大質量星の重力崩壊におけるブラックホール形成	 
関口雄一郎（京都大学基礎物理学研究所）	 

BH formation and Long GRBs	 
 Collapse of massive stellar core to BH + Disk

Promising theoretical candidate of central engine ofLong Gamma-ray Bursts (LGRBs)
Time	  Proﬁle	  HighlyTime-variable	

BH0	  	  	  	  	  	  10	  	  	  	  	  20	  	  	  	  	  30	  	  	  	  	  40	  	  	  	  	  50	  	  	  [s]	  1050erg/s < Lγ < 1052erg/s
Most	  violent	  explosion	  in	  the	  universe	  To produce LGRB	 

If the LGRB jet is produced by neutrino pair annihilation
 Huge amount of neutrinos should be emitted
 in a highly time-variable mannerBasic Equations and Physics	 
 Solve Einstein’s equation with eqs. for source fields

 All four known interactions play important roles
 Gravity : GR, BH formation, ISCO, etc
 Strong : EOS (equation of state ) of dense nuclear/hadronic matter EM: MHD phenomena, EOS of dense matter Weak : Electron capture, Neutrino production
 99% gravitational binding energy released is carried away byneutrinos in SNePrevious full GR numerical studies	  実はあまりシミュレーションは行われていない 球対称 w. Boltzmann transfer & microphysics

Sumiyoshi et al. 2006,2007,2008,2009,2010
 Nakazato et al. 2006,2007,2010,2011
 Fisher et al. 2009
BH 形成後の時空は追えない 軸対称 w.o. microphysics

Shibata & Shapiro 2002
 Sekiguchi & Shibata 2005, 2007
 Liu et al. 2007
BH 形成後の時空を多少追跡 Long term 軸対称 w. GR neutrino leakage & microphysics
Sekiguchi & Shibata 2011,

in prep.

 3D w.o. microphyics
Ott et al. 2011

BH 形成後の時空を多少追跡	Remark	  一般相対論的強重力の重要性	

Chandrasekhar 1964, 1965

回転重力崩壊からの重力波	 
Dimmelmeier et al (2002) A&A 393, 523

GRNewtonRotational frequency increases during collapse

Stable in Newtonian if thecentrifugal term is dominant 	

Qualitative difference in collapse dynamics and thus in waveforms

電子捕獲反応(弱い相互作用)	 
 電子の縮退圧：P ∝ ρ4/3 ニュートン重力では臨界安定


電子捕獲反応によって、実効的に γ < 4/3

 弱い相互作用への応答が違う

ニュートリノトラッピング

衝撃波のエネルギー∝ バウンス時のコアの大きさ ∝ バウンス時の (Ye)2

Shock energy @ bounce (1052 erg)

電子捕獲反応率
 ニュートリノ散乱断面積Takahara & Sato (1984) PTP 72, 978

状態方程式(強い相互作用)	 
核力： P 〜 K ρ2-3 ∝ ρ2-3 と非常に「かたい」


ニュートン重力では 2.5 も 3 も十分に「かたい」ため違いが顕著に現れない	
Van Riper (1988) ApJ 326, 235
Shock velocity @ 300 km (1000km/s)Results	 
 アニメーションを中心に

BH formation 2005	 
 Initial condition : rotating polytrope Microphysics was not included The simulation was terminated shortly after the BHformation

BH formation 2005	 

BH

BH formation 2010-2011: Setting	  (Very) Brief summary of simulation code
 Initial conditions	Summary of Code	  Sekiguchi (2010) Progress of Theoretical Physics 124, 331
 Einstein’s	  equa@ons:	  	  Puncture-­‐BSSN	  formalism	   4th	  order	  ﬁnite	  diﬀerence	  in	  space,	  4th	  order	  Runge-­‐KuNa	  @me	  evolu@on	  	   Gauge	  condi@ons	  :	  1+log	  slicing,	  dynamical	  shiR	   GR	  Hydrodynamics	  with	  GR	  Leakage	  Scheme	  (Sekiguchi 2010)	  EOM	  of	  Neutrinos	  and	  Lepton	  Conserva@ons	   Nuclear-­‐theory-­‐based	  EOS	  (Shen et al. 1998)	   Weak	  Interac@ons	  

e± captures (Fuller et al 1985), e± pair annihilation (Cooperstein et al. 1986)
 plasmon decay (Ruffert et al. 1996)
 Bremsstrahlung (Burrows et al. 2006)


Neutrino	  opaci@es	  (Burrows	  et	  al.	  2006)	  	  (n,p,A)-­‐scaNering	  and	  absorp@on	   Ion-­‐ion	  screening,	  nucleon	  recoil	  High-­‐resolu@on-­‐shock-­‐capturing	  scheme	  BH	  excision	  technique	  Code Validity (球対称崩壊)
 1D GR Boltzmann solver (Liebendorfer et al. 2004) と比較	 Ye, エントロピー, Lνともに良く一致	

Disk 構造、Lνのよい計算ができると期待 	Three Initial Models	 Evolution path is characterized by central entropy (mass)	
More Massive
Stellar Core	
Radiation pressure
dominant	Gas pressure
dominant	
Electron degeneratepressure dominant	
Ordinary SNCore	Three Initial Models	 

Rotational Profiles are added by hand	

500 Msolar Model :Ohkubo et al. (2006)
Mcore ~20Msolar	
Simplified Model
Nakazato et al, (2007)
Mcore ~ 10Msolar	

100 Msolar Model :Umeda & Nomoto (2008)
Mcore ~ 3Msolar	Comments on progenitor model of GRB	  中心動力源：ブラックホール＋ディスク

高速回転する親星コアが必要Type-Ic SN の付随: H, He を失うに伴いに回転が遅くなる 要：角運動量を保持しつつ外層をなくす特殊な親星モデル
 He星合体モデル (Fryer & Heger 2005)
 Tidal spin up モデル (van den Huevel & Yoon 2007)
 Chemically homogeneous evolution (Woosley & Heger 2006, Yoon et al. 2006)

 いずれのモデルも親星コアでの強いmixingを伴う
 中心エントロピーが高くなる傾向がある

 GRBの親星コアは 高いエントロピー を持つ可能性を示唆高速回転・高エントロピーコアの重力崩壊	 
 高エントロピーコアの重力崩壊で期待されること

高エントロピー大質量のコア
BH形成
 鉄の光分解がより進んだコア 主な衝撃波冷却源である重元素光分解が少ない
 1051 erg per 0.1Msolar Fe より(大質量で)コンパクトなコア
高い質量降着率 高エネルギー爆発（Hypernova）が期待できる？ (山田さん’talk) 高速回転で期待されること

ディスク形成 衝撃波面の変形 (斜め衝撃波) 実際のところは数値相対論シミュレーションをしてみるしかないBH formation 2010-2011: Results	  Sekiguchi & Shibata , ApJ737, 6 (2011)
 Sekiguchi & Shibata , in preparation
 Without

Magnetic fields

 Please see how NR simulations have been developed since 2005.

Collapse of 100Msolar presupernova model 	  Two	  rota4on	  models	  (Moderate	  and	  Rapid	  rota4on)	

Radiation pressure
dominant	Gas pressure
dominant	
Electron degeneratepressure dominant	Collapse of 100Msolar presupernova model 	  ‘Moderately’	  rota@ng	  model	  (Ωc=1.2 rad/s, ΩFe=0.6 rad/s)	

1053 erg/s	

280km	Collapse of 100Msolar presupernova model	  ‘Moderately’	  rota@ng	  model	  (Ωc=1.2 rad/s, ΩFe=0.6 rad/s)	

Collapse of 100Msolar presupernova model	  ‘Rapidly’	  rota@ng	  model	  (Ωc=1.2 rad/s, ΩFe=1.2 rad/s)	

1053 erg/s	

980km	Collapse of 100Msolar presupernova model	  ‘Rapidly’	  rota@ng	  model	  (Ωc=1.2 rad/s, ΩFe=1.2 rad/s)	

Rotational Profile of PNS	  Rota@onal	  proﬁles	  of	  Proto-­‐Neutron	  Star	  are	  similar	  	   Small	  diﬀerence	  in	  rota4onal	  proﬁle	  of	  outer	  region	  results	  in	  large	  diﬀerence	  in	  dynamics	  rapid
moderate	

PNS	
Rapid	
Moderate	Oblique Shock	  Rapid	  rota4on:	   Torus-­‐structured	  shock	   Infalling	  materials	  are	  accumulated	  into	  the	  PNS	  due	  to	  the	  oblique	  shock	   Thermal	  energy	  is	  eﬃciently	  stored	  near	  the	  pole	  of	  PNS	   Ram	  pressure	  is	  decreasing	  Oualow	  Neutrino Luminosity (PNS Phase)	 
 Moderate	  rota4on	  Higher	  luminosity	   Time	  variability	  due	  to	  convec@ve	  ac@vity	  

Neutrino
burst	

Core bounce	 Rapid	  rota4on	  Lower	  luminosity	   Neutrino	  pair	  produc@on	  processes	  are	  dominant	  Neutrino Luminosity (BH Phase)	 
 Moderate	  rota4on	  Ltot ~ 1051-52 erg/s	   No	  @me	  variability	  	   Rapid	  rota4on	  Ltot ~ 1051-52 erg/s
 Violent	  4me	  variability	   Preferable	  feature	  for	  GRB	  Collapse of 500Msolar PopIII stellar core	  Rapidly	  rota4ng	  model	  (Ωc=0.5 rad/s)	  :	  Direct	  BH	  forma@on	

105353erg/s	
10 erg/s	
Radiation pressure
dominant	Gas pressure
dominant	
Electron degeneratepressure dominant	

480km	Collapse of 500Msolar PopIII stellar core	  Rapidly	  rota4ng	  model	  (Ωc=0.5 rad/s)	  :	  Direct	  BH	  forma@on	Outflow appears even when BH is formed	 Infalling	  materials	  are	  accumulated	  into	  the	  central	  region	  due	  to	  the	  oblique	  shock	  At	  a	  later	  phase	  BH	  is	  surrounded	  by	  	  shock	  waves	  	  1900km	

Advec@on	  of	  energy	  into	  BH	  becomes	  less	  eﬃcient	   Thermal	  energy	  is	  stored	  Entropy per Baryon	

Oualow	  Moderately High entropy core	  Moderate	  rota4on	  (Ωc=0.5 rad/s)	

1053 erg/s	Radiation pressure
dominant	Gas pressure
dominant	
Electron degeneratepressure dominant	

600km	

Moderately High entropy core	  Moderate	  rota4on	  (Ωc=0.5 rad/s)	

Neutrino luminosity	 Thick	  torus	  emits	  L ~ 1054 erg/s,	  @me-­‐variable	

Thin	  disk	  emits	  	  L ~ 5・1053 erg/s	KH instability

Infalling matterv ~ 0.3c	

Convective eddy
~ 0.1c	Summary	  The	  ﬁrst	  full	  GR	  simula@ons,	  incorpora@ng	  microphysics,	  of	  stellar	  core	  collapse	  are	  performed,	  adop@ng	  various	  ini@al	  models	   BH	  forma4on	  process	  is	  quite	  dynamical,	  accompanying	  oblique	  shock,	  convec4on,	  KH	  instability	  and	  ouNlows	  The	  dynamics	  is	  very	  sensi@ve	  to	  the	  ini@al	  rota@onal	  proﬁle	  which	  is	  poorly	  known	   Accumula@on	  of	  material	  (energy)	  into	  the	  pole	  region	  of	  the	  central	  object	  is	  a	  key	  feature	  for	  driving	  an	  oualow	   OuNlows	  can	  be	  driven	  even	  when	  BH	  is	  directly	  formed	   The	  resul@ng	  system	  has	  preferable	  features	  for	  LGRBs	  Connec@on	  between	  varia@on	  in	  LGRB	  and	  varia@on	  in	  dynamics	  ?	  	今後の展望	  GR leakage scheme からの脱却、Thorne’s moment formalismに基づくGR-Radiation-Hydrodynamics へ
定式化：Shibata et al. 2011
 BH -Disk の GR-Rad-MHD (simplified) : Shibata & Sekiguchi 2012
 Detailed microphysics ver. ほぼ完成しつつある


爆発計算 ⇒ SN groups のシミュレーションコードの図に参入

 3D simulation

BH 周りの massive disk の Papaloizou-Pringle不安定性
Kiuchi et al. 2011 Torus-Shaped shock の SASI は起こるか？
 Convection と MHD processes (e.g. MRI) の関係

競合する可能性も (c.f. CDAF) GR-Rad/ν-MHD

今後の展望	  より“現実的な”初期条件/回転プロファイルの採用	Collapse of 500Msolar PopIII stellar core	  Slowly	  rota4ng	  model	  (Ωc=0.3 rad/s)	

Geometrically-thin disk phase Heating source : Shocks at the surface of disk
 Cooling source : neutrinos, Advection vs neutrinos

advection
Infalling matter
Large ram pressure	 R >> H ⇒ tdiff << tadv

sub-Keplerian disk	

Disk expansion
 As a result of subsequent accretion of high-angular-momentum matter…
 Density ↑ optical depth ↑ thermal energy ↑ H increases
 When tdiff = tadv ,neutrinos are ‘trapped’ Ram pressure ↓ Disk expand to begeometrically-thicktorus	

sub-Keplerian disk	

Convective activities
Accretion disk in collapsar is convectively unstable !
• Point: the disk is, effectively, “heated from below”• In inner region…
•Shock heating isstronger• Neutrino cooling isless efficient•••

Negative entropy
gradient
SN component ?
•Convective luminosity issufficient(Milosavljevic et al.2010)

Importance of BH spin
Efficiency of exchange of gravitational binding energy : ~0.01 (a=0) ⇒ ~ 0.4 (a=1)
 Disk properties : no neutrino trapping for a=0

Efficient cooling ⇒ no/very-weak negative entropy gradient
No convective activities, no time variability 	
trapped	a = 0	
Chen & Beloborodov (2007)	

a = 0.95	

Rapidly rotating modelCentrifugally supported, geometrically thick torus isimmediately formed because of rapid rotation

Copious neutrino emissions (~ 1054 erg/s ) from the torusConvection is suppressed due to stabilizing epicyclic mode
With epicyclic mode	

Unstable Solberg-Hoiland
frequency [Hz] 	

Without epicyclic mode	

Neutrino emission
from the torus