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The moon at first quarter

Best Observing Season: any Level:  Introductory Learning Goals: The student will use geometry to measure the height of craters from images taken on earth. Terminology: crater, tangent, terminator Software: MaxIm Archive Image Directory: lunar Archive Image List: firstqtr.fts, thirdqtr.fts References: Price, Fred. 1988, The Moon Observer’s Handbook. (Cambridge University Press, Cambridge).

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Background and Theory

            It is possible to calculate the height of various lunar features from measuring the shadows they cast.  Measurements of this sort were made by NASA and USAF as part of planning the Apollo missions to the Moon.  The USAF surveys were typically done from lunar orbiting satellites which took high resolution pictures of the lunar surface, but with some minor modifications, the same techniques can be used for ground based observations. The first figure below shows the geometry that makes such measurements possible.  A feature of height h casts a shadow of length l.  The angle Φ is the angle the shadow makes with the lunar surface, so

tan Φ = h/l

The next figure shows the directions of the Earth and Sun from the Moon. The triangle CDE is similar to triangle ABC, so that the angle Φ CBA = Φ . Therefore, from triangle ABC we obtain

sin Φ  =  d/R

 where d  is the projected distance from the terminator to the crater as viewed from Earth and R is the Moon’s radius (1,738 km). Now if the crater is near the terminator, Φ will be quite small (say <10 Φ ), so that sin Φ ~ tan Φ   ~ Φ where Φ is measured in radians. We can then set the right hand side of the last two equations equal to each other and solve for the crater height h

 

 

To obtain the height h in km, we need to convert both l and d into km.  To convert l and d, we first multiply the number of pixels by the image scale (1.23 arcsec for the IRO) to get the angular sizes θ_d and θq_l respectively, and then we use the small angle equation:

 

 

where D_km is the Moon’s distance from the Earth at the time of observation. This can be determined using a sky display programs such as Starry Night.

 

A lunar crater showing the wall height h, the shadow length l, and the altitude f of the Sun at the shadow edge

Procedure

Observations

Schedule an observation of the moon within two days of first or third quarter moon. Use Starry Night (or an internet source) to determine the lunar phase for the current month. A blue filter is recommended. Exposure times for lunar imaging will depend on the exact lunar phase, but 0.1 sec should be about right. 

Image Analysis

1.      Examine the lunar image using MaxIm. Choose three features (craters or mountains) near the terminator where shadows will be longest and most obvious.  Be sure you can identify them on the lunar map found with the Lunar Topography Exercise. (A larger, more detailed map can be found in the lab - ask your instructor). You will want to choose areas far enough away from the terminator that all the shadow will be visible and will not overlap with other shadows. Chapter 4 of The Moon Observer’s Handbook (by F. Price) will help you to identify the lunar features best visible at the exact phase you will be observing.

2.      Determine the length of each shadow (l) in pixels and convert to km (L).

3.      To determine the distance d between the crater and the terminator, use a transparency with a ‘T’ on it (actually any right angle will do!).  Line up the bar of the ‘T’ along the terminator, and place the post over the crater.  Move your cursor over the crater, and read off the x and y positions from the top of the window.  Move your cursor over the spot where the post of the ‘T’ meets the bar.  Record these x and y coordinates.

4.      Find the total distance between the two points (in pixels), using the Pythagorean Theorem:

5.      Using the method described in the background section above, compute the height (in km) of each lunar crater. 

6.      Determine the percent uncertainty in the crater height h in the following way.  First, estimate the uncertainties (in pixels) in the measured shadow length Dl and terminator distance Dd. Then the uncertainty of the result is:

7.      Compare with craters on the Earth (e.g. Winslow Crater, Arizona: height 100 m.). Why are Lunar crater walls are so much higher?