= Solution
For steady spherical <polytropic flow>, write $u=u_r$ and use the <polytropic equation of state> $p=K\rho^\gamma$, with $K>0$. Mass conservation and radial <Euler momentum equation> are
$$
\frac{d}{dr}(r^2\rho u)=0,\qquad uu'=-\frac{p'}\rho-\frac{GM}{r^2},\qquad c_s^2=\frac{dp}{d\rho}=\gamma K\rho^{\gamma-1}.
$$
The mass equation gives $\rho'/\rho=-2/r-u'/u$. Substitute it into momentum balance to obtain
$$
\boxed{\frac{u^2-c_s^2}{u}u'=\frac{2c_s^2}{r}-\frac{GM}{r^2}.}
$$
At a <sonic point> the derivative coefficient vanishes. A smooth finite-slope solution must make the numerator vanish there too:
$$
\boxed{u_c^2=c_{s,c}^2,\qquad r_c=\frac{GM}{2c_{s,c}^2}.}
$$
Finally $dp/\rho=d[\gamma K\rho^{\gamma-1}/(\gamma-1)]$. Integrating momentum gives the <Bernoulli equation>
$$
\boxed{\frac{u^2}{2}+\frac{\gamma p}{(\gamma-1)\rho}-\frac{GM}{r}=C.}
$$
This is <kinetic energy> plus <specific enthalpy> plus gravitational potential per unit mass. The <polytropic stellar wind> selects a particular transonic branch of these equations.
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