= Isothermal sphere
{title2=$r^{-2}[r^2(\ln\rho)\prime]\prime=-4\pi G\rho/K$}
An <isothermal sphere> is a spherically symmetric self-gravitating gas equilibrium with <equation of state> $p=K\rho$ and constant $K$. The <hydrostatic equilibrium> equations $p'=-GM\rho/r^2$ and $M'=4\pi r^2\rho$ yield the title equation. Its regular finite-central-density solution is distinct from the <singular isothermal sphere>. A finite outer radius requires confinement because the regular density does not vanish at finite radius.
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