() () () 15%85% ( 10 9 kg m 3 ) (10 21 kg m 3 ) C C C C... () Instroduction : 15 2

email: shibata@sci.kj.yamagata-u.ac.jp URL: http://astr-www.kj.yamagata-u.ac.jp 27 9 29 / Introduction() () () / 1

() () () 15%85% ( 10 9 kg m 3 ) (10 21 kg m 3 ) C C C C... () http://astr-www.kj.yamagata-u.ac.jp/~shibata/ Instroduction : 15 2

(A4 2 )(100 ) 30 3

1 1.1 : km 4m 4m (1 AU ) 1 5000 km (tab1) AU 1/107 0.4 0.7 1.0 1.5 5.2 9.6 19.2 30.1 10000 αcen 2.7 10 5 (4.3 ) 30 km 1 1? π 10 7 sec 3.15 10 7 sec 4

1 1 2πR /4 = 1 km (1) 1AU () = 1 5 km (2) 1ly () = c 1 (3) = 9.46 10 15 m (4) 1pc () = 3.26ly (5) 1 100 1/100 1/100 100 1 2 4m 4 km (1) cm (2) cm (3) cm () 5

3 1. (1AU) 2. 3. () 100bun 6

1.2 () ( = ) ( = ) 4 () kouseiq (kouseiq) 2000 7

137 138 1.3 () 10m 10km 100km 1000km 10000km 8

5 10 m 120km 6.4 10 3 km 6.96 10 8 m 1.5 10 11 m αcen 4.3 ly 30pc 8.5kpc 50kpc 6.7 10 2 kpc 6647km/s 150 9

() 1 km = 1000 m = 10 3 m 1 cm = 1 100 = 10 2 m 1 AU = 1.496 10 11 m ( ) 1 pc = 3.086 10 16 m = 1 AU / 1 =3.26 ly 1 ly = 9.46 10 15 m = tera T = 10 12 giga G = 10 9 mega M = 10 6 kilo k = 10 3 hecto h = 10 2 deca da = 10 deci d = 10 1 centi c = 10 2 milli m = 10 3 micro µ = 10 6 nano n = 10 9 pico p = 10 12 : = π = 3.1416 : 1 = 60 = 3600 1 =1 1 = 1 : l = rθ () : = 365.26 days = 3.16 10 7 s : = c = 3.00 10 8 m/s : = 100 50 (km/s)/mpc x log x 1.00 0.00000 2.00 0.30103 3.00 0.47712 4.00 0.60206 5.00 0.69897 6.00 0.77815 7.00 0.84510 8.00 0.90309 9.00 0.95424 10.00 1.00000 x = 10 a x a log x = 10 a a = log x. (6) 10

1.4 () v = H 0 d (7) v d H 0 71(km/sec)/(Mpc) (hubble/p.tex)! 11

Mpc km /s 0.032 170 0.034 290 NGC 6822 0.214 130 NGC 598 0.263 70 NGC 221 0.275 185 NGC 224 0.275 220 NGC 5457 0.45 200 NGC 4736 0.5 290 NGC 5194 0.5 270 NGC 4449 0.63 200 NGC 4214 0.8 300 NGC 3031 0.9 30 Mpc km /s NGC 3627 0.9 650 NGC 4826 0.9 150 NGC 5236 0.9 500 NGC 1068 1.0 920 NGC 5055 1.1 450 NGC 7331 1.1 500 NGC 4258 1.4 500 NGC 4151 1.7 960 NGC 4382 2.0 500 NGC 4472 2.0 850 NGC 4486 2.0 800 NGC 4649 2.0 1090 (Proc. Nat. Acad. Sci., 15, 168) [4] 12

1000 km/sec 0 300 1 2 Mpc () 13

1.5 ( )1/2 = θ = l D (8) l:d: 6 1 radian arcmin 1. : 1.74 10 6 m 3.84 10 8 m 2. : 7.0 10 5 km 1.5 10 8 km 3. : 25 ly 1.3 10 3 ly 4. M13: 98 ly 2.2 10 4 ly 5. M31: 50 kpc 6.7 10 2 kpc 6. : 1 Mpc 6647 km/sec () 1.6 (parallax) 1 / 14

0 ( annual parallax) (arc sec) p D = 1 p pc (9) 1.7 file = LandF.tex ( [Watt/m 2 ] [erg/sec cm 2 ]) (Flux) ( [W] [erg/sec]) (luminosity) F L F = D L 4πD 2 (10) 7 L = 4 10 26 W (flux) () 2mm 15

() LandF-Q f = L/4πD 2 4 10 26 W 4π(1.5 10 11 ) 2 m 2 = 1.4 103 W/m 2 πr 2 r = 1 10 3 4.4 10 3 W 1.8 file = scale.tex () () (magnitute) b = 10 a a = log b (11)..., 1/10, 1, 10, 100, 1000......, 1, 0, 1, 2, 3,... (12) : 2 1/12 = 1.05946... f/f 0 2 0/12 1.00000 Do 1 2 1/12 1.05946 2 2/12 1.12246 Re 9/8 = 1.125 2 3/12 1.18921 2 4/12 1.25992 Mi 5/4 = 1.25 2 5/12 1.33484 Fa 4/3 = 1.333 2 6/12 1.41421 2 7/12 1.49831 Sol 3/2 = 1.5 2 8/12 1.58740 2 9/12 1.68179 La 5/3 = 1.667 2 10/12 1.78180 2 11/12 1.88775 Si 15/8 = 1.875 2 12/12 2.00000 Do 16

= n = 12 log 2 f f 0 (13) f f 0 = 2 n/12 (14) M = 2 3 log 10 E 11.8 (15) f[hz] (Do, Re, Mi,..) () F [W/m 2 ] m 6 () 5 100 (magnitude) = m 1 m 0 = 5 2 log 10 F 1 F 0 = 10 2 5 (m 1 m 0 ) F 1 F 0 = 5 2 log 10 () (16) (17) Linear Scale Log Scale () 0 () F ν F ν = 3.6 10 26 [W/m 2 Hz] (AB ) 8?? 17

() scale 9 f do f re f mi f fa f sol f la f si f DO (1) f sol f do > f DO f sol (2) f sol f do f DO f sol (1) f sol f do < f DO f sol () doremi 1.9 file = mag.tex 10pc () () L (luminosity) Joul/sec = W M (absolute magnitude) 10pc () f (flux) Joul/m 2 sec m (apparent magnitude) 4.7 m M m M = 5 log D(in pc ) 5 (18) 18

L = 4πD 2 f (19) = 4π10 2 f 10pc (20) (21) f f 10pc = L/4πD2 L/4π10 2 = 102 D 2 (22) ( ) f log 10 f 10pc = 2 log 10 D + 2 (23) (24) ( ) 5 2 log f = m M (25) f 10pc (23) (25) (M m) 2 5 = 2 log 10 D + 2 (26) M m = 5 log 10 D + 5 (27) M L ( ) 26.74 L M M = 4.74 5 ( ) L 2 log (28) L L = 3.85 10 26 W = 3.85 10 33 erg/sec (Bolometric magnitude, Bolometric luminosity) ( ) () ( CCD) () bolometric () (18) 19

(Type I SNR) () M snr (m snr ) () (magniq)? (α CMa) m = 1.5, d = 2.7pc (α Ori) m = 0.8, d = 150pc 20

2 2.1 file = col.tex ( ) (photon) ν( [Hz]) = 1/P ; P ( [sec]) ω = 2πν; λ( [m] or [cm], [nm],...); k = 2π/λ k c = 3.00 10 10 cm/sec () (...) ϵ = hν = hω h 6.626 10 27 erg sec p = hk h = h/2π p = h/λ () cp = c ν = λ (29) x E(x, t) = E 1 sin(ωt kx) (30) B(x, t) = (k E 1 ) sin(ωt kx) (31) 4 k µ (k 0, k 1, k 2, k 3 ) = (ω/c, k x, k y, k z ) (32) 21

1 1 (3Volt) ev = 1.60 10 19 C 1V = 1.60 10 19 J = 1.60 10 12 erg (33) MeV m e c 2 = (1/2)MeV (76.2MHz FM ) 4m X 3eV kev MeV, GeV, TeV,......Hz 1000GHz...m 0.3mm 390 (550) 720nm () () () (colq) λ, ν, hν 390nm () 720nm 22

() Hz, ev 10 x V x ω n = (n x, n y, n z ) k (k x, k y, k z ) = (kn x, kn y, kn z ) ω (k x, k y, k z) ω, k x, k y, k z ω, k x, k y, k z (Doppler effect) (abberation) () dopplerq 2.2 telsc.tex CCD 2.2.1 2.2.2 () 1 1 () 23

θ θ 1.22 λ D (34) D ( VLBI )VLA(), VSOP () () E/E λ/λ 10msec () F ()O F P F l = P F OF b2 f 2π l/λ OF = f, OP = b O O P Φ(b) b 2π l + Φ(b) = 0 (36) λ F P Q P Q = f 2 + (b + x) 2 (37) OF P Q OF = f { 1 + (35) } 1 2 (b + x)2 f 2 f (38) = b2 f + bx f = l + bx f (39) 24

O F 2π l λ bx bx + Φ(b) + 2π = 2π λf λf b Q a(x) = A(b) P 2πb/λ = Y, x/f = x a(x) = (40) A(b)e (2πbx/λf)i db (41) A(Y )e XY i dy (42) A(b) A(b) δ - A(b) θ XY 1, πd λ x f = θ (43) 11 f 1 f 2 12 () diffrq 25

2.3 :flux Instenisty fi.tex () de da t t + dt ν ν + dν F ν = de dt da dν () flux [erg/sec cm 2 Hz] dω (44) I ν = de dt da dν dω (45) Specific Intensity ( Brightness) [erg/sec cm 2 Hz str ] F ν = I ν cos θdω (46) θ net flux 3 2 = 6 m box (6 ) (3 π/180) (2 π/180) = π 2 /5400str () dθ dφ sin θdφ!) dω = sin θdθdφ (47) 26

: Instensity : 6 (x, y, z, p x, p y, p z ) (snap shot ) dn = f ph (r, p)dxdydzdp x dp y dp z (48) = h 3 f ph c 3 ν2 dνdωdv (49) = f ν (r, θ, φ)dνdωdv (50) (p x, p y, p z ) (p, θ, ϕ) p = hν/c f ν (θ, φ) dω ( ν ν + dν ) da dt : I ν = cf ν hν (51) D R I ν de R dω D 2 de dω D 2 ( ) I ν I ν = (52) ν ν + dν F ν dν, I ν dν, L ν dν, 27

F ν, I ν, L ν ν F = F ν dν, I = I ν dν, L = L ν dν, 13 1. 1m 2 (Instensity) ν ν + dν I ν flux F ν ( πi ν ) 2. flux F ν ( 0) () fluxq 14 30 () I = 0 I ν dν 1.4 10 6 erg/sec cm 2 1.4 kw/m 2 flux F = 0 F ν dν () intensityq 15 F ν [W/m 2 Hz] 1 AU () I ν [W/m 2 Hz str] F ν I ν () telscq 2.4 intf.tex 28

D λ/d D λ/d D D D D 2 D web pages 16 VLA (Rreen Bank, West Virgnia, USA, National Radio Astronomical Observatory). 27km 1cm 10cm () intfa θ = 1.22 λ D () intfq = 1.22 (1 10)cm 2.7 10 6 cm = (0.45 4.5) 10 6 rad = 0.09 0.9arcsec (53) π 180 3600 (arcsec) 2.5 spec.tex : ( ) () 29

E(t) E(t) = Ê(ω) = 1 E(t)e iωt dt (54) 2π flux () F ω = c Ê(ω) 2 flux F = (c/4π) E(t) 2 dt Parseval s theorem E(t) 2 dt = 2π Ê(ω) 2 dω4π 0 Ê(ω) 2 dω (55) F = 0 F ω dω (56) () Dopper effect (Dark matter ) web pages 2.6 () black.tex T T I ν T B ν I ν = B ν (T ) (57) Planck function I ν = 2hν3 /c 2 e hν/kt 1 B ν(t ) (58) 30

planck () hν max 2.82kT (59) B ν ν B λ (T ) = 8πhc λ 5 1 e hc/λkt 1 (60) λ max hc/4.97kt hν kt Raylei-Jeans law hν kt Wean law I ν 2hν3 c 2 I ν 2ν2 kt (61) c2 ( exp hν ) kt (62) hν max 2.82kT F ν = I ν cos θdω = πi ν = πb ν (63) 2π str F = π 0 B ν (T )dν = 2π5 k 4 15c 2 h 3 T 4 σt 4 (64) 0 x 3 π4 e x dx = 1 15 Stefan-Boltzmann law F = σt 4 σ = 2π5 k 4 31 (65) 15c 2 h 3 T 4 = 5.67 10 5 erg/cm 3 deg 4 sec (66)

17 p = u/3 u () T ds = du + pdv ds = V du T dt dt + 4 u dv (67) ( ) 3 T ( ) S S = dt + dv (68) T V 2 S/ T V V u = at 4 Stefan-Boltzmann law (69) V U T () StefanQ (StefanQ: ) du = d(v u) = V du + udv p = u/3 ( T ( ( ) S V T ( ) ) S V T V T ds = V du + 4 udv (70) 3 V ) T = T = ( ) V du V T dt ( ) 4 u = 4 3 T 3 T du dt = 1 T T ( 1 T 2 u + 1 T ) du dt (71) (72) 4dT/T = du/u ln T 4 = ln u + const., or u = at 4 (73) 18 ν advance () blackq 2.7 efftemp.tex 32

() F F obs R D ( ) 2 D F = F obs (74) R σt 4 (75) F DR L = 4πR 2 F 4πR 2 σt 4 (76) (effective temperature) T eff LRT; L = 4πR 2 σt 4 eff (77) T eff () F U,B,V,R,I () m 2 m 1 = 5 2 log ( Bλ1 B λ1 B λ1 = λ5 2 e hc/λ2kt 1 B λ1 λ 5 1 e hc/λ 1kT 1 ) 5 2 log λ5 2 λ 5 + hc ( 1 1 ) = a + b 1 kt λ 2 λ 1 T (78) (79) a,b 2.8 HR HRzu.tex 33

O A... O B A F G S K M R N (Russell s students in Princeton O Be A Fine Girl Kiss Me Right Now, Smack!) () HR (Hertzsprung-Russel Diagram) HR (77) HR = () HR HR HR HR HR 19 L T R 34

() efftempq 20 HR () HRzuQ 21 HR () HR2Q (HR2Q: ) HR ( ) ( ) T F = σt 4 σ R L = 4πR 2 σt 4 HR 22 (α CMa) m = 1.5 2.7pc M (α Ori) m = 0.8 150pc M () startypeq (startypeq: ) M = m 5 log D + 5 m pc D M = 1.5 5 log(2.7) + 5 = 1.3 (or 1.4) (80) log 2.7 0.301 + (0.477 0.301) 0.7 = 0.38 1.4() (81) log 2.7 = log 3 + log 9 1 = 0.431() (82) 35

M = 0.8 5 log(150) + 5 = 5.1 (83) log 150 = log 1.5 + 2 0.301 0.5 + 2 2.15() (84) log 150 = log 3 + log 5 + 1 = 2.176() (85) 10pc 15 10pc 15 2 = 225 100 5 2.5 1 6 0.8 6 = 5.2 10pc 1/3.7 3.7 2 = 14 2.5 2.5 2.5 = 15.6 3 1.5 + 3 = 1.5 4.7 L L = 4πσR 2 T 4 (86) (1) A 1 (10000/6000) 4 = 7.7 (5/2) log 7.7 = 2.2 3 (2) M 3000 10 1 100 ( ) 400 HR 1500 100 36

(M core > M ch ) (He ) (RGB) () L M 3.5 He 2 3α CO (AGB:Asymptotoic Giant Branch) CO < 8M 10M CO H He CO 56 Fe () H,Si,(O,Ne,Mg),(C,O),He,H 1.3 2M () 56 Fe 13 4 He + 4n 124.4MeV (87) 4 He 2p + 2n 28.3MeV (88) 161MeV 0.1 37

1: (10 8 K) (20M ) H pp CNO 4 He 0.15-0.2 1 3 4 He 12 C He 12 C + 12 C 1 6O + γ 14 C 16 O 1.5 C 12 C + 12 C 23 Na + p 12 C + 12 C 20 Ne + α Ne, Na Mg, Al 7 100 Ne 20 Ne + γ 16 O + α 20 Ne + α 24 Mg + γ O Mg 15 O 16 O + 16 O 28 Si + α 16 O + 16 O 28 P + p Si, P, S Cl, Ar, Ca 3 28 Si + γ 24 Mg + α Si 24 Mg + γ 23 Na + p 24 Mg + γ 20 Ne + α Cr, Mn Ge, Co, Ni, Cu 40 M /m F e = 2 10 55 1.2 10 52 erg GM /R 2 NR = 3 10 53 erg R NS 10km 2.9 radtransfer.tex Intensity I ν I ν = B ν (T ) T 6000K I ν = B ν (T room ) T room 300K 38

(absorption) (emission) (radiation)( ) () ( ) Radiation emission [] ν σ ν n dl κ ν = nσ ν dl [] (emission coefficient) j [erg/cm 3 sec str]: ν ν + dν monochromatic emission coefficient j ν (Intensity)I di ν ds = α νi ν + j ν (89) s (radiation transfer equation) di ν /ds = 0I ν = B ν (T ) j ν /α ν = B ν (T ) T j ν α ν = B ν (T ) (90) Kirchoff s Law L α ν ()( I 0 ) I ν = I 0 e ανl (91) 39

l ν l ν = κ 1 ν τ ν = α ν ds (92) () τ ν 1 τ ν 1 (Source function) S ν = j ν α ν (93) : di ν dτ ν = I ν + S ν (94) I ν > S ν I ν (95) I ν < S ν I ν (96) I ν τ ν I ν S ν, (97) S ν B ν (T ) 23 F ν [W/m 2 Hz] 1 AU () I ν [W/m 2 Hz str] F ν I ν () telscq 2.10 α ν l ν ν E = hν 0 (T 1 ) (T 2 ) 40

ν 0 I ν0 = B ν0 (T 1 ) I ν0 = B ν0 (T 2 ) I ν0 = B ν0 (T 2 ) ν 0 I ν = B ν (T 1 ) I ν I ν0 () = B ν0 (T 2 ) (98) I ν () = B ν (T 1 ) (99) T 1 > T 2 B ν (T 1 ) > B ν0 (T 2 ) (line absorption) ν 0 (line emission) (line absorption) (line emission) 10 6 K 6000K () 6000K 41

3 : : : SN1087A : (): :... 3.1 () crossing time () t cross = R v (100) R v V 250km/s 50kpc t coss 10 8 yr 35 ( ) m 1 m 2 () r F = G m 1m 2 r 2 (101) G = 6.773 10 8 dyn/cm 2 g 2 42

1 : F 1 = F 2 + F 2 + F 3 +... + F n (102) () 24 g :T = 27.3 :R = 3.84 10 8 m g/g g = 9.8m/s 2 2 R (R = 6.37 10 6 ) 2 (R/R ) 2 () 2 inv2q g = R(2π/T ) 2 = 2.72 10 3 m/s g/g = 3.63 10 3 (R/R ) 2 = 3.60 10 3 3.2 Virial () () I 1 2 N m i ri 2 (103) i=2 d 2 I/dt 2 d 2 I = W + 2T (104) dt2 T = 1 2 m ivi 2 = (105) W = G m im j (106) r ij allpairs 43

W + 2T W + 2T = 0 M = m i v 2 = 2T/M (): r g W = GM 2 /r g v 2 = GM r g (107) : 4 25 1. 9.8 m/s 2 2. 3. () newtonjq 26 () () () kyoriq (kyoriq: ) () θ R 0 D = θr 0 44

() ()( ) ( ) () M m ( D [pc] m M = 5 log D 5 ) (): v D v = H d H 71 km/s Mpc v 27 () iroirotelq 28 6000 (K) () suntempq 29 () globq 30 85% (dark matter) () darkmq ( darkmq) () v 2 /R R (dark matter) 45

31 A B (1) A B (2) A (3) B (4) A B (5) () expq 32 () impressq 46

() x log x 1.00 0.00000 2.00 0.30103 3.00 0.47712 4.00 0.60206 5.00 0.69897 6.00 0.77815 7.00 0.84510 8.00 0.90309 9.00 0.95424 10.00 1.00000 x = 10 a x a log x = 10 a a = log x. (108) log(4 10 4 ) log 4 + 4 log 10 = 4.60206 log 6.4 0.4(log 7 log 6) + log 6 = 0.805 log 6.4 = log(64/10) = 2 log 8 1 = 0.80618 π = 3.14159263 G = 6.67 10 8 dyncm3/g 2 c = 3.00 10 10 cm/s h = 1.05 10 27 ergs e = 4.80 10 10 esu m p = 1.67 10 24 g m e = 9.11 10 28 g σ SB = 5.67 10 5 erg/scm 2 K 4 1eV = 1.60 10 12 erg M = 5.97 10 27 cm M = 1.99 10 33 g L = 3.90 10 33 erg/s 1AU = 1.50 10 13 cm 1pc = 3.08 10 18 cm 1ly = 0.946 10 18 cm (109) σ SB Stefan Boltzman const. 47