A week or so back I asked a question about the gravitational potential field $$\phi=\frac{-Gm}{r}, \qquad r\neq 0, $$ and how to show the Laplacian of $\phi$ equals zero for $r\neq 0$? Eventually, (it took a while) I was able to understand that
$$\nabla\cdot\nabla\phi=Gm\left(\frac{2x^{2}-y^{2}-z^{2}+2y^{2}-x^{2}-z^{2}+2z^{2}-x^{2}-y^{2}}{\left(x^{2}+y^{2}+z^{2}\right)^{5/2}}\right)~=~0, \qquad r\neq 0,$$ which was a revelation. But now I'm wondering why Poisson's equation $$\nabla\cdot\nabla\phi=\nabla^{2}\phi=4\pi G\rho$$ doesn't always equal zero as well? Obviously it doesn't, so I'm assuming that inside a mass the gravitational potential field cannot be given by $$\phi=\frac{-Gm}{r}, \qquad r\neq 0.$$ Is that correct? Also, is there a comparably easy formula for gravitational potential inside a mass or does it vary (horribly?) depending on the shape and density of the mass?