## Analyzing the Photographs

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After you have downloaded them, select the best low power and high magnification photographs. Sharpen them and adjust the brightness and contrast to represent the moon as you see it visually in the telescope. Make a full-page print of each of the best.

Use the print showing the entire moon to measure the diameters of a few of the larger craters. Then use those values to calibrate the high magnification prints from which shadow measurements will be made.

Here is the procedure: To measure the diameter of the moon, use a metric scale on the low power print and then measure the diameter of a few large dominant craters near the terminator; the larger the crater the smaller the error. The Moon's diameter is 3476 kilometers. If D is the diameter of the Moon in millimeters on the print and X, the diameter of the crater in millimeters, then the diameter of the crater in kilometers will be given by d = 3476 (X/D)

Figures 7.1 and 7.2 are examples of low and high magnification photographs taken with an 80-mm f/11 refractor using a 5-megapixel camera. Figure 7.1 was a 1/50 second exposure with a 20-mm eyepiece that has a 60° apparent field. For Figure 7.2, the camera was used at maximum zoom with a 9-mm, 66° apparent field eyepiece. The exposure time was 1/5 second.

The height of a lunar mountain or crater wall can be found by measuring the length its shadow casts on the lunar surface or crater floor. In Figure 7.3, the distance L is the length of the shadow of an object, as seen from Earth, which has a height h.

The angle ^ is the angle between the object casting the shadow and the terminator. The terminator is the dividing line between the sunlit and dark areas of the moon. Since ^ is equal to d we can write h = L sin ^

Selenographic longitudes are measured from 0° to +90° if east, and 0° to -90° if west of the Moon's mean central meridian. Selenographic colongitude is the selenographic longitude of the morning terminator. From new Moon to first quarter, it goes from 270° to 360°. After the first quarter it continues to move eastward from 0° to 90°, 180° and back to 270° at the first quarter.

If the selenographic colongitude is designated by S then from new Moon to first quarter the longitude of the terminator is equal to 360° - S. From first quarter to full Moon the terminator's longitude is equal to the colongitude. From full Moon to last quarter, the terminator is at 180° - S. From the last quarter to the new Moon the terminator's longitude is S - 180°.

The angle ^ is the difference between the longitude of the object and the longitude of the terminator. The selenographic colongitude for any time on any date can be found in the lunar ephemeris on Guide 8.0. It can also be found in the Astronomical Almanac or other astronomical ephemeredes. From this

Figure 7.1. A low magnification digital photo of the Moon taken with an 80-mm refractor.
Figure 7.2. A digital photo of the crater Copernicus taken with an 80-mm f/11 refractor.
Figure 7.3. The geometric relationships for measuring lunar topography.

the longitude of the terminator can be determined. The longitude of the object casting the shadow can be found on a lunar map.

Use the high magnification photographs to measure vertical dimensions on the lunar surface. After bringing up the image, sharpen it and improve its brightness and contrast as needed. Use one of the craters already measured on low power to determine an image scale for the print. This will be equal to the crater diameter in kilometers divided by its diameter in millimeters. Measure the length of the shadow cast by the object of interest and convert it to kilometers using the scale. Determine the angle ^ and calculate h.

The best measurements of lunar topography can be obtained near the Moon's equator and central meridian near the first and last quarter phases. There are a number of interesting surface features to measure. For starters, find and measure the heights of the Lunar Alps, the Lunar Apennines, the width of the Alpine Valley, the height of Mount Piton, the diameters of the craters Theophilus and Copernicus and the width of the Hyginus rill. It was suggested at one time that the craters Messier A and B located in Mare Fecunditatis near 50° east longitude, with rays streaming from them toward the west, were created by the entrance and exit of a single object. Measuring the distance between them should answer that question. Measure the width of the smallest clearly discernable feature on the print.

### Measurements on the Computer Monitor

An alternative to measuring a print is to scan and download the linear scale that can be saved to a disk (Figure 7.4). First bring the photo to be measured, then bring up the scale and drag it to overlay the image. Fade the scale to transparency, move

Figure 7.4. A scale for measuring images on the computer monitor.

it to the location on the image you want to measure, then stretch or reduce it to the appropriate size. Zooming in on the object of interest will enlarge the scale by the same amount. Consequently, the calibration of the scale will not be changed.

Measure a crater on the lunar surface with a known diameter to calibrate the scale. Then you can measure any other shadow or object by moving and rotating the scale as desired. With this method one does not need to enlarge sections of the Moon by cropping the print. The scale can be moved around the Moon and zoomed in on any object of interest. In Figure 7.5 the scale is superimposed on the crater Copernicus.

### Measuring Lunar Libration

Because the Moon travels in a slightly elliptical orbit, its orbital speed varies while its rotation rate remains constant, hence we can see slightly more than 50% of its surface in longitude. This is called libration in longitude. Libration

Figure 7.5. The linear scale superimposed on the crater Copernicus.

in latitude is caused by the inclination of the Moon by approximately 6.7° to the plane of its orbit. This combination of longitudinal and latitudinal librations permits us to see about 59% of the entire lunar surface.

Longitudes on the Moon are measured east or west from the mean center. They are positive from 0° to 90° if east; and negative, 0° to -90°, if west. Latitudes are measured from the lunar equator, positive if north; negative if south.

Lunar libration is measured by using the lunar coordinate grid in Figure 7.6. The procedure is the same as with the linear scale. Scan the grid, download it and then drag it to overlay the Moon's image. Fade it to transparency and then stretch it to fit horizontally and vertically. The lunar image will most likely be in a partial phase. In order to stretch the grid to a circle that fits the Moon's image, move it through four 90° rotations and refit it at each rotation. Adjust the rotation of the grid so that the terminator is parallel to the longitudinal grid lines.

The grid is a fixed reference frame against which east-west and north-south changes in the position of craters can be measured. Take photographs of the Moon as frequently as you can. It is best to photograph it after the first quarter or before the last. For each photograph measure the latitudes and longitudes of

Figure 7.6. A coordinate grid for measuring lunar libration.

some select craters near the center and along its limb. Measurements made over a period of a few months will disclose the subtle dynamics of the Earth-Moon gravitational interaction.