Stellar electron capture rates on neutron-rich nuclei and their impact on stellar core collapse

During the late stages of gravitational core-collapse of massive stars, extreme isospin asymmetries are reached within the core. Due to the lack of microscopic calculations of electron-capture (EC) rates for all relevant nuclei, in general simple analytic parametrizations are employed. We study here several extensions of these parametrizations, allowing for a temperature, electron density, and isospin dependence as well as for odd-even effects. The latter extra degrees of freedom considerably improve the agreement with large-scale microscopic rate calculations. We find, in particular, that the isospin dependence leads to a significant reduction of the global EC rates during core collapse with respect to fiducial results, where rates optimized on calculations of stable $fp$-shell nuclei are used. Our results indicate that systematic microscopic calculations and experimental measurements in the $N\approx 50$ neutron-rich region are desirable for realistic simulations of the core collapse.

Figure 1

Thermodynamic conditions explored during collapse by the central elements of two progenitors, a $15M_{\odot }$ and $25M_{\odot }$ one, from Ref. [15]. (a) Baryon number density and (b) temperature are plotted as a function of the electron density $n_e$. The grid in panel (b) indicates the temperature-electron density grid of the tables for weak interaction rates from Ref. [13].

Figure 2

Results of the extended NSE model of Ref. [16] using thermodynamic conditions reported in Ref. [15] of Fig. 1. (a) Baryonic and total energy per baryon, (b) entropy per baryon, (c) and pressure are plotted as a function of baryonic density.

Figure 3

(a) Average electron-capture $Q$ value as a function of baryon number density for heavy $\left(A\ge 20\right)$ and light $\left(2\le A<20\right)$ clusters, (b) the same versus average isospin asymmetry $\langle (N-Z)/A\rangle$, and (c) $\langle (N-Z)/A\rangle$ versus baryon number density. The thermodynamic conditions correspond to the central element of the core during collapse, taken from Ref. [15], and the nuclear abundances are calculated within the extended NSE model of Ref. [16]. Vertical and horizontal error bars correspond to the standard deviations of the $Q$ and $I$ distributions. The color legend is the same as in Fig. 1. Solid stars in panel (b) mark the nuclei which have been identified in Ref. [12] as playing, from their individual rates, the dominant role for electron capture in CCSN simulations.

Figure 4

Evolution of $\mathrm{\Delta }{E}^{\left(1\right)}$ as a function of (a) $n_e$ and (c) $T$ for constant values of $T$ and, respectively, $n_e$. Panels (b) and (d) illustrate the evolution of electron chemical potential ${\mu }_e$ under the same conditions as in panels (a) and (c).

Figure 5

Electron-capture rates: comparison between predictions of Eq. (1) with different prescriptions for the average transition energy $\mathrm{\Delta }E$ (see text) and values of the tables from Ref. [13]. $\mathrm{\Delta }{E}^{\left(0\right)}$ stands for the original parametrization of Ref. [10]. The thermodynamic conditions, chosen from the grid points of weak interaction rates of Ref. [13], are mentioned on each panel. The symbol and line legend is the same on the three panels.

Figure 6

The residual differences between ${log}_{10}$ of the shell-model rates of Ref. [13] and the approximate rates for each nucleus in the weak interaction library calculated according to different recipes. The thermodynamical conditions, specified on each panel, are the same as in Fig. 5. To increase readability, for models (1) and (3) the $Q$ values have been shifted by 0.2 and, respectively, 0.4 MeV. In all cases, the ${\chi }^2$ values are mentioned in the key legend (see also Table 1).

Figure 7

$T=1.4$ MeV, $n_B=3.52\times {10}^{-4}{\mathrm{fm}}^{-3},Y_e=0.32$. Ratio between individual EC rates corresponding to models (0) and (3) (color levels) and isotopic multiplicities per unit volume (contours). For both $({\lambda }_{EC}^{\left(0\right)}/{\lambda }_{EC}^{\left(3\right)})$ and $n(N,Z),{log}_{10}$ scales are employed.

Figure 8

Ratio between NSE-averaged EC rates using models (2), (3), and (4) for the effective transition-energy parameter $\mathrm{\Delta }E$ (see Sec. 3) and the fiducial parametrization of Ref. [10]. The thermodynamic conditions correspond to the central element during core collapse of two progenitors with $15M_{\odot }$ and $25M_{\odot }$ [15]. The average rate is calculated only on heavy nuclei $\left(A\ge 20\right)$ in panel (a) and on all nuclei in panel (b).