Large-radius asymptotics of an isothermal Parker wind ($u(r)\sim 2c\sqrt{\ln (r/r_s)}$)

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 $w/(4\ln x)\to 1$ 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.