Three-dimensional orbit of AC Her determined: Binary-induced truncation cannot explain the large cavity in this post-AGB transition disk

Some evolved binaries, namely post-asymptotic giant branch binaries, are surrounded by stable and massive circumbinary disks similar to protoplanetary disks found around young stars. Around 10% of these disks are transition disks: they have a large inner cavity in the dust. Previous interferometric measurements and modeling have ruled out the cavity being formed by dust sublimation and suggested that the cavity is due to a massive circumbinary planet that traps the dust in the disk and produces the observed depletion of refractory elements on the surface of the post-AGB star. In this study, we test alternative scenario in which the large cavity could be due to dynamical truncation from the inner binary. We performed near-infrared interferometric observations with the CHARA Array on the archetype of such a transition disk around a post-AGB binary: AC Her. We detect the companion at ten epochs over 4 years and determine the 3-dimensional orbit using these astrometric measurements in combination with the radial velocity time series. This is the first astrometric orbit constructed for a post-AGB binary system. We derive the best-fit orbit with a semi-major axis $2.01 \pm 0.01$ mas ($2.83\pm0.08$ au), inclination $(142.9 \pm 1.1)^\circ$ and longitude of the ascending node $(155.1 \pm 1.8)^\circ$. We find that the theoretical dynamical truncation and dust sublimation radius are at least $\sim3\times$ smaller than the observed inner disk radius ($\sim21.5$ mas or 30 au). This strengthens the hypothesis that the origin of such a cavity is due to the presence of a circumbinary planet.