The Math of Sunrise and Sunset

I knew since I was ten (we had quite a comprehensive curriculum at school) that the shortest day of the year falls on December 22nd. What I didn’t ponder until very recently was whether it was also the day of the latest sunrise (and, consequently, the earliest sunset).

While it may seem like a natural consequence of December 22nd being the shortest day, it doesn’t necessarily have to be true. If we model the time of sunrise and of sunset as two sine waves, \(a(t) = A \text{sin}(t+α)+K\) and \(b(t) = B \text{sin}(t+β)+L\) such that \(t_0\) minimizes the difference (we can drop the constants):

\[B \text{sin}(t+β) - A \text{sin}(t+α)\]

This means that

\[\frac{d(b(t)-a(t)}{dt} = 0 \text{at} t_0 \Rightarrow B \text{cos}(t_0+β) - A \text{cos}(t_0+α) = 0\]

We need to show that this equality may hold (for some values of \(α\), \(β\), \(A\) and \(B\)) even if one of the waves is not minimized at \(t_0\). Let

\[\begin{align}\frac{da(t)}{dt} = P \neq 0 \text{at} t_0 \Rightarrow A \text{cos}(t_0+α) & = P\\ B \text{cos}(t_0+β) = A \text{cos}(t_0+α) &= P\\ \text{cos}(t_0+β) = \frac{P}{B}, \text{where} \frac{A}{B} \leq \frac{P}{B} \leq \frac{A}{B}\end{align}\]

We can always find some values of \(A\) and \(B\) such that \(\frac{P}{B}\) is between -1 and 1, and hence the equation will be satisfied for some values of \(t_0\) and \(β\).

We can also take a short route and recall that a linear combination of two sine waves of the same frequency but not necessarily the same phase is still a sine wave. Its phase is a function of the difference in phases of the two waves. The value of \(t_0\) that minimizes the resulting wave is not necessarily going to minimize any of the two input waves because the three phases are different (and not different by a multiple of π).

In fact, if you look at the sunrise and sunset times in Connecticut around this December, the shortest day, unsurprisingly, falls on December 22nd, but the latest sunrise is on January 4th 2010 and the earliest sunset is on December 8th.

This is great news–it means that starting on December 8th (and not the 22nd), it will finally start getting darker later and later!