Telling Time by Shadow Length
In a recent blog entry, I mentioned the idea of telling time based on the length of a shadow.

Monday was Mabon, the Vernal Equinox, so I spend last weekend camping at festival in observation of the holiday. I had time to perform measurements based on my calculation, which are displayed to the right.

Before collecting data, I had determined that I would not have a greater accuracy than 5 minutes. A 6 foot measuring device accurate to an inch leaves only 72 distinct measurements to predict the time over 6 hours. Spread evenly apart, that means that one inch has a resolution of 5 minutes. I'm taking a few liberties here[1], but this back of the envelope calculation gave me a rough idea of whether a number represented an error in my tools or in my equations.

As you can see by the graph, the time I predicted follows the same general trend as clock time, but it isn't yet terribly accurate. Further, the accuracy improves right at noon, and is better in the early and later part of the day.

I know why this happened, but first I should explain the equation I'm using.


As the sun rises and sets throughout the day, the length of a shadow cast by an object changes size. As the sun is low in the sky during the early morning and late afternoon, shadows are long. Near noon shadows are short, as the sun is higher in the sky.

This fact can be used to determine at what angle the sun is in the sky. If you presume the sun is at 0 degrees on the horizon, at noon on the equator it will be at 90 degrees. The current angle is:

sun angle = arctan( pole / shadow )

where pole is the length of your measuring device, and shadow is the length of the shadow cast by the object when it is held upright on level ground.

But having the sun angle is not useful by itself, as you need to know how high the sun will get during the day. This is a function of your latitude and what is called the sun's declination. Declination is the seasonal difference in sun angle. At the summer solstice the declination is +23.5 degrees, while at the winter solstice declination is -23.5 degrees. At the equinoxes, declination is 0, but I consider it anyway in the equations below.

An inaccurate but simple equation for declination is given by:

declination = 23.5 * cos( 360 / 365 * ( N + 10 ) )

Where N is the day number (January 1st=1, February 1st=32 March 1st=59 [60 on leap year], &c). We add 10 to it in this equation because the date is relative to the winter solstice, which is generally December 21st.

I haven't yet determined whether this equation is accurate enough for my purposes. It is least accurate near the equinoxes, but other errors currently prevent me from determining whether I will need to use a more complex definition.

Once you've determined your declination, you also must determine you latitude. You can do this by measuring the angle from the horizon to Polaris at night, if you have something like a celtic cross. I used Google Earth for this experiment, as my cross is 8' tall and a bit tough to transport.

Having these two things, the maximum angle of the sun for a particular day is:

max sun angle = 90 - latitude + declination

Knowing the current sun angle and the maximum sun angle, you know how far the sun angle is from the noon sun angle, but you don't know how long it will take to reach noon. For this, you need to know how long the sun will take to go from 0 degrees (the horizon) to the maximum sun angle, noon. That is determined by the sunrise equation, a simple form of which is given here:

hours to noon = arccos( -tan( latitude ) * tan( declination )) / 15

You divide by 15 in this equation to convert the angle to a number of hours. Since there are 24 hours in a day, which covers a 360 degree rotation, 360 / 24 = 15.


Having made this calculation, you know how many hours it will take the sun to travel from 0 degrees to the maximum sun angle at noon.

Currently, I take the current sun angle and divide by the maximum sun angle to determine the ratio of the day that has elapsed:

sun time = current sun angle / maximum sun angle * hours to noon.

To convert this to clock time, I have to adjust for daylight savings time as well as the difference between my timezone and local noon. My timezone (UTC-7) is based on the mean solar time at the 105th degree, while I was at 106 degrees and some minutes. 1 degree is equal to four minutes, because there are 15 degrees per hour (calculated above) meaning that there are four minutes per degrees (60 / 15). This worked out to 6 minutes, and since we are observing daylight savings time, I need to add 1 hour, 6 minutes to my final time to make it agree with clock time.

clock time = 12 - hours to noon + sun time + 1:06

This equation holds until noon, and requires you to rearrange signs after noon.

With that, finally, you have determined the clock time based on the length of a shadow cast. In reality, I rearranged several of these equations to create a daily constant, meaning the bulk of the calculation was computed in the morning and the intermediate result multiplied by each of my measurements.

This doesn't account for the error between my model and reality, as shown in the chart at the top of this entry. I have made the assumption that the sun rises in the sky at a constant rate, going from 0 degrees at sunrise to the maximum angle at noon in a steady progression. This is not true, as the sun is also moving east to west as well as north to south, so that the apparent motion is eliptical rather than linear. I need to account for this when determining sun time. The sun's apparent motion is a sin wave rather than a straight line.

I haven't worked out the equation precisely, as I have yet to put these equations on the computer so that I can play with them more freely.

I'll want to have that worked out before turning this into a nomogram, which will make the instantaneous calculation of time much simpler than what I've presented here.


For Mabon next year, I'd like to build a sundial in our ritual field, with a marker for each event laid out along the sundial so that when the gnomon casts its shadow over the object, the event is scheduled to start.

Performing this experiment also got me some positive attention, which I'll talk about in a later entry.

[1]: Specifically, the shadow is longer than the measuring device for the early and late part of the day, improving my resolution, and the length of a day happens to be roughly 12 hours now, but that isn't true closer to either solstices.