Poisson equation for Newtonian gravity (${\nabla }^2\Phi =4\pi G\rho$)

For mass density $\rho$, the Newtonian gravitational potential satisfies the Poisson equation ${\nabla }^2\Phi =4\pi G\rho$. The gravitational acceleration is $-\nabla \Phi$. An isolated system also requires the exterior vacuum solution and decay at infinity; solving the interior equation alone does not fix a globally physical field.

Suppose the shell $2^{-n}<r<2^{1-n}$ has mass density $2^{n-1}{\rho }_0$. Its mass is $(14\pi {\rho }_0/3)4^{-n}$, by integrating $4\pi r^2\rho dr$. Summing the geometric series gives total mass $14\pi {\rho }_0/9$. At $r=2^{-N}$ the enclosed shells have indices $n\ge N+1$, so the enclosed mass is the total mass times $r^2$. The magnitude of the Newtonian gravitational field at every such boundary is therefore the same, $14\pi G{\rho }_0/9$. This does not assert that the field is constant throughout each shell.