Einasto profile

Example

Let $\alpha > 0$, $\rho_{-2} > 0$ and $r_{-2} > 0$. The Einasto profile is the spherical density $$\rho(r) = \rho_{-2} \exp\left( -\frac{2}{\alpha} \left[ \left( \frac{r}{r_{-2}} \right)^{\alpha} - 1 \right] \right).$$ Its logarithmic slope is $$\frac{d \ln \rho}{d \ln r} = -2 \left( \frac{r}{r_{-2}} \right)^{\alpha},$$ so the slope varies continuously with radius and equals $-2$ exactly at $r = r_{-2}$, where $\rho = \rho_{-2}$.

The mass inside radius $r$, with $x = r/r_{-2}$, is $$M(r) = \frac{4 \pi \rho_{-2} r_{-2}^3 e^{2/\alpha}}{\alpha} \left( \frac{\alpha}{2} \right)^{3/\alpha} \gamma\left( \frac{3}{\alpha}, \frac{2}{\alpha} x^{\alpha} \right),$$ where $\gamma$ is the lower incomplete gamma function. Check: substituting $u = \frac{2}{\alpha} s^\alpha$ in $\int_0^x e^{-2 s^\alpha/\alpha} s^2 \, ds$ gives $s^2 \, ds = \frac{1}{\alpha} \left( \frac{\alpha}{2} \right)^{3/\alpha} u^{3/\alpha - 1} \, du$.