Sonic-point obstruction at the Eddington luminosity

For an optically thick diffusion-controlled wind, put $\beta =P_{\mathrm{gas}}/(P_{\mathrm{gas}}+P_{\mathrm{rad}})$. Diffusion gives $(\log T{)}^{\prime }=-g\Gamma \beta /[4a_g^2(1-\beta )]$. At $\Gamma =1$, the numerator of the gas and radiation pressure stellar wind equation is $2a_g^2/r+g\beta /[4(1-\beta )]>0$ for finite-temperature gas and $0<\beta <1$. A subsonic branch then decelerates and cannot pass a regular sonic point. This statement is specific to subsonically launched winds in this transport model, not to every supersonic solution or every form of stellar-wind driving.