Possible constraints on density dependence of …kkotake/sotani_fu.pdf• physics in NS crust •...

超新星から期待される重力波:NS研究からの知見

祖谷元 (国立天文台)

滝脇知也（国立天文台）

watermelon

• how to know the best time to eat a watermelon ?– inside can not be checked before cutting

• “empirical rule”– to check the best time, knock a watermelon

• high frequency “KIN-KIN” ; too young

• “BAN-BAN” ; best time !

• low frequency “BON-BON” ; too old

– need many years to get this ability

• one could see the interior with specific sound from object.

– asteroseismology !!

1. Sep 2016 セミナー@福岡大学 1

Seismology, Helioseismology

©NAOJ©東大地震研究所

• Seismic waves tell usinformation inside the Earth (seismology)

• The interior of the Sun can be probed through the wave pattern on the surface (helioseismology)


Asteroseismology in Neutron Stars

Andersson & Kokkotas (1998)

• via the observations of GW frequencies, one might be able to see the properties of NSs

f-mode

w1-mode

w f » 0.78 +1.64M

1.4M⊙

æ

èç

ö

ø÷10km

R

æ

èçö

ø÷

3é

ëê

ù

ûú

1/2

ww »10km

R

æ

èçö

ø÷20.92 - 9.14

M

1.4M⊙

æ

èç

ö

ø÷10km

R

æ

èçö

ø÷é

ëê

ù

ûú

average density

ωf

Rω

w

compactness

1. Sep 2016

determination of (M, R)

セミナー@福岡大学 3

asteroseismology 2• imprints of inside NSs

– In deeper region of NS, quarkmatter could appear.

– one might be probe the existenceof density discontinuity

– one might see the surface tensionin hadron-quark mixed phase


HS, Yasutake, Maruyama & Tatsumi 2011 HS, Maruyama & Tatsumi 2013

torsional oscillationsin HQ mixed phase

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海老名市小田原市

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リックします。

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1 / 1 14/02/07 17:18

Properties of Neutron Stars

1. Sep 2016

R ~12kmM/M⨀ ~ 1.4 – 2.0ρ ~1015 g/cm3

rs = 2.68 ´1014 g / cm3<nuclear density>


How to construct NSs

• Tolman-Oppenheimer-Volkoff (TOV) equation gives density profile of the spherically symmetric equilibrium of cold NSs.

1. Sep 2016

dP

dr= -r

Gm

r21+

4pr3P

mc2

æ

èçö

ø÷1+

P

rc2

æ

èçö

ø÷1-

2Gm

c2r

æ

èçö

ø÷

-1

P = P(r)

dm

dr= 4pr2r

dP

dr= -r

Gm

r2

relativistic correction

equation of state (EOS)

R = r @P = 0

M = m(R)

EOS is essential to construct the stellar model, but still uncertain.

R

M

lnr

lnP

r

m(r)

rc

(M,R)


too many EOSs suggested…

1. Sep 2016

Demorest et al. (2010)

NS observations can make a constraint on EOS!!


nuclear saturation

• radius of an atomic nucleus with mass number A

• binding energy

which are independent of atomic nuclei.

• density of atomic nuclei

1. Sep 2016

r »M

R3=

mA

r0

3A=

m

r0

3

saturation density = 2.68×1014 g/cm3

r »M

R3=

mA

r0

3A=

m

r0

3@ rs


neutron stars

• Structure of NS

– solid layer (crust)

– nonuniform structure (pasta)

– fluid core (uniform matter)

• Crust thickness ≲ 1km

• Determination of EOS for high density (core) region could bequite difficult on Earth

• Constraint on EOS via observationsof neutron stars

– stellar mass and radius

– stellar oscillations (& emitted GWs)

“(GW) asteroseismology” ∆R[km]

pasta

crust core

0 0.2 0.4 0.6 0.8lo

g P

Lorenz et al. (1993)

Oyamatsu (1993)

crust

core

1. Sep 2016

r » rs

rs

4 ´1011g / cm3

Neutrondrip


(P)NS - EOS

• physics in NS crust

• low-mass NSs

(1) TOV equation(2) equation of state

- model- nuclear interaction- composition

constraints from the terrestrialnuclear experiments

≀≀properties around

the saturation density

1. Sep 2016

physics in NS (core)

high density region


oscillations in NSs


Oscillations & Instabilities

13.05.2016 NAOJ 16

Themostpromisingstrategyforconstrainingthephysicsofneutronstarsinvolvesobservingtheir“ringing”(oscillationmodes)

Ø f-mode:scaleswithaveragedensityØ p-modes:probesthesoundspeedthroughoutthestarØ g-modes:sensitivetothermal/composition gradientsØ w-modes:oscillationsofspacetime itself.Ø s-modes:ShearwavesinthecrustØ Alfvènmodes:duetomagneticfieldØ i-modes:inertialmodesassociatedwithrotation(r-mode)

TypicallySMALLAMPLITUDEoscillations–>weakemissionofGWsUNLESS

theybecomeunstableduetorotation(r-mode &f-mode)

slide by Kokkotas

Protoneutron stars (PNSs)

• Unlike cold neutron stars, to construct the PNS models, one has to prepare the profiles of Ye and s.– for example, with LS220 and s =1.5 (kB/baryon), but Ye =0.01, 0.1, 0.2, and

0.3


strategy

• calculate the 1D simulation of core-collapse supernova (by Takiwaki)– time evolutions of radius and mass of PNS are determined

– radius and mass of PNS are fitted by simple formula

• PNS models are constructed in such a way that the radius and mass of PNS are equivalent to the expectation from the fitting– with the assumption that the PNS is quasi-static at each time step

– with the profiles of Ye and s

• calculate the eigen-frequencies via the eigen-value problems on PNS models– dependence of the frequencies on the profiles of Ye and s

– dependence on the average density of PNS

– dependence on the progenitor models

• LS220 (Mpro/M⊙ = 11.2, 15, 27, 40), Shen (Mpro/M⊙ = 15)


evolutions of mass and radius

• fitted with

100 200 300 400 500 600 700 80020

30

40

50

60

70

80

t (msec)

RP

NS (

km

)

1012 g/cm3

1011 g/cm3

fitting

1. Sep 2016 セミナー@福岡大学 14Scheck et al. (2006)

LS220M15.0

Ye and s profiles• the snap shot at t=100, 200, and 500ms after bounce

1. Sep 2016 セミナー@福岡大学 150 MPNS

0

1

2

3

4

5

6

m(r)

s (k

B/b

aryo

n)

s = sm(t1)

s = sm(t2)

2MPNS/3

s0(t1)

s0(t2)

0 MPNS0

0.1

0.2

0.3

0.4

m(r)

Ye

s0 is given from numerical results,

but sm should be determined to fit

comparison with other results• results by Roberts (2012), where he has done the 1D simulations

for long-term.


Ye s

0 MPNS0

0.1

0.2

0.3

0.4

m(r)

Ye

2MPNS/3 0 MPNS0

1

2

3

4

5

6

m(r)

s (k

B/b

aryo

n)s = sm(t1)

s = sm(t2)

2MPNS/3

again, sm should bedetermined to fit (M,R).

PNS models• adopting two different profiles of Ye and s inside the PNS, we

construct the PNS models.– unknown parameter: εc & sm

– to reproduce the PNS models with given (M, R), εc and sm are fixed.

• evolutions of εc and sm depend strongly on the profiles of Ye and s.


LS220M15.0

oscillations in PNS

• with relativistic Cowling approximation

• omitting the entropy variation

• frequencies depend on mass and radius of PNS, but weakly depend on (Ye, s) profiles.

• in the early stage, the typical frequencies of f-mode is ~ a few hundred hertz, which is good for gravitational wave detectors.


0 200 400 600 800 1000200

400

600

800

1000

1200

1400

t (msec)

f f (

Hz)

Fig. 4Fig. 5

0 200 400 600 800 1000800

1200

1600

2000

2400

t (msec)

f p1 (

Hz)

Fig. 4Fig. 5

LS220M15.0

T.T.L.R.

T.T.L.R.

characterized by average density • frequencies of f-mode for cold neutron stars:

• Similarly, frequencies for PNS can be characterized by average density, but obviously different from those for neutron stars.


Andersson & Kokkotas (1998)

T.T.L.R.

T.T.L.R.

dependence on progenitor models• results for LS220 with Mpro/M⊙=11.2, 15, 27, and 40, for Shen with

Mpro/M⊙=15

• progenitor model dependence is quite weak.


comparison with g-modes• as characteristic GWs from core-collapse supernova, the

excitation of g-modes around PNS has been reported (Muellar et al. (2013); Cerda-Duran et al. (2013))

– due to the convection and the standing accretion-shock instability.


mn: neutron mass

: mean energy of electron antineutrinos

time (s)

lum

inosi

ty (

10

52

erg

/s)

Muellar & Janka (2014) black: electron neutrinosred: electron antineutrinosblue: μ/τ neutrinos

Muellar et al. (2013) Muellar et al. (2013)

comparison with g-modes

• careful observing the gravitational wave spectra after core-collapse supernova, one might see the different sequences in spectra – which tells us the radius and mass of PNS


0 200 400 600 800 10000

500

1000

1500

t (msec)

f (H

z)

f modeg mode

conclusion• We examine the frequencies of gravitational waves radiating from PNS

after bounce.

• The PNS models are constructed in such a way that the mass and radius obtained from 1D simulation are reconstructed.– two different profiles of Ye and s are considered

• frequencies of gravitational waves are almost independent from the profiles of Ye and s, but sensitive to the mass and radius.– characterized by average density

– different from that expected for cold neutron stars

• progenitor dependence is quite weak.– find an universal relation for frequencies of f- and p-modes as a function of

average density

– different dependence for g-mode around PNS

• one might be possible to determine the mass and radius of PNS via careful observations of time evolution of gravitational wave spectra.


Ye and s profiles

• the snap shot at t=100, 200, and 500ms after bounce

• distribution close to the surface at 1011 g/cm3 abruptly increase.

• region for ~1011 - 1012 g/cm3 could be relatively dilute


Possible constraints on density dependence of …kkotake/sotani_fu.pdf• physics in NS crust •...

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