Is Kepler's constant the same for moons orbiting a planet as it is for planets orbiting the Sun? Why?

Question

vincent-lyon.fr · Answer

What do YOU think?

anonymous · Answer

I think it is, because I'm fairly certain that the same formula is used in both cases. But I need more validation and explanation if could provide it?

anonymous · Answer

Mmmm I don't think so since the constant value depends on the mass of the central object..

anonymous · Answer

So what do you think it is?

anonymous · Answer

Hmmm?

anonymous · Answer

Do you think that the constant is the same for moons orbiting a planet as it is for planets orbiting the Sun? I think the answer is an obvious "yes", hence the term "constant"... but why?

anonymous · Answer

Okay so the law states:
$$R ^{3}/T ^{2}$$
This constant is for one system only, and each of them will have a distinct constant. 

So think about the gravitational force in our solar system: 

$$F = GMm/R ^{2}$$

G is gravitational constant, M is the sun's mass, m is a planet's mass, and R is the distance between the two masses.

$$GMm/R ^{2} = mv ^{2}/R$$ 

m and R cancels out on both sides...

$$GM/R = v$$

v is the revolution velocity of the planet. 
The velocity of revolution is the circumference $$2 \pi R $$divided by T (period of revolution). 

$$GM/R = (2\pi R/T)^2 = 4\pi^2R^2/T^2 $$

Rearranging the equation gives you 

$$R^3/T^2 = GM/(4\pi^2) $$

On both sides, G, M, and pi is constant.
Therefore, right hand side is constant, and the value of R^3/T^2 is constant for all planets in the solar system.

However, M, the mass of the central object, will be different depending on the primary planet and such. 

That took a while. Hehe ^^

anonymous · Answer

Wow! Thanks for the response, lots of help