Hyperon equation of state for supernovae andneutron stars with the variational method
H. Togashi (RIKEN)
E. Hiyama (RIKEN) , M. Takano (Waseda University)

Outline
1：Introduction
2：Nuclear EOS for core-collapse simulations
3：Extension to hyperonic nuclear matter
4：Summary
4th DTA Symposium @ NAOJ, Japan, May 14, 2016

1： Introduction
The nuclear equation of state (EOS) plays important roles for astrophysical studies.
At zero temperature, a variety of nuclear EOSs have been applied to neutron stars.
2.5

EOS by APR (A. Akmal et al., PRC58(1998)1804)

PRe

<∞

AP3

ity AP4
salauC

2.0

1.

PAL1

2.

MS2

J1614-2230

SQM1

1.5MS0MPA1ENG

SQM3FSU
J1903+0327

Mass (M()

Potential: AV18+UIX
Wave function: Jastrow wave function
Method: Fermi Hypernetted chain

GR

4.

GM3

PAL6

J1909-3744

3.

MS1

5.

GS1

Double neutron stars sysystems

6.

1.0

7.
8.

9.

tion

0.5

Rota

100.0

789101112
Radius (km)131415| Neutron star mass–radius
The plot shows
P.Figure
B. 3Demorest
et al., diagram.
NATURE
467non-rotating(2010)
mass versus physical radius for several typical EOSs : blue, nucleons; pink,

111227

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 thismeasurement. In particular, most EOS curves involving exotic matter, such askaon 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.

The hyperon mixing in neutron stars
has been studied with various nuclear theories.
••••Relativistic mean field theory(C. Ishizuka et al., J. Phys. G 35 (2008) 085201)
Relativistic Hartree-Fock theory (T. Miyatsu, et al.,common
PRC
88of models(2013)01802)featurethat include
the appearance of ‘exotic’
hadronic matter such as hyperons or kaon condensates at densities
of a few times
the nuclear
saturation
density (n035801)), for example models
Brueckner-Hatree-Fock theory (H. Schulze, T. Rijken,
PRC
84(2011)
GS1 and GM3 in Fig. 3. Almost all such EOSs are ruled out by ourresults. Our mass measurement does not rule out condensed quark
Variational many-body theory(D. Lonardoni et al.,114of the(2015)
matterPRL
as a componentneutron star092301)
interior , but it strongly
4,53s

6,21

constrains quark matter model parameters12. For the range of allowed
EOS lines presented in Fig. 3, typical values for the physical parameters

1314

15

16

1718

1920

21
22

232425

EOS table for core-collapse simulations
Nuclear EOS at finite temperature → Core-collapse simulations
1. Lattimer-Swesty EOS : The Skyrme-type interaction (NPA 535 (1991) 331)
2. Shen EOS : The Relativistic Mean Field Theory (NPA 637 (1998) 435)

Hyperon EOS

SN-EOS list by M. Hempel

There are no SN EOSs based on the microscopic many-body theory.

Our Plan to Construct the EOS for SN Simulations
Uniform Nuclear Matter
EOS constructed with the cluster variational method
Non-uniform Nuclear Matter
EOS constructed with the Thomas-Fermi calculation

Completion of a Nuclear EOS table for SN simulations★We are here.★

Density ρΒ : 105.1 ≤ ρB ≤ 1016.0g/cm3

110 point

Temperature T : 0 ≤ T ≤ 400 MeV

92 point

Proton fraction Yp : 0 ≤ Yp ≤ 0.65

66 pointOur Plan to Construct the EOS for SN Simulations
Uniform Nuclear Matter
EOS constructed with the cluster variational method
Non-uniform Nuclear Matter
EOS constructed with the Thomas-Fermi calculation

Completion of a Nuclear EOS table for SN simulations★We are here.★

We extend the nuclear EOS table to considerΛ hyperon mixing.

2. Nuclear EOS for core-collapse simulations
Potential: AV18+UIX

Wave function: Jastrow wave function

Method : Cluster variational method

Neutron mattern0[fm-3]

E0[MeV]K [MeV]Esym[MeV]

0.16

-16.1

245

30.0

Our EOS : NPA902 (2013) 53
APR : PRC58(1998)1804
GFMC : PRC56(1997)1720

Thomas-Fermi calculation for isolated atomic nuclei⊿M = MTF-Mexp

MTF : Mass by the Thomas-Fermi calculation
Mexp : Experimental data

(G. Audi et al., Nucl. Phys. A 729 (2003) 337)

RMS deviation (for 2226 nuclei)
2.99 MeV (Variational EOS)

42.9 MeV (Shen EOS)

β-stability line and drip lines

Our results are in good agreement withthe experimental data and the sophisticated atomic mass formula.

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)

Our EOS : NPA902 (2013) 53
FHNC : A. Mukherjee, PRC 79(2009) 045811

Numerical results of Thomas-Fermi Calculation

Shen EOS (APJ Suppl.197 (2011) 20)

Application to neutron stars
Application of the EOS at zero temperature to neutron stars

Figure by M. Hempel

Radius of a 1.4 M◉ neutron star: R1.4 = 11.56 km(Variational)

R1.4 = 11.89 km (SFHo)J0348+0432: Science 340 (2013) 1233232
J1614-2230: Nature 467 (2010) 1081
Shaded region is the observationally suggested region by Steiner et al.
(Astrophys. J. 722 (2010) 33)

3. Extension to hyperonic nuclear matterTwo-body HamiltonianTwo-body potential

Three-body Hamiltonian

Three-nucleon potential : UIX• NN potential: AV18 two-nucleon potential(PRC 51 (1995) 38)• ΛN and ΛΛ potentials: Central potentials(E. Hiyama et al., PRC 74 (2006) 054312) (E. Hiyama et al., PRC 66 (2002) 024007)• The ab initio variational calculations for Λ hypernuclei
reproduce their experimental eigenvalues.

The experimental data of hypernuclei give no informationon the odd-state part of the ΛΛ interactions.

The odd-state part of the ΛΛ interaction
We prepare four different models for the odd-state part of the ΛΛ interaction.
Type 1 : The most attractive
Type 2 : Less attractive
Type 3 : Slightly repulsive
Type 4 : The most repulsive
The repulsion strength of Type 4 is comparable to thatof the odd-state repulsion of ΛN interaction.

The repulsive effect increases monotonically from Type 1 to Type 4.

• We investigate the effects of
the odd-state part of bare ΛΛ interactions
on the compact astrophysical objects.

Energy of hyperonic nuclear matter
Energy per baryon
E2 : The expectation value of H2 with the Jastrow wave functionin the two-body cluster approximation

ΦF: Fermi-gas wave function

E3N : Three-nucleon energy
Based on the expectation value of H3
with the Fermi-gas wave function(NPA902 (2013) 53)
Energy of hyperonic nuclear matter (Type 1)

Hyperon EOS 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 F is expressed by the average occupation probabilities.
The average occupation probability

εi(k): Single particle energy�
(i = p, n, Λ)

mi*: Effective mass of baryons�Free energies are minimized
with respect to mi*
Free energy of hyperonic nuclear matter

Application to Neutron Star
Maximum mass of neutron stars
Type 1

1.52 M◉Type 21.60 M◉Type 31.65 M◉Type 41.73 M◉

without Y

2.22 M◉

The maximum mass increases.
(1.52 M◉ → 1.73 M◉)
Mass-radius relations of neutron stars
J0348+0432: Science 340 (2013) 1233232

J1614-2230: Nature 467 (2010) 1081

Shaded region is the observationally suggested region by Steiner et al.
(Astrophys. J. 722 (2010) 33)

Application to supernova matter
We calculate the onset density of Λ hyperons in hot dense matterwith the equilibrium condition µn = µΛ.

4. Summary
We construct the EOS of nuclear matterincluding Λ hyperons at zero and finite temperaturesby the variational method.
•The obtained thermodynamic quantities are reasonable.

•The repulsion in the odd-state ΛΛ interaction raises
the maximum mass of neutron star. (1.52 M◉ → 1.73 M◉)•

The onset density of Λ is insensitive to the odd-state ΛΛ interaction.

•TBF shifts the onset density of Λ to the higher density region at low temperatures.

Future Plans• Construction of the EOS table for core-collapse simulations• Taking into account mixing of other hyperons (Σ-, Σ0, Σ+, Ξ0, Ξ-)• Employing more sophisticated baryon interactions (e.g. Nijmegen)