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

Rotating collapse to a BH

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

BH mass and spin

1.315

1.32

1.325

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

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

⇒ 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

neutrino loss large

neutrino loss small

Sack et al. 1980

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)