Forbidden electron capture on ${}^{24}\mathrm{Na}$ and ${}^{27}\mathrm{Al}$ in degenerate oxygen-neon stellar cores

Background: Stars with an initial mass of $\approx 7$–11 solar masses form degenerate oxygen-neon cores following carbon burning. Electron captures in such cores can trigger runaway oxygen burning, resulting in either a collapse or a thermonuclear explosion. Previous work constrained the contribution of the forbidden $0_{\text{g.s.}}^{+}\to 2_{\text{g.s.}}^{+}$ transition to the ${}^{20}\mathrm{Ne}(e^{-},{\nu }_e){}^{20}\mathrm{F}$ rate and discussed its significance for the evolution of the core.

Purpose: We provide a detailed description of the formalism used in previous work and apply it to two further forbidden transitions that are relevant to degenerate oxygen-neon cores: the $4_{\text{g.s.}}^{+}\to 2_1^{+}$ transition in ${}^{24}\mathrm{Na}(e^{-},{\nu }_e){}^{24}\mathrm{Ne}$ and the $5/2_{\text{g.s.}}^{+}\to 1/2_{\text{g.s.}}^{+}$ transition in ${}^{27}\mathrm{Al}(e^{-},{\nu }_e){}^{27}\mathrm{Mg}$.

Method: The relevant nuclear matrix elements are determined through shell model calculations and constraints from the conserved vector current (CVC) theory. We then investigate the astrophysical impact using the stellar evolution code mesa (Modules for Experiment in Stellar Astrophysics) and through timescale arguments.

Results: In the relevant temperature range, the forbidden transitions substantially reduce the threshold densities for ${}^{24}\mathrm{Na}(e^{-},{\nu }_e){}^{24}\mathrm{Ne}$ and ${}^{27}\mathrm{Al}(e^{-},{\nu }_e){}^{27}\mathrm{Mg}$. In the mesa models, ${}^{24}\mathrm{Na}(e^{-},{\nu }_e){}^{24}\mathrm{Ne}$ now occurs immediately following the onset of ${}^{24}\mathrm{Mg}(e^{-},{\nu }_e){}^{24}\mathrm{Na}$. The impact on the overall evolution is uncertain; this is due to known difficulties in accounting for convective instabilities triggered by the $A=24$ electron captures. The transition between ${}^{27}\mathrm{Al}$ and ${}^{27}\mathrm{Mg}$ may have a minor effect on the early evolution but is unlikely to affect the outcome.

Conclusions: The studied transitions should be included when calculating weak interaction rates between ${}^{24}\mathrm{Na}$ and ${}^{24}\mathrm{Ne}$ for temperatures ${log}_{10}\left(T\left[\text{K}\right]\right)\lesssim 8.5$ and between ${}^{27}\mathrm{Al}$ and ${}^{27}\mathrm{Mg}$ for ${log}_{10}\left(T\left[\text{K}\right]\right)\lesssim 8.8$.

Figure 1

Energy level diagram illustrating the ${}^{24}\mathrm{Mg}(e^{-},{\nu }_e){}^{24}\mathrm{Na}$ and ${}^{24}\mathrm{Na}(e^{-},{\nu }_e){}^{24}\mathrm{Ne}$ reactions. We list excitation energies and, for ${}^{24}\mathrm{Na}$ and ${}^{24}\mathrm{Ne}$, the ground-state energy relative to the ground state of the parent nucleus. All energies are in units of MeV. The nuclear data have been taken from Refs. [18, 41]. Relevant transitions are shown with arrows, with the forbidden transition in red.

Figure 2

Electron capture rate on ${}^{24}\mathrm{Na}$ as a function of density with Coulomb corrections included. We plot the total rate as well as the individual contributions from the four transitions that dominate the capture rate. The upper panel corresponds to ${log}_{10}\left(T\left[\text{K}\right]\right)=8.4$ and the lower panel to ${log}_{10}\left(T\left[\text{K}\right]\right)=7.8$.

Figure 3

Energy level diagram illustrating the transitions that determine the rate of the ${}^{27}\mathrm{Al}(e^{-},{\nu }_e){}^{27}\mathrm{Mg}$ electron capture as well as the reverse $\beta$ decay. The forbidden transition is in red. We list excitation energies as well as the ground-state energy of ${}^{27}\mathrm{Mg}$ relative to the ground state of ${}^{27}\mathrm{Al}$. Both are in units of MeV. The nuclear data have been taken from Ref. [44].

Figure 4

Electron capture rates on ${}^{27}\mathrm{Al}$ (solid) and the reverse ${\beta }^{-}$ decay rates of ${}^{27}\mathrm{Mg}$ (dashed) as a function of density with Coulomb corrections included. We have assumed a temperature of ${log}_{10}\left(T\left[\text{K}\right]\right)=8.3$. In the upper panel, we only include allowed transitions: This reproduces the rate from Ref. [21]. In the lower panel, we show the impact of the forbidden transition. Note that the total electron capture and ${\beta }^{-}$ decay rates are plotted assuming the $\mathrm{SM}+\mathrm{CVC}$ rate.

Figure 5

Central temperature as a function of central density in MESA simulations with three different treatments of the forbidden transition between ${}^{24}\mathrm{Na}$ and ${}^{24}\mathrm{Ne}$: included using the SM rate, using the $\mathrm{SM}+\mathrm{CVC}$ rate, or ignored completely. The onset of each electron capture is labeled with the parent nucleus. Note that for simulations including the forbidden transition capture on ${}^{24}\mathrm{Na}$ does not occur separately from capture on ${}^{24}\mathrm{Mg}$.

Figure 6

Temperature and electron fraction profiles at ignition for the models in Fig. 5. Note that we plot the quantities as a function of density as opposed to radius. We label the approximate locations where electron capture on ${}^{24}\mathrm{Mg}$ and (when occurring separately) ${}^{24}\mathrm{Na}$ is taking place. The circles mark the center of the core: Note the correspondence to the final temperatures and densities in Fig. 5.

Figure 7

Evolution of central temperature and density as in Fig. 5, but with the core having a residual carbon mass fraction of $X\left({}^{12}\mathrm{C}\right)=0.01$. The dotted gray line is the carbon ignition curve taken from Fig. 6 in Ref. [48]. Along this curve, energy generation from carbon burning equals thermal neutrino losses.

Figure 8

Ratio between the probabilities of occupying the $1_1^{+}$ and $4_{\text{g.s.}}^{+}$ states as a function of the density. Solid lines corresponds to the steady-state expression (D5) and dashed lines to the thermal-equilibrium limit (D7). Note that there is no dashed blue line as (D7) is negligible (${log}_{10}(P_2/P_1)<-38$) for ${log}_{10}\left(T\left[\text{K}\right]\right)=7.8$.

Figure 9

Electron capture rates on ${}^{24}\mathrm{Na}$ at ${log}_{10}\left(T\left[\text{K}\right]\right)=7.8$ assuming that the relative population of the $1_1^{+}$ state is given by (D5).

Figure 10

Neutrino loss rates for the electron capture on ${}^{24}\mathrm{Na}$ at ${log}_{10}\left(T\left[\text{K}\right]\right)=7.8$, assuming that the relative population of the $1_1^{+}$ state is given by (D5).