Gamma Ray Heating and Neutrino Cooling Rates due to Weak Interaction Processes on sd-shell Nuclei in Stellar Cores

Gamma ray heating and neutrino cooling rates, due to weak interaction processes, on $sd$-shell nuclei in stellar core are calculated using the proton neutron quasiparticle random phase approximation theory. The recent extensive experimental mass compilation of \citep{Wang12}, other improved model input parameters including nuclear quadrupole deformation \citep{Ram01}, \citep{Mol16} and physical constants are taken into account in the current calculation. The purpose of this work is two fold, one is to improve the earlier calculation of weak rates performed by \citep{Nabi99} using the same theory. We further compare our results with previous calculations. The selected $sd$-shell nuclei, considered in this work, are of special interest for the evolution of O-Ne-Mg core in 8-10 M$_\odot$ stars due to competitive gamma ray heating rates and cooling by URCA processes. The outcome of these competitions is to determine, whether the stars end up as a white dwarf \citep{Nabi08}, an electron-capture supernova \citep{Jones13} or Fe core-collapse supernova \citep{Suz16}. The selected $sd$-shell nuclei for calculation of associated weak-interaction rates include $^{20,23}$O, $^{20,23}$F, $^{20,23,24}$Ne, $^{20,23-25}$Na, and $^{23-25}$Mg. The cooling and heating rates are calculated for density range ($10 \leq \rho($\;g.cm$^{-3}) \leq$ 10$^{11}$) and temperature range ($0.01\times10^{9}$ $\leq$ $\;T(K)$ $\leq$ $30\times10^{9}$). The calculated gamma heating rates are orders of magnitude bigger than the shell model rates (except for $^{25}$Mg at low densities). At high temperatures the gamma heating rates are in reasonable agreement. The calculated cooling rates are up to an order of magnitude bigger for odd-A nuclei.