Measuring the Moon's Orbit – Another At Home Lab
Almost a year ago to the day, I was gearing up to teach my first course at UCLA. With only a few weeks before the start of the Spring 2020 quarter, COVID forced all of us to move our lives online. I remember scrambling with my Astro-3 TAs adapt our lectures, homeworks, labs, and exams to remote instruction. I don't think any of us imagined we'd still be at it a full year later.
But here we are, and I'm gearing up to teach our advanced undergraduate lab. This is an important course in the astronomy major because students actually have to get their hands dirty with astronomical instruments and data. They have to answer vexing questions like: "why isn't the telescope pointing where I want it?" and "do I have to wake up at 3am to observe star Y?"
For some of the labs, we will distribute data from previous years and emphasize numerical methods and data analysis. But I would feel like the students would be missing out if they were just using canned data. So two of our four labs will involve some good-old-fashion data-taking and analysis.
One of them is to measure the circumference of the Earth following in the footsteps of Eratosthenes. I wrote about that lab in a previous post, but I've attached a manual that has been adapted for an upper-division Astronomy major.
The other is to measure the shape and orientation of the Moon's orbit with a smartphone camera. Students will take pictures of the Moon over a full orbit and measure its position relative to reference stars with cataloged positions. This can be a bit challenging given the light pollution that many of us contend with and the high contrast between the Moon and stars, but I managed to do it with an iPhone in Los Angeles. Students build up a plate model for their smartphone image using standard astrometric techniques and estimate errors with a jackknife resampling. I was pleasantly surprised with the accuracy of these measurements. With just a few reference stars, I could measure the moon's position ~0.1 deg or about a 1/4 or the moon's angular diameter.
Over the course of the month, students compile a time series of lunar position (in ecliptic coordinates). They then model this with Keplerian (code provided) and derive period, eccentricity, argument of periapsis, inclination, longitude of ascending node, and time of periapsis; everything except semi-major axis.
I workshopped this lab over a month and experienced significant losses due to overcast skies. Even so, with partial phase coverage (<50%) I was able to get pretty respectable constraints on the shape and orientation of the Moon's orbit. In particular, I was able to measure eccentricity to half a percent and inclination to 0.1 degrees. (In fact, I was somewhat relieved that my measurements weren't more accurate because I'd have to account for parallax and non-Keplerian effects).
Overall, I'm excited that the students will have a challenging hands-on lab that touches on a range of topics from astrometry, to non-linear optimization, to lunar nodal regression. They may also develop newfound appreciation for what Tycho Brahe achieved before the invention of the telescope. His measurements of planetary positions, good to about an arcminute, paved the way for Kepler's Law's planetary motion, and a complete revolution in astronomy.
If you're an instructor, please feel free to use/adapt this lab. I've attached the manual. Upon request, I can provide jupyter notebooks with useful code to guide students and a completed lab.