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. 

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.