M. Takano

Nuclear equation of state for core-collapse supernova simulations with realistic nuclear forces

Abstract A new table of the nuclear equation of state (EOS) based on realistic nuclear potentials is constructed for core-collapse supernova numerical simulations. Adopting the EOS of uniform nuclear matter constructed by two of the present authors with the cluster variational method starting from the Argonne v18 and Urbana IX nuclear potentials, the Thomas–Fermi calculation is performed to obtain the minimized free energy of a Wigner–Seitz cell in non-uniform nuclear matter. As a preparation for the Thomas–Fermi calculation, the EOS of uniform nuclear matter is modified so as to remove the effects of deuteron cluster formation in uniform matter at low densities. Mixing of alpha particles is also taken into account following the procedure used by Shen et al. (1998, 2011). The critical densities with respect to the phase transition from non-uniform to uniform phase with the present EOS are slightly higher than those with the Shen EOS at small proton fractions. The critical temperature with respect to the liquid–gas phase transition decreases with the proton fraction in a more gradual manner than in the Shen EOS. Furthermore, the mass and proton numbers of nuclides appearing in non-uniform nuclear matter with small proton fractions are larger than those of the Shen EOS. These results are consequences of the fact that the density derivative coefficient of the symmetry energy of our EOS is smaller than that of the Shen EOS.

Application of the nuclear equation of state obtained by the variational method to core-collapse supernovae

The equation of state (EOS) for hot asymmetric nuclear matter which is constructed with the variational method starting from the Argonne v18 and Urbana IX nuclear forces is applied to spherically symmetric core-collapse supernovae (SNe). We first investigate the EOS of isentropic beta-stable SN matter, and find that the matter with the variational EOS is more neutron-rich than that with the Shen EOS. Using the variational EOS for uniform matter supplemented by the Shen EOS of non-uniform matter at low densities, we perform general-relativistic spherically symmetric simulations of core-collapse SNe with and without neutrino transfer, starting from a presupernova model of 15 solar mass. In the adiabatic simulation without neutrino transfer, the explosion is successful, and the explosion energy with the variational EOS is larger than that with the Shen EOS. In the case of the simulation with neutrino transfer, the shock wave stalls and then the explosion fails, as in other spherically symmetric simulations. The inner core with the variational EOS is more compact than that with the Shen EOS, due to the relative softness of the variational EOS. This implies that the variational EOS is more advantageous for SN explosions than the Shen EOS.

Shell effects in hot nuclei and their influence on nuclear composition in supernova matter

We calculate nuclear composition in supernova (SN) matter explicitly taking into account the temperature dependence of nuclear shell effects. The abundance of nuclei in SN matter is important in the dynamics of core-collapse supernovae and, in recently constructed equations of state (EOS) for SN matter, the composition of nuclei are calculated assuming nuclear statistical equilibrium wherein the nuclear internal free energies govern the composition. However, in these EOS, thermal effects on the shell energy are not explicitly taken into account. To address this shortfall, we calculate herein the shell energies of hot nuclei and examine their influence on the composition of SN matter. Following a simplified macroscopic-microscopic approach, we first calculate single-particle (SP) energies by using a spherical Woods-Saxon potential. Then we extract shell energies at finite temperatures using Strutinsky method with the Fermi distribution as the average occupation probability of the SP levels. The results show that at relatively low temperatures, shell effects are still important and magic nuclei are abundant. However, at temperatures above approximately 2 MeV, shell effects are almost negligible, and the mass fractions with shell energies including the thermal effect are close to those obtained from a simple liquid drop model at finite temperatures.

Variational calculations with explicit energy functionals for fermion systems at zero temperature

The variational method with explicit energy functionals (EEFs) is applied to infinite neutron matter. Starting from the Argonne v8 two-body potential and the repulsive part of the Urbana IX three-body potential, the energy per neutron is expressed explicitly with the spin-dependent central, tensor, and spin-orbit distribution functions. This EEF is constructed as an extension of the previously proposed one for the Argonne v6 potential, including central and tensor forces. The Euler-Lagrange equations derived from this EEF are solved numerically. The obtained fully minimized energy with this EEF is in good agreement with that obtained from the auxiliary field diffusion Monte Carlo calculation.

Nuclear equation of state for core-collapse supernovae with realistic nuclear forces

H. Togashi∗1,2, Y. Takehara3, S. Yamamuro3, K. Nakazato3, H. Suzuki3, K. Sumiyoshi4, M. Takano2,5 1 Institute for Physical and Chemical Research (RIKEN), 2-1 Hirosawa, Wako, Saitama 351-0198, Japan 2 Research Institute for Science and Engineering, Waseda University, 3-4-1 Okubo Shinjuku-ku, Tokyo 169-8555, Japan 3 Department of Physics, Faculty of Science and Technology, Tokyo University of Science, Yamazaki 2641, Noda, Chiba 278-8510, Japan 4 Numazu College of Technology, Ooka 3600, Numazu, Shizuoka 410-8501, Japan 5 Department of Pure and Applied Physics, Waseda University, 3-4-1 Okubo Shinjuku-ku, Tokyo 169-8555, Japan E-mail: hajime.togashi@riken.jp

Explicit energy functional for infinite nuclear matter with the tensor force

We have applied the variational method using explicit energy functionals (EEFs) to energy calculations of infinite nuclear matter. In EEFs, the energy per nucleon is explicitly expressed with spin-isospin-dependent two-body distribution functions, which are regarded as variational functions, and fully minimized energies are conveniently calculated with the EEF. A remarkable feature of this approach is that EEFs guarantee non-negativeness of structure functions. In this study, we extend the EEF variational method so as to consider state- independent three-body forces for neutron matter at finite temperatures following the procedure proposed by Schmidt and Pandharipande. For neutron matter, the free energies obtained with the Argonne v4 two-body potential and the repulsive part of the Urbana IX (UIX) three- body potential are quite reasonable. Furthermore, we improve the EEF of nuclear matter using the two-body central and tensor forces by considering the main three-body cluster terms and guaranteeing non-negativeness of tensor structure functions. In addition, healing distances are introduced for two-body distribution functions so that Mayers condition is satisfied. The obtained energies per neutron of neutron matter with the Argonne v6 two-body potential and the repulsive part of the UIX potential are in good agreement with those obtained by auxiliary field diffusion Monte Carlo calculations.

Nuclear equation of state with the variational method and its application to supernova simulations

We report on an equation of state (EOS) of hot asymmetric nuclear matter constructed using the variational method and its application to hydrodynamic simulations of core-collapse supernovae. This nuclear EOS is based on the AV18 two-body potential and UIX three-body potential, and the energy per nucleon at zero temperature is constructed with the cluster variational method. At finite temperatures, the free energies per nucleon are calculated with an extension of the variational method devised by Schmidt and Pandharipande. This EOS is in good agreement with that by the Fermi hypernetted chain variational calculations at zero and finite temperatures, and the structure of neutron stars calculated with this EOS is consistent with recent observational data. Using this nuclear EOS, we perform a spherically symmetric general-relativistic adiabatic simulation of the SN explosion. The explosion energy calculated with our EOS in the present simulation is larger than that obtained with the Shen EOS, implying that the variational EOS is softer than the Shen EOS.

Equation of state for nuclear matter in core-collapse supernovae by the variational method

We construct a new nuclear equation of state (EOS) for core-collapse supernova (SN) simulations using the variational many-body theory. For uniform nuclear matter, the EOS is constructed with the cluster variational method starting from the realistic nuclear Hamiltonian composed of the Argonne v18 two-body potential and the Urbana IX three-body potential. The masses and radii of neutron stars calculated with the obtained EOS at zero temperature are consistent with recent observational data. For non-uniform nuclear matter, we construct the EOS in the Thomas-Fermi approximation. In this approximation, we assume a functional form of the density distributions of protons, neutrons, and alpha-particles, and minimize the free energy density in a Wigner-Seitz cell with respect to the parameters included in the assumed density distribution functions. The phase diagram of hot nuclear matter at a typical temperature is reasonable as compared with that of the Shen EOS.