Project # 4 Height of Lunar Mountains

The purpose of this project is to estimate the height of lunar mountains based on the lengths of the shadows they cast. To do this you will need to
  1. acquire images of lunar mountains
  2. locate the mountains on the moon
  3. measure apparent length of shadows
  4. correct apparent length of shadows for tilt of lunar surface
  5. determine the altitude of the sun over mountains at the time of the image
  6. calculate height of mountains

Preparation

Find out when the moon will be at first quarter. Your observations should be 4 nights before or four nights after that. (Why are other times not good?) Get the ra and dec for the moon using http://www.heavens-above.com. While you are there, record the distance to the moon [Range]. The moon is easy to find but it moves quickly in the sky so coordinates help. [Do not use the moon to reset the encoders on the telescope - its position is too approximate.]

Observations

Look at the moon in the erfle eyepiece first. Pick a good set of mountains with a good shadow cast on a flat surface. Pick mountains near the terminator. Start out with very short exposure times (maybe .02 sec). You might try different colors to see if it helps your contrast. Do a quick sketch of what the moon looks like and where your picture was taken. If you did not get several craters in our picture move over and get one with a few good sized craters. You will also need a good shot of the terminator (although you may have it in the other images.

Reduction and Analysis

Using your sketch, locate your feature on a lunar map. Measure the latitude and longitude of your mountain to the nearest degree. Use the convention that Eastern longitudes are positive, and Western longitudes are negative. Also find the longitude of the terminator by identifying features the terminator runs through and finding them on your map.

Record any names for your feature.

Open up the lunar image and measure the length of a shadow. Just use the x,y coordinate. Convert the pixel to angular distance in the sky and then into kilometers using the distance to the moon and the small angle equation. (Use the Range recorded before.)

To check your conversion from pixel to distance on the lunar surface is correct, pick several craters in the image with sharp, well-defined rims or walls. Measure and record the size of these craters in their longest dimension (which will probably be top-to-bottom). Calculate their sizes. Compare them with a list of known sizes.

If your image covers an area near the center of the lunar disk (i.e. a spot midway between the northern and southern limbs, and midway between the eastern and western limbs), then you should find your crater sizes to be close to that listed, but if your image shows an area far from the middle of the disk, then it will be tilted -- the lunar surface is not face-on to the camera. In that case, you will have to make a correction for the fact you are seeing the crater tilted.

How do we correct for this tilt? Envision the moon and you looking at it:

In the picture below you are viewing two shadows. Each shadow is the same size [42 pixels]. You see the entire 42 pixel length of the one in the center of the moon but the one closer to the edge (limb) of the moon measures only 35 pixels. Thus a picture of center of the lunar disk has little or foreshortening because the lunar surface is roughly face-on to the camera. But a picture of any other area will be tilted with respect to the camera; the closer the area is to the limb of the moon, the larger the tilt.

You will need to correct for this foreshortening.

Use the lunar latitude and longitude of your mountain in the following formula:

tilt correction factor = 1/ (cos(latitude) x cos(longitude))

The correction factor is always larger than 1. Multiply your shadow length (in km) by this factor, to get the corrected shadow length (also in km).

As a check, use this same factor to correct the measured diameters of craters in your image; compare to the known diameters of the craters. How close are your measurements to the listed diameters?

Now you know the size of the shadow you can find the height of the mountain, - if you know the angle of the Sun. See the figure below


The simplest way to the sun angle is to measure the number of degrees between the feature and the terminator (as long as they are less than 90 degrees apart) and multiply by 0.01*(100 - |feature latitude) as a substitute for the solar angle.

Sun angle = (feature longitude -terminator longitude) x .01 x (100 - |feature latitude|)

Shadows are prominent (and most easily measured) when the local solar angle is about 15 degrees or less.

Calculate the height of your feature.

height = shadow length x tan (Sun angle)

Can you find another source, which lists the height of your mountain, or other pictures of your mountain? See How does this height compare to the height of mountains here on Earth? Compare your lunar mountain to Brasstown Bald in Georgia, the tallest mountain in the USA, and the tallest mountain on Earth.

Write-up

Along with the usual log and description of your observations, you should include your lunar sketches. For your crater measurements you might use a table with the crater name, longitude and latitude, your measured size (pixels) in both directions, the size in km and the official size. When you do a table make sure to tell what each column is.

The description of your mountain should have its location, name (if you know it) and a run-down on how you go its height. You might even include a sketch.

Be sure to answer all the questions above.

When you find the official values or other information, like the height of Brasstown Bald, you need to say where you got the information - reference, URL, etc.