Integration time adjusted completeness

Abstract

Abstract. Future, large-scale, exoplanet direct-imaging missions will be capable of discovering and characterizing Earth-like exoplanets. These mission designs can be evaluated using completeness, the fraction of planets from some population that are detectable by a telescope at an arbitrary observation time. However, the original formulation of completeness uses instrument visibility limits and ignores additional integration time and planetary motion constraints. Some of the sampled planets used to calculate completeness may transit in and out of an instrument’s geometric and photometric visibility limits while they are being observed, thereby causing the integration time agnostic calculation to overestimate completeness. We present a method for calculating completeness that accounts for the fraction of planets that leave the visibility limits of the telescope during the integration time period. We define completeness using the aggregate fraction of an orbital period during which planets are detectable, calculated using the specific times that planets enter and leave an instrument’s visibility limits and the integration time. To perform this calculation, we derive analytical methods for finding the planet-star projected separation extrema, times past periastron that these extrema occur, and times past periastron that the planet-star projected separation intersects a specific separation circle. We also provide efficient numerical methods for calculating the planet-star difference in magnitude extrema and times past periastron corresponding to specific values Δmag. Our integration time adjusted completeness shows that, for a planned star observation at 25 pc with 1-day and 5-day integration times, integration time adjusted completeness of Earth-like planets is reduced by 1% and 5% from the integration time agnostic completeness, respectively. Integration time adjusted completeness calculated in this manner also provides a computationally inexpensive method for finding dynamic completeness—the completeness change on subsequent observations.