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Tokyo institute of technology Yu-ichirou Sekiguchi University of Tokyo Masaru Shibata

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

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Tokyo institute of technology Yu-ichirou Sekiguchi University of Tokyo Masaru Shibata

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  1. 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

  2. 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 • neutrinoemission • electron capture • nuclear physics • equation of state (EOS) of dense matter • Macro Physics • hydrodynamics • rotation, convection • general relativity • magnetic field • magnetohydrodynamics

  3. Importance of GR Dimmelmeier et al (2002) A&A 393, 523 • Rotation increases strongly during collapse • Newtonian : hard to reach nuclear density⇒ multiple-spike waveform • GR : stronger gravitational attraction⇒ burst-like waveform GR Newton Qualitative difference in collapse dynamics and in waveforms

  4. Hot disk Importance of microphysics • Strong interactions: nuclear EOS • Maximum neutron star (NS) mass • Dynamics of proto-neutron star (PNS) • Weak interactions : • Drivehydrodynamic instabilities • Convection, SASI • Neutrino heating mechanism in SN explosion • Realistic calculation of GWs • GRBs (collapsar scenario) YS & Shibata (2007)

  5. 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

  6. Rotating collapse to a BH

  7. Collapsar model Woosley (1993); MacFadyen & Woosley (1999) • Central engine of GRBs : BH + Disk • Energy source : • Gravitational energy of accretion matter ⇒ neutrino annihilation ( ) • BH spin ⇒ electromagnetic flux • E.g. via Blandford-Znajek process MacFadyen & Woosley 1999

  8. 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

  9. BH + Disk formation YS & Shibata (2007) • massive core：4.2Msun • spin parameter = 0.98 (rigid rotation) • Simplified EOS • BH + Disk formation • Shock wave formation at Disk • BH : 90~95% mass • Disk : 5~10% mass • BH spin ~ 0.8 Slightly before the AH formation Density contour log(g/cm^3)

  10. BH + Disk formation YS & Shibata (2007) • massive core：4.2Msun • spin parameter = 0.98 (rigid rotation) • Simplified EOS • BH + Disk formation • Shock wave formation at Disk • BH : ~95% mass • Disk : ~5% mass • BH spin ~ 0.8 Slightly before the AH formation Larger region Density contour log(g/cm^3)

  11. 1.315 1.32 1.325 BH mass and spin

  12. density Outcome • Convenient for GRB fireball • Low density region • Shock heating • Large neutrino luminosities • Less Pauli blocking by electrons • Thick Disk • Preconditioning: Subsequent evolution on viscous time-scale temperature

  13. Neutrino emission • Disk structure: • High temperature (10^11K) due to shock • Small density along the rotational axis • Neutrino luminosity • Pair annihilation rate(Setiawan et al. (2005)) • Notes • No mechanism for time variation • More sophisticated studies are required Full GR study with microphysics required

  14. Full GR simulation with microphysics

  15. ○ ○ 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

  16. 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 • ⇒constraint violation • One have to add the cooling in terms of the energy momentum tensor

  17. Energy momentum tensor • Energy momentum tensor • Neutrino part : streaming neutrino • Fluid part : baryons, e/e+, radiation, trapped neutrino • Basic equations:

  18. Lepton conservations • Lepton evolution : In Beta equilibrium

  19. Neutrino emission Cross sections by Burrows et al. (2003) • Neutrino Leakage Scheme • “Cross sections” : • “Opacities” : • “Optical depth” : • Diffusion time : • Neutrino energy and number diffusion :

  20. Equations of state Shen et al. (1998) • Baryons • EOS table based on relativistic mean field theory (Shen et al. (1998)) • Sound velocity does not exceed the velocity of light • Electrons and positrons • Ideal Fermi gas • Charge neutrality condition (Yp=Ye) • Radiation EOS table is constracted • Neutrinos : ideal Fermi gas

  21. 201.3 ms 202.8 ms 197.8 ms 206.7 ms 217.3 ms 199.7 ms 215.5 ms Ye 211.9 ms 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 contours

  22. 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

  23. 80 0 110 The previous study Muller and Janka (1997) A&A 317, 140 Spherical model No neutrino transfer • amplitude : h ～ 3×10-21 @ 10 kpc • frequency : 100－1000 Hz • The hydrostatic condition is imposed at PNS surface • Convective motions are suppressed near the boundary • Smaller • Amplitude • frequency 115 km

  24. Notes • Gravitational wave amplitude • Due to convection • Cf. Due to core bounce • No effects to suppress the convective activities • Neutrino transport will flatten the existing negative gradients • The GW amplitude is the maximum estimates

  25. 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

  26. 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

  27. Hot, thick Disk Low density region What to explore further • BH + Disk formation • Disk structure • Shock strength • Neutrino luminosity • Time variability in Lν • Mass, angular momentum dependence • Magnetic field • Metallicity dependence

  28. Einstein’s equation • 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))

  29. Simplified EOS • Equation of State • parametric EOS : • idealized EOS : microphysics is treated only qualitatively • maximum allowed mass of EOS : • c.f. the maximum pulsar mass :(Nice et al. 2005) • parameters of EOS

  30. 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 • The gravitational energy released :

  31. Disk formation → shock wave formation (1) • The disk height H increases as the thermal energy is stored (balance relation) • temperature and density of the disk increase to be • While the ram pressuredecreases：

  32. Disk formation → Shock wave formation (2) • The disk expands escaping the gravitational bound • ：strong shock waves are formed and propagated • Shock waves are mildly relativistic ～ 0.5c • does neutrino cooling work ?

  33. 1.315 1.32 1.325 • condition that thermal energy be stored is • The present results show • Unless the conversion efficiency α is too low (<<0.1), the thermal energy is stored • In the a few millisecond,

  34. Sack et al. 1980 neutrino loss small neutrino loss large

  35. 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) : • Initial shock energy (input): • accretion power (input): • Photo-dissociation (loss): ～ 1.5×1051 erg per 0.1 Msolar • neutrino cooling (loss) :

  36. PNS Convection • Vigorous convective motion • Shock wave is pushed outward • Enhancement in neutrino luminosity Contours of electron fraction 197.8 ms 199.7 ms 201.3 ms 202.8 ms 206.7 ms 211.9 ms 215.5 ms 217.3 ms

  37. amb blob Energy available in convection • Exchange of fluid element via ⊿h • Free energy available per unit mass • Convection of mass ⊿M

  38. 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

  39. Gravitational wave signal • Gravitational waves : Type-I waveform • Comparison with Ott et al. (2006) : Second peak is surppressed • Due to deceleration along z-direction • Spectrum is similar • GW is mainly due to bounce motion This peak is associated with non-axisymmetric instabilities Ott et al. (2006)

  40. Neutrino emission Cross sections by Burrows et al. (2003) • Neutrino Leakage Scheme • “Cross sections” : • “Opacities” : • “Optical depth” : • Diffusion time-scale : • Neutrino energy and number diffusion :

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