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1 (UWM) / 35
2 1 2 Corotation Irrotation 3 4 (UWM) / 35
3 10 100/yr (D = 300 Mpc) 1 40/yr (D = 300 Mpc) GWs 1 700/yr (D = 300 Mpc) X BH NS LIGO (UWM) / 35
4 (UWM) / 35
5 r GM Ω d h rad s f GW Ω π 1160[Hz] C 0.2 C GMNS M = m c m 2 R NS i C 1/2 3RNS 3/2 10 km 0.2 d R NS 1/2 3RNS d 3/2 10 km R NS Observer m 2 r Ω Z d Y m 1 X (UWM) / 35
6 r GM Ω d h rad s f GW Ω π 1160[Hz] C 0.2 h 2Gm1m2 c 2 rm C 0.2 C GMNS M = m c m 2 R NS i C 1/2 3RNS 3/2 10 km 0.2 d R NS GMΩ 2/3 c 3 2 RNS 10 km 1/2 3RNS d 3RNS d 3/2 10 km R NS 100 Mpc r Observer m 2 r Ω Z d Y m 1 X (UWM) / 35
7 r GM Ω d h rad s f GW Ω π 1160[Hz] C 0.2 h 2Gm1m2 c 2 rm C 0.2 C GMNS M = m c m 2 R NS i C 1/2 3RNS 3/2 10 km 0.2 d R NS GMΩ 2/3 c 3 2 RNS 10 km 1/2 3RNS d 3RNS L = de X dt = c Q 5G ij Q ij h erg s i,j i C 0.2 d 5 3RNS 5 d 3/2 10 km R NS 100 Mpc r Q 2 32c5 5G Observer m 2 r Ω Z d Y Q ij = R ρx ix jd 3 x m1m 2 2 GMΩ 10/3 M 2 c 3 (UWM) / 35 m 1 X
8 BH NS ν BH γ ray _ ν NS NS Disk Swift (UWM) / 35
9 (UWM) / 35
10 GRB b Bloom et al., ApJ., 638, 354 (2006) GRB Berger et al., Nature 438, 988 (2005) (UWM) / 35
11 GRB b Bloom et al., ApJ., 638, 354 (2006) GRB Galaxy Coalescence Berger et al., Nature 438, 988 (2005) NSNS formation v 1000 km/s = 100 kpc (UWM) / 35
12 (UWM) / 35
13 Inspiraling NS NS 1 ( ) 1 ( ) (UWM) / 35
14 Inspiraling NS NS 1 ( ) 1 ( ) Intermediate NS NS 1 ( ) 3 ( ) (UWM) / 35
15 Inspiraling NS NS 1 ( ) 1 ( ) Intermediate NS NS 1 ( ) 3 ( ) Merging 1 ( ) 1 ( ) (UWM) / 35
16 (UWM) / 35
17 Inspiraling BH NS ( MBH ) 2/3( MNS ) 2.5 6M BH M NS R NS 1 ( ) 1 ( ) (UWM) / 35
18 Inspiraling BH NS ( MBH ) 2/3( MNS ) 2.5 6M BH M NS R NS 1 ( ) 1 ( ) Intermediate BH NS 1 ( ) 1 ( ) d > 6M BH M BH /M NS > 3 d < 6M BH M BH /M NS 3 d 6M BH (UWM) / 35
19 Inspiraling BH NS ( MBH ) 2/3( MNS ) 2.5 6M BH M NS R NS 1 ( ) 1 ( ) Intermediate BH NS 1 ( ) 1 ( ) d > 6M BH M BH /M NS > 3 d < 6M BH M BH /M NS 3 d 6M BH Merging BH NS 1 ( ) < 1 ( ) (UWM) / 35
20 (UWM) / 35
21 ds 2 = α 2 dt 2 + γ ij (dx i + β i dt)(dx j + β j dt) γ ij = ψ 4 γ ij Conformally flat: Non-conformally flat: γ ij = η ij γ ij η ij α n µ δ t t µ Σ t+δt n µ µ β β µ δ t Σ t (UWM) / 35
22 ds 2 = α 2 dt 2 + γ ij (dx i + β i dt)(dx j + β j dt) γ ij = ψ 4 γ ij Conformally flat: Non-conformally flat: γ ij = η ij γ ij η ij α n µ δ t t µ Σ t+δt Hamiltonian constraint Momentum constraint Evolution equation for the extrinsic curvature Evolution equation for the spatial metric n µ µ β β µ δ t Σ t (UWM) / 35
23 ds 2 = α 2 dt 2 + γ ij (dx i + β i dt)(dx j + β j dt) γ ij = ψ 4 γ ij Conformally flat: Non-conformally flat: γ ij = η ij γ ij η ij α n µ δ t t µ Σ t+δt Hamiltonian constraint Momentum constraint Evolution equation for the extrinsic curvature Evolution equation for the spatial metric (UWM) / 35 n µ µ β β µ δ t Σ t
24 ds 2 = α 2 dt 2 + γ ij (dx i + β i dt)(dx j + β j dt) γ ij = ψ 4 γ ij Conformally flat: Non-conformally flat: γ ij = η ij γ ij η ij α n µ δ t t µ Σ t+δt Hamiltonian constraint Momentum constraint Evolution equation for the extrinsic curvature Evolution equation for the spatial metric n µ µ β P = P(ρ, ǫ, T) (UWM) / 35 β µ δ t Σ t
25 Corotation Irrotation Corotation Irrotation (UWM) / 35
26 Corotation Irrotation Corotation Irrotation t v i + v j j v i = 1 ρ ip i φ t ρ + i (ρv i ) = 0 (UWM) / 35
27 Corotation Irrotation Corotation Irrotation t v i + v j j v i = 1 ρ ip i φ t ρ + i (ρv i ) = 0 v i = V i + k i t v i + L k v i = 0 k i = (Ω R) i = Ω( Y, X, 0) (UWM) / 35
28 Corotation Irrotation Corotation Irrotation t v i + v j j v i = 1 ρ ip i φ t ρ + i (ρv i ) = 0 v i = V i + k i t v i + L k v i = 0 k i = (Ω R) i = Ω( Y, X, 0) i (h 1 ) 2 v2 + v j V j + φ + V j ( j v i i v j ) = 0 i (ρv i ) = 0 h (UWM) / 35
29 Corotation Irrotation Corotation Irrotation t v i + v j j v i = 1 ρ ip i φ t ρ + i (ρv i ) = 0 v i = V i + k i t v i + L k v i = 0 k i = (Ω R) i = Ω( Y, X, 0) i (h 1 ) 2 v2 + v j V j + φ + V j ( j v i i v j ) = 0 i (ρv i ) = 0 h Corotation: V i = 0 Irrotation: v i = i Ψ (UWM) / 35
30 Corotation Irrotation Corotation Irrotation Corotation Kochanek, ApJ. 398, 234 (1992) Bildsten & Cutler, ApJ. 400, 175 (1992) v i = (Ω R) i (UWM) / 35
31 Corotation Irrotation Corotation Irrotation (UWM) / 35
32 Corotation Irrotation Corotation Irrotation circulation (UWM) / 35
33 Corotation Irrotation Corotation Irrotation circulation C = v dl = ( v) ds = l S (UWM) / 35
34 Corotation Irrotation Corotation Irrotation circulation C = v dl = ( v) ds = l S v i = (Ω R) i + (ω r) i Ω ω C 2πRNS 2 (Ω + ω) (UWM) / 35
35 Corotation Irrotation Corotation Irrotation circulation C = v dl = ( v) ds = l S v i = (Ω R) i + (ω r) i Ω ω C 2πRNS 2 (Ω + ω) Ω initial + ω initial = C = Ω final + ω final Ω final 4000 rad/s (UWM) / 35
36 Corotation Irrotation Corotation Irrotation circulation C = v dl = ( v) ds = l S v i = (Ω R) i + (ω r) i Ω ω C 2πRNS 2 (Ω + ω) Ω initial + ω initial = C = Ω final + ω final Ω initial Ω final ω initial Ω final Ω final 4000 rad/s ω final Ω final (UWM) / 35
37 Corotation Irrotation Corotation Irrotation circulation C = v dl = ( v) ds = l S v i = (Ω R) i + (ω r) i Ω ω C 2πRNS 2 (Ω + ω) Ω initial + ω initial = C = Ω final + ω final Ω initial Ω final ω initial Ω final C = 0 Irrotation Ω final 4000 rad/s ω final Ω final (UWM) / 35
38 Corotation Irrotation Corotation Irrotation Irrotation v i i [(Ω d) j X j ] (Ω d) i (Ω R) i (Ω [R d]) i (UWM) / 35
39 Corotation Irrotation Corotation Irrotation Irrotation v i i [(Ω d) j X j ] (Ω d) i (Ω R) i (Ω [R d]) i V i (Ω [R d]) i (Ω r) i (UWM) / 35
40 (UWM) / 35
41 P = κρ Γ κ Γ Γ = n, n (UWM) / 35
42 P = κρ Γ κ Γ Γ = n, n Piecewise Polytropic EOS P = κ 0 ρ Γ 0 (0 ρ < ρ 0 ) P = κ 1 ρ Γ 1 (ρ 0 ρ) P [dyn / cm 2 ] Akmal, Pandharipande, and Ravenhall (1998) Piecewise Polytrope (Γ 1 = 3.0, log 10 P 1 = 13.45) n B [1 / fm 3 ] (UWM) / 35
43 Gravitational Mass [M sol ] APR BBB2 BPAL12 FPS Glend SLy4 pw. poly Circumferential Radius [km] (UWM) / 35
44 (UWM) / 35
45 Excision BH Apparent horizon (apparent horizon) Cook & Pfeiffer, PRD 70, (2004) (UWM) / 35
46 Excision BH Apparent horizon (apparent horizon) Cook & Pfeiffer, PRD 70, (2004) Puncture puncture Apparent horizon ψ = 1 + M P 2r BH + φ φ Brandt & Brügmann, PRL 78, 3606 (1997) BH (UWM) / 35
47 γ ij Conformally flat: γ ij = η ij Non-conformally flat: Irrotation Corotation Excision Puncture Piecewise Polytrope (UWM) / 35
48 (UWM) / 35
49 1997 Baumgarte, Cook, Scheel, Shapiro, & Teukolsky, PRL 79, 1182 (1997) 1998 Baumgarte, Cook, Scheel, Shapiro, & Teukolsky, PRD 57, 7299 (1998) Marronetti, Mathews, & Wilson, PRD 58, (1998) 1999 Bonazzola, Gourgoulhon, & Marck, PRL 82, 892 (1999) Marronetti, Mathews, & Wilson, PRD 60, (1999) 2000 Usui, Uryū, & Eriguchi, PRD 61, (2000) Uryū & Eriguchi, PRD 61, (2000) Uryū, Shibata, & Eriguchi, PRD 62, (2000) 2001 Gourgoulhon,Grandclément,K.T.,Marck,&Bonazzola, PRD63,064029(2001) 2002 Usui & Eriguchi, PRD 65, (2002) K.T. & Gourgoulhon, PRD 66, (2002) 2003 Marronetti & Shapiro, PRD 68, (2003) K.T. & Gourgoulhon, PRD 68, (2003) 2005 Bejger, Gondek-Rosińska, Gourgoulhon, Haensel, K.T., & Zdunik, A&A 431, 297 (2005) 2006 Uryū, Limousin, Friedman, Gourgoulhon, & Shibata, PRL 97, (2006) 2009 Uryū, Limousin, Friedman, Gourgoulhon, & Shibata, arxiv: Tichy, arxiv: K.T. & Shibata, in preparation (UWM) / 35
50 1999 Shibata, PRD 60, (1999) 2000 Shibata & Uryū, PRD 61, (2000) 2002 Shibata & Uryū, PTP 107, 265 (2002) 2003 Duez, Marronetti, Shapiro, & Baumgarte, PRD 67, (2003) Shibata, K.T., and Uryū, PRD 68, (2003) 2004 Miller, Gressman, & Suen, PRD 69, (2004) 2005 Shibata, K.T., & Uryū, PRD 71, (2005) 2006 Shibata & K.T., PRD 73, (2006) 2008 Anderson, Hirschmann, Lehner, Liebling, Motl, Neilsen, Palenzuela, & Tohline, PRD 77, (2008); PRL 100, (2008) Liu, Shapiro, Etienne, & K.T., PRD 78, (2008) Yamamoto, Shibata, & K.T., PRD 78, (2008) Baiotti, Giacomazzo, & Rezzolla, PRD 78, (2008) 2009 Kiuchi, Sekiguchi, Shibata, & K.T., PRD 80, (2009) Giacomazzo, Rezzolla, & Baiotti, arxiv: (UWM) / 35
51 Conformally flat, Irrotation (UWM) / 35
52 Conformally flat, Irrotation Bonazzola, Gourgoulhon, & Marck (1999) Marronetti, Mathews, & Wilson (1999) Uryū & Eriguchi (2000) Uryū, Shibata, & Eriguchi (2000) Gourgoulhon, Grandclément, K.T., Marck, & Bonazzola (2001) K.T. & Gourgoulhon (2002, 2003) Bejger, Gondek-Rosińska, Gourgoulhon, Haensel, K.T., & Zdunik (2005) K.T. & Shibata, in preparation (2009) (UWM) / 35
53 Conformally flat, Irrotation Bonazzola, Gourgoulhon, & Marck (1999) Marronetti, Mathews, & Wilson (1999) Uryū & Eriguchi (2000) Uryū, Shibata, & Eriguchi (2000) Gourgoulhon, Grandclément, K.T., Marck, & Bonazzola (2001) K.T. & Gourgoulhon (2002, 2003) Bejger, Gondek-Rosińska, Gourgoulhon, Haensel, K.T., & Zdunik (2005) K.T. & Shibata, in preparation (2009) K.T. & Gourgoulhon (2002, 2003) K.T. & Shibata, in preparation (2009) (UWM) / 35
54 Non-conformally flat, Irrotation (UWM) / 35
55 Non-conformally flat, Irrotation Uryū, Limousin, Friedman, Gourgoulhon, & Shibata, (2006) Uryū, Limousin, Friedman, Gourgoulhon, & Shibata, (2009) (UWM) / 35
56 Non-conformally flat, Irrotation Uryū, Limousin, Friedman, Gourgoulhon, & Shibata, (2006) Uryū, Limousin, Friedman, Gourgoulhon, & Shibata, (2009) (UWM) / 35
57 1.15M vs 1.55M Piecewise Polytrope Γ = 3.0, log P 1 = C MNS ADM R NS = C MNS ADM R NS = (UWM) / 35
58 1.35M vs 1.35M E b / M Piecewise Polytrope Equal-mass : 1.35 M sol M sol Γ 1 = 3.0, log 10 P 1 = Γ 1 = 3.0, log 10 P 1 = Γ 1 = 3.0, log 10 P 1 = Γ 1 = 3.0, log 10 P 1 = Γ 1 = 3.0, log 10 P 1 = PN approximation M 0 Ω E b M ADM M 0 ADM M ADM = 1 2π i ψds i (UWM) / 35
59 1.35M vs 1.35M J / M Piecewise Polytrope Equal-mass : 1.35 M sol M sol Γ 1 = 3.0, log 10 P 1 = Γ 1 = 3.0, log 10 P 1 = Γ 1 = 3.0, log 10 P 1 = Γ 1 = 3.0, log 10 P 1 = Γ 1 = 3.0, log 10 P 1 = PN approximation M 0 Ω J i = 1 16π ǫ ijk (X j K kl X k K jl )ds l (UWM) / 35
60 (UWM) / 35
61 2001 Miller, arxiv:gr-qc/ Baumgarte, Skoge, and Shapiro, PRD 70, (2004) 2005 K.T., Baumgarte, Faber, and Shapiro, PRD 72, (2005) 2006 K.T., Baumgarte, Faber, and Shapiro, PRD 74, (R) (2006) Grandclément, PRD 74, (2006); Erratum 75, (E) (2007) [Shibata and Uryū, PRD 74, (R) (2006)] 2007 [Shibata and Uryū, CQG 24, S125 (2007)] K.T., Baumgarte, Faber, and Shapiro, PRD 75, (2007) 2008 K.T., Baumgarte, Faber, and Shapiro, PRD 77, (2008) Foucart, Kidder, Pfeiffer, and Teukolsky, PRD 77, (2008) 2009 Kyutoku, Shibata, and K.T., PRD 79, (2009) (UWM) / 35
62 2006 Faber, Baumgarte, Shapiro, K.T., and Rasio, PRD 73, (2006) Faber, Baumgarte, Shapiro, and K.T., Astrophys. J. 641, L93 (2006) Sopuerta, Sperhake, and Laguna, CQG 23, S579 (2006) Shibata and Uryū, PRD 74, (R) (2006) 2007 Shibata and Uryū, CQG 24, S125 (2007) 2008 Etienne, Faber, Liu, Shapiro, K.T., and Baumgarte, PRD 77, (2008) Shibata and K.T., PRD 77, (2008) Yamamoto, Shibata, and K.T., PRD 78, (2008) Duez, Foucart, Kidder, Pfeiffer, Scheel, and Teukolsky, PRD 78, (2008) 2009 Etienne, Liu, Shapiro, and Baumgarte, PRD 79, (2009) Shibata, Kyutoku, Yamamoto, and K.T., PRD 79, (2009) (UWM) / 35
63 Conformally flat, Irrotation Excision (UWM) / 35
64 Conformally flat, Irrotation Excision Grandclément, (2006, 2007) K.T., Baumgarte, Faber, & Shapiro (2007, 2008) Foucart, Kidder, Pfeiffer, & Teukolsky (2008) (UWM) / 35
65 Conformally flat, Irrotation Excision Grandclément, (2006, 2007) K.T., Baumgarte, Faber, & Shapiro (2007, 2008) Foucart, Kidder, Pfeiffer, & Teukolsky (2008) [Etienne,Liu,Shapiro,&Baumgarte(2009) ] (UWM) / 35
66 Conformally flat, Irrotation Puncture Kyutoku, Shibata, & K.T. (2009) (UWM) / 35
67 Non-conformally flat, Irrotation (UWM) / 35
68 Non-conformally flat, Irrotation Excision [K.T., Baumgarte, Faber, & Shapiro (2006)] [Foucart, Kidder, Pfeiffer, & Teukolsky (2008)] (UWM) / 35
69 Non-conformally flat, Irrotation Excision [K.T., Baumgarte, Faber, & Shapiro (2006)] [Foucart, Kidder, Pfeiffer, & Teukolsky (2008)] [Foucart, Kidder, Pfeiffer, & Teukolsky (2008)] (UWM) / 35
70 Non-conformally flat, Irrotation Excision [K.T., Baumgarte, Faber, & Shapiro (2006)] [Foucart, Kidder, Pfeiffer, & Teukolsky (2008)] [Foucart, Kidder, Pfeiffer, & Teukolsky (2008)] Puncture (UWM) / 35
71 Conformal factor 3 : 1 J BH /(MADM BH )2 = 0.5 Γ = 2.0 Excision M irr BH = M ADM BH = M B NS = M ADM NS = C MNS ADM R NS = ADM M 0 MADM BH + MNS ADM (UWM) / 35
72 M B NS 5 : 1 = 0.15 E b / M J BH / M BH = J BH / M BH = J BH / M BH = J BH / M BH = J BH / M BH = Ω M 0 E b M ADM M 0 ADM M ADM = 1 i ψds i 2π (UWM) / 35
73 Critical mass ratio Γ = ISCO Mass-shedding Compaction ( C ) 0.270C 3/2 (1 + ˆq) 1 + 1ˆq 1/2 h = `1 3.54C 1/3 i ˆq 0.25 Mass-shedding ISCO Mass ratio ( M ADM BH / MADM NS ) (UWM) / 35
74 1990 (UWM) / 35
75 1990 Piecewise Polytrope irrotation corotation (UWM) / 35
76 1990 Piecewise Polytrope irrotation corotation γ ij = η ij γ ij η ij (UWM) / 35
77 1990 Piecewise Polytrope irrotation corotation γ ij = η ij γ ij η ij Marronetti & Shapiro (2003) Baumgarte & Shapiro (2009) (UWM) / 35
78 2006 (UWM) / 35
79 2006 Γ = 2 M B NS = irrotation (UWM) / 35
80 2006 Γ = 2 M B NS = irrotation Excision K.T.,, Puncture,K.T. S/(M BH ) 2 = (UWM) / 35
81 2006 Γ = 2 M B NS = irrotation Excision K.T.,, Puncture,K.T. S/(M BH ) 2 = M BH ADM /MNS ADM = 1 10 γ ij = η ij (UWM) / 35
82 2006 Γ = 2 M B NS = irrotation Excision K.T.,, Puncture,K.T. S/(M BH ) 2 = M BH ADM /MNS ADM = 1 10 γ ij = η ij (UWM) / 35
2009 2 26 1 3 1.1.................................................. 3 1.2..................................................... 3 1.3...................................................... 3 1.4.....................................................
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