Astrometric Methods for Detecting Exomoons Orbiting Imaged Exoplanets: Prospects for Detecting Moons Orbiting a Giant Planet in $\alpha$ Centauri A's Habitable Zone

Nearby giant exoplanets offer an opportunity to search for moons (exomoons) orbiting them. Here, we present a simulation framework for investigating the possibilities of detecting exomoons via their astrometric signal in planet-to-star relative astrometry. We focus our simulations on $\alpha$ Centauri A, orbited by a hypothetical giant planet consistent with candidate detections in Very Large Telescope and James Webb Space Telescope observations. We consider a variety of observatory architectures capable of searching for exomoons, including upcoming facilities and also a hypothetical dedicated facility $-$ e.g., a purpose-built space telescope with diameter = 3m, central observing wavelength of 500 nm, and contrast-limited performance of $\sim$10$^{-9}$ in 1 hr observations. We find that such a facility would be capable of detecting $\sim$Earth-mass moons in a five year campaign, assuming a Saturn-mass planet. More generally, we simulate expected detection limits for a variety of levels of astrometric precision. We find that moons as small as $\sim$0.2 M$_\oplus$ on orbital periods of 4$-$30 days can be detected with astrometric precision of 0.1 mas and observing cadence of 1 hr over a five year campaign. Additionally, we find that a 39m ground-based telescope can detect Earth-sized exomoons orbiting the same hypothetical planet with a more modest observing cadence of one day. We discuss these results as motivation for a dedicated space observatory as well as a more detailed study of the physical parameters of a greater variety of star-planet-moon systems.