Central Bodies Part Two - The Earth and Jupiter

Last time, I introduced the concept of orbital factor - the change in velocity and orbital period of a body if it orbited a different central body with a different mass. In this blog post I will show that the orbital factor of Jupiter is 17 times greater than Earth and how that relates to their respective masses.

Let's say that Io orbits the Earth at the same distance as it orbits Jupiter - 421,700 km. Using Kepler's third law to compare the moon and Io :

D^3 = T^2

(384,000/421,700)^3 = T^2

T = 0.868

So, if the moon takes 27.3 days to orbit Earth, then Io will take 1.132 (1-0.868) times longer, viz, 31 days.

We can then use the distance squared law to calculate the new velocity for Io.

D^2 = T/V

0.91^2 = 0.868 / V

0.828 = 0.868 / V

V = 0.868 / 0.828 = 1.048

This means that the moon's velocity will be 1.048 times that of Io. Given that the moon's velocity is circa 3,683 km/hr , this means Io's new Earth bound velocity would be 3,514 km/hr or 0.976 km/s.

	Io in Jupiter orbit	Io in Earth orbit	Difference ∆  (Orbital Factor)
Distance	421,700	421,700	1
Orbit	42 hours	31 days / 744 hrs	17.71
Velocity	17.3 km/s	0.976 km/s	17.72

From Part One :

The square root of the ratio of the masses of two central bodies is equal to the orbital factor of their orbital bodies.

Jupiter has a mass of 317.83 times that of Earth :

The square root of 317.83 is 17.82 (a small rounding difference with 17.72 in the above table).

In Part One I showed that the square root of the ratio of the masses also works for Jupiter and The Sun :

	Io in Jupiter orbit	Io in Sun’s orbit	Difference ∆  (Orbital Factor)
Distance	421,700	421,700	1
Orbit	42 hours	1.3 hrs	32.3
Velocity	17.3 km/s	561 km/s	32.4

The Sun has a mass of 1047.36 times that of Jupiter.

The square root of 1047.36 is 32.36.

Therefore, we would expect that the Sun would have an orbital factor of 576.6 times that of Earth (17.82 x 32.36).

NASA states that the Sun has a mass of 333,000 times that of Earth. 

The square root of 333,000 is 577.