Steady photoevaporating-disk mass flux ($F_{\mathrm{acc}}={\dot{M}}_d-{\dot{M}}_w\sqrt{r_g/r}$)

For a sink $W=W_0(r/r_g{)}^{-5/2}$ outside $r_g$, mass conservation gives $d{F}_{\mathrm{acc}}/dr=2\pi rW$ 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 ${\dot{M}}_w=4\pi W_0{r}_g^2$. 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.

With constant kinematic viscosity and zero inner viscous torque in an accretion disk, $\nu \Sigma \sqrt{r}=(6\pi {)}^{-1}{\int }_0^r{F}_{\mathrm{acc}}(r^{\prime })/\sqrt{r^{\prime }}d{r}^{\prime }$. For the steady photoevaporating-disk mass flux, the density is $({\dot{M}}_d-{\dot{M}}_w)/(3\pi \nu )$ inside $r_g$ and the displayed logarithmic profile outside. At equal feeding and wind rates the inner density vanishes, and the outer normalized profile rises from zero with zero slope and tends to one. Its near-edge expansion is $(r/r_g-1{)}^2/8$.