collapse of rapidly rotating massive stellar core to a black hole in full gr tokyo institute of...
TRANSCRIPT
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
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
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)
e e
ee
Hot disk
YS & Shibata (2007)
Contents of my talk
Rotating collapse to a BH with simplified EOSCollapsar scenarioBH + Disk formation
Full GR simulation with microphysicsSummary of implementationGWs from proto-neutron star (PNS) convection
Summary and Future works
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
e e 2BH Disk
GRB, DiskISCO
~ 0.42 v
GM ME M c
R
2 2GRB, GRB, BH GRB, BH( ) 0.29B B BE f q M c M c
Woosley (1993); MacFadyen & Woosley (1999)
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
cold thP P P
1 nuccold
2 nuc
1 2
th th
,
,
4 / 3, 2.0
( 1)
KP
K
P
BH + Disk formationYS & Shibata (2007)
massive core :4.2Msun
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
BH + Disk formationYS & Shibata (2007)
massive core :4.2Msun
spin parameter = 0.98 (rigid rotation)
Simplified EOS
BH + Disk formation Shock wave
formation at Disk BH : ~95% mass Disk : ~5% mass BH spin ~ 0.8Density contour log(g/cm^3)
Slightly before the AH formation
Larger region
Outcome
Convenient for GRB fireball Low density region
Shock heatingLarge neutrino luminositiesLess Pauli blocking by electrons
Thick Disk
Preconditioning: Subsequent evolution on viscous time-scale
density
temperature
2 2vis ~ , , Q L L
[ ]
[ ]e e
e e L L
n p e L
colisionrate 1 cos
Disk structure:High temperature (10^11K) due to shockSmall density along the rotational axis
Neutrino luminosity
Pair annihilation rate (Setiawan et al. (2005))
NotesNo mechanism for time variationMore sophisticated studies are required
Neutrino emission
1 2253 disk
11 17 25 10 erg/s
3 10 K 10 g/cm 70km
N RTL
2
5253
erg/s5 10 erg/s
5 10
LL
Full GR study with microphysics required
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
⇒ constraint violation
One have to add the cooling in terms of the energy momentum tensor
0T Q
Energy momentum tensor
Neutrino part : streaming neutrinoFluid part : baryons, e/e+, radiation, trapped neutrino
Basic equations:
Energy momentum tensor
,stream
( ) : perfect fluid
( )
Fluid ab
ab a b a b b a ab
T
T En n F n F n P
,stream
( )
( )
Fluid aa b b
aa b b
T Q
T Q
includes :
e capture (Fuller et al. (1985))
/ capture (Fuller et al. (1985))
e -annihilation (Cooperstein et al. (1986))
plasmon decay (Ruffert et al. (1996))
bremsstrahlung (Burrows et al.
bQ
(2004))
neutrino leakage (described later)
tot ,trap ,stream
,stream
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
Matter M
Fluid
T T T T T T
T T
leak loceff
leak loc
loc diff
leak
( short )
(otherwise)
( . . Ruffert et al. (1996);
Rosswog & Liebendoerfer (2002))
b b b
Q QQ Q u u
Q Q
Q for T
Q
c f
Lepton conservations
Lepton evolution :
e-cap ep-capedY
dt
e-cap pair plasmon leak
( )e
ed Y
dt
ep-cap pair plasmon leak
( )e
ed Y
dt
pair plasmon leak
( )x
xd Y
dt
e-cap/ep-cap
pair
plasmon
leak
: Fuller et al.(1985)
: Cooperstein et al. (1986)
: Ruffert et al. (1996)
: neutrino leakage
(explained later)
, , , Ye e e x
leak
( )ll
d Y
dt
In Beta equilibrium
Neutrino emission
Neutrino Leakage Scheme“Cross sections” :
“Opacities” :
“Optical depth” :
Diffusion time :
Neutrino energy and number diffusion :
diff dyn~T T
t( )T
s( )T
2( )i iE E
2( ) ( )iE E E
2( )E ds E
diff 21diff
diff0diff
ˆ( ) ( )
( )
ˆ( ) ( )
( )
E n EQ dE T F
T E
n ER dE T F
T E
2diff 2( )
( ) ( )x E
T E E Ec c
ˆ( )n n E dE
Cross sections by Burrows et al. (2003)
A A
e
e
p ne
p p n pe
n n
e e
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
4 / 3, 3 /r r r rP a T P
Neutrinos : ideal Fermi gas
Shen et al. (1998)
EOS table is constracted
PNS convection (using old ver. leakage)
Ye 197.8 ms199.7 ms201.3 ms202.8 ms
206.7 ms211.9 ms215.5 ms217.3 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)
Gravitational wavesYS (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
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
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
22
nonsphe4 2 2
2nonsphe omp20
2 1 2 ~ ~
10kpc ~ 10
0.1 0.3 10km 0.1
ijd QG GM R vh
c D dt c R D c
C R v
D c
2nonsphe20 10kpc
~ 100.1 1.4 10km 1kHz
M R f
D M
Summary
Rotating collapse to a BHBH + Disk formation (with simplified EOS)
Shock occurs at the diskOutcome: low density region, high temperature thick disk
New full GR code with microphysicsBrief description of the implementation
neutrino radiation energy momentum tensorleakage scheme for neutrino coolingnuclear EOS by Shen et al. (1998)
GWs from PNS convectionAs 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
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
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 conditionApproximate maximal slicing (Balakrishna et al. (1996); Shibata (1999))Dynamical shift (Shibata (2003))
4
8ab ab
GG T
c
1
6t K L
2t ij ijA L
TF4
4 2 8 / 3
t ij ij i j
kij ik j ij ij
A e R D D
KA A A e S S
L
2 / 3
4
k ijt k ij
h
K D D A A K
S
L
2jl jl jlt i l ij l ij ij lF A A L
Equation of State parametric EOS :
idealized EOS : microphysics is treated only qualitativelymaximum allowed mass of EOS :
c.f. the maximum pulsar mass : (Nice et al. 2005)
parameters of EOS
cold thP P P
1 nuccold th th
2 nuc
, , ( 1)
,
KP P
K
( 4 / 3) 1.31 1.325 14 3
2 nuc2.45 2.6, 2 10 g/cm
max,EOS 2M M
2.1 0.2M M
th 1
Simplified EOS
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 : 52BH disk
ISCO
4 9 10 ergGM M
ER
2disk ISCO BH0.1 0.2 , 4 5M M R Gc M
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 :
3
disk ram BH BH2 2 3/ 2 3ISCO ISCO
2
31 2disk ram 11
ISCOg/cm
( )
10 dyn/cm10
s
s
P P GM H GM H
H R H R
HP P
R
12 3 11 31 2disk disk disk10 g/cm , 10 K 10 dyn/cmT P
32 30 10 2ram f f f g/cm10 ( /10 ) dyn/cmP v
BH ISCO
1/ 2f (2 / ) 0.4 0.5v GM R c
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 ?
231disk ram disk ISCOdyn/cmNow 10 , , then / 1P P P H R
disk ram BHBH BHdisk ram2 2 3/ 2 2
ISCO
( )
s
s
P P GMGM H GMP P
H R H H H
disk ramP P
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,
53BH
ISCO
1
erg/s
10
sm
GM mL
R
M
110 sm M
331disk diskerg/cm10 P
1.315
1.32
1.325
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) : ~ 1.5×1051 erg per 0.1 Msolar
neutrino cooling (loss) :
251 core infall
shock, init 6 10 erg 0.4
M vE
M c
2 353 shock infall infall
hydro 9 31.4 10 erg/s
100km 10 g/cm 0.2
R vL
c
253 shock infall infall
diss 9 31.1 10 erg/s
100km 10 g/cm 0.2
R vL
c
4 253~ 10 erg/s block
10 MeV 50 km
T RL
23/ 2 4 /3
1/3core init init init
init ,init B
~ ~ 0.6 , (34
l l
l
Y YK hcM M M M K
K Y m
PNS Convection
197.8 ms 199.7 ms 201.3 ms 202.8 ms
206.7 ms 211.9 ms 215.5 ms 217.3 ms
Vigorous convective motion Shock wave is pushed outward Enhancement in neutrino luminosity
Contours of electron fraction
Exchange of fluid element via ⊿h
Free energy available per unit mass
Convection of mass M ⊿
Energy available in convection
blob blob,
amb amb amb amb, , ,
( ) ( )
( ) ( ) ( ) ( )
e
e e
s Y
es Y P Y e s P
d dPP
d dP ds dYP s Y
blob amb( ) ( )dP dP 1
eff amb blob amb
1 1
amb ambeff
, , ,,
( ( )
( ) ( )ln ln ln ln
ln ln ln lne
Ye e es Ye s Yes
w g d d
ds dYP P P Pg h
s s Y Y
51 PNS| | | | 50km10 ergs ,
0.3 10kme
e
Y MM h sW
M Y s r M
,
,
,
[( ln / ln
( ln / ln
( ln / ln
(1)]
s Ye
Ye
e s
P
P s
P Y
O
blob
amb
hblob blob
, ,
( ) ( )e
eP Y e s P
ds dYs Y
blob amb
,
amb amb amb amb, , ,
( ) ( )
( ) ( ) ( ) ( )
e
e e
s Y
es Y P Y e s P
d dPP
d dP ds dYP s Y
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 2 zz xxA I I
Spectrum is similar GW is mainly due
to bounce motion
This peak is associated with non-axisymmetric instabilities
Ott et al. (2006)
Neutrino emission
Neutrino Leakage Scheme“Cross sections” :
“Opacities” :
“Optical depth” :
Diffusion time-scale :
Neutrino energy and number diffusion :
diff dyn~T T
t( )T
s( )T
2( )i iE E
2( ) ( )iE E E
2( )E ds E
diff 21diff 3 2
diff0diff 3 2
ˆ( ) 4( ) ( )
( ) ( )
ˆ( ) 4( ) ( )
( ) ( )
B
B
E n E cgQ dE k T F
T E hc
n E cgR dE k T F
T E hc
2diff 2( )
( ) ( )x E
T E E Ec c
ˆ( )n n E dE
Cross sections by Burrows et al. (2003)