# That’s Maths: Darkening mornings and brightening evenings around Christmas

## The unsteady path of the earth around the sun means we see a stretch in the evenings weeks before the mornings start to become noticeably brighter

Today is the winter solstice, the shortest day of the year. We might expect that the latest sunrise and earliest sunset also occur today. In fact, the earliest sunset, the darkest day of the year, was on December 13th, over a week ago, and the latest sunrise is still more than a week away. This curious behaviour is due to the unsteady path of the earth around the sun. Our clocks, which run regularly at what is called mean time, move in and out of synchronisation with solar time.

The difference between clock time and solar time is encapsulated in a mathematical expression called the Equation of Time. The extent of the discrepancy between apparent solar time, following the sun, and mean solar time, with noons always 24 hours apart, can be up to about 15 minutes. Technically, the difference is due to the eccentricity of the earth’s orbit and the obliquity of the ecliptic. In plain language, the orbit of earth is not perfectly circular, but is slightly elongated and the axis of rotation of earth has a tilt.

Imagine that the path of earth around the sun were a perfect circle, and the axis of rotation were perpendicular to the plane of the orbit. Then there would be a symmetry in the day: the time between sunrise and noon would equal the time between noon and sunset. Thus, if sunrise is at 7am, five hours before noon, sunset should be at 5pm, five hours after noon. But suppose now that your watch is one hour fast. Then sunrise is at 8am by your watch, four hours before noon. And sunset is at 6pm, six hours after noon. The symmetry is broken.

As a result of this asymmetry, noon is not normally halfway between sunrise and sunset. This happens on only four days every year, when mean time and solar time agree. One of those days is Christmas Day. Just before Christmas, the sun is running fast so that, by our clocks, both sunrise and sunset seem slightly early.

## The benefits of being among the cyborg generation

The Astronomical Almanac gives the times of sunrise and sunset in Dublin on the winter solstice as 08:38 and 16:08. We must adjust Universal Time (formerly Greenwich Mean Time) depending on longitude to get local mean time. Since Dublin is about 6.25 degrees west of the prime meridian, local noon is 25 minutes later than at Greenwich. Allowing for this correction, the time from sunrise to local noon is three hours 47 minutes, whereas the time from noon to sunset is slightly less, at three hours 43 minutes.

The practical upshot of all this is that, while we may start to see a stretch in the evenings from the beginning of the new year, we have to wait another fortnight or so before the mornings start to become noticeably brighter. For January 1st 2018, the times of sunrise and sunset are 08:40 and 16:17. For February 1st, they are 08:09 and 17:08. Therefore, the change in the time of sunrise over the month of January is just over 30 minutes, whereas the change in the time of sunset is over 50 minutes.

Peter Lynch is emeritus professor at the School of Mathematics and Statistics, University College Dublin – he blogs at thatsmaths.com