Despite my initial enthusiasm, however, I still made a few mistakes. I will discuss them here.
On problem 1, I lost two points-- for what I now realize was a careless mistake. The question gave the RA of a star--13 hours-- and asked when in the year the star would be directly overhead at midnight (suposing we are at the latitude corresponding to the star's declination. I drew the following picture
which is my automatic first step every time I work on an RA problem. I knew, based on my diagram, that the Southern Cross would be directly overhead at midnight when the Sun was at an RA of 1. Because I know that each month changes the RA of the sun by 2 hours, I knew this would happen half a month after the Vernal Equinox. Here comes the careless part. I forgot that the Vernal Equinox is a week and a half before the end of March (and not at the end), so half a month after that would be the first week of April, not the last.
My second mistake came in number 4, and was little more than my lack of ability to do math well without a calculator (and a little bit because I'm apparently really bad at copying what's right in front of me). The problem asked me to calculate the semi-major axis of Io's orbit using Kepler's Third Law. I forgot to carry the exponent over the $\pi$ from one equation to the next. This threw off all of my subsequent calculations. Also, when I had to take the cube root of my (incorrect) quotient, I got a little scared because I didn't get an integer. The person who graded my exam helpfully pointed out that rough guess-and-check would have been sufficient.
The biggest chunk of my missed points came from number 6. I knew going into this exam that Fourier Transforms were a weak spot. Still, I thought I had a (loose) grasp on top hat functions. As it turns out, I had a pretty strong grasp on the general concept. The height of the top hat corresponds to amplitude while its width corresponds to the frequency of the function.
I misinterpreted the widths, however, and thought that a wider top hat corresponded to lower frequencies. At the time, I believe I reasoned it by saying that a wider top hat means more space to distribute the waves, and therefore a lower frequency? I know now that wider top hats correspond to higher frequencies
as if the x-axis of these graphs were in Hertz.
On my exam, I completely switched the two graphs corresponding to the two top hats.
My last missed point came from problem 7. It was, again, another careless mistake that came about because I...am really careless. I actually don't know how to explain what went wrong. I would like to say I'll never make this mistake
$Pv=2\pi \Rightarrow r=\frac{2\pi}{Pv}$
or anything like it again, but I know I'd be lying.
Nice job Moiya! Don't worry, careless mistakes happen to everyone! That's where solid methodology and units checks come in :) Keep up the good work!
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