Boundary dependence of an isotropic self-gravitating cusp ($\rho =A{r}^{-k},\rho {\sigma }_r^2=P_b+\frac{4\pi G{A}^2}{3-k}{\int }_r^b{s}^{1-2k}ds$)

For an isotropic self-gravitating collisionless stellar system with $k<3$ and no central point mass, $M(r)=4\pi A{r}^{3-k}/(3-k)$. Integrating the Spherical Jeans equation to an outer radius $b$ gives the displayed pressure, with $P_b=\rho (b){\sigma }_r^2(b)$. For $k<1$ the inner pressure is set by outer radii and ${\sigma }_r^2\sim C{r}^k$, with $C$ depending on the boundary. An unbounded pure power law has divergent pressure in this case. For $1<k<3$, inner radii dominate and ${\sigma }_r^2\sim 2\pi GA{r}^{2-k}/[(3-k)(k-1)]$. At $k=1$ a logarithm appears. For $k\ge 3$ the enclosed central mass diverges unless the density law is modified at small radii.