Equinoxes and Solstices

The ecliptic longitude of the sun when it reaches the vernal equinox is $\lambda = 0^\circ$. It follows, from Eq. (32), that the altitude of the sun on the day of the equinox is given by $\sin a = \cos L\,\cos \alpha_0$. Thus, the sun rises when $\alpha_0=-90^\circ$, culminates at an altitude of $90^\circ - \vert L\vert$ when $\alpha_0 = 0^\circ$, and sets when $\alpha_0=90^\circ$. We conclude that the length of the equinoctial day is $180$ time-degrees, which is equivalent to 12 hours (since $15^\circ$ of right ascension cross the meridian in one hour). Thus, day and night are equally long on the day of the vernal equinox. It is easily demonstrated that the same is true on the day of the autumnal equinox.

The ecliptic longitude of the sun when it reaches the summer solstice is $\lambda = 90^\circ$. It follows that the altitude of the sun on the day of the solstice is given by $\sin a = \sin L\,\sin \epsilon + \cos L\,\cos\epsilon\,\sin \alpha_0$. Thus, the sun rises when $\alpha_0=-\sin^{-1}(\tan L\,\tan\epsilon)$, culminates at an altitude of $90^\circ - \vert L- \epsilon\vert$ when $\alpha_0=90^\circ$, and sets when $\alpha_0=180^\circ + \sin ^{-1}(\tan L\,\tan\epsilon)$. We conclude that the length of the longest day of the year in the earth's northern hemisphere (which, of course, occurs when the sun reaches the summer solstice) is $180 +2\,\sin ^{-1}(\tan L\,\tan\epsilon)$ time-degrees. Likewise, the length of the shortest night (which also occurs at the summer solstice) is $180-2\, \sin^{-1}(\tan L\,\tan\epsilon)$ time-degrees. These formulae are only valid for northern latitudes below the arctic circle. At higher latitudes, the sun never sets for part of the year, and the longest ``day" is consequently longer than 24 hours. It is easily demonstrated that the shortest day in the earth's northern hemisphere, which takes place when the sun reaches the winter solstice, is equal to the shortest night, and the longest night (which also occurs at the winter solstice) to the longest day. Moreover, the sun culminates at an altitude of $90^\circ - \vert L + \epsilon\vert$ on day of the winter solstice. Again, at latitudes above the arctic circle, the sun never rises for part of the year, and the longest ``night" is consequently longer than 24 hours.

Consider an observation site on the earth's surface of latitude $L$ which lies above the northern arctic circle. The declination of the sun on the first day after the spring equinox on which it fails to set is $\delta = 90^\circ - L$. According to Eq. (15), its ecliptic longitude on this day is $\sin ^{-1} (\cos L/\sin \epsilon)$. Likewise, the declination of the sun on the day when it starts to set again is $\delta = 90^\circ - L$, and its ecliptic longitude is $180^\circ - \sin^{-1}(\cos L/\sin \epsilon)$. Assuming that the sun travels around the ecliptic circle at a uniform rate (which is approximately true), the fraction of a year that the sun stays above the horizon in summer is $0.5-\sin^{-1}(\cos L/\sin \epsilon)/180^\circ$. It is easily demonstrated that the fraction of a year that the sun stays below the horizon in winter is also $0.5-\sin^{-1}(\cos L/\sin \epsilon)/180^\circ$.