2025AJ....170..354S

Science yield studies will drive the development of future direct imaging telescopes, such as the Habitable Worlds Observatory. These studies rely on a metric called completeness, which represents the fraction of planets from an assumed planet population that can be detected for a given observing scenario. Completeness is often calculated by comparing the brightness of planets in the population to the “photometric constraint,” or the dimmest planet detectable in a given observing scenario. The photometric constraint has also been used to calculate the probability of directly imaging a planet detected by the radial velocity method. This work shows how to numerically and analytically invert the analytic exposure time calculator used by the Nancy Grace Roman Space Telescope's coronagraph instrument to calculate a precise photometric constraint that accounts for planet-star separation, integration time, zodiacal light brightness during observation, and assumed exozodiacal light. This refined photometric constraint is then used to calculate completeness and probability of detection in an efficient manner. Finally, we show that the probability of detection values calculated using the refined photometric constraint closely tracks the planet's true detectability. We validate our approach by generating realistic planetary systems, simulating an extreme-precision radial velocity survey, performing orbit fits, and computing the probability of directly imaging the fitted planets with a telescope design similar to the future Habitable Worlds Observatory. Our validation tests show that the probability of detection values predicts the number of direct imaging detections to within 4% when scheduling observations at high probability of detection values.