Akmal-Pandharipande-Ravenhall equation of state for simulations of supernovae, neutron stars, and binary mergers

Differences in the equation of state (EOS) of dense matter translate into differences in astrophysical simulations and their multimessenger signatures. Thus, extending the number of EOSs for astrophysical simulations allows us to probe the effect of different aspects of the EOS in astrophysical phenomena. In this work, we construct the EOS of hot and dense matter based on the Akmal, Pandharipande, and Ravenhall (APR) model and thereby extend the open-source SROEOS code which computes EOSs of hot dense matter for Skyrme-type parametrizations of the nuclear forces. Unlike Skrme-type models, in which parameters of the interaction are fit to reproduce the energy density of nuclear matter and/or properties of heavy nuclei, the EOS of APR is obtained from potentials resulting from fits to nucleon-nucleon scattering and properties of light nuclei. In addition, this EOS features a phase transition to a spin-isospin ordered state of nucleons, termed a neutral pion condensate, at supranuclear densities. We show that differences in the effective masses between EOSs have consequences for the properties of nuclei in the subnuclear inhomogeneous phase of matter. We also test the new EOS of APR in spherically symmetric core-collapse of massive stars with $15M_{\odot }$ and $40M_{\odot }$, respectively. We find that the phase transition in the EOS of APR speeds up the collapse of the star. However, this phase transition does not generate a second shock wave or another neutrino burst as reported for the hadron-to-quark phase transition. The reason for this difference is that the width of the coexistence region and the latent heat in the EOS of APR are substantially smaller than in the quark-to-hadron transition employed earlier, which results in a significantly smaller softening of the high density EOS.

Figure 1

Pressure of SNM (top) and PNM (bottom) for the EOS's of APR, ${\mathrm{APR}}_{\mathrm{LDP}}$, NRAPR, and SkAPR. The SNM and PNM results are compared to the pressure of nuclear matter deduced from analysis of heavy ion collision experiments by Danielewicz et al. [67]. The PNM results are also compared to results from chiral effective field theory supplemented by piecewise polynomials by Tews et al. [68].

Figure 2

Mass radius relationships for the models of APR, ${\mathrm{APR}}_{\mathrm{LDP}}$, NRAPR, and SkAPR computed for charge neutral and temperature beta-equilibrated matter at zero temperature. Results are compared to the maximum NS mass observed by Antoniadis et al. [45], the mass radius relationship of Nättilä et al. [47], and the radius of a $1.4M_{\odot }$ NS inferred by Most et al. [4].

Figure 3

Neutron (left) and proton (right) effective masses $m_t$ normalized by the neutron vacuum mass $m_n$ as a function of density $n$ and proton fraction $y$ for the APR (top) and NRAPR (bottom) models.

Figure 4

Neutron (left) and proton (right) effective masses for the APR (thick) and NRAPR (thin) EOSs for proton fractions $y=0.10$ (red) and 0.50 (black).

Figure 5

Surface properties ${\sigma }^{\prime }(y_i,T)$ computed for the APR model (top), its best fit $\sigma (y_i,T)$ using Eqs. (21) and (22) (center), and ratio between the computed and best fit ${\sigma }^{\prime }(y_i,T)/\sigma (y_i,T)$ (bottom). White regions are places where matter is unstable against phase coexistence.

Figure 6

Plots of normalized surface tension $\sigma (y_i,T)/{\sigma }_s$ for APR (top), NRAPR (center), and SkAPR (bottom) EOSs. White regions indicate areas of the parameter space where there is no coexistence of matter with two different densities and proton fractions.

Figure 7

Nuclear mass number $A$, charge $Z$ and the volume fraction $u$ occupied by the dense phase for the APR model.

Figure 8

Same as Fig. 7, but for the NRAPR model.

Figure 9

Same as Fig. 7, but for the SkAPR model.

Figure 10

From top to bottom: number fraction of neutrons $x_n$, protons $x_p$, alpha particles $x_{\alpha }$, and heavy nuclei $x_h$ for proton fraction $y=0.01,0.10,0.30$, and 0.50 for the APR EOS.

Figure 11

From top to bottom: sizes of nuclei for the APR (top), NRAPR (center), SkAPR (bottom) EOSs for proton fraction $y=0.01,0.10,0.30$, and 0.50.

Figure 12

Pressure per baryon $P/n$ for APR EOS (top) and APRLDP (bottom) for proton fraction $y=0.01,0.10,0.30$, and 0.50.

Figure 13

Chemical potential splitting $\widehat{\mu }={\mu }_n-{\mu }_p$ for APR EOS (top) and APRLDP (bottom) for proton fraction $y=0.01,0.10,0.30$, and 0.50.

Figure 14

From top to bottom: central density ${\rho }_c$, central temperature $T_c$, shock radius $R_{\mathrm{shock}}$, and PNS radius $R_{12}$ for the core collapse of the $15M_{\odot }$ presupernova progenitor of Woosley, Heger, and Weaver [80].

Figure 15

Proton fraction $y$ (top) and temperature $T$ (bottom) as a function of density inside the PNS radius $R_{12}$; see Fig. 14 for the core collapse of the $15M_{\odot }$ presupernova progenitor of Woosley, Heger, and Weaver [80]. For clarity, we only plot the APR and ${\mathrm{APR}}_{\mathrm{LDP}}$ EOSs at times $t-t_{\mathrm{bounce}}=50$, 100, 150, 200, 300, 400, 600, and $800\mathrm{ms}$. We also show in light gray in the top plot the coexistence region. Core-collapse evolution predicted by both APR and ${\mathrm{APR}}_{\mathrm{LDP}}$ EOSs match up to $t-t_{\mathrm{bounce}}\simeq 180\mathrm{ms}$ and diverge afterwards once the coexistence region is crossed by the run using the APR EOS. Although the phase transition affects the inner regions of the PNS during collapse, that does not translate to a significant change in the outer regions of the star, $n\lesssim n_{\mathrm{sat}}$.

Figure 16

Pressure as a function of density (top) and as a function of energy density (bottom) for SNM and PNM at $T\simeq 0\mathrm{MeV}$. Phase coexistence regions are indicated by the shaded areas. Due to the lack of temperature dependence in the APR potential, Eq. (10), the coexistence region is barely affected by temperature effects; see Fig. 32 in Ref. [48]. Thick lines were computed directly from our APR EOS table which spans 60 points per decade in density at $T=0.01\mathrm{MeV}$ and are compared to analytical results at $T=0$, thin lines. The EOS table used to make this plot has twice the density resolution of other EOS tables often used in CCSN simulations [16, 74]. Nevertheless, very few points in the transition region are still probed by this table.

Figure 17

Root mean square energy $\sqrt{\langle {\epsilon }_{\nu }^2\rangle }$ (top) and luminosity $L_{\nu }$ (bottom) for electron neutrinos ${\nu }_e$ (thick lines) and electron antineutrinos ${\nu }_{\overline{e}}$ (thin lines) for the core collapse of the $15M_{\odot }$ presupernova progenitor of Woosley and Weaver [80].

Figure 18

Same as Fig. 14, but for the $40M_{\odot }$ presupernova progenitor of Wooseley and Heger [81].

Figure 19

Root mean square energy $\sqrt{\langle {\epsilon }_{\nu }^2\rangle }$ (top) and luminosity $L_{\nu }$ (bottom) for electron neutrinos ${\nu }_e$ (thick lines) and electron antineutrinos ${\nu }_{\overline{e}}$ (thin lines) for the core collapse of the $40M_{\odot }$ presupernova progenitor of Woosley, Heger, and Weaver [80].