Noon day project - September, 2012

Twice a year I give a shout out to all that can hear about my favorite collaborative project called the Noonday Project which twice a year has students from around the world recreate measuring the circumference of the earth similar to what Eratosthenes did 2200 years ago. The measurements are done at the time of the Equinoxes approximately March 21 and September 21. (That's because the sun is directly overhead at the equator at those times.) Since I was in Aruba on vacation in August, I thought I would "partner" with my location in Aruba and determine the "north-south" distance between there and my house in White Plains, NY. Here's my data. I know that on August 19th the sun in Aruba when it reached its zenith point in the day, cast no shadow. That means that the sun was directly overhead or another way to put is that the solar altitude was 90º. Last March I measured the shadow to be 40.2º here in White Plains.

The sun angle is formed by the sun's rays and the stick. Getting the shadow length just right is a fuzzy business because not only is the shadow "fuzzy" but it's constantly changing. Getting it when it is at its longest is the goal. That's when the sun is highest in the sky (reaches its zenith.) So with the data I have so far I want to determine the distance between Aruba and White Plains not as the crow flies, but rather when the sun is highest in the sky in both places i.e. north-south distance. Because the earth turns from west to east the sun will be highest in sky first in Aruba, then in White Plains. To find out the north-south distance we will need to find the measurement of the central angle.  (see figure below.)

The central angle equals the difference 

between the sun angles at city 1 and city 2.

The north-south distance is the distance of the arc between city 1 and city 2. (1º of arc is approximately 111 km since the circumference of the earth is approximately 40,000 km and there are 360º in a full circle.) So 28º is 3,108 km which is the north-south distance from my hotel in Aruba and my home in White Plains.
The central angle is the difference between the sun angles in Palm Beach, Aruba and White Plains, NY (City 2.) Why? See this description.

Not too late to join the project. Sign up your class here.

Also, Eratosthenes measurements, September 21.9.2012, Medvode, Slovenia on Youtube.