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.

中文翻译：

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积分时间调整的完整性

未来的大规模系外行星直接成像任务将能够发现和表征类地系外行星。可以使用完整性来评估这些任务设计，即在任意观测时间，望远镜可探测到的某些群体中行星的比例。然而，完整性的原始公式使用仪器可见性限制并忽略额外的积分时间和行星运动约束。一些用于计算完整性的采样行星在被观测时可能会进出仪器的几何和光度能见度限制，从而导致积分时间不可知计算高估完整性。我们提出了一种计算完整性的方法，该方法考虑了在积分时间段内离开望远镜能见度限制的行星的比例。我们使用行星可检测到的轨道周期的总分数来定义完整性，使用行星进入和离开仪器的能见度限制的特定时间和积分时间计算。为了执行此计算，我们推导出了用于寻找行星-恒星预计分离极值、这些极值发生的越过近天体的时间以及行星-恒星预计分离与特定分离圈相交的越过近天体的时间的分析方法。我们还提供了有效的数值方法来计算对应于特定值 Δmag 的行星-恒星的极值差异和经过近天体的时间。我们的积分时间调整完整性表明，对于计划在 25 pc 处进行的具有 1 天和 5 天积分时间的恒星观测，类地行星的积分时间调整完整性比积分时间不可知的完整性降低了 1% 和 5% ， 分别。以这种方式计算的积分时间调整的完整性还提供了一种计算成本低廉的方法来寻找动态完整性——后续观测的完整性变化。