Revisiting the hardening of the cosmic-ray energy spectrum at TeV energies
aa r X i v : . [ a s t r o - ph . H E ] A ug Mon. Not. R. Astron. Soc. , (0000) Printed 18 March 2018 (MN L A TEX style file v2.2)
Revisiting the hardening of the cosmic-ray energy spectrum at TeVenergies
Satyendra Thoudam ⋆ and J ¨org R. H ¨orandel Department of Astrophysics, IMAPP, Radboud University Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands
ABSTRACT
Measurements of cosmic rays by experiments such as ATIC, CREAM, and PAMELA in-dicate a hardening of the cosmic-ray energy spectrum at TeV energies. In our recent work(Thoudam & H¨orandel 2012a), we showed that the hardening can be due to the effect ofnearby supernova remnants. We showed it for the case of protons and helium nuclei. In thispaper, we present an improved and more detailed version of our previous work, and extendour study to heavier cosmic-ray species such as boron, carbon, oxygen, and iron nuclei. Un-like our previous study, the present work involves a detailed calculation of the backgroundcosmic rays and follows a consistent treatment of cosmic-ray source parameters between thebackground and the nearby components. Moreover, we also present a detailed comparisonof our results on the secondary-to-primary ratios, secondary spectra, and the diffuse gamma-ray spectrum with the results expected from other existing models, which can be checked byfuture measurements at high energies.
Key words: cosmic rays — diffusion — ISM: supernova remnants — ISM: general
Recent measurements of cosmic rays by the ATIC (Panov et al.2007), CREAM (Yoon et al. 2011), and PAMELA (Adriani et al.2011) experiments have indicated the presence of hardening in theenergy spectra of protons, helium and heavier nuclei at TeV ener-gies. The observed hardening does not seem to be in good agree-ment with general theoretical predictions. Based on the simple lin-ear theory of diffusive shock acceleration (DSA) of cosmic rays(Krymskii 1977; Bell 1978), and the nature of cosmic-ray transportin the Galaxy (see e.g., Ginzburg & Ptuskin 1976), the cosmic-rayspectrum is expected to follow a single power-law at least up to ∼ PeV, the so-called “cosmic-ray knee”.Different explanations for the spectral hardening have beenproposed. Most of these explanations suggest either hardening inthe cosmic-ray source spectrum or changes in the propagationproperties in the Galaxy. Possible scenarios that can produce ahardened source spectrum have been suggested in Biermann et al.2010, Ohira et al. 2011, Yuan et al. 2011, and Ptuskin et al. 2013,and models based on propagation effects have been discussed inAve et al. 2009, Tomassetti 2012, and Blasi et al. 2012. Various in-terpretations of the spectral hardening including the effect of localsources can be found in Vladimirov et al. 2012.It is generally believed that nearby sources can affect the prop-erties of cosmic rays observed at the Earth, and they might also ac-count for the observed spectral hardening. In Erlykin & Wolfendale2012, it was suggested that the hardening can be due to a steep lo- ⋆ E-mail: [email protected] cal component dominating up to ∼ GeV/n, and a flatter back-ground dominating at higher energies. Another scenario is that thehigh-energy spectrum might be dominating by the local componentand the low-energy region is dominated by the background. Thisscenario was proposed in our recent work (Thoudam & H¨orandel2012a, hereafter referred to as Paper I), in which we considerednearby supernova remnants within kpc in the Solar vicinity. Inaddition to explaining the spectral hardening, we could also ex-plain the observed hardening of the helium spectrum at relativelylower energy/nucleon with respect to the protons. Moreover, weshowed that the hardening may not continue beyond few tens ofTeV/nucleon.In Paper I, we concentrated only on protons and helium nu-clei. Moreover, we did not perform a detailed calculation for thebackground component. The background was obtained by fittingthe measured low-energy data in the range of (20 − GeV/n.Considering that this is the energy region where both the contri-bution of the local sources and the effect of the Solar modula-tion are expected to be minimum, the background thus obtainedseems to represent fairly well the averaged cosmic-ray backgroundpresent in the Galaxy. However, the background should be con-sistent with the observed data down to lower energies say up tosub-GeV or ∼ GeV energies. This consistency was not checkedin Paper I. In addition, if we assume a common cosmic-ray sourcethroughout the Galaxy, then the source parameters for the back-ground component such as the spectral index and the cosmic-rayinjection power should also be consistent with those of the nearbysources. To check whether the spectral hardening can still be ex-plained by maintaining such consistencies, a detailed calculationshould be carried out for both components together. This is the c (cid:13) Satyendra Thoudam and J¨org R. H¨orandel main motivation of our present paper. A similar study was recentlypresented in Bernard et al. 2012. The main difference between theirwork and ours is on the treatment of the nearby sources. They as-sumed supernova remnants as instantaneous point-like sources in-jecting cosmic rays in an energy-independent manner, while weconsider finite-sized remnants producing cosmic rays of differentenergies at different stages in their lifetime. For distant sources,the much simpler energy-independent point-source approximationmay represent a valid approximation, but for nearby sources, it ismore realistic to take into account their finite sizes and the nature ofcosmic-ray injection in the Galaxy (Thoudam & H¨orandel 2012b).In the present work, we will further extend our study to heav-ier cosmic-ray species namely boron, carbon, oxygen, and iron.We will also present a detailed comparison of our results on thesecondary-to-primary ratios, secondary spectrum and the diffusegamma-ray spectrum with those expected from other existing mod-els. Our model is described in section 2. Then, our calculations forthe cosmic-ray spectrum from a nearby source and for the back-ground spectrum are presented in sections 3 and 4, respectively, andin section 5, the various interaction cross-sections and the matterdensity that will be used for our study will be presented. In section6, constraints on the contribution of the nearby sources imposedby the secondary cosmic-ray spectrum are discussed. In section 7,the results on the heavy nuclei are presented, and in section 8, theresults on protons and helium nuclei are described. In section 9,we discuss and compare our results with the predictions of othermodels.
Although there is no direct evidence yet that proves that supernovaremnants are the major source of Galactic cosmic rays, they rep-resent the most probable candidates both from the theoretical andthe observational point of views. Theoretically, it has been estab-lished that the DSA process (Krymskii 1977; Bell 1978) inside su-pernova remnants can produce a power-law spectrum of particlesup to very high energies with a spectral index close to . This valueof spectral index agrees nicely with the values determined from ra-dio observation of supernova remnants (Green 2009). Moreover, if ∼ (10 − of the total supernova kinetic energy is channeledinto cosmic rays, supernova remnants can easily account for thetotal amount of cosmic-ray energy contained in the Galaxy. Ob-servationally, the presence of high-energy particles inside super-nova remnants is evident from detections of non-thermal X-raysand TeV γ -rays from a number of remnants (Parizot et al. 2006;Aharonian et al. 2006, 2008). The non-thermal X-rays are mostlikely synchrotron radiations produced by high-energy electrons inthe presence of magnetic fields. The TeV γ -rays are produced eitherby inverse compton interactions of high energy electrons with am-bient low energy photons or by the decay of neutral pions, whichare produced by the interaction of hadronic cosmic rays, mainlyprotons with the surrounding matter. Irrespective of the nature ofproduction, the detection of TeV γ -rays suggests the presence ofcharged particles with energies larger than few TeV inside super-nova remnants.Based on these observational evidences and our current theo-retical understanding, we assume that supernova remnants are themain sources of cosmic rays in the Galaxy. We further assume thatcosmic rays observed at the Earth consist of two components: asteady background which dominates most of the spectrum and alocal component which is produced by nearby sources. The back- Table 1.
List of supernova remnants with distances < kpc considered inour study. Name Distance (kpc) Age (yr)Geminga .
15 3 . × Loop1 .
17 2 . × Vela .
30 1 . × Monogem .
30 8 . × G299.2-2.9 .
50 5 . × Cygnus Loop .
54 1 . × G114.3+0.3 .
70 4 . × Vela Junior .
75 3 . × S147 .
80 4 . × HB9 .
80 6 . × HB21 .
80 1 . × SN185 .
95 1 . × ground is considered to be produced by distant sources which fol-low a uniform and continuous distribution in the Galactic disk, andthe local component is assumed to be contributed by supernovaremnants with distances within kpc from the Earth. A list of su-pernova remnants that will be considered for our study are given inTable 1.The two cosmic-ray components are treated with differentpropagation models. The background component is treated in theframework of a steady state diffusion model, and the local compo-nent in a time dependent model. For the background component, weadopt the model described in Thoudam 2008 where the cosmic-raydiffusion region is taken to be a cylindrical disk with infinite radialboundary and finite vertical boundaries ± H . For a typical valueof the radial boundary which is ∼ kpc or more, our assump-tion of infinite radial boundary represents a good approximationfor cosmic rays at the Earth. It is because for large radial bound-ary, cosmic-ray escape is dominated by escape through the verticalboundary and the effect of the radial boundary on the cosmic-rayflux becomes negligible. Regarding the size of H , different modelsadopt different values in the range of ∼ (2 − kpc (Strong et al.2010), and for the present study, we assume H = 5 kpc. Further-more, both the background sources and the interstellar matter areassumed to be distributed in the Galactic plane in an infinitely thindisk of radius R . And we take R = 20 kpc for our study.For the local component, we assume a diffusion region withinfinite boundaries in all directions. This assumption is made, con-sidering that the spectrum of cosmic rays from nearby sources donot depend much either on the radial or the vertical boundaries be-cause of their very short propagation time to the Earth relative tothe escape times from the Galactic boundaries (Thoudam 2007).Strictly speaking, the propagation does depend on the boundary be-cause of the dependence of the diffusion coefficient on the verticalboundary. But, once the diffusion coefficient is fixed, one can ne-glect the dependencies on the diffusion boundaries and determinethe cosmic-ray flux using infinite boundaries. The propagation of cosmic rays from a nearby source is governedby the diffusive transport equation, ∇ · ( D ∇ N ) + Q = ∂N∂t (1) c (cid:13) , evisiting the hardening of the cosmic-ray energy spectrum at TeV energies The first and the second terms on the left represents the diffusionand the source terms respectively, with N ( r , E, t ) representing theparticle number density, E the kinetic energy/nucleon, r the dis-tance from the center of the source and t representing the time. Thediffusion coefficient D is assumed to be a function of the particlerigidity ρ as D ( ρ ) = D β ( ρ/ρ ) δ , where δ is the diffusion in-dex, β = v/c with v denoting the particle velocity and c is thevelocity of light. For a nuclei carrying charge Ze and mass num-ber A , the rigidity can be written as ρ = AP c/Ze , where P de-notes the momentum/nucleon of the nuclei. Thus, for the same P ( ≈ E for all the energies of our interests here), all heavy nuclei withcharge Z > diffuse relatively faster than the protons by a factor of ( A/Z ) δ . In Eq. (1), we neglect the effect of interactions with the in-terstellar matter as the time taken for cosmic rays to reach the Earthis expected to be much less than the nuclear interaction timescales.Also, we do not include effects which are important only at low en-ergies such as ionization losses, convection by the Galactic wind,and a possible re-acceleration by the interstellar turbulence. Withthese approximations, we assume that Eq. (1) effectively describesthe propagation of cosmic rays with energies above GeV/n.As mentioned before, an important feature of our model (asalso in Paper I) is the assumption of finite-sized sources whichinject cosmic rays in an energy-dependent manner. Our modelis based on the current basic understandings of the DSA theory(Krymskii 1977; Bell 1978) inside supernova remnants. In DSAtheory, charged particles are accelerated each time they cross thesupernova shock front. During the acceleration process, a smallfraction of the particles are advected downstream of the shockand cannot be further accelerated, while a major fraction diffuseupstream which can be again taken over by the expanding shockand continue their acceleration. Thus, efficient acceleration can beachieved when particles are effectively confined near the shock inthe upstream region. It is now generally accepted that the confine-ment can be provided by magnetic scattering due to the turbulencegenerated ahead of the shock. Particles can remain confined as longas their upstream diffusion length is much less than the shock ra-dius, i.e., l d ≪ R s . The diffusion length is related to the upstreamdiffusion coefficient D s and the shock velocity u s as l d = D s /u s .Then, for D s that scales with energy, for instance D s ∝ E in theBohm diffusion limit, l d ∝ E/u s . This shows that high-energyparticles can escape at early stages during the evolution of the rem-nant while the shock is still strong. For the lower energy particles,they can escape only at later stages when the shock becomes weak.Based on this general understanding, and at the same time consid-ering that the actual energy dependence of D s is not clearly under-stood because of many complicated processes involved during theacceleration process such as magnetic field amplification and back-reaction of the particles on the shock, the cosmic-ray escape timeis parameterized in a simple form as (Paper I) t esc ( ρ ) = t sed (cid:18) ρρ m (cid:19) − /α (2)where ρ represents the particle rigidity, t sed represents the onset ofthe Sedov phase, ρ m the maximum rigidity, and α is a positive con-stant which is taken as a parameter in the present study. The max-imum particle energy is assumed to scale with its charge as ZU m ,where U m denotes the maximum kinetic energy for protons. Wetake U m = 1 PeV (Berezhko 1996), which corresponds to ρ m = 1 PV. Expressing Eq. (2) in energy/nucleon, we get t esc ( E ) = t sed (cid:18) AEZeρ m (cid:19) − /α (3) Eq. (3) shows that for the same E , all heavy nuclei escape atrelatively early times compared to the protons by a factor of ( A/Z ) − /α . This early escape of heavier nuclei in our model wasthe key to explain the observed spectral hardening of helium atlower energy/nucleon with respect to the protons in Paper I.At some later stage when the supernova shock becomes tooweak to accelerate particles, the turbulence level in the upstreamregion goes down and the remnant can no longer hold any particles.At this point, all low-energy particles which remained confined un-til this stage escape into the interstellar medium. We assume this tooccur when the supernova age becomes yr. Then, the completecosmic-ray escape time is taken to be T esc ( E ) = min (cid:2) t esc ( E ) , yr (cid:3) (4)Details about the nature of particle escape from supernova rem-nants can be found in the literature (Ptuskin & Zirakashvili 2005,Caprioli et al. 2009, etc.). Having parameterized the cosmic-ray es-cape times, the corresponding escape radii can be calculated as, R esc ( E ) = 2 . u t sed "(cid:18) T esc t sed (cid:19) . − . (5)where u is the initial velocity of the shock.Assuming that the supernova remnant is spherically symmet-ric, the source term in Eq. (1) is taken as Q ( r , E, t ) = q ( E ) A esc δ ( t − T esc ) δ ( r − R esc ) (6)where A esc ( E ) = 4 πR esc denotes the area of the remnantcorresponding to the escape time of cosmic rays of kinetic en-ergy/nucleon E , r represents the radial variable, and q ( E ) = Aq ( U ) is the source spectrum with q ( U ) given by, q ( U ) = κ ( U + 2 Um ) − (Γ+1) / ( U + m ) (7)where U = AE is the total kinetic energy of the particle, Γ is thesource spectral index, m denotes the rest mass energy of the par-ticle, and κ is a constant which denotes the cosmic-ray injectionefficiency. In Eq. (7), we assume that particles are produced with apower-law momentum spectrum. It should be noted that non-linearDSA theory predicts a slight deviation from pure power-laws witha somewhat flatter spectrum at high energies (Berezhko et al. 1994;Ptuskin et al. 2010). In Ptuskin et al. 2013, this non-linear effecthas been considered to explain the observed cosmic-ray spectralhardening at TeV energies. In the present model, we neglect suchnon-linear effects, and assume a pure power-law source spectrum.With all the above ingredients, the solution of Eq. (1) is ob-tained as, N ( r s , E, t ) = q ( E ) R esc r s A esc p πD ( t − T esc ) exp " − (cid:0) R esc + r s (cid:1) D ( t − T esc ) × sinh (cid:18) r s R esc D ( t − T esc ) (cid:19) (8)Eq. (8) gives the cosmic-ray spectrum at a distance r s (measuredfrom the center) from a supernova remnant injecting cosmic raysof different energies at different times during its evolution. Us-ing the properties e x → and sinh( x ) ≈ x for very small x ,it can be checked that at high energies where the diffusion radius r diff = p D ( t − T esc ) is much larger than ( r s , R esc ) , Eq. (8)yields N ( E ) ∝ q ( E ) /D / ∝ E − (Γ+3 δ/ . c (cid:13) , Satyendra Thoudam and J¨org R. H¨orandel
The background component for a primary cosmic-ray species,hereafter denoted by p , can be calculated from the steady statediffusion-loss equation: ∇ · ( D p ∇ N p ) − ηv p σ p δ ( z ) N p = − Q p (9)The terms on the left represent diffusion and losses due to inelasticcollisions, where η is the averaged surface density of interstellarmatter on the Galactic disk, v p is the particle velocity and σ p ( E ) is the collision cross-section. We consider the diffusion region as acylindrical disk bounded in the vertical directions at z = ± H andunbounded in the radial direction, as mentioned in section 2. Forsources uniformly distributed in the thin Galactic disk, the sourceterm is represented by a delta function as Q p ( r , E ) = Sq ( E ) δ ( z ) ,where S denotes the surface density of supernova explosion rate inthe Galactic disk. Eq. (9) can be solved analytically as described in(Thoudam 2008), and the solution at r = 0 is given by N p ( z, E ) = RSq ( E )2 D p F p (10)where, F p = Z ∞ sinh[ K ( H − z )]sinh( KH ) h K coth( KH ) + ηv p σ p D p i × J ( KR ) dK In Eq. (10), J denotes the Bessel function of order 1, and R is theradial size of the supernova remnant distribution. Taking z = 0 ,Eq. (10) gives the cosmic-ray spectrum at the Earth. This is reason-able considering the fact that the position of the Earth ( ∼ . kpcfrom the Galactic center) is well contained within the size of thesource distribution which is taken to be kpc, and also that themajority of the cosmic rays that reach the Earth are produced bysources within ∼ kpc, the size of the vertical halo boundary. It isimportant to mention that in Eq. (10), q ( E ) is taken to be the sameas in the case of the local component given in the previous section,thus maintaining the same source spectrum between the local andthe background components. Cosmic-ray secondaries are produced as spallation products fromthe interaction of heavier primaries with the interstellar matter dur-ing their propagation through the Galaxy. For matter distributionon the thin Galactic disk, the secondary production rate can be cal-culated as, Q s ( r , E ) = Z ∞ E ηv p N p ( r , E ′ ) δ ( z ) ddE ′ σ ps ( E, E ′ ) dE ′ (11)where s denotes secondary species, N p represents the primarynumber density, and dσ ps ( E, E ′ ) /dE ′ represents the differentialcross-section for the production of a secondary nucleus with an en-ergy/nucleon E from the spallation of a primary nucleus with en-ergy/nucleon E ′ . Assuming that the energy/nucleon is conservedduring the spallation process, the differential cross-section can bewritten as, ddE ′ σ ps ( E, E ′ ) = σ ps δ ( E ′ − E ) , (12)where σ ps is the total spallation cross-section of the primary to thesecondary. Eq. (11) then reduces to Q s ( r , E ) = ηv p σ ps N p ( r , E ) δ ( z ) (13) The propagation of secondaries follows a similar equation thatdescribes their primaries as given by Eq. (9), with the source termreplaced by Eq. (13). Their differential number density is given by(Thoudam 2008) N s ( z, E ) = ηv p σ ps N p (0 , E ) R D s F s (14)where N p (0 , E ) is given by Eq. (10), and F s = Z ∞ sinh[ K ( H − z )]sinh( KH ) h K coth( KH ) + ηv s σ s D s i × J ( KR ) dK By taking z = 0 , Eq. (14) gives the background spectrum of sec-ondary cosmic rays at the Earth. From Eq. (14), it can be shownthat the ratio of the secondary to the primary densities for the back-ground component gives a good measure of the cosmic-ray diffu-sion coefficient as, N s N p ∝ D s (15)which is a well-known result in cosmic-ray propagation studies. In this section, we present the various interaction cross-sections andthe interstellar matter distribution that will be used in our calcula-tions. For protons, the inelastic interaction cross-section is takenfrom the simple parameterization given in Kelner et al. 2006: σ P ( T ) = (cid:0) . . L + 0 . L (cid:1) " − (cid:18) T th T (cid:19) mb (16)where T is the total energy of the cosmic-ray proton, L =ln( T / and T th = 1 . GeV is the threshold energy forthe production of π mesons. For helium and other heavier nuclei,the spallation cross-sections are taken from Letaw et al. 1983: σ A ( E ) = σ h − . e − E/ . sin (cid:0) . X − . (cid:1)i (17)where A represents the mass number of the nuclei, E is the kineticenergy/nucleon in GeV/n, X = E/ . and σ = 45 A . [1 + 0 .
016 sin (5 . − .
63 ln A )] mb (18)Both Eqs. (16) and (17) represent a good approximation to the mea-sured cross-section data down to sub-GeV energies. For helium, assuggested by Letaw et al. 1983, we further include a correction fac-tor of . in Eq. (17).For the secondary boron production, we consider only the C and O primaries as they dominate the total boron produc-tion in the Galaxy. Through spallation, they produce ( B, B) and( C, C) isotopes. The latter further decay into ( B, B) therebycontributing to the production of boron. For our calculations, weuse the tabulated production cross-sections of these isotopes givenin Heinbach & Simon 1995.
Since we assume that the interstellar matter is distributed in thethin Galactic disk, it is more relevant for our study to determinethe surface matter density on the disk rather than the actual number c (cid:13) , evisiting the hardening of the cosmic-ray energy spectrum at TeV energies density. For a given distribution of atomic hydrogen in the Galaxy n HI ( r, z ) , the surface density at the Galacto-centric radius r can beobtained as n HI ( r ) = R ∞−∞ n HI ( r, z ) dz . Similarly, for the molec-ular hydrogen distribution, we can calculate the surface density as n H ( r ) = R ∞−∞ n H ( r, z ) dz . The total surface density of atomichydrogen is then obtained as n H ( r ) = n HI ( r ) + 2 n H ( r ) . Be-cause cosmic rays arriving at the Earth are mostly produced withina distance equivalent to the halo height H = 5 kpc, only the in-terstellar matter distributed within a circle of radius kpc is impor-tant for our study. For our calculations, we use the averaged surfacedensity determined for this circle.The distribution of atomic hydrogen is taken fromGordon & Burton 1976 and Cox et al. 1986, while that of themolecular hydrogen is taken from Bronfman et al. 1988. Fromthese distributions, we obtain the averaged surface density ofatomic hydrogen within our kpc circle as . × atomscm − , and that of molecular hydrogen as . × moleculescm − . This gives a total averaged surface density of atomichydrogen of . × atoms cm − which is finally used for thepresent study. In addition, it is assumed that the interstellar matterconsists of helium. Before we proceed, we will first determine the diffusion coeffi-cient of cosmic rays in the Galaxy. As already mentioned in sec-tion 4.2, the diffusion coefficient D ( ρ ) = D β ( ρ/ρ ) δ can bedetermined from the secondary-to-primary ratio. For our study,we choose the boron-to-carbon ratio since this is the most well-measured and well-studied ratio. The boron-to-carbon ratio calcu-lated using Eq. (14) is compared with the measured data in Figure1. The solid line represents our calculation, and the data are takenfrom HEAO (Engelmann et al. 1990), CRN (Swordy et al. 1990),CREAM (Ahn et al. 2008) and TRACER (Obermeier et al. 2011).We find that choosing values of D = 1 . × cm s − , ρ = 3 GV and δ = 0 . produces a good fit to the data. Ourcalculation takes into account the effect of solar modulation usingthe force field approximation with modulation parameter φ = 400 MV (Gleeson & Axford 1968).Both the values of D and δ obtained in this work are lowerthan the values adopted in Paper I for the case of pure diffusionmodel . We used D = 2 . × cm s − and δ = 0 . in Paper I.These values were taken from Thoudam 2008 which were deter-mined using a slightly larger value of interstellar hydrogen density,and based on earlier measurements before CREAM and TRACERdata became available. In Figure 1, it can be noticed that our newvalue of D nicely agrees with the measurements up to ∼ (1 − TeV/n.In our model which considers the effect of nearby sources onthe observed cosmic rays, it is important to note that nearby sourcescan produce noticeable affects mostly on the primary spectrum. Theeffect on the secondaries can be neglected. This is because for cos-mic rays produced by nearby sources located within ∼ kpc from For the rest of this paper, unless otherwise stated, any comparison withPaper I will be always with the pure diffusion model (Model A of Paper I).Comparison for the re-acceleration model (Model B of Paper I) will not bepresented here as the comparison will look similar to what we obtain in thecase of the pure diffusion model. -2 -1 B / C E (GeV/n)
HEAOCRNCREAMTRACER
Figure 1.
Boron-to-carbon ratio. Data: HEAO (Engelmann et al. 1990),CRN (Swordy et al. 1990), CREAM (Ahn et al. 2008) and TRACER(Obermeier et al. 2011).
Thick-solid line : Our calculation. the Earth, the nuclear spallation time is much longer than the propa-gation time to the Earth. Therefore, the primaries do not get enoughtime for spallation before they reach us. By the time they undergospallation with the interstellar matter, they have already travelledso far that the resulting secondary flux reaching the Earth is al-most negligible. A detailed calculation on the resulting secondaryflux can be found in Thoudam 2008. Thus, we assume that the sec-ondary cosmic rays that we observe are produced entirely by thebackground primary cosmic rays. With this assumption, the sec-ondary spectrum can be used to determine the contribution of thebackground cosmic rays to the observed primary spectrum. Oncethe background contribution has been fixed, we can then set a limiton the contribution of nearby sources since the observed spectrumis taken to be equal to the background plus the local component.Thus, the secondary spectrum can put a constraint on the contribu-tion of the nearby sources to the observed primary spectrum. Thiswill be demonstrated in the following, taking boron and their pri-mary nuclei, carbon and oxygen, as an example.Figure 2 shows the background spectra of carbon (top) andoxygen (bottom) calculated using Eq. (10). The dashed lines rep-resent the results of our calculations, and the data are takenfrom CREAM (Ahn et al. 2009), ATIC (Panov et al. 2007), CRN(M¨uller et al. 1991), HEAO (Engelmann et al. 1990) and TRACER(Obermeier et al. 2011). The calculations assume the same sourceparameters for both elements. The source spectral index and thesource power are chosen such that the resulting boron spectrum bestexplain the measured boron data up to ∼ GeV/n where the un-certainties in the measurements are small. The boron spectrum isshown in Figure 3, where the line represents our calculation andthe measurements are taken from HEAO (Engelmann et al. 1990),CRN (Swordy et al. 1990) and TRACER (Obermeier et al. 2011).We find that taking the source index of
Γ = 2 . , and the sourcepower Sf C ( O ) = 4 . × ergs Myr − kpc − produces a goodfit to the boron data up to few hundred GeV/n, where S representsthe surface density of the supernova explosion rate as introducedin section 4.1, f is the cosmic-ray injection efficiency in units of ergs which is defined as the amount of supernova explosionenergy converted into the primary species, and the subscript C ( O ) denotes carbon (oxygen). In the present model where secondariesare assumed to be produced only in the interstellar medium, it isdifficult to explain the apparent hardening of the boron spectrumin the TeV/n region indicated by the highest energy point fromTRACER. This hardening might be an effect of secondary produc- c (cid:13) , Satyendra Thoudam and J¨org R. H¨orandel -4 -3 -2 E . × I n t en s i t y [ c m - s r - s - ( G e V / n ) . ] E (GeV/n)
Carbon
CREAMATIC CRNHEAO TRACER
Background -4 -3 -2 E . × I n t en s i t y [ c m - s r - s - ( G e V / n ) . ] E (GeV/n)
Oxygen
CREAMATIC CRNHEAO TRACER
Background
Figure 2.
Carbon (top) and oxygen (bottom) energy spectra ( × E . ).Data: CREAM (Ahn et al. 2009), ATIC (Panov et al. 2007), CRN(M¨uller et al. 1991), HEAO (Engelmann et al. 1990) and TRACER(Obermeier et al. 2011). Thick-dashed line : Background spectrum. tion inside supernova remnants or re-acceleration of backgroundsecondaries by strong supernova shock waves (Wandel et al. 1987;Berezhko et al. 2003), which produces an additional hard compo-nent of secondaries.Although we did not normalize our calculations either on thecarbon or the oxygen data, it can be seen from Figure 2 that boththe background spectra are already in very nice agreement with therespective measurements up to ∼ TeV/n. This good agreementbetween the data and the background component has already set avery tight constraint on the contribution of nearby sources at leastbelow ∼ TeV/n. If we allow the maximum contribution of nearbysources to be ∼ of the total observed spectrum at GeV/n(see next section), the carbon or oxygen injection efficiency of su-pernova remnants is constraint to a value of f C ( O ) = 0 . , andthe supernova surface density required to maintain the backgroundis obtained to be S = 7 . Myr − kpc − . The latter corresponds toa supernova explosion rate of . per century in the Galaxy.It should be noted that different nearby supernova remnantsmay contribute at different energies. The constraint on the cosmic-ray injection efficiency that we just derived is based on the con-tribution of those remnants contributing at low energies taken at GeV/n. But, if we assume an equal injection efficiency for allthe supernova remnants in the Galaxy, the same constraint canalso apply to those nearby remnants contributing at higher ener-gies, thereby also putting a limit on their contribution to the ob-served cosmic rays. Under this constraint, the total contribution ofthe nearby supernova remnants listed in Table 1 to the observed -4 -3 -2 E × I n t en s i t y [ c m - s r - s - ( G e V / n ) ] E (GeV/n)
Boron
HEAOCRNTRACER
Background
Figure 3.
Boron energy spectrum ( × E ). Data: HEAO (Engelmann et al.1990), CRN (Swordy et al. 1990) and TRACER (Obermeier et al. 2011). Thick-solid line : Background spectrum. cosmic rays will be calculated. In all our calculations hereafter, thesupernova rate will be taken to be the same as given above, whilethe injection efficiencies and the spectral indices will be allowed tovary for different cosmic-ray species and optimized based on theirrespective data.
In Figure 2, we can see that although the background componentsagree nicely with the data up to ∼ TeV/n, at higher energiesthere is a discrepancy between the data and the calculations. Thedata seem to show some excess above ∼ TeV/n. This excess orhardening in the spectrum can be explained if we include the con-tribution of the nearby supernova remnants as shown in Figure 4.In the figure, the thin-solid lines represent the total contribution ofthe nearby supernova remnants, the thick-dashed lines represent thebackground spectrum and the thick-solid lines represent the totalnearby plus background spectrum. It can be noticed that the nearbyremnants contribute mostly above ∼ GeV/n, explaining theobserved spectral hardening.In Figure 4, the thin-dashed lines show the contribution ofthe dominant supernova remnants at different energies. At ener-gies below ∼ GeV/n, the nearby contribution is dominated byLoop1 and Monogem, while above, the main contributions comefrom Vela and G299.2-2.9. The cut-off at low energies in the caseof Vela and G299.2-2.9 is largely due to energy-dependent cosmic-ray escape from the remnants with some effect of slow propagationat those energies, and the high energy fall-off is mainly because ofthe energy-dependent propagation effect (see section 3).The model parameters used in our calculation for the localcomponent are discussed as follows. The value of t sed depends onthe initial shock velocity of the supernova remnant, the initial ejectamass, and the gas density of the surrounding interstellar medium.Typical values fall in the range of ∼ (100 − ) yr, and we con-sider t sed = 500 yr for the present work. For the initial shock ve-locity, we assume a value of u = 10 cm/s. These values give thecosmic-ray escape times in the range of t esc = (500 − ) yr, and R esc in the range of ∼ (5 − pc. The cosmic-ray escape param-eter α is kept as a free parameter, but its value is assumed to be thesame for all the cosmic-ray species. For a given cosmic-ray species,we take the same injection efficiency and the same source index as c (cid:13) , evisiting the hardening of the cosmic-ray energy spectrum at TeV energies -4 -3 -2 E . × I n t en s i t y [ c m - s r - s - ( G e V / n ) . ] E (GeV/n)
Carbon
Loop1 M onoge m Vela G . - . CREAMATICCRNHEAO TRACER
Total nearby componentBackground
Total: Nearby + Background -4 -3 -2 E . × I n t en s i t y [ c m - s r - s - ( G e V / n ) . ] E (GeV/n)
Oxygen
Loop1 M onoge m Vela G . - . CREAMATICCRNHEAO TRACER
Total nearby componentBackground
Total: Nearby + Background
Figure 4.
Carbon (top) and oxygen (bottom) energy spectra ( × E . ). Thick-dashed line : Background spectrum.
Thin-solid line : Total nearbycomponent.
Thick-solid line : Total nearby plus background.
Thin-dashedlines : Dominant nearby supernova remnants. those used in the calculation for the background component. And,all the nearby supernova remnants considered in our study are as-sumed to have the same set of model parameters.For the results shown in Figure 4, Γ C ( O ) = 2 . , f C ( O ) =0 . , S = 7 . Myr − kpc − and α = 2 . . This value of α gives particle escape times of t esc = (500 − ) yr for particles ofenergies . PeV/n to . GeV/n. The present value of α is slightlylarger than the value of . adopted in Paper I. A larger value is re-quired due to the smaller value for the diffusion coefficient used inthe present study. Physically speaking, larger α means shorter con-finement within the remnant, while smaller D means longer timefor cosmic rays to reach the Earth. Therefore, in order to explainthe spectral hardening above ∼ GeV/n, which in our model isdue to the contribution of the nearby sources, a larger α is requiredto compensate the effect of a smaller D . This will become moreclear in the next section, when we compare our present results forprotons and helium nuclei with those obtained in Paper I.For iron, the result is shown in Figure 5. We take the samesource index of Γ F e = 2 . as in the case of carbon and oxygen,and an injection efficiency of f F e = 0 . . All other model pa-rameters remain the same as in Figure 4. Also, in the case of iron,we can see that the nearby supernova remnants produce a spectralhardening above ∼ GeV/n. Future sensitive measurements athigh energies can provide a crucial check of our prediction. -5 -4 -3 E . × I n t en s i t y [ c m - s r - s - ( G e V / n ) . ] E (GeV/n)
Iron
Loop1 M onoge m Vela G . - . CREAMATICCRNHEAO TRACER
Total nearby componentBackground
Total: Nearby + Background
Figure 5.
Iron energy spectrum ( × E . ). Data: CREAM (Ahn et al.2009), ATIC (Panov et al. 2007), CRN (M¨uller et al. 1991), HEAO(Engelmann et al. 1990) and TRACER (Obermeier et al. 2011). Thick-dashed line : Background spectrum.
Thin-solid line : Total nearby compo-nent.
Thick-solid line : Total nearby plus background.
Thin-dashed lines :Dominant nearby supernova remnants.
In Figure 6, we show our results for protons (top) and helium nu-clei (bottom). The thick-dashed and the thin-solid lines representthe background and the total nearby contributions respectively, andthe thick-solid line shows the total sum of the background and thenearby components. The contributions from the nearby dominantsources are also shown by the thin-dashed lines. The measurementsdata are from CREAM (Yoon et al. 2011), ATIC (Panov et al.2007), PAMELA (Adriani et al. 2011), and AMS (Alcaraz et al.2000; Aguilar et al. 2002). The results look very similar to thoseobtained for the heavier nuclei in section 7. Except for the sourceindex and the injection efficiency, all other model parameters aretaken to be the same as for the heavier nuclei. For protons, we findthat taking Γ P = 2 . and f P = 17 . produces a good fit tothe data, while for helium the best fit parameters are found to be Γ He = 2 . and f He = 1 . . For comparison, these values arelisted in Table 2 along with those obtained for carbon, oxygen andiron. In general, our present results are quite similar to the resultspresented in Paper I. For instance, the nearby supernova remnantsshow dominant contribution at energies & (0 . − TeV/n, therebyexplaining the observed spectral hardening. Moreover, the heliumspectrum shows a hardening at lower energy/nucleon with respectto the proton spectrum which, in our model, is attributed mainly tothe early escape times of helium nuclei from the supernova rem-nants relative to the protons. It might be recalled from Paper I, andalso discussed in section 3 of this paper that in our model, sucha spectral hardening at lower energies/nucleon is expected for allheavier primaries ( A > whose escape times are shorter thanthe time for protons by a factor of ( A/Z ) − /α at the same en-ergy/nucleon. Also, both our present and the previous studies showthat the main contribution at high energies comes from the Vela andG299.2-2.9 remnants, and that the spectral hardening does not con-tinue up to very high energies. Available measurements also seemto support these results.However, some basic differences can be noticed. First, the Data taken from the compilation by Strong & Moskalenko 2009.c (cid:13) , Satyendra Thoudam and J¨org R. H¨orandel -1 E . × I n t en s i t y [ c m - s r - s - G e V . ] E (GeV)
Proton
Loop1 M onoge m Vela G . - . CREAMATICAMSPAMELA
Total nearby componentBackground
Total: Nearby + Background -3 -2 -1 E . × I n t en s i t y [ c m - s r - s - ( G e V / n ) . ] E (GeV/n)
Helium
Loop1 M onoge m Vela G . - . CREAMATICAMSPAMELA
Total nearby componentBackground
Total: Nearby + Background
Figure 6.
Proton ( × E . , top) and helium ( × E . , bottom) energy spec-tra. Data: CREAM (Yoon et al. 2011), ATIC (Panov et al. 2007), PAMELA(Adriani et al. 2011), and AMS (Alcaraz et al. 2000; Aguilar et al. 2002). Thick-dashed line : Background spectrum.
Thin-solid line : Total nearbycomponent.
Thick-solid line : Total nearby plus background.
Thin-dashedlines : Dominant nearby supernova remnants.
Table 2.
Source spectral indices Γ and injection efficiencies f for the vari-ous cosmic-ray nuclei considered in our study.Nuclei Γ f ( × ergs)Proton .
27 17 . Helium .
21 1 . Carbon .
31 0 . Oxygen .
31 0 . Iron .
31 0 . background spectrum is steeper in the present case. At high en-ergies, the proton background follows a spectral index of . andthe helium background has an index of . . To be compared, thebackground indices were obtained as . and . respectivelyin Paper I. It may be recalled that in Paper I, the backgrounds wereobtained by fitting the measured spectra between (20 − GeV/nand their consistency with the low energy data below ∼ GeV/nwas not checked. From Figure 6, it can be noticed that the back-ground adopted in the present study agrees nicely (in fact, evenbetter when the small local component has been added) with thedata down to GeV/n.Second, in Paper I, the source index for the nearby sourceswere obtained as (Γ b − δ ) , where Γ b is the background index and δ is the diffusion index. So, for the proton background of Γ b = 2 . and δ = 0 . adopted in Paper I, the source index was found to be . . This is flatter than the proton source index of . adopted inthe present study. Adopting a steeper source index suppresses thecontribution from nearby sources at high energies. For the sameamount of total energy injected into protons, a source with an in-dex of . produces ∼ . times less number of source particles at ∼ TeV than a source with an index of . . A similar differenceis also expected in the case of helium. On the other hand, taking asmaller value of D enhances the flux from a nearby source. Thisis clear from the discussion on Eq. (8), given in section 3 whichshowed that at very high energies, the particle spectrum dependson D as N ∝ D − / . After detailed investigation, we find that theincrease in the particle flux in the present study due to smaller D is almost equal to the decrease in the flux due to the steeper sourceindex. Because of these two almost equal and opposite effects, wecan still explain the spectral hardening of helium with an injectionefficiency very close to that used in Paper I. For protons, we needan injection efficiency of . in the present study, which is ap-proximately twice the value obtained in Paper I. This difference isbecause of the lower proton background in the present study at en-ergies above ∼ TeV as compared to the background in Paper I.For helium, the background does not differ too much at TeV ener-gies in the two studies.Another difference results from the difference in the cosmic-ray escape parameter and the diffusion index. A smaller α producesa sharper low-energy cut-off and a larger δ leads to a steeper fall-off in the high-energy spectrum from a nearby source. In Paper I,where we took α = 2 . and δ = 0 . , this led to sharper peaksin the individual contributions of the nearby remnants, thereby re-sulted into stronger structures in the resultant total spectrum. In thepresent study, the slightly larger value of α = 2 . and the smallervalue of δ = 0 . produce broader peaks in the individual contri-butions leading to weaker structures in the overall total spectrum. We have shown that the spectral hardening of cosmic rays at TeVenergies recently observed by the ATIC, CREAM, and PAMELAexperiments can be due to nearby supernova remnants. Consider-ing that cosmic rays escape from supernova remnants in an energy-dependent manner, we also show that heavier elements should ex-hibit spectral hardening at relatively lower energies/nucleon withrespect to protons, and that the hardening might not continue up tovery high energies. These results also seem to agree with the mea-sured data.In general, the results obtained in this paper agree very wellwith those presented in Paper I. Our present study involves a de-tailed calculation of the background cosmic rays unlike in Paper I,and also follow a consistent treatment of the cosmic-ray sourcespectrum for the background and the nearby sources. Our resultsare found to be consistent with the observed data over a wide rangein energy from GeV/n to ∼ GeV/n for a reasonable set ofmodel parameters. Our calculation requires a supernova explosionrate of ∼ per century in the Galaxy, and cosmic-ray injection ef-ficiencies of f P = 17 . for protons, f He = 1 . for heliumnuclei which is exactly of the proton value, f C ( O ) = 0 . for carbon and oxygen, and f F e = 0 . for iron. The re-quired source index for protons is Γ P = 2 . and for helium nu-clei, Γ He = 2 . . For carbon, oxygen and iron, we determinedthe same source index of . . The required source indices of c (cid:13) , evisiting the hardening of the cosmic-ray energy spectrum at TeV energies -1 E . × I n t en s i t y [ c m - s r - s - G e V . ] E (GeV)
Proton
Model IModel IIOur Model (Total)Our Model (Background)
Figure 7.
Proton energy spectrum under different models.
Thick-dashedline : Model I.
Thin-solid line : Model II.
Double-dashed line : Total back-ground plus nearby spectrum in our model.
Thin-dashed line : Our back-ground spectrum. Data as given in Figure 6 (top). ∼ (2 . − . in the present study are slightly steeper than a valueof Γ = 2 . − . predicted by DSA theory. Actually, even a largervalue of Γ ∼ (2 . − . is favored by the high level of cosmic-ray isotropy observed between around and TeV which inturn suggests a smaller value of the diffusion index of δ ∼ . to . (Ptuskin et al. 2006). This discrepancy between observationand theory is still not clearly understood.Our model predictions are expected to be different in many re-spects from those of other existing models. Models which are basedeither on the hardening in the source spectrum or changes in thediffusion properties of cosmic rays at high energies will producea spectrum that remains hard up to very high energies (Ohira et al.2011, Yuan et al. 2011, Ave et al. 2009). But, although not very sig-nificant, the CREAM data seems to indicate that the spectral hard-ening does not continue beyond few tens of TeV/n, which in generalis in good agreement with our prediction. In Figure 7, we compareour result for proton spectrum (double-dashed line) with the pre-dictions of other models: Model I (thick-dashed line) and Model II(thin-solid line). Model I represents models with a hardened sourcespectrum above a certain energy, but assumes the same diffusioncoefficient as in our model. Model II represents models which in-corporate a break or hardening in the diffusion coefficient. Thesource spectrum in Model II is taken to be the same as in our model.Also shown in Figure 7 for reference is the background spectrumobtained in our model. To reproduce the same measured spectrum,Models I and II are chosen to have their respective breaks in thesource spectrum or in the diffusion coefficient at the same energy E b = 850 GeV/n. It can be noticed that our result shows significantdifference mostly at energies above ∼ . PeV. Thus, if the cosmic-ray spectrum exhibit an exponential cut-off below ∼ . PeV, ourresult will not be significantly different from the others. However,detailed studies on the origin of the cosmic-ray knee suggest a cut-off at energies around − PeV (H¨orandel 2003).Even more different between the different models will be thesecondary-to-primary ratios and the secondary spectra. In the stan-dard model of cosmic-ray propagation, the secondary-to-primaryratio is independent of the source parameters and gives a good mea-sure of the cosmic-ray diffusion coefficient in the Galaxy. However,in our model, which considers the effect of the nearby sources,the ratio may deviate from the standard result. This is because thenearby sources can affect only the primary spectrum, and their ef-fect on the secondaries is negligible. Thus, we expect to see a steep- -2 -1 B / C E (GeV/n)
Model IModel IIOur Model
Figure 8.
Boron-to-carbon ratio under different models.
Thick-dashed line :Model I.
Thin-solid line : Model II.
Double-dashed line : Our model. Data asgiven in Figure 1. -4 -3 -2 E × I n t en s i t y [ c m - s r - s - ( G e V / n ) ] E (GeV/n)
Boron
Model IModel II
Our Model
Figure 9.
Boron energy spectrum under different models.
Thick-dashedline : Model I.
Thin-solid line : Model II.
Double-dashed line : Our model.Data as given in Figure 3. ening in the ratio in the energy region where the nearby contribu-tion on the primary spectrum is significant. This is shown in Figure8 for the boron-to-carbon ratio. Notice the significant steepeningin our model above ∼ GeV/n with respect to Model I eventhough both the models assume the same diffusion coefficient. Theresult for Model II is even more flatter at high energies, reflectingthe harder value of diffusion coefficient above E b = 850 GeV/nassumed in the model.For an equilibrium primary spectrum N p ∝ E − γ , and a dif-fusion coefficient D ∝ E δ , the secondary spectrum in the Galaxyfollows N s ∝ E − ( γ + δ ) . This shows that for a fixed diffusion coef-ficient, the shape of the secondary spectrum is determined by theshape of the primary spectrum. Therefore, models that considerthe same diffusion coefficient, but different N p will produce dif-ferent N s . Since the background primary spectrum above E b inour model is steeper than in Model I (see Figure 7, thin dashedand thick-dashed lines), the secondary spectrum is expected to besteeper above E b in our model. This is shown in Figure 9 for theboron spectrum, where the thick-dashed line represents Model I,and the thin-dashed line represents our model. The difference isexpected to be even more significant if we compare with Model II,which assumes a harder diffusion index above E b . The thin-solidline in Figure 9 represents Model II. Similar differences are also c (cid:13) , Satyendra Thoudam and J¨org R. H¨orandel -9 -8 -7 -6 -5 -1 E γ × I n t en s i t y [ c m - s r - s - G e V ] E γ (GeV) FERMI
Model IModel IIOur Model
Figure 10.
Galactic diffuse gamma-ray spectrum under different mod-els.
Thick-dashed line : Model I.
Thin-solid line : Model II.
Double-dashedline : Our model. The data represents the gamma-ray intensity measured byFERMI for Galactic latitudes | b | > ◦ (Abdo et al. 2010). expected in other types of secondary nuclear species such as sub-iron, and anti-protons as shown in Vladimirov et al. 2012, whichalso discussed the effects of different models on various observedproperties of cosmic rays including the secondary-to-primary ratiosand the diffuse gamma-ray emissions.The Galactic diffuse gamma-ray spectrum in our model is alsoexpected to be different from those calculated using other models.If the diffuse emission is dominated by gamma-rays produced fromthe decay of π mesons, then their spectrum at high energies wouldlargely follow that of the primary protons. As the background spec-trum in our model is steeper above E b than the spectrum adoptedin other models, our diffuse gamma-ray spectrum will be steeperabove E b . This is shown in Figure 10 where the thin-dashed linerepresents our model, and the thick dashed and thin-solid lines rep-resent Models I and II respectively. The data represents the gamma-ray intensity for Galactic latitudes | b | > ◦ measured by theFERMI experiment (Abdo et al. 2010). All results plotted in Fig-ure 10 are normalized to the data at . GeV. The calculations usethe different proton spectra shown in Figure 7, and the gamma-rayproduction cross-section is taken from Kelner et al. 2006. In Figure10, we can see that Models I and II give almost the same result, andproduce a harder gamma-ray spectrum with respect to our modelabove ∼ GeV. This difference can be checked by future mea-surements at high energies, and can distinguish our model fromothers. The data in Figure 10 show some excess above the modelpredictions between ∼ (10 − GeV. Although it is not the aimof this paper to perform a detailed modeling of the FERMI data, itcan be mentioned that the excess is most likely due to additionalcontributions from other processes such as bremsstrahlung, inversecompton and unresolved point sources, which are neglected in ourcalculations. Detailed calculations involving all the possible con-tributions have shown that the diffuse gamma-rays up to ∼ GeV measured by FERMI from different regions in the Galaxy canbe explained using a single power-law cosmic-ray spectrum with-out any break above a few GeV (Ackermann et al. 2012, see alsoVladimirov et al. 2012).
10 CONCLUSION
We have presented a detailed and improved version of our previouswork presented in Paper I (Thoudam & H¨orandel 2012a) where we showed that the spectral hardening of cosmic rays observed at TeVenergies can be a local effect due to nearby supernova remnants.Unlike in Paper I where the cosmic-ray background was obtainedby merely fitting the low-energy data, the present work involvesa detailed calculation of both, the background and the local com-ponents, considering consistent cosmic-ray source parameters be-tween the two components.In addition to the results for protons and helium nuclei (whichwere also studied in Paper I), we have also presented results forheavier cosmic-ray species, such as boron, carbon, oxygen, andiron nuclei. Unlike in other existing models, we have shown thatheavier nuclei should exhibit hardening at a lower energy/nucleonas compared to protons, and also that the hardening may not con-tinue up to very high energies for all the species. Although notvery significant, the available data seem to suggest our findings.Moreover, we have also shown that our results on the secondary-to-primary ratios, the secondary spectra, and the Galactic diffusegamma-ray spectrum at high energies are expected to be differentfrom the predictions of other models. Future sensitive high-energymeasurements on these quantities can differentiate our model fromothers.
REFERENCES
Abdo, A. A., et al. 2010, PRL, 104, 101101Ackermann, M., et al. 2012, arXiv:1202.4039Adriani, O., et al. 2011, Science, 332, 69Aguilar, M., et al. 2002, Physics Reports, 366, 331Aharonian, F. A., et al. 2006, ApJ, 636, 777Aharonian, F. A., et al. 2008, A & A, 477, 353Ahn, H. S., Allison, P. S., Bagliesi, M. G., et al., 2008, APh, 30,133Ahn, H. S., Allison, P. S., Bagliesi, M. G., et al. 2009, ApJ, 707,593Alcaraz, J., et al. 2000, Physics Letters B 490, 27Ave, M. et al., 2009, ApJ, 697, 106Bell, A. R., 1978, MNRAS 182, 147Berezhko, E. G. 1996, APh, 5, 367Berezhko, E. G., Yelshin, V. K., & Ksenofontov, L. T., 1996, APh,2, 215Berezhko, E. G., Ksenofontov, L. T., Ptuskin, V. S., et al. 2003,A&A, 410, 189Bernard, G., Delahaye, T., Keum, Y. -Y., Liu, W., Salati, P., Taillet,R., 2012, arXiv:1207.4670Biermann P. L., Becker J. K., Dreyer J., Meli A., Seo E., StanevT., 2010, ApJ, 725, 184Blasi, P., Amato, E., & Serpico, P. D., 2012, PRL, 109, 061101Bronfman, L., Cohen, R. S., Alvarez, H., May, J., Thaddeus, P.,1988, ApJ, 324, 248Caprioli, D., Blasi, P., & Amato, E. 2009, MNRAS, 396, 2065Cox, P., Kr¨ugel, E., & Mezger, P. G., 1986, A&A, 155, 380Engelmann, J. J., Ferrando, P., Soutoul, A., Goret, P., & Juliusson,E. 1990, A&A, 233, 96Erlykin, A. D. & Wolfendale, A. W., 2012, APh, 35, 449Ginzburg, V. L., & Ptuskin, V. S., 1976, RvMP, 48, 161Gleeson, L. J., & Axford,W. I., 1968, ApJ, 154, 1011Gordon M. A., & Burton W. B., 1976, ApJ, 208, 346Green, D. A., 2009, BASI, 37, 45Heinbach, U., & Simon, M., 1995, ApJ, 441, 209H¨orandel, J. R., 2003, APh, 19, 193 c (cid:13) , evisiting the hardening of the cosmic-ray energy spectrum at TeV energies Kelner, S. R., Aharonian, F. A., & Bugayov, V. V., 2006, PRD 74,034018Krymskii, G. F., 1977, Akad. Nauk SSSR Dokl., 234, 1306Letaw, J. R., Silberberg, R., & Tsao, C. H., 1983, ApJS, 51, 271M¨uller, D., Swordy, S. P., Meyer, P., L’Heureux, J., & Grunsfeld,J.M., 1991, ApJ, 374, 356Obermeier, A., Ave, M., Boyle, P., et al., 2011, ApJ, 742, 14Ohira, Y., Murase, K., & Yamazaki, R., 2011, MNRAS, 410, 1577Panov, A. D. et al. 2007, Bull. Russ. Acad. Sci., Vol. 71, No. 4,pp. 494Parizot, E., Marcowith, A., Ballet, J., & Gallant, Y. A. 2006, A&A,453, 387Ptuskin, V. S., & Zirakashvili, V. N. 2005, A&A, 429, 755Ptuskin, V. S., Jones, F. C., Seo, E. S., & Sina, R., 2006, AdvancesSpace Res., 37, 1909Ptuskin, V. S., Zirakashvili, V., & Seo, E. S., 2010, ApJ, 718, 31Ptuskin, V. S., Zirakashvili, V., & Seo, E. S., 2013, ApJ, 763, 47Strong, A. W., Porter, T. A., Digel, S. W., 2010, ApJ, 722, L58Strong, A. W. & Moskalenko, I. V., 2009, th ICRC, LODZSwordy, S. P., M¨uller, D., Meyer, P., L’Heureux, J., & Grunsfeld,J. M., 1990, ApJ, 349, 625Thoudam, S., 2007, MNRAS, 380, L1Thoudam, S., 2008, MNRAS, 388, 335Thoudam, S., & H¨orandel, J. R. 2012a, MNRAS, 421, 1209 (Pa-per I)Thoudam, S., & H¨orandel, J. R. 2012b, MNRAS, 419, 624Tomassetti, N. 2012, ApJ, 752, L13Vladimirov, A. E, J´ohannesson, G., Moskalenko, I. V. & Porter, T.A., 2012, ApJ, 752, 68Wandel, A., Eichler, D., Letaw, J. R., et al. 1987, ApJ, 316, 676Yoon, Y. S. et al. 2011, ApJ, 728, 122Yuan, Q., Zhang, B., & Bi, X. -J, 2011, Phys. Rev. D 84, 043002 c (cid:13)000