On Trapped Modes In Variable White Dwarfs As Probes Of The ${}^{12}$C($\alpha ,\gamma$)${}^{16}$O Reaction Rate

We seek signatures of the current experimental ${}^{12}$C$(\alpha ,\gamma {)}^{16}$O reaction rate probability distribution function in the pulsation periods of carbon-oxygen white dwarf models. We find that adiabatic g-modes trapped by the interior carbon-rich layer offer potentially useful signatures of this reaction rate probability distribution function. Probing the carbon-rich region is relevant because it forms during the evolution of low-mass stars under radiative helium burning conditions, mitigating the impact of convective mixing processes. We make direct quantitative connections between the pulsation periods of the identified trapped g-modes in variable WD models and the current experimental ${}^{12}$C$(\alpha ,\gamma {)}^{16}$O reaction rate probability distribution function. We find an average spread in relative period shifts of $\mathrm{\Delta }P\mathrm{/}P\simeq \pm$ 2% for the identified trapped g-modes over the $\pm$ 3$\sigma$ uncertainty in the ${}^{12}$C$(\alpha ,\gamma {)}^{16}$O reaction rate probability distribution function – across the effective temperature range of observed DAV and DBV white dwarfs and for different white dwarf masses, helium shell masses, and hydrogen shell masses. The g-mode pulsation periods of observed white dwarfs are typically given to 6-7 significant figures of precision. This suggests that an astrophysical constraint on the ${}^{12}$C$(\alpha ,\gamma {)}^{16}$O reaction rate could, in principle, be extractable from the period spectrum of observed variable white dwarfs.