Part 3: Measure the Height of Lunar Craters

Panorama of East Crater during Apollo 11 mission.
lunarcrater

Sizes of Lunar Craters

It is possible to approximate the height of various surface features (like impact craters), by measuring the shadows they cast.

As shown in the diagram to the right, a crater with a height, H, casts a shadow with a length (as seen from above), L. The anglθ is the angle that the shadow makes with the lunar surface.

The relationship between the height and the length then can be expressed as:

tanthet
earthmoonsuntri

Theаterminatorа(the division between the illuminated and dark halves of the Moon) forms another triangle with the Sun and the Earth. As shown in the diagram to the right.

Here,аθ isаthe angleаformed between the line from the terminator to the Eath, and the line from the Moon's center to the crater, which can be expressed as:

sinthet




Where R is the radius of the Moon, and d is the linear distance from the crater to the terminator.

As a crater comes closer and closer to the terminator line, both angles become smaller. For small angles (<10 degrees) we can approximate the values as:


tansinthet



Making this small angle approximation, we are able to determine the height of a feature on the moon using images.

Height of Lunar Craters

By measuring the angular size of the crater's shadow, and the angular distance from the crater to the terminator, using the method for converting angular to linear distances learned in the parallax lab - We can now approximate the height of the crater as:


happrox



Using the small angle formula, we can solve for the height. Here D is the average surface-to-surface distance from the Earth to the Moon, about 375,900 km.

termdist




Excercise

Using the method above, calculate the height of three craters in the lunar imagesаprovided.

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