Numerical relativity simulation with Microphysics

National Astronomical Observatory of JapanYuichiro Sekiguchi

Masaru Shibata (YITP)Kenta Kiuchi (YITP)

Koutaro Kyutoku (YITP)Keisuke Taniguchi (Tokyo)

Introduction• Exploring phenomena in strong, dynamical gravity

– Black hole (BH) formation, Merger of compact objects, Collapse of massive star, etc.

– Gravitational waveforms– Test of GR in strong gravity

• High energy astrophysical phenomena– Gamma-ray bursts (GRBs), Supernovae etc

• Theoretical study ⇔ Observation

• Einstein equations : nonlinear partial differential equation– Numerical simulation will be unique approach to the problem

⇒　Numerical Relativity

Targets of Numerical Relativity

• Collapse of massive stellar core, NS/BH-NS/BH merger

• Gravitational waveform• EOS of dense matter• High energy astrophysical phenomena

What is required to explore the phenomena and what is the

problems ?

GR effects

22GR crit, )1(78.23

4 ~ 78.2

3

4

Rc

GMO

c

P

Chandrasekhar 1964, 1965

GR and EOS

Sh

ock v

elo

cit

y @

300 k

m

(1000

km

/s)

Incompressibility K(sym) (MeV)

22GR crit, 78.23

4 ~ 78.2

3

4

Rc

GM

c

P

Van Riper (1988) ApJ 326, 235

GR and weak rates

trapping)- capture,-(e rateson weak depends :~

3/4

initlepton,

bouncelepton,

Y

Yd

Sh

ock e

nerg

y @

bou

nce (

10

52

erg

)

Log

(S

hock e

nerg

y @

eje

cti

on

)

Takahara & Sato (1984) PTP 72, 978

Collapse of massive star

• Dense (hot) matter region– 　　　　　　　　⇒ neutrinos drive the thermal / che

mical evolution of the core• Neutrinos and weak interaction must be included• Strong dependence of weak rates on temperature ⇒ a

finite temperature EOS is required– Currently, Shen EOS and LS EOS available

– 　　　　　　　　⇒ β-equilibrium may be achieved• Very different two timescales• Numerically, very ‘ stiff ’ source terms appear

– Generally, implicit schemes are necessary– In my study, sophisticated GR leakage scheme is adopted

to solve in an explicit manner

dynweak

e ,

Merger of NS-NS/BH

• Late inspiral phase : NS is ‘ cold ‘– kBT/ EF << 1 : NS will be described well with zero temperatu

re EOS (Cold EOS)– Extension to finite temperature

• Meger phase : Compression, shock heating– kBT/ EF ~ O(0.1) : a finite temperature EOS is required– Currently, Shen EOS and LS EOS

• Prompt BH formation and small disk– Effects of finite temperature may be miner (Cold EOS may b

e sufficient)• HMNS formation or massive disk formation

– Shock heating and neutrino emission, etc. are important (finite temperature EOS required)

• Cauthy problem in GR is constrained system– There are constraint equations (similar to no-monopole conditi

on and gauss’low in EM)

• Everything is in terms of energy-momentum tensor– All equations of source field are obtained from

• One can not add any source terms to the system – If added, constraint violations will lead to termination of simul

ations

• Neutrino energy momentum should be considered

Problems in NR ①

• Existence of ut (Lorentz factor) – There is a procedure to solve nonlinear equations for ut

• Total energy (depends on ut) is evolved – There is a procedure to recover T or (P) from the evolved total en

ergy• The above two procedure couples in a complex manner

Problems in NR ②

)],,ˆ( ,~ˆ[ ˆ TYuuuu eittt

),,(

),,(

),,( ˆ

),,( ˆ

e

e

e

e

QTYSQ

QTYSY

QTYSu

QTYSe

Yee

ui

e

Evolved quantities

Argument quantities

Nonlinear eq. with EOS table search

ta

a uuu 1

e depends on ut

Nonlinear eq. with EOS table search

• Due to these complexity, solving the equation implicitly is very hard in NR– Iteration includes two loops : no guarantee for

convergence – Explicit scheme is required

• A resolution : GR leakage scheme– Utilizing the fact that ‘ leakage timescale ’ is much

longer than the weak timescale

– Approximate treatment of neutrino cooling based on ‘ leakage time scale ’

dynleakweak ~)/( cR

Problems in NR ②

GR leakage scheme (hydro)

• Basic equation :

• Energy-momentum tensor of neutrinos :– ‘Trapped neutrino’ and

‘Streaming neutrino’ parts

• Trapped neutrino part is included into Fluid part

• The equation to be solved

)stream,()trap,()( ababab TTT

ba

ba

ba

ba

QT

QT

)(

)(fluid

(leak)stream) ,(

(leak) trap),(

stream) ,( trap),( )(

ba

ba

bba

ba

ba

ba

ba

QT

QQT

QTT

)trap,()fluid( ababab TTT

ababbabaab

abbaab

PnFnFnEnT

PguhuT

)stream,(

(leak)stream),(

(leak)

ba

ba

ba

ba

QT

QT

0)( Total aba T

abab EP 3

1

Only leakage timescale appears

• Source terms: – local rates (electron capture, pair processes : weak ti

mescale)– leak out of trapped neutrinos to be streaming neutrinos (le

akage timescale)

• Problem :– How to treat the local rates– andβ-equilibrium

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

GR leakage scheme (Lepton conservation)

• In the hot matter region, weak timescale becomes too short and the source term becomes too large– We introduce some limiters to the source terms– Assumption: Ynu’s cannot exceed the corresponding values at β-eq

uilibrium

• First, trial evolution of total lepton fraction Yl– Note that the source term is in leakage timescale

– Under the assumption of β-equilibrium, Ynubeta’s are calculated. These provide the limiters

• Second, evolution of lepton fractions– If the local rates are below the limiters, we simply evolve them– On the other hand, if the local rates exceeds the limiters, the values

at β-equilibrium are adopted

GR leakage scheme (Lepton conservation)

leak

( )ll

d Y

dt

• Important issues :– Use the EOS table with arguments (ρ,Yl, T)– In this case, only one dimensional search is required– Otherwise two dimensional search (Yl, e) ⇒ (Ye, T) required,

which in general may be convergent

GR leakage scheme (Lepton conservation)

Summary of microphysics

• EOS: Tabulated EOS can be used– Currently Shen EOS + electrons + radiation

• Weak rates– Electron capture: FFN1985, rate on NSE back gr

ound – e±annihilation: Cooperstein et al. 1985, Itoh et al. 1996– plasmon decay: Ruffert et al. 1996, Itoh et al. 1996– Bremsstrahlung: Burrows et al. 2006, Itoh et al. 1996

• Neutrino leakage– Opacity based on Burrows et al. 2006

• (n, p, A) scattering– Including correction such as ion-ion correlation

• (n, p, A) absorption

GR leakage works well

• Neutrino luminosities consistent with result by 1D GR radiation hydro (Liebendoefer et al. 04) – Collapse of 15 Msun model by WHW02– Besides convection induced modulation in luminosities

• Neutrino luminosities in BNS merger and GRB will be estimated

Liebendoerfer et al. (2004)Results by Sekiguchi (2009)

GR leakage works well

• Results consistent with Liebendorfer et al. 2004

Convective activities

(unstable) 0 :criterion Ledoux ,,

dr

ds

sdr

dY

YlYP

l

sPl

Applications : PopIII core collapse

High energy astrophysics: GRB

central engines: BH+Disk Stellar core collapse NS-NS/BH mergerγ線

0 10 20 30 40 50 [s]

Time Profile

1051erg/s<Liso<1054erg/sMost violent explosion in the universe

BHDisk

Jet

Gamma-ray burst by neutrino pair annhilation

ee

Hot disk

PopIII core collapse

• BH formation with microphysics– black hole excision technique for hydrodynamics &

microphysics– puncture evolution for geometry

• Initial condition– Simplified model (S = Ye = const core)– S=7kB, 8kB; Ye=0.5

density log( g/cm3 )

Ye entropy per baryon ( kB )

Collapse dynamics : Weak bounce

• Do not directly collapse to BH – Weak bounce

• At bounce– ρ ~ 1013 g/cm3

• subnuclear ! – T ~ 18 MeV– Ye ~ 0.2

Bounce due to gas pressure

• He → 2p + 2n– Gas pressure (Γ=5

/3) increase• Indeed Γth >4/3

MeV1832

gas

4MeV18

31rad

32deg

102~

101~

101~

TP

TP

P

• Gas pressure dominates at ρ~1013g/cm3, T~18 MeV

• EOS becomes stiffer ⇒ weak bounce

Collapse dynamics : Disk formation

Neutrino emission rate [erg/cm3/s]

Collapse dynamics : Disk formation

Neutrino emission rate [erg/cm3/s]

Final state of the simulation• Neutrino torus is formed• Density along the rotational axis > 108 g/cm3

– Higher for the formation of GRB fireball via ν-annihilation

density

Final state of the simulation

• Some fluctuation can be seen in Ye• Heavy elements are completely dissociated

Ye

Neutrino emission

AH formation

Neutrino emission from the torus

Expected neutrino pair annihilation

Setiawan et al. (2005)

352

disk

40km(efficiency) 10

10 erg/s 10MeVe e

e e x

E L

L L L R

2 2

E CL L

• Neutrino luminosity ~ 1054 erg/s• Average energy ~ 20-30MeV• According to the results by Setiawan et al. pair annih

ilation luminosity of >1052 erg/s is expected

To estimate the pair annihilation rates more accurately, Ray-tracing calculations are planned (Harikae, Se

kiguchi, Takiwaki, Kotake)

~300km

Neutrino interaction is important

The results in which first order correction to the neutron / proton magnetic moment is considered

Evolution of BH mass

• Assuming Kerr BH geometry – BH mass = 6~7 Msolar

– Rotational energy = MBH – Mirr ～ 1054 erg– If strong magnetic field exists, the rotational energy can be extract

ed• Mass accretion rates is still large as > several Msolar/s

Summary

• Effects of GR cannot be ignored• In NR, to treat weak interactions such as electron ca

pture and neutrino cooling is difficult• We developed GR leakage scheme in which these ca

n be treated approximately• GR leakage scheme works well

• We applied the GR leakage code to collapse of PopIII core

• Neutrino luminosity is sufficient to produce the GRB fireball by neutrino pair annihilation

Very preliminary result (just started)

• Simulations are ongoing with electron capture and GR neutrino leakage

• Some room for improvement in EOS construction, atmosphere treatments, etc

• If you have good EOS, let us use it !

Density profile