Single-visit photometric and obscurational completeness

We report a method that uses "completeness'' to estimate the number of extrasolar planets discovered by an observing program with a direct-imaging instrument. We develop a completeness function for Earth-like planets on "habitable'' orbits for an instrument with a central field obscuration, uniform sensitivity in an annular detection zone, and limiting sensitivity that is expressed as a "delta magnitude'' with respect to the star, determined by systematic effects ( given adequate exposure time). We demonstrate our method of estimation by applying it to our understanding of the coronagraphic version of the Terrestrial Planet Finder (TPF-C) mission as of 2004 October. We establish an initial relationship between the size, quality, and stability of the instrument's optics and its ability to meet mission science requirements. We provide options for increasing the fidelity and versatility of the models on which our method is based, and we discuss how the method could be extended to model the TPF-C mission as a whole to verify that its design can meet the science requirements.