As a non-mathematical analysis, and I may have the wrong end of the stick here so I'll beg your pardon in advance, but isn't it the case that as you move further south in the southern summer then the position of sunset moves ever further north until after passing the Tropic of Capricorn the sunset has moved so far north that the sun doesn't set at all? Indeed, the sun at its lowest point during the 24 hours of daylight will define true north. (And the reverse for the northern summer) Surely the secret lies in the angle of tilt of our home planet?
**********************************
Through difficulties to the cinema