= Solution
Let
$$
a_i^2=\frac{P_g}{\rho}=\frac{\mathcal R T}{\mu}
$$
be the local <isothermal sound speed> squared. Steady spherical <mass conservation> gives
$$
4\pi r^2\rho v=\dot M,
\qquad
\frac{d\log\rho}{dr}=-\frac2r-\frac1v\frac{dv}{dr}.
$$
The momentum equation, with radiative acceleration represented by the radiation-pressure gradient, is
$$
v\frac{dv}{dr}
=-\frac1\rho\frac{dP_g}{dr}
-\frac{Gm_r}{r^2}
+\frac{\kappa L}{4\pi cr^2}.
$$
<Radiative diffusion in a star> gives
$$
\frac{dP_{\rm rad}}{dr}
=-\frac{\kappa\rho L}{4\pi cr^2}.
$$
Using $P_{\rm rad}/P_g=(1-\beta)/\beta$ and
$$
\Gamma\equiv\frac{L}{L_{\rm crit}},
\qquad
L_{\rm crit}=\frac{4\pi cGm_r}{\kappa},
$$
this becomes
$$
\frac{d\log T}{dr}
=-\frac{\beta}{4(1-\beta)}
\frac{Gm_r}{r^2a_i^2}\Gamma.
$$
Substitution of $dP_g/dr=a_i^2d\rho/dr+\rho a_i^2d\log T/dr$ and the continuity equation yields the wind equation
$$
\boxed{\left(v-\frac{a_i^2}{v}\right)\frac{dv}{dr}
=\frac{2a_i^2}{r}
-\frac{Gm_r}{r^2}
\left[1-\Gamma\frac{4-3\beta}{4(1-\beta)}\right]}.
$$
Its topology is that of the <Parker wind equation>. The coefficient of $dv/dr$ vanishes at the sonic line $v=a_i$. A smooth transonic solution must pass through a critical point where the right-hand side also vanishes; generic subsonic solutions are breezes or turn back, while generic supersonic branches cannot be joined smoothly to a quasi-static stellar atmosphere.
At the critical point,
$$
\Gamma_c=
\frac{4(1-\beta_c)}{4-3\beta_c}
\left(1-\frac{2a_{i,c}^2r_c}{Gm_c}\right).
$$
The assumed inequality $a_i^2\ll Gm_r/r$ makes the second factor positive and close to one. Since $0<\beta<1$,
$$
\frac{4(1-\beta)}{4-3\beta}<1.
$$
Thus acceleration through a regular sonic point requires radiation to cancel most, but not all, of the effective gravity and in particular
$$
\boxed{\frac{L}{L_{\rm crit}}<1}.
$$
If the local luminosity reached or exceeded $L_{\rm crit}$ in this diffusion model, the numerator would have the wrong sign for the subsonic branch to cross the sonic line smoothly under the cold-wind assumption.
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