# Why should the Gnomon be set Parallel to the Earth's Axis?

The Sun in its apparent daily journeys across the sky gives us a means of telling the time and dividing our daylight hours into convenient intervals, by sundials. Sundials were used in ancient Egypt and were often little more than sticks stuck in the sand, or vertical pillars.These were not at all accurate because the Sun varies its track across the sky according to the season of the year. A rod stuck in the ground can be used to show these variations from month to month by measuring the length and direction of the shadow it makes in sunlight. We cannot mark on the ground the shadow angle, or time by the Sun without tables or mathematical calculations becausethe shadow angle is dependent on the declination of the Sun which changes with the seasons throughout the year.

A great advance in accuracy and convenience was made by the Arabs who had the bright idea of tilting the gnomon such that it points to the north celestial pole (NCP) and lies parallel to the Earth's axis. To explain why the gnomon has to be parallel to the Earth's axis, we look at the case whereby we set the gnomon at an angle to the Earth's axis.

When the gnomon is not parallel to the Earth's axis,

1. The shadow cast by the gnomon at a given hour points in different directions depending on the seasons.
2. The angle covered by the shadow during a certain time interval depends on the seasons.

These inconsistencies make it impossible to calibrate sundials with a gnomon that is not parallel to the Earth's axis. We shall illustrate these two inconsistencies - the change in direction of the shadow and the different shadow angles resulting from the same interval of time on different days, with the gnomon set at 90º to the ground, at latitude 45ºN.

## Change in direction of the shadow of the gnomon every day at given hour

At latitude 45ºN, if we set the gnomon at 90º to the ground, and fix time at 1p.m., the gnomon casts a shadow, which changes direction every day as depicted in the animation above.

However, when the gnomon is set parallel to the Earth's axis and the time fixed at 1p.m., the gnomon casts shadows toward a fixed direction every day as depicted in the animation above.

By having the same direction every day at given hour, the calibration of the hour lines on our sundials can then be consistent.

## Variable shadow angles resulting from the same interval of time on different days

Referring to the above animation, on 21st June (summer solstice at the northern hemisphere) at 9a.m., the gnomon casts a shadow at A, and at 12noon, the shadow is at B. On 21st December (winter solstice at the northern hemisphere) at 9a.m., the gnomon casts a shadow at C and at 12 noon, the shadow is again at B. Hence, given the same duration of time (9a.m. to 12 noon) on both days, the shadow has to turn a greater angle on 21st June than on 21st December. Thus, the shadow has a faster angular speed on 21st June than on 21st December. Applying the same reasoning to the other days, the speed of the shadow angle of the gnomon varies every day, when the gnomon is at an angle to the Earth's axis.

However, when the gnomon is placed parallel to the Earth's axis, the gnomon casts a shadow that sweeps out the same angle given the same interval of time on different days.

From the above animations, we note that the length of the shadow varies. The length of the shadow gets longer as the magnitude of the Sun's declination increases.

Also, in the morning and evening, when the angle that the sun rays make with the horizon is the smallest, the shadow is at its longest. On the other hand, at noon, when the Sun crosses the meridian, the shadow is at its shortest.