Height of mountain on Moon. Using equipment not much more sophisticated than our

Question

Height of mountain on Moon. Using equipment not much more sophisticated than our own rooftop telescopes, one could measure the height of mountains on the Moon. This exercise will illustrate one method. Suppose you are observing the Moon through a good 12” telescope at exactly first quarter – so the terminator (dividing line of light and shadow) appears as a vertical great circle. It just so happens to be a night with exceptional seeing (little atmospheric turbulence) so you are very close to achieving the resolution your telescope is capable of. As you switch lenses to achieve maximum magnification, you begin to notice that, in addition to craters, there are mountains all throughout the surface. Suddenly, an idea strikes you: you could actually figure out how tall the mountains are just by measuring the lengths of their shadows, and a few other measurements! (Also, you read the introductory paragraph, above.) Gripped by an insatiable curiosity, you mount a camera on your telescope and you take images of the Moon (either CCD images or photographs — up to your imagination) . You later make a very large print such that the whole Moon measures 1m across on your (poster-size) print. You decide you will measure the height of a mountain on the apparent equator (the great circle perpendicular to the terminator). You measure the distance of this mountain to be 200 mm from the terminator. You then make another print of the area surrounding the mountain, and this print is 5 times larger than the first one. In this second print, you measure the length of the shadow to be 9 mm. How tall is the mountain, in km? Draw and explain the geometrical relations you use. (Hint: keep things as simple as possible.)

Explanation / Answer

ANSWER: Today, we have lots of tools at our disposal to examine the topography of our nearest neighbor, but measurements of lunar mountains were being recorded long before the development of satellites, space travel, and photography. With a keen understanding of light and shadow and a whole lot of geometry. the shifting pattern of light and dark at the Moon’s terminator (no, not a futuristic killer robot, but the line separating “day” from “night” that sweeps across the lunar disk, creating the familiar phases of the Moon) is caused by sunlight interacting with peaks and valleys. If a bright spot appears beyond the terminator, surrounded by blackness, it must mean that a mountain stands there, its height allowing the peak to catch the sun’s rays while everything around it is shadowed. We see a disk half light and half dark, and a ray of light, DBC, just skims the edge of the Moon to hit the peak of a mountain at C. This geometry is special because the ray of light is perpendicular to our line of sight. That means that the distance between the terminator and the mountain (BC) can be directly compared to the diameter of the Moon because you’re seeing it straight on, while at any other angle this distance would be somewhat foreshortened. Once you have that ratio – the distance BC over the lunar diameter – you know the lengths of two sides of the right triangle, and all you have to do is calculate the hypotenuse, CA, and subtract the lunar radius to get the height of the mountain itself. when the Moon was not in her quadrature” but at nearly any phase. All it takes is recognizing that the distance measured between the mountain and the terminator – now viewed at an angle and therefore appearing shorter than it is – can be related to other known distances that make up the sides of similar triangles. In Herschel’s drawing, the similar triangles are oOL and rLM, and the distance you want to measure is LM, the actual distancebetween the terminator and the mountain peak, so:

LO is just the radius of the Moon, on (the same as rL) is the apparent distance that you measured, and Lo is simply related to the angle of the Sun at that particular lunar phase – something you’d look up in a table, or ephemeris, for the given date. Once you have LM, you can make a right triangle with LM, OL, and OM and use the Pythagorean Theorem again to find the height of the mountain.

CCD imaging of celestial objects has supplanted the use of photographic techniques in a variety of astronomical applications. The images captured by the electronic eye of the Charge Coupled Device (CCD) camera are adroitly handled by computers, and can be collected, manipulated and viewed in a matter of minutes instead of the many hours typical of emulsion based photography. CCD cameras are particularly good at imaging faint objects, but can be used with equal effectiveness for brighter objects such as the moon and planets.

You should enhance your images using the IMDISP commands until you are confident that the details you are interested in are sufficiently prominent to accurately measure.

1.Run the IMDISP program and load a CCD image for analysis. The command HELP will assist you in selecting appropriate tools.

2.Enhance and/or process the original image in order to bring out the details. Measure shadow length.

3.Run the ICE program to determine the angles of the Sun and the Earth with respect to the moon.

4.Measure the position of the object on a map of the moon, and determine its lunar longitude.

5.Calculate the height of the feature.

Run the IMDISP image processing program. The instructor will provide details of how to locate and run this program. Use the DISPLAY AUTO command to display the image. The image should appear in the upper left of your screen. Enhancing the image. PITON (and PICO) is very small. Use the finder chart) to identify it on the image. Use the CURSOR command to place a small cross on the image. Use the mouse to position the crossover PITON and click the left mouse button. Now, zoom in on the feature with the command: DISPLAYZOOM 5 CENTER. An enlarged image of the PITON should appear centered on the screen. carefully to make sure you have properly identified the object. Piton casts a shadow to the lower-right because sunlight is shining from the upper-left. Identify it’s shadow . Then use the command: DISPLAY ZOOM 15 CENTER. To make the image bigger still. Determine the length of the shadow, in pixels, by counting them. Since the shadow falls at an angle, it will be necessary to use the Pythagorean theorem to determine the actual length. Then enter the result of your shadow length calculation on the worksheet.

Run the ICE(Interactive Computer Ephemeris).ICE is a computerized version of the Astronomical Almanac, which can be used if you do not have access to ICE. From the initial menu screen of ICE, press F1to set the time and date. Enter the date and time the CCD image was taken in UTC time. Now press F4 for PHYSICAL EPHEMERIS. Then enter MOON as the object name and select ILLUM .When you press ENTER, the computer will calculate the position of the moon in the sky at the time the CCD Image was taken.

On the worksheet, record the ANGLE. The ANGLE is the angle formed by the EARTHMOON-SUN. Press ENTER to return to the ICE menu, and finally F10 to quit ICE.

Measure the position of the object on the moon map. Find the object on the map, and measure the perpendicular distance, in millimeters, from the object to the centerline of the moon (distance D). Measure the radius of the moon in millimeters (distance L). Enter D and L on the worksheet. Measure, to the nearest degree, the number of degrees longitude (ß) the object sits east or west of the moon’s centerline). Enter the longitude on the worksheet. Complete the calculations on the worksheet to determine the actual height of the object.

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