Astronomy at the Beach

This post is a little overdue, as this experiment occurred last week, but whatever. Last Friday, most of the Ay 20 class journeyed to Santa Monica beach, not merely for the cool ocean breeze or the calming sound of waves breaking on the sand, but for science. Read on for the details.




Abstract
The goal of this experiment is to calculate the radius of the Earth to a reasonable amount of accuracy by observing the setting of the Sun. I worked with one of three groups, each taking measurements from different heights. My group observed from the beach, which, for the purposes of our calculations, we define as the surface of the Earth. Though this was essentially a project for the whole class, I specifically worked with Iryna, Nathan, Mee, and David.

Procedure
Data collection for this experiment was relatively simple. Upon arriving at the beach and securing a good vantage point (where the pier wasn't blocking the Sun), we simply waited for the Sun to begin setting. One person (Iryna) observed while lying down. When she first saw the bottom of the Sun touch the horizon, she stood up and we began timing; the timers stopped when she saw the Sun touch the horizon again. We were, however, unsatisfied with this measurement, as the atmosphere made it difficult to tell when exactly the Sun first touched the horizon. So we started our timers again, this time until the Sun disappeared. From this data, we could then calculate the radius of the Earth!

Solution

Starting at the Earth's center, you can draw a line to the point on the Earth's surface to the point of the observer, and another to the point at the observer's horizon. Call this distance R. We can then make a right triangle with the lines R, R + h (where h is the height of the observer when standing up), and a line connecting the point on the horizon with the top of the observer's head. The goal is to calculate R.

To do this, we'll need the value of the angle created by the two lines from the Earth's center. Call this Theta. We can use the measured time to calculate Theta, if we think of one rotation of the Earth in terms of time. The time we measured while the Sun was setting represents a small fraction of the 24-hour full rotation of the Earth, and it's related to the distance to the horizon, which in on our triangle. What else is on our triangle? The radius of the Earth.

So, both of our measured times were on the order of ten seconds, which we'll use for simplicity. The ratio of the time we measured to the time in one rotation of the Earth should be equal to the ratio of Theta to one full turn around the Earth in degrees:



Using 10 s for t, we can solve for Theta in degrees:



We then have an equation relating Theta to R and h by way of cosine, since cosine of Theta represents the ratio of the side of a triangle adjacent to Theta to the triangle's hypoteneuse:



Assuming h = 2 meters (definitely an overestimate, as none of us are very tall), then R is the only unknown variable. So, with some simple algebra, we can solve for the radius:



This value is in meters, and is equal to 6670 kilometers. How close is this to the real value? The Astronomical Almanac puts the Earth's radius at 6378.1 km. Not too bad!