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 19.49After trying this for 3 years, I am convinced that the most confident (and accurate) data are taken when the phase of the Moon is waxing gibbous or waning gibbous, and it is twilight. You need the right contrast between the Moon and the sky.

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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