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1 ( 517, ) u.ac.jp 1 γ <0.1Å X 0.1Å 6Å X 6Å 100Å 100Å 3000Å 3000Å 1µm 1µm 5µm 5µm 20µm 20µm 300µm 300µm 1mm 1mm 1cm 1cm 10cm >10cm λ (cm) ν (Hz) λ = c/ν (1) E (erg, ergs) T (K) E = hν = kt (2) 1
2 Planck c = e+10 cm sec 1 h = e 27 erg sec Boltzmann k = e 16 erg K 1 1eV = e 12 erg 13.6eV? 2 Surface brightness I ν (erg sec 1 cm 2 Hz 1 Sr 1 ) = I ν dνdadtdω (3) de = I ν dνdadtdω = I ν dν da dt dω (4) dadω = da dω = dada r 2 (5) dν = dν I ν = I ν (6) Flux density F ν (erg sec 1 cm 2 Hz 1, Jy) F ν = 1 Jy = 1 e 23 erg sec 1 cm 2 Hz 1 I ν cosθ dω (7) 2
3 Momentum flux density Π ν (dyn cm 2 Hz 1 ) Π ν = 1 c I ν cos 2 θ dω (8) Total flux F (erg sec 1 cm 2 ) F = Pressure p (dyn cm 2 ) ν2 ν 1 F ν dν (9) p = ν2 Intensity I (erg sec 1 cm 2 Sr 1 ) I = ν 1 Π ν dν (10) ν2 ν 1 I ν dν (11) Emissivity j ν (erg sec 1 cm 3 Hz 1 Sr 1 ) j ν = di ν ds Absorption coefficient α ν (cm 1 ) (12) di ν ds = α νi ν (13) 3 diν ds = j ν α ν I ν (14) 3
4 I ν (s) = I ν (0)+ s 0 j ν(s ) ds (15) I ν (0) = 0, j ν = const. I ν (s) = j ν s (16) Optical depth τ ν I ν (s) = I ν (0)e τ ν (17) τ ν s 0 α ν(s ) ds (18) Source function S ν (19)(20) (14) dτ ν = α ν ds (19) S ν j ν α ν (20) di ν dτ ν +I ν = S ν (21) e τ ν I ν = I ν (0)e τ ν + τν 0 e (τ ν τ ν ) S ν (τ ν) dτ ν (22) S ν = const. I ν (τ ν ) = S ν +e τ ν [I ν (0) S ν ] (23) τ ν I ν S ν 4
5 dω = da, dv = dads (24) r2 (7) cosθ 1 (12)(24) F ν = 1 r 2 Luminosity L ν (erg sec 1 Hz 1 ) j ν dv (25) P νv (25)(26) L ν = Total luminosity L tot (erg sec 1 ) j ν = 1 4π P νv (26) P νv dv = 4πr 2 F ν (27) L tot = 4πr 2 F ν dν (28) ( 0.1,1Jy@1.4GHz, F ν ν 0.7, 10GHz cutoff) Total Luminosity? ( Mpc, 1pc=3.09e+18 cm) 0 5
6 4 Plank law B ν (erg sec 1 cm 2 Hz 1 Sr 1 ) B ν = 2hν3 c 2 1 exp(hν/kt) 1 (29) T ((3) ) (1) (1) dν = c/λ 2 dλ B λ = 2hc2 λ 5 1 exp(hc/λkt) 1 R ( ) I ν = B ν (7) 2π F ν = B ν dφ 0 = πb ν sin 2 θ c = πb ν ( R r )2 θc 0 sinθcosθdθ r = R (30) (31) F ν = πb ν (32) 6
7 Stefan-Boltzmann law (9)(32) total flux F = 0 πb ν dν = 2πh ( kt )4 x 3 dx c 2 h 0 e x 1 = σt 4 (33) 0 x 3 dx e x 1 = π4 15 Stefan-Boltzmann Rayleigh-Jeans law (hν kt) exp σ 2π5 k 4 15c 2 h 3 (34) = e 5ergsec 1 cm 2 K 4 ( ) hν 1 = hν kt kt I RJ Wien law (hν kt) I W ν +... (35) ν (T) = 2ν2 kt (36) c2 ( 2hν3 (T) = exp hν ) c 2 kt (37) 7
8 Wien B ν ν ν max B ν ν ν=νmax = 0 (38) x = hν max /kt x = 3(1 e x ) x 2.82 ν max T = e+10 Hz K 1 (39) (λ max T = 5100 µm K ) B λ λ λ max B λ λ λ=λmax = 0 (40) y = hc/λ max kt y = 5(1 e y ) y 4.97 λ max T = 2898 µm K (41) S ν = B ν (23) I ν B ν 8
9 Brightness Temperature T b (K) I ν = B ν (T b ) (42) (31) B ν Rayleigh-Jeans law T b (23) T b = c2 2ν 2 k I ν = T(1 e τ ν ) (43) T b Color Temperature T c (K) Effective Temperature T eff (K) T b Total flux (9) Stefan-Boltzmann law (33) F = I ν cosθ dωdν = σt 4 eff (44) 9
10 Vega T c 1e+4 K Vega arcsec? (Vega = e+10 cm 1arcsec = 1/3600 F ν λ µm )
11 5 Einstein Mean intensity J ν = 1 4π Line profile function I ν dω (45) 0 φ(ν) dν = 1 (46) E E+hν 0 intensity J = 0 J ν φ(ν) dν (47) Einstein A A 21 (sec 1 ) 2 1 (Spontaneous emission) Einstein B B 12, B 21 B 12 J 1 2 (Absorption) B 21 J 2 1 (Stimulated emission) Einstein 1,2 11
12 n 1, n 2 n 1 B 12 J = n 2 A 21 +n 2 B 21 J (48) J = A 21 /B 21 (n 1 /n 2 )(B 12 /B 21 ) 1 (49) n 1 /n 2 n 1 n 2 = g 1 exp( E/kT) g 2 exp[ (E +hν 0 )/kt] ( ) hν0 kt = g 1 g 2 exp (50) g 1, g 2 (49)(50) J = g n = 2 n 1 (2l+1) = 2n 2 (51) l=0 A 21 /B 21 (g 1 B 12 /g 2 B 21 )exp(hν 0 /kt) 1 (52) Line profile function φ(ν) ν 0 (52) ν 0 (29) g 1 B 12 = g 2 B 21 (53) A 21 = 2hν3 0 c 2 B 21 (54) 12
13 1 j ν = hν 0 4π n 2A 21 φ(ν) (55) α ν = hν 0 4π (n 1B 12 n 2 B 21 )φ(ν) (56) (20) S ν = n 2 A 21 n 1 B 12 n 2 B 21 (57) (56)(57) (53)(54) α ν = hν 0 4π n 1B 12 S ν = 2hν3 0 c 2 ( ) 1 g 1n 2 φ(ν) (58) g 2 n 1 ( ) g2 n (59) g 1 n 2 (50) (58)(59) α ν = hν [ ( )] 0 4π n hν0 1B 12 1 exp φ(ν) kt (60) S ν = B ν0 (T) (61) (50) n 1 g ( ) 1 hν0 exp (62) n 2 g 2 kt 13
14 ( Maxwell ) (50) n 2 g 1 n 1 g 2 = exp ( hν0 kt ) < 1 (63) n 1 g 1 > n 2 g 2 (64) n 1 g 1 < n 2 g 2 (65) (58) 6 Scattering coefficient σ ν (cm 1 ) (coherent/elastic/monochromatic scattering) mean intensity (45) j ν = σ ν J ν (66) 14
15 (14) di ν ds = σ ν(i ν J ν ) (67) ( ) j ν = α ν S ν = α ν B ν (14)(67) di ν ds = α ν(i ν B ν ) σ ν (I ν J ν ) (68) = (α ν +σ ν )(I ν S ν ) (69) S ν = α νb ν +σ ν J ν α ν +σ ν (70) Extinction coefficient α ν +σ ν (cm 1 ) (19) dτ ν = (α ν +σ ν )ds (71) Mean free path l ν (cm) τ ν = (α ν +σ ν )l ν = 1 (72) l ν = 1 α ν +σ ν (73) 15
16 Albedo 1 ǫ ν ǫ ν ǫ ν = α ν α ν +σ ν (74) 1 ǫ ν = σ ν α ν +σ ν (75) single-scattering albedo (70) Diffusion length l (cm) S ν = (1 ǫ ν )J ν +ǫ ν B ν (76) N Mean free path l ν N ǫ ν l = Nl ν = l ν ǫν = 1 αν (α ν +σ ν ) (77) diffusion/thermalization length effective mean path Rosseland (78) (69) ds = dz cosθ = dz µ I ν (z,µ) = S ν 16 µ I ν α ν +σ ν z (78) (79)
17 2 mean free path I ν I ν 0 2 (70) I (0) ν = S (0) ν = B ν (80) (79) (7) I (1) ν (z,µ) = B ν F ν (z) = = 2π Total flux(9) I ν (1) +1 1 I(1) ν µ B ν α ν +σ ν z (z,µ)cosθ dω (z,µ)µ dµ = 2π B ν α ν +σ ν z 4π B ν = 3(α ν +σ ν ) T +1 1 µ2 dµ T z (81) (82) F(z) = 0 F ν (z) dν = 4π T 3 z 0 = 16σT3 T 3α R z 1 α ν +σ ν B ν T dν (83) 17
18 Rosseland mean absorption coefficient 1 α R (33) 0 1 B ν α ν +σ ν T dν 0 B ν T dν (84) B ν 0 T dν = B ν dν = 4σT3 (85) T 0 π (83) Rosseland approximation equation of radiative diffusion T C (core) T S (shell) ν 0 α ν A, B T C > T S, T C < T S Tc A Ts B 18
19 7 P, P (erg sec 1 ) P = e2 6πε 0 c 3 r2 (86) P = e2 x 2 0ω 4 12πε 0 c 3 (87) e = e 19 C ε 0 = e 21 g 1 cm 3 sec 2 C 2 m = e 28 g Radiation reaction force F rad (dyn) F rad ṙ = P (88) 1 t 1 t 2 t2 t2 F rad ṙ dt = e2 r r dt t 1 6πε 0 c 3 t 1 = e2 ( [ r ṙ] t ) t2 2 6πε 0 c 3 t1 ṙ ṙ dt t 1 (89) r ṙ(t 1 ) = r ṙ(t 2 ) F rad = e2 6πε 0 c 3ṙ = mτṙ (90) Radiation reaction time scale τ (sec) τ 2r 0 3c = e 24 sec (91) 19
20 r 0 (cm) e 2 r 0 = = e 13 cm (92) 4πε 0 mc2 Radiation reaction F = m( r τṙ ) = mω 2 0 r+ee 0e iωt (93) x τẋ +ẍ+ω0x 2 = ee 0 m eiωt (94) x = x 0 e iωt = x 0 e i(ωt+δ) (95) x 0 = ee 0 1 m ω 2 ω0 2 iω 3 τ (96) tanδ = ω3 τ ω 2 ω0 2 (97) (87) P = e2 x 0 2 ω 4 12πε 0 c 3 e 4 E0 2 ω 4 12πε 0 m 2 c 3 (ω 2 ω0) 2 2 +(ω0τ) 3 (98) 2 ( τω 1 ω ω 0 ω ω 0 ) flux (Poynting vector S ) S (erg sec 1 cm 2 ) S = 1 2 ε 0cE 2 0 (99) 20
21 Scattering cross section σ (cm 2 ) σ P S = σ ω 4 T (ω 2 ω0) 2 2 +(ω0τ) 3 (100) 2 Thomson cross section σ T (cm 2 ) σ T = 8π 3 r2 0 = e 25 cm 2 (101) Thomson scattering X ω ω 0 (100) Rayleigh scattering σ σ T (102) ω ω 0 (100) σ σ T ( ω ω 0 ) 4 (103) ω ω 0 ω 2 ω0 2 = 2ω 0(ω ω 0 ) ω = ω 0 (100) Γ σ πcr 0 (104) (ω ω 0 ) 2 +(Γ/2) 2 Classical damping constant Γ (rad 2 sec 1 ) Γ ω 2 0τ (105) 21
22 Oscillator strength f ij (104) ω ω 0 (100) (104) 0 σ dν 1 2π σ dω = πcr 0 = e2 4ε 0 mc (106) f ij 0 σ dν = e2 4ε 0 mc f ij = hν 0 4π B ij (107) (56) 1 n r n r = c c = ε µ = ε r µ r (108) ε 0 µ 0 µ r 1 (n r 1) n n r = ε r = (96)(105) n r = nex ε 0 E 1+ nex 2ε 0 E ne 2 2ε 0 m(ω 2 0 ω 2 +iγω) 22 (109) (110)
23 F = m r = ee 0 sinωt (111) (101) (Θ r ) dp dω = e2 r 2 Θ (112) 16π 2 ε 0 c 3sin2 Thomson scattering (θ ) dσ dω = 1 2 r2 0(1+cos 2 θ) (113) θ ( x y ) y x π 2 θ π 2 θ z 23
24 8 Fourier ˆf(ω) = 1 2π f(t) = Parseval f2 (t) dt = 2π f(t)eiωt dt (114) ˆf(ω)e iωt dω (115) ˆf(ω) 2 dω (116) W (erg) W = P dt = 0 dw dω dω (117) (86) (116) W = = = 0 e 2 6πε 0 c 3 r2 dt e 2 3ε 0 c 3 ˆ r 2 dω 2e 2 3ε 0 c 3 ˆ r 2 dω (118) dw/dν (erg Hz 1 ) dw dν = 2πdW dω = 4πe2 3ε 0 c 3 ˆ r 2 (119) 24
25 (114) b (cm) ˆ r = 1 2π reiωt dt = 1 2π r dt, ωb v 0, ωb v (120) Ze Ze2 dt = r 4πε 0 m (120)(121) (119) dw dν = b dt Ze2 = (b 2 +v 2 t 2 ) 3/2 2πε 0 mbv (121) Z 2 e 6 12π 3 ε 3 0 c3 m 2 b 2 v 2, ωb v 0, ωb v (122) b b+db n e n i 2πvb db (122) d 3 W dνdvdt = d2 P dνdv bmax dw(b) = n e n i 2πv b db b min dν e 6 = 6π 2 ε 3 0c 3 m 2 v n en i Z 2 ln ( ) bmax b min (123) 25
26 b max b max = v 2πν (124) b min b min = Gaunt factor g ff Ze 2 1 2πε 0 mv, 2 2 mv2 13.6Z 2 ev h mv, 1 2 mv2 13.6Z 2 ev (125) g ff = ( ) 3 π ln bmax b min (123) g ff (126) d 2 P dνdv = e 6 6 3πε 3 0m 2 c 3 v n en i Z 2 g ff (127) Maxwell v 2 exp( mv 2 /2kT) dv (127) 1 2 mv2 hν v min = (2hν/m) 1/2 vmin d 2 d P 2 P dνdv = dνdv v2 exp( mv 2 /2kT) dv 0 v 2 exp( mv 2 /2kT) dv ( ) m 3/2 d 2 P = 4π 2πkT v min dνdv v2 exp mv2 dv 2kT = e 6 2π 6π 2 ε 3 0mc 3 3mkT n en i Z 2 e hν/kt ḡ ff (128) 26
27 (ḡ ff velocity averaged Gaunt factor) 0 e ax2 dx = x 2 e ax2 dx = 4π π a ( a 3/2 1 π) (129) (130) xe ax2 dx = 1 x 0 2a e ax2 0 (131) S ν = B ν (20) (26) j ν = 1 d 2 P 4πdνdV (29) j ν = α ν B ν (132) (133) e 6 α ν = 2π n e n i Z 2 48π 3 ε 3 (1 e hν/kt )ḡ 0mhc 3mkT ν 3 ff (134) hν kt e hν/kt 0 hν kt e 6 α ν = n e n i Z 2 48π 3 ε 3 ḡ 0mkc 3mkT 3 ν 2 ff (135) = n en i Z 2 T 3/2 ν 2ḡff (136) 27
28 L (cm) (18) Emission measure EM (cm 6 pc) 1pc = 3.086e+18 cm τ ν = α ν L (137) (Hii ) Z = 1, n i = n e EM n 2 e L (138) (137) EM (136) τ ν = (5.5e+16) EM T 3/2 ν 2ḡff (139) τ ν = 1 ν c EM ν c = (2.3e+8) ḡff (140) (43) T 3/4 I ν = 2kν2 T c 2 b = (3.07e 37)ν 2 T(1 e τ ν ) (3.1e 37)ν 2 T, ν ν c = (1.7e 20) EM (141) T ḡ ff, ν ν c ν 2 28
29 9 γm v = ev B (142) γ = (1 β 2 ) 1/2, β = v c (143) Gyration frequency ω B (rad Hz) v B α (142) v v rω B = v γmrωb 2 = ev B = vcosα = const. = vsinα = const. (144) ω B = eb γm (145) r = γm vsinα eb (146) Lorentz a = γ 3 a = 0 a = γ2 a = γ 2 ω B vsinα = γeb m vsinα (147) (86) P = e2 6πε 0 c 3 γ 2 e 2 B 2 m 2 v 2 sin 2 α = ε 0 σ T c 3 β 2 γ 2 B 2 sin 2 α (148) 29
30 sin 2 α 1 4π sin 2 α dω = 2 3 (149) (148) P = 2 3 ε 0σ T c 3 β 2 γ 2 B 2 = 4 3 σ Tcβ 2 γ 2 U B (150) U B (erg cm 3 ) U B = B2 2µ 0 = 1 2 ε 0c 2 B 2 (151) 1/γ t v v B (142) v r ω B γmr ω B 2 = const. = v = evbsinα (152) ω B = ω B sinα (153) r = r sinα (154) 30
31 t ω B t = 2 γ (155) t = 2 γω B sinα (156) t t A v c t A = ( 1 v c c t A = c t v t (157) ) t t 2γ 2 = 1 γ 3 ω B sinα (158) 1/ t A v c F dp 3 dν = e 3 ( Bsinα ν F (159) 4πε 0 mc νc) ν c = 3 4π γ3 ω B sinα = 3e 4πm γ2 Bsinα (160) F(x) = x = x K 5/3 (η) dη (161) ( x 2 ) 4π 1/3 3Γ = 2.13x ( 3) 1/3, x 1 1 (πx/2) 1/2 e x, x 1 F(x) dx = 8π 0 9 (162) 3 (159) ν (148)(β = 1) 31
32 (150) β = 1 E = γmc 2 P = de dt = 4σ T 3m 2 c 3U BE 2 (163) E(t) = E(0) 1+ t 1 (164) t 1/2 t 1/2 E(t) = 1 E(0) 2 t 1/2 = 3m2 c 3 UB 1 E(0) 1 (165) 4σ T spectral index p N(E) E p (166) dp tot dν = N(E) dp 0 dν de 0 γ p BF ( ν νc) dγ (167) x = ν/ν c (160) ν c γ 2 B ν x = ν c γ 2 B (168) ν x2dx γbdγ (169) (167) dp tot dν ν (p 1)/2 B (p+1)/2 x (p 3)/2 F(x) dx 0 ν (p 1)/2 B (p+1)/2 (170) 32
33 F ν ν s s = p 1 2 (171) (56) α ν = hν 0 4π [N(E hν 0)B 12 N(E)B 21 ]φ(ν)dν 0 de (172) φ(ν) = δ(ν ν 0 ) dp α ν = hν 4π [N(E hν)b 12 N(E)B 21 ] de (173) dν (55) dp dν = (54)(174) hν 0 A 21 φ(ν) dν 0 = hνa 21 (174) B 21 = c2 2hν 3A 21 = c2 dp 2h 2 ν 4 dν (175) (53) g p p+dp g(e)de = 4πp 2 dp = 4πE2 c 3 de (176) 33
34 (53)(176) B 12 = = g(e) g(e hν) B 21 = g(e) ( 1+ 2hν E ) g(e) hν dg(e) de 1 B 21 B 21 (177) N(E hν) = N(E) hν dn(e) de = N(E) 1 hν dn(e) (178) N(E) de (175)(177)(178) (173) α ν = c2 8πhν 3 = c2 8πν 2 ν 2 γ (p+1) BF (168)(169) N(E) 2hν E hν N(E) E 2 d N(E) dp de E 2 dν de ( ν νc) dn(e) de dp dν de dγ (179) α ν ν (p+4)/2 B (p+2)/2 x (p 2)/2 F(x) dx 0 ν (p+4)/2 B (p+2)/2 (180) (20)(170)(180) S ν = 1 4πα ν dp tot dν ν5/2 B 1/2 (181) (165) 3µG 5GeV t 1/2 (1G=0.1 g sec 1 C 1 ) 34
35 10 Compton scattering (ǫ 0,p 0 ) (ǫ,p), (mc 2,0) (γmc 2,p e ) p 0 = p+p e p e 2 = p 0 p 2 ( ) 2 ( ) (γmβc) 2 ǫ0 ǫ 2 ( )( ) ǫ0 ǫ = + 2 cosθ c c c c (γmc 2 ) 2 = (mc 2 ) 2 +ǫ 2 0 +ǫ 2 2ǫ 0 ǫcosθ (182) (182)(183) mc 2 +ǫ 0 = γmc 2 +ǫ (γmc 2 ) 2 = (mc 2 +ǫ 0 ǫ) 2 (183) ǫ = 1+ ǫ 0 mc (1 cosθ) 2 Compton wavelength λ c (cm) (184) ǫ 0 (184) λ = λ λ 0 = λ c (1 cosθ) (185) 35
36 λ c h mc = e 10 cm (186) Klein-Nishina formula dσ dω = r2 0ǫ 2 2 ǫ 2 0 ( ǫ0 ǫ + ǫ ) sin 2 θ ǫ 0 x = ǫ 0 /mc 2 (184) Ω σ = σ T ( 3 8 σ T 1 x Doppler effect 1 2x+ 26x2 ( ln2x+ 1 2 Inverse Compton scattering (187) ) +, x 1 5 ) (188), x 1 c ν = γ(c vcosθ) ν (189) ǫ = ǫγ(1 βcosθ) (190) n(ǫ 0 )dǫ 0 ǫ 0 ǫ 0 + dǫ 0 (cm 3 ) ndǫ 0 /ǫ 0 Lorentz ( ) ndǫ 0 ǫ 0 = n dǫ 0 ǫ 0 (191) Lorentz 36
37 ( ) de dt = de = cσ ǫ n dǫ dt 0 (192) Thomson σ = σ T, ǫ = ǫ 0 (192) (191)(190) de dt = cσ T ǫ 2n dǫ 0 0 ǫ 0 = cσ T ǫ 2ndǫ 0 0 ǫ 0 = cσ T γ 2 (1 βcosθ 0 ) 2 ǫ 0 n dǫ 0 (193) (193) (1 βcosθ 0 ) 2 = β2 (194) de dt = cσ Tγ 2 (1+ 1 ) 3 β2 U ph (195) U ph (erg cm 3 ) U ph = ǫ 0 n dǫ 0 (196) de 0 dt = cσ T ǫ 0 n dǫ 0 = cσ T U ph (197) 37
38 (195)(197) γ 2 1 = γ 2 β 2 [ ( P = cσ T U ph γ ] ) 1 3 β2 = 4 3 σ Tcβ 2 γ 2 U ph (198) (150) P synch P compt = U B U ph (199) Compton Thomson (190) ǫ = ǫ 0 ǫ 0 = ǫ 0 γ(1 βcosθ 0 ) ǫ = ǫ γ(1+βcosθ ) (200) ǫ = ǫ 0 γ 2 (1 βcosθ 0 )(1+βcosθ ) < 4γ 2 ǫ 0 (201) Compton ǫ 0 n 38
39 Compton dp dν = 3hcσ Tnxf(x) (202) x ν 4γ 2 ν 0 (203) f(x) = 2 (1 x) (204) 3 Tomson (113) f(x) = 2xlnx+x+1 2x 2 (205) (201) 0 < x < 1 (202) ν P = 3hcσ T n = 12σ T cγ 2 ǫ 0 n xf(x) 4γ 2 ν 0 dx xf(x) dx = 4 3 σ Tcγ 2 ǫ 0 n (206) ǫ 0 n = U ph (198)(β = 1) (199) (165) U B U ph t 1/2 = 3m2 c 3 4σ T U 1 ph E(0) 1 (207) 39
40 spectral index p N(E) E p (208) dp tot dν = N(E) dp 0 dν de γ p nxf(x) dγ (209) 0 (203) ν ν 0 x γ2 (210) (209) ν ν 0 x2dx γdγ (211) ( ) dp tot ν (p 1)/2 dν 1 n ν 0 0 x(p 1)/2 f(x) dx ν (p 1)/2 ν (p 1)/2 0 n (212) ν (p 1)/2 0 n ν (p 1)/2 0 n(ν 0 ) dν 0 (213) F ν ν s s = p 1 (214) 2 (171) 40
41 11 Maxwell E,D,H,B e i(k r ωt) divd = ρ ik D = ρ (215) divb = 0 ik B = 0 (216) roth = j+ D ik H = j iωd (217) t rote = B ik E = iωb (218) t D = ε 0 E, B = µ 0 H (219) ( ) E e i(k r ωt) m v = ee (220) v = ee iωm Condactivity σ (Ω 1 cm 1 ) n (221) j = nev = σe (222) σ = ine2 ωm (223) 41
42 Maxwell ( (217) k ) (222)(225) divj+ ρ t = 0 (224) k j = ωρ (225) ρ = σk E ω (226) (215)(217) (222)(226) iεk E = 0 (227) ik H = iωεe (228) ε ( ) ε = ε 0 ( 1 σ = ε 0 iε 0 ω 1 ω2 p ω 2 ω p (rad Hz) ) (229) ω 2 p = ne2 ε 0 m (230) 42
43 n r (108) n r = ε ε 0 = v ph (cm sec 1 ) 1 ω2 p ω 2 (231) v ph ω k = c n r (232) k = n rω = 1 ω c c 2 ωp 2 (233) ω < ω p k E exp ω p 1 ω2 r e iωt (234) c ω 2 p c/ω p = λ p /2π v g (cm sec 1 ) (233) ω = c 2 k 2 +ω 2 p v g = ω k = cn r (235) Dispersion measure DM (cm 3 pc) ω ω p L (cm) t p t p = L v g L c ω2 p 2ω 2 (236)
44 ω t p t p = Lω2 p nle2 ω = cω3 ε 0 mcω3 ω (237) DM nl (238) (L pc) ν (MHz) t p = (4.15e+3)DM (ν 2 ) (239) DM ( ) B 0 (220) m v = e(e+v B 0 ) (240) B 0 (z) (xy ) / 2 E = Ee iωt (e x ie y ) B 0 = B 0 e z v = ae (240) iωmae = e(e+ae B 0 ) 44 (241)
45 = e(e iab 0 E) e v = im(ω ±ω B ) E (242) ω B cyclotron frequency (145)(γ = 1) ω B = eb 0 m E B 0 (243) (e x ie y ) e z = e y ie x = i(e x ie y ) (244) (242) (221) (223) ω ω ± ω B (229) ε ( ) ωp ε R,L = ε (245) ω(ω ±ω B ) Faraday rotaion θ (rad) ω ω p, ω ω B (233) ω 2 p k R,L = ω 1 c ω(ω ±ω B ) ω 1 ω2 ( p 1 ω ) B c 2ω 2 (246) ω L (cm) / 45
46 ( θ) θ = 1 2 (k R k L )L = ω2 p ω B 2cω 2 L Rotation measure RM (rad cm 2 ) (247) Depolarization = nlb 0e 3 2ε 0 m 2 cω 2 (247) θ = RMλ 2 (248) RM = nlb 0e 3 8π 2 ε 0 m 2 c 3 = 81.2nLB 0 (249) (L pc, B 0 gauss) Faraday rotation θ π λ d λ d = 0.20(nLB 0 ) 1/2 (250) 46
47 12 1 Schrödinger h ψ 2m j=1 x 2 +(E V) = 0 (251) j V = Ze2 4πε 0 r (252) 1 ( m a ) E n (erg) Rydberg E n = RhcZ2 n 2 (253) R = R ( 1 m m a ) (254) R = me4 8ε 2 0h 3 c = e+5 cm 1 (255) 1 λ nn (cm) 1 λ nn ( 1 = RZ 2 n 1 ) 2 n 2 (256) Schrödinger h ψ 2m i j=1 x 2 +(E V) = 0 (257) ij V = e2 4πε 0 i Z r i + i 47 1 i <i r ii +H (258)
48 (H ) n : 1,2,3,4,5,6,7,... K,L,M,N,O,P,Q l : 0,1,2,3,...,n 1 n s,p,d,f,... m l : l,...,+l 2l+1 s : 1/2 m s : 1/2,+1/2 2 ( ) j : l+s,..., l s L : l 0,1,2,3,... S,P,D,F,... M L : m l L,...,+L 2L+1 S : s M S : m s S,...,+S 2S +1 J : L+S,..., L S (L,S 0 ) 48
49 m s m l (Zeeman ) L,S (2L+1)(2S +1) J 2J +1 i, +,2+,... ii,iii,... (Hi, Heii, Oiii ) (selection rule) L = ±1,0 ( L = 0 0 ) S = 0 ( ) J = ±1,0 ( J = 0 0 ) (permitted line) (forbidden line) ( [ ]) 2p 3 4 S3/2 o 2 n = 2 (L ) p 3 l = S +1 = 4 (S=3/2), 4 S L = 0 3/2 J = 3/2 o ( ) 49
50 Hi, Heii : 1s,(L,S) = (0,1/2) 1s 2 S 1/2 M L \M S +1/2 1/2 0 (0 ) (0 ) (0 m l,m s = 0,+1/2 ) : 2p,(L,S) = (1,1/2) 2p 2 P 3/2,1/2 Hei M L \M S +1/2 1/2 +1 (+1 ) (+1 ) 0 (0 ) (0 ) 1 ( 1 ) ( 1 ) : 1s 2,(L,S) = (0,0) 1s 2 1 S 0 M L \M S (0,0 ) ( ) : 1s2s,(L,S) = (0,1),(0,0) 1s2s 3 S 1, 1 S 0 M L \M S (0,0 ) (0,0 ),(0,0 ) (0,0 ) ( ) : 1s2p,(L,S) = (1,1),(1,0) 1s2p 3 P 2,1,0, 1 P 1 M L \M S (0,+1 ) (0,+1 ),(0,+1 ) (0,+1 ) 0 (0,0 ) (0,0 ),(0,0 ) (0,0 ) 1 (0, 1 ) (0, 1 ),(0, 1 ) (0, 1 ) 50
51 Nii, Oiii 1s 2 2s 2 2p 2 1s 2 2s 2 1 ( ) (L,S) = (2,0),(1,1),(0,0) 2p 2 1 D 2, 3 P 2,1,0, 1 S 0 M L \M S (+1,+1 ) +1 (+1,0 ) (+1,0 ),(+1,0 ) (+1,0 ) 0 (+1, 1 ) (+1, 1 ),(0,0 ),(+1, 1 ) (+1, 1 ) 1 ( 1,0 ) ( 1,0 ),( 1,0 ) ( 1,0 ) 2 ( 1, 1 ) Oii, Sii 1s 2 2s 2 2p 3,(L,S) = (2,1/2),(1,1/2),(0,3/2) 2p 3 2 D 5/2,3/2, 2 P 3/2,1/2, 4 S 3/2 M L \M S +3/2 +1/2 1/2 3/ (+1,+1,0 ) (+1,+1,0 ) +1 (+1,0,0 ) (+1,0,0 ) (+1,+1, 1 ) (+1,+1, 1 ) 0 (+1,0, 1 ) (+1,0, 1 ) (+1,0, 1 ) (+1,0, 1 ) (+1,0, 1 ) (+1,0, 1 ) (+1,0, 1 ) (+1,0, 1 ) 1 ( 1,0,0 ) ( 1,0,0 ) ( 1, 1,+1 ) ( 1, 1,+1 ) 2 ( 1, 1,0 ) ( 1, 1,0 ) 3 51
52 n+ n n n n Lyman Balmer Paschen Brackett Pfund Ly H Pa Br Pf n = 1,2,3,... α,β,γ,... n n (Å) Lyα Lyβ Lyγ Ly limit Hα Hβ Hγ H limit Paα Paβ Paγ Pa limit (256) 52
53 OIII NII 1 S0 1 S A 1 D2 TA D N P P 3/2 1/ OII P A 3/2 1/2 SII 2 P 5/2 3/2 2 D 5/2 3/2 2 D N TA S3/2 4 S3/2 [Oiii]λ5007, [Oii]λλ3727 (λλ 2 ) N/A/TA nebular/auroral/trans-auroral line 53
54 E e ( Λ,Σ )/ E v ( v)/ E r ( J) ( ) ( E v = hcω e v + 1 ) +... (259) 2 E r = h2 J(J +1) = hbj(j +1) (260) 8π 2 I (ω e :, I:, B: ) J = 2, 1,0,+1,+2 S,R,Q,P,O branch (v,j) = (1,4) (0,2) H 2 v=1 0S(2) ( 2 J ) 21cm (3 ) (1 ) 54
55 13 (50) i n i g i exp( E i /kt) n i n = g ie Ei/kT U (261) n = n i (262) U = g i e E i/kt (263) U Saha (50) ( v) (n = 1) χ dn + 1(v) n 1 = g+ 1 g e g 1 exp = 8πm3 n e h 3 g + 1 g 1 exp (χ+ 1 2 mv2 ) kt g e = 2dx 1dx 2 dx 3 dp 1 dp 2 dp 3 h 3 = 2 1 4πp 2 dp h 3 n e (χ+ 1 2 mv2 ) kt v 2 dv (2 : spin) (264) = 8πm3 v 2 dv n e h 3 (265) 55
56 (264) (130) v n + 1n e n 1 = ( 2πmkT h 2 )3/2 2g + 1 g 1 e χ/kt (266) (261) n + ( n e 2πmkT )3/2 n = 2U(T) + h 2 U(T) e χ/kt (267) (267) j j Hii Hii α nl ( nl) A n L,nL n L nl Einstein A nl n p n e α nl (T)+ n >n (266) L n n L A n L,nL = n nl )3/2 n 1 n =1 L A nl,n L (268) ( n p n e 2πmkT = e χ/kt (269) n 1S h 2 b n (50) n nl n 1S = b n (2L+1)e hν n1/kt (270) 56
57 n χ n = χ hν n1 (269)(270) n nl = b n (2L+1) (268) h2 2πmkT ( α nl (T) 2πmkT )3/2 (2L+1) h 2 3/2 e χ n/kt e χ n/kt n p n e (271) + b n A n L 2L +1,nL e (χ n χ n)/kt n >n L 2L+1 n 1 = b n (272) A nl,n L n =1 L T b n n b n = 1 n b 1 b n (271) n nl j nn = hν nn 4π n 1 L=0L =L±1 n nl A nl,n L = hν nn 4π n pn e α eff nn (273) (α eff nn effective recombination coefficient ) Case A Lyα 57
58 n = 1 n = 1 ((268)(272) n = 2 ) Case B Hii Case B Case A T (K) α eff Hβ (e-14 cm3 sec 1 ) Lyα/Hβ Hα/Hβ Hγ/Hβ Case B T (K) α eff Hβ (e-14 cm 3 sec 1 ) Lyα/Hβ Hα/Hβ Hγ/Hβ
59 n e, v ( ) 1 2 σ 12 n e vσ 12 n e Maxwell q 12 = 0 σ 12 v 3 exp( mv 2 /2kT) dv 0 v 2 exp( mv 2 /2kT) dv (274) 1,2 ( ) n 1,n 2 (50)(275) n e n 1 q 12 = n e n 2 q 21 (275) q 21 q 12 = n 1 n 2 = g 1 g 2 exp ( ) hν0 kt (276) (276) n e n 1 q 12 = n e n 2 q 21 +n 2 A 21 (277) 4π j = n 2 A 21 = n en 1 q 12 A 21 hν 0 n e q 21 +A 21 = n g en 1 A 2 21 g 1 exp ( hν 0 kt n e + A 21 q )
60 = Critical density n c (cm 3 ) n e n 1 q 12 (n e n c ) g n 1 A 2 21 g 1 exp ( hν ) 0 kt (ne n c ) (278) n c = A 21 q 21 (279) n e (278) i critical density n c = Doppler Broadening A 21 j<ia ij j iq ij (280) Doppler z (189) c = γ(c+vcosθ ) (281) ν ν vcosθ = v z, γ = 1, ν = ν 0 v z = c(ν ν 0) ν 0 (282) dv z = cdν ν 0 (283) 60
61 ν ν +dν (282)(283) exp m avz 2 dv z exp m ac 2 (ν ν 0 ) 2 2kT 2ν0kT 2 dν (284) line profile function φ(ν) φ(ν) = 1 e (ν ν 0) 2 / νd 2 (285) ν D π ν D = ν 0 c Gaussian 2kT (286) m a Natural Broadening Collisional Broadening (n ) Natural Broadening Collisional/Pressure Broadening line profile function (104) Lorentzian ν col (Hz) Γ/4π 2 φ(ν) = (ν ν 0 ) 2 +(Γ/4π) 2 (287) Γ = A nn +2ν col (288) n <n 61
62 Gaussian Lorentzian 3 broadening line profile function (282) Doppler ν 0 (1+v z /c) (287) ν 0 Maxwell φ(ν) = Γ/4π 2 [ν ν 0 (1+v z /c)] 2 +(Γ/4π) 2 exp( mv 2 z/2kt) dv z exp( mv 2 z /2kT) dv z = Voigt function 1 ν D π H(a,u) (289) a = Γ 4π ν D, u = ν ν 0 ν D (290) Gaussian Lorentzian H(a,u) a π e y2 dy a 2 +(u y) 2 (291) H(a,u) du = π (292) 62
63 14 r ρ M r φ (erg g 1 ) dm r dr = 4πr2 ρ (293) dφ dr = GM r r 2 (294) G = e 8 erg cm g 2 Poisson (293)(294) 1 d r 2 dr ( r 2dφ dr ) = G r 2 dm r dr r P = 4πGρ (295) dp dr = GM rρ r 2 (296) µ, m p m p = e 24 g P = ρ µm p kt (297) 63
64 R, M (293) M r = 4 3 πr3 ρ, M = 4 3 πr3 ρ (298) (296)(298) r = R P = 0 dp dr = 4 3 πrgρ2 (299) P = 2 3 πgρ2 (R 2 r 2 ) (300) P c T c (297)(298) P c = 2 3 πgρ2 R 2 = 3GM2 8πR 4 (301) T c = µm pgm 2kR (302) ( X,Y,Z ) ( /( +1)) 1/2 4/ µ = 2X Y Z = (303) 64
65 M = e+33 g, R = e+10 cm (302) T c = 0.74 e+7 K ( T c = 1.5 e+7 K) (294)(298) dφ dr = φ = GM 4 3 (r > R) r 2 MG πgρr = R r (r < R) 3 GM r (r > R) GM 2R 3 (3R 2 r 2 ) (r < R) (304) (305) (r = R ) W = 1 2 i j i Gm i m j = 1 m i φ i (306) r ij 2 i (306) R W = 1 ρφ4πr 2 dr = 3GM2 (307) 2 0 5R 3 1 kt 2 1/µm p (297)(300) U = R 0 3kT 2µm p ρ4πr 2 dr = R P4πr2 dr = 3GM2 10R = 1 2 W (308) 65
66 E = U +W = 1 2 W (309) L = de dt = 1 dw 2 dt = du dt (310) Eddington luminosity L E (erg sec 1 ) 1 GMmp r 2 = σ TL E 4πr 2 c (311) L E = 4πGMm pc σ T = (1.3 e+38) M M erg sec 1 (312) = (3.3 e+4) M M L ( L =3.85 e+33 erg sec 1 ) 66
67 15 Population synthetic model population synthetic model Star formation rate SFR (M yr 1 ) Instantaneous burst : SF R δ(t) (313) ( Exponential burst : SFR exp t ) (314) τ Constant formation : SF R = const. (315) Initial mass function IMF IMF M 1.35 (Salpeter) (316) M model SFR, IMF 67
68 0.1Gyr 0.3Gyr 1Gyr 3Gyr SFR: Instantaneous burst, IMF: Salpeter 1e+11M Balmer jump Balmer ( 3700Å) 4000Åbreak Ca 68
69 V-band (5500Å) A V (mag) A V = 2.5log ( ) Iobserved I intrinsic (317) B-band (4400Å) E(B V) (mag) E(B V) = A B A V = A V (318) R R dust size R 3.1 ( 4 6) dust (Mie ) (LMC) (SMC) dust A λ A V 0.5λ 1.1 UV(SMC),Optical 0.4λ 1.6 NIR Optical depth (17)(317) (319) I observed I intrinsic = 10 A λ/2.5 = e τ λ (320) e A λ τ λ 69
70 Screen/Slab dust A λ dust (Screen dust) (320) dust (Slab dust) I observed I intrinsic = ( A λx/2.5) dx = 1 10( A λ/2.5) ln10a λ /2.5 = 10 A λ /2.5 (321)
71 Hii SFR Hα, [Oii]λλ3727 L(Hα), L([Oii]) erg sec 1 SFR = (7.9 e 42)L(Hα) M yr 1 (322) SFR = (1.4 e 41)L([OII]) M yr 1 (323) Case B Hα/Hβ = 2.86 (10000K) (319)(320) A V log Hα/Hβ 2.86 A V ( ) A V 8 log Hα/Hβ 2.86 (324) 71
4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5.
A 1. Boltzmann Planck u(ν, T )dν = 8πh ν 3 c 3 kt 1 dν h 6.63 10 34 J s Planck k 1.38 10 23 J K 1 Boltzmann u(ν, T ) T ν e hν c = 3 10 8 m s 1 2. Planck λ = c/ν Rayleigh-Jeans u(ν, T )dν = 8πν2 kt dν c
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[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
B 1 B.1.......................... 1 B.1.1................. 1 B.1.2................. 2 B.2........................... 5 B.2.1.......................... 5 B.2.2.................. 6 B.2.3..................
QMI_10.dvi
... black body radiation black body black body radiation Gustav Kirchhoff 859 895 W. Wien O.R. Lummer cavity radiation ν ν +dν f T (ν) f T (ν)dν = 8πν2 c 3 kt dν (Rayleigh Jeans) (.) f T (ν) spectral energy
IA
IA 31 4 11 1 1 4 1.1 Planck.............................. 4 1. Bohr.................................... 5 1.3..................................... 6 8.1................................... 8....................................
. ev=,604k m 3 Debye ɛ 0 kt e λ D = n e n e Ze 4 ln Λ ν ei = 5.6π / ɛ 0 m/ e kt e /3 ν ei v e H + +e H ev Saha x x = 3/ πme kt g i g e n
003...............................3 Debye................. 3.4................ 3 3 3 3. Larmor Cyclotron... 3 3................ 4 3.3.......... 4 3.3............ 4 3.3...... 4 3.3.3............ 5 3.4.........
4 2 Rutherford 89 Rydberg λ = R ( n 2 ) n 2 n = n +,n +2, n = Lyman n =2 Balmer n =3 Paschen R Rydberg R = cm 896 Zeeman Zeeman Zeeman Lorentz
2 Rutherford 2. Rutherford N. Bohr Rutherford 859 Kirchhoff Bunsen 86 Maxwell Maxwell 885 Balmer λ Balmer λ = 364.56 n 2 n 2 4 Lyman, Paschen 3 nm, n =3, 4, 5, 4 2 Rutherford 89 Rydberg λ = R ( n 2 ) n
V(x) m e V 0 cos x π x π V(x) = x < π, x > π V 0 (i) x = 0 (V(x) V 0 (1 x 2 /2)) n n d 2 f dξ 2ξ d f 2 dξ + 2n f = 0 H n (ξ) (ii) H
199 1 1 199 1 1. Vx) m e V cos x π x π Vx) = x < π, x > π V i) x = Vx) V 1 x /)) n n d f dξ ξ d f dξ + n f = H n ξ) ii) H n ξ) = 1) n expξ ) dn dξ n exp ξ )) H n ξ)h m ξ) exp ξ )dξ = π n n!δ n,m x = Vx)
006 11 8 0 3 1 5 1.1..................... 5 1......................... 6 1.3.................... 6 1.4.................. 8 1.5................... 8 1.6................... 10 1.6.1......................
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The Structure and Evolution of Stars I. Basic Equations. M r r =4πr2 ρ () P r = GM rρ. r 2 (2) r: M r : P and ρ: G: M r Lagrange r = M r 4πr 2 rho ( ) P = GM r M r 4πr. 4 (2 ) s(ρ, P ) s(ρ, P ) r L r T
(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν k = ω/c k x ωt ( ω ) k α c, k k x ωt η αβ k α x β diag( + ++) x β = (ct, x) O O x
![(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν k = ω/c k x ωt ( ω ) k α c, k k x ωt η αβ k α x β diag( + ++) x β = (ct, x) O O x](/thumbs/104/162595168.jpg "(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν k = ω/c k x ωt ( ω ) k α c, k k x ωt η αβ k α x β diag( + ++) x β = (ct, x) O O x")
Compton Scattering Beaming exp [i k x ωt] k λ k π/λ ω πν k ω/c k x ωt ω k α c, k k x ωt η αβ k α x β diag + ++ x β ct, x O O x O O v k α k α β, γ k γ k βk, k γ k + βk k γ k k, k γ k + βk 3 k k 4 k 3 k
18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α
18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α 2 ), ϕ(t) = B 1 cos(ω 1 t + α 1 ) + B 2 cos(ω 2 t
30
3 ............................................2 2...........................................2....................................2.2...................................2.3..............................
1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2
2005 9/8-11 2 2.2 ( 2-5) γ ( ) γ cos θ 2πr πρhr 2 g h = 2γ cos θ ρgr (2.1) γ = ρgrh (2.2) 2 cos θ θ cos θ = 1 (2.2) γ = 1 ρgrh (2.) 2 2. p p ρgh p ( ) p p = p ρgh (2.) h p p = 2γ r 1 1 (Berry,1975) 2-6
(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0
1 1 1.1 1.) T D = T = D = kn 1. 1.4) F W = F = W/ = kn/ = 15 kn 1. 1.9) R = W 1 + W = 6 + 5 = 11 N. 1.9) W b W 1 a = a = W /W 1 )b = 5/6) = 5 cm 1.4 AB AC P 1, P x, y x, y y x 1.4.) P sin 6 + P 1 sin 45
5 5.1 E 1, E 2 N 1, N 2 E tot N tot E tot = E 1 + E 2, N tot = N 1 + N 2 S 1 (E 1, N 1 ), S 2 (E 2, N 2 ) E 1, E 2 S tot = S 1 + S 2 2 S 1 E 1 = S 2 E
5 5.1 E 1, E 2 N 1, N 2 E tot N tot E tot = E 1 + E 2, N tot = N 1 + N 2 S 1 (E 1, N 1 ), S 2 (E 2, N 2 ) E 1, E 2 S tot = S 1 + S 2 2 S 1 E 1 = S 2 E 2, S 1 N 1 = S 2 N 2 2 (chemical potential) µ S N
1 1 1 1-1 1 1-9 1-3 1-1 13-17 -3 6-4 6 3 3-1 35 3-37 3-3 38 4 4-1 39 4- Fe C TEM 41 4-3 C TEM 44 4-4 Fe TEM 46 4-5 5 4-6 5 5 51 6 5 1 1-1 1991 1,1 multiwall nanotube 1993 singlewall nanotube ( 1,) sp 7.4eV
The Physics of Atmospheres CAPTER :
The Physics of Atmospheres CAPTER 4 1 4 2 41 : 2 42 14 43 17 44 25 45 27 46 3 47 31 48 32 49 34 41 35 411 36 maintex 23/11/28 The Physics of Atmospheres CAPTER 4 2 4 41 : 2 1 σ 2 (21) (22) k I = I exp(
II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
![II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2](/thumbs/101/149159646.jpg "II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2")
II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh
) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4
![) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4](/thumbs/92/107866707.jpg ") ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4")
1. k λ ν ω T v p v g k = π λ ω = πν = π T v p = λν = ω k v g = dω dk 1) ) 3) 4). p = hk = h λ 5) E = hν = hω 6) h = h π 7) h =6.6618 1 34 J sec) hc=197.3 MeV fm = 197.3 kev pm= 197.3 ev nm = 1.97 1 3 ev
http://www.ns.kogakuin.ac.jp/~ft13389/lecture/physics1a2b/ pdf I 1 1 1.1 ( ) 1. 30 m µm 2. 20 cm km 3. 10 m 2 cm 2 4. 5 cm 3 km 3 5. 1 6. 1 7. 1 1.2 ( ) 1. 1 m + 10 cm 2. 1 hr + 6400 sec 3. 3.0 10 5 kg
ma22-9 u ( v w) = u v w sin θê = v w sin θ u cos φ = = 2.3 ( a b) ( c d) = ( a c)( b d) ( a d)( b c) ( a b) ( c d) = (a 2 b 3 a 3 b 2 )(c 2 d 3 c 3 d
A 2. x F (t) =f sin ωt x(0) = ẋ(0) = 0 ω θ sin θ θ 3! θ3 v = f mω cos ωt x = f mω (t sin ωt) ω t 0 = f ( cos ωt) mω x ma2-2 t ω x f (t mω ω (ωt ) 6 (ωt)3 = f 6m ωt3 2.2 u ( v w) = v ( w u) = w ( u v) ma22-9
ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 +
2.6 2.6.1 ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.121) Z ω ω j γ j f j
QMI13a.dvi
I (2013 (MAEDA, Atsutaka) 25 10 15 [ I I [] ( ) 0. (a) (b) Plank Compton de Broglie Bohr 1. (a) Einstein- de Broglie (b) (c) 1 (d) 2. Schrödinger (a) Schrödinger (b) Schrödinger (c) (d) 3. (a) (b) (c)
I ( ) 1 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT
I (008 4 0 de Broglie (de Broglie p λ k h Planck ( 6.63 0 34 Js p = h λ = k ( h π : Dirac k B Boltzmann (.38 0 3 J/K T U = 3 k BT ( = λ m k B T h m = 0.067m 0 m 0 = 9. 0 3 kg GaAs( a T = 300 K 3 fg 07345
.2 ρ dv dt = ρk grad p + 3 η grad (divv) + η 2 v.3 divh = 0, rote + c H t = 0 dive = ρ, H = 0, E = ρ, roth c E t = c ρv E + H c t = 0 H c E t = c ρv T
NHK 204 2 0 203 2 24 ( ) 7 00 7 50 203 2 25 ( ) 7 00 7 50 203 2 26 ( ) 7 00 7 50 203 2 27 ( ) 7 00 7 50 I. ( ν R n 2 ) m 2 n m, R = e 2 8πε 0 hca B =.09737 0 7 m ( ν = ) λ a B = 4πε 0ħ 2 m e e 2 = 5.2977
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Contents a 6 6 6 6 6 6 6 7 7. 8.. 8.. 8.3. 8 Part. 9. 9.. 9.. 3. 3.. 3.. 3 4. 5 4.. 5 4.. 9 4.3. 3 Part. 6 5. () 6 5.. () 7 5.. 9 5.3. Γ 3 6. 3 6.. 3 6.. 3 6.3. 33 Part 3. 34 7. 34 7.. 34 7.. 34 8. 35
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II 28 5 31 3 I 5 1 7 1.1.......................... 7 1.1.1 ( )................ 7 1.1.2........................ 12 1.1.3................... 13 1.1.4 ( )................. 14 1.1.5................... 15
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m(ẍ + γẋ + ω 0 x) = ee (2.118) e iωt P(ω) = χ(ω)e = ex = e2 E(ω) m ω0 2 ω2 iωγ (2.119) Z N ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.120)
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6 2 T γ T B (6.4) (6.1) [( d nm + 3 ] 2 nt B )a 3 + nt B da 3 = 0 (6.9) na 3 = T B V 3/2 = T B V γ 1 = const. or T B a 2 = const. (6.10) H 2 = 8π kc2
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127 3 II 3.1 3.1.1 Φ(t) ϕ em = dφ dt (3.1) B( r) Φ = { B( r) n( r)}ds (3.2) S S n( r) Φ 128 3 II S 1, S 2 Φ 1, Φ 2 Φ 1 = { B( r) n( r)}ds S 1 Φ 2 = { B( r) n( r)}ds (3.3) S 2 S S 1 +S 2 { B( r) n( r)}ds
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.1, θ, ϕ d = A, t dt + B, t dtd + C, t d + D, t dθ +in θdϕ.1.1 t { = f1,t t = f,t { D, t = B, t =.1. t A, tdt e φ,t dt, C, td e λ,t d.1.3,t, t d = e φ,t dt + e λ,t d + dθ +in θdϕ.1.4 { = f1,t t = f,t {
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215 11 13 1 2 1.1....................... 2 1.2.................... 2 1.3..................... 2 1.4...................... 3 1.5............... 3 1.6........................... 4 1.7.................. 4
2009 2 26 1 3 1.1.................................................. 3 1.2..................................................... 3 1.3...................................................... 3 1.4.....................................................
E 1/2 3/ () +3/2 +3/ () +1/2 +1/ / E [1] B (3.2) F E 4.1 y x E = (E x,, ) j y 4.1 E int = (, E y, ) j y = (Hall ef
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I A A441 : April 15, 2013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida )
I013 00-1 : April 15, 013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida) http://www.math.nagoya-u.ac.jp/~kawahira/courses/13s-tenbou.html pdf * 4 15 4 5 13 e πi = 1 5 0 5 7 3 4 6 3 6 10 6 17
positron 1930 Dirac 1933 Anderson m 22Na(hl=2.6years), 58Co(hl=71days), 64Cu(hl=12hour) 68Ge(hl=288days) MeV : thermalization m psec 100
positron 1930 Dirac 1933 Anderson m 22Na(hl=2.6years), 58Co(hl=71days), 64Cu(hl=12hour) 68Ge(hl=288days) 0.5 1.5MeV : thermalization 10 100 m psec 100psec nsec E total = 2mc 2 + E e + + E e Ee+ Ee-c mc
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A04-164 2008 2 13 1 4 1.1.......................................... 4 1.2..................................... 4 1.3..................................... 4 1.4..................................... 5 2
5 1.2, 2, d a V a = M (1.2.1), M, a,,,,, Ω, V a V, V a = V + Ω r. (1.2.2), r i 1, i 2, i 3, i 1, i 2, i 3, A 2, A = 3 A n i n = n=1 da = 3 = n=1 3 n=1
4 1 1.1 ( ) 5 1.2, 2, d a V a = M (1.2.1), M, a,,,,, Ω, V a V, V a = V + Ω r. (1.2.2), r i 1, i 2, i 3, i 1, i 2, i 3, A 2, A = 3 A n i n = n=1 da = 3 = n=1 3 n=1 da n i n da n i n + 3 A ni n n=1 3 n=1
( ; ) C. H. Scholz, The Mechanics of Earthquakes and Faulting : - ( ) σ = σ t sin 2π(r a) λ dσ d(r a) =
1 9 8 1 1 1 ; 1 11 16 C. H. Scholz, The Mechanics of Earthquakes and Faulting 1. 1.1 1.1.1 : - σ = σ t sin πr a λ dσ dr a = E a = π λ σ πr a t cos λ 1 r a/λ 1 cos 1 E: σ t = Eλ πa a λ E/π γ : λ/ 3 γ =
9 1. (Ti:Al 2 O 3 ) (DCM) (Cr:Al 2 O 3 ) (Cr:BeAl 2 O 4 ) Ĥ0 ψ n (r) ω n Schrödinger Ĥ 0 ψ n (r) = ω n ψ n (r), (1) ω i ψ (r, t) = [Ĥ0 + Ĥint (
9 1. (Ti:Al 2 O 3 ) (DCM) (Cr:Al 2 O 3 ) (Cr:BeAl 2 O 4 ) 2. 2.1 Ĥ ψ n (r) ω n Schrödinger Ĥ ψ n (r) = ω n ψ n (r), (1) ω i ψ (r, t) = [Ĥ + Ĥint (t)] ψ (r, t), (2) Ĥ int (t) = eˆxe cos ωt ˆdE cos ωt, (3)