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The circular Keplerian orbit has speed $v_K=(GM/r{)}^{1/2}$. Equating it with the heated gas sound speed givesThis is the photoevaporative gravitational radius, where thermal and orbital binding energies have the same order of magnitude. A circular orbit has specific mechanical energy $-GM/(2r)$. Heating adds thermal energy and, in a fluid outflow, available specific enthalpy of order $c^2$. For example, if $c$ is the adiabatic sound speed, an ordinary ideal gas has enthalpy $c^2/(\gamma -1)$; for $\gamma =5/3$ this is $3c^2/2$. At $r\ge r_g$, this more than compensates the circular-orbit binding energy. Equivalently the hot hydrostatic scale height satisfies $H_{\mathrm{hot}}/r\sim c/v_K\gtrsim 1$, so a thin bound surface layer cannot be maintained. With continued irradiation, the gas can expand into a thermal wind: photoevaporation removes disk material.

The condition is a thermal binding scale, rather than an assertion that the sound speed equals the ballistic escape speed, which is $\sqrt{2}{v}_K$. Detailed wind launching can change the numerical critical radius by factors of order unity. Using exactly the supplied numerical estimates, $c=10^6\mathrm{cm}{\mathrm{s}}^{-1}$, and therefore

The wind removes $2\pi rW(r)dr$ of mass per unit time from an annulus; $W$ is already the surface-density loss term in the supplied mass conservation equation, so there is no additional two-face factor. Integrating the photoevaporation profile givesThe convergence at infinity is important: the loss is concentrated near the photoevaporative gravitational radius. Using $r_g=1.4\times 10^{14}\mathrm{cm}$ yields ${\dot{M}}_w\simeq 2.46\times 10^{17}\mathrm{g}{\mathrm{s}}^{-1}$, or about $3.9\times 10^{-9}{M}_{\odot }\mathrm{y}{\mathrm{r}}^{-1}$. The initial disk mass is $2\times 10^{30}\mathrm{g}$, so the wind-only depletion time isThis estimate treats the heated area and wind normalization as fixed and neglects additional removal through stellar accretion. Once the disk shrinks, its wind rate and geometry need not remain constant.

Radiation heats gas sufficiently to drive an escaping thermal outflow. It can remove material from protoplanetary disks, planetary atmospheres and irradiated clouds. In a heated Keplerian disk, the photoevaporative gravitational radius compares the sound speed with the orbital speed. The mass-loss rate depends on the heating, density and wind geometry rather than radiation pressure alone.