Polytropic stellar wind ($\dot{M}=4\pi r^2\rho u$)

A steady spherical polytropic flow escaping a point-mass potential conserves the outward mass loss rate and the Bernoulli function $C=u^2/2+c_s^2/(\gamma -1)-GM/r$. For positive terminal speed, a finite-radius smooth sonic point requires $C>0$ and $1<\gamma <5/3$. Its sound speed obeys $c_{s,c}^2=2C(\gamma -1)/(5-3\gamma )$, and evaluating the conserved flux there selects the transonic mass loss rate. Other parameter limits need separate treatment.

Put $y=\rho r^{4/(\gamma +1)}$ and $\alpha =4(\gamma -1)/(\gamma +1)$. The polytropic stellar wind equations become $F(r)=C{r}^{\alpha }+GM{r}^{\alpha -1}=G(y)=\gamma K{y}^{\gamma -1}/(\gamma -1)+{\dot{M}}^2/(32{\pi }^2{y}^2)$. At a regular sonic point both derivatives vanish. Matching their minima allows the two positive-$y$ branches to join with finite slope; unequal minima either leave a forbidden radial interval or separate the subsonic and supersonic branches. This reduces a singular differential-equation crossing to a geometric minimum comparison.