Ultrahigh-energy cosmic rays: possible origin and spectrum

A. V. Uryson

P. N. Lebedev Physical Institute, Russian Academy of Sciences, 117924 Moscow, Russia~Submitted 30 June 1997!Zh. Eksp. Teor. Fiz.113, 12–20~January 1998!

The complicated shape of the cosmic ray spectrum recorded by giant arrays in the energy range101721020 eV is analyzed. It is shown that in the energy region;101821019 eV thespectrum probably coincides with the injection spectrum whose exponent is equal approximatelyto 3.223.3. The flatter component in the energy region (3.225.0)31019 eV is due tobraking of extragalactic protons on primordial photons~the cosmic background radiation!. Atenergies exceeding 3.231019 eV the spectrum does not have a blackbody cutoff. Thepossibility of determining the distances at which cosmic rays originate and investigating theevolution of their sources on the basis of ultrahigh-energy cosmic ray data is discussed. ©1998American Institute of Physics.@S1063-7761~98!00201-7#

1. INTRODUCTION galactic in the energy regionE,1019 eV and extragalactic19

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The origin of cosmic rays of ultrahigh-energy,E.1017

eV, is still unclear. The experimental data indicate that cmic rays with energiesE.431019 eV are probably extragalactic in origin.1–3 If this is so, then their spectrum may hava blackbody cutoff:4,5 the recorded particle flux with energ631019 eV will be twice as small as expected from thpower-law extrapolation of the spectrum as a consequencthe interaction of the cosmic rays with primordial photo~the cosmic background radiation! in intergalactic spaceHowever, if the proton sources are not farther away fromthan 40–50 Mpc the blackbody cutoff will be absent sinprotons of energies up toE'1022 eV traverse such distancealmost freely.6 In Ref. 7 it was shown that the main protosources with energiesE.Ebb'3.231019 eV are probablythe nuclei of active galaxies no farther from us than 40 Mif the Hubble constant is equal to 75 km/(s Mpc). In thcase the proton spectrum does not have a blackbody cuAt present the experimental data obtained at differdetectors—Yakutsk,8 Akeno and AGASA,9 ‘‘Fly’s Eye,’’ 10

Haverah Park,11, Sydney,12 and Volcano Ranch13—neitherconfirm nor refute its presence.

The origin of cosmic rays in the energy regio1017,E<1019 eV is determined not only on the basistheir spectrum, but also their anisotropy and chemicomposition.1–3 However, the available experimental daare not sufficiently unequivocal to determine whether cosrays of such energies are galactic or extragalactic.

Different models have been considered in attemptsexplain the shape of the spectrum in the energy regE.1017 eV. According to the results of Refs. 14–17, thspectrum can have a complicated shape if it is formedextragalactic protons whose sources are hundreds of mparsecs from us. On the other hand, modeling of chargparticle trajectories in galactic magnetic fields has shothat cosmic rays in the energy region 101721018 eV are ga-lactic or are of mixed origin—they are accelerated in tGalaxy and in the Local Supercluster.18,19

The present paper proposes two models to explainproton spectra. The first model assumes that cosmic rays

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for E.10 eV. In the second model they are assumed toextragalactic for energiesE.1017 eV. Using these two mod-els the paper discusses the possibility of investigatingevolution of sources of ultrahigh-energy cosmic rays.

2. EXPERIMENTAL DATA

The cosmic ray spectrum in the energy regionE.1017

eV has a complicated shape:3,8,10 for E'531017 eV theslope of the spectrumg grows from g'3.023.1 tog'3.223.3 ~the error in the determination ofg is0.0220.06), while in the energy regionE'1019 eV it de-creases tog'2.622.7, i.e., a flatter component appearsthe spectrum. The error in the determination of the slopethe flatter component is 0.1.~Spectral slopes are not providein Refs. 11–13.! Cosmic ray spectra measured at differedetectors8–12 and energy-normalized in the same way asRef. 3 are plotted in Fig. 1.

3. COSMIC RAY SPECTRUM FOR E<1019 eV IN THEGALACTIC MODEL

The propagation of cosmic particles in the Galaxy candescribed in the diffusion approximation if their energy donot exceed 101721018 eV ~Ref. 18!. In addition, it wasshown in Refs. 20–22 that particles with chargeZ cease topropagate diffusively if their energy exceeds some vaE0Z, such that in the energy regionE.E0Z the particlespectrum coincides with the injection spectrum andslopes of the spectra are equal tog5g0. ~Note that this resultwas obtained in Refs. 20–22 by different methods: in Re20 and 21 it is due to drift of ultrahigh-energy cosmic rayslarge-scale magnetic fields, while in Ref. 22 it is due totransition to collisionless propagation of particles in a mdium where they excite MHD waves.! An estimate of theenergyE0 was obtained by numerical simulation of the paticle trajectories in the Galactic magnetic field:1 E0'231018

eV.The chemical composition of cosmic rays in the ener

region 101821019 eV is still unclear. According to the resultof measurements reported in Ref. 23, the proton fraction

605$15.00 © 1998 American Institute of Physics

tic

FIG. 1. Cosmic ray spectrum forE.1017 eV, energy-normalized inthe same way as in Ref. 3; measurement data at:d — Yakutsk,8

3 — Akeno and AGASA,9 1 — ‘‘Fly’s Eye,’’ 10 s — HaverahPark.11 Solid line — theoretical spectrum calculated in the galacmodel forE,Ebb and in the extragalactic model forE.Ebb .

creases systematically, starting from;1016 eV, so that fore

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energiesE.1018 eV protons predominate. According to thdata of Ref. 10, the composition varies in the energy ra4310172431019 eV in the following way: to start with,iron nuclei predominate, and at the other end there are oprotons.

We will assume that at energiesE>1018 eV protonspredominate. Then the spectrum of the protons coinciwith their injection spectrum at energiesE>231018 eV.

The regionE>231018 eV is the region in which theslope of the measured spectrum grows. This means apently that at energiesE.1018 eV the slope of the protoninjection spectrum g0 is roughly equal to 3.223.3:g0'3.223.3.

Particles with energyE.Ebb probably accelerate mainlfrom sources no farther away from us than 40–50 Mpc,7,24,25

and as a consequence their spectrum does not have a bbody cutoff. If this is so, then the exponent of the spectrin this region coincides with the exponentg0 of the injectionspectrum. We assume that in the regionE.Ebb the injectionspectrum is the same as forE>231018 eV. Then the slopeof the spectrum in the regionE.Ebb is equal tog53.223.3.

Particles with energiesE.Ebb , propagating fromsources closer than 40 Mpc, will interact with the cosmbackground radiation until their energy fallsE'(3.225.0)31019 eV. Particles with such energies cannundergo any interactions in intergalactic space since tmean free paths in the cosmic background radiation fielquite large:l.1000 Mpc~Ref. 6!. This leads to the resulthat protons with energiesE.3.231019 eV ‘‘pump’’ intothe regionE'(3.225.0)31019 eV, and as a result the slopof the spectrum in this region changes fromg.3.1 to a valueg1 such that

EEbb

E2gdE5EEbb

E3E2g1dE,

whereE3 is the upper limit of the energy range of the flattcomponent. We findg1 from the experimental data.3,8–13Themeasured value ofE3 is approximately 431019 eV, and theenergy of the particles is determined with an error of aproximately 20–30%~Refs. 8 and 10!. Therefore we esti-mated the exponentg1 for several values ofE3 in the inter-val 431019,E3,531019 eV and several values ofg in the

7 JETP 86 (1), January 1998

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in Table I. The energy range of the flatter component wallowance for the 30% error in the determination of the eergy isE'(225)31019 eV. It agrees with the measuremeresults of Refs. 8, 9, 11–13 and does not contradictsingle-measurement data of Ref. 10.

The possible existence in the spectrum of a flattencomponent of such a nature was predicted in Refs. 14–1

The theoretical spectrum based on the proposed modplotted in Fig. 1. It is normalized to the measured intensityE'1019 eV. In addition, the calculations assumed that tenergy region of the flatter componentE'(3.225.0)31019 eV. The theoretical spectrum agreewith the measurements within the limits of error.

Let us consider the slopes of the spectra in the regE>1019 eV. It is clear from the table that the proposemodel yields values ofg1 in agreement with the slope of thflatter component.

To estimate the slope of the measured spectrum inregionE.Ebb we make use of the summary of the expemental data in Ref. 3: in 1993 only 881 events with enerE>1019 eV were recorded, and only 7 withE>1020 eV and2 with E>231020 eV. For a power-law spectrum whereNis the number of particles with energy greater thanE,N(>E), the relation N1(>E1)/N2(>E2)5(E1 /E2)g11

holds, and from it we obtaing53.120.110.2 for E151019 eV,

E251020 eV.Some of the 881 events have energy in the inter

'(1.023.2)31019 eV and make up the flattened compnent. Therefore, in the regionE.3.231019 eV the exponentof the spectrum will be greater than the estimate:g.3.1 and,consequently,g0.3.1.

TABLE I. Calculated exponentg1 of the flatter component for differenvalues~within the limits of experimental error! of its upper limit E3 andspectral slopeg.

g E3, eV g1

3.0 4.931019 2.63.05 4.831019 2.653.05 4.931019 2.73.1 4.531019 2.63.2 4.531019 2.73.3 4.231019 2.7

7A. V. Uryson

If the blackbody cutoff is absent, then the proton spec-s

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trum coincides with the injection spectrum in two region231018<E,1019 eV andE.3.231019 eV. Estimates of theslope of the spectrum in these intervals agree with eother: g53.223.3 andg.3.1; consequently, the measurespectrum apparently does not have a blackbody cutoff.~Theauthors of Ref. 3, on the basis of these same experimedata, concluded that its existence was possible. Theysumed that if there is no cutoff, then the slope forE.Ebb

must coincide with the slope of the flatter component.!

Cosmic ray injection spectrum

Let us consider how the cosmic ray injection spectrvaries between the different energy intervals, making usthe results presented above.

In the regionE,1017 eV the proton spectrum is relateto the injection spectrum by the relation1 N(.E)}E2g02m,where the parameterm describes the dependence of the dfusion coefficient on the energy,D}Em. The measurementof Ref. 26 yield a value ofm the range 0.320.7 for energiesof a few GeV/nucleon, the measurements of Ref. 27 yim50.6 for energies'1 TeV/nucleon, and analysis of thdiffusion model18 yields m50.1520.20 in the energy rangeE510921017 eV. The slopeg of the cosmic ray spectrumfor E,331015 eV is equal to approximately 2.75 and henthe exponent of the injection spectrum in this regiong0'2.2 for m50.6 andg0'2.6 for m50.1520.2.

The spectral indexg0 in the region 33101521018 eV ishard to determine since it is still not clear for what reasothe slope of the cosmic ray spectrum varies forE.331015

eV. Particles with chargeZ are accelerated to energieE<1015Z21 eV, apparently, in supernova bursts.28 Accord-ing to Refs. 18, 29, and 30, the slope of the spectrum vaas a consequence of the propagation and subsequent aeration of the particles in the Galaxy. In addition, itpossible3 that high-energy protons accelerate in other~notyet established! processes, and their injection spectruchanges.

Thus, if protons predominate in the composition of comic rays in the energy regionE.1018 eV ~Ref. 23!, then it ispossible that the slope of the injection spectrum varies infollowing way: it increases to a valueg0'3.223.3 forE.1018 eV in comparison with the regionE,331015 eV,where the slope does not exceed 2.6, 2.2<g0<2.6.

4. EXTRAGALACTIC MODEL OF THE ORIGIN OF COSMICRAYS IN THE ENERGY REGION E>1017 eV

In this model we assume that the particles with eneE.1018 eV are mainly extragalactic, that their spectrum hthe single exponentg'3.023.1, found in the regionE'(224)31017 eV,3,8,10 and that for energiesE.1018 eVthe spectrum is distorted as a result of interaction ofparticles with the fossil radiation~cosmic background radiation! in intergalactic space.

A possible change in the shape of the spectrum inregion E<3.231019 eV was noted in Ref. 14 and investgated in Refs. 15–17. According to the results of these sies, the spectrum can have a complicated shape if it is formby extragalactic protons: by interacting with the cosm

8 JETP 86 (1), January 1998

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the red shift, in processes ofe e pair formation if theirenergy satisfiesE,1019 eV, and by photo-generation opions for E.Ebb . As a result, the differential spectrum othe protons emitted by an isolated source can exhibit a huand a valley frome1e2 pair formation, a photo-pion humpand a blackbody cutoff. However, if the sources uniformfill the Universe, the hump and valley frome1e2 pair for-mation will be weakly expressed. The sources should fisphere of radius corresponding to the red shiftz'0.2. Thephoto-pion hump~without a valley! will be present in thespectrum if the proton sources uniformly fill a sphere ofdius corresponding toz<0.085.

Hence we may assume that for a nonuniform sourcetribution the spectrum will have not only a photo-pion humbut also other peculiarities. Thus, the measured spectcould be explained by varying the shape of the source disbution. Observation data indeed indicate that the nucleactive galaxies are distributed nonuniformly on scales reaing ;102 Mpc ~Ref. 31! and that they are most likely thmain sources of ultrahigh-energy protons.1,7

Let us analyze the spectrum in accordance with thispothesis. If, in accordance with Refs. 7, 24, and 25, the pticles with E.Ebb are accelerated mainly in sources seprated from us by distances not exceeding 40250 Mpc, thentheir spectrum does not have a blackbody cutoff. Thus,exponent of the spectrum in this region is equalg'3.023.1. This is just the rough estimate of the slopethe regionE.Ebb obtained above. It coincides with thslope for E'(125)31017 eV and, consequently, in thimodel the measured spectrum does not have a blackbcutoff.

The shallow component at energiesE'(3.225.0)31019

eV in this model is also due to ‘‘pumping’’ of protons intthis region having energiesE.Ebb . Values of the spectraindex g1 of the flattened component forg'3.023.1 areshown in the table.

In the spectrum of particles accelerated in sources wz'0.2, a notch can appear in the regionE,1019 eV as aresult of particles with energiesE'2310182331019 eVlosing energy by creatinge1e2 pairs in the background radiation field.16

This model can be verified by calculating the protspectra of sources distributed nonuniformly at distanr .40 Mpc from us with a nonuniformity scale of up t;100 Mpc.

5. POSSIBLE CONSTRAINTS ON COSMIC RAY SOURCES

Particles with energiesE.1018 eV probably propagatealong straight-line paths in the Galaxy21,22 and beyond it.32

The energy of a proton emitted at the epoch with rshift z falls as it propagates in intergalactic space due tored shift and formation ofe1e2 pairs and pions.1 Let E0(z)be the energy that a proton should have at its epoch of geration in order for its energy atz50 to beE. We note thatthe luminosity and density of sources in the accompanyvolume increase with growth of their red shiftz. Thus, theenergy density of extragalactic particles in the interv(E,E1dE) is equal to

8A. V. Uryson

w ~E!dE5Ezmaxn ~z!L @E ~z!#t~z!dE dz,

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not only in the region of the flattened component, but also18 19 re

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where neg(z) is the density of extragalactic sourceLCR(E0)dE is their cosmic-ray luminosity in the interva(E0 ,E01dE), andt(z) is the propagation time of such paticles. The lower integration limitzmin corresponds to thedistance the particles can traverse essentially without losAccording to Ref. 6, this distance is;10 kpc, and thereforezmin'0.003. The upper limitzmax is probably'324.33,34

The energy densityweg(.E) can be found from thecosmic ray spectrum:

weg~.E!54p

c E ~E!EdE,

whereI (.E) is the total intensity of cosmic rays with enegies.E andc5331010 cm/s is the speed of light.~Energyrequirements on ultrahigh-energy particle sources arecussed in Ref. 1.!

At the present time, models of the cosmological evotion of sources are not exact enough35,36 to allow one toextract from them estimates of the density and luminositysourcesneg(z) and LCR(z). It is also unclear whether thenergetics of the sources is not connected in some waythe efficiency of the particle acceleration. It has still not bepossible to identify the most powerful extragalactic souras possible sources of cosmic protons withE'1020 eV ~Ref.32!. On the contrary, in Ref. 7 I identified the nuclei of activegalaxies, emitting moderate fluxes in the radio and x-ranges, as the sources of such protons. From the propmodels we obtain the estimate

weg~.E!5E neg~z!LCR@E0~z!#t~z!dz.

For example, according to Fig. 1, forE5Ebb we haveI (Ebb)Ebb

3 '1024.5 (m2•s•sr•eV22)21, and hence

weg(.Ebb)'4310221 erg/cm3. In the second modeweg(.E) can be estimated at lower energies: from Fig. 1follows for E'231018 eV thatI •(231018 eV)•(231018 eV)3'1024.7 (m2

•s•sr•eV22)21

and henceweg(.E)'1310219 erg/cm3.It is possible that the slope of the flattened compon

reflects how distant the proton sources are that form it. Tfarther the source is located from us, the larger the enethat the proton loses on average traversing intergalaspace. The dimensions of the voids between galaxies amto (2.52100)h21 Mpc, and between clusters of galaxies a(1002250)h21 Mpc for the Hubble constanH5100h km•s21

•Mpc21 ~Ref. 31!. Therefore, if thesources beyond the Local Supercluster are located at atancesr>100 Mpc, then the slope of the flattened compnent will be greater than forr<100 Mpc. Thus, by studyingthe flattened component it is possible to estimate thetances from which the protons are arriving. So far suchanalysis has been difficult to carry out because of the laexperimental error in the slope of the spectrum forE.Ebb .If the second model is confirmed, then it will be possibleobtain from it an estimate of the distances from which thultrahigh-energy cosmic rays are arriving using the spect

9 JETP 86 (1), January 1998

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for E'2310 210 eV, where the experimental errors asignificantly less.

6. CONCLUSION

I have proposed two models of the origin of ultrahigenergy cosmic rays. In the first model the cosmic raysassumed to be extragalactic forE>331019 eV and galacticat lower energies. In the second model they are assumebe extragalactic starting atE.1017 eV. It follows from bothmodels that the measured spectrum has apparently no bbody cutoff and that the flattened component in the eneregion (325)31019 eV is due to braking of protons on primordial photons.

Moreover, it follows from the first model that there ispossible changeover from galactic to extragalactic rays inregion of the notch. The data on anisotropy and chemcomposition in this energy region are still not sufficientdefinite to reliably confirm this conclusion.

In the first model we also found that the injection spetrum of cosmic rays with energiesE.1018 eV is differentfrom that at lower energies. Its exponent is largg0'3.223.3 whereas for 1010,E,331015 eV it lies in theinterval 2.2<g0<2.6. The proton spectrum has exponeg0'3.223.3 in the energy region;101821018 eV. This re-sult was obtained from the measurements of Ref. 23, frwhich it follows that forE>1018 eV protons predominate inthe composition of cosmic rays.

To check the second model it is necessary to calcuthe spectra of protons withE.1017 eV from sources at dis-tancesr .40 Mpc from us and nonuniformly distributed oscales up to;100 Mpc.

From the ultrahigh-energy cosmic ray data it is possito form a picture of the evolution of the sources and estimthe distances from which cosmic rays are arriving. Calcutions of the proton spectra with allowance for the evolutiof the sources were performed in Refs. 1 and 16. A simdependence of the source density and luminosity onz wasadopted. It was then shown that the evolution of the sourhas a more complicated form.35,36 It follows from the pro-posed models that the evolution of the sources can be intigated employing the extragalactic particle spectrum

I ~.E!4p

cE5E neg~z!LCR@E0~z!#t~z!dz.

The distances from which the cosmic rays are arrivcan be estimated by analyzing the slope of the flattened cponent. The dimensions of the voids between galaxies(2.52100)h21 Mpc, and between clusters of galaxie(1002250)h21 Mpc ~Ref. 31!. The slope of the flattenedcomponent will be greater if the distance to the sourcesyond the Local Supercluster exceeds 100 Mpc than incaser ,100 Mpc. However, so far it has been difficult tcarry out such an analysis due to the large error in the slof the measured spectra forE.Ebb . If the second model isconfirmed, then it can be used to obtain an estimate ofdistances from which cosmic rays are arriving using

9A. V. Uryson

spectrum not only in the region of the flattened component,18 19 l

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ex

.

,

.

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14A. M. Hillas, Can. J. Phys.21, 1016~1968!.15C. T. Hill and D. N. Schramm, Phys. Rev. D31, 564 ~1985!.

r.

.

but also forE'2310 210 eV, where the experimentaerrors are significantly less.

The above conclusions can be verified in further msurements of cosmic ray spectra at energiesE.1017 eV atthe detectors described in Refs. 8–11 and also in Refsand 38, and at the ShAL-1000 detector,39 all of which havesignificantly better resolution. Projected new detectorstended for recording cosmic rays with energiesE.1019 eVare described in Ref. 40.

ACKNOWLEDGMENTS

I am grateful to S. I. Nikol’ski� and G. B. Christiansenfor discussion of the experimental data, V. A. Dogel’ aV. S. Ptuskin for discussion of the models of propagationgalactic cosmic rays, and Yu. N. Vetukhnovski�, B. V. Ko-mberg, and O. K. Sil’chenko for some remarks abouttragalactic sources.

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Translated by Paul F. Schippnick

10A. V. Uryson