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Measuring the height of Lunar Mountains using data from the Liverpool Telescope

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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.

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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.

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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.

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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’.

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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.

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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.

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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. SH R T or where H will be the approximate height of the feature we measured. Sunlight

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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.

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Moonsaic Use the included moonmap.jpg file to determine where each section of the moonsaic JPEGs should go. Have fun !

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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. MeasurementValue Shadow Length (S) Terminator Distance (T) Moon Radius (R) Mountain Height (H) Make sure you use the same units when measuring

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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

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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?

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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?

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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.

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