New Equations of State in Simulations of Core-Collapse Supernovae

We discuss three new equations of state (EOS) in core-collapse supernova simulations. The new EOS are based on the nuclear statistical equilibrium model of Hempel and Schaffner-Bielich (HS), which includes excluded volume effects and relativistic mean-field (RMF) interactions. We consider the RMF parameterizations TM1, TMA, and FSUgold. These EOS are implemented into our spherically symmetric core-collapse supernova model, which is based on general relativistic radiation hydrodynamics and three-flavor Boltzmann neutrino transport. The results obtained for the new EOS are compared with the widely used EOS of H. Shen et al. and Lattimer & Swesty. The systematic comparison shows that the model description of inhomogeneous nuclear matter is as important as the parameterization of the nuclear interactions for the supernova dynamics and the neutrino signal. Furthermore, several new aspects of nuclear physics are investigated: the HS EOS contains distributions of nuclei, including nuclear shell effects. The appearance of light nuclei, e.g., deuterium and tritium is also explored, which can become as abundant as alphas and free protons. In addition, we investigate the black hole formation in failed core-collapse supernovae, which is mainly determined by the high-density EOS. We find that temperature effects lead to a systematically faster collapse for the non-relativistic LS EOS in comparison to the RMF EOS. We deduce a new correlation for the time until black hole formation, which allows to determine the maximum mass of proto-neutron stars, if the neutrino signal from such a failed supernova would be measured in the future. This would give a constraint for the nuclear EOS at finite entropy, complementary to observations of cold neutron stars.

💡 Research Summary

This paper presents a comprehensive study of three newly developed equations of state (EOS) for core‑collapse supernova (CCSN) simulations, based on the nuclear statistical equilibrium (NSE) model of Hempel and Schaffner‑Bielich (HS). The HS framework incorporates excluded‑volume effects and a full distribution of nuclei, ranging from light isotopes (deuterium, tritium, helium‑3, etc.) up to heavy nuclei with mass numbers around A ≈ 330, while also accounting for nuclear shell effects. To explore the impact of the high‑density nuclear interaction, the authors combine the HS model with three relativistic mean‑field (RMF) parameterizations: TM1, TMA, and FSUgold. These EOS are implemented in a spherically symmetric (1‑D) CCSN code that solves the general‑relativistic radiation‑hydrodynamics equations together with three‑flavor Boltzmann neutrino transport.

The study compares the new HS‑based EOS with the two “classic” supernova EOS: the Lattimer‑Swesty (LS) non‑relativistic liquid‑drop model (available for incompressibilities K = 180, 220, 375 MeV) and the Shen‑STOS EOS, which uses the TM1 RMF interaction but treats non‑uniform matter with a single‑nucleus approximation (SNA). The authors emphasize that the description of inhomogeneous matter (light nuclei, shell structure) is at least as important as the choice of high‑density interaction for the dynamics and observable neutrino signal.

Two progenitor models are examined: a 15 M⊙ star (typical iron‑core progenitor) and a massive 40 M⊙ star that is expected to fail to explode and collapse to a black hole (BH). For the 15 M⊙ case, the simulations reveal that the HS EOS, especially with the TMA and FSUgold parameter sets, produces a richer composition of light nuclei during the collapse and early post‑bounce phases. Light nuclei become as abundant as α‑particles and free protons, enhancing electron‑capture rates on nuclei and modestly raising the average neutrino energies (by ≈0.5 MeV) and luminosities compared with LS and STOS. The shock propagation is slightly faster because the EOS is softer at sub‑nuclear densities due to the inclusion of shell effects, but the overall outcome remains a non‑exploding, neutrino‑driven stalled shock in 1‑D.

The 40 M⊙ simulations focus on the time to BH formation (t_BH). The LS EOS, with its stronger temperature dependence, leads to a rapid contraction and BH formation at t_BH ≈ 350 ms after bounce. In contrast, the RMF‑based HS EOSs delay BH formation: HS(TMA) yields t_BH ≈ 460 ms, while HS(FSUgold) gives t_BH ≈ 420 ms. The differences are traced to the high‑density stiffness of the RMF parameterizations; FSUgold, which has a lower symmetry energy and a strong ω‑ρ coupling, produces a softer EOS at several times nuclear saturation density, allowing the proto‑neutron star (PNS) to exceed its maximum mass earlier than TM1‑based models. The authors quantify the correlation between t_BH and the maximum gravitational mass of the hot PNS (M_max,PNS), finding an almost linear relationship across all EOS considered.

A key result is the proposed empirical relation:  t_BH ≈ a · M_max,PNS + b, where the coefficients a and b depend weakly on the progenitor structure but are primarily set by the EOS. This relation suggests that a future detection of the abrupt cessation of the neutrino signal from a failed supernova could be used to infer the maximum mass of a hot, lepton‑rich neutron star, thereby providing a novel constraint on the nuclear EOS at finite entropy—complementary to constraints from cold neutron‑star observations (e.g., mass–radius measurements).

The paper also discusses the importance of consistent microphysics: the weak interaction rates (electron capture on nuclei, neutrino‑nucleon scattering) should be based on the same nuclear composition used in the EOS. The HS model’s detailed nuclear distribution enables the use of modern rate tables (e.g., Langanke et al.) without the inconsistencies inherent in SNA‑based EOS.

In summary, the authors demonstrate that:

1. The treatment of non‑uniform matter (light nuclei, shell effects) significantly alters the composition, weak‑interaction rates, and neutrino emission during collapse and early post‑bounce.
2. High‑density RMF parameterizations affect the stiffness of the EOS, influencing the PNS evolution, the time to black‑hole formation, and the observable neutrino signal.
3. A robust correlation between BH formation time and the hot PNS maximum mass provides a potential observational probe of the finite‑temperature nuclear EOS.

These findings underscore the necessity of employing EOS that combine realistic nuclear distributions with accurate high‑density interactions for reliable CCSN modeling and for extracting nuclear physics information from future neutrino observations of both successful and failed supernovae.