Possibility to restrict on neutron star
matter by using asteroseismology

Hajime Sotani (YITP)
K. Nakazato, K. Iida, & K. Oyamatsupossibility toto restrict
restrict onon neutron
neutron starstar
possibility
matter byby using
using asteroseismology
asteroseismology
matterHajime SOTANI (YITP) collaborated with K. Nakazato, K. Iida, & K. Oyamatsu

QPOs
QPOs inin giant
giant flares
flaresRecent observations of the quasi-periodic oscillations(QPOs) in giant flares from soft gamma repeaters(SGRs) suggest that long-awaited evidence for theneutron star (NS) oscillations. The QPOs withfrequencies in the range from tens Hz up to kHz havebeen discovered. Many theoretical attempts to explain
the observed evidences have been done...
©NAS
A

Magnetars

SGRs are promising candidate of magnetar, which is a
NS with strong magnetic field, such as B ≳ 1014G. Sinceinterior region of NS is quite high density, NS is suitable
laboratory to see the properties for high density region.

Watts & Strohmayer (2006)

CrustCrust EOS
EOS

We adopt the crust EOS suggested by Oyamatsu & Iida(03,07), which is based on the extended Thomas-Fermi
theory. As in their article, we adopt the parameter
range 0 < L < 160 MeV & 180 ≦ K0 ≦ 360 MeV.

Considering the magnetic field strength of central
object in SGRs, observed QPOs could be due to thecrust torsional oscillations.
Comparing QPOs with frequencies of crust torsional
oscillations, one can make a constraint on the crust
EOS parameters. (asteroseismology)
B ≈ 1015 G

LTorsional(+Alfven)

Alfvenmagnetic strength

Constraint
Constraint 22

Constraint
Constraint 11

We find that the fundamental frequencies
independent of the incompressibility K0.

areIn order to explain the observed QPOs with the crust
torsional oscillations, at least, the expected lowest
frequency, 0t2, should be lower than the lowest
observed QPO frequency, 18Hz.
Concluding that we can make a constraint on L as L ≳50 MeV.

Taking into account the effect of neutron superfluidity inthe inner crust, the effective enthalpy should reduceand the shear velocity increases. As a result, thefrequencies of torsional oscillations also increases,
because the frequencies are proportional to shear
velocity.
Comparing the observed QPOs with torsional modes,
we can get the constraint on L as 100 MeV ≲ L ≲ 130
MeV.
HS+12bL ≳ 50 MeV

HS+12a16/Dec./2012

LQUCS2012 @ Nara1

possibility toto restrict
restrict on
on neutron
neutron star
star
possibility
matter byby using
using asteroseismology
asteroseismology
matterHajime SOTANI (YITP) collaborated with K. Nakazato, K. Iida, & K. Oyamatsu

QPOs
QPOs inin giant
giant flares
flaresRecent observations of the quasi-periodic oscillations(QPOs) in giant flares from soft gamma repeaters(SGRs) suggest that long-awaited evidence for theneutron star (NS) oscillations. The QPOs withfrequencies in the range from tens Hz up to kHz havebeen discovered. Many theoretical attempts to explain
the observed evidences have been done...

astronomicalobservations+calculate
torsional
oscillations

Magnetars

SGRs are promising candidate of magnetar, which is a
NS with strong magnetic field, such as B ≳ 1014G. Sinceinterior region of NS is quite high density, NS is suitable
laboratory to see the properties for high density region.

Watts & Strohmayer (2006)

CrustCrust EOS
EOS

We adopt the crust EOS suggested by Oyamatsu & Iida
(03,07), which is based on the extended Thomas-Fermi
theory. As in their article, we adopt the parameter
range 0 < L < 160 MeV & 180 ≦ K0 ≦ 360 MeV.

Considering the magnetic field strength of central
object in SGRs, observed QPOs could be due to thecrust torsional oscillations.
Comparing QPOs with frequencies of crust torsional
oscillations, one can make a constraint on the crust
EOS parameters. (asteroseismology)
B ≈ 1015 GL

Torsional(+Alfven)

Alfvenmagnetic strength

Constraint
Constraint 22

Constraint
Constraint 11

We find that the fundamental frequencies
independent of the incompressibility K0.

areIn order to explain the observed QPOs with the crust
torsional oscillations, at least, the expected lowest
frequency, 0t2, should be lower than the lowest
observed QPO frequency, 18Hz.
Concluding that we can make a constraint on L as L ≳50 MeV.

Taking into account the effect of neutron superfluidity inthe inner crust, the effective enthalpy should reduceand the shear velocity increases. As a result, thefrequencies of torsional oscillations also increases,
because the frequencies are proportional to shearvelocity.
Comparing the observed QPOs with torsional modes,
we can get the constraint on L as 100 MeV ≲ L ≲ 130
MeV.
HS+12bL ≳ 50 MeV

constraint on L
HS+12a16/Dec./2012©NASA

LQUCS2012 @ Nara2

• Structure of NS

Neutron Stars– solid layer (crust)– nonuniform structure (pasta)– fluid core (uniform matter)crust??

• Thickness of pasta ~ 70m• Determination of EOS forhigh density region is quitedifficult on Earth• Constraint on EOS via observationsof NS“(GW) asteroseismology”• NS can be considered as a“Rosettastone” to see physics in ultra-highdensity region.
16/Dec./2012

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Oyamatsu (1993)
Lorenz et al.
(1993)

crustlog P– stellar mass and radius– stellar oscillations (& emitted GWs)

corecore
pasta

00.2

0.4

∆R[km]

0.60.83

QPOs in giant flares 1••Magnetars : B > 1014G
Candidates of magentars––•Anomalous X-ray pulsars （AXPs）
Soft gamma repeaters（SGRs）~ sporadic emission with X and γ-rays (~ 1041 erg/s)

Giant flares from SGRs (1044-1046 ergs/s)–––SGR 0526–66 in March.5.1979
SGR 1900+14 in August.27.1998
SGR 1806–20 in December.27.2004
Watts & Strohmayer (2006)

γ-ray
X-ray

Strohmayer & Watts (2006)
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QPOs in giant flares 2•

Afterglow of giant flares → quasi periodic oscillations(QPOs)
 Barat et.al. (1983); Israel et.al. (2005);–––Watts & Strohmayer (2005, 2006)
SGR 0526-66 : 23ms (43Hz), B ∼ 4 ×1014G
SGR 1900+14 : B > 4 ×1014G, 28, 54, 84, 155 Hz
SGR 1806–20 : B ~ 8 ×1014G, L ∼ 1046 ergs/s
18, 26, 30, 92.5, 150, 626.5, 1837 Hz + something ?

• Theoretical attempts to explain…

– torsional oscillations in neutron star crust.
– magnetic oscillations (Alfven oscillations)
Torsional(+ Alfven)B ≈ 1015 G
AlfvenMagnetic strengthc.f.) fundamental modes of NS is order of kHz
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Torsional oscillations in curst• Frequencies of torsional oscillations:– For Newtonian star; Hansen & Cioffi (1980), McDermott et al.
(1998), Carroll et al. (1986), Storhmayer (1991), …
 in Newtonian case,shear speed: vsl=2

16/Dec./2012• incompressible• without stellar deformation• without density deviation-2v
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6

EOS for crust region

Oyamatsu & Iida (2003), (2007)• Bulk energy per nucleon near the saturation point of symmetric nuclear
matter at zero temperature;symmetry parameter
incompressibility• Calculations of the optimal density distribution of stable nuclei within
Thomas Fermi theory.
- Obtain the value of w0, n0, and S0 for given
L & K0 by fitting Z, mass, & charge radiusthat can be calculated from the optimal
density distribution to the empirical datafor stable nuclei.
- To constrain in L & K0 with experiments
on Earth is quite difficult.
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L (gradient)experiments forstable nuclei
7

What we do• EOS for core region is still uncertain.
• To prepare the crust region, we integrate from r=R.
– M, R : parameters for stellar properties– L, K0 : parameters for curst EOS (Oyamatsu & Iida (2003), (2007))→ For L ≳ 100MeV, pasta structure almost disappears• In crust region, torsional oscillations are calculated.
– considering the shear only in spherical nuclei.
– frequency of fundamental oscillation ∝ vs (vs2 ~ μ/H )– calculated frequencies could be lower limit
corecrustfor bcc latticeρs
16/Dec./2012

pasta nucleidensity

ρpQUCS2012 @ Narani : number density of quark droplet
Z : charge of quark droplet
a : Wigner-Seitz radius8

0t2 without superfluidity• For M=1.4M⊙ & R=12km,calculated frequencies 0t2• 0t2 is almost independentof the value of K0• For R=10~14 km andM/M⊙=1.4~1.8，
similar dependence on K0• One can write fitting line
• Focus on L dependence of 0t2• 0t2 becomes smaller with larger R and M.

16/Dec./2012

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HS+2012a9

Constraint on L
• For R=10km~14km & M/M⊙=1.4~1.8,
0t2 are calculated• Assuming that the observed QPOs
would come from torsional oscillations• 0t2 is the smallest frequency among
a lot of torsional oscillations

HS+2012a– 0t2 should be equal to or smaller than
the smallest observed QPOs frequency• Consequently, L ≳ 47.6 MeV.

– For L ≳ 47.6 MeV, pasta region could be very narrow– Modification due to the pasta effect should be small– This is first constraint in the symmetry parameter withastronomical observations

16/Dec./2012

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Effect of superfluidity

HS+2012bFor ρ ≳ 4 × 1011 g cm-3, neutron could drip from nuclei•• Some of dripped neutron play a role as superfluid• Effective enthalpy affecting on the shear oscillations could bereduced- shear speed (vs2 ~ μ/H ) increases due to the effect of superfluidity• 0tl could also increase due to the effect of superfluidity• While, the fraction of superfluid
neutron in dripped neutron is
Ns /Nd ↑unknown…
- Chamel (2012): superfluid neutron
are not so much (~10-30%?)•0tl with using a parameter of

Ns /Nd for R=14km & M=1.8M⊙
16/Dec./2012

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Comparison with QPO frequencies
HS+2012b• Comparison of frequencies of torsional oscillations withthe QPO frequencies observed in SGR 1806-20

L = 127.1 MeV- less than 5% accuracy
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12

expected values of L HS+2012b• Comparison with the observations in SGR 1806-20– independently of the stellar models, we can get a constraint on L
as 100 MeV ≲ L ≲ 140 MeV.
– with the help of the observation of (M,R) of oscillating star,
one may make a severer constraint in the value of L.

16/Dec./2012

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expected values of L HS+2012b• Comparison with the observations in SGR 1806-20– independently of the stellar models, we can get a constraint on L
as 100 MeV ≲ L ≲ 140 MeV.
– with the help of the observation of (M,R) of oscillating star,
one may make a severer constraint in the value of L.

allowed region
from SGR 1806-20
observation.

16/Dec./2012

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Allowed values of L

HS+2012b• Comparisons with the both observations in SGR 180620 & SGR 1900+14
130 MeVallowed101 MeV

16/Dec./2012

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Conclusion• QPOs are found in afterglow of giant flares• We examined the frequencies of torsional oscillations asvarying the stellar and EOS parameters with/without theeffect of superfluidity.
• Fundamental frequencies are almost independent of theincompressibility K0, depend only on L.
• Assuming the QPOs are associated with crustal oscillations,
we can make a constraint on EOS as– L ≲ 50 MeV– 100 MeV ≲ L ≲ 130 MeV• We are successful to constrain L with the astronomicalobservations.

– This is completely different approach from the experiments on Earth16/Dec./2012

QUCS2012 @ Nara16•We are hoping to observe nest giant flare & QPOs–––SGR 0526–66 in March.5.1979
SGR 1900+14 in August.27.1998
SGR 1806–20 in December.27.2004• Next observation might be possible soon...

16/Dec./2012

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another possibility• Otherwise, to realize 60 MeV ≲ L ≲ 80 MeV,
one needs to consider another oscillation mechanismwith l = 10??

16/Dec./2012

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Outlook• Examine overtones of crust torsional oscillations• Consider the more realistic μ– effect of nonuniform nuclear (pasta) structure• Take into account the ratio of Ns /Nd• γ-ray emission mechanism• Estimate the emitted GWs

16/Dec./2012

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1st overtone 1t2• For M=1.4M⊙& R=10, 12, 14 km,1t2 are calculated• Unlike 0t2, 1t2 depends not onlyon L, but also K0 & R.
• Dependence of R :– 0 t2 ∝ 1/ΔR
– ΔR ∝ R2(Ravenhall & Pethick (1994))• Dependence of K0 :– Dependence of K0 to determine
the density of pasta phase?
– Depend on shear in pasta phase?

• now making an analysis…
16/Dec./2012

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Compared with otherconstraints• Lattimer & Lim (2012)

16/Dec./2012

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Pasta structure

Oyamatsu & Iida (2003), (2007)• Adopt the values of (L , K0), with which can be
reproduced the mass and radius data for stable.
– 0 < L < 160MeV, 180MeV ≦ K0 ≦ 360MeV• Whether pasta phase exists or not depends strongly on L.
• For L ≳ 100MeV, pasta structure almost disappears.
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Asteroseismology• Via the observations of stellar oscillations→ One can get the interior information (asteroseismology)e.g., helioseismology for Sun• With this technique, the possibilities to get theinformation of NSs have been suggested.

withwithout

Andersson & Kokkotas (1998)
16/Dec./2012

HS+2011
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0t2 without superfluidity• For M=1.4M⊙ & R=12km,
calculated frequencies 0t2• 0t2 is almost independentof the value of K0• For R=10~14 km andM/M⊙=1.4~1.8，
similar dependence on K0• One can write fitting line
• Focus on L dependence of 0t2• 0t2 becomes smaller with larger R and M.

16/Dec./2012

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HS+2012aM↑
R↑24