Seismic Signatures of the ${}^{12}$C($\alpha$, $\gamma$)${}^{16}$O Reaction Rate in White Dwarf Models with Overshooting

We consider the combined effects that overshooting and the ${}^{12}$C($\alpha$, $\gamma$)${}^{16}$O reaction rate have on variable white dwarf stellar models. We find that carbon-oxygen white dwarf models continue to yield pulsation signatures of the current experimental ${}^{12}$C($\alpha$, $\gamma$)${}^{16}$O reaction rate probability distribution function when overshooting is included in the evolution. These signatures hold because the resonating mantle region, encompassing $\simeq$,0.2,\Msun\ in a typical $\simeq$,0.6,\Msun\ white dwarf model, still undergoes radiative helium burning during the evolution to a white dwarf. Our specific models show two potential low-order adiabatic g-modes, $g_2$ and $g_6$, that signalize the ${}^{12}$C($\alpha$, $\gamma$)${}^{16}$O reaction rate probability distribution function. Both g-mode signatures induce average relative period shifts of $\mathrm{\Delta }P\mathrm{/}P=0.44\mathrm{\%}$ and $\mathrm{\Delta }P\mathrm{/}P=1.33\mathrm{\%}$ for $g_2$ and $g_6$ respectively. We find that $g_6$ is a trapped mode, and the $g_2$ period signature is inversely proportional to the ${}^{12}$C($\alpha$, $\gamma$)${}^{16}$O reaction rate. The $g_6$ period signature generally separates the slower and faster reaction rates, and has a maximum relative period shift of $\mathrm{\Delta }P\mathrm{/}P=3.45\mathrm{\%}$. We conclude that low-order g-mode periods from carbon-oxygen white dwarfs may still serve as viable probes for the ${}^{12}$C($\alpha$, $\gamma$)${}^{16}$O reaction rate probability distribution function when overshooting is included in the evolution.