 Solution

Letbe the local isothermal sound speed squared. Steady spherical mass conservation givesThe momentum equation, with radiative acceleration represented by the radiation-pressure gradient, isRadiative diffusion in a star givesUsing $P_{\mathrm{rad}}/P_g=(1-\beta )/\beta$ andthis becomesSubstitution of $d{P}_g/dr=a_i^{2}d\rho /dr+\rho a_i^{2}d\log T/dr$ and the continuity equation yields the wind equation

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,The assumed inequality $a_i^2\ll G{m}_r/r$ makes the second factor positive and close to one. Since $0<\beta <1$,Thus acceleration through a regular sonic point requires radiation to cancel most, but not all, of the effective gravity and in particularIf the local luminosity reached or exceeded $L_{\mathrm{crit}}$ 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.