Komori, Y

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29th International Cosmic Ray Conference Pune (2005) 3, 177180Propagation of Cosmic Ray Electrons Including the Source Region I :Leaky Box ModelY. KomoriKanagawa University of Human Services, Kanagawa 238-8522, JapanPresenter: Y. Komori (komori-y@kuhs.ac.jp), jap-komori-Y-abs2-og13-posterWe treat the propagation of Galactic electrons including the source region, which is currently considered to besupernova remnants(SNRs). The simplest model is formulated in which the source region is considered as aleaky box and is characterized by the escape rate of electrons into the interstellar space. It becomes clear thatthe energy index of the escape lifetime in SNRs influences the high energy part of the interstellar spectrum,and makes it possible to explain the observed data when the source spectral index is smaller than 2.3 that isexpected from the radio spectral index in SNRs. The escape lifetime of electrons in SNRs is also discussedusing the ratio of radio flux from SNRs and the Galactic background radio flux. The result shows that theelectron lifetime in SNRs is  yr around 1GeV, which corresponds to the SNR age in the Sedov phase.1. IntroductionThe supernova remnants(SNRs) are most likely sources of cosmic ray electrons because the non-thermal ra-dio and X-ray emission from SNRs indicate the existence of high energy electrons in SNRs. The averageradio index of 0.5   0.6 in SNRs[1] implies the source index of 2.0   2.2. For a global view of the cos-mic ray propagation the source region is regarded as a leaky box and distinguished from the interstellarspace in this paper. The propagation parameters in a SNR include the energy loss coefficient of electrons	 ﬁﬀﬂﬃ! #"$&%('*),+.-/+.0132,	4576*8 9:;	<=>@?in the average magnetic field2and theescape lifetime (inversion of the escape rate). The average escape rate from SNRs is estimated from the ratioof the total radio flux from SNRs to the Galactic background radio emission. Duric et al.[2] have investigatedthe escape rate in M33 and concluded that SNRs are enough to be the candidate of cosmic ray sources.2. FormulationIf each SNR is specified by the subscript A and has the source spectrum BC$D6(B;EFCﬃHGI, the electronlifetime JC$D6J*EFCﬃLKMIand the energy loss coefficientC , the electron spectrum NCsatisfies the leakybox model(LBM) equation,NC$D6JC$D618NDC$D6M?OB;C$D6P(1)and the solution[3] is given byNC$D6QSR EBC$ T6U9*VUWX8YZR [\C$	TH]6T]TH]?TCT_^B`EaCCG,bc dP1\ﬃ! C(2)where\CXeCJ*C$6means the (escape/energy-loss) lifetime ratio in the SNR( A ). The spectral index changesfrom$d1gfC6to$d1h6at the break energyC satisfying di\ﬃO C, namely JC$C6 	CC ,Cj8$dC6kClJEaC?ﬃ! .m*no @ﬃLKMI&p. The approximate expressions are estimated asNDC$D6 B;C$D6JC$D6rq ﬃXnYGIb!KIphstC(3)NDC$6 B;C$D6u	C$dC6vq ﬃ7nwGIlbX .pCsﬂ178 Y. KomoriThere are many SNRs in the Galaxy from which we suppose that Galactic electrons are injected into theinterstellar space. Thus the source spectrum in the Galaxy BDx$D6is represented by the summation of thesolution NC of Eq. (2) multiplied by the escape rate of	JyC$D6,B x$D6PSzCN C$D6J*C$D6{zCJ*C$D6`|B EaCCG I bX R ETGIMﬃ9*VUWX8P1T @ﬃLK I\C$D6$.fC6?}TThe spectral index changes fromdC to$dC13fC6at each break energyC .If the Galaxy is treated as one leaky box with the lifetime Jyx$D6~JEﬃLK#, the interstellar electron spectrumNDx$D6.also satisfies the LBM equation and is given by the solution of Eq. (2) with the source spectrum Bx .Substituting the above expression, we obtain the interstellar spectrum,NDx$D6Q R EzCNDC$	T6J C$	T69VW78w R [T]	T]\x$	TL]6?TxTizCB EaC\C$D6.xG I bX R ETR ET]TGIMﬃ! aﬃKMIT]GIﬃ|9VW!8T aﬃKMI*$1T] @ﬃLKMI6\C$D6$#fC6?9*VW78Q1T @ﬃLKMI\x$D6$#fx6?(4)^zCB EFC\C$D6kxGI/bc dCP1\ﬃO CdCfC1\ﬃO x(5)in which the energy-loss coefficient in the interstellar space is assumed to bex ﬀﬃO ."8 9:ﬃ! 9<ﬃO ?[4] and the lifetime ratio becomes\xŁeJx$D6.x. The solution is easily calculated using the numericalintegration technique.The interstellar spectrum NDx has two break energy:8$dCfC6kx,JE?ﬃO km*no aﬃK#apis given by d CfC{\ﬃ! xandC8$dC6kClJEaC?ﬃ! .m*no @ﬃLKMI&pis given by d C\ﬃO C. The break energy in the interstellar spaceis almost independent of A and has the value of3GeV when d C  ,fC   andJx$1GeV6ﬀ_yyr ( =9.4g	cm[5] for hydrogen density 7 cmﬃ>). On the other hand,C dependson the SNR type, age and so on, and is larger thanbecause ofCJCsxJx . Thus the spectrum changes asNx$D6  CBC$D6Jx$6q ﬃ7nwGIb!Kphs(6)Nx$D6 CBC$D6	x$dCfC6vqﬃ7nwGIbX .psﬂsCNx$D6  CBC$D6c	CxD$dC6$dCfC6JC$D6rq3ﬃXnYGIbﬃKIpCsﬂThe spectral index in the highest energy region depends on the SNR parameterfC and is steeper than the indexdC1of the simple LBM byfC . It is important to notice that whenfC7 ,CX ¢¡ and Eq. (4) is reducedto the simple LBM solution with the single break energy. Another point to notice is that iffCfxf, theintegral in Eq. (4) is reduced to the difference of two simple LBM solutions as£ ETTGI#ﬃ,	$\ﬃO x¤\ﬃO C6|¥9VW!8$P1T aﬃK6F	$\C$#f6.6?L¦9*VW78$P1T aﬃK6F	$\x$#f6.6M?&§3. CalculationsWe investigate the electron lifetime in SNRs from the electron number ratio of two regions: interstellar spaceand SNRs. The ratio CNC	NDx is calculated from Eq. (2) divided by Eq. (5). If major SNRs, namely shell-typeSNRs, have the same parameters of Jy¨ and¨ , the ratio is given byCNC$6Nx$D6J¨$D6Jx$D6$.P1©$df¨6k\x6Propagation of Cosmic Ray Electrons Including the Source Region I : Leaky Box Model 179At the low energy around 1GeV (vs), the Galactic parameter\x (1GeV) becomes  ªwith the valuesofx and J*x given in Sec. 2. Therefore, the electron number ratio approximates the lifetime ratio, namelyCNC$D6k	Nx$D6J¨$D6k	Jx$D6. In the case that Galactic SNRs have different types and ages, this relationis also derived from the definition of the average rate	J,¨$D6( J¨$D6QJEﬃLK#«), namely	J¨$D6eCN C$D6F	J C$D6CN C$D6If we use NDxJxCB;CJxCNDC	J*C obtained from Eq. (3) and Eq. (6), we get CNDC	NDxJ¨	Jx .If the observed radio flux ¬$®­6has the spectral index¯around the frequency­, the electron spectrumproducing this flux in the magnetic field2is expected to be N$D6#° ±nª²*p³nµ´p/²·¶¸2 ﬃ7nµ´bc #pﬃXn´bX .pwhere ¹$¯c6only includes¯and does not sensitive to¯. This formula gives the relationship between theescape lifetime ratio J ¨	J x and the radio flux ratio ¬  	¬ , in which ¬  $®­6q­Lﬃº´«is the total radio flux fromGalactic SNRs, and ¬ $&­6Pq­ ﬃH´is the whole background radio flux in the Galactic disk.J ¨Jx$#9*:6¬ $®­¨6¬$®­x6$2¨2x6ﬃ! ¬ $&­¨6¬$&­¨6$2¨2x6ﬃXnµ´·FbX .p^ |$2¨2x6ﬃXno a» "a¼X @» Fp(7)where 1GeV electrons emit the radiation of the frequency­¨ and­x in the magnetic field2¨ of SNRs, and inthe interstellar magnetic field2x respectively. In addition we used the relationship of­¨	­x2¨	2x andassume that¯¨ and¯x do not change between­¨ and­x . The Galactic radio index¯x  º½.The flux ratio ¬  	¬ is estimated in the following way. ¬  ¬O¾O¿ﬀ¨ : ¬O¾O¿ is the average flux of 51SNRs with the known distance in the Green’s catalog[1] and has the value of ¬ ¾O¿$#½¤ À6ﬀÁU #W	Hzat 1GHz. The total number of SNRs, X¨ , has the expression of X¨OE5!Â$OE 6within the distanceÂ[kpc] from the solar system as shown in Fig. 1. It is estimated from the accumulated number of 54 SNRs inGreen’s catalog[1] and of 125 SNRs in Milne’s catalog [6]. Green’s and Milne’s data sets are consistent within3kpc. 231 SNRs are currently detected[1], however we consider that almost nearby SNRs within several kpcare counted in 54 SNRs. ¬ ©Ã'$®­6$/À5X	ÄP6ﬀ(Åx©45!*Â`*Ã'$®­6:Ã'$#`ÆÇ,6º È_ﬀŁﬃFÉ{	·=kÊÆÇis the intensity in the polar direction [7][4] with the distance of the line of sight,Ä[kpc]. The volume of thepropagation regionÅx^5!ÂﬀÁÄis substituted. The flux ratio finally has the following expression and thevalue.¬ ¬$.`ÆÇ,6¬!¾O¿ﬀE45!Ã' ﬀﬃAt the different frequency of 408MHz, the average flux ¬X¾O¿$&4Ëµ½6ﬀ #É{	·ÆÇ[8] and the Galacticradio fluxÃ'$lÀ4ÌÁÆÇy6½U ﬀ¦yﬃF@É{	·=kÊÆÇgive the ratio ¬  	¬$lÀ4ÌÁÆÇ6 Èﬀﬃ.The value of2¨ is not well known, however, the strength for several SNRs are investigated from the measure-ments of X-ray and gamma ray emissions. In the case of typical shell-type SNR, SN1006, HESS has recentlyreported the lower limit of 25 Í G[9]. The arguments of particle acceleration using the X-ray morphology re-quire 10-100 Í G[10][11]. As2¨ represents the value in radio emission region, it may be smaller than that inacceleration site. Since 80% of Galactic SNRs are shell-type,2¨ seems to be below 100 Í G and2¨	2xÎy.If the magnetic field ratio is2¨	2xy, Eq. (7) gives the lifetime ratio Jy¨	Jx$®ÏS6ﬀÐﬃH. Thus themean lifetime of 1GeV electrons in SNRs becomes J·¨ﬀŁyr as the lifetime J*x_ﬀËyyr. Conse-quently, the electrons around 1GeV stay in SNRs for the rather long period of  Uyr which corresponds tothe SNR age in the Sedov phase.Using the result of Eq. (7), we compare the interstellar solution of Eq. (4) with the observed data. The curvesof Eq. (4) with different energy indexf¨ are shown in Fig. 2. It is clearly shown that the value off¨ influencesthe high energy spectrum and is consistent with the ECC data [12] whenf¨Áª$d6. These smallvalues off¨ indicates that the high-energy electrons in SNRs are intensively confined.180 Y. KomoriÑÑ#ÒÑ#ÒaÓÑ#ÒkÔÕ Ö×Ø Ù ÚÛÜ ÝÞØ ßàááÒâ Ñ Ñ Ñ#ÒÑ#Ò Óã(äµåæ,ç#èéêHëÑaì íaîï Ó ðFigure 1. Accumulated number of SNRs within the dis-tance ñ from the sun. Two data sets are given: the thicklineis made up of 54 SNRs in Green’s catalog [1], the thinlineis early data set of 125 SNRs in Milne’s[6]. Both data setsare consistent within 3kpc. The regression line is assumedto be proportional to ñò .ó*ô õ}ö ö ÷Yöµø ùúû#ükü@ý þªß}ß Fü®ý 	/þYõªö  ý þªß}ß}ß ﬀﬁﬂﬁ ﬃﬃ !" #$% &'%)(#*(+,-     . /0214365 798:9;=<>@?BA CEDGFCIH?JD?LKMNOQPSRUTVWJH?@KM)XOQP2Y[Z \[V]W^D_ 2_ 2_ 2_ `abaW/cd e`f8Lg /WeLhb.`idfkj^D?bA Figure 2. Calculated local spectra from the model with dif-ferent values of lnm . Those lines are compared with the ob-served data[12]. The best t curve has the break energy ofoqpsrutnv GeV, and oxw@rzy{v{v GeV. The dashed line normal-ized at 10GeV shows the approximate expression of Eq (5).4. ConclusionsWe have discussed the propagation of Galactic electrons treating each SNR as a leaky box. This is the globalmodeling of electron propagation including the source region. The key parameter, the electron escape lifetimein SNRs has been estimated from the radio flux from SNRs and the Galactic background radio emission. Ifthe magnetic field in SNRs is nearly 10 times the strength of the Galactic magnetic field, the lifetime of 1GeVelectrons in SNRs becomes  yyr, which corresponds to the SNR age in the Sedov phase. The escapelifetime from SNRs influences the interstellar electron spectrum above a few hundred GeV. The observed dataindicate that the energy index of the escape rate is small, if the source spectral index is  that isexpected from the average radio index of SNRs.References[1] D.A. Green, http://www.mrao.cam.ac.uk/surveys/snrs/, 2004[2] N. Duric et al., ApJ, 445, 173, 1995[3] R.F. Silverberg, R. Ramaty, Nature, 243, 134, 1973[4] Y. Komori, 28th ICRC, 4, 1997, 2003; 28th ICRC, 7, 3847, 2003[5] V.S. Ptuskin et al., 27th ICRC, 1947, 2001[6] D.K. Milne, Aust. J. Phys. (32) 83, 1979[7] W.R. Webber et al., ApJ., 236, 448, 1980; J.D. Peterson, et al., 26th ICRC, 4, 251, 1999[8] D.H. Clark and F.L. Caswell, Mon. Not. Roy. Astron. Soc., 174, 267, 1976[9] F. Aharonian, et al., A&A, 437, 135, 2005[10] R. Yamazaki, et al., A&A, 416, 595, 2004[11] Berezhko, E.G., et al. A&A, 412, L11, 2003[12] T. Kobayashi, J. Nishimura, Y. Komori et al., 26th ICRC, 3, 61, 1999

Propagation of Cosmic Ray Electrons Including the Source Region I: Leaky Box Model

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