A possible approach to three-dimensional cosmic-ray propagation in the Galaxy IV. Electrons and electron-induced gamma-rays
aa r X i v : . [ a s t r o - ph . H E ] O c t A POSSIBLE APPROACH TO THREE-DIMENSIONALCOSMIC-RAY PROPAGATION IN THE GALAXY. IV.ELECTRONS and ELECTRON-INDUCED γ -RAYS(To appear in ApJ, December 2010) T. Shibata, T. Ishikawa, and S. Sekiguchi
Department of Physics and Mathematics, Aoyama-Gakuin University, Kanagawa 229-8558, Japan
ABSTRACT
Based on the diffusion-halo model for cosmic-ray (CR) propagation, includ-ing stochastic reacceleration due to collisions with hydromagnetic turbulence,we study the behavior of the electron component and the diffuse γ -rays (D γ ’s)induced by them. The galactic parameters appearing in these studies are es-sentially the same as those appearing in the hadronic CR components, whilewe additionally need information on the interstellar radiation field, taking intoaccount dependences on both the photon energy, E ph , and the position, r . Wecompare our numerical results with the data on hadrons, electrons and D γ ’s,including the most recent results from FERMI, which gives two remarkable re-sults; 1) the electron spectrum falls with energy as E − e up to 1 TeV, and does notexhibit prominent spectral features around 500 GeV, in contrast to the dramaticexcess appearing in both ATIC and PPB-BETS spectra, and 2) the EGRETGeV-excess in the D γ spectrum is due neither to an astronomical origin (muchharder CR spectrum in the galactic center) nor a cosmological one (dark matterannihilation or decay), but due to an instrumental problem. In the present paper,however, we focus our interest rather conservatively upon the internal relationbetween these three components, using common galactic parameters. We findthat they are in reasonable harmony with each other within both the theoreticaland experimental uncertainties, apart from the electron-anomaly problem, whilesome enhancement of D γ ’s appears in the high galactic latitude with | b | > ◦ inthe GeV region. Subject headings: cosmic rays — Galaxy: structure — electrons: diffuse back-ground 2 –
1. Introduction
Although the electron component is only a small fraction of all cosmic-ray (CR) compo-nents, around 1% of the proton intensity around 10 GeV, it plays a key role in understandingthe structure of our Galaxy and the galactic phenomena occurring within it. This is becauseelectrons have electromagnetic interactions with the interstellar radiation field, such as pho-tons and magnetic fields, resulting in drastic energy loss during propagation through theGalaxy, in contrast to the hadronic component.This peculiar nature yields valuable information for the study of CR astrophysics, whichcan not be obtained by the hadronic components alone. Namely, due to the rapid energy-lossrate, proportional to E e in the high energy region, from the inverse Compton scattering offphotons and synchrotron radiation in magnetic fields, the life-time of TeV electrons is at most10 yr, indicating that detected electrons have originated in nearby sources, less than 1 kpcfrom the solar system (SS). Therefore, accurate observations of TeV electrons will providea direct signature of nearby CR sources as well as the mechanism of the CR acceleration,while depending on the release time from supernova remnants and their distance from theSS. Qualitative studies of such a possibility have been performed by many authors (Shen1970; Nishimura et al. 1979; Cowsik & Lee 1979; Berezinskii et al. 1990; Aharonian et al.1995; Ptuskin & Ormes 1995; Pohl & Esposito 1998; Kobayashi et al. 2004; Delahaye etal. 2010), with Kobayashi et al. and Delahaye et al. presenting explicitly several candidatesfor nearby sources of high energy CR electrons, based on the most recent data for the ageand distance of each supernova remnant near the SS, although the statistics of high energyelectron data are currently too poor to identify sources definitely.Particle identification and the energy determination of high energy electrons is, however,quite difficult, while direct observation of low energy electrons is relatively easy using, forinstance, magnetic spectrometers, and has been performed by several groups (Golden et al.1994; Boezio et al. 2000; DuVernois et al. 2001; Aguilar et al. 2002).Although the statistics are not sufficient, the only group that succeeded in observing directly TeV electrons is Nishimura et al. (1980; see also Kobayashi et al. 1999) with the useof the balloon-borne emulsion chamber. It should be noted that they actually observe eventby event the vertex point of the electron with subsequent e ± -pair due to bremsstrahlung γ ,with no uncertainty from proton contamination. The precision in the energy determinationis approximately 10% for electrons in the energy region larger than 50 GeV, based on boththe three-dimensional cascade theory (Nishimura 1964) and the simulations (Kasahara 1985;Okamoto & Shibata 1987), which have been well established by the use of accelerator beams 3 –(Hotta et al. 1980; Sato & Sugimoto 1979).Recent development in high energy electron observations is indeed remarkable, partic-ularly those of ATIC (Chang et al. 2008) and PPB-BETS (Torii et al. 2006), which showedan anomaly in the electron spectrum with a significant bump around 500 GeV. Both groupspoint out that the excess indicates either a nearby source of energetic electrons, or thosecoming from the annihilation of dark matter particles.On the other hand, the most recent results obtained by the FERMI Large Area Telescope(FERMI-LAT; Abdo et al. 2009) present no prominent excess, with the electron spectrumfalling with energy as E − . e up to 1 TeV, which is not inconsistent with the emulsion cham-ber data (Kobayashi et al. 1999) within the statistical errors. The H.E.S.S. ground-basedtelescope (Aharonian et al. 2008, 2009) also shows no indication of structure in the electronspectrum, but rather a power-law spectrum with E − . ± . ± . e (0.1: stat. error, 0.3: syst.error), albeit this being an indirect observation.Nevertheless, looking carefully FERMI data around the anomaly-energy, they still showsystematically an enhancement as large as 30% compared to the numerical results (Abdo etal. 2009; Strong et al. 2004; see also Figure 14 in this paper), so that we can not excludethe possibility of an additional component such as local sources and/or the dark matterscenario, while strength of the anomaly compared to the background diffuse electrons is notas dramatic as presented by ATIC and PPB-BETS.In any case, both observational and theoretical studies for high energy electrons arebecoming increasingly important not only for astrophysics, but also for particle physics andcosmology. It is, therefore, desirable to find a reasonable model for electron propagationin the Galaxy, which must explain consistently and simultaneously all CR observables andnot just electrons, using common galactic parameters with the smallest number of variablespossible. In the sense, the recent review article by Strong, Moskalenko, & Ptuskin (2007)is a useful survey of both the theory and relevant experimental data for the propagation ofCRs, comprehensively summarizing the current landscape and open questions, although itwas published just before the anomaly problem mentioned above.Under these situations, we have studied the three-dimensional CR propagation modelanalytically, and found excellent agreement with the experimental data for various hadroniccomponents, stable primaries, secondaries such as boron and sub-iron elements ( Z =21–23),isotopes such as Be, and antiprotons as well, in four papers, (Shibata et al. 2004, 2006,2007a, 2008), hereafter referred to as Papers I, II, III and IV, respectively.We have applied our model further to the studies of diffuse γ -rays (D γ ’s) (Shibata,Honda, & Watanabe 2007b; hereafter Paper V), and found that all these components are 4 –generally in agreement with each other using the same galactic parameters, within the uncer-tainties in the experimental data and various kinds of cross-sections used for the numericalcalculations. However, in Paper V, we use the simulation results for electron-induced γ -raysprovided by Hunter et al. (Bertsch et al. 1993; Hunter et al. 1997), where the modelingof CR propagation and the galactic parameters assumed are somewhat different from ours.So we have yet to see complete internal consistency among all CR components — hadrons,electrons and D γ ’s — using the same galactic parameters in our propagation model.In the present paper, we extend it to the electron component, based on the diffusion-halomodel proposed by Ginzburg, Khazan & Ptuskin (1980), taking the reacceleration processinto account. However, we focus in the present work on diffuse electrons in the steadystate without discriminating those produced by nearby sources from those of distant ones,and present the intensity of the D γ ’s produced by them in the energy range, E γ = 30 MeV–100 GeV, covered by EGRET and FERMI. Comparison with radio and TeV- γ data will bereported separately in the near future.In order to apply our model to the electron component and electron-induced D γ ’s, weneed information on the interstellar radiation field (ISRF) in addition to the interstellar mat-ter (ISM), particularly their spatial gradients for the study of the ( l, b )-distribution of D γ ’s( l : galactic longitude; b : galactic latitude). Nowadays the most advanced and standard codefor the ISM and ISRF models is GALPROP, extensively developed by Strong & Moskalenko(1998), incorporating the latest survey data in the very wide wavelength range from ultra-violet to radio. In the present work, we assume empirical density distributions for the ISMand ISRF, smoothing the numerical data given by GALPROP available most recently (Porteret al. 2008), in order to combine with our analytical solution for electron-induced D γ ’s.In §
2, we discuss the interstellar environment provided by GALPROP, focusing onthe spatial distribution of both matter (atomic, molecular, and ionized hydrogen) and pho-tons (ultraviolet, visible, infrared, mid- and far-infrared, and cosmic microwave background[CMB] radiations), and in § §
4, we present the diffusion equation, and give its solution explicitly in the steadystate, N e ( r ; E e ), where the Klein-Nishina effect is quite important in the electron energyspectrum in the high energy region, > ∼
10 GeV. In §
5, we present the emissivity of electron-induced γ ’s, q γ ( r ; E γ ), with use of realistic spatial distributions of ISM and ISRF as discussedin §
2, and show the numerical results at several observational points, where those of thehadron-induced γ ’s are presented as well. In §
6, we first summarize the galactic parametersand their explicit values expected from the CR data, and then compare our numerical results 5 –of electron flux and D γ ’s with recent observational data, including those most recently ob-tained by FERMI and H.E.S.S. Finally in §
7, we summarize the results, and discuss severalremaining open questions, while we do not touch upon the so called electron-anomaly.
2. Interstellar environment of our Galaxy2.1. Interstellar matter
First we consider the ISM for two processes, ionization and bremsstrahlung. In Figure 1we plot histograms of column density for H I and H in the galactic plane (GP) given byGALPROP, where we also plot the empirical curves used in the present work, − ln ρ HI ( r ) = P (0) HI + P (1) HI r + P (2) HI ln r + P (3) HI r , (1a) − ln ρ H ( r ) = P (0) H + P (1) H r + P (2) H ln r + P (3) H r , (1b)with r in kpc, and ρ h (“ h ” ≡ H I, H ) in 10 H atoms cm − . The numerical values of thecoefficients are summarized in Table 1. However, the choice of above empirical form is notcritical, and other choices may be possible.The H gas is strongly confined to the GP and its vertical structure is modeled by agaussian distribution with a width of approximately 70 pc, while the H I gas lies in a flatlayer with a FWHM of 230 pc in 3.5 kpc < r < r ⊙ (=8.5 kpc), and is approximated by thesum of two gaussians and an exponential tail (Ferriere 2001; Moskalenko et al. 2002). Taking -1 r (kpc) c o l u m n d e n s i t y ( H a t o m s c m - ) H gas HI gas : GALPROP code: our empirical (( Fig. 1.— Column density of interstellar hydrogen. Curves are empirical ones given byequation (1) with the parameterization summarized in Table 1. 6 –Table 1. Summary of the numerical values of the coefficients appearing in equations (1a)and (1b) in units of 10 H atoms cm − , where “( ± m )” denotes the multiplication of 10 ± m . P (0) HI P (1) HI P (2) HI P (3) HI P (0) H P (1) H P (2) H P (3) H . . − − . − − . . . − − . . − Table 2. Summary of functions for ISM gas density, H I and H , where r , r ⊙ (=8.5kpc), z ,and z are all in units of kpc, and n ⊙ h in units of H atoms cm − . “ h ” n ⊙ h Ξ h ( r, z )H I 0.57 10 . √ π + 0 . ( . (cid:20) − (cid:16) z . (cid:17) (cid:21) + 0 . (cid:20) − (cid:16) z . (cid:17) (cid:21) + 0 . (cid:16) − z . (cid:17)) H . √ π + 0 . z ( exp (cid:20) − (cid:16) z . (cid:17) (cid:21) + 0 . zz exp (cid:16) − zz (cid:17)) ; z ( r ) = 0 . (cid:16) r r ⊙ (cid:17) n h ( r ) n ⊙ h = Ξ h ( r, z ) Ξ h ( r ⊙ , ρ h ( r ) ρ h ( r ⊙ ) , (“ h ” ≡ H I, H ) , (2)where n ⊙ HI ( n ⊙ H ) is the gas density of H I (H ) at the SS with typically n ⊙ HI ≈ n ⊙ H ≈ .
5H atoms cm − . See Table 2 for the explicit forms of Ξ HI and Ξ H .For the ionized hydrogen gas, H II, we use the two-component model of Cordes et al.(1991), n HII ( r ) = n (1) HII ( r ) + n (2) HII ( r ) , and both components are modeled by a gaussian-typedistribution for the radial structure, and by a simple exponential one for the vertical struc-ture. The explicit values of the two components at the SS, [ n (1) HII ( r ⊙ ) , n (2) HII ( r ⊙ )], are [0.025,0.013] cm − respectively (Cordes et al. 1991; Strong et al. 1998). So the contribution of H IIis much smaller than those of H I and H and is not important in the present work. First we consider the medium — virtual photons induced by the static magnetic field— for the synchrotron process. It is approximately given by an exponential-type gradient,while the scale height is not yet clear. Practically, for the study of synchrotron radiation,we need the energy density of virtual photons at r , ǫ B ( r ), and assume in the present work ǫ B ( r ) = ǫ B , exp[ − ( r/r B + | z | /z B )] , (3)with ǫ B , = B / π, where B is the magnetic field at the galactic center (GC), and ǫ B , is its energy density, forinstance ǫ B , ≈ − for B = 6 µ G, and typically [2 r B , z B ] ≈ [10, 2] kpc (Strong et al.2000).On the other hand, the photon gas for the IC process is somewhat different from thosediscussed above. Namely, we need the number density of the photon gas in the ISRF, n ph ( r ; E ph ), as a function of the target photon energy E ph at r . Separating it into twoparts, a r -dependent energy-density term, ǫ ph ( r ), and a r -independent term, W ph ( k ) with k = E ph / [ k B T ph ], we rewrite n ph ( r ; E ph ) as 8 – E ph n ph ( r ; E ph ) dE ph = ǫ ph ( r ) W ph ( k ) d ln k, (4)where k B is the Boltzmann constant, and T ph is the characteristic temperature of the ISRF.There are three main radiation sources in the photon gas, (i) the 2.7 K CMB radiation,(ii) stellar radiation with wavelengths of 0.1–10 µ m (ultraviolet–visible–near-infrared), and(iii) re-emitted radiation from dust grains at 10-1000 µ m (mid-to-far–infrared).We classify them further into six wavelength bands, each labeled with i = 0 for (i), i = 1,2, 3 for stellar-1, -2, -3 in (ii), and i = 4, 5 for dust-1, -2 in (iii) (see Fig. 2). Needless tosay, there is no spatial gradient in the CMB ( i = 0), which is distributed uniformly in space, ǫ ph ( r ) ≡ ǫ (0)ph = 0 .
261 eVcm − , and the normalized spectrum, W (0)ph ( k ), is given by the familiarPlanck formula with T ph = 2.73 K.On the other hand, for (ii) and (iii) in the wavelength range λ = 0.1–1000 µ m, the energydensity, ǫ ( i )ph ( r ) ( i = 1–5), must depend on r , and W ( i )ph ( k ) is unlike the simple CMB spectrum,and is very complicated. In the following discussions, we often omit the suffix i for simplicityunless otherwise specified.In the present work, we assume a gaussian-type distribution in ln k for W ph ( k ), W ph ( k ) = 1 √ πσ e − (ln k ) / (2 σ ) ; k = λ /λ, (5)so that the mean radiation intensity, I ph , is given by4 πλI ph ( r ; λ ) cǫ ph ( r ) = 1 √ πσ exp (cid:20) − [ln( λ /λ )] σ (cid:21) , (6)where λ is the peak wavelength for each radiation with k B T ph = 2 πc ~ /λ .In Figure 2, we present examples of the mean radiation intensity (multiplied by 4 πλ ) forthe maximal metalicity gradient ( filled symbols ) and no metalicity gradient ( open symbols )at two radial distances, r = 0 ( squares ) and 8 kpc ( circles ) in the GP given by GALPROP,where also drawn are curves expected from the right-hand side of equation (6) for W ph ( k ),assuming ǫ ph ( r ) = ǫ ph , exp[ − ( r/r ph + | z | /z ph )] , (7)for r ≥ λ ( i )0 , T i , σ i ; ǫ ( i )ph,0 ] ( i = 0–5) with r ph = 3.2 kpc irrespective of the population i , andalso presented are those of ǫ ( i )ph ( r ) for r ≤ r except ǫ (2)ph ( r ).Let us demonstrate the energy density separately for the stellar and the dust radiation, P i =1 ǫ ( i )ph and P i =4 ǫ ( i )ph respectively, against the galactocentric distance r in Figure 3, where 9 – -4 -3 -2 -1 -1 pl I ph ( l , r ) i n [ er g c m - s - ] wavelength; l ( m m) stellar dust CMB r :: 0 (GC): 8 kpc (GALPROP code) i = i = = = = = Fig. 2.— Interstellar radiation field (ISRF) at two galactocentric distances obtained byGALPROP, r = 0 (GC; square symbols ) and 8 kpc (near SS; circle symbols ). Open markscorrespond to maximum metalicity gradient, and filled ones to the minimum metalicitygradient. Dotted curves are given by equation (6) with parameters summarized in Table 3for each population i , while the solid ones are those superposing them. CMB radiation ( solidcurve ) is also shown for reference.we plot also numerical data given by Mathis et al. (1983; filled grey symbols ). Two curves forthe stellar emission and the dust re-emission are drawn by the use of the parameterizationsummarized in Table 3, where we do not take the difference in the choice of metalicitygradient into account, as it is effective only near the GC for the dust re-emission and isapproximately one order of magnitude smaller than the stellar radiation.For the latitudinal scale height, z ph , in equation (7), we assume z ph ≈ r ph / r are not sufficient to construct a reliablemodel.
3. Energy loss and gain3.1. Energy loss in ISM and ISRF
The energy loss processes for the electron component are dramatically different fromthose for the hadronic components, with four main processes: bremsstrahlung ( ≡ “rad”), 10 – -1 e n er gy d e n s i t y o f ph o t o n ga s [ e V c m - ] galactocentric distance r (kpc) CMBexp[- r /(3.2kpc)] stellar (GALPROP code): maximum metalicity grad.: minimum metalicity grad.: Mathis et al. parametrization dustexp[- r /(2.0kpc)] Fig. 3.— ISRF energy density as a function of galactocentric distance r at z = 0 for thestellar radiation and the re-emission from dust grains, each with the maximum metalicitygradient ( filled black circles ) and minimum metalicity gradient ( open circles ), where alsoshown are those obtained by Mathis et al. (1983) ( filled grey circles ). Solid curves are theempirical ones obtained by equation (7).ionization ( ≡ “ion”), synchrotron and IC (together ≡ “sic”). For the bremsstrahlung (Koch& Motz 1959; Gould 1969; Ginzburg 1979), − E e D ∆ E e ∆ t E rad ≃ n ( r ) w rad ( E e ) h O (cid:16) n HII n (cid:17)i , (8)with n ( r ) = n HI ( r ) + n HII ( r ) + n H ( r ) , (9)where w rad ( E e ≫ m e c ) ≡ w ( ∞ )rad ≈ . × − cm s − , independent of E e with the completescreening cross-section in the high energy region; see Appendix A for the explicit forms of w rad ( E e ), and § n h , (“ h ” ≡ H I, H II, H ).Similarly for the ionization, we use the Bethe-Bloch formula (Ginzburg 1979) − D ∆ E e ∆ t E ion ≃ n ( r ) w ion ( E e ) h O (cid:16) n HII n (cid:17)i , (10)with w ion ( E e ) = w (0)ion n ln[ E e / GeV] + 13 . o , and w (0)ion = 0 . × − cm s − . 11 – Electron energy; E e (GeV) : interstellar radiation: reemission from dust: 2.7 K CMB radiationThomson limit all integrated : synchrotron + IC'sall IC's synchrotron DE e D t ph ( r , ; E e ) [ - G e V - s - ] E e Fig. 4.— Energy losses per unit time of CR electrons in ISRF at SS ( r ⊙ = 8.5 kpc), shownseparately for four components, synchrotron ( heavy dotted line ), IC’s by stellar radiation( broken curve ), by re-emission from dust grains ( dotted curve ), and by CMB ( broken dottedcurve ), together with the sums, IC-all ≡ IC-stellar + IC-dust + IC-CMB ( thin solid curve ),and synchrotron + IC-all ( heavy solid curve ).On the other hand, the energy losses due to the synchrotron (abbreviated as “SY” forsubscripts appearing in the following equations) and IC are rather complicated, in additionto the energy dependent cross-section of the Klein-Nishina formula, D ∆ E e ∆ t E sic = D ∆ E e ∆ t E SY + D ∆ E e ∆ t E IC , (11)where − w T E e D ∆ E e ∆ t E SY = ǫ B ( r ) , (12a) − w T E e D ∆ E e ∆ t E IC = X i =0 ǫ ( i )ph ( r ) Λ ( E e , T i ) , (12b)with w T = 1 . × − cm s − . See § T i ; ǫ ( i )ph ( r )] ( i = 0–5), and Λ ( E e , T i )is given by equation (A6) in Appendix A, which comes from the Klein-Nishina cross-section.In Figure 4, we present the energy loss divided by E e at the SS against E e separately forindividual (virtual) photon fields as well as for superposed ones, −h ∆ E e / ∆ t i ⊙ sic /E e , wherewe assume B ⊥ = 5 µ G, corresponding to ǫ ⊙ B = 0.93 eVcm − , for the magnetic field, and use ǫ ( i )ph ( r ⊙ ) presented in the second line from the bottom of Table 3 with r = r ⊙ for the photongas field. 12 – -1 Electron energy; E e (GeV) [ G e V s - ] DE e D t ( r , ; E e ) : synchrotron + IC without K-N effect n = atoms cm -3 for ion., EB, rea. ionizationreaccelerationbremsstrahlungsynchrotron + IC : empirical synchrotron + IC - - - - - - - e = cm -3 for synchrotron + IC Fig. 5.— Energy losses per unit time of CR electrons in ISRF and ISM at SS( r = r ⊙ ) as a function of electron energy, shown separately for four processes, syn-chrotron + IC, bremsstrahlung, reacceleration, and ionization. We present three curves forsynchrotron + IC, with the solid one from the Klein-Nishina cross-section, the broken onefrom the Thomson cross-section, and the dotted one from the empirical one.For E e < ∼ Λ ( E e , T i ) ≈
1, i.e., the Thomson cross-section is valid, so that equation(11) is separable in r and E e , leading to a simple expression, ǫ ( r ) w T E e , with ǫ ( r ) = ǫ B ( r ) + P i =0 ǫ ( i )ph ( r ). In practice, we find it is well reproduced by the following form over a wideenergy range − D ∆ E e ∆ t E sic ≃ ǫ ( r ) w T E − δe ; δ = 0 . , (13)while ǫ ( r ) depends on E e very weakly.In Figure 5 we demonstrate the energy loss of individual processes separately, thosedue to “rad”, “ion” and “sic” at the SS against the kinetic energy of the electron E e with ǫ ⊙ = 2 eVcm − and n ⊙ = 1 H atoms cm − , where we plot the above empirical relationship (13)( dotted curve ) and the energy gain due to the reacceleration ( ≡ “rea”; see next subsection)together. One finds that it reproduces satisfactorily the exact one (11) with equations (12a)and (12b). In Paper II, we present the energy gain per unit time due to the reacceleration1 E e D ∆ E e ∆ t E rea = n ( r ) w rea [ E e / GeV] − α , (14) 13 –with w rea = cζ ; ζ ≈ v M n ∗ cD ∗ , (15)where w rea = 15.0 × − cm s − in the case of, for instance, ζ = 50 millibarn (mbarn), corre-sponding to the choice of a parameter set with v M = 20–30km s − (Alfv´en velocity), n ∗ = 0.06–0.14H atoms cm − , and D ∗ = 2 × cm s − . The smallness of the gas density with n ∗ ≪
1H atoms cm − indicates that the reacceleration process occurs even at some distance fromthe GP.The fluctuation in the energy gain due to the reacceleration is given (Gaisser 1990;Paper II) by 1 E e D ∆ E e ∆ t E rea = 12 n ( r ) w rea [ E e / GeV] − α . (16) As discussed in the last two subsections, we have the total average energy-loss and theenergy-gain per unit time − D ∆ E e ∆ t E all = n ( r ) W n ( E e ) + ǫ ( r ) W ǫ ( E e ) , (17)where W n ( E e ) = w rad ( E e ) E e + w ion ( E e ) − w rea E − αe , (18a)and W ǫ ( E e ) ≃ w T E − δe ; δ = 0 . . (18b)One might note from equation (18a) that there exist two energies, E − c and E + c , at whichthe first term proportional to n ( r ) in the right-hand of equation (17) becomes null. In Figure6, we demonstrate W n ( E e ) against several choices of ζ , and find [ E − c , E + c ] ≈ [0.1, 7] GeV inthe case of ζ = 50 mbarn. Namely, the synchrotron-IC is dominant for E e > ∼ E + c , while thereacceleration is effective for E − c < ∼ E e < ∼ E + c , and the ionization for E e < ∼ E − c .As discussed in §
2, the total number density of the ISM gas, n ( r ), and the total energydensity of the ISRF, ǫ ( r ), have complicated spatial distributions coming from local irregu-larities, which are not yet well established. On the other hand, in our previous papers, wehave assumed a simple exponential-type form for n ( r ), smearing out the local irregularities,¯ n ( r ) = ¯ n exp[ − ( r/r n + | z | /z n )] , (19) 14 – -1 E e (GeV) [ - c m s - ] z = 50 mb40 mb w r a d ( E e ) E e + w i o n ( E e ) - w re a E e - a
45 mb 55 mb
Fig. 6.— Numerical values of W n ( E e ) ≡ w rad ( E e ) E e + w ion ( E e ) − w rea E − αe against E e for ζ = 40–55 mbarn, where the screening effect for w rad is taken into account.greatly simplifying the complicated distributions given by equations (1), (2) with Tables 1,2, where ¯ n is the ( interpolated ) average gas density with approximately 1.5 H atoms cm − atthe GC, and [ r n , z n ] ≈ [20, 0.2] kpc.In spite of such a simplification, we have found that our model reproduces remarkablywell the experimental data on hadronic components . This tells us that charged CR com-ponents are well mixed during their propagation in the Galaxy over a residence time ofapproximately 10 yr, effectively smearing the local inhomogeneous structure of the ISM. Infact, it is well established that the anisotropy amplitude of CRs is of the level of at most 10 − at energies of 1–100 TeV (Sakakibara 1965; Nagashima et al. 1989; Cutler & Groom 1991).This is the reason why even the simplest leaky-box model and/or the simplified diffusionmodel such as, for instance, constant gas density and constant diffusion coefficient withoutspatial gradient, reproduces the CR hadronic components so well (Berezinskii et al. 1990).Now, corresponding to the simplification (19) for n ( r ), we assume the following simpleexponential type form for ǫ ( r ) as well¯ ǫ ( r ) = ¯ ǫ exp[ − ( r/r ǫ + | z | /z ǫ )] , (20)where ¯ ǫ is the ( interpolated ) average energy density of the ISRF at the GC, and two param-eters, r ǫ and z ǫ , correspond to the scale heights for the spatial gradients, almost independentof the energy. Typically [¯ ǫ ; r ǫ , z ǫ ] ≈ [16 eVcm − ; 4 kpc, 0.75 kpc] (Ishikawa 2010).However, while the simplifications given by equations (19) and (20) are applied for elec-trons (and hadrons), we stress here that those presented in § γ ’sas discussed in §
6, namely n h ( r ) with “ h ” ≡ H I, H II, H for n ( r ), and ǫ ( i )ph ( r ) ( i = 0–5) for 15 – ǫ ( r ) with weak energy dependences in ǫ ( i )ph,0 as presented in Table 3. This is because D γ ’sproduced by CR hadrons and electrons are directly affected by the environment of ISM andISRF around the birth site of the produced γ ’s.
4. Diffusion equation for electron component4.1. Basic equation
The transport equation for the electron density, N e ( r ; E e , t ), is given by (Berezinskii etal. 1990), h ∂∂t − ∇ · D ( r ; E e ) ∇ + ∆ E i · N e ( r ; E e , t ) = Q ( r ; E e , t ) , (21)with ∆ E = ∂∂E e (cid:28) ∆ E e ∆ t (cid:29) all − ∂ ∂E e (cid:28) ∆ E e ∆ t (cid:29) rea , (22)see equations (19) and (20) for the average energy-loss (-gain) in the all processes, withthe replacement of [ n ( r ) , ǫ ( r )] in equation (17) by [¯ n ( r ) , ¯ ǫ ( r )], and equation (16) for thefluctuation of the energy gain in the reacceleration process respectively. For the diffusioncoefficient and the source spectrum, we assume (note v ≈ c , and R e ≈ E e ) D ( r ; E e ) = E αe D ( r ) , Q ( r ; E e , t ) = E − γe Q ( r ; t ) , (23)with D ( r ) = D exp( r/r D + | z | /z D ) , (24 a ) Q ( r ; t ) = Q ( t ) exp[ − ( r/r Q + | z | /z Q )] . (24 b )In Table 4, we summarize parameters related to the scale heights, r D , z D , . . . , which oftenappear in the present paper.Now, remembering W ǫ ( E e ) ≫ W n ( E e ) in the high energy (HE) region, say, E e > ∼ E + c ( ≈ E e < ∼ E + c , the energy loss given byequation (17) is written as − (cid:28) ∆ E e ∆ t (cid:29) all ≃ ¯ ǫ ( r ) W ǫ ( E e ) + O [¯ n ( r ) W n ( E e )]; E e > ∼ E + c , (25a)¯ n ( r ) W n ( E e ) + O [¯ ǫ ( r ) W ǫ ( E e )]; E e < ∼ E + c , (25b) 16 –Table 3. Summary of the numerical values of [ λ ( i )0 ( µ m), T i (K), σ i ], and those of ǫ ( i )ph,0 (eVcm − ) for r ≥ ǫ ( i )ph (eV cm − ) for r ≤ λ (5)0 , T (K), and σ have weak r -dependence as shownin remarks with r in kpc, while ǫ ( i )ph are independent of r except ǫ (2)ph . The numerical value of13.9 in ǫ (2)ph corresponds to the energy density of the stellar radiation for the population i = 2 at GC (see Figure 2). See also equation (7) for ǫ ( i )ph,0 and ǫ ( i )ph ( r ) for r ≥ i = 0 i = 1 i = 2 i = 3 i = 4 i = 5 remarks ( r -dependence) λ ( i )0 . . −
1) 1 . . . . λ (5)0 ( r ) = 90e r/ . T i . . . . . . T ( r ) = 160e − r/ . σ i ———— 3 . −
1) 6 . −
1) 2 . −
1) 2 . −
1) 4 . −
1) ; σ ( r ) = 0 . r/ . ǫ ( i )ph,0 . −
1) 5 . −
1) 7 . . −
1) 3 . −
1) 4 . ǫ ( i )ph ( r ) = ǫ ( i )ph,0 e − r/ . ǫ ( i )ph . −
1) 2 . −
1) 1 . . −
1) 1 . −
1) 1 . ǫ (2)ph ( r ) = 13 . − r/ . Table 4. Summary of parameters often appearing in our propagation model, classifyingthem into two groups, one related to the gas density of ISM, ¯ n ( r ), and another to theenergy density of ISRF, ¯ ǫ ( r ). Parameters for ISM Typical values for ISM Parameters for ISRF Typical values for ISRF ν = 11 + z D /z n ν = 0 . − . κ = 11 + z D /z ǫ κ = 0 . − . U ν = 2 p ν + ν U ν = 0 . − . U κ = 2 p κ + κ U κ = 0 . − . r n = 12 (cid:18) r D + 1 r n (cid:19) ¯ r n = [20 − r ǫ = 12 (cid:18) r D + 1 r ǫ (cid:19) ¯ r ǫ = [5 − z n = 12 (cid:18) z D + 1 z n (cid:19) ¯ z n = [0 . − . z ǫ = 12 (cid:18) z D + 1 z ǫ (cid:19) ¯ z ǫ = [1 . − . ω ν = (cid:18) z Q − z n (cid:19) ¯ z n ω ν = 0 . − . ω κ = (cid:18) z Q − z ǫ (cid:19) ¯ z ǫ ω κ = 4 . − .
17 –so that in the following discussion, we give first the solution of the diffusion equation (21) inthe HE region, regarding ¯ n W n as a perturbative term, where we can neglect the fluctuationterm due to the reacceleration. Next we give the solution in the LE region, regarding ¯ ǫ W ǫ as a perturbative term by contrast, which is completely the same as the former one afterreplacing [¯ ǫ, W ǫ ] with [¯ n, W n ] (and vice versa), while we have to take the fluctuation term, h ∆ E e i rea , into account in this case.Thus for the steady state ( ∂/∂t = 0), the solution of equation (21) in the HE region isdevided into three N e,ǫ ≃ N (0) e,ǫ + ˜ N (0) e,n + N (1) e,ǫ , (26a)where the first term is a principal one coming from ¯ ǫ W ǫ , the second term corresponds to theperturbative term from ¯ n W n , and the third term to the fluctuation due to the reaccelerationgiven by the second term of the right-hand in equation (22), while it is negligible in practice, N (1) e,ǫ ≈ ǫ with n (andvice versa), but we can not neglect the fluctuation term N (1) e,n in contrast, N e,n ≃ N (0) e,n + ˜ N (0) e,ǫ + N (1) e,n . (26b)The first term in equation (26a) is written immediately as N (0) e,ǫ ( r ; E e ) = Z ∞ Π (0) ǫ ( r ; y ) f (0) ǫ ( y ; E e ) dy, (27)where Π (0) ǫ and f (0) ǫ satisfy, h ¯ ǫ ( r ) c ∂∂y − ∇· D ( r ) ∇ i · Π (0) ǫ ( r ; y ) = Q ( r ) δ ( y ) , (28a) h cE αe ∂∂y − ∂∂E e W ǫ ( E e ) i · f (0) ǫ ( y ; E e ) = 0 , (28b)with f (0) ǫ (0; E e ) = E − γ − αe . It is possible to solve exactly equation (28a) with use of the procedure presented inPaper I, after replacing ¯ n ( r ) by ¯ ǫ ( r ), and we present here only the critical term related to( r, z ; y ), omitting constant terms such as Q and ¯ ǫ (see Appendix B for the full form), Π (0) ǫ ( r ; y ) ∝ exp[ − ¯ s r y − | z | /z D ] , (29) 18 –¯ s r ≃ D r ¯ ǫ r cz D (cid:16) κ (cid:17) ; κ = 11 + z D /z ǫ , (30)with D r ≡ D ( r, ǫ r ≡ ¯ ǫ ( r, Π (0) ǫ is of the form of e − ¯ s r y , the Laplace transform of f (0) ǫ with respect to y , F (0) r,ǫ ( E e ), is sufficient for our purpose to obtain the electron density, F (0) r,ǫ ( E e ) = Z ∞ e − ¯ s r y f (0) ǫ ( y ; E e ) dy, thus we have immediately from equation (28b) F (0) r,ǫ ( E e ) = c W ǫ ( E e ) Z ∞ E e dE E − γ e − Y r,ǫ ( E e ,E ) , (31)with Y r,ǫ ( E e , E ) = c ¯ s r Z E E e E α W ǫ ( E ) dE. In the HE limit, E e ≫ F (0) r,ǫ ( E e ) ≃ cE − ( γ +1 − δ ) e (1 − α − δ ) w T (cid:20) − γ + α − δ − c ¯ s r E αe /w T + . . . (cid:21) , giving a spectral index with γ + 1 − δ , where δ (= 0 . − E e /E cut in the electron injection spectrum somewhere around20 TeV (Reynolds & Keohane 1999; Hendrick & Reynolds 2001; Yamazaki et al. 2006).Now the principal term, N (0) e,ǫ , in equation (26a) for the electron density in the HE region, E e > ∼ E + c , is given by N (0) e,ǫ ( r ; E e ) ∝ F (0) r,ǫ ( E e ) e −| z | /z D , (32)while the perturbative term, ˜ N (0) e,n , is obtained by the use of the iteration method as presentedin Appendix B1, giving ˜ N (0) e,n /N (0) e,ǫ ∼
10% with the first iteration for E e > ∼ N e,ǫ ( r ; E e ) is given by equation (B3).The numerical procedure in the LE region is similar to that in the HE region mentionedabove by replacing the suffix “ ǫ ” with “ n ” (and vice versa), while we have to take intoaccount the third term in equation (26b), N (1) e,n , corresponding to the fluctuation. We findagain that the perturbative term, ˜ N (0) e,ǫ , is obtained by the use of the iteration method aspresented in Appendix B2, giving ˜ N (0) e,ǫ /N (0) e,n ∼
10% with the first iteration for E e < ∼
10 GeV,as shown in Figure 24b. 19 – -2 -1 Electron energy; E e (GeV) z = 50 mb s = 180 mb . : 0 eV: 4 eV: 8 eV N ( ) / [ N ( ) + N ( ) ] ~ e , n e , n e , e h * Fig. 7.— Contributions of the fluctuation in the reacceleration with [¯ σ ⊙ , ζ ] = [180, 50] mbarnat SS for several sets of the free parameter ¯ η ∗ given by equation (B6).On the other hand, the numerical procedure in the fluctuation effect due to the reaccel-eration is a little bit cumbersome, which is presented in Appendix B3. We give an example ofthe ratio, N (1) e,n / [ N (0) e,n + ˜ N (0) e,ǫ ], at SS for the first iteration in Figure 7, where ¯ η ∗ is the effective ratio of the energy density to the gas density defined by equation (B6), approximately with2eV. One finds that it is significant around 0.3–1.5 GeV in the case of ζ = 50 mbarn, boost-ing the solution without the fluctuation, N (0) e,n + ˜ N (0) e,ǫ , by approximately 25%. So we performonly the first iteration also for N (1) e,n in the LE region, as the contribution coming from thesecond and higher iterations is at most of the magnitude of a few % or less (Ishikawa 2010).Full form of N e,n ( r ; E e ) is given by equation (B15).Finally, we give the electron density covering all energies so that it continues smoothlyat the energy E c between the HE and LE regions at the SS ( r = r ⊙ ), with E c ≈ E + c inpractice, but not always E c = E + c , N e ( r ; E e ) N e, = F r,ǫ ( E e ) F ⊙ ,ǫ ( E c ) e −| z | /z D : for E e ≥ E c , (33a) F r,n ( E e ) F ⊙ ,n ( E c ) e −| z | /z D : for E e ≤ E c , (33b)with ∂∂E e F ⊙ ,ǫ ( E e ) F ⊙ ,ǫ ( E c ) (cid:12)(cid:12)(cid:12)(cid:12) E e = E c = ∂∂E e F ⊙ ,n ( E e ) F ⊙ ,n ( E c ) (cid:12)(cid:12)(cid:12)(cid:12) E e = E c , see equations (B4) and (B13) for F r,ǫ ( E e ) and F r,n ( E e ) respectively, and N e, is determinedby the normalization with the experimental data as discussed in §
5. 20 –
In Figure 8 we show the numerical results of N e ( r ⊙ ; E e ) /N e, in two cases, (a) [ ζ , ¯ σ ⊙ ,¯ s ⊙ ] = [50, 180, 30 eV − ]mbarn with α = (reacceleration with Kolmogorov-type spectrumin hydromagnetic turbulence), and (b) [0, 90, 15 eV − ]mbarn with α = (no reaccelerationwith Kraichnan-type spectrum) for ¯ ǫ ⊙ / ¯ n ⊙ = 2 eV with β ( ≡ γ + α ) = 2.6, 2.7 and 2.8, seeequation (30) with r = r ⊙ for ¯ s ⊙ , where we assume E cut = 20 TeV (Reynolds & Keohane 1999;Hendrick & Reynolds 2001; Yamazaki et al. 2006) in the electron injection spectrum with E − γe e − E e /E cut , and the results show the use of both Klein-Nishina ( solid curves ) and Thomson( dotted curves ) cross-sections.We find two critical points in Figure 8. First, those by the former cross-section giveapproximately 40–50% (20–30%) larger than those by the latter at 1 TeV (100 GeV), wherethe density is normalized at E e = 10 GeV, leading to significantly harder spectra than thosewith the Thomson cross-section, as expected. Similar results are also recently reported byDelahaye et al. (2010), while their main purpose is to study the nearby sources of electronand the positron excess problem as well, which are outside the range of the present paper.Second, the reacceleration effect is significant in the energy region less than 10 GeV ascompared to the curves without the reacceleration process. Unfortunately, however, it isdifficult to observe such a signal in the direct experimental data on the electron component -1 Electron energy; E e (GeV) E e N e ( r ; E e ) ( a r b i t r a r y un i t s ) )) : Thomson cross-section: Klein-Nishina cross-section(a) reacceleration ( z = mb) (b) no reacceleration ( z = 0 ) b : Fig. 8.— Numerical results of the electron density, N e ( r ⊙ ; E e ), at the SS ( r = r ⊙ ) in the caseof (a) reacceleration with [ α ; ζ , ¯ σ ⊙ ] = [ ; 50 mbarn, 180 mbarn], and (b) no reaccelerationwith [ ; 0, 90 mbarn], for β = 2.6, 2.7, and 2.8, where the vertical axis is multiplied by E e ,and normalized to E e = 10 GeV. We show results for two cross-sections in IC process fromthe Thomson ( dotted curves ) and Klein-Nishina ( solid curves ) formulae. 21 – Quad.-IQuad.-IIQuad.-IIIQuad.-IV (Hunter et al.) N e ( E e ; r ) N e ( E e ; r ) / (a) electron density normalized at SS Strong et al. SS E e = GeV10010
Galactocentric distance: r (kpc) 0 5 10 15 20 25 30 35 Quad.-IQuad.-IIQuad.-IIIQuad.-IV N p ( E p ; r ) N p ( E p ; r ) / Galactocentric distance: r (kpc)(b) proton density normalized at SS (Hunter et al.) E p = GeV101100Strong et al. SS Fig. 9.— CR densities of (a) electrons, N e ( r ; E e ), and (b) protons, N p ( r ; E p ), as a functionof the galactocentric distance r for several energies, where the vertical axis is normalized tothe density at SS ( r = r ⊙ ). We present also those given by Hunter et al. (1997) (four kinds of square symbols ), and by Strong et al. (1988) ( thin filled histograms ), where the CR densitiesare averaged azimuthally in each galactocentric quadrant by Hunter et al. (1997), assumingthat they are coupled to the gas density of ISM independent of the energy, while azimuthalsymmetric γ -ray emissivity is used by Strong et al. (1988).because of the modulation effect in the low energy region < ∼ z from the GP with the latitudinal scale heightof the diffusion coefficient, z D = 2–4 kpc, independent of the energy E e . This is the sameresult as in the case of the proton density, N p ( r ; E p ) (Paper V), namely the ratio of electrondensity to the proton density is independent of z .Contrary to the latitudinal behavior, the longitudinal behavior of the CR densities, N e ( r ; E e ) and N p ( r ; E p ), are somewhat complicated, both of which depend on the energy,and appear implicitly in the form of ¯ s r and ¯ σ r (see Paper I for ¯ σ r and its physical meaning).We present these in Figure 9 against the radial distance r for the (a) electron and (b)proton components, both normalized at the SS for four energies, 0.1, 1, 10, and 100 GeV,with β = 2.7, where the scale heights are set as [¯ r n , ¯ r ǫ ] = [30, 8] kpc and [ z D ; z n , z ǫ ] = [3; 0.2,0.75] kpc (see Table 4 for ¯ r n and ¯ r ǫ ). We plot the results of Hunter et al. (1997; square symbols )and Strong et al. (1988; thin filled histogram ) together, where the former are based on theassumption that the CR density is coupled to the density of ISM, and plotted separately forfour galactocentric quadrants, I, II, III, and IV. We find that the radial dependence of theelectron density, N e ( r ; E e ), is much stronger than that of the proton density, N p ( r ; E p ), in 22 –the energy region of 1–100 GeV as expected, while the other two authors assume no spatialdependence in the energy spectrum, namely the shape of the energy spectrum at the SS isthe same everywhere in the Galaxy.
5. Electron-induced γ -ray spectrum For convenience in the following discussion, we summarize two cross-sections in Table 5, σ EB ( E e , E γ ) and σ IC ( E e , E γ ; E ph ), each for the bremsstrahlung (abbreviated as “EB” forsubscript attached here and in the following) and the IC processes respectively, where E ph isthe energy of target photon before scattering. In these cross-sections, we take into accountthe screening effect for the bremsstrahlung (Koch & Motz 1959; Gould 1969), and the Klein-Nishina cross-section (Jones 1965, 1968; Blumenthal & Gould 1970) for IC. In the followingdiscussion, we put F r ( E e ) ≡ F r,ǫ ( E e ) for E e ≥ E c , and F r ( E e ) ≡ F r,n ( E e ) for E e ≤ E c inequation (33) for simplicity.First we consider the emissivity of γ ’s from the bremsstrahlung at the position r , whichis immediately written down as q EB ( r ; E γ ) = Z ∞ E γ N e ( r ; E e )[ n ( r ) cσ EB ( E e , E γ )] dE e , (34)where the electron density, N e ( r ; E e ), is given by equation (33). For the numerical calculationof equation (34), we need the absolute electron density at r . To do so, we use the observationaldata on the electron intensity at the SS, dI ⊙ e /dE e , which is related to the electron densityby dI ⊙ e dE e ( E e ) = c π N e ( r ⊙ ; E e ) . In practice, we normalize the electron density at E s = 10 GeV with use of the mostrecent data (see Fig. 14), where the solar modulation effect is negligible, cN ⊙ s ≡ cN e ( r ⊙ ; E s ) = 2 .
26 m − s − GeV − , corresponding to dI ⊙ e /dE e = 0 .
180 m − sr − s − GeV − at E e = 10 GeV in Figure 14, while E s = 100 GeV (per nucleon) for the hadron-induced γ ’s ( π → γ ) with cN p ( r ⊙ ; E s ) =6.16 m − s − GeV − (Paper V). One should keep in mind that the uncertainty in the normal-ization is of the magnitude as large as 10%. 23 –Table 5. Summary of the production cross-sections of γ -rays in the bremsstrahlung andthe IC processes with x = E γ /E e , where E e is the incident energy of electron, and E γ is theenergy of the produced γ ’s, and E ph the energy of the target photon before electronscattering. For the bremsstrahlung process, we present the cross-section in the case of onlyone-electron atoms ( Z = 1), see Gould (1969) for two-electron atoms ( Z = 2). bremsstrahlung (EB) inverse Compton (IC) σ EB ( E e , E γ ) dE γ = σ (0) EB φ EB ( x, E γ ) dxx σ IC ( E e , E γ ; E ph ) dE γ = σ (0) IC φ IC ( x, q ) dxXσ (0) EB = 4 α f Z ( Z + 1) (cid:16) e m e c (cid:17) ; α f = 1137 σ (0) IC = 3 σ T = 8 π (cid:18) e m e c (cid:19) φ EB ( x, E γ ) = n − x ) o φ ( χ ) −
23 (1 − x ) φ ( χ ) φ IC ( x, q ) = 2 q ln q + (1 − q ) (cid:18) q + 12 x x (cid:19) φ ( χ ) = 1 + Z χ φ ( y ) (cid:18) − χy (cid:19) dyy q ≡ q ( x, X ) = x − x Xφ ( χ ) = 56 + Z χ φ ( y ) (cid:26) χ y (cid:18) χ y (cid:19) − χ y (cid:27) dyy X ≡ X ( E e , E ph ) = k Θ e = E ph E e [ m e c / φ ( y ) = 1 − y / (2 α f Z ) ] k ≡ k ( E ph , T ph ) = E ph k B T ph χ ≡ χ ( x, E γ ) = x − x m e c E γ Θ e ≡ Θ e ( E e , T ph ) = m e c / p k B T ph E e
24 –Thus taking care of the terms related to r , we have q EB ( r ; E γ ) n ( r ) w (0)EB N ⊙ s = e −| z | /z D Z φ EB ( x, E γ ) F r ( E x ) F ⊙ ( E s ) dxx , (35)for E γ ≥ E c with E x = E γ /x , and w (0)EB = cσ (0)EB = 1.39 × − cm s − for the hydrogen gas( Z = 1), where one should take care of the energy range E e ≤ E c in the case of E γ ≤ E c .Next we consider the emissivity of γ ’s coming from the IC process, which is somewhatcomplicated, as there are several kinds of target photons with different energy density aswell as with different scale heights in the spatial gradient. Here we present a result only,taking into account the six wavelength bands in ǫ ( i )ph ( r ) ( i = 0–5) (see eq. [7] and Table 3), q ( i ) IC ( r ; E γ ) ǫ ( i )ph ( r ) w T N ⊙ s = e −| z | /z D Z Φ ( i ) IC ( x, E γ ) F r ( E x ) F ⊙ ( E s ) dxx , (36)where Φ ( i ) IC ( x, E γ ) is given by equation (C2), see Appendix C for the details.Let us show the numerical results for two cases of emissivity in Figure 10, (a) r/r ⊙ =0 . , , z = 0 in the GP, and (b) z = 0 . , . , . r = r ⊙ normal to theGP at SS, assuming β ≡ γ + α = 2.7, where we present separately those coming from π ( solid curves ), EB ( broken curves ), and IC ( dotted curves ). One finds that EB- γ ’s and π - γ ’sare comparable around 50 MeV, and IC- γ ’s and π - γ ’s around two energies, ∼
20 MeV and ∼ π , q π ( r ; E γ ), while we use more realisticgas density, n ( r ), in the present paper. Note also in q π that the semi-empirical productioncross-section of γ ’s, σ pp → γ ( E p , E γ ), in proton-proton collision we use is valid over very wideenergy ranges, 1 GeV–1 PeV, reproducing nicely various kinds of physical quantities such aspsuedo-rapidity, energy spectrum, multiplicity, etc, obtained by both the accelerator and CRexperiments with local target layer (Suzuki, Watanabe & Shibata 2005).Once we have the emissivity of γ ’s induced by the interaction between the electrons andthe media of ISM and ISRF, we can obtain immediately the intensity of γ ’s observed at theSS ( r = r ⊙ ), coming from the direction θ ( l, b ) d I ⊙ γ ( θ ; E γ ) dE γ dld (sin b ) = 14 π Z ∞ q γ ( r ; E γ ) ds, with q γ ( r ; E γ ) = q EB ( r ; E γ ) + X i =0 q ( i ) IC ( r ; E γ ) ,
25 –Fig. 10.— Emissivity of γ ’s (a) at three radial distances, r/r ⊙ = 0.5, 1.0, 1.5 in the galacticplane ( z = 0), and (b) at three latitudinal distances, z = 0.2, 0.4, 0.6 kpc with r = r ⊙ , wherethe three components for γ ’s emission, π ( solid curves ), EB ( broken curves ), and IC ( dottedcurves ) are shown separately.where the integration with respect to s is performed along the arrival direction of γ ’s, θ ( l, b ),at the SS, and r ( r, z ) is bound to ( s ; l, b ) as follows, r ( s ; l, b ) = q r ⊙ + s cos b − r ⊙ s cos b cos l,z ( s ; b ) = s sin b.
6. Comparison with the observational data6.1. Critical parameters
We assume that the source distribution of electron component, Q ( r ; E e ), is the sameas that of the hadronic component except for the cutoff electron energy, for instance E cut ≈
20 TeV, with the supernova remnants as the main energy supply, while the pulsars and pulsarwind nebulae might contribute to them as well, particularly to positrons and electrons (forinstance, Delahaye et al. 2010). So the galactic parameters used in the present work areessentially the same as those appearing in Papers I–V, and we summarize them briefly inthe following.The recent observational data on the energy spactra of CR hadronic components giveindices with 2 . ± .
08 for proton (Derbina et al. 2005), and with a common value of ∼ . ± .
05 by JACEE (Asakimori et al. 1998) in contrast to2 . ± .
20 by RUNJOB (Derbina et al. 2005). Note that PAMELA (Picozza et al. 2007)reports recently a common index of 2.73 in both the proton and helium spectra, albeit theenergy region is limited below 500 GeV. Any way, the spectrum index β of proton, must liewell within 2.7–2.8 in the high energy region at the SS, which is the most effective elementfor the hadron-induced D γ ’s. See Paper V for the contribution of helium and nuclei toD γ ’s, which is taken into account by introducing the enhancement factor with 1.53. So inthe present paper we use the critical parameter β in place of γ (source index of the energyspectrum) with β = γ + α , and consider three values of β ; 2.6, 2.7, and 2.8, each for α = (Kolmogoroph-type spectrum) and (Kraichnan-type spectrum).There are three galactic parameters, [ D ( r ), ¯ n ( r ), Q ( r )], in our approach to the CRpropagation, and six scale heights for longitudinal and latitudibal directions correspoding toeach one, [ r D , r n , r Q ] and [ z D , z n , z Q ], respectively. In practice, however, explicit parametersneeded to compare with the experimental data appear in two critical ones alone, ¯ σ r and ζ ,besides [ α , β ] mentioned above, while the parameter, ¯ µ r ≡ z D / √ ¯ τ D r , is also important forthe study of the CR isotopes (¯ τ : normalized life time of an isotope with 10 yr).For electron components, the additional parameter newly appears, ¯ s r , given by equation(30), physical meaning of which is essentially the same as ¯ σ r ; i.e., while the inverse of ¯ σ r gives the average path length, ¯ x r , in units of cm − in ISM as discussed in Paper I, that of¯ s r corresponds to the average path length, ¯ y r , in units of eVcm − in ISRF, namely the totalamount of photon-gas energy that CR has passed through the ISRF.Now from equation (33), one should remark that there appear only three critical pa-rameters, [ ζ , ¯ σ ⊙ , ¯ s ⊙ ], in F r,ǫ ( E e ) and F r,n ( E e ) needed to compare with the observationaldata, aside from two critical indices, [ α , β ], note that various galactic parameters such asthe diffusion constant, gas density, energy density, their scale heights, etc are all involvedimplicitly in these three ones. As we have presented the experimental results on CR hadron components in the pastpapers (Papers I–IV), we give here only three kinds of secondary-to-primary ratio with newdata, B/C, sub-Fe/Fe, and ¯ p/p , that have since become available. See Paper III for thesecondary unstable nuclei, while new data are still not available. 27 –In Figure 11, we present B/C and sub-Fe/Fe, plotted together with new ones fromCREAM (Ahn et al. 2008) and TRACER (M¨uller 2009), where we plot also RUNJOB(Derbina et al. 2005) data for reference, while the data quality is rather poor with large -3 -2 -1 -1 ACE/CRISHEAO-3SANRIKURUNJOBATIC-07 kinetic energy : E N [GeV/nucleon] s ec o nd a r y -t o - p r i m a r y r a t i o Boron/Carbon = 50 mb with a = 1 / z (a) mb 200 300 150 200 300 s : _ . ( x (with reacceleration) TRACERCREAM -3 -2 -1 -1 ACE/CRISHEAO-3SANRIKURUNJOBATIC-07 75 mb 100 150 s : _ .
100 150 Boron/Carbon = 0 mb with a = 1 / z (b) ( x kinetic energy : E N [GeV/nucleon] s ec o nd a r y -t o - p r i m a r y r a t i o TRACERCREAM
Fig. 11.— Energy dependence of the secondary-to-primary ratio for boron/carbon andsub-iron/iron. See Paper II and references therein for the experimental data, while CREAM(Ahn et al. 2008) and TRACER (M¨uller 2009) data are newly plotted. Numerical curvesare demonstrated for two cases; (a) reacceleration with ( α, ζ ) = ( , 50 mbarn) and (b) noreacceleration with ( , 0). -6 -5 -4 -3 -1 BESS95+97BESS1998BESS1999BESS2000BESS2002BESS2004CAPRICE98MASS91HEAT00PAMELA10 red marks : A < : A > kinetic energy : E P (GeV) a n t i p r o t o n -t o - p r o t o n r a t i o (with reacceleration) f (GV) = 50 mb with a = 1 / z Fig. 12.— Energy dependence of the antiproton-to-proton ratio, where we assume thereacceleration model with α = for six modulation parameters, 0.2, 0.3, 0.5, 0.7, 1.0, and1.5 GV, and A <
A >
0) corresponds to positive (negative) polarity state in heliosphericmagnetic field, although the present calculations do not take the effect into account. SeePaper IV and the references therein for the data other than PAMELA (Adriani et al. 2010). 28 – -1 HEAT-94HEAT-LIS BETSAMS Golden94PPB-BETS ATICECC = 50 mb with a = / z (with reacceleration) E e d I d E e [ m - s r - s ec - G e V ] / b : Electron energy; E e (GeV) before FERMI : s = eV -1 mb : s = eV -1 mb a) with reacceleration _ .. _ -1 HEAT-94HEAT-LIS BETSAMS Golden94PPB-BETS ATICECC = 0 with a = / z (no reacceleration) 2.82.7 b : Electron energy; E e (GeV) before FERMI b) no reacceleration E e d I d E e [ m - s r - s ec - G e V ] / : s = eV -1 mb : s = eV -1 mb .. __ Fig. 13.— Electron energy spectra in two models, (a) with reacceleration and (b) noreacceleration, compared with the measurements, where the vertical axis is multiplied by E e . All numerical values are normalized to E e = 10 GeV, and several sets of ( β , ¯ s r ) areassumed. See text for references for individual experimental data.atmospheric correction. We compare our numerical results with the data for two models,(a) reacceleration with the set of [ ζ , ¯ σ ⊙ ] = [50, 150–300] mbarn, for α = , and (b) no reac-celeration with [0, 75–150] mbarn, for α = . It is still not clear which model reproducesthe experimental data more satisfactorily. As is well known, the advantage of the formerexplains naturally the drop of the ratio in the lower energy region around ACE/CRIS (Daviset al. 2000) without assuming an ad hoc drop in the path length distribution.Next we present ¯ p/p in Figure 12, plotted together with new data from PAMELA(Adriani et al. 2010), where we present numerical curves with several sets of modulationparameters, 0.2-1.5 GV, for the reacceleration model shown in Figure 11a. One finds thatour result is in good agreement with the PAMELA in the high energy region around 100 GeV,where the modulation effect is absolutely negligible. Let us present the electron data separately before and after FERMI, where “electron”denotes both electron and positron. First in Figure 13 we present the electron energy spec-trum before FERMI, where the experimental data are presented for those reported in theperiod from 1994 to 2008 alone (Golden et al. 1994; Kobayashi et al. 1999; DuVernois etal. 2001; Torii et al. 2001, 2006; Aguilar et al. 2002; Chang et al. 2008), and also plottedare the data ( filled purple squares ) for reference after applying a demodulated correction to 29 –HEAT data, HEAT-LIS, (DuVernois et al. 2001) using the force-field approximation withthe modulation parameter of 755 MV (670 MV) for the 1994 (1995) data.The numerical curves are normalized at 10 GeV with two indices, β = 2.7, 2.8, assumingtwo models, (a) reacceleration and (b) no reacceleration each with the same parameter setsas those used in Figure 11, while we assume additionally two cases of ¯ s ⊙ , [20, 30] mbarn forthe reacceleration (a), and [10, 15] mbarn for no reacceleration (b). Aside from the prominentspectral features around 500 GeV appearing in ATIC (Chang et al. 2008) and PPB-BETS(Torii et al. 2006) data, our model with the reacceleration reproduces the data well in thehigher energy region, > ∼
10 GeV, in Figure 13a, where the solar modulation effect is small.On the other hand, the model without reacceleration in Figure 13b is somewhat difficult tofit to the demodulated HEAT-LIS data.Now, in Figure 14 we present the most recent data obtained by FERMI (Abdo et al.2009, 2010b) and H.E.S.S. (Aharonian et al. 2009) together with those presented in Figure 13,where numerical curves are the same as shown in Figure 13a. We find that both FERMI andH.E.S.S. data do not exhibit the prominent bump around 500 GeV reported by ATIC andPPB-BETS, with both giving a spectrum falling with energy as E − up to 1 TeV, which isnot inconsistent with emulsion chamber data (Kobayashi et al. 1999) within the statisticalerrors. Looking Figure 14, however, we find that FERMI and H.E.S.S. data seem to deviatesystematically from numerical curves with an enhancement by 20–30% around 500 GeV,indicating still some additional local sources of high energy CR electrons, which will bediscussed again in § -1 HEAT-94HEAT-LIS BETSAMS Golden94PPB-BETS ATICECC E e d I d E e [ m - s r - s ec - G e V ] / b : Electron energy; E e (GeV) FERMI; HESS-LE analysis, HESS-HE analysis (with reacceleration): s = 30 mb: s = 20 mb .. __ Fig. 14.— Same as Figure 13a, but with FERMI (Abdo et al. 2009, 2010b) and H.E.S.S.(Aharonian et al. 2009), where drawn are numerical curves with the reacceleration shown inFigure 13a. 30 – -90 -60 -30 0 30 60 90Galactic Latitude; b : EB: IC: p : background: total(EB + IC + p + BG) I n t e n s i t y [] - c m - s - s r -
0 < l < 360 D E g = [300-500] MeV = 50 mb; a = 1/3, b = 2.7 z : EGRET Fig. 15.— An example of the estimation of the BG ( black horizontal line ) using the latitu-dinal D γ ’s data from EGRET (Hunter et al. 1997) with the energy interval of 300–500 MeVaveraged over the whole radial direction, l = 0 ◦ – 360 ◦ . γ -ray component γ -rays D γ ’s near the GP are mainly hadron-induced ( π → γ ) and electron-induced (EB + IC).In addition to these two components, we have isotropic background γ ’s (BGs) with variousorigins such as extragalactic sources (EGs), unidentified sources, instrumental sources, darkmatter (DM), etc, so that the BGs depend on individual detectors with different sensitivityin energy and the angular resolution, while depending on the propagation model as well.Therefore it is not easy task to estimate the extragalactic D γ , while its origin is one of thefundamental problems in astrophysics, studied in so many papers with various candidates;unresolved blazers (e.g. Stecker & Salamon 1996; Chiang & Mukherjee 1998; M¨ucke & Pohl2000), intergalactic shocks produced by the assembly of large-scale structures (e.g. Loeb &Waxman 2000; Totani & Kitayama 2000; Miniati et al. 2000; Gabici & Blasi 2003), darkmatter annihilation (e.g. Bergstr¨om 2000; Ullio et al. 2002; Ahn et al. 2007), etc. In thepresent paper, however, we use the acronym “BGs” all together for D γ ’s other than thoseinduced by π , EB (bremsstrahlung) and IC, while acknowledging EGRET and FERMIteams have estimated very carefully the EG- γ intensity.In Figure 15, we present an example of EGRET data ( histogram ; source subtracted)(Hunter et al. 1997) together with numerical curves on the latitudinal distribution aver-aged over full longitude ranges, 0 ◦ –360 ◦ with the energy interval 300–500 MeV, where wegive the contributions of D γ ’s separately from π ( solid red ), EB ( dotted red ), IC ( bro-ken red ), BG ( solid black ), and total flux, π +EB+IC+BG ( heavy solid red ), assuming 31 – -5 -4 -3 -2 COMPTEL(Kappadath et al.)EGRET(Sreekumar et al.)EGRET (Strong et al.)FERMI-LAT(Abdo et al.) d I / d E g [] c m - s - s r - M e V E g g -ray energy; E g (MeV) FERMI (Fig.
17b in this paper)EGRET (Figs.
15, 17a in this paper) : Abdo et al. (2010): used in this work
Fig. 16.— The BG spectrum obtained by COMPTEL (Kappadath et al. 1996), EGRET(Sreekumar et al. 1998), EGRET (revised; Strong et al. 2004), and FERMI (Abdo et al.2010a), where also plotted are those estimated by the present work using EGRET andFERMI data (see Figs. 15 and 17). Dotted line is given by Abdo et al., and the solid curveis used in the present work (see text), modifying it slightly in the low energy region.[ ζ , ¯ σ ⊙ , ¯ s ⊙ ] = [50, 180, 30 eV − ]mbarn with [ α, β ] = [ , 2.7]. Here we draw a horizontal linefor BG by the use of the least square method so that the histogram is well reproduced,where the fitting is applied for | b | ≤ ◦ as there remain considerable uncertainties in thelatitudinal distribution for both the ISM and ISRF far distant from the GP, see Ξ h ( r, z )(“ h ” ≡ H I, H ) in equation (2) and the scale height z ph in equation (7). It is remarkablethat the numerical curve is in good agreement with the data not only in shape, but also inabsolute value, except the high latitude around the galactic pole.In Figure 16, we summarize the intensity of BGs obtained by past works, Kappadathet al. (1996) for COMPTEL, Sreekumar et al. (1998) for EGRET, Strong et al. (2004)for EGRET (revised), and Abdo et al. (2010a) for FERMI, where also plotted are thoseestimated in this work (see Figs. 15 and 17) for the reference, six points ( open circles ) forEGRET and one point ( filled circle ) for FERMI. We draw a broken line given by Abdo et al.with dI BG /dE γ = 9 . · − × E − . γ in units of [cm − s − sr − MeV − ] with E γ in MeV, anda solid curve with dI ∗ BG /dE γ = dI BG /dE γ × [1 − exp ( − . E . γ )], used in the present work,slightly modifying the FERMI result in the low energy region, while the modification doesnot affect any change for the results. Significant difference between EGRET and FERMI,with the former giving much harder spectrum than the latter, might be due to a diffrentmodel in CR propagation as well as those in ISM and ISRF. 32 – -90 -60 -30 0 30 60 901.00.10.10.01 I n t e n s i t y [] - c m - s - s r -
0 < l < 360 (source subtracted) a) EGRET BG D E g = [300-500] MeV BG b) FERMI [350-480] MeV Galactic Latitude; b )) : b = 2.6 Fig. 17.— Latitudinal distributions of D γ ’s obtained by a) EGRET (Hunter et al. 1997)with the energy range 300–500 MeV, and b) FERMI (Porter 2009) with the energy range350–480 MeV, both averaged over the the whole radial direction, l = 0 ◦ –360 ◦ , where drawnare three curves with β = 2.6 ( green ), 2.7 ( red ), and 2.8 ( blue ), taking the BG contributionsinto account. We present two examples of the latitudinal distributions for EGRET and FERMI (Porter2009) with the energy interval around [300–500] MeV in Figures 17a and 17b respectively,together with our numerical results taking the BG contribution ( broken-dotted lines ) intoaccount mentioned above, dI ∗ BG /dE γ , where plotted are three curves for each figure with β = 2.6 ( green ), 2.7 ( red ), and 2.8 ( blue ). One finds the agreement between the data andthe curves is excellent except the high latitude | b | > ∼ ◦ . In these calculations, we take theangular resolution (PSF) effect with the energy dependence into account, for instance, with7 ◦ (HWHM) at 30–50 MeV (Hunter et al. 1997).Corresponding to the latitudinal distributions as shown in Figures 17a and 17b, wedemonstrate the longitudinal distributions near the GP in Figures 18a and 18b, where nu-merical curves are shifted by ∆ l = +10 ◦ in both EGRET and FERMI so that experimentaldata are reproduced more satisfactorily. Again we find the numerical results are in niceagreement with the data in both shape and absolute value, and consistent with β ∼ .
7. 33 – -180 -120 -60 0 60 120 1801.0 I n t e n s i t y [] - c m - s - s r - )) l D E g = [350-480] MeV a) EGRETb) FERMI -5 < b < 5 (source subtracted) D E g = [300-500] MeV : b = 2.6 Fig. 18.— Same as Figure 17, but for longitudinal distribution with the same condition,where BG contributions are not presented. -4 -3 -2 -1 -3 -2 -1 EGRETFERMICOMPTEL g -ray energy; E g (GeV) E g d I / d E g [] c m - s - s r - M e V -60 < l < | b | < total p IC BG EB a) with reacceleration: b = 2.6 -4 -3 -2 -1 -3 -2 -1 EGRETFERMICOMPTEL g -ray energy; E g (GeV) E g d I / d E g [] c m - s - s r - M e V total p IC BG EB b) no reacceleration: b = 2.6 -60 < l < | b | < Fig. 19.— Differential energy spectra of D γ ’s averaged over the whole radial direction with | b | < ◦ obtained by COMPTEL (Kappadath et al. 1996), EGRET (Hunter et al. 1997) andFERMI (Abdo et al. 2010b). Numerical curves are demonstrated for two cases, a) with thereacceleration and b) no reacceleration, each presented separately for individual components. First, in Figure 19a we present the energy spectrum of D γ ’s averaged over the field ofview with − ◦ < l < ◦ and | b | < ◦ , where numerical curves with the reacceleration arealso presented separately for those coming from π , EB, and IC (all with colored thin solidcurves ), BG ( heavy black solid curve ), and total ( colored heavy solid curves ). Here and in 34 – -4 -3 -2 EGRETFERMI total p IC EB E g d I / d E g [] c m - s - s r - M e V BG < l < < | b | < a) low latitude -4 -3 -2 EGRETFERMI E g d I / d E g [] c m - s - s r - M e V total p EBIC BG b) mid latitude < l < < | b | < -5 -4 -3 -2 -2 -1 EGRETFERMI E g d I / d E g [] c m - s - s r - M e V g -ray energy; E g (GeV)total p EBIC BG c) high latitude < l < < | b | < Fig. 20.— Same as Figure 19a, but those averaged over three different ranges in the galacticlatitude, a) low latitude with 10 ◦ < | b | < ◦ , b) mid latitude with 20 ◦ < | b | < ◦ , and c)high latitude with 60 ◦ < | b | < ◦ . Parameter sets for the numerical calculations are thesame as those used in Figure 19a with the reacceleration.the following we omit EGRET data in GeV region, because of the instrumental problem indetection of γ ’s (Stecker et al. 2008). One finds that the curve with β = 2.7 ( red ) is in goodagreement with the data in the energy region below 1 GeV, while they deviate slightly fromthe curve above 1 GeV, with approximately 20% enhancement.Figure 19b reproduces Figure 19a, but for curves without reacceleration, correspondingto Figures 11b and 13b (see also Fig. 8b). The fit is not as good as for the reaccelerationmodel, particularly in the low energy region, E γ < ∼
200 MeV, with ∼
40% enhancement, whilewith ∼
20% in the high energy region, E γ > ∼ γ ’s for different skyviews (Abdo et al. 2010a; see also supplementary material at http://link.aps.org/supplement-al/10.1103/PhysRevLett.101101).Figure 20 shows those averaged over independent galactic latitude ranges covering low, midand high galactic latitudes, a) 10 ◦ < | b | < ◦ , b) 20 ◦ < | b | < ◦ and c) 60 ◦ < | b | < ◦ respec- 35 – -4 -3 -2 E g d I / d E g [] c m - s - s r - M e V total p IC BG EB a) Hemispheres for Galactic latitude with | b | > 10South hemisphereNorth hemisphere -4 -3 -2 E g d I / d E g [] c m - s - s r - M e V total p IC BG EB b) Hemisphere centered at the Galactic center with | b | > 10 , and 270 < l < 90 (inner Galaxy) -4 -3 -2 -2 -1 E g d I / d E g [] c m - s - s r - M e V g -ray energy; E g (GeV)total p IC BG EB c) Hemisphere centered at the Galactic anticenter with | b | > 10 , and 90 < l < 270 (outer Galaxy) Fig. 21.— Same as Figure 20, but those averaged over different hemispheres on the sky forthe galactic latitudes with | b | > ◦ , which are centered at a) the north galactic pole ( opensquare ) and the south galactic pole ( filled square ), b) the galactic center with 270 ◦ < l < ◦ (inner Galaxy), and c) the anticenter 90 ◦ < l < ◦ (outer Galaxy).tively. Figure 21 shows those averaged over different hemishperes, which are, a) centeredat the north ( b ≥ ◦ ; open squares ) and south ( b ≤ ◦ ; filled squares ) galactic poles, b) thegalactic center (270 ◦ ≤ l ≤ ◦ ), and c) anticenter (90 ◦ ≤ l ≤ ◦ ), all with the galacticlatitudes excluding | b | < ◦ . In these figures, we subtract γ ’s coming from point sourcesbased on the FERMI catalog.It is remarkable in Figure 20 that EGRET and FERMI data agree pretty well witheach other, overlapping nicely around 0.2–1GeV, in all latitude ranges. One finds that thenumerical curves with β = 2.7 reproduce generally well both the EGRET and FERMI data inFigures 20 and 21 but 20c, taking account of the uncertainties in various galactic parameters,particularly in those related to the ISM and ISRF.On the other hand, in Figure 20c for the high latitude, | b | > ◦ , we have a noticeableenhancement in FERMI with approximately 70% as compared to the numeical curves. Tosee the deviation more clearly, we present them all together in Figure 22, where we show ad- 36 –Fig. 22.— Same as Figure 20, but two cases of numerical curves, a) with the BG- γ given byFERMI (Abdo et al. 2010a) ( solid curves ), and b) with the BG- γ by EGRET (Sreekumaret al. 1998) ( dotted curves ).Fig. 23.— Same as Figure 21, but two cases of numerical curves, a) with the BG- γ given byFERMI (Abdo et al. 2010a) ( solid curves ), and b) with the BG- γ by EGRET (Sreekumaret al. 1998) ( dotted curves ).ditionally numerical curves ( dotted colors ) using EGRET-BG obtained by Sreekumar (1998)for reference (see Fig. 16). Figure 23 reproduces Figure 21 with numerical curves usingEGRET-BG ( dotted colors ) in addition to those using FERMI-BG ( solid colors ).One finds the spectrum shapes with EGRET-BG are quite different from the data inthe high energy region, although the enhancement is rather improved in the energy region < ∼
7. Discussion and summary
We have studied the diffusion-halo model with stochastic reacceleration, comparingit with the most recent data on hadronic, electronic and D γ components. We have twoparticular interests: to find an unified model for the CR acceleration and propagation fromthe viewpoint of astrophysics, and to search for a signal of novel sources such as PBHand/or DM from the viewpoints of particle physics and cosmology. Both are of courseclosely connected with each other in the sense that the knowledge of the former is decisivein confirming the latter. While several groups (Torii et al. 2006; Chang et al. 2008) havereported the possibility of annihilation and/or decay of DM particles, giving a significantbump in electron flux around 500 GeV, FERMI (Abdo et al. 2009) and H.E.S.S. (Aharonianet al. 2009) give a rather flat spectrum up to 1 TeV without the prominent excess. In thepresent paper, however, we have focussed our interest rather conservatively on the internalconsistency among various CR components from the view point of astrophysics, leaving thepuzzle of the possible electron/positron-excess to further observations and mutual cross-checks in data analysis among individual groups.In our past works on the hadronic component, we concluded that the diffusion-halomodel with the reacceleration with the parameter set, [ ζ , ¯ σ ⊙ ] = [50, 180] mbarn with [ α, β ] =[ , 2.7–2.8], is in harmony with the CR hadron data presently available. The most recentdata on the B/C ratio by CREAM and TRACER (Fig. 11) as well as on the ¯ p/p ratio byPAMELA (Fig. 12) also support the present model. However, it is worth mentioning herethat our interpretation for the energy dependence of the B/C ratio is somewhat differentfrom that by CREAM (Ahn et al. 2008) and TRACER (M¨uller 2009).They claim that the index α favors 0.5–0.6 instead of , resulting in a rapid decrease withenergy for the interstellar propagation path length. In contrast to their interpretation, wewould like to point out that the value of 0.5–0.6 is not fundamental , but is rather accidental due to the reacceleration effect, namely it is boosted upward around the GeV region bythe energy gain, resulting coincidentally in the soft slope with 0.5–0.6 in the energy region1–100 GeV. The intrinsic one must be (Kolmogorov-type for wave number spectrum inhydromagnetic turbulence), leading to 1) a natural drop in path length distribution in the lowenergy region < ∼ − in TeV region nowadays established experimentally.We apply the diffusion-halo model with and without the stochastic reacceleration for theelectron and D γ components. Apart from the electron-anomaly around 500 GeV, we find thatthe parameter set with the reacceleration, [ α, β ; ζ , ¯ σ ⊙ ], expected from hadron componentreproduces rather well both the spectrum shape and the absolute value in both the electron 38 –(Fig. 13a) and D γ (Figs. 17–21) components, assuming the additional parameter ¯ s ⊙ with20–30 eV − mbarn. Physical meanings of the numerical set with [ ζ , ¯ σ ⊙ ] = [50, 180] mbarnare discussed in Papers I–III in connection with the diffusion constant D ⊙ , gas density ¯ n ⊙ ,their scale heights, z D , z n , etc, giving reasonable values matched with the observtional data.Let us consider the physical meaning of 20–30 eV − mbarn in ¯ s ⊙ . The relation between¯ σ ⊙ and ¯ s ⊙ is given by ¯ s ⊙ ¯ σ ⊙ = ¯ n ⊙ ¯ ǫ ⊙ /κ /ν = ¯ n ⊙ ¯ ǫ ⊙ z D /z ǫ z D /z n , see § s ⊙ and ¯ σ ⊙ with r = r ⊙ , and Table 4 for ν and κ . Namely, it is closely related tothe ratio of the energy density ¯ ǫ ⊙ to the gas density ¯ n ⊙ at the SS, for the smeared energydensity, smeared gas density respectively, and three latitudinal scale heights, z ǫ , z n and z D . As discussed in §§ n ⊙ = n ⊙ HI + n ⊙ H + n ⊙ HII = 1 .
14H atoms cm − , and ǫ ⊙ = ǫ ⊙ B + ǫ ⊙ ph ≈ . − , leading to ǫ ⊙ /n ⊙ ≈ z n , z ǫ ; z D ] = [0.2, 0.75; 3.0] kpc, we find ¯ s ⊙ = 32 eV − mbarn for¯ σ ⊙ = 180 mbarn, giving a consistent result, while the latter with 180 mbarn is expected fromthe relation, ¯ σ ⊙ ≃ D ⊙ / [¯ n ⊙ cz D z n ] with a reasonable set [ D ⊙ , ¯ n ⊙ ] = [3 × cm s − , 1 cm − ]and [ z D , z n ] = [3, 0.2] kpc as discussed in Papers I, II.As mentioned above, the electron spectra currently available are generally in agreementwith those expected from the hadron spectra, considering the uncertainties inherent in boththe experimental data and the numerical parameters, but not quite satisfactory, with theFERMI data giving the excess by 20–30% around several hundred GeV compared to thenumerical results as seen in Figure 14. It might be related to the positron excess around 10–100 GeV observed by PAMELA (Adriani et al. 2009), indicating some nearby sources and/orexotic ones from DM annihilation or decay, while beyond the subject of the present paper. Infact most recently Delahaye et al. (2010) show that the electron spectra with FERMI, HESSand PAMELA are reproduced rather well by the standard astrophysical processes, assumingtwo sources separately, the distant and local nearby ones, whereas they stress that thereremain too large theoretical uncertainties to build a standard model for CR electrons. So itis critical to study the D γ ’s and diffuse radio emissions simultaneously in order to reduce theuncertainties inherent in the galactic parameters assumed for the numerical calculations.We compared our numerical results on the energy spectrum of D γ ’s with EGRET andFERMI data for several sets of the field of view (Figs. 17–21), and found that overall,the CR data, hadron (Fig. 6 in Paper V) and electron (Fig. 14) components, reproducerather satisfactorily D γ ’s for both EGRET and FERMI, considering the fact that we haveuncertainties with at least 10–20% in the galactic parameters assumed here as well as in theflux normalization of the hadron and electron components. Small enhancements of D γ ’s in 39 –GeV region (Figs. 19, 20), albeit they are still within the uncertainties, may indicate thosefrom nearby sources such as the supernova remnants, pulsars, and pusar wind nebulae.We found, however, that FERMI data give the significant excess with approximately70% or more in the high latitude (Fig. 22), well beyond the uncertaities, against the numericalresults in GeV region. This result may indicate a signature of very large electron-halo fardistant from the GP, with, for instance, as large as 25kpc (Keshet et al. 2004), and/orsomething else coming from the cosmological origin. We are also concerned if the exess herediscussed relates to those appearing in the electron spectrum between 100 and 1000GeVobserved by FERMI and HESS (Fig. 14) and in the positron spectrum around several tensGeV by PAMELA. To make clear the correlation between these excesses, D γ ’s in the highlatitude and the electrons/positrons around several tens to hundred GeV, crucially importantis the anisotropy study for the high energy electron, which will be discussed elsewhere in thenear future.Finally we briefly argue the electron spectrum obtained by FERMI from the observa-tional point of view, aside from the prominent bumps indicated by ATIC and PPB-BETS.FERMI is indeed excellent in the observation for γ -rays, we have some concerns about theseparation of electrons from hadrons as well as their energy determination in the high energyregion, while acknowledging the team have studied very carefully the reliability from variouskinds of checks, with both beam tests and the simulational analyses.Nevertheless, one should keep in mind that FERMI is not purely-direct observations forelectrons, but quasi-direct ones in the sense that electron events are selected by statistical analysis based on simulations for the spread of electron showers, where a small number ofelectrons are statistically selected from a large proton background. In contrast to these quasi-direct experiments, the PAMELA apparatus consists of a permanent magnetic spectrometerwith a silicon tracking system, providing good identification between electrons and positrons,though limited to a maximum detectable rigidity (MDR) of 100 GV.Anyway, we await further studies and mutual cross-checks among the groups from var-ious points of view to get a firm conclusion for the electron-excess around 500GeV, whilenot so prominent as given by ATIC and PPB-BETS. So results from the AMS program(Bindi 2009), AMS-02, will be of particular interest. This program aims at high precisionmeasurements of CR (both electron and hadron) and γ -ray fluxes from a few hundred MeVto a few TeV using a super-conducting magnet , with the space shuttle launch scheduled forSeptember 2010. We also look forward to D γ ’s data from ground-based telescopes currently After submitting the present paper, we find that they decided to use the permanent magnet in place ofthe super-conducting magnet (Kounine 2010).
40 –operating such as H.E.S.S., MAGIC, and VERITAS, as well as the CTA-program now underconsideration (Caballero et al. 2008), the threshold energies of which are now overlappingwith the FERMI satellite data.We are very grateful to P.G. Edwards (CSIRO Astronomy and Space Science) for hiscareful reading of the manuscript and valuable comments.APPENDIX AENERGY LOSS OF ELECTRONS IN ISM and ISRFThe energy-loss rate due to the bremsstrahlung in the gas density n ( r ) is given by − D ∆ E e ∆ t E rad = Z E e E γ [ n ( r ) cσ EB ( E e , E γ )] dE γ = n ( r ) w rad ( E e ) E e , (A1)where w rad ( E e ) = w (0) EB Z φ EB ( x, E e x ) dx, with w (0) EB = cσ (0) EB = 4 cα f Z ( Z + 1) (cid:16) e m e c (cid:17) = 1 . × − cm s − , for hydrogen atoms, and see the left-hand side of Table 5 for φ EB ( x, E γ ).For E e ≫ m e c , we can use the complete screening cross-section, leading to the well-known result − D ∆ E e ∆ t E rad = n ( r ) w ( ∞ )rad E e ; w ( ∞ )rad = 7 . × − cm s − . On the other hand the energy-loss rate due to the IC is given, taking into account theenergy spectrum of the target photon at r , n ph ( r ; E ph ), by − D ∆ E e ∆ t E IC = Z ∞ dE ph Z E M E ph E γ [ n ph ( r ; E ph ) cσ IC ( E e , E γ ; E ph )] dE γ , (A2)with E M ≡ E M ( E e , E ph ) = E e X X ; X ≡ X ( E e , E ph ) = 4 E ph E e ( m e c ) , (A3)see the right-hand side of Table 5 for σ IC ( E e , E γ ; E ph ). Here we omit the suffix i introducedin § E e ≫ m e c , equivalently X ≫
1, one finds a reasonable result, E M ≈ E e , leading to E ph ≤ E γ ≤ E e . 41 –From equation (4) in the text n ph ( r ; E ph ) dE ph = (cid:20) ǫ ph ( r ) k B T ph (cid:21) W ph ( k ) dkk ; k = E ph k B T ph , where W ph ( k ) is the Planck function for the 2.7 K CMB, and the gaussian function given byequation (5) for the stellar radiation and the re-emission from the dust grains.The integration with respect to E γ is given (Jones 1965, 1968) by, (see eq. [A3] for X ) Z E M E ph E γ σ IC ( E e , E γ ; E ph ) dE γ ≈ σ (0) IC E e S IC ( X ) /X , with σ (0) IC = 3 σ T = 2 . × − cm , ( σ T : Thomson cross-section) , and S IC ( X ) = (cid:16) X X (cid:17) ln(1 + X ) − X /
12 + 6 X + 9 X + 4(1 + X ) − − Z X ln(1 + t ) t dt. One should note that the approximation used above is only E e ≫ m e c , readily satisfyingthe condition in the energy region of interest, E e > ∼
10 MeV.Now, we have the energy-loss rate due to the IC scattering in a compact form afterintegrating over the energy E ph of the target photon in equation (A2), − D ∆ E e ∆ t E IC = ǫ ph ( r ) w T Λ ( E e , T ph ) E e , (A4)where E e is in units of GeV and ǫ ph in eVcm − , and w T = 43 cσ T × − [ m e c / GeV] = 1 . × − cm s − , (A5) Λ ( E e , T ph ) ≡ Λ ( Θ e ) = 9 Z ∞ S IC ( X ) W ph ( Θ e X ) dX/X , (A6)with Θ e ≡ Θ e ( E e , T ph ) = m e c / p ( k B T ph ) E e . (A7)The above discussions are applicable also for the synchrotron radiation, since it is causedby the collision between an electron and the virtual photon induced by the magnetic field.Practically, however, we have the condition ~ ω c Γ e ≪ m e c ( Γ e : Lorentz factor of elec-tron) with ω c = eH ⊥ /m e c , and we can use the Thomson scattering cross-section, namely Λ ( E e , T ph ) →
1. Hence we obtain equation (12a). 42 –APPENDIX BCONTRIBUTION OF PERTURBATIVE TERMS IN THE TRANSPORT EQUATIONB1. HIGH ENERGY REGION E e > ∼ E + c Since we can neglect the fluctuation due to the reacceleration in the HE region, we takehere the second term in equation (25a) alone, omitting the second term in equation (22).The transport equation for the electron density in the HE region, N (0) e,ǫ ( r ; E e , t ), without theperturbative term, is given by h ∂∂t − ∇ · D ( r ; E e ) ∇ − ¯ ǫ ( r ) ∂∂E e W ǫ ( E e ) i · N (0) e,ǫ ( r ; E e , t ) = Q ( r ; E e , t ) . As discussed in § N (0) e,ǫ ( r ; E e , t ) as the solution of the first order approxi-mation for equation (21), so that we have the following equation with the perturbative term,¯ n ( r )[ W n ( E e ) N (0) e,ǫ ( r ; E e , t )] ′ , moving it to the right-hand side, h ∂∂t −∇· D ( r ; E e ) ∇− ¯ ǫ ( r ) ∂∂E e W ǫ ( E e ) i · N e,ǫ ( r ; E e , t ) = Q ( r ; E e , t )+¯ n ( r ) ∂∂E e W n ( E e ) N (0) e,ǫ ( r ; E e , t ) . Now, we rewrite the solution N e,ǫ ( r ; E e , t ) = N (0) e,ǫ ( r ; E e , t ) + ˜ N (0) e,n ( r ; E e , t ) , leading to h ∂∂t − ∇ · D ( r ; E e ) ∇ − ¯ ǫ ( r ) ∂∂E e W ǫ ( E e ) i · ˜ N (0) e,n ( r ; E e , t ) = ¯ n ( r ) ∂∂E e W n ( E e ) N (0) e,ǫ ( r ; E e , t ) . Thus for the steady state ( ∂/∂t = 0), we have the solution of the second order approximation˜ N (0) e,n ( r ; E e ) = Z ∞ ˜ Π (0) ǫ ( r ; y ) ˜ f (0) n ( y ; E e ) dy, with h ¯ ǫ ( r ) c ∂∂y − ∇· D ( r ) ∇ i · ˜ Π (0) ǫ ( r ; y ) = ˜ Q (0) ( r ) δ ( y ) , h cE αe ∂∂y − ∂∂E e W ǫ ( E e ) i · ˜ f (0) n ( y ; E e ) = 0 , where ˜ Q (0) ( r ) and ˜ f (0) n (0; E e ) are given by replacing Q (0) ( r ) [ ≡ Q ( r )] and f (0) ǫ (0; E e ) (seeeqs. [24b] and [28b]) with Q (0) ( r ) = Q e − r/r Q −| z | /z Q ⇒ ˜ Q (0) ( r ) = ¯ n ( r )e −| z | /z D Q ¯ ǫ J κ ( ω κ , U κ ) J κ ( U κ ) ,
43 – f (0) ǫ (0; E e ) = E − γ − αe ⇒ ˜ f (0) n (0; E e ) = E − αe c ∂∂E e h W n ( E e ) F (0) r,ǫ ( E e ) i , with J κ ( ω, U ) = Z t ω J κ ( U t ) dt, see Table 4 for κ, ω κ , and U κ , and J κ ( U ) is the Bessel function of the index with κ (Paper I).Corresponding to the replacement of Q (0) ( r ) ⇒ ˜ Q (0) ( r ), the scale heights in the source, r Q and z Q , must be replaced as1 r Q ⇒ r n , z Q ⇒ z n + 1 z D = 2¯ z n , leading to the following replacements, ω κ = (cid:16) z Q − z ǫ (cid:17) ⇒ (cid:16) z n − z ǫ (cid:17) ≡ ˜ ω κ , while the radial scale height in the source, r Q , doesn’t appear explicitly in this procedure.Now the Laplace transform of ˜ f (0) n ( y ; E e ) is immediately given (see eq. [31]) by˜ F (0) r,n ( E e ) F (0) r,ǫ ( E e ) = Z ∞ E e dE [ W n ( E ) F (0) r,ǫ ( E )] ′ W ǫ ( E e ) F (0) r,ǫ ( E e ) e − Y r,ǫ ( E e ,E ) , (B1)where [ · · · ] ′ denotes the differential with respect to E , and see § Y r,ǫ ( E e , E ), and weobtain ˜ N (0) e,n ( r ; E e ) N (0) e,ǫ ( r ; E e ) = 2¯ η J κ (˜ ω κ , U κ ) J κ ( U κ ) ˜ F (0) r,n ( E e ) F (0) r,ǫ ( E e ) , (B2)with ¯ η r = ¯ ǫ r / ¯ n r = ¯ η e − r (1 / ¯ r ǫ − / ¯ r n ) ; ¯ η = ¯ ǫ / ¯ n . In Figure 24a, we show ˜ N (0) e,n ( r ; E e ) /N (0) e,ǫ ( r ; E e ) against E e at the SS with [¯ η ⊙ , ¯ r ǫ ] = [2 eV,8 kpc], corresponding to [¯ ǫ ⊙ , ¯ n ⊙ ] = [2 eVcm − , 1 cm − ]. Then one finds that the perturbativecontribution due to the energy change in proportion to the gas density, ¯ n ( r ), is less than10% in the energy region E e > ∼ E + c .We finally obtain N e,ǫ ( r ; E e ) = 2 Q ¯ ǫ J κ ( ω κ , U κ ) J κ ( U κ ) F r,ǫ ( E e )e −| z | /z D , (B3)with F r,ǫ ( E e ) = F (0) r,ǫ ( E e ) + 2¯ η J κ (˜ ω κ , U κ ) J κ ( U κ ) ˜ F (0) r,n ( E e ) . (B4) 44 –B2. LOW ENERGY REGION E e < ∼ E + c In the LE region, the fluctuation term due to the reacceleration , h ∆ E e / ∆ t i rea , becomesnow effective as compared to the average energy-loss term due to the synchrotron-IC effect, h ∆ E e / ∆ t i sic , in proportion to the energy density, ¯ ǫ ( r ). So equation (25b) is approximatelywritten as ¯ n ( r ) W n ( E e ) + O [¯ ǫ ( r ) W ǫ ( E e )] ≃ ¯ n ( r )[ W n ( E e ) + ¯ η ∗ W ǫ ( E e )] , (B5)with ¯ ǫ ( r ) / ¯ n ( r ) ≈ h ¯ ǫ ( r ) / ¯ n ( r ) i eff ≡ ¯ η ∗ , (B6)where the effective value of ¯ η ∗ is of the magnitude of [1–5] eV, and for instance ¯ η ∗ = ¯ η ⊙ ≈ § ν , U ν ) N (0) e,n ( r ; E e ) = 2 Q ¯ n c J ν ( ω ν , U ν ) J ν ( U ν ) F (0) r,n ( E e )e −| z | /z D , (B7)with F (0) r,n ( E e ) = c |W n ( E e ) | Z E max E min dE E − γ e − Y r,n ( E e ,E ) , (B8)and Y r,n ( E e , E ) = c ¯ σ r Z E E e E α W n ( E ) dE. (B9) Electron energy; E e (GeV) N e , n ( r ; E e ) N e , e ( r ; E e ) ( ) / ( ) z : z n = kpc ( n = z D = kpc z e = kpc ( k = .. .. e /n = eV(a) contribution of perturbation term in HE region ~ -1 Electron energy; E e (GeV) N e , e ( r ; E e ) N e , n ( r ; E e ) ( ) / ( ) .. z = 50 mb s = 180 mb . (b) contribution of perturbation term in LE region ~ h * = Fig. 24.— Numerical results of the contribution of the perturbative terms in the (a) HEregion and (b) LE region. 45 –Now putting W ∗ n ( E e ) = W n ( E e ) + ¯ η ∗ W ǫ ( E e ) , the electron density with the synctrotron-IC effect (perturbative term here) is immediately N ∗ (0) e,n ( r ; E e ) = 2 Q ¯ n c J ν ( ω ν , U ν ) J ν ( U ν ) F ∗ (0) r,n ( E e )e −| z | /z D , (B10)where F ∗ (0) r,n ( E e ) is given by replacing W n with W ∗ n in equations (B8) and (B9), and it isrelated to the perturbvative term, ˜ N (0) e,ǫ , discussed in § N ∗ (0) e,n ( r ; E e ) = N (0) e,n ( r ; E e ) + ˜ N (0) e,ǫ ( r ; E e ) . (B11)Here one should be careful of the integral range of E in equation (B8), [ E min , E max ], sincewe have two zero points in W ∗ n ( E e ) at two energies, E − c ∗ and E + c ∗ , for instance, [ E − c ∗ , E + c ∗ ] ≈ [0 . ,
1] GeV for ζ = 50 mbarn and ¯ η ∗ = 2 eV, and[ E min , E max ] = [ E e , E − c ∗ ] for E e < E − c ∗ ,[ E − c ∗ , E e ] for E − c ∗ < E e < E + c ∗ ,[ E e , + ∞ ] for E e > E + c ∗ .We present ˜ N (0) e,ǫ /N (0) e,n against E e for several sets of [¯ η ∗ ; z n , z D ] at the SS in Figure 24b,and one finds it is much less than 10% in the low energy region E e < ∼ E + c .B3. CONTRIBUTION FROM THE FLUCTUATION IN THE REACCELERATIONOnce we confirm that the contribution of O [¯ ǫ ( r ) W ǫ ( E e )] is approximately given byequation (B5) with the energy loss proportional to ¯ n ( r ), it is possible to use the path lengthdistribution, Π n ( r ; x ), as presented in Paper I, Π n ( r ; x ) ≃ Q ¯ n c J ν ( ω ν , U ν ) J ν ( U ν ) e − ¯ σ r x −| z | /z D , (B12)but the slab equation is now slightly cumbersome, h c ¯ σ r E αe − ∂∂E e W ∗ n ( E e ) − cζ ∂ ∂E e E − αe i · F r,n ( E e ) = cE − γe . Remembering that F ∗ (0) r,n ( E e ) is a solution of the equation h c ¯ σ r E αe − ∂∂E e W ∗ n ( E e ) i · F ∗ (0) r,n ( E e ) = cE − γe ,
46 –we can write F r,n ( E e ) as F r,n ( E e ) = F ∗ (0) r,n ( E e ) + F (1) r,n ( E e ) , (B13)with h c ¯ σ r E αe − ∂∂E e W ∗ n ( E e ) i · F (1) r,n ( E e ) = cζ ∂ ∂E e E − αe F ∗ (0) r,n ( E e ) . (B14)The solution for equation (B14) is immediately obtained after the following replacementin equation (28b) cE − γe ⇒ cζ ∂ ∂E e E − αe F ∗ (0) r,n ( E e ) , and the explicit form is given by F (1) r,n ( E e ) F ∗ (0) r,n ( E e ) = cζ Z ∞ E e dE [ E − α F ∗ (0) r,n ( E )] ′′ |W ∗ n ( E e ) | F ∗ (0) r,n ( E e ) e − Y ∗ r,n ( E e ,E ) , where Y ∗ r,n ( E e , E ) is given by replacing W n ( E e ) in equation (B9) with W ∗ n ( E e ).Now, from equations (B12) and (B13), the electron density in the LE region is given by N e,n ( r ; E e ) = 2 Q ¯ n J ν ( ω ν , U ν ) J ν ( U ν ) F r,n ( E e )e −| z | /z D , (B15)where F r,n ( E e ) is given by equation (B13).Corresponding to equation (26b), we rewrite equation (B15), dividing into three termsas (see also eq. [B11]) N e,n ( r ; E e ) = N (0) e,n ( r ; E e ) + ˜ N (0) e,ǫ ( r ; E e ) + N (1) e,n ( r ; E e ) , and in Figure 7 in the text, we present N (1) e,n / [ N (0) e,n + ˜ N (0) e,ǫ ] against E e . One finds that thecontribution of the fluctuation is effective around GeV region, boosting the electron densitywithout the fluctuation by approximately 25%.APPENDIX CEMISSIVITY of γ ’s COMING FROM INVERSE COMPTON PROCESSIn this appendix, we omit the suffix i for simplicity. The production rate of γ ’s per unittime due to the bremsstrahlung, P EB ( r ; E e , E γ ) ≡ n ( r ) cσ EB ( E e , E γ ) in equation (34), mustbe replaced by P IC ( r ; E e , E γ ) ≡ Z ∞ E m n ph ( r ; E ph ) cσ IC ( E e , E γ ; E ph ) dE ph , (C1) 47 –where E ph is the energy of the target photon before electron scattering, and E m is given bysolving equation (A3) with E M ≡ E γ with respect to E ph ( ≡ E m ), E m ≡ E m ( E e , E γ ) = k B T ph Θ x , with Θ x ≡ Θ x ( E γ , T ph ; x ) = m e c / p ( k B T ph ) E γ x √ − x ; x = E γ E e , and σ IC ( E e , E γ ; E ph ) is the production cross-section of γ ’s due to IC scattering, which issummarized in the right-hand side of Table 5.Remarking that the integral range, E m ≤ E ph ≤ ∞ , in equation (C1) corresponds to0 ≈ ( m e c / E e ) ≤ q ≤
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