Collapse of Differentially Rotating Supermassive Stars: Post Black Hole Formation Stage Motoyuki Saijo (Rikkyo University, Japan) Ian Hawke (University of Southampton, UK) CONTENTS 1. Introduction 2. General Relativistic Hydrodynamics 3. Collapse of Differentially Rotating Stars 4. Summary 1
1. Introduction Supermassive Objects Our galaxy (Sgr A*) @Chandra Formation theory of supermassive black holes Still uncertain (Rees 24) 2
Supermassive Star Collapse 1. Uniformly Rotating Stars high viscosity strong magnetic field t ev > t vis,t mag (1) maintains uniform rotation (1) SMS contracts until mass shedding, conserving the angular momentum (2) SMS evolves quasi-stationary along the mass shedding limit, releasing mass and angular momentum (3) Reaches the critical onset of collapse because of relativistic gravitation Black hole and a disk forms after the gravitational collapse a/m ~.75 BH disk stable (2) (3) unstable (Shibata & Shapiro 2) 3 9% 1%
2. Differentially Rotating Supermassive Stars t ev < t vis,t mag low viscosity magnetic braking differential rotation (1) Spherically symmetric star + angular momentum (2) Star contracts along the zero-viscosity sequence, conserving the angular momentum 1. post-newtonian gravitational instability 2. bar mode instability Possible! (investigation of the zero-viscosity sequence in Newtonian gravitation) (New & Shapiro ) 4
2. Differentially Rotating Supermassive Stars t ev < t vis,t mag low viscosity magnetic braking differential rotation (1) Spherically symmetric star + angular momentum (2) Star contracts along the zero-viscosity sequence, conserving the angular momentum 1. post-newtonian gravitational instability 2. bar mode instability Possible! (investigation of the zero-viscosity sequence in Newtonian gravitation) (New & Shapiro ) Star may collapse because of post-newtonian gravitational instability during the contraction 4
2. General Relativistic Hydrodynamics Einstein s Field Equations normal line t coordinate line 3+1 Decomposition proper distance coordinate distance 3D hypersurface BSSN Formalism 17 spacetime variables for time evolution Lapse (Generalised hyperbolic K-driver) Shift (Generalised hyperbolic driver) t β i β j j β i = 3 4 αbi 5 t B i β j j B i = t Γi β j j Γi ηb i
Relativistic Hydrodynamics Perfect fluids with Gamma-law EOS Generalised Roe method for the shocks Space and Time Refinement To maintain resolution at the central core, we introduce 4-1 levels of spatial refinement (increase the refinement level during the collapse) Enable to adjust the timestep for each refinement levels 6
3. Collapse of Differentially Rotating Stars Collapse Condition Require radially unstable, differentially rotating stars for collapse Evolution is necessary to determine the radial stability of the star 1% pressure deplete to trigger collapse! max 1-1 1-2 1-3 1-4 I II III IV 1-5 2 4 6 8 1 t / t dyn Unstable Central density has an exponentially growth during the evolution Stable Central density remains oscillation around its equilibrium value 7 I, II: Unstable, III, IV: Stable
M (BH) / M 8J(BH)/ J 1.8.6.4.2.8.6.4.2 J (BH) / M (BH) 2 Growth of a black hole 1 2 3 4 t / t dyn 1.8.6.4.2 I II 1 2 3 4 t / t dyn Dynamical Horizon Spacelike world tube of the apparent horizon Compute the mass and the angular momentum of the hole locally Numerical Results Mass and angular momentum of the hole grows monotonically through evolution Final Kerr parameter of the holes are >.96 Unfortunately computation of the mass and angular momentum is not sufficiently accurate
/ M M (disk) Disk Formation 1.8.6.4.2 1 2 3 4 t / t dyn Definition of the disk mass Rest mass outside the apparent horizon Numerical results Disk mass monotonically decreases after black hole formation Reaches the quasi-stationary state (has a plateau) Density Snapshot in meridional plane z / M 1 8 6 4 2 Horizon radius ~.2 M Matter exists very close to the hole Indicates the existence of a rapidly rotating hole -1-8 -6-4 -2 2 4 6 8 1 x / M 9
r M Re[! 4 ] r M Re[! 4 ] Gravitational Waveform.2.1 -.1 -.2 2 4 6 8 t / M.1 6e-5-6e-5 Meaning of Represents outgoing gravitational waves at spatial infinity Observer at r~4m in the equatorial plane Burst and ringdown can be seen in the gravitational waveform Amplitude seems to grow at the ringdown when the hole is very close to the maximum Kerr Resonance of the disk through ringdown?! BH " kep 1 -.1 2 4 6 8 1 t / M
4. Summary We investigate the formation of the massive disk through the collapse of differentially rotating unstable stars by means of hydrodynamic simulations in General Relativity When the newly formed black hole is very close to the maximum Kerr with sufficient surrounding matter, various instability of the disk might occur (gravitational wave sources) Further evolution (and resolution) is necessary to confirm the above statement pxpxpxpxpxpx ( ) ( ) h burst 2 1 19 M 1Gpc 1 6 M d pxpxpxpxpxpx f QNM 2 1 2 ( 1 6 M M ) [Hz] 11