Angular Width Of The Moon

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

Here is the bones idea. Ptolemy'due south model of the motion of the Moon implied that it varies in distance by a gene of two. Even a casual observer of the Moon tin can run across that the range is not so peachy. The question arises: during ancient or medieval times, were any measures of the Moon's angular size fabricated which have come up downwardly to usa. The reply is: very few. Run into the article in from the American Periodical of Physics.

Then it occurred to me that I should encounter if I can measure out regular variations of the Moon's angular bore. I fashioned a simple cantankerous staff using a ruler and the bottom of a cardboard box, which allows usa to place some other piece of cardboard with a vi.2 mm diameter hole in information technology along the ruler.

My left eye is better than my right eye, so I accept taken all the observations listed below with my left eye. The ruler is placed confronting my cheek bone just under my left heart and I movement the cantankerous piece out until the Moon neatly fits in the hole. Then I move the cross piece out to the far end and move information technology in toward my eye until I go another mensurate of a good fit. These two measures should ideally exist very close (within ten mm is skilful). Sometimes it is better. Sometimes it is worse. Then I place the cross staff nigh the mid-point between the ii distances and see if I should arrange the distance a little bit either way.

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

          cavalcade 1 = mean solar day and month column two = Universal Fourth dimension (hh:mm) cavalcade iii = Julian Date - 2,450,000 column iv = true angular diameter of Moon in arc minutes, as viewed            from the center of the Earth (interpolated from Astronomical Almanac) column v = place holder column vi = seven (twilight or daylight observations of 2009/x)          = iv (nighttime observations of 2009/ten)          = 24 (observations of 2011/12) column seven = altitude along ruler that 6.two mm pigsty was from my eye column eight = 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 four 794    vi.56 14Jul  11:15  5026.9688  xxx.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 four  816    six.96 04Aug  02:08  5047.5889  29.42 0 4  859   12.98 07Aug  12:01  5051.0007  29.72 0 seven  846.5 16.39 10Aug  11:58  5053.9986  thirty.forty 0 vii  846.five 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  ten:00  5060.9167  32.96 0 4 805    26.31 27Aug  02:48  5070.6167  30.26 0 4 810     6.lxx 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 vii 795    20.07 16Sep  11:46  5090.9903  32.lxxx 0 7 804    27.07 27Sep  23:30  5102.4792  29.54 0 seven 848     9.20 29Sep  23:42  5104.4875  29.68 0 7 865    xi.21 03Oct  03:51  5107.6604  xxx.41 0 4 859.5  14.38 08Oct  03:40  5112.6528  31.73 0 iv 841    19.37 08Oct  thirteen:00  5113.0417  31.81 0 7 802    nineteen.76 15Oct  eleven:54  5119.9958  32.23 0 7 761.5  26.72 23Oct  01:27  5127.5604  29.95 0 7 765     iv.83  outlier? 23Oct  23:29  5128.4785  29.74 0 seven 840     5.75 24Oct  23:42  5129.4875  29.60 0 7 843     half-dozen.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.xxx 05Nov  12:42  5141.0292  32.26 0 7 773    xviii.xxx 06Nov  12:55  5142.0382  32.35 0 vii 789.5  xix.31 10Nov  13:15  5146.0521  32.11 0 vii 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  fourteen:16  5171.0944  32.79 0 seven 760    eighteen.79 09Dec  xiv:xi  5175.0910  31.79 0 7 804    22.79 20Dec  23:08  5186.4639  29.45 0 seven 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 vii 793    12.47 30Dec  23:34  5196.4819  33.00 0 7 748.five  xiv.48 21Jan  23:38  5218.4847  30.16 0 seven 817.5   6.69  2010 26Jan  23:xviii  5223.4708  32.53 0 7 781.5  eleven.67 06Feb  xiii:57  5234.0812  30.67 0 vii 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 vii 745   13.xc  28Feb  04:28  5255.6861  33.36 0 four 784   fourteen.07  25Apr  00:49  5311.5340  32.53 0 7 774.5 10.51  02May  12:sixteen  5319.0111  thirty.42 0 7 824   17.99 23May  01:07  5339.5465  32.fifteen 0 7 760   9.00 28May  02:44  5344.6139  30.99 0 4 764  fourteen.07   30May  04:38  5346.6931  30.34 0 four 788  xvi.15   30May  11:10  5346.9653  thirty.26 0 7 798  sixteen.42   30May  12:08  5347.0056  xxx.25 0 7 819  sixteen.46   31May  xi:42  5347.9875  29.98 0 seven 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   ix.58   23Jun  01:39  5370.5688  31.14 0 seven 790  10.sixty 24Jun  01:41  5371.5701  thirty.87 0 7 795  11.60   28Jun  11:32  5375.9806  29.lxxx 0 7 835  xvi.01   04Jul  13:12  5382.0500  29.92 0 7 830  22.51    14Feb  00:11  5606.5076  31.44 0 24 763 10.xc  2011 25Feb  12:51  5618.0354  31.24 0 24 769 22.43   15Mar  01:03  5635.5438  31.85 0 24 823 x.18 16Mar  23:58  5636.4986  32.35 0 24 785 11.thirteen   19Mar  00:46  5639.5319  33.44 0 24 780 14.17   24Mar  12:22  5645.0153  31.87 0 24 790 nineteen.65   11Apr  23:18  5663.4778  31.65 0 24 758  eight.37   18Apr  02:02  5669.5847  33.29 0 24 780 xiv.48   14May  01:02  5695.5431  32.85 0 24 760 ten.76   15May  00:52  5696.5361  32.97 0 24 770 eleven.75   26May  15:38  5708.1514  29.53 0 24 829 23.37   11Jun  00:57  5723.5396  32.49 0 24 762  nine.xvi   13Jun  01:14  5725.5514  32.49 0 24 790 11.17   24Jun  13:xvi  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  seven.90   10Sep  00:27  5814.5187  30.09 0 24 777 xi.89   12Sep  12:00  5817.0000  29.63 0 24 825 xiv.37 17Sep  12:26  5822.0181  29.60 0 24 830 xix.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.xv   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  xi:40  6085.9861  32.xvi 0 24 807 17.50   09Jun  eleven:33  6087.9813  31.15 0 24 815 xix.49

After trying this for three years, I am convinced that the most confident (and accurate) information 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'due south eyeball. Why? Because the size of the hole y'all are fitting in the Moon in is comparable to the size of your pupil. Here'southward how you can calibrate your eye. Have a circle 91 mm in diameter and record it to a wall 10 meters away. It subtends an angle of a little over 31 arc minutes, merely about the hateful athwart bore of the Moon. Simple geometry stipulates where I should place the hole along the ruler, but I really have to place the cross piece at a different location. I accept to identify information technology about 17% further abroad than simple geometry would suggest, so my correction factor for scaling my derived angular sizes is 1.17. (My iii most trustworthy straight determinations of this calibration factor are one.205, 1.169, and 1.178.) Your eye volition be dissimilar. 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 truthful athwart diameter to the unscaled value [theta_obs = (6.ii/distance_along_ruler)*(180/pi)*threescore.0].

From the 100 observations given above I notice a perigee to perigee period of 27.5042 +/- 0.0334 days and an epoch of minimum angular size (i.e. apogee) of Julian Date ii,455,682.4479 (xxx 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 truthful value is larger (0.0549), but the Moon actually gets closer than (1 - 0.0549) of its hateful distance and further than (1 + 0.0549) of its hateful distance. This is due to the gravitational issue of the Lord's day. Run across below.

The upper figure beneath 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 nosotros make more accurate or less acccurate measures at item phases of the Moon's monthly bike? A plot of the difference of the truthful 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 gradient that is not-cypher at the 2.four-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 middle doctor showed that my right eye might actually be better than my left middle, for the first fourth dimension since I went to an centre doc at age 16.

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

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

Since we know the size of the World (equatorial radius = 6378.1 km) and the size of the Moon (radius = 1738.2 km), it is like shooting fish in a barrel 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 Annual , the Moon was at apogee on January 2, 2012, at 20 hours UT. We apply that and an accurate value of the anomalistic month (27.55455 d) to fold the data similar we did to our naked eye observations higher up.

Because of the dual attraction on the Moon past the Earth and the Sun, the Moon'south orbit is not a simple elllipse. In fact, if nosotros calculate the radial amount that the Moon differs from the simple ellipse, we terminate upwards with the penultimate effigy. The bones ellipse has been shrunken to a circumvolve of radius 2 R_Earth. Nosotros plot how much farther or closer the Moon is than that ellipse, for the twelvemonth 2012. We see that the Moon'south orbit bulges out about 1.1 R_Earth (or 7000 km) at tertiary quarter for the outset three months of the year, then at get-go 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, nosotros evidence something related to the motion of the Moon around the Earth. Ptolemy (ca. 150 AD) was familiar with the beginning two anomalies of the Moon's motion. (Co-ordinate to the ancient Greeks, all celestial motions were supposed to be along perfect circles.) The start bibelot is explained by Kepler's First Police and Second Police of Planetary Move. The orbit is an ellipse. And the area swept out in equal times is a constant. The Second Law can exist stated every bit follows: r^2 time (d theta/dt) = abiding = h. For a semi-major axis a and an eccentricity of due east the range of distance is a(1-eastward) to a(one+e). Therefore the angular rate of motility from h/sqrt[a(i-e)] to h/sqrt[a(1+e)].

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

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

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

Last revised on 21 January 2021.

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