Wednesday, 21 December 2016

How short is the shortest day?

Happy Yule!  As you probably know, Yule (Jól) was the pagan Nordic celebration of the bleak midwinter.  And it's the winter solstice today - the shortest day of the year.

Ring of Brodgar (from @VisitScotland via @HistoryNeedsYou)
What does that mean?  The word solstice means "stopped Sun" in Latin - so it's not really a whole day, but the moment when the Sun appears to stand still at its most southern point (so, furthest away from us in the northern hemisphere), before returning back towards the north. To put it another way, it's the moment in our annual journey around the Sun when the Earth's axis is tilted directly away from it (or directly towards it if you're reading this in Madagascar).

Notice that the first explanation I just gave - and the word solstice - is a geocentric view of things: we're talking about the Sun moving, rather than the Earth.  But that's what makes most intuitive sense (and of course it doesn't make any practical difference, since these motions are only relative).  Still, I was interested to see that that was the explanation the Met Office used in the "6 facts about the winter solstice" they published this morning.

One of their 6 facts was that the shortest day is "nine hours darker" than the longest day.  That made me wonder: Where?  Of course you know that the winter days are shorter in Aberdeen than they are in Aberystwyth - so at what latitude is it true that, as the Met Office say, the shortest day is 7 hours and 50 minutes long?

Peterhouse, Cambridge MS 75.I, "The Equatorie of the Planetis", f. 63v
That was a question that interested medieval astronomers too.  In the days before electricity, they were naturally more aware than us of the receding and returning daylight.  And there's lots of evidence of their scientific approach to the matter.  To take just one example, my favourite manuscript (from Peterhouse in Cambridge) features this table (right).

The title (in Latin) is "Table of the increase of the longest day over the equinoctial day, for all the habitable earth".  As you can see, there are two columns, which repeat 3½ times.  They are headed "altitude of the pole" and "half addition".

That's simpler than it may sound!  The altitude is that of the celestial pole - the height of the pole star above the horizon.  That height - an angle on the sphere of the sky - is equal to your latitude.  When the pole star is directly overhead, you're at the north pole.  So the "habitable earth" in this table is from 1° to 60° North (sorry, Icelanders).  The "half addition" tells you that what we are actually being given is the difference in the length of the afternoon (or morning) of the longest day - from noon to sunset, compared with the equinox.  It's given in degrees. To find the difference in hours, you divide by 15 (360° ÷ 24 hrs = 15).

The fact that we are given altitude rather than latitude, and additions in degrees rather than hours, reminds us that these medievals were astronomers and mathematicians.  They were interested in scientific questions, not just mundane practicalities.  And those questions involved some complex science.  To draw up a table like this, showing the different day lengths at different latitudes, you don't just need to know spherical trigonometry.  You also need to have an estimate for the axial tilt of the Earth.

And medieval astronomers did. (Though because they didn't think the Earth spun on an axis, it was called the obliquity of the ecliptic - the angle between the celestial equator and the Sun's annual path between the tropics of Cancer and Capricorn.)  Estimates varied - the obliquity itself has too, over time - between about 23½° and 24°.  So historians trying to find the sources of science through the ages can check these tables to see what parameters are being used, and work out where they came from.  The table above uses an obliquity of 23° 35', which was a value popularised by the 9th-century Arab astronomer Al-Battani, and used by a number of Europeans in the middle ages.

So can we use this table (produced by a 14th-century monk, John Westwyk) to work out where in the UK the Met Office's numbers are true?  You bet!  Their blog post gives 16 hours 38 minutes as the length of the longest day.  On the equinox, of course, it's exactly 12 hours.  So we're looking for a difference of 4 hours and 38 minutes.  Of course the table gives half-additions, in degrees; 4h38 ÷ 2, x 15 = 34.75, or 34° 45'.  The table above gives a value very close to that (34° 43') for a latitude of 52° 30' N - the latitude of Birmingham.  The Met Office headquarters is actually in Exeter (latitude 50° 43') - so maybe they didn't want to rub in the fact that they have a few minutes more light today than the rest of us...

May your days be merry and bright!

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