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• Kristjan Salev

# HOW HORIZONTAL AND EQUATORIAL SUNDIALS WORK

Updated: Jul 19, 2022

In this post, we take a look at an ancient gadget that helped us read the time when mechanical clocks hadn't been invented yet. Since I'm planning to make one such sundial myself, it seemed just the right time to write this post and find out how this sundial works and how to make it properly. In an earlier post "Atomic wall clock" we could read, that in the heart of every clock we can find a device that oscillates at a certain frequency. This device, which oscillates at a certain frequency, helps us to measure time. In the case of the sundial, such a periodically oscillating object is the Earth. Although the Earth does not oscillate like a clock pendulum, it rotates around its axis periodically, making one full revolution in 24 hours. Thus the Earth revolves around its axis at a certain rotational frequency, and by taking advantage of this knowledge, we can build a clock that shows us the time according to the apparent position of the Sun in the sky.

Sundials have a wide variety of designs, but they all work in a similar way. Every sundial has an element, that creates a shadow. The apparent movement of the Sun creates a changing shadow, which helps us measure time. There are horizontal dials, vertical dials, equatorial dials, polar dials, analemmatic dials, reflected ceiling dials, and portable dials. Let's take a closer look at the two most common types of sundials - horizontal and equatorial sundials. The image below shows the main components of a typical horizontal sundial.

The simplest and most basic sundial would be just a stick bushed to the ground, which would be very similar to a horizontal sundial. When the Sun moves in the sky, the shadow of the stick moves accordingly and we can measure time by the movement of this shadow around the stick. Unfortunately, such a clock would not measure time accurately enough, because the axis of rotation of the Earth is tilted. That's why the shadow casting element of the horizontal sundial which is called a gnomon style, is usually at some angle to the horizontal dial plate.

The image above shows, that the axis of rotation of the Earth is an angle of 23.4 degrees to the orbital axis, which is at right angles to the plane of orbit. This tilted axis causes the seasons to change on Earth. When the axis of rotation is towards the Sun, this hemisphere has summer, and if away from the Sun, winter. The upper picture shows winter in the northern hemisphere and summer in the southern hemisphere at the moment.

The gnomon style of the horizontal sundial must be parallel to the axis of rotation of the Earth and point true north in the northern hemisphere or true south in the southern hemisphere so that the sundial is accurate all year around. To set the gnomon style parallel to the Earth's axis of rotation, we need to know the latitude of our location. Latitude is the angle, that varies from 0 to 90 degrees north and south of the equator.

For example, I live in Estonia, which is located about 60 degrees north. Using simple geometry, it can be shown that if the gnomon is at 60 degrees to the horizontal and points true north, then it is parallel to the axis of rotation of the Earth.

It should be added, that the geographical north pole and south pole do not coincide with the magnetic north and south poles. True north and true south are fixed points on the globe, around which the Earth rotates, the magnetic poles however change their position over time. Therefore, the compass does not indicate true north and true south directions, but the direction of the Earth's magnetic poles. In the northern hemisphere, we are fortunate to have a bright star (Polaris) at almost the right point in the sky, which shows us the direction of the true north and around which the apparent rotation of the starry sky takes place.

The equatorial sundial differs slightly in its construction from the horizontal sundial. The image below shows one such equatorial sundial.

The equatorial sundial has a dial plate parallel to the Earth's equator, which s why it is called an equatorial sundial. Dial plate that receives a shadow, is perpendicular to the gnomon's style. The image below illustrates the position of the equatorial sundial at my latitude, which is 60 degrees north.

Again, it is easy to show that if the gnomon style is perpendicular to the dial plate, points true north, and forms an angle with the horizontal, which is always equal to the latitude of our location, then the dial plate is also parallel to the equatorial plane of the Earth and gnomon style is parallel to the Earth axis. In the southern hemisphere, however, the gnomon style must point in the direction of the true south. The equatorial sundial, unlike the horizontal sundial, can have two sides. In summer, when the Sun is above the celestial equator, it casts a shadow on one side of the dial plate, and in the winter, when the Sun is below the celestial equator, it casts a shadow on the other side of the plate. You can visualize the celestial equator as a plane you get when you stretch the dial plate infinitely far away in all directions. Half a year the Sun is below that plane and the other half a year is above that plane.

On September 21 and March 23, the Sun is exactly on the celestial equator, in these two days of the year, it does not cast a shadow on either side of the dial plate. In the northern hemisphere, between September 21 and March 23, the Sun casts a shadow on the bottom side of the dial plate, and between 23 March and 21 September, on the upper side of the dial plate.

The hour lines on the horizontal sundial are not evenly distributed, since the shadow-creating side of the gnomon is not at right angles to the dial plate and the shadow does not move quite evenly. The equatorial sundial has the shadow-creating side of the gnomon perpendicular to the dial plate and hour lines are evenly distributed. The image below shows the dial plates of the horizontal and equatorial sundials, created with Shadows software. This software can be a great help if you want to design your own sundial and you don't want to go into too much geometry.

On the right, you can see the dial plate of the horizontal sundial with the hour lines and on the left is the dial plate of the equatorial sundial with the hour lines.

It takes about 24 hours for the Earth to complete one rotation around its axis. This means, that the apparent movement of the Sun in the sky takes place at a rate of 15 degrees per hour (one full revolution is 360 degrees, and takes place in 24 hours, which makes 360/24 = 15 degrees in an hour). For this reason, the hour lines on the dial plate are 15 degrees apart. When the sun rises from the east, gnomon casts a shadow on the west side of the dial plate. As the Sun moves across the sky, the shadow moves to the northern edge of the dial plate and as the Sun sets in the west, it casts a shadow on the east side of the dial plate.

All sundials measure apparent solar time, based on the apparent position of the Sun in the sky. The Sun does not recognize man-made changes in the measurement of time, It may differ slightly from the time, shown on your regular clock. The globe is divided into 24 time zones, each about 15 degrees wide. Each time zone has the same local time. The Sun moves from east to west, which means that the sundial at the east side of the same time zone disagrees with the sundial in the western part of the same time zone, even though our regular clock shows the same time everywhere in the same time zone. For example, let's suppose the Sun is at the highest point in the sky at the eastern side of the local time zone. That is, there is solar noon at this point. However, in the middle of the same time zone, the second sundial shows 11.30 AM, as the Earth rotates 15 degrees an hour and the solar noon arrives 30 minutes later at this location.

The second difference from the time shown by an ordinary clock follows from the fact that the axis of rotation of the Earth is tilted and the Earth does not move around the Sun in a completely circular orbit but in an elliptical orbit so that its orbital velocity varies slightly during the year. That's why the solar time may again vary from clock time by a small amount.

The graph shows the total difference (red line) from the time shown by the normal clock if we add the effects caused by the tilted axis (green line) and the elliptical orbit (blue line) of the Earth. Here you can find a calculator, which helps to convert local time to solar time or vice versa.