Unravelling Kepler's Law: How F = m/r^2 is Reached

[Miike012]

what is it exactly?

and in my book they had an equation

F = (4)(pi)^2(m)(r)/T^2

then they said by using keplers law... they arrived to a new equation that relates the gravitational force exerted by the sun which is...
F = m/r^2

If Keplers law says T = r^3/2 how in the heck did they go from F = (4)(pi)^2(m)(r)/T^2
to F = m/r^2 by substituting T for r?
Where did (4)(pi)^2(r) go ?

 

[gneill]

It would appear that they started with the centripetal force:
$$F = m\frac{v^2}{r}~~~~~\text{where:}~~~~v = \frac{2 \pi r}{T}$$
$$F = \frac{4 \pi^2 m r}{T^2}$$
Now, Kepler's Third Law states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. For a circular orbit we identify the semi-major axis with the orbital radius. Note that the law as stated does not give us the constant of proportionality, so we write: $r^3 \propto T^2$. So we can replace the T² in the force equation with r³ but without a precise constant of proportionality it doesn't make sense to retain the others, so that:
$$F \propto \frac{4 \pi^2 m r}{r^3} \propto \frac{m}{r^2}$$

 

[NascentOxygen]

For different planets, Kepler's third Law says (R^3)/(T^2) = constant
and equals unity when using units of years and astronomical units (A.U.).

 

[Miike012]

Ok so I see that I r will divide out and it will equal 4(pi)^2m/r^2... where did the 4 pi^2 go?