Day Lab: Summary and Analysis

The ultimate goal of the day lab was to observationally calculate the distance from the Earth to the Sun, or the Astronomical Unit (AU).  In the three parts of the lab, we found the velocity of each side of the sun relative to Earth, the rotational period of the sun, and the "orbital" angular velocity of the sun.  We can use these three measurements to calculate the AU.

\(P_{Orbit}\) is orbital period
\(P_{Rotation}\) is rotational period
\(\omega_{Orbit}\) is orbital angular velocity
\(R\) is solar radius
\(AU\) is the Astronomical Unit
\(\theta\) is the sun's angular diameter
\(V\) is rotational velocity of a particle of the sun











Let's get started.  First we can find the solar radius.
\[V=\frac{2\pi R}{P_{Rot}}\]
\[R=\frac{V \times P_{Rot}}{2 \pi}=\frac{1.75 km/s \times 24.92 days}{2\pi}=\boxed{6\times 10^{10} cm}\]

Now that we know the solar radius, we can find the AU using the sun's angular diameter and some trigonometry.
\[sin(\theta)=\frac{2R}{AU}\]
For small angles, \(sin(\theta) \approx \theta\), so:
\[\theta \approx \frac{2R}{AU}\]

\[AU=\frac{2R}{\theta}=\frac{2\times (6\times 10^{10}cm)}{0.55^{\circ}}=\boxed{1.25 \times 10^{13} cm}\]

I estimate the uncertainty of this value to be \(\pm 5\times 10^{12} cm\) because all of our calculations were performed to about 2 significant figures, and no experiment seemed like it could introduce an extremely skewing amount of error.

The actual measure of the AU is \(1.50\times 10^{13} cm\), placing our calculation only \(2.5 \times 10^{12} cm\) off.  This equates to about 17% error.

This error could be introduced by relatively crude means of tracking sunspots on a grid overlay using a marker, (not the highest degree of accuracy), inexact measurement of the time it took the sun to pass through it's diameter (we stopped the time when it looked like it had fully crossed a line), or our spectral data could be skewed from dust on the CCD or unforeseen alternate light sources.

This experiment was conducted with the Tuesday at 1:00PM lab group.