True story: In a tough astronomy Oral Exam, my grinning professors posed this problem:
"How would you determine the density of Jupiter using only a pair of binoculars? Assume you don't know the planet's distance."
"Umm..." But, to my surprise, I figured it out.
Once outside the exam room, I boasted of my accomplishment to my friend Rico. "Impossible!" he proclaimed. I bet him 10$ and won. How did I do it?
Great! I posted the solution, including deriving Kepler's 3rd law from fundamental principles (which I had to do on the blackboard, as few of us remember the proportionality constant, which includes the all important central body's mass, but just that P^2 ~ a^3). Newton derived it more exactly later, including ellipses and the minor body's mass. But, using Kepler's empirical approximation (M>>m, circular orbits) and guessing that the moon plane's inclination is near zero (5 deg) gives an excellent result. Actually, the plane's inclination to our LOS is not relevant, as all tilts have the same greatest elongation! (But, your missed factor of 3 would have yielded a density indicating a planet made of rock. :-) )
In case anyone cares, Jupiter stands out brilliantly in the Southern sky these summer evenings!
Edited on July 5, 2018, 8:42 pm