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Our latitude in Shawnee is 35 degrees north, and we find Polaris 35 degrees above the northern horizon. Similarly, to sail west at a constant latitude Columbus kept the north star at a constant altitude above the horizon.
Since the atmosphere bends the rays of the Sun, the polar day is longer than the polar night, and the area that is affected by polar night is somewhat smaller than the area of midnight sun. The polar circle is located at a latitude between these two areas, at the latitude of approximately 66.5 degrees.