Monday Morning Math: the Winter Solstice


December 12 was going to be the final Monday Morning Math of the semester, but finals had started two days earlier and with one thing and another (well, really just one thing – the aforementioned final exams), it didn’t happen.  


It felt a bit odd to take a break without announcement, however, so here is one final MMM for 2022.


(Arriving on a Monday at least, even though it’s not quite morning anymore.) And the timing is perhaps good for a math-adjacent topic: the Winter Solstice, which happened on Wednesday, December 21.  This is the shortest day of the year in the northern hemisphere, with just under 9 hours of daylight (technically 8 hours, 59 minutes, and 10 seconds) here in Rochester.  But there are two things about the solstice that I find interesting mathematically.

The first is that if you google “When is the Winter Solstice?” you get not just a day but a time: 4:47pm here.  This feels a little weird to me if I think about it being a day, but it has to do with northern hemisphere being tilted as far away as possible from the rays of the sun, as in this tweet from NASA below:

Or, if you want to envision the Earth with its axis vertical, it’s moving along a plane that is not horizontal, as explained on NASA’s blog

This is the image that makes the idea of a moment for the solstice make sense to me: it’s at the very peak of the ellipse, and that happens at one particular moment rather than a full day.

The second math adjacent thing is about subtraction.  You might think that the shortest day must have the latest sunrise and the earliest sunset, but actually neither is true: the lastest sunrise doesn’t happen until January 3, 2023, at 7:42am, which is about 3 minutes later than it rose on Dec 21.  So that’s kind of a bummer, in terms of how dark the mornings are.  But that’s compensated by the fact that the earliest sunset happened several weeks ago, back on December 9.  The sun set at 4:35pm that day, about 3 minutes earlier than it set on the solstice.  The sunrises and sunsets don’t quite change symmetrically, and that’s why the shortest day is about halfway in between. 

Happy Solstice and Happy Holidays to everyone! We’ll start again in about a month – see y’all in 2023!


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