= Subsonic isothermal breeze
{title2=$u\propto r^{-2},\quad\rho\to\rho_\infty>0$}
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 $u\sim ce^{-C/2}x^{-2}$. 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.
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