Dark thoughts

Last night, a concert in the Spitalfields Music Winter Festival, entitled “This year’s midnight”, given by the viol group Fretwork and the mezzo-soprano Clare Wilkinson, with readings by the actor Simon Callow.

The title is taken from a poem “A Nocturnal upon St Lucy’s Day” by John Donne, which opened the concert. The poet compares the shortest day of the year to the midnight of the day, when all seems dark. Now St Lucy’s day is 13 December. If you notice an inconsistency here you have to remember that, when he wrote, Britain had not yet adopted the Gregorian calendar reform, and so the winter solstice occurred about ten days earlier in the Julian calendar as it was then. The programme said that 13 December was “thought in Donne’s time to be the shortest day”.

Actually, the programme, which had not been proof-read, was not much use, since the concert took place in darkness, the only light being on the performers’ music-stands. So I had plenty of opportunity to let my thoughts wonder as the words and music sounded out.

A diversion: a recent exchange in Private Eye sent me back to the Moody Blues’ second album, “Days of Future Passed”. Not entirely dissimilar to the concept of this concert, this album describes a typical day, from before dawn to the coming of night, with two spoken poems bracketing a suite of songs. However, it seems that they were thinking of a summer Tuesday, not of the shortest day of the year.

More recently, on 12 December, Diamond Geezer remarked that this marked the earliest sunset of the year, even though, counter-intuitively, it is not the shortest day. It seems a little pointless to try to add to his clear explanation; but let me do so anyway.

The earth spins on its axis at, to a first approximation, constant angular speed, relative to the distant stars. (There is a difficulty here: this fact seems to contract Einstein’s relativity principle, by invoking a fixed frame of reference. One physicist who has thought deeply about this is Julian Barbour, and I recommend his writings, especially The End of Time.) The time taken for one revolution is a sidereal day, and is approximately 23 hours and 56 minutes.

At the same time, the earth revolves around the sun, with constant angular momentum, and hence (as Newton realised) in such a way that the earth-Sun line sweeps out equal areas in equal times (Kepler’s second law). So the time taken for one revolution is also approximately constant, and is around 366.25 sidereal days.

The earth’s rotation and revolution are in the same direction. So, measuring from the time the sun is overhead on one day, by the end of a siderial day the earth will have moved on in its orbit, and will have to rotate a bit further until the sun is overhead again. So the solar day is slightly longer than the sidereal day. Over an entire year, we have one fewer solar day than sidereal days. So the average length of a solar day is (366.25)/(365.25) times the length of a sidereal day, that is, 24 hours. (Of course, some reverse engineering has gone on in this calculation!)

But since the earth’s orbit is not a circle but an ellipse, and the earth moves at different speeds at different parts of the orbit (Kepler’s first two laws), the length of a solar day is not constant; so if our clocks keep time by the average solar day, the actual time when the sun is overhead will be sometimes ahead of noon, sometimes behind. This is described by the equation of time or secular equation. The difference can be as much as a quarter of an hour. This is why sundials require correction to tell the “clock” time.

The other complication is that the earth’s axis is tilted with respect to its orbital plane, by about 23.5 degrees. This fact gives rise to the seasons: in summer, the sun is higher in the sky, and the earth is warmer (since by geometry, the energy flux per square metre of surface area is greater), and the reverse in winter. So recording the position of the sun in the sky at midday (by the clock) every day in the year gives a 2-dimensional curve called the analemma.

The picture shows a sundial on the campus of Monash University in Melbourne, Australia. There is one analemma for each hour of the clock, so correct time can be read off by interpolating between consecutive hours (but you have to remember which side of the curve to use). Remember too that this is a southern-hemisphere analemma.

So, for the next couple of weeks, both sunrise and sunset in the northern hemisphere will be getting later, as the day first shortens to its minimum at the solstice, then lengthens again.