Vacuum boundary obstruction for an isothermal sphere ($d\ln \rho /dr=-GM(r)/(c^2{r}^2)$)

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: $\rho =c^2/(2\pi G{r}^2)$ and $p=c^4/(2\pi G{r}^2)$ never vanish at finite radius, and its mass grows as $2c^{2}r/G$ without bound at infinity.