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

Stars with an initial mass of $\sim 7$-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. 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 $4^+_\text{g.s.}\rightarrow{}2^+_1$ transition in $^{24}\text{Na}(e^-,ν_e)^{24}\text{Ne}$ and the ${5/2^+_{\text{g.s.}} \rightarrow{} 1/2^+_{\text{g.s.}}}$ transition in $^{27}\text{Al}(e^-,ν_e)^{27}\text{Mg}$. The relevant nuclear matrix elements are determined through shell model calculations and constraints from CVC theory. We then investigate the astrophysical impact using the stellar evolution code MESA and through timescale arguments. In the relevant temperature range, the forbidden transitions substantially reduce the threshold densities for $^{24}\text{Na}(e^-,ν_e)^{24}\text{Ne}$ and $^{27}\text{Al}(e^-,ν_e)^{27}\text{Mg}$. In the MESA models, $^{24}\text{Na}(e^-,ν_e)^{24}\text{Ne}$ now occurs immediately following the onset of $^{24}\text{Mg}(e^-,ν_e)^{24}\text{Na}$. 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 $^{27}\text{Al}$ and $^{27}\text{Mg}$ may have a minor effect on the early evolution but is unlikely to affect the outcome. The studied transitions should be included when calculating weak interaction rates between $^{24}$Na and $^{24}$Ne for temperatures $\log_{10}(T[\text{K}])\lesssim8.5$ and between $^{27}$Al and $^{27}$Mg for $\log_{10}(T[\text{K}])\lesssim8.8$.