Forbidden electron capture on Na24 and Al27 in degenerate oxygen-neon stellar cores

Abstract

Background: Stars with an initial mass of $\ensuremath{\approx}7$--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 ${0}_{\text{g.s.}}^{+}\ensuremath{\rightarrow}{2}_{\text{g.s.}}^{+}$ transition to the $^{20}\mathrm{Ne}({e}^{\ensuremath{-}},{\ensuremath{\nu}}_{e})^{20}\mathrm{F}$ 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 ${4}_{\text{g.s.}}^{+}\ensuremath{\rightarrow}{2}_{1}^{+}$ transition in $^{24}\mathrm{Na}({e}^{\ensuremath{-}},{\ensuremath{\nu}}_{e})^{24}\mathrm{Ne}$ and the $5/{2}_{\text{g.s.}}^{+}\ensuremath{\rightarrow}1/{2}_{\text{g.s.}}^{+}$ transition in $^{27}\mathrm{Al}({e}^{\ensuremath{-}},{\ensuremath{\nu}}_{e})^{27}\mathrm{Mg}$.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 $^{24}\mathrm{Na}({e}^{\ensuremath{-}},{\ensuremath{\nu}}_{e})^{24}\mathrm{Ne}$ and $^{27}\mathrm{Al}({e}^{\ensuremath{-}},{\ensuremath{\nu}}_{e})^{27}\mathrm{Mg}$. In the mesa models, $^{24}\mathrm{Na}({e}^{\ensuremath{-}},{\ensuremath{\nu}}_{e})^{24}\mathrm{Ne}$ now occurs immediately following the onset of $^{24}\mathrm{Mg}({e}^{\ensuremath{-}},{\ensuremath{\nu}}_{e})^{24}\mathrm{Na}$. 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 ${log}_{10}(T[\text{K}])\ensuremath{\lesssim}8.5$ and between $^{27}\mathrm{Al}$ and $^{27}\mathrm{Mg}$ for ${log}_{10}(T[\text{K}])\ensuremath{\lesssim}8.8$.