The Jovian Keplerian system

Updated 11 May 26

The discovery of Jupiter's moons through the newly-invented telescope back in the early 1600s must have caused great excitement at the time and a wish to investigate further to see if the Jovian system follows Kepler's laws.

However, imagine trying to track these moons individually over many days and weeks in an attempt to measure the orbital radius and orbital period for each!

The orbital radius could be estimated crudely (being imprecise and prone to human error) using the diameter of the planet as the base unit, e.g. the innermost moon being found to orbit at around 3 Jovian diameters from the centre of the planet.

I've tried this myself on just a few occasions with binoculars and it was difficult! Doing this to an acceptable level of accuracy and precision in the early 17th century would have demanded the best optics of the time, a solid mount, good skies over many weeks and an experienced and patient observer. And then no doubt repeating the exercise again and again. 

The orbital period could be measured with more precision by timing the intervals between successive identical locations in a moon's orbit (e.g. maximum elongation), although we should bear in mind that this was before the invention of the pendulum clock.

(Would we have to allow for the movement of the Earth introducing errors due to parallax?)

See the table at the bottom of the page for an investigation. (The table had to be imported and in this case, if not placed there, messes up the font of the text following it!). The moons are numbered according to their proximity to Jupiter.

It turns out that the orbital periods of moons I, II and III are in the ratio 1:2:4, so I'm using the orbital period of moon I as the unit of time here.

The average estimate of the Jovian Kepler 'constant' (last column) is 26, but with a large range of values, no doubt due to the lack of precision in measuring orbital radii in whole Jovian diameters, which are then cubed. (In practice, there would be even greater variation with this method, because, unlike the case here, estimates would likely be inaccurate, as well as imprecise). However, these values are of a similar order of magnitude and a more precise and more accurate method should yield results that are more consistent*. Therefore, we tentatively assert that these moons do indeed obey Kepler's third law.

(* If instead we were to estimate accurately using the radius of Jupiter as our measuring unit, then we should find that the Jovian Kepler constant has an average of 202 with a proportionately smaller range of 182 to 216. However, I expect this advantage would have been easily removed by increased human error, in estimating, say, that Callisto orbits at 26 units from Jupiter! I imagine that the later invention of the reticule would have been revolutionary for making accurate and precise measurements). 

Moon	Orbital radius (r) in Jovian diameters	r^3	Period (t) as a multiple of that of moon I	t^2	r^3/t^2
I. Io	3	27	1	1	27
II. Europa	5	125	2	4	31
III.Ganymede	7	343	4	16	21
IV. Callisto	13	2197	9.4	89	25