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2 l ηµν hµν g µν = η µν + h µν, h µν 1 l Transvers-Traceless (TT) h TT 0µ = 0, η ij i h TT jk = 0, η ij h TT ij = 0, η µν µ ν h TT ij = 2 c 2 t + Δ 2 h TT = 0 ij 2
3 Polarization l From the TT condition, h TT 0µ = 0, η ij h TT ij = 0, η ij i h TT jk = 0, non-zero components are ( ) h TT xx = h TT yy = h TT + = A TT + cos ω(t z /c) h TT xy = ( ) h TT yx = h TT = A TT cos ω(t z /c) l There are only two independent components; h + and h 3
4 Polarization l h+ or h 4
5 l the + (plus) mode h + TT 0, h TT = 0 Polarization l the (cross) mode h + TT = 0, h TT 0 If only + mode or x mode wave comes, each particle oscillates along a straight line. The x mode is the same as + mode if rotated by 45 degrees. These modes correspond to the linear polarization of light. 5
6 l the circular polarization h + TT = h TT 0 Polarization In case of wave containing both + and x modes with the same amplitude, each particle moves along a circle. It corresponds to the circular polarization. l If their amplitude are not same, each particle moves along a elliptic. It corresponds to the elliptic polarization. 6
7 l Ø Ø Ø ~1/r l 4 cf. アインシュタインの一般相対論以外の重力理論では, これとは異なる 重力波 の放射もあり得る 7
8 l Ø Ø quadrupole formula h ij TT (t, x) = 1 r 2G c 4 Q ij TT (t r /c) 8
9 l h ij TT (t, x) = 1 r Q ij = d 3 xρ(t, x) x i x j 1 3 r2 δ ij ; Q TT ij = Λ ij,kl Q kl Λ ij,kl P ik P jl 1 2 P P, P ij kl ij = δ ij n i n j, n i = x i luminosity) de GW dt = c 3 r 2 32πG dω h TT hij TT ij i dj GW dt 2G c 4 = 2G 5c 5 ikl Q ij TT (t r /c) Q ka Q la = G 5c 5 Q ij Q ij r 9
10 l Ø M 2a Ø fb l de GW dt = G 5c 5!!! Q ij!!! Q ij = erg/s ( W) = 128G M 2 a 4 ( 2π f ) 6 5c 5 b M 10 3 kg 2 2a 10 m 4 f b 10 Hz 6 10
11 l Ø photon Ø graviton l ν GW E GW = h ν GW ν GW = 2 f b, E GW = 2h f b de GW dt = 128G M 2 a 4 ( 2π f ) 6 5c 5 b 11
12 l de GW /dt = 4π E GW 5c 5 h ( ) 6 G = 1 4 /s M 2 a 4 5 f b M 10 3 kg 2 2a 10 m 4 f b 10 Hz 5 l 1GHz, 1W /s 12
13 l h ij TT = 1 r 2G c 4!! Q ij TT ~ 8G c 4 r h l Ø Ø monopole 2 Ma ( ) 2 2π f b r 13
14 h ij TT = 1 r 2G c 4 Q TT ij ~ 8G c 4 2 Ma ( ) 2 2π f b r l f b =10Hz f GW = 2f b = 20Hz l 15,000km ~ c = c f GW 2 f b l r > h < 64π 2 G c 5 M Ma 2 f 3 b = kg 2a 10 m 2 f b 10 Hz 3 14
15 vs. 陽子に働く力 1 e 2 F e = 4πe 0 r, 2 F = Gm 2 p ; g r 2 F g F e = Gm p 2 e 2 / 4πε l : Ø Ø l : Ø Ø 15
16 l TT Q ij Ø Ø Ø v v 16
17 l 10 Ø (Compact Binary Coalescence = CBC) (NS) (BH) NS-NS, NS-BH, BH-BH Ø (Supernova = SN) Ø Ø 17
18 l Ø Ø l Ø BICEP Ø 18
19 l Ø Ø l Ø Ø l deterministic stochastic 19
20 CBC CBC: Compact Binary Coalescence 20
21 CBC l (inspiral phase) l (merger phase) l BH NS (ringdown phase) 21
22 Compact Binary l (NS) (BH) Ø Ø l the innermost stable circular orbit (ISCO) 22
23 CB l m 1, m 2 a P b f b = 1/P b M = m 1 + m 2 µ = m 1 m 2 /(m 1 + m 2 ) de GW dt l de GW dt = 32G 5c µ 2 a 4 ( 2π f ) 6 5 b h = 4G = 32G 7/3 5c 5 GM = (2π f b ) 2 a 3 Mc 2 µa c 4 M 10/3 ( c 2π f ) 10/3 h = 4G5/3 b c 4 ( ) 2 2π f b r M 5/3 ( c 2π f ) 2/3 b r M c µ 3/5 M 2/5 = q 3/5 M, q = µ / M 23
24 CBC l Ø a da dt = 64G 3 µm 2 a 3 = 64G 3 5c 5 5c 5 M 5/3 c M 4/3 a 3 Ø P b f b df b dt = 48G5/3 5πc 5 dp b dt = 192G5/3 5c 5 M 5/3 ( c 2π f ) 11/3 b 2π 5/3 P M b c 5/3 24
25 CBC l spiral orbit ( 25
26 chirp signal l Chirp Signal 26
27 l dp b dt M = 0.01 c 1.2M 5 3 P b 0.01s 5 3 = 1 N c N c : l l 1.4M 2 (Mc = 1.2M ) τc ~ 0.3 s P b ~ 0.01s (f GW 200Hz) Nc ~100 Ø spiral orbit l τc ~ 1.5 ms, P b ~ 1.3 ms (f GW 1.5 khz) Nc ~3 the innermost stable circular orbit (ISCO) 27
28 inspiral phase (ISCO ) l ISCO M τ c = 2.2 s c 1.2M 5 3 f GW 100Hz 8 3 M = 2.2 s c 1.2M 5 3 P b 0.02 s 8 3 l ISCO M N cyc = 340 c 1.2M 5 3 f GW 100Hz 5 3 M = 340 c 1.2M 5 3 P b 0.02 s
29 l e de GW dt = de GW dt e=0 f (e); f (e) = (1 e 2 ) 7/2 24 e e4 l a da dt = da dt e=0 f (e) = 64 5 G 3 µm 2 c 5 a 3 f (e) de dt = G 3 µm 2 c 5 a 4 e (1 e 2 ) 5/2 304 e2 29
30 l da de = 12a (73/ 24)e 2 + (37 / 96)e 4 e(1 e 2 )[1 + (121/ 304)e 2 ] a(e) = c 0 g(e), g(e) = e12/ e e2 870/2299 c0: a=a0 when e=e0 l e e0 e a g(e a 0 ) 0 19/12 30
31 e a g(e a 0 ) , g(e 0 ) = e 12/ e e l the Hulse-Taylor binary pulsar (PSR B ) a m, e a = O(100R NS ) 10 3 km f b 0.3Hz e l = 31
32 CBC l (inspiral phase) l (merger phase) l BH NS (ringdown phase) 32
33 Inspiral Phase l inspiral phase = chirp signal l ( ) quadrupole formula l + post-newtonian effects 33
34 chirp signal l inspiral phase chirp signal l matched filter 34
35 Merger Phase l merger phase: l Ø (EOS) Ø Ø 35
36 Ringdown Phase l l hyper-massive NS l BH BH 36
37 CBC l CBC Ø NSs or stellar mass BHs: f = 10 ~ 10 3 Hz (λ = 300 ~ 30,000 km) Ø supermassive BHs f = 10-4 ~ 10-1 Hz (λ = ~ km) 0.02 ~ 20AU ² f >30 khz (λ < 10 km) ² 音波 ( 可聴域 ) f =20 ~ 20,000 Hz (λ = 1.5 cm ~ 15 m) l CBC 37
38 1.5M + 1.5M 100M + 100M 10M M 10M M ; e =
39 (merger rate) l NS-NS merger rate: NS-NS 10 l 5 1 PSR t mrg /Gyr M 1 /M M 2 /M J field (double PSR) B C cluster J field B field J field B field 39
40 event rates l 1 merger rate: 3~ yr -1 l AdvLIGO 1 : 7~400 yr -1 l KAGRA event rate 20 yr -1 (Kim et al. 2010) 40
41 event rates l BH-BH or BH-NS l NS-NS いろいろなモデルで推定した AdvLIGO の event rate 数字は, もっとも楽観的な値 括弧内は, 一番もっともらしい値 (Belczynski 2013) 41
42 event rates l event rates for CBC with AdvLIGO N low yr -1 N re yr -1 N high yr -1 NS-NS NS-BH BH-BH (Adadie et al. 2010) 445Mpc for NS-NS 927Mpc for NS-BH 2187Mpc for BH-BH 42
43 CBC l l dynamical gravity or Ø BICEP Ø l l l cf. 43
44 l Ø l Ø ; standard candle Ø 44
45 Chirp Signals as Standard Sirens l B. Schutz (1986): CBC chirp signal l standard sirens: cf. standard candles l GW from CBC: h + (t) = A r h (t) = A r 2/3 π f gw (τ ) 1+ cos 2 ι c cosφ(τ ) 2 π f gw (τ ) c 2/3 cosιsin Φ(τ ) (inclination) inclination: the angle between the line of sight and the direction normal to the orbit 45
46 Chirp Signal h + (t) = A r h (t) = A r where A = 4 GM c c 2 5/3 f gw (τ ) = 1 5 π 256τ Φ(τ ) = τ 0 τ τ = t coal t; 2/3 π f gw (τ ) 1+ cos 2 ι c cosφ(τ ) 2 π f gw (τ ) c 3/8 2/3 GM c c 2 2π f gw (τ )dτ = 2 cosιsin Φ(τ ) 5/8 5GM c c 2 t coal : time at coalescence 5/8 τ 5/8 + Φ 0 46
47 Standard Siren f gw = 96 5 π 8/3 GM c c 3 5/3 l The chirp mass M c f gw 11/3 f GW and f GW h + (t) = A r 2/3 π f gw (τ ) 1+ cos 2 ι c cosφ(τ ), h (t) = A 2 r π f gw (τ ) c 2/3 cosιsin Φ(τ ) l h + and h the distance r the inclination l M c (1+z) M c ; f gw f gw /(1+z) z: the distance r the luminosity distance D L (z). 47
48 Measurement of H 0 l GW observation the luminosity distance independently of the distance ladder l Uniquely clean and powerful way to measure the Hubble constant H 0 l Need redshift of the source l Short gamma-ray bursts = binary NS-NS mergers: Ø determine H 0 within ~3% (D. Holz 2012) 48
49 l core collapse Ib, Ic, II Ø M 8M O+Ne+Mg ( 56 Fe 13 4 He + 4n) Ø (p + e - n + e ) Ø Ø 49
50 50
51 超新星残骸 SN
52 SN1987A 52
53 l Betelgeus Ø (1920) - M = 7.7 ~ 20M, R = 950 ~ 1200R D = 197 ± 45pc (643 ± 146 ) Ø 100 Type II SN 53
54 l Ø Ø delayed explosion Ø 54
55 l Ø Ø Ø r-mode non-radial pulsation Ø (anisotropic emission) 55
56 l Ø Ø unknown micro-physics EOS, l matched filter 56
57 57
58 58
59 K. Kotake arxiv:
60 l 8.5 kpc l cf. h rss ~ Hz 1/2, h rss h(t) 2 dt E toto 10 9 ~ 10 5 M c 2 = ~ erg Ø SN erg Ø erg Ø erg 99% 60
61 連星中性子星合体 (200Mpc) 超新星爆発 (10kpc) 回転パルサ - ( 上限値 ) 12
62 l l ( 伊藤洋介, 理論懇 2013) 62
63 l Ø Ø A. Stuver/LIGO 63
64 l h = 16πG c 4 εq zz f D ε = Q zz g cm 2 f 100 Hz D 1kpc 1 D: ε = ( Q Q ) Q xx yy zz 64
65 l Ø l Ø Crab Pulsar h <
66 Crab Vela Arxiv:
67 l l ü ü ü l Ø 67
68 stochastic waves l Ø Ø 1 (Pop III) 宇宙背景輻射 CMB 宇宙背景重力波 68
69 Planck time: t p =!G c 5 インフレーション s = s CMB 38 万年 69
70 l CMB b-mode BICEP2 70
71 BBO: Big-Bang Obs. (astro-ph/ ) ρ GW ( f ) : density of GW; ρ c : the critical density Ω GW ( f ) : density parameter; Ω GW ( f ) = 1 ρ c dρ GW ( f ) d ln f 71
72 stochastic waves l Ø detector Ø 72
73 Multi-Messenger Observations l Ø CBC: red-shift Hubble const. Ø supernova, gamma-ray burst: l triggered search Ø l follow-up observation Ø 73
74 l l 3 KAGRA 4 D1 l 74
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l l 1916 Ø 2016/12/10 日本物理学会新潟支部 2 l l 1916 Ø l 2016/12/10 日本物理学会新潟支部 3 l 2015 9 14 UTC Ø Advanced LIGO l 2016 2 11 2 12 Ø LIGO & Virgo https://losc.ligo.org/events/gw150914/ http://media1.s-nbcnews.com/
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Title ブラックホールと重力波天文学 Author(s) 長峯, 健太郎 Citation 高大連携物理教育セミナー報告書. 28 Issue Date Text Version publisher URL DO
Title ブラックホールと重力波天文学 Author(s) 長峯, 健太郎 Citation 高大連携物理教育セミナー報告書. 28 Issue Date 2017-03 Text Version publisher URL http://hdl.handle.net/11094/60516 DOI rights Osaka University Knowledge Archive : OUKA
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(NMR) 0 (NMR) 2µH hω ω 1 h 2 1 1-1 NMR NMR h I µ = γµ N 1-2 1 H 19 F Ne µ = Neh 2mc ( 1) N 2 ( ) I =1/2 I =3/2 I z =+1/2 I z = 1/2 γh H>0 2µH H=0 µh I z =+3/2 I z =+1/2 I z = 1/2 I z = 3/2 γh H>0 2µH H=0
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
SN 2007bi Yoshida, T. & Umeda, H., MNRAS 412, L78-L82 (2011)
SN 2007bi Yoshida, T. & Umeda, H., MNRAS 412, L78-L82 (2011) SN 2007bi SN 2007bi (Gal-Yam et al. 2009) 2007 4 6.5 Type Ic subluminous dwarf galaxy Z ~ (0.2-0.4) Z (Young et al. 2010) 36 (Young et al. 2010)
50 2 I SI MKSA r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq
49 2 I II 2.1 3 e e = 1.602 10 19 A s (2.1 50 2 I SI MKSA 2.1.1 r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = 3 10 8 m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq F = k r
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中央大学セミナー.ppt
String Gas Cosmology References Brandenberger & Vafa, Superstrings in the early universe, Nucl.Phys.B316(1988) 391. Tseytlin & Vafa, Elements of string cosmology, Nucl.Phys.B372 (1992) 443. Brandenberger,
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2007-Kanai-paper.dvi
19 Estimation of Sound Source Zone using The Arrival Time Interval 1080351 2008 3 7 S/N 2 2 2 i Abstract Estimation of Sound Source Zone using The Arrival Time Interval Koichiro Kanai The microphone array
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19 σ = P/A o σ B Maximum tensile strength σ % 0.2% proof stress σ EL Elastic limit Work hardening coefficient failure necking σ PL Proportional
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meiji_resume_1.PDF
β β β (q 1,q,..., q n ; p 1, p,..., p n ) H(q 1,q,..., q n ; p 1, p,..., p n ) Hψ = εψ ε k = k +1/ ε k = k(k 1) (x, y, z; p x, p y, p z ) (r; p r ), (θ; p θ ), (ϕ; p ϕ ) ε k = 1/ k p i dq i E total = E
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液晶の物理１：連続体理論（弾性，粘性）
The Physics of Liquid Crystals P. G. de Gennes and J. Prost (Oxford University Press, 1993) Liquid crystals are beautiful and mysterious; I am fond of them for both reasons. My hope is that some readers