公募研究「重力波観測を用いた中性子星内部における物質構造への制限の可能性 」	

3-5/Dec./2011

素核宇融合による計算基礎物理学の進展研究会 @ 合歓の郷0NSs and asteroseismology	•In neutron stars (NSs), the nuclear structure forms 	crust• Body center cubic (bcc) lattice in crust region	

• Uniform matter in core region	

•It is also considered that the nonuniform nuclear 
structure could exist at the basis of crust.	

core• so-called “Pasta”.	

The thickness of pasta phase is quite small,
which is ~ 100 m.	

•It is quite difficult to see the properties ofpasta via the observations of M and/or R.	

•One could know those via the observationsof neutron star oscillations (or GWs).	

crustlog P•

Lorenz et al. (1993)

pasta

0

• “asteroseismology” or “GW asteroseismology”	

3-5/Dec./2011

core

0.2

0.4

∆r [km]素核宇融合による計算基礎物理学の進展研究会 @ 合歓の郷

0.6

0.8

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Watts & Strohmayer (2006)

3-5/Dec./2011

素核宇融合による計算基礎物理学の進展研究会 @ 合歓の郷 Strohmayer & Watts 2(2006)

Crustal Oscillations without Pastavs
QPO Frequencies	

HS, K.Kokkotas, N.Stergioulas (2007)• SGR 1900+14	f [Hz]285484

155

torsional

3t0

6t0

9t0

17t0

• SGR 1806-20	

f [Hz]

18263092.5

150626.5

1837

torsional

2t0???

3t09t015t0

lt1lt4• Missing frequencies exist	

• Difficulty to explain two QPO frequencies 26 and 30 Hzwith a stellar model 	• What’s problem ??	

3-5/Dec./2011

素核宇融合による計算基礎物理学の進展研究会 @ 合歓の郷

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Shear modulus in crust region	•isotropic shear modulus : μ	•

Body center cubic (bcc) is assumed	

: number density of ions
+Ze： average charge of ions: average space between ions
Strohmayer, et al. (1991)

3-5/Dec./2011

素核宇融合による計算基礎物理学の進展研究会 @ 合歓の郷4

Shear modulus in Pasta phase 1	

•Nonuniform structure “pasta” exists		

− shear modulus in pasta could be differentfrom that with bcc structure.	

•Pethick & Potekhin (1998)		

− behavior like “liquid crystal”		

− smaller shear modulus	•Density that pasta structure appears		– ~ 1013 g/cc (ex. Lorenz et al. (1993))	

•Realistic shear modulus in pasta phase is
still unknown…	

©Dany P. Page	

3-5/Dec./2011

素核宇融合による計算基礎物理学の進展研究会 @ 合歓の郷5

Shear modulus in Pasta phase 2	

•We assume toy model for the shear modulus in pasta phase	HS (2011)	

– ρp : density that non-uniform nuclear structure appears		– ρp = 1013 – 1014 g/cm3		– ρc : density between the crust and core region (ρc = 1.24 × 1014 g/cm3)		

– ρ < ρp : nuclear structure forms bcc lattice		– ρp < ρ < ρc : non-uniform nuclear structure (pasta phase)		

– ρ > ρc : core fluid region	

corecrustρc3-5/Dec./2011pastaρρp

素核宇融合による計算基礎物理学の進展研究会 @ 合歓の郷

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Comparison with QPO frequencies	•For ρp = 1 × 1014 g/cm3		

− some of QPOs are impossible to explain		

− similar difficulty to the previous studywithout pasta structure	•For ρp = 2 × 1013 g/cm3		

− possible to explain the all QPO frequencies	

•One needs to consider the effect of pasta 	•Do more systematic analysis	

3-5/Dec./2011

素核宇融合による計算基礎物理学の進展研究会 @ 合歓の郷

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Crust EOSs & Pasta Structure	Oyamatsu & Iida (2003), (2007)•Bulk energy per nucleon;	

• w0, n0, and S0 might be fixed experimentally. 	

•Whether the pasta phase appear or not
depends strongly on L.	

• L ≳ 100MeV, pasta phase could vanish.	

•How do the parameters L & K0 affect
on the stellar oscillations ? 	

3-5/Dec./2011

nb [fm-3]	

• adopt two parameters; L & K0.	

0	

100	

L [MeV]	

素核宇融合による計算基礎物理学の進展研究会 @ 合歓の郷

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Strategy	

HS, Nakazato, Iida, Oyamatsu, in prep.

•EOS for inner core is still uncertain.	

•We construct the stellar model by integrating TOV eqs. from r = R inward.	

• M and R become parameters	

• L and K0 are parameters for curst EOS 	•In the crust region, we calculate the shear oscillations	• We will see the relation between the frequencies and parameters.	

• It might be possible to know something…	

corecrust* Frequencies correspond tothe lower limit
ρc3-5/Dec./2011

ρρp

素核宇融合による計算基礎物理学の進展研究会 @ 合歓の郷

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Fundamental l=2 oscillations	•Fundamental l=2 torsional frequencywith M=1.4M⊙ & R=12km.	

•Frequencies of torsional oscillations 
could be independent of K0.	

•One can derive the fitting formula;	

• Expected frequencies for the stellar model with 1.4M⊙ ≤ M ≤ 1.8M⊙ and
10km ≤ R ≤ 14km.	

• l=2 fundamental oscillation is the lowest frequency in torsional oscillations.	

• Via the observed QPO frequencies, L should be more than 50 MeV 
independent of the stellar models.
3-5/Dec./2011

素核宇融合による計算基礎物理学の進展研究会 @ 合歓の郷

10Summary	•Asteroseismology is powerful tool to see the interior properties of star.	

•In the giant flares, the QPO frequencies are discovered.	

• It is impossible to explain the observed QPO frequencies theoretically by using
only crustal oscillations without pasta phase.	

•We show the possibility to explain the all observed QPO frequencies withconsidering the pasta structure.	

•It could be possible to see the properties of crust EOS via stellar oscillations.	

Future works	

•Consideration of realistic shear modulus μ	•

Effect of Magnetic fields	

•Mechanism of gamma-ray burst	

•Possibility to detect GWs from magnetars	

3-5/Dec./2011

素核宇融合による計算基礎物理学の進展研究会 @ 合歓の郷11