Transcript Numerical study for prompt black hole formation

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
§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
§1 Introduction
§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 ?
§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
§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)
§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)
§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))
Δ
§3 Setting
§3.1 Initial conditions
§3.2 Parametric equations of state
§3.1 Setting -- Initial conditions --
Initial conditions
– Rotating iron cores of massive stars
– Modeled by   4 / 3 rotating polytrope in
equilibrium
Central density
Mass range
angular momentum
c  1.0 1010 g/cm3
M / M  2.0 3.0
cJ
q
 0 1.1 (spherical to mass shedding limit)
2
GM
§3.1 Setting -- Initial conditions -Rotation law (Komatsu et al. MNRAS 239, 153 (1989) )
ut u   d2 (0  )
– In Newtonian limit
Cylindrical rotation
– Differential rotation parameter : A
A   d /  e  , 1.0, 0.5
 e : radius at equator
A  , rigid rotation
A  0, larger degree of differential rotation
§3.2 Setting -- Parametric EOS (1) -Parametric equations of state P  Ppoly  Pth
Ppoly 
K1   (   nuc )
K 2  2 (   nuc )
    4 / 3 Unstable due to the photo-dissociation
and electron capture
1  2  2.0 Sudden stiffening due to nuclear force
Pth  ( th  1)  th
Thermal and shock heating effects
Parameters of EOS : (1 , 2 , nuc , th )
– Parameters of EOS are so chosen that the maximum
mass of the cold spherical polytropes is almost identical
– We set  th  1 for simplicity
M max NS  1.6M
§3.2 Setting -- Parametric EOS (1) -We set  th  1 for simplicity
Note : Collapse dynamics is less sensitive
to the value of  .
th
– As long as
1.3   th  5 / 3
Shibata & Sekiguchi PRD 69, 084024 (2004)
§3.2 Setting -- Parametric EOS (2) -Parameters of EOS
(1 , 2 , nuc / 14 ),  th  1
14  1014 g/cm3
EOS-a : (1.32, 2.25, 2.0)
EOS-b : (1.30,2.5,2.0)
EOS-c : (1.30, 2.22,1.0)
EOS-d : (1.28,2.75,2.0)
M max NS  1.6M
Note: EOS-c is
stiffer than EOS-b
§4 General Feature of Collapse
Infall phase
Bounce phase
Ring-down phase
§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
§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
§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
§5 Black Hole Formation
A criterion for black hole formation
Dependence on parameters
Dependence of EOS
Effects of shocks
Effects of rotation
Effects of differential rotation
Predicting the final system
§5.1 A criterion for prompt black hole formation
－ in M-q plane －
a : (1.32, 2.25, 2.0)
b : (1.30,2.5,2.0)
,d c : (1.30, 2.22,1.0)
d : (1.28,2.75,2.0)
■ : BH for all EOS
☆ : BH for EOS-b (-d)
× : BH for EOS-a
□ : NS for all EOS
§5.2 Black hole formation － Dependence on EOS －
EOS-a
M  2.7 M
q  1.0
a : (1.32, 2.25, 2.0)
b : (1.30,2.5,2.0)
c : (1.30, 2.22,1.0)
d : (1.28,2.75,2.0)
Direct Collapse
(1 ,  2 ,  nuc /1014 )  (1.32, 2.25, 2.0), th  1.32
M  2.7 M , cJ / GM 2  1.0
Neither sudden stiffening
of EOS nor Rotational
effect cannot halt the
collapse
Direct collapse to a BH
§5.2 Black hole formation － Dependence on EOS －
EOS-a : (1 , 2 , nuc / 14 )  (1.32, 2.25, 2.0),  th  1.32
A black hole is formed directly without any distinct
bounce
Mass of the inner core
at bounce is large
M inner core  M NS max
No shock propagates
outward
BH is more likely to
be formed
M  2.4M
§5.2 Black hole formation － Dependence on EOS －
EOS-b M  2.7M
q  1.0
a : (1.32, 2.25, 2.0)
b : (1.30,2.5,2.0)
c : (1.30, 2.22,1.0)
d : (1.28,2.75,2.0)
Fallback Induced Collapse (1)
(1 ,  2 ,  nuc /1014 )  (1.3, 2.5, 2.0), th  1.3
M  2.7 M , cJ / GM 2  1.0
The shock wave propagate outward ……, however,
Fallback Induced Collapse (2)
(1 ,  2 ,  nuc /1014 )  (1.3, 2.5, 2.0), th  1.3
M  2.7 M , cJ / GM 2  1.0
The shocked matters
fall back to the inner
core and a black hole
is eventually formed
§5.2 Black hole formation － Dependence on EOS －
EOS-b : (1 , 2 , nuc / 14 )  (1.3, 2.5, 2.0),  th  1.3
EOS-d : (1 , 2 , nuc / 14 )  (1.28, 2.75, 2.0),  th  1.28
Inner cores experience a bounce before BH formation
・Mass of the inner
core at bounce is large
M inner core  M NS max
・Shocks propagate
outward
Pth
Contributes to
support the core
Threshold mass of prompt
BH formation is larger than
for the cases with EOS-a
EOS-b
§5.2 Black hole formation － Dependence on EOS －
Dependence on 
– For larger 
Cores collapse more homologously |   4 / 3| 0
– 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
§5.2 Black hole formation － Dependence on EOS －
Dependence on 2
– For smaller  2
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
§5.2 Black hole formation － Dependence on EOS －
EOS-a : (1 , 2 , nuc / 14 )  (1.32, 2.25, 2.0),  th  1.32
A black hole is formed directly without any distinct
bounce
M  2.4M
 is larger
Mass of the inner core
at bounce is large
M inner core  M NS max
Pth No shock
propagates outward
BH is more liable to
be formed
§5.2 Black hole formation － Dependence on EOS －
EOS-b : (1 , 2 , nuc / 14 )  (1.3, 2.5, 2.0),  th  1.3
EOS-d : (1 , 2 , nuc / 14 )  (1.28, 2.75, 2.0),  th  1.28
Inner cores experience a bounce before BH formation

is smaller
M inner core  M NS max
Shocks propagate
outward
Pth Contributes to
support the core
Threshold mass of prompt
BH formation is larger than
for the cases with EOS-a
EOS-b
§5.2 Black hole formation － Dependence on EOS －
a : (1.32, 2.25, 2.0)
b : (1.30,2.5,2.0)
c : (1.30, 2.22,1.0)
d : (1.28,2.75,2.0)
■ : BH for all EOS
☆ : BH for EOS-b (-d)
× : BH for EOS-a
□ : NS for all EOS
§5.2 Black hole formation － Dependence on EOS －
EOS-b : (1 , 2 , nuc / 14 )  (1.3, 2.5, 2.0),  th  1.3
EOS-c : (1 , 2 , nuc / 14 )  (1.3, 2.22,1.0),  th  1.3
The pressure near nuclear density is larger for EOS-c
EOS-c
EOS-b
Shocks are stronger for EOS-c
Threshold mass is larger
§5.3 Black hole formation － Effect of shocks －
Contribution of Pth
Maximum mass of the
cold spherical polytrope
M max NS  1.6M
For spherical models
M crit sphe  2.1 2.3M
Thermal effects increase
the threshold mass by 20
～40 %
Effect of shock is
stronger for EOS-c
§5.4 Black hole formation － Effects of 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)
M crit rot  M crit sphe  Crot J 2
Rotational effects increases
the threshold mass at most
by 17 ～ 20 %
§5.5 Black hole formation－ Effect of differential rotation －
As the degree of differential rotation increases, a black
hole is less liable to form
q  0.89, NS, rigid
M  2.4M
q  0.79, BH, rigid
The threshold for BH
formation locates
between these curves
q  0.63, NS, A  0.5
q  0.54, BH, A  0.5
The inner region which
is responsible to black
hole formation “rotates”
more rapidly
§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 j  jISCO
will fall into the black hole
ISCO
・ If jISCO increases as a
result of the accretion, more
fluid elements fall into the
BH
BH
・ Thus, if evolution of jISCO has a maximum, the
dynamical growth of BH will terminate there
§5.6 Estimation of mass of disk
・ Define mass and spin parameter in terms of the specific
angular momentum : q(j) and m(j)
J ( j)
q ( j ) 
m ( j ) 2
・ 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)
§5.6 Estimation of mass of disk
Mass of the formed
disk will be < 10% of
the initial mass
§6 Neutron star Formation
6.1 Dependence of EOS
6.2 Gravitational waves
– Waveforms
– Energy spectra
§6.1 Dependence on EOS
The collapse dynamics depends strongly on the
adopted EOS
The effects are reflected in Gravitational waves
EOS-a : (1 , 2 , nuc / 14 )  (1.32, 2.25, 2.0),  th  1.32
EOS-b : (1 , 2 , nuc / 14 )  (1.3, 2.5, 2.0),  th  1.3
EOS-d : (1 , 2 , nuc / 14 )  (1.28, 2.75, 2.0),  th  1.28
§6.1 Dependence on EOS
EOS-a
M  2.5M
q  1.0
rig. rot.
§6.1 Dependence on EOS
EOS-a
M  2.5M
q  1.0
rig. rot.
§6.1 Dependence on EOS
EOS-b
M  2.5M
q  1.0
rig. rot.
§6.1 Dependence on EOS
EOS-d
M  2.5M
q  1.0
rig. rot.
§6.1 Dependence on EOS
EOS-a : (1 ,  2 ,  nuc /1014 )  (1.32, 2.25, 2.0)
M  2.5M , q  1.0
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
§6.1 Dependence on EOS
EOS-a : (1 ,  2 ,  nuc /1014 )  (1.32, 2.25, 2.0)
M  2.5M , q  1.0
Since the density of
matters in front of the
shock “pole” is much
smaller,
The shock velocity
is higher in this
direction
§6.1 Dependence on EOS
EOS-b : (1 ,  2 ,  nuc /1014 )  (1.3, 2.5, 2.0)
M  2.5M , q  1.0
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
§6.1 Dependence on EOS
EOS-b : (1 ,  2 ,  nuc /1014 )  (1.3, 2.5, 2.0)
M  2.5M , q  1.0
A slightly prorate
shock wave is
generated
§6.1 Dependence on EOS
EOS-d : (1 ,  2 ,  nuc /1014 )  (1.28, 2.75, 2.0) M  2.5M , q  1.0
Since the shock wave generated at the bounce is rather
weak, the shocked matters around the rotational axis fall
and beat the inner core.
The inner core oscillates and subsequent shock waves are
generated.
§6.2 Gravitational Waves
– dependence on EOS -
EOS-a
M  2.5M
q  1.0
rig. rot.
BH
h  3 10
20
 I zz  I xx   10 kpc  2
 sin 


 1000 cm   r 
§6.2 Gravitational Waves
EOS-a
M  2.5M
q  1.0
rig. rot.
1/ 2
 dE / df 
heff  5 1020  46

10
erg/Hz


 10kpc 
20

h
~
5

10
at r  10kpc
eff


 r 
§6.2 Gravitational Waves
EOS-b
M  2.5M
q  1.0
rig. rot.
h  3 10
20
 I zz  I xx   10 kpc  2
 sin 


 1000 cm   r 
§6.2 Gravitational Waves
EOS-b
M  2.5M
q  1.0
rig. rot.
1/ 2
 dE / df 
heff  5 1020  46

10
erg/Hz


 10kpc 
20

h
~
5

10
at r  10kpc
eff


 r 
§6.2 Gravitational Waves
EOS-d
M  2.5M
q  1.0
rig. rot.
h  3 10
20
 I zz  I xx   10 kpc  2
 sin 


 1000 cm   r 
§6.2 Gravitational Waves
EOS-d
M  2.5M
Several
oscillation modes
q  1.0
rig. rot.
§6.2 Dependence on Mass
EOS-b : (1 ,  2 ,  nuc /1014 )  (1.3, 2.5, 2.0)
q  cJ / GM 2  1.0
M  2.7 M
M  2.2M
Amplitudes of GW at
bounce increase as the
inner core mass
becomes larger
Due to large fraction of
outer core falling into
inner core, the
oscillation of inner core
is prevented
M inner core / M outer core
§6.2 Effect of rotation
The amplitudes of GW increase as the value of q does
EOS-b : (1 ,  2 ,  nuc /1014 )  (1.3, 2.5, 2.0)
M  2.5M
q  0.66
q  0.89
No saturation is observed
in the amplitudes
q  1.0
§6.2 Effect of rotation
EOS-d : (1 ,  2 ,  nuc /1014 )  (1.28, 2.75, 2.0)
M  2.5M
q  1.0
q  0.89
No saturation is observed
in the amplitudes
q  0.66
In the Newtonian simulation……..
Yamada and Sato (1994) ApJ 434, 268; (1995) ApJ 450, 245
The amplitude of GW spike saturates for q=0.5
∵ quadrupole formura
rotation ↑
h
1 I zz  I xx
r ( char ) 2
Mass and radius of inner core at bounce ↑
| I zz  I xx |
The core less contract due to centrifugal
force
 char
The value of h saturates for a certain value of q
In the present simulation…………
Mass of the cores are much larger
Gravity is not Newtonian but full general relativity
1   bounce
(GM / Rc 2 ) bounce  0.3
Due to the stronger gravity, the inner core
contract more.
This leads to the higher central density and
the more deformation of the inner core.
Thus, no saturation is observed for q<1
Indeed, the modulation of the waveforms is
more outstanding in the GR simulation
§7 Summary and Discussion
In rotating stellar core collapse to a black hole
– Thermal effects (in particular shock) increase the
threshold mass by 20 ～ 40 %
– Rotational effects increase the threshold at most
by 17 ～ 20 %
– These effects depend sensitively of the equations
of state
Direct black hole formation and fallback-induced
collapse
– Differential rotation further increases the
threshold
Black hole + Disk system
– The predicted mass of the disk is at most ~10% of
the initial mass
– BH excision technique
§7 Summary and Discussion
In rotating stellar core collapse to a
neutron star
– The collapse dynamics and the shock
generation and propagation strongly depend
on EOS
– Reflecting this, the gravitational waveforms
also depend on EOS
– No amplitude saturation is shown in the
present simulations ⇔ the previous
Newtonian simulation
§7 Summary and Discussion
Possibility for onset of dynamical
nonaxisymmetric instabilities ?
– which occur for T/W > 0.27 or highly differentially
rotating case (even for T/W < 0.1)
– Unlikely to occur for rigidly and moderately diff.
rot. cases
– Set in during the collapse ?
Yes, but the conditions for onset is
– Initially rapidly rotating, highly differentially
rotating (A<0.1)
– Also necessary is large initial pressure
depletion
Γ1<～1.28
－Dynamical Instability－
Relations of initial and maximum value of T/W
Candidates
for the onset
of dynamical
instabilities
0.27
Max
Models of diff.rot.
0.14
Models of rigid rot.
Initial
YS and Shibata 05
－ Dynamical Instability －
A=0.1 models
T /W
0.27
(T/W)_init
increases
Candidates
EOS-d
T / W |init  0.0127
M  2.5M
A  0.1
強微分回転
EOS-d
T / W |init  0.0177
M  2.5M
A  0.1
強微分回転
－ Dynamical Instability －
EOS-d
T / W |init  0.0127
Gauge inv.
M  2.5M
A  0.1
Quadrupole
formula
h  10
21
 R ,   10Mpc 
18  R ,   10kpc 
 0.31km   r   3 10  0.1km   r 






－ Dynamical Instability －
EOS-d
T / W |init  0.0177
Gauge inv.
M  2.5M
A  0.1
Quadrupole
formula
h  10
21
 R ,   10Mpc 
18  R ,   10kpc 
 0.31km   r   3 10  0.1km   r 






－ Dynamical Instability －
~ 1 kHz
EOS-d
T / W |init  0.0177, 0.0127
M  2.5M
A  0.1
1/ 2
heff  2 10
22
 dE / df 
 1047 erg/Hz 


 10Mpc 
18

h
~
3

10
at r  10kpc
eff


 r 
－ Dynamical Instability －
非軸対称変形
により、Axi.
Sym. の～１０
倍以上の振幅
M  1.5M
M  2.5M
§2.1 ３＋１decomposition (1)
Foliation of the spacetime 4D manifold into a family of
3D hypersurface Σ
： unit normal to Σ
： lapse function
： shift vector
： metric on Σ
timelike coordinate
vector
§2.1 ３＋１decomposition (2)
： lapse function
Represent gauge freedom
： shift vector
： 3D metric on Σ
： extrinsic curvature of Σ
which describes how Σ is embedded in the spacetime
Corresponds to the “velocity” of the 3D metric
Dynamical variable is the pair (γ, K)
Cf. York (1979) in Sources of Gravitational Radiation
§2.1 ３＋１decomposition (3)
Hamiltonian constraint
momentum constraint
(3
Evolution equation for
Definition of
Evolution equation for
§2.1 ３＋１decomposition (4)
Decomposed Einstein equations
Constraint equations
Evolution equations
§2.2 Shibata-Nakamura reformulation
Cf. Shibata and Nakamura (1995) PRD 52, 5428
Decompose geometrical variables
Add
as a independent variable