Derive kepler's third law from 2 other equations

Answer 1

The two equations you are looking for are relating centripetal force with gravitational force between the objects involved. F=mrw² and F=G(mM)/r² are the two equations in question.

m=mass of smaller object

M=mass of object enforcing gravitational pull (in our case the Sun)

G=gravitational constant

r=radius

w=rotational velocity

So we set them equal to one another and solve for the period(T) of rotation using w=2π/T

mr(2π/T)²=G(mM)/r²

r^3(4π^2)/GM=T²

This relation allows us to see the relationship between the period and the radius of rotation stated in Kepler's third law.