Astrophysical weak-interaction rates for selected $A=20$ and $A=24$ nuclei

We have evaluated the electron capture rates on ${}^{20}$Ne, ${}^{20}$F, ${}^{24}$Mg, ${}^{24}$Na and the $\beta$ decay rates for ${}^{20}$F and ${}^{24}$Na at temperature and density conditions relevant for the late-evolution stages of stars with $M=8$-12 M${}_{\odot }$. The rates are based on recent experimental data and large-scale shell model calculations. We show that the electron capture rates on ${}^{20}$Ne, ${}^{24}$Mg and the ${}^{20}$F, ${}^{24}$Na $\beta$-decay rates are based on data in this astrophysical range, except for the capture rate on ${}^{20}$Ne, which we predict to have a dominating contribution from the second-forbidden transition between the ${}^{20}$Ne and ${}^{20}$F ground states in the density range $\log \rho Y_e({\mathrm{g}\mathrm{c}\mathrm{m}}^{-3})=9.3$-9.6. The dominance of a few individual transitions allows us to present the various rates by analytical expressions at the relevant astrophysical conditions. We also derive the screening corrections to the rates.