A New Equation of State for Astrophysical Simulations

Abstract: We generate a new complete equation of state (EOS) of nuclear matter for a wide range of temperatures, densities, and proton fractions ready for use in astrophysical simulations of supernovae and neutron star mergers. Our previous two papers tabulated the EOS at over 180,000 grid points in the temperature range $T$ = 0 to 80 MeV, the density range $n_B$ = 10$^{-8}$ to 1.6 fm$^{-3}$, and the proton fraction range $Y_P$ = 0 to 0.56. In this paper we combine these data points using a suitable interpolation scheme to generate a single equation of state table on a finer grid. This table is thermodynamically consistent and conserves entropy during adiabatic compression tests. We present various thermodynamic quantities and the composition of matter in the new EOS, along with several comparisons with existing EOS tables. Our equation of state table is available for download.

• The paper presents a new EOS that couples a Virial expansion at low densities with a relativistic mean field approach at higher densities.
• Its novel interpolation scheme rigorously maintains thermodynamic consistency and stability, validated by precise adiabatic compression tests.
• The EOS predicts key astrophysical properties, including a 2.77 M☉ neutron star maximum mass, and refines neutrino interaction models in supernova environments.

A Thermodynamically Consistent, Comprehensive Equation of State for Nuclear Astrophysics

Overview

The paper "A New Equation of State for Astrophysical Simulations" (1101.3715) presents a rigorously constructed, thermodynamically consistent EOS for nuclear matter across extensive ranges of temperature ($T$), baryon density ($n_B$), and proton fraction ($Y_P$). The EOS is specifically designed for use in state-of-the-art simulations of core-collapse supernovae, proto-neutron star evolution, and neutron star mergers. The authors couple a Virial expansion for nonideal nuclear and nuclear cluster interactions at low densities with relativistic mean field (RMF) theory at intermediate and high densities, incorporating nuclear shell effects and a broad distribution of nuclear species. A sophisticated interpolation and smoothing procedure ensures the resulting EOS table respects thermodynamic consistency and stability, maintaining entropy conservation under adiabatic flows.

Methodology

Microscopic Modeling

At low densities and temperatures, the EOS leverages a Virial expansion employing experimental nuclear masses and elastic nucleon-nucleon/nucleon-cluster scattering phase shifts, accurately modeling nonideal nuclear matter while including Coulomb corrections crucial for neutrino-matter interactions. The inclusion of thousands of heavy nuclei from the FRDM mass evaluation and explicit treatment of nuclear partition functions allows detailed accounting for compositional effects.

For higher densities and temperatures, the EOS transitions to a relativistic mean field (RMF) description based on the NL3 interaction, which is known for its high nuclear incompressibility and stiffness. Nonuniform nuclear matter at sub-saturation densities is modeled using a Hartree mean field approach with spherical Wigner-Seitz cells, preserving shell and pairing physics. At the highest densities and temperatures, a uniform, interacting nucleon gas is adopted.

Numerical Interpolation and Thermodynamic Consistency

The EOS data are computed on a finely sampled grid (over 180,000 points, spanning $T = 0$–80 MeV, $n_B = 10^{-8}$–$1.6\,\mathrm{fm}^{-3}$, $Y_P = 0$–0.56) and are postprocessed to generate a single, high-resolution EOS table. Crucially, the interpolation retains:

• Exact monotonicity and smoothness for primary thermodynamic quantities (free energy, pressure, entropy, internal energy)
• Enforcement of convexity and the appropriate adiabatic derivatives (e.g., positivity of $(\partial S/\partial T)_{n_B, Y_P}$ and $(\partial P/\partial n_B)_{T,Y_P}$)
• Adiabatic entropy conservation validated to the per mille level in controlled compression tests.

A hybrid of monotonic cubic Hermite and bicubic interpolation is employed. No unphysical oscillations or numerical artifacts are introduced, ensuring the EOS’s usability in hydrodynamical codes sensitive to thermodynamic derivatives.

Key Results

EOS Properties and Composition

The computed EOS resolves the phase diagram of nuclear matter, identifying the boundaries among free nucleons, nuclear clusters, and uniform matter, as functions of $T$, $n_B$, and $Y_P$. Strong shell effects manifest in stepwise increases in mean mass and proton numbers of heavy nuclei, especially at low $T$. The broad compositional spectrum at low densities and moderate $Y_P$ results in entropy and composition profiles that differ substantially from the Lattimer-Swesty (L-S) and H. Shen, Toki, Oyamatsu, and Sumiyoshi (S-S) EOSs, both of which represent nuclei by averaged single-species approximations and neglect shell effects.

The EOS returns detailed mass fractions for free neutrons, protons, alpha particles, and heavy nuclei, critically impacting electron capture and neutrino opacity calculations required for supernova modeling.

High-Density and Neutron Star Implications

At supra-nuclear densities, the use of the stiff NL3 RMF interaction results in high central pressures. This stiffness yields a maximum nonrotating neutron star mass of $2.77\,M_\odot$ (radius $13.3$ km), significantly exceeding the maximum masses produced by L-S or S-S EOSs with less repulsive short-range nuclear forces. The proton fraction evolution with density at $T=0$ is non-monotonic, reflecting the relatively large symmetry energy in NL3 at high densities, which favors symmetric matter.

Validation and Comparison

The EOS is extensively benchmarked:

• Adiabatic Consistency: Adiabatic compression tests confirm entropy conservation to better than 1%.
• Thermodynamic Stability: $(\partial P/\partial n_B)_T\geq 0$ is preserved globally.
• Cross-Comparison: For $T=1$ MeV and $Y_P=0.05$, the EOS pressure agrees with S-S and L-S up to saturation density; at higher density, NL3 yields the largest pressure. For composition, the EOS reports a broader distribution of heavy nuclei and smaller mean mass numbers than L-S/S-S, which impacts computed entropies and predicted neutrino-matter coupling.

Implications

Practical Impact for Simulations

The presented EOS enables current-generation supernova, neutron star merger, and proto-neutron star cooling simulations to:

• Employ a thermodynamically consistent, high-resolution EOS valid over all relevant astrophysical regimes.
• Quantify sensitivities to nuclear composition, shell physics, and phase transitions, particularly in low-density, neutron-rich environments where neutrino transport and nucleosynthesis occur.
• Explore the dependence of macroscopic objects (e.g., neutron star maximum mass, radii) on EOS stiffness, symmetry energy, and clusterization.

Theoretical and Future Perspectives

The table serves as a benchmark for further development. Notably:

• By releasing the EOS in tabular form, it standardizes benchmarking among different hydrodynamics codes and enables systematic study of EOS-systematics by varying underlying nuclear interactions.
• The methodology accommodates alternative RMF parameterizations, allowing construction of 'softer' EOS tables, which are necessary to explore constraints from nuclear experiment and neutron star observations.
• The detailed compositional treatment is critical for studies of neutrino-matter interaction rates, electron capture, and nucleosynthesis in explosive astrophysical environments.

Conclusion

The authors' EOS (1101.3715) provides the field with a high-fidelity, consistent tool for astrophysics simulations, improving upon prior models by incorporating microscopic nuclear physics—especially at low density—while enforcing the rigorous thermodynamic properties required for hydrodynamic accuracy. Its public availability and modular structure facilitate future adoption, direct comparison with alternative nuclear models, and the systematic reduction of EOS-related uncertainties in neutron star and supernova theory.