The size of our planet



We all know how Eratosthenes measured the size of the Earth by combining observations made from two different locations in Egypt. There is a more elegant way: we can estimate the size of our planet from any beach facing west, without traveling long distances, using just a tape measure and a stopwatch. We do not know if Eratosthenes ever imagined this method. Anyway he did not have good clocks; he did the best he could with the technology at his disposal. And indeed his measurement, very nearly correct, was one of the most remarkable achievements of naked-eye astronomy. But let us consider what we can do if we have a good watch.

Imagine two observers at the beach, where they can enjoy a clear view of the Sun setting on the ocean. Put them at different heights; one as exactly as possible at sea level, lying on the beach, and another several meters higher. If the Earth were flat, then both observers would see the last piece of the Sun disappear at the same time. But of course we know the Earth is spherical, and we know it rotates; therefore the observer at sea level will see the Sun disappear at some time t1, while the observer at height h will see the same thing at some later time t2. During the time between t1 and t2 the Earth has rotated by an angle a. The situation is described in the two figures that follow.

How do we make our measurement? We need to measure the height h of the second observer (that is why we need a tape measure), and we need to measure the time interval between t1 and t2. For example, the first observer gives a signal at t1, and the second observer starts the chronometer, which he then stops at t2. It is very simple. If you have a chance, do it. We will describe the derivation of the size of the Earth in class. The basic idea is that we can calculate the angle a in degrees knowing that the Earth needs 24 hours to rotate 360 degrees. Given a height h, the larger the Earth, the smaller is the angle a. For an infinitely large Earth we get a=0. You may think that h needs to be very large for a significant delay to exist between t1 and t2. Surprisingly, a few meters are enough for a rough estimate. The Earth is not so big, after all.

Additional work

If you like, you can try to derive the formula for the radius of the Earth as a function of h and the angle a. Assume that the Sun sets vertically, which is very nearly correct for a place like Hawaii. Extra credit: what would happen if the first observer is not exactly at sea level? For example, if the first observer is 10 meters above sea level and the second is 20 meters above sea level? Make a drawing to illustrate the new situation and try to describe what is the effect on the time interval between t1 and t2.


Roberto H. Méndez (mendez@ifa.hawaii.edu)

Last modified: September 27, 2005
http://www.ifa.hawaii.edu/~mendez/ASTRO110LAB05/earthsize.html