Equation of state effects in the core collapse of a $20\text{−}{M}_{\odot }$ star

Uncertainties in our knowledge of the properties of dense matter near and above nuclear saturation density are among the main sources of variations in multimessenger signatures predicted for core-collapse supernovae (CCSNe) and the properties of neutron stars (NSs). We construct 97 new finite-temperature equations of state (EOSs) of dense matter that obey current experimental, observational, and theoretical constraints and discuss how systematic variations in the EOS parameters affect the properties of cold nonrotating NSs and the core collapse of a $20\text{−}{M}_{\odot }$ progenitor star. The core collapse of the $20\text{−}{M}_{\odot }$ progenitor star is simulated in spherical symmetry using the general-relativistic radiation-hydrodynamics code GR1D where neutrino interactions are computed for each EOS using the NuLib library. We conclude that the effective mass of nucleons at densities above nuclear saturation density is the largest source of uncertainty in the CCSN neutrino signal and dynamics even though it plays a subdominant role in most properties of cold NS matter. Meanwhile, changes in other observables affect the properties of cold NSs, while having little effect in CCSNe. To strengthen our conclusions, we perform six octant three-dimensional CCSN simulations varying the effective mass of nucleons at nuclear saturation density. We conclude that neutrino heating and, thus, the likelihood of explosion is significantly increased for EOSs where the effective mass of nucleons at nuclear saturation density is large.

Figure 1

Plots for variations in the effective mass $m^{★}$ and effective mass splitting $\mathrm{\Delta }{m}^{★}$ of (a) the pressure of SNM and (d) of PNM as a function of density, of (b) the mass-radius relations for cold beta-equilibrated NSs and (e) the NS baryonic mass above critical proton fraction, $y_{\mathrm{crit}}=0.11$, as a function of the total gravitational NS mass, and of (c) the density and (f) proton fraction as function of the radius for a canonical $1.4\text{−}{M}_{\odot }$ NS. Effective masses are computed in units of the neutron vacuum mass $m_n$. Nuclear matter pressures are compared to results of Danielewicz et al., Ref. [47]. For PNM there are two bands in Ref. [47] based on a strong (top band) and weak (bottom band) density dependence of the symmetry energy proposed in Ref. [61]. PNM pressure is also compared to chiral effective field theory results of Tews et al., Ref. [62]. Mass-radius relations are compared to the mass of a NS observed by Antoniadis et al., Ref. [50], the mass-radius relations obtained from observations of x-ray bursts by Nättilä et al., Ref. [52], and the radius of a $1.4\text{−}{M}_{\odot }$ NS computed from the limits of tidal deformability of NSs by Most et al., Ref. [63]. Note that the outer $\simeq 1\mathrm{km}$ of canonical $1.4\text{−}{M}_{\odot }$ NSs have densities below ${10}^{14}\mathrm{g}{\mathrm{cm}}^{-3}$. All quantities plotted show only minor dependence with respect to variations in the effective mass at nuclear saturation density $m^{★}$ and the neutron-proton effective mass splitting $\mathrm{\Delta }{m}^{★}$.

Figure 2

Same as Fig. 1 but for variations in the symmetry energy ${\epsilon }_{\mathrm{sym}}$ and the slope of the symmetry energy $L_{\mathrm{sym}}$. Both quantities are shown in units of $\mathrm{MeV}{\mathrm{baryon}}^{-1}$. Because only the two lowest order isospin asymmetry terms are varied, the pressure of SNM (top left) is unchanged while the effects on the pressure of PNM (bottom left) are more pronounced in the region $n\lesssim 2n_{\mathrm{sat}}$. These changes impact the mass radius relationship of NSs more significantly for low mass NSs (top center). Meanwhile, the inner NS composition is affected even for massive NSs (bottom center). The difference in compositions can also be seen for canonical $1.4\text{−}{M}_{\odot }$ NSs, which have similar density profiles in their core (top right) but proton fractions that may differ by a factor of 2 (bottom right).

Figure 3

Same as Fig. 1 but for variations in the isoscalar and isovector incompressibilities $K_{\mathrm{sat}}$ and $K_{\mathrm{sym}}$, respectively, measured in $\mathrm{MeV}{\mathrm{baryon}}^{-1}$. Because of the lower uncertainty in $K_{\mathrm{sat}}$ relative to $K_{\mathrm{sym}}$ the variations in the pressure of SNM (top left) are smaller than those of PNM (bottom left). Due to the imposed constraints the pressures of both SNM and PNM match at $n=n_{\mathrm{sat}}$ and $n=4n_{\mathrm{sat}}$. For NSs with masses lower than $\simeq 2.0M_{\odot }$ there is a direct correlation between increasing incompressibility and NS radius (top center) and inverse correlation with phase space available for direct Urca processes (bottom center). These correlations are inverted for NSs with masses higher than $2.0M_{\odot }$. Canonical $1.4\text{−}{M}_{\odot }$ NSs are more compact for lower incompressibilities (top right) and the core proton fraction is impacted almost exclusively by the isovector incompressibility (bottom right).

Figure 4

Same as Fig. 1 but for variations in the pressure of symmetric nuclear matter and pure neutron matter at $n=4n_{\mathrm{sat}}$. Pressure values are given in $\mathrm{MeV}{\mathrm{fm}}^{-3}$. In the first column we plot the pressure of SNM (top left) and PNM (bottom left). Higher pressures allow for higher NS masses (top center). Proton fraction in the core is higher for lower (higher) pressure of SNM (PNM) (bottom center). Meanwhile, canonical $1.4\text{−}{M}_{\odot }$ NSs are more compact if the pressure at high densities is lower (top right). Again, the proton fraction in the core is higher for lower (higher) pressure of SNM (PNM) (bottom right).

Figure 5

Neutrinosphere (a) radius, (b) density, (c) temperature, and (d) proton fraction for electron neutrinos ${\nu }_e$ (left), electron antineutrinos ${\overline{\nu }}_e$ (center), and heavy neutrinos ${\nu }_x$ (right) for the spherical core collapse of the $20\text{−}{M}_{\odot }$ star of Woosley and Heger [32]. We observe that increasing the EOS effective mass $m^{★}$ leads to smaller neutrinosphere radii and densities as well as higher neutrinosphere temperatures and proton fractions. The only exception is the ${\nu }_x$ neutrinosphere density which has the opposite behavior. Increasing the effective mass splitting $\mathrm{\Delta }{m}^{★}$ has the same qualitative effect as increasing the effective mass, but to a lower order.

Figure 6

Time evolution of (a) neutrino RMS energies and (b) luminosities for ${\nu }_e$ (left), ${\overline{\nu }}_e$ (center), and ${\nu }_x$ (right) as a function of variations in the effective masses in the EOS for the spherical core collapse of the $20\text{−}{M}_{\odot }$ star of Woosley and Heger [32]. We observe that increasing the EOS effective mass $m^{★}$ leads to higher neutrino RMS energies and luminosities. Increasing the EOS effective mass splitting $\mathrm{\Delta }{m}^{★}$ leads to the same qualitative effect as increasing the effective mass $m^{★}$, but to a lower order.

Figure 7

Plot of PNS (a) central density ${\rho }_c$, (b) central temperature $T_c$, (c) shock radius $R_{\mathrm{shock}}$, and (d) radius $R_{12}$ where $\rho ={10}^{12}\mathrm{g}{\mathrm{cm}}^{-3}$ for the spherical core collapse of the $20\text{−}{M}_{\odot }$ star of Woosley and Heger [32] for variations in the effective mass of SNM at saturation density $m^{★}$ and the neutron-proton effective mass splitting in the PNM limit $\mathrm{\Delta }{m}^{★}$.

Figure 8

Plot of PNS (a) density, (b) temperature, and (c) proton fraction profiles at $500\mathrm{ms}$ after core bounce for the $20\text{−}{M}_{\odot }$ star of Woosley and Heger [32] for variations in the effective mass of SNM at saturation density $m^{★}$ and the neutron-proton effective mass splitting in the PNM limit, $\mathrm{\Delta }{m}^{★}$.

Figure 9

Plot of PNS (a) central density ${\rho }_c$, (b) central temperature $T_c$, (c) shock radius $R_{\mathrm{shock}}$, and (d) radius $R_{12}$ where $\rho ={10}^{12}\mathrm{g}{\mathrm{cm}}^{-3}$ for the spherical core collapse of the $20\text{−}{M}_{\odot }$ star of Woosley and Heger [32] for variations in the isoscalar and isovector incompressibilities $K_{\mathrm{sat}}$ and $K_{\mathrm{sym}}$, respectively.

Figure 10

Plot of PNS (a) density, (b) temperature, and (c) proton fraction profiles at $500\mathrm{ms}$ after core bounce for the $20\text{−}{M}_{\odot }$ star of Woosley and Heger [32] for variations in the isoscalar and isovector incompressibilities $K_{\mathrm{sat}}$ and $K_{\mathrm{sym}}$, respectively.

Figure 11

Plot of (a) neutrino RMS energies, $\sqrt{\langle {\epsilon }_{\nu }^2\rangle }$, and (b) luminosities, $L_{\nu }$, for ${\nu }_e$ (left), ${\overline{\nu }}_e$ (center), and ${\nu }_x$ (right) for our octant runs. After $\simeq 100\mathrm{ms}$ after core bounce a clear trend appears and we observe that simulations using EOSs with higher $m^{★}$ lead to higher neutrino RMS energies and neutrino luminosities.

Figure 12

Shock radius $R_{\mathrm{shock}}$ (solid lines, left axis), and accretion rates at $400\mathrm{km}$, ${\dot{M}}_{400}$ (dashed lines, right axis), for our octant simulations. Thick solid line shows the average shock radius while thin lines show the maximum and minimum shock radius. Accretion rates are mostly independent of the EOS and are only plotted up to the point where shock radius reaches $400\mathrm{km}$. The shock radius is very sensitive to the EOS used in the simulation, particularly after it crosses the Si/Si-O interface $\simeq 220\mathrm{ms}$ after core bounce. EOSs with a higher effective mass $m^{★}$ predict longer expansion of the shock radius with the LS220 EOS predicting shock runaway.

Figure 13

Plots of (a) neutrino heating rate $\dot{Q}$, (b) mass in the gain layer $M_{\mathrm{gain}}$, (c) total gain-layer-integrated turbulent kinetic energy $E_{\mathrm{turb}}$ across radial and angular directions, (d) heating efficiency $\eta =\dot{Q}/(L_{{\nu }_e}+L_{{\overline{\nu }}_e})$, (e) ratio ${\tau }_{\mathrm{adv}}/{\tau }_{\mathrm{heat}}$ between the mass advection time scale ${\tau }_{\mathrm{adv}}$ and neutrino heating time scale ${\tau }_{\mathrm{heat}}$, and (f) PNS compactness $\left(G{M}_{\mathrm{PNS}}\right)/\left(R_{\mathrm{PNS}}{c}^2\right)$. We observe a clear correlation between quantities plotted and the effective mass $m^{★}$ of the EOS used in a given simulation. EOSs with larger $m^{★}$ lead to simulations with higher neutrino heating rates, higher heating efficiency, more mass in the gain region, a larger ratio between the advection and the heating time scales, which favors shock runaway, as well as a more compact PNS. Total gain-layer-integrated turbulent energy is anisotropic on large scales, i.e., $E_{\mathrm{turb},\mathrm{r}}\simeq E_{\mathrm{turb},\theta +\varphi }$, nearly EOS independent up to $\simeq 240\mathrm{ms}$, and depends on the shock radius behavior at late times; see discussion in text.