# Solstice Musings

On one of the last ski lift rides of Christmas Day, the question came up, “I wonder how much longer today is than the Solstice?”

Between the two of us, we knew two things. One was that a friend had posted on the Solstice, “Happy solstice!!!! 4 extra seconds of daylight tomorrow, then 10 the next day…”  The other was that on January 29, the day is about an hour longer.

Susan’s intuitive guess was about a minute. How could we make an informed guess before the end of the chair ride?

### “Differential method”

The approximation I made was that near a turning point, any smooth function is reasonably well approximated by a parabola. Since I wasn’t sure when the solstice occurred during the 21st, then the 6 second difference between the 22nd and the 23rd would have to do to determine the curvature of the parabola.

I made the guess that the length of the day would increase like $6 t^2$, where t is measured in days from the actual solstice. As the 25th is three to four days after the solstice, I guessed $6 \times 9 = 54$ to $6\times 16 = 96$ seconds. (It would have been “better” to use the 14 second increase over two days to figure out the change, just like the next method below.)

### January 29 method

We didn’t do this on the chair, but we could have used our other fact, that January 29 had about an hour of extra daylight, to determine the curvature of the parabola.

January 29 is 39 days after the solstice. There are 3600 seconds in an hour. Our fact is a different way of saying, $3600 = C \times (39)^2$. If we approximate 39 as 40, then $C = \frac{3600}{1600}=9/4=2.25$.  Using that method, then we’d guess that the day would be $2.25 \times 16 = 36$ seconds longer.

The actual value is 52 seconds.

It turns out that the day is 1:02:13 = 3733 seconds longer on January 29. Once we’re home, there’s no need to do the $39 \approx 40$ approximation. Doing so would yield $C = 2.45$

Doing the parabolic fit for two days and 14 seconds gain yields $C=14/4=3.5$.

Let’s plot! So, my original guess barely covered day 4 (Christmas). The two-day/14 second method does a little better. But why didn’t the January 29 method win? After all, we had a huge baseline over which to measure the curvature of the parabola…. Well, over longer times, the January 29 method wins. By design, it’s guaranteed to win on day 39. What kept the January 29 method from working perfectly is that the length of the day isn’t parabolic in time; they’re more properly sinusoidal (days must, after all, get shorter again in the autumn). A month is a big-enough fraction of the time from a solstice to an equinox that the parabolic approximation isn’t “perfect”.

There are a few ways to improve upon these approaches. Can you do better?

NB: Susan’s guess of 60 seconds was quite accurate. One more reason to trust a ski partner?