  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.
