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Astrophysics and Space ScienceAn International Journal of Astronomy,Astrophysics and Space Science ISSN 0004-640XVolume 342Number 1 Astrophys Space Sci (2012) 342:131-136DOI 10.1007/s10509-012-1138-y

Propagation of H and He cosmic rayisotopes in the Galaxy: astrophysical andnuclear uncertainties

Nicola Tomassetti

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Astrophys Space Sci (2012) 342:131–136DOI 10.1007/s10509-012-1138-y

O R I G I NA L A RT I C L E

Propagation of H and He cosmic ray isotopes in the Galaxy:astrophysical and nuclear uncertainties

Nicola Tomassetti

Received: 8 May 2012 / Accepted: 5 June 2012 / Published online: 14 June 2012© Springer Science+Business Media B.V. 2012

Abstract Observations of light isotopes in cosmic rays pro-vide valuable information on their origin and propagationin the Galaxy. Using the data collected by the AMS-01 ex-periment in the range ∼0.2–1.5 GeV nucleon−1, we com-pare the measurements on 1H, 2H, 3He, and 4He with cal-culations for interstellar propagation and solar modulation.These data are described well by a diffusive-reaccelerationmodel with parameters that match the B/C ratio data, indi-cating that He and heavier nuclei such as C–N–O experiencesimilar propagation histories. Close comparisons are madewithin the astrophysical constraints provided by the B/C ra-tio data and within the nuclear uncertainties arising fromerrors in the production cross section data. The astrophysi-cal uncertainties are expected to be dramatically reduced bythe data upcoming from AMS-02, so that the nuclear uncer-tainties will likely represent the most serious limitation onthe reliability of the model predictions. On the other hand,we find that secondary-to-secondary ratios such as 2H/3He,6Li/7Li or 10B/11B are barely sensitive to the key propagationparameters and can represent a useful diagnostic test for theconsistency of the calculations.

Keywords Cosmic rays · Acceleration of particles ·Nuclear reactions, nucleosynthesis, abundances

1 Introduction

Secondary Cosmic Ray (CR) isotopes such as 2H, 3He andLi–Be–B are believed to be produced as a results of nuclearinteractions primary CRs such as 1H, 4He or C–N–O with

N. Tomassetti (�)INFN—Sezione di Perugia, 06122 Perugia, Italye-mail: nicola.tomassetti@pg.infn.it

the gas nuclei of the interstellar medium (ISM). The sec-ondary CR abundances depend on the intensity of their pro-genitors nuclei, their production rate and their transport inthe turbulent magnetic fields (Strong et al. 2007). Secondaryto primary ratios such as 2H/4He, 3He/4He or B/C are usedto study the CR propagation processes in the Galaxy. TheB/C ratio is widely used to determine the key parametersof propagation models. In fact the B/C ratio is measuredby several experiments between ∼100 MeV and ∼1 TeVof kinetic energy per nucleon. The CR propation physics isalso connected with the indirect search of dark matter parti-cles. In this context the CR propagation models, once tunedto agree with the B/C ratio, are used to compute the sec-ondary production for other rare species such as p or d ,that provides the astrophysical background for the search ofnew physics signals (Donato et al. 2008; Evoli et al. 2011;Salati et al. 2010). Clearly, understanding the CR propa-gation processes is crucial for modeling both the CR sig-nal and the background. Furthermore, these studies assumethat all the CR species experience the same propagation ef-fects in their journey throughout the ISM (Putze et al. 2010;Trotta et al. 2011). It is therefore important to test the CRpropagation with nuclei of different mass-to-charge ratios.This issue of the universality of CR propagation historieswas also studied in Webber (1997) and, recently, in Costeet al. (2012).

In this work we use the recent AMS-01 observations forthe 2H/4He and 3He/4He ratios and compare them withthe expected ratios based on interstellar and heliosphericpropagation calculations. The aim of this work is to de-termine whether the AMS-01 observations are consistentwith the propagation calculations derived from heavier nu-clei (mainly from B/C data). This consistency is inspectedwithin two classes of model uncertainties: the astrophysicaluncertainties, which are related to the knowledge of the CR

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132 Astrophys Space Sci (2012) 342:131–136

Fig. 1 Left: energy spectra of CR proton, deuteron (divided by 20),3He and 4He. Right: isotopic ratios 2H/4He and 3He/4He. Calculationsare compared with the data from AMS-01 (Aguilar et al. 2002, 2011),IMAX (Reimer et al. 1998; De Nolfo et al. 2000), SMILI (Ahlen et al.

2000), BESS (Wang et al. 2002), Hatano et al. (1995), Leech andO’Gallagher (1978), Webber and Yushak (1983)

transport parameters given by the B/C ratio, and the nuclearuncertainties, which arise from the 2H and 3He productioncross sections.

2 Observations

The AMS-01 experiment operated successfully in the STS-91 mission on board the space shuttle Discovery. The spec-trometer was composed of a cylindrical permanent mag-net, a silicon micro-strip tracker, time-of-flight scintillatorplanes, an aerogel Cerenkov counter and anti-coincidencecounters. The performance of AMS-01 is described else-where (Aguilar et al. 2002). Data collection started on 1998June 3 and lasted 10 days. AMS-01 observed cosmic raysat an altitude of ∼380 km during a period, 1998 June, ofrelatively quiet solar activity. Results on isotopic spectrahave been recently published in Aguilar et al. (2011) withthe ratios 2H/4He, 3He/4He, 6Li/7Li, 7Be/(9Be + 10Be) and10B/11B in the range ∼0.2–1.5 GeV of kinetic energy pernucleon. Figure 1 shows the AMS-01 energy spectra of pro-ton, deuteron, helium isotopes, and the ratios 2H/4He and3He/4He. The other data come from balloon borne exper-iments IMAX (Reimer et al. 1998; De Nolfo et al. 2000),SMILI (Ahlen et al. 2000), BESS (Wang et al. 2002), Hatanoet al. (1995), Leech and O’Gallagher (1978), Webber andYushak (1983).

The AMS-01 observations are made in a period, 1998June, of relatively quiet solar activity, and the data recorded

are free from any atmospheric induced background. Further-more, the AMS-01 material thickness between the top ofthe payload and the active detector amounted to ∼5 g cm−2

which is considerably less than that of previous balloonborne experiments (∼9–20 g cm−2 of top-of-instrument ma-terial plus ∼5 g cm−2 of residual atmosphere). Also impor-tant, for the aims of this work, is to realize that high preci-sion data are currently flowing in from two active projectsPAMELA and AMS-02, both operating in space. In par-ticular, the AMS-02 data are expected to provide a dra-matic improvement in our understanding of the CR transportprocesses and interactions (Tomassetti and Donato 2012;Coste et al. 2012; Oliva 2008).

3 CR transport and interactions

Galactic CR nuclei are believed to be accelerated by dif-fuse shock acceleration mechanisms occurring in galacticsites such as supernova remnants (SNRs). Their propaga-tion in the ISM is dominated by particle transport in theturbulent magnetic field and interactions with the matter,that is generally described by a diffusion-transport equa-tion including source distribution functions, magnetic dif-fusion, energy losses, hadronic interactions, decays, diffu-sive reacceleration and convective transport (the latter is notconsidered in this work). Models of CR propagation in theGalaxy employ fully analytical (Thoudam 2008; Tomassetti2012), semi-analytical (Jones et al. 2001; Putze et al. 2010),

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Astrophys Space Sci (2012) 342:131–136 133

Table 1 Propagation parameter set

Parameter Name Value

Injection, break value RB [GV] 9

Injection, index below RB ν1 1.82

Injection, index above RB ν2 2.36

Diffusion, magnitude D0 [cm2 s−1] 5.75 · 1028

Diffusion, index δ 0.34

Diffusion, ref. rigidity R0 [GV] 4

Reacceleration, Alfvén speed vA [km s−1] 36

Galactic halo, radius R [kpc] 20

Galactic halo, height zh [kpc] 4

Solar modulation parameter φ [MV] 500

or fully numerical calculation frameworks (Di Bernardoet al. 2010; Strong et al. 2007). The present work re-lies on the diffusive-reacceleration model implemented withGALPROP-v50.1p,1 which numerically solves the cosmicray propagation equation for a cylindrical diffusive regionwith a realistic interestellar gas distribution and source dis-tribution.

3.1 Diffusive-reacceleration model

The propagation equation of the diffusive-reaccelerationmodel for a CR species j is given by:

∂Nj

∂t= q tot

j + �∇ · (D �∇Nj ) − NjΓtotj

+ ∂

∂pp2Dpp

∂

∂pp2 Nj − ∂

∂p(pj Nj ) (1)

where Nj = dNj/dV dp is the CR density of the species i

per unit of total momentum p.The source term, q tot

j = qprij + qsec

j , includes the primaryacceleration spectrum (e.g. from SNRs) and the term aris-ing from the secondary production in the ISM or decays.The primary spectrum is q

prij = q0(R/RB)−ν , is normalized

to the abundances, q0j , at the reference rigidity RB . The in-

jection spectral indices are the same for all the primary el-ements. The source spatial distribution in the galactic discis extracted from SNR observations. The secondary produc-tion term, qsec

j = ∑k NkΓk→j , describes the products of de-

cay and spallation of heavier CR progenitors with numerdensity Nk . For collisions with the interstellar gas:

Γk→j = βkc∑

ism

∫ ∞

0nismσ ism

k→j

(E,E′)dE′, (2)

1http://galprop.stanford.edu.

where nism are the number densities of the ISM nuclei, nH ≈0.9 cm−3 and nHe ≈ 0.1 cm−3, and σ ism

k→j are the fragmen-tation cross sections for the production of a j -type speciesat energy E from a k-type progenitor of energy E′ in H orHe targets. Γ tot

j = βj c(nH σ totj,H + nHeσ

totj,He) + 1

γj τjis the

total destruction rate for inelastic collisions (cross sectionσ tot) and/or decay for unstable particles (lifetime τ ). Thespatial diffusion coefficient D is taken as spatially homo-geneous and rigidity dependent as D(R) = βD0(R/R0)

δ ,where R = pc/Ze is the rigidity, D0 fixes the normaliza-tion at the reference rigidity R0, and the parameter δ spec-ifies its rigidity dependence. Diffusive reacceleration is de-scribed as diffusion process acting in momentum space. It isdetermined by the coefficient Dpp for the momentum spacediffusion:

Dpp = 4p2v2A

3δ(4 − δ2)(4 − δ)D, (3)

where vA is the Alfvén speed of plasma waves movingthrough the ISM. The last term describes Coulomb and ion-ization losses by means of the momentum loss rate pj .GALPROP solves the steady-state equation ∂Nj /∂t = 0

for a cylindrical diffusion region of radius rmax and half-thickness L with boundary conditions Nj (r = rmax) = 0 andNj (z = ±L) = 0. The local interstellar spectrum (LIS) isthen computed for each species at the solar system coordi-nates r = 8.5 kpc and z = 0:

ΦLISj (E) = cA

4πNj (r,p), (4)

where A is the mass number and the flux ΦLIS is givenin units of kinetic energy per nucleon E. For the descrip-tion we adopt the “conventional model” which finely repro-duces the CR elemental fluxes at intermediate energies of∼100 MeV–100 GeV per nucleon.

3.2 Heliospheric propagation

CRs in the solar neighbourhood undergo convection, dif-fusion and energy changes as results of the expansion ofthe solar wind. To describe the solar modulation effect, weadopt the so-called force-field approximation that arise fromthe a spherically symmetric description of the heliosphere(Gleeson and Axford 1968). The correspondence betweenthe (modulated) top-of-atmosphere spectrum, ΦTOA, and the(unmodulated) LIS spectrum of 4, ΦLIS, is expressed by theeffective parameter φ (GV) through:

ΦIS(EIS) =

(pIS

p

)2

ΦTOA(ETOA)

, (5)

where the LIS and TOA energies per nucleon are related byEIS = ETOA + Z

Aφ. The main parameters of the model are

listed in Table 1. The remaining specifications are as in thefile galdef_50p_599278 provided with the package.

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134 Astrophys Space Sci (2012) 342:131–136

Fig. 2 Left: Cross section parametrizations (Cucinotta et al. 1993) forthe channels 4He → 3He, 4He → 3H, 4He → 2H and p + p → π + 2H(multiplied by 20). The data are encoded as ME72: Meyer (1972),WE90: Webber (1990), NI72: Nicholls et al. (1972), GL93: Glagolevet al. (1993), AB94: Abdullin et al. (1994), BL01: Blinov et al. (2001),

LM49: Lebowitz and Miller (1969), Right: B/C ratio from the CR prop-agation model of Table 1 and 1-σ uncertainty band. Data are fromTRACER (Obermeier et al. 2011), AMS-01 (Aguilar et al. 2010), CRIS(De Nolfo et al. 2003), CREAM (Ahn et al. 2008), HEAO (Engelmannet al. 1990), Orth et al. (1978), and Lezniak and Webber (1978)

3.3 Fragmentation cross sections

To compute the secondary nuclei production rate from thedisintegration of the heavier CR nuclei, a large amount ofcross section estimates is required. The accuracy of the cal-culated secondary spectra therefore depends on the relia-bility of the production and destruction cross sections em-ployed.

The production of 2H and 3He isotopes is mainly dueto collision of 4He nuclei. The 3He isotopes are also pro-duced via decay of trithium (3H → 3He) which, in turn,is predominantly created by 4He spallation. The most rel-evant projectile → fragment processes for the 3He abun-dance are 4He → 3He and 4He → 3H. The main deuteronproduction channel is 4He → 2H. Low energy deuterons arealso created by the fusion reaction p + p → π + 2H act-ing between ∼300 and ∼900 MeV of the proton energy(Meyer 1972). Although the p–p fusion cross section is verysmall (σ ∼ 3 mb), this reaction contributes appreciably tothe 2H abundance because of the large CR proton flux. Spal-lation of heavier nuclei (C, O, Fe) give a minor contribu-tion, roughly �10 % of their fractional abundance. For allthese channels the total inclusive reaction can be realizedin a number of possible final states. We employ the phe-nomenological parametrization of Cucinotta et al. (1993).These parametrizations are shown in Fig. 2 together withthe accelerator data. For 2H and 3He, the partial contribu-tions of break-up (B) and stripping (S) reactions are shownseparately.

For all these processes we have assumed that the frag-ment is ejected with the same the kinetic energy per nu-

cleon E of the projectile, E′. This straight-ahead approxi-mation, expressed by σ(E,E′) ≈ σ(E)δ(E − E′), has beenvalidated within some percent of accuracy for reactions in-volving Z > 2 nuclei (Kneller et al. 2003) and for lighterspecies (Z < 3) (Cucinotta et al. 1993). For the p–p fu-sion channel, the kinetic energy per nucleon is not conserveddue to the kinematic of the reaction: the energy of ejecteddeuterons is systematically lower than that of the proton ofa factor ∼ 4 on average. Using a straight-ahead fashion, wewrite σ(E,E′) ≈ σ(E)δ(E − ξE′), where ξ ∼= 4 is the av-erage inelasticity of the 2H. The p–p fusion contributes tothe 2H flux at energies below ∼250 MeV nucleon−1. For nu-clear reactions involving heavier (Z > 2) nuclei, we use thedefault cross section parametrization of GALPROP. To ac-count for CR collisions with the interstellar helium (∼10 %of the ISM) the parametrization of Ferrando et al. (1988)is used. Calculations of CR spectra and ratios 2H/4He and3He/4He are shown in Fig. 1 for the modulation intensity ofφ = 500 MV.

4 Model uncertainties

We consider two classes of uncertainty in the model esti-mates. The astrophysical uncertainties are those associatedwith the transport parameters constrained by the B/C ra-tio. The relevant parameters for the secondary productionsare δ, vA and the ratio D0/L. We perform a grid scan inthe parameter space {δ, vA,D0/L} by running GALPROPmultiple times. The other parameters, e.g. the source pa-rameters and the modulation potential, are kept fixed. For

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Fig. 3 Astrophysical (left) and nuclear (right) uncertainty bands forthe predicted ratios 2H/4He, 3He/4He and 2H/3He in comparison withthe AMS-01 data. Other data are from IMAX (Reimer et al. 1998;

De Nolfo et al. 2000), SMILI (Ahlen et al. 2000), BESS (Wang et al.2002), Hatano et al. (1995), and Webber and Yushak (1983)

each parameter configuration, we select the B/C-compatiblemodels within one sigma of uncertainty in the χ2 statis-tics. We use B/C data from HEAO (Engelmann et al. 1990),CREAM (Ahn et al. 2008), AMS-01 (Aguilar et al. 2010;Orth et al. 1978). Figure 2 illustrates the uncertainty bandderived by this procedure. Note that this method has severelimitations and allows to simultaneously explore only a fewparameters. More robust strategies require advanced statis-tical tools, see e.g. Trotta et al. (2011), Putze et al. (2010)and in particular Coste et al. (2012). However the purposein this work is estimating the parameter uncertainties ratherthan determining their exact values.

The nuclear uncertainties on the 2H and 3He calculationsare those arising from uncertainties in their production crosssections. In order to estimate the cross section uncertaintiesusing the information from the measurements, we re-fit theparametrizations with the data to determine their overall nor-

malizations and associated errors. The uncertainty bands areshown in Fig. 2 for the main reactions of 2H and 3He produc-tion. These uncertainties can be directly translated into errorbands for the predicted ratios 2H/4He and 3He/4He. Theseuncertainty bands are shown in Fig. 3.

The AMS-01 data agree well with calculations within theastrophysical band, indicating consistency with the propa-gation picture arising from the B/C analysis. It is also clearthat Z ≤ 2 nuclear ratios carry valuable information on thetransport parameters, i.e., they can be in principle used totighten the constraints given by the B/C ratio. On the otherhand, the nuclear uncertainties represent an intrinsic limi-tation on the accuracy of the model predictions, as reportedin the right panels of the figure. Only precise cross sectiondata or more refined calculations may pin down these un-certainties. Unaccounted errors or systematic biases in crosssection estimates cause errors on the predicted ratios which,

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136 Astrophys Space Sci (2012) 342:131–136

in turn, may lead to a mis-determination of the CR trans-port parameters. Given the level of precision expected fromPAMELA or AMS-02, systematic errors in the cross sec-tion data may represent the dominant source of uncertaintyfor the model predictions of light CR isotopes. A strategyto check the model consistency with CR data is the use ofsecondary to secondary ratios such as 2H/3He. In fact, the2H and 3He isotopes have the same astrophysical origin andsimilar progenitors (mainly 4He). Thus, their ratio is almostunsensitive to the propagation physics and can be used toprobe the net effect of the nuclear interactions. In fact, a mis-consistency between calculations and 2H/3He data would in-dicate the presence of systematic biases in the cross sectionsthat cannot be re-absorbed by the propagation parameters.As illustrated in the bottom panels of Fig. 3, the tight astro-physical uncertainty band (left) indicates a little discrepancybetween data and model which can be understood if one con-sider the nuclear uncertainty (right), which is dominant forthe 2H/3He ratio. Similarly, the use of other ratios such as6Li/7Li, Li/Be or 10B/11B can represent a useful diagnostictool for testing the overall consistency of the model.

5 Conclusions

We have compared new observations of the 2H/4He and3He/4He ratios in CRs made by the AMS-01 experimentswith standard calculations of secondary production in theISM. These ratios are well described by propagation mod-els consistent with B/C ratio under a diffusive-reaccelerationscenario, suggesting that He and heavier nuclei such as C–N–O experience similar propagation histories. The accuracyof the secondary CR calculations relies on the accuracy ofthe cross sections employed. Given the level of precision ex-pected from AMS-02, the errors in the cross section data willlikely represent the dominant source of uncertainty for themodel predictions of rare CR isotopes such as 2H or 3He.Similar issues may concern for 6,7Li, 7,9,10Be, or 10,11B.These errors may be reduced using more refined calculationsor more precise accelerator data. For example, future obser-vations may require the departure from the straight-aheadapproximation which is generally assumed in the CR prop-agation studies for light nuclei. Possible consistency checksfor propagation models can be done using the secondary tosecondary ratios, which are less sensitive to the astrophys-ical aspects of the interstellar CR propagation. The use ofratios such as 2H/3He, 6Li/7Li or 10B/11B can represent a use-ful diagnostic test for the reliability of the calculations: anyCR propagation model, once tuned on secondary to primaryratios, must correctly reproduce the secondary to secondaryratios as well. Another model limitation concerns the solarmodulation effect. Any more refined modeling requires toleave the force field approximation, that may be too simpleto finely reproduce the future data of different A/Z isotopes.

Our understanding of the CR heliospheric propagation maybe dramatically improved by the AMS-02 long-term obser-vations of different CR species.

Acknowledgements I thank my many colleagues in AMS for helpfuldiscussions and F. Focacci for her bibliographic support. This workis supported by Agenzia Spaziale Italiana under contract ASI-INFN075/09/0.

References

Abdullin, S.K., et al.: Nucl. Phys. A 569, 753 (1994) (AB94)Aguilar, M., et al.: Phys. Rep. 366, 331–405 (2002) (AMS-01)Aguilar, M., et al.: Astrophys. J. 724, 329–340 (2010) (AMS-01)Aguilar, M., et al.: Astrophys. J. 746, 105 (2011) (AMS-01)Ahlen, S.P., et al.: Astrophys. J. 534, 757–769 (2000) (SMILI-II)Ahn, H.S., et al.: Appl. Phys. 30(3) 133–141 (2008) (CREAM)Blinov, A.V., Chadeyeva, M.V., Grechko, V.E., Turov, V.F.: Phys. At.

Nucl. 64, 907 (2001) (BL01)Coste, B., Derome, L., Maurin, D., Putze, A.: Astron. Astrophys. 539,

88 (2012)Cucinotta, M.G., Townsend, L.W., Tripathi, R.K.: NASA-TR 3285

(1993)De Nolfo, G.A., et al.: In: Mewaldt, R.A., et al. (eds.) Proc. ACE2000

Symp. AIP, vol. 528, p. 425. AIP, New York (2000) (IMAX)De Nolfo, G.A., et al.: In: Proc 28th (Tsukuba), vol. 2, pp. 1667–1670

(2003) (CRIS)Di Bernardo, G., Evoli, C., Gaggero, D., Grasso, D., Maccione, L.:

Astropart. Phys. 34(5), 274–283 (2010)Donato, F., Fornengo, N., Maurin, D.: Phys. Rev. D 78, 043506 (2008)Engelmann, J.J., et al.: Astrophys. J. 233, 96–111 (1990) (HEAO3-C2)Evoli, C., Cholis, I., Grasso, D., Maccione, L., Ullio, P.: arXiv:1108.

0664 [astro-ph] (2011)Ferrando, P., et al.: Phys. Rev. C 37(4), 1490–1502 (1988)Glagolev, V.V., et al.: Z. Phys. C 60, 421 (1993) (GL93)Gleeson, L.J., Axford, W.I.: Astrophys. J. 154, 1011 (1968)Hatano, Y., Fukada, Y., Saito, Y., Oda, H., Yanagita, T.: Phys. Rev. D

52, 6219–6223 (1995)Jones, F.C., Lukasiak, A., Ptuskin, V., Webber, W.: Astrophys. J. 547,

264 (2001)Kneller, J.P., Phillips, J.R., Walker, T.P.: Astrophys. J. 589, 217–224

(2003)Lebowitz, E., Miller, J.M.: Phys. Rev. 177, 1548 (1969). (LM69)Leech, H.W., O’Gallagher, J.J.: Astrophys. J. 221, 1110 (1978)Lezniak, J.A., Webber, W.R.: Astrophys. J. 223, 676–696 (1978)Meyer, J.P.: Astron. Astrophys. Suppl. Ser. 7, 417–467 (1972)Nicholls, J.E., et al.: Nucl. Phys. A 181, 329 (1972) (NI72)Obermeier, A., Ave, M., Boyle, P.J., Höppner, C., Hörandel, J.R.,

Müller, D.: Astrophys. J. 742, 14 (2011) (TRACER)Oliva, A.: Nucl. Instrum. Methods A 588, 255–258 (2008) (AMS-02)Orth, C.D., et al.: Astrophys. J. 226, 1147–1161 (1978)Putze, A., Derome, L., Maurin, D.: Astron. Astrophys. 516, 66 (2010)Reimer, O., et al.: Astrophys. J. 496, 490 (1998) (IMAX)Salati, P., Donato, F., Fornengo, N.: arXiv:1003.4124 [astro-ph] (2010)Strong, A.W., Moskalenko, I.V., Ptuskin, V.S.: Annu. Rev. Nucl. Part.

Sci. 57, 285–327 (2007)Thoudam, S.: Mon. Not. R. Astron. Soc. 388, 335–346 (2008)Tomassetti, N.: arXiv:1204.4492 (2012)Tomassetti, N., Donato, F.: arXiv:1203.6094 (2012)Trotta, R., et al.: Astrophys. J. 729, 106–122 (2011) (GALPROP)Wang, J.Z., et al.: Astrophys. J. 564, 244–259 (2002) (BESS)Webber, W.R.: Adv. Space Res. 19(5), 755–758 (1997)Webber, W.R., Yushak, S.M.: Astrophys. J. 275, 391–404 (1983)Webber, W.R.: In: Jones, W.V. et al. (eds.) AIP Conf. Proc., vol. 203

(1990)

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