Day Lab (Part 1): Angular Size of the Sun

The Astronomical Unit (AU) is a fundamental unit of measurement that measures the distance between the Earth and the Sun. The AU is the measure-stick by which many astronomical distances are based-off, therefore, it is imperative that we have an accurate measurement of the AU.

In this lab, we will be using the Doppler shift technique, using the rotational speed of the Sun and its angular diameter in the sky, to measure the AU.

The first thing we need is the angular diameter of the Sun. The angular diameter is angular measurement describing how large an object appears from a given point of view. In this case, our point of view is the Earth, and we're measuring the size of the Sun as it appears in the sky.

In order to measure the size of the Sun as it appears in the sky, we can use a proportion with the rotation of the Earth. We know that the Earth rotates a full 360° in 24 hours, or an Earth day, in which the Sun makes a full 360° shift and is back to it's original position in the sky. Therefore, one way to measure how large the Sun appears in the sky is to see how long it takes the Sun to shift across the sky by the length of its own diameter. Then, we can relate the time it takes for the Sun to move across its own diameter to the time it takes the Earth to rotate a full 360° to figure out what portion of the 360° the Sun takes up in the sky.

The proportion to figure out the angular diameter of the Sun, $\theta_{\odot}$ can be set up as follows:

\begin{align}
\frac{360°}{\text{Time for full rotation of Sun in the sky}} &= \frac{\theta_{\odot}}{\text{Time for Sun to shift across the sky by its diameter}}\\
\theta_{\odot} &= \frac{360° \times \text{Time for Sun to shift across the sky by its diameter}}{\text{Time for full rotation of Sun in the sky}} \\
\theta_{\odot} &= \frac{360° \times t}{24 \text{ hours}} \\
\theta_{\odot} &= \frac{360° \times t}{24 \text{ hours} \times \frac{60 \text{ minutes}}{\text{1 hour}} \times \frac{\text{60 seconds}}{\text{1 minute}}}\\
\theta_{\odot} &= \frac{360° \times t}{86400 \text{ seconds}}
\end{align}

Now that we have an equation for finding the angular diameter, we just need to measure how long it takes the Sun to move across its diameter. In this lab, we had the following data:

Trial 1 : 130.28 seconds

Trial 2 : 133.22 seconds

Trial 3 : 134.62 seconds

Trial 4 : 131.50 seconds

Trial 5 : 136.06 seconds

Average: 133.136 seconds

The average time for the Sun to move along its diameter is 133.136 seconds. We can plug in this value into the equation for the angular diameter of the Sun for the $t$ value.

\begin{align}
\theta_{\odot} &= \frac{360° \times t}{86400 \text{ seconds}}\\
\theta_{\odot} &= \frac{360° \times 133.136 \text{ seconds}}{86400 \text{ seconds}}\\
\theta_{\odot} &=  0.5547°
\end{align}

Therefore, the angular diameter of the Sun is $\theta_{\odot} =  0.5547°$.