A mathematical model for the change in the duration of daylight as we progress from solstice to equinox

This is an interesting problem to try and figure out. We will propose a trigonometric approach: since the seasons of the year are cyclic events, we can model daylight oscillations after a sine or cosine function.

Caveat: this model is not universal and depends on the actual sunrise and sunset times for a given place. Nevertheless, this actually makes sense, since the distribution of daylight across the globe is not uniform, but a function of the latitude and current season of said location.

First, let’s pick a place for our analysis, let’s say Seattle, WA in the US. Then let’s define the amplitude and period for the function based on the selected location.

Finding the amplitude

Knowing that the solstices are the high and low points of daylight, we look for the daylength of these two points of the year. We will use data from http://www.timeanddate.com:

Then,

Daylength 21 Jun = $DL_A = 16$ hrs

Daylength 21 Dec = $DL_B = 8.42$ hrs

Amplitude = $\frac{DL_A - DL_B}{2}$ = $\frac{16 - 8.42}{2}$ = $3.79$ hrs

The average daylight throughout the year is given by $average(DL_A, DL_B)$ which is 12.21 hrs. This way we have $12.21 \pm 3.79$ hrs.

Finding the period

The period of the sine/cosine is $2\pi$ , but since we need to express this in days we divide $2\pi$ by $365t$ to actually have the proportion of the period covered by a time t within one cycle of the curve (effectively a year).

Daylight equation

Defining the highest daylight point (June 21) as $t=0$ , we can now have an equation that models the change from peak to minimum daylight for a time t and a given location, in this case Seattle.