Poisson equation and gravitational potential

Question

In Newtonian gravity, what is the difference between the Poisson equation $\Delta(\Phi)$ and the expression $\Phi = -GM/r$ for the gravitational potential $\Phi$? Are both somewhat related?

Answer 1

The solution $$-\frac{G}{|\mathbf{r}-\mathbf{r}'|}$$ is the Green's function of the Poisson's equation $$\nabla^2 \Phi = 4\pi G\rho.$$ For a differential equation $L\Phi = \rho$, where $L$ is a linear differential operator, the Green's function is the function $F(\mathbf{r},\mathbf{r}')$ satisfying $LF(\mathbf{r},\mathbf{r}') = \delta(\mathbf{r}-\mathbf{r}')$ with the appropriate boundary conditions. It tells us the solution at $\mathbf{r}$ produced by a point source located at $\mathbf{r}'$. The linearity then enables the solution for any source to be found using superposition by integrating over $\mathbf{r}'$: $$\Phi(\mathbf{r}) = \int F(\mathbf{r},\mathbf{r}') \rho(\mathbf{r}') \mathrm{d}\mathbf{r}'.$$ In the case of a point mass $M$, the mass distribution is just a single delta function multiplied by $M$, so the potential is $M$ times the Green's function itself, giving $-GM/|\mathbf{r}-\mathbf{r}'|$.

So this is what's happening behind the scenes every time you integrate over the mass/charge distribution to find the gravitational/electric potential.