Büsching, I

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29th International Cosmic Ray Conference Pune (2005) 3, 201204The influence of discrete Cosmic Ray sources on the secondary toprimary ratioI. Bu¨sching , A. Kopp, M. Pohland R. Schlickeiser(a) School of Physics, Potcheftroom Campus, North-West University, Venter Street, Potchefstroom 2520, South Africa(b) Theoretische Physik IV, Ruhr-Universit¤at Bochum, 44780 Bochum, Germany(c) Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USAPresenter: I. Bu¨sching (fskib@puk.ac.za), saf-buesching-I-abs1-og13-oralThe recent discovery of direct evidences for the acceleration of high energetic particles at the shell supernovaremnant RXJ1713.7-3946 underlined the need to calculate the cosmic ray (CR) distribution in the Galaxy on aspatial grid fine enough to resolve the changes in the CR density due to these kind of objects. To analyze theimpact of the discreteness of the CR sources on the CR spectrum and also on the secondary to primary ratio,we developed a method that enabled us to calculate the Galactic CR distribution with high spatial and temporalresolution. Our calculations for CR hadrons show that the density of the primary CR component varies inspace and time, however, the secondary component shows only minor variations, approaching the steady statedistribution. These findings imply that the secondary to primary ratio, which is widely used to determine theparameters of CR propagation models is also changing with space and time. In particular, we expect a decreaseof the ratio in the vicinity of an active source.1. IntroductionThe recent discovery of direct evidences for the acceleration of high energetic particles at the shell supernovaremnant RXJ1713.7-3946 [1] underlined the need to calculate the cosmic ray (CR) distribution in the Galaxyon a spatial grid fine enough to resolve the changes in the CR density due to these kind of objects.As the necessity of high spatial resolution leads to a huge numerical grid, we developed a method to calculatethe CR density with high spatial and temporal resolution with reasonable computational effort, avoiding thehuge numerical grid by a series ansatz[4].We start from the time dependent diffusion equation with catastrophic losses,	  ﬁﬀﬁﬂﬃ  "!#ﬀﬂ(1)which is suitable for nuclei with charge number $&%(' . Here, N is the differential CR number density,  thespatial diffusion coefficient,  ﬀﬂ)ﬃ  the total spallation cross section (taken from Letaw et al.[11]) and ! ﬀﬁﬂ theparticle velocity. Eq. (1) has to be solved in a diffusive volume in the shape of a disk with radius *  ',+ kpc(cf. Webber et al. [16]) and height '#- /. kpc, with free escape at the boundaries.We further assumed the density of the interstellar gas to decrease with distance 0 from the galactic plane  1324657980;:< (2)with12>=,?'. cm @A , : < CB +EDGF 4 @3H . This corresponds to a column density of IKJ ? 'ML = +GN2PO6Q@N , whichis consistent with Webber et al. [16] and Berezinskii et al.[3]. The chemical composition of the interstellarmatter was taken into account by multiplying the gas density Eq. (2) by a factor of 1.3 [12]. The spatial202 I. B¤usching et al.diffusion coefficient  2RSTSU2WVYXX[Z\2^] _ `5badcMefc22 V XX[Z\@2g] hji `5,adcMklc2(3)with 2+?'#J,DGF4N6mona@PH , c2p. GV/c, depends on the the particle rigidity c rq3s"t , with particlemomentum q and charge t . This form was chosen to reproduce the observed boron to carbon ratio. The modelwas checked by comparing secondary to primary ratios computed for the axisymmetric steady state case withboron to carbon and beryllium data.2. Results from the CalculationsWe performed calculations for point sources distributed arbitrarily in the galactic disc. This means we havefor the source term in Eq. (1) a sum of contributions of many sources, each of which has the same temporalform for which we assume a linear increase with an exponential cutoff. The source spectra are taken as powerlaws with spectral index u which in the calculations was fixed at u v ' ?w= . The sources were distributedstochastically in the Galactic disk, with the radial distribution given by Case & Bhattacharya [5]. The numberof sources was chosen to reproduce the local supernova rate of 20 kpc @N Myr @PH (cf. Grenier [9]) A detaileddescription of the model and the method used for the calculation is given in Bu¨sching et al. [4].2.1 Cosmic Ray primary componentWe investigated the CR primary component considering HxN C as example. The results are shown in Fig-ures 1,3,5. Figure 1 shows the temporal variation of the CR primary component taking H9N C as an example.The density is highly variable in time and strongly influenced by recent local sources. This is also seen inFigure 3 where we plot four arbitrarily chosen distributions (marked by different line styles) perpendicular tothe Galactic plane at a galactocentric distance of 8.5 kpc. The CR distributions only differ in and near theGalactic plane, where the sources are placed. These findings imply that also the local CR spectrum is subjectedto variations. We also investigated in which range one would expect possible local CR primary spectra. Theresults of these investigation are shown in Figure 5. Here, the white dashed line marks the averaged spectrum,whereas the dark red band shows the 68% containment probability range at each given energy and the orangeband gives the 95% range. Superimposed are the data given by Engelmann et al.[6], Mu¨ller et al.[13], Orthet al.[14] and Simon et al.[15]. In the calculated spectra the effect due to solar modulation was taking intoaccount using the force filed approximation [8]. The data fits quite well in the calculated range of possiblespectra.2.2 Cosmic Ray secondary componentThe CR secondary component was studied considering HyH B as an example. The source term for HzH B wascalculated from the H9N C density and the distribution of the interstellar gas, using the cross section given byWebber [17]. The results of these calculations are shown in Figures 2,4,6. Figure 2 corresponds to Figure 1 andshows the HzH B density over time. The temporal variations are negligible compared to that of the HxN C primarycomponent. This is due to the fact that the secondary component is produced in the whole volume occupiedby interstellar gas whereas the CR primary component is assumed to originate from point like sources. Thisfinding is supported when looking at the CR density perpendicular to the galactic plane as shown in Figure 4The inuence of discrete Cosmic Ray sources ... 203Figure 1. Temporal variation of the CR densityfor {| C at a galactocentric radius of 8.5 kpcFigure 2. Same as Figure 1 but for {[{ BFigure 3. CR densities ( {| C) perpendicular tothe Galactic plane at }&~,  kpcFigure 4. Same as Figure 3 but for {[{ BFigure 5. Range of possible{|C spectra at theposition of the Earth (cf. text).Figure 6. Same as Figure 5 but for {[{ B204 I. B¤usching et al.the different distributions (marked by different line styles) plotted here only differ marginally from the steadystate distribution. It is therefore not surprising to find in Figure 6 only a very narrow band of possible spectraat the Earth, the dark red band showing the 68% containment probability range for each energy and the orangeband giving the 95% range almost coincide with the averaged spectrum marked with the white dashed line.3. ConclusionSince there are now observational evidences for SNR as sources of hadronic CR, the Galactic CR density andspectrum has to be examined with a spatial and temporal resolution that is high enough to resolve changes dueto these objects. The numerical solution of the propagation equation thus has to be performed on a huge spatialgrid. This was avoided by applying a series ansatz. Our calculations of the CR density with high spatial andtemporal resolution show that there are indeed spatial and temporal variations in the CR primary density andspectrum due to the stochastic appearance of SNR, but this is not seen for the CR secondary component. Inthat way, the secondary to primary ratio may vary depending on the local source history, thus the widely usedmethod of deriving the parameters of CR propagation models from this ratio is at least tainted, as long as thelocal CR history is not fully known. In particular, it is important to know whether or not we live in the vicinityof a recent supernova, as has been proposed by some authors (see [7] and subsequent papers). It thereforeseems advisable to investigate the local SN history and also search for other methods for the determination ofCR propagation parameters.4. AcknowledgementsIB acknowledges support by the Bundesministerium fu¨r Bildung und Forschung through DLR grant 50 OR0006 and by the Deutsche Forschungsgemeinschaft through Sonderforschungsbereich 591.References[1] F.A. Aharonian et al., Nature 432, 75 (2004)[2] G.E. Allen et al., ApJ 487, L97 (1997)[3] V.S. Berezinskiı˘ et al., Astrophysics of cosmic rays, Amsterdam, North-Holland Elsevier Science Pub-lishers B.V. (1990)[4] I. Bu¨sching et al., ApJ. 619, 314 (2005)[5] G. Case and Bhattacharya, A&AS 120, 437 (1996)[6] J.J. Engelmann et al., A&A 233, 96 (1990)[7] D.A. Erlykin and A.W. Wolfendale, APh 7, 1 (1997)[8] L.J. Gleeson and W.I. Axford, ApJ 154, 1011 (1968)[9] I. Grenier, A&A, 364, L93 (2000)[10] K. Koyama et al., Nature 378, 255 (1995)[11] J.R. Letaw et al., ApJS 51, 271 (1983)[12] K. Mannheim and R. Schlickeiser, A&A 286, 983 (1994)[13] D. Mu¨ller et al. ApJ 374, 356 (1991)[14] C.D. Orth et al., ApJ 226, 1147 (1978)[15] M. Simon et al., ApJ 239, 712 (1980)[16] W.R. Webber et al., ApJ 390, 96 (1992)[17] W.R. Webber et al., Phys. Rev. C 41, 566 (1990)

The influence of discrete cosmic ray sources on the secondary to primary ratio

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