A binary with a $\delta$~Scuti star and an oscillating red giant: orbit and asteroseismology of KIC9773821

We study the $\delta$ Scuti -- red giant binary KIC9773821, the first double-pulsator binary of its kind. It was observed by \textit{Kepler} during its four-year mission. Our aims are to ascertain whether the system is bound, rather than a chance alignment, and to identify the evolutionary state of the red giant via asteroseismology. An extension of these aims is to determine a dynamical mass and an age prior for a $\delta$ Sct star, which may permit mode identification via further asteroseismic modelling. We determine spectroscopic parameters and radial velocities (RVs) for the red giant component using HERMES@Mercator spectroscopy. Light arrival-time delays from the $\delta$ Sct pulsations are used with the red-giant RVs to determine that the system is bound and to infer its orbital parameters, including the binary mass ratio. We use asteroseismology to model the individual frequencies of the red giant to give a mass of $2.10^{+0.20}_{-0.10}$ M$_{\odot}$ and an age of $1.08^{+0.06}_{-0.24}$ Gyr. We find that it is a helium-burning secondary clump star, confirm that it follows the standard $\nu_{\rm max}$ scaling relation, and confirm its observed period spacings match their theoretical counterparts in the modelling code MESA. Our results also constrain the mass and age of the $\delta$ Sct star. We leverage these constraints to construct $\delta$ Sct models in a reduced parameter space and identify four of its five pulsation modes.