectrum for the two extragalactic mod-els are shown in Figs.2, 3 and 4. In the case ofthe proton model, a discontinuity appears when thetwo spectra are added, regardless of the lower limitadopted for the extragalactic component: 108 GeV(Fig.2) or 5 × 107 GeV (Fig.3). The latter cor-

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30TH INTERNATIONAL COSMIC RAY CONFERENCE

Figure 2: Galactive and Extragalactic spectrummatching for the proton models for a EG lowerenergy limit of 108 GeV . The sum of the renor-malized diffusive Galactic spectrum and differentextragalactic spectrum models (ΦEG) is shown fordifferent renormalizations of the heavy component(ΦHG ). The CNO group (ΦCNOG ) of the diffusiveGalactic spectrum (ΦG) has been renormalized bya factor 2.2. Different cases of local overden-sity/deficit of sources are considered [2]: (1) uni-versal spectrum, (2) and (3) overdensity of sources,(4) and (5) deficit of sources .

responds to cosmic accelerators operating for theentire Hubble time.For both, proton and mixed-composition models,there is a flux deficit above 108 GeV. The problemis much stronger for the mixed-composition modelwhere, regardless of the luminosity evolution ofEG CR sources, the total spectrum presents a largedeficit of flux between 108 and ≈ 3 × 109 GeV(Fig.4).In order to solve this flux deficit, the only wayout seems to be the introduction of an additionalGalactic component. We estimate this component

Figure 3: Idem to Fig. 2 but for the proton modelwith a lower energy limit of 5 ∗ 107 GeV.

Figure 4: Idem to Fig.3 but for the Galactive andExtragalactic spectrum for the mixed-compositionmodel. The CNO group (ΦCNOG ) of the diffusiveGalactic spectrum (ΦG) has been renormalized bya factor 2. Three source evolution models are con-sidered [3]: (a) strong, (b) SFR and (c) uniform.

by subtracting the sum of the diffusive Galacticand extragalactic fluxes from a smooth fit to theworld data. This method confirms us the need ofthe renormalization of the CNO group by a fac-tor ≈ 2.2 and ≈ 2 for the proton and mixed-composition models, respectively. In the case ofthe proton models (Figs.5, 6), the observed deficitcan be resolved with an additional heavy compo-nent, while in the mixed-composition models thisis not enough and we need one more additionalheavy component (Fig.7). The additional compo-nent common to both families of models is ob-tained with a shift in energy of a factor ≈ 1.5 ofthe diffusive galactic heavy component, renormal-ized by a factor ≈ 0.04 in the case of the mixed-composition models and . 0.03 for the protonmodels respectively. The second additional com-ponent is obtained in an analogous way but with anenergy-shift factor of ≈ 1.8 and a renormalizationby a factor 5 × 10−4. The corresponding spectraare shown in Figs.5, 6 and 7.

Discussion

We have analyzed the matching conditions of theGalactic and extragalactic components of cosmicrays along the second knee and the ankle.It seems clear that an acceptable matching ofthe Galactic and extragalactic fluxes can only beachieved if the Galaxy has additional accelerators,besides the regular SNR, operating in the interstel-

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Figure 5: Galactive and Extragalactic spectrumfor the proton model for an EG lower limit of108 GeV : the additional heavy component φHGAand the total heavy component (φHG + φHGA) arealso shown.

Figure 6: Idem to Fig.5 but for an EG lower limitof 5 × 107 GeV : the additional heavy componentφHGA and the total heavy component (φHG + φHGA)are also shown.

lar medium. In the particular case of the protonmodel, only one additional component is required,and this could well represent the contribution fromcompact and highly magnetized SNR, like thoseoccurring in the central, high density regions ofthe Galactic bulge or the dense cores of molecularclouds. It must be noted, however, that a perfectlysmooth match seems unrealistic and that some dis-continuity, whose magnitude depends mainly onthe time depth from which EG CR are able to ar-rive at our Galaxy, should be eventually observablein the combined spectrum.The matching of the mixed-composition model ismore complicated. The Galactic spectrum has tobe extended up to the middle of the dip and this re-quires, besides the previous additional component,another high energy Galactic component. The ori-gin of these cosmic rays pushes even further the

Figure 7: Galactive and Extragalactic spectrum forthe mixed-composition model: the two additionalheavy components φHGA, φHGA2 and the total heavycomponent (φHG + φHGA + φHGA2) are shown.

acceleration requirements imposed on the Galaxyand probably rapidly spinning inductors, like neu-tron stars, could be invoked to fill in the gap. If thiswere the case, it is very likely that photon emis-sion at TeV energies should uncover the sources.Changes in propagation regime inside the Galaxyat these high energies should manifest as a dipolaranisotropy if enough statistics were available.

Acknowledgements

CDD thanks ICN-UNAM for hosting a long stayand Universita degli Studi di Milano for a PhDgrant. GMT thanks PAPIIT/CIC-UNAM for sup-port.

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