= Large-radius asymptotics of an isothermal Parker wind
{title2=$u(r)\sim2c\sqrt{\ln(r/r_s)}$}
On the escaping supersonic <isothermal Parker wind> branch, $w=u^2/c^2\to\infty$. Its integrated equation gives $w=4\ln x+\ln w-3+4/x$, so $w/(4\ln x)\to1$ and the title result follows. A further iteration yields $w=4\ln x+\ln(4\ln x)-3+o(1)$. The leading logarithmic growth holds for any supersonic branch with a fixed finite integration constant, but does not describe a <subsonic isothermal breeze>. The unbounded speed is a formal consequence of indefinite isothermal heating in a prescribed central gravitational potential.
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