Download Collapse of rapidly rotating massive stellar core to a black hole

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
 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
Importance of GR
Dimmelmeier et al (2002) A&A 393, 523
GR
Newton
 Rotation increases strongly during collapse
 Newtonian : hard to reach nuclear density
 GR : stronger gravitational attraction
⇒ multiple-spike waveform
⇒ burst-like waveform
Qualitative difference in collapse dynamics and in waveforms
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
ee


SN explosion
Hot disk
 Realistic calculation of GWs
 GRBs (collapsar scenario)
   e  e

YS & Shibata (2007)
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
Rotating collapse to a BH
Collapsar model
Woosley (1993); MacFadyen & Woosley (1999)
 Central engine of GRBs : BH + Disk
 Energy source :
 Gravitational energy of accretion
matter
neutrino annihilation
(   e  e)
EGRB, ~  v
⇒
GM BH M Disk
 0.42  M Disk c 2
RISCO
 BH spin ⇒ electromagnetic flux
 E.g. via Blandford-Znajek process
EGRB,B
GRB,B f (q)M BHc2  0.29GRB,B M BHc2
MacFadyen & Woosley 1999
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
P  Pcold  Pth


K

 1 ,    nuc
Pcold  


 K 2  ,    nuc
1  4 / 3,  2  2.0
Pth  ( th  1) 
YS & Shibata (2007)
BH + Disk formation
 massive core：4.2Msun
 spin parameter = 0.98
(rigid rotation)
 Simplified EOS
Slightly before the
AH formation
 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)
YS & Shibata (2007)
BH + Disk formation
 massive core：4.2Msun
 spin parameter = 0.98
(rigid rotation)
 Simplified EOS
Slightly before the
AH formation
Larger region
 BH + Disk formation
 Shock wave
formation at Disk
 BH : ~95% mass
 Disk : ~5% mass
 BH spin ~ 0.8
Density contour
log(g/cm^3)
BH mass and spin
1.315

1.32
1.325
density
Outcome
 Convenient for GRB fireball
 Low density region
 [   e  e ]  L L

[n  e  p  e]   L e
temperature
 Shock heating
 Large neutrino luminosities
 Less Pauli blocking by electrons
 Thick Disk
 rate  1  cos colision
 Preconditioning:
Subsequent evolution on viscous
time-scale Q ~ 2 , L   , L
vis


2
Neutrino emission
Disk structure:
High temperature (10^11K) due to shock
Small density along the rotational axis
Neutrino luminosity
1
  Rdisk 

 N  T  
L  5 1053 erg/s     11   17

2  
 3   10 K   10 g/cm   70km 
2
2
Full GR study with microphysics required
Pair annihilation rate (Setiawan et al. (2005))
L
L


5 1052 erg/s 

53
 5 10 erg/s 
2
Notes
No mechanism for time variation
More sophisticated studies are required
Full GR simulation with
microphysics
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
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

  T   Q  0 ⇒ constraint violation
 One have to add the cooling in terms of the
energy momentum tensor
Energy momentum tensor
leak loc
Q
Q
eff
Qb  Q ub  leak
u
loc b
 Energy momentum tensor
Q Q
 ,stream
(T tot )  (T Matter )  (T  )  (T M )Qloc(T  ,trap
)

(
T
))
 short T diff
( for


 leak
 (T
)  (T
)
(otherwise)
Q
 Neutrino part : streaming(cneutrino
. f . Ruffert et al. (1996);
Fluid
 ,stream
 Fluid part : baryons, e/e+, Rosswog
radiation, trapped
neutrino (2002))
& Liebendoerfer
 Basic equations:
a (T Fluid )ba  Qb
a (T  ,stream )ba  Qb
Qb includes :
e capture (Fuller et al. (1985))
 / capture (Fuller et al. (1985))
e -annihilation (Cooperstein et al. (1986))
plasmon decay (Ruffert et al. (1996))
(T Fluid )ab : perfect fluid
bremsstrahlung (Burrows et al. (2004))
(T  ,stream )ab  Ena nb  F a nb  F b na  P ab
neutrino leakage (described later)
Lepton conservations
 Lepton evolution : Ye,  e,  e,  x
dYe
  e-cap   ep-cap
dt
d (Y e )
  e-cap   pair   plasmon    eleak
dt
d (Y e )
  ep-cap   pair   plasmon    e leak
dt
d (Y x )
  pair   plasmon    xleak
dt
d (Yl )
  lleak
dt
In Beta equilibrium
 e-cap/ep-cap : Fuller et al.(1985)
 pair : Cooperstein et al. (1986)
 plasmon : Ruffert et al. (1996)
  leak : neutrino leakage
(explained later)
Neutrino emission
 Neutrino Leakage Scheme
Cross sections by
Burrows et al. (2003)
 “Cross sections” :  i ( E )   i E2
 “Opacities”
A
p
n
e
2

(
E
)


(
E
)


E


i


:
2

(
E
)


ds


E


 “Optical depth” :

 Diffusion time
:
diff
T
 A e p
 p  en
n
e
x( E )
2 2
( E ) 
 ( E ) 
E
c
c
(T  s )
diff
n   nˆ ( E )dE
pe

 Neutrino energy and number diffusion :
E nˆ ( E )
Q    diff
dE  T 2 F1 ( )
T ( E )
nˆ ( E )
diff
 R    diff
dE  T F0 ( )
T ( E )
ne

(T  t )
Tdiff ~ Tdyn
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
EOS table is constracted
 Electrons and positrons
 Ideal Fermi gas
 Charge neutrality condition (Yp=Ye)
 Radiation
4
 Pr  arT / 3,  r  3Pr / 
 Neutrinos : ideal Fermi gas
Shen et al. (1998)
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
202.8
201.3
ms
197.8
199.7 ms
Ye contours
215.5ms
217.3
ms
206.7
ms
211.9
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
The previous study
Muller and Janka (1997) A&A 317, 140
 amplitude : h ～ 3×10-21 @ 10 kpc
 frequency : 100－1000 Hz
Spherical model
No neutrino transfer
 The hydrostatic condition is
imposed at PNS surface
 Convective motions are suppressed
near the boundary
80
 Smaller
 Amplitude
 frequency
0
110
115 km
Notes
 Gravitational wave amplitude
2
2
d
Q
2
G
1
2
GM
R
v


ij
 Due to convection h ~
~  nonsphe 2
 
c 4 D dt 2
c R D c
2

C



10kpc
R
v




nonsphe
omp
~ 1020 





 0.1  0.3   D  10km  0.1c 
 Cf. Due to core bounce
~ 10
20
  nonsphe

 0.1
2
  10kpc   M   R  f 





D
1.4
M
10km
1kHz








 No effects to suppress the convective activities
 Neutrino transport will flatten the existing negative gradients
 The GW amplitude is the maximum estimates
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
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
What to explore further
ee



 BH + Disk formation
 Disk structure
 Shock strength
 Neutrino luminosity
Hot, thick Disk
 Time variability in Lν
 Mass, angular momentum
dependence
Low
density
region
 Magnetic field
 Metallicity dependence
Einstein’s equation
 Gab 
8 G
Tab
4
c
 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))



t

 t Fi   jl  l  L   ij   2  jl l Aij  Aij jl l
1
L      K
6
t
 L    ij  2 Aij
t
 L   Aij  e 4  Rij  Di D j 

TF

  KAij  2 Aik Akj  8  e 4 Sij   ij S / 3

t

 L   K   D k Dk   Aij Aij  K 2 / 3
 4  h  S 


Simplified EOS
 Equation of State
 parametric EOS : P  Pcold  Pth

Pcold


 K1   ,    nuc

, Pth  ( th  1) 


 K 2  ,    nuc
 idealized EOS : microphysics is treated only qualitatively
 maximum allowed mass of EOS : M max,EOS  2M
 c.f. the maximum pulsar mass : M  2.1  0.2M
 parameters of EOS
 ( 4 / 3)  1.31 1.325
2  2.45  2.6, nuc  2 1014 g/cm3
 th  1
(Nice et al. 2005)
 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
GM BH M disk
 The gravitational energy released :
E
 4  9 1052 erg
RISCO
M disk
0.1  0.2M , RISCO
4  5Gc 2 M BH
 Disk formation
→ shock wave
formation (1)
 The disk height H increases as the thermal energy is stored (balance relation)
Pdisk  Pram
s H
GM BH H
2
( RISCO
 H 2 )3/ 2
GM BH H
3
RISCO
2
 Pdisk  Pram


 H 
2
1031  11 s

 dyn/cm

 10 g/cm   RISCO 
3
 temperature and density of the disk increase to be
disk 1012 g/cm3 , Tdisk 1011 K  Pdisk
 While the ram pressure decreases： Pram
vf
1031 dyn/cm 2
f vf2  1030 ( f /1010 g/cm ) dyn/cm 2
3
(2GM BH / RISCO )1/ 2
0.4  0.5c
 Disk formation
→ Shock wave
formation (2)
 Now Pdisk
1031 dyn/cm 2 , Pram
Pdisk  Pram
s H
GM BH H
2
( RISCO
 H 2 )3/ 2
Pdisk , then H / RISCO
GM BH
H2
 Pdisk  Pram
1
GM BH  s
H
 The disk expands escaping the gravitational bound
 Pdisk
Pram ：strong shock waves are formed and propagated
 Shock waves are mildly relativistic ～ 0.5c
 does neutrino cooling work ?
 condition that thermal energy be stored is
GM BH m

L 1053 erg/s
RISCO
 m

 M
s 1
 The present results show
m
10M s 1
1.315
 Unless the conversion efficiency
is too low (<<0.1),
thermal energy is stored
 In the a few millisecond,
 disk 1031 erg/cm
3
Pdisk

α1.32
the
1.325
Sack et al. 1980
neutrino loss small
neutrino loss large
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) :
M core
 K 


 Kinit 
3/ 2
M init
 Y
~ l
Y
 l ,init
2

 M init ~ 0.6M init ,

 Initial shock energy (input):
 accretion power (input):
 Photo-dissociation (loss) :
～ 1.5×1051 erg per 0.1 Msolar
Eshock, init
Lhydro
Ldiss
Y 
hc
K
(3 1/ 3  l 
4
 mB 
 M core   vinfall 2
 6 10 erg 


M
0.4
c




51
 R

1.4 10 erg/s  shock 
 100km 
53
 R

1.110 erg/s  shock 
 100km 
53
4
 neutrino cooling (loss) :
4/3
2
2
 infall   vinfall 
 9

3 
 10 g/cm   0.2c 
 infall   vinfall 
 9

3 
 10 g/cm   0.2c 
2
 T
  R 
L ~ 1053 erg/s 
 
  block
 10 MeV   50 km 
3
PNS Convection
 Vigorous convective motion
 Shock wave is pushed outward
 Enhancement in neutrino luminosity
Contours of electron fraction
197.8 ms
206.7 ms
199.7 ms
211.9 ms
201.3 ms
215.5 ms
202.8 ms
217.3 ms
Energy available in convection
 Exchange of fluid element via ⊿h
  
  
  
(d  ) blob    (dP)amb
(ds ) blob  
blob  
 (dYe ) blob


s

Y
 P  s ,Ye

 P ,Ye
 e s,P
  
  
  
(d  )amb    (dP)amb    (ds)amb  
 (dYe )amb

Y
 P  s ,Ye
 s  P ,Ye
 e s,P
 Free energy available per unit mass
1
w  g eff amb
(d  blob  (d  )amb 
  g eff
h
blob
(dP)blob  (dP)amb
1
  ln P    ln P 1 (ds)




(dYe )amb 

ln
P

ln
P
amb


 h

 
 


  ln s   ,Ye   ln   s ,Ye s
  ln Ye   , s   ln   s ,Ye Ye 
 Convection of mass ⊿M
W
amb
 M   h   | Ye | | s |   50km   M PNS 
1051 ergs 
,





0.3
M
10km
Y
s
r
M




 e





[( ln P /  ln  s ,Ye
 ( ln P /  ln s   ,Ye
 ( ln P /  ln Ye s , 
 O (1)]
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
Gravitational wave signal
 Gravitational waves : Type-I waveform
 Comparison with Ott et al. (2006) : Second peak is surppressed
 Due to deceleration along z-direction A2  I zz  I xx
 Spectrum is similar
 GW is mainly due
to bounce motion
Ott et al.
(2006)
This peak is
associated with
non-axisymmetric
instabilities
Neutrino emission
 Neutrino Leakage Scheme
 “Cross sections” :  i ( E )   i E2
 “Opacities”
Cross sections by
Burrows et al. (2003)
:  (E )  i (E )   E2
2
 “Optical depth” :  ( E )    ds   E
 Diffusion time-scale : T
diff
x( E )
2 2
( E ) 
 ( E )  E
c
c
 Neutrino energy and number diffusion :
E nˆ ( E )
4 cg 
diff
2
Q    diff
dE 
(
k
T
)
F1 ( )
B
3
2
T ( E )
(hc) 
nˆ ( E )
4 cg 
diff
 R    diff
dE 
(k BT ) F0 ( )
3
2
T ( E )
(hc) 
n   nˆ ( E )dE


(T  t )
Tdiff ~ Tdyn
(T  s )