XXV European Cosmic Ray Symposium, Turin, Sept. 4-9 2016 1 On the inverse problem for extragalactic cosmic ray nuclei with energies 1018 to 1020 eV V.N.Zirakashvili, S.I.Rogovaya, V.S.Ptuskin, E.G.Klepach Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radiowave Propagation, 108840 Moscow Troitsk, Russia The inverse problem of cosmic ray transport of ultra-high energy cosmic rays is considered. The analysis of Auger data on energy spectrum, energy dependence of mean logarithm of atomic mass numberanditsvarianceallowsdefiniteconclusionsontheshapeofthesourcespectrumintheframe- works of the inverse problem approach. The discussion on regularization procedure for considered ill-posed problem is presented. 7 1 0 I. INTRODUCTION ing from the observedat the Earthspectrum without 2 ad hoc assumptions about the shape of source spec- n TheoriginofcosmicrayswithenergiesE >1018eV trum. Simple casesofthe sourcecompositionthatin- a cludes protons and Iron nuclei were considered. The is a key problem of cosmic ray astrophysics. The ob- J more realistic chemical composition including other servedsuppressionofcosmicrayfluxatenergiesabove 2 ∼5×1019 eVseemsconfirmthe presenceofthe GZK nuclei is considered in the present work. ] cutoffpredictedin[1,2]althoughthesuppressiondue E to the acceleration limits in cosmic ray sources can II. SOLUTION OF INVERSE PROBLEM FOR H not be excluded [3, 4]. The occurrence of the GZK A SYSTEM OF COSMIC-RAY TRANSPORT h. suppression and the high isotropy of the highest en- EQUATIONS ergy cosmic rays are indicative of their extragalactic p origin. The list of potential sources which could give - We use the following transport equation for cosmic o the observed cosmic ray flux includes active galactic r nuclei, gamma-raybursts, fast spinning newborn pul- rayprotonsandnucleiintheexpandingUniversefilled st sars,interacting galaxies,large-scalestructure forma- with the background electromagnetic radiation (see a [12] for detail): tion shocks and some other objects, see reviews [5–7] [ and references therein. ∂ H(z) 1 1 The present knowledge about the highest energy − ε + F(A,ε,z) v ∂ε(cid:18) (cid:18)(1+z)3 τ(A,ε,z)(cid:19) (cid:19) cosmic rayswasmainly acquiredfromthe HighReso- 0 2 lution Fly’s Eye Experiment (HiRes), Pierre Auger 8 Observatory (Auger), Telescope Array experiment ∂ F(A,ε,z) 0 (TA), and from the Yakutsk complex EAS array, see −H(z)(1+z) +ν(A,ε,z)F(A,ε,z) ∂z (cid:18) (1+z)3 (cid:19) 0 [5, 8, 9]. The mass composition of these cosmic rays . 1 remains uncertain. The interpretation of HiRes and 0 TA data favors predominantly proton composition at = ν(A+i→A,ε,z)F(A+i,ε,z)+q(A,ε)(1+z)m 7 energies 1018 to 5×1019 eV, whereas the Auger data 1 i=X1,2... indicate that the cosmic ray composition is becoming : (1) v heavier with energies changing from predominantly The system ofeqs. (1) for allkinds of nucleiwith dif- i protonat1018eVtomoreheavycompositionatabout X ferentmass numbersAfromIronto Hydrogenshould 5 × 1019 eV. The mass composition interpretation be solved simultaneously. The energy per nucleon r a of the measured quantities depends on the assumed ε = E/A is used here because it is approximately hadronicmodelofparticleinteractionswhichisbased conservedin a processofnuclearphotodisintegration, on not well determined extrapolation of physics from F(A,ε,z)isthecorrespondingcosmic-raydistribution lower energies. function, z is the redshift, q(A,ε) is the density of The energy spectrum in extragalactic sources is cosmic-ray sources at the present epoch z = 0, m commonly determined by the trail-and-error method characterizesthesourceevolution(theevolutionisab- whenonemakesthecalculationsoftheexpectedatthe sent for m = 0), τ(A,ε,z) is the characteristic time Earthcosmicrayintensityassumingsomeshapeofthe of energy loss by the production of e−e+ pairs and source energy spectrum and the source composition. pions, ν(A,ε,z) is the frequency of nuclear photo- The calculations follow cosmic ray propagation from disintegration, the sum in the right side of eq. (1) the source to the observer, e. g. [10]. The standard describes the contribution of secondary nuclei pro- assumptionisthatthesourcespectrumisapowerlaw duced by the photodisintegration of heavier nuclei, on magnetic rigidity up to some maximum rigidity. H(z) = H0((1+z)3Ωm +ΩΛ)1/2 is the Hubble pa- Inour previouswork[11] weshowedhow to inverse rameter in a flat universe with the matter density theprocedureandcalculatethesourcefunctionstart- Ωm(=0.3) and the Λ-term ΩΛ(=0.7). eConf C16-09-04.3 2 XXV European Cosmic Ray Symposium, Turin, Sept. 4-9 2016 Acomprehensiveanalysisofcosmicraypropagation the followingregularizationprocedure[14]forthis set in the intergalactic space was presented in [13]. of equations below. The numerical solution of cosmic-ray transport Let introduce the functional L equations follows the finite differences method. The 2 variables are the redshift z and log(E/A). 1 Let us introduce solution G(A,ε;As,εs) of eqs. (1) L= 1− N SijQj +εR (Qj−1−2Qj+Qj+1)2 at z = 0 for a delta-source q(A,ε) = δAAsδ(ε−εs). Xi  i Xj  Xj This source function describes the emission of nuclei (6) with mass number A and energy ε from cosmic ray Here ε is the regularization parameter. The first s s R sources distributed over all z up to some zmax. The terminthisequationissimplythesumofsquaredrel- general solution of eqs. (1) at the observer location ative deviations from the observable spectrum N(E). z =0 can now be presented as For ε = 0 this functional is minimized by solutions R of Eqs. (5) and and its value equals to zero. ′ ′ ′ ′ ′ Renormalized set of equations is found from the F(A,ε,z =0)= dεG(A,ε;A,ε)q(A,ε). Z condition ∂L/∂Q =0: XA′ j (2) 1 The observed all-particle spectrum is deter- SRQ =NR, NR = S , SR = mined by the summation over all types of nuclei Xj kj j k k Xi Ni ik kj F(A,E/A,z =0)/A that is A P 1 N(E)= A−1 dε′G(A,E/A;A′,ε′)q(A′,ε′). N2SikSij+εR(6δkj−4δk,j−1−4δk.j+1+δk,j−2+δk.j+2), Z Xi i AX,A′ (7) (3) Weshallassumebelowthatsourcespectraofnuclei can be expressed in terms of one function on rigidity: III. APPROXIMATION OF EXPERIMENTAL DATA q(A,ε)=k(A)Q(εA/Z) (4) To simplify calculations and damp the spread of data points in the measured at the Earth cosmic ray Here Q(ε) is the source proton spectrum and coeffi- spectrum, we use its analytical approximations. cients k(A) determine the source chemical composi- The formula tion. The set of discrete values of particle energy εi is J(E)∝E−3.23,E <5×1018eV; defined to solve the transport equation numerically. J(E)∝E−2.63×[1+exp(log(E/1019.63eV)/0.15)]−1× The gridwithconstant△ε/εandwith25energybins per decade is used in our calculations. Eq. (3) in the exp(−(E/(1.5×1020eV))4),E >5×1018eV.(8) discrete form is is used in our calculations to approximate the Auger N = S Q , (5) data[15]. This formulaissimilartothe equationsug- i ij j Xj gestedbytheAugerteambutcontainsexp(−(E/1.5× 1020eV)4) factor of cosmic rayflux suppressionat en- ergies >∼1.5×1020 eV. ′ ′ Z(A)k(A) ′ ′ ′ S = △ε G (A,E /A;A,ε Z(A)/A), ij A′A j ij i j AX,A′ IV. RESULTS where the subscript indexes i and j denote the corre- sponding energies ε and ε . The minimal value 10−3 − 10−2 of the parame- i j The source term Q can be found from this set ter ε was adjusted to provide the smooth positive j R of linear eqs. (5) if the observed all particle spec- sourcespectrumQ . Wefoundthatthismethoddoes j trum N(E) and chemical composition of the source not work for any chemical composition. However the are known. We have already considered the case of rangeofthechemicalcompositionisstronglyextended protons and iron nuclei in the source [11]. It was in comparison with the exact solution of Eq. (5). found that the solutions of equation(5) havea physi- The results obtained for light and heavy composi- calmeaning only for a limited rangeof protonto iron tion of cosmic ray non-evolutionary sources (m = 0) ratio. In addition the solution can be unstable rela- are shown in Figures 1-4. The maximum redshift tive small deviations of the left hand side of Eq. (5) zmax = 3 was used. The coefficients k(A) are given sothattheinverseproblemisill-posed. Weshallused in Table I. The light composition corresponds to the eConf C16-09-04.3 XXV European Cosmic Ray Symposium, Turin, Sept. 4-9 2016 3 FIG.1: CalculatedsourcespectraofIroninarbitraryunits FIG. 3: Calculated value of hln(A)i for light (solid line) obtained for light (solid line) and heavy (dash line) com- and heavy (dashed line) composition together with corre- position. sponding Auger data (dots and gray regions which char- acterizes errors in determination of hln(A)i in the EPOS LHCinteraction model). FIG.2: Calculatedspectraofprotons(solid),He(dashed), C (short dashed), Si (dotted), Fe (dot-dashed line) for heavy composition of sources. The all particle spectrum (thick solid) and theanalytical approximation Auger cos- mic ray spectrum (gray solid line) are also shown. FIG.4: Calculatedvarianceofhln(A)iforlight(solidline) and heavy (dashed line) composition together with corre- sponding Auger data (dots and gray regions which char- composition of Galactic cosmic rays. We adjust the acterizes errors in determination of variance of hln(A)i in theEPOS LHC interaction model). heavycompositiontoreproducetheAugerdataonen- ergy dependence of the mean logarithmofthe atomic mass number hlnAi calculated in the EPOS-LHC model of particle interactions in the atmosphere [15]. equations (1) are ill-posed in the general case that Itisevidentthatourmodelreproducestheobserved manifests itself in the instability of derived solutions. all particle spectrum and measured mean logarithm Toavoidthisproblemweusetheregularizationproce- hlnAi. dure(Eqs. 6,7)andperformcalculationsforarealistic chemical composition. In addition the same spectral function on the rigidity for the source spectra of dif- V. DISCUSSION AND CONCLUSION ferent nuclei was assumed. We found that assumption of heavy composition We showed how one can find average spectrum of permits to explain the Auger data [15]. The Auger extragalactic sources from the cosmic ray spectrum data favor the transition from a proton source com- observed at the Earth. This task was formulated as position to the heavier one as the energy is rising. an inverse problem for the system of transport eqs. With our heavy source composition, this case is most (1)thatdescribethepropagationofultra-highenergy closely reproduced by the calculations illustrated in cosmic rays in the expanding Universe filled with the figure2. Theobtainedsourcespectra(seefigure1)re- background electromagnetic radiation. semblethe results [16, 17]basedonthe analysisofdi- Mathematically, the inverse problems for transport recttransportproblemswithapowerlawsourcespec- eConf C16-09-04.3 4 XXV European Cosmic Ray Symposium, Turin, Sept. 4-9 2016 The study of inverse transport problem is a use- TABLE I: Coefficients k(A) describing the chemical com- ful tool for the investigation of ultra high energy cos- position of sources mic rays allowing the abandonment of the standard H He C O Mg Si Fe assumption of power law source spectrum with an A 1 4 12 16 24 28 56 abrupt cutoff at some maximum magnetic rigidity as light 1 0.2 0.05 0.05 0.015 0.04 0.004 it is usually assumed when the direct problem is con- sidered. heavy 1 6 0.65 0.2 0.1 0.12 0.015 trum. The maximum energy of accelerated particles Acknowledgments (3...5)Z×1018 eVisrelativelylowinthiscasethatal- leviates the problem of cosmic ray acceleration. 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