Neptune is the eighth planet and the farthest planet away from the Sun. Neptune also has thirteen moons that are known so far. I decided to do make a movie of its biggest moon, Triton, rotating around Neptune. Individual images are centered on Neptune. Notice how stars drift up and to the left (north and east) relative to Neptune over time due to Neptune's orbital motion. Triton circles Neptune in a counterclockwise fashion. Analyzing Triton's movement we can tell the period of the moon orbiting around Neptune along with its angular separation. Using this and the distance to Neptune we can calculate the mass of Neptune using Kepler's Third Law Revised by Newton.

Specifically, in the first graph we show the measured orientation of Triton with respect to Neptune as a function of time. Approximating this with a straight line, we found the rate of angular motion to be 61.107 degrees per day, corresponding to an orbital period of 5.89 days. In ths second graph we show the separation of Triton from Neptune as a function of time in pixels on the camera. A typical number is 11 pixels, corresponding to a linear separation of 308,000 km (using the known distance to Neptune and the plate scale of 1.31 arcseconds per pixel). This would be a good estimate of Triton's orbital size if the orbit is face on. In fact it is an underestimate as the orbit is tilted.

The computations below show 1) Kepler's third law revised, 2) how to calculate the orbital period, 3) how to calculate the orbital separation, and 4) how to calculate Neptune's mass by putting these things together.