Day Lab Part 4: Putting it all together

After conducting our experiments, we were ready to put all our data together in order to come up with our final value of the Astronomical Unit. We had determined that:
Angular Diameter of the Sun \(\:=\:0.567^{\circ}\:=\:9.89\:\times\:10^{-3} radians\)
Rotational Period of the Sun \(\:=\:26.5\:days\:=\:2.3\:\times\:10^{6}\:seconds\)
Rotational Velocity\(\:=\: 1.6\:\times\:10^{5}\:\frac{cm}{s}\)

Then, using relatively simple geometry, we were able to calculate the astronomical unit. We know that \(v\:=\:\frac{P}{2\pi r}\), where v is velocity, P is the period, and r is the radius. We also know by the small angle theorem that \(\theta\:\approx\:sin(\theta)\). Using these equations and the below diagram we were able to solve for the AU.

Using this equation and the previously collected data, we determined that the AU is equal to \(1.21\:\times\:10^{13}\) centimeters.  The known value of the AU in centimeters is about \(1.5\:\times\:10^{13}\). This gives us a percent error of 19.3%. Considering we were within the same order of magnitude, I'd say we did a pretty good job with what we had to work with. There were numerous possibilities for error throughout our experiments. In the first experiment, we were using our eyes and reflexes to estimate when the circle of light had moved one diameter. We easily could have made a slight mistake in our recording there. In the second experiment, we were again going off of what we could see and mark with a pen on the screen. The sun spots were sometimes hard to see and did not move perfectly along the lines of latitude. Additionally, the rotational period of the sun is not the same at each latitude and thus our average is not entirely accurate. In the third experiment, we were not working with perfect data and had to do a lot of estimating when fitting our curves. All in all, we did a pretty good job and got pretty close to getting the real value of such an important unit in astronomy!