## Tuesday, March 22, 2011

### Changes in the Amount of Daylight Throughout the Year

Yesterday was the Vernal Equinox at which time the amount of daylight and nighttime are equal to each other and this day marks the beginning of spring. The seasons are characterized by the amount of sunlight each day receives and the amount of warming that results. Ptolemy combined astronomy with geography noted the connection between latitude and the amount of the daylight. Like Aristotle before him who divided the world into five climate zones Ptolemy was concerned with the habitable parts of the world. But Ptolemy also noticed that longest and shortest periods of daylight could be used to determine a location's latitude. As an example he notes that for Rhodes the latitude is 36° and the longest period of daylight is about 14½ hours (at the summer solstice).

I thought it might be interesting to derive a formula to find the amount of daylight throughout the year. In what follows θ and φ indicate the latitude and longitude of the Sun and a location on the Earth's surface. The subscripts, S and L are used to distinguish the two sets of coordinates. Z indicates a location and its zenith direction. P is the pole of the Earth's rotation, M a point on the Equator through which the location's meridian passes and M' is another meridian which is 90° to the east of M.

Using these quantities we can specify both the position of some location and the relative position of the Sun. The Sun is above the horizon when the projection of the two directions onto each other is positive. This condition gives in an equation involving three angles.

The times of sunrise and sunset depend on the latitudes of Sun, θ_S, and the location, θ_L, and the difference between these times, Δt, is known if Δφ is known. For the day with longest daylight the Sun is over the Tropic of Cancer and its latitude is 23.44°. At the latitude of Rhodes, 36°, this gives the maximum amount of daylight as being 14.4 hours which is in good agreement with Ptolemy's value. One can also reverse the procedure and estimate the latitude if the maximum amount of daylight is known.