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# The Analemma and Time

'Noon', solar time, on his diurnal way
Great Helios can once again proclaim,
But mean time in Greenwich: twelve-o-six.

- William Messent Jones "The Winter Sundial"

## Overview

• Understand what the analemma is.
• Understand how it affects timekeeping

## Introduction

You may have noticed the strange, lopsided figure 8 pattern on most Earth globes and occasionally on maps. That figure is called the analemma, and it represents the true position of the Sun every 24 hours.

The figure below is the first known photograph of the analemma. It was taken by Dennis DiCicco in 1978-9. Although sketches and maps of the analemma go back thousands of years, it is very difficult to photograph because of the time and planning involved.

Figure 1: Photograph of the analemma by Dennis DiCicco:

Part of this shape is easy to understand. The Earth's axis is tilted 23.5º, which means the Sun appears to travel 23.5º above and below the equator (see the Seasons activity). Between the winter solsice and the summer solstice, the Sun will travel 47º, from -23.5º to +23.5º. This 47º accounts for the height of the analemma. Of course, if that were all there was to it, it would just be a straight line. To get the rest of the figure 8, we have to do some timekeeping.

Figure 2: Diagram illustraiting the difference between solar and sidereal time

You may recall that the Earth actually rotates 360º in 23 hours 56 minutes: the sidereal day (see the Timekeeping and Telescopes at the Detroit Observatory activity). However, because the Earth also travels a tiny bit around the Sun, there is a difference for a given position to rotate around to the Sun. The time for a given position to rotate around to the Sun is the Solar day. The length of a solar day is the time it takes to rotate 360º plus or minus the time it takes to make up for the Earth’s motion around the Sun. On average, it is 24 hours, but there is some variation, because of the eccentricity of Earth’s orbit. This accounts for the shape of a figure 8, rather than just a line. To really understand the shape, we'll need to look more closely at the effects of the eccentricity.

The Earth does not move at a constant speed as it orbits the Sun. Kepler’s second law tells us that Earth moves fastest when it is closest to the Sun. If you did the Seasons activity, you saw the effect of this on our calendar: there are fewer days between the autumnal and vernal equinox because we closest to the Sun in January.

Figure 3: The effect of an elliptical obit on the length of the day.

Because it is moving faster, Earth moves a greater distance along its orbit in winter (see figure 3). The bigger step size means that the deviation from 24 hours is also bigger in winter. In the previous paragraph we said this was “the time it takes to make up for the Earth’s motion around the Sun.” You can see in figure 3 that the angle β, when Earth is closer to the Sun (winter in the north), is bigger than α, so it will take longer for the same point on earth to turn back to the Sun. The bigger angle means a bigger variation from the average, so the figure 8 must be wider on the winter end than on the summer end.

This effect is amplified on Mars, which has a similar tilt, but a greater eccentricity. The “winter” end of the lobe is so much bigger that there is no cross-over point: it’s a tear-drop shape instead of a figure 8. There is a picture at http://antwrp.gsfc.nasa.gov/apod/ap061230.html.

The difference is never more than a couple of minutes, so you would really have to be paying careful attention to notice it. The effect more people are likely to notice is the time of sunset (or sunrise). For example, the summer solstice falls roughly on June 21, so you might think the earliest sunrise and latest sunset of the year will be that day. However, the earliest sunrise is about a week before the solstice, and the latest sunset actually occurs about a week after it.

The most significant effect is actually in reading a sundial. In fact, the word “analemma” is the Latin word for sundial. A sundial shows the local solar time. To get the current civil time (i.e. the time on you watch), you have to adjust your sundial reading based the analemma, and the time zone. The time zone parties easy: simply add or subtract a set number of minutes based on how far you are from the center of the time zone. For Ann Arbor, add 32 minutes. To adjust for the analemma, you have to add or subtract an amount based on the how far the Sun deviates from the meridian at noon. More on this latter in the activity.

For the planetarium, the analemma has a more important function. It is effectively the pattern the Sun follows over the course of a year if you observed at the same civil time every day. Watching the analemma is effectively like holding your watch to a constant time, but letting the calendar run forward. You can see how the positions of the stars, planets, Sun and Moon all change from day-to-day without having to wait for everything to rise and set.

## Activities:

Updated: 9/30/10 by SAM