# Teacher Guide to Determining the Distance to the Moon Ronald

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Teacher Guide to “Determining the Distance to the Moon”
Ronald G. Probst
National Optical Astronomy Observatory

1. Introduction

This activity can accompany the laboratory “Observing an Eclipse of the Moon”. That lab is a
nature study exercise. This one is a more traditional science lab, using measurement and
calculation to obtain a numerical result. The writeup appears lengthy, but half the length is in
simple diagrams and figures. I assume familiarity with basic algebra (use of symbols and
balancing equations) and with enough geometry to know what a sine is. The instructor could
walk the class through the calculation part if necessary. I also assume some familiarity with
doing laboratory science: making precise measurements, recording them in an organized way,
and so forth.

The activity is in two parts, with two worksheets. The first part can be done by itself, dropping
the second part, if desired. The second part uses results from the first part. This part might be
suitable as an extra credit activity for better students to do on their own. While the writeup
assumes students are working individually outside of class hours, everything except the actual
night time observations of the eclipse could be done in a lab class. The pinhole camera portion is
a daytime activity well suited to carrying out in groups or with the entire class. The activities
could be shortened by simply giving students the critical numerical input values at certain steps.

The underlying thrust of this activity is remote sensing—how we can use our minds and things
we can measure and calculate to discover the properties of distant objects we can’t go to and
examine up close. Key to this is the concept of a model of the real world, which may deliberately
simplify reality in order to get a quick, approximate answer. The value of such a first order result
can be explained. The individual or team results can be compared in a discussion of
experimental error, where the scientific use of “error” means uncertainty, not mistake.
Comparison of the two models is an opportunity to discuss one kind of systematic error which
can’t be reduced by more, or more precise, repetition of the same measurement. The pinhole
camera illustrates how experimental apparatus is built to satisfy a specific need, and doesn’t
have to be fancy or complex to do so. The stated value for the lunar month, and the use of an
atlas, in the first part shows how science advances in part by building on previous knowledge,
combined with new ideas.

Teachers working within a multidisciplinary integrated curriculum can connect science with
history. The October 27, 2004 eclipse for which this lab was produced occurs about when world
history courses are discussing Greek civilization. I’ve attempted to point out that people who
lived more than 2000 years ago were just as curious about nature, and just as clever and
inventive, as we are. I’ve included some other tidbits from the history of science in the following
discussion.

The National Optical Astronomy Observatory – Education Outreach
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2. Notes to the “cylindrical shadow model” activity

The heart of this activity is timing the two moments during the eclipse when the umbra just
crosses the center of the Moon. This is done by eye estimation. Timing to the nearest minute is
adequate to get reasonable results, and about the best that one can do with the unaided eye.
Observing the Moon with binoculars makes the timing easier, and perhaps more precise. The
times when this happens are 6:48:30 pm and 9:19:30 pm MST for the October 27, 2004 eclipse.
In case it’s cloudy, students can be given these numbers to complete the worksheet.

The umbral shadow line is curved. Emphasize that the moment to be timed is when the curved
shadow crosses the center of the Moon. The Moon will not appear to be cut in two by a straight
line. Diagrams that show the geometry from above, like Figure 3, can give rise to this
misconception. Figure 5 shows the correct, face-on appearance of the shadow line on the Moon.

The different timings can be compared in class to get an idea of the experimental error and its
impact on the result. As an extension of this idea, students can be asked to estimate their own
uncertainty for the exact moment of shadow crossing—one minute? five minutes?—and go
through the math to work out the error of their individual results. It probably needs to be
emphasized repeatedly that “error” in this context means uncertainty, not mistake.

For the map-reading part of the activity, in Arizona I suggest using Tucson and Winslow. They
are very nearly on a north-south line, separated by 191 miles or 2.79 degrees of latitude. I used
the Rand McNally Road Atlas map of Arizona. This has a scale in miles and kilometers, and
indicates latitude along the sides of the state map. Some interpolation will be necessary since
neither city lies on a marked latitude line. Latitude given in degrees and minutes must be
converted to fractions of a degree. This part of the activity may be done in class with a larger
scale map to walk through these complications.

The major approximation in this model is that the Earth’s shadow is a cylinder. Another
assumption we are making is that the Moon’s path crosses the shadow through the middle (on its
diameter). This is not necessarily so, and in fact isn’t quite true for the October 27 eclipse.

A second, subtle source of error is that the Earth is moving around the Sun, so its shadow cone is
slowly moving eastward through space at the same time that the Moon is moving more rapidly
eastward on its orbit to go through the shadow. Some students may wonder why the Moon
moves through the shadow from west to east, while it’s obviously crossing the sky from east to
west during the eclipse. This can be the opener for a class discussion on orbital motion and what
we see from our moving platform in space.

To prepare students for the second part, have them make a drawing that superposes the conical
model gives a distance that is systematically too long. In fact, it is about 40% too long. The main
thing to be learned from the cylindrical shadow model is that the Moon is distant by many times
the size of the Earth.

The National Optical Astronomy Observatory – Education Outreach
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Some historical notes. Aristarcos of Samos (c. 310-230 BCE) was the first Western thinker to
conclude that the Earth orbits the Sun, not vice versa. From a measurement of the angle between
the Sun and the Moon at first quarter Moon, he concluded that the Sun was 19 times farther away
than the Moon. He combined this with observations of the curvature of Earth’s shadow on the
Moon during a lunar eclipse to conclude that the Sun was also several times larger than the
Earth. So it seemed logical to him that the smaller body would orbit the larger one. His
mathematics was very limited (he didn’t have the concept of the sine of an angle, for example),
his numerical results were way off, and his physics wasn’t very physical, but his work marks the
opening up of scientific astronomy. He was able to imagine the Solar System from the outside,
and to combine careful numerical observation with geometrical models to get results that were
intended to represent the actual physical world. This is in sharp contrast to the rationalism of
Plato and Pythagoras, who regarded the natural world as full of change, uncertainty, and illusion,
and sought ultimate truth by thought alone.

Eratosthenes (276-194 BCE) was one of the first directors of the great library at Alexandria. So
an accurate measurement of the size of the Earth was first made by a librarian! He noted that on
the longest day of the year, a vertical column in the city of Syene in southern Egypt cast no
shadow; so the Sun was directly overhead1. At the same time, a column in Alexandria to the
north cast a shadow, which he measured to get the angle between the Sun and overhead. So he
determined the angular (latitude) difference between the two locations. Then he benefited from
the fact that the king of Egypt had an excellent royal mail service. Runners carried the king’s
commands up and down the Nile on a regular route, rather like the US Pony Express. So he knew
the distance between Alexandria and Syene was 5000 stadia, measured by the time the king’s
messengers took to run it. The stadium was a length of about 157 meters—about the size of a
modern football or soccer stadium—and his result for the diameter of the Earth was about 7800
miles.

Hipparcos (c. 162-126 BCE) lived and worked mostly on the island of Rhodes, which at that
time was a rival of Alexandria as a center of intellectual life. He used the long historical record
of Babylonian astronomers to investigate astronomical phenomena that take place over long
periods of time2. Combining these records with his own careful observations and calculations, he
determined the length of the lunar month and the year, and discovered the precession of the
equinoxes, a 23000 year cycle caused by the wobble of Earth’s spin axis. He also compiled the
first catalog of star positions from his own observations. His measurements were so precise that
they were used for 1500 years, up to the invention of the telescope. Hipparcos observed several
lunar eclipses between 146 and 135 BCE and used them to do the calculations we are doing in
the second part of this lab. And all of this without a hand calculator!

1
Some accounts state that the Sun in Syene was visible from the bottom of a deep well, so was
directly overhead. I’m skeptical of this version. How likely is it that somebody living next to a
big river would dig a dry well that deep?
2
The Babylonians were careful observers and recorders of astronomical events, but never
developed a geometrical model to literally picture what was going on. It was all just tables of
numbers to them.

The National Optical Astronomy Observatory – Education Outreach
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3. Notes to the conical shadow model and the pinhole camera

The geometrical model we are using was developed and used by Hipparcos. So was the
approximation that the Sun is so far away that the angle corresponding to this distance (the solar
parallax) can be set to zero. Hipparcos had Aristarcos’ result that the Sun was 19 times more
distant than the Moon, but he wasn’t content to just use this figure. He used various methods to
try to measure the solar parallax, or at least put an upper limit on it, but eventually decided that
this angle was so small and the distance so great that he couldn’t come to any meaningful result.
Knowing when you can’t solve a problem is an important part of scientific thinking.

The equations involving angles in Figure 6 show the power of geometry and algebra together.
This is not the usual situation involving equal angles and parallel lines.

Hipparcos attempted to measure the angular diameter of the Sun with a different apparatus, a
kind of adjustable scale. This involved staring directly at the Sun, which can be harmful to the
eyes. The pinhole camera uses an image of the Sun which is much dimmer. There’s no record
that Hipparcos ever used a pinhole camera, but he could have if he had thought of it. We’re using
only the technology available to him, no mirrors or lenses.

Students can construct their own pinhole cameras at home. It’s helpful to show them one in class
before they do. The pinhole camera is also suitable for a daytime group activity in construction
and use, if the class can go outdoors on a sunny day. A room with a south-facing window can be
turned into a giant camera by blocking the window (except for a small hole) and projecting the
solar image onto a card near the floor. An image of everything else outside will be formed on the
walls. Rooms built especially for this purpose were popular during the Renaissance. They were a
kind of primitive movie theater called a camera obscura, Latin for “dark room”.

For camera construction, the emphasis should be on simplicity. I built mine using two discarded
boxes and a scrap length of 2x4, in about twenty minutes. This is an opportunity to show by
example that scientific apparatus can be simple, even downright crude, as long as the necessary
measurement can be carried out with sufficient precision. Hipparcos did a great job with only
simple hand tools available to build his equipment.

While using the camera, students will see the Sun’s image moving steadily across the screen due
to the rotation of the Earth. It takes about two minutes to move by its own diameter.

If this activity is too lengthy or complicated, the teacher can supply the value of the Sun’s
angular diameter so students can go through the mathematics. The angular diameter of the Sun
on Oct. 27, 2004 is 0.54 degrees.

The coincidence of the angular size of the Sun and Moon has given rise to much speculation.
This is a temporary phenomenon. Exchange of angular momentum between Earth and Moon
means that the day is getting longer, and the Moon is getting further away. Our distant
descendants won’t be able to enjoy the spectacle of a total solar eclipse.

The National Optical Astronomy Observatory – Education Outreach

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