Comparing the Forces of Two Central Bodies - Part One

So far, I have shown that the mass of the orbiting body does not matter in the motion of the planets. The velocity and the orbital period change according to the distance from the central body only. But what happens if the orbiting body orbits a different central body ?

First, let's have another look at my new Kepler based equation for gravity :

Force 𝞪 Velocity / Time

This is applicable in any orbiting system for any two orbiting bodies orbiting the same central body.

Now let's look at what happens if say, Io a moon of Jupiter, orbits the Sun instead. This should give us an insight into the relative forces of the two largest central bodies in our Solar System - Jupiter and the Sun. The Sun has a mass of 1,047 times that of Jupiter so we should expect Io to be subject to a greater force and therefore orbit at a greater velocity than it does around Jupiter.

First, we will set Io's distance from the Sun as the same as it's distance from Jupiter, viz, 421,700km. This will allow us to determine the orbital factor - the increase in velocity and decrease in the orbital period from orbiting a central body with a greater mass. Next, we will compare it with the closest planet, Mercury. Mercury is 137.32 times further out from the Sun. Using Kepler III :

D^3 = T^2

2,589,509 = T^2

T = 1609

Mercury takes 88 days to orbit the Sun. We know that Io should take a lot less because it's closer to the Sun, 88 / 1609, to be exact. So Io's new orbit period around the Sun will be 0.05 days or just 1.3 hours.

Now, using my first formula, to figure out the velocity :

• Distance squared = Time / Velocity

(137.32)^2 = 1609 / V

18856 x V = 1609

V = 0.0853

This means that the velocity of Mercury, 47.9 km/s, will be just 8.5% of the new velocity of Io, viz, 561 km/s. This would make Io the fastest body in the solar system by far.

I could also have used my velocity formula to calculate the same thing :

• V = 1 / √ D

V = 1 / √ 137.32 = 0.0853

Force of the Sun relative to Jupiter

Io orbits Jupiter in 42 hours compared to 1.3 hours if it orbited the Sun at the same distance.

The velocity of Io is 17.3 km/s in the orbit of Jupiter compared to 561 km/s if it orbited the Sun.

This is a factor of about 32.3 times for both (the orbital factor).

How does the orbital factor relate to the Sun / Jupiter mass ratio ?

As stated above, the Sun has 1,047 times the mass of Jupiter.

The square root of 1,047 is 32.3.

Therefore:

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