The Saturnian Keplerian system

Decades after the discovery of Jupiter's moons, Saturn was observed to have its own satellites: the first being Titan, the largest, in 1655.

Let's try to compute the Kepler value for the three largest, using the same method as for Jupiter, noting however, that accuracy and precision would likely have been low in the early days, so the figures below (especially its density) may not have been achievable until much later.

There is no orbital resonance with these, so the table shows the period in days.

(Rhea's orbit is actually about 4.5 Saturnian diameters. Were we able to estimate to this precision, then we should expect the Kepler value to be ~4.5).

We see that the values in the last column suggest a Kepler 'constant' for Saturn of ~ 4 and hence that Saturn, as well as the Sun and Jupiter, also has its own gravitational system obeying the same laws.

Converting this to Solar system units:

• as Saturn's angular diameter at opposition is ~ 1/3 minute of arc and it's 8.4 AUs away, it's diameter is ~ 0.0008 AUs
• we cube this:  ~ 5 x 10^(-10) and multiply our Kepler constant by it
• 1 day is 1/365 years, so we square this ~  7.5 x 10^(-6) and divide the above result by it

Hence, the average Kepler 'constant' for Saturn is ~ 0.0003.

We deduce that its gravitational pull is ~ 0.0003 times that of the Sun and ~ 0.3 times that of Jupiter. 

We suspect its mass is also in the same proportions. That being the case, as we know its diameter is ~ 0.8 of Jupiter's, its volume is therefore around half of Jupiter's and its density ~ 0.6 times Jupiter's, indicating a different composition.

Moon	Orbital radius (r) in Saturnian diameters	r^3	Period (t) days	t^2	r^3/t^2
I. Rhea	4	64	4.5	20	3.2
II. Titan	10	1000	16	256	4
III. Iapetus	30	27000	79	6241	4.3