Analytic relations assessing the impact of precursor knowledge and key mission parameters on direct imaging survey yield

The Habitable Worlds Observatory will attempt to image Earth-sized planets in Habitable Zone orbits around nearby Sun-like stars. In this work we explore approximate analytic yield calculations for a future flagship direct imaging mission for a survey sample of uniformly distributed set of identical Sun-like stars. We consider the dependence of this exoplanet detection yield on factors such as eta_Earth, telescope diameter, total on-sky time, orbital phase and separation, inner working angle, flux contrast, desired signal-to-noise ratio, spectral resolution, and other factors. We consider the impact on yield and survey efficiency in the absence of and with precursor knowledge of the Earth-size analog exoplanets. In particular, for precursor knowledge we assume the exoplanet orbital phase at the time of observation can be optimized so as to only image the Earth-size analog exoplanet when it is outside the inner working angle. We find that the yield of flagship direct imaging missions such as Habitable Worlds Observatory will be inner-working angle limited for the estimated exoplanet yields, and will not be impacted by precursor knowledge given our assumptions presented herein. However, we find that the survey efficiency will be enhanced by precursor knowledge. We benchmark our analytic approximations against detailed simulations for coronagraphs and starshades carried out for the HabEx and LUVOIR missions concept studies, and find consistent conclusions. Our analytic relations thus provide quick estimates and derivatives of the impact of key mission parameter choices on exo-Earth yield when considering design trades that can supplement existing computational simulations.