An article by Kevin Krisciunas published in the American Journal of Physics (volume 78, pp. 834-838, August 2010) can be downloaded by clicking here. In the caption to Figure 3, you should note that instead of "aphelion to aphelion" it should read "apogee to apogee".

Here is the basic idea. Ptolemy's model of the motion of the Moon implied that it varies in distance by a factor of two. Even a casual observer of the Moon can see that the range is not so great. The question arises: during ancient or medieval times, were any measures of the Moon's angular size made which have come down to us. The answer is: very few. See the article in from the American Journal of Physics.

So it occurred to me that I should see if I can measure regular variations of the Moon's angular diameter. I fashioned a simple cross staff using a ruler and the bottom of a cardboard box, which allows us to place another piece of cardboard with a 6.2 mm diameter hole in it along the ruler.

My left eye is better than my right eye, so I have taken all the observations listed below with my left eye. The ruler is placed against my cheek bone just under my left eye and I move the cross piece out until the Moon neatly fits in the hole. Then I move the cross piece out to the far end and move it in toward my eye until I get another measure of a good fit. These two measures should ideally be very close (within 10 mm is good). Sometimes it is better. Sometimes it is worse. Then I place the cross staff near the mid-point between the two distances and see if I should adjust the distance a little bit either way.

The first 36 observations are discussed in the AJP article. The 27th observation on 23 October 2009 is basically an outlier. Here are 100 observations made between 21 April 2009 and 9 June 2012.

column 1 = day and month
column 2 = Universal Time (hh:mm)
column 3 = Julian Date - 2,450,000
column 4 = true angular diameter of Moon in arc minutes, as viewed
           from the center of the Earth (interpolated from Astronomical Almanac)
column 5 = place holder
column 6 = 7 (twilight or daylight observations of 2009/10)
         = 4 (nighttime observations of 2009/10)
         = 24 (observations of 2011/12)
column 7 = distance along ruler that 6.2 mm hole was from my eye
column 8 = days since previous new Moon

21Apr  11:23  4942.9743  30.69 0 7 874   25.80  2009
06May  03:55  4957.6632  31.03 0 4 840   11.02
29May  01:33  4980.5646  32.52 0 4 810    4.56
31May  01:33  4982.5646  31.82 0 4 794    6.56
14Jul  11:15  5026.9688  30.65 0 7  828   21.65
16Jul  12:02  5029.0014  31.65 0 7  812   23.69
29Jul  01:33  5041.5646  30.62 0 4  816    6.96
04Aug  02:08  5047.5889  29.42 0 4  859   12.98
07Aug  12:01  5051.0007  29.72 0 7  846.5 16.39
10Aug  11:58  5053.9986  30.40 0 7  846.5 19.39
13Aug  12:04  5057.0028  31.48 0 7  796   22.40
14Aug  11:57  5057.9979  31.90 0 7  817   23.39
15Aug  12:08  5059.0056  32.31 0 7  816   24.40
17Aug  10:00  5060.9167  32.96 0 4 805    26.31
27Aug  02:48  5070.6167  30.26 0 4 810     6.70
30Aug  03:00  5073.6250  29.70 0 4 870.5   9.71
02Sep  01:12  5076.5500  29.56 0 4 899    12.63
07Sep  02:48  5081.6167  30.83 0 4 811    17.70
09Sep  11:45  5083.9896  31.23 0 7 795    20.07
16Sep  11:46  5090.9903  32.80 0 7 804    27.07
27Sep  23:30  5102.4792  29.54 0 7 848     9.20
29Sep  23:42  5104.4875  29.68 0 7 865    11.21
03Oct  03:51  5107.6604  30.41 0 4 859.5  14.38
08Oct  03:40  5112.6528  31.73 0 4 841    19.37
08Oct  13:00  5113.0417  31.81 0 7 802    19.76
15Oct  11:54  5119.9958  32.23 0 7 761.5  26.72
23Oct  01:27  5127.5604  29.95 0 7 765     4.83  outlier?
23Oct  23:29  5128.4785  29.74 0 7 840     5.75
24Oct  23:42  5129.4875  29.60 0 7 843     6.76
25Oct  22:37  5130.4424  29.56 0 7 868     7.71
27Oct  23:43  5132.4882  29.75 0 7 819     9.76
31Oct  23:28  5136.4778  30.98 0 7 817    13.75
03Nov  12:51  5139.0354  31.85 0 7 792    16.30
05Nov  12:42  5141.0292  32.26 0 7 773    18.30
06Nov  12:55  5142.0382  32.35 0 7 789.5  19.31
10Nov  13:15  5146.0521  32.11 0 7 800    23.32
26Nov  23:40  5162.4861  30.31 0 7 791    10.18
04Dec  13:05  5170.0451  32.85 0 7 777    17.74
04Dec  14:16  5171.0944  32.79 0 7 760    18.79
09Dec  14:11  5175.0910  31.79 0 7 804    22.79
20Dec  23:08  5186.4639  29.45 0 7 840     4.46
25Dec  23:37  5191.4840  30.65 0 7 777     9.48
27Dec  23:04  5193.4611  31.63 0 7 786    11.46
28Dec  23:20  5194.4722  32.16 0 7 793    12.47
30Dec  23:34  5196.4819  33.00 0 7 748.5  14.48
21Jan  23:38  5218.4847  30.16 0 7 817.5   6.69  2010
26Jan  23:18  5223.4708  32.53 0 7 781.5  11.67
06Feb  13:57  5234.0812  30.67 0 7 756.0  22.28
22Feb  00:03  5249.5021  31.35 0 7 763.5   7.88
28Feb  00:24  5255.5167  33.38 0 7 745   13.90 
28Feb  04:28  5255.6861  33.36 0 4 784   14.07 
25Apr  00:49  5311.5340  32.53 0 7 774.5 10.51 
02May  12:16  5319.0111  30.42 0 7 824   17.99
23May  01:07  5339.5465  32.15 0 7 760   9.00
28May  02:44  5344.6139  30.99 0 4 764  14.07  
30May  04:38  5346.6931  30.34 0 4 788  16.15  
30May  11:10  5346.9653  30.26 0 7 798  16.42  
30May  12:08  5347.0056  30.25 0 7 819  16.46  
31May  11:42  5347.9875  29.98 0 7 808  17.44  
04Jun  11:56  5351.9972  29.59 0 7 838  21.45  
22Jun  01:16  5369.5528  31.42 0 7 789   9.58  
23Jun  01:39  5370.5688  31.14 0 7 790  10.60
24Jun  01:41  5371.5701  30.87 0 7 795  11.60  
28Jun  11:32  5375.9806  29.80 0 7 835  16.01  
04Jul  13:12  5382.0500  29.92 0 7 830  22.51  

14Feb  00:11  5606.5076  31.44 0 24 763 10.90  2011
25Feb  12:51  5618.0354  31.24 0 24 769 22.43  
15Mar  01:03  5635.5438  31.85 0 24 823 10.18
16Mar  23:58  5636.4986  32.35 0 24 785 11.13  
19Mar  00:46  5639.5319  33.44 0 24 780 14.17  
24Mar  12:22  5645.0153  31.87 0 24 790 19.65  
11Apr  23:18  5663.4778  31.65 0 24 758  8.37  
18Apr  02:02  5669.5847  33.29 0 24 780 14.48  
14May  01:02  5695.5431  32.85 0 24 760 10.76  
15May  00:52  5696.5361  32.97 0 24 770 11.75  
26May  15:38  5708.1514  29.53 0 24 829 23.37  
11Jun  00:57  5723.5396  32.49 0 24 762  9.16  
13Jun  01:14  5725.5514  32.49 0 24 790 11.17  
24Jun  13:16  5737.0528  29.57 0 24 845 22.68  
14Jul  02:05  5756.5868  31.37 0 24 819 12.72  
16Aug  11:55  5789.9965  29.66 0 24 831 16.72  
18Aug  11:42  5791.9875  29.34 0 24 850 18.71  
06Sep  00:42  5810.5292  31.30 0 24 780  7.90  
10Sep  00:27  5814.5187  30.09 0 24 777 11.89  
12Sep  12:00  5817.0000  29.63 0 24 825 14.37
17Sep  12:26  5822.0181  29.60 0 24 830 19.39  
15Oct  12:27  5850.0187  29.63 0 24 832 18.05
19Oct  12:32  5854.0222  30.80 0 24 788 27.05  
04Nov  23:33  5870.4813  29.83 0 24 792  9.15  
13Nov  13:16  5879.0528  29.99 0 24 817 17.72  
07Dec  22:49  5903.4507  29.59 0 24 847 12.69  
11Feb  11:59  5969.0826  32.46 0 24 827 19.26  2012
05Mar  00:26  5991.5181  31.29 0 24 772 12.08  
10Apr  11:55  6027.9965  32.73 0 24 785 18.89
04May  01:41  6051.5701  32.68 0 24 777 12.77
12May  11:42  6059.9875  31.05 0 24 772 21.18  
29May  01:22  6076.5569  31.55 0 24 770  8.07  
02Jun  01:20  6080.5556  33.13 0 24 750 12.06  
07Jun  11:40  6085.9861  32.16 0 24 807 17.50  
09Jun  11:33  6087.9813  31.15 0 24 815 19.49  

As described in the paper, one do some calibration of one's eyeball. Why? Because the size of the hole you are fitting in the Moon in is comparable to the size of your pupil. Here's how you can calibrate your eye. Take a circle 91 mm in diameter and tape it to a wall 10 meters away. It subtends an angle of a little over 31 arc minutes, just about the mean angular diameter of the Moon. Simple geometry stipulates where I should place the hole along the ruler, but I actually have to place the cross piece at a different location. I have to place it about 17% further away than simple geometry would suggest, so my correction factor for scaling my derived angular sizes is 1.17. (My three most trustworthy direct determinations of this scale factor are 1.205, 1.169, and 1.178.) Your eye will be different. When my students have done this, their derived scale factors have ranged from 0.7 to 1.3.

Here is a plot of the ratios of the true angular diameter to the unscaled value [theta_obs = (6.2/distance_along_ruler)*(180/pi)*60.0].

From the 100 observations given above I find a perigee to perigee period of 27.5042 +/- 0.0334 days and an epoch of minimum angular size (i.e. apogee) of Julian Date 2,455,682.4479 (30 April 2011 at 22:45 UT). The true value of the anomalistic month is 27.55455 days, so we are 1.5 standard deviations from the true value. My value for the eccentricity of the Moon's orbit is 0.039 +/- 0.004. The true value is larger (0.0549), but the Moon actually gets closer than (1 - 0.0549) of its mean distance and further than (1 + 0.0549) of its mean distance. This is due to the gravitational effect of the Sun. See below.

The upper figure below shows the individual points folded by our derived period. The lower figure shows the binned data. The number of data points that went into each bin ranged from 7 to 14.

Do we make more accurate or less acccurate measures at particular phases of the Moon's monthly cycle? A plot of the difference of the true angular diameters of the Moon (from the Astronomical Almanac ) minus our observed (and scaled) values versus the number of days since new Moon is shown next. There is no strong trend here. The least-squares line shown has a slope that is non-zero at the 2.4-sigma level of significance. A non-linear fit seems unjustified. Taken at face value the plot hints that I systematically measure the Moon to be smaller in angular size at evening twilight than during morning twilight. Recently (29 November 2012) an appointment at the eye doctor showed that my right eye might actually be better than my left eye, for the first time since I went to an eye doctor at age 16.

A histogram of the measurement errors is given next. The standard deviation of the distribution is +/- 1.02 arc minutes. That should be considered to be the accuracy of an individual observation.

The next figure shows the actual variation of the Moon's distance from the Earth over the course of a whole year. The data are from the 2012 volume of the Astronomical Almanac . The perigee distance ranges from 56 to 58 Earth radii, but the apogee distance is closer to being constant.

Since we know the size of the Earth (equatorial radius = 6378.1 km) and the size of the Moon (radius = 1738.2 km), it is easy to calculate the true angular size of the Moon for a hypothetical observer at the center of the Earth. According to p. D1 of the 2012 Astronomical Almanac , the Moon was at apogee on January 2, 2012, at 20 hours UT. We use that and an accurate value of the anomalistic month (27.55455 d) to fold the data like we did to our naked eye observations above.

Because of the dual attraction on the Moon by the Earth and the Sun, the Moon's orbit is not a simple elllipse. In fact, if we calculate the radial amount that the Moon differs from the simple ellipse, we end up with the penultimate figure. The basic ellipse has been shrunken to a circle of radius 2 R_Earth. We plot how much further or closer the Moon is than that ellipse, for the year 2012. We see that the Moon's orbit bulges out about 1.1 R_Earth (or 7000 km) at third quarter for the first three months of the year, then at first quarter a few months later. But it does not do this at full Moon or new Moon. In this figure the Sun is off the right of all the diagrams.

Finally, we show something related to the motion of the Moon around the Earth. Ptolemy (ca. 150 AD) was familiar with the first two anomalies of the Moon's motion. (According to the ancient Greeks, all celestial motions were supposed to be along perfect circles.) The first anomaly is explained by Kepler's First Law and Second Law of Planetary Motion. The orbit is an ellipse. And the area swept out in equal times is a constant. The Second Law can be stated as follows: r^2 time (d theta/dt) = constant = h. For a semi-major axis a and an eccentricity of e the range of distance is a(1-e) to a(1+e). Therefore the angular rate of motion from h/sqrt[a(1-e)] to h/sqrt[a(1+e)].

It turns out that the Moon's ecliptic longitude varies from 5 degrees ahead of its motion to 5 degrees behind at full Moon and new Moon, but it can be 7 1/2 degrees ahead or behind at first quarter or third quarter. This is known as the evection. Tycho Brahe discovered four more inequalities in the Moon's motion, two in longitude, and two in latitude. This is beyond the scope of this web page, but we give two references below.

Gutzwiller, Martin C., "Moon-Earth-Sun: The oldest three-body problem," Reviews of Modern Physics, 70, no. 2, April 1998, pp. 589-639.

Swerdlow, N. M., "The Lunar Theories of Tycho Brahe and Christian Longomontanus in the Progymnasmata and Astronomia Danica," Annals of Science, 66, no. 1, January 2009, pp. 5-58.

Last revised on 21 January 2021.

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