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We have modeled the run of brightness of the sky during twilight in order to estimate the yearly amount of nightime available for visual and infrared observations ($0.55 - 8.0 \mu m$) with the Polar Stratospheric Telescope (POST). We have also applied the model to the estimation of the brightness of scattered moonlight. Our calculations were motivated by the sensitive dependence on solar depression angle of the duration of twilight for ground-based observers in the polar regions. The twilight model represents singly and doubly Rayleigh scattered sunlight and the full three-dimensional geometry of the earth's atmosphere and shadow. The onset of nightime was determined using an additional model of the brightness of the night sky which incorporates the thermal emission of the atmosphere, scattered and thermal emissivity of the zodiacal light, and the thermal emission of POST. We find at $0.55 \mu m$ for an observer at an altitude of $13 km$ that $ZA_{\sun}$ is $0.7 \deg$ less than for a sea-level observer to attain the same twilight sky brightness. The difference decreases to zero at $6 - 8 \mu m$. In itself this will have little effect on the available nightime for POST. However, the predicted brightness of the $2.5 \mu m$ sky as observed with POST is three orders of magnitude fainter than for a ground-based observer, and as a consequence the sensible twilight is prolonged to $ZA_{\sun}=110 \deg$. In contrast, the great brightness of the sky (although less than for a ground-based observer) in the range $3.5 - 8.0 \mu m$ reduces the sensible duration of twilight. At the end of twilight $ZA_{\sun}=101 \deg$ at $3.5 \mu m$, $ZA_{\sun}=97 \deg$ at $6.0 \mu m$, and $ZA_{\sun}=95 \deg$ at $8.0 \mu m$. The number of hours of darkness per year for an arctic site (latitude $= 65 \deg$) is 2000 at $0.55 \mu m$, 1850 at $2.5 \mu m$, 2700 at $3.5 \mu m$, 3300 at $6.0 \mu m$, and 3600 at $8.0 \mu m$. Finally, we find the brightness of atmospherically scattered moonlight is six times less for a POST observer than for a ground-based observer. Since the brightness of scattered moonlight exceeds other sources of background light in the $2 - 3.5 \mu m$ band and shortward of $0.7 \mu m$, the available dark-time will be greater for stratospheric observations in those wavebands than for ground-based observations.
