Day Lab Conclusion: Calculating AU

Finally, we can put everything we have done together. We have found (to a degree of accuracy) the angular size, rotational speed, and rotational period of the Sun. Now we can put it all together to find the distance between the Earth and the Sun, a foundational unit of measurement in astronomy, also known as the Astronomical Unit or AU.

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The first step is to find the radius of the Sun from the information we found. We can use the formula: \[v = \frac{2 \pi R}{P} \] to get \[R = \frac{v \cdot P }{2 \pi }  \] Plugging in our data gives: \[R = \frac{1.442 \times 10^5 \cdot 2.29 \times 10^6}{2 \pi } =  5.256 \times 10^{10} cm\] 

Now that we have the radius, we can set up some simple trigonometry to find AU.

\[ tan(\theta/2) = \frac{R}{d} \] We can use the small angle approximation, but only if our angle is in radians. Converting angular diameter to radians gives: \[ \theta = .5750^{\circ} \pm 0.01192^{\circ} \times \frac{\pi}{180^{\circ}} =  1.004 \times 10^{-2} \pm 0.00208 \times 10^{-2} \: rad\] So we can now isolate d and plug in what we know. \[ d = \frac{2R}{\theta} \] \[ d = \frac{2 \times 5.256 \times 10^{10} }{1.004 \times 10^{-2} \pm 0.00208 } \] \[AU =  1.047 \times 10^{13} \pm 5.05 \times 10^{15} cm\]  In reality, an AU is \(1.496 \times 10^{13} cm \) giving us a 42% error (incidentally, the answer to life, the universe, and everything)

Nerdy references aside, 42% error is not amazing. On one level it is pretty cool that we can get a measurement for AU within the correct order of magnitude, but it also means we have significant sources of error. This can come from various steps in our lab:

• Step 1 - Angular Size: Our calculation of the angular size of the Sun was decently accurate, but not perfect. Error in this step could have come from slow reaction times when using the timer, ambiguity with the thickness of the line we used to mark the edge of the Sun. 
• Step 2 - Rotational Speed: When taking data, our images were not as clear as they could have been, which could be seen in the relatively thick bands imaged for the NaD lines. This can also be seen in the shift of the Telluric lines, which we tried to correct for.
• Step 3 - Rotational Period: There are a few possible sources of errors. First, we are averaging data that has already been averaged, each time cutting off significant figures and decreasing the accuracy of the result. Additionally, the data shows that the third measurement (the bottom sunspot) was significantly different from the other two, which indicates something may have gone wrong when taking the data. this could result from the tendency of sunspots to appear to group together at the edges of the Sun, making them hard to tell apart and track.
  
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• Finally, we are also assuming that the sunspots are rotating at exactly the speed of the Sun. While they are a good estimate, they are not going at exactly the same period of the Sun.

All done! Thanks again to my lab group.