Diameter and Depth of Lunar Craters

Home , Abulfeda (crater), Archytas (crater), Barrow (crater), Descartes (crater), Dollond (crater), Fernelius (crater)

Student Name Alan Brockman HET602 Student ID 1611739 Project Supervisor Barry Adcock [email protected]

SAO Project Cover Page

Project 14 Project Title Diameter and Depth of Lunar Craters

Submitted 15 June 2008

All of the work contained in this project is my own original work, unless otherwise clearly stated and referenced.

I have read and understood the SAO Plagiarism Page “What is Plagiarism and How to Avoid It” at http://astronomy.swin.edu.au/sao/students/plagiarism/

Introduction

Galileo was the first to attempt measuring mountains of the Moon using a light-tangent method that estimates a peak‟s distance from the lunar terminator at the moment the peak is first (or last) illuminated. He calculated heights of over 7,000 metres, a vast overestimation. Using the same method in 1787, William Herschel used an eyepiece micrometer to determine that lunar hills range in height between 800 and 2,400 metres1.

During the same time period, Johann Schroter introduced a new “shadow method” for height determination which was modified by Heinrich Olbers for improved accuracy7. This method has since been used to measure thousands of lunar hills and mountains being replaced only since the advent of remote sensing by lunar probes.

AIM

The aim of this project was to measure the size and depth of features on the lunar surface using one or more techniques. Two methods are described which utilize digital imagery and trigonometry to calculate the size and depth of selected lunar craters. It is proposed to use these methods and compare their accuracy to published data.

METHOD

Digital imaging of the Moon This was accomplished using a permanently mounted 14” f/10 Schmidt Cassegrain Telescope and a simple Philips ToUcam web camera (see Figure 1). The large scale image of the moon was accomplished by use of a 4” Takahashi FSQ-106 refractor and a SBIG STL- 11000 CCD camera. Images were taken of selected targets along the Lunar terminator when the Moon was 20 days old. All images were taken in the early hours of May 26 2008 from my observatory in Exmouth, Western Australia.

Objects chosen showed good surface structure and were close to the Lunar terminator at time of imaging for maximum relief and shadow length.

Statistical analysis comparing paired measurements non-parametrically were performed with the Wilcoxin Signed Ranks Test using SPSS 11.0 for Windows.

Figure 1. My equipment set up consisting of a Tak FSQ106 piggy-backed onto a 14” f/10 SCT and mounted on a Paramount ME inside a Sirius Dome.

Method 1 for determination of the size (diameter) of Lunar craters

The time and date was recorded and used to determine the correct distance and the angle of the Sun to the Lunar surface.

Heights of lunar features were calculated by measuring the length of the shadow of the feature and determining the angle of the Sun at the Moon with the help of an Astronomical Almanac.

Calculating the height of a lunar feature required solving a series of plane and spherical triangles that related the positions of the Sun and Earth to the position of the feature. One of these triangles involves the point on the Moon where the Earth is directly overhead, which the Astronomical Almanac tabulates as the Earth‟s selenographic latitude and longitude. The Sun‟s selenographic colongitude and latitude were also recorded.

Steps 1. A number of potential targets on the lunar surface are selected showing good surface structure and relief.

2. Images are taken and the precise time noted to enable distance and angle of the Sun on the lunar surface. 3. The Sun‟s selenographic longitude is taken from published ephemerides to find the length of the target‟s shadow on the lunar surface. 4. The official crater size and latitude and longitude are obtained and compared.

Method 2 for determining the heights or depths of Lunar features2

Regions were chosen near the lunar terminator so that the mountains cast long, prominent shadows.

Determine the selenographic location of the lunar features The selenographic colongitude is the longitude of the morning terminator on the Moon, as measured in degrees westward from the prime meridian. The morning terminator forms a half-circle across the Moon where the Sun is just starting to rise. As the Moon continues in its orbit, this line advances in longitude and the value of the selenographic colongitude increases from 0° to 359° in the direction of the advancing terminator3.

The approximate lunar latitude and longitude (to the nearest degree) of the mountains were found using a lunar atlas3. The selenographic longitude of the morning terminator measured westward from the prime or central meridian is known as the co-longitude.

The relationship between selenographic longitude, λ, and co-longitude, c, is as follows:

Λ = selenographic longitude of morning terminator = 360º - c = sunrise on the Moon.

Measure the apparent length of the shadows Using CCDSoft image processing software, the number of pixels from the rim of a crater to the end of its cast shadow were determined.

Several craters were chosen with well-defined rims or walls. The size of these craters were recorded in a line perpendicular to the terminator shadow.

Convert the apparent length from pixels to kilometres The image scale per pixel in arc seconds was determined and converted to degrees Correct the apparent length of shadows for tilt of the lunar surface (foreshortening) Tilt correction factor = 1/(cos (latitude) + cos (longitude)). See Figure 2. This correction factor assumes that the observer is directly above the point on the lunar disk which can also be corrected for greater accuracy. Multiply the shadow length by this factor to get the corrected shadow length (km).

Figure 2. Correcting for angle tilt4

Formulae used for determination of crater length

Image scale, or angular size per pixel, υpixel, υpixel = 206265 x dpixel/F [arcseconds], (Eqn. 1) -3 υpixel = 206265 x 5.55 x 10 /3500 = 0.327 = 0.33 arcsec/pixel

Converting apparent length from pixels to kilometres, 1 pixel = 0.33 arc sec = 0.33 arcsec/3600 = 9.16 x 10-5 degrees Earth-Moon distance at 26 May 2008 = D = 395,633.44 km.

Linear distance on the Moon covered by one pixel , ldpixel, -5 ldpixels = D [km] x tan 9.16 x 10 [degrees], (Eqn. 2) -5 ldpixels = 395,633.44 x tan (9.16 x 10 degrees) = 0.6325 km.

Correction factor for foreshortening due to lunar curvature, cf = 1/[cos (latitude) x cos (longitude)] (Eqn. 3)

Distance between two pixels at coordinates (x1, y1) and (x2, y2) can be determined using the 2 2 2 Pythagorean Theorem for 3 points forming a right triangle where (x1 - x2) + (y1 – y2) = z . 2 So that uncorrected length of crater, clupixels = √ z (Eqn. 4)

Corrected length of crater, cl = clupixels x cf

Length of crater, L (km) = cl x ldpixels (Eqn. 5)

Determining a mountains height from the length of its shadow

Determine the altitude of the Sun over the mountains at the time of the image Checking an ephermerides using the date and time of the observation will determine the solar colongitude (Co) and the subsolar point Latitude (Bo). The angle of the Sun above the mountain can be calculated by: x = sin(Bo) x sin(latitude) + cos(Bo) x cos(latitude) x sin(Co + longitude) angle θ = arcsin(x)

Calculate the height of the mountains using simple trigonometry Now that the shadow length (L) and the angle θ has been determined, the height of the lunar peak can be calculated using simple trigonometry where, cot θ = L/H6 Therefore, height (H) = cot θ/L. See Figure 3.

Figure 3. Calculating heights using simple trigonometry4

Optical specifications

14” SCT at f/10 gives a focal length, F, = 3500 mm. With a Phillips ToU webcam, -3 Pixel size, dpixel, = 5.5 um x 5.6 um = 5.55 um (average) = 5.55 x 10 mm Array size 659 x 494 pixels Chip size 2.8 x 3.7 mm Field of view = 2.7 x 3.6 arcmin

RESULTS

Table 1. Moon Ephemeris for 06:30, 26 May 20084.

Figure 4 shows the position of the terminator for the 20 day old Moon in the early hours of May 26 taken with the FSQ-106 at f/5.

Figure 4. Day 20 Moon showing position of the terminator and major features

Figures 5, 6 and 7 describe some of the more obvious Lunar craters and other features along the terminator.

Figure 5. 90°N to 15°N

Figure 6. 17°N to 24°S

Figure 7. 22°S to 90°S

SECTION 1. Determining Crater Size

Twelve craters were selected targets for crater size estimation covering varying established size ranges and latitudes. UT date at the start of the imaging session was yyyymmdd.hhmm: 20080525.2241. This enabled determination of the distance to the Moon, D, as 395,633 km.

See Figure 8 for the selected craters, W. Bond and Archytas. Time of image: 26 May 2008, 06:53 am. 1. W. Bond [65.3°N, 3.7°E]. Walled plain (158 km)1. cf = 1/(cos 65.3 x cos 3.7) = 1/(0.4179 x 0.9979) = 2.3979

Length of crater is defined by two points with coordinates (xi, yi) of (308, 415) and (300, 292) as determined using the program CCDSoft. 2 2 clupixels = √(308 - 300) + (415 - 292) = √(64+ 15129) = 123.260 L = 123.260 x 2.3979 x 0.6325 = 186.94 km.

Diameter of Crater W. Bond = 187 km

2. Archytas [58.7°N, 5.0°E]. Crater (32 km)1. cf = 1/(cos 58.7 x cos 5.0) = 1/(0.5195 x 0.9962) = 1.9323 2 2 clupixels = √(474 - 451) + (424 - 407) = √(529 + 289) = 28.601 L = 28.601 x 1.9323 x 0.6325 = 34.96 km.

Diameter of Crater Archytas = 35 km

Figure 8. Area surrounding the North Pole and the eastern border of Mare Frigoris

06:53

+ Goldschmidt

+ Barrow Epigenes +

W. Bond Timaeus +

Archytas

K3CCD.0007 Protagoras

See Figure 9 for the selected craters, Aristoteles and Eudoxus. Time of image: 26 May 2008, 06:41 am.

3. Aristoteles [50.2°N, 17.4°E]. Crater with terraced walls (87 km)1. cf = 1/(cos 50.2 x cos 17.4) = 1/(0.6401 x 0.9542) = 1.6372 1 pixel = 0.6905 km

Length of crater is defined by two points with coordinates (xi, yi) of (174, 200) and (244, 252) as determined using the program CCDSoft. 2 2 clupixels = √(174 - 244) + (200 - 252) = √(4900 + 2704) = 87.201 L = 87.201 x 1.6372 x 0.6325 = 90.30 km.

Diameter of Crater Aristoteles = 90 km

4. Eudoxus [44.3ºN, 16.3ºE]. Prominent crater with terraced walls (67 km)1. cf = 1/(cos 44.3 x cos 16.3) = 1/(0.7157 x 0.9598) = 1.4557 2 2 clupixels = √(358 - 416) + (313 - 361) = √(3364 + 2304) = 75.286 L = 75.286 x 1.4557 x 0.6325 = 69.32 km.

Diameter of Crater Eudoxus = 69 km

Figure 9. The Eastern part of Mare Frigoris with the striking companion craters Aristoteles and Eudoxus

See Figure 10 for the selected craters, Menelaus and Silberschlag. Time of image: 26 May 2008, 06:51 am.

5. Menelaus [16.3°N, 16.0°E]. Prominent crater with a sharp rim and central peaks (27 km/3010 m)1. cf = 1/(cos 16.3 x cos 16.0) = 1/(0.9598 x 0.9613) = 1.0839 2 2 clupixels = √(254 - 290) + (228 - 203) = √(1296 + 625) = 43.829 L = 43.829 x 1.0839 x 0.6325 = 30.05 km.

Diameter of Crater Menalaus = 30 km

6. Silberschlag [6.2°N, 12.5°E]. Circular crater (13.4 km/2530 m)1. cf = 1/(cos 6.2 x cos 12.5) = 1/(0.9942 x 0.9763) = 1.0303 2 2 clupixels = √(569 - 592) + (443 - 421) = √(529 + 484) = 31.828 L = 31.828 x 1.0303 x 0.6325 = 20.74 km.

Diameter of Crater Silberschlag = 20.7 km

Figure 10. The western part of Mare Serenitatis bordered by the Haemus mountain range

23. Linne 06:51 MARE SERENITATIS + Sulpicius Gallus + Bobillier

+ Menelaus

+ Manilius LACUS LENITATIS

+ Silberschlag

K3CCD.0006

See Figure 11 for the selected craters, Julius Caesar and Dionysius. Time of image: 26 May 2008, 06:47 am.

7. Julius Caesar [9.0ºN, 15.4ºE]. Flooded crater with a wide wall and a dark floor (90 km)1. cf = 1/(cos 9.0 x cos 15.4) = 1/(0.9877 x 0.9641) = 1.0502 2 2 clupixels = √(145 - 232) + (171 - 261) = √(7569 + 8100) = 125.176 L = 125.176 x 1.0502 x 0.6325 = 83.15 km.

Diameter of Crater Julius Caesar = 83 km

8. Dionysius [2.8°N, 17.3°E]. Circular crater (17.6 km)1. cf = 1/(cos 2.8 x cos 17.3) = 1/(0.9988 x 0.9548) = 1.0486 2 2 clupixels = √(450 - 468) + (366 - 383) = √(324 + 289) = 24.759 L = 24.759 x 1.0486 x 0.6325 = 16.42 km.

Diameter of Crater Dionysius = 16.4 km

Figure 11. A region with a prominent radial structure containing the rille Rima Ariadaeus.

34 HYGINUS 06:47

+ Boscovich + Julius Caesar + Sosigenes

Silberschlag +

Whewell + + Ariadaeus + Cayley

+ Dionysius + Ritter

See Figure 12 for the selected craters, Abulfeda and Almanon. Time of image: 26 May 2008, 06:41 am.

9. Abulfeda [13.8°S, 13.9°E]. Crater (62 km/3110 m)1. cf = 1/(cos 13.8 x cos 13.9) = 1/(0.9711 x 0.9707) = 1.0608 2 2 clupixels = √(212 - 248) + (217 - 291) = √(1296 + 5476) = 82.292 L = 82.292 x 1.0608 x 0.6325 = 55.21 km.

Diameter of Crater Albufeda = 55 km

10. Almanon [16.8°S, 15.2°E]. Crater (49 km/2480 m)1. cf = 1/(cos 16.8 x cos 15.2) = 1/(0.9573 x 0.9650) = 1.0825 2 2 clupixels = √(327 - 358) + (313 - 363) = √(961 + 2500) = 58.830 L = 58.830 x 1.0825 x 0.6325 = 40.28 km.

Diameter of Crater Almanon = 40 km

Figure 12. A „continental‟ area showing the landing site of Apollo 16 which landed some distance to the north of the crater Descartes.

Apollo 16 +

+ Andĕl + Dollond

+ Descartes

Abulfeda +

Almanon +

See Figure 13 for the selected craters, Stöfler and Licetus. Time of image: 26 May 2008, 06:55 am.

11. Stöfler [41.1°S, 6.0°E]. Walled plain with flooded floor (126 km/1880 m)1. cf = 1/(cos 41.1 x cos 6.0) = 1/(0.7536 x 0.9945) = 1.3343 2 2 clupixels = √(400 - 340) + (299 - 170) = √(3600 + 16641) = 142.271 L = 142.271 x 1.3343 x 0.6325 = 120.07 km.

Diameter of Crater Stöfler = 120 km

12. Licetus [47.1°S, 6.7°E]. Crater (75 km)1. cf = 1/(cos 47.1 x cos 6.7) = 1/(0.6807 x 0.9932) = 1.4792 2 2 clupixels = √(483 - 460) + (444 - 372) = √(529 + 5184) = 58.830 L = 58.830 x 1.4792 x 0.6325 = 55.04 km.

Diameter of Crater Almanon = 55 km

Figure 13. A dense crater field in the vicinity of the prime meridian on the southern hemisphere of the Moon.

06:55

Kaiser +

+ Fernelius

Miller Stöfler

Licetus +

K3CCD.0008

Table 2. Published and estimated crater size determinations

Crater name Latitude Longitude Measured Published Difference diameter diameter (km) (%) (km) (km)1 W. Bond 65.3°N 3.7°E 187 158 +29 (18%) Archytas 58.7°N 5.0°E 35 32 +03 (9%) Aristoteles 50.2°N 17.4°E 90 87 +03 (3%) Eudoxus 44.3ºN 16.3ºE 69 67 +02 (3%) Menelaus 16.3°N 16.0°E 30 27 +03 (11%) Julius Caesar 9.0ºN 15.4ºE 83 90 -07 (8%) Silberschlag 6.2°N 12.5°E 20.7 13.4 +07.3 (54%) Dionysius 2.8°N 17.3°E 16.4 17.6 -01.3 (7%) Abulfeda 13.8°S 13.9°E 55 62 -07 (11%) Almanon 16.8°S 15.2°E 40 49 -09 (23%) Stöfler 41.1°S 6.0°E 120 126 -06 (5%) Licetus 47.1°S 6.7°E 55 75 -20 (27%)

Sources of error and accuracy of estimation of crater diameters

Where you decide to measure the length of the crater from rim to rim has a significant effect on the final result and can cause the calculated diameter to vary by as much as 100%. This variability is due to the observed foreshortening. Given the small sample size of n = 12, the data was assumed to be non-normally distributed so the Wilcoxin Signed Ranks Test was used to compare the two groups of measurements. No significant difference between the measured and published values for the crater diameters were found, P = 0.75 (Table 2). From Figure 14, it is also evident that there was no trend in under- or over-estimating crater diameter based on the actual size of the crater.

Figure 14. Difference (Estimated – Published) vs Average of Crater Diameter (km)

SECTION 2. Determining Mountain or Crater Heights

Richmond 2003 Determine the altitude of the Sun over the mountains at the time of the image Checking an ephermerides using the date and time of the observation will determine the solar colongitude (Co) and the subsolar point Latitude (Bo). The angle of the Sun above the mountain can be calculated by: x = sin(Bo) x sin(latitude) + cos(Bo) x cos(latitude) x sin(Co + longitude) angle θ = arcsin(x)

Colongitude of the Sun (Co) = 160.88 Subsolar point latitude (Bo) = 1.5

Angle of the Sun above the crater Step 1: x = sin (Bo) x sin (latitude) + cos (Bo) x cos (latitude) x sin (Co + longitude) Step 2: angle theta (θ) = arcsin (x).

Knowing the length of the shadow (L) and the angle (θ) will allow calculation of the height of the Lunar peak using simple trigonometry.

Colongitude of the Sun (Co) = 160.88 Subsolar point latitude (Bo) = 1.5 Time of image: 26 May 2008, 06:41 am. UT date as yyyymmdd.hhmm: 20080525.2241. This enables determination of the distance to the Moon, D, as 395,633 km.

1. Menelaus [16.3°N, 16.0°E]. See Figure 10. Prominent crater with a sharp rim and central peaks (27 km/3010 m)1. Tilt correction factor = 1/(cos 16.3 x cos 16.0) = 1/(0.9598 x 0.9613) = 1.0839 2 2 shadowlupixels = √(274 - 297) + (232 - 207) = √(529 + 625) = 33.971 L = 1.0839 x 33.971 x 0.6325 = 23.289 km x = sin (Bo) x sin (latitude) + cos (Bo) x cos (latitude) x sin (Co + longitude) x = sin 1.5 x sin 16.3 + cos 1.5 x cos 16.3 x sin (160.88 + 16.0) x = 0.0262 x 0.2807 + 0.9997 x 0.9598 x 0.0544 x = 0.0074 + 0.0522 x = 0.0596 angle theta (θ) = arcsin (x) = asin 0.0596 = 3.4167° height (H) = cot θ/L = cot 3.4167 / 23.289 = 3.164 km

Height of Crater Menelaus = 3164 m

2. Manilius [14.5°N, 9.1°E]. See Figure 10. Very prominent crater with terraces and central peaks (39 km/3050 m). Tilt correction factor = 1/(cos 14.5 x cos 9.1) = 1/(0.9681 x 0.9874) = 1.0461 2 2 shadowlupixels = √(91 - 106) + (489 - 474) = √(225 + 225) = 21.213 L = 1.0461 x 21.213 x 0.6325 = 14.036 x = sin (Bo) x sin (latitude) + cos (Bo) x cos (latitude) x sin (Co + longitude) x = sin 1.5 x sin 14.5 + cos 1.5 x cos 14.5 x sin (160.88 + 9.1) x = 0.0262 x 0.2504 + 0.9997 x 0.9681 x 0.1740 x = 0.0066 + 0.1684 x = 0.1750 angle theta (θ) = arcsin (x) = asin 0.1750 = 10.0786° height (H) = cot θ/L = cot 10.0786 / 14.036 = 6.008 km

Height of Crater Manilius = 6008 m

3. Abulfeda [13.8°S, 13.9°E]. See Figure 12. Crater (62 km/3110 m)1. Tilt correction factor = 1/(cos 13.8 x cos 13.9) = 1/(0.9711 x 0.9707) = 1.0608 2 2 shadowlupixels = √(199 - 225) + (292 - 267) = √(676 + 625) = 36.069 L = 36.069 x 1.0608 x 0.6325 = 24.20 km. x = sin (Bo) x sin (latitude) + cos (Bo) x cos (latitude) x sin (Co + longitude) x = sin 1.5 x sin 13.8 + cos 1.5 x cos 13.8 x sin (160.88 + 13.9) x = 0.0262 x 0.2385 + 0.9997 x 0.9711 x 0.0910 x = 0.0062 + 0.0883 x = 0.0945 angle theta (θ) = arcsin (x) = asin 0.0945 = 5.4251° height (H) = cot θ/L = cot 5.4251 / 24.20 = 3.287 km

Height of Crater Albufeda = 3287 m

4. Almanon [16.8°S, 15.2°E]. See Figure 12. Crater (49 km/2480 m)1. Tilt correction factor = 1/(cos 16.8 x cos 15.2) = 1/(0.9573 x 0.9650) = 1.0825 2 2 shadowlupixels = √(319 - 337) + (363 - 345) = √(1936 + 324) = 47.5395 L = 47.5395 x 1.0825 x 0.6325 = 32.550 km. x = sin (Bo) x sin (latitude) + cos (Bo) x cos (latitude) x sin (Co + longitude) x = sin 1.5 x sin 16.8 + cos 1.5 x cos 16.8 x sin (160.88 + 15.2) x = 0.0262 x 0.2890 + 0.9997 x 0.9573 x 0.0684 x = 0.0076 + 0.0655 x = 0.0731 angle theta (θ) = arcsin (x) = asin 0.0731 = 4.1921° height (H) = cot θ/L = cot 4.1921 / 32.550 = 2.3528 km 20

Height of Crater Almanon = 2353 m

5. Stöfler [41.1°S, 6.0°E]. See Figure 13. Walled plain with flooded floor (126 km/1880 m)1. Tilt correction factor = 1/(cos 41.1 x cos 6.0) = 1/(0.7536 x 0.9945) = 1.3343 2 2 shadowlupixels = √(287 - 299) + (278 - 269) = √(144 + 81) = 15.000 L = 1.3343 x 15.000 x 0.6325 = 12.659 km x = sin 1.5 x sin 41.1 + cos 1.5 x cos 41.1 x sin (160.88 + 6.0) x = 0.0262 x 0.6574 + 0.9997 x 0.7536 x 0.2270 x = 0.0172 + 0.1710 x = 0.1882 angle theta (θ) = arcsin (x) = asin 0.1882 = 10.8487° height (H) = cot θ/L = cot 10.8487 / 12.659 = 6.6935 km

Height of crater Stöfler = 6694 m

Table 3 presents a summary of the estimated and published heights of selected crater walls.

Table 3. Published and estimated crater height determinations

Crater Latitude Longitude Estimated Published Difference name height of height of (m) (%) crater wall crater (m)1 wall (m) Menelaus 16.3°N 16.0°E 3164 3010 +154 (5%) Manilius 14.5°N 09.1°E 6008 3050 +2958 (97%) Abulfeda 13.8°S 13.9°E 3287 3110 +177 (6%) Almanon 16.8°S 15.2°E 2353 2480 -127 (5%) Stöfler 41.1°S 06.0°E 6694 1880 +4814 (256%)

Sources of error and accuracy of procedure for estimating heights of Lunar formations

Despite the observed differences in measurements of crater rim heights with the small sample set of observations (n = 5), there was no significant difference between the measured and published values for crater heights using the Wilcoxin Signed Ranks Test, P = 0.08. However, there was a trend for overestimating the heights which was seen for four of the five observations. This became evident particularly for the largest height measurements (Figure 15).

Figure 15. Difference (Estimated – Published) vs Average of Crater Height (m)

Conclusion

In all, 12 craters spread over a wide range of Lunar latitudes were photographed and their size estimations determined to range from 16 – 187 km in diameter. Five craters with significant rims were also selected and ignoring the two outliers, their heights ranged from 2400 – 3300 m.

As a result of measuring the dimeters and heights of craters using digital Lunar images, the following conclusions may be drawn: The images need to be of high quality and with sufficient image scale to enable features to be seen in suitable detail to make accurate estimations of shadow and feature lengths. Inaccuracies resulted from measurement error when dealing with a small image scale and small shadow lengths. The elevation of the Sun will wholly influence the length of the cast shadow or relief of the crater. Therefore Lunar features on or close to the terminator were preferentially selected. The measurement of long shadows was also susceptible to error with a large uncertainty in height determination due to the shadows falling across ground which itself varied with height. Despite the use of trigonometrical principles to counter for foreshortening, greater errors resulted from taking measurements when the object was far from the direct line of sight. The results of crater diameter estimation obtained in this project did not differ significantly from published values. Greater variability was found for estimating heights of selected craters with rims. It could be that my over-estimation was in part due to the cast shadows falling in depressions on the crater floor so that the shadow length was extended. So what I was actually measuring was a combination of the crater height and the crater depth. Published values as described in this project would have been referring only to crater wall heights.

Future studies could be to take repeat measures of the same Lunar features during different elevations of the Sun over the Lunar surface. This would provide a greater estimation of the estimated error of measurement.

References

1. Davis B. (1998). The Mountains of the Moon. Sky & Telescope, November 1998. Sky Publishing Corporation, MA, USA.

2. Richmond M. (2003). Height of Lunar Mountains. http://stupendous.cis.rit.edu/classes/phys236/moon_mount/moon_mount.html

3. http://en.wikipedia.org/wiki/Selenographic_coordinates

4. Burnett K. (2000). http://www.lunar-occultations.com/rlo/ephemeris.htm

5. Rukl A. (1990). Hamlyn Atlas of the Moon. Paul Hamlyn Publishing.

6. Gibilisco S. (2003). Trigonometry DeMystified. A Self-Teaching Guide. McGraw Hill.

7. North G. (2000). Observing the Moon. The Modern Astronomer’s Guide. Cambridge University Press.

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