A planet is a homogeneous ball of radius 'R' having mass 'M'. It is surrounded by a dense atmosphere having volume density   $\rho =\frac{{\sigma }_0}{r}$. Where ${\sigma }_0$  is constant and ‘r’ is distance from the centre of planet. Then

If  ${\sigma }_0=\frac{M}{2\pi R^2}$, then acceleration due to gravity is constant throughout the atmosphere of planet

If  ${\sigma }_0=\frac{M}{\pi R^2}$, then acceleration due to gravity is constant throughout the atmosphere of planet

For any value of  ${\sigma }_0$  acceleration due to gravity can be constant throughout the atmosphere of planet

If  ${\sigma }_0=\frac{M}{2\pi R^2}$, then the weight experienced by a body of mass 'm' at altitude  $\frac{R}{2}$ from the surface of planet will be $\frac{GMm}{R^2}$