Gas and radiation pressure stellar wind equation ($(v-a_g^2/v)v^{\prime }=2a_g^2/r-a_g^2(\log T{)}^{\prime }-g(1-\Gamma )$)

In an optically thick steady spherical stellar wind of constant mean molecular weight ${\mu }_{\mathrm{mol}}$, let $v$ be the outward speed and $a_g^2=k_{B}T/({\mu }_{\mathrm{mol}}{m}_u)$ the isothermal sound speed squared. Mass conservation gives ${\rho }^{\prime }/\rho =-2/r-v^{\prime }/v$, where a prime denotes differentiation with respect to radius $r$. Use $P_{\mathrm{gas}}=\rho a_g^2$ and radiative diffusion to write $P_{\mathrm{rad}}^{\prime }/\rho =-g\Gamma$, where $g=G{M}_r/r^2$ and $\Gamma =L_r/L_{\mathrm{Edd}}$ uses the local Eddington luminosity $L_{\mathrm{Edd}}=4\pi cG{M}_r/\kappa$ and opacity $\kappa$. Substituting into the momentum equation yields the displayed wind equation. Smooth sonic point crossing requires both its denominator and numerator to vanish. The radiation-pressure gradient already supplies radiative acceleration; adding that force separately would count it twice.

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.