= Isothermal Parker wind
{title2=$w-\ln w=4\ln x+4/x-3$}
A steady spherical <Parker wind> with constant <isothermal sound speed> $c$ and negligible wind <self-gravity> obeys $4\pi r^2\rho u=\dot M$ and $(u-c^2/u)u'=2c^2/r-GM/r^2$. A regular <sonic point> has $u=c$ at $r_s=GM/(2c^2)$. With $x=r/r_s$ and $w=u^2/c^2$, integration gives the displayed equation for the smooth accelerating <transonic branch>. Its slope at the sonic point is $u'(r_s)=c/r_s$. Maintaining constant gas <temperature> during expansion generally requires heat input, so this model is distinct from an adiabatic wind with $\gamma>1$.
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