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Variational study of nuclear equation of state
for core-collapse supernovae
and hyperonic neutron stars
H. Togashi (RIKEN)
Outline
1：Introduction
2：Variational EOS for core-collapse supernovae
3：Variational EOS for hyperonic neutron stars
4：Summary
International Workshop on Neutrino Physics and Astrophysics
@ Mimar Sinan Fine Arts University, Istanbul, Turkey, March 16, 2015
1. Introduction
Nuclear Equation of State (EOS) plays important roles
for astrophysical studies.
(Neutron stars, core-collapse supernovae (SNe), black hole formations)
Nuclear EOS available for SN simulation
Thermodynamic quantities in a wide range of ρ, T, Yp
SN matter contains uniform and non-uniform nuclear matter.
1. Lattimer-Swesty EOS : The Skyrme-type interaction (NPA 535 (1991) 331)
2. Shen EOS : The Relativistic Mean Field Theory (NPA 637 (1998) 435)
Those EOSs are based on phenomenological models for uniform matter.
There is no nuclear EOS based on the microscopic many-body theory.
We construct a new SN-EOS with the variational method
starting from the realistic nuclear forces.
EOS with Variational Method
Fermi Hypernetted Chain (FHNC) method
Zero temperature : APR
A. Akmal et al., PRC58(1998)1804
Finite temperature : AM
A. Mukherjee, PRC 79(2009) 045811
Potential: AV18+UIX
Wave function: Jastrow wave function
For Symmetric Nuclear Matter and Pure Neutron Matter
SN-EOS table must cover in the following wide ranges.
Density ρ : 105.1 ≤ ρm ≤ 1016.0g/cm3
110 point
Temperature T : 0 ≤ T ≤ 400 MeV
92 point
Proton fraction Yp : 0 ≤ Yp ≤ 0.65
66 point
We have to treat Asymmetric Nuclear Matter directly.
SN-EOS for Uniform Nuclear Matter is constructed
with the simplified cluster variational method.
Our Plan to Construct the EOS for SN Simulations
Collaborators : M. Takano (Waseda Uniersity) ,
Y. Takehara, S. Yamamuro, K. Nakazato, H. Suzuki (Tokyo Univ. of Science)
K. Sumiyoshi (Numazu College of Tech.)
Uniform Nuclear Matter
EOS is constructed with the cluster variational method
Non-uniform Nuclear Matter
EOS is constructed with the Thomas-Fermi calculation
★We are here.★
Completion of a Nuclear EOS table for SN simulations
Density ρ : 105.1 ≤ ρm ≤ 1016.0g/cm3
110 point
Temperature T : 0 ≤ T ≤ 400 MeV
92 point
Proton fraction Yp : 0 ≤ Yp ≤ 0.65
66 point
2. Variational EOS for core-collapse supernovae
EOS for uniform nuclear matter at zero temperature Two-body Hamiltonian
Three-body Hamiltonian
The AV18 two-body nuclear potential
Jastrow wave function
ΦF: The Fermi-gas wave function
fij : Correlation function
Central
The UIX three-body nuclear potential
Ptsµ: Spin-isospin projection operators�
Tensor
Spin-orbit
E2/N is the expectation value of H2 in the two-body cluster approximation.
E2/N is minimized with respect to fCtsµ(r), fTtµ(r) and fSOtµ(r)
with the appropriate constraints. Two-Body Energy E2/N
APR : PRC58(1998)1804
Our results are in good agreement with the results by
APR (FHNC method).
Three-Body Energy
UIX potential
Three-Body Energy
: 2π exchange part
: Repulsive part
Correction term
Expectation value with the Fermi-gas wave function
α, β, γ, δ : Parameters in E3/N
・ Total energy per nucleon E/N = E2/N + E3/N reproduce
the empirical saturation properties.
・ Thomas-Fermi calculation of isolated atomic nuclei with E/N
reproduces the gross feature of the experimental data.
13/32
Total Energy per Nucleon E/N
Our EOS : NPA902 (2013) 53
APR : PRC58(1998)1804
n0[fm-3]
E0[MeV]
K [MeV]
Esym[MeV]
0.16
-16.1
245
30.0
Application of EOS to Neutron Stars
J0348+0432: Science 340 (2013) 1233232
J1614-2230: Nature 467 (2010) 1081
Shaded region: Steiner et al.,
Astrophys. J. 722 (2010) 33
The NS mass-radius relation is consistent with observational data.
EOS of Uniform Nuclear Matter at Finite Temperature
We follow the prescription proposed by Schmidt and Pandharipande.
(Phys. Lett. 87B(1979) 11)
(A. Mukherjee et al., PRC 75(2007) 035802)
Free energy per nucleon at T=20MeV
Our EOS : NPA902 (2013) 53
Free energy per nucleon at T=30MeV
FHNC : A. Mukherjee, PRC 79(2009) 045811
EOS for Non-uniform Nuclear Matter
1013)
We follow the Thomas-Fermi (TF) method by Shen et. al. (PTP100(1998)
(APJS 197(2011) 20)
Free energy in the Wigner-Seitz (WS) cell
Bulk energy
Gradient energy
Coulomb energy
Local free energy density : f = fN + fα
F0 = 68.00 MeV fm5 a : Lattice constant
f N: Free energy density of uniform nuclear matter
Parameter
Minimum
log10(T) [MeV]
-1.08
Maximum
1.52
Number
66 + 1
Yp
0.0
0.5
213
nB [fm-3]
0.000001
0.18
1980
67×213×1980
= 28256580 points
F/Vcell is minimized with respect to parameters in np (r), nn (r), nα (r)
for various densities, temperatures and proton fractions.
Thermodynamic Quantities
Mass Number and Proton Number
Mass number A
Proton number Z
Shen EOS (APJ Suppl.197 (2011) 20)
Application to supernova simulation
1.  Fully GR spherically symmetric simulation�(1D)
2. �Adiabatic collapse�(Ye is fixed during simulation.)
3. Progenitor: Woosley Weaver 1995, 15M◉
Astrophys. J. Suppl. 101 (1995) 181
SN simulation numerical code: K. Sumiyoshi, et al., NPA 730 (2004) 227
Density profiles at the bounce
Adiabatic indices at the bounce
Variational EOS is softer than Shen EOS in the adiabatic simulation.
Compositions of Central Core
Mass, neutron and proton numbers of nuclei
The mass fraction of particles
Xα and Xp at the initial profile are larger than those with the Shen EOS.
Our new SN-EOS table will be available soon!
3. Variational EOS for hyperonic neutron stars
GR
2.5
Hyperons (Λ, Σ, Ξ ) mixing soften
an EOS of neutron star matter.
<∞
AP3
y
it
sal
au
C
2.0
AP4
2
MS2
J1614-2230
SQM1
1.5
1
PAL1
ENG
SQM3
FSU
J1903+0327
Mass (M()
HYPERON PUZZLE
(HYPERON CRISIS?)
P
R
MS0
MPA1
4
GM3
PAL6
J1909-3744
3
MS1
5
GS1
Double neutron star
s sy
systems
6
1.0
7
8
9
tion
0.5
Rota
1
0.0
7
8
9
10
11
12
Radius (km)
13
14
15
(P. B. Demorest et al., NATURE 467 (2010)
Figure 3 | Neutron star mass–radius diagram. The plot shows non-rotating
mass versus physical radius for several typical EOSs27: blue, nucleons; pink,
nucleons plus exotic matter; green, strange quark matter. The horizontal bands
show the observational constraint from our J1614-2230 mass measurement of
(1.97 6 0.04)M[, similar measurements for two other millisecond pulsars8,28
and the range of observed masses for double neutron star binaries2. Any EOS
line that does not intersect the J1614-2230 band is ruled out by this
measurement. In particular, most EOS curves involving exotic matter, such as
kaon condensates or hyperons, tend to predict maximum masses well below
2.0M[ and are therefore ruled out. Including the effect of neutron star rotation
increases the maximum possible mass for each EOS. For a 3.15-ms spin period,
this is a =2% correction29 and does not significantly alter our conclusions. The
grey regions show parameter space that is ruled out by other theoretical or
observational constraints2. GR, general relativity; P, spin period.
There is no modern hyperon EOS with the variational method.
•  Our cluster variational method for asymmetric nuclear matter
is extended to that for hyperonic nuclear matter. common feature of models that include the appearance of ‘exotic’
hadronic matter such as hyperons4,5 or kaon condensates3 at densities
of a few times the nuclear saturation density (ns), for example models
GS1 and GM3 in Fig. 3. Almost all such EOSs are ruled out by our
results. Our mass measurement does not rule out condensed quark
matter as a component of the neutron star interior6,21, but it strongly
constrains quark matter model parameters12. For the range of allowed
EOS lines presented in Fig. 3, typical values for the physical parameters
of J1614-2230 are a central baryon density of between 2ns and 5ns and a
radius of between 11 and 15 km, which is only 2–3 times the
Schwarzschild radius for a 1.97M[ star. It has been proposed that
the Tolman VII EOS-independent analytic solution of Einstein’s
equations marks an upper limit on the ultimate density of observable
cold matter22. If this argument is correct, it follows that our mass mea-
•  We study contributions from bare hyperon interactions
to neutron star structure.
Collaborators : E. Hiyama (RIKEN) ,
M. Takano (Waseda University), Y. Yamamoto (RIKEN)
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
Cluster Variational Method for hyperon matter
Two-body Hamiltonian
Two-body potential YN ・YY potential: Central potential
(µ = Λp, Λn, ΛΛ, Σ-p, Σ-n)
Jastrow wave
function
Two-body correlation function
YN・YY correlation : Central component
(µ = Λp, Λn, ΛΛ, Σ-p, Σ-n)
<H2> is calculated in the two-body cluster approximation.
ΛΛ Interaction and EOS of Neutron Star Matter
ΛΛ potential（Triplet-odd state）
Pressure of neutron star matter
As the odd-state part of ΛΛ interaction becomes repulsive,
the pressure of neutron star matter increases.
Particle Fraction in Neutron Star Matter
Λ threshold density nthΛ : 0.46 fm-3
Σ- threshold density nthΣ : 0.92 fm-3 → 0.82 fm-3 → 0.77 fm-3 → 0.72 fm-3
Gravitational Mass of Neutron Stars
As the odd-state part of ΛΛ interaction becomes repulsive,
maximum mass of neutron star increases about 10%.
1.45M◉ → 1.50M◉→ 1.55M◉→ 1.60M◉　EOS of Neutron Star Matter with Three Baryon Force
We supplement an phenomenological Three Baryon Forces (TBF)
�as a density dependent two-body effective potential. Y. Yamamoto et al., PRC 90 (2014) 045805
Λ threshold density nthΛ : 0.59 fm-3
Maximum mass of NS : 2.12 M◉
4. Summary
1. New nuclear EOS for SN simulations is constructed
with the variational method.
•  Nuclear Hamiltonian composed of realistic Nuclear forces (AV18 + UIX).
•  Variational EOS is softer than Shen EOS in the adiabatic simulation.
Our EOS is advantageous for SN explosion! Future Plan
•  SN simulations with the neutrino transfer
2. EOS for hyperonic neutron star is constructed
with the extended variational method.
•  Cluster variational method for hyperonic nuclear matter is developped .
•  Odd-state part of ΛΛ interaction affect the structure of neutron stars.
Future Plan
•  Modern NY and YY interaction (e.g. Nijmegen)