DIAMETER & DEPTH OF LUNAR CRATERS

A paper by
Andre James Clayden

of the
Springbrook Research Observatory

Introduction

 This project (measuring the diameter & depth of craters on the lunar surface) will show a method to calculate the sizes and height of features on the lunar surface using a photographic technique and how to acquire, process, measure, and calculate these features.

Aim

To measure the size and depth of features on the lunar surface and. devise a measuring technique based on timing the passage of the features past the datum. The method will utilise digital photographic technique, and will show how to calculate the size and depth of the object from a digital image.

 Materials used

C14 working @F11 ST7E CCD Camera Ruler
     
Lunar map Scientific calculator Stopwatch

 Good astronomical data base program that has been synchronized with the computer and the
atomic clock.

Method used 

A)     A number of targets were chosen from a lunar map. These objects show good surface structure, Theophilus, Cyrillus fabricius Piccolomini, Steinheil, Vlacaq

Latitude and longitude of the objects had to been noted. This helped determine the correction factor (moon being round) Camera had to be focused and a number of imagestaken to be sure the sharpest possible images had been acquired 

B)     At the precise time of the image being taken, the time had to been noted. This enabled determination of the distance and the angle of the sun on the lunar surface. A print out from the astronomical database program (Sky version 5) was taken, the distance of the moon from earth at the precise time of the image being taken. Example below

Moon

      Earth-Moon distance: 397250.1 km (246840.4 miles)

     True Equatorial RA: 19h 30m 27.2s Dec: -2342'58"

     Topocentric coordinates: RA: 19h 30m 42.5s  Dec: -2338'09"

     Rise: 10:01  Transit: 17:10  Set:      ------ 

            Angular diameter: 0030'34"

            Azm: 3700'15” Alt: +8340'29"  (with refraction: 8340'35”)

      Physical ephemeris

            Phase: 42.03 %, Phase angle: 99.17

      Position angle of the bright limb: 261.5

      True ecliptical coordinates:  Lambda: 29037'25” Beta: 0152'53"

      Parallax: 0055'11. 875"

      Geocentric angular diameter: 0030'05"

      Optical libration: l': 5.458 , b': 2.423

      Physical libration: l":  0.025, b":  0.055

      Total libration: l: 5.483, b: 2.477

      Position angle: -10.24

      Rates ra: 0.3299 Dec: 0.0339 (arc-secs/sec) 

C)       From the web http://www.xylem.demon.co.uk/kepler/jsmoon.html a print out of the  "Sun's selenographic colongitude" was also taken which was used to find the length of was the shadow.

D)     The crater size and the latitude, longitude was obtained from http://www.fourmilab.ch/earthview/lunarform/cratnear.html

E)      The image size 765*510 pixels from the st7eto process the image using Photoshop

F)       Unsharp mask was high light the feature in the image and saved as a jpg image.

G)     To measure the size of the crater, the image was printed out maintaining the pixel size and the pixels then converted into km. This was done by using the method below.  Using the Crater Theophilus as an example - lat 11.4s long 26.4e and referenced size110km

H)                                          1 pixel =0.35 arcsec      0.35 =000098

                                                                  3600

Distance of the Moon from Earth = 397 251.3

1 pixel =D (397251.3) *tan (0.000098= 1.710422)  

1 pixel =0.67946728 km

Image size = 765 * 0.67946728 km =519.792

Pixels           510 * 067946728 km =346.528

Image size in mm 141mm *90mm

141mm/519.7927354 km  = 3.850 km to 1mm

90mm/346.5284903 km  = 3.686    km to 1mm

        Average the two figures   = 3.768 km to 1 mm

         

I)        Now we have the image, the latitude, longitude, time of the image when it was taken, the distance from Earth to the moon and the sun selenographic colongitude, size of the image in pixels and the diameter of the crater and the length of the shadow

 selenographic.gif (4326 bytes)

 

                      

a)       Close up of Theophilus and Cyrillus showing the dark area around bottom of the rim of the crater this is the shadow that we need to measure the length by obtaining the sun’s selenogrphic colongitude this will give use angle theta. 

J)       Measure the size of the crater by using a ruler. The size per mm is = 3.768   km to 1 mm, and in this instance it will change with every image so the same method used above will need to be repeated. To calculate size use the method below 

Correction factor for moon surface

1
_______________________

Cos (latitude) *Cos (longitude) =CF

1
_____________

11.4   *      26.4

 

(0.980271174) *( 0.89571176) = CF 1.138899735 this allows for curviture of the moon

 Size of the crater in mm =25.5mm diameter

 

        25.5mm * 3.7683949 km *CF (1.138899735)

         

        Diameter of Crater Theophilus  = 109.429 km

 


The method used to determine the height of craters or mountains

Measuring the length of the shadow in mm multiply that by 3.76839493 then

By the CF (1.12342645) as shown below

selenographic2.gif (2157 bytes)

 

 

Shadow = L                            Theta = 10.79deg

L= shadow length in mm3.768 * km to 1 mm) *CF (1.123) 

Theta = tan (angle of the sun) 10.24deg (0.190) 

H
______________________________________

Height  =Theta (angle) 10.79Tan (0.190579324) 

L
___________________________________________________

                 H= L (6.5mm)* 3.76839493 km *CF (1.12342645) *Theta (0.190)

 

The Height of Theophilus is = 5244.3m

 

 

Time 19.39.02 Date 10/23/2001 Earth Moon Distance 397729.5.3km Angle –10.79deg

Image size 765pixels*510 pixels telescope C14 camera st7e  f11

“Crater Fabricus lat 40.3s long 43.3e Ref size 78km”

1 pixel =0.35 arcsec      0.35 =000098

                                      3600

Distance of the Moon from Earth = 397729.5.3km
1 pixel =D (397729.5km) *tan (0.000098)
                                      tan = 1.710422669

1 pixel = 0.680285552km
Image size = 765 *0.680 km = 520.418 km
 Pixels           510 *0.680 km = 346.945 km
image size in mm 141mm *90mm
141mm/520.4184474 km  = 3.690km to 1mm
  90mm/346.9456315 km  = 3.854km to 1mm
Averaged the two figures   = 3.772 km to 1 mm
 Correction factor for moon surface

                     1

Cos (latitude) *Cos (longitude) =CF
                 42.9 * 42.0
     (0.732) *(0.743) = CF 1.836
 Size of the crater in mm = 11.5mm diameter
11.5mm * 3.772 km  *CF (1.836)

Diameter of Crater Fabricus  = 79.7   km

 http://www.fourmilab.ch/earthview/lunarform/cratnear.html     

 

 

Prosidonius           

Latitude 31.8N    Longitude 29.9E Image was taken on 10/23/2001 at18.32.13
From the Springbrook Observatory Using C14 @f11 st7e CCD camera
Image size is 750 x 510 pixels
Each pixel =0.35 arcsec     3600 arc seconds in a degree
1 pixel = 0.35 arcsec
                  …………………………    =0.000098 degrees
                3600 arcsec/degrees
Determine the distance of the moon from earth = D
1 pixel = D (km) x tan (0.000098 degrees)
Reference of distances from sky version 5
Earth –moon distance =397536.8 km
A)    1 pixel = D (397536.8km) x tan (0.000098 1.710422667) = 0.679955953 km to pixel
B)     1 pixel = 0.6799553
C)    Image size = 765 wide x o.6799553 = 520.1658045
                                 510 height x o.6799553 =346.777203
D)                        520.1658045 divided by 141mm   3.68911045
                                 346.777203 divided by 90mm      3.853080033 
                                                                
                 Add together
                                               Averaged 3.771099607
                                  Factor       x1.153539913 
                                            4.350113912
Size of the crater in mm =22 x   4.350113912  

Prosidonius Diameter  = 95.702 km

 Time 16.54.12  Date 10/23/2001 Earth Moon Distance 397251.3km Angle –10.24deg

Image size 765pixels*510 pixels telescope C14 camera st7e  f11

Crater Theophilus lat 11.4s long 26.4e Ref  size110km

1 pixel =0.35 arcsec      0.35 =000098

                                      3600

Distance of the Moon from Earth = 397251.3

1 pixel =D (397251.3) *tan (0.000098)

                                       tan = 1.710422

1 pixel =0.67946728 km

Image size = 765 * 0.67946728 km =519.792

 Pixels           510 * 067946728 km =346.528

Image size in mm 141mm *90mm

141mm/519.7927354 km  = 3.8503 km to 1mm

  90mm/346.5284903 km  = 3.6864    km to 1mm

Averaged the two figures   = 3.7683    km to 1 mm

 Correction factor for moon surface

                     1

Cos (latitude) *Cos (longitude) =CF

                      1

            11.4   *      26.4

(0.980271174) *( 0.89571176) = CF 1.138899735

Size of the crater in mm =25.5mm diameter

 

25.5mm * 3.7683949 km *CF (1.138899735)

Theophilus diameter = 109.441 km  

fabricus.gif (1450 bytes)

                            Shadow = L                            Theta 10.24deg

L= shadow length in mm *(3.772931218 km to 1 mm) *CF( 1.138899735)
Theta = tan (angle of the sun) 10.24deg (0.180649223)
   H
………=Theta(angle)10.24Tan (0.180649223)
    L
H= L(1.5mm)*3.77291218 km *CF (1.138899735) *Theta (0.80649223)

Fabricus Height = 3465.47m

theophillus.gif (1260 bytes)

                            Shadow = L                            Theta 10.79deg

L= shadow length in mm3.76839493 * km to 1 mm) *CF(1.12342645)
Theta = tan (angle of the sun) 10.24deg (0.190579324 )
   H
………=Theta(angle)10.79Tan (0.190579324)
    L
H= L(6.5mm)* 3.76839493 km *CF (1.12342645) *Theta (0.190579324 )

Theophilus Height = 5244.33m

 

Example

                   Enter UT date in yyyymmdd.hhmm format: 200011023.1654                                             

          Results: Physical Ephemeris of Moon
          Selenographic longitudes towards Mare Crisium are taken as positive. All figures here are based on geocentric coordinates.

                                 Libration in Latitude (B):     3

                              Libration in Longitude (L):   5.1

                              Colongitude of the Sun (Co):  350.4
                                Subsolar point Lat (Bo):      1.3

                                Subsolar point Long (Lo):     99.6

                      Selenographic Longitude of terminator:  9.6

                         Illuminated fraction of Moon's disc:   .461

                              Position angle of bright limb:  259.3

                               Position angle of polar axis:  -12.2

          Results: Time

                                    Days since J2000.0:   661.2042
                                    Julian Day Number:  2452206.2042
                                                                    

          Results: Sun's position
                                        Distance (AU):  .9948
                                       Right Ascension:   13.884
                                           Declination:  -11.6

          Results: Moon's position
                                   Distance (Earth radii):  62.42

                                       Right Ascension:  19.88
                                           Declination:  -23.25

 
Conclusion

As a result from photographic measuring of craters the following conclusions were drawn

b)       The images need to be of high quality, with features clearly defined to enable accurate measurements.

c)       Inaccuracies can flow from limitations in measuring to tenths of a millimeter; atmospheric conditions can vary by up to 10 pixels plus.

d)       The time is very important, as one or two minutes difference will make the distance of the moon from Earth and the suns angle on the lunar surface vary to a greater or lesser extent.

e)       A firm grasp of trigonometry is necessary to calculate the size and depth.

f)         A reliable source of data is required to determine the sun’s selenogrphic colongitude

g)       The results that are gained from this method were acceptable with in 300 to 700m, but there are more accurate methods such as using a filar micrometer, or timing the passage from one side of the crater to the other using a stop watch and illuminated redical (drift method)

h)       This project examined over twenty major craters our finding found that crater diameters ranged between 50km to 110km and the height of 1km to 5km

Crater name

Diameter

Latitude

Longitude

Vlacq

89km

53.3S

38.8E

Rosenberger

95km

55.45S

43.1E

Watt

66km

49.55S

48.6E

Steinneil

67km

48.6S

46.5E

Metius

87km

40.3S

43.3E

Rheita

70km

37.1S

47.2E

Piccolomini

87km

29.7S

32.2E

Neander

50km

31.5S

39.9E

 

i)         The photographic method is almost as accurate as the drift method.  The filar micrometer is more accurate.

j)         It is also noted during the processing of the image that one pixel equaled one kilometer if we did the measuring from Photoshop.  This is worthy of further investigation.

 
 

Reference

 

Moon numbers

By: Keith Burnett

http://www.xylem.demon.co.uk/kepler/jsmoon.html

 

Height of Lunar Mountains

By Michael Richmond.

http://www.tass-survey.org/classes/phys236/moon_mount/moon_mount.html

 

Named Lunar Formations

By John Walker

http://www.fourmilab.ch/earthview/lunarform/lunarform.html

 

Lunar and Planetary Institute, 2001

Kin Leung

http://www.lpi.usra.edu/

 

The Lunar Orbiter Photographic Atlas of the Moon

By Bowker and Hughes (NASA SP-206)

http://cass.jsc.nasa.gov/pub/research/lunar_orbiter/index.html

 

SIZES AND DISTANCES IN THE SUN-EARTH-MOON SYSTEM:

                              Webwork 2, Astronomy 211

Written by Joachim Stadel and Luis Mendoza

1998/99, Astronomy Department, University of Washington

http://www.astro.washington.edu/astro201/eratosthenes2/rung3_djn.html

 

Sun or Moon Altitude/Azimuth Table for One Day

 

U.S. Naval Observatory

http://aa.usno.navy.mil/data/docs/AltAz.html#formb

 

Lunar and Planetary Maps

NASA Official: Dr. Joseph H. King, Head, NSSDC

Version 2.0, 24 June 1998

http://nssdc.gsfc.nasa.gov/nmc/map.html


 Reference books used

 

A complete manual of Amateur Astronomy

By P.Clay Sherrod 1974 page118

New concise Atlas of Universe

By Patrick Moore 1970 page 53