The moon has long been one of the favorite subjects for study by both amateur and professional astronomers. Over 400 billion years old, the lack of an active geology and atmospheric conditions make it an excellent arena for studying impact craters. Many craters on the moon have remained relatively unchanged for thousands of years, offering us the unique opportunity to learn about conditions on the moon, the evolution of the solar system, and deduce the possibility of additional impacts in the future. Craters are the oldest common features found throughout the solar system [16], and in fact, the largest known crater in the system is known to be on the moon. This is the Pole-Aitken Basic at the lunar south pole, which is some 1300 miles across.
The study of craters and lunar geology in general was at it's height during NASA's lunar missions in the 1960's. Telescopic observations of craters in regards to width, depth, and structure played a key role in choosing landing sites. [31] A number of methods were developed by which to extrapolate information regarding the craters, and also the objects that created them. Today, we can determine the size, mass, direction, and velocity of objects based on our knowledge gained from crater research. [14] This also helps scientists to understand the earth's own history of impacts, and the potential risk of future impacts. In an effort to observe and study lunar geology, this project will cover the following objectives:
· Conduct observations of lunar features
· Devise measurement techniques to measure the diameter and height of lunar objects
· Research and provide the altitude of the sun at the time of the observations
· Compare results with published material
There are several types of features on the moon, such as craters, mares, and mountains. For the purpose of this study, this project shall focus particularly on features with notable shadows such as crater rims and mountains.
The moon always keeps the same face towards the earth because it's rotation is the same period that it takes to orbit the earth. However, the moon turns unevenly on it's axis and it's speed varies along the path of the elliptical orbit, causing a libration in longitude to occur. The result is that the moon swings back and forth approximately 7° along the east-west axis during each lunation. This movement allows astronomers to observe up to 59% of the surface rather then merely 50%. [28]
Coordinates on the moon are called selenographic. Selenographic latitude is measured from the sub-earth point [6] going northward, and negative going southward [27], while the mean center of the lunar disk is selenographic 0°. Although libration affects the positions of features on the surface, the selenographic system allows astronomers to refer to their mean apparent positions as a reference.
The terminator corresponds to where the sun is setting on the moon, while the sun's selenographic colongitude represents the position of the morning terminator on the lunar surface. Colongitude is approximately 270° at the new moon, 0° at the first quarter moon, 90° at full moon, and 180° at the last quarter. [27] In many lunar ephemerides, it is often measured in relation to the mean center of the lunar disk not accounting for the affects of libration. This may cause the observer to notice a difference between the position of the terminator in relation to a lunar feature as dictated by the ephemeris and the actual observation. [28]
The most common and obvious process that affects the lunar surface is cratering. These are the result of impacts by objects such as comets and asteroids, and can range in size from microcraters which are not visible in telescopes, to more then 1000km across. [14] The size of a crater is related to the energy of the impacting object, as the energy increases, so does the size of the crater. [4] Objects can strike at a wide range of speeds, with the average being 20km per second. An impact of this nature would typically produce a crater that is 10 - 20 times the diameter of the object itself. [7] Generally speaking, a 1km wide crater will be formed on the moon every 50,000 years, a 10km crater every 5,000,000 years, and a 100km crater ever 500,000,000 years. [12]
There are three general groups of craters. First is the primary crater site, which are randomly distributed across the lunar surface. In some instances, impacts can even occur along a line forming a crater chain. Second, there are smaller craters often seen surrounding a larger site. These are typically the result of ejected material falling onto the surface as a result of the impact. The third type of crater is of volcanic origin, which are generally found along fractures in the surface. [14] Impact craters fall into two categories, simple and complex. Simple craters are bowl shaped, and are the most common craters on the moon. Generally they have diameters no larger then 15km in diameter. [15] Normally a simple crater will be found to have bright ejecta, circular raised rims, and a concave floor. [29]
Complex craters on the other hand have diameters larger then 15km across, and typically have shallow, flat floors. In larger craters of the 20 - 175km size, there is often a central peak or peaks. Craters larger then 175km are found to have complex ring structures, while craters larger then 300km are no longer classified as craters but as impact basins. [15]
The equipment used for observations was comprised of an 8inch Celestron Dobsonian Starhopper, and three eyepieces: 25mm, 9mm, and a 12mm micrometric. Photographs were taken using an Olympus D460Z 1.3 megapixel camera. Images were processed using Camedia software, and in some instances, they were given increased contrast to highlight the shadows of the terrain.
Two sets of images were taken during each observation, an image of the entire moon using the 25mm eyepiece, and close up images of craters using the 9mm. The primary objective of the observations was to capture images of the moon during different angles of sunlight exposure. Under such lighting conditions, surface features are more distinctive, [28] while under full moon lighting conditions, the light appears to wash out the details and flatten the surface. Craters on or partially in the terminator were not selected for observation, as the shadows near the terminator can change rapidly and appear deeper then they actually are. [28] Both full and higher magnification images were used in measuring the diameter and height of objects. After all the images had been processed, they were compared to the Hatfield Photographic Lunar Atlas and identified.
Compare these two images of the moon. The first was taken during a full moon on 10/02/01 using a moon filter. Notice that the angle of the sun causes the surface details to appear flattened. Sun's colongitude 85.74. In the second image taken on 10/5/01, the surface features stand out. Sun's colongitude 122.23.
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A number of methods on how to measure heights and diameters of lunar objects were found in astronomy books and Internet research, all of which proposed very different methods of measuring the diameter and depth of lunar objects. To touch on a few of these :
· A geometric method advises the student to determine how many "bases" of a triangle can fit across the base of a crater in order to develop a scale to determine the height of the walls. [17]
· One method known as stereo photography includes the use of two photos taken from different perspectives, which are then combined to form a stereo image from which the measurements can be taken. [21]
· The late astro-geologist Gene Shoemaker developed a complex method for calculating crater diameters based on his study of craters formed in nuclear tests. His formulas involve gravity, density, a crater collapse factor, and the projectile's kinetic energy. [25]
· Gerald North offers a method which uses the sun's colongitude to measure the crater. His method also includes advice on correcting the results for the effects of parallax. [28]
· One method involves photographing the lunar structure, then counting the pixels forming the shadow. The measurements are then converted into realistic units. [3]
Two methods of measurement were used in this project. The first was based upon the method Galileo used to measure the height of the lunar mountains in the 1600's. His method was based upon calibrating the tick marks in an eyepiece and timing the passage of a lunar object along the east-west axis, then converting these measurements into a realistic units such as miles or kilometers. [31] After experimenting with this method, I found it more productive to use an updated version of this procedure. Instead of counting tick marks, photographs were taken and measured, however the end result is the same. In this particular method, which can be found in detail in "Calculating the Height of Piton" [9], the example provided was measurement of the height of the mountain Piton, an isolated peak which borders a mare. To compare my own results to the example, I also measured the height of Piton with images taken on 11/7/01. This is a step by step example of the first method used.
This method requires that two images have been taken during the same observation, one of the entire moon, and one of the object. An alternative would be to 'blow-up' a section of the full image to arrive at a close up and more detailed image of the object to be measured. In this method, three basic measurements are required.
· radius of the moon in the full image
· length of the shadow in the close up image
· distance of the object to the terminator in the full image
In this method, principles of geometry are used to solve the problem, as demonstrated in the following images[9]:
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In this example, the smaller triangle is similar to the larger triangle. Using the property that corresponding sides of similar triangles are proportional, we are able to make the necessary calculations to determine the size of the object. The following applies:
height of object/length of shadow = distance to terminator/radius
or
height of object = (distance to terminator x length of shadow)/radius
1. Using Image A, I measured the radius, distance of the object to the terminator, and measured the distance from the object to a different object to be used in computing a scaling factor. In this instance, I measured the distance from the tip of Piton to the top of Plato. I also computed the diameter of the moon using the radius.
Image of Mons Piton taken on 11/7/01 using 9mm
Piton Latitude 41° N, Longitude 1° W, 25km wide
Solar Colongitude 164.16 [34]
Age in days 21.2
radius 16.5cm
distance of Piton's shadow to the lunar terminator 2cm
measurement for scaling factor 3cm
Note: The radius must be taken from the center of the moon. This is the sub-Earth point, the closest point of the moon to the earth.
2. The next step was to use Image B and take a similar measurement for the scaling factor, and measure the length of the objects shadow. The measurements taken for the scaling factor are important, because we must make a conversion factor by which to compare measurements from the two photographs. This is a determination called plate scale [3].
length of shadow .5cm
measurement for scaling factor 6.5cm
3. Next, calculate the scaling factor between Images A and B.
measurement B/measurement A = scaling factor
6.5cm/3cm = 2.17
4. In this step, the length of the shadow in Image B is converted to the same plate scale as Image A.
length in Image B/scaling factor = length in Image A
.5cm/2.17 = .23cm
5. The height of the object in Image A can now be calculated.
(distance to terminator * length in Image A)/radius = height in Image A
(2cm * .23cm)/16.5cm = .028cm height in Image A
6. Next, the height of the object in Image A must be scaled up to the actual size of the moon so we derive another scaling factor.
lunar diameter/diameter in Image A = scaling factor
2180/33cm = 66.06
7. The height of the object on the actual moon can be calculated now.
height in Image A * scaling factor = height on moon
.028cm * 66.06 = 1.8 miles estimated height
The actual height of Piton is 1.6 miles
8. An estimation was made of the object diameter. This was done by measuring the diameter of a known width and then converting this into the millimeter scale used for measurement. In the image used for Piton, 100km equals 1cm. Piton measured 25% of a cm, hence the estimate is that Piton is approximately 25km wide which is correct.
In the second method, I used formulas which were provided by the Astronomical League [36]. This is a more scientific method, which also provided the value for the sun's altitude, and angle of the earth and sun as seen from the center of the moon. The following values are required for this method, and were taken from the ALPO Ephemeris [34]. Note that latitude and longitude coordinates must be converted into decimal degrees for use in the formula.
MLAT - lunar object's selenographic latitude 41°
MLONG - lunar object selenographic longitude 1°
SLAT - solar selenographic latitude 1.08
SCOL - solar selenographic colongitude 164.16
ELAT - earth's selenographic latitude -2.98
ELON - earth's selenographic longitude -3.95
DIST - distance to the moon at the time of the observation in km 370,959
1. The altitude of the sun is calculated using the formula:
(asin represents inverse sin)
asin(sinMLAT sinSLAT + cosMLAT * cosSLAT *sin(SCOL + MLON)) = Altitude 11.87°
2. Next, the angle between the earth and sun as seen from the lunar center is calculated using: (acos represents inverse cos)
acos(sinSLAT * sinELAT + cosSLAT * cosELAT * sin(SCOL + ELON)) = Angle 70.3°
3. The size of the object is adjusted by:
(measurement * distance) / focal length = Shadow height 32.64
(.11 * 370,959 / 1250in)
The measurement and focal length are converted into the same units.
4. The height of the object is then calculated and converted into realistic units:
Shadow * (sinAltitude/sinAngle)-2.88x10^-Shadow^2*(cosAltitude/sinAngle)^2 = 7.1295
7.1295 * 1000 = 7129.53ft = Estimated height 1.35 miles
Crater wall heights used for comparisons were taken from Astronomica [35].
4/28/01
21% Illuminated
Earth Longitude -3.54
Earth Latitude 0.74
Sun's Colongitude 328.28
Selen Latitude -1.37
Phase Angle 126.10
Semi-diameter 15.98'
Catharina
Latitude 18° 6' 00"S Longitude 23° 24' 00" E
Depth 2743m
Diameter 104km
Height 5000m (16,404ft)
Sun's altitude -8.33
Sun angle 125.27
Shadow 80.11
Estimated height 14,211ft
Estimated diameter 100k
Cyrillius
Latitude 13° 12' 00"S Longitude 24° 00' 00" E
Depth 3657m
Diameter 98k
Height 3550m (11,646.98ft)
Sun's altitude -7.83
Sun angle 125.27
Shadow 53.41
Estimated height 8,910ft
Estimated diameter 100km
Theophilus
Latitude 11° 24' 00"S Longitude 26° 24' 00" E
Depth 6858m
Diameter 110km
Height 5500m (18,044.6ft)
Sun's altitude -5.48
Sun angle 125.27
Shadow 80.11
Estimated height 9,381ft
Estimated diameter 101km
10/5/01
94% Illuminated
Earth Longitude -4.73
Earth Latitude 5.17
Sun's Colongitude 122.23
Selen Latitude 1.57
Phase Angle 28.10
Semi-diameter 15.13'
Hercules
Latitude 46° 42' 00"N Longitude 39° 06' 00" N
Depth 3810m
Diameter 69km
Height 3200m (10,498ft)
Sun's altitude 13.85
Sun angle 27.68
Shadow 21.65
Estimated height 11,555ft
Estimated diameter 50km
Atlas
Latitude 46° 42' 00" N Longitude 39° 00' 00" E
Depth 3048m
Diameter 87km
Height 3000m (9842ft)
Sun's altitude 13.92
Sun angle 27.68
Shadow 18.56
Estimated height 9600ft
Estimated diameter 102km
Democritius
Latitude 62° 18' 00´ N Longitude 35° 00' 00" E
Depth N/A
Diameter 39km
Height 1950m (6397ft)
Sun's altitude 11.77
Sun angle 27.68
Shadow 15.46
Estimated height 6790ft
Estimated diameter 50km
As a result of this project, I was able to make a number of determinations. These are:
· Object shadows near the terminator are extended.
· Shadows change quickly near the terminator.
· Objects near the limb of the moon are being viewed at an angle.
· The terminator is jagged so measurement of the distance of the object to the terminator is difficult.
· The use of a digital camera may introduce errors as the images may be undersampled, extending the area of the shadow.
· Shadows should be measured against flat terrain, avoiding crater bottoms where possible.
· Measuring shadows across the basin of small craters may increase errors as the bottom is bowl shaped, whereas larger more complex craters have flatter bottoms.
· Small shadows are likely to incur more errors in measurement.
· Measurements may have been taken at points with heights that vary from that of published data, which may be averages or best case scenarios.
· Measurements should be repeated to reduce the human error factor.
In general, crater diameter was easier to estimate then height, as the suns angle at the time of the observation dictates the length of the shadow. Measuring the true depth of craters is increasingly difficult as one must make a determination between the measurement from the lunar surface to basin of the crater rather then the from the height of the crater rim.
The study of lunar geology continues to benefit science by allowing us to better understand the evolution of bodies in the solar system. In addition, as our knowledge grows regarding impacts, we are better prepared to assess our own risk of encounters with objects, and the consequences of such natural disasters.