SOME ASPECTS OF THE PROPAGATION OF SUPER-HIGH ENERGETIC COSMIC RAYS IN THE GALAXY

International Journal of Modern Physics A

V ol.20,No.29(2005)6825–6827

c Worl

d Scienti?c Publishing Company

SOME ASPECTS OF THE PROPAGATION OF SUPER-HIGH ENERGETIC

COSMIC RAYS IN THE GALAXY

J¨ORG R.H¨ORANDEL,1,?NIKOLAI N.KALMYKOV2and ALEKSEI V.TIMOKHIN2 1University of Karlsruhe,Institut f¨u r Experimentelle Kernphysik,PO3640,Karlsruhe,D-76021Germany

?hoerandel@ik.fzk.de

2Skobeltsyn Institute,Moscow State University,Leninskie Gory1,Moscow,RU-119992Russia

Received25October2004

The origin of the knee in the energy spectrum of cosmic rays is one of the central questions of high-

energy astrophysics.One possible explanation is the energy dependent leakage of nuclei from the

Galaxy due to their propagation.The latter is investigated in a combined method using numerical

calculations of trajectories and the diffusion approximation.The life time of cosmic rays in the Galaxy

and the corresponding pathlength are presented.The resulting energy spectra as observed at Earth are

discussed and compared to experimental data.

Keywords:Cosmic rays;propagation;diffusion model;knee;pathlength

1.INTRODUCTION

The origin of super-high energy cosmic rays(CRs)is one of the most important problems in astrophysics.Since the spectrum of sources is not identical to the spectrum of CRs at Earth,the study of sources is closely connected to the study of the propagation processes in the Galaxy.The validity of various concepts is veri?ed by calculations of the primary energy spectrum making some assumptions about the density of CR sources,the CR source spectrum and the con?guration of the galactic magnetic?elds.The diffusion model can be used in the energy range E<1017eV,where the CR energy spectrum is calculated using the diffusion equation for the density of CRs in the Galaxy.Another method is based on the simulation of the motion of particles in the magnetic?eld of the Galaxy(numerical calculation of trajectories).The calculation of the spectrum in the energy range1014?1019eV has been performed with a combined approach:the use of the diffusion model and the numerical integration of CR particle trajectories.For details of the assumptions on the magnetic?elds and the corresponding diffusion equation see.1,2

2.RESULTS

The calculated spectra for protons are shown in Fig.1(left)for the diffusion model(solid line)and for the numerical calculation of trajectories(dotted line).Both methods give iden-tical results up to about30PeV.At higher energies there is a sharp decrease in the diffusion

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J.R.H¨o randel,N.N.Kalmykov &A.V .Timokhin

Energy [GeV]F l u x 10-410-310-210-1110102103107108109101010

1010Energy E 0 [GeV ]F l u x d Φ/d E 0 ? E 0 2.5 [m -2 s r -1 s -1 G e V 1.5]Fig.1.Calculated spectra for protons for the diffusion model (solid line)and the numerical trajectory calcula-tions (dotted line)(left).Proton ?ux as obtained from various measurements,for references see Ref.3,compared to models,see text (right).

solution,a consequence of a strong rise in the diffusion coef?cient that results in a large leakage of particles from the Galaxy.It is possible to accept the energy 108GeV as the conventional boundary to apply the diffusion model.At this energy the results of the two methods differ by a factor of two and for higher energies the diffusion approximation be-comes invalid.The predicted spectra are compared to a compilation of data in Fig.1(right).The solid and dotted lines represent the two approaches as described above.The steepen-ing of the spectrum due to the propagation process solely is not as strong as indicated by the data,which agree with the steep decrease as described by the Poly-Gonato model.4A steeper cut-off may be obtained through an additional structure in the source spectrum,e.g.by introducing a maximum energy reached in the acceleration process.

Within the model the residence time τof protons in the Galaxy has been calculated as shown in Fig.2(left)as dashed line.In the knee region the energy dependence can not be described with a simple law as τ(E )∝E ?α,since the logarithmic slope of the curve is not constant.In the region of the largest slope αis about 0.7to 0.9.At higher energies the life time decrease slows down.At such energies particles are barely retained in the Galaxy and they cover approximately the same distance.To extrapolate the results to lower energies,where a value of τ=15Myr at 0.4GeV has been obtained,5with a simple assumption τ∝E ?αa relatively small value α=0.05is needed,indicated by the dashed-dotted line.

A further topic considered is the matter traversed by CRs.Results for primary protons are displayed in Fig.2(left)as solid line.The path length for heavier nuclei is obtained by scaling the energy with their respective charge number.At the corresponding knees,the amount of traversed material is less than 1g/cm 2.The dotted line indicates a trend at lower energies according to Λ∝E ?δ.Values around 10g/cm 2as obtained at 1GeV 6lead to a relatively small slope δ=0.17–much lower than the value usually assumed δ=0.6.7Using the latter to extrapolate data from 10GeV results in extremely short pathlengths.For example,the residual pathlength model 7with Λr (E )=λ0·(R /R 0)?δ+λr ,λ0=6g/cm 2,

Some Aspects of the Propagation of Super-High Energetic Cosmic Rays in the Galaxy6827

Fig.2.Left:Pathlength(left scale)and residence time(right scale)in the Galaxy for protons.Right:Fraction of nuclei surviving without interaction in the Galaxy for different elements.

R0=10GV,δ=0.6,andλr=0.013g/cm2yields values forΛwhich are at1PeV about two decades below the values predicted by the diffusion solution,see Fig.2.Such a value would be smaller than the traversed matter along a straight line from the center of the Galaxy to the solar system.

The interaction probability for different nuclei has been calculated based on the ob-tained pathlength and interaction parameters according to the QGSJET model.8Nuclear fragmentation is taken into account in an approximate approach.9It should be pointed out that a nuclear fragment conserves the trajectory direction of its parent if Z/A in question is the same as for the primary nucleus and for most stable nuclei the ratio Z/A is close to1/2. The resulting fraction of nuclei which survive without an interaction is presented in Fig.2 (right)for selected elements.It turns out that at the respective knees(～Z·4.5PeV)more than～50%of the nuclei survive without interactions,even for the heaviest elements.This is an important result,since the Poly-Gonato model relates the contribution of ultra-heavy CRs rays to the second knee in the CR all-particle spectrum around400PeV.4

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