Applying the principle of homogeneity of dimensions, determine which one is correct, where $$T$$ is time period, $$G$$ is gravitational constant, $$M$$ is mass, $$r$$ is radius of orbit.

Use dimensional analysis.

Assume the relation:

$$T\propto G^aM^br^c$$

Write dimensions:

$$[T]=T$$

$$[G]=M^{-1}L^3T^{-2}$$

$$[M]=M$$

$$[r]=L$$

Substitute:

$$T=(M^{-1}L^3T^{-2})^a\cdot M^b\cdot L^c$$

$$=M^{-a+b}\cdot L^{3a+c}\cdot T^{-2a}$$

Now compare powers with LHS$$[T^1]$$:

For mass:

$$-a+b=0$$

For length:

$$3a+c=0$$

For time:

$$-2a=1\Rightarrow a=-\frac{1}{2}$$

Substitute aaa:

$$b=a=-\frac{1}{2}$$

$$3a+c=0\Rightarrow c=\frac{3}{2}$$

Final relation:

$$T\propto G^{-1/2}M^{-1/2}r^{3/2}$$

$$T\propto\sqrt{\frac{r^3}{GM}}$$

Correct expression:

$$T=k\sqrt{\frac{r^3}{GM}}$$

$$T^2 = \frac{k r^3}{GM}$$