Union College
Spring 2014
Astronomy 50 Lab 2 (c) Diameter of the Earth:
Aristotle knew that the Earth was spherical and not flat. There were several observations that
Aristotle used to argue this fact. One such observation was that the position of the North Pole
star above the horizon changes as one moves around the Earth. In this lab, you will use this
effect with the Sun not only to prove that the Earth is spherical, but how large a sphere it is. The
principle here is that because the surface of the Earth is curved, as you view a celestial object at
different positions on the Earth the direction of that object relative to the ground will change.
Consider Figure 1 below, where two observers are located at different places on the Earth and
view the Sun at the same time.
Figure 1: Elevation angle of Sun, , at two locations on the Earth.
The curve of the Earth's surface causes the orientation of the ground to vary and so the direction
of the Sun relative to the ground must vary as well. This is an easily measurable effect and
depends solely on the sharpness of the curvature of the Earth's surface, which is related to the
radius, or diameter, of the Earth. This effect, therefore, can be used to measure the size of the
Earth. This was realized by an Egyptian astronomer named Eratosthenes about 200 BC.
There is a complicating factor that we need to account for. The position of the Sun in the sky
changes throughout the day and during the year and so using the Sun to measure this effect
involves special timing coordination with observers at two different locations. Eratosthenes was
able to accomplish this measurement because of prior knowledge about the elevation of the Sun
in one location at exactly solar noon on the exact day of the summer solstice and so needed only
to measure its elevation in another city at this exact moment. For this lab, we will coordinate
with other astronomy students at Swarthmore College (slightly southwest of Philadelphia).
Let’s first discuss how the elevation angles of the Sun at different locations relates to the
diameter of the Earth, and then we’ll discuss how to best obtain the data. Imagine, for a
moment, that the two locations and the Sun all lie in the same plane and therefore fit on a page,
as in Figure 2 below. The angles 1 and 2 are the elevation angles of the Sun and the angle 
is the angle between the two locations as seen from the center of the Earth.
Figure 2: The angular North-South separation of two locations on the Earth,  and the
elevation angles, , of the Sun.
Now note that the separation between the two locations is a fraction/ 360o of the way around
the Earth. Likewise, since the circumference of the Earth is 2R, where R is the radius of the
Earth, then their separation distance, s, can also be said to be a fraction s / 2R of the Earth’s
circumference. Since these fractions must be equal, / 360o = s / 2R or
R = 360o s / 2.
(1)
So, if we know s (the distance between the two locations), we need only to figure out how 
relates to the elevation angles, 1 and 2, and we can determine the radius of the Earth.
To see how  relates to 1 and 2, consider the other two angles labelled in Figure 2. Since
the ground makes a 90o angle with the vertical at each location, the angle 90o is the angle
between the line pointing to the Sun and the extension of the line pointing to the center of the
Earth, as indicated. And, since the Sun is, essentially, infinitely far away, the lines pointing to
the Sun from two locations on Earth are parallel. Now, try the following exercise (taken from
“Stephen Hawkin’s Universe Teacher’s Guide”).
1. On a piece of lined paper, draw any two intersecting lines, as shown below.
2. With a protractor, measure the angle each line makes with one of the parallel printed lines
(angles ‘a’ and ‘b’ in the diagram). The printed lines represent the parallel lines pointing to a
celestial object, like the Sun.
3. Subtract one angle from the other.
4. Measure the angle between the two lines where they intersect (angle ‘c’ in figure).
5. Compare angle c with the difference between a and b.
Figure 3: Relation of angles between two lines and the angles they make with a set of a
parallel lines.
Compare the situation of the lines and angles drawn on the lined paper and that in Figure 2
above. From this analogy, can you see that in Figure 2, we should have

With just a little algebra, we get that
 






[Here’s a geometrical proof of this equation, for those of you not satisfied. First, add to Figure 2
a line from the center of the Earth to the Sun, as shown in Figure 3, below. The lines pointing to
the Sun from each location and from the center of the Earth are all essentially parallel, so by the
geometrical rule of alternate interior angles, the angles 90o-1 and 90o-2 are equal to the
angles between this new line and the lines from the center of the Earth to each city, as shown.
Figure 4: Relation between  and 1 and 2.
Then, by examining the angles at the center of the Earth, we can see that
 = (90o  1)  (90o  2),
and hence  = 2  1.]
In this lab, we'll measure 1 and 2 directly to get  via Equation 2 and then by determining s
you will use Equation 1 to get the Earth's radius.
Measuring the elevation angle of the Sun.
A simple and fairly accurate method to obtain the elevation angle of the Sun is to measure
the length of the shadow of a vertical object. Consider Figure 5, below. The angle  is the
elevation angle of the Sun and is equal to the inverse tangent of the height of the object divided
by the length of the shadow,
 = tan-1(H/L),
where H is the height of the shadow caster and L is the length of its shadow.
Figure 5. Elevation angle, , of the Sun and length of a shadow.
The trick, now, is to reduce the observing situation to the one described above in Figure 4, i.e.
with the two locations and the Sun all in the same plane. We’ll accomplish this by:
1. Measuring the elevation angle of the Sun, specifically, when the Sun is due South. When the
Sun is due South, is also when it is highest in the sky. We get the elevation angle of the Sun
when it is due south, then, by finding the object’s shortest shadow length.
2. Only considering the North-South distance between the two locations. Since we’ll only use
the elevation angle of the Sun when it is due South, the East-West distance between the locations
are irrelevant. Consider the true separation between Schenectady and Swarthmore, as shown in
Figure 6, and consider the location that is due East of Swarthmore and due South of Schenectady
(which is in the ocean just off the coast from Toms River, NJ). If someone at this location were
to measure the elevation angle of the Sun at solar noon, how would it compare to what observers
in Swarthmore measure. They should measure the same solar-noon elevation angle, since all that
matters for the Sun’s solar noon elevation angle is how far above the southern horizon it is.
Figure 6: (Take from Google Maps)
Therefore, we can use the measurement of the solar noon elevation angle of the Sun in
Swarthmore just as if it had been measured from this other location near Toms River, provided
we also only use the North-South distance between in Schenectady and Swarthmore for s in
Equation 1.
To measure this distance you can either use a flat map, if you have the opportunity, Do not
use a globe or make use of the latitudes; either of these methods implicitly assume prior
knowledge of the Earth's radius and would be cheating.
Procedure:
Target date: April 21.
Needed supplies. Plumb-bob (any mass hanging from a string), a carpenter’s level (if none is
available, bring a ball), and a meter stick or tape measure.
1. Make the following measurements on three separate days, over a period which includes the
target date. Making this measurement on three separate days improves the results for this lab in
three ways:
a. Since the Sun’s position in the sky changes day to day, we need measurements of the Sun’s
solar noon elevation angle in the two locations on the same day. But, there is no guarantee
that the weather will cooperate and provide clear skies at both locations at solar noon on
any day in the small period of time that we have for this lab. By obtaining measurements
on three separate days, though, we will be able to interpolate the results to infer the
elevation angle of the Sun for any one day.
b. The quality of any measurement improves when we average many trials. By measuring
the elevation vs. time and plotting the values vs. date and interpolating, we are making use
of all three measurements, effectively, averaging the three.
c. we will obtain an additional result, that being the change of the Sun’s elevation angle as we
move toward summer.
2. Find a vertical object with a sharp corner or point that can be seen in its shadow, and which
has flat, level ground to its north, where the noon-time shadow is seen. (If you can’t find one
with a pointy top, you can place a small sharp object on top during your observation to give the
shadow such a point.) Use a plumb-bob to check the verticality of the object, and a level to
check the horizontality of the ground. If you don’t have access to a level, and the ground is
smooth, place a ball on the ground and see if starts to roll on its own. Or, you can pour some
water on the ground and see if it flows more prevalently in one direction than another.
3. Measure the height (in cm), from the ground to the top (corner or point) of the shadow-caster.
We’ll call this ‘H.’ Make a data table and write the height of the shadow caster at the top.
4. For each day, record data in a pair of columns, one for time and another for shadow length.
5. On the first clear day, at 12:40 pm, measure the length of the shadow (to the corner or point at
the top), in cm. Write 12:40 in the first column and the shadow length in the second column.
6. Measure the shadow length every 5 minutes, recording the time and shadow length in your
data table, until 1:20 pm. (You should then have a total of 11 shadow lengths.) You may notice
that the edge of the shadow is not sharp – instead of the dark part of the shadow making an
instant transition to lighted ground, the shadow fades over a noticeable width. Thinking of the
edge of the shadow as like a border to a picture, measure the length to the middle of the border.
7. Repeat on two more days within a week.
Data Analysis (to be done in the lab, with instructor):
1. For each day of data, use Excel to make a “scatter” graph with the shadow length (in cm) on
the y-axis and time (in minutes) on the x-axis.
2. “Insert trendline,” select “polynomial,” of order 2.
3. Read the y-value of the lowest point of the trendline curve. Record this length for each day.
Call this ‘Lmin.’
4. For each day, calculate the elevation angle of the Sun by taking the inverse tangent of the
height of the shadow caster divided by the length of the shortest shadow.
 = tan-1 (H/Lmin)
5. Make a new table with a column for date, and another for , and input your 3 dates and ’s.
6. Make a new graph with  on the y-axis and date on the x-axis. “Insert a trendline,” and read
the value of  for the date of 21-April.
7. Repeat steps 1-6 for the Swarthmore data.
8. On the map provided by your instructor, use a ruler to draw a horizontal line going through
Swarthmore (indicated by the ‘S’) and vertical line going through Union College (the ‘U’).
Measure the length of the vertical line from the ‘U’ to where it meets the horizontal line.
9. Use the scale of the map to convert to km. Call this value ‘s’.
10. Use Equation 2 to calculate , and Equation 1 to get R. Multiply by 2 to get the diameter of
the Earth in km.
Additional questions to consider discussing: How do your results prove that the Earth is not flat?