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Collapse of rapidly rotating massive stellar core to a black hole in full GR Tokyo institute of technology Yu-ichirou Sekiguchi University of Tokyo Masaru Shibata AIU @ KEK 13/03/2008 Introduction Collapse of stellar cores Association with supernova explosion (SN) Association with long GRBs (BH + Disk formation) Main path of stellar-mass BH formation A wide variety of observable signals (GWs, neutrinos, EM radiation) Observations of GWs and neutrinos can prove the innermost part All known four forces play important roles Microphysics • weak interactions — neutrino emission — electron capture • nuclear physics — equation of state (EOS) of dense matter Macro Physics • hydrodynamics — rotation, convection • general relativity • magnetic field — magnetohydrodynamics Importance of GR Dimmelmeier et al (2002) A&A 393, 523 GR Newton Rotation increases strongly during collapse Newtonian : hard to reach nuclear density GR : stronger gravitational attraction ⇒ multiple-spike waveform ⇒ burst-like waveform Qualitative difference in collapse dynamics and in waveforms Importance of microphysics Strong interactions : nuclear EOS Maximum neutron star (NS) mass Dynamics of proto-neutron star (PNS) Weak interactions : Drive hydrodynamic instabilities Convection, SASI Neutrino heating mechanism in ee SN explosion Hot disk Realistic calculation of GWs GRBs (collapsar scenario) e e YS & Shibata (2007) Contents of my talk Rotating collapse to a BH with simplified EOS Collapsar scenario BH + Disk formation Full GR simulation with microphysics Summary of implementation GWs from proto-neutron star (PNS) convection Summary and Future works Rotating collapse to a BH Collapsar model Woosley (1993); MacFadyen & Woosley (1999) Central engine of GRBs : BH + Disk Energy source : Gravitational energy of accretion matter neutrino annihilation ( e e) EGRB, ~ v ⇒ GM BH M Disk 0.42 M Disk c 2 RISCO BH spin ⇒ electromagnetic flux E.g. via Blandford-Znajek process EGRB,B GRB,B f (q)M BHc2 0.29GRB,B M BHc2 MacFadyen & Woosley 1999 What is done Collapse simulation of rapidly rotating, massive core in full GR (Einstein eq. : BSSN formalism) (Gauge condition : 1+log slicing, Dynamical shift) (hydrodynamics : High-resolution central scheme) (A BH excision technique (Alcubierre & Brugmann (2001))) Simplified EOS (e.g. Zwerger & Muller (1997)) Qualitative feature can be captured Rigidly rotating polytrope (Γ=4/3) at mass shedding limit Formation of BH + Disk formation Mass (BH : Disk), BH spin Disk structure Estimates of neutrino luminosity P Pcold Pth K 1 , nuc Pcold K 2 , nuc 1 4 / 3, 2 2.0 Pth ( th 1) YS & Shibata (2007) BH + Disk formation massive core:4.2Msun spin parameter = 0.98 (rigid rotation) Simplified EOS Slightly before the AH formation BH + Disk formation Shock wave formation at Disk BH : 90~95% mass Disk : 5~10% mass BH spin ~ 0.8 Density contour log(g/cm^3) YS & Shibata (2007) BH + Disk formation massive core:4.2Msun spin parameter = 0.98 (rigid rotation) Simplified EOS Slightly before the AH formation Larger region BH + Disk formation Shock wave formation at Disk BH : ~95% mass Disk : ~5% mass BH spin ~ 0.8 Density contour log(g/cm^3) BH mass and spin 1.315 1.32 1.325 density Outcome Convenient for GRB fireball Low density region [ e e ] L L [n e p e] L e temperature Shock heating Large neutrino luminosities Less Pauli blocking by electrons Thick Disk rate 1 cos colision Preconditioning: Subsequent evolution on viscous time-scale Q ~ 2 , L , L vis 2 Neutrino emission Disk structure: High temperature (10^11K) due to shock Small density along the rotational axis Neutrino luminosity 1 Rdisk N T L 5 1053 erg/s 11 17 2 3 10 K 10 g/cm 70km 2 2 Full GR study with microphysics required Pair annihilation rate (Setiawan et al. (2005)) L L 5 1052 erg/s 53 5 10 erg/s 2 Notes No mechanism for time variation More sophisticated studies are required Full GR simulation with microphysics Current status No full GR, multidimensional simulations including realistic EOS, electron capture, and neutrino cooling Necessary for rotating BH formation, GRBs, and GW Electron capture with not self-consistent manner Ott et al. (2006); Dimmelmeier et al. (2007) Recently, I constructed a code including all the above for the first time (the following 2nd part of my talk) ○ ○ sophisticated Difficulty in full GR simulation To treat the neutrino cooling in numerical relativity If one adds a cooling term into the right-hand side of the matter equation T Q 0 ⇒ constraint violation One have to add the cooling in terms of the energy momentum tensor Energy momentum tensor leak loc Q Q eff Qb Q ub leak u loc b Energy momentum tensor Q Q ,stream (T tot ) (T Matter ) (T ) (T M )Qloc(T ,trap ) ( T )) short T diff ( for leak (T ) (T ) (otherwise) Q Neutrino part : streaming(cneutrino . f . Ruffert et al. (1996); Fluid ,stream Fluid part : baryons, e/e+, Rosswog radiation, trapped neutrino (2002)) & Liebendoerfer Basic equations: a (T Fluid )ba Qb a (T ,stream )ba Qb Qb includes : e capture (Fuller et al. (1985)) / capture (Fuller et al. (1985)) e -annihilation (Cooperstein et al. (1986)) plasmon decay (Ruffert et al. (1996)) (T Fluid )ab : perfect fluid bremsstrahlung (Burrows et al. (2004)) (T ,stream )ab Ena nb F a nb F b na P ab neutrino leakage (described later) Lepton conservations Lepton evolution : Ye, e, e, x dYe e-cap ep-cap dt d (Y e ) e-cap pair plasmon eleak dt d (Y e ) ep-cap pair plasmon e leak dt d (Y x ) pair plasmon xleak dt d (Yl ) lleak dt In Beta equilibrium e-cap/ep-cap : Fuller et al.(1985) pair : Cooperstein et al. (1986) plasmon : Ruffert et al. (1996) leak : neutrino leakage (explained later) Neutrino emission Neutrino Leakage Scheme Cross sections by Burrows et al. (2003) “Cross sections” : i ( E ) i E2 “Opacities” A p n e 2 ( E ) ( E ) E i : 2 ( E ) ds E “Optical depth” : Diffusion time : diff T A e p p en n e x( E ) 2 2 ( E ) ( E ) E c c (T s ) diff n nˆ ( E )dE pe Neutrino energy and number diffusion : E nˆ ( E ) Q diff dE T 2 F1 ( ) T ( E ) nˆ ( E ) diff R diff dE T F0 ( ) T ( E ) ne (T t ) Tdiff ~ Tdyn Equations of state Baryons EOS table based on relativistic mean field theory (Shen et al. (1998)) Sound velocity does not exceed the velocity of light EOS table is constracted Electrons and positrons Ideal Fermi gas Charge neutrality condition (Yp=Ye) Radiation 4 Pr arT / 3, r 3Pr / Neutrinos : ideal Fermi gas Shen et al. (1998) PNS convection (using old ver. leakage) Using S15 model of Woosley et al. (2001) Neutrino burst emission Shock passes the neutrino sphere ⇒ Copious neutrino emission from hot region behind the shock ⇒ shock stalls ⇒ negative lepton/entropy gradients ⇒ convectively unstable Ye 202.8 201.3 ms 197.8 199.7 ms Ye contours 215.5ms 217.3 ms 206.7 ms 211.9 Gravitational waves YS (2007) Amplitude : h ~ 6-9×10-21 @10 kpc ~rotational core bounce frequency : 100-1000 Hz Convection timescale : 1~10 ms Convective eddies penetrate PNS Core bounce The previous study Muller and Janka (1997) A&A 317, 140 amplitude : h ~ 3×10-21 @ 10 kpc frequency : 100-1000 Hz Spherical model No neutrino transfer The hydrostatic condition is imposed at PNS surface Convective motions are suppressed near the boundary 80 Smaller Amplitude frequency 0 110 115 km Notes Gravitational wave amplitude 2 2 d Q 2 G 1 2 GM R v ij Due to convection h ~ ~ nonsphe 2 c 4 D dt 2 c R D c 2 C 10kpc R v nonsphe omp ~ 1020 0.1 0.3 D 10km 0.1c Cf. Due to core bounce ~ 10 20 nonsphe 0.1 2 10kpc M R f D 1.4 M 10km 1kHz No effects to suppress the convective activities Neutrino transport will flatten the existing negative gradients The GW amplitude is the maximum estimates Summary Rotating collapse to a BH BH + Disk formation (with simplified EOS) Shock occurs at the disk Outcome: low density region, high temperature thick disk New full GR code with microphysics Brief description of the implementation neutrino radiation energy momentum tensor leakage scheme for neutrino cooling nuclear EOS by Shen et al. (1998) GWs from PNS convection As large amplitude as GWs from rotational core bounce Future works Formation of Kerr BH Association of GRBs (BH+Disk formation) Initial conditions based on stellar evolution are now available (Yoon et al (2006); Woosley & Heger (2006)) PopIII star collapse GWs from it Realistic calculation of gravitational waveforms Effects of magnetic fields Fruitful scientific results will be reported near feature What to explore further ee BH + Disk formation Disk structure Shock strength Neutrino luminosity Hot, thick Disk Time variability in Lν Mass, angular momentum dependence Low density region Magnetic field Metallicity dependence Einstein’s equation Gab 8 G Tab 4 c BSSN reformulation (Shibata & Nakamura (1995); Baumgarte & Shapiro (1999)) Cartoon method (Alcubierre et al (2001) )is adopted to solve equations in the Cartesian coordinate Gauge condition Approximate maximal slicing (Balakrishna et al. (1996); Shibata (1999)) Dynamical shift (Shibata (2003)) t t Fi jl l L ij 2 jl l Aij Aij jl l 1 L K 6 t L ij 2 Aij t L Aij e 4 Rij Di D j TF KAij 2 Aik Akj 8 e 4 Sij ij S / 3 t L K D k Dk Aij Aij K 2 / 3 4 h S Simplified EOS Equation of State parametric EOS : P Pcold Pth Pcold K1 , nuc , Pth ( th 1) K 2 , nuc idealized EOS : microphysics is treated only qualitatively maximum allowed mass of EOS : M max,EOS 2M c.f. the maximum pulsar mass : M 2.1 0.2M parameters of EOS ( 4 / 3) 1.31 1.325 2 2.45 2.6, nuc 2 1014 g/cm3 th 1 (Nice et al. 2005) BH formation → Disk formation mass of the (inner) core is larger than the maximum allowed mass → prompt BH formation matter with large angular momentum forms a thin disk around the BH kinetic energy is converted into thermal energy at the disk surface by shocks GM BH M disk The gravitational energy released : E 4 9 1052 erg RISCO M disk 0.1 0.2M , RISCO 4 5Gc 2 M BH Disk formation → shock wave formation (1) The disk height H increases as the thermal energy is stored (balance relation) Pdisk Pram s H GM BH H 2 ( RISCO H 2 )3/ 2 GM BH H 3 RISCO 2 Pdisk Pram H 2 1031 11 s dyn/cm 10 g/cm RISCO 3 temperature and density of the disk increase to be disk 1012 g/cm3 , Tdisk 1011 K Pdisk While the ram pressure decreases: Pram vf 1031 dyn/cm 2 f vf2 1030 ( f /1010 g/cm ) dyn/cm 2 3 (2GM BH / RISCO )1/ 2 0.4 0.5c Disk formation → Shock wave formation (2) Now Pdisk 1031 dyn/cm 2 , Pram Pdisk Pram s H GM BH H 2 ( RISCO H 2 )3/ 2 Pdisk , then H / RISCO GM BH H2 Pdisk Pram 1 GM BH s H The disk expands escaping the gravitational bound Pdisk Pram :strong shock waves are formed and propagated Shock waves are mildly relativistic ~ 0.5c does neutrino cooling work ? condition that thermal energy be stored is GM BH m L 1053 erg/s RISCO m M s 1 The present results show m 10M s 1 1.315 Unless the conversion efficiency is too low (<<0.1), thermal energy is stored In the a few millisecond, disk 1031 erg/cm 3 Pdisk α1.32 the 1.325 Sack et al. 1980 neutrino loss small neutrino loss large Stall of shock wave Note that the shock stalls due to insufficient energy input bounce core mass (Goldreich & Weber (1980) ApJ. 238, 991; Yahil (1983) ApJ. 265, 1047) : M core K Kinit 3/ 2 M init Y ~ l Y l ,init 2 M init ~ 0.6M init , Initial shock energy (input): accretion power (input): Photo-dissociation (loss) : ~ 1.5×1051 erg per 0.1 Msolar Eshock, init Lhydro Ldiss Y hc K (3 1/ 3 l 4 mB M core vinfall 2 6 10 erg M 0.4 c 51 R 1.4 10 erg/s shock 100km 53 R 1.110 erg/s shock 100km 53 4 neutrino cooling (loss) : 4/3 2 2 infall vinfall 9 3 10 g/cm 0.2c infall vinfall 9 3 10 g/cm 0.2c 2 T R L ~ 1053 erg/s block 10 MeV 50 km 3 PNS Convection Vigorous convective motion Shock wave is pushed outward Enhancement in neutrino luminosity Contours of electron fraction 197.8 ms 206.7 ms 199.7 ms 211.9 ms 201.3 ms 215.5 ms 202.8 ms 217.3 ms Energy available in convection Exchange of fluid element via ⊿h (d ) blob (dP)amb (ds ) blob blob (dYe ) blob s Y P s ,Ye P ,Ye e s,P (d )amb (dP)amb (ds)amb (dYe )amb Y P s ,Ye s P ,Ye e s,P Free energy available per unit mass 1 w g eff amb (d blob (d )amb g eff h blob (dP)blob (dP)amb 1 ln P ln P 1 (ds) (dYe )amb ln P ln P amb h ln s ,Ye ln s ,Ye s ln Ye , s ln s ,Ye Ye Convection of mass ⊿M W amb M h | Ye | | s | 50km M PNS 1051 ergs , 0.3 M 10km Y s r M e [( ln P / ln s ,Ye ( ln P / ln s ,Ye ( ln P / ln Ye s , O (1)] Applications : rotational core bounce Deformation of neutrino sphere due to the rotation will play an important role Shock propagate in z-direction suffered more from the neutrino burst Deceleration of motion along the rotational axis GWs are also modifeid Contours of electron fraction Deformed neutrino sphere Gravitational wave signal Gravitational waves : Type-I waveform Comparison with Ott et al. (2006) : Second peak is surppressed Due to deceleration along z-direction A2 I zz I xx Spectrum is similar GW is mainly due to bounce motion Ott et al. (2006) This peak is associated with non-axisymmetric instabilities Neutrino emission Neutrino Leakage Scheme “Cross sections” : i ( E ) i E2 “Opacities” Cross sections by Burrows et al. (2003) : (E ) i (E ) E2 2 “Optical depth” : ( E ) ds E Diffusion time-scale : T diff x( E ) 2 2 ( E ) ( E ) E c c Neutrino energy and number diffusion : E nˆ ( E ) 4 cg diff 2 Q diff dE ( k T ) F1 ( ) B 3 2 T ( E ) (hc) nˆ ( E ) 4 cg diff R diff dE (k BT ) F0 ( ) 3 2 T ( E ) (hc) n nˆ ( E )dE (T t ) Tdiff ~ Tdyn (T s )