The Analemmatic Sundial
THAT'S MATHS: IF YOU ARE in Dún Laoghaire, take a stroll out the East Pier and you will find an analemmatic sundial.
In most sundials, the gnomon, the part that casts the shadow, is fixed and the hour-lines radiate outward from it to a circle. In an analemmatic sundial the hour-points are on an ellipse, or flattened circle, the horizontal projection of a circle parallel to the equator.
You yourself form the gnomon, and the point where you stand depends on the time of year. This is shown on a date scale set into the dial. Your shadow, falling somewhere on the ellipse, indicates the hour.
Advances in mathematics and astronomy have gone hand-in-hand for millennia. As civilisation developed, accurate time measurement became essential. More precise observations of the stars and planets called for more exact mathematical descriptions of the universe.
The similarity between how we divide up angles and hours of the day arises from the use of astronomical phenomena to measure time. The division of a circle into 360 degrees, with each degree divided into 60 minutes and each minute into 60 seconds of arc, dates back to the Babylonians.
Tycho Brahe, the great Danish astronomer, made observations more precise than ever before. They enabled Johannes Kepler to deduce that the form of the Earth’s orbit around the Sun is an ellipse. When the Earth is closer to the Sun it moves faster and when farther away it moves slower. As a result, the length of a solar day varies through the year. Further complications arise from the tilt of the Earth’s axis, the obliquity of the orbit.
The unequal length of solar days is inconvenient. To simplify everyday life, we use mean time, with a fixed length of day equal to the average solar day. As a result, the Sun is not due south at clock noon but sometimes running ahead and sometimes behind. The mathematical expression for this discrepancy is the “Equation of Time”. The position of the Sun at mean-time noon falls on a curve called an analemma.
Mathematically, the analemma is a plot of the Sun’s altitude (angle above the horizon) versus its azimuth (angle from true north), and it has the form of a great celestial figure-of-eight.
Three adjustments must be made to get mean time from sun-dial time. First, since Dún Laoghaire is just over six degrees west of Greenwich, 25 minutes must be added. Next, a seasonal correction must be made. This is complicated to calculate, but help is at hand: it can be read from a graph of the Equation of Time, conveniently plotted on a bronze plaque. Finally, an extra hour must be added during Irish Summer Time.
The Dún Laoghaire Harbour Company is to be commended for installing the analemmatic sundial, a feature of artistic and scientific interest for all who use the amenities of the Harbour.
Local authorities elsewhere might follow this example. The sundial is a rich source of ideas for students, giving rise to many questions on geometry, trigonometry and astronomy, ranging from elementary problems to matters that have taxed the greatest minds.
The significance of the mathematical and astronomical theory involved in the Equation of Time is not confined to the design of sundials, but is important in many scientific and engineering contexts. It is used for the design of solar trackers and heliostats, vital for harnessing solar energy, which will one day be our main source of power.
Peter Lynch is professor of meteorology, School of Mathematical Sciences at University College Dublin. Visit his blog, thatsmaths.com