Download Measuring the Earth`s Diameter

Naked-Eye Astronomy Lab
Diameter of the Earth
Needed Math: geometry and trigonometry
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 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 differ. 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
The curve of the Earth's surface causes the orientation of the ground for the two
observers to differ and so the direction of the Sun relative to the ground must differ.
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 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.
However, the Sun moves around the sky during the year and moves with the sky
throughout the day and so using the Sun to measure this effect involves some special
timing coordination with observers at two different locations. This measurement is
much easier if one uses a celestial object that doesn’t move at all. Since the whole sky
rotates around the North Celestial Pole, the North Pole star (which, actually, is a little
bit away from the pole and so does move a little) is the best object for this measurement.
You can determine the size of the Earth, then, by measuring the angular distance
between the horizon and the North Pole star (the angular distance above the horizon is
known as the “elevation angle” or “altitude”) at two different locations. If you make
this measurement at about the same time of night and about the same time of year, the
small movement of the North Pole star will be insignificant. The measurement method
is quite simple, so you can make the second measurement during a break in the term if
you go someplace far enough away. Alternatively, if you know someone else at
another location whom you trust and who is willing to do a small favor for you, you
can ask them to make the second measurement.
The relation between the elevation angle of the North Pole star at two different
locations and the size of the Earth is as follows. First, imagine that the two locations
that you use are located such that they and the North Pole star all lie in the same plane
and therefore fit on a page, as in Figure 2 below. (Notice that in Figure 2 the North Star
is drawn to the lower right, rather than at the top of the page. This figure is a correct
picture. It is drawn this way partly to remind you that North is not really “up.”) The
angles α1 and α2 are the elevation angles of the North Pole Star and the angle θ is the
angle between the two locations as seen from the center of the Earth.
Figure 2
Now note that the separation between the two locations is a fraction θ/360o of the way
around the Earth. Likewise, since the circumference of the Earth is 2πR, where R is the
radius of the Earth, if the two locations are separated by a distance s, then they are also
a fraction s/2πR of the way around the Earth. Since these fractions must be equal,
θ/360o=s/2πR or
R = 360o s/2πθ
πθ.
So, if we know s (the distance between the two locations), we need only to figure out
how θ relates to the elevation angles, α1 and α2, and we can determine the radius of the
Earth.
To see how θ relates to α1 and α2, 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-α is
the angle between the line pointing to the North Pole star and the extension of the line
pointing to the center of the Earth, as indicated. And, since the North star is,
essentially, infinitely far away, the lines pointing to the North star 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 drawn 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 North Star.
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
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
θ = (90o-α1) - (90o-α2)?
With just a little simple algebra, you should be able to then discover that
θ = α2 - α1.
Here’s a geometrical proof of this equation, for those of you not satisfied (hold on and
follow closely if you feel weak in geometry). First, we make use of a little trick. Add to
Figure 2 a line from the center of the Earth to the North Pole star, as shown in Figure 3,
below. The lines pointing to the star 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-α1 and 90o-α2 are equal to the angles between this new line and the lines
from the center of the Earth to each city, as shown. Then, by examining the angles at
Figure 4
the center of the Earth, you should be able to see that θ = (90o-α1) - (90o-α2), and hence
θ = α2 - α1. In this lab, you'll measure α1 and α2 directly and then by determining s
you can use the equation for R, above, and get the Earth's radius.
The trick, now, is to reduce your observing situation to the one described above, i.e.
that the two locations and the North Pole star all lie in the same plane. Since the North
Celestial Pole is due North of any location on the Earth, you get the desired situation by
picking two locations that are due North and South of each other. It is unlikely,
however, that you’ll be able to pick a second location that is, conveniently, due North or
South of your first location. That’s OK, though, provided you account for the East-West
separation properly, as follows. First, imagine if another measurement was made in a
town due East or West of your second location and due North or South of your first
location. Let's call this other (fictitious) town Dorksville just for discussion-sake. What
First Location
N o rth
West
East
S o u th
Second Location
D o rksville
Figure 5
is the elevation angle of the North Pole star in Dorksville compared to what you
measure in your second location? Answer: the same. All that matters for the North
Star’s elevation angle is how far above the northern horizon it is. To an observer
standing at the North Pole, the North Star will be straight overhead, and the so the
elevation angle of the North Star will be 90o. As one moves towards the equator, the
North Star appears closer to the northern horizon. Movement East and West, though,
does not change the position of the North Star. Therefore, two observers that are due
East and West of each other measure the same elevation angle of the North Star.
Therefore, you can use your measurement of the elevation angle of the North Star in
your second location as if you had measured it from Dorksville. But, you need to
follow through with the rest of the analysis as if this measurement was made in
Dorksville. This means that you must also use the distance, s, between the two
locations due North and South of each other, i.e. from your first location to Dorksville.
This distance, though, is also just the North-South component of the total distance
between your two locations. So, you get what you want by measuring just the N-S
distance between your two locations, not the total distance. To measure this distance
you can either use a flat map or, if you have the opportunity, try cloking it on the
odometer when you go (that would be the truly honest approach). Do not use a globe
or make use of the latitudes; either of these methods would be cheating since they
implicitly assume prior knowledge of the Earth's radius.
Lastly, you need a way of measuring the elevation angle of the North Pole star. A
good and easy method involves a easily made device known as a “quadrant (see figure
on next page):” Thumbtack a protractor to the side of a stick so that its flat edge is
parallel with the edge of the stick. Attach some string to the same thumbtack so that
when a weight is attached, it points straight down from the center of the protractor.
Now place one end of the stick to your cheek, just under your eye and sight the North
Star along the length of the stick, so that the stick points at the star. While holding the
stick in this position, have a friend read what angle the string crosses on the protractor.
However, this number alone is not the correct answer. You also need to "calibrate"
your instrument. When the stick is horizontal (pointing to the horizon, for example),
the elevation angle is 0o. You should find, though, that it does not read 0o when it is
horizontal. What does it read? This is your zero-point. Now, when you point to the
North Star, you need to compare your reading with the zero-point reading to infer the
angle of elevation.
Procedure:
1. Make a quadrant from a piece of wood with a straight edge, a protractor, thumbtack,
string, and weight.
2. Measure the elevation angle of the North Pole star.
3. Travel to another location (at least 100 miles North or South) and measure the
elevation angle of the North Pole star there around the same time of night.
4. Determine the N-S distance between your two locations.
5. Use your data and the equations above to determine the radius of the Earth.
Additional questions to consider discussing:
1. Can you think of how to use the above data to determine the latitude of each
location? (Latitude is the distance from the equator of a particular location defined in
terms of the angular distance as seen from the center of the Earth (see figure below).)
What are your measured latitudes?
2. How do your results prove that the Earth is not flat?