= Steady photoevaporating-disk mass flux
{title2=$F_{\mathrm{acc}}=\dot M_d-\dot M_w\sqrt{r_g/r}$}
For a sink $W=W_0(r/r_g)^{-5/2}$ outside $r_g$, <mass conservation> gives $dF_{\mathrm{acc}}/dr=2\pi rW$ in <steady state>. Remote feeding gives $F_{\mathrm{acc}}=\dot M_d-\dot M_w$ inside $r_g$, and $\dot M_d-\dot M_w\sqrt{r_g/r}$ outside, where $\dot M_w=4\pi W_0r_g^2$. The inward flux decreases towards the star because mass is diverted into the wind. A finite feeding radius or outer truncation supplies an additional boundary condition.
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