Jupiter vs. the Sun

To compare the Jovian Kepler constant with the Solar one, we must convert it to the same units (AUs cubed over years squared):

• we determine the diameter of Jupiter is ~ 0.001 AUs by estimating* its angular diameter is about 3/4 of one minute of arc at opposition when we know the planet is 4.1 AUs away (its orbital radius minus that of the Earth's)
• we cube this: ~ 10⁻⁹ and multiply our Jovian Kepler constant by it
• we determine the period of moon I (Io) is ~1.8/365.25 or 0.0049 years
• we square this: ~ 2.4 x 10⁻⁵ and divide the above result by it

(* If there were two stars of known angular separation visible in the eyepiece of the telescope at the same time as Jupiter, their presence could provide a useful yardstick against which to make a reasonably accurate estimate).  

Hence, the average Kepler constant for the Jovian system is ~  0.001, i.e. 1/1000th of the value (1) for the Solar System. This means that for two different objects orbiting at the same distance from the Sun and from Jupiter, in the former case the object's speed is 10√10 times that of the latter. (Similarly, if they have the same orbital period, then the object orbiting the Sun does so at 10 times the distance of that orbiting Jupiter). 

We conclude that the Sun's gravitational pull is much stronger than Jupiter's and that the ratio of their Kepler constants is a measure of this.

Given its angular diameter at opposition is as stated earlier, then we conclude that at the distance of the Sun (1 AU), it would be 4.1 times larger, i.e. ~ 0.05°, compared with the Sun's  ~ 0.5°. That is, Jupiter's diameter is about 10% that of the Sun and its volume is about one-thousandth, i.e. about the same as the ratio of its Kepler constant to the Sun's.

Here then is a first inkling about what may be behind the strength of a body's gravitational pull. Volume seems an unlikely candidate as we shouldn't expect a diffuse cloud of gas to have the same effect as a solid object of the same size. Perhaps the gravitational pull is simply proportional to its mass then? If so, then Jupiter's density is about the same as the Sun's. And yet, what different bodies they seem to be!