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Yu-ichirou Sekiguchi (Univ. of Tokyo) 関口　雄一郎

Axisymmetric collapse simulations of rotating massive stellar cores in full general relativity: Numerical study for prompt black hole formation. Yu-ichirou Sekiguchi (Univ. of Tokyo) 関口　雄一郎. §1 Introduction §2 Numerical Implementation §3 Setting §3.1 Initial conditions

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Yu-ichirou Sekiguchi (Univ. of Tokyo) 関口　雄一郎

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  1. Axisymmetric collapse simulations of rotating massive stellar cores in full general relativity:Numerical study for prompt black hole formation Yu-ichirou Sekiguchi (Univ. of Tokyo) 関口　雄一郎

  2. §1 Introduction • §2 Numerical Implementation • §3 Setting • §3.1 Initial conditions • §3.2 Parametric Equations of state • §4 General Feature of Collapse • §5 Black Hole Formation • §5.1 Criterion for prompt black hole formation • §5.2 Dependence on parameters • §5.3 Prediction of the final system • §6 Neutron Star Formation • §6.1 Collapse dynamics • §6.2 Gravitational Waves • §7 Summary

  3. §1 Introduction

  4. §1 Introduction ① In this talk, let me talk about ……. • Results of simulations for rotating stellar core collapse in full GR • Highly nonlinear and dynamical phenomena • Numerical simulation in full GR is the unique approach • Numerical relativity as a powerful tool of exploring astrophysical phenomena • Black hole formation ? • Neutron star formation ?

  5. §1 Introduction ② • Black hole formation via massive rotating stellar core collapse • Candidate for the central engine of the long duration GRBs • Known as the collapsar model (Woosley ApJ 405, 273 (1993)) • A source of the gravitational radiation • Quasi-normal ringing • Neutron star formation via massive rotating stellar core collapse • Study extensively in Newtonian gravity • e.g. Zwerger and Muellar A&A 320, 209 (1997) • A promising source of the GW

  6. §1 Introduction ③ • We consider a criterion of black hole formation in the collapse of stellar iron cores • Performing fully general relativistic simulations • On assumption of Axial symmetry • Putting emphasis on clarifying the dependence of black hole formation on mass, angular momentum, rotational velocity profile of iron cores, and equations of state • For systematic investigation, a parametric equation of state (e.g. Dimmelmeir et al. A&A 393, 523 (2002)) is adopted (which will be introduced later)

  7. §2 Numerical Implementation • York in “Sources of gravitational radiation” (1979) • Baumgarte & Shapiro Phys.Rep. 376, 41 (2003) • Lehner Class. Quantum Grav. 18, R25 (2001) • Font Living Rev. Relat. 6, 4 (2003)

  8. §2 Numerical Implementation • Einstein equations • ADM (3+1) decomposition of the spacetime • e.g. York in Sources of gravitation (1979) Cambridge; Baumgarte & Shapiro Phys.Rep. 376, 41 (2003) • Shibata-Nakamura (BSSN) reformulation • Shibata and Nakamura PRD 52, 5428 (1995), Baumgarte & Shapiro PRD 59, 024007 (1999) • Cartoon method (solving 2D problem in Cartesian grid) • Alcubierre et al. Int.J.Mod.Phys. D10, 273 (2001) • Gauge conditions • Approximate maximal slicing condition (Shibata Prog.Theor.Phys. 101, 251 (1999)) • Dynamical gauge (shift) condition (Shibata ApJ 595, 992 (2003)) • e.g. Baumgarte & Shapiro (2003) • Apparent horizon finder(Shibata PRD 55, 2002 (1997))

  9. Δ

  10. §3 Setting • §3.1 Initial conditions • §3.2 Parametric equations of state

  11. §3.1 Setting -- Initial conditions -- • Initial conditions • Rotating iron cores of massive stars • Modeled by rotating polytrope in equilibrium • Central density • Mass range • angular momentum

  12. §3.1 Setting -- Initial conditions -- • Rotation law (Komatsu et al. MNRAS 239, 153 (1989)) • In Newtonian limit • Cylindrical rotation • Differential rotation parameter : A

  13. §3.2 Setting -- Parametric EOS (1) -- • Parametric equations of state • Parameters of EOS : • Parameters of EOS are so chosen that the maximum mass of the cold spherical polytropes is almost identical • We set for simplicity Unstable due to the photo-dissociation and electron capture Sudden stiffening due to nuclear force Thermal and shock heating effects

  14. §3.2 Setting -- Parametric EOS (1) -- • We set for simplicity • Note : Collapse dynamics is less sensitive to the value of . • As long as • Shibata & Sekiguchi PRD 69, 084024 (2004)

  15. §3.2 Setting -- Parametric EOS (2) -- • Parameters of EOS Note: EOS-c is stiffer than EOS-b

  16. §4 General Feature of Collapse Infall phase Bounce phase Ring-down phase

  17. §4 General feature of the collapse (1) • Infall phase : • Core becomes unstable due to sudden softening of EOS • Photo-dissociation, electron capture Outer core : The outer region in which the matter falls at supersonic velocity Inner core : The inner region which collapses at subsonic velocity

  18. §4 General feature of the collapse (2) • Bounce phase : • Sudden stiffening of EOS decelerates the inner core at supra-nuclear density • (a) mass of the inner core is very large → collapse to a black hole • (b) mass of the inner core is not too large → bounce • Part of stored internal energy at bounce is released • The shock wave is generated at the outer edge of the inner core (a) (b) BH IC

  19. §4 General feature of the collapse (3) • Ringdown phase : (after the bounce) • The inner core oscillation damps via PdV works (This process powers shocks) • (a) Shock is strong enough → a neutron star is left • (b) Shock is not strong enough → fallback induced collapse to a black hole (b) (a) Fall back NS NS BH

  20. §5 Black Hole Formation • Acriterion for black hole formation • Dependence on parameters • Dependence of EOS • Effects of shocks • Effects of rotation • Effects of differential rotation • Predicting the final system

  21. §5.1 A criterion for prompt black hole formation－ in M-q plane － ,d ■ : BH for all EOS ☆ : BH for EOS-b (-d) × : BH for EOS-a □ : NS for all EOS

  22. §5.2 Black hole formation － Dependence on EOS －

  23. Direct Collapse Neither sudden stiffening of EOS nor Rotational effect cannot halt the collapse Direct collapse to a BH

  24. §5.2 Black hole formation － Dependence on EOS － • A black hole is formed directly without any distinct bounce Mass of the inner core at bounce is large No shock propagates outward BH is more likely to be formed

  25. §5.2 Black hole formation － Dependence on EOS －

  26. Fallback Induced Collapse (1) The shock wave propagate outward ……, however,

  27. Fallback Induced Collapse (2) The shocked matters fall back to the inner core and a black hole is eventually formed

  28. §5.2 Black hole formation － Dependence on EOS － • Inner cores experience a bounce before BH formation ・Mass of the inner core at bounce is large ・Shocks propagate outward Contributes to support the core Threshold mass of prompt BH formation is larger than for the cases with EOS-a

  29. §5.2 Black hole formation － Dependence on EOS － • Dependence on • For larger • Cores collapse more homologously • Mass of the inner core at the bounce is larger • Shocks (if generated) heat less fraction of the core • Degree of overshooting at the bounce is larger • For smaller • The initial pressure reduction is lager (in particular at central region) • The central region collapses first • Mass of the inner core at bounce is smaller • Shocks heat larger fraction of the core

  30. §5.2 Black hole formation － Dependence on EOS － • Dependence on • For smaller • Equation of state for proto-neutron star is softer • Degree of overshooting is larger • Larger inner core mass • Larger degree of overshooting • Compactness at maximum compression is larger BHs are more liable to form promptly

  31. §5.2 Black hole formation － Dependence on EOS － • A black hole is formed directly without any distinct bounce is larger Mass of the inner core at bounce is large No shock propagates outward BH is more liable to be formed

  32. §5.2 Black hole formation － Dependence on EOS － • Inner cores experience a bounce before BH formation is smaller Shocks propagate outward Contributes to support the core Threshold mass of prompt BH formation is larger than for the cases with EOS-a

  33. §5.2 Black hole formation － Dependence on EOS － ■ : BH for all EOS ☆ : BH for EOS-b (-d) × : BH for EOS-a □ : NS for all EOS

  34. §5.2 Black hole formation － Dependence on EOS － • The pressure near nuclear density is larger for EOS-c Shocks are stronger for EOS-c Threshold mass is larger

  35. §5.3 Black hole formation－ Effect of shocks － Contribution of Maximum mass of the cold spherical polytrope For spherical models Thermal effects increase the threshold mass by 20 ～40 % Effect of shock is stronger for EOS-c

  36. §5.4 Black hole formation － Effectsof rotation － Rotational effects (i) Effectively supply additional pressure (ii) Reduce the amount of matters which eventually fall into inner core Threshold mass for rotating models may be written as (Shibata (2000) PThP 104, 325) Rotational effects increases the threshold mass at most by 17 ～ 20 %

  37. §5.5 Black hole formation－ Effectof differential rotation － As the degree of differential rotation increases, a black hole is less liable to form The threshold for BH formation locates between these curves The inner region which is responsible to black hole formation “rotates” more rapidly

  38. §5.6 Estimation of mass of disk Cf. Shibata and Shapiro ApJ 572, L39 (2002) ・ Consider the innermost stable circular orbit (ISCO) around a formed BH Fluid elements of smaller specific angular momentum will fall into the black hole ISCO ・ If increases as a result of the accretion, more fluid elements fall into the BH BH ・ Thus, if evolution of has a maximum, the dynamical growth of BH will terminate there

  39. §5.6 Estimation of mass of disk ・ Define mass and spin parameter in terms of the specific angular momentum : q(j) and m(j) ・Approximating the spacetime as Kerr spacetime, Jisco can be expressed by m(j) and q(j) e.g. Shapiro and Teukolsky Chap.12 Search the maximum of Jisco (j)

  40. §5.6 Estimation of mass of disk Mass of the formed disk will be < 10% of the initial mass

  41. §6 Neutron star Formation • 6.1 Dependence of EOS • 6.2 Gravitational waves • Waveforms • Energy spectra

  42. §6.1 Dependence on EOS • The collapse dynamics depends strongly on the adopted EOS • The effects are reflected in Gravitational waves

  43. §6.1 Dependence on EOS

  44. §6.1 Dependence on EOS

  45. §6.1 Dependence on EOS

  46. §6.1 Dependence on EOS

  47. §6.1 Dependence on EOS Since mass of the inner core and the degree of overshooting of the Inner core at bounce is larger….., A steep density gradient is formed around the rotational axis Aspherical shock wave generation and propagation

  48. §6.1 Dependence on EOS Since the density of matters in front of the shock “pole” is much smaller, The shock velocity is higher in this direction

  49. §6.1 Dependence on EOS Difference of the centrifugal force between along the rotational axis and around the equator is not very large The density gradient along the rotational axis is not outstanding A slightly prorate shock wave is generated

  50. §6.1 Dependence on EOS A slightly prorate shock wave is generated

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