How to derive gravitational potential from Navier-Stokes equation?

Question

Starting from the Navier-Stokes equation I want to be able to derive the gravitational potential using the Poisson equation but am unsure how to do it in spherical polar coordinates.
This is what I have so far $$\partial\rho +\nabla\cdot(\rho{\bf v})=0$$ This is satisfied as in my case it is not dependent on time and the system is stationary so ${\bf v}=0$. $$\frac{dv}{dt}=-\frac{1}{\rho}\nabla p-\nabla \phi$$ since $\frac{dv}{dt}=0$ we have $$\nabla \phi=-\frac{1}{\rho} \nabla p$$ I want to integrate the Poisson Equation $\nabla ^2\phi=4\pi G\rho$ in spherical polar coordinates to be able to set it equal to the above equation but I'm really not sure how to go about this so any help would be appreciated!