Measuring the height of Lunar Mountains

using data from the Liverpool Telescope

The Project

The following project details a method of estimating the height of a mountain on the Moon by measuring the

length of the shadow it casts across the lunar surface.

The Project - Discussion

Whilst there are more accurate ways of measuring the height of objects on the Moon’s surface, such as radar

mapping by an orbiting spacecraft, the methods we will use provide a simple way of estimating the size of a distant mountain using a little brainpower and tools

available in the typical classroom.

Background

The Moon is our nearest neighbour in space, and the only celestial body on which we can see surface detail

without the need for a telescope.

Look more closely, however, and it is soon apparent that the lunar surface is not smooth, but hosts a

variety of dark and bright regions, mountain ranges and thousands upon thousands of craters.

In this exercise we will examine image data of the Moon taken by the Liverpool Telescope, and use it to

estimate the size of any lunar mountains we find.

Shadows on Earth

If we measure the shadow length of an object of known height on Earth, we can use the information to estimate the

height of a different object just by measuring its shadow.

The method relies on the Sun remaining at the same angle during both measurements, and the application of a bit of simple

geometry, known as ‘similar triangles’.

Lunar Shadows

When the phase of the Moon is full (see top image), the Sun is right behind us, and thus sunlight falls straight down

onto the lunar surface. As a result, we do not see any shadows being cast by tall

objects or crater walls.

However, when the Moon is close to first quarter or last quarter phase (see bottom image), the angle at which

sunlight falls onto features close to the terminator (the line between light and dark) means that shadows will be cast.

Prediction

Given our existing knowledge, one might predict that lunar

mountains are of a similar size to those found on Earth, i.e. somewhere between 1000

and 8000 metres.

However, the mountains on Earth formed through active volcanism and tectonic plate activity – both of which are not seen on the Moon. It is believed that mountains and craters on the Moon are the result of many asteroid impacts over millions of years. On the other hand, however, it could be argued that the lack of atmospheric erosion and lower gravity may allow lunar mountains to be higher than on Earth.

The Geometry – A Rough Calculation

The red and white (exaggerated) triangles can be treated as similar triangles because the top lines of each are parallel, and

S (shadow length) is at right-angles to H (height). With small terminator distances (T), R is effectively the lunar radius.

S H

R

T

R

T

S

H

R

TSH

or

where H will be the approximate height of the feature we measured.

Sunlight

Assembling the Moonsaic

Now that we know the geometry, we need to assemble a large mosaic of the accompanying 20 Moon images so that

we can find a few examples of lunar mountains to measure. The image data has been converted to JPEG format so that

you can print them out and stick them together – like a jigsaw puzzle. Note that each image overlaps slightly, which will help

to match the edges and glue them securely.

Moonsaic

Use the included moonmap.jpg file to

determine where each section of the moonsaic

JPEGs should go. Have fun !

Making your Measurements

Once the moonsaic is complete, find a mountain near the centre of the Moon and fairly close to the terminator. We

can now measure the distances of S (shadow length), T (distance to terminator) and R (radius of the Moon) using a ruler or tape measure. Write the values in a table and then

calculate H using the equation we saw earlier.

Measurement Value

Shadow Length (S)

Terminator Distance (T)

Moon Radius (R)

Mountain Height (H)

Make sure you use the same units when measuring

Calibrating the Result

We now need to calibrate the result, so that we can express the answer in units that we can better understand, such as

kilometres. The way we do this is by using some simple algebra and by finding out what the radius of the Moon

really is. There are various methods for calculating R, which you can discuss now, but for the purposes of this exercise

we shall tell you that the radius of the moon (R) is

Radius of Moon = 1738 km

)(

)(

)(

)(

unitsmyR

unitsmyH

kmR

kmH

Now for the algebra

Always check your result

So we finally have an answer but, as with all forms of research, we need to check whether the answer sounds

reasonable. For example, it would be impossible to measure a height of 0.002 km (20m) on the lunar surface using the

techniques described here, whereas 2000km would be greater than the Moon’s radius – thus clearly not right.

As a final check, the highest mountain on the visible side of the Moon is around 4700 metres (4.7 km).

So …. does your answer

still make sense?

Discussion

Our initial prediction suggested that heights may be similar to mountains on Earth – how does that fit with our results?

Of course, the method we have just used will only ever give us a rough estimate of the true height of the mountains

that we have measured. Can you think of any areas of the process where errors may have crept in?

Can you think of any other ways in which we could measure the height of lunar mountains, whether it be from Earth,

using a telescope, or with a spacecraft?

Questions, Exercises & Tasks

Now that you have measured a mountain or two on the lunar surface, you may want to investigate the depth of crater walls, or even see how surface features change in

different parts of the Moon.

You may want to explore the process where mountains are created in the centre of craters following an impact.

Look at the Moonsaic again, and then try to work out whether it was taken at first or last quarter phase. Try to establish in which direction the Moon orbits the Earth.