Obscurational completeness

We study the selection effects imposed by a central field obscuration on direct searches for companions to stars. We analyze both "specific" versions of obscurational completeness, which depend explicitly on the semimajor axis and eccentricity of companion orbits, and "ensemble" versions, which integrate over a companion population represented by probability distributions for semimajor axis and eccentricity. In the case of effectively nonmoving companions (orbital periods significantly longer than the observing program) or for a single observation, we find that high completeness requires obscurations much smaller than the smallest orbit size of interest. For example, to achieve 95% obscurational completeness for a particular orbit size on a single visit, the obscuration must be less than similar to1/3 that orbit size. For relevant ensembles of terrestrial planets, an observing program can find more than 95% of all companions in less than three visits over 3 months - if the obscuration is smaller than the smallest orbit of interest. If the obscuration is larger, but not larger than the largest orbit of interest, then the design obscurational completeness is reduced, but 95% of all the planets that can ever be found are found by less than six visits over 2-5 yr. Nevertheless, achieving such high completeness when the ensemble is only partly resolved calls for several low-yield visits.