 Isothermal sphere

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 $p^{\prime }=-GM\rho /r^2$ and $M^{\prime }=4\pi r^2\rho$ 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.