状態方程式がニュートリノ駆動型爆発に及ぼす影響o)nationsctralg

On the Importance of the Equation of State for the NeutrinoDriven Supernova Explosion Mechanism

Yudai SUWA / 諏訪 雄大(YITP, Kyoto University)

pt

ld a
s
ock

Collaboration with
T. Takiwaki, K. Kotake (NAOJ)、M. Liebendörfer (U. Basel)、T. Fischer (GSI)、K. Sato (NINS)[Vol. 62,
2011年12月4日日曜日

From progenitor to supernova
Final phase ofstellar evolution

Fe
Si
O,Ne,Mg
C+O
HeH

neutrinosphere formation(neutrino trapping)neutron star formation(core bounce)neutrinosphereneutronstar

ρc~1011g cm-3

ρc~1014g cm-3

ρc~109g cm-3

shock stallshock revival

SUPERNOVA!

FeSi
O,Ne,Mg
C+O
HeH
2011/12/4
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2 /14

From progenitor to supernova
Final phase ofstellar evolution

Fe
Si
O,Ne,Mg
C+O
HeH

neutrinosphere formation(neutrino trapping)neutron star formation(core bounce)neutrinosphereneutronstar

ρc~1011g cm-3

ρc~1014g cm-3

shock front

ρc~109g cm-3
gain radius

shock stallshock revival

SUPERNOVA!

PNS

Fe
Si
O,Ne,Mg
C+O
HeH
2011/12/4
2011年12月4日日曜日

素核宇融合による計算基礎物理学の進展@合歓の郷

2 /14

temperature can e.g. be found in Nadyozhin and Yudin
0+ξfRMF(T, nn /ξ, np /ξ) ,(2005); Potekhin et al. (2009); Potekhin and Chabrier(2010). However, we leave this for future studies as the
The first term in Eq. (6) is the summed ideal g
Coulomb energy becomes only important at low temperpression of the nuclei. The Coulomb free energy
atures so that the simplest expression is sufficient for our
nuclei appears in addition. The second line in Eq
purposes.
the direct contribution from the excluded volum(1985)
temperature leads to the population of Hillebrandt
excitedcause&of Wolﬀ
this term,
as long as nuclei are present, tLattimer states&FiniteSwesty(LS)(1991)of the nuclei. Here we use the temperature depenenergy density goes to infinity when approachingdent degeneracy function of Fái and Randrup (1982).Hartree-FockItration density,
because the free volume of nuclei gcalculation
based oncompressible
drop
model
is thesame analyticliquid
expression
as inthe original reference
zero, κ → 0. Thus, nuclei will always disappearof the HS model (Hempel and Schaffner-Bielich 2010),
saturation density is reached. The RMF contribu
G.Shenetal. (2010,2011)
0
nowK=180,
we consider
energies below thevariantsbut
with
220,only
andexcitation
375 MeV
the nucleons fRMF
is weighted with their filling fa
binding energy of the corresponding nucleus, in order toas the free energy is an extensive quantity. If nuc
representthat the2011)
excited states still have to be boundrelativistic
mean
(NL3,
FSUGold)
absent,ξ = ﬁeld
1, andtheory
we get theunmodified
RMF dH.Shen etal.
(1998,(see, e.g., Röpke (1984)). We note that the inclusion oftion, as it should be. The excluded volume correct
excited states up to infinite energies had only aHempel
minor
the al.
nucleirepresents a hard-core repulsion of theet(2011)
relativistic
meanonﬁeld
theory (TM1)
influencethe composition
but would lead to an unat large densities close to saturation density. Instephysically large contribution of the excited states to themodification of the free energy of the unbound nu
relativisticmean
ﬁeld theory
(TM1,
TMA,
energydensitycomponent
and entropy (~2011)at very large temperatures.
including
hyperonis purely
geometric
and just
describes
that the nu
We describe nuclear matter as a chemical mixtureFSUGold)offill only a fraction of the total volume. In this sen
the different nuclear species and nucleons. As we distintwo aforementioned model assumptions for the ex
guish
between nuclei andthe surroundingslope
interacting
nuvolume are essential, as they lead to the desired li
incompressibility
symmetry energy
of symmetry energy
cleons we still have to specify how the system is composed
behavior of the EOS.
K
[MeV]
J
(S)
[MeV]
L
[MeV]
of the different particles. Our thermodynamic model is
2.4. EOS characteristics & constraints
built on two main assumptions: First, we assume for unboundthat they are
not allowed to be situatedLS
180, nucleons
220, 375
29.3--Table 2 lists some characteristic saturation pro
inside of nuclei, whereas nuclei are described as uniformof uniform bulk nuclear matter for the three di0
hard spheres at saturation density nB . Second, for nuRMF parameterizations. We also include the L
mass number A ≥36.9
2) we assume that they 111
mustwith the compressibility of K = 180 MeV in the
HShen clei (with281
not overlap with any other baryon in the system (nuclei
The quantities shown in Table 2 correspond to t
or unbound nucleons). Thus we take the volume whichefficients of the following power-series expansion
is available
to be the part of the totalbinding energy per baryon at T = 0 around the s
HW
263for the nucleons
32.9--volume of the system which is not excluded by nuclei.
tion point:
This is described by the filling factor of the nucleons271.5 (NL3)
37.29 (NL3)
118.2 (NL3)
1
1
2
!
GShen
E(x,
β)
=
−E
+
Kx
+
K ! x3 + ...
00
162
230.0 (FSU)ξ = 1 −32.59
60.5 (FSU)
A (FSU)nA,Z /nB ,
(3)
" 18
#
1
A,Z
318 (TMA)
30.7
(TMA)
90 (TMA)+β 2 J + Lx + ... + ... ,Hempel3
230
32.6we(FSU)
60 (FSU)(here
and(FSU)in the following,mean A ≥ 2). The
freevolume in which a nucleus can move is the total volume
with x = nB /n0B − 1 denoting the relative deviatio
2011/12/4
3 param/14
素核宇融合による計算基礎物理学の進展@合歓の郷
minus the volume filled
by nuclei and nucleons. This is
the saturation density, and the asymmetry

Finite temperature EOSs

2011年12月4日日曜日

Equation of state

O’Connor & Ott 10

Imcompressibility [MeV]The “standard” equation of states (EOSs) in supernova community
400
・Lattimer & Swesty EOS (liquid drop)
LS375
350
・Shen EOS (relativistic mean ﬁeld)
300

Shen

250

LS220

200
150

LS180
20

25

30

35

40

Symmetry Energy [MeV]

GR

2.5

P

<∞

AP3

C

2.0
Mass (M()

MS2

J1614-2230

SQM3
FSU
J1903+0327

SQM1

1.5

PAL1

ENG

ity AP4
sal
u
a

MS0

MPA1

GM3

PAL6

J1909-3744

MS1

GS1

Double neutron stars sysystems

1.0

tion

0.5

0.0

2011/12/4
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Rota

Demorest+ 107

素核宇融合による計算基礎物理学の進展@合歓の郷

8

9

10

11
12
Radius (km)

13

14

15

4 /14

able that the luminosities and average energies increase bya factor of 2 or more toward the formation of black hole,Studies on EOS dependence

4

17

200

10

Thompson+ 03
LS180 & LS375

Sumiyoshi+ 06
LS & Shen

Luminosity [1053 erg/s]

16

T. Fischer et al.: The neutrino signal from proto

10

15

10

14

10

ρb [g/cm3]
at center(a)Fischer+ 09
LS & Shen

6

5
150
4
3
1002
1
0
50

0

0.2

0.4 0.6 0.8
1Time After Bounce [s]T [MeV] (b)at Mb=0.6Msolar

40
0
35
10
0.0
0.5
1.0
1.5
0.0
0.5
1.0
1.5
30
t pb [sec]t pb [sec]25per) and luminositiesNew(lower)of of state in core-collapse supernova simulations 20
equations21
15
and !"=# (dot-dashed line) as a
FIG. 3. Baryon
mass
densityatcenter(left) and temperature at
15.5
10
dels 15LS (thin line) and SH (thickmodels LS (thinoff-center (right) as(a)a function of time
(tpb ) inSTOS
5
LS
15
0
0.2 0.4 0.6 0.81line) and SH (thickline).
HSTime(TMA)
After Bounce [s]1.4

rms Energy [MeV]

1.2

14
13
12

091101-3
Hempel+11
RMF

11

Baryon Density, log (! [g/cm3])
10

There are several works, which investigated theEOS dependence with 1D simulation

3

Central Baryon Density, log10(!c [g/cm ])13

e Neutrinoe Antineutrinoµ/τ Neutrinos

1.2

1.4

Since
1D simulations fail toHSproduce
explosion,(FSUGold)
14.5
Fig. 2. Luminosities and mean energies during the post bounce phase of
the14representablephysicalquantities
in thesea core collapse
simulationof a 40 M ! progenitor
model from Woosley& Weaver (1995). Comparing eos1 (thick lines) and eos2 (thin lines).
studiesare
13.5
thetimelow density and low temperature regime is coupled to theBH
13 formation

1
EoS
for1.5hot and2 dense2.5
nuclear 3matter above temperatures of
T = 0.5 MeV, where nuclei are assumed to be in nuclear sta10
neutrino
luminosity/spectrum
evolution
0.4
tistical equilibrium (NSE). In addition, there are free nucleons
and light nuclei. The transition from NSE to bulk nuclear matter9
0.3(free nucleons only) above the neutron drip line is handled via−0.2
0
0.2 0.4 0.6 0.8
1
1.2Time After Bounce [s]the EoSs intrinsically.
0.2
In the following paragraphs, we compare the soft EoS from
Fig.2011/12/4
16.— Evolution of the central density for thecollapse of
5 /14
素核宇融合による計算基礎物理学の進展@合歓の郷
e 40 M! progenitor model from Woosley and Weaver (1995),
Lattimer & Swesty (1991) (eos1) with the compressibility of
0.1

2011年12月4日日曜日
mparing
the different EOS under investigation.

(b)

0.5

e

0.5

Electron Fraction, Y

STOSLS
HS (TMA)HS (FSUGold)How about the explosion?

Neutrino-driven explosion
Recently, we have successful exploding models driven by neutrino heating
800

Buras+ 06

600

Marek & Janka 09
Physical time: t=700 ms

40

30

200
0

20

-200
-400

10

-600
-800
-800 -600 -400 -200 0f0
200 400 600 800

r [km]

Suwa+ 10, 11

2011/12/4
2011年12月4日日曜日

s[kB/ baryon]

r [km]kB

400

All of these simulations
employ LS180, which,however, cannot support a
2M⦿ NS. How about stiﬀer
EOS?
The Astrophysical Journal, 730:70 (20pp), 2011 April 1

Figure 1. Baryonic (left) and gravitational (right) neutron mass–radius relations
for various hot nuclear EOS. The temperature is taken to be constant throughout
the star at T = 0.1 MeV and the electron fraction is determinedthrough6/14
素核宇融合による計算基礎物理学の進展@合歓の郷
neutrinoless β-equilibrium with an imposed minimum of 0.05 due to table
constraints.

EOS and shock evolution
from H.-Th. Janka’s presentation!"#$%&'()*(+)#(,#--.!#/)0+)123#4##56)1#(,#$(7#8+,'06+96!

!

2011/12/4
2011年12月4日日曜日!"#$%&'"()*+,(-+6'7088%$%13'19:;%#$'<4=>?'@#330A%$'('=B%>3C'!@(=5,'*0;;%D$#173'('E4;88'!*(E5,'=F%1'%3'#;G.!(,'/-0+123!4+5(!+&)6+'7$*+#(-/+8(#9&8'+:;!+%/&6*+'(+/&-%"/-+/<9%(*"()++!"#$%&'(')*+,'-../,'01'2$%2#$#30415
素核宇融合による計算基礎物理学の進展@合歓の郷

=>?@A

7 /14

EOS and shock evolution
from H.-Th. Janka’s presentation!"#$%&'()*(+)#(,#--.!#/)0+)123#4##56)1#(,#$(7#8+,'06+96!

!

2011/12/4
2011年12月4日日曜日!"#$%&'"()*+,(-+6'7088%$%13'19:;%#$'<4=>?'@#330A%$'('=B%>3C'!@(=5,'*0;;%D$#173'('E4;88'!*(E5,'=F%1'%3'#;G.!(,'/-0+123!4+5(!+&)6+'7$*+#(-/+8(#9&8'+:;!+%/&6*+'(+/&-%"/-+/<9%(*"()++

what is “soft”?
what is “stiﬀ”?

!"#$%&'(')*+,'-../,'01'2$%2#$#30415
素核宇融合による計算基礎物理学の進展@合歓の郷

=>?@A

7 /14

Numerical simulation
400

Axisymmetric simulation (ZEUS-2D; Stone & Norman 92)

Imcompressibility [MeV]

EOS: LS180, (LS220,) LS375, and Shen

LS375

Hydrodynamics + Neutrino transfer

350
300

Shen

250

LS220

200
150

LS180
20

25

30

35

40

Symmetry Energy [MeV]

� �
��
� �
�
�
d ln ρ 3v
df∂f
d ln ρ 3v
∂f
v
∂f
+µ
+ µ
+
(1 − µ2 )
+ µ2
+
−
D
cdt
∂r
cdt
cr
∂µ
cdt
cr
cr
∂E
�
�
�
�
E2
= j(1 − f ) − χf +(1 − f ) Rf � dµ� − f R(1 − f � )dµ�3c(hc)(Lindquist 1966; Castor 1972; Mezzacappa & Bruenn 1993)

Isotropic Diﬀusion Source Approximation (Liebendörfer+ 09)
electron-type neutrino/antineutrino

progenitor: 15 M⦿ (Woosley & Weaver 95)
2011/12/4
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研究の立ち位置
空間次元
３次元

Blondin+, 07

Iwakami+, 08

Mikami+, 08

Scheidegger+, 08
Nordhaus+, 10

Burrows+, 06

２次元 (軸対称)

Yamada & Sato, 94

Kotake+, 03

Blondin & Mezzacappa, 03

Ohnishi+, 06

Obergaulinger+, 05

Murphy+, 08

Buras+, 06

Takiwaki+, 09

Rampp & Janka, 00
Liebendörfer+, 01

１次元 (球対称)

Thompson+, 03
Sumiyoshi+, 05

なし(断熱)

冷却のみ
or
手で加熱

輻射輸送

ニュートリノ2011/12/4
2011年12月4日日曜日

素核宇融合による計算基礎物理学の進展@合歓の郷

9 /14

研究の立ち位置
空間次元
３次元

Blondin+, 07

Iwakami+, 08

Mikami+, 08

Scheidegger+, 08
Nordhaus+, 10

Burrows+, 06

２次元 (軸対称)

Yamada & Sato, 94

Kotake+, 03

Buras+, 06

Blondin & Mezzacappa, 03

Ohnishi+, 06

Obergaulinger+, 05

Murphy+, 08

Suwa+, 10

Takiwaki+, 09

Rampp & Janka, 00
Liebendörfer+, 01

１次元 (球対称)

Thompson+, 03
Sumiyoshi+, 05

なし(断熱)

冷却のみ
or
手で加熱

輻射輸送

ニュートリノ2011/12/4
2011年12月4日日曜日

素核宇融合による計算基礎物理学の進展@合歓の郷

9 /14

研究の立ち位置
この領域だけがニュートリノ

空間次元

駆動爆発の正否を議論できる

３次元

Blondin+, 07

Iwakami+, 08

Mikami+, 08

Scheidegger+, 08
Nordhaus+, 10

Burrows+, 06

２次元 (軸対称)

Yamada & Sato, 94

Kotake+, 03

Buras+, 06

Blondin & Mezzacappa, 03

Ohnishi+, 06

Obergaulinger+, 05

Murphy+, 08

Suwa+, 10

Takiwaki+, 09

Rampp & Janka, 00
Liebendörfer+, 01

１次元 (球対称)

Thompson+, 03
Sumiyoshi+, 05

なし(断熱)

冷却のみ
or
手で加熱

輻射輸送

ニュートリノ2011/12/4
2011年12月4日日曜日

素核宇融合による計算基礎物理学の進展@合歓の郷

9 /14

研究の立ち位置
この領域だけがニュートリノ

空間次元

駆動爆発の正否を議論できる

３次元

Blondin+, 07

Iwakami+, 08

Mikami+, 08

Scheidegger+, 08
Nordhaus+, 10

Burrows+, 06

２次元 (軸対称)

Yamada & Sato, 94

Kotake+, 03

Buras+, 06

Blondin & Mezzacappa, 03

Ohnishi+, 06

Obergaulinger+, 05

Murphy+, 08

Suwa+, 10

Takiwaki+, 09

Rampp & Janka, 00
Liebendörfer+, 01

１次元 (球対称)

No explosion

Thompson+, 03
Sumiyoshi+, 05

なし(断熱)

冷却のみ
or
手で加熱

輻射輸送

ニュートリノ2011/12/4
2011年12月4日日曜日

素核宇融合による計算基礎物理学の進展@合歓の郷

9 /14

研究の立ち位置
この領域だけがニュートリノ

空間次元

駆動爆発の正否を議論できる

３次元

Blondin+, 07

Iwakami+, 08

Mikami+, 08

Scheidegger+, 08
Nordhaus+, 10

Burrows+, 06

２次元 (軸対称)

Yamada & Sato, 94

Kotake+, 03

Buras+, 06

Blondin & Mezzacappa, 03

Ohnishi+, 06

Obergaulinger+, 05

Murphy+, 08

Suwa+, 10

Explosion!

Takiwaki+, 09

Rampp & Janka, 00
Liebendörfer+, 01

１次元 (球対称)

No explosion

Thompson+, 03
Sumiyoshi+, 05

なし(断熱)

冷却のみ
or
手で加熱

輻射輸送

ニュートリノ2011/12/4
2011年12月4日日曜日

素核宇融合による計算基礎物理学の進展@合歓の郷

9 /14

研究の立ち位置
この領域だけがニュートリノ

空間次元

駆動爆発の正否を議論できる

３次元

Blondin+, 07

Iwakami+, 08

Mikami+, 08

Scheidegger+, 08

Takiwaki, Kotake, & YS (2011)

Nordhaus+, 10

Burrows+, 06

２次元 (軸対称)

Yamada & Sato, 94

Kotake+, 03

Buras+, 06

Blondin & Mezzacappa, 03

Ohnishi+, 06

Obergaulinger+, 05

Murphy+, 08

Suwa+, 10

Explosion!

Takiwaki+, 09

Rampp & Janka, 00
Liebendörfer+, 01

１次元 (球対称)

No explosion

Thompson+, 03
Sumiyoshi+, 05

なし(断熱)

冷却のみ
or
手で加熱

輻射輸送

ニュートリノ2011/12/4
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9 /14

Entropy evolution
LS180

2011/12/4
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Shen

素核宇融合による計算基礎物理学の進展@合歓の郷

10/14

Shock radiusmaximum
averageminimum

LS180 and LS375 succeed the explosion
Shen EOS fails

2011/12/4
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11/14

! " #%1/2$2
1 π
2 0 M(r, θ) − M(r) sin θdθM(r)Dispersionofthemomentwhere

M(r, θ) ≡ ρ(r, θ)vr2 (r, θ) + P (r, θ),&1 πM(r) ≡
M(r, θ) sin θdθ.
2 0

,

(8)

(9)
(10)

M corresponds to total pressure including the ram presLS180
sure of infalling material and should be conserved beyond
and behind the shock at the shock frame (i.e., the momentum part of Rankine-Hugoniot equation). Therefore,
M is a useful indicator for the shock expansion, that is,
if M is larger behind the shock than beyond the shock,the shock wave propagates outward, and vice versa. In
fact, the dispersion of M is significantly small (∼ 0.1)
irrespective to the radius before 100 ms after the bounce(especially for Shen; see Figure 15). After that the pressure perturbation increases in the shocked region and theshock wave pushed outward. As for LS180, the pressure
perturbation continues to glow and from ∼ 200-300 ms
after the bounce a strong pressure wave gives momentum to the shock, which is absent for Shen. As a result,the pressure behind the shock wave overwhelms the rampressure beyond the shock so that the shock begins topropagate outward and never recedes. In addition, it isobvious that the pressure wave is produced around the

In Figure 18, we show the time evolution of the N
dius. One can see that Shen has smaller radius thEOSs at the early phase. At ∼ 200 ms after the bothat of LS180 gets smaller than Shen, but LS37
remains larger than Shen. The explosions of LS18
LS375 are initiated at ∼ 300 ms after the bounce,
the radius of NS of Shen locates between those ofand LS375. Thus, the radius of NS, which is often
as a guidepost of the energy budget available fromcollapse, is not a good indicator for the successf
plosion, andthe same is equally true of the compa
Shen(GMNS /c2 RNS ).
In Figure 19, we show the prompt convection ac
The top panel present the tim evolution" 2 of the kine
ergy of the lateral (θ) direction, ρvθ /2 dV . LS18
LS375 show the larger amplitude than Shen for tpb
ms, which is triggered by the "prompt convection.
bottom panel, which shows ( vθ2 d cos θ)1/2 , the cotive motion is apparent for LS180 (top half in thisfor 30 km! r ! 100 km at that time, which is wethe case of Shen (bottom half). After the promp
vection, both LS180 and LS375 still have larger kenergy than Shen.
In Figure 20, we show the ante-sonic condition
posed by Pejcha & Thompson (2011), which ind
2that if max(c2s /vesc) " 0.2,11 where cs is the sound11 This critical value depends on the microphysics.

cf.
2011/12/4
2011年12月4日日曜日

素核宇融合による計算基礎物理学の進展@合歓の郷

12/14

Time after Bounce [ms]Dispersion of the moment

early phase (tpb !100 ms after the bounce) LS
Fig. 14.—largeramplitude than Shen. After that the am
are similar for these EOSs for 100 ! tpb ! 300 ms
early phase (tpb !100 ms after the bounce) LS
LS180, the shock wave begins to expand for tpb ∼larger amplitude than Shen. After that the am
soare
that
the amplitudeis decreasing
similarfor these EOSs
for 100 !after
tpb !that
300 tim
ms
Shen
shows
the
increasing
amplitude
becausetheLS180, the shock wave begins to expand for tpb ∼only
oscillating
withoutisexpansion.
Finally,so thatthe amplitudedecreasing after
thatLS18
tim
the
shock
expansion
with thedominanceof the
!=
Shen
shows
the increasing
amplitude
because(redline).
On thewithout
other while,
Shen Finally,shows thede
only
oscillating
expansion.
LS18
amplitude
tpb " 500
the morpholog
the shockfor
expansion
withmstheanddominanceof ! =
shockto thespherical,
bydecw(red waveline). isOnapproaching
the other while,Shenshows thecaneasily see
It shamplitude
forthetpb failure" 500 ofmsthe
andexplosion.
the morphology
notedmode analysis
by theshock byradiu
shockthatwavetheis approaching
to thespherical,
wha can
properindicator
the explosion.
easilysee the of
failureof the explosion. It sh
Next, that
we discusstheanalysis
EOS dependence
SAS
noted
the modeby the shockofradiu
proper indicator of the
explosion.
itya (acoustic-vorticity
cycle).
We show the devi
Next,forwe LS180
discussand
the Shen
EOS dependenceof whicSASpressurein Figure 15,
ity (acoustic-vorticity
termined
as follows: cycle). We show the devipressure!for" LS180
and Shen in $Figure 15,
whic
%1/2
#
2
π
termined 1as follows:
2 0 M(r, θ) − M(r) sin θdθ! " #%1/2 ,$
2π
1
M(r)2 0 M(r, θ) − M(r) sin θdθ
,
where
M(r)

M(r, θ) ≡ ρ(r, θ)vr2 (r, θ) + P (r, θ),& π
1≡ ρ(r, θ)v 2 (r, θ) + P (r, θ),M(r,θ)
r θ) sin θdθ.
M(r) ≡
M(r,&21 0 πM(r) ≡
M(r, θ) sin θdθ.
2 0pressure including13theM corresponds to total/14ra
素核宇融合による計算基礎物理学の進展@合歓の郷
sureinfalling material
shouldbe conserved
M ofcorresponds
to total andpressure
including
the rawhere

2011/12/4
2011年12月4日日曜日

Summary and discussion
We perform axisymmetric simulations of a core-400

collapse supernova driven by the neutrino

350

equation of state
Lattimer & Swesty EOS: explosion

300

Shen

250

LS220

200
150

Shen EOS: failure

fail
LS375

Imcompressibility [MeV]

heating and investigate the dependence on the

explosion

LS180
20

25

30

35

40

Symmetry Energy [MeV]The symmetry energy would have greater impact than the incompressibility
The diﬀerence of the incompressibility does not aﬀect the dynamics very muchat least with the current setup
In order to make the complete understanding of EOS impacts, a more
systematic study is strongly required!

2011/12/4
2011年12月4日日曜日

素核宇融合による計算基礎物理学の進展@合歓の郷

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