2.2 1 2.2 2.2.1 (LSR: Local Standard of Rest) 2.1 LSR R 0 LSR Θ 0 (Galactic Constant) 1985 (IAU: International Astronomical Union) R 0 =8.5 kpc, Θ 0 = 220 km s 1. (2.1) R 0 7kpc 8kpc Θ 0 180 km s 1 270 km s 1 (2.1) LSR 1985 (U,V,W ) = (10.0, 15.4, 7.8) km s 1, (2.2) 2.1 X U Y V Z W 2.2.2 LSR (Solar Circle R 0 2.1
2 X Y Z 2.1 (LSR) LSR (Inner Galaxy) (Outer Galaxy) 2.1 2.2.3 LSR V r V t V r V t LSR (2.2) V r = Θ sin(180 φ) Θ 0 sin l. (2.3)
2.2 3 Θ R θ 180 φ φ R0 D l Θ0 2.2 LSR V t = Θ cos(180 φ) Θ 0 cos l. (2.4) Θ sin(180 φ) = sin φ cos(180 φ) = cos φ sin φ/r 0 = sin l/r (2.2) R 0 cos l = D R cos φ (2.3) (2.4) ( Θ V r = R Θ ) 0 R 0 sin l, (2.5) R 0 V t = ( Θ R Θ ) 0 R 0 cos l Θ D, (2.6) R 0 R LSR R 2 = D 2 + R 2 0 2DR 0 cos l, (2.7) D R 0 R l
4 D = R 0 cos l ± R 2 R 2 0 sin2 l. (2.8) 0 R = ±R 0 sin l LSR (tangent point) 2.1 R 0 (2.8) 2.2.4 (2.5) (2.6) (J. Oort) D R 0 R R R 0 D cos l, (2.9) (2.5) V r [ d ( Θ )] (R R 0 ) R 0 sin l dr R R 0 (2.9) A [ A R d ( Θ )] = 1 [ Θ 2 dr R R 0 2 R dθ ]. (2.10) dr R 0 V r AD sin 2l. (2.11) A D l (2.11) (2.6) V t [ d ( Θ )] (R R 0 )R 0 cos l Θ 0 D, dr R R 0 R 0
2.2 5 100 (=Vt/D) 50 0-50 -100 0 100 200 300 2.3 µ V t /D l Feast (1997) (2.9) (2.10) B [ B 1 d 2R dr ]R (RΘ) = 1 [ Θ 0 2 R + dθ ]. (2.12) dr R 0 V t (A cos 2l + B)D. (2.13) A B D l (2.13) A B 1927 2.3 (2.13) µ V t /D 180 (2.13) A B R 0 Θ 0
6 A B = Θ 0 R 0 Ω 0, (2.14) (A + B) = ( dθ ). (2.15) dr R 0 Ω 0 LSR 1985 IAU km s 1 kpc 1 A B Oort (1927) 19±3-24±5 OB IAU (1985) 14.4±1.2-12.0±2.8 (1993) 12.2±0.6-8.2±0.6 KM Feast (1997) 14.8±0.8-12.0±0.6 Mignard (2000) 14.5±1.0-11.5±1.0 KM 2.2.5 LSR (epicycle) (epicyclic approximation) (R, φ, z) Φ(R, z)
2.2 7 R = V 2 R Φ R, (2.16) L z (2.16) L z = R 2 φ = const. (2.17) R = Φ eff R, Φ eff = Φ + L 2 z 2R 2, (2.18) Φ eff (effective potential) (2.18) Ṙ 1 2Ṙ2 + Φ eff = E (= const.), (2.19) E 1) R 2.4 (R 1, R 2 ) (2.19) Ṙ =0 E 0 E Φ eff R E Φ eff guiding center R g x R R g Φ eff R g (2.18) [ 2 Φ ] eff ẍ = x R 2. (2.20) R g 1) z 0
8 2.4 0 >E>Ψ eff [ Φ eff / R] Rg =0 (2.20) x = X 0 cos(κt + θ 0 ), (2.21) X 0 κ (epicyclic frequency) θ 0 κ (2.16) Φ/ R = V 2 /R = RΩ 2 Ω κ 2 = d [ RΩ 2] + 3L2 z dr R 4 =4Ω2 +2RΩ dω dr = 4BΩ 0. (2.22) B B B κ 2
2.2 9 κ B κ (2.20) (Ω = const.) κ =2Ω κ = Ω 2Ω κ Ω κ κ 1.4Ω 0 y φ = L z R 2 = ΩR2 ( g (R g + x) 2 Ω 1 2x ). (2.23) R g y (2.21) (2.23) y y = 2Ω κ X 0 sin(κt + θ 0 ) (2.24) (2.21) (2.24) 2.5 2Ω/κ κ 1.4Ω y 2.5 2.2.6 (K. Schwarzschild) n 0 d v [ ( v 2 f( v)d v = exp R (2π) 3/2 σ R σ φ σ z 2σR 2 + v2 )] φ 2σφ 2 + v2 z 2σz 2. (2.25)
10 1 0.5 0 x -0.5 y -1-1 -0.5 0 0.5 1 2.5 1 X 0 =0.1 (κ = Ω) σ R σ φ σ z (velocity ellipsoid) R 0 R g x = R 0 R g LSR y v y v y = R 0 φ R0 Ω 0, (2.26) (2.23) [ v y = x 2Ω + R dω ] =2Bx, (2.27) dr R 0
2.2 11 x (2.21) <vy 2 > <vx 2 > = 4B2 <x 2 > κ 2 <x 2 > = 4B2 κ 2 = B A B, (2.28) (2.25) σ y B = σ x A B, (2.29) 2.2.7 HI HI 21cm 1420MHz 1951 HI 21cm HI 2.6 - Position-Velocity diagram PV b =0 HI l LSR V r 2) 2) PV l l V
12 2.6 HI l LSR V r l (terminal velocity) PV 2.2.8 2.6 PV V term HI (2.5) l V r / D =0 (2.5) V r D R( = Θ ) R R D R 0 sin l =0. (2.30) R/ D (2.7) R D = D R 0 cos l. (2.31) R
2.2 13 80 60 40 V term Radial Velocity (km/s) 20 0-20 -40-60 -80 l = 135 l = 45-100 -120 0 2 4 6 8 10 12 14 16 18 20 Distance (kpc) 2.7 D V r Θ(R) =Θ 0 = 220 km s 1 R 0 =8.5 kpc l =45 l = 135 V term Θ D R 0 cos l =0 (Tangent point) 2.1 V term Θ(R) R = R 0 sin l, V term = Θ(R) Θ 0 sin l. (2.32) 2.1 (2.32)
14 2.2.9 Rotation Curve ( Θ V r = R Θ ) 0 R 0 sin l. (2.33) R 0 R 0 Θ 0 V r R Θ(R) R V t V t HI CO PV HI (M.Merrifield) HI (2.33) ( Θ V r = W (R) sin l, W(R) R Θ ) 0 R 0, (2.34) R 0 W l HI b a ( z0 ) ( ) z 0 b a = arctan = arctan D R 0 cos l + R 2 R 2 0 sin l, (2.35)
2.2 15 5 4.5 4 apparent disk thickness 3.5 3 2.5 2 1.5 1 0.5 0 180 160 140 120 100 80 60 40 20 0 Galactic Longitude 2.8 HI z R/R0 Θ(R) z 0 D R (2.8) 2.35 R/R 0 z 0 R 0 R/R 0 (2.34) W (R) Θ(R) 2.9 10 km s 1 30 km s 1 2.9 Θ 0 220, 200, 180 km s 1 Θ 0
16 300 200 100 0 0 0.5 1 1.5 2 2.5 3 2.9 (1997) R R 0 R 0 Θ 0 R 0 Θ 0 2.2.10
2.3 17 VLBI VERA JASMINE SIM GAIA 21 2.3 2.3.1 Φ(R, z) K z = Φ/ z 4πGρ(z) 2 Φ z 2 = K z z, (2.36) z σz 2 1 (nσz 2) = Φ n z R = K z (2.37) n z σz 2 ρ(z)
18 ρ 0 = ρ(z =0) (Oort Limit) ρ 0 =0.1 0.2 M pc 3 ρ obs 0.1 M pc 3 2.3.2 (Enclosed Mass) M r vc 2 r = GM r r 2, (2.38) v c v r M r = rv2 c G (2.39) M r ρ(r) M r = 4πr 2 ρ(r)dr. (2.40) M r ρ(r) ρ r α α = 2 M r r 1 (2.39) v c ρ r 2
2.3 19 2.10 2.9 M r Θ 0 =180, 200, 220 km s 1 R 0 8.5 kpc 2.3.3 (2.39) R kpc V km s 1 M r M (=1.99 10 30 kg) ( M r =2.32 10 5 R )( )2 V kpc km s 1 M, (2.41) 2.9 M r 2.10 M r M r Θ 0 1 3
20 10 11 M Θ 0 (2.41) I = I 0 exp( R/h) h = 3 kpc 10 kpc 85% 2.10 10 kpc 2.10 (dark matter) 2.3.4 HI HI 20 kpc 10 kpc 100 kpc (2.41) 100 kpc 4 10 11 1 10 12 M 2.10 HI halo, dark halo)
2.3 21 2.3.5 (Mass-to-Light ratio) M/L M/L = (M/M ) (L/L ), (2.42) M/L=1 V M/L V M/L M/L L V 1.4 10 10 L M =1 3 10 11 M M/L M/L 7 21 (R <2R 0 ) M/L M/L, M/L 30 70 (R <100 kpc) 2.3.6
22 D LS D S D L 2.11 3) MACHO: Massive Astrophysical Compact Halo Object 4) 2.11 θ β θ β = α D LS D S. (2.43) α α = 4GM c 2 b, (2.44) 3) 4) MACHO (WIMPs: Weakly Interacting Massive Particles) macho wimp
2.3 23 M b R E R E ( 4GM )1/2 D L D LS R E c 2, r D L θ, u D L β, (2.45) D S R E R E (2.43) r 2 ur 1=0, (2.46) 2 r 1,2 = u ± u 2 +4 2 (2.47) u 1 r 1,2 1 (2.45) R E D L =10kpc D S =50kpc M =1M R E 8AU θ E R E /D L =0.8 mas A A =1 A 1,2 = r 1,2dr 1,2 r = 4 1,2 udu r1,2 4 1. (2.48) 1, 2 (2.47)
24 A = A 1 + A 2 = u2 +2 u u 2 +4. (2.49) u =0 A u 1 u =0 A 100 u = A =1 u =1 A =3/ 5=1.34 3 2.3.7 (2.49) µ β β = β0 2 +(µt)2, (2.50) β 0 t 0 (2.45) u = u 2 0 +(t/t E) 2, (2.51) t E t E R E µd L (2.52) (2.51) u 0 t E (2.49) (2.51) (2.12) t E (2.52) (2.45)
2.3 25 2.12 MOA MOA M =1M D L =10kpc v µd L = 200 km s 1 t E 70 2.3.8 (optial depth) πr 2 E τ = πr 2 ρ E dd, (2.53) M D M ρ
26 R E (2.45) (2.53) M τ = 4πG c 2 ρ(d) D L(D S D L ) dd, (2.54) D S ρ 200 km s 1 ρ r 2 τ exp 5 10 7 200 MACHO EROS OGLE MOA MACHO 1992 1999 5.7 17 τ =1.2 10 7 τ exp 20% MACHO Γ MACHO 0.5M MACHO MACHO MOA OGLE