Astronomy Lab 3
Measuring the Size of the Earth
To be completed after reading Arny Chapter 1
Name: _________________
Introduction: By approximately 200 B.C. a Greek scholar named Erastothenes had
already come up with an astonishingly accurate calculation of the size of the earth. He
knew that at noon on a certain day, shadows in a town a fixed distance south were
directly below the objects casting them. He measured the angle of the shadows in his city
on that same day, and with that angle, and the distance between the two cities, he was
able to compute the size of the earth. In this lab you will calculate the size of the earth
using a very similar technique, but using the star Polaris rather than the Sun.
Theory: Two cities on the globe, one of which lies directly north of the other, can be
idealized and portrayed as simply two points on the edge of a circle with the distance
between them being the arc length, s. With this simplification, it is possible to apply the
geometrical relationship between the interior angle and the arc length of a circle to
calculate the radius of this idealized earth. Once the interior angle, α, is determined, the
following proportion can be used to discover the radius, R:
s
Δθ
C
s
=
o
2R
360
R
(1)
Fig.
1
To find the radius of the earth, R, using Eq. (1), we have to find some way to measure the
angle α. This is really not all that difficult, as this angle is simply the difference in the
latitudes of the two cities. So all we really have to do to find the radius of the earth is to
find a way to determine the latitude of a given point on the earth.
To NCP
Fig 2
O = location of observer.
h = horizon of observer.
C = center of the earth.
ℓ = latitude of observer.
θ = altitude of NCP
θ
h
θ
90°-ℓ
or
90°-θ
C
equator
O
ℓ
One of the most important ideas in
Celestial Navigation is that the altitude
of the North Celestial Pole (Polaris) is
equal to the latitude of the observer. Fig
2 illustrates the geometry of this
concept. It is not particularly difficult to
prove ℓ, the latitude of the observer, is
equal to angle θ, the altitude of the
North Celestial Pole (NCP). Thus, it is
only necessary to measure the angle
between the horizon and the star Polaris
to find the latitude of a particular point
on Earth.
The proof that θ equals ℓ involves simply realizing that the two lines headed off to the
NCP are parallel lines. Then it is obvious that angle θ is equal to angle ℓ, because
90°- ℓ equals 90°- θ.
Altitude θ= l
The figure above shows how Polaris (the North Celestial Pole) would appear from a
latitude, l. In this lab, you will use Starry Night to measure the altitude of Polaris from
two different cities a known distance, s, apart. This will give the size of the earth.
Procedure:
Open the program Starry Night, go to the tab Labels, and click on Show All. Then go to
the tab View/Celestial Guides and click on Celestial Poles. Then set your location by
using the tab Options, and clicking on Viewing Location. Scroll to find Birmingham, AL
and double-click to select and go to that location. Change the time until it is night and
then drag your field of view until you are looking directly north. Choose your cursor tool
to be Angular Separation using the button at the far left. Measure the angular separation
between the North point on the horizon straight up to the North Celestial Pole, and record
this value. Starry Night will display the angular separation in degrees of arc, then
minutes of arc, then seconds of arc. Record only the degrees and minutes. Note that the
star Polaris is within a degree of the North Celestial Pole.
Altitude of Polaris at Birmingham = Latitude of Birmingham:
θB = lB =______ deg ____ _____ min
Convert latitude to decimal degrees, lB = _______.______ degrees
(divide minutes by 60, then add the decimal to the degrees)
Use the Go tab again to set your viewing location to South Bend, IN (nearly due north of
Birmingham) by typing in its zip code 46601, and again measure the angular separation
between Polaris and the horizon.
Altitude of Polaris at South Bend = Latitude of South Bend:
θSB = lSB =______ deg ____ _____ min
Convert latitude to decimal degrees, lSB = _______.______ degrees
(divide minutes by 60, then add the decimal to the degrees)
Calculate Δθ, which is the difference in the two latitude values
Δθ= lSB – lB = _____.____
360 o s
Solving Eq. (1) for R we get: R =
2
The odometer of a car could be used to determine the distance, s, between South Bend
and Birmingham to be approximately s ≈ 1000 kilometers.
360 o s
Calculate R =
=______________ km
2
Compare your value with the generally accepted value of REarth = 6,378 kilometers.
Calculate your percent error:
% error =
R − 6,378
100% = _______________ %
6,378
What uncertainties in this experiment could lead to this amount of error?
...

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