JAXA Sep., 2010 p.1/36
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- あきみ さわい
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1 JAXA Sep., 2 p./36
2 Contents Sep., 2 p.2/36
3 ffi(e;v) ) =3 dv (E=W M X ψ( ;u) W=W du P X ψ(w;u) ) =3 du (W=W n3:9 = 6 (H=H c ) 2=3 GeV =3 o =cm =W =3 ; with H c =4:4 3 G M X Comparison of the cross-sections radiation pair X in ffi(e; v)dv matter ψ(w;u)du X Φ(E; v)dv ffi(»;v) X in E=W dv photon gas P Ψ(W;u)du in X magnetic field M where and u W=E and v E=W =4ff(N=A)Z 2 r 2 ln(83z =3 ) X P =(3ff T =4)n =» X Sep., 2 p.3/36
4 Homogenious ffi(»; v) functions, ψ( ; and u), of the cross-section for Inverse Compton (left) and photon-photon pair production (right) are indicated below: φ(κ,v) κ=. κ=.2 κ=.5 κ=. κ=2. κ=5. κ= κ=2 κ=5 κ= ψ(λ,u) λ=.5 λ=3 λ=5 λ= λ=3 λ= v = ε γ / ε e Figure :»! " e =.,.2,.5,,, from left to right u = ε e / ε γ Figure 2:! " fl =.,.2,.5,,, from bottom to top. Sep., 2 p.4/36
5 Homogenious ffi(e;v) functions, and ψ(w;u), of the cross-section for photon radiation (left) and pair production (right) under the magnetic fields are indicated below: phi(x) psi(x) ϕ(v) ϕ(v) Figure 3: v. +( v)2 ( v) =3 2=3 v Figure 4: v u2 +( u) 2. ψ(u) =3 = u) =3 u ( Sep., 2 p.5/36 ffi(v) =
6 Number of electrons e+7 e+6 cascades in matter Our E /E = ^2 Our E /E = ^4 Our E /E = ^6 Our E /E = ^8 CasA E /E = ^2 CasA E /E = ^4 CasA E /E = ^6 CasA E /E = ^ cascade length (t) Number of electrons... cascades in photon field Our E /E = ^ Our E /E = ^2 Our E /E = ^3 Our E /E = ^4 AP s E /E = ^ AP s E /E = ^3 AP s E /E = ^ cascade length (t) Number of electrons cascades in magnetic field Our E /E = ^2 Our E /E = ^4 Our E /E = ^6 Our E /E = ^8 AP s E /E = ^2 AP s E /E = ^4 AP s E /E = ^6 AP s E /E = ^ cascade length (t) Figure 5: Transition curves of shower electron developing in matter (left), photon fields (middle), and magnetic fields (right). Our results in matter (lines) are compared with the analytical results in Nishimura, and our results in photon» fields = with 3 and in magnetic fields with cascade length defined by ( H 6 2 3:9 ) 3 x cm =( W GeV ) 3 c (lines) are compared with those indicated H in Aharonian and Plyasheshnikov (dots). Sep., 2 p.6/36
7 d ß(»; t) = ß(»)» dt» Z ffi(»; v)dv + Z Z Cascades in photon gas Diffusion equation ;» ffi(» )ß(» )» d»»»» ;» ψ( )fl( ) d ; +2» d dt fl( ; t) = Z» Z ffi(» ; )ß(» ) d»» fl( ) ψ( ; u)du;»» for the ß(»; t) differential fl( ; energy spectra, and t), with»! " e and! " fl ; where " e, " fl, and! denote the energies of the shower electron, shower photon, and background photon in units of mc 2, and» denotes» or of the incident particle. Sep., 2 p.7/36
8 Approximating the homogenious functions φ(κ,v) κ=. κ=.2 κ=.5 κ=. κ=2. κ=5. κ= κ=2 κ=5 κ= ψ(λ,u) λ=.5 λ=3 λ=5 λ= λ=3 λ= v = ε γ / ε e u = ε e / ε γ Figure 6: ffi(»; v) '. Figure 7: ψ( ; u) '. Sep., 2 p.8/36
9 Applying Mellin transforms t) M(s; @t fl(»; t) M(s; s d» ß(»; t) ; M(s ;t) = A +) 2=(s +) s=(s ; A N (s; t) : A (s; t) N we have the differential-difference equations, fl(»; t) with N (s; t) N (s ;t) R(s) and the differential spectra of shower particles become =(s +) t) ß(»; ff+i Z ff i»s ds M(s; t) Sep., 2 p.9/36 2ßi
10 N k (s) N n (s) N k (s) k R [k] (s) N k (s ) N (s k) ; A t; A We derive the approximated solution by dividing t with n equal stepsizes, t t=n, M k (s) = A M k (s ) + A k =; 2; ;n; with M (s) =and N (s) =. Then we have n X nc M t k A = n k= M (s where R [] (s) and R [k] (s) R(s) R(s ) R(s k +): Sep., 2 p./36
11 Applying the inverse Mellin transforms, we have the approximated differential electron spectrum ß n (»; t) as»ß n (»; t) = nx nc k t k nc k t k Z ff+i» s [k] ;2 (s)ds; R k 2 [k] 2) + ρ[k] ;2 () +» ρ ρ [k] ;2 ;2 ( ) ff ; (k» 2ßi k n k= ff i» R where denotes the,2 element of (s). R[k] (s) [k] ;2 [k] (s) have poles at s = -,, k-2, so we have the differential energy R spectrum of electron,»ß n (»; t) = nx k ρ»» n k= using ρ residues [k] (s) at ;2 residuess ρ[k] sṡ ;2 (s) ( ) k+ 2k(k +)=3 ( ) k 2k(k )=3 k 2 ( ) k+ 2k=3 Sep., 2 p./36
12 »t e» t=» ;» Π(»; t) = 2 3 t ρ(2 t)(»» )e t + te t ln» ρ t)(» (2 )e t + te t ln»»»t e» t=» ;» Exact solutions for the differential spectra and the transition curves At the limit of n!, we have the exact solution for the differential electron spectrum ß(»; t). The differential spectra and the integral spectra of electron components and photon components for gamma-initiated shower so obtained are»ß(»; t) = 2 3 t 2 t te t + 2 3» =»»fl(»; t) =»ffi(»» )e t + t 3 t 2 t e t 2 3» =» + 2(t)» E» E 2 (» t» ) ff ;» (»; t) =e t + t 3 where E 2 (z) denotes the exponential integral function, +2E E 2(t) 2» 2 (» t» ) ff ;»» E 2 (z) Z e zt t 2 dt: Sep., 2 p.2/36
13 Differential spectra of electrons and photons for cascades in photon gas.. κ π(κ,t)... e-5 e κ/κ t=. t=.2 t=.5 t=. t=.2 t=.5 t=. t=2. t=5. λ γ(λ,t)... t=. t=.2 t=.5 t=. t=.2 t=.5 t=. t=2. t=5. e-5 e λ/κ Figure 8:»-weighted differential energy spectrum of electrons for gamma-initiated showers of energy». Figure 9: -weighted differential energy spectrum of photon for gamma-initiated showers of energy». Sep., 2 p.3/36
14 Transition curves of electrons and photons for cascades in photon gas 5 4 Anal. κ /κ= Anal. κ /κ= 2 Anal. κ /κ= 3 Anal. κ /κ= 4 Num. κ /κ= Num. κ /κ= 2 Num. κ /κ= 3 Num. κ /κ= Anal. κ /κ= Anal. κ /κ= 2 Anal. κ /κ= 3 Anal. κ /κ= 4 Num. κ /κ= Num. κ /κ= 2 Num. κ /κ= 3 Num. κ /κ= 4 Π(κ/κ,t) 3 2 Γ(κ/λ,t) radiation length (t) radiation length (t) Figure : Transition curves of electrons with =,,, for 3 4» =» gamma-initiated showers 2 of energy. Analytical» results (linrs) are compared with numerical results(dots). Figure : Transition curves of photons with» = =, 2, 3, 4 for gammainitiated showers of energy». Analytical results (linrs) are compared with numerical results(dots). Sep., 2 p.4/36
15 @ fl(w;t) Z W Z W Z ffi(v)dv + Z W Z Cascades in magnetic field Akhiezer equation ß(E ß(E;t) = ß(E;t) ) =3 (E=W ) =3 de E ffi( E E ) E E ψ( ) fl(w ;t) (W=W ) =3 dw W W ; +2 E ß(E ;t) (E=W ) =3 de fl(w;t) (W=W ) =3 E ffi( W E ) ψ(u)du; W Sep., 2 p.5/36
16 We approximate the homogenious functions as ffi(v) =ψ(u) =a: phi(x) phic(x) psi(x) psic(x) φ(v) ψ(u) v u Figure 2: ffi(v) =2. Figure 3: ψ(u) =2. Sep., 2 p.6/36
17 M(s; N (s; t) M(s =3;t) = A s=(s +) 2=(s =(s +) ; A A The differential-difference equations for cascades in the magnetic fields are with N (s =3;t) R(s) a Sep., 2 p.7/36
18 X W s [k;=3] ;2 (s)ds; R ; A Then we have the Mellin transform functions, M(s; = A k t R[k;=3] (s) k! M(s k=3; where N (s; t) k= N (s k=3; ) R [;=3] (s) ; and R [k;=3] (s) R(s) R(s =3) R(s [k ]=3): Applying the inverse Mellin transforms, we have the differential electron spectrum ß(E;t) for gamma-initiated shower as X Z ff+i Eß(E;t)= k t k! ff i k= 2ßi E R where (s) denotes the,2 element of R (s). [k;=3] [k;=3] ;2 Sep., 2 p.8/36
19 (k 6)=3 ( ) k+ (2=3)a k k(k )(k 2)=6 (k 5)=3 ( ) k (2=3)a k (k +)k(k )=3 (k 4)=3 ( ) k+ (2=3)a k (k +2)(k +)k=6 residues ρ [k;=3] ;2 (s) in magnetic fields [k;=3] ;2 (s) have poles at s = ; 2=3; ; 2=3 and R s =(k 6)=3; (k 5)=3; (k 4)=3. s ρ residues [k;=3] (s) ;2 ( ) k+ (2=3)a k k(k +) (k +5)=36 2=3 ( ) k (2=3)a k (k )k (k +4)=72 =3 ( ) k+ (2=3)a k (k 2)(k ) (k +3)=36 ( ) k (2=3)a k (k 3)(k 2) (k +2)=36 =3 ( ) k+ (2=3)a k (k 4)(k 3) (k +)=72 2=3 ( ) k (2=3)a k (k 5)(k 4) k=36 Sep., 2 p.9/36
20 Exact solutions for the differential spectra and the transition curves The differential spectra and the integral spectra of electron components and photon components for gamma-initiated shower so obtained are Eß(E;t) = (2=3)(W =E) x(72 8x + 2x 2 3x 3 +3x 4 x 5 )e x =36 + (2=3)(W =E) 2=3 x 2 (36 48x + 8x 2 24x 3 + x 4 )e x =72 + (2=3)(W =E) =3 x 3 (2 9x +8x 2 x 3 )e x =36 + (2=3)x 4 (3 2x + x 2 )e x =36 + (2=3)(W =E) =3 x 5 (6 x)e x =72 + (2=3)(W =E) 2=3 x 6 e x =36 + (2=3)(W =E) x 3 e y =6 + (2=3)(W =E) x 2 (3 y)e y =3 + (2=3)(W =E) x(6 6y + y 2 )e y =6; where we define x at and y at(w =E) =3. Sep., 2 p.2/36
21 = Wffi(W W )e at (2=3)(W =W ) x 3 e y =6 (2=3)(W =W ) x 2 (3 y)e y =3 (2=3)(W =W ) x(6 6y + y 2 )e y =6; Wfl(W;t) + (=3)(W =W ) x(72 8x +2x 2 3x 3 +3x 4 x 5 )e x =36 + (=3)(W =W ) 2=3 x 2 (36 48x +8x 2 24x 3 + x 4 )e x =72 + (=3)(W =W ) =3 x 3 (2 9x +8x 2 x 3 )e x =36 + (=3)x 4 (3 2x + x 2 )e x =36 + (=3)(W =W ) =3 x 5 (6 x)e x =72 + (=3)(W =W ) 2=3 x 6 e x =36 Sep., 2 p.2/36
22 (E=W )x(2 + 2x 4y + x 2 =3 2xy +5y 2 x 2 y=6 +2xy 2 + x 2 y 2 =6)e y =3 x 4 (3 + 2x + x 2 )(E (x) E (y))=8; Π(E;t) = (2=3)( E=W )x(72 8x + 2x 2 3x 3 +3x 4 x 5 )e x =36 + f (E=W ) 2=3 gx 2 (36 48x + 8x 2 24x 3 + x 4 )e x =72 + 2f (E=W ) =3 gx 3 (2 9x +8x 2 x 3 )e x =36 + (2=3) ln(w =E)x 4 (3 2x + x 2 )e x =36 + 2f(W =E) =3 gx 5 (6 x)e x =72 + f(w =E) 2=3 gx 6 e x =36 + x(2 2x +2x 2 +x 3 + x 4 )e x =8 Sep., 2 p.22/36
23 = e at x(2 2x +2x 2 +x 3 + x 4 )e x =8 (W;t) + (=3)( W=W )x(72 8x +2x 2 3x 3 +3x 4 x 5 )e x =36 + (=2)f (W=W ) 2=3 gx 2 (36 48x + 8x 2 24x 3 + x 4 )e x =72 + f (W=W ) =3 gx 3 (2 9x +8x 2 x 3 )e x =36 + (=3) ln(w =W )x 4 (3 2x + x 2 )e x =36 + f(w =W ) =3 gx 5 (6 x)e x =72 + (=2)f(W =W ) 2=3 gx 6 e x =36 + (W=W )x(2 + 2x 4y + x 2 =3 2xy +5y 2 x 2 y=6 +2xy 2 + x 2 y 2 =6)e y =3 + x 4 (3 + 2x + x 2 )(E (x) E (y))=8; E where (x) R t e xt dt denotes the exponential integral function. Sep., 2 p.23/36
24 Differential spectra of electrons and photons for cascades in magnetic field E π(e,t).... e-5 e-6 e-8e-7e-6e E/E t=. t=.2 t=.5 t=. t=.2 t=.5 t=. t=2. t=5. W γ(w,t).... e-5 e-6 e-8e-7e-6e W/E t=. t=.2 t=.5 t=. t=.2 t=.5 t=. t=2. t=5. Figure 4: E-weighted differential energy spectrum of electrons for gamma-initiated showers of W energy. Figure 5: W-weighted differential energy spectrum of photon for gamma-initiated showers of W energy. Sep., 2 p.24/36
25 Transition curves of electrons and photons for cascades in magnetic field Anal. E /E= 2 Anal. E /E= 4 Anal. E /E= 6 Anal. E /E= 8 Anal. E /W= 2 Anal. E /W= 4 Anal. E /W= 6 Anal. E /W= 8 Π(E /E,t) Γ(E /W,t) radiation length (t) radiation length (t) Figure 6: W =E = W Transition curves of electrons with,,, for gamma-initiated showers of energy. Figure 7: Transition curves of photons with W =W =, 2, 3, 4 for gamma-initiated showers of energy W. Sep., 2 p.25/36
26 IACTs 5=3 s =2=3 ICRC2 P. Colin Id:92 F. Aharonian Invited review talk Sep., 2 p.26/36
27 ES28+34 SED: GeV TeV Preliminary The intrinsic spectrum between GeV and TeV can be well described by a pure power-law (index ~-.9)
28 RXJ TeV γ-rays and shell type morphology: acceleration of p or e in the shell to energies exceeding TeV can be explained by γ-rays from pp ->π o ->2γ HESS: dn/de=k E α exp[-(e/eo) β ] α=2. Eo=7.9 TeV β= α=.79 Eo=3.7 TeV β=.5 with just right energetics: Wp= 5 (n/cm -3 ) - erg/cm 3 (e.g. Berezhko et al, Blasi et al 27+) but IC models generally are more preferred because of TeV-X correlations (?) IC origin of g-rays cannot indeed excluded, but this is not a good argument
29 ( E π(e,t)... t=. t=.2 t=.5. t=. t=.2 t=.5 e-5 t=. t=2. t=5. e-6 e-8e-7e-6e E/E W γ(w,t)... t=. t=.2 t=.5. t=. t=.2 t=.5 e-5 t=. t=2. t=5. e-6 e-8e-7e-6e W/E Figure 8: SED ( ) SED( ) Sep., 2 p.29/36
30 ).. κ π(κ,t)... e-5 e κ/κ t=. t=.2 t=.5 t=. t=.2 t=.5 t=. t=2. t=5. λ γ(λ,t)... t=. t=.2 t=.5 t=. t=.2 t=.5 t=. t=2. t=5. e-5 e λ/κ Figure 9: SED ( ) SED( ) Sep., 2 p.3/36
31 ( E π(e,t). t=. t=.2 t=.5. t=. t=2. t=5.. t=. t=2. t=5.. e-8e-7e-6e W γ(w,t). t=. t=.2 t=.5. t=. t=2. t=5.. t=. t=2. t=5.. e-8e-7e-6e E/E W/E Figure 2: SED ( ) SED( ) Sep., 2 p.3/36
32 Sep., 2 p.32/36
33 5 4 Anal. κ/κ= Anal. κ/κ= 2 Anal. κ/κ= 3 Anal. κ/κ= 4 Num. κ/κ= Num. κ/κ= 2 Num. κ/κ= 3 Num. κ/κ= Anal. κ/κ= Anal. κ/κ= 2 Anal. κ/κ= 3 Anal. κ/κ= 4 Num. κ/κ= Num. κ/κ= 2 Num. κ/κ= 3 Num. κ/κ= 4 Π(κ/κ,t) 3 2 Γ(κ/λ,t) radiation length (t) radiation length (t) Figure 2: ( ) ( ) ( ) ( ) W =E W =W,,, Sep., 2 p.33/36
34 E π(e,t)... W γ(w,t)..... e-5 e-5 e-6 e-8e-7e-6e e-6 e-8e-7e-6e E/E W/E Figure 22: SED ( ) SED( ) ( ) ( ) t =.,.2,.5,.,.2,.5,., Sep., 2 p.34/36
35 E π(e,t). W γ(w,t) e-8e-7e-6e e-8e-7e-6e E/E W/E Figure 23: SED ( ) SED( ) ( ) Nishimura ( ) t =.,.2,.5,., 2., 5., 2, 5 Sep., 2 p.35/36
36 Sep., 2 p.36/36
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