Use Moon shadows to measure crater peaks

How a simple formula gives you a deeper understanding of the physical form of lunar craters

The lunar surface is at its most striking when there are well-defined shadows. As well as being visually pleasing, the study of shadows cast by lunar mountains and craters can tell us a lot about these features. When we see a lunar crater from above it’s easy to think that the crater wall is uniform around the entire perimeter. However, if we look at the shape of the shadows cast by a crater wall, we can get a clearer sense of the peaks and troughs along the rim; and if the crater has a central peak its shadows will reveal its nature too.

Here, we will show you a straightforward method to measure these shadows. By using some simple rightangled triangle trigonometry we can calculate the height of the lunar feature casting the shadow. Our equation is: O = tanθ x A; where O = ‘opposite’ (ie the height of the feature), tanθ = the tangent of the Sun angle, and A = ‘adjacent’ (ie the length of the shadow).

Timing is important

To get started all you need is a ruler, a calculator and a photo of a crater that has clear shadows. You need the time and date the photo was taken, so you can use the Lunar Terminator Visualization Tool (bit.ly/3taEmyf) to find out what the Sun angle was at that time, at any point on the lunar surface. We used a photo of crater Theophilus taken (by Alessandro Bianconi) at 03:48 UT on 18 September 2011.

It is important to mention that our process here has been simplified. This method relies on a published crater diameter so we can scale up our shadow measurement. Most craters are not perfectly circular, so the published figure is an average; we only took one diameter measurement. If you use this calculation for an isolated feature, you’ll need to know the pixel/ kilometre ratio for the equipment used to take photo.

Additionally, our process doesn’t take the curvature of the lunar surface into account. Remember, the Moon is a sphere, so if you chose a crater that is quite central, the foreshortening effects are less apparent. It can also be difficult to know exactly where the shadow starts and ends if it’s located in a complex region. We are only using a single measurement of a complex crater at one Sun angle and comparing that to a published figure, which will be an average value.

Our method brings a sense of scale to an otherwise abstract landscape. Even though it has been simplified to make the maths easier, it still yields results that are close to the published figure. This project will help you to gain a deeper awareness of the lunar features you choose to analyse.

What you’ll need

A high-resolution digital photo of a lunar crater that has clearly defined shadows; we used a photo of the crater Theophilus.

A ruler; we used it to measure the shadow lengths on the computer screen, but you can use Photoshop’s measuring tool or a printed photo.

A scientific calculator with a tan button. If you use Excel for these calculations, convert the Sun angle from degrees to radians first.

A copy of the Lunar Terminator Visualization Tool (VLT), which you can download from bit.ly/3taEmyf.

The published crater diameter figure for the crater you’re measuring from this online list of near-side craters: bit.ly/3GQNFYb

Step by step

Step 1

Open your crater photo and view it full screen. Next, use a ruler to measure the diameter of the crater. We measured our example, Theophilus, at the widest point in the same direction that the shadows lie and got a figure of 11.5cm.

Step 2

Measure the crater wall shadow; we got a value of 3.5cm. Calculate the real length of the shadow by taking the published diameter (110,000m) and dividing by the diameter on screen (11.5cm), then multiply by the shadow length on screen (3.5cm) to get 33,478m.

Step 3

Measure the crater’s central peak shadow; we got a value of 2.3cm. Calculate the real length of the shadow by taking the published diameter (110,000m) and divide by the one on screen (11.5cm), then multiply by the shadow length on screen (2.3cm) to get 22,000m.

Step 4

Open the Lunar Terminator Visualization Tool (bit.ly/3taEmyf) and input the observation’s date and time. Point the mouse at the point you began the crater wall shadow measurement and read the Sun angle; we got 4.87°. Repeat for the central peak; we got 3.67°.

Step 5

To calculate the height of the crater wall, use O = tanθ x A, where tanθ = tan4.87° and A = shadow length of 33,478m. When we applied this to the crater wall in our example, we calculated a height of 2,852m compared to the published value of 3,200m.

Step 6

To calculate the height of the crater’s central peak, use O = tanθ x A, where tanθ = tan3.67° and A = shadow length of 22,000m. In our example, we calculated a height of 1,411m, compared to the published value of 1,400m.

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