Dates of the beginning and end of polar night and polar day at a given latitude
This online calculator calculates the start and end dates of polar night and polar day at a given latitude. The latitude of the observation site must be north of the north polar circle or south of the south polar circle.
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A polar day is a period when the Sun does not go below the horizon for more than 24 hours (i.e., more than a day). In other words, polar day is the period when the observed Sun at the time of the lower culmination does not fall below the horizon.
And polar night is the period when the Sun does not appear from behind the horizon for more than 24 hours (i.e., more than a day). In other words, polar night is the period when the observed Sun does not rise above the horizon at the time of the upper culmination. Why "observed" Sun - because one must also consider the radius of the solar disk, 16′ or ~0.3°, and the magnitude of atmospheric refraction, which at sea level averages 35′ or ~0.6°. Thus, for the polar night it is necessary that the Sun's altitude at the time of the upper culmination should not be higher than -0.9°, and for the polar day - that the Sun's altitude at the time of the lower culmination should not be lower than -0.9°.
The Sun's altitude at the upper and lower culmination can be found by the Sun's declination in the second equatorial coordinate system, which we can calculate for any latitude on any date, thanks to the Declination and maximum altitude of the Sun above the horizon on a given date calculator. Finding the dates of polar day and polar night becomes a matter of enumerating dates and calculating the Sun's declination on each date for a given latitude - as soon as the declination at the upper culmination becomes less than -0.9° - we fix the onset of polar night, and, accordingly, as soon as the declination at the lower culmination becomes greater than -0.9° - we fix the onset of polar day. The longest polar day is observed at the poles - more than 6 months, and there the longest polar night - but a little less than 6 months - thanks to refraction.
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