Finite-temperature electron-capture rates for neutron-rich nuclei near $N=50$ and effects on core-collapse supernova simulations

The temperature dependence of stellar electron-capture (EC) rates is investigated, with a focus on nuclei near $N=50$, just above $Z=28$, which play an important role during the collapse phase of core-collapse supernovae (CCSN). Two new microscopic calculations of stellar EC rates are obtained from relativistic and nonrelativistic finite-temperature quasiparticle random-phase approximation approaches, for a conventional grid of temperatures and densities. In both approaches, EC rates due to Gamow-Teller transitions are included. In the relativistic calculation, contributions from first-forbidden transitions are also included and add strongly to the EC rates. The new EC rates are compared with large-scale shell-model calculations for the specific case of ${}^{86}\mathrm{Kr}$, providing insight into the finite-temperature effects on the EC rates. At relevant thermodynamic conditions for core collapse, the discrepancies between the different calculations of this paper are within about one order of magnitude. Numerical simulations of CCSN are performed with the spherically symmetric gr1d simulation code to quantify the impact of such differences on the dynamics of the collapse. These simulations also include EC rates based on two parametrized approximations. A comparison of the neutrino luminosities and enclosed mass at core bounce shows that differences between simulations with different sets of EC rates are relatively small ($\approx 5\%$), suggesting that the EC rates used as inputs for these simulations have become well constrained.

Figure 1

Schematic representations of (a) least and (b) most computationally expensive contour integrations as discussed in the main text. Poles on the real and imaginary axes are black markers, with circles representing poles from the exponential prefactor in Eq. (16) and crosses representing poles from the QRPA strength function.

Figure 2

Five quantities are plotted against the excitation energy in ${}^{86}\mathrm{Kr}$, based on shell-model and electron-capture rate calculations with the code nushellx [83] and ecrates [59, 86, 88], respectively. (a) The average occupation of the $g_{9/2}$ shell in ${}^{86}\mathrm{Kr}$, (b) the total GT strength of each initial state in ${}^{86}\mathrm{Kr}$, and the logarithm of the electron-capture rates of each initial state in ${}^{86}\mathrm{Kr}$ without (c) and with (d) weighting with the probabilities of occupying the state $i$ in ${}^{86}\mathrm{Kr}$. (e) The logarithm of the cumulative electron-capture rates including thermal population weighting. The contributions from spins of states are represented by different symbols and the parities are distinguished by red (positive) or black (negative) color. The results are obtained at $T=10$ GK and $\rho Y_e={10}^9$ $\mathrm{g}{\mathrm{cm}}^{-3}$. These conditions are representative of the start of CCSN and high density burning during SN1a.

Figure 3

Electron-capture rate of ${}^{86}\mathrm{Kr}$ as a function of the temperature at $\rho Y_e={10}^{11}\mathrm{g}{\mathrm{cm}}^{-3}$, for the shell-model calculation (SM), and the different temperature-dependent QRPA calculations (FT-QRPA, FT-PNRQRPA GT, and FT-PNRQRPA $\mathrm{GT}+\mathrm{FF}$ including GT and first-forbidden transitions) of this paper, as well as for the approximation from Ref. [30] and the third version of the modified approximation from Ref. [43].

Figure 4

Gamow-Teller strength distribution of ${}^{86}\mathrm{Kr}$ as a function of the transition energy $E_{if}$ (defined in the text), for FT-PNRQRPA, FT-QRPA, and shell-model calculations, respectively, from top to bottom. The red dashed line indicates the ground-state threshold (8.1 MeV for ${}^{86}\mathrm{Kr}$).

Figure 5

Electron-capture rate as a function of the isospin asymmetry $(N-Z)/A$ at $T=10$ GK and $\rho Y_e={10}^{11}\mathrm{g}{\mathrm{cm}}^{-3}$. Results from various electron-capture rate prescriptions are compared: the shell-model calculation (SM), and the different temperature-dependent QRPA calculations (FT-QRPA, FT-PNRQRPA GT, and FT-PNRQRPA $\mathrm{GT}+\mathrm{FF}$ including GT and first-forbidden transitions) of this paper, as well as for the approximation from Ref. [30] and the third version of the modified approximation from Ref. [43].

Figure 6

Region of the nuclear chart with nuclei dominating the electron-capture rate during core-collapse supernovae, as defined in Ref. [5]. The dashed lines distinguish the shell closures $Z=28$ and $N=50$. The color scale represents a ratio of electron-capture rates from different prescriptions, in (a) the FT-QRPA, over the FT-PNRQRPA, with GT transitions only, and in (b) the FT-PNRQRPA with GT over the FT-PNRQRPA with GT and first-forbidden transitions. The rates are obtained at $T=10$ GK and $\rho Y_e={10}^{11}\mathrm{g}{\mathrm{cm}}^{-3}$.

Figure 7

Electron-capture rate as a function of the temperature ($T$) for $\rho Y_e={10}^{11}\mathrm{g}{\mathrm{cm}}^{-3}$ (a) and as function of the density $\rho Y_e$ for $T=10$ GK (b). Results from various electron-capture rate prescriptions are compared: the different temperature-dependent QRPA calculations (FT-QRPA, FT-PNRQRPA GT, and FT-PNRQRPA $\mathrm{GT}+\mathrm{FF}$ including GT and first-forbidden transitions) of this paper, as well as for the approximation from Ref. [30] and the third version of the modified approximation from Ref. [43].

Figure 8

Core-collapse simulations results obtained from gr1d [93, 94] and nulib [5] codes, with s15WW95 [95] progenitor and SFHo [96] equation of state. Different finite-temperature electron-capture calculations are compared; see labels in panel (a), where the three first are the finite-temperature microscopic calculations introduced in this paper, “approx.” corresponds to the parametrization [30], and “approx. mod.” stands for its modified version (third parametrization in Ref. [43]). (a) The electron-fraction $Y_e$ as a function of central baryon density $\rho$, (b) the electron neutrino luminosity as measured at a radius of 500 km as a function of time after bounce, and (c) the central velocity as a function of the enclosed mass.