Effect of $^{12}C+$ $^{12}C$ Reaction & Convective Mixing on the Progenitor Mass of ONe White Dwarfs

Stars in the mass range ~8 - 12 $M_{\odot }$ are the most numerous massive stars. This mass range is critical because it may lead to supernova (SN) explosion, so it is important for the production of heavy elements and the chemical evolution of the galaxy. We investigate the critical transition mass ($M_{up}$), which is the minimum initial stellar mass that attains the conditions for hydrostatic carbon burning. Stars of masses < $M_{up}$ evolve to the Asymptotic Giant Branch and then develop CO White Dwarfs, while stars of masses $\geqslant $ $M_{up}$ ignite carbon in a partially degenerate CO core and form electron degenerate ONe cores. These stars evolve to the Super AGB (SAGB) phase and either become progenitors of ONe White Dwarfs or eventually explode as electron-capture SN (EC-SN). We study the sensitivity of $M_{up}$ to the C-burning reaction rate and to the treatment of convective mixing. In particular, we show the effect of a recent determination of the $^{12}C+$ $^{12}C$ fusion rate, as well as the extension of the convective core during hydrogen and helium burning on $M_{up}$ in solar metallicity stars. We choose the 9$M_{\odot }$ model to show the detailed characteristics of the evolution with the new C-burning rate.