Sveshnikova, L G

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29th International Cosmic Ray Conference Pune (2005)  3, 145-148  The low limit  of  the number of individual sources giving the main contribution to the CR flux around the knee.  L.G. Sveshnikova Skobeltsyn Institute of Nuclear physics of Moscow State University, Moscow 119992, Russia Presenter: L.G.Sveshnikova (sws@dec1.sinp.msu.ru), rus-sveshnikova-LG-abs2-og12-poster  We check  the hypothesis   whether only  a small class of  peculiar Supernovae  gives the main contribution  to the flux of cosmic rays  (CR) in the knee  region.  As compared with previous work, where CR flux  aver-aged over Supernova (SN) explosion energy, type and conditions  around the explosion was calculated,  the effects of CR propagation were included in  calculations. Besides, new formulae for the maximal energy of accelerated particles in one SN remnant (SNR) were used.  It is shown that the idea, that only small fraction of SNRs provides the  main  contribution to the cosmic ray flux around  PeV- region obviously contradicts to a smooth  character of the observed CR spectrum and to the measured anisotropy  due  to large fluctuations of  individual  SNR contribution.  But in frames of the applied  simplified  approach  even  in the case, when all SNRs can accelerate CR particles up to the knee energy  the  calculated value of  <A2>1/2   more or less agrees with  the experimentally measured anisotropy  (at E > 100 TeV) only if the chemical composition becomes heavier around the knee region,  that compensates the D(E) increase, and   nearby SNRs  with RSNR < 700 pc  for some reason do not contribute to CR flux.  But it looks preferentially to suggest that the num-ber of sources providing the knee region is larger than the number of SNRs.   1. Introduction  While recent H.E.S.S. observations show that the new Galactic VHE gamma-ray emitters cluster close to the Galactic plane and this is a clear indication that they see a population of Galactic (rather than extragalactic) sources [1], in a contrast with the generally accepted opinion [2] the wide variety of CR sources classes has been discovered, most of them are not SNRs.  So the question of CR origin is still open. In this paper we continue to study the effects of averaging contributions of different SNRs in the cosmic ray flux observed near the Earth.  In [3], [4]   CR flux averaged over SN explosion energy , type and circum-stellar conditions  was calculated  and  it was shown  that may be only a limited class of  most energetic SNRs  (Gypernovae) with extremely high energy  gives the main contribution to the flux around  1 PeV energy.  But in our calcu-lations   we  did not take into account  the propagation effects and relatively new achievement  in the theory of a diffusive shock acceleration: the  charge particles accelerated in usual  SNR  (with 1051 erg kinetic en-ergy)  can  easily  reach  the energy Emax as large as the energy of the knee due to a nonlinear amplification of random magnetic field around the shock front [5,6] caused by cosmic ray streaming instability.  And even some experimental evidence of this amplification in  six SNRs   Cas A, SN 1006, Ticho’s , Kepler’s, RCW 86, RX J1713.7-3946  can be found  [7].  So we do not need to attract the Gypernovae to explain the CR spectrum up to the knee energy, may be they are needed at E > 1017 eV.  But if consider  the expected maximum energy  of accelerated particles [5,6,7], one can see that maximum energy depends on many pa-rameters: square velocity of  the shock,  density of the circum-stellar  gas,  strength  of magnetic field  sur-rounding SNR, injection efficiency and so on.  As a result one may expect, that  the number of SNR giving the main contribution to the knee region will be less than the total number of SNR providing CRs in  GeV-TeV region.  But it means that fluctuations of CRs near the Earth should increase.  What is the low limit of SNRs giving the main contribution to the CR flux around the knee,  that does not contradict to the measured anisotropy we try to estimate below.  In comparison with [3,4] we  take into account  a propagation  of CRs in the Galaxy, because propagation effects  mainly determine  CR fluctuations and  an anisotropy   [2,8,9,10] near the Earth. Practically,  the anisotropy measured  experimentally (see reference in [9-11]) in the range  146      L.G. Sveshnikova  around 100  TeV occurred  to be as small as if  it would be  fully  produced   by Compton-Getting effect   by the amplitude and phase [11]. Expected energy dependence which follows the D(E) is not seen at all.   2. The model of CR flux near the Earth  As was done in [3] we chose the spectrum of instantly injected CRs in every individual SNR (sited in a point ri at time ti in the Galactic plane):  Ni (E, γs, Emax, ri , ti) ~  E-γs  (with γs=2.4 or 2.15) extended up to maximal value Emax, and  at E> Emax the spectrum  becomes much steeper   Ni ~  E-5  [6], (also we take into account  slight  flattening  of the spectrum before the  Emax ). In every SNR  five groups of different nuclei with charge Z was simulated  with the source abundance R(j=1,5): protons (0.59), helium (0.29), CNO (0.06),  Ne-Ca  and Fe  with rigidity dependent value of Emax(Z)=Emax(p)*Z and the same values of γs.  In accordance with new ideas [5,6] Emax can be as large as  3 PeV.   The absolute intensity of the CRs in every SNR  was normalized by the condition that  10% of kinetic energy (1050 erg)  is transformed to CRs  [2]. The ensemble N0 of SNe explosions in our Galaxy was calculated  by Monte-Carlo method (as in [12,13]) suggesting   the SNe uniformly occurred in the  Galactic plane with the rate 3 per century during the time T =108 years, cor-responding to the survival time  of CRs  with energy  1 GeV in our Galaxy (N0 = 3×106).  The diffusion co-efficient  D was chosen close to the value used in [9]  D ~ 3×1029 см2/с × (ETeV/Z)0.3. More strong energy dependence of  D drastically contradicts to experimental anisotropy [9]. The total flux of all CR nuclei  near the Earth I(r , t ) at the position r in time t was expressed through source Grin function of CR density G(t-ti,r-ri, E), as it is often done  [2, 8, 13].  I(r , t )=Σj=1,5 (с/4π)∑i=1, N0 R(j) Ni (E, γs , Emax(j), ri , ti) G (t-ti,r-ri, E)                      (1) G (t-ti,r-ri, E)= ρ(r-ri, Rd,E)=(1/8π3/2 Rd3 ) exp(-((r-ri)/2Rd)2) exp{-(t-ti)/τ(E)}        (2) Rd=(D(E)⋅t-ti)1/2 ,         τ(E)=108 (EGeV/z)-0.3                                                                                                       (3)  It is necessary to note that this approach with Gaussian Grin function corresponds to simplified solution [2] of diffusion equations: boundaryless Galaxy,   τ(E) is inverse to D(E), the CR age function has exponential form characteristic to Leaky-box model [2].  But just this approach usually is used for an investigation of fluctuations and an anisotropy of CRs [2, 8, 9, 10].    Observed anisotropy for every SN ensemble was calculated as A=3D×(∂I(r,t)/∂r)/c/I(r,t)  [2] , where c – velocity if the light.  Because we are interested in fluctuations of CRs, we calculated 100 ensembles of SNRs that means 100 different Galaxies like our Milky Way. Then we calculated the fluctuation of anisotropy as   (<A2>)1/2  [9,10] and fluctuations  of intensity  δI  by 100 samples.  3. Results   I variant. In the first  variant we calculated the spectrum of  CRs near the Earth, suggesting that the every SNR ejects instantly the  spectrum Ni (E, γs=2.4, Emax(p)=3 PeV). Obtained spectra are presented in the up-per part of Figure 1.   As was expected [2] γobs ~ 2.7 = 2.4 +  0.3.  In this case in spite of the propagation ef-fects the observed spectrum looks practically as in one SNR, only one can see that  relative abundances of heavy  nuclei  are stressed in a comparison with source abundances R(j) due to rigidity dependent diffusion coefficient D(E/Z).  The investigation shows that the contribution of the remote SNRs at the distance RSNR>4 kpc is only 10 %, while the contribution of nearby SNRs with  200 pc <RSNR< 700 pc  is ~ 20 %. But just these nearby sources determine the fluctuations and an anisotropy as many times have been  discussed ear-lier [2, 8-10], and where it has been  proposed to introduce the cut off  parameter (τ0) rejecting  nearby and  recent sources to overcome the  divergence in  the integral at analytical solution:                                                 <A2>1/2 ~  D(E) (σSN τ0)-1/2                                     (4)         The low limit of the number of …                            147  Here σSN – is SN rate, and  τ0 ~  (σSN D(E))-1/2 artificially changes   the anisotropy  <A2>1/2 ~  D5/4(E) σSN -1/4 . In  Monte-Carlo calculations we also faced with the problem of nearby sources. In the bottom part of  Fig-ure1  the value <A2>1/2  is presented for SNRs at a distance RSNR>200 pc and for  SNRs at a distance RSNR>700 pc.  One can see that only last variant more or less agrees with experimental points.  Practically may be this variant   reproduces  the picture  that  the Earth  is located in the relatively quiet  place of the Galaxy between arms at some distance from the  field of active star formation zones  mostly sited in  Galac-tic arms.  But when   we calculate  the anisotropy caused by  really known SNRs, we get the same results as in [9]:  the contribution of nearby known sources in different energy range can be by order larger than ob-served anisotropy: at the energy around 1-10 TeV the main input is from  Vela,  while at the energies around 1 PeV   enough recent (t ~ (2−4)×103 years ago)  sources located at distance around 800 pc should give the noticeable individual input.  So the  idea of  Erlykin &Wolfendale[12] that the one nearby source determines the  bump in  the knee in CR spectrum looks not as the exotic one, but vise versa  more than  normal one.    On the Figure 1 the very useful   effect can be seen:  in the knee region the anisotropy stops to rise with in-creasing energy due to gradual increase of the content of heavier nuclei,  for which the diffusion coefficient is less than for protons, and as a results the anisotropy also is less.    In our Monter - Carlo approach anisot-ropy depends on the number of SNRs providing CRs flux as σSN -1/2 (4),  so it is obviously that in all models when  only  fraction of  total number of  SNRs  provide CR flux,   the anisotropy will be  larger  than in Fig-ure1 and will begin to contradict to  measured anisotropy.                                                                                                                                                                                                                                                                                          104 105 106 107 10810-410-310-210-1100  ~C-OFeAllPHe<A2>1/2dN/dE E2.7  cm-2 c-1 sr -1 GeV 1.7E, GeV Figure 1. The upper part:  spectra of all CRs  and different nuclear components near the Earth if one pro-poses that all SNRs accelerate particles to 3 PeV with power low spectrum ~ E-2.40. The lower part: the value of average anisotropy  <A2>1/2  for all SNRs with RSNR>200 pc (♦-♦-♦)  and SNRs at the distance                  RSNR > 700 pc far from the Erath (***).  • - experimental points  (ref. from [9]).   In Figure 2 we presents II variant  of  CR spectrum  near the Earth, suggesting as in [4],  that  Emax  can vary from SNR to SNR by randomly.  We chose  the simple dependence   F(Emax) ~ Emax−1.15 , which provides  the needed  exponent of observed spectra  2.7 near the Earth, if we propose the spectrum in sources  Ni ~ E−2.15  predicted by the theory of acceleration of CRs in SNRs [2,6,7].  In this case the softening of the spectrum on their way to the Earth  is caused not only by  the energy dependent diffusion coefficient D ~ E0.3  but also by the decreasing number of SNRs giving the main contribution to the high energy CRs. The observed spectrum looks much more smoothed in a comparison with the spectrum in one SNR. But anisotropy obviously con-tradicts to the experimental one even if RSNR >700 pc. Of course, one may admit that the applied approach is too simplified [2] and a more complicated picture of the propagation (as, for example, in [14]) should be  148      L.G. Sveshnikova  investigated. But may be SNRs are not main players at the knee energy range.                                                                                                                                                                                                                                                                                          104 105 106 107 10810-410-310-210-1100  ~C-OFeAllPHe~<A2>1/2dN/dE E2.7  cm-2  c-1 sr -1 GeV 1.7E, GeV Figure 2. In the upper part: spectra of CRs in the variant when not all SNRs accelerate particles to Emax=3 PeV:  F(Emax) ~ Emax−1.15,  and  the source spectrum in one SNR N(E)~E−2.15, D~E0.3. In the lower part: <A2>1/2 for  SNRs with  RSNR>200 pc ( -♦-♦-♦)  and SNRs at the distance R> 700 pc far from the Erath (***). • - experimental points  (ref. from [9]).   4.Conclusions   The idea, that only small fraction of SNRs provides the main contribution to the CR flux obviously contra-dicts to a smooth character of the observed CR spectrum and measured anisotropy due to large fluctuations of  individual SNR contribution.  But in frames of the applied  simplified approach  even  in the case, when all SNRs can accelerate CR particles up to the knee energy  the  calculated value of <A2>1/2   more or less agrees with  the experimentally measured anisotropy  (at E > 100 TeV) only if the chemical composition becomes heavier around the knee region  (that compensates the D(E) increase) and  if  nearby SNRs  with RSNR < 700 pc  for some reason do not contribute to CR flux.  But it looks preferentially to suggest that the number of sources providing the knee region is larger than the number of SNRs  (as it follows from H.E.S.S. observation [1]).  I thank A.D. Erlykin  who initiated this discussion.  This work is  supported by RFBR 05-02-16781.   References  [1] F. Aharonian  et all, astro-ph/0504380 (2005) [2] V.S.Berezinskii et al, Astrophysic of cosmic rays (ed. V.L.Ginzburg), Amsterdam, Noth Holland (1990). [3] L.G.Sveshnikova,  A&A 409, 799-807 (2003) [4] L.G.Sveshnikova, Astronomy Letters, Vol.30, No 1,   41 (2004) [5] A.R.Bell & S.G.Lucek, MNRAS, 321, 433 (2001) [6]  V.S.Ptuskin & V.N.Zirakashvili, astro-ph/0408025 (2004) [7] H.J.Volk, E.G.Berezhko, L.T.Ksenofontov, astro-ph/0409453 (2004) [8] M.A.Lee,  Astrophys. Journal, 229, 424 (1979). [9] V.S.Ptuskin, F.C.Jones, E.S. Seo, R.Sina,  28 ICRC, v. OG, p. 1933. [10] A.A.Lagutin and Yu.A.Nikulin, JETP, 81, 825 (1995). [11] V.A.Kozyarivskii, A.S.Lidvanskii  et all, Izv. RAN, ser. fiz., 68, N11, 1596 (2004) in russian [12] A.D.Erlykin, A.W.Wolfendale, Astro-ph/0404530. [13] A.D.Erlykin, A.A.Lagutin, A.W.Wolfendale,  Astropart. Phys. 19, 351 (2003).  [14] T. Shibata et al. ApJ, 612, 238 (2004).                                                                                                                                                          

The low limit of the number of individual sources giving the main contribution to the CR flux around the knee

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