Statistical Model for a Complete Supernova Equation of State

Abstract: A statistical model for the equation of state (EOS) and the composition of supernova matter is presented with focus on the liquid-gas phase transition of nuclear matter. It consists of an ensemble of nuclei and interacting nucleons in nuclear statistical equilibrium. A relativistic mean field model is applied for the nucleons. The masses of the nuclei are taken from nuclear structure calculations which are based on the same nuclear Lagrangian. For known nuclei experimental data is used directly. Excluded volume effects are implemented in a thermodynamic consistent way so that the transition to uniform nuclear matter at large densities can be described. Thus the model can be applied at all densities relevant for supernova simulations, i.e. rho=10^5 - 10^15 g/cm^3, and it is possible to calculate a complete supernova EOS table. The model allows to investigate the role of shell effects, which lead to narrow-peaked distributions around the neutron magic numbers for low temperatures. At larger temperatures the distributions become broad. The significance of the statistical treatment and the nuclear distributions for the composition is shown. We find that the contribution of light clusters is very important and is only poorly represented by alpha-particles alone. The results for the EOS are systematically compared to two commonly used models for supernova matter which are based on the single nucleus approximation. Apart from the composition, in general only small differences of the different EOSs are found. The differences are most pronounced around the (low-density) liquid-gas phase transition line where the distribution of light and intermediate clusters has an important effect. Possible extensions and improvements of the model are discussed.

• The paper presents a comprehensive statistical model that integrates RMF parameterization and combined experimental and theoretical nuclear mass data to enhance supernova EOS simulation.
• It highlights the significant impact of light nuclear clusters like deuterons and α-particles on energy density and entropy, particularly at intermediate densities.
• Comparisons with LS and Shen EOSs reveal improved accuracy at low temperatures, offering refined predictions for core-collapse supernovae and related astrophysical phenomena.

Statistical Model for a Complete Supernova Equation of State

The study presented in this paper offers a comprehensive statistical model for deriving the equation of state (EOS) pertinent to supernova matter, emphasizing the intricate aspects of the liquid-gas phase transition within nuclear matter. This work by Hempel and Schaffner-Bielich integrates a nuanced treatment of both nuclei and their interactions, setting a foundational platform appropriate for simulating supernovae across a range of densities between $10^5$ and $10^{15}$ g/cm$^3$.

At its core, the model aligns with a relativistic mean field (RMF) approach to handle nucleons, employing a parameterization consistent with the TMA set. Distinctively, the nuclear mass data incorporated into the model harness both experimental observations from the Audi-Wapstra-Thibault mass table and results from theoretical nuclear structure calculations based on RMF. This dual methodology enables the model to surpass previous EOS frameworks that predominantly rely on the Single Nucleus Approximation (SNA) and which often downplay the complexity and variability observed in nuclear compositions.

A seminal aspect of the model arises in its phenomenological treatment of excluded volume effects. This is a pivotal factor ensuring thermodynamic consistency while acting to constrain the heavy nuclei contributions as the density approaches saturation ($\sim2/3$ of nuclear saturation density). This structural design guarantees that the transition to uniform nuclear matter is recognized, even amidst the formation of a complex nuclear mixture with pronounced shell effects at low temperatures.

One of the central findings from their systematic simulations is the importance of light nuclear clusters, particularly the deuterons and $\alpha$-particles, which are shown to significantly impact energy density and entropy at intermediate densities. Crucially, the distribution of nuclear species is not merely a Gaussian but evolves from a steep exponential to a broader power-law with increasing density. Such behavior is indicative of dynamics akin to those found in multifragmentation models for low-energy nuclear reactions.

Notably, comparisons between this model and the benchmark EOSs by Lattimer and Swesty (LS) and Shen et al. reveal key insights. For low temperatures, the agreement is substantial, indicating the efficacy of the nuclear structure-driven approach in these regimes. Nevertheless, at higher temperatures and around the onset of the liquid-gas phase transition, discrepancies emerge. These are posited to arise from the light cluster impacts inadequately captured in SNA-based models.

In terms of implications, this research not only enhances the accuracy of core-collapse supernova simulations where precise nucleosynthesis pathways critically depend on subtle EOS features, but it also opens avenues for refined predictions in many astrophysical phenomena, including neutron star mergers and proto-neutron star cooling. Future iterations might integrate advanced partition function analyses or extend the nuclear mass tables to encompass super-heavy elements, thereby increasing the model's applicability to diverse astrophysical and terrestrial contexts.

Overall, this paper introduces a robust statistical model that successfully tackles the pivotal challenge of defining a thermodynamically consistent and complete EOS for supernova simulations, holding potential to reshape existing paradigms in nuclear astrophysics simulations. Future work would do well to explore the multidimensional nature of nuclear interactions further, thereby progressively refining the predictive power of such models.