The Jovian Keplerian system

Imagine being among the first to view Jupiter through the newly-invented telescope in the early 1600s!

How to make sense of the 4 small objects moving about it and how long before we recognize this is a Keplerian system in its own right, I wonder? 

Imagine trying to track the individual 'moons' over days, weeks and months in an attempt to measure the orbital radius and orbital period for each - no easy task!

The former could be estimated by comparing distances visually with the diameter of the planet, perhaps starting with the innermost moon's orbit and then using that as a yardstick in turn. These would very much be estimates, to the nearest Jovian diameter  'unit'.

The latter we could of course determine with more precision by timing the intervals between successive identical locations in a moon's orbit (e.g. maximum elongation).

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

After a while we might observe the orbital periods of moons I, II and III are in the ratio 1:2:4, so, why not use the orbital period of moon I as our unit instead?

Did these newly-discovered moons obey the same laws as the planets going about the Sun? See the table at the bottom of the page (the table had to be imported and in this case, if not placed there, messes up the font of the text following it!).

The average Jovian Kepler 'constant' (last column) is 202. To convert this to the same units for the Solar Kepler constant (AUs cubed over years squared):

• we determine the diameter of Jupiter is ~ 4.8 x 10^(-4) AUs by estimating its angular diameter at opposition when we know the planet is 4.1 AUs away (its orbital radius minus that of the Earth's)
• we cube this: ~ 1.11 x10^(-10) and multiply our Jovian Kepler constant by this
• we determine the period of Io is ~1.8/365.25 or 4.9 x 10^(-3) years
• we square this: ~ 2.4 x 10^(-5) and divide our Jovian Kepler constant by this

Hence, the average Kepler constant for the Jovian system is ~  9 x 10^(-4), ~ 1/1000th of the value (1) for the Solar System, from which we can deduce using Newton, that the mass of Jupiter is also ~ 1/1000th of that of the Sun.

Moon	Orbital radius (r) in Jovian diameters	r^3	Period (t) in Ionian units	t^2	r^3/t^2
I. Io	6	216	1	1	216
II. Europa	9	729	2	4	182
III.Ganymede	15	3375	4	16	211
IV. Callisto	26	17576	9.4	89	197