# Size of the Earth

Project for Elementary Astronomy

It is relatively easy to measure the size of the Earth using simple tools at your disposal: straws, string, brains, and feet. In this activity you will follow in the footsteps of Eratosthenes, a Greek astronomer of the third century BC. Eratosthenes' method involved observations of the Sun on a specific day (the summer solstice); you will instead use the fixed position of the star Polaris.

This activity can be done only if you are going to be traveling 150 miles or more due north or south this semester--the farther the better. Neither Price nor Pocatello is quite far enough, but Richfield or Rexburg is adequate. This is a great project if you are planning such a trip already. You'll also need to have clear weather at night when at your destination (and at home).

To conduct this observing project, follow this procedure:

• Construct your low-tech equivalent of an astrolabe as shown in the figure below. You'll need a sheet of heavy cardboard or posterboard (about the size of a sheet of paper), a drinking straw, string, a heavy washer or other weight, and some tape.
• Tie one end of your string about 2/3 of the distance down the straw. Tape the straw firmly to the top edge of the cardboard, allowing the string to hang down on one side. Tie the heavy weight to the free end of the string, so the string hangs vertically when the cardboard is held vertically with the straw on top.

A cousin to the astrolabe that you can build in your kitchen.

• You will be observing the star Polaris (the North Star) from your two different locations. The two observations should be done at about the same time of night, and within a week or two of each other (Polaris does in fact move a small amount; keeping the observations similar will reduce error from the motion of Polaris).
• To make your observations, on a clear night hold the cardboard vertically, and sight Polaris through the straw. Once you are certain you are looking at Polaris, mark the position of the string without moving the cardboard; you can either clamp down on the string so you can mark it yourself, or get a trustworthy assistant to help you mark it. Label the line with your location, and the time and date of your observation.
• Repeat this observation at both the northern and southern locations. When finished, you should have two straight lines that cross each other where the string is tied to the straw. Your scientific data is the angle between these two lines.
• Your measurements can give surprisingly good results if you're very careful in the sighting and marking. Depending on how far north or south you travel, the difference in angles will be very small; even small errors will lead to major difference between your value and the true value of the size of the Earth. Don't fret over this too much; just be aware that great care is needed for accurate measurements.
To complete the project, include the following in your analysis:
• Using a map, measure the north-south distance between your two observing locations. This will be less than your actual travel distance, unless you traveled exactly straight north or south, which is highly unlikely since we're not in Kansas and roads weave about. (If we were in Kansas and you drove directly north or south, you could just use your car's odometer for this measurement and you wouldn't need a map.) Attach a copy of your map, suitably marked, to your report. Be sure that your copy shows the map scale.
• Using a protractor, measure the angle between the two vertical lines on your device. The lines cross because the direction of "vertical", measured relative to the stars, has changed as you moved around Earth's surface. The angle between the lines is the number of degrees of latitude that you have moved.
• To figure the circumference of the Earth, divide 360° by the angle between your lines. This is the ratio by which the full circumference of the earth is bigger than the distance you traveled. Multiply the north-south distance you traveled by this ratio; the result should be close to the circumference of the Earth. Round off your answer appropriately (to the nearest 500-1000 miles, depending on how accurate you think your measurements were). Show and explain your calculations in your report.
• Look up the circumference of the Earth. Note that it might be easier to find the radius or diameter, from which you can compute the circumference using π. By what percentage does your result differ from the official value? (Show this calculation.) How could you make your answer agree better?
• Please attach your measuring device to your report, along with the notes and lines you drew on it. (You may remove the weight.)
• Be sure to follow the general instructions applicable to all projects.