 Subsonic isothermal breeze

The steady isothermal Parker wind differential equation also has positive solutions describing subsonic flow at every radius. Write their first integral as $w-\ln w=4\ln x+4/x+C$, where $w=u^2/c^2$ and $x=r/r_s$. On the small-$w$ branch at large radius, $w\sim e^{-C}{x}^{-4}$, so $u\sim c{e}^{-C/2}{x}^{-2}$. Constant spherical mass flow rate then gives nonzero limiting mass density and hence finite external pressure. This distinguishes a pressure-confined breeze from an escaping wind into vanishing pressure. Specifying outward motion alone does not select the supersonic asymptotic branch.