 Gas and radiation pressure stellar wind equation

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.