Objective

By using the model of a round earth orbiting the sun at a large distance you can easily find the size of the earth by triangulating with shadows in different latitudes. In this lab, you will measure (with surprising accuracy!) the circumference of the earth in much the same way Eratosthenes did in 240BC.

Time Required

Two days when you can make observations of Polaris's altitude at about the same time of night. One of these days should be in Seattle and the other day at least 150 miles north or south of Seattle, say in Portland or Vancouver, B.C. You need not go exactly due north or south, but the experimental results are simpler to interpret if you don't wander too far east or west. Basically you need to go into Canada or south of the state -- Boise or San Francisco are O.K., but Spokane or Walla Walla are not. If necessary, the observations can be made several days apart (or even weeks apart).

Materials

Homemade quadrant - the same tool you used for the celestial navigation lab. Car with lots of fuel

Procedure (Overview)In this lab you will basically do the latitude portion of Lab 1 in two different locations. It's your choice as to whether you get the latitudes via the sun or Polaris. You will then combine the results and calculate the circumference of the earth. More detailed instructions are below.

Procedure (Using Polaris)

- Find Polaris.
- Use your quadrant to determine the latitude at each site using Polaris. The advantages with Polaris are that the observations are very quick and are not complicated by doing them on separate days, although they still should be carried out at the same time of night, since Polaris is not exactly at the north celestial pole.
- Estimate the distance between the sites using a map or your car's odometer (if the two places are directly North-South of each other). If the second site is not directly north or south of Seattle, calculate the North-South separation between the sites (ie- the distance between Seattle and a place directly north/south of seattle at the same latitude as the city you took your measurements from).

CalculationsThe sketch below lays out the geometry you will use for your determination of the Earth's circumference. Let's call the latitude of the first site

aand the latitude at the second site_{1}a. Simple geometrical considerations will show you that the ratio of the north-south distance,_{2}d, to the entire circumference of the earth,C, is equal to the ratio of the angle,a(which you have measured), to a full circle of 360 degrees. In other words:_{1}- a_{2}

d / C = (a_{1}-a_{2}) / 360^{o}

- Calculate the circumference of the earth.
- Use your estimates of uncertainty in latitude to estimate an uncertainty in your value of the Earth's cirumference. Check your measured value of
awith the difference in the latitudes of the two sites and compare with the uncertainties you estimated for the two latitudes. Most good maps will show latitude or you could look for internet resources that will tell you the latitudes of your locations._{1}- a_{2}- Compare your result for circumference with the value in the book or from a trusted online source. Use your estimate of uncertainty in your comparison.
- Eratosthenes used this method in around 240BC to estimate the circumfrance of the Earth. He found that the Earth was about 250,000
stadia, a measurement used by the Greeks. Now the exact length of a Greek stadia at the time of Eratosthenes is still debated by scholars, but assuming 1 stadia is 160 meters compare your result to that found by Eratosthenes.