Plummer model

Example

Let $M > 0$ be a total mass, $a > 0$ a scale length and $G$ the gravitational constant. Let $r$ be the distance from the centre. The Plummer model has density and potential $$\rho(r) = \frac{3M}{4\pi a^3} \left( 1 + \frac{r^2}{a^2} \right)^{-5/2}, \qquad \Phi(r) = -\frac{G M}{\sqrt{r^2 + a^2}}.$$

Check of Poisson's equation $\Delta \Phi = 4\pi G \rho$: here $r^2 \Phi'(r) = G M r^3 (r^2 + a^2)^{-3/2}$, whose derivative is $3 G M a^2 r^2 (r^2 + a^2)^{-5/2}$. Dividing by $r^2$ gives $\Delta \Phi = 3 G M a^2 (r^2 + a^2)^{-5/2} = 4 \pi G \rho$. The mass inside radius $r$ is $M(r) = M r^3 / (r^2 + a^2)^{3/2}$, which tends to $M$ as $r \to \infty$.

It is a polytrope of index $5$: $\rho \propto (-\Phi)^5$, and $\theta(\xi) = (1 + \xi^2/3)^{-1/2}$ solves the Lane–Emden equation of index $5$.