Question

Imagine a universe in which the law of gravitation has the inverse - cube form
$\underset{F}{\to }=-G^{\prime }\frac{Mm}{r^3}\underset{r}{\to }.$
where G' is a universal constant. Find the form that Kepler's third law would take in this universe for circular orbits. (Note : Non-circular orbits are not possible in such a universe, and ever circular orbits are unstable in the sense that the slightest perturbation makes them "blow up" into spirals.)

Answer 1

When something moves in orbit then centripetal force is equal to the gravitational force.

$\frac{m{v}^2}{r}=\frac{GMm}{r^3}$

$\Rightarrow v=\frac{\sqrt{GM}}{r}$

$\Rightarrow \frac{2\pi r}{T}=\frac{\sqrt{GM}}{r}$

$\frac{2\pi }{\sqrt{GM}}{r}^2=T$

$\therefore T\propto r^2$

$\frac{T_1}{T_2}=\frac{r_1^2}{r_2^2}$

$\frac{T_1^2}{T_2^2}=\frac{r_1^4}{r_2^4}$