= Gas and radiation pressure stellar wind equation
{title2=$(v-a_g^2/v)v'=2a_g^2/r-a_g^2(\log T)'-g(1-\Gamma)$}
In an optically thick steady spherical <stellar wind> of constant <mean molecular weight> $\mu_{\rm mol}$, let $v$ be the outward speed and $a_g^2=k_BT/(\mu_{\rm mol}m_u)$ the <isothermal sound speed> squared. Mass conservation gives $\rho'/\rho=-2/r-v'/v$, where a prime denotes differentiation with respect to radius $r$. Use $P_{\rm gas}=\rho a_g^2$ and <radiative diffusion> to write $P_{\rm rad}'/\rho=-g\Gamma$, where $g=GM_r/r^2$ and $\Gamma=L_r/L_{\rm Edd}$ uses the local <Eddington luminosity> $L_{\rm Edd}=4\pi cGM_r/\kappa$ 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.
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