Ⅰ. HIGHLIGHTS OF THE YEARRIKEN Accel. Prog. Rep. 51 (2018)

Supernova equation of state based on realistic nuclear forces†
H. Togashi,∗1,∗2 K. Nakazato,∗3 Y. Takehara,∗4 S. Yamamuro,∗4 H. Suzuki,∗4 and M. Takano∗2,∗5
Table 1. Ranges of temperature T , proton fraction Yp , andbaryon mass density ρB in the table of the variational
EOS. At the top of the last column, “+1” represents
the case at T = 0 MeV.

The equation of state (EOS) for hot nuclear matteris one of the main ingredients in astrophysical simulations, such as core-collapse supernovae, cooling of
protoneutron stars, black hole formations, and binary
neutron star mergers. In order to describe the properties of nuclear matter in these simulations, a nuclear
EOS table containing various thermodynamic quantities in wide ranges of the baryon number density nB ,proton fraction Yp , and temperature T is necessary.
Therefore, it is not an easy task to construct a nuclear
EOS suitable for the astrophysical simulations. In fact,
a limited number of nuclear EOSs are currently available for the core-collapse simulations.1) Furthermore,high-density uniform matter is treated with the phenomenological models in those EOSs. Under these circumstances, we have recently constructed a new nuclear EOS table based on realistic nuclear forces.2)
For uniform nuclear matter, the EOS is constructedwith the cluster variational method. At zero temperature, we calculate the energy per nucleon as an expectation value of the realistic nuclear Hamiltonian composed of the Argonne v18 two-body and Urbana IX
three-body potentials with the Jastrow wave functionin the two-body cluster approximation. The obtainedenergies for pure neutron matter and symmetric nuclear matter are in good agreement with the resultsof the more sophisticated Fermi hypernetted chain(FHNC) variational calculations by Akmal et al.3) Atﬁnite temperature, the free energy per nucleon for uniform nuclear matter was calculated with an extensionof the variational method by Schmidt and Pandharipande.4) The obtained results for pure neutron matterand symmetric nuclear matter are also in good agreement with those obtained using FHNC calculations.5)
For non-uniform nuclear matter, we adopt theThomas-Fermi approximation following the method byShen et al.6) In this method, non-uniform nuclear matter is assumed to be a mixture of free neutrons, freeprotons, alpha particles, and a single species of heavy
nuclei that is located at the center of a Wigner-Seitz
cell in a body-centered cubic lattice. We then calculate the free energy per nucleon for non-uniform matterby minimizing the average free energy density of theWigner-Seitz cell with respect to the parameters specifying the density distributions of particles in the cell.
†∗1
∗2
∗3
∗4
∗5

Parameterlog10 (T ) [MeV]
Yplog10 (ρB ) [g/cm3 ]Min−1.00
05.1

Max
2.60
0.65
16.0

Mesh
0.04
0.01
0.1

Number91 + 1
66
110

Finally, we determine the thermodynamically favorable state for each nB , Yp , and T , by comparing thefree energy for non-uniform nuclear matter with thatfor uniform nuclear matter. The obtained free energy
and related thermodynamic quantities are tabulated inranges of nB , Yp , and T , as shown in Table 1. Here,the baryon mass density ρB is deﬁned as ρB = mu nB
with mu being the atomic mass unit.
The obtained EOS for various nB , Yp , and T is reasonable as compared with the Shen EOS.6) It is also
found that the critical temperature with respect to theliquid–gas phase transition decreases more moderately
with Yp for our EOS. Furthermore, masses of heavy nuclides are slightly larger than those of the Shen EOS in
neutron-rich nuclear matter. As reported in Refs. 2, 7),
those results are caused by the fact that the density
derivative coeﬃcient of the symmetry energy L of ourEOS is smaller than that of the Shen EOS.
To the best of our knowledge, this is the ﬁrst nuclear
EOS for astrophysical simulations based on realistic
nuclear forces. In the near future, we will extend thepresent EOS so as to take into account the additional
hyperon degrees of freedom at ﬁnite temperature, because the hyperon mixing is expected to strongly aﬀect
the EOS of dense matter at high densities. In fact, wehave already calculated the EOS of hyperonic matterat zero temperature by using the cluster variationalmethod.8)
The EOS table constructed in this study is open forgeneral use in the studies of nuclear physics and astrophysics. The complete EOS table is available on theWeb at http://www.np.phys.waseda.ac.jp/EOS/.
References
1) M. Oertel et al., Rev. Mod. Phys. 89, 015007 (2017).
2) H. Togashi et al., Nucl. Phys. 961, 78 (2017).
3) A. Akmal et al., Phys. Rev. C 58, 1804 (1998).
4) K. E. Schmidt, V. R. Pandharipande, Phys. Lett. B 87,11 (1979).
5) A. Mukherjee, Phys. Rev. C 79, 045811 (2009).
6) H. Shen et al., Astrophys. J. Suppl. Ser. 197, 20 (2011).
7) K. Oyamatsu, K. Iida, Phys. Rev. C 75, 015801 (2007).
8) H. Togashi et al., Phys. Rev. C 93, 035808 (2016).

Condensed from the article in Nucl. Phys. A 961, 78 (2017).
RIKEN Nishina Center
Waseda Research Institute for Science and Engineering,
Waseda University
Faculty of Arts and Science, Kyushu University
Department of Physics, Tokyo University of Science
Department of Pure and Applied Physics, Waseda University- 10 -