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Moon Crater Lab

  The goal of this lab was to measure the height of a lunar crater.  We did so by applying geometry to a flat picture of the moon, and by looking up the diameter of the moon in our textbook.  The diameter of the moon is 3,476 km.  Here is the picture of the moon that we used:

  The first step was to find the scale between the real moon and this image.  The process for doing so went something like this:

  According to a standard geometry theorem, x*x = y*z.  Also, for the real moon, y + z = 3476, as y + z compose a diameter of the moon (in case this is not clear from the diagram).  Given this,
144 = 4.6z
31.3 cm = z

y + z = diameter in cm
4.6 + 31.3 = 35.9 = diameter in cm

3476km / 35.4cm = 96.82 km/cm

  The average scales for the class came out to be very nearly 100 moon km per map cm., so that is the ratio that we used as a class.  Given this, three pieces of information were needed to find the height of the crater: the length of the crater's shadow, the distance from the crater to the edge of the visible moon (the edge beyond which the sun does not reach), and the distance from this edge to the center of the moon.  Because this edge is relatively close to the center of the moon, the distance from it to the center can be converted directly into arclengths and subtracted from the resulting angle, without doing a rigorous calculation using it.

  Plugging the total distance, 1200km, into the following diagram, gives an initial angle:

cosqi = 1200/1738
qi = 0.8248 (radians)

  Now, to subtract the angle between the moon's center and the light border:
a = 330/1738
a = 0.1899 (that's why I used radians)

qi - a = q = 0.6349

  Now, I have the angle between a tangent to the surface of the moon and the light coming in at the crater.  This angle lets me find the height of the crater in the next diagram, given that the length of the crater's shadow, as measured from the map, is 10km:

tan q = height / shadow length
height = 7.4 km

  So the height of the crater, as calculated, is roughly 7 km.  This measurement admittedly has very little accuracy; the length of the shadow on the crater was only a few pixels across on the analyzed image, and the image was stored using an imprecise image format, meaning that the exact arrangement of pixels is not guaranteed to be accurate.  Also, it was measured at 1 mm in length; this was the smallest unit of measurement detectable using our ruler, so it could easily be off by a factor of 2 or more.  Other than that, though, most measurements were quite accurate.  This lab was quite effective for demonstrating how to find the height of a crater, or, for that matter, any information about features on the surface of a stellar object; it may not have been perfect for processing this specific crater, but it was still quite good.