Report10.dvi

[76 ] Yuji Chinone - t t t = t t t = fl B = ce () - Δθ u u ΔS /γ /γ observer

= fl t t t t = = =fl B = ce - Eq.() t ο t v ο fl ce () c v fl fl - S = r = r fl = v ce S =c t t t ο t S c = ce ce v c = ce v c = t v ο = fl ce c ce fl -3 <ν<fl ce (3) -3 t ν > < ν <fl ce

-4 [μg] [TeV] [GeV] Hz ev -4 K E K u K = E m = m + p = m (4) E = m + p = (m + K) = m + mk + K ; ) p = K + mk (5) u = @E @p = p E ; u = p E = K + mk (K + m) (6) Lorentz factor fl = u = K + mk (K + m) = m + K m = + K m (7) 3 ν ν = 3 4 fl ce (8) ν = :39 ce = 7:588 fl 4 ψ B [μg] ψ B [μg] [Hz] ψ fl [GHz] ß 4 B :4 [μg] [GHz] (9) :4 [GHz] = [cm] Eq.(7) K = [TeV] ν = 5 7 [Hz]; hν = 3 [ev] K = [GeV] ν = 5 9 [Hz]; hν = 5 [ev] 3

- (=emissivity: ffl ff ν ) kev 3 cm 3 - ffl ff ν = dw(t;ν) dνdvdt ψ = 5 ße 6 = g ff T =ne 3mc 3 3k B m exp hν () ffl ff ν = = ß 3 3 3 = ß 3 3 3 = = 6:35 48 g ff = 6:35 48 g ff 3= ff 3 ( [kev]) = 6 [cm 6 ] :599898 6 [ev] = g ff [kev] 3 [cm 3 ] exp hν 3= i 3 ff 3 :599898 8 6 ] h() :98644544 :67646 3 9 7 = [kev] [kev] = = exp hν [ergsec Hz cm 3 ] g ff [kev] 3 [cm 3 ] exp hν [ergsec Hz cm 3 ] 3 [cm 3 ] exp hν 3 [cm 3 ] [erg sec Hz cm 3 ] () - kev 3 cm 3 Mpc 4

- (=bolometric luminosity: L) emissivity Z Z L = dw = dνdv dw = dνdv ffl ff ν dt dνdvdt ( 4 ) 3 = 6:35 48 3 ß 3:85677587 4 = Z V g ff dν exp [kev] 3 [cm 3 ] 4 3 ß [Mpc] 3 ( 4 = 6:35 48 3 ß = :8899 45 g ff = :89 45 g ff 3:85677587 4 3 ) hν [erg sec ] [kev] 4:356677 8 [kev] = V g ff [kev] 3 [cm 3 ] 4 [ergsec ] 3 ß [Mpc] 3 = V [kev] 3 [cm 3 ] 4 [erg sec ] 3 ß [Mpc] 3 = V [kev] 3 [cm 3 ] 4 [erg sec ] () 3 ß [Mpc] 3-3 = Λ kev = 3 [cm 3 ] [Mpc] = 3 4 [cm] [Mpc] [kev] [cm ] R [kev] x e x dx = :8 : -3 ffl ff ν hν hν = 3 [cm 3 ]; = [kev]; V = 4ß Z dw dν = dv = L = dn dt dνdvdt hν = :4684 53 [sec ] [kev] [Mpc] 3 =3 6:35 48 4 3 Z 6:666876 34 7 3 ß 3:85677587 4 x e x dx : (3) Z x e x dx = :8 : 5

g ff ß d = [Mpc] hν [kev] L = 4ßd I I = dn dtds = L 4ßd :4684 53 = = :7947 [cm 4ß 6 3:85677587 8 sec ] = :8 [cm sec ] (4) 3 RM RM Mpc L = Mpc = 3 cm 3 = kev 3-6

3- Mpc L = Mpc = [rad] = 8 6 [arcmin] ο 34 [arcmin] ß (b) ο 35 arcmin RM 35 ο arcmin RM RM 3-3- Rotation Measure RM = :8 Z d=[pc] ψ Bk ψ [μg] [cm 3 ] dl [rad m ] (5) ψ ψ RM = [rad m ] :8 Bk [μg] [cm 3 ] ψ L [pc] (6) L plasma (a) RM ο 4 rad m plasma B = 4 (:8) 3 6 = 4:9 [μg] 3-3 coherent length l = kpc l RM 3-3 random walk random walk l A hai = ; DA E = l ha + A + + A n i = ; E D E D(A + A + + A n ) = A + A + + A + A n A + A A 3 + E = n DA = nl = ff r ) ff = p L nl= l l = (Ll)= 7

Rotation Measure ψ ffirm = [rad m ] :8 ffib [μg] ψ [cm 3 ] ψ l [pc] r L l = :8 ψ ffib [μg] ψ [cm 3 ] (random) ffirm = 4 (:8) 3 3 6 = = :6[μG] ψ l L [pc] [pc] = (7) 4 Compton Scattering y Pγ, Pγ e - θe θ x Pe, 4- (ffl) (ffl ) 4- P μ fl ffl = c ; ffl ; ; c P μ = ffl fl; c ; ffl c cos ; ffl c sin ; (8) (9) P μ e = ( m e c ; ; ) () P μ = ( flm ec; flm e; e v cos e ; flm e v sin e ; ) () P μ fl + P μ e = P μ fl; + Pμ e; =) P μ e; = Pμ fl + P μ e P μ fl; () 8

P μ e; P e;μ = m e c = P μ flp flμ + P μ e P eμ + P μ fl; P fl;μ + P μ flp eμ P μ e P fl;μ P μ flp fl;μ = m e c + ( m e ffl) ( m e ffl ) fflffl c ( + cos ) ) ffl = ffl + ffl m e c ( cos ) (3) ffl = hν = hc= = + c ( cos ) ; c = h m e c = :463 [cm] = :463 [Å] (4) 5 K e - Pγ, K θ - θ e θ Pγ, θ Pγ Pγ 3 Compton Scattering K K (prime) K ffl fi m e c v = fic Lorentz factor fl = fi = x 5- K v t ο t + dt K dt obs dt dt obs 5- t obs; = t + n R c ; t obs; = t + n R c n v c (t t ) ; ) dt obs = t obs; t obs; = ( n fi) dt (5) 9

K R n/c t dt obs v n/c x dt 4 5- K dt K dt 5- Lorentz K K fl dt = fl dt ; fl (6) 5-3 dt ffl = fflfl ( fi cos ) ; ffl = ffl fl + fi cos (7) 5-3 Eq.(5),(6) dt obs = ( n fi) dt = h ; dt = dt = fl dt = h ; ) obs ffl ffl ffl = flh = fl ( n fi) ffl (8) dt Eq.(7) 5-4 Taylor

5-4 ffl = ffl + ffl ( cos ) = ffl ffl ( cos ) ο ffl ; * ffl fi m e c (9) m e c m e c K K y y z φ n θ x z φ n θ x 5 K n ; n n = cos ; sin cos ffi ; sin sin ffi n = cos ; sin cos ffi ; sin sin ffi (3) (3) n n = cos = cos cos + sin sin cos ffi cos ffi + sin sin sin ffi sin ffi = cos cos + sin sin cos ffi ffi ) cos =cos cos + sin sin cos ffi ffi (3) 5-5 ffl ο fl ( fi cos ) + fi cos ffl (33) = ß; = ffl = ffl ffl ffl 5-5 Eq.(7) Eq.(33) = ß; = ffl ο fl ( + fi) ffl (34)

5-6 Lorentz factor fl fl 4fl 5-6 ffl ffl ο fl ( + ) = 4fl (35) 5-7 fi fi ffl = fiffl 5-7 ffl = fl ( + fi) ffl = + fi ( + fi) ffl ffl = fiffl + O(fi 4 ) (36) 5-8 ffl = 3 4 ev Λ ffl ο kev Λ Lorentz factor 5-8 O( 8 ) fl fl Eq.(35) r 3 fl = r ffl ffl = 3 4 = 886:75 ο 3 3 (37) 5-9 ffl = 3 4 ev Λ kb T = 5 kev Λ 5-9 Λ Λ 5 kev fi m e = :5 3 kev Eq.(36) ψ = ψ = 3kB T ffl ο ffl = 3 4 3 5 3 Λ = :899 ο 4 ev m e c :5 6 (38)

6 Bessel Function J n (z) = ß Bessel Function cos (n' z sin ') d' (39) 6- n z J n(z) = J n (z) + J n+(z) (4) d dz J n(z) = J n (z) J n+(z) (4) 6- RHS of Eq.(4) = ß fcos ([n ] ' z sin ') + cos ([n + ] ' z sin ')g = ß n' z sin ' = ο dο = nd' z cos ' d'; = ß cos (n' z sin ') cos ' d' = n z ß = n z J n(z) + = LHS of Eq.(4) dο + nd' ) cos ' d' = ; z cos (n' z sin ') d' cos (n' z sin ') cos ' d' ' ß zß ο nß Z nß cos ο dο LHS of Eq.(4) = ß f sin (n' z sin ')g ( sin ') d' = ß = RHS of Eq.(4) fcos ([n ] ' z sin ') cos ([n + ] ' z sin ')g d' = J n (z) J n+(z) cos A + cos B = cos A + B cos A B ; cos A cos B = sin A + B sin A B 6- Bessel Function: J n (z) d dz y n (z) + z d dz y n(z) + ψ n y n (z) = (4) z 3

6- d dz J n (z) = d dz (J n (z) J n+(z)) = 4 (J n (z) J n (z)) 4 (J n(z) J n+(z)) = 4 (J n (z) J n (z) + J n+(z)) (43) d z z z dz J n(z) = z (J n (z) J n+(z)) = z (n ) (J n (z) + J n (z)) z (n + ) (J n(z) + J n+(z)) 4 n [(n + ) J n (z) + J n (z) (n ) J n+(z)] (44) J n (z) = z n (J n (z) + J n+(z)) = z n z z (n ) (J n (z) + J n (z)) + z n z (n + ) (J n(z) + J n+(z)) = 4n n [(n + ) J z n (z) + (n ) J n+(z)] + n J n(z) ψ z n =) J n n (z) = z n J z n(z) = 4n n [(n + ) J n (z) + (n ) J n+(z)] =) J n (z) = z n Φ n z Ψ f(n + ) J n (z) + (n ) J n+(z)g (45) Eq.(43),(44),(45) Eq.(4) + J n (z) + 4 4 (n + ) 4 4 (n + ) = 4 + 4 (n ) n 4 (n ) J n (z) + ψ J n+(z) + + 4 + 4 (n + ) n 4 (n + ) ψ n + d dz J n(z) = J n (z) J n+(z) = n z J n(z) J n+(z) J n+(z) = n =) ψ d + n J n (z) = J n+(z) dz z J n+(z) = n z J n (z) z J n(z) J n (z) 6-3 d dz J n(z) = J n (z) J n+(z) = J n (z) n z J n(z) + J n (z) = n ) =) ψ d dz + n z J n (z) = J n (z) ψ d + n + ψ d + n dz z dz z J n (z) = J n (z) Eq.(4) z J n(z) + J n+(z) J n (z) = = i n Z Z e iz sin ' in' d' (46) e iz cos '+in' d' (47) 4

6-3 J n (z) = ß = = = cos (n' z sin ') d' = Z e iz sin ' in' d' + e iz sin ' in' d' + e iz sin ' in' d' Z ß Z ß e iz sin ' in' + e iz sin '+in' e +iz sin ο inο dο; = ο + ; e +iz sin in d d' = ο ß ß e iz sin ' in' d' + Z ß e +iz sin ο inο dο J n (z) = = i n = i n = i n Z Z Z e iz sin ' in' d' = Z 3= ß= f 3=g ß= e iz cos ο+inο dο = i n 3= ψz Z e iz cos ο+inο dο = + = e iz cos '+in' d' = Z ß= e iz cos ο+inο i n ( dο) ; ; ' = ß ο; ' ψz + e iz cos '+in' d' = e iz cos ο+inο dο = + = Z 3= e iz cos ο+inο dο e iz cos '+in' d'; ο + = '; ο ß= 3= ο 3= ' ß= 6-4 e iz sin ' = +X n= J n (z)e +in' (48) 6-4 Fourier +X f (x) = f (x + n ) = n= +X =) f (') = e iz sin ' = e iz = sin('+) f (' + ) = c +in(= )x ; c n = n= ) e iz sin ' = Z c +in' ; c n = +X n= J n (z)e +in' f (x)e in(= )x dx: Z e iz sin ' in' d' = J n (z); * Eq.(46) 5