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
T
Sunlight
H T

S R
R
or
S T
H
R
where H will be the approximate
height of the feature we measured.
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
Now for the algebra
H (km) H (m y units)

R(km) R(m y units)
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.
So …. does your answer
still make sense?
As a final check, the highest mountain on the visible side of
the Moon is around 4700 metres (4.7 km).
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.