Internet Project

Simultaneously Photographing of the Moon 
and Determing its Distance

Solutions

Detailed Problems

  1. Consider, for instance, December 9th, 2000, 21.00 UT:
    1. In an astronomical almanach for amateurs, you may find:
      2. date rectascension  declination
      Jupiter  December 5th  4h14min  20.3°
      December 10th  4h11min  20.2°
      December 9th (calc.)  4h11min  20.2°
      Saturn  November 30th  3h40min  17.2°
      December 15th  3h35min  16.9°
      December 9th (calc.)  3h37min  17.0°

      You can find the calculated values by linear interpolation.

    2. With an astronomical computer program you can find the following values:
      3. rectascension declination
      Jupiter  4h11min 4.7s  62.770°  20°10'51" 20.181°
      Saturn  3h36min28.7s  54.120°  17°00'55"  17.017°
  2. Determination of angular distances
    1. Measurement of the angular distance between Jupiter and Saturn:
    2. Calculation of the angular distance between Jupiter and Saturn:
      1. For the use of Pythagoras' theorem, you have to take into account the spherical kind of the coordinates by multiplying the rectascension alpha by the cosine of the declination delta.

        2. The angular distance d can therefore be calculated by
        d2 = (alpha*cos(delta))2 + delta2.

        By this way, you will get the following result:

        dJupiter, Saturn = 8.86°
      2. Using spherical trigonometry, you have to use the "theorem for the cosines of sides" ("Seitencosinussatz", in German, I don't know the English terminus!):
        3. cos(d)  = cos(90°-deltaJ)*cos(90°-deltaS)+
        sin(90°-deltaJ)*sin(90°-deltaS)* cos(alphaJ-alphaS)
        = sin(deltaJ)*sin(deltaS)+
        cos(deltaJ)*cos(deltaS)*cos(alphaJ-alphaS) 

        Using this formula, you get the following result:

        dJupiter, Saturn = 8.79°,

        i.e., with an accuracy of about one promille, the same result as above!

  3. For instance, we will now evaluate the two pictures taken from Koblenz and Namibia at 21.00 UT.

    4. Koblenz Namibia

    With an arbitrary graphic program (PhotoStudio, for instance), you can determine the pixel coordinates of Jupiter, Saturn and the moon:
     
     

       
    x
    y
    d (J - S)
    Koblenz
    Jupiter
    263
    272
    342,18
     
    Saturn
    605
    261
     
    moon
    540
    166
    Namibia
    Jupiter
    507
    226
    416,83
     
    Saturn
    92
    265
     
    moon
    186
    320

    Therefore, the scales of the pictures are:
     

    Koblenz 0,0257 o/Pixel
    Namibia 0,0211 o/Pixel

     
  4. With the scales known, the angular distances between the moon and either planets follow immediately from the above pixel coordinates:

    5.  
     
    d (m - J)
    d (m - S)
     
    pixels
    degrees
    pixels
    degrees
    Koblenz
    		  296,59  7,62  115,11  2,96
    Namibia
    		  334,48  7,06  108,91  2,30

     
  5. Determination of the moon´s parallactic displacement:
           
    1. With ruler and compasses:

      You can draw the either triangles into the same drawing scaled 1o/ cm, for instance.

    This construction is in good coincidence with the combinated picture in our evaluation document.

    By measuring the distance of the either positions of the moon one gets the following result:

    The moon's parallactic displacement due to Koblenz and Namib desert:

    1,2°

  • The distance between the observers:
    1. On the large 21.00 UT picture of the earth Koblenz and Windhoek have the following pixel coordinates:

      x y d
      Koblenz 322 165 459
      Windhoek 351 623
                

      Because of the earth´s diameter of 768 pixels the distance, therefore, is

      d (Koblenz, Namibia) = 1.195 RE = 7619 km

                 This value means the projection of the distance as seen by the moon.
    2. Our globe has a perimeter of 105.5 cm. For the length of the piece of string connecting Koblenz and Windhoek we meassured 21.8 cm. The angular distance between the either locations is, therefore:

      delta = 360°*21.8/105.5 = 74.4o.

      The straight line connecting them, therefore, has the length

      d = 2 RE sin (delta/ 2) = 1.209RE=7709 km
    3. The geographical coordinates are

      latitude longitude
      Koblenz 50o11´02´´ N 07o32´16´´ E
      Namib Desert 22o28´43´´ S 14o56´59´´ E

      With the formula given in 2. b. ii. we get the following angular distance:

      delta = 73.0o .
      Therefore, the straight distance is
      d = 1.190 RE = 7588 km.

     

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    last update: January 10th, 2020