Dependence of weak interaction rates on the nuclear composition during stellar core collapse

We investigate the influences of the nuclear composition on the weak interaction rates of heavy nuclei during the core collapse of massive stars. The nuclear abundances in nuclear statistical equilibrium (NSE) are calculated by some equation of state (EOS) models including in-medium effects on nuclear masses. We systematically examine the sensitivities of electron capture and neutrino-nucleus scattering on heavy nuclei to the nuclear shell effects and the single-nucleus approximation. We find that the washout of the shell effect at high temperatures brings significant change to weak rates by smoothing the nuclear abundance distribution: the electron capture rate decreases by $\sim 20\%$ in the early phase and increases by $\sim 40\%$ in the late phase at most, while the cross section for neutrino-nucleus scattering is reduced by $\sim 15\%$. This is because the open-shell nuclei become abundant instead of those with closed neutron shells as the shell effects disappear. We also find that the single-nucleus description based on the average values leads to underestimations of weak rates. Electron captures and neutrino coherent scattering on heavy nuclei are reduced by $\sim 80\%$ in the early phase and by $\sim 5\%$ in the late phase, respectively. These results indicate that NSE like EOS accounting for shell washout is indispensable for the reliable estimation of weak interaction rates in simulations of core-collapse supernovae.

Figure 1

Temperature (red solid line), electron fraction (blue dashed line), and average energy of electron-type neutrinos (green dashed-dotted line) as a function of density at the center of the collapsing core in the reference supernova simulation of an 11.2 ${\mathrm{M}}_{\odot }$ progenitor [36].

Figure 2

Abundances of elements as a function of the mass number for models WS (red solid lines), FS (blue dashed-dotted lines), and NS (green dashed lines) at $({\rho }_B\left[{\mathrm{g}/\mathrm{cm}}^3\right],T\left[\mathrm{MeV}\right],Y_e)=(2.0\times {10}^{10},0.63,0.41)$, ($2.0\times {10}^{11}$, 0.90, 0.36), and ($2.0\times {10}^{12}$, 1.25, 0.29) from top to bottom.

Figure 3

Nuclear abundances in the $(N,Z)$ plane at ${\rho }_B=2.0\times {10}^{12}{\mathrm{g}/\mathrm{cm}}^3$, $T=1.25\mathrm{MeV}$, and $Y_e=0.29$ for models FS (top), WS (middle), and NS (bottom) from top to bottom. The cyan circled dot and green cross indicate the average values and most probable nucleus, respectively. Dotted lines are neutron and proton magic numbers.

Figure 4

Mass fraction of heavy nuclei with $Z\ge 6$ as a function of central density for models WS (red solid lines), FS (blue dashed-dotted lines), and NS (green dashed lines) in the top panel. The bottom panel displays the average mass and proton numbers, $\overline{A}$ and $\overline{Z}$ (solid lines), and those of the most probable nuclei, $A_{\mathrm{mp}}$ and $Z_{\mathrm{mp}}$ (dashed lines), for models WS (red), FS (blue), and NS (green).

Figure 5

The nuclear species for which the data of FFN (blue squares), ODA (cyan squares), LMP (green squares), or LMSH (red squares) or the approximation formula (yellow squares) for electron capture rate is adopted. Black circles represent stable nuclei and dashed black lines display the neutron and proton drip lines (dashed black lines), which are estimated by the KTUY mass formula [42]. A dashed-dotted line indicates the trajectory of average neutron and proton numbers of heavy nuclei in the center of collapsing cores, which is calculated by model WS and shown in Fig. 4.

Figure 6

The nuclear species with the largest electron capture contribution, $n_i{\lambda }_i/n_B$, which account for the top $50\%$ (black squares), $50\text{–}90\%$ (red circles), $90\text{–}99\%$ (blue triangles), and $99\text{–}99.9\%$ (green diamonds) of the total electron capture rate per baryon ${\lambda }^{\mathrm{ec}}$ for model WS at ${\rho }_B=2.0\times {10}^{10}{\mathrm{g}/\mathrm{cm}}^3$ (top panel) and $2.0\times {10}^{11}{\mathrm{g}/\mathrm{cm}}^3$ (bottom panel). Dotted lines are neutron and proton magic numbers.

Figure 7

The nuclear species with the largest electron capture contribution, $n_i{\lambda }_i/n_B$, which account for the top $50\%$ (black squares), $50\text{–}90\%$ (red circles), $90\text{–}99\%$ (blue triangles), and $99\text{–}99.9\%$ (green diamonds) of the total electron capture rate per baryon ${\lambda }^{\mathrm{ec}}$ for models FS (top panel), WS (middle panel), and NS (bottom panel) at ${\rho }_B=2.0\times {10}^{12}{\mathrm{g}/\mathrm{cm}}^3$. Dotted lines are neutron and proton magic numbers.

Figure 8

Electron capture rate per baryon of heavy nuclei, ${\lambda }^{\mathrm{ec}}={\sum }_i{n}_i{\lambda }_i/n_B$, as a function of central density for models WS (red solid lines), FS (blue solid lines), and NS (green solid lines). The red dashed line shows the result for model WS in which the approximation formula is adopted for all nuclei and weak rate data are not utilized.

Figure 9

Ratio of electron capture rates per baryon ${\lambda }^{\mathrm{ec}}$ (left panel) and those per nuclei, ${\overline{\lambda }}^{\mathrm{ec}}={\sum }_i{n}_i{\lambda }_i/\left({\sum }_i{n}_i\right)$, (right panel) for models WS (red solid lines) and NS (green dashed lines) to those for model FS (blue dashed-dotted lines) as a function of central density.

Figure 10

Ratio of cross sections of neutrino-nucleus scattering per baryon ${\sigma }^{\mathrm{sc}}={\sum }_i{n}_i{\sigma }_i/n_B$ (left panel) and those per nuclei ${\overline{\sigma }}^{\mathrm{sc}}={\sum }_i{n}_i{\sigma }_i/\left({\sum }_i{n}_i\right)$ (right panel) for models WS (red solid lines) and NS (green dashed lines) to those for model FS (blue dashed-dotted lines) as a function of central density.

Figure 11

Electron capture rate per baryon of heavy nuclei as a function of central density for model WS based on the multinucleus description with the parametrized rate provided by Langanke et al. [32] (red solid lines) and the Bruenn rate [34] (red dashed lines). Those of the average nucleus are shown for the former rate (cyan solid lines) and the latter (cyan dashed lines). The black dashed-dotted line shows that of the most probable nucleus for the former rate.

Figure 12

Ratio of electron capture rates in the single-nucleus descriptions to those of the multinucleus descriptions for models WS (red solid lines), FS (blue solid lines), and NS (green solid lines). Left and right panels display those of the average nucleus and the most probable one.

Figure 13

Ratio of cross sections of neutrino-nucleus scattering in the single-nucleus descriptions to those of the multinucleus descriptions for models WS (red solid lines), FS (blue solid lines), and NS (green solid lines). Left and right panels display those of the average nucleus and the most probable one.

Figure 14

Dispersion of mass number normalized by the average mass number squared, $(\overline{A^2}-{\overline{A}}^2)/\overline{A^2}$, as a function of central density for models FS (blue dashed-dotted line), WS (red solid line), and NS (green dashed line).