= Vacuum boundary obstruction for an isothermal sphere
{title2=$d\ln\rho/dr=-GM(r)/(c^2r^2)$}
An <isothermal sphere> with constant positive <isothermal sound speed> $c$ obeys the title <hydrostatic equilibrium> equation. Between two positive finite radii with bounded enclosed mass, its right-hand side has a finite integral. A positive starting density therefore remains positive at any proposed finite outer radius. It cannot meet a vacuum boundary with $p=c^2\rho=0$ there without an additional confining structure or a change of equation of state. The <singular isothermal sphere> makes the obstruction explicit: $\rho=c^2/(2\pi Gr^2)$ and $p=c^4/(2\pi Gr^2)$ never vanish at finite radius, and its mass grows as $2c^2r/G$ without bound at infinity.
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