Role of symmetry energy at subnuclear densities in protoneutron star crusts

TL;DR

The paper investigates how subnuclear-density nuclear matter properties, encoded by the symmetry-energy slope $L$ and the saturation symmetry energy $S_0$, influence protoneutron-star cooling and crust crystallization. It extends the Oyamatsu–Iida macroscopic nuclear model to finite temperatures via a finite-temperature Thomas–Fermi framework and builds a suite of EOSs spanning different $L$ (with fixed $K$) and $S_0$, subsequently linking them to inhomogeneous phases and crust formation. The study finds that smaller $L$ raises the crystallization temperature and biases heavy nuclei toward larger $Z$ and $A$, which can shorten crust-formation times despite slower overall cooling; late-time neutrino signals are shown to carry imprints of subnuclear EOS properties. Supranuclear-density parameters, parameterized by $S_{00}$, have limited impact on crust timing, suggesting that late-time neutrino observations could probe subnuclear matter properties while being relatively robust to high-density uncertainties. The work highlights prospects for using neutrino data from protoneutron stars to constrain the symmetry energy’s density dependence and outlines avenues for including pasta phases and multi-component effects in future models.

Abstract

The impact of matter properties at subnuclear densities on the evolution of protoneutron stars is investigated. Several models of nuclear equation of state (EOS) are constructed with varying saturation parameters, particularly the symmetry energy $S_0$ and its density slope $L$. Using the Thomas--Fermi approximation, the mass and proton numbers of heavy nuclei at subnuclear densities are systematically evaluated, along with their dependence on the EOS. Cooling simulations of protoneutron stars reveal that EOSs with smaller $L$ values lead to a longer cooling timescale and higher average neutrino energies. This behavior is attributed to the enhanced neutrino scattering caused by larger mass numbers, which increases the thermal insulation. Furthermore, the crystallization temperature, marking the onset of crust formation, is found to be higher for EOSs with smaller values of $L$. This is due to the enhanced Coulomb energy associated with larger proton numbers. As a result, despite slower cooling, crust formation occurs earlier for smaller-$L$ EOSs. These findings indicate that the timing of crust formation is sensitive to the EOS and highlight the importance of late-time neutrino observations as probes of the matter properties at subnuclear densities.

Figures (8)

• Figure 1: Phase diagrams of matter with $Y_{\rm p} = 0.04$ for (a) EOS B, (b) EOS E, (c) EOS Z, and (d) EOS H. The plotted regions represent the inhomogeneous phase, with the color scale indicating the value of $\Gamma$. Regions with fully black plots denote $\Gamma > 300$, and regions with black-outlined plots denote $\Gamma > 175$.
• Figure 2: Mass-radius relations of cold neutron stars for (a) reference models, (b) models of EOS E, and (c) models of EOS Z. In (a), orange, green, blue, and red lines correspond to the models B60, E50, Z40, and H30, respectively. In (b) and (c), darker lines correspond to models with smaller values of $S_{00}$.
• Figure 3: Luminosity (upper) and average energy (lower) of neutrinos as a function of time for (a) $\nu_e$, (b) $\bar{\nu}_e$, and (c) $\nu_x$. The line colors follow the same definitions as in Fig. \ref{['fig:MR']}(a).
• Figure 4: Same as Fig. \ref{['fig:nulc']} but for the models of EOS E. The line colors follow the same definitions as in Fig. \ref{['fig:MR']}(b).
• Figure 5: Same as Fig. \ref{['fig:nulc']} but for the models of EOS Z. The line colors follow the same definitions as in Fig. \ref{['fig:MR']}(c).