P^2 = {4 pi^2 a^3} / {G (M1 + M2)}
where G is the gravitational constant (6.673 x 10^-8 cm^3/gm/sec^2). Astronomers use this relationship to measure the mass of certain objects, for example stars and planets, by observing the motions of satellites (e.g., planets and moons) orbitting around them.
In 1990, I became interested to see how accurately I could measure the mass of the planet Jupiter by observing the motions of the four Galillean moons, Io, Europa, Ganymede and Callisto. I took a series of 45 photographs (typically 2 sec exposures on Kodak Tri-X film) using the 8 inch, f/16 "Reed" refractor at Yale University (the small white dome mounted on the SOM building, for those of you with local knowledge). The photos were taken over almost two months in Jan and Feb of 1990; I tried to get 2-3 observations each clear night. The pictures themselves aren't much to look at. Jupiter is an over-exposed blur, and the 4 moons are tiny dots.
I measured the photos by placing them in a photographic enlarger and projecting the images onto a flat surface, where I measured the distance between each moon and the center of Jupiter. I did the same thing for an image of the Pleiades star cluster. I looked up the distances between all the Pleiades stars, so this gave me a measurement of the scale of the enlargement, in arc seconds on the sky per millimeter on the projection. Thus, I could convert the individual mm measurements to arc seconds: 11.05 arcsec/mm. I also had a short (1/30 sec) exposure of Jupiter which was not over-exposed, which I used to measure the diameter of Jupiter: 38 +/- 3 arcsec. Click here for an ASCII table containing the angular distances and times of observation for this data set. Here is a PostScript plot of the observations of the 4 moons (26 kb). The points mark individual observations of the moons, and the horizontal lines mark the limb of Jupiter.
The Kepler's Law equation above needs an estimate of the orbital semi-major axis in physical, not angular units, i.e., kilometers, not arc seconds. To achieve this conversion, we need to know the distance to Jupiter. I looked it up in the "Sky & Telescope" Celestial Calendar (click here for the data) and fitted a smooth curve to it. For any given observation, the physical distance from a moon to Jupiter, d, is related to the angular distance, delta, by the formula
d = D_J tan(delta),
where D_J is the distance from the Earth to Jupiter. This allowed me to plot the moon's orbits around Jupiter. Here is a PostScript plot of the orbits of the 4 moons (78 kb). The points mark the individual orbits, and the smooth curves are sine curves I fit to the data. Here is a PostScript plot of the super-imposed sine curves (40 kb).