AAsstteerroosseeiissmmoollooggyy    wwiitthh  pprroottoonneeuuttrroonn  ssttaarrss  
Hajime Sotani (NAOJ)
Tomoya Takiwaki (NAOJ)

nd 6, respectively. From this figure, we can find that the frequencies
depend strongly on 1063
the stellar mas
Towards gravitational wave asteroseismology
endence on the (Ye , s) distributions seems to be weak. We remark that the frequencies of f-mode oscilland a few hundred of hertz at the early stage after bounce, which could be very good signal for ground-b
detectors.
e the f-, p1 -, and p2 -modes oscillations are acoustic waves, at least for cold neutron stars, it is known• oosscciillllaattiioonn  ssppeeccttrraa  aarree  iimmppoorrttaanntt  iinnffoorrmmaattiioonn  ffoorr  eexxttrraaccttiinngg  tthhee  such oscillationsiinntteerriioorr  iinnffoorrmmaattiioonn  ooff  ssttaarr  
can be characterized by the average density of the star, which is defined as (M/R3 )1/2
us (R) of cold neutron stars [5]. In fact, in Ref. [5], they found that the f-mode frequencies can be expr
fluid oscillation modes of a star, and we consequently expect thatapparent in Fig. 2, and one can easily see that a typical value for theqf , r̄1=2 . That is, we should normalize the f mode frequency with
Towards gravitational wave asteroseismology
1061 damping time of the f mode is a tenth of a second.
For the damping rate of the p modes the situation is not so
the average density of the star. The result of doing this is shown in
Fig. 1. From
this figure
it is apparent
that thefind
relation
betweenthe fpulsars favourable. This is because the damping of the p modes is more
Although some of these EOS might be outdated, none of them is
Similarly,
van Kerkwijk
et al. (1995)
that data
for X-raymode frequencies
and<themean
is almostlinear,and atwolinearruled out by present observations. Furthermore, the range of
1:88. The
data usedin thesestudies is sensitive to changes in the modal distribution inside the star. Thus,
indicate 1:04
M=M
( <density
fitting leads
to thesimple relation
stiffness of the EOS listed by Arnett & Bowers is still relevant
actually
consistent
with (if one includes error bars) M < 1:44 M( . different EOS lead to rather different p mode damping rates, cf.
today. This is important for the present study. In order for our
We now recall thatvarious EOS that have been proposed by Fig. 3. Previous evidence for polytropes (Andersson & Kokkotas
✓ the
◆1=2
M̄ cananalysis to be robust it is necessary that our sample of EOS spans thebe(i) 1997) actually indicate that this would be the case. Clearly, an
qf ðkHzÞtheoretical< 0:78 þ physicists
1:635; divided into two major categories:
ð5Þ
anticipated range of stiffness. However, we have also included threethe ‘soft’ EOS, whichR̄3 typically lead to neutron star models with empirical relation based on the data in Fig. 3 would not be very
more modern EOS: one of the models of Wiringa, Ficks &maximummasses around
1:4 M( and radii
usually smaller than 10 robust.
where wehave introduced
the dimensionlessvariables
Fabrocini (1988) and two models from Glendenning (1985). Forkm, and (ii) the ‘stiff’ EOS with the maximum values M , 1:8 M(
The situation is slightly better if we consider the oscillation
M
R& Bowers 1977). From this one can deducethe EOS that were also considered by Lindblom & Detweiler (1983)
15 kmR̄(Arnett
M̄ ¼ and R , and¼:
ð6Þ
frequency of the p mode. From the data shown in Fig. 4 we can
1:4that,M(even though the
10 constraintkm
we have chosen identical stellar models to facilitate a comparison ofput on the neutron star mass by deduce a relation between the p mode frequency and the parameters
the results. Finally, we have only included stellar models the masses
present-day
observations
seems
strong, fit mode
does notrule out manyof of the star,
From equation(5) itfollows thatthe typicalfrequencyisand radii of which are within the limits accepted by current
the kHz.
proposed EOS. In order to arrive at a more useful result we✓◆around 2.4
1M̄
observations (Finn 1994; van Kerkwijk, van Paradijs & Zuiderwijk To deduce
are likely
to need detailedof ratethe ofstellar1:75 þ 5:59;
ð9Þ
a corresponding
relation observations
for the damping
the f radius qp ðkHzÞ <M̄
R̄1995).
also.canUnfortunately,
informationmode, weuse the roughavailable
estimate data
givenprovide
by the little
quadrupole
about
the
radius.
The
recentobservationsofquasiperiodic
oscillaand we see that a typical p mode frequency is around 7 kHz.
formula. That is, the damping time should follow fromtions in low-mass X-ray binaries✓ ◆3 indicate that R < 6M, but again Although the data for several EOS deviate significantly from (9) it is
2 W H AT C A N W E L E A R N F R O M
oscillation
R Although a number of attempts havethisis not a energy
severe constraint.
still a useful result. Stellar masses and radii deduced from it will not
tf ,
:
ð7Þ
,R
O B S E R VAT I O N S ?
beenemitted
made, using
either X-ray
power
in GWsM observations (Lewin, van Paradijs & be as accurate as ones based on f mode data, but on the other hand, if
Our present understanding of neutron stars comes mainly fromTaam 1993) or the limiting spin period of neutron
3 stars
4 ¹1 (Friedman,
Using this normalization we plot the functional ðtf M =R Þ
as aM and R are obtained in some other way (say, from a combination of
X-ray and radio-timing observations. These observations provide
Ipser & Parker 1986), to put constraints on the mass–radiusfunction of the stellar compactness, cf. Fig. 2. The data shown inobserved f - and w modes) the p mode can be used to deduce thesome insight into the structure of these objects and the properties ofrelation, we do not yet have a method which can provide the desired
this figure
lead to a relation between the damping time of the f mode
relevant EOS.
supranuclear matter. The most commonly and accurately observed
answer.
and the stellar parameters M and R,
That empirical relations based on p mode data would be lessparameter is the rotation period, and we know that radio pulsars can✓◆robust and useful than those for the f mode was expected, since the pspin very fast (the shortest observed period being the 1.56 ms of 1
M̄ 3 A detection scenario
M̄
2.1
:
ð8Þ
modes are sensitive to changes in the matter distribution inside thePSR 1937+21). Another basic observable, that can be obtained (in ta ðsÞ < 4 22:85 ¹ 14:65 R̄R̄f
few cases) with some accuracy from present day observations, is theIn view of this situation, any method that can be used to infer star. In contrast, the gravitational wave w modes should lead to
The smalldeviation
the numerical
data fromthe above
mass of the neutron star. As Finn (1994) has shown, theneutronstar ofparameters
is a welcome
addition.
Offormulaspecific isinterest very robust results. It is well known (Kokkotas & Schutz 1992;
observations of radio pulsars indicate that 1:01 < M=M( < 1:64.
may be the new possibilities offered once gravitational wave
observations become reality. An obvious question is the extent towhich one can solve the inverse problem in gravitational wave

GGrraavviittaattiioonnaall  wwaavvee  aasstteerroosseeiissmmoollooggyy  

"−3/2

M
R
(NS)
(kHz) ≈ 0.78 + 1.635,• ffsseevveerraall  aatttteemmppttss  hhaavvee  bbeeeenn  ddoonnee  ffoorr  ccoolldd  nneeuuttrroonn  ssttaarrss  
1.4M⊙
10 km

– oonnee  ccaann  ggeett  tthhee  sstteellllaarr  mmaassss,,  rraaddiiuuss,,  EEOOSS,,  aanndd  mmaaggnneettiicc  pprrooppeerrttiieess…�  

Downloaded from http://mnras.oxfordjournals.org/ at National AstronomicalObservatory, Japan on January 14, 2016

Downloaded from http://mnras.oxfordjournals.org/ at National AstronomicalObservatory, Japan on January 14, 2016

– sseeiissmmoollooggyy  oonn  tthhee  EEaarrtthh  – hheelliioosseeiissmmoollooggyy  oonn  tthhee  SSuunn  !
"1/2 !
– GGWW  aasstteerroosseeiissmmoollooggyy  oonn  ccoommppaacctt  ssttaarrss  

RRωww  

ωff  dependent from the EOS with which the neutron star model is constructed. So, we check the frequencie
vviiaa  tthhee  oobbsseerrvvaattiioonnss  ooff  ff  aanndd  ww  Ss. In Fig. 9, for the PNS model with Mpro = 15M⊙ and LS220,
we show the frequencies of f-, p1 mmooddee   oosscciillllaattiioonnss,,   oonnee   ccoouulldd  
1/2
ction of the normalized average density, which is defined as (M(RPNS/10km)−3/2 . T
PNS /1.4M⊙ )MM  aanndd  
ddeetteerrmmiinnee  tthhee  
RR  wwiitthhiinn  ~~1100%%  aaccccuurraaccyy..
   for PNSs. We remark
equencies of f-, p1 -, and p2 -modes are proportional to the averagedensity evenon the late phase after the bounce become massive with small radius
as shown in Fig. 2, the PNSs with
AAnnddeerrssssoonn  &&  KKookkkkoottaass  ((11999988))  
rrespond to the late phase. Furthermore, in the left panel of Fig. 9, we also plot the f-mode frequeneutron stars, which are calculated with Eq. (18), with the dottedline. From this panel, one can see tha
HHooww  aabboouutt  pprroottoonneeuuttrroonn  ssttaarrss??  
s for cold neutron stars obviously deviate from those for PNSs.- iitt  ccoouulldd  bbee  ppoossssiibbllee  iiff  oonnee  ccaann  
ccoonnssttrruucctt  sstteellllaarr  mmooddeellss  e the frequencies of f-,models of PNSs shown in Table I33))and
11//22   p2 -modes for the various progenitor
1 -,((MMp//RR
MM//RR  
1133  MMaayy  22001166  p the calculations
at 940 msec for the PNS44tthh  DDTTAA  ssyymmppoossiiuumm@@NNAAOOJJ  
constructed with Mpro = 40.0M⊙ for LS22011  EOS, becauss more than the maximum mass expected with LS220 around 950 msec. Then, the obtained frequencies oFigure 5. The functional Rqw as a function of the compactness of the star (MFigure 1. The numerically obtained f mode frequencies plotted as functions
Figure 2. The normalized damping time of the f modes as functions of the Figure 6. The functional M=tw as a function of the compactness of the starin kHz).
and R arestellar
in kmcompactness
and qw mode (M(M and R are in km and tw mode in ms).
of the mean stellar density (M and R are in km and qf mode in kHz).
and R are in km and tf mode in s).
q 1998 RAS, MNRAS 299, 1059–1068

q 1998 RAS, MNRAS 299, 1059–1068

PPrroottoonneeuuttrroonn  ssttaarrss  ((PPNNSSss))  
•UUnnlliikkee  ccoolldd  nneeuuttrroonn  ssttaarrss,,  ttoo  ccoonnssttrruucctt  tthhee  PPNNSS  mmooddeellss,,  oonnee  hhaass  ttoo  pprreeppaarree  tthhee  pprrooffiilleess  ooff  YYee  aanndd  ss..  
– ffoorr  eexxaammppllee,,  wwiitthh  LLSS222200  aanndd  ss  ==11..55  ((kkBB//bbaarryyoonn)),,  bbuutt  YYee  ==00..0011,,  00..11,,  
00..22,,  aanndd  00..33  

2.0

J1614-2230

M/M

1.5
1.0

0.3

0.5
0
10

0.2
0.01
12

0.1
14

16

18

20

R (km)
1133  MMaayy  22001166  

44tthh  DDTTAA  ssyymmppoossiiuumm@@NNAAOOJJ  

22  

ssttrraatteeggyy  •

ccaallccuullaattee  tthhee  11DD  ssiimmuullaattiioonn  ooff  ccoorree--ccoollllaappssee  ssuuppeerrnnoovvaa  ((bbyy  TTaakkiiwwaakkii))  
– ttiimmee  eevvoolluuttiioonnss  ooff  rraaddiiuuss  aanndd  mmaassss  ooff  PPNNSS  aarree  ddeetteerrmmiinneedd  
– rraaddiiuuss  aanndd  mmaassss  ooff  PPNNSS  aarree  ffiitttteedd  bbyy  ssiimmppllee  ffoorrmmuullaa  •PPNNSS  mmooddeellss  aarree  ccoonnssttrruucctteedd  iinn  ssuucchh  aa  wwaayy  tthhaatt  tthhee  rraaddiiuuss  aanndd  mmaassss  
ooff  PPNNSS  aarree  eeqquuiivvaalleenntt  ttoo  tthhee  eexxppeeccttaattiioonn  ffrroomm  tthhee  ffiittttiinngg  
– wwiitthh  tthhee  aassssuummppttiioonn  tthhaatt  tthhee  PPNNSS  iiss  qquuaassii--ssttaattiicc  aatt  eeaacchh  ttiimmee  sstteepp  
– wwiitthh  tthhee  pprrooffiilleess  ooff  YYee  aanndd  ss  •ccaallccuullaattee  tthhee  eeiiggeenn--ffrreeqquueenncciieess  vviiaa  tthhee  eeiiggeenn--vvaalluuee  pprroobblleemmss  oonn  PPNNSS  
mmooddeellss  – ddeeppeennddeennccee  ooff  tthhee  ffrreeqquueenncciieess  oonn  tthhee  pprrooffiilleess  ooff  YYee  aanndd  ss  – ddeeppeennddeennccee  oonn  tthhee  aavveerraaggee  ddeennssiittyy  ooff  PPNNSS  – ddeeppeennddeennccee  oonn  tthhee  pprrooggeenniittoorr  mmooddeellss  
• LLSS222200  ((MMpprroo//MM⊙  ==  1111..22,,  1155,,  2277,,  4400)),,  SShheenn  ((MMpprroo//MM⊙  ==  1155))  
1133  MMaayy  22001166  

44tthh  DDTTAA  ssyymmppoossiiuumm@@NNAAOOJJ  

33  

1.7

10
-100 -50

0

50 100 150 200 250 300-100 -50
LLSS222200MM1155..00  

080
50 100150 200 250 300
12

eevvoolluuttiioonnss  ooff  mmaassss  aanndd  rraaddiiuuss  
t (msec)
10

10 g/cm3g/cm3
me evolutions of1.5mass shell and shock radius of LS220M15.0. The position of the shock front10is11indicatedby the tﬁttingon of mass shells are plotted by thin-solid lines (∆M = 0.01M60
whose enclosed ma⊙ ). In particular, the mass-shell
1.4line.
h the thick-dotted50
1.3
12 LS220M15.0.
me evolutions of mass shell and shock radius10of
The position of the shock front is indicated by the t
40
g/cm3
1.2
on of mass shells are plotted by thin-solid lines
1011(∆M
g/cm3 = 0.01M⊙ ). In particular, the mass-shell whose enclosed ma
30
1.1 line.
h the thick-dottedﬁtting20Ri
1.0
100 200
100 200 300 400
500
600
700
800
# 300 $400
R(t) =, 500 600 700 800

t (msec)70

RPNS (km)MPNS/M

1.6

PNSt (msec)!
"t
1 + 1 − exp(− τ )Rit (msec)
Rf − 1

ffiitttteedd  wwiitthh  Ri
and Rf are the• radiiof the PNSs R
at t =(t)
0 and
∞,
while
τ
denotes
for the evolution o#a typical$ time-scale
=
,
PNS
!
"
Ri
t
we also obtain the fitting formula of the evolution
theexp(−
PNS mass
1 + of
1−
) as
−1
τ

Rf

MPNS (t)c0=c1 t + c2a. typical time-scale for the evolution oand Rf are the radii of the PNSs at t = 0 and ∞, while
τ+denotes
M⊙t
we also obtain the fitting formula of the evolutionof the PNS mass as
SScchheecckk  eett  aall..
44tthh  DDTTAA  ssyymmppoossiiuumm@@NNAAOOJJ  
hat t in Eqs. 1133  MMaayy  22001166  (1) and (2) is in the unit of msec.
The fitting parametersin Eqs.  ((22000066))  
(1) and (2)44   for various s
MPNS (t)c012in Tables II, where the radius of PNS correspondsg/cm= to the+ cposition
1 t + c2 . that the density becomes 10
11
3
M⊙
t

YYee  aanndd  ss  pprrooffiilleess  •

tthhee  ssnnaapp  sshhoott  aatt  tt==110000,,  220000,,  aanndd  550000mmss  aafftteerr  bboouunnccee  s = 1012 g/cm3

Ye

0.2

0.1

0
0

100 ms200 ms
500 ms
0.2

0.4

0.6

0.8

s (kB/baryon)

0.3

1.0

8
7
6
5
4
3
2
1
0
0

100 ms200 ms
500 mss = 1012 g/cm3
0.2

s (kB/baryon)

ss00  iiss  ggiivveenn  ffrroomm  nnuummeerriiccaall  rreessuullttss,,
   bbuutt  
6
ssmm  sshhoouulldd  bbee  ddeetteerrmmiinneedd  ttoo  ffiitt  ((MM,,  RR))..  

0.3

Ye

0.6

0.8

1.0

m/MPNS

m/MPNS
0.4

0.4

0.2
0.1

5
4
3
2

s = sm(t1)s = sm(t2)s0(t2)

1 s (t )0 1

001133  MMaayy  22001166  

m(r)

MPNS

0
0

44tthh  DDTTAA  ssyymmppoossiiuumm@@NNAAOOJJ  2MPNS/3

m(r)

MPNS

55  

100

150

30
100

T (MeV)

ccoommppaarriissoonn  wwiitthh  ootthheerr  rreessuullttss  
The Astrophysical Journal, 755:126 (20pp), 2012 August 20
20

50

rreessuullttss  bbyy  RRoobbeerrttss  ((22001122)),,  wwhheerree  hhee  hhaass  ddoonnee  tthhee  11DD  ssiimmuullaattiioonnss  
50
ffoorr  lloonngg--tteerrmm..
    
200
0.4
0.8
1.2
Enclosed Baryonic Mass (Solar Units)

0

1.6

0
0

0.4

0.3

40

6
0.3

-3

ss  

nB (fm )
s

0.2

0.2

0.4
0.8
1.2
Enclosed Baryonic Mass (Solar Units)
0.4
0.8
1.2
Enclosed Baryonic Mass (Solar Units)

t=0
t=1
t=5
t = 10t = 20t = 30

1.6

0

1.6

0

0
En

0

0
En

0.3

0.2

150

3

0.1

0

0

Ye

0

9

1.6

50

10

0

100

50

0.1

30
0

0.1

0.4
0.8
1.2
Enclosed Baryonic Mass (Solar Units)

1.6

0
0
0
0

µeq (MeV)•
1.6µeq (MeV)µn - µp (MeV)µeq (MeV)

t = 1
t = 5
t = 10t = 20t = 30150
40

T (MeV)

1.2
Units)

150

YYYeee  1.2
Units)

t = 20t = 30

100

0.4
0.8
1.2
0.4
0.8
1.2
Enclosed
EnclosedBaryonic
BaryonicMass
Mass(Solar(SolarUnits)
Units)

1.6
1.6

0

selected times in the fiducial simulation. The temperature, equilibrium
electronneutrinochemical
proton neutron
Figure1. Internal
structureof thepotential,
PNS for selected
times in the fiducial sim
chemicalpotentialdifference,
entropyperbaryon, electron
ess entropy per baryon, electron fraction, and baryon density are plottedat times
0 s, 1s, 5 s, 10dimensionlesss, 20 s, and 30s, from
leftright 6andbottom.
Thehorizontal
axes
show
the enclosed baryol axes show the enclosed
baryon number in units of the number oftobaryons
in topthe to
Sun
(N⊙ ≡
12.04
× 1056). Thisfigure0.4
can abedirectlycompared
Figure 9ofofstatePonsandet al.
(1999), as it was prodmm
ons et al. (1999), as it was produced using the same initial model andvery
similarnucleartoequation
neutrino
opacity set.
s = sm(t1)(A color5version of this figure is available in the online journal.)
in the online journal.)

20

Yeplies that the supernova
10shockis reasonably0.2
consistent withson et al. (2003). The shocked
o the core and extends to large
ity of 10−5 fm−3 is found atcontribution 0.1
to the pressure.
hock-heated mantle rapidly0
r so of the simulation.0 This
0 and leptond loss of energy
0 due to then readily escape1133  MMaayy  22001166  9d long interaction meanfree
and thermal energy reduces
and the mantle responds by
ative to the cooling timescales (kB/baryon)

0.3

aaggaaiinn,,   ss    sshhoouulldd   bbee  ddeetteerrmmiinneedd  ttoo  ffiitt  ((MM,,  RR))..  

50

4 mass of 1.0 M⊙ . This implies that the supernova sh
enclosed
which significantly
alters therate of energy andleptons=sm(t2) number consistent w
was bornat aroundet
0.6
M⊙, whichis reasonablytransport in the PNS(cf. Robertsal.
2012).
Additionally,
3 accretionresults
the core-collapseof Thompson
et al.
(2003). The shoc
there might be significant
luminosity
over thisperiodmantle is at low density relative to the core and extends to l(although this is approximately accounted for by the mantle).
radius 2(material that is at a density of 10−5 fm−3 is foun
Therefore, especiallygiven the older provenance of the initial
99 km), mainly due to the thermal contribution to the press
conditions, the results
fromthis
this initialperiod state,shouldthe
be shock-heatedtaken as only mantle rap
From
1
qualitatively correct.
contracts over the first secondor so of the simulation. T
0is contracting,ν̄byandrapidνx s lossaree s the
0.4 While the
0.8 mantle
1.2
1.6
0
0.4 being
0.8 and lep
contraction
is drivenof energy
0 via neutrinos,
transported
down
thepositive
radialtemperature
gradient
into
numberwhich
canreadilyescapedue(SoltoEnclosed
Baryonic(SolarUnits)
Enclosed
Baryonic
Mass
MPNSMass
2M
0 being transported
M
/3
2MPNS
PNS/3
PNSinteraction
thecore while
the lowνe s areoutwarddown
thedensityof the envelopeand longmean
44tthh  DDTTAA  ssyymmppoossiiuumm@@NNAAOOJJ  
66  
large equilibrium paths.
chemical
gradient.
This
resultsin energy redu
The potential
loss of lepton
numberandthermala net heat flux into
the core
and ainnet
flux
out
ofmantle
thepressure
supportthelepton
mantle,andtheresponds
core. This has beenreferred
to as “Joule
heating”
of thecore
rapidly
contracting(i.e., rapid
relativeto thecooling timesof the core,
not rapid 1986).
comparedto the dynamicaltimescalin previous work (Burrows& Lattimer
Additionally,
the

m(r)

m(r)

PPNNSS  mmooddeellss  
•

aaddooppttiinngg  ttwwoo  ddiiffffeerreenntt  pprrooffiilleess  ooff  YYee    aanndd  ss  iinnssiiddee  tthhee  PPNNSS,,  wwee  
ccoonnssttrruucctt  tthhee  PPNNSS  mmooddeellss..  
– uunnkknnoowwnn  ppaarraammeetteerr::  εcc  &&  ssmm  
– ttoo  rreepprroodduuccee  tthhee  PPNNSS  mmooddeellss  wwiitthh  ggiivveenn  ((MM,,  RR)),,  εcc  aanndd  ssmm  aarree  ffiixxeedd..  

4

5

3
4

2
1
0
0

s0
LLSS222200MM1155..00  
200

400

600

800

3
1000

5.4

c

5.2

7

5.0

6

4.8

5

4.6

4

4.4
4.2
0

200

400

600

800

3
1000

t (msec)t (msec)•smc (1014 g/cm3)

c

sm

8

5.6

sm (kB/baryon)

56c (1014 g/cm3)

s0, sm (kB/baryon)

6eevvoolluuttiioonnss  ooff  εcc  aanndd  ssmm  ddeeppeenndd  ssttrroonnggllyy  oonn  tthhee  pprrooffiilleess  ooff  YYee  aanndd  ss..    

1133  MMaayy  22001166  

44tthh  DDTTAA  ssyymmppoossiiuumm@@NNAAOOJJ  

77  

oosscciillllaattiioonnss  iinn  PPNNSS  
••wwiitthh  rreellaattiivviissttiicc  CCoowwlliinngg  aapppprrooxxiimmaattiioonn  
oommiittttiinngg  tthhee  eennttrrooppyy  vvaarriiaattiioonn  
2400

1400

1000

LLSS222200MM1155..00  

2000

fp (Hz)

800

1600

1

ff (Hz)

1200

600
Fig.
TT..TT4..  
Fig.
LL..RR5..  

400
200
0

200

  
•

400

600

800

1000

1200
800
0

TT..TT4..  
Fig.
Fig.
LL..RR5..  
200

400

600

800

1000

t (msec)t (msec)

ffrreeqquueenncciieess  ddeeppeenndd  oonn  mmaassss  aanndd  rraaddiiuuss  ooff  PPNNSS,,  bbuutt  wweeaakkllyy  
ddeeppeenndd  oonn  ((YYee,,  ss))  pprrooffiilleess..  
• iinn  tthhee  eeaarrllyy  ssttaaggee,,  tthhee  ttyyppiiccaall  ffrreeqquueenncciieess  ooff  ff--mmooddee  iiss  ~~  aa  ffeeww  
hhuunnddrreedd  hheerrttzz,,  wwhhiicchh  iiss  ggoooodd  ffoorr  ggrraavviittaattiioonnaall  wwaavvee  ddeetteeccttoorrss..  
1133  MMaayy  22001166  

44tthh  DDTTAA  ssyymmppoossiiuumm@@NNAAOOJJ  

88  

t the dependence on the (Ye , s) distributions seems to be weak. We remark that the frequencies of f -mode osc
ted around a few hundred of hertz at the early stage after bounce, which could be very good signal for groun
nal wave detectors.
ion, since the f -, p1 -, and p2 -modes oscillations are acoustic waves, at least for cold neutron stars, it is kno
ncies of such oscillations can be characterized by the average density of the star, which is defined as (M/R3 )and radius (R) of cold neutron stars [5]. In fact, in Ref. [5], they found that the f -mode frequencies can be excchhaarraacctteerriizzeedd  bbyy  aavveerraaggee  ddeennssiittyy    •ffrreeqquueenncciieess  ooff  ff--mmooddee  ffoorr  ccoolldd  nneeuuttrroonn  ssttaarrss::  
!
"1/2 !
"−3/2
M
R
(NS)
ff
(kHz) ≈ 0.78 + 1.635
,
1.4M⊙
10 km

1

ff (Hz)fp (Hz)AAnnddeerrssssoonn  &&  KKookkkkoottaass  ((11999988))  
lmost independent from the EOS with which the neutron star model
is constructed. So, we check the frequen
SSiimmiillaarrllyy,,  ffrreeqquueenncciieess  ffoorr  PPNNSS  ccaann  bbee  cchhaarraacctteerriizzeedd  bbyy  aavveerraaggee  f the PNSs. In• Fig.
9, for the
PNS model with Mpro = 15M⊙ and LS220, we show the frequencies of f -,ddeennssiittyy,,
  bbuutt  oobbvviioouussllyy  ddiiffffeerreenntt  ffrroomm  tthhoossee  ffoorr  nneeuuttrroonn  ssttaarrss..
as a function of thenormalized
average density, which is defined as (MPNS /1.4M⊙ )1/2   (RPNS /10km)−3/2
1400 of f -, p -, and p -modes are proportional
2400to the average density even for PNSs. We remat the frequencies
1
2models on the
1200late phase after the bounce become massive with small radius as shown in Fig. 2, the PNSs w
Cold NSs
2000panel of Fig. 9, we also plot the f -mode freq
ensity correspond to the late phase. Furthermore, in the left
1000
or cold neutron stars, which are calculated with Eq. (18), with the dotted line. From this panel, one can see
1600for PNSs.
equencies for800
cold neutron stars obviously deviate from thoseo examine the 600
frequencies of f -, p1 -, and p2 -modes for the various progenitor models of PNSs shown in Tabl
1200 with Mpro = 40.0M⊙ for TT..
at we stop the calculations at 940 msec for the
PNSconstructed
LS220
TT4..   EOS, bec
TT..
TT
..
  
Fig.
4
Fig.
400
becomes more than the maximum mass expected
LS220 around 950 msec. Then, the obtained
Fig.
LL..RR5..   frequenci
Fig.
LL..RR5..with
  
200

0.10

0.15

0.20

0.25

0.30

0.35

(MPNS/1.4M)1/2(RPNS/10 km)-3/2
1133  MMaayy  22001166  

800

0.10

0.15

0.20

0.25

0.30

0.35

(MPNS/1.4M)1/2(RPNS/10 km)-3/2

44tthh  DDTTAA  ssyymmppoossiiuumm@@NNAAOOJJ  

99  

1/2(R
-3/2
)
/10
km)
!
PNS

(MPNS/1.4M!)1/2(RPNS/10 km)-3/2(MPNS/1.4M

ddeeppeennddeennccee  oonn  pprrooggeenniittoorr  mmooddeellss  
ious progenitor models, the frequencies of f -, p -, and p -modes are shown as a function o12

1/2  ffoorr  SShheenn  • rreessuullttss  ffoorr  LLSS222200  wwiitthh  MMpprroo//MMby,,  1155,,/1.4M
  2277,,  aanndd  4400,,⊙==1111..
ere the normalized
average density is defined(M22PNS(RPNS /10km)−3/2 . T⊙)
wwiitthh  MMpprroo//MM⊙==1155  the universal relation
shown as Eq. (19).

1400

2300

LS220M11.2
LS220M15.0
LS220M27.0
LS220M40.0
ShenM15.0

1000

1900

1

fp (Hz)ff (Hz)or the various progenitor models are shown in Fig. 10, where the frequencies are cal
1500
the star as in Fig. 6. In this figure, LS220M11.2,
LS220M15.0, LS220M27.0,
and L
LS220M11.2
LS220M15.0 , and 40
ned with the600progenitor models with Mpro =1100
11.2M⊙ , 15.0M⊙ , 27.0M⊙
LS220M27.0
LS220M40.0
ShenM15.0 is the results obtained with the progenitor model with Mpro
= 15.0M⊙ f
ShenM15.0
700the same line as a function of the aerve that the200frequencies
of 0.25
PNSs
almost on
0.10 0.15 0.20
0.30are0.35
0.10 0.15 0.20 0.25 0.30 0.35
1/2
-3/2
1/2we
-3/2universs are almost (Mindependent
fromtheprogenitor
models.
Thus,
can
get
an
/1.4M
)
(R
/10
km)
(M
/1.4M
)
(R
/10
km)
PNS

PNS
PNS

PNS
e PNSs and the average density of PNSs, such as•

pprrooggeenniittoorr  mmooddeell  ddeeppeennddeennccee  iiss  qquuiittee  wweeaakk..  
(PNS)fi(Hz) ≈ c0i + c1i

1133  MMaayy  22001166  

!

MPNS
1.4M⊙

"1/2 !

44tthh  DDTTAA  ssyymmppoossiiuumm@@NNAAOOJJ  

RPNS
10 km

"−3/2

,
1100  

p1 , and p2 for f -, p1 , and p2 -modes, and c0i and c1i are some constants irrespective oneutrino mean energy [f−29.483690 has also been reported [32, 33], which is associatedon and the standing accretion-shock
instability
10
ons around (and above) the psurfaceof PNSs.
343.9
5352In fact,
10 the frequencies can be well-expressed by using
1
With respect to the characteristic gravitational waves radiating after bounce of core-collapse
Ss asp2
640.8 7435
20
signalthe1/2convection
and the standing
20! due to"!
"2 accretion-shock instability has also been reported
s15s7b2
1 GMPNS with1.1m
GM
z9.6
n
PNS
the
g-modeoscillations
around(and above)fg ≈1−, the surface of PNSs. In fact, the frequencies(20)ca2
2• aass  cchhaarraacctteerriissttiicc  GGWWss  ffrroomm  ccoorree--ccoollllaappssee  ssuuppeerrnnoovvaa,,
  tthhee  eexxcciittaattiioonn  
15
2π RPNS⟨Eν̄e ⟩c RPNS15
theradius
and
mass
of
PNSs
as
acteristic gravitational
waves radiating after bounce of core-collapse supernovae, the evidence ofooff  gg--mmooddeess  aarroouunndd  PPNNSS  hhaass  bbeeeenn  rreeppoorrtteedd  ((MMuueellllaarr  eett  aall..  ((22001133));;  
noteandstanding
accretion-shock
instability
has also
been reported
[32,
33],
which
is associatedthetheneutronmass
and the mean  ((22001133))))  
energyof electron
antineutrinos
[32].
That
is,
the frequencies!
"!
"2 esCCeerrddaa--DDuurraann  eett  aall..
1/2
10
2
3
10 thefrequencies
around(and above)the surfaceof PNSs.
In ffact,
frequencies
can1.1mbe nwell-expressedby using
1 GMPNS
GM/R
different
from the-mode
dependingon (MPNS)1/2
PNSPNS
Ss/R
PNS , which is–completely
dduuee  ttoo  tthhee  ccoonnvveeccttiioonn  aanndd  tthhee  ssttaannddiinngg    fg ≈1 −supernovae,, one
srefullyasobserving thefrequencies of gravitationalradiating
from
the
PNSs
in
2
2
aaccccrreettiioonn--sshhoocckk  iinnssttaabbiilliittyy..
   waves
2π RPNS ⟨Eν̄e ⟩c RPNS
20
20
mine the mass and radius of !
PNSs via"Eqs.
(19)
and
(20).
For
example,one
might
observethe time!
"2s25
1/2s11.2
1 the
GMPNS
GM
wave spectra from
Mpron = 15M⊙ and
LS220,
PNS for1.1m
PNS as shown in Fig. 11. We remark that, tofg ≈1−, and the mean energy of electron antineutrinos
(20)
where
m
and
⟨E
⟩
denotetheneutron
mass
[32
15
n
ν̄
2 adopt2
e
15
uencies with Eq. 2π(20),Rwethe⟨E⟩
distribution
given
by
ν̄
⟨E
⟩
c
R
e
ν̄e
PNS
2 PNS
sentially
dependon
M/R
PNS PNS , which is completely different from the f-mode frequencies dep#
MMuueellllaarr  eett  aall..  ((22001133))  
MMuueellllaarr  eett  aall..  ((22001133))  
mmnn::  nneeuuttrroonn  mmaassss  
10
e the neutron massand themean
antineutrinos
[32].
That
is, the waves
frequencies
es3t/400+energy
13above.
(0 of≤telectron
≤
400
msec)
as
shown
Thus,
carefully
observingthefrequenciesof gravitationalradiating(21)
from
10
⟨E
⟩
=
,
ν̄e   mmeeaann  eenneerrggyy  ooff  eelleeccttrroonn  aannttiinneeuuttrriinnooss  
2
3
1/2
RPNS
, which is            ::completely
from
the
f
-mode
frequencies
depending
on
(M
/R
)
PNS
16 different
(400
msec
≤
t)
PNS
might be possible to determine the mass and radius of PNSs via Eqs. (19) and (20). For exampl
neutrino mean energy [MeV]lluummiinnoossiittyy  ((11005522  eerrgg//ss))    

ccoommppaarriissoonn  wwiitthh  gg--mmooddeess  

20 bbllaacckk::
efully observing
the frequencies
ofgravitational
waves
radiating
from
the
PNSs
in andsupernovae,
onei
20
  eelleeccttrroonn  nneeuuttrriinnooss  MMuueellllaarr  &&  JJaannkkaa  ((22001144))  
evolution of gravitational
wave
spectra
from
the
PNS
for
M
=
15M
LS220,
as
shown
pro
s27
rreedd::  eelleeccttrroonn  aannttiinneeuuttrriinnooss  
mine the mass and radius of PNSs via Eqs.s15s7b2(19) and (20).
For example, one
might⊙ observe the time
frequencies
withEq.  μ//τ  nneeuuttrriinnooss  
we adopt⟨Eν̄e11.
⟩ distribution
given bythat, tobblluuee::
ave spectra from the PNS forcalculate
Mprothe=g-mode
15M⊙
and15LS220,
as(20),
shownin theFig.
We remarkncies with Eq. (20), we adopt the ⟨Eν̄e ⟩ distribution given by #3t/400 + 13 (0 ≤ t ≤ 400 msec)#
10,
3t/400 + 13 (0 ≤ t ≤ 400 msec)⟨Eν̄e ⟩ = 16(400 msec ≤ t)
⟨E0ν̄e ⟩ = 0.2
,
(21)
.6
0.8
0.4
0.6
0.8
20
16
(400
msec ≤ t)
1133  MMaayy  22001166  
44tthh  DDTTAA  ssyymmppoossiiuumm@@NNAAOOJJ  
1111  
time after bounce [s]ttiimmee  ((ss))    

s25umn), and mean energies, ⟨E⟩ (right column), of the emitted neutrinos for models
15 u8.1, z9.6, s11.2,

ccoommppaarriissoonn  wwiitthh  gg--mmooddeess  •

ccaarreeffuull  oobbsseerrvviinngg  tthhee  ggrraavviittaattiioonnaall  wwaavvee  ssppeeccttrraa  aafftteerr  ccoorree-ccoollllaappssee  ssuuppeerrnnoovvaa,,  oonnee  mmiigghhtt  sseeee  tthhee  ddiiffffeerreenntt  sseeqquueenncceess  iinn  
ssppeeccttrraa    – wwhhiicchh  tteellllss  uuss  tthhee  rraaddiiuuss  aanndd  mmaassss  ooff  PPNNSS  

f (Hz)

1500

1000

500
f modeg mode
0
0

200

400

600

800

1000

t (msec)
1133  MMaayy  22001166  

44tthh  DDTTAA  ssyymmppoossiiuumm@@NNAAOOJJ  

1122  

various progenitor
modelsshown
10,various
where
the frequencies
are calcu
E the
III: Coefficientsin the universal
relationareshown
as Eq. in(19)Fig.
for theprogenitor
models of PNSs.

he star as in Fig. 6. In this figure,modes LS220M11.2,
c0i (Hz) c1i (Hz) LS220M15.0, LS220M27.0, and LS2
d with the progenitor modelsf with
Mpro3690= 11.2M⊙ , 15.0M⊙ , 27.0M⊙ , and 40.0
−29.48henM15.0 is• theresults obtained
the progenitor
model with Mpro = 15.0M⊙ forWWee  eexxaammiinnee  tthhee  ffrreeqquueenncciieess  ooff  ggrraavviittaattiioonnaall  wwaavveess  rraaddiiaattiinngg  ffrroomm  
p1 with
343.9
5352
PPNNSS  aafftteerr  bboouunnccee..
ve that the frequencies
of PNSs
almoston the same line as a function of the avep  2 are640.8
7435
TThhee  PPNNSS  mmooddeellss  aarree  ccoonnssttrruucctteedd  iinn  ssuucchh  aa  wwaayy  tthhaatt  tthhee  mmaassss  aanndd  
are almost •independent
from the progenitor models. Thus, we can get an universal
  
PNSs and therraaddiiuuss  oobbttaaiinneedd  ffrroomm  11DD  ssiimmuullaattiioonn  aarree  rreeccoonnssttrruucctteedd..
average density of PNSs, such as

ccoonncclluussiioonn  

ttwwoo  ddiiffffeerreenntt  pprrooffiilleess  ooff  
YYee  aanndd  ss  aarree  ccoonnssiiddeerreedd  
he characteristic –gravitational
waves radiating
after
bounce of core-collapse supernovae, the evid
!
"−3/2
1/2 !
nvection and• theffrreeqquueenncciieess  ooff  ggrraavviittaattiioonnaall  wwaavveess  aarree  aallmmoosstt  iinnddeeppeennddeenntt  ffrroomm  
standing accretion-shock instability
has"also
been reported[32, 33], which is ass
M
R
PNS
(PNS)
0 ss,,  of
1PNSs. PNS
YY≈  aanndd  
bbuutt  sseennssiittiivvee  ttoo  tthhee  mmaassss  aanndd  rraaddiiuuss..
   , well-expressed bcillations aroundtthhee  pprrooffiilleess  ooff  (andthesurface
In
fact,
the
frequencies
can befi above)
(Hz)
c
+
c
ee
i
i
1.4M⊙
10 km
of PNSs as– cchhaarraacctteerriizzeedd  bbyy  aavveerraaggee  ddeennssiittyy  
– ddiiffffeerreenntt  ffrroomm  tthhaatt  eexxppeecctteedd  ffoorr  ccoolldd  nneeuuttrroonn  ssttaarrss  
!
"1/2 !
"2
0
1
1 GM -modes,
1.1m
n c and c GM-,fgp≈
andsome, constants irrespective of
1 , and p2 for
1 , and p2PNS• fpprrooggeenniittoorr  ddeeppeennddeennccee  iiss  qquuiittee  wweeaakk..
  i arePNS
i
1
−
2
2π
R
c2 R
ν̄e ⟩ III and the
PNSients in this relation
are shown
Table
universal
relations obtained here a
PNS in⟨E– ffiinndd  aann  uunniivveerrssaall  rreellaattiioonn  ffoorr  ffrreeqquueenncciieess  ooff  
ff--  aanndd  pp--mmooddeess  aass  aa  ne.
Note thethatneutron
one ffuunnccttiioonn  ooff  aavveerraaggee  ddeennssiittyy  
canseeandthethedeviationof thefrequencies
from therelation[Eq.
(19)] i⟩ denotemass
mean
energyofelectronantineutrinos
[32].
That
is,
the
frequen– ddiiffffeerreenntt  ddeeppeennddeennccee  ffoorr  gg--mmooddee  aarroouunndd  PPNNSS    
2 an effect
3
may
be
of
the
mass
accretion
from
the
outer
region
of
PNS.
M
/R
,
which
is
completely
different
from
the
f
-mode
frequencies
depending
on
(M
/R
PNS
PNS
PNS
P

oonnee  mmiigghhtt  bbee  ppoossssiibbllee  ttoo  ddeetteerrmmiinnee  tthhee  mmaassss  aanndd  rraaddiiuuss  ooff  PPNNSS  
hus, carefully• observing
the frequencies of gravitational waves radiating from the PNSs in supernov
vviiaa  ccaarreeffuull  oobbsseerrvvaattiioonnss  ooff  ttiimmee  eevvoolluuttiioonn  ooff  ggrraavviittaattiioonnaall  wwaavvee  
o determine the mass
and radius of PNSs via Eqs. (19) and (20). For example, one might observe t
ssppeeccttrraa..  
tionalwave spectra
the PNS relation
for Mpro= 15M
and(19)LS220,as shown
in progenitor
Fig. 11.
Wemodels
remarko
III: Coefficientsin from
the universal
shown
as⊙Eq.
for thevarious1133  MMaayy  22001166  
44tthh  DDTTAA  ssyymmppoossiiuumm@@NNAAOOJJ  
1133  
e frequencies with Eq. (20), we adopt the ⟨Eν̄e ⟩ distribution given bymodes c0i (Hz) c1i (Hz)#