= Radially anisotropic isothermal halo asymptotics
{title2=$\rho\sim\sigma^2/[4\pi G r^2\log(r/r_*)]$}
For a self-consistent <Gaussian angular-momentum suppression in a stellar distribution>, put $y=[\psi(r)-\psi(0)]/\sigma^2$ and $s=\log(r/r_a)$. At large radius the <Poisson equation for Newtonian gravity> reduces to $y_{ss}+y_s\sim-\lambda e^y$, where $\lambda=4\pi G\rho_c r_a^2/\sigma^2$. The asymptotic balance $y_s\sim-1/s$ gives $e^y\sim1/(\lambda s)$, the displayed density, and <circular speed> $v_c^2\sim\sigma^2/\log(r/r_*)$. A sufficiently small regular core can have an intermediate isotropic <singular isothermal sphere> envelope with $\rho\simeq\sigma^2/(2\pi Gr^2)$ and $v_c^2\simeq2\sigma^2$. At the actual regular centre, however, $\rho\to\rho_c$ and $v_c\propto r$; the intermediate power law must not be mistaken for a central limit.
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