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

4. 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 SN explosion  Realistic calculation of GWs  GRBs (collapsar scenario) Hot disk 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  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 Woosley (1993); MacFadyen & Woosley (1999) 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 ： 4.2M sun  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 Density contour log(g/cm^3) Slightly before the AH formation

10. BH + Disk formation YS & Shibata (2007)  massive core ： 4.2M sun  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 Density contour log(g/cm^3) Slightly before the AH formation Larger region

11. BH mass and spin 1.315 1.32 1.325

12. 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 density temperature

13.  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 Neutrino emission 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 2 nd 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  Neutrino part : streaming neutrino  Fluid part : baryons, e/e+, radiation, trapped neutrino  Basic equations: Energy momentum tensor

18. Lepton conservations  Lepton evolution : In Beta equilibrium

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

20. 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  Electrons and positrons  Ideal Fermi gas  Charge neutrality condition (Yp=Ye)  Radiation   Neutrinos : ideal Fermi gas Shen et al. (1998) EOS table is constracted

21. PNS convection (using old ver. leakage) Ye 197.8 ms 199.7 ms 201.3 ms 202.8 ms 206.7 ms 211.9 ms 215.5 ms 217.3 ms Ye contours  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 Using S15 model of Woosley et al. (2001)

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. The previous study  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 Muller and Janka (1997) A&A 317, 140 115 km 110 0 80 Spherical model No neutrino transfer

24.  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 Notes

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

29. 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))

30.  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 Simplified EOS

31.  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 :

32.  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 pressure decreases ：

33.  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 ?

34.  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, 1.315 1.32 1.325

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

36. 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) : ～ 1.5×10 51 erg per 0.1 M solar  neutrino cooling (loss) :

37. PNS Convection 197.8 ms199.7 ms201.3 ms202.8 ms 206.7 ms211.9 ms215.5 ms217.3 ms  Vigorous convective motion  Shock wave is pushed outward  Enhancement in neutrino luminosity Contours of electron fraction

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

39. 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

40. 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)

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