= Boundary dependence of an isotropic self-gravitating cusp
{title2=$\rho=Ar^{-k},\quad\rho\sigma_r^2=P_b+\frac{4\pi GA^2}{3-k}\int_r^bs^{1-2k}\,ds$}
For an isotropic self-gravitating <collisionless stellar system> with $k<3$ and no central point <mass>, $M(r)=4\pi Ar^{3-k}/(3-k)$. 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 $\sigma_r^2\sim Cr^k$, 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 $\sigma_r^2\sim2\pi GA r^{2-k}/[(3-k)(k-1)]$. At $k=1$ a logarithm appears. For $k\geq3$ the enclosed central <mass> diverges unless the density law is modified at small radii.
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