= Terminal speed of a constant-mass dusty shell
{title2=$v_\infty$}
A fixed-mass shell with ultraviolet opacity $\kappa_{\rm UV}$ has <dust transparency radius> $R_{\rm UV}=\sqrt{\kappa_{\rm UV}M_{\rm sh}/(4\pi)}$. For an initially expanding shell driven by constant ultraviolet luminosity, a sharp opaque-to-transparent approximation gives
$$
v_\infty^2=v_0^2+\frac{2GM}{R_0}\left[\Gamma_{\rm ss}(2X-1)+\Gamma_{\rm IR}(1-X^{-1})-1\right],
\qquad X=R_{\rm UV}/R_0.
$$
Here $\Gamma_{\rm ss}=LR_0^2/(GM M_{\rm sh}c)$ is the initial single-scattering force ratio and $\Gamma_{\rm IR}=\kappa_{\rm IR}L/(4\pi GMc)$. The trajectory must actually reach $R_{\rm UV}$. Large initial ultraviolet <optical depth> enhances the radiative work at fixed initial single-scattering ratio; the same conclusion does not follow unconditionally at fixed electron-scattering luminosity.
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