Sundials

You’re probably familiar with the idea of sundials. A shadow falls on a scale, which then indicates the time of day. They may seem like a primitive technology, but there is a lot of science behind them, and each sundial is in fact a model of the planet Earth.

This article describes some of the background to sundials and descriptions of some sundials that you can build yourself. The four different sundials described in this article are shown in the following image.

A selection of printable sundials

Just show me the sundials

If you’re eager to build your own sundials, there are many templates that you can download for free, print out, and assemble. For example, the Sunny Day U website alone has “21 types of sundials available in 74 models”.

Note: I found the Sunny Day U website confusing to navigate initially. Select the door images to navigate to other pages and to download files.

However, I think that the easiest model to start with is the paper pop-up sundial that you can download from blocklayer.com. You can print it on a single piece of paper and then fold it into the final shape. You can then use glue or a paper clip to hold it together. This paper model also provides your first lesson in sundials, which is that you need to configure them for a specific latitude. On the pop-up sundial page, you can select What’s my latitude, and then either allow the browser to access your location or manually mark your location on a map. Follow the other instructions on the page to create your sundial.

You can then skip the science and go straight to the placement section.

What the sundial represents

A key part of a sundial is the thing that casts the shadow, which is called the gnomon. The gnomon needs to have a very specific angle for the latitude where the sundial will be placed. The reason is because the gnomon needs to be parallel to the Earth’s axis of rotation.

To make the following diagrams easier, we’ll ignore the fact that the Earth’s axis of rotation is tilted. In reality, the Earth’s rotation is tilted by 23.5° relative to the plane of the solar orbit.

Imagine that the top of the Earth was removed and a pole added for the axis of rotation. We could then add hour markers to the flat surface. It takes approximately 24 hours for the Earth to rotate, so 360/24 = 15 degrees between each hour marker. As the Earth rotates, the shadow cast by the pole would fall on each of the hour markers in turn.

A cross section view of the Earth with hour markers

Now imagine that we created a smaller but identical object, and attached it at the equator. For example, the 0 on the smaller object to the 12 on the Earth. As the Earth rotated, the two shadows would fall on the same numbers.

Earth dial aligned with a smaller dial

Earth dial rotated to 3 o’clock, with small dial showing the same

As you’ve likely guessed, the smaller object represents a sundial. By aligning the gnomon with the Earth’s axis of rotation, we make it easier to mark out the face of the sundial. You can simply divide a circle into 15 degree segments.

If you’ve only ever seen sundials in your latitude, then you may have thought that the gnomon was always at the same angle. In reality, since the gnomon is always aligned with the Earth’s axis of rotation, it’s angle relative to you depends on where you’re standing on the curve of the Earth’s surface.

If you’re at the North pole, the gnomon would point directly upwards. If you’re at the equator, the gnomon would need to be horizontal with the ground. Everywhere else requires an angle somewhere between vertical and horizontal.

Gnomons aligned with the Earth’s axis of rotation at different latitudes

Calculating the angle for a gnomon

It’s easy to determine the angle of the gnomon for any location on the Earth. It’s simply the latitude of that location. Why? Because latitude is a measure of the angle from the equator. Remember that at the equator, the gnomon would need to be horizontal – parallel with the floor. As you move north or south, your angle from the equator increases, so your gnomon needs to rotate in the opposite direction by the same amount.

At the risk of making things more complicated than they need to be, you can use geometry to proves that the gnomon angle needs to match the latitude.

Proving gnomon angle with geometry

On the left, is a representation that shows the angle of latitude equal to the angle of the gnomon. On the right, is the proof that both the gnomon and latitude angles are complementary angles to θc.

Placement

It’s worth stating the obvious that you should place a sundial in a location that receives direct sunlight for as much of the day as possible. However, the sundials described in this article are intended to be temporary and would not survive long in wet or windy weather conditions. For these reasons, I attached a small cardboard platform adjacent to a south-facing window.

Cardboard platform for sundials

Aligning true north

You need to align a sundial along a line that runs true south to true north. I explain why that is and what the true part means later in the article.

Sunny Day U provide a snourth o’meter, which you can use to work out the north-south line. It feels like a bit of a cheat, since you need to use a clock, but you only need to work out the direction once.

The premise is that you find out the time when the sun will be at it’s highest point in the sky, which is called solar noon. At solar noon, the sun will either be directly true south, if you’re in the Northern hemisphere, or directly true North if you’re in the Southern hemisphere.

To find the time of solar noon in your area, perform a Google search for “solar noon <your location>”. For example, “solar noon Cork” or “solar noon New York”. You then have the following two options:

• If you see a strong shadow on the snourth o’meter at solar noon, mark the location of the shadow. The line from the centre of the snourth o’meter to the marked point is the true north-true south line.
• If you don’t think you’ll see a strong shadow at solar noon, for example because of forecast clouds, then you need to mark two times that are equal intervals before and after solar noon. You then draw a line between those two points, find the point halfway between them, and draw a line from the centre of the snourth o’meter to the halfway point. This line is the true north – true south line.

Using a snourth o’meter

If you live in a cloudy area like I do, then the real way that you use the snourth o’meter is to mark the shadow whenever you see it in the morning. You hope that there’s a clear shadow at solar noon, and if not, you keep aiming to catch a clear shadow in the afternoon at a time that’s an equal interval from a time that you recorded in the morning.

Marking the location and time of a shadow on the snourth o’meter

Drawing the true north – true south line

When you determine the true north – true south line, you’re ready to print out and assemble a sundial, and then align it with the true north – true south line.

True north Vs magnetic north

Most of the time, when we talk about north, we mean magnetic north. This is the direction that compasses point to. Magnetic north is created by the Earth’s magnetic field, but it has a couple of problems.

• It doesn’t line-up with the Earth’s centre of rotation.
• It isn’t consistent across the surface of the Earth.
• It changes over time.

You can actually correct for all of these problems if you know the current magnetic declination at your location. Magnetic declination is, conveniently, a measure of the difference between magnetic north and true north. It indicates how many degrees you need to add or subtract from your compass reading to find true north.

Map of magnetic declination in 2015 – Wikimedia

Because the magnetic field isn’t consistent across the planet, the chart looks complicated. And, because the location of magnetic north changes over time, you’ll need to use a recent map.

Find you location on the map, and then look for a line that either runs through that location or is close to it. The look at nearby lines with numbers, and count from those to find your local declination. In the example image below, a line conveniently runs through my location in Cork. By identifying the 0 and -10 lines, I can identify the declination at my location as -4 degrees. So I would subtract 4 degrees from the compass reading to find true north.

Identifying declination for Cork, Ireland

Positive numbers are also referred to as westerly declination (“west is best”), and negative numbers as easterly declination (“east is least”).

Aligning a sundial with a (broken) compass and -4 degree correction

If you want to see how much the position of magnetic north has changed over time, you can go to https://www.ncei.noaa.gov/maps/historical_declination/, and select the Modeled Historical Track of Poles check box.

Why your sundial shows the wrong time

Maybe I should have broken this news to you earlier, but your sundial will show the wrong time. More accurately, the time shown on the sundial will almost certainly be different to the time on any clocks that you have. This is because of the following reasons:

• Daylight saving time: Most sundials show standard time. If you are currently observing daylight saving in your region, then the sundial will be one hour behind.
• Time zones: Regions are grouped together into time zones that share the same time. This means that your sundial is actually more accurate, since it shows your local time, and not the agreed regional time. You can adjust based on a longitude offset from the centre of your time zone.
• Earth’s tilt and orbit eccentricity: The Earth is tilted relative to the plane of orbit, it wobbles slightly, and the orbit is more of an oval than a circle. These variations mean that sometimes the time shown on a sundial will be ahead of the time shown on a clock, and sometimes it’ll be behind. You can account for all of this by using the grand sounding equation of time, described below.
• Practical limitations: It’s also worth remembering that you’re not working with a piece of precision equipment. It’s just printed out paper.

I’ll address the main issues separately, but you can see a full explanation of how to convert from local time to local solar time here: https://susdesign.com/popups/sunangle/time-basis.php.

Time zone offset

As the Earth rotates, solar time slowly creeps across the face of the planet. Your local solar time will be behind locations to the east of you, and ahead of locations to the west of you. To be precise, each degree of longitude means a 4 minute difference in solar time: 24 hours multiplied by 60 minutes in an hour, divided by 360 degrees.

In contrast, time zones are standardised across vast swathes of land. Each time zone is defined by its offset from 0 degrees longitude, in Greenwich, England. There are exceptions, but in general, there’s a new time zone every 15 degrees of longitude. How much your clock’s time is different from your local solar time depends on how far away you are from the middle of your time zone – called the local standard time meridian. To the east you’ll be ahead, to the west you’ll be behind.

You can calculate the difference in time caused by your time zone with the following equation:

difference = 4 minutes * (local longitude – ( time difference from UTC * 15 degrees ))

For me in Cork, Ireland, the longitude is approximately 8.5 degrees west (so -8.5 degrees), and my local time zone, when not observing daylight saving, is UTC+0. So, my difference is:

4 * (-8.5 – (0 * 15) ) = 4 * (-8.5 – 0) = 4 * -8.5 = -34

So the time shown on my clock is 34 minutes behind my local solar time.

Equation of Time

The equation of time takes into account multiple characteristics of the Earth’s movement through space, and I don’t have a detailed enough understanding to explain it. However, the output is simple to understand. It creates a graph.

The x axis is the day of the year. The y axis indicates a sundial’s error compared to a clock. A positive number means the sundial would be ahead, while a negative number means the sundial would be behind.

You can take an approximate reading from this graph, and other more high resolution versions on the internet. However, there is a good online calculator on PLANETCALC where you can select a date and the calculation is done for you: https://planetcalc.com/9198/

Pedro Sánchez / Wikimedia Commons

Example application

The three separate readings of the sundial shown in the images below average out at 1hr and 17 minutes behind the time that was shown on my clock. Based on the time that I took the pictures, I need to make the following adjustments to the sundial time:

• Add one hour for daylight saving being in effect.
• Add 34 minutes because of my longitude difference from the centre of my time zone.
• Subtract 12 minutes because of the Equation of Time.

With these combined factors, the sundial was out by an average of 5 minutes. Which I think it pretty good for a paper model.

Photo taken at 10:28, showing ~9:07: -1 hr 21 min

Photo taken at 12:09, showing ~10:55: -1hr 13 min

Photo taken at 15:17, showing 14:00: -1hr 17 min

A selection of sundials

In terms of maximum sundial for your effort, you can’t beat the four in one sundial. A single gnomon is used for horizontal, vertical, equatorial, and polar sundials. It’s perfect to show how the different types of sundial relate to each other. The sundial also includes an image of the Equation of Time graph for handy reference.

However, this sundial is only useful at latitudes between 30 and 60 degrees. It’s also a bit awkward to assemble. There aren’t any tabs to hold the various parts together. Imagine trying to stick a flat sheet of cardboard at a right angle to another flat sheet of cardboard. You just need to use copious amounts of glue. Or create your own tabs.

Four in one combined sundial

The simple equatorial sundial has an elegant design and supports latitudes between 0 and 60 degrees, both north and south of the equator. The ability to rotate the dial in the stand supports, deliberately or not, adjusting the sundial for daylight saving and your distance from centre of your time zone. The design also helps you to identify an equinox, although mainly because it will be difficult to use your sundial on those days.

More than any of the other designs discussed in this article, it’s an obvious model of the planet Earth. The cocktail stick is the Earth’s axis of rotation, and the circular plate is a cross section of the Earth through the Equator.

On the negative side, because the shadow was cast on the underside of the dial, I found it awkward to take a reading.

Simple equatorial sundial

The spider sundial incorporates both the Equation of Time correction and daylight saving adjustments. It uses a series of concentric rings – one for each month. Each squiggly line is the Equation of Time mapped out over the year for a particular hour. One colour of line represents standard time, while the other colour represents daylight saving. You’ll still need to adjust for your distance from the centre of your local time zone, but you only need to calculate that once. The added functionality makes it more difficult to read the time, but it just requires some practice.

Spider sundial