Internet Project

Simultaneously Photographing of the Moon
and Determining its Distance

Solutions

The moon's proper motion (Example: December 9^th, 2000, 21.00-21.45 UT, Tenerife)

Determination of the moon's topocentric positions for Tenerife yields the following results:

Time Right ascension declination

21.00 UT 3h47m37.626s 15°34'53".836

21.45 UT 3h49m10.664s 15°45'26".560

Δ 24'45".220=0.41256°

Time	Right ascension	declination
21.00 UT	3h47m37.626s	15°34'53".836
21.45 UT	3h49m10.664s	15°45'26".560
Δ	24'45".220=0.41256°

From this result, you can derive the moon's proper motion to be 0.55008°/h. That means that it takes T_sid = 27d6h27min for a complete revolution of 360°!

Example: The moon's distance on December 9th, 2000, 21.00 UT

The equatorial positions of Jupiter and Saturn:

In an astronomical almanach for amateurs, you may find:

	date	right ascension	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.

With an astronomical computer program you can find the following values:

	right ascension		declination
Jupiter	4h11min 4.7s	62.770°	20°10'51"	20.181°
Saturn	3h36min28.7s	54.120°	17°00'55"	17.017°

Determination of the angular distance between Jupiter and Saturn

For the use of Pythagoras' theorem, you have to take into account the spherical kind of the coordinates by multiplying the right ascension alpha by the cosine of the declination delta.

d

d² = (alpha*cos(delta))² + delta².

By this way, you will get the following result:

d_{Jupiter, Saturn} = 8.86°

Using spherical trigonometry, you have to use the "theorem for the cosines of sides" ("Seitencosinussatz", in German, I don't know the English terminus!):

cos(d)	=	cos(90°-delta_J)*cos(90°-delta_S)+
		sin(90°-delta_J)sin(90°-delta_S) cos(alpha_J-alpha_S)
	=	sin(delta_J)*sin(delta_S)+
		cos(delta_J)cos(delta_S)cos(alpha_J-alpha_S)

Using this formula, you will get the following result:

d_{Jupiter, Saturn} = 8.79°,

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

Scaling the pictures (Example: Koblenz and Namib Desert)

Koblenz	Namib Desert

With an arbitrary graphic program, 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

Namib Desert Jupiter 507 226 416,83

Saturn 92 265

moon 186 320

Therefore, the scales of the pictures are:

Koblenz 0,0257 ^o/Pixel

Namib Desert 0,0211 ^o/Pixel
With the scales known, the angular distances between the moon and either planets follow immediately from the above pixel coordinates:

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

Determination of the moon's parallactic displacement:
1. With ruler and compasses: You can draw the either triangles into the same drawing scaled to 1°/cm, for instance.
2. This construction can be replaced by combinating the both pictures:
3. 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 is Π=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
  
  Δ(Koblenz, Namib Desert) = 1.195 R_E = 7622 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 = 74.4°.
  The straight line connecting them, therefore, has the length
  
  d = 2 R_E sin (delta/ 2) = 1.209R_E=7712 km
3. The geographical coordinates are
  
  latitude longitude
  
  Koblenz 50°11'02" N 07°32'16" E
  
  Namib Desert 22°28'43" S 14°56'59" E
  
  With the formula given in 2. b. we get the following angular distance:
  delta = 73.0°. Therefore, the straight distance is d = 1.190 R_E = 7588 km.
4. Only the result Δ in a. takes into account that the connection line between Koblenz and Namib Desert is not perpendicular to the line from the earth's center to the moon.
The moon's distance
With the known values of Δ and the corresponding parallactic displacement Π, the distance r_M to the moon can easily be determined:
1. A quite good approximation results from sin(Π/2)=(Δ/2)/r_M: r_M=(Δ/2)/sin(Π/2)=363800 km.
2. An even better can be derived by taking into account that the line Koblenz-Namib Desert doesn't meet the earth's center. By adding the square root of R_E²+(Δ/2)² we get, finally the following result:
  rM=368900 km = 57.8 R_E
3. If we took into account the angle between the lines moon - earth's center and observer 1 - observer 2 we would get an even better result.
4. An astronomical computer program tells us that the correct distance at the considered moment is r_M = 57.74 R_E!

Some more results

The following tabular shows the best results which could be derived by evaluating the pictures of the different project days:

Date Combination r_M correct value

December, 9^th, 21.00 UT Koblenz - Namib Desert 57.8 r_E 57.7 r_E

Koblenz - Tenerife 57.2 r_E

Smolyan - Tenerife 55.6 r_E

Smolyan - Namib Desert 57.4 r_E

Tenerife - Namib Desert 60.0 r_E

December, 9^th, 21.15 UT Osnabrück - Tenerife 59.1 r_E

Osnabrück - Windhoek 58.7 r_E

Tenerife - Windhoek 59.3 r_E

January, 6^th, 21.00 UT Süderdeich - Windhoek 58.5 r_E 57.8 r_E

March, 1^rst, 20.00 UT Koblenz - Namib Desert 63.8 r_E 60.1 r_E

The results would be worth to be controlled by another project participant!

Date	Combination	r_M	correct value
December, 9^th, 21.00 UT	Koblenz - Namib Desert	57.8 r_E	57.7 r_E
Koblenz - Tenerife	57.2 r_E
Smolyan - Tenerife	55.6 r_E
Smolyan - Namib Desert	57.4 r_E
Tenerife - Namib Desert	60.0 r_E
December, 9^th, 21.15 UT	Osnabrück - Tenerife	59.1 r_E
Osnabrück - Windhoek	58.7 r_E
Tenerife - Windhoek	59.3 r_E
January, 6^th, 21.00 UT	Süderdeich - Windhoek	58.5 r_E	57.8 r_E
March, 1^rst, 20.00 UT	Koblenz - Namib Desert	63.8 r_E	60.1 r_E

Conclusions

It is possible to "see" the moon's proper motion with a time intervall of about one hour.

For determining the angular velocity of the moon satisfyingly, the time interval of one hour is too short.

The moon is not infinitely far away. Simultaneously taken photos visualize the moon's parallactic shift - the more evidently the larger the distance between the observers.

It is possible to determine the distance to the moon quite precisely. But it has become evident that for this goal

the photos should be exposured as correctly as possible,

the moon and the reference objects should be pictured as large scaled as possible,

the demand of perfectness reduces with increasing distance between the observers.

Some of our pictures were too small scaled and/or too long exposured (or the weather has been too bad!) to yield satisfying results for the distance to the moon.

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last update: June 22nd, 2008

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

	latitude	longitude
Koblenz	50°11'02" N	07°32'16" E
Namib Desert	22°28'43" S	14°56'59" E

Koblenz	0,0257 ^o/Pixel
Namib Desert	0,0211 ^o/Pixel

	x	y	d
Koblenz	322	165	459
Windhoek	351	623	459