The Emptiness Between the Earth and the Moon

The space between Earth and its satellite is 1.3 light seconds, three days in an Apollo flight, ten times the Earth’s circumference, or more than the triple of the distance that an average American travels by feet in a lifetime. To help you to visualize this, a Moon-Earth family picture can have length 30 times longer than its width. At a mean distance of 384,400 km, the Moon oscillates between 405,696 km and 363,104 km. A number within this interval is not hard to measure with simple geometry. The Greek dudes have done it, using their crude instrument and trigonometry. Various methods have been demonstrated back in the ancient time. Among the most famous ones, Aristarchus of Samos relied on lunar phase and his solar distance estimation to obtain a ratio of the lunar distance and the Earth radius. Hipparchus, roughly 100 years later, after the first experimental measurement of the Earth’s circumference, used a hypothetical solar parallax to estimate a range of lunar distance. But my favorite one is Hipparchus’ alternative method which is essentially founded on the pure coincidence that the Moon’s shadow ends on the surface of the Earth.

It is quite simple to prove that the Moon’s shadow ends on the Earth surface. By observation of solar eclipse, we can see that the Moon and the Sun have the same apparent size. In addition to no observable lunar shadow on Earth, we obtain this diagram:

Not convinced yet? You can directly measure lunar shadow by applying Thales’ theorem, discovered in 600 B.C., stating that any line parallel to a segment of a triangle and intersecting the other segments creates a similar triangle to the first. By holding different sizes of pieces of paper towards the Sun until they match the apparent solar size, we can average the shadow length and paper diameter proportion.

Suppose that you are observing at point O, the Sun is centered at A and the piece of paper, at B, then measuring the distance OB, divided by the diameter B’B”, you’ll get around 108.

This means that any spherical object at one astronomic unit will have a shadow 108 times its diameter. It also implies by Thales’ theorem that the Sun is at a distance of 108 times its diameter.

Since the Moon-Earth system is so small compared to an astronomical unit, we can assume that the Moon also has a shadow of 108 times its diameter. To verify this, you can also hold pieces of paper towards the Moon and see if it matches the number 108. It should be obvious since the lunar and solar apparent sizes are more or less the same. But this kind of procedure as you may doubt is unlikely to give precise or even accurate results. I have tried this myself, but not with pieces of paper (I can’t trust my skills in cutting circles), with fake Chinese square holed coins, and buttons. The holes are small enough for me to match the Moon’s apparent size without the need to holding it at a distance greater than my arm length. But it turns out that the data aren’t really satisfying my expectations.

The smaller holes produced data points that seem too low from the trendline. As further investigations follow, I figured out that the holes might have been small enough to create light diffraction, thus causing the Moon to appear bigger in the hole than what it should be.

So I amended my procedure, now using a plastic sheet stuck on the window and a Sharpie to trace the Moon diameter at specific distance. This method is a bit harder to obtain accurate result. I have to admit that I have selected my data to make the graph work. You can’t blame me for this~ Imprecision, lack of ruler longer than 15 cm, atmospheric condition, Moon illusion, bla bla bla. There just too many factors that can disqualify a data point. Of course, I’m not the only one who select data. For example, Ptolemy has done it in his experiment with lunar parallax. You may understand what kind of challenge he was dealing with if you have ever try this only with a protractor, a compass, some paper and pen.

So here it is, approximately 108, proven with 2 data points. I’m sure any of you can make a better graph than me. But back to the subject matter, which was how to measure the distance of the Moon. Now that we know it’s about 108 times its diameter (3,475 km), we just plug in the number and get 375,300 km. Plus one Earth radius, we get 381,671 km (0.710% difference with the mean distance, but it falls between the lunar apogee and perigee; therefore it is a valid distance). The problem with this equation is the requirement of numerical knowledge of lunar size. What if we don’t know it?

The key is to use the other eclipse, lunar total eclipse. By measuring the time it takes to travel in Earth’s shadow, we can deduce how big is the width of Earth’s shadow at one lunar distance in terms of lunar diameter, 2.7. Now use this configuration:

If triangle OεE” is the same as triangle E’E”O, and triangle E”M’M” is similar to triangle OεE”, then E’E”M”’M” is a parallelogram with E’E” = M”’M” = 3.7 lunar diameter. Thus, if we know the Earth’s diameter (E’E” = 12,742 km), all we need to do is E’E”/3.7 * 108 = 371 929 km. Plus one Earth radius, and behold 378,300 km (1.69 % difference with actual mean distance).

REFERENCES:
Beaudry, Pierre. “How The Greeks Measured The Invisible.” Science LaRouche Pac. Lyndon Larouche Political Action Committee .
Copley, Chris. “Have You Walked Your 10,000 Steps Today?” The Herald-Mail. HeraldMail, 08 June 2012.
Hoyert., Donna L., and Jiaquan Xu. “Deaths: Preliminary Data for 2011.” Centers for Disease Control and Prevention, 10 Oct. 2012.
Inglis-Arkell, Esther. Earth and Moon shadow. Digital image. How to Measure the Distance from the Earth to the Moon. IO9, 13 Nov. 2001.
Maeyama, Y. “Determination of the Sun’s Orbit: Hipparchus, Ptolemy, Al-Battānī, Copernicus, Tycho Brahe.” Archive for History of Exact Sciences. 1st ed. Vol. 53. 1998. 1-49. JSTOR.
O’Meara, Stephen James. “Don’t Let Psychological Tricks Fool You When Determining the Moon’s Size.” Astronomy. Vol. 38. Kalmbach, 2010. 58-59. Ser. 3. Stephen James O’Meara’s Secret Sky: Measure the Moon. Astronomy, 25 Jan. 2010.
O’Neill, Ian. “How Long Does It Take to Get to the Moon?” Universe Today RSS. WordPress.com, 10 Apr. 2008.
Sigismondi, Costantino. “Measuring the Earth-Sun Distance During a Lunar Eclipse.” Cornell University Library. Cornell University, 5 July 2011.
“Solar System Exploration: Planets.” Solar System Exploration: Planets. NASA.
Toomer, G. J. “Hipparchus on the Distances of the Sun and Moon.” Archive for History of Exact Sciences 14.2 (1974): 126-42. JSTOR. Web. 8 Dec. 2012.