Futo, Y

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29th International Cosmic Ray Conference Pune (2005) 3, 185188A new approach to the cosmic-ray propagation in the Galaxy in thewide energy range 1 GeV – 1 PeVY. Futo , M. Hareyama, N. Honda , T. Shibata and J. Watanabe (a) Department of Physics and Mathematics, Aoyama-Gakuin University, Kanagawa 229-8558, Japan(b) Major in Pure and Applied Physics, Science and Engineering, Waseda University, Tokyo 169-8555, JapanPresenter: T. Shibata (shibata@phys.aoyama.ac.jp), jap-shibata-T-abs1-og13-oralWe extend the model of three dimensional cosmic-ray propagation recently developed without the energychange to that with the energy change due to the reacceleration and the ionization energy loss. It is possible toapply the weighted slab technique for the transport equation even including the energy change process, if thediffusion coefficient 	 is separable in the energy ( 	 : rigidity) and the space, i.e., 		assumed in the present work, and combining it with the first pole approximation for the path length distribution,we can obtain the analytical solution rather easily. We find the spectral indexes of the secondary componentin rigidity are ﬁﬀ in the high energy limit, and ﬃﬂﬁﬀ in the low energy limit, while those of the primarycomponent are  !ﬀ and "ﬂ#ﬀ , respectively. This means that the rigidity dependence of the secondary-to-primary ratio behaves as 	$ in the high energy region, and as 	% in the low energy region. We compare ournumerical results with the experimental data in both the low and the high energy regions, and find our modelreproduces very nicely all components over the wide energy ranges, 1 GeV/nucleon & 100 TeV/nucleon, byadopting appropriate scale heights, and the reacceleration efficiency.1. IntroductionRecently the experimental data on the cosmic-ray (CR) spectrum and composition become available with goodstatistics and with good accuracy for the various kinds of elements in the very wide energy ranges. Particu-larly, AMS [1] and BESS [2] groups have reported excellent data on the proton and the helium component,covering the energy ranges 0.2–100 GeV/nucleon. RUNJOB [3] group reports most recently their final re-sults on the nuclear components, proton & iron, covering the very high energy region 10–500TeV for proton,2–100 TeV/nucleon for helium, and 1–30 TeV/nucleon for heavy elements, and also presents the secondary-to-primary ratio in the energy region around TeV/nucleon, while the data quality is limited due to the largeatmospheric correction.In addition to these stable nuclear components, rare elements such as the antiparticle ( '( and )+* ) and isotopes(Be ,- , Al .0/ , and so on) are also available, while the statistics and the energy ranges are still limited for thelatter elements, at most 1 GeV/nucleon.It is also remarkable that with the recent development of the techniques in both space-based and the ground-based  -ray observation, we can study quantitatively those coming from both point sources and the diffusiveones, covering very wide energy ranges MeV & TeV. It might be worth mentioning that the energy regionsfor the diffusive  -rays covered by the space-based and the ground-based experiments will overlap each othersomewhere around 100 GeV in the near future.All elements mentioned above, stable rich CR elements, rare CR elements, and  -ray component, relate eachother, and give us important information for both particle physics and astrophysics, in particular those ofantiparticle and  -rays. It is quite essential nowadays in physics to identify either these rare elements comefrom the secondary products due to the CR interactions in the Galaxy, or from some novel sources such asthose due to the annihilation products in the dark matter. Of course, in order to induce a definite conclusion forthese challenging problems, we need good experimental data with good statistics and with good accuracy, and186 Y. Futo et al.also reliable physical parameters related to the CR propagation in the Galaxy, such as the diffusion coefficient,gas density, source distribution and so forth.Under these motivations, one of the present authors (T. S.) have proposed an approach for the three dimensionalCR propagation in the Galaxy ([4]; hereafter referred to Paper I), taking a rather realistic structure of the Galaxyinto account, while our model is based on the diffusion model developed extensively by Ginzburg group [5].Namely we assume no boundary in both the radial spread of the disk and the halo space in our Galaxy, and threecritical parameters, the diffusion coefficient  , the gas density 1 and the CR source 2 have spatial distributionwith the exponential-type, while the current models have assumed rather uniform-type, typically the leaky-boxmodel. These assumptions make possible to give the analytical solution in a quite simple form despite thecomplicated process in the CR propagation in the Galaxy.In Paper I, however, we didn’t touch upon the energy change process during the CR propagation, such as thereacceleration and the ionization energy loss, focusing on the high energy region. In the present paper, westudy these effects in order to extend our model to the low energy region down to 1 GeV/nucleon and below.It is not so easy to obtain the analytical solution with the energy change effects, while rather easy in the caseof no energy change as presented by Paper I. Fortunately, however, Ptuskin et al. [6] have pointed out that wecan apply the weighted slab techniques for the diffusion problem even including the energy change effects, ifthe diffusion coefficient is separable in the energy (rigidity) and the space.Moreover, as was presented in Paper I, we found the path length distribution 345 is well described by thefirst-pole approximation, while the exact one is quite complicated, namely given by the simple exponentialfunction, 3457698;:=<>ﬂ'?A@ 4 , where '?B@ depends on the position of the observer in the Galaxy as well as thescale heights of the diffusion coefficient (see Paper I for the detail). One might think that it is nothing but theform assumed by the leaky-box model, but our model includes the spatial dependence in '? @ closely related tothe Galactic parameters, i.e., depending on the position of observer  , while the current model assumes it isindependent of the position of the observer.Combining the weighted slab techniques given by Ptuskin et al. with the first pole approximation developed byPaper I, we can obtain rather easily the analytical solution with the energy change effect. Due to the limitedspace, we don’t present explicitly the procedures of the derivation of the solution in the present paper, whichwill appear in a separate paper soon.2. Basic assumptionsWe assume that there is no boundary in both radial spread of the disk and the latitudinal spread of the halo, andthat the three critical parameters, the diffusion coefficient  , the gas density 1 , and the cosmic-ray source den-sity 2 , depend on the space position  in the form of 5 = -8;:=<C DﬁEFDHGJI KI EFKLG5M , 1N5 = 1-8;:=<COﬂ%DPEFDRQSI KI EﬁKQTM , and 2U = 2-8;:=<COﬂ%DPEVDTW XI K5I EFKLW5HM , respectively, where we omit the rigidity-dependent term forY5 and 2UA . The set of the scale heights ( D G[Z D Q\Z DHW ) correspond to the radial size of the disk, and the scaleheight of the diffusion coefficient K G to the thickness of the halo.The rigidity dependences for  and 2 are given by the following equations	N!]^	Zwith ﬀ_X`FEFabﬂ`VEﬁZ`dce2U	Nf2UA>	$AgZ with hf=ij`kﬂlmi n Z `dopwhere ] is the velocity of the particle in units of the velocity of the light.It is well known that the average energy gain and its dispersion per interaction with magnetic turbulence in theA new approach to the cosmic-ray propagation ... 187case of three dimensional scattering model are given byqTrsutrea vna].w(EV] Zqrxs.trea va].w(.Z y+where ] w is the velocity of the magnetic turbulence, and ( the particle momentum per nucleon. Thus the energygain and the dispersion per unit of time are given byzrsrb{x|reav1N>}-(	Zzrxs>.rx{~|reav1N5}-](.P	 with }-n 7].w`1 eff  effZ a^where }-corresponds to the reacceleration efficiency, and we introduced two effective parameters, 1 eff and eff, denoting the effective gas density and the effective diffusion coefficient. }-has the dimensions of thecross section, and for a typical set of parameters, ] w = 20 km/sec, 1 eff = 1 cm $/ ,  eff = `R+. cm . /sec, we find}-i a mb.The ionization energy losses per nucleon in the interstellar medium is given by the Bethe’s formula,z%s{x|ionﬂ71NT]^u( Zwith u ( nP)dŁ]..jj ]e.'>`ﬂ_].ﬂ].LZ neand 'is the mean ionization potential of the interstella medium (mostly composed of hydrogen with approxi-mately 10% helium).2.1 Transport equationLet us consider a case that a nucleus with initial kinetic energys-is produced at a position -D-ZK- in thecylindrical coordinate system. The transport equation for the CR propagation, k{ZZ-sZs- , is given by,taking account of the energy change mentioned in the last section,{ﬂuj	1NAT]?sUBrsrb{breas{bion ﬂ`.s.rs.rb{~rea =sﬂs- =Uﬂ-ﬁ\D-Zy+where ? is the inelastic collision cross section with nuclei of the interstellar gas, and ¡\ipiRi£¢ ion denotes theionization loss term.According to Ptsukin et al., in the case of separable diffusion coefficient in space and the energy (rigidity) asin the case of Eq. (1 c ), it is possible to apply the weighted slab technique by introducing a new path lengthparameter ¤ , and we can write down  for the steady state (NE{! )kZ-sZs-N9¥§¦-3¤Z->¨[¤sZs-¤Zy+where 3 and ¨ satisfy ©1N5¤ﬂYªA>u«+3¬! =¤m = ﬂ-F\D-Z­+c=©]N®d	¤?\¯s®V}-(	ﬂ_]^¯ﬂ`.s.®d}-](.ﬁ	¯«P¨h! =¤m =sﬂs-]-	-i ­^o;188 Y. Futo et al.3. Numerical results and discussionWhile it seems to be cumbersome to perform the above integration, we can obtain rather easily the CR densityin the analytical form at the Galactic plane ( K° ) with use of the results presented in Paper I. However, aswe have no space here to present the procedure of the derivation and the explicit form of the solution, we showonly two examples of numerical results on the proton and the helium components. Other examples and thecomparison with the experimental data will be presented orally in the conference.In Fig. 1, we demonstrate the energy spectra of proton and the helium component in the energy range0.1 GeV/n – 1 PeV/n, for two sets of }-, 0, and 50 mb, with the modulation parameter of ± = 0.6, 1.2GV,where experimental data are also plotted. One finds that the reacceleration effect boost the intensity consider-able around the energy region & 10 GeV/n, and }-is as large as 50 mb. Detail discussions will be presented atthe conference, including the heavier components as well as secondary-to-primary ratio.10-110010110210310410-1 100 101 102 103 104 105 106 107E 02.5 dI  dE0 [m-2sr-1sec-1(GeV/n)1.5 ]/kinetic energy :  E0 [GeV/nucleon]: SOKOL:  IMAX: JACEEprotonhelium:  BESS:  AMS : RUNJOB=  600 MV:1200 MV: 0  0 mb 50 mb = 1/3, = 2.43Figure 1. Examples of the numerical calculation on the energy spectra of the proton and helium components. See Derbinaet al. [3] and references therein for the data.References[1] J. Alcaraz et al., Phys. Lett. B, 490, 27 (2000); B494, 193 (2000).[2] S. Haino et al., Phys. Lett. B, 545, 1135 (2004).[3] V.A. Derbina et al., ApJ Letters, in press.[4] T. Shibata et al., ApJ, 612, 238 (2004).[5] V. Berezinskii et al., Astrophysics of Cosmic Rays (Amsterdam: North Holland 1990).[6] V.A. Ptsukin et al., ApJ, 465, 972 (1996).

A new approach to the cosmic-ray propagation in the Galaxy in the wide energy range 1 GeV - 1 PeV

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