|The Moon is not robed in a smooth and polished surface but is ... rough and uneven, covered everywhere, just like the earth's surface, with huge prominences, deep valleys, and chasms.|
|-- Galileo, c. 1610|
Everyone is familiar with the fact that as the sun gets lower in the sky, shadows get progressively longer. Just before sunset your shadow can be significantly longer than you are tall. The same thing happens on the moon, and we can actually use this effect to determine how tall features on the surface of the moon are.
The bright part of the moon is where it is "day time" and the dark part is where it is "night time". So the line between these two regions, called the "terminator" corresponds to where the sun is setting on the moon. Thus, it is here that the shadows will be the longest, so this is where it is easiest to find the heights of the features. Astronomers used this very technique to determine the topography of the moon, which was critical in being able to safely land the Eagle on the moon in 1969.
Measuring the size of the shadows is fairly straightforward. But how do you convert these distances into heights of the features? Figure 1 shows you a schematic diagram of the moon. The sunlight is coming from the right side of the picture. M is the location of some mountain on the moon. T is the location of the terminator. And O is the center of the moon. Since the light is coming from the right, you can imagine horizontal lines representing sun rays. Thus, if you extend a horizontal line from the top of the mountain to the left, the place it hits the surface of the moon is where the end of the shadow will be.
Figure 2 shows this more explicitly. We've added a few points to the diagram: P is the peak of the mountain, and S is the end of the shadow. So the shadow is line SP. (Not SM, since the peak also casts a shadow on the mountain itself.)
Now the geometry. (If you want, you can skip all this and just use the result, but if you remember some basic geometry, you might want to follow it. It's not really that bad.) There are two similar triangles in Figure 2: OTM and SMP. Thus, we can setup ratios of corresponding sides:
TM / OM = MP / SP
OK, so what are each of these really? OM is the radius of the moon (Know it -- 1738 km). TM is the distance of the mountain from the terminator (We can measure it). SP is the size of the shadow (We can measure it, too). MP is the height of the feature (This is what we want). So let's rearrange that equation and rename things for what they are:
height = (shadow length) x (distance to terminator) / (radius of moon) .
Step One: Callibrating the Eyepiece
In order to measure the sizes of the shadows, we will need to callibrate the tick marks in an eyepiece. That is, we need to know how to covert the size of a shadow in ticks to something more physical like arcseconds or even kilometers. So this callibration is the first step.
The easiest way to do this is to use the Earth's rotation to move an object across the field of view and time how long it takes. Since we know how fast the Earth is turning, we can use the old distance = rate x time equation to figure out the spacing. To do this we need the rate, though.
At the celestial equator, objects move through the sky at the rate of 360 degrees every 24 hours, which is 15 degrees per hour or 15 arcminutes per minute or 15 arcseconds per second. But this is only at the equator. For instance, Polaris moves at the rate of 0 arcseconds per second (that is, it doesn't move). So what about stuff in between? Basically it just depends on the cosine of the declination:
rate = 15 * cos(dec) arcsec/sec .
So, we are ready to do the callibration. First find the moon in your telescope. The callibration lines are easiest to see when backlit by the moon, so pick some easy to recognize feature on the moon somewhere. Before doing the timing run, you need to figure out which way the moon moves when you turn off the tracking. So watch the moon in the eyepiece as you unplug the tracking motor, and see which way it moves. (Then plug the tracking back in to set up for the real run.)
You want the feature you picked to move from one end of the callibration marker to the other, so reposition the object on the appropriate side of the field of view and turn the eyepiece so the callibration marks extend in this direction. When you think you have it set, you should unplug the tracking again to make sure. The feature should move right along the line of tick marks. Then reposition the feature again for the timing run.
Now, time how long it takes from the time you unplug the tracking for the feature to reach the other end of the callibration line. The angle between tick marks is then
scale = (rate x time) / (# tick marks traversed) .
(Hint: Don't count the mark where you started.)
Step Two: Find the height of a mountain
There are two things we need to measure: the size of the shadow and the distance of the shadow from the terminator. But first we need to pick a feature to work with. Any crater or mountain with a measurable shadow will work. You don't want one right next to the terminator, since the shadows often disappear off the edge of the terminator. Nor do you want something far from the terminator, since then the shadow will be too small. So pick something in between where you can measure the shadow, but you won't get confused about where the end of it is.
Now just line up the callibration marks over your shadow and measure how many ticks (plus fractions of a tick) it is. Be as precise as you can. Also measure how many ticks it is from the terminator to the far side of your shadow. Granted, the terminator is not very well defined, since there are shadows all over the place, but do your best.
Use your scale factor from step one to convert to arcsec. Then we want to convert this to kilometers. To do this we need to use the small angle formula. Let X be the linear size of either the shadow or the distance to the terminator, A be the angular size, and d be the distance to the moon. Then:
X / (2 * pi * d) = A / (360-degrees)
or, X = (d) * (A [in arcsec]) / 206265 .
d = 384,400 km on average (which is good enough for us), so d / 206265 = 1.864 km. Thus,
X = (1.864 * A [in arcsec]) km .
Actually, this is not exactly right. The equation we started with is only accurate if we are looking at something straight on -- that is, if it's pointed perpendicular to the direction we are looking. If you look at Figure 1 again, you'll see that shadows near the terminator are only perpendicular to us if it is exactly first (or third) quarter moon, when the suns light is coming exactly from the side.
If it's not exactly first quarter, the shadows will be at a bit of an angle to us. What this means is that a shadow will appear smaller than it really is. If you hold a pen at arm's length and turn it a bit, the angular size will get smaller, even though the pen is still the same length. If you know some trig, you can convince yourself that the apparent size of the pen is smaller by the cosine of the angle that you turned it. (If you don't know trig or don't want to bother, you can take my word for it.)
So, we have to correct for this projection effect by dividing the observed angular size by the cosine of this angle. Let's call it T (since we already used A and this angle is based on the time from first quarter). Then, since there are 29.5 days between new moons,
T = (360 degrees) * (# days from first quarter) / 29.5 .
And our complete formula for X is
X = 1.864 * A[in arcsec] / cos(T) km .
Now you can find the height of the mountain using the equation given in the introduction.